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 $T=T^*$ in the bulk, e.g. the bulk order parameter $M_b$ jumps to zero at the melting point. However, the surface order parameter may nevertheless behaves continuously like $M_s \simeq \vert T-T^*\vert^{\beta_1}$.

A $d$-dimensional semi-infinite system with a $(d-1)$ dimensional surface is considered. The coordinate perpendicular to the surface is denoted by $z$. As a result of the broken translation symmetry (translation invariance) at the surface, the order parameter $M$ depends on $z$: $M=M(z)$. The Landau expansion for the free energy (per unit area) has the generic form:

$$f\{M \}=\int_0^\infty dz\left[\frac{1}{2}\left(\frac{dM }{dz }\right)^2+f(M)+\delta(z)f_1(M) \right]$$ (3.5)

where $(\frac{dM }{dz })^2$ corresponds to an increase of the free energy due to inhomogeneity of the order parameter, and the bulk term $f(M)$ is given by the well-known expression for a system with a bulk tricritical point:

$$f(M)=\frac{1}{2}a(T)M^2+\frac{1}{4}uM^4+\frac{1}{6}vM^6$$ (3.6)

with $v>0$ and $u<0$, which leads to first-order bulk transition at $a(T^*)=a^*=3u^2/16v$, and to a jump of the order parameter from $M_B=(3\vert u\vert/4v)^{1/2}$ to zero. The additional term $\delta(z)f_1(M)$ mimics the microscopic changes of the interaction parameters near the free surface. If $f_1(M)$ is expanded in powers of of $M$ (up to second order), one obtains:

$$f_1(M)=\frac{1}{2}a_1M^2$$ (3.7)

where $a_1>0$ 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 $\delta F/ \delta M = 0$ one obtains a differential equation for the order parameter:

$$\frac{dM}{dz}=\left[2f(M)-2f(M_B) \right]^{1/2}$$ (3.8)

together with the implicit equation for the surface layer:

$$\frac{\partial f_1(M_1)}{\partial M_1}=\left[2f(M_1)-2f(M_B) \right]^{1/2}$$ (3.9)

where $M_1 \equiv M(z=0)$. From the last equation one finds the temperature dependence of order parameter $M_1$ as $T \rightarrow T^*$ from below:

$$M_1 \simeq \left\{ \begin{array}{rcl} const,~~~~~~~~ a_1 < \... ...-T^*\vert^{1/2} ~~~~~~~ a_1 > \sqrt{a^*}\\ \end{array}\right.$$ (3.10)

where $a_1$ is inverse extrapolation length, and $a^*$ is a Landau coefficient at $T=T^*$. Thus, two different types of phase transition are obtained, referred as $O_1$ and $O_2$. (See Fig. )

Figure: Phase diagram, from ref. [28].

At the transition $O_1$ (when the inverse extrapolation length is small $a_1 < \sqrt{a^*}$ ), the surface order parameter $M_1$ is discontinuous like the bulk order parameter $M_B$. However, at the transition $O_2$ (when the inverse extrapolation length is large $a_1 > \sqrt{a^*}$ ) and at the the tricritical point $\bar{s}$ ( $a_1 = \sqrt{a^*}$ ), the surface order parameter goes continuously to zero with the surface exponents $\beta_1=1/4$ (for $O_2$) and $\beta_2=1/2$ (for $\bar{s}$). 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 $T^*$ is approached from below (See Fig. ).

Figure: Order-parameter profile $M(z)$ as $O_2$ and $\bar{s}$ are approached from $T<T^*$, from ref. [28].

Hence, an interface appears at $z=\bar{l}$, which separates the disordered surface layer from the ordered phase in the bulk. As $T^*$ is approached, this interference becomes delocalized since $\bar{l}\sim \vert ln(T^*-T)\vert$. 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 $O_2$ transition may take place one has to estimate the magnitude of the inverse extrapolation length $a_1$. The parameter $a_1$ should be calculated relying on microscopic model. As a first step toward this goal, the semi-infinite $g$ state Potts model on a lattice has been investigated by means of mean-field theory. The Hamiltonian of the $g$ state Potts model is given by:

$$H=\sum_{<ij>}^gJ\delta_{s_i,s_j}+J_1\delta_{s_i,s_j}$$ (3.11)

here $J_1$ is the coupling constant for a pair of spins at the surface, and $J$ is the coupling constant for a pair of spins in the bulk, and $g$ 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 $g=3$ the new continuous transition occurs, provided $J_1 \le 1.1 J$. It seems very likely that the interaction parameters of real systems fulfill this inequality.