Insight into open hydrodynamic problems by atomistic simulations [original]

Insight into open hydrodynamic problems by atomistic

simulations

vorgelegt von

Dipl.-Ing.

René Spencer Chatwell

ORCID: 0000-0002-2692-1460

an der Fakultät III - Prozesswissenschaften

der Technischen Universität Berlin

zur Erlangung des akademischen Grades

Doktor der Naturwissenschaften

- Dr. rer. nat. -

genehmigte Dissertation

Promotionsausschuss:

Vorsitzender: Prof. Dr.-Ing. habil. Jens Repke

Gutachter: Prof. Dr. rer. nat. Michael Pﬁtzner

Gutachter: Prof. Dr.-Ing. habil. Jadran Vrabec

Tag der wissenschaftlichen Aussprache: 08. November 2021

Berlin 2022

Prologue

"Die Thermodynamik ist ein komisches Fach! Das erste Mal, wenn man sich

damit befasst, versteht man nichts davon. Beim zweiten Durcharbeiten denkt

man, man hätte nun alles verstanden, mit Ausnahme von ein oder zwei kleinen

Details. Das dritte Mal, wenn man den Stoﬀ durcharbeitet, bemerkt man, dass

man fast garnichts davon versteht, aber man hat sich inzwischen so daran

gewöhnt, dass es einen nicht mehr stört."

Arnold Sommerfeld

iii

Acknowledgement

This work was facilitated by the Collaborative Research Center (SFB) 75 of the Deutsche

Forschungsgemeinschaft (DFG) and was funded in parts under grants VR 6/9-2 and VR

6/11. The work was additionally carried out under the auspices of the Boltzmann-Zuse

Society (BZS) and all computations were performed either on the HPC Clusters OCuLUS

and Noctua at the Paderborn Center for Parallel Computing (PC2) or on the Cray XC40

system Hazel Hen and the HPE Apollo system Hawk at the High Performance Computing

Centre (HLRS) contributing to the project MMHBF2.

Abstract

The results leading towards the present thesis are presented in a cumulative format, i.e.

each investigated problem is brieﬂy motivated in a separate section and in order to avoid

unnecessary redundancy its main results are not repeated here, yet are to be found in

the respective original publication that is enclosed at the end of each section. This work

is characterised by theoretical investigations that are complemented by atomistic simula-

tions aiming at further insight into currently unresolved hydrodynamic problems, of which

Stefan’s problem,Stokes’ hypothesis and Widom’s conjecture were selected.

A Stefan problem emerges during a ﬁrst-order phase transition, since the boundary sepa-

rating both aggregate states, i.e. the domain Ωtor its subdomain, is subject to temporal

changes. Consequently, the determination of the position of this freely moving boundary

∂Ωthas to be part of the solution. In a pioneering eﬀort, a physical model of continuous

phase transition across a planar liquid surface is proposed. The model incorporates both

phases’ genuine thermodynamic non-idealities and its analytical solution allows to predict

the ﬁlm’s thickness as well as its molar composition over time. As the proposed model was

successfully validated by large scale molecular dynamics simulations, it was also demon-

strated that the classical hydrodynamic formalism is applicable to length scales of around

l=O(15) nm.

Stokes’ hypothesis reduces the complexity of Cauchy’s stress tensor ﬁeld τij by postulat-

ing that form invariant changes of a ﬂuid’s local volume, i.e. compressions or dilatations,

are not associated with the dissipation of linear momentum, which is synonymous with

the bulk viscosity to be zero. Despite being challenged by theoretical and experimental

evidence, the hypothesis is still widely applied throughout all realms of ﬂuid mechanics.

This work addresses the bulk viscosity of monatomic liquids. On the basis of a multi-

mode relaxation ansatz, an equation of state is proposed, exposing the bulk viscosity to

be a two-parametric power function of reduced density and temperature, with parameters

showing a conspicuous saturation behaviour.

Widom conjectured the maxima of speciﬁc thermodynamic response functions to con-

ﬂuence towards a single line in close vicinity to a substance’s critical point. Widom

understood these lines to be an extrapolation of the vapour pressure curve into the super-

critical region, partitioning parts of the latter into liquid-like and gas-like subsections. In

this work, the Widom line was observed to aﬀect ﬁnitely diluted mixtures containing only

traces of one component, with the diluted component forming aggregated clusters due to

its hydrogen bonding interactions.

The present results were made possible by substantial advances in high performance com-

puting facilitating the development of atomistic simulations into a suitable tool to examine

the dynamics at hydrodynamic length and time scales. While static macroscopic observ-

ables, such as temperature, pressure and density, are accurately sampled in ensembles

containing on the order of N=O(102)particles, dynamic observables, such as the bulk

viscosity, require ensemble sizes of order N=O(103). On the other hand, sampling

evaporation dynamics with reasonably good statistics requires particle ensembles of order

N=O(105), which is only obtainable with highly scalable simulation programmes.

Zusammenfassung

Um unnötige Redundanz zu vermeiden, werden die Ergebnisse dieser Dissertation kummu-

lativ präsentiert. Jedes der betrachteten Probleme wird dabei in einem separaten Kapitel

motiviert und mit der jeweiligen Fachpublikation abgeschlossen. Die vorliegende Arbeit

ist theoretischer Natur, ergänzt durch den Einsatz atomistischer Simulationsmethoden mit

dem Ziel Lösungsvorschläge für bisher ungelöste Strömungsmechnikprobleme zu unterbre-

iten. Hierzu wurden die drei Probleme das Stefan Problem, die Stokessche Hypothese und

die Widomsche Vermutung ausgewählt.

Unter einem Stefan Problem wird die Formulierung einer Diﬀererenzialgleichung auf einer

zeitveränderlichen Domäne Ωtverstanden. Der Bereich, auf dem die Randbedingungen

deﬁniert sind, ∂Ωtist dabei nicht ortsfest und muss somit Teil des Lösungskalküls sein.

In der vorgelegten grundlegenden Arbeit, wird das Stefan Problem eines mehrkompo-

nentigen Flüssigﬁlms analytisch gelöst. Der Film verdunstet dabei kontinuierlich über

eine planare Grenzﬂäche in eine dichte Gasphase. Das aufgestellte physikalische Mod-

ell beschreibt die ausgeprägte thermodynamische Nichtidealität beider Phasen vollständig

und die daraus erzeugte Lösung ermöglicht die Vorhersage der Filmdicke, sowie der mo-

laren Zusammensetzung beider Phasen als Funktion der Zeit. Dabei wurde aufgezeigt,

dass der klassische hydrodynamische Formalismus bis auf Längenskalen der Größenord-

nung l=O(15) nm angewendet werden kann.

Über die Stokessche Hypothese wird postuliert, dass lokale forminvariante Änderungen

eines Fluidvolumens keine Dissipation des linearen Impulses zur Folge haben, woduch sich

die Komplexität des Causchy Stresstensorfeldes τij reduziert, was gleichbedeutend ist mit

einer verschwindenden Volumenviskosität. Die Hypothese wird, experimenteller Belege

zum Trotz, immernoch in sehr vielen Bereichen der Strömungsmechanik angewendet. Im

Rahmen dieser Dissertation wird die Volumenviskosität der Edelgase in ﬂüssiger Phase

diskutiert. Auf Basis eines multimodalen Relaxationsansatzes wird eine zweiparametrige

Zustandsgleichung vorgeschlagen, wobei die Parameter ein auﬀälliges Sättigungsverhalten

zeigen.

Unter der Widomschen Vermututung wird der Ansatz verstanden, dass die Ortskurven der

Maxima verschiedener thermodynamischer Responsefunktionen in der Nähe des kritischen

Punkts einer Substanz zusammenfallen und die sogenannte Widomsche Linie ausbilden.

Widom hat diese Ortskurven als Verlängerung der Dampfdruckkurve in den überkritischen

Bereich hinein verstanden, um diesen in die zwei Teilbereiche ﬂüssigähnlich und gasähn-

lich zu unterteilen. Im Rahmen dieser Dissertation wurde der Einﬂuss der Widomlinie auf

eine stark verdünnten Mischung untersucht, wobei beobachtet wurde dass die verdünnte

Komponente aufgrund von Wasserstoﬀbrückenbindungen zur Clusterbildung neigt.

Die Ergebnisse dieser Arbeit wurden nur durch Fortschritte im Hoch- und Höchstleis-

tungsrechnen möglich, wodurch atomistische Simulationsprogramme auf hydrodynamis-

che Problemstellungen angewendet werden konnten. Während statische Observablen, wie

Temperatur, Druck und Dichte, über atmostische Simultionen in Teilchenensembles der

Größenordnung N=O(102)bestimmt werden können, bedarf es für die Volumenviskosität

bereits Ensembles der Größe N=O(103). Um die Verdunstungsdynamik an einer Gren-

zﬂäche atomistisch sinnvoll aufzulösen, bedarf es jedoch weit größerer Teilchenensembles.

vii

Contents

Abstract vi

List of Symbols x

1 Introduction 1

1.1 Historicity ..................................... 1

1.2 Small scale hydrodynamics . .......................... 4

1.3 Microscopic picture of a ﬂuid .......................... 8

1.4 Continuum framework and Newtonian ﬂuids . . . ............... 9

2 Original publications 10

2.1 List of original publications . .......................... 10

2.2 Stefan’s problem ................................. 11

2.2.1 Arctic ice accretion . ........................... 11

2.2.2 Droplet evaporation . .......................... 14

2.2.3 Film evaporation . . ........................... 18

2.2.4 Interphase dynamics . .......................... 19

2.2.5 Hydrodynamic description . ....................... 20

2.2.6 Solution strategy . . ........................... 21

2.2.7 Diﬀusion limited evaporation of a binary liquid ﬁlm . ........ 24

2.3 Stokes’ hypothesis . ............................... 36

2.3.1 Isotropic stress tensor ﬁeld . ...................... 36

2.3.2 Sound attenuation . ........................... 38

2.3.3 Acoustic approximation . ........................ 39

2.3.4 Bulk viscosity of liquid noble gases ................... 49

2.4 Widom’s proposition . .............................. 60

2.4.1 Widom region of supercritical carbon dioxide . ............ 61

2.4.2 Diﬀusion of the carbon dioxide - ethanol mixture in the extended

critical region . .............................. 63

Further publications 75

Appendix 77

Bibliography 125

viii

List of Symbols

N- Number of particles

- mathematical sum

O() - order of magnitude

δij - Kronecker delta

lm unit of length

ts unit of time

Ωtm3generic time dependent domain

∂Ωtm2closure of a generic time dependent domain

∂j() m−1spatial derivative with respect to j

∂t() s−1time derivative

∇2() m−2Laplace operator

Bt- material point

P() m position vector of a material point

n- generic orientation vector

x,zm generic position vector

v,wms

−1velocity vector, i.e. v=dt(x)

vims

−1i-th component of the velocity vector

mjkg mass of the j-th particle

Mikg mol−1molar mass of the i-th component

ρmol dm−3molar density

pJm

−3thermodynamic pressure

TK thermodynamic temperature

J s reduced Planck constant, i.e. =1.054571628 ·10−34 Js

kBJK

−1Boltzmann constant, i.e. kB=1.38064852 ·10−23 JK

−1

RJ (mol K)−1universal gas constant, i.e. R=8.3144 J mol−1K−1

ψJ generic wave function

VqJ quantised inter-atomic interaction potential

VcJ classical inter-atomic interaction potential

V2J interaction potential truncated to two-body interactions

Vrep J repulsive contribution to the truncated interaction potential

Vdisp J dispersive contribution to the truncated interaction potential

λth m thermal de Broglie wave length

¯m average distance between particles

λm generic wave length

λi,γ

iW (mK)−1thermal conductivity of the i-th component

rij m distance vector between particles iand j

r0m eﬀective particle diameter

α, β - generic ﬁtting parameters

σm Lennard-Jones length parameter

J Lennard-Jones energy parameter

N- natural numbers

R- real numbers

C- complex numbers

i- imaginary number, i.e. i=√−1

φi- generic conserved quantity

ΘiK temperature of the i-th component

Θ0K ambient air temperature

ΘK sea water freezing point

Θ∞K deep sea temperature

s(t)m ice sheet thickness

s0m initial ice sheet thickness, i.e. s0=s(t=0)

cpJ (mol K)−1speciﬁc heat capacity at constant pressure

cvJ (mol K)−1speciﬁc heat capacity at constant volume

ΔhsJ mol−1speciﬁc enthalpy of solidiﬁcation

ΔhvJ mol−1speciﬁc enthalpy of evaporation

ΔhrJ mol−1speciﬁc enthalpy of combustion

erf() - error function

erfc() - complemented error function

amt

−1

2ice accretion constant

SA- sea water salinity

ymol mol−1vapour phase’s molar composition

yimol mol−1mole fraction of the i-th component in the vapour phase

d(t)m droplet diameter

Km2s−1mass transfer rate

Dm2s−1Fick diﬀusion coeﬃcient

BM- Spalding mass transfer number

νi- stochiometric coeﬃcient of the i-th component

fz,i kg1

2J−1

2z-component of the Maxwell distribution of the i-th component

gJ mol−1speciﬁc Gibbs function

ij s−1strain-rate tensor

Ωij s−1rigid-body rotation tensor

Dij - orthogonal transformation matrix

Iij - unity matrix

τij kg s−1stress tensor ﬁeld

tr( ) - trace of a tensor valued function

ijk - Levi-Civita symbol

ψ,μ Jsm

−3general material constants

μsJsm

−3shear viscosity

μbJsm

−3bulk viscosity

z0m Fresnel parameter

c0ms

−1thermodynamic speed of sound

βK−1inverse temperature

κ- heat capacity ration, i.e. κ=cp/cv

αpK−1thermal expansivity

βTm3J−1isothermal compressibility

ξ- perturbation parameter

uJ mol−1speciﬁc internal energy

sJ (mol K)−1speciﬁc entropy

DvJ s mol−1viscous diﬀusivity

Dhm2s−1thermal diﬀusivity

ωs−1angular frequency, i.e. ω=2πf

F−1s−1inverse Fourier transformation

km−1wave number

τhs thermal relaxation time

τvs viscous relaxation time

τss shear viscous relaxation time

τbs bulk viscous relaxation time

xii

K0molms

−2zero-frequency bulk modulus, i.e. K0=ρc2

Γp- propagation sound mode

Γh- thermal sound mode

αm−1sound wave attenuation coeﬃcient

αλ- sound wave attenuation coeﬃcient per wavelength, i.e. αλ=α·λ

BR,i - Kohlrausch-Williams-Watts functions of the i-th relaxation mode

Ci,β

i- Kohlrausch ﬁtting parameters

xiii

1 Introduction

1.1 Historicity

Three centuries worth of research had been conducted by mathematicians, physicists and

aerodynamicists alike to develop a conclusive "theory of ﬂuids". The journey to explore

the ﬂowing motion of continuous matter had begun in the eighteenth century by a small

group of French and Swiss polymaths. First formal methods and concepts had emerged in

Johann Bernoulli’s "Hydraulics" (1737) [1], Daniel Bernoulli’s "Hydrodynamica" (1738) [2]

and Jean le Rond d’Alembert’s "Traité de l’équilibre et du mouvement des ﬂuides" (1744)

[3]. The fundamental ideas presented in these works were formalised mathematically in

Leonard Euler’s "Principes généraux du mouvement des ﬂuides" (1755) [4] establishing him

as the founder of rational ﬂuid mechanics. Euler had limited his focus on inviscid ﬂuids

and despite facing substantial diﬃculties developing of a satisfactory theory [5], he be-

lieved to have principally reduced ﬂuid mechanics to a mathematical-physical science. The

eighteen’s century research, however, was characterised by a division into the two traits

hydrodynamics and hydraulics. Hydrodynamics belonging to the domain of mathematics

and theoretical physics and hydraulics being based on empirical rules and correlations, as

succinctly outlined in Wilhelm Wien’s "Lehrbuch der Hydrodynamik" (1900) [6]

"Indessen ist gerade die Hydrodynamik am wenigsten abgeschlossen. Hier stim-

men die tatsächlichen Vorgänge mit den theoretischen Folgerungen vielfach

so ungenügend überein, dass die Technik sich für ihre Zwecke eine besondere

Behandlungsweise hydrodynamischer Aufgaben, die meistens den Namen Hy-

draulik führt, zurecht gemacht hat. Bei dieser lassen nun sowohl die Grund-

lage als auch die Schlussfolgerung soviel an strenger Methode zu wünschem

übrig, dass meistens Ergebnisse keinen höheren Wert als den rein empirischer

Formeln mit sehr beschränkter Gültigkeit besitzen."

D’Alembert’s treatise titled "Paradoxe proposé aux Géometres sur la Résistance des Flu-

ides" (1768) [7] asserting that a body moving through an ideal ﬂuid does not experience

a resistive force had become the epitome of this division. Both traits were attempted to

be uniﬁed by incorporating viscous eﬀects into the description, as ﬁrst oﬀered in Claude

Louis Marie Henri Navier’s works "Sur les lois des mouvements des ﬂuides, en ayant égard

á l’adhesion des molecules" (1821) [8] and "Mémoire sur les lois du mouvement des ﬂuides"

(1823) [9]. Navier’s work, however, went largely unnoticed and the governing equations

describing viscous ﬂuid motion were subsequently rediscovered four additional times, i.e.

by Augustin-Louis Cauchy (1823) [10], Siméon Denis Poisson (1829) [11], Adhémar Jean

Claude Barré de Saint-Venant (1837) [12] and George Gabriel Stokes (1845) [13, 14]. Each

author had used his own rationale in the derivation of the equations and judged diﬀerently

the scenarios to which they might apply.

Saint-Venant was the ﬁrst to partition the ﬂuid domain Ωinto various regularly shaped

subdomains, i.e. Ω=Ωiand insinuated each subdomain to propagate with a hydro-

dynamic velocity being the average over the respective particles’ irregular motion. Stokes

had reﬁned this approach by deﬁning the physical quantities at each material point P(z,t),

i.e. at any position zand time twithin the ﬂuid domain Ω. These physical quantities

are thereby derived from the behaviour of all molecules in the respective material point’s

vicinity. This results in two conclusions. First, a physically sound notion of density ρand

hydrodynamic velocity vis introduced and, second since the properties of neighbouring

material points are constructed similarly, there exists a cut set of molecules generating

properties for both material points. Consequently, a causal relation between diﬀerent ma-

terial points arises forming the basis of ﬂuid mechanics as a linearised ﬁeld theory.

The progress achieved during the nineteenth century, which also included the introduc-

tion of the concept of stress and its formulation as a tensor ﬁeld

τij =ωδij +∂ivj+∂jvi,(1)

had tempted British theoreticians to believe that the theory of ﬂuids had been completed

and following the correspondence between William Thomson (Lord Kelvin) and George

Gabriel Stokes it was even believed that this new theory possesses an inherent universality

(cite)

"Now I think hydrodynamics to be the root of all physical science, and is at

present second to none in the beauty of its mathematics".

In fact, hydrodynamics had been conceived as an underlying explanation in one way or

another for light, electricity, magnetism, heat and even gravity [15].

The theoretical advancements in hydrodynamics, however, had just widened the gap to

hydraulics. The crucial era of reconciliation between these two traits began in the ﬁrst

half of the twentieth century and was primarily driven by the "Göttinger Schule" around

Ludwig Prandtl’s "boundary layer theory" [16]

"Ich habe mir nun die Aufgabe gestellt, systematisch die Bewegungsgesetze

einer Flüssigkeit zu durchforschen, deren Reibung als sehr klein angenommen

wird. Die Reibung soll so klein sein, daß sie überall vernachlässigt werden

darf, wo nicht etwa große Geschwindigkeitsunterschiede auftreten oder eine

akkumulierende Wirkung der Reibung stattﬁndet."

In his seminal work, Prandtl had assumed the viscous term, representing the inﬂuence of

stress in the Navier-Stokes equations, to be nonzero in close vicinity to an obstacle and

concluded the resulting ﬂow to divert into a friction dominated transition layer underly-

ing a frictionless free stream. This newly developed theory had further enabled a causal

explanation for the separation of ﬂow from the obstacle’s surface. While having initially

received almost no recognition outside Germany, the boundary layer theory constitutes

the theoretical framework to successfully amalgamate hydrodynamics and hydraulics into

ﬂuid mechanics’ modern formulation.

The transcendental character that has been attributed to ﬂuid mechanics is the result

of a relatively large hydrodynamic scale, which despite being delimited by dominating

quantum eﬀects at the lower and emerging self-gravitation at its higher end encompasses

at least 1029 orders of magnitude, cf. Tab. 1. Theoretical and experimental investigations

on quantum vortices emerging in superﬂuid 3He which had started in the 1950s and con-

tinued in the 1970s with 4He suggested the hydrodynamic scale’s lower boundary to be of

order O(10−6) m. Arrays of quantum vortices, however, are not only conﬁned to spatial

extensions of about one micrometre, but also to low temperatures at which the helium

atoms transition into their superﬂuid state, i.e. below the λ-point at around T≤2.17 K,

by Bose-Einstein condensation. Quantum vortices in superﬂuid helium appear to be one

of the smallest observable hydrodynamic structures, whose spatio-temporal evolution of

momentum, however, is not governed by the Navier-Stokes equations.

Table 1. While ﬂuid mechanics is most certainly not the root of all physical science,

its unique position indeed is reﬂected by the plethora of hydrodynamic

phenomena spanning a multitude of length scales.

Phenomenon typical length scale / m References

Quantum vortex 10−6[17, 18, 19]

Quantum Kelvin wavelength 10−4[20]

Vapour-Liquid interface 10−4[21]

Shock wave thickness 10−4-10

−1[22, 23, 24]

Blasius’ boundary layer thickness 10−3-10

−2[25, 26]

Mach shock cell 10−3-10

1[27, 28]

Rayleigh-Bénard convection cell 10−3-10

6[29, 30]

Taylor-Couette vortex cell 10−2[31]

Görtler vortex 10−2[32]

Av. length of a Tollmien-Schlichting wave 10−2-10

0[33]

Av. width of von Kármán vortex street 10−2-10

4[34, 35]

Vortex Crow instability 101[36]

Rogue waves 101[37, 38]

Planetary boundary layer 102-10

3[39]

Oceanic Rossby wave 105-10

6[40]

North polar vortex 106[41]

Atmospheric Rossby wave 106-10

7[42]

Jupiter’s great red spot 107[43]

Cosmic web 1023 [44]

Contrastingly, the hydrodynamic scale’s upper boundary is much more diﬃcult to be de-

termined precisely. The self-gravitation caused by vast amounts of matter gives rise to

larger clusters that eventually evolve into galaxies spanning various Megaparsec O(1025)

m and constitute the largest observed hydrodynamic structures thus far.

Experimental observations of supernova luminosity [45, 46], cosmic microwave back-

ground peaks [47], baryon acoustic oscillations [48] and large scale structures [49] indicate

not only the existence of dark matter, but also its abundance in the universe. More specif-

ically, the universe’s present state is dominated by the so-called "dark sector", which is an

umbrella term for dark matter and dark energy. Various explanations for the universe’s

acceleration have been proposed, ranging from the infamous introduction of a cosmologi-

cal constant, via modiﬁed matter content models [50] better known as "quintessence" and

descriptions of modiﬁed gravity [51, 52, 53] to statistical distribution of matter [54, 55, 56],

yet none of these models is able to produce satisfactory solutions, but is stricken with ﬁne-

tuning problems or arising instabilities. In contrast, the universe’s present state as well

as its current acceleration is successfully and elegantly described by the so-called viscous

cosmological model, i.e. by a homogeneous, isotropic and ﬂat universe ﬁlled with a bulk

viscous ﬂuid [57, 58]. The paradigm of a bulk viscous cosmological ﬂuid does additionally

oﬀer a physically sound uniﬁcation ansatz for dark matter and dark energy [59, 60].

1.2 Small scale hydrodynamics

The hydrodynamic scale was long considered to be unattainable by atomistic simulations,

since Navier-Stokes hydrodynamics were formalised around length scales of order O(100)

m. The smallest scale at which the hydrodynamic formalism was applicable remained

unclear until advances in experimental methods had speciﬁed this so-called hydrodynamic

limit to be around 1 nanometre [61], a limit that emerges naturally as the lower boundary

below which surface eﬀects become dominant over volumetric eﬀects. The robustness of

the Navier-Stokes equations at such small scales came as a surprise given the gradients

in microscopic scenarios greatly exceed those encountered in the respective macroscopic

scenarios by orders of magnitude and astonishingly even hydrodynamic structures were

observed to emerge at the nanoscale, cf. Tab. 2. Atomistic methods, such as molecular

Table 2. Overview of hydrodynamic phenomena that have been observed at the

microscale.

Phenomenon length scale / nm Reference

Array of Taylor-Couette vortices 10 [62]

Evaporation of a liquid ﬁlm 14 [63]

Stefan problem of a multicomponent liquid ﬁlm 15 [64]

Single Rayleigh-Bénard convection cell 32 [65]

Fundamental λof a Richtmyer-Meshkov instability 80 [66]

von Kármán vortex street 100 [67]

Rayleigh breakup length of a nanojet 170 [68]

Riemann shock tube scenario 200 [69]

dynamics simulations, represent a framework to describe a ﬂuid at a certain level of detail.

The most basic description of an ensemble containing Nparticles is given by the non-

relativistic time-dependent Schrödinger equation

i∂tψ(x,t)+



j=1

2mj∇2+Vqψ(x,t)=0,(2)

with the reduced Planck constant and an appropriate inter-atomic interaction poten-

tial Vq. In many cases, however, this full generality is unnecessary, since each particle’s

quantum-mechanical wave packet, as measured by the thermal de Broglie wave length λth,

is small compared to the average distance between the particles λ

¯[70]

B=λth

¯=3

√n2π2

mkBT,B∈R>0,(3)

with number density nand Boltzmann’s constant kB. Consequently, quantum eﬀects

become negligible after passing the classical limit [71], which is synonymous with B1,

and particle positions and velocities that are random quantities under the Schrödinger

equation can be replaced by their respective mean values yielding classical Newtonian

mechanics

Table 3. The ratio of the thermal de Broglie wave length to the average distance be-

tween particles Bwas found to be small compared to unity. Consequently,

each presented scenario could be investigated by classical methods.

Scenario λth /nm λ

¯/nm L/nm B/-

Stefan’s problem 0.038 - 0.059 0.583 - 0.329 30-15 0.066 - 0.179

Stokes’s hypothesis 0.024 - 0.026 0.403 - 0.350 6.447 - 5.596 0.060 - 0.074

Widom’s conjecture 0.020 - 0.021 0.652 - 0.455 9.403 - 6.561 0.031 - 0.047

dttxj(t)+1

mj∇Vc(x1, ..., xN)=0 ,∀j∈[1,N],(4)

with xj(t)being the j-th particle position at time t. The j-th particle velocity is conse-

quently determined by the gradient of a potential function Vc(x1, ..., xN). Determining

the j-th particle potential energy within the ensemble constitutes a N-body problem,

for which no closed form solution exists. However, the total potential energy Vcof the

ensemble can be expressed as the sum of increasing n-body contributions

Vc=



i<j

V2(rij)+



i<j<k

V3(rij,r

ik,r

jk,φ

ij,φ

ik,φ

jk)+... , (5)

with the two-body term V2being a function of the particle’s distance rij =xj−xiand

rij =|rij|being its vector norm. The three-body term V3is additionally a function of

the angles φij between adjacent particles. While for most systems in either liquid or

vapour phase, four-body interactions can safely be neglected, the three-body contribution

at moderately dense states and in suﬃcient distance from the substance’s critical point is

also vanishingly small. It is consequently suﬃcient to focus on the two-body interaction

energy V2(rij), which partitions into an uncorrelated and a correlated part

V2(rij)=Vrep. +Vdisp. ,(6)

with the former being alternatively interpreted as repulsive and the latter as dispersive

contribution. While the repulsive contribution decays rapidly, an attractive force is present

at large distance even between electro statically neutral particles. Temporary charge dis-

placements within a particle’s outer electron shell occur randomly and propagate through

the medium by inducing other temporal dipoles in the surrounding particles’ outer elec-

tron shells. Although this process is highly dynamic, it can reasonably be approximated

by a static potential function.

Building on the classical lattice theory of solids [72], John Lennard-Jones [73] had suc-

cessfully explained the molecular origin of the shear viscosity’s temperature variation in

gases by assuming that both parts in (6) follow an inverse power law of the form

V2(r)=αr−n−βr−m,α,β∈R>0,n>m, (7)

with r=rijr−1

0being the particles’ distance reduced with an eﬀective particle diameter r0.

The introduction of this eﬀective diameter is thereby a progression of the classical kinetic

gas theory that is based on hard spheres, by modelling a rudimentary outer electron shell

that allows for a marginal penetration during particle interactions. While Lennard-Jones

had originally determined the repulsive contribution’s exponent to be m=3on the basis

of experimentally available shear viscosity data for argon, he later reﬁned both exponents

by ﬁtting (7) to argon’s second virial coeﬃcient data [74]

V2(r)=αr−9−βr−5.(8)

Due to a paradigm shift from the kinetic theory of gases towards quantum mechanics, both

contributions to the potential function (6) had been reﬁned. A second-order approximation

of the perturbation energy due to the particles’ interaction leads to an interaction matrix

that can be developed into an inverse power function of particle distance [75] under the

Born-Oppenheimer approximation. This approach is strictly speaking valid for spherical,

monatomic particles only [76]

Vdisp. =−β1r−6+β2r−8+β3r−10 +...,β

i∈R>0,(9)

with dispersion coeﬃcients βi. The particle distance r=rijr−1

mis thereby reduced by the

position of the potential minimum rm, in contrast to the reduction used in the context of

the kinetic gas theory (7). Further experimental work on the lattice constant of crystals

indicated that the repulsive contribution must follow an exponential function rather than

an inverse power law, yielding the so-called Born-Mayer potential [77]

Vrep. =Aexp−αr,(10)

with ﬁtting parameters A, α. The combination of a truncated version of (9) with (10) is

known in the literature as the Hartree-Fock plus dispersion (HFD) potential

V2(r)=Aexp−αr−β1r−6+β2r−8+β3r−10,(11)

for which various alterations [78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93]

exist and the most prominent are given in Tab. 4.

Table 4. Most prominent variations of the HFD potential, each suited for diﬀerent

substances and thermodynamic states. The particle distance rij is reduced

with the position of the potential minimum rmvia r=rijr−1

mand the

exponentially decaying weighting functions fi(r).

Name Composed two-body potential energy V2

Buckingham [79] Aexp−αr−βr−6

Ahlrichs et al. [83] Aexp−αr−β1r−6+β1r−8+β3r−10·f(r)

Aziz-Chen [84] Aexp−α1r+α2r2−β1r−6+β2r−8+β3r−10·f(r)

Gilbert et al. [92] Arγexp−αr−β1r−6+β2r−8+β3r−10·f(r)

Tang-Toennies [93] Aexp−αr−f1β1r−6+f2β2r−8+f3β3r−10

The inﬂuence of three-body interactions V3on the thermophysical properties, primarily

vapour-liquid equilibrium densities and pressures, has been studied extensively. Discrep-

ancies between results obtained with two-body and three-body potentials [94, 95] were

found to be relevant at densities above three times the substance’s critical density and

ﬁve times its critical temperature [96], as well as in the vicinity of the critical point. For

noble gases in the liquid state at moderate density outside the extended critical region, the

Lennard-Jones potential produces most relevant thermophysical properties with suﬃcient

accuracy [97]. In contrast to its original form (7), the Lennard-Jones potential is most

commonly used in its computational friendly form using London’s dispersion term to ﬁrst

order, i.e. α=β=4,n=12and m=6

V2(rij)=4r−12 −r−6,r=rijr−1

0<r

c∀i, j ∈[1,N],i=j. (12)

Since the dispersive term r−6decays rapidly with distance r, its contribution can be

neglected after a cut-oﬀ radius, i.e. all particles outside a ﬁctitious circle with r≥rcare

excluded from any interaction with the j-th particle and the function’s gradient yields the

force fij between the j-th and i-th particle

fij =48

r0r−14 −1

2r−8r,∀i, j ∈[1,N],i=j. (13)

The Lennard-Jones potential in the form (12), together with its resulting inter-particle

force (13) has been used for all atomistic simulations within this thesis.













    













 

Figure 1. Lennard-Jones 12,6-potential as a function of particle distance r(solid

line) and its two contributions, i.e. short-rang repulsion (dashed-line) and

long-rang dispersion (dashed-dotted line). The zero crossing of the inter-

particle force that arises from the potential function via (13) (solid grey

lines) is marking the transition from dispersive to repulsive interaction.

1.3 Microscopic picture of a ﬂuid

The conﬁguration of electrons primarily within a particle’s outmost shell is responsible

for the repulsive and dispersive interactions between diﬀerent nonpolar particles and the

sum nof all Nparticle interactions eventually leads to the macroscopically perceived

homogeneity and isotropy of the respective ﬂuid

2n=N·(N−1) ,N∈N≥0.(14)

Due to these relatively short-termed interactions, all particles involved exchange potential

for kinetic energy and vice versa. The ﬂuid domain Ωcan be partitioned into ﬁnitely

many arbitrarily shaped subdomains containing suﬃciently many particles to represent

their own thermodynamic systems.

The subdomains have historically been termed material bodies Btand are used to

construct the material points P(r,t), following the concept ﬁrst introduced by Maxwell

and then prominently advocated by Stokes. Since each material body constitutes its

thermodynamic system, the macroscopic properties, such as density, pressure, viscosity

etc., are subsequently derived by averaging over all particles within Bt. These macroscopic

properties are then attributed to a spatial position within the material body, the so-called

material point P(r,t), where each material point’s macroscopic properties are generated

from the individual particles occupying its surroundings. The properties attributed to

adjacent material points Pi,Pjresult from diﬀerent Bi,Bj, however, there exists a cut set

of particles that belong to both material bodies causing the macroscopic properties to be

causally linked and thus allowing to construct a physically sound linearised ﬁeld theory.

Stokes had initially applied this concept to the hydrodynamic velocity, i.e. the individual

particle velocities at any point in the ﬂuid are not simply grouped symmetrically around a

state of rest, but "cluster" instead around a velocity that is identiﬁed with the macroscopic

velocity of the ﬂuid at the respective material point P(r,t)[13]

"In the ﬁrst place, the expression the velocity of a ﬂuid at any particular point

will require some notice. If we suppose a ﬂuid to be made up of ultimate

molecules, it is easy to see that these molecules must, in general, move along

one another in an irregular manner [...]. Let Pbe any material point in the

ﬂuid, and consider the instantaneous motion of a very small element Bof the

ﬂuid about P. This motion is compounded of a motion of translation, the same

as that of P, and of the motion of the motion of several points of Brelative to

P".

Table 5. The number of maximum interactions nat a given time is a function of

the ensemble’s particle number N, cf. equation (14). The dispersive term

in the pair potential function decays rapidly with distance resulting in a

sensible cut-oﬀ radius beyond which all interactions are considered on the

basis of a mean-ﬁeld approach and consequently the number of actually

computed interactions is substantially lower than n.

N32 108 256 500 864 1372 2048 2916 4000

n496 5778 32640 124750 372816 940506 2096128 4250070 7998000

Mathematically speaking, the particle velocities are "drifting Maxwellian" distributed with

the drift being identiﬁed as hydrodynamic velocity.

These material bodies are conceptually similar to the particle ensembles used in atom-

istic simulations. The ensemble size, however, varies signiﬁcantly for diﬀerent macroscopic

properties, i.e. the accurate determination of static observables such as density, pressure

and temperature by atomistic simulations necessitates ensemble sizes of about 32 to 256

particles depending on the molecule’s complexity. Contrastingly, according to the Kubo

formalism [98, 99, 100, 101], transport properties are time-autocorrelations of Green func-

tions requiring ensembles sizes up to 4000 to yield reasonable statistics. Particle ensembles

of this size are associated with a multitude of particle interactions, cf. Tab. 5, and thus

require substantial computational resources.

1.4 Continuum framework and Newtonian ﬂuids

Partitioning the ﬂuid domain Ωinto various material bodies Biand subsequently into

material points Piallows to geometrise the ﬂow ﬁeld. This geometrisation in addition with

a local thermal equilibrium hypothesis [102] eventually leads to evolution equations for

mass, momentum, energy and entropy density. Evolution equations have a characteristic

form, i.e. the time evolution of a generic conserved quantity φiis balanced via ﬂux terms

across the joint boundary of adjacent material bodies

Deﬁnition 1.1 (Evolution equation).

dtBt

dV ρi(xn,t)φi(xn,t)=∂Bt

dAJj(xn,t)

 

ﬂux term

·nj+Bt

dV ψ

ˆi(xn,t)

 

supply term

+φ

ˆi(xn,t)

 

source term.

(15)

While the mass-, momentum- and energy density ﬁelds exchange information exclusively

via the supply term, e.g. a decrease in kinetic energy gives rise to either potential or

internal energy and vice versa, a genuine source term only exists for entropy density, i.e.

entropy is the sole quantity that can be generated ex nihilo. The material derivative on

the left hand site of equation (15) can be resolved by Reynold’s transport theorem

Deﬁnition 1.2 (Reynold’s transport theorem).

dtBt

dV ρiφi(xn,t)=Bt

dV ∂tρiφi(xn,t)+



j=1

∂jρivj,i φi(xn,t),(16)

given in d-dimensions. Another central element of the continuum framework is the tran-

sition from a pure substance to a mixture, which is realised by superpositioning the com-

ponent’s extensive properties at each material point [103]

ρ(x,t)=



i=1

ρi(x,t),(17)

as given here exemplary for the ﬂuid’s density. Equations (15) to (17) form the basis of

all hydrodynamic equations used throughout this thesis

2 Original publications

2.1 List of original publications

Stefan’s problem

R.S. Chatwell, M. Heinen and J. Vrabec, "Diﬀusion limited evaporation of a binary liquid

ﬁlm", J. Heat and Mass Transf.,132, 1296-1305 (2019)

DOI: https://doi.org/10.1016/j.ijheatmasstransfer.2018.12.030

Stokes’ Hypothesis

R.S. Chatwell and J. Vrabec, "Bulk viscosity of liquid noble gases", J. Chem. Phys.,

152, 094503 (2020)

DOI: https://doi.org/10.1063/1.5142364

Widom’s proposition

R.S. Chatwell, G. Guevara-Carrion, Y. Gaponenko, V. Shevtsova and J. Vrabec, "Dif-

fusion of the carbon dioxideethanol mixture in the extended critical region", Phys. Chem.

Chem. Phys.,23, 3106 (2021)

DOI: https://doi.org/10.1039/D0CP04985A

2.2 Stefan’s problem

2.2.1 Arctic ice accretion

An initial value problem in which regions of diﬀerent aggregate states are separated by

a freely moving boundary is commonly referred to as a Stefan problem. The ﬁrst free

boundary problem discussed in the literature was Lamé and Clapeyron’s work on solidi-

ﬁcation of liquid droplets due to freezing [104]. A proper deﬁnition, however, ﬁrst arose

from Stefan’s 1889 publications [105, 106, 107, 108], among which his predictions of Arctic

sea ice variations during the seasons "Über die Theorie der Eisbildung, insbesondere über

die Eisbildung im Polarmeere" had drawn the most attention and was subsequently repub-

lished [109]. Stefan had based his investigations on experimental data that were gathered

during an Arctic mission to ﬁnd a sailable North-West passage, cf. Tab. 6. Stefan was able

to predict the ice sheet thickness s(t)as function of a driving temperature gradient, i.e.

the diﬀerence between the sea water freezing point Θand the ambient air temperature

Θ0. The set of partial diﬀerential equations governing the problem is readily derived from

the principle of energy conservation. The problem is most sensibly deﬁned piecewise, i.e.

separately for each aggregate state, with the additional assumption of Fourier type heat

conduction

∂t(Θ1)−k1∂zz (Θ1)=0 ,0≤z≤s(t),

∂t(Θ2)−k2∂zz (Θ2)=0 ,z≥s(t),(18)

with ki=γi/(ρcp,i)being the respective phase’s thermal diﬀusivity. In the following, any

change of volume during phase transition will be neglected, which is synonymous with ρ

= const., and reminiscent of Stefan’s original work.

Table 6. Original measurements of mean ambient air temperature Θ0and ice sheet

thickness s, which have been conducted at the Gulf of Boothia during a

mission intended to ﬁnd a sailable North-West passage across the Arctic

sea between the years 1829 and 1832 [110].

Date Mean air temperature Θ0/◦C ice sheet thickness s/m

31.10.1831 -12.83 0.48

30.11.1831 -24.44 0.84

31.12.1831 -31.11 1.22

03.02.1832 -33.11 1.52

31.03.1832 -36.11 2.13

The solution of this system of partial diﬀerential equations has to satisfy certain bound-

ary conditions. The essential diﬃculty of Stefan problems is that parts of the domain’s

boundary are not known a priori, yet have to be determined as part of the solution. The

interface between ambient air and ice sheet z=0represents the stationary boundary ∂Ω0

on which the average air temperature Θ0constitutes a physically sound Dirichlet-type

condition. Additionally, the sea water’s temperature Θ2is asymptotically constraint to

what is commonly referred to as deep sea temperature Θ∞

Θ1(z,t)=Θ

0,z=0,

Θ2(z,t)=Θ

∞,z→∞.(19)

The interface between ice and sea water z=s(t)constitutes the free boundary ∂Ωton

which the water’s freezing point Θacts as Dirichlet-type condition and the rate of ice

accretion as the so-called Stefan condition

Θ1(z,t)=Θ

2(z,t)=Θ



−λ1∂z(Θ1)+λ2(Θ2)=ρΔhsdt(s)at z=s(t),t>0,(20)

with Δhsbeing the enthalpy of solidiﬁcation. In order to solve this set of partial diﬀerential

equations, appropriate initial conditions have to be chosen

Θ1(0,t)=Θ

0,z∈[0,s

Θ2(z,t)=Θ

∞,z∈(s0,∞)

s(t)=s0

⎫

⎪

⎬

⎪

⎭

for t=0.(21)

While the ice sheet initially occupies the space z∈[0,s

0]at ambient air temperature Θ0,

the semi-half space z∈(s0,∞)is ﬁlled with sea water at deep sea conditions Θ∞. The

solution of equations (18) with boundary (19) - (20) and initial conditions (21) is given

for the respective domain by either the error function (erf) or the complemented error

function (erfc)

Θ1=Θ

0−Θ0−Θ

erf α

2√k1·erf z−s0

2√k1t,z∈[s0,s(t)]

Θ2=Θ

∞−Θ∞−Θ

erfc α

2√k2·erfc z−s0

2√k2t,z∈[s(t),∞)

⎫

⎪

⎬

⎪

⎭

for t>0.(22)

with the accretion constant abeing an implicit function of all thermophysical properties

[111, 112, 113, 114]. Both temperature ﬁelds Θiare coupled at the freely moving boundary,

whose position is proportional to the square root of time s(t)=s0+a√t.

The accuracy of equation (22) will become readily apparent when applied to the icing

of the Arctic sea, as reported in Tab. 6. Specifying accurate thermophysical properties

at Arctic conditions, however, is challenging since each phase’s thermodynamic behaviour

depends strongly on its salinity.

Table 7. While each phase’s thermodynamic properties are highly dependent on

salinity SA, commonly given in parts per thousand (ppt), the chosen values

accurately represent the Arctic conditions. For the anticipated tempera-

ture interval, an averaged isobaric heat capacity has been determined for

the ice sheet.

SA/ ppt cp/kJkg

−1K−1γ/Wm

−1K−1k/10

−7m2s−1

sea ice 8 56.944 [115] 2.03 0.360

sea water 35 3.996 [116] 0.56 [117] 1.416





























Figure 2. Succession of temperature variation across the ice sheet (red) and sea wa-

ter (blue) at diﬀerent instances of time. The ice block that occupies the

space z∈[0,s

0]is kept at uniform averaged air temperature Θ

¯0and is

growing by cooling down the water from deep sea Θ∞to freezing con-

ditions Θ. After 151 days, the ice sheet is predicted to have reached a

thickness of s=2.29 m, which is remarkably close to the observed value

of sobs =2.13 m.

In addition to the properties disclosed in Tab. 7, the eﬀective density ρ= 990 kg m−3

[118], sea water freezing point Θ= 271.35 K [118], deep sea temperature Θ∞= 272.25

K (cite) and enthalpy of solidiﬁcation Δhs= 256.48 kJ kg−1[115] were chosen.

The observed ice sheet thickness as reported in Tab. (6) is readily reproduced, given

the ambient air temperature Θ

¯0= 243.66 K is averaged over the period between October

1831 and March 1832. Both temperature ﬁelds and the respective ice thickness s(t)are

graphically depicted in Fig. 2. The Stefan condition (20) at the phase boundary lies the

heart of this remarkably accurate prediction, whose quality can even be increased by more

reﬁned boundary conditions, i.e. by specifying a more physically sound temperature proﬁle

across the initial ice sheet at z∈[0,s

0]and incorporating the density jump associated with

the phase transition.

Stefan, in contrast, solved a simpliﬁed version of equations (18), by assuming a linear

temperature variation across the ice sheet [109]

"Zu diesem Gesetze und auch zu einem angenäherten Werthe der in ihm en-

thaltenen Constanten führt aber auch eine ganz elementare Betrachtung. Es

genügt, anzunehmen, dass die Kälte innerhalb der Eisdecke von dem Werthe,

den sie in der Oberﬂäche hat, bis zum Gefrierpunkte an der unteren Begren-

zungsebene des Eises nach dem Gesetze einer geraden Linie abfalle".

Despite this simpliﬁcation, Stefan in full accordance with the exact solution above also

concluded the ice’s thickness to be proportional to the square root of time. However,

in his case, the accretion constant arises as explicit function of the ice’s thermophysical

properties

alin. =2k1cp,1

Δhs

(Θ−Θ0).(23)

The linearised accretion constant alin. =6.7·10−4mt−1/2is noticeably close to the exact

model’s a=5.0·10−4mt−1/2that was used above.

2.2.2 Droplet evaporation

Over the years, a multitude of free boundary problems have emerged of which spray

atomisation, i.e. the complete disintegration of a liquid jet into ﬁnely dispersed droplets,

constitutes one of the most prominent. A coaxial jet that is injected into a quiescent atmo-

sphere exhibits diﬀerent disintegration phenomenologies, ranging from dripping, breakup,

wavy or ﬂipping jets to spray formation, depending on the scenario’s hydrodynamic pa-

rameters, thermodynamic states and geometry. A droplet’s full atomisation is either

achieved through high pressure injection or by the succession of diﬀerent stages. An ini-

tially thin ﬁlm can disintegrate consecutively into ligaments, drops and eventually into

ﬁnely dispersed droplets. Each of these droplets either vaporises or evaporates into the

atmosphere, depending on the scenario’s conditions. While vaporisation is a volume phe-

nomenon in which small vapour bubbles, that constantly form within the droplet, have a

suﬃciently high pressure to sustain themselves and grow continuously (nucleate boiling).

Evaporation, in contrast, is a surface phenomenon that occurs substantially below boiling

conditions and consequently on much larger time scales.

Under natural conditions, the phenomenon of mass transfer between a droplet and its

surrounding atmosphere is extremely complex. Due to the non-stationary nature of the

process, the mass transfer rate is strictly speaking a function of time. Additionally, the

droplet moves irregularly in the surrounding atmosphere with non-uniform temperature

and emerging internal circulation ﬂuxes. A holistic theory of such processes is necessarily

very complex and consequently simpliﬁcations have to be made.

Droplet evaporation is mathematically less sophisticated than vaporisation and techni-

cally speaking an inverse Stefan problem, i.e. the domain under consideration regresses

over time. The uniform temperature and pressure ﬁelds require considering mass conser-

vation only, which is most sensibly used in its integral form

dtΩt

dV ρl(r, t)+∂Ωt

dAjc·n=0.(24)

While the convective molar ﬂux jcis decomposed into advective and diﬀusive contribu-

tions, the latter is readily related to Fick’s constitutive law. The solution of equation (24)

has to comply with the Stefan condition at the domain’s boundary ∂Ωt, i.e. the droplet

surface, and additionally with the Dirichlet condition in suﬃcient distance from the droplet

−Ddrye

ya,s =dts,r=s(t)

ya,s

=ye,∞

aa,s

,r→∞ ⎫

⎪

⎬

⎪

⎭

for t>0,(25)

with Dbeing the Fick diﬀusion coeﬃcient and ye,y

athe evaporate’s and atmosphere’s

mole fractions respectively, which are necessarily constraint ye+ya=1. The convective

molar ﬂux that is complying with both boundary conditions (25) is given by [119]

jc=2Dρv

dln1+ye,s −ye,∞

ya,s .(26)

Inserting this convective molar ﬂux (26) into the conservation equation (24), with the

additional assumption of constant liquid density ρl= const., yields

6ρldtd3(t)+πd2(t)jc=0.(27)

Integrating the equation above with respect to time and observing the initial condition

d=d0at t=0, yields an equation of the type

d2(t)=d2

0−Kt , (28)

with the mass transfer rate Kbeing an exclusive function of thermodynamic properties

K=8

Dρv

ρl

ln1+ye,s −ye,∞

ya,s ,B

M=ye,s −ye,∞

ya,s

.(29)

BMconstitutes the so-called Spalding mass transfer number [120] and Krepresents the

accretion constant’s analogue of Stefan’s original work.

As evident from equation (29), the mass transfer rate is determined by the diﬀerence

between the evaporate’s mole fraction at the surface and in suﬃcient distance from the

droplet, necessitating to determine the vapour phase’s molar composition at the droplet’s

surface ys=(ye,s,y

a,s). Having originated from Maxwell [121], it is widespread to pur-

port that both phases remain in vapour-liquid equilibrium (VLE) throughout the entire

process, an assumption that is only true if the droplet’s radius is signiﬁcantly larger than

the mean free path of the vapour molecules. A central obstacle to verify such an assertion

about the phases’ molar composition lies in the inability to suﬃciently resolve the inter-

face’s vicinity experimentally. Over the years, a variety of techniques has been developed

to study liquid vaporisation into quiescent atmospheres, each of which comes with diﬀerent

eﬀorts and shortcomings. The most primitive device is a ﬁbre support that is mechanically

intrusive and thus enables parasitic heat ﬂuxes to and primarily from the droplet [122].

In an eﬀort to minimise such eﬀects, a cross-ﬁbre support was introduced to successfully

reduce the apparatus’ inﬂuence on the droplet [123]. After realising that mechanical de-

vices necessarily have an unwanted inﬂuence on the object, alternative methods became

inevitable. Acoustic levitation allows for a less-intrusive measurement, yet acoustically

stimulated toroidal vortices prohibit static boundary conditions [124]. A more favourable

methodology is optical levitation in combination with Lorenz-Mie scattering [125], where

the droplet is balanced on a focused laser beam, allowing stable conditions with mea-

surement times up to 20 min. [126]. All methods above are additionally inﬂuenced by

buoyancy eﬀects, which ,however, can be minimised under microgravity conditions.

The d-squared law (28) has been validated experimentally over a substantial range of

thermodynamic states as well as for a variety of non-sooting monocomponent droplets

through a wider spectrum of chemical complexity, ranging from n-heptane [127, 128,

129, 130, 131, 132] over biofuels [133] to petroleum based fuels [134] and even to quasi-

Table 8. The d-squared law has been validated for a variety of non-sooting mono-

component and quasi-multicomponent droplets for a substantial range of

thermodynamic states und various atmospheric conditions using diﬀerent

experimental methods.

substance T/K p/MPa d0/mm K/mm

2s−1atmosphere

n-heptane 298 - 371 0.04 - 0.1 0.50 - 1.20 0.69 - 0.79 air, O2/He

bio diesel (RME) 473 - 973 0.1 0.70 - 1.45 0.02 - 1.09 air

kerosene 774 - 1274 0.10 - 3.0 1.52 0.32 - 1.96 N2

monocomponent droplets where the species share suﬃciently similar physico-chemical

properties, such as kerosene [135], cf. Tab. 8.

The combustion of n-heptane droplets in an oxygen-rich atmosphere is a representative

example of a physically complex system that obeys the d-squared law in the form (28).

Combustion is a multistage process, i.e. the fuel initially vaporises and accumulates in

the vicinity of the droplet, with a mass transfer rate according to equation (29). Once a

threshold saturation is reached, the combustion process initiates, which is characterised

by an exothermic reaction of fuel with the atmosphere’s oxygen, that is provoking a stark

temperature increase of the vapour surrounding the droplet. The droplet temperature,

however, is assumed to be at boiling condition during the entire process, i.e. at Ts= 371.6

K. The combustion regime is characterised by a substantially larger mass transfer rate, i.e.

(31) and is followed in the case of incomplete combustion, by an extinction phase that is

characterised by the residual droplet regressing according to equation (28) again, cf. Fig.

Analogous to the previous problem, specifying the necessary thermophysical properties

is posing a formidable challenge. While the equation of state for n-heptane is limited to

temperatures T≤600 K [136], the isobaric heat capacity in this high temperature scenario

has to be determined by so-called NASA polynomials [137]

R=a1+a2T+a3T2+a4T3+a5T4,(30)

with the universal gas constant Rand coeﬃcients aigiven in Tab. 9. The vapour phase’s

thermal conductivity is understood as sum of the respective conductivities of fuel γeand

ambient air γaweighted by the fuel’s mole fraction in the vapour phase, i.e. γ=yeγe+

(1−ye)γawith γe=0.3180 W/(m K)−1[138], γa=0.0805 W/(m K)−1and ye= 0.13 mol

mol−1. In many combustion scenarios the thermal and concentration boundary layers are

of the same order, as represented by a Lewis number of unity, i.e. Dρv=γc−1

p, yielding

Table 9. NASA polynomial coeﬃcients aiof the vapour phase’s isobaric heat capac-

ity cpfor higher temperatures [137].

a1/- a2/K

−1a3/K

−2a4/K

−3a5/K

−4

cp(T<1000 K) 3.015 0.055 2.181 ·10−5-5.423 ·10−82.081 ·10−11

cp(T>1000 K) 22.819 0.033 -1.112 ·10−51.713 ·10−9-9.621 ·10−14











        























 

Figure 3. Multistage combustion process of a n-heptane droplet in an oxygen-rich

atmosphere under microgravity conditions. The open symbols denote ex-

perimental data from the literature [129]. While the dashed lines indicate

the initial (I) and ﬁnal stage (III) of the process that follows the mass

transfer rate according to equation (29), the combustion regime (II) (solid

line) is characterised by high mass transfer according to equation (31).

the combustion regime’s mass transfer Krate

K=8 λ

cpρl

ln1+ cp

ΔhvT∞−Ts+Δhr

ΔhvνeMe

νOMOyO,∞,(31)

with properties given in Tab. 10 and additionally with liquid phase density ρl=6.1224 dm3

mol−1[136], enthalpy of evaporation Δhv=31.07 kJ mol−1[129], enthalpy of combustion

Δhr= 4452.74 kJ mol−1[129], temperature and oxygen mole fraction in suﬃcient distance

from the droplet T∞= 298 K, yO,∞=0.232 mol mol−1, stochiometric coeﬃcients νe=1,

νO=11as well as molar masses Me= 100.21 g mol−1and MO=31.99 g mol−1.

Contrasting the mass transfer rate of regimes (I) and (III) K=0.04848 mm2s−1with

that for regime (II) K=0.7748 mm2s−1clearly indicates the rapid vaporisation during

combustion as a consequence of the heat transfer from the surrounding to the droplet.

Table 10. Thermodynamic properties used to determine the mass transfer rates (29)

and (31).

phase T/K λ/W(mK)

−1cp/ J (mol K)−1ye/ mol mol−1

liquid 371.6 [136] 0.1009 [136] 256.368 [136] 1.00

vapour 1338 [129] 0.1114 417.799 0.13

2.2.3 Film evaporation

The prominent examples listed above illustrate the pervasion of free boundary problems

in ﬂuid mechanics. The Stefan condition, i.e. equations (20) and (25), lies at the heart

of each scenario and the quality of its solution particularly depends on the accuracy of

the thermophysical properties involved. Atomistic simulations, in contrast to contem-

porary experimental techniques, provide a framework to determine the thermodynamic

state, transport coeﬃcients, the thermodynamic non-idealities as well as the spatial and

temporal resolution of each phase’s molar composition unconstrainedly.

In the following original publication, an inverse Stefan problem is investigated, by com-

paring the predictions of the classical hydrodynamic description with large scale molecular

dynamics (MD) simulations. More speciﬁcally, a two-component liquid ﬁlm is considered

that continuously forfeits matter into an adjacent inert gas dominated vapour phase. Dur-

ing evaporation, the ﬁlm does not only regress, but also changes its molar composition, cf.

Fig 4. Evaporation in this context is understood as phase transition below the mixture’s

respective boiling conditions that manifests at the ﬁlm’s surface and is driven by a spa-

tial chemical potential gradient. The latter is established by substituting all evaporated

particles entering a control volume, representing the conditions at suﬃcient distance from

the ﬁlm, with inert gas particles.







   







   



Figure 4. Snapshot of the investigated Stefan problem taken from the performed

large scale MD simulation at two successive instants of time t2>t

1. The

considered two-component ﬁlm (I), z∈[0,z

l], regresses by continuously

forfeiting matter into an adjacent vapour phase (III), z∈[zs,z

δ]. While

all phase transitions are moderated by an interphase (II), z∈(zl,z

s), the

evaporation dynamics were initiated and maintained by a spatial chemical

potential gradient that was realised by substituting all molecules arriving

at a control volume (IV) with inert gas particles. The control volume’s

rate of expansion was matched with the ﬁlm’s regression rate in order to

maintain a constant vapour phase’s thickness over time, i.e. δ=δ(t).

2.2.4 Interphase dynamics

The transfer of matter between a liquid and its adjacent vapour phase classiﬁes as a

ﬁrst-order transition [139], which in the present case was considered evaporatively, i.e.

substantially below the mixtures boiling conditions. As mentioned before, evaporation

is a surface phenomenon that is initiated and maintained by spatial variations in molar

composition, or technically speaking by a chemical potential gradient. The latter is rela-

tively weak and the particles’ random thermal motion is hardly perturbed, i.e. the system

remains in local equilibrium. On the basis of simulations with small particle numbers,

various authors have erroneously concluded such processes to depart into non-equilibrium,

however, the present work’s large scale simulations with particle numbers N=2·105fa-

cilitate much better statistics and clearly indicate Maxwell distributed particle velocities.

The region coupling both bulk phases is commonly referred to as interphase [140] and

is characterised by a steep yet continuous variation of density from liquid to vapour. This

interphase moderates all phase transitions by imposing a selection criterion on each par-

ticle. Those at the high-velocity tail of the Maxwell distribution are more likely to either

evaporate or condense (cite). Phase transitions under thermodynamic equilibrium are

characterised by drift free Maxwellian distributed particle velocities and the interphase

consequently selects in both ways equally.

In the performed large scale MD simulations the evaporation dynamics were initiated

by introducing a control volume at suﬃcient distance from the ﬁlm’s surface. This sub-

stitution of particles gives rise to an hoc disruption starting from the control volume and

propagating through the vapour phase to introduce a drift uito the particles’ otherwise

random thermal motion wi

vi=ui+wi,i∈[1,3] ,(32)

and the total particle velocity viadopts a so-called drifting Maxwellian distribution (cite)

fz,i =mi

2πkBTexp −mi(vz−uz)2

2kBT.(33)

The perturbation is subsequently attenuated at the interphase preventing a further per-

meation into the liquid ﬁlm. The drift reduces the number of particles likely to condense

into the ﬁlm and simultaneously increases those to evaporate into the vapour, a fortiori

the greater the drift. This established asymmetry causes the net transfer of mass resulting

in the ﬁlm’s regression.

The presence of local thermal equilibrium facilitates the deﬁnition of thermodynamic

variables as functions of position and time. The following strategy to solve the liquid ﬁlm’s

composition x(t)and its regression rate ξ

˙(t)is exploiting the fact that both bulk phases

remain in local thermal equilibrium throughout the entire process, as characterised by the

balance of the Gibbs energy across the interphase

gliq(zl(t)) !

=gvap(zs(t)) .(34)

2.2.5 Hydrodynamic description

The problem’s hydrodynamic description follows from the principle of mass conservation

dtVt

dV ρ(z,t)+∂Vt

dAjc(z,t)·n=0,(35)

which applied to the liquid ﬁlm is merely stating that any changes of the ﬁlm’s mass is

the result of a net particle ﬂux into the adjacent vapour phase. Equation (35) is readily

modiﬁed to describe a two-component liquid ﬁlm regressing quasi one-dimensionally

dtΩt

dV ⎡

⎣ρl,1(z,t)

ρl,2(z,t)⎤

⎦+∂Ωt

dA ⎡

⎣jc

z,1(z,t)

z,2(z,t)⎤

⎦=⎡

⎣0

0⎤

⎦.(36)

The set of equations (36) is coupled via the ﬁlm’s overall density ρl,i =xiρl, with mole

fractions xibeing constraint to one, that has to be solved over the domain Ωt={z∈

R≥0|z∈[0,z

l(t)]}. Since the interphase is additionally assumed to be incompressible,

the molar ﬂuxes across its speciﬁc boundaries are balanced and consequently allow to

specify the Stefan conditions at ∂Ωt={z∈R≥0|z=zs(t)}

dzz (bi3)=dzz (bi3)2

bi3(zs)=bi3,s ⎫

⎬

⎭

at z=zs(t),t>0.(37)

In contrast to liquid and vapour phase mole fractions xi(t),yi(z,t), the mass transfer

number bi3

bi3=yi

y3≈const. ,i∈[1,2] ,(38)

was observed in the large scale atomistic simulations to remain suﬃciently constant at the

interphase throughout the entire process. An additional Dirichlet-type boundary condition

is enforced at the control volume that represents the conditions in suﬃcient distance from

the ﬁlm’s surface

b33(z,t)=1

dz(bi3)(z,t)=0 for z≥zδ,t>0,i∈[0,2] .(39)

The formulation of the problem is completed by specifying proper initial conditions

zl(0) = z0

xi(t)=xi,0

yi(z,t)=yi,0

⎫

⎪

⎬

⎪

⎭

at t=0.(40)

The set of coupled partial diﬀerential equations (36) with boundary (37), (39) and initial

conditions (40) is solved with an appropriate diﬀusion ansatz for the component’s molar

ﬂux jc

z,i(z,t), while the vapour phase’s molar composition is treated as a classical boundary

layer problem.

2.2.6 Solution strategy

Formulating the Stefan problem of an evaporating two-component liquid ﬁlm leads to a

characteristic set of coupled ordinary diﬀerential equations (ODEs). While an abbreviated

version of the solution strategy is to be found in the subsequent original publication, the

following more detailed discussion will focus on the strategy’s mathematics and hence the

set of ODEs is formulated in terms of generic variables

h(t)⎡

⎣c2(t)a20

0c2(t)a1⎤

⎦·⎡

⎣g˙1(t)

g˙2(t)⎤

⎦+h

˙(t)⎡

⎣c(t)+φ10

0c(t)+φ2⎤

⎦·⎡

⎣g1(t)

g2(t)⎤

⎦=⎡

⎣0

0⎤

⎦,

i.e. each term’s physical interpretation is omitted. Consequently, the four unknown quan-

tities c, h, giare considered as generic functions of time and ai,φ

iare considered as con-

stants. The number of unknowns, however, can be reduced by exploiting that both gi(t)

are constraint

g1(t)+g2(t)=1,(41)

and that the function c(t)is a linear combination of the gi(t)

a1g1(t)+a2g2(t)= 1

c(t).(42)

Additionally, a series of algebraic manipulations and a separating of variables ansatz will

disclose the set of ODEs to be solvable by quadrature.

First, a multiplication of the ODEs’ ﬁrst line of with a1and its second line with a2

yields after adding the results

˙(t)+g1(t)·a1φ1−a2φ2+a2φ2=0,(43)

˙(t)+a2φ2·g1(t)a1φ1

a2φ2−1+1

=0.(44)

Secondly, the multiplication of the ODEs’ ﬁrst line with g2(t)and its second line with

g1(t)leads to

g˙1(t)+ 1

h(t)·g3

1(t)·a2−a1+g2

1·a1−2a2+g1a2·φ1−φ2=0,(45)

g˙1(t)−a2φ2

h(t)·g3

1(t)·1−a1

a2+g2

1(t)·a1

a2−2+g1·1−φ1

φ2=0.(46)

Dividing equation (44) by equation (46) and exploiting the approach known as "cancelling

of the dots" leads to the ﬁrst preliminary result

dg =h(t)·

g11−a1φ1

a2φ2−1

1−φ1

φ2·g3

1(t)1−a1

a2+g2

1a1

a2−2+g1.(47)

Separating the variables in equation (47) after lightening the notation =a1/a2,ν=

φ2/φ1yields

h(t)=ν

ν−1·g1(t)1−

ν−1

g1(t)·g2

1(t)·1−+g1(t)−2+1

·dg , ν =1,(48)

h(t)=1

1− ν

ν−1·g1(t)1−

ν−1

g1(t)·g1(t)−1·g1(t)−1

1−·dg ,  =1.(49)

A partial fraction decomposition of the quotient function partitions (49), in the case of

three real-valued denominator polynomial roots, leads to

1

1− ν

ν−1·g1(t)1−

ν−1

g1(t)·g1(t)−1·g1(t)−1

1−=A

g1(t)+B

g1(t)−1+C

g1(t)−1

1−

1

1− ν

ν−1·g1(t)1−

ν−1

g1(t)·g1(t)−1·g1(t)−1

1−=ν

1−ν·g1(t)+1

ν−1·g1(t)−1+1

g1(t)−1

1−

The ﬁrst quadrature emerges by inserting the result above into equation (49) with hbeing

divided by its initial value h0, i.e. h=hh−1

0, in order to simplify the initial conditions

h(t)

h(t0)

dh1

h(t)=g1(t)

g1(t0)

dg1ν

1−ν·g1(t)+g1(t)

g1(t0)

dg11

ν−1·g1(t)−1

+g1(t)

g1(t0)

dg11

g1(t)−1

1−.(50)

The integration of integration (50) is straightforward with initial conditions h(t0)=1

and g1(t0)=g10

lnh(g1(t))=ν

1−ν·lng1(t)

g10−1

1−ν·lng1(t)−1

g10−1+ln

g1(t)−1

1−

g10−1

1−,

h(g1(t)) = g1(t)

g10ν

1−ν·1−g10

1−g1(t)1

1−ν·g1(t)−1

1−

g10−1

1−,(51)

and the complexity of eq. (51) is readily reduced by exploiting the constraints (41), (42)

h(g1(t)) = g1(t)

g10ν

1−ν·g20

g2(t)1

1−ν·c0

c(t).(52)

While this ﬁrst quadrature (52) solves has a function of g1, the second quadrature will

solve g1as a function of t

dh(g1(t))

dt =dh(g1)

dg1·dg1(t)

dt .(53)

The derivative of hwith respect to time is readily available from equation (44) and the

derivative of hwith respect to g1is obtained from equation (52).

Separating the variables in equation (53) will eventually lead to the problem’s second

quadrature

t

dt =g1

g10

dg1dh

dg1

/dh

dt .(54)

The derivative of hwith respect to g1is much easier to compute if all constant terms in

equation (52) are re-grouped into h(g10)

h(g1(t)) = h(g10)·gν

1−g11

1−ν·g1−1

1−,(55)

yielding

dh

dg1

=h(g10)·g

2ν−1

1−ν

1·1−g1ν−2

1−ν·g1+ν

1−ν·g1−1

1−+gν

1−g11

1−ν.

While assuming the ratio νto be small compared to unity, i.e. ν1, the equation above

simpliﬁes

dh

dg1≈h

¯(g10)·g1+ν·g1−1

1−

g1·1−g12+1

1−g1,(56)

with h

¯being the approximation of hunder the aforementioned assumption, which also

simpliﬁes equation (44)

dh

dt ≈−a2φ1

h0·g1·−ν.(57)

Dividing equation (56) by equation (57) directly leads to the integrand in equation (54)

dt =−h

¯h0

a2φ1·g1+ν·g1−1

1−

g1·1−g12·g1−γ+1

1−g1·g1−ν·dg1,(58)

which partitions under a partial fraction decomposition into

g1+ν·g1−1

1−

g1·1−g12·g1−ν=A

1−g1+C

1−g12+D

g1+ν.(59)

Inserting equations (58) and (59) into equation (54) eventually leads to the second quadra-

ture, with the negative sign on the right-hand side resulting from dg1<1

t

dt =−h

¯h0

a2φ1g

g10

d|g1|1

1−·1

g1−ν2−ν −2ν +ν−

−1−ν2·1

1−g1

ν +

−1−ν·1

1−g12+ν3−ν +3+2

−1−ν2·1

g1−ν

1

−ν·1

1−g1

+

ν−·1

g1−ν.(60)

2.2.7 Diﬀusion limited evaporation of a binary liquid ﬁlm

R.S. Chatwell, M. Heinen and J. Vrabec, International Journal of Heat and Mass Trans-

fer,132, 1296-1305 (2019)

Reprinted from the International Journal of Heat and Mass transfer with the permis-

Mass Transf.,132, 1296-1305 (2029) and my be found at https://doi.org/10.1016/j.

ijheatmasstransfer.2018.12.030.

In the following publication, the Stefan problem for a binary mixture, i.e. the set of

equations (36) with the respective boundary and initial conditions (37), (39), (40), is

solved analytically The liquid phase is initially set up as an equimolar mixture of two

species with diﬀerent volatility, which is understood in this context as the species’ endeav-

our to transition from liquid to vapour phase. Throughout the entire process, the adjacent

vapour phase is dominated by a dense non-condensable inert gas and local thermal equi-

librium was found to be present in every domain, i.e. all particle velocities were observed

to follow the drifting Maxwellian (33). Both phases were deliberately speciﬁed to exhibit a

strongly non-linear behaviour. The analytical solution predicts both phases’ mole fractions

xi,y

i, as well as the liquid ﬁlm’s thickness ξover time. Subsequently, large scale molecu-

lar dynamics simulations were employed to verify the physical model as well as its solution.

The leading author, R.S. Chatwell, has developed the physical model, as well as its an-

alytical solution. The leading author has also determined all required thermodynamic

non-idealities by atomistic simulations and evaluated the results, in order to verify the as-

sumptions made during the derivation of the physical model. The co-author, M. Heinen,

has performed all large scale simulations and post-processed their results. The manuscript

was revised and redacted by J. Vrabec.

Diffusion limited evaporation of a binary liquid ﬁlm

René Spencer Chatwell, Matthias Heinen, Jadran Vrabec

⇑

Thermodynamics and Process Engineering, Technical University of Berlin, 10587 Berlin, Germany

article info

Article history:

Received 9 October 2018

Received in revised form 3 December 2018

Accepted 4 December 2018

Keywords:

Evaporative mass transfer

Analytical modeling

Large scale molecular dynamics simulations

abstract

An analytical solution of a model ﬂuid’s time behavior, known as the Stefan problem, is presented.

A scenario is investigated in which a planar two-component liquid ﬁlm is continuously evaporating into

a thermodynamically non-ideal vapor phase. Evaporation is initiated and maintained by a spatial

chemical potential gradient, while its rate is limited by the components’ diffusion ﬂuxes across the

vapor-liquid interface. Local thermodynamic equilibrium is found to be present throughout the process.

In contrast to the classical approach relying on equations of state, all required non-idealities are

formulated in relation to the Gibbs energy and are determined by molecular simulations. Initially, the

liquid is an equimolar mixture of two components of different volatility, whereas the adjacent vapor

phase is dominated by a dense inert gas. To validate the analytical model and verify all exploited

assumptions, the results are contrasted to large scale molecular dynamics simulations.

1. Introduction

Investigations of phase transitions for spherical and planar

geometries are methodically similar. Historically, research was

focused on droplet evaporation indicating a relationship between

the current square diameter d

ðtÞand an initial value d

to exist,

which for pure liquids is linear in time t

ðtÞ¼d

Kt:ð1Þ

For droplets immersed into a quiescent atmosphere, the evapora-

tion rate Kis exclusively a function of the thermodynamic state.

The literature refers to this equation as ‘‘d-squared law” [1]. The

problem’s more thorough theoretical treatments originated from

around the early twentieth century [2,3], whereas a ﬁrst estimation

of Khad already been conducted by Maxwell [4] in his 1877 work

on ”Diffusion” and was presented in its current form by Fuchs [5]

K¼8D

e;s

y

e;1



;ð2Þ

where y

e;i

represent the mole fraction of evaporate at either surface

or in sufﬁcient distance from the droplet. This mole fraction

difference initiates and maintains the evaporation process while a

diffusion coefﬁcient Din combination with the vapor-liquid density

ratio q

1

determines its rate. Maxwell’s approach was later

reﬁned, among others by Spalding [6], to account for substantial

evaporation rates that are indicative of an evaporate’s high vapor

pressure and results in its accumulation in the interface region

between liquid and vapor. Consequently, a more detailed descrip-

tion of the droplet’s surface composition became indispensable

and necessitated Spalding to take the quiescent atmospheric gas’

mole fraction y

a;s

into account

K¼8D

ln 1 þy

e;s

y

e;1

a;s

:ð3Þ

In fact, Maxwell’s description (2) represents the ﬁrst-order approx-

imation of Spalding’s formulation (3) for the case of a dominating

atmospheric gas at droplet surface, i.e. y

a;s

1. Maxwell and Spald-

ing assumed both bulk phases to be constantly in vapor-liquid equi-

librium that due to mass transport is only sustainable via

continuous evaporation.

The d-squared law in its form (1) has been experimentally val-

idated over a substantial range of thermodynamic states as well as

for a variety of non-sooting monocomponent droplets through a

wider spectrum of chemical complexity [7–16]. Multicomponent

droplets, however, offer a conspicuously different behavior owed

to the continuous variation in molar composition of both liquid

xðtÞand coexisting vapor yðtÞ. Species with genuinely dissimilar

properties lead to evaporation rates that vary substantially over

time [17–19]. Complementing theoretical and experimental work,

multicomponent evaporation has also been successfully addressed

by atomistic simulations [20,21].

https://doi.org/10.1016/j.ijheatmasstransfer.2018.12.030

⇑Corresponding author.

E-mail address: [email protected] (J. Vrabec).

International Journal of Heat and Mass Transfer 132 (2019) 1296–1305

Contents lists available at ScienceDirect

International Journal of Heat and Mass Transfer

journal homepage: www.elsevier.com/locate/ijhmt

2. Object under study

Complementary to the literature’s focus on spherical droplets,

this work solves the Stefan problem [22,23] for a multicomponent

liquid ﬁlm analytically. This analytical solution, representing the d-

squared law’s (1) analogue for planar surfaces, is then validated by

large scale molecular dynamics (MD) simulations. Consequently, a

two-component liquid ﬁlm’s change in molar composition xðtÞand

thickness nðtÞis investigated during evaporation into an inert gas

dominated vapor phase. Each phase is composed of components

with deliberately speciﬁed characteristics, cf. Appendix A. Due to

these characteristics, in combination with the selected thermody-

namic state, outlined in Table 1, both bulk phases are thermody-

namically non-ideal. An adequate description of the establishing

evaporation dynamics requires all component’s diffusion coefﬁ-

cients, liquid phase activity coefﬁcients and vapor phase fugacities

to be known as functions of the present state.

Molecular simulations are employed for two different reasons.

For one, the sampling of all required thermodynamic observables

by dedicated MD and Monte Carlo (MC) simulations requires sub-

stantially less approximation than modeling resting on equations

of state. On the other hand, the present hydrodynamic formulation

addresses the phases’ molar composition xðtÞ;yðtÞdirectly. Large

scale MD simulations offer an alternative access to determine a

phase’s molar composition with a spatial and temporal resolution

that is orders of magnitude greater than what is currently achiev-

able by experiment. To be comparable to hydrodynamic length and

time scales, both a sufﬁciently large ensemble and a substantial

simulated time is necessary. Fig. 1 outlines an atomistic represen-

tation of the investigated scenario containing 2:510

particles

with an initial liquid ﬁlm and vapor phase thickness of

ðt

Þ¼15 nm and d¼z

ðtÞz

ðtÞ¼30 nm, respectively. Various

separate simulation series were carried out, to either sample ther-

modynamic observables or to pursue large scale MD simulations,

cf. Appendix A.

The present hydrodynamic formulation rests on a ﬂuid domain

decomposition into four separate regions, as outlined in Fig. 1. The

liquid phase (I) initially consists of two equimolarly mixed compo-

nents (1 and 2) with different volatility, that in this context should

be understood as an attribute for the component’s endeavor to

transition from the liquid and to remain in the coexisting vapor

phase (III). The latter was chosen to be dominated by an inert

gas (component 3), which was speciﬁed not to be condensable

and be barely miscible in the liquid ﬁlm. Both bulk phases are

physically coupled by an interface region (II) that is addressed as

interphase in the following. A control volume (IV) representing

the invariant atmospheric conditions completes the ﬂuid domain.

2.1. Local thermodynamic equilibrium

Under vapor-liquid equilibrium the particles’ entire motion is

thermal and consequently drift-free Maxwellian. The applied

chemical potential gradient initiates evaporation dynamics by

introducing a collective drift, i.e. hydrodynamic contribution u

to the i-th component’s particle velocity

, that superimposes to

the random thermal motion w

¼u

þw

:ð4Þ

Local thermal equilibrium exists in every spatial domain (I) - (IV) if

all particles’ thermal velocities are Maxwellian distributed. The cor-

responding velocity distribution function f

z;i

for the relevant mass

transport direction zis most sensibly displayed in its contracted

form [24,25]

z;i

¼ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

rexp m

u

ðÞ

:ð5Þ

Particle interactions identify as mechanism that drive a system

towards local equilibrium [26]. A dense vapor phase is indicative

of such dominating interactions, realized in the present scenario

by a high system pressure p¼6:34 MPa ensuring the particles’

thermal velocities to be Maxwellian distributed naturally in each

domain, even the narrow interphase. To conﬁrm the anticipated

Maxwellian distribution (5), additional stationary large scale MD

simulations for the liquid ﬁlm being prepared and maintained at

¼ð0:48;0:52Þmol mol

1

were carried out. The results are pre-

sented in Fig. 2 and fully conﬁrm local equilibrium to exist. Similar

results were reported in the literature on interfacial mass transfer,

Table 1

The system was initially prepared in a thermodynamic state with temperature T¼80 K and pressure p¼6:34 MPa, where both phases are under vapor-liquid equilibrium (VLE)

with the given liquid xiand vapor phase mole fractions yias well as partial vapor qv;iand pure component molar densities q0;i. The thermodynamic non-idealities, i.e. the fugacity

coefﬁcient’s excess contribution #i, the activity coefﬁcient ciand consequently the diffusion ﬂux’s entire thermodynamic contribution /i, albeit functions of composition, were

considered constant and evaluated under these VLE conditions.

0;i

mol mol

1

mol dm

3

– mol mol

1

mol dm

3

– mol cm

2

1

1 0.13 1.10 4.09 0.5 47.77 1.07 1.41

2 0.03 0.23 20.07 0.5 42.33 1.06 0.13

3 0.84 7.12 – 0 8.33 – –

Fig. 1. Snapshot taken from one of the present large scale MD simulations depicting the scenario in vapor-liquid equilibrium, i.e. prior to evaporation. The studied domain is

decomposed into four regions that are connected physically on microscopic length scales. The central role is attributable to the interphase (II), characterized by a steep

density decline ranging from liquid (I) to vapor (III). A chemical potential gradient is established by substituting evaporated particles that originated from the liquid phase by

inert gas particles within the control volume (IV). The scenario’s symmetry ensures the net molar ﬂux to be zero at the origin z¼0, while selective thermostatisation

establishes a spatially and temporally constant temperature T¼80 K which is accompanied by a pressure of p¼6:34 MPa for the present mixture.

R.S. Chatwell et al. / International Journal of Heat and Mass Transfer 132 (2019) 1296–1305 1297

for both stationary [27,28] and instationary [29,30] cases.

Non-Maxwellian distributed particle velocities, in contrast, have

conclusively been demonstrated to establish during evaporation

into rareﬁed gas phases or vacuum [31–33].

3. Hydrodynamic description

The present formalism models the vapor phase as a classical

boundary layer and predicts the ﬁlm’s evaporation rate, i.e. its

regression, as well as the change of both phases’ molar composi-

tion over time, while resting exclusively on particle conservation

formulated in integral form.

A domain’s change in mass is determined by an imbalance of

particle ﬂuxes j

z;i

through its boundary [34] and yields for the bulk

liquid’s domain (I) z2½0;z

ðtÞ, while invoking the inert gas charac-

teristics not to be condensable and be barely miscible in this liquid

l;1

ðz;tÞ

l;2

ðz;tÞ

"#!

þI

dA j

z;1

ðz;tÞ

z;2

ðz;tÞ

¼0



:ð6Þ

A component’s particle density decomposes into mole fraction x

and overall molar density q

of the respective phase q

l;i

¼x

whereas the particle ﬂux factorizes to j

z;i

¼x

z;i

, with the convec-

tive molar averaged velocity u

z;i

towards the interphase. Assuming

spatial homogeneity of the liquid ﬁlm’s molar composition

–x

ðzÞand density q

–q

ðzÞwith the additional observation that

the particle ﬂuxes are invariant across the domain’s surface leads to

a straightforward integration, where VðtÞ¼z

ðtÞA

ðtÞ

ðtÞz

ðtÞ

ðtÞz

ðtÞ



ðtÞx

ðtÞu

z;1

ðz;tÞ

ðtÞu

z;2

ðz;tÞ



¼0



:ð7Þ

The interphase z2½z

ðtÞ;z

ðtÞ balances the molar particle ﬂuxes

between both bulk phases, where j

z;i

¼x

ðtÞu

z;i

ðz

;tÞindicates the

i-th component’s molar ﬂux exiting the bulk liquid phase and

z;i

¼y

ðz

;tÞu

z;i

ðz

;tÞindicating the ﬂux entering the adjacent bulk

vapor phase



ðtÞx

ðtÞu

z;1

ðz

;tÞ

ðtÞu

z;2

ðz

;tÞ



y1ðzs;tÞuz;1ðzs;tÞ

y2ðzs;tÞuz;2ðzs;tÞ



¼0



:ð8Þ

It is by bringing together (7) and (8) that the connection between

the liquid phase’s mass and the ﬂuxes in the vapor phase (III)

z2½z

ðtÞ;z

ðtÞ is achieved

ðtÞ

ðtÞz

ðtÞ

ðtÞz

ðtÞ



ðtÞy

ðz

;tÞu

z;1

ðz

;tÞ

ðz

;tÞu

z;2

ðz

;tÞ



¼0



:ð9Þ

Although the inert gas is assumed not to be soluble in the liquid, its

presence in the interphase and vapor has to be accounted for. Each

component’s collective drifting motion y

, as deﬁned in (4),is

decomposed further into an advective y

uand a diffusive contribu-

tion [35] y

¼y

uþU

ðÞ:ð10Þ

The evaporate’s total convective ﬂux is related to its total diffusive

ﬂux, cf. supplementary material, with all velocities being formu-

lated in the mean molar reference frame

i¼1

z;i

¼y

z;1

þy

z;2

;ð11Þ

and it is this correlation that renders the evaporation process diffu-

sion driven. It is not necessarily trivial to assume the vapor phase

mole fraction quotients to be time invariant

ðzÞ¼y

ðz;tÞ

ðz;tÞ;ð12Þ

yet this simpliﬁcation still allows to reasonably predict the vapor

phase’s molar composition. To ease the notation L

z;i

¼b

z;i

is set

for each component’s diffusive ﬂux.

3.1. Diffusive motion

A general description of diffusive motion in an inhomogeneous

medium is given by the Fokker-Planck diffusivity law [36]. In the

present case, this ﬂux has to be computed adjacent to the inter-

phase, i.e. at z¼z

ðtÞ

z;1

ðz

z;2

ðz



¼@

ðyÞD

ðyÞ

ðyÞD

ðyÞ



b

ðzÞ



!

;ð13Þ

wherein the Fick diffusion coefﬁcients D

express the phenomeno-

logically postulated proportionality between ﬂux and respective

mole fraction gradient [37]. For multicomponent systems, however,

Fig. 2. Contrasting the results between drifting Maxwell distributed (solid line), as

deﬁned in (5), and the sampled MD velocities for the relevant volatile (triangles)

and less-volatile (circles) components does not disclose any disparities. These data

were evaluated at temperature T¼80 K and three different positions ranging from

bulk liquid to bulk vapor. The large difference between both component’s

distribution functions is due to their different molar mass. (a) Values sampled in

the bulk liquid phase at position z6z

ðtÞin close vicinity to the interphase. (b)

Values sampled in the interphase at position z2½z

ðtÞ;z

ðtÞ. (c) Values sampled in

the bulk vapor phase at position zPz

ðtÞin vicinity to the interphase.

1298 R.S. Chatwell et al. / International Journal of Heat and Mass Transfer 132 (2019) 1296–1305

each diffusive ﬂux is the result of all driving gradients. The diffusion

matrix, containing Fick’s coefﬁcients D

, is separable into a kinetic

and a thermodynamic contribution [38]



¼B



1





;ð14Þ

the former being the Maxwell-Stefan and the latter the thermody-

namic factor matrix, describing a mixture’s departure from ideality

and being composition derivatives of the excess Gibbs energy [39]

. In this work, both matrices were considered constant throughout

the entire evaporation process. Assuming the kinetic part to be

invariant is justiﬁable, as for the thermodynamic part, depending

strongly on composition, this simpliﬁcation was conﬁrmed by the

performed MD simulations, cf. supplementary material. Bringing

together (9), (11),(13) and (14) leads to

ðtÞ

ðtÞz

ðtÞ

ðtÞz

ðtÞ







1





d

ðzÞ





¼0



ð15Þ

The diffusion matrix couples both ﬂuxes, its coupling strength is

measured by how much the eigenvector matrix Tdiffers from the

identity matrix I, and can spectrally be decomposed



1





¼TD



T

1

:ð16Þ

The eigenvalues D

have to be understood as effective diffusion

coefﬁcients that map the weighted action of both driving forces

onto each ﬂux

1

d

ðtÞ

ðtÞz

ðtÞ

ðtÞz

ðtÞ





D10

0D2



T1dz

b13ðzÞ

b23ðzÞ



z

¼0



ð17Þ

In the attempt to pursue ﬁrst order effects, the appearing spatial

gradient d

ðb

Þis linearized. The vapor domain d¼z

ðtÞz

ðtÞcon-

stitutes a classical boundary layer problem with exemplary bound-

ary conditions out of which the following curvature-gradient

correlation (18) is a statement of particle conservation, cf. supple-

mentary material

z¼z

ðtÞ:b

ðz

Þ¼b

i3;s

ðb

Þj

¼d

ðb

Þj



;

z¼z

ðtÞ:b

ðz

Þ¼b

i3;1

ðb

Þj

¼0:ð18Þ

It has already been established that Fick diffusion originates from

a spatial mole fraction gradient comprising the stagnant gas pres-

ence, which historically arisen has been termed mass transfer

number [6]

¼b

i3;1

b

i3;s

¼y

i;1

y

i;s

3;s

:ð19Þ

The nonlinearity in the proposed boundary conditions (18) necessi-

tates a different ansatz to approximate the differential operator by a

difference quotient. The simplest function that complies with this

set of boundary conditions is a third order polynomial, ﬁrst intro-

duced by Pohlhausen [40] and later applied to mass transfer prob-

lems by Spalding [41]

ðzÞb

i3;s

zz

ðtÞ



þb

zz

ðtÞ



zz

ðtÞ



ð20Þ

A set of nonlinear algebraic equations emerges that interrelates

those coefﬁcients

1¼

þb

;ð21aÞ

0¼

2

;ð21bÞ

0¼

þ2b

þ3

;ð21cÞ

with two sets of solutions out of which only the ﬁrst yields physi-

cally sensible boundary layer proﬁles

¼2

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

1þ3

r1

;ð22aÞ

¼3þ4

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

1þ3

rþ1

;ð22bÞ

¼2

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

1þ3

r1

2:ð22cÞ

The binomial theorem allows to approximate square roots to a con-

venient degree of accuracy, given the fact the transfer number is

small B

1, and the anticipated linearization is attained by trun-

cating the series approximation to ﬁrst order

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

1þ3

r1þ3

:ð23Þ

It is by considering this approximation in the coefﬁcients a

that (20) exposes the proper derivative’s substitution by a differ-

ence quotient

ðb

Þj



i3;1

b

i3;s

d2

i;1

y

i;s

3;s

:ð24Þ

The proportionality coefﬁcient a

in front of the otherwise simple

difference quotient is being mandated by the physically required

boundary conditions. The present scenario was set up such that

an inert gas component dominates the vapor phase, permitting to

set y

3;s

1 even at the surface. Moreover, the composition of the

surrounding atmosphere, represented by the control volume, was

speciﬁed to be y

i;1

¼0. Inserting the attained linearization into

(17), while assuming weak ﬂux coupling TI, leads to

ðtÞ

ðtÞz

ðtÞ

ðtÞz

ðtÞ



þ3

D10

0D2



y1ðzs;tÞ

y2ðzs;tÞ



0



:ð25Þ

During local equilibrium, liquid and vapor phase mole fractions x

are entangled and as such cannot be determined independently.

3.2. Thermodynamic non-ideality

The central element when formulating phase equilibria is the

extensive Gibbs energy Gthat is constructed by the weighted arith-

metic mean out of every component’s chemical potential l

in the

mixture and is reduced g¼G=ðnRTÞwith the universal gas con-

stant Rand the respective phase’s overall mole number n

g¼X

ðT;p;xÞ:ð26Þ

The classical framework of irreversible thermodynamics offers to

use its concepts locally and to further understand the Gibbs energy

as a function of both position and time [42]. It has already been

demonstrated that local diffusive equilibrium exists across the

interphase, which is represented by the equality of the phase’s

chemical potentials

liq

¼@g

liq

ðtÞ¼@g

vap

ðz;tÞ¼

vap

;i2½1;2:ð27Þ

A formulation that is true, due to the extensive Gibbs energy’s scal-

ing behavior as a homogeneous function of ﬁrst-order in the respec-

tive mole numbers. The ideal gas acts as the lower physical

R.S. Chatwell et al. / International Journal of Heat and Mass Transfer 132 (2019) 1296–1305 1299

boundary of every vapor phase given a sufﬁciently low pressure p

At elevated system pressure p, the transition to a real gas mixing

behavior leads to

vap

¼X

i¼1

0;i

ðT;p

Þþln p



þlnðy

Þþln

ðT;p;yÞðÞ



;ð28Þ

in which all non-ideal contributions to the chemical potential are

summarized into the fugacity coefﬁcient

ðT;p;yÞ¼w

ðT;pÞ#

ðT;p;yÞ;ð29Þ

that factorizes into a residual w

and an excess part #

, fully covering

the component’s non-ideal behavior and the deviation from ideal

mixing at system pressure p, respectively. This form allows segre-

gating the ideal from the non-ideal mixing contribution to the vapor

phase’s Gibbs energy

vap

¼X

i¼1

0;i

ðT;pÞþlnðy

Þþln #

ðT;p;yÞðÞ



:ð30Þ

Liquid phases should be described differently. The components in

their pure liquid state at system pressure are chosen to be the ref-

erence, while the activity coefﬁcient c

accounts for all non-ideal

mixing behavior

liq

¼X

i¼1

0;i

ðT;pÞþlnðx

Þþln

ðT;p;xÞðÞ



:ð31Þ

The combination of (27), (30) and (31) exposes the correlation

between liquid and vapor phase composition, given both are in local

equilibrium

ðz

;tÞ¼x

ðtÞ

ðT;p;xÞ

ðT;p;yÞ:ð32Þ

It is evident from the excess parts’ functional dependence on their

respective phase composition that a proper substitution of the mole

fractions y

to x

requires c

and #

to be constant, which has already

been utilized in (15). Introducing (32) into (25), together with the

diffusion ﬂux’s entire thermodynamic contribution

¼3

2d#

;ð33Þ

leads to the substitution of the vapor phase mole fractions in favor

of the liquid phase’s.

3.3. Analytical solution

A set of non-linear differential equations arises that can be

solved by quadrature

ðtÞ

ðtÞz

ðtÞ

ðtÞz

ðtÞ



þ/



x

ðtÞ



0



:ð34Þ

Multicomponent behavior is not inevitably the mere consequence

of pure component properties. The simultaneous presence of all

species generates excess contributions, which can inﬂuence the

thermal and caloric observables differently. A multicomponent liq-

uid may behave with respect to its thermal properties often times

nearly ideal, the density is then a sum of pure component molar

densities q

0;i

proportional to the current molar composition x

ðtÞ,

cf. supplementary material

ðtÞ

1

X

i¼1

ðtÞ

0;i

ðT;pÞ

1

:ð35Þ

Vanishing volumetric excess does thereby in no way entail ideal

caloric behavior. Applying the derivative and utilizing the assumed

ideal behavior (35) reveals a seemingly proper decoupled system

1

0;2

1

0;1





þ/



x



0



;

ð36Þ

where the liquid’s total molar composition enforces a constraint on

the mole fractions

ðtÞþx

ðtÞ¼1:ð37Þ

Two thermodynamic parameters can be identiﬁed that determine

the evaporation rate with the thermodynamic non-ideality quotient

kbeing the more decisive



0;2

0;1

;k¼/

:ð38Þ

Multiplying the ﬁrst line of (36) with q

1

0;1

and the second line with

1

0;2

allows to fully utilize the constraint (37) and yields upon sum-

mation of the resulting equations

ðtÞþ

1

0;2

ðtÞ



k1



þ1



¼0:ð39Þ

Similarly, the multiplication of the ﬁrst line of (36) with x

ðtÞand

the second one with x

ðtÞyields upon subtraction

ðtÞ /

0;2

ðtÞx

ðtÞ1



ðÞþx

ðtÞ



2ðÞþx

ðtÞ



11



¼0:

ð40Þ

The ﬁlm thickness z

ðtÞhas consistently been formulated as a func-

tion of time and so has the liquid mole fraction x

ðtÞ. The division of

(39) by (40) discloses the alternative formulation of ﬁlm thickness

as a function of composition z

ðx

ðtÞÞ

ðtÞ

ðtÞ¼z

ðtÞx

ðtÞ1





1

1



ðtÞ1



ðÞþx

ðtÞ



2ðÞþx

ðtÞ



:ð41Þ

This functional relationship will be resolved successively, where the

description of ﬁlm thickness as a function of composition z

ðx

Þis

established ﬁrst and the correlation between mole fraction x

and

time tsubsequently. It is expedient to factorize the denominator

polynomial

ðtÞ

ðtÞ¼z

ðtÞ

1



k1

ðtÞ1





1



ðtÞx

ðtÞ1ðÞx

ðtÞ

1





;



;k–1;ð42Þ

with the exclusion of pure component behavior, being represented

by a mixture of identical particles ;k¼1ðÞ. While all denominator

roots are evidently real valued, the use of a partial fraction decom-

position conveniently partitions the occurring rational function into

a primitive sum

1k

k1

ðtÞ1





1



ðtÞx

ðtÞ1ðÞx

ðtÞ

1



¼k

1kðÞx

ðtÞþ1

k1ðÞx

ðtÞ1ðÞ

þ1

ðtÞ

1

ð43Þ

A variable separation of (42) in combination with the given decom-

position (43) renders the emerging equation integrable

nðtÞ

nðt

dn1

nðtÞ



¼Z

ðtÞ

ðt

1kðÞx

ðtÞþ1

k1ðÞx

ðtÞ1ðÞ

þ1

ðtÞ

1



;

ð44Þ

with a non-dimensional ﬁlm thickness nðtÞ¼z

ðtÞz

ðt

1

that

allows to formulate a simpler initial condition

nðt

Þ¼1;x

ðt

Þ¼x

1;0

:ð45Þ

An integration reveals the anticipated description of ﬁlm thickness

as a function of composition and unfolds its physical interpretation

1300 R.S. Chatwell et al. / International Journal of Heat and Mass Transfer 132 (2019) 1296–1305

as being the product of an exponentially weighted composition and

liquid density ratio

nðx

Þ¼ x

1;0



1k

2;0



1k

ðt

ðtÞ:ð46Þ

To lighten the notation and simplify further calculations, all invari-

ant terms are grouped into one constant nðx

1;0

Þand the liquid den-

sity is formulated in terms of molar composition

nðx

Þ¼nðx

1;0

Þx

1x



1k

1

1





:ð47Þ

The description of liquid composition over time x

ðtÞcannot easily

be constructed from already formulated functions, however, the

inverse solution of time as a function of composition tðx

Þis readily

attainable

dnðx

ðtÞÞ

dt ¼dnðx

ðtÞÞ

ðtÞ

dt ;ð48Þ

where the derivative on the left hand side is given in (39) and the

ﬁrst derivative on the right hand side can be computed from (47)

dx1

¼nðx1;0Þx

2k1

1k

11x1

ðÞ

k2

1k

x1þk

1k



x11

1





þxk

1x1



1k

ð49Þ

Eqs. (39) and (49) contain negligible terms. A reduction in complex-

ity is easily obtainable by exploiting the strongly different volatili-

ties k1 of the two components in the liquid, which physically

determines the more volatile component to undergo phase transi-

tion preferentially, a characteristic that was built into the particle

model used for all simulations

d

nðx

1;0

Þx

þkðÞx



1





1x

ðÞ

þ1

1x

;ð50aÞ

d

dt /

0;2

l;0



þkðÞ:ð50bÞ

The required relationship between composition and time is then

determined by an additional separation of variables ansatz in (48)

resulting in a quotient function

dt ¼Z

1;0

=dn



;ð51Þ

that in combination with the attained approximations (50a) and

(50b) generates an analytically solvable integral equation

dt ¼Z

1;0



0;2

l;0

þkðÞx



1





1x

ðÞ



þkðÞ

þ1

1x

ðÞ



þkðÞ

! !

;

ð52Þ

where the emerging rational function similarly partitions into a

primitive sum, given all denominator roots being real valued

þkðÞx



1





1x

ðÞ



þkðÞ

¼a



;kÞ

þa



;kÞ

1x

þa



;kÞ

1x

ðÞ



;kÞ



þk;

ð53aÞ

1x

ðÞ



þk

ðÞ

¼a



;kÞ

1x



;kÞ



þk:ð53bÞ

All occurring integration constants show the already proclaimed

dependence on the thermodynamic parameters ;kand do thereby

signiﬁcantly contribute to the ﬁlm’s evaporation rate



;kÞ¼ 1



1;a



;kÞ¼k



þkþk



1ðÞ



þkðÞ

;



;kÞ¼ k



1ðÞ



þkðÞ



;kÞ¼k



k



þkðÞ

;



;kÞ¼ a



;kÞ¼ 1



þk:

ð54Þ

As a result, the functional relationship between composition and

time is decomposed into four linear terms that are straightfor-

wardly integrable to yield

tðx

Þ¼

0;2

l;0

ln x

1;0



þa

ln x

2;0



þa

x

2;0

þa



1;0

þk



þk



;ð55Þ

with a

¼a

þa

and a

¼a

þa

The solution of a liquid ﬁlm’s Stefan problem, i.e. the d-squared

law’s (1) planar surface analogue for both cases, the two-

component and pure liquid phase is readily obtained. The former

is given by the combination of (47) and (55) relating ﬁlm thickness

nðx

Þto time tðx

Þvia the liquid’s mole fraction x

. The latter is a

special case for ;k¼1, that is not entailed in (47) and (55), yet

is alternatively obtained by integration of (39) with the addition

of (32) to substitute the vapor phase’s non-ideality #

with the

evaporate’s mole fraction y

e;s

near the surface

ðtÞ¼z

l;0

3D

e;s

t:ð56Þ

Comparing pure component liquid ﬁlms to droplets offers a salient

difference in the order to which both cases evaporate with respect

to their characteristic length z

ðtÞand dðtÞ. In this ﬁrst-order

approximation (56) the liquid ﬁlm evaporates linearly in time,

where the spherical droplet’s analogue for the same level of approx-

imation, as given by Maxwell’s equation (2), regresses quadratically

ðtÞ¼d

8D

e;s

t:ð57Þ

A high evaporation rate, i.e. short liquid phase lifetime, is facilitated

by a high diffusivity or low liquid density, respectively. For planar

surfaces, the boundary layer thickness dremains in the description

as a ﬁnite size effect, which is in full accordance with classical

boundary layer theory. Small values of dcorrespond to steep chem-

ical potential gradients that inevitably lead to faster evaporation

rates.

4. Results

Under vapor-liquid equilibrium (VLE), the mass ﬂuxes between

both bulk phases balance and consequently no net mass transfer

occurs. During evaporation, the liquid ﬁlm is constantly forfeiting

mass in the attempt to dispose of the developed chemical potential

gradient and restore a VLE. The establishing evaporation dynamics

affects each bulk phase’s molar composition differently. Due to the

volatile component’s characteristic to evaporate preferentially, the

less-volatile component successively enriches within the liquid, as

depicted in Fig. 3 for the interphase’s vicinity at z¼z

ðtÞ. The pro-

cess causes the ﬁlm not only to regress but also increases its overall

density, since the less-volatile component has a higher saturated

liquid density, cf. Table 1 and Fig. 4. The bulk vapor phase’s molar

composition in vicinity to the interphase does marginally change

during evaporation. While the volatile component depletes quite

rapidly, the inert gas accumulates at the surface. The less-

volatile’s mole fraction, in contrast, remains almost stagnant, cf.

Fig. 5 evaluated at z¼z

ðtÞ.

R.S. Chatwell et al. / International Journal of Heat and Mass Transfer 132 (2019) 1296–1305 1301

The combination of (47) and (55) allows to compute a two-

component ﬁlm’s dimensionless thickness nðtÞas function of time,

if the phases’ non-idealities #

;

and the components’ Fick diffu-

sion coefﬁcients D

are known. Similarly, (56) determines a mono-

component ﬁlm’s regression rate, given the evaporate’s diffusion

coefﬁcient Dand molar composition adjacent to the interphase

e;s

are known. All required thermodynamic data were sampled

by molecular simulations, cf. Appendix A, and are listed in Table 1,

Appendix B and the enclosed supplementary material. Three differ-

ently composed liquid ﬁlms were investigated. Ranging from pure

volatile over an initially equimolarly composed two-component

mixture to pure less-volatile. The monocomponent ﬁlms’ evapora-

tion rates expectedly envelop those of all possible binary mixtures,

cf. Fig. 6. The equimolarly prepared ﬁlm regresses initially quite

rapidly and then, due to a limitation embedded in (55) that pre-

dicts the volatile component’s full evaporation to be reached only

asymptotically lim

tðx

Þ¼1, transitions into an apparently

stagnant and ﬁnite thickness for small values of x

, i.e. x

60:01

mol mol

1

. In contrast, the large scale MD simulations indicate a

transition into a constant rate that closely resembles, yet not

equals, the pure less-volatile component’s evaporation rate

KðT;p;xÞK

ðT;pÞ, since residual volatile particles still remain in

the vapor phase for an extended period of time.

5. Conclusion

The Stefan problem for a multicomponent planar ﬁlm was

solved. An analytical model was presented, describing evaporation

dynamics of a liquid ﬁlm into a dense non-ideal vapor phase. The

outlined formalism requires speciﬁed bulk phase non-idealities

, as well as Fick diffusion coefﬁcients D

at the interphase’s

Fig. 3. Time evolution of the molar fractions of the volatile (red) and less-volatile

(blue) components in the liquid phase. The dashed lines represent the analytical

solution and the solid lines are large scale MD simulation data, evaluated on the

liquid side of the interphase at z

ðtÞ. The liquid is successively enriched with the

less-volatile component. (For interpretation of the references to color in this ﬁgure

legend, the reader is referred to the web version of this article.)

Fig. 4. Time evolution of the partial density proﬁles of both components in the

liquid mixture as sampled by the present large scale MD simulations. Two opposing

density gradients emerge, resulting in two counter-oriented currents, where the

volatile component (red, 1) propagates towards the interphase and preferentially

evaporates, while the less-volatile component (blue, 2) evaporates at a lower rate

and partly regresses towards the liquid ﬁlm’s center. A local volatile component’s

enrichment within the interphase is observable. The respective overall density is

depicted in black. The liquid ﬁlm’s density increase over time is a mixture effect due

to the enrichment of component 2 that has a higher saturated liquid density

l;1

l;2

Þ. (For interpretation of the references to color in this ﬁgure legend, the

reader is referred to the web version of this article.)

Fig. 5. Time evolution of the vapor phase mole fraction on the vapor side of the

interphase at z

ðtÞ, where the dominating inert gas’ behavior (cyan) is prominent.

(For interpretation of the references to color in this ﬁgure legend, the reader is

referred to the web version of this article.)

Fig. 6. Time evolution of dimensionless ﬁlm thickness nðtÞ¼z

ðtÞ=z

ðt

Þ, where the

dotted lines represent the analytical solution while the solid lines the large scale

MD simulations. The evaporation rate is given by the slope of these curves, where

the equimolar liquid case (yellow) is enveloped by the pure volatile’s (red) and pure

less-volatile’s (blue) rate K

ðT;pÞ>KðT;p;xÞ>K

ðT;pÞ. The strong difference

between the evaporation rates of the two pure ﬂuids K

is intentional and a

consequence of the chosen intermolecular force ﬁeld model. (For interpretation of

the references to color in this ﬁgure legend, the reader is referred to the web version

of this article.)

1302 R.S. Chatwell et al. / International Journal of Heat and Mass Transfer 132 (2019) 1296–1305

vicinity, i.e. at z¼z

ðtÞ. Consequently, its solutions (47),(55) and

(56) allow to determine the time evolution of both phases’ molar

compositions xðtÞ;yðtÞas well as the ﬁlm’s thickness nðtÞquantita-

tively. The components’ deliberately speciﬁed characteristics, i.e.

strongly different volatilities k1, weak diffusion ﬂux coupling

TIand the inert gas behavior, facilitate expedient simpliﬁcations

in the derivation of the model and its solutions.

The limiting quantity of a two-component ﬁlm’s evaporation

rate _

nðtÞis identiﬁed as the volatile component’s diffusion ﬂux

z;1

, wherein all driving inﬂuences, i.e. the activity coefﬁcient

the fugacity coefﬁcient’s excess contribution #

, the vapor phase

thickness dand the effective diffusivity D

, appear linearly and

consequently affect the ﬁlm’s behavior equally.

The analytical model was intentionally contrasted to large scale

MD simulations serving three purposes. First, the phases’ molar

compositions were assessed in an unconstrained manner. Second,

the existence of local thermodynamic equilibrium was veriﬁed

unambiguously in every domain and third it was demonstrated

that the hydrodynamic formalism can justiﬁably be applied to such

a nanoscale scenario. The MD simulations fully conﬁrm that

neglecting the ﬁlm’s internal dynamics, by assuming spatial homo-

geneity x

–x

ðzÞ,asin(6) and (7), does not compromise the accu-

rate prediction of its composition xðtÞand thickness nðtÞover time.

Consequently, the hydrodynamic ﬁeld description is applicable to

system sizes of the order of 50 nm, given the thermophysical prop-

erties are known a priori. Atomistic simulations prove to be an ade-

quate methodology determining all required properties without

constraints. The symbiosis of both approaches offers a promising

route for further studies on evaporation.

Conﬂict of interest

The authors declared that there is no conﬂict of interest.

Acknowledgements

The present work contributes to the Collaborative Research

Center (SFB) 75 of Deutsche Forschungsgemeinschaft (DFG) and

was funded under the grant VR 6/9-2. All computations were per-

formed either on the HPC cluster OCuLUS at the Paderborn Center

for Parallel Computing (PC

) or on the Cray XC40 system Hazel

Hen at the High Performance Computing Center Stuttgart (HLRS)

with resources allocated according to grant MMHBF2. This work

was carried out under the auspices of the Boltzmann-Zuse society.

Appendix A. Molecular dynamics simulations

The model ﬂuid’s thermodynamic behavior is determined by

deliberately speciﬁed Lennard-Jones parameters as outlined in

Table A.2. The component’s distinct volatilities were realized here

by different energy parameters 

. The inert gas characteristic to be

barely miscible in the liquid was modeled primarily by the Berth-

elot parameter f

–1.

All particles in the investigated scenario interact via the full

Lennard-Jones potential with 3:5rcut-off radius and analytic long

range corrections [43]. The chosen Lennard-Jones parameters

approximately describe a liquid ﬁlm composed of either nitrogen

+ oxygen or argon + krypton. The large scale MD simulations were

carried out in a cuboid volume with dimensions L

¼L

¼15 nm

and L

¼90 nm, an integrator time step of

t¼5 fs and periodic

boundary conditions in xand ydirections.

The present large scale MD simulations are, technically speak-

ing, nonequilibrium molecular dynamics (NEMD) simulations,

which does not inevitably lead to non-Maxwellian distributed par-

ticle velocities, but allows possible local equilibria to arise natu-

rally, as demonstrated in Section 2.1. In order to investigate

evaporation dynamics that are driven exclusively by a spatial

chemical potential gradient the system was selectively ther-

mostated. Consequently, the less-volatile particles in the liquid

z6z

ðtÞand the inert gas particles in the vapor z2½z

ðtÞ;z

ðtÞ were

thermostated in xand ydirections only, avoiding artiﬁcial interfer-

ence with the investigated mass transport in the relevant z

direction.

The necessary concentration gradient to initiate and maintain

evaporation dynamics was realized by substituting all particles of

components 1 and 2 by inert gas particles (component 3) within

the control volume, i.e. maintaining a molar composition of

¼1atz>z

ðtÞ. During evaporation, the control volume was

expanded to the same extent as the liquid ﬁlm regressed, keeping

the vapor phase thickness d–dðtÞconstant, cf. Fig. A.7.

The symmetry of the ﬂuid system was exploited to generate

considerably smoother sampling proﬁles by calculating the arith-

metic mean between both sides’ density and molar composition

proﬁles.

Appendix B. Diffusion properties

A thermodynamically non-ideal system can display a conspicu-

ous disparity between multicomponent and collective pure com-

ponent behavior, as measured by the Gibbs energy’s excess

contribution

ðT;p;yÞ¼X

i¼1

lnð#

Þ:ðB:1Þ

The fugacity coefﬁcient’s excess contribution #

can alternatively be

calculated from the species’ chemical potentials

(Appendix C)

and in contrast to the conventional approach, utilizing equations

of state, were computed here via dedicated MC simulations

lnð#

Þ¼

lnðy

Þ

0;i

:ðB:2Þ

Non-ideal behavior, however, is not merely described by the excess

Gibbs energy’s numerical value but also by its derivatives [46],

where the thermodynamic factor is determined by the curvature

¼d

þy

@g

X

k¼1

@g

! !

:ðB:3Þ

Table A.2

The thermodynamic behavior of the ﬂuid sampled by simulation was determined by this set of Lennard-Jones parameters that were chosen to produce a strongly non-ideal vapor,

yet retain a fairly ideal liquid phase. The length parameter rdetermines a particle’s effective diameter while the energy parameter is a measure for a particle’s dispersive

interaction strength. Additionally, a modiﬁcation of the Berthelot combining rule via fki allows to alter the interaction energy between unlike particle species. The inert gas was in

a highly supercritical state and consists of mainly repulsive interacting heavy particles. While all simulations sampling thermodynamic observables were carried out with the

open source molecular simulation tool ms2[44], the large scale MD calculations were performed with the massively parallel open source code ls1mardyn [45].

Component i

/nm

i

/K m

/u f

/– f

/–

Volatile 1 0.3 75 2 – 1

Less-volatile 2 0.3 100 20 1 –

Inert gas 3 0.3 10 30 0.3 0.3

R.S. Chatwell et al. / International Journal of Heat and Mass Transfer 132 (2019) 1296–1305 1303

A discrete data set of g

values for a varying molar composition

y¼ðy

Þwas produced by comprehensive MC simulations

(see supplementary material) and the computation of the necessary

derivatives in Appendix B.3 was made possible by selecting an

appropriate g

model. An appropriate ansatz is a Margules type

polynomial that is the result of an empirically motivated Wohl’s

expansion [47] truncated to desired order. As data generated by

simulations were made sufﬁciently precise and plentifully available,

an advanced fourth-order ansatz [48] became feasible

¼y

þA

C

ðÞ

þy

þA

E

ðÞ

þy

þA

E

ðÞ

þy

þA

E

ðÞ;ðB:4Þ

where the ﬁrst term models the genuine ternary contribution with

parameters C

and the following three terms describe the respective

binary subsystems’ inﬂuence with parameters A

. All parameters

ðA

Þwere determined with a standard nonlinear least squares

ﬁt to the respective excess Gibbs energy data obtained from MC

simulations.

The Fick diffusion matrix Dcan be decomposed into a kinetic

1

and a thermodynamic Ccontribution, which are functions of

the vapor phase’s molar composition. It has been argued above that

both matrices were considered constant and will only be evaluated

at equilibrium condition y



¼ð0:13;0:03;0:84Þmol mol

1

Cðy



Þ¼ 0:82 0:18

0:03 1:0



:ðB:5Þ

Determining the kinetic contribution B

1

, however, is a two-step

procedure. The Maxwell-Stefan diffusivities Ð

have to be com-

puted ﬁrst, which for this ternary mixture are readily accessible

via MD simulations

¼171:010

9

1

;

¼58:010

9

1

;

¼34:510

9

1

;

ðB:6Þ

and subsequently the matrix Bwith its diagonal B

and off-diagonal

elements B

has to be calculated [39]

¼y

þX

k¼1–i

;ðB:7aÞ

¼y

1



;i;j2½1;2:ðB:7bÞ

The Maxwell-Stefan diffusivities Ð

quantify the relative motion

between the i-th and j-th component in the ternary mixture and

are incorporated reciprocally in the description. Consequently, it

is this matrix’s inverse that is part of the thermodynamic contribu-

tion to relate the driving thermodynamic force, i.e. the concentra-

tion gradient to its corresponding diffusion ﬂux, and is given as

1

ðy



Þ¼ 155:78 2:62

25:02 53:69



10

9

1

:ðB:8Þ

The Fick diffusion matrix D¼B

1

Ccouples the simultaneous

inﬂuence of both gradients to each molar diffusion ﬂux L

z;i

, as spec-

iﬁed in (13). A spectral decomposition facilitates the decoupling by

mapping both gradients’ inﬂuence onto an effective diffusion coef-

ﬁcient D

that is given as the system’s i-th eigenvalue

Dðy



Þ¼ 120:97 0

055:88



10

9

1

;ðB:9Þ

where the deviation of the eigenvector matrix Tfrom the identity

matrix Iis a measure of coupling strength and the arguably present

mild coupling is neglected by setting

T¼10:35

0:26 1



I:ðB:10Þ

The simpliﬁcation introduced in (50a) and (50b) was based on the

components’ strongly different volatilities k1 and the exclusion

of a pseudo mixture –1

k¼/

¼0:093;



0;2

0;1

¼1:18:ðB:11Þ

Appendix C. Thermodynamic non-ideality

The vapor phase’s Gibbs energy could be decomposed into the

sum of a variety of physically interpretable contributions

vap

ðT;p;yÞ¼X

i¼1

0;i

ðT;pÞþg

res

0;i

ðT;pÞþy

lnðy

Þþg

ðT;p;yÞ;

ðC:1Þ

with the ﬁrst term being the ideal gas contribution of the pure com-

ponent, given at system temperature and pressure, and the second

its residual contribution. The third term describes the ideal mixing

term and the fourth its non-ideal mixing correction. The Gibbs

energy can also be decomposed into the weighted sum of its com-

ponents’ chemical potentials, as stated in (26)

Fig. A.7. Snapshot taken from the present large scale MD simulations depicting two successive instants of time t

during evaporation. While the liquid ﬁlm (I) regresses,

the control volume (IV) was simultaneously expanded to keep the vapor phase’s (III) thickness constant d–dðtÞ.

1304 R.S. Chatwell et al. / International Journal of Heat and Mass Transfer 132 (2019) 1296–1305

vap

ðT;p;yÞ¼X

i¼1

ðT;p;yÞ:ðC:2Þ

The ideal gas contribution is readily understood as the ideal gas

chemical potential g

0;i

¼y

0;i

. The residual and excess contribu-

tion’s mathematical form has, historically arisen, been deﬁned to

be a weighted logarithmic function akin to the ideal gas mixing

entropy

vap

ðT;p;yÞ¼X

i¼1

0;i

ðT;pÞþln w

ðT;pÞðÞþlnðy

Þþln #

ðT;p;yÞðÞ



ðC:3Þ

Generally, all pure component contributions were grouped together

to ease notation

0;i

ðT;pÞ¼

0;i

ðT;pÞþlnðw

ðT;pÞÞ;ðC:4Þ

which consequently leads to the vapor phase’s Gibbs energy

vap

ðT;p;yÞ¼X

i¼1

0;i

ðT;pÞþlnðy

Þþlnð#

ðT;p;yÞÞ



:ðC:5Þ

Bringing (C.2) and (C.5) together leads to description (B.2) for the

case of known chemical potentials

, which were sampled by here

by dedicated MC simulations

lnð#

Þ¼

lnðy

Þ

0;i

:ðC:6Þ

Alternatively, if the mixture’s Gibbs excess energy is known, e.g.

from a g

model, the component’s excess contribution to the fugac-

ity coefﬁcient can be computed [46]

lnð#

Þ¼g

þ@g

X

k¼1

:ðC:7Þ

Appendix D. Supplementary material

Supplementary data associated with this article can be found, in

the online version, at https://doi.org/10.1016/j.ijheatmasstransfer.

2018.12.030.

References

[1] C. Law, Recent advances in droplet vaporization and combustion, Energy

Combust. Sci. 8 (1982) 171–201.

[2] H. Morse, On evaporation from the surface of a solid sphere. Preliminary note,

Proc. Amer. Acad. Sci. 45 (1910) 363–367.

[3] N. Fuchs, Über die Verdampfungsgeschwindigkeit kleiner Tröpfchen in einer

Gasatmosphäre, Phys. Z. Sowjetunion 6 (1934) 224–243.

[4] J.C. Maxwell, in: W.D. Niven (Ed.), The Scientiﬁc Papers of James Clark

Maxwell, vol. 2, Cambridge Library Collection, 2011, pp. 625–646.

[5] N. Fuchs, Evaporation and Droplet Growth in Gaseous Media, Pergamon Press,

1959.

[6] D. Spalding, The combustion of liquid fuels, Fourth Symp. Comb. 4 (1953) 847–

864.

[7] S. Kumagai, T. Sakai, S. Okajima, Combustion of free fuel droplets in a freely

falling chamber, Proc. Combust. Inst. 13 (1971) 779–785.

[8] S. Okajima, S. Kumagai, Further investigations of combustion of free droplets in

a freely falling chamber including moving droplets, Proc. Combust. Inst. 15

(1975) 401–407.

[9] C.T. Avedisian, J.C. Yang, C.H. Wang, On low-gravity droplet combustion, Proc.

R. Soc. Lond. 420 (1988) 183–200.

[10] I. Gökalp, C. Chauveau, J. Richard, M. Kramer, W. Leuke, Observations on the

low temperature vaporization and envelope or wake ﬂame burning of n-

heptane droplets at reduced gravity during parabolic ﬂights, Proc. Combust.

Inst. 22 (1989) 2027–2035.

[11] C. Chauveau, I. Gökalp, Droplet combustion under microgravity conditions, in:

B. Kaldreich (Ed.), VIIIth European Symposium on Materials and Fluid Sciences

in Microgravity, vol. 2, European Space Agency, Paris, 1992, pp. 467–472.

[12] C. Chauveau, X. Chesnau, I. Gökalp, Burning characteristics of n-heptane

droplets under different regimes, in: 31st Aerospace Science Meeting, 1993,

pp. 824–832.

[13] H. Nomura, Y. Ujiie, H.J. Rath, J. Sato, M. Kono, Experimental study on high-

pressure droplet evaporation using microgravity conditions, Proc. Combust.

Inst. 26 (1996) 1267–1273.

[14] V. Nayagam, J.B. Haggard, R.O. Colantonio, A.J. Marchese, F.L. Dryer, B.L. Zhang,

F.A. Williams, Microgravity n-heptane droplet combustion in oxygen-helium

mixtures at atmospheric pressure, AIAA J. 36 (1998) 1369–1378.

[15] C. Morin, C. Chaveau, I. Gökalp, Droplet vaporization characteristics of

vegetable oil derived biofuels at high temperatures, Exp. Therm. Fluid. Sci.

21 (2000) 41–50.

[16] H. Ghassemi, S.W. Baek, Q.S. Khan, Experimental study on evaporation of

kerosine droplets at elevated pressures and temperature, Combust. Sci.

Technol. 178 (2006) 1669–1684.

[17] V. Glushkov, O. Todes, V. Fedoseev, Evaporation kinetics of drops of binary

mixtures of organic liquids, Kolloidn Zh. 32 (1967) 678–683.

[18] E. Ravindran, J. Davis, Multicomponent evaporation of single aerosol droplets,

J. Coll. Int. Sci. 85 (1982) 278–288.

[19] G. Brenn, L. Deviprasath, F. Durst, C. Fink, Evaporation of acoustically levitated

multi-component liquid droplets, Int. J. Heat Mass Transf. 50 (2007) 5073–

5086.

[20] S. Sumardiono, J. Fischer, Molecular dynamics simulations of mixture droplet

evaporation, in: J. Pagliarini, S. Rainieri (Eds.), Eurotherm Seminar 77, Heat

and mass transfer in food processing, Edizione ETS Parma, Parma, 2005, p. 323.

[21] A. Frezzotti, L. Gibelli, D.A. Lockerby, J.E. Sprittles, Mean-ﬁeld kinetic approach

to evaporation of a binary liquid into vacuum, Phys. Rev. Fluids 3 (2018)

054001.

[22] J. Stefan, Über die Theorie der Eisbildung, Monatsh. f. Math. Phys. 1 (1890) 1–6.

[23] J. Stefan, Über die Verdampfung und die Auﬂösung als Vorgänge der Diffusion,

Ann. Phys. 277 (1890) 725–747.

[24] W. Baumjohann, R. Treusmann, Basic Space Plasma and Physics, Imperial

College Press., 2012.

[25] A. Lofti, J. Vrabec, J. Fischer, Evaporation from a free liquid surface, Int. J. Heat

Mass Transf. 73 (2014) 303–317.

[26] R. Schunk, A. Nagy, Ionospheres: Physics, Plasma Physics, and Chemistry,

Cambridge University Press, 2009.

[27] B. Hafskjold, S. Kjelstrup, Criteria for local equilibrium in a system with

transport of heat and mass, J. Stat. Phys. 78 (1995) 463–494.

[28] I. Inazoli, S. Kjelstrup, D. Bedeaux, J. Simon, Thermodynamic properties of a

liquid-vapor interface in a two-component system, Chem. Eng. Sci. 65 (2010)

4105–4116.

[29] J. Xu, S. Kjielstrup, D. Bedeaux, A. Røsjorde, L. Rekvig, Veriﬁcation of Onsager’s

reciprocal relations for evaporation and condensation using non-equilibrium

molecular dynamics, J. Colloid Interface Sci. 299 (2006) 452–463.

[30] S. Kjelstrup, D. Bedeaux, I. Inazoli, J.-M. Simon, Criteria for validity of

thermodynamic equations from non-equilibrium molecular dynamics

simulations, Energy 33 (2008) 1185–1196.

[31] T. Ishiyama, T. Yano, S. Fujikawa, Molecular dynamics study of kinetic

boundary condition at an interface between argon vapor and its condensed

phase, Phys. Fluids 16 (2004) 2899–2906.

[32] A. Frezzotti, L. Gibelli, S. Lorenzani, Mean ﬁeld kinetic theory description of

evaporation of a ﬂuid into vacuum, Phys. Fluids 17 (2005) 012102.

[33] A. Frezzotti, Boundary conditions at the vapor-liquid interface, Phys. Fluids 23

(2011) 030609.

[34] L.D. Landau, E. Lifshitz, Course of Theoretical Physics. Fluid Mechanics,

Pergamon Press, 1987.

[35] S. de Groot, P. Mazur, Non-Equilibrium Thermodynamics, Dover Press, 1984.

[36] B. van Milligen, P. Bons, B. Carreras, R. Sánchez, On the applicability of Fick’s

law to diffusion in inhomogeneous systems, Eur. J. Phys. 26 (2005) 913–925.

[37] A. Fick, Über Diffusion, Ann. Phys. 170 (1855) 59–86.

[38] C. Truesdell, Mechanical basis of diffusion, J. Chem. Phys. 37 (1962) 2336–

2344.

[39] R. Taylor, R. Krishna, Multicomponent Mass Transfer, Wiley, 1993.

[40] K. Pohlhausen, Zur näherungsweisen Integration der Differentialgleichung der

laminaren Grenzschicht, ZAMM 1 (1921) 252–290.

[41] D. Spalding, Mass transfer in laminar ﬂow, Proc. R. Soc. A 221 (1954) 78–99.

[42] G. Lebon, D. Jou, J. Casas-Vázquez, Understanding Non-Equilibrium

Thermodynamics, Springer, 2008.

[43] J. Janec

ˇek, Long range corrections in inhomogeneous simulations, J. Phys.

Chem. B 110 (2006) 6264–6269.

[44] C.W. Glass, S. Reiser, G. Rutkai, S. Deublein, A. Köster, G. Guevara-Carrion, A.

Wafai, M. Horsch, M. Bernreuther, T. Windmann, H. Hasse, J. Vrabec, ms2: a

molecular simulation tool for thermodynamic properties, new version release,

Comp. Phys. Commun. 185 (2014) 3302–3306.

[45] C. Niethammer, S. Becker, M. Bernreuther, M. Buchholz, W. Eckhardt, A.

Heinecke, S. Werth, H. Bungartz, C. Glass, H. Hasse, J. Vrabec, M. Horsch, ls1

mardyn: the massively parallel molecular dynamics code for large systems, J.

Chem. Theory Comput. 10 (2014) 4455–4464.

[46] R. Taylor, H. Koojiman, Composition derivatives of activity coefﬁcient models

(for the estimation of thermodynamic factors in diffusion), Chem. Eng.

Commun. 102 (1991) 87–106.

[47] J. Prausnitz, R. Lichtenthaler, E. de Azvedo, Molecular Thermodynamics of

Fluid-Phase Equilibria, third ed., Prentice Hall, 1999.

[48] E. Hála, J. Pick, V. Fried, O. Vilím, Vapor-Liquid Equilibrium, Pergamon Press,

1967.

R.S. Chatwell et al. / International Journal of Heat and Mass Transfer 132 (2019) 1296–1305 1305

2.3 Stokes’ hypothesis

2.3.1 Isotropic stress tensor ﬁeld

The mathematical formalisation of the concept of stress as a tensor ﬁeld was one of ﬂuid

mechanic’s most important achievements. During the ﬁrst half of the nineteenth century,

Euler’s hydrodynamic equations constituting a ﬁrst approximation, were successfully com-

plemented to entail the experimentally observed decaying momentum in propagating ﬂu-

ids, as well as the resistance suﬀered by submerged bodies. In 1823, Cauchy constitutively

introduced a stress tensor ﬁeld Pij, as the dissipative momentum’s underlying principle,

in his general momentum equation

ρDtvi−ρgi+∂jPij =0.(61)

The stress ﬁeld’s contemporary formulation, however, arose from Poisson’s 1829 publi-

cation in which he combined the static pressure p0of Navier’s description with the two

material constants μ, ψ that originated from Cauchys work

Pij =p0−ψtr ()−2μij ,(62)

and identify as ﬁrst and second coeﬃcients of viscosity, respectively [141]. Equation (62),

evidently expressed in a more modern notation, constitutes the second approximation of

the hydrodynamic equations and relates the stresses linearly to its causing strains ij.

In 1845, Stokes argued geometrically that the reduction of both viscosity coeﬃcients to

one μbwould issue a dynamical contribution to the thermodynamic pressure, which he

conjectured must be exclusively a function of state variables, i.e. density and temperature

[13]

"Of course we may at once put μb=0if we assume that in the case of a

uniform motion of dilatation the pressure at any instant depends only on the

actual density and temperature at that instant, and not on the rate at which

the former changes with time".

A relationship that later became known as the Stokes’ hypothesis and, despite of his orig-

inal recommendation of a rather tentative use, has been employed quite extensively.

As mentioned above, a central element in the hydrodynamic formalism is the concept

of material points P(xi,t)and consequently the deﬁnition of local macroscopic observ-

ables that are continuously distributed in space and time. The macroscopic observables

are thereby a representation of the ensemble average of discrete particles in the vicinity

of each material point. Assuming the correlation between any two spatially adjacent ve-

locity vectors to be Markovian, while additionally enforcing its linearisation, leads to the

displacement ﬁeld’s ﬁrst-order representation

ui(xn+Δxn)=ui(xn)+ij +Ω

ijΔxn,(63)

that enables a Helmholtz-Hodge decomposition into elementary, kinematically interpretable

contributions. Mind that the Jacobian ∂juihas already been recognised as the superpo-

sition of a symmetric straining contribution 2ij =∂jui+∂iujand a rigid body rotation

2Ωij =∂jui−∂iuj.

Local variations in the displacement ﬁeld generate a spatial redistribution of momentum,

leading to restore global equilibrium. This transfer occurs irreversibly and is macroscop-

ically described by the concept of stress. The stress tensor ﬁeld is constructed Galilean

invariantly and is therefore identiﬁed as a function of the displacement ﬁeld’s straining

contribution Pij(ij)only. This imposed Galilean symmetry induces the stress ﬁeld’s form

invariance under arbitrary passive transformation, as realised by an orthogonal transfor-

mation matrix Dij ∈SO(3) [142]

D−1

il Plk(lk)Dkj =Pij D−1

il lkDkj.(64)

Any tensor valued function that satisﬁes this condition and is restricted to ﬁrst-order

eﬀects is of the form

Pij =a0δij +a1ij ,(65)

with coeﬃcients aithat are either scalar-valued functions of the tensor’s ﬁrst principle

invariant or constant. Splitting the stress ﬁeld into its static and dynamic viscous contri-

butions constitutes a basic conjecture of classical irreversible thermodynamics [102]

Pij =Pstat.

ij (p0)+Pdyn.

ij (ij),(66)

that facilitates to investigate both parts individually. It is sometimes erroneously claimed

that the dynamic part would be a measure of the ﬂuid’s departure from thermal equi-

librium. However, the particles’ velocities are still Maxwellian distributed, i.e. the ﬂuid

is in local equilibrium. In the presence of purely random thermal motion, i.e. ij =0,

the conservation of linear momentum (61) yields a balance of short-range dissipative and

long-range gravitational forces

∂jPstat.

ij −ρgi=0.(67)

In postulating the stress ﬁeld’s equilibrium contribution to be a spherically symmetric

tensor, as represented by an isotropic tensor with thermodynamic equilibrium pressure p0

acting as exclusive eigenvalue, the general form (65) reduces with a0=1/3 tr(p0I)to

Pstat.

ij (p0)=1

3tr (p0I)δij =p0δij .(68)

A formulation that readily reproduces the consistently validated description of hydrostatic

and barometric pressure. The stress ﬁeld’s viscous part is equally constructed on the basis

of equation (65) with a0=−ψtr()and a1=−2μs, yielding

Pdyn.

ij (ij)=−ψtr()+2μs,(69)

where ψis known in the literature as dilatational and μsas shear viscosity (cite). Bringing

together both contributions (68), (69) into equation (66) yields

Pij =p0−ψtr()δij −2μsij .(70)

Since the stress tensor ﬁeld, as given in equation (70) is symmetric, it is further decom-

posable into an isotropic and a deviatoric part (cite)

Pij =Piso.

ij +Pdev.

ij .(71)

This symmetry also allows to decompose the stress tensor’s dynamic part (66) into its

isotropic and deviatoric contributions. While the stress tensor’s static contribution (68)

is purely isotropic, its dynamic contribution’s isotropic part is computed by

Piso.

ij =1/3 tr(P)δij ,(72)

Piso.

ij =p0−ψ+2

3μstr(),(73)

yielding for its deviatoric part

Pdev.

ij =−2μsij −1

3tr()δij.(74)

Identifying the bulk viscosity coeﬃcient to be μb=ψ+2/3μsleads to the stress ﬁeld’s

contemporary form (cite)

Pij =p0−μbtr()δij −2μsij −1

3tr()δij.(75)

Equation (75) represents Cauchy’s stress tensor ﬁeld for Newtonian ﬂuids and the two

empirical viscosity coeﬃcients μb,μ

sare a measure of the ﬂuid’s linear momentum dis-

sipation due to local compressions and shearing motions respectively. Consequently, the

bulk viscosity has to be understood as a hydrodynamic contribution to the otherwise static

thermodynamic pressure p0and invoking Stokes’ hypothesis, i.e. μb=0,is equivalent to

state that local isotropic dilatations do not cause viscous stresses and therefore do not

dissipate momentum.

2.3.2 Sound attenuation

The bulk viscosity constitutes an indirectly measurable property and as such only its

ramiﬁcations on other phenomena can be observed. Despite its contribution to various

hydrodynamic phenomena, its eﬀect on a sound wave’s attenuation is the easiest to access

experimentally. While a transducer is emitting a deﬁned sound pulse into a quiescent ﬂuid,

the attenuation of the wave is readily determined by its amplitude decay over propagated

distance. The acoustic ﬁeld produced by the transducer partitions into two distinct regions.

The near-ﬁeld, or Fresnel region, is located directly in vicinity of the transducer’s surface

and is characterised by a cylindrical acoustic ﬁeld with a spatial expansion estimated by

the inverse Fresnel parameter z0

z0=d2

4λ=d2f

4c0

.(76)

The near-ﬁeld is followed by the far-ﬁeld, or Fraunhofer region, that is characterised by a

conical acoustic ﬁeld caused by the sound pulse’s dispersion, or beam spread. The angle

θof the pulse’s dispersion in the horizontal plane is estimated by

2θ= arcsinξλ

d,ξ∈R,(77)

with a ﬁtting parameter ξ. Due to the near-ﬁeld’s non-linearity, only the acoustic ﬁeld

in the Fraunhofer region can be determined by an acoustic approximation of the Navier-

Stokes equations.

2.3.3 Acoustic approximation

Acoustic waves are the result of local pressure perturbations that are periodically intro-

duced into an otherwise quiescent atmosphere. These disturbances propagate away from

their source by displacing the atmosphere’s particles and result in a temporary change of

its thermodynamic state. This mechanism is inherently dissipative and responsible for the

wave’s attenuation, which is adequately measured by an attenuation coeﬃcient α.

A description of αas a function of the ﬂuid’s thermodynamic state in the Fraunhofer re-

gion is readily obtained by introducing a perturbation ansatz into the governing equations

for mass, momentum and energy. The latter are most suitably considered in Eulerian

coordinates and read in Einstein notation

∂t(ρ)+



j=1

∂jρvj=0,(78)

∂t(ρvi)+



j=1

∂jρvivj+



j=1

∂jpδij −Πij=0,(79)

∂t1

2ρv2

i+ρu+



j=1

∂j1

2ρv2

ivj+ρuvj+



j=1

∂jqj+pδijvi−Πijvi=0,(80)

with all operators being projected onto d-dimensional Euclidean space. Within the hy-

drodynamic framework, the dissipation of linear momentum and consequently the wave’s

attenuation is described by the stress tensor’s Πij divergence

∂j(Πij)=



j=1

∂jμb−2

3μsd



n=1

∂n(vnδij)+μs∂j(vi)+∂i(vj).(81)

The problem is simpliﬁed by treating the viscosity coeﬃcients as constants μs,μ

b∈R>0

∂j(Πij)=μb+4

3μsd



j=1

∂jd



n=1

∂n(vn)δij−μsd



j=1

∂jd



n=1

∂n(vnδij)−



j=1

∂jj (vi),

and using the Grassmann identity to reformulate the vector Laplacian

∂j(Πij)=μb+4

3μsd



j=1

∂jd



n=1

∂n(vnδij)−μs



n=1



l=1



m=1

jknjlm∂n∂l(vm).(82)

Proposition 1 (Classical perturbation scheme). In the classical perturbation scheme,

each macroscopic observable ψ(x,t,ξ)factorises into

ψ(x,t,ξ)=ψ0+ξ·ψ1(x,t)+ξ2·ψ2(x,t)+... =



n=0

ξn·ψn(x,t),(83)

the sum of its unperturbed contribution ψn(x,t)and a perturbation variable ξnof order

n. Due to the assumed small amplitudes, each contribution ψidecreases with increasing

order, i.e. ψ0ψ1... ψn.

Restricting each observable’s perturbation to ﬁrst-order

T=T0+ξ·T1,

p=p0+ξ·p1,

ρ=ρ0+ξ·ρ1,

u=u0+ξ·u1,

vi=ξ·vi,1,

results in the acoustic approximation of the respective conservation equation (78) to (80).

Note, that the velocity vector ﬁeld viis entirely caused by the acoustic wave and its zeroth-

order perturbation is necessarily zero v0,i =0. A Helmholtz-Hodge decomposition of the

acoustic velocity ﬁeld vifurther separates the momentum equation’s (79) irrotational virr

1,i

from its solenoidal vsol

1,i contribution

v1,i =ijk∂i(ψj)+∂i(φ)!

=virr

1,i +vsol

1,i .(84)

While the heat ﬂux vector is assumed to be proportionate to the spatial changes of the per-

turbed temperature ﬁeld qj=−γ∂

j(T1), the Grassmann identity is applied to the velocity

vector, which eventually leads to the conservation equations’ acoustic approximations

∂t(ρ1)+



j=1

∂jρ0virr

1j=0,(85)

∂tρ0virr

1i+



j=1

∂j(p1)−μb+4

3μsd



j=1

∂j



m=1

∂mvirr

1mδij=0,(86)

∂tρ0vsol

1i−μs



j=1

∂jjvsol

1j=0,(87)

∂tρ0u1−γ



j=1

∂jj (T1)+



i=1



j=1

∂jp0δijvirr

1i=0.(88)

The linear momentum equation’s solenoidal part (87) describes rapidly decaying shear

waves, which are exclusively controlled by μsand are consequently neglected.

Substituting the ﬁrst-order density ρ1with pressure p1and temperature T1via an

acoustic equation of state yields a description of a sound wave’s propagation in terms of

directly measurable ﬁeld variables. Consequently, the density ρ=ρ(T,p)is expanded in

a Taylor series around the initial quiescent state

ρ−ρ0=∂ρ

∂pT·p−p0,+∂ρ

∂T p·T−T0.(89)

The diﬀerentials are calculated at ρ=ρ0

∂ρ

∂pT=cp

cv·∂ρ

∂ps=κ

,(90)

∂ρ

∂T p=−∂p

∂ρ−1

T·∂p

∂T =−βκ

,(91)

and are universally valid, i.e. no ideal gas assumptions have been made.

Result 1 (Acoustic equation of state). Retaining all ﬁrst-order terms after inserting

equations (90), (91) into equation (89) leads to the so-called acoustic equation of state

ρ1−κ

0p1−βT1=0.(92)

Taking the divergence of the linear momentum’s irrotational part (86) in combination with

conservation of mass (85) and the acoustic equation of state (92) results in the so-called

modiﬁed wave equation



j=1

∂jj(p1)−κ

0∂tt

−Dv∂t



j=1

∂jj

p1−βT1=0,(93)

with the viscous diﬀusivity Dv=(μb+4/3μs)ρ−1

0. The internal energy u1in equation

(88) is substituted with pressure and temperature via a ﬁrst-order Taylor expansion of

u=u(s, ρ)around the unperturbed equilibrium conditions

u−u0=∂u

∂sρu0·s−s0+∂u

∂ρsu0·ρ−ρ0,(94)

while exploiting the universally valid diﬀerentials

∂u

∂sρ=T0,(95)

∂u

∂ρs=p0

ρ2

.(96)

Again, retaining only ﬁrst-order terms, yields the acoustic energy equation

u1=T0·s1+p0

ρ2

0·ρ1,(97)

which inserted into equation (88) gives the ﬁrst intermediate result

ρ0T0∂t(s1)−γ



j=1

∂jj(T1)=0.(98)

Likewise, the entropy s=s(T,p)is expanded in a ﬁrst-order Taylor series around the

unperturbed equilibrium conditions

s−s0=∂s

∂T ps0·T−T0,+∂s

∂pTs0·p−p0,(99)

with the following diﬀerentials, which do not succumb to any ideal gas assumptions

∂s

∂T p=cp0

,(100)

∂s

∂pT=1

ρ2

0·∂ρ

∂T p.(101)

Retaining ﬁrst-order results after inserting equations (100), (101) into equation (99) leads

to the second intermediate result

∂t(T1)+ T0

ρ2

0cp0∂ρ

∂T p∂t(p1)−γ

ρ0cp0



j=1

∂jj(T1)=0,(102)

which in combination with the thermodynamic relation for the diﬀerential quotient

ρ2

0·∂ρ

∂T p=−κ−1

κβ ,(103)

eventually leads to the conservation of acoustic energy as a function of acoustic pressure

and density.

Result 2 (First-order perturbations). The dynamics of any ﬂuid under the acoustic

approximation is fully described by the modiﬁed wave equation (93) and energy conserva-

tion



j=1

∂jj(p1)−κ

0∂tt

−Dv∂t



j=1

∂jj

p1−βT1=0,(104)

∂tT1−κ−1

κβ p1−Dh



j=1

∂jj(T1)=0,(105)

with Dh=γ/(ρ0cp0)being the ﬂuid’s thermal diﬀusivity. This set of two diﬀerential

equations is suﬃcient to determine the two unknown quantities T1,p1unambiguously.

All parameters involved μs,μ

b,γ,κ,β as well as the equilibrium values ρ0,c

p0and c0are

considered constant and additionally assumed to be known a priori.

Transforming the ﬁrst-order perturbations (104), (105) from time to frequency domain

results in the diﬀerential equations’ reduction from partial to ordinary that are addi-

tionally of Helmholtz type. Sound signals that originate from plane transducers can be

viewed in a ﬁrst-order approximation as two-dimensional waves that propagate quasi one-

dimensionally in time.

Proposition 2 (Plane harmonic waves). The sound wave is assumed to propagate one-

dimensionally, harmonic and mono-frequent through the quiescent atmosphere, necessitat-

ing the following Fourier transform between time and frequency domain

h(xj,t)=F−1h

ˆ(xj,ω)=1

√2π∞

−∞

dωeiωt ·h

ˆ(xj,ω),ω∈R\{0}.(106)

Deﬁnition 2.1 (Propagation constant). Representing the wave equation in the fre-

quency domain reduces the Laplace operator to its eigenvalue, a solution strategy that has

originally been applied to the inviscid, one-dimensional case

√2π∞

−∞

dωeiωt−ω

c02∂zz ()+k2pˆ1=0,(107)

yet is straightforwardly generalised to viscous ﬂuids. The Fourier transform above moti-

vates the deﬁnition of the dimensionless propagation constant Γ

−c0

ω2∂zz ()=Γ

2,(108)

which is related to the wave number k, as evident from comparing equation (107) and

(108). Non-trivial solutions for p1exist, if and only if

Γ=c0

ωk. (109)

Fourier transforming the system of partial diﬀerential equations (104), (105) results in a

single algebraic equation that is straightforwardly solvable. To lighten the notation once

again, the viscous and thermal relaxation times τv,τ

hare introduced

√2π∞

−∞

dωeiωtω

c02(−1)Γ2−κ+τviωκΓ2pˆ1−κβ−1+τviωΓ2T

ˆ1=0,(110)

√2∞

−∞

dωeiωt1−τhiωΓ2T

ˆ1−κ−1

κβ pˆ1=0.(111)

While the acoustic pressure in Fourier space is eliminated in favour of the acoustic tem-

perature, non-trivial solutions for T

ˆ1and pˆ1are obtained if the terms in brackets vanish

leading to an algebraic equation

ˆ1Γ4ω2κτvτh−iωτh+Γ

21+κiωτh+τviω−1=0,(112)

whose solutions correspond to two diﬀerent propagational modes

Γ2=−i

2ωτh·1+κiωτh+iωτv∓D

1+κiωτv,(113)

with the discriminant Dgiven by

D2=1+ω22κτvτh−κ2τ2

h−τ2

v+2ωiκτh+τv−2τh.(114)

Exact solutions of equation (114) are unnecessary diﬃcult to obtain, however, assuming

small relaxation times τv,τ

h1will facilitate reasonable approximations.

Proposition 3 (Small relaxation times). The relaxation times in classical ﬂuids are

usually small compared to unity

τv=τb+τs=1

K0μb+4

3μs1,(115)

τh=1

K0κ−1γ

cp01,(116)

with the zero-frequency bulk modulus K0=ρ0c2

0, allowing to approximate the discriminant

(114) and consequently the propagation constant (113).

Theorem 1 (Square root approximation). The binomial theorem allows to approxi-

mate square roots of complex numbers to a convenient degree of accuracy

1+zm=∞



k=0 m

kzk≈m

0z0+m

1z1+m

2z2,z∈C,(117)

given the fact that zis small compared to unity, i.e. z1.

The approximation of equations (114), (117) is achieved by setting z=D2−1and m=1/2,

which then leads to a simple algebraic expression

D≈1+1

2D2−1−1

8D2−12,(118)

to approximate the square roots of the complex number D.

Proposition 4 (Acoustic approximation). Taking everything that has been proposi-

tioned so far into account leads to the approximation of the discriminant when only ﬁrst

order terms in ωτvand second-order in ωτhare considered

D≈1+2κω2τhτv−2κω2τ2

h−2ω2τhτv+2ω2τ2

h+iωκτh+ωτv−2ωτh.

Result 3 (Sound modes). Equation (113) has two solutions for the dimensionless prop-

agation constant, with the ﬁrst solution being known as the propagation sound mode Γ2

and the second one as the thermal sound mode Γ2

h. The propagation sound mode cor-

responds with the negative sign in equation (113) while observing small relaxation times

(115), which is equivalent with 1+κiωτv≈1and eventually leads to

Γ2

p≈1−iωτv+κ−1ωτh.(119)

The result (119) solves both the modiﬁed wave equation and the energy conservation

equation in Fourier space to ﬁrst-order and is related to the propagation constant, as in

equation (109), yielding

kp=ω

Γp.(120)

Since the relaxation times are assumed to be small, certain approximations of the tangent

function and its inverse are valid, i.e. tan(x)≈xand arctan(x)≈x, which in turn after

applying de Moivre’s formula to compute the square roots of a complex number leads to

kp≈ω

c01−1

2iωτv+(κ−1)ωτh!

=ω

c01+iα,(121)

where only the solution with a positive real part has been considered. The speed of sound

in viscous and heat conducting ﬂuids is frequency independent and identical with the

thermodynamic speed of sound, i.e. c0=c(ω), and the losses due to transport only aﬀect

the attenuation part of the propagation constant.

Result 4 (Sound attenuation coeﬃcient). The imaginary part of the propagation mode’s

constant determines the sound attenuation coeﬃcient α

α=−Im(kp),(122)

which can be formulated exclusively in terms of transport properties by taking equations

(115) and (116) into account

αλ=ωπ

K04

3μs+(κ−1) γ

cp0

+μb=ω·ζ(T,ρ),(123)

and is considered here per wavelength, i.e. αλ=α·λ. Under the paradigm of a mono-

frequent and harmonic wave propagating through an otherwise quiescent atmosphere, the

absorption coeﬃcient αλfactorises into a frequency contribution and a function that ex-

clusively depends on the thermodynamic state, as validated by experimental data, cf. Fig.



 



















Figure 5. The proposed linearity between frequency and the attenuation coeﬃcient

(123) becomes evident by evaluating experimental sound attenuation data

at various frequencies. All experimental data were obtained at states along

the liquid binodal line for diﬀerent noble gases starting from T/Tc=0.62

(red) to T/Tc=0.95 (yellow). An incipient non-linear behaviour, emerges

at temperatures around T/Tc=0.97 (pink).

Under the restrictions presented, the acoustic approximation of classical hydrodynamics,

i.e. the linearised Navier-Stokes equation, is capable to adequately predict a sound wave’s

attenuation. In addition to the constraints leading to equation (123), the sound wave’s

frequency is also restricted to moderate ultrasonics, i.e. f≤280 MHz [143], since for

frequencies above this threshold, the wave experiences artiﬁcial attenuation in the form

of so-called high frequency blockage due to the particles’ inertia. The linear ansatz has

further been proven insuﬃcient in vicinity of the critical point, where the presence of

diﬀerent relaxation modes as well as couplings thereof have been observed [144, 145], as

indicated at temperatures around T/Tc≈0.95 in Fig. 5.

As mentioned above, there is no experimental method to determine the bulk viscosity

directly and sound attenuation measurements in the linear regime provide the most suit-

able alternative. Sound attenuation measurements are most adequately performed via the

two-channel pulse-echo technique [146]. A transducer is emitting a deﬁned sound pulse

into two distinctly sized channels. Each wave experiences diﬀerent attenuation depending

on the propagated distance, cf. Fig 6. Both amplitude peaks decay exponentially following

an envelope that is determined by the attenuation coeﬃcient αλvia [147]

A(z2)

A(z1)= exp−αλ

λz2−z1,z

2>z

1,α

λ≥0.(124)



      

















Figure 6. Vertical signal’s reconstruction (fast Fourier transform) of diﬀerently at-

tenuated sound waves as detected at the device’s transducer [148]. While

the ﬁrst amplitude corresponds to the less attenuated signal, having prop-

agated through the shorter channel, the second signal corresponds to the

device’s longer channel. Both amplitude’s maxima follow an envelope that

is determined by the attenuation coeﬃcient αλvia equation (124).

The combination of equations (123) and (124) eventually leads to the bulk viscosity’s

determining equation

μb=λK0

πωz2−z1lnA(z1)

A(z2)−4

3μs+κ−1γ

cp0,(125)

that evidently requires substantial information about the respective ﬂuid’s thermodynamic

properties. Since each contribution is subject to either statistical or systematic error, de-

termining μbvia equation (125) leads to large uncertainties Δμb. Following from equation

(123) with equations (115) and (116), the attenuation coeﬃcient can alternatively be un-

derstood in terms of relaxation times

αλ=ωπ ·τs+τh+τb.(126)

Likewise, the bulk viscosity can be interpreted in terms of its relaxation time τb

μb=K0τb,(127)

that factorises into diﬀerent relaxation modes

μb=Kr

Ciτi.(128)

Each mode is thereby an improper time integral of a relaxation function

τi=1

lim

t→∞t

dtBR,i(t),(129)

and the combination of equations (128) and (129) leads to the alternative interpretation

of bulk viscosity in terms of its fundamental relaxation modes

μb=Kr

lim

t→∞t

dtBR,i(t).(130)

Each mode BR,i follows a so-called Kohlrausch-William-Watts [149, 150] or stretched ex-

ponential function. While for associating ﬂuids, such as water [151] and methanol, only

two modes are present and the slow mode is non-degenerate, i.e. ω=0

BR(t)=Cfexp−t

δfβf·cosωt+Csexp−t

δsβs,ω∈(0,2π],(131)

with constrained weights Cf+Cs=1, for non-associating liquids three diﬀerent relaxation

modes have been observed (cite)

BR(t)=Cfexp−t

δfβf+Cmexp−t

δmβm+Csexp−t

δsβs,(132)

with constrained weights Cf+Cm+Cs=1. Each relaxation ansatz (131) and (132) can be

ﬁtted to a time auto-correlation function obtained from equilibrium molecular dynamics

simulations. In the following original publication, the ansatz (130) in combination with

the relaxation mode’s sum (132) resulted in an equation of state for liquid noble gases.









    

















Figure 7. The bulk viscosity of associating liquids, such as methanol, can be deter-

mined by the superposition of two relaxation modes (131).









   





















Figure 8. The bulk viscosity of non-associating liquids, like argon, in contrast was

observed to be determined by the superposition of three relaxation modes

(132).

2.3.4 Bulk viscosity of liquid noble gases

R.S. Chatwell and J. Vrabec, Journal of Chemical Physics,152, 094503 (2020)

Reprinted from the Journal of Chemical Physics with the permission of the American

use only. Any other use requires prior permission of the author and AIP Publishing. The

following article appeared in J. Chem. Phys.,152, 094503 (2020) and my be found at

https://doi.org/10.1063/1.5142364.

In the following publication, an equation of state for the bulk viscosity of liquid noble

gases is proposed. The poor availability of experimental bulk viscosity data for each noble

gas can be circumvented by exploiting the corresponding states principle. The resulting

consolidated data set indicates an increasing bulk viscosity when approaching the liquid

binodal, an eﬀect that intensiﬁes with rising temperature. However, due to the large un-

certainties in the available experimental data, no physically sound equation of state can

be determined. Consequently, a substantial set of bulk viscosity data was determined via

atomistic simulations to cover larger sections of the liquid state. This synthetic data set

is subject to much smaller uncertainties Δμband in addition allows to interpret μbas

superposition of diﬀerent relaxation modes, as given in equation (132). Evaluating this

multi-mode relaxation ansatz yields a two-parametric power function of μbwith both pa-

rameters showing a conspicuous saturation behaviour.

The leading author R.S. Chatwell has consolidated all available experimental data, per-

formed the atomistic simulations and formulated the equation of state. The manuscript

was revised and redacted by J. Vrabec.

The Journal

of Chemical Physics ARTICLE scitation.org/journal/jcp

Bulk viscosity of liquid noble gases

Cite as: J. Chem. Phys. 152, 094503 (2020); doi: 10.1063/1.5142364

Submitted: 12 December 2019 •Accepted: 13 February 2020 •

Published Online: 3 March 2020

René Spencer Chatwell and Jadran Vrabeca)

AFFILIATIONS

Thermodynamics and Process Engineering, Technische Universität Berlin, 10587 Berlin, Germany

a)Author to whom correspondence should be addressed: [email protected]

ABSTRACT

An equation of state for the bulk viscosity of liquid noble gases is proposed. On the basis of dedicated equilibrium molecular dynamics

simulations, a multi-mode relaxation ansatz is used to obtain precise bulk viscosity data over a wide range of liquid states. From this dataset,

the equation of state emerges as a two-parametric power function with both parameters showing a conspicuous saturation behavior over

temperature. After passing a temperature threshold, the bulk viscosity is found to vary signiﬁcantly over density, a behavior that resembles

the frequency response of a one pole low-pass ﬁlter. The proposed equation of state is in good agreement with available experimental sound

attenuation data.

Published under license by AIP Publishing. https://doi.org/10.1063/1.5142364

., s

I. INTRODUCTION

In contrast to prevailing opinion, bulk viscous effects have

widely been explored in ﬂuid mechanics and even share a somewhat

controversial history. In 1845, Stokes1had argued geometrically that

Cauchy’s stress tensor ﬁeld,2

Πij =(μb−2

3μs)∂nvnδij +μs(∂jvi+∂ivj), (1)

entails not only the well-known shear viscosity μsbut also the bulk

viscosity μb, which he concurrently postulated to be zero,

μb

=0. (2)

Stokes, whose proof can hardly be considered rigorous, had

limited his argument to incompressible ﬂuids and remained skep-

tical that his ﬁnding (2) would emerge as universally valid. For

more than a century, this hypothesis was afﬂicted with miscon-

ceptions until the Royal Society hosted “a discussion on the ﬁrst

and second viscosities of ﬂuids,”3–18 reinvigorating investigations

of bulk viscous effects on a wide variety of ﬂuid mechanical

problems.

Riemann’s solution19 of the Euler equation20 was among the

ﬁrst. Retaining the bulk viscosity in the stress tensor ﬁeld21 conclu-

sively solved the problem of insensibly small wave thicknesses that

were predicted22,23 in descriptions limited to shear viscosity.24 The

conjectured increase in wave thickness was qualitatively conﬁrmed

in successive atomistic simulations of non-ideal liquids25,26 and in

numerical investigations of rareﬁed gases,27–29 given that the shock

structure is symmetric.30

Bulk viscous effects have long been considered mere higher-

order contributions.31 However, including the bulk viscosity in

the acoustic approximation32,33 straightforwardly explained the

observed second-order ﬁelds associated with ultrasonic waves. The

vorticity that is generated across a propagating wave’s free sur-

face induces a counter-oriented circulatory ﬂow34–39 that is also

known as acoustic streaming or quartz-wind. In hypersonic bound-

ary layer approximations, bulk viscous effects are promoted to

second-order in the pressure distribution40 and even ﬁrst-order

in the outer and inner ﬂow velocities of high Reynolds number

ﬂows.41

In ﬂuids conﬁned to capillaries, the bulk viscosity contributes

to ﬁrst-order in radial pressure and to second-order in the axial

velocity.42–47

Bulk viscous behavior was also investigated for more complex

scenarios. An increased shock wave thickness affects the outer ﬂow’s

adverse pressure gradient and consequently suppresses the shock

induced boundary layer separation,48 while additionally the shock

wave’s location and strength are much more accurately predicted

when bulk viscous effects are included.49 Likewise, bulk viscous

damping has been observed in compressible turbulent ﬂows.50 A

non-zero bulk viscosity enhances kinetic energy dissipation while

additionally inhibiting the energy transfer between translational and

conﬁgurational energy, thus rendering the ﬂow effectively incom-

pressible.51

J. Chem. Phys. 152, 094503 (2020); doi: 10.1063/1.5142364 152, 094503-1

Published under license by AIP Publishing

The Journal

of Chemical Physics ARTICLE scitation.org/journal/jcp

Each of the aforementioned investigations, however, has suf-

fered from missing, incomplete, or unreliable bulk viscosity data and

was consequently restricted to predominantly qualitative results.

II. SOUND ATTENUATION

Since its introduction, the bulk viscosity was closely associ-

ated with linear acoustics. Stokes had originally proposed that sound

attenuation measurements could conﬁrm his hypothesis (2) experi-

mentally. A propagating wave’s linear momentum is dissipated by a

surrounding atmosphere, which leads to the exponential decay of its

amplitude A(z) over traveled distance Δz=z2−z1and is measured

by an attenuation coefﬁcient α,

A(z2)

A(z1)=exp(−αλ

Δz

λ), (3)

given here per wavelength λ, i.e., αλ=αλ. In a ﬁrst estimate of αλ

for one-dimensional motion, limited to shear viscosity μs, Stokes

found that the attenuation factorizes into frequency ω=2πfand a

transport function ξ,

αλ=ωξStokes =ωπ

3μs, (4)

where K=ρc2is the ﬂuid’s low-frequency modulus of compres-

sion, ρits density, and cits thermodynamic speed of sound. Kirch-

hoff52 advanced the discussion by including heat conduction and

established that both effects superimpose linearly in the transport

function,

αclassical

λ=ω(ξStokes +ξKirchhoff)

=ωπ

K(4

3μs+(κ−1)γ

cp), (5)

where κ=cp/cvis the ratio of speciﬁc heats and γis the thermal

conductivity. The attenuation predicted by this classical theory cor-

responds ostensibly well with low frequency acoustic measurements.

Subsequently performed ultrasonic measurements,53–57 how-

ever, disclosed large deviations from the classical description (5)

and motivated Herzfeld58 and Kneser59,60 to introduce an additional

mechanism,

αλ=αclassical

λ+αexcess

λ. (6)

Both connected classical hydrodynamics to relaxation theory and

attributed the excess attenuation αexcess

λto a time lag that occurs dur-

ing the transfer of energy between the molecules’ translational and

internal degrees of freedom. In contrast, Tisza61 refrained from any

physical interpretation and incorporated the Herzfeld mechanism

into the transport function ξby force ﬁtting the complex mechanism

of relaxation into a scalar valued bulk viscosity,62

αλ=ω(ξStokes +ξKirchhoff +ξTisza)

=ωπ

K(4

3μs+(κ−1)γ

+μb). (7)

This extended theory has been validated experimentally up to mod-

erate ultrasonic frequencies, i.e., f≤280 MHz,63 over a wide range

of thermodynamic states.

In the extended vicinity of the critical point, however, an

anomalously high attenuation was observed64–66 that was conclu-

sively attributed to long-range density correlations,67–73 which are

alterations of structural relaxation effects.74–77 The total attenuation

partitions into the already established background part (7) and a

critical contribution,78,79

αtotal

λ=αλ+αcritical

λ. (8)

Acoustic dispersion, i.e., the frequency dependent speed of sound

c(ω), conﬁnes this critical attenuation to an extended critical region

in the thermodynamic state space, cf. supplementary material,

αcritical

λ=πI(˜

ω)

J(˜

ω)([c(ω)

c]2−1), (9)

where I(˜

ω),J(˜

ω)are improper integrals of a characteristic

frequency ˜

ω.

III. EXPERIMENTAL DATA

At present, the bulk viscosity is determined experimen-

tally either by non-resonant Rayleigh–Brillouin scattering80–90 or

ultrasonic attenuation measurements.91–121 Both techniques have

successfully been applied to a variety of substances, yet each mea-

surement series was restricted to selective thermodynamic states.

A substantially larger dataset, however, can be obtained for

liquid noble gases by utilizing their self-similar behavior. While all

available sound attenuation measurements for neon, argon, kryp-

ton, and xenon were evaluated, data subject to critical attenuation

FIG. 1. Overview of thermodynamic states at which experimental sound atten-

uation data are available for neon106 (circles), argon98–101,110,112,113,115,116

(squares), krypton115–117,119 (triangles), and xenon79,115,116,118,119,122 (diamonds).

Highlighted: Seven isotherms were selected T/Tc= 0.759, 0.76, 0.83, 0.86,

0.863, 0.91, and 0.93, while T/Tc= 0.759 and 0.863 were omitted in the plot

for visibility reasons, along which atomistic simulations were performed to com-

plement and extend the dataset to higher pressures p/pc≤21. All thermodynamic

states were reduced with the respective ﬂuid’s critical pressure pcand density ρc.

The extended critical region was constructed on the basis of sound dispersion

measurements from the literature and is delimited by the dashed line.

J. Chem. Phys. 152, 094503 (2020); doi: 10.1063/1.5142364 152, 094503-2

Published under license by AIP Publishing

The Journal

of Chemical Physics ARTICLE scitation.org/journal/jcp

TABLE I. The following parameters for the full, i.e., untruncated Lennard-Jones poten-

tial σ,, and atomic mass mwere used in the canonical transformation130 to reduce

the bulk viscosity and time t⋆=t√/(mσ2), where kBis Boltzmann’s constant.

/kB(K) σ(Å) m(u)

Ne 33.92 2.801 20.180

Ar 116.79 3.395 39.948

Kr 162.58 3.627 83.798

Xe 226.14 3.949 131.293

were identiﬁed and discarded, cf. Fig. 1. The bulk viscosity was cal-

culated on the basis of the extended classical theory (7) and in con-

trast to previous works, more accurate thermodynamic data were

available by resorting to most recent equations of state.123–127 Sub-

sequently, μbwas reduced by a canonical transformation resting on

the Lennard-Jones potential, i.e., μ⋆

b=μbσ2/√m, with the required

set of parameters speciﬁed in Table I.

Each contribution to bulk viscosity is afﬂicted by a different

uncertainty. While the uncertainty of most recent thermodynamic

data are well established to range from Δρ/ρ= 0.005 to Δμs/μs

= 0.3 depending on substance and state, the attenuation coefﬁcient

αλhas been determined by single measurements at the respective

state point, and thus, only its absolute maximum errors Δαλhave

been estimated. Moreover, it is not assured that systematic measure-

ment errors in the literature data, speciﬁcally diffraction effects,128

were properly accounted for. Consequently, the reduced bulk vis-

cosity’s uncertainty Δμbwas determined to be linearly affected129 by

its various contributions Δαλ,Δc,Δμs,Δγ,Δcv,Δcp,

Δμb=∣K

ωπ ∣Δαλ+⋯+∣(κ−1)γ

p∣Δcp. (10)

FIG. 2. Overview of bulk viscosity data calculated from the extended classical the-

ory on the basis of experimental sound attenuation data, taken from the literature at

the state points indicated in Fig. 1.Highlighted: Experimental data at the selected

isotherms T/Tc= 0.76, 0.83, 0.86, 0.91, 0.96 with their respective uncertainties,

calculated from Eq. (10), indicating a monotonic decline of μbas a function of

density.

After selecting and evaluating the experimental data, a decline of

the bulk viscosity along each isotherm, i.e., from saturation line

toward higher density, can qualitatively be inferred. The effect tends

to increase with temperature, however, due to large errors neither

the function’s gradient nor the curvature can accurately be speciﬁed,

cf. Fig. 2.

IV. MOLECULAR DYNAMICS SIMULATION

To interpret and extend the experimental dataset to higher

densities, additional bulk viscosity data were sampled by equi-

librium molecular dynamics (EMD) simulations along seven

selected isotherms T/Tc= 0.759, 0.76, 0.83, 0.86, 0.863, 0.91,

0.93, cf. Fig. 1. The bulk viscosity was determined microscopi-

cally by time-autocorrelation functions of local small-scale, tran-

sient pressure ﬂuctuations131,132 that are intrinsic in any ﬂuid under

equilibrium.133 The Lennard-Jones interaction potential was used,

which has demonstrated to resolve such small-scale dynamics ade-

quately and hence successfully describes macroscopic transport in

liquid noble gases.134–138

In the present work, the bulk viscosity’s autocorrelation func-

tion BAwas sampled in the microcanonical (NVE) ensemble, uti-

lizing the fully open source program ms2.139 The necessary average

energies ⟨E⟩were determined by preceding canonical (NVT) ensem-

ble simulations for the given pair of temperature and density. Finite

size effects were minimized by placing N= 4096 particles in cubic

volumes with periodic boundary conditions and choosing a sufﬁ-

ciently large cutoff radius rc≥5.5σ.140–142 The employed particle

number is well chosen, as simulations containing N= 12000 par-

ticles yielded virtually identical results, cf. supplementary material.

In order to adequately resolve both the existing small-scale dynam-

ics and also the slowly decaying pressure ﬂuctuations,143 a reduced

integrator time step Δt⋆=5⋅10−4was speciﬁed and each autocor-

relation function BAwas sampled over a reduced time period of at

least t⋆≥14.6.

A. Relaxation ansatz

The ﬂuid’s intrinsic small-scale pressure ﬂuctuations have con-

clusively been established to relax in different modes.144–146 Each

mode decays exponentially over time following a Kohlrausch–

Williams–Watts function.147,148 For all of the investigated state

points, three superimposing relaxation modes were found to be

present, leading to the relaxation model’s analytical form,

BR(t)=Cfexp(−(t

δf)βf)+Cmexp(−( t

δm)βm)

+Csexp(−(t

δs)βs). (11)

The ﬁrst term describes the fast, and the subsequent terms describe

the intermediate and slow modes, respectively. The weighting fac-

tors are constraint Cf+Cm+Cs= 1, and the Kohlrausch param-

eters δi,βiare a measure of relaxation time scale and distortion

from the exponential function, respectively. The eight independent

parameters of Eq. (11) were determined by ﬁtting the relaxation

model BRto the sampled autocorrelation function BAat each state

point independently.

J. Chem. Phys. 152, 094503 (2020); doi: 10.1063/1.5142364 152, 094503-3

Published under license by AIP Publishing

The Journal

of Chemical Physics ARTICLE scitation.org/journal/jcp

Each mode’s average relaxation time τiis properly deﬁned

as integral mean value of its respective contribution BR,ito the

relaxation model,149

τi=1

lim

t→∞ ∫t

0dt(BR,i(t)). (12)

As originally proposed by Maxwell, the bulk viscosity μbis propor-

tional to the cumulative averaged relaxation time,

μb=Kr∑

Ciτi, (13)

with the proportionality constant Krbeing the ﬂuid’s relaxation

modulus.150

V. RESULTS

A. Relaxation times

The present results are exemplary discussed for states along

the isotherm T/Tc= 0.86 yet are qualitatively similar for all other

investigated state points as disclosed in the supplementary material.

The sampled autocorrelation function BApartitions into three

segments, with each segment being dominated by a different mode,

cf. Fig. 3. The two dents in the sampled signal readily indicate the

fast and intermediate mode’s decay. Accompanying the latter, incip-

iently small scale oscillations are ampliﬁed, giving rise to substantial

noise contributions in the slow mode. After reconstructing its sig-

nal by post-processing, the good agreement between the sampled

autocorrelation function BAand the proposed relaxation model BR

becomes apparent.

More importantly, in contrast to the sampled autocorrelation

function that is plagued by noise, the employed relaxation model’s

time integral properly converges to a deﬁnite value, thus allowing to

determine the bulk viscosity unambiguously at each state point, cf.

Fig. 4.

FIG. 3. Comparison of sampled autocorrelation function and relaxation model

including all three relaxation modes. The gray line constitutes BAsampled at T/Tc

= 0.86 and ρ/ρc= 2.06, while the solid black line represents the relaxation model

(11), and its fast, intermediate, and slow modes are depicted by dashed, short-

dashed, and dashed–dotted lines, respectively. The post-processed simulation

signal (dotted line) is in good agreement with the relaxation model.

FIG. 4. Comparison of the integrated autocorrelation function BAwith the relaxation

model’s (11) integral at T/Tc= 0.86 and ρ/ρc= 2.06. Due to noise contribu-

tions to BA, the bulk viscosity μbis difﬁcult to determine precisely by molecular

dynamics simulation (gray line). In contrast, the proposed relaxation model (solid

black line) converges toward an unambiguous value at ﬁnite times. The dashed,

short-dashed, and dotted–dashed lines represent the fast, intermediate, and slow

modes, respectively.

The average reduced relaxation times τiwere found to decline

exponentially with density for each mode and to differ roughly

by one order of magnitude among the modes up to ρ/ρc≤2.2.

While all relaxation modes are increasingly damped with rising den-

sity, facilitating shorter relaxation times, the slow mode is damped

disproportionately, cf. Fig. 5.

FIG. 5. Distribution of average reduced relaxation times τifor all three modes

along the isotherm T/Tc= 0.86. While each mode is identiﬁed to relax

exponentially, the slow mode is predominantly affected by increasing den-

sity. The symbols represent the relaxation times of the fast τs(white circles),

the intermediate τm(black triangles), and the slow mode τs(gray squares),

respectively.

J. Chem. Phys. 152, 094503 (2020); doi: 10.1063/1.5142364 152, 094503-4

Published under license by AIP Publishing

The Journal

of Chemical Physics ARTICLE scitation.org/journal/jcp

B. Equation of state

The most recognized equation of state, of the very few that

have actually been discussed in the literature,151,152 relates the bulk

viscosity to a power function of density,

μb∝ρα, (14)

for which special case solutions with ﬁxed exponents

2α=−1, 0, 1, 3 exist.153–156 This proportionality, which was origi-

nally proposed in the context of viscous cosmological ﬂuids, is much

more universal and also applies to liquid noble gases if the observed

temperature dependence is included.

After consolidating all relaxation results, a threshold tempera-

ture Tt/Tc∼0.74 emerges below which the reduced bulk viscosity

indicates virtually no variation with density.

In contrast, at higher temperatures, μbincreases progressively

toward saturated liquid states, an effect that intensiﬁes with rising

temperature, cf. Fig. 6.

TABLE II. Best ﬁt parameters ai,bi,ciof Eq. (16).

aibici

α1−0.93 5.91 8.67

α2−0.53 −1.49 7.86

While any physically sound equation of state must necessar-

ily establish a unique one-to-one relation between μband each state

point, i.e., being bijective, entropy constraints additionally restrict

this equation to be non-negative157,158 and the present results fur-

ther specify it to be convex with monotonically decreasing gradient

and curvature. Satisfying all conditions, a reduced two-parametric

power function is proposed

μ⋆

b=(ρ

ρc−1)α1

+α2, (15)

FIG. 6. Outline of bulk viscosity variations from the saturation line toward high density along four isotherms (a) T/Tc= 0.759, (b) T/Tc= 0.83, (c) T/Tc= 0.86, and (d) T/Tc

= 0.863. The gray shaded areas represent the experimentally determined bulk viscosity, according to Eq. (7), including its absolute maximum error, i.e., μb±Δμb, according

to Eq. (10). The symbols indicate bulk viscosity data obtained by the employed relaxation ansatz (11) on the basis of the present EMD simulations. The solid line constitutes

the present equation of state (15).

J. Chem. Phys. 152, 094503 (2020); doi: 10.1063/1.5142364 152, 094503-5

Published under license by AIP Publishing

The Journal

of Chemical Physics ARTICLE scitation.org/journal/jcp

FIG. 7. Variation of parameters α1(circles) and α2(squares) over temperature.

Above the threshold temperature T/Tc∼0.74 indicated by the vertical line, the

bulk viscosity shows a substantial density dependence.

with parameters α1,α2depending exclusively on temperature,

αi=ai+bi⋅tanh(ci(T

Tc−1)). (16)

Both parameters, as speciﬁed in Table II, offer a salient saturation

behavior that closely resembles the frequency response of a one

TABLE III. Comparison of the bulk viscosity from the present equation of state

(15) with literature data that have either been obtained by molecular dynamics

simulation140,159 or by theoretical calculations.160

μbσ2/√mμ

bσ2/√m

T/Tcp/pcρ/ρcEq. (15) literature

0.795 0.545 2.196 1.17 1.20 ±0.37140

... 3.616 2.353 0.99 0.93 ±0.05

... 8.727 2.510 0.92 0.87 ±0.05

... 16.558 2.667 0.88 0.89 ±0.05

0.875 1.118 2.039 1.42 1.10 ±0.36

... 3.274 2.196 0.98 1.03 ±0.05

... 6.966 2.353 0.79 0.79 ±0.04

0.674 0.473 2.448 1.00 0.97 ±0.10159

0.679 0.707 ... 1.00 0.98 ±0.10

0.826 7.875 ... 0.89 0.75 ±0.08

0.699 17.065 2.773 0.94 1.10 ±0.11

0.763 21.558 ... 0.91 1.00 ±0.11

0.802 24.239 ... 0.87 0.96 ±0.10

0.605 3.117 2.640 0.96 1.05160

0.652 6.149 ... 0.96 0.93

0.667 7.126 ... 0.96 1.01

0.678 7.800 ... 0.96 1.00

0.747 12.086 ... 0.93 0.86

0.792 14.845 ... 0.89 0.86

FIG. 8. Illustration of the present equation of state (15). The solid lines constitute

the bulk viscosity’s variation over density at constant temperature, and the dashed

lines constitute its variation at constant pressure. The present results straightfor-

wardly explain the experimental results in Fig. 2, which already suggested the

bulk viscosity to be a monotonically declining function of density with increasing

gradient for higher temperatures.

pole low-pass ﬁlter, cf. Fig. 7, causing the ﬂuid to become increas-

ingly more bulk viscous after passing the threshold temperature.

This observed temperature dependence is caused by a transition

from short-range order, that is present at low temperatures,150 to the

long-range density correlations that are intrinsic within the extended

critical region. The present equation of state is in good agreement

not only with experimental sound attenuation data but also with

concurrent MD simulations, cf. Table III.

VI. CONCLUSION

An equation of state for the bulk viscosity of liquid noble

gases is proposed. The bulk viscosity originates microscopically

from relaxations of small-scale pressure ﬂuctuations which were

found to decay in three different modes following stretched expo-

nential functions. The slow mode was observed to be dispropor-

tionately affected by high density and the average relaxation times

τibetween the modes to differ roughly by one order of magni-

tude. Each mode was determined on the basis of an autocorrelation

function that was straightforwardly sampled by EMD simulations

at the respective state point. In order to adequately resolve the slow

mode, considerably long autocorrelation functions were necessary.

The equation of state emerges as a two-parametric power func-

tion with parameters depending exclusively on temperature, cf.

Fig. 8. This temperature dependence is attributed to a transition

from short-range order that is present at high densities to long-range

density correlations that arise when approaching the extended criti-

cal region. After a threshold T/Tc∼0.74 is passed, the ﬂuid becomes

increasingly more bulk viscous. This effect causes sound attenua-

tion to rise progressively with temperature, closely resembling the

frequency response of a one pole low-pass ﬁlter. In addition, both

bulk viscosity coefﬁcients were observed to exhibit opposing behav-

ior, i.e., an increase in bulk viscosity corresponds to a decline of

shear viscosity and vice versa, causing the viscosities’ ratio to peak

J. Chem. Phys. 152, 094503 (2020); doi: 10.1063/1.5142364 152, 094503-6

Published under license by AIP Publishing

The Journal

of Chemical Physics ARTICLE scitation.org/journal/jcp

μb/μs∼5 at the highest investigated temperature close to the satura-

tion line.

SUPPLEMENTARY MATERIAL

See the supplementary material for the construction of the crit-

ical region, as well as for a full disclosure of all data associated with

this manuscript.

ACKNOWLEDGMENTS

This work was carried out under the auspices of the

Boltzmann–Zuse Society (BZS). All simulations were performed

either on the HPC clusters OCuLUS and Noctua at the Paderborn

Center for Parallel Computing (PC2) or on the Cray CX40 system

Hazel Hen at the High Performance Computing Centre Stuttgart

(HLRS) with resources allocated according to Grant No. MMHBF2.

REFERENCES

1G. G. Stokes, “On the theories of the internal friction of ﬂuids in motion, and

of the equilibrium and motion of elastic solids,” Cambridge Philos. Soc. 8, 287

(1845).

2A. L. Cauchy, “Recherches sur l’equilibre et le mouvement intérieur des corps

solides ou ﬂuides. Élastiques ou non élastiques,” Bull. Soc. Philomath. 2, 9 (1823).

3L. Rosenhead, “Introduction. The second coefﬁcient of viscosity: A brief review

of fundamentals,” Proc. R. Soc. London, Ser. A 226, 1 (1954).

4P. E. Doak, “Vorticity generated by sound,” Proc. R. Soc. London, Ser. A 226,7

(1954).

5E. G. Richardson, “Acoustic experiments relating to the coefﬁcient of viscosity of

various liquids,” Proc. R. Soc. London, Ser. A 226, 16 (1954).

6R. O. Davies, “Kinetic and thermodynamic aspects of the second coefﬁcient of

viscosity,” Proc. R. Soc. London, Ser. A 226, 24 (1954).

7G. I. Taylor, “The two coefﬁcients of viscosity for an incompressible ﬂuid

containing air bubbles,” Proc. R. Soc. London, Ser. A 226, 34 (1954).

8G. I. Taylor, “Notes on the volume viscosity of water containing bubbles,” Proc.

R. Soc. London, Ser. A 226, 38 (1954).

9R. O. Davies, “A note on Sir Geoffrey Taylor’s paper,” Proc. R. Soc. London,

Ser. A 226, 39 (1954).

10H. O. Kneser, “Transport and relaxation phenomena,” Proc. R. Soc. London,

Ser. A 226, 40 (1954).

11J. E. Piercy and J. Lamb, “Acoustic streaming in liquids,” Proc. R. Soc. London,

Ser. A 226, 43 (1954).

12J. H. Andreae and J. Lamb, “Ultrasonic measurements and the second viscosity

of carbon disulphide,” Proc. R. Soc. London, Ser. A 226, 51 (1954).

13J. Meixner, “On the thermodynamic theory of the second viscosity,” Proc. R.

Soc. London, Ser. A 226, 51 (1954).

14S. M. Karim, “Experimental determination of the second viscosity,” Proc. R. Soc.

London, Ser. A 226, 56 (1954).

15J. G. Oldroyd, “Note on the hydrodynamic and thermodynamic pressures,”

Proc. R. Soc. London, Ser. A 226, 57 (1954).

16C. A. Truesdell, “The present status of the controversy regarding the bulk

viscosity of ﬂuids,” Proc. R. Soc. London, Ser. A 226, 59 (1954).

17E. N. da C. Andrade, “Review of discussion,” Proc. R. Soc. London, Ser. A 226,

65 (1954).

18R. O. Davies and L. Rosenhead, “The two viscosities of ﬂuids,” Nature 173, 1209

(1954).

19B. Riemann, “Über die Fortpﬂanzung ebenener Luftwellen von endlicher

Schwingungsbreite,” in Collected Works of Bernhard Riemann, edited by H. Weber

(Dover, 1953), p. 157.

20L. Euler, “Principes généraux du mouvement des ﬂuides,” Mém. Acad. Sci.

Berlin 11, 274 (1755).

21D. Gilbarg and D. Paolucci, “The structure of shock waves in the

continuum theory of ﬂuids,” J. Ration. Mech. Anal. 2, 617 (1953), see

https://www.jstor.org/stable/24900350.

22R. Becker, “Stoßwelle und Detonation,” Z. Phys. 8, 321 (1921).

23H. Grad, “The proﬁle of a steady shock wave,” Commun. Pure Appl. Math. 5,

257 (1952).

24W. Tollmien, H. Schlichting, H. Görtler, and F. W. Riegels, “Zur Theorie des

Verdichtungsstoßes,” in Ludwig Prandtl Gesammelte Abhandlungen, edited by F.

W. Riegels (Springer, Berlin, 1961).

25W. G. Hoover, “Structure of a shock-wave front in a liquid,” Phys. Rev. Lett. 42,

1531 (1979).

26B. L. Holian, W. G. Hoover, B. Moran, and G. K. Straub, “Shock-wave structure

via nonequilibrium molecular dynamics and Navier-Stokes continuum mechan-

ics,” Phys. Rev. A 22, 2798 (1980).

27G. Emanuel and B. M. Argrow, “Linear dependence of the shock wave thickness

on bulk viscosity,” Phys. Fluids 6, 3203 (1994).

28T. G. Elizarova, A. A. Khokhlov, and S. Montero, “Numerical simulation of

shock wave structure in nitrogen,” Phys. Fluids 19, 068102 (2007).

29A. V. Chikitkin, B. V. Rogov, G. A. Tirsky, and S. V. Utyuzhnikov, “Effect of bulk

viscosity in supersonic ﬂow past spacecraft,” Appl. Numer. Math. 93, 47 (2015).

30S. Taniguchi, T. Arima, T. Ruggeri, and M. Sugiyama, “Overshoot of the non-

equilibrium temperature in the shock wave structure of a rareﬁed polyatomic gas

subject to the dynamic pressure,” Int. J. Nonlinear Mech. 79, 66 (2016).

31M. van Dyke, “Second-order compressible boundary layer theory with applica-

tion to blunt bodies in hypersonic ﬂow,” in Hypersonic Flow Research, edited by

F. R. Riddell (Academic Press, 1962), p. 37.

32P. M. Morse and K. U. Ingard, Theoretical Acoustics (Princeton University Press,

1986).

33M. Trusler, Physical Acoustics and Metrology of Fluids (CRC Press, 1991).

34C. Eckart, “Vortices and streams caused by sound waves,” Phys. Rev. 73,68

(1948).

35L. N. Liebermann, “Second viscosity of liquids,” Phys. Rev. 75, 1415 (1949).

36J. J. Markham, “Second-order acoustic ﬁelds: Streaming with viscosity and

relaxation,” Phys. Rev. 86, 497 (1952).

37P. J. Westervelt, “The theory of steady rotational ﬂow generated by a sound

ﬁeld,” J. Acoust. Soc. Am. 25, 60 (1953).

38W. L. Nyborg, “Acoustic streaming due to attenuated plane waves,” J. Acoust.

Soc. Am. 25, 68 (1953).

39J. Lighthill, “Acoustic streaming,” J. Sound Vib. 61, 391 (1978).

40G. Emanuel, “Effect of bulk viscosity on a hypersonic boundary layer,” Phys.

Fluids 4, 491 (1992).

41M. S. Cramer and F. Bahmani, “Effect of large bulk viscosity on large-Reynolds-

number ﬂows,” J. Fluid Mech. 751, 142 (2014).

42H. R. van den Berg, C. A. ten Seldam, and P. S. van der Gulik, “Compressible

laminar ﬂow in a capillary,” J. Fluid Mech. 246, 1 (1993).

43J. C. Harley, Y. Huang, H. H. Bau, and J. N. Zemel, “Gas ﬂow in micro-

channels,” J. Fluid Mech. 284, 257 (1995).

44Y. Zohar, S. Y. K. Lee, W. Y. Lee, L. Jiang, and P. Tong, “Subsonic gas ﬂow in a

straight and uniform microchannel,” J. Fluid Mech. 472, 125 (2002).

45D. C. Venerus, “Laminar capillary ﬂow of compressible viscous ﬂuids,” J. Fluid

Mech. 555, 59 (2006).

46E. G. Taliadorou, M. Neophytou, and G. C. Georgiou, “Perturbation solutions

of Poiseuille ﬂows of weakly compressible Newtonian liquids,” J. Non-Newtonian

Fluid Mech. 163, 25 (2009).

47D. C. Venerus and D. J. Bugajsky, “Compressible laminar ﬂow in a channel,”

Phys. Fluids 22, 046101 (2010).

48F. Bahmani and M. Cramer, “Suppression of large shock-induced separation in

ﬂuids having large bulk viscosities,” J. Fluid Mech. 756, R2 (2014).

49S. Bhola and T. K. Sengupta, “Roles of bulk viscosity on transonic shock-

wave/boundary layer interactions,” Phys. Fluids 31, 096101 (2019).

50S. Chen, X. Wang, J. Wang, M. Wan, H. Li, and S. Chen, “Effects of bulk viscosity

on compressible homogeneous turbulence,” Phys. Fluids 31, 085115 (2019).

51S. Pan and E. Johnsen, “The role of bulk viscosity on the decay of compressible,

homogeneous, isotropic turbulence,” J. Fluid Mech. 833, 717 (2017).

J. Chem. Phys. 152, 094503 (2020); doi: 10.1063/1.5142364 152, 094503-7

Published under license by AIP Publishing

The Journal

of Chemical Physics ARTICLE scitation.org/journal/jcp

52G. Kirchhoff, “Über den Einﬂuß der Wärmeleitung in einem Gas auf die

Schallbewegung,” Ann. Phys. 210, 177 (1868).

53N. Neklepajev, “Über die Absorption kurzer akustischer Wellen in der Luft,”

Ann. Phys. 340, 175 (1911).

54T. P. Abello, “Absorption of ultrasonic waves by various gases,” Phys. Rev. 31,

1083 (1928).

55W. H. Pielemeier, “The Pierce acoustic interferometer as an instrument for the

determination of velocity and absorption,” Phys. Rev. 34, 1184 (1929).

56E. Großmann, “Schallabsorptionsmessung in Gasen bei hohen Frequenzen,”

Ann. Phys. 405, 681 (1932).

57R. W. Curtis, “An experimental determination of ultrasonic absorption an

reﬂexion coefﬁcients in air and in carbon dioxide,” Phys. Rev. 46, 811 (1934).

58K. F. Herzfeld and F. O. Rice, “Dispersion and absorption of high frequency

sound waves,” Phys. Rev. 31, 691 (1928).

59H. O. Kneser, “Schallabsorption in mehratomigen Gasen,” Ann. Phys. 408, 337

(1933).

60H. O. Kneser, “Schallabsorption und -dispersion in Flüssigkeiten,” Ann. Phys.

424, 277 (1938).

61L. Tisza, “Supersonic absorption and Stokes’ viscosity relation,” Phys. Rev. 61,

531 (1942).

62W. E. Meador, G. A. Miner, and L. W. Townsend, “Bulk viscosity as a relaxation

parameter: Fact or ﬁction?,” Phys. Fluids 8, 258 (1996).

63R. A. Rapuano, “Ultrasonic absorption from 75 to 280 Mc/sec,” Phys. Rev. 72,

78 (1947).

64W. G. Schneider, “Sound velocity and sound absorption in the critical region,”

Can. J. Chem. 29, 243 (1951).

65A. G. Chynoweth and W. G. Schneider, “Ultrasonic propagation in xenon in the

region of its critical temperature,” J. Chem. Phys. 20, 1777 (1952).

66H. D. Parbrook and E. G. Richardson, “Propagation of ultrasonic waves in

vapours near the critical point,” Proc. Phys. Soc., Sect. B 65, 437 (1952).

67M. Fixman, “Ultasonic attenuation in the critical region,” J. Chem. Phys. 33,

1363 (1960).

68W. Botch and M. Fixman, “Sound absorption in gases in the critical region,”

J. Chem. Phys. 42, 199 (1965).

69M. Fixman, “Transport coefﬁcients in the gas critical region,” J. Chem. Phys. 47,

2808 (1967).

70K. Kawasaki, “Correlation-function approach to the transport coefﬁcients near

the critical point. I,” Phys. Rev. 150, 291 (1966).

71K. Kawasaki, “Sound attenuation and dispersion near the liquid-gas critical

point,” Phys. Rev. A 1, 1750 (1970).

72L. P. Kadanoff, W. Götze, D. Hamblen, R. Hecht, E. A. S. Lewis, V. V.

Palciauskas, M. Rayl, and J. Swift, “Static phenomena near critical points: Theory

and experiment,” Rev. Mod. Phys. 39, 395 (1967).

73L. P. Kadanoff and J. Swift, “Transport coefﬁcients near the liquid-gas critical

point,” Phys. Rev. 166, 89 (1968).

74L. Hall, “The origin of excess ultrasonic absorption in water,” Phys. Rev. 71, 318

(1947).

75L. Hall, “The origin of ultrasonic absorption in water,” Phys. Rev. 73, 775

(1948).

76D. Sette, “Ultrasonic absorption in liquid mixtures and structural effects,”

J. Chem. Phys. 21, 558 (1953).

77K. F. Herzfeld and T. A. Litovitz, Absorption and Dispersion of Ultrasonic Waves

(Academic Press, 1959).

78C. W. Garland, D. Eden, and L. Mistura, “Critical sound absorption in xenon,”

Phys. Rev. Lett. 25, 1161 (1970).

79P. E. Mueller, D. Eden, C. W. Garland, and R. C. Williamson, “Ultrasonic

attenuation and dispersion in xenon near its critical point,” Phys. Rev. A 6, 2272

(1972).

80P. A. Fleury and J.-P. Boon, “Brillouin scattering in simple liquids: Argon and

neon,” Phys. Rev. 186, 244 (1969).

81A. S. Pine, “Velocity and attenuation of hypersonic waves in liquid nitrogen,”

J. Chem. Phys. 51, 5171 (1969).

82B. Y. Baharudin, D. A. Jackson, P. E. Schoen, and J. Rouch, “Bulk viscosity of

liquid argon, krypton and xenon,” Phys. Lett. A 51, 409 (1975).

83X. Pan, M. N. Shneider, and R. B. Miles, “Coherent Rayleigh-Brillouin scatter-

ing,” Phys. Rev. Lett. 89, 183001 (2002).

84J. Xu, X. Ren, W. Gong, R. Dai, and D. Liu, “Measurement of the bulk viscosity

of liquid by Brillouin scattering,” Appl. Opt. 42, 6704 (2003).

85X. Pan, M. N. Shneider, and R. B. Miles, “Power spectrum of coherent Rayleigh-

Brillouin scattering in carbon dioxide,” Phys. Rev. A 71, 045801 (2005).

86A. S. Meijer, A. S. de Wijn, M. F. E. Peters, N. J. Dam, and W. van de Water,

“Coherent Rayleigh-Brillouin scattering measurements of bulk viscosity of polar

and nonpolar gases, and kinetic theory,” J. Chem. Phys. 133, 164315 (2010).

87X. He, H. Wei, J. Shi, J. Liu, S. Li, W. Chen, and X. Mo, “Experimental mea-

surement of bulk viscosity of water based on stimulated Brillouin scattering,”

Opt. Commun. 285, 4120 (2012).

88Z. Gu and W. Ubachs, “A systematic study of Rayleigh-Brillouin scattering in

air, N2, and O2gases,” J. Chem. Phys. 141, 104320 (2014).

89Z. Gu, W. Ubachs, and W. van de Water, “Rayleigh-Brillouin scattering of

carbon dioxide,” Opt. Lett. 39, 3301 (2014).

90Y. Ma, H. Li, Z. Gu, W. Ubachs, Y. Yu, J. Huang, B. Zhou, Y. Wand, and

K. Liang, “Analysis of Rayleigh-Brillouin spectral proﬁles and Brillouin shifts in

nitrogen gas and air,” Opt. Express 22, 2092 (2014).

91G. W. Pierce, “Piezoelectric crystal oscillators applied to the precision measure-

ment of the velocity of sound in air and CO2at high frequency,” Proc. Am. Acad.

Arts Sci. 60, 271 (1925).

92M. Kohler, “Schallabsorption in Mischungen einatomiger Gase,” Ann. Phys.

431, 209 (1941).

93J. M. M. Pinkerton, “The absorption of ultrasonic waves in liquids and its

relation to molecular constitution,” Proc. Phys. Soc., Sect. B 62, 129 (1949).

94M. Pancholy, “Temperature variation of velocity and absorption coefﬁcient of

ultrasonic waves in heavy water,” J. Acoust. Soc. Am. 25, 1003 (1953).

95T. A. Litovitz and E. H. Carnevale, “Effect of pressure on sound propagation in

water,” J. Appl. Phys. 26, 816 (1955).

96W. Tempest and H. D. Parbrook, “The absorption of sound in argon, nitrogen

and oxygen,” Acustica 7, 354 (1957).

97D. G. Naugle and C. F. Squire, “Ultrasonic attenuation in liquid argon,” J. Chem.

Phys. 42, 3725 (1965).

98D. G. Naugle, “Excess ultrasonic attenuation and intrinsic volume viscosity in

liquid argon,” J. Chem. Phys. 44, 741 (1966).

99D. G. Naugle, J. H. Lunsford, and J. R. Singer, “Volume viscosity in liquid argon

at high pressure,” J. Chem. Phys. 45, 4669 (1966).

100D. S. Swyt, J. F. Havlice, and E. F. Carome, “Ultrasonic absorption in liquid

argon,” J. Chem. Phys. 47, 1199 (1967).

101W. M. Madigosky, “Density dependence of bulk viscosity in argon,” J. Chem.

Phys. 46, 4441 (1967).

102J. R. Singer and J. H. Lunsford, “Ultrasonic attenuation and volume viscosity

in liquid nitrogen,” J. Chem. Phys. 47, 811 (1967).

103J. R. Singer, “Excess ultrasonic attenuation and volume viscosity in liquid

methane,” J. Chem. Phys. 51, 4729 (1969).

104A. E. Victor and R. T. Beyer, “Ultrasonic absorption in liquid oxygen and

nitrogen,” J. Chem. Phys. 52, 1573 (1970).

105J. Allegra, S. Hawley, and G. Holton, “Pressure dependence of the ultrasonic

absorption in toluene and hexane,” J. Acoust. Soc. Am. 47, 144 (1970).

106E. V. Larson, D. G. Naugle, and T. W. Aldair, “Ultrasonic velocity and

attenuation in liquid neon,” J. Chem. Phys. 54, 2429 (1971).

107S. K. Kor, O. N. Awasthi, G. Rai, and S. C. Deorani, “Structural absorption of

ultrasonic waves in methanol,” Phys. Rev. A 3, 390 (1971).

108S. K. Kor, S. C. Deorani, and B. K. Singh, “Origin of ultrasonic absorption in

methanol,” Phys. Rev. A 3, 1780 (1971).

109S. K. Kor, S. C. Deorani, B. K. Singh, R. Prasad, and U. Tandon, “Ultrasonic

study of structural relaxation in ethanol,” Phys. Rev. A 4, 1299 (1971).

110J. A. Cowan and R. N. Ball, “Temperature dependence of bulk viscosity in

liquid argon,” Can. J. Phys. 50, 1881 (1972).

111G. Rai, B. K. Singh, and O. N. Awasthi, “Structural absorption of ultrasonic

waves in associated liquids,” Phys. Rev. A 5, 918 (1972).

112J. A. Cowan and P. W. Ward, “Ultrasonic attenuation of the bulk viscosity of

liquid argon near the critical point,” Can. J. Phys. 51, 2219 (1973).

J. Chem. Phys. 152, 094503 (2020); doi: 10.1063/1.5142364 152, 094503-8

Published under license by AIP Publishing

The Journal

of Chemical Physics ARTICLE scitation.org/journal/jcp

113P. W. Ward, J. A. Cowan, and R. K. Pathria, “Critical attenuation of ultrasound

in argon,” Can. J. Phys. 53, 29 (1975).

114G. J. Prangsma, A. J. Alberga, and J. J. M. Beenakker, “Ultrasonic determination

of the volume viscosity of N2, CO, CH4and CD4between 77 and 300 K,” Physica

64, 278 (1973).

115S. A. Mikhailenko, B. G. Dudar, and V. A. Shmidt, “Volume viscosity and

relaxation times in monoatomic classical ﬂuids,” Fiz. Nizk. Temp. 1, 224 (1975).

116P. Malbrunot, A. Boyer, E. Charles, and H. Abachi, “Experimental bulk viscosi-

ties of argon, krypton, and xenon near their triple point,” Phys. Rev. A 27, 1523

(1983).

117J. A. Cowan and R. N. Ball, “Ultrasonic attenuation and bulk viscosity in liquid

krypton,” Can. J. Phys. 58, 74 (1980).

118J. A. Cowan and J. W. Leech, “Ultrasonic attenuation and bulk viscosity of

liquid xenon,” Can. J. Phys. 59, 1280 (1981).

119J. A. Cowan and J. W. Leech, “Critical region ultrasonic attenuation in the

condensed inert gases,” Can. J. Phys. 61, 895 (1983).

120A. S. Dukhin and P. J. Goetz, “Bulk viscosity and compressibility measurement

using acoustic spectroscopy,” J. Chem. Phys. 130, 124519 (2009).

121M. J. Holmes, N. G. Paker, and M. J. W. Povey, “Temperature dependence

of bulk viscosity in water using acoustic spectroscopy,” J. Phys.: Conf. Ser. 269,

012011 (2011).

122J. Thoen and C. W. Garland, “Sound absorption and dispersion as a function

of density near the critical point of xenon,” Phys. Rev. A 10, 1311 (1974).

123C. Tegeler, R. Span, and W. Wagner, “A new equation of state for argon cov-

ering the ﬂuid region for temperatures from the melting line to 700 K at pressures

up to 1000 MPa,” J. Phys. Chem. Ref. Data 28, 779 (1999).

124E. W. Lemmon and R. T. Jacobsen, “Viscosity and thermal conductivity

equations for nitrogen, oxygen, argon and air,” Int. J. Thermophys. 25,21

(2004).

125E. W. Lemmon and R. Span, “Short fundamental equations of state for 20

industrial ﬂuids,” J. Chem. Eng. Data 51, 785 (2006).

126M. L. Huber, “Models for viscosity, thermal conductivity, and surface tension

of selected pure ﬂuids as implemented in REFPROP v10.0,” Technical Report No.

8209, National Institute for Standards and Technology, 2014.

127R. Katti, R. T. Jacobsen, R. B. Stewart, and M. Jahangiri, “Thermodynamic

properties of neon for temperatures from the triple point to 700 K at pressures to

700 MPa,” in Advances in Cryogenic Engineering, edited by R. W. Fast (Springer,

1986), pp. 1189–1197.

128L. Claes, L. M. Hülskämper, E. Baumhögger, N. Feldmann, R. S. Chatwell,

J. Vrabec, and B. Henning, “Acoustic absoprtion measurement for the determi-

nation of the bulk viscosity of pure ﬂuids,” TM - Tech. Mess. 86, 2–6 (2019).

129J. R. Taylor, Introduction to Error Analysis: The Study of Uncertainties in

Physical Measurements, 2nd ed. (University Science Books, 1997).

130G. Rutkai, M. Thol, R. Span, and J. Vrabec, “How well does the Lennard-Jones

potential represent the thermodynamic properties of noble gases?,” Mol. Phys.

115, 1104 (2017).

131M. S. Green, “Markoff random process and the statistical mechanics of time-

dependent phenomena. II. Irreversible processes in ﬂuids,” J. Chem. Phys. 22, 398

(1954).

132R. Kubo, “Statistical-mechanical theory of irreversible processes. I. General

theory of simple applications to magnetic and conductive problems,” J. Phys. Soc.

Jpn. 12, 570 (1957).

133R. Kubo, “The ﬂuctuation-dissipation theorem,” Rep. Prog. Phys. 29, 255

(1966).

134L. Verlet, “Computer “experiments” on classical ﬂuids. I. Properties of

Lennard-Jones molecules,” Phys. Rev. 159, 98 (1967).

135D. Levesque and L. Verlet, “Computer “experiments” on classical ﬂuids. III.

Time-dependent self-correlation functions,” Phys. Rev. A 2, 2514 (1970).

136D. Levesque, L. Verlet, and J. Kürkijarvi, “Computer “experiments” on classical

ﬂuids. IV. Transport properties and time-correlation functions of the Lennard-

Jones liquid near its triple point,” Phys. Rev. A 7, 1690 (1973).

137S. V. Lishchuk, “Role of three-body interactions in formation of bulk viscosity

in liquid argon,” J. Chem. Phys. 136, 164501 (2012).

138F. Jaeger, O. K. Matar, and E. A. Müller, “Bulk viscosity of molecular liquids,”

J. Chem. Phys. 148, 174504 (2018).

139G. Rutkai, A. Köster, G. Guevara-Carrion, T. Janzen, M. Schappals,

C. W. Glass, M. Bernreuther, A. Wafai, S. Stephan, M. Kohns, S. Reiser,

S. Deublein, M. T. Horsch, H. Hasse, and J. Vrabec, “ms2: A molecular simula-

tion tool for thermodynamic properties, release 3.0,” Comput. Phys. Commun.

221, 343 (2017).

140K. Meier, A. Laesecke, and S. Kabelac, “Transport coefﬁcients of the Lennard-

Jones model ﬂuid. III. Bulk viscosity,” J. Chem. Phys. 122, 014513 (2005).

141V. G. Baidakov and S. P. Protsenko, “Metastable Lennard-Jones ﬂuids. III. Bulk

viscosity,” J. Chem. Phys. 141, 114503 (2014).

142A. Zaragoza, M. A. Gonzales, L. Joly, I. López-Montero, M. A. Canales,

A. L. Benavides, and C. Valeriani, “Molecular dynamics study of nanocon-

ﬁned TIP4P/2005 water: How conﬁnement and temperature affect diffusion and

viscosity,” Phys. Chem. Chem. Phys. 21, 13653 (2019).

143G. S. Fanourgakis, J. S. Medina, and R. Prosmiti, “Determining the bulk

viscosity of rigid water models,” J. Phys. Chem. A 116, 2564 (2012).

144G.-J. Guo, Y.-G. Zhang, K. Refson, and Y.-J. Zhao, “Viscosity and stress

autocorrelation function in supercooled water: A molecular dynamics study,”

Mol. Phys. 100, 2617 (2002).

145G. Delgado-Barrio, P. Villareal, G. Winter, J. S. Medina, B. González, J. V.

Alemán, J. L. Gomez, P. Sangrá, J. J. Santana, and M. E. Torres, “Viscosity of

liquid water via equilibrium molecular dynamics simulations,” in Frontiers in

Quantum Systems in Chemistry and Physics, edited by S. Wilson, P. J. Grout,

G. Delgado-Barrio, and P. Piecuch (Springer, 2008), pp. 351–361.

146J. S. Medina, R. Prosmiti, P. Villarreal, G. Delgado-Barrio, G. Winter,

B. González, J. V. Alemán, and C. Collado, “Molecular dynamics simulations of

rigid and ﬂexible water models: Temperature dependence of viscosity,” Chem.

Phys. 388, 9 (2011).

147R. Kohlrausch, “Theorie des elektrischen Rückstandes in der Leidner Flasche,”

Ann. Phys. 167, 179 (1854).

148G. Williams and D. C. Watts, “Non-symmetrical dielectric relaxation

behaviour arising from a simple empirical decay function,” Trans. Faraday Soc.

66, 80 (1970).

149R. D. Mountain and R. Zwanig, “Shear relaxation times of simple ﬂuids,”

J. Chem. Phys. 44, 2777 (1966).

150T. A. Litovitz and C. M. Davis, “Structural and shear relaxation in liquids,”

in Peroperties of Gases, Liquids and Solutions, Physical Acoustics, edited by

W. P. Mason (Academic Press, 1965), Vol. 2, pp. 281–349.

151G. L. Murphy, “Big-bang model without singularities,” Phys. Rev. D 8, 4231

(1973).

152V. A. Belinskii and I. M. Khalatnikov, “Inﬂuence of viscosity on the character

of cosmological evolution,” Zh. Eksp. Theor. Fiz. 69, 401 (1975).

153I. Brevik, E. Elizalde, S. Nojiri, and S. D. Odintsov, “Viscous little rip cosmol-

ogy,” Phys. Rev. D 84, 103508 (2011).

154R. Colistete, J. C. Fabris, J. Tossa, and W. Zimdahl, “Bulk viscous cosmology,”

Phys. Rev. D 76, 103516 (2007).

155M. Cruz, N. Cruz, and S. Lepe, “Accelerated and decelerated expansion in a

causal dissipative cosmology,” Phys. Rev. D 96, 124020 (2017).

156B. Li and J. D. Barrow, “Does bulk viscosity create a viable uniﬁed dark matter

model?,” Phys. Rev. D 79, 103521 (2009).

157S. R. de Groot and P. Mazur, Non-Equilibrium Thermodynamics (Dover, 1962).

158D. Jou, J. Casas-Vázquez, and G. Lebon, Extended Irreversible Thermodynamics

(Springer, 2010).

159P. Borgelt, C. Hoheisel, and G. Stell, “Exact molecular dynamics and kinetic

theory results for thermal transport coefﬁcients of the Lennard-Jones argon ﬂuid

in a wide range of states,” Phys. Rev. A 42, 789 (1990).

160K. Tankeshwar, “Bulk viscosity and the relation between transport coefﬁ-

cients,” Phys. Chem. Liq. 24, 91 (1991).

J. Chem. Phys. 152, 094503 (2020); doi: 10.1063/1.5142364 152, 094503-9

Published under license by AIP Publishing

2.4 Widom’s proposition

All known matter in the universe that is not in its quantum or high energy state can

be categorised as either solid, liquid, gaseous or supercritical. Each of these aggregate

states is thereby distinguishable by a characteristic microscopic local order that is most

adequately determined by a radial distribution function. While it is well-established that

some substances can exist in diﬀerent solid states, i.e. there are 18 known crystalline forms

of water [152, 153, 154, 155, 156, 157, 158, 159, 160], four of solid hydrogen [161, 162, 163]

and three forms of solid helium [164, 165], it is lesser known that substances can also exist

in diﬀerent liquid states, i.e. supercooled water was observed to transition from a state

of high- (HDL) to low-density (LDL) [166]. It is, however, hardly recognised that the

supercritical region also partitions into diﬀerent liquid-like and gas-like subregions, being

divided by so-called crossover lines. While several such crossover lines have been proposed,

i.e. Frenkel [167, 168, 169], Fisher-Widom [170, 171], Nishikawa [172, 173] and Widom

[174, 175] each characterising a diﬀerent transition, the Widom line is arguably the most

prominent [175]

"It is generally recognized that a ﬂuid in the neighborhood of its critical point

diﬀers in many important respects from the classical ﬂuid which obeys an equa-

tion of state of van der Waals type. Near the critical point of a real ﬂuid, the

densities of the coexisting phases, the shape of the compressibility, and the na-

ture of the singularity in the constant-volume speciﬁc heat, are all nonclassical".

The Widom line can be seen as an extrapolation of the vapour-pressure line into the su-

percritical region and its inﬂuence was observed to be limited to pressures up to three

times the critical pressure, i.e. p3pc[168, 176]. While the Widom line has originally

been deﬁned by the maxima of thermodynamic correlation lengths [177], it can alterna-

tively and more practically be approximated by the maxima of certain thermodynamic

response functions. These response functions’ maxima, i.e. speed of sound c, isothermal

compressibility βT, thermal expansivity αp, isobaric and isochoric heat capacities cp,c

[178, 179], conﬂuence towards the so-called "Single Widom Line" in close vicinity of the

critical point and diverge at higher temperatures to span a delta shaped region, the so-

called "Widom Delta" [180], cf. Fig. 9. The thermodynamic response functions cp,αp,βT

can also be formulated in relation the the Gibbs free energy and the SWL can alternatively

be determined by its curvature

cp=−T∂2g

∂T2p

,α

p=1

∂2g

∂p∂T p

,β

T=−1

∂2g

∂p2T

.(133)

Widom, however, never considered these anomalous lines, that where later named in his

honour [181], to be special.

While the Widom region, i.e. the Single Widom Line and the Widom Delta, partitions

sections of the supercritical domain into liquid-like and gas-like subdomains, the Frenkel

line in contrast separates a rigid from a non-rigid liquid at relatively high pressures [168].

Crossing the Widom region closely resembles subcritical phase transition, however, without

ﬁrst-order singularities in the respective thermodynamic observables and is consequently

referred to as pseudo-boiling. At the boundary between rigid and non-rigid transition, the

liquid’s relaxation time equals the minimal period of transverse quasi-harmonic waves and

the ﬂuid consequently loses all its shear stiﬀness.

2.4.1 Widom region of supercritical carbon dioxide

As mentioned above, the response functions’ maxima conﬂuence to a Single Widom Line

in close vicinity to the critical point and diverge into the Widom Delta at higher pressures.

The inﬂuence of this Widom region on the supercritical ﬂuid’s thermodynamic behaviour

eventually vanishes for pressures above three times the critical pressure, i.e. p3pc.Itis

quite common, however, to approximate the SWL by a logarithmic function, the so-called

Plank-Riedel equation [182]

ln(pr)=a+b

+cln(Tr), a,b,c∈R,(134)

with ﬁtting parameters a, b, c and reduced pressure pr=pp−1

cand temperature Tr=TT−1

Fitting equation (134) to the geometric mean of the response functions’ maxima for carbon

dioxide yields a=−41.8798,b =41.90074,c =46.66034 and partitions the supercritical

region near carbon dioxide’s critical point into liquid-like and gas-like subregions, as is

depicted in Fig. 9.





    



























Figure 9. Carbon dioxide has been observed to exist in various aggregate states,

ranging from solid (I), liquid (II) gaseous (III) to supercritical (IV). The

Widom Delta partitions the supercritical region in close vicinity to the

critical point into liquid-like (IVa) and gas-like (IVb) subdomains. The

locus of the various response functions’ maxima are depicted in diﬀerent

colours, i.e. in cpin blue, vvin green, βTin pink and cin red. These

maxima span a delta shaped area that conﬂuences at around 10 % above

the critical pressure to a single line, the so-called "single Widom line".

Equation (134) was ﬁtted here to the geometric mean of the response

functions’ respective maxima and given by the dashed black line. Across

the Widom Delta, supercritical CO2undergoes a liquid-like to gas-like

transition that is alternatively known as pseudo-boiling.

The liquid- to gas-like transition across the Widom region occurs over a ﬁnite temperature

range ΔT, which increases with rising pressure, cf. Fig. 10.

In binary mixtures, the Widom region’s inﬂuence also extends to the thermodynamic

factor Γand consequently to the Fick diﬀusion coeﬃcient D[183]. The thermodynamic

factor as function of temperature drops characteristically when transitioning across the

Widom region, thus substantiating the latter’s interpretation as a extrapolation of the

vapour-pressure curve into the supercritical region. The eﬀect is also observable in ﬁnitely

diluted binary mixtures containing only traces of one component.



   















































Figure 10. Top: Variation of enthalpy over reduced temperature. Bottom: Variation

of density over reduced temperature. The transition from liquid-like to

gas-like behaviour spreads over a ﬁnite temperature range that increases

with rising pressure. The Widom Delta is enveloped by the locus of

maxima of the isobaric heat capacity cp(blue line) and the speed of

sound c(red line).

2.4.2 Diﬀusion of the carbon dioxide - ethanol mixture in the extended critical region

R.S. Chatwell, G. Guevara-Carrion, Y. Gaponenko, V. Shevtsova and J. Vrabec, Physical

Chemistry Chemical Physics,23, 3106-3115 (2021)

Reprinted from Physical Chemistry and Chemical Physics with the permission of the

Royal Society of Chemistry. This article is licensed under a Creative Commons license

(CC BY 3.0, https://creativecommons.org/licenses/by/3.0). The following arti-

cle appeared in Phys. Chem. Chem. Phys,32, 3106-3115 (2021) and my be found at

https://doi.org/10.1039/D0CP04985A.

In this work, the inﬂuence of the Widom region on a ﬁnitely diluted binary mixture is

investigated. Consequently, the thermodynamic properties and particularly the transport

diﬀusion coeﬃcient of a mixture containing traces of ethanol within supercritical carbon

dioxide is studied. While the transport diﬀusion coeﬃcient is studied twofoldedly, i.e.

by experimental Taylor dispersion measurements and atomistic simulations, the mixture’s

thermodynamic behaviour is studied exclusively by molecular simulations. The latter

allows to investigate the mixture’s microscopic structure unconstrainedly and revealed in-

cipient clustering of ethanol molecules within the Widom region.

The leading author R.S. Chatwell has performed all atomistic simulations, evaluated and

interpreted their results. The manuscript was written conjoinedly by R.S. Chatwell and

G. Guevara-Carrion. Y. Gaponenko performed the Taylor dispersion measurements and

post-processed their results. This work has been initiated by V. Shevstova and J. Vrabec

who both redacted and revised the manuscript.

3106 |Phys. Chem. Chem. Phys.,2021,23, 3106--3115 This journal is ©the Owner Societies 2021

Cite this: Phys. Chem. Chem. Phys.,

2021, 23,3106

Diﬀusion of the carbon dioxide–ethanol mixture

in the extended critical region†

Rene

´Spencer Chatwell,

Gabriela Guevara-Carrion,

Yuri Gaponenko,

Valentina Shevtsova

and Jadran Vrabec *

The eﬀect of traces of ethanol in supercritical carbon dioxide on the mixture’s thermodynamic

properties is studied by molecular simulations and Taylor dispersion measurements. This mixture is

investigated along the isobar p= 10 MPa in the temperature range between T= 304 and 343 K. Along

this path, the mixture undergoes two transitions: First, the Widom line is crossed, marking the transition

from liquid-like to gas-like conditions. A second transition occurs from the supercritical gas-like domain

to a subcritical gas. The Widom line crossover entails inflection points for most of the studied properties,

i.e. density, enthalpy, shear viscosity, Maxwell–Stefan and intradiffusion coefficients. On the other hand,

the transition between the super- and subcritical regions is found to be generally smooth, an observation

that is qualitatively confirmed by experimental Taylor dispersion measurements. Dedicated atomistic

simulations show the presence of microheterogeneities due to ethanol self-association along the

investigated path, which lead to the mixture’s anomalous behavior in its extended critical region.

1 Introduction

Investigating the thermophysical properties of supercritical

carbon dioxide (scCO

) has grown to a topic of great

interest.

1–4

While the majority of work had focused on pure

fluids in sufficient distance from the critical point,

5,6

only some

special case scenarios with respect to a second component have

been examined under near-critical conditions.

7–9

is commonly used in various technological applications,

ranging from environmental, mechanical, chemical, geothermal

to pharmaceutical industries.

It constitutes an attractive alter-

native to organic or aqueous solvents by having a remarkably

mild critical point, being non-toxic, non-flammable, widely

available and largely inert. However, due to its low dielectric

constant in combination with a zero dipole moment,

it is a

poor solvent for polar substances. Yet, the solubility of high

molecular weight solutes is readily enhanced by adding small

amounts of strongly polar entrainers.

12,13

Consequently, ethanol

emerges as a widespread commodity in supercritical extraction

processes. Further, ethanol forms a non-ideal mixture with CO

due to the presence of strong solute-solute interactions and

hydrogen bonding making this mixture particularly interesting.

In contrast to the classical perception of the supercritical

region as a featureless domain, experimental evidence

14–17

and

molecular dynamics simulations

3,4,9,18–20

indicate a partition

into several subdomains. The continuous dynamic crossover

between different domains occurs across transition lines, i.e.

Fisher–Widom,

Nishikawa,

14,16

Frenkel

22,23

and Widom.

For moderate pressures, i.e. up to three times the critical

pressure

pr3p

, the crossover between gas-like and liquid-

like regimes is accomplished over a delta shaped area

20,26,27

that is delimited by the loci of extrema of particular thermo-

dynamic response functions, i.e. isobaric and isochoric heat

capacities c

, thermal expansion a

, isothermal compressi-

bility b

, density rand speed of sound c. These response

functions eventually confluence to a single line in close vicinity

to the critical point, the so-called Widom line, which can be

considered as an extension of the vapor pressure curve. Crossing

the Widom line is additionally associated with a minimum of

the thermodynamic factor and with large density fluctuations.

While the critical point exhibits critical opalescence,

its

extended vicinity is characterized by emerging microscopic

clusters that have been observed experimentally

and with

equilibrium molecular dynamics simulations.

30–34

The present

work contributes to the understanding of how ethanol diluted

into CO

affects the mixture’s microscopic structure and

Thermodynamics and Process Engineering, Technische Universita

¨t Berlin,

10587 Berlin, Germany. E-mail: [email protected]; Tel: +49 30 314 22755

Microgravity Research Center, Universite

´Libre de Bruxelles, 1050 Bruxelles,

Belgium

†Electronic supplementary information (ESI) available: It includes a schematic

of the employed Taylor dispersion apparatus, graphical results for the enthalpy h,

specific volume v,c

and b

, as well as the center-of-mass radial distribution

function and average coordination number of the CO

–ethanol pair. Predictions

of the Fick diffusion coefficient of the CO

+ ethanol mixture with the regarded

equations are shown in comparison with experimental and simulation results.

The numerical data from equilibrium molecular dynamics simulation are also

listed. See DOI: 10.1039/d0cp04985a

Received 21st September 2020,

Accepted 17th January 2021

DOI: 10.1039/d0cp04985a

rsc.li/pccp

PCCP

PAPER

This journal is ©the Owner Societies 2021 Phys. Chem. Chem. Phys.,2021,23, 3106--3115 | 3107

consequently its transport properties in the extended critical

region. The CO

+ ethanol mixture was examined twofold.

While the microscopic structure was investigated exclusively

by equilibrium molecular dynamics (EMD) simulations, mutual

diffusion was sampled atomistically and was measured in the

lab with the Taylor dispersion technique. The present mixture

has already been investigated experimentally

35–37

and by

molecular simulations. In particular, the vapor–liquid equilibrium

(VLE),

solubility parameters,

microscopic structure

31,32,40–42

and diffusion coefficients

have been studied by molecular simu-

lation. However, the effects of structural changes on mutual

diffusion across the Widom line in the extended critical region

of this mixture have not yet been considered neither experimen-

tally, nor by molecular simulation. This is of special interest for the

understanding and description of supercritical extraction pro-

cesses which are performed in this region.

2 Method

2.1 Molecular simulation

Classical molecular simulations are based on molecular

force fields models that are typically optimized to describe

the pure component phase behavior. The transferability to

a given mixture is validated by comparing simulation predic-

tions to experimental data along the VLE curves. While the

employed force fields were based on the Lennard-Jones

potential for both components, a predictive mode was used

to describe the interactions between unlike Lennard-Jones sites,

i.e. the Lorentz–Berthelot combining rules were assumed. The

force field for ethanol consisted of three Lennard-Jones sites

with three superimposed point charges.

In contrast, the two

Lennard-Jones sites of carbon dioxide’s force field included a

superimposed point quadrupole.

States along the VLE curves

were sampled by Monte Carlo (MC) simulations with the grand

equilibrium method containing N= 1372 molecules for the

liquid phase and N= 460 molecules for the gas phase

simulations.

The MC simulation results for the VLE curves are in excellent

agreement with experimental literature data

46–52

over the entire

investigated temperature and pressure ranges, cf. Fig. 1. In order

to properly appreciate the simulation results, the coexistence

curves were additionally computed by three equations of state,

allowing for a temperature-dependent binary parameter k

,cf.

Table 1. While the employed cubic equations of state, i.e. Soave–

Redlich–Kwong

and Peng-Robinson,

tend to overestimate

the mixture’s vapor pressure at high CO

mole fractions, the

PCP-SAFT

equation of state overestimates the vapor pressure at

low CO

mole fractions.

The intra- D

and Maxwell–Stefan



diffusion coefficients

as well as the shear viscosity were sampled directly with

EMD simulation and the Green–Kubo formalism.

57,58

The

working equations for the determination of these transport

properties have been published e.g. in ref. 59 and are not

repeated here. EMD simulations for transport properties

were made in two steps. In the first step, a simulation in the

isobaric-isothermal (NpT) ensemble was performed to calculate

the density at the desired temperature and pressure. In

the second step, EMD simulations were performed at this

temperature and density in the canonical (NVT) ensemble

containing N= 3000 molecules in cubic volumes with periodic

boundary conditions and specifying an integrator time step of

Dt= 1 fs. The associated finite size effects were corrected with a

modified Yeh–Hummer approach

60,61

employing the sampled

shear viscosity values. To improve statistics, a total of 2.5 10

correlation functions was averaged. The thermodynamic

factor Gwas sampled directly with Kirkwood–Buff integration

based on the methodology proposed by Ganguly and van der

Vegt,

which was found to be the most adequate in previous

work.

Extrapolation to the thermodynamic limit was not

necessary. All molecular simulations were performed with the

fully open source software ms2.

Statistical uncertainties were

calculated with the error propagation law and a coverage factor

of k= 1. Throughout, the cut-off radius was set to r

=17.5Å.

Electrostatic long-range corrections were made using the

Fig. 1 Vapor–liquid phase diagram of CO

+ ethanol. The open symbols

indicate experimental literature data along three isotherms depicted in

black T= 312 K (bullets,

squares,

triangles,

stars,

male symbols

blue T= 325 K (hexagons

) and green T= 333 K (diamonds,

inverted

triangles,

crosses,

female symbols

). The solid symbols are present

molecular simulation results. The solid lines represent the binodals

according to the Peng–Robinson and the dashed lines according to the

PCP-SAFT equation of state. The solution according to the Soave–Red-

lich–Kwong equation was omitted to maintain readability. The red

bullet symbolizes the molar composition x

CO2

= 0.97 mol mol

1

and

pressure p= 10 MPa at which all other molecular dynamics simulations

were carried out.

Table 1 Each of the employed equations of state was fitted to experi-

mental VLE data, resulting in a linear temperature dependence of the

binary interaction parameter, i.e. k

=m(T/K 305) + C

Equation of state m/10

4

Peng–Robinson 1.7 0.0879

Soave–Redlich–Kwong 1.0 0.0863

PCP-SAFT 3.0 0.0506

Paper PCCP

3108 |Phys. Chem. Chem. Phys.,2021,23, 3106--3115 This journal is ©the Owner Societies 2021

reaction field technique with conducting boundary conditions

=N).

2.2 Taylor dispersion technique

The employed Taylor apparatus has been optimized for the use of

scCO

as described in preceding works.

35,65,66

The experimental

determination of the mixture’s mutual diﬀusion coeﬃcient D

constitutes a four stage procedure utilizing the Taylor dispersion

technique. The experimental schematic is disclosed in the ESI.†

During the first step, the pure CO

carrier fluid which was stored

under VLE conditions, i.e. at T= 288.15 K, pB5MPawitha

purity 0.99998 mol mol

1

(Air Liquide), was initially liquefied

through a cryostat reducing its temperature to T= 269.15 K.

Liquid CO

was subsequently pumped above the critical pressure

and eventually heated to its target temperature before it reached

the injection valve (Knauer model D-14163). In the second step,

scCO

was delivered to the dispersion tube with a constant flow

rate. The carrier stream was thermostated by means of a heat

reservoir ensuring a constant target temperature ranging from

exp

=304to343Kwithanaccuracyof0.1 K and barostated

with a high pressure pump at p

exp

=10MPawith0.05 MPa

accuracy. The ethanol sample (purity 99.9% (GC) in volume

fraction, CAS 64-17-5; purchased from VWR) adopts to the

respective target temperature T

exp

within the valve’s loop prior

to its injection into the scCO

stream. In order to ensure

temperature homogenization in this section of the experiment,

the dispersion tube and injection valve were placed inside a

polyurethane foam insulated housing with an additional air

fan. In the third step, the sample was injected into the scCO

stream. The resulting strongly diluted mixture was fed to a l=

30.916 0.001 m long dispersion capillary with a circular cross

section of radius r= 0.375 mm. The capillary was coiled around a

grooved, hollow aluminum cylinder with radius R

= 0.175 m

providing stability and fixation. The cylinder was additionally

thermostated with an internal circular flow ensuring a tempera-

ture stabilization of 0.1 K. In order to minimize pressure and

density disturbances during injection, an ethanol sample volume

of V

=210

6

was selected, with smaller volumes having a

negative eﬀect on the signal-to-noise ratio.

35,65

Thepressureof

the system was controlled by a back pressure regulator (Jasco BP-

2080) and measured by means of a pressure sensor (JUMO dTrans

p30) with an accuracy of 0.05 MPa. In the experiment’s final

step, the Taylor peak was monitored at the outlet of the disper-

sion tube by means of a FT-IR spectrometer (Jasco FT-IR 4100)

with 0.01 cm

1

accuracy and 4 cm

1

resolution. In contrast to

its nominal operation, the employed FT-IR was equipped with a

custom-built high pressure demountable cell (Harrick) that was

optimized for the best possible signal-to-noise ratio.

The ZnSe

cell had a thickness 150 mm

andallowedforamaximum

working pressure of 25 MPa. The data generated by the FTIR

were digitally read out by a specific software (Spectra Manager by

Jasco) and the variation of the solute concentration over time was

monitored through the absorbance spectra at wavenumbers

corresponding to diﬀerent vibration modes. The procedure to

select the working wavenumbers, the experimental protocol and

the fitting procedure have been reported in ref. 65 and 66.

3 Results

3.1 Critical line and Widom line

The CO

+ ethanol mixture was investigated by EMD simulation

along the isobar p= 10 MPa in the temperature range between

T= 305 and 340 K with a composition x

CO2

= 0.97 mol mol

1

,cf.

Fig. 2. This composition was chosen as the closest to the

infinite dilution limit that allowed for adequate statistics. Along

this path, the mixture undergoes two transitions, indicated as

I and II in Fig. 2. First, the Widom line is crossed at TB323 K

(point I) marking the transition from liquid-like to gas-like

conditions, as indicated by the maxima of the response func-

tions c

,cf. Fig. S2–S4 in the ESI.†Each function’s

inflection point corresponds to a maximum of a thermody-

namic response function and consequently determines the

mixture’s Widom line. The inflection points of the enthalpy

and density, which correspond to maxima of the isobaric heat

capacity c

and the thermal expansion a

, respectively, occur

according to the employed force field at T= 317 K for pure CO

and T= 323 K for the mixture, cf. Fig. 3. Thus, the addition of a

small amount of ethanol shifts the Widom line up by B6K.To

assess the capability of the employed force field to predict the

studied properties, the temperature dependence of density and

enthalpy of pure CO

was compared to the Span–Wagner

equation of state,

which is of reference quality. In general,

the employed CO

force field is able to predict the density of

in the studied temperature and pressure ranges with a

good accuracy. The average deviation between simulation

results and the Span–Wagner equation of state

is 5.4%.

Fig. 2 Pressure–temperature projection of both components’ vapor

pressure curves (green lines). The critical line of the mixture (dashed line)

was determined on the basis of experimental literature data

68,69

(black

crosses). The Widom line (dashed blue line) extends the influence of the

present mixture’s (x

CO2

= 0.97 mol mol

1

) critical point (red cross) to

higher temperatures and pressures. The red line represents the studied

isobar in the homogeneous region. Points I and II represent the crossing of

the Widom line and the transition from the super- to the subcritical

regimes. The inset shows the critical line (dashed line) as a function of

mole fraction as given in Table 2 (black crosses). The red bullet symbolizes

the molar composition x

CO2

= 0.97 mol mol

1

and pressure p= 10 MPa at

which all molecular dynamics simulations were carried out.

PCCP Paper

This journal is ©the Owner Societies 2021 Phys. Chem. Chem. Phys.,2021,23, 3106--3115 | 3109

However, the predicted inflection points, which define the

Widom line, occur approximately 2 K below those according

to the Span–Wagner equation of state.

Therefore, the related

uncertainty of the present simulation results is expected to have

a similar magnitude, which is marked as a shaded region

around the calculated transition points I and II.

After crossing the Widom line, the mixture undergoes a

transition from the supercritical gas-like domain to a gas at

subcritical conditions at TB328 K (point II), as indicated by the

critical line of the mixture, cf. Table 2. More specifically, while for

T= 318 K all states above p=8.64MPaandx

CO2

=0.938molmol

1

are located in the supercritical region, for T=328Konlystates

above p= 10.09 MPa and x

CO2

=0.863molmol

1

lie in the

supercritical region. This can be clearly seen in the inset of

Fig. 2, while for temperatures below TB328 K the simulated

pressure-composition pair is clearly located above the critical

line, for higher temperatures, the simulated pressure is below

that of the critical line. The pressure–temperature projection of

the vapor pressure curves for the CO

+ ethanol mixture is shown

in Fig. 2. The critical line of the mixture was determined by

joining the mixture’s critical points from experimental data.

67–70

A distinctive feature of the curve is that it goes through a

pronounced pressure maximum at pB15.5 MPa despite the

relatively small difference between the critical pressures of pure

and pure ethanol, p

= 7.4 and 6.3 MPa, respectively. The

large temperature range of that curve (304 K to 515 K) is simply

due to the difference between the critical temperatures of the

pure substances. Note that the composition varies along the

critical line such that the pressure-composition pair that is

focussed on does not cross the two-phase region.

3.2 Intradiﬀusion coeﬃcients

The dynamical crossover between the supercritical high density

and the low density regimes across the Widom line has been

related to the presence of inflection points or extrema of some

transport coeﬃcients.

Therefore, the analysis of the diﬀusion

behavior of the mixture along the regarded isobar oﬀers insight

into the underlying transition dynamics. The temperature

dependence of the intradiﬀusion coeﬃcient of CO

is sigmoidal,

with an initially rather weak temperature dependence followed

by a strong stepwise increase at temperatures between T=320

and 330 K, cf. Fig. 4. This curve is similar to that observed for the

enthalpy and shows an inflection point at the Widom line, i.e. at

TB323 K. The random propagation of molecules is not only

dependent on the thermodynamic state point, but is also

strongly aﬀected by other factors like molecular size and polarity.

Thus, within a mixture, the components generally propagate

with diﬀerent velocity distributions leading to diﬀerent intra-

diﬀusion coeﬃcients. In fact, the intradiﬀusion coeﬃcient of the

bulkier ethanol molecules is on average 40% lower than that of

the smaller CO

molecules. Although both curves show a sig-

moidal behavior, the stepwise increase of the intradiﬀusion

coeﬃcient of ethanol is less pronounced than the one of CO

which can be linked to the presence of local density inhomo-

geneities caused by self-association. Further, the inflection point

Fig. 3 EMD simulation results for enthalpy h(black) and density r(blue) of

pure CO

(triangles) and the CO

–ethanol mixture (circles) to determine

the Widom line at pressure p= 10 MPa. The lines represent the properties

of pure CO

calculated with the Span–Wagner equation of state.

Statistical uncertainties are within symbol size. The temperatures I and II

represent the crossing of the Widom line and the transition from the

super- to the subcritical regimes. The shaded areas indicate their expected

uncertainty.

Table 2 Selected critical points of the studied mixture from experimental

data

67–70

T/K p/MPa x

/mol mol

1

304.13 7.38 1.0

312.82 8.15 0.967

318.24 8.64 0.938

328.36 10.09 0.863

333.82 10.88 0.832

350.62 12.80 0.769

514.71 6.27 0.0

Fig. 4 EMD simulation results for the intradiﬀusion coeﬃcients of ethanol

(red triangles) and CO

(blue triangles) as well as the Maxwell–Stefan

diffusion coefficient (black symbols) along the isobar p= 10 MPa. The

statistical uncertainties of the intradiffusion coefficients are within symbol

size. The dashed line serves as a guide to the eye. The temperatures I and II

represent the crossing of the Widom line and the transition from the

super- to the subcritical regimes. The shaded areas indicate their expected

uncertainty.

Paper PCCP

3110 |Phys. Chem. Chem. Phys.,2021,23, 3106--3115 This journal is ©the Owner Societies 2021

of the curve is observed at the transition between the super- and

subcritical regimes TB328 K.

3.3 Microscopic structure

To elucidate the relevant microscopic structure aspects, the

center-of-mass radial distribution functions of the ethanol–

ethanol g

EtOH–EtOH

(r), CO

–ethanol g

CO2–EtOH

(r) and CO

–CO

CO2–CO2

(r) pair interactions were analyzed, cf. Fig. 5 and Fig. S5

of the ESI.†The main peak of g

EtOH–EtOH

(r) with a maximum

located at rB4.3 Å is related to ethanol self-association

through hydrogen bonding. This first peak is followed by a

well defined shoulder, which is more pronounced at the lowest

studied temperature and indicates an overlap with the second

solvation shell. The location of the main peak does not change

significantly along the studied isobar, suggesting that ethanol

remains structured at short intermolecular distances. On the

other hand, g

CO2–CO2

(r) shows a well defined first peak followed

by a considerably smaller second peak, which becomes weaker

with increasing temperature and disappears at TB330 K. This

reduction of the long-range structure is expected because of

the transition from supercritical gas-like to compressed gas

conditions. The observed differences in peak intensity and the

relatively small main peaks observed in the radial distribution

functions at lower temperatures are mainly a result from

statistical standardization, i.e. more ethanol or CO

molecules

can be found in the far range of the simulation volume.

The

–ethanol pair interaction, shown in Fig. S5 in the ESI,†

exhibits a relatively small peak with a shoulder located between

rB3.0 and 6.3 Å, which partially lies within the range of

hydrogen bonding interactions and suggests the occurrence of

relatively strong CO

–ethanol association.

However, because

of the rather small peak magnitude, these interactions are

expected to occur rather sporadically.

A quantification of the local inhomogeneities can be achieved

by monitoring the average coordination numbers for the

first coordination shell defined as Nxy¼4pryÐrc

0r2gxyðrÞdr.

Therein, xstands for the central interaction site surrounded by

interaction sites of type y,r

is the bulk density of interaction

sites of type y,g

xy

(r) represents the radial distribution function

of the pair involved in the running average number calculation

and r

is the radius of the first coordination shell, i.e. the location

of the first minimum of the regarded radial distribution function

xy

(r). The average coordination number N

CO2–CO2

shows a

similar behavior as the bulk density. At temperatures between

T= 315 and 330 K, the coordination number decreases rapidly to

approximately one third of its initial value and remains more

stable in the compressed gas region, cf. Fig. 6. The strong

decrease of the coordination number is mirrored in the observed

stepwise increase of the CO

intradiffusion coefficient. Similarly,

the inflection point of the coordination number was also found

to be located at the Widom line, i.e. at TB323 K. The average

coordination number N

EtOH–EtOH

, indicating the amount of

ethanol self-association, shows a stronger temperature depen-

dence in the supercritical liquid-like region. At T= 305 K, each

ethanol molecule is associated on average with 1.4 alcohol

molecules, but after an increase of 10 K in temperature this

value is reduced to 1.23. This implies a reduction of about 12%

in the hydrogen bonded structures and explains the related

increase of the ethanol intradiffusion coefficient between

Fig. 5 (a) Ethanol–ethanol and (b) CO

–CO

radial distribution functions

at T= 310 K (black), 320 K (blue), 330 K (green) and 340 K (red) along the

isobar p= 10 MPa.

Fig. 6 Average coordination number of the ethanol–ethanol and CO

–

pairs as a function of temperature along the isobar p= 10 MPa. The

dashed lines serve as a guide to the eye. The temperatures I and II

represent the crossing of the Widom line and the transition from the

super- to the subcritical regimes. The shaded areas indicate their expected

uncertainty.

PCCP Paper

This journal is ©the Owner Societies 2021 Phys. Chem. Chem. Phys.,2021,23, 3106--3115 | 3111

T= 305 and 315 K. In the region between 315 and 328 K, the

decrease of the average ethanol–ethanol coordination number is

much less pronounced than that of CO

, suggesting the presence

of enhanced ethanol hydrogen bonded structures that are

disrupted to a lesser extent by the strong density reduction.

These rather stable ethanol hydrogen bonded structures are

linked to mixture microheterogeneities observed in the domain

influenced by the Widom line and the super- to subcritical

transition. The relatively small values of the thermodynamic

factor as well as the milder sigmoidal increase of the ethanol

intradiffusion coefficient can also be explained with these

microscopic structures. At temperatures above TB328 K, the

coordination number decreases almost stepwise, suggesting a

steady breakup of the hydrogen bonded structures with tem-

perature in the compressed gas region.

The presence of ethanol self-association and microheterogene-

ities can be visually corroborated when simulation snapshots are

analyzed. Although a snapshot represents only one microstate of a

molecular system, in case of mixtures with associating compo-

nents, a single microstate may well represent all possible micro-

states, since they are permutations of the segregation patterns.

Fig. 7 shows snapshots of simulation volumes for four tempera-

tures. In spite of the low alcohol content, ethanol molecules tend

to self-associate and form hydrogen bonded networks. At low

temperatures up to the Widom line, most of the ethanol mole-

cules are part of clusters that form segregated domains. As the

temperature is increased, the ethanol clusters become smaller

and are more uniformly distributed in the simulation volume.

This observation corresponds to the observed values of the

intradiﬀusion coeﬃcient of ethanol and the relatively low values

of the thermodynamic factor.

3.4 Mutual diﬀusion coeﬃcients

The Maxwell–Stefan diffusion coefficient calculated from

the Onsager phenomenological coefficients

of the studied

mixture is shown in Fig. 4. As can be seen for temperatures

below T= 328 K, the Maxwell–Stefan diffusivity curve runs

almost parallel to and above the one of CO

intradiffusion.

Higher values of the Maxwell–Stefan diffusion coefficient with

respect to the intradiffusion coefficient are related to cluster

formation due to alcohol self-association.

In the regime

following the transition to the compressed gas, the Maxwell–

Stefan diffusion coefficient becomes lower than the CO

intra-

diffusion coefficient, which is in line with the strong reduction

of the ethanol hydrogen bonded structures indicated by the

decrease of the average ethanol–ethanol coordination number.

The inflection point of the Maxwell–Stefan diffusivity curve is

located in the vicinity of the Widom line, i.e. at TB324 K.

The Maxwell–Stefan diffusion coefficient obtained from

EMD simulation can be straightforwardly related to the Fick

diffusion coefficient Dthrough the so-called thermodynamic

factor G,i.e. D =G



, which is a measure of the mixture’s non-

ideality. Because of the nature of the CO

+ ethanol mixture and

the presence of microheterogeneities, the sampled thermo-

dynamic factor reaches relatively low values GB0.45. The

expected minimum of the thermodynamic factor in the proxi-

mity of the Widom line is predicted clearly by the employed

equations of state. The Kirkwood–Buff integration results pre-

dict a weaker minimum shifted to higher temperatures, cf.

Fig. 8. For the sake of consistency and because of the rather

large differences in Gobtained from the different equations of

state, the thermodynamic factor from Kirkwood–Buff integra-

tion was employed here to calculate the Fick diffusion coeffi-

cient. Since the values of the sampled thermodynamic factor do

Fig. 7 Snapshots of the present EMD simulations at selected temperatures.

The CO

solvent molecules were graphically removed to reveal clustering

among ethanol molecules.

Fig. 8 Thermodynamic factor Gof the CO

+ ethanol mixture as a

function of temperature along the isobar p= 10 MPa. The solid lines

represent results from equations of state, i.e. Peng–Robinson (green),

Soave–Redlich–Kwong (red) and PCP-SAFT (blue). The symbols indicate

simulation results from Kirkwood–Buff integration. The dashed line serves

as a guide to the eye. The temperatures I and II represent the crossing of

the Widom line and the transition from the super- to the subcritical

regimes. The shaded areas indicate their expected uncertainty.

Paper PCCP

3112 |Phys. Chem. Chem. Phys.,2021,23, 3106--3115 This journal is ©the Owner Societies 2021

not show a strong variation in the range of the studied

thermodynamic conditions, the temperature dependence of

the sampled Fick diffusion coefficient is similar to that

observed for the Maxwell–Stefan diffusion coefficient, cf.

Fig. 9. However, its inflection point is shifted to a higher

temperature, i.e. T B329 K, which can be explained with the

changes observed for the ethanol–ethanol average coordination

number near this temperature. In general, values of the Fick

diffusion coefficient sampled by EMD are comparable with

present measurements within the statistical uncertainties, cf.

Table 3. However, the experimental values exhibit a smoother

temperature dependence with an inflection point located at

TB327 K. The comparably lower values of the predicted Fick

diffusion coefficient in the supercritical liquid-like region, i.e.

at To320 K, could be due to an overestimation of ethanol self-

association by molecular simulation. Approaching the Widom

line, where density variations are large, simulation and experi-

mental results agree quite well up to TB332 K, i.e. right after

the transition into the subcritical region. At the highest studied

temperatures, the Fick diffusion coefficient predicted by EMD

is overestimated when compared with the experimental data.

This suggests that the actual breakup of ethanol hydrogen

bonded structures in the compressed gas region occurs more

gradually than predicted by EMD simulation. It should be

noted that contrary to EMD simulations, where the mixture

concentration is exactly known, the experimental concentration

can only be estimated. Therefore, the differences between

simulation and experimental values can partially be explained

with the difference in ethanol concentration, which has been

proven to have a strong influence on the Fick diffusion coeffi-

cient even in the dilution limit of scCO

mixtures.

66,75

This

significant influence of concentration on the Fick diffusion

coefficient is mainly related to the strong concentration depen-

dence of the thermodynamic factor given by the proximity

of the critical point.

A direct comparison of present

experimental results with data from previous studies

36,37

not possible because of they were measured for different state

points.

The Stokes–Einstein based equations by Wilke-Chang,

Sassiat,

Tyn-Calus,

Scheibel,

Reddy-Doraiswamy

and

Lai-Tan

as well as the free volume based equations by Catch-

pole and King,

He and Yu

as well as Funazukuri et al.

were

tested with respect to their ability to predict the Fick diffusion

coefficient at infinite dilution of the CO

+ ethanol mixture.

However, none of these equations was able to capture neither

qualitatively nor quantitatively the present experimental

Fick diffusion coefficient values, cf. Fig. S7 of the ESI.†The

predictive He and Yu

equation yields the best results among

the tested equations, however, its agreement with the present

experimental results is quite poor in the supercritical region, cf.

Fig. 9(a). The strong overestimation of the Fick diffusion coeffi-

cient in regions with a higher density, where ethanol self-

association plays a decisive role for molecular mobility, as well

as the better agreement with experiments in the compressed gas

region, indicate a failure of these models to adequately consider

the hydrogen bonded structures in the mixture.

3.5 Shear viscosity

Theimportanceofconsideringinformationonhydrogenbonding

dynamics for the prediction of transport properties can be clearly

Fig. 9 (a) Fick diﬀusion coeﬃcient predicted by EMD simulation (crosses)

and obtained experimentally (squares) as a function of temperature along

the isobar p= 10 MPa. The red line represents the predictive equation by

He and Yu.

(b) Shear viscosity predicted by EMD simulation (crosses)

compared to the shear viscosity of pure CO

according to the correlation

by Laesecke and Muzny

(cyan line). The dashed lines serve as a guide to

the eye. The temperatures I and II represent the crossing of the Widom line

and the transition from the super- to subcritical regimes. The shaded areas

indicate their expected uncertainty.

Table 3 Experimental Fick diﬀusion coeﬃcient of the CO

+ ethanol

mixture along the isobar p=100.05 MPa. The listed values represent the

average Dand standard deviation sof typically ten diﬀerent measure-

ments. The given temperatures have an uncertainty of 0.1 K

T/K D/10

8

1

s/10

8

1

304 1.55 0.06

308 1.64 0.07

313 1.75 0.07

318 1.97 0.08

320 2.07 0.09

323 2.43 0.23

328 2.86 0.25

333 3.34 0.28

338 3.62 0.31

343 3.97 0.36

PCCP Paper

This journal is ©the Owner Societies 2021 Phys. Chem. Chem. Phys.,2021,23, 3106--3115 | 3113

observed when the shear viscosity is analyzed. Fig. 9(b) shows the

strong diﬀerence between the shear viscosity of pure CO

and that

of the mixture. In the liquid-like region, where intermolecular

forces dominate the viscous eﬀects, the shear viscosity of the

mixture exhibits an increment of approximately 70% from

the value of pure CO

. This enhancement can be ascribed to the

presence of microheterogeneities caused by hydrogen bonding

networksdespitethesmallamountofethanolpresentinthe

mixture. Here, ethanol self-association is expected to play

the main role in the formation of microscopic structures because

of the low values of the CO

–ethanol coordination number, i.e.

–EtOH

o0.3. As higher temperatures are reached, the momen-

tum transfer due to molecular thermal motion makes an increas-

ingly important contribution to the shear viscosity and hydrogen

bonds break, reducing the viscosity enhancement to less than

15% in the compressed gas region. Further, the shear viscosity

curve of the mixture shows an inflection point located at TB

320 K, which cannot be observed for pure CO

.Viscosityand

diffusion curves exhibit an opposing behavior, as expected from

the Stokes–Einstein equation.

4 Conclusions

A study on the dynamic behavior of ethanol diluted in –CO

in the temperature range from T= 304 to 343 K along the isobar

p= 10 MPa was conducted employing complementary

approaches, i.e. experiment and molecular simulation. Along

this path, the studied mixture goes through a transition in the

supercritical region by crossing the Widom line and from

super- to subcritical regimes. The Fick diffusion coefficient

was measured with the Taylor dispersion technique,

while several EMD simulation methods were employed. Shear

viscosity, intra- and Maxwell–Stefan diffusion coefficients were

sampled with equilibrium molecular dynamics, employing

rigid, non-polarizable force fields based on Lennard-Jones sites

and superimposed point charges or quadrupoles, and

the Green–Kubo formalism. The thermodynamic factor was

calculated with Kirkwood–Buff integration. In this way, the

Fick diffusion coefficient was determined by EMD simulations

consistently on the basis of the selected force fields. In order to

validate the force fields employed to describe the mixture, the

VLE of the mixture was calculated at three temperatures

and compared successfully with experimental data from the

literature. Further, to gain insight into the microscopic

structure of the mixture, center-of-mass radial distribution func-

tionsaswellasaveragedcoordinationnumberswereinvestigated.

In the region where the crossover between the liquid-like

and gas-like regimes occurs, several thermophysical properties

become very sensitive to temperature and pressure variations

and can therefore change drastically along supercritical paths.

Along the studied isobar, the presence of inflection points

in the temperature dependence of density, enthalpy, shear visc-

osity, intra- and transport diﬀusion coeﬃcients were reported.

Mostly, the inflection points were observed at temperatures

between TB321 and B324 K, a range that specifies the region

influenced by the Widom line. Moreover, the calculated values of

the center-of-mass average coordination numbers up to the first

solvationshellalsoexhibitaninflectionpointinthisregion,

shedding light on the underlying microscopic structural back-

ground of the observed macroscopic property behavior.

The transition between the supercritical gas-like and the

subcritical compressed gas regimes at TB328 K is smooth for

most of the studied properties, i.e. density, enthalpy, shear

viscosity, Maxwell–Stefan and CO

intradiffusion coefficients.

However, the intradiffusion coefficient of ethanol and the

Fick diffusion coefficient showed an inflection point in the

proximity of this transition. It was noticed that the averaged

coordination number of ethanol–ethanol pairs, which is an

indicator of ethanol self-association, shows an almost stepwise

change in this region. This was related to a relatively strong

decrease of ethanol self-association when moving from super-

critical to subcritical states, which could also be observed by

analyzing snapshots of the simulation volumes.

AsatisfactoryagreementwasfoundbetweenpredictiveEMD

simulation data and experimental results for the Fick diﬀusion

coeﬃcient, especially in the temperature range from the Widom

line to the super- to subcritical transition. Both experiment and

simulation exhibit a sigmoidal behavior along the studied isobar,

however, experiments showed an inflection point at a slightly

lower temperature. Possible reasons for the diﬀerence between

simulation and experimental results were thoroughly discussed.

An analysis of the shear viscosity of the mixture corrobo-

rated the strong influence of microheterogeneities given by

hydrogen bonded networks on the macroscopic transport prop-

erties of the mixture.

Conflicts of interest

There are no conflicts to declare.

Acknowledgements

All authors greatly acknowledge the support of Prof. Joachim Gross

(University of Stuttgart) regarding the handling of the PCP-SAFT

equation of state. The authors Y. G. and V. S. kindly appreciate

financial support by the PRODEX program of the Belgian Federal

Science Policy Oﬃce and the European Space Agency (ESA). The

authors R. C., G. G.-C. and J. V. want to acknowledge the support

by – Deutsche Forschungsgemeinschaft (DFG) under Grant No. VR

6/11. This work was carried out under the auspices of the

Boltzmann-Zuse Society (BZS). Equilibrium molecular dynamics

simulations were performed either on Cray’s CS500 system Noctua

at the Paderborn Center for Parallel Computing (PC

)oronthe

HPE Apollo system Hawk at the High Performance Computing

CentreStuttgart(HLRS)contributingtotheprojectMMHBF2.

Notes and references

1 M. Mukhopadhyay, Natural extraction using supercritical

carbon dioxide, CRC Press, 1st edn, 2000.

Paper PCCP

2 R. Gupta and J.-J. Shim, Solubility in supercritical carbon

dioxide, CRC Press, 1st edn, 2007.

3 F. A. Gorelli, T. Bryk, M. Kirsch, G. Ruocco, M. Santoro and

T. Scopigno, Sci. Rep., 2013, 3, 1203.

4 J. Losey and R. J. Sadus, J. Phys. Chem. B, 2019, 123,

8268–8273.

5 K. Karalis, C. Ludwig and B. Niceno, Sci. Rep., 2019,

9, 15731.

6 Y. D. Fomin, V. N. Ryzhov, E. N. Tsiok and V. V. Brazhkin, Phys.

Rev. E: Stat., Nonlinear, Soft Matter Phys., 2015, 91, 022111.

7 A. Imre, G. Hazi, A. Horvath, C. Maraczy, V. Mazur and

S. Artemenko, Nucl. Eng. Des., 2011, 241, 296–300.

8 A. Imre, C. Ramboz, U. Deiters and K. Kraska, Environ. Earth

Sci., 2015, 73, 4373–4384.

9 M. Raju, D. T. Banuti, P. C. Ma and M. Ihme, Sci. Rep., 2017,

7, 3027.

10 N. Polikhronidi, R. Batyrova, A. Aliev and I. Abdulagatov,

J. Therm. Sci., 2019, 28, 394–430.

11 P. Raveendran, Y. Ikushima and S. L. Wallen, Acc. Chem.

Res., 2005, 38, 478–485.

12 T. Tassaing, P. Lalanne, S. Rey, F. Cansell and M. Besnard,

Ind. Eng. Chem. Res., 2000, 39, 4470–4475.

13 I. Skarmoutsos and E. Guardia, J. Phys. Chem. B, 2009, 111,

8887–8897.

14 K. Nishikawa and I. Tanaka, Chem. Phys. Lett., 1995, 244,

149–152.

15 H. Nakayama, K.-I. Saitow, M. Sakashita, K. Ishii and

K. Nishikawa, Chem. Phys. Lett., 2000, 320, 323–327.

16 N. Nishikawa and T. Morita, Chem. Phys. Lett., 2000, 316,

238–242.

17 G. G. Simeoni, T. Bryk, F. A. Gorelli, M. Kirsch, G. Ruocco,

M. Santoro and T. Scopigno, Nat. Phys., 2010, 6, 503–507.

18 S. Hans and C. C. Yu, Phys. Rev. E: Stat., Nonlinear, Soft

Matter Phys., 2012, 85, 051201.

19 P. Gallo, D. Corradini and M. Rovere, Nat. Commun., 2014, 5, 5806.

20 M. Y. Ha, T. J. Yoon, T. Tlusty, Y. Jho and W. B. Lee, J. Phys.

Chem. Lett., 2018, 9, 1734–1738.

21 M. Fisher and B. Widom, J. Chem. Phys., 1969, 50,

3756–3772.

22 V. Brazhkin, Y. Fomin, A. Lyapin, V. Ryzhov and

K. Trachenko, Phys. Rev. E: Stat., Nonlinear, Soft Matter

Phys., 2012, 85, 031203.

23 D. Bolmativ, M. Zhernenkov, D. Zav’yalov, S. Tkachev,

A. Cunsolo and Y. Cai, Sci. Rep., 2015, 5, 15850.

24 L. Xu, P. Kumar, S. Buldyrev, S.-H. Chen, P. Poole,

F. Sciortino and H. Stanley, Proc. Natl. Acad. Sci. U. S. A.,

2005, 102, 16558–16562.

25 D. Banuti, M. Raju and M. Ihme, J. Supercrit. Fluids, 2020,

165, 104895.

26 T. J. Yoon, M. Y. Ha, W. B. Lee and Y.-W. Lee, J. Chem. Phys.,

2018, 149, 014502.

27 T. J. Yoon, M. Y. Ha, W. B. Lee and Y.-W. Lee, J. Chem. Phys.,

2019, 150, 154503.

28 E. Mareev, V. Aleshkevich, F. Potemkin, V. Bagratashvili,

N. Minaev and V. Gordienko, Opt. Express, 2018, 26,

13229–13238.

29 B. Chu, Ber. Bunsen-Ges., 1972, 76, 202–215.

30 M. M. Hoﬀmann and M. S. Conradi, J. Phys. Chem. B, 1998,

102, 263–271.

31 M. Saharay and S. Balasubramanian, J. Phys. Chem. B, 2006,

110, 3782–3790.

32 W. Xu, J. Yang and Y. Hu, J. Phys. Chem. B, 2009, 113,

4781–4789.

33 T. Aida, T. Aizawa, M. Kanakubo and H. Nanjo, J. Phys.

Chem. B, 2010, 114, 13628–13636.

34 B. A. McGuire, M.-A. Martin-Dumel and M. C. McCarthy,

J. Phys. Chem. A, 2017, 121, 6283–6287.

35 Y. Gaponenko, V. Gousselnikov, C. I. A. V. Santos and

V. Shevtsova, Microgravity Sci. Technol., 2019, 31, 475–486.

36 D. Mei, H. Li and W. Wang, Huagong Xuebao, 1995, 46,

357–364.

37 C. Y. Kong, T. Funazukuri and S. Kagei, J. Supercrit. Fluids,

2006, 37, 359–366.

38 Y. Houndonougbo, K. Kuczera, B. Subramaniam and

B. B. Laird, Mol. Simul., 2007, 33, 861–869.

39 L. Hui, W. Rui, F. Weiyu, L. Zhaomin, M. Tao and

N. Guozhi, Acta Pet. Sin., 2015, 31, 78–83.

40 I. Skarmoutsos, E. Guardia and J. Samios, J. Chem. Phys.,

2010, 133, 014504.

41 W. Xu and J. Yang, J. Phys. Chem. A, 2010, 114, 5414–5428.

42 S. Reiser, N. McCann, M. Horsch and H. Hasse, J. Supercrit.

Fluids, 2012, 68, 94–103.

43 Z. Li, S. Lai, W. Gao and L. Chen, Russ. J. Phys. Chem. A,

2018, 92, 1332–1337.

44 T. Schnabel, J. Vrabec and H. Hasse, Fluid Phase Equilib.,

2005, 223, 134–143.

45 J. Vrabec, J. Stoll and H. Hasse, J. Phys. Chem. B, 2001, 105,

12126–12133.

46 K. Suzuki, H. Sue, M. Itou, R. L. Smith, H. Inomata, K. Arai

and S. Saito, J. Chem. Eng. Data, 1990, 35, 63–66.

47 J. S. Lim, Y. Y. Lee and H. S. Chun, J. Supercrit. Fluids, 1994,

7, 219–230.

48 J. W. Ziegler, J. G. Dorsey, T. L. Chester and D. P. Innis, Anal.

Chem., 1995, 67, 456–461.

49 L. A. Galicia-Luna, A. Ortega-Rodriguez and D. Richon,

J. Chem. Eng. Data, 2000, 45, 265–271.

50 I. Tsivintzelis, D. Missopolinou, K. Kalogiannis and

C. Panayiotou, Fluid Phase Equilib., 2004, 224, 89–96.

51 H.-Y. Chiu, M.-J. Lee and H.-M. Lin, J. Chem. Eng. Data,

2008, 53, 2393–2402.

52 C.-N. Han and C.-H. Kang, Korean J. Chem. Eng., 2017, 34,

1781–1785.

53 G. Soave, Chem. Eng. Sci., 1972, 27, 1197–1203.

54 D.-Y. Peng and D. P. Robinson, Ind. Eng. Chem. Fundam.,

1976, 15, 59–64.

55 J. Gross and J. Vrabec, AIChE J., 2006, 52, 1194–1204.

56 D. W. Jennings, R. J. Lee and A. S. Teja, J. Chem. Eng. Data,

1991, 36, 303–307.

57 M. S. Green, J. Chem. Phys., 1954, 22, 398–414.

58 R. Kubo, J. Phys. Soc. Jpn., 1957, 12, 570–586.

59 Y. M. Mun

˜oz-Mun

˜oz, G. Guevara-Carrion and J. Vrabec,

J. Phys. Chem. B, 2018, 122, 8718–8729.

PCCP Paper

60 I.-C. Yeh and G. Hummer, J. Phys. Chem. B, 2004, 108,

15873–15879.

61 S. H. Jamali, L. Wolﬀ, T. M. Becker, A. Bardow, T. J. H. Vlugt

and O. A. Moultos, J. Chem. Theory Comput., 2018, 14,

2667–2677.

62 P. Ganguly and N. F. A. van der Vegt, J. Chem. Theory

Comput., 2013, 9, 1347–1355.

63 R. Fingerhut and J. Vrabec, Fluid Phase Equilib., 2019, 485,

270–281.

64 G. Rutkai, A. Ko

¨ster, G. Guevara-Carrion, T. Janzen,

M. Schappals, C. W. Glass, M. Bernreuther, A. Wafai,

S. Stephan, M. Kohns, S. Reiser, S. Deublein, M. Horsch,

H. Hasse and J. Vrabec, Comput. Phys. Commun., 2017, 221,

343–351.

65 S. Ancherbak, C. Santos, J. Legros, A. Mialdun and

V. Shevtsova, Eur. Phys. J. E: Soft Matter Biol. Phys., 2016,

39, 111.

66 G. Guevara-Carrion, S. Ancherbak, A. Mialdun, J. Vrabec and

V. Shevtsova, Sci. Rep., 2019, 9, 8466.

67 R. Span and W. Wagner, J. Phys. Chem. Ref. Data, 1996, 25,

1509–1596.

68 K. W. Hutcheson and N. R. Foster, Innovations in Super-

critical Fluids: Science and Technology, American Chemical

Society, 1995.

69 S.-D. Yeo, S.-J. Park, J.-W. Kim and J.-C. Kim, J. Chem. Eng.

Data, 2000, 45, 932–935.

70 J. A. Schroeder, S. G. Penoncello and J. S. Schroeder, J. Phys.

Chem. Ref. Data, 2014, 43, 043102.

71 C. Oldiges, K. Wittler, T. To

¨nsing and A. Alijah, J. Phys.

Chem. A, 2002, 106, 7147–7154.

72 M. Pozˇar, B. Lovrinc

ˇevic

´, L. Zoranic

´, T. Primorac

´, F. Sokolic

and A. Perera, Phys. Chem. Chem. Phys., 2016, 18,

23971–23979.

73 R. Krishna and J. A. Wesselingh, Chem. Eng. Sci., 1997, 52,

861–911.

74 P. W. M. Rutten, Diﬀusion in liquids, Delft University Press,

1992.

75 H. Nishiumi and T. Kubota, Fluid Phase Equilib., 2007, 261,

146–151.

76 C. R. Wilke and P. Chang, AIChE J., 1955, 1, 264–270.

77 P. R. Sassiat, P. Mourier, M. H. Caude and R. H. Rosset,

Anal. Chem., 1987, 59, 1164–1170.

78 M. T. Tyn and W. F. Calus, J. Chem. Eng. Data, 1975, 20,

106–109.

79 K. A. Reddy and L. K. Doraiswamy, Ind. Eng. Chem. Fundam.,

1967, 6, 77–79.

80 C.-C. Lai and C.-S. Tan, Ind. Eng. Chem. Res., 1995, 34,

674–680.

81 O. J. Catchpole and M. B. King, Ind. Eng. Chem. Res., 1994,

33, 1828–1837.

82 C.-H. He and Y.-S. Yu, Ind. Eng. Chem. Res., 1998, 37,

3793–3798.

83 T. Funazukuri, C. Y. Kong and S. Kagei, Ind. End. Chem. Res.,

2000, 39, 835–837.

84 A. Laesecke and C. D. Muzny, J. Phys. Chem. Ref. Data, 2017,

46, 013107.

Paper PCCP

Futher publications

Other publications not contained in this thesis

• L. Claes, R.S. Chatwell, J. Vrabec and B. Henning, Spectral approach to acoustic

absorption measurements, Proccedings-Sensor, 304-309, Nürnberg (2017)

• L. Claes, L. M. Hülskämper, E. Baumhögger, N. Feldmann, R.S. Chatwell, J. Vrabec

and B. Henning, Acoustic absorption measurement for the determination of the bulk

viscosity of pure ﬂuids, TM-Technisches Messen, 86, S1-S5 (2019)

• R.S. Chatwell, R. Fingerhut, G. Guevara-Carrion, M. Heinen, T. Hitz, Y. Mauricio

Muñoz-Muñoz, C.-D. Munz and J. Vrabec, Atomistic simulations: The driving force

behind modern thermodynamic research, High Performance Computing in Science

and Engineering ’19, Springer Berlin, in press (2021)

• M. Heinen, R.S. Chatwell, S. Homes, G. Guevara-Carrion, R. Fingerhut, M. Kohns,

S. Stephan, M.T. Horsch and J. Vrabec, Molecular modelling and simulations: model

development, thermodynamic properties, scaling behavior and data management,

High Performance Computing and Engineering ’20, Springer Berlin, in press (2021)

Scientiﬁc talks

• M. Heinen, R.S. Chatwell and J. Vrabec, Insight into the nature of evaporation

processes enabled by massively parallel molecular dynamics simulations, IEEE 23rd

International Conference on High Performance Workshops (HiPCW), Hyderabad

(2016)

• R.S. Chatwell, M. Heinen and J. Vrabec, Insight into the dynamics of an evaporat-

ing liquid ﬁlm enabled by MD simulations,7

th Internation Symposium on Physical

Sciences in Space (ISPS) and 25th European Low Gravity Association (ELGRA)

Biennial Symposium and General Assembly, Juan-les-Pins (2017)

• R.S. Chatwell, M. Heinen and J. Vrabec, Insight into interfacial evaporation en-

abled by molecular dynamics simulations,12

th European Fluid Mechanics Conference

(EFMC), Vienna (2018)

• R.S. Chatwell, E. Baumhögger and J. Vrabec, Bestimmung der Volumenviskosität

einfacher Fluide mittels atomistischer Simulation, Thermodynamik-Kolloquium, Kas-

sel (2018)

• R.S. Chatwell, G. Guevara-Carrion, V. Shevtsova and J. Vrabec, Transport diﬀusion

in the critical region,14

th International Conference on Two-Phase Systems for Space

and Ground Applications (ITTW) and 26th European Low Gravity Association (EL-

GRA) General Assembly and Symposium, Granada (2019)

Poster presentations

• R.S. Chatwell, M. Heinen and J. Vrabec, Diﬀusion driven evaporation of a binary

mixture: Fluid dynamics vs. Molecular Dynamics simulations, Thermodynamik-

Kolloquium, Bochum (2015)

• S. Homes, R.S. Chatwell, R. Fingerhut, G. Guevara-Carrion, M. Heinen, T. Hitz,

Y. Mauricio Muñoz-Muñoz, C.-D. Munz and J. Vrabec, Atomistic simulations: The

driving force behind modern thermodynamic research, High Performance Computing

in Science and Engineering - 22nd Results and Review Workshop of the HLRS,

Stuttgart (2019)

Appendix

In the following each electronic supplementary material (ESM) of the respective original

publication is attached. The ESM are used to disclose and archive all generated data.

.BzmbBQM HBKBi2/ 2pTQ`iBQM Q7  KmHiB+QKTQM2Mi HB[mB/ }HK

_2Mû aT2M+2` *?ir2HH- Jii?Bb >2BM2M- M/ C/`M o`#2+R- V

h?2`KQ/vMKB+b M/ 1M2`;v h2+?MQHQ;v- lMBp2`bBiv Q7 S/2`#Q`M- jjyN3 S/2`#Q`M-

:2`KMv

h?Bb bmTTH2K2Mi`v Ki2`BH /Bb+HQb2b HH +H+mHi2/

i?2`KQ/vMKB+ /i 7Q` i?2 T`2b2Mi MHviB+H bQHmiBQM

iQ;2i?2` rBi? i?2 geKQ/2H `2HvBM; QM /i bKTH2/

#v J. Q` J* bBKmHiBQMbX oHm2b 7`QK i?2 MHviB+H

bQHmiBQM `2 BM/B+i2/ #v i?2 bm#b+`BTi M- pHm2b 7`QK

i?2 geKQ/2H `2 /2MQi2/ #v KQ/ M/ bBKmHi2/ /i `2

bT2+B}2/ #v bBKX AM i?2 7QHHQrBM; i#H2b- HH [mMiBiB2b

`2 mM/2`biQQ/ b 7mM+iBQMb Q7 2Bi?2` HB[mB/ [xi]=KQH

KQH−1Q` pTQ` T?b2 [yi]=KQH KQH−1KQH2 7`+iBQMb- B7

MQi Qi?2`rBb2 bT2+B}2/X h?2 :B##b 2t+2bb 2M2`;v geM/

HH +?2KB+H TQi2MiBHb μi`2 ;Bp2M BM MQM@/BK2MbBQMH

7Q`K- BX2X ge=Ge/nRTM/ μi=¯μi/RT

HH iBK2@BM/2T2M/2Mi i?2`KQ/vMKB+ T`QT2`iB2b r2`2

/2i2`KBM2/ #v J* bBKmHiBQMb BM i?2 BbQi?2`KH@BbQ#`B+

2Mb2K#H2 +QMiBMBM; N4 kyyy T`iB+H2bX h`MbTQ`i

T`QT2`iB2b r2`2 /2i2`KBM2/ #v J. bBKmHiBQMb BM i?2

+MQMB+H 2Mb2K#H2 +QMiBMBM; N4 9NR9 T`iB+H2b M/

M BMi2;`iQ` iBK2 bi2T Q7 Δt4 yX33 7bX oHm2b r2`2

Q#iBM2/ 7i2` #Qmi 9·108iBK2 bi2Tb- r?B+? `2T`2b2Mib

 bKTHBM; iBK2 Q7 yX3 μbX

+QMiBMBM; N4 9NR9 T`iB+H2b- rBi? pHm2b #2BM; Q#@

iBM2/ 7i2` #Qmi 9·108iBK2 bi2Tb- r?B+? `2T`2b2Mib 

bKTHBM; iBK2 Q7 yX3 μbX

h?2 b2H2+i2/ geKQ/2H U7Qm`i? Q`/2` J`;mH2b ivT2

TQHvMQKBHV rb }ii2/ rBi?  biM/`/ MQMHBM2` H2bi

b[m`2b H;Q`Bi?K iQ KBMBKBx2 i?2 `QQi K2M b[m`2 2``Q`

#2ir22M bBKmHiBQM /i M/ i?2 KQ/2H `QmM/ i?2 2[mB@

HB#`BmK pTQ` T?b2 +QKTQbBiBQM y∗=(0.13,0.03,0.84)

M/ vB2H/2/ 7Q` i?2 T`K2i2`b Aij,E

ij M/ Ci

⎡

⎣

ĜA12 A13

A21 ĜA23

A31 A32 Ĝ⎤

⎦=⎡

⎣

Ĝ−39.91.7

4.2Ĝ3.3

3.80.7Ĝ⎤

⎦URV

⎡

⎣

ĜE12 E13

ĜĜE23

ĜĜ Ĝ

⎤

⎦=⎡

⎣

Ĝ−0.01 1.63

ĜĜ−1.82

ĜĜ Ĝ

⎤

⎦,UkV

⎡

⎣

C3⎤

⎦=⎡

⎣−50.6

−126.3

−40.9⎤

⎦.UjV

hQ /2b+`B#2 i?2 HB[mB/ T?b2Ƕb MQM@B/2HBiB2b rBi? ;`2i2`

T`2+BbBQM- M //BiBQMH }i `QmM/ x∗=(0.495,0.505,0)

rb T2`7Q`K2/ iQ vB2H/ /Bz2`2Mi T`K2i2`b 7Q` i?2 bK2

VD/`MXp`#2+!mT#X/2

6A:X RX :B##b i2`M`v THQi BHHmbi`iBM; +QKTQbBiBQMb r?2`2

i?2`KQ/vMKB+ /i r2`2 ;2M2`i2/ #v J* bBKmHiBQMbX lM@

/2` 2[mBHB#`BmK- i?2 pTQ` T?b2Ƕb +QKTQbBiBQM Bb ;Bp2M b

y=(0.13,0.027,0.843) KQH KQH−1U`2/ #mHH2iV r?BH2 i?2 HB[@

mB/ T?b2Ƕb Bb ;Bp2M b x=(0.495,0.505,0) KQH KQH−1U;`22M

#mHH2iVX h?2 /b?2/ HBM2b b?Qr i?2 pTQ` M/ HB[mB/ #BMQ/HbX

6A:X kX a2;K2Mi Q7 i?2 :B##b i2`M`v THQi /2TB+iBM; /Bz2`@

2Mi Ti?b 7Q` i?2 pTQ` T?b2 +QKTQbBiBQM /m`BM; 2pTQ`iBQM

7`QK bBKmHiBQM U#Hm2V M/ MHviB+ bQHmiBQM UpBQH2iVX

KQ/2H TTHB2/ iQ i?2 #BM`v bm#bvbi2K QMHv

⎡

⎣

A21

A12

E12 ⎤

⎦=⎡

⎣

0.26

0.25

−0.01 ⎤

⎦.U9V

oTQ` T?b2 +QMp2+iBp2 KQiBQM

h?2 HBM2` KQK2MimK Q7  KmHiB+QKTQM2Mi KBtim`2 Bb

+QMbi`m+i2/ Qmi Q7 i?2 p2+iQ` bmK Q7 Bib +QMbiBim2MiǶb

HBM2` KQK2Mi

ρvuz=



i=1

ρu,ivz,i .U8V

h?2 i@i? +QKTQM2MiǶb T`iBH /2MbBiv Bb ;Bp2M #v ρv,i =

yiρvX h?2 i?B`/ +QKTQM2Mi Bb b22M b  bi;MMi ;b

uz,3≈0- BX2X MQ +QHH2+iBp2 KQiBQM BM  bTiBH /B`2+@

iBQM Bb K2bm`#H2- Hi?Qm;? `M/QK i?2`KH KQiBQM Bb

T`2b2MiX h?2 KQH` p2`;2/ p2HQ+Biv uzBM i?2 `2H2pMi

z/B`2+iBQM Bb i?2M ;Bp2M b  T`QTQ`iBQMi2 bmK Q7 2p2`v

+QKTQM2MiǶb +QMp2+iBp2 p2HQ+Biv vz,i

uz=



i=1

yiuz,i ≈



i=1

yiuz,i .UeV

h?2 +QMp2+iBp2 KQiBQM uz,i +M #2 /2+QKTQb2/ BMiQ M

/p2+iBp2 M/  /BzmbBp2 +QMi`B#miBQM uz,i =uz+Uz,i



i=1

yiuz,i =



i=1

yiuz+Uz,i.UdV

am#biBimiBM; i?2 KQH` p2`;2/ KQiBQM uzrBi? i?2 `2@

bmHi H`2/v Q#iBM2/ 7`QK UeV vB2H/b



i=1

yiuz,i =



i=1

yi2



i=1

yiuz,i +Uz,i,U3V

r?B+? 7i2`  bBKTH2 H;2#`B+ KMBTmHiBQM +QMM2+ib

2+? +QKTQM2MiǶb +QMp2+iBp2 yivz,i iQ Bib /BzmbBQM ~mt

yiUz,i



i=1

yiuz,i1−



i=1

yi=



i=1

yiUz,i .UNV

AM  Hbi bi2T- i?2 KBtim`2Ƕb iQiH KQH` +QKTQbBiBQM +QM@

bi`BMi y1+y2+y3=1Bb miBHBx2/ iQ `2iBM i?2 /2b+`BTiBQM

mb2/ BM 2[miBQM URRV Q7 i?2 KMmb+`BTi



i=1

yiuz,i =



i=1yi

y3Uz,i =



i=1

bi3Uz,i .URyV

"QmM/`v +QM/BiBQM

AM 2[miBQM UR3V Q7 i?2 KMmb+`BTi-  #QmM/`v

+QM/BiBQM rb mb2/ i?i M22/b  KQ`2 /2iBH2/ +H`B}@

+iBQM +QM+2`MBM; Bib HBKBiiBQMbX FBM iQ i?2 T`iB+H2

/2MbBiv +QMb2`piBQM 2[miBQM

∂tρv,i(z,t)+∂zρv,i(z,t)vz,i(z,t)=0,URRV

 +QMb2`piBQM 2[miBQM 7Q` i?2 r2B;?i2/ KQH2 7`+iBQMb

bi3+M #2 +QMbi`m+i2/

∂tbi3(z)+∂zbi3(z)vz,i(z,t)=0,URkV

r?2`2 miBHBxBM; bbmKTiBQM URkV Q7 i?2 KMmb+`BTi `2@

/m+2b i?2 +QKTH2tBiv bB;MB}+MiHvX Ai b?QmH/ #2 MQi2/

i?i i?2 T`K2i2`b bi3`2 BM/22/ 7mM+iBQMb Q7 iBK2-

v2i i?2B` p`BiBQM BM iBK2 rb +QMbB/2`2/ M2;HB;B#H2X 

7Q`KmHiBQM 7Q` i?Bb /2`BpiBp2 i i?2 BMi2`T?b2 z=zs

M22/b iQ #2 7QmM/

∂zbi3(z)vz,i(z,t)zs

=0.URjV

.2+QKTQbBM; i?2 +QMp2+iBp2 p2HQ+Biv p2+iQ` vz,i =uz+

Uz,i BMiQ Bib /p2+iBp2 M/ /BzmbBp2 T`ib ;BM vB2H/b

∂zbi3(z)uz+Uz,izs

=0.UR9V

h?2 /BzmbBQM ~mt ?b H`2/v #22M B/2MiB}2/ iQ #2 /2@

b+`B#2/ #v 6B+FǶb /BzmbBQM Hr i?i rBi? i?2 2z2+iBp2

/BzmbBQM +Q2{+B2Mi Di`2/b

bi3Uzi=−Didzbi3(z),UR8V

7i2` M2;H2+iBM; bi3dzuzzs7Q` #2BM; bKHH +QKT`2/

iQ i?2 Qi?2` i2`Kb

uzdzbi3(z)zs−Didzzbi3(z)zs

=0.UReV

AM  Hbi bi2T- i?2 +QMM2+iBQM #2ir22M i?Bb pTQ` T?b2

`;mK2MiiBQM M/ i?2 HB[mB/ T?b2Ƕb bm`7+2 `2;`2bbBQM

p2HQ+Biv M22/b iQ #2 7QmM/X h?2 Hii2` Bb B/2MiB}2/ b i?2

M2;iBp2 +QMp2+iBp2 KQiBQM Q7 2pTQ`i2 i i?2 BMi2`T?b2

M/ Bb ;Bp2M

uz,liq(zs)=−uz(zs)=



i=1 Didzbi3(z)zs

.URdV

7i2` BMb2`iBM; URdV BMiQ UReV M/ 2tTM/BM; i?2 bmK- 

7m`i?2` bbmKTiBQM ?b iQ #2 K/2

D1dzb13(z)zs+D2dzb23(z)zsdzbi3(z)zs

−Didzzbi3(z)zs=0.UR3V

HH +`QbbQp2` i2`Kb `2 bbmK2/ iQ #2 M2;HB;B#H2 Qp2` i?2

MQM@+`QbbQp2` i2`Kb

Djdzbj3(z)zsdzbi3(z)zsD

idzbi3(z)zs2,∀j=i.

URNV

PM2 +M }MHHv +QM+Hm/2 i?2 T`Q#H2KǶb #QmM/`v +QM/B@

iBQM i }HKǶb bm`7+2 iQ #2

dzbi3(z)zs2=dzzbi3(z)zs.UkyV

h"G1 AX Sm`2 +QKTQM2Mi bBKmHiBQM `2bmHib i?i ;Bp2 i?2 #2?pBQ` Q7 2+? +QKTQM2MiǶb +?2KB+H TQi2MiBH μ0,i M/ `2bB/mH

+QMi`B#miBQM ψib  7mM+iBQM Q7 T`2bbm`2 i +QMbiMi i2KT2`im`2 T4 3y EX h?2 bBKmHiBQMb vB2H/2/  mMB7Q`K KQH` /2MbBiv

Q7 ρ4 yXyR8R KQH /K−37Q` 2+? +QKTQM2Mi T`2T`2/ BM i?2 bii2 T43yE-p4 yXyR JS i?i Bb T`2+Bb2Hv i?2 B/2H ;b

2[miBQMǶb `2bmHi M/ /2i2`KBM2b i?2 B/2H ;b +?2KB+H TQi2MiBHb iQ #2 μ0

0=−(7.86,7.86,7.85)X aBM+2 i?2 `2bB/mH +QMi`B#miBQM

ψi- ++Q`/BM; iQ UkdV- K2bm`2b  +QKTQM2MiǶb /2T`im`2 7`QK i?2 B/2H ;b #2?pBQ`- i?2 +QKTQM2Mib R M/ k `2 HB[mB/ r?BH2

+QKTQM2Mi j Bb ;b2Qmb 7Q` i?2 bvbi2K #2BM; T`2T`2/ BM i?2 bii2 T43yE-p4 eXj9 JSX

pfJS μ0,1μ0,2μ0,3ψ1ψ2ψ3

yXyR @dX3e @dX3e @dX38 R R R

yXRy @8X8e @8X8d @8X88 yXNNy yXN3j RXyyk

RXyy @jXjd @9X3y @jXkj yX338 yXk9d RXykR

kXyy @jXy8 @9Xe3 @kX8R yXde9 yXRky RXy99

jXyy @jXyR @9Xe8 @kXyN yX9k8 yXy3k RXyed

jXk8 @jXyy @kXyy yXjNd RXydj

jX8y @kXNN @RXNk yXjdj RXydN

jXd8 @kXN3 @RX38 yXj8y RXy38

9Xyy @kXNd 9Xek @RXd3 yXjjk yXye9 RXyNR

9Xk8 @kXNe @RXdR yXjR8 RXyNd

9X8y @kXN8 @RXe8 yXjyy RXRyj

9Xd8 @kXN9 @RX8N yXk3d RXRyN

eXyy @kXNj @9X8N @RX8j yXkd8 yXy8j RXRR8

8Xk8 @kXNk @RX93 yXke8 RXRkk

8X8y @kXNR @RX9k yXk88 RXRk3

8Xd8 @kXNR @RXjd yXk9e RXRj9

eXyy @kXNy @RXjj yXkj3 RXR9R

eXk8 @kX3N @RXk3 yXkjy RXR9d

eXj9 @kX3N @9X8e @RXke yXkk3 yXy9j RXR8y

6A:X jX :`T?B+H `2T`2b2MiiBQM Q7 i?2 BM/BpB/mH Tm`2 +QKTQM2MiǶb +?2KB+H TQi2MiBHb μ0,i b  7mM+iBQM Q7 T`2bbm`2 i

+QMbiMi i2KT2`im`2X h?2 i?`22 +m`p2b- b i?2v b?QmH/- TT`Q+? B/2MiB+H pHm2b 7Q` i?2 B/2H ;b +b2X

h"G1 AAX h?2 #BM`v bm#bvbi2K i?i Bb +QKTQb2/ Q7 pQHiBH2 U+QKTQM2Mi RV M/ H2bb@pQHiBH2 U+QKTQM2Mi kV T`iB+H2b `2KBMb

HB[mB/ 7Q` HH TQbbB#H2 +QKTQbBiBQMbX h?2 7Qm`i? Q`/2` J`;mH2b ivT2 TQHvMQKBH +Qp2`b HbQ i?2 `2bT2+iBp2 BM}MBi2 /BHmiBQM

+iBpBiv +Q2{+B2Mib γ∞

i- b /2TB+i2/ BM 6B[XU9VX

x1x2y3ge

sim ge

mod γ1,sim γ2,sim γ1,mod γ2,mod

y R y y y @ R RXk38 R

yXyk8 yXNd8 y yXyR yXyR RXkeN RXyy9 RXkdy RXyyy

yXy8y yXN8y y yXyk yXyR RXk83 RXyyd RXk8e RXyyR

yXyd8 yXNk8 y yXyk yXyk RXkj3 RXyyR RXk9k RXyyR

yXRyy yXNyy y yXyk yXyk RXkkk RXyyR RXkk3 RXyyk

yXRk8 yX3d8 y yXyk yXyj RXky9 yXNN3 RXkR8 RXyy9

yXR8y yX38y y yXyj yXyj RXRN9 RXyyj RXkyk RXyy8

yXRd8 yX3k8 y yXy9 yXy9 RXRN9 RXyR9 RXRNy RXyyd

yXkyy yX3yy y yXyj yXy9 RXRe3 RXyyj RXRd3 RXyRy

yXkk8 yXdd8 y yXy8 yXy9 RXRRe RXyR9 RXRee RXyRk

yXk8y yXd8y y yXy8 yXy8 RXR83 RXykR RXR88 RXyRe

yXkd8 yXdk8 y yXy8 yXy8 RXR99 RXyky RXR98 RXyRN

yXjyy yXdyy y yXye yXy8 RXRjd RXykd RXRj8 RXykj

yXjk8 yXed8 y yXy8 yXye RXRky RXykj RXRk8 RXykd

yXj8y yXe8y y yXye yXye RXRR3 RXyj9 RXRR8 RXyjR

yXjd8 yXek8 y yXye yXye RXRy9 RXyjk RXRye RXyje

yX9yy yXeyy y yXye yXye RXyNN RXy9j RXyN3 RXy9R

yX9k8 yX8d8 y yXye yXye RXy3N RXy98 RXyNy RXy9e

yX98y yX88y y yXye yXye RXy3R RXy8y RXy3k RXy8k

yX9d8 yX8k8 y yXye yXye RXyd8 RXy83 RXyd9 RXy8N

yX9N8 yX8y8 y yXye yXye RXydy RXyeR RXyeN RXye9

yX8k8 yX9d8 y yXye yXye RXyeR RXydR RXyeR RXydk

yX88y yX98y y yXye yXye RXy8e RXydN RXy89 RXy3y

yX8d8 yX9k8 y yXyd yXye RXy8k RXyNy RXy93 RXy33

yXeyy yX9yy y yXye yXye RXy9R RXyNR RXy9j RXyNe

yXek8 yXjd8 y yXye yXye RXy9R RXRy8 RXyjd RXRy8

yXe8y yXj8y y yXye yXye RXyj9 RXRRj RXyjj RXRR9

yXed8 yXjk8 y yXye yXye RXykN RXRkj RXyk3 RXRk9

yXdyy yXjyy y yXy8 yXy8 RXykk RXRk3 RXyk9 RXRj9

yXdk8 yXkd8 y yXy8 yXy8 RXyky RXR9k RXyky RXR99

yXd8y yXk8y y yXy8 yXy8 RXyRd RXR8j RXyRe RXR88

yXdd8 yXkk8 y yXy9 yXy8 RXyRj RXRej RXyRj RXRed

yX3yy yXkyy y yXy9 yXy9 RXyRk RXRdN RXyRR RXRdN

yX3k8 yXRd8 y yXy9 yXy9 RXyyN RXRNk RXyy3 RXRNk

yX38y yXR8y y yXyj yXyj RXyy8 RXkyj RXyye RXky8

yX3d8 yXRk8 y yXyj yXyj RXyy9 RXkR3 RXyy9 RXkR3

yXNyy yXRyy y yXyk yXyk RXyyj RXkjj RXyyj RXkjj

yXNk8 yXyd8 y yXyk yXyk RXyyk RXk8y RXyyR RXk9d

yXN8y yXy8y y yXyR yXyR RXyyR RXkee RXyyR RXkej

yXNd8 yXyk8 y yXyR yXyR RXyyy RXk3k RXyyy RXkdN

R y y y y R @ R RXkN8

h"G1 AAAX .i 7Q` i?2 #BM`v bm#bvbi2K i?i Bb +QKTQb2/ Q7 pQHiBH2 U+QKTQM2Mi RV M/ BM2`i ;b U+QKTQM2Mi jV T`iB+H2bX

h?2`KQ/vMKB+ bi#BHBiv MHvbBb `2Hi2b i?2 i?2`KQ/vMKB+ 7+iQ`Ƕb H;2#`B+ bB;M iQ 2Bi?2`  bi#H2 UΓ>0V Q` M mMbi#H2

UΓ<0V bii2X h?Bb #BM`v KBtim`2 Kmbi 2t?B#Bi  pTQ`@HB[mB/ T?b2 b2T`iBQM #2ir22M yX98 <y

1<yXNkX "Qi? i?2 :B##b

2t+2bb 2M2`;v M/ i?2 7m;+Biv +Q2{+B2Mi ϑ1`2 r2HH /2b+`B#2/ BM i?2 pTQ` T?b2 #v i?2 b2H2+i2/ geKQ/2HX

y1y2y3ge

sim ge

mod ϑ1,sim ϑ1,mod Γ1

yyR y y @ 8 R

yXyRy y yXNNy yXyk yXyk 8XRy 8XRe yXN3

yXyky y yXN3y yXyj yXyj 8Xyk 8Xyd yXNe

yXyk8 y yXNd8 yXy9 yXy9 9XN3 8Xyj yXNe

yXyjy y yXNdy yXy8 yXy8 9XN9 9XN3 yXN8

yXy9y y yXNey yXye yXye 9X3d 9XNy yXNj

yXy8y y yXN8y yXy3 yXy3 9XdN 9X3k yXNk

yXyey y yXN9y yXRy yXRy 9Xdk 9Xd9 yXNR

yXydy y yXNjy yXRR yXRR 9Xe8 9Xed yX3N

yXy3y y yXNky yXRj yXRj 9X8N 9Xey yX33

yXyNy y yXNRy yXR9 yXR9 9X8k 9X89 yX3d

yXRyy y yXNyy yXRe yXRe 9X98 9X9d yX3e

yXR8y y yX38y yXkj yXkj 9XR8 9XR3 yX3R

yXkyy y yX3yy yXjy yXjy jX33 jXNj yXd8

yXk8y y yXd8y yXje yXje jXej jXe3 yXed

yXkd8 y yXdk8 yXjN yXjN jX8k jX8e yXej

yXjyy y yXdyy yX9k yX9k jX9y jX99 yX83

yXj8y y yXe8y yX93 yX93 jXky jXky yX9e

yX9yy y yXeyy yX8j yX8j jXyy kXN8 yXjk

yX98y y yX88y yX8d yX8d kX3R kXdy yXRe

yX8yy y yX8yy yXeR yXey kXej kX98 @yXyR

yX88y y yX98y yXej yXej kXjR kXkR @yXR3

yXeyy y yX9yy yXe9 yXe9 kXyR RXN3 @yXj8

yXejN y yXjek yXe9 yXe9 RX3e RX3R @yX9e

yXe8y y yXj8y yXej yXe9 RX3j RXde @yX9N

yXdyy y yXjyy yXek yXek RXe9 RX8d @yXey

yXd8y y yXk8y yX8N yX83 RX8y RX9R @yXe9

yX3yy y yXkyy yX89 yX8k RXjj RXkd @yXey

yX38y y yXR8y yX9e yX99 RXkj RXR8 @yX99

yXNyy y yXRyy yXjd yXjk RXRk RXyd @yXR8

yXNRy y yXyNy yXj8 yXjy RXRy RXye @yXyd

yXNky y yXy3y yXjk yXkd RXy3 RXy8 yXyk

yXNjy y yXydy yXkN yXk9 RXye RXy9 yXRR

yXN9y y yXyey yXke yXkR RXy9 RXyj yXkR

yXN8y y yXy8y yXkj yXR3 RXyk RXyk yXjk

yXNey y yXy9y yXR3 yXR8 RXyR RXyR yX99

yXNdy y yXyjy yXR9 yXRR RXyR RXyR yX8d

yXN3y y yXyky yXyN yXy3 RXyy RXyy yXdy

yXNNy y yXyRy yXy8 yXy9 RXyy RXyy yX38

Ryy y y R R R

h"G1 AoX .i 7Q` i?2 #BM`v bm#bvbi2K i?i Bb +QKTQb2/ Q7 H2bb@pQHiBH2 M/ BM2`i ;b T`iB+H2bX h?Bb bm#bvbi2K 2t?B#Bib 

rB/2` T?b2 i`MbBiBQM BM i2`Kb Q7 KQH` p`BiBQM i?M i?2 bvbi2K BM i#H2 UAAAVX h?Bb bm#bvbi2K Kmbi 2t?B#Bi  pTQ`@HB[mB/

T?b2 b2T`iBQM BM i?2 `M;2 0.175 <y

2<0.8X

y1y2y3ge

sim ge

mod ϑ2,sim ϑ2,mod Γ2

y y R y y @ j8Xj9 R

y yXyRy yXNNy yXyj yXy9 keXjj jkXNj yXN8

y yXyky yXN3y yXyd yXyd k8XdR jyXd9 yXNy

y yXyjR yXNeN yXRy yXRR k8Xye k3X88 yX3j

y yXy9y yXNey yXRj yXR9 k9X89 keXNR yXd3

y yXy9j yXN8d yXR9 yXR8 k9XjR keX8y yXdd

y yXy8y yXN8y yXRe yXRd kjXN3 k8Xkj yXdk

y yXRyy yXNyy yXjk yXjj kRX9y R3XeN yX9R

y yXR8y yX38y yX9d yX9d RNXRj R9Xk3 yXRy

y yXRd8 yX3k8 yX89 yX8j R3Xyd RkXek @yXye

y yXkyy yX3yy yXeR yX8N ReXeR RRXkj @yXkR

y yXk88 yXdd8 yXek yXe8 NXkj RyXy8 @yXje

y yXk8y yXd8y yXee yXdy dXe8 NXy9 @yX9N

y yXjyy yXdyy yXd9 yXd3 eXyN dX9k @yXdj

y yXj8y yXe8y yX3R yX38 8Xkd eXky @yXNR

y yX9yy yXeyy yX3d yX3N 9X8j 8Xk8 @RXyk

y yX98y yX88y yXNy yXNR jX33 9X8y @RXyd

y yX8yy yX8yy yXNj yXNR jX9y jX3N @RXy8

y yX88y yX98y yXN9 yX3N kXN8 jXj3 @yXNe

y yXeyy yX9yy yXNj yX38 kX8e kXN8 @yX3R

y yXe8y yXj8y yXNR yXdN kXRN kX83 @yXey

y yXdyy yXjyy yX33 yXdk RX3N kXke @yXj8

y yXd8y yXk8y yX3k yXej RXej RXN3 @yXyd

y yX3yy yXkyy yXd8 yX8k RX99 RXdj yXkk

y yX38y yXR8y yXe9 yX9y RXjy RX8R yX9N

y yXNyy yXRyy yX8R yXk3 RXRj RXjk yXd9

y yXN8y yXyy8 yXjR yXR9 RXy9 RXR8 yXNk

yR y y y R R R

6A:X 9X "BM`v HB[mB/ KBtim`2Ƕb U+QKTQM2Mib R M/ kV +iBpBiv +Q2{+B2Mib b ;Bp2M BM i#H2 UAAAVX h?2 HBM2b `2T`2b2Mi i?2

J`;mH2b TQHvMQKBH KQ/2H r?BH2 i?2 #mHH2ib `2T`2b2Mi bBKmHiBQM /iX h?2 +iBpBiv +Q2{+B2Mib i BM}MBi2 /BHmiBQM γ∞

Ub[m`2bV `2 r2HH +Qp2`2/ #v i?2 +?Qb2M J`;mH2b ivT2 TQHvMQKBHX

h"G1 oX .i 7Q` i?2 i2`M`v bvbi2K- r?2`2 i?2 }`bi +QKTQM2MiǶb KQH2 7`+iBQM Bb p`B2/ M/ i?2 b2+QM/Ƕb Bb F2Ti +QMbiMi

i y2=0.025 KQH KQH−1X

y1y2y3ge

sim ge

mod ϑ1,sim ϑ2,sim ϑ1,mod ϑ2,mod Γ11 Γ12 Γ21 Γ22

y yXyk8 yXNd8 yXy3 yXy3 @ k8X9R 9XNe k8X93 R y @yXye yXN8

yXyk8 yXyk8 yXN8y yXRk yXRk 9Xd8 k9Xk8 9Xd9 k9Xk8 yXN8 @yXy8 @yXy8 yXN3

yXy8y yXyk8 yXNk8 yXRe yXRe 9X83 kjXRe 9X88 kjXkk yXNk @yXyN @yXy9 RXyy

yXydy yXyk8 yXNy8 yXRN yXky 9X98 kkXjj 9X9R kkX8R yX3N @yXRR @yXy9 RXyR

yXyd8 yXyk8 yXNyy yXky yXky 9X9k kkXRj 9Xj3 kkXj9 yX33 @yXRR @yXy9 RXyR

yXRyy yXyk8 yX3d8 yXkj yXk9 9Xke kRXRe 9Xkj kRX83 yX38 @yXR8 @yXy9 RXyR

yXRRy yXyk8 yX3e8 yXk8 yXke 9Xky kyXdN 9XRd kRXkN yX39 @yXRe @yXyj RXyR

yXRky yXyk8 yX388 yXke yXkd 9XR8 kyX9k 9XRR kRXyk yX3j @yXRd @yXyj RXyR

yXRk8 yXyk8 yX38y yXkd yXk3 9XRk kyXk9 9XyN kyX33 yX3k @yXR3 @yXyj RXyy

yXRjy yXyk8 yX398 yXk3 yXkN 9XyN kyXyd 9Xye kyXd8 yX3k @yXR3 @yXyj RXyy

yXR9y yXyk8 yX3j8 yXkN yXjy 9Xyj RNXdk 9XyR kyX9N yX3R @yXRN @yXyj yXNN

yXR8y yXyk8 yX3k8 yXjy yXjR jXN3 RNXj3 jXNe kyXk9 yXdN @yXky @yXyj yXN3

yXRey yXyk8 yX3R8 yXjk yXjj jXNj RNXy9 jXNR RNXNN yXd3 @yXkR @yXyj yXNd

yXRdy yXyk8 yX3y8 yXjj yXj9 jX3d R3XdR jX3e RNXd9 yXdd @yXkj @yXyj yXNe

yXRd8 yXyk8 yX3yy yXj9 yXj8 jX38 R3X88 jX3j RNXek yXde @yXkj @yXyj yXN8

yXR3y yXyk8 yXdN8 yXj9 yXje jX3k R3XjN jX3R RNX9N yXde @yXk9 @yXyj yXN9

yXRNy yXyk8 yXd38 yXje yXjd jXdd R3Xy3 jXde RNXk9 yXd9 @yXk8 @yXyj yXNk

yXkyy yXyk8 yXdd8 yXjd yXj3 jXdk RdXde jXdR R3XNN yXdj @yXke @yXyj yXNR

yXkk8 yXyk8 yXd8y yX9y yX9k jXey RdXyR jX8N R3Xje yXeN @yXjy @yXy9 yX38

yXk8y yXyk8 yXdk8 yX9j yX98 jX9N ReXkN jX93 RdXeN yXe8 @yXj8 @yXy9 yXdN

yXkdy yXyk8 yXdy8 yX98 yX9d jX9y R8Xd9 jXjN RdXRj yXek @yXj3 @yXy9 yXdj

yXkd8 yXyk8 yXdyy yX9e yX93 jXj3 R8Xey jXjd ReXN3 yXeR @yXjN @yXy9 yXdR

yXjyy yXyk8 yXed8 yX9N yX8R jXkd R9XN9 jXk8 ReXkj yX8e @yX98 @yXy8 yXej

yXjRy yXyk8 yXee8 yX8y yX8k jXkj R9Xe3 jXkR R8XNR yX89 @yX9d @yXy8 yX8N

yXjky yXyk8 yXe88 yX8R yX8j jXRN R9X9k jXRe R8X8N yX8k @yX9N @yXy8 yX88

yXjk8 yXyk8 yXe8y yX8R yX8j jXRd R9XkN jXR9 R8X9k yX8R @yX8R @yXye yX89

yXj9y yXyk8 yXej8 yX89 yX88 jXRR RjXNR jXyd R9XNR yX9d @yX89 @yXye yX9d

yXj8y yXyk8 yXek8 yX88 yX8e jXyd RjXee jXyk R9X8e yX98 @yX8d @yXye yX9j

yXjey yXyk8 yXeR8 yX8e yX8d jXyj RjX9k kXN3 R9XkR yX9k @yXey @yXye yXjN

yXjdy yXyk8 yXey8 yX8e yX83 kXNN RjXRd kXNj RjX39 yX9y @yXey @yXyd yXj9

yXj3y yXyk8 yX8N8 yX83 yX8N kXN8 RkXNj kX33 RjX9d yXjd @yXe8 @yXyd yXkN

yXjNy yXyk8 yX838 yX8N yXey kXNk RkXeN kX39 RjXyN yXj9 @yXe3 @yXyd yXk9

yX9yy yXyk8 yX8d8 yXeR yXey kX33 RkX98 kXdN RkXdR yXjk @yXdR @yXy3 yXRN

yX9k8 yXyk8 yX88y yXe9 yXek kXdN RRX38 kXed RRXdj yXk8 @yXdd @yXyN yXye

yX9d8 yXyk8 yX8yy yXe8 yXee kX8N RyX99 kX99 NXdk yXRy @yXNy @yXRy @yXk9

yX8k8 yXyk8 yX98y yXee yXe3 kXRe eXRy kXkk dXd9 @yXye @RXyy @yXRk @yX83

yX8d8 yXyk8 yX9yy yXe8 yXeN RXNe 9XN9 kXyy 8XNR @yXky @RXyj @yXR8 @yXN8

yXed8 yXyk8 yXjyy yXej yXed RXej jX8y RXek kXNN @yX9y @yXdd @yXRN @RX3R

yXdk8 yXyk8 yXk8y yXey yXej RX93 kXNj RX9d RXN3 @yX9k @yX9y @yXkk @kXkN

yX3k8 yXyk8 yXR8y yX93 yX9N RXkk kXyN RXkj yXd8 @yXRd RXRR @yXkd @jXj8

yX3d8 yXyk8 yXRyy yXj3 yXjd RXRR RXde RXRe yX9j yXR9 kXje @yXkN @jXN9

yX338 yXyk8 yXyNy yXje yXj8 RXyN RXdy RXR8 yXj3 yXkj kXee @yXkN @9Xye

yX3N8 yXyk8 yXy3y yXjj yXjk RXyd RXe9 RXR9 yXj9 yXjk kXNd @yXjy @9XR3

yXNy8 yXyk8 yXydy yXjy yXkN RXy8 RX8N RXRj yXjy yX9k jXjR @yXjy @9Xjy

yXNR8 yXyk8 yXyey yXkd yXke RXy9 RX89 RXRk yXke yX8k jXee @yXjR @9X9j

yXNk8 yXyk8 yXy8y yXk9 yXkj RXyk RX93 RXRk yXkj yXej 9Xyj @yXjR @9X88

yXNj8 yXyk8 yXy9y yXRN yXky RXyR RX99 RXRR yXkR yXde 9X9k @yXjk @9Xe3

yXN88 yXyk8 yXyky yXRy yXRj RXyy RXj8 RXRR yXRe RXyj 8Xkd @yXjj @9XN9

yXNe8 yXyk8 yXyRy yXy8 yXyN RXyy RXjR RXRR yXR9 RXR3 8Xdk @yXjj @8Xyd

yXNd8 yXyk8 y yXyR yXy8 RXyy RXk3 RXRR yXRk RXjj eXky @yXjj @8Xky

h"G1 oAX *QMiBMmiBQM Q7 i#H2 UoV 7Q` y2=0.050 KQH KQH−1X

y1y2y3ge

sim ge

mod ϑ1,sim ϑ2,sim ϑ1,mod ϑ2,mod Γ11 Γ12 Γ21 Γ22

y yXy8y yXN8y yXRe yXRe @ kjXN3 9Xd3 kjX3d R y @yXy9 yX39

yXyk8 yXy8y yXNk8 yXky yXkR 9X89 kkXNy 9X83 kjX98 yXN8 @yXyk @yXy8 yXNy

yXy8y yXy8y yXNyy yXk9 yXk8 9Xj3 kRX33 9X9R kkXNj yXNR @yXye @yXye yXNj

yXydy yXy8y yX33y yXkd yXk3 9Xke kRXRR 9Xk3 kkX9k yX33 @yXyd @yXyd yXNj

yXyd8 yXy8y yX3d8 yXkd yXkN 9Xkj kyXNk 9Xk8 kkXk3 yX33 @yXyd @yXyd yXNj

yXRyy yXy8y yX38y yXjR yXjj 9XyN kyXyk 9XRy kRX8R yX39 @yXRd @yXyN yXNk

yXRRy yXy8y yX39y yXjk yXj9 9Xyj RNXed 9Xy8 kRXRe yX3j @yXky @yXyN yXNR

yXRky yXy8y yX3jy yXj9 yXje jXN3 RNXjj jXNN kyXd3 yX3k @yXk9 @yXRy yXNy

yXRk8 yXy8y yX3k8 yXj9 yXjd jXN8 RNXRe jXNe kyX8N yX3R @yXk8 @yXRy yX3N

yXRjy yXy8y yX3ky yXj8 yXjd jXNk R3XNN jXN9 kyXjN yX3y @yXkd @yXRR yX33

yXR9y yXy8y yX3Ry yXje yXjN jX3d R3Xee jX33 RNXNd yXdN @yXjR @yXRk yX3e

yXR8y yXy8y yX3yy yXj3 yX9y jX3k R3Xj9 jX3j RNX8k yXd3 @yXj9 @yXRk yX39

yXRey yXy8y yXdNy yXjN yX9k jXdd R3Xyj jXd3 RNXye yXde @yXj3 @yXRj yX3k

yXRdy yXy8y yXd3y yX9y yX9j jXdk RdXdk jXdj R3X83 yXd8 @yX9k @yXR9 yXdN

yXRd8 yXy8y yXdd8 yX9R yX99 jXeN RdX8e jXdR R3Xjj yXd9 @yX98 @yXR8 yXdd

yXR3y yXy8y yXddy yX9R yX99 jXed RdX9R jXe3 R3Xyd yXdj @yX9d @yXR8 yXd8

yXRNy yXy8y yXdey yX9j yX9e jXde ReX3k jXe9 RdX88 yXdk @yX8R @yXRe yXdk

yXkyy yXy8y yXd8y yX99 yX9d jX8d ReX3k jX8N RdXyR yXdy @yX8e @yXRd yXe3

yXkk8 yXy8y yXdk8 yX9d yX8y jXje ReXRy jX9d R8Xey yXee @yXeN @yXRN yX8d

yXk8y yXy8y yXdyy yX8y yX8j jXj8 R8X9R jXj8 R9XRk yXek @yX3k @yXkk yX99

yXkdy yXy8y yXe3y yX8k yX8e jXkd R9X33 jXke RkXNR yX83 @yXNj @yXk9 yXjj

yXkd8 yXy8y yXed8 yX8k yX8e jXk8 R9Xd8 jXk9 RkXey yX83 @yXNe @yXk8 yXkN

yXjyy yXy8y yXe8y yX88 yX8N jXR9 R9XRR jXRk RRXyN yX8j @RXRy @yXk3 yXRj

yXjRy yXy8y yXe9y yX8e yXey jXRy RjX38 jXy3 RyX9N yX8y @RXRe @yXkN yXye

yXjky yXy8y yXejy yX8d yXeR jXye RjXey jXyj NXNy yX93 @RXkR @yXjy @yXyk

yXjk8 yXy8y yXek8 yX83 yXek jXy9 RjX93 jXyR NXeR yX9d @RXk9 @yXjR @yXye

yXj9y yXy8y yXeRy yX8N yXej kXNN RjXRR kXN9 3Xd8 yX99 @RXjj @yXjj @yXR3

yXj8y yXy8y yXeyy yXey yXe9 kXN8 RkX3e kX3N 3XRN yX9k @RXj3 @yXj9 @yXke

yXjey yXy8y yX8Ny yXeR yXe8 kXNR RkXek kX38 dXe8 yXjN @RX9j @yXj8 @yXj8

yXjdy yXy8y yX83y yXek yXee kX3d RkXj3 kX3y dXRk yXjd @RX9N @yXjd @yX99

yXj3y yXy8y yX8dy yXej yXee kX3j RkXRj kXde eXek yXj8 @RX89 @yXj3 @yX8j

yXjNy yXy8y yX8ey yXej yXed kX3y RRX33 kXdR eXRj yXjk @RX8N @yXjN @yXej

yX9yy yXy8y yX88y yXe9 yXe3 kXde RRXek kXed 8Xee yXjy @RXej @yX9R @yXdk

yX98y yXy8y yX8yy yXee yXdR kXj8 dXRj kX98 jXe9 yXR3 @RX3j @yX93 @RXke

yX8yy yXy8y yX98y yXed yXdj kXRR 8X8R kXk9 kXR3 yXye @RXNk @yX8e @RX3e

yX88y yXy8y yX9yy yXe3 yXdj RXNk 9X8N kXy9 RXky @yXy9 @RX3e @yXe9 @kX89

yXeyy yXy8y yXj8y yXed yXdk RXdd jXNj RX3e yXeR @yXRy @RXeR @yXdk @jXk3

yXe8y yXy8y yXjyy yXe8 yXeN RXek jXj8 RXdy yXkN @yXRk @RXRR @yX3y @9Xy3

yXdyy yXy8y yXk8y yXeR yXe9 RX9e kX3R RX8d yXRk @yXye @yXk3 @yX33 @9XN8

yXd8y yXy8y yXkyy yX8e yX8e RXjj kXjN RX9e yXy8 yXRy yXNj @yXNe @8X3N

yX3yy yXy8y yXR8y yX9N yX9e RXkR kXyj RXjN yXyk yXj3 kXey @RXy9 @eX33

yX38y yXy8y yXRyy yXjN yXjk RXRR RXdk RXj8 yXyR yX3R 9X3k @RXRk @dXN9

yX3dy yXy8y yXy3y yXj9 yXke RXyd RXeR RXj8 yXyy RXyj 8X3d @RXR8 @3Xj3

yX33y yXy8y yXydy yXjR yXkj RXy8 RX8e RXj8 yXyy RXRe eX99 @RXRd @3Xey

yX3Ny yXy8y yXyey yXk3 yXRN RXyj RX8R RXj8 yXyy RXkN dXy9 @RXR3 @3X3k

yXNyy yXy8y yXy8y yXk8 yXRe RXyk RX9e RXj8 yXyy RX9j dXee @RXky @NXy8

yXNRy yXy8y yXy9y yXky yXRk RXyR RX9R RXje yXyy RX83 3XjR @RXkR @NXk3

yXNky yXy8y yXyjy yXRe yXy3 RXyy RXjd RXjd yXyy RXd9 3XNN @RXkj @NX8R

yXNjy yXy8y yXyky yXRR yXy9 RXyy RXjj RXj3 yXyy RXNy NXdy @RXk9 @NXd8

yXN9y yXy8y yXyRy yXye y RXyy RXkN RXjN yXyy kXy3 RyX98 @RXke @NXN3

yXN8y yXy8y y yXyR @yXy8 RXyy RXkd RX9R yXyy kXkd RRXkk @RXkd @RyXkk

h"G1 oAAX *QMiBMmiBQM Q7 i#H2 UoAV 7Q` y2=0.075 KQH KQH−1X

y1y2y3ge

sim ge

mod ϑ1,sim ϑ2,sim ϑ1,mod ϑ2,mod Γ11 Γ12 Γ21 Γ22

yXyk8 yXyd8 yXNyy yXk3 yXkN 9Xj8 kRXe9 9X8j kRXN3 yXN9 y @yXyj yXde

yXy8y yXyd8 yX3d8 yXjR yXjj 9XRN kyXeN 9Xjk kRX3y yX3N @yXy9 @yXyd yXd3

yXydy yXyd8 yX388 yXj9 yXje 9Xy3 RNXNd 9XRe kRX9R yX38 @yXRy @yXRR yXd3

yXyd8 yXyd8 yX38y yXj8 yXjd 9Xy8 RNX3y 9XRk kRXkd yX39 @yXRR @yXRk yXd3

yXRyy yXyd8 yX3k8 yXj3 yX9R jXNk R3XN9 jXN8 kyX9R yX3y @yXkk @yXRd yXd8

yXRRy yXyd8 yX3R8 yX9y yX9j jX3e R3Xek jX33 RNXN3 yXd3 @yXke @yXRN yXdj

yXRky yXyd8 yX3y8 yX9R yX99 jX3R R3XkN jX3R RNX8y yXde @yXjk @yXkR yXdy

yXRk8 yXyd8 yX3yy yX9R yX98 jXdN R3XR9 jXd3 RNXk9 yXde @yXj9 @yXkk yXeN

yXRjy yXyd8 yXdN8 yX9k yX9e jXde RdXN3 jXd8 R3XNd yXd8 @yXjd @yXk9 yXed

yXR9y yXyd8 yXd38 yX9j yX9d jXdR RdXed jXeN R3X9y yXdj @yX9j @yXke yXe9

yXR8y yXyd8 yXdd8 yX98 yX9N jXee RdXje jXek RdX3y yXdR @yX8y @yXk3 yXey

yXRey yXyd8 yXde8 yX9e yX8y jXek RdXye jX8d RdXR8 yXdy @yX8d @yXjR yX8e

yXRdy yXyd8 yXd88 yX9d yX8k jX8d ReXdd jX8R ReX93 yXe3 @yXe9 @yXjj yX8R

yXRd8 yXyd8 yXd8y yX93 yX8k jX88 ReXek jX93 ReXR9 yXed @yXe3 @yXj9 yX9N

yXR3y yXyd8 yXd98 yX93 yX8j jX8k ReX93 jX98 R8Xd3 yXed @yXdR @yXj8 yX9e

yXRNy yXyd8 yXdj8 yX8y yX89 jX93 ReXRN jXjN R8Xye yXe8 @yXdN @yXj3 yX9R

yXkyy yXyd8 yXdk8 yX8R yX8e jX9j R8XNR jXj9 R9Xjj yXej @yX3d @yX9y yXj8

yXkk8 yXyd8 yXdyy yX89 yX8N jXjk R8Xkj jXkR RkX9e yX8N @RXy3 @yX9d yXR3

yXk8y yXyd8 yXed8 yX8e yXeR jXkk R9X8e jXy3 RyX8N yX88 @RXjy @yX89 @yXyR

yXkdy yXyd8 yXe88 yX83 yXe9 jXR9 R9Xy8 kXN3 NXR8 yX8R @RX93 @yX8N @yXRN

yXkd8 yXyd8 yXe8y yX8N yXe9 jXRk RjXNk kXNe 3X3y yX8R @RX8k @yXeR @yXkj

yXjyy yXyd8 yXek8 yXeR yXee jXyk RjXkN kX39 dXR9 yX9e @RXd9 @yXe3 @yX93

yXjRy yXyd8 yXeR8 yXek yXed kXN3 RjXy9 kX3y eX8k yX99 @RX3j @yXdR @yX83

yXjky yXyd8 yXey8 yXej yXe3 kXN9 RkXd3 kXd8 8XNj yX9j @RXNk @yXdj @yXeN

yXjk8 yXyd8 yXeyy yXe9 yXe3 kXNk RkXee kXdj 8Xe8 yX9k @RXNe @yXd8 @yXd8

yXjjy yXyd8 yX8N8 yXe9 yXeN kXNy RkX8j kXdR 8Xj3 yX9R @kXyy @yXde @yX3R

yXj9y yXyd8 yX838 yXe8 yXdy kX3e RkXkR kXee 9X38 yXjN @kXyN @yXdN @yXNk

yXj8y yXyd8 yX8d8 yXee yXdy kX3R RRXek kXek 9Xje yXj3 @kXRd @yX3k @RXy9

yXjey yXyd8 yX8e8 yXee yXdR kXd9 RyX3d kX83 jXNR yXje @kXk9 @yX38 @RXRd

yXjdy yXyd8 yX888 yXed yXdk kXe8 NXed kX89 jX93 yXj9 @kXjk @yX3N @RXjy

yXjd8 yXyd8 yX88y yXed yXdk kX8N 3XNj kX8R jXk3 yXjj @kXj8 @yXNy @RXje

yXj3y yXyd8 yX898 yXed yXdk kX88 3X89 kX9N jXyN yXjj @kXjN @yXNk @RX9j

yXjNy yXyd8 yX8j8 yXed yXdj kX9d dXdR kX98 kXd9 yXjR @kX9e @yXN8 @RX8d

yX9yy yXyd8 yX8k8 yXed yXdj kXj3 eXN8 kX9R kX9R yXkN @kX8k @yXN3 @RXdR

yX9k8 yXyd8 yX8yy yXe3 yXd9 kXk8 eXye kXjk RXdk yXke @kXe8 @RXye @kXy3

yX9d8 yXyd8 yX98y yXeN yXd9 kXye 8Xy8 kXR9 yX3R yXky @kXdN @RXkk @kX3N

yX8d8 yXyd8 yXj8y yXe3 yXdy RXd9 jXdj RX39 yXRj yXkR @kXje @RX88 @9XdN

yXek8 yXyd8 yXjyy yXee yXe8 RXey jXkj RXdj yXy9 yXjy @RXe9 @RXdk @8X33

yXed8 yXyd8 yXk8y yXej yX83 RX98 kXdj RXe8 yXyR yX93 @yX9d @RX3N @dXy8

yXdk8 yXyd8 yXkyy yX83 yX93 RXjk kXjj RXey yXyy yXd3 RXkk @kXye @3XjR

yXdd8 yXyd8 yXR8y yX8y yXj9 RXkR RXNN RXey yXyy RXkR jX8k @kXkk @NXee

yXdN8 yXyd8 yXRjy yX9d yXk3 RXRd RX3e RXeR yXyy RX9j 9Xek @kXkN @RyXkR

yX3k8 yXyd8 yXRyy yX9y yXR3 RXRy RXe3 RXe9 yXyy RX3k eX88 @kXj3 @RRXy3

yX3j8 yXyd8 yXyNy yXjd yXR9 RXy3 RXek RXee yXyy RXNd dXk8 @kX9k @RRXj3

yX398 yXyd8 yXy3y yXj8 yXRy RXyd RX83 RXe3 yXyy kXRk dXN3 @kX98 @RRXed

yX388 yXyd8 yXydy yXjk yXye RXy8 RX8k RXdy yXyy kXk3 3Xd8 @kX93 @RRXNd

yX3e8 yXyd8 yXyey yXkN yXyR RXyj RX93 RXdj yXyy kX9e NX88 @kX8R @RkXk3

yX3d8 yXyd8 yXy8y yXk8 @yXyj RXyR RX9j RXde yXyy kXe9 RyXjN @kX89 @RkX83

yXNy8 yXyd8 yXyky yXRk @yXRd RXyy RXjR RX3d yXyy jXk9 RjXR9 @kXej @RjX8k

yXNR8 yXyd8 yXRyy yXyd @yXkk RXyy RXk3 RXNR yXyy jX9e R9XRj @kXee @RjX39

yXNk8 yXyd8 y yXyk @yXk3 RXyy RXk8 RXNd yXyy jXeN R8XRe @kXeN @R9XRe

h"G1 oAAAX aTiBHHv `2bQHp2/ pTQ` T?b2 KQH` +QKTQbBiBQM bi3/iX h?2 MQ`KHBx2/ 7Q`K (bi3−bi3,s)fBi3+M #2 +QKTmi2/

rBi? i?2 7QHHQrBM; aTH/BM; i`Mb72` MmK#2`b B13 4 @yXR89 M/ B23 4 @yXyyjX

(z−zs)fδb

13(z)b23(z)(b13 −b13,s)fB13 (b23 −b23,s)fB23

y yXR88 yXyjk y y

yXyR yXR8k yXyjR yXyRe yXyk8

yXyj yXR9d yXyjy yXy93 yXyde

yXy8 yXR9k yXyk3 yXydN yXRk8

yXyd yXRj3 yXykd yXRRR yXRd9

yXyN yXRjj yXyk8 yXR9k yXkkk

yXRR yXRk3 yXyk9 yXRdj yXkeN

yXRj yXRkj yXykk yXky9 yXjR9

yXR8 yXRR3 yXykR yXkj9 yXj83

yXRd yXRR9 yXyky yXke9 yX9yR

yXRN yXRyN yXyR3 yXkN9 yX99j

yXkR yXRy8 yXyRd yXjkj yX93j

yXkj yXRRy yXyRe yXj8k yX8kR

yXk8 yXyNe yXyR8 yXj3R yX883

yXkd yXyNR yXyRj yX9yN yX8N9

yXkN yXy3d yXyRk yX9jd yXek3

yXjR yXy3j yXyRR yX9e8 yXeey

yXjj yXydN yXyRy yX9Nk yXeNR

yXj8 yXydR yXyyN yX8R3 yXdky

yXjd yXyed yXyy3 yX899 yXd9d

yXjN yXyej yXyy3 yX8dy yXddk

yX9R yXyeR yXyyd yX8N8 yXdNe

yX9j yXy8N yXyye yXeRN yX3RN

yX98 yXy8e yXyy8 yXe9j yX39y

yX9d yXy8k yXyy8 yXeee yX38N

yX9N yXy93 yXyy9 yXe3N yX3de

yX8R yXy98 yXyy9 yXdRR yX3Nk

yX88 yXyj3 yXyyj yXd8j yXNky

yX8d yXyj8 yXyyk yXddj yXNjk

yX8N yXyjk yXyyk yXdNk yXN9j

yXeR yXykN yXyyk yX3RR yXN8k

yXej yXykd yXyyR yX3k3 yXNeR

yXe8 yXyk9 yXyyR yX39e yXNe3

yXed yXykR yXyyR yX3ek yXNd9

yXeN yXyRN yXyyR yX3dd yXN3y

yXdR yXyRd yXyy8 yX3Nk yXN39

yXd8 yXyRj yXyyy yXNRN yXNNR

yXdd yXyRR yXyyy yXNjR yXNNj

yXdN yXyyN yXyyy yXN9k yXNN8

yX3R yXyyd yXyyy yXN8k yXNNd

yX3j yXyye yXyyy yXNeR yXNN3

yX38 yXyy8 yXyyy yXNdy yXNNN

yX3d yXyy9 yXyyy yXNdd yXNNN

yX3N yXyyj yXyyy yXN39 RXyyy

yXNR yXyyk yXyyy yXN3N RXyyy

yXNj yXyyR yXyyy yXNNj RXyyy

yXN8 yXyy8 yXyyy yXNNd RXyyy

yXNd yXyyk yXyyy yXNNN RXyyy

yXNN yXyyy yXyyy RXyyy RXyyy

Ryy R R

6A:X 8X h?2 MQ`KHBx2/ pTQ` T?b2 KQH` +QKTQbBiBQM rb bbmK2/ iQ #2  +HbbB+H #QmM/`v Hv2` T`Q#H2K M/ ?2M+2 +M

#2 +QMbB/2`2/ MHQ;Qmb iQ i?2 p2HQ+Biv /Bbi`B#miBQM BM  +QKT`2bbB#H2 #QmM/`v Hv2`X

6A:X eX *Qm`b2 Q7 i?2 r2B;?i2/ i@i? +QKTQM2Mi KQH2 7`+iBQM bi3(z)=yi(z,t)/y3(z,t)b  7mM+iBQM Q7 TQbBiBQMX "Qi? pHm2b

TT`Q+? bi3=07Q` z=zδ/m2 iQ i?2 2M7Q`+2/ #QmM/`v +QM/BiBQMX

h"G1 AsX .i 7Q` i?2 HB[mB/ xM/ pTQ` yT?b2 KQH` +QKTQbBiBQMb- b r2HH b i?2 }HK i?B+FM2bbǶ iBK2 2pQHmiBQMX aBM+2

i?2 /BbT`Biv #2ir22M i?2 MHviB+H bQHmiBQM M/ i?2 L1J. bBKmHiBQMb ζsim <ζ

ana #2+K2 bm#biMiBH- /Bb+HQbBM; 7m`i?2`

/i 7i2` t>kRjX8y Mb rb +QMbB/2`2/ Q#bQH2i2X

ifMb x1,sim x2,sim x1,ana x2,ana y1s,sim y2s,sim y3s,sim y1s,ana y2s,ana y3s,ana ζsim ζana

y yX9N8 yX8y8 yX9N8 yX8y8 yXRjy yXykd yX39j yXRjy yXykd yX39j R R

RXk8 yX9N8 yX8y3 yX9N8 yX8y8 yXyN3 yXykj yX3d3 yXRkN yXykd yX399 yXNN RXyy

jXk8 yX9N8 yX8Rd yX938 yX8R8 yXyd3 yXykR yXNyR yXRkd yXykd yX39e yXNd yXNd

8X8y yX9d yX8j yX9d yX8j yXydk yXyky yXNy3 yXRkj yXyk3 yX39N yXN9 yXN9

dX8y yX9e yX89 yX9e yX89 yXydy yXyky yXNRy yXRky yXykN yX38R yXNk yXNk

NX8y yX98 yX88 yX98 yX88 yXye3 yXyRN yXNRk yXRR3 yXykN yX38j yXNy yX3N

RRX8y yX99 yX8e yX99 yX8e yXyed yXyky yXNRj yXRR8 yXyjy yX388 yX33 yX3d

RjX8y yX9j yX8d yX9j yX8d yXye8 yXyky yXNR8 yXRRk yXyjy yX38d yX38 yX38

R8X8y yX9k yX83 yX9k yX83 yXyej yXykR yXNRe yXRRy yXyjR yX3ey yX3j yX3j

RdXk8 yX9R yX8N yX9R yX8N yXyek yXykk yXNRd yXRyd yXyjR yX3ek yX3R yX3R

RNXk8 yX9N yXey yX9y yXey yXy8N yXykk yXNRN yXRy8 yXyjk yX3e9 yXdN yX3y

kRXk8 yXjN yXeR yXjN yXeR yXy8d yXykR yXNkk yXRyk yXyjk yX3ee yXd3 yXd3

kjXyy yXj3 yXek yXj3 yXek yXy8e yXykR yXNkj yXyNN yXyjj yX3e3 yXde yXde

k8Xyy yXjd yXej yXjd yXej yXy89 yXykR yXNk8 yXyNd yXyjj yX3dy yXd9 yXd9

keXd8 yXje yXe9 yXje yXe9 yXy8k yXykR yXNkd yXyN9 yXyj9 yX3dk yXdj yXdj

k3X8y yXj8 yXe8 yXj8 yXe8 yXy8y yXykR yXNk3 yXyNk yXyj9 yX3d9 yXdR yXdR

jyX8y yXj9 yXee yXj9 yXee yXy9N yXykk yXNkN yXy3N yXyj8 yX3de yXeN yXdy

jkXk8 yXjj yXed yXjj yXed yXy9N yXykR yXNjy yXy3e yXyj8 yX3d3 yXe3 yXe3

j9Xyy yXjk yXe3 yXjk yXe3 yXy9d yXykR yXNjR yXy39 yXyje yX33y yXee yXed

jeXyy yXjR yXeN yXjR yXeN yXy9e yXykk yXNjk yXy3R yXyje yX33k yXe8 yXe8

jdXd8 yXjy yXdy yXjy yXdy yXy99 yXykj yXNjj yXyd3 yXyjd yX338 yXej yXe9

jNX8y yXkN yXdR yXkN yXdR yXy9k yXykj yXNj8 yXyde yXyj3 yX33d yXek yXej

9RX8y yXk3 yXdk yXk3 yXdk yXy9R yXykj yXNje yXydj yXyj3 yX33N yXeR yXeR

9jXk8 yXkd yXdj yXkd yXdj yXyjN yXykj yXNj3 yXydR yXyjN yX3NR yX8N yXey

98Xk8 yXke yXd9 yXke yXd9 yXyjd yXyk9 yXNjN yXye3 yXyjN yX3Nj yX83 yX8N

9dXyy yXk8 yXd8 yXk8 yXd8 yXyje yXyk9 yXNjN yXye8 yXy9y yX3N8 yX8d yX83

9NXyy yXkj yXdd yXk9 yXde yXyj8 yXyk9 yXN9R yXyej yXy9y yX3Nd yX8e yX8d

8RXyy yXkk yXd3 yXkj yXdd yXyjj yXyk9 yXN99 yXyey yXy9R yX3NN yX88 yX88

8jXyy yXkR yXdN yXkk yXd3 yXyjR yXykj yXN9e yXy83 yXy9R yXNyR yX8j yX89

88Xyy yXky yX3y yXkR yXdN yXykN yXykj yXN93 yXy88 yXy9k yXNyj yX8k yX8j

8dXyy yXRN yX3R yXky yX3y yXyk3 yXyk9 yXN93 yXy8k yXy9k yXNy8 yX8R yX8k

8NXyy yXR3 yX3k yXRN yX3R yXykd yXyk9 yXN9N yXy8y yXy9j yXNy3 yX8y yX8R

eRXk8 yXRd yX3j yXR3 yX3k yXyk8 yXyk9 yXN8R yXy9d yXy9j yXNRy yX9N yX8y

ejX8y yXRe yX39 yXRd yX3j yXykj yXyk9 yXN8k yXy99 yXy99 yXNRk yX93 yX9N

e8Xd8 yXR8 yX38 yXRe yX39 yXykk yXyk8 yXN8j yXy9k yXy99 yXNR9 yX9d yX93

e3Xk8 yXR9 yX3e yXR8 yX38 yXyky yXyk8 yXN88 yXyjN yXy98 yXNRe yX9e yX9d

dyXd8 yXRj yX3d yXR9 yX3e yXyRN yXyk8 yXN8e yXyjd yXy98 yXNR3 yX98 yX9e

djXk8 yXRk yX33 yXRj yX3d yXyRd yXyk9 yXN83 yXyj9 yXy9e yXNky yX99 yX98

deXyy yXRR yX3N yXRk yX33 yXyRe yXyk8 yXN8N yXyjR yXy9d yXNkk yX9j yX99

dNXyy yXRy yXNy yXRR yX3N yXyR9 yXyke yXNey yXykN yXy9d yXNk9 yX9k yX9j

3kXyy yXyN yXNR yXRy yXNy yXyRj yXyk8 yXNek yXyke yXy93 yXNke yX9R yX9k

38X8y yXy3 yXNk yXyN yXNR yXyRR yXyk8 yXNe9 yXyk9 yXy93 yXNk3 yX9y yX9R

3NXk8 yXyd yXNj yXy3 yXNk yXyRy yXyke yXNe9 yXykR yXy9N yXNjy yXjN yX9y

NjXk8 yXye yXN9 yXyd yXNj yXyy3 yXyk8 yXNee yXyR3 yXy9N yXNjj yXj3 yXjN

N3Xyy yXy8 yXN8 yXye yXN9 yXyyd yXyke yXNed yXyRe yXy8y yXNj8 yXjd yXj3

RyjXk8 yXy9 yXNe yXy8 yXN8 yXyy8 yXykd yXNe3 yXyRj yXy8y yXNjd yXj8 yXje

RyNXd8 yXyj yXNd yXy9 yXNe yXyy9 yXyk8 yXNdk yXyRy yXy8R yXNjN yXj9 yXj8

RRdXd8 yXyk yXN3 yXyj yXNd yXyyj yXykd yXNdy yXyy3 yXy8R yXN9R yXjj yXj9

Rk3Xd8 yXyR yXNN yXyk yXN3 yXyyk yXyk3 yXNdR yXyy8 yXy8k yXN9j yXjR yXjk

R9dXk8 yXyR yXNN yXyR yXNN yXyyR yXykd yXNdk yXyyj yXy8k yXN98 yXk3 yXkN

R8yXk8 yXyye yXNN9 yXyyN yXNNR yXyyR yXykd yXNdk yXyyk yXy8k yXN98 yXkd yXkN

ifMb x1,sim x2,sim x1,ana x2,ana y1,sim y2,sim y3,sim y1,ana y2,ana y3,ana ζsim ζana

R8jXk8 yXyye yXNN9 yXyy3 yXNNk yXyyy yXykd yXNdj yXyyk yXy8k yXN98 yXkd yXkN

R8eXd8 yXyy8 yXNN8 yXyyd yXNNj yXyyy yXykd yXNdj yXyyk yXy8k yXN9e yXke yXk3

Re8Xd8 yXyy9 yXNNe yXyy8 yXNN8 yXyyy yXykd yXNdj yXyyR yXy8j yXN9e yXk8 yXkd

RdRX8y yXyy8 yXNN8 yXyy9 yXNNe yXyyy yXykd yXNdj yXyyR yXy8j yXN9e yXk9 yXke

R3NX8y yXyy9 yXNNe yXyyk yXNN3 yXyyy yXykd yXNdj yXyyR yXy8j yXN9d yXkk yXk8

kydXd8 yXyyj yXNNd yXyyR yXNNN yXyyy yXykd yXNdj yXyyy yXy8j yXN9d yXR3N yXkkN

kRyX8y yXyyj3 yXNNe yXyyyN yXNNNR yXyyy yXyk3 yXNdk yXyyy yXy8j yXN9d yXR38 yXkke

kRjX8y yXyyjk yXNNd yXyyy3 yXNNNk yXyyy yXyke yXNd9 yXyyy yXy8j yXN9d yXR3R yXkkj

h"G1 sX *QKT`BM; i?2 /2pBiBQM #2ir22M i?2 MHviB+H bQHmiBQM HB[mB/ /2MbBiv iBK2 2pQHmiBQM iQ i?2 L1J. bBKmHiBQMǶb-

DmbiB}2b i?2 bbmKTiBQM i?i ?b #22M K/2 BM UjjVX

ifMb ρl,ana ρl,sim /2pX BM %

yXjR 9eXyNd 9eXe9k RXRd

kX8e 9eXRdk 9eXeN9 RXRk

9Xdd 9eXk9e 9eXd8d RXyN

NXyd 9eXjN8 9eXNyj RXy3

kRXjj 9eX39k 9dXj93 RXyd

k8Xkd 9eXNNR 9dX9N8 RXye

kNXRe 9dXR9y 9dXe9j RXye

j3X3e 9dX8Rj 93XyR3 RXy8

93Xde 9dX338 93Xjd8 RXyR

8yX3R 9dXNey 93X993 RXyR

89XNN 93XRyN 93X8dd yXNe

8dXR9 93XR3j 93Xeej yXNN

8NXj9 93Xk83 93XdRN yXN8

eRXey 93Xjjk 93Xdde yXNR

ejXNR 93X9yd 93X38N yXNk

eeXjy 93X93R 93XNRj yX33

e3Xdd 93X88e 93XNd9 yX38

dRXj9 93Xejy 9NXy8j yX3e

d9Xyk 93Xdy8 9NXRy9 yX3R

3eXjR 9NXyyj 9NXj9e yXdy

3NXN8 9NXyd3 9NX9yd yXed

NjXNj 9NXR8k 9NX993 yXey

RyjXjj 9NXjyR 9NX8eR yX8k

RyNXyN 9NXjde 9NXeRy yX9d

RReXyy 9NX98y 9NXeje yXjd

Rk9XdR 9NX8k8 9NXedk yXjy

RjeXdj 9NX8NN 9NXeNN yXky

R8eX38 9NXed9 9NXdkj yXRy

RejXk8 9NXe3N 9NXdjy yXy3

RdRX9d 9NXdyj 9NXdky yXyj

RdeXed 9NXdRR 9NXdR3 yXyR

R3jXyR 9NXdR3 9NXd9R yXy9

RNRXRd 9NXdke 9NXd9y yXyj

kykXe8 9NXdjj 9NXd9y yXyR

amTTH2K2Mi`v Ki2`BH iQ,

"mHF pBb+QbBiv Q7 HB[mB/ MQ#H2 ;b2b

_2Mû aT2M+2` *?ir2HH M/ C/`M o`#2+a)

h?2`KQ/vMKB+b M/ S`Q+2bb 1M;BM22`BM;- h2+?MBb+?2 lMBp2`bBii "2`HBM- Ry83d "2`HBM- :2`@

KMv

HH #mHF pBb+QbBiv /i T`2b2Mi2/ BM i?2 KMmb+`BTi `2 /Bb+HQb2/ BM i?Bb bmTTH2K2Mi`v Ki2`BHX

h?2 #mHF pBb+QbBiv rb +QKTmi2/ 7`QK 1J. bBKmHiBQMb pB BKT`QT2` iBK2 BMi2;`iBQM Q7

i?2 miQ+Q``2HiBQM 7mM+iBQM Q7 i?2 KB+`Qb+QTB+ bi`2bb i2MbQ`Ƕb /B;QMH 2H2K2Mib (R)

μχχ

b=1

BT∞

dtJχχ

p(t)−pV (t)·Jχχ

p(0) −pV (0),URV

r?2`2 χχ =xx, yy M/ zzX "2ii2` biiBbiB+b r2`2 +?B2p2/ #v +QMbB/2`BM; i?2 p2`;2 pHm2 Q7

HH i?`22 BM/2T2M/2Mi bTiBH 2H2K2Mib

μb=1

3μχχ

b.UkV

h?2 2ti2M/2/ +`BiB+H `2;BQM rb +QMbi`m+i2/ QM i?2 #bBb Q7 bQmM/ /BbT2`bBQM K2bm`2K2Mib

7`QK i?2 HBi2`im`2X h?2 /Bb+`2TM+v #2ir22M i?2 i?2`KQ/vMKB+ M/ 7`2[m2M+v@/2T2M/2Mi

bT22/ Q7 bQmM/- BX2X c=c(ω)- /2HBKBib i?2 2ti2M/2/ +`BiB+H `2;BQMX

h"G1 AX *`BiB+H /i i?i r2`2 mb2/ iQ `2/m+2 i2KT2`im`2- T`2bbm`2 M/ /2MbBivX

TcfE pcfJS ρcf KQHf/K3_272`2M+2

L2 99X9y kXeek k9XRy (k)

` R8yXeN 9X3ej RjX9R (j)

E` kyNX93 8X8k8 RyX38 (9)

s2 k3NXdj 8X39k 3X9y (9)

6A:X RX GBi2`im`2 /i Q7 7`2[m2M+v@/2T2M/2Mi bT22/ Q7 bQmM/ K2bm`2K2Mib BM t2MQM HQM; i?2

bim`i2/ pTQ` M/ HB[mB/ HBM2bX h?2 7`2[m2M+B2b r2`2 p`B2/ 7`QK f=0.4J>x U/BKQM/bV (8)- 0.55

J>x Ui`BM;H2bV (8)- 0.6J>x Ub[m`2bV (e)- 1J>x U+B`+H2bV (8- e) iQ 3J>x U+`Qbb2bV (8- e)X h?2 b[m`2/

`iBQ c(ω)/c2rb Q#b2`p2/ iQ 7QHHQr  :mbbBM 7mM+iBQM M/ Bb T`2b2Mi2/ 7Q` f=0.55 J>x U/b?2/V

M/ 1J>x UbQHB/VX h?2 2ti2M/2/ +`BiB+H `2;BQM rb +QMb2[m2MiHv 7QmM/ iQ #2 /2HBKBi2/ iQ ρ/ρc=0.4-

p/pc=0.738 QM i?2 bim`i2/ pTQ` M/ ρ/ρc=1.70-p/pc=0.710 QM i?2 bim`i2/ HB[mB/ HBM2X

6A:X kX 1pHmiBQM Q7 //BiBQMH t2MQM K2bm`2K2Mib (e) HQM; i?2 BbQi?2`Kb T/Tc=1.00009 UbQHB/V-

1.00055 U/b?2/V- 1.00113 U/b?@/Qii2/V i f=0.6J>x 7m`i?2` +QM}MBM; i?2 2ti2M/2/ +`BiB+H `2;BQM

#2ir22M ρ/ρ =0.52-p/pc=0.933 M/ ρ/ρc=1.13-p/pc=1.318X

6A:X jX //BiBQMH /i HQM; i?2 +`BiB+H BbQ+?Q`2 rBi? i?2 bK2 bvK#QHb b BM 6B;X R rBi? //BiBQMH

f=5J>x UBMp2`i2/ i`BM;H2bV (8) M/ f=7J>x U?2t2/2`bV (8) K2bm`2K2MibX h?2b2 `2bmHib 7m`i?2`

+QM}M2 i?2 +`BiB+H `2;BQM iQ #2 i p/pc<1.3X

h"G1 AAX h?2 +`BiB+H `2;BQMǶb #QmM/`v rb +QMbi`m+i2/ b  4th@Q`/2` TQHvMQKBH g(x)=ax4+bx3+

cx2+dx +eX h?2 T`K2i2`b r2`2 }ii2/ iQ i?2 #bBb TQBMib BM/B+i2/ BM 6B;bX R iQ j- b bmKK`Bx2/

#2HQr- iQ vB2H/ a=−0.8751-b=−0.2817-c=−0.9633-d=0.2704 M/ e=0.3X

T/Tcp/pcρ/ρc

yXN9N yXdj3 yX9y

RXyyR yXNjj yX8k

RXy8y RXjyy RXyy

RXyyR RXjR3 RXRj

yXN9j yXdRy RXdy

h"G1 AAA, Pp2`pB2r Q7 #mHF pBb+QbBiv /i /2i2`KBM2/ 7`QK 2tT2`B@

K2MiH bQmM/ ii2MmiBQM K2bm`2K2Mib 7`QK i?2 HBi2`im`2 7Q` L2QM

(d)X

T/Tcp/pcρ/ρcffμbαλΔαλμ∗

b,2tTX Δμ∗

b,2tTX

yX88 yXy9R kX8e3 jy yXyyy9k yXyyyyk RX9N RXyN

yX88 yXyk3 kX8e3 jy yXyyy9k yXyyyyk RX8k RXRy

yX88 yXykk kX8ed jy yXyyy9k yXyyyyk RX8R RXRy

yX88 yXyRd kX8ed jy yXyyy9k yXyyyyk RX8R RXyN

yX8d yXR9N kX88j jy yXyyy9R yXyyyyk RX9R RXy8

h"G1 AAA, +QMiBMm2/

T/Tcp/pcρ/ρcffμbαλΔαλμ∗

b,2tTX Δμ∗

b,2tTX

yX8d yXyd3 kX88R jy yXyyy9R yXyyyyk RX9R RXy8

yX8d yXye8 kX88R jy yXyyy9R yXyyyyk RX9y RXy8

yX8d yXy8k kX88y jy yXyyy9R yXyyyyk RXje RXy8

yX8d yXyjN kX88y jy yXyyy9k yXyyyyk RX98 RXy8

yX8d yXyke kX88y jy yXyyy9k yXyyyyk RX99 RXy8

yX8d yXyky kX89N jy yXyyy9k yXyyyyk RX99 RXy8

yX8d yXRRe kX89y jy yXyyy9R yXyyyyk RXjR RXyk

yX8d yXyd3 kX8jN jy yXyyy9R yXyyyyk RX9R RXyk

yX8d yXy8k kX8j3 jy yXyyy9k yXyyyyk RX9j RXyk

yX8d yXyjk kX8jd jy yXyyy9k yXyyyyk RX9k RXyk

yX8d yXykj kX8jd jy yXyyy9k yXyyyyk RX99 RXyk

yX83 yXR3R kX8jR jy yXyyy9y yXyyyyk RXk9 yXNN

yX83 yXRjy kX8jy jy yXyyy9y yXyyyyk RXR3 yXNN

yX83 yXyd3 kX8k3 jy yXyyy9y yXyyyyk RXkR yXNN

yX83 yXyjN kX8kd jy yXyyy9y yXyyyyk RXkk yXNN

yX83 yXyk8 kX8kd jy yXyyy9R yXyyyyk RXk9 yXNN

yX8N yXRy9 kX8Rd jy yXyyy9k yXyyyyk RX9k yXN3

yX8N yXyd3 kX8Re jy yXyyy9k yXyyyyk RX99 yXN3

yX8N yXy8k kX8Re jy yXyyy9k yXyyyyk RX9k yXN3

yX8N yXyjN kX8R8 jy yXyyy9k yXyyyyk RX99 yXN3

yX8N yXyk3 kX8R8 jy yXyyy9k yXyyyyk RX9d yXN3

yXey yXj3N kX8R8 jy yXyyy9j yXyyyyk RXe3 yXNN

yXey yXk8N kX8RR jy yXyyy9j yXyyyyk RXed yXN3

yXey yXR88 kX8y3 jy yXyyy9k yXyyyyk RX9d yXNe

yXey yXyNd kX8ye jy yXyyy99 yXyyyyk RXdR yXN3

yXey yXy8k kX8y9 jy yXyyy9k yXyyyyk RX9d yXNe

yXey yXy9y kX8y9 jy yXyyy9R yXyyyyk RXj8 yXN8

yXey yXyjN kX8y9 jy yXyyy9j yXyyyyk RXek yXNd

yXey yXyjR kX8y9 jy yXyyy9k yXyyyyk RX8R yXNe

yXey yXk8N kX8y9 jy yXyyy9k yXyyyyk RX98 yXN8

yXey yXR88 kX8yy jy yXyyy9k yXyyyyk RX9k yXN8

yXey yXyd3 kX9N3 jy yXyyy9k yXyyyyk RX9j yXN8

yXey yXyjN kX9Nd jy yXyyy9k yXyyyyk RX9N yXN8

yXey yXyjj kX9Nd jy yXyyy9k yXyyyyk RX9e yXN8

yXey yXkek kX9Nd jy yXyyy99 yXyyyyk RXdd yXNe

yXey yXkk3 kX9Ne jy yXyyy99 yXyyyyk RXdd yXNe

yXey yXRNj kX9N8 jy yXyyy99 yXyyyyk RXd8 yXNe

yXey yXy99 kX9Ny jy yXyyy99 yXyyyyk RXdk yXN8

yXey yXyj8 kX9Ny jy yXyyy99 yXyyyyk RXdN yXNe

yXey yX9ee kX8yk jy yXyyy9j yXyyyyk RXdN yXNd

yXey yXk8N kX9N8 jy yXyyy99 yXyyyyk RXdk yXNe

yXey yXy8k kX93N jy yXyyy99 yXyyyyk RXe3 yXN8

yXey yXyj8 kX933 jy yXyyy99 yXyyyyk RXed yXN8

yXeR yX99y kX9Nj jy yXyyy9y yXyyyyk RXjR yXNk

yXeR yXjRR kX93N jy yXyyy9y yXyyyyk RXk9 yXNk

yXeR yXR3R kX938 jy yXyyy9R yXyyyyk RXk3 yXNR

yXeR yXRjy kX93j jy yXyyy9R yXyyyyk RXk8 yXNR

yXeR yXyd3 kX93R jy yXyyy9R yXyyyyk RXkd yXNR

yXeR yXy8k kX93y jy yXyyy9R yXyyyyk RXjj yXNR

yXeR yXyj3 kX93y jy yXyyy9R yXyyyyk RXjj yXNR

yXeR yXR88 kX9de jy yXyyy9k yXyyyyk RX9d yXNR

yXeR yXyNR kX9d9 jy yXyyy9k yXyyyyk RX9j yXNR

yXeR yXy98 kX9dj jy yXyyy9j yXyyyyk RX93 yXNR

yXeR yXy9R kX9dj jy yXyyy9j yXyyyyk RX93 yXNR

h"G1 AAA, +QMiBMm2/

T/Tcp/pcρ/ρcffμbαλΔαλμ∗

b,2tTX Δμ∗

b,2tTX

yXek yX8dy kX93y jy yXyyy9k yXyyyyk RX8R yXNR

yXek yXj3N kX9d8 jy yXyyy9j yXyyyyk RX8e yXNR

yXek yXkyd kX9eN jy yXyyy9k yXyyyyk RX9j yXNy

yXek yXRy9 kX9e8 jy yXyyy9j yXyyyyk RX8d yXNy

yXek yXy8k kX9ej jy yXyyy9j yXyyyyk RX8e yXNy

yXek yXy99 kX9ej jy yXyyy9j yXyyyyk RX88 yXNy

yXej yXR88 kX98e jy yXyyy9j yXyyyyk RX8e yX3N

yXej yXRy9 kX989 jy yXyyy9j yXyyyyk RX88 yX3N

yXej yXy8k kX98j jy yXyyy9j yXyyyyk RX9d yX33

yXej yXy93 kX98k jy yXyyy9j yXyyyyk RX9N yX33

yXej yXNk3 kX9dk jy yXyyy9y yXyyyyk RXjN yX3N

yXej yX99y kX98e jy yXyyy9k yXyyyyk RX9k yX33

yXej yXjRR kX98k jy yXyyy9k yXyyyyk RXj9 yX3d

yXej yXR3R kX99d jy yXyyy9k yXyyyyk RXjR yX3e

yXej yXRjy kX99e jy yXyyy9k yXyyyyk RXj8 yX3e

yXej yXyd3 kX999 jy yXyyy9k yXyyyyk RXjk yX3e

yXej yXy8R kX99j jy yXyyy9k yXyyyyk RXjy yX3e

yXe9 yXj3N kX99e jy yXyyy99 yXyyyyk RXej yX33

yXe9 yXkjj kX99R jy yXyyy9j yXyyyyk RX8R yX3e

yXe9 yXRy9 kX9je jy yXyyy99 yXyyyyk RX9N yX3e

yXe9 yXye8 kX9j9 jy yXyyy9j yXyyyyk RX9d yX3e

yXe9 yXy88 kX9j9 jy yXyyy9j yXyyyyk RX9e yX38

yXe8 RXyje kX98e jy yXyyy9R yXyyyyk RX8R yX3d

yXe8 yXk8N kX9kN jy yXyyy9j yXyyyyk RX98 yX39

yXe8 yXRjy kX9k8 jy yXyyy9j yXyyyyk RX98 yX39

yXe8 yXye8 kX9kk jy yXyyy99 yXyyyyk RX9j yX39

yXe8 yXy8N kX9kk jy yXyyy99 yXyyyyk RX9j yX39

yXe8 yXNNN kX998 jy yXyyy9j yXyyyyk RXe3 yX3d

yXe8 yXd3y kX9j3 jy yXyyy9k yXyyyyk RX98 yX38

yXe8 yX8y8 kX9k3 jy yXyyy9j yXyyyyk RX9d yX39

yXe8 yXkek kX9RN jy yXyyy9j yXyyyyk RX9y yX3j

yXe8 yXydN kX9Rk jy yXyyy99 yXyyyyk RX9R yX3k

yXe8 yXyd3 kX9Rk jy yXyyy9e yXyyyyk RXeN yX39

yXe8 yXyde kX9Rk jy yXyyy9j yXyyyyk RXj8 yX3k

yXe8 yXye9 kX9Rk jy yXyyy99 yXyyyyk RX9y yX3k

yXed yX8R3 kX9y8 jy yXyyy9j yXyyyyk RX98 yX3R

yXed yXjjd kXjN3 jy yXyyy98 yXyyyyk RX8k yX3R

yXed yXkjj kXjN9 jy yXyyy98 yXyyyyk RX9N yX3R

yXed yXRjy kXjNy jy yXyyy98 yXyyyyk RX8k yX3y

yXed yXyd3 kXj33 jy yXyyy98 yXyyyyk RX8k yX3y

yXed yXyd9 kXj33 jy yXyyy98 yXyyyyk RX8k yX3y

yXed RXyke kX9ky jy yXyyy99 yXyyyyk RXdk yX39

yXed yX8R3 kX9yR jy yXyyy99 yXyyyyk RX8e yX3R

yXed yXRjy kXj3e jy yXyyy98 yXyyyyk RX93 yX3y

yXed yXydd kXj3j jy yXyyy98 yXyyyyk RX9e yXdN

yXe3 RXyje kX9yd jy yXyyy9j yXyyyyk RX88 yX3R

yXe3 yXNjj kX9y9 jy yXyyy9j yXyyyyk RX9e yX3R

yXe3 yXed9 kXjN9 jy yXyyy9j yXyyyyk RXj3 yXdN

yXe3 yX9R8 kXj3j jy yXyyy99 yXyyyyk RXjN yXdN

yXe3 yXk38 kXjd3 jy yXyyy99 yXyyyyk RXj3 yXd3

yXe3 yXR88 kXjdj jy yXyyy98 yXyyyyk RXj3 yXd3

yXe3 yXRy9 kXjdy jy yXyyy98 yXyyyyk RXj3 yXd3

yXe3 yXy39 kXjdy jy yXyyy98 yXyyyyk RXjN yXd3

yXeN RXyyN kXjdN jy yXyyy99 yXyyyyk RX8j yXd3

h"G1 AAA, +QMiBMm2/

T/Tcp/pcρ/ρcffμbαλΔαλμ∗

b,2tTX Δμ∗

b,2tTX

yXeN yXd3y kXjeN jy yXyyy98 yXyyyyk RX8R yXd3

yXeN yX8yy kXj83 jy yXyyy98 yXyyyyk RX98 yXde

yXeN yXkek kXj9d jy yXyyy98 yXyyyyk RXkd yXd8

yXeN yXRyd kXj9y jy yXyyy9d yXyyyyk RX9j yXd8

yXeN yXRyR kXj9y jy yXyyy9d yXyyyyk RX99 yXd8

yXdR RXykj kXj9N jy yXyyy9e yXyyyyk RX8N yXde

yXdR yXddd kXjj3 jy yXyyy9e yXyyyyk RX8j yXd8

yXdR yX8R3 kXjkd jy yXyyy93 yXyyyyk RX88 yXd9

yXdR yXk8N kXjR8 jy yXyyy9N yXyyyyk RX88 yXdj

yXdR yXRjy kXjy3 jy yXyyy8y yXyyyyk RX8j yXdj

yXdR yXRky kXjy3 jy yXyyy8y yXyyyyk RX8k yXdj

yXdj RXyje kXjRN jy yXyyy9d yXyyyyk RX8e yXdj

yXdj yXddd kXjye jy yXyyy93 yXyyyyk RX88 yXdk

yXdj yX8R3 kXkN9 jy yXyyy9N yXyyyyk RX8R yXdR

yXdj yXk8N kXk3y jy yXyyy8R yXyyyyk RX88 yXdR

yXdj yXR88 kXkd8 jy yXyyy8j yXyyyyk RXek yXdR

yXdj yXR9j kXkd9 jy yXyyy8j yXyyyyk RX8N yXdR

yXd8 RXyj8 kXk3d jy yXyyy9N yXyyyyk RX8j yXdy

yXd8 yXNy8 kXk3R jy yXyyy93 yXyyyyk RX9k yXeN

yXd8 yXdRN kXkdR jy yXyyy8y yXyyyyk RX8k yXeN

yXd8 yX8ye kXk8N jy yXyyy8R yXyyyyk RX98 yXe3

yXd8 yXjRk kXk93 jy yXyyy8j yXyyyyk RX8j yXe3

yXd8 yXR3R kXk9R jy yXyyy8j yXyyyyk RX9e yXed

yXd8 yXRe3 kXk9y jy yXyyy89 yXyyyyk RX8y yXe3

yXde RXyje kXk8k jy yXyyy8R yXyyyyk RX8d yXe3

yXde yXddd kXkj3 jy yXyyy8j yXyyyyk RX8N yXe3

yXde yX8R3 kXkkj jy yXyyy8e yXyyyyk RXe9 yXed

yXde yXjRR kXkRy jy yXyyy83 yXyyyyk RXek yXee

yXde yXkyd kXkyj jy yXyyy8N yXyyyyk RXee yXee

yXde yXRNd kXkyk jy yXyyy8N yXyyyyk RXe8 yXee

yXd3 RXyje kXkRd jy yXyyy89 yXyyyyk RX8e yXee

yXd3 RXyyN kXkR8 jy yXyyy8j yXyyyyk RX9j yXe8

yXd3 yXddN kXkyR jy yXyyy8e yXyyyyk RX8e yXe8

yXd3 yXddd kXkyR jy yXyyy8e yXyyyyk RX89 yXe8

yXd3 yX8R3 kXR39 jy yXyyy8N yXyyyyk RX8d yXe9

yXd3 yX9NR kXR3k jy yXyyy8d yXyyyyk RX9e yXe9

yXd3 yXjj3 kXRdR jy yXyyyeR yXyyyyk RXe9 yXe9

yXd3 yXk9e kXRe8 jy yXyyyeR yXyyyyk RX8R yXej

yXd3 yXkjy kXRej jy yXyyyek yXyyyyk RX88 yXej

yX3R RXyje kXR8d jy yXyyyey yXyyyyk RXee yXej

yX3R yXddd kXRj3 jy yXyyye9 yXyyyyj RXe3 yXek

yX3R yX8R3 kXRRd jy yXyyyed yXyyyyj RXed yXeR

yX3R yXkN3 kXyN3 jy yXyyydj yXyyyyj RXdN yXeR

yX3R yXkNy kXyNd jy yXyyydj yXyyyyj RXdd yXeR

yX39 RXyje kXRyR jy yXyyyed yXyyyyj RXeN yXeR

yX39 yXddd kXydN jy yXyyydR yXyyyyj RXdk yXey

yX39 yX8R3 kXy89 jy yXyyyd3 yXyyyyj RXd3 yX8N

yX39 yXjej kXyj3 jy yXyyy3k yXyyyyj RX3R yX8N

yX39 yXj8R kXyj9 jy yXyyy3j yXyyyyj RX3R yX8N

h"G1 Ao, "mHF pBb+QbBiv /i /2i2`KBM2/ 7`QK 2tT2`BK2MiH bQmM/

ii2MmiBQM K2bm`2K2Mib 7`QK i?2 HBi2`im`2 7Q` `;QM (3)X

T/Tcp/pcρ/ρcffμbαλΔαλμ∗

b,2tTX Δμ∗

b,2tTX

yX8e yXyR9 kXe98 9k yXyyy8j yXyyyyk RX3y RXj3

yX8e yXyR9 kXe9j 9k yXyyy8j yXyyyyk RXdN RXj3

yX8e yXyRe kXejk 9k yXyyy8j yXyyyyk RXdj RXj8

yX8d yXyR3 kXeky 9k yXyyy8j yXyyyyk RXdR RXjk

yX83 yXykk kX8Nd 9k yXyyy8k yXyyyyk RXek RXkd

yXey yXykd kX8d9 9k yXyyy8k yXyyyyk RX8j RXkk

yXek yXyjd kX8jN 9k yXyyy8j yXyyyyk RX9e RXR8

yXej yXy99 kX8R8 9k yXyyy8j yXyyyyk RX98 RXRk

yXe9 yXy8k kX9Ny 9k yXyyy88 yXyyyyk RX8R RXyN

h"G1 o, "mHF pBb+QbBiv /i /2i2`KBM2/ 7`QK 2tT2`BK2MiH bQmM/

ii2MmiBQM K2bm`2K2Mib 7`QK i?2 HBi2`im`2 7Q` `;QM (N)X

T/Tcp/pcρ/ρcffμbαλΔαλμ∗

b,2tTX Δμ∗

b,2tTX

yX8d RXR8j kXe98 8y yXyyyek yXyyyyk RXNR RXj3

yX83 RX8Ne kXe9R 8y yXyyyej yXyyyyk kXRy RXjN

yX8N RX3d8 kXe9k 8y yXyyyej yXyyyyk kXkj RX9y

yX8N kXkde kXe9k 8y yXyyyeR yXyyyyk kXyk RXj3

yXey kXdky kXe9k 8y yXyyy83 yXyyyyk RXeN RXj9

yXey jXRjy kXe9R 8y yXyyy83 yXyyyyk RX39 RXj8

yX83 RXjjj kXej9 8y yXyyye9 yXyyyyk kXRN RXjN

yX8N RX8de kXej9 8y yXyyyej yXyyyyk kXR8 RXj3

yX8N RXNde kXej8 8y yXyyyeR yXyyyyk kXyy RXje

yXey kXd99 kXej9 8y yXyyyeR yXyyyyk kXyk RXje

yXeR jXRkN kXej9 8y yXyyy8N yXyyyyk RXNy RXj9

yX8e yX8ek kXe98 8y yXyyyed yXyyyyk kXj3 RX99

yX83 yXN93 kXej8 8y yXyyyej yXyyyyk RXNN RXjd

yX83 RXjRy kXej9 8y yXyyyek yXyyyyk RX3d RXj8

yX8N RX8Nj kXej8 8y yXyyyeR yXyyyyk RX38 RXj8

yX8N RX3jj kXej9 8y yXyyyey yXyyyyk RXd9 RXj9

yXey kXkd8 kXejk 8y yXyyy8N yXyyyyk RXdj RXjj

yXey kX8yR kXej9 8y yXyyy8N yXyyyyk RX3y RXj9

yXeR jXyy8 kXejk 8y yXyyy83 yXyyyyk RXdy RXjk

yX83 yXj39 kXekk 8y yXyyydR yXyyyyk kXdk RX9k

yX83 yXdy8 kXeRN 8y yXyyydR yXyyyyk kXd3 RX9k

yX8N RXRdy kXeRN 8y yXyyydy yXyyyyk kX3j RX9k

yX8N RXkNR kXeRd 8y yXyyyeN yXyyyyk kXd9 RX9R

yXey RXNN9 kXekR 8y yXyyye3 yXyyyyk kXdk RX9y

yXdR RXjNj kX9kR 8y yXyyye3 yXyyyyk RXe9 yX33

yXdk RXd8k kX9Re 8y yXyyyed yXyyyyk RXeR yX3d

yXdk kXR9y kX9R3 8y yXyyyee yXyyyyk RX8d yX3d

yXdj kX8kR kX9R9 8y yXyyye9 yXyyyyk RX93 yX3e

yXd8 kXN38 kX9Rj 8y yXyyye9 yXyyyyk RX8e yX3e

yXd8 jXRk8 kX9R8 8y yXyyyej yXyyyyk RX8R yX3e

yXe3 yXke9 kX9ky 8y yXyyyd8 yXyyyyj RXNk yXNR

yXdy yX3ee kX9ky 8y yXyyydk yXyyyyj RX39 yXNy

yXdy RXkdy kX9kj 8y yXyyydy yXyyyyk RXdN yXNy

yXdk RXNjj kX9R3 8y yXyyye3 yXyyyyk RXe3 yX33

yXdj kX9eR kX9R3 8y yXyyyed yXyyyyk RXeN yX33

yXd8 jXR9e kX9R8 8y yXyyye9 yXyyyyk RX8N yX3e

yXdy yX3y3 kX9Rk 8y yXyyydR yXyyyyk RXe8 yX3d

yXdk RX3jd kX9Ry 8y yXyyye9 yXyyyyk RXjj yX39

yXdj kX9yR kX9Rj 8y yXyyye9 yXyyyyk RX93 yX38

h"G1 o, +QMiBMm2/

T/Tcp/pcρ/ρcffμbαλΔαλμ∗

b,2tTX Δμ∗

b,2tTX

yXd9 kX9yy kX9yN 8y yXyyye9 yXyyyyk RX9j yX38

yXd8 kXd99 kX9y9 8y yXyyye9 yXyyyyk RX98 yX39

yXd8 jXRe9 kX9y9 8y yXyyyek yXyyyyk RXjd yX3j

yXdy yXe98 kXjN3 8y yXyyy3d yXyyyyj kXd8 yXNd

yXdk RX88R kXjNd 8y yXyyy3k yXyyyyj kXej yXNe

yXd9 kXyd3 kXjN8 8y yXyyy3y yXyyyyj kX8d yXN8

yXd8 kX89k kXjNj 8y yXyyydd yXyyyyj kXje yXNj

yXde jXR9j kXjNR 8y yXyyyd8 yXyyyyj kXje yXNk

yX3d yX898 RXN38 jy yXyyRye yXyyyy9 kX8k yXdj

yX3N RXR8y RXNNd jy yXyyy3N yXyyyyj kXRk yXdy

yXNk RX3j8 kXyyj jy yXyyyd3 yXyyyyj RX3d yXed

yXN8 kX8eR RXNNN jy yXyyydj yXyyyyj RX39 yXed

yXNe jXy39 kXyyy jy yXyyye3 yXyyyyk RXe3 yXe8

yX3d yX8k9 RXN3k jy yXyyRye yXyyyy9 kX9d yXdk

yX3N yXN9N RXN3N jy yXyyyNe yXyyyyj kXjR yXdR

yXNy RXk8y RXN3N jy yXyyyNy yXyyyyj kXR8 yXeN

yXNj kXRd3 RXNN8 jy yXyyy3y yXyyyyj kXy3 yXeN

yXN8 kX89R RXNNd jy yXyyydd yXyyyyj kXyd yXeN

yXNd jXR38 RXNNd jy yXyyyd8 yXyyyyj kXRk yXeN

h"G1 oA, "mHF pBb+QbBiv /i /2i2`KBM2/ 7`QK 2tT2`BK2MiH bQmM/

ii2MmiBQM K2bm`2K2Mib 7`QK i?2 HBi2`im`2 7Q` `;QM (Ry)X

T/Tcp/pcρ/ρcffμbαλΔαλμ∗

b,2tTX Δμ∗

b,2tTX

yXey yXykd kX8d9 88 yXyyydy yXyyyyk RXe3 RXRd

yXey yX9Rd kX838 88 yXyyyeN yXyyyyk RXdy RXRN

yXey yX3jj kX8Ne 88 yXyyye3 yXyyyyk RXeN RXkR

yXey RXk8y kXeyd 88 yXyyyee yXyyyyk RXed RXkk

yXee yXyed kX98j 88 yXyyydd yXyyyyk RXdR yX3d

yXee yX9Rd kX9ee 88 yXyyyd8 yXyyyyk RXee yX3d

yXee yX3j9 kX93y 88 yXyyydk yXyyyyk RX8N yX33

yXee RXk8y kX9N9 88 yXyyyeN yXyyyyk RX89 yX33

yXdj yXRjd kXjky 88 yXyyy3N yXyyyyj RXe9 yXdd

yXdj yX9Rd kXjj9 88 yXyyy3e yXyyyyj RXey yXdd

yXdj yX3j9 kXj89 88 yXyyy3k yXyyyyk RX83 yXd3

yXdj RXk8y kXjdj 88 yXyyydN yXyyyyk RXek yX3y

yX3y yXk9N kXRdR 88 yXyyRRe yXyyyyj RXd8 yXeN

yX3y yX9Rd kXR39 88 yXyyRRR yXyyyyj RXdk yXdy

yX3y yX3jj kXkRk 88 yXyyRyk yXyyyyj RXed yXdR

yX3y RXk8y kXkj3 88 yXyyyN8 yXyyyyj RXej yXdk

yX3e yX9Rd RXNN9 88 yXyyRd3 yXyyyyd kXRk yXdj

yX3e yX3j9 kXy9j 88 yXyyR99 yXyyyye RX39 yXdj

yX3e RXy9k kXye9 88 yXyyRjR yXyyyy8 RXdR yXdk

yX3e RXk8y kXy3j 88 yXyyRkk yXyyyy8 RXej yXdk

yXNy yX8k8 RX33N 88 yXyyk9j yXyyyRy kX9j yXdk

yXNy yX3jj RXN9y 88 yXyyR3N yXyyyy3 kXyN yXdR

yXNy RXy9k RXNe3 88 yXyyRee yXyyyyd RXNR yXdy

yXNy RXk8y RXNNj 88 yXyyR9N yXyyyye RXd9 yXeN

yXNj yXe8R RXdek 88 yXyy9yj yXyyyRe jXjj yXdd

yXNj yX3jj RX3RR 88 yXyyjye yXyyyRk kX3e yXd9

yXNj RXy9k RX388 88 yXyykjj yXyyyyN kXkd yXeN

yXNj RXk8y RX3Ny 88 yXyyRNe yXyyyy3 kXyy yXe3

h"G1 oA, +QMiBMm2/

T/Tcp/pcρ/ρcffμbαλΔαλμ∗

b,2tTX Δμ∗

b,2tTX

yXNe yX3yy RX8N8 k8 yXyy9Rj yXyyyk8 8Xk9 RXRd

yXNe yX3jj RXeRd j8 yXyy9jR yXyyyke jXNN RXyy

yXNe RXy9k RXdyd 88 yXyyjee yXyyyR8 kXeR yXe3

yXNe RXk8y RXde9 88 yXyykeN yXyyyRR kXRj yXe8

RXyy RXy9k RX99d k8 yXyy8NR yXyyye8 8Xkj RXdj

RXyy RXR9e RX8j3 j8 yXyy9jk yXyyyke jX93 yXNy

RXyy RXk8y RX8N9 j8 yXyyjy8 yXyyyRk kXde yXee

h"G1 oAA, "mHF pBb+QbBiv /i /2i2`KBM2/ 7`QK 2tT2`BK2MiH bQmM/

ii2MmiBQM K2bm`2K2Mib 7`QK i?2 HBi2`im`2 7Q` `;QM (RR)X

T/Tcp/pcρ/ρcffμbαλΔαλμ∗

b,2tTX Δμ∗

b,2tTX

yXNj yXe99 RXdek 88 yXyy98y yXyyykd jXNe RXy8

yXNj yXdky RXd3k 88 yXyyjdR yXyyykk jXjR yXNd

yXNj yX3kj RX3yN 88 yXyyjR3 yXyyyRN jXyy yXN9

yXNj yXNk8 RX3jk 88 yXyyk3k yXyyyRd kX3y yXNk

yXN8 yXdjR RXeeN 88 yXyye8k yXyyyjN 9X8N RXRy

yXNe yXdNR RX8N8 88 yXyRyyy yXyyyey 8XN8 RXkd

yXNe yX3kj RXeRy 88 yXyydN8 yXyyy93 9X3y RXRR

yXNe yX339 RXe99 88 yXyyeR8 yXyyyjd 9XyN RXyk

yXNe RXyk3 RXdyk 88 yXyyjNN yXyyyk9 kXN9 yX3N

yXNe RXkj9 RXdeR 88 yXyykNe yXyyyR3 kX9N yX38

yXN3 yX3d9 RX8Rj 88 yXyRk3d yXyyydd 8X3k RXkk

yXN3 yXNk8 RX8ey 88 yXyyNRy yXyyy88 9XNy RXRR

yXN3 RXyk3 RXekR 88 yXyy8e9 yXyyyj9 jX8k yXN9

yXN3 RXkj9 RXeNN 88 yXyyj9j yXyyykR kX9d yX3k

yXNN RXyk3 RX89j 88 yXyy38y yXyyy8R 9X9N RXy9

yXNN RXkj9 RXe9d 88 yXyy9Rj yXyyyk8 kXdy yX3j

yXNN RXyk3 RX9d9 88 yXyRkyN yXyyydj 8Xk3 RXRj

yXNN RXyNy RX8k3 88 yXyy3yN yXyyy9N 9XRk yXNN

yXNN RXkj9 RXey3 88 yXyy93R yXyyykN kXNj yX38

RXyy RXyk3 RX9kN 88 yXyR933 yXyyRe9 8Xe9 RX3R

RXyy RXRjR RX8k3 88 yXyydey yXyyy9e jX3N yXNe

RXyy RXkj9 RX83e 88 yXyy8jk yXyyyjk jXR9 yX3d

h"G1 oAAA, "mHF pBb+QbBiv /i /2i2`KBM2/ 7`QK 2tT2`BK2MiH bQmM/

ii2MmiBQM K2bm`2K2Mib 7`QK i?2 HBi2`im`2 7Q` `;QM (Rk)X

T/Tcp/pcρ/ρcffμbαλΔαλμ∗

b,2tTX Δμ∗

b,2tTX

yX8e yXRed kXe9d ekX8 yXyyydN yXyyyy8 yXj9 RXd9

yX83 yXRed kXeRj ekX8 yXyyyde yXyyyy8 yXj9 RXey

yXey yXRed kX8d3 ekX8 yXyyydd yXyyyy8 yXj8 RX8R

yXd9 yXRed kXkNj ekX8 yXyyRyN yXyyyyd yX8R RXyd

h"G1 As, "mHF pBb+QbBiv /i /2i2`KBM2/ 7`QK 2tT2`BK2MiH bQmM/

ii2MmiBQM K2bm`2K2Mib 7`QK i?2 HBi2`im`2 7Q` `;QM (Rj)X

T/Tcp/pcρ/ρcffμbαλΔαλμ∗

b,2tTX Δμ∗

b,2tTX

RXyR RXRe3 RX9kN 8 yXyyRee yXyyyR3 3X9e kXee

RXy9 RXk98 yXN3R 8 yXyyjkk yXyyyj8 eXkR RXNe

RXy9 RXk8R RXyyy 8 yXyyjjd yXyyyjd eXdN kXRy

RXy9 RXk8d RXykj 8 yXyyjjN yXyyyjd dXRd kXky

RXy9 RXkdj RXydj 8 yXyyjR3 yXyyyj8 dX9y kXkd

RXye RXjjj yXNRy 8 yXyyReR yXyyyR3 kXej RXy3

RXye RXj8R yXN8k 8 yXyyRdy yXyyyRN jXy3 RXky

RXye RXje8 yXN38 8 yXyyR3k yXyyyky jXek RXj9

RXye RXjdR RXyyy 8 yXyyR3d yXyyykR jX3d RX9R

RXye RXjdd RXyRk 8 yXyyR38 yXyyyky jXNy RX9k

RXye RXj3N RXyjN 8 yXyyRd3 yXyyyky jXNe RX99

RXye RX9Rk RXy3d 8 yXyyRd9 yXyyyRN 9Xj9 RX88

h"G1 s, "mHF pBb+QbBiv /i /2i2`KBM2/ 7`QK 2tT2`BK2MiH bQmM/

ii2MmiBQM K2bm`2K2Mib 7`QK i?2 HBi2`im`2 7Q` F`vTiQM (R9)X

T/Tcp/pcρ/ρcffμbαλΔαλμ∗

b,2tTX Δμ∗

b,2tTX

yX8d yXyRN kXe8e 88 yXyyyNR yXyyyy9 RXjy RXd3

yX8d yXjek kXee8 88 yXyyyNR yXyyyy9 RXjd RX3R

yX8d yXdk9 kXed9 88 yXyyyNy yXyyyy9 RX9k RX39

yX8d yXNy8 kXedN 88 yXyyyNy yXyyyy9 RX98 RX3e

yX8d RXRdd kXe3e 88 yXyyy3N yXyyyy9 RX9N RX33

yXed yXydy kX9dN 88 yXyyRyk yXyyyy9 RXjR RXk8

yXed yXjek kX9Ny 88 yXyyRyy yXyyyy9 RXj8 RXkd

yXed yXdk9 kX8y9 88 yXyyyNd yXyyyy9 RXjd RXkN

yXed yXNy8 kX8Ry 88 yXyyyNe yXyyyy9 RXjd RXjy

yXed RXRde kX8ky 88 yXyyyN8 yXyyyy9 RX9R RXjk

yXdk yXRRN kXj3k 88 yXyyRR9 yXyyyy8 RXje RXy3

yXdk yXR3R kXj38 88 yXyyRRy yXyyyy9 RXRN RXyd

yXdk yXjek kXjN9 88 yXyyRy8 yXyyyy9 RXy3 RXy3

yXdk yXdk9 kX9RR 88 yXyyRyR yXyyyy9 RXye RXRy

yXdk yXNy8 kX9RN 88 yXyyyN3 yXyyyy9 RXy9 RXRR

yXdk RXRde kX9jR 88 yXyyyNd yXyyyy9 RXyd RXRj

yXde yXR33 kXkdd 88 yXyyRj8 yXyyyy8 RX9d yXN8

yX3R yXk3k kXRek 88 yXyyReN yXyyyyd RXee yX38

yX3e yX9y8 kXyjy 88 yXyykje yXyyyyN kXyj yXdd

yX3e yX89j kXy9d 88 yXyykRj yXyyyyN RX3R yXde

yX3e yXdk9 kXye3 88 yXyyRNk yXyyyy3 RXek yXdd

yX3e yXNy8 kXy3d 88 yXyyRdN yXyyyyd RX8d yXd3

yX3e RXRde kXRRj 88 yXyyRe9 yXyyyyd RX9N yXdN

yXNR yX8e9 RX3dk 88 yXyyj33 yXyyyRe kXe3 yXdR

yXNR yXdk9 RXNy8 88 yXyyjjy yXyyyRj kX9d yXdR

yXNR yXNy8 RXNje 88 yXyyk3y yXyyyRR kXRN yXdR

yXNR RXy3e RXNej 88 yXyyk9d yXyyyRy kXyy yXdk

yXNR RXRde RXNde 88 yXyykjk yXyyyyN RX3N yXdk

yXNj yXe83 RXdde 8 yXyyy89 yXyyyyk jX83 yXdR

yXNj yXdk9 RXdNe 8 yXyyy9d yXyyyyk jXk3 yXdy

yXNj yXNy8 RX39j 8 yXyyyj9 yXyyyyR kX9N yXe3

yXNj RXy3e RX3dN 8 yXyyyk3 yXyyyyR kXR9 yXe3

yXNj RXRde RX3N8 8 yXyyyke yXyyyyR kXyd yXe3

yXNe RXkR3 RXdN8 8 yXyyyjN yXyyyyk kX3N yXe3

yXNe yXNj3 RXdRk 8 yXyyy8e yXyyyyk jXk8 yXee

yXNe yX389 RXedd 8 yXyyy3j yXyyyyj 9X3R yXd9

h"G1 s, +QMiBMm2/

T/Tcp/pcρ/ρcffμbαλΔαλμ∗

b,2tTX Δμ∗

b,2tTX

yXNe yXd3R RXejd 8 yXyyRRk yXyyyy9 8Xe9 yXdd

yXNd RXked RXdeR 8 yXyyy99 yXyyyyj jXRe yXd3

yXNd RXykR RXe3R 8 yXyyye3 yXyyyy9 9Xy9 yX3R

yXNd yX3dk RXeyk 8 yXyyRj3 yXyyyy3 eX3R RXyy

yXNd yX3jk RX8dy 8 yXyyRdN yXyyyky dX39 RX88

yXN3 RXkj3 RXe3R 8 yXyyy89 yXyyyyj jXRN yXd8

yXN3 RXydN RXeRk 8 yXyyRyy yXyyyye 8Xjd yX3N

yXN3 RXyjd RX83d 8 yXyyRRd yXyyyyd 8X3j yXNk

yXN3 yXNdy RX8je 8 yXyyR3y yXyyyRR dXej RXy9

yXNN RXkj9 RXej3 8 yXyyyej yXyyyy9 jXj9 yXd9

yXNN RXyeR RX89y 8 yXyyR9y yXyyyy3 eXRy yXNk

yXNN RXyy8 RX939 8 yXyyk9e yXyyyR8 NXkj RXR8

yXNN yXNdk RX9j8 8 yXyyj3e yXyyykj RkXkk RXj3

yXNN yXN8k RXj38 8 yXyydkR yXyyydN RNX9N jXyR

h"G1 sA, "mHF pBb+QbBiv /i /2i2`KBM2/ 7`QK 2tT2`BK2MiH bQmM/

ii2MmiBQM K2bm`2K2Mib 7`QK i?2 HBi2`im`2 7Q` F`vTiQM (R8)X

T/Tcp/pcρ/ρcffμbαλΔαλμ∗

b,2tTX Δμ∗

b,2tTX

yXNR yX8e9 RX3dk dX8 yXyyye yXyyyy9 kXNN yXN9

yXNR yX8e9 RX3dk Ry yXyyyd yXyyyye kX3d yXNj

yXNR yX8e9 RX3dk R8 yXyyRR yXyyyyN kXN9 yXNj

yXNj yXe83 RXdde 8 yXyyy3 yXyyyye eXjj RXkk

yXNj yXe83 RXdde dX8 yXyyy3 yXyyyyd jXde yXNe

yXNj yXe83 RXdde Ry yXyyRR yXyyyyd jXdj yX39

yXNj yXe83 RXdde R8 yXyyRe yXyyyRy jXed yX3j

yXN8 yXdej RXe83 8 yXyyRR yXyyyy3 8XdR RXyN

yXN8 yXdej RXe83 dX8 yXyyR8 yXyyyyN 8XR3 yX3N

yXN8 yXdej RXe83 Ry yXyykR yXyyyRk 8X89 yXNk

yXN8 yXdej RXe83 R8 yXyyjR yXyyyR3 8X8R yXNk

yXNe yXd3R RXejd 8 yXyyRk yXyyyyN 8XNj RXRy

yXNe yXd3R RXejd R8 yXyyj9 yXyyykR 8X38 yXN9

yXNe yX3yN RXeyk 8 yXyyR8 yXyyyRk dXyy RXRN

yXNe yX3yN RXeyk dX8 yXyykk yXyyyRj eX3y RXyy

yXNd yX3jk RX8dy 8 yXyyky yXyyyRe 3XNk RXjd

yXNd yX3jk RX8dy dX8 yXyyjy yXyyyR3 3X3e RXRe

yXNd yX3jk RX8dy Ry yXyy9y yXyyyk9 3X39 RXR8

yXNd yX3jk RX8dy R8 yXyy8e yXyyyjj 3XRd RXRy

RXy9 RXkRk RXyyy 8 yXyy8j yXyyy9k RyX9k RXj9

RXy9 RXkRk RXyyy dX8 yXyyd9 yXyyy99 NXey RXy9

RXy9 RXkRk RXyyy Ry yXyyNy yXyyy89 3Xd9 yXNd

RXy9 RXkkd RXyyy 8 yXyy9e yXyyyjd NXRk RXkR

RXy9 RXkkd RXyyy Ry yXyy3R yXyyy9N dXN8 yXNR

RXy9 RXk9R RXyyy 8 yXyy9R yXyyyjj 3XRe RXRR

RXy9 RXk9R RXyyy dX8 yXyyey yXyyyje 3Xyy yXNR

RXy9 RXk9R RXyyy Ry yXyyde yXyyy9e dX8y yX3d

RXy9 RXk88 RXyyy 8 yXyyjdj yXyyyjy dX98 RXy9

RXy9 RXk88 RXyyy Ry yXyyeNj yXyyy9k eX38 yX3k

RXy8 RXkeN RXyyy 8 yXyyj8j yXyyyk3 dXR9 RXyR

RXy8 RXkeN RXyyy Ry yXyyejk yXyyyj3 eXk3 yXdd

100

h"G1 sAA, "mHF pBb+QbBiv /i /2i2`KBM2/ 7`QK 2tT2`BK2MiH bQmM/

ii2MmiBQM K2bm`2K2Mib 7`QK i?2 HBi2`im`2 7Q` t2MQM (R8)X

T/Tcp/pcρ/ρcffμbαλΔαλμ∗

b,2tTX Δμ∗

b,2tTX

yX8e yXyR9 kXe3N 9k yXyyyd9y yXyyyyj RXj9 yXe3

yX8e yXyR8 kXe3e 9k yXyyydjN yXyyyyj RXjj yXe3

yX8e yXyR8 kXe3y 9k yXyyydjN yXyyyyj RXjR yXed

yX8d yXyRd kXee3 9k yXyyyd9y yXyyyyj RXk3 yXee

yX8N yXykj kXejd 9k yXyyyd93 yXyyyyj RXk8 yXej

yXey yXyjy kXeye 9k yXyyydd8 yXyyyyj RXj8 yXeR

yXek yXyj3 kX8d8 9k yXyyy3jd yXyyyyj RXed yXey

h"G1 sAAA, "mHF pBb+QbBiv /i /2i2`KBM2/ 7`QK 2tT2`BK2MiH bQmM/

ii2MmiBQM K2bm`2K2Mib 7`QK i?2 HBi2`im`2 7Q` t2MQM (3)X

T/Tcp/pcρ/ρcffμbαλΔαλμ∗

b,2tTX Δμ∗

b,2tTX

yX8d yXyRd kXee3 88 yXyyRRyy yXyyyy9 kXRe yXd9

yX8d yXR98 kXedR 88 yXyyRyN9 yXyyyy9 kXRe yXd8

yX8d yX8yR kXe3y 88 yXyyRy33 yXyyyy9 kXk8 yXde

yX8d yX3e3 kXe3N 88 yXyyRyd3 yXyyyy9 kXjy yXdd

yX8d RXRjk kXeNe 88 yXyyRydk yXyyyy9 kXj9 yXd3

yX8N yXykj kXejd 88 yXyyRRy9 yXyyyy9 kXy9 yXdR

yXey yXyjy kXeye 88 yXyyRyNe yXyyyy9 RX39 yXed

yXek yXyj3 kX8d8 88 yXyyRyNy yXyyyy9 RXe9 yXej

yXek yXyN3 kX8dd 88 yXyyRy3k yXyyyy9 RXek yXej

yXek yX99N kX833 88 yXyyRydd yXyyyy9 RXdk yXe9

yXek yXd3k kX8N3 88 yXyyRy39 yXyyyy9 RX33 yXee

yXek RXRRj kXeye 88 yXyyRyey yXyyyy9 RX3j yXee

yXeN yXy3N kX99k 88 yXyyRk9d yXyyyy8 RXej yX89

yXeN yXyN3 kX99j 88 yXyyRkke yXyyyy8 RX88 yX89

yXeN yX9yR kX988 88 yXyyRRdd yXyyyy8 RX93 yX89

yXeN yXd9R kX9eN 88 yXyyRRjN yXyyyy8 RX9e yX88

yXeN RXyyk kX9dN 88 yXyyRRRe yXyyyy9 RX9e yX88

yXd9 yXR8k kXjj9 88 yXyyR9dk yXyyyye RXde yX8y

yXde yXRdN kXkN8 88 yXyyR83R yXyyyye RX3j yX9N

yXde yXke9 kXjyy 88 yXyyR8Re yXyyyye RXeN yX93

yXde yX9Rd kXjyN 88 yXyyR9ey yXyyyye RXeR yX93

yXde yXd8d kXjk3 88 yXyyRje8 yXyyyy8 RX8R yX93

yXde RXyR8 kXj9k 88 yXyyRjyy yXyyyy8 RX9j yX93

yXd3 yXkyN kXk88 88 yXyyRe3j yXyyyyd RX3j yX9d

yX3j yXjky kXRk8 88 yXyykRd3 yXyyyyN RXNd yX99

yX3j yX9je kXRje 88 yXyykyRN yXyyyy3 RXdd yX9j

yX3j yXeyd kXR8k 88 yXyyRNy9 yXyyyy3 RXdR yX9j

yX3j yXded kXRe8 88 yXyyR3yR yXyyyyd RXek yX9j

yX3j yXNNe kXR3j 88 yXyyRe39 yXyyyyd RX89 yX9j

yX3e yX9R8 kXykd 88 yXyykNyj yXyyyRk kXjN yX9j

yX33 yX9eN RXNdj 88 yXyyjkNe yXyyyRj kX9j yX9k

yXNy yX8k3 RXNR9 88 yXyy9yd3 yXyyyRe kX3y yX9j

yXNy yXej9 RXNj9 88 yXyyjk8d yXyyyRj kXyd yXjN

yXNy yX3yd RXNek 88 yXyykNkd yXyyyRk kXyR yXjN

yXNy yXNdk RXN3e 88 yXyyked9 yXyyyRR RXN9 yX9y

yXNy RXy88 RXNNd 88 yXyyk888 yXyyyRy RX33 yX9y

yXNR yX8Nk RX38R 88 yXyy8jy8 yXyyykR jXjk yX98

yXNR yXejd RX3eR 88 yXyy99R8 yXyyyR3 kXey yX9R

yXNR yX3Re RX3NN 88 yXyyjee9 yXyyyR8 kXje yX9y

yXNR yXNde RXNkd 88 yXyyjRd9 yXyyyRj kXRj yXjN

yXNR RXy8R RXNjN 88 yXyykN3k yXyyyRk kXyk yXjN

101

h"G1 sAAA, +QMiBMm2/

T/Tcp/pcρ/ρcffμbαλΔαλμ∗

b,2tTX Δμ∗

b,2tTX

yXNj yXeeR RXddN 8 yXyyyd8d yXyyyyk 9X3e yXjN

yXNj yXeeR RXddN dX8 yXyyRyN9 yXyyyyk 9XeR yXj3

yXNj yXeeR RXddN Ry yXyyR899 yXyyyyN 8Xyy yXe3

yXNj yXedk RXd3j 88 yXyyejy9 yXyyyk8 jXk3 yX9j

yXNj yXedk RXd3j 8 yXyyydRe yXyyyy3 9X8N yXNd

yXNj yX3kk RX3k8 88 yXyy9dk9 yXyyyRN kXee yX9y

yXNj yX3kk RX3k8 8 yXyyy8R9 yXyyyye jX8N yX38

yXNj yXN39 RX3eR 88 yXyyj33j yXyyyRe kXj8 yXjN

yXNj RXyed RX3d3 88 yXyyj8je yXyyyR9 kXR3 yXjN

yXNj RXyed RX3d3 8 yXyyyjNk yXyyyy9 jXRj yX3R

yXN8 yXdjd RXeNd 88 yXyRR833 yXyyRkd 8X98 RXye

yXN8 yXdjd RXeNd 8 yXyyRR3R yXyyyRj eXjj RXR3

yXN8 yXdjd RXeNd dX8 yXyyRe9j yXyyyR3 8Xd9 RXRy

yXN8 yXdjd RXeNd Ry yXyykkyR yXyyyRj 8Xdd yXdk

yXN8 yX39e RXd9k 88 yXyyed88 yXyyykd jXk3 yX9k

yXN8 yX39e RXd9k 8 yXyyyd98 yXyyyy3 9XjN yXN9

yXN8 RXyyR RXd3N 88 yXyy9N9j yXyyyky kXej yXjN

yXN8 RXyyR RXd3N 8 yXyyy89j yXyyyye jX8N yX38

yXN8 RXy39 RX3Ry 88 yXyy9jee yXyyyRd kX9y yXj3

yXN8 RXy39 RX3Ry 8 yXyyy9N9 yXyyyy8 jX93 yX39

yXNe yXdNj RXekN 8 yXyyRd3y yXyyyky 3Xye RXjN

yXNe yXdNj RXekN dX8 yXyyk93j yXyyykd dXj3 RXjy

yXNe yXdNj RXekN Ry yXyyj9Nk yXyyyj3 dX3d RXje

yXNe yX38e RXeee 8 yXyyRkRk yXyyyRj eXRk RXR9

yXNe RXykd RXdj8 8 yXyyydke yXyyyy3 9X9d yXN8

yXNe RXRRj RXdeR 8 yXyyye8j yXyyyyd 9X9R yXN8

yXNd yX399 RX8eR 8 yXyyjyR8 yXyyyjj RRX99 RX3k

yXNd yX399 RX8eR dX8 yXyy9R3e yXyyy9e RyX9d RXeN

yXNd yX399 RX8eR Ry yXyy8NkN yXyyye8 RRXkk RXdN

yXNd yXN9R RXejk 8 yXyyRR33 yXyyyRj 8X9R RXy9

yXNd RXykd RXed9 8 yXyyyN88 yXyyyRR 8Xy3 RXyR

yXNd RXRRj RXdyd 8 yXyyy33R yXyyyRy 8X9k RXye

yXN3 yXN9R RX88k 8 yXyykRN9 yXyyyk9 3Xk9 RX9y

yXN3 RXykd RXeRj 8 yXyyRRd9 yXyyyRj 8XR3 RXyR

yXN3 RXRRj RXe88 8 yXyyy3ee yXyyyRy 9Xjy yXNy

yXNN RXykd RX88k 8 yXyyRdkR yXyyyRN eX8e RXR3

yXNN RXRRj RXey3 8 yXyyRy83 yXyyyRk 9Xe8 yXN9

yXNN RXykd RX9N8 8 yXyykedR yXyyykN 3X3y RX9e

yXNN RXRRj RX8ed 8 yXyyRjye yXyyyR9 8XkR RXyy

yXNN RXykd RX9dk 8 yXyyjRde yXyyy8R NX3R kXRe

yXNN RXRRj RX88k 8 yXyyRj9e yXyyykk 8XyN RXjk

h"G1 sAo, "mHF pBb+QbBiv /i /2i2`KBM2/ 7`QK 2tT2`BK2MiH bQmM/

ii2MmiBQM K2bm`2K2Mib 7`QK i?2 HBi2`im`2 7Q` t2MQM (Re)X

T/Tcp/pcρ/ρcffμbαλΔαλμ∗

b,2tTX Δμ∗

b,2tTX

yXNj yXeeR RXddN 8 yXyyydRd yXyyyy9 9X8y yXe9

yXNj yXeeR RXddN 3 yXyyRydj yXyyyye 9X9N yXej

yXNj yXeeR RXddN Ry yXyyR8Ne yXyyyRy 8Xk9 yXdy

yXNj yXeeR RXddN R8 yXyykye3 yXyyyRk 9Xk8 yXek

yXN9 yXede RXde9 8 yXyyy3RN yXyyyy8 8XRy yXe3

yXN9 yXede RXde9 Ry yXyyRej3 yXyyyRy 8XRy yXe3

yXN9 yXede RXde9 R8 yXyykk9k yXyyyRj 9X93 yXej

yXN9 yXeNR RXd93 8 yXyyy39y yXyyyy8 9XN8 yXed

102

h"G1 sAo, +QMiBMm2/

T/Tcp/pcρ/ρcffμbαλΔαλμ∗

b,2tTX Δμ∗

b,2tTX

yXN9 yXeNR RXd93 Ry yXyyRdjN yXyyyRy 8XRN yXeN

yXN9 yXeNR RXd93 R8 yXyyk9j3 yXyyyR8 9Xdj yXe8

yXN9 yXdye RXdjR 8 yXyyRykR yXyyyye eXyk yXd8

yXN9 yXdye RXdjR Ry yXyyR3R3 yXyyyRR 8XRe yXe3

yXN9 yXdye RXdjR R8 yXyyke9e yXyyyRe 9XN8 yXee

yXN8 yXdkR RXdR9 Ry yXyykyyy yXyyyRk 8X9d yXdy

yXN8 yXdkR RXdR9 R8 yXyykNRd yXyyyR3 8Xkd yXe3

yXN8 yXdjd RXeNd 8 yXyyRRey yXyyyyd eXRN yXde

yXN8 yXdjd RXeNd dX8 yXyyRek8 yXyyyRy 8Xee yXdR

yXN8 yXdjd RXeNd Ry yXyykRNj yXyyyRj 8Xd8 yXdk

yXN8 yXdjd RXeNd R8 yXyyjkyR yXyyyRN 8X88 yXdy

yXN8 yXd8k RXedN 8 yXyyRk3d yXyyyy3 eX8d yXd3

yXN8 yXd8k RXedN Ry yXyyk99e yXyyyRy eXR8 yX8N

yXN8 yXd8k RXedN R8 yXyyjekk yXyyyR9 eXy8 yX83

yXNe yXd38 RXe9y 8 yXyyReRd yXyyyRy dX93 yX38

yXNe yXd38 RXe9y dX8 yXyyk83y yXyyyRy 3Xye yXdy

yXNe yXd38 RXe9y Ry yXyyj9k9 yXyyyR9 3Xyk yXdy

yXNe yXd38 RXe9y R8 yXyy9eee yXyyyRN dXRk yXe9

yXNe yXdNj RXekN 8 yXyyRdNR yXyyyRR 3XRj yXNy

yXNe yX3yk RXeRN Ry yXyyj3jN yXyyyR8 3X9d yXdj

yXNd yX3RN RX8Nd Ry yXyy98jN yXyyyR3 NX93 yXd3

yXNd yX3je RX8dj 8 yXyyk8k3 yXyyyR8 NX3y RXyj

yXNd yX3je RX8dj dX8 yXyyj33e yXyyykj RyXyN RXy8

yXNd yX3je RX8dj Ry yXyy89jR yXyyykk RyXe8 yX38

yXNd yX3je RX8dj R8 yXyyd8Ny yXyyyjy NX3R yX3y

yXNd yX38j RX893 8 yXyyjRek yXyyyRN RRX8y RXRd

yXNd yX38j RX893 dX8 yXyy9e3k yXyyyk3 RRXjj RXR8

yXNd yX38j RX893 Ry yXyye3ke yXyyykd RkX88 yXNd

yXNd yX38j RX893 R8 yXyyNRjN yXyyyjd RRXyk yX3d

h"G1 so, _2bmHib Q7 i?2 T`2b2Mi `2HtiBQM Mbix 1[X URyVX

T/Tcp/pcρ/ρccfδfβfcmδmβmcsδsβsμ∗

b,`2HX Δμ∗

b,`2HX

yX3j yXjk kXyN yXeRy yXy93 kXykk yXjRN yXRjN RXRRk yXydR yX9R9 yXe9e RXj8 yXy3

yX3j yX99 kXRy yX8N9 yXy93 kXyej yXjeN yXR9R RXyky yXyjd yX8e3 yX83N RXj8 yXy3

yX3j yXeR kXRk yXeRd yXy9d kXyj3 yXj99 yXR9j RXy88 yXyjN yX8k8 yXeye RXkN yXyd

yX3j yXdd kXRj yXeyy yXy9e kXyNk yXj9e yXRjk RXRkd yXy89 yX9j9 yXejR RXke yXyN

yX3j RXyy kXR8 yXejR yXy9d kXyRk yXjkj yXRj8 RXRRk yXy98 yX89k yXe9R RXkk yXy3

yX3e yX9R kXyy yX8eR yXy93 kXRj9 yX9y8 yXRjd RXykd yXyj9 RXyR9 yX8jy RXdk yXky

yX3e yXdk kXy9 yXek8 yXy9N RXNde yXjkj yXR99 RXy83 yXy8k yX88y yXeyj RXje yXRR

yX3e yXNy kXye yX8dk yXy93 kXRk9 yXjy3 yXRk3 RXR3j yXRky yXRNy yX89N RXjk yXRR

yX3e RXj9 kXRy yX833 yXy93 kXRej yXj83 yXRj9 RXy83 yXy89 yXRy9 yXjd9 RXRe yXyd

yX3e RX89 kXRk yXdj3 yXy9N RX3NR yXkRN yXk88 RX3dj yXy9j yX9yR yXeyd RXRR yXR8

yX3e RX38 kXR9 yXeRR yXy98 kXye8 yXj8j yXRkd RXRjN yXyje yX9jN yXeej RXRy yXye

yX3e kX9d kXR3 yXe8d yXy9e kXyRR yXjke yXRjR RXRy8 yXyRd yX3kj yXde8 RXye yXye

yX3e jXy3 kXkR yXeee yXy98 kXyRN yXjkj yXRjy RXRy9 yXyRy yXN98 8X8jy yXN3 yXy3

yX3e jXdy kXk8 yXe9y yXy99 kXyd9 RXRRj yXRRN yXj8k yXyy3 yX93e RRXj93 yXNj yXy8

yX3e 9XN9 kXjy yXe3N yXy9k kXyk9 yXk38 yXRRk RXj8N yXyke yX9j8 9Xk9k yX39 yXy8

yX3e 8Xde kXjj yXe8d yXy9j kXRRk yXjky yXRy9 RXy8k yXykj yXj8k yX3R8 yX3j yXy3

yX3e dXd3 kX9y yXddR yXy9R RXN8N yXRNR yXRk8 RX8Nd yXyj3 yXjjR RX9Nk yXdd yXy3

yX3e RRX8e kX8y yXd89 yXy9y kXy8k yXky8 yXRy3 RXk8e yXy9k yXk89 yXNNd yX3y yXy9

yX3e ReX98 kXey yX3j9 yXyj3 kXyRe yXRRR yXRkk RXNk9 yXy88 yXjjy RXyNd yX3R yXy3

yX3e kkXe9 kXdy yXN9N yXyj3 kXyRe yXRRR yXy3j yXNN9 yXy8R yXje8 RXj9N yX3y yXyd

yX3ej yX9k RXNN yX8dy yXy8y kXyd3 yXj3d yXR8e yXN8e yXy9j yXd9y yX8kj RXee yXjy

103

h"G1 so, +QMiBMm2/

T/Tcp/pcρ/ρccfδfβfcmδmβmcsδsβsμ∗

b,`2HX Δμ∗

b,`2HX

yX3ej kX9d kXRd yXej9 yXy98 kXyjR yXjyR yXRR3 RXk9e yXyee yXjR8 yXdNj RXyj yXye

yX3ej jXy3 kXkR yXe9d yXy9e kXyey yXj9e yXRk9 RXy8j yXyye yXN99 yXeRR yXN3 yXyN

yX3ej 9XN9 kXjy yXeN9 yXy9j kXyy9 yXkNR yXRk8 RXRNy yXyR8 yX8RR RXkej yX38 yXy3

yX3ej dXNj kX9y yX39j yXy9j RXN9k yXRyy yXR9N jX89y yXy8d yXjjk RXj9y yX38 yXy3

yX3ej RRXdj kX8y yXdeN yXy9R kXy99 yXkR3 yXRRd RXkke yXyRj yX838 RXe8N yXd8 yXy8

yX3ej ReXe9 kXey yX3jN yXyjN kXykd yXRyy yXRkd RX893 yXyeR yXk39 RXyRk yXdd yXRj

yX3ej kkX38 kXdy yX3j8 yXyj3 kXy83 yXR8N yXRj8 yX3dy yXyye RX93R yX3dd yX3j yXyN

6A:X 9X .m2 iQ i?2 bBKBH`Biv Q7 #Qi? miQ+Q``2HiBQM 7mM+iBQMb 7Q` bBKmHiBQMb +QMiBMBM; N= 4096

U;`2vV M/ N= 12000 T`iB+H2b U#H+FV- i?2 `2bmHib 7Q` i?2 T`2b2Mi `2HtiBQM Mbix `2 pB`imHHv

B/2MiB+HX

(R) EX aBM;2` M/ EX 1X :m##BMb- ǳh?2`KH i`MbTQ`i +Q2{+B2Mib 7Q` /2Mb2 ~mB/b-Ǵ BM aiiBbiB+H

J2+?MB+bX oQHmK2 R- 2/Bi2/ #v EX aBM;2` Uh?2 _QvH aQ+B2iv Q7 *?2KBbi`v- RNdjV TTX RN9Ĝk8jX

(k) _X EiiB- _X hX C+Q#b2M- _X "X ai2r`i- M/ JX C?M;B`B- ǳh?2`KQ/vMKB+ S`QT2`iB2b Q7

L2QM 7Q` h2KT2`im`2b 7`QK i?2 h`BTH2 SQBMi iQ dyy E i S`2bbm`2b iQ dyy JS-ǴBM/pM+2b BM

*`vQ;2MB+ 1M;BM22`BM;- 2/Bi2/ #v _X qX 6bi UaT`BM;2`- RN3eV TTX RR3NĜRRNdX

(j) *X h2;2H2`- _X aTM- M/ qX q;M2`- CX S?vbX *?2KX _27X .i k3- ddN URNNNVX

(9) 1X qX G2KKQM M/ _X aTM- CX *?2KX 1M;X .i 8R- d38 UkyyeVX

(8) SX 1X Jm2HH2`- .X 1/2M- *X qX :`HM/- M/ _X *X qBHHBKbQM- S?vbX _2pX  e- kkdk URNdkVX

(e) CX h?Q2M M/ *X qX :`HM/- S?vbX _2pX  Ry- RjRR URNd9VX

(d) 1X oX G`bQM- .X :X Lm;H2- M/ hX qX H/B`- CX *?2KX S?vbX 89- k9kN URNdRVX

(3) SX JH#`mMQi- X "Qv2`- 1X *?`H2b- M/ >X #+?B- S?vbX _2pX  kd- R8kj URN3jVX

(N) .X :X Lm;H2- CX >X GmMb7Q`/- M/ CX _X aBM;2`- CX *?2KX S?vbX 98- 9eeN URNeeVX

(Ry) CX X *QrM M/ _X LX "HH- *MX CX S?vbX 8y- R33R URNdkVX

(RR) CX X *QrM M/ SX qX q`/- *MX CX S?vbX 8R- kkRN URNdjVX

(Rk) .X :X Lm;H2- CX *?2KX S?vbX 99- d9R URNeeVX

(Rj) SX qX q`/- CX X *QrM- M/ _X EX Si?`B- *MX CX S?vbX 8j- kN URNd8VX

(R9) CX X *QrM M/ CX qX G22+?- *MX CX S?vbX eR- 3N8 URN3jVX

(R8) CX X *QrM M/ _X LX "HH- *MX CX S?vbX 83- d9 URN3yVX

(Re) CX X *QrM M/ CX qX G22+?- *MX CX S?vbX 8N- Rk3y URN3RVX

104

Supplementary material to:

Diusion of the carbon dioxide - carbon mixture in the extended critical

region

René Spencer Chatwell

, Gabriela Guevara-Carrion

, Yuri Gaponenko

, Valentina Shevtsova

and Jadran Vrabec

Thermodynamics and Process Engineering, Technische Universität Berlin, 10587 Berlin,

Germany

Microgravity Research Center, Université de Bruxelles, 1050 Bruxelles, Belgium

CONTENTS

This supplementary material includes a schematic of the employed Taylor dispersion apparatus described in the

manuscript. The enthalpy

, specic volume

as well as the response functions

αv

and

βT

along the studied

isobar are also graphically shown. The center-of-mass radial distribution function and average coordination number of

the CO

-ethanol pair are depicted to complement the remaining molecular pairs of the mixture. Further, predictions

of the Fick diusion coecient of the CO

+ ethanol mixture with the regarded equations are shown in comparison

with experimental and simulation results. The numerical data from molecular simulation are also listed here.

EXPERIMENTAL SET-UP

The Taylor dispersion apparatus that was used in this work is depicted in Figure S1. It consisted of four modules:

a carrier uid conditioning device, the CO

delivery system with a solute injection valve, the air bath thermostat

housing the diusion capillary and a FT-IR detector.

FIG. S1. Schematic of the high pressure Taylor dispersion apparatus.

Electronic Supplementary Material (ESI) for Physical Chemistry Chemical Physics.

105

RESPONSE FUNCTIONS

Thermodynamic response functions were obtained from equilibrium molecular dynamics simulations in the

NpT

and

NVT

ensembles performed with the simulation program

2. Extensive equilibrium molecular dynamics simulations

were performed in both ensembles during this work. In the

NVT

ensemble, the isochoric heat capacity

was

determined from uctuations of the residual potential energy and the virial. In the

NpT

ensemble, the residual

isobaric heat capacity

, the isothermal compressibility

βT

and the volume expansivity

αv

are functions of other

ensemble uctuations. For a detailed information the interested reader is encouraged to read our publications on the

features of the

2 software [1].

FIG. S2. Enthalpy

and isobaric heat capacity

of CO

(blue) and the mixture CO

+ ethanol (black) along the isobar

p=10

MPa. The green line represents the properties of pure CO

calculated with the Span-Wagner equation of state [2].

106

FIG. S3. Specic volume

and thermal expansion

αv

of CO

(blue) and the mixture CO

+ ethanol (black) along the isobar

p=10

MPa. The green line represents the properties of pure CO

calculated with the Span-Wagner equation of state [2].

107

FIG. S4. Isothermal compressibility

βT

of CO

(blue) and the mixture CO

+ ethanol (black) along the isobar

p=10

MPa.

The green line represents the isothermal compressibility of pure CO

calculated with the Span-Wagner equation of state [2].

MICROSCOPIC STRUCTURE

























FIG. S5. CO

-ethanol radial distribution function at

T= 310

K (black),

320

K (blue),

330

K (green) and

340

K (red) along

the isobar

p=10

MPa.

108

FIG. S6. Average coordination number of the CO

-ethanol pair along the isobar

p=10

MPa. The dashed line serves as a

guide to the eye.

109

PREDICTIVE EQUATIONS

Several equations were tested on their ability to predict the diusion coecient of ethanol innitely diluted in

. Fig. S7 shows their predictions in comparison with present experimental measurements and molecular dynamics

simulation data.

FIG. S7. Fick diusion coecient at innite dilution of the CO

+ ethanol mixture along the isobar

= 10 MPa. Experimental

measurements (blue squares) and simulation results (black bullets) are compared with selected predictive equations: Wilke-

Chang [3] (black line), Catchpole and King [4] (cyan line), Funazukuri et al. [5] (yellow line), Lai-Tan [6] (green line), He and

Yu [7] (blue line), Vaz et al. [8] (red line) and Scheibel [9] (pink line). The dashed lines serve as a guide to the eye.

110

NUMERICAL SIMULATION RESULTS

TABLE S1. Density

, enthalpy

, internal energy

, isobaric heat capacity

, isochoric heat capacity

, thermal expansion

αv

and isothermal compresibility

βT

of the CO

+ ethanol mixture with

xCO2

= 0.97 mol

mol

−1

along the isobar

=10MPa.

The numbers in parentheses denote the statistical uncertainty in the last digits

Tρ h u c

pcvαvβT

K mol dm

−3

kJ mol

−1

kJ mol

−1

J mol

−1

J mol

−1

MPa

−1

305 17.6431 -11.363 (1) -9.392 (1) 103 (4) 15.0 (2) 0.0111 (3) 0.021 (1)

306 17.3900 -11.264 (1) -9.291 (1) 108 (5) 15.1 (2) 0.0117 (3) 0.024 (1)

307 17.2078 -11.146 (1) -9.169 (1) 112 (5) 14.8 (2) 0.0128 (4) 0.026 (1)

308 16.9733 -11.038 (2) -9.063 (2) 110 (5) 15.4 (2) 0.0139 (5) 0.026 (1)

309 16.7440 -10.919 (1) -8.942 (1) 132 (6) 15.5 (2) 0.0145 (6) 0.034 (2)

310 16.5099 -10.787 (1) -8.811 (2) 133 (7) 15.5 (2) 0.0154 (5) 0.037 (2)

311 16.2407 -10.658 (1) -8.683 (1) 134 (6) 15.6 (2) 0.0162 (6) 0.039 (2)

312 15.9789 -10.525 (1) -8.553 (2) 133 (7) 15.7 (3) 0.0179 (8) 0.043 (3)

313 15.6723 -10.369 (1) -8.398 (1) 147 (8) 15.7 (2) 0.0204 (7) 0.051 (3)

314 15.3408 -10.209 (1) -8.242 (2) 163 (8) 17.0 (3) 0.0211 (7) 0.059 (3)

315 15.0566 -10.040 (2) -8.082 (2) 187 (11) 17.2 (3) 0.0247 (11) 0.076 (5)

316 14.7274 -9.831 (2) -7.881 (2) 193 (13) 17.4 (3) 0.0278 (13) 0.082 (6)

317 14.1996 -9.639 (2) -7.701 (2) 263 (17) 18.1 (3) 0.0325 (16) 0.126 (9)

318 13.7597 -9.379 (2) -7.458 (2) 224 (14) 19.1 (4) 0.0340 (19) 0.116 (8)

319 13.2006 -9.130 (2) -7.231 (2) 257 (17) 19.3 (4) 0.0509 (26) 0.155 (10)

320 12.4577 -8.838 (2) -6.967 (2) 406 (32) 20.3 (5) 0.0557 (30) 0.297 (27)

321 11.8604 -8.479 (2) -6.648 (2) 330 (30) 20.9 (5) 0.0647 (31) 0.263 (28)

322 11.0933 -8.092 (2) -6.310 (2) 360 (27) 21.6 (6) 0.0714 (35) 0.311 (23)

323 10.3431 -7.676 (2) -5.953 (2) 363 (32) 22.3 (5) 0.0683 (31) 0.336 (29)

324 9.6679 -7.295 (2) -5.631 (2) 317 (26) 21.6 (6) 0.0657 (33) 0.337 (26)

325 9.1395 -6.991 (3) -5.377 (3) 307 (23) 22.3 (7) 0.0560 (34) 0.359 (27)

326 8.6622 -6.671 (3) -5.115 (3) 305 (26) 21.5 (7) 0.0484 (27) 0.390 (32)

327 8.3640 -6.471 (3) -4.950 (3) 242 (21) 21.3 (6) 0.0454 (23) 0.318 (28)

328 7.9564 -6.197 (2) -4.726 (2) 215 (16) 18.7 (5) 0.0406 (20) 0.272 (24)

329 7.7105 -6.031 (2) -4.592 (2) 197 (16) 19.0 (5) 0.0365 (18) 0.295 (26)

330 7.4506 -5.851 (3) -4.449 (2) 170 (13) 19.6 (5) 0.0315 (15) 0.299 (20)

331 7.2152 -5.677 (3) -4.309 (3) 152 (11) 19.5 (6) 0.0283 (12) 0.226 (19)

332 7.0533 -5.545 (3) -4.208 (3) 136 (13) 18.4 (6) 0.0253 (11) 0.237 (21)

333 6.8707 -5.409 (2) -4.096 (2) 169 (12) 16.5 (4) 0.0231 (10) 0.330 (24)

334 6.7101 -5.282 (2) -3.995 (2) 134 (14) 16.7 (5) 0.0236 (11) 0.254 (28)

335 6.5657 -5.175 (2) -3.912 (2) 101 (9) 16.5 (4) 0.0212 (9) 0.196 (18)

336 6.4372 -5.064 (2) -3.824 (2) 110 (4) 15.5 (4) 0.0207 (9) 0.218 (9)

337 6.3047 -4.952 (2) -3.736 (2) 101 (4) 14.9 (3) 0.0195 (7) 0.213 (8)

338 6.2109 -4.881 (2) -3.682 (2) 96 (4) 14.6 (4) 0.0181 (7) 0.202 (10)

339 6.0860 -4.783 (2) -3.606 (2) 97 (4) 13.8 (3) 0.0186 (7) 0.210 (8)

340 5.9914 -4.690 (2) -3.533 (2) 85 (2) 14.5 (5) 0.0166 (5) 0.195 (6)

111

TABLE S2. Fick diusion coecient

, intradiusion coecients

DCO2

DEtOH

, thermodynamic factor

and shear viscosity

of the CO

+ ethanol mixture with

xCO2

= 0.97 mol

mol

−1

at the given temperature

and pressure

. The numbers in

parentheses denote the statistical uncertainty in the last digits.

Tp D

CO2 DEtOH DΓη

KMPa

10−8

−110−8

−1

−4

Pas

305 9.96 (1) 1.947(2) 1.204 (7) 1.12(18) 0.452 (8) 1.09 (4)

306 9.96 (1) 1.961(2) 1.251 (7) 1.11(19) 0.465 (8) 1.07 (4)

307 9.92 (1) 2.010(2) 1.292 (7) 1.24(19) 0.479 (7) 1.00 (4)

308 9.97 (1) 2.062(2) 1.326 (7) 2.24(20) 0.485 (8) 1.00 (4)

309 9.95 (1) 2.119(2) 1.361 (7) 1.31(20) 0.491 (7) 0.97 (4)

310 9.94 (1) 2.200(2) 1.398 (8) 1.23(20) 0.483 (8) 0.86 (3)

311 9.95 (1) 2.260(2) 1.445 (8) 1.23(22) 0.487 (8) 0.84 (3)

312 9.97 (1) 2.320(2) 1.498 (8) 1.44(24) 0.502 (8) 0.89 (3)

313 9.92 (1) 2.414,(3) 1.527 (9) 1.30(21) 0.473 (9) 0.83 (3)

314 9.909 (9) 2.509(3) 1.609 (9) 1.33(22) 0.499 (9) 0.77 (3)

315 9.954 (9) 2.603(3) 1.656 (9) 1.48(24) 0.487(10) 0.73 (3)

316 9.913 (9) 2.721(3) 1.745 (9) 1.50(23) 0.480 (9) 0.74 (3)

317 9.935 (8) 2.845(3) 1.797 (9) 1.62(25) 0.500 (9) 0.70 (3)

318 9.918 (8) 3.018(3) 1.918(10) 1.54(25) 0.475(10) 0.65 (3)

319 9.965 (7) 3.190(3) 1.990(10) 1.71(27) 0.477(10) 0.67 (2)

320 9.980 (6) 3.407(3) 2.132(11) 1.78(30) 0.481 (8) 0.59 (2)

321 9.990 (6) 3.705(3) 2.257(12) 2.05(31) 0.465 (9) 0.52 (2)

322 9.996 (5) 4.050(4) 2.438(14) 2.01(32) 0.469(10) 0.47 (2)

323 10.000 (4) 4.447(4) 2.653(14) 2.15(35) 0.447(10) 0.40 (2)

324 9.998 (3) 4.826(4) 2.829(15) 2.58(40) 0.482(10) 0.41 (2)

325 10.011 (3) 5.161(5) 3.060(16) 2.46(36) 0.437(10) 0.37 (2)

326 9.994 (3) 5.548(5) 3.243(18) 2.61(42) 0.437(10) 0.34 (1)

327 10.024 (3) 5.784(5) 3.413(19) 2.78(42) 0.425(11) 0.32 (1)

328 9.992 (3) 6.139(5) 3.653(20) 2.86(47) 0.455(11) 0.31 (1)

329 9.999 (3) 6.359(5) 3.778(20) 2.92(46) 0.441(13) 0.31 (1)

330 9.999 (3) 6.613(6) 3.936(21) 3.31(50) 0.497(11) 0.31 (1)

331 9.989 (2) 6.834(6) 4.099(22) 3.12(51) 0.477(15) 0.32 (1)

332 10.005 (2) 7.065(6) 4.295(23) 3.33(54) 0.489(14) 0.27 (1)

333 9.999 (2) 7.294(6) 4.486(22) 3.79(58) 0.512(11) 0.26 (1)

334 9.995 (2) 7.497(6) 4.654(25) 3.92(63) 0.545(14) 0.27 (1)

335 9.992 (2) 7.664(6) 4.786(26) 4.03(63) 0.523(13) 0.29 (1)

336 9.999 (2) 7.866(6) 4.961(25) 4.15(64) 0.564(12) 0.26 (1)

337 9.999 (2) 8.057(7) 5.147(26) 4.45(70) 0.588(13) 0.26 (1)

338 10.010 (2) 8.185(7) 5.140(27) 4.52(76) 0.599(10) 0.28 (1)

339 9.998 (2) 8.386(7) 5.357(27) 4.60(76) 0.583(12) 0.27 (1)

340 10.004 (2) 8.530(7) 5.521(27) 4.94(76) 0.613(14) 0.27 (1)

112

TABLE S3. Average coordination number

Nx−y(r)

of the CO

-CO

, ethanol-ethanol and CO

-ethanol pairs for the rst

coordination shell along the isobar

p=10

MPa

NCO2−CO2 (r)NEtOH−EtOH (r)NCO2−CO2 (r)

305 9.50 1.42 0.314

306 9.47 1.38 0.312

307 9.50 1.35 0.310

308 9.48 1.33 0.308

309 9.49 1.31 0.304

310 9.32 1.30 0.306

311 9.13 1.27 0.307

312 9.04 1.26 0.302

313 9.06 1.25 0.301

314 8.90 1.24 0.300

315 8.65 1.23 0.300

316 8.73 1.24 0.299

317 8.22 1.19 0.290

318 8.31 1.22 0.280

319 7.97 1.19 0.283

320 7.82 1.14 0.269

321 7.53 1.18 0.263

322 7.17 1.14 0.255

323 6.84 1.16 0.247

324 6.50 1.09 0.240

325 6.13 1.14 0.228

326 6.21 1.12 0.224

327 5.85 1.11 0.222

328 5.87 1.07 0.220

329 5.52 1.06 0.212

330 5.42 0.979 0.210

331 5.53 0.981 0.209

332 5.28 0.938 0.201

333 5.18 0.891 0.202

334 5.16 0.836 0.202

335 5.04 0.837 0.202

336 5.11 0.795 0.203

337 4.84 0.757 0.203

338 4.84 0.768 0.190

339 4.83 0.747 0.202

340 4.83 0.726 0.202

[1] G. Rutkai, A. Köster, G. Guevara-Carrion, T. Janzen, M. Schappals, C. W. Glass, M. Bernreuther, A. Wafai, S. Stephan,

M. Kohns, S. Reiser, S. Deublein, M. Horsch, H. Hasse, and J. Vrabec, Comput. Phys. Commun.

221

, 343 (2017).

[2] R. Span and W. Wagner, J. Phys. Chem. Ref. Data

, 1509 (1996).

[3] C. R. Wilke and P. Chang, AIChE J.

, 264 (1955).

[4] O. J. Catchpole and M. B. King, Ind. Eng. Chem. Res.

, 1828 (1994).

[5] T. Funazukuri, C. Y. Kong, and S. Kagei, Ind. End. Chem. Res.

, 835 (2000).

[6] C.-C. Lai and C.-S. Tan, Ind. Eng. Chem. Res.

, 674 (1995).

[7] C.-H. He and Y.-S. Yu, Ind. Eng. Chem. Res.

, 3793 (1998).

[8] R. V. Vaz, A. L. Magalhaes, and C. M. Silva, J. Supercrit. Fluids

, 24 (2014).

[9] E. G. Scheibel, Ind. Eng. Chem.

, 2007 (1954).

113

References

[1] J. Bernoulli. Opera omnia, tam antea sparsim edita, quam hactenus inedita. Marci-

Michaelis Bousquet et Sociorium, Geneva, 1742.

[2] D. Bernoulli. Hydrodynamica, sive de viribus et motibus ﬂuidorum commentarii.

Georgii Deckeri, Basel, 1738.

[3] J. l. R. d’Alembert. Traité de l’équilibre et du mouvement des ﬂuides: Pour servir

de suite au Traité de dynamique. David, Paris, 1744.

[4] L. Euler. Principes généraux du mouvement des ﬂuides. Mémoires de l’acamémie

des sciences de Berlin, 11:274–315, 1755.

[5] C. A. Truesdell. Editor’s Introduction to Euleri Opera Omnia. In Rational Fluid

Mechanics, 1687-1765. Orell Füssli, Zürich, 1954.

[6] W. Wien. Lehrbuch der Hydrodynamik. S. Hirzel, Leipzig, ﬁrst edition, 1900.

[7] J. l. R. d’Alembert. Paradoxe proposé aux Géometres sur la Résistance des Fluides.

Memoir XXXIV, Opuscules Mathématiques, 5:132–138, 1768.

[8] C.-L.-M.-H. Navier. Sure les lois des mouvemens des ﬂuides, en ayant égard á

l’adhésion des molécules. Ann. de Chimie, 19:244–260, 1821.

[9] C.-L.-M.-H. Navier. Mémoire sur les lois du mouvement des ﬂuides. Mémoires de

l’Académie Royal des Sciences de l’Institute de France, pages 389–440, 1823.

[10] A.-L. Cauchy. Recherches sur l’équilibre et le mouvement intérieur des corps solides

ou ﬂuides, élastique ou non-élastique. Bull. Soc. Philomath., 9:9–13, 1823.

[11] S.-D. Poisson. Mémoire sur les équations générales de l’équilibre et du mouvement

des corps solides élastiques et des ﬂuides. J. Ecole Poly., 13:1–174, 1829.

[12] A.-J.-C. B. de Saint-Vernant. Note á joindre au mémoire sur la dynamique des

ﬂuides. C. R. Acad. Sci. Paris, 17:1240–1243, 1837.

[13] G. G. Stokes. On the theories of the internal friction of ﬂuids in motion and of the

equilibrium and motion of elastic solids. Trans. Cambr. Phil. Soc., 8:287–319, 1845.

[14] G. G. Stokes. On the eﬀect of the internal friction of ﬂuids on the motion of pendu-

lums. Trans. Cambr. Phil. Soc., 9:8–106, 1851.

[15] M. Eckert. The Dawn of Fluid Dyamics. A Discipline between Science and Technol-

ogy. Wiley, Weinheim, 2006.

[16] L. Prandtl. Über Flüssigkeitsbewegung bei sehr kleiner Reibung. In Verhandlungen

des III. Internationalen Mathematiker-Kongresses, Leipzig, 1904. Teubner.

[17] G. Nardin, G. Grosso, Y. Léger, B. Pietka, F. Morier-Genoud, and B. Deveaud-

Plédran. Hydrodynamic nucleation of quantized vortex pairs in a polariton quantum

ﬂuid. Nat. Phys., 7:635–641, 2011.

[18] K. Sasaki, N. Suzuki, and H. Saito. Bénard-von Kármán vortex street in a Bose-

Einstein condensate. Phys. Rev. Lett., 104:150404, 2010.

114

[19] W. J. Kwon, J. H. Kim, S. W. Seo, and Y. Shin. Observation of von Kármán vortex

street in an atomic superﬂuid gas. Phys. Rev. Lett., 117:245301, 2016.

[20] E. Fonda, D. P. Meichle, N. T. Ouellette, S. Hormoz, and D. P. Lathrop. Direct

observation of Kelvin waves excited by quantized vortex reconnection. Proc. Natl.

Acad. Sci., 111:4707–4710, 2014.

[21] D. K. Lubensky and R. E. Goldstein. Hydrodynamics of monolayer domains at the

air-water interface. Phys. Fluids, 8:843–854, 1996.

[22] B. Schmidt. Electron beam density measurements in shock waves in argon. J. Fluid.

Mech., 39:361–373, 1969.

[23] F. Robben and L. Talbot. Measurement of shock wave thickness by the electron

beam ﬂuorescence method. Phys. Fluids, 9:633–643, 1966.

[24] R. A. Gross. Thickness of a shock wave in hydrogen. Phys. Fluids, 10:1853–1855,

1967.

[25] A. M. Pavlenko, B. Yu. Zanin, and M. M. Katasonov. Features of ﬂow around the

ﬂying wing model at various attack and slip angles. AIP Conf. Proc., 1893:030098,

2017.

[26] E. J. Anderson, W. R. McGillis, and M. A. Grossenbaugh. The boundary layer of

swimming ﬁsh. J. Exp. Biol., 204:81–102, 2001.

[27] C. Zhang, P. Weng, Z. Yao, Y. Yuan, and X. Fan. Visualization of ﬂow separation

inside cut kerf during laser cutting of thick sections. J. Laser Appl., 28:022204, 2016.

[28] A. N. Karpetis. Miniature supersonic burner for the study of combuation at extreme

conditions II: External ﬂow. J. Energ. Eng., 144:04018058, 2018.

[29] M. Krishnan, V. M. Ugaz, and M. A. Burns. PCR in a Reyleigh-Bénard convection

cell. Science, 298:793, 2002.

[30] G. Ahlers. Turbulent convection. Physics, 2:74–81, 2009.

[31] C.D. Andereck, R. Dickman, and H. L. Swinney. New ﬂows in a circular Couette

system with co-rotating cylinders. Phys. Fluids, 26:1395–1401, 1983.

[32] H. Mitsudharmadi, S.H Winoto, and D. A. Shah. Splitting and merging of Görtler

vortices. Phys. Fluid, 17:124102, 2005.

[33] T. J. Mueller, R. C. Nelson, J. T. Kegelman, and M. V. Morkovin. Smoke visual-

ization of boundary-layer transition on a spinning axisymmetric body. AIAA J., 19:

1607–1608, 1981.

[34] P. Roushan and X. L. Wu. Universal wake structures of Kármán vortex streets in

two-dimensional ﬂows. Phys. Fluids, 17:073601, 2005.

[35] K. P. Chopra and L. F. Hubert. Kármán vortex street in Earth’s atmosphere. Nature,

4952:1341–1343, 1964.

[36] S. C. Crow. Stability theory for a pair of trailing vortices. AIAA Journal, 8:2172–

2179, 1970.

115

[37] S. Haver. A possible freak wave event measured at the Drauper Jacket January 1

1995. Proc. of Rogue Waves, pages 1–8, 2004.

[38] T. A. A. Adcock, P. H. Taylor, S. Yan, Q. W. Ma, and P. A. E. M. Janssen. Did

the Drauper wave occur in a crossing sea? Proc.R.Soc.A., 467:3004–3021, 2011.

[39] L. Yiqi and X. Zhou. Soil Respiration and the Environment. Academic Press, ﬁrst

edition, 2006.

[40] D. B. Chelton and M. G. Schlax. Global observations of oceanic Rossby waves.

Science, 272:234–238, 1996.

[41] S. M. Cavallo and G. J. Hakim. Potential vorticity diagnosis of a Tropopause polar

cyclone. Mon. Wea. Rev., 137:1358–1371, 2009.

[42] Y. Wang, A. Goolchiani, Y. Ashkenay, Y. Berezin, O. Guez, and S. Havlin. Dominant

imprint of Rossby waves in the climate network. Phys. Rev. Lett., 111:138501, 2013.

[43] A. A. Simon, F. Tabataba-Vakili, R. Cosentino, R. F. Beebe, M. H. Wong, and G. S.

Orton. Historical and contemporary trends in the size, drift, and color of Jupiter’s

Great Red Spot. Astron. J., 155:151–164, 2018.

[44] J. Onorbe, F. B. Davies, Z. Lukić, J. F. Hennawi, and D. Sorini. Inhomogeneous

reionization models in cosmological hydrodynamical simulations. Mon. Notices Royal

Astron. Soc., 486:4075–4097, 2019.

[45] A. G. Riess, A. V. Filippenko, P. Challis, A. Clocchiatti, A. Diercks, P. M. Gar-

navich, R. L. Gilliland, C. J. Hogan, S. Jha, R. P. Kirshner, B. Leibundgut, M. M.

Phillips, D. Reiss, B. P. Schmidt, R. A. Schommer, R. Chris Smith, J. Spyromilio,

C. Stubbs, N. B. Suntzeﬀ, and J. Tonry. Observational evidence from supernovae for

an accelerating universe and a cosmological constant. Astrophys. J., 116:1009–1038,

1998.

[46] S. Perlmutter, G. Aldering, G. Goldhaber, R. A. Knop, P. Nugent, P. G. Castro,

S. Deustua, S. Fabbro, A. Goobar, D. E. Groom, I. M. Hook, A. G. Kim, M. Y.

Kim, J. C. Lee, N. J. Nunes, R. Pain, C. R. Pennypacker, R. Quimby, C. Lidman,

R. S. Ellis, M. Irwin, R. G. McMahon, P. Ruiz-Lapuente, N. Walton, B. Schaefer,

B. J. Boyle, A. V. Filippenko, T. Matheson, A. S. Fruchter, N. Panagia, H. J. M.

Newberg, W. J. Couch, and The Supernova Cosmology Project. Measurements of ω

and λfrom 42 high-redshift supernovae. Astrophys. J., 517:565–586, 1999.

[47] E. Komatsu, K. M. Smith, J. Dunkley, C. L. Bennett, B. Gold, G. Hinshaw,

N. Jarosik, D. Larson, M. R. Nolta, L. Page, D. N. Spergel, M. Halpern, R. S.

Hill, A. Kogut, M. Limon, S. S. Meyer, N. Odegard, G. S. Tucker, J. L. Weiland,

E. Wollack, and E. L. Wright. Seven-year Wilkinson microwave anisotropy probe

(WAMP) observations: Cosmological interpretation. Astrophys. J. Suppl., 192:18–

65, 2011.

[48] D. J. Eisenstein, I. Zehavi, D. W. Hogg, R. Scoccimarro, M. R. Blanton, R. C.

Nichol, R. Scranton, H.-J. Seo, M. Tegmark, Z. Zheng, S. F. Anderson, J. Annis,

N. Bahcall, J. Brinkmann, S. Burles, F. J. Castander, A. Connolly, I. Csabai, M. Doi,

M. Fukugita, J. A. Frieman, K. Glazebrook, J. E. Gunn, J. S. Hendry, G. Hennessy,

116

Z. Ivezić, S. Kent, G. R. Knapp, H. Lin, Y.-S. Loh, R. H. Lupton, B. Margon, T. A.

McKay, A. Meiksin, J. A. Munn, A. Pope, M. W. Richmond, D. Schlegel, D. P.

Schneider, K. Shimasaku, C. Stoughton, M. A. Strauss, M. SubbaRao, A. S. Szalay,

I. Szapudi, D. L. Tucker, Br. Yanny, and D. G. York. Detection of the baryon

acoustic peak in the large-scale correlation function of SDSS luminous red galaxies.

Astrophys. J., 633:560–574, 2005.

[49] M. Tegmark, D. J. Eisenstein, M. A. Strauss, D. H. Weinberg, M. R. Blanton, J. A.

Frieman, M. Fukugita, J. E. Gunn, A. J. S. Hamilton, G. R. Knapp, R. C. Nichol,

J. P. Ostriker, N. Padmanabhan, W. J. Percival, D. J. Schlegel, D. P. Schneider,

R. Scoccimarro, U. Seljak, H.-J. Seo, M. Swanson, A. S. Szalay, M. S. Vogeley,

J. Yoo, I. Zehavi, K. Abazajian, S. F. Anderson, J. Annis, N. A. Bahcall, B. Bassett,

A. Berlind, J. Brinkmann, T. Budavari, F. Castander, A. Connolly, I. Csabai, M. Doi,

D. P. Finkbeiner, B. Gillespie, K. Glazebrook, G. S. Hennessy, D. W. Hogg, B. Jain,

D. Johnston, S. Kent, D. Q. Lamb, B. C. Lee, H. Lin, J. Loveday, R. H. Lupton, J. A.

Munn, K. Pan, C. Park, J. Peoples, J. R. Pier, A. Pope, M. Richmond, C. Rockosi,

R. Scranton, R. K. Sheth, A. Stebbins, C. Stoughton, I. Szapudi, D. L. Tucker, D. E.

VandenBerk, B. Yanny, and D. G. York. Cosmological constraints from the SDSS

luminous red galaxies. Phys. Rev. D, 74:123507, 2006.

[50] R. R. Caldwell, R. Dave, and P. J. Steinhardt. Cosmological imprint of an energy

component with general equation-of-state. Phys. Rev. Lett., 80:1582–1585, 1998.

[51] J.-P. Uzan. Cosmological scaling solution of non-minimally coupled scalar ﬁelds.

Phys. Rev. D, 59:123510, 1999.

[52] T. Chiba. The gravitational constant and gravity. Phys. Rev. D, 60:083508, 1999.

[53] L. Amendola. Coupled quintessence. Phys. Rev. D, 62:043511, 2000.

[54] S. Räsänen. Dark energy from back-reaction. J. Cosmol. Astropart. Phys., 2:003,

2004.

[55] E. W. Kolb, S. Matarrese, and A. Riotto. On cosmic acceleration without dark

energy. New J. Phys., 8:322, 2006.

[56] T. Buchert. Dark energy from structure - A status report. Gen. Rel. Grav., 40:

467–527, 2008.

[57] R. Maartens. Dissipative cosmology. Class. Quantum Grav., 12:1455–1465, 1995.

[58] W. Zimdahl. Bulk viscous cosmology. Phys. Rev. D, 53:5483–5493, 1996.

[59] R. Colistete, J. C. Fabris, J. Tossa, and W. Zimdahl. Bulk viscous cosmology. Phys.

Rev. D, 76:103516, 2007.

[60] B. Li and J. D. Barrow. Does bulk viscosity create a viable uniﬁed dark matter

model? Phys. Rev. D, 79:103521, 2009.

[61] L. Bocquet and E. Charlaix. Nanoﬂuidics, from bulk to interfaces. Chem. Soc. Rev.,

39:1073–1095, 2010.

117

[62] D. Hirshfeld and D. C. Rapaport. Moleular dynamics simulation of Taylor-Couette

vortex formation. Phys. Rev. Lett., 80:5337–5340, 1998.

[63] V. V. Zhakhovsky, A. P. Kryukov, V. Y. Levashov, I. N. Shishkova, and S. I. Anisi-

mov. Mass and heat transfer between evaporation and condensation surfaces: Atom-

istic simulation and solution of boltzmann kinetic equation. Proc. Natl. Acad. Sci.,

116:18209–18217, 2019.

[64] R.S. Chatwell, H. Heinen, and J. Vrabec. Diﬀusion limited evaporation of a binary

liquid ﬁlm. Int. J. Heat and Mass Transf., 132:1296–1305, 2019.

[65] D. C. Rapaport. Hexagonal convective patterns in atomistically simulated ﬂuids.

Phys. Rev. E., 73:025301, 2006.

[66] P. Sun, J. Ding, S. Huang, X. Luo, and W. Cheng. Microscopic Richtmyer-Meshkov

instability under strong shock. Phys. Fluids, 32:024109, 2020.

[67] D. C. Rapaport and E. Clementi. Eddy formation in obstructed ﬂuid ﬂow: A

molecular-dynamics study. Phys. Rev. Lett., 57:695–698, 1986.

[68] M. Moseler and U. Landman. Formation, stability, and breakup of nanojets. Science,

289:1165–1169, 2000.

[69] T. Hitz, M. Heinen, J. Vrabec, and C.-D. Munz. Comparison of macro- and micro-

scopic solutions of Riemann problem I. Supercritical shock tube and expansion into

vacuum. J. Comput. Phys., 402:109077, 2020.

[70] J. P. Hansen and I. R. McDonald. Theory of simple liquids. Elsevier, London, third

edition, 2006.

[71] R. Shankar. Principles of Quantum Mechanics. Plenum Press, New York, second

edition, 1994.

[72] M. Born. Atomtheorie des festen Zustandes. Dynamik der Kristallgitter.

Vieweg+Teubner, Wiesbaden, ﬁrst edition, 1923.

[73] J. E. Jones. On the determination of molecular ﬁelds I. From the variation of the

viscosity of a gas with temperature. Proc. Royal Soc. A., 106:441–462, 1924.

[74] J. E. Jones and S. Chapman. On the determination of molecular ﬁelds II. from the

equation of state of a gas. Proc. Royal Soc. A., 106:463–477, 1924.

[75] F. London. Zur Theorie und Systematik der Molekularkräfte. Z. Physik, 63:245–279,

1930.

[76] W. J. Deal. The long-range interaction between two hydrogen atoms. Intern. J.

Quantum Chem., 6:593–597, 1972.

[77] M. Born and J. E. Mayer. Zur Gittertheorie der Ionenkristalle. Z. Physik, 75:1–18,

1932.

[78] F. London. The general theory of molecular forces. J. Chem. Soc., Faraday Trans,

33:8–26, 1937.

118

[79] R. A. Buckham. The classical equation of state of gaseous helium, neon and argon.

Proc. Royal Soc. A., 168:264–283, 1938.

[80] J. H. Dymond and B. J. Alder. Pair potential for argon. J. Chem. Phys., 51:309–320,

1969.

[81] J. A. Barker, R. A. Fisher, and R. O. Watts. Liquid argon: Monte Carlo and

molecular dynamics simulations. Mol Phys., 21:657–673, 1971.

[82] J. P. Toennies. On the validity of a modiﬁed Buckingham potential for the rare gas

dimers at intermediate distances. Chem. Phys. Letters, 20:238–241, 1973.

[83] R. Ahlrichs, R. Penco, and G. Scoles. Intermolecular forces in simple systems. Chem.

Phys., 19:119–130, 1977.

[84] R. A. Aziz and H. H. Chen. An accurate intermolecular potential for argon. J.

Chem. Phys., 67:5719–5726, 1977.

[85] R. A. Aziz, V. P. S. Nain, J. S. Carley, L. W. Taylor, and G. T. McConville. An

accurate intermolecular potential for helium. J. Chem. Phys., 70:4330–4342, 1979.

[86] R. A. Aziz. An improved potential for krypton. Mol Phys., 38:177–190, 1979.

[87] K. T. Tang and J. P. Toennies. An improved simple model for the van der Waals

potential based on universal damping functions for the dispersion coeﬃcients. J.

Chem. Phys., 80:3726–3741, 1984.

[88] R. A. Aziz and M. J. Slaman. On the Xe-Xe potential energy curve and related

properties. Mol Phys., 57:825–840, 1986.

[89] R. A. Aziz and M. J. Slaman. The argon and krypton interatomic potentials revis-

ited. Mol Phys., 58:679–697, 1986.

[90] R. A. Aziz, F. R. W. McCourt, and C. C. K. Wong. A new determination of the

ground state interatomic potential for He2.Mol Phys., 61:1487–1511, 1987.

[91] R. A. Aziz. A highly accurate interatomic potential for argon. J. Chem. Phys., 99:

4518–4525, 1993.

[92] T. L. Gilbert, O. C. Simpson, and M. A. Williamson. Relation between charge and

force parameters of closed-shell atoms and ions. J. Chem. Phys., 63:4061–4071, 1975.

[93] K. T. Tang and J. P. Toennies. The van der Waals potentials between all the rare

gas atoms from He to Rn. J. Chem. Phys., 118:4976–4983, 2003.

[94] B. M. Axilrod and E. Teller. Interaction of the van der Waals type between three

atoms. J. Chem. Phys., 11:299–300, 1943.

[95] F. H. Stillinger and T. A. Weber. Computer simulation of local order in condensed

phases of silicon. Phys. Rev. B, 31:5262–5271, 1985.

[96] F. del Río, E. Díaz-Herrera, O. Guzmán, J. A. Moreno-Razo, and J. E. Ramos.

Analytical equation of state with three-body forces: Application to noble gases. J.

Chem. Phys., 139:184503, 2013.

119

[97] G. Rutkai, M. Thol, R. Span, and J. Vrabec. How well does the Lennard-Jones

potential represent the thermodynamic properties of noble gases? Mol. Phys., 115:

1104–1121, 2017.

[98] M. S. Green. Markoﬀ random processes and the statistical mechanics of time-

dependent phenomena II. Irreversible processes in ﬂuids. J. Chem. Phys., 22:398–

414, 1954.

[99] R. Kubo. Statistical-mechanical theory of irreversible processes. I. General theory

and simple applications to magnetic and conduction problems. J. Phys. Soc. Jpn.,

12:570–586, 1957.

[100] R. Zwanzig. Time-correlation functions and transport coeﬃcients in statistical me-

chanics. Annu. Rev. Phys. Chem., 16:67–102, 1965.

[101] R. Kubo. The ﬂuctuation-dissipation theorem. Rep. Prog. Phys., 29:255–284, 1966.

[102] S. de Groot and P. Mazur. Non-Equilibrium Thermodynamics. Dover, New York,

ﬁrst edition, 2008.

[103] I. Müller and T. Ruggeri. Rational extended thermodynamics. Springer, New York,

second edition, 1998.

[104] G. Lamé and B. P. Clapeyron. Mémoire sur la solidiﬁcation par refroidissement d’un

globe solide. Ann. Chem. Phys., 47:250–256, 1831.

[105] J. Stefan. Über einige Probleme der Therie der Wärmeleitung. S.-B. Wien. Acad.

Mat. Natur., 98:473–484, 1889.

[106] J. Stefan. Über die Diﬀusion von Säuren und Basen gegen einander. S.-B. Wien.

Acad. Mat. Natur., 98:616–634, 1889.

[107] J. Stefan. Über die Theorie der Eisbildung, insbersondere über die Eisbildnung im

Polarmeere. S.-B. Wien. Acad. Mat. Natur., 98:965–983, 1889.

[108] J. Stefan. Über die Verdampfung und die Auﬂösung als Vorgange der Diﬀusion.

S.-B. Wien. Acad. Mat. Natur., 98:1418–1442, 1889.

[109] J. Stefan. Über die Theorie der Eisbildung, insbersondere über die Eisbildnung im

Polarmeere. Ann. Phys. Chem., 42:269–286, 1891.

[110] R. H. Scott, editor. Contributions to our knowledge of the Meteorology of the arctic

regions, pages 216–253. Meteorological Oﬃce of Great Britain, 1880.

[111] H. S. Carslaw and J. C. Jaeger. Conduction of Heat in Solids. Clarendon press,

Oxford, second edition, 1959.

[112] J. Crank. Free and moving boundary problems. Clarendon press, Oxford, ﬁrst edition,

1984.

[113] L. I. Rubinstein. The Stefan Problem. In Translations of Mathematical Monographs

Vol. 27. Americal Physical Society, Providence, 1971.

[114] S. C. Gupta. The classic Stefan Problem. Elsevier, Amsterdam, ﬁrst edition, 2018.

120

[115] P. Schwedtfeger. The thermal properties of sea ice. J. Glaciol., 4:789–807, 1963.

[116] T. McDougall, R. Feistel, F. Millero, D. R. Jackett, D. Wright, B. King, G. Mar-

ion, C.-T. A. Chen, and P. Spitzer. The International Thermodynamic Equation

of Seawater 2010 (TEOS-10): Calculation and Use of Thermodynamic Properties.

2010.

[117] M. H. Scharqawy, J. H. Lienhard, and S. M. Zubair. Thermophysical properties

of seawater: a review of existing correlations and data. Desalin. Water Treat., 16:

345–380, 2010.

[118] M. Nakawo and N. K. Sinha. Growth rate and salinity proﬁle of ﬁrst-year sea ice in

the high arctic. J. Glaciol., 27:315–330, 1981.

[119] K. Kuo. Principles of Combustion. Wiley, New Jersey, second edition, 2005.

[120] D. B. Spalding. The combustion of liquid fuels. Proc. Combust. Inst., 4:847–864,

1953.

[121] J. C. Maxwell. Diﬀusion, pages 625–646. Cambridge Libarary Collection, Cam-

bridge, 2011.

[122] J.-R. Yang and S.-C. Wong. On the discrepancies between theoretical and experi-

mental results for microgravity droplet evaporation. Int. J. Heat Mass Transf., 44:

4433–4443, 2001.

[123] C. Chauveau, M. Birouk, and I. Gökalp. An analysis of the d2-law departure during

droplet evaporation in microgravity. Int. J. Multiph. Flow, 37:252–259, 2011.

[124] A. L. Yarin, G. Brenn, O. Kastner, D. Rensink, and C. Tropea. Evaporation of

acousticaly levitated droplets. J. Fluid Mech., 399:151–204, 1999.

[125] A. Ashkin. Application of laser radiation pressure. Science, 210:1081–1088, 1980.

[126] A Frohn and N. Roth. Dynamics of Droplets. Springer, ﬁrst edition, 2000.

[127] S. Kamagi, T. Sakai, and S. Okajima. Combustion of free fuel droplets in a freely

falling chamber. Proc. Combust. Inst., 13:779–785, 1971.

[128] S. Okajima and S. Kumagai. Further investigations of combustion of free droplets

in a freely falling chamber including moving droplets. Proc. Combust. Inst., 15:

401–407, 1975.

[129] C. T. Avedisian, J. C. Yang, and C. H. Wang. On low-gravity droplet combustion.

Proc. Royal Soc. A, 420:183–200, 1988.

[130] I. Gökalp, C. Chauveau, J. R. Richard, M. Kramer, and W. Leucke. Observations on

the low temperature vaporization and envelope of wake ﬂame burning of n-heptane

droplers at reduced gravity during parabolic ﬂights. Proc. Combust. Inst., 22:2027–

2035, 1989.

[131] H. Nomura, Y. Ujie, H. J. Rath, J. Sato, and M. Kono. Experimental study on

high-pressure droplet evaporation using microgravity conditions. Proc. Combust.

Inst., 26:1267–1273, 1996.

121

[132] V. Nayagam, J. B. Haggard, R. O. Colantonio, A. J. Marchese, F. L. Dryer, B. L.

hang, and F. A. Williams. Microgravity n-heptane droplet combustion in oxygen-

helium mixtures at atmospheric pressure. AIAA, 38:1369–1378, 1998.

[133] C. Morin, C. Chauveau, and I. Gökalp. Droplet vaporization characteristics of

vegetable oil derived biofuels at high temperatures. Exp. Therm. Fluid Sci., 21:

41–50, 2000.

[134] T. Kristyadi, V. Deprédurand, G. Castanet, F. Lemoine, S. S. Sahin, A. Elwardany,

E. M. Sazhina, and M. R. Heikal. Monodisperse monocomponent fuel droplet heating

and evaporation. Fuel, 89:3995–4001, 2010.

[135] H. Ghassemi, S. W. Baek, and Q. S. Khan. Experimental study on evaporation of

kerosene droplets at elevated pressures and temperatures. Combust. Sci. Technol.,

178:1669–1684, 2006.

[136] O. Kunz and W. Wagner. The GERG-2008 wide-range equation of state for natural

gases and other mixtures: An expansion of GERG-2004. J. Chem. Eng. Data, 57:

3032–3091, 2012.

[137] B. J. McBride, S. Gordon, and M. A. Reno. Coeﬃcients for Calculating Thermo-

dynamic and Transport Properties of Individual Species. Technical report, NASA

Lewis Research Center, Cleveland, 1993.

[138] T. W. Park, S. K. Aggarwal, and V. R. Katta. Gravity eﬀects on the dynamics of

evaporating droplets in a heated jet. J. Propul Power, 11:519–528, 1995.

[139] P. Ehrenfest. Phasenumwandlung im üblichen und erweiterten Sinn, klassiﬁziert

nach den entsprechenden Singularitäten des thermodynamischen Potentials. Proc.

Royal Acad. Amsterdam, 36:153–157, 1933.

[140] R. W. Schrage. A Theoretical Study of Interphase Mass Transfer. Columbia Uni-

versity Press, New York, 1953.

[141] S. M. Karim and L. Rosenhead. The second coeﬃcient of viscosity of liquids and

gases. Rev. Mod. Phys., 24:108–116, 1952.

[142] C. Eck and H. Garcke. Mathematische Modellierung. Springer, Berlin, ﬁrst edition,

2008.

[143] R. A. Rapuano. Ultrasonic absorption from 75 to 280 Mc/sec. Phys. Rev., 72:78–79,

1947.

[144] L. P. Kadanoﬀ, W. Götze, D. Hamblen, R. Hecht, E. A. S. Lewis, V. V. Palci-

auskas, M. Rayl, and J. Swift. Static phenomena near critical points: Theory and

experiment. Rev. Mod. Phys., 39:395–431, 1967.

[145] L. P. Kadanoﬀ and J. Swift. Transport coeﬃcients near the liquid-gas critical point.

Phys. Rev., 166:89–101, 1968.

[146] K. Meier and S. Kabelac. Speed of sound instrument for ﬂuids with pressures up to

100 MPa. Rev. Sci. Instrum., 77:123903, 2006.

122

[147] H. D. Parbrook and E. G. Richardson. Propagation of ultrasonic waves in vapours

near the critical point. Proc. Phys. Soc. B, 65:437–444, 1952.

[148] F. H. Dubberke, E. Baumhögger, and J. Vrabec. Burst design and signal processing

for the speed of sound measurement of ﬂuids with the pulse-echo technique. Rev.

Sci. Instrum., 86:054903, 2015.

[149] R. Kohlrausch. Theorie des elektrischen Rückstandes in der Leidner Flasche. Ann.

Phys., 167:56–82, 1854.

[150] G. Williams and D. C. Watts. Non-symmetrical dielectric relaxation behaviour

arising from a simple empirical decay function. Trans. Faraday Soc., 66:80–85, 1970.

[151] G. S. Fanourgakis, J. S. Medina, and R. Prosmiti. Determining te bulk viscosity of

rigid water models. J. Phys. Chem. A, 116:2264–2570, 2012.

[152] N. Bjerrum. Structure and properties of ice. Science, 115:385–390, 1952.

[153] L. G. Dowell and A. P. Rinfret. Low-temperature forms of ice as studies by x-ray

diﬀraction. Nature, 188:1144–1148, 1960.

[154] P. V. Hobbs. Ice Physics. Oxford press, New York, ﬁrst edition, 1974.

[155] M. F. Chaplin. Structure and Properties of Water in its Various States, pages 1–19.

American Cancer Society, 2019.

[156] S. Klotz, J. M. Besson, G. Hamel, R. J. Nelmes, J. S. Loveday, and W. G. Marshall.

Metastable ice VII at low temperature and ambient pressure. Nature, 398:681–684,

1999.

[157] T. Matsuo, Y. Tajima, and H. Suga. Calorimetric study of a phase transition in

D2O ice Ih doped with KOD: Ice XI. J. Phys. Chem. Solids, 47:165–173, 1986.

[158] C. Lobban, J. L. Finney, and W. F. Kuhs. The structure of a new phase of ice.

Nature, 391:268–270, 1998.

[159] C. G. Saltzmann, P. G. Radaelli, E. Mayer, and J. L. Finney. Ice XV: a new

thermodynamically stable phase of ice. Phys. Rev. Lett., 103:105701, 2009.

[160] A. Falenty, T. C. Hansen, and W. F. Kuhs. Formation and properties of ice XVI

obtained by emptying a type sii clathrate hydrate. Nature, 516:231–233, 2014.

[161] H.-K. Mao and R. J. Hemley. Ultrahigh-pressure transitions in solid hydrogen. Rev.

Mod. Phys., 66:671–692, 1994.

[162] M. I. Eremets and I. A. Troyan. Conductive dense hydrogen. Nat. Mater., 10:

927–931, 2011.

[163] R. T. Howie, C. L. Guillaume, T. Scheler, A. F. Goncharov, and E. Gregoryanz.

Mixed molecular and atomic phase of dense hydrogen. Phys. Rev. Lett., 108:125501,

2012.

[164] J. Wilks. The properties of liquid and solid helium. Clarendon Press, New York,

ﬁrst edition, 1967.

123

[165] W. E. Keller. Helium-3 and helium-4. Springer, New York, ﬁrst edition, 1969.

[166] K. H. Kim, K. Amann-Winkel, N. Giovambattista, A. Späh, F. Perakis, H. Pathka,

M. L. Parada, C. Yang, D. Mariedahl, T. Eklund, T. J. Lane, S. You, S. Jeong,

M. Weston, J. H. Lee, I. Eom, M. Kim, J. Park, S. H. Chun, P. H. Poole, and

A. Nilson. Experimental observation of the liquid-liquid transition in bulk super-

cooled water under pressure. Science, 370:978–982, 2020.

[167] J. I. Frenkel. Kinetic theory of liquids. Oxford University Press, New York, ﬁrst

edition, 1946.

[168] V. V. Brazhkin, Y. D. Fomin, A. G. Lyapin, V. N. Ryzhov, and K. Trachenko. Two

liquid states of matter: a dynamic line on phase diagram. Phys. Rev. E, 85:031203,

2012.

[169] D. Bolmatov, M. Zhernenkov, D. Zav’yalov, S. Tkachev, A. Cunsolo, and Y. Cai.

Structural evolution of supercritical CO2across the Frenkel line. J. Phys. Chem.

Lett., 5:2785–2790, 2015.

[170] M. Fisher and B. Widom. Decay of correlations in linear systems. J. Chem. Phys.,

50:3756–3772, 2015.

[171] R. Evans, J. R. Henderson, D. C. Hoyle, A. O. Parry, and Z. A. Sabeur. Asymptotic

decay of liquid structure: oscillatory liquid-vapour density proﬁles and the Fisher-

Widom line. Mol Phys., 80:755–775, 1993.

[172] K. Nishikawa and I. Tanaka. Correlation lengths and density ﬂuctuations in super-

critical states of carbon dioxide. Chem. Phys. Lett., 244:149–152, 1995.

[173] K. Nishikawa and T. Morita. Inhomogeneity of molecular distribution in supercritical

ﬂuids. Chem. Phys. Lett., 316:238–242, 2000.

[174] B. Widom. Degree of the critical isotherm. J. Chem. Phys., 41:1633–1634, 1964.

[175] B. Widom. Equation of state in the neighborhood of the critical point. J. Chem.

Phys., 43:3898–3905, 1965.

[176] D. T. Banuti. Crossing the Widom-line - supercritical pseudo-boiling. J. Supercrit.

Fluids, 98:12–16, 2015.

[177] G. G. Simeonie, T. Bryk, F. A. Gorelli, M. Krisch, G. Ruocco, M. Santoro, and

T. Scopigno. The Widom line as the crossover between liquid-like and gas-like

behavior in supercritical ﬂuids. Nature Phys., 6:503–50, 2018.

[178] L. Xu, P. Kumar, S. Buldyrev, S.-H. Chen, P. Poole, F. Sciortino, and H. Stanley.

Relation between the Widom line and the dynamic crossover in systems with a

liquid-liquid phase transition. Proc. Natl. Acad. Sci., 102:16558–16562, 2005.

[179] L. Liu, S.-H. Chen, A. Faraone, C.-W. Yen, and C.-Y. Mou. Pressure dependence

of fragile-to-strong transition and a possible second critical point in supercooled

conﬁned water. Phys. Rev. Lett., 95:117802, 2005.

[180] M. Y. Ha, T. J. Yoon, T. Tlusty, Y. Jho, and W. B. Lee. Widom delta of supercritical

gas-liquid coexistence. J. Phys. Chem. Lett., 9:1734–1738, 2018.

124

[181] H. E. Stanley. Introduction to phase transitions and critical phenomena. Clarendon

Press, New York, ﬁrst edition, 1971.

[182] F. Gorelli, M. Santoro, T. Scopigno, M. Kirsch, and G. Ruocco. Liquid-ike behavior

of supercritical ﬂuids. Phys. Rev. Lett., 97:24702, 2006.

[183] R. S. Chatwell, G. Guevara-Carrion, Y. Gaponenko, V. Shevtsova, and J. Vrabec.

Diﬀusion of the carbon-dioxide - ethanol mixture in the extended critical region.

Phys. Chem. Chem. Phys., 23:3106–3155, 2021.

125