Minimal physical requirements for crystal growth self-poisoning

Abstract

Self-poisoning is a kinetic trap that can impair or prevent crystal growth in a wide variety of physical settings. Here we use dynamic mean-field theory and computer simulation to argue that poisoning is ubiquitous because its emergence requires only the notion that a molecule can bind in two (or more) ways to a crystal; that those ways are not energetically equivalent; and that the associated binding events occur with sufficiently unequal probability. If these conditions are met then the steady-state growth rate is in general a non-monotonic function of the thermodynamic driving force for crystal growth, which is the characteristic of poisoning. Our results also indicate that relatively small changes of system parameters could be used to induce recovery from poisoning.

FIG. 1.

Graphical construction used to determine the phase diagram of the mean-field model of growth poisoning (see Fig. 2(a)). The solutions of Eq. (17) give the solid compositions at which the growth rate vanishes. The horizontal dotted line shows a value of p/(1 − p) for which three such solutions exist; the associated values n_B, n_A and n_R lie on the “solubility,” “arrest,” and “precipitation” lines shown in Fig. 2(a).

FIG. 2.

Dynamic mean-field theory predicts that crystal growth rate is a non-monotonic function of concentration. (a) Mean-field phase diagram in the temperature (T)-concentration (C) plane derived from Equations (17) and (22). The line marked “solubility” shows the concentration at which the crystal (the “blue” solid) neither grows nor shrinks; the line marked “precipitation” is the same thing for the impure (“red”) precipitate. The line marked “arrest” shows where the growth rate of the (impure) crystal goes to zero. (b) Growth rate V and (c) crystal quality n as a function of concentration at the three temperatures indicated in the left panel (line colors correspond to arrow colors), obtained from Equations (12) and (13). At the solubility line the crystal does not grow; upon supersaturation it grows with finite speed V and becomes less pure. Consequently, its growth rate begins to decline for sufficiently large C, going to zero at the arrest line. Beyond the precipitation line the precipitate grows rapidly (see inset in (b), drawn as in the main panel but with C extended to just beyond the precipitation line). Parameters: p = 10⁻²; Δ/T ≡ (ϵ_s − ϵ_n)/T = 2/T; ϵ_n/T = 1/T.

FIG. 3.

Simulation snapshots taken after fixed long times (5 × 10⁹ MC sweeps) for a range of concentrations c (increasing from left to right) bear out the key prediction of mean-field theory: growth rate is a non-monotonic function of the driving force for crystal growth. Growth rate first increases and then decreases with concentration, because the growing structure becomes less pure (more red). The right-hand snapshot lies beyond the precipitation line, where the impure solid grows rapidly. Parameters: p = 10⁻²; ϵ_s = 3.5; ϵ_n = 1.4. From left to right, values of c are 0.008, 0.0083, 0.009 875, 0.0119, 0.014 225, 0.0149, 0.015 12.

FIG. 4.

Simulations show a non-monotonic growth rate and decline of crystal quality as the driving force for crystal growth is increased. Number of deposited layers L (a) and crystal quality n (b) after 5 × 10⁹ MC sweeps as a function of concentration c, for simulations run at various values of the nonspecific interaction parameter ϵ_n (plot legends show values of ϵ_n). The spike in growth rate at large c signals the passing of the precipitation line. Other parameters: p = 10⁻²; ϵ_s = 3.5.

FIG. 5.

Simulations satisfy detailed balance and so eventually evolve to equilibrium. Here, we show a time-ordered series of snapshots from a simulation done within the precipitation regime. Fluctuations allow the eventual emergence of the thermodynamically stable crystal structure. Parameters: p = 10⁻²; ϵ_s = 3.5; ϵ_n = 1.2; c = 0.0274.

FIG. 6.

Solubility and arrest line calculated from mean-field theory as in Fig. 2 (with no precipitation line drawn), with a second solution, the larger loop to the right, drawn for the case of diminished nonspecific binding energy ϵ_n → 3ϵ_n/4 (with Δ unchanged). This change greatly enlarges the region of phase space in which crystal growth can happen.