Models of 3D confluent tissue as under-constrained glasses

Abstract

The dynamics of glassy materials slows down upon cooling, typically showing either Arrhenius or super-Arrhenius behavior. However, it was recently shown that 2D cell-based models for biological tissues can be continuously tuned between Arrhenius and sub-Arrhenius dynamics. In previous work, using the 2D Voronoi model, we proposed that such atypical dynamical behavior could be a generic feature of the broad class of mechanically under-constrained materials. Our earlier study had left two important points open: (1) many 2D systems are affected by long-wavelength fluctuations and the 2D melting scenario, and (2) the 2D Voronoi model sits exactly at the isostatic point, making it a marginal case rather than a strictly under-constrained one. Both points complicate the interpretation of our 2D Voronoi model results and their generalization to other systems; to remedy this, here we use large-scale simulations to study the glassy behavior of the 3D extension of the Voronoi model. We first show that the structural relaxation time $τ_α$ of the 3D Voronoi model can be tuned between sub-Arrhenius and Arrhenius behavior, like the 2D Voronoi model. We then establish that the four-point susceptibility, the structure factor, and the model's mechanical properties all display trends consistent with the 2D Voronoi model. These results provide strong evidence that sub-Arrhenius glassy dynamics are a generic feature of under-constrained materials across dimensions. Our work thus broadens the class of disordered materials known to have highly unusual glassy phenomenology.

Figures (6)

• Figure 1: The 3D Voronoi model exhibits tunable sub-Arrhenius relaxation dynamics. Angell plot of the $\alpha$-relaxation time, $\tau_\alpha$, for four values of $s_0$. Each curve represents a different $s_0$ from high to low (dark red to light blue), and for each curve, $T_g$ is determined separately as the temperature where $\tau_\alpha=10^4$.
• Figure 2: $\alpha$- and $\beta$-relaxation from the self-intermediate scattering function. (Top) $F_s(t)$ from the trajectories across all $(s_0,T)$ parameter pairs, plotted with time scaled by $\tau_\alpha$, showing collapse of all curves in the $\alpha$-relaxation regime. The black line is the fitting result. The inset shows $F_s(t)$ with the time axis scaled by $\tau_\beta$. (Bottom) The measured $\tau_\beta$ as a function of $T$ for several values of $s_0$; the dotted black line is a guide to the eye with slope $-1/2$. The inset shows the ratio of the $\alpha$ and $\beta$ relaxation times as a function of temperature; a clear cross-over is observed from a regime at large $s_0$ in which the time scales have the same temperature dependence to a more normal regime at low $s_0$ in which $\tau_\alpha$ grows much more rapidly as the temperature is decreased.
• Figure 3: The temperature dependence of dynamical heterogeneity is controlled by $s_0$. Four-point dynamical susceptibility $\chi_4(t)$ for $s_0=5.39$ (Top) and $s_0=5.50$ (Bottom) at different temperatures (temperature decreases from dark red to light blue).
• Figure 4: The temperature dependence of peak height of static structure factor is controlled by $s_0$. Static structure factor $S(k)$ for $s_0=5.39$ (Top) and $s_0=5.50$ (Bottom) at different temperatures (temperatures decrease from red to blue). The inset in the top panel magnifies the region corresponding to the first peak in the main panel; the inset in the bottom panel shows the temperature dependence of the first peak height of $S(k)$ for different $s_0$.
• Figure 5: The isotropic tension increases with temperature and matches the theoretical prediction for generic under-constrained systems. (Top) isotropic tension as a function of temperature for several $s_0$, showing a transition from $T^{1/2}$ to approximately linear $T^1$ scaling. Inset: shear relaxation modulus $G(t)$ for $s_0=5.425$, displaying an intermediate-time plateau that decreases monotonically with temperatures (from red to blue, temperature decreases). (Bottom) When plotting isotropic tension $\Sigma_{\textrm{iso}}$ versus isotropic strain $\epsilon$, a rescaling by $\sqrt{T}$ leads to a collapse of the curves for the different state points. The dashed lines correspond to Eq. \ref{['eq:isotropicTension']} with parameter values as given in the main text.