Stress Relaxation of Solutions of Flexible Macromolecules

Abstract

For stress relaxation after cessation of steady shearing flow with $v_x{\mathrm{=\kappa }}_{0}y$ , the following relation has been proposed for connecting the normal stress difference ${\tau }_{\mathrm{xx}}^{-}{\mathrm{-\; \tau }}_{\mathrm{yy}}^{-}$ during steady state flow and the shear stress ${\tau }_{\mathrm{yx}}^{+}$ during the relaxation process after t=0: $${\tau }_{\mathrm{xx}}^{-}{\mathrm{-\; \tau }}_{\mathrm{yy}}^{-}{\text{=2κ}}_0{\displaystyle {\int }_0^{\infty }}{\tau }_{\mathrm{yx}}^{+}\mathrm{dt.}$$ It is shown here that this relation can be derived for a dilute suspension of flexible macromolecules represented as a set of N beads joined by N — 1 connectors which may be nonlinear springs; equilibrium‐averaged hydrodynamic interaction is included in the theory. A generalization of the above formula may be derived for the stress relaxation following any steady homogeneous flow. In the derivation use is made of an expression for the stress tensor which differs from that of Giesekus in that hydrodynamic interaction has been included. From the latter formula it is particularly easy to rederive the Lodge‐Wu constitutive equation for the Zimm model with Gaussian springs.