Crystal nucleation along an entropic pathway: Teaching liquids how to transition

We combine machine learning (ML) with Monte Carlo (MC) simulations to study the crystal nucleation process. Specifically, we use ML to infer the canonical partition function of the system over the range of densities and temperatures spanned during crystallization. We achieve this on the example of the Lennard-Jones system by training an artificial neural network using, as a reference dataset, equations of state for the Helmholtz free energy for the liquid and solid phases. The accuracy of the ML predictions is tested over a wide range of thermodynamic conditions, and results are shown to provide an accurate estimate for the canonical partition function, when compared to the results from flat-histogram simulations. Then, the ML predictions are used to calculate the entropy of the system during MC simulations in the isothermal-isobaric ensemble. This approach is shown to yield results in very good agreement with the experimental data for both the liquid and solid phases of argon. Finally, taking entropy as a reaction coordinate and using the umbrella sampling technique, we are able to determine the Gibbs free energy profile for the crystal nucleation process. In particular, we obtain a free energy barrier in very good agreement with the results from previous simulation studies. The approach developed here can be readily extended to molecular systems and complex fluids, and is especially promising for the study of entropy-driven processes.

Figure 1

Structure of the ANN used in this work for the determination of the Helmholtz free energy $A$ for a given temperature $T$ and density $\rho$, in which each neuron is labeled with a number between 1 and 5 for the first hidden layer, between 1 and 4 for the second hidden layer, and in which $b$ designates a bias node for each of the first three layers.

Figure 2

Reduced Helmholtz free energy as a function of the reduced density. Results from the JZG EOS are shown as filled circles for a reduced temperature of $T^{*}=2$ (in black), $T^{*}=3$ (in red), and $T^{*}=4$ (in green) for the liquid, while results from the VDH EOS are plotted for reduced temperatures of $T^{*}=0.75$ (in blue) and $T^{*}=1.35$ (in purple). The ML predictions for the corresponding conditions are shown as open triangles connected through dashed lines.

Figure 3

Product of the reduced excess Helmholtz free energy by $\beta =1/\left(k_{B}T\right)$ against density for the Lennard-Jones system. Same legend as in Fig. 2.

Figure 4

Logarithm of the reduced canonical partition functions $Q^{*}(N,V,T)$ against $N$, the number of LJ particles. Filled circles are the results from prior work using EWL simulations [46], for reduced temperatures $T^{*}=0.75$ (in blue), $T^{*}=1.35$ (in purple), $T^{*}=2$ (in black), $T^{*}=3$ (in red), and $T^{*}=4$ (in green). Open triangles connected through dashed lines are the ML predictions.

Figure 5

Monte Carlo NPT simulations of liquid argon at $T=95$ K and $P=5$ bar. Variation of the interaction energy (top) and of entropy (bottom) as a function of the number of MC steps. In both plots, the red line indicated the average value.

Figure 6

Monte Carlo NPT simulations of solid argon at $T=80$ K and $P=100$ bar. Same legend as in Fig. 5.

Figure 7

Variation of (a) the reduced density, (b) the reduced interaction energy, and (c) $Q_6$ as a function of the reduced entropy for the successive NPT-S simulations.

Figure 8

(a) Crystal nucleation of the Lennard-Jones system at $T^{*}=0.86$ and $P^{*}=5.68$: Gibbs free energy of nucleation as a function of the reduced entropy $S^{*}$. (b) Snapshot of a crystal nucleus (in red) of a critical size formed within a supercooled liquid (in cyan) of Lennard-Jones particles.