Athermal Fluctuations in Disordered Crystals

We analyze the fluctuations in particle positions and interparticle forces in disordered crystals composed of jammed soft particles in the limit of weak disorder. We demonstrate that such athermal systems are fundamentally different from their thermal counterparts, characterized by constrained fluctuations of forces perpendicular to the lattice directions. We develop a disorder perturbation expansion in polydispersity about the crystalline state, which we use to derive exact results to linear order. We show that constrained fluctuations result as a consequence of local force balance conditions, and are characterized by non-Gaussian distributions, which we derive exactly. We analytically predict several properties of such systems, including the scaling of the average coordination with polydispersity and packing fraction, which we verify with numerical simulations using soft disks with one-sided harmonic interactions.

Figure 1

A section of a disordered crystal composed of jammed soft particles. The particles are colored according to their incremental size $\mathrm{\Delta }{\sigma }_i={\sigma }_i-{\sigma }_0$, where ${\sigma }_i$ are their radii and ${\sigma }_0=1/2$. When $\mathrm{\Delta }{\sigma }_i=0$ the system settles into a triangular lattice. The black arrows represent the change in the interparticle forces from their values in the pure crystal in response to the change in radii. For small disorder (polydispersity) the forces fluctuate primarily along the lattice directions.

Figure 2

(a) Two-dimensional distributions of the forces $p(f_x,f_y)$ in the thermal crystal. (b) Scaled distributions of the fluctuations of the components of forces along ($\delta f_{||}$) and orthogonal ($\delta f_{\perp }$) to the original lattice directions. Both display Gaussian fluctuations (marked with solid lines) with a variance proportional to the temperature $T$. (c),(d) In contrast $p(f_x,f_y)$ for the athermal system displays highly constrained fluctuations orthogonal to the original lattice directions as the polydispersity ($\eta$) is increased. Here $\varphi =0.92$ and the number of particles is $N=2500$.

Figure 3

The distribution of $f_{\perp }$ in the athermal system, displaying near-perfect agreement with the theoretical prediction in Eq. (12) (solid lines). Here $\varphi =0.92$ and $N=2500$. This distribution exhibits pronounced non-Gaussian behavior as the disorder strength is increased, in contrast to the thermal system in Fig. 2. (Inset) The distribution of $f_{||}$ displays Gaussian behavior. The predictions from the theory are displayed with solid lines.

Figure 4

Variation of the average coordination with the strength of the disorder for different packing fractions ($\varphi$). The points represent data from simulations, and the solid lines represent the theoretical prediction in Eq. (13). (Inset) The scaling collapse of the average coordination with the predicted scaling variable $\eta /\mathrm{\Delta }\varphi$. Here $N=2500$.