Diffusion limited growth

To deal with this more complex problem, we follow Sekerka et al. [2] and Sekerka and Wang [3] and first establish a general analysis that allows for these changes of volume. The previous melting problem was solved by first obtaining independent solutions to the diffusion equation in each phase and then coupling them via the Stefan flux condition at the interface, similar approach can be employed for the present problem. To accomplish this, it is necessary to identify suitable frames for analyzing the diffusion in each phase and then to find the relations between them necessary to construct the Stefan condition. 

Framework for Describing the Diffusion. We again make the acceptable approximation that the atomic volume of each component within a given phase is independent of concentration. No volume changes will therefore occur within each phase as a result of diffusion within the phase. As shown in Sections 3.1.3 and 3.1.4, chemical diffusion within each phase can then be described by employing a V-frame for that 

Stefan Condition at an a/(3 Interface. Consider the interface between the moving a and (3 phases shown in Fig. 20.26. The a and 3 phases (along with their l -frames) will be bodily displaced with respect to each other, and the Stefan condition can 

2Note that the use of a y-frame for diffusion within a phase merely requires that an equation such as Eq. 3.21 is satisfied. The fact that the volume of the phase may be changing due to the gain or loss of atoms at its interfaces is irrelevant. 

There is then no overall volume change or bodily movement between phases. 

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In this case, however, if the step separation i is small, the consecutive steps are coupled via the terrace diffusion held, and the width cannot increase as fast as but increases slowly, like the edge-diffusion-limited growth, as [68]... 

Size-dependent crystal growth is Included in the model because it can be important to describe diffusion limited growth rates or crystal attrition. As discussed in (6,X), the size reduction by attrition can be modelled by an effective growth rate G (L,t) which is the difference between the kinetic growth rate G, (L,tT and an attrition rate G (L,t) ... 

This is the special form which f. in Eq. (10) must take for y. to be constant during diffusion-limited growth. It corresponds to a unimodal particle size distribution characterized by the two constants k/B and A/B. ... 

Cylinders, Ellipsoids, and Elliptical Paraboloids. The diffusion-limited growth of particles whose planar intercepts are conic sections can also be analyzed by the scaling method. For example, the scaling function appropriate for a cylinder is T] = r/y/t.4 The solution for the growth of a cylinder is obtained in Exercise 20.5. 

Figure 20.7 Edgewise diffusion-limited growth of 9-phase platelet in an a matrix (or,...

Spherical Particle during Diffusion-Limited Growth in an Isothermal Binary Solid. This problem was analyzed by Mullins and Sekerka who found expressions for the rate of growth or decay of shape perturbations to a spherical H-rich /3-phase particle of fixed composition growing in an a matrix as in Section 20.2.1 [9]. Perturbations are written in the form of spherical harmonics. Steps to solve this problem are ... 

In spite of these theoretical results, dendritic-type forms for solid-state precipitation processes are the exception rather than the rule. This may happen because the theory is for pure diffusion-limited growth. Interface-limited growth tends to be stabilizing because the composition gradient close to a growing precipitate is less steep when the reaction is partly interface-limited. Thus, G is smaller, and Eq. 20.73 shows that this is a stabilizing effect. This and several other possible explanations for the paucity of observations for unstable growth forms in solid-state... 

Metal deposition on the silicon surface may follow an instantaneous or a progressive nucleation process followed by a diffusion-limited growth of the nuclei. The growth of nuclei can be either kinetically limited, diffusion limited, or under a mixed control. The current transients measured by Oskam el at various potentials of... 

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