A constitutive equation for Rouse model modified for variations of spring stiffness, bead friction, and Brownian force intensity under flow

We derived a constitutive equation for the Rouse model (the most frequently utilized bead-spring model) with its spring constant κ, bead friction coefficient ζ, and the (squared) Brownian force intensity B being allowed to change under flow. Specifically, we modified the Langevin equation of the original Rouse model by introducing time (t)-dependent κ, ζ, and B (of arbitrary t dependence), which corresponded to the decoupling and preaveraging approximations often made in bead-spring models. From this modified Langevin equation, we calculated time evolution of second-moment averages of the Rouse eigenmode amplitudes and further converted this evolution into a constitutive equation. It turned out that the equation has a functional form, $σ ( t ) = ∫ − ∞ t d t ′ { κ ( t ) / κ ( t ′ ) } M ( t , t ′ ) C − 1 ( t , t ′ )$⁠, where $σ ( t )$ and $C − 1 ( t , t ′ )$ are the stress and Finger strain tensors, and $M ( t , t ′ )$ is the memory function depending on $κ ( t ′ ) , ζ ( t ′ )$⁠, and $B ( t ′ )$ defined under flow. This equation, serving as a basis for analysis of nonlinear rheological behavior of unentangled melts, reproduces previous theoretical results under specific conditions, the Lodge–Wu constitutive equation for the case of t-independent κ, ζ, and B [A. S. Lodge and Y. Wu, “Constitutive equations for polymer solutions derived from the bead/spring model of Rouse and Zimm,” Rheol. Acta 10, 539 (1971)], the finite extensible nonlinear elastic (FENE)-Peterlin mean-Rouse formulation for the case of t-dependent changes of the only κ reported by Wedgewood and co-workers [L. E. Wedgewood et al., “A finitely extensible bead-spring chain model for dilute polymer solutions,” J. Non-Newtonian Fluid Mech. 40, 119 (1991)], and analytical expression of steady state properties for arbitrary $κ ( t ) , ζ ( t )$⁠, and B(t) reported by ourselves [H. Watanabe et al., “Revisiting nonlinear flow behavior of Rouse chain: Roles of FENE, friction reduction, and Brownian force intensity variation,” Macromolecules 54, 3700 (2021)]. It is to be added that a constitutive equation reported by Narimissa and Wagner [E. Narimissa and M. H. Wagner, “Modeling nonlinear rheology of unentangled polymer melts based on a single integral constitutive equation,” J. Rheol. 64, 129 (2020)] has a significantly different functional form and cannot be derived from the Rouse model with any simple modification of the Rouse–Langevin equation.