Minimal phase-field crystal modeling of vapor-liquid-solid coexistence and transitions

A phase-field crystal model based on the density-field approach incorporating high-order interparticle direct correlations is developed to study vapor-liquid-solid coexistence and transitions within a single continuum description. Conditions for the realization of the phase coexistence and transition sequence are systematically analyzed and shown to be satisfied by a broad range of model parameters, demonstrating the high flexibility and applicability of the model. Both temperature-density and temperature-pressure phase diagrams are identified, while structural evolution and coexistence among the three phases are examined through dynamical simulations. The model is also able to produce some temperature and pressure related material properties, including effects of thermal expansion and pressure on equilibrium lattice spacing, and temperature dependence of saturation vapor pressure. This model can be used as an effective approach for investigating a variety of material growth and deposition processes based on vapor-solid, liquid-solid, and vapor-liquid-solid growth.

Figure 1

Coexistence of vapor and liquid phases if not considering the solid state. (a) The free-energy density of the uniform phase as a function of $\overline{n}$ at different $C_0$ ($=\epsilon -1$). (b) The $C_0\text{−}\overline{n}$ phase diagram for vapor and liquid states, where the boundary of vapor-liquid coexistence is plotted as solid lines and the spinodal line is plotted as dashed lines. $D_0=0$ is set in both (a) and (b). (c) The density of vapor-liquid coexistence (${\overline{n}}_{\mathrm{coexist}}$, solid curves) or the spinodal density (${\overline{n}}_{\mathrm{spinodal}}$, dashed curves) as a function of $D_0$ at different values of $C_0$. Other parameters are $B_0=0$ and $E_0=-6$.

Figure 2

Equilibrium free-energy density of uniform (blue) and solid (red) phases as a function of $\overline{n}$ at different values of $C_0$ in one-mode approximation, where the solid phase is of (a–c) 1D stripe or (d–f) 2D hexagonal (Hex) symmetry. The parameters used are $B_0=C_6=D_0=D_{11}=E_{1122}=0, C_2=-2, C_4=-1$, and $E_0=-6$.

Figure 3

Variation of ${\overline{n}}_{\mathrm{supercool}}$ with $D_{11}$. Calculations are based on Eq. (15) shown as solid curves. The dashed lines correspond to $3C_2+2D_{11}\overline{n}=0$, while arrows point to the region of $3C_2+2D_{11}\overline{n}<0$. Regions for the solid phase are shown in shadow. The parameters used are $B_0=C_6=D_0=E_{1122}=C_0=0, C_4=-1, E_0=-6$, and (a) $C_2=-2$ and (b) $C_2=2$.

Figure 4

Equilibrium free-energy density of uniform (blue) and stripe (red) phases as a function of $\overline{n}$ in one-mode approximation, for various values of $E_{1122}$. The result of the uniform phase is not affected by $E_{1122}$, while for stripes the value of the free-energy density increases with the decrease of $E_{1122}$, i.e., $E_{1122}=0,-0.22,-0.24,-0.26,-0.28,-0.30,-0.32,-0.34,-0.36$ (red curves, from bottom to top). Other parameters are $B_0=C_6=D_0=0, C_0=0.5, C_2=2, C_4=-1, E_0=-6$, and $D_{11}=-8$.

Figure 5

The fluid-state structure factor $S\left(q\right)$ at different values of $C_6$, evaluated from the analytic result derived in Appendix pp1. The curves are plotted with four sets of parameters, including $(C_6,C_2,C_4)=(0,-10.6,-2.88), (8,3.63,18.5), (16,17.8,39.8)$, and $(32,46.3,82.5)$, with other parameters $C_0=-5.75, D_0=-9, E_0=-6, D_{11}=-34.2, E_{1122}=-52.1$, and $\overline{n}=-0.15$ remaining the same for each set. Each parameter set would lead to a state of vapor-liquid-solid coexistence under one-mode approximation of 2D hexagonal structure when ${\overline{n}}_{\mathrm{vapor}}=-2.5, {\overline{n}}_{\mathrm{liquid}}=-0.5, {\overline{n}}_{\mathrm{solid}}=0$, and $A=0.2$. As $C_6$ increases the peak position of $S\left(q\right)$ approaches the value $q=2/\sqrt{3}$ used in one-mode approximation.

Figure 6

The broad range of model parameters yielding three-phase coexistence, under the condition of fixed values of $B_0=-1.875, C_0=-5.75, D_0=-9$, and $E_0=-6$ such that vapor-liquid coexistence occurs at ${\overline{n}}_{\mathrm{vapor}}=-2.5$ and ${\overline{n}}_{\mathrm{liquid}}=-0.5$. (a) Values of solid-phase coexistence density ${\overline{n}}_{\mathrm{solid}}$ and one-mode amplitude $A$ for the 2D hexagonal phase that can lead to existence of solutions for vapor-liquid-solid or vapor-solid-liquid coexistence [across all possible combinations of $(C_2,C_4,D_{11},E_{1122})]$, as indicated by the shaded region. The results are generated for $C_6=16$, with very similar outcomes for other choices of $C_6>0$. (b) The allowed values of parameter set $(C_2,C_4,D_{11},E_{1122})$ to achieve three-phase coexistence at different ${\overline{n}}_{\mathrm{solid}}$ when $C_6=16$ and $A=0.2$. (c) The free-energy density curves of liquid (blue) and solid (red) phases corresponding to four of the parameter sets in (b) that give ${\overline{n}}_{\mathrm{solid}}=-0.25,0,0.25,0.5$, respectively. The vapor-phase free-energy density (not shown here) is minimized at $\overline{n}=-2.5$ and forms a common tangent with these liquid- and solid-phase curves. The procedure of calculations under one-mode approximation is given in Appendix pp2.

Figure 7

Free-energy density profiles of uniform (solid curves) and 2D hexagonal (dashed) phases at different temperatures $\mathrm{\Delta }T=-0.514$ (blue), $\mathrm{\Delta }T=0$ (green), and $\mathrm{\Delta }T=0.486$ (red), using model parameters listed in Table 1. Results are obtained from numerical solution of the full dynamical Eq. (18). Here all the free-energy density curves have been tilted by a factor of $-\beta \overline{n}$ with $\beta =2.596$ for a better illustration.

Figure 8

Saturation vapor pressure at vapor-liquid coexistence as a function of temperature $\mathrm{\Delta }T$. Upper red line, $B_0=-4.5-3\mathrm{\Delta }T$ as set in Table 1; bottom blue line, $B_0=0$.

Figure 9

Vapor-liquid-solid phase diagrams calculated numerically from the full PFC model, where the solid state is of the 2D hexagonal phase. (a) Temperature-density phase diagram, where the phase boundaries for vapor, liquid, and solid states are plotted in green, blue, and red, respectively; the vapor-liquid spinodal is plotted as the black dashed curve, and the linear instability of the homogeneous state is indicated by the purple dashed line. Star symbols refer to the five parameter locations used in our numerical simulations shown in Fig. 11. (b) Temperature-pressure phase diagram. The model parameters listed in Table 1 are used.

Figure 10

Thermal and pressure effects on the lattice constant. (a) Thermal expansion of equilibrium lattice spacing $a_{\mathrm{eq}}=2\pi /q_{\mathrm{eq}}$ under a constant pressure $P=3.87$. (b) The pressure-induced variation of $a_{\mathrm{eq}}$ under a constant temperature $\mathrm{\Delta }T=0$. Parameters in Table 1 for the 2D hexagonal phase are used in numerical calculations.

Figure 11

Sample results from PFC simulations at $\mathrm{\Delta }T=-0.264$, corresponding to the locations marked in the phase diagram of Fig. 9 below the triple-point temperature. They include five characteristic cases for which the initial configuration is set up as a crystalline nucleus of $\overline{n}=0.1$ and grid size $48\times 48$ embedded in a uniform phase of (a) $\overline{n}=-2.5$, (b) $\overline{n}=-2$, (c) $\overline{n}=-1$, and (d) $\overline{n}=-0.5$, or by (e) a random initial condition of $\overline{n}=-0.1$ in the whole system. Left column: Spatial structure at the steady state of phase coexistence (a–d) or at the late stage of the polycrystalline state (e). An early-stage configuration having liquid-vapor phase separation is also shown in (b) and (c) for initial $\overline{n}$ within the spinodal. The brighter (darker) regions correspond to higher (lower) values of density field $n$. Middle column: The corresponding circularly averaged structure factor, with the diffraction pattern shown as an inset. Right column: The $y$-averaged density of the final state across the grid points along the $x$ direction.

Figure 12

PFC simulation results of vapor-liquid-solid coexistence at $\mathrm{\Delta }T=0$, using the model parameters given in Table 1. (a) The initial state with half of the system in solid (with $\overline{n}=-0.012$) and the other half in a uniform state (with $\overline{n}=-1.5$). (b) The time-evolving state with vapor-liquid-solid coexistence. (c) The corresponding $y$-averaged density along the $x$ direction.