Landau-Ginzburg free energy

The Landau-Ginzburg free energy functional in the form given by Gompper and Schick is as follows ... 

The functional derivative in Eq. (60) represents deterministic relaxation of the system toward a minimum value of the free-energy functional E[< )(r, f)], which is usually taken to have the form of the coarse-grained Landau-Ginzburg free energy... 

Now — L is the Landau-Ginzburg free energy, where m2 = a(T — Tc) near the critical temperature, is a macroscopic many-particle wave function, introduced by Bardeen-Cooper-Schrieffer, according to which an attractive force between electrons is mediated by bosonic electron pairs. At low temperature these fall into the same quantum state (Bose-Einstein condensation), and because of this, a many-particle wave function (f> may be used to describe the macroscopic system. At T > Tc, m2 > 0 and the minimum free energy is at = 0. However, when T [Pg.173]

The phase behaviour of blends of homopolymers containing block copolymers is governed by a competition between macrophase separation of the homopolymer and microphase separation of the block copolymers. The former occurs at a wavenumber q = 0, whereas the latter is characterized by q + 0. The locus of critical transitions at q, the so-called X line, is divided into q = 0 and q + 0 branches by the (isotropic) Lifshitz point. The Lifshitz point can be described using a simple Landau-Ginzburg free-energy functional for a scalar order parameter rp(r), which for ternary blends containing block copolymers is the total volume fraction of, say, A monomers. The free energy density can be written (Selke 1992)... 

It is well known that there is an interesting analogy between spontaneous symmetry breaking of the vacuum and the Landau-Ginzburg free energy in superconductors. The latter is obtained from the locally invariant Lagrangian... 

Analytic techniques often use a time-dependent generalization of Landau-Ginzburg free-energy functionals. The different universal dynamic behaviours have been classified by Hohenberg and Halperin [94]. In the simple example of a binary fluid (model B) the concentration difference can be used as an order parameter m. A gradient in the local chemical potential p(r) = 5T75m(r) gives rise to a current j... 

While the theory is simple to solve for spatially uniform phases, it has been extremely difficult to obtain solutions with a given desired symmetry such as that of the hexagonally packed cylinders. What has been done in the past is to further approximate the mean-field free energy of the system by expanding it in terms of the Fourier components of the local A and B densities, (/> (k) - (f)g k) = [Pg.95]

Landau-Ginzburg free-energy density of superconductors [44]-[47] ... 

Ferroelectricity is an electrical phenomenon and also an important property in solids. It arises in certain crystals in terms of spontaneous dipole moment below Curie temperature [1], The direction of this moment can be switched between the equivalent states by the application of an external electric field [2-4], It is observed in some crystal systems that undergo second-order structural changes below the Curie temperature, which results in the development of spontaneous polarization. This can be explained by Landau-Ginzburg free energy functional [3, 4, 9]. The ferroelectric behavior is commonly explained by the presence of domains with uniform polarization. This behavior is nonlinear in terms of hysteresis of polarization (P) and electric field (E) vectors. Phenomenological models of ferroelectrics have been developed for engineering computation and for various applications. 

Now, we carry out the simulation by using the above model in two dimensions. In order to concentrate our efforts on examining the effect of dynamical asymmetry, we simplify the model assuming that F is the symmetric Ginzburg-Landau-type free energy and ta and G are... 

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. 

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. 

Within this contimiiim approach Calm and Flilliard [48] have studied the universal properties of interfaces. While their elegant scheme is applicable to arbitrary free-energy fiinctionals with a square gradient fomi we illustrate it here for the important special case of the Ginzburg-Landau fomi. For an ideally planar mterface the profile depends only on the distance z from the interfacial plane. In mean field approximation, the profile m(z) minimizes the free-energy fiinctional (B3.6.11). This yields the Euler-Lagrange equation... 

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-... 

When fluctuations are present, the system is inhomogeneous in space and (j)(r) is a function of the spatial coordinate r. Also any local gradients V(j)(r) cost in free energy. Now, the change in free energy to excite a fluctuation / = ( — c o (where cj)o is the equilibrium value) is given by the Landau-Ginzburg form... 

Ginzburg-Landau Free Energy for Inhomogeneous Gels. 75... 

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. 

Our purposes require a Ginzburg-Landau free energy for generally inhomogeneous gels [12,19-21]. First, Fmix + Fio is expressed as a functional of space-dependent variable tj>,... 

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) ... 

Here M is a mobility coefficient, which is assumed to be constant and r/(r.t) is the random thermal noise term, which for a system in equilibrium at temperature T satisfies the fluctuation-dissipation theorem. The free energy functional is taken to be of a Ginzburg-Landau form. In the notation of Qi and Wang (1996,1997) it is given by... 

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