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Space-resolved stress correlations and viscoelastic moduli for polydisperse systems: the faces of the stress noise

Jörg Baschnagel, Alexander N. Semenov

TL;DR

This work develops a space-resolved rheological framework for polydisperse amorphous liquids and solids by linking wavevector-dependent relaxation moduli to static structural correlations and to stress-noise fluctuations via fluctuation–dissipation relations. It derives the generalized compressibility equation $L_e(q)=\frac{c_0 T}{S_2(q)}$ and, for polydisperse systems, expresses the equilibrium moduli $L_e(q)$ and $M_e(q)$ through structure factors $S_2(q)$, $S_1(q)$ and $\tilde S(q)$, providing a practical route to compute these moduli from equal-time data; it also presents a novel, constraint-free derivation of the C–E relation linking the stress-correlation tensor $C$ to the elasticity tensor $E$. The paper further analyzes the stress-noise $\sigma^n$ and reduced deviatoric stress $\sigma^{rd}$, showing their autocorrelations coincide but that the two processes are not generally jointly stationary, with convergence at long times, highlighting subtle links between Newtonian and Zwanzig–Mori projected dynamics. Overall, the results establish a comprehensive, mass-polydispersity–aware framework for predicting equilibrium moduli from structural correlations and clarify the roles of stress fluctuations in the viscoelastic response of complex fluids.

Abstract

Several advances in the theory of space-resolved viscoelasticity of liquids and other amorphous systems are discussed in the present paper. In particular, considering long-time regimes of stress relaxation in liquids we obtain the generalized compressibility equation valid for systems with mass polydispersity, and derive a new relation allowing to calculate the wavevector-dependent equilibrium transverse modulus in terms of the generalized structure factors. Turning to the basic relations between the spatially-resolved relaxation moduli and the spatio-temporal correlation functions of the stress tensor, we provide their new derivation based on a conceptually simple argument that does not involve consideration of non-stationary processes. We also elucidate the relationship between the stress noise associated with the classical Newtonian dynamics and the reduced deviatoric stress coming from the Zwanzig-Mori projection operator formalism. The general relations between the stress noise and the tensor of relaxation moduli are discussed as well.

Space-resolved stress correlations and viscoelastic moduli for polydisperse systems: the faces of the stress noise

TL;DR

This work develops a space-resolved rheological framework for polydisperse amorphous liquids and solids by linking wavevector-dependent relaxation moduli to static structural correlations and to stress-noise fluctuations via fluctuation–dissipation relations. It derives the generalized compressibility equation and, for polydisperse systems, expresses the equilibrium moduli and through structure factors , and , providing a practical route to compute these moduli from equal-time data; it also presents a novel, constraint-free derivation of the C–E relation linking the stress-correlation tensor to the elasticity tensor . The paper further analyzes the stress-noise and reduced deviatoric stress , showing their autocorrelations coincide but that the two processes are not generally jointly stationary, with convergence at long times, highlighting subtle links between Newtonian and Zwanzig–Mori projected dynamics. Overall, the results establish a comprehensive, mass-polydispersity–aware framework for predicting equilibrium moduli from structural correlations and clarify the roles of stress fluctuations in the viscoelastic response of complex fluids.

Abstract

Several advances in the theory of space-resolved viscoelasticity of liquids and other amorphous systems are discussed in the present paper. In particular, considering long-time regimes of stress relaxation in liquids we obtain the generalized compressibility equation valid for systems with mass polydispersity, and derive a new relation allowing to calculate the wavevector-dependent equilibrium transverse modulus in terms of the generalized structure factors. Turning to the basic relations between the spatially-resolved relaxation moduli and the spatio-temporal correlation functions of the stress tensor, we provide their new derivation based on a conceptually simple argument that does not involve consideration of non-stationary processes. We also elucidate the relationship between the stress noise associated with the classical Newtonian dynamics and the reduced deviatoric stress coming from the Zwanzig-Mori projection operator formalism. The general relations between the stress noise and the tensor of relaxation moduli are discussed as well.
Paper Structure (20 sections, 149 equations)

This paper contains 20 sections, 149 equations.

Table of Contents

  1. Introduction
  2. The basic equations and definitions of the fields encoding liquid dynamics
  3. Density, velocity and stress fields, and the generalized relaxation moduli
  4. Stress correlation functions, deterministic stress and stress noise
  5. Equilibrium elastic moduli of a liquid or amorphous solid
  6. The generalized compressibility equation (GCE)
  7. GCE for polydisperse systems
  8. Cross-correlations between the longitudinal stress and concentration fluctuations
  9. Relation between the equilibrium transverse modulus and the structure factors
  10. Discussion on equilibrium moduli
  11. The general relation between the elasticity tensor and the stress correlation functions
  12. The relation for longitudinal components
  13. The general case
  14. The stress noise vs. the reduced deviatoric stress of the projected dynamics: do they coincide
  15. General relations for the stress noise and the reduced deviatoric stress
  16. ...and 5 more sections