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.Graphical abstract
Highlights
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]
Employing the projection operator formalism, we decomposed the stress autocorrelation in Brownian systems into a structure that formally agrees with the one previously obtained in Newtonian [6, 7] or Langevin systems [5]
In those systems the dynamical coupling between stresses and momentum currents was considered, while particle momenta are not among the dynamical variables in the Brownian case. This interpretation is based on the fact that in the final expression for the correlation function (32) a matrix Kαβ appears which is identical to the autocorelation of the current in the overdamped Langevin system
Summary
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]. The linear response of the local stress tensor σ(r, t) to an external velocity field vext(r , t ) at a distant space-time point is investigated in such a model of an overdamped colloidal system. Since the generalized viscosity should be defined as the response of the local stress to the local current gradients [16], we have to reintroduce the flux as a dynamical variable. This is done via a linear response and a hydrodynamic approach. Starting on the Brownian level appears more efficient than overdamping a calculation containing particle momenta
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