A time-development operator formulation of the kinetic theory of dilute polymeric solutions

Abstract

For dilute solutions of polymeric molecules, the distribution function in the configuration space of a single molecule is determined by the so-called ‘‘diffusion equation.’’ In the present discussion, we transform the problem of solving this diffusion equation to that of solving a different linear differential equation involving a Hermitian operator ℋ associated with the unperturbed system and an operator 𝒲=κ:ℳ̂, which describes the effects of the velocity gradient κ°. It is shown that the eigenvalues of ℋ are the reciprocals of the time constants associated with the decay of perturbations of the system from equilibrium. It is then shown that the stress tensor may be written simply in terms of matrix elements of an operator ℳ and a time-development operator 𝒰 with respect to the eigenfunctions of ℋ as the bases. This development then leads to an expression for the relaxation modulus of linear viscoelasticity, which is of the form of the ‘‘generalized Maxwell model.’’ A formal expression for the relaxation modulus involves an exponential operator which may easily be expanded leading to a series in powers of the time variable. This then leads to quite general expressions for the high frequency limiting values of the real and imaginary parts of the complex viscosity. These expressions are evaluated for the special case of the finitely extensible nonlinear elastic (FENE) dumbbell and the results shown to be consistent with earlier asymptotic results.