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Numerical interpretation of the concentration dependence of micelle diffusion coefficients
Numerical Interpretation of the Concentration Dependence of Micelle Diffusion Coefficients* GEORGE D. J. PHILLIES Department o f Physics, Worcester Polytechnic Institute, Worcester, Massachusetts 01609 Received September 8, 1986; accepted December 3, 1986 The numerical interpretation of the concentration dependence of the micelle mutual diffusion coefficient D in terms ofintermicellar forces is discussed. The conflict in the literature between different expansions for D as D = Do(1 + ac), c being the micelle concentration, is resolved. Use of the correct form for a, rather than the forms previously used to analyze experimental data, leads to modestly different predictions for a as a function o f intermicellar potential. © 1987AcademicPress,Inc. INTRODUCTION
The size, shape, and interactions of micelles, and the dependences of these properties on amphiphile concentration, temperature, and ionic strength, have been the subject of extensive investigation. Besides their fundamental physico-chemical interest, micelle properties may also have practical consequences. Micelles have been proposed as carriers for pharmaceutical agents; the formation (or nonformation) of cholesterol gallstones has been related to the ability of cholesterol, lecithin, and bile salts to form mixed miceUes (1). Experimental studies based on quasi-elastic light scattering spectroscopy (QELSS) have recently played a significant part in these investigations. QELSS measures directly the mutual diffusion coefficient D. For extremely dilute solutions of spheres, D is expected to obey the Stokes-Einstein equation kBT D - 67r~lR'
[1]
k~ being Boltzmann's constant, T the absolute temperature, ~ the viscosity, and R the sphere's hydrodynamic radius. Equation [ 1] is not valid for measurements with QELSS on systems of interacting macroparticles (2, 3). In nondilute * This work supported by the National Institutes of Health under Grant GM36270-01.
systems, variations in D may be due either to changes in micelle size or to changes in intermicellar interactions, or to both simultaneously. Electrostatic interactions between model bodies (protein molecules) having the size and charge of small ionic micelles give D a strong concentration dependence (4), even at ionic strengths as high as 0.15 M. For interacting systems, D has often been parameterized as O -- D0(1 + (ka + k~)4~+ O(~b2)),
[2]
where q~ is the micelle volume fraction, ka is a purely thermodynamic term, and kh reflects hydrodynamic interactions between micelles. A variety of expressions for kh and kd exist (5-8). Recently, there has been interest in inverting Eq. [2], so as to compute micellar properties such as the fractional ionization from ka and kh (9, 10). This inversion process requires a good knowledge of the connection between kh and the intermicellar potential. Most workers have used the formulae for D of Batchelor (5) and Felderhof (6), without discussing the uncertainty in the analytical form used to compute kh. As several different forms for kh have been proposed, the potential systematic error in their analyses may have been underemphasized. The following note treats the uncertainty in kh.
518 0021-9797/87 $3.00 Copyright© 1987by AcademicPress,Inc. All fightsof reproductionin any formreserved.
Journalof Colloidand InterfaceScience,Vol. 119,No. 2, October1987
519
MICELLE DIFFUSION
DISCUSSION
It has been shown (7, 8) that the correct form for D is
D=D0{1+cofdrg(r)[i~.(b+W).f~ +=~-~(b i[W" +T)]eik.r}/S(k), [3] where Do is the diffusion coefficient at zero concentration, Co is the solute concentration, g(r) is the solute-solute radial distribution function, k is the scattering vector (with k = Ikl and [~ the unit vector), and b and T are the self and distinct hydrodynamic interaction tensors, respectively. Experimental evidence confirming this form is noted below (11). The derivation of this form will not be repeated (but see Refs. (7, 8)). The Appendix examines alternative approaches to D, showing why different expressions for D were obtained by other authors. For spheres of radius a, the tensors b and T are (12)
()
1/a'6
15 a 4 ~ + i _ ~ r ) (105~-- 171) [4a] b =--~- r and 3a _1[a/3 75 [a~7^^ T = ~ r (I + f~) + 2 \ r] (! - 3ff) + -~- ~r) rr, [4b] where f is the unit vector pointing from particle i to particle j. T is conveniently divided between an Oseen ((a/r) 1) component To~ and a shorter range ((a/r)n, n >~3) component Tsr. As described in (7), the integral in Eq. [3] of Tos over go can evidently be made convergent by applying the sum rule
= q~, the volume fraction of solute in the system. In Eq. [3] the integral over the (a/r) 3term of T~r is convergent only because the factor exp(ik, r) is present (the limit k --~ 0 then being taken); without the exponential, this integral would be improper. Equation [3] differs from those used in previous analyses of the micelle diffusion coefficient through the q5l term (introduced via the sum rule, Eq. [5]) and through the presence of the divergence term f drg(r)exp(ik, r)il~V : (b + T). The sum rule, which describes the displacement of solvent by moving solute particles, is equivalent to the reference frame correction of Kirkwood et al. (13). The existence of reference frame corrections in diffusion has been amply demonstrated by experiment. For example, they are necessary if studies on diffusion in mixed-salt systems are to agree with the Onsager reciprocal relations (I 4-16). Early work on D did not reveal the divergence terms, largely because it was assumed (2) that b and T, arising as they do from the flow of an incompressible fluid, are divergenceless. This assumption is correct for loworder (in a/r) approximations to b and T. However, b and T describe the flow of suspended particles, not the flow of fluid. As first recognized by Felderhof (17), while the fluid flow is incompressible, the particles move with respect to the fluid, so the particle flow may have a nonzero divergence. For hard spheres, one obtains D = D0(1 - 8.949)/S(k) = D0(1 - 0.940.
In Eq. [6], S(k) = I + Co f dr(g(r) - 1) is the static structure factor; 1/S(k) = 1 + ka4). The divergence and reference frame corrections are not small; without them, for hard spheres one would obtain D = Do(1 + 1.6q5).
fdr[k,'T'.f~eik'r+cht] =0,
[51
where T' includes corrections to T due to the finite volume of the container, and the particle size a satisfies ka ~ 1. Here ~t is the volume fraction of a single particle, so that Cof dr4~/
[61
[7]
The most recent experiments on the diffusion of a model system for the hard sphere suspension are in excellent agreement with Eq. [6], and in substantial disagreement with Eq. [7]. While these are difficult measurements, experiments appear to confirm the validity of Journal of Colloid and Interface Science, Vol. 119, No, 2, October 1987
520
GEORGE
D. J. P H I L L I E S
Eqs. [3] and [6]. Namely, Mos et al. (11) measured the diffusion of stearylated silica spheres (18) in toluene and xylene, finding (their Figs. 5 and 6) kd + kh = --0.9 to --1.2, which is very close to the theoretical value -0.9 given in Eq. [6]. For polystyrene latex spheres in water, Mos et al. (11) found ka + kh of 0.0. As the polystyrene spheres are slightly charged, their kd + kh would be expected to be slightly larger than the -0.9 found for simple hard spheres, as was observed. Mos's et al. results (11) are not in good agreement with the previous studies of KopsWerkhoven and Fijnaut (19), who found kd + kh near + 1.6. The experiments differ in that Kops-Werkhoven and Fijnaut (19) used QELSS, while Mos et al. (11) used both QELSS and homodyne coincidence spectroscopy (HCS) (20, 21), a two-detector technique which is immune to multiple scattering artifacts. Mos et al. compared QELSS and HCS data on the same silica sphere suspensions, finding with QELSS that D increases with sphere concentration, while with HCS D falls with increasing sphere concentration. This disagreement between QELSS and HCS results
from multiple scattering, which artifactually increases the QELSS value for D. When the techniques disagree, the HCS value for D is correct (20, 21). From Mos's et al. data, D for hard spheres falls gradually with increasing c, as predicted by Eq. [6]. To treat micelles, which are not simple hard spheres, the radial distribution function is usefully divided between a hard-sphere term go (go -- 0, r < 2a; go -- 1, r > 2a) and a perturbation term 6g, so that g = go,
r < 2a
[Sa]
g=go+rg(r), r>2a,
[8b]
where fg describes the effect of screened electrostatic, van der Waals, and other forces, which are all assumed to be short-range. For charged particles in a salt solution, the linearized Debye-Huckel model gives fig =
A exp(-Kr)
,
[9]
r
where K is the Debye-Huckel inverse screening length. The methods which gave Eq. [6] for hard spheres give
D = D0(1- 8.9q~+ c 0 f ~ dt (12r6-45r3+93r+450)fg(r))/S(k)4r 5
[10]
in the presence of an arbitrary perturbing potential. Through S(k), repulsive interactions tend to increase D, while attractive interactions tend to reduce D. If the spheres are charged and screened, Eq. 10 becomes D = D0(1 - 8.9~b+
OA[120(Ka)4- 184(Ka)3+122(Ka)2+568(Ka)l-87] e - z K a 2 5 6 + --~-a
32
where Ei is the exponential integral evaluated for a specific argument, [,~ e-Kax
Ei= J2 --;-dx,
tl21
and where S(k) must be computed from g(r), not from 6g(r). The divergence terms, which have been omitted in previous treatments of the micelle interaction problem, are not a trivial correction. If one calculated D for a hard-sphere + screened electrostatic potential, but omitted the divergence terms, D would be changed by an amount Journal of Colloid and Interface Science, Vol. 119, No. 2, October 1987
MICELLE DIFFUSION AD=
521
+1.47(p-chA[(200(Ka)4-312(Ka)3+206(Ka)z+704Ka-21) 512
e-ZKa
_(50(Ka)'-53(K4)4+240(Ka)2)Ei] In Eq. [13], the +1.47 term arises from the average of the divergence terms over the hardsphere potential, while the term in A represents the average of these terms over a Debye potential. The significance of Eq. [13] is shown in Table I, which compares the electrostatic contribution to D in the presence of divergence terms (Eq. [ 11]) to the modification o l d by the electrostatic divergence terms (Eq. [ 13]). The ratio of the electrostatic parts of Eqs. [11] and [13] is independent of A. At Ka ~ 1, the divergence terms are not important, since with long-range interactions present the integral f drk.Tos.[~rg(r) completely dominates. However, with added salt the electrostatic interactions become shorter in range, under which conditions the form of the hydrodynamic interactions at small distances (where V . (T + b) is large) become more substantial. The effect of the V. (T + b) terms on kh, while never dominant, is numerically identifiable for the potentials considered here. The experimental kd + kh includes both hard-sphere and electrostatic forces. Even if the electrostatic part of Eqs. [11] and [13] is not significant, neglecting the hard-sphere
[13]
component of the divergence terms can significantly perturb data analysis. For example, if one neglected the divergence terms, a hardsphere system would be predicted to have a diffusion coefficient which increased weakly with increasing concentration. An experimental system in which D increased slowly with concentration would then appear to be hard-sphere-like, implying that attractive van der Waals and repulsive electrostatic forces must nearly have canceled in their effect on D. From Eq. [6], though, a system in which van der Waals and electrostatic interactions nearly cancel (in their effect on D) will actually have a D which falls with increasing qS. If D actually rose with increasing qS, Eqs. [6] and [ 11] indicate that long-range repulsive interactions would need to predominate over longrange attractive interactions, in their effect on D. APPENDIX
The purpose of this Appendix is to compare the methods used by Batchelor (5), Felderhof (6), and this author (7, 8) to compute D. It will be argued that all four references are fundamentally correct as calculations of some diffusion coefficient, but that only Refs. (7 and 8) calculated the diffusion coefficient measured TABLEI by QELSS. ElectrostaticContributionI (from Eq. [11]), Expanded Batchelor (5) obtained D by evaluating the as D = D0(l - 8.9q~ + epAI)/S(k)to the Concentration Dependenceof D, and the FractionalContribution of the flux of particles due to an applied steady (therElectrostatic Component of the Divergence Terms (Eq. modynamic) force, the flux being related to [ 131) to I. the diffusion coefficient by Ka
I
0.2 0.5
9.98 2.21 0.417 0.0297 0.0032
1.0
2.0 5.0
flay : (T +h)exp(ikr'r)dr/l
0.6% 2.5% 6%
11% 19%
J = -DVc.
[A. 1]
D was obtained by generalizing the Einstein expression D = kBT(b) [A.2] for D, where b is the mobility tensor. In Einstein's original argument, la was a constant. In Journal of Colloid and Interface Science, Vol. 119,No. 2, October 1987
522
GEORGE
D . J. P H I L L I E S
Batchelor's generalization, b is given by a microscopic expression, which depends on the relative positions of the particles in the system. In contrast, D is a macroscopic quantity which does not depend explicitly on particle positions. To compute the macroscopic D from a microscopic b, Batchelor took an ensemble average of b over possible particle configurations, so that D = kBT(b).
[A.3I
Experimentally, light scattering spectroscopy is directly sensitive to particle positions, not to the particle current J. Specifically, the instantaneous scattered field is proportional to the instantaneous value of the/cth spatial Fourier component Ek(t) of the local index of refraction, where ek(t) is in turn proportional to the kth spatial Fourier component ak(t) of the scatterer concentration. Here k is the scattering vector selected by the source and detector positions. In light scattering spectroscopy, D is obtained from the temporal evolution of a~(t), namely
croscopic representation of a~(t) as a function of particle positions, N
ak(t) = ~ e ik''j,
[A.71
j=l
the sum being over all N particles in the system. D is then obtained from an ensemble average over the short-time limit of (a-k(O)$ak(t)). Equation [A.6] manifestly contains terms in V. [O], which a full calculation shows to be nonvanishing. While the calculation of D as kBT(b) is mathematically correct, QELSS does not determine D by measuring d and Vc directly. Rather, QELSS yields directly the quantity D defined by Eq. [A.4]. The prescription accompanying Eqs. [A.5]-[A.7] therefore gives a form for D which is preferable to the one obtained from Eq. [A.3]. Felderhof's calculation (6) of D is also based on the Smoluchowski equation, set in the form
Oa( rl , t) - - Do~71 • (~Tla(rl, t) Ot
lim (d/dt)(a_k(O)a~(t)) D ~ t---~o
_k2(a_k(O)ak(O))
[A.4]
References (7, 8) made assumptions equivalent to assuming that particle motions are correctly described by the Smoluchowski sedimentation equation
+ Col3f x71qSga(r2, t)dr2) + coVl- f bgdrz • Via(r1, t) + CoVl" ~ T. gV2a(r2, t)dr2 + 0(c2).
[A.8]
d
dc (x, t) dt
$c(x,
t),
[A.5]
where the Smoluchowski operator is S = V. (D. V + D. F/kBT).
[A.6]
Here O is the diffusion tensor (as distinct from the inferred diffusion coefficient D) and F is the applied force. Just as Eq. [A.2] was applied to the problem by interpreting b as a microscopic mobility tensor, Eqs. [A.5] and [A.6] were applied in (Refs. 7, 8) by giving D, F, and c(x, t) their microscopic interpretations. In particular, c(x, t) was replaced with the miJournal of Colloid and InterfaceScience, Vol. 119, No. 2, October 1987
where a(r~, t) is the local density at r~, q~is the intermicellar potential, g = g(rl - r2) is the equilibrium pair distribution function, and Vi is the gradient with respect to particle i, the particles being the particle of interest 1 and a neighboring particle 2. As interacting particles are relatively close together, the concentration gradients at their locations should be roughly equal, i.e., Vla(rl, t) ~ V2a(r2, t)
[A.9]
for a pair of interacting particles. (This is a small-k approximation.) Rearrangement of Eq. [A.8] gives
MICELLE DIFFUSION
da(rlt) dt
= DoVI" (V~a(rl, 0
+ CoBf dr2Vl(4~)go(r2 - rl)nl(r2, t)) + Cof (b + T) : V~a(rl, t)go(r2 - rl)dr2 [A. I0] as evaluated in Ref. (6), plus such terms as
c o f V l " [T]g0" V2a(r2,t)dr2
[A.1 1]
which vanish because f V , Tdr is odd in r. Reference (6) thus does not find nonzero terms in V . (T + b). References (6-8) or Eqs. [A.10] and [3], disagree because they evaluate different functions in order to determine D. In Eq. [3], D was obtained from the average
(a_~(O)Sak(t))
[A. 12]
while Eqs. [A.10] and [A.11] obtain D from an ensemble average over the algebraic kernel o f S, with no factors ofa~ included within the average. The microscopic S does give the concentration changes to be expected from a given microscopic particle configuration, so ( S ) does give an average rate of change for ak(t); also, ( S ) has no nonzero divergence terms. However, QELSS obtains the light-scatteringintensity weighted-average temporal evolution of ak(t), not the unweighted average. States which scatter no light make no contribution to the observed temporal evolution of a~(t). The z-weighting is due to the factors ak(0) and ak(t). For a z-weighted average, these factors must be included in the ensemble average. Including these factors in the average replaces, e.g., Eq. [A.I 1] by
co f d r 2 V , . ( e i k ' r , o v . ( t T ] ) ,
[A.13]
as seen in Eq. [3]. Unlike term [A, 1 1], the average l~V. T e x p ( i k , r) of [A. 13] is not odd in r and does not vanish. QELSS measures (a(O)da(t)/dt), not J, so the appropriate mi-
523
croscopic average for D is that of Eq. [3], not that ofEqs. [A.3] or [A. 10], though [A.3] and [A.10] do represent diffusion coefficients which some (hypothetical) experiment could measure. The failure of D (from Eq. [A.3]) and D (from Eq. [A.6]) to equal each other does not mean that the Onsager regression hypothesis has failed. The calculations of ( S ) and of(ak(O)Sak(t)) were m a d e using the same D in Eq. [A.6]. However, O is a function of particle positions; if D is averaged repeatedly over an ensemble, using different weighting functions each time, the averages over D can yield different numerical values. REFERENCES I. Carey, M. C., and Small, D. M., J. Clin. Invest. 61, 998 (1978). 2. Altenberger,A. R., and Deutch, J. M., J. Chem. Phys. 59, 894 (1973). 3. Phillies, G. D. J., J. Chem. Phys. 60, 976 (1974). 4. PhiUies,G. D. J., Benedek, G. B., and Mazer, N. A, J. (?hem. Phys. 65, 1883 (1976). 5. Batchelor,G. K., J. FluidMeeh. 74, 1 (1976). 6. Felderhof, B. U., J. Phys. A. 11,929 (1978). 7. Phillies, G. D. J., J. Chem. Phys. 77, 2623 (1982). 8. Carter, J. M., and Phillies, G. D. J., J. Phys. Chem. 89, 5118 (1985). 9. Corti, M., and Degiorgio, V. J., Phys. Chem. 85, 711 (1981). 10. Biresaw, G., McKenzie, D. C., Bunton, C. A., and Nicoli, D. F., J. Phys. Chem. 89, 5144 (1985). 11. Mos, H. J., Pathmamanoharan, C., Dhont, J. K. G., and de Kruif, C. G., J. Chem. Phys. 84, 45 (1986). 12. Mazur, P., and van Saarloos, W., Physica A 115, 21 (1982). 13. Kirkwood,J. G., Baldwin, R. L., Dunlop, P. J., Gosting, L. J., and Kegeles,G., J. Chem. Phys. 33, 1505 (1960). 14. Fujita, H., and Gosting, L. J., J. Amer. Chem. Soe. 78, 1099 (1956). 15. Dunlop, P. J., and Gosting, L. J., J. Phys. Chem. 63, 86 (1959). 16. Dunlop, P. J., J. Phys. Chem. 63, 612 (1959). 17. Felderhof, B. U., Physica A 85, 509 (1985). 18. Stober,W., Fink, A., and Bohn, E., Z ColloidInterface Sci. 81, 196 (1968). 19. Kops-Werkhoven, M. M., and Fijnaut, H. M., J. Chem. Phys. 74, 1618 (1981). 20. Phillies, G. D. J., J. Chem. Phys. 74, 260 (1981). 21. PhiUies,G. D. J., Phys. Rev. A 24, 1939 (1981).
Journal of Colloid and Interface Science, Vol. 119, No. 2, October 1987
The size, shape, and interactions of micelles, and the dependences of these properties on amphiphile concentration, temperature, and ionic strength, have been the subject of extensive investigation. Besides their fundamental physico-chemical interest, micelle properties may also have practical consequences. Micelles have been proposed as carriers for pharmaceutical agents; the formation (or nonformation) of cholesterol gallstones has been related to the ability of cholesterol, lecithin, and bile salts to form mixed miceUes (1). Experimental studies based on quasi-elastic light scattering spectroscopy (QELSS) have recently played a significant part in these investigations. QELSS measures directly the mutual diffusion coefficient D. For extremely dilute solutions of spheres, D is expected to obey the Stokes-Einstein equation kBT D - 67r~lR'
[1]
k~ being Boltzmann's constant, T the absolute temperature, ~ the viscosity, and R the sphere's hydrodynamic radius. Equation [ 1] is not valid for measurements with QELSS on systems of interacting macroparticles (2, 3). In nondilute * This work supported by the National Institutes of Health under Grant GM36270-01.
systems, variations in D may be due either to changes in micelle size or to changes in intermicellar interactions, or to both simultaneously. Electrostatic interactions between model bodies (protein molecules) having the size and charge of small ionic micelles give D a strong concentration dependence (4), even at ionic strengths as high as 0.15 M. For interacting systems, D has often been parameterized as O -- D0(1 + (ka + k~)4~+ O(~b2)),
[2]
where q~ is the micelle volume fraction, ka is a purely thermodynamic term, and kh reflects hydrodynamic interactions between micelles. A variety of expressions for kh and kd exist (5-8). Recently, there has been interest in inverting Eq. [2], so as to compute micellar properties such as the fractional ionization from ka and kh (9, 10). This inversion process requires a good knowledge of the connection between kh and the intermicellar potential. Most workers have used the formulae for D of Batchelor (5) and Felderhof (6), without discussing the uncertainty in the analytical form used to compute kh. As several different forms for kh have been proposed, the potential systematic error in their analyses may have been underemphasized. The following note treats the uncertainty in kh.
518 0021-9797/87 $3.00 Copyright© 1987by AcademicPress,Inc. All fightsof reproductionin any formreserved.
Journalof Colloidand InterfaceScience,Vol. 119,No. 2, October1987
519
MICELLE DIFFUSION
DISCUSSION
It has been shown (7, 8) that the correct form for D is
D=D0{1+cofdrg(r)[i~.(b+W).f~ +=~-~(b i[W" +T)]eik.r}/S(k), [3] where Do is the diffusion coefficient at zero concentration, Co is the solute concentration, g(r) is the solute-solute radial distribution function, k is the scattering vector (with k = Ikl and [~ the unit vector), and b and T are the self and distinct hydrodynamic interaction tensors, respectively. Experimental evidence confirming this form is noted below (11). The derivation of this form will not be repeated (but see Refs. (7, 8)). The Appendix examines alternative approaches to D, showing why different expressions for D were obtained by other authors. For spheres of radius a, the tensors b and T are (12)
()
1/a'6
15 a 4 ~ + i _ ~ r ) (105~-- 171) [4a] b =--~- r and 3a _1[a/3 75 [a~7^^ T = ~ r (I + f~) + 2 \ r] (! - 3ff) + -~- ~r) rr, [4b] where f is the unit vector pointing from particle i to particle j. T is conveniently divided between an Oseen ((a/r) 1) component To~ and a shorter range ((a/r)n, n >~3) component Tsr. As described in (7), the integral in Eq. [3] of Tos over go can evidently be made convergent by applying the sum rule
= q~, the volume fraction of solute in the system. In Eq. [3] the integral over the (a/r) 3term of T~r is convergent only because the factor exp(ik, r) is present (the limit k --~ 0 then being taken); without the exponential, this integral would be improper. Equation [3] differs from those used in previous analyses of the micelle diffusion coefficient through the q5l term (introduced via the sum rule, Eq. [5]) and through the presence of the divergence term f drg(r)exp(ik, r)il~V : (b + T). The sum rule, which describes the displacement of solvent by moving solute particles, is equivalent to the reference frame correction of Kirkwood et al. (13). The existence of reference frame corrections in diffusion has been amply demonstrated by experiment. For example, they are necessary if studies on diffusion in mixed-salt systems are to agree with the Onsager reciprocal relations (I 4-16). Early work on D did not reveal the divergence terms, largely because it was assumed (2) that b and T, arising as they do from the flow of an incompressible fluid, are divergenceless. This assumption is correct for loworder (in a/r) approximations to b and T. However, b and T describe the flow of suspended particles, not the flow of fluid. As first recognized by Felderhof (17), while the fluid flow is incompressible, the particles move with respect to the fluid, so the particle flow may have a nonzero divergence. For hard spheres, one obtains D = D0(1 - 8.949)/S(k) = D0(1 - 0.940.
In Eq. [6], S(k) = I + Co f dr(g(r) - 1) is the static structure factor; 1/S(k) = 1 + ka4). The divergence and reference frame corrections are not small; without them, for hard spheres one would obtain D = Do(1 + 1.6q5).
fdr[k,'T'.f~eik'r+cht] =0,
[51
where T' includes corrections to T due to the finite volume of the container, and the particle size a satisfies ka ~ 1. Here ~t is the volume fraction of a single particle, so that Cof dr4~/
[61
[7]
The most recent experiments on the diffusion of a model system for the hard sphere suspension are in excellent agreement with Eq. [6], and in substantial disagreement with Eq. [7]. While these are difficult measurements, experiments appear to confirm the validity of Journal of Colloid and Interface Science, Vol. 119, No, 2, October 1987
520
GEORGE
D. J. P H I L L I E S
Eqs. [3] and [6]. Namely, Mos et al. (11) measured the diffusion of stearylated silica spheres (18) in toluene and xylene, finding (their Figs. 5 and 6) kd + kh = --0.9 to --1.2, which is very close to the theoretical value -0.9 given in Eq. [6]. For polystyrene latex spheres in water, Mos et al. (11) found ka + kh of 0.0. As the polystyrene spheres are slightly charged, their kd + kh would be expected to be slightly larger than the -0.9 found for simple hard spheres, as was observed. Mos's et al. results (11) are not in good agreement with the previous studies of KopsWerkhoven and Fijnaut (19), who found kd + kh near + 1.6. The experiments differ in that Kops-Werkhoven and Fijnaut (19) used QELSS, while Mos et al. (11) used both QELSS and homodyne coincidence spectroscopy (HCS) (20, 21), a two-detector technique which is immune to multiple scattering artifacts. Mos et al. compared QELSS and HCS data on the same silica sphere suspensions, finding with QELSS that D increases with sphere concentration, while with HCS D falls with increasing sphere concentration. This disagreement between QELSS and HCS results
from multiple scattering, which artifactually increases the QELSS value for D. When the techniques disagree, the HCS value for D is correct (20, 21). From Mos's et al. data, D for hard spheres falls gradually with increasing c, as predicted by Eq. [6]. To treat micelles, which are not simple hard spheres, the radial distribution function is usefully divided between a hard-sphere term go (go -- 0, r < 2a; go -- 1, r > 2a) and a perturbation term 6g, so that g = go,
r < 2a
[Sa]
g=go+rg(r), r>2a,
[8b]
where fg describes the effect of screened electrostatic, van der Waals, and other forces, which are all assumed to be short-range. For charged particles in a salt solution, the linearized Debye-Huckel model gives fig =
A exp(-Kr)
,
[9]
r
where K is the Debye-Huckel inverse screening length. The methods which gave Eq. [6] for hard spheres give
D = D0(1- 8.9q~+ c 0 f ~ dt (12r6-45r3+93r+450)fg(r))/S(k)4r 5
[10]
in the presence of an arbitrary perturbing potential. Through S(k), repulsive interactions tend to increase D, while attractive interactions tend to reduce D. If the spheres are charged and screened, Eq. 10 becomes D = D0(1 - 8.9~b+
OA[120(Ka)4- 184(Ka)3+122(Ka)2+568(Ka)l-87] e - z K a 2 5 6 + --~-a
32
where Ei is the exponential integral evaluated for a specific argument, [,~ e-Kax
Ei= J2 --;-dx,
tl21
and where S(k) must be computed from g(r), not from 6g(r). The divergence terms, which have been omitted in previous treatments of the micelle interaction problem, are not a trivial correction. If one calculated D for a hard-sphere + screened electrostatic potential, but omitted the divergence terms, D would be changed by an amount Journal of Colloid and Interface Science, Vol. 119, No. 2, October 1987
MICELLE DIFFUSION AD=
521
+1.47(p-chA[(200(Ka)4-312(Ka)3+206(Ka)z+704Ka-21) 512
e-ZKa
_(50(Ka)'-53(K4)4+240(Ka)2)Ei] In Eq. [13], the +1.47 term arises from the average of the divergence terms over the hardsphere potential, while the term in A represents the average of these terms over a Debye potential. The significance of Eq. [13] is shown in Table I, which compares the electrostatic contribution to D in the presence of divergence terms (Eq. [ 11]) to the modification o l d by the electrostatic divergence terms (Eq. [ 13]). The ratio of the electrostatic parts of Eqs. [11] and [13] is independent of A. At Ka ~ 1, the divergence terms are not important, since with long-range interactions present the integral f drk.Tos.[~rg(r) completely dominates. However, with added salt the electrostatic interactions become shorter in range, under which conditions the form of the hydrodynamic interactions at small distances (where V . (T + b) is large) become more substantial. The effect of the V. (T + b) terms on kh, while never dominant, is numerically identifiable for the potentials considered here. The experimental kd + kh includes both hard-sphere and electrostatic forces. Even if the electrostatic part of Eqs. [11] and [13] is not significant, neglecting the hard-sphere
[13]
component of the divergence terms can significantly perturb data analysis. For example, if one neglected the divergence terms, a hardsphere system would be predicted to have a diffusion coefficient which increased weakly with increasing concentration. An experimental system in which D increased slowly with concentration would then appear to be hard-sphere-like, implying that attractive van der Waals and repulsive electrostatic forces must nearly have canceled in their effect on D. From Eq. [6], though, a system in which van der Waals and electrostatic interactions nearly cancel (in their effect on D) will actually have a D which falls with increasing qS. If D actually rose with increasing qS, Eqs. [6] and [ 11] indicate that long-range repulsive interactions would need to predominate over longrange attractive interactions, in their effect on D. APPENDIX
The purpose of this Appendix is to compare the methods used by Batchelor (5), Felderhof (6), and this author (7, 8) to compute D. It will be argued that all four references are fundamentally correct as calculations of some diffusion coefficient, but that only Refs. (7 and 8) calculated the diffusion coefficient measured TABLEI by QELSS. ElectrostaticContributionI (from Eq. [11]), Expanded Batchelor (5) obtained D by evaluating the as D = D0(l - 8.9q~ + epAI)/S(k)to the Concentration Dependenceof D, and the FractionalContribution of the flux of particles due to an applied steady (therElectrostatic Component of the Divergence Terms (Eq. modynamic) force, the flux being related to [ 131) to I. the diffusion coefficient by Ka
I
0.2 0.5
9.98 2.21 0.417 0.0297 0.0032
1.0
2.0 5.0
flay : (T +h)exp(ikr'r)dr/l
0.6% 2.5% 6%
11% 19%
J = -DVc.
[A. 1]
D was obtained by generalizing the Einstein expression D = kBT(b) [A.2] for D, where b is the mobility tensor. In Einstein's original argument, la was a constant. In Journal of Colloid and Interface Science, Vol. 119,No. 2, October 1987
522
GEORGE
D . J. P H I L L I E S
Batchelor's generalization, b is given by a microscopic expression, which depends on the relative positions of the particles in the system. In contrast, D is a macroscopic quantity which does not depend explicitly on particle positions. To compute the macroscopic D from a microscopic b, Batchelor took an ensemble average of b over possible particle configurations, so that D = kBT(b).
[A.3I
Experimentally, light scattering spectroscopy is directly sensitive to particle positions, not to the particle current J. Specifically, the instantaneous scattered field is proportional to the instantaneous value of the/cth spatial Fourier component Ek(t) of the local index of refraction, where ek(t) is in turn proportional to the kth spatial Fourier component ak(t) of the scatterer concentration. Here k is the scattering vector selected by the source and detector positions. In light scattering spectroscopy, D is obtained from the temporal evolution of a~(t), namely
croscopic representation of a~(t) as a function of particle positions, N
ak(t) = ~ e ik''j,
[A.71
j=l
the sum being over all N particles in the system. D is then obtained from an ensemble average over the short-time limit of (a-k(O)$ak(t)). Equation [A.6] manifestly contains terms in V. [O], which a full calculation shows to be nonvanishing. While the calculation of D as kBT(b) is mathematically correct, QELSS does not determine D by measuring d and Vc directly. Rather, QELSS yields directly the quantity D defined by Eq. [A.4]. The prescription accompanying Eqs. [A.5]-[A.7] therefore gives a form for D which is preferable to the one obtained from Eq. [A.3]. Felderhof's calculation (6) of D is also based on the Smoluchowski equation, set in the form
Oa( rl , t) - - Do~71 • (~Tla(rl, t) Ot
lim (d/dt)(a_k(O)a~(t)) D ~ t---~o
_k2(a_k(O)ak(O))
[A.4]
References (7, 8) made assumptions equivalent to assuming that particle motions are correctly described by the Smoluchowski sedimentation equation
+ Col3f x71qSga(r2, t)dr2) + coVl- f bgdrz • Via(r1, t) + CoVl" ~ T. gV2a(r2, t)dr2 + 0(c2).
[A.8]
d
dc (x, t) dt
$c(x,
t),
[A.5]
where the Smoluchowski operator is S = V. (D. V + D. F/kBT).
[A.6]
Here O is the diffusion tensor (as distinct from the inferred diffusion coefficient D) and F is the applied force. Just as Eq. [A.2] was applied to the problem by interpreting b as a microscopic mobility tensor, Eqs. [A.5] and [A.6] were applied in (Refs. 7, 8) by giving D, F, and c(x, t) their microscopic interpretations. In particular, c(x, t) was replaced with the miJournal of Colloid and InterfaceScience, Vol. 119, No. 2, October 1987
where a(r~, t) is the local density at r~, q~is the intermicellar potential, g = g(rl - r2) is the equilibrium pair distribution function, and Vi is the gradient with respect to particle i, the particles being the particle of interest 1 and a neighboring particle 2. As interacting particles are relatively close together, the concentration gradients at their locations should be roughly equal, i.e., Vla(rl, t) ~ V2a(r2, t)
[A.9]
for a pair of interacting particles. (This is a small-k approximation.) Rearrangement of Eq. [A.8] gives
MICELLE DIFFUSION
da(rlt) dt
= DoVI" (V~a(rl, 0
+ CoBf dr2Vl(4~)go(r2 - rl)nl(r2, t)) + Cof (b + T) : V~a(rl, t)go(r2 - rl)dr2 [A. I0] as evaluated in Ref. (6), plus such terms as
c o f V l " [T]g0" V2a(r2,t)dr2
[A.1 1]
which vanish because f V , Tdr is odd in r. Reference (6) thus does not find nonzero terms in V . (T + b). References (6-8) or Eqs. [A.10] and [3], disagree because they evaluate different functions in order to determine D. In Eq. [3], D was obtained from the average
(a_~(O)Sak(t))
[A. 12]
while Eqs. [A.10] and [A.11] obtain D from an ensemble average over the algebraic kernel o f S, with no factors ofa~ included within the average. The microscopic S does give the concentration changes to be expected from a given microscopic particle configuration, so ( S ) does give an average rate of change for ak(t); also, ( S ) has no nonzero divergence terms. However, QELSS obtains the light-scatteringintensity weighted-average temporal evolution of ak(t), not the unweighted average. States which scatter no light make no contribution to the observed temporal evolution of a~(t). The z-weighting is due to the factors ak(0) and ak(t). For a z-weighted average, these factors must be included in the ensemble average. Including these factors in the average replaces, e.g., Eq. [A.I 1] by
co f d r 2 V , . ( e i k ' r , o v . ( t T ] ) ,
[A.13]
as seen in Eq. [3]. Unlike term [A, 1 1], the average l~V. T e x p ( i k , r) of [A. 13] is not odd in r and does not vanish. QELSS measures (a(O)da(t)/dt), not J, so the appropriate mi-
523
croscopic average for D is that of Eq. [3], not that ofEqs. [A.3] or [A. 10], though [A.3] and [A.10] do represent diffusion coefficients which some (hypothetical) experiment could measure. The failure of D (from Eq. [A.3]) and D (from Eq. [A.6]) to equal each other does not mean that the Onsager regression hypothesis has failed. The calculations of ( S ) and of(ak(O)Sak(t)) were m a d e using the same D in Eq. [A.6]. However, O is a function of particle positions; if D is averaged repeatedly over an ensemble, using different weighting functions each time, the averages over D can yield different numerical values. REFERENCES I. Carey, M. C., and Small, D. M., J. Clin. Invest. 61, 998 (1978). 2. Altenberger,A. R., and Deutch, J. M., J. Chem. Phys. 59, 894 (1973). 3. Phillies, G. D. J., J. Chem. Phys. 60, 976 (1974). 4. PhiUies,G. D. J., Benedek, G. B., and Mazer, N. A, J. (?hem. Phys. 65, 1883 (1976). 5. Batchelor,G. K., J. FluidMeeh. 74, 1 (1976). 6. Felderhof, B. U., J. Phys. A. 11,929 (1978). 7. Phillies, G. D. J., J. Chem. Phys. 77, 2623 (1982). 8. Carter, J. M., and Phillies, G. D. J., J. Phys. Chem. 89, 5118 (1985). 9. Corti, M., and Degiorgio, V. J., Phys. Chem. 85, 711 (1981). 10. Biresaw, G., McKenzie, D. C., Bunton, C. A., and Nicoli, D. F., J. Phys. Chem. 89, 5144 (1985). 11. Mos, H. J., Pathmamanoharan, C., Dhont, J. K. G., and de Kruif, C. G., J. Chem. Phys. 84, 45 (1986). 12. Mazur, P., and van Saarloos, W., Physica A 115, 21 (1982). 13. Kirkwood,J. G., Baldwin, R. L., Dunlop, P. J., Gosting, L. J., and Kegeles,G., J. Chem. Phys. 33, 1505 (1960). 14. Fujita, H., and Gosting, L. J., J. Amer. Chem. Soe. 78, 1099 (1956). 15. Dunlop, P. J., and Gosting, L. J., J. Phys. Chem. 63, 86 (1959). 16. Dunlop, P. J., J. Phys. Chem. 63, 612 (1959). 17. Felderhof, B. U., Physica A 85, 509 (1985). 18. Stober,W., Fink, A., and Bohn, E., Z ColloidInterface Sci. 81, 196 (1968). 19. Kops-Werkhoven, M. M., and Fijnaut, H. M., J. Chem. Phys. 74, 1618 (1981). 20. Phillies, G. D. J., J. Chem. Phys. 74, 260 (1981). 21. PhiUies,G. D. J., Phys. Rev. A 24, 1939 (1981).
Journal of Colloid and Interface Science, Vol. 119, No. 2, October 1987