TDGL theory

Using field-based models, it is more difficult to provide information about the chain conformation on the surface however, attempts have been made to understand phase separation and mechanical properties of composites. Shou et al. combined SCF/DFT techniques with lattice spring model (LSM) to study the effects of the spatial distribution and aspect ratio of particles (rods and spheres) on the mechanical properties of the composite. Buxton and Balazs combined TDGL theory for polymer blends with BD for nanorods in the simulations of nanocomposites. i A percolating network of nanorods was identified in the minority phase of a bicontinuous structure. Clancy and Gates developed a hybrid model for CNTs in a bulk poly(ethylene vinyl acetate) matrix. Molecular structures of... 

Dynamical Self-Organization. When the parameter X passes slowly through X (l),the bifurcation picture of the previous section accurateiy describes the system. However, in Fucus, and probably in many other examples, this time scale separation between the characteristic time on which X varies and the time to obtain the patterned state does not hold. Thus a dynamical theory allowing for the interplay of these two time scales is required to characterize the developmental scenario. A natural formalism to describe this process is that of time dependent Ginzburg-Landau (tdgl) equations used successfully in other contexts of nonequilibrium phase transitions (27). 

TDGL Formalism. Our object is to obtain a dynamical equation describing the onset of a pattern. The TDGL method has two main features it (l) identifies a few variables that specify the pattern and (2) develops relatively simple equations for these variables in the limit where the system is not too far from the point of marginal stability. We shall develop these ideas specifically in terms of the Fucus-type theory of the previous section and in particular near conditions wherein the X = 1 disturbance just becomes unstable. 

Figure 3b. Dynamical TDGL-type theory compared to popular bifurcation theory.

One particularly interesting application of these ideas is to the problem of double rhizoid formation in Fucus (19). Indeed do these arise directly from a l = 2 bifurcation from the spherically symmetric state or after the egg passes through the normal 1 - l pattern In this context it is useful to combine the TDGL method and the Keener theory ( o) to produce a formalism capable of describing the dynamics of the problem as we discussed in the section on dynamical self-organization. 

Typically developing systems are in a situation of dynamical patterning. A bifurcation picture is not relevant unless there is time scale separation thus a TDGL formalism is more appropriate as discussed above in dynamicsl self-organization. We are developing such a theory to study lability in Fucus-like polarization phenomena. 

Mesoscopic methods include several field-based approaches such as cell dynamical systems (CDS), mesoscale density functional theory (DFT), and self-consistent field (SCF)" theory. Most of these methods are related to the time-dependent Ginzburg-Landau equation (TDGL) ... 

The mesoscopic regime lies between discrete particles and finite element representations of a continuum. Examples of mesoscopic field-theoretic methods are complex Langevin technique (CLT), time-dependent Ginzburg-Landau (TDGL) approach, and dynamic density functional theory (DDFT) method. 

TDGL time-dependent Ginsburg-Eandau theory... 

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