ESPCI Parismichael/pdf/Evaporation08.pdf · 2008. 11. 20. · Created Date: 11/19/2008 12:22:46 PM

EPJ manuscript No.(will be inserted by the editor)

Modeling phase behavior for quantifying micro-pervaporationexperiments

Michael Schindler and Armand Ajdari

Laboratoire PCT, UMR “Gulliver” CNRS-ESPCI 7083, 10 rue Vauquelin, 75231 Paris cedex 05

Received: date / Revised version: date

Abstract. We present a theoretical model for the evolution of mixture concentrations in a micro-pervaporation device, similar to those recently presented experimentally. The described device makes useof the pervaporation of water through a thin PDMS membrane to build up a solute concentration profileinside a long microfluidic channel. We simplify the evolution of this profile in binary mixtures to a one-dimensional model which comprises two concentration-dependent coefficients. The model then providesa link between directly accessible experimental observations, such as the widths of dense phases or theirgrowth velocity, and the underlying chemical potentials and phenomenological coefficients. It shall thus beuseful for quantifying the thermodynamic and dynamic properties of dilute and dense binary mixtures.

PACS. 47.61.-k Micro- and nano- scale flow phenomena – 47.61.Jd Multiphase flows – 64.75.-g Phaseequilibria – 82.60.Lf Thermodynamics of solutions – 05.70.Ln Nonequilibrium and irreversible thermody-namics

1 Introduction

The thermodynamic and kinetic behavior of dense solu-tions is of central interest in a number of industrial andscientific activities. The so-called “formulation” of multi-component systems for the production of cosmetics, paintsand also comestible goods are industrial examples [1]. Sci-entific interests range from mobilities of macromoleculesin cells to the statistical physics and material propertiesof colloidal suspensions [2,3] and other mixtures. In allthese applications it is important to know the diffusivityor mobility of solutes, their permeability with respect to asolvent or other solutes and similar non-equilibrium quan-tities. Eventually, dense mixtures are complex substanceswhich are likely to undergo a number of thermodynamicand dynamic transitions of state, which makes their anal-ysis a challenge.

The thermodynamic and dynamic properties of diluteand dense mixtures are usually measured by experimen-tal techniques such as sedimentation/centrifugation, ul-tra-filtration or reverse osmosis [4–6]. Depending on theproperties of the solution under investigation, one or theother technique proves to be advantageous. Recently, apromising new microfluidic tool for the analysis of aque-ous solutions has been introduced, the microevaporator [7,8]. The employed method is applicable to a wide rangeof different solutes, such as electrolytes, surfactants, col-loids, or polymers. It makes use of the pervaporation ofthe solvent water through a thin membrane (short verticalarrows in Fig. 1a). The solutes cannot pass the membraneand build up a concentration profile in an underlying mi-

crofluidic channel (black dots in Fig. 1a). This profile of so-lute concentration contains information on the transportcoefficients of the mixture. In experiments, the methodhas been successfully employed for the semi-quantitativescreening of equilibrium phase diagrams and the determi-nation of transport coefficients near equilibrium [9]. As atool for determining phase diagrams, the microevaporator

air flow

reservoirPDMSmembrane

eh

x=0 x=L x

(a)

h

e

L

(b)

Fig. 1. (a) Schematic side view of the micro-evaporator de-vice. It comprises a thin PDMS membrane (thickness e≈20µm)which allows water to pervaporate out of the lower channelwhich is filled by a solution (h≈100µm, L≈1cm). (b) Moredetails of the flow pattern: The pervaporating solvent inducesalso a concentration profile of trapped solute at the end ofthe channel, here in several dense phases as indicated by thedifferent patterns.

2 Michael Schindler, Armand Ajdari: Modeling phase behavior for quantifying micro-pervaporation experiments

resembles the functionality of the “phase chip” [10,11].However, the microevaporator allows to go one step fur-ther towards the controlled analysis of out-of-equilibriumsituations.

The quantitative interpretation of the experimentalobservations in the microevaporator, that is the extractionof the transport coefficients, requires a theoretical model.The observed quantities are spatio-temporal profiles of so-lute densities, the texture of occurring phases, and thegrowth dynamics of dense phases which present a mov-ing phase interface. The link between these observableson one side and the underlying thermodynamic equationof state, the transport coefficients of the mixture, and pos-sibly more detailed kinetic parameters on the other sidecan only be achieved by a sufficiently detailed theoreticalmodel. At the moment, no theoretical model exists for theconcentration process caused by pervaporation. We herestart to fill this gap.

A model for the phase behavior in microevaporationis subject to the following constraints: On one hand, themicroevaporator device can be applied to very differenttypes of solutes, such as colloids, surfactants and elec-trolytes. Consequently, a sufficiently general theoreticalmodeling is required which includes phase transitions andout-of-equilibrium phenomena namely metastable phasesand precursor phases. It must further take into accountthe influences of the driving conditions and of the de-sign parameters of the channels. On the other hand, themodel should turn the microevaporator into a quantitativetool, making the extraction of thermodynamic and kineticproperties of mixtures possible in a systematic and unam-biguous way. This aim requires a sufficiently simple modelwhich permits to approach several limits analytically, andwhich lends itself to a fast numerical solution of the equa-tions.

The model proposed here can thus be seen as a firstcompromise between the complexity of the subject, includ-ing out-of-equilibrium properties, and the requirement toallow a fast and direct comparison of its solutions withexperimental observations.

In order not to overload the description, we restrictit to binary mixtures (water plus a single solute) whichare described by two spatio-temporal fields, namely theconcentration of solute φ(x, t) and a velocity v(x, t) of themixture. The evolution of these two variables is governedby equations of the convection–diffusion type, which com-prise two transport coefficients, namely D(φ) for inter-dif-fusion of solute and water, and q(φ) for the pervaporation.Both coefficients are state-dependent, which is indicatedby their dependence on the concentration profile φ.

The structure of the paper is the following: In the restof the current section we give a rough and qualitative de-scription of the physics at work, exemplified for a typicalpattern of dense phases such as the one in Fig. 1b. Thisspecific example will be taken up again in the last sectionof the paper. Section 2 is dedicated to the detailed deriva-tion of the model proposed in this paper. It addressesin two distinct subsections the modeling of the mixturebehavior in the microfluidic channel and the pervapora-

tion through the membrane. All steps and assumptionsmade during the derivation of the final equations are ex-plained in the subsections. After the derivation of the gen-eral equations we continue to explore several limit cases.The section ends with a summary of the model equationsand the underlying assumptions. In Sec. 3 we focus onsituations without phase change, where we will study thesmooth concentration profile of solute in the dilute andthe dense limits. In the following Sec. 4 we then treat in-terfaces between different phases occurring in the mixture.The motion of a single moving interface is described first,with the aim to extract its dependence on the propertiesof the mixture. Then, we return to the introductory ex-ample and analyze the thicknesses of several phase slabs.This last analysis is done in a quasi-stationary approxima-tion. Both sections 3 and 4 employ numerically obtainedsolutions of the model equations and their analytical ap-proximations. We finish with a summary and an outlookin Sec. 5.

1.1 The experimental parameters

The experimental technique and the production of themicroevaporator device are detailed in Refs. [7–9]. In thetheoretical treatment in the present paper, we refer onlyto the most essential properties of the device. They aredisplayed in Fig. 1. We use three geometrical parameters,the length L of the PDMS membrane, its thickness e andthe height h of the channel containing the solution. Themixture enters the channel from a reservoir with a givensolute concentration φL. This concentration is fixed fromoutside by the experimenter.

The mixture is driven out of equilibrium by the appli-cation of an air flow above the PDMS membrane (thickarrows in Fig. 1a). The air flow guarantees that the perva-porated water is constantly removed from the membraneand that the permeation continues until the solute “dries”out. The out-of-equilibrium situation is thus limited bythe slow pervaporation. The experimenter’s second con-trol parameter is the humidity in the air flow. In the caseof nearly saturated wet air no pervaporation takes place,because the chemical potentials of water on both sidesof the PDMS membrane will become equal. In general,the two external “knobs,” namely the reservoir concentra-tion φL and the air humidity need not be constant butcan be changed according to a temporal protocol.

Due to the slow pervaporation process, a typical ex-periment takes hours to days. According to Ref. [7], theintrinsic time scales of the device are in the range of sec-onds (for the evaporation and the mechanical adjustmentof the membrane) up to fifteen minutes (for the diffusion).We may therefore hope that the mixture is not too “farfrom equilibrium,” which is a necessary requirement forthe following description.

During the evaporation, several quantities can be mea-sured, such as the total amount of mixture which passesthrough the device during the experiment. It gives someinformation on the amount of solute in the channel andon the rate of water pervaporation. Within the channel,

Michael Schindler, Armand Ajdari: Modeling phase behavior for quantifying micro-pervaporation experiments 3

more information on the mixture can be gathered: If thereare several phases in the mixture and if they can be dif-ferentiated e.g. by their texture, then the position of theinterfaces can be tracked as a function of time. This hasbeen done for the case of the surfactant AOT (docusatesodium salt) in water in Ref. [8] and for a salt solution inRef. [7]. More information is obtained if the full densityprofile can be measured as a function of time [7]. Thesetypes of observables are in the focus of the present theo-retical description.

1.2 An intuitive example

The central interest of the present theoretical descriptionis the precise functional forms of two coefficients, one ofwhich is the inter-diffusion coefficient D(φ) of solute andsolvent, the other one q(φ) quantifies the pervaporation ofsolvent through the PDMS membrane. Both coefficientsdo depend on the local concentration φ(x, t) of solute. Allthermodynamic and kinetic properties of the mixture areimplicitly contained in these two coefficients. In particular,they can be expressed in terms of the chemical potentialsand of the kinetic properties of the mixture, both of whichdepend on φ.

Suppose the situation in Fig. 1b, which has been ex-plored experimentally using the surfactant AOT (docusatesodium salt) in water [8]: The channel is filled by four dif-ferent phases of the mixture, which are ordered by theirsolute concentration. Three of them are dense, indicatedby different patterns in Fig. 1b, the fourth phase is dilute.The pervaporation of water, quantified by introducing thecoefficient q(φ) gives rise to a flow from the reservoir (longcurved arrows) which compensates the water loss at thePDMS membrane. On its way through the phases the wa-ter drags the solute with it. It thereby induces the concen-tration of solute, and, at higher and higher concentration,the appearance of the three dense phases which would beabsent without flow.

The final concentration profile is thus a dynamic equi-librium between convective and diffusive processes: Withineach phase slab the concentration gradient gives rise todiffusion which in turn tries to shrink the phases. Whilethe diffusion emerges together with the concentration gra-dients, the convective transport, or “drag” is reduced bythe solute becoming dense. The solvent has to permeatethrough the dense phases of solute before it reaches themembrane and pervaporates. The denser the solute, theless solvent may pass. Furthermore, the more the waterflow tries to compress, or to “squeeze” the dense phases,the harder it becomes to squeeze them even further. Oneor the other of the mentioned mechanisms may remind thereader of terms such as “osmotic compressibility,” “diffu-sion,” “mobility,” “permeability,” depending on the likingand the experience of the reader. In fact, all these effectsare linked to one another, and we will provide the connec-tion between some of these terms in Sec. 2.3. We therebyalso show the equivalence of these terms, in so far as theycan expressed one by the other. In our model we abandonthe use of terms such as mobility, osmotic compressibility

and permeation in favor of the diffusion coefficient D(φ)and the coefficient of pervaporation q(φ), which are ex-pressed in terms of the chemical potential of water and ofthe phenomenological coefficient for inter-diffusion.

An important observation in the situation of Fig. 1bis the following: As more and more solute is transportedfrom the reservoir, all three dense phases grow. It hasbeen observed experimentally that the two interior “sand-wiched” phases apparently grow at constant thickness.They loose as much solute as they gain, while the last,the most dense phase continues to grow. Keeping in mindthe dynamic equilibrium between convective and diffusiveprocesses within each phase we therefore find in the thick-nesses of these middle phases a fingerprint of the diffusioncoefficient and of the pervaporation coefficient.

One may now ask whether a large diffusion coefficientwill finally lead to a thinner or to a thicker slab, as com-pared to one with a smaller diffusion coefficient? On onehand, the diffusion coefficient scales the gradients of theconcentration profile which try to dissolve the phases. Itshould thus lead to small thicknesses of the quencheddense phases. On the other hand, for fixed values of theconcentration at the phase interfaces, a large diffusion co-efficient will render the profiles flat and therefore lead tothick phases. This may appear paradoxical on a first sight,but it only reflects the same feedback mechanism as be-tween flow of solvent and permeability. The resolution isthat we really need the full concentration profile and thevelocity field in order to understand the phase thicknesses.

2 Detailed derivation of the model

In this section we derive a set of equations which modelthe evolution of a binary system, reduced to one spatialdimension x along the evaporation channel. As variables ofstate we use a concentration φ(x, t) of solute and a mixturevelocity v0(x, t). Special focus will be put on their precisemeaning in the evolution equation that will be derived inthe following sections,

∂xv0 = −νwq(φ), (1)

∂tφ = −∂x[φv0 −D(φ)∂xφ

]. (2)

Here, νw denotes the specific mass of water; ∂t and ∂x arepartial derivatives with respect to time and space. Thephysical content of the equations is the independent bal-ance of solvent and solute mass. The variables are chosensuch that q(φ) reflects the loss of solvent (water) via per-vaporation, and D(φ) is a state-dependent inter-diffusioncoefficient, which contains information on the thermody-namic and dynamic properties of the mixture. The expres-sions for D(φ) and q(φ) will be provided by equations (31)and (46) below.

2.1 Diffusion model for the binary solution

The aim here is to state clearly all underlying assumptionswhich lead to the model equations (1) and (2), starting


from mass conservation. This procedure will allow to pointout the restrictions of the model. We employ the theory oflinear out-of-equilibrium thermodynamics as summarizedin the book by de Groot and Mazur [12]. We try to adoptthe notation used there, also for an easier comparison withthe results by Peppin et al. [4] for the link to sedimenta-tion, ultra-filtration, and reverse osmosis.

2.1.1 Balance equations for binary mixtures

The balance equations for the mass densities ρs(r, t) andρw(r, t) of solute (index s) and solvent (index w) read

∂tρk = −div(jk), with k ∈ {s, w}. (3)

The mass current densities jk are split into an advectedand a diffusive part,

js = ρsvs = ρsv + Js, (4)jw = ρwvw = ρwv + Jw. (5)

The Ji denote the “diffusive currents”, simply defined bythe difference ρi(vi−v). A common choice for the mixturevelocity is the mass-average of the individual constituentvelocities vk, weighted by the mass fractions ck,

v :=∑

i

civi (6)

with ci :=ρiρ

and ρ :=∑

k

ρk. (7)

This choice makes the two diffusive currents opposite toeach other,

Jw = −Js. (8)Formally, the mass densities are taken to be functions

of the local thermodynamic variables of state as ρi(T, P, cs).The use of local temperature and pressure fields showsthat here and in the following we will assume local ther-modynamic equilibrium. The evolution equations (3) areaccompanied by the following constraint, which is an im-mediate consequence of the chemical potentials being in-tensive quantities (or of the Euler relation),

1 = νsρs + νwρw. (9)

The specific volumes per mass, νi, are defined as the deriva-tives of the chemical potentials per mass, γi(T, P, cs), withrespect to the pressure,

νi(T, P, cs) :=∂γi∂P

∣∣∣∣T,cs

. (10)

γi(T, P, cs) :=µi(T, P, cs)

mi. (11)

Here, µi is the chemical potential per molecule, and mithe mass of such a molecule. The derivative in Eq. (10)equals the change of volume induced by adding an in-finitesimal amount of mass of type i upon keeping temper-ature, pressure and all remaining masses constant. Equa-tion (9) can be used to introduce the volume fractions

φi(T, P, cs) := νiρi. Below, equation (9) will also be thepoint of entrance for the notion of simple mixtures and ofincompressibility of the constituents. Apart from Eq. (9),another consequence of the intensivity of chemical poten-tials is the following relation between the derivatives of thetwo chemical potentials with respect to the mass fractionof solvent, cs,

cs∂γs∂cs

∣∣∣∣T,P

= −(1− cs)∂γw∂cs

∣∣∣∣T,P

. (12)

This equation establishes the standard logarithmic behav-ior of the chemical potentials in the limits cs → 0 andcs → 1.

Together, equations (9) and (12) reduce the number ofrelevant chemical potentials to one, as one can always beexpressed by the other. This fact is important for the num-ber of kinetic coefficients within the theory of linear non-equilibrium thermodynamics. Applying this framework inthe present description, the diffusive current Js is takento be proportional to the gradients of chemical potentials.The proportionality factor is called the phenomenologicalcoefficient L. Using the relation of Gibbs-Duhem, only thedifference of chemical potentials can occur, and we receivefor the diffusive current

Js = −L∇(γs − γw). (13)

We do not present the details of the derivation of thisequation here, as it implies a rather tedious considerationof all possible symmetries in the entropy production. In-stead, we refer to Eq. (IV.15) in the book by de Grootand Mazur [12] or to the summary in the paper by Pep-pin et al. [4]. Note that external forces such as gravity can-cel out and do not appear in equation (13). This fact isdue to the chemical potentials being defined as energy permass rather than per number of particles. Furthermore, wehave omitted a term proportional to the temperature gra-dient, as from now on we take the temperature as uniformand constant. We will omit it also from the argument listsof thermodynamic functions. The phenomenological coef-ficient L may in principle depend on all thermodynamicvariables of state as well as on their derivatives, and evenon the mixture velocity v and its derivatives. However,if the assumption of local thermodynamic equilibrium iswell satisfied, the dependence on the mixture velocity fieldshould be negligible, such that we use a function L(P, cs)for the moment. The chemical potentials are also func-tions of the two thermodynamic parameters: γs(P, cs) andγw(P, cs). This functional dependence allows to separatethe influence of pressure gradients from concentration gra-dients in Eq. (13),

Js = −L[(νs − νw)∇P +

∂(γs − γw)∂cs

∣∣∣T,P

∇cs]. (14)

With the aid of relation (12) the diffusive second term canbe rephrased using either of the chemical potentials,

∂(γs − γw)∂cs

∣∣∣T,P

=1

1− cs∂γs∂cs

∣∣∣T,P

= − 1cs

∂γw∂cs

∣∣∣T,P

. (15)


Equation (14) has been used by Peppin et al. [4, Eq. (28)]to demonstrate that Fick’s and Darcy’s laws are limitingcases of the same equation. Below, we will use it to de-fine the diffusion coefficient and to make the connectionbetween the different methods micro-pervaporation, sedi-mentation, and ultra-filtration.

2.1.2 Reduction to 1D

We shall now disregard the details of the flow pattern inthe channel and reduce the three-dimensional evolutionequations (3) to one dimension. We do this by averagingover the two spatial dimensions perpendicular to the chan-nel axis, as depicted in Fig. 2. For a given uniform height hand width b of the channel we obtain from integration overa slice Ω with area |Ω| = bh the one-dimensional counter-parts of (3),

∂tρs = −∂xjs (16)∂tρw = −∂xjw − q. (17)

The new densities and currents turn out to be

ρk(x, t) :=1|Ω|

∫Ω

dy dz ρk(x, y, z, t), (18)

jk(x, t) :=1|Ω|

∫Ω

dy dz ex · jk(x, y, z, t). (19)

The water loss due to pervaporation through the PDMSmembrane is all included in q(x, t). It represents the inte-gral of water current in normal direction over the bound-ary of the channel,

q(x, t) :=1|Ω|

∫∂Ω

N · jw(x, y, z, t) (20)

≈ 1b h

∫ b0

N · jw(x, y, h, t) dy. (21)

The pervaporation has a dominant contribution from theupper boundary where the thin layer of PDMS allows asteady water pervaporation. In addition to equations (16)and (17) we expect an averaged version of constraint (9)to hold. In the same manner as above in Eqs. (4) and (5)we may subdivide the total currents into advective anddiffusive parts.

2.1.3 Incompressibility and simplicity of mixtures

We now assume both constituents of the mixture to beindividually incompressible, having constant specific vol-umes νi. This assumption will allow to reduce the set ofthree equations (16), (17), and (9) to only two equations.The assumption, however, is quite strong as it implies themixture to be both incompressible and simple,

∂1/ρ∂P

∣∣∣∣T,cs

= 0,∂1/ρ∂cs

∣∣∣∣T,P

= νs − νw = const. (22)

x

y

z

b

h

jw = j̄w

js, jw

Ω

xjs(x, t), jw(x, t)

q(x, t)

(a)

(b)

Fig. 2. Visualization of the averaging used for reducing thebalance equations to one dimension. (a) The area of integra-tion Ω in three dimensions. (b) The one-dimensional represen-tation.

The term simple mixture here means that the total massdensity is a linear function of the mass fraction. It implieslinear relations between mass density, mass fraction andvolume fraction. We will therefore use the terms “density”,“fraction” and “concentration” interchangeably. Simplemixtures are for example glycerol in water [13] or rigidspheres in an incompressible solvent of much smaller par-ticles. The constraint (9) allows to write one density as afunction of the other, ρw = ρw(ρs). The evolution equa-tion (17) for ρw(ρs) then becomes(

ρw(ρs)−∂ρw∂ρs

ρs

)∂xv + ∂xJw −

∂ρw∂ρs

∂xJs = −q. (23)

This equation can be understood as a differential equa-tion for the velocity field, similar to the one we are look-ing for (1). However, the occurrence of the diffusive cur-rents Js and Jw makes it unfavorably dependent on theconcentration variable, see equations (14) and (8). A moreconvenient equation is obtained by using from the begin-ning the volume-averaged mixture velocity

v0 :=∑

i

φivi (24)

instead of the mass-averaged one. The total currents arethen written with modified diffusive currents,

jk = ρkv0 + J0k , k ∈ {s, w}. (25)

With the volume-averaged mixture velocity, and by set-ting the specific volumes constant, one obtains as relationbetween the new diffusive currents,

J0w = −νsνw

J0s (26)

instead of equation (8). This property makes the diffusivecurrents vanish from the differential equation for v0(x, t),which now comes close to the form we seek (1),

∂xv0 = −νwq(P, φ). (27)

In terms of the volume-averaged mixture velocity, andwith constant νs, the volume fraction of solute evolves as

∂tφs = −∂x[φsv

0 +(νwφs + νs(1−φs)

)Js

]. (28)


Or, omitting the index s and collecting also equations (14)and (15), we finally arrive at

∂tφ = −∂x[φv0 −K(P, φ)∂xP −D(P, φ)∂xφ

], (29)

with the shortcuts

K(P, φ) = [νs + φ(νw − νs)] L(P, φ) (νs − νw), (30)

D(P, φ) = − [νs + φ(νw − νs)]2

νwφL(P, φ) ∂γw

∂φ

∣∣∣T,P

. (31)

Equations (27) and (29) with the coefficients q and Dfrom (20) and (31) come close to the model we announcedin the introduction. They nicely separate the effect of per-vaporation, which is all contained in the coefficient q, fromthe bulk thermodynamic and kinetic properties of the mix-ture. These properties are implicitly contained in the func-tional dependence of L and γw on P and on φ.

2.1.4 The role of the pressure gradient

With equations (27) and (29) we have found two equa-tions for yet three variables v0, φ, and P . We thus haveto seek for a closure of the above set of equations. Thecontribution by the pressure requires some further atten-tion. The pressure occurs as an argument in several func-tions, but also its gradient adds a contribution to the dif-fusive current in Eq. (29). This gradient appears also inthe (compressible) Stokes equation. In principle, it wouldhave been possible to assume local mechanical equilibriumalready in the 3D theory, such as the x-component of thecompressible Stokes equation,

0 = −∂xP + η∆vx + (λ + η/3)∂x div v (32)

and then reduce it to one dimension together with fit-ting boundary conditions. The shear and bulk viscositiesare denoted by η and λ, respectively. However, this proce-dure raises fundamental difficulties: We showed above thatthe natural description of pervaporation of incompressiblemixtures involves the volume-averaged velocity v0. Thenatural formulation of the Stokes equation uses the mass-averaged velocity. Only for the special case νs = νw bothvelocities do coincide—but in this very case the term with∂xP vanishes due to K = 0. When expressing v by v0, theterm with ∂xP in Eq. (29) will contribute to both otherterms in the expression for the diffusive current in (29).We do even expect a coupling of velocity and density vari-ables, as well as a second derivative ∂2xφ due to the secondderivatives in the Stokes equation (32).

The precise analysis of this splitting is beyond thescope of the present paper. Here, we have to restrict thevalidity of the model to cases where the influence of thepressure gradient is negligible. An estimation [14] indicatesthat this restriction is indeed weak: In the dilute limit,using values from the experiment [7], the term with thepressure gradient in Eq. (29) is several magnitudes smallerthan the diffusion term with the concentration gradient.In what follows we therefore use the simplistic model with

a homogeneous pressure variable (which we omit in thenotation from now on):

∂xv0 = −νwq(φ), (33)

∂tφ = −∂x[φv0 −D(φ)∂xφ

]. (34)

with the evaporation coefficient q(φ) and the diffusion co-efficient D(φ), both depending only on the volume frac-tion φ.

As we take the microevaporation channels to be hor-izontal, our assumption of setting the pressure gradientto zero coincides with the assumptions employed by Pep-pin et al. [4] and also in the context of sedimentation [3]where the pressure gradient balances gravitational forces.It therefore opens the possibility to consistently comparewith their model equations, see Sec. 2.3.

2.1.5 Boundary conditions

Equations (33) and (34) are to be accompanied by anappropriate number of boundary conditions. For the hy-perbolic differential equation (33) we require one, for theparabolic equation (34) we need two boundary conditions.At the end of the channel, both the mixture velocity andthe total solute current vanish,

v0(0, t) = 0, (35)∂xφ(0, t) = 0. (36)

As a third boundary condition we use a fixed concentra-tion at the inlet of the channel, reflecting the concentra-tion in the reservoir,

φ(L, t) = φL. (37)

The model equations could equally handle a time-depen-dent reservoir concentration φL(t). In a setup with a fixedtotal amount of solute in the channel, it is mathematicallyalso possible to replace the latter boundary condition byan integral condition setting

∫ L0

φ(x, t) dx to a given value.

2.1.6 Moving interfaces between phases

In the introductory example we mentioned that the po-sitions of phase interfaces and their velocities are conve-nient observables in the pervaporation experiments. Suchexperimental observations have been reported in Ref. [7]and in more detail in Ref. [8] for a solution of the surfac-tant AOT (docusate sodium salt) in water. The essentialobservation in the latter reference is the decrease of theinterface velocity with time, which will be investigated inmore detail below in section 4.1. A quantitative analysisof this decrease can give insight into the mechanisms atwork, especially on the relative importance of diffusionand advection caused by pervaporation.

Throughout this paper, we model the interfaces withinthe framework of local equilibrium thermodynamics, which

Michael Schindler, Armand Ajdari: Modeling phase behavior for quantifying micro-pervaporation experiments 7γ

s−

γw

0 φ(r) φ(`) 1

0

φ(r)

φ(`)

1

X(t) x

Fig. 3. Sketch of the modeling of sharp interfaces betweendifferent phases. The concentration values left and right of theinterface are fixed by a Maxwell construction on the chemicalpotential (upper panel), the two regions shaded in gray haveequal area.

implies that the interfaces are infinitely sharp and that foreach such interface there are two unique concentration val-ues at its left-hand and its right-hand sides. These valuesare determined by a Maxwell construction on the localchemical potential function. Figure 3 sketches this con-struction and the resulting concentration profile aroundthe interface. The fixed concentration variables serve astwo Dirichlet boundary conditions for the evolution equa-tion (34),

φ(X(`), t

)= φ(`), φ

(X(r), t

)= φ(r). (38)

The superscripts (`) and (r) indicate the left and rightlimits towards the interface position X(t).

The conservation of solute mass, which is valid alsoacross the interface, provides us with a continuity condi-tion at the interface. If the boundary does not move, thesolute currents at the left-hand and at the right-hand sidesof the interface must coincide. Any possible difference be-tween these incoming and outgoing currents must lead toa growth of one phase and thus to the movement of the in-terface position X(t). Its velocity times the concentrationjump at the surface must equal the difference between thecurrents,

Ẋ(t) =js

(X(`), t

)− js

(X(r), t

)φ(`) − φ(r)

. (39)

We further require continuity conditions for the veloc-ity field. As we did in the derivation of the bulk equations,we also here want the mixture velocity to reflect only theinfluence of the pervaporation. As we do not expect the

pervaporation to diverge near phase interfaces, the veloc-ity field should be continuous across such an interface,

v0(X(`), t) = v0(X(r), t). (40)

This continuity equation is consistent with the conserva-tion of solvent mass across the interface.

2.2 Pervaporation through the PDMS membrane

The pervaporation of water through the PDMS layer canbe described on the same level as the advective and diffu-sive currents in the evaporation channel. We now focus onthe binary mixture of water and PDMS, as solutes cannotenter the PDMS domain. The notation is such that theproperties inside the PDMS membrane will be denoted byan overbar. As we did above, we also here split the totalcurrent density j̄w of water into convective and diffusiveterms. This time it is more convenient to take the velocityof PDMS, which is zero, as the reference velocity of themixture. The mass current density of water in the layerthen becomes

j̄w =1

1− c̄wJ̄w, (41)

where c̄w is the mass fraction of water inside the layer,and J̄w is the corresponding diffusive current. In com-plete analogy to equation (13), the diffusive current canbe written in terms of a phenomenological coefficient L̄,

j̄w = −L̄(P, c̄w)1− c̄w

∇(γ̄w − γ̄PD). (42)

with the chemical potential of PDMS denoted by γ̄PD. Weunderstand the pervaporation as a diffusive process andagain omit the pressure-dependence of the chemical po-tentials. Then, the water current is proportional to thegradient of water concentration. The diffusion process isfurther assumed to be sufficiently established that the wa-ter current does not vary within the membrane. In thedilute limit, where the chemical potential exhibits a loga-rithmic behavior,

γ̄w(T, P, c̄w) = γ̄∗w(T, P ) +kT

mwln c̄w, (43)

the uniform water current then corresponds to a constantgradient of water concentration inside the membrane. Thewater current through the membrane is then proportionalto the concentration difference across the membrane ofthickness e,

N · j̄w =c̄w(z=h+e)− c̄w(z=h)

elim

c̄w→0

L̄(P, c̄w)(1− c̄w)2

∂γ̄w∂c̄w

.

(44)To make the connection with the conditions in the ad-

jacent channels of the mixture and of the air flow, weassume that the chemical potentials at both sides of bothboundaries of the PDMS membrane have matching values,

γ̄w(z=h) = γw and γ̄w(z=h+e) = γairw . (45)


The evaporation coefficient q can now be expressed as afunction of the chemical potential γw(φ) in the mixturechannel. From equations (44), (43), and its definition (21)we finally obtain the following dependence of the pervap-oration coefficient on the volume fraction,

q(φ) =1νw

D̄

eh

[exp

{mwkT

γw(φ)}− exp

{mwkT

γairw

}]. (46)

We have collected several of the above constants in theshortcut D̄,

D̄ = νw exp{−mw

kTγ̄∗w

}lim

c̄w→0L̄ ∂γ̄w

∂c̄w. (47)

Expression (46) contains the further assumption that thechemical potential directly at the interface between mix-ture channel and PDMS layer is the same as the averagechemical potential in the mixture which we obtained fromthe reduction to one spatial dimension. In the presentframework of a one-dimensional description, there is noalternative to this approximation.

At the present point we have completed the deriva-tion of the model, consisting of the two differential equa-tions (33) and (34) for the volume fraction φ and the vol-ume-averaged velocity field v0, together with the concen-tration-dependent diffusion coefficient (31) and the evap-oration coefficient (46).

2.3 Connection with sedimentation and ultra-filtration

We now make the connection between the two coefficientsD(φ) and q(φ) and other pairs of coefficients. This will al-low to compare different techniques of measuring chemicalpotentials and phenomenological coefficients and to verifythe consistency of the outcomes. We will not make use ofthese connections in the present paper. The equations inthis section can rather be seen as a convenient referencefor the reader working on different techniques. However,the comparisons give some more intuition on the variablesand coefficients which are used. Above, in equations (31)and (46) we have already found the link to the pair γ(φ)and L(φ). We continue by expressing also the coefficientsof sedimentation and of permeability in terms of γ(φ) andL(φ). Each of these two coefficients forms together withthe diffusion coefficient a pair which allows to determineγ(φ) and L(φ) and thus to make the link to our originalcoefficients D(φ) and q(φ).

2.3.1 Sedimentation factor and osmotic compressibilityfactor

In the context of sedimentation of colloids, Russel, Sav-ille, and Schowalter [3] express the thermodynamic anddynamic properties in terms of the sedimentation factorKRSS(φ) and the osmotic compressibility factor ZRSS(φ).Both can be introduced by rewriting Eq. (29) as follows [3,compare Eq. 12.5.6],

∂tφ = −div(φv0 + U0φKRSS(φ)−D(φ)∇φ

)(48)

with the diffusion coefficient

D(φ) = D0KRSS(φ)d

dφ

[φZRSS(φ)

](49)

and with the shortcuts

D0 :=kT

6πη a, U0 := −

2a2

9η

( 1νw

− 1νs

)g (50)

with η denoting the viscosity of pure solvent, a the radiusof a colloidal sphere, and g the acceleration by gravity. Thevolume-averaged velocity in Eq. (48) vanishes by construc-tion of the sedimentation experiment in a closed container.In the comparison with Eq. (29) the remaining two termsthus lead us to the sedimentation coefficient KRSS givenas

KRSS(φ) =|∇P |ρg

9η2a2

ν2s

(1 + φ

νw−νsνs

)2L(φ)φ

. (51)

The osmotic compressibility factor, originally defined bythe osmotic pressure of solute Π(φ) can equivalently beexpressed in terms of the chemical potential of water,

ZRSS(φ) =3

4π a3Π(φ)φkT

=3

4π a3γw(0)− γw(φ)

νwφkT. (52)

Peppin, Elliott, and Worster [4] define the sedimenta-tion coefficient SPEW in a marginally different way. Theirdefinition leads to the following variant of the transportequation (48) (see Eq. (32) of Ref. [4]),

∂tφ = −div(φv0 + gφKPEW(φ)−D(φ)∇φ

)(53)

In their treatment, the pressure gradient is assumed tobalance the force density by gravitation, ∇P = ρg. In ournotation of Eq. (30), their sedimentation coefficient reads

SPEW(φ) =K(φ)φνw

(1 + φ

νw − νsνs

). (54)

2.3.2 Permeability

For the coefficient of permeability kPEW Peppin et al. givethe result (Eq. (37) of Ref. [4])

kPEW

η=L(φ)νsφ2ν2w

(1 + φ

νw − νsνs

)2. (55)

Except for the dilute limit, the phenomenological coef-ficient L(φ) can be understood as a direct measure ofthe permeability of the solute with respect to the solvent.This will be useful below when choosing a model functionfor L(φ). In particular, Eq. (55) helps choosing the valueof L(1), which has influence on the qualitative behaviorof the solutions, see the discussion of Fig. 8. In the oppo-site limit, Eq. (51) indicates that L(φ) ∝ φ for small φ.This last requirement leads to a finite non-zero diffusioncoefficient in the dilute limit, see below in Fig. 13.


2.4 Summary of the model equations, and theirpreparation for a numerical implementation

We here summarize the model equations (1), (2), (31),and (46) for their further use in the following sections. Werepeat their boundary conditions and the condition forthe interface movement. We also reformulate the modelequations in such a way that they may be discretized andimplemented more easily. The dimensionless equations ofevolution (1) and (2) read

∂x′v0′ = −q′(φ), (56)

∂t′φ = −∂x′j′ = −∂x′[v0′φ− D

′(φ)Pe

∂x′φ]. (57)

They describe the evolution of an incompressible binarymixture in one effective spatial dimension under the fur-ther assumption of homogeneous temperature and pres-sure. The variables carrying a prime are made dimension-less by the following scales for length (x), time (t) andthe mixture velocity (v0). The volume fraction φ needs nofurther rescaling. The scales have been chosen such thatall dimensionless variables are of the order unity as far aspossible,

x = Lx′, t = t′/(νwq(0)

), v0 = Lνwq(0) v0′, (58)

q(φ) = q(0) q′(φ), D(φ) = D(0)D′(φ), (59)

Pe =L2νwq(0)

D(0). (60)

Here, the Peclet number Pe is a global Peclet number ofthe system. The scales for the diffusion coefficient and thecoefficient of evaporation, which are simply the coefficientsevaluated at zero concentration density (D(0) and q(0))are chosen such that Pe presents an upper bound for thetrue (local) Peclet number Pe loc(x, t) which can only bedetermined a posteriori from the solution. In numericalsolutions we have found the local Peclet number to bealways maximal at the entrance of the channel and unityat the end. However, it does not need to be monotonous.

The dimensionless version q′(φ) of the pervaporationcoefficient is, according to the expression we found in (46),

q′(φ) =exp

{mwkT

(γw(φ)− γairw

)}− 1

exp{

mwkT

(γw(0)− γairw

)}− 1

. (61)

It depends on the volume fraction φ only via the chemicalpotential of water γw(φ), all other parameters are con-stants or external driving parameters. The dimensionlessdiffusion coefficient reads, according to equation (31),

D′(φ) =[1 + φ(νw/νs − 1)]2

φ

L(φ) ∂φγw(φ)∂φγw(0) lim

ϕ→0L(ϕ)/ϕ

. (62)

It obtains it dependence on φ from γw(φ) and from thephenomenological coefficient L(φ) for inter-diffusion.

The boundary conditions (35), (36), (37) read in theirrescaled version,

v0′(x′, t′) = 0 at x′ = 0, (63)

∂x′φ(x′, t′) = 0 at x′ = 0, (64)

φ(x′, t′) = φL at x′ = 1, (65)

as well as the boundary/continuity conditions at movingphase interfaces (38) and (40). The velocity (39) of aninterface becomes

∂t′X′(t) = v0′(X ′, t)

+−D(φ(`))∂x′φ(X ′(`), t) + D(φ(r))∂x′φ(X ′(r), t)

Pe [φ(`) − φ(r)](66)

The following sections will require solutions for themodel equations, partly for arbitrary functions q(φ) andD(φ). Such solutions are available only by numerical meth-ods. For the numerical discretization we transform equa-tions (56) and (57) into time-dependent systems of co-ordinates ξ(x′, t′) which range from 0 to 1 in each phase.x′(ξ, t′) interpolates linearly between the (moving) bound-aries,

x′(ξ, t′) = X ′(`)(t′) + ξ[X ′(`)(t′)−X ′(r)(t′)

](67)

Here, X ′(`) and X ′(r) are the positions left and right ofthe phase in question. We further use the shortcut ∆X ′ :=X ′(r)−X ′(`). The coordinate transformation adds a secondadvection-like term with velocity ∂tx′(ξ, t′) to the evolu-tion equation, such that we obtain for the model equationsin each phase,

∂ξv0′(ξ, t′) = −∆X ′q′(φ), (68)

∂t′φ(ξ, t′) =∂tx

′(ξ, t′)∆X ′

∂ξφ− ∂ξ[

v0′φ

∆X ′− D

′(φ)Pe (∆X ′)2

∂ξφ

].

(69)

Equations (66), (68), and (69) are then solved using themethod of lines, with a finite-volume discretization andlinear interpolation of φ in space. The time-stepping isdone by a standard explicit fourth-oder Runge–Kutta al-gorithm [15].

3 Prototype solutions: a single phase

In this and in the following sections we discuss severaldifferent solutions of the above model equations, whichare (1) and (2), completed by the coefficient functions q(φ)

v0(x, t)φ(x, t)

φ

x

φ(x, t)

Fig. 4. Sketch of the typical single-phase situation describedin Sec. 3

.


and D(φ) from Eqs. (31) and (46)—Or, in their dimen-sionless versions, Eqs. (68), (69), (62), and (61). We herechoose two functions q(φ) and D(φ) and solve the equa-tions numerically, accompanied by analytical approxima-tions. The aim is to display typical solutions and to ex-plore the variety of solutions by using different functionsq(φ) and D(φ). These solutions are meant to guide fu-ture experiments and their interpretation, where the finaltarget is to employ the model for extracting the functionsq(φ) and D(φ) from experimental data and to invert themto obtain the chemical potentials and the phenomenologi-cal coefficients. This will then require to solve the inverseproblem of what we do in the present paper. We startwith the description of three typical situations with a sin-gle phase and smooth concentration profiles, see also Fig. 4for a sketch.

3.1 Solvent only

For setups containing only water and no solutes, the waterloss is the constant q(0), which has been measured experi-mentally by the total consumption of water per time [8]. Asituation with water only therefore permits to identify theconstants in expression (46) for the pervaporation coeffi-cient. A corresponding series of experiments with varyingparameters can be used to verify that the assumptions wemade for (46) are indeed fulfilled. Possible variations caneither be geometrical such as changing the length L orthe height h of the channel, or the membrane thickness e.Other variations concern the driving parameters, such asthe solute concentration φL in the reservoir, or the princi-pal control parameter during the experiment, which is theexternal chemical potential γairw of water in the air flow.This chemical potential is controlled by the humidity ofthe air circulating through the top channel. Equation (46)predicts a linear dependence of the pervaporation on thespecific humidity cairw , with an offset,

q(0) =1νw

D̄

eh

[exp

{mwkT

γw(0)}− cairw exp

{mwkT

γair∗w

}],

(70)where the slope of the line and the offset may in prin-ciple depend on the pressure. Quantitative informationon this dependence would thus offer a validation of thecurrent model and yield insight into the thermodynamicmechanisms inside the PDMS layer. Previous studies [16]assumed that the pervaporation is driven by a gradient ofconcentration and not of pressure, as we did it in Sec. 2.2.

3.2 Dilute solutions

The situation with dilute solutions, in which both theevaporation coefficient q and the diffusion coefficient D areessentially constant, has already been explored in Ref. [7].Two different regimes were identified in the solution, first ahyperbolic ramp corresponding to a steady uniform currentat high Peclet numbers, and second a Gaussian growinglinearly in time, corresponding to the accumulation zone

Pe loc � 1, ∂tφ ≈ 0

Pe loc & 1, j ≈ 0q(φ) = q(0)

D(φ) = D(0)

time

0.001

0.002

0.005

0.01

0.02

0.05

0.1

φ(x

′ ,t′)

0 0.2 0.4 0.6 0.8 1x′

Fig. 5. Growth of a dilute solute density according to the tworegimes “Gaussian” and “hyperbolic ramp.” The plot showsthe numerical solution with parameters (q′ = 1, D′ = 1, Pe =100, solid curves) together with the analytical approximationfrom Eqs. (71) and (72) (dashed curves), see also Ref. [7].

at Peclet number equal one. Both regimes can be identi-fied in Fig. 5 where a numerical solution of equations (1)and (2) is depicted together with the analytical approxi-mation from Ref. [7]. These approximations are given bythe solution of

j′(1) = j′(x′) ≈ v0′φ = −q′x′φ

=⇒ φ(x′) ≈ −j′(1)q′x′

(71)

for the hyperbolic ramp, and by the solution of

0 = j′(x′) = −q′x′φ− D′

Pe∂x′φ (72)

for the Gaussian profile. The width of the Gaussian humpthus scales as L/

√Pe. The values given in Ref. [7] indicate

a Peclet number of 100.

3.3 From dilute to dense single-phase solutions

The mere model equations (1) and (2) include no mech-anism to prevent unphysical values φ(x, t) > 1. We thushave to ask what mechanisms the model provides to en-sure φ ≤ 1 in the dense limit when φ approaches unity. Fordense solutions, both coefficients q(φ) and D(φ) exhibitnon-trivial dependences on φ which provide two indepen-dent physical mechanisms to keep φ below unity. Thesemechanisms are described in the following paragraphs.Numerical examples for both are depicted in Fig. 6.

3.3.1 Vanishing pervaporation

One mechanism for keeping the volume fraction boundedis governed by the pervaporation coefficient q(φ). Equa-tion (46) for this coefficient implies an equilibrium value φeqat which the pervaporation vanishes. If the concentrationexceeds this value, the pervaporation direction is inverted,


which stabilizes φ around the equilibrium value, see Fig. 7.In combination with the boundary condition (35) for thevelocity field we find a flattening of the concentration pro-file at φ = φeq, at least in an interval starting with x = 0.In this interval, the velocity remains approximately zero.A numerical example of such a profile is given in Fig. 6b.

3.3.2 Diverging diffusion coefficient

The other mechanism leading to volume fractions φ(x, t)which strictly cannot exceed unity is that the chemical po-tential of water diverges at φ → 1. In this case, the solventbecomes dilute in the solute, leading to the standard log-arithmic form of the chemical potential of water in the so-lute concentration (cf. Eq. (12) or Eq. (43)). Together withthe chemical potential also its derivative diverges in thelimit φ → 1. In other words, the osmotic compressibilityof the solute phase diverges as well: The more the solute isconcentrated, the more difficult it becomes to concentrateit even further—and it cannot be concentrated beyond avalue φtd which is given by thermodynamics as the pointwhere the chemical potential diverges. φtd will be unity inmost cases.

Mathematically speaking, the diffusive currents tendto flatten the spatial concentration profile. In the limitof an infinite diffusion coefficient—which is proportionalto the derivative of the chemical potentials—the volumefraction φ(x, t) become arbitrarily flat. It will then notgrow beyond the value φtd at which the diffusion coeffi-cient diverges. A numerical example for such a situationis depicted in Fig. 6a. This flattening poses a boundingmechanism which is absent if the diffusion coefficient staysfinite or even vanishes.

It depends on the values of φeq and φtd which of thetwo mechanisms takes place first in a given mixture. Inthe present description, however, which is based on ther-modynamics, the logarithmic divergence must be locatedat φtd = 1 which is always larger than the equilibriumvalue φeq. In a mixture of small solutes, which are ofmolecular size, the value φtd = 1 should indeed be ap-proachable. It is not approachable in a system of rigidspheres in a much smaller solvent, where there is a close-packing limit φtd < 1. In such systems with large soluteparticles, the solvent can still pass through the holes be-tween the spheres, and the pervaporation never ceases [17].We acknowledge that such a system cannot be describedby our model—a deficiency which stems from the treat-ment of the pressure, and which is therefore shared byany other thermodynamic model that assumes mechani-cal equilibrium in form of the pressure gradient balancingall external forces. In the following, we will therefore fo-cus the discussion on the case where the pervaporationvanishes before the diffusion coefficient diverges,

φeq < φtd. (73)

For our treatment of q and D, this implies that the di-vergence of D is not reached, because q vanishes first. Wethus expect the effect of drying out at least for very high

densities. (Strictly speaking, this is true only for non-di-verging L(φ).) In dense solutions, especially at the end ofthe pervaporation channel, we thus expect a flat concen-tration profile with a nearly vanishing pervaporation anda large but finite diffusion coefficient.

We have to add a technical point here, as the diffusioncoefficient depends not only on the osmotic compressibilitybut also on the phenomenological coefficient L(φ). Thereare three different possibilities, which are all sketched inFig. 8: If L(1) is non-zero, the diffusion coefficient divergesas φ → 1. This is the case which we have discusses sofar. In the cases with vanishing phenomenological coeffi-cient L(1) = 0 the diffusion coefficient remains finite oreventually vanishes.

4 Prototype solutions: several phases

After the treatment of smooth concentration profiles suchas the one in Fig. 4, we now continue with situations com-prising several distinct thermodynamic phases, see Figs. 9and 12. As more and more solute is transported from thereservoir into the channel the concentration rises and maylead to a phase transition. The position of such a phaseinterface and its movement in time is a convenient observ-able in experiments. The velocity of such an interface isgoverned both by the advective and the diffusive currentsaround it, and it thus presents a fingerprint of the mix-ture properties which have here been reduced to the twocoefficients D(φ) and q(φ). It does not provide as much in-formation on these coefficients as for example the full con-centration profile. The interesting question is how muchinformation on the coefficients can be extracted from thevelocity of an interface as a function of time.

In addition to the model Eqs. (1) and (2) we nowtake the positions Xi(t) of the interfaces into account asunknown variables. This type of problem is known as afree-surface or Stefan problem. The interface movement isgoverned by Eq. (39), or in its dimensionless variant byEq. (66). The typical situation with one moving interfaceis sketched in Fig. 9.

4.1 Two phases: Moving phase interfaces

To start with, we repeat the argument in Ref. [8] on theexperiment of solution of AOT in water. It led to an expo-nential decrease of the interface velocity, or equivalentlyto an interface position of the form

X(t)L

= 1− exp{−φLνwq(0)

∆φ(t− t0)

}(74)

with an effective nucleation time t0. This result is based onseveral assumptions: first, that all incoming solute currentis transferred into the interface movement. This meansthat the total solute current j(x, t) is a step-function, be-ing zero in the dense phase (left-hand side) and a non-zero constant in the dilute phase (right-hand side). Equa-tion (74) assumes further that the velocity profile vanishes


q′(φ) = 1D′(φ) = 1/(1− φ)

(a)

time

0.001

0.01

0.1

1

φ(x

′ ,t′)

0 0.2 0.4 0.6 0.8 1x′

q′(φ) = 1− φD′(φ) = 1

(b1)

time

0.001

0.01

0.1

1

φ(x

′ ,t′)

(b2)

−1

−0.75

−0.5

−0.25

0

v0′ (

x′ ,

t′)

0 0.2 0.4 0.6 0.8 1x′

Fig. 6. Two different mechanisms to keep the density pro-file bounded from above: Panel (a) shows the influence of adiverging diffusion coefficient D′(φ) = 1/(1−φ) while the per-vaporation coefficient is constant (q′ = 1, Pe = 103). In thiscase, the velocity profile is linear for all times (not shown).In the two lower panels (b) the diffusion coefficient is constantand the pervaporation coefficient varies with φ as q′(φ) = 1−φ(D′ = 1, Pe = 103). This leads to a time-dependent velocityprofile.

everywhere left of the interface and that it is linear oth-erwise, v0(x) = −νwq(0)

(x − X(t)

). With these two as-

sumptions, the solute current which enters the channel atthe reservoir (x = L) is

j(L, t) = φLv0(L, t) = −νwq(0)φL[L−X(t)]. (75)

The amount of solute coming from the reservoir thus de-pends on the position of the interface. As all solute currentarrives at the phase interface, we find its movement from

γairw

γw(φ)

φeq φtd

pervaporationγw(φ) > γairw

refilling from outsideγw(φ) < γairw

Fig. 7. A sketch of the chemical potential γw(φ) of water withthe maximal concentration permitted by thermodynamics, φtd.The pervaporation vanishes at the equilibrium value φeq be-cause of the equilibrium γw(φ

eq) = γairw .

0

L(φ)

0 1φ

permeableD →∞

impermeableD → const

impermeableD → 0

Fig. 8. Three possible functions L(φ) for the phenomenologicalcoefficient of inter-diffusion: They lead to qualitatively differentbehavior of the diffusion coefficient at φ → 1.

Eq. (66) to be

Ẋ(t) = −j(X(t), t)∆φ

=νwq(0)φL

∆φ(L−X(t)). (76)

This differential equation leads to the solution (74).The two assumptions are quite strong, since the tem-

poral change of the volume fraction φ does lead to a spatialchange of solute current. Nevertheless, the solution (74)has been successfully fitted to experimental data, at least

v0(x, t)φ(x, t)Ẋ(t)

x

φ(x, t)φ(`)

φ(r)

X(t)

γs−

γw

φ(r) φ(`)

Fig. 9. Sketch of the typical situation with one moving phaseboundary, described in Sec. 4.1

.


for short times. For longer times, however, it over-esti-mated the velocity. It is this deviation which we like toexplore in the following.

For a more detailed analysis of the interface movementwe have to develop a global view on the spatial soluteprofile. In general, whatever value the solute current hasat the entrance of the channel (i. e. the concentration φLtimes the velocity there), it drops to zero at the end of thechannel. We may therefore identify three major possiblegrowth mechanisms due to the spatial change of the solutecurrent along the channel: One possibility is the case wediscussed around Eq. (74), where the solute current is con-stant in both phases with a discontinuity at the interface.The current leads to the growth of the dense phase and tothe movement of the interface. The two other possibilitiesare losses in the bulk of the dilute and the dense phases,leading to a temporal change of the concentration profilesthere.

Independent of the total solute current, the mixture ve-locity v0 may be zero or not in the dense and dilute phases.In the following three subsections we discuss three dif-ferent combinations of vanishing/non-vanishing currents/velocities left and right of the interface.

4.1.1 Solute accumulation at the phase boundary: thespatial profile in the dilute phase

We first investigate the spatial concentration profile whichresults from the assumptions taken above for Eq. (74),namely that all incoming solute is accumulated at thephase interface. The dense phase plays no role, i. e. v0(x) =0, j(x) = 0 for x < X(t). The velocity of the interfacethen depends only on the concentration profile in the di-lute phase,

Ẋ(t) =D∂xφ

∆φ. (77)

In a system of coordinates moving together with the inter-face the evolution equation for the density reads (comparewith Eq. (69)),

∂tφ = Ẋ∂xφ− ∂xj. (78)

A steady solution in this moving frame corresponds to auniform current j0 = −∆φẊ in the reference frame. Theuniformity of the current corresponds to the above men-tioned assumption that all solute entering the dilute phaseleaves it at the interface. We then find the profile station-ary with respect to the moving frame as the solution of

(v0 − Ẋ)φ−D∂xφ = j0 (79)

As this equation is to be solved in the dilute phase, we mayassume constant coefficients q(φ) ≈ q(0) and D(φ) ≈ D(0)which allows to find the spatial profile

φ(X(t) + y) = exp{−

(y + βα

)2}×[

φ(r) exp{β2

α2

}+ 2∆φ

β

α

[G

(y + βα

)−G

(βα

)]], (80)

0

0.1

0.2

φ(r)

0.4

0 0.2 0.4 0.6 0.8 1x′ −X ′(t′)

φ(x′−X ′(t′))asymptotic

Fig. 10. Visualization of the solution (80) which is stationaryin the frame moving with the interface. The parameters arearbitrary (Pe = 103, Ẋ ′ = 0.05, φ(r) = 0.3, φ(`) = 0.8).

with the shortcuts

α :=

√2Dνwq

= L

√2Pe

, β :=Ẋ

νwq, G(x) :=

∫ x0

eξ2dξ.

(81)An example of this solution is shown in Fig. 10. It exhibitsthe same two regimes as the dilute equation describedabove in Sec. 3.2 and in Fig. 5: There is a hump dominatedby diffusion and a hyperbolic ramp.

The solution from Eq. (80), depicted in Fig. 10 givesinsight into the implications of the taken assumptions: Itshows that the diffusive hump exceeds the binodal con-centration value φ(r). Evidently, there is a zone of higherconcentration which moves together with the phase inter-face, and which continuously feeds the dense phase. Aslong as both the velocity and the total solute current van-ish at the interface, the diffusive hump must exceed thevalue φ(r) since there must be a negative current causingthe interface to move to the right. The only contributionto the solute current is the one caused by the concentra-tion gradient, thus having to be positive. The assump-tions j(X, t) = 0 and v0(X, t) = 0 create a situation inwhich it is in principle possible that the peak which isclearly visible in Fig. 10 reaches a spinodal value, whichis larger than φ(r) but smaller than φ(`). As the concen-tration reaches the spinodal, it will undergo local phasechanges and exhibit local blobs of a metastable phase.Such blobs have occasionally been observed in the exper-iments using a KCl solution [7].

The diffusive hump in Fig. 10 appears pronounced dueto the large Peclet number chosen in this example. ThePeclet number renders the length scale α small, which re-duces the lateral extension of the hump and increases itsmaximum value. In experimental realizations, its height islimited by several factors, first by the overall Peclet num-ber, second by the spinodal concentrations which enforcea local phase transition instead of the smooth maximum,and third by the pervaporation in the dense phase. For anon-vanishing velocity v0(X, t) the necessity of having apositive slope ∂xφ at x = X(t) gradually vanishes, allow-ing the height of the maximum to decrease or to vanishcompletely.


We continue to discuss the solution (80) by showinghow the result (74) follows from the spatial form of thesolute concentration. The hyperbolic ramp, which is theasymptotic solution for large x, here reads (in rescaledunits)

φ(X(t) + y) ∼ ∆φ1 + y′/Ẋ ′

. (82)

We may use the asymptotic form to calculate an approxi-mate velocity of the interface: Of course, this result corre-sponds to a stationary solution in a reference frame mov-ing at constant velocity. Therefore, the input concentra-tion at the end of the channel must be increased consis-tently in order to find this solution. Reversely, if the inputconcentration is held constant at φ(L) = φL, the currentarriving at the phase interface decreases in time. As longas the diffusive hump does not reach the inlet of the chan-nel, this change of current is small and we may use φ(L)as a slowly varying parameter. Setting φ(L) = φL in theasymptotic solution (82) then yields essentially the sameexpression for the evolution of the interface position as inequation (74),

X ′(t) = 1− exp{− φL

∆φ− φL(t′ − t′0)

}. (83)

4.1.2 Solute accumulation at the phase boundary: thevelocity field in the dense phase

As a next approximation we relax the assumption of avanishing velocity left of the interface. However, we con-tinue to keep the total current zero in the dense phase.Also the dilute phase is passive, such that all incomingsolute is accumulated at the interface.

As discussed above in section 3.3 the velocity and den-sity profiles in dense phases can behave in two qualita-tively different ways: One corresponds to the vanishingof q(φ) and the other to the divergence of D(φ). We makethe same distinction here. The first case, with vanish-ing q(φ), is characterized by a flat velocity profile near theend of the channel. It is likely to lead to a density profilewhich has maximum density from the end of the channelup to the interface, where it reaches a small region withnon-vanishing velocity. After a transient time, the den-sity profile in a small region could become stationary in areference frame moving together with the interface. Thissolution supports a mixture velocity at the interface thatis constant in time. The position of the interface is thengiven by

X(t)L

=(1− v(X)

Lνwq(0)

)(1− exp

{−φLνwq(0)

∆φ(t− t0)

}).

(84)The dilute phase is here characterized by a constant q andD.

In the second case, in which the diverging diffusioncoefficient leads to a bounded density profile, the veloc-ity field needs not to vanish except at the very end ofthe channel. For simplicity, we adopt a constant non-zero

value for the pervaporation in the dense phase, q(φ) ≈q(1) < q(0). This solution leads to a piecewise linear mix-ture velocity and to the interface position

X(t)L

=q(0)

q(0)− q(1)

(1−exp

{−φLνwq(0)

∆φ(t−t0)

}). (85)

None of the solutions (84) and (85) can explain thevelocity decrease found in the AOT experiments. Instead,both velocities are larger than the initial proposition solu-tion (74). Note that we still kept the assumption j(x) = 0in the dense phase, which corresponds to a large diffusioncoefficient. It must therefore be concluded that in the caseof AOT either the diffusion coefficient is not sufficientlylarge, or the velocity decrease stems from the dilute phase.

4.1.3 Solute accumulation also in the dilute phase

In both preceding subsections we have kept the assump-tion that all incoming solute contributes to the movementof the interface. We now continue with the more generalcase where there may be a temporal change of the con-centration profile such that there is a gradient of solutecurrent in the dilute phase. The dense phase is still as-sumed to be passive (no pervaporation).

This more general situation cannot be described ana-lytically but has to be treated numerically. A numericaltest of the situation with a dense phase growing into a di-lute one exhibits the limitations of the assumption takenabove in Sec. 4.1.1. Figure 11 shows the full numerical so-lution of the model. The velocity in the dense part hasbeen kept zero (q = 0), leading to a flat density profilethere. The dilute part of the solution agrees qualitativelywith the graph given in Fig. 10. There are, however, devi-ations. The plot shows the solutions of Eq. (80) with thetrue position and the true velocity of the interface takenfrom the numerical solution. Panel b quantifies the differ-ence between the approximation from Sec. 4.1.1 and thetrue solution. It shows the position of the interface as afunction of time. A fit with the exponential function (74)with the known value for φLνwq/∆φ and t0 as the fittingparameter proves that the approximation does not workin this example. A second fit for both parameters indi-cates that the resulting function is still exponential, butwith a different timescale than assumed in the approxima-tion (74). In both fittings only the data in the gray-shadedarea has been used. The fit quantifies that around 60% ofthe incoming solute is accumulated at the interface andleads to the growth of the dense phase. The rest remainsin the dilute phase. Panel 11c visualizes this ratio directly.Evidently, the current has not the form of a step function,but takes on different values at the interface and at thechannel entry. This clearly falsifies the assumption of auniform current in the dilute phase.

Upon varying the parameters of the example, we foundthat the concentration φ(r) right of the interface has thelargest influence on the deposition of current. For φ(r) =0.05 nearly all of the incoming current was found to ac-cumulate at the phase boundary—despite a well devel-oped diffusive hump similar to the one in Fig. 11a, and


0

0.25

0.5

0.75

1

φ(x

′ ,t′)

0 0.2 0.4 0.6 0.8 1x′

numericalEq. (80)

0

0.2

0.4

0.6

0.8

X′ (

t′)

0 20 40 60 80t′

numericalexponential fit via t0full exponential fit

(b)

−0.01

−0.0075

−0.005

−0.0025

0

j′(x

′ ,t′)

0 0.2 0.4 0.6 0.8 1x′

(c)

(a)

Fig. 11. Numerically determined density profile φ(x′, t′) con-taining a single moving phase interface. The binodal densityvalues at the interface are φ(`) = 0.8 and φ(r) = 0.3; at thechannel entry φL = 10

−2; Pe = 103. Panel (a) shows the den-sity profile, panel (b) the position of the interface as a functionof time, and panel (c) the evolution of the spatial current pro-file.

despite the fact that the solute current still exhibited apronounced hump. The current apparently overflows thediffusive hump completely, if φ(x, t) has already reachedthe value φ(r) before the hump starts. We may quantifythis by stating that approximation (74) is valid if

φ(r) ≤√

PeL−X

LφL. (86)

Here, we assumed the width of the diffusive hump to scaleas L/

√Pe , see the definition of α in (81), and also Ref. [7].

4.2 Several phases: Phase slab thicknesses instationary solutions

We now return to the introductory example of Sec. 1.2with four phases, one of them dilute. This situation issketched in Fig. 12. The aim here is to present an examplehow the thicknesses of phase slabs are governed by thethermodynamic and dynamic properties of all the phases.

For the solution of AOT in water, a remarkable ob-servation has been reported in Ref. [8]: First, the samephases as in equilibrium have been found in the evapora-tion channel. They were identified by their polarizabilityand birefringence, showing hexagonal, cubic and lamellarinternal structures for the dense phases, while the dilutephase was isotropic. The spatial pattern showed that thetwo enclosed slab thicknesses (of the cubic and the lamel-lar phases) were of comparable extension. This result isremarkable because these two phases occupy a very dif-ferent range of φ-values in the equilibrium phase diagram.There, the lamellar phase appears much broader than thecubic phase, see the supplementary material of Ref. [8].We conclude that there is apparently a much larger con-centration gradient in the lamellar phase than in the cu-bic phase. This larger concentration gradient was stablein the experiment, even when all dense phases were con-tinuously fed with more incoming solute. After a shorttransient time, the interfaces bounding the cubic and thelamellar phase moved together at constant distance.

We now show that our model possesses qualitativelythe same solutions. We also observe stable thicknesses ofcomparable size for the lamellar and the cubic phases, ifwe choose the chemical potential and the phenomenolog-ical coefficients well. The strategy here is to adopt simplemodel functions for γw(φ) and L(φ). They are then trans-ferred into diffusion coefficient D(φ) and pervaporationcoefficient q(φ) using Eqs. (61) and (62) which are thenused in the stationary model equations.

We here adopt the most simple models for the chem-ical potential and for the phenomenological coefficients.The functions used below are displayed in Fig. 13. Thevalues of φ are taken from the equilibrium phase diagramof AOT (see supplementary material of Ref. [8]). The re-gions shaded in gray indicate phase coexistence, wherethe chemical potentials are constant. Within the mostdense phase we use a logarithmic behavior of the formγ′w(φ) ∝ ln(1 − φ), whereas in the other three phases welinearly connect some coexistence values which have beenchosen as model parameters. The phenomenological coef-

v0(x, t)φ(x, t)

φ

x∆X2(t) ∆X3(t)

Fig. 12. Sketch of the situation with several phase slabs ex-hibiting characteristic extents, as described in Sec. 4.2

.


ficient has been chosen constant in each phase except inthe dilute phase where a linear function of φ is required(see Sec. 2.3.2). The coefficients q′(φ) and D′(φ) followfrom equations (61) and (31), see Fig. 14.

Figure 15 shows a typical profile of the volume frac-tion of solute, φ(x′), as it results from the model functionsgiven in Figs. 13 and 14. The lamellar and cubic phasesare indeed found to have approximately the same extent.It is mainly the different slope of the chemical potentialin Fig. 13a which has been adjusted manually to achievethe comparable extent of the two phases. The physicalcontent of these slopes are the very different osmotic com-pressibilities of these two phases. Of course, also the phe-nomenological coefficient plays a role. We do not claimthat the presented chemical potentials are the only oneswhich may lead to comparable phase extensions. However,it is evident in this very example that the osmotic com-pressibilities of the chemical potential directly influencethe extension of the phase slabs. The profile in Fig. 15 hasbeen calculated as the stationary solution of the modelequations (56) and (57). The use of the stationary equa-tions will be explained in the following.

4.2.1 Stationary and quasi-stationary solutions

The full evolution of the phase thicknesses in time is sub-ject to many parameters including not only the modelparameters for D(φ) and q(φ) and the incoming volumefraction and the Peclet number. It depends also on theinitial state, or in other words, on the precise conditionsof phase creation at the end of the channel. We do notwant to treat the issues of phase creation here, the moreas a reasonably fast numerical method for this parameterregime is not at hand. Instead, we employ a quasi-station-ary treatment, which is focused on the main features ofthe phase thicknesses. The idea is to assume a very smallinput current and to assume at any time an equilibratedstationary profile. Such a stationary profile is shown inFig. 15 for the model functions of Fig. 13. Note the slabthicknesses of the lamellar and the cubic phases which areof the same extent. This behavior has been achieved bychoosing the slopes of the chemical potential in Fig. 13,which is 20 times larger in the cubic phase than in thelamellar one. Of course, the variation of the chemical po-tential also has its influence on the evaporation. We thuscannot clearly separate between thickness due to a varia-tion of diffusion and due to modified pervaporation. Thisambiguity is a systematic issue in the interpretation of theexperimental results, and it is reflected in our model bythe two coefficients D and q.

In order to obtain the ensemble of all boundary po-sitions, which is given in Fig. 16, we parametrized thesolutions by the total amount of solute

∫φ(x′) dx′. In a

quasi-stationary setting with extremely small input cur-rent, this amount may be understood as time multiplied bythe input current. A numerical test of the full time-depen-dent equations revealed similar lines as a function of time(not shown). It can be seen that the interior slab thick-nesses relax to constant values, and that only the last, the

0

0.25

0.5

0.75

γ′ w(φ

)−

γ′a

irw

lamellar

cubi

c

hexa

gona

l

0

0.25

0.5

0.75

1

L′ (

φ)

0 0.2 0.4 0.6 0.8 1φ

Fig. 13. Simple model functions for the rescaled chemical po-tential difference of water across the PDMS layer and for thephenomenological coefficient, both as functions of the volumefraction φ. Note the different slopes of the chemical potentialin the lamellar and the cubic phases which lead to the sameslab thickness in Fig. 15.

0

0.25

0.5

0.75

1

q′(φ

)

lamellar

cubi

c

hexa

gona

l

0.01

0.1

1

10

D′ (

φ)

0 0.2 0.4 0.6 0.8 1φ

Fig. 14. The rescaled coefficients for evaporation q′(φ) and fordiffusion D′(φ) resulting from the model in Fig. 14.

hexagonal phase continues to grow. This behavior stemsfrom the vanishing pervaporation in the hexagonal phase


0

0.25

0.5

0.75

1

φ(x

′ )

0.81 0.82 0.83x′

lam

ella

r

cubi

chexagonal dilute

Fig. 15. An example solution of the stationary equations inwhich the thickness of phase slabs is controlled by the slope ofchemical potential in Fig. 13. Note the equal thickness of thelamellar and the cubic phases.

0

0.05

0.1

0.15

Xi

0 0.05 0.1 0.15R 10 φ(x

′)dx′

(a)

dilute

hexagonal

0

0.05

0.1

0.15

Xi

0 0.05 0.1 0.15R 10 φ(x

′)dx′

(b)

dilute

hexagonal

Fig. 16. The development of phase thicknesses for (a) themodel given in Fig. 13; and in (b) for a chemical potentialin the cubic phase having the same slope as the one in thelamellar phase.

which leads at the interfaces to a mixture velocity con-stant in time. If the velocity does not vanish in the densephase, then the interior thicknesses do not truly becomestationary.

In order to demonstrate the effect of the slopes ofthe chemical potential on the slab thicknesses, Fig. 16bpresents the result for a modified model chemical poten-tial. We took the curve in Fig. 13a and changed the slopein the cubic phase to the slope in the lamellar phase.This situation implies the same osmotic compressibilities

in both phases, and also a comparable spatial concentra-tion gradient. As a result, the cubic phase has shrunk andis hardly visible in Fig. 16b.

The numerical solutions in Figs. 15 and 16 have beencreated by a Runge–Kutta algorithm solving the station-ary equations (1) and (2). On the analytical side, they maybe solved formally by integration over φ. The stationaryequations can be rewritten such that

d

dx(v0)2 = −D(φ)∂xφ

φνwq(φ) and (87)

dx

dφ=

D(φ)v0φ

. (88)

The formal solution of the problem then reads

v0(x, x0) = −[(

v0(x0))2 + 2νw φ(x0)∫

φ(x)

D(ξ)q(ξ)ξ dξ

] 12

(89)

x = x0 +

φ(x0)∫φ(x)

D(ϕ)ϕ

[(v0(x0)

)2 + 2νw φ(x0)∫ϕ

D(ξ)q(ξ)ξ dξ

]− 12dϕ.

(90)

This expression describes in principle any stationarysolution, but in the unusual way x(φ). This formulation isespecially convenient for the determination of phase slabthicknesses. For the first phase at the end of the evapora-tion channel, the expression simplifies due to the bound-ary condition v0(0) = 0. In case of the phase interfacex = X(t) the lower bound of the integrals is fixed to thebinodal value φ(`) of the phase transition. We then find

v0(X(t)) = −[2νw

∫ φ(0,t)φ(`)

D(ξ)q(ξ)ξ

dξ

] 12

(91)

X(t) =∫ φ(0,t)

φ(`)

D(ϕ)ϕ

[2νw

∫ φ(0,t)ϕ

D(ξ)q(ξ)ξ

dξ

]− 12dϕ

(92)

4.2.2 Phase nucleation

The treatment of the phase thickness in the present sec-tion resembles to some extent the treatment of nucleationas it has been presented by Evans, Poon, and Cates [18].Both include the assumption of local equilibrium, leadingto a diffusion-type equation, while the total system is outof equilibrium. Both allow the occurrence of metastablephases which are absent in equilibrium. Here, these phasesare the dense ones growing because of the non-zero solventflow; in Ref. [18] it is a freshly nucleated metastable phasewhich grows because the whole system is quenched in tem-perature.

We have not included the details of phase nucleationnear the end of the channel, as it would require a moreadvanced numerical treatment. The general framework,however, should be applicable to the nucleation in a sim-ilar manner as in Ref. [18].


5 Summary

We have derived a model for the behavior of binary solute–solvent mixtures in recently developed microevaporatordevices. The final target of the model is to allow an inter-pretation of experimentally observed data in terms of well-defined thermodynamic and dynamic properties of themixture which are measurable also by different means. Asa first step towards this goal we have developed the modelequations (1) and (2) which relate the phase behavior inthe microevaporator to an inter-diffusion coefficient D andanother coefficient q quantifying the pervaporation of so-lute. We have shown in Eqs. (31) and (46) how these twocoefficients depend on the chemical potentials in the mix-ture and on the phenomenological coefficient for inter-dif-fusion. This relation has then been extended to make alsothe connection to other quantities such as the coefficientsof sedimentation, permeability and osmotic compressibil-ity.

The presented model is focused on the transport equa-tions of solute and solvent mass in binary mixtures, usingthe hypothesis of local thermodynamic equilibrium. Weprovide in detail all the necessary assumptions to reducethe description to two variables, one of which is the volumefraction of solute, the other one is the volume-averagedmixture velocity. We further assume incompressibility ofthe mixture and reduce the spatial dimensions to the oneparallel to the evaporation channel.

We solved the model equations in a number of typicallimit cases which still allow analytical approximations andcompared these with full numerical solutions. In particularwe found good agreement in the case of dilute solutions,where the diffusion and pervaporation coefficients may betreated as constants. In dense binary mixtures, where thisis no longer the case, we were able to show that there aretwo qualitatively different mechanisms which both lead toa saturation of the concentration at its maximal possiblevalue. One of them is due to vanishing evaporation at anequilibrium concentration. The other case comes from thesingular behavior of the chemical potential of water in thislimit.

We have further analyzed the growth velocity of adense phase into a dilute one and have found several ex-pressions which allow to extract quite some information onthe properties of the mixture only from tracking a singleinterface between two phases. Further, we have analyzedthe resulting thicknesses of phase slabs in the microevapo-rator in a stationary setting. In particular, the experimen-tally observed case of equal phase thicknesses has beenregarded. We were able to explained this remarkable factas the result of very different osmotic compressibilities inthe two phases.

In total, the results presented here show that the in-terpretation of the experimental results is possible but farfrom trivial. In all cases, there is some ambiguity in theinterpretation of the observed behavior: As the physicalproperties of the mixture are summarized in two coeffi-cients, also the reasons for the flattening at high soluteconcentrations are twofold. In a similar manner, the thick-nesses of the phase slabs are not only governed by their

thermodynamic properties, but also by the phenomeno-logical coefficient containing dynamic information.

It is precisely this twofold information contained inthe microevaporator data as well as in the two coeffi-cients D and q which presents the core of the presentwork. We are convinced that the presented model estab-lishes a good first compromise between the complexity ofthe phenomenon being out of equilibrium and the needfor simplicity in the equations. Our model will help onthe experimental plan to identify sets of measurementswhich allow to lift the ambiguity in the coefficients. It isthe detailed derivation of the model and the numericaltreatment of the equations which allows to improve theinterpretation of the experimentally measured quantitiesand to extract from them the chemical potentials and thetransport coefficients as full functions of the solute con-centration.

Acknowledgements: We are grateful to J. Leng for valu-able discussions on the experiments. For comments on themanuscript we thank K. Sekomoto and D. Lacoste. Thiswork was financed by the German DFG via grant Schi/1-1and by the French ANR via project “Scan 2”.

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