Interface shape

Since taking simply ionic or van der Waals radii is too crude an approximation, one often rises basis-set-dependent ab initio atomic radii and constnicts the cavity from a set of intersecting spheres centred on the atoms [18, 19], An alternative approach, which is comparatively easy to implement, consists of rising an electrical eqnipotential surface to define the solnte-solvent interface shape [20],... 

M. C. Liang, C. W. Fan. Three-dimensional thermocapillary and buoyancy convections and interface shape in horizontal Bridgman crystal growth. J Cryst Growth 180 5% , 1997. 

The vapor pressure, density and temperature practically do not change along the evaporation region in physieally realistic systems. The latter allows one to simplify the system of governing equations and reduce the problem to a successive solution of the shortened system of equations to determine the velocity, liquid pressure and gaseous phases as well as the interface shape in a heated capillary. 

Hoffman R (1975) A study of the advancing interface. I. Interface shape in liquid gas system. J Colloid Interface Sci 50 228-241... 

A difiiculty with this mechanism is the small nucleation rate predicted (1). Surfaces of a crystal with low vapor pressure have very few clusters and two-dimensional nucleation is almost impossible. Indeed, dislocation-free crystals can often remain in a metastable equilibrium with a supersaturated vapor for long periods of time. Nucleation can be induced by resorting to a vapor with a very large supersaturation, but this often has undesirable side effects. Instabilities in the interface shape result in a degradation of the quality and uniformity of crystalline material. 

The mapping (7) introduces the unknown interface shape explicitly into the equation set and fixes the boundary shapes. The shape function h(x,t) is viewed as an auxiliary function determined by an added condition at the melt/crystal interface. The Gibbs-Thomson condition is distinguished as this condition. This approach is similar to methods used for liquid/fluid interface problems that include interfacial tension (30) and preserves the inherent accuracy of the finite element approximation to the field equation (27)... 

Steady-state solutions are found by iterative solution of the nonlinear residual equations R(a,P) = 0 using Newton s methods, as described elsewhere (28). Contributions to the Jacobian matrix are formed explicitly in terms of the finite element coefficients for the interface shape and the field variables. Special matrix software (31) is used for Gaussian elimination of the linear equation sets which result at each Newton iteration. This software accounts for the special "arrow structure of the Jacobian matrix and computes an LU-decomposition of the matrix so that qu2usi-Newton iteration schemes can be used for additional savings. 

Figure 7. Families of cellular interfaces computed for System I with k = 0.865 as a function of increasing P in a A /2 sample size. The cells are represented by the dimensionless arc length. The letters refer to sample interface shapes shown in Figure 8.

Figure 8. Sample interface shapes for System I with k = 0.865 for the parameter values marked on Figure 7.

Figure 13. Sample interface shapes from each of the families shown in Fig. 12.

Figure 17. Sample interface shapes for System III for increasing P and A = 1.0 as computed using the mixed cylindrical/cartesian representation.

For every attribute in the type specification, a function (read-only operation) is written that yields its value in any state of the implementation. This retrieval can be written in executable code for debugging or test purposes, but its execution performance is not important. The functions are private. They are useful for testing but not available to clients. interface Shape ... 

Figures 5.6a and 5.6 present the calculated and measured shape and location of the solid-gas interface at selected time instants. Due to the uneven heat flux distribution along the surface, the interface shape changes in time. For example, the rear portion of the solid is consumed faster than the front portion. There are also nonsmooth spots developed at various time instants, reflecting a locally concentrated heat flux.

A major complication in the analysis of convection and segregation in melt crystal growth is the need for simultaneous calculation of the melt-crystal interface shape with the temperature, velocity, and pressure fields. For low growth rates, for which the assumption of local thermal equilibrium is valid, the shape of the solidification interface dDbI is given by the shape of the liquidus curve Tm(c) for the binary phase diagram ... 

Figure 6. Schematic of driving forces for flows in Czochralski crystal growth system, which shows the regions where the driving forces will produce the strongest motions. The shape functions describing the unknown interface shapes are listed also.

Although the balance equations are linear, in the absence of bulk convection, the unknown shape of the melt-crystal interface and the dependence of the melting temperature on the energy and curvature of the surface make the model for microscopic interface shape rich in nonlinear structure. For a particular value of the spatial wavelength, a family of cellular interfaces evolves from the critical growth rate VC(X) when the velocity is increased. 

Figure 19 shows sample isotherms and interface shapes predicted by the QSSM for calculations with decreasing melt volume in the crucible, as occurs in the batchwise process. Because the crystal pull rate and the heater temperature are maintained at constant values for this sequence, the crystal radius varies with the varying heat transfer in the system. Two effects are noticeable. First, decreasing the volume exposes the hot crucible wall to the crystal. The crucible wall heats the crystal and causes the decrease in... 

Figure 19. Sample isotherms and interface shapes computed for the QSSM for CZ crystal growth by Atherton et al. (153). The model includes detailed radiation between the surfaces of the melt, crystal, and crucible. Isotherms are spaced at 10 K increments in the melt and 30 K increments in the other...

Figure 20. Isotherms and interface shapes for the time t = 1.0 for batchwise simulations of CZ growth. Results are shown for (a) uncontrolled, (b-d) integral control, and (e) proportional-integral control simulations. Isotherms are spaced as described for Figure 19. The figure is taken from Atherton et al. (153).

Figure 21. Isotherms and interface shapes for selected times during the batch-wise simulations of GaAs crystal in LEC growth. The figure is taken from...

Figure 1.149 Interface shapes obtained from particle tracking for K = 10 (upper row),...

Mathematically, this problem bears some resemblance to those considered above. The governing partial differential equation is still Eq. (6), and on the surfaces boundary conditions of constant potential, constant charge density or linear regulation [i.e., Eq. (45)] must be imposed. However, a further constraint arises from the need to satisfy mechanical equilibrium at the interface, and it is this new condition that provides the mathematical relation needed to calculate the interface shape. The equation is the normal component of the surface stress balance, and it is given by [12]... 

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