# Effect of Inertia on Linear Viscoelasticity of Harmonic Dumbbell Model

## TL;DR

This paper investigates how weak inertia influences the linear viscoelastic behavior of the harmonic dumbbell model, revealing kinetic coupling between momentum and bond relaxation modes and deriving an overdamped Langevin equation that accounts for inertia effects.

## Contribution

It introduces a theoretical analysis showing inertia's impact on bond relaxation and derives an overdamped Langevin equation incorporating weak inertia effects.

## Key findings

- Inertia can affect bond relaxation when momentum relaxation is slow.
- Derived an overdamped Langevin equation with inertia effects.
- Predicted bond relaxation dynamics accurately with weak inertia.

## Abstract

The overdamped (inertialess) dumbbell model is widely utilized to study rheological properties of polymers or other soft matters. In most cases, the effect of inertia is merely neglected because the momentum relaxation is much faster than the bond relaxation. We theoretically analyze the effect of inertia on the linear viscoelasticity of the harmonic dumbbell model. We show that the momentum and bond relaxation modes are kinetically coupled and the inertia can affect the bond relaxation if the momentum relaxation is not sufficiently fast. We derive an overdamped Langevin equation for the dumbbell model, which incorporates the weak inertia effect. Our model predicts the bond relaxation dynamics with the weak inertia effect correctly. We discuss how the weak inertia affects the linear viscoelasticity of a simple harmonic dumbbell model and the Rouse model.