Differential constitutive equation for entangled polymers with partial strand extension

Beginning with a formal statement of the conservation of probability, we derive a new differential constitutive equation for entangled polymers under flow. The constitutive equation is termed the Partial Strand Extension (PSE) equation because it accounts for partial extension of polymer strands in flow. Partial extensibility is included in the equation by considering the effect of a step strain with amplitude E on the primitive chain contour length. Specifically, by a simple scaling argument we show that the mean primitive chain contour length after retraction is L=L0E1/2, not the equilibrium length L-0 as previously thought. The equilibrium contour length is infact recovered only after a characteristic stretch relaxation time lambda(s) that is bounded by the reptation time and longest Rouse relaxation time for the primitive chain. The PSE model predictions of polymer rheology in various sheer and extensional flows are found to be in good to excellent agreement with experimental results from several groups.