Stretched-Exponential Aging Governs Nonequilibrium Precipitate Patterns

Abstract

Localized growth in driven materials is often governed by intermittent failure, yet how a material's history biases failure sites remains poorly understood. Using pause-restart experiments on chemical precipitate membranes, we quantify the probability of age-dependent breaching. We show that the kinetics follow a stretched-exponential aging law with parameters that obey one-parameter scaling. As the system approaches a critical concentration, the stretching exponent $β$ tends to zero, signaling a crossover to scale-free, power-law behavior. A stochastic cellular automaton based on this aging rule reproduces the emergent filaments and their concentration-dependent thickening. Our findings identify aging-controlled failure with long-lived but decaying memory as a general route to pattern formation in far-from-equilibrium systems.

Figures (4)

• Figure 1: Photos of precipitate patterns formed when pink $\mathrm{CoCl_2}$ solution is injected into a Hele-Shaw cell filled with clear sodium silicate solution. The black disks are injection ports connected to tubing. Initial concentrations: [$\mathrm{CoCl_2}$] = 0.625 M and (a,b) [silicate] = 4.0, 5.0 M, respectively. Image heights: 10.0 cm. Injection rate: (a) 1.15 mL/min and (b) 0.15 mL/min.
• Figure 2: (a) Measurement protocol: the pattern is pre-grown, injection is paused for a wait time $t_w$, and growth is restarted while breach sites are optically detected. (b) Example of a paused pattern (left) and subsequent regrowth at the breach site (arrow). (c) Color-coded precipitate age maps for representative patterns at silicate concentrations $c=3$ M and 5.25 M; blue and red indicate fresh and old material, respectively.
• Figure 3: (a) Age-dependent breach probabilities after pauses in $\mathrm{CoCl_2}$ injection at varying silicate concentrations $c$ between 1 M and 5 M. Top row: experimental probability distributions $P(t|t_w)$ per 10-s bin for different breach formation times $t$ and wait times $t_w$. Each column is individually normalized. Bottom row: stretched-exponential fits with respective $\beta$ values. (b,c) Stretched exponential fit parameters $\beta$ and $k$ versus $c$. Error bars represent profile-likelihood-based confidence intervals. Fits (red curves) yield the indicated square root laws. Inset: $k$ versus $\beta$ follows $k\,\beta$ = 2.8. The 5 M data point was excluded from fits due to poorly constrained values (flat profile likelihood).
• Figure 4: Stochastic cellular automaton simulations with an age-dependent breach probability $P(t)\propto \exp(-k t^{\beta})$. (a) Representative morphologies for different $(\beta,k\,\beta)$ values showing filamentary conduits of increasing average width, irregular patterns, and rugged fronts. Blue represents the injected solution; orange and black mark newly formed and old precipitate, respectively. (b,c) Simulated restart experiments reproducing age/wait-time probability maps for $k\,\beta=1$ and $\beta = 0.5, 0.25$, respectively. Color denotes the column-normalized probability that the first breach after a wait time $t_w$ occurs in material of age $t$.