mutual diffusion governed by kinetics and thermody- · cal temperature of ˘319k. the spinodal...

Journal Name

Mutual diffusion governed by kinetics and thermody-namics in the partially miscible mixture methanol +cyclohexane†

Tatjana Janzena, Shi Zhangb,c, Aliaksandr Mialdunb, Gabriela Guevara-Carriona, JadranVrabec∗a, Maogang Hec and Valentina Shevtsova∗b

To gain an understanding of the transport and thermodynamic behavior of the highly non-idealmixture methanol + cyclohexane, three complementary approaches, i.e. experiment, molecularsimulation and predictive equations, are employed. The temperature and composition depen-dence of different diffusion coefficients is studied around the miscibility gap at ambient pressure.On the one hand Fick diffusion coefficients are measured experimentally by interferometric prob-ing and on the other hand Maxwell-Stefan diffusion coefficients and intradiffusion coefficients aresampled by equilibrium molecular dynamics simulation at five temperatures below the upper criti-cal temperature of∼319 K. The spinodal curve is determined from extrapolation of the experimen-tal Fick diffusion coefficient data and compared to predictions from excess Gibbs energy models.It is found that these models are not capable to correctly describe the activity coefficients overthe whole composition range of the studied mixture. Thus, different parameter sets for a modifiedWilson model are used for calculations of the thermodynamic factor, which is needed to transformMaxwell-Stefan into Fick diffusion coefficients and vice versa. Further, predictive equations for theMaxwell-Stefan diffusion coefficient, which are based on intradiffusion coefficients, are comparedto simulation results. Using different approaches provides a clearer understanding of the relationsbetween kinetic and thermodynamic properties contributing to the diffusion behavior of partiallymiscible mixtures.

1 IntroductionThe behavior of partially miscible liquids is an intriguing scientificproblem, wherein diverse processes, like separation and nucle-ation, are governed by mass diffusion. To understand and modelthese phenomena, mutual diffusion coefficients, which are tem-perature and composition dependent, must be known. Becausethe according measurements are challenging and time consum-ing, numerous predictive approaches exist, relying on differentdiffusion coefficient types. However, the interrelation between

a Thermodynamics and Energy Technology, University of Paderborn, 33098 Paderborn,Germany. E-Mail: [email protected] Microgravity Research Center, Université Libre de Bruxelles (ULB), CP–165/62, Av.F.D. Roosevelt, 50, B–1050 Brussels, Belgium. E-mail: [email protected] MOE Key Laboratory of Thermal Fluid Science and Engineering, School of Energy andPower Engineering, Xi’an Jiaotong University, Xi’an 710049, China† Electronic Supplementary Information (ESI) available: Includes experimental andsimulation data for molar excess volume and shear viscosity, summery of thermo-dynamic relations, comparison of LLE and ACID calculated with five gE models, hy-drogen bonding statistics and simulation snapshots, chemical potentials from MonteCarlo simulation, Darken relation from velocity correlation functions, and numericalsimulation data. See DOI: 10.1039/xxxxxxxx/

the different diffusion coefficients and the thermodynamic behav-ior of a liquid is still not fully understood. This topic is not only in-teresting from the scientific point of view, but also highly relevantfor process design in chemical engineering. While liquid-liquidequilibrium (LLE) data and correlations thereof are available formany binary and ternary mixtures, studies reporting their diffu-sion coefficients are rather scarce.

The mixture methanol + cyclohexane was studied in thiswork around its demixing region at ambient pressure. Temper-atures and compositions considered in this work are illustratedin Fig. 1, together with LLE data measured by other authors1,2.In the framework of the European Space Agency (ESA) pro-gram DCMIX (Diffusion and Thermodiffusion Coefficients Mea-surements in Ternary Mixtures), research groups conduct experi-ments on board of the International Space Station6–8 (ISS) to val-idate ground experimental techniques and various theories. Theternary mixture examined in the DCMIX2 campaign, toluene +methanol + cyclohexane, contains a miscibility gap. Accordingly,this study of its binary subsystem methanol + cyclohexane is alsomotivated by processing and understanding the results of the ex-periments performed on the ISS. Only at 298.15 K, Fick diffusion

Journal Name, [year], [vol.],1–18 | 1

0 0.2 0.4 0.6 0.8 1285

290

295

300

305

310

315

320

325

293.55 K

298.15 K

303.38 K

313.21 K

315.30 K

xmethanol

/ mol mol−1

T /

K

Fig. 1: Temperatures and compositions studied in this work around theexperimental binodal curve 1,2 of methanol + cyclohexane at 0.1 MPa.

coefficient data of methanol + cyclohexane were already mea-sured by other authors14,15.

The present study combines experimental work, molecular dy-namics simulation and predictive equations to determine differ-ent diffusion coefficients and to examine their interrelation aswell as their connection to the thermodynamic behavior of thismixture. Only few other studies on diffusion coefficients in bi-nary liquid mixtures with LLE phase separation are available inthe literature. Vitagliano et al.16 carried out measurements forthe mixture water + triethylamine and extrapolated their data toobtain the spinodal curve16. Clark and Rowley followed, studyingthe mixture methanol + n-hexane17. Krishna and van Baten18

studied the same mixture by means of molecular simulation andthe NRTL (nonrandom, two-liquid) excess Gibbs energy model,which was already used by Clark and Rowley17. In another exper-imental work19, Fick diffusion coefficient data of several binarymixtures exhibiting a miscibility gap were measured and com-pared to different correlation equations.

The goal of the experimental part of the present study was tomeasure the Fick diffusion coefficient of methanol + cyclohexaneas close as possible to the binodal. We have considerable experi-ence in measuring Fick diffusion coefficients using the Taylor dis-persion technique9–12. However, when conducting experimentsnear the LLE, it is necessary to ensure that the fluid is in a singlephase, avoiding the formation of the second phase and the de-velopment of turbidity. Thus, optical digital interferometry (ODI)was employed here, which was also used for diffusion-controlledexperiments on board the ISS13. The unique feature of the ODImethod is that it traces the transient path of the system over theentire 2D cell cross-section throughout the whole diffusion pro-cess.

While the LLE binodal is often measured, the metastable regionis hardly accessible by experiments. In order to reach the spinodalregion of the phase diagram, the metastable region or the criticalpoint must be transversed. Diffusion dynamics near the criticalpoint has been studied theoretically in an extensive manner, seereviews and other work3–5. In real systems, as a result of fluctu-

ations, the spinodal curve is not a sharp boundary and it can notbe measured directly because in this region even arbitrarily smallcomposition or density fluctuations grow exponentially over time,leading to a diffusion controlled phase seperation. Since the Fickdiffusion coefficient must be zero at the spinodal, extrapolationsof the measured Fick diffusion coefficient to zero were carried outin this work to determine spinodal compositions.

In addition to the experimental approach, equilibrium molecu-lar dynamics simulations were carried out to compute Maxwell-Stefan (MS) and intradiffusion coefficients, with the aim to studythe kinetics contributing to the diffusion behavior and to analyzethe interrelation between different diffusion coefficients. Manypredictive approaches for mutual diffusion are based on intra-diffusion coefficients or diffusion coefficients at infinite dilution.Four selected approaches for the MS diffusion coefficient werecompared to simulation results. Diffusion coefficients at infinitedilution were obtained by extrapolation of simulation and exper-imental data.

To consider the thermodynamic behavior of the studied mix-ture, a modified Wilson excess Gibbs energy (gE) model was em-ployed. Such models describe the composition dependence of theactivity coefficients, thus the model parameters can be fitted toexperimental binodal compositions or to activity coefficients atinfinite dilution. That gE model also allows for a prediction of thespinodal curve and the calculation of the thermodynamic factor.

The Fick diffusion coefficient was further predicted from theMS diffusion coefficient simulation data and the thermodynamicfactor from the gE model for a comparison with experimental re-sults. This set of methods allows for an examination of the dif-fusion behavior from different points of view and promotes theunderstanding of the behavior of partially miscible mixtures.

This article is organized as follows: First, a theoretical explana-tion is given for the different diffusion equations and coefficients,their interrelation and connection to the thermodynamic behav-ior in the vicinity of a miscibility gap. Second, the experimentalmethodology is described in detail, followed by an outline of theemployed simulation techniques and predictive equations. Sub-sequently, the experimental and simulation results are presented,analyzed and compared with each other, comprising Fick, MS andintradiffusion coefficients. Extrapolated infinite dilution diffusioncoefficients and spinodal loci are also examined in light of a com-parative analysis. Finally, conclusions are drawn.

2 Theoretical considerations

2.1 Diffusion coefficients

The most common approach to describe mutual diffusion is Fick’slaw that expresses the diffusive molar flux Ji of component i in abinary mixture with respect to the volume averaged velocity uV

JVi = xiρ(ui−uV ) =−D∇ci , (1)

with mole fraction xi and velocity ui of component i, mixture mo-lar density ρ and Fick diffusion coefficient D. Here the molar con-centration gradient ∇ci is acting as the driving force. This form isused in experimental work because the concentration gradient isaccessible by measurements.

2 | 1–18Journal Name, [year], [vol.],

Alternatively, the molar flux can be expressed in terms of themolar averaged velocity uM and the gradient of the mole fraction∇xi acting as the driving force

JMi = xiρ(ui−uM) =−ρD∇xi . (2)

The molar fluxes of both components in a binary mixture are con-strained by ∑JM

i = 0 in the molar reference frame or by ∑viJVi = 0

in the volume reference frame, where vi is the partial molar vol-ume of component i. In case of a binary mixture, there is only asingle Fick diffusion coefficient D, which is identical for both fluxequations (1) and (2).

The MS equation is another widely used approach to describediffusion. Originally derived from the kinetic theory of gases, itwas generalized to liquids. Therein, the relative velocity betweenthe two components (ui−u j) is expressed in relation to the chem-ical potential gradient ∇µi acting as the driving force

x j(ui−u j) =−Ð1

RT∇µi , (3)

with MS diffusion coefficient Ð, molar gas constant R and tem-perature T . Because chemical potentials can not be measured, itis not possible to obtain MS coefficients experimentally.

Irreversible thermodynamics20 has shown that the chemicalpotential gradient is the intrinsic driving force for diffusion un-der isothermal and isobaric conditions. This is a consequence ofthe second law of thermodynamics, i.e. diffusion occurs due toentropy production. In this context, a diffusive molar flux withrespect to the molar averaged velocity can be expressed as

JMi =−Lii∇µi−Li j∇µ j , (4)

with the phenomenological coefficients Li j, also called Onsagercoefficients21. These coefficients are restricted by three types ofconstraints22: Onsager’s reciprocal relation Li j = L ji, the vanish-ing total flux and the Gibbs-Duhem relation, leading to ∑Li j = 0.As a consequence of these constraints, Eq. (4) can be transformedto

JMi =−Li∇µi . (5)

The same driving force is present in Eqs. (3) and (5) so that thediffusion coefficients are directly related to each other by

Li = Ðρxi

RT. (6)

The transformation between the different driving force gradi-ents is23

− xi

RT∇µi =−

xi

RT∂ µi

∂xi

∣∣∣∣T,p

∇xi . (7)

Note that the partial derivative of the chemical potential mustbe evaluated under the constraint x1 + x2 = 1. This term is thethermodynamic factor Γ and can be written as

Γ =xi

RT∂ µi

∂xi

∣∣∣∣T,p

= 1+ xi∂ lnγi

∂xi

∣∣∣∣T,p

, (8)

with the activity coefficient γi of component i. For ideal mixturesγi = 1 and Γ = 1.

Inserting Eqs. (6) and (8) into Eq. (5) yields

JMi =−ρÐΓ∇xi . (9)

A comparison of Eqs. (2) and (9) shows the relation between theFick and the MS diffusion coefficients

D = ÐΓ . (10)

This expression can be interpreted as a separation of the Fick dif-fusion coefficient into a kinetic (Ð) and a thermodynamic (Γ) con-tribution. In the composition limits xi→ 0 and xi→ 1, the thermo-dynamic factor is unity so that MS and Fick diffusion coefficientscoincide there.

To calculate the thermodynamic factor, the non-ideality of amixture must be known, which is expressed by the activity coef-ficients γi of both components. These can be obtained e.g. fromactivity coefficient models, also called excess Gibbs energy mod-els. These models typically consider only temperature depen-dence and neglect pressure dependence of gE so that they shouldonly be used under conditions for which they were parametrized.

In addition to mutual diffusion, which occurs due to a gradi-ent, intradiffusion coefficients Di can be defined, which describethe mobility of molecules of species i in the absence of drivingforce gradients. These coefficients are also composition depen-dent. The same phenomenon is quantified in a pure fluid bythe self-diffusion coefficient, although this term is also often usedfor mixtures. Another property is the so-called tracer diffusioncoefficient, which is measured with isotopic tracers in mixturesor pure fluids. It is assumed to be identical to the intradiffu-sion or self-diffusion coefficients, neglecting any isotopic effects.Einstein derived intradiffusion from kinetic theory as a randommovement which is given by the mean squared displacement ofsingle molecules24. In the two infinite dilution limits of a binarymixture, the MS and Fick diffusion coefficients coincide with theintradiffusion coefficient of the diluted component. Different pre-dictive approaches for the MS coefficient are based on intradiffu-sion coefficients, e.g. the well known Darken relation25.

2.2 Liquid-liquid equilibrium

Liquid mixtures with a miscibility gap exhibit two liquid phaseswith different compositions. These LLE compositions are repre-sented by the binodal and can be measured experimentally. Atconstant temperature and pressure, the binodal is defined by thecondition that the chemical potentials of all components i areequal in the coexisting phases I and II

µIi = µ

IIi , (11)

resulting in the isoactivity criterion for phase equilibrium

(xiγi)I = (xiγi)

II . (12)

Being within the binodal, the metastable region is limited bythe spinodal curve beyond which the unstable region lies. Thecriterion for phase stability in a binary mixture can be expressedin terms of the chemical potential at constant temperature and


pressure26–28(∂ µi

∂xi

)T,p≥ 0 stable and metastable ,

(∂ µi

∂xi

)T,p

< 0 unstable .

(13)

The spinodal curve indicates the stability limit where thederivative of the chemical potential with respect to the mole frac-tion vanishes. This condition can be rewritten as(

1xi+

∂ lnγi

∂xi

)T,p

= 0 . (14)

It follows that the thermodynamic factor Γ as defined in Eq. (8)is also zero at the spinodal curve and negative in the unstableregion. According to Eq. (10), the same applies to the Fick dif-fusion coefficient, resulting in a flux of the components againsttheir own concentration gradient within the unstable region, i.e.during spinodal decomposition.

2.3 Thermodynamic non-ideality

The activity coefficients γi of all components represent the devia-tion from ideal mixing behavior. They constitute the molar excessGibbs energy by

gE

RT= ∑xi lnγi . (15)

Different gE models rely on this relation to describe phase be-havior, like LLE, with algebraic expressions for the excess Gibbsenergy. As every other excess property, gE vanishes in the infinitedilution limits. However, this is not the case for the activity co-efficients. The activity coefficient at infinite dilution γ∞

i (ACID) isan important thermodynamic property that is often used for mod-eling separation processes because it is related to the behavior ofan infinitely dilute solute in a solvent. ACID are relevant for diffu-sion as they determine the slope of the thermodynamic factor atlow concentrations of one component and therefore also governthe composition dependence of the Fick diffusion coefficient.

3 Experimental approach3.1 Instrument, liquids and experimental procedure

For the experimental measurement of the Fick diffusion coeffi-cient, a diffusion cell with interferometric probing was used. Be-fore the experiment, mixtures of methanol and cyclohexane wereprepared gravimetrically with an accuracy of ±0.005 g. Two so-lutions with a mass fraction difference of 1 % were treated asone pair for the diffusion measurement. Reagent grade solventsMethanol, 99.99 % (CAS Number: 67-56-1) and Cyclohexane,99.99 % (CAS Number: 110-82-7) supplied by Fisher Scientificwere used without further purification.

At the beginning of each experimental run, a two-layer liquidsystem was formed inside a diffusion cell by the simultaneousinjection of two solutions through the inlets, the more dense mix-ture at the bottom and lighter one on top, cf. Fig. 2. This diffusioncell type was earlier called "counterflow cell"29. By proper manip-ulation, it provides an interface of quite good sharpness. The de-

Fig. 2: Schematic of the diffusion cell.

Fig. 3: Schematic of the entire set-up including interferometric probing.


Fig. 4: Transient diffusion fields: the two pictures on the left are fringeimages at t = 0 and 5 min, the following two pictures show the wrappedphase difference at t = 5 and 30 min after the beginning of the experi-ment, and the picture on the right shows the unwrapped phase differencebetween t = 5 and 30 min.

sign of the present diffusion cell yielded a geometric path for theprobing beam in the bulk liquid with a total length L=5.3 mm.

To study elevated temperatures, the existing set-up9,29 wasmodified, a sketch of the upgraded instrument is shown in Fig. 3.All of its major parts, including the diffusion cell, were main-tained inside a thermally insulated box equipped with an activethermal control system. The temperature inside the box could beregulated between 290 K to 320 K with residual fluctuations be-low 0.1 K. The liquid supply tubes and both syringes were alsomaintained at the specified temperature.

A Mach-Zehnder interferometer was employed to sample con-centration changes inside the diffusion cell. The light source wasan expanded and collimated He-Ne laser beam with a wavelengthof λ=632.8 nm. The resulting interferogram was recorded by aCCD camera with a 1280×1024 pixels sensor. The resolution ofthe imaging system was around 15.73 µm/pixel, while the imageacquisition time step was varied from 10 s at the beginning of theexperiment to 300 s at its end.

3.2 Image processing and optical contrast factorsThe extraction of an optical phase in the interferogram was imple-mented via 2D Fourier transformation, details are described else-where30. Fig. 4 shows original fringe images of the experiment,corresponding to a variation of the refractive index at differenttime instances. The first picture demonstrates that the interfaceshape is initially sharp. Distortion at the fringes of the secondpicture clearly indicates that a diffusion process occurs. The twofollowing pictures in Fig. 4 show the wrapped phase differenceat t=5 and 30 min after the initiation of the experiment. Notethat the wrapped phase map in the third picture was obtainedfrom the fringe image in the second picture (at the same timeinstance). The last picture in Fig. 4 shows the unwrapped phasedifference between t=5 and 30 min.

For a proper estimation of an optical phase, a reference image

Fig. 5: Transient diffusion fields: the two pictures on the left show thereference interferogram and the fringe image at t = 30 min, while thetwo pictures on the right show the wrapped and unwrapped phase att = 30 min.

was needed. Ideally, it should provide a pure background withno traces of concentration change, which is difficult to achievein practice. The presence of tiny concentration imperfections inthe reference image can lead to data corruption, which will prop-agate to the full data set. We thus suggest to use the very lastimage of the experiment as the reference. When image process-ing is carried out for that reference interferogram, the phase dis-tribution image looks as shown in Fig. 5. The first two picturesin Fig. 5 show the reference interferogram and the fringe imageat t=30 min. The last two pictures show the wrapped and un-wrapped phase at t=30 min.

In the present experiments, it was preferred to deal with di-rectly accessible composition variables, i.e. mass fractions wi,where w1 and (1−w1) are the mass fractions of methanol andcyclohexane, respectively. To recover the concentration field fromthe phase map, it is necessary to know the optical contrast fac-tor (∂n/∂w)T at the temperatures of interest, where n(w,T ) is therefractive index. There are two routes to determine the opticalcontrast factor (∂n/∂w)T . It can either be found by curve fittingof experimental phase difference data, or be estimated throughthe boundary condition of phase distribution at the beginning ofthe experiment. The equation for the estimation of the contrastfactor is

ϕ(−∞)−ϕ(∞) =2π L

λ

(∂n∂w

)T

∆w , (16)

where ϕ(±∞) is the optical phase corresponding to the boundary,λ is the wavelength and ∆w is the mass fraction difference (∆w =

wb−wτ ) between two solutions, where wb and wτ are the initialmass fractions at the bottom and top of the diffusion cell.


3.3 Working equations

Four different routes were followed for data analysis, comparingtheir results. In each case, the mathematical description of thisclassic experiment is different. Throughout, the unwrapped phasedifference was used from the beginning of the experiment.

All four methods led to accurate fitting results of the employedequations to the sampled data so that the resulting Fick diffusioncoefficient values did not deviate significantly from each other.Thus, the average value of the Fick diffusion coefficient obtainedby the four different methods was used as the final result.

3.3.1 Long-time model

When the diffusion time is long, the diffusing fronts reach theends of the cell, and the problem should be considered as oc-curring in a finite medium. The spatio-temporal evolution of theindependent variable mass fraction w can be written as9

w(z, t) = wτ +∆w

[hH

+2π

∞

∑n=1

1n

sin(

nπ hH

)×

cos(

nπ z+nπ hH

)exp(−n2π2

H2 Dt)]

, (17)

where h is the distance between the initial interface and the bot-tom of the diffusion cell, H the height of the cell, D the Fick dif-fusion coefficient and t the real diffusion time. Through imageprocessing, the phase difference at different time instances afterthe beginning of the experiment (t0) is obtained. The relationshipbetween phase difference and mass fraction difference is31

w(z, t)−w(z, t0) =λ

2π L[ϕ(z, t)−ϕ(z, t0)]/(∂n/∂w)T . (18)

The Fick diffusion coefficient is obtained by minimization of thedeviation between experimental and calculated mass fraction dis-tribution

Φ = ∑i, j

[wcalc(zi, t j)−wcalc(zi, t0)+wexp(zi, t j)

]2. (19)

In this minimization, both Fick diffusion coefficient and contrastfactor are determined. The initial distribution of the refractiveindex recorded in the experiment and naturally assigned to t = 0does not have a perfect stepwise shape at the interface. We thusintroduce time t0, which is the first time instance at which the the-oretical profile coincides with the experimental one9. However,the initial time of the diffusion process is hard to be estimatedwithout the other parameters. Therefore, three unknown param-eters were concurrently fitted: D, (∂n/∂w)T and t0.

3.3.2 Short-time model

When the diffusion time is short and the concentration changedoes not reach the top and bottom of the diffusion cell, the prob-lem can be considered as occurring in an infinite medium. Thissolution corresponds to the short time model52,53. In this casethe mass fraction distribution is

w(z, t) =wb +wτ

2− ∆w√

π

∫ z/√

4Dt

0exp(−η

2)

dη . (20)

Fig. 6: Illustration of an averaged composition solution.

The fitting procedure for extracting the coefficients is similar tothe long-time model.

3.3.3 Averaged composition model

Here, the method used for the sliding symmetric tubes tech-nique32 was adopted. In this approach, the advancing diffusionfront is described by a self-similar solution on the short-time scale,i.e. before it reaches the far ends of the cell. Within the self-similar regime the time derivative is neglected and the mass frac-tion in the cell is determined as32

w = wτ +∆w2

erfc(η) (21)

where erfc(η) is the complementary error function of η = z/2√

Dt,which varies from η →−∞ (bottom) to η → ∞ (top) and η = 0corresponds to the position of the initial interface between theliquids. The solution for the mean mass fraction w̄ of each partof the cell can be obtain by integrating Eq. (21) over half of thecell, and then dividing it by its height H/2. For example, the massfraction averaged in the upper half can be written in the from

w̄τ =2H

∫ H/2

0w(z, t)dz = wτ +

∆wH

∫ z/√

H/2

0erfc

(z

2√

Dt

)dz

= wτ +∆wH×2√

Dt∫

∞

0erfc(η)dη = wτ +

2∆wH

√Dtπ

. (22)

An analogous expression can be written for the bottom half of thecell. Larrañaga et al.32 emphasized that the solution of Eq. (22)remains valid until the diffusion front reaches the ends of the cell.

When the mass fraction difference between two solutions issmall, there is a linear relationship between mixture compositionand optical phase. Then a similar equation will be valid for themean phase distribution

ϕ̄ =

∫ H/20 ϕdz

H/2= ϕ

τ +2∆ϕ

H

√Dtπ

, (23)

where ∆ϕ = ϕ(−∞)−ϕ(+∞). In this case, the fitting equation for


the experimental data is

ϕ̄

∆ϕ=

∫ N0 ϕdN

N/∆ϕ = ϕ

τ +2H

√Dπ×√

(tc− t0) , (24)

where N is the measured number of pixels and tc is the recordingtime (t = tc− t0). Fig. 6 illustrates the use of Eq. (24), where theslope (2/H)(D/π)1/2 is determined by curve fitting. On that basis,the Fick diffusion coefficient can be obtained.

3.3.4 Linear interpolation model

This model is applied in three steps. First, the optical contrastfactor is determined through the boundary condition. Second,the dot product of diffusion coefficient and diffusion time (Dt)can be obtained through curve fitting of the phase distribution.Finally, the Fick diffusion coefficient is determined by means ofthe linear relationship between (Dt) and t. Because of the linearrelationship between composition and optical phase, the workingequation for the phase distribution is

ϕ(z, t) =−2√

π L∆wλ

(∂n∂w

)T

∫ z/√

4Dt

0exp(−η

2)

dη +ϕb +ϕτ

2. (25)

In this approach, each experimental phase distribution curveleads to a (Dt) at one time instance. Recording numerous val-ues of D over a long time interval, the Fick diffusion coefficientwas determined by linear fitting.

4 Predictive approachMS and intradiffusion coefficients were sampled by equilibriummolecular dynamics (EMD) simulations and the Green-Kubo for-malism. Different classical approaches for the prediction of theMS diffusion coefficient are based on the intradiffusion coeffi-cients, thus four selected ones were compared to MS diffusioncoefficient simulation data. The activity coefficients and the ther-modynamic factor were considered by excess Gibbs energy mod-els that were parametrized to experimental data. By combiningsimulation results for the MS diffusion coefficient with calcula-tions based on a modified Wilson gE model, the Fick diffusioncoefficient was predicted.

4.1 Molecular simulation

4.1.1 Molecular models

The employed molecular force field models for methanol and cy-clohexane were developed in preceding work33,34 using experi-mental vapor-liquid equilibrium data only in the case of methanoland additionally the self-diffusion coefficient at ambient pres-sure in the case of cyclohexane. These models are rigid, non-polarizable and of united-atom type with a geometry based onab initio data. Cyclohexane was modeled by six Lennard-Jones(LJ) sites and methanol by two LJ sites and three point chargesfor electrostatics and hydrogen bonding33. Interactions betweenunlike LJ sites were specified by the Lorentz-Berthelot combiningrules. These two models have successfully been used for the pre-diction of transport properties, i.e. diffusion coefficients, shearviscosity and thermal conductivity, of numerous other binary liq-uid mixtures34.

4.1.2 Correlation functions

According to the Green-Kubo formalism, the intradiffusion coeffi-cient of component i is given by the time integral of the velocityauto-correlation function of single molecules averaged over all Ni

molecules that belong to that species

Di =1

3Ni

∫∞

0

⟨ Ni

∑k=1

vki (0) ·vk

i (t)⟩

dt · (26)

Here, vki (t) is the center of mass velocity vector of molecule k of

species i at some time t and the brackets <...> denote the en-semble average. This expression is equivalent to that derived byEinstein24 in terms of the mean squared displacement. The inte-gration time has to be chosen such that the correlation functiondecays to zero and the integral reaches a stationary value.

Irreversible thermodynamics relates mutual diffusion to phe-nomenological Onsager coefficients. The according coefficientscomputed in this work are related to the molar flux with a massaveraged velocity um

Jmi /ρ = xi(ui−um) =− 1

RT

(Λii∇µi +Λi j∇µ j

). (27)

In EMD simulation, the mass averaged velocity (um = ∑wiui) istypically set to zero when the net momentum is set to zero. Then,the coefficients can be sampled by the net velocity correlationfunction

Λi j =1

3N

∫∞

0

⟨ Ni

∑k=1

vki (0) ·

N j

∑l=1

vlj(t)⟩

dt , (28)

where N is the total number of molecules. Therein, cross-correlations between different molecules are explicitly taken intoaccount. Because the coefficients defined in this way belong toa molar flux with a mass reference velocity, they are constrainedby ∑i MiΛi j = 0, with molar mass Mi of species i. According toOnsager’s reciprocal relation, the coefficients are symmetric, i.e.Λi j = Λ ji. Finally, the MS diffusion coefficient of a binary mixtureis given by35

Ð =x j

xiΛii +

xi

x jΛ j j−Λi j−Λ ji . (29)

4.1.3 Simulation details

All molecular simulations were carried out with the programms236,37. A cubic volume was assumed with periodic boundaryconditions containing 4000 molecules. Intermolecular interac-tions were explicitly evaluated within a cutoff radius of 17.5 Å,considering the LJ long-range corrections beyond the cutoff ra-dius with the angle-averaging method of Lustig38 and the long-range electrostatic interactions by means of the reaction fieldmethod39.

The simulations were conducted in the canonic (NV T ) ensem-ble, first equilibrated over 4×105 time steps followed by produc-tion runs of 1 to 1.5 ×107 time steps. Newton’s equations of mo-tion were solved with a fifth-order Gear predictor-corrector nu-merical integrator and an integration time step of ∼ 1 fs. Theintra- and MS diffusion coefficients were calculated by averagingup to 1.6× 105 independent time origins of the correlation func-


Table 1: Predictive equations for the mole fraction dependence ofthe Maxwell-Stefan diffusion coefficient of binary mixtures.

Maxwell-Stefan diffusion coefficient ref.

Ð = xi ·D j + x j ·Di Darken 25

Ð = (D∞i )

x j · (D∞j )

xi Vignes 40

Ð = Di ·φ j · v/v j +D j ·φi · v/vi(a) Li et al. 41

Ð = (D∞i )

φ j ·v/v j · (D∞j )

φi·v/vi (a) Zhou et al. 42

(a)φi = xi/(xi + x jAi j), with parameters Ai j according to Eq. (31). vi

is the partial molar volume and v the molar volume of the mixture.

tions with a sampling length of 19 ps for the individual correlationfunctions.

4.2 Predictive Maxwell-Stefan equations

Different predictive equations can be found in literature for theMS diffusion coefficient, which are mainly based on intradiffusioncoefficients. The four approaches listed in Table 1 were chosenfor a comparison to MS diffusion coefficient data from EMD sim-ulations. Darken and Vignes are classical interpolation methods,whereas the other two approaches are extensions thereof basedon the local composition concept as it is used in the Wilson gE

model. In a recent study on numerous other mixtures contain-ing methanol and cyclohexane34, the latter two equations werefound to yield convincing results in many cases.

4.3 Excess Gibbs energy models

The thermodynamic factor was calculated in this work by an ex-cess Gibbs energy gE model. For an appropriate description of thethermodynamic behavior of the present mixture, two propertiesare essential: The LLE delimits the metastable region where thethermodynamic factor is already close to zero. However, the ACIDgovern the slope of the thermodynamic factor in the diluted re-gions, which can be very steep, especially in the case of hydrogenbonding molecules such as methanol.

Several models were tested beforehand, namely Van Laar,Margules, NRTL (nonrandom, two-liquid), UNIQUAC (universalquasi-chemical) and a modified Wilson model43,44. Their modelparameters were fitted to experimental LLE data (binodal curve),detailed results can be found in the supplementary material. Thespinodal curve predicted by these five models does not vary much,however, all of them significantly underestimate the ACID, espe-cially for methanol. Because the modified Wilson model showedthe best results for the ACID, it was used subsequently.

Modified Wilson model

In contrast to the original Wilson model, the modified Wilsonmodel43,44 is capable to describe a miscibility gap. It defines thedimensionless excess Gibbs energy by

Q =gE

RT=−x1 ln

(x1 +A12x2

x1 +(v2/v1)x2

)− x2 ln

(A21x1 + x2

(v1/v2)x1 + x2

),

(30)

with

Ai j =v j

viexp

(−

∆λi j

RT

), (31)

where λi j is a Wilson interaction parameter.

Compared to vapor-liquid equilibria, it is much more challeng-ing to reproduce the temperature dependence of LLE data with gE

models. Therefore, as suggested by Matsuda et al.1, temperaturedependent parameters were used

λi j = ai j +bi j(Tc−T )+ ci j(Tc−T )2 +di j(Tc−T )3 , (32)

where Tc is the upper critical temperature. It was observed thatthe differences between the partial molar volumes and pure com-ponent volumes are negligible in case of the present mixture, thustemperature dependent molar volumes of the pure componentswere used instead

vi = ai +biT + ciT 2 , (33)

with parameters ai, bi and ci of Matsuda et al.1.

The activity coefficient is the partial molar excess Gibbs energyand can therefore be obtained from partial derivatives with re-spect to the mole fraction

lnγi = Q+Qi− x1Q1− x2Q2 , (34)

with

Qi =∂Q∂xi

∣∣∣∣x j

. (35)

Note that the partial derivative of Q with respect to the mole frac-tion xi has to be evaluated while keeping the mole fraction ofthe second component x j constant in this case. The thermody-namic factor contains the derivative of lnγi with respect to themole fraction and can therefore be expressed in terms of secondorder partial derivatives of Q

Γ = 1+ x1x2(Q11 +Q22−2Q12) , (36)

with

Qik =∂Qi

∂xk

∣∣∣∣x j

. (37)

Note that the second derivative Qik has to be evaluated withoutusing the constraint x1 + x2 = 1.

5 Results and discussion

5.1 Experimental results

As detailed in Section 3.3, four models were used to determinethe Fick diffusion coefficient from experimentally sampled data.Fitting results are exemplarily discussed for the state point at T =

293.55 K and w1 = 0.895 kg/kg.

Fig. 7 (a) shows phase distribution curves at two time in-stances, one of which (the black curve) corresponds to the pro-file at an early stage of diffusion, and the other one corresponds


tc = 3 min tc = 54.88 min

t0 = 0.88 min

t0 = 0.80 min

t0 = 0.81 min

t0 = 0.90 min

w1 /

kg

kg-1

(a) (b)

(c) (d)

Fig. 7: Characteristic experimental dependencies used for fitting different models for the Fick diffusion coefficient of the mixture methanol + cyclohexaneat T= 293.55 K and w1 = 0.895 kg/kg: (a) short-time model; (b) long-time model; (c) averaged composition model; (d) linear interpolation model. t0is the first time instance at which experimental and theoretical profiles coincide.

w1 / kg kg-1

0.6 0.7 0.8 0.9 1.0

D /

10-9

m2 s-1

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5293.55 K

Linear fitting

313.21 K303.38 K 298.15 K

315.30 K

w1 / kg kg-1

0.00 0.01 0.02 0.03 0.04

D /

10-9

m2 s-1

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

315.30 KFitting

313.21 K

303.38 K298.15 K293.55 K

308.10 K

(a) (b)

Fig. 8: Measured Fick diffusion coefficient at different temperatures: (a) cyclohexane-rich region; (b) methanol-rich region.


Table 2: Measured Fick diffusion coefficient D of the mixture methanol + cyclohexane at ambient pressure.

methanol-rich region cyclohexane-rich regionT w1 x1 D w1 x1 D

K kg kg−1 mol mol−1 10−9m2s−1 kg kg−1 mol mol−1 10−9m2s−1

0.995 0.998 2.240 0.025 0.063 0.6750.895 0.957 1.665 0.015 0.038 0.9030.795 0.910 1.185 0.005 0.013 1.575

293.55 0.705 0.862 0.751 0.0025 0.0065 2.3470.695 0.857 0.6940.685 0.851 0.6470.675 0.845 0.5770.990 0.996 2.320 0.035 0.087 0.630

298.15 0.885 0.953 1.75 0.010 0.026 1.2300.700 0.859 0.750 0.005 0.013 1.7100.665 0.839 0.640 0.0025 0.0065 2.4950.995 0.998 2.714 0.025 0.063 1.1170.895 0.957 2.135 0.015 0.038 1.451

303.38 0.795 0.910 1.567 0.005 0.013 2.1730.685 0.851 0.957 0.0025 0.0065 2.9080.645 0.826 0.713

308.10 0.0025 0.0065 2.9870.995 0.998 3.124 0.025 0.063 1.418

313.21 0.895 0.957 2.526 0.015 0.038 1.7790.795 0.910 1.907 0.005 0.013 3.0530.685 0.851 1.2270.995 0.998 3.167 0.0025 0.063 3.571

315.30 0.895 0.957 2.6120.795 0.910 1.957

to the validity limit of the short-time model. The concentrationfront reached the top and bottom of the cell at the observationtime tc=54.88 min, which means that the experimental data atlater times can only be used for the long-time model. Note thatthe diffusion experiments lasted for about 10 h at each state point.

Fig. 7 (b) illustrates the fitting result of the long-time model.The successive curves show the time evolution of the methanolmass fraction along the cell height. Panel (b) can be consideredas an extended version of the diffusion process shown in panel(a), but here mass fraction profiles are given instead of the phasedistribution. The fitting results obtained by the short-time andlong-time model are very similar, i.e. D = [1.655 and 1.656]×10−9m2s−1, respectively.

Fig. 7 (c) shows the mean value of the phase change with thesquare root of time. The phase, averaged over either the lower orupper half of the cell, indicates a linear behavior. Values for theFick diffusion coefficient obtained in different parts of the cell areidentical, they are written along the curves.

Fig. 7 (d) presents the variation of the measured quantity (Dt)with time from which the Fick diffusion coefficient was evaluatedby linear interpolation. It is important to point out that the lattertwo models are only valid for the time interval tc ≤54.88 min.

According to the values for the Fick diffusion coefficient indi-cated in Fig. 7, it could be concluded that the results from thedifferent models agree within 0.5 %. However, it has to be notedthat these data are given for the mixture composition w1= 0.895

kg/kg, which is far from the binodal curve. The dispersion of theFick diffusion coefficient data obtained by the four models slightlyincreases when approaching the binodal. The largest deviationof up to 5 % was found for the short-time model, while the dis-crepancy between other three models did not exceed 1.5 %. Theaverage value of the Fick diffusion coefficient obtained by thesefour models was considered as the final measurement result.

Averaged Fick diffusion coefficient data are listed in Table 2and illustrated in Fig. 8. All processing of the raw experimentaldata was carried out with the compositions expressed in terms ofmass fractions. However, for comparison with the present EMDsimulations it is convenient to switch to mole fractions. Accord-ingly, compositions in Table 2 are given both in mass and molefractions. The general trend in Fig. 8 is that the Fick diffusion co-efficient, as expected, significantly decreases towards the binodalat all considered temperatures.

Cyclohexane-rich region. The Fick diffusion coefficient has avery strong mass fraction dependence in this region, cf. Fig. 8(a). In its core part, it can be described by the power law

D(w1,T ) = D∗(T )w−γ

1 , (38)

where D∗ and γ are fitting parameters, with γ varying between0.535 (T=293.55 K) and 0.479 (T=313.21 K). However, such adependence cannot provide the asymptotic behavior neither to-wards w1→ 0 nor towards the spinodal.

To measure the diffusion coefficient close to infinite dilution


of methanol w1 →0, we implemented a dedicated experimentalseries with a very small composition difference between the twosolutions in contact. The smallness of the composition differencewas limited mainly by the optical resolution of the phase change.The minimum difference for the optical phase between the twosolutions was set to ϕ(−∞)−ϕ(∞) = 3× 2π rad. Using Eq. (16)and literature values of the optical contrast factor15, the neces-sary mass fraction difference between the two solutions was esti-mated as ∆w1 = 0.005 kg/kg, which is by a factor of two smallerthan for typical experiments. Filling the lower part of the cell withpure cyclohexane and the upper one with w1 = 0.005 kg/kg, thesmallest average mass fraction at which the Fick diffusion coeffi-cient was measured was w1=0.0025 kg/kg.

Methanol-rich region. In this region, the Fick diffusion coef-ficient decreases linearly towards the miscibility gap, cf. Fig. 8(b). A careful examination indicates that the mass fraction de-pendence for the lowest measured temperature T=293.55 K canbe approximated by a weakly quadratic polynomial function, butfor the other temperatures it follows a linear function

D(w1) = D(w1,re f )+α(w1−w1,re f ) . (39)

Here, any measured point can be selected as the reference. Theslope of the straight lines α [in 10−9m2s−1] varies between 5.08(T=293.55 K) and 6.13 (T=313.21 K). Furthermore, the temper-ature dependence of the slope can also be described by a linearfunction

α = 0.0551 ·T /K−11.123, 293.55K≤ T ≤ 313.21K . (40)

Applying these relationships to the data provided in Table 2, de-viations to individual values are about 2-3 %. This gives confi-dence that the Fick diffusion coefficient can be represented withEqs. (39) and (40) over a range of T and w1.

5.1.1 Infinite dilution diffusion coefficients

Because of the strongly different mass fraction dependence of Don both sides of the binodal curve, the estimation of the infinitedilution diffusion coefficient D∞

i requires an individual approachfor each side, cf. Fig. 8.

Infinite dilution of methanol (as a tracer) in cyclohexane. Inorder to accurately characterize the behavior of D(w1 → 0), theexperimental data points have to be described by an analyticalfunction and the limit of this function at w1→ 0 has to be consid-ered. The power law (38) describes the Fick diffusion coefficientin the core part well, but it diverges for w1 → 0. Thus, a sec-ond order polynomial was fitted to the measured Fick diffusioncoefficient data in the methanol-poor region and extrapolated tow1→ 0. These D∞

1 values are presented in Fig. 9 (top).Infinite dilution of cyclohexane (as a tracer) in methanol. The

linear behavior of D(w1) at the considered temperatures in themethanol-rich region allows for a straightforward extrapolationto the infinite dilution limit w1 →1. The results are shown inFig. 9 (bottom).

The extrapolated experimental data for the infinite dilution dif-fusion coefficient D∞

i were compared to well-known predictiveempirical equations summarized in Ref.46. The considered op-

D1∞

/ 10

-9 m

2 s-1

0

1

2

3

4

5

SimulationExperimental Wilke-Chang

T / K

290 295 300 305 310 315 320

D2∞

/ 10

-9 m

2 s-1

0

1

2

3

4

5

Fig. 9: Infinite dilution diffusion coefficient of methanol (as a tracer) incyclohexane (top) and of cyclohexane (as a tracer) in methanol (bottom).Experimental and simulation results are compared to the Wilke-Changequation 45.

tions were the Wilke-Chang, Tyn-Calus and Hayduk-Minhas cor-relations. The required input data, including the pure fluid molarvolume at normal boiling temperature for methanol47 and cyclo-hexane48, data and correlations for dynamic shear viscosity49,parachor50 and surface tension51 were taken from the literature.

Surprisingly, the best agreement with experimental data onboth sides of the binodal curve was found for the older Wilke-Chang correlation

D∞i = 7.4 ·10−8 (φ jM j)

0.5Tη jv0.6

i, i 6= j, i, j = 1,2 . (41)

Therein M j is the molar mass (g/mol), vi the molar volume(cm3/mol) at the normal boiling point, η j the shear viscosity(mPas), and T the temperature (K). The recommended value forthe association factor was used, which is different on both sides:for methanol φ = 1.9, for cyclohexane φ = 1.0. The results ofthe Wilke-Chang correlation are shown in Fig. 9. As expected,the agreement is better for infinite dilution of cyclohexane inmethanol.


wmethanol / kg kg-1

0.0 0.2 0.4 0.6 0.8 1.0

T /

K

270

280

290

300

310

320

Binodal experimental, literature dataSpinodal, extrapolated from Fick diffusionSpinodal, modified Wilson (LLE fit)

Fig. 10: Spinodal curve reconstructed on the basis of the experimentaldata; the dashed curve is a guide for the eye. The binodal curve wasmeasured by Matsuda et al. 1.

5.1.2 Spinodal reconstruction

As discussed in Section 2.2, the spinodal defines the limit of localstability, where the Fick diffusion coefficient vanishes D(w1,sp) =

0. On the basis of the present experimental data, a spinodal pointcan be found by assuming that the measured trend of isother-mal D(w1) data towards the binodal will continue until the Fickdiffusion coefficient becomes zero. The spinodal curve can thenbe reconstructed by applying this procedure on both sides of thetwo-phase region for each temperature. Since the Fick diffusioncoefficient does not vanish on the opposing sides of the miscibilitygap with the same rate, independent fits of the data on both sideswere necessary.

The methanol-rich region is straightforward for such a fit due tothe linear mass fraction dependence of D(w1) and a larger numberof well-distributed data points. Thus, the spinodal mass fraction(w1,sp) can be found by Eqs. (39) and (40).

As discussed above, the cyclohexane-rich region is more chal-lenging. The power law (38) cannot provide an asymptotic be-havior towards a vanishing diffusion coefficient. Moreover, dueto the experimental challenges discussed above, the measuredpoints are not well-distributed. For the missing points at eachtemperature, we have calculated diffusion coefficients using thepower law (38) up to the mass fraction w1=0.035 kg/kg. Then,a linear fit was applied to the last three points with the purposeto determine the slope towards the demixing gap. The spinodalmass fraction was evaluated according to that slope. These as-sumptions increase the uncertainty of the spinodal determined inthis region. It should be noted that the slope towards the spinodalon the cyclohexane-rich side varies from -11.2 (T=293.55 K) to-18.5 (T=313.21 K) [in 10−9m2s−1] , which is 2-3 times morepronounced than on the methanol-rich side. The result of thespinodal reconstruction is shown in Fig. 10.

5.2 Predictive results

5.2.1 Simulation results

EMD simulations were carried out at different compositions in thestable and metastable regions as well as at equimolar composi-tion. Even for compositions in the unstable region of the mixtureno phase separation could be observed during the simulations,which is a consequence of the small system size and the shortsimulation time of ∼ 15 ns.

Chemical potential data obtained by MC simulations indicateda too wide miscibility gap of the molecular mixture model, cf.supplementary material. When considering to adjust the unlikeinteractions of the molecular mixture model according to the ex-perimental LLE data, it was observed that the chemical potentialsare much more sensitive to such adjustments55 than the trans-port properties. Thus, no readjustment of the unlike interactionsbetween different molecule species was considered in this work.

Simulation results for three of the five studied isotherms areshown in Fig. 11. Numerical simulation data for density, intradif-fusion and MS diffusion coefficients at all temperatures are tabu-lated in the supplementary material.

Intradiffusion coefficients

The composition dependent intradiffusion coefficients ofmethanol and cyclohexane in their mixture are shown in Fig. 11.The intradiffusion coefficients of both components have the samemagnitude in almost the entire composition range, although themethanol molecule is significantly smaller and lighter than thecyclohexane molecule. This can be explained by the associationof methanol molecules, i.e. cluster formation by means ofhydrogen bonds, which slows down their mobility. The stronglyincreasing intradiffusion coefficient for diluted methanol is re-lated to hydrogen bond breaking because only very few methanolmolecules are present at such compositions. Methanol monomershave a much higher mobility, which is a common effect in dilutemixtures of an associating and a non-associating component.Hydrogen bonding statistics confirm this explanation as can beseen from EMD simulation data in the supplementary material.

Another interesting aspect of the computed intradiffusion coef-ficients is that no indication of the miscibility gap can be observed.The thermodynamic non-ideality of this mixture, which leads to avanishing Fick diffusion coefficient, does not affect the mobility ofthe molecules. This means that no direct conclusion can be drawnfrom the intradiffusion coefficients for mutual diffusion. The ab-sence of anomalous behavior was also confirmed for temperaturedependent intradiffusion coefficients around the critical point byother molecular simulation studies56,57.

Maxwell-Stefan coefficient

The MS diffusion coefficient as a function of mole fraction isshown in Fig. 11. The larger statistical uncertainties of the MScoefficient data compared to the intradiffusion coefficient dataare a consequence of the collective nature of the correspondingvelocity correlation functions. Extensive simulation runs of up to15 ns were carried out, which would not have been necessary forthe sampling of intradiffusion coefficients.

In the infinite dilution limits, the MS and the intradiffusion co-


D1

/ 10

-9 m

2 s-1

0

1

2

3

4 293.55 K303.38 K313.21 K

D2

/ 10

-9 m

2 s-1

0

1

2

3

4

xmethanol / mol mol-10.0 0.2 0.4 0.6 0.8 1.0

Ð /

10-9

m2 s

-1

0

2

4

6

8

10

12

14

Fig. 11: Intradiffusion coefficient of methanol (top), intradiffusion coef-ficient of cyclohexane (center) and MS diffusion coefficient (bottom) ofmethanol + cyclohexane at 0.1 MPa predicted by EMD simulation.

efficient of the diluted component must coincide. As expected,this condition is satisfied by the simulation results. The MS dif-fusion coefficient values strongly increase towards the miscibilitygap and show a significant and asymmetric mole fraction depen-dence. The steeper increase at low methanol concentrations isrelated to the hydrogen bonding interaction of methanol.

The persistent assumption that the MS diffusion coefficient isless composition dependent than the Fick diffusion coefficient28

even in non-ideal liquid mixtures has already been disproved, e.g.based on experimental58,59 and molecular simulation data60.The present results are consistent with that view.

In a simulation study of a binary Lennard-Jones mixture by Daset al.56 significant size effects were found for the Onsager coeffi-cients in the vicinity of the critical point. Simulations with moreparticles led to higher coefficients, which was explained by longrange fluctuations around the critical point. It can be presumedthat size effects also do affect the present simulation results to acertain extent, although 4000 particles were sampled.

Infinite dilution diffusion coefficient

Although simulations were also carried out at very low mole frac-tions (10−3 mol/mol), the diffusion coefficients at infinite dilu-tion were determined by extrapolation to xi → 0 and xi → 1. Inthe infinite dilution limit of a binary mixture the Fick, MS and in-tradiffusion coefficients must coincide. Because the simulated in-tradiffusion coefficients have smaller statistical uncertainties thanthe MS diffusion coefficient, the former were chosen for this ex-trapolation, while in the experimental part, the infinite dilutioncoefficients were extrapolated from the measured Fick diffusioncoefficient, cf. Section 5.1.1.

The less composition dependent intradiffusion coefficient of cy-clohexane was extrapolated by a linear fit towards infinitely di-luted cyclohexane (x1 → 1). For this fit, simulation results formole fractions of methanol between 0.8 and 1 mol/mol wereused. The results are slightly below the extrapolation of exper-imental data, cf. Fig. 9 (bottom).

The intradiffusion coefficient of methanol in the cyclohexane-rich region is much stronger composition dependent, similar tothe Fick diffusion coefficient. Towards the infinite dilution limit ofmethanol (x1 → 0) the intradiffusion coefficient exhibits a strongincrease to much higher values than for other compositions, cf.Fig. 11. Thus, a fifth order polynomial was fitted to the sim-ulated intradiffusion coefficient, employing data points for molefractions of methanol between 0 and 0.06 mol/mol. The resultinginfinite dilution coefficient of methanol is shown in Fig. 9 (top).This extrapolation of the intradiffusion coefficient of methanolyields higher values than D∞

1 obtained by extrapolation of themeasured Fick diffusion coefficient, while both extrapolations arehigher than the prediction from the Wilke-Chang equation.

5.2.2 Predictive Maxwell-Stefan equations

Four predictive equations were compared to simulation data forthe MS diffusion coefficient at 298.15 K, cf. Fig. 12. The cal-culations were based on simulated intradiffusion coefficients andtheir extrapolated values in the infinite dilution limit. For the twomodels based on local composition, the modified Wilson parame-



Ð /

10-9

m2 s

-1

0

2

4

6

8

10

12

14SimulationDarkenVignesLi et al.Zhou et al.

Fig. 12: Predictive equations based on the intradiffusion coefficients com-pared to simulation data for the MS diffusion coefficient at 298.15 K and0.1 MPa.

ters fitted to ACID data were used, cf. Table 3.It is evident from Fig. 12 that the classical models by Darken

and Vignes are not able to predict the non-ideal behavior of theMS diffusion coefficient. The Darken relation can be derived fromthe velocity correlations by neglecting all cross-correlations, cf.supplementary material. This assumption is not valid for associ-ating liquids where the motion of molecules is collective due tothe formation clusters61.

The expressions for the MS diffusion coefficient by Li et al.41

and Zhou et al.42 are more promising, whereas the former modelbased on the Darken equation exhibits an unusual behavior atvery low methanol concentrations. It leads to a minimum of theMS diffusion coefficient at a very low methanol mole fraction,which is a consequence of the steep decrease of the methanolintradiffusion coefficient. Moreover, there is a maximum at themethanol mole fraction of ∼0.18 mol/mol, but the predicted MSdiffusion coefficient is significantly below the simulation results.The model by Zhou et al., being based on the intradiffusion coef-ficients at infinite dilution, exhibits a maximum at the same com-position as the model by Li et al., but predicts higher values thansimulation. Both models show a more asymmetric compositiondependence of the MS diffusion coefficient than the simulationdata.

5.2.3 Thermodynamic behavior

The thermodynamic behavior of the studied mixture was ana-lyzed with a modified Wilson gE model43,44. For the calculationof the thermodynamic factor, its parameters fitted to experimen-tal LLE data by Matsuda et al.1 were used in a first step. Due toshortcomings in the prediction of ACID, a second parameter setwas fitted to experimental ACID data, cf. Table 3. It was foundfor five different gE models that it is not possible to fit a param-eter set which simultaneously represents the LLE and ACID in aquantitative manner. This was confirmed by other studies62,63

that found the UNIQUAC model to be unable to properly describe


Γ

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

293.55 K313.21 K

T /

K

280

290

300

310

320

330

340

Experiment

modified Wilson, binodalmodified Wilson, spinodal

T / K

280 300 320 340

γ i∞

0

50

100

150

200

250

Experiment, methanolExperiment, cyclohexanemodified Wilson, methanolmodified Wilson, cyclohexane

Fig. 13: Properties calculated with the modified Wilson model with pa-rameters fitted to experimental LLE data (black) or parameters fitted toexperimental ACID data (red) are compared to experimental data 1,2,64–72

at 0.1 MPa. Top: Activity coefficients at infinite dilution. Center: Tem-perature dependent liquid-liquid equilibrium. Bottom: Thermodynamicfactor.


Table 3: Parameters for the modified Wilson model as defined byEq. (32), the upper critical temperature is Tc = 319.13 K1.

Source ai j/R bi j/R ci j/R di j/RK1 K0 K−1 K−2

Fitted to LLE 1

12 866.348 12.2606 -0.39739 0.00769421 -19.6981 6.13503 -0.161120 0.002199

Fitted to ACID12 1230.53 7.54729 -0.07081 021 80.0109 2.55109 -0.00235 0

the composition dependence of activity coefficients in mixturescontaining associating components.

Results for the ACID, LLE and thermodynamic factor calculatedwith two different parameter sets of the modified Wilson model(Table 3) are compared in Fig. 13. The predicted spinodal curve,obtained form the LLE fit, is also compared to the extrapolatedexperimental data in Fig. 10. The LLE calculations based on theACID parameters lead to a too wide miscibility gap and a morenon-ideal thermodynamic factor. Moreover, the composition de-pendence of the thermodynamic factor is more asymmetric com-pared to the LLE fit, which seems to be consistent with the asym-metric behavior of the MS diffusion coefficient obtained by simu-lation.

5.2.4 Predicted Fick diffusion coefficient

The Fick diffusion coefficient was predicted by combining the sim-ulation results for the MS diffusion coefficient with the thermody-namic factor from the modified Wilson model with two differentparameter sets. The results are shown together with the experi-mental data in Fig. 14.

Calculations based on the LLE parameters lead to a false max-imum of the Fick diffusion coefficient at very low methanol con-centrations. This can be explained by the underestimated ACID,which leads to an incorrect slope of the thermodynamic factorat low concentrations. More notably, the calculations based onthe ACID parameters are in good agreement with the experimen-tal data. However, these parameters yield spinodal compositionsthat are even outside of the two-phase region at temperaturesabove 300 K.

It was observed that the slopes of the composition dependentMS diffusion coefficient and the thermodynamic factor are in-terrelated. The steep increase of the MS diffusion coefficient atlow methanol concentrations must be accompanied by a similarlysteep decrease of the thermodynamic factor to yield a moderatedecrease of the Fick diffusion coefficient. This interrelation showsthe stronger dependence of Fick diffusion coefficient on the ther-modynamic factor of this mixture. Kinetics, which is quantifiedby the intradiffusion and MS diffusion coefficients through themolecular velocity correlation functions, influence the Fick diffu-sion coefficient mostly in the infinite dilution limits of this mix-ture. This is confirmed by the agreement of the intradiffusion co-efficients at infinite dilution obtained by simulation and the trendof the measured Fick diffusion coefficient, cf. Fig. 14, as discussedabove.

An investigation similar to the present study was carried outby Krishna and van Baten18 for the partially miscible mixturemethanol + n-hexane. They calculated the MS coefficient fromintradiffusion coefficients obtained by EMD simulation with theDarken model and the thermodynamic factor from the NRTLmodel. It was found that the Darken relation is not capable topredict MS coefficients of this highly non-ideal mixture and thatdifferent gE models are not able to correlate the composition de-pendence of the activity coefficients correctly. Therefore we sup-pose that the shortcomings of both models may cancel each otherout, such that good results can be obtained for the Fick diffusioncoefficient with this combination of models.

6 ConclusionsA comprehensive study of diffusion coefficients of the binary liq-uid mixture of methanol + cyclohexane was conducted. Differentcomplementary approaches, i.e. experiment, molecular simula-tion and predictive equations, were considered in this work togain an understanding of the behavior of this highly non-idealmixture around its liquid-liquid equilibrium (LLE) at ambientpressure.

The Fick diffusion coefficient was measured in a diffusion cellwith interferometric probing at different temperatures and com-positions around the binodal. To obtain reliable results, four dif-ferent models were used to process the experimentally sampleddata. As expected, the Fick diffusion coefficient increases towardshigher temperatures and decreases towards the miscibility gap.The Fick diffusion coefficient in the methanol-rich compositionrange shows a linear mass fraction dependence, whereas it hasa more pronounced dependence in the cyclohexane-rich compo-sition range that was correlated by a power law. The measuredFick diffusion coefficient data were further analyzed by extrapo-lations. First, the coefficients were extrapolated to both infinitedilution limits to obtain them for infinitely diluted methanol andcyclohexane. A second extrapolation was carried out into the mis-cibility gap to determine the spinodal compositions, where theFick diffusion coefficient is zero. While the extrapolation on themethanol-rich side could be done straightforwardly, the extrapo-lation in the cyclohexane-rich region was much more difficult sothat different functions were fitted towards this limit.

Intradiffusion and Maxwell-Stefan (MS) diffusion coefficientswere computed by equilibrium molecular dynamics (EMD) simu-lations with the Green-Kubo formalism. Both intradiffusion coef-ficients have a similar magnitude and composition dependence,except for low methanol concentrations, where the intradiffu-sion coefficient of methanol rises significantly due to hydrogenbond breaking. The intradiffusion coefficients do not exhibit anotable change around the miscibility gap. The extrapolated in-finite dilution diffusion coefficients are in reasonable agreementwith the experimental data. The simulated MS diffusion coeffi-cient is strongly composition dependent in case of this non-idealmixture and not as well behaved as it is often assumed. Classicalpredictive equations for the MS diffusion coefficient, i.e. Darkenand Vignes, are not able to predict this composition dependence,while newer approaches based on local composition models alsoshow a non-ideal behavior of the MS diffusion coefficient, as ob-


D /

10-9

m2 s-1

0

1

2

3

4

5

6

xmethanol / mol mol-10.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

D /

10-9

m2 s-1

0

1

2

3

4

5

6

xmethanol / mol mol-10.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

D /

10-9

m2 s-1

0

1

2

3

4

5

6

Metastable region (modified Wilson, LLE fit) Unstable region (modified Wilson, LLE fit) Fick coefficient (experimental)MS coefficient × thermodyn. factor (LLE fit) MS coefficient × thermodyn. factor (ACID fit) Intradiffusion at infinite dilution (simulation) Lapeira et al. (2017)Story and Turner (1996)

315.30 K

303.38 K

313.21 K

298.15 K

293.55 K

Fig. 14: Fick diffusion coefficient at different temperatures. Experimental results from this work and the literature 14,15 are compared to predictedcoefficients from simulated MS diffusion coefficients multiplied with the thermodynamic factor obtained from the modified Wilson model that was eitherfitted to experimental LLE data or experimental ACID data. The intradiffusion coefficient of the diluted component must coincide with the Fick diffusioncoefficient in the infinite dilution limits.


served by molecular simulation.

To consider the thermodynamic behavior of the studied mix-ture, a modified Wilson excess Gibbs energy (gE) model was used.It was not possible to find parameters, which are able to repre-sent the activity coefficients at infinite dilution and the LLE be-havior of this mixture consistently, neither with the modified Wil-son model, nor with four other gE models. Because the modifiedWilson model is not capable to yield the correct composition de-pendence of the activity coefficients, it is not surprising that thepredicted spinodal curve does not coincide with the extrapolationfrom experimental data.

The thermodynamic factor, which is needed for the transforma-tion between MS and Fick diffusion coefficients, was calculatedwith the modified Wilson model employing two different parame-ter sets. The predicted Fick diffusion coefficient obtained with theactivity coefficient at infinite dilution (ACID) fit of the modifiedWilson model coincides better with measured values than the LLEfit. This emphasizes the strong connection between the composi-tion dependence of the activity coefficients and the Fick diffusioncoefficient. The analysis shows that the Fick diffusion coefficientof a partially miscible mixture is governed by the kinetics of sin-gle particles, i.e. intradiffusion coefficients, only in the infinitedilution limits. Apart from these compositions, the Fick diffusioncoefficient is governed by the thermodynamic non-ideality of thepresent mixture.

Conflicts of interest

There are no conflicts to declare.

Acknowledgements

The authors AM and VS would like to acknowledge the PRODEXprogram of the Belgian Federal Science Policy Office. The teamof Xi’an Jiaotong University (SZ and MH) acknowledges supportby the National Science Fund for Distinguished Young Scholarsof China (No. 51525604). The molecular simulation work wascarried out under the auspices of the Boltzmann-Zuse society andwas funded by Deutsche Forschungsgemeinschaft under the grantVR 6/11-1. The simulations were carried out on the Cray XC40system (Hazel Hen) at the High Performance Computing CentreStuttgart (HLRS) within the project MMHBF2.

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Journal Name

Supplementary material to:

Mutual diffusion governed by kinetics and thermodynamics in

the partially miscible mixture methanol + cyclohexane

by

Tatjana Janzena, Shi Zhangb,c, Aliaksandr Mialdunb, Gabriela Guevara-Carriona, Jadran Vrabec∗a,Maogang Hec and Valentina Shevtsova∗b

A Molar excess volume S 2

B Shear viscosity S 3

C Thermodynamic relations S 4

D Comparison of excess Gibbs energy models S 6

E Hydrogen bonding statistics and simulation snapshots S 7

F Chemical potentials from Monte Carlo simulation S 8

G Darken relation from velocity correlation functions S 9

H Numerical simulation data S 10

a Thermodynamics and Energy Technology, University of Paderborn, 33098 Paderborn, Germany. E-Mail: [email protected] Microgravity Research Center, Université Libre de Bruxelles (ULB), CP–165/62, Av. F.D. Roosevelt, 50, B–1050 Brussels, Belgium. E-mail:[email protected] MOE Key Laboratory of Thermal Fluid Science and Engineering, School of Energy and Power Engineering, Xi’an Jiaotong University, Xi’an710049, China

S 1

A Molar excess volume

The molar excess volume vE was calculated from the measured specific mixture density ρm by

vE =2

∑i=1

xiMi

(1

ρm− 1

ρ0m,i

), (1)

where Mi is the molar weight and ρ0m,i is the specific density of the pure component i.


vE /

cm3 m

ol-1

0.0

0.2

0.4

0.6

0.8

Experiment, this workSimulation, this workLapeira et al.Canosa et al.

Fig. 1: Excess molar volume of methanol + cyclohexane at 298.15 K and 0.1 MPa compared to experimental data fromthe literature1,2.

S 2

B Shear viscosity

Simulation

The shear viscosity η can be obtained from equilibrium molecular dynamics simulations by means of the corre-lation function of the off-diagonal elements of the stress tensor Jxy

p

η =1

V kBT

∫∞

0dt⟨Jxy

p (t) · Jxyp (0)

⟩, (2)

where V is the total volume. The component Jxyp of the microscopic stress tensor Jp is given by3

Jxyp =

N

∑k=1

mkvxkvy

k−12

N

∑k=1

N

∑l 6=k

rxkl

∂u(rkl)

∂ rykl

. (3)

Here, k and l denote different molecules of any species. The upper indices x and y stand for the spatial vectorcomponents, e.g. for velocity vx

k or site-site distance rxkl. Five independent terms of the stress tensor, i.e. Jxy

p , Jxzp ,

Jyzp , (Jxx

p − Jyyp )/2 and (Jyy

p − Jzzp )/2, were considered to improve statistics4.


η / 1

0-4 P

a s

0

2

4

6

8

10

Simulation, this workLapeira et al.

Experiment, this work

Fig. 2: Shear viscosity of methanol + cyclohexane at 298.15 K and 0.1 MPa compared to experimental data from theliterature1.

S 3

C Thermodynamic relations

Chemical potential

The chemical potential µi of component i in a binary mixture is defined as the partial derivative of the Gibbsenergy with respect to the mole number ni at constant temperature T , pressure p and mole number of thesecond component n j

µi =

(∂G∂ni

)T,p,n j

. (4)

It can be subdivided into three parts

µi = µ0i +RT ln(xi)+RT ln(γi) , (5)

with the chemical potential of the pure component µ0i , the mole fraction xi and the activity coefficient γi.

Gibbs energy

The molar Gibbs energy at constant temperature and pressure is

g = G/n = ∑i

xiµi ,

= ∑i

xiµ0i +RT ∑

ixi ln(xi)+RT ∑

ixi ln(γi) = ∑

ixiµ

0i +RT ∑

ixi ln(xiγi) ,

= gideal,0 +gideal,mix +gE = gideal,0 +gmix .

(6)

Liquid-liquid equilibrium

In addition to thermal and mechanical equilibrium, the chemical equilibrium condition for two phases I and IImust hold for all components i at the binodal

µ(I)i = µ

(II)i ,

(xiγi)(I) = (xiγi)

(II) ,(∂gmix

∂xi

)(I)

T,p=

(∂gmix

∂xi

)(II)

T,p.

(7)

Fig. 3 shows the mole fraction dependence of the activities ai = xiγi, molar Gibbs energy of mixing gmix andthermodynamic factor Γ of a partially miscible liquid mixture at constant temperature and pressure. The binodalcondition is fulfilled in Fig. 3 at the compositions marked by the outer limits of the light gray areas, whichindicate the metastable region. Stability ceases to exist at the spinodal, where

(∂ µi

∂xi

)T,p

= 0 ,

(∂ ln(xiγi)

∂xi

)T,p

= 0 ,

S 4

(∂ 2gmix

∂x2i

)T,p

= 0 , (8)

which is fulfilled in Fig. 3 at the compositions marked by the outer limits of the dark gray area, which indicatesthe unstable region. The thermodynamic factor Γ is zero at the spinodal compositions.

ai

0.88

0.90

0.92

0.94

0.96

0.98

1.00

1.02

component 2component 1

gm

ix /

RT

-0.10

-0.08

-0.06

-0.04

-0.02

0.00

x / mol mol-10.0 0.2 0.4 0.6 0.8 1.0

Γ

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

Fig. 3: Mole fraction dependence of activities (top), molar Gibbs energy of mixing (center) and thermodynamic factor(bottom) of a system with liquid-liquid equilibrium at constant temperature and pressure.

S 5

D Comparison of excess Gibbs energy models

Five excess Gibbs energy gE models were assessed for their ability to describe the composition dependent activitycoefficients in the mixture methanol and cyclohexane: Van Laar, Margules, NRTL, UNIQUAC and a modifiedWilson model. Initially, all models were fitted to experimental LLE data, resulting in an adequate reproductionof the binodal curve. The spinodal curve predicted by this set of models shows a moderate variance, cf. Fig. 4.However, all models significantly underestimate the activity coefficients at infinite dilution (ACID), especially formethanol, cf. Fig. 5. Note that the upper critical temperature is not accurately met by several models. Becausethe modified Wilson model showed the best results for the ACID it was used for subsequent calculations of theFick diffusion coefficient.


T /

K

280

290

300

310

320Experiment, Atik et al.Experiment, Matsuda et al.modified Wilson, binodalmodified Wilson, spinodalVan Laar, spinodalMargules, spinodalNRTL, spinodalUNIQUAC, spinodal

Fig. 4: Temperature dependent liquid-liquid equilibrium of methanol + cyclohexane at 0.1 MPa. Experimental data forthe binodal5,6 and spinodal curves calculated with five gE models fitted to experimental LLE data are shown.

T / K

280 300 320 340

γ i∞

0

50

100

150

200

250Experiment, methanolExperiment, cyclohexanemodified Wilson, methanolmodified Wilson, cyclohexaneVan Laar, methanolVan Laar, cyclohexaneMargules, methanolMargules, cyclohexaneNRTL, methanolNRTL, cyclohexaneUNIQUAC, methanolUNIQUAC, cyclohexane

Fig. 5: Temperature dependent activity coefficients at infinite dilution at 0.1 MPa. Experimental data7–15 and calculateddata with five gE models are shown.

S 6

E Hydrogen bonding statistics and simulation snapshots

Hydrogen bonding statistics

Hydrogen bonding statistics were obtained by molecular simulation on the basis of geometric criteria. FollowingHaughney et al.16, two methanol molecules are considered as hydrogen bonded when the distance between anoxygen (acceptor) site and a hydrogen (donor) site of two molecules is below 2.6 Å, the distance between theoxygen (acceptor) sites is less than 3.5 Å and the angle between the acceptor acceptor axis and the acceptordonor axis of one molecule is below 30◦.


frac

tion

of m

etha

nol m

olec

ules

0.01

0.1

1

monomerdimertrimertetramer

Fig. 6: Fractions of methanol molecules in different association states due to hydrogen bonding in the mixture methanol+ cyclohexane at 298.15 K and 0.1 MPa.

Simulation snapshots

The simulation snapshots in Fig. 7 indicate that the methanol molecules do not only constitute dimers totetramers through hydrogen bonding, but also form larger clusters depending on the methanol concentration.

Fig. 7: Simulation snapshots of methanol + cyclohexane at 298.15 K and 0.1 MPa for methanol mole fractions of 0.05,0.2 and 0.5 mol mol−1. Cyclohexane molecules are not visible, while methanol molecules are fully depicted.

S 7

F Chemical potentials from Monte Carlo simulation

Chemical potentials were computed by thermodynamic integration17,18 in the isobaric-isothermal (N pT ) en-semble by Monte Carlo simulation. With these simulations the configurational part of the reduced chemicalpotentials µ̃i = µ∗i /(RT ) was obtained. The activity coefficients can be calculated from these data by

lnγi = µ̃i− µ̃0i − lnxi , (9)

where µ̃0i is the chemical potential of pure component i at the same temperature and pressure.

γ i

1

10

100

1000Cyclohexane (mod. Wilson, LLE fit)Methanol (mod. Wilson, LLE fit)

Methanol (simulation)Cyclohexane (simulation)

Methanol (mod. Wilson, ACID fit) Cyclohecane (mod. WIlson, ACID Fit)


gE /

RT

0.0

0.2

0.4

0.6

0.8

1.0

1.2mod. Wilson, LLE fit

Simulationmod. Wilson, ACID fit

µ i

-13

-12

-11

-10

-9

-8

-7

Methanol Cyclohexane

˜

Fig. 8: Chemical potentials, activity coefficients and excess Gibbs energy of methanol + cyclohexane at 298.15 K and0.1 MPa from simulation compared to the modified Wilson model based on parameters fitted to experimental LLE data orexperimental ACID data.

S 8

G Darken relation from velocity correlation functions

The Darken relation for the Maxwell-Stefan (MS) diffusion coefficient can be derived from velocity correlationfunctions. As discussed in the article, the MS coefficient Ð was computed in this work from19

Ð =x j

xiΛii +

xi

x jΛ j j−Λi j−Λ ji , (10)

with

Λi j =1

3N

∫∞

0

⟨ Ni

∑k=1

vki (0) ·

N j

∑l=1

vlj(t)⟩

dt . (11)

The phenomenological coefficients Λi j contain the velocity cross-correlations for particles of different species iand j. On the other hand, the coefficients Λii can be expanded to two types of terms: the auto-correlation func-tions, which correlate the velocity of one particle with itself, and cross-correlation functions between differentparticles belonging to the same species i. Considering this, the MS coefficient can be rewritten as20–22

Ð = xiDaj + x jDa

i + xix j(Dcii +Dc

j j−Dci j−Dc

ji) . (12)

Superscript a denotes auto-correlations of individual particles and these coefficients coincide with the intra-diffusion coefficients, i.e. Di = Da

i . Superscript c denotes cross-correlations between different particles. Thereare two types of cross-correlations: between different particles of the same species, denoted by lower indicesii and j j, and correlations between particles of different species, denoted by lower indices i j and ji, whereDc

i j = Dcji. Neglecting the cross-correlations entirely leads to the Darken relation

Ð = xiD j + x jD j . (13)

It can be concluded that the Darken relation is valid for mixtures where the velocity of single molecules is onlycorrelated to itself, which is fulfilled e.g. in the case of gases when molecules hardly interact with each othergeneral. The Darken relation is also valid when the different cross-correlations cancel each other out, e.g. in anearly ideal liquid (or solid) mixture of very similar molecules.

S 9

H Numerical simulation data

Table 1: Simulation results for molar density ρ, intradiffusion coefficients Di and Maxwell-Stefan diffusioncoefficient Ð in methanol + cyclohexane at 0.1 MPa.

x1 ρ err D1 err D2 err Ð errmol mol−1 mol l−1 10−9m2s−1 10−9m2s−1 10−9m2s−1

T = 293.55 K0.000 9.2728 0.0003 1.457 0.0020.001 9.2769 0.0000 2.919 0.067 1.458 0.002 3.11 0.150.005 9.2984 0.0000 2.374 0.029 1.464 0.002 3.75 0.180.010 9.3263 0.0004 2.002 0.020 1.468 0.002 4.70 0.220.020 9.3821 0.0004 1.782 0.014 1.469 0.002 5.05 0.230.030 9.4402 0.0004 1.731 0.011 1.476 0.002 5.47 0.260.040 9.4992 0.0004 1.682 0.010 1.484 0.002 6.32 0.310.050 9.5589 0.0004 1.638 0.009 1.487 0.002 5.69 0.290.060 9.6192 0.0002 1.610 0.008 1.490 0.002 6.05 0.280.100 9.8676 0.0005 1.594 0.006 1.502 0.002 6.31 0.310.150 10.1995 0.0004 1.602 0.005 1.518 0.002 6.92 0.330.200 10.5604 0.0005 1.617 0.005 1.535 0.002 8.10 0.380.250 10.9533 0.0006 1.643 0.005 1.538 0.002 10.17 0.470.500 13.3866 0.0000 1.770 0.004 1.638 0.003 12.17 0.550.750 17.2878 0.0000 1.923 0.003 1.771 0.004 7.04 0.300.799 18.4190 0.0011 1.929 0.004 1.799 0.004 6.07 0.270.850 19.6178 0.0000 1.980 0.003 1.844 0.004 4.52 0.190.910 21.3601 0.0007 2.065 0.003 1.923 0.005 2.98 0.120.951 22.7354 0.0008 2.135 0.004 1.960 0.007 2.43 0.120.960 23.0793 0.0012 2.150 0.004 1.942 0.007 2.38 0.110.970 23.4508 0.0014 2.170 0.004 1.981 0.008 2.36 0.100.980 23.8420 0.0016 2.189 0.003 1.952 0.010 2.08 0.100.990 24.2529 0.0013 2.218 0.004 1.958 0.014 2.18 0.101.000 24.6802 0.0014 2.240 0.004T = 298.15 K0.000 9.2189 0.0003 1.568 0.0020.001 9.2246 0.0003 3.150 0.077 1.568 0.002 3.72 0.180.005 9.2452 0.0003 2.497 0.031 1.572 0.002 4.65 0.220.010 9.2734 0.0003 2.174 0.021 1.576 0.002 4.93 0.220.020 9.3296 0.0003 1.937 0.015 1.585 0.002 5.03 0.250.030 9.3875 0.0003 1.863 0.012 1.589 0.002 5.73 0.260.040 9.4462 0.0003 1.830 0.010 1.594 0.002 6.07 0.270.050 9.5050 0.0003 1.790 0.009 1.599 0.002 6.35 0.290.060 9.5627 0.0002 1.789 0.008 1.609 0.002 7.34 0.320.100 9.8149 0.0004 1.720 0.007 1.619 0.002 7.10 0.330.150 10.1486 0.0004 1.726 0.006 1.638 0.002 9.21 0.400.200 10.5031 0.0004 1.751 0.005 1.659 0.002 10.10 0.440.250 10.8869 0.0005 1.756 0.005 1.660 0.003 9.53 0.450.500 13.3289 0.0006 1.903 0.004 1.758 0.003 11.37 0.490.750 17.2244 0.0007 2.051 0.003 1.894 0.004 6.95 0.280.800 18.3089 0.0009 2.074 0.003 1.915 0.004 5.41 0.23

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Table 1 – continued from previous page


0.850 19.5554 0.0009 2.135 0.004 1.943 0.004 4.29 0.190.910 21.2301 0.0007 2.255 0.003 2.062 0.005 3.40 0.140.950 22.5869 0.0011 2.330 0.004 2.103 0.007 2.66 0.120.960 22.9475 0.0012 2.356 0.004 2.116 0.008 2.52 0.110.970 23.3209 0.0012 2.367 0.004 2.151 0.010 2.46 0.110.980 23.7028 0.0013 2.400 0.004 2.143 0.011 2.41 0.110.990 24.1134 0.0013 2.415 0.004 2.136 0.016 2.14 0.101.000 24.5170 0.0012 2.447 0.004T = 303.38 K0.000 9.1583 0.0003 1.709 0.0030.001 9.1628 0.0000 3.474 0.075 1.701 0.002 3.63 0.170.005 9.1823 0.0000 2.871 0.036 1.709 0.002 4.31 0.200.010 9.2094 0.0004 2.533 0.023 1.720 0.003 5.29 0.250.020 9.2653 0.0004 2.150 0.016 1.720 0.002 6.02 0.280.030 9.3233 0.0004 2.068 0.013 1.729 0.002 6.07 0.280.040 9.3794 0.0004 2.005 0.012 1.737 0.002 5.97 0.290.050 9.4390 0.0004 1.957 0.010 1.733 0.003 6.81 0.340.060 9.4988 0.0002 1.940 0.009 1.744 0.002 6.74 0.290.100 9.7435 0.0004 1.889 0.007 1.764 0.002 7.60 0.340.150 10.0723 0.0005 1.899 0.006 1.781 0.003 8.78 0.390.200 10.4227 0.0006 1.910 0.006 1.793 0.003 10.21 0.470.250 10.8172 0.0006 1.918 0.005 1.794 0.003 10.42 0.480.500 13.2123 0.0000 2.088 0.004 1.919 0.003 12.27 0.520.751 17.1049 0.0006 2.254 0.004 2.065 0.004 6.42 0.290.790 17.9511 0.0006 2.287 0.004 2.090 0.005 5.80 0.260.850 19.3600 0.0000 2.379 0.004 2.175 0.005 5.14 0.220.910 21.0882 0.0000 2.478 0.004 2.240 0.006 3.49 0.140.950 22.4319 0.0013 2.550 0.004 2.277 0.008 2.87 0.130.960 22.7978 0.0012 2.584 0.004 2.290 0.009 2.72 0.130.970 23.1670 0.0014 2.603 0.004 2.304 0.009 2.46 0.110.980 23.5592 0.0017 2.639 0.004 2.339 0.012 2.54 0.120.990 23.9660 0.0014 2.663 0.004 2.347 0.016 2.62 0.111.000 24.3869 0.0015 2.690 0.004T = 313.21 K0.000 9.0419 0.0004 1.975 0.0030.001 9.0468 0.0000 3.970 0.084 1.975 0.003 4.14 0.180.005 9.0666 0.0000 3.720 0.037 1.982 0.003 4.80 0.200.010 9.0932 0.0004 2.983 0.026 1.988 0.003 5.77 0.270.020 9.1478 0.0004 2.666 0.019 1.994 0.003 6.66 0.300.030 9.2029 0.0004 2.475 0.015 2.004 0.003 7.14 0.310.040 9.2593 0.0004 2.400 0.013 2.008 0.003 7.23 0.350.050 9.3188 0.0004 2.341 0.011 2.020 0.003 8.26 0.370.060 9.3765 0.0002 2.313 0.011 2.022 0.003 8.18 0.370.100 9.6184 0.0005 2.254 0.008 2.044 0.003 8.57 0.400.150 9.9411 0.0005 2.253 0.007 2.069 0.003 11.38 0.490.200 10.2873 0.0005 2.249 0.006 2.087 0.003 12.23 0.50

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Table 1 – continued from previous page


0.250 10.6613 0.0000 2.274 0.005 2.111 0.003 11.01 0.480.500 13.0388 0.0000 2.454 0.005 2.229 0.003 13.25 0.540.740 16.6920 0.0006 2.639 0.004 2.353 0.005 8.22 0.350.790 17.6996 0.0006 2.737 0.005 2.435 0.005 7.49 0.330.840 18.8957 0.0007 2.802 0.005 2.494 0.006 5.80 0.260.910 20.8199 0.0008 2.955 0.005 2.602 0.007 4.16 0.180.950 22.1495 0.0013 3.051 0.004 2.665 0.008 3.30 0.140.960 22.5125 0.0013 3.085 0.005 2.677 0.009 3.28 0.140.970 22.8824 0.0014 3.114 0.005 2.704 0.010 2.85 0.120.980 23.2699 0.0015 3.148 0.005 2.722 0.013 3.09 0.130.990 23.6701 0.0013 3.172 0.005 2.720 0.018 2.88 0.121.000 24.0885 0.0014 3.208 0.005T = 315.31 K0.000 9.0175 0.0004 2.038 0.0030.001 9.0226 0.0000 4.093 0.086 2.037 0.003 3.98 0.190.005 9.0419 0.0000 3.514 0.041 2.039 0.003 4.74 0.240.010 9.0684 0.0004 3.140 0.027 2.043 0.003 5.77 0.250.020 9.1227 0.0004 2.780 0.020 2.059 0.003 7.00 0.310.030 9.1780 0.0004 2.620 0.015 2.063 0.003 6.82 0.310.040 9.2346 0.0004 2.500 0.013 2.070 0.003 7.62 0.340.050 9.2926 0.0004 2.438 0.012 2.077 0.003 7.75 0.340.060 9.3500 0.0002 2.407 0.010 2.085 0.003 8.08 0.340.100 9.5907 0.0004 2.307 0.009 2.106 0.003 9.04 0.440.150 9.9123 0.0005 2.312 0.007 2.129 0.003 9.83 0.450.200 10.2570 0.0005 2.340 0.006 2.157 0.003 11.46 0.520.250 10.6477 0.0006 2.334 0.006 2.144 0.003 12.03 0.540.500 13.0021 0.0000 2.544 0.005 2.293 0.004 11.83 0.520.750 16.8022 0.0000 2.784 0.004 2.484 0.004 8.47 0.350.810 18.0907 0.0006 2.863 0.004 2.549 0.005 7.05 0.290.850 19.0533 0.0000 2.942 0.004 2.622 0.006 5.58 0.220.910 20.7610 0.0008 3.061 0.004 2.682 0.007 4.06 0.170.950 22.0907 0.0014 3.175 0.005 2.763 0.009 3.63 0.150.960 22.4619 0.0014 3.184 0.005 2.735 0.010 3.16 0.140.970 22.8141 0.0014 3.228 0.005 2.789 0.011 3.25 0.140.980 23.2044 0.0015 3.258 0.005 2.787 0.013 3.13 0.130.990 23.6076 0.0014 3.299 0.005 2.839 0.018 3.01 0.131.000 24.0242 0.0013 3.326 0.005

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mutual diffusion governed by kinetics and thermody- · cal temperature of ˘319k. the spinodal...

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