Effective-medium theory for elastic systems with correlated disorder

TL;DR

The paper addresses rigidity transitions in disordered elastic networks with spatially correlated disorder, a regime not captured by traditional i.i.d. disorder theories. It develops a Correlated Coherent Potential Approximation (CCPA) by introducing an F-matrix and a generalized self-consistency ⟨F⟩ = 0 to account for multi-defect correlations, and applies it to a rigidity-percolation model of soft gels. The analysis shows that correlations effectively increase the average coordination ⟨z⟩, which lowers the apparent critical packing fraction φc while preserving the isostatic critical point at zc = 2d; a simple linear relation k(c,φ) = (1/2)⟨z⟩(c,φ) − 2 emerges, with near-threshold scaling that matches simulations. The framework provides a versatile EMT tool for predicting mechanical response in a broad class of correlated disordered elastic materials and guides interpretation of sub-isostatic-like behavior observed in experiments and simulations.

Abstract

Correlated structures are intimately connected to intriguing phenomena exhibited by a variety of disordered systems such as soft colloidal gels, bio-polymer networks and colloidal suspensions near a shear jamming transition. The universal critical behavior of these systems near the onset of rigidity is often described by traditional approaches as the coherent potential approximation - a versatile version of effective-medium theory that nevertheless have hitherto lacked key ingredients to describe disorder spatial correlations. Here we propose a multi-purpose generalization of the coherent potential approximation to describe the mechanical behavior of elastic networks with spatially-correlated disorder. We apply our theory to a simple rigidity-percolation model for colloidal gels and study the effects of correlations in both the critical point and the overall scaling behavior. We find that although the presence of spatial correlations (mimicking attractive interactions of gels) shifts the critical packing fraction to lower values, suggesting sub-isostatic behavior, the critical coordination number of the associated network remains isostatic. More importantly, we discuss how our theory can be employed to describe a large variety of systems with spatially-correlated disorder.

Figures (5)

• Figure 1: Schematic representation of the Coherent Potential Approximation. (a) Regular networks in which bonds (springs) are randomly removed mimics the microscopic structure of diverse disordered solids. (b) In traditional CPA, the randomly-diluted network depicted in (a) is mapped into a homogeneous network with effective spring constant $k$, which is determined as the solution to a self-consistent equation originating in the linear response to a perturbation on a single bond (dotted gray line). (c) We extend traditional CPA to include the combined effect of multiple defects, so that the elastic spring constants $k'_i$ are sampled from a distribution that captures spatial correlations.
• Figure 2: Contrast between correlated and uncorrelated disordered structures. Simulated configurations following the protocol of Ref. Zhang2019 for a rigidity-percolation model of gels at target volume fraction $\phi_l=0.67$, and correlation strength $c=0$ (left), $0.3$ (center) and $0.6$ (right), representing a bias to add particles where there are more neighbors.
• Figure 3: Rigidity-percolation model for gels in the CCPA framework. (a) We start with a triangular lattice unit cell with no occupied bonds and no occupied sites. (b) A set of $N_{nn}$ particles is then added at the outer sites with uncorrelated probability $p_{nn}$. (c) Finally, a particle is added at the center with correlated probability $f(c, \phi, N_{nn})$. Panels (d) - (f) show an example in which two particles are added at the outer sites (d), followed by occupation or not of the central site [(e) and (f), respectively].
• Figure 4: (a) Effective elastic constant $k$ as a function of packing fraction $\phi$ for several values of the correlation strength $c$. The uncorrelated case ($c=0$, solid line) shows standard rigidity-percolation behavior, whereas the transition of correlated systems ($c>0$, dashed lines) are shifted to lower values of $\phi$. The inset shows that all curves collapse when $k$ is plotted as a function of average coordination number $\langle z \rangle$. (b) Phase diagram showing in terms of packing fraction $\phi$ and the inverse of the correlation strength $c^{-1}$. The continuous black line represent the phase boundary between the floppy and the rigid states and is obtained analytically using the generalized coherent potential approximation.
• Figure 5: Scaling collapse plot in terms of $k / (\alpha + \beta c)$ and $\phi - \phi_c$, for several values of the correlation parameter $c$ and packing fraction $\phi$. All curves collapse onto a straight line (dashed line) in the vicinity of the critical packing fraction.