Lattice theory

The first microscopic theory of surface melting have been developed by A. Troyanov and E. Tossati [36]. For a microscopic theory, one would need for a start a simple and accurate model, capable of accounting for the bulk phase diagram including solid, liquid and vapor phases, the triple point and the critical point of any substance. Such a complete theory has not yet been developed and the very basic building block for a microscopic surface melting theory is missing.

Nevertheless A. Troyanov and E. Tossati [36] developed a theory of surface melting, based on the fact that at the price of introducing a discrete reference lattice and using a drastic simplification like the mean-field approach, the partition function, $Z_N$, of a system of particles interacting via a pairwise potential can be calculated. In order to calculate the free energy $F$ of an assembly of $N$ atoms they introduced a reference lattice. The volume $v_0$ of each cell is chosen to be so small, that possibility of multiple occupancy of a lattice cell can be neglected.

The partition function $Z_N$ is:

$$Z_N=v_0^N \sum_{\{p_i \}} \int_{v_0} .... \int_{v_0} exp\lef... ...\sum p_i p_j U(r_i,r_j) \right] dr_1dr_2 ...dr_N=e^{-\beta F}$$ (3.12)

where $U(r_i,r_j)$ is a pair-wise interaction potential of a Lennard-Jones type, $\beta=(kT)^{-1}$ is the inverse temperature, and $p_i=0,1$, i.e. it is zero if a considered lattice site is empty and unity otherwise. The summation $\sum_{\{p_i \}}$ is taken over all possible configurations of $N$ atoms on the lattice sites. The main approximation of the model is splitting of $Z_N$ onto a separate lattice sum, $Q$, and a free volume term $\Omega$, e.g. $Z_N \approx Q\Omega$. The lattice term is an Ising-like spin lattice model state sum which is evaluated using the saddle-point approximation. The second term $\Omega$ depends on cooperative motion of the atoms and it is a complicated function of their positions, which calculated using the special effective volume method [36].

In order to describe order-disorder transition two order parameters - ``crystallinity'', $c_l$, and average density, $\rho_0$, are defined and calculated for each layer of the reference lattice separately. The ``crystallinity'' in each layer $l$ is given by:

$$c_l=\frac{<p_{0,l}>-<p_{1,l}>}{<p_{0,l}>+<p_{1,l}> }$$ (3.13)

where $l$ is the layer number, $<p_{0,l}>$ is the average occupation of the first reference lattice, and $<p_{1,l}>$ is the average occupation of the second one. The free energy of the system is expressed in term of density $\rho_l$ and crystallinity $c_l$. The minimization of the free energy with respect to these variables leads to a set of nonlinear algebraic equations which is supplemented by the boundary conditions for the solid bulk. This system of a large number of coupled layer-by-layer equations ($M\approx 400$) is solved iteratively, with an initial guess for the set of $\{c_l^0,\rho_l^0 \}$. It was found that the solution of the set of those equations is unique, i.e. it does not depend on the initial guess of $\{c_l^0,\rho_l^0 \}$.

According to the calculation, surface melting takes place both on the (100) and (110) surfaces. A thin liquid-like film gradually appears at the surface region. The crystallinity inside the quasiliquid film drops rapidly to zero, and also density is jumping rather abruptly from liquid-like to solid-like (See Fig.), but this drastic change of the density is seem to be due to the mean-field approximation, e.g. due to ignoring of fluctuations. Molecular dynamics studies have shown substantially broader transition from a solid-like to liquid-like region in surface premelting.

Figure: Density profiles at the solid-to-liquid and liquid-to-gas interfaces at various temperatures $T/T_m$, from ref. [36].

The thickness of the quasiliquid film increases very fast as $T_m$ is neared. The growth behavior is depended on the range of inter-atomic interactions, and in this case of rare-gas solids the interactions are considered to be long-ranged (van der Waals potential), and therefore the power law of the thickness growth with temperature was expected $\bar{l}\sim t^{-p}$, where $\bar{l}$ is the thickness of the quasiliquid film, $t=1-T/T_m$ is reduced temperature and the exponent $p=1/3$. In the frame of this model, $\bar{l}$ is defined to be equal to the number of layers whose crystallinity is less than $1/2$ (the interface with the solid) and the density is no less than $90\%$ of the bulk liquid density (the interface with the vapor). The theory confirmed the power-law dependence of the thickness on reduced temperature (See Fig. ).

Figure: Dependence of the quasilquid layer thickness upon the reduced temperature, from ref. [36].

The surface free energy $\gamma$ decreases with temperature very fast when $T_m$ is neared (See Fig. ). The anisotropy of $\gamma$ diminishes with temperature, and both the (100) and (110) surfaces free energies merge at about $0.9T_m$.

Figure: Temperature dependence ($T/T_m$) of the surface free energy $\gamma$ of the (110) and (100) LJ faces, from ref. [36].

The overall decrease is remarkably large, about $200 \%$ form $0.5T_m$ to $T_m$, in contrast to roughening, where the same decrease of the surface free energy $\gamma$ is about $1\%$. In this model the surface transition appears to be continuous. At $T_m$, $\gamma$ has a vertical tangent and jumps to the value of the surface free energy of the liquid-vapor interface.

In order to understand the influence of the form of the inter-atomic potential on surface melting A. Troyanov and E. Tossati [36] investigated the phenomenon when the sign of the tail of the LJ potential was reversed. The physical origin of such a weak repulsive at long-range distance was not proposed and the reversion was merely considered as a tool for better understanding of the phenomenon. But it seems plausible, that many-body interactions might effectively lead to a long-range repulsion.

It turns out that a small change of the interatomic potential has a dramatic effect on surface melting. The surface starts to melt, i.e. few layers become disordered with the increase of temperature, but further increase of temperature does not increase the number of molten layers. This phenomenon is called ``blocked surface melting'' (an analog of incomplete wetting). This ``blocked surface melting'' was predicted by phenomenological theories of surface melting [37] and also was observed experimentally at the Ge(111) surface [38]. It was concluded that the mode of growth of the quasiliquid layer is extremely fragile and sensitive to the range and sign of the interatomic potential.

This theory excludes fluctuations, which may play an important rôle in surface melting. For example, both the solid-liquid and the liquid-vapor interfaces are expected to execute very close to $T_m$ a joint ``meandering'', i.e. out-of-plane fluctuations, which is typical for all surfaces about their roughening temperature. It is not clear that effect could roughening have on the results predicted by the model. Surface melting and roughening are, respectively, short-range and long-range phenomena, and need not necessarily interfere one with another. It may happen that locally, on a short-range scale, the physics is described by surface melting, while globally only roughening will matter, irrespective of whether surface melting taking place or not. If in-plane fluctuations are considered, then there is a possibility of a ``first layer melting'' [39] at temperatures of order $0.7 T_m$. This transition is acting as a kind of ``gateway'' to the subsequent development of a quasiliquid layer. The surface layers become shear unstable due to anharmonic effects and diffusion of the surface atoms sets in.