CONCENTRATION DEPENDENCY OF SELF- ANDMUTUAL DIFFUSION COEFFICIENTS INBINARY LIQUID MIXTURES- A CONFIGURATIONAL RELAXATION MODEL APPROACH

Makoto HARADA*,Masataka TANIGAKIand Wataru EGUCHIInstitute of Atomic Energy, Kyoto University, Uji, 611

To predict the concentration dependency of diffusion coefficients in binary liquid mixtures,a configurational relaxation model proposed by Gray was developed with reference to the recentinvestigation for the micro-kinetics of molecular motion. The physical supposition of this modelis that the evolution in the pair configuration space obeys a relaxation equation and that the con-figurational relaxation takes place when the free vacancy is located at the neighborhood of thefirst shell molecule. An important relation between shear viscosity and self-diffusion coefficientswas derived by eliminating the relaxation rate coefficient from the equations for them. Fromthis model, the concentration dependencies of self- and mutual diffusion coefficients for non-associating binary liquid mixtures are predictable.

Introduction

Isothermal and isobaric diffusion processes in a

binary liquid mixture can be described by three diffu-sion coefficients or three friction coefficients. Bear-man^derived general phenomenological expressionsbased on statistical mechanics for these diffusion

coefficients. For their actual evaluation, however,the distortion of the distribution in the configura-

tional space has to be known.Rahman's molecular dynamics investigations1^ haveprovided valuable insight into the nature of molecularmotion. His results of the velocity auto-correlationfunction for liquid argon displayed Gaussian decay atthe earlier stage of diffusion in singlet space andthereafter became negative. Takeuchi22) recentlystudied the micro-kinetics for atomic motion of liquidsodium. The velocity auto-correlation function dis-played a more distinct damped oscillatory pattern.The initial Gaussian decay shows that the moleculeexecutes a quasi-Brownian motion in the fluctuatingmean force field in the cage interior. Thereafter, themolecule executes a quasi-elastic momentumexchangewith the cage boundary, because the cage boundary islittle deformed. This leads to backscattering or re-

bound motion of the diffusing molecule. The negativepart or the dampedoscillatory pattern of the velocity

auto-correlation function arises from the difficulty ofcage boundary movement.The above microkineticnature anticipates that the evolution of the configura-tion is important in understanding the nature of dif-fusion processes.

Gray5) proposed the configurational relaxationmodel, taking into account the evolution of the con-figuration. His starting point for evaluating the dis-tortion of distribution function was the equation ofcontinuity in pair space. The authors7} have pointed

out that the relaxation rate of the pair configurationaldistribution function was closely connected with theexistence of a vacancyaround the cage molecules andthen could be related to the density of the fluid.The aim of the present work is to develop Gray'sconfigurational relaxation model to the diffusion

processes in binary liquids with the use of the author'sdensity-dependent relaxation rate and Bearman's

phenomenological expression and then to propose therelation for predicting the concentration dependency

of the diffusion coefficients.

Friction Coefficients and Shear Viscosity

Weconsider the classical ensemble composedof atwo-component dense fluid of the species A and B, inwhich each molecule has three dimensional freedoms.A general theorem which leads to the deduction of thetransport equations was derived by Kirkwood9).

The equation of continuity in pair space can be de-rived from this theorem as

Received July 16, 1975.Presented in part at the 39th Annual Meeting of The Soc.Chem. Engrs., Japan, at Kobe, 1974.

VOL 9 NO. 2 1976 85

(I=A or B) (1)where n{2i is the configurational density function inpair space and

7iV.i= (-^i?'1'^, r,, puP%)dPldp2J W/

(2)

Jai.i is the flow of the species A at rx when /is locatedat r2. /?! and p2 are the momentumof molecules atthe positions, rx and r2, respectively. /j/>2) is the pairdistribution function. m7is the molecular mass ofspecies /.Eq. (1) can be written in terms of the mean positionof the pair molecules and the relative coordinates,which are defined as

r= (r2-ri)(3)

Eq. (1) thus becomes

dt-ttS+Vr 'JAI+V*'*=<>i ;(2) »(2)

JAl-JAI.2 JAl,l

J- 9 v7^/,2~tJai,i)(4)

(5)

/ is the flow for the mean position of the pair mole-cules and jAI is the relative flow. It is assumed that

VR'Jcan be replaced by the form:VR -J=jAM21-n^n(6)

where n{2ie) is the value in equilibrium. The aboveequation is the essence of Gray's configurational

relaxation model5\ For the stationary state, Eqs. (4)and (6) are reduced to

n%=n%»--Pr-jAI(7)

Let us consider the diffusion process in binary liquidmixtures. In this case, jiV.i can be expressed by

;(2) f/(2) M(2)

(8)

Uai,i is the macroscopic velocity of A at r± when Aand / are located at rx and r2, respectively. This

would be written with the aid of macroscopic flowvelocity in singlet space, ul9 as2)

UAi,i=uA(r1)+2AI- (h7-ua)(9)

Then the relative flow can be written asJai= -nTi [uA.(rd -«i(fi)](10)

In the derivation of the above equation, the VriuTterm was assumed to be negligible.

«/(»*2) =«7(»<i)+»# à" VriuArd -å  =uM) (ll)

Introduction of Eq. (10) into Eq. (7) yields, replacingnil ^«£6) in Eq. (10),

w (2) w(2,e) x /pw(2,e)\ /.-Mai-Mai ~m Vt^ai )'Kui~

Tai-uA)(12)

Bearman2) derived the following phenomenologicalexpression.

nTFA= - Z nTn^Ai{uA-Ul)

=-e(--2Vjrdr

-WlK^ ^-ri^ rKdr

(13)

FA is the meanforce acting on a molecule ofA species,which arises from the diffusional motive force. nAX)is the number density of the species A in singlet space.<f>Ai is the intermolecular pair potential function be-tween A and /. n^^AI is the friction coefficient be-tween A and / molecules. Introduction of Eq. (12)into Eq. (13) yields

il2>FA= zf-1fnd<pA dn{2ie)

LTai.drdr dr-(UA -Uj)j (14)

With the aid of partial integration, Eq. (14) is reducedto

ti{1)F=-z\

iu-[v^Ain^driu^ti;)] (15)rJ J

S ince» <?.«>=»'.1'm'1'

=rtFriPgAl{r )(16)

Cai can be obtained as

^AiA v"<t>AjgAidr\ZjA1(17)

where gAI is the radial distribution function. As is clear from Eq. (6), the term jAI is the ratecoefficient of the relaxation process of~nl

j}. It is

expected that j{2i is in inverse proportion to the re-laxation time. For a single-component liquid, Eq.

(17) is reduced to

^AA~'

3r,-\v^AAgAAdr (18)

The authors7) have proposed a relaxation model forsingle-component liquids, by supposing that theconfigurational relaxation took place when the free

vacancies were located near the pair molecules. Thearrangement of the equation thus derived yields

^>AA-mAA(2DMkT(2ff I

:4-i>igAAdr

(19)

where mAAis the reduced mass for A andA. DA'\sthe self-diffusion coefficient. The term, /, is the prob-ability of finding a free vacancy around the firstshell molecule which is located in the direction of themotion of the diffusing molecule. Equating Eq. (19)to Eq. (18), jAA is written as

mAA(DA+DA) (\ 2 VZU/

Ta a kT(2ff

This equation shows that the configurational relaxa-tion time is equal to the product of the relaxation

86 JOURNAL OF CHEMICAL ENGINEERING OF JAPAN

time for free diffusion and (2/)~2. It is, therefore,

anticipated for binary mixture that

TaimAJ(DA +Dx)

kT(2ff (21)

From Eqs. (21) and (17), Cai can be expressed asc^=kT(2ff I3

\^AigAId^ (22)

Rice et al.16tl7) proposed the small step diffusion

model for binary mixtures with the assumption thatthe pair distribution function was approximated asthe product of the density distribution function andthe local equilibrium momentum distribution func-tion, and that the pair diffusion tensor was given bythe direct sum of the self-diffusion coefficients in singletspace independently of the relative distance betweenthe pair molecules. Arrangement of their resultleads to

1 _mAI{DA+DI)Tax kT

(21')

This equation differs from Eq. (21) by only (2/) 2and hence the term is considered as a correctionfactor arising from the coupling motion between thediffusing and the neighboring molecules, which wasneglected in the small step diffusion model.

The self-diffusion coefficient of / species is re-

lated to2)DA=kTfCA

^nTZAA+n^QAB(23)

The mutual diffusion coefficient in isothermal andisobaric process can be obtained as follows when thevolume change due to diffusion is negligible.

D=DJdlnaJd\nxA)p,T, Du=vkT^AB (24)wherev is the meanvolumeoccupied by a molecule.aA and xA are the activity and the mole fraction of thespecies A.Let us now consider the shear viscosity. Its deri-vation follows the steps shown by Gray5) in greatdetail, and we shall here show the result of derivation.The intermolecular contribution to the shear viscosityfor binary liquid mixtures is given by

30kr Tu J Wr2 +r dr )gljCtrK^}

Generally, the kinetic contribution is negligible indense fluids. Introducing Eq. (21) into Eq. (25), theshear viscosity is finally expressed by

å M^^W' «Extension of the Model

In the preceding section, the relations for the fric-tion coefficients and the shear viscosity were derived

as a function of/. In general, the numerical value of/ is difficult to estimate in liquid mixtures. The

concentration dependency of the diffusion coefficientswould be calculable by eliminating / from the frictioncoefficients and the shear viscosity.

We shall derive, first, the relation between theshear viscosity and the self-diffusion coefficients.Since d^^uldr2 is much greater than (^/rXdfiu/dr),

the following relation holds approximately :;^+7^>^4vi*'*''* (2?)

Rowlinson's approximation2^ holds for the liquid

interacting through Lennard-Jones potential function.\r^4,IjglJdr^ah^V^jgjjdv (28)

where aI3 is the separation diameter when the poten-tial becomes zero, which is assumed to be the arith-metic meanof the two molecular diameters.

oij =(oii+ojj)/2 (29)

Using Eqs. (17), (23), (24) and Eqs. (27) to (29), Eq.(25) can be rewritten as

10»L Dx Xa+ Db Xb 2Du~XaXb(30)

For single-component liquids, the shear viscosity isrelated to the self-diffusion coefficient as5>7)

Vt^kTeyiOtfD?(31)

The superscript, *, denotes the values for the single

component liquid. Then, Eq. (30) can be rewritten asd* D* 01* D*

^~ v DA^>aXa^ v DBv*-bXbkT

-(Paa - <*bb¥xaXb (32)20Didv

Since DA, DBand Did are numerically of the sameorder, the third term of the right-hand side in Eq. (32)can be neglected unless aAAand aBB are far differentfrom each other. Then, Eq. (32) is simplified as

rh~-v DAv DB

,BXB(33)

It is important to note that the above relation isderived by eliminating the term j.

Friction coefficients Cj. and C# can be expressed

from Eqs. (22) and (23) asr -('Xa\mAA p /JL_l J_

1 +4-(xB\ mAB

(xB\mBB (1 1

V(2/)2

VOL. 9 NO. 2 1976 87

where

Bu = ^j-\v2fajgudr (35)

Eliminating / from Eq. (34) with the aid of Eq. (23),the ratio of (DA/DB) is obtained as

(DAIDB)\6AxA +xB) + (DAIDB)(xB -xA)-(dBxB+xA)=O (36)

where6A=2mAABAAl(mABBAB) (37)

The diffusion coefficient, Did, can be obtained fromEqs. (22), (24) and (34) as

Did=DA[xs+xA6Al{l +(DB/DA)}] (38)The mutual diffusion coefficient at infinite dilution isthen given by

(D%A^= (DA%A^ = (DB)XA_^dB/{ l + (DJDB%A^}}{D)XB^= (DB%B^= {DA)XB^dAl{\ + {DBIDA)XB^}\

(39)

TheneA = (DBIDA\B^[l + (DBIDA)XB^]eB = (DA/DB)XA^[l + (DA/DB)XA^](40)

6A and 6B are independent of mole fraction by Rice'sargument for the corresponding states17\

The concentration dependency of self-diffusioncoefficients can be evaluated from Eqs. (33), (36) and(40), employing the diffusion coefficients at x=0 and1 and also using the viscosity of liquid mixture. Themutual diffusion coefficient can be evaluated fromEqs. (24) and (38).When / is known, the diffusion coefficients can becalculated without using shear viscosity. Eliminatingthe contribution of unlike molecules, BAB,from Eqs.(22) and (23), the following equation is obtained.

/xA_xB\ 1

\DA Dj (kTfv(2fyX[mABAAx\DA-mBBBBxl]DB (41)

Self-diffusion coefficient for a single componentliquid is given from Eqs. (19) and (23) as

(D*f = (kT)\2fffv*/(mABAA) (42)Eqs. (41) and (42) yield

uA uAJJ A JJbft

fDA 0

D*A 6UB 2

A

(43)

ft is given by7/j* =zO* -u*o)M

(44)

where z is the effective site number existing in thedirection of the diffusive movement and v% is themolecular volume when diffusion coefficient becomeszero. Similarly, it is assumed for a binary mixturethat

f=z(v- vo)/v(45)

Further, it is assumed for simplicity that (v-v0) isgiven bv the formgiven by the form

(v-vo)=(vi-v%)xA+(v%-v%0)xB+AvxAxB (46)

The terms (vf-vf0) are related to self-diffusion co-efficients for single-component liquids as

D%(v% dB v*A (47)

Introduction of Eqs. (44) to (47) into Eq. (43) yieldsr ^ J^A- J!L.\\ v-Z^-A. y2UB VB \

^Ab ^^ \V4/L ^a uA uAj/

[-+-(f)(fl7)I"+-T «where

w= dv/(vi -v%0)(48')

The self-diffusion coefficients can be calculated fromEqs. (36) and (48) with the diffusion coefficients at

x=0 and 1, without using the shear viscosity. Themutual diffusion coefficient can be calculated fromEq. (38) with the values of the self-diffusion coef-ficients.

Discussion

In the preceding section, Eq. (33), which relatesthe shear viscosity for binary mixture to its self-diffusion coefficients, was derived. It does not in-clude the term ju explicitly and hence is expected tobe widely applicable. Viscosity values calculated fromEq. (33) for several binary mixtures are comparedwith the experimental ones in Fig. 1. The datasources of self-diffusion coefficients and averagemolar volumes for binary mixtures used for the calcu-lation are shown in the figure. For simple liquidmixtures, the calculated values agree quite well withthe experimental ones, as is clear from Fig. 1(A). Evenfor some binary mixtures with association or com-pound formation the agreement is fairly good, asshown in Fig. 1(B). This agreement seems to originatefrom the characteristics of Eq. (33), that is, it doesnot include explicitly the molecular masses and theconfigurational relaxation rate, ju.

The diffusion coefficients are strongly affected bythe interaction between like and unlike molecules.These interactions, i.e., 0A and 0B, are difficult toestimate strictly from the intermolecular potentials

alone. Hence, these terms are evaluated by Eq. (40)with the observed mutual diffusion coefficients atinfinite dilution and the self-diffusion coefficients forsingle-component liquids. Figs. 2 to 4 show thecomparison of the calculated diffusion coefficientsfrom Eqs. (33), (40), (36), (38) and (23), (24) (solid lines)with the observed values. The data sources for shearviscosity, mean molar volume, d\naAldlnxA, DIjX={)and DI>X=1 necessary for the calculation are listed in

88 JOURNAL OF CHEMICAL ENGINEERING OF JAPAN

1.61 1 1 1 1 -r\ (b) -

1.2-/ \

0.8 ' X

I M 1 1 1 1--& (a) /

Io - L*i - Keys:observed / Solid lines : calculation V / 1) cyclohexane-benzene (A)^ "\ / " data:y18\vi8\D^ \v / a 2) «-hexane-«-dodecane (A)

0.6- >V^_^ data:,«",v*\D/» J 3) water-acetone (A) / data: rf\ v*\ D/v -/ - 4) Chloroform-acetone(A)

/ data: 5y4), v*\ D^

0.2I 1 1 1 1 0 0.2 04 0.6 0.8 1 XA {-)

Fig. 1 Comparison of the calculated values by Eq. (33)with the observed viscosity, at 25°C

I I I 1

o

I I I I

0 0.2 0.4 0,6 0.8 1

Xa C-)

Keys : observedSolid lines : calculated with viscosityDotted line: calculated with w=0Broken lines: calculated with w= -0.4

Data: D21\ Dj21\ r]21\ v21\ (In aA/ln xAf^Fig. 2 The calculated and the observed diffusion co-efficients for nearly ideal solution, //-dodecane (A)-n-

hexane (B) at 25°C

these figures. Reasonable agreement between calculat-ed and observed values are obtained for both self-

and mutual diffusion coefficients.For benzene-cyclohexane binary mixture, Mc-

Laughlin12) compared the values calculated fromseveral models with the observed ones. Darken's

3| , , , ,

t-\

^ " ^r,- j-,, DA

10 0.2 0.4 0.6 0.8 1

Keys : observedSolid lines : calculated with viscosity

Dotted line: calculated with w=0Broken lines : calculated with w=0.7

Data: Dlo\ Dju\ ^\ vls\ (In aA/ln xA)lg»Fig. 3 The calculated and the observed diffusion co-

efficients for non-ideal solution, benzene (A)-cyclohexane (B)at25°C

4 å - , , , ,

°« " d ^^rm^^PQ

Q

°il . . 1 i I

0 0.2 tt4 0.6 0.8 1XA <-J

Keys : observed

Solid lines : calculated with viscosityDotted line: calculated with w=0Broken lines : calculated with w=0.1

Data: D8\ D/\ 7jn\ v6\ (In aA/\n xA)8)Fig. 4 The calculated and the observed diffusion co-

efficients for quasi-globular and chain molecules, benzene(A)-w-heptane (B) at 25°C

equation could not fit the observed values. Theimprovement by the Rice-Allnatt model was onlymarginal. In this model, the relation between shearviscosity and self-diffusion coefficients is used, andthis leads to a marked improvement on their model.The comparison for acetone-water system is shownin Fig. 5. The agreement of the self- and the mutualdiffusion coefficients is rather poor. This would

originate from association or compoundformation.VOL. 9 NO. 2 1976 89

fl-r-'-' - ^

0 0.2 0.4 0.6 0.8 1

XA C-]

Keys : observedSolid lines : calculated with viscosityData: Dn\ Z>/3), v'\ ^\ On aA/ln xAy^

Fig. 5 The calculated and the observed diffusion co-

efficients for acetone (A)-water (B) at 25°C

Mutual diffusion coefficients can be calculated byEqs. (38) and (24) with the self-diffusion coefficientsobtained by Eqs. (23), (36) and (48), when the valueof w is known. The values calculated by assumingw=0 are shown in Figs. 2 to 4 by the dotted lines.The agreement between the calculated and the ob-

served values is less successful. Whenthe w values areselected as shown in these figures, the agreement isexcellent (the calculated values are shown by the

broken lines). This proves the validity of the presentapproach more directly. Unfortunately, the way toestimate the w values is unknownat present.

The elimination of the unknown term, w, i.e., /from the friction coefficients with aid of the shearviscosity yields a simple equation similar to the hydro-dynamic relation. This equation can be also derivedfrom the hard-sphere approach. McLaughlin10) derivedthe viscosity relation for hard-sphere molecules.

-£r(2xkTy'*±-Z: UxIXjm¥fahgij(<rij)] (49)1J 0 I J

Friction coefficients for hard-sphere molecules were

also given by McLaughlin^ asr

8 2/lizkT\1/2^Ij = -7rmIJ<Th( - SijOi/) (50)

where oh is the separation diameter when the mole-cules of / and / are in contact and guipu) is thevalue of the radial distribution function at r=Gu.From the above two equations and Eq. (23), theexactly same equation as Eq. (32) can be obtained.Eqs. (49) and (50) do not include the contribution ofbackscattering for a dense hard-sphere ensemble.This contribution, however, would be cancelled outin the derivation of Eq. (32).

ConclusionThe configurational relaxation model by Gray wasdeveloped for binary liquid mixtures. The present

approach was based on the supposition that theevolution of the configurational distribution obeys

the relaxation equation and that the configuration ofthe pair molecules relaxes in the neighborhood of the

free vacancy around the molecules. The derived

relations contain the term /, which represents thecontribution of the rebound motion of the diffusingmolecule. Since the numerical value of the term isdifficult to estimate, this is eliminated from friction

coefficients with the aid of the shear viscosity ofbinary liquid mixtures. The concentration depend-encies of the self- and the mutual diffusivities can bepredicted with good accuracy for non-associating

binary liquid mixtures, when using the shear viscosityfor binary mixture, self-diffusion coefficients of singlecomponent liquids and the mutual diffusion coef-

ficients at infinite dilution. The problem of predictingw value is left to future work.

Nomenclature

aT = activity of / speciesBu = interaction parameter between / and J, defined

by Eq. (35)= mutual diffusion coefficient

= self-diffusion coefficient for species /= diffusion coefficient defined by Eq. (24)= mean force acting on a molecule of species /= probability for a molecule to diffuse freely= distribution function for the pair molecules, /

and/= radial distribution function for / and / molecules

aiJ) = radial distribution function for hard spheres incontact

= flow of the pair molecules at the mean positionL = flow defined by Eq. (2)

= relative flow defined by Eq. (5)= Boltzmannconstant

= molecular massof /-species= reducedmass

= number density in singlet space= number density in pair space

= momentum

=pressure= mean position of pair molecules= relative position vector

=absolute temperature=time

= macroscopic mean velocity for /-speciesM(_2) i = macroscopic mean velocity for /-species when /

and /, respectively, are located at #*i and r2= meanmolecularvolume= mean molecular volume whendiffusion coef-

ficient becomes zero= mole fraction

= parameter defined by Eq. (48')= effective site numberexisting in the direction of

diffusion movement

= relaxation rate coefficient

= friction coefficient for /-species= contribution of /-/ interaction to C/

90 JOURNAL OF CHEMICAL ENGINEERING OF JAPAN

v) - shear viscosity#j = interaction parameter defined by Eq. (40)<p = intermolecular pair potential

<Subscripts>I,J,A,B = species

0 = intermolecular potential force contributionSuperscrip ts)* = single component

e = equilibrium

Literature Cited1) Al-Chalabi, H. A. and E. McLaughlin: Molecular Phys.,

19, 703 (1970).

2) Bearman, R. J. and J. G. Kirkwood: /. Chem. Phys., 28,136 (1958).

3) Fort, R. J. and W. R. Moore: T. Faraday Soc, 61, 2102(1965).

4) Fort, R. J. and W. R. Moore: ibid., 62, 1112 (1966).5) Gray, P.: Molecular Phys., 7, 235 (1964).6) Halm, R. L. and L. I. Stiel: AIChE/., 16, 3 (1970).7) Harada, M., M. Tanigaki, A. Nishimura and W. Eguchi:

J. Chem. Eng. Japan, 7, 407 (1974).8) Harris, K. R., C. K. N. Pua and P. J. Dunlop: /. Phys.

Chem., 74, 3518 (1970).

9) Irving, J. H. and J. G. Kirkwood: /. Chem. Phys., 18,

817 (1950).

10) Kandiyochi, R. and E. McLaughlin: Molecular Phys., 17,643 (1970).

ll) Konobeev, B. I. and V. V. Lyapin: Zh. Prikl. Khim., 43,803 (1970).

12) Lofline, T. and E. McLaughlin: J. Phys. Chem., 73, 186(1969).

13) McCall, D. W. and E. W. Anderson: ibid., 70, 601 (1966).14) Mills, R.: ibid., 69, 3116 (1965).

15) Rahman, A.: Phys. Rev., 136, A405 (1964); /. Chem.

Phys., 45, 2585 (1966).

16) Rice, S. A. and J. G. Kirkwood: /. Chem. Phys., 31, 901(1959).

17) Rice, S. A. and A. R. Allnatt: ibid., 34, 409 (1961).18) Ridgway, K. and P. A. Butler: /. Chem. Eng. Data, 12,

509 (1965).

19) Rodwin, L., J. A. Harpst and P. A. Lyons: /. Phys. Chem.,69, 2783 (1965).

20) Rowlinson, J. S.: Molecular Phys., 7, 477 (1964).21) Shieh, J. C. and P. A. Lyons: /. Phys. Chem., 73, 3258

(1969).

22) Tanaka, M., Y. Fukui and S. Takeuchi: Nippon KinzokuGakkaishi, 37, 907 (1973).

VOL 9 NO. 2 1976 91