4084 J. Phys. Chem. 1992, 96, 4084-4085

Light Scattering by Solutions of Associating Polymers at Equlllbriumt

W. H. Stockmayer

Department of Chemistry, Dartmouth College, Hanover, New Hampshire 03755 (Received: December 2, 1991)

The intensity of zereangle static Rayleigh light scattering by a polymer solution containing asscciated species, all in equilibrium with each other, can be calculated by introducing the thermodynamic condition for equilibrium either at the start or at the end of the calculation. Provided all species have the same value of the specific refractive index increment, it is shown for Flory-Huggins systems that the same formula is obtained by both routes.

During the past several years, Burchard and his collaboratorslJ have measured both integrated and dynamic light scattering from solutions of self-associating polymers, including galactosidase, cellulose-2.5-acetate, and end-tagged polystyrene lithium sulfonate. A question that has arisen in this connection is the following: if the various associated species are all in chemical equilibrium with each other, so that the system contains only two components (though many species), is it legitimate to use the standard for-

for scattering from a many-component system and then introduce the condition of chemical equilibrium only at the end, or must one introduce the equilibrium condition at the very start and thus perhaps obtain a different scattering formula? Since the mean-square concentration fluctuations that enter the relation for the scattered intensity are quadratic quantities, the answer did not seem obvious to the writer. On the one hand, one might suppose that the standard formula, derived by treating each solute species as an independently fluctuating component, might over- shoot the correct result if in fact the species were interdependent by virtue of the association equilibrium; on the other hand, this very interdependence means that the various species drag each other along, thus perhaps producing greater net fluctuations of refractivity. Since the writer received no definite assurance from several randomly chosen passersby (but not including M. Fixman) to whom he posed the question, a discussion is offered here. It turns out that if the specific refractive index increment (dnldc) is the same for all the species, the condition of equilibrium can be introduced at any stage (at least for Flory-Huggins systems) without altering the final result, and so the question may after all have been trivial? However, if the specific refractive increment (or neutron scattering length) differs from species to species, the two approaches do not yield identical predictions. This allegation is most easily checked by performing the algebra for the simplest case of two polymeric species, say monomer and dimer, with eqs 1, 7, and 8.

Specific Example: Quasi-Bury Flory-Huggins System. A 'quasi-binary" system obeying standard Flory-Huggins thermo- dynamics is defined as one in which all polymer species are de- scribed by the same interaction parameter x . Using the subscript 0 to denote the solvent and k for polymer species k, we have the following expression for the free enthalpy of mixing per mole of lattice sites:

A G / N R T = $0 In 40 + cril'$k In '#'k + XdJod (1) k> 1

where the ratios of molar volumes are rk = vk/ vo and the volume fractions are

dJk = rknk/N dJ = C4k = 1 - 40 k l I

with N = n,, + xkrknk, the n's being numbers of moles. The corresponding chemical potentials are given by7

(CCO - w8)/RT = In ( 1 - 4) + dJ(1- ( r ) ; ' ) + x4' ( 2 )

(pk - Pa)/RT = In 4 k + 1 - rk(dO + (r)i14) + rkx& (3) where the number-average chain length is ( r ) , = '$/xkrlldJk. If

' Dedicated to Marshall Fixman for his sixtieth birthday.

the polymer is a t an association equilibrium, with all the species being aggregates of species 1, we have rk = krl and & = k p l , so that from eq 3 we find the equilibrium relation

$k = Kk& (4)

with

In Kk = k - 1 + (kpp - & / R T

When this condition is applied, we now have a binary system, for which the zero-angle Rayleigh ratio Ro (with neglect of the contribution due to density fluctuations) is inversely proportional to the so-called osmotic modulus (concentration derivative of the osmotic pressure). The result of straightforward calculation with eqs 2 and 4 is

1/Ro - (1/4RT)(-aClo/a4) = [4(&1-l + 42 - 2% ( 5 )

where the average sizes are

We now take the alternative route and start from the multi- component scattering formula in the form handiest to eq 1:

where

Gkl [@(AG/NRT) / & k a@/] T,p,otherg, (7b)

Here Ak/ is the cofactor of Gkl in the above "spinodal determinant"* and [ k is the speclfic refractive increment of the kth species. From eq 1 , expressing the free enthalpy as a function of the solute species concentrations, we get

Gkl = ( 6 k / / r k 6 k ) + - 2% (8)

For the case of interest, with &k having the same value for all the solute species, we then arrive a t exactly eq 5 , but with ( r ) , yet to be specified. This is the well-known9 standard result for a

Burchard, W. Makromol. Chem., Macromol. Symp. 1990, 39, 179. Burchard, W.; Schulz, L.; Auersch, A.; Littke, W. Polym. Prepr. (Am.

. Soc., Diu. Polym. Chem.) 1990, 31(2), 402. Brinkman, H. C.; Hcrmans, J . J. J . Chem. Phys. 1949, 17, 574. Kirkwood, J. G.; Goldberg, R. J. J . Chem. Phys. 1950, 18, 54. Stockmayer, W. H. J . Chem. Phys. 1950, 18, 5 8 .

(6) Definition of this term is left to the reader. (7) See, for example: Flory, P. J. Principles of Polymer Chemistry;

Cornell University Press: Ithaca, NY, 1953; Chapter XII. (8) Gordon, M.; Chermin, H. A. G.; Koningsveld, R. Macromolecules

1969, 2, 207. (9) See, for example: Benoit, H.; Benmouna, M.; Wu, W. Macromolecules

1990,23,1511.

0022-3654/92/2096-4084%03.00/0 0 1992 American Chemical Society

J. Phys. Chem. 1992, 96, 4085-4093 4085

multicomponent quasi-binary Flory-Huggins solution. We can now finally introduce the equilibrium value of (r) , , , and thus amve at complete agreement with eqs 5 and 6, obtained by a different sequence of operations.

General Case. The grand partition function of a system of the type being considered is

z(T,v,k) = CQ(T,V,N)exp[B(w% i- CPkNk)] (9)

where Nk is the number of molecules of the kth species, Q is the petit canonical partition function, and B = l /kB T. Introduction of the equilibrium condition and the mass balance

IM k > l

P k = kPi E,kNk = N (10)

reduces eq 9 to

showing that the fluctuations of the solute component concen- tration (Le., of N) can be found directly from the chemical po- tential p’ of that component or, equally well, via the Gibbs-Duhem relation, from the potential of the solvent. When the specific refractive index increment is the same for all the solute species, these fluctuations directly control the Rayleigh scattering.

Unfortunately we have not managed as yet to follow the al- ternative route (deferment of the equilibrium specification until the end) to a final demonstration in the general case, and must leave this for a future exercise.

Solvation Free Energies and Solvent Force Constantst

Teresa Fonseca,* Branka M. Ladanyi,* Department of Chemistry, Colorado State University, Fort Collins, Colorado 80523

and James T. Hynes* Department of Chemistry and Biochemistry, University of Colorado, Boulder, Colorado 80309 (Received: January 3, 1992)

A theoretical formulation for the solvent force constant k,, which gauges electrical potential fluctuations for an ion in solution and whose charge dependence is a measure of nonlinear aspects of solvation, is presented in terms of the solute charge (9 ) variation of the solvation free energy. This formulation allows the calculation of k, via integral equation theories. This is illustrated by a series of calculations for ionic solutes in model dipolar-quadrupolar solvents via the reference hypernetted chain (RHNC) integral quation approach. It is found that the q variation of k, can be comprehended in terms of the cooperative (or competing) contributions of the solvent dipole and quadrupole to the acceleration of the solvation free energy. By contrast, traditional notions of dielectric saturation prove to be of much less direct relevance, due in part to the importance of competing electrostriction effects. The formalism is also applied to available simulation and integral equation solvation free energy studies of aqueous ionic solvation to infer the behavior of k,. The extensions for the formalism to more complex solutes (e.g., ion pairs), to higher order fluctuations (e.g., electric field), and to the solvent frequency and effective mass are briefly indicated.

I. Introduction The idea that an ionic or polar solute can induce nonlinear

electrostatic effects in a surrounding polar solvent is an old one. The concept is most familiar in the context of the dielectric continuum description of the solvent, in which one speaks of “dielectric saturation” and a dielectric constant dependent on the solutesolvent distance.’ Such a continuum characterization has a number of difficulties, h o w e ~ e r , ~ ~ ~ and a more microscopic level description is desirable.

A convenient molecular measure of nonlinear electrostatic effects is the solvent force constant ke4 As shown by Carter and Hynes,s k is given by

in terms of the Boltzmann constant kB, the temperature T, and the mean-squared fluctuations in AE, the electrostatic contribution to the solutesolvent potential energy. In the simplest case of a spherical solute ion, AE represents the sum of Coulombic inter- actions between the solute charge q and partial charges qa on sites a within the solvent molecules:

This paper is dedicated to Professor Marshall Fixman on the occasion of his 60th birthday.

*Deceased December 16, 1991.

where r,, is the distance between the ion and a t h site of the ith solvent molecule. For polyatomic solute ions or molecules, eq 1.2 would contain a sum over solute interaction sites as well. In such cases, AE serves as a microscopic solvent coordinate for electron transfer6 and other charge-transfer reactions.’

According to eq 1.1, the smaller the solvent electrical fluctu- ations, the larger the solvent force constant. In a simplified harmonic description, the Helmholtz free energy along this solvent coordinate is given by5

(1) See, e.g.: Laidler, K. J. Chemical Kinetics, 3rd ed.; Harper and Row:

(2) Morita, T.; Ladanyi, B. M.; Hynes, J. T. J . Phys. Chem. 1989, 93, New York, 1987; Chapter 6.

1386. (3) Jarayam, B.; Fine, R.; Sharp, K.; Honig, B. J . Phys. Chem. 1989.93,

4320. (4) Kakitani, T.; Mataga, N. Chem. Phys. 1985,93,381; J. Phys. Chem.

1985,89,4152; 1985,89,8; 1986, 90, 993; 1987,91,6211; Chem. Phys. Lett. 1986, 124, 437. Hatano, Y.; Saito, M.; Kakitani, T.; Mataga, N. J . Phys. Chem. 1988, 92, 1008.

(5) Carter, E. A,; Hynes, J. T. J . Phys. Chem. 1989, 93, 2184. (6) (a) Warshel, A. J . Phys. Chem. 1982,86,2218. Warshel, A.; Hwang,

J. K. J. Chem. Phys. 1986, 84, 4938. Hwang, J. K.; Warshel, A. J. Am. Chem. Soc. 1987,109,715. Halley, J . W.; Hautman, J. Phys. Reu. B 1988, 38, 11704. Kuharski, R. A.; Bader, J. S.; Chandler, D.; Sprik, M.; Klein, M.; Impey, R. W. J. Chem. Phys. 1988,3248. Benjamin, I. J . Phys. Chem. 1991, 95, 6675. (b) Zichi, D. A.; Ciccotti, G.; Hynes, J. T.; Ferrario, M. J. Phys. Chem. 1989, 93, 6261.

(7) (a) Kim, H. J.; Hynes, J. T. J . Am. Chem. Soc., submitted. Kierstead, W.; Wilson, K. R.; Hyncs, J. T. J . Chem. Phys. 1991, 95, 5256. (b) Borgis, D.; Hynes, J. T. Ibid. 1991, 94, 3619.

0022-365419212096-4085%03.00/0 0 1992 American Chemical Society