Crystal side surface

Internal radiation through the crystal depends largely on the absorption coefficient and the refraction index. The first parameter determines the radiative heat absorption and emission inside the crystal, while the second determines reflection and refraction of radiation at the crystal side surface. The absorption coefficient of a melt is generally much greater than that of a crystal. Therefore, radiation is crucial in heat removal from the melt through the crystal/melt interface RTH within the crystal can even lead to instability of the crystallization front [5]. The refraction... 

The radiation problem has been solved using a new approach already mentioned above. The following assumptions have been made the melt is opaque, the absorption coefficient of the crystal is wavelength dependent, the crystallization front is black, the crucible waU is diffusely reflective, while the crystal side surface and the free melt surface can be either diffusely or specularly reflective. Scattering, as a rule, was neglected. 

Simulation of heat transfer was performed for an experimental setup [14] that allows growing crystals up to 80 mm in diameter. A schematic diagram of the setup is presented in Fig. 8.1. The temperature distribution along the cmcible wall was given and the meniscus shape near the triple point was neglected, the crystal cone angle was equal to 45. The crystal side surface and free melt surface could be either diffusely or specularly reflective. 

The diffuse reflection coefficient of the crystal side surface was taken equal to = 0.18 for radiation coming from the gas and = 0.823 for radiation coming from the crystal. The first value was obtained by averaging of the Fresnel reflection coefficient over an incident angle, while the second follows from relation between and p, Eq. (8.8). The specular reflection coefficient was calculated by the Fresnel formulae. The reflection coefficient of a free surface of a melt is unknown at present and was taken equal to p both for diffuse and specular reflection that can be justified by a small difference between refraction index of solid and hquid phases. The latter, along with opacity of the melt, permits consideration of the... 

Evolution of the Crystal/Melt Interface Figure 8.4 shows the evolution of the crystal/melt interface and the temperature field in the crystal as it is pulled for diffuse and specular crystal side surfaces. 

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. 

Fig. 8.5 Effect of perturbation of only cylindrical (left) or only conical (right) parts of the crystal side surface on the radiation heat-flux distribution at the crystal/melt interface. The unperturbed part of the side...

At this point a third intermediate approach deserves mentioning. It is due to Allegra [43] who proposed that polymer crystallization is controlled by a metastable equilibrium distribution of intramolecular clusters, the so-called bundles , forming in the liquid phase. These subsequently aggregate to the side surfaces of the crystals, driven by van der Waals interactions. The lamellar thickness is determined by the average contour length of the loops within the bundles. Although the model can... 

The growth of thin lamellae takes place at their side surfaces, where polymer chains partially adsorbed to the surface are continually being taken in the basic elementary process in the conventional polymer crystallization theory is the completion of a single patch of two-dimensional lamella on the growth surface. We first consider the polymer crystallization in 2D space assuming that the whole molecule is strongly adsorbed on the growth surface (substrate), the potential on which is represented by in Eq. 4. The... 

We have so far considered the substrate to be infinitely extended both in the x- and in the y-axis directions. Actually, polymers form thin lamellae and the crystallization takes place on the narrow side surface of the lamellae. 

Fig. 18 Schematic picture of the system for simulating polymer crystallization from the dense melt. Polymer chains that should be crystallized are sandwiched between parallel side surfaces of the lamellae made of the same polymer chains. The z-axis is taken normal to the substrate, while the y-axis is along the chain direction of the substrate crystals...

Figure 36 is a three dimensional representation of the order parameter P at 350 K after 19.2 ns of simulation, where about 25% of the system has transformed into the crystalline state. The black regions near both side surfaces correspond to the crystalline domains with higher P values, while the white regions are in a completely isotropic state of P = 0. Detailed inspection of these data has shown that no appreciable order is present in the melt. A simple interface model between the crystal and the isotropic melt seems to be more plausible at least in this case of short chain Cioo-... 

TVansfer the sin e crystal quickly to the measurement cell while holding a droplet of quenching water on the electrode surface. The single crystal electrodes were lifted to make meniscus, which prevented the side surface of the electrode from touching the electrolyte. 

Fig. 5 Chain deposition on the side surface of a polymer crystal, a and ae are side-surface and end- (fold-) surface free energies, and b is the width of the chain (after [16])...

Fig. 33 Determination of side-surface free energyfrom chainlength dependence of extended-chain crystal growth rates G, (left) and Ga (right) from the melt using Eq. 5 [30]...

Fig. 16. Surface termination of the rutile TiO2(001) surface. Only one possibility exists to cut a Ti02 crystal in this direction, see the side view at the left side. Surface Ti atoms are 4-fold coordinated and surface O atoms 2-fold coordinated.

Fig. 24. SEMs of the side surfaces of A508C1.2 CERT specimens tested in high-purity water containing 8 ppm of oxygen at 250 °C. (A) As-tested surface appearance (B) high magnification of the cone-shaped hematite in (A) (C) hematite crystal straddling crack in steel (D) after descaling of (A) in Clark s solution (E) the depression observed after 12.5 h (F) the depression across a crack after 28 h [58], The bars indicate the scde (pm). Reproduction firam Corrosion J. 38, 136 (1982) by permission of the Editor.

Crystallization theory and equilibrium thermodynamics of mixed alkanes have been discussed by Lauritzen, Passaglia, and DiMarzio173 (LPD) and Asbach and Kilian.174 In the LPD theory, the excess end-surface energy comes from exposed side surface due to nonuniformity in chain length. In contrast, Asbach et al. treat the enthalpy of mixing in more general terms. LPD theory has been applied to melting and the solid-state transition of binary mixtures of /2-alkanes.175... 

According to the secondary nucleation theory, polymer crystals grow by deposition of layers of stems on the side surface of the lamella183 (see Figure 33). Each new layer needs to be nucleated ( secondary or surface nucleation), after which it spreads comparatively rapidly. The main barrier to secondary nucleation is the excess side surface free energy 2blo... 

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