Adaptive elastic networks as models of supercooled liquids

The thermodynamics and dynamics of supercooled liquids correlate with their elasticity. In particular for covalent networks, the jump of specific heat is small and the liquid is strong near the threshold valence where the network acquires rigidity. By contrast, the jump of specific heat and the fragility are large away from this threshold valence. In a previous work [Proc. Natl. Acad. Sci. USA 110, 6307 (2013)], we could explain these behaviors by introducing a model of supercooled liquids in which local rearrangements interact via elasticity. However, in that model the disorder characterizing elasticity was frozen, whereas it is itself a dynamic variable in supercooled liquids. Here we study numerically and theoretically adaptive elastic network models where polydisperse springs can move on a lattice, thus allowing for the geometry of the elastic network to fluctuate and evolve with temperature. We show numerically that our previous results on the relationship between structure and thermodynamics hold in these models. We introduce an approximation where redundant constraints (highly coordinated regions where the frustration is large) are treated as an ideal gas, leading to analytical predictions that are accurate in the range of parameters relevant for real materials. Overall, these results lead to a description of supercooled liquids, in which the distance to the rigidity transition controls the number of directions in phase space that cost energy and the specific heat.

Figure 1

Illustration of rigidity transition. Blue, green, and red color the floppy, isostatic, and stressed clusters, respectively.

Figure 2

(a) and (b) Illustration of the frozen network model [41]; (c) and (d) illustrate the adaptive network model [42]. In the latter case, the triangular lattice is systematically distorted in a unit cell of four nodes shown in the inset of (c). We group nodes by four, labeled as A, B, C, and D in Fig. 2. One group forms the unit cell of the crystalline lattice. Each cell is distorted identically in the following way: node A stays, while nodes B, C, and D move by a distance $\delta$, B along the direction perpendicular to BC, C along the direction perpendicular to CD, and D along the direction perpendicular to DB. $\delta$ is set to $0.2$ with the lattice constant as unity. Weak springs connecting (b) six nearest neighbors without strong springs and (d) six next-nearest neighbors are indicated in straight cyan lines, emphasized for the central node. (c) Illustration of an allowed step, where the strong spring in red relocates to a vacant edge indicated by a dashed blue line.

Figure 3

Illustration of configuration energy of the adaptive network model ($\delta z=0.27$). Solid lines are springs, colored according to their extensions: from red to purple, the springs go from being stretched to being compressed, with spring extensions shown in the unit of $\epsilon$. Left: Nodes sit at lattice sites, so the color shows the rest length mismatches of the springs $\left\{{\epsilon }_{\gamma }\right\}$. Right: Nodes are relaxed to mechanical equilibrium. Most links appear in green, indicating that most of the elastic energy is released. The configuration energy is defined by the residual energy.

Figure 4

Relaxation time $\tau$ in log-scale versus inverse temperature $1/T$ for different coordination numbers $\delta z$ and $\alpha =0.0003$. The solid black line indicates a power-law relation between $\tau$ and $T$: $\tau \sim T^{-1/2}$.

Figure 5

Left: Shear modulus of adaptive networks at temperature $T$ rescaled by $G$ at $T=\infty G(z,T)/G(z,\infty ), \alpha =0.0003$. The temperature $T$ is rescaled by $T_g$. Right: Correlation between transition temperature $T_g$ and shear modulus $G$ in the frozen network model [41].

Figure 6

Thermodynamics of the adaptive network model without weak constraints $\alpha =0$. (a) Energy $E/N_s$ vs. temperature; (b) specific heat $C/N_s$ vs. temperature; (c) excess number density of redundant constraints $n_{\mathrm{ex}}$ extracted using the pebble game algorithm $vs$ temperature. Symbols are numerical data; solid lines are theoretic predictions.

Figure 7

Top: Specific heat $c(z,T)$ vs. scaled temperature $T/T_g$ for networks with average coordination numbers near and away from the isostatic on both floppy and rigid sides. The strength of the weak constraints is given by $\alpha =0.0003$. Bottom: Specific heat at temperature $T_g, c(z,T_g)$, vs. coordination number $\delta z$ for $\alpha =0,0.0003,0.003,0.03.$ The inset shows the transition temperature $T_g$ for different $z$ and $\alpha$. Symbols are numerical results, and lines are theoretical predictions: dashed lines are for frozen network model and solid lines are for the new model derived in Sec. 4.

Figure 8

Theoretical predictions for the jump of specific heat. For vanishingly weak springs $\alpha \to 0$, it is predicted that the jump is essentially constant for $z<z_c$ and then drops to zero a $z_c$. For larger $z$, it behaves as $z-z_c$. As $\alpha$ grows this sharp curve becomes smooth, but a minimum is still present near $z=z_c$.

Figure 9

(a) $z<z_c$, localized redundant constraints (red) in a floppy sea (blue); (b) $z>z_c$ localized floppy modes (blue) in a rigid sea (red and green).

Figure 10

Left: Excess number density of redundant constraints $n_{\mathrm{ex}}(z,\beta )$. Right: Fluctuation of the number density of redundant constraints ${\left(\mathrm{\Delta }{n}_r\right)}^2$. The solid black lines show the predictions from the approximate entropy Eq. (9).

Figure 11

Changes of density of states $D(\omega ,T)$ with temperature for the same $z=-0.055, \alpha =0.0003$. Left: density of states in log-log scale. Right: density of states normalized by its $T=\infty$ value, emphasizing its difference under cooling. Inset: participation ratio $P(\omega ,T)$ variation under cooling.

Figure 12

Density of states $D(\omega ,T)$ for adaptive networks with different $z$. (a) Random diluted networks $T=\infty$; a power law $D\left(\omega \right)\sim {\omega }^{-0.25}$ is shown in the low-frequency range of networks near $z_{\text{cen}}$. (b) Adaptive networks without weak constraints ($\alpha =0$) at $T=0.0003$; power laws with different exponents are shown for networks in the rigidity window: $D\left(\omega \right)\sim {\omega }^{-0.25}$ for $\delta z=-0.055, D\left(\omega \right)\sim {\omega }^{-0.5}$ for $\delta z=0.0$. (c) Adaptive networks with weak constraints ($\alpha =0.0003$) at $T\approx \alpha$; away from isostatic, the densities of states are gapped between zero frequency and Boson peak, where $D\left(\omega \right)\sim {\omega }^0$. Inset (d) is the participation ratio $P(\omega ,T)$ at $T=\infty$; see text for definition.

Figure 13

Vector plots of vibrational modes in randomly diluted networks, $N=100\times 100$. (a) A typical Debye mode, $\delta z=0.501, \omega =0.017$. (b) A typical anomalous mode on boson peak, $\delta z=-0.049, \omega =0.011$. (c) A typical fracton, $\delta z=-0.049, \omega =0.0007$.

Figure 14

Correlation between a low-frequency fractal mode and isostatic clusters. A network configuration ($\delta z=-0.042$) is shown with its springs in the over-constrained regions colored in red, in the isostatic regions colored in green, and in the floppy regions colored in blue. A typical fracton ($\omega =5\times {10}^{-4}$) specified in this configuration is plotted on top.