Ginzburg-Landau model

B3.6.2.3 SELF-CONSISTENT FIELD APPROACH AND GINZBURG-LANDAU MODELS... 

As already mentioned in the Introduction, phenomenological models for amphiphilic systems can be divided into two big classes Ginzburg-Landau models and random interface models. 

Langevin simulations of time-dependent Ginzburg-Landau models have also been performed to study other dynamical aspects of amphiphilic systems [223,224]. An attractive alternative approach is that of the Lattice-Boltzmann models, which take proper account of the hydrodynamics of the system. They have been used recently to study quenches from a disordered phase in a lamellar phase [225,226]. 

Keywords superconductivity, fractal dimensions, Ginzburg — Landau model, non-berturbative approach... 

In a Ginzburg-Landau model the chemical potential p is related to the free energy functionalF[0(r,/)] by... 

Phase transitions in which the square of the soft-mode frequency or its related microscopic order parameter goes to zero continuously with temperature can be defined as second order within the framework of the Ginzburg-Landau model [110]. The behavior is obviously classical and consistent with mean field... 

In homeotropic cells, however, in-plane rotations of the director are reflected in a net azimuthal rotation of the optical axis (and the light polarization) across the cell which has allowed a detailed exploration of the characteristics of the NR-AR transition. Experiments have shown an excellent agreement with the predictions of generalized Ginzburg-Landau models [36],... 

The origin of the Ginzburg-Landau approach lies in the study of the thermal behavior near critical points, which is characterized by a set of universal critical exponents. One of the advantages of this approach is that many techniques that have been developed in this context can be applied to Ginzburg-Landau models of ternary amphiphilic systems as well. 

Ginzburg-Landau models can be derived in a straightforward way from all microscopic lattice models of microemulsions. This has been done explicitly for the Widom model [43], for the three-component model [44], for vector models [45], and for the charge-frustrated Ising model [37]. In the case of the three-component model of Eqs. (2) and (3), the derivation shows, for example, that... 

The parameter Zy in these equations is the fugacity of the amphiphiles, z, = exp( /< J. For balanced systems, z, = 2pg. With these results, the dependence of the parameters in the Ginzburg-Landau model on the experimental variables such as amphiphile concentration p and chain length / is now explicit. 

Until now, most of the work on Ginzburg-Landau models has been aimed at exploring their parameter space and obtaining an overview of their general properties. 

Ginzburg-Landau models that in addition to the scalar order parameter for the concentrations of oil, water, and amphiphile contain a vector order parameter for the amphiphile orientation have also been studied [41, 45, 49-53]. 

The series of lyotropic phases that are characteristic of the binary water-amphiphile system are well described for the most part by Ginzburg-Landau models [50,51] and polymer chain models [29] (see Fig. 2). The most complicated lyotropic phase, the cubic gyroid phase, was the last phase to be determined theoretically, and it was found first [77] in models of long ffexible diblock copolymers, as described in Sec.V.C.1. Quite recently, it has also been obtained in simulations of the short polymer chain model [78]. 

Figure 2 Phase diagram of a binary amphiphile-water mixture obtained from a Ginzburg- Landau model with a vector order parameter for the amphiphile orientation (50,51]. The phases L and L2 are micellar liquids, is a lamellar phase. H and H denote hexagonal and inverse hexagonal phases, respectively, I is an fee crystal of spherical micelles, and V is a simple cubic bicontinuous phase. (From Ref. 51.)...

The scattering intensity in bulk contrast can be calculated easily in the Ornstein-Zernike approximation for all lattice [15, 90-92] and Ginzburg-Landau models. In the limit of wave vector q < q — n/a, one obtains in all cases the Teubner-Strey form... 

One of the motivations for introducing the anharmonic Ginzburg-Landau model, Eq. (16), was that it should be possible to describe interfaces between different phases. The simplest case is the interface between an oil-rich and a water-rich phase. A mean-field calculation [42] shows that the interfacial free energy is given by... 

The dynamical behavior of Ginzburg-Landau models is described by Langevin equations. In the simplest case, the equation of motion for a conserved order parameter field 0(r, /), which now depends on time / in addition to r, reads... 

Predictions (61) and (62) have been qualitatively confirmed in experiments for a symmetrical diblock copolymer melt [147] (see Fig. 12). We show in Sec.V.B that these systems are indeed closely related to ternary amphiphilic systems, and, in the weak segregation regime, can be described by the same Ginzburg-Landau models. 

Finally, a quench into the one-phase region of the microemulsion has been investigated. An analysis based on a Ginzburg-Landau model for a single, conserved order parameter predicts [160] that the equal-time structure factor, Eq. (65), approaches its equilibrium form S(k) algebraically for long times t. 

B. Self-Consistent Field Approximation and Ginzburg-Landau Models... 

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