Sphere isotropic growth

For a time-independent constant growth rate u and assuming isotropic growth (i.e., spheres), the radius of the sphere after time t will be... 

In the isotropic growth, the three motion velocity components of the interfaces are identical and thus the formed solid can have the shape of a portion of sphere (Figure 9.5). 

Ultimately, it is noted that in all the cases, except that of the isotropic growth on a sphere (also true for a cylinder) with inward development and the rate-determining step located at the internal interface, the space function of growth of a nucleus is a monotonous function of time, either constantly increasing or constantly decreasing, or constant in time if the active surface is of invariable area. 

The fact of observing small islands of B on A suggests a phenomenon of slow nucleation and isotropic growth on spherical particles, which excludes the diffusion as the rate-determining step, and thus, we adopt a mode of growth limited by a step taking place at the internal interface, with inward development. We will thus use the Mampel model for spheres. 

Isotropic growth models for which the growth is the same in all three dimensions so grains of 5 are portions of spheres on the surface A (Figure 14.7b) created from seeds at different times. 

The formation of a 3D lattice does not need any external forces. It is due to van der Waals attraction forces and to repulsive hard-sphere interactions. These forces are isotropic, and the particle arrangement is achieved by increasing the density of the pseudo-crystal, which tends to have a close-packed structure. This imposes the arrangement in a hexagonal network of the monolayer. The growth in 3D could follow either an HC or FCC struc-... 

With regard to the attachment and detachment energies, the corners of a crystal or a rough interface that is constructed by kinks alone are sites where the process proceeds most quickly, whereas the low-index crystal faces, corresponding to smooth interfaces, represent the direction with the minimum rate of normal growth and dissolution. As a result, if a single crystalline sphere is dissolved in an isotropic environmental phase, a dissolution form bounded by both flat and curved crystal faces appears. This is called the dissolution form, which is not the same as the growth form. 

Coalescence of mesophase is often said to be determined by the mesophase viscosity. This aspect requires much further investigation. However, it is clear that, amongst other factors, the rheological behaviour (including viscoelastic effects) of each phase is important in mesophase growth and coalescence. Diffusion of molecular species through the isotropic pitch to the mesophase spheres is likely to be related to the viscosity of the isotropic med i urn. 

Transmission electron microscopy of deformed SB diblocks with spherical PB domains showed clear evidence for PB sphere cavitation in the crazed zones In fact, in contr t to the thick crazes of the cylindrical morphology materials, the crazes in the spherical morphology were exceedingly thin, approaching the dimension of the PB sphere diameter. Because of the simplifications offered by the isotropic nature of the spherical morphology, it was possible to develop a precise model for the rate of growth of crazes for this system. 

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