Polymorph selection during crystallization of a model colloidal fluid with a free energy landscape containing a metastable solid

The free energy landscape responsible for crystallization can be complex even for relatively simple systems like hard sphere and charged stabilized colloids. In this work, using the hard-core repulsive Yukawa model, which is known to show complex phase behavior consisting of fluid, FCC, and BCC phases, we studied the interplay between the free energy landscape and polymorph selection during crystallization. When the stability of the BCC phase with respect to the fluid phase is gradually increased by changing the temperature and pressure at a fixed fluid-FCC stability, the final phase formed by crystallization is found to undergo a switch from the FCC to the BCC phase, even though FCC remains thermodynamically the most stable phase. We further show that the nature of local bond-orientational order parameter fluctuations in the metastable fluid phase as well as the composition of the critical cluster depend delicately on the free energy landscape and play a decisive role in the polymorph selection during crystallization.

Figure 1

Computed phase diagram of the $k\sigma =5$ hard-core repulsive Yukawa system. The phase diagram consists of BCC-fluid, FCC-fluid, and FCC-BCC coexistence lines along with two triple points. The asterisks (marked with $\mathrm{I}$, $\mathrm{II}$, and $\mathrm{III}$) indicate the representative state points where we have performed simulations to explore the pathways of crystallization from the metastable fluid. As reported in Table 1, at these state points, the chemical potential difference between the fluid and the FCC solid ($\beta \mathrm{\Delta }{\mu }_{\mathrm{FCC}}$) is fixed at $-0.28$, and the stability of the BCC phase with respect to the fluid phase ($|\beta \mathrm{\Delta }{\mu }_{\mathrm{BCC}}|$) gradually increases on moving from $\mathrm{I}$ to $\mathrm{III}$.

Figure 2

(top) The variation of the fraction of BCC (blue), FCC (orange), and HCP (yellow) -like (in the grayscale: dark, intermediate, and light gray, respectively) particles ($x_{\mathrm{i}}$) with the total number of particles in the largest solidlike cluster ($n_{\mathrm{sol}}^l$) at state points I, II, and III in the phase diagram. $x_i$ is defined as $n_{\mathrm{i}}/n_{\mathrm{sol}}^l$, where $n_{\mathrm{i}}$ is the number of particles of the $i\mathrm{th}$ polymorph in the largest cluster. Note the crossover from the FCC-dominated to the BCC-dominated largest cluster. (bottom) Nucleation free energy profiles: free energy cost ($\beta \mathrm{\Delta }G$) for the formation of BCC (blue or dark gray lines), FCC (orange or light gray lines), and solidlike (dashed black lines) clusters of size $n$ is shown. Note the crossover of the nucleation free energy barriers of FCC and BCC phases on moving from I to III.

Figure 3

(top) The radial distribution functions (RDFs) of the final solids formed after crystallization from the metastable fluid phase (black lines) along with the RDFs of the equilibrated pure FCC and BCC phases (indicated by dashed orange and dash-dotted blue lines, respectively) at thermodynamic conditions of I, II, and III. $\zeta =r{\rho }^{1/3}$ is the scaled distance. Note that, at I and III, the RDFs are similar to that of the pure thermally equilibrated FCC and BCC phases, respectively. (bottom) Scatter plot of the final solid phase formed from the metastable fluid (black crosses) in the ${\overline{w}}_4\text{−}{\overline{w}}_6$ plane at I, II, and III. The blue diamonds and orange circles denote the BCC and FCC phases, respectively. The horizontal dashed black lines at ${\overline{w}}_6=0$ separate BCC-like particles from FCC- and HCP-like particles. Note the formation of FCC and HCP (${\overline{w}}_6<0$), FCC and HCP-BCC mixture, and BCC (${\overline{w}}_6>0$) solids at I, II, and III, respectively.

Figure 4

(top) The composition of the critical cluster at state points I, II, and III in the phase diagram. The blue, orange, and yellow (or, in the grayscale: dark, intermediate, and light gray, respectively) spheres denote the BCC-, FCC-, and HCP-like particles, respectively, in the cluster. (bottom) Composition profiles (number of particles of $i\mathrm{th}$ solid phase in a shell of radii $r$ and $r+\mathrm{\Delta }r$ divided by the total number of solidlike particles in that shell) for FCC (orange circles), BCC (blue diamonds), and HCP (yellow squares) -like particles in the critical cluster as a function of the distance from the center of the cluster. The dashed black lines indicate the variation of the solidlike particles as a function of distance from the center of the cluster at I, II, and III, respectively, and the positions of the vertical dotted black lines are the radii of the respective clusters. Fraction of solidlike particles at a distance $r$ from the center of the cluster is computed by taking the ratio of the number of solidlike particle and the total (both solid and fluid) particles in a shell of radii $r$ and $r+dr$.

Figure 5

(a) The probability distribution of the local ${\overline{w}}_6$ order parameter in the metastable fluid phase at I (solid orange line), II (dashed green line), and III (dash-dotted blue line) for particles with ${\overline{q}}_6>0.27$. On moving from I to III, note the gradual shift of the distribution maximum from the negative towards the positive ${\overline{w}}_6$ value, suggesting suppression of FCC-like fluctuations. (b) The conditional probability distribution of local ${\overline{w}}_4$ order parameter for particles with ${\overline{w}}_6<0$ and ${\overline{q}}_6>0.27$ in metastable fluid is shown. The asymmetry in the distributions suggests that the fluctuations are dominated by FCC-like (${\overline{w}}_4\le 0$) local structures.

Figure 6

(top) The probability distribution of the local ${\overline{w}}_6$ order parameter for the particles in the fluid phase with ${\overline{q}}_6^{\min }<{\overline{q}}_6<{\overline{q}}_6^{\max }$ where ${\overline{q}}_6^{\min }=0.27$, and ${\overline{q}}_6^{\max }=0.28$ (yellow), 0.29 (orange), 0.30 (green), 0.31 (blue), and 0.34 (black) at I, II, and III. The black arrow (or, the gradual shift from light gray to black in the grayscale) indicates the direction of the increase of ${\overline{q}}_6^{\max }$. The vertical dashed lines separate FCC- or HCP- and BCC-like local environments. The gradual shift of the distribution towards FCC- or HCP-like local structures ($w_6<0$) on increasing ${\overline{q}}_6$ is suppressed as one moves from I to III. (bottom) The conditional probability distributions of order parameter ${\overline{w}}_4$ for fluid particles with ${\overline{w}}_6\le 0$ and for the same ${\overline{q}}_6$ ranges as in the top panel.

Figure 7

The fraction of solidlike particles in the critical cluster formed via direct and indirect routes at state points I, II, and III in the phase diagram. BCC-direct (FCC-direct) denotes the BCC-like (FCC-like) particles formed directly from their fluidlike environment and BCC (FCC) -indirect denotes BCC (FCC) -like particles formed through an indirect solid-solid transition mediated route — $\mathrm{fluid}\to \mathrm{FCC}\to \mathrm{BCC}$ ($\mathrm{fluid}\to \mathrm{BCC}\to \mathrm{FCC}$). Note that at I, the FCC-direct path is the dominant contributor, and at III the BCC-direct path is the dominant contributor to the formation of solidlike particles in the critical cluster. At II we observe competitive direct appearance of FCC- and BCC-like particles from respective fluidlike environments. The transition fractions were obtained by averaging over 100 independent trajectories. The lower panel depicts a schematic representation where gray, orange, and blue spheres (marked with f, F, and B, respectively) indicate fluid, FCC-, and BCC-like particles, respectively, and the width of the arrow for each step is proportional to its weight.