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# Landau model of surface premelting

Historically, the first model of surface melting in the framework of Landau-Ginsburg theory was proposed by Lipowsky [28,29,30]. The phenomenological model considered above, is a particular case of this type of models, which are more abstract and general. As it is well-known Landau theory is based on a power series expansion in the order parameter for the phase transition of interest. In the framework of the theory it is assumed that the order parameter is small'', so that only the lowest order terms required by symmetry are kept. It is most useful in the vicinity of second-order phase transitions, where the order parameter is guaranteed to be small. It, however, can be used with care to first - order transitions. Lipowsky considered a semi-infinite system which undergo a first-order phase transition at in the bulk, e.g. the bulk order parameter jumps to zero at the melting point. However, the surface order parameter may nevertheless behaves continuously like .

A -dimensional semi-infinite system with a dimensional surface is considered. The coordinate perpendicular to the surface is denoted by . As a result of the broken translation symmetry (translation invariance) at the surface, the order parameter depends on : . The Landau expansion for the free energy (per unit area) has the generic form:

 (3.5)

where corresponds to an increase of the free energy due to inhomogeneity of the order parameter, and the bulk term is given by the well-known expression for a system with a bulk tricritical point:
 (3.6)

with and , which leads to first-order bulk transition at , and to a jump of the order parameter from to zero. The additional term mimics the microscopic changes of the interaction parameters near the free surface. If is expanded in powers of of (up to second order), one obtains:
 (3.7)

where is a constant called extrapolation length, which is independent of temperature. Inclusion of the higher-order terms in this expression does not change significantly the results deduced from the theory.

By minimizing the free energy one obtains a differential equation for the order parameter:

 (3.8)

together with the implicit equation for the surface layer:
 (3.9)

where . From the last equation one finds the temperature dependence of order parameter as from below:
 (3.10)

where is inverse extrapolation length, and is a Landau coefficient at . Thus, two different types of phase transition are obtained, referred as and . (See Fig. )
At the transition (when the inverse extrapolation length is small ), the surface order parameter is discontinuous like the bulk order parameter . However, at the transition (when the inverse extrapolation length is large ) and at the the tricritical point ( ), the surface order parameter goes continuously to zero with the surface exponents (for ) and (for ). This is rather surprising since there are no corresponding bulk exponents. An additional unexpected feature, discovered by Lipowsky, is that a layer of disordered phase intervenes between the free surface and the ordered bulk as the critical point is approached from below (See Fig. ).
Hence, an interface appears at , which separates the disordered surface layer from the ordered phase in the bulk. As is approached, this interference becomes delocalized since . Within Landau theory, such a logarithmic divergence has also been found in wetting transition.

In order to decide whether the new type of continuous surface melting, i.e. the transition may take place one has to estimate the magnitude of the inverse extrapolation length . The parameter should be calculated relying on microscopic model. As a first step toward this goal, the semi-infinite state Potts model on a lattice has been investigated by means of mean-field theory. The Hamiltonian of the state Potts model is given by:

 (3.11)

here is the coupling constant for a pair of spins at the surface, and is the coupling constant for a pair of spins in the bulk, and is the number of possible spin orientations, and the sum includes the nearest neighbors. It was found that in a three dimensional Potts model with the new continuous transition occurs, provided . It seems very likely that the interaction parameters of real systems fulfill this inequality.

Next: Layering effect Up: Surface melting Previous: Phenomenological thermodynamics model
2003-01-15