Landau theory free energy

In fact, there are several other theories proposed to explain the mechanisms by which microbial adhesion on a surface can take place. These include, but may not be limited to, the Derjaguin Landau Verwey and Overback (DLVO) theory (which involves considering the effects of hydrophobicity and surface charge) and the theory of thermodynamics of attachment (which involves surface-free energy). These theories and their different aspects have been explained elsewhere [50] and we will not introduce those details here. 

An essential feature of mean-field theories is that the free energy is an analytical fiinction at the critical point. Landau [100] used this assumption, and the up-down symmetry of magnetic systems at zero field, to analyse their phase behaviour and detennine the mean-field critical exponents. It also suggests a way in which mean-field theory might be modified to confonn with experiment near the critical point, leading to a scaling law, first proposed by Widom [101], which has been experimentally verified. 

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-... 

The construction of the phase diagram of a heteropolymer liquid in the framework of the WSL theory is based on the procedure of minimization of the Landau free energy T presented as a truncated functional series in powers of the order parameter with components i a(r) proportional to Apa(r). The coefficients of this series, known as vertex functions, are governed by the chemical structure of heteropolymer molecules. More precisely, the values of these coefficients are entirely specified by the generating functions of the chemical correlators. Hence, before constructing the phase diagram of the specimen of a heteropolymer liquid, one is supposed to preliminarily find these statistical characteristics of the chemical structure of this specimen. Here a pronounced interplay of the statistical physics and statistical chemistry of polymers is explicitly manifested. 

To demonstrate that equation (3) describes a mean-field theory of gas-liquid transitions it will be shown how it can be obtained by minimizing a Landau free-energy function. This objective is achieved by working backwards. 

Salje (1985) interpreted overlapping (displacive plus Al-Si substitutional) phase transitions in albite in the light of Landau theory (see section 2.8.1), assigning two distinct order parameters Q n and to displacive and substitutional disorder and expanding the excess Gibbs free energy of transition in the appropriate Landau form ... 

According to the well-known Landau theory, the eigenvector of the order parameter in any second order solid-solid phase transition transforms according to an irreducible representation of the space group of the parent phase state. Furthermore, the free energy F=U -TS can be expanded around the transition temperature Tc in terms of the scalar order parameter p, which... 

To investigate these problems, we should first devise a Ginzburg-Landau free energy and then set up dynamic equations for network and solvent taking account of both nonlinear elasticity and inhomogeneous fluctuations. Therefore, the aims of this paper are firstly to introduce such a theory [19-21], secondly to review consequences of the theory obtained so far, and thirdly to give new results. Such efforts have just begun and many problems remain unsolved. 

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. 

Near the ODT, the composition profile of ordered microstructures is approximately sinusoidal (Fig. 2.1).The phase behaviour in this regime, where the blocks are weakly segregated, can then be modelled using Landau-Ginzburg theory, where the mean field free energy is expanded with reference to the average composition profile. The order parameter for A/B block copolymers may be defined as (Leibler 1980)... 

Within Landau-Ginzburg theory, the free energy functional near a second-order or weakly first-order phase transition is expanded in terms of an order parameter rj>(q) ... 

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