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]. 

W. Gozdz, R. Hotyst. Triply periodic surfaces and multiply continuous structures from the Landau model of microemulsions. Phys Rev E 54 5012-5027, 1996. 

It was shown by Wilson [131] that the Kadanoff procedure, combined with the Landau model, may be used to identify the critical point, verify the scaling law and determine the critical exponents without obtaining an exact solution, or specifying the nature of fluctuations near the critical point. The Hamiltonian for a set of Ising spins is written in suitable units, as before... 

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... 

Figure 7 The fraction polymerized material fas a function of the dimensionless time Ty according the kinetic Landau model discussed in the main text, with h the nucleation rate. Shown are results valid in the limit where the nucleation reaction is rate limiting, for a quench to X/Xp = 2 where in equilibrium f— 0.5. Depolymerization is much faster than polymerization.

M. M. Menon and U. Landau, Modeling of Electrochemical Cells Including Diffusion, Migration, and Unsteady-State Effects, J. Electrochem. Soc., 134, No. 9, 2248-2253 (1987). 

The Landau model for phase transitions is typically applied in a phenomenological manner, with experimental or other data providing a means by which to scale the relative terms in the expansion and fix the parameters a, b, c, etc. The expression given in Equation (9) is usually terminated to the lowest feasible number of terms. Hence both a second-order phase transition and a tricritical transition can be described adequately by a two term expansion, the former as a 2-4 potential and the latter as a 2-6 potential, these figures referring to those exponents in Q present. 

Once convection starts in either of the two layers, it drives a fluid motion in the other. Using a certain model (the Landau model), Lienhard and Catton [299] predicted the heat transfer in the Rayleigh number range slightly greater than critical. Use of Eq. 4.78 for both layers, with the Rac the one relevant to thicker layer, is also tentatively recommended. 

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],... 

With d being different from a" for both polymers XII and XIII, the relationship between P and 0 is nonlinear. Such behavior is typical of ferroelectric liquid crystal materials with high P, and can be explained on the basis of the generalized Landau model for the free energy density. A complete treatment is available for polymers XII and XIII and the different calculated coefficients. 

Detailed stress-optical measurements have been analyzed to yield further information [4]. In Fig. 10 the birefringence (order parameter) was plotted as a function of reduced temperature for several nominal stresses <7 . These results were combined with the predictions of the Landau model and static stress-strain curves and led to a number of interesting consequences. In Fig. 11 the shift in the phase transition temperature is plotted as a function of nominal stress and shifts of up to 7.5 K were found compared to maximum displacements by electric and magnetic fields of about 5 mK in low molecular weight materials. In Fig. 12 the birefringence An is shown as a function of strain X=L/Lq at constant nominal stress f7n = 2.11xlO Nmm. A strictly... 

Quite recently electromechanical and electro-optic effects have been studied in some detail for the SmA-SmC transition in sidechain LCE [40]. The authors account for their observation using a Landau model, which contains an additional elastic energy associated with the tilt, when compared to the description of low molecular weight materials. 

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... 

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