Simon Brandon

Partial wetting of chemically patterned surfaces: the effect of drop size.

Partial wetting of chemically heterogeneous substrates is simulated. Three-dimensional sessile drops in equilibrium with smooth surfaces supporting ordered chemical patterns are considered. Significant features are observed as a result of changing the drop volume. The number of equilibrated drops is found either to remain constant or to increase with growing drop volume. The shape of larger drops appears to approach that of a spherical cap and their three-phase contact line seems, on a larger scale, more circular in shape than that of smaller drops. In addition, as the volume is increased, the average contact angle of drops whose free energy is lowest among all equilibrium-shaped drops of the same volume appears to approach the angle predicted by Cassie. Finally, contrary to results obtained with two-dimensional drops, contact angle hysteresis observed in this system is shown to exhibit a degree of volume dependence in the advancing and receding angles. Qualitative differences in the wetting behavior associated with the two different chemical patterns considered here, as well as differences between results obtained with two-dimensional and three-dimensional drops, can possibly be attributed to variations in the level of constraint imposed on the drop by the different patterns and by the dimensionality of the system.

Heat transfer in vertical Bridgman growth of oxides: Effects of conduction, convection, and internal radiation

Abstract The vertical Bridgman growth of an oxide crystal with properties chosen to resemble those of yttrium aluminum garnet (YAG) is investigated. Internal radiation and heat conduction are accounted for in the crystalline phase, while transport through the melt (which is assumed opaque) is dominated by convection and conduction. Heat is also conducted through the ampoule walls, whose outer surface exchanges energy with the furnace via combined natural convection and enclosure radiation. A quasi-steady-state, axisymmetric Galerkin finite element method is employed for the calculation of thermal fields, melt flow patterns and melt/crystal interface shapes and positions for different parameter values. Results indicate that heat transfer through the system is strongly affected by the optical absorption coefficient of the crystal and that convective heat transfer through the melt is unimportant for this small-scale system. Coupling of internal radiation through the crystal with conduction through the ampoule walls promotes melt/crystal interface shapes which are highly deflected near the ampoule wall. This radiative interface effect is much more pronounced than that observed in the Bridgman growth of opaque crystals, where the interface deflection at the ampoule wall is attributed to the thermal conductivity mismatch between ampoule and charge. Calculations demonstrate that a flatter overall interface shape can be achieved through optimization of ampoule properties and furnace temperature profiles.

Modeling the vertical Bridgman growth of cadmium zinc telluride. I: Quasi-steady analysis of heat transfer and convection

A quasi-steady-state analysis of the vertical Bridgman growth of large-diameter, cadmium zinc telluride is conducted using a finite element model which accounts for the details of heat transfer, melt convection, and solid/liquid interface shape. Large radial gradients are shown to dominate the temperature field in the solid, while convection flattens the radial temperature distribution in the melt. Concave interface shapes are predicted to arise from the thermal conductivity mismatch between solid and liquid. The shape of the solid/liquid interface is sensitive to the growth rate due to the importance of latent heat release.

Internal radiative transport in the vertical Bridgman growth of semitransparent crystals

Abstract A quasi-steady-state model for the vertical Bridgman growth of semitransparent (partially absorbing and emitting) crystals is developed. For the first time, internal radiative heat transfer through the crystal (considered to be a participatine medium) is calculated rigorously, taking into account the full three-dimensional shape of the crystal, including curvature of the melt/crystal interface. The crystal is assumed to be grey, and the ampoule walls, as well as the crystal/melt interface, are assumed to be grey and diffuse. Heat transfer through the melt is considered to be dominated by conduction. A Galerkin finite element method is employed to determine the position and shape of the interface and the temperature field in the crystal and melt. Results for a model oxide growth system demonstrate the sensitivity of melt/crystal interface shape, position, and interfacial gradients to changes in optical properties. The total heat flow through the melt and crystal also depends strongly on the optical parameters of the system. The phenomenon of radiative supercooling is not observed for the system considered here.

Modeling the vertical Bridgman growth of cadmium zinc telluride II. Transient analysis of zinc segregation

Abstract A transient analysis of the vertical Bridgman growth of large-diameter, cadmium zinc telluride is conducted using a finite element model which accounts for the details of heat transfer, melt convection, solid/liquid interface shape, and zinc segregation. Significant axial and radial segregation is produced by convective mixing in the melt; the system is far from the diffusion-controlled limit. Previous experimental reports of “anomalous” axial segregation are explained by solid-state diffusion mechanisms. Lowering the growth rate is predicted to slightly increase axial segregation but markedly reduce radial segregation. The geometry of the ampoule cone region is shown to significantly affect the initial growth transient.

Analysis of interrupted growth strategies for cadmium telluride in an unseeded vertical Bridgman system

A transient, non-dilute finite element model is employed to study a novel, interrupted growth strategy which has been employed for the unseeded, vertical Bridgman growth of cadmium telluride. Computations clearly show the time-dependent translation strategy causes solute diffusion layers in the melt to successively grow and die in time, thus providing a means to mix the cadmium rejected at the growth interface into the bulk. This strategy stabilizes the solid-liquid interface by delaying the onset of constitutional supercooling, thus allowing the use of growth rates for grain selection which are higher than would be possible using continuous translation.

Facetting during directional growth of oxides from the melt: coupling between thermal fields, kinetics and melt/crystal interface shapes

Implicit in most large-scale numerical analyses of crystal growth from the melt is the assumption that the melt/crystal interface shape and position are determined by transport phenomena. Although reasonable for many materials under a variety of growth conditions, this assumption is incorrect for a number of practical systems under realistic growth conditions. Specifically, the behavior of systems (e.g. certain oxides) which tend to develop facets along the melt/crystal interface is often affected both by transport phenomena and by interfacial attachment kinetics. We present a new modeling approach which accounts for interfacial kinetic effects during melt growth of large single crystals. The isotherm condition, typically employed at the melt/crystal interface, is replaced by an equation accounting for undercooling due to interface kinetics. A finite-element algorithm, designed to accommodate its numerical mesh to the appearance of facetted interfaces, is applied to this problem. Results are presented for the simulated directional growth of oxide slabs. The interplay between evolving thermal fields and anisotropic interface kinetics is investigated. In particular, the evolution of facets and the dependence of their size on growth conditions is explored. Trends reported here are in qualitative agreement with those appearing in the literature. Discrepancies between quantitative predictions of facet sizes using a theory (see Ann. Rev. Mater. Sci. 3 (1973) 397 and references within) and those calculated in this paper can, in a number of cases, be attributed to the simplifications on which this theory is based.

PHASE FIELD EQUATIONS WITH MEMORY: THE HYPERBOLIC CASE*

We present a phenomenological theory for phase transition dynamics with memory which yields a hyperbolic generalization of the classical phase field model when the relaxation kernels are assumed to be exponential. Thereafter, we focus on the implications of our theory in the hyperbolic case, and we derive asymptotically an equation of motion in two dimensions for the interface between two different phases. This equation can be considered as a hyperbolic generalization of the classical flow by mean curvature equation, as well as a generalization of the Born--Infeld equation. We use a crystalline algorithm to study the motion of closed curves for the generalized hyperbolic flow by mean curvature equation our hyperbolic generalization of flow by mean curvature and present some numerical results which indicate that a certain type of two-dimensional relaxation damped oscillation may occur.

Interface shapes and thermal fields during the gradient solidification method growth of sapphire single crystals

We present a finite-element model describing the melt-growth of cylindrical sapphire single crystals via the gradient solidification method. The advantage of this model lies in its ability to accurately capture complex physical phenomena associated with heat transfer through the system, while remaining modest in its computational requirements. Internal radiative heat transport through the transparent crystalline phase is accounted for in our formulation, as are details of flow fields evolving in the melt during growth. Both buoyancy and surface-tension-gradient (Marangoni) driven convection effects are considered. Results show a strong dependence of the thermal field in the charge and of melt/crystal interface shapes on operating parameters such as crystal growth rate and furnace temperature gradient. Specifically, the large latent heat value associated with this system, coupled with enhanced radiative cooling through the crystalline phase, causes a dramatic reduction in interface curvature and position for relatively high growth rates and shallow furnace gradients. In addition, effects of fluid flow on the thermal field are shown to be unimportant in this system, even when considering growth in relatively large-diameter crucibles. Trends reported here are in general agreement with experimental observations.

Large-scale numerical analysis of materials processing systems: High-temperature crystal growth and molten glass flows

Abstract We present Galerkin finite element computations of heat transfer and fluid dynamics in high-temperature materials processing systems. Examples are presented of how large-scale numerical simulations have been used to gain insight into the workings of several processes, specifically the growth of large, single crystals and the flow characteristics of molten glass. These systems are characterized by nonlinear interactions between field and interfacial phenomena — the transport of momentum, heat and mass and effects of solidification and capillarity. Additional complications are caused by the effects of internal radiative energy transport within participating media (internal radiation).