Polymorph selection and nucleation pathway in the crystallization of Hertzian spheres

The crystallization process of Hertzian spheres is studied by means of molecular dynamics simulations in an NPT ensemble where the total number of particles N, the pressure P, and the temperature T are kept constant. It has been observed that the bond orientational ordering rather than the translational ordering (density) plays a primary role. The crystal polymorphs are determined by the state points. Under the conditions of small supercooling, the system is likely to be nucleated into crystals that have a preference for the metastable bcc structure, which can be regarded as a manifestation of the Alexander-McTague mechanism. In contrast, small nuclei are found to have a preference for fcc symmetry under conditions of a high degree of supercooling. Prestructured precursors that act as seeds and wet on the nuclei during nucleation always have a high degree of bcc-like ordering, despite different state points. The results above may provide a clue to the understanding of the crystallization process in core-softened particles.

Figure 1

Probability distribution of $Q_6$ for liquid and solid particles in metastable fluid. The state point is $T=5\times {10}^{-3}$ and $P=0.25$. The black line represents the curve for solid particles and the red line represents the curve for liquid particles.

Figure 2

Structures of liquid particles that have different $Q_6$ threshold $Q_6^{\text{thr}}$ in a metastable fluid. The state point is $T=5\times {10}^{-3}$ and $P=0.25$. The arrows indicate increasing $Q_6^{\text{thr}}$. Top: $W_4$ probability distribution with different threshold values, $Q_6^{\text{thr}}=0.24$, 0.26, 0.28, 0.3, 0.31, 0.32. Middle: $W_6$ probability distribution with different threshold values, $Q_6^{\text{thr}}=0.24$, 0.26, 0.28, 0.3, 0.31, 0.32. Bottom: Dependence of the average number of fcc-like, hcp-like, and bcc-like particles on $Q_6^{\text{thr}}$.

Figure 3

Average number (left panels) and fraction (right panels) of different solid particles in the crystalline as a function of the cluster size $n$ at $T=5\times {10}^{-3}$. $\overline{f}=\overline{n}/n$, where $\overline{n}$ is the average number of solid particles for each polymorph in a crystalline. Red, green, and blue lines represent the average number or fraction of fcc, hcp, and bcc particles, respectively. (a) $P=0.25$. (b) $P=0.4$. (c) $P=0.5$.

Figure 4

Top: Evolution of the number of crystal particles for each polymorph during a typical crystallization process. Bottom: The configuration obtained when the crystallization is completed. The state point is $T=5\times {10}^{-3}$ and $P=0.25$.

Figure 5

Snapshots of a typical crystallization at the state point of $T=5\times {10}^{-3}$ and $P=0.25$. The precursors (liquid particles with $Q_6\ge 0.3$) are depicted with gray spheres. Red, green, and blue spheres represent fcc, hcp, and bcc particles, respectively. (a) $t=2\times {10}^5\delta t$. (b) $t=2.7\times {10}^5\delta t$. (c) $t=2.8\times {10}^5\delta t$. (d) $t=2.9\times {10}^5\delta t$.

Figure 6

Relation between the total number of solid particles and the average density for liquid particles with $Q_6<0.3$, precursors, and crystals, respectively. The state point is $T=5\times {10}^{-3}$ and $P=0.25$.

Figure 7

Temporal evolution of the solid bond number $\zeta$ in the $(Q_6,\rho )$ plane at the state point $T=5\times {10}^{-3}$ and $P=0.25$. $\zeta$ grows from 0 to 14, represented by the color of each symbol. Notice that a particle with $\zeta \ge 8$ is identified as a crystal.

Figure 8

Average number (left panels) and fraction (right panels) of different solid particles in the crystalline state as a function of the cluster size $n$ at $T=2.5\times {10}^{-3}$. The representation of the lines is the same as that in Fig. 3. (a) $P=0.1$. (b) $P=0.45$. (c) $P=0.5$.

Figure 9

The number of precursor particles with different $Q_6$ threshold $Q_6^{\text{thr}}$ in a metastable fluid. The supercooling temperature is $T=2.5\times {10}^{-3}$. Top: $P=0.1$. Middle: $P=0.45$. Bottom: $P=0.5$.

Figure 10

Nucleation rate calculated by the means of mean first-passage time (MFPT). Red circles, green squares, and blue diamonds represent the nucleation rate of fcc $\left(J_{\text{fcc}}\right)$, hcp $\left(J_{\text{hcp}}\right)$, and bcc $\left(J_{\text{bcc}}\right)$ particles, respectively. The inner panel displays the nucleation rate $J$ of total solid particles. (a) $T=5\times {10}^{-3}$. (b) $T=2.5\times {10}^{-3}$.