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Measurements of molecular and thermal diffusion coefficientsin ternary mixtures

Alana Leahy-Diosa�

Department of Chemical Engineering, Yale University, New Haven, Connecticut 06520-8286

Mounir M. Bou-Alib�

Mondragon Unibertsitatea, Loramendi, 4 Apartado 23, Mondragon 20500 Spain

Jean K. Plattenc�

General Chemistry Service, University of Mons-Hainaut, Mons, Avenue du Champs de Mars 24, Belgium

Abbas Firoozabadid�

Department of Chemical Engineering, Yale University, New Haven, Connecticut 06520-8286

�Received 2 February 2005; accepted 1 April 2005; published online 17 June 2005�

Thermal diffusion coefficients in three ternary mixtures are measured in a thermogravitationalcolumn. One of the mixtures consists of one normal alkane and two aromatics�dodecane-isobutylbenzene-tetrahydronaphthalene�, and the other two consist of two normal alkanesand one aromatic �octane-decane-1-methylnaphthalene�. This is the first report of measured thermaldiffusion coefficients �for all species� of a ternary nonelectrolyte mixture in literature. The results internary mixtures of octane-decane-1-methylnaphthalene show a sign change of the thermal diffusioncoefficient for decane as the composition changes, despite the fact that the two normal alkanes aresimilar. In addition to thermal diffusion coefficients, molecular diffusion coefficients are alsomeasured for three binaries and one of the ternary mixtures. The open-end capillary-tube methodwas used in the measurement of molecular diffusion coefficients. The molecular and thermaldiffusion coefficients allow the estimation of thermal diffusion factors in binary and ternarymixtures. However, in the ternaries one also has to calculate phenomenological coefficients fromthe molecular diffusion coefficients. A comparison of the binary and ternary thermal diffusionfactors for the mixtures comprised of octane-decane-1-methylnaphthalene reveals a remarkabledifference in the thermal diffusion behavior in binary and ternary mixtures. © 2005 AmericanInstitute of Physics. �DOI: 10.1063/1.1924503�

I. INTRODUCTION

Molecular and thermal diffusion coefficients provide theproportionality relation between diffusion flux in mixturesunder concentration and temperature gradients, respectively.These coefficients, especially the thermal diffusion coeffi-cients, are very sensitive to the interaction potentials betweendifferent molecules.

There have been extensive experimental and theoreticalstudies of thermal diffusion coefficients in binary mixtures inliterature. To the best of our knowledge, there is no report ofmeasured thermal diffusion coefficients in ternary nonelec-trolytes and higher mixtures; Leaist and Hui1 have measuredthe Soret coefficients of ternary mixtures of electrolytes us-ing a conductometric technique, which is suitable for elec-trolyte systems. The technique works well for dilute solu-tions. Using a forced Rayleigh scattering technique, Ganset al.2 have measured the Soret coefficients of a polymer anda colloid in ternary mixtures of a polymer and a colloid in awater-ethanol solvent, treating the ternary mixture as a

pseudobinary; Gans et al.3 and Kita et al.4 have measured thepolymer Soret coefficient in ternary mixtures of the samepolymer in a water-ethanol solvent mixture. They reported asign change in the Soret coefficient as the composition of thesolvent mixture changes. The work reported in Refs. 2–4 isbased on the assumptions of �1� a negligible cross-moleculardiffusion and �2� a fluid phase ideality, in the sense that thereis no volume change on mixing.

There is more complexity in the diffusion coefficients�thermal, molecular, and pressure� of ternary mixtures incomparison to those of binary mixtures. As an example, inthe calculation of thermal diffusion factors in a ternary mix-ture, the phenomenological coefficients of molecular diffu-sion are required, in addition to thermal diffusioncoefficients.5 Even for molecular diffusion, the appearance ofcross-molecular diffusion complicates the coefficients in ter-nary and higher mixtures in comparison to those in binarymixtures.

The basic goal of this work is the measurement of ther-mal diffusion coefficients and thermal diffusion factors internary mixtures. Because molecular diffusion coefficientsare required in the evaluation of thermal diffusion factors, wehave also measured molecular diffusion coefficients for one

a�Electronic mail: [email protected]�Electronic mail: [email protected]�Electronic mail: [email protected]�Author to whom correspondence should be addressed. Electronic mail:

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THE JOURNAL OF CHEMICAL PHYSICS 122, 234502 �2005�

0021-9606/2005/122�23�/234502/12/$22.50 © 2005 American Institute of Physics122, 234502-1

of the three ternary mixtures to be described shortly. Thesemeasurements are believed to provide a base for the theoret-ical work in thermal diffusion.

There are three major methods which have been used inthe measurement of thermal diffusion coefficients in binarymixtures: �1� the two-bulb approach, �2� the thermogravita-tional column, and �3� the optical methods. The use of thefirst two methods has been limited to binaries. Optical meth-ods have been used for binaries and for only one species internary mixtures. The working equations for the measure-ment of thermal diffusion coefficients in ternary mixtures canbe found in the literature for the thermogravitational-columntechnique.6,7 The thermogravitational-column technique is,therefore, selected in this work because it provides a measur-able separation for all species, the availability of workingequations, and a well-developed theory.

We used two types of ternary mixtures in our study. Inone system, a ternary mixture of n-dodecane, isobutylben-zene, and tetrahydronaphthalene, with equal mass fractionfor all components, was used. Different techniques have beenused for the measurement of the molecular and thermal dif-fusion coefficients of the binary mixtures for the above ter-nary system; the results from these various techniques are ingood agreement.8 In the second ternary system, we have se-lected a mixture comprised of two similar molecules,n-octane and n-decane, and a third molecule of a differentshape, 1-methylnaphthalene; n-decane and 1-methyl-naphthalene have similar molecular weights but differentdensities �different molecular shapes�. Two different compo-sitions in the second ternary system have been selected. Inone mixture, all components have equal mass fractions. Inthe second mixture, the mass fraction of1-methylnaphthalene is four times greater than that of thetwo normal alkanes. In addition to the thermal diffusion co-efficients in ternaries, we also measured the thermal diffu-sion coefficients of the three binaries of n-octane, n-decane,and 1-methylnaphthalene. These measurements are con-ducted to examine the additive rules for the estimation ofternary thermal diffusion coefficients from binary thermaldiffusion coefficients, as suggested by Larre et al.9 We alsomeasured the molecular diffusion coefficients of the binaryand ternary mixtures of n-octane, n-decane, and1-methylnaphthalene to evaluate thermal diffusion factors.

This paper is organized as follows: In Sec. II we brieflydescribe the materials and equipment. We also provide thetechnique for compositional analysis and measurements ofthe molecular and thermal diffusion coefficients. In Sec. IIIwe present the mathematical expressions for the determina-tion of the molecular and thermal diffusion coefficients. Sec-tion IV provides the results of the measurements and a briefdiscussion. We conclude the work in Sec. V.

II. EXPERIMENTS

A. Materials and equipment

Table I shows the binary and ternary mixtures used inthis work. The normal alkanes are referred to by their num-ber of carbons atoms �nC8, nC10, and nC12�; 1-methyl-naphthalene is referred to as MN, isobutylbenzene as IBB,

and tetrahydronaphthalene as THN. All reagents were pur-chased from Acros Organics, with 99% purity or higher, ex-cept for 1-methylnaphthalene �97% purity� and tetrahy-dronaphthalene �98% purity�; all fluids were used withoutfurther refining. Measurements for the ternary mixture 1�nC12-THN-IBB� were performed at 298.15 K �temperatureused in previous work10 with the corresponding binaries� andatmospheric pressure. Measurements for the other binariesand ternaries were performed at 295.65 K and atmosphericpressure.

The density was measured by the Anton PAAR densim-eter DMA 5000, with an estimated temperature fluctuation of±0.005 K and an accuracy of ±2�10−6 g /cm3. The refrac-tive index measurements were performed using aBellingham+Stanley refractometer RFM 340 of accuracy±4�10−5, with a water bath for temperature control of±0.01 K. The samples were weighted on a digital scale witha precision of ±0.0001 g. The viscosity measurements wereperformed on a Haake Falling-Ball viscometer. The waterbaths used with the thermogravitational column had a tem-perature control of ±0.01 K.

B. Composition analysis

In the determination of both molecular and thermal dif-fusion coefficients, the compositional analysis of the fluidmixture is required. Most analytical methods are not suffi-ciently sensitive for the determination of composition to ourdesired accuracy. A very accurate compositional analysis canbe inferred from the calibration curves for density in binarymixtures and calibration planes for density and refractive in-dex in ternary mixtures, provided that the density and refrac-tive index of the individual components are significantly dif-ferent from each other. For our working conditions, the idealliquid mixture density and refractive index expressions, �= ��i��i /�i

0��−1 and n=�ini0xi, do not provide a reliable ap-

proach for the calculation of the composition from densityand refractive index measurements due to nonideality ��i

0 isthe density of pure component i, ni

0 is the refractive index ofpure component i, xi is the mole fraction of component i, �i

is the mass fraction of component i, and � and n are themixture mass density and refractive index, respectively�.However, we can express the density and refractive index aslinear functions of the composition. In a ternary mixture,from �=��1 ,�2� and n=n��1 ,�2�, we can write

TABLE I. Composition of binary and ternary mixtures.

Mixture Weight ratio of components

nC12-IBB-THN 1 : 1 : 1 �mixture 1�nC8-nC10-MN 1 : 1 : 1 �mixture 2�

nC8-MN 1 : 1nC10-MN 1 : 1nC8-nC10 1 : 1

nC8nC10-MN 1 : 1 : 4 �mixture 3�nC8-MN 1 : 4nC10-MN 1 : 4

234502-2 Leahy-Dios et al. J. Chem. Phys. 122, 234502 �2005�

� � �0 +��

��1��1 − �1

0� +��

��2��2 − �2

0� , �1�

n � n0 +�n

��1��1 − �1

0� +�n

��2��2 − �2

0� , �2�

where �0=�0��10 ,�2

0� and n0=n0��10 ,�2

0� are the referencemass density and refractive index, respectively, and �1

0 and�2

0 are the mass fraction of components 1 and 2 for the mix-ture of interest. For small composition changes, the deriva-tives �� /��i and �n /��i �i=1,2� are constant, and Eqs. �1�and �2� can be written as

� = a0 + a1�1 + a2�2, �3�

n = b0 + b1�1 + b2�2. �4�

In Eqs. �3� and �4�, ai and bi �i=0,1 ,2� are constant param-eters. From the measurements of � and n at various massfractions close to �1

0 and �20, we obtain the constants ai and

bi. Then, for a sample of unknown composition, the massfractions �1 and �2 are computed from � and n by

�1 =b2�� − a0� + a2�b0 − n�

a1b2 − a2b1, �5�

�2 =a1�n − b0� − b1�� − a0�

a1b2 − a2b1. �6�

The mass fraction of the third component is obtained from�3=1−�1−�2. For a binary mixture, we can use Eq. �3�with �=a0+a1�1; therefore, density measurements are suffi-cient for calibration.

For every binary we measure the density of several mix-tures with a composition close to the mixture of interest.Then the constants a0 and a1 are determined from the dataset of � and �1. Our measurements show that the mass ex-pansion coefficient �= �1/�0�� /��1� is constant in therange of our calibration curve from Eq. �3� for a binary mix-ture. For a ternary mixture we require two calibration planes�that is, Eqs. �3� and �4��. We prepared about thirty sampleswith a composition close to the reference sample for compo-sitional analysis. As an example, for the ternary mixture 2�see Table I�, where the mass fraction of each components is0.333, 33 samples of known composition were prepared. Themass fraction of each component was varied from 0.303 to0.363 with increments of 0.01 between the mixtures.

C. Measurement of molecular diffusion coefficients

The experimental setup and technique for the measure-ment of the binary and ternary molecular diffusion coeffi-cients were the same as those used by Dutrieux et al.,11

which we briefly outline. Capillary tubes, of 2.91-cm heightand 1.5-cm3 total volume filled with a homogeneous solutionof mass fraction �1,0 �for the binary mixture�, are immersedin a bath of large volume �approximately 400 cm3� with ahomogeneous solution of mass fraction �1,� �for the binarymixture�. The solution inside the tubes is more concentratedin the heavier component and denser, and therefore bulk flowdue to gravity is avoided as the solution inside the tubes

diffuses. After approximately every 8 h, a tube is taken outof the bath and its average mass fraction ��1� is determinedby density measurement. Double-jacketed baths were used tocirculate water with a water bath of 0.1-K temperature fluc-tuations, maintaining a constant temperature.

For the ternary mixtures, we measure the refractive in-dex in addition to the density of the liquid in the tubes. Thefollowing analytical expressions for the time variation of thedensity and refractive index of the ternary liquid mixtures inthe tubes provide the composition change of all three spe-cies, as we will discuss later.

��t� = c0 + c1 exp�− c2t� , �7�

n�t� = d0 + d1 exp�− d2t� . �8�

The parameters c and d are constants. To the best of ourknowledge, this is the first time that the open-end capillary-tube technique is extended for the measurement of moleculardiffusion coefficients for ternary mixtures.

D. Measurement of thermal diffusion coefficients

The experimental setup and technique for the measure-ment of binary and ternary thermal diffusion coefficientswere the same as those used by Dutrieux et al.;11 they will bediscussed here briefly. The column used is shown schemati-cally in Fig. 1. For binary mixtures, we determine the com-position of the five samples taken, at steady state, along thecolumn by density measurements.

For the ternary mixtures, we measure both the densityand refractive index. In our experiments the compositionvariation is small in the column �thus Eqs. �3� and �4� arevalid�, and the density and refractive index are both observedto be linear functions of the column height; therefore, thecomposition variation inside the column is also linear. Thedensity and refractive index variations with height can bewritten as

��z� = h�z + c�, �9�

n�z� = hnz + cn. �10�

We combine Eqs. �3� and �4� and Eqs. �9� and �10� to elimi-nate � and n and obtain expressions for the composition at

FIG. 1. Schematic of thermogravitational column for the measurement ofthermal diffusion coefficients; dimensions are Lx=1.59±0.02 mm, Ly

=3.0 cm, Lz=53.0 cm.

234502-3 Diffusion in ternary mixtures J. Chem. Phys. 122, 234502 �2005�

each sampling point as a function of the sampling pointheight z and the constants h�, c�, hn, and cn

�1 =z�b2h� − a2hn� + b2�c� − a0� − a2�cn − b0�

a1b2 − a2b1, �11�

�2 =z�a1hn − b1h�� − b1�c� − a0� + a1�cn − b0�

a1b2 − a2b1. �12�

The mass fraction of the third component is found by massbalance �3=1−�1−�2.

III. MATHEMATICAL FORMULATION

In this section we first present the expressions that areused to evaluate the molecular diffusion coefficients in bothbinary and ternary mixtures in a one-dimensional tube. Wethen present the expressions that can be used in the evalua-tion of the thermal diffusion coefficients and factors in bi-nary and ternary mixtures.

A. Molecular diffusion coefficient

1. Binary mixtures

For a binary mixture in the one-dimensional tube, themass balance of the heavier component is written as

��1��t

=�

�zD�

��1

�z , �13�

where D is the molecular diffusion coefficient. When densityis constant, Eq. �13� is simplified to

��1

�t= D

�2�1

�z2 . �14�

The initial and boundary conditions are, respectively,

�1�z,0� = �1,0, �1�0,t� = �1,�,� ��1

�z�

z=L= 0. �15�

In the above equation, �1,0 is the initial mass fraction ofcomponent 1 and �1,� is the mass fraction of the same com-ponent at the tube outlet �the composition in the large con-tainer�.

The integration of Eq. �14� and the use of the boundaryand initial conditions given in Eq. �15� yield the concentra-tion

�1 − �1,� =4��1,0 − �1,��

�

��m=0

� sin�m + 1/2��

Lzexp−

�2�m + 1/2�2

L2 Dt�2m + 1�

.

�16�

The average mass fraction of component 1 inside the tube attime t is given in

��1� − �1,�

= �8��1,0 − �1,��2 �

m=0

� exp−�2�m + 1/2�2

L2 Dt�2m + 1�2 .

�17�

After a long enough period of time, the dominant term of theseries in Eq. �17� is m=0; therefore, using m=0 and rear-ranging Eq. �17�, we obtain the working equation

4L2

�2 ln 8��1,0 − �1,��2��1� − �1,�� = Dt . �18�

The slope of �4L2 /�2�ln�8��0−�� /�2��−�� versus tgives D for the binary mixtures.

In some of our experiments, the mass fraction at the tubeoutlet �1,� changes slightly due to evaporation. Also, thedensity varies with both time and position, and therefore Eq.�14� is not strictly valid. For these reasons, we will use Eq.�18� to obtain a first estimate for the molecular diffusioncoefficients and solve Eq. �13� using finite difference to ob-tain a more precise value for D. Our finite-difference schemeconsists of 1000 nodes and about 700 equal time steps.

2. Ternary mixtures

The governing equations for the molecular diffusion of aternary system can be written in matrix form as

�

�t��1

��2� = �D11 D12

D21 D22� �

�z�

�

�z��1

�2� . �19�

When density is constant, we obtain

��1

�t

��2

�t = �D11 D12

D21 D22��

�2�1

�z2

�2�2

�z2 � . �20�

The initial and boundary conditions for components 1 and 2are the same as the ones for the binary mixture, given in Eq.�15�.

The system of Eq. �20� can be solved by decoupling thetwo equations, as done by Cussler,12 and applying the resultsof the binary mixtures for each decoupled equation. The ana-lytical solution to Eq. �20� is given in the Appendix. Theaverage mass fraction inside the tubes is given in

��1� − �1,� =4

��2 − �1��1,0 − �1,��2 − D11�S̄1 + ��2

− D22�S̄2� + ��2,0 − �2,��D12�S̄2 − S̄1�� , �21�

��2� − �2,� =4

��2 − �1��1,0 − �1,��D21�S̄2 − S̄1�

+ ��2,0 − �2,��2 − D11�S̄2

+ ��2 − D22�S̄1�� , �22�

where


S1 =2

��m=0

� exp−�2�m + 1/2�2

L2 �1t�2m + 1�2 , �23�

S2 =2

��m=0

� exp−�2�m + 1/2�2

L2 �2t�2m + 1�2 . �24�

There are two working expressions, Eqs. �21� and �22�,and four unknown molecular diffusion coefficients. A non-linear regression is used to determine the molecular diffusioncoefficients for the ternary mixture from average concentra-tions measured in the tubes at various times.

Similar to the binary mixtures, Eqs. �21� and �22� areused to obtain the initial values for the molecular diffusioncoefficients, and Eq. �19� is solved numerically by a finite-difference scheme with 500 nodes and about 240 equal timesteps. A nonlinear regression is again used to minimize theresidue between the experimental values of compositions andthose obtained numerically by the finite-difference scheme.

In the nonlinear regression, the coefficients are generallycalculated using a least-squares approach.13 Nonphysicalmolecular diffusion coefficients may arise from the calcula-tions; therefore, constraints are implemented to avoid un-physical coefficients. Taylor and Krishna14 provide the fourrestrictions on the molecular diffusion coefficients

D11 � 0, �25�

D22 � 0, �26�

D11D22 − D12D21 � 0, �27�

�D11 − D22�2 + 4D12D21 0. �28�

We use a least-squares procedure that follows the sim-plex method; it does not require analytical or numerical gra-dients. The minimized variable is the root mean square �rms�of the difference between the calculated and measured valuesfor the average composition of components 1 and 2 in thetubes.

B. Thermal diffusion coefficient

In a multicomponent mixture, the mass diffusion flux isgenerally given in

Ji = − ��i=1

n−1

Dij � � j + DiT � T . �29�

In Eq. �29� Dij is the mass-based molecular diffusion coeffi-cient �the mole-based molecular diffusion coefficients will bepresented later� and Di

T is the thermal diffusion coefficient ofcomponent i �it is also mass based�.

There is confusion in the definition of terms for the ther-mal diffusion coefficients in literature. Sometimes the secondterm of the flux expression is assigned a negative sign arbi-trarily. The sign convention adopted for binary mixtures bysome authors often conflicts with thermodynamic stabilityanalysis.5 In binary mixtures when the thermal diffusion co-

efficient �also the thermal diffusion factor and ratio, as wellas the Soret coefficient� is negative, the component shouldsegregate to the hot side. For ternary and higher mixtures,the sign of the thermal diffusion coefficient does not neces-sarily determine the direction of component migration. In aternary mixture when cross-molecular diffusion coefficient isnonzero, the composition gradient of, say, species one is re-lated to the composition gradient of species two, and thesame is true for the temperature gradient. Depending on themagnitude of the thermal diffusion and cross molecular dif-fusion coefficients, species one may segregate to the coldside with either positive or negative thermal diffusion coef-ficient.

Many authors use a different form of Eq. �29�, and vari-ous authors use different symbols; they define a so-called

thermal diffusion coefficient D– 12T which relates to D1

T in abinary mixture by

D1T = �1�1 − �1�D– 12

T . �30�

In order to relate binary and ternary thermal diffusion coef-ficients, Larre et al.9 suggested an additive rule. In their for-mulation, the thermal diffusion coefficients D1

T and D2T of

species 1 and 2 in a ternary mixture are given in

D1T = D– 12

T �1�2 + D– 13T �1�3, �31�

D2T = D– 21

T �2�1 + D– 23T �2�3. �32�

Furry et al.15 developed a theory for determining thethermal diffusion coefficients of binary gas mixtures in athermogravitational column, and Majumdar16 generalized itto concentrated solutions. More recently, Marcoux andCharrier-Mojtabi6 have extended the theory to ternary mix-tures, and Haugen and Firoozabadi7 made a generalizationfor multicomponent systems. The simplifying assumptionsare that the aspect ratio Lz /Lx of the thermogravitational col-umn is very high �so that the horizontal velocity componentis negligible�, there is no vertical temperature gradient, thelinear Boussinesq approximation is valid, and the density isonly a function of T �the so-called “Forgotten effect”�. In ourexperiments, the column dimensions allow us to neglect themolecular diffusion in the vertical direction in comparison tothe vertical convection. The following approximate equa-tions give the thermal diffusion coefficients for component iin a binary and in a ternary mixture,6,7 respectively,

D– iT = −

�0gLx4

504��i,0�1 − �i,0��i

Lzfor a binary mixture,

�33�

DiT = −

�0gLx4

504�

��i

Lzfor a ternary mixture. �34�

Note that D– iT and Di

T represent different definitions of thesecoefficients. In Eqs. �33� and �34�, is the thermal expansioncoefficient �=−�0

−1�� /�T�, g is the gravitational accelera-tion, Lx is the column width, � is the viscosity, ��= ��top

−�bottom�, and Lz is the height of the column. Once the mo-lecular and thermal diffusion coefficients are determined for


a binary mixture, the thermal diffusion factor 1T can be de-

termined in

1T =

D– 1TT

D. �35�

The thermal diffusion factor is perhaps the most usefulparameter in comparing the thermal diffusion effect in binarymixtures. It is dimensionless and has less compositional de-pendency than the thermal diffusion ratio and thermal diffu-sion coefficient Di

T. While in a binary mixture the thermaldiffusion factor can be calculated without the need for thephenomenological coefficients, this is not generally the casein a ternary mixture. In a ternary system, the thermal diffu-sion factor i

T is given in5

iT =

DiTMTcMiMc

MRLii, i = 1,2, �36�

where c is the molar density of the mixture, Mi and Mc arethe molecular weights of component i and the reference com-ponent, respectively, M is the average molecular weight ofthe mixture, R is the universal gas constant, Di

TM is the mole-based thermal diffusion coefficient, which is related to themass-based thermal diffusion coefficients from Eq. �34� byDi

TM =DiTM, and Lii is the phenomenological coefficient,

which is related to the mole-based molecular diffusion coef-ficients in a ternary mixture by17

L11 =cM3x3

R�c2c3 − c1c4��c2D11

M − c4D12M � , �37�

L12 =cM3x3

R�c2c3 − c1c4��c2D21

M − c4D22M � , �38�

L21 =− cM3x3

R�c2c3 − c1c4��c1D11

M − c3D12M � , �39�

and

L22 =− cM3x3

R�c2c3 − c1c4��c1D21

M − c3D22M � . �40�

The parameters c1–c4 are defined in

c1 =M1x1 + M3x3

M1f12 + x2f22, �41�

c2 = x1f12 +M2x2 + M3x3

M2f22, �42�

c3 =M1x1 + M3x3

M1f11 + x2f21, �43�

c4 = x1f11 +M2x2 + M3x3

M2f21. �44�

In Eqs. �41�–�44� fkj = �� ln fk /�xj�, which is the deriva-tive of the fugacity of component k with respect to mole

TABLE II. Composition and other relevant data for the binary mixtures, at 295.65 K and 1 atm. Component 1is MN, except in the last row, where it is nC10. The parameters a0 and a1 are the coefficients for the calibrationcurves in the binaries �see Eq. �3��. The accuracy of the viscosity measurements is 0.7% on average.

Mixture �1

a0

�g/cm3�a1

�g/cm3� �

�10−3 K−1��0

�g/cm3��

�10−3 Pa s�

nC8-MN 0.50 0.678 31 0.311 69 0.373 66 −0.9518 0.834 289 0.947nC10-MN 0.50 0.707 28 0.289 10 0.339 39 −0.8977 0.851 688 1.320nC8-MN 0.80 0.635 06 0.377 80 0.403 08 −0.8242 0.937 313 1.669nC10-MN 0.80 0.671 93 0.341 73 0.361 50 −0.8026 0.945 24 1.965nC8-nC10 0.50 0.700 55 0.027 47 0.038 46 −1.0911 0.714 246 0.678

FIG. 2. Calibration planes for �a� density and �b� refractive index of theternary mixture 2 at T=295.65 K. The apparent solid line is a plane that fitsthe data. The three-dimensional �3D� view of the planes has been chosen toshow the small data dispersion. Similar calibration planes were obtained forthe ternary mixtures 1 and 3.


fraction of component j. The related term � ln fk /� ln xj pro-vides a measure of nonideality.18 The relationship betweenthe matrices of the mole-based Dij

M=

and the mass-based Dij=

molecular diffusion coefficient is given in

Dij= = DijM=

�MAiBi� + I=�Xij= , �45�

where Ai= ��i /Mi�, Bi= �Mi /Mc−1�, Xi== � ij /Mij� and I= is theidentity matrix.

IV. RESULTS AND DISCUSSION

In this section we will first present the expansion coef-ficients and other relevant data of binary and ternary mix-tures followed by the molecular diffusion coefficients ofnC8-nC10, nC8-MN and nC10-MN. Then, the molecular dif-fusion coefficients of ternary mixture 2 are reported. Next,we present the thermal diffusion coefficients of thenC8-nC10, nC8-MN, and nC10-MN binaries and those of thethree ternary mixtures.

A. Expansion coefficients and calibration curves/planes

Table II shows the expansion coefficients, viscosity, andother parameters for the binary mixtures. In the determina-tion of viscosity, we conducted several measurements foreach sample.

For the ternary mixtures, from the density and refractiveindex measurements of about 30 samples, the equations ofthe two calibration planes are obtained by the least squares.The maximum errors �absolute value of the maximum differ-ence between the calculated and the experimental values� ofthe density calibration planes were 1.141�10−4 g /cm3 formixture 1, 4.046�10−4 g /cm3 for mixture 2, and 9.823�10−5 g /cm3 for mixture 3; the maximum errors for therefractive index calibration plane were 6.710�10−5 g /cm3

TABLE III. Thermal expansion coefficient, density, viscosity, and linearcoefficients ai and bi �i=0,1 ,2� for the calibration planes in Eqs. �3� and �4�for the ternary mixtures. For ternary mixture 1, T=298.15 K, component 1is THN and component 2 is nC12. For ternary mixtures 2 and 3, T=295.65 K, component 1 is MN and component 2 is nC8. The accuracy ofthe viscosity measurements is 0.6% on average.

Mixture 1 Mixture 2 Mixture 3

�1 0.333 0.333 0.666�2 0.333 0.333 0.167

a0 �g/cm3� 0.848 83 0.721 99 0.693 17a1 �g/cm3� 0.103 37 0.252 08 0.312 61a2 �g/cm3� −0.117 76 −0.031 05 −0.038 25

b0 1.484 64 1.406 53 1.387 74b1 0.048 05 0.173 71 0.213 82b2 −0.073 51 −0.018 03 −0.020 79

�10−3 K−1� 0.934 4 0.981 7 0.861 9� �g/cm3� 0.844 214 0.795 668 0.895 133

� �10−3 Pa s� 1.370 0 0.933 3 1.424 4

TABLE IV. Mass-based D and mole-based DM molecular diffusion coeffi-cients, phenomenological coefficients L, and the derivatives � ln f i /� ln xj forthe ternary mixture 2. Component 1 is nC8 and component 2 is nC10; thereference component is 1-methylnaphthalene, T=295.65 K.

Subscriptsij

Dij

�10−9 m2/s�DM

�10−7 m2 g/s /gmol�Lij

�10−6 g K s/m3� � ln f i /� ln xj

1 1 1.99 2.36 8.04 0.871 2 −0.93 −1.24 −4.08 −0.122 1 −0.42 0.13 −4.98 −0.202 2 2.40 3.67 9.34 0.82

FIG. 3. �a� Plot of data for the estimation of molecular diffusion coefficientsin the binary mixtures; �b� composition profile for the three binary mixtures:DnC8-nC10=2.19�10−9 m2/s, DnC8-MN=1.10�10−9 m2/s, and DnC10-MN

=0.853�10−9 m2/s, at T=295.65 K.

FIG. 4. Density and refractive index of liquid mixture in capillary tubes atdifferent times for the ternary mixture 2 at 295.65 K. Note that the refractiveindex data is shown on the left axis. The coefficients for the expressions ofdensity and refractive index as functions of time in Eqs. �7� and �8� are c0

=0.7866 g/cm3, c1=0.0130 g/cm3, and c2=0.0108 h−1 and d0=1.4522, d1

=0.0089, and d2=0.0106 h−1. The maximum relative error is less than 0.1%for both density and refractive index.


for mixture 1, 2.335�10−4 g /cm3 for mixture 2, and 1.101�10−4 g /cm3 for mixture 3. Figure 2 shows the density andrefractive index planes for mixture 2. Similar results wereobtained for mixtures 1 and 3 �not shown here for the sake ofbrevity�. Table III presents the linear coefficients for the cali-bration planes of ternary mixtures 1, 2, and 3 and other im-portant parameters.

B. Molecular diffusion coefficients

1. Binary mixtures

Figure 3�a� shows a plot of �4L2 /�2�ln�8��1,0

−�1,�� /�2��1�−�1,�� versus time for the binary mixturesnC8-nC10, nC8-MN, and nC10-MN, all with 50 wt % of eachcomponent. The slopes of the lines give the approximatevalue of D, in m2/s. Note that because we have used only thefirst term of the series in Eq. �17�, the plot does not passthrough the origin. Figure 3�b� depicts the results for thefinite-difference scheme which yields the smallest error be-tween the experimental and numerical values; the moleculardiffusion coefficients are shown in the same figure. The un-certainty of the measurements is estimated to be of the orderof 5%.

Lo19 measured the molecular diffusion coefficients forthe binary mixtures nC7-nC10 and nC8-nC14 at 25 °C forvarious mole fractions. Interpolating the results in Ref. 19,the molecular diffusion coefficients at equal mass fractionsare 2.12�10−9 m2/s and 1.08�10−9 m2/s for the mixtures

nC7-nC10 and nC8-nC14, respectively. We expect the molecu-lar diffusion coefficient of the binary nC8-nC10 to be close tothat of nC7-nC10 and within the range above. Our value of2.19�10−9 m2/s �Fig. 3�b�� agrees well with the value fornC7-nC10, within the slight temperature difference and ex-perimental error.

2. Ternary mixtures

Figure 4 shows the measured density and refractive in-dex of the liquid in the tubes versus time for ternary mixture2. The fit of Eqs. �7� and �8� to the data in this figure is alsoshown as dashed lines. These equations describe the mea-sured data well. From the density and refractive index at timet, we determine the mass fraction of the three componentsusing Eqs. �5� and �6�. The mass-based molecular diffusioncoefficients using constrained nonlinear regression for thefinite-difference scheme are listed in Table IV. The mole-based molecular diffusion coefficients DM �from Eq. �45��and the phenomenological coefficients L �from Eqs.�37�–�44�� are shown in Table IV. The maximum rms for thenonlinear regression is 0.0050. The cross-phenomenologicalcoefficients L12 and L21 have similar values, which is ex-pected by the Onsager reciprocal relation. Figure 5 shows thevariation of the average composition for all the components,from experiments and from calculations using the finite-difference scheme. There is good agreement between the ex-perimental average composition data and those obtained bythe linear regression. We estimate the error of the moleculardiffusion coefficient calculation to be in the order of 30%. In

TABLE V. Thermal diffusion coefficients, thermal diffusion factors and other relevant data for the binarymixtures, at 295.65 K. Component 1 is MN, except for the 3rd row, where it is nC10.

Mixture �1

�� /�z�10−4 g /cm3/cm�

��1 /�z�10−4 cm−1�

D– 12T

�10−12 m2 s−1 K−1� T

nC8-MN 0.50 −0.8412±0.0079 −2.698±0.025 11.26±0.19 3.01nC10-MN 0.50 −0.7233±0.0055 −2.502±0.019 7.21±0.12 2.50nC8-nC10 0.50 −0.0079±0.0003 −0.286±0.012 1.64±0.09 0.22nC8-MN 0.80 −0.8781±0.0140 −2.324±0.037 8.37±0.17 ¯

nC10-MN 0.80 −0.7011±0.0074 −2.052±0.022 6.16±0.09 ¯

FIG. 5. Average composition variation in the capillary tube in the ternarymixture 2 at 295.65 K. The initial composition inside the capillary tubes is33.5% nC8 and 30.9% nC10; the composition of the mixture outside thecapillary tubes is 36.0% nC8 and 32.4% nC10.

FIG. 6. Composition vs height in thermogravitational column for the binarymixture nC10-MN at 295.65 K. Results are for duplicate runs.


this work we used the Peng–Robinson equation of state, withappropriate critical properties and shift parameters, to calcu-late the derivatives fkj =� ln fk /�xj in Eqs. �41�–�44�. The de-rivatives are also shown in Table IV. Because � ln fk /� ln xj

and � ln fk /� ln xk deviate from zero and one, respectively,the ternary mixture 2 does not behave ideally.

C. Thermal diffusion coefficients

1. Binary mixtures

Figure 6 depicts the density variation in the thermogravi-tational column at steady state with height for the nC10

-MN mixture; note that MN segregates to the bottom of thecolumn. We performed duplicate runs with excellent agree-ment. Similar results were obtained for the other four binarymixtures �not shown for brevity�. Table V shows the slopesof the density and mass fraction for all binary mixtures,along with the thermal diffusion coefficients from Eq. �33�and the thermal diffusion factors from Eq. �35� for the threemixtures for which we determined the molecular diffusioncoefficients. The mixture nC8-nC10 has a small thermal dif-fusion coefficient and a large molecular diffusion coefficient,therefore T is very small for this system; that is, a tempera-ture gradient does not have a large separation effect for the

nC8-nC10 mixture. The thermal diffusion factors for the mix-tures MN-nC8 and MN-nC10 are large and do not differ sig-nificantly, since the two n-alkane molecules differ only bytwo carbon atoms.

2. Ternary mixtures

Figure 7 shows the variation of the density and refractiveindex along the column for ternary mixture 2. The linear

FIG. 7. �a� Density and �b� refractive index vs height of thermogravitationalcolumn for the ternary mixture 2 at 295.65 K; results are for duplicate runs.Similar results were obtained for the ternary mixtures 1 and 3.

FIG. 8. Composition vs height of thermogravitational column for �a� ternarymixture 1 �T=298.15 K�, �b� ternary mixture 2 �T=295.65 K�, and �c� ter-nary mixture 3 �T=295.65 K�. Note that for �c� the mass fraction of MN isshown on the left axis.


variation validates the expressions of the density and refrac-tive index as linear functions of height, as described in Eqs.�9� and �10�. Similar results were obtained for mixtures 1 and3. Figure 8 depicts the essence of our measurements for thethree ternary mixtures, where the change in composition isshown for all the components for mixtures 1, 2, and 3. Notethat the thermogravitational-column technique can detectsmall changes in composition, such as for IBB shown in Fig.8�a�. The results presented in Fig. 8 for the three ternarymixtures reveal that the thermogravitational column is a vi-able tool for the measurements of thermal diffusion coeffi-cients in ternary mixtures. We believe this technique can alsobe used in multicomponent mixtures with more than threecomponents; the challenge is the compositional analysis forsuch mixtures.

In ternary mixtures 2 and 3, the component with inter-mediate density, nC10, has a positive slope in mixture 2, be-having as a light component and segregating to the top of thecolumn, and a negative slope in mixture 3, behaving as aheavy component and segregating to the bottom of the col-umn. The sign change in slope results in a sign change of thethermal diffusion coefficient of nC10. That is, a change in thecomposition of the mixtures causes a sign change in the ther-mal diffusion coefficient of nC10. Despite similar propertiesof nC8 and nC10, the ternary mixture nC8-nC10-MN does notbehave as a pseudobinary. The change in sign of the thermaldiffusion coefficients has been widely reported in the litera-ture for binary mixtures.20,21 References 2–4 report a signchange in the Soret coefficient of polymers and colloids withcomposition change of the solvent mixture, for mixtures ofpolymers and colloids in two solvents. For poly�ethylene ox-ide� �PEO� mixtures, the thermal diffusion coefficient ofPEO changes from negative to positive as the water contentincreases; for the colloidal boehmite ��-AlOOH� rods, the

thermal diffusion coefficient changes from positive to nega-tive with increasing water content. Note that for the ternarymixtures used in this work we cannot define a Soret coeffi-cient in the same way it is defined in Refs. 2–4 due to fluidnonideality �volume change on mixing� and nonzero cross-molecular diffusion coefficients.

For ternary mixture 3, which has greater difference inmass fractions between the components, the separation isgreater for nC8 and nC10 than for ternary mixture 2. A com-parison of the segregation results for the binary and ternarymixtures for the species nC8, nC10, and MN suggests signifi-cant differences in the behavior of thermal diffusion. Refer-ence 1, which is the only literature source for the measure-ment of thermal diffusion coefficients �more specificallySoret coefficients� in ternary mixtures �for electrolytes� to thebest of our knowledge, also reveals a remarkable differencebetween the thermal diffusion in binary and ternary mixturesin electrolytes. In the NaOH-water and NaCl-water binariesand the NaOH-NaCl-water ternary, NaOH and NaCl segre-gate to the cold side in the binary mixtures, but NaCl segre-gates to the hot side in the ternary mixture.

Table VI summarizes the results for the ternary mixtures,including the thermal diffusion factor i

T for ternary mixture2. Note that there is a large difference between the thermaldiffusion factors of nC8 �nC8

T =−3.01� and nC10 �nC10T =

−2.50� in the binary mixtures nC8-MN and nC10-MN andthose of the same components in the ternary mixture 2�nC8

T =−0.64;nC10T =−0.46�.

In Table VII we compare the measured thermal diffusioncoefficients to those obtained using the additive rule de-scribed in Eqs. �31� and �32�. The binary data used for thenC12-THN-IBB mixture was obtained from a previouswork.10 For the ternary mixture 1, the additive rule resultsare in agreement with the data. For the ternary mixture 2, theagreement deteriorates. For ternary mixture 3, the rule pre-dicts poorly the thermal diffusion coefficients of nC8 andnC10. Even the sign from Eq. �32� is incorrect for the thermaldiffusion coefficient of nC10.

V. CONCLUSIONS

We have measured the thermal diffusion coefficients fortwo types of ternary mixtures using the thermogravitational-column technique. This is the first report of such measure-ments in a ternary mixture in the literature for a nonelectro-lyte system. The results demonstrate that thermal diffusioncoefficients can be measured in a thermogravitational col-umn.

TABLE VII. Comparison of the measured and calculated thermal diffusioncoefficients in the three mixtures. The calculated thermal diffusion coeffi-cients are based on the additive expression from Larre et al.9 For ternarymixtures 1 �T=298.15 K� component 1 is nC12 and component 2 is THN.For ternary mixtures 2 and 3 �T=295.65 K�, component 1 is nC8 and com-ponent 2 is nC10. The reference component is IBB �ternary mixture 1� andMN �ternary mixtures 2 and 3�.

DternaryT

�10−12 m2 s−1 K−1�

Mixture 1 Mixture 2 Mixture 3

Literaturetheory Data



Component 1 −1.07 −1.02 −1.43 −1.19 −0.97 −2.99Component 2 0.97 0.87 −0.62 −0.98 −0.64 1.29

TABLE VI. Thermal diffusion coefficients and relevant data for the three ternary mixtures. For ternary mixture1 �T=298.15 K�, component 1 is nC12, component 2 is THN, and the reference component is IBB. For ternarymixtures 2 and 3 �T=295.65 K�, component 1 is nC8, component 2 is nC10, and the reference component isMN.

Mixture �1 �2

D1T

�10−12 m2 s−1 K−1�D2

T

�10−12 m2 s−1 K−1� 1T 2

T

1 0.333 0.333 −1.02±0.33 −0.87±0.43 ¯ ¯

2 0.333 0.333 −1.19±0.66 −0.98±0.06 −0.64 −0.463 0.167 0.167 −2.99±0.59 1.29±0.58 ¯ ¯


We have also measured the molecular diffusion coeffi-cients in a ternary mixture using the capillary-tube method.This is also the first report of such measurements in a ternarymixture in the open-ended capillary-tube method. The resultsand the working equations show the reliability of measuringthe diagonal and cross-diagonal molecular diffusion coeffi-cients in ternary mixtures by this method.

Ternary molecular diffusion coefficients are used to cal-culate the phenomenological coefficients related to molecu-lar diffusion. These coefficients and thermal diffusion coef-ficients are used to calculate the thermal diffusion factors inone of the ternary mixtures. The results show that the binaryand ternary thermal diffusion factors for the same species arevery different.

A comparison of the measured thermal diffusion coeffi-cients of the binary mixtures nC8-MN and nC10-MN andternary mixture nC8-nC10-MN show that the ternary mixtureis not a pseudobinary mixture. In the ternary mixture, thethermal diffusion coefficient of nC10 changes sign as the con-centration of the mixture changes.

ACKNOWLEDGMENTS

This work was supported by a Yale University fellow-ship to A.L.-D. and by the member companies of the Reser-voir Engineering Research Institute �RERI� in Palo Alto,California. The experimental part of the work was done atthe University of Mons-Hainaut, Belgium.

APPENDIX: ANALYTICAL SOLUTION TO TERNARYMOLECULAR DIFFUSION

In this appendix, the analytical solution of Eq. �20� isobtained. The matrix D= in Eq. �20� can be diagonalized suchthat

D= = P= E= P= −1. �A1�

Where E= is the eigenvalue matrix and the matrix P= containsthe eigenvectors

E= = ��1 0

0 �2� , �A2�

P= = � 1 D12

�2 − D11

�1 − D11

D12 1

= � 1 �2 − D22

D21

D21

�1 − D22 1 , �A3�

P= −1 =�2 − D11

�2 − �11� 1 D12

D11 − �2

D11 − �1

D12 1

=�2 − D11

�2 − �11� 1 D22 − �2

D21

D21

D22 − �1 1 . �A4�

The eigenvalues are given in

�1 =D11 + D22 + ��D11 − D22�2 + 4D12D21

2, �A5a�

�2 =D11 + D22 − ��D11 − D22�2 + 4D12D21

2. �A5b�

Substituting Eq. �A1� into Eq. �20� we get

�t�� = P= E= P= −1�2�� , �A6�

where �� is the column vector of mass fractions, �t=� /�t and�2=�2 /�z2. Equation �A6� can be multiplied by P= −1 to give

�tP=−1�� = E= �2P= −1�� . �A7�

Defining a new concentration variable � as

�� = P= −1�� A8�

decouples the two governing equations

��1

�t

��2

�t = ��1 0

0 �2��

�2�1

�z2

�2�2

�z2 . �A9�

The new initial and boundary conditions are given in

�0� = P= −1�0, �� = P= −1��, � ��z�

z=L= 0� . �A10�

The boundary and initial conditions are the same as the onesused for the binary mixtures; with the results from the binarysystem we obtain

�1 − �1,� =4��1,0 − �1,��

�

��m=0

� sin�m + 1/2��

Lzexp−

�2�m + 1/2�2

L2 �1t�2m + 1�

�A11�

and


�2 − �2,� =4��2,0 − �2,��

�

��m=0

� sin�m + 1/2��

Lzexp−

�2�m + 1/2�2

L2 �2t�2m + 1�

.

�A12�

The conversion of expressions �A11� and �A12� into massfractions is given by in

�1* = K�1

* +D12

�1 − �2�2

* �A13�

and

�2* =

D12

�1 − �2�1

* + K�2*, �A14�

where

K =�2 − D11

�2 − �1. �A15�

The symbol * stands for initial � 0�, boundary � ��, or aver-age concentration. Substituting Eqs. �A13� and �A14� into�A11� and �A12�, we get

K��1 − �1,�� +D12

�1 − �2��2 − �2,��

=4

��K��1,0 − �1,�� +

D12

�1 − �2��2,0 − �2,��S1

�A16�

and

D12

�1 − �2��1 − �1,�� + K��2 − �2,��

=4

�� D12

�1 − �2��1,0 − �1,�� + K��2,0 − �2,��S2,

�A17�

where

S1 = �m=0

� sin�m + 1/2��

Lzexp −

�2�m + 1/2�2

L2 �1t�2m + 1�

,

�A18�

S2 = �m=0

� sin�m + 1/2��

Lzexp−

�2�m + 1/2�2

L2 �2t�2m + 1�

.

�A19�

From Eqs. �A16� and �A17�, we obtain

�1 − �1,� =4

��2 − �1��1,0 − �1,��2 − D11�S1 + ��2

− D22�S2� + ��2,0 − �2,��D12�S2 − S1�� ,

�A20�

�2 − �2,� =4

��2 − �1��1,0 − �1,��D21�S2 − S1� + ��2,0

− �2,��2 − D11�S2 + ��2 − D22�S1�� . �A21�

Equations �A20� and �A21� collapse to the Eq. �16� for thebinary mixture. The average composition inside the tubescan be obtained by the integration of Eqs. �A20� and �A21�.The results provide Eqs. �21� and �22� of the text.

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