Ginzburg-Landau free energy functional

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... 

Let us apply the idea of the Cahn-Hilliard approach to a diblock copolymer, where (pA and 4>b are now the reduced local densities of monomers A and B which are chemically bonded in the diblock-copolymer hnear chain molecule. As before, we shall assume that 4>a) = a—4>b i 4>) = 0) as the order parameter. It has been shown [33]-[35] that the long-range interaction of monomers in a copolymer chain can be described by an additional nonlocal term in the Ginzburg-Landau free energy functional ... 

The optimized wavelength can be calculated from the Ginzburg-Landau free energy functional considering concentration fluctuations (Ginzburg and Landau 1950). The functional adds the interfacial free energy onto the mean-field free energy, i.e. 

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>,... 

The results embodied in (6.4) and (6.5) are obviously too formal to be directly useful, so we assume further that the probability / [t ] = n,P(o,). We can therefore take for / [o] form we have developed thus far, (3.3) as modified in Section V to include cluster surface energies. The F(v) depends only on the probability distribution P(v), the cluster size distribution C, (p), and p. Thus (6.5) can be converted into a functional integral over P(v), Q/p), and p, and F is replaced by F[P,C,p], a Landau-Ginzburg-Wilson free-energy functional... 

Here the first term suppresses local density fluctuations [73,74] and the rest is an analog of the Ginzburg-Landau free energy associated with the instantaneous tensorial fields Q and B. Phenomenological parameters V,]i and A entering this functional are normally chosen such that the thermodynamic state of interest, e.g., a biaxial-nematic mesophase, is reproduced. In this respect, mean-field estimates can help to limit the physically adequate parameter ranges. Positive isothermal compressibility, for example, requires k >V + Ji + A [64]. 

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... 

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... 

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... 

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)... 

The RPA can be improved on by the Landau-Ginzburg (LG) formalism [47] appropriate in a quasistatic regime. One introduces a complex order parameter i[f( ) (dimensions of energy) associated with Apld(jc), which can also be related to the amplitude of the lattice distortion [Eqs. (4 and 5)] qt oc e,2fc, vjf(jtj) + e 2kF i (xi). It is complex because the phase of the CDW or BOW at +2kF is independent of the one at -2kF. TTie partition function is expressed as a functional integral weighing all fluctuations in the order parameter Z = J3)i ie-p/w, where the free-energy functional is... 

In the previous section, we have seen that it cannot suffice to consider the order parameter alone. A crucial role is played by order parameter fluctuations that are intimately connected to the various singularities sketched in fig. 11. We first consider critical fluctuations in the framework of Landau s theory itself, and return to the simplest case of a scalar order parameter (j ) with no third-order term, and u > 0 [eq. (14)], but add a weak wavevector dependent field <5 H(x) = SHqexp(iq x) to the homogeneous field H. Then the problem of minimizing the free energy functional is equivalent to the task of solving the Ginzburg-Landau differential equation... 

With the form of free energy functional 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 term is ignored) the model B equation is often referred to as the Cahn-Hilliard-Cook equation, and as the Cahn-Hilliard equation in the absence of the noise term. 

Within this continuum approach Cahn and Hilliard [481 have studied the universal properties of interfaces. While their elegant scheme is applicable to arbitrary free-energy functionals with a square gradient form we illustrate it here for the important special case of the Ginzburg-Landau form. For an ideally planar interface the profile depends only on the distance z from the interfacial plane. In mean field approximation, the profile m(z) minimizes the free-energy functional (B3.6.11). This yields the Euler-Lagrange equation... 

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... 

In order to describe the diffusive dynamics of composition fluctuations in binary mixtures one can extend the time-dependent Ginzburg-Landau methods to the free energy functional of the SCF theory. The approach relies on two ingredients a free energy functional that accurately describes the chemical potential of a spatially inhomogeneous composition distribution out of equilibrium and an Onsager coefficient that relates the variation of the chemical potential to the current of the composition. 

Minimization of this free energy functional gives what is often denoted as a Ginzburg-Landau equation, using the derivative/ of the function/(p) ... 

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 — 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]

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