Order parameter for structural heterogeneity in disordered solids

We construct a structural order parameter from the energy equipartition of normal modes of vibration to quantify the structural heterogeneity in disordered solids. The order parameter exhibits strong spatial correlations with low-temperature dynamics and local structural entropy. To characterize the role of particles with the most defective local structures identified by the order parameter, we pin them and measure the system response. It turns out that particles with the largest value of the order parameter are responsible for the quasilocalized low-frequency vibration, instability, softening, and nonaffinity of disordered solids. The order parameter thus crucially links the heterogeneous structure to low-temperature dynamics and mechanical properties of disordered solids.

Figure 1

Spatial distributions of (a) order parameter $\Psi$, (b) LDWF $\alpha$, and (c) LSE $S_2$ for a 2D system. (d) Order parameter $\Psi$ dependence of the number of modes $N_m$ (squares) and weighted frequency ${\omega }_m$ (circles) defined in the text for 3D glasses, with the solid curves to guide the eye.

Figure 2

(a) Reduced density of states $D\left(\omega \right)/{\omega }^2$ and (b) participation ratio $P\left(\omega \right)$ of 3D $T=0$ glasses. The circles, diamonds, and triangles are for pinning $3\%$ particles with the largest $\Psi$, randomly pinning $3\%$ particles, and no pinning, with the solid curves to guide the eye.

Figure 3

(a) Dependence of the cage-jump temperature $T_j$ on the fraction of pinned particles $N_p/N$ for 3D systems. (b) Distributions of $T_j$ for pinning $3\%$ particles with the largest $\Psi$ (circles), randomly pinning $3\%$ particles (diamonds), and no pinning (triangles). The solid curves are to guide the eye.

Figure 4

Dependence of (a) bulk modulus $B$, (b) shear modulus $G$, (c) nonaffinity of compression ${\mu }_c$, and (d) nonaffinity of shear ${\mu }_s$ on the fraction of pinned particles $N_p/N$ for 3D $T=0$ glasses. The circles and diamonds are for the pinning of particles with the largest $\Psi$ and random pinning, with the solid curves to guide the eye.