Density functional theory of surface melting

The first density functional theory of surface melting was proposed by R. Ohnegson et al. [40]. In the density functional approach the central quantity is the grand canonical free energy functional $\Omega[\rho]$ of an inhomogeneous system with local density $\rho(\vec r)$, temperature $T$, and chemical potential $\mu$. The grand canonical free energy functional is given by:

$$\Omega[\rho]=F_{exc}[\rho]+\int d^3r \rho(r)\left \{V_{ext}(r)-\mu +k_BT\left(ln(\Lambda^3\rho(r))-1\right) \right\}$$ (3.14)

where $\Lambda$ denotes the thermal wavelength $\Lambda \sim \left(\frac{\hbar^2 }{mk_BT }\right)^2$, $V_{ext}(r)$ is an external potential, and $F_{exc}[\rho]=F[\rho]-F_{id}[\rho]$ is an excess free energy functional. In general, the explicit form of $F_{exc}[\rho]$ is not known and one has to rely on approximations. For example, R. Ohnegson et al. [40] used the analytical Percus-Yevick expressions [41] for the excess free energy functional of particles interacting via Lennard-Jones potential. The main problem is to find the equilibrium density $\rho_{eq}(r)$ which minimizes the grand canonical free energy functional $\delta \Omega[\rho_{eq}(r)]/\delta \rho=0$. The minimization of the free energy functional was done numerically, using the simulated quenching [40], which is similar to conjugated gradient method, but considered to be more efficient than the last one.

Surface melting and especially the onset of anisotropic surface disordering was also investigated theoretically for the first time (in the framework of density functional theory). This issue can not be addressed with a phenomenological approach where the existence of a wetting film viewed as an undercooled liquid is taken for granted, but requires a fully microscopic theory. Surface melting is visible for each orientation, with a clear anisotropy in the structure of interface (See Fig. ). The more loosely packed (110) and (100) planes of a fcc Lennard-Jones crystal are more disordered than the dense (111) plane.

Figure: Density profile $\rho(z)$ vs $z$ for a LJ system obtained from hard-sphere perturbation theory at the reduced temperature $t=1-T/T_m$, from ref. [40].

The theory predicts the logarithmic growth law for the quasiliquid film $\bar{l} \sim -ln(t)$, which is determined by a short-ranged interaction between the particles. Besides this, for the (100) surface orientation the hysteresis effect in minimizing the density functional was observed for increasing and decreasing temperatures which hints layer-by-layer growth of the quasiliquid layer via a first-order surface phase transition.