Variational bounds for polymer transport coefficients

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Abstract
Upper and lower variational bounds are derived for certain polymer transport coefficients, such as the sedimentation constant and intrinsic viscosity. These bounds require the equilibrium averages of complicated functions of configuration, and are generally calculable only from equilibrium computer simulations. The upper bounds contain the steady state bead velocities as trial parameters. When these velocities are taken to arise from rigid body motion, Zimm’s algorithms [Macromolecules 13, 592 (1980)] for the sedimentation coefficient and intrinsic viscosity [η] are recovered. Zimm’s formula for [η] has previously been shown to be an upper bound by Wilemski and Tanaka [Macromolecules (in press, 1981)]. The lower bounds are of two forms. The first is obtained from perturbation theory, and contains any positive-definite and symmetric bead diffusion matrix, e.g., a preaveraged diffusion matrix, as the variational parameters. The second form is derived similarly to the lower bound on the sedimentation constant obtained by Rotne and Prager [J. Chem. Phys. 50, 4831 (1969)], for which the variational parameters specify the perturbed distribution function. Their approach is generalized to include the intrinsic viscosity. Numerical illustrations for Gaussian chains with fluctuating hydrodynamic interaction are reported in the following paper.