Hybrid multiscale method for polymer melts: analysis and simulations

TL;DR

This work develops a hybrid multiscale framework to model flow of dense blends of flexible and semiflexible ring polymers near confining walls by coupling molecular dynamics–derived rheology to a macroscopic $CHNS$ system. The macroscopic model uses the $\ ext{Irving-Kirkwood}$ stress to inform a viscosity $\eta(\dot{\gamma},\phi)$ and a mobility $M(\phi)$, with dynamic boundary conditions to capture wall effects and phase behavior. The authors prove mass conservation and energy stability for a semi-implicit finite element scheme and establish solvability via Schaefer's fixed point theorem, then demonstrate 2D channel-flow simulations that reproduce MD-observed flow-induced segregation. The approach provides a rigorous, data-driven route to predict and control segregation phenomena in microfluidic devices and dense polymer melts, contributing to multiscale modeling of complex rheology in soft matter.

Abstract

We model the flow behaviour of dense melts of flexible and semiflexible ring polymers in the presence of walls using a hybrid multiscale approach. Specifically, we perform molecular dynamics simulations and apply the Irving-Kirkwood formula to determine an averaged stress tensor for a macroscopic model. For the latter, we choose a Cahn-Hilliard-Navier-Stokes system with dynamic and no-slip boundary conditions. We present numerical simulations of the macroscopic flow that are based on a finite element method. In particular, we present detailed proofs of the solvability and the energy stability of our numerical scheme. Phase segregation under flow between flexible and semiflexible rings, as observed in the microscopic simulations, can be replicated in the macroscopic model by introducing effective attractive forces.

Figures (2)

• Figure 1: (a) Shear viscosity $\eta$ as function of shear rate $\dot{\gamma}$ for dense ($\rho = 0.8$) binary blends of flexible ($\kappa=0$) and stiffer ($\kappa=10$) ring polymers, corresponding to different proportions ($\chi_0$) of flexible rings . Each ring consists of $N = 15$ monomers. Corresponding zero-shear viscosities, $\eta_\text{GK}$ (calculated by the Green-Kubo relation) are shown on the $y$-axis. Note that the value of $\eta_\text{GK}$ corresponding to $\kappa=10$ as exhibited on the $y$-axis has been obtained from a least square fitting and not from MD simulations, cf. Appendix \ref{['sec_fitting']}. (b) Zero-shear viscosity from the Green-Kubo relation $\eta_\text{GK}$ as a function of $\chi_0$ in a binary mixture of ring polymers consisting of flexible and stiff ($\kappa=10$) rings. A box size of $15\times15\times15$ was considered for all the simulations. Figure \ref{['fig1']}b has been adapted with permission from R. Datta, F. Berressem, F. Schmid, A. Nikoubashman, and P. Virnau. Viscosity of Flexible and Semiflexible Ring Melts: Molecular Origins and Flow-Induced Segregation. Macromolecules, 56(18):7247–7255, 2023. Copyright 2023 American Chemical Society Datta_2023. All lines are guides for the eye.
• Figure 2: (a) An equilibrated binary mixture of flexible ($\kappa=0$, yellow) and stiffer ($\kappa=10$, red) rings at $\chi_0=0.5$, confined within a channel bounded by particle-based walls (blue). (b) The same mixture under flow, induced by applying a constant force $f_x=0.095$ along the $x$-axis to all particles. (c) Density profiles of the respective components across the channel cross-section. (The inset depicts the velocity profile in the flow direction, confirming the implementation of no-slip boundary conditions.) Adapted with permission from R. Datta, F. Berressem, F. Schmid, A. Nikoubashman, and P. Virnau. Viscosity of Flexible and Semiflexible Ring Melts: Molecular Origins and Flow-Induced Segregation. Macromolecules, 56(18):7247–7255, 2023. Copyright 2023 American Chemical Society

Theorems & Definitions (22)

• Remark 3.1
• Theorem 3.2
• Remark 3.4
• Lemma 3.5: energy inequality
• proof
• Corollary 3.6
• proof
• Corollary 3.7: a priori estimates
• proof
• Remark 3.8