Nucleation Expansion Model

They also applied their technique to nanocrystalline magnets, and discussed the difference between the Nucleation Model and the Nucleation Expansion Model . Their conclusion was that the difference between the models, if it exists, corresponds to a very small energy, and has almost no meaning in real magnets. 

To model this, Duncan-Hewitt and Thompson [50] developed a four-layer model for a transverse-shear mode acoustic wave sensor with one face immersed in a liquid, comprised of a solid substrate (quartz/electrode) layer, an ordered surface-adjacent layer, a thin transition layer, and the bulk liquid layer. The ordered surface-adjacent layer was assumed to be more structured than the bulk, with a greater density and viscosity. For the transition layer, based on an expansion of the analysis of Tolstoi [3] and then Blake [12], the authors developed a model based on the nucleation of vacancies in the layer caused by shear stress in the liquid. The aim of this work was to explore the concept of graded surface and liquid properties, as well as their effect on observable boundary conditions. They calculated the hrst-order rate of deformation, as the product of the rate constant of densities and the concentration of vacancies in the liquid. 

It is observed that the normal craze fibril structure can be observed just behind the craze tip where the craze is as thin as 5—lOnm . This observation was difficult to reconcile with early models of craze tip advance which postulated that this occurred by repeated nucleation and expansion of isolated voids in advance of the tip. One problem was to explain how the void phase became interconnected while the craze was still so thin. Another was that the predicted kinetics of craze growth appeared to be incorrectly predicted indeed since this mechanism almost involves the same steps as the original craze nucleation, it is hard to understand how craze growth could be so much faster than craze nucleation as observed experimentally. 

Weber, M. Russell, L.M. Debenedetti, P.G. Mathematical modeling of nucleation and growth of particles formed by the rapid expansion of a supercritical solution under subsonic conditions. J. Supercrit. Fluids 2002, 23, 65-80. 

The mechanical interaction between the different epitaxial layers may result in the formation of misfit dislocations. Nucleation and propagation of cracks can ensue if the mismatch in thermal expansion coefficient is relatively large. The defects significantly influence the physical properties of the thin films. Examples from different material combinations and models of how to predict the numbers for critical thicknesses are provided in Section 14.4. 

The first three terms on the right follow from Helfrich s free energy expansion in the curvature [13] as discussed in Sec. Ill of this chapter and are identical to the right-hand side of Eq. (4). The last term quantifies the finite size effect as mentioned at the end of Sec. V. A. It was introduced by Fisher [35] in his treatment of condensation and is widely used in phenomenological theories of nucleation (see, e.g.. Refs. 36 and 37). r has an estimated value on the order of 1. We note that the calculation of z from a model is far from trivial see, for example. Ref 38 for a discussion of the relatively simple case of on average flat interfaces. 

The self-similar solution of an unsteady rarefaction wave in a gas-vapour mixture with condensation is investigated. If the onset of condensation occurs at the saturation point, the rarefaction wave is divided into two zones, separated by a uniform region. If condensation is delayed until a fixed critical saturation ratio Xc > 1 is reached, a condensation discontinuity of the expansion type is part of the solution. Numerical simulation, using a simple relaxation model, indicates that time has to proceed over more then two decades of characteristic times of condensation before the self-similar solution can be recognized. Experimental results on heterogeneous nucleation and condensation caused by an unsteady rarefaction wave in a mixture of water vapour, nitrogen gas and chromium-K)xide nuclei are presented. The results are fairly well described by the numerical rdaxation model. No plateau formation could be observed. 

The criteria for cavitation in polymers modified with rubbers were modeled by Lazzeri and Bucknall (Bucknall et al. 1994 Lazzeri and Bucknall 1993, 1995). They are based on energy release rate principles similar to those used in fracture mechanics. Void nucleation and expansion in elastomer particles are accompanied by the formation of a new surface, significant stretching of the surrounding layers of elastomer, and the stress relaxatimi in the adjacent matrix. All of these are driven by... 

Classical nucleation theory has been applied to model particle formation during the rapid expansion of supercritical solutions (RESS) [33, 34]. Since in most RESS applications involving small-molecule solutes the supercritical fluid does not condense (see [35] for a discussion of the polymeric case), the unary nucleation expressions provide an adequate starting point for modeling. However, fluid-phase nonideality must be taken into account [33]. In addition, little is known about the interfacial tension between solids and supercritical fluids, and so far this quantity has been used as an adjustable parameter in the calculations. This is clearly an important problem that deserves experimental and theoretical attention. 

The Avrami equationhas been extended to various crystallization models by computer simulation of the process and using a random probe to estimate the degree of overlap between adjacent crystallites. Essentially, the basic concept used was that of Evans in his use of Poisson s solution of the expansion of raindrops on the surface of a pond. Originally the model was limited to expansion of symmetrical entities, such as spheres in three dimensions, circles in two dimensions, and rods in one, for which n = 2,2, and 1, respectively. This has been verified by computer simulation of these systems. However, the method can be extended to consider other systems, more characteristic of crystallizing systems. The effect of (a) mixed nucleation, ib) volume shrinkage, (c) variable density of crystallinity without a crystallite, and (random nucleation were considered. AH these models approximated to the Avrami equation except for (c), which produced markedly fractional but different n values from 3, 2, or I. The value varied according to the time dependence chosen for the density. It was concluded that this was a powerful technique to assess viability of various models chosen to account for the observed value of the exponent, n. 

Moreover, it should be kept in mind that such a simplistic model for the nucleation of cavities would predict a simultaneous expansion of all existing cavities at the same identical applied hydrostatic stress. This is contrary to experimental results which show that cavities appear sequentially at a range of applied stresses. Furthermore, a difficult outstanding question is that of the nature... 

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