Field theory Landau free energy model

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... 

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)... 

As a final topic of this section, we consider the free energy of droplets in metastable phases. Metastable phases are very common in nature, and also readily predicted by approximate theories such as the Landau theory. Consider c.g. the transition of an Ising model at T < Tc as a function of magnetic field H (fig. 37). From eq. (14) one obtains... 

In this section we consider a general model that has broad applicability to phase transitions in soft materials the Landau theory, which is based on an expansion of the free energy in a power series of an order parameter. The Landau theory describes the ordering at the mesoscopic, not molecular, level. Molecular mean field theories include the Maier-Saupe model, discussed in detail in Section 5.5.2. This describes the orientation of an arbitrary molecule surrounded by all others (Fig. 1.5), which set up an average anisotropic interaction potential, which is the mean field in this case. In polymer physics, the Flory-Huggins theory is a powerful mean field model for a polymer-solvent or polymer-polymer mixture. It is outlined in Section 2.5.6. 

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