Elongated microswimmers: influence of hydrodynamics [original]

Elongated Microswimmers:

Influence of Hydrodynamics

vorgelegt von

Master of Science

Arne Wolf Zantop

ORCID: 0000-0002-6537-3292

an der Fakult¨

at II – Mathematik und Naturwissenschaften

der Technischen Universit¨

at Berlin

zur Erlangung des akademischen Grades

Doktor der Naturwissenschaften

– Dr. rer. nat. –

genehmigte Dissertation

Promotionsausschuss:

Vorsiztender: Prof. Dr. Dieter Breitschwerdt

Erster Gutachter: Prof. Dr. Holger Stark

Zweiter Gutachter: Prof. Dr. Roland Netz

Termin der wissenschaftlichen Aussprache: 14. Dezember 2022

Berlin 2023

Abstract

Our nature is full of microscopic organisms such as algae, which are at the basis

of the marine food chain and thus establish entire ecosystems, or bacteria, which

are important for our nutrition and health, and also find a broad application in

industrial processes. Thus, a broad research field is concerned with the many

aspects of the biology, chemistry and physics of microorganisms. Physicists are

particularly interested in the swimming motion of organisms such as algae, bacteria,

spermatozoa and others. In collective motion, these microswimmers exhibit a diverse

behavior such as the formation of swarms, vortices, convection rolls and plumes, and

active turbulence. Therby, most microswimmers are elongated, and are otherwise

characterized by different swimming mechanisms. Consequently, their dynamics

arises from the interplay of different interactions, i.e., direct collisions and long-range

flow fields. It is subject of current research to determine the role of the different

interactions for the fascinating collective dynamics.

In this context, we consider the squirmer rod, a model microswimmer that

consists of several spherical squirmers, in order to create an elongated body shape.

To realize the swimming mechanisms of bacteria and algae, i.e., pushers and pullers,

we concentrate the surface slip-velocity field on the rear end or front side of the rod.

This generates the force dipole, which these swimming mechanisms exert on the

fluid. Using the simulation method of multi-particle collision dynamics (MPCD),

we analyze the flow fields of the squirmer rods both in the bulk fluid as well as in

the Hele-Shaw geometry, i.e., in narrow confinement between two parallel plates.

Using hydrodynamic multipole expansions, we categorize the different multipole

contributions created by neutral, and pusher squirmer rods in the bulk fluid as well

as in the Hele-Shaw geometry. Thereby, we show how the confinement changes

the radial decay of the flow fields of the force or source multipoles, and hence the

characteristic flow fields compared to the bulk fluid.

Further, we present a detailed study of the collective dynamics of neutral squirmer

rods moving in the midplane of the Hele-Shaw geometry. From small to a large

aspect ratio and density, we observe a disordered state, dynamic swarms, a single

swarm, and a jammed cluster state and characterize them accordingly. We also

investigate a wide range of aspect ratios and densities for pushers and pullers and

provide corresponding state diagrams. The flow field of pushers destabilizes ordered

structures and favors the disordered state at small densities and aspect ratios. As

soon as geometric interactions become relevant for longer squirmer rods, we observe

a turbulent state, as well as a dynamic cluster, while a single swarm and jammed

clusters reappear at large aspect ratios. The power spectrum of the turbulent state

shows two distinct energy cascades at small and large wave numbers, which follow

power laws with non-universal exponents. Pullers show a strong tendency to form

swarms so that no disordered state occurs at the investigated densities. For larger

aspect ratios, the single swarm and jammed cluster are observed again.

Another part of this work deals in more detail with the multi-particle collision

dynamics (MPCD) method, which is widely used in soft matter physics to simulate

fluid flows at micrometer scale. In general, in this model the fluid exhibits the

equation of state of an ideal gas, making it highly compressible. This is in contrast to

real fluids, which are incompressible for velocities well below the speed of sound.

Therefore, we propose a modified collision rule that leads to a MPCD algorithm

with a non-ideal equation of state and a significantly reduced compressibility. At

the same time, our algorithm requires less computational resources compared to

conventional MPCD algorithms. To further establish the algorithm, we provide

analytic expressions for the equation of state and shear viscosity, which show a good

agreement with simulations of the pressure in a fluid at rest, the shear viscosity in

linear shear flow, and the velocity field of a Poiseuille flow. Using two exemplary

squirmer rod systems, we further compare the results of the dynamics under the

extended MPCD method to those with the established MPCD version with Andersen

thermostat. Thereby, we investigate the dynamic swarm state and single swarm

state, which create large pressure gradients due to the sum of the many individual

squirmer-rod flow fields. For the single swarm state the extended MPCD fluid

shows more homogeneous fluid density, and we make the interesting observation

that dynamic swarms are more pronounced and exhibit a higher polar order for the

extended MPCD method.

Zusammenfassung

Unsere Natur ist voll von mikroskopisch kleinen Organismen wie Algen, die an der

Basis der marinen Nahrungskette stehen und damit ganze

Okosysteme begr

unden,

oder Bakterien, die f

ur unsere Ern

ahrung und Gesundheit wichtig sind und auch in

industriellen Prozessen eine breite Anwendung finden. Daher befasst sich ein breites

Forschungsfeld mit den vielf

altigen Aspekten der Biologie, Chemie und Physik

von Mikroorganismen. Physiker:innen interessieren sich dabei besonders f

ur die

Schwimmbewegungen von Organismen wie Algen, Bakterien, Spermatozoen und

Anderen. In kollektiver Bewegung zeigen diese Mikroschwimmer ein vielf

altiges

Verhalten, wie die Bildung von Schw

armen, Wirbeln, Konvektionsrollen und Plumes

sowie aktive Turbulenz. Dabei sind die meisten Mikroschwimmer l

anglich, und

zeichnen sich sonst durch verschiedene Schwimmmechanismen aus. Ihre Dynamik

entsteht folglich durch das Zusammenspiel der verschiedenen Wechselwirkungen

also durch direkte Zusammenst

oße und weitreichenden Flussfelder. Die Rolle der

verschiedenen Wechselwirkungen f

ur die faszinierende kollektive Dynamik zu

bestimmen, ist Gegenstand aktueller Forschung.

In diesem Kontext betrachten wir den Squirmer-Rod, ein Mikroschwimmer-

modell, das wir aus mehreren kugelf

ormigen Squirmern zusammensetzen, um

eine l

angliche K

orperform zu erzeugen. Um die Schwimmmechanismen von Bak-

terien und Algen zu realisieren, d.h. Pusher und Puller, konzentrieren wir das

Oberfl

achengeschwindigkeitsfeld auf die R

uckseite oder die Vorderseite des Rods

und k

onnen so den Kraft-Dipol dieser Schwimmmechanismen generieren. Mit Hilfe

der Simulationsmethode der Vielteilchenstoßdynamik (MPCD) analysieren wir die

Str

omungsfelder dieser Squirmer-Rods sowohl in der freien Fl

ussigkeit, sowie in

der Hele-Shaw-Geometrie, d.h. eingeschlossen zwischen zwei dichten parallelen

Platten. Mittels der hydrodynamischen Multipolentwicklung kategorisieren wir

die unterschiedlichen Anteile der Flussfelder von neutralen und Pusher Squirmer-

Rods in der freien Fl

ussigkeit, sowie eingeschlossen zwischen zwei Platten. Dabei

zeigen wir wie der Einschluss in der Hele-Shaw-Geometrie die radiale Reichweite

der Flussfelder der Kraft- oder Quellmultipole, und somit der charakteristischen

Flussfelder, im Vergleich zur freien Fl¨

ussigkeit ver¨

andert.

Weiter pr

asentieren wir eine ausf

uhrliche Studie der kollektiven Dynamik neu-

traler Squirmer-Rods, die sich in der Mittelebene einer Hele-Shaw-Geometrie bewe-

gen. Von einem kleinen bis zu einem großen L

angenverh

altnis und einer großen

Dichte beobachten wir einen ungeordneten Zustand, dynamische Schw

arme, einen

einzelnen Schwarm, sowie ein blockiertes Cluster und charakterisieren diese ent-

sprechend.

Auch f

ur Pusher und Puller untersuchen wir eine breite Spanne von L

angen-

verh

altnissen und Dichten und liefern entsprechende Zustandsdiagramme. Das

Str

omungsfeld von Pushern destabilisiert dabei geordnete Strukturen und beg

unstigt

den ungeordneten Zustand bei kleinen Dichten und L

angenverh

altnissen. Sobald

iii

geometrische Wechselwirkungen bei l

angeren Squirmer-Rods relevant werden,

beobachten wir einen turbulenten Zustand, sowie ein dynamisches Cluster, w

ahrend

bei großen L

angenverh

altnissen wieder ein einzelner Schwarm und blockierte Clus-

ter auftreten. Die spektrale Leistungsdichte des turbulenten Zustands zeigt dabei

zwei unterschiedliche Energiekaskaden bei kleinen und großen Wellenzahlen die

Potenzgesetzen mit nicht-universellen Exponenten folgen. Puller hingegen zeigen

eine starke Tendenz zur Bildung von dynamischen Schw

armen, sodass bei den dabei

untersuchten Dichten gar keine ungeordneten Zust

ande auftreten. Bei gr

oßerem

angenverh

altnis tritt wieder ein einzelner Schwarm oder ein blockiertes Cluster

auf.

Ein weiterer Teil dieser Arbeit befasst sich eingehender mit der Methode der

Vielteilchenstoßdynamik (MPCD), welche in der Physik der weichen Materie h

aufig

verwendet wird, um Hydrodynamik im Mikrometerbereich zu simulieren. In der

Regel weist in diesem Modell die Fl

ussigkeit die Zustandsgleichung eines ide-

alen Gases auf, wodurch sie stark kompressibel ist. Dies steht jedoch im Wider-

spruch zu realen Fl

ussigkeiten, welche f

ur Geschwindigkeiten weit unterhalb der

Schallgeschwindigkeit inkompressibel sind. Wir schlagen daher eine modifizierte

Kollisionsregel vor, die zu einem MPCD-Algorithmus mit einer nicht idealen Zus-

tandsgleichung und einer deutlich reduzierten Kompressibilit¨

at f¨

uhrt. Gleichzeitig

ben

otigt unser Algorithmus im Vergleich zu herk

ommlichen MPCD-Algorithmen

weniger Rechenressourcen. Um dem Algorithmus ein theoretisches Fundament

zu geben, liefern wir analytische Ausdr

ucke f

ur die Zustandsgleichung und die

Scherviskosit

at, die eine gute

Ubereinstimmung mit Simulationen des Drucks in

einem ruhenden Fluid, der Scherviskosit

at bei linearer Scherstr

omung und dem

Geschwindigkeitsfeld einer Poiseuille-Str

omung zeigen. Anhand zweier exem-

plarischer Squirmer-Rod-Systeme vergleichen wir außerdem die Ergebnisse der

Dynamik der Squirmer-Rods unter der erweiterten MPCD-Methode mit denen der

etablierten MPCD-Version mit Andersen-Thermostat. Dabei betrachten wir die

Zust

ande der dynamischen Schw

arme und des einzelnen Schwarms, die durch

die Summe der vielen Flussfelder einzelner Squirmer-Rods große Druckgradienten

erzeugen. F

ur den Zustand des einzelnen Schwarms zeigt die erweiterte MPCD-

Methode eine homogenere Fluiddichte, und wir machen die interessante Beobach-

tung, dass die dynamischen Schw

arme bei der erweiterten MPCD-Methode st

arker

ausgepr

agt sind und eine h

ohere polare Ordnung aufweisen f

ur die erweiterte

MPCD-Methode.

Contents

Abstract i

Zusammenfassung iii

1 Introduction 1

2 Physics of swimming microorganisms 7

2.1 Hydrodynamicflows............................ 7

2.1.1 Hydrodynamic transport equations . . . . . . . . . . . . . . . 8

2.2 Newtonianfluids .............................. 10

2.2.1 Heatflux............................... 11

2.2.2 The fluid stress tensor . . . . . . . . . . . . . . . . . . . . . . . 11

2.2.3 The Navier-Stokes equations . . . . . . . . . . . . . . . . . . . 12

2.2.4 The Stokes equations . . . . . . . . . . . . . . . . . . . . . . . . 13

2.2.4.1 Fundamental solution of Stokes flow . . . . . . . . . 14

2.2.5 Higher order singularities . . . . . . . . . . . . . . . . . . . . . 15

2.2.5.1

Multipole expansion for axisymmetric microswimmers

2.3 Swimming at low Reynolds number . . . . . . . . . . . . . . . . . . . 18

2.3.1 Flow field of a moving sphere . . . . . . . . . . . . . . . . . . . 18

2.3.2 Biological microswimmers . . . . . . . . . . . . . . . . . . . . 19

2.3.3 Flow fields of elongated particles . . . . . . . . . . . . . . . . . 20

2.4 Swimming in confinement . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.4.1 Stokes flow near a plain wall . . . . . . . . . . . . . . . . . . . 21

2.4.2 The Hele-Shaw geometry . . . . . . . . . . . . . . . . . . . . . 22

2.4.3 Multipole expansion in the Hele-Shaw geometry . . . . . . . . 25

2.5 Hydrodynamic interactions . . . . . . . . . . . . . . . . . . . . . . . . 26

2.5.1 Fax´

en’slaw ............................. 27

2.5.2 Jeffery orbits of elongated microswimmers . . . . . . . . . . . 28

2.5.3 Hydrodynamic interaction between microswimmers . . . . . 28

2.6 The squirmer model for microswimmers . . . . . . . . . . . . . . . . . 30

2.6.1 The squirmer-rod model . . . . . . . . . . . . . . . . . . . . . . 31

2.7 Brownianmotion .............................. 32

3 Multi-particle collision dynamics (MPCD) 35

3.1 Introduction................................. 35

3.2 Streamingstep................................ 36

3.3 Collisionstep ................................ 37

3.3.1 Stochastic rotation dynamics (SRD) . . . . . . . . . . . . . . . 37

3.3.2 MPCD with Andersen thermostat . . . . . . . . . . . . . . . . 38

3.3.3 Equationofstate .......................... 39

3.3.4 Shearviscosity ........................... 41

3.4 Extended MPCD method with non-ideal equation of state . . . . . . 43

3.5 Boundary conditions and solute particles . . . . . . . . . . . . . . . . 44

3.6 Molecular dynamics of solute particles . . . . . . . . . . . . . . . . . . 45

3.7 Efficient implementation of MPCD on graphic cards (GPUs) . . . . . 48

4 Published Articles of the Thesis 51

4.1 Synopsis ................................... 53

4.2

Squirmer rods as elongated microswimmers: flow field and confinement

4.3 Multi-particle collision dynamics with a non-ideal equation of state. I 70

4.4

Multi-particle collision dynamics with a non-ideal equation of state.

II. Collective dynamics of elongated squirmer rods . . . . . . . . . . . 84

4.5

Emergent collective dynamics of pusher and puller squirmer rods:

swarming, clustering, and turbulence . . . . . . . . . . . . . . . . . . 96

5 Conclusions and Outlook 109

References II

Danksagung XIX

CHAPTER 1

Introduction

Despite often unnoticed in our everyday life, our worlds flora and fauna consist of

microscopic organisms to a major fraction. Their omnipresence becomes especially

obvious when regarding the plant biomass that performs photosynthesis in the

carbon cycle. In fact, microscopic swimming algae amount to about 50% of the

total biomass involved in the carbon cycle, as estimated by satellite measuring of

chlorophyll concentration [1–3].

Bacteria exist right next to us and even in our bodies. For example, the normal

strains of the well-known bacterium Escherichia coli are a part of our gut microbiota

[4], where it plays an important role for nutrition, our health, our immune system

[5–8] and even performs tasks such as the production of vitamins [9].

Also in industrial processes, microorganisms find a broad application. In biotech-

nological and medical industry, straits of the E. coli and Bacillus subtilis bacteria are

used for the synthesis of different biochemicals. Examples are the production of bio

fuels [10–12], enzymes for the food and other branches of industry [13–16], ecological

plant protectives [17, 18], pharmaceuticals [19, 20] or supplemental vitamins [21].

Another important application is found in the process of waste water recycling [22,

23]. As this collection implies, a broad research field is concerned with the biology,

chemistry and physics of microorganisms and the many aspects of their behavior is

subject of current research.

For physicists, these self-propelled organisms – algae, bacteria, spermatozoa

and others – are particularly interesting because they swim in aqueous solutions.

Rather than only passively diffusing around, they perform an active swimming

motion to explore their environment [24, 25]. For this reason, physicists speak of

microswimmers. The microswimmers’ active motion introduces a whole variety of

new fascinating phenomena that go beyond the physics of passive particles [26, 27].

We illustrate this with a few examples from the dynamics of microswimmers.

In order to steer or navigate, microswimmers have developed various strategies

like the run-and-tumble motion pattern [28, 29]. Through the modulation of the time

1. Introduction

between tumbling events in response to chemical concentrations microswimmers

are able to approach nutrients [30–34]. For sperm cells, steering in response to the

rheology of the fluid [35] and boundary-following navigation [36] is observed.

In collective motion, microswimmers exhibit a particularly interesting and di-

verse dynamic behavior that arises form their complex interactions [37, 38]. Ex-

amples are the formation of swarms [39–42] or vortices [43–46], convection rolls

and plumes [47], fluid pumps [48, 49], active nematic patterns [50–52], and active

turbulence [53–57].

Thereby, the phenomenon of active turbulence is currently receiving special

attention, because it forms an interesting counterpart to the classical turbulence in

passive fluids. While turbulence in passive fluids is caused by external driving,

active turbulence occurs due to the motion of the self-propelled particles. In both

cases, the supplied energy is transported and dissipated over different length scales.

Thereby, active turbulence exhibits a specific length scale, which appears as the

characteristic size of vortices and patterns, whereas classical turbulence is scale

invariant [58–60]. Furthermore, the power spectrum of the velocity fluctuation is no

longer universal but depends on the specific type of active particles and parameters

[60–62]. Systems, which exhibit active turbulence, include bacteria [53], bio-filaments

within the cytoplasm [50, 56] and cells in growing tissue [63, 64].

On the one hand, microswimmers interact through steric collisions, which typ-

ically cause an alignment due to their mostly elongated body shapes [24]. On the

other hand, swimming microorganisms interact via long-ranged flow fields created

by their swimming motion [65]. This second type of interaction depends funda-

mentally on the biological propulsion strategy of the species under consideration.

While bacteria propel with flagella that extent from the back of the cell body, also

other propulsion strategies exist [65, 66]. For example, algae like Chlamydomonas

reinhardtii propel with two short flagella that extent from the front of their body

and perform a breast stroke to swim forward [67, 68]. Other organisms like Parame-

cium are covered entirely with hair-like appendages called cilia, which perform a

synchronized wave-like motion to propel [69, 70]. On the microscopic scale, where

hydrodynamic flows are dominated by viscous dissipation, these mechanisms gener-

ate entirely different long-ranged flow fields, which results in the impressive variety

of observed phenomena [71–76]. Besides experiments with biological entities like

bacteria, synthetic microswimmers like the famous Janus particles [25, 77] or active

emulsion droplets [78, 79] are employed. However, the physics of active matter is

not limited to microswimmers, but is also found with slightly different mechanisms

or interactions in herds of sheep, flocks of birds, school of fish or human crowds.

On the theoretical side, a range of different particle-based models are employed.

Examples range from the famous Vicsek model that employs coarse-grained interac-

tions [80–82], to Langevin dynamics simulation with more realistic steric interactions

[83–88] and implicit hydrodynamic interactions [89, 90]. Models which explicitly

simulate the flow field use the method of multi-particle collision dynamics [41, 47,

73, 74, 76, 91–97] or the lattice Boltzmann approach [98–101].

A suspension of microswimmers can also be considered in the continuum limit,

where they become the constituents of an active fluid [102, 103]. Continuum models

for polar active particles, which combine elements of the Toner-Tu [104] and Swift-

Hohenberg [105] equations, are able to reproduces active turbulence [58, 106] or

clustering [107]. In general, continuum models are distinguish between polar or

nematic symmetry, and dry or wet systems. For example, continuum models for wet

active nematic fluids include equations of motion for the order-parameter field and

extend the constitutive relation of stress tensor by an active contribution [108].

Nevertheless, it is still subject of current research to precisely determine the

influences of both steric and hydrodynamic interactions of the constituting particles

on the swarm formation, active turbulence, or other phenomena [75, 109–113]. In

this context, we introduced the elongated squirmer rod as a model for elongated mi-

croswimmers in their fluid environment. The squirmer rod thereby extend the well

known squirmer model swimmer [69, 74, 114] for spherical microswimmers such

as synthetic Janus particles [115–117] or biological organisms such as Volvox [118].

Commonly, squirmers propel themselves by imposing an axisymmetric surface-

velocity field on the surrounding fluid [74, 119, 120], which models the propulsion

through the beating motion of cilia in biological systems. The spherical squirmer

has been employed in various hydrodynamic studies of the collective behavior of

microswimmers [73, 74, 92, 94, 100, 121–124]. To model the hydrodynamic flow of

the solvent, we use the method of multi-particle collision dynamics (MPCD) [125].

In general, MPCD is well suited for numerical simulations of highly viscous hydro-

dynamic flows in the context of microscopic solute particles. As stochastic solver,

it includes thermal fluctuations [126, 127] and also complex geometric boundaries

can be implemented precisely with this method [128–131]. Furthermore, the MPCD

algorithms are well suited for parallel computing on graphic cards, such that large

systems can be simulated efficiently already on desktop computers [132, 133]. Be-

sides the application to model solvent dynamics in the context of microswimmers,

MPCD algorithms have been used to study colloidal suspensions [134–138], poly-

mers [139–141], blood cells [142], the African trypanosome as the causative agent of

the sleeping sickness [143], or fish schools [144]. The MPCD method itself has been

extended to multi component fluids [145], liquid crystals [146–148], or fluids with

chemical reactions [149].

Scientific questions

The large variety of different microswimmers described so far is mainly character-

ized by different body shapes and the flow fields generated by specific locomotion

mechanisms. Despite the multitude of microswimmers now known to researchers, it

remains an open question to identify the respective influences of mutual interaction

by hydrodynamic flow fields and steric collisions on collective behavior [109, 110,

1. Introduction

Figure 1.1: Different interactions between colliding microswimmers. (a) Dominated

by steric interaction, the collision leads to an alignment of the particles. (b) Hydrody-

namic interactions can cause a torque on both particles, which leads to a deflection

[65].

112].

With the introduction of the elongated squirmer rod, we therefore first want to

explore the direct relationship between the flow field and the cell body shape. We

also investigate the influence of narrow confining walls in a Hele-Shaw geometry

on the hydrodynamic flow fields of the squirmer rods. This geometry enables us

to better understand the physics in experimental systems such as Petri dishes or

microscope slides. Furthermore, we also want to use our model to investigate the

respective influences of the hydrodynamic and steric interactions on the collective

dynamics of many microswimmers.

As depicted in Fig. 1.1(a), purely steric interactions mainly lead to an alignment

of elongated microswimmers such that they preferentially form swarming clusters

[150]. In contrast, hydrodynamic interactions can lead to an aligning or anti-aligning

torque depending on the type of swimmer and relative position, as shown for

pushers in Fig. 1.1(b) [65, 94].

Outline

This thesis is written in cumulative form, and organized as follows. Chapter 2

introduces the theoretical background of the physics of swimming microorganisms.

To begin with, the hydrodynamic transport equations that satisfy the continuity

equations for mass, momentum and energy, are discussed. After motivating New-

tonian fluids, the limit of highly viscous Stokes flow is discussed. On the basis of

hydrodynamic multipoles, the swimming strategies of different microorganisms are

reviewed and the interactions due to their characteristic flow fields are discussed.

Lastly, the squirmer-rod model for elongated microswimmers is introduced.

Chapter 3 introduces the multi-particle collision dynamics (MPCD) method, a

stochastic solver for hydrodynamic flows. Here, different variants of the original

algorithm are compared. Subsequently, the coupling to solute particles and inte-

gration of their equation of motion are discussed. We end with some notes on the

efficient implementation of MPCD on modern computer graphic cards.

Since this thesis is written in cumulative form, the results of the research will be

presented in the form of their publication in scientific journals. The relevant four

articles are introduced and set into context in chapter 4. The first article in Sec. 4.2

introduces the squirmer-rod model and explores the resulting flow field for different

aspect ratios in the bulk fluid and in confinement between two parallel plain walls.

In Sec. 4.3 we introduce an extended variant of the MPCD method with a non-ideal

equation of state for the pressure, and we present the application to simulations of

squirmer rods in Sec. 4.4. Thereby, we present an in-depth study of the collective

dynamics of neutral squirmer rods for different aspect ratios and area fractions. In

the last article in Sec. 4.5 we then investigate the collective dynamics of pusher and

puller-type squirmer rods. Chapter 5 draws conclusions from the combination of

the articles and gives responses to the scientific questions that were posed above.

1. Introduction

CHAPTER 2

Physics of swimming

microorganisms

As we saw at the beginning of chapter 1, the large variety of motile microscopic

lifeforms predominantly lives in aquatic habitats in the oceans, lakes or other bodies

of water. Consequently, we approach the subject of active microswimmers starting

from the physics of fluids in motion. This will lead us from the more general

hydrodynamic transport equations to the Newtonian fluids, which are described by a

linear relation of the fluid stress and the strain rate. The dynamics of microswimmers

is further characterized by the limiting case of highly viscous flow. In the context

of this so-called Stokes flow, we introduce the theoretical framework to understand

the hydrodynamic interaction of microswimmers. As we will see in the following

sections, the fluid environment introduces fascinating hydrodynamic interactions

between organisms. Together with the active motion of the particles, this leads to a

particularly rich collective behavior.

2.1 Hydrodynamic flows

The physical properties of the matter that surrounds us — gases, liquids and solids

— can be described at different levels of detail. In our everyday life matter appears as

continuum because the complex microscopic state and dynamics smear out on the

macroscopic scale of the physical processes relevant for us. From our macroscopic

perspective, the number of the constituting particles is so large that even for a small

parcel of matter the microscopic dynamics can be described with the framework of

statistical mechanics. Thus, we may work with the dynamics of an ensemble average

because the microscopic detail vanishes in the thermodynamic limit. For the relevant

physical phenomena, this suggests a classical continuum description of matter, for

the dynamics of the fields of macroscopic variables like mass, momentum and energy.

In the case of gases and liquids, the constituting particles are not bound to a fixed

2. Physics of swimming microorganisms

configuration such that their continuum description deals with deformation, flows

and transport. Due to this similarity, gases and liquids are typically combined under

the terminology of fluids.

Hydrodynamic flows are most commonly described by the so-called Navier-

Stokes equations, which apply to conventional length scales much larger than the

inter-particle distance. In addition to the simpler Euler equations, they take into

account internal friction in the fluid. In their mathematical form, they are a contin-

uum formulation of the basic physical principle of the conservation of momentum

with the addition of constitutive relations for the viscous stress tensor for the class of

Newtonian fluids. These equations governing the motion of fluids can be derived

starting from different basic premises. Most fundamentally, starting from the physics

of collisions between the constituting particles in the kinetic theory of gases, the

famous Chapman-Enskog approach can be used to derive hydrodynamic equations

of motion to different order in the Knudsen number from the Boltzmann equation

[151, 152]. The Knudsen number is thereby defined as the ratio of the mean free path

for the fluid particles to a characteristic physical length scale. Here, we follow a more

direct approach that starts with continuity equations as first principles and directly

leads to the hydrodynamic transport equations following Ref. [153]. In Section 2.2,

we then introduce the constitutive relations for the stress tensor and heat flux in a

Newtonian fluid, which then leads us to the Navier-Stokes equations.

2.1.1 Hydrodynamic transport equations

In the following, we derive the hydrodynamic transport equations following the

seminal work of Irving and Kirkwood [153]. In this approach, we derive continuity

equations for the fluid density, momentum and energy. For the momentum and

energy, pressure and viscous forces will become relevant.

We begin by considering a small stationary volume

inside the fluid of mass

density

ρ(r

. The local velocity of the fluid is described by the vector field

u(r

The rate of change in the total mass of the volume is due to transport of fluid mass

by the velocity field u(r,t)through the surface BVof the volume V

Btm(r,t) = BtżV

ρ(r,t)d3r=´żBV

ρ(r,t)u(r,t)¨dA. (2.1)

Here, d

is an infinitesimal oriented surface element normal to

. For simplicity

and clarity, we will omit writing the dependence of the fields

and

on the position

and time

in the following. According to Gauß’ theorem, the surface integral can

be rewritten as a volume integral

żBV

ρu¨dA=żV

∇¨(ρu)d3r. (2.2)

2.1 Hydrodynamic flows

Since the choice of the integration volume

is arbitrary, the equality also applies to

the integrand, so that with Eq. (2.1) we obtain the continuity equation for the mass

density

Btρ=´∇¨(ρu). (2.3)

If we consider an incompressible fluid (

∇¨u=

)

, we can use the so-called sub-

stantial or co-moving derivative

D(˝)

Dt= (Bt+u¨∇) (˝)

to rewrite Eq. (2.3) as

Dρ

Dt=

Put in words, this means that as we move with a fluid parcel, the density of the

incompressible fluid remains constant.

To obtain the continuity equation for the momentum transport, we consider the

rate of change in total momentum

BtżV

(ρv)d3r(2.4)

in the fluid volume

. As for mass transport, we consider the transport of the

three components of momentum with the velocity field through the surface of the

stationary volume

. In vector notation this momentum flux through the surface is

żBV

(ρubu)dA=żV

∇(ρubu)d3r(2.5)

However, in contrast to the conserved mass regarded before, the momentum of

the fluid is also changed by forces that act on the fluid inside

and at the surface

. These are (i) external forces such as gravity, that act in the whole volume,

and (ii) pressure and frictional surface forces arising from the interaction with the

neighboring fluid. The total external force (i) follows from the integration over the

whole volume of the force densities fext

żV

fext(r,t)d3r. (2.6)

The surface forces (ii) are described by the symmetric stress tensor

σ(r

that

assigns every infinitesimal surface element d

an infinitesimal surface force d

fA=

σ(r

. The specific form of the stress tensor

depends on the physical properties

of the fluid under consideration, and will be addressed later. To obtain the total

surface force

FA(t)

, the infinitesimal surface force is integrated over the whole

surface BV

FA(t) = żBV

dfA=żBV

σ(r,t)dA=żV

∇σ(r,t)d3r, (2.7)

where we have used Gauß’ theorem again. In total, the change of momentum in

Eq. (2.4) is equal to the sum of Eq. (2.5), (2.6) and (2.7). Again using the arbitrary

choice of the integration volume

, we obtain Cauchy’s equation of motion for the

2. Physics of swimming microorganisms

momentum transport

Bt(ρv) + ∇(ρubu) = ∇σ(r,t) + fext(r,t). (2.8)

Here, the external force acts as a source or sink of momentum, and can be, for

example, a uniform gravitational field fext =ρg.

To complete the description of the fluid flow, we also have to derive an equation

for the rate of change of energy. As for mass and momentum, we consider the energy

E(r

in an arbitrary stationary volume

in the fluid. This energy changes due to

advective transport, heat conduction, heat production and transfer of mechanical

work through the surface

. Similar to before, integrals of flows across the surface

are transformed using Gauß’ theorem, such that we arrive at the equation for the

energy transport

BtE+∇¨(Eu+q´σu) = 0. (2.9)

Here,

is the advective flow of energy,

is the heat current yet to be defined. The

term

∇(σu)

describes the heat production due to friction and work performed by

the pressure. For isothermal systems, where the energy transport is not relevant for

the dynamics, the evaluation of Eq. (2.9) can be omitted. This is also the case for the

hydrodynamic flows around bacteria, which we consider in this work.

As became apparent in this section, the present hydrodynamic transport equa-

tions are quite general requiring only basic physical conservation laws and basic

vector calculus. However, we still need expressions for the viscous stress tensor

the heat flux

and the thermodynamic pressure

, to close the set of equations [154].

These so-called constitutive relations are defined by the transport phenomena of the

physical fluid under consideration. In general they can be very model specific. In

the next section we derive the constitutive relations for the most common class of

the Newtonian fluids based on simple assumptions on the underlying materials.

2.2 Newtonian fluids

In the most simple case that applies to most ordinary liquids, the shear stress and heat

flux both depend linearly on the shear rate and temperature gradient, respectively.

This common behavior is expressed in the constitutive relations of the so-called

Newtonian fluids [151].

The hydrodynamic behavior of the Newtonian Fluids is described by the Navier-

Stokes equations, which combines these constitutive relations and the transport

equation derived above. In other words, Newtonian fluids are characterized in how

they react to deformations in the velocity field. To give a few counter-examples,

non-Newtonian fluids can exhibit shear stress which is non-linear in the strain rate

or viscoelastic stress that depends on the history of the deformation [154].

In the following, we first rationalize the constitutive relation of the heat flux. We

2.2 Newtonian fluids

then describe how stress arises due to internal friction and pressure in the fluid and

express the dependence of the viscous part of the stress tensor on the velocity field.

Inserting the constitutive relation of the stress tensor into the Cauchy momentum

equations, we are then able to derive the Navier-Stokes equations.

2.2.1 Heat flux

Our intention is to express the conduction of heat in the most simple, linear depen-

dence on the temperature gradient. To justify this at the microscale, we have to

consider the transport of heat between two neighboring fluid elements due to molec-

ular motion and interactions. Clearly, the distribution of thermal energy has to vary

sufficiently smoothly in order to describe the fluxes resulting from inhomogeneities

as linear function locally. In view of the molecular processes, this requires that the

temperature gradient

∇T

itself does not vary significantly over the length scale of

the molecular interaction

,i.e.

|BxT|/|B2

xT| " λ

, which is usually satisfied in practice

[155]. In addition, the second law of thermodynamics requires that the heat flux

vanishes in the uniform distribution. Given these assumptions, the heat current can

be described by Fourier’s law of heat conduction

q=´κ∇T

[154]. Here

denotes

the scalar thermal conductivity that applies for isotropic materials [155]. It is worth

noting that, with this definition, the energy transport equation describes diffusion of

thermal energy in the absence of flow, u=0.

2.2.2 The fluid stress tensor

The constitutive relation for the stress tensor can be derived as follows [156, 157].

First, we separate out the thermodynamic pressure

which is a material property of

the fluid at equilibrium and not related to the derivatives of the velocity field

v(r)

Thus, we write

σ=´pI+σ1,

where

is the identity matrix and the second term

σ1

the viscous stress tensor. In contrast to the pressure contribution, viscous stress

arises from the friction of two neighboring fluid layers moving relative to each other.

The viscous stress tensor is therefore only proportional to derivatives of the velocity

field and vanishes when the fluid is in uniform motion or at rest. For small velocity

gradients it is sufficient to consider only the first order derivatives. The fact that

no internal friction occurs in uniform rotational fields further constrains

σ1

to the

symmetric part of the velocity gradient. All together, we obtain the following for the

viscous stress tensor

σ1=η[︃(∇bu)T+∇bu´2

3I(∇¨u)]︃+η1I(∇¨u). (2.10)

Here, the first term in square brackets is the traceless rate of deformation tensor

while the second term describes pure isotropic compression given by the trace of the

symmetric velocity gradient. The viscosity coefficients

and

η1

are thereby intrinsic

2. Physics of swimming microorganisms

properties of the fluid and independent of the fluid motion, but depend on the

density

and temperature

of the fluid. The viscosity

η1

related to volume changes

is generally referred to as bulk viscosity and is sometimes written as

η1=γ´

η/

with the new parameter

, to rearrange Eq. (2.10) [157]. Inserting the stress tensor

into Eq. (2.9) for the hydrodynamic transport of energy one obtains the conditions

that

ηě

0 and

η1ě

0 to guarantee the positivity of the entropy production rate in

accordance with the second law of thermodynamics [154].

In the case of incompressible fluids (

∇¨u=

0) the viscous stress tensor reduces

σ1=η[︂(∇bu)T+∇bu]︂. (2.11)

This simplification is justified if the typical flow speed

|u|

is much smaller than the

speed of sound

usound

and the Mach number M

=|u|/usound !

1 is small. In this

case, the motion of the fluid

is not subject to compressibility effects, and that the

fluid appears incompressible [154].

2.2.3 The Navier-Stokes equations

Using the continuity equation (2.3), we may first transfer the Cauchy momentum

equation from its conservation form to the convective form

ρ(Btu+u¨∇u)=∇σ+fext. (2.12)

Since the typical flow velocity in the context of microscopic self-propelled particles is

very small compared to the speed of sound, we restrict ourselves to the case of incom-

pressible fluids

∇¨u=

0 in the following. Consequently, also the density

ρ(r

t) = ρ

becomes a constant. Inserting the stress tensor in the above equation, we note that

the divergence of the first term

η(∇bu)T

vanishes due to the incompressibility

condition, such that we arrive at the Navier-Stokes equations for incompressible

fluids

ρ(Btu+u¨∇u)=´∇p+η∇2u+fext. (2.13)

In this form, a moving fluid parcel is accelerated due to gradients of pressure and

external force densities. Additionally, it exchanges momentum with the rest of the

fluid via momentum diffusion, described by the second order derivative

η∇2u

. To

solve the Navier-Stokes equations for the unknown velocity field

u(r)

, additional

boundary conditions are necessary. In the case of solid walls this is the so-called

no-slip condition, which states that

u(r) =

0 at the wall. Note that the co-moving

derivative on the left hand side of the equation is non-linear in the velocity which

greatly complicates solving it in practice [158].

To further characterize the flow, we require a knowledge of the geometry and the

typical flow velocity such that characteristic scales of the length

, flow velocity

and time

can be specified. Using these characteristic parameters along with the

2.2 Newtonian fluids

density

and viscosity

the Navier-Stokes equation can be converted into a non-

dimensional form that contains only a few dimensionless parameters that further

characterize the flow [154, 159]. In this manner we use the characteristic length

flow velocity

and time

T=L/U

to introduce dimensionless variables according

˜”u/U,t

˜”tU/L,∇

Ă”L∇,p

˜”pL/ηU,f

˜”fextL2/ηU.

For a flow inside a channel,

is the maximum velocity in the center, while in an

unbounded flow it is chosen as the flow velocity at infinity. For a microswimmer

corresponds to the self-propulsion velocity. These choices are not unique, especially

the choice of the inherent time scale

T=L/U

applies to stationary flow problems,

and the non-dimensional pressure may also be defined as

˜=p/ρU2

[154]. Inserting

these definitions into the Navier-Stokes equations, we obtain the dimensionless form

Re (︃Bu

˜+u

˜¨∇

Ău

˜)︃=´∇

˜p

˜+∇

Ă2u

˜+f

˜. (2.14)

Here, we have introduced the Reynolds number

Re ”ρLU

η, (2.15)

which describes the ratio between the inertial forces (numerator) and viscous forces

(denominator) of the Navier-Stokes equations [160]. An important implication for

systems with the same Reynolds number is that their flows are similar and can be

transformed into each other by a simple re-scaling of the lengths, velocities, and time

[156]. As a very illustrative consequence, aerodynamic studies can be performed

using miniaturized models immersed in water. In this particular example the change

in length scale

and viscosity

cancel in the Reynolds number and the two flows

are called similar. For large Reynolds numbers the non-linear term

u¨∇u

is relevant

and makes the general solution of the Navier-Stokes equation complicated. In this

case, steady flows are unstable to small disturbances, which may lead to turbulent

flows [154, 160]. Of particular interest for us is the case of low Reynolds number

Re !

1, where the Navier-Stokes equation can be further simplified to the so-called

Stokes equation.

2.2.4 The Stokes equations

For hydrodynamic flows in the context of bacteria and other microscopic particles,

length scales and typical flow velocities are small. Both of these conditions imply

that the Reynolds numbers

Re !

1 is small and thus the hydrodynamic flows are

dominated by the viscous terms on the right-hand side of

Eq. (2.14)

of the Navier-

Stokes equation. For this reason microscopic flows are termed strongly viscous [157].

Given these circumstances, the inertial terms on the left hand side of Eq. (2.14) can

2. Physics of swimming microorganisms

be neglected, and we arrive at the so-called Stokes equations for creeping motion

[159]

η∇2u´∇p+f=0

∇¨u=0 (2.16)

in terms of the dimensional variables. However, one should note that low Reynolds

numbers may also occur in very dilute aerodynamic flows [157].

The Stokes equations (2.16) manifest some important implications for hydro-

dynamic flows at low Reynolds numbers. Most crucially, without the convective

derivative in the inertial term the Stokes equation becomes a linear partial differ-

ential equation of the velocity and the pressure, such that flow problems can be

solved using Green’s formalism and boundary integral methods [159]. Another

consequence is the absence of a time derivative, such that solutions of the Stokes

equations are instantaneous and steady flows.

Note that there is also the so-called transient Stokes flow which also comprises the

acceleration term

ρBtu

. This hydrodynamic regime appears when the characteristic

acceleration time does not occur on the slow time scale

L/U

chosen in Eq. (2.14)

before, but over the much faster time scale

L2ρ/η

that is required for the vorticity

to diffuse over the characteristic length

. In this case the acceleration term

ρBtu

of the same order as the viscous terms and only the nonlinear term

u¨∇u

can be

neglected for the Navier-Stokes equations [154, 158].

2.2.4.1 Fundamental solution of Stokes flow

The fundamental solution or Green’s function for Stokes flow is the flow field

generated by a point force acting on the fluid. It can be derived from the Fourier

transformation of the Stokes Eq. (2.16) and the incompressibility condition [158].

To account for an arbitrary orientation of the force the most general form of this

solution has the tensorial form

O(r) = G(r)

8πη =1

8πη (︃I

r+rbr

r3)︃, (2.17)

for a point force located at the origin [161]. Thereby, the term

O(r)

is called the

Oseen tensor, but often we use

G(r)

in calculations without the factor

(

πη)´1

for

simplicity. The associated pressure field has the form

p(r) = 1

4π

r3. (2.18)

The solution

uFM(r) = O(r)F

and

p=p(r)¨F

for a given point force

is called

Stokeslet. Due to its appearance in response to a point force, the Stokeslet is also

called hydrodynamic force-monopole. The flow field for an arbitrary distribution of

2.2 Newtonian fluids

forces F(r)can then be obtained by convolution with the Oseen tensor [161]

u(r) = żV

O(r´r1)F(r1)d3r1(2.19)

and the associated pressure field

p(r) = żV

p(r´r1)¨F(r1)d3r1. (2.20)

Additional to the Stokeslet, the Stokes equations have another branch of solutions

for constant pressure

. In this case, the Stokes equation takes the form of the Laplace

equation with the vectorial solution [159]

uSM(r) = cr

r3, (2.21)

which describes a sink or a source with a flux of fluid into or out of the origin,

depending on the sign of the constant c.

2.2.5 Higher order singularities

Similar to electrostatics, the flow field of an arbitrary object can the be expressed

as a series of multipoles. To do so, the Oseen tensor in Eq. (2.19) for an arbitrary

disturbance is expressed in a Taylor series

G(r´r1) =

n=0

(´r1¨∇)n

n!G(r). (2.22)

We may then separate the primed and unprimed variables. The integration over

the primed variable yields the multipole coefficients of

F(r)

, while the higher-order

gradients of

G(r)

represent the Stokes multipole moments, which are also solutions

to the Stokes equations. Note that in the context of the multipole expansion the factor

(

πη)´1

is integrated into the multipole coefficients so that we write

ui(r) = fjGij(r

˜)

with fj=Fj/(8πη)for the Stokeslet, and so on.

In the most general form, the Stokes multipoles moments are directional deriva-

tives

śn

i=1(d

ˆi¨∇1)n

that differentiate in the direction of the unit vectors

ˆi

with

respect to the position of the singularity. In the following we will write these deriva-

tives as gradients of the force and source monopole fields such that a singular solu-

tion of

-th order is described by a tensor of rank

(n+

)

. The flow fields then follow

from the contraction with a tensorial multipole coefficient

d1,...,i=d1bd2b. . . bdn

of rank-(n+1)that describes the directional derivatives [158, 159].

Applying one directional derivative to the source monopole results in the flow

field of two sources with opposite sign, separated by the vector

. Denoting the

distance to the singularity with

r=|r

˜|

, where

˜=r´r1

and

is the position of the

2. Physics of swimming microorganisms

singularity, the resulting source dipole is described by the tensor

GSD

ij (r

˜) = B

Br1

˜j

˜3=´δij

˜3+3r

˜ir

˜j

˜5, (2.23)

and the resulting flow field is

uSD

i(r) = djGSD

ij (r

˜)

(using the Einstein sum conven-

tion). The source dipole is also related to the force monopole through the relation

GSD

ij (r

˜) = ´1

2∇12Gij(r

˜).

Higher order solutions derived from the Oseen tensor are the force dipole

GFD

ijk (r

˜) = B1

kGij(r

˜) = δijr

˜k´δikr

˜j´δjkr

˜i

˜3+3r

˜ir

˜jr

˜k

˜5, (2.24)

and the force quadrupole

GFQ

ijkl(r

˜) = B1

lGFD

ijk (r

˜)

. Two other notable solutions are

a uniform rotation of the flow field — also called rotlet — which follows from

the combination of two force dipoles, i.e., anti-symmetric part of the gradient of

the Stokeslet

ij (r

˜) = ´1

2ϵkljBlGik(r

˜) = ϵijk r

˜k

˜3

and its gradient, the rotlet dipole

GRD

ijk (r) = B1

kϵijl r

˜l

˜3

[159]. The latter of these two is encountered in the context of

microswimmers for the rotating flagella of bacteria [162]. For microswimmers,

additional symmetry arguments are used to simplify the tensorial forms as we show

in the following.

2.2.5.1 Multipole expansion for axisymmetric microswimmers

We investigate the multipole expansion of the flow field

u(r)

of our model mi-

croswimmers to provide an understanding of their hydrodynamic pair interactions.

To introduce the multipole expansion in the context of microswimmers, a few simpli-

fications are made. First of all, we consider microswimmers as axisymmetric around

their direction of swimming

[65, 74]. In this case, the symmetry requires that all gra-

dients in Sec. 2.2.5 reduce to directional derivatives of the Stokeslet

ˆ¨∇1)n[G(r

˜)e

ˆ]

and source dipole

ˆ¨∇1)n[GSD(r

˜)e

ˆ]

and all multipole coefficients reduce to scalars.

Furthermore, self-propelled particles are typically free of external forces and do not

consume fluid such that the force and source monopole moments are not present in

their flow fields [65, 162]. In Fig. 2.1 we show a collection of the first axisymmetric

multipoles beginning with the Stokeslet [Fig. 2.1(a)] and its first two derivatives

[Figs. 2.1(b) and (c)], along with the source monopole in Fig. 2.1(d) and its first two

derivatives [Figs. 2.1(e) and (f)].

At large distances to the microswimmer located at the origin of the coordi-

nate system, the flow field can then be written as a sum of the velocity fields of a

force dipole (uFD)

, source dipole (

uSD

), force quadrupole (

uFQ

), and

rotlet dipole (uRD)

u(r) = uFD(r) + uSD(r) + uFQ(r) + uRD(r) + O(r´4)(2.25)

2.2 Newtonian fluids

Figure 2.1: Flow fields of different singular solutions of Stokes flow in the laboratory

frame of reference. The top row shows the solutions based on point forces (a)

Stokeslet/hydrodynamic force monopole (

uFM

), (b) force dipole (

uFD

), (c) force

quadrupole (

uFQ

). The bottom row shows the potential flow solutions for constant

pressure

, namely (d) source monopole (

uSM

), (e) source dipole (

uSD

), (f) source

quadrupole (uSQ)

up to 3-rd order in 1

. Among these moments, the force-dipole field

|uFD|9

/r2

has the longest range. If it is present for a type of microswimmer, it is usually the

dominant mode for pair interactions in the far field [74]. In the following, we omit

the rotlet dipole since it does not appear for the microswimmer model used in this

work.

In the case of axisymmetric flow fields, the multipoles can be expressed using

spherical coordinates with the swimmer located at the origin of the coordinate

system [69, 74, 114]. We introduce the polar angle

relative to the swimmer axis

according to

cos θ=e

ˆ¨r/r

. Then, in spherical coordinates

, and

the respective

radial and polar components of the multipole expansion of the velocity field become

ur(r,θ) =

n=1[︂Anr´n+Bnr´n´2]︂Pn(cos θ), (2.26)

uθ(r,θ) =

n=1[︄(︃n

2´1)︃Anr´n+n

2Bnr´n´2]︄Vn(θ). (2.27)

2. Physics of swimming microorganisms

Here, Pn(cos θ)are the ordinary Legendre polynomials and

Vn(θ) = 2 sin θ

n(n+1)P1

n(cos θ), (2.28)

where

denotes the derivative with respect to

cos θ

. The coefficients

and

describe the strength of the respective

nth

-order force and source multipoles, e.g.,

belongs to the source dipole. Due to the restriction to axisymmetric flow fields

ur(r

θ)

and

uθ(r

θ)

are independent of the azimuthal angle

. Furthermore, the

azimuthal velocity component

uϕ

vanishes since the swimmer is torque-free and we

do not consider a rotlet-dipole or derivatives of it.

The form of Eq. (2.26) may be used to decompose a flow field

u(r)

measured in

experiments or simulations into the series of multipole moments

and

[163].

To determine the terms

and

, we calculate the

nth

Legendre coefficient of the

radial component ur(r,θ)from the integral

ur,n(r)”2n+1

2żπ

ur(r,θ)Pn(cos θ)sin θdθ

=Anr´n+Bnr´n´2, (2.29)

due to the completeness relation of the Legendre polynomials. With the knowledge

of a flow field and, in particular, the radial decay of the

nth

Legendre coefficient

ur,n(r)

, we can then infer the multipole coefficients

and

. Force and source

multipole moments can thereby be distinguished using their different radial decay.

Note that for a spherical rigid body each pair of

and

for

ną

1 is not

independent because the radial velocity

ur(r

θ)

has to vanish on the surface of the

body, due to the no-slip boundary condition.

2.3 Swimming at low Reynolds number

2.3.1 Flow field of a moving sphere

With regard to the hydrodynamic flow fields of arbitrary microswimmer models,

which will be considered in the following, it is instructive to look at the flow field of

a rigid sphere moving in a bulk fluid with the velocity

. To do so, we consider the

multipole expansion in spherical coordinates given in Eqs. (2.26) and (2.27). With

P1(cos θ) = cos θ

and

V1(θ) = sin θ

we see that only terms of order

1 are suitable

to fulfill the boundary condition of a rigid moving sphere. For a sphere of radius

the no-slip boundary condition gives

ur(R

, 0

) = |U|

and

uθ(R

π/

) = ´|U|

. For the

terms in the square brackets in Eqs. (2.26) and (2.27) we then find

A1=

|U|/

2 and

2.3 Swimming at low Reynolds number

B1=´R2|U|/2. In Cartesian coordinates we obtain the velocity field

u(r) = 6πηU(︄1+R2

6∇12)︄O(r´r1)(2.30)

as combination of the Stokeslet with magnitude

and a source dipole with magni-

tude

for a sphere located at

, where we used the relation

GSD(r

˜) = ´

∇12G(r

˜)

[158]. Due to the different radial decay of the two hydrodynamic multipoles, the

Stokeslet dominates the flow field at large distances

r"R

. Furthermore, we can

deduce the force acting on the moving sphere

F=6πηRU(2.31)

which follows from the prefactor of the Stokeslet term. This force is related to the

inhomogeneity in the Stokes equation fext which drives the sphere.

2.3.2 Biological microswimmers

The fact that microorganisms swim in the regime of low Reynolds number governed

by the Stokes equation, has important implications for their propulsion strategies.

Phenomenologically speaking, low Reynolds numbers imply the dominance of

viscous friction such that microswimmers are not able to exploit their own inertia

during their propulsion as human swimmers do. More precisely, since the Stokes

equation is linear and contains no time derivative, its solutions are instantaneous

and exhibit kinetic reversibility. This means that for a solution

∇p

and

fext

, also

the reverse flow

´u

is a solution for

´∇p

and

´fext

[159]. As a consequence, the

propulsion strategies of microswimmers must use non-reciprocal deformations to

achieve a net displacement [164], which is know by the term Scallop theorem [165].

To solve this constraint on swimming strokes, nature has found a number of

different strategies. Organisms as Opalina, Paramecium or Volvox [70, 166] are en-

tirely covered by hair-like appendages, called cilia, which perform a non-reciprocal

beating motion [see Fig. 2.2(a)]. On a large scale, neighboring cilia synchronize

via hydrodynamic interactions such that metachronal waves form and push the

swimmer forward [69]. Pusher-type organisms as the bacteria E. coli or B. subtilis

[28, 38] possess a number of flagella joined in a bundle, which are shaped similar to

a cork-screw [see Fig. 2.2(b)]. The rotation of this flagella bundle pushes the fluid

backward similar to a propeller, and thus pushes the swimmer forward through the

fluid. Puller-type organisms as the Chlamydomonas algae [167, 168] have developed

two shorter flagella extending from the front of the cell body performing a beating

pattern similar to a breast stroke as depicted in see Fig. 2.2(c).

The three swimming strategies described so far differ decisively in their flow

fields. As we have seen in Sec. 2.3.1, a moving sphere free of external forces generates

the flow field of a source dipole. Even more, the source dipole is in fact present for

2. Physics of swimming microorganisms

Figure 2.2: Sketches of the three fundamental types of biological microswimmer

propulsion. (a) Ciliated organisms such as Opalina, Paramecium or Volvox [70, 166]

are entirely covered by hair-like cilia, which transport fluid along their surface. (b)

Pusher type organisms as the bacteria E. coli or B. subtilis [28, 38] use a bundle of

rotating flagella to push themselves through the fluid. (c) Puller type organisms as

the Chlamydomonas algae [167] use two beating flagella that fulfill a breast-stroke-like

motion to pull themselves through the fluid [168]. Dashed blue lines are used to

sketch the effective stroke averaged body shape of the swimmers including their cilia

or flagella. As depicted in the sketches, the cell bodies of all three microswimmer

types are rather elongated than spherical.

all self-propelled particles [162]. Fascinatingly, the surface slip-velocity of the source

dipole closely resembles the stroke averaged biological cilia beating patterns such

that the source dipole corresponds to ciliates like Opalina or Paramecium [114].

The swimming patterns of pushers and pullers generate an asymmetric distribu-

tion of forces, which results in the flow field of a force dipole in leading order. For

pushers the flagella create a backward pushing force, while the cell body pushes

fluid forward, for pullers the situation is reversed [74]. Consequently the flow field

of pushers and pullers possesses a force dipole moment of opposite sign as shown in

Fig. 2.1(b). The force dipole decays as

„r´2

with the distance to the swimmer, such

that the flow fields of pushers and pullers are more long-ranged than the source

dipole

„r´3

characteristic for ciliates. More complex shapes, such as elongated

particles, result in higher order multipole moments which are necessary to fulfill the

boundary conditions of the surface of the swimmer.

2.3.3 Flow fields of elongated particles

Different analytical approaches exist for the treatment of the flow field of elongated

particles. To begin with, the ordinary multipole expansion given in Eqs. (2.41), (2.26)

and (2.27) may be applied. This description results in higher-order terms to fulfill the

no-slip boundary condition on the non-spherical surface of the elongated particle,

and as we will see also the distribution of the surface slip velocity plays a role. A

more detailed picture of microswimmers and resulting boundary conditions will be

2.4 Swimming in confinement

given in Sec. 2.6.

Another approach exists for ellipsoidal particles, which are relatively simple to

treat mathematically. In this case, the Stokes equation can be solved in elliptical

coordinates resulting in a multipole expansion in ellipsoidal harmonics [169–171].

Relating this solution to the ordinary multipole series the flow field may be written

in terms of a distribution of Stokeslets. However, the formulation of these solutions

are quite intricate, for more details we refer to Ref. [158].

Lastly, the so-called slender body theory is applied in the limit of thin elon-

gated particles as, for example, rigid polymers or fibers. In a nutshell, instead of

considering a volume distribution of Stokeslets as in Eq. (2.19), the distribution of

the disturbance forces is approximated by a line distribution of Stokeslets, source

dipoles, and force dipoles. The resulting velocity field is then obtained from a line

integral over the distribution of the singularities. In the limit of thin spheroids, the

ellipsoidal solution is consistent with the slender body theory [158].

2.4 Swimming in confinement

Although many microswimmers live in vast habitats that resemble a bulk fluid,

confined geometries are also a common environment. For example, on solid surfaces

or at air interfaces [172, 173]. Furthermore, Stokes flow in channels is relevant in the

context of blood flow [131, 142, 143], and also experimental setups using droplets,

Petri dishes or microscope slides [174].

In the presence of solid walls or interfaces, the solution of the Stokes equations

has to be adjusted such that it fulfills the boundary conditions, i.e., the no-slip

condition on the walls. In the following, we will therefore first consider the case of a

Stokeslet in the presence of a single plane no-slip wall. Then, we will proceed to the

Hele-Shaw geometry of a microswimmer between two parallel plane no-slip walls,

which is used throughout this work.

2.4.1 Stokes flow near a plain wall

A showcase of swimming in confinement is in the presence of a single plain wall. In

this case, analytical formulas have been derived by Blake et al. for a Stokeslet [175]

and other multipoles [178]. In the presence of a wall, the so-called Lorentz image

technique is used to fulfill the boundary condition for the velocity field similar to

electrostatics [158]. For a Stokeslet the mirrored image Stokeslet is accompanied by a

force dipole and a source dipole in order to fulfill the no-slip boundary condition at

the wall [175]. The resulting flow field for a point force pointing towards the wall

is shown in Fig. 2.3(a). The combined image system fulfills the no-slip boundary

condition and is often called Blake tensor. For the Stokeslet located at

with the

position

mirrored at the plane perpendicular to the

-direction it takes the form

2. Physics of swimming microorganisms

Figure 2.3: Image systems for a Stokeslet in the presence of solid walls. (a) In the

presence of a single infinite wall with no-slip boundary conditions, the image system

consisting of a Stokeslet, force dipole and source dipole is known as the so-called

Blake tensor [175, 176]. (b) In the Hele-Shaw geometry, the system is confined by two

parallel infinite walls with no-slip condition. In this case, the image system consists

of an infinite recursion of reflections, which results in a different characteristic of

hydrodynamic flow fields [177].

[159, 175, 176]

Bij(r,r2(r1)) = (´δjk ´2r2

zδk3∇2

j+ (r2

z)2Mkj∇22)Oik(r´r2)(2.32)

where

M=diag(

1, 1,

)

. For a plane that lies in the origin of the coordinate

system

r2=r1´

(r1¨z

ˆ)

. The total flow field at

due to a point force

then reads

u(r)=[O(r´r1) + B(r

r2)]f

. In Eq. (2.32), we can recognize the reflected Stokeslet,

force dipole and source dipole from left to right. The additional image system near

the plane wall affects the far field of the force monopole such that it behaves as a

force dipole if it is oriented parallel to the wall, or like a source dipole if it oriented

normal to the wall. Likewise, a rotlet near a plane wall appears as a rotlet dipole

[178].

2.4.2 The Hele-Shaw geometry

The geometry which we use throughout this work is the so-called Hele-Shaw ge-

ometry where the fluid is confined between two parallel plane walls with small

distance

∆z

as shown in Fig. 2.3(b). This geometry describes for instance the fluid

between microscope slides or a thin fluid film in Petri dishes. Similar to the presence

of a single plane wall presented before, image singularities are used recursively to

solve the boundary conditions. However, an infinite number of recursive image

singularities are necessary in order to fulfill the boundary conditions at both walls.

As Liron and Mochon showed, an analytical solution can also be derived in this

geometry [177]. The solution involves a Fourier transformation in two dimensions,

where the infinite series of image singularities can be separated into an algebraic far

field contribution and an exponentially decaying near field term. The approach was

recently generalized to fluids in a thin film, i.e., with one no-slip and one slip wall,

2.4 Swimming in confinement

and higher order singularities by Mathijssen et al. [176]. For a point force located at

position r1the modified Green’s tensor at the position ris given by

Gij(r,r1) = ´24∆zrz

∆z(︃1´crz

∆z)︃r1

∆z(︄1´cr1

∆z)︄[︄δαβ

2ρ

˜2´r

˜αr

˜β

˜4]︄δiαδjβ+O(︂e´ρ

˜/∆z)︂.

(2.33)

Here,

˜=r´r1

denotes the distance to the point force,

˜=br

˜2

x+r

˜2

, and the

coefficient

distinguishes the parallel plates geometry (

1) and thin film case

with one slip boundary condition (c=1/2). The corresponding pressure reads

Pj(r,r1) = ´12η

∆z

∆z(︄1´cr1

∆z)︄r

˜α

˜2δjα+O(︂e´ρ

˜/∆z)︂. (2.34)

Here, the Greek indices

denote the first two dimensions parallel to the confining

walls.

As we see here, the solution contains a leading order far-field term with an

algebraic radial decay if the force is oriented parallel to the walls (square brackets)

and an additional near field term with exponential decay. The structure of this

far-field term thereby behaves as a two-dimensional source dipole in the direction

parallel to the walls [176], and has a parabolic or shape in the direction perpendicular

to the walls. If the force is oriented perpendicular to the walls, the entire flow field

decays exponentially.

As in the bulk fluid, higher-order force moments follow from applying the

directional derivative

ˆ¨∇1)

to the Green’s function [176]. Likewise, applying the

Laplace operator

∇12

to the Green’s function flow field yields the flow field of a

source dipole, which here is up to a factor identical to the Green’s function [176].

Higher source-multipole moments are again generated by directional derivatives.

For an axisymmetric microswimmer oriented in the direction of

the contributions

to the flow field are [162, 176]

uFD(r) = κ(e

ˆ¨∇1)G(r,r1)e

ˆ(2.35)

uFQ(r) = ν(e

ˆ¨∇1)2G(r,r1)e

ˆ(2.36)

uSD(r) = ´σ1

2∇12G(r,r1)e

ˆ(2.37)

uSQ(r) = ´χ(e

ˆ¨∇1)GSD(r,r1)e

ˆ(2.38)

where ∇1denotes the derivative with respect to the position of the singularity at r1.

In general, the multipole coefficients can be transferred from the bulk flow since the

same strength of the singularity is considered for the derivation of e.g., Eq. (2.33). In

order to discuss the implications of the modified multipole terms, we calculate the

expressions for the force dipole and source dipole terms. In the following, we restrict

ourselves to the case where the microswimmer is oriented parallel to the confining

2. Physics of swimming microorganisms

walls. In the Hele-Shaw geometry, the flow field of the source dipole resulting from

Eq. (2.37) reads

GSD

ij =´24c

∆z

∆z(︃1´crz

∆z)︃[︄δij

2ρ

˜2´r

˜jr

˜i

˜4]︄. (2.39)

Here, we neglected the exponentially decaying terms of the near field in Eq. (2.33),

such that this far-field limit of the flow field is independent of the swimmer height

z. For the force dipole we obtain

GFD

ij (r,r1) = ´24∆zrz

∆z(︃1´crz

∆z)︃r1

∆z(︄1´cr1

∆z)︄⎡

⎣

2δijr

˜j

˜4+r

˜i

˜4´4r

˜ir

˜2

˜6⎤

⎦. (2.40)

As we can read from Eqs. (2.33), (2.39) and (2.40) an important characteristic of

the Hele-Shaw geometry is that the flow fields decay with different exponents in

compared to the bulk fluid. For the force multipoles, this means that the velocity

fields decay more rapidly with the order of

G9ρ

˜´2

and

GFD9ρ

˜´3

, whereas in the

bulk fluid

G9r

˜´1

and

GFD9r

˜´2

. At the same time, the source dipole and other

source multipoles become more long ranged such that the velocity fields decay as

GSD9ρ

˜´2

compared to the bulk case where

GSD9r

˜´3

. This transformation of the

radial decay radically changes the nature of the flow fields and also hydrodynamic

interactions in comparison to the flow fields in bulk [179].

Independent of the geometry, the source dipole is the lowest-order multipole

term for microswimmers free of external forces, which occurs for all self propelled

particles [162].

However, in the bulk flow the force dipole which is more long-ranged and

therefore always the dominant term of the far field, if it exists [179]. As we saw,

this changes in the Hele-Shaw geometry such that the source dipole becomes the

dominant term of the flow field, being most long-ranged. Although this seems to

implicate that it becomes irrelevant to distinguish between pusher and puller type

microswimmers, this is not the case. As we show, the force dipole still dominates

the intermediate range of the flow field, while the source dipole dominates the far

field [125].

Comparing the source dipole in Eq. (2.39) and the Green’s function in Eq. (2.33),

which are of the same order ρ

˜´2, note that they both have the same shape

[δij/2 ´r

˜ir

˜j/ρ

˜2]

of a two-dimensional source dipole. Furthermore, in the Hele-Shaw

geometry force and source singularities of equal order

also possess the same radial

decay ρ

˜´(n+1).

Consequentially, the pairs of singularities of order

only differ in their de-

pendence on the confinement strength

∆z

, which can be identified from the re-

maining factors

∆z

and 1

/∆z

in Eqs. (2.33) and (2.39), respectively. For the force

monopole in Eq. (2.33) and higher-order force multipoles, the factor

∆z

is factored

out from the hyperbolic profiles in

and

. For the source multipoles, the identity

2.4 Swimming in confinement

uSD(r)9∇12(G(r,r1)e

ˆ)results in a factor of 1/∆zinstead.

In other words, the magnitude of the force multipoles increases with the height

of the Hele-Shaw geometry. At the same time, the magnitude of the source multipole

increases with the strength of the confinement. Thus, force and source multipoles

of the same order in

may only be identified in a given flow field by using their

dependence on ∆z.

2.4.3 Multipole expansion in the Hele-Shaw geometry

With the multipoles presented in Sec. 2.4.2 it is possible to construct a series ex-

pansion for the flow field of a microswimmer similar to the bulk fluid. Again, we

consider the relevant flow fields of the source dipole (

uSD

), force dipole (

uFD

), force

quadrupole (u

uSD), and source octupole (u

uSD) in the Hele-Shaw geometry

u(r) = u

uSD(r) + u

uFD(r) + u

uFQ(r) + u

uSO(r) + . . . . (2.41)

Although the source quadrupole behaves equally to the force dipole in the Hele-

Shaw geometry, it is omitted here because we neither observe in the bulk fluid nor in

the Hele-Shaw geometry. This series is converted into a form similar to the bulk case,

such that a measured or simulated flow field can be decomposed into its multipole

moments and thereby determine the contributions in Eq. (2.41).

First, we switch to cylindrical coordinates

(ρ

. We eliminate the

-dependence

by averaging over the

coordinate, restricting the height of the singularity

z=∆z/

2, and then obtain the average radial component from the projection on

ˆρ

˜ρ(ρ,φ)”e

ˆρ

∆z¨ż∆z

u(ρ,φ,z)dz. (2.42)

Applying this transformation to the flow fields of the multipoles presented in Sec.

2.4.2 we identify that the radial coordinates of the velocity can be written in terms of

Chebyshev polynomials of the first kind

Tn(cos φ) = cos(︁nφ)︁

[125]. For example,

for the force monopole we obtain

˜FM

ρ(ρ,φ) = A1ρ´2T1(cos φ)(2.43)

and for the source dipole

˜SD

ρ(ρ,φ) = B1ρ´2T1(cos φ)(2.44)

for a motion along the

axis,

ˆ=e

ˆx

. This form again shows the force monopole

and source dipole have the same dependence on

and

, resulting in the same

factor

ρ´2T1(cos φ)

, similar to other multipoles of the same order. Here, we have

integrated the

∆z

dependence and remaining factors into the coefficients such that

they scale as

A19∆z

and

B19∆z´1

, respectively. Applying the identities (2.35) to

2. Physics of swimming microorganisms

(2.38) repeatedly also to higher order multipoles, one obtains a multipole series

˜ρ(ρ,φ) =

n=1

An+Bn

ρn+1Tn(cos φ). (2.45)

Here,

An„∆z

and

Bn„∆z´1

are the respective coefficients of the force and source

multipole moments which all show the same scaling with the confinement strength

„∆z

and

„∆z´1

, respectively. Again we can recognize that in case of a force-free

swimmer,

A1=

0, the source dipole becomes the most long-ranged multipole for

microswimmers in the Hele-Shaw geometry.

We may use the orthogonality relations of

cos(︁nφ)︁=Tn(cos φ)

to extract the

multipole moments from Eq. (2.45) by projecting

˜ρ(ρ

φ)

on the Chebyshev polyno-

mials,

˜ρ,n(ρ)”1

πż2π

˜ρ(ρ,φ)Tn(cos φ)dφ, (2.46)

which yields

˜ρ,n(ρ) = An+Bn

ρn+1. (2.47)

We again stress that, in contrast to the bulk fluid [Eq. (2.29)], force and source

multipoles of same order

cannot be distinguished in a given flow field in the form

of Eq. (2.42) at constant ∆z. Principally, the different scaling with the slab width ∆z

may be used to infer the coefficients

and

, although in practice our simulation

data presented in Section 4.2 is not always sufficiently accurate to distinguish both

cases.

However, the relevant multipole coefficients may be transferred from the bulk

fluid, which can be rationalized as follows. On the one hand, both multipole ex-

pansions capture the disturbance created by a microswimmer in leading order, and

the characteristic distribution of force and source multipoles should remain the

same in the Hele-Shaw geometry. On the other hand, in contrast to a point-like

singularity higher-order terms enter to fulfill the boundary conditions on the surface

of a voluminous microswimmer and on the bounding plates. However, these are

short-ranged and do not contribute to the far field.

2.5 Hydrodynamic interactions

Hydrodynamic interactions emerge through the advection of particles in the flow

fields created by other particles, or the by the effective flow field resulting through

the reflection at a boundary as in, for example, Eq. (2.32). In the first case, the particle

interacts with another particle, in the second case with a wall.

In this section we will give a brief overview over the dynamical response of

spherical and elongated particles to arbitrary flow fields. Subsequently, we will

2.5 Hydrodynamic interactions

argue how this affects interactions between particles. We start with Fax

en’s law

for spherical particles which describes velocity and reorientation resulting from a

background flow. We will then describe the generalization to the case of prolate

spheroids, which will also lead us to the Jeffery orbits of elongated particles in shear

flow.

2.5.1 Fax´en’s law

Fax

en’s law describes the advection

and rotation

of a particle in a background

flow

u(r)

. The formalism uses a formulation of the Lorentz’ reciprocal theorem, an

integral theorem that follows from Green’s second identity and the structure of the

stress tensor [158]. The formulation

¿S

u(σ1¨dS)´żV

u(∇¨σ1)d3r=¿S

u1(σ¨dS)´żV

u1(∇¨σ)d3r. (2.48)

relates two different solutions

and

σ1

of the Stokes equation. Choosing

different solutions for

and

, the reaction of different particles shapes to specific

flows can be calculated. To specifically obtain the force acting on a moving sphere of

radius

, the first solution

is the flow field of the sphere moving with velocity

which is yet to be determined. As second solution

σ1

serves the flow field of a

sphere in the flow of a point force acting on the fluid. For a sphere at the position

the resulting hydrodynamic force is given by [157, 180]

v=F

6πηR+[︄1+R2∇2

6]︄u(r). (2.49)

Here, the first term coincides with Stokes-law for the velocity of a sphere due to the

external force

in the case of

u(r) =

0. The first term in the square brackets is the

leading order direct advection in the flow field, while the additional term

∇2u=

∇p/η

can be understood as proportional to the pressure gradient. Equivalently, the

torque on a sphere is obtained by considering the flow field of a rotating sphere as

first solution u,σ, which results in [158]

ω=1

2∇ˆu(r)´T

8πηR3(2.50)

For elongated particles the multipole solution of Stokes flow for prolate spheroidal

particles can be inserted into the reciprocal theorem resulting in Fax

en’s law for

prolate spheroids. The resulting advection velocity has a similar structure to Eq.

(2.49), but contains a line integral over the spheroid axis. This can be understood

recalling that the flow field of prolate spheroids can also be written as a superpo-

sition of lowest-order singularities. Likewise, an expression for the torque on a

2. Physics of swimming microorganisms

prolate spheroid may be calculated. Refer to [158] for more details and complete

expressions.

2.5.2 Jeffery orbits of elongated microswimmers

In contrast to spherical particles, which only rotate in response to a vortex in the

velocity field, anisotropic particles also react to a shearing motion of the solvent

depending on their instantaneous orientation. As a result, force-free anisotropic

particles perform a periodic tumbling motion in shear flow, known as Jeffery orbits.

The equations of motion may be derived using resistance tensors for the spheroid

and the velocity of the solvent [171], or the corresponding Fax

en law for the rotation

of a prolate ellipsoid [158]. For uniaxial spheroids with director

, the corresponding

equation for the angular velocity ω(t)is

ω(t) = 1

2∇ˆu(r) + γ␣e

ˆ(t)ˆ[︁E(r)e

ˆ(t)]︁(, (2.51)

where

E(r) = [(∇bu)T+∇bu]/

2 is the local strain rate tensor at the position

which we already encountered in the context of the fluid stress tensor in Eq. (2.11)

[171, 181]. Further, the geometric factor

γ=α2´1

α2+1P[

0, 1

]

depends on the aspect ratio

of the elongated particles [158]. The equation of motion for the director

of the

spheroid is

Bte

ˆ=ω(t)ˆe

ˆ(t)

. The resulting solution for

ˆ(t)

describes orbits on a

unit sphere, called Jefferey orbits. As we can also read from Eq. (2.51), in the case of a

spherical particle

α=

1 only a fluid vortex with vorticity

∇ˆu

results in a rotation

of the particle.

2.5.3 Hydrodynamic interaction between microswimmers

Hydrodynamic interactions between pairs of microswimmers are decisively char-

acterized by the dominant hydrodynamic multipoles of their flow fields and their

shapes or aspect ratios. Together with their velocities, these enter Fax

en’s law and

describe the resulting interaction by advection and reorientations. Additionally,

short-ranged interactions are described by the so-called lubrication theory [158]. We

begin with the most fundamental types of interactions, which can be understood

already in the context of spherical microswimmers. Considering emergent advection

and reorientations described by Eqs. (2.49) and (2.50), we realize that advection

resulting from the friction force 6

πηR(u(r)´U)

is most long ranged and dominates

the interaction for the flow field

u(r)

of a given multipole. The reason is that this

term decays with the inter-particle distance just as the multipole itself, while all

other terms contain derivatives of

u(r)

, which decay faster. While the leading order

advection in the flow field always occurs, the pressure gradient

∇2u=∇p

vanishes

for source multipoles, which fulfill the harmonic equation ∇2u=0.

2.5 Hydrodynamic interactions

Figure 2.4: Hydrodynamic interactions between different types of microswimmers

adapted from [65]. Sketch of the flow fields of (a) pusher-, (b) puller- and (c) neutral-

type microswimmers. Both propulsion mechanisms create a pair of opposing forces

(red arrows) distinguished by a change of sign. Advection along the stream lines

(blue arrows) is the main manifestation of hydrodynamic interaction. Hydrodynamic

reorientations for pairs of (d) pusher and (e) puller type microswimmers resulting

from the vorticity of the flow fields shown in (a) and (b). Two pushers on an ap-

proaching trajectory are reoriented towards the swimming in parallel configuration.

For pullers, the rotation is opposite such that they are reoriented towards a perpen-

dicular configuration. In contrast, neutral microswimmer types do not impose a

rotation on each other since the flow field of the source dipole has zero vorticity. For

elongated microswimmers, Jeffery orbits arise due to the local strain rate tensor

E(r)

according to Eq. (2.51). The radial decay of the interactions is denoted in powers of

(bulk fluid) and ρ(Hele-Shaw geometry).

For pushers and pullers, which impose the flow field of a force dipole as shown

in Fig. 2.4(a) and (b), this has the following implications. Pushers swimming side-

by-side in parallel attract each other while pullers repel each other in the same

configuration, following the stream lines. Swimming behind one another pushers

repel while pullers attract each other [65]. The additional term

„∇2u

is more short

ranged. For neutral swimmers, for which the source dipole is the leading-order

multipole contribution, the mutual advection in the side-by-side and head-to-tail

configurations cancels and therefore does not introduce a characteristic attraction or

repulsion [Fig. 2.4(c)].

Regarding the reorientations, we look at the vorticity of the flow fields of the

force dipole, as shown in Fig. 2.4(a) and (b). As the streamlines indicate, the sign

2. Physics of swimming microorganisms

of the vorticity alternates in four quadrants. For pairs of swimmers, this affects a

converging trajectory in two different ways. As depicted in Fig. 2.4(d), pushers on a

converging path are reoriented towards a configuration swimming side-by-side in

parallel. If the force dipole changes the sign for pullers, swimmers approaching each

other are reoriented even more towards each other, as depicted in Fig. 2.4(e). Lastly,

the source-dipole flow field does not possess a vorticity, such that neutral spherical

microswimmers do not reorient. For elongated microswimmers, the local strain rate

tensor E(r)creates to Jeffery orbits according to Eq. (2.51).

However, compared to the reorientation arising from the vorticity in the Fax

theorem, the effect of these periodic oscillations about a fixed axis is small.

Distinguishing between the bulk fluid and the Hele-Shaw geometry, we note that

the general shape of the flow fields remain the same such that interactions qualita-

tively remain the same. On the other hand, the radial decay of the hydrodynamic

multipoles change in the Hele-Shaw geometry such that the source dipole becomes

the most-long ranged contribution of the flow field. However, as we elaborated

before, the interaction via advection and reorientation mainly arise from the force

dipole such that the Hele-Shaw geometry primarily implies more short-ranged

interactions compared to the bulk fluid.

2.6 The squirmer model for microswimmers

The squirmer is a model for spherical microswimmers of radius

, which covers,

among others, the major microswimmer types presented in Sec. 2.3. For the ciliary

propulsion, the squirmer model enforces the flow velocity of a source dipole as slip

velocity on surface of the microswimmer [69, 114]. This resembles the analytical

solution of a moving sphere without external forces in Eq. (2.30). For spherical pusher

- and puller-type microswimmers, a pair of the force dipole and source quadrupole,

where the second term ensures that the normal velocity

ur(r

θ)

Eq. (2.26)

vanishes

on the squirmer surface [120]. In the same manner, pairs of higher order multipoles

can be added to model, for example, Janus particles [182]. However, for simplicity,

we restrict ourselves to terms up to

2 in Eq. (2.27) in the following. All together,

we obtain the slip-velocity at the surface of the spherical microswimmer

vs(x

ˆs) = Bs

1(1+βe

ˆ¨x

ˆs)[︁(e

ˆ¨x

ˆs)x

ˆs´e

ˆ]︁(2.52)

that acts on the surrounding fluid [73, 119]. Here,

ˆs

is the unit vector pointing from

the squirmer center to the respective surface point, and the unit vector

indicates the

orientation of the squirmer. The squirmer parameter

is used to control the swimmer

type by changing the ratio between source-dipole and force-dipole moment. Thereby,

a value of

βă

0 results in a pusher, where the slip velocity is concentrated at the

rear of the sphere. Likewise, values of

βą

0 concentrate the slip velocity at the

front of the sphere and result in a puller. The absence of a force dipole,

β=

2.6 The squirmer model for microswimmers

results in a neutral squirmer, as depicted in Fig. 2.5(a). For a neutral squirmer, the

induced flow field in the bulk fluid is given by the flow field of a pure source-dipole

singularity with strength

B1=Bs

1R3

. It can be shown that the squirmer parameter

also determines the swimming speed

v0=

[114]. The resulting surface slip-

velocity field may then be used in computer simulations by enforcing the boundary

condition within the MPCD method [74, 120], the Lattice-Boltzmann method [183]

or the boundary-element method [184].

Figure 2.5: (a) Schematic of a single squirmer of radius

and with orientation

given by the unit vector

. The surface slip-velocity field of a neutral squirmer is

indicated by blue arrows. (b) Schematic of the squirmer-rod model consisting of

nsq =

10 spherical squirmers placed on a straight line separated by the distance

All squirmer orientations

are aligned with the rod axis. (c) Realization of a pusher

type squirmer rod. The surface slip velocity on the rod surface is multiplied with the

scalar envelope function

f(x˚

s¨e

ˆ)

given by Eq. (2.53), such that the slip-velocity is

concentrated on the rear of the rod.

2.6.1 The squirmer-rod model

To pursue our main objective and investigate how the interplay of shape anisotropy

of microswimmers and their hydrodynamic interactions determine their collective

motion, we introduce the new squirmer-rod model [125]. Squirmer rods consist of a

number of

Nsq =

10 spherical squirmers [cf. Fig. 2.5(a)] arranged on a line to form a

rigid body as shown in Fig. 2.5(b). The resulting shape mimics the actual rod shape

of many biological microswimmers such as bacteria or Paramecium. Furthermore, the

model can be extended to flexible shapes by introducing bending rigidity between

the beads in future works, see chapter 5. To form rods of different aspect ratios

α=lS/

, where

is the rod length, we vary the squirmer distance

[cf. Fig. 2.5(b)].

To ensure a smooth surface and enable swimmers to slide past each other easily,

we do not exceed a maximum squirmer distance of

d«

0.8

, which amounts to a

maximum aspect ratio of

α=

5. This closely resembles the aspect ratio of bacteria

such as E. coli or B. subtilis. The squirmer rod propels itself via the axisymmetric

2. Physics of swimming microorganisms

surface slip-velocity field of the neutral squirmer, which is imposed on the surface

of its individual squirmers to the surrounding fluid [74, 119, 120].

To encompass pusher and puller-type microswimmers in the squirmer rod model,

we concentrate the surface slip velocity either to the back or the front of the squirmer

rod as shown in Fig. 2.5(c). This is achieved by multiplying the magnitude of the

slip velocity by using the envelope function

f(x˚

s¨e

ˆ,χ) = 1+χtanh(10x˚

s¨e

ˆ/lS)(2.53)

where

x˚

is the vector that points from the rod center to a location on the rod surface

[cf. Fig. 2.5(c)] and

lS/

10 is the step width of the envelope function [185]. The

parameter

χP[´

1, 1

]

determines the swimmer type and force-dipole strength, such

that for

χă

0 a pusher-type swimmer is realized and likewise a puller-type swimmer

for

χą

0. Choosing either

χ=´

1 or

χ=

1 results in one completely passive half of

the rods, while for intermediate values the relative contributions of the source dipole

and force dipole to the flow field vary. For

χ=

0, the model resembles the neutral

squirmer rod [185]. With the definition that ensures that the average

xf(x

χ)yx=

is independent of

, also the swimming velocity is nearly independent of

. Note

that the squirmer rod model is also related to the ellipsoidal squirmer model, which

is also elongated [184, 186]. However, it features a more realistic rod shape [187, 188]

and the surface-slip velocity is distributed evenly over the cell body independent of

the aspect ratio [186].

2.7 Brownian motion

Additional to the self-propelled motion, microswimmers are also subject to a dif-

fusive Brownian motion, which describes a random walk through the fluid. This

motion is a result of the random collisions with the solvent particles. The combi-

nation of the diffusive and active motion is described by the so-called Langevin

equation [152, 189]. In the overdamped limit of high viscosity, the particles velocity

vand angular velocity ωare given by [150, 161]

v(t) = v0e

ˆ+µT[︁´∇V(x,e

ˆ) + frand(t)]︁and

ω(t) = µR[︁´∇e

ˆV(x,e

ˆ) + τrand(t)]︁(2.54)

Here,

µT

and

µR

are the respective translational and rotational mobility tensors. In

this picture, the particle moves with the self-propulsion velocity

v0e

and is subject to

two forces. The first deterministic force due to a potential

V(x,e

ˆ)

corresponds to an

external force or the interaction with other particles. The second force and last term

frand(t)

describes a random force, which results from the random thermal motion of

the solvent particles. With the mobility

µT

we obtain the stochastic velocity

η(t) =

µTfrand(t)

. Likewise, the angular velocity is given by the deterministic torque due to

2.7 Brownian motion

the potential

V(x,e

ˆ)

, and a stochastic torque

τrand(t)

transferred from the solvent.

The dynamic of the particle orientation then follows from Bte

ˆ(t) = ω(t)ˆe

ˆ(t).

The stochastic velocity

η(t)

has zero mean

xη(t)y=

0 and is delta-correlated

in time with the variance

xη(t)bη(t1)y=

DTδ(t´t1)

, where

DT=kBTµT

the translational diffusion tensor given by the Stokes-Einstein relation [152]. A

similar expression applies for the stochastic angular velocity

ζ(t) = µRτrand(t)

which also has zero mean

xζ(t)y=

0 and is delta-correlated in time

xζ(t)bζ(t1)y=

DRδ(t´t1)

. For the rotational diffusion tensor, the Stokes-Einstein relation reads

DR=kBTµR

. These relations for the autocorrelation of the stochastic noise terms

are also a consequence of the fluctuation-dissipation theorem, because they relate

the fluctuations of the random force and torque to the dissipation described by the

friction tensor γT=D´1

T/kBTand γR=D´1

R/kBT, respectively.

For spherical particles as, for example, the squirmers, the friction and hence

diffusion tensors are rotationally symmetric and given by DT=1kBT/(6πηR)and

DR=1kBT/(8πηR3). If rod-like particles move through a fluid, they are subject to

an anisotropic friction force, so that also the diffusion becomes anisotropic as a result

of the Stokes-Einstein relation. In this case, the diffusion tensors are given by

DT=kBT[︂µ∥(e

ˆbe

ˆ) + µK(1´e

ˆbe

ˆ)]︂and

DR=kBTµR, (2.55)

with the mobilities [190]

µ∥=µ0

2π[︃ln(α)´0.207 +0.980

α´0.133

α2]︃,

µK=µ0

4π[︃ln(α)+0.839 +0.185

α+0.233

α2]︃, (2.56)

µR=3µ0

πα2[︃ln(α)´0.662 +0.917

α´0.050

α2]︃.

Here,

denotes the aspect ratio of the rod-like particles and

µ0

is the Stokes mobility

of a spherical particle with radius

. The rotational mobility

µR

thereby relates to

rotation perpendicular to the rod axis e

ˆ.

To discuss the resulting dynamics, we restrict ourselves to the case of a spherical

active Brownian particle free of external influences

V(x,e

ˆ) =

0. In this case, the

random torque leads to an exponential decay of the orientational autocorrelation

[26]

ˆ(t)¨e

ˆ(0)y=e´t/τR,

where

τR= [(d´

)DR]´1

is the orientational persistence time in

dimensions. As

a result, the persistent motion becomes irrelevant at large time scale. The mean

square displacement is obtained by integration of the equations of motion Eq. (2.54),

2. Physics of swimming microorganisms

resulting in

x|r(t)´r(0)|2y=2dDTt+2v2

0τrt´2v2

0τ2

r(1´e´t/τr). (2.57)

Here, we can identify two special cases depending on the time scale of

. At times

shorter that the persistence time

t!τR

, the mean square displacement describes

a persistent ballistic motion due to the active swimming velocity

, such that the

mean square displacement

x|r(t)´r(0)|2y= (v0t)2+2dDTt

grows quadratically in time

9t2

. At long time scales

t"τR

, the motion becomes

diffusive due to the stochastic reorientation of the particles, such that the mean

square displacement

x|r(t)´r(0)|2y=2t(dDT+v2

0τR)

grows linear in time

. For a discussion of the motion of elongated particles we

refer to ten Hagen et al. [191]. In order to compare the relative influences of the

active persistent motion and diffusive motion to the dynamics of the active particle,

another dimensionless quantity is introduced. The so-called P´

eclet number

Pe =td

=2Rv0

(2.58)

describes the ratio of the time scales

td= (

R)2/DT

and

ta=

R/v0

, required for

active Brownian particles to travel the length of the particle diameter 2

due to

diffusion or active motion, respectively. Hence, active motion is negligible if

Pe !

while diffusive motion is not relevant for Pe "1

Also hydrodynamic interactions can be integrated into the Langevin equation via

the mobility or diffusion tensors within the Rodne-Prager framework for spherical

particles [161] or rods [192]. Thereby, the Fax

en theorem of Eq. (2.49) is used to calcu-

late the acting hydrodynamic forces in pairs of particles within the so-called method

of reflection. Lastly, we note that the Langevin equation is physically equivalent to

the Smoluchowski equation for the evolution of the probability distribution function

of the active Brownian particles.

CHAPTER 3

Multi-particle collision

dynamics (MPCD)

3.1 Introduction

When it comes to the numerical simulations of hydrodynamic flows, there are

numerous approaches at different levels of detail and abstraction. Most commonly,

differential equations that describe hydrodynamic flows are discretized and then

solved using finite elements or finite volume schemes [193]. For limiting cases there

are specific schemes like the boundary element method for Stokes flow that exploits

the Lorentz reciprocal theorem to describe the entire flow at the boundary of the

domain [159].

Another approach is followed in the so-called mesoscopic schemes, which in-

cludes the methods of multi-particle collision dynamics (MPCD), dissipative-particle

dynamics [194, 195], smoothed-particle hydrodynamics [194, 196] and the lattice-

Boltzmann method [197]. Common for this class is the utilization of coarse-grained

fluid particles that evolve according to simple dynamic rules [74, 194]. These rules

are defined such that the resulting hydrodynamic flow field fulfills the Navier-Stokes

equations [126, 198] In contrast, the lattice-Boltzmann method begins at a lower level

of description and discretizes the Boltzmann equation for the velocity distribution

function of the solvent particles [197, 199].

Since their introduction in 1999 [126], algorithms belonging to the method of

multi-particle collision dynamics (MPCD) have become a standard tool for the

simulation of fluid flows in the field of soft-matter physics [74, 200, 201]. In particular,

MPCD algorithms have been used extensively in the context of microswimmers [41,

47, 73, 76, 91–96, 125, 202, 203], to cite only a few examples. Other studies with

MPCD cover polymers [139–141], colloidal suspensions [134–138], blood cells [142],

the African trypanosome as the causative agent of the sleeping sickness [143], and

even schools of fish [144]. Also extensions to binary fluid mixtures [145], liquid

3. Multi-particle collision dynamics (MPCD)

Figure 3.1: Sketch of the particle motion and collision rule in the MPCD method.

The fluid is represented by point like-particles (back dots) with continuous positions

and velocities (back arrows). The two steps mimic the ballistic motion and collision

of particles. In the streaming step (a) particles move ballistically for the duration of

∆t

. For the collision step (b) the space is divide in a cubic lattice with lattice constant

to form groups of particles. The particles in these cells then exchange momentum

(green arrows) and energy according to a collision rule, while conserving the total

momentum

m0nξvξ

. Some methods resemble the micro-canonical ensemble, such

that also the energy in the cell is conserved.

crystals [146–148], and chemically reacting systems[149] exist.

In our simulations, we employ a specific MPCD method that incorporates

a so-called Andersen thermostat and features angular-momentum conservation

(MPCD-AT+a)

[73, 204]. Furthermore, we developed an extended version of MPCD

with a non-ideal equation of state and thus a reduced compressibility that enables

accurate simulations of dense microswimmer configurations [205, 206]. By solving

the Navier-Stokes equations, the method treats hydrodynamic interactions between

active squirmer rods and also with the confining walls of the Hele-Shaw geometry

[126, 201]. Furthermore, thermal fluctuations are included in the method.

In concrete, the MPCD method considers point-like particles with number density

and defines simplified rules for the two dynamic stages in their motion, which are

ballistic motion and collisions, as sketched in Fig. 3.1. These rules are then applied

in alternating steps [126]. In the following we will first introduce the streaming step

which is common to all MPCD variants. We will then give a brief overview over the

most common collision rules and discuss their specific properties.

3.2 Streaming step

During the streaming step the point particles with masses

, positions

xi(t)

, and

velocities vi(t)move ballistically during the time ∆taccording to

xi(t+∆t) = xi(t) + vi(t)∆t. (3.1)

3.3 Collision step

This motion represents the free motion of a particle in the fluid before a collision

occurs, which is represented by the collision step. During the streaming step, par-

ticles collide with confining walls or moving objects such as squirmers in our case

[47, 73, 74, 76, 94, 125] and thereby transfer both linear and angular momentum,

in particular, to the squirmers. The so-called bounce-back rule [126, 128] enforces

the no-slip boundary condition at confining walls or the slip-velocity field at the

squirmer surfaces [125].

3.3 Collision step

The key of the MPCD collision step is to define a simplified multi-particle collision

rule, which ensures momentum conservation. It can be shown that this is sufficient

to guarantee that the model fulfills the Navier-Stokes equations [126]. The resulting

methods are efficient for the numerical solution of hydrodynamic flows, and enable

the usage of powerful parallel computing architectures [132]. Furthermore, model

parameters can be derived by analytical calculations [194, 207].

To approach the MPCD collision rule, we begin with the very intuitive variant of

the so-called stochastic rotation dynamics (SRD), which was formulated by Male-

vanets and Kapral [126]. We will then cover variants with different thermodynamic

and mechanical properties as the utilization of a thermostat and the incorporation of

angular momentum conservation. We will then approach more complex collision

rules that generate different thermodynamic equations of state as a result of a tailored

inherent momentum transport mechanism [205, 206, 208]. However, we stress again

that all collision rules conserve momentum, and therefore obey the Navier-Stokes

equations.

Generic for all collision rules, the simulation volume is first divided into a cubic

lattice and the fluid particles are grouped into cubic unit cells of linear size

and

centered around

as depicted in Fig. 3.1(b). Thereby, the total number of particles

is chosen such that, on average, a number of

particles resides in each cell of unit

volume

. Then, for the

nξ

particles in each cell volume

Vξ=a3

around

the mean

velocity

vξ

and center-of-mass position

xξ

are determined. At this point one of the

following collision rules is applied to the particles within each cell. Lastly, before

the following collision step, the cubic lattice is randomly shifted to ensure Galilean

invariance [208].

3.3.1 Stochastic rotation dynamics (SRD)

The collision rule of the SRD method consists of a rotation of the thermal velocities

vi´vξ

around a randomly chosen axis

ˆξ

, which is different for each cell

and time

step, by the constant angle

[126]. With the rotation matrix

R(n

ˆξ

α)

the collision

3. Multi-particle collision dynamics (MPCD)

step can be summarized as

vnew

i=vξ+R(n

ˆξ,α)(︂vi´vξ)︂. (3.2)

As can be easily proven, this rule conserves the momentum

m0nξvξ

, thermal energy

Eξ=m0

2nξřxiPVξ(vi´vξ)2

and, trivially, the number of particles

nξ

within each

collision cell. With these properties, it can be shown that the resulting flow field obey

the Navier-Stokes equation and the hydrodynamic equation of energy transport.

Furthermore, it can be proven that the dynamics fulfills the Boltzmann H-theorem

such that the system evolves towards thermodynamic equilibrium under the SRD

algorithm [126]. The model parameter

decisively controls the local redistribution

of momentum and energy and therefore describes other parameters of the SRD fluid,

as the viscosity [207]. Typically, values are chosen around 120

. In two dimensions,

the rotation axis is perpendicular to the

-plane, and the rotation angle randomly

selected between ˘α[207].

Due to the rotation of the relative velocities, the method does not conserve angular

momentum locally at the scale of the collision cells. This leads to incorrect angular

diffusion of immersed solute particles [209]. Angular-momentum conservation

can be restored by declaring the collision angle within each cell

αξ(txi|xiPVξu)

function of the instantaneous position of the particles. This extension was proposed

by Ryder for SRD in two dimensions [201, 210].

To realize simulations in the canonical ensemble, a thermostat can be added to

the SRD method. In this case, the total kinetic energy in each cell

Eξ

relative to

vξ

calculated. Then, a random new thermal energy

is generated from the appropriate

gamma distribution, which follows from the Maxwell-Boltzmann distribution for

the 3

(nξ´

)

velocities. Lastly, all thermal velocities are rescaled with the factor

ξ/Eξ

[211]. In the canonical ensemble the SRD method has further been extended

to restore angular-momentum conservation of the collision rule. In this case, a

correction is added to the new velocities following from Eq. (3.2) and the rescaling

with the thermostat [212]. We elaborate this procedure in detail in the context of the

MPCD method with Andersen thermostat in Sec. 3.3.2.

3.3.2 MPCD with Andersen thermostat

The MPCD algorithm with Andersen thermostat

(MPCD-AT)

describes fluids in the

thermodynamic canonical ensemble. In the collision step, fluid particles are assigned

new random thermal velocities

δvi

relative to the center-of-mass velocity which are

drawn from a Boltzmann-distribution with variance ?kBT0/m0[74, 125, 204].

To restore overall momentum conservation, the change in linear momentum per

particle,

m0∆vξ=m0

nξřxiPVξδvi

, is subtracted from each particle velocity, such that

3.3 Collision step

the final new velocity reads

vnew

i=vξ+δvi´∆vξ. (3.3)

The collision step of the MPCD-AT method can be adapted to realize angular mo-

mentum conservation, and is then denoted by

(MPCD-AT+a).

To achieve this, a

correction term is added to Eq. (3.3) containing the angular momentum before the

collision

Lξ=m0

nξřxiPVξxi,cˆvi

, and the angular momentum introduced by the

random velocities

∆Lξ=m0

nξřxiPVξxi,cˆδvi

. Here

xi,c

denotes the particle position

relative to the center-of-mass position

xξ

. With the additional correction term, the

collision step can be summarized by

vnew

i=vξ+δvi´∆vξ´xi,cˆI´1

ξ(Lξ´∆Lξ), (3.4)

Here,

Iξ

is the moment-of-inertia tensor of the configuration of particles inside

Vξ

relative to the center-of-mass, such that

∆ω=I´1

ξ(Lξ´∆Lξ)

corresponds to a

correcting contribution in the angular velocity. Similar to the stochastic rescaling of

the thermal energy in the SRD method in Sec. 3.3.1, rescaling the thermal energy in

the MPCD-AT to the value before the collision can be used to perform simulations

in the micro-canonical ensemble [194].

In general, the MPCD-AT and SRD versions of MPCD show a similar perfor-

mance in simulations, and can be applied almost interchangeably. However, one

difference is the relaxation time to thermodynamic equilibrium. For SRD the relax-

ation time increases with the mean density

, while it decreases for MPCD [201].

At the same time, the numerical implementation poses different challenges. For

MPCD efficient and fast random number generators are required to generate 3

nξ

random numbers for the normal distribution. In contrast, SRD requires only 2 ran-

dom floating point numbers to generate a rotation axis, but generating a random

energy

, which follows the gamma distribution, is numerically intensive. In our

experience the computational cost of the two different collision steps with thermo-

stat and angular momentum conservation are rather similar. In implementations

on graphic cards, which have immense numerical capabilities, the performance is

instead dominated by the efficient organization of the data transfer between the

memory and the processor. This is common for all MPCD algorithms, which require

multiple gigabytes of data to describe the state of the fluid particles.

3.3.3 Equation of state

In this section, we address how the macroscopic thermodynamic pressure of the

MPCD fluid can be derived from the microscopic dynamics of the streaming and

collision step. To calculate the equation of state for the pressure, we use the ex-

perimental definition of pressure as the normal component of the momentum flux

3. Multi-particle collision dynamics (MPCD)

through an arbitrarily oriented plane for the fluid in thermal equilibrium, i.e., with-

out gradients in the velocity field [207]. This is important to avoid contributions to

the normal component of the momentum flux which arise from the bulk viscosity in

the presence of gradients in the velocity field. In general, both the MPCD streaming

and collision step contribute to the pressure:

P=Pstr +Pcoll . (3.5)

During the streaming step, particles do not interact and simply transport momentum

by moving across any plane considered. This results in the ideal gas contribution

Pstr =n0kBT/a3

0, which is common for all MPCD methods [145].

To evaluate the contribution

Pcoll

from the collision step, we consider the momen-

tum flux across a plane with unit area

, which lies inside a collision cell. Without

loss of generality, we choose the plane perpendicular to the

-axis at position

and

then average over all

. During the collision step momentum is transported from

the region

yăy0

across the plane into the region

yąy0

during time

∆t

. Thus, for

the pressure as momentum transfer per area and time we obtain

Pcoll =m0

0∆tBy

ˆ¨ÿ

ti|yiąy0u

(vnew

i´vi)F. (3.6)

Here,

is restricted to all particles

yiąy0

and

m0y

ˆ¨(vnew

i´vi)

is the change in the

normal momentum component of particle

during collision as given in, for example,

Eq. (3.4). The change in momentum is thereby transferred to the particles

yiăy0

such that these particles are included in this consideration implicitly. The average

goes over all possible collisions, particle configurations, and positions

. For the

traditional MPCD methods presented above, the collisional contribution vanishes

since there is no correlation between the velocity change

(vnew

i´vi)

and the positions

. In consequence, traditional MPCD possesses the thermodynamic equation of state

of the ideal gas [145]. In simulations of Stokes flow far from mechanical equilibrium

this may cause artifacts, because large density inhomogeneities arise to resolve the

Stokes pressure in an ideal gas [41, 206]. This, in turn, leads to inhomogeneous

material properties, such as the viscosity of the fluid [206].

As Ihle, T

utzel, and Kroll demonstrated, introducing position and velocity de-

pendent collision rules in two dimensions results in different material properties,

in particular, a non-ideal equation of state [213]. As depicted in Fig. 3.2, pairs of

neighboring collision cells connected by red arrows are chosen randomly between

the configurations (i)-(iv). Using a stochastic collision with a probability that de-

pends on the densities and relative velocities in the pairs of cells, the method mimics

hard-sphere collisions. Most importantly, the collision is only initiated if the particle

groups move towards each other. Exchanging the relative velocities of the two

particle groups, the modified collision rule introduces a directed momentum flux at

equilibrium [213].

3.3 Collision step

Figure 3.2: Sub-divided collision cells used to implement a position, velocity and

density dependent collision rule in two dimensions adapted from Ref. [213].

3.3.4 Shear viscosity

Similar to the pressure and equation of state, the shear viscosity consists of two

contributions from the streaming and collision step, respectively,

η=ηstr +ηcoll . (3.7)

The dynamic shear viscosity thereby relates the off-diagonal component of the

viscous stress tensor

σ1

xy =ηByvx=ηγ

in the linear shear flow

v(y) = γ

˙yx

with

the constant shear rate

. As for the pressure in Sec. 3.3.3, we may use that the

momentum flux through a plane in the fluid is equivalent to components of the

viscous stress tensor

σ1

to determine the shear viscosity of the MPCD method [207].

The contribution of the streaming step is determined by the correlation between

the velocity components

xvxvyy

of a fluid particle, since only correlated velocity

components result in a flux of the

component of momentum along the

direction.

The velocity correlation function

xvxvyy

changes during both the streaming and

collision step, but should recover its initial state after one step cycle in the steady

state. Using this self-consistency condition, one can ultimately determine

xvxvyy

and

the contribution to σ1

xy (see Refs. [205] and [207] for more details).

For the contribution of the collision step, we again consider momentum trans-

ported within a collision cell from the region

yăy0

across the plane into

yąy0

during time

∆t

. Compared to Eq. (3.6) for the pressure, we now need the transfer of

the xcomponent of momentum per area and time, thus

σ1

xy =´m0

0∆tBx

ˆ¨ÿ

i|yiąy0

vnew

i´viF. (3.8)

To obtain

ηcoll

, Eq. (3.8) is then evaluated in the linear shear flow using that

σ1

xy =ηγ

The time step

∆t

naturally determines the ratio of the two contributions to the

viscosity. The collisional contribution,

ηcoll „

/∆t

, scales with the frequency of the

collision, while in the streaming step,

ηstr „∆t

. Large mean-free paths

λ„∆t

imply

that the models behave more like a gas. Thus,

∆t!

1 are applied to realize highly

3. Multi-particle collision dynamics (MPCD)

viscous flows in the context of microswimmers. Then, the collisional contribution

ηcoll dominates.

For the SRD method with or without thermostat one obtains the

expressions [207]

ηSRD

str =nkBT∆t

a3[︄5n

(n´1+e´n)[4´2 cos(α)´2 cos(2α)]´1

2]︄and

ηSRD

coll =m0(︁n´1+e´n)︁

18a0∆t(︁1´cos(α))︁. (3.9)

in the case of three spatial dimensions.

For the MPCD method with Andersen thermostat, one obtains [194]

ηMPCD´AT

str =n0kBT∆t

m0(︃n0

n0´1+e´n0´1

2)︃and

ηMPCD´AT

coll =a2

0(︁n0´1+e´n0)︁

12∆t. (3.10)

It is important to note that for MPCD-AT the collisional contribution to the stress

tensor is not symmetric, which means the violation of angular-momentum conser-

vation [201]. To restore the symmetry of the stress tensor of a Newtonian fluid,

the antisymmetric part of the velocity gradient

r1=σ1+η

r[︂(∇bu)T´∇bu]︂

added. Here,

is the antisymmetric part of the viscosity, which vanishes in angular

momentum conserving systems

˜=

0. With the Cauchy momentum equation Eq.

(2.8) and assuming an incompressible fluid, this results in [209]

ρ(Btu+u¨∇u)=´∇p+ (η+η

r)∇2u+fext. (3.11)

Although flow fields are not affected, this leads to spurious torques which influence

the dynamics of solute particles [209, 214]. Using a kinetic theory, one can show that

both contributions to the viscosity need to be equal,

ηcoll =η

rcoll

[215]. This can be

understood considering that the angular momentum of the particles in the collision

cell is induced by the vorticity of the shear flow, which generates the angular velocity

ω= (∇ˆv)/

=´(γ

˙/

. Hence, preserving the associated velocity

ωˆxi,c

during the collision step essentially reduces the viscous momentum transport by a

factor of two by eliminating η

rcoll.

This is reflected in the expressions for the collisional viscosity of the collision rule

MPCD-AT+a with conserved angular momentum [194]. For sufficiently large

, the

3.4 Extended MPCD method with non-ideal equation of state

streaming and collisional viscosity contributions read

ηMPCD´AT+a

str =n0kBT∆t

m0[︃n0

n0´(d+2)/4 ´1

2]︃and

ηMPCD´AT+a

coll =a2

0(n0´7/5)

24∆t, (3.12)

where ddenotes the number of spatial dimensions.

For the SRD method with thermostat and angular-momentum conservation,

values for the viscosity are obtained by simulations [41]. As elaborated before, the

angular momentum conservation reduces the collisional viscosity by a factor of two.

For common collision angles around

α=

130

, the values of the collisional viscosity

are comparable to MPCD-AT+a.

Figure 3.3: (a) Side view of a collision cell divided by a plane (red line) through the

center-of-mass position

xξ

and with the unit normal vector

. (b) Normal vectors

of the 13 possible collision directions. They point to the corners, as well as the

centers of the edges and faces of the collision cell [205].

3.4

Extended MPCD method with non-ideal equation

of state

In Ref. [205] we suggested an alternative MPCD algorithm compared to the original

SRD [126] or MPCD-AT method [74]. In the following, we briefly describe the most

important properties of the algorithm based on Ref. [205]. The main difference of this

extended MPCD method is that the collision rule depends on both the positions and

velocities of the fluid particles. This mechanism generates a momentum flux, which

contributes to the pressure in the isotropic part of the stress tensor and, thereby,

generates a non-ideal equation of state. The collision rule thus generalizes the

method introduced in Ref. [213] to three dimensions.

In particular, collisions occur among two groups of particles at random, with

a probability that depends on the local density and velocities, and only if the two

3. Multi-particle collision dynamics (MPCD)

groups move towards each other. To select two groups of particles in three dimen-

sions, we divide each collision cell into two halves

and

by a plane with the unit

normal vector

and containing the center-of-mass position

xξ

[see Fig. 3.3(a)]. At

each collision step

is chosen at random from a discrete set of 13 possible orienta-

tions for each cell, as shown in Fig. 3.3(b). This enables the analytic calculation of

transport properties [205]. Alternatively,

may also be chosen randomly on the unit

sphere, which even increases the viscosity of the model slightly.

To introduce a momentum flux during the collision, the particle groups exchange

the relative momentum along the unit vector

. To define the rate of the random

collisions, we use the mean velocities

¯A

and

¯B

along the normal vector

on both

sides of the plane to define the relative velocity

∆u=v

¯B´v

¯A

. Thereby, we define

such that it always points towards region

, and

∆uą

0 when the groups approach

each other. With the numbers of particles

in the two groups and the

scattering cross section

, the collision rate

c∆u nAnB

is similar to the rate of collision

in two clouds of hard-core particles and can be derived from the collision term of

the Boltzmann equation [216]. Accordingly, we introduce the collision probability

p(∆u)”Θ(∆u)c∆u nAnB(3.13)

«Θ(∆u)[︃1´exp(︂´c∆u nAnB)︂]︃, (3.14)

where the exponential ensures that

p(∆u)ď

1, assuming a sufficiently small

Further, the Heaviside step function

Θ(∆u)

ensures that collisions only occur when

the groups approach each other.

3.5 Boundary conditions and solute particles

If boundaries like walls or solute particles are immersed in the MPCD fluid, they are

coupled to the fluid in both the streaming and collision step. During the streaming

step, fluid particles collide with the boundaries and thereby exchange momentum,

in particular, with the solute particles. To implement a no-slip boundary condition

for the fluid flow, the so-called bounce-back rule is applied. This means that the

velocity of a fluid particle is reversed

vnew

i=´vi

upon collision with a solid wall, as

shown in Fig. 3.4(a).

In the case of a moving colloid or with the imposed slip velocity field on the

surface of a squirmer vs(x

ˆi)given in Eq. (2.52), the bounce-back rule implies that

vnew

i=´vi+2[︁p/m+ωˆ(xi´r) + vs(x

ˆi,s)]︁. (3.15)

Here,

and

p/m

are the center-of-mass position and velocity of the solute particle

and

ˆi,s

is a unit vector that points from the center of the squirmer to the position of

the fluid particle

[see Fig. 3.4(a)]. The change of momentum of the fluid particle is

3.6 Molecular dynamics of solute particles

Figure 3.4: Coupling of the MPCD fluid to immersed boundaries. (a) During the

streaming step, particles collide with the solid walls according to the bounce-back

rule and exchange momentum with solute particles. (b) In the collision step, ghost

particles are distributed in the region of the boundaries with density

. These

participate in the collision and their momentum change is assigned to the respective

solute particles.

thereby subtracted from the solute particle, such that the total linear and angular

momenta of solvent and solute particles are conserved.

In the MPCD collision step, the walls and solute particles contain so-called “ghost”

or “virtual” particles, which are randomly distributed in the collision cells which

overlap with the boundaries, as shown in blue in Fig. 3.4(b). The ghost particles

thereby guarantee that the number density remains equal to

. Before the collision

step, their velocities are initialized as the local velocity of the translating and rotating

solute particles with the addition of a random thermal velocity

δvMB

drawn from

the Boltzmann distribution

vghost =p/m+ωˆ(xghost ´r) + vs(x

ˆghost,s) + δvMB. (3.16)

Here,

ˆghost,s

is a unit vector that points from the center of the squirmer to the

nearest point on the surface. In case of solid walls, only a random thermal velocity is

assigned. After the collision step, the changes in linear and angular momentum of the

ghost particles are assigned to the relevant squirmer, which ensures the conservation

of linear and angular momentum.

On the one hand, the use of ghost particles improves the validity of the no-

slip condition, on the other hand, they prevent that the density dependent fluid

properties do not deviate due to partially empty collision cells at boundaries [128].

3.6 Molecular dynamics of solute particles

Treating solute particles immersed in the MPCD fluid requires additional integration

methods known from molecular dynamics (MD) or event driven simulations. In our

case, interactions between solute particles are described by smooth potentials such

3. Multi-particle collision dynamics (MPCD)

that MD methods apply to solve the equations of motion for the solute positions

and orientations e.

In particular, repulsive forces act between pairs of squirmers that were introduced

in Sec. 2.6, and also when they belong to different squirmer rods introduced in Sec.

2.6.1. The forces are described by the Weeks-Chandler-Andersen (WCA) potential,

which is a purely repulsive modification of the well-known Lennard-Jones potential

[217]. The WCA potential is defined as

VWCA(r) = $

4ϵ[︂(σ/r)12 ´(σ/r)6]︂+ϵif r/σă6

?2,

0 else, (3.17)

where

is the particle-particle distance,

describes the interaction energy, and

the

interaction range. The first part of the potential thereby is the Lennard-Jones potential

shifted by

, which has a minimum at the distance

rmin =σ6

with

VWCA(rmin) =

The step-wise definition guarantees that the potential is continuously differentiable

and goes to zero at the cutoff distance. For squirmer rods, we set

σ=

Rsq/6

the diameter of the squirmer and choose a large interaction energy

ϵ„

4kBT

mimic hard-core interactions [205].

Because the squirmer rods are considered as rigid bodies, these squirmer interac-

tions result in respective forces and torques acting on the rods [218],

F=´∇ÿ

VWCA(rij)and τ=´ÿ

ri,cˆ[︂∇VWCA(rij)]︂. (3.18)

Here,

iterates over squirmers in the rod under consideration,

over all squirmers

of different rods and

ri,c

denotes the squirmer position in the center-of-mass frame

of the rod.

The motion of the squirmer rods is therefore characterized by the forces

, torques

, their mass

, and their moment-of-inertia tensor

I=diag(I1

I3)

, written in

their center-of-mass frame aligned along the principle axes. The rotational symmetry

of the squirmer rods implies

I1=I2

with

ˆ=e

ˆz

in the body-fixed frame [125, 219].

The equations of motion for momentum

and angular momentum

of the squirmer

rods are dp

dt=Fand dL

dt=τ. (3.19)

Further, the orientation of the squirmer rod evolves as

˙=ωˆe

, where

ω=I´1L

To integrate these dynamics into our simulation method, we alternate between the

MPCD steps and the MD integration. During the MPCD steps, the fluid particles ex-

change energy and momentum with the solute particles (squirmer rods) as described

in Sec. 3.5. During the MD integration the exchanged momenta are considered

together with the acting forces and torques. One cycle of the combined integration

scheme consist of

3.6 Molecular dynamics of solute particles

1. the MPCD streaming step,

2. the MPCD collision step,

3. the MD integration with the refined time step δt=∆t/10,

4. resolving the overlap of solutes and fluid particles.

The MD integration 3is performed with a symplectic splitting method for rigid

body dynamics, which provides a high accuracy through a precise conservation of

energy [219]. To tune the precision of the MD integration independent of the MPCD

time step

∆t

, which determines the fluid properties, we introduce the MD time step

δt=∆t/

10. The coupling to the fluid, which is subject to the Andersen thermostat,

guarantees that the whole system is in the canonical ensemble.

In the following, we provide a brief overview of the symplectic splitting method

for the integration of the rigid-body positions

r(t)

and orientations

ˆ(t)

according

to Ref. [219]. The MD method uses the leapfrog variant of the well-known velocity

Verlet method to integrate the translational degrees of freedom [218] and includes

additional steps to integrate the orientations

ˆ(t)

of, for example, the squirmer rods

[219]. The MD method is initiated by advancing the linear and angular momentum

by half a time step δt/2

p(t+δt/2) = p(t) + f(t)δt/2,

L(t+δt/2) = L(t) + τ(t)δt/2. (3.20)

Then, the positions are advanced by a complete time step δt

r(t+δt) = r(t) + p(t+δt/2)δt/m, (3.21)

Before advancing the momenta to the full time step

t+δt

with the updated forces

and torques in the second step, we integrate the angular degrees of freedom, in

particular, the orientations e

ˆ(t)of the squirmer rods.

The scheme by Dullweber et. al [219] thereby considers the dynamics of the

rotation matrix

Q(t)

, that describes the reorientation

ˆ(t) = QT(t)e

ˆ0

that the rigid

body has undergone with respect to a fixed reference frame

ˆ0=z

. For our case of a

rotationally symmetric rigid body, the update of Q(t)is

Q(t+δt) = Q(t)RT

zRT

L(t), (3.22)

where

L(t)

is the rotation matrix about the axis

L(t)

by the angle

δt|L(t)|/I1

[219].

For our case of a truly rotationally symmetric rigid body, we may omit the factor

in the update of

Q(t)

and directly apply the change of the transformation matrix

on the orientation vector of the rigid body

e(t+δt) = RL(t)e(t)

. Note that in the

absence of external forces the integration scheme is is exact [219]. Lastly, the method

is completed by advancing the linear and angular momentum to the full time step

3. Multi-particle collision dynamics (MPCD)

δt, respectively,

p(t+δt) = p(t+δt/2) + f(t+δt)δt/2 and

L(t+δt) = L(t+δt/2) + τ(t+δt)δt/2. (3.23)

In the last step 4, overlaps between fluid particles and the solute resulting from

the MD step 3are corrected by correcting the motion of the fluid particles similar to

Ref. [220]. Then, another cycle begins at step 1.

3.7

Efficient implementation of MPCD on graphic cards

(GPUs)

Over the recent years, computer graphic cards and their centerpiece, the graphics

processing unit (GPU), have become a very powerful tool that can replace small

computer clusters for applicable algorithmic problems. Their strength compared to

traditional computer clusters lies in their low price and superior efficiency regarding

electric power consumption [221].

However, the way GPUs process instructions and data is entirely different from

multi-core computers or clusters, and therefore require different programming

approaches or even algorithms [222]. In particular, processor cores combined in the

so-called vector processing units in GPUs are not independent and always apply the

same instruction on multiple data (SIMD) in contrast to multi-core computers which

apply multiple instructions on multiple data (MIMD). Otherwise, modern GPUs

possess multiple levels of hierarchies such that multiple SIMD vector processors can

work independently on different tasks, as shown in Fig. 3.5.

The collision rule of MPCD — which is simultaneously applied to millions

of fluid particles almost independently — is a good example for an algorithm

suitable for a GPU. Although already elementary implementations of MPCD on

GPUs achieve immense performance benefits compared to conventional computers,

a set of optimizations can again provide large performance increases.

The difficulty in a MPCD collision step is that in a naive implementation the

fluid particles have to be loaded into the processor repeatedly, in order to calculate

the center of mass, mean velocity, angular momentum, and moment of inertia

tensor. If the fluid particles that belong to one collision cell are located in different

memory locations, the GPU hardware cannot access their memory in a combined

read instruction, and instead has to use separate read instructions on different

memory locations [223]. To solve this issue, the memory array containing the fluid

particles is frequently sorted based on the index of the collision cell in which the

fluid particles lie [125, 132]. We thereby use the counting sort algorithm [224] by

calculating a parallel prefix sum [225]. This way, memory accesses can be combined

and the performance increases by a factor of two.

3.7 Efficient implementation of MPCD on graphic cards (GPUs)

Figure 3.5: Sketch of a computer system consisting of a graphic card (GPU) and

main processor (CPU). When running programs, the CPU mainly orchestrates the

GPU, without computing itself. The GPU can access and process the large number

of MPCD particles fast, due to the high bandwidth of its memory (

800GiB/s) and

large number of vector-processor cores („103´104) [222].

To avoid the repeated memory accesses in the collision step, which further limit

the performance, we make use of the small shared data caches inside of the GPU’s

vector processors. However, to be able to store all fluid particles residing in one

collision cell into this memory, the number of collision cells is typically lower than

the number of vector-processor cores (threads). To maximize performance, we

dynamically allocate shared memory and form groups of threads that work on each

collision cell. To synchronize work in the groups of threads (SIMD), we use parallel

summation and prefix sum algorithms.

3. Multi-particle collision dynamics (MPCD)

CHAPTER 4

Published Articles of the

Thesis

In this chapter, I will present the four articles published in scientific journals as part

of this thesis. To begin with, I will give a brief overview of the overall content and

place the different articles in line. The articles will then be presented in the following

sections.

During my thesis work, I have developed the new squirmer rod model for elon-

gated microswimmers, as well as an extended version of the MPCD algorithm to

improve simulations of dense configurations of squirmer rods. Using both models,

we investigated the flow fields of individual squirmer rods and the collective dy-

namics of squirmer rods as a function of the aspect ratio, density and swimmer type.

Our first two articles are thereby concerned with the introduction of the squirmer

rod (1., see below) and the extended MPCD method (2.). In the following articles,

we concentrated on the collective dynamics of neutral squirmer rods (3.) as well as

pusher and puller-type squirmer rods (4.).

Below, I provide an overview over the relevance of the articles.

1. A. W. Zantop

and H. Stark. Squirmer rods as elongated microswimmers: flow field and

confinement. Soft Matter, 16, 6400-6412, (2020).

doi: https://doi.org/10.1039/D0SM00616E

Contribution: First author, performed research and wrote of the manuscript,

see Sec. 4.2 for more details.

Relevance:

This work introduces the squirmer-rod model for elongated mi-

croswimmers and its implementation in the MPCD method. Its flow fields

in the bulk fluid and Hele-Shaw geometry are investigated by means of a

hydrodynamic multipole decomposition for multiple aspect ratios and

confinement strengths.

Status: published in peer reviewed journal.

4. Published Articles of the Thesis

2. A. W. Zantop

and H. Stark. Multi-particle collision dynamics with a non-ideal equation

of state. I. The Journal of Chemical Physics, 154, 024105, (2021).

doi: https://doi.org/10.1063/5.0037934

Contribution: First author, performed research and wrote of the manuscript,

see Sec. 4.3 for more details.

Relevance:

This work introduces an extended MPCD method with a collision

rule that results in a non-ideal equation of state. Analytic expressions

for the equation of state and the shear viscosity are derived and show

excellent agreement with results from simulations.

Status: published in peer reviewed journal.

3. A. W. Zantop

and H. Stark. Multi-particle collision dynamics with a non-ideal equation

of state. II. Collective dynamics of elongated squirmer rods. The Journal of Chemical

Physics, 155, 134904, (2021).

doi: https://doi.org/10.1063/5.0064558

Contribution: First author, performed research and wrote of the manuscript,

see Sec. 4.4 for more details.

Relevance:

This work provides a study of the collective dynamics of squirmer

rods at different aspect ratio and density. Thereby, it is shown that the

extended MPCD method improves the precision of simulations in dense

configurations of squirmer rods and the flow field shows excellent agree-

ment with the analytical solution.

Status: published in peer reviewed journal.

4. A. W. Zantop

and H. Stark. Emergent collective dynamics of pusher and puller

squirmer rods: swarming, clustering, and turbulence. Soft Matter, 18, 6179-6191,

(2022).

doi: https://doi.org/10.1039/D2SM00449F

Contribution: First author, performed research and wrote of the manuscript,

see Sec. 4.5 for more details.

Relevance:

This work provides an in-depth analysis of the collective dynam-

ics of pusher and puller-type squirmer rods as a function of the aspect

ratio, density, and force dipole strength. It is found that swarming states

dominate the behavior of pullers, while pushers exhibit turbulence.

Status: published in peer reviewed journal.

4.1 Synopsis

In our first article presented in Sec. 4.2, we introduce the squirmer-rod model for

elongated microswimmers motivated by the shape of biological microscopic organ-

isms [226, 227]. The model covers the different biological propulsion strategies of

ciliates, as for example Paramecium [70, 166], pushers like the E. coli or B. subtilis

bacteria [28, 228, 229], or pullers as the algae Chlamydomonas

reinhardtii [38, 167].

model the hydrodynamic flow of the solvent, we employ the multi-particle collision

dynamics (MPCD) method [74, 120].

Thereby, we focus on the flow fields created by single squirmer rods in the bulk

fluid and Hele-Shaw geometry, where the squirmer rod swims between two parallel

plates. The flow fields are characterized using a decomposition into hydrodynamic

multipoles to identify the dominant contributions. For the bulk fluid, the decom-

position into hydrodynamic multipoles was introduced by de Graaf et al. [163]. In

our work, we extend their idea further to the Hele-Shaw geometry based on the

appropriate Green’s function of the Stokes equations derived by Liron and Mochon

[177] and Mathijssen

et al. [176].

This allows us to formulate an alternative multipole

expansion for the far field in the Hele-Shaw geometry. Thereby, all multipoles are

modified with the transition from the bulk fluid to the confinement in the Hele-Shaw

geometry [176, 177, 179]. More specifically, this means that the algebraic radial de-

cays of the multipoles change; the force-multipoles become more short-ranged, while

the source-multipoles become more long-ranged. At the same time, the distance of

the confining plates controls the absolute strength of the multipoles. This has the

fundamental consequence that the source dipole dominates the far field of the fluid

flow of every type of self-propelled particle [74, 162].

For the neutral squirmer rod, we observe that the flow field is accurately de-

scribed by solely adding a force-quadrupole term compared to the spherical squirmer.

This additional term possesses the same 1

/r3

radial decay as the source dipole and,

therefore, it does not alter the far-field behavior. The strength of this additional term

increases linearly with the aspect ratio of the rod. For the pusher-type squirmer rod

the modified surface slip-velocity results in the expected characteristic force-dipole

moment in the flow field, which is more long-ranged and, therefore, dominates flow

field in the bulk fluid.

In the Hele-Shaw geometry, we verify both the different radial decay of the flow

fields of the source and force multipoles as well as the scaling with the distance of the

confining walls. In particular, the characteristic source-dipole moment in the flow

field of neutral squirmer rods becomes more long-ranged, while the force-dipole

moment in the flow field of pushers becomes more short-ranged. The appropriate

multipole expansion for the Hele-Shaw geometry fits well in the far field and in

strong confinement. For larger separations of the walls, deviations appear at short

distances due to terms neglected from the full form of the appropriate Green’s func-

tion of the Stokes equations. For the pusher-type squirmer rod in the Hele-Shaw

4. Published Articles of the Thesis

geometry an interesting crossover of the source-dipole to the force-dipole appears as

a function of the confinement strength.

In our second article presented in Sec. 4.3, we introduce an extended MPCD

method with improved thermodynamic properties for simulations of the collec-

tive dynamics of squirmer rods. This work is separated into two articles. In the

present first article, we introduce the extended MPCD method and derive analytic

expressions for the equation of state and the shear viscosity, which are compared

with simulations. In the second article, we evaluate the new method in the context

of squirmer rods and perform an in-depth study of the phase diagram of neutral

squirmer rods.

Our starting point is that the traditional MPCD fluid has the equation of state

of the ideal gas and is thus very compressible [126, 145, 230]. On this basis, we

formulated a new MPCD collision step, which extends the concept of T

uzel, Ihle,

and collaborators, who integrated geometric properties of collisions between hard-

core particles into a two-dimensional variant of MPCD [231, 232].

The definition of the extended collision rule resembles properties of collisions in

clouds of hard-core particles. This is realized as follows. Additionally to dividing

MPCD particles into collision cells, each cell is divided into two groups using a

randomly oriented plane. To mimic geometric properties of hard-core particles,

collisions exchange the relative momentum of these groups if they approach each

other. Moreover, the collision frequency in the MPCD collision cell increases with the

density and relative velocity of the two particle groups. To realize different collision

frequencies for each MPCD cell, collisions occur at random with a probability that

depends both on the local density and velocities of the particles. This is motivated

by the collision term in the Boltzmann equation for clouds of particles [216] and

appears in a similar form for chemical reactions of second order [233].

The extended MPCD method creates an inherent momentum flux that adds

to the pressure term of the stress tensor. As we show, this results in a non-ideal

equation of state of the model fluid and achieves a low compressibility at a high

numeric efficiency. Furthermore, we derive analytic expressions for pressure and the

shear viscosity, which are in good agreement with values obtained from numerical

simulations of the bulk fluid, constant linear shear and Poiseuille flow.

In our third article presented in Sec. 4.4 we continue our work on the extended

MPCD method [205], assess its properties and performance, and apply it for an

in-depth study of the collective dynamics of neutral squirmer rods.

As we are able to show, the extended MPCD method accurately reproduces the

analytic solution of the flow field of a pusher-type spherical squirmer in both the

bulk flow as well as in the Hele-Shaw geometry.

We further compare the extended MPCD method to the traditional MPCD-AT+a

method in simulations of a dense configuration of squirmer rods with large aspect

4.1 Synopsis

ratio

, for the same viscosity

and P

eclet number

. Simulations of the collective

dynamics of squirmer rods are thereby performed in a Hele-Shaw geometry, where

the squirmer rods are confined to move in the center plan between the confining

walls. In both MPCD methods, the squirmer rods form a large stable swarm, but as

expected the extended MPCD fluid indeed behaves more incompressible, and the

properties of the fluid remain more uniform. In a second comparison, we consider

squirmer rods with small aspect ratio

, which exhibit the dynamic formation of

small swarms with a finite lifetime for both MPCD methods. However, in this case,

the collective dynamics differs and the emergent swarms in the extended MPCD

method have larger velocities and reach larger sizes.

We further determine the state diagram of neutral squirmer rods for variations of

the aspect ratio

and the density, described by the two-dimensional area fraction

in the center plane. Overall, four different states emerge. In dilute systems, where

distances between particles are large, a disordered state without any correlations

between squirmer rods occurs. For higher aspect ratios, αą3.25, the rods exhibit a

dynamic swarming state, in which small to medium size swarms frequently form

and dissolve. If the aspect ratio is further increased, swarms become stable, and

ultimately a single big swarm forms. Lastly, at large density the rod configuration

reaches a jammed state with a large stationary cluster.

In our latest article presented in Sec. 4.5, we extend our study to the collective

dynamics of pusher and puller-type squirmer rods. In this regard, we adapt the

squirmer-rod model to include arbitrary values of the force-dipole strength, in order

to smoothly vary from strong pusher to strong puller type. We then investigate

the state diagram of squirmer rods in the Hele-Shaw geometry, as a function of the

force-dipole parameter χ, the aspect ratio α, and the density or area fraction ϕ.

For pusher-type squirmer rods we observe a new turbulent state alongside with

a dynamic cluster state, which emerges intermediate between active turbulence and

jammed clustering. We find that the dynamic states emerge in an interplay between

hydrodynamic interactions, described by the force-dipole strength

, and the steric

interactions, described by the combination of

and

. In dilute systems, where

particles collide less and steric interactions are less important, the hydrodynamic

interactions of pushers result in a complete destabilization of the swarming state

that emerges for neutral squirmer rods. Instead, we observe a disordered state that

exists also for higher aspect ratios

and densities

. For larger

and

, we observe

that short-ranged steric interactions that align rods are more important and the

turbulent state emerges in a competition with the more long-ranged hydrodynamic

disordering interactions. This results in positive and negative velocity correlations

that alternate on the length scale of a few swimmer lengths. The corresponding

power spectral density exhibits two distinct regimes with power-law scaling and

non-universal exponents and a maximum, indicating a characteristic patter length-

scale. For larger aspect ratios and denser systems, the steric interactions dominate

4. Published Articles of the Thesis

and induce the single swarm and jammed cluster states already found for neutral

squirmer rods.

In contrast, the hydrodynamic interactions of pullers enhance the formation of

dynamic swarms, which then occur for all aspect ratios already at low densities.

Accordingly, also the single-swarm state and the jammed cluster state are shifted

towards smaller densities.

The variation of the swimmer-type parameter

supports the previous findings.

For pullers, we observe that the dynamic swarming state dominates at low densities

for also small dipole strengths. The turbulent state is again found for intermediate

strengths of both hydrodynamic and steric interactions, and we observe that it is

destroyed if hydrodynamic interactions dominate.

6400 |Soft Matter, 2020, 16, 6400--6412 This journal is ©The Royal Society of Chemistry 2020

Cite this: Soft Matter, 2020,

16,6400

Squirmer rods as elongated microswimmers: flow

fields and confinement

Arne W. Zantop * and Holger Stark *

Microswimmers or active elements, such as bacteria and active filaments, have an elongated shape,

which determines their individual and collective dynamics. There is still a need to identify what role

long-range hydrodynamic interactions play in their fascinating dynamic structure formation. We

construct rods of diﬀerent aspect ratios using several spherical squirmer model swimmers. With the help

of the mesoscale simulation method of multi-particle collision dynamics we analyze the flow fields of

these squirmer rods both in a bulk fluid and in Hele-Shaw geometries of diﬀerent slab widths. Based on

the hydrodynamic multipole expansion either for bulk or confinement between two parallel plates, we

categorize the diﬀerent multipole contributions of neutral as well as pusher-type squirmer rods. We

demonstrate how confinement alters the radial decay of the flow fields for a given force or source

multipole moment compared to the bulk fluid.

1 Introduction

Swimming at the micron scale and understanding how micro-

organisms overcome the constraints of low-Reynolds-number

hydrodynamics have attracted a lot of attention among physicists.

1–3

Intriguing features such as active turbulence,

4,5

swarming,

6,7

polar

patterns,

and vortex formation,

8,9

which arise from the active nature

of the fluid, still pose challenges to scientists. The fluid environment

and the flow fields initiated by microswimmers strongly determine

both their isolated motion and their appealing collective patterns,

which they develop in non-equilibrium.

But also the elongated

body shapes

4,7,8

and geometric confinement

11–15

of microswimmers

significantly contribute to their single and collective dynamics.

However, it is still a matter of current debate to completely identify

the individual contributions of direct steric and long-range hydro-

dynamic interactions.

2,16,17

By introducing an active squirmer rod in

this article, we aim to contribute to the discussion with a detailed

analysis of its flow field in bulk and Hele-Shaw geometry.

Artificial microscopic swimmers with various locomotion

mechanisms have recently been constructed to study the principal

properties of their individual and collective motion. Examples

mostly include spherical microswimmers.

6,18–21

Elongated and

flexible active constituents are realized by polar biofilaments in

motility assays

or at a fluid–fluid interface,

while bacteria provide

a natural realization of active rods.

Combining elements of the Toner–Tu

and Swift–Hohenberg

theories, continuum models for bacterial microswimmers have

been developed that are able to reproduce pattern formation such

as vortices and active turbulence.

4,24–26

Alternative continuum

theories for active matter based on liquid-crystal hydrodynamics

reproduce the creation, annihilation, and motion of liquid-crystal

defects

orareabletodescribethedynamicsoftheintracellular

cytoskeleton gel.

2,28

In contrast, particle-based simulations are employed to study

the collective dynamics of active rods. In the famous Vicsek

model coarse-grained aligning interactions are able to reproduce

dynamical states such as flocking and swarming.

29–31

Langevin

dynamics simulations of active rods

4,32,33

or active filaments

34,35

suggest that many properties might already emerge from short-

ranged steric interactions. However, also implicit hydrodynamic

pair interactions are included in such simulations to provide

more realistic models with novel dynamic states.

36,37

An exten-

sion to single active filaments exists.

38,39

Explicit hydrodynamic

simulation schemes have been applied, as well. For example, the

lattice Boltzmann method was used to investigate the properties

of microswimmers with various shapes

40–42

while with the

method of multi-particle collision dynamics collective phases of

spherical

43–45

and ellipsoidal microswimmers

46,47

were studied,

as well as realizations of pusher and puller-type swimmers.

48,49

In this work, we introduce and characterize single squirmer

rods with the idea to model elongated microswimmers in their

fluid environment. They then can be used to disentangle the

eﬀect of direct steric and long-range hydrodynamic interactions

in their collective behavior. The squirmer model swimmer

45,50,51

is a good model for spherical artificial microswimmers such

as Janus particles

52–54

and also biological organisms such as

Volvox.

Squirmers have in common that they propel themselves

by an axisymmetric surface–velocity field, which acts on the

surrounding fluid.

45,56,57

In biological systems this is realized

Institut fu

¨r Theoretische Physik, Technische Universita

¨t Berlin, Hardenbergstraße 36,

10623 Berlin, Germany. E-mail: a.zantop@tu-berlin.de, holger.stark@tu-berlin.de

Received 8th April 2020,

Accepted 9th June 2020

DOI: 10.1039/d0sm00616e

rsc.li/soft-matter-journal

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by cilia, located all over the cell surface, which perform synchronized

collective non-reciprocal motions. The squirmer has frequently been

used in hydrodynamic simulations of the collective behavior of

microswimmers

42,58,59

and also by our own group,

43–45,60–62

where we rely on the mesoscale method of multi-particle

collision dynamics.

45,63,64

In this article we use squirmers to build rigid rod-shaped

microswimmers of diﬀerent aspect ratios and perform large-

scale simulations with multi-particle collision dynamics. We

explore their hydrodynamic flow fields both in a bulk fluid and

in a Hele-Shaw geometry with varying cell width, where we keep

the squirmer rod in the midplane to mimic the fluid interface

in the experiments of ref. 7 (cf. Fig. 1).

In our analysis, we rely on the hydrodynamic multipole expansion

both in the bulk fluid

10,40,50,65

and between two parallel plates.

12–14

This will enable us to categorize the different multipoles of both

neutralaswellaspusher-typesquirmer rods and to determine the

different radial decays of their multipole flow fields.

The paper is organized as follows. In Section 2 we introduce

the squirmer rod and the methods to generate and analyze its

flow fields. We present the results of our analysis in Section 3

and conclude in Section 4.

2 System and methods

The hydrodynamics of the squirmer rod at small Reynolds

numbers is determined by the Stokes equations including the

incompressibility condition,

urp+f=0 ru= 0. (1)

Here, Zis the viscosity of the fluid, u(r) the fluid velocity, p(r)

the pressure, and f(r) denotes a body force at point r. In the

following we introduce the squirmer-rod model, the funda-

mental solutions of the Stokes equations both in bulk and a

Hele-Shaw cell, in order to characterize its hydrodynamic

multipole moments, and details of the method of multi-

particle collision dynamics (MPCD) to simulate the flow fields

together with the relevant parameters.

2.1 Squirmer rod model

To investigate the influence of shape anisotropy of microswimmers

and their hydrodynamic interactions on collective motion, we

introduce a new model of squirmingactiverods.Toconstructthem,

we arrange several spherical squirmers [cf. Fig. 2(a)] on a line to form

a rigid body as shown in Fig. 2(b). On the one hand, this closely

resembles the actual rod shape of many biological microswimmers

such as bacteria, on the other hand the model can be extended to a

flexible swimmer body by introducing bending rigidity in future

works. In concrete, we always take N

=10sphericalsquirmersand

vary the squirmer distance d[cf. Fig. 2(b)] to form rods of diﬀerent

aspect ratios a=l

/2R,wherel

is the rod length. We do not go

beyond a maximum squirmer distance of dE0.8R, which amounts

to a maximum rod length l

=9.5R, so that the surface is still smooth

enough that swimmers can slide past each other. Each of the

individual squirmers of a rod propels itself by an imposed axisym-

metric surface slip-velocity field of a neutral squirmer,

45,56,57

[(e

ˆxˆ

)xˆ

e

ˆ] (2)

that acts on the surrounding fluid. Here, xˆ

is the unit vector

along x

, which points from the squirmer center to its surface,

and unit vector e

ˆindicates the orientation and swimming

direction of the squirmer. In the case of a single squirmer,

the induced flow field in a bulk fluid agrees with the velocity

field of a pure source-dipole singularity with strength B

as explained in the following section. The squirmer parameter

also determines the swimming speed v

, which amounts to

= 2/3B

for a single squirmer.

Having the surface velocity field distributed over the whole

squirmer rod [cf. Fig. 2(b)], our model resembles ciliated

microorganisms such as Paramecium. In contrast, bacteria like

Fig. 1 Perspective view of a squirmer rod moving in the midplane (in light

gray) of a Hele-Shaw cell with slab width Dz. Periodic boundary conditions

along the xand ydirections are used.

Fig. 2 (a) Schematic of a single squirmer of radius Rand with orientation

given by the unit vector e

ˆ. The surface slip-velocity field of a neutral

squirmer is indicated by blue arrows. (b) Schematic of the squirmer rod

model. Always, n

= 10 spherical squirmers are placed on a straight line

with distance dto form active rods. All squirmer orientations e

ˆare aligned

with the rod axis. (c) Implementation of a pusher type squirmer rod. The

surface slip velocity on the rod surface is multiplied by the envelop

function f(x

*e

ˆ) of eqn (3) such that the slip-velocity field is concentrated

on the rear of the rod.

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E. coli propel themselves with a bundle of rotating flagella that

pushes fluid backwards while the cell body does not have any

surface velocity field. To implement squirmer rods of such a

pusher or also of puller type, we use the envelope function

fðxs^

eÞ¼1tanh 10xs^

e=lS

ðÞ

2:(3)

Here, x

* is a vector from the center of mass of the rod to any

point on the surface [cf. Fig. 2(c)] and 10/l

is the step width of

the envelope function. Multiplying the surface velocity field

with f(x

*e

ˆ), concentrates the slip velocity field on the rear half

of the rod for the – sign in fand thereby a pusher squirmer rod

is realized, while the + sign generates a puller.

2.2 Fundamental solutions of Stokes flow in bulk

To provide an understanding of the flow field u(r)thatsquirmerrods

initiate in a bulk fluid and how they interact hydrodynamically, we

review the multipole expansion of u(r).

50,66

It is composed of singular

solutions of the Stokes equations including the Stokeslet as the flow

field of the leading force monopole, which has to vanish for force-

free microswimmers, the flow field of the source dipole, and their

higher-order derivatives. We introduce the polar angle yrelative to

the squirmer-rod axis via cos y=e

ˆr/r. Then, in spherical coordinates

r,f,andythe respective radial and polar components of the velocity

field can be written as

50,66

urðr;yÞ¼X

n¼1

AnrnþBnrn2



Pnðcos yÞ;

uyðr;yÞ¼X

n¼1

21



Anrnþn

2Bnrn2

VnðyÞ:

(4)

Here, P

(cos y) are the ordinary Legendre polynomials and

VnðyÞ¼ 2siny

nðnþ1ÞPn0ðcos yÞ;(5)

where 0means derivative with respect to cos y. Due to the

rotational symmetry of a squirmer rods about its axis, the flow

field is independent of the azimuthal angle fand the azimuthal

velocity component u

vanishes. The multipole coeﬃcients A

and B

describe the strength of the nth-order force and source

multipoles, where B

belongs to the source dipole. To determine

them, we calculate the nth expansion or Legendre coeﬃcient of

the radial component of the flow field in eqn (4),

ur;nðrÞ2nþ1

2ðp

urðr;yÞPnðcos yÞsin ydy;(6)

using the completeness relation of the Legendre polynomials,

which gives

r,n

(r)=A

n

n2

. (7)

Thus, from the simulated flow fields of the squirmer rods, in

particular, from the radial decay of u

r,n

(r), we can infer their

leading force and source multipole moments. We will demon-

strate this in Section 3.1. Note that the nth force and source

monopole can be clearly distinguished by their radial decay.

2.3 Fundamental solutions of Stokes flow in Hele-Shaw

geometry

In a seminal paper Liron and Mochon

determined the Stokeslet

solution between two infinitely extended parallel plates with no-slip

boundary condition (cf. Hele-Shaw geometry in Fig. 1) using

an infinite series of image forces. For a force parallel to the plates

(xyplane), the x,ycomponents of the resulting Stokeslet flow

field are long-range and consist of a Poiseuille flow profile g(z)

along the plate normal times the flow field of a two-dimensional

source dipole in the plane (cf. Appendix A). As in the bulk fluid

one can generate the flow fields of higher-order force moments by

applying the directional derivative e

ˆr

to the Stokeslet flow field.

Here, r

is the nabla operator acting on the position of the

Stokeslet singularity and the unit vector e

ˆgives the direction of

the force multipole in the xyplane, which we assume to be

uniaxial for simplicity. Likewise, by applying the Laplace operator

to the Stokeslet flow field, one obtains the flow field of a source

dipole, which here is up to a factor identical to the Stokeslet flow

field.

Higher source multipole moments are again generated by

directional derivatives.

Relevant for the squirmer rod will be the flow fields of the

source dipole (u

), force dipole (u

), force quadrupole (u

and source octupole (u

):†

u(r)=u

(r)+u

(r)+v

(r)+.... (8)

To determine these contributions from the simulated flow field

of the squirmer rod, we first average over the zcoordinate and

then treat the resulting velocity field u

˜(r,j) in the x–yplane

using polar coordinates r,j. As we demonstrate in Appendix A,

for a microswimmer oriented along the xaxis, e

ˆ=e

, the

multipole expansion for the radial component of the flow

field gives

urðr;jÞ¼X

n¼1

AnþBn

rnþ1Tnðcos jÞ:(9)

Here, T

(cos j) = cos(nj) are the Chebyshev polynomials of the

first kind and A

are the respective coefficients of the force

and source multipole moments. They start with A

for the

force monopole, which does not exist for a force-free swimmer,

and B

for the source dipole, and so on.

It immediately becomes obvious that in contrast to the bulk

fluid one cannot distinguish the flow fields of force and source

multipoles with the same angular dependence (same order n)

by the radial decay. We note that the coeﬃcients scale diﬀer-

ently with the slab width Dz. In Appendix A we motivate A

Dzand B

p1/Dz. However, our simulated data are not always

suﬃciently accurate to discriminate both cases. So, we assign

the relevant multipole assuming that it is preserved from the

bulk fluid. The multipole expansion captures the distribution

†Although the source quadrupole shares the same radial dependence as the

force dipole in the Hele-Shaw geometry, it is not considered in this list for two

reasons. First, we solely observe a force dipole in the bulk fluid but no source

quadrupole. Second, the strength of u

r,2

(r) grows with the slab width Dzin our

simulations, which identifies it as a force dipole rather than a source quadrupole.

We will elaborate on this further below.

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of force and source multipoles in leading order. This distribution

generated by the rod surface should remain the same in the

Hele-Shaw geometry. However, since we deal with a voluminous

rod in contrast to a point-like source, higher-order terms enter to

fulfill the boundary conditions on both the rod surface and the

bounding plates. The flow fields of these terms will decay faster

than the leading bulk moments at some distance from the rod.

Finally, exploiting the orthogonality relations of cos nj=

(cos j), we extract the multipole moments from eqn (9) by

projecting u

(r,j) on the Chebyshev polynomials,

~ur;nðrÞ1

pð2p

~urðr;jÞTnðcos jÞdj;(10)

which yields

ur;nðrÞ¼AnþBn

rnþ1:(11)

In both the bulk fluid and Hele-Shaw geometry we can restrict

our focus on a particular set of Stokes flow singularities. These

will be the source dipole (B

), force dipole (A

), force quadru-

pole (A

), and source octupole (B

2.4 MPCD fluid model

To model the fluid in our simulations, we apply the method of

multi-particle collision dynamics together with the Andersen

thermostat and angular momentum conservation (MPCD-AT+a).

The method solves the Navier–Stokes equations and thereby can

also treat hydrodynamic interactions between the active squirmer

rod and the confining walls of the Hele-Shaw geometry.

63,64

Furthermore, it also includes thermal fluctuations. The MPCD

method considers point-like particles that represent the fluid and

defines rules for (i) their motion and (ii) their collisions such that

the resulting hydrodynamic flow field fulfills the Navier–Stokes

equations. These rules are applied in alternating steps. We shortly

introduce the essential features of the MPCD method and refer to

ref. 45 for more details.

During the streaming step (i) the point particles with masses

, positions x

(t), and velocities v

(t) move ballistically during

time Dt,

(t+Dt)=x

(t)+v

(t)Dt. (12)

During this step fluid particles collide with confining walls or

the surface of the squirmer rod. By applying the so-called

bounce-back rule

44,45

the collisions either enforce the no-slip

boundary condition on confining walls or the slip-velocity field

on the squirmer-rod surface. In addition, the collisions also

transfer both linear and angular momentum, in particular, to

the squirmer rod. This changes the center-of-mass and angular

momentum of the rigid rod, which is calculated relative to the

center-of-mass. In Appendix B we present formulas for the mass

rod

and moment-of-inertia tensor I

rod

of the rigid squirmer rod.

For the collision step (ii), the simulation volume is divided

by a cubic lattice and the fluid particles are grouped into the

cubic unit cells of linear size a

and centered around x. First, for

the n

particles in each cell with volume V

the mean velocity v

and center-of-mass position x

are determined. Then, in the

center-of-mass frame the fluid particles are assigned new

random velocities dv

from a Boltzmann-distribution with tem-

perature T

. To restore overall momentum conservation, the

total change in linear momentum, m0Dvx¼m0

nxP

xi2Vx

dvi, has to

be subtracted from the new velocities, while an additional term

containing the diﬀerence of the angular momentum before

the collision Lx¼m0

nxP

xi2Vx

xi;cvi, and the change in angular

momentum, DLx¼m0

nxP

xi2Vx

xi;cdvi, is added to preserve

angular momentum. Here x

i,c

denotes the particle position

relative to the center-of-mass position x

. Thus, the collision

step can be summarized by

new

+dv

Dv

x

i,c

I

x1

DL

), (13)

Here, I

is the moment-of-inertia tensor of the distribution of

particles inside V

calculated in the center-of-mass frame.

During this step immersed boundaries are represented by

so-called ‘‘ghost’’ particles. They are added to the collision cells

divided by the boundaries and interact with the other fluid

particles. These ghost particles are assigned the local velocity of

the translating and rotating squirmer rod plus a random

thermal velocity drawn from a Boltzmann distribution. The

changes of linear and angular momentum of the ghost particles

during the collision step are then added to the squirmer rod to

ensure linear and angular momentum conservation. Finally,

before performing each collision step, the lattice is randomly

shifted to ensure Galilean invariance.

Given the center-of-mass and angular momentum of the

rigid squirmer rod, its location and orientation are updated 10

times during each collision step using a standard leapfrog

algorithm.

The rod is treated as single rigid body the position

and orientation vector of which is updated following ref. 67. To

keep the squirmer rod in the midplane of the Hele-Shaw cell,

we only use the xand ycomponent of the fluid force acting on

the rod to integrate its motion in time throughout the simula-

tion. Due to the symmetry about the midplane only Brownian

forces are acting on the squirmer normal to the midplane. Note

due to this constraint we do not observe any oscillatory trajec-

tories between the plates as observed in ref. 68.

We have implemented the MPCD model fluid together with

the squirmer-rod model in C++ and CUDA to enable the use of

graphic cards. Because the main performance bottleneck in

MPCD is memory access, we integrate a sorting algorithm and

lookup table following ref. 69. Additionally, we use variable size

cooperative thread groups to add dynamic load balancing to

our MPCD collision routines in CUDA.

2.5 Geometry and fluid parameters

For the MPCD fluid we use a density of n

=10/a

fluid particles

per cell and the same mass density r

for the immersed

squirmer rod. With time step Dt¼0:02a0ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

m0=kBT0

pone obtains

a dynamic fluid viscosity of Z¼16:05 ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

m0kBT0

pa02.

Through-

out this work, the radius and the squirner parameter of the

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single squirmer are chosen as R=3a

and Bs

1¼0:1ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

kBT0=m0

respectively.

To determine how the hydrodynamic moments of the squirmer

flow field in a bulk fluid varies with the aspect ratio ain

Section 3.1, we simulate single rods in a cubic box of linear

size L= 100a

using periodic boundary conditions in all three

dimensions. For all other bulk simulations in Sections 3.1 and

3.3, we use box sizes of L= 180a

with periodic boundary

conditions. We begin by simulating the system for a time 10

to equilibrate the MPCD fluid flow and then average the flow

fields over the time interval from 10

Dtto 5 10

Dt.To

determine the radial component v

(r,y) of the velocity field,

we also exploit the rotational symmetry of the flow fields by

averaging about the rod axis. Using eqn (6), we then obtain the

expansion coeﬃcients u

r,n

(r) for diﬀerent multipole order nand

extract the strengths of the hydrodynamic moments by fitting

r,n

(r) from eqn (7) to the curves determined from the simulated

flow fields.

For the simulations in Hele-Shaw geometry in Sections 3.2

and 3.3, we use box sizes of L= 200a

with no-slip walls and

periodic boundary conditions along the xand ydirections. For

the slab width we investigate the three values Dz/R= 2.7, 6.0,

and 9.3. When determining the flow fields in the Hele-Shaw

geometry, the total simulation time is increased up to 10

Dtto

compensate for the fact that the flow field cannot be averaged

about the rod axis.

3 Results

In the following, we present our results. We first discuss the

flow field of a neutral squirmer rod in the bulk fluid and how it

depends on the aspect ratio a. We then study neutral squirmer

rods in Hele-Shaw geometries of diﬀerent slab widths and

illustrate the diﬀerent far-field behavior compared to the bulk

fluid. Finally, we look at a squirmer rod of pusher type moving

in both bulk fluid and the Hele-Shaw geometry.

3.1 Neutral squirmer rods in bulk fluid

In the bulk fluid, rods of diﬀerent aspect ratio acreate flow

fields as shown in Fig. 3. In all panels (a), (b), and (c) only a part

of the simulated fluid volume is shown. In the case of the

spherical squirmer, a= 1, in Fig. 3(a) we observe a perfect

match with the theoretical prediction for the flow field of a

source dipole moment. For a short squirmer rod with a= 1.75

the flow field already becomes stretched along the rod

[cf. Fig. 3(b)]. At higher aspect ratio, a= 4.0, the flow field

shows a strong deviation from the one of a source dipole

moment, with the streamlines buckling inwards at the sides

of the rod. Thus, we expect higher-order moments to contribute

strongly to squirmer rods with higher aspect ratios a.

To provide a quantitative analysis, we apply eqn (6) and

decompose the flow field into its diﬀerent angular contribu-

tions with the nth-order Legendre coeﬃcients u

r,n

(r). As an

example, we show in Fig. 4(a) the measured u

r,n

(r) as data

points for a squirmer rod of a= 4 and also include fitted

polynomials in r

1

as solid lines.

All radial velocity components u

r,n

(r) either show distinct

power law behavior as for n= 1 and 3 or vanish. The first-order

coeﬃcient u

r,1

(r) clearly shows a pure 1/r

decay, indicating a

zero force monopole moment A

while the source dipole

moment B

is present, as expected. This is consistent with the

fact that we consider microswimmers free of external forces.

Consistent with the head–tail symmetry of the forces, the

squirmer rods exert on the fluid, we do not observe a force

dipole moment, A

= 0, since the second-order coefficient u

r,2

(r)

vanishes. Furthermore, the third-order contribution u

r,3

(r)is

also present and decays with 1/r

, which indicates an additional

force quadrupole moment A

. We elaborate on its origin further

below. Both curves u

r,1

(r) and u

r,3

(r) fall off from the theoreti-

cally predicted power law at large rdue to the finite size of the

simulation box. They also show small deviations from the

power law, which we attribute to the flow fields of the image

swimmers introduced by the use of periodic boundary condi-

tions. All other coefficients u

r,n

(r) are too small to be distin-

guished from noise and hence can be neglected in the

following. Thus, the flow field of a neutral squirmer rod is a

superposition of a source dipole moment (B

a0), and a force

quadrupole moment (A

a0), which both decay as 1/r

.In

Fig. 4(b) we plot the two moments normalized by the squirmer

parameter B

versus the aspect ratio a. They both increase

roughly linearly in a. While the source-dipole moment B

shows

Fig. 3 Simulated flow fields of an active squirmer rod for diﬀerent aspect ratios a= 1 (a), a= 1.75 (b), and a= 4.0 (c) shown in the laboratory frame. The

color-coded magnitude of the velocity field and streamlines are presented. In each case only a part of the simulated fluid volume is shown.

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a modest increase in astarting from B

for a= 1, the force

quadrupole moment A

increases much more strongly from

zero and clearly dominates beyond a=2.

The additional force quadrupole does not break the head–

tail symmetry of the squirmer rod, which we therefore denote

as neutral. However, the corresponding flow field clearly affects

the shape of the overall flow field as we saw in Fig. 3. Since its

angular dependence is governed by the third-order Legendre

polynomial P

(cos y) compared to P

(cos y) of the source dipole,

it will influence the hydrodynamic interactions with other

squirmer rods, also because its flow field shows the same radial

decay, 1/r

To shed some light on the origin of the force quadrupole, we

determined the local force F

hyd

), with which the spherical

squirmer component iplaced at y

along the rod axis acts on the

surrounding fluid. Of course, the rod is force free, thus all the

forces F

hyd

) add up to zero, P

Nsq

i¼1

FhydðyiÞ¼0. The forces from

the terminal squirmers i= 1 and i=N

are mainly determined

by fluid pressure. They cancel each other up to some remaining

force, F

hyd,p

hyd

)+F

hyd

). In the following, we refer the

force of the other squirmers on this remaining force, dF

hyd

)F

hyd,p

/(N

2), so that they sum up to zero:

Nsq1

i¼2

dFhydðyiÞ¼0. In Fig. 5(a) we plot the relative force dF

hyd

the schematic in Fig. 5(b) rationalizes the diﬀerent signs of

hyd

). In the center of the rod (y= 0), the relative local force

is negative due to the surface velocity of the squirmer rod

pushing fluid backwards, while at both ends of the rod the fluid

is dragged with the rod. This generates the observed force

quadrupole.

Finally, as we show in Fig. 4(c), the swimming speed v

rod

the rod increases with aand then saturates. With increasing

aspect ratio the swimming speed is more and more determined

by the maximum surface velocity around the equator of the

constituent squirmers and thus the speed increases. At large a

the cap regions of the squirmer rod become irrelevant and the

swimming speed saturates. To calculate an analytic expression

for the swimming speed, we average the surface slip-velocity

field v

of the squirmer rod over the whole surface with area

rod

For the component along the squirmer axis e

ˆwe obtain

vrod ¼ 1

SrodðSrod

evsdS(14)

¼v0

21

2a1

54 þ1

162aþa

54 a2

162



:(15)

The derivation is sketched in Appendix C. Note that the first

two terms on the right-hand side of eqn (15) result when

a perfect spherocylinder with the maximum slip velocity

=3v

/2 on the straight region is assumed. For a=1we

reproduce the swimming velocity v

of the spherical squirmer.

Fig. 4(c) shows the analytic expression overestimates the swim-

ming speed. The reason is that the surface average of eqn (14)

to calculate the swimming speed only applies to spherical

shapes since then the hydrodynamic surface stress for a passive

sphere is constant along the surface.

This is used in the

derivation of eqn (14) and obviously no longer valid for the

passive squirmer rod.

3.2 Neutral squirmer rods in Hele-Shaw geometry

To discuss the flow fields of a squirmer rod in the confining

Hele-Shaw geometry, we start with Fig. 6 where we show the

color-coded strength of the flow field and streamlines in the

midplane of the cell (upper panel) and the cross-sectional plane

Fig. 4 (a) Radial Legendre coeﬃcients u

r,n

(r) of the simulated flow field of

a neutral squirmer rod with aspect ratio a= 4.0 calculated from eqn (6)

(symbols) and with fitted power laws (solid lines). SD and FQ stand

for source dipole and force quadrupole, respectively. (b) Hydrodynamic

multipole moments in units of the squirmer parameter B

plotted versus

aspect ratio a. (c) Swimming speed v

rod

of a squirmer rod plotted versus a.

rod

is normalized by the swimming speed v

=2/3B

of a single squirmer at

a= 1. Simulation data (blue dots) and analytical values as given by eqn (15)

(dashed black line).

Fig. 5 (a) Local relative force dF

hyd

), with which the squirmer rod acts

on the fluid, plotted along the long axis yof the squirmer rod. The aspect

ratio is a= 4. The dots correspond to the axial positions y

of the

constituent squirmers and the line is a guide to the eye. Note, in contrast

to the rest of the article, N

= 20 squirmers are used to compose the rod

in order to generate a smoother surface. (b) Sketch of dF

hyd

(y) (blue arrows)

to illustrate the force quadrupole. The grey arrow indicates the swimming

velocity.

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(lower panel). For a squirmer rod of a= 4.0 and similar to the

bulk fluid [cf. Fig. 2(c)], we observe how the streamlines at the

sides of the rod buckle inwards (upper panel of Fig. 6) due to

the flow field of the force quadrupole moment. However, as

explained in Section 2.3 and detailed in Appendix A, in the

Hele-Shaw geometry the flow field of the source dipole with

moment B

is of longer range than the one of the force

quadrupole with moment A

. Indeed, already at a distance l

from the rod the buckling of the flow lines is much weaker

compared to the bulk fluid [cf. Fig. 3(c)] and the flow field

assumes the shape of a pure source dipole.

In the lower panel of Fig. 6, further away from the rod

(|x/l

|41) the streamlines are approximately parallel to the

bounding plates in agreement with the Poiseuille flow profile of

the single force and source multipoles as detailed in Appendix

A. However, in the direct vicinity of the rod, |x/l

|o3/4, near-

field flow along the zdirection occurs, which is due to terms

with exponential decay in the full hydrodynamic solution in a

slab geometry.

We now present a more quantitative analysis of the simu-

lated flow fields in Fig. 7 and 8. As for the bulk fluid, we

decompose the flow fields into the diﬀerent angular contribu-

tions given by the Chebyshev polynomials T

(cos j) using

eqn (10). Since from the radial decay of the expansion coeﬃ-

cients u

r,n

(r) we cannot distinguish between the force multi-

pole of nth order and the source multipole of n+ 1th order, we

are guided by the analysis in the bulk fluid from Section 3.1 and

attribute the expansion coeﬃcients u

r,1

(r) and u

r,3

(r), plotted

in Fig. 7 and 8, to a source dipole (SD) and force quadrupole

(FQ), respectively.

In Fig. 7 we plot u

r,1/3

(r) averaged over the cell height for

diﬀerent slab widths Dz. For a small width Dz= 2.7R= 0.34l

the

coeﬃcients are in very good agreement with the expected

power-law decay: 1/r

for the source dipole and 1/r

for the

force quadrupole. Both power laws fit well over the whole range

of the radial distance. For slab width Dz=6R= 0.75l

the

measured coeﬃcients u

r,n

(r) approach the predicted power-law

decays at a radial distance of approximately r/l

= 1. This

is larger than the slab width, where we expect the far-field

solutions of the diﬀerent multipoles to become valid. At this

distance the corresponding streamlines shown in the lower

panel of Fig. 6 are parallel to the bounding plates as predicted

by the multipole flow fields. Finally, increasing the slab width

further to Dz= 9.3R= 1.2l

the measured coeﬃcients u

r,n

(r)

show clear deviations from the expected power-law decay over

the whole range of the radial distance r. All in all, when we vary

the slab width, the force quadrupole moment stays the same,

while the source dipole moment increases with decreasing Dz.

In Fig. 8 we choose the smallest slab width Dz/R= 2.7, where

we obtained the best agreement in the previous figure and plot

the coeﬃcients u

r,1/3

(r) for diﬀerent aspect ratios a. Again there

is very good agreement between the coeﬃcients and the

expected power law decay. Only at radial distance smaller than

one realizes deviations, as expected. This is, in particular,

visible in plot (a).

3.3 Pusher-type squirmer rods

As explained in Section 2.1 we also implemented a pusher-type

squirmer rod with aspect ratio a= 4.0 and show our analysis in

Fig. 9. In the bulk fluid, the force dipole now dominates the

flow field as the new coeﬃcient u

r,2

(r) demonstrates, which

decays with 1/r

starting from rEl

. Compared to the neutral

squirmer rod and the dominating force dipole, the flow field of

the source dipole is roughly an order of magnitude smaller.

We also observe a deviation from the expected 1/r

decay

(as indicated by the solid line) to a decay of roughly 1/r

.One

explanation might be the hydrodynamic interaction of the

squirmer rod with its images due to the periodic boundary

conditions, which especially aﬀects the weaker multipole

moments. Compared to the neutral rod, the force-dipole field

is more long-ranged and therefore hydrodynamic interactions

between the images are stronger, which especially influences the

field of higher-order multipoles. Very diﬀerent from the neutral

squirmer rod, the third-order coeﬃcient u

r,3

(r) changes sign at a

radial distance of rEl

from positive to negative with increasing

r. This is why we plot the magnitude of the coeﬃcients u

r,n

(r). It

is tempting to fit the 1/r

decay of a source octupole with positive

moment B

at distances r/l

o1, while at r/l

41 we indicate the

decay of a force quadrupole with negative moment A

.Again,the

latter is probably strongly disturbed by the periodic images,

resulting in a decay of roughly 1/r

, similar to our observations

before. In conclusion, since the flow fields of both the source

dipole and force quadrupole are one order of magnitude weaker

compared to the force dipole, the overall flow field is well

described by the latter.

In Fig. 10 we analyze the flow field of the pusher-type

squirmer rod in the Hele-Shaw geometry for diﬀerent slab

widths. In addition to the findings for the neutral squirmer

rod (cf. Fig. 7), we also observe the force dipole. Compared to

the bulk fluid, it now has a stronger radial decay, u

r,2

(r)p1/r

as described by theory. At distances r/l

o1.3 the strong but

shorter ranged force-dipole flow field dominates, while for

r/l

41.3 the slower decay of the source-dipole field takes over.

Fig. 6 Simulated flow field of a squirmer rod with aspect ratio a= 4.0 in a

Hele-Shaw geometry with slab width Dz=6R=18a

shown in the

laboratory frame. The color-coded magnitude of the velocity field vand

streamlines are presented. Upper panel: Flow field in the midplane; lower

panel: flow field in the cross-sectional plane indicated by the red dashed

line in the upper panel.

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Increasing the slab width, the flow field of the force dipole always

dominates in the given radial range [cf. Fig. 10(b) and (c)]. The

reason is the green curve shifts upwards while the blue curve

shifts downwards with increasing Dz. Thus, the dipole-force

moment increases with Dzin qualitative agreement with A

pDz

(cf. Section 2.3), while the source-dipole moment decreases with Dz

again in qualitative agreement with B

p1/Dz.

The third-order coeﬃcient u

r,3

(r)p1/r

does not change

very significantly in one direction with increasing Dz.Wetakethis

as an indication that both the force quadrupole and source

octupole contribute to the flow field. A more detailed analysis of

and B

is not feasible with the hydrodynamic MPCD method

since due to thermal fluctuations it requires long averaging in order

to obtain smooth flow lines at large distances. Therefore, larger

system sizes would require an immense computational eﬀort.

4 Summary and conclusion

In this paper we introduced the squirmer rod as a new model

for elongated microswimmers. By varying aspect ratio and

surface slip-velocity field, it is able to describe artificial and

biological microswimmers of diﬀerent shape and propulsion

type, such as pushers, pullers, and neutral swimmers. To

quantify the generated hydrodynamic flow fields of the squirmer

rod, we determined the hydrodynamic multipole moments both

in the bulk fluid and the Hele-Shaw geometry, by projecting the

simulated flow fields on Legendre or Chebyshev polynomials,

respectively. The corresponding expansion coeﬃcients showed

the expected radial decay.

The flow field of the neutral squirmer rod in the bulk fluid

shows the expected source dipole, while a force quadrupole

moment develops linearly with increasing aspect ratio and

Fig. 7 Multipole analysis of the simulated flow field of a neutral squirmer rod with aspect ratio a= 4 for diﬀerent slab widths of the Hele-Shaw cell:

(a) Dz=2.7R,(b)Dz=6.0Rand (c) Dz=9.3R. As in the bulk fluid the radial decays of the expansion coeﬃcients u

r,n

(r) feature a source dipole (n= 1, blue

dots) and force quadrupole (n= 3, red pluses). The solid lines show fits with the corresponding power laws of r

2

and r

4

, respectively. Grey vertical bar:

the coeﬃcients u

r,n

(r) were determine for radial distances rZl

/2. The gray vertical dashed line indicates r=Dz.

Fig. 8 Multipole analysis of the simulated flow field of a neutral squirmer rod confined in a Hele-Shaw cell with slab width Dz= 2.7Rfor diﬀerent aspect

ratios: (a) a= 1.75, (b) a= 3.25, and (c) a= 4.0. Otherwise, the same description as in Fig. 7 is used.

Fig. 9 Absolute values of the radial Legendre coeﬃcients |u

r,n

(r)| of the

simulated flow field of a pusher-type squirmer rod with aspect ratio a=4.0

calculated from eqn (6) (symbols) and with fitted power laws (solid lines).

FD, FQ stand for force dipole and quadrupole, while SD, SO mean source

dipole and octupole, respectively.

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becomes dominant beyond a= 2. It is due to a non-uniform

distribution of the force, with which the rod acts on the fluid.

By taking an average of the surface slip-velocity field over the

rod surface, the actual swimming velocity is overestimated. In

the Hele-Shaw geometry the radial decay of the multipole flow

fields changes as predicted by theory. Especially at low slab

width Dzwe find a good match with our simulations. The flow

field of the source dipole now decay as 1/r

and dominates at

radial distances r4Dzfor all aand Dzover the field of the

force quadrupole, which now decays as 1/r

For the pusher-type squirmer rod with noticeable elongation

we observe that the flow field is composed of four hydro-

dynamic moments: force dipole, source dipole, force quadru-

pole, and source octupole. In bulk the force dipole completely

dominates the flow field and determines the radial decay with

1/r

. However, in the Hele-Shaw geometry the radial decay

changes to 1/r

and is less long-ranged. Nevertheless, for larger

slab widths and the recorded radial distances it dominates the

flow field, while for small slab widths we see a cross over to

the longer-ranged source-dipole field. This is in qualitative

agreement with the expected behavior of the strength of the

multipole moments with increasing slab width: the force dipole

becomes stronger while the source dipole weakens. Finally, in

the Hele-Shaw geometry the flow fields of force quadrupole and

source octupole have the same radial decay. Varying slab width

suggests that they both contribute.

Our work shows how elongated microswimmers generate

additional hydrodynamic multipole moments compared to

swimmers of spherical shape, which leads to a more complex

appearance of the generated flow field. Since rods experience

additional torques in a non-uniform flow field via the strain-

rate tensor

this will generate diﬀerent behavior in suspen-

sions of squirmer rods compared to their pure steric interac-

tions. Furthermore, the source dipole is predicted to be the

dominant hydrodynamic moment in a Hele-Shaw geometry,

which we confirm for neutral squirmer rods for varying slab

width. However, for pusher and puller rods the dominance of

the force dipole depends on the slab width and radial distance.

Therefore, in the continuation of this work, we plan to

investigate the collective dynamic behavior of the squirmer rods

introduced in this work. Thereby we will gain an understanding

how hydrodynamic interactions through the self-generated flow

fields contribute to diﬀerent types of dynamic behavior such as

swarming in active nematics or active turbulent phases. As a

further extension of the presented model, we plan to introduce

bending rigidity between the diﬀerent components of the squir-

mer rod to model active flexible filaments used, for example, in

ref. 7 and 8.

Conflicts of interest

There are no conflicts to declare.

A Force and source multipole

moments in Hele-Shaw geometry

In their seminal paper Liron and Mochon derived Green’s

function Gfor solving the Stokes equations in a Hele-Shaw cell

of width Dz(cf. Fig. 1) using an infinite series of image point

forces.

For a point force along the zaxis normal to the

bounding plates, the Stokeslet decays exponentially and the

same applies to the flow field component u

(r) along the z

direction. In the following we concentrate on the case, where

the point force is directed along unit vector e

ˆin the x–yplane

and only flow in this plane is monitored. Then the main feature

of the Stokeslet is the long-range flow field of a two-

dimensional hydrodynamics source dipole, which appears in

the second line,

Gij ^

ej/3Dz

Dz1r3



rsw

Dz1rsw



1

dij

2^

ri^



ej:

(16)

Here, the indices i,jA{1, 2} belong to the x–yplane, ris the

corresponding radial distance from the point force, and r

ˆthe

radial unit vector. In the first line on the right hand side a

Poiseuille flow profile is visible, which vanishes at the plate

locations r

= 0 and Dz, and an equivalent term with the z

coordinate of the swimmer, r

, appears.

In the Stokes flow regime any flow field surrounding a solid

body can be written as the sum of hydrodynamic multipoles or

Fig. 10 Multipole analysis of the simulated flow field of a pusher-type squirmer rod with aspect ratio a= 4 for diﬀerent slab widths of the Hele-Shaw cell:

(a) Dz=2.7R,(b)Dz=6.0Rand (c) Dz=9.3R. In addition to the source dipole (n= 1, blue dots) and force quadrupole/source octupole (n= 3, red pluses),

the force dipole (n= 2, green crosses) is observed. Otherwise, the same description as in Fig. 7 is used.

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singularities, which can be derived from the Stokeslet in eqn (16).

For the flow field of the squirmer rod we list the relevant terms:

u(r)=u

(r)+u

(r).... (17)

Since the squirmer rod is force free, there only appears the flow

field of a force dipole [u

(r)] and, in addition, the flow fields of

a force quadrupole [u

(r)], a source dipole [u

(r)], and a

source octupole [u

(r)].

14,65

Following ref. 14 and 65 we derive the flow fields of force

multipoles with uniaxial symmetry from the Stokeslet in

eqn (16) by applying the directional derivative e

ˆr

ntimes on

, where r

acts on the location of the point force. For n=1

and 2 we thus obtain the respective flow fields of a hydrody-

namic force dipole,

ui;FDðrÞ¼24A2

Dz1r3



rsw

Dz1rsw



2dij^

ri^

ejþ^

ri4^

ri^

rj2^



;

(18)

and force quadrupole,

ui;FQðrÞ¼24A3

Dz1r3



rsw

Dz1rsw



dij 4dij^

ri24^

ri^

rjþ8^

ri^

rj3



ej:

(19)

Note that we allocated a factor Dzfrom the Stokeslet to the force-

multipolemoments,sothatwehaveA

pDz. Again following

ref. 14 and 65, we derive the flow field of a source dipole by the

operation, u

i,SD

(r)pr

. While the second derivative with

respect to r

removes the term with r

and generates a factor

1/Dz

, the remaining two-dimensional Laplace operator acting

on the source dipole term gives zero. Thus, one arrives at

vi;SDðrÞ¼6B1

Dz1r3



dij

2^

ri^



ej(20)

which is nearly identical to the Stokeslet. Flow fields of higher

source multipoles are generated by the directional derivate e

ˆr

acting on v

(r). Of relevance is the second derivative, which

gives the flow field of a source octupole:

vi;SOðrÞ¼6B3

Dz1r3



dij 4dij^

ri24^

ri^

rjþ8^

ri^

rj3



ej:

(21)

Again, we have allocated a factor 1/Dzto the source-multipole

moments, so that we have B

p1/Dz. In the following, we

restrict ourselves to a squirmer rod oriented along the x-axis and

swimming in the midplane of the Hele-Shaw cell, i.e.,e

ˆ=e

and

rsw

3¼Dz

2. To convert the flow fields in the Hele-Shaw geometry to

a form similar to the one used in free Stokes flow, we switch to

cylindrical coordinates (r,j,z). We eliminate the z-dependence of

the multipole flow fields by integrating over the Poiseuille profile

along z, project the resulting velocity field on the radial direction,

and arrive at

~urðr;jÞ^

DzðDz

uðr;j;zÞdz:(22)

The resulting radial velocity component can also be decomposed

into cell-height averaged multipole moments u

r,n

. For the rele-

vant multipole moments we have

(r,j)= u

r,FD

(r,j)+u

r,FQ

(r,j)+u

r,SD

(r,j)+u

r,SO

(r,j)+....

(23)

which we calculate using eqn (18)–(21). For the force dipole and

quadrupole, we obtain

ur;FDðr;jÞ¼A2

T2ðcos jÞ

r3(24)

and

ur;FQðr;jÞ¼A3

T3ðcos jÞ

r4;(25)

respectively. The source dipole and octupole give

ur;SDðr;jÞ¼B1

T1ðcos jÞ

r2(26)

and

ur;SOðr;jÞ¼B3

T3ðcos jÞ

r4(27)

respectively, where the T

(cos j) = cos(nj) are Chebyshev poly-

nomials of the first kind. These formulas agree with the general

result given in eqn (9) in the main text. Using eqn (10) and (11),

we can then determine the force and source moments A

, and B

by a numerical fit of the power law r

n

to u

r,n

(r).

B Moment-of-inertia tensor of the squirmer rod

The moment-of-inertia tensor I

rod

of the squirmer rod can be

calculated applying Steiner’s theorem to the constituent parts

of the squirmer rod. First, we calculate the masses m

mid

and

cap

as well as the moment-of-inertia tensors I

mid

and I

cap

the middle segments and end caps relative to their respective

centers of mass. Due to the rotational symmetry around the rod

axis, I

rod

mid

and I

cap

have two degenerate eigenvalues in the

xyplane perpendicular to the rod axis and a diﬀerent eigen-

value along the rod axis e

ˆ=zˆ.

In each of the constituent parts of the squirmer rods, we use

cylindrical coordinates (r,j,z), where z= 0 is the center of each

spherical squirmer. The constituent parts are sections of

spheres of radius Rthat extend from z=h

min

to z=h

max

Integration over the enclosed volume with constant mass

density r

, gives the mass

m¼r0pR2zz3



hmax

hmin

:(28)

For the middle segments of the rod, h

min

max

=d/2, and for

the end caps h

min

=d/2 and h

max

=R.

Based on the known formula for the moment-of-inertia

tensor for a solid body with uniform mass density r

Iij ¼r0Ððx2xixjÞdV, we first calculate the I

components of

the constituent parts. Integrating r

over the enclosed

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volume yields

Izz ¼pr0

4R4zþ2R2z3

23z5



hmax

hmin

(29)

For the component I

the integrand becomes r0ðz2þr2sin2jÞ,

which yields

Ixx0¼pr0

2R4z2R2z3

2þz5



hmax

hmin

:(30)

We attach a dash here since for the end caps the mass

distribution is not symmetric about z= 0 and therefore the

center of mass of the caps does not coincide with the reference

point x=0,forwhichI

0was calculated. For further use below, we

needthemomentofinertiaI

for a rotation axis perpendicular to

the zaxis through the center of mass and calculate it using

Steiner’s theorem: I

0mz

com2

.Thedistancez

com

between

z= 0 and the center of mass follows by integrating zr

/mover the

enclosed volume,

zcom ¼pr0

z2R2

2z3



hmax

hmin

:(31)

For the middle segments the mass distribution is symmetric

about z= 0 and hence I

We now use the previous results to calculate the relevant

moments of inertia I

rod,ij

for the squirmer rod. For the moment

rod,zz

the relevant rotational axis along zˆ goes through all the

centers of mass of the n

segments. Thus, we can just add up

their moments of inertia:

rod,zz

=2I

cap,zz

+(n

2)I

mid,zz

. (32)

For the remaining non-zero moments, I

rod,xx

rod,yy

, we have to

shift the respective moments of the middle segments and end

caps, I

mid,xx

and I

cap,xx

, using their distances from the rod’s

center of mass and Steiner’s theorem:

Irod;xx ¼2X

ns=21

i¼1

Imid;xx þmmidd2i1



"#(

þIcap;xx þmcap

ns1



2);

(33)

for an even number of squirmers per rod n

C Swimming velocity

In principle, following ref. 70 the swimming velocity of a

squirmer of arbitrary shape can be calculated analytically using

the reciprocal theorem.

If (v,s) are the velocity and stress

tensor that solve the Stokes equation for the force-free swim-

mer and (v0,s0) the corresponding fields for a passive object of

the same shape pulled now by a force F0, then

F0U¼

ðS

ns0vsdS:(34)

Here Uis the swimming velocity, n

ˆthe surface normal, v

the

slip velocity at the swimmer surface and Sthe surface of the

swimmer’s body. For a sphere of radius Rthe acting force is

F0=6pZRU0, where U0is the translational velocity, and the

surface force density ^

ns0¼3Z

2RU0is constant and propor-

tional to U0. Thus, for a sphere the swimming velocity Uis

simply the negative average of the surface slip velocity v

U¼ 1

4pR2ðS

vsdS:(35)

To provide an estimate for the swimming velocity of the

squirmer rod, we apply the approximation that the surface

force density is constant along the surface of the passive rod,

when pulled by force F0along its long axis. The estimate for the

swimming velocity is thus the average of the surface slip

velocity v

as given in eqn (14).

To determine the swimming velocity, we calculate the sur-

face of the squirmer rod,

Srod ¼2pR2ðp

sin ydyþðns1Þðymax

ymin

sin ydy



;(36)

which adds up the surface of one complete sphere and n

1

middle segments of a sphere. The integration limits for the

polar angles of the middle segments are ymin ¼arccos a1

ns1



and y

max

=py

min

. We obtain S

rod

=4pR

a. Similarly, we

integrate the scalar product of the surface velocity v

and the

orientation vector e

e

ˆ=B

(cos

y1) (37)

over the surface of one complete sphere and n

1 middle

segments,

ðSrod

vs^

edS¼2pR2Bs

1ðp

vs^

ðÞ

sin ydy



þðns1Þðymax

ymin

vs^

eðÞsin ydy;

(38)

and obtain

ðSrod

vs^

edS¼4pR2Bs

1a1

3ns1

a1

n1



:(39)

The swimming velocity of the squirmer rod is ultimately given

by eqn (15).

Acknowledgements

We thank Felix Ru

¨hle and Josua Grawitter for helpful discus-

sions on the topic of the manuscript. We also acknowledge

financial support from the Collaborative Research Center 910

funded by Deutsche Forschungsgemeinschaft.

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The Journal

of Chemical Physics ARTICLE scitation.org/journal/jcp

Multi-particle collision dynamics

with a non-ideal equation of state. I

Cite as: J. Chem. Phys. 154, 024105 (2021); doi: 10.1063/5.0037934

Submitted: 17 November 2020 •Accepted: 16 December 2020 •

Published Online: 8 January 2021

Arne W. Zantop and Holger Starka)

AFFILIATIONS

Institute of Theoretical Physics, Technische Universität Berlin, Hardenbergstraße 36, 10623 Berlin, Germany

a)Author to whom correspondence should be addressed: [email protected]

ABSTRACT

The method of multi-particle collision dynamics (MPCD) and its different implementations are commonly used in the field of soft matter

physics to simulate fluid flow at the micron scale. Typically, the coarse-grained fluid particles are described by the equation of state of an ideal

gas, and the fluid is rather compressible. This is in contrast to conventional fluids, which are incompressible for velocities much below the

speed of sound, and can cause inhomogeneities in density. We propose an algorithm for MPCD with a modified collision rule that results

in a non-ideal equation of state and a significantly decreased compressibility. It allows simulations at less computational costs compared to

conventional MPCD algorithms. We derive analytic expressions for the equation of state and the corresponding compressibility as well as

shear viscosity. They show overall very good agreement with simulations, where we determine the pressure by simulating a quiet bulk fluid

and the shear viscosity by simulating a linear shear flow and a Poiseuille flow.

(http://creativecommons.org/licenses/by/4.0/). https://doi.org/10.1063/5.0037934

., s

I. INTRODUCTION

Since their introduction in 1999,1algorithms belonging to the

method of multi-particle collision dynamics (MPCD) have become

a standard tool to simulate fluid flows in the field of soft matter

physics.2–4 In particular, MPCD algorithms have commonly been

used to model solvent dynamics in the context of microswim-

mers,5–18 where we can only cite a few examples. Further stud-

ies address colloidal suspensions,19–23 polymers,24–26 blood cells,27

the African trypanosome as the causative agent of the sleeping

sickness,28 and even fish schools.29 Also, extensions to binary and

ternary fluid mixtures,30–32 liquid crystals,33–35 and chemically react-

ing systems36 exist. MPCD methods are particularly suited to sim-

ulate solvent flow on the microscopic scale because they solve the

Navier–Stokes equations but also incorporate the omnipresent ther-

mal fluctuations.1,37 The particle-based strategy of MPCD makes the

implementation of no-slip boundary conditions in complex geome-

tries very straightforward.38 Furthermore, the collision rules for the

coarse-grained fluid particles are well suited for the implementation

on parallel computer hardware39,40 so that extensive simulations can

also be performed on desktop computers with graphic cards.

Although MPCD methods are often used to simulate the

dynamics of incompressible solvents, one has to be aware that the

coarse-grained fluid particles follow the equation of state of an ideal

gas.1,32,41 Therefore, the fluid is rather compressible and has a low

speed of sound cs.21 This is tolerable for typical flow velocities well

below cs. In contrast, in the presence of large pressure gradients, pro-

nounced inhomogeneities in the fluid density can occur due to the

high compressibility. For example, such a situation has recently been

observed in strongly clustered microswimmers,12 where the overlap-

ping flow fields of many microswimmers are responsible for strong

pressure gradients. While variations in fluid density are, in principal,

necessary to generate pressure gradients, these variations need to be

small to stay close to the limit of an incompressible fluid. Thus, the

compressibility needs to be sufficiently small. For the MPCD fluid

with its ideal-gas equation of state, this can be achieved by increas-

ing the number n0of fluid particles per collision cell and thereby

density.12 However, such an approach causes an immense increase

in the simulation time proportional to the square of the fluid den-

sity n2

0if the system size should be kept constant at an equal Péclet

number.

In this paper, we follow a different strategy to decrease com-

pressibility. All the MPCD algorithms consist of a sequence of col-

lision and streaming steps. Here, we propose a new collision rule

that results in a non-ideal equation of state for the MPCD fluid.

Note that such non-ideal equations of state are required and have

J. Chem. Phys. 154, 024105 (2021); doi: 10.1063/5.0037934 154, 024105-1

The Journal

of Chemical Physics ARTICLE scitation.org/journal/jcp

already been introduced in the context of simulating fluid mix-

tures within MPCD.31,32 Thus, compressibility is reduced for the

constant particle number n0, and the computational efficiency is

enhanced compared to conventional MPCD algorithms, which need

to employ a larger particle density. Our approach extends ideas of

Tüzel, Ihle, and collaborators, who included geometric properties

of hard-core particles in two dimensions into the collision rule to

control momentum transport in the fluid.42,43 This approach has

also been extended to the simulation of fluid mixtures.30 In contrast

to conventional MPCD algorithms, where collisions take place at a

fixed rate 1/Δt, collisions instead occur stochastically with a proba-

bility that depends on the local density and velocities. In the present

work, we extent the approach of Refs. 42 and 43 to three dimensions

and strongly modify the geometric rules of the collision so that they

can be implemented in an existing MPCD code more easily. Further-

more, our new collision rule allows us to keep the typical canonical

thermostat and also to take care of angular momentum conservation

during collisions, which is particularly important for the simulation

of colloids and active particles.44

This article is structured as follows: In Sec. II, we introduce

the extended MPCD method with its new collision rule including

three possible collision probabilities. Then, we derive approximate

analytic expressions for the equation of state and the associated com-

pressibility in Sec. III as well as the shear viscosity in Sec. IV. For

the shear viscosity, we consider both contributions that arise from

the streaming and collision step of the extended MPCD method.

In Sec. V, we compare these analytic expressions with the results

from simulations and obtain very good agreement for the equation

of state. In particular, we demonstrate the reduction of the com-

pressibility for reasonable particle densities. We measure the shear

viscosity by determining the collisional and streaming viscosities in

a linear shear flow geometry. The total viscosity agrees very well with

values determined from simulating a Poiseuille flow and also with

the analytic expression above a density of ∼n0= 20. We close with

conclusions and an outlook in Sec. VI.

II. ALGORITHM OF THE EXTENDED MPCD METHOD

Our method shares the common features typical for the group

of MPCD algorithms.1,3,45 Like all MPCD algorithms, it consid-

ers point-like particles that represent the fluid at a mesoscopic

level of description. They perform a sequence of streaming and

collision steps. Since the latter conserves linear momentum, the

resulting hydrodynamic flow fields fulfill the Navier–Stokes equa-

tions.1While we perform the steaming step as in other MPCD

algorithms, we alter the collision step as already mentioned in the

Introduction. We now explain the extended MPCD method in more

detail.

During the streaming step (i), the point particles with masses

m0, positions xi(t), and velocities vi(t) move ballistically during

time Δt,

xi(t+Δt)=xi(t)+vi(t)Δt. (1)

They collide with confining walls or moving objects such as model

microswimmers called squirmers4,8,9,16–18 and thereby transfer both

linear and angular momentum to these moving objects. By apply-

ing the so-called bounce-back rule,1,38 the collisions either enforce

the no-slip boundary condition at confining walls and passive

colloids or the slip-velocity field, which are present at squirmer

surfaces.

For the collision step (ii), we suggest an alternative algorithm

compared to the original SRD method1or the collision operator

based on the Andersen thermostat.4As in all MPCD algorithms, the

simulation volume is divided by a cubic lattice and the fluid parti-

cles are grouped into the cubic unit cells of linear size a0centered

around ξand with volume Vξ. Each cell then contains nξparticles

with the mean velocity vξand center-of-mass position xξ. Addition-

ally, each cell is divided into two halves Aand Bby a plane Pxξ,ˆ

through the center-of-mass position xξ46 and with an orientation

defined by the unit normal vector ˆ

n[see Fig. 1(a)]. For each cell,

nis randomly drawn from a discrete set of 13 possible orientations

at each collision step. By definition, ˆ

nalways points to region A. The

number of particles on each side of the plane is denoted by nAand

nB, respectively. Their mean velocities ¯

vAand ¯

vB, respectively, along

the normal vector ˆ

nare given relative to vξ.

The main idea of the new collision step is that the particles in

region A and B only collide when they move toward each other.

Then, they stochastically exchange a momentum m0δvialong ˆ

nboth

with particles in the same half A,Band also on the other side of the

plane. The latter mechanism generates momentum flux across the

randomly oriented plane and thereby contributes to pressure, which

belongs to the isotropic part of the stress tensor.

The collision step can be summarized by

vnew

i=vi+χ(Δu){ˆ

n[ˆ

n⋅(vξ−vi)+δvi]

−I−1

ξm0∑

xj∈Vξ[xj,c×ˆ

n(δvj−ˆ

n⋅vj)]×xi,c}, (2)

where xi,cdenotes the position vector of particle irelative to

the center-of-mass position xξ. As we explain below, the collision

between the particles in region A and B occurs with a certain prob-

ability. To initiate a collision, the stochastic variable χ(Δu) is set to

one; otherwise, it is zero. The term following χ(Δu) in the square

brackets sets the normal velocity components of all particles ito

the normal component of the center-of-mass velocity, ˆ

n⋅vξ. Then,

new values for the relative velocity component δviare assigned as

explained below. They all add up to zero in order to preserve the total

momentum. The second term in the curly brackets is added to con-

serve the angular momentum. Thus, the value Lξ=m0∑xi∈Vξxi×vi

FIG. 1. (a) Side view of a collision cell with the dividing plane Pxξ,ˆ

nthrough the

center-of-mass position xξand unit normal vector ˆ

n. (b) 13 possible collision or

normal vectors ˆ

nthat point to the corners as well as centers of surfaces and edges

of the collision cell.

J. Chem. Phys. 154, 024105 (2021); doi: 10.1063/5.0037934 154, 024105-2

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of Chemical Physics ARTICLE scitation.org/journal/jcp

before the collision is preserved. Here, Iξis the moment-of-inertia

tensor of all particles in the cell relative to the center of mass.

We have already introduced the mean values of the normal

velocity components ¯

vAand ¯

vBon either side of the collision plane.

Then, collisions between the particles of region A and B occur, when,

on average, they move toward each other, meaning that the relative

velocity

Δu=¯

vB−¯

vA(3)

is positive. Furthermore, collisions between the particle clouds in A

and B occur with the rate cΔunAnB, where cquantifies the scatter-

ing cross section. A similar term has been used for the collision rate

of two clouds of hard-core particles or in chemical reactions of the

second order47 and can be motivated by the collision term in the

Boltzmann equation.48 Thus, the probability that a collision occurs

or that the stochastic variable χ(Δu) is set to one becomes

pχ(Δu)≡Θ(Δu)cΔu nAnB(4)

≈Θ(Δu)[1−exp(−cΔu nAnB)]. (5)

Here, Θ(Δu) is the Heaviside step function so that collisions only

occur for Δu>0. In the second line, assuming a sufficiently small

c, we have introduced the exponential that guarantees pχ(Δu)≤1.

Another possibility to fulfill this constraint using Eq. (4) is30

pχ(Δu)={Θ(Δu)cΔu nAnBfor pχ(Δu)≤1

1 else. (6)

We will explore also this form in Sec. V B when we calculate the

pressure in the MPCD simulations.

Finally, we introduce the changes δviin the velocity component

along the normal ˆ

n. It consists of two contributions: δvi=δvt

i+δvs

The first term transfers momentum from region Bof the cell to

particles iin the region Aand vice versa,

δvt

i≡nB/A

nA/B

vB/A. (7)

Here, the first indices apply to particles iin region A that take over

the momentum m0¯

vBfrom region B and the second indices apply

to particles iin region B. The ratios nB/nAand nA/nBguarantee the

overall momentum conservation, meaning the total momenta from

regions A and B are just swapped. The second contribution,

δvs

i≡δvMB

i−ΔvA/B, (8)

assigns each particle a random velocity δvMB

idrawn from a

Maxwell–Boltzmann distribution at temperature T, which serves as

a thermostat for the fluid. We subtract the mean random velocity

ΔvA/B=1

nA/B∑

{xi∈VA/B}

δvMB

i(9)

to preserve total momentum in both regions A and B, separately.

In particular, the introduction of the momentum transfer in

Eq. (7) and the transfer rate Eq. (5) defines the equation of state. As

shown in Sec. III, it contains a term proportional to n2

ξresembling a

virial expansion and thus extends the ideal gas term.

As in other MPCD algorithms, immersed boundaries are repre-

sented by the so-called “ghost” particles during the collision step.38

These are added to the collision cells to interact with the other

fluid particles. In simulations with squirmers, the ghost particles are

assigned the local velocity of the translating and rotating squirm-

ers plus a random thermal velocity drawn from a Boltzmann dis-

tribution. Then, the changes in linear and angular momentum of

the ghost particles following from step (ii) are assigned to the rele-

vant squirmer, which ensures that linear and angular momentum are

conserved. Finally, before performing each collision step, the lattice

is randomly shifted to ensure Galilean invariance.49

III. EQUATION OF STATE

To calculate the equation of state, we use the definition of pres-

sure as the normal component of the momentum flux through an

arbitrarily oriented plane.50 In the extended MPCD method, both

streaming (i) and collision step (ii) contribute to the pressure,

P=Pcoll +Pstr. (10)

During the streaming step (i), particles do not interact and sim-

ply transport momentum across a plane. This results in the ideal

gas contribution Pstr =nkBT/a3

0, which we already know from the

conventional MPCD methods.32

To evaluate the contribution Pcoll from the collision step (ii),

we consider the momentum flux across a plane with area a2

0that lies

in a single collision cell. Without loss of generality, we choose the

plane Pˆ

y,y0perpendicular to the ˆ

yaxis at position y0and then aver-

age over all y0(see Fig. 2). During the collision step, momentum is

transported from the region y<y0across the plane Pˆ

y,y0into y>y0

during time Δt. Thus, for the pressure as momentum transfer per

area and time, we obtain

Pcoll =m0

0Δt⟨ˆ

y⋅∑

{i∣yi>y0}

(vnew

i−vi)⟩. (11)

Here, iis restricted to all particles above Pˆ

y,y0and m0ˆ

y⋅(vnew

i−vi)is

the change in the normal momentum component of particle iduring

FIG. 2. To derive the equation of state, we consider the momentum transferred to

the region above the plane Pˆ

y,y0with unit normal vector ˆ

yand at position y=y0.

Note that to evaluate sign(x⋅ˆ

n)in Eq. (15), one has to distinguish particles that

are located in the green region as part of region A relative to the collision plane

with normal ˆ

nand particles in the blue region as part of B.

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collision and given in Eq. (2). The average goes over all possible col-

lisions, particle configurations, orientations ˆ

nof the collision planes,

and positions y0. The term added to Eq. (2) in the second line to

preserve angular momentum does not contribute to Pcoll since it van-

ishes when averaging over all possible collisions. Furthermore, since

particle iis either in region A and B and we average over all particle

velocities relative to vξwith identical velocity distributions, we can

ultimately replace ˆ

n⋅(vξ−vi)in Eq. (2) by the mean velocities −¯

vA/B

in region A or B of the collision cell. Note that ¯

vA/Bare given rela-

tive to ˆ

n⋅vξ. The choice of index A or B depends on the location of

particle i. Thus, we can simplify Eq. (11) to

Pcoll =m0

0Δt⟨χ(Δu)ˆ

y⋅ˆ

n∑

{i∣yi>y0}

δvi−¯

vA/B⟩. (12)

The stochastic contribution δvs

iof δvigiven in Eq. (8) obeys a Gaus-

sian distribution with zero mean and therefore vanishes on average.

The remaining part δvt

igiven in Eq. (7) becomes ¯

vB/Ausing nA/B

≈nξ/2. Thus, with the definition of the collision velocity Δuin

Eq. (3), we can finally replace δvi−¯

vA/Bby ¯

vB/A−¯

vA/B=Δusign

(xi⋅ˆ

n). The factor sign(xi⋅ˆ

n)comes in since the first index in

vB/A−¯

vA/Bapplies if particle iis in region A, while the second index

refers to a particle iin B (see Fig. 2). Noting also that Δuand xi

are independent stochastic variables, we can factorize the average in

Eq. (12) and rewrite it as

Pcoll =m0

0Δt⟨χ(Δu)Δu⟩⟨ˆ

y⋅ˆ

n∑

{i∣yi>y0}

sign(xi⋅ˆ

n)⟩, (13)

where

⟨χ(Δu)Δu⟩=

∞

∫

Δu pχ(Δu)p(Δu)dΔu. (14)

Here, p(Δu) is the probability distribution for Δuand pχ(Δu) is the

probability for a collision to take place, as introduced in Sec. II.

We now calculate the two averages of Eq. (13). In the second

average, we replace the conditional sum by a volume integral intro-

ducing the factor nξ/a3

0Θ(y−y0)and average over y0. We write the

second average as nξαP, where we identify the purely geometrical

factor

αP≡⟨ˆ

y⋅ˆ

0∫Vξ∫a0/2

−a0/2sign(x⋅ˆ

n)Θ(y−y0)dy0dV⟩ˆ

. (15)

It is the difference between the green and blue volume in Fig. 2 aver-

aged over all ˆ

nand y0and weighted by the projection of ˆ

non ˆ

y. The

integrals can be calculated for each of the 13 normal vectors ˆ

nso that

we obtain in total

αP=1

26[1

2+ 4(1

3√2+13

48√3)]≈0.08. (16)

For the second average, we need the probability distribution for

Δu=¯

vB−¯

vA. Since the components of the single-particle velocities

are Gaussian distributed with variance kBT/m0, also Δuis Gaus-

sian distributed with variance 4kBT/(m0nξ), as shown in Appendix A

using nA=nB≈nξ/2. Taking the collision probability from Eq. (4),

we then have

⟨χ(Δu)Δu⟩=c nAnB⟨Θ(Δu)Δu2⟩=c nξ

kBT

2m0

. (17)

Thus, in total, we obtain from Eq. (13) for the pressure contri-

bution of the collision step,

Pcoll =cαP

2a2

0ΔtkBTnξ2, (18)

which is quadratic in the particle density nξ. Hence, up to second

order in density, the full equation of state reads

Pa3

0=(Pstr +Pcoll)a3

0=nξkBT(1 + ca0αP

2Δtnξ). (19)

This gives a compressibility

β=1

nξ

∂nξ

∂P=a3

nξkBT

1 + cαPa0nξ/Δt, (20)

where the ideal gas contribution from the streaming step, βid

=a3

0/(nξkBT), is diminished by the second-order contribution

from the collision step. This means that the MPCD fluid is less

compressible.

We add two comments. First, if we take for the collision proba-

bility pχ(Δu) the expression from Eq. (5), which we use as one option

in the simulations, one can still evaluate ⟨χ(Δu)Δu⟩and then obtain

the pressure contribution from the collision step,

Pcoll =αPc kBTnξ2

2a2

0Δtexp(c2kBT

8m0

nξ3)erfc⎛

⎝√kBT

2m0

cnξ3

2⎞

⎠

=cαP

2a2

0ΔtkBTnξ2⎡⎢⎢⎢⎢⎣1−c√kBTnξ3

2π⎤⎥⎥⎥⎥⎦+O(nξ5). (21)

We will use this form when comparing the pressure in the simula-

tions to the analytic result. Second, in deriving Pc, we have always

set nA=nB≈nξ/2, thus neglecting fluctuations in the particle num-

bers in regions A and B. For sufficiently large particle numbers, these

fluctuations are small. When we compare our analytic results to sim-

ulations, we obtain good agreement and the approximation seems to

be reasonable.

IV. SHEAR VISCOSITY

To derive an expression for the dynamic shear viscosity η, we

consider the linear shear flow

v(y)=˙

γyˆ

x(22)

with constant shear rate ˙

γ(see Fig. 3). We also note that the non-

vanishing component of the viscous stress tensor, σxy =η∂yvx=η˙

γ,

describes the negative flux of the xcomponent of momentum along

the ydirection. Similar to the derivation of the equation of state in

Sec. III, the viscosity consists of two contributions from the collision

and streaming step, respectively,

η=ηcoll +ηstr. (23)

While the derivation of the collisional viscosity ηcoll requires simi-

lar steps used in calculating the collisional contribution of pressure,

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FIG. 3. To derive the shear viscosity, we

apply the shear flow v(y)=˙

γyˆ

xand

consider the momentum transferred to

the region above the plane Pˆ

y,y0with unit

normal vector ˆ

yand at position y=y0.

Examples for collision planes with yA= 0

(a) and yA≠0 (b) are shown.

the streaming viscosity ηstr needs special attention. We start with the

derivation of ηcoll.

A. Collisional viscosity

Similar to our derivation of the pressure starting from Eq. (11),

we consider the momentum transported during the collision step

from the region y<y0across the plane Pˆ

y,y0into y>y0during

time Δt(see Fig. 3). However, for σxy, we need the transfer of the

xcomponent of momentum per area and time, and thus,

σxy =−m0

0Δt⟨ˆ

x⋅∑

i∣yi>y0

vnew

i−vi⟩. (24)

When evaluating the term in the angular bracket using Eq. (2), we

concentrate on the first line of Eq. (2) and then at the end following

Ref. 45 to include angular momentum conservation, which is the ori-

gin of the second line. Replacing δviin Eq. (2) by ¯

vB/Aas before in

Sec. III, we obtain

σxy =−m0

0Δt⟨χ(Δu)ˆ

x⋅ˆ

n∑

i∣yi>y0

[¯

vB/A+ˆ

n⋅(vξ−vi)]⟩.

On average, the relative velocity Δu=¯

vB−¯

vAis equally distributed

on the mean velocities of regions A and B so that we can use Δu/2

≈¯

vA=−¯

vB. Furthermore, on average, vξcan be replaced by vi/nξ,51

and we arrive at

σxy =−m0

0Δt⟨χ(Δu)ˆ

x⋅ˆ

n∑

i∣yi>y0

sign(xi⋅ˆ

n)Δu

2−ˆ

n⋅vi(1−1

nξ)⟩.

(25)

As explained in Sec. III, sign(xi⋅ˆ

n)is necessary to distinguish

between particle ibeing in either region A or B.

Now, we need to introduce the shear rate ˙

γfrom the applied

linear shear profile of Eq. (22). It comes in by setting viequal to

its deterministic part ˙

γyiˆ

xand through the Gaussian distribution of

Δu.InAppendix B, we show that the conditional distribution for

Δu, given the collision vector ˆ

nand fixed position xiof particle i, is

Gaussian with the mean value

μi,ˆ

n=⟨Δu⟩i,ˆ

n=−2˙

γˆ

x⋅ˆ

nyA(nξ−1)+ sign(xi⋅ˆ

n)yi

nξ

. (26)

The first factor originates from the orientation of the collision plane

with collision vector ˆ

nrelative to the shearing direction ˆ

xand the

second factor from keeping particle iat fixed height yi. The quantity

yAis the ycoordinate of the center of mass of region A defined by

the collision plane. For the different collision vectors ˆ

n, we give them

in Table I. We need this conditional mean value μi,ˆ

n=⟨Δu⟩i,ˆ

nwhen

averaging over Δusince in Eq. (25), we also average over the posi-

tion of particle i. Now, to evaluate the shear viscosity, it is sufficient

to only consider the terms of σxy linear in the shear rate ˙

γ. As we

demonstrate in the following, they result from either thermal fluc-

tuations of Δuor the deterministic part of viequal to ˙

γyiˆ

x. Thermal

fluctuations of vican be neglected since they produce higher-order

terms in ˙

γ.

To perform the average over Δuin Eq. (25), we first evaluate the

required averages using the conditional distribution p(Δu−μi,ˆ

n)(see

Appendix C). Since the shear-induced shift μi,ˆ

nis small compared to

the width of the distribution, m0μ2

i,ˆ

n/kBT≪1, we can always lin-

earize in μi,ˆ

n∝˙

γ. First, for the mean conditional collision rate using

Eq. (4) for pχ(Δu), we obtain

⟨χ(Δu)⟩=∫∞

0p(Δu−μi,ˆ

n)pχ(Δu)dΔu

=c√kBTnξ3

8πm0

+O(μi,ˆ

n)≡Γ(nξ,c)+O(μi,ˆ

n). (27)

Only the contribution of the zeroth order in μi,ˆ

nis required, since

the last term in Eq. (25) already contributes the required term linear

in ˙

γby setting vi=˙

γyiˆ

x. For the second necessary mean value, we

obtain

TABLE I. Values of |yA| and ∣αη,ˆ

n∣for all collision vectors ˆ

n. Both quantities yA

and αη,ˆ

nhave the same sign equal to −sign(ˆ

n⋅ˆ

y)and only appear as product in

Eq. (30). Furthermore, note that only collision vectors with ˆ

n⋅ˆ

x≠0are relevant for

the evaluation of Eq. (30).

n|yA|∣αη,ˆ

n∣

x,ˆ

z,(ˆ

x±ˆ

z)/√2 0 0

y1/4 1/2

(ˆ

x±ˆ

y)/√2, (ˆ

y±ˆ

z)/√2 1/6 1/3

(ˆ

x±ˆ

y±ˆ

z)/√3 13/96 13/48

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⟨χ(Δu)Δu⟩=∫∞

Δu p(Δu−μi,ˆ

n)pχ(Δu)dΔu

=c⎛

⎝kBTnξ

2m0

+√kBTnξ3

2πm0

μi,ˆ

n⎞

⎠+O(μ2

i,ˆ

≡Ξ(nξ,c)+ Ω(nξ,c)μi,ˆ

n+O(μ2

i,ˆ

n)(28)

up to the linear order in μi,ˆ

n∝˙

γ. One can show that the zeroth-

order term Ξ(nξ,c) does not contribute to σxy, as it should be. The

contribution vanishes for collision planes with yA= 0 or in combi-

nation with two collision vectors. Using Eqs. (27) and (28) in the

expression (25) and only collecting all terms linear in ˙

γ, we arrive at

σxy =m0˙

0Δt nξ⟨(ˆ

x⋅ˆ

n)2∑

yi>y0

yiΩ + (nξ−1)

×[yiΓ+ sign(xi⋅ˆ

n)yAΩ]⟩. (29)

Here, the remaining average ⟨⋯⟩goes over xi, the offset y0of the

plane Pˆ

y,y0, and the collision vector ˆ

As in the derivation of the equation of state, we replace the aver-

age over all particles and the conditional sum by a volume integral

over nξ/a3

0Θ(y−y0)and also average over y0. With only the average

over the collision vector ˆ

nremaining, we obtain

σxy =m0˙

0Δt⟨(ˆ

x⋅ˆ

n)2{Ω/12 + (nξ−1)[Γ/12 + (αη,ˆ

nyA)Ω]}⟩, (30)

where

αη,ˆ

n≡1

0∫Vξ∫a0/2

−a0/2sign(x⋅ˆ

n)Θ(y−y0)dy0dV. (31)

In Table I, we give the values αη,ˆ

nand yAfor all collision vectors ˆ

Averaging over all of them, we obtain for the collisional viscosity

without taking into account angular momentum conservation in the

collision rule of Eq. (2),

η−A

coll =m0

78 a2

0Δt{13

6[Γ(nξ−1)+ Ω]+361

576 Ω(nξ−1)}. (32)

For our choice of c= 1/100 and nξ= 20, we obtain Ω ≈5/14 and

Γ= Ω/2. If instead of Eq. (4), we use Eq. (5) to have a bounded colli-

sion probability pχ(Δu), we still can evaluate the averages of Eqs. (27)

and (28) and expand into ˙

γ. The resulting expressions for Ω and Γ

are given in Appendix C. For c= 1/100 and nξ= 20, we then obtain

Ω≈1/4 and Γ≈9/65.

So far, we did not consider the term due to angular momen-

tum conservation in our collision rule (2) when evaluating σxy. We

follow here Ref. 45 to take into account two additional terms. The

essential contribution is the rotational motion of the particles in the

collision cell induced by the vorticity of the shear flow, which gener-

ates the rotational velocity ω=∇ × v/2=−˙

γ/2ˆ

z. The velocity ω×

xi,cof particle idue to this rotational flow is removed during the ran-

dom collision, and we have to add it to vnew

i−viconsidered so far to

preserve angular momentum. More precisely, our collision rule (2)

only considers the component normal to the collision plane, and

when we average over all collision vectors, we realize that only the

xcomponent of ω×xi,cis needed. Hence, in total, we need to add to

the last term in Eq. (25) the normal velocity component (ˆ

n⋅ˆ

x)˙

γyic/2.

When averaging over all particle positions j≠i, we can set yi,c=yi

−yξ=yi(1 −1/nξ) following a similar reasoning as in footnote 1.

Thus, a careful inspection of Eq. (25) and the following steps show

that we have to subtract half of the last term in Eq. (25). Ultimately,

this replaces Γin Eq. (32) by Γ/2.

A minor contribution comes from the random velocity changes

δvjduring collision. We mention it here since it gives a near perfect

agreement with the simulation results we will present in Fig. 5(a).

The random changes δvjadd the angular momentum ∑jxj,c×ˆ

nδvj

to the cell, for which we have to subtract a term in the second

line of Eq. (2) in order to restore angular momentum conserva-

tion. This velocity also has to be considered in ˆ

x⋅(vnew

i−vi)

when starting with Eq. (24) for σxy, and then, the relevant steps

carefully have to be repeated. As before, we only need the zcom-

ponent of the angular momentum. Dividing by the moment of

inertia for the relevant zdirection, Iξ,zz/m0=∑jx2

j+y2

j, and

taking the average over Δu, we obtain the mean angular velocity

⟨ωδvi

z⟩Δu=ˆ

x⋅ˆ

n⟨χ(Δu)∑jyj,cΔu/2 sign(xj⋅ˆ

Iξ,zz/m0⟩. Then, evaluating this aver-

age and introducing the mean moment of inertia ⟨Iξ⟩zz/m0=a2

(nξ−1)/6 gives ⟨ωδvi

z⟩Δu=−12(ˆ

x⋅ˆ

n)2y2

A˙

γΩ/a2

0, where we neglected

correlations in the product yjΔuand used ⟨χ(Δu)Δu⟩≃−2Ωˆ

x⋅ˆ

nyA˙

γ.

Finally, from the angular velocity, one calculates the mean xcompo-

nent of the velocity correction, ⟨⟨ωδvi

z⟩Δuyic⟩. Performing the remain-

ing averages, one ultimately realizes that this changes the prefactor of

Ω in Eq. (32) from 361/567 to 0.5034 ≈1/2. Together with the correc-

tion from the previous paragraph, we then obtain the final formula

for the shear viscosity,

η+A

coll =m0

78 a2

0Δt{13

6[Γ

2(nξ−1)+ Ω]+Ω

2(nξ−1)}. (33)

B. Streaming viscosity

To determine the streaming viscosity based on the linear shear

flow of Eq. (22), we follow the work of Kikuchi et al.50 They deter-

mined the shear stress component σxy from the momentum along

the xdirection transported through the plane y= 0 during the

streaming time Δt. They showed that this results in the expression

σxy =m0nξ

0(˙

γΔt

2⟨v2

y⟩−⟨vxvy⟩), (34)

where vxand vyare velocity components of the fluid particles. The

average is performed at the beginning of the streaming step. In the

steady state, we can immediately use ⟨v2

y⟩=kBT/m0. However, as

we explain now, the velocity correlation ⟨vxvy⟩changes when we

cycle once through the streaming and collision step. However, in the

steady state, it should be back to the value at the start of the cycle.

Using this self-consistency condition, one can ultimately determine

⟨vxvy⟩and therefore σxy.

First of all, if p(vx,vy) is the velocity distribution of the par-

ticles at the beginning of the streaming step, it will evolve toward

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the distribution p(vx+˙

γvyΔt,vy)at the end of the streaming step

since particles in the shear flow acquire additional speed along the

xaxis when moving along the ydirection. Based on this altered dis-

tribution, the velocity correlation at the end of the streaming step

becomes50

⟨vxvy⟩new

str =⟨vxvy⟩−˙

γΔt⟨v2

y⟩. (35)

In other words, the value of ⟨vxvy⟩decreases by a constant value dur-

ing the streaming step. Both Eqs. (34) and (35) are common to all

MPCD algorithms.45

In a second step, ⟨vxvy⟩new

str is altered during the subsequent

collision step. This depends on the detailed collision rule. As

we demonstrate below and in Appendix D, the velocity correla-

tions change by a constant factor during collision. Thus, ⟨vxvy⟩new

coll

=(1−b)⟨vxvy⟩new

str . Inserting Eq. (35) and using the self-consistency

condition ⟨vxvy⟩new

coll =⟨vxvy⟩as explained above, we can solve

⟨vxvy⟩=(1−1

b)˙

γΔt⟨v2

y⟩. (36)

We insert this result into the expression (34) for σxy and use ⟨v2

y⟩

=kBT/m0to finally arrive at

σxy =˙

γnξkBTΔt

0(1

b−1

2). (37)

Thus, after determining the factor 1 −bfor our collision rule, we will

have an expression for the streaming viscosity.

In order to write ⟨vxvy⟩new

coll in a compact way, we abbreviate

in the collision rule of Eq. (2) the term added to restore angu-

lar momentum conservation by Aand use for the other term

Bi=ˆ

n[ˆ

n⋅(vξ−vi)+δvi]. Furthermore, right before the collision, the

velocity correlation is ⟨vxvy⟩new

str so that we have

⟨vxvy⟩new

coll =⟨vxvy⟩new

str

+⟨χ(Δu)[vi,xBi,y+Bi,xvi,y+vi,xAy+Axvi,y

+AxAy+Bi,xBi,y+AxBi,y+AyBi,x]⟩. (38)

Since the value of χ(Δu) is either 0 or 1, we have set χ(Δu)2=χ(Δu).

Note that in δvi=δvs

i+δvt

i, we can drop δvs

isince it is zero, on

average, and also set δvt=¯

vB/Ausing nA=nB≈nξ/2 in Eq. (7).

Hence, we will always use δvi=¯

vB/Ain the following. For the first

term in Eq. (38), we demonstrate here how it is evaluated and refer

to Appendix D for the evaluation of all the other terms. We obtain

with δvi=¯

vB/A

⟨χ(Δu)vi,xBi,y⟩=⟨χ(Δu)vi,xny[¯

vB/A+ˆ

n⋅(vξ−vi)]⟩.

Here, we recognize that ¯

vB/A+ˆ

n⋅vξ=2/nξ∑{j∈VB/A}ˆ

n⋅vjafter

using the respective definitions of ¯

vB/Aand vξ. Since the construction

particles iand jlie on different sides of the collision plane and are

therefore different, this term vanishes under the typical molecular

chaos assumption ⟨vi,xvj,y⟩= 0. For the remaining term, we realize

that it involves the projector ˆ

n⊗ˆ

n, which when averaging over all ˆ

gives the unit matrix 1/3. Hence, we ultimately have

⟨χ(Δu)vi,xBi,y⟩=−⟨χ(Δu)vi,xvi,y⟩/3.

≈−⟨χ(Δu)⟩⟨vi,xvi,y⟩/3. (39)

In the last line, we used again the molecular chaos assumption and

neglected higher correlations for particle i.

For the derivation of the remaining terms in Eq. (38), we refer

to Appendix D. Finally, putting all terms in Eqs. (39),(D3),(D1),

(D4), and (D8) into Eq. (38), we obtain

⟨vxvy⟩new

coll =⟨vxvy⟩new

str [1−⟨χ(Δu)⟩14 −13nξ+ 8nξ2

18nξ2], (40)

from which we read off the factor bas the second term in the brack-

ets. Using it in Eq. (37) together with ⟨χ(Δu)⟩≈Γ(nξ,c) from Eq. (27)

and dividing by the shear rate, we obtain the streaming viscosity

ηstr =nξkBTΔt

0[18nξ2

(14 −13nξ+ 8nξ2)Γ(nξ,c)−1

2]. (41)

The sum of this equation and the collisional viscosity from

Eq. (33) gives the complete shear viscosity in this new version of

MPCD,

η=ηstr +η+A

coll. (42)

V. COMPARISON WITH SIMULATIONS

In this section, we compare the derived analytic expressions

(21) for the pressure and (33) and (41) for the collision and stream-

ing viscosities with values obtained from simulations. To calculate

the collisional contribution to the pressure, we use with Eq. (11) the

same formula with which we started the analytic calculations. Like-

wise, for the collisional and streaming viscosities, we set up the linear

shear–flow profile v(y)=˙

γyˆ

xand then explore Eqs. (24) and (34),

respectively, to evaluate the viscosities from η=σxy/˙

γ. Finally, to

test our method in a realistic situation, we also simulate a Poiseuille

flow profile and measure the total viscosity from the maximum flow

velocity. We start with some computational details.

A. Computational details

To calculate the collisional contribution to the pressure equa-

tion of state, we perform MPCD simulations in a box with edge

length Lusing periodic boundary conditions and the parameters

introduced further below. The setups of the shear flow profile and

the Poiseuille flow need more comments. All the simulations are

performed with the bounded collision probability of Eq. (5). For the

pressure, we will also show results for the alternative form of Eq. (6).

1. Linear shear flow profile

To generate a steady shear flow profile with constant shear rate

∂yvx=˙

γin a cubic simulation box with edge length L, we use the

so-called Lees–Edwards boundary conditions. We introduce them

shortly.50,52 In the directions along the ˆ

xand ˆ

zaxes, perpendicular to

the shear gradient, regular periodic boundary conditions are applied.

However, along the direction of the shear gradient, the boundary

conditions are modified such that the periodic images of the sys-

tem move with velocity ±L˙

γ. This means that a particle receives a

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shift in position and velocity when crossing the boundaries along

the ydirection. If the particle crosses the lower boundary at time t

and position (x,y=−L/2, z), it re-enters the system at the upper

boundary at position ((x+L˙

γt)mod L,y=+L/2, z)with velocity

(vx+L˙

γ,vy,vz). If it crosses the corresponding upper boundary at

time tand position (x,y= + L/2, z), it re-enters the system at the

lower boundary at position ((x−L˙

γt)mod L,y=−L/2, z)with velocity

(vx−L˙

γ,vy,vz).

2. Poiseuille flow profile

To generate a Poiseuille flow profile, we do not introduce

bounding walls but simulate a driven system with periodic bound-

aries in all three dimensions and two profiles with opposing flow

directions along the ˆ

xaxis.53 For this, we introduce a pressure dif-

ference Δpby acting with a constant body force on all the particles.

Particles with positions yi<0 experience the force −m0gˆ

x, and for

particles with positions yi>0, the force points in the opposite direc-

tion, m0gˆ

x. This setup with the box dimensions L×2L×Lpro-

duces two opposing Poiseuille flow profiles and thereby avoids the

implementation of any solid boundaries. With the resulting pres-

sure gradient Δp/L=m0n0g/a3

0, the viscosity then follows from the

maximum flow velocity vmax =ΔpL

8η.54

3. Parameters

For all simulations, we use the edge length L= 64a0, the colli-

sion parameter c= 1/100, and, in MPCD units, set kBT= 1 and mass

m0= 1. For the Lees–Edwards simulations, the shear rate is cho-

sen as ˙

γ=∂yvx=0.006 25 in units of the inverse MPCD time scale

t0=a0√m0/kBT.

Each system is initialized by randomly distributing N=n0L3

particles in the volume L3and by choosing their velocities from

the Maxwell–Boltzmann distribution. For the Lees–Edwards simu-

lations, a local offset for the mean velocity component along the x

direction is chosen, ⟨vx⟩=˙

γy. To equilibrate the system at the begin-

ning, we simulate it for 105time steps Δt. Then, we sample Eqs. (11),

(24), and (34) during a simulation time of 5 ×105Δt. When simu-

lating the Poiseuille flow, we average the flow profile over the same

amount of time but use an increased equilibration time of 5 ×105Δt

to assure that the flow has reached its maximum velocity.

Our goal is to perform the MPCD simulations with defined

values of the parameters, which we keep constant throughout the

simulations. The collision parameter cintroduced in Eq. (4) has to

be sufficiently small so that we can explore the dependence on Δu,

nA, and nB. It turns out that c= 1/100 and an average number of

n0= 20 particles per cell is a suitable choice, which yields a collision

rate of ⟨χ(Δu)⟩≈0.14.

Together with a time step of Δt=t0/200, our set of param-

eters is particularly interesting because it yields a total viscosity

η=ηstr+η+A

coll ≈16a0√kBT/m0, which is commonly used for simulat-

ing microswimmers with MPCD.8,9,12,17 Hence, in the following, we

focus on densities between n0= 7 and n0= 35 and investigate how

pressure and viscosities behave in this range of densities centered

around n0= 20.

B. Equation of state

Figure 4 shows the simulated total pressure P, normalized by

the ideal gas pressure Pid =n0kBT/a3

0, as a function of density n0for

FIG. 4. (a) Total pressure Prelative to the ideal gas pressure Pid =n0kBT/a3

0plot-

ted vs density n0for three different values of the time step Δt. Circle symbols show

data points obtained from simulations using Eq. (11) together with the bounded

collision probability of Eq. (5), and dashed lines show the corresponding theory

curves from Eq. (21). The dotted lines show the theory curves from Eq. (19), and

the squares show data points obtained from simulations using the bounded colli-

sion probability of Eq. (6) for Δt=t0/200. (b) The corresponding compressibility β

relative to the ideal gas value βid =a3

0/(nξkBT)plotted vs n0.

different time steps Δt. The colored circle symbols are the numerical

results using the bounded collision probability from Eq. (5). They are

in very good agreement with the analytic result of Eq. (21) plotted as

dashed lines. In particular, the simulations confirm the relation Pcoll

∝1/Δtaccording to which a smaller Δtresults in a larger pressure.

This makes sense since the collision probability is independent of

the time step Δt. Hence, there are more collisions in the same time

interval when Δtdecreases. As dotted lines, we also show the pres-

sure of Eq. (19) calculated with the unbounded collision probability

of Eq. (4). They are in very good agreement with the simulated pres-

sure only until ∼n0= 10. With the idea to enhance the pressure in

the simulations further, we also used the alternative bounded col-

lision probability of Eq. (6). Indeed, for the example of Δt=t0/200,

we obtain a larger pressure (square symbols) since we keep the linear

dependence in ΔunAnBuntil the probability becomes one. It starts to

deviate from the dotted line not until n0= 20.

In Fig. 4(b), we plot the corresponding compressibility as a

function of n0relative to its ideal-gas value βid = 1/Pid. Using

P=Pid[1+f(n0)], the compressibility

β=1

n0(∂P

∂n0)−1

=βid

1+f+∂f

∂n0

(43)

can be directly related to the deviation of pressure from Pid. The

dashed and dotted lines in Fig. 4(b) represent the analytic results cal-

culated from the formulas for pressure, while the derivative ∂f/∂n0

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for the numerical results was determined with the standard Python

toolchain. Relative to the ideal-gas value, compressibility is further

reduced and, in particular, βalso decreases with the decrease in Δt.

For example, at Δt=t0/200 and with n0= 20 as a reasonable den-

sity, compressibility is reduced to 0.4βid. Now, applying the bounded

collision probability of Eq. (6), the compressibility is down to 0.3βid.

To obtain such a reduction with conventional MPCD methods and

the ideal-gas pressure, one would need to increase the particle num-

ber per cell by a factor of three. Thus, the new collision rule with its

non-ideal equation of state reduces the computational efforts.

C. Shear viscosity

We first discuss the collisional viscosity. Figure 5(a) shows the

simulated collisional viscosity η+A

coll in MPCD units m0/(a0t0) as a

function of the density n0for three values of Δt. The circle symbols

show data points from simulations using the bounded collision rule

Eq. (5). Over a wide range of densities, the values are in very good

agreement with the analytical result of Eq. (33) shown as dashed

lines. Hence, there is a quantitative agreement between simulations

and theory. Similar to the pressure, the simulations confirm the scal-

ing η+A

coll ∝1/Δt. For the two larger time steps Δt=t0/100Δt=t0/200,

we see a deviation at densities n0≤10. We attribute this to the fol-

lowing reasons: first, our collision rule is not constructed for small

numbers of particles, and second, to derive Eq. (33), we neglected

FIG. 5. (a) Collisional viscosity η+A

coll in MPCD units m0/(a0t0) plotted vs density n0

for three values of the time step Δt. Circle symbols show data points obtained from

simulations using Eq. (24), and the dashed lines show the corresponding analytical

values as given by Eq. (33). (b) Streaming viscosity ηstr in MPCD units m0/(a0t0)

plotted vs density n0for the same values of Δt. Here, the circle symbols show data

points obtained from the same simulations using Eq. (34), and the dotted lines

refer to the analytic expression of Eq. (41).

fluctuations of the center of mass position, which also requires a

higher number of particles.

We now continue with the streaming viscosity ηstr that we

extract from the same simulations. Figure 5(b) shows ηstr in MPCD

units m0/(a0t0) as a function of the density n0and for different val-

ues of Δt. Again, the circle symbols show data points for the bounded

collision rule from Eq. (5), while dotted lines refer to the analytic

values given by Eq. (41). Although we observe an approximate quan-

titative agreement of the simulated values for ηstr with Eq. (41) for

larger densities n0, there are clear differences. First, Eq. (41) predicts

an increase in the streaming viscosity toward smaller n0, which then

falls sharply to zero at n0= 0 (not shown). The simulated stream-

ing viscosities only show a slight increase for Δt=t0/100; other-

wise, they are roughly independent of n0. Second, while we do not

reproduce the predicted scaling ηstr ∝Δt, we observe a clear increase

in the streaming viscosity with Δt, and thus, the expected trend is

reproduced qualitatively. As a main reason for the disagreement of

the simulated viscosities with Eq. (41), we consider the approxima-

tion ⟨χ(Δu)vi,xvi,y⟩≈⟨χ(Δu)⟩⟨vi,xvi,y⟩made during the derivation

of Eq. (41). Nevertheless, factoring out the collision rate ⟨χ(Δu)⟩in

the previous expression provides a rough quantitative estimate of the

streaming viscosity ηstr as demonstrated.

FIG. 6. (a) Poiseuille flow profiles vxvs lateral channel position ydetermined

in simulations for different densities n0. The velocity unit is the thermal velocity

vT=√kBT/m0. (b) Total viscosity in MPCD units m0/a0t0. Blue circle symbols

show data points for the viscosity determined from the flow profiles in (a). The red

triangle symbols refer to data points resulting from the sum of viscosities, ηstr+η+A

coll,

determined in Sec. V C. The black dashed line shows the analytical expression of

Eq. (42).

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D. Poiseuille flow

To simulate the Poiseuille flow profiles, we used the time step

Δt=t0/200. After averaging the velocity field over the time 5 ×104Δt,

the final flow profiles are generated by also averaging along the xand

zdirections. The resulting profiles vxas a function of the yposition

are shown in Fig. 6(a) for different densities n0. The two oppos-

ing profiles are clearly visible, and in both regions y<0 and y>0,

we observe excellent agreement with the expected parabolic shape.

The decrease in the flow velocity toward higher densities n0already

indicates an increase in the total viscosity ηwith n0. This increase

is even more pronounced since also the pressure difference Δp

increases with n0because we always use the same body force per fluid

particle.

In Fig. 6(b), we plot the total viscosity ηas a function of the

density n0. The blue circle symbols show data points obtained by

extracting vmax from the Poiseuille flow profiles shown in Fig. 6(a)

and using η=ΔpL/(8vmax). The red triangle symbols show the total

viscosity η+A

coll +ηstr consisting of the collisional and streaming viscos-

ity, which we determined in Sec. V C from the simulated linear shear

profile. The numerical data are compared to the analytical result of

Eq. (42), which is shown as the dashed line.

First of all, the values for the viscosities, determined in the sim-

ulations by analyzing either momentum transfer in linear shear flow

or the maximum flow speed of the Poiseuille profile, are in excellent

agreement over the whole range of densities. In addition, we also

observe a good agreement with the analytical expression of Eq. (42)

for densities n0≥10, where the collisional viscosity η+A

coll dominates.

For low densities n0<10, the deviations occur due to ηstr as discussed

before.

VI. CONCLUSIONS AND OUTLOOK

The new collision rule in our extended MPCD method provides

the fluid with a non-ideal equation of state by introducing stochas-

tic collisions between two particle clouds in the collision cell. In

short, the collision frequency is quadratic in density and collisions

only occur if the particle clouds move toward each other. In con-

trast to prior approaches, the extended MPCD method is designed

for three dimensions, conserves angular momentum, and features

a thermostat. The main goal of our method is to guarantee a low

fluid compressibility for simulations in which significant pressure

gradients occur. Since with the reduced compressibility, we can keep

the particle number per collision cell at reasonably small values, our

method requires significantly less simulation time compared to rais-

ing the fluid density in classical MPCD algorithms.12 We provide an

example in footnote.55 At the same time, our method saves computer

memory necessary to store MPCD particles so that we do not need

to reduce the system size. We will explore this more in a planned

second publication.

Based on the new collision rule, we have derived the equa-

tion of state and also demonstrated the impact of different collision

probabilities. Indeed, in the regime where the collision probability

is quadratic in density, we observe the nonlinear quadratic varia-

tion of pressure with density. For larger densities, where the colli-

sion frequency is bounded by the maximum value 1/Δt, the pres-

sure again becomes linear in density, albeit at a higher value, which

increases with 1/Δt. For typical values of Δt/t0= 1/200 and n0= 20

together with the most effective collision probability, compressibil-

ity is reduced by a factor three compared to the ideal-gas value at

n0= 20. Overall, we find very good agreement with values obtained

from simulations in the regime where our analytic expressions

apply.

Moreover, for the shear viscosity, we have derived analytic

expressions for the contributions of the collision and streaming step.

For the collisional viscosity, we find very good agreement with the

values obtained from simulating a linear shear flow and determining

momentum transport, while for the streaming viscosity, the ana-

lytic expression only provides a rough estimate. However, for density

values n0above 10, the collisional viscosity starts to dominate and

we obtain a very good agreement with the simulated values. This

is also demonstrated by simulating a Poiseuille flow and extracting

viscosity from the maximum flow velocity.

In a planned second publication, we will use our extended

MPCD method for selected flow problems to demonstrate its appli-

cability. Furthermore, we intend to apply it to dense systems of

microswimmers, where large pressure fields arise naturally. Prelim-

inary simulations of such systems show that the extended MPCD

method keeps the inhomogeneities in fluid density small. This will

help us to obtain reliable insight into how hydrodynamic flow

fields influence the collective dynamics of clustering and swarming

microswimmers.

ACKNOWLEDGMENTS

We thank Josua Grawitter and Christian Schaaf for helpful dis-

cussions on the topic of the manuscript. We also acknowledge finan-

cial support from the Collaborative Research Center 910 funded by

Deutsche Forschungsgemeinschaft.

APPENDIX A: GAUSSIAN DISTRIBUTION FOR Δu

For the derivation of the pressure, the fluid is considered at

rest. Assuming molecular chaos, the velocities vjof the individ-

ual particles all obey the Maxwell–Boltzmann distribution. For the

scalar product with the collision vector ˆ

n, this implies the probability

density

p(vj⋅ˆ

n)=√m0

2πkBTexp(−m0(vj⋅ˆ

n)2

2kBT). (A1)

The probability distribution for Δu=¯

vB−¯

vAthen follows from

p(Δu)=∫δ⎛

⎝Δu−⎡⎢⎢⎢⎢⎣∑

{xj∈VB}

vj⋅ˆ

−∑

{xj∈VA}

vj⋅ˆ

nA⎤⎥⎥⎥⎥⎦⎞

⎠

∏

j=1

p(vj⋅ˆ

n)d(vj⋅ˆ

n), (A2)

where δ(⋯) stands for the delta function. Eliminating it by integrat-

ing over one normal velocity component and then carefully per-

forming the rest of the N−1 integration finally gives the Gaussian

distribution

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p(Δu)=1

√2π⟨Δu2⟩exp(−

Δu2

2⟨Δu2⟩). (A3)

It has zero mean, ⟨Δu⟩= 0, and variance

⟨Δu2⟩=kBT

nAm0

+kBT

nBm0

=4kBT

m0nξ

, (A4)

where we used the approximation nA/B≈nξ/2 in the last step. Thus,

the sum over Gaussian distributed random numbers follows again a

Gaussian distribution.

APPENDIX B: CONDITIONAL DISTRIBUTION OF Δu

IN SHEAR FLOW

For deriving the collisional shear viscosity in Sec. IV A, we need

the distribution for Δuunder the condition that the collision vector

nand the position xiof particle iare given. We again start with the

single-particle velocity distributions. Relative to the applied shear

flow, the velocities still follow the Maxwell–Boltzmann distribution,

p(vj⋅ˆ

n,yj)=√m0

2πkBTexp(−m0[(vj−˙

γyjˆ

x)⋅ˆ

n]2

2kBT). (B1)

Following the same reasoning as in Appendix A, this implies that

the conditional distribution for Δuis again Gaussian with the same

variance as before: ⟨(Δu−⟨Δu⟩)2⟩= 4kBT/(m0nξ).

However, the conditional mean of Δuunder the applied shear

flow and for fixed ˆ

nand xiis non-zero. Starting from the definition of

Δuand averaging over all particle velocities and all positions besides

particle i, one obtains

⟨Δu⟩i,ˆ

n=˙

γˆ

x⋅ˆ

n⎡⎢⎢⎢⎢⎣∑

{xj∈VB/xi}⟨yj⟩

−∑

{xj∈VA/xi}⟨yj⟩

nA⎤⎥⎥⎥⎥⎦

−sign(xi⋅ˆ

n)yi

nA/B

, (B2)

where the subscripts i,ˆ

nindicate the conditions that the particle i

resides at xiand the collision vector takes the value ˆ

n. When we

introduce the center-of-mass in the respective regions ⟨yj⟩=yA/B

for xj∈VA/Band use yA=−yBand nA/B≈nξ/2, we can ultimately

write the conditional mean as

⟨Δu⟩i,ˆ

n=−2˙

γˆ

x⋅ˆ

nyA(nξ−1)+ sign(xi⋅ˆ

n)yi

nξ

≡μi,ˆ

n. (B3)

Based on the conditional distribution p(Δu−μi,ˆ

n), we can now cal-

culate the required mean values ⟨χ(Δu)⟩and ⟨χ(Δu)Δu⟩in shear

flow.

APPENDIX C: MEAN VALUES ⟨χ(Δu)⟩AND ⟨χ(Δu)Δu⟩

IN SHEAR FLOW

We start with the unbounded form of the collision rate pχ(Δu)

in Eq. (4) and find for the mean collision rate

⟨χ(Δu)⟩=∫∞

0p(Δu−μxi,ˆ

n)pχ(Δu)dΔu

=cnξ

2⎧

⎪

⎨

⎪

⎩√kBTnξ

2πm0

exp(−μ2

i,ˆ

nm0nξ

4kBT)(C1)

+μi,ˆ

nnξ

8[1 + erf(μi,ˆ

n√m0nξ

8kBT)]⎫

⎪

⎬

⎪

⎭

=c√kBTnξ3

8πm0

+O(μi,ˆ

n)≡Γ(nξ,c)+O(μi,ˆ

n), (C2)

where in the last line, we show the relevant zeroth-order term after

expansion in μi,ˆ

n. For the second mean value, we obtain

⟨χ(Δu)Δu⟩=∫∞

Δu p(Δu−μxi,ˆ

n)pχ(Δu)dΔu

=cnξ

2{(kBT

+μ2

i,ˆ

nnξ

4)[1 + erf(μi,ˆ

n√m0n

8kBT)]

+μi,ˆ

n√kBTnξ

2πm0

exp(−μ2

i,ˆ

nm0nξ

8kBT)⎫

⎪

⎬

⎪

⎭(C3)

and, after expanding to first order in μi,ˆ

⟨χ(Δu)Δu⟩=c⎛

⎝kBTnξ

2m0

+√kBTnξ3

2πm0

μi,ˆ

n⎞

⎠+O(μ2

i,ˆ

≡Ξ(nξ,c)+ Ω(nξ,c)μi,ˆ

n+O(μ2

i,ˆ

n). (C4)

For the bounded form of the collision rate pχ(Δu) in Eq. (5), we

can also calculate the mean values. The mean collision rate becomes

⟨χ(Δu)⟩=∫∞

0p(Δu−μxi,ˆ

n)pχ(Δu)dΔu

2⎧

⎪

⎨

⎪

⎩1 + erf(μi,ˆ

n√m0nξ

8kBT), (C5)

−exp[cnξ2

8(cnξ

kBT

−2μi,ˆ

n)]

×erfc⎡⎢⎢⎢⎢⎣√nξ

8⎛

⎝cnξ√kBT

−μi,ˆ

n√m0

kBT⎞

⎠⎤⎥⎥⎥⎥⎦⎫

⎪

⎬

⎪

⎭

=1−exp(c2kBTnξ

8m0)erfc⎛

⎝c√kBTnξ3

8m0⎞

⎠+O(μi,ˆ

≡Γ(nξ,c)+O(μ2

i,ˆ

n), (C6)

where the last line shows the relevant zeroth-order term after expan-

sion in μi,ˆ

n. The second mean value becomes

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⟨χ(Δu)Δu⟩=∫∞

Δu p(Δu−μxi,ˆ

n)pχ(Δu)dΔu

2⎧

⎪

⎨

⎪

⎩μi,ˆ

n[1 + erf(μi,ˆ

n√m0nξ

8kBT)]

+(cnξ

kBT

−μi,ˆ

×exp[cnξ2

8(cnξ

kBT

−2μi,ˆ

n)]

×erfc⎡⎢⎢⎢⎢⎣√nξ

8⎛

⎝cnξ√kBT

−μi,ˆ

n√m0

kBT⎞

⎠⎤⎥⎥⎥⎥⎦⎫

⎪

⎬

⎪

⎭,

and after expanding to first order in μi,ˆ

n, one has

⟨χ(Δu)Δu⟩=ckBTnξ

2m0

exp(c2kBTnξ3

8m0)erfc⎛

⎝c√kBTnξ

8m0⎞

⎠

+μi,ˆ

2⎡⎢⎢⎢⎢⎣1 + c√kBTnξ3

2πm0

−(1 + c2kBTnξ3

4m0)

×exp(c2kBTnξ3

8m0)erfc⎛

⎝c√kBTnξ3

8m0⎞

⎠⎤⎥⎥⎥⎥⎦+O(μ2

i,ˆ

≡Ξ(nξ,c)+ Ω(nξ,c)μi,ˆ

n+O(μ2

i,ˆ

n). (C7)

APPENDIX D: VELOCITY CORRELATION DURING

COLLISIONS

To derive of the streaming viscosity in Sec. IV B, we consider

the evolution of the velocity correlation ⟨vi,xvi,y⟩new

coll during a col-

lision step. In the main text, we have already evaluated the term

⟨χ(Δu)vi,xBi,y⟩. Here, we calculate the remaining terms ⟨χ(Δu)vxAy⟩,

⟨χ(Δu)AxAy⟩,⟨χ(Δu)Bi,xBi,y⟩, and ⟨χ(Δu)AxBi,y⟩.

We begin by applying some transformations on the abbrevia-

tions Biand Athat we introduced to write Eq. (38) in a compact

way. First, we note that we may drop the stochastic part δvs

iof

δvi=δvt

i+δvs

ibecause it averages to zero. Furthermore, we replace

δvt

i=¯

vA/Busing nA=nB≈nξ/2. With the definitions of vξand ¯

vA/B,

the quantity Bireads

Bi=−ˆ

n⋅vi+2ˆ

nξ∑

{j∈VB/A}

n⋅vj.

Furthermore, we can insert the quantity Bjinto A,

A≡−I−1

ξm0∑

xj∈Vξ[xj,c×ˆ

n(δvj−ˆ

n⋅vj)]×xi,c

=−I−1

ξm0∑

xj∈Vξ(xj,c×Bj)×xi,c,

and using that, we are free to add the constant velocity vξinside the

round brackets.

With these simplifications, we now begin considering the next

most simple term ⟨χ(Δu)Bi,xBi,y⟩of the new velocity correlation

⟨vi,xvi,y⟩new

coll .

We first note that the single term and the sum in Bialways con-

tain different particles i≠j. Hence, the product of these terms van-

ishes under the usual molecular chaos assumption ⟨vi,xvj,y⟩= 0. If we

further use that particles are interchangeable so that ⟨χ(Δu)vi,xvi,y⟩

=⟨χ(Δu)vj,xvj,y⟩, we obtain

⟨χ(Δu)Bi,xBi,y⟩=(2

nξ

+ 1)⟨χ(Δu)nxny(ˆ

n⋅vi)2⟩.

Finally, averaging over ˆ

nand neglecting higher correlations for

particle i, we arrive at

⟨χ(Δu)Bi,xBi,y⟩≈2

9(2

nξ

+ 1)⟨χ(Δu)⟩⟨vi,xvi,y⟩. (D1)

In the next term ⟨χ(Δu)vi,xAy⟩, we note that the quantity A

also depends on the positions of the particles so that these must

be included in the average. We may perform this average over the

particle positions separately on Ato obtain

⟨A⟩=−⟨I−1

ξm0∑

xj,c∈Vξ[(xi,c⋅xj,c)Bj−(Bj⋅xi,c)xj,c]⟩

=−⟨I−1

ξm0[x2

i,cBi−(xi,c⋅Bi)xi,c]⟩

=−2⟨I−1

ξ⟩m0⟨x2

i,c⟩Bi=−Bi

nξ

, (D2)

where xi,cdenotes any of the components of xi,c. In the last line,

we assumed that the contribution of the single particle iis low so

that we can average ⟨Iξ⟩=(nξ−1)m0a2

0I/6 separately. Furthermore,

we used that ⟨x2

i,c⟩=a2

0/12(1−1/nξ)for any of the components of

the position and that different components of the position xi,care

uncorrelated.

Putting Eq. (D2) into ⟨χ(Δu)vi,xAy⟩and using Eq. (39), we

obtain

⟨χ(Δu)vi,xAy⟩=−⟨χ(Δu)vi,xBi,y⟩

nξ

≈⟨χ(Δu)⟩⟨vi,xvi,y⟩

3nξ

. (D3)

We proceed with the term ⟨χ(Δu)Bi,xAy⟩. Since the term Bi,xdoes

not depend on the position of the particle, we can immediately apply

Eq. (D2) and insert Eq. (D1) to arrive at

⟨χ(Δu)Bi,xAy⟩≈−2

9nξ(2

nξ

+ 1)⟨χ(Δu)⟩⟨vi,xvi,y⟩. (D4)

The last term to calculate is

⟨χ(Δu)AxAy⟩

=⟨χ(Δu)⎡⎢⎢⎢⎢⎣I−1

ξm0∑

xj,c∈Vξ(xi,c⋅xj,c)Bj−(Bj⋅xi,c)xj,c⎤⎥⎥⎥⎥⎦x

×⎡⎢⎢⎢⎢⎣I−1

ξm0∑

xk,c∈Vξ(xi,c⋅xk,c)Bk−(Bk⋅xi,c)xk,c⎤⎥⎥⎥⎥⎦y⟩. (D5)

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Similarly, we separately average ⟨I2

ξ⟩=(19/360 + (nξ−1)/36)(nξ

−1)m0a2

0I≈(nξ−1)2m0a2

0I/36 based on the assumption of molec-

ular chaos and that the contribution of single pairs of particles is

small. Multiplying out yields three terms,

⟨χ(Δu)AxAy⟩

=⟨I−2

ξ⟩m2

0⟨χ(Δu)∑

xj,c,xk,c∈Vξ⎡⎢⎢⎢⎢⎣Bj,xBk,y∑

i,c,αxj,c,αxk,c,α

−2∑

α,β

xi,c,αxj,c,αBj,xBk,βxi,c,βxk,c,y⎤⎥⎥⎥⎥⎦⟩, (D6)

of which the last is zero because ⟨xj,xxk,y⟩= 0 for all j,k. We con-

tinue with the averages over the positions inside the sums. In the

first summand of Eq. (D6), we recognize that ⟨xj,c,αxk,c,α⟩=⟨x2

j,c,α⟩δkj,

and in the second one, we may rewrite ⟨xi,c,αxi,c,β⟩=⟨x2

i,c,α⟩δαβ and

⟨xj,c,αxk,c,y⟩=⟨x2

j,c,α⟩δαyδkj. This follows from the usual molecular

chaos assumption that different particles are uncorrelated and the

assumption that the components of a position are also uncorrelated.

Performing the sums with these replacements, we obtain

⟨χ(Δu)AxAy⟩

=⟨I−2

ξ⟩m2

0⟨χ(Δu)3Bj,xBj,y[(nξ−1)⟨x2

i,c⟩2+⟨x4

i,c⟩]

−2Bi,xBi,y[(nξ−1)⟨x2

i,c⟩2+⟨x4

i,c⟩]⟩

=⟨I−2

ξ⟩m2

0[(nξ−1)⟨x2

i,c⟩2+⟨x4

i,c⟩]⟨χ(Δu)Bi,xBi,y⟩. (D7)

For the term ⟨χ(Δu)Bj,xBj,y⟩, we may refer to Eq. (D1). If we

approximate ⟨x4

i,c⟩≈a4

0/80 and nξ−1

4nξ2≈1

4nξ, we arrive at

⟨χ(Δu)AxAy⟩

=⟨I−2

ξ⟩a4

0m2

0[(nξ−1)3

144nξ2+1

80]⟨χ(Δu)Bi,xBi,y⟩

=[1

4nξ

20(nξ−1)2]⟨χ(Δu)Bi,xBi,y⟩

≈1

18nξ(2

nξ

+ 1)⟨χ(Δu)⟩⟨vi,xvi,y⟩. (D8)

In the last line, we have furthermore neglected the term 9

20(nξ−1)2,

which is small compared to 1

4nξ.

DATA AVAILABILITY

The data that support the findings of this study are available

from the corresponding author upon reasonable request.

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55To compare our method to classical MPCD in an example, we consider how the

compressibility is reduced by a factor of six at equal Péclet numbers. To achieve

this reduction in classical MPCD, the particle number has to be increased by a

factor of six from n0= 10 to 60. However, this also increases the viscosity by a

factor of six. Therefore, to keep the Péclet number constant, the active velocities of

microswimmers have to be reduced by a factor of six. As a result, the time required

for a simulation is increased by a factor of 6 ×6 = 36.

This is in contrast to our method, where we increase the particle number from

n0= 10 to 20.To compensate that the collision rate is effectively lowered due to the

stochastic collision rule, we further divide the time step by four and typically take

Δt/t0= 1/200, which also keeps the viscosity at the same value as the reference case

of classical MPCD. Thus, we achieve a decrease in compressibility by a factor of 2

×3 = 6 at an equal Péclet number. In total, the time required for a simulation is

increased by a factor of 2 ×4 = 8. Additionally, the implementation of our collision

rule only resulted in 5% computational overhead compared to classical MPCD,

Therefore, our new method reduces the computational effort by a factor of 4.5.

J. Chem. Phys. 154, 024105 (2021); doi: 10.1063/5.0037934 154, 024105-14

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Multi-particle collision dynamics with a non-ideal

equation of state. II. Collective dynamics

of elongated squirmer rods

Cite as: J. Chem. Phys. 155, 134904 (2021); doi: 10.1063/5.0064558

Submitted: 24 July 2021 •Accepted: 14 September 2021 •

Published Online: 6 October 2021

Arne W. Zantopa) and Holger Starkb)

AFFILIATIONS

Institute of Theoretical Physics, Technische Universität Berlin, Hardenbergstraße 36, 10623 Berlin, Germany

a)Author to whom correspondence should be addressed: [email protected]

b)[email protected]

ABSTRACT

Simulations of flow fields around microscopic objects typically require methods that both solve the Navier–Stokes equations and also include

thermal fluctuations. One such method popular in the field of soft-matter physics is the particle-based simulation method of multi-particle

collision dynamics (MPCD). However, in contrast to the typically incompressible real fluid, the fluid of the traditional MPCD methods obeys

the ideal-gas equation of state. This can be problematic because most fluid properties strongly depend on the fluid density. In a recent article,

we proposed an extended MPCD algorithm and derived its non-ideal equation of state and an expression for the viscosity. In the present work,

we demonstrate its accuracy and efficiency for the simulations of the flow fields of single squirmers and of the collective dynamics of squirmer

rods. We use two exemplary squirmer-rod systems for which we compare the outcome of the extended MPCD method to the well-established

MPCD version with an Andersen thermostat. First, we explicitly demonstrate the reduced compressibility of the MPCD fluid in a cluster of

squirmer rods. Second, for shorter rods, we show the interesting result that in simulations with the extended MPCD method, dynamic swarms

are more pronounced and have a higher polar order. Finally, we present a thorough study of the state diagram of squirmer rods moving in

the center plane of a Hele-Shaw geometry. From a small to large aspect ratio and density, we observe a disordered state, dynamic swarms, a

single swarm, and a jammed cluster, which we characterize accordingly.

(http://creativecommons.org/licenses/by/4.0/). https://doi.org/10.1063/5.0064558

I. INTRODUCTION

Simulations of hydrodynamic flows in the context of micro-

scopic colloidal and active particles share a long history with the

method of multi-particle collision dynamics (MPCD).1–3 Different

modifications of the original algorithm4have been applied in stud-

ies of colloidal suspensions,5–9 microswimmers,10–23 polymers,24–26

and even complex deformable bodies such as red blood cells27,28

or the African trypanosome,29,30 the causative agent of the sleep-

ing sickness, to name a few examples. Our particular interest lies

in the collective dynamics of elongated microswimmers and active

filaments for which interesting states, such as swarming,21,31–33 vor-

tex formation,34–37 active nematic patterns,38–40 and active turbu-

lence,41–45 have been observed. Many of these features can already

be described by continuum models46–49 that combine elements of

the Toner–Tu50 and Swift–Hohenberg51 equations. However, it is

still a matter of current debate to identify the importance of purely

steric and hydrodynamic interactions on the scale of the constitut-

ing particles.52–57 In this context, we introduced elongated squirmer

rods as a model for elongated bacteria using MPCD to simulate their

flow fields.17 Our model thereby adds to the topmost layer of detail

in a wide range of models ranging from coarse-grained interactions

in the famous Vicsek model58–60 over Langevin dynamics simula-

tion41,61–67 with implicit hydrodynamic interactions68,69 to explicit

hydrodynamic simulations using the MPCD and lattice Boltzmann

approach.70–73

In this context, solving the Navier–Stokes equations at vanish-

ing Reynolds numbers with the particle-based mesoscale method of

MPCD has few advantages. It inherently includes the omnipresent

thermal fluctuations,4,74 and complex geometries are straightfor-

ward to implement with this method.28,75–77 Furthermore, the col-

lision rules of MPCD algorithms suit well for parallel computing on

graphic cards so that also on desktop computers, one can perform

extensive simulations.78,79

J. Chem. Phys. 155, 134904 (2021); doi: 10.1063/5.0064558 155, 134904-1

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However, a property that most MPCD algorithms share is that

the MPCD fluid particles obey the equation of state of an ideal

gas.4,80,81 This makes the model fluid rather compressible and results

in a low speed of sound cs,5which is in conflict with simulating

the dynamics of incompressible solvents at a low Reynolds num-

ber. It is a matter of current debate how this affects the dynamics

of immersed particles,20 especially for strongly inhomogeneous sys-

tems, such as clustered microswimmers.21 To render the MPCD

fluid more incompressible, different strategies can be used. For the

ideal-gas models, increasing the density of fluid particles n0natu-

rally decreases the compressibility.21 However, this comes with an

immense increase in simulation time proportional to n2

0if the system

size is held constant. Alternatively, auxiliary field equations can be

coupled to the dynamics of the MPCD particles to alter the equation

of state of the model.81

In a recent publication,82 we introduced an extended version

of MPCD with a modified collision step that transfers momentum

across a randomly oriented plane within the collision cell in order

to generate a non-ideal equation of state. In addition, the collision

probability depends on the local density and velocities. This is dif-

ferent from traditional MPCD algorithms, where collisions occur at

a fixed rate. Our approach is motivated by the collision rate in the

Boltzmann equation83 and has the advantage that it can easily be

integrated into an existing MPCD program. A similar approach has

been formulated for two-dimensional systems in Refs. 84 and 85.

While we have already derived expressions for the equation of state

and the viscosity in our previous publication,82 in the present work,

we demonstrate its application to the dynamics of single squirmers

and squirmer rods, but most importantly to explore their collective

dynamics.17

In Sec. II, we start by shortly reviewing the extended MPCD

method82 and the squirmer-rod model for elongated microswim-

mers17 introduced by us in earlier publications. We then demon-

strate the applicability of the extended MPCD method and, in par-

ticular, compare it to the traditional MPCD version. First, in Sec. III,

we show that our new method accurately reproduces the leading

hydrodynamic multipoles contributing to the flow field of a single

squirmer in the bulk fluid and in confinement. Second, in Sec. IV,

we explicitly demonstrate the lower compressibility of the extended

MPCD method considering the specific example of a cluster of

squirmer rods. Then, for shorter rods, we illustrate quantitative dif-

ferences in their swarming behavior between both MPCD methods.

Finally, in Sec. V, we employ the extended MPCD method to study

in-depth the collective dynamics of squirmer rods. We present a

state diagram of squirmer rods for variable aspect ratio and den-

sity, which includes dynamic swarms, single swarms, and jammed

clusters, and we discuss their structural and dynamic properties. We

finish with concluding remarks and an outlook in Sec. VI.

II. METHODS

A. Extended MPCD method

As common to all MPCD algorithms, the fluid is modeled by

point-like particles, the dynamics of which proceeds in two alter-

nating steps: ballistic motion along the mean free path followed by a

collision. Because the collision rule conserves linear and also angular

momentum, the simulated fluid flow solves the Navier–Stokes equa-

tions.4In MPCD, these steps occur simultaneously for all particles.

During the streaming step (i), particles with masses m0, positions

xi(t), and velocities vi(t)move ballistically for the duration of the

time step Δt,

xi(t+Δt)=xi(t)+vi(t)Δt. (1)

Moreover, the particles collide with bounding walls, colloids, or,

in our case, squirmers3,13,14,16–18 and transfer linear and angular

momenta. To enforce either the no-slip boundary condition or a pre-

defined slip-velocity at a squirmer surface, the so-called bounce-back

rule is applied.4,75

Our extended MPCD method82 now modifies the collision step

(ii). As usual, the simulation volume is divided by a cubic lattice

with lattice constant a0and all particles are grouped into cubic cells

with volume 𝒱ξcentered around position ξ. Before each collision

step, the lattice is randomly shifted against the previous lattice to

ensure Galilean invariance.86 The nξparticles inside a cell have the

mean velocity vξ, center-of-mass position xξ, and moment-of-inertia

tensor Iξdefined relative to xξ.

Now, in our modified collision step, each cell is divided into

two halves Aand Bby a plane Pˆ

n,xξthat contains the center of mass

xξand has the normal unit vector ˆ

n, which, by definition, always

points toward A. For each cell and time step, ˆ

nis randomly drawn

from a discrete set of 13 possible orientations. The modified colli-

sion step transfers momentum across the randomly oriented plane

and thereby generates a non-linear equation of state. We denote the

respective number of particles on each side of the plane by nAand

nBand their respective mean velocity components along ˆ

ntaken

relative to vξby ¯

vAand ¯

vB. We also introduce the relative velocity

Δu≡¯

vB−¯

vAfor further use. The collision step introduces for each

particle velocity viin cell 𝒱ξa new value vnew

iand is defined by

vnew

i=vi+χ(Δu)⎧

⎪

⎨

⎪

⎩

n[ˆ

n⋅(vξ−vi)+δvi]

−I−1

ξm0∑

xj∈𝒱ξ[xj,c×ˆ

n(δvj−ˆ

n⋅vj)]×xi,c⎫

⎪

⎬

⎪

⎭

, (2)

where xi,cdenotes the position vector of particle irelative to the

center-of-mass position xξ. In contrast to other MPCD methods, the

collision only occurs with a certain probability as explained below.

To initiate a collision, the stochastic variable χ(Δu)is set to one,

otherwise it is zero. The term in square brackets in the first line sets

the normal velocity component of particle irelative to ˆ

n⋅vξto a

new random value δvi, which we introduce below. It mainly trans-

fers momentum across the collision plane Pˆ

n,xξ, and all values add

up to zero in order to preserve the total momentum. Finally, the

term in the second line is added to preserve the angular momentum

Lξ=m0∑xi∈𝒱ξxi×viduring collision.

Note that the momentum transfer across the randomly oriented

plane is responsible for the decreased compressibility of the MPCD

fluid. As we showed in Ref. 82, it generates a term proportional to

ξin the equation of state, which thereby deviates from the ideal-

gas equation of the classical MPCD fluid. On average, a collision

between particles in regions Aand Boccurs when the two clouds

move toward each other meaning Δu≡¯

vB−¯

vA>0. Following the

collision term in the Boltzmann equation, we take for the collision

rate cΔu nAnB, where cis the scattering cross section.83 Thus, we set

the probability for a collision to occur to

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pχ(Δu)≡Θ(Δu)[1−exp(−cΔu nAnB)], (3)

where Θ(Δu)is the Heaviside step function, and we use the expo-

nential function to guarantee pχ(Δu)≤1.

Finally, we introduce the new velocity component along the

normal ˆ

δvi≡nB/A

nA/B

vB/A+δvMB

i−ΔvA/B. (4)

Here, the first term transfers momentum between regions Aand

B. The first index applies to particles iin region A. They take

over the momentum m0¯

vBfrom region Band vice versa. The

prefactor nB/A/nA/Bguarantees that the overall momentum is con-

served. Essentially, the total momenta from regions A and B are just

swapped. The second term assigns each particle a thermal velocity

δvMB

idrawn from a Maxwell–Boltzmann distribution at temperature

T, which thereby serves as a thermostat for the fluid. The last term

subtracts the mean thermal velocities in each region (Aand B),

ΔvA/B=1

nA/B

∑

{xi∈𝒱A/B}

δvMB

i, (5)

in order to conserve momentum.

As in all MPCD algorithms, the so-called “ghost” particles

are introduced to participate in the collision step. They are added

where overlap with boundaries leads to partially empty collision

cells, thereby improving the validity of the no-slip boundary condi-

tion.75 After the collision step, the changes in the linear and angular

momentum of the ghost particles are assigned to movable objects,

which in our case are squirmers. This ensures that the linear and

angular momenta are conserved.

B. Squirmer and squirmer rods

To investigate the flow fields of microswimmers and their

hydrodynamic interactions in collective motion, we use spherical

squirmers and build squirmer rods from them. Spherical squirm-

ers propel by an imposed axisymmetric slip-velocity field on their

surfaces,3,87,88

vs=Bs

1[(1+βˆ

e⋅ˆ

rs)(ˆ

e⋅ˆ

xs)ˆ

xs−ˆ

e], (6)

which initiates flow in the surrounding fluid. Here, the unit vector

xspoints from the squirmer center toward the squirmer surface and

eindicates the orientation and swimming direction of the squirmer.

The resulting flow field in the bulk fluid is a superposition of mul-

tipole solutions of the Stokes equations. In particular, it contains a

source dipole singularity with strength B1=Bs

1R3and a force dipole

singularity with strength A2=βBs

1R2. The latter is controlled by the

parameter β, which determines the squirmer type, and Rdenotes the

radius of the squirmer. For different values of β, the squirmer repre-

sents either a pusher (β<0), a neutral swimmer (β=0), or a puller

(β>0). The squirmer parameter Bs

1also determines the swimming

speed v0=2/3Bs

To model the actual rod shape of many biological microswim-

mers, we use the squirmer rod. It is constructed by arranging sev-

eral neutral spherical squirmers on a line such that they overlap

and form a single rigid body. By varying the squirmer distance

d, rods of different aspect ratios α=lS/2Rare realized where lS

is the rod length. In this work, we always use Nsq =10 spherical

squirmers and a maximum squirmer distance of d≈0.8Rso that the

surface is still smooth enough that swimmers can slide past each

other. This amounts to a maximum aspect ratio of α=4.75. To

model steric interactions between pairs of squirmer rods, we employ

the Weeks–Chandler–Andersen potential89 that acts between two

squirmers from different rods. The force constant is chosen strong

enough (εWCA ≈104kBT) to ensure that there is no significant over-

lap between two rods.

The squirmer rod model described so far resembles ciliated

micro-organisms such as Paramecium. To model bacteria, such as

E. Coli, which are driven by a bundle of rotating flagella extending

from its rear, one can concentrate the surface velocity on the back

of the rod by multiplying the slip velocity with an envelope or step

function.17

C. Parameters

We choose the parameters for the extended MPCD method

such that the total viscosity η=ηstr +η+A

coll ≈16√m0kBT/a2

0(see

Ref. 82) given in MPCD units matches previous studies where other

MPCD methods are employed.13,14,17,21 In particular, the collision

parameter c=1/100 and the average number density n0=20 are

chosen such that the average collision rate ⟨χ(Δu)⟩≈0.14 is low

enough to be in the quadratic regime of the equation of state.82

Together with a time step of Δt=0.005a0√kBT/m0, the total vis-

cosity takes the prescribed value. For the squirmers, we use a radius

of R=3a0and a source dipole strength of Bs

1=0.1, which results in a

Péclet number of Pe =Rv0/DT≈354.6, where DTis the translational

diffusion coefficient of the spherical squirmer.14

To initialize a simulation, we distribute the fluid particles ran-

domly and choose thermal velocities from the Maxwell–Boltzmann

distribution. In addition, the squirmer rods are randomly distributed

in the midplane of the simulation box for area fractions ϕbelow 0.6.

For ϕ≥0.6, we initiate the simulation by arranging the swimmers in

a rectangular pattern and by choosing their orientations at random

along the longer side of the rectangular unit cell.

We use the following geometries and simulation times in our

computer studies. To simulate the flow fields of pusher-type squirm-

ers, we use periodic boundary conditions for the bulk fluid with an

edge length of L=150a0of the cubic simulation box. In the Hele-

Shaw geometry, the confining parallel plates have a separation of

Δz=8a0and an edge length of L=300a0, and the squirmers are

constrained to move in the midplane of the cell (see Fig. 1) by set-

ting the normal component of the velocity vzand the components

ωx,ωyof the angular velocities of the squirmer rods to zero after

each time step. The flow fields are first equilibrated during 50 000

time steps Δt, and 20 independent simulation runs are used to aver-

age the flow fields over a total of 107Δt. To simulate the collective

motion of squirmer rods, we choose a Hele-Shaw geometry with a

FIG. 1. Schematic of the Hele-Shaw geometry in a perspective view. The squirmers

and rods are restricted to move in the midplane between two parallel walls with

separation Δz. In the directions parallel to the walls, periodic boundary conditions

are applied.

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plate separation of Δz=18a0, otherwise the edge length L=300a0is

the same. The system is equilibrated during 107time steps Δt, and

data are sampled during another 107Δt.

III. FLOW FIELDS IN THE EXTENDED MPCD METHOD

In this section, we present a first test for the extended MPCD

method by demonstrating the accuracy with which it simulates flow

fields of microswimmers. We compare the simulation results of the

flow field of a pusher-type squirmer with the analytic multipole

expression both in the bulk fluid and in confinement between two

parallel plates.

We briefly outline the underlying theory. In the Stokes regime,

the flow field generated by an object moving through the bulk fluid

can be written as a series of hydrodynamic multipoles. This series

contains at the lowest order the Stokeslet or hydrodynamic force

monopole, which is not allowed for autonomous swimmers, the

source dipole, and their higher-order derivatives. For swimmers

with radial symmetry about their swimming direction ˆ

e, the flow

fields of these multipoles can be expanded into a Legendre polyno-

mial Pn(cos θ). In spherical coordinates r,ϕ, and θwith ˆ

z=ˆ

e, the

radial velocity component of the multipole expansion reads

ur(r,θ)=

∞

∑

n=1[Anr−n+Bnr−n−2]Pn(cos θ). (7)

We use the orthogonality of the Legendre polynomials to extract the

coefficients ur,n(r)=Anr−n+Bnr−n−2from the simulated flow field

with the radial component ur(r,θ)and then determine the coef-

ficients of the force multipoles Anand source multipoles Bnby a

polynomial fit in r−nand r−n−2.

When the object moves in a fluid confined between two parallel

plates with distance Δz(Hele-Shaw geometry), a multipole expan-

sion for the flow field averaged along the plate normal exists.17,90

Using polar coordinates ρand φin the plane, the radial velocity

component reads

uρ(ρ,φ)=

∞

∑

n=1(𝒜n+ℬn)ρ−n−1cos(nφ), (8)

where φis measured against the swimming direction ˆ

e. Along the

normal or zdirection, the flow field is parabolic, which is inte-

grated out and included into the coefficients 𝒜nand ℬn. Similar

to the bulk case, we use the orthogonality of the basis functions

cos(nφ)to extract the coefficients ˜

uρ,n(ρ)=(𝒜n+ℬn)ρ−n−1. The

relative strength of the amplitudes 𝒜nand ℬncan be deduced

either by extrapolating from the bulk case or by their different

scaling behavior 𝒜n∝Δzand ℬn∝1/Δz. Most interesting about

Eq. (8) is the radial decay of the different multipoles compared to the

bulk case. For a force-free microswimmer (A1=𝒜1=0), the source

dipole moment ℬ1becomes the dominant term in the Hele-Shaw

geometry, while in the bulk fluid the force dipole A2dominates.

In Fig. 2, we demonstrate the multipole analysis of the flow

field of the pusher squirmer simulated with the extended MPCD

method. Figure 2(a) shows all non-vanishing radial velocity coef-

ficients ur,1(r)and ur,2(r)in the bulk fluid for the squirmer type

with β=−1. In agreement with the analytic expression, each follows

a distinctive power law specific for the multipole moment. The radial

decay of ur,1(r)clearly has the exponent −3 corresponding to the

FIG. 2. Radial decay of the radial velocity coefficients of the hydrodynamic mul-

tipoles obtained from the Legendre–Fourier decomposition of the flow field of a

pusher squirmer. (a) Bulk fluid and squirmer type β=−1. (b) Hele-Shaw geometry

with Δz=8a0and squirmer type β=−3.

source dipole moment B1, and ur,2(r)decays with an exponent −2

in the farfield as expected for the force dipole moment A2. However,

for ur,2(r)close to the squirmer, we also recognize the importance

of the short-ranged flow field of the source quadrupole moment B2,

which has the exponent −4. Thus, the polynomial fit deviates from

the straight line. The additional source quadrupole moment with a

negative value is necessary to balance the force dipole A2such that

the correct surface velocity field of the pusher squirmer is imple-

mented. In contrast, the flow field of the source dipole moment B1

obeys the boundary condition by itself so that ur,1(r)follows a pure

power law.

In the Hele-Shaw geometry, we choose β=−3 to get a stronger

signal from the force dipole. We observe the corresponding radial

coefficients ˜

uρ,1(ρ)and ˜

uρ,2(ρ), which perfectly show the expected

radial decay [see Fig. 2(b)]. However, note that due to the confine-

ment, the exponents of the moments of force and source dipole are

swapped, and hence, the source dipole ultimately dominates the far

field. Note that, unlike in the bulk fluid, force and source multipoles

in ˜

uρ,n(ρ)decay with the same power law in a Hele-Shaw geometry.

Here, the deviation of the force dipole field from theory at ρ<Δz

(gray dashed line) is due to short-ranged contributions to the flow

field that are neglected in Eq. (8).90,91

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IV. COMPARISON OF TRADITIONAL

AND EXTENDED MPCD

A. Fluid density in squirmer-rod systems

In this section, we demonstrate how the low compressibility

of the extended MPCD algorithm results in a much more uniform

density of the MPCD particles compared to traditional MPCD meth-

ods in systems where the fluid is strongly forced. In particular, we

consider a dense suspension of squirmer rods that are constrained

to move in the center plane between two confining parallel plates,

which we will further study in Sec. V. We choose squirmer rods with

an aspect ratio of α=4.0 at an area fraction of ϕ=0.6.

As a reference, we also perform simulations with the standard

MPCD-AT+A algorithm with angular momentum conservation and

an Andersen thermostat.3We choose the simulation parameters

such that the relevant physical quantities characterizing the system,

the fluid viscosity and the Péclet number, match in both MPCD

algorithms. Hence, for the standard MPCD method, we use n′

0=10

and Δt′=0.02a0√kBT/m0, which results in the same viscosity

η≈16√m0kBT/a2

0.14,82 Because the two MPCD algorithms use dif-

ferent time steps, we choose 20 ×106and 5 ×106time steps, respec-

tively, to simulate the same period with the extended and conven-

tional MPCD method.

With both algorithms, the initial random distribution of

squirmer rods evolves into a stable compact swarm, which moves

through the system [insets of Figs. 3(a) and 3(b)]. Above and below

the swarm, the fluid experiences strong forcing due to the slip

velocity fields on the rod surfaces. This generates a strong pres-

sure gradient between the region of the cluster and the rest of

the system. However, the MPCD fluid densities simulated with the

traditional and extended MPCD algorithms behave differently, as

shown in the main parts of Figs. 3(a) and 3(b). While the relative

fluid densities ⟨ρ/ρ0⟩zand ⟨ρ/ρ′

0⟩z, averaged over the zdirection,

become non-uniform in both simulations, the deviations from the

mean density when using the extended MPCD algorithm are much

smaller.

For the traditional MPCD algorithm [cf. Fig. 3(a)], the pressure

gradient creates regions sparse of fluid particles above and below the

swarming cluster as well as between the rods. These fluid particles

are pressed into regions outside of the cluster, where the density

increases. Note that the density field also includes the ghost parti-

cles of the squirmer rods, which are clearly visible in white because

their density is held constant at n0by definition. In the vicinity of

the swarming cluster, the relative density drops to values of about

60% and it rises to 140% in regions outside the cluster. In simula-

tions with stronger confinement, i.e., smaller Δz, this effect becomes

even stronger.

In contrast, for the extended MPCD algorithm, the deviations

in relative density are significantly lower with around ±6% [cf.

Fig. 3(b)]. Thus, the density is only mildly decreased in the vicinity

of the cluster and increased in the remaining regions of the sys-

tem. Note that the fluid density directly between squirmer rods in

the midplane follows the trend of the height-averaged density in

Figs. 3(a) and 3(b), but the deviations are stronger. With the tradi-

tional MPCD method, the relative density drops to 30%, while with

the extended MPCD method, the minimum relative density is 90%.

All this is in accordance with the low compressibility of the fluid

in the extended MPCD method, which is about five times lower

FIG. 3. Density field of the fluid in systems with squirmer rods of an aspect ratio

of α=4.0 and an area fraction of ϕ=0.6 and in a Hele-Shaw geometry with a

plate distance of Δz=18a0. The relative density is averaged over the height of

the system and shown in top view. The insets show snapshots of the configuration

of squirmer rods. Results obtained with the traditional MPCD-AT+A algorithm (a)

and with the extended MPCD algorithm (b).

compared to the simulations with traditional MPCD.82 However,

note that this comes with an increase in simulation time.

Thus, the extended method better satisfies the incompressibility

condition ∇ ⋅ v=0. Furthermore, also the viscosity, which depends

on the local density, is more uniform in the extended MPCD

method and the simulation thus closer to the Stokes limit. This is

particularly important for simulating microswimmers, where typ-

ically a constant Péclet number Pe ∝ηis assumed for the entire

system.

To be more quantitative, we show in Fig. 4 the pressure ⟨p/p0⟩z

and viscosity fields ⟨η/η0⟩zderived from the density fields shown

in Fig. 3 using the respective analytical formulas of the traditional

method and our extended MPCD method.2,82 Note that, since both

pressure and viscosity depend on density ρ, they are not independent

of each other. While the viscosity of the traditional MPCD method

is linear in the density and varies accordingly by ±40% [cf. Fig. 4(c)],

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FIG. 4. Fluid pressure (top row) and viscosity (bottom row) fields of the systems

shown in Fig. 3 in top view. Both quantities are averaged over the height of the

system and smoothed with a Gaussian filter using σ=3a0. Results obtained with

the traditional MPCD-AT+A algorithm in the left panels [(a) and (c)] and with the

extended MPCD algorithm in the right panels [(b) and (d)]. Note that p0is different

for the two methods.

variations are much smaller for the extended method. Here, the rela-

tion between density and viscosity is more complex and only results

in a variation of ±10% [cf. Fig. 4(d)]. The pressure fields show a sim-

ilar behavior. A closer inspection shows that the deviations from the

mean value are marginally smaller compared to the viscosity vari-

ations, as one can see in Fig. 4(a) for the traditional method and

Fig. 4(b) for the extended method.

Now, with the significant increase in the local Péclet number

Pe ∝ηoutside of the cluster for the traditional MPCD method,

we would assume enhanced clustering compared to the much less

compressible fluid, which has been reported for spherical squirm-

ers.21 However, clustering in the two simulations reported here is

very similar, suggesting that steric interactions between the elon-

gated rods dominate. In Subsection IV B, we will therefore compare

two simulations with shorter rods such that the alignment due to the

elongated shape is less pronounced and hydrodynamic interactions

between the rods become more important.

B. Influence on hydrodynamic interactions

To investigate how the lower compressibility of the extended

MPCD method affects the hydrodynamic interactions between the

rods and ultimately their clustering, we choose a lower aspect ratio

of α=2.5 and an area fraction of ϕ=0.4 to reduce the effect of steric

interactions. Otherwise, we use the same system parameters as in

Sec. IV A and perform simulations with the two different MPCD

algorithms.

In the temporal evolution of both systems, small clusters form

due to direct collisions. However, only with the extended MPCD

method, they begin to form swarms that grow to a significant

size, while with the traditional MPCD method the collision clusters

remain small and dissolve rather quickly. The snapshots of Figs. 5(a)

and 5(b) as well as videos V1 and V2 illustrate this difference. To

quantify it, we identify individual rod clusters such that all spheri-

cal squirmers that are separated by a center-to-center distance less

than Δx<1.02Rbelong to one cluster. The cluster size Ncthen is the

number of rods within a cluster determined by the ratio of squirmer

number in the cluster to squirmers per rod. Additionally, we define

the cluster velocity as the instantaneous average of rod velocities over

the cluster.

Figure 5(c) shows the mean cluster velocity ⟨vc⟩relative to the

squirmer velocity v0, and Fig. 5(d) shows the probability density for

a rod to lie in a cluster with size Nc. Note that the swimming speed of

single squirmer rods is ∼1.1v0.17 The difference between both meth-

ods is immediately visible. For small groups N<10, we observe sim-

ilar velocities vc≈1.1v0in Fig. 5(c) close to the value of a single rod

in the bulk fluid. These rods form a dilute background, which exists

equally in both systems, as the identical probability distributions of

cluster sizes with Nc<10 in Fig. 5(d) show.

FIG. 5. Snapshots of simulations with (a) the traditional MPCD method and (b) the

extended MPCD method. Both snapshots show a typical cluster/swarm, which is

marked by a red circle. (c) Mean velocity of clusters with size Ncgiven in units of

the squirmer velocity v0for squirmer rods with an aspect ratio of α=2.5 at a dilute

area fraction of ϕ=0.4. (d) Probability density for a squirmer rod to lie in a cluster

of size Nc. The system size is equal to Fig. 3 and also the systems used in Sec. V.

The results are obtained with the traditional MPCD-AT+A algorithm (blue) and the

extended MPCD algorithm (green).

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Large clusters also exist in both systems, but they behave differ-

ently. Visual inspection of video V1 shows that clusters in simula-

tions with traditional MPCD are formed rather by collisions between

rods or other clusters and dissolve quickly. As a result, the mean

cluster velocity vcin Fig. 5(c) decreases drastically with the cluster

size Nc. Furthermore, Fig. 5(d) shows that the clusters do not exceed

a size of roughly Nc=200.

In contrast, the formation of stable swarms in simulations with

the extended MPCD method leads to the plateau value of the mean

cluster velocity at vc≈0.6v0for Nc>30 [Fig. 5(c)]. In addition, the

distribution P(Nc)in Fig. 5(d) shows the existence of larger clusters.

Both results indicate that these swarms are more ordered compared

to the simulations with the traditional MPCD method. They accu-

mulate new members during the collective swimming so that the

polar order remains constant. Visual inspection of video V2 shows a

dynamic swarming behavior where swarms still merge and dissolve,

but overall they are more stable.

This is an interesting result because it shows that the extended

MPCD method with its lower compressibility stabilizes clusters

in the current system. Moreover, we conclude that the extended

method mediates hydrodynamic interactions better such that rods

align and form swarms.

V. COLLECTIVE DYNAMICS OF SQUIRMER RODS

We now present a thorough investigation of the collective

dynamics of squirmer rods varying aspect ratio αand 2D area frac-

tion ϕ. Overall, we observe four different dynamic states, which

we briefly describe here based on visual inspections before turn-

ing to a detailed analysis in Subsections V A and V B. The state

diagram together with exemplary snapshots is given in Fig. 6, and

videos V3–V6 in the supplementary material illustrate the differ-

ent dynamic states and correspond to the respective snapshots

[Figs. 6(b)–6(e)].

At small densities and also at larger densities below a critical

value of the aspect ratio, αmin ≈2, we observe a disordered state

with little variations in the local density. Thereby, we support the

conclusion of Ref. 21 that the low fluid compressibility reduces

clustering. Increasing αabove 2 and also for sufficiently large den-

sity, swarms of squirmer rods form. They consist of clusters with

inherent polar order such that the squirmer rods move collectively

in one direction. However, as described already in Sec. IV B, the

swarms are dynamic, meaning that they are not stable in time but

frequently break apart and reform creating an interesting dynamics

reminiscent of dynamic clustering found for spherical active par-

ticles in the presence of a chemical field.92,93 Crossing the dashed

black line in the state diagram by increasing density and/or aspect

ratio further, the dynamic swarms merge into one large swarm,

which is stable in time. Depending on the individual history of

such a swarm, it can show either continuous translational or rota-

tional motion. Further increasing area fraction ϕ, the global swarms

grow bigger until they span over the whole system. In this situation,

the cluster is jammed, where all movements inside the cluster are

blocked and only some drift velocity of the whole cluster remains.

Our findings generally show parallels to the surface dynamics of

B. subtilis reported in Ref. 94. In particular, the dynamic swarms,

single swarm, and the jammed cluster are observed. For systems of

FIG. 6. (a) State diagram of squirmer rods of different aspect ratios αand at differ-

ent area fractions ϕ. Red circles: disordered system (b), green circles: dynamic

swarms (c), black stars: single moving swarm (d), and black squares: single

jammed or drifting cluster (e). [(b)–(e)] Snapshots of the respective states for the

points indicated in the state diagram.

elongated spheroidal squirmers, the disordered and single swarm

states have been reported as well.21

For a detailed quantitative analysis of these dynamic states, we

first study in Sec. V A the distribution of the local area fraction ϕloc as

well as the cluster size and velocity. To analyze the alignment of the

squirmer rods in the dilute and also cluster states, we use the spatial

orientational pair-correlation function in Sec. V B.

A. Distributions of local area fraction, cluster size,

and velocity

The observed swarms are clustered squirmer rods with addi-

tional polar order. Thus, to differentiate swarms from dilute regions

in the system and to characterize the different states, we determine

the distribution of local area fraction as well as cluster sizes and their

velocities. To assign each squirmer rod a local area fraction ϕloc, we

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use a Voronoi tessellation following our earlier work.14 The con-

cept is extended to the squirmer rods as follows and illustrated in

Fig. 7(b). First, the ordinary Voronoi tessellation is calculated for

all spherical squirmers. Here, periodic images are added around the

central simulation box to achieve correct results across the periodic

boundary conditions. Then, the Voronoi cells of the squirmers that

belong to the same rod form the Voronoi cell of the rod, which

are no longer convex polygons. Finally, the local area fraction fol-

lows from the ratio of the squirmer rod’s cross section and the area

of the Voronoi cell. In addition, we identify individual rod clusters

and determine their velocity as described in Sec. IV B. To calculate

the distribution of local area fractions P(ϕloc)and to identify clus-

ters, we use all snapshots of the system after an equilibration time of

teq =10 ×106Δtin steps of 5 ×104Δt.

We now characterize the different dynamic states using

Fig. 7(a), where we plot the normalized distribution functions

P(ϕloc)for different global or mean area fractions ϕ, always for

squirmer rods with an aspect ratio of α=3.25. In addition, Fig. 7(c)

shows individual clusters with their size Ncand velocity vcso that for

each area fraction a dynamic state is represented by a cloud of points.

A dashed line is added for each area fraction ϕ, indicating the max-

imum possible cluster size equal to the total number of rods. Note

that the distribution in the velocity is naturally broader for small

clusters, which give rise to larger fluctuations.

Increasing ϕfrom small to large values, we observe the follow-

ing characteristics. At ϕ=0.1, the distribution P(ϕloc)is narrow and

strongly peaked at the global value, which indicates the absence of

pronounced clustering in the system. The distribution of cluster sizes

in Fig. 7(c) reflects this behavior indicating only small groups of col-

liding squirmer rods. This is where we locate the disordered state,

which is homogeneous in the local area fraction. At the larger area

fraction of ϕ=0.2, P(ϕloc)shows the same trend but now the peak

at low ϕloc is broader and a pronounced tail toward higher values

of ϕloc is visible. Most of the system is still dilute, but small denser

clusters are present, as shown in Fig. 7(c). This indicates that we are

at the transition to the state of dynamic swarms. However, Fig. 7(c)

indicates that these clusters are still mainly caused by collisions and

not by longer lasting swarms with intrinsic rod alignment because

their velocities are low.

At ϕ=0.4, we are right in the subsequent state of dynamic

swarms. Here, the distribution P(ϕloc)becomes very broad and, in

particular, bimodal. The bimodal distribution of area fractions has

also been found for swarming B. Subtilis.94 This signature is caused

by parts of the system forming a dilute disordered background, while

also dense dynamic swarms are visible. Their preferred area frac-

tion indicated by the right peak of P(ϕloc)is close to the maximum

ϕloc =1. Thus, the clusters in the swarms are compact. Note that

the peak for the dilute disordered background becomes more pro-

nounced when decreasing the global area fraction and/or the aspect

ratio toward the disordered state. The very broad distribution of

cluster sizes and velocities in Fig. 7(c) also reflects this dynamic state

of swarms. However, the size of the clusters always remains well

below the total number of squirmer rods.

At the global aspect ratio of ϕ=0.6, we are in the state where

the squirmer rods form a single moving swarm. Here, the distribu-

tion P(ϕloc)only shows one peak close to ϕloc ≈1 and the dilute

background is nearly absent. The distribution of cluster sizes in

FIG. 7. (a) Distribution of local area fraction P(ϕloc)obtained from the Voronoi

tessellation of the configurations of squirmer rods with an aspect ratio of α=3.25

at different global area fractions ϕ. (b) Demonstration of a Voronoi tessellation for

squirmer rods with α=3.25 and at ϕ=0.4. The black dots mark the positions of

the spherical squirmers, and the green circles mark their cross sections. Note that

because always ten squirmers lie on a line, each squirmer rod creates nine parallel

lines in the Voronoi diagram. (c) Cluster velocity vcin units of the squirmer velocity

v0plotted against the cluster size Nc. Each point represents a cluster determined

from the snapshots. The colored dashed lines mark the total number of rods for

the given area fraction ϕ. The black symbols ⊗indicate the mean cluster velocity

for clusters with size Nc>50, which is chosen arbitrarily to exclude small collision

groups.

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Fig. 7(c) is narrow, meaning that this cluster is highly stable and

its size almost reaches the total number of rods in the system. The

jammed cluster at the highest aspect ratio ϕ=0.77 shows almost the

same trend. However, the maximum of P(ϕloc)slightly moved to

the left and the distribution is broader compared to ϕ=0.6. This is

in agreement with visual inspections, which show that the jammed

clusters are less dense. When they form, the density is so high that

they cannot all align properly as in the swarms, which we will con-

firm in Sec. V B. As shown in Fig. 7(c), the size of the cluster now

reaches the total number of squirmer rods and the velocity of the

jammed cluster is significantly smaller compared to the single swarm

state.

For squirmer rods with other aspect ratios α, we observe the

same behavior, with differences only in the quantitative descrip-

tion. At smaller aspect ratios α<3.25, the distributions P(ϕloc)are

broader and the bimodality in the swarming state is less pronounced.

For higher aspect ratios α>3.25, the trend is inverted and also the

peaks of P(ϕloc)are more pronounced. Equally, the distribution of

cluster sizes becomes more compact. This is a clear consequence of

the fact that the rods tend to align better at larger aspect ratios α,

which stabilizes the swarms.

B. Orientational correlations

As a second measure to characterize the different states of our

system, we use the orientational pair-correlation function ⟨ˆ

ei⋅ˆ

ej⟩,

where ˆ

eiand ˆ

ejare the orientation vectors of two squirmer rods with

a distance Δx. It helps us to distinguish the different swarming and

jamming states by quantifying the degree of alignment in the clus-

ters. Figure 8(a) presents the orientational pair-correlation function

for different global area fractions for squirmer rods with an aspect

ratio of α=3.25. The plot uses the same parameters as Figs. 7(a) and

7(c) in Sec. VAand supplements the results reported there.

For dilute rod suspensions with an area fraction of ϕ=0.1, we

observe that the squirmer rods are indeed disordered. Although the

orientational pair correlations have a pronounced peak at the side-

by-side contact distance of two squirmer rods, they quickly decay to

almost zero at two rod lengths. The strong correlations at a short

distance make sense since the squirmer rods collide with each other

even in the disordered state. Because of their active movement,

pairs of squirmer rods are more likely to approach and collide if

their swimming directions are already nearly aligned to each other.

Moreover, a side-by-side configuration is more persistent, which

causes the first peak with ⟨ˆ

ei⋅ˆ

ej⟩≈1. At Δx=lS, we observe a sec-

ond smaller peak in the orientational pair correlations, which is due

to two squirmer rods swimming behind each other. This configu-

ration persists for longer times than a head on collision, and there-

fore, the correlation function is positive. At ϕ=0.2, we are close to

the dynamic-swarm state. Still, the pair correlations are similar to

ϕ=0.1, but overall they are increased. While the height of the first

peak is almost the same, in particular, the second peak is more pro-

nounced. This indicates the presence of small clusters with squirmer

rods swimming side by side and also behind each other.

At ϕ=0.4, the orientational pair-correlations clearly change.

They are of longer range, and the system is in the dynamic-swarm

state. For example, the correlation only decays slowly to ⟨ˆ

ei⋅ˆ

ej⟩

≈0.25 at 4lS. The peak at Δx=lSfor two squirmer rods swim-

ming behind each other increases, and a careful inspection of the

FIG. 8. (a) Orientational pair-correlation function ⟨ˆ

ei⋅ˆ

ej⟩of two squirmer rods

plotted vs their distance Δxfor different global area fractions ϕ. (b) Results of

simulation runs with different initial conditions at ϕ=0.6 and α=3.25.

curve also shows an additional peak at Δx=0.62, twice the side-by-

side distance. This agrees with visual inspections, which show that

squirmer rods tend to swim in successive rows within a swarm. Fur-

thermore, the very weak peaks at 2lSand 3lSagree with the approx-

imate extension of RS≈3lSof the dynamic swarms. Note that the

alignment is not perfect within clusters, and hence, the correlation

reduces with increasing distance.

For the higher area fraction ϕ=0.6, the squirmer rods now

form a single big swarm. Here, we see a further increase in the

strength and range of the orientational correlations. Interestingly,

Fig. 8(b) shows that the shape of the correlation function depends on

the initial conditions. As described in Sec. II C, for ϕ≥0.6, we only

change the orientation randomly along one direction. As a result,

in each simulation run, the cluster assumes a different shape and

behaves differently. In particular, in the first simulation run the clus-

ter of squirmer rods rotates rather than performing a straight motion

as observed in simulation run 2. Thus, the rods at opposite sides of

the cluster are anti-parallel to each other, causing the negative corre-

lations ⟨ˆ

ei⋅ˆ

ej⟩at distances Δx>6lS. In simulation runs 3 and 4, the

configuration of the cluster creates both straight motion and rotation

so that it moves on a circle. Thus, the orientational pair-correlation

function has an intermediate characteristic.

Increasing the area fraction further to ϕ=0.77, the cluster is

jammed. Compared to the swarming cluster, the orientational corre-

lations are weaker and of shorter range. This indicates that both local

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and global alignments are reduced and swarming does not occur.

The system is too dense to allow alignment during swimming, which

initiated the swarm formation at lower densities. For α=3.25, the

jammed cluster is not entirely stable and can partially break up and

reform again, as observed by visual inspection. As a result, the orien-

tational correlation function looks quite similar to that for the swarm

state at ϕ=0.4. For the larger aspect ratios α=4.0 and α=4.75, the

jammed cluster is entirely stable and the orientational correlation

function decays to zero at even shorter distances.

For aspect ratios different from α=3.25, the results are qualita-

tively similar. For smaller aspect ratios, the orientational correlations

are expectably lower and decay faster than for higher aspect ratios.

For α>3.25, the orientational correlations are stronger at short dis-

tances Δx<2lS, especially in the dynamic-swarm state. However, in

the single-swarm state, the squirmer rods cannot align as much as

before, which results in lower cluster velocities.

VI. CONCLUSIONS AND OUTLOOK

In this article, we have tested the extended MPCD method

introduced in our previous article82 against known cases and also

used it to explore the collective dynamics of squirmer rods. As we

have demonstrated in Sec. III, the extended MPCD method accu-

rately solves the Navier–Stokes equations in the Stokes limit. In

particular, the hydrodynamic multipoles that occur due to the pre-

scribed surface flow field of the pusher-type squirmer are correctly

represented in the simulated velocity field. In both, the bulk fluid

and Hele-Shaw geometry, the hydrodynamic source dipole and force

dipole decay with the correct power law. In the simulations of the

bulk fluid, we could also identify the source quadrupole, which

becomes noticeable close to the squirmer.

Due to the non-linear equation-of-state, the MPCD fluid of the

extended method is less compressible and, therefore, the density,

viscosity, and pressure are significantly more uniform compared to

traditional MPCD methods. Small variations in the fluid density

remain since also in the extended model the fluid density encodes

fluid pressure. Nevertheless, squirmer rods with a high aspect ratio

α=4 show very similar clustering in both methods, which suggests

that steric interactions due to the strongly elongated shape are dom-

inant. To reduce the tendency to align and thereby increase the

influence of hydrodynamic interactions, we lowered the aspect ratio

to α=2.5. Interestingly, the observed swarming of dynamic clus-

ters is more pronounced with the extended MPCD method. They

have higher polar order, are more stable, and form larger clusters.

Our understanding is that in a less compressible fluid where the

fluid density is not reduced between nearby rods as in the tradi-

tional MPCD, the hydrodynamic flow reduces the occurrence of

hard collisions, and thereby, the rods have the possibility to better

align along each other. This favors the formation of swarms due to

the prolonged duration of polar encounters.

Finally, we systematically investigated the collective dyna-

mics of squirmer rods and found four different dynamic states with

increasing aspect ratio αand area fraction ϕ, namely, a disordered

state, dynamic swarms, a single swarm, and a jammed cluster. Sim-

ilar states are identified for B. Subtilis in the experiments of Ref. 94.

The disordered state and a global swarm also exist for spheroidal

squirmers.21 To characterize the dynamic states quantitatively, we

determined the distribution of local area fraction P(ϕloc)as a mea-

sure for the degree of clustering in the system. Additionally, we iden-

tified clusters of squirmer rods and determined their mean velocities,

while the intrinsic structure of the swarms was analyzed with the

orientational pair-correlation function.

For low density ϕand aspect ratio α, we observe a disordered

state with homogeneous density in which squirmer rods only inter-

act during frequent collisions. Above a critical value of α≈2 and at

larger densities, the squirmer rods form dynamic swarms, where the

system clearly separates into a dilute background and very dynamic

polar clusters with a broad size distribution. When the density is

increased further, the dynamic swarms turn into a single stable

swarm. The dilute background of squirmer rods disappears, and the

swarm keeps a high polarization. Depending on the initial condition,

the swarm can exhibit directed motion, rotation, or a combination

of both, which results in a circular trajectory. At the highest density

ϕ=0.77 in our simulations, the system is so crowded that the forma-

tion of highly ordered swarms is inhibited. The rod cluster becomes

jammed and moves with a significantly lower velocity compared to

the swarms. Our findings show similarities with previous studies64–67

of dry active rods with different penetrability. In particular, the dis-

ordered and also motile cluster or swarming state is found in all

systems. These states appear to originate from the rod shape and fre-

quent collisions rather than the particular details of the steric inter-

actions. Additionally, Refs. 64–67 found states of giant clusters of

different nature. If particles cannot cross each other, also immobile

giant clusters are observed.65,67 In contrast, the giant clusters in our

study are more polar and therefore more motile, which we attribute

to the hydrodynamic interactions present in our work. In the case

when rods are allowed to cross each other, the tendency to jam is

reduced.66 However, the mechanism for forming giant clusters is

now different. Reference 65 adds an interesting study of hydrody-

namically interacting flagella of pusher type. Thus, their collective

dynamics is less comparable to our system.

In future investigations, we plan to extend our study to the

collective dynamics of pusher- and puller-type squirmer rods as

introduced in Ref. 17. In addition, it is straightforward to extend

the squirmer-rod model to active flexible filaments by adding bend-

ing rigidity. This will provide valuable insights how hydrodynamic

interactions influence the collective dynamics of active filaments and

bridge between dry models33,95 and models with implicit hydrody-

namic interactions,96,97 on the one hand, and continuum descrip-

tions,98,99 on the other hand. Clearly, the extended MPCD method

will prove beneficial for simulations of dense suspensions of active

filaments by providing a less compressible fluid environment in the

regime of vanishing Reynolds numbers.

SUPPLEMENTARY MATERIAL

See the supplementary material for additional video files

V1–V6, which show the dynamic evolution of the systems shown

in Figs. 5(a),5(b), and 6(b)–6(e), in the order of their occurrence.

ACKNOWLEDGMENTS

We thank Josua Grawitter for helpful discussions on the topic

of the manuscript. We also acknowledge financial support from

J. Chem. Phys. 155, 134904 (2021); doi: 10.1063/5.0064558 155, 134904-10

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of Chemical Physics ARTICLE scitation.org/journal/jcp

the Collaborative Research Center 910 funded by the Deutsche

Forschungsgemeinschaft.

AUTHOR DECLARATIONS

Conflict of Interest

The authors have no conflicts to disclose.

DATA AVAILABILITY

The data that support the findings of this study are available

from the corresponding author upon reasonable request.

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Cite this: Soft Matter, 2022,

18, 6179

Emergent collective dynamics of pusher and

puller squirmer rods: swarming, clustering,

and turbulence†

Arne W. Zantop * and Holger Stark *

We study the interplay of steric and hydrodynamic interactions in suspensions of elongated micro-

swimmers by simulating the full hydrodynamics of squirmer rods in the quasi two-dimensional geometry

of a Hele-Shaw cell. To create pusher or puller-type squirmer rods, we concentrate the surface

slip-velocity field more to the back or to the front of the rod and thereby are able to tune the rod’s

force-dipole strength. We study a wide range of aspect ratios and area fractions and provide

corresponding state diagrams. The flow field of pusher-type squirmer rods destabilizes ordered

structures and favors the disordered state at small area fractions and aspect ratios. Only when steric

interactions become relevant, we observe a turbulent and dynamic cluster state, while for large aspect

ratios a single swarm and jammed cluster occurs. The power spectrum of the turbulent state shows two

distinct energy cascades at small and large wave numbers with power-law scaling and non-universal

exponents. Pullers show a strong tendency to form swarms instead of the disordered state found for

neutral and pusher rods. At large area fractions a dynamic cluster is observed and at larger aspect ratio a

single swarm or jammed cluster occurs.

1 Introduction

Microscopic unicellular organisms make up a major fraction of

all life forms on our planet. They are involved in important

natural processes such as photosynthesis

1–3

or industrial pro-

cesses such as the production of enzymes

4,5

or biofuels,

6–8

the recycling of wastewater.

9,10

Especially in photosynthesis,

commonly viewed as the basis of life, unicellular phytoplankton

accomplish about half of the worldwide natural turnover.

2,3

However, still many aspects of the behavior of unicellular

organisms is not completely understood. In fact, many of these

life forms are self-propelling microswimmers,

11–13

as, for exam-

ple, the algae C. reinhardtii

14,15

or the bacterium E. coli.

16–18

From the perspective of physicists, their active motion gives rise

to very interesting new collective phenomena. In this article we

study the rich emergent collective dynamics of rod-shaped

model microswimmers, which we can tune between the pusher

and puller type.

Active motion is always performed in non-equilibrium and,

therefore, gives rise to new and interesting phenomena. For

example, specially designed boundaries can rectify the random

motion of active particles,

19–23

or under gravity active particles

develop polar order and even show inverted sedimentation

profiles when they are bottom-heavy.

24,25

In addition, micro-

swimmer suspensions are subject to long-range hydrodynamic

interactions with characteristic power-law decay.

13,26,27

But also

short-range steric interactions play an important role, in parti-

cular, for elongated particles, which align along each other.

The combination of these interactions gives rise to numerous

interesting dynamic patterns. Common examples are the for-

mation of swarms or flocks,

28–32

convection rolls and plumes,

fluid pumps,

33,34

vortices,

35–38

active nematic patterns,

39–41

and

the emergence of the so-called active turbulence,

42,43

termed in

analogy to classical inertial turbulence.

44,45

However, contrary to

classical turbulence, where fluid flow is driven on the macroscopic

scale, active turbulence is generated at the microscopic scale of

the self-propelled particles and then energy is dissipated on larger

scales. This mechanism causes a characteristic length scale for

the formation of vortices and patterns, which is in contrast to

the scale invariance of classical turbulence.

46–48

In particular, the

specific model parameters now determine the scaling of the

velocity power spectrum, which is no longer universal.

48–50

Active

turbulence is found for microswimmers,

51,52

active bio-filaments,

which exist in the cytoplasm,

39,53

and in growing tissue.

54,55

Institut fu

¨r Theoretische Physik, Technische Universita

¨t Berlin, Hardenbergstraße 36,

10623 Berlin, Germany. E-mail: a.zantop@tu-berlin.de, holger.s[email protected]

†Electronic supplementary information (ESI) available: We provide eight videos

of diﬀerent collective dynamic states in the top view. Videos 1–3 show pusher rods

in the turbulent (1), single swarm (2) and dynamic cluster state (3). Videos 4–8

show puller rods in the swarming (4 and 8), single swarm (5), jammed cluster (6),

and dynamic cluster state (7). See DOI: https://doi.org/10.1039/d2sm00449f

Received 8th April 2022,

Accepted 30th June 2022

DOI: 10.1039/d2sm00449f

rsc.li/soft-matter-journal

Soft Matter

PAPER

On the theoretical side, the dynamics of active particles has

been investigated at diﬀerent levels of description. While the

celebrated Vicsek model

56–58

uses coarse-grained alignment

rules, also models with explicit steric interactions exist.

42,51,59–64

Thesemodelshavealsobeengeneralizedtoactivefilaments

31,65,66

and models, which implicitly include hydrodynamic inter-

actions.

67,68

Models, which directly simulate fluid flow and

thereby explicitly include hydrodynamic interactions, use the

method of multi-particle collision dynamics (MPCD)

25,30,32,69–79

or the lattice-Boltzmann method.

80–83

Continuum models

combine elements of the Toner-Tu

and Swift-Hohenberg

equations to generate hydrodynamic equations for active

suspensions.

46,86,87

Microswimmers are also distinguished by

their swimming mechanisms and the flow field, they generate

in the surrounding fluid.

11,13

While some microorganisms

propel with cilia located all over their surface, common bacteria

and algae propel with flagella that extent from the front or back

of the cell body. In the first case, the flow field of a source

dipole is realized, which decays as r

3

, while bacteria and algae

are termed pusher or puller-type swimmers that generate a

long-range force-dipole flow field, which decays as r

2

13,27

The

specific form of these hydrodynamic multipole flow fields deter-

mines the collective dynamics of the microswimmers.

32,69,74,88–90

In this context, we proposed in ref. 76 the squirmer rod as

a realistic microscopic model for elongated microswimmers.

It consists of overlapping squirmers and thereby extends the

well-known spherical squirmer model for ciliary propulsion

26,91

and its implementation in MPCD.

Restricting the surface slip-

velocity to the front or the back of the squirmer rod, puller and

pusher-type squirmer rods can be realized, respectively. The

hydrodynamic flow field is simulated using an eﬃcient imple-

mentation of MPCD with a reduced compressibility such that

also collective dynamics in large and dense systems can be

studied.

Most recently, we presented the state diagram of

neutral squirmer rods and identified with increasing area

fraction and depending on the aspect ratio of the rods the

disordered state, dynamic swarms, a single swarm, and ultimately

ajammedcluster.

In this article we generalize the squirmer-rod model to

pushers and pullers with tunable force-dipole strength in the

flow field. Compared to spheroidal squirmers it has the advan-

tage that it better approximates real rodlike microswimmers

such as E. coli. Using this model, we provide a comprehensive

study of the state diagram for hydrodynamically interacting

microswimmers over a wide range of aspect ratios, densities,

and force-dipole strengths as the state diagrams in Fig. 3(a),

7(a) and 9(a) show. Hereby, we go well beyond previous works,

which focused mainly on more dilute systems

95,96

or a single

aspect ratio.

Thus, our work provides an overall view how

hydrodynamics and shape determine the dynamic states of

microswimmers. For example, our particle-based model nicely

illustrates that the active turbulent state occurs as a compro-

mise between the disordering hydrodynamic pusher–pusher

interactions and aligning steric interactions. We also demon-

strate that the two distinct energy cascades at low and large

wave numbers in the power spectral density of velocity

fluctuations exhibits non-universal exponents. Moreover, we

find a dynamic cluster state at large densities. Besides this

state all other states are also found in dry active rods.

But, in

our case, they have a specific contribution from hydrodynamics

as, for example, the turbulent and swarming states show. The

overall appearance of the states in our state diagrams can be

summarized as follows. For pushers we observe that the swarm-

ing states of neutral squirmer rods are destabilized. Instead, for

smaller aspect ratio between the disordered and dynamic

cluster state, we observe the turbulent state as already men-

tioned. At high aspect ratios, where steric interactions become

more relevant, we recover the single swarm and jammed cluster

state of neutral squirmer rods. For pullers, hydrodynamic

interactions stabilize the swarming state even for our smallest

area fraction. Thus, compared to neutral squirmer rods, states

are shifted towards lower densities. Variation of the force-dipole

strength at constant aspect ratio supports all these findings.

The article is organized as follows. In Section 2 we give a

brief overview of the methods used in this paper. Section 3

provides a detailed study of the diﬀerent dynamic states of the

squirmer rods as a function of their area fraction, aspect ratio,

and force-dipole strength. We end with a summary and con-

clusions in Section 4.

2 System and methods

We first introduce the squirmer-rod model and then summarize

some details of the method of multi-particle collision dynamics,

which we use to simulate the flow fields generated by the

squirmer rods.

2.1 Model of the squirmer rod

To model shape-anisotropic microswimmers, we employ the

squirmer rod model as introduced in our previous work.

Squirmer rods consist of N

overlapping spherical squirmers

of radius R

, arranged on a line to form a single rigid body [see

Fig. 1(a)]. By varying the distance dbetween neighboring

squirmers, we can tune the aspect ratio of the squirmer rod,

a=l

/2R, where l

is the rod length. However, we do not exceed

a distance of dE0.8Rso that the surface of the rod is

suﬃciently smooth. With a number of N

= 10 squirmers in

this work, this amounts to a maximum aspect ratio of aE5,

which closely resembles the aspect ratio of bacteria such as

E. coli or B. subtilis.

The squirmer rods propel through the axisymmetric and

tangential slip velocity field at the surface of individual sphe-

rical squirmers,

[(e

ˆxˆ

)xˆ

e

ˆ], (1)

which is imposed on the surrounding fluid.

26,92

Here, e

ˆis the

rod axis and xˆ

the unit vector pointing from the center of a

squirmer to a point on the squirmer surface. This generates a

source-dipole flow field, which is a higher-order singular

solution of the Stokes equations Zr

v=rptogether with the

incompressibility condition rv= 0, that govern fluid flow at

Paper Soft Matter

the microscale. Here, vand pare the respective fluid velocity

and pressure fields, and Zis the dynamic shear viscosity. The

strength B

controls the swimming velocity v

= 2/3B

of the

spherical squirmer and hence of the squirmer rod. In ref. 76

we showed that the swimming velocity of the squirmer rods

vE1.2v

slightly exceeds the velocity of a single spherical

squirmer. Additionally, the velocity of the rods varies by 10% in

the range of aspect ratios used in this article. Although the

increased velocity at the larger aspect ratios might augment

clustering, we assume the eﬀect to be negligible.

As described so far, the surface slip velocity of the squirmer

rod resembles ciliated microorganisms such as Paramecium.

In this realization, the profile of the slip velocity generates a

flow field in the surrounding fluid, the far field of which can be

described by a source dipole and an additional force quadru-

pole singularity, which both decay with |u

|, |u

|Br

3

,aswe

show in ref. 76.

However, other prominent microswimmers such as E. coli

bacteria or Chlamydomonas algae propel by rotating or beating

flagella that extent from the back or the front of their bodies,

respectively. These modes of propulsion create a pair of oppos-

ing forces that generate the more long-ranged force-dipole flow

field |u

|Br

2

. To generalise the squirmer rod model to these

pusher and puller-type microswimmers, we concentrate the

surface slip velocity either to the back or the front of the

squirmer rod [see Fig. 1(a)]. This is done by multiplying

the surface flow field with the envelope function

fðx

s^

e;wÞ¼1þwtanhð10x

s^

e=lSÞ;(2)

where x

spoints from the rod center to a location on the rod

surface. The parameter wA[1, 1] determines the swimmer

type and force-dipole strength, such that for wo0 a pusher-

type swimmer is realised and likewise a puller-type swimmer

for w40. For either w=1orw= 1 this modification leads to a

completely passive half of the rods [see Fig. 1(a)], while for

intermediate values the relative contributions of the source

dipole and force dipole to the flow field vary. For w= 0, the

model again resembles the neutral squirmer rod.

In the present work, we consider the collective dynamics of

squirmer rods confined between two parallel walls. In this

so-called Hele-Shaw cell the radial decays of hydrodynamic

multipoles are modified compared to the bulk fluid, such that

theconfinedsourcedipoleandforcedipoledecayas|u

|Br

2

and |u

|Br

3

, respectively, where ris the polar distance

76,99

(see

appendix of ref. 99). As a consequence, the source dipole has the

longest range in the flow field and ultimately dominates the far

field. However, already in our previous work we realized that a

distance of Dz=6Rbetween the walls, which we will use in our

simulations, alters the relative strength of the source and force

dipoles.

As a consequence, the force dipole dominates the flow

field at short and medium distance as we will demonstrate in

Section 3.2.

2.2 Method of multi-particle collision dynamics

To model the fluid flow in our simulations, we employ the

meso-scale simulation technique of multi-particle collision

dynamics (MPCD).

100–102

The MPCD method is particularly

suited for solving the Navier–Stokes equations at the micro-

scale, because it includes thermal fluctuations and is straight-

forward to implement boundary conditions for complex

geometries.

The MPCD method uses a sequence of streaming and

collision steps of the point-like fluid particles. In the streaming

step the fluid particles move ballistically with their velocities

during time step Dt. Then, the simulation box is divided into

cubic cells with edge length a

and the velocities of the fluid

particles in one cell are modified randomly but keeping the

mean velocity or linear momentum fixed. In this work, we use a

collision rule optimized to achieve a low compressibility of the

fluid.

As in other MPCD methods, it includes angular

momentum conservation and a thermostat. By choosing the

MPCD fluid density n

= 20/a

,i.e., on average 20 fluid particles

per cell, and the time step Dt¼0:005a0ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

m0=kBT0

p, we obtain a

fluid viscosity of Z¼16:05 ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

m0kBT0

p=a0

2, which is comparable

to previous work.

74,76,96

Here, m

is the mass of one fluid

particle.

The immersed squirmer rods are modeled with the mass

density of the fluid r

. The radius of the constituting

squirmersischosenasR

=3a

and the strength of the surface slip

velocity as B

=0.1,whichresultsinaPe

´clet number of Pe E350.

Steric repulsion is realized with a Weeks–Chandler–Andersen

Fig. 1 (a) Sketch of the surface slip velocity of three squirmer rods with

respective swimmer-type parameters w=1, 0, and 1 (from left to right).

The arrows within the squirmer rods indicate the swimming direction e

ˆ.

The parameter wA{1, 1} can be used to smoothly vary the type of the

squirmer rod from pusher to neutral to puller. (b) Force-dipole coeﬃcient

(w) (blue) and active velocity v(w)/v(0) (red) as a function of the swimmer

parameter w. (c) Schematic state diagram for neutral squirmer rods adapted

from ref. 94.

Soft Matter Paper

potential

103

that acts between two squirmers of different rods.

To ensure that there is no significant overlap between two

squirmer rods, we choose a strong force constant e

WCA

E10

The squirmer rods move because they acquire momentum

from the surrounding fluid. In the MPCD streaming step, linear

and angular momentum is transferred to the squirmer rod by

collisions with the fluid particles. We achieve this by applying

the so-called bounce back rule, which we also modify accord-

ingly to implement the slip velocity in eqn (1) on the surface of

the squirmer rods.

100,101

Squirmer rods also contain so-called

‘‘ghost’’ particles, which improve the implementation of no-slip

boundary conditions.

101

During the MPCD collision step, they

exchange momentum with the fluid, which is ascribed to the

squirmer rod. Lastly, the equations of motion for the squirmer

rods are integrated with a refined time step of dt

=Dt/3

using a symplectic splitting algorithm for rigid body molecular

dynamics.

104

For our simulations we use two diﬀerent geometries.

To determine how the force dipole strength A

depends on

the swimmer-type parameter w, we use a cubic box of linear size

L= 100a

with periodic boundary conditions along all spatial

directions. In this case we use the time 10

Dtto equilibrate the

MPCD fluid flow fields and then average the fluctuating flow

fields over additional 5 10

Dttime steps during simulations.

To simulate the collective dynamics of the squirmer rods, we

consider the quasi two-dimensional geometry of a Hele-Shaw

cell of linear size L= 300a

in the xand ydirection. Along the

zdirection the system is confined by walls separated by a

distance Dz=6Rto mimic the experimental setups using

microfluidic chambers

51,105

or liquid–oil interfaces.

106,107

It has

also been used in previous work.

94,97

Additionally, this realization

guarantees a strong contribution of the force-dipole interaction

in the near field, as we will show in Section 3.2 and Fig. 2. In the

xand ydirection periodic boundary conditions are employed.

In this geometry, the Nsquirmer rods are confined to only

move in the midplane of the Hele-Shaw cell by a strong

harmonic potential. The rods’ initial positions are generated

randomly for area fractions f=NA

o0.6, where A

is the

two-dimensional cross section of one squirmer rod. For fZ0.6

squirmer rods are placed on a rectangular lattice, all with a

randomly chosen orientation either parallel or anti-parallel to

one of the major axis of the unit cell. We simulate for a time of

Dtwhile saving snapshots every 2500Dtfor further analysis.

To ensure that the system is equilibrated, we omit the first

100 snapshots from the analysis.

To improve statistics in the study of the emergent turbulent

patterns of pusher-type squirmer rods, we perform two addi-

tional simulation runs for all turbulent states and their neigh-

boring points in the (a,f) parameter space. Furthermore, for all

these cases we also perform three simulation runs with an

increased system size of L= 600a

to investigate finite-size eﬀects.

3 Results

In the following we report on our simulation results. First, we

show that the anisotropy parameter wof the surface slip-velocity

field is directly proportional to the strength of the hydro-

dynamic force-dipole field and we illustrate the flow field of a

single pusher-type squirmer rod in the Hele-Shaw cell. Then, we

thoroughly discuss the state diagrams of the strongest pusher

rod (w=1) and the strongest puller rod (w= 1) depending on

aspect ratio aand area fraction f. We describe the diﬀerent

states using the velocity pair-correlation function, the power

spectral density of the velocity fluctuations, and the orienta-

tional autocorrelation function. Finally, for a specific aspect

ratio a, we show the state diagram in the space of wversus f.

3.1 Variation of the swimmer-type parameter v

To extract the force-dipole coeﬃcient A

from the flow field of

the squirmer rod in the 3D bulk fluid, we follow the method

described in detail in our previous article ref. 76. To do so, we

consider the expansion of an axisymmetric flow field into a

series of hydrodynamic multipoles u(r)=u

(r)+u

(r)+

(r)+..., where the leading-order multipoles are the force

dipole, source dipole, and force quadrupole, respectively. The

radial velocity component with the general form urðr;yÞ¼

n¼1

AnrnþBnrn2



Pnðcos yÞis measured from the simula-

tions and then projected on the second Legendre polynomial

(cosy). From the resulting polynomial 5

2Ðp

0urðr;yÞP2ðcos yÞ

sin ydy¼A2r2þB2r4, we determine the force-dipole

coeﬃcient A

by a polynomial fit in r

1

Indeed, we find a linear relation of the swimmer-type para-

meter wand the force-dipole coeﬃcient wBA

[Fig. 1(b)]. This

is expected since the terminal values of the envelope function

f(x) in eqn (2), which determine the strength of the force

dipole, are linear in w. Furthermore, we find that due to the

definition of the envelope function with hf(x)i= 1, the swim-

ming velocity is nearly independent of w[cf. Fig. 1(b), red curve].

3.2 Force-dipole flow fields in the Hele-Shaw geometry

As already explained, our study of the collective dynamics of the

squirmer rods is performed in a Hele-Shaw cell, which alters

Fig. 2 Hydrodynamic flow field around a pusher-type squirmer rod

(w=1.0) swimming in the Hele-Shaw geometry with cell height Dz=6R.

(a) Flow field in the mid plane of the Hele-Shaw cell. The force dipole clearly

dominates the near field. (b) Radial components u

r,n

(r)oftheleadingsource-

dipole (blue) and force-dipole (red) flow fields. They are normalized by the

thermal velocity of fluid particles, vth ¼ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

kBT=m0

Paper Soft Matter

the radial decay of the multipole far fields but also their

strengths depends on the cell height Dz.

76,99

Thus, the multi-

pole expansion for the radial component of the flow field

becomes ~

urðr;jÞ¼P

n¼1ðAnþBnÞrðnþ1ÞcosðnjÞfor a micro-

swimmer oriented along the x-axis.

Force multipoles dissipate

momentum at the bounding walls, which increases for smaller

Dz,i.e., when they are closer to the walls.

99,108

In our notation

this means that the coeﬃcient of the force dipole in the

Hele-Shaw cell scales as A

Dzwith respect to the bulk

coeﬃcient A

. In contrast, the fluid mass flux initiated by a

point source in the Hele-Shaw geometry is distributed in a

volume that is proportional to the cell height Dz. Hence, the

coeﬃcient of the source dipole, as all the other source multi-

poles, scales as B

/Dz.

76,99

Fig. 2(a) shows the flow field of a pusher-type squirmer rod

(w=1.0, a= 3.25) swimming in Hele-Shaw geometry with wall

distance Dz=6R. The force-dipole field visibly dominates the

flow field with its characteristic outwards directed streamlines

along the rod and inwards directed streamlines at the side.

From the simulated flow field, we extracted the radial parts

of diﬀerent hydrodynamic multipoles u

r,n

(r) following our

previous work,

and arrived at the curves shown in Fig. 2(b).

The radial part of the force dipole Br

3

(red symbols) domi-

nates the flow field up to a distance of circa 8l

, where it is

exceeded by the more long-ranged field of the source dipole

2

(blue symbols). For smaller cell heights Dz, this crossover

occurs at smaller distances. Since we are interested in exploring

the eﬀect of the hydrodynamic force dipole, we keep Dz=6Rfor

the rest of this work, such that the force dipole dominates the

flow field close to the squirmer rod.

3.3 Pushers-type squirmer rods

3.3.1 State diagram. For the minimum dipole strength

w=1, i.e., the pusher-type squirmer rods with the largest

force dipole, we study the collective dynamical states as a

function of the aspect ratio aand area fraction f.Acomparison

between the state diagrams of neutral squirmer rods [cf. Fig. 1(c)]

and pusher rods [cf. Fig. 3(a)] already illustrates the strong

influence of the long-ranged hydrodynamic dipole–dipole inter-

actions. Compared to neutral squirmer rods the transition line

from the disordered to other states is shifted to larger area

fractions and new dynamic states arise such as active turbulence

and dynamic clustering. We provide an overview of the observed

dynamic states before presenting a more quantitative analysis in

Section 3.3.2.

Most interestingly, compared to neutral squirmer rods the

flow field of the pusher rod obviously suppresses the formation

of dynamic swarms and impedes the single swarm state, which

only occurs for large aZ4.0. This is in agreement with findings

of Saintillan and Shelley,

who employ slender-body theory to

show that polar and nematic order in systems of elongated

pusher microswimmers is destroyed by their hydrodynamic

flow fields. Likewise, explicit hydrodynamic simulations of

collective dynamics of spherical squirmers show that pushers

create disordered homogeneous systems,

13,109

while pullers

show swarming.

109

An argument for this diﬀerence in the

collective dynamic behavior is found in both implicit

and

explicit

hydrodynamic simulations, which show that pushers

are deflected during collision, while pullers align.

Instead of the suppressed swarming states for ar3.25,

we find a very dynamic or turbulent state. A typical snapshot of

a rod configuration is shown in Fig. 4(a), a video is provided

in the Video 1 (ESI†). Here, the hydrodynamic dipole–dipole

interactions compete with steric interactions that favor the

formation of single swarms at large area fraction fin the case

of neutral rods.

Towards lower fthe turbulent state transi-

tions to the disordered state and towards higher aspect ratio a

to cluster or swarm states.

Similar to other examples of active turbulence in theory and

experiments,

47,49–51,97

we find two cascades in the power spec-

tral density of the squirmer velocities, as we will show in detail

in Section 3.3.2. For the turbulent state at a= 3.25 and f= 0.6

we construct a continuous velocity field v(x,y) to visualize the

turbulent flow pattern and its vortices. A snapshot of the system

and the resulting flow field are shown in Fig. 4(a) and (b),

respectively. We also calculate the vorticity o=(rv)

, which

is shown in Fig. 4(c). To easily obey the periodic boundary

conditions, the vorticity was determined via a Fourier trans-

formation. In all turbulent states, we observe that squirmer

rods show a local alignment, which extends over short streaks,

where squirmer rods swim side by side and head to tail.

However, these streaks buckle and dissolve frequently, leaving

a chaotic pattern of streaks and vortices.

Fig. 3 (a) State diagram of the strongest pusher-type squirmer rods with

w=1 in the parameter space aspect ratio aversus area fraction f.

(b) Snapshot of a single swarm state at a= 4.75 and f= 0.6, (c) snapshot of

a dynamic cluster state at a= 4.0 and f= 0.7 as indicated in the state

diagram. The color of the individual squirmer rods encode their orientation

in the xy-plane.

Soft Matter Paper

For large aspect ratio aZ4.0 we still find the single swarm

state [Fig. 3(b) and Video 2, ESI†], where a stable cluster moves

through the system, and also a jammed cluster state. Since they

are also found for neutral squirmer rods,

we deduce that they

mainly arise from steric interactions. Indeed, they are also

found for dry active rods.

For aspect ratios below ao4.75

and large area fractions clusters still form, but they deform

dynamically while squirmer rods join or leave constantly

[Fig. 3(c) and Video 3, ESI†]. However, the power spectral

densities of this dynamic cluster state lacks the characteristic

scaling behavior of the turbulent state, as we will show in

Section 3.3.2. The absence of two power-law regimes (energy

cascades) in the power spectra separated by a maximum, which

defines a characteristic length scale, is our main criterion to

distinguish the dynamic cluster state from the turbulent state.

Hence, this state is intermediate between the jammed cluster

and turbulent state. In the jammed cluster state observed for

a= 4.75 and f= 0.77 the average swimmer velocity is low,

e

iE0.18v

, and rises to hv

e

iE0.4v

for the aspect ratio

ar4 in the dynamic cluster state.

As already mentioned, the disordered state extends to higher

densities compared to neutral squirmer rods. Here, velocity

correlations between squirmer rods are short-ranged and the

power spectral density decays without any algebraic behavior as

we will show in Section 3.3.2. In other words, no patterns

emerge because energy is not transported to larger scales.

All together, our results imply that the pusher-type flow

fields inhibit or destabilize the steric alignment of swimmers

that has been found for neutral squirmer rods. In intermediate

regions of the state diagram, the competition of both eﬀects

lead to new dynamic states.

3.3.2 Velocity pair correlations and power spectral density.

To investigate and classify the emergent dynamic states of

pushers-type squirmer rods, we employ the velocity pair-

correlation function C

(R) and the power spectral density of

velocity fluctuations, E(k). While the velocity pair correlation

allows us to examine emerging patterns, the power spectral

density quantifies the distribution of kinetic energy over

diﬀerent length scales 1/k, which we will use to classify

turbulent dynamics. To calculate the velocity pair-correlation

function,

CvðRÞ¼ P

iaj

vivjdðjrijjRÞ

iaj

dðjrijjRÞ

;(3)

we use a histogram for the scalar product v

v

with distance R

between rods iand j, assuming our systems are isotropic,

i.e.,C

(R)=C

(R). Here, hi means an average over time. The

power spectral density is formally defined as h|v

˜(k)|

i, where

˜(k) is the Fourier transform of the velocity field of the squirmer

rods. According to the Wiener–Khinchin theorem, the power

spectral density is related to the Fourier transform of the

velocity pair-correlation function, h|v

˜(k)|

i=F(C

)(k). Due to

the isotropy of the velocity fluctuations, we introduce the

spectrum as a function of wave number kfollowing ref. 51

and 110, EðkÞ¼ k

2pj~

vðkÞj2



. Then, using the zeroth-order Bessel

function of the first kind, we arrive at

EðkÞ¼ k

2pðL2

CvðRÞeikRd2R¼kðRmax

CvðRÞJ0ðkRÞRdR;(4)

where R

max

=Lis the system size.

111

Since for small R,C

(R)

only exists for rods positioned side by side and thereby is highly

anisotropic, we shift C

(R) by a length l

/2 to smaller distances

Rwhen calculating E(k), similar to ref. 97.

In Fig. 4 we compare the normalized velocity pair-correlation

functions C

(R)=C

(R)/hv

iof pusher-type squirmer rods for

aspect ratio a= 3.25 at diﬀerent area fractions f. A more

complex behavior is observed for an area fraction in the

range fA[0.4, 0.77], which allows for competing steric and

hydrodynamic interactions and for which we show C

(R)

in Fig. 5. The respective states are characterized as disordered

(f= 0.4/0.5), turbulent (f= 0.6/0.7), and dynamic cluster

f= 0.77, as we detail also further below. Generally, for larger f,

Fig. 4 Turbulent state of pusher-type squirmer rods with dipole strength w=1, an aspect ratio a= 3.25, and for an area fraction of f= 0.6. (a) Snapshot

of the system with individual squirmer rods. Their orientation angles jin the x/y-plane are color-coded to help identifying small groups with the same

orientation. (b) Streamlines of the velocity field v(x,y) of the squirmer rods constructed from the snapshot in (a) with color-coded swimming velocity

v=|v(x,y)|. To construct the velocity field, each rod is represented by an elliptic Gaussian function with the standard deviations s

and s

matching the

shape of the squirmer rods. Each point on a regular grid is then assigned the average velocity of the surrounding rods weighted with the Gaussians.

Paper Soft Matter

the correlations extend to larger distances, which indicates

that the size of ordered patterns increases with density.

Distinctive for pushers we find that C

(R) is not purely positive

but also shows distances where it becomes negative. In more

detail, we observe the following characteristics. For distances,

where steric interactions occur and dominate, Rol

(R)is

positive. With increasing Rit decays to zero around R=l

R=3l

and becomes negative for area fractions fup to 0.7.

In the turbulent states for f= 0.6 and 0.7, C

(R) returns to

positive values at more than twice the first zero-crossing dis-

tance [inset of Fig. 5]. For f= 0.6 we even observe a third zero

crossing. In the disordered states for f= 0.4 and 0.5C

(R) does

not show a second zero crossing but approaches zero from the

negative region. Lastly, at f= 0.77 the system is in the dynamic

cluster state with densely packed squirmer rods. This is in

agreement with the pair-correlation function, which has the

longest range and only exhibits anti-correlations for R46l

However, we find that the orientational autocorrelation func-

tion he

ˆ(t)e

ˆ(t+dt)i(not shown) decays after the cluster moved a

distance of 3–5l

, considering its mean velocity of hv

e

iE0.4v

for a= 4.0. This clearly indicates that the clusters are dynamic.

For other aspect ratios aonly small quantitative changes occur.

Especially, the correlation functions for the turbulent and

disordered states are representative for all aand f.

The power spectral densities E(k) calculated from C

(R) are

in accordance with the findings so far. In Fig. 6(a) we compare

the power spectral densities E(k) for three diﬀerent aspect ratios

a= 1.75, 2.5, and 3.25 in dense systems. The velocity autocorrela-

tion functions C

(R) show the same oscillating decay as already

observed above for a=3.25andf= 0.6. We called these states

active turbulence since the power spectral density E(k)showsa

broad peak, which separates two regions from each other with

power-lawdecaystowardslowandhighk.Thisischaracteristicof

active turbulence. Close to the characteristic wave number k

=2p/l

where energy is inserted from the active motion of the squirmer

rods, E(k) decays to zero. The maxima of the power spectral density

all roughly occur at 0.2k

, indicating that the pattern size scales

with l

, which makes sense when the density is roughly the same.

As reported in other experimental

and theoretical

studies

on polar active fluids, the scaling exponents for the power laws

in kare not universal. Towards smaller length scales (larger k),

the exponent shows only a weak dependence on the aspect ratio

awith values of 1.1 and 1.3, while towards larger length

scales we observe a stronger variation of the scaling exponent

with aspect ratio. For short squirmer rods with a= 1.75, the

power spectral density decays more rapidly with E(k)Bk

1.9

and

for a larger aspect ratio a= 3.25, the power spectral density

scales with E(k)Bk

. This indicates that for larger aspect ratio

patterns at large length scales (small k) are observed more

frequently.

For disordered states the power spectra do not follow power

laws [cf. Fig. 6(b)]. Instead, they quickly decay towards smaller

k, which indicates again that energy is not transported to larger

length scales. In contrast, for the dynamic cluster state energy

is concentrated at the larger length scales (small k) as expected,

while E(k) decays strongly towards smaller length scales

[cf. Fig. 6(c)]. To investigate finite size eﬀects, we compare the

regular systems of size L= 300a

to systems with the increased

system size of L= 600a

performed for a= 4 and f= 0.7. In the

bigger system the maximum of E(k) shifts to smaller values

of kroughly proportional to the change in Lwhich suggests

that the maximum correlation length always corresponds to the

system size, and no characteristic length scale emerges. At the

same time, we observe a decrease of max[E(k)] indicating

that the dynamic of the cluster slows down as Lincreases.

Fig. 5 Normalized velocity pair-correlation function C

(R) as a function of

the rod distance Rfor pusher-type squirmer rods (w=1) with aspect ratio

a= 3.25 at diﬀerent area fractions f.Inset: Plot of |C

(R)| versus R.

Fig. 6 (a) Power spectral density E(k) in the turbulent state calculated

from the velocity pair-correlation function C

(R). Light and dark colors

correspond to small and large system sizes L, respectively. (b) Power

spectral density E(k) in the disordered state and (c) in the dynamic cluster

state.

Soft Matter Paper

This behavior clearly distinguishes the dynamic cluster state

from active turbulence, which has an intrinsic characteristic

length scale. However, we note that data for larger systems

would be necessary to show that the maximum correlation

length 1/argmax[E(k)] diverges for infinite system size. Though,

this currently exceeds our computational capabilities. Note, the

irregular behavior in the curve for L= 300a

is due to the fact

that with increasing aclose to the jammed cluster state the

dynamics of the clusters slows down and longer simulation

times are needed to smooth the curve.

3.4 Puller-type squirmer rods

3.4.1 State diagram. For strong puller-type squirmer rods

with w= 1, we also investigate the system for varying aspect ratio

aand area fraction fand generate the state diagram as shown

in Fig. 7(a). Here, we observe a completely diﬀerent behavior

compared to the pusher-type squirmer rods. Only for dense

systems and at large aspect ratios, where steric interactions

dominate, we observe the same single swarm/jammed cluster

states for pusher and puller rods. Thus, the behavior resembles

that of dry self-propelled rods,

although diﬀerences for

pushers and pullers still remain. Otherwise, the state diagram

is more similar to that of the neutral squirmer rods [Fig. 1(c)].

Only the disordered state is completely suppressed and replaced

by the swarming state. This also means that the occurrence of all

states are shifted towards smaller area fractions f.

The most prominent feature of puller-type squirmer rods is

that their flow fields strongly promote the formation of small

swarms. Even for dilute systems with area fractions down to 0.2,

the hydrodynamic interactions between the squirmer rods

favor small swarms against the disordered state. Diﬀerent from

the purely steric alignment responsible for swarming neutral

squirmer rods,

pairs of puller-type squirmer rods form

T-shaped configurations, although deformed, when they col-

lide. Examples are indicated in the snapshot of Fig. 7(b). This

leads to small groups, where several squirmer rods follow a

central leader of the group, arranged in a ‘‘comet tail’’ and

pointing inwards [Fig. 7(b)]. A similar behavior is also found in

ref. 95 and for dry active rods.

However, we observe that for

the hydrodynamically interacting squirmer rods the swarming

behavior extends to smaller densities and aspect ratios com-

pared to dry self-propelled rods.

61,64

Often the swarms have an

asymmetric shape, which then induces curved trajectories

(Video 4, ESI†). With increasing area fraction, fZ0.4, swarms

also increase in size. However, the overall alignment in the

swarms rather decreases. This is due to the low polarity of the

small swarms, which then merge in an unordered fashion with

increasing f.

For aspect ratios aZ3 we also observe the formation of a

single swarm, which collects all squirmer rods in the system

[Fig. 7(d) and Video 5, ESI†]. For area fractions fZ0.6 and

suﬃciently large aspect ratios the single swarm becomes

jammed and forms a static cluster state [Fig. 7(e) and Video 6,

ESI†]. Reducing the aspect ratio at f=0.77toar2.5, the jammed

cluster becomes dynamic similar to the state observed in pusher-

type squirmer rods [Fig. 7(c) and Video 7, ESI†].

3.4.2 Velocity pair correlations. For a quantitative study of

the emergent states of the puller-type squirmer rods, we use

again the velocity pair-correlation function. In Fig. 8(a) we

compare results for the representative aspect ratio a= 4, which

exhibits most of the emergent states. We immediately notice

the diﬀerence to the pusher-type squirmer rods; C

(R) does not

become negative. This underlines the tendency of pullers to

form two swarming states. To thoroughly distinguish these

states, which both give very similar velocity correlation functions,

we will further examine the orientational autocorrelations in

Section 3.4.3.

We have already seen, that even in dilute systems with

f= 0.2, the puller-type flow fields locally align the squirmer

rods and thereby cause swarming. This is also visible in the

velocity correlation function C

(R), which is more long-ranged

compared to pusher-type rods [Fig. 5(a)]. Thus, squirmer rods

are significantly correlated for distances up to R=2l

.As

described in Section 3.4.1, the size of swarms increases with

density. For the subsequent area fraction of f= 0.4 the system

is already in the single swarm state at the chosen aspect ratio of

4.0. Hence, the velocity correlation function C

(R) becomes

Fig. 7 (a) State diagram of the puller-type squirmer rods with w= 1 in the

parameter space aspect ratio aversus area fraction f. Snapshots of the

system in (b) the swarming state at f= 0.2 and a= 2.5 (only a part of

the system is shown), (c) the dynamic cluster state at f= 0.77 and a= 2.5,

(d) the single swarm state at f= 0.4 and a= 4.0, and (e) the jammed cluster

state at f= 0.77 and a= 4.0.

Paper Soft Matter

more long-ranged. At f= 0.6 and f= 0.77, when the system is

noticeably in the jammed cluster state, we observe two diﬀerent

features. For f= 0.6, C

(R) reaches a plateau, which indicates

that the jammed cluster drifts in one direction since the

squirmer rods are aligned, on average. At the highest density

f= 0.77 the cluster formation takes place more quickly result-

ing in a disordered structure without drift. Here, the velocity

correlation function C

(R) decreases and does not exhibit a

plateau until R/l

=5.

3.4.3 Orientational autocorrelations. To further characterize

and distinguish the single swarm, dynamic cluster, and jammed

cluster states from each other, we employ the orientational

autocorrelation function C

(dt)=he

(t)e

(t+dt)i.InFig.8(b)we

compare C

(dt) for diﬀerent area fractions fat constant aspect

ratio a= 4, and in Fig. 8(c) for diﬀerent aat constant f= 0.77.

In dilute systems with f= 0.2, C

(dt) decays to zero at around

dt=15l

, meaning that swarms swim a distance of around 7l

until they either rotate, merge with other clusters or dissolve

(Video 8, ESI†). For the single swarm state at f= 0.4, we observe

a more long-ranged autocorrelation. After a swarm has formed

it remains stable but also goes through configurational changes

so that partially straight motion is observed interrupted by

rotations or turns of the swarm (Video 5, ESI†). As a result,

(dt) exhibits a damped oscillation, where the first minimum

at dtE37l

means that the swarm has rotated by 1801. In the

jammed cluster state, we again observe two diﬀerent features.

For f= 0.6, the cluster drifts due to its overall polar order.

However, also configurational changes occur, which results in a

decay of C

(dt) towards a non-zero value meaning that the

cluster has reached some stable configuration within the

observation window. At the higher density f= 0.77, the cluster

now has a more random but static arrangement of the rods.

Therefore, it hardly drifts and it only rotates very slowly, which

causes a very slow decrease of C

(dt).

Comparing dense systems (f= 0.77) of diﬀerent aspect

ratios to each other, we observe the following characteristics

from high to low a[Fig. 8(c)]. For the largest aspect ratio a= 4.75

steric interactions between the rods are most dominant and we

find an entirely jammed configuration with constant C

(dt)=1.

Lowering the aspect ratio, the jammed cluster becomes more

and more interrupted by occasional configurational changes,

which results in the noticeable but slow decay of C

(dt) for

a= 3.25. In contrast, the dynamic cluster states at a= 1.75 and

a= 2.5 show a fast exponential decay of C

(dt) due to the

dynamic rearrangements within the cluster. The decay is faster

for the lower aspect ratio, where the rods reorient more easily.

3.5 Variation from pusher to puller-type squirmer rod

In this section, we compare squirmer rods at a constant aspect

ration a=3.25andvarytheswimmertypebytheforce-dipole

strength w. Furthermore, we performed simulations at diﬀerent

area fractions f. The resulting state diagram is shown in Fig. 9(a).

States of neutral squirmer rods are adapted according to ref. 94.

For puller-type squirmer rods with w40, we observe that

already weak pullers with w= 0.25 show swarming at the small

area fraction of f= 0.2, i. e., the same state we observed in

Section 3.4.1 for the highest strength w= 1.0. Obviously, the

force-dipole field, which attracts nearby rods along the rod axis,

promotes swarming. However, with increasing f, swarming

also extends to weak pusher rods. At higher area fractions

fZ0.6, we find single swarm and jammed cluster states due to

steric interactions between the rods, where neutral (w=0)and

puller-type squirmer rods show qualitatively similar behavior.

For pusher-type squirmer rods, wo0, we observe a more

diverse behavior, which we already noted for w= 1 in Sec-

tion 3.3.1. While at f= 0.2 only the disordered state occurs,

at f= 0.4 we find a transition from swarming to turbulent to

disordered state with increasing pusher strength. The analysis

of the power spectral density E(k) reveals that the system with

w=0.5 is indeed in the turbulent state. Thus, the pusher flow

field destabilizes swarming clusters. As a compromise between

swarming and disordered states, active turbulence occurs in

conjunction with steric repulsion. Increasing density further to

f= 0.6, steric repulsion stabilizes swarming clusters and active

turbulence is shifted to the largest pusher strength w=1.

Finally, at f= 0.77 the pusher flow field destabilizes the

Fig. 8 (a) Normalized velocity pair-correlation function C

(R) plotted

versus distance Rfor puller-type squirmer rods at constant aspect ratio

of a= 4.0. (b) Orientational autocorrelation function C

(dt) for constant

aspect ratio a= 4.0. (c) Orientational autocorrelation function C

(dt)in

dense systems at area fraction f= 0.77.

Soft Matter Paper

jammed cluster of neutral and puller-type squirmer rods, which

then becomes dynamic.

3.5.1 Velocity pair correlations and power spectral density.

For a more quantitative analysis, we use the normalized velocity

pair-correlation function C

(R) as defined in eqn (3). In Fig. 9(b)

we compare C

(R) in dilute systems with f= 0.2 for diﬀerent

force-dipole strengths wto each other. The collective dynamics

is mainly governed by hydrodynamic interactions. Fig. 9(c)

shows C

(R) for the larger area fraction f= 0.6, where a

transition to the turbulent state occurs at w=1.

In the dilute systems, all the velocity pair-correlation func-

tions in the swarming state of puller-type swimmers (w40) are

nearly identical and show correlations up to circa 2l

[Fig. 9(b)].

This indicates that the mechanism behind swarm formation is

fully established already at small dipole strength. Pusher-type

squirmer rods (wo0) only exhibit the disordered state and no

swarms, so C

(R) is more short-ranged compared to pullers.

Pushers prefer to order side by side and therefore pronounced

peaks at closest distance R=l

/aare observed compared to

puller rods. Furthermore, we observe anti-correlations with

(R)o0 starting at distances Rbetween l

and 2l

. They

belong to pusher rods approaching each other head on.

At the higher area fraction f= 0.6, we observe the single

swarm, swarming, and turbulent states [Fig. 9(c)]. In the single

swarm state observed for weak pullers (w= 0.25), C

(R) is long-

ranged and shows significant correlations for all recorded

distances up to R=5l

. In the swarming state that emerges

for pushers at w=0.25 and 0.5 the correlation length of C

(R)

decreases with increasing force-dipole strength. At w=0.5 it is

around R=2l

. In the turbulent state at w=1, C

(R) becomes

even more short-ranged and negative at R=l

due to the

occurrence of vortices.

In Fig. 9(d) we compare the power spectral densities E(k) for

three systems in the turbulent state at a= 3.25. All three spectra

show a regular cascade towards large k(small scales) and a

second cascade towards small k(large scales). At large kall

three power spectral densities show roughly the same scaling

behavior E(k)Bk

1.3

. Furthermore, the curves for (w,f)=

(0.5, 0.4) and (w,f)=(1.0, 0.6) coincide in the range from

k= 0.2k

to k

, which includes the maximum of both spectra.

For the second cascade, diﬀerent scaling exponents are

observed. At w=1 the power spectral density decays more

rapidly in the system with f= 0.6 following a scaling E(k)Bk

while in the denser system (f= 0.7) the scaling is E(k)Bk

Furthermore, the maximum of E(k) shifts to a smaller wave

number k, which implies a larger intrinsic length scale of the

turbulent pattern. For the system with the smaller pusher

strength, w=0.5, the scaling follows E(k)Bk

1.4

, thus, it is

situated between the two cases just discussed.

4 Summary and conclusion

In this article, we investigated the dynamic states of pusher and

puller-type squirmer rods. Varying the head-to-tail anisotropy

parameter wof the surface slip-velocity field of the squirmer

rod, we are able to smoothly tune the force-dipole strength of

the resulting flow field between a pusher and puller.

Fig. 9 (a) State diagram of the squirmer rods for diﬀerent dipole-

strengths wand area fraction fat an aspect ratio of a= 3.25. The labeled

vertical lines indicate the states for which the velocity correlation function

is shown in (b and c). (b) Normalized velocity pair-correlation function

(R) for dilute systems with f= 0.2 and diﬀerent force-dipole strengths w.

Red curves belong to systems in the swarming state and green curves to

the disordered state. (c) Normalized velocity pair-correlation function

(R)forf= 0.6. The black curve belongs to the system in the single

swarm state, green to the swarming state, and yellow to the turbulent case

(d) Power spectral density E(k) for the turbulent state at diﬀerent w,f.

Paper Soft Matter

For pushers with largest force-dipole strength w=1, we

observe a new turbulent state along with other modifications in

the state diagram compared to neutral squirmer rods. In dilute

systems, where steric interactions are less important, the

pusher flow field suppresses the formation of the swarming

states so that the disordered state extends to larger area frac-

tions and aspect ratios. When steric interactions and thereby

steric alignment of the rods become more important with

increasing area fraction, the turbulent state emerges at small

aspect ratios. To observe it for weaker pusher strengths at

constant aspect ratio, one has to decrease the area fraction,

which confirms the importance of steric interactions for the

turbulent state. The velocity pair-correlation function decays and

exhibits negative regions after one or two swimmer lengths,

which indicates a characteristic length scale. At the same time,

the power spectral density of the velocity fluctuations shows

two energy cascades at small and large wave numbers with

power-law scaling and non-universal exponents as reported

in other works on active turbulence in polar fluids.

49,50,97

Increasing the area fraction further, a transition to the dynamic

cluster state occurs at medium aspect ratios, which is reminis-

cent of the turbulent state but without the characteristic energy

cascades. This is in contrast to soft active rods, where jammed

states occur,

42,63

which are, however, destroyed for strong

enough self-propulsion.

112

At larger aspect ratios, steric inter-

actions become more important. Here, pushers resemble the

dynamics of neutral squirmer rods

and dry active rods,

61,62

such that, instead of the turbulent and dynamic swarm state, a

single swarm and jammed cluster are observed.

For pullers at all studied force-dipole strengths wand in the

most dilute system at f= 0.2, hydrodynamic interactions

already promote the swarming state instead of the disordered

state observed for neutral rods. At smaller aspect ratios and the

largest area fraction, the dynamic cluster state occurs, while

for larger aspect ratios swarming and dynamic clustering are

replaced by a single swarm and jammed cluster state.

Thus, our study clearly indicates the importance of the type

of the swimmer flow field for the occurring states. Although

comparable states are often found for dry self-propelled rods,

for squirmer rods their occurrence in the state diagram cru-

cially depends on the specific hydrodynamic flow field. Speci-

fically, active turbulence is only found for pushers and puller

type flow fields greatly enhance the formations of swarms.

Additionally, the diﬀerent scaling exponents in the turbulent

state show that steric interactions, tuned by rod density, play an

important role for this state. All in all, the turbulent state

occurs as a compromise between disordering hydrodynamic

pusher flow fields and aligning steric interactions. Therefore,

with our work we contribute to the insights how various bio-

logical propulsion strategies determine the collective motion of

microswimmers.

In future work, it will be interesting to investigate collective

dynamics of pusher and puller-type squirmer rods also in the

bulk fluid, where the hydrodynamic force-dipole has the most

long-range or dominant flow field. Likewise, implementing a

liquid–air interface using a slip-boundary condition at one wall

of the Hele-Shaw geometry, would provide a realistic modeling

of recent experiments, where bacteria move close to a fluid–air

interface.

113

Conflicts of interest

There are no conflicts to declare.

Acknowledgements

We thank Felix Ru

¨hle, Reinier van Buel and Josua Grawitter for

helpful discussions on the topic of the manuscript. We also

acknowledge financial support from the Collaborative Research

Center 910 funded by Deutsche Forschungsgemeinschaft.

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Soft Matter Paper

CHAPTER 5

Conclusions and Outlook

Conclusions

Biological microswimmers exist in various shapes and employ different propulsion

strategies to explore their environment. To model the particle shape of microswim-

mers, which is mostly elongated, we have introduced the squirmer-rod model. The

squirmer rod consists of several spherical squirmers arranged on an straight line

to form a rigid body. The surface slip-velocity of the squirmer thereby models the

ciliary propulsion found for organisms such as Paramecium. To model also pusher

and puller-type microswimmers, the surface slip velocity of the individual squirmers

is modified such that the surface velocity is concentrated to either the back or the

front of the rod. These realizations serve as model for E. coli bacteria and C. reinhardtii

algae, respectively. To model the fluid surrounding the squirmer rods, we use the

method of multi-particle collision dynamics (MPCD). This method solves the full

Navier-Stokes equation including thermal fluctuations.

Performing simulations in the bulk fluid, we find that the neutral squirmer rod

mainly results in the addition of a force quadrupole moment to the basic flow field

of the spherical squirmer, which is a source dipole. The force quadrupole possesses

the same radial decay

|vFQ| „ r´3

such that the overall radial decay of the flow field

remains unchanged. For pusher-type squirmer rods we observe the additional flow

field of a force dipole due to concentrating the surface velocity to the back of the

rod. Consequently, the overall flow field of the pusher-type squirmer rod decays as

|v| „ r´2in leading order.

In addition, we considered the Hele-Shaw geometry in which the fluid and mi-

croswimmer is confined in a narrow slit between to parallel plates. In this geometry

the hydrodynamic multipoles are crucially altered and as a result, the flow fields

show different characteristics. Most importantly the radial decay of the different mul-

tipoles is altered such that the source dipole, which now exists for all self-propelled

109

5. Conclusions and Outlook

particles becomes the most long-ranged multipole field,

|vSD| „ ρ´2

. At short and

intermediate distances, however, we observe that the force dipole moment still

dominates. Moreover, the force and source multipoles are affected by the confine-

ment strength in an opposite way. Hence, the confinement strength introduces an

interesting variation between the two hydrodynamic multipoles.

To create a less compressible model fluid, we introduced an extended collision

rule for the MPCD method. The model generates an inherent momentum flux within

each collision cell that contributes to the pressure. We showed that this extended

rule results in a non-ideal equation of state and also derived analytic expressions

for the shear viscosity. We find that these predictions are in good agreement with

measurements obtained from equilibrium, constant shear, and Poiseuille flow sim-

ulations. Likewise, simulations of the flow fields of squirmers in the bulk fluid

and Hele-Shaw geometry show good agreement with the analytic prediction. In

simulations of the collective dynamics of squirmer rods, where the MPCD fluid

is subject to strong forces, the comparison of the traditional MPCD-AT+a and the

extended MPCD method shows a more homogeneous fluid density and viscosity in

the context of clustering. For swarming squirmer rods we observe that the extended

MPCD results in an increase in the size and velocity of emergent swarms compared

to the traditional MPCD method.

In an in-depth study of neutral squirmer rods, we investigate the influence of

their aspect ratio

and the density, given by the two-dimensional area fraction

We observe four different states going from small to large aspect ratio

. For short

squirmer rods and low density we observe a disordered state with no correlations

between the particles. At higher aspect ratio a state of dynamic swarms emerges,

which is characterized by the frequent formation and break up of medium sized

swarms. As the aspect ratio or density is further increased, the swarms become

stable and the systems exhibits a single swarm state. For very high density the

system becomes stationary and forms a jammed configuration.

The dynamic behavior becomes more diverse for pusher and puller-type squirmer

rods. Here, we observe a new turbulent state for the pusher-type squirmer rods,

as well as a dynamic clustering state, which is intermediate between turbulence

and clustering. In dilute systems, where hydrodynamic interactions dominate, the

pusher-type flow fields completely suppress the formation of swarms observed for

neutral squirmer rods. Instead, a disordered state is found also for higher aspect

ratio and density.

When steric interactions become more important at higher densities and aspect

ratios, the turbulence state emerges as a compromise between the aligning and

disordering interactions. In the turbulent state we observe a positive velocity pair-

correlation at short distances which becomes negative at the length scale of a few

110

swimmer lengths. Then, again, it becomes positive after a few swimmer lengths, then

negative again, and so on. The power spectral density of the velocity fluctuations

shows two energy cascades at small and large wave numbers with power law scaling

and non-universal exponents. When steric interactions dominate at high density

and aspect ratio, the clustering and single-swarm states of neutral squirmer rods are

recovered.

For puller-type squirmer rods we observe an increased tendency to form dynamic

swarms for all aspect ratios. Here, the dynamic swarming state occurs already for

very low densities, and the single-swarm state is shifted to lower densities and

aspect ratios.

Investigating the swimmer-types for intermediate values of the force-dipole

moment confirms our previous findings. Accordingly, also pushers rods exhibit

the turbulent state, which, however, disappears for large values of the force-dipole

strength. For pullers the tendency to form swarms sets in immediately when the

force dipole is introduced.

Overall, we conclude that our study of the squirmer rod-model clearly indicates

the importance of both the shape and hydrodynamic characteristic of the flow

fields for the dynamics of elongated microswimmers. Although studies of dry self-

propelled rods exhibit often similar states, our results clearly confirm previous work

that the occurrence of, for example, active turbulence or the swarming state crucially

depends on the type of hydrodynamic flow field [61]. As such, active turbulence is

a characteristic of pushers, while swarms prevail for puller-type microswimmers.

Furthermore, we observe that the scaling exponents, which characterize the turbulent

state, depend on the aspect ratio and density and thus the strength of the interactions.

This stresses again that the turbulent state emerges through a balance between

aligning steric and disordering pusher-type hydrodynamic interactions. In contrast,

we observe that neutral squirmer rods mainly exhibit swarming and jamming, which

is more similar to dry active rods. Therefore, with our work we contribute to the

understanding how biological propulsion strategies determine the collective motion

of microswimmers.

Outlook

Below, we discuss a number of extensions for the squirmer rod models, which

would be of interest for future studies. We have already implemented some of these

extensions, such that we can discuss some preliminary results.

Rotlet dipole

A notable extension of the pusher-type squirmer rod is to improve the model for

the actual propulsion mechanism of a rotating flagellum. For bacteria, the motor of

the flagellar apparatus creates a torque that rotates the flagellum. At the same time,

111

5. Conclusions and Outlook

the reverse torque rotates the cell body in the opposite direction. Resulting from

the opposing torques, the flow field possesses the additional flow field of the rotlet

dipole, as indicated in Sec. 2.2.5.1. This rotlet dipole results in circular trajectories

when microswimmers are close to solid walls or liquid-air interfaces, where the

orientation of rotation depends on the slip condition [234, 235].

To introduce the rotating flagellum in the squirmer-rod model, we add an az-

imuthal contribution to the surface slip velocity, which mimics the rotation of the

flagellum. Preliminary research, which has been performed by the bachelor student

Phillip Eisenhuth, showed that indeed circular trajectories emerge in the proximity

of solid walls or liquid-air interfaces. We assume that this circular motion will be

interesting in the context of the active turbulence that we observed for pusher-type

squirmer rods. In particular, we assume an interaction between the radius of the

circular trajectories of single squirmer rods and the characteristic vortex size of the

active turbulence. However, more detailed simulations are required to investigate

the statistics of the turbulent patters observed in the collective dynamics of these

chiral pusher-type squirmer rods.

Flexible filaments

Another interesting extension of the squirmer-rods model is the realization of flexible

filaments, as a model for polymer chains [43, 50], flagella [228], or elongated bacteria

[236, 237] or the nematode worm C. elegans [238]. Our approach thereby provides an

accurate description with explicit hydrodynamic interactions.

Similar work on filaments has been done using overdamped Langevin dynamics

and implicit hydrodynamic interactions via the Rotne-Prager approximation [239–

241], or inside the MPCD fluid without volume exclusion [242]. We have already

conducted preliminary work on this topic, which we will discuss in the following.

We begin with outlining the details of the flexible filament model and how it is

coupled to the fluid.

To implement flexible filaments, the rigid-body constraint of the squirmer rod is

partially released so that the constituting squirmers now move individually. In this

way, the configuration of squirmers can deform and thereby model the deformation

of a filament. To hold the squirmers in place and model the flexibility of filaments, we

introduce additional potential forces between the individual squirmers. To introduce

bending rigidity, we use the additional bending potential

Vbend(r1, ..., rNsq ) = ´ϵbend

Nsq´1

i=2[︂cos(θi)+θ12

i]︂(5.1)

where

θi=>(ri´1

ri+1)

is the angle between the connecting vectors in the triplets

of spherical squirmer, similar to Refs. [239, 240]. The additional

θ12

term is added

to avoids sharp angles above 90

in the configuration. For small

θďπ/

4 this term

112

0 100 200 300

x/a0

100

200

300

y/a0

(a)

0 100 200 300

x/a0

100

200

300

y/a0

(b)

Figure 5.1: Snapshots of two simulations of flexible filaments for different bending

energy

ϵbend

(a) For

a bending rigidity

ϵbend „

the filaments are rather soft and

deform when they touch each other due to their active swimming. The softness

thereby reduces the alignment due to steric interactions and hinders the formation

of swarms. Moreover, filaments are able to escape dense configuration due to their

flexibility. (b) For more stiff filaments with bending rigidity

ϵbend „

we see again

the formation of swarms.

is negligible, and thus mostly does not affect the configuration. For long filaments

the WCA potential of Eq. (3.17) is also applied between pairs of squirmers within

a filament, when the index difference is greater than 10, which guarantees that the

repulsion does not interfere with the bending potential. This avoids overlap of

the filament with itself in compact configurations. To hold the bond length at the

prescribed value, we use the constraint solver algorithm P-LINCS [243, 244], which

is known from the simulation of molecules, e.g., with the GROMACS package [245].

In other words, the P-LINCS algorithm keeps the distance between neighboring

squirmers inside the filaments constant. Masses and moment-of-inertia tensors of

the segments, which are required by P-LINCS, are taken from Ref. [125] presented in

Sec. 4.2.

To couple the motion of the MPCD fluid to the flexible filament, we apply the

momentum transfer of the squirmer

on the same squirmer. To apply the angular

momentum transfer of the squirmer

to the filament, we consider triplets that

include the two neighboring squirmers and are treated as rigid bodies during the

angular momentum transfer. Furthermore, the orientation vector of the squirmer

is set to

ˆi= (ri´1,i´ri,i+1)/|ri´1,i´ri,i+1|

, such that we may define the surface

slip-velocity of the spherical squirmer for every bead.

In Fig. 5.1 we compare two simulations of flexible filaments using different

bending energies

ϵbend

. In both systems of size

300

and thickness

∆z=

the filaments are constrained to move on in the center between the confining walls,

similar to [206]. Filaments consist of

Nsq =

40 squirmers of radius

2 and an

113

5. Conclusions and Outlook

aspect ratio of

α«

20. The area fraction is given by

ϕ=

0.6 and individual squirmers

propel with the typical slip-velocity parameter Bsq

1=0.1.

For the soft filaments shown in Fig. 5.1(a), the flexibility leads to strong shape

changes upon collisions of the flexible filaments. Therefore, filaments are rather

reoriented, than aligned, which suppresses the formation of swarms for this bending

energy

ϵbend

. Likewise, the flexibility reduces the formation of clusters because

filaments can escape dense regions in the system.

For the more rigid filaments shown in Fig. 5.1(b), we observe the formation of

swarms again. Here, the increased rigidity allows an alignment of the filaments

generally, while the deformations enables more aligned configurations and therefore

more efficient formation of swarms. However, the deformation of filaments within

swarms also leads to curved trajectories of the swarms.

Obstacles

In theory [106, 246] and experiments [174] with E. coli bacteria, it has been shown

that the emerging active turbulence can be stabilized in regular patterns through the

placement of obstacles in the domain of the microswimmers. It would be interesting

to simulate our microscopic squirmer-rod model interacting with such pillars, since

in the experiments microswimmers close to the pillars are not accessible for the

microscopic imaging technique. [106].

In this context we performed preliminary research which, however, predates our

latest articles including the investigation of active turbulence [185] and the more

incompressible MPCD method [205, 206]. It would therefore be interesting to revisit

this topic and explore pillars with structured surfaces, which prohibit the sliding

motion of the microswimmers around the pillars. In our preliminary simulations, we

immersed cylindrical pillars in the flow, which are realized with the WCA potential

and no-slip boundary condition for the fluid. We show two exemplary simulations

of pusher-type squirmer rods within the unit cell of a quadratic lattice in Fig. 5.2.

Since active turbulence exhibits a characteristic length scale, the emergent dynamics

is expected to couple to a specific lattice size [174], which is given by the different

system sizes in Fig. 5.2(a) and (b).

114

0 100 200 300

x/a0

100

200

300

y/a0

(a)

0 200 400 600

x/a0

200

400

600

y/a0

(b)

Figure 5.2: Snapshots of two simulations of pusher-type squirmer rods in the unit

cell of a periodic square lattice with a pillar radius of

Rpillar =

1.25

at different

system sizes.

(a) Small

system size

300

and (b) large system size

600

The system height is ∆z=18a0in both cases.

5. Conclusions and Outlook

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XVIII

Danksagung

Zuallererst m

ochte ich mich bei Prof. Holger Stark f

ur die durchg

angige Unterst

utzung

und Motivation

uber viele Jahre hinweg bedanken. Er war stets offen f

ur Ratschl

age,

Anregungen und Diskussionen und hat mir erm

oglicht an vielen Tagungen und

Fortbildungen teilzunehmen. Ebenso m

ochte ich mich bei Prof. Roland Netz von der

Freien Universit

at Berlin f

ur die Beguachtung meiner Arbeit und der Teilnahme an

der wissenschaftlichen Aussprache bedanken, sowie bei Prof. Dieter Breitschwerdt,

den Vorsitzenden des Promotionsausschusses.

Außerdem bedanke ich mich bei der Deutschen Forgschungsgemeinschaft (DFG),

welche mich im Rahmen des SFB 910 finanziell unterst¨

utzt hat.

Weiter gilt mein Dank auch allen derzeitigen und ehemaligen Mitgliedern der Ar-

beitsgruppe, wobei ich auch die gemeinsamen Mittagessen in der Mensa, Barabende

oder Klettersessions hervorheben m

ochte. F

ur die angenehme Zusammenarbeit und

interessanten sowie hilfreichen Diskussionen bedanke ich mich insbesondere bei

meinen B

uronachbarn Josua Grawitter sowie Reinier van Buel, Maximilian Seyrich

und Christian Schaaf, ebenso wie f¨

ur das Korrekturlesen bei Felix R¨

uhle.

Zu guter Letzt m

ochte ich mich auch bei meine Familie bedanken, insbesondere

meiner Partnerin Alice Faust, die mich w

arend dieser Zeit sehr viel unterst

utzt und

motiviert hat und stets ein offenes Ohr f¨

ur mich hatte.