Free-energy landscape and nucleation pathway of polymorphic minerals from solution in a Potts lattice-gas model

Metastable minerals commonly form during reactions between water and rock. The nucleation mechanism of polymorphic phases from solution are explored here using a two-dimensional Potts model. The model system is composed of a solvent and three polymorphic solid phases. The local state and position of the solid phase are updated by Metropolis dynamics. Below the critical temperature, a large cluster of the least stable solid phase initially forms in the solution before transitioning into more-stable phases following the Ostwald step rule. The free-energy landscape as a function of the modal abundance of each solid phase clearly reveals that before cluster formation, the least stable phase has an energetic advantage because of its low interfacial energy with the solution, and after cluster formation, phase transformation occurs along the valley of the free-energy landscape, which contains several minima for the regions of three phases. Our results indicate that the solid-solid and solid-liquid interfacial energy contribute to the formation of the complex free-energy landscape and nucleation pathways following the Ostwald step rule.

Figure 1

Occurrence of silica precipitates from silica-supersaturated aqueous solutions at $430{}^{\circ }\mathrm{C}$ and 30 MPa (modified after Ref. [6]). (a) SEM image of spherical opal-A particles covered by opal-C. (b) Photomicrograph of a thin section showing quartz crystals embedded in opal-C.

Figure 2

Temperature-density $T/J\text{−}\rho$ phase diagram of the two-dimensional Potts lattice-gas model used in this study.

Figure 3

Temperature dependence of (a) the energy density $e$ and (b) the specific heat $c$ of the system with $\rho =0.09$ in equilibrium for lattices $L=16$, 32, 48, 64, and 128.

Figure 4

Temperature dependence of the equilibrium modal abundances of the solid phases (a) $\langle M_1\rangle$, (b) $\langle M_2\rangle$, and (c) $\langle M_3\rangle$ and (d) the maximum cluster size $S_{\max }$ for lattices $L=16$, 32, 48, 64, and 128. The gray dashed curves in (a)–(c) indicate the approximated modal abundances calculated by the low-density expansion.

Figure 5

Typical configuration of the system with $L=64$ and $\rho =0.09$ in equilibrium at $T/J=0.50$, 0.80, 0.92, 0.99, and 1.52.

Figure 6

Time evolution of the modal abundances of solid particles $M_1,M_2$, and $M_3$ and the maximal cluster size $S_{\max }$ with the insets showing the configuration of the system $L=64$ at $T/J=0.80$ for (a) $0\text{–}7000\mathrm{MCS}$ and (b) $\left(0\text{–}2\right)\times {10}^5\mathrm{MCS}$. Initial states and the positions of solid particles are chosen at random.

Figure 7

Time evolution of the modal abundances of solid particles $M_1,M_2$, and $M_3$ and the maximal cluster size $S_{\max }$ of the system $L=64$ at (a) $T/J=1.52 \left(0\text{–}10000\mathrm{MCS}\right)$, (b) $T/J=0.92$ [$\left(0\text{–}1.0\right)\times {10}^5\mathrm{MCS}$], and (c) $T/J=0.50$ [$\left(0\text{–}1.0\right)\times {10}^5\mathrm{MCS}$].

Figure 8

Free-energy landscape of the system with $L=64$ and $\rho =0.09$ at $T/J$ equal to (a) 3.00, (b) 1.52, (c) 0.99, (d) 0.92, (e) 0.80, and (f) 0.50 in the ternary diagram of solid phases ($s_1,s_2$, and $s_3$).

Figure 9

(a) Nucleation pathway (semitranslucent white line) of the system shown in Fig. 6 on the free-energy landscape at $T/J=0.80$. Arrows indicate the direction of the pathway. The system evolves from the intermediate modal abundances of the solid phases (point $A$) to the $s_1$-dominant (point $B$), then $s_2$-dominant local minima of $F$ (point $C$), and finally to the lowest-$F$ value at the $s_3$-dominant region (point $D$). (b) Cross section of the free-energy landscape at $T/J=0.80$ along the nucleation pathway $A\text{−}B\text{−}C\text{−}D$ (solid line) shown in Fig. 9. Here $E_{a,B\text{−}C}$ and $E_{a,C\text{−}D}$ indicate the activation energy density at $T/J=0.80$ for the forward reactions from $s_1$ to $s_2$ and $s_2$ to $s_3$, respectively; $E_{a,C\text{−}B}$ and $E_{a,D\text{−}C}$ indicate their respective reverse reactions. For comparison, a cross section of the free-energy landscape at $T/J=0.92$ [Fig. 7] along the nucleation pathway is also shown (light gray symbols). The nucleation pathway at $T/J=0.92$ is different from that at $T/J=0.80$, but it also has the local minima at the $s_1$-, $s_2$-, and $s_3$-dominant regions. Therefore, in Fig. 9 the horizontal axis (reaction coordinate) at $T/J=0.92$ is set by roughly adjusting the positions of the local minima for $s_1,s_2$, and $s_3$ to $B,C$, and $D$, respectively.

Figure 10

Free-energy landscapes at $T/J=0.80$ for smaller lattices with $\rho =0.09$: (a) $L=16$, (b) $L=32$, and (c) $L=48$.