Landau-Ginzburg model equation

MSI) that uses the same time-dependent Ginzburg Landau kinetic equation as CDS, but starts from (arbitrary) bead models for polymer chains. The methods have been summarized elsewhere. Examples of recent applications include LB simulations of viscoelastic effects in complex fluids under oscillatory shear,DPD simulations of microphase separation in block copoly-mers ° and mesophase formation in amphiphiles, and cell dynamics simulations applied to block copolymers under shear. - DPD is able to reproduce many features of analytical mean field theory but in addition it is possible to study effects such as hydrodynamic interactions. The use of cell dynamics simulations to model non-linear rheology (especially the effect of large amplitude oscillatory shear) in block copolymer miscrostructures is currently being investigated. ... 

To elucidate the spatiotemporal emergence of crystalline structure and liquid-hquid phase separation in these polyolefin blends, we employ the time dependent Ginzburg-Landau (TDGL) equations pertaining to the conserved concentration order parameter and the nonconserved crystal order parameter. The spatiotemporal evolution of the nonconserved order parameter i/f, known as TDGL model-A equation (31,32), may be expressed as... 

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. 

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... 

Basically, the time-dependent Ginzburg-Landau (TDGL) equation [12] relates the temporal change of a phase order parameter to a local chemical potential and a nonlocal interface gradient. With respect to a non-conserved phase field order parameter, the TDGL model A equation is customarily described as ... 

With the fomi of free energy fiinctional 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 temi is ignored) the model B equation is often referred to as the Calm-Flilliard-Cook equation, and as the Calm-Flilliard equation in the absence of the noise temi. 

In the previous sections, we briefly introduced a number of different specific models for crystal growth. In this section we will make some further simplifications to treat some generic behavior of growth problems in the simplest possible form. This usually leads to some nonlinear partial differential equations, known under names like Burgers, Kardar-Parisi-Zhang (KPZ), Kuramoto-Sivashinsky, Edwards-Anderson, complex Ginzburg-Landau equation and others. 

Abstract Pattern formation is a widespread phenomenon observed in different physical, chemical and biological systems on varions spatial scales, including the nanometer scale. In this chapter discussed are the universal features of pattern formation pattern selection, modulational instabilities, structure and dynamics of domain walls, fronts and defects, as well as non-potential effects and wavy patterns. Principal mathematical models used for the description of patterns (Swift-Hohenberg equation, Newell-Whitehead-Segel equation, Cross-Newell equation, complex Ginzburg-Landau equation) are introduced and some asymptotic methods of their analysis are presented. 

Perhaps a sign of the growing importance of this approach is a recent review article in the journal Reviews of Modern Physics under the title The world of the complex Ginzburg-Landau equation (by I. Aranson and L. Kramer), just on solutions and properties of a particular type of order parameter model. 

In these equations k, a, ct, d ( = Pu in eq 10.29), and g are system-dependent coefficients with g being related to the inverse of the Ginzburg number AIq. Slightly different versions for the crossover function R q) have also been used. In the critical limit 0 one recovers the linear-model parametric equation in Section 10.2.2 with coefficients a and k. In the classical limit q-rcc. Ad becomes an analytic function of AT and Ap. For a comparison of this phenomenological parametric crossover equation with the crossover Landau models the reader is referred to some previous publications. " " ... 

Note that if Z) is a scalar Z>o, then d is real and equal to Dq. Generally, d is positive. This is because the uniform steady solution is stable below criticality by assumption. In Appendix B, the derivation of the Ginzburg-Landau equation (in particular, the explicit calculation of Ai, d, and g) is illustrated for a hypothetical reaction-diffusion model. 

Equation (2.4.18) is not like the usual reaction-diffusion equations since the diffusion matrix has an antisymmetric part. This seemingly peculiar property is actually a general consequence of contracting the usual reaction-diffusion equations, for which the diffusion matrix may be a diagonal matrix of positive diffusion constants. On account of its sound physical basis, we shall use the Ginzburg-Landau equation in later chapters in preference to the A-co model. In particular, the existence of the Cj terms will turn out to be crucial to the destabilization of uniform oscillations (see Appendix A) and hence to the occurrence of a certain type of chemical turbulence. 

It would be instructive here to illustrate the theory presented above with a simple reaction-diffusion model. A suitable model would be the Ginzburg-Landau equation in the form of (2.4.13). As noted in Sect. 2.4, it is expressed as a two-component reaction-diffusion system, although the diffusion matrix D then involves an antisymmetric part ... 

In Sect. 3.5, we calculated explicitly and for the Ginzburg-Landau equation. It is now possible to calculate for the same model. Noting that D is independent of 0 for this particular model, and using the eigenvectors and eigenvalues obtained in Sect. 3.5, we easily get... 

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