3148
J . Phys. Chem. 1990, 94. 3 148-3 152
Referring now to Figure 2 we may note an inversion of the R dependence and in terms of practical recommendation we would be quite certain of the applicability of eq 20 for the solvent density below 0.7 and skeptical above for size ratio R greater than say 0.85. However, we do not see the study of the expansion (20) in terms of practical applicability alone. I t is clear from the physical picture that one has two very different situations for a small solute and for a large solute. I n the first case the small solute particle may fit into the existing cavities of the host solvent fluid. Then an average over unperturbed configurations of the solvent may be reasonably considered and it may even produce relatively accurate results, barring extreme cases (such as of a very strong ab interaction, t a b >> t b b ) . I n the case of large solute, clearly the immediate neighborhood of the solute particle is strongly deformed, even beyond recognition, and new methods must be devised. It is to be regretted that we could not compare our results with the simulation results.2d However, no two state points agree exactly and, moreover, the potentials are in fact different. We have used the complete Lennard-Jones potential and the cutoff
imposed by Haile and GuptaI3 at r = 6 u whereas in simulation one uses most often a potential cutoff at a much shorter distance, r =2.5~ typically. The smallness of the final contribution of the difference A resulting from 8,' minus the superposition approximation is to be noted. An attractive idea, exploited in many early theories, is to divide the process of insertion of the foreign molecule in two stages: a formation of a cavity in the host fluid, followed by the insertion:
In the context of the calculation of section 2 such a subdivision does not impose itself at all. The "formation of a cavity" may be more natural for large solutes. Differentiation with respect to (3 (cf. section 2) produces the energy of dissolution. Acknowledgment. The author is indebted to Professor W. A. Steele who made available to him the M S c . thesis by J. Gupta, and to Dr. B. Malesinska for help with drawings and tabular data.
A Fluctuation Solution Theory of Activity Coefficients: Phase Equilibria in Associating Molecular Solutions Esam Z. Hamad? and G. Ali Mansoori* Department of Chemical Engineering, University of Illinois, Box 4348. Chicago, Illinois 60680 (Received: July 17, 1989; In Final Form: October I I , 1989)
A new analytic statistical mechanical fluctuation solution theory for activity coefficients in multicomponent mixtures is developed. This theory is based on the newly formulated exact relations among the mixture direct correlation function integrals and the closures for cross direct correlation function integrals. One major advantage of this theory is its independence from the nature of intermolecular interaction potentials in solutions which are generally unknown for complex molecules. The theory is successfully used for vapor-liquid equilibria, liquid-liquid equilibria, and phase splitting prediction and correlation of fluid mixtures consisting of polar and associating molecules.
Introduction Definition of activity coefficients provides a convenient way of describing the nonidealities encountered in solution thermod y n a m i c ~ . ' - ~ J ~However, >'~ there exists very limited theoretical understanding through which one can derive expressions for activity coefficients using the principles of statistical mechanics. As a result, most of the available analytic expressions for activity coefficients are empirical which are based on classical thermodynamic definition of excess properties. Lattice theories of statistical mechanics were successful in providing us with analytic expressions for activity coefficient^.^,^ There exist numerous group-contribution expressions for activity coefficients which are based on the lattice theories of statistical mechanics. The high degree of combinatorial approximations which are used in the lattice theories have made their resulting activity coefficient expressions only a limited success. Another statistical mechanical route for the development of analytic expressions of activity coefficients is the use of Gibbs, or NPT, e n ~ e m b l e . ~Gibbs ensemble, joined with the conformal solution theory of statistical mechanics, have provided us with the formalism in systematically deriving analytic expressions of activity coefficients. The success of this technique is also limited to simple molecular fluid mixture for which the intermolecular potential energy functions are available. In the case of polar mixtures and mixtures possessing hydrogen-bonding species the route through 'Present address: Chemical Engineering Department, Kuwait University,
P.O.Box 5969, Safat 13060, Kuwait.
the Gibbs ensemble will be at a loss. The third statistical mechanical route to activity coefficients is the Kirkwood-Buff solution theory of statistical mechani~s.~J~J' The activity coefficient, yi, of component i in a mixture is defined by the following relation In (XiTi) = ( p i - bip)/kT (1) where F ; is the chemical potential of component i in the mixture, wuipis the standard-state chemical potential of i, k is the Boltzmann constant, and Tis the absolute temperature. By utilizing the above definition, activity coefficients in solutions are obtained from the relations between the chemical potentials, F ~ and , the fluctuation integrals, Cis. According to the Kirkwood-Buff solution theory5 the expression for the chemical potential, pi, of component i in a binary mixture takes the following form: x i [ a ~ i / a x i l=~ k, ~ T / [ 1 + X I X Z P ( G+I IG22 - 2G12)1 (2) where G, =
L-[a(')
- 1]4ar2 dr
(3)
xi is the mole fraction of component i, P is the pressure, p is the (1) Manswri, G. A . Fluid-Phase Equilib. 1980, 4 , 197-209.
(2) Chao, K. C.; Greenkorn, R . A . Thermodynamic of Fluids; Marcel Dekker: New York, 1975. ( 3 ) Kreglewski. A. Equilibrium Properties of Fluids and Fluid Mixtures; Texas A & M University Press: College Station, TX, 1984. (4) Manswri, G . A. Fluid-Phase Equilib. 1980, 4 , 61-69. ( 5 ) Kirkwood, J. G.; Buff, F. P. J . Chem. Phys. 1951, 19, 744.
0022-3654/90/2094-3 148$02.50/0 G I990 American Chemical Society
The Journal of Physical Chemistry, Vol. 94, No. 7, 1990 3149
Theory of Activity Coefficients
"
density, and gij(r) is the radial distribution function of the pair of molecules i and j . Expressions for [api/dxilP,,are also written in terms of direct correlation integrals6
x i [ a ~ i / d x i l p=. ~
where
and the direct correlation function, cij(r),is defined with respect to the radial distribution function, gij(r), in an n-component mixture, by
Finally, the activity coefficient, yi, of component i is obtained by integrating eqs 2 or 4 with respect to the mole fraction. Equation 1 then becomes
This equation, in principle, could be used to calculate the activity coefficients provided information about Gij integrals are available. One major advantage of this formalism is the fact that there is no need for direct information about the intermolecular potential energy functions of species of the mixture. However, the lack of sufficient knowledge about G , (or Cv) integrals and their relationships has limited the utility of this theory. For mixtures of species with model intermolecular potential energy functions it has been possible to calculate G , (or Cij) integrals and as a result the activity coefficients from statistical mechanics. Equation 7 has some utility for complex solutions (mixtures consisting of highly asymmetric, polar, and hydrogen-bonding species) only at infinite dilution. The theory reported here provides us with a strong tool for developing analytic expressions for activity coefficients of multicomponent mixtures by using eq. 7 and a number of newly discovered characteristics of direct correlation function integrals in mixtures. This theory is shown to be applicable to complex multicomponent mixtures at finite concentrations as well as solutions at infinite dilutions.
Theory of Multicomponent Activity Coefficients Let us consider a single-phase multicomponent solution consisting of n components. The chemical potential, p i . of every species in this mixture is a function of n + 1 independent variables
d [ a p i / d x j ] / 8 P= a [ a p i / a P ] a x j= [ ~ 3 D , / d x ~ ] ~ ,i,, j = 1,
= a[api/axk]/dx,
... n (8) (
i, j , k =
I , ..., n and j # k (9)
I n the above equations &,/axj and partial molar volume, Di, are related to the direct correlation function integrals by6 Ni(dpi/aNj) = kT[6ij - Xi11
By replacing eqs IO and 11 in eqs 8 and 9 we will derive a number of expressions between Ciis of the mixture. In an n-component mixture there exist n(n + 1)/2 direct correlation function integrals which are considered as unknowns in eqs 8 and 9. Note that there are n(n - 1) relations of the type in eq 8 and n(n - I)(n - 2)/2 relations of the type in eq 9. Of all these equations only n(n 1)/2 - 1 of them are independent. Since there are n(n 1)/2 unknowns, one additional relation will be needed to determine all the unknowns regardless of the number of components involved. The resulting equations for direct correlation function integrals consist of a set of nonlinear partial differential equations. A general analytic solution of this set of equations for multicomponent mixtures does not seem feasible at the present time. Numerical solutions of these equations are generally involved and lengthy and could require the use of multivariable finite difference techniques. In what follows we report the analytic solution of these equations for mixtures at low to moderate pressures. Analytic Solutiow for Low to Moderate Pressures. For liquids at low to moderate pressures it can be assumed that the activity coefficient is not dependent on pressure. In mathematical form using eq 1 this means
+
[a In ~ i / a P l x ,=, ~[ a ~ i / a P l x , , ~ / =k T0 As a result of this approximation eq 8 reduces to the following expression [ d ~ , / d x , =] ~0 ,or~ 8, = ui, where ui is the molar volume of pure component i. As a result of this approximation eq 11 can be written in the following form: n
n
n
By considering that
- x~XiX,pcij= ) (kTKT)-'
p(1
(13)
Equation 12 then becomes n
EXjpCi, = 1 - Ui/(kTKT) J=
i = 1, ..., n
1
(14)
where the mixture isothermal compressibility is now given by KT
= -xxi(aui/ap)/cxiui
(15)
The set of relations in eq 9 do not simplify upon making the assumption of low pressure. To overcome this difficulty and to provide the one additional relation needed, the following closure relation is assume to be valid for all cross direct correlation function integrals (C,) i # j)'.* Cij = ajiCii
= ~~(T,P,x~,x~...,x~-I)
Using this expression one can write the following equations between the mixed second derivatives of the chemical potential:
a[api/aXj]/dXk
n
+
Cij = L m c i j ( r ) 4 m 2d r
l i
n
+ aijCjj
i, j = 1, ..., n and i # j
(16)
where ajiand aij are binary interaction parameters. In an earlier publication' we have shown this form of the closure expression is quite satisfactory in representing the cross direct correlation function integrals of a variety of nonpolar, polar, and hydrogenbonding binary mixtures. Substituting eq 16 in eq 14 gives the following set of linear equations (Ljajl)PCII
+ X2a12PC22+ ... + XnalnPCnn =
1 - ul/(kTKT)
X I ~ Z I P C+I(IZ x j a j ~ ) P C + 2 ~... + XnaZnPCnn = 1 - U ~ / ( ~ T K T )
n
+ P c i j - PEXk(Cik + Cjk) + k=l
XlanlPCIl
+ X2an2PC22 + .*. + (CXjajn)PCn, = I
where summations run from j = 1 and ( 6 ) Pearson, F. J.; Rushbrooke, G. S. Proc. R. SOC.Edinburgh, S e r f . A 1957, A64, 305. See also: OConnell, J . P. Mol. Phys. 1971, 20, 27.
-
- Un/(kTKT)
(17)
n, and aii = I . The above
(7) Hamad, E. Z.; Mansoori, G. A,; Matteoli, E.; Lepri, L. Z . Phys. Chem. Neue Folge 1989, 162, S.27-45. (8) Hamad. E. Z. Ph.D. Thesis, University of Illinois, Chicago, 1988.
3150
The Journal of Physical Chemistry, Vol. 94, No. 7, 1990
Hamad and Mansoori
relations represent a set of n linear equations in the unknowns pCiP To solve for pCii it is more convenient to use matrix notations A-C = b
(18)
where A is a matrix with elements Aii = x x l a l i ,Aij = xjaij,and b and c are vectors with elements bi = 1 - Ui/kTKT and ci = pcii, respectively. The solution of eq 18 is n
PCkk = xIAlikbi/lA/
(19)
i= I
where lAlik is the cofactor of the element Aik in the matrix A, and IAl is the determinant of A. By using the direct correlation function integrals calculated above in the following equation one can derive the expression for the activity coefficients of a species in a multicomponent mixture in the following form
N ( a In r k / a N k ) = 1 - PCkk
+ uk2[kTxXi(aUi/ap)]-' i
(20)
As an example of application of this theory it will be used for vapor-liquid and liquid-liquid equilibria calculation of binary mixtures. Binary Mixture Calculations. For binary liquid mixtures at low to moderate pressures the following expressions for the direct correlation function integrals were derived
pc..=
+ Wxi(aijxi+ X j ) U i / U j + ( 1 - aij)xj+ q j x i + xixj + ajix/2
W( 1 - aiju,/ui)Xj2
0.0
0.1
0.2
0.3
0.4
03
0.6
0.7
0.8
0.9
1.0
2 1
Figure 1. Vapor-liquid composition (x-y) diagram for methanol ( I ) + water (2) at 60 OC. The squares are the experimental data: the solid curve is the present theory, and the dashed curve is Wilson correlati~n.~.'~
(Yi,X?
( i , j = 1, 2; i # j ) ( 2 1 )
+ al2XIX2u2/uI + a21x22) + + ~ 2 1 X 2 1 / [ ~ 1 2 X 1+2 X l X 2 + a21x221-' ( 2 2 )
PCl2 = [W(al2Xl2+ a21xIx2uI/u2 (YI2XI
I n these expressions oZ1and
cyI2 are
the binary parameters, and
W = - [ k T ( X l K ~ l / 0 2-b XzKn/ul)]-'
(23)
where K T ~is the isothermal compressibility of the pure components i.
Given the direct correlation function integrals, activity coefficients can be obtained by substituting the Cijsin eq 20, which for binary mixtures can also be written as
Substituting for p C , , from eq 22 and performing the integration results in the analytical expression for activity coefficient
+
+
In y I = In [ x I ( L 2 / L I ) x 2 ] ( I / 2 ) ( L l 71) In [(a/.12)X22 + (a12-I - 2)x2 + 1 1 + ( 1 / 2 ) [ ( 2 a , , L ,L1)7 + 1 - 2a121/9 In [ ( I + ~ 2 / r - ) / ( 1+ x2/r+)l ( 2 5 ) where Ll = ~ T K T I / U Lz ~ , = kTKn/UI, a =
0.0
+ Cy12 - I , 9 = ( 1 - 4a2lCf12)"~
Cy21
r* = ( 1 / 2 ) ( 1 - 2 a I 2f q ) / a 7
=
(a21u1/u2
+ ~ I Z U ~ / U-I 1)/(a21Ll2 + a l A 2- L1.U
(26)
The expression for In y 2 is obtained by interchanging subscripts 1 and 2 in eq 25 and in the expression for 9.The quantities a, 9 , and T remain unchanged under this operation. To test the activity coefficient expressions, they are used for vapor-liquid and liquid-liquid equilibria calculations of a number of binary systems. Applications and Results Applications of Vapor-Liquid Phase Equilibria. To test va-
por-liquid equilibrium (VLE) prediction by the new activity coefficient expressions a number of systems are c h ~ s e n . ~The vapor phase is assumed to be an ideal gas and the vapor pressure ( 9 ) Vapor-Liquid Equilibrium Data Collection; DECHEMA: Frankfurt,
1977.
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Figure 2. Vapor-liquid composition (x-y) diagram for I-propanol ( I ) + water ( 2 ) at 30 OC. The squares are the experimental data,9the solid curve is the present theory, and the dashed curve is Wilson correlati~n.~*'~
of the pure components, p p , is represented by the Antoine e q ~ a t i o n .Two ~ pure liquid properties are needed in the activity coefficient equation: Specific volumeg and isothermal compressibility.1° The VLE data are fitted to the activity coefficient expressions by minimizing the following objective function
s = x(P - P e x p P / S p+ ccv - Y,,,)*/s,
(27)
where Spand S, are the sum of squares resulting from minimizing E@- pexp)'and C(y - yeXp)* individually." The results of calculations are shown in Table 1. Comparison is made with the Wilson c o r r e l a t i ~ n ~which ? ' ~ is selected because (IO) Handbook of Chemistry and Physics; CRC Press: Boca Raton, FL, 1979. ( 1 1) Van Ness, H. C.; Abbott, M. M. Classical Thermodynamics of nelectrolyte Solutions; McGraw-Hill: New York, 1982.
No-
The Journal of Physical Chemistry, Vol. 94, No. 7, 1990 3151
Theory of Activity Coefficients
TABLE I: Parameters' of the Present Activity Coefficient Model and Comparisons with Wilson C~rrelation~~~* eq 25 system methanol water ethanol water ethanol + water ethanol water I-propanol water ethyl acetate ethanol ethanol + benzene ethanol benzene ethanol + benzene acetone chloroform
+
+
+
+
TJ'C
a21
a12
SP
60 40
0.3869 0.2900 0.2920 0.2942 0.2392 0.5105 0.1028 0.0974 1 0.09688 0.2907
0.3562 0.3561 0.3799 0.4048 0.2722 0.2989 0.6393 0.6458 0.6498 0.6404
2.0 1.4 0.7 1.6 0.07 0.03 0.009 0.2 0.9 0.04
55 70 30 40 25 40
+
+
55
+
25
Wilson
SP
SY 0.10 0.09 0.14 0.1 1 0.46 0.04 0.008 0.07 0.13 0.24
SY 0.07 0.06 0.13 0.13 0.64 0.04 0.01 0.05 0.13 0.24
1 .o 1.1 0.4 0.6 0.09 0.02 0.004 0.1 0.5 0.04
In this table Sp = 1 0 - 2 ~ ( p - pexp)2and S, = 1 0 2 ~ (-yy,xp)2.
ff
; 4 u ?&
;
; 9
L 0.0
0.1
0.2
0.3
0.5
0.4
0.7
0.6
0.8
0.9
1.0
0.0
0.1
0.2
0.3
0.5
0.4
5 1
0.8
0.7
0.8
0.9
1.0
5 1
Figure 3. Vapor-liquid composition (x-y) diagram for ethanol ( I ) + Figure 4. Composition versus temperature (x-T) diagram for liquidbenzene (2) at 55 OC. The squares are the experimental data? the solid liquid equilibrium of methanol ( I ) + heptane (2). The squares are the curve is the present theory, and the dashed curve is Wilson c ~ r r e l a t i o n . ~ * l ~ experimental datal2 and the solid curve is the present theory.
it gave the best overall fit to the large number of mixtures studied in VLE calculation^.^ Table I also includes parameters cyzl and a12,and optimized values of S, and S,. Figures 1-3 show the vapor-liquid equilibria compositions (x - y ) diagrams for the systems methanol water, I-propanol water, and ethanol benzene, respectively. According to Table I and Figures 1-3 it can be concluded that the present theory can fit the VLE of the systems studied as good as the Wilson equation. Applications f o r Liquid-Liquid Phase Equilibria. Phase splitting in liquid mixtures occurs when two phases have a lower total Gibbs free energy than one phase at constant temperature and pressure. A necessary condition for the stability of a twocomponent homogeneous mixture is1'
+
+
dpi/dx,
>0
i = 1, 2
In terms of the activity coefficients, eq 28 becomes d(ln yi)/dxi + 1 /xi > 0
+
(28) (29)
Substituting for In y i in this equation from eq 25 gives the following result provided parameters azl and a12are positive: (a12x1 + a2lx2)(L,xl
+ L2x2) -
XIX2(~21UI/~Z+ a I z u 2 / U l - 1)
'0 (30)
For liquid-liquid equilibria (LLE) the left-hand side of inequality (30) should be negative. Considering the magnitudes of the different quantities one concludes that the present activity coefficient model is capable of predicting LLE. The Wilson on the other hand, is known to be unable to predict c~rrelation,~J*
0.0
0.1
0.2
0.3
0.4
0.5
0.8
0.7
0.8
0.D
1.0
5 1
Figure 5. Composition versus tempreature (x-7") diagram for hid-liquid equilibrium of formic acid ( I ) benzene (2). The squares are the experimental data'* and the solid curve is the present theory.
+
J . Phys. Chem. 1990, 94, 3 152-3 156
3152
TABLE II: LLE Parameters4 of the Present Activity Coefficient Model for Different Binary Mixtures system @,I" a,,, s x 103
methanol + heptane -1070.9 methanol + cyclohexane -980.29 formic acid + benzene -748.15 phenol + octane -1077.0 methanol + carbon disulfide -797.20 nitrobenzene + hexane -975.97 I ,3-dihydroxybenzene + benzene -937.02
1.0243 1.0191
0.97983 1.0031 1.0186
1.0027 1.0032
0.12 0.12 1.7
7.3 0.45 2.0 5.5
LLE, because it always satisfies inequality.29 The present activity coefficient theory is tested versus experimental LLE data.I2 For the variation of the isothermal compressibilities with temperature, which is needed in this calculation, a three-parameter equation'j is used. The temperature and pure component volume dependence of parameters and a 1 2for a number of organic binary systems ( 12) Liquid-Liquid Equilibrium Data Collecrion; DECHEMA: Frankfurt, 1979. (13) Brostow. W.: Maynadier, P. High Temp. Sci. 1979, / I , 7. (14) McQuarrie, A. Statistical Mechanics, Harper and Row: New York, 1975. (15) Haile, J. M., Mansoori, G. A,, Eds. Molecular-Based Study of Fluids; American Chemical Society: Washington, DC, 1983. (16) Mazo, R. M. J . Chem. Phys. 1958, 29, 1122. (17) Hamad, E. Z.; Mansoori, G. A.; Ely, J. F. J . Chem. Phys. 1987,86,
1478.
(18) Wilson, G. M. J . Am. Chem. Soc. 1964, 86, 127.
exhibiting LLE are represented by the following expressions. a21 = ( U * / U I ) exp(a,,,/kT) (31) = 0'20(4/~2) (32) In which a210 and a I Z O are constants independent of temperature and composition. The experimental LLE data is fitted to the activity coefficient equation with eqs 31 and 32 for a2,and c y l 2 by minimizing the difference between the activities in the two liquid phases a12
where superscripts (1) and (2) are for phases 1 and 2, respectively. and aI2, and optimum Table I1 shows values of parameters a z I O S for a number of binary systems. The variation of the compositions of two LLE systems (methanol heptane and formic acid benzene) with temperature is shown in Figures 4 and 5. According to these figures the agreement with the data is satisfactory. The largest deviation of theory from experimental data occurs near the upper critical solution temperature. Overall the present technique is capable of formulating analytic expressions for activity coefficients in mixtures. For this purpose it is necessary to define closure expressions for the cross direct correlation function integrals. Application of the activity coefficient expression resulting from this technique for vapor-liquid and liquid-liquid equilibria calculations has been as successful as the other activity coefficient expressions available.
+
+
Acknowledgment. This research is supported by the Gas Research Institute Contract No. 5086-260-1244.
Relationship between Microscopic and Macroscopic Orientational Relaxation Times in Polar Liquidst Amalendu Chandra and Biman Bagchi* Solid State and Structural Chemistry Unit, Indian Institute of Science, Bangalore 560 012, India (Received: May I , 1989; In Final Form: October 12, 1989)
Microscopic relations between single-particle orientational relaxation time ( T , ) , dielectric relaxation time ( T ~ ) and , many-body orientational relaxation time ( T ~ of) a dipolar liquid are derived. We show that both T~ and T~ are influenced significantly by many-body effects. In the present theory, these many-body effects enter through the anisotropic part of the two-particle direct correlation function of the polar liquid. We use mean-spherical approximation (MSA) for dipolar hard spheres for explicit numerical evaluation of the relaxation times. We find that, although the dipolar correlation function is biexponential, the frequency-dependent dielectric constant is of simple Debye form, with T~ equal to the transverse polarization relaxation time. The microscopic T~ falls in between Debye and Onsager-Glarum expressions at large values of the static dielectric constant.
Introduction
The relationship between single-particle orientational relaxation time, T,, and many-body orientational relaxation time, T ~ of, dipolar molecules in a dense liquid has been a subject of much discussion in the recent past.'-4 Another related question is the relationship between T , and dielectric relaxation time T~ of a pure dipolar liquid.5-" It is clear that the dielectric relaxation time contains some amount of many-body effects, although the precise , T~ have never been made clear. relations between T,, T ~ and Following Kivelson and Madden,Iv2 we shall call a relation between a microscopic relaxation time, such as T " , and a macroscopic relaxation time, such as T M ,a macro-micro relation. The first such relation was proposed by D e b ~ e ' who , ~ used his continuum 'Contribution No. 567 from Solid State and Structural Chemistry Unit. 0022-3654/90/2094-3 1 52$02.50/0
theory of dielectric relaxation to propose the following relation n2 + 2 7, = (1) co 2TD
+
where co is the static dielectric constant and n is the refractive ( 1 ) Kivelson, D.; Madden, P. Mol. Phys. 1975, 30, 1749. (2) Madden, P.: Kivelson, D. Adu. Chem. Phys. 1984, 56, 467. (3) Evans, M . ; Evans, G. J.; Coffey, W. T.; Grigolini, P. Molecular Dynamics and Theory of Broad Band Spectroscopy; Wiley: New York, 1984; Chapter 3. (4) Berne, B. J . Chem. Phys. 1975, 62, 1154. Berne, B.; Pecora, R. Dynamic Light Scattering; Wiley: New York, 1976. (5) Hubbard, J. B.; Wolynes, P. G. J . Chem. Phys. 1978, 69, 998. (6) Hill, N. E. In Dielectric Properties and Molecular Eehauiour; Hill, N. E., Vaughan, W. E., Prince, A. H., Davies, M., Eds.; van Nostrand: London, 1989.
0 1990 American Chemical Society
