Next: Layering effect
Up: Surface melting
Previous: Phenomenological thermodynamics model
Historically, the first model of surface melting in the framework of LandauGinsburg 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 wellknown 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
secondorder 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 semiinfinite
system which undergo a firstorder 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 semiinfinite 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 wellknown expression for a system with
a bulk tricritical point:

(3.6) 
with and , which leads to firstorder 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 higherorder 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. )
Figure:
Phase diagram, from ref. [28].

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. ).
Figure:
Orderparameter profile as and are approached from , from ref. [28].

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 semiinfinite state Potts model
on a lattice has been investigated by means of meanfield 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
20030115