Stress auto-correlation tensor in glass-forming isothermal fluids: From viscous to elastic response

Abstract

We develop a generalized hydrodynamic theory, which can account for the build-up of long-ranged and long-lived shear stress correlations in supercooled liquids as the glass transition is approached. Our theory is based on the decomposition of tensorial stress relaxation into fast microscopic processes and slow dynamics due to conservation laws. In the fluid, anisotropic shear stress correlations arise from the tensorial nature of stress. By approximating the fast microscopic processes by a single relaxation time in the spirit of Maxwell, we find viscoelastic precursors of the Eshelby-type correlations familiar in an elastic medium. The spatial extent of shear stress fluctuations is characterized by a correlation length which grows like the viscosity or time scale ∼, whose divergence signals the glass transition. In the solid, the correlation length is infinite and stress correlations decay algebraically as in d dimensions.

Stress correlation function and linear response of Brownian particles

“…Our ansatz naturally leads to the complete coupling of the tensorial stress fluctuations to the vectorial particle displacements. In the end, we regain the formally identical expression for the stress autocorrelation and therefore for the linear response of the stress tensor as in systems where velocities are kept as dynamical variables [5,7]. In the final section, we argue that the irreducible memory kernel indeed is the correct one since it can be related to the Green-Kubo transport coefficients, viz.…”

“…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 e-mail: matthias.fuchs@uni-konstanz.de long-ranged correlations in the supercooled state expected from the Newtonian case [6,7]. Only based on this projection, the overdamped case and consequently the formation of colloidal solids could be considered.…”

Analytical prediction of logarithmic Rayleigh scattering in amorphous solids from tensorial heterogeneous elasticity with power-law disorder

“…In this paper, by developing a fully tensorial replica field theory for athermal amorphous systems with power-law decay in elastic constant correlations (or in stress fields), we reveal the origin of the enhanced phonon attenuation, especially where the logarithmic enhancement is prompted. The analytical theory shows that the logarithmic enhancement is either due to the long-range power-law correlations of elastic constants 21,22 or (as shown in the Appendix B) to long-range power-law correlations of the internal stresses (with no fluctuations in the elastic constants) 23,24 , which is the key ingredient in our framework leading to the prediction of the logarithmic enhancement. Some previous works dealing with mean-field theory confirm the Rayleigh scattering law without the logarithmic factor.…”

“…Here we instead assume that elastic constants C αβ κ χ have no fluctuations, while fluctuations exist in the internal stresses, a situation encountered in glasses 24 and granular materials 23 . Writing p i j = −(1/2)V i j (r i j )r i j , the local stress tensor can be decomposed at the pair level as:…”

“…All other elastic constants like C αβ κ χ or σ 1 , σ 2 are short-ranged and hence can be regarded as constant when r is large. The long-range decay in shear stress correlations has been derived using generalized hydrodynamic theory in 24 . Then the elastic wave equation becomes…”