Shear-induced diffusivity in supercooled liquids

TL;DR

This work addresses why supercooled liquids exhibit a linear-in-shear enhancement of molecular diffusivity rather than the quadratic Taylor–Aris behavior observed in normal liquids. By solving the Smoluchowski diffusion-convection equation with a glassy energy barrier and applying Kramers’ escape theory, the authors derive a shear-corrected diffusivity that reduces to $D_{\mathrm{eff}} = D \left(1 + \dfrac{\mu_0 a \Delta r^2}{k_B T}\dot{\gamma}\right)$ for small shear rates. Molecular dynamics simulations of supercooled mW water and LJ liquids validate the linear relationship and reveal the predicted dependencies on temperature and zero-shear viscosity, supporting a potential universal mechanism tied to glassy dynamics. The findings offer a quantitative link between diffusivity, viscosity, and temperature under shear, with broad implications for mass transport in nonequilibrium glassy systems and applications to shear-melting phenomena. The theory remains agnostic to the specific barrier details, underscoring a robust, universal aspect of shear diffusion in crowded liquids.

Abstract

The Taylor-Aris theory of shear diffusion predicts that the effective diffusivity of a tracer molecule in a sheared liquid is enhanced by a term quadratic in the shear rate. In sheared supercooled liquids, instead, the observed enhancement is linear in the shear rate. This is a fundamental observation for the physics of nonequilibrium liquids. Here, we derive a formula for the effective molecular diffusivity in supercooled liquids under shear flow based on the underlying Smoluchowski equation with shear (Smoluchowski diffusion-convection equation) with an energy barrier due to the crowded energy landscape. The obtained formula recovers the effective diffusivity with a correction term linear in the shear rate, in reasonable agreement with results from numerical simulations of different liquids as well as with earlier experimental results on shear melting of colloidal glass. The theory predictions are compared with molecular simulations of supercooled water and supercooled Lennard-Jones liquids. The comparison suggests that the predicted enhancement of diffusivity is inversely proportional to temperature and directly proportional to the zero shear viscosity.

Figures (8)

• Figure 1: Schematic illustration of an event by which a particle abandons its original quasi-equilibrium position in the cage formed by its nearest-neighbours and jumps under the influence of thermal fluctuations to a new quasi-equilibrium position just outside the cage. The energy barrier $V_{\mathrm{max}}$ can be estimated as the elastic energy needed to accommodate the particle in the cavity, which for simplicity is taken to be spherical. This leads to a quantitative estimate of $V_{\mathrm{max}}$ according to e.g. Dyre.
• Figure 2: The variation in zero-shear diffusivity (a) and viscosity (b) of supercooled mW water with temperature, measured from NEMD simulations. The symbols represent the simulation data whereas the solid lines are linear fits to the simulated data.
• Figure 3: The variation in zero-shear diffusivity (a) and viscosity (b) of supercooled LJ particles with temperature, measured from NEMD simulations. The symbols represent the simulation data, whereas the solid lines are linear fits to the simulated data.
• Figure 4: The plot of ($D_{\mathrm{eff}}/D$) Vs shear, for supercooled mW water (a) and supercooled LJ particles (b), respectively, at different supercooled temperatures. The symbols represent the measured values from NEMD simulations, whereas the solid lines are best fits of the diffusivity ratio to the theoretically derived Eq. 15 in the paper.
• Figure 5: The ratio of viscosity ($\mu_{\mathrm{eff}}/\mu_{0}$) in presence and absence of shear, for supercooled mW water (a) and LJ particles (b) at different shear rates, measured from NEMD simulations.