[Paper Review] Hard discs under steady shear: comparison of Brownian dynamics simulations and mode coupling theory

Oliver Henrich, Fabian Weysser+2 more|arXiv (Cornell University)|Jan 19, 2010

Material Dynamics and Properties21 references47 citations

TL;DR

This study presents the first fully quantitative comparison between Brownian dynamics simulations and mode coupling theory (MCT-ITT) for two-dimensional bidisperse hard discs under steady shear. It demonstrates that while MCT-ITT qualitatively captures the non-linear response and lack of linear response in yielding glasses, it quantitatively overestimates shear-induced stresses and structural anisotropies by up to a factor of 10, primarily due to underestimated structural relaxation rates under shear.

ABSTRACT

Brownian dynamics simulations of bidisperse hard discs moving in two dimensions in a given steady and homogeneous shear flow are presented close to and above the glasstransition density. The stationary structure functions and stresses of shear-melted glass are compared quantitatively to parameter-free numerical calculations of monodisperse hard discs using mode coupling theory within the integration through transients framework. Theory qualitatively explains the properties of the yielding glass but quantitatively overestimatesthe shear-driven stresses and structural anisotropies.

Motivation & Objective

• To perform a fully quantitative, parameter-free comparison between mode coupling theory (MCT-ITT) and Brownian dynamics simulations for sheared hard discs.
• To test whether MCT-ITT accurately predicts the stationary microstructure and rheological response of yielding glasses under steady shear.
• To investigate the origin of discrepancies between theory and simulation, particularly regarding structural anisotropy and stress overestimation.
• To validate the MCT-ITT framework in the context of a homogeneous, disordered, shear-melted glass state without hydrodynamic interactions.
• To assess the validity of the non-linear response and absence of linear response regime predicted by MCT-ITT in the glassy state.

Proposed method

• Perform Brownian dynamics simulations of a 2D bidisperse hard disc system using Lees-Edwards boundary conditions and a thermostat to maintain homogeneous shear flow.
• Use a semi-event-driven algorithm to simulate particle collisions and compute time-averaged stresses via momentum transfer during collision events.
• Compute the equal-time structure factor Sq(˙γ) and its shear-induced distortion (Sq(˙γ) − Sq)/Sq using angular-averaged pair correlations.
• Apply the MCT-ITT framework to solve for stationary stress and structure factor responses in monodisperse hard discs under shear.
• Compare simulation results directly with MCT-ITT predictions at identical shear rates and densities, using consistent normalization and averaging procedures.
• Characterize transient density correlation functions crucial for MCT-ITT calculations, especially in the intermediate-time regime.

Experimental results

Research questions

• RQ1 Does MCT-ITT quantitatively predict the shear stress and normal stress in a sheared yielding glass state?
• RQ2 How accurately does MCT-ITT describe the shear-induced structural anisotropy in the microstructure of a 2D hard disc glass?
• RQ3 Why does MCT-ITT overestimate the magnitude of structural distortions and stresses compared to simulations?
• RQ4 Does the absence of a linear response regime in the glass state, as predicted by MCT-ITT, hold in the simulation?
• RQ5 To what extent do discrepancies arise from the approximations in MCT-ITT, such as the mean-field decoupling and neglect of hydrodynamic interactions?

Key findings

• MCT-ITT qualitatively reproduces the non-analytic behavior of stationary properties and the absence of a linear response regime in the shear-melted glass state.
• The theory overestimates the shear-induced structural anisotropy by up to a factor of 10, particularly along the compressional and extensional axes.
• At ϕ = 0.79 and Pe₀ = 2×10⁻⁴, the relative distortion (Sq(˙γ) − Sq)/Sq shows a 20% deviation from equilibrium in the glassy state, consistent with theory’s prediction of non-linear response.
• The simulation shows isotropic deviations in the low-q wing of the primary peak, while MCT-ITT predicts a stronger anisotropic suppression along the compressional axis (qx = −qy).
• The overestimation in MCT-ITT arises primarily from too-slow transient correlation functions, leading to excessive strain accumulation before structural relaxation occurs.
• Discrepancies in the equilibrium structure factor Sq between the bidisperse simulation and monodisperse theory may contribute to the mismatch, but are not the dominant source of error.