Suppression of crystalline fluctuations by competing structures in a supercooled liquid

We propose a geometrical characterization of amorphous liquid structures that suppress crystallization by competing locally with crystalline order. We introduce for this purpose the crystal affinity of a liquid, a simple measure of its propensity to accumulate local crystalline structures on cooling. This quantity is explicitly related to the high-temperature structural covariance between local fluctuations in crystal order and that of competing liquid structures: favoring a structure that, due to poor overlap properties, anticorrelates with crystalline order reduces the affinity of the liquid. Using a lattice model of a liquid, we show that this quantity successfully predicts the tendency of a liquid to either accumulate or suppress local crystalline fluctuations with increasing supercooling. We demonstrate that the crystal affinity correlates strongly with the crystal nucleation rate and the crystal-liquid interfacial free energy of the low-temperature liquid, making our theory a predictive tool to determine which amorphous structures enhance glass-forming ability.

Figure 1

(a) An illustration of the local energy landscape where the crystalline structure (XS) and a number $n$ of competitors have an energy of $-1$, while other structures have zero energy. (b) Two sets of $n=6$ competitors. (c) Average energy per site as a function of temperature during a slow annealing [parameters: system size ${30}^3$, cooling rate ${10}^5$ Monte Carlo (MC) steps per unit $T$ with lines corresponding to the legend of (d)]. The set 1 liquid crystallizes similarly to the reference liquid without competitors (lower inset). The set 2 liquid does not crystallize, but arrests into an amorphous state (upper inset shows a slice). (d) The concentration of the crystalline structure as a function of the inverse temperature for these three systems. Dotted lines indicate the leading order in $1/T$ with slope $Q$, the liquid's affinity to the crystalline structure, as predicted from Eq. (2).

Figure 2

(a) Nearby structures on the lattice overlap and share sites (blue dots), giving rise to athermal correlations. (b) Structural covariance between the crystalline structure (all up) and the 217 other structures (blue crosses) as a function of the number of spin flips required to convert this structure into the crystalline one. Black circles indicate average values. (c), (d) Two structures with five down spins, the former being an agonist [cyan square in (b)] while the latter is an antagonist (red square).

Figure 3

(a) Influence of the crystal affinity $Q$ [Eq. (3)] of liquids with $n=7$ competitors on the concentration of XS, at low temperature $T=0.6$ where nucleation is the fastest. To prevent the liquid from crystallizing, the XS was not favored in this specific plot. (b) Fastest nucleation time of the crystal as a function of $Q$ (see the Appendix for details on the methods). The red error bar indicates no nucleation in ${10}^{12}$ MC steps. The 13 systems presented here are selected to uniformly span the range of $Q$ values.

Figure 4

(a) Nucleation rates of liquids with varying crystal affinity $Q$ and number $n$ of frustrated competitors, showing that it is the value of $Q$, rather than the number of competitors, that determines the crystallization rate. Black crosses indicate ${\tau }_{\mathrm{nucl}}>{10}^{12}$ MC steps. (b)–(d) Correlation between $Q$ and the three parameters of classical nucleation theory: liquid relaxation time ${\tau }_{\alpha }$ (computed as the autocorrelation time of the energy, in MC steps per site), liquid-crystal free energy difference $\mathrm{\Delta }F$ (computed by thermodynamic integration), and liquid-crystal surface tension $\sigma$ [inferred using Eq. (5) and the measured nucleation times]. The legend indicates the Pearson $r$ coefficient and $p$ value. The only statistically significant correlation is between $Q$ and $\sigma$.

Figure 5

The covariance matrix of the FLS model at infinite temperature, showing all coefficients $C_{i,j}$ defined in Eq. (4).

Figure 6

The average covariance between any two structures as a function of their geometrical distance, complementing the plot in Fig. 2 where only the all-up structure was shown. Error bars show standard deviation. The covariances are clearly monotonically decreasing over the first seven spin flips, though with large standard deviation reflecting the importance of the geometrical distribution of the spins. The increase for distance above seven spin flips can be explained by symmetry effects (structures very far apart tend to have higher symmetry, which results in lower absolute value of the covariances as the mean concentrations are lower) and lack of data.

Figure 7

The 19 choices of local structures used to generate the liquids studied in Figs. 3 and 4, with their label as in Ref. [13] and the corresponding ground state energy per site (minus the maximum fraction of sites that can be in the corresponding structure). Each column represents the spin- and mirror-inverted variants of the same structures, which on their own have identical properties, but have different packing properties when combined with other structures. Structures 62 and 63 have an additional symmetry (respectively spin inversion, and spin inversion combined with mirror symmetry) and thus have only two variants. Structures 26–29 have four spins down; structures 43–55 have five; and structures 62–73 have six spins up and six down. Extensive details about these structures can be found in the supplementary information of Ref. [13].

Figure 8

Correlations between the liquid-crystal surface tension and bulk free energy difference are negligible within our data set.

Figure 9

Correlations between number of competing structures, $n$, and parameters of the classical nucleation theory that influence the nucleation time. The data set is the same as in Fig. 4.