Critical comparison of general-purpose collective variables for crystal nucleation

The nucleation of crystals is a prominent phenomenon in science and technology that still lacks a full atomic-scale understanding. Much work has been devoted to identifying order parameters able to track the process, from the inception of early nuclei to their maturing to critical size until growth of an extended crystal. We critically assess and compare two powerful distance-based collective variables, an effective entropy derived from liquid state theory and the path variable based on permutation invariant vectors using the Kob-Andersen binary mixture and a combination of enhanced-sampling techniques. Our findings reveal a comparable ability to drive nucleation when a bias potential is applied, and comparable free-energy barriers and structural features. Yet, we also found an imperfect correlation with the committor probability on the barrier top which was bypassed by changing the order parameter definition.

Figure 1

(a)–(d) Temporal evolution of the collective variables during metadynamics simulations. In panels (a) and (c) and in panels (b) and (d), the biasing is made respectively using PIV-based and entropy-based collective variables. (e), (f) Corresponding temporal evolution of the metadynamics instantaneous bias that results from successive Gaussians depositions. Each color corresponds to an independent simulation. (g) Typical images of the observed nucleation event along metadynamics trajectories using the two variables. Color coding is based on the value of the averaged Steinhardt parameters [14, 15] taken in their sixth order $\overline{q_6}$ and particles with $\overline{q_6}$ smaller than 0.25 are shown with a smaller size [see Supplemental Material [68] (SI.A therein) for more information].

Figure 2

(a), (b) Free-energy barriers obtained with umbrella sampling using (a) PIV.s and (b) $S$ as CV. The dotted lines indicate the transition-state CV values as obtained from CPA. (c), (d) Commitment probability for the two CVs (c) PIV.s and (d) $S$: the dotted lines indicate the critical value deduced from a fit of the data set. (e), (f) Commitment distribution extracted from all of the obtained transition-state configurations with (e) PIV.s (300 samples) and (f) $S$ (300 samples).

Figure 3

(a), (b) Commitment probability as a function of $N_{\mathrm{crys}}$ at fixed critical values of (c) PIV.s and (d) $S$. The black lines indicate a hyperbolic tangent fit of the whole data set. (c), (d) Commitment probability distribution obtained with (c) PIV.s and (d) $S$ when also constraining the value of $N_{\mathrm{crys}}\in [305:345]$. (e), (f) Typical aspect of the critical nucleation cluster, defined as having $P_{\mathrm{crys}}=0.5$. Color coding is based on the value of $\overline{q_6}$ and particles with $\overline{q_6}$ smaller than 0.25 are shown with a smaller size.