Predicting the outcome of the growth of binary solids far from equilibrium

The growth of multicomponent structures in simulations and experiments often results in kinetically trapped, nonequilibrium objects. In such cases we have no general theoretical framework for predicting the outcome of the growth process. Here we use computer simulations to study the growth of two-component structures within a simple lattice model. We show that kinetic trapping happens for many choices of growth rate and intercomponent interaction energies, and that qualitatively distinct kinds of kinetic trapping are found in different regions of parameter space. In a region in which the low-energy structure is an “antiferromagnet” or “checkerboard,” we show that the grown nonequilibrium structure displays a component-type stoichiometry that is different from the equilibrium one but is insensitive to growth rate and solution conditions. This robust nonequilibrium stoichiometry can be predicted via a mapping to the jammed random tiling of dimers studied by Flory, a finding that suggests a way of making defined nonequilibrium structures in experiment.

Figure 1

(a) Schematic of the lattice model and Monte Carlo protocol we use in this paper to study growth. (b) Distinct kinds of kinetic trapping can be found for different combinations of red-blue interaction energies (see definitions of $J$ and $h$ in the text). In this paper we focus on the region of phase space to the left of the dotted line, where the low-energy structure is a red-blue checkerboard. We also comment on growth at points $\blacksquare$ [13], $▴$ [8], and $⧫$ [10], considered in previous studies.

Figure 2

The outcome of (a) growth and (b) growth-and-maturation simulations for symmetric (top panel: ${\epsilon }_{\mathrm{br}}<0\equiv {\epsilon }_{\mathrm{rr}}\equiv {\epsilon }_{\mathrm{bb}}$) and asymmetric (bottom panel: ${\epsilon }_{\mathrm{br}}<0\equiv {\epsilon }_{\mathrm{rr}}\ll {\epsilon }_{\mathrm{bb}}$) interaction energy hierarchies reveals the existence of “mature” nonequilibrium structures whose stoichiometry is insensitive to growth rate [(b), top panel] and growth rate and solution stoichiometry [(b), bottom panel]. Here $f_{\mathrm{b}}$ is the fraction of colored components in the grown structures that are blue, and $\mu$ is a chemical potential: the larger is $\mu$, the more rapid is the rate of growth. Growth simulations were done using three distinct solution fractions of blue components, $f_{\mathrm{b}}^{\mathrm{s}}=$ 0.2, 0.5, and 0.8 [red (lightly shaded), black (solid), and blue (unfilled) circles, respectively]. Panels (c) and (d) show that near-equilibrium and far-from-equilibrium regimes are separated by a regime of large fluctuations of color and energy (measured using ${10}^3$ independent simulations at each value of $\mu$).

Figure 3

Fast growth followed by maturation results in “magic number” structures. (a) The blue fraction $f_{\mathrm{b}}$ for freshly grown structures varies smoothly with growth rate between equilibrium and far-from-equilibrium limits. If allowed to evolve further, structures grown at a range of rates evolve to nonequilibrium structures that possess the same magic number stoichiometry. (b) and (c) show “grown” and “mature” structures corresponding to the points indicated in the top panel [see also Figs. S4, S5, and S6 (artificially disallowing additional nucleation at high $\mu$ also resulted in a similar magic number plateau; Fig. S7 [20])]. Maturation was stopped at ${10}^4$ Monte Carlo cycles in (a); stopping the simulations between ${10}^2$ and ${10}^5$ cycles yield similar plateaus [(d); Fig. S8 20].

Figure 4

(a), (b) In $d=1$ the inherent structures of the lattice model with no white sites are, for the energetic hierarchy ${\epsilon }_{\mathrm{br}}<0\equiv {\epsilon }_{\mathrm{rr}}\ll {\epsilon }_{\mathrm{bb}}$, equivalent to those produced by RSA of dimers on a lattice. (c), (d) This equivalence does not hold in higher dimensions, but there we can nonetheless use the jamming result to approximate the inherent structure result via the graphical construction shown (see text). The resulting prediction, Eq. (1), is in reasonable accord with inherent structure results for $d\le 3$ (see Fig. 5). Because the long-time outcome of growth simulations for this energetic hierarchy are inherent structures of the lattice model with no white sites, the same magic numbers are seen in our growth simulations, i.e., in Fig. 2 (bottom) and Fig. 3. Thus, the nonequilibrium stoichiometry resulting from growth can be predicted via a mapping to a jammed system of dimers.

Figure 5

Our approximate extrapolation of Flory's dimer-packing result, Eq. (1) (red line), matches with reasonable precision the nonequilibrium magic number stochiometries seen in inherent structures of the lattice model in $d\le 3$ dimensions when component interactions satisfy the hierarchy ${\epsilon }_{\mathrm{br}}<{\epsilon }_{\mathrm{rr}}\ll {\epsilon }_{\mathrm{bb}}$. The plateaus seen in growth simulations in Fig. 2 (bottom) and Fig. 3 have numerical values similar to the point at $d=2$ here. For dimensions $d\ge 4$ the analytic and numerical results deviate.