Micro-rheology of a particle in a nonlinear bath: Stochastic Prandtl-Tomlinson model

TL;DR

This paper introduces an extended Prandtl-Tomlinson model to study the nonlinear rheology of a particle in a viscoelastic bath, revealing velocity-dependent oscillations, rupture transitions, and resonance effects that align with experimental observations.

Contribution

It develops a novel extension of the Prandtl-Tomlinson model to capture complex nonlinear bath behaviors and predicts new phenomena like velocity-induced oscillations and rupture transitions.

Findings

Position oscillations depend on velocity and trap stiffness.

Oscillation amplitude and frequency increase with driving velocity.

A rupture transition occurs at a critical velocity, with frequency scaling as (v0 - v0*)^{1/2}.

Abstract

The motion of Brownian particles in nonlinear baths, such as, e.g., viscoelastic fluids, is of great interest. We theoretically study a simple model for such bath, where two particles are coupled via a sinusoidal potential. This model, which is an extension of the famous Prandtl Tomlinson model, has been found to reproduce some aspects of recent experiments, such as shear-thinning and position oscillations [J. Chem. Phys. {\bf 154}, 184904 (2021)]. Analyzing this model in detail, we show that the predicted behavior of position oscillations agrees qualitatively with experimentally observed trends; (i) oscillations appear only in a certain regime of velocity and trap stiffness of the confining potential, and (ii), the amplitude and frequency of oscillations increase with driving velocity, the latter in a linear fashion. Increasing the potential barrier height of the model yields a rupture transition as a function of driving velocity, where the system abruptly changes from a mildly driven state to a strongly driven state. The frequency of oscillations scales as $(v_0-v_0^*)^{1/2}$ near the rupture velocity $v_0^*$, found for infinite trap stiffness. Investigating the (micro-)viscosity for different parameter ranges, we note that position oscillations leave their signature by an additional (mild) plateau in the flow curves, suggesting that oscillations influence the micro-viscosity. For a time-modulated driving, the mean friction force of the driven particle shows a pronounced resonance behavior, i.e, it changes strongly as a function of driving frequency. The model has two known limits: For infinite trap stiffness, it can be mapped to diffusion in a tilted periodic potential. For infinite bath friction, the original Prandtl Tomlinson model is recovered. We find that the flow curve of the model (roughly) crosses over between these two limiting cases.