Landau free energy model

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. 

While the existence of the hydration force is undisputed, its origin is still a matter of debate. Marcelja and Radic showed that the exponential repulsion observed experimentally can be obtained if a suitable Landau free energy density, dependent on an unknown order parameter , is associated with the correlation of the water molecules in the vicinity of the surface.11 Later, Schiby and Ruckenstein12 and Gruen and Marcelja13 presented two different models, both involving the polarization of the water molecules. 

Lynden-Bell RM (1995) Landau free energy, Landau entropy, phase transitions and limits of metastability in an analytical model with a variable number of degrees of freedom. Mol. Phys. 86 1353-1374... 

Figure 3. (a) The Landau free energy for methane confined in a model silica pore. The three minima correspond to three different phases, (b) The structure of phase B, showing that the contact layer is a fluid while the inner layers are frozen. 

To analyse the phenomenon of domain size growth in a quantitative way, let us consider a simpler physical system, a metallic alloy. There are two kinds of atoms, A and B, with volume fractions (j)A and 4>b, respectively. For the sake of simplicity, assume that the averaged volume fractions (pA) and 4>b) are equal. There exists a temperature Tc such that for T > Tc the fractions are mixed, i.e. the order parameter (p = (pA — thermodynamically stable phases, one with (p > 0 ( A-rich phase") and the other with (p < 0 ( B-rich phase"). A mathematical model of this phenomenon has been suggested by Cahn and Hilliard [25]. From the point of view of thermodynamics, phase separation can be described by means of the Ginzburg-Landau free energy functional... 

Woo, C.H., Zheng, Y. Depolarization in modeling nano-scale ferroelectrics using the Landau free energy functional. Appl. Rhys. A 91, 59-63 (2007)... 

On a more phenomenological level, the relation between Ps and 0 is described by Landau theories. Since the predictions of Landau models will be used in several of the following sections, a short introduction is given here. The basic part gg of the Landau free energy g consists, for chiral and nonchiral compoimds, of a series of powers of the tilt order parameter... 

In terms of these fields, the Landau free energy of the model reads ... 

The Landau type free energy expression can be supplemented by the free volume contribution to produce a model free energy for the composite system. In general, the standard Landau free energy density for SmC -SmA transition can be described by taking the tilt angle 0 as the primary order parameter (Musevic et al. 2002) and... 

At the largest length scales, field theoretic and continuum mechanics models are able to predict the equiUbriirm stmcture of multicomponent systems and macroscopic flow response of a polymer system. However, these models do not contain molecular detail. They are based either on a phenomenological description of the free energy of the system, such as the Flory-Hu ns or Landau free energies, or on actual hydrodynamic parameters such as viscosity. 

With the fomi of free energy fiinctional prescribed in equation (A3.3.52). equation (A3.3.43) and equation (A3.3.48) respectively define the problem of kinetics in models A and B. The Langevin equation for model A is also referred to as the time-dependent Ginzburg-Landau equation (if the noise temi is ignored) the model B equation is often referred to as the Calm-Flilliard-Cook equation, and as the Calm-Flilliard equation in the absence of the noise temi. 

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] ... 

The first step in studying phenomenological theories (Ginzburg-Landau theories and membrane theories) has usually been to minimize the free energy functional of the model. Fluctuations are then included at a later stage, e.g., using Monte Carlo simulations. The latter will be discussed in Sec. V and Chapter 14. 

The basic idea of a Ginzburg-Landau theory is to describe the system by a set of spatially varying order parameter fields, typically combinations of densities. One famous example is the one-order-parameter model of Gompper and Schick [173], which uses as the only variable 0, the density difference between oil and water, distributed according to the free energy functional... 

Random interface models for ternary systems share the feature with the Widom model and the one-order-parameter Ginzburg-Landau theory (19) that the density of amphiphiles is not allowed to fluctuate independently, but is entirely determined by the distribution of oil and water. However, in contrast to the Ginzburg-Landau approach, they concentrate on the amphiphilic sheets. Self-assembly of amphiphiles into monolayers of given optimal density is premised, and the free energy of the system is reduced to effective free energies of its internal interfaces. In the same spirit, random interface models for binary systems postulate self-assembly into bilayers and intro-... 

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,... 

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