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

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 a 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 [R. Jain et al., “Two step micro-rheological behavior in a viscoelastic fluid,” J. Chem. Phys. 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.