PERSPECTIVE

Comments on “The General Theory of Irreversible Processes in Solutions of Macromolecules,” by John G. Kirkwood, 1. Polym. Sci, XII, 1 (1954)

JACK DOUGLAS

Polymers Division, National Institutes of Standards and Technology, Gaithersburg, MD 20899

The problem of describing the transport prop- erties of polymer chains which are dynamically evolving through a wide range of configurations through Brownian motion of the chain segments and the chain as a whole is cast in terms of a diffusion equation acting within molecular configuration space. Kirkwood’s elegant formulation of polymer chain dynamics, utilizing methods of Riemann ge- ometry, generalized Kramers’ previous formulation of polymer chain dynamics in the absence of intra- chain hydrodynamic interactions.’ The hydrody- namic interaction modeling of polymers, based on the idealized Oseen tensor, also had a precedent in the work of Burgers’ so that Kirkwood’s contribu- tion represents a natural logical synthesis of earlier work. Kirkwood’s review article spans a wide range of applications-viscoelastic response of polymer solutions to applied stresses, flow birefringence and the Kerr effect, and dielectric polarization and dis- persion.

Although the general conceptual framework de- veloped by Kirkwood and its earlier precedents pro- vide an appealing framework for treating a wide range of hydrodynamic problems involving suspen- sions of flexible particles, this theory is difficult to apply in its complete generality to concrete polymer models3 and a variety of approximations were nec- essarily introduced by Kirkwood for reasons of mathematical expediency. Kirkwood’s calculations, for example, involved a number of “preaveraging” approximations which are still active subjects of in- vestigation and frustration because adequate ana- lytic methods to avoid certain of these approxima-

Journal of Polymer Science: Part B: Polymer Physics, Vol. 34, 595-596 (1996) 0 1996 John Wiley & Sons, Inc. CCC 0887-6266/96/040595-02

tions remain elusive. The preaveraging approxi- mations included an “angular averaging” approximation of the Oseen tensor to reduce the tensorially defined hydrodynamic interaction to a scalar interaction.‘Recent calculations4 suggest that this approximation is actually rather minor in com- parison with the “configurational preaveraging” approximati~n~?~ which can be on the order of a 10% error for common transport properties such as the intrinsic viscosity and the diffusion coefficient. It is also now appreciated that Kirkwood‘s often cited expression for the diffusion coefficient of a flexible polymer [eq. (9) of Kirkwood’s paper] is incorrect due to a faulty transformation between the config- uration space and coordinate space representations of the hydrodynamically interacting polymer.’ Ak- casu showed’ that Kirkwood‘s expression for the dif- fusion coefficient is actually correct for the “short time” diffusion coefficient (assuming configurational and angular preaveraging approximations) and found that the error involved in Kirkwood’s “ap- proximation” is less than 2% for flexible chains. Wang et a1.6 later showed that Kirkwood’s approx- imate expression for the diffusion coefficient results from an additional “contour preaveraging” approx- imation within the configurationally preaveraged Kirkwood-Riseman (rigid body) theory. The various preaveraging approximations introduced by Kirk- wood remain an active topic of investigation and the configurational preaveraging approximation remains a quantitative limitation on the accuracy of analytic calculations.

The subtle technical difficulties in Kirkwood’s calculations reflect the complexity of his general Riemann geometry formulation of polymer dynam- ics and it was no doubt a great relief to many polymer

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science theorists when Zimm recognizedg that Kirk- wood’s general theory could be recast into a more tractable form using chain normal coordinates sim- ilar to those introduced shortly before by Rouse” for an idealized “bead-spring’’ model of polymer chains without intramolecular hydodynamics inter- actions. The Rouse theory in turn had its antecedent in terms of Kuhn’s coarse grained modeling of poly- mer chains in terms of statistical segments which naturally led to a dynamical analog involving chains of harmonic oscillators to represent the dynamical polymer chain segments. Zimm’s specialized form of the general Kirkwood theory in terms of a normal coordinate representation has become a cornerstone of polymer solution theory and the influence of Kirkwood‘s Journal of Polymer Science article is ev- ident from Zimm’s prominent citation of this work. Bixon3 and Zwanzig” later established the foun- dations of Zimm’s important conceptual advance of the Kirkwood theory and these developments have recently led to more molecularly faithful modeling of polymers through the incorporation of statistical information related to the detailed chain structure through the Rouse-Zimm force constant matrix. This type of “optimized Rouse-Zimm” model has recently been refined in applications to protein dy- namics involving inhomogeneous polymer structure, complicated bond rotation potentials, excluded vol- ume interactions, and other physically important features such as mode-coupling effects.12 The Kirk- wood formalism, as modified by Zimm, continues to evolve into an increasingly realistic model of polymer dynamics which should ultimately have applications to protein folding, the dynamics of polymer adsorp- tion, polymer collapse in poor solvents, and other important dynamical processes of polymers in so- lution. It should also be mentioned that the Kirk- wood theory also developed into a tractable theory of rigid particle hydrodynamic^'^ and justified ear- lier, more heuristic, treatments of rigid body hydro- dynamics by Kirkwood and Riseman.14 Yamakawa and Yamaki13 have discussed the conceptual foun- dations of this rigi body approach to polymer hy-

tions. Interestingly, the configurational preaveraging errors seem to be different in the Rouse-Zimm the- ory of ideally flexible chains and the Kirkwood- Riseman theory of perfectly rigid chains, suggesting that the degree of configurational preaveraging error depends on the degree of dynamic rigidity of the polymer chain [see ref. 6b].

Although the full Riemann geometry formulation of polymer hydodynamics developed by Kirkwood

drodynamics, whic i has also led to many applica-

is rather uncommonly utilized in the polymer lit- erature at present, this formalism has recently been revived by Brenner and co-~orkers’~ in an effort to avoid angular and configurational preaveraging ap- proximations and also the point source (“bead”) model of intrachain hydrodynamic interactions. These calculations, which are restricted by their technical difficulty to simple models such as a pair of tethered spheres, indicate that preaveraging cal- culations miss important Taylor-dispersion-type phenomena associated with the dynamical fluctua- tions of flexible bodies. This work shows that the original Kirkwood formalism continues to have im- portance in developing a greater understanding of the dynamics of polymer solutions.

REFERENCES

1. a) H. A. Kramers, J. Chem. Phys., 14,415 (1946). b) J. G. Kirkwood, Rec. Trau. Chim., 68,649 (1949).

2. J. M. Burgers in Second Report on Viscosity and Plas- ticity, North Holland, Amsterdam, 1938, p. 113-184.

3. M. Bixon, J. Chem. Phys., 58, 1459 (1973). 4. J. F. Douglas, H.-X. Zhou, and J. B. Hubbard, Phys.

Rev. E, 49, 5319 (1994). 5. a) B. H. Zimm, Macromolecules, 13,592 (1980). b) J. G. de la Torre, A. Jiminez, and J. Friere, Macromol- ecules, 15, 148 (1982).

6. a) S.-Q. Wang, J. F. Douglas, and K. F. Freed, J. Chem. Phys., 85, 3674 (1986), 87, 1346 (1987). b) J. F. Douglas and K. F. Freed, Macromolecules, 27, 6088 (1994).

7. a) Y. Ikeda, Kobayashi Rigaku Kenkyusho Hokoku, 6, 44 (1956). b) R. Zwanzig, J. Chem. Phys., 45, 1858 ( 1966).

8. A. Z. Akcasu, Macromolecules, 15, 1321 (1982). 9. B. H. Zimm, J. Chem. Phys., 24,269 (1956). 10. P. E. Rouse, Jr., J. Chem. Phys., 21, 1272 (1953). 11. a) R. Zwanzig, J. Chem. Phys., 60,2717 (1974). b) A.

Perico and M. Guenza, J. Chem. Phys., 83, 3103 (1985); 84,510 (1986).

12. a) X. Y. Chang and K. F. Freed, J. Chem. Phys., 99, 8016 (1993). b) Y. Hu, J. M. Macinnis, B. J. Cherayil, G. R. Fleming, K. F. Freed, A. Perico, J. Chem. Phys., 93,822 (1990).

13. H. Yamakawa and J. Yamaki, J. Chem. Phys., 58, 2049 (1973).

14. a) J. G. Kirkwood and J. Riseman, J. Chem. Phys., 16, 565 (1948). b) J. G. Kirkwood, Macromolecules, P. L. Auer (ed.), (Gordon and Breach, New York, 1967.

15. a) A. Nadim and H. Brenner, Phys. Chem. Hyd., 11, 315 (1989). b) H. Brenner, A. Nadim, S. Haber, J. Fluid. Mech., 183,511 (1987). c) S. Haber, H. Bren- ner, M. Shapiro, J. Chem. Phys., 92,5569 (1990).