Hybrid multiscale method for polymer melts: analysis and simulations

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.

💡 Research Summary

This paper presents a comprehensive hybrid multiscale framework for modeling the flow behavior of dense melts composed of flexible and semiflexible ring polymers confined by solid walls. The approach bridges atomistic molecular dynamics (MD) simulations with a continuum Cahn‑Hilliard‑Navier‑Stokes (CHNS) system, thereby capturing both microscopic rheology and macroscopic phase‑separation phenomena.

In the microscopic component, the authors employ coarse‑grained bead‑spring chains based on the Kremer‑Grest model. Non‑bonded interactions are described by the Weeks‑Chandler‑Andersen (WCA) potential, while bonded pairs follow a finitely extensible nonlinear elastic (FENE) potential. Semiflexibility is introduced via an angular bending term Vθ = κ(1 + cos θ), with κ = 0 for fully flexible rings and κ = 10 for semiflexible rings. Simulations are performed at a number density ρ = 0.8 and polymerization degree N = 15 in a cubic box of size 15³ using LAMMPS. Shear flow is imposed along the x‑direction through SLLOD equations combined with Lees‑Edwards periodic deformation, and temperature is maintained at T = 1 by a Nosé‑Hoover thermostat.

The Irving‑Kirkwood expression is used to compute the off‑diagonal stress component σ_xy, from which a shear‑rate‑dependent viscosity η(γ̇, χ₀) = σ_xy/γ̇ is extracted. Zero‑shear viscosity η_GK is independently obtained via the Green‑Kubo integral. The MD results reveal pronounced shear‑thinning for both polymer types, with the zero‑shear viscosity decreasing sharply as the fraction χ₀ of flexible rings increases. These data provide a quantitative rheological map that depends on both shear rate and composition.

The macroscopic model consists of a coupled CHNS system. The phase‑field variable φ denotes the volume fraction of the stiffer component, while u and p represent the mixture velocity and pressure. The Cahn‑Hilliard equation reads ∂_t φ + ∇·(φu) = ∇·