Crystallization front shape

Impossibility to control the crystallization front shape and bring to perfection the structure of the growing crystal ... 

It was often found that, contrary to the theoretical prediction, the value of n is noninteger (Avrami 1939). The Avrami model is based on several assumptions, such as constancy in shape of the growing crystal, constant rate of radial growth, lack of induction time, uniqueness of the nucleation mode, complete crystallinity of the sample, random distribution of nuclei, constant value of radial density, primary nucleation process (no secondary nucleation), and absence of overlap between the growing crystallization fronts. These assumptions are often not met in polymer (blend) crystallization. Also, erroneous determination of the zero time and an overestimation of the enthalpy of fusion of the polymer at a given time can lead to noninteger values for n (Grenier and Prud homme 1980). 

Although the sample shown in Fig. 11.7 was crystallized under almost the same conditions as the one in Fig. 11.6, the fact that it could grow at a higher rate (due to the higher number of available molecules the film was about 11 nm thicker the crystal front moved faster) the crystal front is more prone to become unstable. Therefore, the square-shape envelope of the crystal is not established. Nonetheless, some features hke the four-fold symmetry and the dominance of the diagonals are reproduced also in this situation. The ripples are also clearly visible. 

A) The radiation problem (Eqs. (8.6) and (8.7)) along with the heat-transfer problem in a crystal (Eqs. (8.3) and (8.4)) and convection and heat transfer in a melt are computed using the shape of the crystallization front from the previous step. 

In the case of diffuse reflection the deflection of the crystallization front toward the melt during the whole process is small and does not exceed 7 mm. These results are similar to those obtained earlier in the case of dominating rotationally driven convection [7-9]. However, they fail to reproduce the observed shapes of the crystal/melt interface in actual LTG Cz growth. Thus, in the case of a diffusely reflective crystal side surface the role of internal radiation is reduced mainly to the increase of the heat removal from the interface, while the formation of the strongly deflected interface toward the melt at the initial stage of the growth and its variations with crystal length is related directly to the specular reflection at the conical part of the crystal side surface. 

In practice, the deflections of the crystal side surface from a circular cylinder or a cone generally lead to three-dimensional distortions of the crystal shape. However, to keep the problem tractable, we have considered the radial perturbations of the crystal side surface R = Ro + sisin(27tn/l)z and z = (Ro — R)cota + L — (62/sin d sin InnlRIRo) for the cylindrical and conical parts of the crystal side surface, respectively. Here, Rq and I are the radius and length of the cyHndiical part of an ideal crystal, a is half of a cone angle, and S12 are perturbation amphtudes. We also assume a flat crystallization front. 

Connection between Transport Processes and Solid Microstructure. The formation of cellular and dendritic patterns in the microstructure of binary crystals grown by directional solidification results from interactions of the temperature and concentration fields with the shape of the melt-crystal interface. Tiller et al. (21) first described the mechanism for constitutional supercooling or the microscale instability of a planar melt-crystal interface toward the formation of cells and dendrites. They described a simple system with a constant-temperature gradient G (in Kelvins per centimeter) and a melt that moves only to account for the solidification rate Vg. If the bulk composition of solute is c0 and the solidification is at steady state, then the exponential diffusion layer forms in front of the interface. The elevated concentration (assuming k < 1) in this layer corresponds to the melt that solidifies at a lower temperature, which is given by the phase diagram (Figure 5) as... 

In region B, back reflectivity contributes to the total reflectivity, but because of macroscopic defects, crystal shape, finite spectral resolution, etc., the front- and back-face reflection components are incoherent, and the reflectivities merely add to give the observed reflectivity. In the... 

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