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§8.6 Molecular Theories for Polymeric Liquids 253

1 2 3
-.6 MOLECULAR THEORIES FOR POLYMERIC LIQUIDS ' '
It should be evident from the previous section that proposing and testing empirical
expressions for the stress tensor in nonlinear viscoelasticity is a formidable task. Recall
that, in turbulence, seeking empirical expressions for the Reynolds stress tensor is
equally daunting. However, in nonlinear viscoelasticity we have the advantage that we
can narrow the search for stress tensor expressions considerably by using molecular
theory. Although the kinetic theory of polymers is considerably more complicated than
the kinetic theory of gases, it nonetheless guides us in suggesting possible forms for the
stress tensor. However, the constants appearing in the molecular expressions must still
be determined from rheometric measurements.
The kinetic theories for polymers can be divided roughly into two classes: network
theories and single-molecule theories:
3
a. The network theories were originally developed for describing the mechanical
properties of rubber. One imagines that the polymer molecules in the rubber are joined
chemically during vulcanization. The theories have been extended to describe molten
polymers and concentrated solutions by postulating an ever-changing network in which
the junction points are temporary, formed by adjacent strands that move together for a
while and then gradually pull apart (see Fig. 8.6-1). It is necessary in the theory to make
some empirical statements about the rates of formation and rupturing of the junctions.
b. The single-molecule theories 1 were originally designed for describing the poly-
mer molecules in a very dilute solution, where polymer-polymer interactions are infre-
quent. The molecule is usually represented by means of some kind of "bead spring"
model, a series of small spheres connected by linear or nonlinear springs in such a way
as to represent the molecular architecture; the bead spring model is then allowed to
move about in the solvent, with the beads experiencing a Stokes' law drag force by the
solvent as well as being buffeted about by Brownian motion (see Fig. 8.6-2a). Then from
the kinetic theory one obtains the "distribution function" for the orientations of the mol-
ecules (modeled as bead spring structures); once this function is known, various macro-
scopic properties can be calculated. The same kind of theory may be applied to
concentrated solutions and molten polymers by examining the motion of a single bead
spring model in the "mean force field" exerted by the surrounding molecules. That is,

Fig. 8.6-1. Portion of a polymer network
formed by "temporary junctions," indi-
cated here by circles.

1 R. B. Bird, C. F. Curtiss, R. C. Armstrong, and O. Hassager, Dynamics of Polymeric Liquids, Vol. 2,
Kinetic Theory, Wiley-Interscience, New York, 2nd edition (1987).
2
M. Doi and S. F. Edwards, The Theory of Polymer Dynamics, Clarendon Press, Oxford (1986);
7
J. D. Schieber, "Polymer Dynamics/ in Encyclopedia of Applied Physics, Vol. 14, VCH Publishers, Inc.
(1996), pp. 415-443. R. B. Bird and H. С Ottinger, Ann. Rev. Phys. Chem., 43, 371^106 (1992).
3
A. S. Lodge, Elastic Liquids, Academic Press, New York (1964); Body Tensor Fields in Continuum
Mechanics, Academic Press, New York (1974); Understanding Elastomer Molecular Network Theory,
Bannatek Press, Madison, Wis. (1999).