Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Topological Quarantine and the Standard Candle for Network-Glass Rigidity

Kejun Liu

Abstract

We study rigidity percolation in random covalent networks to establish the pure topological baseline of the floppy-to-rigid transition. Using a generating-function mean-field theory on configuration-model graphs, we report three results. First, we prove -- within the locally tree-like (mean-field) approximation -- that the onset of the topological giant rigid component (GRC) coincides with the mechanical Maxwell isostatic point ($\langle r\rangle_c = 2.4$), providing a clean topological lower bound from which pebble-game deviations in finite-dimensional glasses can be measured. Second, finite-size scaling extended to $N = 32\,000$ locates a sharp internal geometric marker at $\langle r\rangle^* = 2.431 \pm 0.002$, residing inside the experimental Boolchand intermediate phase window $[2.35, 2.47]$, where the rigid backbone crosses the classical ER percolation threshold $S_\infty = 1/8$. Third, we demonstrate via controlled perturbation simulations that generic enthalpic stress avoidance delays but cannot compress this window, while chemical medium-range order (MRO) is the necessary ingredient -- a conclusion directly supported by recent NMR evidence that homopolar defect fractions in GeSe$_2$ fall well below the predicted rigidity-rescue threshold.

Topological Quarantine and the Standard Candle for Network-Glass Rigidity

Abstract

We study rigidity percolation in random covalent networks to establish the pure topological baseline of the floppy-to-rigid transition. Using a generating-function mean-field theory on configuration-model graphs, we report three results. First, we prove -- within the locally tree-like (mean-field) approximation -- that the onset of the topological giant rigid component (GRC) coincides with the mechanical Maxwell isostatic point (), providing a clean topological lower bound from which pebble-game deviations in finite-dimensional glasses can be measured. Second, finite-size scaling extended to locates a sharp internal geometric marker at , residing inside the experimental Boolchand intermediate phase window , where the rigid backbone crosses the classical ER percolation threshold . Third, we demonstrate via controlled perturbation simulations that generic enthalpic stress avoidance delays but cannot compress this window, while chemical medium-range order (MRO) is the necessary ingredient -- a conclusion directly supported by recent NMR evidence that homopolar defect fractions in GeSe fall well below the predicted rigidity-rescue threshold.

Paper Structure

This paper contains 1 equation, 3 figures, 1 table.

Figures (3)

  • Figure 1: GRC fraction $S_\infty$ (solid) and locally rigid fraction $p_r$ (open) vs. $\langle r\rangle$ ($N=5000$, 20 trials). Dashed line: $S_\infty = 1/8$ (ER reference level); dotted: Maxwell point $\langle r\rangle_c = 2.4$. Shaded region: Boolchand intermediate phase [2.28, 2.46].
  • Figure 2: Finite-size scaling of the $S_\infty = 1/8$ GRC marker. (a) $S_\infty$ vs. $\langle r\rangle$ for $N \in [500, 32\,000]$. (b) $\langle r\rangle^*(N)$ vs. $N^{-1/\nu}$ with power-law fit. (c) Extrapolated limit $\langle r\rangle^*_\infty = 2.431 \pm 0.002$ lying within the Boolchand window zeidler2017.
  • Figure 3: Physical perturbations beyond the ideal topological baseline. (a) Metropolis growth with rigid-rigid bond penalty $\varepsilon=1$ at five temperatures. Lower $T$ shifts $\langle r\rangle^*$ rightward, widening the intermediate phase window. (b) Two independent rescue mechanisms from the topological quarantine (fixed $\langle r\rangle = 2.42$, $P_{\rm AA}=0$ baseline): chemical defects ($P_{\rm AA}^* = 0.410$, circle) and ring-closure MRO ($P_{\rm ring}^* = 0.496$, square).