Multi-scale lattice Boltzmann and mode-coupling theory calculations of the flow of a glass-forming liquid

Abstract

We present a hybrid-lattice Boltzmann (LB) algorithm for calculating the flow of glass-forming fluids that are governed by integral constitutive equations with pronounced nonlinear, non-Markovian dependence of the stresses on the flow history. The LB simulation for the macroscopic flow fields is combined with the mode-coupling theory (MCT) of the glass transition as a microscopic theory, in the framework of the integration-through transients formalism. Using the combined LB-MCT algorithm, pressure-driven planar channel flow is studied for a schematic MCT model neglecting spatial correlations in the microscopic dynamics. The cessation dynamics after removal of the driving pressure gradient shows strong signatures of oscillatory flow both in the macroscopic fields and the microscopic correlation functions.

Nonlinear mechanical response of supercooled melts under applied forces

“…the nonlinear Maxwell model suggested by White and Metzner which agrees with an α-scaling law by MCT for driven fluids), which rationalize particle based simulations of viscoelastic fluids. For the channel flow, this verifies that a local theory, connecting stress and strain at the [91]. Right panel: corresponding results from BD simulations using a 2D glass-forming hard-sphere mixture in channels of different widths as given in the legend; see there also for packing fraction and force.…”

“…In order to couple to the Navier-Stokes equations history-dependent nonlinear response entering the constitutive equation, the LB simulation needs to be coupled to an integral-equation solver capable of solving equation (3) (or equations of similar type). Such a hybrid-LB algorithm extends each spatial LB lattice node by a time axis in order to store the history of local flow [90]; for the inclusion of MCT, further memory is needed to store the local correlation functions and memory kernels needed to integrate equation (3) [91]. In order to limit the computational demand, it is assumed that the nonlinear response function becomes slowly varying in time once the time interval t − t becomes large; this assumption is inherent to all numerical approaches to MCT and potentially limits the evaluation of the response to external drive that varies slowly over arbitrarily large time intervals.…”

“…In order to limit the computational demand, it is assumed that the nonlinear response function becomes slowly varying in time once the time interval t − t becomes large; this assumption is inherent to all numerical approaches to MCT and potentially limits the evaluation of the response to external drive that varies slowly over arbitrarily large time intervals. For a description of the adaptive time-domain grids, see references [90,91].…”

“…Formally, the Finger tensor can be expressed in terms • E(t − δt, t ). In the case of advection-free flow where v(t) • ∇ ≡ 0, these expressions simplify considerably and allow for an efficient integration of the flow history [91].…”

“…In this case, the combined MCT-LB approach allows to study the interplay of microscopic stress relaxation and macroscopic flow cessation. After cessation of pressure-driven planar channel flow, LB-MCT predicts an oscillatory decay of both the velocity profile and the stresses towards zero [91].…”