Theory of Crystallization versus Vitrification

Abstract

The competition between crystallization and vitrification in glass-forming materials manifests as a non-monotonic behavior in the time-temperature transformation (TTT) diagrams, which quantify the time scales for crystallization as a function of temperature. We develop a coarse-grained lattice model, the Arrow-Potts model, to explore the physics behind this competition. Using Monte Carlo simulations, the model showcases non-monotonic TTT diagrams resulting in polycrystalline structures, with two distinct regimes limited by either crystal nucleation or growth. At high temperatures, crystallization is limited by nucleation and results in the growth of compact crystal grains. At low temperatures, crystal growth is influenced by glassy dynamics, and proceeds through dynamically heterogeneous and hierarchical relaxation pathways producing fractal and ramified crystals. To explain these phenomena, we combine the Kolmogorov-Johnson-Mehl-Avrami theory with the field theory of nucleation, a random walk theory for crystal growth, and the dynamical facilitation theory for glassy dynamics. The unified theory yields an analytical formula relating crystallization timescale to the nucleation and growth rates through universal exponents governing glassy dynamics of the model. We show that the formula with the universal exponents yields excellent agreement with the Monte Carlo simulation data and thus, it also accounts for the non-monotonic TTT diagrams produced by the model. Both the model and theory can be used to understand structural ordering in various glassy systems including bulk metallic glass alloys, organic molecules, and colloidal suspensions.

Figures (19)

• Figure 1: TTT diagram for $\mathrm{Zr}_{41.2}\mathrm{Ti}_{13.8}\mathrm{Cu}_{12.5}\mathrm{Ni}_{10.0}\mathrm{Be}_{22.5}$ showcasing non-monotonic and cross-over behavior. The crystallization time scale $t_\mathrm{x}$ is measured from the time in which the release of enthalpic heat is first observed. The liquidus temperature is $1024\,$K. Adapted from Ref. Masuhr1999.
• Figure 2: An illustration of the coarse-graining concept leading to the Arrow-Potts model. A supercooled liquid consists of regions with low/high mobility and crystalline order, which are identified on the first layer. The grey and white colors indicate mobile liquid and immobile liquid regions, respectively, while other colors indicate crystal clusters in different orientations. On the second layer, we associate each mobile region with its direction of motion, which dictates its direction of facilitation. On the third layer, we finally obtain a coarse-grained lattice model, which consists of immobile/mobile liquid states and crystal states with arrows representing direction of facilitation.
• Figure 3: A schematic of facilitated dynamics. (Top) Motion leads to more motion, and so the facilitation arrow indicates where the next set of motion is going to occur. (Bottom) An illustration of possible moves for mobile-immobile liquid transitions elucidating the kinetic constraint on a lattice. Dashed circle and $\times$ indicate sites where a transition is allowed or forbidden, respectively.
• Figure 4: An illustration of the emergent relaxation behavior due to facilitation mechanism. The system relaxes when a mobile state (labeled as ②) changes its spin using another mobile state (labeled as ①) $\ell_\mathrm{eq}$ away. The minimal energy pathway corresponds to a chain of excitations of length $\ell_\mathrm{eq}$, which starts from excitation ① to excitation ②. Note that the pathway may generate holes, relaxing a few excitations, to achieve a lower energy barrier at the transition state. Once this transition state is achieved, a cascade of events can be initiated, starting from the excitation pair boxed in red, where the chain retracts back to excitation ①.
• Figure 5: (Left) The decay of the persistence function $P(t)$ as a function of time, where red to blue indicates higher to lower equilibrium temperatures. (Right) Equilibrium relaxation time $\tau_\mathrm{eq}$, measured in Monte Carlo sweeps (MCS). Parameters: $J_0=0.25$ and $k_\mathrm{B}=1$.