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.
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Funding