Stress correlation function and linear response of Brownian particles
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THE EUROPEAN PHYSICAL JOURNAL E
Regular Article
Stress correlation function and linear response of Brownian particles Florian Vogel and Matthias Fuchsa University of Konstanz - D-78457 Konstanz, Germany Received 30 July 2020 / Received in final form 5 October 2020 / Accepted 8 October 2020 Published online: 16 November 2020 c The Author(s) 2020. This article is published with open access at Springerlink.com Abstract. We determine the non-local stress autocorrelation tensor in an homogeneous and isotropic system of interacting Brownian particles starting from the Smoluchowski equation of the configurational probability density. In order to relate stresses to particle displacements as appropriate in viscoelastic states, we go beyond the usual hydrodynamic description obtained in the Zwanzig-Mori projection-operator formalism by introducing the proper irreducible dynamics following Cichocki and Hess, and Kawasaki. Differently from these authors, we include transverse contributions as well. This recovers the expression for the stress autocorrelation including the elastic terms in solid states as found for Newtonian and Langevin systems, in case that those are evaluated in the overdamped limit. Finally, we argue that the found memory function reduces to the shear and bulk viscosity in the hydrodynamic limit of smooth and slow fluctuations and derive the corresponding hydrodynamic equations.
1 Introduction Stress fluctuations play an important role in viscoelastic fluids, and understanding their spatio-temporal patterns remains an open question when starting from first principles [1]. A system of interacting Brownian particles can be used to model the dynamics of concentrated colloidal dispersions [2]. While instantaneous solvent mediated interactions are neglected, the collective effects arising from steric particle interactions can be analyzed [3]. In the present work, the linear response of the local stress tensor σ(r, t) to an external velocity field v ext (r , t ) at a distant space-time point is investigated in such a model of an overdamped colloidal system. The main question is, whether precursors of the elastic properties of a colloidal glass already arise in the underlying fluid-like dynamics. The elastic response decays as 1/|r − r |3 [1, 4], while the fluid one is short-ranged. This question was already considered in [5]. There, a set of Langevin’s equations of motion for the individual colloidal particles was investigated, which leads to a time evolution of the probability distribution function that is governed by the Klein-Kramers equation. It describes the dynamics in the phase space of the positions and velocities of the colloidal particles. When applying a Zwanzig-Mori projection formalism, it was argued that the coupling of the shear stress to the transverse current flow has to be taken into account, to obtain the correct long-lived and a
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long-ranged correlations in the supercooled state expected from the Newtonian case [6,7]. Only based on this projection, the over
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