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NASA NTRS1982

Heat transfer analysis of the Bridgman-Stockbarger configuration for crystal growth. Part 1: Analytical treatment of the axial temperature distribution

All first order effects on the axial temperature distribution in a solidifying charge in a Bridgman-Stockbarger configuration for crystal growth are analyzed on the basis of a one dimensional model whose validity can be verified through comparison with published finite difference ana;uses of two dimensional models. The model presented includes an insulated region between axially aligned heat pipes and considers the effects of charge diameter, charge motion, thickness, and thermal conductivity of a confining crucible, thermal conductivity change at the crystal-melt interface, generation of latent heat at the interface, and finite charge length. Results are primarily given in analytical form and can be used without recourse to computer work for both improve furnace design and optimization of growth conditions in a given thermal configuration.

Jasinski, T. J.↗
NASA NTRS1984

Models of Rapid Solidification

Laser annealing studies provide much information on various consequences of rapid solidification, including the trapping of impurities in the crystal, the generation of vacancies and twins, and on the fundamental limits to the speed of the crystal-melt interface. Some results obtained by molecular dynamics methods of the solidification of a Lennard-Jones liquid are reviewed. An indication of the relationship of interface speed to undercooling for certain materials can be derived from this model. Ising model simulations of impurity trapping in silicon are compared with some of the laser annealing results. The consequences of interface segregation and atomic strain are discussed.

Gilmer, G. H.↗
NASA NTRS1984

The undercooling of liquids

The formation by melt quenching of such metastable structures as glassy or microcrystalline solids and highly supersaturated solutions is made possible by the extreme resistance of most melts to homophase crystal nucleation at deep undercooling. This nucleation resistance contrasts sharply with the very low kinetic resistance to the movement of crystal-melt interfaces, once formed, in metals and other fluid systems at even minute undercooling. The methods of nucleation study which have proven especially effective in bypassing nucleation by heterophase impurities thereby exposing the high resistance of melts to homophase nucleation may be summarized as follows: observation of the crystallization behavior of dispersed small droplets; drop tube experiments in which liquid drops solidify, under containerless conditions, during their fall in the tube; and observation of the crystallization of bulk specimens immersed in fluxes chosen to dissolve or otherwise deactivate (e.g., by wetting) heterophase nucleants. This method has proven to be remarkably effective in deactivating such nucleants in certain pure metals.

Turnbull, D.↗
NASA NTRS1983

Seeded crystal growth of pseudobinary systems with large liquidus-solidus separation

The design and execution of seeded melt growth involving concentrated alloys is strongly dependent on the crystal-melt interface temperature and its change during seeding and growth. Thermodynamic instabilities during the seeding process can be controlled by the choice of the seed and melt composition and by the axial temperature gradient of the growth apparatus. These basic factors controlling seeded melt growth of concentrated alloys are of major importance for planning space experiments.

Bourret, E. D.↗
NASA NTRS1985

Simulation of the temperature distribution in crystals grown by Czochralski method

Production of perfect crystals, free of residual strain and dislocations and with prescribed dopant concentration, by the Czochralski method is possible only if the complex, interacting phenomena that affect crystal growth in a Cz-puller are fully understood and quantified. Natural and forced convection in the melt, thermocapillary effect and heat transfer in and around the crystal affect its growth rate, the shape of the crystal-melt interface and the temperature gradients in the crystal. The heat transfer problem in the crystal and between the crystal and all other surfaces present in the crystal pulling apparatus are discussed at length. A simulation and computer algorithm are used, based on the following assumptions: (1) only conduction occurs in the crystal (experimentally determined conductivity as a function of temperature is used), (2) melt temperature and the melt-crystal heat transfer coefficient are available (either as constant values or functions of radial position), (3) pseudo-steady state is achieved with respect to temperature gradients, (4) crystal radius is fixed, and (5) both direct and reflected radiation exchange occurs among all surfaces at various temperatures in the crystal puller enclosure.

Dudokovic, M. P.↗
NASA NTRS1986

Mathematical model for the Bridgman-Stockbarger crystal growing system

In a major technical breakthrough, a computer model for Bridgman-Stockbarger crystal growth was developed. The model includes melt convection, solute effects, thermal conduction in the ampule, melt, and crystal, and the determination of the curved moving crystal-melt interface. The key to the numerical method is the use of a nonuniform computational mesh which moves with the interface, so that the interface is a mesh surface. In addition, implicit methods are used for advection and diffusion of heat, concentration, and vorticity, for interface movement, and for internal gracity waves. This allows large time-steps without loss of stability or accuracy. Numerical results are presented for the interface shape, temperature distribution, and concentration distribution, in steady-state crystl growth. Solutions are presented for two test cases using water, with two different salts in solution. The two diffusivities differ by a factor of ten, and the concentrations differ by a factor of twenty.

Roberts, G. O.↗
NASA NTRS1987

Coupled morphological and convective instabilities in a microgravity environment

The role of convection during crystal growth from the melt is discussed in terms of dimensionless groups of material parameters and control variables. The limitations of such analyses and of models in which the crystal-melt interface is treated as a rigid boundary are reviewed. The interaction of thermosolutal convective instabilities with morphological instabilities is discussed for unidirectional solidification, emphasizing suppression of convective instabilities in a microgravity environment and/or a static magnetic field.

Sekerka, R. F.↗
NASA NTRS1986

Simulation of jet cooling effects on Czochralski crystal growth

The effects of cooling the crystal side surface by blowing a jet of an inert gas are examined in detail for Czochralski crystal growth. A combined model of the crystal + melt, which incorporates the detailed radiation calculations, the shape of the melt-gas meniscus, predicts the growth rate and the crystal-melt interface shape, is used for this study. The convective heat transfer coefficient for the jet is estimated from the correlation available in the literature. The effect of the jet cooling on the interface shape and the pulling rate is significant. The crystal diameter as well as the interface shape tend to be more stable in the environment of the rapid cooling of the crystal by the jet. The crystal diameter or the interface shape can be easily controlled by adjusting the gas flow rate through the jet. This gives the Czochralski pulling an additional degree of freedom facilitating the control of crystal diameter and interface shape.

Srivastava, R. K.↗
NASA NTRS1987

Thermosolutal convection during directional solidification. II - Flow transitions

The influence of thermosolutal convection on solute segregation in crystals grown by vertical directional solidification of binary metallic alloys or semiconductors is studied. Finite differences are used in a two-dimensional time-dependent model which assumes a planar crystal-melt interface to obtain numerical results. It is assumed that the configuration is periodic in the horizontal direction. Consideration is given to the possibility of multiple flow states sharing the same period. The results are represented in bifurcation diagrams of the nonlinear states associated with the critical points of linear theory. Variations of the solutal Rayleigh number can lead to the occurrence of multiple steady states, time-periodic states, and quasi-periodic states. This case is compared to that of thermosolutal convection with linear vertical gradients and stress-free boundaries.

Mcfadden, G. B.↗
NASA NTRS1987

Directional solidification in an imperfect furnace

The upward directional solidification of a binary alloy in which the rejected solute is more dense than the solvent is considered. The system is stably stratified both thermally and solutally. If heat is lost through the sidewalls, horizontal thermal gradients drive a steady double-diffusive convection. Solidification cells with small aspect ratios (width/length) A are considered, and asymptotic solutions to the steady double-diffusive system are obtained as A tends to 0 under the assumption heat losses are also small. When the thermal Grashof number is large, buoyancy boundary layers are at the sidewalls. The shape of the crystal-melt interface is determined by the coupled effects of local thermodynamic equilibrium, diffusion and convection.

Brattkus, K.↗
NASA NTRS1984

Effect of a forced Couette flow on coupled convective and morphological instabilities during unidirectional solidification

The effect of a forced Couette flow, parallel to a horizontal crystal-melt interface during directional solidification of an alloy of lead containing tin, on the onset of convective and morphological instabilities, is calculated numerically via a linear stability analysis. Such a flow does not affect perturbations with wave vectors perpendicular to the flow. For perturbations with wave vectors parallel to the flow, the onset of morphological instability is somewhat suppressed and thermosolutal convection is greatly suppressed. When instabilities occur, they are oscillatory and correspond to travelling waves. For values of the crystal growth velocity for which mixed morphological and convective modes occur, the presence of a forced flow produces sufficient decoupling to allow otherwise degenerate branches to be identified.

Coriell, S. R.↗
NASA NTRS1988

Growth of needle-shaped crystals in the presence of convection

The motion of the freezing front between a needle-shaped crystal and a supercooled liquid is analyzed for situations where there is forced convection aligned with the crystal axis. It is shown that in the absence of capillary effects the shape of the crystal-melt interface is a paraboloid of revolution, similar to that found in situations where diffusion is the sole heat-transfer mechanism. A relation between the supercooling, the product of tip velocity and tip radius, and the strength of the flow is derived which reduces to the well-known Ivantsov theory in the absence of convection.

Saville, D. A.↗
NASA NTRS1987

Experimental studies of crystal-melt differentiation in planetary basalt compositions

An important process that controls the evolution of magmas on and within planetary bodies is crystal-melt differentiation. Experimental studies of silicate melt solidification were performed on several planetary and terrestrial melt compositions, and experiments on one of these compositions in the microgravity environment of the space station would provide an opportunity to understand the factors that control crystal growth and crystal-melt exchange processes at crystal-melt interfaces during solidification. Experimental requirements are presented.

Grove, T. L.↗
NASA NTRS1989

Ginzburg-Landau theory for the solid-liquid interface of bcc elements. II - Application to the classical one-component plasma, the Wigner crystal, and He-4

The previously developed Ginzburg-Landau theory for calculating the crystal-melt interfacial tension of bcc elements to treat the classical one-component plasma (OCP), the charged fermion system, and the Bose crystal. For the OCP, a direct application of the theory of Shih et al. (1987) yields for the surface tension 0.0012(Z-squared e-squared/a-cubed), where Ze is the ionic charge and a is the radius of the ionic sphere. Bose crystal-melt interface is treated by a quantum extension of the classical density-functional theory, using the Feynman formalism to estimate the relevant correlation functions. The theory is applied to the metastable He-4 solid-superfluid interface at T = 0, with a resulting surface tension of 0.085 erg/sq cm, in reasonable agreement with the value extrapolated from the measured surface tension of the bcc solid in the range 1.46-1.76 K. These results suggest that the density-functional approach is a satisfactory mean-field theory for estimating the equilibrium properties of liquid-solid interfaces, given knowledge of the uniform phases.

Zeng, X. C.↗
NASA NTRS1989

An ion microprobe study of the intra-crystalline behavior of REE and selected trace elements in pyroxene from mare basalts with different cooling and crystallization histories

The effects of crystallization interaction on the trace element zoning characteristics of pyroxenes are analyzed using electron and ion microprobe techniques. Four pigeonite basalts with similar isochemical composition, but different cooling rates and crystallization histories are studied. Pyroxene quadrilaterals displaying crystallization trends are presented. The crystal chemical rationalization of REE zoning, pattern shapes, and abundances are examined. The data reveal that the trace element zoning characteristics in pyroxene and the partitioning of trace elements between pyroxene and the melt are related to the interaction between the efficiency of the crystallization process, the kinetics at the crystal-melt interface, the kinetics of plagioclase nucleation and the characteristics of the crystal chemical substitutions in the pyroxene and the associated crystallizing phase.

Shearer, C. K.↗
NASA NTRS1990

Instability during directional solidification - Gravitational effects

After an elementary introduction to solidification, recent research that attempts to understand the interaction of fluid flow with a crystal-melt interface under gravitational influence using linear stability analysis is reviewed. Recent numerical calculations of fluid flow and interface morphology during alloy solidification are briefly discussed.

Coriell, S. R.↗
NASA NTRS1984

The effect of anisotropic crystal-melt surface tension on grain boundary groove morphology

The shape of a stationary solid-liquid interface in a temperature gradient near a grain boundary in a pure material is calculated for anisotropic crystal-melt surface tension and equal thermal conductivities of crystal and melt. Results are compared with those for the well-known problem of the two-dimensional equilibrium shape of a crystal. For small anisotropy, the resulting interface shapes have continuously turning tangents but differ in detail from the grain boundary groove shapes that have been calculated for isotropic surface tension. For larger anisotropy, the interface shapes have discontinuities in slope as a result of missing orientations.

Voorhees, P. W.↗
NASA NTRS1985

Crystal Melts Under Reduced Gravity

To date, fluid motion due to thermocapillary convection and centrifugal buoyancy has been studied theoretically and numerically, by adapting specialized programs for rotating viscous fluids. It has been shown that by rotating the crystal melt, thermocapillary flow can be confined to a thin layer at the melt-gas interface even when the interfacial velocity is large enough for non-linear effects to be important. Theoretical analysis of the basic processes proceeds, but a much larger effort is now being devoted to the numerical simulation of the actual motion in a rotating crystal melt. A numerical program is under development for the study of the fully nonlinear problem. Although the flow is viewed, to start, as one of rapid rotation (in order that certain boundary layer simplifications can be invoked), a more general approach will allow consideration of the situations and conditions of real practical interest.

Greenspan, H. P.↗