Ginzburg-Landau functional

In order to study microemulsion and ordered phases which appear in systems containing surfactants the following Landau-Ginzburg functional was proposed [11,12] ... 

Van der Waals was first to realize that the density varies continuously across a fluid-fluid interface. The fact that interfaces vary smoothly suggests that interfacial properties can be calculated with the Landau-Ginzburg functional. The following approach is originally due to van der Waals, but was subsequently reformulated by Landau and Lifshitz, and later was rediscovered and extended by Cahn and Hilliard. ... 

The chemical potential difference in the inhomogeneous liquid mixture is the derivative of the Landau-Ginzburg functional, which in this case describes the local Gibbs energy. 

Fluctuations are aecounted for in a Lan-dau-Ginzburg expansion of a local Hamiltonian. Once again even powers of vr only are permitted. Including gradient terms and fluctuations of the nematic director 5nj =n(r)-no(no= ) yields the following Landau-Ginzburg functional ... 

With non-zero expression (Eq. (20)) is very similar to the Landau-Ginzburg functional describing the normal-superconductor transition [1, 22, 23] ... 

A modulus 6=0 corresponds to the SmA state. A Landau-Ginzburg functional similar to Eq. (20) with 5n = 0, and therefore to the superfluid-normal helium problem, can be constructed to describe the SmA-SmC transition. 

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]

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... 

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) ... 

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)... 

Extension of the classical Landau-Ginzburg expansion to incorporate nonclassical critical fluctuations and to yield detailed crossover functions were first presented by Nicoll and coworkers [313, 314] and later extended by Chen et al. [315, 316]. These extensions match Ginzburg theory to RG theory, and thus interpolate between the lower-order terms of the Wegner expansion at T -C Afa and mean-field behavior at f Nci-... 

A primary focus of our work has been to understand the ferroelectric phase transition in thin epitaxial films of PbTiOs. It is expected that epitaxial strain effects are important in such films because of the large, anisotropic strain associated with the phase transition. Figure 8.3 shows the phase diagram for PbTiOs as a function of epitaxial strain and temperature calculated using Landau-Ginzburg-Devonshire (lgd) theory [9], Here epitaxial strain is defined as the in-plane strain imposed by the substrate, experienced by the cubic (paraelectric) phase of PbTiOs. The dashed line shows that a coherent PbTiOs film on a SrTiOs substrate experiences somewhat more than 1 % compressive epitaxial strain. Such compressive strain favors the ferroelectric PbTiOs phase having the c domain orientation, i.e. with the c (polar) axis normal to the film. From Figure 8.3 one can see that the paraelectric-ferroelectric transition temperature Tc for coherently-strained PbTiOs films on SrTiOs is predicted to be elevated by 260°C above that of... 

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... 

Description of the Landau-Ginzburg model 5.2.1 Green s functions and diagrams... 

Let us note that as a consequence of the invariance of the Landau-Ginzburg Lagrangian for the transformation body Green s function is... 

This quantity u plays an essential role in the theory. In fact, eqn (12.3.23) expresses rR(8,0,8,5) as a function of the characteristic length which defines the size of the system in the critical domain. Therefore, eqn (12.3.23) is a scaling law and u a genuine physical quantity. Since the renormalized model exists even in the critical domain, as a consequence of the fact that all the divergences are eliminated by renormalization, the value of u always remains finite. Thus, whereas b aR 1,2 - oo when one reaches the critical point, the quantity u has a finite limit u. Moreover, as the Landau-Ginzburg model becomes classical above four dimensions (the mean field theory applies in this case), we expect that u = 0 for d > 4. Thus the variable u is a very good expansion parameter. 

For Landau-Ginzburg theory as for polymer theory, the space dimension d = 4 plays a pivotal role. Indeed, for d > 4, the corresponding systems have a classical behaviour, and accordingly for d > 4, the critical exponents v, y, and co have the classical values v = 1/2, y = 1, and co = 0. For d < 4, the behaviour changes and with it, the values of the critical exponents which become functions of d. These functions can be expanded, in the vicinity of d - 4, in powers of 6 = 4 — d. 

In practice, to study the properties of P(r) and especially of f(x), it is convenient to use the correspondence existing between polymer theory and field theory (see Chapter 11). The Green s function (k, — k a) of the Landau-Ginzburg model (zero component field) is connected to the partition function of an isolated chain by the Laplace transform introduced by de Gennes. 

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... 

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... 

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