Melt growth parameters

In the ideal case, the thermal conductivity of the crucible should be smaller than the thermal conductivity of the crystal and of the melt Then, all the heat coming from the melt and generated at the solid/liquid interface due to the release of latent heat would flow through the crystal, resulting in a flat or even convex interface depending on the growth parameters (compare also Eq. (5.6)). 

The approach used in these studies follows idezus from bifurcation theory. We consider the structure of solution families with a single evolving parameter with all others held fixed. The lateral size of the element of the melt/crystal interface appears 2LS one of these parameters and, in this context, the evolution of interfacial patterns are addressed for specific sizes of this element. Our approach is to examine families of cell shapes with increasing growth rate with respect to the form of the cells and to nonlinear interactions between adjacent shape families which may affect pattern formation. 

The next most importtmt parameters in Czochralski growth of crystals are the heat flow and heat losses in the system. Actually, aU of the parameters (with the possible exception of 2 and 9) are strongly ciffected by the heat flow within the crystal-pulling system. A tj pical heat-flow pattern in a Czochralski sjretem involves both the crucible and the melt. The pattern of heat-flow is important but we will not expemd upon this topic here. Let it suffice to point out that heat-flow is set up in the melt by the direction of rotation of the cr5rstal being pulled. It is also ctffected by the upper surface of the melt and how well it is thermally insulated from its surroundings. The circular heat flow pattern causes the surface to radiate heat. The crystal also absorbs heat and re-radiates it... 

They were evaluated from our analysis of the primary nucleation and lateral growth rates and that of the l dependence to the melting temperature Tm using the Gibbs-Thomson equation. Insertion of the parameters given by Eq. 20 into Eq. 6 shows that the shape of a nucleus is a long thin rectangular parallelepiped with the ratio of... 

For the solubility of TPA in prepolymer, no data are available and the polymer-solvent interaction parameter X of the Flory-Huggins relationship is not accurately known. No experimental data are available for the vapour pressures of dimer or trimer. The published values for the diffusion coefficient of EG in solid and molten PET vary by orders of magnitude. For the diffusion of water, acetaldehyde and DEG in polymer, no reliable data are available. It is not even agreed upon if the mutual diffusion coefficients depend on the polymer molecular weight or on the melt viscosity, and if they are linear or exponential functions of temperature. Molecular modelling, accompanied by the rapid growth of computer performance, will hopefully help to solve this problem in the near future. The mass-transfer mechanisms for by-products in solid PET are not established, and the dependency of the solid-state polycondensation rate on crystallinity is still a matter of assumptions. 

Even when the principles of interface reaction and diffusion are thought to be understood, the integrated results may still require major new work. For example, the growth rate of an individual crystal in an infinite melt can be predicted if parameters are known, but the growth rates of many crystals (and different minerals), i.e., the kinetics of crystallization of a magma, is not quantitatively understood. 

Figure 4-6 Interface reaction rate as a function of temperature, pressure, and composition. The vertical dashed line indicates the equilibrium condition (growth rate is zero), (a) Diopside growth and melting in its own melt as a function of temperature with the following parameters Te= 1664K at 0.1 MPa, A5m-c = 82.76J mol K , E/R —30000 K, 4 = 12.8 ms K, and AV c l. l x 10 m /mol. The dots are experimental data on diopside melting (Kuo and Kirkpatrick, 1985). (b) Diopside growth and melting in its own melt as a function of pressure at 1810 K (Tg = 1810 K at 1 GPa from the equilibrium temperature at 0.1 MPa and the Clapeyron slope for diopside). (c) Calcite growth and dissolution rate in water at 25 °C as a function of Ca " and CO concentrations.

By the above definition, b is positive for crystal dissolution, and negative for crystal growth. During convective crystal dissolution, the dissolution rate u is directly proportional to b. During diffusive crystal dissolution, the dissolution rate is proportional to parameter a, which is positively related to b. Hence, for the dissolution of a given mineral in a melt, the size of parameter b is important. The numerator of b is proportional to the degree of undersaturation. If the initial melt is saturated, b = 0 and there is no crystal dissolution or growth. The denominator characterizes the concentration difference between the crystal and the saturated... 

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