A model of edge-defined film-fed crystal growth (EFG) is developed to study melt growth of cylinders of the scintillator crystal cesium iodide (CsI). The model, which includes fluid flow, heat transfer, and gravitational effects, is solved numerically by the finite element method to predict crystal diameter as a function of growth rate and die geometry. The model computes the location and shape of the crystal-melt interface and the meniscus to satisfy a self-consistent formulation of phase change and capillary dynamics. A sharp interface formulation solved on a deforming finite element mesh ensures accurate conservation of mass, energy, and momentum at the interfaces.
This system is characterized by strongly nonlinear interactions of heat transfer, capillarity, and die geometry that give rise to multiple stationary solution states under a single set of operating conditions. We use bifurcation theory and parameter continuation techniques to identify these solution states and characterize their stability.
Purely capillary instabilities are identified that result from the interaction of surface tension with gravity, and also the interaction of in-plane and out-of-plane (hoop stress) curvatures of the meniscus. These give rise to two solution families distinguished by a difference in gap width between die face and growth interface. The narrow gap solutions are shape stable, and the wide gap solutions are shape unstable, though these can be stabilized by heat transfer in a manner similar to Czochralski growth.
Additionally we identify several instabilities of convective heat transfer, which generally are of two types. One type is mostly convective in nature and largely independent of the interface or meniscus shapes. In this type the solution families are distinguished by different flow structures within the capillary bore. The other type is characterized by a strong interaction of convection with the geometry of the growth interface. Stronger convection moves the interface to a higher position, forming a larger gap. The larger gap allows a stronger flow to form, which enhances convection. This reinforcing behavior enables a family of solutions to exist in which the growth interface shape is concave in the central region, though it is convex elsewhere. Finally, we address the possible mechanism leading to the formation of longitudinal pores of cylindrical shape observed in crystals grown by experiment.
Supported in part by U.S. Department of Homeland Security, HSHQDC-11-C-00094, the content of which does not necessarily reflect the position or policy of the United States Government, and no official endorsement should be inferred.