Time-dependent Ginzburg-Landau model

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]. 

Abstract Dynamic response of microemulsions to shear deformation on the basis of two-order-parameter time dependent Ginzburg-Landau model is investigated by means of cell dynamical system approach. Time evolution of anisotropic factor and excess shear stress under steady shear flow is studied by changing shear rate and total amount of surfactant. As the surfactant concentration is increased. 

Abstract We numerically investigate nonlinear regimes of shear-induced phase separation in entangled polymer solutions. For the purpose a time-dependent Ginzburg-Landau model describing the two-fluid dynamics of polymer and solvent is used. A conformation tensor is introduced as a new dynamic variable to represent chain deformations. Its variations give rise to a large viscoelastic stress. Above the coexistence curve, a dynamical steady state is attained, where fluctuations are enhanced on various spatial... 

One way to sample the fluctuations in the order parameter, and thus model tiieir effect upon the phase transition, is to propose a stochastic model for the order parameter field such as the time dependent Ginzburg-Landau model A (TDGL-A) dynamics ... 

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. 

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... 

When one applies an external flow to the microemulsion system, its mechanical response is deeply affected by its internal structure. Using a single-order-parameter time-dependent Ginzburg-Landau (TDGL) model, Mundy et al. have investigated rheological properties of microemulsions theoretically [2]. In their model, the order parameter represents the concentration difference between oil and water, and the presence of surfactants is taken into account through the surface tension parameter. Their work has been extended by Patzold and Dawson, and it was shown that the microemulsions behave in an essentially non-Newtonian manner [3]. 

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 ... 

Fig. 2.53 Computer simulation results, using lime-dependent Ginzburg-Landau dynamics, of a lattice model of an asymmetric copolymer forming a hex phase subject to a step-shear along the horizontal axis (Ohta et al. 1993), The evolution of the domain pattern after the application of the step-shear is shown, (a) t = 1 (the pattern immediately after the shear is applied) (b) t = 5000 (c) t = 10000 (d) t = 15 000. The time-scale corresponds to the characteristic time for motion of an individual chain, t = R M. 

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... 

Analytic teclmiques often use a time-dependent generalization of Landau-Ginzburg ffee-energy fiinctionals. 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) = 8F/ m(r) gives rise to a current j... 

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