Free energy Landau expansion

This fomi is called a Ginzburg-Landau expansion. The first temi f(m) corresponds to the free energy of a homogeneous (bulk-like) system and detemiines the phase behaviour. For t> 0 the fiinction/exliibits two minima at = 37. This value corresponds to the composition difference of the two coexisting phases. The second contribution specifies the cost of an inhomogeneous order parameter profile. / sets the typical length scale. 

Undoubtedly the most successful model of the nematic-smectic A phase transition is the Landau-de Gennes model [201. It is applied in the case of a second-order phase transition by combining a Landau expansion for the free energy in tenns of an order parameter for smectic layering with the elastic energy of the nematic phase [20]. It is first convenient to introduce an order parameter for the smectic stmcture, which allows both for the layer periodicity (at the first hannonic level, cf equation (C2.2A)) and the fluctuations of layer position ur [20] ... 

A first step toward quantum mechanical approximations for free energy calculations was made by Wigner and Kirkwood. A clear derivation of their method is given by Landau and Lifshitz [43]. They employ a plane-wave expansion to compute approximate canonical partition functions which then generate free energy models. The method produces an expansion of the free energy in powers of h. Here we just quote several of the results of their derivation. 

The simplification, which enables reducing expression (Eq. 42) into (Eq. 58) remains in force in considering the generating function of the arbitrary chemical correlator. This means that in order to use expression (Eq. 54) for the calculation of the dependence of the vertices of the Landau free energy expansion on wave vectors at region Qs <[Pg.164]

Such continuous phase transitions are conveniently described in a phenomenological Landau free-energy expansion of the order parameter. Since we... 

H2SQ reported in the earlier NMR studies [17-19], it was clear that the plot in Fig. 8 resembles that of an order parameter. The plot of Fig. 8 was therefore fitted to the Landau free energy expansion ... 

Whether the phase transition is first- or second-order depends on the relative magnitudes of the coefficients in the Landau expansion, Eq. 17.2. For a first-order transition, the free energy has a discontinuity in its first derivative, as at the temperature Tm in Fig. 17.1a, and higher-order derivative quantities, such as heat capacity, are unbounded. In second-order transitions, the discontinuity occurs in the second-order derivatives of the free energy, while first derivatives such as entropy and volume are continuous at the transition. 

Order parameters may also refer to underlying atomic structure or symmetry. For example, a piezoelectric material cannot have a symmetry that includes an inversion center. To model piezoelectric phase transitions, an order parameter, r], could be associated with the displacement of an atom in a fixed direction away from a crystalline inversion center. Below the transition temperature Tc, the molar Gibbs free energy of a crystal can be modeled as a Landau expansion in even powers of r (because negative and positive displacements, 77, must have the same contribution to molar energy) with coefficients that are functions of fixed temperature and pressure,... 

An important step in developing the mean-field concept was done by Landau [8, 10]. Without discussing the relation between such fundamental quantities as disorder-order transitions and symmetry lowering, we just want to note here that his theory is based on thermodynamics and the derivation of the temperature dependence of the order parameter via the thermodynamic potential minimization (e.g., the free energy A(r),T)) which is a function of the order parameter. It is assumed that the function A(rj,T) is analytical in the parameter 77 and thus near the phase transition point could be expanded into the series in 77 usually it is a polynomial expansion with temperature-dependent coefficients. Despite the fact that such a thermodynamical approach differs from the original molecular field theory, they are quite similar conceptually. In particular, the r.h.s. of the equation of state for the pressure of gases or liquids and the external field in ferromagnetics, respectively, have the same polynomial form. 

A phenomenological treatment of the hydration repulsion, based on a Landau expansion of the free energy density, was proposed by Marcelya and Radic.9 They showed that, if the free energy density is a function of an order parameter that varies continuously from the surface, and if only the quadratic terms in this parameter and its derivative are nonnegligible, an exponential decay... 

At a simple phenomenological level, the hydration force can be described via an exponentially decreasing force, additive to (and independent of) the DLVO double layer and van der Waals forces [10]. An alternative phenomenological description is to consider the existence of an order parameter" and a Landau-like expansion of the free energy in that parameter. When only some of the expansion terms are retained in the latter expansion, both phenomenological descriptions lead to similar behaviors for the hydration forces [11]. 

At small k(long-wave approximation) in a onemode pole approximation this dependence has the form 4>(k) = [4tt/(1 — 1/e)] (l + Ak2) where A is the correlation length. This form corresponds to a gradient expansion of the Landau free energy functional... 

For the magnetic system in zero external field, the Landau expression for the free energy can be written as an expansion in

[Pg.218]

Table 1. Phase transitions in minerals for which elastic constant variations should conform to solutions of a Landau free-energy expansion (after Carpenter and Salje 1998).

Figure 4. Variation with temperature of the complete set of elastic constants for LaPsOn at the mmm < Hm transition (from Carpenter and Salje 1998, after Errandonea 1980). Solid curves are solutions derived from a Landau free energy expansion for a pseudo proper ferroelastic transition.

The Landau free energy expansion to describe the 3 a P6T12 o P3i21) transition in quartz has been constructed in the same manner to include lowest order coupling terms for all the possible strain components (Grimm and Domer 1975, Bachheimer and Dolino 1975, Banda et al. 1975, Dolino and Bachheimer 1982). It is reproduced here from Carpenter et al. (1998b) ... 

As with spontaneous strains, the elastic constants of a crystal have symmetry properties. Symmetry-adapted combinations of the elastic constants are obtained by diagonalising the elastic constant matrix for a given crystal class, and the eigenvalues are then associated with different irreducible representations of the point group. Each is then also associated with a particular symmetry-adapted strain. Manipulations of this type only need to be done once for all possible changes in crystal class and the results are available in tabulated form in the literature (e g. Table 6 of Carpenter and Salje 1998). In practice, the most important process is the derivation of the Landau free energy expansion for a... 

Figure 6. The behaviour of two order parameters relative to each other. The numbers on the graph refer to the strength of the coupling. Both order parameters have the same value for the a Landau coefficient in the expansion of the free energy. The transition temperature for Q2 is twice that of Qi.

When concentrations of substitutional atoms are high, the presumption generally has been that transition temperatures will vary in a simple linear fashion with dopant concentration. This supposition can be rationalized by a Landau-Ginzburg excess Gibbs free-energy expansion (Salje et al. 1991). A simple second order phase transition for phase A with the regular free energy expression... 

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