Active emulsion droplets
driven by Marangoni flow
vorgelegt von
Diplom-Ingenieur
Maximilian Schmitt
geboren in Wissen (Sieg)
Von 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:
Vorsitzender Prof. Dr. Martin Schoen
Erster Gutachter Prof. Dr. Holger Stark
Zweiter Gutachter Prof. Dr. Uwe Thiele
Tag der wissenschaftlichen Aussprache: 04.05.2016
Berlin 2017
2
Contents
1 Introduction 7
2 Microswimmers 11
2.1 Hydrodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.1.1 Navier-Stokes equation . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.1.2 Reynolds number . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.1.3 Stokes equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.1.4 The squirmer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.2 Brownian motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.3 Active Brownian particles . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3 The active emulsion droplet 21
3.1 Definition and distinction . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.2 Microfluidics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.3 The fluid-fluid interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.3.1 Surface tension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.3.2 Surfactants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.3.3 Bromination reaction . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.3.4 Light sensitive surfactants . . . . . . . . . . . . . . . . . . . . . . . 28
3.3.5 Interfacial equation of state . . . . . . . . . . . . . . . . . . . . . . 30
3.3.6 Diffusion-Advection-Reaction equation . . . . . . . . . . . . . . . . 32
3.4 The Marangoni effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.4.1 Early observations of surface tension driven flow . . . . . . . . . . 38
3.4.2 Causes for Marangoni flow . . . . . . . . . . . . . . . . . . . . . . . 39
3.4.3 Plane Marangoni flow . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.4.4 Marangoni number . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.4.5 Droplet Marangoni flow . . . . . . . . . . . . . . . . . . . . . . . . 42
3.5 Experiments, models, and theories on active emulsion droplets . . . . . . 47
3.5.1 Early attempts on droplet Marangoni flow . . . . . . . . . . . . . . 47
3.5.2 Current state of research on active emulsion droplets . . . . . . . . 47
4 Publications 51
4.1 Synopsis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.2 Swimming active droplet: A theoretical analysis . . . . . . . . . . . . . . . 55
4.3 Marangoni flow at droplet interfaces:
Three-dimensional solution and applications . . . . . . . . . . . . . . . . . 63
3
5
6
1 Introduction
An active emulsion droplet is a swimming micron-sized liquid droplet immersed in a
bulk liquid. It is active in a sense that the swimming motion is not driven by external
forces, but rather sustained under a force-free condition. Such a droplet is therefore a
typical example of a synthetic active particle and belongs to the class of active matter [1].
Active matter systems rely on a continual supply of energy, which makes these systems
inherently nonequilibrium in character. They share this property with living systems,
such as e.g. cells [2]. In fact, the boundary between active emulsion droplets and living
cells is starting to blur. By confining an active gel, which is composed of microtubules,
the protein streptavidin, and kinesin, to the interior of a water-in-oil emulsion, it is
possible to render the streaming used by cells to circulate their fluid content as well as
the active motion of cells [3, 4, 5]. Therefore, gathering knowledge about the mechanics
and locomotion of droplets can help to understand various micron-sized liquid swimming
objects, especially protist cells [6].
A hallmark of active matter is the emergence of large-scale patterns in a crowd of
interacting active “agents”. This can be found in schools of fish, flocks of bacteria, or
the collective motion of so-called Janus colloids, which are artificial microswimmers [7].
In contrast to active droplets, Janus colloids have an inherently broken symmetry in
their material properties between two faces. Note that recently swarming behavior was
also observed in experiments of active emulsion droplets [8].
Droplets also play a key role in the growing field of microfluidics, which set itself to
miniaturize biological, chemical, and medical processes in a fluid to the microscale [9].
For instance, one can realize chemical reactions inside emulsion droplets. Such a system
is sometimes called a microreactor [10]. Other digital fluidic operations that can be
combined on a so-called lab-on-a-chip are droplet generation, transport, fusion, fission,
mixing, and sorting of droplets. However, most of these operations are performed with
passive droplets, which move due to an external flow field. A promising next step is
to extend these operations to active droplets, which could open up a plethora of new
applications, e.g., in drug release problems [11, 12].
Consequently, it is worthwhile to study active emulsion droplets, both from a theoret-
ical viewpoint as well as from a technical viewpoint.
A first question, which arises naturally in the study of active emulsion droplets, is how
the onset of motion is realized. In contrast to Janus colloids, active droplets are initially
homogeneous in their physical and chemical properties. Therefore, the onset of motion
is due to a spontaneous symmetry breaking at the droplet interface. Here, typically the
surfactant concentration at the interface of the droplet evolves from being uniform to a
shape which is characterized by two domains with different surfactant concentrations.
Since the surfactant concentration directly influences the surface tension at the droplet
7
interface, a gradient in surface tension emerges. This leads to a jump in shear stress
at the interface, which in turn drags adjacent fluid layers in direction of increasing
surface tension. Ultimately, this leads to propulsion of the droplet. This effect is called
Marangoni effect [13]. In the case of the active droplet realized by Thutupalli et al.
in Ref. [8], the interface of the droplet is covered by two surfactant types, which differ
by their surface tension due to a chemical reaction with bromine. Here, the symmetry
breaking can be seen as a phase separation of the two surfactant types. Thus, the onset
of motion can be studied in the framework of coarsening at a fluid-fluid interface.
Once the droplet has reached a steady swimming state, one classifies it by means of
swimmer types. Typical swimmer types are pushers and pullers, which resemble the flow
field around bacteria and algae, respectively. The swimmer type greatly influences the
hydrodynamic interactions with other droplets, obstacles, and walls, and is therefore of
great practical interest [14]. Apart from simple forward propulsion, oscillatory motion,
where the active emulsion droplet moves forward and backward, is also possible [15, 16].
We confirm these findings in Ref. [17], which can be found in Sect. 4.2.
Without considering thermal noise or boundary conditions, active emulsion droplets
swim forward in a straight line. This is certainly not the case in real systems, where
the finite temperature at the droplet interface leads to fluctuations in the surfactant
mixture, which in turn randomize the swimming direction. As a result, the active droplet
performs a persistent random walk. Here, the amplitude of the fluctuations directly sets
the persistence of the swimming path, and with that the effective diffusion constant
of the droplet. We quantify this systematically by averaging over many trajectories of
active emulsion droplets in Ref. [18], which is given in Sect. 4.4.
As is typical for active matter, swimming droplets are open systems, i.e., they are
subject to a steady inflow and outflow of matter or energy. Here, “matter” means
surfactants: They adsorb at the front of the swimming droplet and desorb back into
the bulk at the posterior end. However, above the critical micelle concentration (cmc),
surfactants tend to form micelles, i.e., spherical aggregates of surfactants. Thus, above
the cmc surfactants adsorb in form of micelles at the front of the droplet. This could
lead to an additional randomization of the droplet trajectory. In a minimal model,
micelles can also explain the onset of motion of an initially surfactant-free droplet. The
Marangoni induced spreading, when a micelle adsorbs at the interface, leads to a motion
of the droplet in direction of the adsorption site. The corresponding outer Marangoni
flow field, however, preferentially advects other micelles towards the existing adsorption
site. Hence, this mechanism can spontaneously break the isotropic symmetry of the
droplet and lead to directed motion. We elaborate on this effect in Sect. 4.3, where we
present Ref. [19].
Another possibility to locally alter the surface tension of a droplet, is to partially
illuminate a droplet, which is covered with light-switchable surfactants [20]. Depending
on the wavelength of the light, the surfactant changes its structure and with that its
surface tension. Hereby, one can locally generate a spot of different surface tension,
which induces Marangoni flow and hence propulsion of the droplet. Here, the emulsion
droplet can be pushed with UV light or pulled with blue light. Furthermore, depending
on the relaxation rate towards the surfactant type in bulk, the droplet shows a plethora
8
of trajectories, including a damped oscillation about the beam axis, an oscillation into
the beam axis, and a motion out of the beam followed by a complete stop [19].
Structure of the thesis
This is a cumulative thesis. As such, it presents the main scientific results in three
publications at the end of the thesis:
•“Swimming active droplet: A theoretical analysis,” Europhys. Lett., vol. 101,
p. 44008, 2013.
•“Marangoni flow at droplet interfaces: Three-dimensional solution and applica-
tions,” Phys. Fluids, vol. 28, p. 012106, 2016.
•“Active brownian motion of emulsion droplets: Coarsening dynamics at the inter-
face and rotational diffusion,” Eur. Phys. J. E, vol. 39, p. 80, 2016.
All publications are written by Maximilian Schmitt and Holger Stark. Prior to this, we
prepare the reader by introducing such concepts as the active Brownian particle, active
emulsion droplets, and Marangoni flow. Furthermore, we integrate our results into the
current state of research. In what follows, we give a brief outline:
Section 2 introduces the physics of microswimmers. We discuss the hydrodynamical
implications of swimming on the microscale and present the squirmer, which is a classical
model of a spherical microswimmer. Furthermore, we shortly address Brownian motion
of passive and active particles.
In Sect. 3, we specialize the discussion to active emulsion droplets. After a clear def-
inition of active emulsion droplets, we dwell on the theoretical description of fluid-fluid
interfaces. We discuss the concept of surface tension, explain how surfactants change the
surface tension, and give a thorough model for the dynamics of surfactants at interfaces.
We also elaborate on Marangoni flow on flat interfaces as well as on droplet interfaces.
Finally, we give an overview of the literature, starting with the discovery of the Maran-
goni effect, up to recent experimental and theoretical results of active emulsion droplets.
Section 4 contains the publications as well as a synopsis of the published results, where
we integrate our findings into the current state of research. We discuss how the spon-
taneous phase separation of a surfactant mixture leads to directed motion of an active
emulsion droplet. Here, we also find oscillatory trajectories. By including thermal fluc-
tuations, we can model the active Brownian motion of swimming droplets. In addition,
we propose a minimal model of symmetry breaking of a droplet due to the adsorption
of micelles. Lastly, we address light-driven droplet propulsion.
Finally, in Sect. 5 we present our conclusions and give an outlook.
9
10
2 Microswimmers
The active emulsion droplet is a typical example of an artificial microswimmer, i.e., a
swimmer whose size is on the order of several µm. We therefore take one step back
and introduce the reader to the fundamental physics of microswimmers, before we dwell
on active emulsion droplets and their unique features in Sect. 3. We start with a short
introduction on the hydrodynamics to the microscale in Sect. 2.1, after which we dis-
cuss the erratic motion of passive particles and microswimmers in Sect. 2.2, and 2.3,
respectively.
2.1 Hydrodynamics
What all microswimmers, including living organisms such as bacteria or algae as well
as artificial microswimmers such as Janus colloids or active droplets, have in common is
that they move in the regime of Stokes flow where viscous forces dominate over inertial
forces. Equivalent expressions are ’creeping flow’ and ’low Reynolds number flow’. This
is for instance the case when micron-sized objects move in a viscous liquid such as water
or oil. A precise definition by means of the Reynolds number will be given in Sect. 2.1.2.
The theory of Stokes flow is a linear theory, which eases the theoretical and numerical
treatment by a great extend. This will be discussed in Sect. 2.1.3 and applied to the
hydrodynamic standard model of microswimmers – the squirmer – in Sect. 2.1.4.
2.1.1 Navier-Stokes equation
We start with the Navier-Stokes equation, which describes the dynamics of viscous fluids.
It is given by Newton’s second law,
F=ma,(2.1)
written for a fluid parcel with density ρ. The acceleration ahas to be written as the
material derivative Du
Dtof the velocity field u(r) of the fluid. The material derivative is
defined as D
Dt=∂t+u· ∇,i.e., it measures the total temporal change of a quantity
flowing with the parcel due to local change ∂tas well as due to advection u· ∇ in the
velocity field u. In addition to body forces F=ρf, a fluid parcel experiences viscous
forces, which are written as the divergence of the stress tensor T:
ρDu
Dt=∇ · T+ρf.(2.2)
For a Newtonian fluid, T= 2ηA−p1, with viscosity ηand strain rate tensor A=
1
2[∇ ⊗ u+ (∇ ⊗ u)T] of the fluid, and pressure field p. Together with the material
11
derivative Du
Dt=∂u
∂t+ (u· ∇)u, one finds the Navier-Stokes equation:
ρ∂u
∂t +ρ(u· ∇)u=η∇2u− ∇p+ρf.(2.3)
The way we wrote down Newton’s second law in Eq. (2.1), we assumed the mass mto
be constant. Similarly, we also assume a constant density ρin Eq. (2.3). However, by
writing out the continuity equation 0 = ∂ρ
∂t+∇ · (ρu) = Dρ
Dt+ρ∇ · ufor the density ρ,
one notices that constant ρtranslates into the constraint
∇ · u= 0 ,(2.4)
for the velocity field u. This is the incompressibility condition. Hence, a velocity field u
is only a correct solution of the Navier-Stokes equation (2.3), if it also satisfies condition
(2.4).
The most striking property of the Navier-Stokes equation is the nonlinearity (u· ∇)u,
which made it impossible to write down a general solution to this day. Therefore one
has to resort to numerical approaches to find an approximative solution for the flow field
u. In what follows we discuss a regime where the nonlinearity can be omitted, which
opens up several possibilities to solve for uexactly.
2.1.2 Reynolds number
To gain some insight into the physics of the Navier-Stokes equation (2.3), we introduce
a typical length scale L, velocity u0, and pressure p0of the physical problem under
consideration. Furthermore, we omit external forces at this point and set f=0. The
Navier-Stokes equation reads in dimensionless quantities (denoted by the tilde):
ρu2
0
L∂˜
u
∂˜
t+ (˜
u·˜
∇)˜
u=ηu0
L2˜
∇2˜
u−p0
L˜
∇˜p , (2.5)
where we used t=L
u0˜
t. This can also be written as
Re ∂˜
u
∂˜
t+ (˜
u·˜
∇)˜
u=˜
∇2˜
u−1
2Eu Re ˜
∇˜p , (2.6)
with the Reynolds number Re and Euler number Eu:
Re = ρu0L
η,Eu = 2p0
u2
0ρ.(2.7)
The Reynolds number compares inertial forces with viscous forces, while Eu is the ratio of
pressure forces to inertial forces. For a microswimmer of size L= 10µm flowing in water
(ρ= 1kg/dm3,η= 1mPa ·s) with velocity u0= 10µm/s, one finds Re = 10−4≪1.
Thus, the dynamics of a microswimmer in water is mainly governed by viscous forces
and not inertia.
Note that the flow fields of two systems that have the same geometry are said to be
similar, when they have equal Reynolds and Euler numbers. This allows to save on
costs and lab time by performing experiments on similar flows in water channels or wind
tunnels, and relate the data to the far field of the actual systems.
12
2.1.3 Stokes equation
The flow of microswimmers, such as bacteria, Janus colloids, or active emulsion droplets,
is in the regime of low Reynolds numbers. We can therefore omit the l.h.s of Eq. (2.3)
and find:
η∇2u− ∇p+f=0,(2.8)
which is called Stokes equation. Here, the constraint of incompressibility ∇ · u= 0 still
holds.
In contrast to the full Navier-Stokes equation, Eq. (2.8) is linear in u. This allows to
find Green’s functions for a given point force f=f0δ(r−r0) for both the velocity field u
as well as for the pressure p. The corresponding flow singularity is called Stokeslet, and
the Green’s function for uis a second-rank tensor called Oseen tensor:
O(r) = 1
8πηr 1−r⊗r
r2,(2.9)
with r=|r|and flow field u(r) = f0·O(r). Note that this flow field decays as u∝1/r.
For instance, the far field around a spherical colloid, which is pulled through a viscous
liquid, is given by the Stokeslet. By integrating the resulting stress of the flow field over
the interface of the pulled colloid, one finds the Stokes drag force
f=−6πηRv,(2.10)
of a spherical colloid with radius R, which moves with velocity vthrough a liquid of
viscosity η. For more complex objects, force and velocity are connected via f=γvwith
the drag tensor γ.
Stokes flow is always laminar, which means that there is no disruption between parallel
fluid layers, and hence no turbulence. It is therefore difficult to mix fluids in the low
Reynolds number regime. Note that even though the flow field around a microswimmer
is laminar, a suspension of microswimmers can still show turbulence [21].
Upon comparing the Navier-Stokes (2.3) with the Stokes equation (2.8), one notices
that the latter is lacking a time derivative. Therefore, reversing an applied force, which
acts on a microswimmer, leads to a perfectly reversed fluid motion. This has striking
consequences for the swimming mechanisms of microswimmers. For instance the algae
Chlamydomonas, which propels itself forward by two beating flagella, which are attached
at the front of the cell, can only swim if forward and backward stroke of the flagella are
not time-reversible. Otherwise, the displacement during forward and backward stroke
would simply cancel each other. This circumstance is also known as Scallop theorem [22].
Swimming in the low Reynolds number regime is discussed extensively in the Reviews
[14, 23, 24].
In any real physical system, the Stokes equation (2.8) is subject to boundary condi-
tions, such as the a no-slip boundary condition u=0on the solid interface of a colloid
or a given pressure difference ∆p in a fluid channel. In Sect. 3.4, we dwell on the hydro-
dynamic conditions at a fluid-fluid interfaces of an emulsion droplet. Another possible
boundary condition is a prescribed flow uat an interface. In the following section, we
discuss a spherical microswimmer with such a prescribed surface velocity.
13
2.1.4 The squirmer
The squirmer is a classic model of an axisymmetric spherical microswimmer [25, 26],
which has been introduced to model the locomotion of microorganisms. Originally the
squirmer was a model for microorganisms that propel themselves by a carpet of beating
cilia on their surfaces. Instead of modeling each cilium separately, one coarse-grains into
a prescribed fluid flow at the surface. Today, the squirmer model is used as a standard
model for microswimmers, especially in simulations of suspensions of microswimmers
[27, 28, 29]. Note that the squirmer has recently been generalized to also model non-
axisymmetric swimmers [30].
The prescribed flow at the interface of an axisymmetric squirmer with radius Ris writ-
ten as an expansion u(R, θ) = P∞
l=1 BlVleθin the polynomials Vl(cos θ) = −2
l(l+1) ∂θPl(cos θ),
where Pl(cos θ) are Legendre polynomials of order l. The resulting flow field around the
squirmer is given by [25, 26]:
ur(r, θ) = 2
3R3
r3−1B1P1+
∞
X
l=2 Rl+2
rl+2 −Rl
rlBlPl,(2.11a)
uθ(r, θ) = 2
3R3
2r3+ 1B1V1+
∞
X
l=2
1
2lRl+2
rl+2 + (2 −l)Rl
rlBlVl.(2.11b)
These expressions are given in spherical coordinates and in the frame of the moving
particle. The squirmer moves in z-direction with velocity vector v=vezand swimming
speed v=2
3B1. The two leading terms of the surface velocity field can also be written
as
uθ|R(θ) = B1sin θ+B2
2sin 2θ , (2.12)
where one defines the squirmer parameter as β=B2
B1. In the following we dwell on the
squirmer parameter, see also Table 2.1.
From Eq. (2.12), one finds that for β < 0 (β > 0), the flow uθ|Rdominates at the
back (front) of the squirmer. Thus, the case β < 0 models a microswimmer, where the
swimming apparatus is located at the back of the swimmer. Since such a swimmer pushes
itself through the liquid, it is called a pusher. Correspondingly a squirmer with β > 0 is
called a puller. A squirmer with B2= 0, hence β= 0, is called a neutral swimmer and
a squirmer with B1= 0, hence β=±∞, is called a shaker. Note that the leading flow
singularity of pusher and pullers is a stresslet, which is a force-free combination of two
Stokeslets [31, 32]. Here, the sign of βexpresses the direction of the two Stokeslets, see
red arrows in Table 2.1. The singularities that account for the neutral swimmer are the
source dipole and the Stokes quadrupole, which is combination of two stresslets. Finally,
one finds from Eq. (2.11), that for β= 0 the velocity field decays as u∝1/r3, while for
β6= 0 the flow is more long-range with u∝1/r2.
In Table 2.1, we also show three real life examples of microswimmers, whose velocity
field can be modeled by the squirmer model. For instance, the dinoflagellate Gonyaulax
has two flagella. One lies in a groove that runs from the center to the posterior end of
the cell and propels the dinoflagellate at the back of the cell. The other flagella is in
14
Swimmer
type
passive
colloid
pusher neutral
swimmer
puller shaker
Squirmer
parameter
β < 0β= 0 β > 0β=±∞
Far field
decay
u∝1/r u∝1/r2u∝1/r3u∝1/r2u∝1/r2
Example Gonyaulax Blepharisma Chlorophyta
g
e
f
Table 2.1: Comparison of the squirmer parameter βand the far field decay of the velocity
field ufor different swimmers. For reference we also include a passive colloid
that is subject to Stokes drag due to gravity, see Eq. (2.10). The illustrations
of the biological examples are taken from Ref. [33]. Red arrows indicate forces
and black arrows indicate direction vectors e=v/|v|.
a groove that runs around the equator of the cell and causes the cell to rotate about
its swimming axis. The cell can change the direction of this rotation and thus tune
its helical swimming path and perform chemotaxis [34]. Since the flagellum, which is
responsible for propulsion, is at the back of the cell, Gonyaulax is a pusher. The protist
cell Blepharisma, on the other hand, is completely covered with a carpet of beating
cilia. Thus, in a coarse-grained picture there is no symmetry breaking between the front
and the back of the cell, and β≈0. Finally, a typical example for a puller is the algae
Chlorophyta, which propels itself with two beating flagella at the front. Here, the forward
and backward strokes have to differ due to the scallop theorem, as discussed in Sect.
2.1.3.
Note that the squirmer parameter βdoes not only differentiate between pushers and
pullers, but also influences the hydrodynamic interactions between swimmers or a swim-
mer and a bounding wall [32, 14, 35]. The interaction between two swimmers that swim
next to each other is attractive if they are pushers and repulsive if they are pullers, see
Fig. 2.1. Accordingly, pushers tend to orient parallel to a wall and pullers perpendic-
ular. The viscosity of a dilute suspension of self-propelled particles also depends on β
[36, 37, 38]. Pushers reduce the viscosity, while pullers increase it.
The squirmer model can also be employed to model artificial microswimmers. Partic-
ularly, the flow field of spherical Janus colloids can be matched to the squirmer flow field
[39]. In general, Janus colloids are colloids with two distinct faces that differ in their
physical or chemical properties [7]. There are various means to construct Janus colloids,
which propel themselves through viscous liquids.
15
(a)
pushers
(b)
pullers
Figure 2.1: Interaction between sides of microswimmers. (a) Pushers experience an
attractive interaction. Red arrows indicate forces, black arrows direction
vectors, and dashed lines trajectories. (b) Pullers experience repulsive
interaction.
For instance, one can construct a Janus colloid where one of the two faces catalyzes a
chemical reaction and the reactants set up a self-diffusiophoretic flow, which propels the
colloid [40]. Janus particles, where the thermal conductivity of both faces differs, can be
heated, which generates a temperature gradient, in which the colloids move. This effect
is called thermophoresis [39]. Also thinkable is the use of electrophoresis in combination
with self-diffusiophoresis. Close to a bimetallic Janus particle in a peroxide solution,
a electrochemical gradient is generated, which propels the colloid [41, 42]. Finally, in
a binary solvent close to its critical point, the liquid around Janus colloids can demix
locally, which induces a self-diffusiophoretic flow [43].
Alternatively, one can realize a squirmer by an active emulsion droplet. In contrast
to a Janus colloid, active emulsion droplets are initially homogeneous in their physical
and chemical properties. The symmetry between the two halves of the droplet breaks
spontaneously. We will come back to this in Sect. 3.
2.2 Brownian motion
Having an understanding of how spherical active particles swim forward in a viscous
environment, we now want to focus on their swimming trajectories. The squirmer, as we
presented it in Sect. 2.1.4, can only swim along the z-direction. A straightforward way to
let the squirmer change its swimming direction is to add forces fto the Stokes equation
(2.8). For instance a buoyancy force in x-direction would add a Stokeslet in x-direction
to the presented solution (2.11). Note that the squirmer model can be generalized to
account for an arbitrary surface velocity field u|R(θ, ϕ), [30]. Thus, by introducing
a time-dependent u|R(θ, ϕ, t), which could represent the carpet of beating cilia, the
squirmer swims in a time-dependent direction v(t). Alternatively one can explicitly
model microswimmers and their swimming mechanism, e.g., the rotating flagella of an
16
(a)
R, γ, m
η, T
r0
r(t)
(b)
101
102
103
101102103104105
t
h[r(t)−r0]2i
t≈m
γ∝t
∝t2
Figure 2.2: (a) Trajectory of a passive colloid from initial position r0=r(0) to position
r(t). The colloid has radius R, drag coefficient γ, and mass m, and is im-
mersed in a fluid with viscosity ηand temperature T. Black dots indicate
fluid particles which randomly kick the colloid. (b) Time-dependence of the
mean squared displacement h[r(t)−r0]2iof the colloid in a log-log plot. The
colloid moves ballistically until t≈m
γand diffusive afterwards.
E. coli bacterium [44], or the attached flagellum of an African trypanosome parasite
[45, 46].
Another effect, which changes the orientation of microswimmers, is due to internal or
external fluctuations. Here, we focus on the latter and discuss internal fluctuations in
Sect. 2.3. External fluctuations, i.e., thermal fluctuations of the surrounding fluid do
not depend on the forward propulsion of the microswimmer. Therefore we first discuss it
by means of a simple spherical colloid, which is immersed in a viscous fluid and subject
to collisions with the fluid molecules, see Fig. 2.2 (a).
In what follows, we want to sketch the erratic motion of a spherical colloid in a fluid
with temperature T. Such a motion is called Brownian motion [47]. Here, we assume
that the motion is in 1D, a generalization to 2D or 3D is straightforward. We start by
writing down Newton’s second law for the velocity v(t) = ˙r(t) of a colloid with mass m:
m˙v(t) = −γv(t) + ζ(t),(2.13)
where γv(t) is the drag force of the particle in the fluid. For a spherical colloid with
radius Rin a fluid with viscosity ηthe drag coefficient is given by γ= 6πηR, see Eq.
(2.10). The second term, ζ(t), on the r.h.s. of Eq. (2.13) is the stochastic force due to
collisions with fluid particles. These collisions vanish on average and are correlated on
a short time scale:
hζ(t)i= 0 ,(2.14a)
hζ(t)ζ(t′)i= 2γkBTδ(t−t′).(2.14b)
17
xy
z
e(t)
e(0)
Figure 2.3: Rotational diffusion of the direction of an active Brownian particle from e(0)
to e(t).
The second relation is called fluctuation-dissipation theorem, as is connects the second
moment of the fluctuations hζ2iwith the drag coefficient γ,i.e., with the dissipation of
the particle. Thus, apart from the material parameter γ, the strength of the fluctuations
is set by the temperature Tof the liquid.
Equation (2.13) can formally be integrated to find velocity v(t) and position r(t) of
the particle, for initial speed v0=v(0) and position r0=r(0). Of more interest is the
mean squared displacement (MSD) w.r.t. the initial position r0. It is given by [48]
h[r(t)−r0]2i=v0m
γ1−e−γ
mt2
+2kBT
γt−2m
γ1−e−γ
mt+m
2γ1−e−2γ
mt.
(2.15)
The MSD is ballistic with h[r(t)−r0]2i=v2
0t2for t≪m
γ. For t≫m
γ, on the other
hand, the MSD is diffusive with h[r(t)−r0]2i=c+ 2dDt, where D=kBT/γ is called
diffusion constant and dthe dimension, in which the motion takes place. Furthermore,
cis constant in time. The MSD is also plotted in Fig. 2.2 (b). The interpretation is as
follows. The colloid starts to move ballistically with velocity v0and is not significantly
disturbed by the individual fluid molecules. When t≈m
γ, the ballistic motion has
relaxed and the motion at later times is solely governed by the individual kicks of the
fluid molecules.
Note that Brownian motion occurs in equilibrium. There is no external driving of the
system. The motion of the colloid is only due to the finite temperature of the fluid.
2.3 Active Brownian particles
In contrast to a passive colloid, an active particle possesses an inherent velocity vector
v(t) = v(t)e(t) with velocity v(t) and direction of motion e(t) with |e(t)|= 1. Thus, the
trajectory of the swimmer is given by the integral r(t) = r0+Rt
0dt′v(t′)e(t′) with initial
position r0.
18
Under the assumption of a constant velocity vone finds for the mean squared dis-
placement
h[r(t)−r0]2i=v2Zt
0
dt′Zt
0
dt′′he(t′)·e(t′′)i.(2.16)
Thus, to calculate the MSD of an active particle, one needs to know the auto-correlation
function he(t′)·e(t′′)iof the swimmer’s direction. In the active Brownian particle model,
the swimming direction e(t) diffuses freely on the unit sphere [49, 50, 51], see Fig. 2.3.
This can be described by the rotational diffusion equation
∂tp(e, t) = Dr∇
s
2p(e, t),(2.17)
for the probability distribution p(e, t). Here ∇
s= (1−er⊗er)∇is the surface gradient
on the unit sphere and Drthe rotational diffusion constant. In this simple model, one
finds for the orientational auto-correlation function [52, 53]
he(0) ·e(t)i= e−t/τr,(2.18)
where the rotational correlation time τr= 1/[(d−1)Dr] with dimension d≥2 is the
characteristic time it takes the swimmer to “forget” about the initial orientation e(0).
Hence, for short times t < τrthe particle moves approximately in the direction of e(0),
while at later times t > τrthe orientation becomes erratic. Together with Eq. (2.16) one
finds for the MSD
h[r(t)−r0]2i= 2(vτr)2t
τr
−1 + e−t/τr.(2.19)
By comparing this with the MSD of the passive particle, one notices the similar shape.
For t≪τr, the active Brownian particle moves ballistic with velocity v, while for t≫τr,
the motion becomes diffusive with h[r(t)−r0]2i= 2dDt and diffusion constant D=
v2τr/3.
An alternative description to the rotational diffusion equation (2.17) is given by the
Langevin equation
∂te=p2Drω×e,(2.20)
for the direction vector e. The random torque ωleads to a random rotation of e. It
fulfills
hω(t)i= 0 ,(2.21a)
hω(t)⊗ω(t′)i=1δ(t−t′),(2.21b)
i.e., it vanishes on average and is correlated on a short time scale. Furthermore, the
components of ωare not cross-correlated.
We want to note that the origin of the random motion of an active Brownian particle
can be manifold. As discussed in Sect. 2.2, the thermal fluctuations, which induce small
translational kicks against the passive colloid, are only due to the temperature of the
fluid. This is a pure equilibrium effect. The random torque ω, which induces a rotation
19
of the direction vector eof an active particle, on the other hand, can also be connected
with its internal swimming mechanism [54].
One should also keep in mind that the presented model of an active Brownian particle
is highly idealized. In real systems, the rotational diffusion of eis usually not perfectly
isotropic due to interactions with obstacles, such as other particles, or walls. Further-
more, if the density of the particle does not match the fluid density, gravity has to be
taken into account [55]. In general, the motion of an active swimmer is also subject
to thermal translational fluctuations, as described for the passive particle in Sect. 2.2.
However, for a sufficiently large velocity vof the swimmer, these become negligible.
Finally, we assumed a constant velocity vof the swimmer. One can show that this is
only a good approximation, if the (indeed always present) velocity fluctuations decay
on a time scale τsimilar to τrof Eq. (2.18) [56]. Otherwise, the MSD can have several
regimes with different power laws.
20
3 The active emulsion droplet
In this section we want to narrow the focus down from microswimmers to active emulsion
droplets. We begin with a definition of active emulsion droplets in Sect. 3.1 and discuss
various applications in the growing field of microfluidics in Sect. 3.2. From a theoretical
physicist’s viewpoint, an active emulsion droplet is a closed fluid-fluid interface which
moves w.r.t. to a lab frame due to the Marangoni effect. We therefore discuss fluid-fluid
interfaces in detail in Sect. 3.3 and introduce the Marangoni effect in Sect. 3.4. Finally,
in Sect. 3.5 we update the reader to the current state of research on active emulsion
droplets.
3.1 Definition and distinction
We start with a definition of the central object of this work, the active emulsion droplet:
‘active’
As already pointed out in Sect. 2.3, active particles differ from passive particles by having
an inherent velocity vector v. Here, ‘inherent’ means that they have the ability to move
on their own and do not require external or stochastic forces to do so. This also applies
to active versus passive droplets. Here, the activity or the motion of active droplets
arises from an inhomogeneous surface tension σalong the interface of the droplet. The
resulting shear stress at the interface drags adjacent fluid layers in direction of increasing
surface tension. This effect is called Marangoni effect and will be treated in Sect. 3.4.
Active droplets rely on a supply of energy, such as a solvent which changes the surface
tension of the fluid-fluid interface. They are therefore not in equilibrium, in contrast to
e.g. the passive colloid, which we discussed in Sect. 2.2. Active droplets are also referred
to as ‘self-propelled droplets’,‘self-driven droplets’, or simply ‘swimming droplets’.
‘emulsion’
An emulsion droplet is a liquid droplet immersed in a second liquid. In general, one
distinguishes between droplets on interfaces and droplets immersed in a fluid. Droplets
on interfaces sit either on a solid or a liquid interface. The top phase can be a gas or
a liquid. Droplets immersed in a fluid are also either in gas or in liquid. In the latter
case, the droplets are called emulsion droplets. A collection of many emulsion droplets is
usually called emulsion. As a side node, gas droplets are called bubbles and a collection
of many bubbles is usually called foam.
21
passive active
droplet on
interface
•friction i.e. lateral adhesion of
sliding droplets [57]
•spreading dynamics [58]
•wettability of surfaces and
polymers [59, 60]
•light-induced Marangoni pro-
pulsion [61, 20]
•Marangoni driven spreading
[62, 63]
emulsion
droplet
•droplet formation, breakup,
and coalescence
•lab-on-a-chip operations [9]
•effect of surfactants on veloc-
ity of sinking droplets [64]
•shape of raindrops in flow [65]
•drop deformation in flow [66]
•mixing within droplets [67]
•swimming bifurcation
•swimming trajectories
•pattern formation on droplet
interface
•light-induced Marangoni pro-
pulsion
Table 3.1: Common classification of liquid droplets and some exemplary research topics.
This work focuses on active emulsion droplets (blue part in the table).
‘droplet’
The term droplet is a diminutive of ‘drop’. It is usually used for drops with diameters
up to 500µm. Sometimes the term ‘liquid colloid’ is used instead.
In Table 3.1, we list a few common research topics of liquid droplets. An in-depth
discussion of the current state of research of active emulsion droplets (blue part in Table
3.1) is given in Sect. 3.5.
3.2 Microfluidics
Similar to the advancements in the 1950s in microelectronics, in the 1980s the multi-
disciplinary field of microfluidics emerged. Here, the goal is to miniaturize biological,
chemical, and medical processes in fluids to the microscale. The implications of minia-
turization to the physics of fluids have already been discussed in Sect. 2.1: Fluid motion
on the microscale takes usually place in the low Reynolds number regime, where the
linear and time-reversible Stokes equation (2.8) holds. A certain analogy to microelec-
tronics can not be denied. However, while a microchip combines independent electronic
components through conducting paths on a single plate of a semiconductor, a lab-on-a-
chip in microfluidics combines several microfluidic devices through microchannels on a
single piece of a moldable elastomeric polymer.
Emulsion droplets play a key role in microfluidics. Ref. [9] summarizes various digital
fluidic operations and applications of droplet-based microfluidics. Basic operations are
droplet generation, transport, fusion, fission, mixing, and sorting of droplets. A key
22
(a)
cII
cI
ci
c
z
(b)
phase II
interface
phase I
x
z
y
(c)
phase II
phase I
Γ(x, y)
x
z
y
Figure 3.1: Coarse-graining of a fluid-fluid interface from (a) a full 3D description of the
interface with volume concentration ci(r) of surfactants to, (b) an interface
with finite thickness to, (c) an idealized interface with zero thickness and
excess concentration of surfactants per area Γ(x, y).
application of droplet microfluidics is the realization of chemical reactions inside emulsion
droplets. Reactants are either dissolved in the droplet fluid or again encapsulated in
smaller emulsion droplets inside the droplet. By carefully timing the droplet breakup
of the sub-droplets, it is possible to exactly trigger the reaction. Such a system is
sometimes called a microreactor [10]. These “matryoshka droplets”, i.e., hierarchically
encapsulated emulsion droplets are also widely used in the synthesis of biomolecules [9].
Some day this might lead the way to an old dream of scientist: the self-assembly of
artificial living cells.
In the future, digital fluidic operations might lead to computers, which consist of
moving droplets. As a matter of fact, it was recently possible to couple magnetic and
hydrodynamic interaction forces between droplets to develop logic gates such as AND,
OR, XOR, as well as a flip-flop [68]. Usually these digital fluidic operations are carried
out by passive emulsion droplets. A possible next step is to extend these tools to active
emulsion droplets. This could also be of interest in drug release applications [11, 12].
3.3 The fluid-fluid interface
An understanding of the fluid-fluid interface is crucial in the study of emulsion droplets,
which is why we discuss such interfaces in detail in this section. Albeit we have a liquid-
liquid interface in mind, the theory of this section holds for fluid-fluid interfaces in
general. Figure 3.1 (a) shows a cut through a typical fluid-fluid interface. The interface
is characterized by sigmoidal concentration profiles cIand cII of the two components that
make up the two phases, e.g., water and oil. Additionally, interfaces are often stabilized
by surface active molecules, co-called surfactants. These have the concentration ciin Fig.
3.1 (a). Here we assume, without loss of generality, that the surfactants are hardly soluble
23
in phase I, highly soluble in phase II, but above all favor the interface. The tendency to
favor interfaces is typical for amphiphilic surfactants, which possess a hydrophilic and a
hydrophobic side. In Sect. 3.3.2 we discuss surfactants in detail.
In the following sections we ignore the finite width of the fluid-fluid interface and
instead idealize it as a sharp interface. This concept of an interface with zero thickness
is called Gibbs diving surface. Figure 3.1 (b)-(c) shows the coarse-graining of the fluid-
fluid interface of Fig. 3.1 (a). This approach is very convenient in studies of the surfactant
dynamics at the interface as it allows to reduce the volume concentration of surfactants
ci(r) to an excess concentration per area Γ(x, y). This is the excess over what would be
present if the bulk concentration continued all the way to the interface, i.e. the excess
over the shaded area in Fig. 3.1 (a). In the following, we call the excess concentration Γ
of surfactants at the interface simply the “surfactant concentration”.
Note that the coarse-graining to a Gibbs diving surface, which we depicted in Fig. 3.1,
can also be done for curved interfaces.
3.3.1 Surface tension
A central physical quantity of a fluid-fluid interface is its surface tension. It plays a key
role in understanding many related fundamental effects. In general, the surface tension
is a property of liquids caused by intermolecular attraction, which makes them minimize
their surface area. In the following we use the term surface tension also for interfaces
between liquids.
Let us assume a film of water in an open container. The individual water molecules
are held together by an attractive potential, e.g., a Lennard-Jones potential U(r) as
depicted in the lower panel of Fig. 3.2 (a). This phenomenon is know as cohesion. At
the interface, however, water molecules have fewer neighboring molecules, which puts
them in a higher energy state. The top panel of Fig. 3.2 (a) shows a simplistic sketch
of the situation. Thus, by increasing the interface of the liquid film, the number of
molecules in the higher energy state increases, which leads to a higher total energy of
the film. The surface tension σis then defined as the ratio of the change in the energy
of the liquid, and the change in the surface area of the liquid. It is measured in energy
per area or force per length: [σ] = N/m.
In equilibrium thermodynamics surface tension is usually formulated in the grand
canonical ensemble. Its thermodynamic variables are temperature T, volume V, and
chemical potential µ. The corresponding energy potential is called grand potential Ω=
Ω(T, V, µ). However, since the system under consideration is an interface, the variable
Vis replaced by area A. The total differential of the grand potential for a surface is
given by:
dΩ=−SdT+σdA−Ndµ , (3.1)
with surface tension
σ=∂Ω
∂AT,µ
.(3.2)
Thus, surface tension measures the increase of energy Ωwith increasing area A, while
temperature Tand chemical potential µare held constant. In the simple case of a clean
24
(a)
r
a
U(r)
(b)
σ=∂Ω
∂AT
dA
(c)
σ=∂Ω
∂AT,µ
dA
Figure 3.2: (a) Top: sketch of water molecules at a water-air interface. Bottom:
Lennard-Jones interaction potential U(r) between water molecules. (b) Il-
lustration of a water film with variable interface area A. (c) Illustration of a
surfactant-laden water film with variable interface area A.
interface without surfactants, i.e. N= 0, one finds σ=∂Ω
∂AT, which is illustrated in
Fig. 3.2 (b). However, the reason for using Ωto define σin the first place, instead of
e.g. the Helmholtz free energy F, is that Ωallows particle exchange, i.e., adsorption
and desorption of surfactants. It is therefore the natural choice for fluid-fluid interfaces,
which are in contact with surfactant enriched phases.
3.3.2 Surfactants
Surfactants (surface active agents) are molecules which accumulate at fluid-fluid in-
terfaces and lower the surface tension. They are amphiphilic, i.e., they consist of a
hydrophilic (water-soluble) head and a hydrophobic (water-insoluble) tail. Therefore,
when a surfactant diffuses to the proximity of a fluid-fluid interface, such as water-air or
water-oil, it will readily adsorb and then stay at the interface. The lowering of surface
tension σcan be understood as follows.
Due to their amphiphilic nature, surfactants are in a lower energy state when adsorbed
at an interface compared to the bulk. This leads to the following thought experiment.
A liquid film is enriched with surfactants and kept at a constant temperature T. Thus,
the interface of the film is coupled to a reservoir of surfactants with chemical potential
µ. Figure 3.2 (c) illustrates the situation. Upon increasing the interface area, more
surfactants can migrate to the interface and thus reach a state of lower energy. This
25
(a)
❜
❜
✧
✧
❜
❜
HO
HO H
❜
❜
✧
✧O
O
❜
❜
✧
✧❜
❜
✧
✧✧
✧❜
❜✧
✧❜
❜✧
✧❜
❜✧
✧❜
❜✧
✧❜
❜✧
✧❜
❜
(b)
hydrophilic head hydrophobic tail
❜
❜
✧
✧
❜
❜
HO
HO H
❜
❜
✧
✧O
O
❜
❜
✧
✧❜
❜
✧
✧✧
✧❜
❜✧
✧❜
❜✧
✧
Br
Br
❜
❜✧
✧❜
❜✧
✧❜
❜❜
❜
✧
✧✧
✧
Figure 3.3: (a) Surfactant molecule monoolein with hydrophilic head and a hydrophobic
tail. (b) Monoolein molecule after halogen addition reaction with bromine.
effect counteracts the general mechanism of energy cost to establish the interface. Thus,
compared to the case without surfactants in Fig. 3.2 (b), increasing the interface has a
less pronounced effect on the energy of the interface. Thus, the surface tension σ, which
measures the effect of increasing area Aon energy Ω[compare Eq. (3.2)] is lower when
surfactants are present:
∂σ
∂Γ <0,(3.3)
with surface concentration Γof surfactants. In Sect. 3.3.5 we use the definition (3.2) of
the surface tension σto derive this relation. Note that σremains always positive, re-
gardless of the bulk concentration of surfactants, or else the interface would be destroyed
by small fluctuations.
There is a plethora of different surfactants which mainly differ by their hydrophilic
head. Most head groups are either anionic, cationic, or nonionic. Fig. 3.3 (a) shows a
typical nonionic surfactant called monoolein. It is one of the most important surfactants
in many applications such as emulsion stabilization, drug delivery, and protein crystal-
lization [69]. The hydrophilic head consists of glycerol with two hydroxyl groups. Thus,
the head of the surfactant is polar and can form hydrogen bonds with water. The C18
alkyl tail features a double bond in the middle and is strongly hydrophobic [70]. In Sect.
3.3.3 we discuss a chemical reaction that changes the structure of the hydrophobic tail
and with that the surface tension of monoolein. Note that monoolein is soluble in oil
but insoluble in water.
In general, surfactants play a crucial role in living cells as well as in industrial appli-
cations such as cosmetics, foam, and in the food industry [71]. Surfactants also act as
oil dispersants in the fracking industry or after an oil spillage to break the oil layer into
small droplets. Apart from these applications, surfactants are interesting objects from a
physicist’s viewpoint since they have the tendency to form self-organized aggregates at
high densities. Figure 3.4 (a) shows the simplest structure: a micelle with hydrophobic
26
(a) (b) (c)
Figure 3.4: Example structures formed by surfactants: (a) micelle, (b) bilayer, and (c)
spherical liposome.
tails inside. Micelles, where the tails point outwards, are called inverse micelles. Mi-
celles are formed above the critical micelle consideration cCMC. For instance, monoolein
molecules form micelles at c≥cCMC = 1mM/l. Other possible structures are bilayers
or liposomes as shown in Fig. 3.4 (b), and (c), respectively. Surfactants have very rich
phase diagrams which include, in addition to the mentioned structures above, cylindrical
micelles, lamellar, hexagonal, and cubic phases [69].
There are various possibilities to change the “surface tension of surfactants”. Here
and in the following, when we speak of the “surface tension of a surfactant”, we mean
the surface tension of an oil-water interface, which is completely covered by the surfac-
tant. By increasing the temperature of a surfactant layer, the surface tension decreases
approximately linearly [72]. Furthermore, many surfactants react with reactants that
are dissolved in the adjacent fluid layers. This can also change the surface tension. This
will be discussed in the following section. Other surfactants are light sensitive, i.e., they
change their surface tension when illuminated with light of a specific wavelength. In
Sect. 3.3.4 we dwell on these so-called photosurfactants.
3.3.3 Bromination reaction
The effect of surfactants on the surface tension of a fluid-fluid interface is due to their
amphiphilic structure. Thus, by changing the structure of head or tail of a surfactant, it
should be possible to increase or decrease its surface tension. In the case of monoolein
this is possible by adding the halogen bromine to one of the adjacent fluid layers [8].
Figure 3.3 (b) shows the structure of a monoolein molecule after it has reacted with
a bromine molecule. By comparing with the pristine monoolein surfactant in Fig. 3.3
(a), one notices that the bromine molecule reacted with the carbon-carbon double bond
in the hydrophobic tail. Such a reaction is called halogen addition reaction and the
corresponding chemical formula reads:
C−
−C + Br2−−→ Br−C−C−Br .(3.4)
27
(a)
❜❜
❜❜
NN
✧✧
❜
❜
✧
✧
❜
❜
✧
✧ ❜
❜
✧
✧
❜
❜
✧
✧
❜
❜
✧
✧
❜
❜
✧
✧
✧✧
N N
✔
✔
✔
✔❚
❚
✔
✔
❚
❚✔
✔
❚
❚
❚
❚
✔
✔❚
❚
✔
✔
❚
❚
✔
✔❚
❚
✲
✛
365nm
450nm
trans cis
(b) hydrophilic tail hydrophobic head
NN
❜
❜
✧
✧
❜
❜
✧
✧
❜
❜✧
✧❜
❜
❜
❜
✧
✧
❜
❜
✧
✧❜
❜✧
✧❜
❜+
N
Br−
Figure 3.5: (a) Photo-isomerization of azobenzene. The trans state can be converted to
the cis state by using ultraviolet light with wavelength 365nm. Blue light
(450nm) can be used to convert the molecule back to the trans state. (b)
AzoTAB surfactant in the trans state.
The actual reaction mechanism of the “bromination” reaction is rather complex [73].
The result is a saturated C−C bond, which weakens the hydrophobic nature of the
alkyl chain of the surfactant. Experiments showed that the surface tension of a droplet
interface covered with monoolein increases from 1.3 to 2.7mN/m [8].
As we will discuss in detail in Sect. 3.4, an inhomogeneous surface tension σat a
fluid-fluid interface causes the adjacent fluid layers to be pulled in direction of ∇
sσ. This
effect is know as Marangoni effect. Thus, a monoolein covered interface which is locally
in contact with a supply of bromine, will experience Marangoni flow. Ref. [8] uses the
Marangoni effect to create active emulsion droplets. We will elaborate on this in detail
in Sect. 4.2.
3.3.4 Light sensitive surfactants
Changing the surface tension of a surfactant by means of a chemical reaction with a
reactant relies on diffusion or advection of the reactant to the interface. It is there-
fore difficult to change the surface tension in a spatially controlled manner with e.g.
bromine. However, there are other surfactants which change their structure and thereby
their surface tension, when illuminated with light. For instance, the photosurfactant
AzoTAB uses the trans-cis isomorphism of azobenzene to change its surface tension
under illumination [74, 75, 61, 76, 77].
28
Figure 3.6: A droplet on an interface is first pushed with UV light (top panel) and then
pulled back with blue light (bottom panel). Taken from Ref. [20].
Figure 3.5 (a) depicts the azobenzene molecule and its trans-cis isomorphism. Azoben-
zene is a molecule composed of two phenyl rings which are connected by a N−
−N double
bond. When illuminated with UV light, the N−
−N double bond changes its orientation
from trans to cis state. Note that the two states have exactly the same atoms but in
a different arrangement. The trans state is the thermodynamically stable state of the
molecule and the cis molecule falls back into it within about an hour. Alternatively, the
cis molecule can be transformed back by using blue light. Photoisomerizable molecules
or so-called smart molecules play an important role in rewritable optical data storage,
such as CDs or DVDs [78].
The surfactant AzoTAB is synthesized by adding alkyl chains to both sides of the
azobenzene molecule and a trimethylammonium hydrophobic head group to one end
[79], see Fig. 3.5 (b). Thus, the trans-cis isomorphism modifies the hydrophobic tail of
the surfactant as in the case of the bromination reaction of monoolein, discussed in the
previous section. The tail of the cis molecule is more polar, i.e., more water-like or less
hydrophobic than the trans molecule. Thus, the surface tension σis higher in the cis
state of AzoTAB. The surface tension is around 7mN/m in the trans state, and around
8mN/m in the cis state, respectively [20].
The photoisomorphism of AzoTAB can be utilized to generate light-driven Marangoni
flow. This effect was first used by Diguet et al. in the group of D. Baigl in Paris to
induce the motion of droplets [20]. Here, the oil droplets are placed on an aqueous
interface. Figure 3.6 shows the reversible motion of a droplet due to partial illumination
with UV light and blue light, respectively. Note that the motion of the droplet is only
due to the local illumination and not due to local heating of the interface.
29
Photosensitive surfactants have also been used to fragment a continuous liquid stream
into droplets [80], to design an optofluidic mixer [81], and to fuse droplets [82]. Ref. [83]
reviews light-driven microfluidics.
Ichikawa et al. also studied light-induced droplet motion [84]. They, however, used
a slightly different setup, where an oil droplet on water is pushed or pulled with green
laser light. Here the direction of motion depends on the presence of surfactants in the
aqueous solution. In contrast to the Baigl system, heat plays a crucial role in the driving
of the droplet.
3.3.5 Interfacial equation of state
In the previous sections, we discussed how the surface tension of a fluid-fluid interface
is modulated by surfactants. In general, the presence of surfactants lowers the surface
tension, but by using either bromine in the case of monoolein or UV light in the case
of AzoTAB the surface tension can be increased again. In order to simulate surfactant
laden interfaces properly it is crucial to have a qualitative understanding of their surface
tension. Thus, we seek an expression for the surface tension as a function of surfactant
concentrations, a so-called interfacial equation of state. Usually an equation of state
σ(Γ) is achieved from an adsorption isotherm, i.e. an equation that relates surface
concentration Γto bulk concentration c. We want to follow a different approach by first
setting up a microscopic model for the surfactants at the interface. This allows us to
model surfactant diffusion in a thermodynamically consistent way in Sect. 3.3.6, [85].
In the following we derive the equation of state for a single surfactant type. A gener-
alization to a mixture of surfactants (e.g. pristine and brominated surfactants) can be
found in Sect. 4.2. Note that the equation of state is defined in thermal equilibrium. We
start by writing down the Helmholtz free energy F=E−TS for a surfactant-laden fluid-
fluid interface of area A. Here, Tis the temperature, Eis the surfactant internal energy
including interaction energy, and Sthe mixing entropy of the surfactants at the inter-
face. In the following model, we consider an ideal mixture of surfactants and therefore
set E= 0. A more elaborate system, which takes surfactant interactions into account,
can be found in Sect. 4.2. The mixing entropy is calculated from the multiplicity ΩM,
S=kBln ΩM=−NtotkB[Γln Γ+ (1 −Γ) ln(1 −Γ)] ,
with surface excess concentration Γ=N/Ntot, which relates the actual number of
surfactants Nto the maximum number of surfactants Ntot =A/ℓ2at the interface.
Here, ℓ2is the area of a surfactant molecule at the interface. The free energy takes the
form
F=NtotkBT[Γln Γ+ (1 −Γ) ln(1 −Γ)] .
For non-ideal, i.e., dense systems, a surfactant interaction term E=ǫΓ 2has to be
added. We define the free energy density as f=F/A or:
f=kBT
ℓ2[Γln Γ+ (1 −Γ) ln(1 −Γ)] .(3.5)
30
−1
−2
−3
Γ
kBT/ℓ2
0.5 1.0
f(Γ)
σ(Γ)
Figure 3.7: Free energy density f(Γ) and interfacial equation of state σ(Γ) from Eq.
(3.5) and Eq. (3.8), respectively. Here we set σ0= 0. The dashed lines show
the dilute limit Γ≪1, i.e., Eqs. (3.10) and (3.11). Note that the diffusive
and advective dynamics of the surfactant laden interface, which we discuss in
Sect. 3.3.6, only depend on derivatives of f(Γ) and σ(Γ), respectively. The
absolute values of f(Γ) and σ(Γ) are therefore irrelevant.
We will use this expression in Sect. 3.3.6 to derive the diffusive dynamics of surfactants
at the interface.
The surface tension σwas defined in Sect. 3.3.1 in terms of the grand potential
Ω(T, A, µ), which is related to F=F(T, A, N) by the Legendre transformation
Ω=F−∂F
∂N N . (3.6)
However, Ωalso follows from the Euler relation U=TS +σA +µN,
Ω=Ω0+σA , (3.7)
where we used the fact that the interface is in thermal equilibrium, i.e., the surface
tension σis homogeneous, and that Tand µare intensive. Thus, using (3.6) in (3.7)
yields:
σ=Ω−Ω0
A=F
A−∂F
∂N
N
A−Ω0
A=σ0+f−∂f
∂Γ Γ ,
with σ0=−Ω0/A. For the ideal free energy density (3.5) one finds the equation of state:
σ=σ0+kBT
ℓ2ln(1 −Γ).(3.8)
Figure 3.7 shows σ(Γ) together with the corresponding free energy density f(Γ) from
Eq. (3.5). Hence, the surface tension is at its maximum for a clean, i.e., surfactant-free
interface σ(Γ= 0) = σ0and decreases with increasing surfactant concentration Γ:
∂σ
∂Γ =kBT
ℓ2
1
Γ−1<0,0≤Γ < 1.(3.9)
31
F=E−TS
entropy & energy
of surfactants
F[Γ(r)]
free energy
functional
σ(Γ)
eq. of state
jD
diffusive current
ζ
thermal noise
u
velocity field
Γ(c)
adsorption
isotherm
jads
adsorption flux
jA
advective current
Legendre transf.
Stokes eq.
Langevin B
Gibbs eq.
fluct.-diss.-theorem
jA=Γu
Figure 3.8: Derivation of thermal noise, diffusion, advection, and adsorption terms in
the dynamic equation for the surfactant density Γat a fluid-fluid interface.
Thus, as illustrated in Sect. 3.3.2, surfactants lower the surface tension of fluid-fluid
interfaces. Equation (3.8) is called Szyszkowski equation [13]. It is the basis for various
complex models of the surface tension of surfactant laden fluid-fluid interfaces [86]. Note
that Eq. (3.8) breaks down for Γ→1.
In the dilute limit Γ≪1, the free energy density (3.5) simplifies to
f=kBT
ℓ2Γ(ln Γ−1) ,(3.10)
and the surface tension becomes
σ=σ0−kBT
ℓ2Γ , (3.11)
or σA ∼ −NkBTwhich is reminiscent of the ideal gas law pV =NkBT. In Fig. 3.7, we
plot the dilute limit of f(Γ) and σ(Γ) as dashed lines.
3.3.6 Diffusion-Advection-Reaction equation
In this section we formulate a dynamic equation for the surfactant density Γat a fluid-
fluid interface. The physical processes that we take into account are (i) diffusion, (ii)
advection within the interface, (iii) adsorption/desorption from/to the bulk, (iv) chemi-
cal reactions, and (v) thermal noise. The evolution of the interfacial surfactant concen-
32
tration is governed by the following equation [87, 13]:
∂Γ
∂t =∇
s·(jD+jA) + jads +jreact +ζ . (3.12)
Here ∇
s= (1−n⊗n)∇is the surface gradient and nthe surface normal. The terms in
the bracket are interfacial diffusion current jDand interfacial advection current jA, while
jads and jreact are the flux of adsorption (desorption) from (to) the bulk and the change
of Γdue to a chemical reaction, respectively. The last term ζincorporates thermal noise.
The groundwork for defining ζ,jD,jA, and jads has been laid in the previous section in
form of the free energy F. The flow chart in Fig. 3.8 gives a preview of the procedure.
In the following we discuss the individual terms.
Diffusion
In the previous sections we considered a fluid-fluid interface in thermal equilibrium. This
implies that the surface tension σis homogeneous. However, the various physical models
that we want to lay the groundwork for all deal with inhomogeneous concentrations Γof
surfactants, and hence with inhomogeneous surface tensions σ. Therefore, these models
are inherently in non-equilibrium. As a result, Marangoni flow, i.e., fluid flow due to an
inhomogeneous surface tension will emerge at the interface. This will be covered in Sect.
3.4 in detail. Another consequence of surfactant concentration gradients is interfacial
diffusion. It can be quantified as follows.
In non-equilibrium, the surface excess concentration Γdepends both on position and
time [88]. Hence, the free energy Fbecomes a functional of Γ(r, t):
F[Γ(r, t)] = Zf(Γ) dA , (3.13)
with free energy density f(Γ) from Eq. (3.5). Note that for more complicated forms of
f(Γ), e.g., a double well potential, a term which penalizes boundaries between regions
of different Γwithin the interface has to be added. Such a term usually has the form
(∇
sΓ)2[88].
The diffusion equation for surfactants at the interface can be written as a continuum
equation ∂Γ
∂t − ∇
s·jD= 0 .(3.14)
The diffusive current is connected to the free energy functional (3.13) via
jD=−λ∇
sµ , (3.15)
with chemical potential µ=δF
δΓand mobility λ, which, in general, is a function of
concentration Γ. Such a dynamics for a conserved quantity is called Langevin model B.
It is widely used to model separating phases of binary mixtures [88]. For the free energy
functional (3.13), we find
jD=−λ∇
s
δF
δΓ =−λ∇
sf′(Γ) = −λf′′(Γ)∇
sΓ . (3.16)
33
The resulting diffusion equation is then
∂Γ
∂t =∇
s·λf′′(Γ)∇
sΓ=∇
s·λ
Γ(1 −Γ)∇
sΓ.
Here, we used the free energy density f(Γ) of the ideal mixture from (3.5) in the second
step. Note that the prefactor (Γ−Γ2)−1diverges for Γ→0 and Γ→1. However, in
the dilute limit Γ→0, one expects linear, or “Fickean”, diffusion with ∂tΓ∝ ∇
s
2Γ. To
account for this one sets λ=DiΓand finds for Γ≪1 linear diffusion with
∂Γ
∂t =Di∇
s
2Γ ,
and interfacial diffusion constant Di.
In Ref. [17], which is given in Sect. 4.2, we extend the presented procedure to mixtures
of brominated and non-brominated surfactants. Since we assume the interface to be fully
covered with surfactants, we have to include energetic interactions between surfactants
in the free energy density.
Advection
Another interfacial transport process is advection, i.e. ,transport of surfactant density
Γdue to tangential fluid flow at the interface. The origin of the interfacial flow uican
be the bulk flow uof an adjacent liquid layer or flow that is generated by the interface
itself. The latter is the case if the surface tension is inhomogeneous, i.e.,σ=σ(r), which
induces Marangoni flow. For the time being we assume that the flow field uiis known.
The resulting surfactant advection current is given by
jA=Γui.
Thus, the dynamics due to diffusion and advection is governed by the continuum equation
(3.14) with the diffusive current replaced by jD+jA.
When surfactants are present at the interface, they are also present in one or both
of the surrounding bulk phases. The corresponding continuum equation for the bulk
concentration cof surfactants is given by
0 = ∂c
∂t +∇ · (cu) = ∂c
∂t +∇c·u
| {z }
Dc
Dt
+c∇ · u,(3.17)
where we omit diffusion for the time being. Thus, Eq. (3.17) illustrates that in in-
compressible fluids with ∇ · u= 0 the material derivative vanishes, Dc
Dt= 0, and the
concentration stays constant within a flowing fluid parcel. Throughout this thesis we
assume that all fluids are incompressible, see also the derivation of the Navier-Stokes
equation in Sect. 2.1. Therefore, for advection equations in bulk, the term ∇ · (cu) can
be written as ∇c·u.
34
(a)
Γ(c)
bulk diff.
advection
n
(b)
0.5
1.0
0.5 1.0
c
Γ
Figure 3.9: (a) Illustration of the two-step adsorption process. In a first step surfactants
are transported via diffusion and/or advection to the adsorption layer. At
the adsorption layer, surfactants aggregate at the interface. The surface
concentration Γis connected to the concentration cin the adsorption layer
via an adsorption isotherm Γ=Γ(c). (b) Adsorption isotherms. Solid
line shows Langmuir adsorption, i.e., Eq. (3.22), dashed line shows Henry
adsorption, i.e., Eq. (3.23). Parameters are b= 1/a = 5.
However, in the diffusion-advection-reaction equation (3.12), the surface divergence
∇
s·acts on jA. While, the incompressibility condition ∇ · u= 0 also holds at the
interface, it does not hold for the surface divergence. It is therefore not possible to
rewrite the term ∇
s·(Γui) as ∇
sΓ·ui. This is owed to the fact that fluid can flow away
and towards the interface from the bulk. In other words, the surfactant density Γat
the interface is not conserved, since surfactants can adsorb and desorb from and into
the bulk. This is expressed by the flux jads in Eq. (3.12). In Ref. [19], which is given
in Sect. 4.3, we show that one can directly relate the surface divergence ∇
s·uito the
radial flow toward or from the droplet interface.
Adsorption
In all surfactant systems, adsorption plays a key role, since the tendency of surfactants
to adsorb at fluid-fluid interfaces is their major property. The adsorption of surfactants
is a process that occurs in two sequential steps, see Fig. 3.9 (a). In a first step surfactants
are transported through diffusion and/or advection to the interface. The corresponding
flux to the interface is given by:
jads =n·(D∇c−cu),(3.18)
with bulk diffusion constant D, surfactant concentration c, and velocity field u. Here,
the surface normal npoints into the surfactant laden bulk fluid. The second step is the
adsorption as such, i.e., aggregation of surfactant molecules at the interface by means
35
of interfacial kinetics. Thus, there is a second relation for the adsorption flux at the
interface [13]:
jads =kac−kdΓ , (3.19)
with adsorption and desorption constants kaand kd.
Here we assume a adsorption layer with a given concentration cof surfactants close
to the interface and focus on the second step of the adsorption process. This process
is typically described via an adsorption isotherm, which relates the surface excess con-
centration Γto the concentration cin the adjacent adsorption layer [13]. Knowledge of
the adsorption isotherm Γ(c) allows to express jads in Eq. (3.19) in terms of Γ. In the
following we describe how the adsorption isotherm can be derived from the surfactant
equation of state σ(Γ).
From the relation Ω=Ω0+σA from Eq. (3.7) for the grand potential Ωone finds
dΩ=Adσ+σdAwhich can be compared with the total differential in Eq. (3.1). For
constant temperature T, we find
dσ=−N
Adµ=−Γ
ℓ2dµ , (3.20)
which is called Gibbs adsorption isotherm or simply Gibbs equation [13]. A generaliza-
tion to multicomponent systems is straightforward. Equation (3.20) can be rewritten
as
−Γ
ℓ2=∂σ
∂µ =∂σ
∂c
∂c
∂µ .
Using the chemical potential of the bulk fluid, µ=µ0+kBTln c
c0, one finds ∂c
∂µ=c
kBT
and thus ∂σ
∂c =−kBT
ℓ2
Γ
c.
By using the chain rule to split up ∂σ
∂c=∂σ
∂Γ ∂Γ
∂cwe find:
∂σ
∂Γ
1
ΓdΓ=−kBT
ℓ2
1
cdc . (3.21)
This equation can be integrated for a given equation of state σ(Γ). For the ideal mixture
we find with Eq. (3.9):
Γ=c
c+a,(3.22)
where ais an integration constant. This widely used relation is called Langmuir ad-
sorption isotherm. It can also be derived by setting up an equilibrium reaction equation
between molecules, empty sites, and occupied sites at the interface while neglecting any
site interaction [72]. In the dilute case Γ≪1, one finds:
Γ=b·c , (3.23)
where bis an integration constant. This relation is called Henry adsorption isotherm
and is the simplest possible adsorption isotherm [72]. In Fig. 3.9 (b), we plot the Lang-
muir isotherm (3.22) as well as the Henry adsorption isotherm (3.23) for b= 1/a = 5.
36
Apparently, the latter lacks any saturation of the surface concentration Γfor large bulk
concentration c, which is certainly unphysical beyond the dilute regime. By comparing
the linear Henry adsorption isotherm (3.23) with the adsorption flux jads of Eq. (3.19),
one notices that linear adsorption corresponds to jads = 0 and b=ka/kd. In that case
the interface is in local equilibrium with the adjacent adsorption layer.
When the concentration cof surfactants in the bulk fluid is above the critical micelle
concentration, the adsorption process is more complicated. Micelles have to attach to
the interface, break open, and then aggregate at the interface. In that case it can be
appropriate to rather model the micelles and their adsorption explicitly. In Ref. [19],
which can be found in Sect. 4.3, we present a possible implementation.
Reaction
Certain surfactants can react with a reactant and reach a state with a lower or higher
surface tension, compare Sect. 3.3.3. In these systems, two surfactant states (pristine
and reacted state) coexist at the interface. If Γdenotes the concentration of the pristine
surfactants, a reaction term can be written as jreact =−kΓ , where infinite supply of the
reactant is assumed. Here k > 0 denotes the reaction constant. The concentration Γ′of
reacted surfactants, on the other hand, increases with jreact =kΓ ′.
In cases where adsorption and a reaction take place, both effects can be combined into
one term in the dynamic equation for Γ, Eq. (3.12) [8]. In Sect. 4.2 we use this approach
to model monoolein surfactants that react with bromine.
Thermal noise
The thermal noise term in Eq. (3.12) represents the effect of thermally induced collisions
between surfactant molecules at the interface. It has the following properties: [88]
hζi= 0 ,(3.24a)
hζ(r, t)ζ(r′, t′)i=−2kBT λ∇
s
2δ(r−r′)δ(t−t′).(3.24b)
Here, the strength of the noise correlations is connected to the mobility λof the diffusive
current in Eq. (3.16) via the fluctuation-dissipation theorem. The first property (3.24a)
ensures that – on average – the noise vanishes, while Eq. (3.24b) expresses that the
noise is delta correlated in time and short-range correlated in space. This is due to
the fact that the thermal noise is connected to the dynamics of the Langevin model B,
which we discussed around Eq. (3.16), and as such it is a conserved quantity. Thus, one
obtains the usual white noise auto-correlation for the noise current jζwhich is defined
via ζ=∇
s·jζ:
hjζ(r, t)⊗jζ(r′, t′)i= 2kBT λ1δ(r−r′)δ(t−t′).
This is equivalent to the expression (3.24b). Hence, the ∇
s
2-operator in Eq. (3.24b)
accounts for the conservation law of the noise. A random noise addition to the sur-
factant concentration Γadded at one site at the interface must be balanced by noise
contributions in the neighborhood of the site [89].
37
r
z
ϕ
t
(a) (b) (c)
r
alcohol
ϕr
∇
sσ
ϕ
g
rϕ
Figure 3.10: Marangoni flow in a glass of wine. Shown are three sequential snapshots,
each with side view and frontal view of the side of a wine glass. (a) Cap-
illary action makes the wine climb up at the glass’ wall. Both alcohol and
water evaporate from the rising film, but the alcohol evaporates faster, due
to its higher vapor pressure. Hence, at the side of the glass the alcohol con-
centration is decreased. (b) Since the surface tension σof alcohol is lower
than that of water, a gradient ∇
sσis initiated. The resulting Marangoni
flow causes more wine to be pulled upwards. (c) The wine forms droplets –
so-called “tears of wine” – which fall back under their own weight.
3.4 The Marangoni effect
The surface tension σis a quantity, which originates in the intermolecular attraction at a
fluid-fluid interface. Molecules in regions with a high surface tension experience a higher
attraction towards each other than molecules in regions with a low surface tension. This
leads to a flow of surfactant molecules and thereby to fluid flow uof the adjacent liquid
in direction of the gradient:
u∼ ∇
sσ . (3.25)
This effect is called Marangoni effect. In this section, we show how Marangoni flow can
be created, derive the flow field for a simple planar geometry, and discuss Marangoni
flow around emulsion droplets. We start with a short introduction to the history of the
Marangoni effect.
3.4.1 Early observations of surface tension driven flow
The Marangoni effect was likely discovered over a glass of wine. James Thomson, who
was the older brother of William Thomson (Lord Kelvin), first mentioned so-called “tears
of wine” in 1855, [90]. Tears of wine describes a phenomenon, where wine climbs up on
the inside of a wineglass and droplets fall back under their own weight. It is illustrated
in Fig. 3.10. The wine is pulled upwards since the alcohol evaporates faster than water
at the side of the glass, but water has a higher surface tension than alcohol. Hence,
38
(a)
u∼ −∇
sΓ
high Γlow Γ
(b)
u∼ ∇
sσ
low σhigh σ
(c)
u∼ −∇
sT
high Tlow T
Figure 3.11: Marangoni effect induced by (a) a gradient in surfactant concentration Γat
the interface, (b) a gradient in surface tension σof the surfactants, and (c)
a gradient in temperature Talong the interface.
tears of wine are the result of Marangoni flow. A quantitative study of the tears of wine
phenomenon can be found in Ref. [91].
Although Thomson discovered that fluids flow in direction of increasing surface ten-
sion, the effect was named after Marangoni, who wrote his Ph.D thesis about the topic in
1865 [92]. A first rigorous theoretical treatment of the subject was conducted by Gibbs
in 1876 [93].
3.4.2 Causes for Marangoni flow
Figure 3.11 depicts three examples of Marangoni flow. A simple scenario that leads
to so-called solutocapillary Marangoni flow, is an inhomogeneous concentration Γof
surfactants at the interface, see Fig. 3.11 (a). As elaborated in Sect. 3.3.5, the surface
tension σof a surfactant-laden interface is in first order proportional to −Γ, see Eq.
(3.11). Therefore, Marangoni flow uis directed away from regions of high surfactant
concentration. The flow u, however, transports the surfactants by means of advection.
Hence, surfactants generally spread at interfaces until the surface tension is homogeneous
[94].
Alternatively one can generate solutocapillary Marangoni flow by directly tuning the
surface tension of surfactants in a fully covered interface, see Fig. 3.11 (b). In Sect.
3.3.3 and 3.3.4 we introduced the bromination reaction of monoolein and the photo-
isomerization of AzoTAB. Both can lead to an inhomogeneous mixture of surfactants
with different surface tensions and thus to Marangoni flow.
The surface tension of a liquid-liquid interface can also be altered without the aid
of surfactants – by changing the temperature. At most interfaces, the surface tension
decreases with increasing temperature [95], as shown in Fig. 3.11 (c). This also holds,
when surfactants are present [96]. Flow, which is generated by gradients in temperature
T, is called thermocapillary Marangoni flow.
In general, several of the above effects occur simultaneously. For instance, an interfa-
cial layer of AzoTAB surfactants, which is partially illuminated with UV light, experi-
ences flow away from the illuminated, i.e.,cis surfactant covered region. The resulting
39
(a)
x
z
y
η
ˆη
σ(x, y)
n
d
d
(b)
η
ˆη
σ(θ, ϕ)
n
R
Figure 3.12: (a) Two immiscible fluid layers with viscosities ηand ˆηand thickness d
confined between plates. The interface in between has a surface tension
σ(x, y). (b) Spherical emulsion droplet of radius Rwith viscosity ˆηim-
mersed in unbounded bulk fluid with viscosity η. The interface in between
the two immiscible fluids has a surface tension σ(θ, ϕ).
advection leads to a thinning of surfactants in the illuminated region. This counteracts
the light induced Marangoni flow. Furthermore, the light beam possibly heats the sur-
factants which also counteracts the light induced Marangoni flow. Thus the physics of
Marangoni flow at fluid-fluid interfaces becomes quite complex for seemingly primitive
systems.
In what follows, we want to define Eq. (3.25) more precisely.
3.4.3 Plane Marangoni flow
Here, we discuss immiscible fluid-fluid stratified flow confined between two infinitely
extended plates. The fluid layers are of height dand have the respective viscosities η
and ˆη, while the interface between the two liquids is subject to a given surface tension
σ(x, y). Figure 3.12 (a) depicts the setup. Only this plane geometry yields the simple
relation u∝ ∇
sσ,i.e., flow in the direction of increasing surface tension, exactly. It
is therefore paradigmatic for the Marangoni effect and serves for several studies of the
stability of stratified flow [97, 98, 99].
At low Reynolds number we have to solve the Stokes equation (2.8) to determine both
the velocity field u(r) above the interface (z > 0) and the field ˆ
u(r) below the interface
(z < 0). We start by collecting the boundary and interface conditions:
u=0, z =±d , (3.26a)
uz= ˆuz= 0 , z = 0 ,(3.26b)
u=ˆ
u, z = 0 ,(3.26c)
∇
sσ=Ps(ˆ
T−T)ez, z = 0 .(3.26d)
40
The first condition (3.26a) takes into account the two confining plates. Furthermore, we
assume an impenetrable interface, see Eq. (3.26b), with continuous tangential velocity,
see Eq. (3.26c). Condition (3.26d) is the essential condition regarding the Marangoni
effect. It states that ∇
sσ,i.e., the gradient in surface tension along the interface, is
balanced by a jump in the fluid shear stresses [13]. Here we introduce the surface
projector Ps=1−n⊗nwith surface normal n=ezand use the notation ∇
s=Ps∇
for the surface gradient. Finally, the viscous part of the Cauchy stress tensor of an
incompressible Newtonian fluid with viscosity ηis given by T=η∇ ⊗ u+ (∇ ⊗ u)T.
Writing out condition (3.26d) in Cartesian coordinates yields the two equations:
∂xσ= ˆη∂zˆux−η∂zux,(3.27a)
∂yσ= ˆη∂zˆuy−η∂zuy.(3.27b)
Next, we employ the ansatz
ux(x, z) = f(x) + zg(x),(3.28a)
uy(y, z) = h(y) + zw(y),(3.28b)
uz= 0 ,(3.28c)
where f, g, h, w are functions to be determined. An analogous ansatz is made for ˆ
uwith
functions ˆ
f, ˆg, ˆ
h, ˆw. This ansatz resembles the ansatz for Couette flow. As such it is
laminar, i.e., there is no interference between layers that are parallel to the x-y-plane.
Note that ansatz (3.28) does already incorporate condition (3.26b). By successively
evaluating the remaining conditions, one finds for the flow field:
u=d−z
η+ ˆη∇
sσ , ˆ
u=d+z
η+ ˆη∇
sσ , (3.29)
and at the interface
u(z= 0) = d
η+ ˆη∇
sσ , (3.30)
which is the hallmark of the Marangoni effect: fluid flow in the direction of increasing
surface tension σ.
3.4.4 Marangoni number
The dimensionless Marangoni number Mcompares the diffusion time scale to the ad-
vection time scale at a fluid-fluid interface. It is an important parameter in all systems
that include Marangoni flow, as it determines its stability [13]. It can be derived from
Eq. (3.30) as follows. We first consider the case of solutocapillary Marangoni flow, i.e.,
flow due a gradient in concentration Γof surfactants, see Fig. 3.11 (a). Hence, one can
write ∇
sσ=∂σ
∂Γ∇
sΓ. A natural rescaling of the surface gradient is ˜
∇
s≡d∇
s,i.e., by the
only length scale in the system. The flow field is rescaled by ˜
u≡τD
du. Here τDis the
41
diffusion time scale of the system, with τD=d2/D and diffusion constant D. Thus, one
finds for concentration driven Marangoni flow at the interface:
˜
u=d
D(η+ ˆη)
∂σ
∂Γ
| {z }
Msol
˜
∇
sΓ . (3.31)
Note that the concentration Γis assumed to be dimensionless. The dimensionless param-
eter Msol is called solutocapillary Marangoni number [13]. It can also be written as the
ratio Msol =τD/τAwhere we introduce the advection time scale τA=d(η+ ˆη)∂σ
∂Γ−1.
The quantity ∂σ
∂Γ is given in Eq. (3.9) for an ideal interface. In Ref. [17], which is given
in Sect. 4.2, we will derive the solutocapillary Marangoni number for a more elaborate
system, where two species of surfactants are present at the interface.
In the case of thermocapillary flow, i.e., flow due to a gradient in temperature Tone
can write ∇
sσ=∂σ
∂T∇
sT. This case is depicted in Fig. 3.11 (c). Again, one rescales
˜
∇
s≡d∇
sand ˜
u≡τD
duwith τD=d2/α and thermal diffusivity α. Thus, one finds
˜
u=d∆T
α(η+ ˆη)
∂σ
∂T
| {z }
Mth
˜
∇
s˜
T , (3.32)
with thermocapillary Marangoni number Mth. Here, we also rescaled temperature ˜
T≡
T/∆T.
Finally, we want to note that these Marangoni numbers are quite general. Albeit
the expression for the flow field ucan become more complex, e.g. around an emulsion
droplet, the prefactor d/(η+ ˆη) and with that the advection time scale τAremain the
same.
3.4.5 Droplet Marangoni flow
Including any boundary conditions in xor ydirection in the plane system of Fig. 3.12
(a), makes the ansatz (3.28) insufficient. Instead, one has to resort to techniques such as
an expansion in orthonormal basis functions in order to fulfill the additional boundary
conditions. For the surface tension σ(θ, ϕ) of the spherical emulsion droplet in Fig. 3.12
(b), spherical harmonics are the natural choice for a basis. We use this approach in Ref.
[19], which is given in Sect. 4.3, to solve for the flow field around a droplet. In contrast
to the plane system, Marangoni flow around a droplet leads to motion of the droplet
w.r.t. a fixed lab frame of reference. Therefore one also has to consider the resulting
hydrodynamic drag of the moving droplet. This is done by superimposing the Marangoni
flow with the drag flow. This approach ensures that the droplet is a force-free swimmer
[14].
First calculations of droplet hydrodynamics date back to the beginning of the 20th
century. In 1911, the famous Stokes drag formula (2.10) was generalized to liquid droplets
42
by Hadamard and Rybczynski [100, 101]. The drag force of a droplet with viscosity ˆη
and radius R, which moves with velocity vthrough a liquid of viscosity ηis
f=−6πηR2η+ 3ˆη
3η+ 3ˆηv.(3.33)
This reduces to Stokes drag, i.e., Eq. (2.10) for ˆη≫η, whereas it predicts a reduced drag
for droplets, due to a finite slip velocity at the droplet interface. This formula, however,
overestimated the velocity, which was found in experiments of sinking emulsion droplets.
The explanation was found by Levich in 1962: While the droplet sinks, the flow field
at the interface advects surface active agents to the posterior end of the droplet and
the induced Marangoni effect reduces the sinking velocity [102]. In other words, the
Hadamard-Rybczynski formula is only correct for perfectly clean droplets.
The first rigorous calculation of the velocity reduction of a sinking surfactant-laden
droplet was presented by Levan and Newman in 1976 in Ref. [64]. Their derivation
uses the Stokes stream function ψ(r, θ) to find the velocity field ur= (r2sin θ)−1∂θψ,
uθ=−(rsin θ)−1∂rψ. Note that the stream function can only be used in two dimensions
or in three dimensions with axisymmetry. The authors found for the interfacial flow field
uθ|R=uθ(R, θ) of a droplet, which is axisymmetric about ez[64]:
uθ|R=ηsin θ v
2(η+ ˆη)+1
2(η+ ˆη)
∞
X
l=1
1
Z
−1
σ(θ)Pldcos θ
∂θPl,(3.34)
and ur|R= 0. Here, Pl(cos θ) are Legendre polynomials of degree l, with P1= cos θ.
The front of the droplet is at θ= 0. This flow field incorporates the set of boundary
conditions in Eqs. (3.26b)-(3.26d), albeit modified to suit spherical coordinates. Note
that u|R=ur|Rer+uθ|Reθis given in the frame of reference of the moving droplet. The
droplet velocity in z-direction is calculated from [64]:
v=−1
2η+ 3ˆη
1
Z
−1
σ(θ) cos θdcos θ . (3.35)
Here, we omitted any external forces such as gravity. Note that the first term of Eq.
(3.34) is the drag flow, while the second term is the pumping Marangoni flow. The latter
can be related to the pumping part of the squirmer flow field (2.11) of Sect. 2.1.4 by
substituting
1
Z
−1
σ(θ)Pldcos θ=−8(η+ ˆη)
l(l+ 1) Bl.
The drag flow of the squirmer (Stokes flow around a colloid) is given by the droplet drag
flow in the limit of infinite internal viscosity ˆη→ ∞ [19].
Albeit Ref. [64], and the subsequent Ref. [103] by Levan, discuss a droplet which moves
due to gravity, the flow field (3.34) also holds for a self-propelled active droplet.
43
(a) (b)
1
−1
πθ
σ˜uθ|R˜
∇
s·˜
u|R
adsorption
desorption
z
θ
uθ|R
v
high
σlow
σ
Figure 3.13: (a) Surface tension σ(θ) = σ0(1 −cos θ) and the resulting surface flow
field ˜uθ|Rand surface divergence ˜
∇
s·˜
u|R. Here, we made u|Rand ∇
s
dimensionless. (b) Illustration of a swimming emulsion droplet. Marangoni
induced droplet propulsion is always accompanied by an inflow of matter
at the front and outflow of matter at the back of the droplet.
In 2000, the velocity field around an emulsion droplet with arbitrary surface tension
σ(θ, ϕ) was found by B lawzdziewicz et al. by using a set of fundamental solutions of the
Stokes equation [104]. They used the formulas to study the rheology of a dilute emulsion
of surfactant laden spherical droplets. In Sect. 4.3 we derive and discuss Marangoni
flow around a spherical emulsion droplet in detail. The full three-dimensional velocity
field was later used to calculate the migration of spherical droplets in Poisseuille flow
[105, 106].
The lowest mode of Eq. (3.34), i.e. l= 1, corresponds to σ(θ)∝cos θ. Note that
this is the only mode that gives v6= 0 in Eq. (3.35), and it is therefore responsible for
propulsion of the droplet. We therefore set σ(θ) = σ0(1 −cos θ) for the time being to
discuss the basic features of droplet Marangoni flow. We find
v=2
3
σ0
2η+ 3ˆηand uθ|R=σ0sin θ
2η+ 3ˆη,(3.36)
which we plot in Fig. 3.13 (a), together with σ(θ). The tangential surface flow uθ|Ris
directed along eθ, which leads to swimming in z-direction with droplet velocity vector
v=vez, see Fig. 3.13 (b).
Note that the surface flow field in the simple case σ=σ0(1−cos θ) can also be written
as
u|R=R
2η+ 3ˆη∇
sσ , (3.37)
where ∇
s=R−1∂θ. This relation resembles Eq. (3.30). Here, however, the character-
istic length scale is given by the droplet radius R. Indeed, this simple relation is an
approximation, which is only valid for the l= 1 mode of Eq. (3.34). It is therefore only
applicable if variations in σare on the order of the droplet radius R. For finer variations
44
in σ, one has to resort to the full expression (3.34). In contrast to the simple relation
(3.36), the full expression prevents the flow field u|Rfrom diverging for sharp steps in σ.
In Fig. 3.14 (b), we compare both expressions for the case when σis given by a smoothed
step function.
The surface divergence ∇
s·u|R, which we introduced in Sect. 3.3.6, is given by
(Rsin θ)−1∂θ(sin θuθ|R) in the case of an axisymmetric flow field . For the simple flow
field given in Eq. (3.36), one finds
∇
s·u|R=2σ0cos θ
(2η+ 3ˆη)Ror ∇
s·u|R∝ −σ(θ),(3.38)
see Fig. 3.13 (a). This illustrates an important property of droplet Marangoni flow. Since
it is the lowest mode in σthat leads to propulsion of the droplet and the lowest mode
leads to ∇
s·u|R6= 0, we conclude that droplet propulsion is always accompanied by
a non-vanishing surface divergence. This is a general property of active particles [107].
Furthermore, at the front of the droplet, the surface divergence ∇
s·u|R>0, i.e., there
is a source of matter. This is realized by a radial flow of matter (usually surfactants)
towards the interface, as depicted in Fig. 3.13 (b) and derived in Ref. [19], which is given
in Sect. 4.3.
Surfactants then adsorb at the interface and move along the streamlines of u|Rtowards
the posterior end of the droplet, where they desorb back into the bulk. Accordingly, the
surface divergence is negative, i.e. ∇
s·u|R<0, at the back of the droplet. This exchange
of matter with the bulk by adsorption and desorption is a necessary condition for droplet
flow driven by the Marangoni effect.
Three slightly more elaborate surface tension profiles are shown together with uθ|R
as well as streamlines of the interior and the surrounding flow in Fig. 3.14. The surface
tension is given by the smoothed step function
σ(θ) = 1
πarctan θ−θ0
ǫ+2
π,(3.39)
for ǫ= 0.1 and (a) θ0=π
4, (b) θ0=π
2, and (c) θ0=3π
2. The streamlines are depicted
in the droplet frame as well as in the lab frame, which is given by uL=u+v. The
lab frame is characterized by uL|r→∞ = 0, while in the droplet frame u|r→∞ =−v.
Furthermore, note that the normal velocity at the interface uL
r|Rdoes not vanish in the
lab frame. The resulting internal flow in the droplet frame develops into two vortices.
Indeed, any surface tension σ, which is not a monotonic function of θ, leads to more
than two vortices. Note that depending on θ0, hence the position of the step in σ(θ),
the droplet is either (a) a puller, (b) a neutral swimmer, or (c) a pusher; compare the
discussion in Sect. 2.1.3.
45
(a) (b) (c)
πθ
σ
πθ
σ
πθ
σ
πθ
uθ|R
πθ
uθ|R
πθ
uθ|R
approximation
(3.37)
droplet frame
lab frame
θ
vθ
vθ
v
Figure 3.14: Surface tension σ(θ), surface velocity flow field uθ|R, and stream lines of
the flow fields in and around the droplet for three different shapes of σ(θ).
(a) The step in σ(θ) is located at the front of the droplet. The droplet is a
puller. (b) The step is located at the equator and the droplet is a neutral
swimmer. (c) The step is located at the back and the droplet is a pusher.
46
3.5 Experiments, models, and theories on active emulsion
droplets
Before discussing several models of active emulsion droplets in Sect. 4, we want to give
the reader an overview of experiments, models, and theories on active emulsion droplets
in the literature.
3.5.1 Early attempts on droplet Marangoni flow
Prior to experimental realizations of active emulsion droplets, researchers started to
study the stability of flat and spherical interfaces. In 1959, Sternling and Scriven showed
that convection cells can be generated in a thin horizontal liquid layer, which is heated
from below [108]. In resemblance to Rayleigh-B´enard cells, these cells are called B´enard-
Marangoni cells. However, the instability, which leads to B´enard-Marangoni cells, is only
due to the Marangoni effect and not due to buoyancy. This hydrodynamic instability
of an interface was then also studied on a spherical interface in 1976 [109] and in 1979
[110].
The first model of a self-propelled emulsion droplet was presented by Ryazantsev in
1985 [111] and refined in a subsequent publication a year later [112]. He studied droplet
motion due to thermocapillary Marangoni flow, where the temperature gradient on the
droplet interface is a consequence of the motion of the droplet itself and not externally
generated, as assumed in former systems, e.g., in Ref. [113]. The model of Refs. [111, 112]
was extended to solutocapillary Marangoni flow in 1990 [114].
In 1993-1995 several articles were published by Ryazantsev, Rednikov, et al., which
refined the model above by including gravity [115, 116] and discussing several instability
thresholds for different swimming modes [117, 118, 119, 120]. Ref. [121] gives an overview
of these publications.
3.5.2 Current state of research on active emulsion droplets
In the following, we classify the publications by denoting citations with superscripts. E
stands for experimental, T for theoretical, S for simulations, and R for review.
Inspired by the growing interest in the physics of active particles, Sumino et al. and
Nagai et al. studied self-running droplets on interfaces in 2005 [122, 123]E. These studies
also initiated a comeback of the active emulsion droplet.
In 2007, the first experimental results of active emulsion droplets since the mid 1990s
were published by Hanczyc et al. [124]E. Oil droplets loaded with a fatty acid precursor,
which are placed into an aqueous fatty acid micelle solution, showed autonomous move-
ment. They also found directional motion of the droplets within chemical gradients and
exhibited a type of chemotaxis. In a follow-up study, Toyota et al. shortly report on sim-
ilar experiments of active oil emulsion droplets, which consume hydrolyzable surfactants
as a “fuel”, which is supported from the bulk phase [125]E.
In what follows, we present recent advancements about swimming emulsion droplets
in chronological order.
47
Figure 3.15: Trajectory of an active emulsion droplet over a duration of 400s, from Ref.
[8]. The scale bar represents 300µm.
2011
Kitahata et al. studied the spontaneous oscillatory droplet motion due to a Belousov-
Zhabotinsky (BZ) reaction medium inside a droplet [15]E,S. They were able to reproduce
their experimental results, which were obtained with a BZ droplet in an oil phase in a
petri dish, with numerical simulations of scroll waves inside the droplet.
At the same time Thutupalli et al. performed experiments on bromine enriched water
emulsion droplets in oil [8]E. The droplets swim due to the bromination reaction of
monoolein, discussed in Sect. 3.3.3. Figure 3.15 shows a path of a bromine water droplet
in oil. Apart from a description of the propulsion mechanism, Ref. [8]Econtains results
of the collective motion of the droplets. The results of Ref. [8]Ehave been extended by
Thutupalli and Herminghaus to the self-propulsion of oscillating droplets [16]E.
2012
Yoshinaga et al. derived an amplitude equation, which explains the drift instability in
the motion of an active emulsion droplet [126]T,S. Numerical simulations support the
obtained critical point at the onset of motion, as well as the characteristic velocity of
the droplet.
Furthermore, Tjhung et al. simulated a droplet filled with actomyosin (“an active
gel whose polarity describes the mean sense of alignment of actin fibres”) to model the
spontaneous symmetry breaking of contractile active cells [5]S. The droplet is found to
be elongated perpendicular to the swimming direction. Stationary swimming as well as
oscillatory motion, and spiral motion is observed.
Yabunaka et al. introduced a model of an active emulsion droplet by means of a phase-
field model that does not consider the interface explicitly [127]T,S. Here, the droplet
48
motion arises due to a reaction of the interface with a chemical product in the bulk
phase. They found a drift bifurcation between a resting and a swimming droplet.
Finally, Sanchez et al. presented experiments of active water-in-oil emulsion droplets,
which contain extensile microtubule bundles [4]E. The flow inside the droplet resembles
the cytoplasmic streaming in fruit fly egg cells. A short note by Marchetti discusses how
the results of Ref. [4]Ecan shed light on the physics of dynamic reorganization, which
occurs inside living cells [3].
2013
Ban et al. presented experimental results of active oil emulsion droplets in an aqueous
phase of NaOH solution [128]E. Here, the swimming of the droplet is due to the depro-
tonation of DEHPA surfactants at the interface. They found that the swimming of the
droplets depends critically on the pH condition of the aqueous bulk phase as well as on
the radius of the droplets.
Albeit Ref. [129]T,Sconsiders a droplet on a surface, we still want to mention it here.
Nagai et al. studied experimentally and theoretically the rotational motion of a flat
droplet induced by Marangoni flow. Here, the droplet rotation is only possible since a
small particle is attached at the interface, which breaks the mirror symmetry about the
anterior-posterior axis.
Finally, Michelin et al. showed that the onset of motion of initially isotropic particles
is a quite general mechanism [130]T,S. The requirements are (i) surface flow and (ii)
advection of a solute to the front of the particle and away from the back of the particle.
Their calculations also show that such a swimmer is a pusher.
2014
Yoshinaga introduced a model of a swimming emulsion droplet, whose shape can deform
from a sphere [131]T,S. He found that the swimming droplet is elongated perpendicular
to the swimming direction and that it is a pusher.
Izri et al. showed that self-propulsion of active droplets is also possible for pure
monoolein covered water droplets (i.e. without bromine) in an oil phase [132]E. Their
droplets shrink while they swim. Furthermore, Izri et al. noticed that inverse micelles
in the oil phase are necessary for propulsion. They, therefore, proposed that water is
solubilized by the inverse micelles, and hence produces a gradient of water outside the
droplet.
A similar approach is taken in Ref. [133]T,E, which was published by Herminghaus et
al.. It explains the locomotion of liquid crystal filled emulsion droplets by means of a
molecular pathway of solubilizing the liquid crystal in micelles surrounding the droplet.
In contrast, a micellar pathway, where micelles are filled directly at the droplet interface
does not lead to locomotion.
Banno and Toyota reported on novel reactive surfactants, which lengthen the loco-
motion time of self-propelled n-heptyloxybenzaldehyde droplets in a cationic reactive
surfactant solution [134]E.
49
Finally, Whitfield et al. presented a continuum level analytical model of a contractile
droplet of active polar fluid consisting of filaments and motors [135]T,S.
2015
Shklyaev studied theoretically the self-propulsion of a so-called Janus droplet, where
only half of the interface is active [136]T,S. Here, the swimming does not depend on a
symmetry breaking instability, which allows to scale down the droplet to micrometers.
2016
Zwicker et al. studied shape instabilities of growing active emulsion droplets that trigger
the devision into two smaller droplets [137]S. Such droplets resemble the proliferation
of living cells and could serve as a model for protocells.
A comprehensive review on swimming droplets by Maass et al. is published [6]R.
50
4 Publications
This section is the centerpiece of the thesis. It puts together knowledge about mi-
croswimmers in general and droplets in particular from Sects. 2 and 3 to study several
applications of active emulsion droplets. Before we proceed with the corresponding pub-
lications, we give a short synopsis where we position our findings within the current state
of research on active emulsion droplets, discussed in Sect. 3.5.
4.1 Synopsis
In the first publication “Swimming active droplet: A theoretical analysis”, which we
present in Sect. 4.2, we develop a model for the active emulsion droplet of Thutupalli
et al. [8]. We start from the mixing free energy of pristine and brominated monoolein
surfactants, along the lines of Secs. 3.3.5-3.3.6, and derive a diffusion-advection-reaction
equation for the surfactant mixture at the droplet interface. For the Marangoni flow
around the droplet and at the interface, we employ the axisymmetric formulas (3.34)
and (3.35) by Levan et al. [64].
Numerical solutions obtained by a Finite difference scheme reveal a stable swimming
regime above a critical Marangoni number M. The bifurcation at the onset of motion is
found to be subcritical, in agreement with the findings in a different model by Yoshinaga
et al. [126]. The swimming emulsion droplet is a pusher, which confirms the experimental
findings in Ref. [8], as well as the general results for swimming particles, which are
initially isotropic, by Michelin et al. [130]. When Mis increased further, the droplet
stops again and becomes a shaker. The flow field in that case corresponds to the flow field
found in early studies on the stability of spherical liquid interfaces [109, 110]. Finally,
for even larger Mthe droplet reaches an oscillating state similar to the experimental
findings in Refs. [15] and [16]. In our simulations, the swimmer type also oscillates
between pusher and puller, see Fig. 4.1 (a). Note that these results are for a droplet,
which is axisymmetric about the swimming axis. Thus, the swimming trajectory is
always one-dimensional.
From Fig. 3.15, which shows the path of a swimming emulsion droplet, it is clear that
these droplets do not swim along straight lines but rather with an erratically changing
direction. A simple model for the random walk of a self-propelled particle is the active
Brownian particle, discussed in Sect. 2.3. However, this raises the question, where the
random torque ω(t) in the corresponding Langevin equation (2.20) originates from. To
tackle this question, we generalize the model of Sect. 4.2. We omit the axisymmetric
constraint and include thermal noise into the diffusion-advection-reaction equation for
the surfactant mixture, akin to Eq. (3.12) in Sect. 3.3.6. To omit the axisymmetric
constraint, we need to use the full three-dimensional Marangoni flow of a spherical
51
(a)
oscillation ❹
z
t
pusher puller
(b)
e
yx
z
(c)
y
x
κ= 4
κ= 20
(d)
κ= 10
blue laser
y
z
κ= 3
blue laser
y
z
Figure 4.1: Samples of droplet trajectories, which are discussed in Secs. 4.2-4.4. (a)
Oscillating axisymmetric droplet. (b) Thermal fluctuations lead to persis-
tent random walk. (c) Micelles can induce directed motion of an initially
surfactant free droplet. (d) Transparent droplet pushed by blue light.
droplet. The solution for the flow field was available in the literature, see Sect. 3.4.5.
However, the derivation is very compact and the results are written in a manner which
is not well suited for our applications. We therefore present and alternative derivation
step by step in the publication “Marangoni flow at droplet interfaces: Three-dimensional
solution and applications”, presented in Sect. 4.3. There, we specifically focus on active
droplets and their swimming kinematics, i.e., the velocity vector vand the squirmer
parameter β. We also discuss two applications, which do not directly concern our model
of the bromine water-in-oil droplet. We will come back to these at the end of this section.
Having at hand the three-dimensional Marangoni flow, we can generalize the dynamic
equation of the surfactant mixture to account for surfactant mixtures with arbitrary
profile on the spherical emulsion droplet. Furthermore, we add thermal noise. This
allows us to focus on two new aspects of droplet dynamics that we could not address
before. They are presented in the third publication “Active Brownian motion of emulsion
droplets: Coarsening dynamics at the interface and rotational diffusion” in Sect. 4.4.
First, we study in detail the dynamics of a droplet with an initially uniform surfactant
mixture towards a stationary uniaxial swimming state, where the surfactant mixture is
phase-separated into the two surfactant types. We quantify the coarsening dynamics
by means of the growth rate of domains. Two steps exist: An initially slow growth
52
of domain size is followed by a nearly ballistic regime, which is reminiscent of coars-
ening in the dynamic model H [138]. Second, we address the random changes in the
swimming direction, which were observed in experiments [8]. The thermally fluctuating
surfactant mixture induces random changes in the swimming direction, and thereby the
emulsion droplet behaves like an active Brownian particle. Figure 4.1 (b) shows a sample
trajectory from a simulation. We characterize trajectories by means of the rotational
correlation time τr, introduced in Sect. 2.3, and discuss how the noise strength of the
thermal fluctuations affects τr.
As noted above, the publication in Sect. 4.3 also contains two other applications of
the three-dimensional droplet Marangoni flow. The simplest way to generate Maran-
goni flow is a non-uniform distribution of a single surfactant type at an interface, as
noted in Sect. 3.4 and depicted in Fig. 3.11 (a). In our first application we consider an
initially surfactant free emulsion droplet immersed in a micelle enriched fluid. When a
micelle adsorbs at the droplet interface, the surfactants are spread by Maragoni flow and
thereby the droplet is propelled in direction of the adsorption site. The corresponding
outer Marangoni flow field, however, preferentially advects other micelles towards the
existing adsorption site. Hence, when the concentration of micelles is sufficiently large,
this mechanism can spontaneously break the isotropic symmetry of the droplet and lead
to directed motion. Figure 4.1 (c) shows the projection onto the x-y-plane of two sam-
ple trajectories for high and low concentration of micelles with parameter κ= 20 and
κ= 4, respectively. The dots indicate the micelle adsorption events. Our explicit hydro-
dynamical treatment of the spontaneous symmetry breaking due to micelle adsorption
could help to understand other systems, in which micelles have been shown to be crucial
[132, 133].
In the second example of Sect. 4.3 we propose light-driven droplet propulsion. By
partially illuminating an emulsion droplet covered by light-switchable surfactants, one
locally generates a spot of different surface tension, which induces Marangoni flow and
hence propulsion of the droplet. So far, light-driven droplet propulsion was only studied
on interfaces. The Baigl group found that droplets on an interface can be pushed with
UV light or pulled with blue light, see Sect. 3.3.4 and Ref. [20]. We find a similar behavior
of the emulsion droplet. Furthermore, depending on the relaxation rate κtowards the
surfactant in bulk, the UV illuminated droplet shows a plethora of trajectories. It can
perform a damped oscillation about the beam axis, oscillate in the beam axis, or leave the
beam and stop. Figure 4.1 (d) shows two sample trajectories. We explore these cases for
strongly absorbing and for transparent droplets and also discuss, how the results depend
on the type of emulsion (water-in-oil vs. oil-in-water).
Table 4.1 summarizes the content of the publications.
53
Sect. 4.2
Swimming active droplet: A theoretical analysis
•Model for Thutupalli et al. droplet from Ref. [8] based on Diffusion-
Advection-Reaction equation for surfactant mixture
•Axisymmetric simulations reveal stable swimming, stopping, and oscil-
lating states
•Study of droplet dynamics in reduced phase space
Sect. 4.3
Marangoni flow at droplet interfaces:
Three-dimensional solution and applications
•Derivation of flow field, velocity vector, stresslet tensor, and squirmer
parameter of droplet with arbitrary surface tension σ(θ, ϕ)
•Spontaneous symmetry breaking and persistent motion of droplet due
to micelle adsorption
•Light-induced Maragoni flow: Pushing/pulling an absorbing or trans-
parent droplet with UV/blue light
Sect. 4.4
Active Brownian motion of emulsion droplets:
Coarsening dynamics at the interface and rotational diffusion
•Refined model for Thutupalli et al. droplet without axisymmetry and
with thermal noise
•Coarsening dynamics of surfactant mixture towards swimming state
studied by means of power spectrum of σand average domain size
•Rotational diffusion of active droplet due to thermal noise
•Perturbation theory confirms scaling law τr∝ξ−2for rotational corre-
lation time τrand noise strength ξ
Table 4.1: Overview of the publications in Sects. 4.2-4.4.
54
56
epl draft
Swimming active droplet: A theoretical analysis
M. Schmitt and H. Stark
Institut f¨ur Theoretische Physik, Technische Universit¨at Berlin - Hardenbergstraße 36, 10623 Berlin, Germany, EU
PACS 47.20.Dr – Surface-tension-driven instability
PACS 47.55.D- – Drops and bubbles
PACS 47.55.pf – Marangoni convection
Abstract – Recently, an active microswimmer was constructed where a micron-sized droplet of
bromine water was placed into a surfactant-laden oil phase. Due to a bromination reaction of the
surfactant at the interface, the surface tension locally increases and becomes non-uniform. This
drives a Marangoni flow which propels the squirming droplet forward. We develop a diffusion-
advection-reaction equation for the order parameter of the surfactant mixture at the droplet
interface using a mixing free energy. Numerical solutions reveal a stable swimming regime above
a critical Marangoni number Mbut also stopping and oscillating states when Mis increased
further. The swimming droplet is identified as a pusher whereas in the oscillating state it oscillates
between being a puller and a pusher.
Introduction. – A rigorous understanding of swim-
ming on the micron scale is crucial for developing microflu-
idic devices such as a lab-on-a-chip [1]. This understand-
ing comes from watching nature, i.e., by studying the loco-
motion of living organisms such as bacteria or algae [2] but
also from designing artificial microswimmers, used for ex-
ample as medical microrobots [3]. Both, real live cells and
man-made microswimmers, have thoroughly been used to
study interaction between swimmers [4], interaction with
walls [5–8], or swarming [9]. One possible design of an
artificial swimmer is an active droplet. Here, we think
of a droplet with a surface where a chemical reaction
occurs. Alternatively, droplets or bubbles can be made
active by having an internal heat source [10]. Droplets
are particularly interesting systems to study since they
are used extensively in microfluidic devices as microreac-
tors in which chemical or biological reactions take place
[11,12]. In the following we give an example of an active
droplet and investigate in detail its propulsion mechanism.
Self-propelled active droplets have been studied in vari-
ous experiments, including droplets on interfaces [13, 14],
droplets coupled to a chemical wave [15], and droplets in a
bulk fluid [16–19]. Theoretical treatments include a model
of droplet motion in a chemically reacting fluid [20], stud-
ies of the stability of a resting droplet [21–25], and simu-
lations of contractile droplets [26] and of droplets driven
by nonlinear chemical kinetics [27].
The swimming active droplet we consider in the follow-
ing is a solution of water and bromine which is placed in a
surfactant-rich oil phase [18]. The resulting water droplet
has a typical radius of 80µm. In order to lower the sur-
face tension and thus the total energy of the system, the
surfactants in the oil phase form a dense monolayer at the
droplet interface, giving the droplet the structure of an
inverse micelle. The observed directed swimming motion
of the droplet with a typical swimming speed of 15µm/s
can be understood as follows [18].
The bromine within the droplet chemically reacts with
the surfactants in the interface which results in a weaker
surfactant. Hence, the ’bromination’ reaction locally leads
to a higher surface tension in the interface. As a conse-
quence local gradients in surface tension will lead to a fluid
flow at the interface and in the adjacent fluid inside and
outside of the droplet in the direction of increasing sur-
face tension. This effect is called Marangoni effect. The
fluid flow then leads in turn to advection of surfactants at
the interface. As a result gradients in surface tension are
enhanced. Thus, the resting state becomes unstable and
the droplet starts to move. Additionally, brominated sur-
factants are constantly replaced by non-brominated sur-
factants from the oil phase by means of desorption and
adsorption. The droplet stops to swim when either the
bromine or the non-brominated surfactants in the oil phase
are exhausted. This was also observed in the experiments
[18].
The active droplet is an interesting realization of the
’squirmer’ [28,29] which has been introduced to model the
locomotion of microorganisms. Often they propel them-
p-1
M. Schmitt et al.
selves by a carpet of beating short filaments called cilia on
their surfaces. Instead of modeling each cilium separately,
one prescribes the fluid flow at the surface initiated by
the beating cilia which then drags the squirmer through
the fluid. Here, for the active droplet the surface flow is
generated by the Marangoni effect.
The swimming active droplet crucially depends on the
dynamics of the mixture of non-brominated and bromi-
nated surfactants at the interface. In this article we
model it by means of a diffusion-advection-reaction equa-
tion based on a free energy functional for the surfactant
mixture. Numerical solutions then show that in a certain
parameter range the resting state of the droplet becomes
unstable and the droplet starts to move. The solutions
reach a stationary state corresponding to a swimming mo-
tion and confirm that the droplet is a pusher [2], as found
in the experiments [18]. In addition, we identify further
patterns of motion. We find that the droplet stops after
an initial motion or that it oscillates back and forth.
Model. – In order to model the droplet propulsion we
set up a dynamic equation for the surfactant mixture at
the droplet interface that includes all processes mentioned
before. We assume that the surfactant completely cov-
ers the droplet interface without any intervening solvent.
We also assume that the area of both types of surfac-
tant molecules (brominated and non-brominated) is the
same. Denoting the brominated surfactant density by c1
and the non-brominated density by c2, we can therefore
set c1+c2= 1. We then take the concentration difference
between brominated and non-brominated surfactants as
an order parameter φ=c1−c2. In other words φ= 1
corresponds to fully brominated and φ=−1 to fully non-
brominated and c1= (1+φ)/2 and c2= (1−φ)/2. Finally,
we choose a constant droplet radius R.
Diffusion-Advection-Reaction equation. The dynam-
ics of φis governed by a diffusion-advection-reaction equa-
tion: ˙
φ=−∇ · (jD+jA)−τ−1
R(φ−φeq),(1)
with diffusive current jDand advective current jAdue
to the Marangoni effect. The third term on the right-
hand side of Eq. (1) is the reaction term and describes the
bromination reaction as well as desorption of brominated
and adsorption of non-brominated surfactants to and from
the outer fluid. τRis the timescale on which these pro-
cesses happen and φeq sets the equilibrium coverage of φ.
In other words, ad- and desorption dominates for φeq <0,
while bromination dominates for φeq >0. Imagine for in-
stance the case φeq = 1, i.e., a droplet with bromination
but without ad- and desorption of surfactants. The reac-
tion term would then always be positive, therefore driving
the droplet to a completely brominated state φ= 1.
The general mechanism of Eq. (1) is as follows. The dif-
fusive current always points ’downhill’, jD∝ −∇φ. How-
ever, we will show below that the opposite is true for jA
since approximately jA∝ ∇φ. Thus, apart from the reac-
tion term, jDand jAare competing and as soon as jA
dominates over jD,φexperiences ’uphill’ diffusion, i.e.
phase separation. As a result the resting state will be-
come instable and the droplet will start to move. We will
now present a careful derivation of jDand jAfrom a free
energy approach. This shows that the diffusive and advec-
tive currents in Eq. (1) are in general non-linear functions
of φ.
Diffusive current. The basis for the following is a free
energy density ffor the droplet interface, which we write
down as a function of concentrations c1and c2. In for-
mulating the free energy density f, we follow the Flory-
Huggins approach [30]. Accordingly, fis composed of the
mixing entropy plus terms mimicking lateral attractive in-
teraction between surfactants:
f=kBT
Ac1ln c1+c2ln c2−b1c2
1−b2c2
2−b12c1c2,
(2)
where Adenotes the area of a surfactant in the inter-
face and b1(b2) is a dimensionless parameter characteriz-
ing the interaction between brominated (non-brominated)
surfactants and b12 the interaction between different kind
of surfactants. In the following we assume for simplicity
b12 = (b1+b2)/2. In terms of the order parameter φwe
obtain:
f(φ) = kBT
Ah1+φ
2ln 1+φ
2+1−φ
2ln 1−φ
2
−3
8(b1+b2)−φ
2(b1−b2)−φ2
8(b1+b2)i.
(3)
The total free energy is then given by the functional
F[φ] = Zf(φ)dA . (4)
For a conserved order parameter field the diffusive cur-
rent is proportional to the gradient of the variation in F
with respect to φ[31]:
jD=−λ∇δF
δφ =−λf00(φ)∇φ(5)
=−λkBT
A1
1−φ2−1
4(b1+b2)∇φ , (6)
with positive mobility λ. Substituting jDinto Eq. (1)
yields a Cahn-Hilliard type equation [32]. Note that the
diffusion constant in Eq. (6) decreases with increasing in-
teraction energy. In fact, the condition jD∝ −∇φis only
fulfilled for a convex free energy with f00(φ)>0, i.e. if
b1+b2<4. In addition, the diffusion coefficient in jD
is smallest for φ= 0. It increases with |φ|and diverges
at |φ|= 1. An alternative approach of deriving diffusion
currents in mixtures is presented in [33,34].
Advective current. The advective current for the order
parameter φis given by
jA=φu,(7)
p-2
Swimmingactivedroplet
whereuisthevelocityofthesurfactantsattheinterface.
1
Since wearestudyingtheactivedropletinanaxisym-
metricgeometry,weassume φ= φ(θ)andu= u
θ
(θ)e
θ
,
wherethefrontofthedropletisat θ=0,seeinsetof
Fig.1(b).Forthisgeometrythereexistsasolutionofthe
Stokesequationforthefluidflowfieldinsideandoutside
ofthedropletaswellasthefluidvelocityattheinterface
[35,36]. Thesolutionattheinterfaceisgivenintermsof
thesurfacetensiongradient:
u
θ
|
r=R =
∞
n=2
n(n−1)
2η
π
0
C
−1/2
n(z)
dσ
dθdθ C
−1/2
n(z)
sinθ ,
(8)
wherez=cos(θ).η=η
i
+η
oisthesumoftheviscosities
insideandoutsideofthedropletandC
−1/2
nare Gegen-
bauerpolynomials. TheyarerelatedtoLegendrepolyno-
mialsby P
n
(z)=−d
dz
C
−1/2
n+1 (z). Equation(8)isnothing
butarepresentationofthethe Marangonieffect.Itessen-
tiallystatesu∝ ∇σ,i.e.,afluidflowinthedirectionof
∇σ.
Thus,inordertocalculateu
θ
, weneedanexpression
fordσ/dθ, whichcanbefoundbyderivinganequation
ofstateforthesurfacetensionσ. Thesurfacetensionσ
isthethermodynamicforceconjugatetothesurfacearea.
Thisgives:
σ=f− ∂f
∂c
1
c
1−∂f
∂c
2
c
2, (9)
which weidentifyastheLegendretransformofthefree
energy(2)tothechemicalpotentialsµ
i=∂f
∂c
i. Hence,
σ= k
BT
Ab
1
c
2
1+b
2
c
2
2+b
12
c
1
c
2,orintermsofφ and
againwithb
12=(b
1+b
2
)/2:
σ(φ)=
k
BT
4A
9
8
(b
1+b
2
)+2(b
1−b
2
)φ+7
8
(b
1+b
2
)φ
2.
(10)
Inordertoobtaintheproperbehavioroftheequation
ofstate,i.e. anincreasingsurfacetension withincreas-
ingφ, weneedtoassurethatσ(φ)> 0. Thisholds
ifb
1>b
2
, meaningthattheinteractionenergybetween
brominatedsurfactantshastobehigherthanbetweenthe
non-brominatedones. Notethatinthelimitofφ→ 0the
equationofstatebecomeslinearinφ. Thegradientofσ
isgivenby
dσ
dθ=σ(φ)
dφ
dθ=k
BT
2A (b
1−b
2
)1+
7
8
b
1+b
2
b
1−b
2
φdφ
dθ.
(11)
BysubstitutingthisintoEq.(8),onecancalculatethe
advectivecurrent(7)foragivenφ(θ).
Eqs.(8)and(11)essentiallystatethatu∝∇φ. There-
fore,whenφ >0,theadvectivecurrentj
A= φuappar-
entlyalwayspoints’uphill’,i.e.,intheoppositedirection
comparedtoj
D. Ontheotherhand, whenφ <0,the
1
Lettheadvectivecurrentsofthetwotypesofsurfactantsbe
j
1
A=c
1
u
1andj
2
A=c
2
u
2
. Undertheassumptionthattheindividual
velocitiesareidenticalu
1=u
2=u,oneobtainsj
A=j
1
Aj
2
A=φu
-1
-0.5
0
0.5
1
0 π/2 π
φ
θ
(a)
init. cond.
M=2.5 (rests) ❶
M=3.0 (swims) ❷
M=4.5 (stops) ❸
.
-0.4
-0.2
0
0.2
0.4
0 π/2 π
u
θ
θ
u
θ
θ
z
(b)
M=2.5 (rests) ❶
M=3.0 (swims) ❷
M=4.5 (stops) ❸
-0.4
-0.2
0
0.2
0.4
0 π/2 π
u
θ
θ
Fig.1:(a)Stationaryorderparameterprofilesafter10
6time
stepsforφ
eq =0.5andseveral MarangoninumbersM. Gray
solidline:Initialcondition.(b)Correspondinginterfaceveloc-
ityprofiles.Inset: Dropletgeometry.
advectivecurrentacts’downhill’,i.e.,inthesamedirec-
tionasj
D. Asaconsequence,the Marangoniflow will
onlydrivethedroplet whenφ > 0. Thisisthecase
whenthereare morebrominatedsurfactantsthannon-
brominatedones.
Together with(6)and(7), Eq.(1)becomesaclosed
equationforφ. WritinggradientsinunitsofR
−1andtime
inunitsofthediffusiontimescaleτ
D= R
2
A(λk
BT)
−1
yields
˙
φ=−∇·(j
D+Mφu)−κ(φ−φ
eq
), (12)
wherethecurrentsj
Dandj
A=Mφuarenowdimension-
lessand
M = (b
1−b
2
)R
λη , (13)
iscalled Marangoninumber. Thisnumbercomparesthe
advectivecurrentduetothe Marangonieffect,whichscales
ask
BT(b
1−b
2
)(RAη)
−1
,tothediffusivecurrent. Ac-
cordingly,κ= τ
Dτ
−1
Ristheratiobetweendiffusionand
reactiontimescale.
Results. – We numerically solve the diffusion-
advection-reactionequationforφwiththeinitialcondi-
tionφ(θ)=φ
eq+δφ(θ),whereδφ(θ)isasmallperturba-
tion[solidlineinFig.1(a)]. Theboundaryconditionsat
θ=0,πaregivenbyavanishingcurrent,j
D+Mφu=0.
p-3
M. Schmitt et al.
-0.8
-0.4
0
0.4
0.8
05*105106
vD
t
(a)
swims ❷
stops ❸
oscillation ❹
t
(b)
0
π/2
π
θ
-0.5 0 0.5 1
φ
oscillation ❹
Fig. 2: (a) Droplet swimming velocity vDfor swimming, stop-
ping, and oscillating droplets. Parameters are the same as in
Fig. 1 and case 4 belongs to M= 10.5. (b) Depiction of the
chemical wave of case 4 in a φ(θ, t) plot. Same timescale as in
(a).
We keep κfixed to a value of 0.1 for all numerical solu-
tions and comment later on the impact of κon the results.
Therefore, we are left with the Marangoni number Mand
φeq as the crucial parameters. To assure a convex free
energy, we set b1+b2= 3.
Order parameter and velocity profiles. Figure 1(a)
shows examples of the stationary order parameter pro-
file for φeq = 0.5 and different values of Mtogether with
the corresponding interface velocity profiles in Fig. 1(b).
Starting with a small Marangoni number of M= 2.5, the
order parameter relaxes into the homogeneous trivial so-
lution φ=φeq of Eq. (1), thus the droplet rests. Above
a critical Marangoni number, the order parameter evolves
to a stationary inhomogeneous profile, as Fig. 1 shows for
M= 3. In parallel, the droplet velocity vDdepicted in
Fig. 2(a) saturates on a non-zero value. The droplet swim-
ming speed is given by vD= (6ηi+ 4ηo)−1Rπ
0sin2θdσ
dθ dθ
[35]. Since C−1/2
2(cos(θ)) = sin2(θ)/2, vDis determined
by the first coefficient of the sum in (8) and thus vD=
8
π
ηi+ηo
6ηi+4ηoRπ
0sin θuθdθ. Note that in our approach vD
reaches a stationary value without having to introduce a
’backward’ Marangoni stress, as suggested in [18]. Further
increasing the Marangoni number to M= 4.5, the droplet
starts to swim but then stops rapidly. The stationary or-
der parameter profile becomes symmetric around θ=π/2
and swimming is not possible. Finally, the droplet reaches
2
4
6
8
10
12
14
0 0.2 0.4 0.6 0.8
M
φeq
❶
❷
❸
❹
rests
swims stops
oscillation
Fig. 3: Phase diagram of the active droplet in (φeq, M) param-
eter space. Examples for the order parameter profiles at the
positions marked with numbers are given in Fig. 1(a) (regime
1-3) and Fig. 2(b) (regime 4).
an oscillating state for even higher Marangoni numbers
where it oscillates back and forth as the droplet swim-
ming speed in Fig. 2(a) demonstrates. In this case the
order parameter φ(θ, t) resembles a chemical wave that
travels back and forward between θ= 0 and θ=π. The
wave is depicted in Fig. 2(b). Note that the frequency of
the oscillation increases with M. Finally, we remark that
from comparing Figs. 1 (a) and (b), it is now apparent
that indeed Eq. (8) essentially gives u∝ ∇φ.
Phase diagram. Figure 3 shows the phase diagram
in (φeq, M) parameter space with the four regimes of the
droplet dynamics: resting, swimming, stopping, and os-
cillating. Since there is no swimming motion possible for
negative φeq, as discussed before, we only show the phase
diagram in the range 0 ≤φeq ≤0.8.2We find similar
phase diagrams for smaller values of κ. For κ= 0.01
the swimming region increases in size and then shrinks
again for κ= 0.001 until for κ= 0 swimming solutions
are no longer possible. The critical Marangoni number at
the onset of the swimming regime stays, however, roughly
constant. On the other hand, for κ= 1 and 10, i.e., in the
limit of fast bromination reaction and exchange of surfac-
tants, only resting, stopping and oscillating solutions but
no stable swimming solutions were found.
Reduced phase space. Due to the axisymmetric geom-
etry we decompose the order parameter φinto Legendre
modes
φ(θ, t) =
∞
X
n=0
Pn(cos(θ))φn(t).(14)
2Due to the φdependent diffusion coefficient in Eq. (6), numerics
requires a much finer grid above φeq = 0.8. However, in several tests
for different values of Mno swimming solutions were found above
φeq = 0.8.
p-4
Swimming active droplet
-0.4
0
0.4
-0.4 -0.2 0
φ2
φ1
M≤2.7
M=2.8
M=4.2
M=4.4
M=9.4
M=4.5
M=3
swims ❷
stops ❸
-0.2
0
0.2
-0.1 0 0.1
oscillation ❹(a)
0
0
(b)
PUS HER
PUL LER
neutralneutral
shakershaker
Fig. 4: Droplet dynamics in the reduced phase space (φ1, φ2).
Black dots show the fixed points for different values of M;
from dot to dot Mincreases by 0.2. The red and the green
line show, respectively, trajectories in the swimming (M= 3)
and stopping (M= 4.5) state. Inset (a): limit cycle in the
oscillating state (M= 10.5). Inset (b): map for the swimmer
type in (φ1, φ2) space classified by the stirring parameter β=
−φ2/|φ1|(see main text).
Together with Eqs. (11) and (8) one obtains an expression
for uθas a function of the mode amplitudes φn.φ1deter-
mines the swimming speed and φn>1corresponds to the
higher modes of uθ. In the following, we use the initial
condition φ0(t= 0) = φeq. In order to investigate the four
regimes of the droplet dynamics, we plot in Fig. 4 the fixed
points in (φ1, φ2) space for increasing Marangoni number
Mat φeq = 0.5. For the cases M= 3 and M= 4.5 the
full trajectories are shown. Note that this illustration is a
projection onto only two modes of infinitely many modes
that make up the full phase space of φ. Starting with the
resting state, one has a stable fixed point at φ1= 0, φ2= 0
for M≤2.7. Via a subcritical bifurcation the droplet en-
ters the swimming state at the critical Marangoni number
M= 2.8. Figure 4 demonstrates that for the chosen initial
condition (φ1, φ2)≈(0,0) both modes φ1and φ2develop
non-zero amplitudes at the same critical Marangoni num-
ber. The trajectory in the swimming state does increase
its size with increasing M, whereas the swimming speed
decreases until the droplet reaches the stopping state at
M= 4.3. As already observed in Fig. 1, the second mode
φ2, which is symmetric around θ=π/2, clearly domi-
nates in the stopping state. In the oscillating regime above
M= 9.5 a stationary solution does not exist. Instead, the
dynamics follows a stable limit cycle as the inset (a) in
Fig. 4 demonstrates for M= 10.5. Finally we remark,
since the bifurcation is subcritical, the critical Marangoni
number for the onset of the swimming state depends on
-8
-6
-4
-2
0
2
4
05*105106
z
t
pullerpusher
oscillation ❹
-∞ ... -10
-5
0
5
10 ... ∞
Fig. 5: Displacement of oscillating droplet plotted versus time.
Color of line shows the value of stirring parameter β.
the chosen initial condition. For example, starting the nu-
merical solution at φ1=φ2=−0.1 the critical Marangoni
number is M= 1.7.
The active droplet as pusher. To describe the ba-
sic features of a squirming swimmer, it is sufficient to
only study the first two modes of its surface velocity field
[2, 28, 29, 37, 38]. While the the first mode φ1determines
the swimming velocity, the dimensionless ’stirring’ param-
eter β=−φ2/|φ1|characterizes the swimmer type. When
βis negative, the flow around the droplet is similar to
the flow around a swimming bacterium such as E. coli.
Such a swimmer is called a ’pusher’ since it pushes fluid
away from itself at the front and at the back. Accord-
ingly, a swimmer with β > 0 is called a ’puller’. The
algae Chlamydomonas is an example for a puller since it
swims by pulling liquid towards itself at the front and at
the back [9]. For β→ ±∞ the droplet becomes a ’shaker’,
i.e., a droplet that shakes the adjacent fluid but does not
swim. If β= 0, the first mode dominates and propels the
droplet, as is the case for Volvox algae [9]. The classifica-
tion of the swimmers according to the ‘stirring’ parameter
βis illustrated in the inset (b) of Fig. 4. Hydrodynamic
interactions between swimmers and with bounding walls
depend on their type (‘stirring’ parameter β) and strongly
influence their (collective) dynamics [7,39]. For instance,
adjacent pushers generally tend to align and swim parallel
to each other, i.e., show a polar velocity correlation [40,41].
In fact this kind of behavior was observed in experiments
of our active droplets [18]. It is therefore of great interest
to determine β. The swimming droplet with φeq = 0.5
is a pusher with βranging from −0.7 for M= 2.8 to
−1.5 for M= 4.2. Similar values from β=−0.5 up to
−7 were observed throughout the whole swimming regime
of the droplet. The stopping droplet is always a shaker
with β=−∞. Since the limit cycle of the oscillating
droplet perambulates all four quadrants of the reduced
phase space, it oscillates in the swimming direction as well
as in β, i.e., between being a pusher and a puller. This is
demonstrated by the droplet displacement plotted versus
time in Fig. 5.
p-5
M. Schmitt et al.
Conclusions. – We have presented a model for an
active squirming droplet with a surfactant mixture at its
interface that drives a Marangoni flow and thereby drags
the droplet forward. Based on a free energy functional
for the mixture, we derived a diffusion-advection-reaction
equation for the mixture order parameter at the droplet
interface. Relevant parameters are the Marangoni number
Mand the reduced relaxation time κ−1with which the
mixture approaches its equilibrium value φeq by bromi-
nation or de- and absorption of the surfactants from the
surrounding.
As predicted from linear stability analysis in [18], nu-
merical solutions of the diffusion-advection-reaction equa-
tion show that above a critical Marangoni number the
resting state of the droplet becomes unstable. The order
parameter develops a non-uniform profile and the droplet
moves with a constant swimming velocity. This only oc-
curs when the relaxation time κ−1(relative to the diffusion
time) is sufficiently large. The negative stirring parameter
βidentifies the droplet as a pusher in agreement with po-
lar velocity correlations found in experiments [18]. A full
parameter study in (φeq, M) space also reveals a stopping
droplet, which is a shaker (β=−∞), and an oscillating
droplet that oscillates between being a puller and a pusher.
We hope that our work initiates further research on the
active droplet which constitutes an attractive realization
of the model swimmer called squirmer.
∗∗∗
We thank S. Herminghaus, U. Thiele, S. Thutupalli
and A. Z¨ottl for helpful discussions and the Deutsche
Forschungsgemeinschaft for financial support through the
Research Training Group GRK 1558.
REFERENCES
[1] Abgrall P. and Gu A.-M.,J. Micromech. Microeng. ,
17 (2007) R15.
[2] Lauga E. and Powers T. R.,Rep. Prog. Phys. ,72
(2009) 096601.
[3] Nelson B. J., Kaliakatsos I. K. and Abbott J. J.,
Annu. Rev. Biomed. Eng. ,12 (2010) 55.
[4] Ishikawa T., Sekiya G., Imai Y. and Yamaguchi T.,
Biophys. J. ,93 (2007) 2217 .
[5] Berke A. P., Turner L., Berg H. C. and Lauga E.,
Phys. Rev. Lett. ,101 (2008) 038102.
[6] Drescher K., Dunkel J., Cisneros L. H., Ganguly
S. and Goldstein R. E.,Proc. Natl. Acad. Sci. U. S. A.
,108 (2011) 10940.
[7] Z¨
ottl A. and Stark H.,Phys. Rev. Lett. ,108 (2012)
218104.
[8] Reddig S. and Stark H.,J. Chem. Phys. ,135 (2011)
165101.
[9] Lauga E. and Goldstein R. E.,Phys. Today ,65 (2012)
30.
[10] Rednikov A. Y., Ryazantsev Y. S. and Velarde
M. G.,J. Colloid Interface Sci. ,164 (1994) 168 .
[11] Teh S.-Y., Lin R., Hung L.-H. and Lee A. P.,Lab
Chip ,8(2008) 198.
[12] Seemann R., Brinkmann M., Pfohl T. and Herming-
haus S.,Rep. Prog. Phys. ,75 (2012) 016601.
[13] Chen Y.-J., Nagamine Y. and Yoshikawa K.,Phys.
Rev. E ,80 (2009) 016303.
[14] Bliznyuk O., Jansen H. P., Kooij E. S., Zandvliet
H. J. W. and Poelsema B.,Langmuir ,27 (2011) 11238.
[15] Kitahata H., Yoshinaga N., Nagai K. H. and Sumino
Y.,Phys. Rev. E ,84 (2011) 015101.
[16] Toyota T., Maru N., Hanczyc M. M., Ikegami T.
and Sugawara T.,J. Am. Chem. Soc. ,131 (2009) 5012.
[17] Hanczyc M. M., Toyota T., Ikegami T., Packard N.
and Sugawara T.,J Am Chem Soc ,129 (2007) 9386.
[18] Thutupalli S., Seemann R. and Herminghaus S.,New
J. Phys. ,13 (2011) 073021.
[19] Banno T., Kuroha R. and Toyota T.,Langmuir ,28
(2012) 1190.
[20] Yabunaka S., Ohta T. and Yoshinaga N.,J. Chem.
Phys. ,136 (2012) 074904.
[21] Rednikov A. Y., Ryazantsev Y. S. and Velarde
M. G.,Phys. Fluids ,6(1994) 451.
[22] Rednikov A. Y., Ryazantsev Y. S. and Velarde
M. G.,J. Non-Equil. Thermody. ,19 (1994) 95.
[23] Velarde M. G., Rednikov A. Y. and Ryazantsev
Y. S.,J. Phys.: Condens. Matter ,8(1996) 9233.
[24] Velarde M. G.,Phil. Trans. R. Soc. Lond. A ,356
(1998) 829.
[25] Yoshinaga N., Nagai K. H., Sumino Y. and Kitahata
H.,Phys. Rev. E ,86 (2012) 016108.
[26] Tjhung E., Marenduzzo D. and Cates M. E.,Proc.
Natl. Acad. Sci. U. S. A. ,109 (2012) 12381.
[27] Furtado K., Pooley C. M. and Yeomans J. M.,Phys.
Rev. E ,78 (2008) 046308.
[28] Lighthill M. J.,Commun. Pur. Appl. Math. ,5(1952)
109.
[29] Blake J. R.,J. Fluid. Mech. ,46 (1971) 199.
[30] Flory P. J.,J. Chem. Phys. ,10 (1942) 51.
[31] Bray A. J.,Adv. Phys. ,51 (2002) 481.
[32] Desai R. C. and Kapral R.,Dynamics of Self-Organized
and Self-Assembled Structures (Cambridge University
Press) 2009.
[33] Nauman E. and He D. Q.,Chem. Eng. Sci. ,56 (2001)
1999 .
[34] Thiele U., Archer A. J. and Plapp M.,Phys. Fluids
, (2012) (submitted).
[35] Levan M. D. and Newman J.,AIChE Journal ,22
(1976) 695.
[36] Mason D. P. and Moremedi G. M.,Pramana ,77
(2011) 493.
[37] Downton M. T. and Stark H.,J. Phys.: Condens. Mat-
ter ,21 (2009) 204101.
[38] Zhu L., Lauga E. and Brandt L.,Phys. Fluids ,24
(2012) 051902.
[39] Evans A. A., Ishikawa T., Yamaguchi T. and Lauga
E.,Phys. Fluids ,23 (2011) 111702.
[40] Ishikawa T., Simmonds M. P. and Pedley T. J.,J.
Fluid. Mech. ,568 (2006) 119.
[41] Ishikawa T. and Pedley T. J.,Phys. Rev. Lett. ,100
(2008) 088103.
p-6
64
PHYSICS OF FLUIDS 28, 012106 (2016)
Marangoni flow at droplet interfaces: Three-dimensional
solution and applications
M. Schmitt and H. Stark
Institute of Theoretical Physics, Technical University Berlin, Hardenbergstraße 36,
10623 Berlin, Germany
(Received 15 June 2015; accepted 15 December 2015; published online 11 January 2016)
The Marangoni effect refers to fluid flow induced by a gradient in surface tension
at a fluid-fluid interface. We determine the full three-dimensional Marangoni flow
generated by a non-uniform surface tension profile at the interface of a self-propelled
spherical emulsion droplet. For all flow fields inside, outside, and at the interface
of the droplet, we give analytical formulas. We also calculate the droplet velocity
vector vD, which describes the swimming kinematics of the droplet, and generalize the
squirmer parameter β, which distinguishes between different swimmer types called
neutral, pusher, or puller. In the second part of this paper, we present two illustrative
examples, where the Marangoni effect is used in active emulsion droplets. First, we
demonstrate how micelle adsorption can spontaneously break the isotropic symmetry
of an initially surfactant-free emulsion droplet, which then performs directed motion.
Second, we think about light-switchable surfactants and laser light to create a patch
with a different surfactant type at the droplet interface. Depending on the setup such
as the wavelength of the laser light and the surfactant type in the outer bulk fluid,
one can either push droplets along unstable trajectories or pull them along straight
or oscillatory trajectories regulated by specific parameters. We explore these cases
for strongly absorbing and for transparent droplets. C2016 AIP Publishing LLC.
[http://dx.doi.org/10.1063/1.4939212]
I. INTRODUCTION
Self-propelled particles swimming in fluids at low Reynolds number have recently gained a lot
of attention.1–4Different methods to construct microswimmers exist. One idea is to generate a slip
velocity field close to the swimmer’s surface by different phoretic mechanisms that drag the particle
forward. A typical example of an artificial swimmer is a nano- or micron-sized Janus colloid. It has two
distinct faces that differ in their physical or chemical properties.5In the simplest realization, one face
catalyzes a chemical reaction and the reactants set up a self-diffusiophoretic flow.6A combination of
self-diffusio- and electrophoresis close to bimetallic Janus particles in a peroxide solution generates
an electrochemical gradient to propel the swimmer.7,8Heating a Janus particle, where the thermal
conductivity of both faces differs, generates a temperature gradient, in which the colloid moves. This
effect is called thermophoresis.9Finally, in a binary solvent close to the critical point, the liquid around
Janus colloids demixes locally, which also induces a self-diffusiophoretic flow.10
Both the individual swimming mechanisms of these Janus particles and other microswimmers as
well as their collective motion have evolved into very attractive research topics.4,11 In fact, the study of
collective motion in non-equilibrium systems has opened up a new field in statistical physics. Recent
studies of collective motion also concentrate on the role of hydrodynamic flow fields.12–17
An alternative realization of a self-propelled particle is an active emulsion droplet. Motivated by
the experimental realization of such a swimming droplet18 and our own work,19 we construct here
first the full three-dimensional solution for the flow field inside and outside of the droplet. It is driven
by a non-uniform surface tension profile at the droplet interface. Then, we present two illustrative
examples, where it is necessary to use this full solution.
When two immiscible liquid phases are mixed, they form emulsion droplets, which are often
stabilized by surfactants. Emulsion droplets can be prepared with a well-defined size. Since they
1070-6631/2016/28(1)/012106/29/$30.00 28, 012106-1 ©2016 AIP Publishing LLC
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012106-2 M. Schmitt and H. Stark Phys. Fluids 28, 012106 (2016)
can enclose very small quantities of matter down to single molecules, they are predestined as mi-
croreactors in which chemical or biological reactions take place. Therefore, they are an important
building block in microfluidic devices.20,21 Droplets are commonly divided into two classes: passive
droplets, which move due to external forces, and active droplets, which swim autonomously without
any external forces. This force-free swimming is a general signature of self-propelled particles.4
Self-propelled active droplets in a bulk fluid have been studied in various experiments18,22–26
including droplets coupled to a chemical wave.27 Theoretical as well as numerical treatments include
deformable and contractile droplets,28,29 droplets in a chemically reacting fluid,30 studies of the
drift bifurcation of translational motion,31–35 droplets driven by nonlinear chemical kinetics,36 and a
diffusion-advection-reaction equation for surfactant mixtures at the droplet interface.19
In the first part of this paper, we derive the flow field around an emulsion droplet initiated by a
non-uniform surface tension at the droplet interface. This phenomenon is known as Marangoni ef-
fect.37 In the proximity of the droplet interface, Marangoni flow is directed towards increasing surface
tension. So far, there have been detailed studies of the flow field around active droplets, where the
surface tension is axisymmetric σ=σ(θ).38,39 Formulas of the non-axisymmetric case have been
mentioned in an extensive study of the rheology of emulsion drops and have been used to explain
cross-streamline migration of emulsion droplets in Poiseuille flow.40–42 Here, we present a detailed
derivation and illustration of the full flow field for an arbitrary surface tension profile σ(θ,ϕ)at the
droplet interface. We provide formulas for the flow fields inside and outside of the droplet, the droplet
velocity vector vD, as well as for the squirmer parameter β, which determines whether a droplet is a
pusher or a puller.
In the second part of this paper, we apply the presented formulas to two illustrative examples.
There are various causes for a non-uniform surface tension field σ(θ,ϕ).A surfactant lowers the sur-
face tension by accumulating at an interface. Thus, the simplest way to generate Marangoni flow is a
non-uniform distribution of a surfactant within an interface. In our first, simple example, we consider
an initially “clean” or surfactant free droplet43 immersed in a fluid, which is enriched by micelles,
i.e., aggregates of surfactant molecules. When a micelle adsorbs somewhere at the droplet interface,
Marangoni flow is induced and propels the droplet in the direction of the adsorption site. Now, the
resulting outer fluid flow preferentially advects other micelles towards the existing adsorption site.
This mechanism can spontaneously break the isotropic symmetry of the droplet, which then moves
persistently in one direction, if the mean adsorption rate of the micelles is sufficiently large. Micelles
have been shown to be crucial in the dynamics of active water as well as liquid crystal droplets.25,26
While we do not attempt to unravel the detailed mechanism for activity in these examples, we present
here a simple idea how micelle adsorption generates directed motion.
In the second example, we use a non-uniform mixture of two surfactant types to induce
Marangoni flow. Such a mixture can be created by a chemical reaction.18 Here, we illustrate a different
mechanism. Light-switchable surfactants exist which change their conformation under illumination
with light.44 So, by shining laser light onto a droplet covered by light-switchable surfactants,44 one
locally generates a spot of different surfactant molecules. Depending on the surfactant type in the bulk
fluid and the wavelength of the laser light, the emulsion droplet is either pushed by the laser beam
or pulled towards it. The first situation is unstable and the droplet moves away from the beam and
then stops. In the second situation, the droplet moves on a straight trajectory along the beam. With
decreasing relaxation rate towards the surfactant in bulk, a Hopf bifurcation occurs and the droplet
also oscillates about the beam axis. We explore these cases for strongly absorbing and for transparent
droplets.
The article is organized as follows. In Sec. II, we derive the flow fields inside and outside an
emulsion droplet induced by a non-uniform surface tension profile. The flow fields depend on the
droplet velocity vector vD, which we evaluate and discuss in Sec. III. Section IV discusses charac-
teristics of the flow field and introduces the squirmer parameter in order to classify active droplets as
pushers or pullers. The Secs. Vand VI contain the illustrative examples. Section Vdemonstrates how
micelle adsorption spontaneously breaks the isotropic droplet symmetry and induces directed propul-
sion. Finally, in Sec. VI, we introduce and discuss the emulsion droplet covered by a light-switchable
surfactant. The article concludes in Sec. VII.
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012106-3 M. Schmitt and H. Stark Phys. Fluids 28, 012106 (2016)
II. VELOCITY FIELD OF A FORCE-FREE ACTIVE EMULSION DROPLET
In the following, we consider a droplet of radius Rwith viscosity ˆηof the inside liquid immersed
in an unbounded bulk fluid with viscosity η. At low Reynolds number, we have to solve the creeping
flow or Stokes equation to determine both the velocity field u(r)outside the droplet (r>R) and the
field ˆ
u(r)inside the droplet (r<R). Solving the problem needs two steps.45
At first, we solve the Stokes equation for a droplet, which is fixed in space, with a given inho-
mogeneous surface tension σat the interface. The resulting flow field of this “pumping problem”
will be called w. Second, we derive the flow field vof a passive droplet swimming with a prescribed
velocity vD, the so-called Hadamard Rybczynski solution.46 The complete flow field of the swimming
droplet is then given by the superposition of both flow fields: u=v+w.This approach ensures that
the swimming droplet is a force-free swimmer.4The droplet velocity vector vDis calculated by means
of the Lorentz reciprocal theorem for Stokes flow in Sec. III.
A. Pumping active droplet
In this section, we fix the active emulsion droplet in space and analyze the velocity fields outside
(w) and inside ( ˆ
w) of the droplet generated by the inhomogeneous surface tension at the fluid interface.
We start with the boundary conditions formulated in spherical coordinates in the droplet frame of
reference,
w=0,r→ ∞,(1)
wr=ˆwr=0,r=R,(2)
w=ˆ
w,r=R,(3)
∇sσ=Ps(ˆ
T−T)er,r=R,(4)
where ris the distance from the droplet center and Rthe droplet radius. These conditions assure that
the droplet is fixed in space (1), has an impenetrable interface (2), and the tangential velocity at the
interface is continuous (3). Condition (4) states that a gradient in surface tension σat the interface
has to be balanced by a jump in the fluid shear stresses. This gradient in surface tension induces
the Marangoni flow close to the interface. Here, we introduce the surface projector Ps=1−n⊗n
with surface normal n=er. Correspondingly, we use the notation ∇s=Ps∇for the surface gradient,
where ∇is the nabla operator. In addition, we assume the droplet to be undeformable, i.e., of constant
curvature ∇·n=2/R, and thus do not need to consider the normal stress balance at the interface.
Hence, we are in the regime of small capillary number Ca =R|∇sσ|/|σ|≪1.33 Finally, the viscous
part of the Cauchy stress tensor of an incompressible Newtonian fluid with viscosity ηis given by
T=η�∇ ⊗ w+(∇⊗w)T�. In spherical coordinates, we find following two equations from condition
(4) for the polar and azimuthal components of ∇sσ:
(∇sσ)θ=ˆη(∂rˆwθ−R−1ˆwθ)−η(∂rwθ−R−1wθ),(5a)
(∇sσ)ϕ=ˆη(∂rˆwϕ−R−1ˆwϕ)−η(∂rwϕ−R−1wϕ).(5b)
Fluid flow at the liquid-liquid interface is always driven by a gradient in σ, whereas pressure only
acts in normal direction.
We now use the set of boundary conditions (1)-(3) and (5) to solve the Stokes equation η∇2w−
∇p=0outside and inside the droplet. Due to the spherical symmetry of our problem and since pres-
sure psatisfies the Laplace equation, the following ansatz for the velocity and pressure fields outside
the droplet are feasible according to Refs. 47 and 48:
w=∞
l=02−l
ηl(4l−2)r2∇pl+1+l
ηl(2l−1)plr+∇Φl+∇×(χlr),
p=∞
l=0
pl.
(6)
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012106-4 M. Schmitt and H. Stark Phys. Fluids 28, 012106 (2016)
Here, χl,Φl, and plare irregular solid harmonics to assure w=0 as r→ ∞ according to Eq. (1),48
pl=r−(l+1)
l
m=−l
αm
lYm
l(θ,ϕ),
Φl=r−(l+1)
l
m=−l
βm
lYm
l(θ,ϕ),
where we give the spherical harmonics Ym
l(θ,ϕ)in Appendix A. We already set the pseudoscalar χl
in Eq. (6) to zero. We will express the flow field as a linear function in the surface tension, which is a
scalar quantity. Due to the isotropic symmetry of the spherical droplet, a preferred direction does not
exist and one cannot construct a term which contains the pseudoscalar χl. Using the solid harmonics
in Eq. (6) results in the following spherical components of the flow field win Eq. (6):
wr=∞
l=1
l
m=−ll+1
(4l−2)η
αm
l
rlYm
l−(l+1)βm
l
rl+2Ym
l,
wθ=∞
l=1
l
m=−l2−l
l(4l−2)η
αm
l
rl∂θYm
l+βm
l
rl+2∂θYm
l,
wϕ=∞
l=1
l
m=−lim(2−l)
l(4l−2)η
αm
l
rl
Ym
l
sin θ+im βm
l
rl+2
Ym
l
sin θ.
The coefficients αm
land βm
lwill be determined in the following. Terms with l=0 do not appear since
the coefficients either vanish due to boundary conditions (1) and (2) (α0
0=β0
0=0).
The ansatz for the interior flow inside the droplet is obtained from Eq. (6) by replacing lby
−(l+1)in the prefactor of each term:47,48
ˆ
w=∞
l=0(l+3)
ˆη(l+1)(4l+6)r2∇ˆpl−l
ˆη(l+1)(2l+3)ˆplr+∇ˆ
Φl+∇×(ˆχlr),
ˆp=∞
l=0
ˆpl,
(7)
with regular solid harmonics, which do not diverge at r=0,
ˆpl=rl
l
m=−l
ˆαm
lYm
l(θ,ϕ),
ˆ
Φl=rl
l
m=−l
ˆ
βm
lYm
l(θ,ϕ).
Again, we can set ˆχl=0. This results in the following spherical components of the flow field ˆ
win
Eq. (7):
ˆwr=∞
l=1
l
m=−ll
(4l+6)ˆηrl+1ˆαm
lYm
l+lrl−1ˆ
βm
lYm
l,
ˆwθ=∞
l=1
l
m=−ll+3
(l+1)(4l+6)ˆηrl+1ˆαm
l∂θYm
l+rl−1ˆ
βm
l∂θYm
l,
ˆwϕ=∞
l=1
l
m=−lim(l+3)
(l+1)(4l+6)ˆηrl+1ˆαm
l
Ym
l
sin θ+imrl−1ˆ
βm
l
Ym
l
sin θ.
Terms with l=0 are not relevant.
In the following, we successively evaluate conditions (2), (3), and (5) to determine all the coef-
ficients αm
l,ˆαm
land βm
l,ˆ
βm
l. The condition of impenetrable interface (2) connects αm
l( ˆαm
l) with
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012106-5 M. Schmitt and H. Stark Phys. Fluids 28, 012106 (2016)
βm
l(ˆ
βm
l),
ˆαm
l=−2 ˆη(2l+3)
R2ˆ
βm
l,αm
l=−2η(1−2l)
R2βm
l.
We eliminate αm
land ˆαm
land use the condition ˆwθ|R=wθ|Rfrom Eq. (3) to relate the interior to the
exterior coefficients:
ˆ
βm
l=−l+1
lR−2l−1βm
l.
Evaluating ˆwϕ|R=wϕ|Ryields the same relations. In the next step, we use final condition (5) to match
the jump in the shear stress to a given profile of the surface tension σ. From Eq. (5) and the coefficients
determined above, we find
(∇sσ)θ=∞
l=1
l
m=−l(η+ˆη)4l+2
l
βm
l
Rl+3∂θYm
l,(8a)
(∇sσ)ϕ=∞
l=1
l
m=−l(η+ˆη)4l+2
l
im βm
l
Rl+3
Ym
l
sin θ.(8b)
We also expand the surface tension into spherical harmonics,
σ(θ,ϕ)=∞
l=1
l
m=−l
sm
lYm
l(θ,ϕ),(9)
with coefficients
sm
l=
2π
0
π
0
σ(θ,ϕ)Ym
l(θ,ϕ)sin θdθdϕ,(10)
where Ym
lis the complex conjugate of Ym
l(see Appendix A). Thus, the l.h.s. of Eqs. (8) are given by
(∇sσ)θ=1
R
∞
l=1
l
m=−l
sm
l∂θYm
l,(11a)
(∇sσ)ϕ=1
R
∞
l=1
l
m=−l
imsm
l
Ym
l
sin θ.(11b)
Comparing Eqs. (8a) and (11a) or alternatively (8b) and (11b), we finally find
βm
l=Rl+2
η+ˆη
l
4l+2sm
l.
This completes the derivation of the velocity field of an active droplet with a given surface tension
profile σ, which is fixed in space.
The fluid flow at the interface is now easily calculated by inserting the coefficients βm
linto the
ansatz for wand setting r=R,
wθ|R=1
η+ˆη
∞
l=1
l
m=−l
sm
l
2l+1∂θYm
l,(12a)
wϕ�R=1
η+ˆη
∞
l=1
l
m=−l
imsm
l
2l+1
Ym
l
sin θ,(12b)
with sm
lfrom Eq. (10).
Comparing the components of Eqs. (11) and (12) with each other, we realize that the expansion
coefficients only differ by a factor 1/(2l+1). Thus, the fluid flow at the interface w|Ris basically
equivalent to a smoothed gradient of the surface tension ∇sσ.
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012106-6 M. Schmitt and H. Stark Phys. Fluids 28, 012106 (2016)
B. Passive droplet
In this section, we will calculate the velocity field vof the viscous flow around a passive sphere
moving with a velocity vD. In the rest frame of the moving sphere, boundary conditions (2) and (3)
from the analysis of the fixed active droplet remain unchanged, while Eqs. (1) and (5) are replaced
by
v=−vD,r→ ∞,(13)
0=Ps(ˆ
T−T)er,r=R.(14)
The second condition is equivalent to ∇sσ=0. It means that the fluid shear stress is continuous across
the droplet interface and hence the droplet is passive. The procedure of calculating the flow field vis
very similar to the case of the active droplet in Sec. II A. We outline it in the following.
We employ the same ansatz for the external (v) and internal (ˆ
v) droplet field, as we did for wand
ˆ
wfor the pumping active droplet in Eqs. (6) and (7). However, in order to satisfy boundary condition
(13), we have to add the three spherical components of −vD,
−er·vD=−v−1
1Y−1
1−v0
1Y0
1−v1
1Y1
1,
−eθ·vD=−v−1
1∂θY−1
1−v0
1∂θY0
1−v1
1∂θY1
1,
−eϕ·vD=−iv1
1
Y1
1
sin θ+iv−1
1
Y−1
1
sin θ,
to the ansatz for v, where we have introduced the coefficients vm
1for vD. This is equivalent to vDin
Cartesian representation,
vD=−3
8���
v1
1−v−1
1
i�v1
1+v−1
1�
−√2v0
1�����
.(15)
From condition (2), we find
ˆαm
l=−2 ˆη(2l+3)
R2ˆ
βm
l,
αm
l=−2η(1−2l)
R2(βm
l+δl,1
R3
2vm
l),
just as in the active case of Sec. II A. Boundary condition (3) relates ˆ
βm
lto βm
l,
ˆ
βm
l=−l+1
lR−2l−1βm
l+δl,1
2vm
l.
Finally, we use boundary condition (14) to derive
ˆ
βm
1=η
2(η+ˆη)vm
1,
while all other coefficients with l≥2 vanish. So, we have related all coefficients to the components
vm
1of vD.
We obtain the axially symmetric velocity field vof a passive droplet, the so called Hadamard-
Rybczynski solution of a creeping droplet,46 which moves with velocity vD. In Appendix B, where
we give the complete velocity field uof the active droplet, one can read offthe flow field vas the
terms that contain vD. These terms either decay as 1/ror 1/r3. In particular, the velocity field at the
interface is
vθ|R=−η
2(η+ˆη)eθ·vD,(16a)
vϕ�R=−η
2(η+ˆη)eϕ·vD.(16b)
We will calculate the droplet velocity vDin Sec. III.
Note that, as ηˆ → ∞, one recovers the usual no-slip boundary condition of a rigid sphere.
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012106-7 M. Schmitt and H. Stark Phys. Fluids 28, 012106 (2016)
C. Complete solution
The complete flow field of the force-free swimming active droplet is the sum of both contribu-
tions, from the fixed active and the passive droplet. The velocity fields inside (ˆ
u=ˆ
v+ˆ
w) and outside
(u=v+w) of the droplet are summarized in Appendix B. The formulas are equivalent to the ones in
Refs. 40–42. The outside flow field is also presented in the lab frame. Here, we mention the velocity
field u|R=w|R+v|Rat the interface with w|Rtaken from Eq. (12) and v|Rfrom Eq. (16),
uθ|R=−η
2(η+ˆη)eθ·vD+1
η+ˆη
∞
l=1
l
m=−l
sm
l
2l+1∂θYm
l,(17a)
uϕ�R=−η
2(η+ˆη)eϕ·vD+1
η+ˆη
∞
l=1
l
m=−l
imsm
l
2l+1
Ym
l
sin θ.(17b)
Before studying the axisymmetric limit, we investigate the role of viscosity. The most commonly
studied droplet emulsions are either oil droplets in water or vice versa, where typical viscosities are
ηwater =1 mPa s and ηoil =36 mPa s.18,25 We will show in Sec. III that vD∝(2η+3 ˆη)−1. Using this
result in Eqs. (17), we find that in the case η≥ˆη, both wand vscale as 1/η. In the opposite case,
η≪ˆη, the pumping solution scales as w∝1/ˆηwhile v∝η/ˆη2. Hence, for an oil drop in water, one
can neglect v, when calculating the velocity field (17) at the interface.
An axisymmetric surface tension σ=σ(θ), where only spherical harmonics with m=0 contri-
bute in Eqs. (9) and (17a), yields
uθ|R=ηsin θvD
z
2(η+ˆη)+1
2(η+ˆη)
∞
l=1����
π
0
σPlsin θdθ����P1
l.(18)
Here, Pl(cos θ)are Legendre polynomials of degree land P1
l(cos θ)=∂θPl(cos θ).
Levan and Newman already solved the case of an axisymmetric swimming droplet,38 where the
Stokes equation can be rephrased to a simpler fourth-order partial differential equation for a scalar
stream function.48 They found the flow field at the interface,
uθ|R=ηsin θvD
z
2(η+ˆη)+1
2(η+ˆη)
∞
l=2
l(l−1)����
π
0
C−1/2
l∂θσdθ����
C−1/2
l
sin θ,(19)
where C−1/2
l(cos θ)are Gegenbauer polynomials of order land degree −1/2. These are connected to
Legendre polynomials by d
dcos θC−1/2
l=−Pl−1. A standard calculation, which uses the properties of
Legendre and Gegenbauer polynomials, shows indeed that Eqs. (19) and (18) are equivalent.
The general solution for the surface flow, Eqs. (17), still contains the unknown droplet velocity
vector vD. We will calculate vDin Sec. III, by relating it to the non-uniform surface tension.
III. DROPLET VELOCITY VECTOR
A central quantity in all studies of swimming droplets is the swimming speed vD. Furthermore,
for droplets without an axial symmetry, the swimming direction eis not obvious. Both together
define the droplet velocity vector vD=vDe. Once this quantity is known, the flow field u|Rin (17) is
completely determined.
In order to derive an expression for vD, we stress that an active particle is force-free.4Accord-
ingly, the total hydrodynamic drag force F=Fa+Fp, acting on the particle, has to vanish. Here, Fa
and Fpare the drag forces of the active pumping droplet and the passive droplet, treated in Secs. II A
and II B, respectively. The drag forces are given by Fa=−4π∇(r3p1|a)and Fp=−4π∇(r3p1|p),
respectively, with solid harmonics p1|aand p1|pof the corresponding flow fields.48 For the passive
droplet, one finds
Fp=−6πηR2η+3 ˆη
3η+3 ˆηvD,(20)
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012106-8 M. Schmitt and H. Stark Phys. Fluids 28, 012106 (2016)
also known as the Hadamard and Rybczynski drag force of a droplet.46,49,50 It reduces to the well
known Stokes drag of a solid sphere for ˆη≫η, whereas it predicts a reduced drag for droplets and
bubbles, due to a finite slip velocity at the interface. The condition Fa+Fp=0 gives the simple
relation αm
1|a+αm
1|p=0 between the coefficients determined in Secs. IIAand II B and ultimately
yields vm
1=−2sm
1/(9 ˆη+6η). Here, vm
1are the coefficients of the velocity vector vDintroduced in
Eq. (15). Thus, one finds41,51
vD=1
6π
1
2η+3 ˆη�����
s1
1−s−1
1
i�s1
1+s−1
1�
−√2s0
1�����
.(21)
An equivalent relation writes vDas the average of flow field wover the droplet surface, see
Appendix C. The droplet velocity vector is solely determined by the dipolar coefficients (l=1) in
the multipole expansion of the surface tension σ. It can be written as vD=vDewith
vD=2�s0
1�2−4s1
1s−1
1
√6π(2η+3 ˆη),(22a)
e=1
2�s0
1�2−4s1
1s−1
1�����
s1
1−s−1
1
i�s1
1+s−1
1�
−√2s0
1�����
.(22b)
Next, we derive an alternative formula for vD. Using the explicit expressions for the sm
1from
Eq. (10) and the Cartesian components of the radial unit vector er, we rewrite Eq. (21) as vD=
−[2πR2(2η+3 ˆη)]−1 σerdA. Finally, extending σinto the droplet with ∂σ/∂r=0 and applying
Gauss’s theorem, we obtain
vD=−1
4πR(2η+3 ˆη) ∇sσdA.
Thus, the droplet velocity vector vDis simply given by the integral of the surface-tension gradient
∇sσover the whole droplet surface. By comparing this with the alternative formula for vDin Eq. (C1),
we realize that for calculating the droplet speed, the following equivalence holds:
w|Rˆ=R
3(η+ˆη)∇sσ.(23)
By using the ˆ=symbol, we stress that this equivalence is only valid in Eq. (C1) and not for the flow
field w|Rin general. However, Eq. (23) illustrates that the surface flow is initiated by a gradient in the
surface tension.
In the axisymmetric case (m=0), vD=−vDezpoints against the zdirection with
vD=1
2η+3 ˆη
π
0
σcos θsin θdθ,
which is equivalent to the swimming speed calculated by Levan and Newman.38 Note that the swim-
ming velocity is independent of the droplet radius R.
Ansätze (6) and (7) for flow and pressure fields can also be used to treat droplets of non-spherical
shape.48 For the torque, which a droplet experiences from the surrounding fluid, one finds M=
−8πη∇(r3χ1).48 However, as explained in Sec. II A, the solid harmonic χ1vanishes and thus M=0.
Therefore, for a spherical droplet, the angular velocity is zero, Ω=0, and the swimming kinematics
is completely determined by vD. Hence, there is no generalization of the Stokes drag torque M=
−8πηR3Ω of a rigid particle to an emulsion droplet.
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012106-9 M. Schmitt and H. Stark Phys. Fluids 28, 012106 (2016)
IV. CHARACTERISTICS OF FLOW FIELD
In this section, we discuss some characteristics of the outside flow field u(r)fully presented in
Appendix B. Flow fields around an active particle can be written as a superposition of flow singular-
ities.52,53 The lowest singularity, the Stokeslet, is the flow field due to a point force fδ(r)apointing
in direction a. It decays as u∝r−1and is only present if external forces act on the particle. In our
analysis, we do not consider external forces.
The leading singularity of a force-free active droplet is the stresslet. An example is the force
dipole constructed from two Stokeslets, which one obtains by taking the derivative of the Stokeslet
along a given direction b. The resulting flow field decays as u∝r−2. In general, the stresslet corre-
sponds to the symmetrized first moment of the force distribution on the particle surface. Thus, it is
characterized by the symmetric tensor S=−2π
3∇⊗∇(r5p2)with solid harmonic p2, which here comes
from the pumping active droplet.54 One obtains
S=6π
5
R2η
η+ˆη����������
s−2
2−2
3s0
2+s2
2i(s2
2+s−2
2)s−1
2−s1
2
i(s2
2+s−2
2)−s−2
2−2
3s0
2−s2
2−i(s−1
2+s1
2)
s−1
2−s1
2−i(s−1
2+s1
2)22
3s0
2
����������
.(24)
For instance, the flow field of two Stokeslets in direction a=±ex, which are connected along b=ey,
is given by component Sxy. Clearly, only the coefficients sm
2account for the stresslet.
The singularities that account for a decay of the velocity field as u∝r−3are both the source dipole
and the Stokes quadrupole. The source dipole is responsible for droplet propulsion and thus related
to the droplet velocity vector vDwith the coefficients sm
1[see Eq. (21)]. Hence, these coefficients are
always non-zero when the droplet is swimming. The Stokes quadrupole is, for example, a combina-
tion of two stresslets. It is related to the second moment of the force distribution, a tensor of rank
three. As can be observed from Appendix B, the coefficients sm
3account for the Stokes quadrupole.
To summarize, the lowest-order decay of the flow field around a swimming active droplet is therefore
either u∝r−3in the case of S=0oru∝r−2if S�0.
A. Generalized squirmer parameter
A useful parameter to quantify the type of a microswimmer driven by surface flow is the
squirmer parameter β. It compares the stresslet strength to the source dipole. The squirmer is a
classic model of an axisymmetric spherical microswimmer.55,56 It has recently been generalized to
the non-axisymmetric case.45 The essential boundary condition of a squirmer is a prescribed flow
field w|Rat the surface of a sphere. In contrast, the flow field w|Rof an active droplet is the result of
a non-uniform surface tension at the fluid interface.
In the following, we calculate the squirmer parameter βfor an active droplet with arbitrary swim-
ming direction as a function of the angular expansion coefficients sm
2of the surface tension. For an
axisymmetric squirmer with surface flow velocity
uθ|R=B1sin θ+B2
2sin 2θ,(25)
one defines the squirmer parameter βas4,55–58
β=B2/|B1|,(26)
where 2/3 |B1| is the swimming speed. When β is positive, the surface flow is stronger in the front
on the northern hemisphere and the flow around the squirmer is similar to the flow field initiated by
a swimming algae such as Chlamydomonas. The swimmer is called a “puller” since it pulls itself
through the fluid. Accordingly, a swimmer with β < 0 is called a “pusher.” For example, the bacterium
E. coli swims by pushing fluid away from itself at the back by a rotating flagellum.4 For β � 0, the
flow field far away from the swimmer is dominated by the hydrodynamic stresslet or force dipole
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012106-10 M. Schmitt and H. Stark Phys. Fluids 28, 012106 (2016)
with u∝r−2. However, in the case β=0 (“neutral swimmer”), the source dipole with u∝r−3domi-
nates. One example for a neutral swimmer is the Volvox algae.4For β→±∞, the swimmer becomes
a “shaker” that shakes the adjacent fluid but does not swim. Note that hydrodynamic interactions
between swimmers as well as between swimmers and walls strongly depend on their type, i.e., on
the squirmer parameter β. Thus, βis a key parameter in the study of individual swimmers as well as
their collective dynamics.13,15,16,19
For squirmers without axisymmetry but still swimming along the z-axis, Eq. (25) also contains
terms depending on the azimuthal angle ϕ. In addition, a multipole expansion for the azimuthal ve-
locity component uϕ|Rhas to be added. Still, the coefficient B1determines the swimming speed and β
the swimmer type since contributions from multipole terms with m�0 vanish when averaging over
ϕ.
So, we first determine the squirmer parameter for an axisymmetric droplet that swims in zdirec-
tion. Since βis related to flow fields decaying like 1/r2, we only have to consider the velocity field
wof the pumping active droplet. For the surface tension profile
σ=s0
1Y0
1+s0
2Y0
2
=3
4πs0
1cos θ+5
4πs0
2(3
2cos2θ−1
2),
where we have only included the relevant two leading modes, we find from Eq. (12a) the following:
wθ|R=−1
η+ˆη��
s0
1
√12πsin θ+1
2
3s0
2
√20πsin 2θ��.
Comparing with Eq. (25), we identify the squirmer parameter of the swimming axisymmetric droplet
as the ratio
β=−27
5
s0
2
|s0
1|.(27)
In Eq. (D1) in Appendix D, we relate all the angular coefficients s0
lto the squirmer coefficients Bl,
which yields the same expression for β.
The ratio of stresslet tensor component Szz =ez·Sezand velocity vDis proportional to βfrom
Eq. (27). Similarly, when projecting the stresslet tensor Sonto an arbitrary swimming direction e, one
averages over the azimuthal angle about e. So, the generalized squirmer parameter to characterize
pushers and pullers becomes
β=−3
4πR2
η+ˆη
(2η+3 ˆη)η
e·Se
vD.(28)
Here, we have set the prefactor such that βagrees with Eq. (27) for e=ez. In Eq. (E1) in Appendix E,
we give the concrete expression for βin terms of sm
1and sm
2. Note that in the non-axisymmetric case,
β=0 does not mean that the stresslet is zero. For this, all components of the stresslet tensor have to
vanish. Only then one can conclude that a flow field with u∝r−2does not exist.
B. Surface divergence
The solution of the Stokes equation for the flow field u, which we presented in Sec. II C, fulfills
the incompressibility condition ∇·u=0 everywhere, i.e., also at the interface. However, this does
not necessarily hold for the surface divergence ∇s·u|R. Using ∇s2Ym
l=−l(l+1)R−2Ym
l, one finds
from Eqs. (17) the following:
∇s·u|R=R−1
η+ˆηηer·vD−∞
l=1
l
m=−l
l(l+1)sm
l
2l+1Ym
l.(29)
Thus, any surface actuation sm
l�0 results in ∇s·u|R�0. In other words, surface divergence is a
necessary condition for propulsion.25,59 In fact, the surface divergence of the pumping field ∇s·w|R,
i.e., the second term on the r.h.s. of Eq. (29), contains the expansion coefficients of σamplified
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012106-11 M. Schmitt and H. Stark Phys. Fluids 28, 012106 (2016)
FIG. 1. Inside (ˆ
u) and outside (u) velocity field streamlines at a cross section through an emulsion droplet for a given
surface tension σ(θ,ϕ). To draw the streamlines, the velocity vectors at the cross section are projected onto the cross section.
Moreover, the surface divergence ∇s·u|Rat the droplet interface is shown.
by a prefactor O(l). Furthermore, comparing Eq. (29) with the radial components of the inside and
outside velocity fields ˆurand urfrom (B1a) and (B1d), respectively, one finds the following. Regions
at the interface with positive divergence, ∇s·u|R>0, are accompanied by radial flows ˆur>0 and
ur<0 towards the interface. On the other hand, regions with convergence, i.e., negative divergence,
∇s·u|R<0, induce radial flows ˆur<0 and ur>0 away from the interface. Figure 1illustrates this.
Depicted is the surface divergence ∇s·u|Ralong with the streamlines of ˆ
uand uat a cross section
through an emulsion droplet with given surface tension field σ(θ,ϕ). In Section V, we will build on
this finding and demonstrate how micelle adsorption spontaneously breaks the isotropic symmetry
of the droplet interface and thereby induces propulsion.
V. SPONTANEOUS SYMMETRY BREAKING BY MICELLE ADSORPTION
For the remainder of this paper, we will discuss possible applications for Marangoni flow initiated
at the interface of an emulsion droplet. In this section, we present a model of a droplet which per-
forms directed motion by adsorbing micelles, i.e., spherical aggregates of surfactant molecules. We
consider a spherical oil droplet in water, the surface of which, initially, is hardly covered by surfactant
molecules. Due to the small ratio of viscosities outside and inside the droplet, η≪ˆη, we neglect the
passive part vof the velocity field and set u=w, as pointed out in Sec. II C. The surrounding water
phase is homogeneously enriched with micelles formed by surfactant molecules. In the following,
we explain how this setup can lead to a persistent swimming motion of the droplet. Once one of the
micelles with radius RMhits the droplet interface, the surfactants will adsorb at the droplet interface
with a probability pand cover a circular region of area 4πR2
M, as illustrated in Fig. 2. Thus, at the
adsorption site, surface tension is lower compared to the surrounding surfactant-free interface. The
resulting Marangoni flow is directed away from the adsorption site and therefore spreads the surfac-
tants over the droplet interface. The interfacial Marangoni flow u|R=w|Rinduces a displacement of
the droplet in the direction of the adsorption site with a velocity given in Eq. (C1). Furthermore, the
flow is accompanied by a positive surface divergence ∇s·u|R>0 and inward radial flow ur<0 at
the front of the droplet, as discussed in Sec. IV B. The flow field initiated by an adsorbed micelle is
illustrated in the inset of Fig. 2. Now, the radial flow towards the interface advects additional micelles
and thereby increases the rate with which surfactants adsorb at the front of the droplet. Following this
train of thought, we expect the droplet to eventually develop a spot with increased surfactant coverage
thereby breaking the isotropic symmetry of the interface. As a result, the droplet performs directed
motion that comes to an end when the interface is fully covered by surfactants.
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012106-12 M. Schmitt and H. Stark Phys. Fluids 28, 012106 (2016)
FIG. 2. Cartoon of a micelle adsorbing at the interface of an emulsion droplet. Marangoni flow u|Rspreads the surfactants
over the interface and propels the droplet in direction etowards the adsorption site. Inset: Flow field and color-coded surface
divergence ∇s·u|Rshortly after a micelle has adsorbed at the droplet interface. Same representation as in Fig. 1.
Note that this model starts with the assumption that the interface of the emulsion droplet initially
is almost surfactant free. Such systems exist and Ref. 43 summarizes recent advances on realizing
surfactant-free emulsion droplets. Furthermore, we do not take into account the detailed kinetics of
the micelle adsorption.60 We rather assume that when micelles adsorb at the interface, they simply
spread their surfactant molecules.
A. Diffusion-advection equation
We propose a simple model for the surfactant dynamics at the droplet interface. The surfactant
concentration Γobeys a diffusion-advection equation with additional source term,
∂tΓ=−∇s·(−Ds∇sΓ+Γu|R)+q.(30)
The two terms in brackets describe transport of surfactants due to diffusion and advection induced by
Marangoni flow, respectively, and Dsis the diffusion constant within the interface. The third term on
the r.h.s. of Eq. (30) represents the bulk current of micelles hitting the droplet interface, where they
are ultimately adsorbed with a mean rate 1/τads and τads is mean adsorption time.
The Marangoni flow u|Rat the droplet interface is generated by a concentration dependent surface
tension, which we assume to be linear in Γ, for simplicity,
σ(Γ)=σ0−σsΓ,(31)
where σ0>σs>0. Here, σ0is the surface tension of the clean or surfactant free droplet (Γ≪1)
and σ0−σsthe surface tension of a droplet, which is fully covered by surfactants (Γ=1). Thus,
for a given surfactant density Γ(θ,ϕ), Eq. (31) yields the field of surface tension, which is expanded
into spherical harmonics with coefficients sm
laccording to Eq. (10). Note that equation of state (31)
typically breaks down at large surface coverage. Here, we mainly focus on the early droplet dynamics,
when the interface is only lightly covered with surfactants.
The micellar source term qhas two contributions. Micelles perform a random walk through the
outer fluid and ultimately hit the droplet interface which acts as a sink for the micelles. This sets up
a diffusive current towards the interface. More importantly, as soon as Marangoni flow is initiated,
micelles are also advected towards the interface as quantified by the radial flow component ur, which
is connected to the surface divergence ∇s·u|Rat the droplet interface, as outlined in Sec. IV B and dis-
cussed in Ref. 61 in more detail. A rigorous study of the full 3D bulk diffusion-advection equation for
the bulk concentration of micelles cis beyond the scope of this paper. Instead, we proceed as follows.
Comparing the time scale of micellar bulk diffusion tD = R2/(6D) = πηR2RM/(kBT) to the
3
time scale
of bulk advection t A = R(η + ηˆ)/σs, we find a Peclet number Pe = tD/t A on the order of 10 . For the
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012106-13 M. Schmitt and H. Stark Phys. Fluids 28, 012106 (2016)
estimate, we took RM=R/20 =50 nm, η=1 mPa s, ˆη=36 mPa s, σs=1 mN/m, and room temper-
ature. Due to the large Peclet number, we neglect bulk diffusion completely and consider advection
only in the following. We view micelle adsorption to occur anywhere at the interface as a Poissonian
process. Adsorption events are independent of each other and the mean adsorption time between the
events is τads as already mentioned. Micelles preferentially adsorb at positions on the interface with
large ∇s·u|R>0, while they do not adsorb at locations with ∇s·u|R<0, where the radial flow is
directed away from the droplet. We will explain the detailed implementation of the adsorption event
in Sec. V B.
We introduce a dimensionless form of the diffusion-advection equation (30) for the droplet inter-
face rescaling lengths by droplet radius Rand times by diffusion time τ=R2/Ds,
∂tΓ=−∇s·(−∇sΓ+MΓu|R)+q.(32)
As one relevant parameter, the so-called Marangoni number M=τ/τAcompares the typical advec-
tion time τA=R(η+ˆη)/σsto τ, where we used u|R=σs/(η+ˆη)to estimate the Marangoni flow.
Furthermore, the reduced adsorption rate becomes κ=τ/τads, which is the most important parameter
in this problem. Although we kept the same symbols, all quantities in Eq. (32), including Γ,t,∇s,
u|R, and q, are from now on dimensionless.
B. Numerical solution
To solve Eq. (32) numerically, we used a finite-volume scheme on a spherical mesh.62 Initially, at
time t=0, the droplet is free of surfactants, Γ=0, and at rest. While diffusive and advective currents
are implemented within the standard finite-volume algorithm, we model the micelle adsorption by a
Poissonian process.63 At each time step ∆tin the numerical scheme, we allow an adsorption event
with probability κ∆t. If it is successful, the micelle is adsorbed with larger probability at positions
were ∇s·u|R>0 is large. To implement this, we introduce the weight function,
f(θ,ϕ)=∇s·u|R
∇s·u|RdΩfor ∇s·u|R>0.(33)
Then, the probability for micelle adsorption during time ∆tand within the solid angle element dΩat
an angular position (θ,ϕ)becomes
p(θ,ϕ,∆t)dΩ=
κ∆t f (θ,ϕ)dΩfor ∇s·u|R>0,
0 for ∇s·u|R≤0.
After a micelle adsorption event at site (θ,ϕ)is determined, we set Γto one in a circular patch with
radius 2RMcentered around (θ,ϕ). In addition, we assume that surfactants stay at the droplet interface
once adsorbed.
In this setup, Marangoni flow and diffusion current act in the same direction along −∇sΓ. So, the
Marangoni number is not the relevant parameter to initiate directed motion and we always set M=1.
However, by tuning the adsorption rate κ, the droplet starts to swim.
Figure 3(a) shows the swimming speed vDof an emulsion droplet for three values of the reduced
adsorption rate κ=τ/τads. For κ=4, the mean adsorption time is too large. The surfactant patch
from a first micellar impact has already spread over the whole interface by diffusion and advection
when a second micelle hits the droplet interface at a different location. As a result, the droplet fol-
lows a random trajectory. Figure 3(b) shows the corresponding swimming trajectory determined from
r(t)=r(0)+t
0dt′vD(t′)e(t′). Increasing κto 11, increases the number of micelles, which adsorb per
unit time, and the swimming speed becomes larger. The swimming trajectory is still irregular albeit
with an increased persistence.
Finally, for κ = 20, mean adsorption time is significantly shorter than the characteristic diffusion
time. Thus, when a second micelle is about to hit the droplet interface, surfactant concentration Γ
and surface divergence ∇s · u|R are still peaked at the impact of the previous micelle. Therefore, the
probability of the following micelle to adsorb at the front of the droplet is increased compared to
the back. This spontaneously breaks spherical symmetry. A defined swimming direction evolves and
the droplet shows directed motion with swimming velocity v D. This is confirmed by the swimming
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012106-14 M. Schmitt and H. Stark Phys. Fluids 28, 012106 (2016)
FIG. 3. (a) Swimming velocity vDplotted versus time tfor an emulsion droplet adsorbing surfactant micelles. The reduced
adsorption rates are κ=4, 11, and 20. Other parameters are M=1 and R/RM=20. The inset shows the long-time limit for
κ=20. (b) Swimming trajectories from the same simulations as in (a) starting from t=0 until the droplets stop when they
are fully covered with surfactants.
trajectory in Fig. 3(b). As the droplet continues to swim, the difference in surfactant concentration
at the adsorption site and the mean concentration at the interface decreases. As a consequence, the
Marangoni flow extenuates and vDdecreases in time [see inset of Fig. 3(a)]. Finally, when the interface
is fully covered, i.e., Γ=1 on the whole interface, the droplet stops.
Figure 4shows the onset of directed swimming by plotting swimming speed versus reduced
adsorption rate κ. Due to amplification of micellar adsorption at a specific spot, the droplet switches
FIG. 4. Swimming speed vDplotted versus reduced adsorption rate κ. For each κ, speed vDis taken at time t=5 and
averaged over 60 simulation runs.
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012106-15 M. Schmitt and H. Stark Phys. Fluids 28, 012106 (2016)
at κ≈12 from slow, random motion with vD≈0.1. . . 0.4 to fast and persistent motion with vD≈
0.9. . . 1.2.
In experiments, κ=τ/τads can be tuned by adjusting the surfactant concentration cS. We equate
the micellar adsorption rate τ−1
ads with the flux j·4πR2of micelles from the bulk to the droplet interface.
The current is advective, j=cMvA, with micelle concentration cMand velocity vA=R/tA. Using
also τ=R2/Ds, one finds κ=cM·4πR4/(DstA). We assume that a micelle consists of 104surfactants
and introduce the degree of micellization as the ratio γbetween micellized surfactants and all surfac-
tants in the system, and this yields cM=γ10−4cS. The ratio γchanges strongly around the critical
micelle concentration cCMC. For example, for cS=0.9·cCMC, i.e., slightly below cCMC, one finds
γ≈5·10−4.64 Together with estimates Ds=10−5cm2/s,65 cCMC =1.5 mmol/l, and values for Rand
tAfrom Sec. V A, we obtain κ≈15, thus around the onset of motion in Fig. 4. However, γ≈10−4
means that micelle adsorption strongly competes with monomer adsorption, which is not contained
in our model to keep it simple. Thus, to observe the onset of droplet motion in experiments, one has
to increase cMby tuning the system closer to cCMC or even above.
Finally, we note that for increasing Marangoni number M, the patch of surfactants spreads faster
due to advection and the crossover in Fig. 4simply shifts towards larger κ.
VI. LIGHT-INDUCED MARANGONI FLOW
Certain surfactants are known to be photosensitive.66–69 For instance, surfactants based on
azobenzene can undergo photoisomerization, where UV light (365 nm) transforms a trans to a cis
configuration and blue light (450 nm) causes a transformation from cis to trans. During the trans-cis
isomerization, subunits within the molecule change their relative orientation. Naturally, a different
molecular structure also affects the surface tension of a surfactant-covered interface. Experiments
showed that surfactants in the cis state cause a higher surface tension compared to the ones in the
trans state.67 This effect has recently been used to generate Marangoni flow.44 Therefore, we suggest
two possible applications of the formulas presented in Sec. II. We first treat light-driven motion of a
strongly absorbing, i.e., “dark,” emulsion droplet and then discuss how the results alter in the case
of a transparent droplet.
A. Pushing an absorbing droplet with UV light
We think of an experiment where a spherical oil droplet of constant radius Ris placed in a water
phase laden with trans surfactants. Initially, the emulsion droplet is in equilibrium with the exterior
phase and thus completely covered with trans surfactants. This corresponds to times t≫κ−1in Sec. V.
An UV laser beam with cross-sectional radius ρ<Ris focused on the center of the droplet. It locally
transforms surfactants at the interface into the cis state and thereby increases surface tension, see
Fig. 5(a).
Here, we assume that the droplet oil phase completely absorbs the incident light beam. Accord-
ingly, the laser beam does not reach the interface opposite to the illuminated side. A thinkable droplet
phase is crude oil, which has a penetration depth of α≈100 µm at wavelength 400 nm.70 On this
length scale, the droplet is still in the low Reynolds number regime and all findings of Secs. II–IV are
valid. Alternatively, one may fabricate a “dark” droplet by enriching the oil phase with soot or black
pigment. In Sec. VI C, we study a transparent droplet.
The initiated Marangoni flow is oriented towards the laser beam and thus the droplet is propelled
away from the laser beam, see Fig. 5(a). Due to the advective current of surfactants towards the laser
beam, the cis surfactants converge at the laser spot on the droplet interface and ultimately leave the
interface. Fresh trans surfactants are adsorbed at the leading front of the droplet, i.e., at the side
opposite to the laser beam.
1. Diffusion-advection-reaction equation
In the following, we review our theoretical approach to describe how the mixture of trans and
cis molecules evolves in time, which then determines the dynamics of the flow field. More details
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012106-16 M. Schmitt and H. Stark Phys. Fluids 28, 012106 (2016)
FIG. 5. (a) UV laser light is aimed at a strongly absorbing oil droplet in water. This increases the surface tension σlocally at
the droplet interface by transforming trans to cis surfactants. The resulting Marangoni flow u|Ris directed towards increasing
surface tension and leads to propulsion in direction eaway from the laser beam. (b) Blue laser light locally decreases the
surface tension σat a droplet interface by transforming cis to trans surfactants. The resulting Marangoni flow u|Rleads to
propulsion in direction etowards the laser beam.
can be found in Ref. 19. We introduce the order parameter field φ(θ,ϕ)with respective values φ= +1
or −1 in regions where all surfactants are either in the cis or trans state, while in mixtures of both
surfactants φis in the range −1<φ<1.
The dynamics of the order parameter φat the droplet interface can be expressed by the diffusion-
advection-reaction equation,
∂tφ=−∇s·(jD+φu|R)−τ−1
eq (φ−φeq),(34)
with diffusive current jDand advective Marangoni current φu|R. The source term couples the order
parameter to the outer fluid laden with trans surfactants, i.e., φeq =−1, by introducing a relaxation
dynamics with time scale τeq.
To derive the diffusive current jD, we use a Flory-Huggins free energy density,19
f(φ)=kBT
ℓ21+φ
2ln 1+φ
2+1−φ
2ln 1−φ
2
−1
4(b1+b2+b12)−φ
2(b1−b2)−φ2
4(b1+b2−b12),(35)
where ℓ2is the head area of a surfactant at the interface. We introduce dimensionless parameters b1and
b2to characterize the respective interactions between either cis or trans surfactants and b12 describes
the interaction between the two types of surfactants. With the total free energy F[φ]=f(φ)dAthe
diffusive current becomes
jD=−λ∇s
δF
δφ =−Ds1
1−φ2−1
2(b1+b2−b12)∇sφ,(36)
where the Einstein relation Ds=λkBT/ℓ2relates mobility λto the interfacial diffusion constant Ds.
Note that the condition jD∝ −∇sφis only fulfilled for a convex free energy with f′′(φ)>0, i.e., if
b1+b2−b12 <2. In the following, we assume for simplicity b12 =(b1+b2)/2.
In order to determine the Marangoni flow at the interface, we need an expression for the surface
tension σ. From free energy (35), we obtain the equation of state for the surface tension,19
σ(φ)=kBT
ℓ2(b1−b2)3
8
b1+b2
b1−b2
+1
2φ+1
8
b1+b2
b1−b2
φ2.(37)
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012106-17 M. Schmitt and H. Stark Phys. Fluids 28, 012106 (2016)
Hence, for a given order parameter profile φ(θ,ϕ), Eq. (37) yields the field of surface tension, which
is expanded into spherical harmonics with coefficients sm
laccording to Eq. (10). Note that in contrast
to equation of state (31) of the single-surfactant model in Sec. V, expression (37) is nonlinear.
In order to make Eq. (34) dimensionless, we rescale lengths by droplet radius Rand time by the
diffusion time scale τ=R2/Dsand obtain
∂tφ=−∇s·(jD+Mφu|R)−κ(φ−φeq).(38)
Here, we introduced the Marangoni number M=τ/τA, where τA=ℓ2R(η+ˆη)[(b1−b2)kBT]−1is
the advection time scale, and κ=τ/τeq. All quantities of Eq. (38) are dimensionless. We numerically
solve Eq. (38) on a spherical domain by the method of finite volumes as explained in detail in Ref. 71.
In all what follows, we set b1=2 and b2=1 as well as M=1, i.e., τ=τA. Furthermore, we choose
κ=1 to illustrate the main behavior but also discuss the system’s dynamics for different values of κ.
2. Stationary solution of pushed droplet
Initially, we set the order parameter φto −1 on the whole interface. We then turn on the UV laser
beam hitting the interface on a circular patch with radius ρ=0.2R. In our numerical scheme, this
is implemented by setting φ=1 in the area of exposure. Furthermore, to couple the droplet to the
outer fluid laden by trans surfactants, we set φeq =−1. Figure 6, case (a) shows a typical stationary
order parameter profile φ, which results from the dynamics of Eq. (38). While φexhibits a step-like
function, the interfacial Marangoni flow uθ|R, also illustrated in Fig. 6, case (a), spreads over the
whole droplet interface. However, since the flow field is concentrated on the northern hemisphere and
directed towards θ=0, the droplet is a pusher. This is confirmed by the formulas for the squirmer
parameter from Sec. IV A, which yield β=−2.8. Increasing κenhances the coupling to φeq =−1 at
the droplet front and the step in the order parameter profile φbecomes steeper, whereas the profile φ
does not significantly depend on the Marangoni number M.
3. Pushing the droplet off-center
So far, the pushed droplet swims with a constant velocity vD=−vDez. Now, we introduce an
offset ∆yof the UV laser beam from the center of the droplet, and study the impact on the droplet
trajectory. Figure 7(a) illustrates the situation. Due to the offset ∆y, the Marangoni flow u|Rpushes
the droplet out of the laser beam. This increases the offset further and the orientation vector etilts
further away from the laser beam.
FIG. 6. Stationary solutions of the order parameter field φand the flow field uθ|Rfor (a) the droplet which is pushed by
UV light and (b) the droplet which is pulled by blue light. In both cases, the laser light hits the droplet interface at θ=0,
compare Fig. 5. Order parameter φ=1 and −1 corresponds to pure cis and trans surfactants, respectively. Further parameters
are M=κ=1.
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012106-18 M. Schmitt and H. Stark Phys. Fluids 28, 012106 (2016)
FIG. 7. (a) UV laser light is aimed at a spot which is offset by ∆yfrom the droplet center. The resulting Marangoni flow
drives the droplet out of the laser beam. (b) Blue laser light is aimed at a spot which is offset by ∆yfrom the droplet center.
The resulting Marangoni flow pulls the droplet back into the laser beam.
Figure 8(a) shows the trajectory of the droplet center for several values of the initial offset ∆y.
For vanishing initial offset, ∆y=0, the droplet swims in a straight line to the left, while in the case
∆y�0, the droplet clearly moves away from the laser beam. As the droplet leaves the laser beam at
y/R=−1, it continues to swim in a straight line until the surface is completely covered with trans
surfactants and the droplet halts. Thus, the swimming of pushed droplets is unstable with respect to
an offset ∆yof the pushing laser beam.
Finally, we discuss how the trajectories are influenced by the reduced relaxation rate κ, with which
the surfactant mixture relaxes towards the equilibrium value φeq. In Fig. 8(a), we also plot trajectories
for κ=2 and 10 in addition to the default case κ=1 for the same initial offset ∆y=−0.2R.In all
three cases, the trajectories lie on top of each other, but for increasing κ, the droplet stops earlier. This
is clear since the surfactant mixture relaxes faster to its equilibrium value, after the droplet has left
the laser beam. Again, changing Marangoni number Mdoes not alter the results significantly.
FIG. 8. (a) Trajectories of a droplet which is pushed by UV light. The droplet initially starts at z=0 and y=∆yand stops
at the positions marked by dots. The laser is positioned at y=0 and shines from right to left [compare Fig. 7(a)]. Parameters
are set to M=1 and κ=1, unless otherwise noted. The trajectories are symmetric with respect to changing the sign of
∆y. (b) Trajectories of a droplet which is pulled by blue light. The droplet initially starts at z=0 and y=∆y. The laser is
positioned at y=0 and shines from right to left [compare Fig. 7(b)]. Again M=κ=1.
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012106-19 M. Schmitt and H. Stark Phys. Fluids 28, 012106 (2016)
B. Pulling an absorbing droplet with blue light
In the following, we present an alternative mechanism to drive an oil droplet by light. Here, the
droplet of constant radius Rinitially is in equilibrium with a water phase laden by cis surfactant. A
blue laser beam with cross-sectional radius ρ<Ris focused on the center of the droplet and locally
transforms the surfactant into the trans state [see Fig. 5(b)], thereby lowering the surface tension of
this region. The resulting Marangoni flow at the interface points away from the laser beam and thus
pulls the droplet towards the laser beam. The advective current moves surfactants away from the laser
beam, which are replenished by cis surfactants from the water phase. Again, the droplet oil phase
completely absorbs the incident light beam. In Sec. VI D, we consider a transparent droplet, which
is pulled by blue light.
1. Stationary solution of pulled droplet
For the numerical solution of Eq. (38), the order parameter φis initially set to φ=φeq =1. The
blue laser beam with its circular patch of radius ρ=0.2Ris implemented by setting φ=−1 in the area
of exposure. Figure 6, case (b) shows the stationary order parameter profile φas well as the interfacial
Marangoni flow uθ|R. Since the maximum of uθ|Ris at the front of the droplet, the droplet is a puller
with β=1.4. Note the different shape of uθ|Rcompared to the pushed droplet. The difference is due
to positive curvature σ′′(φ)>0 of the nonlinear equation of state (37). Again, for increasing κ, the
step in the order parameter profile φbecomes steeper.
2. Pulling the droplet back to center
In Sec. VI A 3, we demonstrated the unstable swimming of the pushed droplet. The droplet pulled
by the blue laser beam shows the opposite behavior. As sketched in Fig. 7(b), the droplet with offset
∆yis pulled into the laser beam. This decreases the offset and the orientation vector etilts towards
and finally aligns along the laser beam. Figure 8(a) shows droplet trajectories for several initial offsets
∆y. For ∆y=0, the droplet swims in a straight line to the right, while in the case ∆y�0, the droplet
position relaxes towards y=0 while performing damped oscillations about the stable swimming
direction. Thus, the straight swimming trajectory along the laser beam is stable with respect to lateral
excursions.
Now, we discuss how the pulled droplet trajectories depend on κ. Figure 9depicts them for an
initial offset of ∆y=−0.9R. For large relaxation rates such as κ=10 (yellow curve in Fig. 9), the
surfactants relax back to the cis conformation as soon as the illuminated region moves out of the
laser beam. Hence, the swimming direction eis always directed towards the illuminated spot [see
FIG. 9. Trajectories of a droplet, which is pulled by blue light for different relaxation rates κ. Initially, the droplet is placed
at z=0 and y=−0.9R. The laser is positioned at y=0 and shines from right to left [compare Fig. 7(b)]. Inset: For κ=0.1,
a stable oscillation with wavelength λand amplitude Adevelops.
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012106-20 M. Schmitt and H. Stark Phys. Fluids 28, 012106 (2016)
FIG. 10. Snapshots of the order parameter profile φand the flow field u|Rat the interface of the pulled droplet for (a)
κ=10, (b) κ=1, and (c) κ=0.1. The area illuminated by the laser beam is shown by a circle and the bold dot indicates the
swimming direction e. The snapshots are taken from a supplemental movie. (Multimedia view) [URL: http://dx.doi.org/10.
1063/1.4939212.1]
snapshot (a) in Fig. 10 (Multimedia view)] and relaxes towards the beam direction as illustrated in
the supplemental movie (see Multimedia view associated with Fig. 10). However, a closer inspection
of the yellow curve in Fig. 9shows that the droplet crosses the zaxis before relaxing towards y=0.
This happens since the surfactant relaxation is not infinitely fast. The effect becomes even clearer for
κ=1 green curve in Fig. 9, where the lateral droplet position performs a damped oscillatory motion
about the laser beam axis. Since the surfactant relaxation (κ=1) is sufficiently slow compared to
the droplet speed M=1, the swimming direction edoes not point towards the illuminated spot at
early times [see snapshot (b) in Fig. 10 (Multimedia view)]. The droplet crosses several times the z
axis before its direction aligns along the laser beam, as the supplemental movie shows. The movie
also demonstrates how the step in the order parameter profile φbecomes steeper with increasing κ
when the stationary state is reached. This was already mentioned before. Interestingly, at very slow
surfactant relaxation (κ=0.1), the droplet performs a stable oscillatory motion about the beam axis,
which is nicely illustrated by the supplemental movie. Increasing the Marangoni number Mincreases
swimming velocity and the oscillations occur at lower κ−1.
Figure 11 plots the amplitude Aof the stable oscillations versus κ−1and reveals a subcritical
Hopf bifurcation. In the parameter range κ−1=7.5–14, both straight swimming (amplitude A=0)
and oscillatory motion (A�0) occur depending on the initial lateral displacement ∆y. Indeed, if
∆yis above the unstable branch of the Hopf bifurcation, plotted as dashed line in Fig. 11, the
droplet assumes the oscillating state. The two swimming regimes are illustrated by phase portraits in
FIG. 11. Amplitude Aand wave number ν=1/λof the oscillating droplet trajectory (see inset of Fig. 9) plotted versus κ−1.
The two insets depict trajectories in (y,ψ)phase space, where ψis the angle between droplet orientation and laser beam axis
(see Fig. 9). Phase trajectories for κ−1=1 and 10 are shown and dots indicate the initial positions.
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012106-21 M. Schmitt and H. Stark Phys. Fluids 28, 012106 (2016)
orientation angle ψ=cos−1(e·ez)versus y. They either reveal a stable fixpoint (inset for κ−1=1)
or a stable limit cycle (inset for κ−1=10). Finally, we also plot the wave number ν=1/λof the
oscillatory swimming motion along the zaxis. It decreases with κ−1since the droplet moves more
persistently and thereby performs longer excursions from the beam axis.
In experiments, κ=τ/τeq can again be tuned by adjusting the surfactant concentration cSin
the bulk phase. We estimate the equilibration rate by τ−1
eq =j·4πR2, where j=ka·cSis the flux of
surfactants from the bulk to the droplet interface and kais a typical adsorption rate constant. Using also
τ=R2/Ds, one obtains κ=ka·cS·4πR4/Ds. With typical values R=50 µm, Ds=10−5cm2/s, and
ka=109m/(mol s), one then finds κ≈105l/mol ·cS.72 Thus, in order to observe the Hopf bifurcation
at κ≈0.1, one has to set up an emulsion with surfactant density cS≈10−3mmol/l. For smaller cS,
we expect oscillations and for larger cSdamped motion.
Finally, we note that we observed the same qualitative behavior as in Figs. 8-11 for a linear diffu-
sive current jD=D∇sφand a linear equation of state for the surface tension σ. Hence, the origin of
the Hopf bifurcation lies clearly in the nonlinear advection term Mφu|Rof Eq. (38).
C. Pushing a transparent droplet with UV light
In the following, we discuss the case of an emulsion droplet with negligible light absorbance.
The laser beam crosses the droplet and also actuates it at a second spot as illustrated in Fig. 12(a).
Here, we focus on a water droplet immersed in a transparent oil phase laden with trans surfactants.
But we will also comment on the inverse case of an oil droplet in water. Due to the different refractive
indices of oil and water, the transmitted beam is refracted at each interface according to the refraction
law nsin α=ˆnsin ˆα. Here, αand ˆαare the respective angles of the beam with respect to the surface
normal in the oil and water phase while nand ˆnare the respective refraction indices. We apply the
refraction law to partial beams of the incident light so that it widens while crossing and leaving the
droplet. In what follows, we use n=1.45 and ˆn=1.35. Note that we neglect any reflection except
for total reflection above the critical angle αmax =arcsin(ˆn/n). For the emulsion droplet, this implies
that laser light is completely reflected if it hits the interface with a lateral distance to droplet center
above ∆ymax/R=ˆn/n≈0.93. The general mechanism for the light-induced Marangoni flow is the
same as in Sec. VI A. It is directed away from each illuminated spot. Since the spots are well sepa-
rated from each other, the droplet velocity vector is a superposition of the vectors induced by each
spot.
FIG. 12. (a) UV laser light is aimed at a transparent water droplet in oil, which is offset by ∆yfrom the laser beam. The
resulting Marangoni flow at the two spots drives the droplet out of the beam. (b) Blue laser light is aimed at a water droplet
in oil, which is offset by ∆yfrom the laser beam. The resulting Marangoni flow at the two spots pulls the droplet back into
the laser beam.
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012106-22 M. Schmitt and H. Stark Phys. Fluids 28, 012106 (2016)
FIG. 13. Trajectories of a transparent droplet which is pushed by UV light. The droplet initially starts at z=0 and y=∆y
and stops at the positions marked by dots. The laser is positioned at y=0 and shines from right to left [compare Fig. 12(a)].
Parameters are set to M=1 and κ=1. The trajectories are symmetric with respect to changing the sign of ∆y.
Again, we start with a laser beam which is aimed at the center of the droplet. Due to refraction,
the transmitted beam widens and the second illuminated spot is slightly larger than the first one. Thus,
the velocity vector induced by the second spot is also slightly larger and slowly pushes the droplet
towards the laser beam. This effect is hardly visible in our simulations. However, as soon as we intro-
duce an offset ∆y, the widening of the laser beam becomes stronger. The resulting velocity vector
with orientation epushes the droplet further away from the laser beam and also against the beam
direction, as illustrated in Fig. 12(a). Ultimately, the droplet leaves the beam completely. Figure 13
shows trajectories for various initial offsets. In the cases ∆y=−0.2Rand −0.5R, the droplet initially
moves in negative yand positive zdirection [see also Fig. 12(a)]. Once the second laser spot has
sufficiently decreased in size, since part of the beam is totally reflected, the droplet moves in negative
zdirection. It leaves the beam and finally stops. Thus, in analogy to the findings of Sec. VI A, the
droplet is pushed out of the beam.
For an oil droplet immersed in water, the transmitted beam becomes more narrower. The droplet
is still pushed out of the beam but the motion along zdirection is reversed. The corresponding trajec-
tories are similar to the ones in Fig. 13, albeit reflected about the y-axis. If droplet and surrounding
phase have equal refractive indices, the motion out of the beam is exactly along the y-axis. In all
cases, the interfacial flow field is concentrated at the back of the droplet and the droplet is a pusher.
D. Pulling a transparent droplet with blue light
Now, we study the effect of a blue light beam aimed at a water droplet, which is suspended in
an oil phase laden with cis surfactants [see Fig. 12(b)]. In this case, the Marangoni flow is directed
away from the illuminated spots. At zero offset, ∆y=0, the droplet slowly moves along the negative
zdirection. Any offset ∆y�0 pulls the droplet back into the beam with the velocity vector slightly
tilted towards −ez[see Fig. 12(b)]. As in Sec. VI B, we use coupling strength κto distinguish between
different regimes of motion.
Figure 14 (Multimedia view) shows trajectories from a supplemental movie (see Multimedia
view A associated with Fig. 14). In the case of strong coupling to the bulk phase, κ=10, the droplet
performs a damped oscillation about y=0. The spatial resolution of our numerical method is not
large enough to resolve the size difference between the two illuminated spots. Therefore, in the supple-
mental movie, the droplet stops and does not move into the negative zdirection. Upon decreasing
the relaxation rate to values below κ=4.5, the droplet undergoes a subcritical Hopf bifurcation and
the droplet starts to oscillate about the laser beam. Figure 14 (Multimedia view) shows the trajectory
from the supplemental movie; the droplet has already left the scene to the left. Figure 15 shows the
subcritical bifurcation in the bottom graph, where amplitude Aand wave number νare plotted versus
κ−1, or in the top phase portraits, where the limit cycle in case 2 is visible. Below κ=2.2, the droplet
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012106-23 M. Schmitt and H. Stark Phys. Fluids 28, 012106 (2016)
FIG. 14. Trajectories of a transparent water droplet suspended in oil, which is actuated by blue light for κ=10,3,1,0.5.
The snapshots are taken from a supplemental movie. Trajectories of a transparent oil droplet suspended in water, which
is actuated by blue light, are shown in a second supplemental movie. In all cases, we set M=1 and used an initial offset
∆y=−0.5[compare Fig. 12(b)]. (Multimedia view) [URL: http://dx.doi.org/10.1063/1.4939212.2] [URL: http://dx.doi.org/
10.1063/1.4939212.3]
FIG. 15. Bottom: Amplitude Aand wave number ν=1/λof the oscillating droplet trajectories plotted versus κ−1. Top:
Trajectories in (y,ψ)phase space at values of κ−1marked by numbers in the bottom plot. Here, ψis the angle between
droplet orientation and laser beam axis, as indicated in Fig. 14 (Multimedia view). Dots indicate the initial positions. Fig. 14
(Multimedia view) shows the corresponding trajectories in (z, y)space.
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012106-24 M. Schmitt and H. Stark Phys. Fluids 28, 012106 (2016)
changes its dynamics completely. After moving along the negative zdirection for a few droplet radii
R, the droplet reverses its swimming direction and reaches a stationary oscillating state. The reversal
occurs because the amplitude of the oscillation is so large that the size of the second spot decreases in
size due to total reflection and the first spot pulls more strongly. This oscillation is characterized by
larger amplitude Aand wave number λcompared to the first oscillation state [see Fig. 15]. Finally, at
relaxation rates belowκ=0.54, the droplet eventually leaves the beam and stops. For the experimental
realization of different values of κ, we refer to the discussion in Sec. VI B 2.
For an oil droplet immersed in water, where total reflection does not occur, we only observe three
states, in which the droplet moves against the laser beam: damped oscillations, stationary oscillations,
and where the droplet ultimately leaves the laser beam. A supplemental movie illustrates the three
cases (see Multimedia view B associated with Fig. 14).
VII. CONCLUSIONS
A non-uniform surface tension profile σat the interface of an emulsion droplet generates flow
fields at the interface and inside as well as outside of the droplet. The flow at the interface is directed
along the gradient of σ. Using this Marangoni effect, the emulsion droplet becomes active. We de-
composed the surface tension profile into spherical harmonics, σ(θ,ϕ)=sm
lYm
l, and for this most
general form of σwe determined the full three-dimensional flow field inside [ˆ
u(r)], outside [u(r)],
and at the interface [u|R(θ,ϕ)] of the droplet as a function of the expansion coefficients sm
l. The swim-
ming kinematics of the droplet follows from the droplet velocity vector vD, which solely depends
on the coefficients sm
1. The flow field outside of the droplet decays either as 1/r3in the case of a
neutral swimmer or as 1/r2in the case of a pusher or a puller. The squirmer parameter β, for which
we derived an expression in terms of the coefficients sm
1and sm
2for arbitrary swimming direction,
enables to distinguish between these cases.
In the second part of this paper, we presented two illustrative examples to demonstrate how
gradients in the surface tension σcan be achieved and studied the resulting droplet motion.
In the first example, we considered an initially surfactant free droplet, which adsorbs micelles
formed by surfactants. The adsorbed micelle not only induces Marangoni flow in the proximity of the
droplet interface but also radial fluid flow towards the adsorption site. The radial flow enhances the
probability that other micelles adsorb at the same site. This mechanism leads to directed propulsion
of an initially isotropic emulsion droplet if the micellar adsorption rate is sufficiently large. Clearly,
the mechanism only works when surfactants are adsorbed through micelles. Single surfactants would
not produce a sufficiently strong radial flow to spontaneously break the isotropic symmetry of the
droplet. Our idealized example stresses the role which micelles play in generating directed motion
in active emulsions. Therefore, it might contribute to understanding the self-propulsion of water and
liquid-crystal droplets, which has been demonstrated in recent publications.25,26
The second example considered a non-uniform mixture of two surfactant types in order to
generate Marangoni flow. We used light-switchable surfactants based on the trans-cis isomerism of
azobenzene to generate a non-uniform surfactant mixture. The analytic formulas for the flow field
together with a diffusion-advection-reaction equation for the mixture order parameter determine
the dynamics of the surfactant mixture and hence the droplet trajectory. We demonstrated that an
emulsion droplet laden with trans surfactants, and either strongly adsorbing or transparent, can be
pushed by a laser beam with UV light. However, the resulting straight trajectory is unstable with
respect to displacing the droplet center relative to the laser beam axis. In contrast, a droplet laden
with cis surfactants can be pulled into a laser beam with blue light. The straight trajectory is stable
against lateral displacements. By decreasing the surfactant relaxation rate, the droplet develops an
oscillatory trajectory about the laser beam via a subcritical Hopf bifurcation.
Having at hand analytic formulas for the full three-dimensional flow field, we are now able to
fully discuss the recently introduced active emulsion droplet, where the surfactant mixture is gener-
ated by a bromination reaction.18,19 We included thermal noise in the diffusion-advection-reaction
equation of the mixture order parameter and currently study the coarsening dynamics of the surfactant
mixture towards the stationary order parameter profile.71 Thermal fluctuations in the composition
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012106-25 M. Schmitt and H. Stark Phys. Fluids 28, 012106 (2016)
of the surfactant mixture are responsible for the rotational diffusion of the swimming direction and
thereby generate a persistent random walk of the active emulsion droplet.
ACKNOWLEDGMENTS
We acknowledge financial support by the Deutsche Forschungsgemeinschaft in the framework
of the collaborative research center SFB 910 and the research training group GRK 1558.
APPENDIX A: SPHERICAL HARMONICS
Throughout this paper, we use the following definition of spherical harmonics:
Ym
l(θ,ϕ)=2l+1
4π
(l−m)!
(l+m)!Pm
l(cos θ)eimϕ,
with associated Legendre polynomials Pm
lof degree l, order m, and with orthonormality,
π
0
2π
0
Ym
lYm′
l′sin θdθdϕ=δl,l′δm,m′,
where Ym
ldenotes the complex conjugate of Ym
l.
APPENDIX B: FLUID FLOW IN THE BULK
Here, we give the complete velocity field inside ˆ
u=ˆ
v+ˆ
wand outside u=v+wof the droplet
in the droplet frame,
ˆur=−η
2(η+ˆη)r2
R2−1er·vD+1
η+ˆη
∞
l=1
l
m=−lrl+1
Rl+1−rl−1
Rl−1l(l+1)sm
l
4l+2Ym
l,(B1a)
ˆuθ=−η
2(η+ˆη)2r2
R2−1eθ·vD+1
η+ˆη
∞
l=1
l
m=−l(l+3)rl+1
Rl+1−(l+1)rl−1
Rl−1sm
l
4l+2∂θYm
l,(B1b)
ˆuϕ=−η
2(η+ˆη)2r2
R2−1eϕ·vD+1
η+ˆη
∞
l=1
l
m=−l(l+3)rl+1
Rl+1−(l+1)rl−1
Rl−1imsm
l
4l+2
Ym
l
sin θ,(B1c)
ur=(−η
2(η+ˆη)R
r−R3
r3−1−3R
2r+R3
2r3)er·vD
+1
η+ˆη
∞
l=1
l
m=−lRl
rl−Rl+2
rl+2l(l+1)sm
l
4l+2Ym
l,(B1d)
uθ=(−η
4(η+ˆη)R
r+R3
r3−1−3R
4r−R3
4r3)eθ·vD
+1
η+ˆη
∞
l=1
l
m=−l(2−l)Rl
rl+lRl+2
rl+2sm
l
4l+2∂θYm
l,(B1e)
uϕ=(−η
4(η+ˆη)R
r+R3
r3−1−3R
4r−R3
4r3)eϕ·vD
+1
η+ˆη
∞
l=1
l
m=−l(2−l)Rl
rl+lRl+2
rl+2imsm
l
4l+2
Ym
l
sin θ.(B1f)
u
Note
v
that
D.
for r = R, one recaptures Eqs. (17) and boundary condition ur = uˆr = 0, while for r → ∞,
= −
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012106-26 M. Schmitt and H. Stark Phys. Fluids 28, 012106 (2016)
Note, by combining the outside field of the pumping solution wwith the first term of the passive-
droplet field v,the stokeslet components, i.e., terms with u∝r−1, cancel each other,
ur=−er·vD(1−R3
r3)+1
η+ˆη
∞
l=2
l
m=−lRl
rl−Rl+2
rl+2l(l+1)sm
l
4l+2Ym
l,
uθ=−eθ·vD(1+R3
2r3)+1
η+ˆη
∞
l=2
l
m=−l(2−l)Rl
rl+lRl+2
rl+2sm
l
4l+2∂θYm
l,
uϕ=−eϕ·vD(1+R3
2r3)+1
η+ˆη
∞
l=2
l
m=−l(2−l)Rl
rl+lRl+2
rl+2imsm
l
4l+2
Ym
l
sin θ.
This shows that the droplet is a force-free swimmer.4Thus, in leading order, the flow field is given by
a stresslet with u∝r−2. The squirmer parameter βcalculated in Sec. IV A determines the sign and
the magnitude of the stresslet. In particular, if the coefficients sm
2vanish, the squirmer parameter also
becomes zero (β=0). Then, the flow field is less long-ranged and decays as u∝r−3.
Finally, by adding the droplet velocity vector to our solution, one arrives at the velocity field
uL=u+vDin the lab frame,
uL
r=R3
r3er·vD+1
η+ˆη
∞
l=2
l
m=−lRl
rl−Rl+2
rl+2l(l+1)sm
l
4l+2Ym
l,
uL
θ=−R3
2r3eθ·vD+1
η+ˆη
∞
l=2
l
m=−l(2−l)Rl
rl+lRl+2
rl+2sm
l
4l+2∂θYm
l,
uL
ϕ=−R3
2r3eϕ·vD+1
η+ˆη
∞
l=2
l
m=−l(2−l)Rl
rl+lRl+2
rl+2imsm
l
4l+2
Ym
l
sin θ.
In this frame, the velocity field satisfies the boundary condition uL|r→∞ =0. Note that in this frame,
the radial component uL
rdoes not vanish at r=R.
APPENDIX C: LORENTZ RECIPROCAL THEOREM
Applying the Lorentz reciprocal theorem to relate the flow fields of the pumping active droplet
from Sec. IIAand the passive droplet from Sec. II B to each other, one arrives at the alternative
expression for the droplet velocity,73
vD=−1
4πR2
3η+3 ˆη
2η+3 ˆη w|RdA.(C1)
Note that this generalizes the expression for rigid active spherical swimmers ( ˆη→ ∞) in Ref. 59.
Using the surface flow field of the pumping droplet from Eqs. (12) in Eq. (C1), one obtains Eq. (21).
APPENDIX D: COMPARISON WITH SQUIRMER MODEL
The presented solution u(r)for the flow field around an active droplet can be related to the axisym-
metric squirmer model introduced by Lighthill55 and later by Blake56 as follows. The squirmer flow
field can also be decomposed into a pumping active and a passive part, usq =wsq +vsq, where vsq is
the usual Stokes flow field of a solid sphere, which we obtain in the limit of infinite internal viscosity:
vsq =lim ˆη→∞v. In order to match wwith the known squirmer field wsq, one has to set
sl=−(η+ˆη)4l+2
l(l+1)4π
2l+1Bl.(D1)
∞
l
This yields the correct flow fi
2
eld
1
of a swimming squirmer with surface v
56
elocity field uθ =
=1 BlVl(cos θ), where Vl = l(l−+1) Pl (cos θ). Here, we used the notation of Blake.
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012106-27 M. Schmitt and H. Stark Phys. Fluids 28, 012106 (2016)
APPENDIX E: WRITTEN-OUT GENERALIZED SQUIRMER PARAMETER
The generalized squirmer parameter in Eq. (28) for a droplet swimming in an arbitrary direction
can be written in terms of the coefficients sm
1and sm
2using Eqs. (22) and (24),
β=−27
5
˜s0
2
|˜s0
1|,(E1a)
˜s0
1=(s0
1)2−2s1
1s−1
1,(E1b)
˜s0
2=(√6�s2
2(s−1
1)2+s−2
2(s1
1)2�−√12s0
1�s1
2s−1
1+s−1
2s1
1�
+2s0
2�(s0
1)2+s1
1s−1
1�)/�2(s0
1)2−4s1
1s−1
1�.(E1c)
1A. Najafiand R. Golestanian, “Simple swimmer at low Reynolds number: Three linked spheres,” Phys. Rev. E 69, 062901
(2004).
2R. Dreyfus, J. Baudry, M. L. Roper, M. Fermigier, H. A. Stone, and J. Bibette, “Microscopic artificial swimmers,” Nature
437, 862 (2005).
3E. Gauger and H. Stark, “Numerical study of a microscopic artificial swimmer,” Phys. Rev. E 74, 021907 (2006).
4E. Lauga and T. R. Powers, “The hydrodynamics of swimming microorganisms,” Rep. Prog. Phys. 72, 096601 (2009).
5A. Walther and A. H. Müller, “Janus particles,” Soft Matter 4, 663 (2008).
6R. Golestanian, T. B. Liverpool, and A. Ajdari, “Propulsion of a molecular machine by asymmetric distribution of reaction
products,” Phys. Rev. Lett. 94, 220801 (2005).
7W. F. Paxton, A. Sen, and T. E. Mallouk, “Motility of catalytic nanoparticles through self-generated forces,” Chem. - Eur.
J. 11, 6462 (2005).
8J. L. Moran and J. D. Posner, “Electrokinetic locomotion due to reaction-induced charge auto-electrophoresis,” J. Fluid.
Mech. 680, 31 (2011).
9T. Bickel, A. Majee, and A. Würger, “Flow pattern in the vicinity of self-propelling hot Janus particles,” Phys. Rev. E 88,
012301 (2013).
10 I. Buttinoni, G. Volpe, F. Kümmel, G. Volpe, and C. Bechinger, “Active Brownian motion tunable by light,” J. Phys.: Con-
dens. Matter 24, 284129 (2012).
11 M. C. Marchetti, J. F. Joanny, S. Ramaswamy, T. B. Liverpool, J. Prost, M. Rao, and R. A. Simha, “Hydrodynamics of soft
active matter,” Rev. Mod. Phys. 85, 1143 (2013).
12 T. Ishikawa and T. J. Pedley, “Coherent structures in monolayers of swimming particles,” Phys. Rev. Lett. 100, 088103
(2008).
13 A. A. Evans, T. Ishikawa, T. Yamaguchi, and E. Lauga, “Orientational order in concentrated suspensions of spherical mi-
croswimmers,” Phys. Fluids 23, 111702 (2011).
14 F. Alarcón and I. Pagonabarraga, “Spontaneous aggregation and global polar ordering in squirmer suspensions,” J. Mol.
Liq. 185, 56 (2013).
15 A. Zöttl and H. Stark, “Hydrodynamics determines collective motion and phase behavior of active colloids in quasi-two-
dimensional confinement,” Phys. Rev. Lett. 112, 118101 (2014).
16 M. Hennes, K. Wolff, and H. Stark, “Self-induced polar order of active Brownian particles in a harmonic trap,” Phys. Rev.
Lett. 112, 238104 (2014).
17 A. Bricard, J.-B. Caussin, N. Desreumaux, O. Dauchot, and D. Bartolo, “Emergence of macroscopic directed motion in
populations of motile colloids,” Nature 503, 95 (2013).
18 S. Thutupalli, R. Seemann, and S. Herminghaus, “Swarming behavior of simple model squirmers,” New J. Phys. 13, 073021
(2011).
19 M. Schmitt and H. Stark, “Swimming active droplet: A theoretical analysis,” Europhys. Lett. 101, 44008 (2013).
20 S.-Y. Teh, R. Lin, L.-H. Hung, and A. P. Lee, “Droplet microfluidics,” Lab Chip 8, 198 (2008).
21 R. Seemann, M. Brinkmann, T. Pfohl, and S. Herminghaus, “Droplet based microfluidics,” Rep. Prog. Phys. 75, 016601
(2012).
22 M. M. Hanczyc, T. Toyota, T. Ikegami, N. Packard, and T. Sugawara, “Fatty acid chemistry at the oil–water interface:
Self-propelled oil droplets,” J. Am. Chem. Soc. 129, 9386 (2007).
23 T. Toyota, N. Maru, M. M. Hanczyc, T. Ikegami, and T. Sugawara, “Self-propelled oil droplets consuming ‘Fuel’ surfactant,”
J. Am. Chem. Soc. 131, 5012 (2009).
24 T. Banno, R. Kuroha, and T. Toyota, “pH-sensitive self-propelled motion of oil droplets in the presence of cationic surfactants
containing hydrolyzable ester linkages,” Langmuir 28, 1190 (2012).
25 S. Herminghaus, C. C. Maass, C. Krüger, S. Thutupalli, L. Goehring, and C. Bahr, “Interfacial mechanisms in active emul-
sions,” Soft Matter 10, 7008 (2014).
26 Z. Izri, M. N. van der Linden, S. Michelin, and O. Dauchot, “Self-propulsion of pure water droplets by spontaneous
marangoni-stress-driven motion,” Phys. Rev. Lett. 113, 248302 (2014).
27 H. Kitahata, N. Yoshinaga, K. H. Nagai, and Y. Sumino, “Spontaneous motion of a droplet coupled with a chemical wave,”
Phys. Rev. E 84, 015101 (2011).
28 E. Tjhung, D. Marenduzzo, and M. E. Cates, “Spontaneous symmetry breaking in active droplets provides a generic route
to motility,” Proc. Natl. Acad. Sci. U. S. A. 109, 12381 (2012).
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions.
012106-28 M. Schmitt and H. Stark Phys. Fluids 28, 012106 (2016)
29 N. Yoshinaga, “Spontaneous motion and deformation of a self-propelled droplet,” Phys. Rev. E 89, 012913 (2014).
30 S. Yabunaka, T. Ohta, and N. Yoshinaga, “Self-propelled motion of a fluid droplet under chemical reaction,” J. Chem. Phys.
136, 074904 (2012).
31 A. Y. Rednikov, Y. S. Ryazantsev, and M. G. Velarde, “Drop motion with surfactant transfer in a homogeneous surrounding,”
Phys. Fluids 6, 451 (1994).
32 A. Y. Rednikov, Y. S. Ryazantsev, and M. G. Velarde, “Active drops and drop motions due to nonequilibrium phenomena,”
J. Non-Equilib. Thermodyn. 19, 95 (1994).
33 M. G. Velarde, A. Y. Rednikov, and Y. S. Ryazantsev, “Drop motions and interfacial instability,” J. Phys.: Condens. Matter
8, 9233 (1996).
34 M. G. Velarde, “Drops, liquid layers and the Marangoni effect,” Philos. Trans. R. Soc., A 356, 829 (1998).
35 N. Yoshinaga, K. H. Nagai, Y. Sumino, and H. Kitahata, “Drift instability in the motion of a fluid droplet with a chemically
reactive surface driven by Marangoni flow,” Phys. Rev. E 86, 016108 (2012).
36 K. Furtado, C. M. Pooley, and J. M. Yeomans, “Lattice Boltzmann study of convective drop motion driven by nonlinear
chemical kinetics,” Phys. Rev. E 78, 046308 (2008).
37 L. E. Scriven and C. V. Sternling, “The Marangoni effects,” Nature 187, 186 (1960).
38 M. D. Levan and J. Newman, “The effect of surfactant on the terminal and interfacial velocities of a bubble or drop,” AIChE
J. 22, 695 (1976).
39 D. P. Mason and G. M. Moremedi, “Effects of non-uniform interfacial tension in small Reynolds number flow past a spherical
liquid drop,” Pramana 77, 493 (2011).
40 J. Bławzdziewicz, P. Vlahovska, and M. Loewenberg, “Rheology of a dilute emulsion of surfactant-covered spherical drops,”
Physica A 276, 50 (2000).
41 J. A. Hanna and P. M. Vlahovska, “Surfactant-induced migration of a spherical drop in Stokes flow,” Phys. Fluids 22, 013102
(2010).
42 J. T. Schwalbe, F. R. Phelan, Jr., P. M. Vlahovska, and S. D. Hudson, “Interfacial effects on droplet dynamics in Poiseuille
flow,” Soft Matter 7, 7797 (2011).
43 T. Sakai, “Surfactant-free emulsions,” Curr. Opin. Colloid Interface Sci. 13, 228 (2008).
44 A. Diguet, R.-M. Guillermic, N. Magome, A. Saint-Jalmes, Y. Chen, K. Yoshikawa, and D. Baigl, “Photomanipulation of
a droplet by the chromocapillary effect,” Angew. Chem., Int. Ed. 121, 9445 (2009).
45 O. Pak and E. Lauga, “Generalized squirming motion of a sphere,” J. Eng. Math. 88, 1 (2014).
46 S. Sadhal, P. Ayyaswamy, and J. Chung, Transport Phenomena with Drops and Bubbles, Mechanical Engineering Series
(Springer, 1996).
47 H. Lamb, Hydrodynamics (Cambridge University Press, 1932).
48 J. Happel and H. Brenner, Low Reynolds Number Hydrodynamics: With Special Applications to Particulate Media, Me-
chanics of Fluids and Transport Processes (Springer, Netherlands, 1983).
49 R. Clift, J. R. Grace, and M. E. Weber, Bubbles, Drops, and Particles (Academic Press, 1978).
50 V. A. Nepomniashchii, M. G. Velarde, and P. Colinet, Interfacial Phenomena and Convection, 1st ed. (Chapman & Hall/CRC,
2002).
51 O. S. Pak, J. Feng, and H. A. Stone, “Viscous Marangoni migration of a drop in a Poiseuille flow at low surface Péclet
numbers,” J. Fluid. Mech. 753, 535 (2014).
52 G. Batchelor, “The stress system in a suspension of force-free particles,” J. Fluid. Mech. 41, 545 (1970).
53 T. Ishikawa, M. P. Simmonds, and T. J. Pedley, “Hydrodynamic interaction of two swimming model micro-organisms,” J.
Fluid. Mech. 568, 119 (2006).
54 S. Kim and S. Karrila, Microhydrodynamics: Principles and Selected Applications, Butterworth—Heinemann Series in
Chemical Engineering (Dover Publications, 2005).
55 M. J. Lighthill, “On the squirming motion of nearly spherical deformable bodies through liquids at very small Reynolds
numbers,” Commun. Pure Appl. Math. 5, 109 (1952).
56 J. R. Blake, “A spherical envelope approach to ciliary propulsion,” J. Fluid. Mech. 46, 199 (1971).
57 M. T. Downton and H. Stark, “Simulation of a model microswimmer,” J. Phys.: Condens. Matter 21, 204101 (2009).
58 L. Zhu, E. Lauga, and L. Brandt, “Self-propulsion in viscoelastic fluids: Pushers vs. pullers,” Phys. Fluids 24, 051902
(2012).
59 H. A. Stone and A. D. Samuel, “Propulsion of microorganisms by surface distortions,” Phys. Rev. Lett. 77, 4102 (1996).
60 I. Griffiths, C. Bain, C. Breward, D. Colegate, P. Howell, and S. Waters, “On the predictions and limitations of the
Becker-Döring model for reaction kinetics in micellar surfactant solutions,” J. Colloid Interface Sci. 360, 662 (2011).
61 V. G. Levich, Physicochemical Hydrodynamics (Prentice-Hall, 1962).
62 Here, we decompose the surface tension up to lmax =60, which has proven to be sufficient to model an adsorbed micelle
without leading to numerical problems such as the Gibbs phenomenon. More details about the simulation method can be
found in Ref. 71.
63 N. Van Kampen, Stochastic Processes in Physics and Chemistry (North-Holland Personal Library, Elsevier Science, 1992).
64 W. Al-Soufi, L. Piñeiro, and M. Novo, “A model for monomer and micellar concentrations in surfactant solutions: Appli-
cation to conductivity, NMR, diffusion, and surface tension data,” J. Colloid Interface Sci. 370, 102 (2012).
65 S. S. Dukhin, G. Kretzschmar, and R. Miller, Dynamics of Adsorption at Liquid Interfaces: Theory, Experiment, Application
(Elsevier, 1995), Vol. 1.
66 O. Karthaus, M. Shimomura, M. Hioki, R. Tahara, and H. Nakamura, “Reversible photomorphism in surface monolayers,”
J. Am. Chem. Soc. 118, 9174 (1996).
67 J. Y. Shin and N. L. Abbott, “Using light to control dynamic surface tensions of aqueous solutions of water soluble surfac-
tants,” Langmuir 15, 4404 (1999).
68 K. Ichimura, S.-K. Oh, and M. Nakagawa, “Light-driven motion of liquids on a photoresponsive surface,” Science 288,
1624 (2000).
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions.
012106-29 M. Schmitt and H. Stark Phys. Fluids 28, 012106 (2016)
69 J. Eastoe and A. Vesperinas, “Self-assembly of light-sensitive surfactants,” Soft Matter 1, 338 (2005).
70 T. Krol, A. Stelmaszewski, and W. Freda, “Variability in the optical properties of a crude oil-seawater emulsion,” Oceanolo-
gia 48, 203 (2006), Available at: http://www.iopan.gda.pl/oceanologia/48Skrol.pdf.
71 M. Schmitt and H. Stark, “Marangoni phase separation of surfactants drives active Brownian motion of emulsion droplet”
(unpublished).
72 B. Binks and D. Furlong, Modern Characterization Methods of Surfactant Systems (CRC Press, 1999), Vol. 83.
73 R. S. Subramanian, “The Stokes force on a droplet in an unbounded fluid medium due to capillary effects,” J. Fluid. Mech.
153, 389 (1985).
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions.
94
96
EPJ manuscript No.
(will be inserted by the editor)
Active Brownian motion of emulsion droplets:
Coarsening dynamics at the interface and rotational diffusion
M. Schmitt and H. Stark
Institut f¨ur Theoretische Physik, Technische Universit¨at Berlin - Hardenbergstraße 36, 10623 Berlin, Germany
Received: date / Revised version: date
Abstract. A micron-sized droplet of bromine water immersed in a surfactant-laden oil phase can swim
[1]. The bromine reacts with the surfactant at the droplet interface and generates a surfactant mixture.
It can spontaneously phase-separate due to solutocapillary Marangoni flow, which propels the droplet.
We model the system by a diffusion-advection-reaction equation for the mixture order parameter at the
interface including thermal noise and couple it to fluid flow. Going beyond previous work, we illustrate the
coarsening dynamics of the surfactant mixture towards phase separation in the axisymmetric swimming
state. Coarsening proceeds in two steps: an initially slow growth of domain size followed by a nearly
ballistic regime. On larger time scales thermal fluctuations in the local surfactant composition initiates
random changes in the swimming direction and the droplet performs a persistent random walk, as observed
in experiments. Numerical solutions show that the rotational correlation time scales with the square of the
inverse noise strength. We confirm this scaling by a perturbation theory for the fluctuations in the mixture
order parameter and thereby identify the active emulsion droplet as an active Brownian particle.
PACS. 47.20.Dr Surface-tension-driven instability – 47.55.D- Drops and bubbles – 47.55.pf Marangoni
convection
1 Introduction
In the past decade autonomous swimming of particles at
low Reynolds number has attracted a tremendous amount
of attention [2–6]. Both, in the study of living organisms
such as bacteria or algae or of artificial microswimmers a
plethora of exciting research subjects has evolved. They
include understanding the swimming mechanism [7–10]
and generic properties of microswimmers [11–14], their
swimming trajectories [15–18], and the study of their in-
teraction with surfaces as well as obstacles [19–22]. The
study of emergent collective motion has opened up a new
field in non-equilibrium statistical physics [23–31].
There are various methods to construct a microswim-
mer. One idea is to generate a slip velocity field close to the
swimmer’s surface using a phoretic mechanism. A typical
example of such an artificial swimmer is a micron-sized
spherical Janus colloid, which has an inherent polar sym-
metry. Its two faces are made of different materials and
thus differ in their physical or chemical properties [32].
For example, a Janus particle with faces of different ther-
mal conductivity moves if exposed to heat. The conversion
of thermal energy to mechanical work in a self-generated
temperature gradient is called self-thermophoresis [33].
Janus colloids also employ other phoretic mechanisms to
become active [34–37].
Send offprint requests to:
A different realization of a self-propelled particle is an
active emulsion droplet. The striking difference to an ac-
tive Janus particle is the missing inherent polar symme-
try. Instead, the symmetry between front and back breaks
spontaneously, for example, in a subcritical bifurcation
[38]. The self-sustained motion of active droplets is due
to a gradient in surface tension, which is usually caused
by an inhomogeneous density of surfactants. The resulting
stresses set up a solutocapillary Marangoni flow directed
along the surface tension gradient that drags the drop-
let through the fluid. An active droplet generates a flow
field in the surrounding fluid typical for the “squirmer”
[39–43]. Originally, the squirmer was introduced to model
the locomotion of microorganisms that propel themselves
by a carpet of short active filaments called cilia beating
in synchrony on their surfaces. The squirmer flow field at
the interface is then a coarse-grained model of the cilia
carpet.
Active droplets have extensively been studied in exper-
iments, including droplets in a bulk fluid [44,45,1,46–50]
and droplets on interfaces [51,52]. Theoretical and numer-
ical studies address the drift bifurcation of translational
motion [53–57], deformable and contractile droplets [58,
59], droplets in a chemically reacting fluid [60], droplets
driven by nonlinear chemical kinetics [61], and the dif-
fusion-advection-reaction equation for the dynamics of a
surfactant mixture at the droplet interface [38]. A com-
prehensive review on active droplets is given in ref. [10].
2 M. Schmitt, H. Stark: Active Brownian motion of emulsion droplets
An active droplet, which swims due to solutocapil-
lary Marangoni flow, has recently been realized [1]. Wa-
ter droplets with a diameter of 50 −150µm are placed
into a surfactant-rich oil phase. The surfactants migrate to
the droplet interface where they form a dense monolayer.
Bromine dissolved in the water droplets reacts with the
surfactants at the interface. It saturates the double bond
in the surfactant molecule and the surfactant becomes
weaker than the original one. Hence, the “bromination”
reaction locally increases the interfacial surface tension.
This induces Marangoni flow, which advects surfactants
and thereby further enhances the gradients in surface ten-
sion. If the advective current exceeds the smoothing diffu-
sion current, the surfactant mixture phase-separates. The
droplet develops a polar symmetry and starts to move
in a random direction, which fluctuates around such that
the droplet performs a persistent random walk. While the
droplet swims with a typical swimming speed of 15µm/s,
brominated surfactants are constantly replaced by non-
brominated surfactants from the oil phase by means of
desorption and adsorption. Finally, the swimming motion
comes to an end when the fueling bromine is exhausted.
In ref. [38] we developed a diffusion-advection-reaction
equation for the surfactant mixture at the droplet inter-
face and coupled it to the axisymmetric flow field initiated
by the Marangoni effect. In a parameter study we could
then map out a state diagram including the transition
from the resting to the swimming state and an oscillating
droplet motion. In this paper we combine our theory with
the full three-dimensional solution for the Marangoni flow,
which we derived for an arbitrary surface tension field at
the droplet interface in ref. [43]. Omitting the constraint
of axisymmetry and adding thermal noise to the dynamic
equation of the surfactant mixture, we will focus on two
new aspects of droplet dynamics that we could not address
in ref. [38]. First, while reaching the stationary uniaxial
swimming state, the surfactant mixture phase-separates
into the two surfactant types. We illustrate the coarsening
dynamics and demonstrate that it proceeds in two steps.
An initially slow growth of domain size is followed by a
nearly ballistic regime. This is reminiscent to coarsening
in the dynamic model H [62]. Second, even in the sta-
tionary swimming state the surfactant composition fluc-
tuates thermally and thereby initiates random changes in
the swimming direction, which diffuses on the unit sphere.
As a result the droplet performs a persistent random walk,
as observed in experiments [1], which we will characterize
in detail.
The article is organized as follows. In sect. 2 we reca-
pitulate our model of the active emulsion droplet from ref.
[38] and generalize it to a droplet without the constraint of
axisymmetry. While sect. 3 explains the numerical method
to solve the diffusion-advection-reaction equation on the
droplet surface, the following two sections contain the re-
sults of this article. Section 4 describes the coarsening
dynamics of the surfactant mixture before reaching the
steady swimming state and sect. 5 characterizes the per-
sistent random walk of the droplet in the swimming state.
The article concludes in sect. 6.
2 Model of an active droplet
In order to model the dynamics of the active droplet, we
follow our earlier work [38]. We use a dynamic equation
for the surfactant mixture at the droplet interface that
includes all the relevant processes. We assume that the
surfactant completely covers the droplet interface without
any intervening solvent. We also assume that the head
area of both types of surfactant molecules (brominated
and non-brominated) is the same. Denoting the bromi-
nated surfactant density by c1and the non-brominated
density by c2, we can therefore set c1+c2= 1. We then
take the concentration difference between brominated and
non-brominated surfactants as an order parameter φ=
c1−c2. In other words φ= 1 corresponds to fully bromi-
nated and φ=−1 to fully non-brominated surfactants
and c1= (1 + φ)/2 and c2= (1 −φ)/2. Finally, we choose
a constant droplet radius R.
2.1 Diffusion-advection-reaction equation
The dynamics of the order parameter φat the droplet
interface can be expressed as [38]:
∂tφ=−∇s·(jD+jA)−τ−1
R(φ−φeq) + ζ(r, t),(1)
which we formulate in the form of a continuity equation
with an additional source and thermal noise (ζ) term.
∇s= (1−n⊗n)∇stands for the directional gradient on a
sphere with radius R, where ∇is the nabla operator and n
the surface normal. The current is split up into a diffusive
part jDand an advective part jA, which arises due to the
Marangoni effect. We summarize them below and in sect.
2.2. The source term describes the bromination reaction
as well as desorption of brominated and adsorption of non-
brominated surfactants to and from the outer fluid. Both
processes tend to establish an equilibrium mixture with
order parameter φeq during the characteristic relaxation
time τR. Ad- and desorption dominate for φeq <0 while
bromination dominates for φeq >0. The source term is
a simplified phenomenological description for the ad- and
desorption of surfactants. A more detailed model would
include fluxes from and to the bulk fluid [63]. We will ex-
plain the thermal noise term further below.
The general mechanism of eq. (1) to initiate steady
Marangoni flow is as follows. The diffusive current jD
smoothes out gradients in φ, while the advective Maran-
goni current jAamplifies gradients in φ. Hence, jDand
jAare competing and as soon as jAdominates over jD,
φexperiences phase separation. As a result, the resting
state becomes unstable and the droplet starts to swim.
We now summarize features of the diffusive current
jD, more details can be found in ref. [38]. We formulate
a Flory-Huggins free energy density in terms of the order
parameter of the surfactant mixture, which includes en-
tropic terms and interactions between the different types
M. Schmitt, H. Stark: Active Brownian motion of emulsion droplets 3
of surfactants:
f(φ) = kBT
ℓ2h1+φ
2ln 1+φ
2+1−φ
2ln 1−φ
2
−1
4(b1+b2+b12)−φ
2(b1−b2)−φ2
4(b1+b2−b12)i,
Here, ℓ2is the head area of a surfactant at the interface.
We introduce dimensionless parameters b1(b2) to charac-
terize the interaction between brominated (non-brominated)
surfactants and b12 describes the interaction between the
two types of surfactants. The diffusive current is now driven
by a gradient in the chemical potential derived from the
total free energy functional F[φ] = RRf(φ) dA:
jD=−λ∇s
δF
δφ =−D1
1−φ2−1
2(b1+b2−b12)∇sφ ,
(2)
where the Einstein relation D=λkBT/ℓ2relates the in-
terfacial diffusion constant Dto the mobility λ. To rule out
a double well form of f(φ), which would generate phase
separation already in thermal equilibrium, we only con-
sider b1+b2−b12 <2. This also means that the diffusive
current jD∝ −∇sφis for all φindeed directed against
∇sφ. In the following we assume b12 = (b1+b2)/2 and
therefore require b1+b2<4.
We formulate the thermal noise term in eq. (1) as
Gaussian white noise with zero mean following ref. [64]:
hζi= 0 ,(3a)
hζ(r, t)ζ(r′, t′)i=−2kBT λ∇s
2δ(r−r′)δ(t−t′).(3b)
Here, the strength of the noise correlations is connected to
the mobility λof the diffusive current via the fluctuation-
dissipation theorem. In order to close eq. (1), we now dis-
cuss the advective Marangoni current jA.
2.2 Marangoni flow
The advective current for the order parameter φis given
by
jA=φu|R,(4)
where u|Ris the flow field at the droplet interface. It is
driven by a non-uniform surface tension σand therefore
called Marangoni flow [65,63]. In our case, we have a non-
zero surface divergence ∇s·u|R6= 0. In fact, it can be
shown that an incompressible surface flow cannot lead to
propulsion of microswimmers [66].
In order to evaluate u|R, one has to solve the Stokes
equation for the flow field u(r) surrounding the spherical
droplet (r > R) as well as for the flow field ˆ
u(r) inside
the droplet (r < R). Both solutions are matched at the
droplet interface by the condition [63],
∇sσ=Ps(T−ˆ
T)err=R,(5)
where Ps=1−er⊗eris the surface projector. Equation
(5) means that a gradient in surface tension σis com-
pensated by a jump in viscous shear stress. Here, T=
η[∇⊗u+ (∇⊗u)T] is the viscous shear stress tensor of a
Newtonian fluid with viscosity ηoutside of the droplet and
the same relation holds for ˆ
Tof the fluid with viscosity ˆη
inside the droplet. We have performed this evaluation in
ref. [43] for a given surface tension field and only summa-
rize here the results relevant for the following. Alternative
derivations are found in ref. [67–70].
In spherical coordinates the Marangoni flow field u|R
at the interface reads [43,67–69]
u|R=−η
2(η+ ˆη)vD+1
η+ ˆη
∞
X
l=1
l
X
m=−l
R sm
l
2l+ 1∇sYm
l,(6)
with spherical harmonics Ym
l(θ, ϕ) given in appendix A.
Here,
sm
l=ZZ σ(θ, ϕ)Ym
l(θ, ϕ) dΩ(7)
are the expansion coefficients of the surface tension, where
Ym
lmeans complex conjugate of Ym
l, and [43,68,70]
vD=vDe=1
√6π
1
2η+ 3ˆη
s1
1−s−1
1
is1
1+s−1
1
−√2s0
1
.(8)
is the droplet velocity vector. It is solely given by the
dipolar coefficients (l= 1) of the surface tension and de-
termines propulsion speed vD≥0 as well as the swimming
direction ewith |e|= 1. Note that by setting m= 0,
eqs. (6)-(8) reduce to the case of an axisymmetric droplet
swimming along the z-direction, as studied in ref. [38].
In ref. [43] we give several examples of flow fields u|R.
In general, Marangoni flow is directed along gradients in
surface tension, i.e. u|Rk ∇sσ. This is confirmed by eq.
(6) and also clear from fig. 2 (b), which we discuss later.
However, according to eq. (6) higher modes of surface ten-
sion contribute with a decreasing coefficient [43]. Note the
velocity field in eq. (6) is given in a frame of reference that
moves with the droplet’s center of mass but the directions
of its axis are fixed in space and do not rotate with the
droplet. Finally, the velocity fields inside (ˆ
u) and outside
(u) of the droplet in both the droplet and the lab frame
can be found in the appendix of ref. [43].
The surface tension necessary to calculate vDand u|R
is connected to the order parameter φby the equation of
state, σ=f−∂f
∂c1c1−∂f
∂c2c2, which gives [38]
σ(φ) = kBT
ℓ2(b1−b2)3
8
b1+b2
b1−b2
+1
2φ+1
8
b1+b2
b1−b2
φ2.
(9)
This implies that for b1> b2>0, ∇sφpoints along ∇sσ.
Moreover, since the Marangoni flow u|Ris oriented along
∇sσ, as noted above, we conclude that for φ > 0 the ad-
vective current jA=φu|Rpoints “uphill”, i.e., in the
direction of ∇sφ, in contrast to jD[38].
This completes the derivation of the surface flow field
u|Ras a function of the expansion coefficients sm
lof the
surface tension. Together with the equation of state σ(φ)
the advective current jAin eq. (4) is specified. Finally,
4 M. Schmitt, H. Stark: Active Brownian motion of emulsion droplets
using the diffusion current jDfrom eq. (2), the diffusion-
advection-reaction equation (1) becomes a closed equation
in φ.
The swimming emulsion droplet is an example of a
spherical microswimmer, a so-called squirmer [39–43]. Squir-
mers are often classified by means of the so-called squirmer
parameter β[5]. When β < 0, the surface flow dominates
at the back of the squirmer, similar to the flow field of
the bacterium E. coli. Since such a swimmer pushes fluid
outward along its major axis, it is called a ’pusher’. Ac-
cordingly, a swimmer with β > 0 is called a ’puller’. The
algae Chlamydomonas is a biological example of a puller.
Swimmers with β= 0 are called ’neutral’.
For an axisymmetric emulsion droplet swimming along
the z-direction, the squirmer parameter is given by
β=−r27
5
s0
2
|s0
1|,(10)
with coefficients sm
lfrom the multipole expansion (7) of
the surface tension σ[43]. A generalization of this for-
mula to droplets without axisymmetry and swimming in
arbitrary directions is derived in ref. [43]. The relevant
expressions are presented in appendix B.
2.3 Reduced dynamic equations and system parameters
In order to write eq. (1) in reduced units, we rescale time
by the characteristic diffusion time τD=R2/D and lengths
by droplet radius R, and arrive at
∂tφ=−∇s·(jD+Mφu|R)−κ(φ−φeq) + ξζ(r, t),(11)
where the Gaussian noise variable fulfills
hζ(r, t)ζ(r′, t′)i=−2∇s
2δ(r−r′)δ(t−t′).(12)
The dimensionless velocity field at the interface and the
droplet velocity vector read, respectively,
u|R=−vD
2+
∞
X
l=1
l
X
m=−l
sm
l
2l+ 1∇sYm
l,(13a)
vD=vDe=1
√6π(2 + 3ν)
s1
1−s−1
1
is1
1+s−1
1
−√2s0
1
.(13b)
All quantities in eqs. (11) and (13), including jD,u|R,t,
∇s,ζ, and vD, are from now on dimensionless, although
we use the same symbols as before. Writing the dynamics
equations in reduced units, introduces the relevant system
parameters M, ν, κ, φeq, and ξ, which we discuss now.
The Marangoni number Mquantifies the strength of
the advective current in eq. (11) and is given by M=
(b1−b2)R
λ(η+ˆη). It is the most important parameter of our model,
as it determines whether the droplet swims. In eq. (13a)
we introduced the ratio of shear viscosities, ν= ˆη/η, for
the fluids inside and outside of the droplet, respectively.
In our study we consider a water droplet suspended in
oil and set ν≈1/36 [1]. The interaction parameters b1
and b2not only appear in Mbut also as b1+b2in the
diffusive current in eq. (2) and in the equation of state
σ(φ) in eq. (9). Therefore, they need to be set individually.
Assuming the head area of a surfactant ℓ2to be on the
order of nm2, we can fit eq. (9) to the experimental values
σ(φ= 1) ≈2.7mN/m and σ(φ=−1) ≈1.3mN/m[1] to
find b1≈0.6 and b2≈0.3. We keep these values fixed
throughout the article.
Parameter κ=τD/τRtunes the ratio between diffu-
sion and relaxation time and the equilibrium order pa-
rameter φeq measures whether ad- and desorption of sur-
factants (φeq <0) or bromination (φeq >0) dominates.
In this study we set κ= 0.1 and φeq = 0.5. A parameter
study for these parameters can be found in [38]. Finally,
the reduced noise strength ξ=ℓ/R ∝1/√N, where Nis
the total number of surfactants at the droplet interface,
connects the the droplet size Rto the molecular length
scale ℓ.
3 Finite volume method on a sphere
To numerically solve the rescaled dynamic equation (11)
for the order parameter field φ, we had to decide on an ap-
propriate method. The most widely used numerical meth-
ods for solving partial differential equations are the fi-
nite difference method (FDM), the finite element method
(FEM), and the finite volume method (FVM) [71,72]. We
ruled out FDM due to numerical complications of its al-
gorithm with spherical coordinates. They are most appro-
priate for the spherical droplet surface but one needs to
define an axis within the droplet. The FEM is also very
delicate when writing a numerically stable code for our
model. This is mainly due to the advective term in eq.
(11), which commonly causes difficulties in FEM routines
[71]. In contrast, the FVM is especially suited for solving
continuity equations. Therefore, it is much more robust for
field equations that incorporate advection and we chose it
for solving eq. (11) on the droplet surface.
In order to generate a two-dimensional FVM mesh that
is as uniform as possible and quasi-isotropic on a sphere,
we chose a geodesic grid based on a refined icosahedron
[73]. An icosahedron has f0= 20 equilateral triangles as
faces and v0= 12 vertices. In each refinement step, each
triangle is partitioned into four equilateral triangles and
the three new vertices are projected onto the unit sphere
enclosing the icosahedron. Hence, after the n-th refine-
ment step, the resulting mesh has fn= 4nf0triangular
faces and vn=vn−1+3
84nf0grid points. 1The “finite
volume” then refers to a small volume (in this case an
area) surrounding each grid point of the mesh. Thus we
have to construct the Voronoi diagram of the triangular
mesh. The Voronoi diagram consists of vnelements, 12 of
1Each face has three edges and every edge belongs to two
faces, hence the number of edges is en=3
2fn. In a refinement
step one new grid point is placed on the middle of each edge
and vn=vn−1+en−1. Thus, vn=vn−1+3
84nf0, withv1=
42, v2= 162, v3= 642, v4= 2562.
M. Schmitt, H. Stark: Active Brownian motion of emulsion droplets 5
Fig. 1. Finite volume element iwith neighboring element j.
The relevant lengths and normal vector are sketched.
which are pentagons associated with the vertices of the
original icosahedron while the rest are hexagons. Unless
otherwise noted we use a Voronoi mesh with v3= 642
FVM elements. The geodesic icosahedral grid is a stan-
dard grid in geophysical fluid dynamics. A comprehensive
review on numerical methods in geophysical fluid dynam-
ics can be found in [74].
In the following we will outline how we convert the
diffusion-advection-reaction equation (11) to a set of or-
dinary differential equations for a vector φcomprising the
values φiof the order parameter field at the center points
of all FVM elements. FVM was developed for treating
current densities in a continuity equation and we illus-
trate the procedure for the diffusion term of eq. (11). We
start by integrating over element iwith area Aiand use
the divergence theorem, where niis the outward normal
at the element boundary:
ZZ
Ai
∇s·jDdA=Z
∂Ai
jD·nidS=
N
X
j=1
jD·nijlij (14a)
=−
N
X
j=1
D(φi, φj)φj−φi
hij
lij =Diφ . (14b)
In the last term of eq. (14a), the line integral is converted
into a sum over the Nstraight element boundaries of
length lij and nij is the normal vector at the correspond-
ing boundary. Figure 1 illustrates the relevant quantities.
In the second line the directional derivative nij ·∇sφre-
sulting from jDin eq. (2) is approximated by a difference
quotient. The prefactor in jD, which we abbreviated by
D(φi, φj) in eq. (14b), also contains φ. It is interpolated
at the boundary between elements iand jby means of
the central differencing scheme as (φi+φj)/2. Finally, we
write the whole term as the product of local diffusion ma-
trix Diand vector φ. After applying this technique to all
elements, the matrices Diare combined into one matrix
Dfor the whole mesh.
The same procedure is carried out for the advective
term in eq. (11) but discretizing jA=Mφu|Rneeds more
care. While u|Ris directly calculated at the boundary be-
tween elements iand j, the order parameter φis treated
differently. If the local Peclet number Pe = hij M|u|R|/D(φi, φj)
is larger than 2, the central differencing scheme fails to
converge. Instead a so-called upwind scheme is used, which
takes into account the direction of flow [71]. For outward
oriented flow, i.e. u|R·nij >0, one uses the element or-
der parameter φi, while for inward flow, i.e. u|R·nij <0,
one uses the order parameter of the neighboring element
φj. In the case Pe <2, φis interpolated by the central
difference (φi+φj)/2.
Finally, the linear terms in φand its time derivative
are simply approximated by φiand ˙
φi. In the end, we are
able to write the discretized eq. (11) as a matrix equation
for the vector φ:
M˙
φ=D φ −MA φ −κM φ−φeq+ 2 ·121/4ξz , (15)
where the diagonal matrix Mcarries the areas of the ele-
ments, and with diffusion matrix D, advection matrix A,
and element noise vector z, which describes typical Gaus-
sian white noise with zero mean and variance one,
hz(t)i= 0 ,(16a)
hz(t)⊗z(t′)i= 1δ(t−t′).(16b)
In appendix C we derive eq. (16b) by integrating eq. (12)
over two FVM elements iand j. Finally, the set of stochas-
tic differential equations are integrated in time by a stan-
dard Runge-Kutta scheme.
In the following we present results obtained with the
described numerical scheme.
4 Dynamics towards the swimming state
This section focuses on the dynamics of the active emul-
sion droplet from an initial resting state with swimming
speed vD= 0 to a stable swimming state with swimming
speed vD>0. After a comparison with the axisymmetric
model of the droplet from our previous work [38], where
we also did not include thermal fluctuations, we investi-
gate the coarsening dynamics of the order parameter φat
the droplet interface while reaching the swimming state.
4.1 Swimming speed vD
In order to test the simulation method, we start our anal-
ysis with a set of parameters, for which we found a swim-
ming state in the inherent axisymmetric model [38]. They
are given by Marangoni number M= 3, reduced reac-
tion rate κ= 0.1, and equilibrium order parameter value
φeq = 0.5. We keep these values fixed throughout the fol-
lowing unless otherwise noted. The initial condition for
solving eq. (15) is an order parameter field that fluctuates
around φeq:φ(θ, ϕ) = φeq +δφ(θ, ϕ). The small fluctua-
tions δφ(θ, ϕ)≪1 are realized by random numbers drawn
from the normal distribution N(φeq, α2) with mean φeq
and variance α2= 10−5and added at the grid points of the
simulation mesh. Furthermore we set the noise strength to
ξ= 10−3. Figure 2(a) shows the droplet swimming speed
vDas a function of elapsed time.
First of all, we notice the good agreement with the
corresponding graph of vaxi
Dof the axisymmetric system of
6 M.Schmitt,H.Stark:ActiveBrownian motionofemulsiondroplets
0
0.2
0.4
0.6
0.8
1
0 5 10 15 20
t
v
D
v
D
axi
∆
(a)
(b)
❶
❷❸
❹
❺
❻
0
0.2
0.4
0.6
0.8
1
0 π/2 π
θ
〈
φ
〉
ϕ
φ
axi
〈u|
R
〉
ϕ
u|
R
axi
Fig.2.(a)Dropletswimmingspeedv
D(t)ofanactivedroplet
fromasimulation withv
4=2562FVMelements. Orderpa-
rameterprofilesφatthetimesteps markedwithnumbersare
giveninfig.4(a).Forcomparison,weplotv
axi
Doftheaxisym-
metricmodeltakenfromfig.2(a)ofref.[38]butonadifferent
scale. Wealsoshowbiaxialityparameter∆oftheorderparam-
eterfielddefinedineq.(18).(b)Orderparameterprofile φϕ
andvelocityfield u|
R ϕ att=20,averagedabouttheswim-
mingaxis easindicatedby ...ϕanddefinedinappendixD.
Thefrontofthedropletcorrespondstothepolarangleθ=0.
Forcomparison,weplotφ
axi andu|
axi
Rfromtheaxisymmetric
modeltakenfromfig.1ofref.[38]. Notethatthe Marangoni
flowu|
Risdirectedalongthegradientsofφandsurfacetension
σ.
ref.[38],whichwealsoplotinfig.2(a).Thesameapplies
totheorderparameterprofileφandthesurfaceveloc-
ityfieldu|
Roftheswimmingstate,whenaveragedabout
theswimmingaxise,seefig.2(b). Thus,thefullthree-
dimensionaldescriptionpresentedinthisworkisconsis-
tentwiththeaxisymmetricmodelofref.[38].Thesameis
trueforthesquirmerparameterβfromeq.(28),forwhich
wefind β≈ −1.2forM =3.Thisisfairlyclosetothe
valueoftheaxisymmetricmodel(β≈−0.8)andconfirms
thattheswimmingactivedropletisapusher.
Westressthatthe Marangoninumber M isthecru-
cialparameterinourmodel,asitdetermineswhetherthe
0
0.2
0.4
0.6
0 2 4 6 8
-2
-1
0
1
M
tr
v
D
β
M
v
D
β
1
2
3
10
-5 10
-3 10
-1
M
tr
α
2
Fig.3. Dropletswimmingspeed v
Dandsquirmerparameter
βplottedversus MarangoninumberM forzerothermalnoise
ξ=0. Atthetransition Marangoninumber Mtr
,v
Djumps
toanon-zerovalueindicatingasubcriticalbifurcation.Inset:
Mtrversusnoisestrengthα
2
,withwhichtheinitiallyuniform
orderparameterprofileisdisturbed. Theswimmingregime
terminatesatanupperbifurcation,seealsoref.[38].
dropletrestsorswims.ForsmallM,thehomogeneous
stateφ=φ
eqisstable,i.e.,anydisturbanceδφoftheini-
tiallyuniformφisdampedbythediffusionandreaction
termsofeq.(11).Asaresult,thedropletrests.Thetransi-
tiontotheswimmingstateoccursatincreasingMarangoni
numberM viaasubcritialbifurcationasillustratedinfig.
3,whichshowsswimmingspeedv
Dandsquirmerparam-
eterβplottedversusM. Weusehereasystemwithout
thermalnoise,i.e.,ξ=0,inordertomonitorthecomplete
transitionregionofthesubcriticalbifurcation.Atatran-
sitionvalueMtrtheadvectivetermofeq.(11)overcomes
thedampingterms.Thehomogeneousstatebecomesun-
stableandthedropletstartstoswimwithafiniteswim-
mingspeed v
D.Asusualforasubcriticalbifurcation,the
transitiontotheswimmingstatetakesplaceinafinitein-
tervalofM.There,thetransition MarangoninumberMtr
dependsontheinitialdisturbancestrengthα
2oftheuni-
formorderparameterprofile.Theinsetoffig.3confirms
thisstatement.Next,wewilldiscussthebiaxialevolution
andthecoarseningdynamicsoftheorderparameterfield,
whichwecouldnotstudyintheaxisymmetricdescription.
4.2Transientbiaxialdynamics
Thegoodagreementoftherotationallyaveragedorderpa-
rameterprofile φϕandtheaxisymmetricφ
axi fromour
earlierwork,bothplottedinfig.2(b),suggeststhatinthe
steadyswimmingstate,thefullthree-dimensionalsolution
isalsonearlyaxisymmetricabouttheswimmingaxise.
However,innon-steadystateweexpectφtodeviatefrom
axisymmetry,whichwequantifybyintroducinganappro-
priate measureforthebiaxialityoftheorderparameter
fieldφ.Inanalogytocharacterizingtheorientationalor-
derofliquidcrystals,wedefinefortheorderparameter
profilethetracelessquadrupolartensor[75]
Q= φ n⊗n−1
3dΩ, (17)
M.Schmitt,H.Stark:ActiveBrownian motionofemulsiondroplets 7
0
2π
0 π
φ
ϕ
θ
❶ t = 0
0
0.5
1
(a)
(b)
0
2π
0 π
ϕ
θ
❷ t = 1
0
2π
0 π
ϕ
θ
❸ t = 2
0
2π
0 π
ϕ
θ
❹ t = 3
0
2π
0 π
ϕ
θ
❺ t = 5
0
2π
0 π
ϕ
θ
❻ t = 9
l
g
l
/P
10
-3
10
0
1 10 20 l
10
-3
10
0
1 10 20 l
10
-3
10
0
1 10 20 l
10
-3
10
0
1 10 20 l
10
-3
10
0
1 10 20 l
10
-3
10
0
1 10 20
Fig.4.(a)Color-codedorderparameterprofileφ(θ,ϕ)atvarioustimestepsinthecoordinateframeofthedroplet,wherethe
frontofthedropletislocatedatθ=0.Linesofequalφaredrawn.Thetimesnapshotsareindicatedinfig.2(a)inthecurve
forv
D(samesimulationrun). Therelevantparametersare:M =3, κ=0.1,φ
eq =0.5,andξ=10
−3
.(b) Thebarcharts
showthenormalizedpolarpowerspectrumg
l
/Pofsurfacetensionσfortheprofilesin(a).Linesdepictg
l
/Paveragedover500
simulationruns.
withsurfacenormaln,unittensor ,andthesurfacein-
tegralisperformedoverthewholedropletinterface.Just
asinthecaseofthe momentofinertiatensor,theeigen-
valuesandeigenvectorsofQ characterizethesymmetries
oftheorderparameterfieldφ.IftwoeigenvaluesofQ
areequal,φissaidtobeuniaxial.Ontheotherhand,if
alleigenvaluesofQ aredistinct,φisbiaxial.Finally,the
caseofthreevanishingeigenvalues,i.e.,Q =0,describes
anisotropicoruniformorderparameterfieldφoratleast
withtetrahedralorcubicsymmetry.Ameasureforthede-
greeofbiaxiality,whichincorporatesthethreementioned
cases,isgivenbythebiaxialityparameter[76,77]
∆=1−6
(trQ3
)
2
(trQ2
)
3. (18)
Iftheorderparameterfieldφisaxisymmetricorisotropic,
∆=0,whilewithincreasingbiaxiality∆ approaches1.
Infig.2(a),weplot∆ asafunctionoftime.Atthe
initialtimet=0,theorderparameterprofileisroughly
uniformwith∆ ≈0(notvisible).Asthedropletspeeds
up,thebiaxialityparameter∆fluctuatesstronglybetween
0and1.Startingatt≈ 3,∆ sharplydecreasestowards
zerobeforetheswimmingspeedbecomes maximal.Fi-
nally,inthesteadyswimmingstate,∆ isnearlyzerobut
stillfluctuatesduetothethermalnoiseintheorderpa-
rameterprofileφ,whichweindicatebytheerrorbarsin
fig.2(b).Hence,duringthespeedupofthedroplet,the
orderparameterfieldφclearlyisnotaxisymmetric.
4.3Coarseningdynamics
Theperiodofstrongbiaxialitygoesinhand withthe
coarseningdynamicsoftheorderparameterprofileto-
wardssteadystate.Figure4(a)showstheorderparameter
profileφ(θ,ϕ)atvarioustimestepsforthesamesimula-
tionrunasinfig.2.Shortlyafterthesimulationstarts
withthenearlyuniforminitialcondition,smallislandsor
domainswithφ>φ
eqandφ<φ
eqemerge,whichrapidly
growuntilt≈1,wherethedroplethardly moves,seefig.
2(a).Thenthecoarseningordemixingprocessisslowed
down.Thedomainscoalesceonlargerscalesandthedrop-
letspeedsupsignificantly.Sincethedropletinterfacearea
isfinite,thedomainscoalesceatsomepointtoonelarge
regionwhichcoversabouthalfoftheinterface.Fromthen
onthedropletinterfaceiscoveredbyonlytworegions
withφ<φ
eqandφ>φ
eq
,whilethedroplethasreached
itstopspeed[comparev
D(t≈ 5)infig.2(a)].Finally,
thedomainwallbetweenthetworegions movestoitsfi-
nalpositionwhilethedropletspeedv
Dslowsdowntoits
stationaryvalue,whichitreachesatt≈9.
Notethatdependingonthefinalpositionofthedo-
mainwallseparatingthetworegions,thedropletiseither
apusherorapuller.Ifthedomainwallwithincreasingφ
issituatedinthesouthernhemisphere(π/2<θ<π),the
dropletisapusher.Ifitislocatedinthenorthernhemi-
sphere(0<θ<π/2),apullerisrealized.Inoursimula-
tionsthedropletisalwaysapusher.However,thesquirmer
parameterβvariesintherange−2<β<0dependingon
8 M.Schmitt,H.Stark:ActiveBrownian motionofemulsiondroplets
Marangoninumber M [seefig.3]andequilibriumorder
parameterφ
eq
.Thisisinagreementwithearlierobserva-
tionsinref.[38].Thetimeframet >9,wheretheswim-
mingspeedfluctuatesarounditssteady-statevalue,will
becoveredinsect.5.
Toquantifyfurtherthespatialstructureoftheorder
parameterprofileduringcoarsening,weexaminethean-
gularpowerspectrum|s
m
l|
2ofthesurfacetension.Itis
relatedtoφineq.(9).Usingtheorthonormalityrelation
ofsphericalharmonicsY
m
l(θ,ϕ),giveninappendixA,one
cancomputethetotalpowerPofthesurfacetensionσ:
P= σ
2
dΩ=
∞
l=1
g
l
=
∞
l=1
l
m=−l
|s
m
l|
2.
Here,thepolarpowerspectrumg
lcharacterizesthevari-
ationofthesurfacetensionandthustheorderparameter
fieldφalongthepolarangleθ.Inparticular,g
lforsmall
lquantifiesthelarge-anglevariationsofσ.Notethatg
1is
directlyrelatedtotheswimmingspeedv
Dcalculatedfrom
eq.(8)inthepolarcoefficientss
m
1.Usings
−1
1=−s
1
1
,we
findg
1=3π[(2+3ν)v
D]
2
.
Figure4(b)depictsthepolarpowerspectrumg
lnor-
malizedbythetotalpower P atthesametimestepsof
thecoarseningdynamicsdiscussedbeforeinFig.4(a). We
alsoshowanensembleaverageofg
l
/P.Attheinitialtime
t=0,thespectrumof g
lissolelycharacterizedbyfre-
quenciesorpolarcontributionsofthenoisyinitialcondi-
tionφ(t=0)=φ
eq+δφ.Thus,the maximumfrequency
orpolarnumberlofthespectrumatt=0issetbythe
levelofrefinementofthesimulationmesh.Duringtheini-
tialperiodoffastcoarseninguntilt=1,thepolarpower
spectrumshiftsfromhightolowfrequenciesindicating
theincreaseofdomainsizes.Thenthehigherfrequencies
vanish moreand morefromthespectrum,asthephases
associatedwithφ <φ
eq andφ >φ
eq separate.Eventu-
ally,thespectrumg
lstronglypeaksatl=1 whilethe
remainingcoefficientsbecomeinsignificantincomparison.
Finally,fromt=5tot=9,thefirstcoefficientg
1ofthe
angularpowerspectrumdecreasesagainwhilethesecond
andthirdcoefficientsg
2andg
3rise.Thisconfirmsthat
inthefinalstagethedropletslowsdownitsvelocityv
D
andtunesitssquirmerparameterβbyshiftingthedomain
wallfurtherawayfromtheequator.
Inordertoquantifyfurtherthetemporalevolution
ofthecoarseningdynamics,wewillnowinvestigatethe
averagedomainsizeasafunctionoftime. Wedefinethe
meanlinearsizeofaphasedomainby
L= v
+
n
v
n
.
Here, v
+
ndenotestheaveragednumberofgridpointsin
aconnectedregion, whereφislargerthanφ
eq
,andv
n
isthetotalnumberofgridpoints.Thus,thedomainsize
lieswithintherange 1/v
n≤L≤1,andL(t)shouldin-
creaseduringthecoarseningdynamicstowardsthesteady
swimmingstate. Thefluctuationsδφoftheinitialpro-
filearenormaldistributed withzero meansuchthatat
10
-1
10
0
10
-3 10
-2 10
-1 10
010
1
µ t
0.1
µ t
0.8
µ t
0.5
µ t
0.1
L
t
ξ
= 0
ξ
= 5×10
-4
ξ
= 1×10
-3
ξ
= 5×10
-3
Fig.5.Meandomainsize Laveragedover200simulationruns
plottedversusreducedtimeinunitsofτ
Dfordifferentnoise
strengthsξ. Adomainisdefinedbyacompactregion with
φ>φ
eq
.Sameparametersasinfig.4areused.
t=0halfofthegridpointshave φ >φ
eq
.Theycannot
allbeisolatedbutratherbelongtosmallconnectedre-
gionswithL≈ 5/v
n
,whereweextractedthefactor
√
5
fromoursimulationsat t=0.Furthermore, weexpect
the maximumlengthtobearoundL≈ 1/2.So,inour
simulationsL(t)liesintheinterval 5/v
n≤L≤ 1/2.
Figure5showsL(t)averagedover200simulationruns
fordifferentnoisestrengthsξ.Theotherparametersare
thesameasbefore. Weclearlyseeaseparationoftime
scalesofthecoarseningdynamicsforbothcases, with
andwithoutnoise.Atearlytimes,wefindinbothcases
apowerlawbehaviorL(t)∝t
0.1
. Withoutnoise,coarsen-
ingquicklyspeedsupatarateL(t)∝t
1/2andthenslows
downagaintoL(t)∝ t
0.1
.Incontrast,thermalfluctua-
tionsintheorderparameterprofilehinderearlycoarsen-
ingandthe meandomainsizecontinuestogrowslowly
with L(t)∝ t
0.1 overseveraldecadesandthencrosses
overtoafastfinalcoarseningwithrateL(t)∝t
0.8
.The
crossovertimeisonlydeterminedbythediffusiontimeτ
D
anddoesnotdependonnoisestrengthξ.Interestingly,a
similarobservationtothesecondcasehasbeen madefor
coarseninginthedynamical model H, wherethe Cahn-
Hilliardequationcouplestofluidflowatlow-Reynolds
numberviaanadvectionterm. Aslowcoarseningrate
L(t)∝t
1/3inadiffusiveregimeatshorttimesisfollowed
byanadvectiondrivenregimewithL(t)∝tatlatertimes
[62,78–80]. Although wecannotsimplyreformulateour
modelasanadvectiveCahn-Hilliardequation,sincethe
phaseseparationinourcaseisdrivenbytheinterfacial
flowu|
Ritself, weobservesimilarcoarseningregimesas
in modelH,whenweincludesomenoise.
5Dynamicsoftheswimmingstate
Wenowconsiderthetimeregime t>9,wherethedroplet
movesinitssteadyswimmingstate.However,ascanbe
observedinfig.2(a),thedropletspeedv
D(t>9)inthe
M.Schmitt,H.Stark:ActiveBrownian motionofemulsiondroplets 9
10
-4
10
-2
10
0
10
2
10
4
10
6
10
-1 110
110
210
3
µ t
2
µ t
〈∆ r
2
〉
t
ξ
= 5×10
-3
ξ
= 1×10
-3
e
Fig. 6. Meansquaredisplacementoftheswimmingactive
dropletfordifferentnoisestrengthsξ. Att=0thedropletis
alreadyintheswimmingstate.Inset:Atypicaltrajectoryr(t)
ofanactivedropletsubjecttonoisewithstrengthξ=5·10
−3
.
Thetrajectoryisreminiscentofanactiveparticle withcon-
stantspeedandrotationallydiffusingorientationvectore(t).
swimmingstatestronglyfluctuatessincewehaveadded
athermalnoisetermtothediffusion-advection-reaction
equation(11)fortheorderparameterfieldφ.Thesefluc-
tuationsalsorandomlychangetheswimmingdirectione
astheinsetoffig.6illustrates,whereweshowanexem-
plaryswimmingtrajectoryr(t)=r(0)+ t
0dt
′
v
D(t
′
)e(t
′
).
Therefore,weexpectthedroplettoperformactiveBrown-
ianmotionorapersistentrandomwalk.Inadropletwith
axisymmetricprofiletheswimmingdirectionisperpen-
diculartothedomainwallseparatingbothphases. When
theorder-parameterprofilefluctuates,wealsoexpectthe
domainwalltofluctuateandtherebytheswimmingdirec-
tione.Therearenootherreasonstochangetheorienta-
tionofe.Inref.[43]weshowedthatasphericalemulsion
droplet with Marangoniflowatitssurfacedoesnotex-
perienceafrictionaltorque,whichcouldalsochangethe
swimmingdirection. Butthisalso meansthatfluctuat-
ingflowfieldsinthesurroundingfluidcannotgeneratea
stochastictorqueactingonthedroplet.
5.1ActiveBrownian motionofthedroplet
TocharacterizetheactiveBrownian motionofthedrop-
let,wefirstdiscussthemeansquareddisplacement(MSD)
∆r
2= [r(t)−r(0)]
2, where weaverageoveranen-
sembleoftrajectories.Here,thedropletisalreadyinthe
swimmingstateatt=0,thusthe MSDdoesnotinclude
thedroplet’saccelerationtowardsthesteadyswimming
stateasdiscussedinsect.4.Figure6showsthe MSDfor
adropletwithnoisestrengthξ=5·10
−3
.Atearlytimes,
thedroplet movesballisticallysincethe MSDgrowsas
∆r
2∝t
2
,whilebetweent=10and t=100itcrosses
overtodiffusivemotionwith ∆r
2∝t.Thismotionper-
sistsast→ ∞. Asexpected,intheabsenceofnoise,
ξ=0,wealwaysobserveballistic motion ∆r
2∝t
2(not
shown).The MSDforξ=10
−3 infig.6doesnotcross
overtodiffusionintheplottedtimerange.Inthefollow-
ing,wewilldiscusstheinfluenceofthenoisestrengthξon
theBrownian motionin moredetailbutwewillfirstin-
troducewhathasbecomethestandardmodelofanactive
Brownianparticle[15,81,11,12].
Ifweassumethedropletspeedv
Dandorientationvec-
toretobeindependentrandomvariables,wecanfactorize
the MSDas
∆r
2=
t
0
dt
′
t
0
dt
′′v
D(t
′
)v
D(t
′′
) e(t
′
)·e(t
′′
).
ForactiveBrownianparticleswithoutanyaligningfield
theswimmingdirectiondiffusesfreelyontheunitsphere,
whichonedescribesbytherotationaldiffusionequation
∂
t
p(e,t) =D
r
∇
s
2
p(e,t). Thustheorientationalcorrela-
tionfunctiondecaysas[82,83]
e(0)·e(t)=e
−t/τ
r. (19)
Here,therotationalcorrelationtimeτ
r=1/(2D
r
)isthe
characteristictimeittakesthedropletto“forget”about
theinitialorientatione(0). Hence,fortimest <τ
rthe
dropletswimsroughlyinthedirectionofe(0), whileat
latertimest>τ
rtheorientationbecomesrandomized.
Undertheassumptionofaconstantswimmingspeed,
i.e.v
D(t
′
)v
D(t
′′
)=(v
D)
2
,onefindsforthe MSD
∆r
2=2(v
Dτ
r
)
2t
τ
r
−1+e
−t/τ
r. (20)
Expression(20)confirmsthefindingsoffig.6: Ballistic
motion ∆r
2=(v
Dt)
2withvelocity v
Datt≪ τ
rand
diffusive motionwith
∆r
2=6D
eff
t and D
eff=(v
D)
2
τ
r
/3 (21)
fort≫ τ
r
.Here,D
eff istheeffectivetranslationaldiffu-
sionconstant.Itneglectsanycontributionfromthermal
translationalmotion,whichiso.k.forsufficientlylargev
D.
Indeed,fortheactivedroplettherotationalcorrela-
tionfunctione(0)·e(t)decaysexponentiallyasdemon-
stratedinfig.7fordifferentnoisestrengthsandbyfitsto
eq.(19).Therotationalcorrelationtimeτ
r
,whichactsas
fittingparameter,isshownintheinsetforvariousvalues
ofnoisestrengthξ.Forξ=5·10
−3
, wefindτ
r≈ 30,
whichisinagreement withthecross-overregionfrom
ballistictodiffusive motioninthe MSDcurveoffig.6.
Furthermore,fromtheasymptoticbehaviorat t≪ τ
r
andt≫ τ
rofthe MSDinfig.6,wefindv
D≈ 0.3and
D
eff≈1,respectively.Thisgivestherotationalcorrelation
timeτ
r=3D
eff
/(v
D)
2≈ 33,whichisclosetothevalue
determinedfromtheorientationalcorrelations.ThusD
eff
andτ
rcomprisethesameinformationaboutthedroplet
trajectoryr(t).However,themeasurementofτ
rinexper-
imentsorsimulationscanbedoneon muchshortertime
scalesthanD
eff
.
Wedonotknowpublishedexperimentaldatafortra-
jectoriesofactivedropletsinanunboundedfluid. How-
ever,fig.1ofref.[1]showsatrajectoryofanactivedrop-
letconfinedbetweentwoglassplates. Onecanestimate
10 M.Schmitt,H.Stark:ActiveBrownian motionofemulsiondroplets
e
-1
10
-1
10
0
010
3210
3
= 510
-4
= 110
-3
= 210
-3
τ
r
e(0)e(t)
t
10
1
10
3
10
5
10
-4 10
-3 10
-2
µ 1/
2
τ
r
Fig.7. Rotationalcorrelationfunctionoftheactivedroplet
andfitstoe
−t/τ
rfordifferentvaluesofnoisestrength. At
therotationalcorrelationtimeτ
r
,onehase(0)·e(τ
r
)=e
−1
,
asillustratedforthecase=2·10
−3
.Inset:τ
rplottedversus
noisestrengthandafitto
−2
.
therotationalcorrelationtimeτ
rtobeontheorderof
100s.Tocomparethisvaluewithour model,werecapitu-
latethenoisestrength=/R,whichconnectssurfactant
headsizewithdropletradiusR,seesect.2.3.Ifweas-
sume, 10
−4...10
−3
,wefindfromfig.7arotational
correlationtimeτ
r10
4giveninunitsofdiffusiontime
τ
D=R
2
/DwithinterfacialdiffusionconstantD.Typical
valuesforD areontheorderof10
−5
cm
2
/s[84].Thus,for
adropletwithRontheorderof10µm,onefindsτ
D0.1s
andtherotationalcorrelationtimeτ
r10
3
s.
Thisisonlyafactor10largerthantheestimatedvalue
of100sfromref.[1]. Givensomeuncertaintiesinoures-
timatesuchadifferencecanbeexpected. Nevertheless,
twocausesforthediscrepancyarethinkable.Firstand
foremost,our modeldropletisallowedto movefreelyin
thebulkfluid,whiletherealdropletofref.[1]isconfined
betweentwoplates,whichlimitsthedegreesoffreedom
andthusaltersτ
r
.Secondly,activeemulsiondropletsare
usuallyimmersedinasurfactantladenfluid wellabove
thecritical micelleconcentration.Hence,thesurfactants
fromthebulkadsorbinformof micelles. Thisleadsto
localdisturbancesinthesurfactant mixtureatthefront
oftheswimmingdroplet,andhencetoanadditionalran-
domizationofthedroplettrajectory. Werecentlymodeled
theadsorptionof micellesexplicitlyinadifferentsystem
[43].
5.2Howfluctuationsrandomizethedropletdirection
Now,wedevelopatheoryhowthenoisestrengthinflu-
encestherotationaldiffusionofthedropletdirection.By
increasinginthediffusion-advection-reactionequation
(11),theorderparameterprofileφissubjecttostronger
fluctuations.Inparticular,thesefluctuationsaffectshape
andorientationofthedomainwallseparatingthetwore-
gionswithφ<φ
eqandφ>φ
eqfromeachother.Thesur-
faceflowfieldislargestinthisdomainwallandthereby
e
z
e
φ
θ
θ
φ
02
0
/2
-/2
(a) (b)
δe
Fig.8.Illustrationofareorientingdroplet.(a)Theblackcurve
aroundthedropletinterfaceshowsthenoisyphaseboundary
denotedinthecoordinatesystem(θ,)ofthedropletwithout
noise.Theredcurveshowsthefirst modeofaFourierexpan-
sion,seetext.(b)Flatrepresentationinthesaidcoordinate
system(θ,).
theorientationofthewallonthedropletinterfacedeter-
minesthedropletswimmingvector e. Thus,increasing
noisestrengthresultsinstrongerfluctuationsofeand
ultimatelya morepronouncedrotationaldiffusion. The
insetoffig.7confirmsthisscenariofortherotationalcor-
relationtimeτ
r
.Interestingly,fornoisestrengthsupto
3·10
−3
,onefitsthedataquitewellbyτ
r∝1/
2
.For
theactiveemulsiondropletitcanbeexplainedbyapply-
ingperturbationtheorytothethermalfluctuationsofthe
orderparameterprofilearounditssteadyprofile.
Asmentionedbefore,smallfluctuationsofthedomain
wallresultinrandomchangesofthedropletdirection.Fig-
ure8showsanexaggeratedillustrationofthesituation.
Plot(a)illustratesatiltintheorientationofthedomain
wallgeneratedbythesinusoidalvariationofthepolaran-
gleθalongtheazimuthalangle.Ingeneral,fluctuations
ofthedomainwallcanbedecomposedintoFouriermodes,
θ= ma
msin[m(−
0m)].Onlythefirst mode,m =1,
ofthisexpansiondeterminesthechangeinorientation,δe,
asillustratedinfig.8(a).Allhighermodescannotchange
theswimmingdirectionsincetheeffectsoftheresulting
surfaceflowfieldonecanceleachother.
Wenowapplyperturbationtheorytothefluctuating
orderparameterprofile,whichdeterminesthesurfaceten-
sionprofileandtherebytheswimmingdirectionaccording
toeq.(8). Weconsideradroplet,whichinitiallyswimsin
z-directionandchangesitsdirectioninxand/ory-direc-
tion,hencee= e
z+δe. Wewritedownaperturbation
ansatzforthesurfacetensionprofile,σ=σ
0
+δσ,withthe
unperturbedaxisymmetricpartσ
0=∞
l=1 s
0
l
0
landthe
perturbationδσ=s
1
1
1
1+s
−1
1
−1
1,whereweonlyinclude
thecoecientss
1
1,whichareresponsibleforchangesδe,
asonerecognizesfromeq.(8).Bylinearizingtheequation
ofstate(9)aroundφ
eq
,onecanconnectthecoecients
s
m
lofσdirectlytotheexpansioncoecientsoftheorder
parameterfieldφ. Writingφ=φ
0
+δφ,whereφ
0describes
theunperturbedsteady-statefieldandδφitsfluctuations,
wefind φ
0= aσ
0andδφ= aδσ, wherethefactorais
giveninappendixE.Similarly,onedecomposesj
Dand
u|
Rintotheirsteady-statefieldsandafluctuatingsmall
M. Schmitt, H. Stark: Active Brownian motion of emulsion droplets 11
perturbation (see appendix E). This allows us to derive
from the field equation (11) of the order parameter, the
dynamic equation linear in the fluctuating perturbations:
∂tδφ =−∇s·[δjD+M(δφu0+φ0δu)] −κδφ +ξζ . (22)
From our study of the coarsening dynamics we know that
the first and second term on the right-hand side describe
a relaxation towards steady state on times t < 10. The
rotational diffusion of the droplet direction occurs on time
scales much larger and can only be due to the noise term.
Extracting from Eq. (22) the coefficients s±1
1relevant for
δe, we obtain
∂ts±1
1≃ξ
aζ±1
1.(23)
A more thorough derivation of Eq. (23) is presented in ap-
pendix F. We have decomposed noise ζinto its multipole
moments, ζ=Pl,m ζm
lYm
l. Projecting the variance of eq.
(12) onto the relevant spherical harmonics, we obtain the
fluctuation-dissipation theorem
hζm
l(t)ζm′
l′(t′)i= 2l(l+ 1)δ(t−t′)δl,l′δm,m′.(24)
Assuming a constant speed vDduring the reorientation
of the droplet, we use eq. (23) in eq. (13b) for the droplet
velocity vector to formulate the stochastic equation for
rotations of the direction vector e:
∂te=ξ
√6πvD(2 + 3ν)aδζ,(25)
where we introduced the rotational noise vector
δζ=
ζ1
1−ζ−1
1
iζ1
1+ζ−1
1
0
.
By comparing eq. (25) with the Langevin equation for
the Brownian motion of a particle’s orientation edue to
rotational noise ηr:∂te=√2Drηr×e[85], we identify
δζ=ηr×eand
ξ
√6πvD(2 + 3ν)a=p2Dr.(26)
Hence, the rotational correlation time τr= 1/(2Dr) scales
as τr∝1/ξ2with noise strength ξ. This confirms the fit
in the inset of fig. 7. Thus, beyond the time scale, the
order parameter profile needs to reach its steady state,
the dynamics of the swimming active emulsion droplet is
equivalent to the dynamics of an active Brownian particle
with constant swimming velocity and rotationally diffus-
ing orientation vector e.
6 Conclusions
In this paper we considered an active emulsion droplet,
which is driven by solutocapillary Marangoni flow at its
interface [1]. A diffusion-advection-reaction equation for
the surfactant mixture at the droplet interface, which we
formulated in ref. [38], is used together with the analytic
solution of the Stokes equation [43]. By omitting the axi-
symmetric constraint and including thermal noise into the
description of the surfactant mixture, we generalized the
model of ref. [38] to a full three-dimensional system and
thereby were able to focus on new aspects.
First, we explored the dynamics from a uniform, but
slightly perturbed surfactant mixture to the uniaxial steady
swimming state, where the two surfactant types are phase-
separated. In between the initial and the swimming state,
the surfactant mixture is not axisymmetric, which we ver-
ified by introducing and evaluating a biaxiality measure.
We then investigated in detail the coarsening dynamics
towards the swimming state by means of the polar power
spectrum of the surface tension σas well as the average
domain size of the surfactant mixture. The coarsening pro-
ceeds in two steps. An initially slow growth of domain size
is followed by a nearly ballistic regime, which is reminis-
cent to coarsening in the dynamic model H [62].
Second, we studied the dynamics of the squirming drop-
let. Due to the included thermal noise, the surfactant
composition fluctuates and thereby the droplet constantly
changes its swimming direction performing a persistent
random walk. Thus, the swimming dynamics of the squirm-
ing droplet is a typical example of an active Brownian par-
ticle. The persistence of the droplet trajectory depends
on the noise strength ξ. It is characterized by the rota-
tional correlation time, for which we find the scaling law
τr∝ξ−2. In fact, we are able to explain this scaling by
applying perturbation theory to the diffusion-advection-
reaction equation for the mixture order parameter. Thus
we can link the dynamics of the surfactants at the molec-
ular level to the dynamics of the droplet as a whole.
Exploring and understanding the swimming mecha-
nisms of both biological and artificial microswimmers is
one of the challenges in the field. Here, we demonstrated
that this task involves new and fascinating physics. Hav-
ing gained deeper insights into these mechanisms can help
to further improve the design of artificial microswimmers
and tailor them for specific needs such as cargo transport.
We acknowledge financial support by the Deutsche Forschungs-
gemeinschaft in the framework of the collaborative research
center SFB 910, project B4 and the research training group
GRK 1558.
A Spherical harmonics
Throughout this paper we use the following definition of
spherical harmonics:
Ym
l(θ, ϕ) = s2l+ 1
4π
(l−m)!
(l+m)! Pm
l(cos θ) eimϕ ,
with associated Legendre polynomials Pm
lof degree l, or-
der m, and with orthonormality:
ZZ Ym
lYm′
l′dΩ=δl,l′δm,m′,
12 M. Schmitt, H. Stark: Active Brownian motion of emulsion droplets
where Ym
ldenotes the complex conjugate of Ym
l.
The spherical harmonics fulfill the following helpful re-
lations:
ZZ Y0
lYm
1Ym′
1dΩ=−1
√20πδl,2δm,m′,(27a)
ZZ ∇sY0
l·∇sYm
1Ym′
1dΩ=−3
√20πδl,2δm,m′,(27b)
where ∇sis the directional gradient defined in sect. 2.1
and evaluated at r= 1.
B Squirmer parameter
The squirmer parameter for a droplet swimming in an
arbitrary direction is given by [43]:
β=−r27
5
˜s0
2
|˜s0
1|,(28a)
˜s0
1=q(s0
1)2−2s1
1s−1
1,(28b)
˜s0
2=√6s2
2(s−1
1)2+s−2
2(s1
1)2−√12s0
1s1
2s−1
1+s−1
2s1
1
+2s0
2(s0
1)2+s1
1s−1
12(s0
1)2−4s1
1s−1
1,(28c)
with coefficients sm
lfrom eq. (7). By setting m= 0, this
reduces to the case of an axisymmetric droplet swimming
along the z-direction.
C Element noise vector
Here, we discretize the thermal noise ζin eq. (11) and
obtain the element noise vector zwith component zifor
the FVM element i. We define the correlation function
between ziand zjby integrating eq. (12) over element
areas Aiand Aj:
hzi(t)zj(t′)i ≡ ZZ
Ai
dAiZZ
Aj
dAjhζ(ri, t)ζ(rj, t′)i(29a)
= 2 Z
∂Ai
dSini·Z
∂Aj
dSjnjδ(ri−rj)δ(t−t′) (29b)
= 2 X
q
liq X
p
ljpδq,pniq ·njpδ(t−t′).(29c)
In eq. (29b) we used the divergence theorem and in eq.
(29c) we converted the line integrals into sums over the
element boundaries. Furthermore, we discretized δ(ri−rj)
by partitioning the surface into rhombi of area A♦(see fig.
1) and defined
δq,p =(1/A♦for q=p ,
0 for q6=p ,
where qand pare the indices of the respective boundaries
of elements iand j. Three cases have to be considered.
First, if the elements iand jare neither identical nor
neighbors, δq,p vanishes in eq. (29c) for all qand p. Second,
for i=j,δq,p = 1/A♦and niq ·njp = 1 for all qand
p. Finally, for neighboring elements there is one common
boundary, where δq,p = 1/A♦and niq ·njp =−1. Thus,
one finds:
hz(t)⊗z(t′)i=2Nl2
A♦1−1
NQδ(t−t′),(30)
where Nis the number of element boundaries. Here, Qij =
1 if elements iand jare neighbors and zero otherwise.
Note that in eq. (30), we assumed the same edge length l
and number of boundaries Nfor all elements. This is rea-
sonable for a refined icosahedron with 642 FVM elements,
as discussed in sect. 3. The form of eq. (30) acknowledges
the conservation law for the noise [86]. However, in simu-
lations we did not observe any effect of the next–neighbor
correlations and therefore simplified the noise to the ex-
pression (16b) in the main text. Furthermore, we take
N= 6 and A♦=p3/4l2, since our grid is mostly hexago-
nal, which explains the prefactor p2Nl2/A♦= 2·121/4in
eq. (15), when we redefine the noise vector by the following
replacement, z→2·121/4z.
D Average over droplet interface
The average
hfiϕ=1
2πZf(θ, ϕ) dϕ ,
is taken over the azimuthal angle ϕin the coordinate frame
whose z-axis is directed along the swimming direction e.
Here, the front of the moving droplet is at θ= 0.
E Perturbation ansatz
The zero and first-order contributions of φ=φ0+δφ,
jD=jD,0+δjD, and u|R=u0+δuare given by:
φ0=a
∞
X
l=1
s0
lY0
l,(31a)
δφ =as1
1Y1
1+s−1
1Y−1
1,(31b)
jD,0=−b∇sφ0,(31c)
δjD=−b∇sδφ , (31d)
u0=cs0
1∇sY0
1+
∞
X
l=2
s0
l
2l+ 1∇sY0
l,(31e)
δu=cs1
1∇sY1
1+s−1
1∇sY−1
1,(31f)
M. Schmitt, H. Stark: Active Brownian motion of emulsion droplets 13
with parameters
a=4(b1−b2)
2(b1−b2) + φeq(b1+b2)≈1.14 ,(32a)
b= (1 −φ2
eq)−1−1
2(b1+b2−b12)≈1.11 ,(32b)
c= (1 + ν)/(2 + 3ν)≈0.49 .(32c)
Here we used the values of sect. 2.3 for b1, b2, b12, φeq and
ν.
F Dynamic equation for s±1
1
To derive a dynamic equation for the expansion coeffi-
cients s±1
1, we project the dynamic equation (22) for the
perturbation δφ onto the spherical harmonics Y±1
1[see
also eq. (31b)]. Employing the orthonormality relation of
the spherical harmonics and using eqs. (27), we ultimately
obtain
∂ts±1
1=s±1
1−2b−3
5−cM
√20πs0
2−κ+ξ
aζ±1
1
(33)
with noise components ζm
ldefined in eq. (24). Due to the
nonlinear advection term Mφu|Rin eq. (11), the coeffi-
cients s±1
1couple to s0
2. The term in square brackets on
the right-hand side describes a relaxational dynamics for
s±1
1. In particular, for the parameters chosen we find the
swimming droplet to be a pusher. Thus, according to eq.
(10) the coefficient s0
2>0 and the term in square brackets
is always negative. On time scales larger than the relax-
ation time, we can ignore the relaxational dynamics and
the time dependence of the order parameter perturbation
is solely determined by the thermal noise term, which con-
firms relation (23).
Note that in the dynamic equation for s0
lequivalent to
eq. (33), the advective term ∝Mis always positive and
triggers for l= 1 and for sufficiently large Mthe onset of
forward propulsion of the droplet (see fig. 3 and ref. [38]).
References
1. S. Thutupalli, R. Seemann, S. Herminghaus, New J. Phys.
13, 073021 (2011)
2. A. Najafi, R. Golestanian, Phys. Rev. E 69, 062901 (2004)
3. R. Dreyfus, J. Baudry, M.L. Roper, M. Fermigier, H.A.
Stone, J. Bibette, Nature 437, 862 (2005)
4. E. Gauger, H. Stark, Phys. Rev. E 74, 021907 (2006)
5. E. Lauga, T.R. Powers, Rep. Prog. Phys. 72, 096601 (2009)
6. J. Elgeti, R.G. Winkler, G. Gompper, Rep. Prog. Phys.
78, 056601 (2015)
7. T. Fenchel, Protist 152, 329 (2001), ISSN 1434-4610
8. T. Qiu, T.C. Lee, A.G. Mark, K.I. Morozov, R. M¨unster,
O. Mierka, S. Turek, A.M. Leshansky, P. Fischer, Nat.
Commun. 5(2014)
9. D. Alizadehrad, T. Kr¨uger, M. Engstler, H. Stark, PLoS
Comput. Biol. 11, e1003967 (2015)
10. C.C. Maass, C. Kr¨uger, S. Herminghaus, C. Bahr, Annu.
Rev. Condens. Matter 7(2016)
11. M. Enculescu, H. Stark, Phys. Rev. Lett. 107, 058301
(2011)
12. P. Romanczuk, M. B¨ar, W. Ebeling, B. Lindner,
L. Schimansky-Geier, Eur. Phys. J.-Spec. Top. 202, 1
(2012)
13. A. Z¨ottl, H. Stark, Phys. Rev. Lett. 108, 218104 (2012)
14. S. Michelin, E. Lauga, D. Bartolo, Phys. Fluids 25, 061701
(2013)
15. J.R. Howse, R.A. Jones, A.J. Ryan, T. Gough,
R. Vafabakhsh, R. Golestanian, Phys. Rev. Lett. 99,
048102 (2007)
16. V. Zaburdaev, S. Uppaluri, T. Pfohl, M. Engstler,
R. Friedrich, H. Stark, Phys. Rev. Lett. 106, 208103 (2011)
17. M. Theves, J. Taktikos, V. Zaburdaev, H. Stark, C. Beta,
Biophys. J. 105, 1915 (2013)
18. F. K¨ummel, B. ten Hagen, R. Wittkowski, I. Buttinoni,
R. Eichhorn, G. Volpe, H. L¨owen, C. Bechinger, Phys. Rev.
Lett. 110, 198302 (2013)
19. G. Volpe, I. Buttinoni, D. Vogt, H.J. Kummerer,
C. Bechinger, Soft Matter 7, 8810 (2011)
20. K. Drescher, J. Dunkel, L.H. Cisneros, S. Ganguly, R.E.
Goldstein, Proc. Natl. Acad. Sci. U. S. A. 108, 10940
(2011)
21. T. Majmudar, E.E. Keaveny, J. Zhang, M.J. Shelley, J. R.
Soc. Interface 9, 1809 (2012)
22. K. Schaar, A. Z¨ottl, H. Stark, Phys. Rev. Lett. 115, 038101
(2015)
23. M.C. Marchetti, J.F. Joanny, S. Ramaswamy, T.B. Liver-
pool, J. Prost, M. Rao, R.A. Simha, Rev. Mod. Phys. 85,
1143 (2013)
24. T. Ishikawa, T.J. Pedley, Phys. Rev. Lett. 100, 088103
(2008)
25. A.A. Evans, T. Ishikawa, T. Yamaguchi, E. Lauga, Phys.
Fluids 23, 111702 (2011)
26. J. Dunkel, S. Heidenreich, K. Drescher, H.H. Wensink,
M. B¨ar, R.E. Goldstein, Phys. Rev. Lett. 110, 228102
(2013)
27. F. Alarc´on, I. Pagonabarraga, J. Mol. Liq. 185, 56 (2013)
28. A. Z¨ottl, H. Stark, Phys. Rev. Lett. 112, 118101 (2014)
29. M. Hennes, K. Wolff, H. Stark, Phys. Rev. Lett. 112,
238104 (2014)
30. O. Pohl, H. Stark, Phys. Rev. Lett. 112, 238303 (2014)
31. A. Z¨ottl, H. Stark, to be published in J. Phys.: Conden.
Matter, arXiv preprint arXiv:1601.06643 (2016)
32. A. Walther, A.H. M¨uller, Soft Matter 4, 663 (2008)
33. T. Bickel, A. Majee, A. W¨urger, Phys. Rev. E 88, 012301
(2013)
34. R. Golestanian, T.B. Liverpool, A. Ajdari, Phys. Rev.
Lett. 94, 220801 (2005)
35. W.F. Paxton, A. Sen, T.E. Mallouk, Chem. Eur. J. 11,
6462 (2005)
36. J.L. Moran, J.D. Posner, J. Fluid. Mech. 680, 31 (2011)
37. I. Buttinoni, G. Volpe, F. K¨ummel, G. Volpe,
C. Bechinger, J. Phys.: Condens. Matter 24, 284129
(2012)
38. M. Schmitt, H. Stark, Europhys. Lett. 101, 44008 (2013)
39. M.J. Lighthill, Commun. Pur. Appl. Math. 5, 109 (1952)
40. J.R. Blake, J. Fluid. Mech. 46, 199 (1971)
41. M.T. Downton, H. Stark, J. Phys.: Condens. Matter 21,
204101 (2009)
14 M. Schmitt, H. Stark: Active Brownian motion of emulsion droplets
42. O. Pak, E. Lauga, J. Eng. Math. 88, 1 (2014), ISSN 0022-
0833
43. M. Schmitt, H. Stark, Phys. Fluids 28, 012106 (2016)
44. M.M. Hanczyc, T. Toyota, T. Ikegami, N. Packard, T. Sug-
awara, J. Am. Chem. Soc. 129, 9386 (2007)
45. T. Toyota, N. Maru, M.M. Hanczyc, T. Ikegami, T. Sug-
awara, J. Am. Chem. Soc. 131, 5012 (2009)
46. H. Kitahata, N. Yoshinaga, K.H. Nagai, Y. Sumino, Phys.
Rev. E 84, 015101 (2011)
47. T. Banno, R. Kuroha, T. Toyota, Langmuir 28, 1190
(2012)
48. T. Ban, T. Yamagami, H. Nakata, Y. Okano, Langmuir
29, 2554 (2013)
49. S. Herminghaus, C.C. Maass, C. Kr¨uger, S. Thutupalli,
L. Goehring, C. Bahr, Soft Matter 10, 7008 (2014)
50. Z. Izri, M.N. van der Linden, S. Michelin, O. Dauchot,
Phys. Rev. Lett. 113, 248302 (2014)
51. Y.J. Chen, Y. Nagamine, K. Yoshikawa, Phys. Rev. E 80,
016303 (2009)
52. O. Bliznyuk, H.P. Jansen, E.S. Kooij, H.J.W. Zandvliet,
B. Poelsema, Langmuir 27, 11238 (2011)
53. A.Y. Rednikov, Y.S. Ryazantsev, M.G. Velarde, J. Non-
Equil. Thermody. 19, 95 (1994)
54. A.Y. Rednikov, Y.S. Ryazantsev, M.G. Velarde, Phys. Flu-
ids 6, 451 (1994)
55. M.G. Velarde, A.Y. Rednikov, Y.S. Ryazantsev, J. Phys.:
Condens. Matter 8, 9233 (1996)
56. M.G. Velarde, Phil. Trans. R. Soc. Lond. A 356, 829 (1998)
57. N. Yoshinaga, K.H. Nagai, Y. Sumino, H. Kitahata, Phys.
Rev. E 86, 016108 (2012)
58. E. Tjhung, D. Marenduzzo, M.E. Cates, Proc. Natl. Acad.
Sci. U. S. A. 109, 12381 (2012)
59. N. Yoshinaga, Phys. Rev. E 89, 012913 (2014)
60. S. Yabunaka, T. Ohta, N. Yoshinaga, J. Chem. Phys. 136,
074904 (2012)
61. K. Furtado, C.M. Pooley, J.M. Yeomans, Phys. Rev. E 78,
046308 (2008)
62. A. Bray, Phil. Trans. R. Soc. Lond. A 361, 781 (2003)
63. V.A. Nepomniashchii, M.G. Velarde, P. Colinet, Interfacial
phenomena and convection, 1st edn. (Chapman & Hall /
CRC, 2002)
64. R.C. Desai, R. Kapral, Dynamics of Self-Organized and
Self-Assembled Structures (Cambridge University Press,
2009)
65. S. Chandrasekhar, Hydrodynamic and hydromagnetic sta-
bility (Oxford Univ. Press, 1961)
66. H.A. Stone, A.D. Samuel, Phys. Rev. Lett. 77, 4102 (1996)
67. J. B lawzdziewicz, P. Vlahovska, M. Loewenberg, Physica
A276, 50 (2000)
68. J.A. Hanna, P.M. Vlahovska, Phys. Fluids 22, 013102
(2010)
69. J.T. Schwalbe, F.R. Phelan Jr, P.M. Vlahovska, S.D. Hud-
son, Soft Matter 7, 7797 (2011)
70. O.S. Pak, J. Feng, H.A. Stone, J. Fluid. Mech. 753, 535
(2014)
71. J.H. Ferziger, M. Peri´c, Computational methods for fluid
dynamics, Vol. 3 (Springer Berlin, 1996)
72. R. Eymard, T. Gallou¨et, R. Herbin, Handbook of Numeri-
cal Analysis, Vol. 7 (2000)
73. J.R. Baumgardner, P.O. Frederickson, SIAM J. Numer.
Anal. 22, 1107 (1985)
74. J.A. Pudykiewicz, J. Comput. Phys. 213, 358 (2006)
75. N.J. Mottram, C.J. Newton, arXiv preprint
arXiv:1409.3542 (2014)
76. L. Longa, H.R. Trebin, Phys. Rev. A 42, 3453 (1990)
77. P. Kaiser, W. Wiese, S. Hess, J. Non-Equil. Thermody. 17,
153 (1992)
78. A.J. Bray, Adv. Phys. 51, 481 (2002)
79. Y. Brenier, F. Otto, C. Seis, SIAM J. Math. Anal. 43, 114
(2011)
80. D.S. Felix Otto, Christian Seis, Commun. Math. Sci. 11,
441 (2013)
81. V. Lobaskin, D. Lobaskin, I. Kuli´c, Eur. Phys. J.-Spec.
Top. 157, 149 (2008)
82. P.S. Lovely, F. Dahlquist, J. Theor. Biol. 50, 477 (1975)
83. M. Doi, S. Edwards, The Theory of Polymer Dynamics
(Oxford University Press, 1986)
84. S. Dukhin, G. Kretzschmar, R. Miller, Dynamics of Ad-
sorption at Liquid Interfaces: Theory, Experiment, Ap-
plication, Studies in Interface Science (Elsevier Science,
1995), ISBN 9780080530611
85. J. Dhont, An Introduction to Dynamics of Colloids (Else-
vier, 1996)
86. K.A. Hawick, D.P. Playne, International Journal of Com-
puter Aided Engineering and Technology 2, 78 (2010)
111
5 Conclusion and Outlook
Active matter is a collection of active agents, which rely on a constant supply of energy
to move or generate mechanical forces. A typical biological example is a suspension of
bacteria, where the individual bacteria cells are the active agents. The hydrodynamics of
swimming bacteria takes place in the low Reynolds number regime, where hydrodynamic
turbulence is absent. This eases the theoretical and numerical treatment to a great
extend and allows to rigorously study the fundamental physics of their collective motion.
It is therefore of great interest to design well-controllable synthetic active agents in the
low Reynolds number regime. A possible candidate is the active emulsion droplet. In
contrast to Janus colloids, active emulsion droplets are “soft” and thereby resemble living
cells. Furthermore, emulsion droplets have already been studied extensively in the past
decade in the context of microfluidics. Most prominently, labs-on-a-chip can combine
several digital fluidic operations such as transport, mixing, and sorting of droplets on a
single chip.
In a first step, we build the theoretical groundwork by deriving a model for active
emulsion droplets. The interface of an emulsion droplet is characterized by a concen-
tration of one or several surfactant species. By carefully including all relevant physical
processes, we gain a diffusion-advection-reaction equation for the surfactant density at
the interface. Here, the advection part of the equation is coupled to the flow field at the
interface. Thus, in order to close the equation, we also need to solve the Stokes equation
to derive the flow field at the interface. The model of an active emulsion droplet, which
we obtain in the end, is quite general and can be applied to many different setups. We
do this by specializing our model droplet and numerically solving the underlying field
equation for the droplet interface. The results are evaluated by employing standard tools
of statistical physics.
In the first model, we study an axisymmetric bromine enriched water droplet in oil.
This setup has been realized experimentally by Thutupalli et al. in Ref. [8]. The droplet
interface is covered with surfactants, which react with the dissolved bromine and as a
result increase the surface tension of the interface. Hence, the “bromination” reaction
locally increases the surface tension, which induces Marangoni flow, i.e., flow in direction
of increasing surface tension. This flow advects surfactants and thereby further enhances
the gradients in surface tension, which ultimately breaks the isotropic symmetry and
leads to stationary swimming of the droplet. We confirm these experimental results by
numerically solving our diffusion-advection-reaction equation. In addition, we identify
further patterns of motion. We find that the droplet stops after an initial motion or
that it oscillates back and forth. The latter also resembles recent experimental results
[15, 16].
112
Due to the axisymmetric constraint, the droplet of our first model can only move
along a straight line. However, in real systems the finite temperature at the droplet
interface leads to fluctuations in the surfactant mixture, which in turn randomize the
swimming direction e. This is analyzed in a second model, where we omit the axisym-
metric constraint and include thermal noise in the diffusion-advection-reaction for the
droplet interface. We find, both in numerical solutions and in a perturbation analysis
that the amplitude ξof the thermal noise is connected to the rotational correlation time
τrof the swimming path via τr∝1/ξ2. The full three-dimensional treatment of the
interface also allows to shed light on the demixing process or phase separation of the two
types of surfactants (pristine and brominated) at the interface during the non-steady
speed up of the droplet. We quantify the coarsening dynamics of the phase-separating
surfactants by means of the growth rate of domain size. Two steps exist: An initially
slow growth of domain size followed by a nearly ballistic regime. Furthermore, we study
the deviation from axisymmetry during the non-steady speed up by introducing an ap-
propriate measure for the biaxiality of the surfactant mixture. Here we find that the
droplet is clearly not axisymmetric during the speed up of the droplet.
In a second model we present a way to spontaneously break the droplet symmetry
without resorting to a chemical reaction. We assume an initially surfactant free emulsion
droplet in a micelle enriched bulk phase. When a micelle adsorbs at the droplet interface,
it spreads due to the Marangoni effect. This spreading also leads to a finite motion of the
droplet in direction of the adsorption site. The corresponding outer flow field, however,
preferentially advects other micelles towards the existing adsorption site. Hence, this
mechanism can spontaneously break the isotropic symmetry of the droplet and lead to
directed motion. We find parameter regimes in which the droplet either performs a
random walk or directed motion.
As an outlook, we want to mention possible extensions of the presented models that we
consider to be of interest. A straightforward extension is the combination of both models,
i.e., to implement the bromination droplet with explicitly modeled micelle adsorption.
As a matter of fact, it has been shown that micelles are crucial in many active droplet
systems [133, 132]. It would be interesting to see the effect of an explicitly modeled
micelle adsorption on the rotational correlation time τr. Furthermore, active droplets
are usually confined between two plates in experiments to ease the observation. However,
our model droplet is allowed to move freely in the bulk fluid. One could overcome this
discrepancy by including appropriate boundary conditions in the Stokes equation for the
flow field around the droplet. Lastly, as mentioned before, active droplets resemble active
agents. Therefore a thorough study of the collective motion of active emulsion droplets
is a worthwhile endeavor. In fact, experiments have shown that active droplets tend
to swim in swarms [16]. However, all computer simulations of the collective motion of
spherical microswimmers so far, assumed a fixed surface velocity field of the swimmer,
i.e., they implement the squirmer model [28, 29]. This could be extended to active
droplets, where the surface velocity field is not a fixed boundary condition but rather
depends dynamically on obstacles and neighboring droplets.
An alternative way to activate an emulsion droplet is to partially illuminate it with
light. If the droplet is covered with light-switchable surfactants, one locally generates a
113
spot of different surface tension, which induces Marangoni flow and with that propul-
sion of the droplet. Since such a droplet is also covered by two types of surfactants, we
can employ a similar model as before. In numerical solutions we find that the emulsion
droplet can be pushed with UV light or pulled with blue light, which resembles recent
experiments done with droplets on interfaces [20]. Furthermore, depending on the relax-
ation rate of the surfactant towards the surfactant in bulk, the droplet shows a plethora
of trajectories, including a damped oscillation about the beam axis, oscillation into the
beam axis, and a motion out of the beam followed by a complete stop. We also dwell
on the dependence of the trajectories on the droplet and bulk fluids. In particular, we
distinguish between strongly absorbing and transparent droplets.
A thinkable extension of our model is the combination of two perpendicular light beams
which intersect at the position of the droplet. By slowly changing the intersection point
of the two beams one could move the emulsion droplet to an arbitrary position in a bulk
medium. Another generalization that might be worthwhile is the addition of external
flows. Here, one could control the position of droplets in microchannel Poiseuille flow by
shining light onto the channel from one side. Similar systems have recently been studied
theoretically in the context of control theory of inertial flow [139, 140].
114
6 Bibliography
[1] S. Ramaswamy, “The mechanics and statistics of active matter,” Annu. Rev. Con-
dens. Matter, vol. 1, pp. 323–345, 2010.
[2] E. Schr¨odinger, What is life? University Press, 1967.
[3] M. C. Marchetti, “Active matter: Spontaneous flows and self-propelled drops,”
Nature, vol. 491, no. 7424, pp. 340–341, 2012.
[4] T. Sanchez, D. T. Chen, S. J. DeCamp, M. Heymann, and Z. Dogic, “Spontaneous
motion in hierarchically assembled active matter,” Nature, vol. 491, no. 7424,
pp. 431–434, 2012.
[5] E. Tjhung, D. Marenduzzo, and M. E. Cates, “Spontaneous symmetry breaking
in active droplets provides a generic route to motility,” Proc. Natl. Acad. Sci. U.
S. A., vol. 109, pp. 12381–12386, 2012.
[6] C. C. Maass, C. Kr¨uger, S. Herminghaus, and C. Bahr, “Swimming droplets,”
Annu. Rev. Condens. Matter, vol. 7, no. 1, 2016.
[7] A. Walther and A. H. M¨uller, “Janus particles,” Soft Matter, vol. 4, pp. 663–668,
2008.
[8] S. Thutupalli, R. Seemann, and S. Herminghaus, “Swarming behavior of simple
model squirmers,” New J. Phys., vol. 13, p. 073021, 2011.
[9] S.-Y. Teh, R. Lin, L.-H. Hung, and A. P. Lee, “Droplet microfluidics,” Lab Chip,
vol. 8, pp. 198–220, 2008.
[10] N. Nam-Trung, Droplet Microreactor. Springer US, 2008.
[11] I. Lagzi, “Chemical robotics—chemotactic drug carriers,” Open Med., vol. 8, no. 4,
pp. 377–382, 2013.
[12] M. M. Hanczyc, “Droplets: Unconventional protocell model with life-like dynamics
and room to grow,” Life, vol. 4, no. 4, pp. 1038–1049, 2014.
[13] V. A. Nepomniashchii, M. G. Velarde, and P. Colinet, Interfacial phenomena and
convection. Chapman & Hall / CRC, 1 ed., 2002.
[14] E. Lauga and T. R. Powers, “The hydrodynamics of swimming microorganisms,”
Rep. Prog. Phys., vol. 72, p. 096601, 2009.
115
[15] H. Kitahata, N. Yoshinaga, K. H. Nagai, and Y. Sumino, “Spontaneous motion of
a droplet coupled with a chemical wave,” Phys. Rev. E, vol. 84, p. 015101, 2011.
[16] S. Thutupalli and S. Herminghaus, “Tuning active emulsion dynamics via surfac-
tants and topology,” Eur. Phys. J. E, vol. 36, no. 8, pp. 1–10, 2013.
[17] M. Schmitt and H. Stark, “Swimming active droplet: A theoretical analysis,”
Europhys. Lett., vol. 101, p. 44008, 2013.
[18] M. Schmitt and H. Stark, “Active brownian motion of emulsion droplets: Coars-
ening dynamics at the interface and rotational diffusion,” Submitted to Eur. Phys.
J. E, 2016.
[19] M. Schmitt and H. Stark, “Marangoni flow at droplet interfaces: Three-
dimensional solution and applications,” Phys. Fluids, vol. 28, p. 012106, 2016.
[20] A. Diguet, R.-M. Guillermic, N. Magome, A. Saint-Jalmes, Y. Chen, K. Yoshikawa,
and D. Baigl, “Photomanipulation of a droplet by the chromocapillary effect,”
Angew. Chem. Int. Edit., vol. 121, no. 49, pp. 9445–9448, 2009.
[21] J. Dunkel, S. Heidenreich, K. Drescher, H. H. Wensink, M. B¨ar, and R. E. Gold-
stein, “Fluid dynamics of bacterial turbulence,” Phys. Rev. Lett., vol. 110, no. 22,
p. 228102, 2013.
[22] E. M. Purcell, “Life at low reynolds number,” Am. J. Phys., vol. 45, no. 1, pp. 3–11,
1977.
[23] G. Subramanian and P. R. Nott, “The fluid dynamics of swimming microorganisms
and cells,” J. Indian Inst. Sci., vol. 91, no. 3, pp. 283–314, 2012.
[24] J. Elgeti, R. G. Winkler, and G. Gompper, “Physics of microswimmers—single
particle motion and collective behavior: a review,” Rep. Prog. Phys., vol. 78,
no. 5, p. 056601, 2015.
[25] M. J. Lighthill, “On the squirming motion of nearly spherical deformable bodies
through liquids at very small reynolds numbers,” Commun. Pur. Appl. Math.,
vol. 5, pp. 109–118, 1952.
[26] J. R. Blake, “A spherical envelope approach to ciliary propulsion,” J. Fluid. Mech.,
vol. 46, pp. 199–208, 1971.
[27] M. T. Downton and H. Stark, “Simulation of a model microswimmer,” J. Phys.:
Condens. Matter, vol. 21, p. 204101, 2009.
[28] A. A. Evans, T. Ishikawa, T. Yamaguchi, and E. Lauga, “Orientational order
in concentrated suspensions of spherical microswimmers,” Phys. Fluids, vol. 23,
p. 111702, 2011.
116
[29] A. Z¨ottl and H. Stark, “Hydrodynamics determines collective motion and phase
behavior of active colloids in quasi-two-dimensional confinement,” Phys. Rev. Lett.,
vol. 112, p. 118101, 2014.
[30] O. Pak and E. Lauga, “Generalized squirming motion of a sphere,” J. Eng. Math.,
vol. 88, pp. 1–28, 2014.
[31] G. Batchelor, “The stress system in a suspension of force-free particles,” J. Fluid.
Mech., vol. 41, no. 03, pp. 545–570, 1970.
[32] T. Ishikawa, M. P. Simmonds, and T. J. Pedley, “Hydrodynamic interaction of two
swimming model micro-organisms,” J. Fluid. Mech., vol. 568, no. 1, pp. 119–160,
2006.
[33] “Biodidac: A bank of digital resources for teaching biology.”
http://biodidac.bio.uottawa.ca/.
[34] T. Fenchel, “How dinoflagellates swim,” Protist, vol. 152, no. 4, pp. 329 – 338,
2001.
[35] I. Llopis and I. Pagonabarraga, “Hydrodynamic interactions in squirmer motion:
Swimming with a neighbour and close to a wall,” J. Non-Newton. Fluid., vol. 165,
no. 17, pp. 946–952, 2010.
[36] Y. Hatwalne, S. Ramaswamy, M. Rao, and R. A. Simha, “Rheology of active-
particle suspensions,” Phys. Rev. Lett., vol. 92, no. 11, p. 118101, 2004.
[37] B. M. Haines, A. Sokolov, I. S. Aranson, L. Berlyand, and D. A. Karpeev, “Three-
dimensional model for the effective viscosity of bacterial suspensions,” Phys. Rev.
E, vol. 80, p. 041922, 2009.
[38] M. Moradi and A. Najafi, “Rheological properties of a dilute suspension of self-
propelled particles,” Europhys. Lett., vol. 109, no. 2, p. 24001, 2015.
[39] T. Bickel, A. Majee, and A. W¨urger, “Flow pattern in the vicinity of self-propelling
hot janus particles,” Phys. Rev. E, vol. 88, p. 012301, 2013.
[40] R. Golestanian, T. B. Liverpool, and A. Ajdari, “Propulsion of a molecular ma-
chine by asymmetric distribution of reaction products,” Phys. Rev. Lett., vol. 94,
p. 220801, 2005.
[41] W. F. Paxton, A. Sen, and T. E. Mallouk, “Motility of catalytic nanoparticles
through self-generated forces,” Chem. Eur. J., vol. 11, pp. 6462–6470, 2005.
[42] J. L. Moran and J. D. Posner, “Electrokinetic locomotion due to reaction-induced
charge auto-electrophoresis,” J. Fluid. Mech., vol. 680, pp. 31–66, 2011.
[43] I. Buttinoni, G. Volpe, F. K¨ummel, G. Volpe, and C. Bechinger, “Active brownian
motion tunable by light,” J. Phys.: Condens. Matter, vol. 24, p. 284129, 2012.
117
[44] T. C. Adhyapak and H. Stark, “Zipping and entanglement in flagellar bundle of
E. coli : Role of motile cell body,” Phys. Rev. E, vol. 92, p. 052701, 2015.
[45] S. B. Babu and H. Stark, “Modeling the locomotion of the african trypanosome
using multi-particle collision dynamics,” New J. Phys., vol. 14, no. 8, p. 085012,
2012.
[46] D. Alizadehrad, T. Kr¨uger, M. Engstler, and H. Stark, “Simulating the complex
cell design of trypanosoma brucei and its motility,” PLoS Comput. Biol., vol. 11,
p. e1003967, 2015.
[47] E. Frey and K. Kroy, “Brownian motion: a paradigm of soft matter and biological
physics,” Ann. Phys., vol. 14, no. 1-3, pp. 20–50, 2005.
[48] J. Dhont, An Introduction to Dynamics of Colloids. Elsevier, 1996.
[49] J. R. Howse, R. A. Jones, A. J. Ryan, T. Gough, R. Vafabakhsh, and R. Golesta-
nian, “Self-motile colloidal particles: from directed propulsion to random walk,”
Phys. Rev. Lett., vol. 99, p. 048102, 2007.
[50] V. Lobaskin, D. Lobaskin, and I. Kuli´c, “Brownian dynamics of a microswimmer,”
Eur. Phys. J.-Spec. Top., vol. 157, pp. 149–156, 2008.
[51] B. ten Hagen, S. van Teeffelen, and H. L¨owen, “Brownian motion of a self-propelled
particle,” J. Phys.: Condens. Matter, vol. 23, no. 19, p. 194119, 2011.
[52] P. S. Lovely and F. Dahlquist, “Statistical measures of bacterial motility and
chemotaxis,” J. Theor. Biol., vol. 50, pp. 477–496, 1975.
[53] M. Doi and S. Edwards, The Theory of Polymer Dynamics. Oxford University
Press, 1986.
[54] P. Romanczuk, M. B¨ar, W. Ebeling, B. Lindner, and L. Schimansky-Geier, “Active
brownian particles,” Eur. Phys. J.-Spec. Top., vol. 202, pp. 1–162, 2012.
[55] M. Enculescu and H. Stark, “Active colloidal suspensions exhibit polar order under
gravity,” Phys. Rev. Lett., vol. 107, p. 058301, 2011.
[56] F. Peruani and L. G. Morelli, “Self-propelled particles with fluctuating speed and
direction of motion in two dimensions,” Phys. Rev. Lett., vol. 99, p. 010602, 2007.
[57] E. Bormashenko, Y. Bormashenko, and G. Oleg, “On the nature of the friction be-
tween nonstick droplets and solid substrates,” Langmuir, vol. 26, no. 15, pp. 12479–
12482, 2010.
[58] A. Yarin, “Drop impact dynamics: splashing, spreading, receding, bouncing,”
Annu. Rev. Fluid Mech., vol. 38, pp. 159–192, 2006.
[59] A. Cassie and S. Baxter, “Wettability of porous surfaces,” T. Faraday Soc., vol. 40,
pp. 546–551, 1944.
118
[60] K. Kawasaki, “Study of wettability of polymers by sliding of water drop,” J. Colloid
Sci., vol. 15, no. 5, pp. 402–407, 1960.
[61] K. Ichimura, S.-K. Oh, and M. Nakagawa, “Light-driven motion of liquids on a
photoresponsive surface,” Science, vol. 288, no. 5471, pp. 1624–1626, 2000.
[62] F. D. Dos Santos and T. Ondar¸cuhu, “Free-running droplets,” Phys. Rev. Lett.,
vol. 75, pp. 2972–2975, 1995.
[63] Y.-J. Chen, Y. Nagamine, and K. Yoshikawa, “Self-propelled motion of a droplet
induced by marangoni-driven spreading,” Phys. Rev. E, vol. 80, p. 016303, 2009.
[64] M. D. Levan and J. Newman, “The effect of surfactant on the terminal and in-
terfacial velocities of a bubble or drop,” AIChE Journal, vol. 22, pp. 695–701,
1976.
[65] K. V. Beard and C. Chuang, “A new model for the equilibrium shape of raindrops,”
J. Atmos. Sci., vol. 44, no. 11, pp. 1509–1524, 1987.
[66] Y. Pawar and K. J. Stebe, “Marangoni effects on drop deformation in an exten-
sional flow: The role of surfactant physical chemistry. i. insoluble surfactants,”
Phys. Fluids, vol. 8, no. 7, pp. 1738–1751, 1996.
[67] F. Blanchette, “Simulation of mixing within drops due to surface tension varia-
tions,” Phys. Rev. Lett., vol. 105, no. 7, p. 074501, 2010.
[68] G. Katsikis, J. S. Cybulski, and M. Prakash, “Synchronous universal droplet logic
and control,” Nat Phys, 2015.
[69] C. V. Kulkarni, W. Wachter, G. Iglesias-Salto, S. Engelskirchen, and S. Ahualli,
“Monoolein: a magic lipid?,” Phys. Chem. Chem. Phys., vol. 13, no. 8, pp. 3004–
3021, 2011.
[70] N. T. Southall, K. A. Dill, and A. Haymet, “A view of the hydrophobic effect,” J.
Phys. Chem. B, vol. 106, no. 3, pp. 521–533, 2002.
[71] S. Glathe and D. Schermer, “Wasser, w¨asche, umwelt: Alles sauber?,” Chem.
unserer Zeit, vol. 37, no. 5, pp. 336–346, 2003.
[72] P. Atkins and J. de Paula, Atkins’ Physical Chemistry. OUP Oxford, 2010.
[73] I. Roberts and G. E. Kimball, “The halogenation of ethylenes,” J. Am. Chem.
Soc., vol. 59, no. 5, pp. 947–948, 1937.
[74] O. Karthaus, M. Shimomura, M. Hioki, R. Tahara, and H. Nakamura, “Reversible
photomorphism in surface monolayers,” J. Am. Chem. Soc., vol. 118, no. 38,
pp. 9174–9175, 1996.
119
[75] J. Y. Shin and N. L. Abbott, “Using light to control dynamic surface tensions
of aqueous solutions of water soluble surfactants,” Langmuir, vol. 15, no. 13,
pp. 4404–4410, 1999.
[76] J. Eastoe and A. Vesperinas, “Self-assembly of light-sensitive surfactants,” Soft
Matter, vol. 1, pp. 338–347, 2005.
[77] E. Chevallier, A. Mamane, H. A. Stone, C. Tribet, F. Lequeux, and C. Monteux,
“Pumping-out photo-surfactants from an air-water interface using light,” Soft Mat-
ter, vol. 7, pp. 7866–7874, 2011.
[78] Y. Zhao and T. Ikeda, Smart light-responsive materials: azobenzene-containing
polymers and liquid crystals. John Wiley & Sons, 2009.
[79] T. Hayashita, T. Kurosawa, T. Miyata, K. Tanaka, and M. Igawa, “Effect of
structural variation within cationic azo-surfactant upon photoresponsive function
in aqueous solution,” Colloid Polym. Sci., vol. 272, no. 12, pp. 1611–1619, 1994.
[80] A. Diguet, H. Li, N. Queyriaux, Y. Chen, and D. Baigl, “Photoreversible fragmen-
tation of a liquid interface for micro-droplet generation by light actuation,” Lab
Chip, vol. 11, no. 16, pp. 2666–2669, 2011.
[81] A. Venancio-Marques, F. Barbaud, and D. Baigl, “Microfluidic mixing triggered by
an external led illumination,” J. Am. Chem. Soc., vol. 135, no. 8, pp. 3218–3223,
2013.
[82] A. Venancio-Marques and D. Baigl, “Digital optofluidics: Led-gated transport
and fusion of microliter-sized organic droplets for chemical synthesis,” Langmuir,
vol. 30, no. 15, pp. 4207–4212, 2014.
[83] D. Baigl, “Photo-actuation of liquids for light-driven microfluidics: state of the art
and perspectives,” Lab Chip, vol. 12, no. 19, pp. 3637–3653, 2012.
[84] M. Ichikawa, F. Takabatake, K. Miura, T. Iwaki, N. Magome, and K. Yoshikawa,
“Controlling negative and positive photothermal migration of centimeter-sized
droplets,” Phys. Rev. E, vol. 88, no. 1, p. 012403, 2013.
[85] U. Thiele, A. J. Archer, and M. Plapp, “Thermodynamically consistent description
of the hydrodynamics of free surfaces covered by insoluble surfactants of high
concentration,” Phys. Fluids, vol. 24, no. 10, p. 102107, 2012.
[86] S. Dukhin, G. Kretzschmar, and R. Miller, Dynamics of Adsorption at Liquid
Interfaces: Theory, Experiment, Application. Studies in Interface Science, Elsevier
Science, 1995.
[87] S. Sadhal, P. Ayyaswamy, and J. Chung, Transport Phenomena with Drops and
Bubbles. Mechanical Engineering Series, Springer, 1996.
120
[88] R. C. Desai and R. Kapral, Dynamics of Self-Organized and Self-Assembled Struc-
tures. Cambridge University Press, 2009.
[89] K. A. Hawick and D. P. Playne, “Modelling, simulating and visualising the cahn-
hilliard-cook field equation,” International Journal of Computer Aided Engineering
and Technology, vol. 2, no. 1, pp. 78–93, 2010.
[90] J. Thomson, “On certain curious motions observable at the surfaces of wine and
other alcoholic liquors,” Philos. Mag., vol. 10, no. 67, pp. 330–333, 1855.
[91] J. Fournier and A. Cazabat, “Tears of wine,” Europhys. Lett., vol. 20, no. 6, p. 517,
1992.
[92] C. Marangoni, Sull’ espansione delle goccie liquide. PhD thesis, University of
Pavia, 1865.
[93] J. W. Gibbs, “On the equilibrium of heterogeneous substances,” Trans. Connect.
Acad. Arts Sci., vol. 3, 1876.
[94] M. Roch´e, Z. Li, I. M. Griffiths, S. Le Roux, I. Cantat, A. Saint-Jalmes, and H. A.
Stone, “Marangoni flow of soluble amphiphiles,” Phys. Rev. Lett., vol. 112, no. 20,
p. 208302, 2014.
[95] J. Berg, An Introduction to Interfaces & Colloids: The Bridge to Nanoscience.
World Scientific, 2010.
[96] S. Phongikaroon, R. Hoffmaster, K. P. Judd, G. B. Smith, and R. A. Handler,
“Effect of temperature on the surface tension of soluble and insoluble surfactants
of hydrodynamical importance,” J. Chem. Eng. Data, vol. 50, no. 5, pp. 1602–1607,
2005.
[97] L. Napolitano, “Plane marangoni-poiseuille flow of two immiscible fluids,” Acta
Astronaut., vol. 7, pp. 461 – 478, 1980.
[98] Z. Mao, P. Lu, G. Zhang, and C. Yang, “Numerical simulation of the marangoni
effect with interphase mass transfer between two planar liquid layers,” Chinese J.
Chem. Eng., vol. 16, pp. 161 – 170, 2008.
[99] X.-Y. You, L.-D. Zhang, and J.-R. Zheng, “Marangoni instability of immiscible
liquid–liquid stratified flow with a planar interface in the presence of interfacial
mass transfer,” J. Taiwan Inst. Chem. E., vol. 45, pp. 772 – 779, 2014.
[100] J. Hadamard, “Mouvement permanent lent d’une sphere liquide et visqueuse dans
un liquide visqueux,” CR Acad. Sci, vol. 152, pp. 1735–1738, 1911.
[101] W. Rybczynski, Ҭ
Uber die fortschreitende bewegung einer ߬ussigen kugel in einem
z¨ahen medium,” Bull. Acad. Sci. Cracovie Ser. A, vol. 1, pp. 40–46, 1911.
[102] V. G. Levich, Physicochemical Hydrodynamics. Prentice-Hall, 1962.
121
[103] M. D. Levan, “Motion of a droplet with a newtonian interface,” J. Colloid Interface
Sci., vol. 83, no. 1, pp. 11–17, 1981.
[104] J. B lawzdziewicz, P. Vlahovska, and M. Loewenberg, “Rheology of a dilute emul-
sion of surfactant-covered spherical drops,” Physica A, vol. 276, pp. 50–85, 2000.
[105] J. A. Hanna and P. M. Vlahovska, “Surfactant-induced migration of a spherical
drop in stokes flow,” Phys. Fluids, vol. 22, p. 013102, 2010.
[106] J. T. Schwalbe, F. R. Phelan Jr, P. M. Vlahovska, and S. D. Hudson, “Interfacial
effects on droplet dynamics in poiseuille flow,” Soft Matter, vol. 7, pp. 7797–7804,
2011.
[107] H. A. Stone and A. D. Samuel, “Propulsion of microorganisms by surface distor-
tions,” Phys. Rev. Lett., vol. 77, p. 4102, 1996.
[108] C. Sternling and L. Scriven, “Interfacial turbulence: hydrodynamic instability and
the marangoni effect,” AIChE Journal, vol. 5, no. 4, pp. 514–523, 1959.
[109] T. S. Sørensen, M. Hennenberg, A. Steinchen, and A. Sanfeld, “Chemical and
hydrodynamical analysis of stability of a spherical interface,” J. Colloid Interface
Sci., vol. 56, no. 2, pp. 191–205, 1976.
[110] T. Sørensen and M. Hennenberg, “Instability of a spherical drop with surface
chemical reactions and transfer of surfactants,” in Dynamics and Instability of
Fluid Interfaces (T. Sørensen, ed.), vol. 105 of Lecture Notes in Physics, pp. 276–
315, Springer Berlin Heidelberg, 1979.
[111] Y. S. Ryazantsev, “Thermocapillary motion of a reacting droplet in a chemically
active medium,” Fluid Dyn., vol. 20, no. 3, pp. 491–495, 1985.
[112] A. Golovin, Y. P. Gupalo, and Y. S. Ryazantsev, “Chemothermocapillary effect
for the motion of a drop in a liquid,” in Soviet Physics Doklady, vol. 31, p. 700,
1986.
[113] Y. K. Bratukhin, “Thermocapillary drift of a droplet of viscous liquid,” Fluid Dyn.,
vol. 10, no. 5, pp. 833–837, 1975.
[114] A. Golovin and Y. S. Ryazantsev, “Drift of a reacting droplet due to the chemo-
concentration capillary effect,” Fluid Dyn., vol. 25, no. 3, pp. 370–378, 1990.
[115] Y. S. Ryazantsev, A. Rednikov, and M. Velarde, “Solutocapillary motion of chemi-
cally reacting drop in microgravity,” in Graz International Astronautical Federation
Congress, vol. 1, 1993.
[116] A. Y. Rednikov, V. N. Kurdyumov, Y. S. Ryazantsev, and M. G. Velarde, “The role
of time-varying gravity on the motion of a drop induced by marangoni instability,”
Phys. Fluids, vol. 7, no. 11, pp. 2670–2678, 1995.
122
[117] A. Y. Rednikov, Y. S. Ryazantsev, and M. G. Velarde, “Drop motion with surfac-
tant transfer in a homogeneous surrounding,” Phys. Fluids, vol. 6, pp. 451–468,
1994.
[118] A. Y. Rednikov, Y. S. Ryazantsev, and M. G. Vel´arde, “Drop motion with sur-
factant transfer in an inhomogeneous medium,” Int. J. Heat Mass Tran., vol. 37,
pp. 361–374, 1994.
[119] A. Y. Rednikov, Y. S. Ryazantsev, and M. G. Velarde, “Active drops and drop
motions due to nonequilibrium phenomena,” J. Non-Equil. Thermody., vol. 19,
pp. 95–113, 1994.
[120] A. Y. Rednikov, Y. S. Ryazantsev, and M. G. Velarde, “On the development of
translational subcritical marangoni instability for a drop with uniform internal
heat generation,” J. Colloid Interface Sci., vol. 164, pp. 168 – 180, 1994.
[121] M. G. Velarde, A. Y. Rednikov, and Y. S. Ryazantsev, “Drop motions and inter-
facial instability,” J. Phys.: Condens. Matter, vol. 8, p. 9233, 1996.
[122] Y. Sumino, N. Magome, T. Hamada, and K. Yoshikawa, “Self-running droplet:
Emergence of regular motion from nonequilibrium noise,” Phys. Rev. Lett., vol. 94,
no. 6, p. 068301, 2005.
[123] K. Nagai, Y. Sumino, H. Kitahata, and K. Yoshikawa, “Mode selection in the
spontaneous motion of an alcohol droplet,” Phys. Rev. E, vol. 71, no. 6, p. 065301,
2005.
[124] M. M. Hanczyc, T. Toyota, T. Ikegami, N. Packard, and T. Sugawara, “Fatty acid
chemistry at the oil-water interface: self-propelled oil droplets,” J. Am. Chem.
Soc., vol. 129, pp. 9386–91, 2007.
[125] T. Toyota, N. Maru, M. M. Hanczyc, T. Ikegami, and T. Sugawara, “Self-propelled
oil droplets consuming “fuel” surfactant,” J. Am. Chem. Soc., vol. 131, pp. 5012–
5013, 2009.
[126] N. Yoshinaga, K. H. Nagai, Y. Sumino, and H. Kitahata, “Drift instability in the
motion of a fluid droplet with a chemically reactive surface driven by marangoni
flow,” Phys. Rev. E, vol. 86, p. 016108, 2012.
[127] S. Yabunaka, T. Ohta, and N. Yoshinaga, “Self-propelled motion of a fluid droplet
under chemical reaction,” J. Chem. Phys., vol. 136, p. 074904, 2012.
[128] T. Ban, T. Yamagami, H. Nakata, and Y. Okano, “ph-dependent motion of self-
propelled droplets due to marangoni effect at neutral ph,” Langmuir, vol. 29,
pp. 2554–2561, 2013.
[129] K. H. Nagai, F. Takabatake, Y. Sumino, H. Kitahata, M. Ichikawa, and N. Yoshi-
naga, “Rotational motion of a droplet induced by interfacial tension,” Phys. Rev.
E, vol. 87, p. 013009, 2013.
123
[130] S. Michelin, E. Lauga, and D. Bartolo, “Spontaneous autophoretic motion of
isotropic particles,” Phys. Fluids, vol. 25, no. 6, p. 061701, 2013.
[131] N. Yoshinaga, “Spontaneous motion and deformation of a self-propelled droplet,”
Phys. Rev. E, vol. 89, p. 012913, 2014.
[132] Z. Izri, M. N. van der Linden, S. Michelin, and O. Dauchot, “Self-propulsion of
pure water droplets by spontaneous marangoni-stress-driven motion,” Phys. Rev.
Lett., vol. 113, p. 248302, 2014.
[133] S. Herminghaus, C. C. Maass, C. Kr¨uger, S. Thutupalli, L. Goehring, and C. Bahr,
“Interfacial mechanisms in active emulsions,” Soft Matter, vol. 10, pp. 7008–7022,
2014.
[134] Banno and T. Taro Toyota, “Design of reactive surfactants that control the loco-
motion mode of cell-sized oil droplets,” Current Phys. Chem., vol. 5, no. 1, 2014.
[135] C. A. Whitfield, D. Marenduzzo, R. Voituriez, and R. J. Hawkins, “Active polar
fluid flow in finite droplets,” Eur. Phys. J. E, vol. 37, p. 8, 2014.
[136] S. Shklyaev, “Janus droplet as a catalytic micromotor,” Europhys. Lett., vol. 110,
no. 5, p. 54002, 2015.
[137] D. Zwicker, R. Seyboldt, C. A. Weber, A. A. Hyman, and F. J¨ulicher, “Growth and
Division of Active Droplets: A Model for Protocells,” ArXiv e-print 1603.01571,
2016.
[138] A. J. Bray, “Theory of phase-ordering kinetics,” Adv. Phys., vol. 51, pp. 481–587,
2002.
[139] C. Prohm, F. Tr¨oltzsch, and H. Stark, “Optimal control of particle separation in
inertial microfluidics,” Eur. Phys. J. E, vol. 36, pp. 1–13, 2013.
[140] C. Prohm and H. Stark, “Feedback control of inertial microfluidics using axial
control forces,” Lab Chip, vol. 14, pp. 2115–2123, 2014.
124
Danksagung
Hiermit m¨ochte ich allen danken, die am Zustandekommen dieser Arbeit beteiligt waren.
Zuallererst danke ich Herrn Prof. Holger Stark f¨ur die Stellung der Aufgabe und die
geduldige Betreuung ¨uber viele Jahre hinweg. Weiterhin danke ich Herrn Prof. Uwe
Thiele f¨ur die Begutachtung dieser Arbeit und Prof. Martin Schoen f¨ur die Leitung des
Promotionsausschusses.
Mein Dank gilt auch allen Mitarbeiten der Arbeitsgruppe von Prof. Holger Stark.
Besonders hervorheben m¨ochte ich die vielen wissenschaftlichen Diskussionen mit Rein-
hard Vogel, Andreas Z¨ottl, Sebastian Reddig und Katrin Wolff. Ich danke weiterhin
Stefan Fruhner f¨ur das Bereitstellen der Meshview Software.
Schließlich danke ich Shashi Thutupalli, dessen Experimente mit schwimmenden Trop-
fen den Grundstein f¨ur diese Arbeit bildeten.
125
Zusammenfassung in deutscher Sprache
Angestoßen durch neue Erkenntnisse auf dem Gebiet der kollektiven Dynamik im Nicht-
gleichgewicht, sowie der Mikrofluidik in sogenannten Chiplabors, erleben aktive Emul-
sionstropfen derzeit eine Renaissance in der Physik der weichen Materie. Aktive Emul-
sionstropfen zeichnen sich dadurch aus, dass sie sich ohne externe Kr¨afte bewegen. Der
Ursprung dieser Bewegung ist eine inhomogene Ober߬achenspannung auf der Grenz߬ache
des Tropfens. Oberfl¨achenspannung basiert auf den molekularen Anziehungskr¨aften
zwischen Fl¨ussigkeitsmolek¨ulen an der Grenzfl¨ache. Folglich wird die Fl¨ussigkeit auf bei-
den Seiten der Grenz߬ache des Tropfens in Richtung steigender Ober߬achenspannung
gezogen. Diesen Effekt nennt man Marangoni-Effekt. Das daraus resultierende Ge-
schwindigkeitsfeld um den Tropfen induziert die Bewegung des Tropfens.
Eine einfache M¨oglichkeit den Marangoni-Effekt zu nutzen um einen Tropfen in Be-
wegung zu setzen, basiert auf der Adsorption kugelf¨ormiger Aggregate aus Tensiden,
sogenannter Mizellen. Tenside sind grenzfl¨achenaktive Molek¨ule, die die Oberfl¨achen-
spannung einer Grenz߬ache verringern. Wenn eine Mizelle an einen Tropfen adsorbiert,
verringert sich lokal die Ober߬achenspannung, was zu einem Marangoni-Fluss weg von
der Adsorptionsstelle f¨uhrt. Das induzierte Geschwindigkeitsfeld um den Tropfen bewegt
den Tropfen in Richtung der Adsorptionsstelle und sorgt f¨ur einen verst¨arkten Fluss zur
Tropfenoberfl¨ache auf der Vorderseite des Tropfens. In dieser Arbeit erkl¨aren wir wie
dieser einfache Mechanismus zu gerichteter Bewegung des Tropfens f¨uhren kann.
Alternativ kann man die molekulare Struktur, und damit die Ober߬achenspannung,
von Tensiden mittels einer chemischen Reaktion mit Brom ¨andern. Ein mit Brom an-
gereicherter Tropfen, dessen Grenz߬ache mit Tensiden belegt ist, kann so eine spontane
Symmetriebrechung erfahren, die zu gerichteter Bewegung f¨uhrt. Auch wenn diese Bewe-
gung gerichtet ist, so zeichnet sie sich doch durch eine auf großen L¨angenskalen zuf¨allige
Richtungs¨anderung des Tropfens aus. Dieses Verhalten l¨asst sich zur¨uckf¨uhren auf das
thermische Rauschen des Tensidfilms auf der Tropfenoberfl¨ache. Diese Arbeit entwickelt
ein Modell f¨ur den Tensidfilm auf der Tropfenoberfl¨ache und erkl¨art sowohl die spontane
Vorw¨artsbewegung als auch die erratische Richtungs¨anderung des Tropfens.
Im Gegensatz zur ¨
Anderung der Ober߬achenspannung mittels einer chemischen Reak-
tion, welche relativ unkontrolliert abl¨auft, l¨asst sich die Oberfl¨achenspannung auch sehr
pr¨azise mit Hilfe von Licht modulieren. Dabei nutzt man den cis-trans Isomorphis-
mus spezieller Tenside aus. Diese ¨andern ihre Struktur und Oberfl¨achenspannung wenn
man sie mit Licht einer bestimmen Wellenl¨ange beleuchtet. So kann man lokal auf der
Tropfenoberfl¨ache die Oberfl¨achenspannung verringern oder erh¨ohen und ein Marangoni-
Flussfeld generieren. In Simulationen solcher Tropfen finden wir eine Vielzahl von Be-
wegungsmustern, angefangen von gerichteter Bewegung in Richtung des Lichtstrahls, bis
hin zu ged¨ampften und stabilen Oszillationen um den Lichtstrahl.
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