Evolution of Covalent Networks under Cooling: Contrasting the Rigidity Window and Jamming Scenarios

We study the evolution of structural disorder under cooling in supercooled liquids, focusing on covalent networks. We introduce a model for the energy of networks that incorporates weak noncovalent interactions. We show that at low temperature these interactions considerably affect the network topology near the rigidity transition that occurs as the coordination increases. As a result, this transition becomes mean field and does not present a line of critical points previously argued for, the “rigidity window.” Vibrational modes are then not fractons but instead are similar to the anomalous modes observed in packings of particles near jamming. These results suggest an alternative interpretation for the intermediate phase observed in chalcogenides.

Figure 1

Three distinct scenarios for the rigidity transition in chalcogenide glasses. Bonds in blue, green, red correspond respectively to floppy (under-constrained), isostatic (marginally constrained), and over-constrained regions; $p(z)$ is the probability that a rigid cluster (made of green and red bonds) percolates, as a function of the valence $z$. (a) Rigidity percolation model where bonds are randomly deposited on a lattice. Percolation occurs suddenly, and $p(z)$ jumps from 0 to 1 at $z_{\mathrm{cen}}<z_c$. At $z_{\mathrm{cen}}$, the rigid network is fractal. (b) The self-organized network model at zero temperature. Over-constrained regions are penalized energetically and are absent for $z<z_z$. For $z\in [z_{\mathrm{iso}},z_c]$, $0<p(z)<1$ even in the thermodynamic limit. (c) Mean-field scenario, where $p(z)$ jumps from 0 to 1 at $z_c$, and where the rigid cluster at $z_c$ is not fractal; $z_{\mathrm{cen}}=3.96$ [34] and $z_{\mathrm{iso}}=3.94$ [38] for triangular lattice.

Figure 2

Model illustration. The triangular lattice is slightly distorted periodically, leading to a unit cell of four nodes shown in the inset of (a). Weak springs connecting all second neighbors are indicated in cyan lines, emphasized for visibility around the central node $A$ in (b). (a) Illustration of allowed moves, where the strong spring in red relocates to a vacant link indicated by a blue dashed line.

Figure 3

$P_{\infty }$ vs $(z-z_c)N^{1/d\nu }$ in (a) the weak-force condition and (b) $T=\infty$. $p$ vs $\delta z\equiv z-z_c$ in (c) the weak-force and (d) strong-force conditions. The black squares are extrapolations of the finite $N$ spline curves, as detailed in the main text. In (c), the gray line is a step function at $z=z_c$, whereas in (d) it corresponds to the result of Ref. [38].

Figure 4

$D(\omega )$ at $z-z_c=-0.05$ and various $T$ indicated in legend for (a) $\alpha =0$ and (b) $\alpha =0.0003$. Gray dashed lines are numerical solutions of mean-field networks generated in Ref. [41]. (c) Boson peak frequency ${\omega }^{*}$ vs coordination $z$ for the weak-interaction regime. ${\omega }^{*}$ is defined as the peak frequency of $D(\omega )/{\omega }^{d-1}$, shown in (d) for coordinations indicated in the legend of (c).

Figure 5

Shear modulus of the strong network $G$ vs $\delta z\equiv z-z_c$ for parameters indicated in the legend. The total shear modulus $G_{\mathrm{tot}}$ including the effect of weak springs is represented for the weak-interaction regime. Inset: same plot in log-log scale, the horizontal axis is $z-z_c$ for low-temperature conditions (blue and green), and $z-z_{\mathrm{cen}}$ at $T=\infty$ (red).