New Density Functional Approach for Solid-Liquid-Vapor Transitions in Pure Materials

A new phase field crystal (PFC) type theory is presented, which accounts for the full spectrum of solid-liquid-vapor phase transitions within the framework of a single density order parameter. Its equilibrium properties show the most quantitative features to date in PFC modeling of pure substances, and full consistency with thermodynamics in pressure-volume-temperature space is demonstrated. A method to control either the volume or the pressure of the system is also introduced. Nonequilibrium simulations show that 2- and 3-phase growth of solid, vapor, and liquid can be achieved, while our formalism also allows for a full range of pressure-induced transformations. This model opens up a new window for the study of pressure driven interactions of condensed phases with vapor, an experimentally relevant paradigm previously missing from phase field crystal theories.

Figure 1

Two-dimensional free energy landscapes (vs average density) for different effective temperatures $r$. Uniform phases: continuous lines, periodic phases: dashed. Black: $r=0.14$, blue: $r\approx 0.148$ (triple point), red: $r=0.17$. Other parameters: $a=50$, $b=-19$, $c=50$, $B_x=0.7$.

Figure 2

(a) Density-temperature-pressure and (b) pressure-temperature phase diagrams of model. Solid thick lines correspond to one-mode calculations. In (a), green: vapor, blue: liquid, red: periodic coexistence values. Dashed line is the vapor-liquid spinodal. In (b), dashed lines show metastability regions. Black dots show average coexistence density [in (a)] or pressure [in (b)] from isothermal simulations. Model parameters as in Fig. 1.

Figure 3

Three-phase dendritic growth. A solid seed (periodic regions) grows into a metastable liquid (gray uniform areas). High depletion areas nucleate vapor pockets (black regions). See [19] for an animation. (a) $t=100$, (b) $t=4177$, (c) $t=10293$, (d) $t=36797$, (e) inset of (d). Scale bar: 20 lattice units. Model parameters: $a=35$, $b=-12.01$, $c=33.5$, $B_x=0.3$, $n_0=0.125$, $N_a/(dx\sqrt{dt})=0.01$, $r=0.145$.

Figure 4

Pressure controlled vapor-liquid [(a), $r=0.155$] and vapor-solid [(b), $r=0.147$] transformations. Thick vertical black line: equilibrium condensation or deposition point. Dashed green, blue, or red lines: average density vs pressure for vapor, liquid, or periodic phases. Continuous orange lines: system under pressure that is continuously increased or decreased (arrows show direction, $N_a/(dx\sqrt{dt})=0.01$). Vertical black arrows: seeded growth of a stable phase out of a metastable phase ($N_a/(dx\sqrt{dt})=0$). Insets show snapshots of the order parameter. Parameters as in Fig. 3.