Nonlinear growth and diffusion

It has been shown that for a planar growth front (97,99) 

This characteristic length was originally introduced by Keith and Padden (99) in their discussion of spherulite growth and texture. The diffusion length is a measure of the distance where there is an appreciable concentration gradient. It is an important concept not only in the present context but in describing spherulite texture.  

The increasing concentration of the noncrystallizing component at the growth front with time serves to reduce the growth rate, as is observed. There are several reasons for the decrease. One is the usual decrease in growth rate with dilution of the melt as is found in blends with constant growth rates. The other, only pertinent in 

The characteristic morphology of such phase separated blends will be discussed in more detail in Volume 3. 

The conventional decrease in growth rate with time has been attributed to the rejection of the noncrystallizing component from the crystallizing species. Thus, the melt becomes richer in the noncrystallizing component with the resultant decrease in the growth rate. For a consistent interpretation of the results shown in 

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The propagation of perturbations under growth and diffusion, described by (4.6) or (4.7) is much faster at long times than the diffusive behavior (2.13). Thus, if present, it will always dominate. Of course, an unlimited exponential growth is not possible in real systems and nonlinearities should be added to (4.1). In Sect. 4.2 we will see that the linear results remain relevant in this case. We stress... 

There are a number of criticisms to this approach. First, the model is incomplete, since once growth begins it continues without limit. Nonlinear saturation and interactions with predators would be needed to stop this. The diffusion coefficient D certainly does not originate from the Brownian motion of the organisms, since this would be irrelevant to these processes above, say, on the millimeter scale. It is rather a turbulent eddy-diffusion coefficient aimed to... 

Interface instabilities, known as myelins, are an example of exotic nonequilibrium behavior present during dissolution in a number of surfactant systems. Although much is known about equilibrium phase behavior much still remains to be understood about nonequilibrium processes present in surfactant dissolution. In this chapter nucleation and growth, self and collective diffusion processes and nonlinear dynamics and instabilities observed in various polymeric systems are reviewed. These processes play an important role in our understanding of myelin instabilities. Kinetic maps and the concept of the free energy landscape provide a useful approach to rationalize some of the more complex behavior sometimes observed. 

In this overview we focus on the elastodynamical aspects of the transformation and intentionally exclude phase changes controlled by diffusion of heat or constituent. To emphasize ideas we use a one dimensional model which reduces to a nonlinear wave equation. Following Ericksen (1975) and James (1980), we interpret the behavior of transforming material as associated with the nonconvexity of elastic energy and demonstrate that a simplest initial value problem for the wave equation with a non-monotone stress-strain relation exhibits massive failure of uniqueness associated with the phenomena of nucleation and growth. 

In conclusion, we have shown that the neutral response approach can be extended to inhomogeneous, space-dependent reaction-diffusion systems. For labeled species (tracers) that have the same kinetic and transport properties as the unlabeled species, there is a linear response law even if the transport and kinetic equations of the process are nonlinear. The susceptibility function in the linear response law is given by the joint probability density of the transit time and of the displacement position vector. For illustration we considered the time and space spreading of neutral mutations in human populations and have shown that it can be viewed as a natural linear response experiment. We have shown that enhanced (hydrodynamic) transport due to population growth may exist and developed a method for evaluating the position of origin of a mutation from experimental data. 

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