Emergence of Long-Ranged Stress Correlations at the Liquid to Glass Transition

Abstract

A theory for the nonlocal shear stress correlations in supercooled liquids is derived from first principles. It captures the crossover from viscous to elastic dynamics at an idealized liquid to glass transition and explains the emergence of long-ranged stress correlations in glass, as expected from classical continuum elasticity. The long-ranged stress correlations can be traced to the coupling of shear stress to transverse momentum, which is ignored in the classic Maxwell model. To rescue this widely used model, we suggest a generalization in terms of a single relaxation time τ for the fast degrees of freedom only. This generalized Maxwell model implies a divergent correlation length ξ∝τ as well as dynamic critical scaling and correctly accounts for the far-field stress correlations. It can be rephrased in terms of generalized hydrodynamic equations, which naturally couple stress and momentum and furthermore allow us to connect to fluidity and elastoplastic models.

Field Theory for Amorphous Solids

“…Similar observations have been made for strain correlations in experiment, both for colloids [12,13] and granular materials [14]. It was argued in [9,11,15] that such correlations could be explained by the dynamical process by which the systems evolve, namely by elastic relaxation of so-called Eshelby transformations [9,11], or by a mode-coupling approach [15]. However, in granular materials the elastic range is extremely small, such that essentially all observed deformation is plastic [16,17], casting doubt on these dynamical explanations.…”

“…One easily sees that all stress correlators involving longitudinal components vanish [21], so that again only the transverse-transverse stress correlator survives, which now, however, is tensorial. Although simple to derive in the gauge formulation, the above results completely prescribe the tensorial structure of the stress correlator, a major aim of previous works [9,11,15,18] . We also see that material isotropy is not important in determining this structure, although it would simplify the implied derivatives.…”

“…This is equivalent to the Beltrami-Michell equation of linear elasticity [48], with an effective Poisson ratio ν = −η/(η + g). Finally, the stress correlator is again determined by (15), where in an infinite domain the solution to a source at the origin is now…”

Green-Kubo stress correlation function at the atomic scale and a long-range bond-orientational ordering in a model liquid

“…In earlier articles on the mode-coupling theory (MCT) it has been demonstrated that, in order to properly describe the decay of the transverse current CF, it is necessary to introduce the coupling between the density fluctuations and the transverse current correlation function [86][87][88][89][90][91][92][93][94][95]. In a recent article, in order to describe the decay of the macroscopic shear stress CF, the coupling between the short-time dynamics and the long-time hydrodynamic modes related to the transverse current correlation function also has been considered [95]. In our view, the features that we observe in the ASCF and which we interpret as shear stress waves should correspond to the transverse current CFs discussed within the MCT.…”

Constructing a Theory for Amorphous Solids

“…It was argued in Refs. [9,11,15] that such correlations could be explained by the dynamical process by which the systems evolve, namely, by elastic relaxation of so-called Eshelby transformations [9,11] or by a mode-coupling approach [15]. However, in granular materials, the elastic range is extremely small, such that essentially all observed deformation is plastic [16,17], casting doubt on these dynamical explanations.…”