Landau-Ginzburg approach

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-... 

Summary. On the basis of phenomenological Ginzburg-Landau approach we investigate the problem of order parameter nucleation in a ferromagnetic superconductor and hybrid superconductor - ferromagnetic (S/F) systems with a domain structure in an applied external magnetic field H. We study the interplay between the superconductivity localized at the domain walls and between the domain walls and show that such interplay determines a peculiar nonlinear temperature dependence of the upper critical field. For hybrid S/F systems we also study the possible oscillatory behavior of the critical temperature TC(H) similar to the Little-Parks effect. 

An excellent account of the symmetry of superconducting states is given in the review article of Sigrist and Ueda [12]. In the first part of this contribution, this article will be followed and consequences of symmetry on the surface of cuprate materials will be derived. In the second part of the contribution, a novel approach to the study of mixed superconducting phases will be introduced and its connection to the Ginzburg-Landau approach briefly discussed. 

An alternative method is to start directly from the Ginzburg-Landau approach [12]. In this method, one introduces (in general complex) order parameters as in Table 8, and expands the free energy in powers of the order parameters. For real order parameters and up to quartic terms one obtains... 

Table IV 1 8. Order parameters in the Ginzburg-Landau approach...

Fig. 3. Computer simulation results using a time-dependent Ginzburg-Landau approach, showing the microstructural evolution after a temperature jump from the lamellar phase to the hexagonal cylinder phase for a moderately asymmetric diblock copolymer. The time units are arbitrary. (Reprinted with permission from Polymer 39, S. Y. Qi and Z.-G. Zheng, Weakly segregated block copolymers Anisotropic fluctuations and kinetics of order-order and order-disorder transitions, 4639-4648, copyright 1998, with permission of Excerpta Medica Inc.)...

Long Range Order in Solids by R. M. White and T. H. Geballe, Academic Press, New York New York, 1979. This book offers a number of perspectives on the generic role of Ginzburg-Landau approaches in condensed matter physics. Chapter II gives a broad-based and interesting discussion of Landau theory with an attempt to show the unity of the approach in the context of a diverse set of applications. 

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. 

A model that is somewhere in between the hard-sphere model, where all three components are described as individual particles, and the Ginzburg-Landau approach, where all components are described by density fields, is the hybrid model, where the oil and water molecules are described by density fields but the discrete nature of the surfactant molecules is retained [60]. 

It has long been known that one of the principal distinctions between the middle phase brought about by a weak amphiphile and the microemulsion brought about by a strong one is that the latter will not wet the interface between the coexisting oil and water phases whereas the former will [107]. One of the successes of the Ginzburg-Landau approach is in explaining this phenomenon in terms of the difference between the bulk structure of the microemulsion and that of an ordinary fluid [42]. This difference is manifest in the correlation function of Eq. (32), Sec. III.Cl,... 

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

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... 

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... 

Comparison of the properties of CSD thin films to the analogous bulk material properties has also received great attention because of the need for high dielectric constant materials for DRAM applications.Basceri et ai 134,135 jj yg thoroughly considered the differences between film and bulk properties from a fundamental perspective and have been able to interpret these differences in terms of stresses present in the films, compositional differences, and the impact of these characteristics on the phenomenological behavior of the material as predicted from a Landau-Ginzburg-Devonshire approach. All observed differences between film and bulk properties were explainable using this approach. 

An alternative approach to such kinematically extended continua is presented by Capriz et al. [9-11] and Svendsen [39, 40]. They introduce scalar-valued order parameters to describe the influence of the microstructure in analogy to the Ginzburg-Landau theory. 

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 ... 

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