Directional Solidification

 

 

Participants: C. B. Clemons (Theo and Applied Math), D. Golovaty (Theo and Applied Math), G. W. Young (Theo and Applied Math)

 

Directional solidification techniques are the most widely used methods for preparing high quality single crystals of metallic, electronic, and opto-electronic materials. In order to obtain optimum microstructure and properties in the solidified material, it is important to maintain a uniform distribution of solute/dopants and a flat solidification interface during a growth process. Such ideal conditions are difficult to achieve in practice because of an unavoidable heat exchange between the crucible and the sample. This heat exchange leads to radial temperature gradients and subsequent fluid flow. The results are non-planar solidification fronts and potentially severe axial and radial solute segregation.

 

Since there is a close relationship between growth conditions and the microstructure and properties of solidified materials, there has been an extensive amount of investigation of sharp-interface, Stefan-type problems and directional solidification configurations.

Exact solutions to these classes of problems are generally restricted to unbounded domains and are subject to limitations on the boundary conditions. For this reason, a variety of approximate analytical and numerical approaches have been developed to examine domains and boundary conditions that more closely simulate actual growth conditions.

 

Our work falls into the category of developing analytical approaches for the system solution. Recent projects include:

 

A model is being developed that describes the steady-state diffusion of heat, and convective-diffusive transport of solute in a directional solidification system. The geometry of the model is described by a two-dimensional Cartesian coordinate system. Heat transfer between the ampoule and melt is possibly non-symmetric between the left and right ampoule walls. The solution procedure involves a coupled asymptotic/numerical approach. The asymptotic expansions are based upon the assumptions that the ampoule aspect ratio, the heat exchange between the ampoule and sample, and the slope of the liquidus line are small. These scalings lead to boundary layer solutions around the solidifying front. The solidifying interfacial shape, thermal, flow, and solutal profiles are analytically evaluated as functions of the heater temperature profile, heater translation rate, and material properties of the system. The buoyancy driven fluid flow is calculated in the outer region as a function of the heat exchange between the ampoule and sample. A slip flow is used to approximate the flow in the boundary layer. Lateral corrections to the shape of the interface are determined by axial and lateral diffusive transport of heat, and diffusive and convective transport of solute. The latter modify the melting point of the interface and the temperature gradients local to the interface. Lateral solute segregation is predicted to increase with increasing flow and to increase with solidification rate. Well-mixed melts are not achieved, so flat interfaces do not necessarily correspond to flat concentration profiles.

 

A model is also developed to describe the time-dependent diffusion of heat and solute in a directional solidification system for a binary alloy. The stagnant film concept is used to include the effects of melt flow and the transport of solute by convection. The geometry of the model is described by an axisymmetric coordinate system. The solution procedure involves a coupled asymptotic/numerical approach. The asymptotic expansions combine quasi-steady limits with a small ampoule aspect ratio. The heat exchange between the ampoule and sample is small, the latent heat is large, the solute diffusivity in the solid phase is small, and the thickness of the stagnant film is small. These scalings lead to boundary layer solutions around the solidifying front. The core melt and solid temperature profiles and the mean position of the solidifying front are calculated numerically as functions of the latent heat, heater profile, and solute profile in the liquid phase. Due to the inclusion of curvature effects on the melting point, the non-planar shape of the interface is described by a superposition of an oscillatory profile onto a parabolic profile. The parabolic profile is set by axial and radial diffusive transport of heat. The oscillatory profile develops in response of the interface to meet the ampoule wall at a prescribed contact angle. This response may lead to an oscillatory interfacial pattern in the case of undercooling. This may signify morphological instability in the interface shape. The above results are categorized as a function of the stagnant film thickness. This thickness is varied to consider melt regimes from diffusion-controlled growth to a well-mixed melt due to strong convective effects. The axial and radial segregation, and morphological stability of the system are discussed as a function of the film thickness.

 

A third model is developed that describes the time-dependent diffusion of heat and solute in a directional solidification system. The geometry of the model is described by an axisymmetric coordinate system. The solution procedure involves a coupled asymptotic/numerical approach. The asymptotic expansions are based upon the assumptions that the ampoule's aspect ratio is small, the heat exchange between the ampoule and sample is small, and that the system Lewis and Stefan numbers are scaled to achieve a well-mixed melt. These scalings lead to boundary layer solutions around the solidifying front. The solidifying interfacial shape, and the thermal and solutal profiles are analytically evaluated in time as functions of the heater temperature profile, heater translation rate, and material properties of the system. Due to the inclusion of curvature effects on the melting point, the shape of the interface is described by a superposition of a Bessel function onto a parabolic profile. The parabolic profile is set by axial and radial diffusive transport of heat. The Bessel function profile is the response of the interface to meet the ampoule wall at a prescribed contact angle. This response may lead to an oscillatory interfacial pattern in the case of undercooling. This may signify morphological instability in the interface shape.

 

 

Publications

  • Asymptotic Solutions for a Time-Dependent, Axisymmetric Directional Solidification System, J. Bonfiglio, J. McHood, C. B. Clemons, D. Golovaty, and G. W. Young, Journal of Crystal Growth, Vol. 285 (2005), pp. 415-426.
  • Asymptotic Solutions for an Axisymmetric, Stagnant Film Model of Directional Solidification C. B. Clemons, D. Golovaty, and G. W. Young, Journal of Crystal Growth, Vol. 289, Issue 2 (2006), pp. 715-726.
  • An Asymptotic Analysis for Directional Solidification of a Binary System, K. Kupchella, C. B. Clemons, D. Golovaty, and G. W. Young, Accepted in Journal of Crystal Growth, (2006).

 

Funding 

  • NASA Glenn Cooperative Agreement for Theory, Modeling, Software and Hardware Development in Computational Materials Science - NASA Grant No. NNC04GB27G, (2004 - 2007), $128,000, G.W. Young, C. B. Clemons, and S. I. Hariharan.