- Geoffrey B. McFadden, ACMD
- Sam R. Coriell, Metallurgy Division
- Robert F. Sekerka, Carnegie Mellon University
- An outstanding problem in solidification theory is to predict
the length scales and time scales that occur during crystal growth.
These scales are crucial in determining the physical properties
of the solidified material; for example, when multicomponent alloys
are produced by directional solidification, instabilities
of the solid-liquid interface can lead to
inhomogeneous solute patterns, or microsegregation, in the solid phase which
are generally undesirable. Predicting the conditions under which
microsegregation occurs, and the associated length scales, is therefore
a problem that receives much attention both experimentally and theoretically.
- Recently M. E. Glicksman, of the Rensselaer Polytechnic Institute, and
colleagues have performed fundamental studies of growth of
a single-component material from a supercooled melt. When the liquid phase is maintained at temperatures below
the equilibrium melting point of the material, the solid that forms
spontaneously has a dendritic, or branch-like structure, with primary stems growing
at constant velocity into the melt. The tip velocities,
the radius of curvature of the tip, and other features of the dendrites
are fundamental properties that Glicksman et al. are able to measure
experimentally, and are used to critically assess various theoretical predictions
for these quantities. In the figure below, a multiple-exposure
photograph taken at equal time intervals illustrates the parabolic tip
and constant tip velocity [S.-C. Huang and M. E. Glicksman, Acta Met. 29 (1981)
717-734].
- An experimental complication is the occurrence of natural convection
in the liquid phase, which is driven by buoyancy forces produced by
the density variations associated with the temperature gradients in the system.
The effect of the convection is to alter the transport of heat away from
the solidifying dendrite, so that the resulting tip velocity and
radius of curvature of the tip are modified.
In order to reduce the effects of this buoyancy-driven convection,
the experiments have been performed in a reduced gravity environment
on board the NASA space shuttle. Examination of typical data for both
terrestrial growth conditions and the microgravity conditions of space
show that in both cases there are significant effects that may be attributed
to natural convection; an example is shown in the figure below.
Here the tip velocity, V, is plotted as a function of
the amount of thermal undercooling, delta T, of the liquid below the
bulk melting point of the material. Data is shown for both
terrestrial conditions (circles) and for microgravity conditions (squares).
- If convection effects were absent, the experimental
data would be expected to fall on
the bottom-most curve shown in the figure, which represents
a theoretical prediction based on a model that includes no convective effects.
Both sets of data show good agreement with the model at large
velocities and large undercoolings, but show systematic deviation from the predicted
behavior at smaller velocities. The agreement is better for the
microgravity data (squares), which can be attributed to the
decreased importance of buoyancy because of the reduced gravity.
Both sets of data eventually deviate from the bottom curve
at low enough undercoolings, implying that even under microgravity conditions
natural convection can play a significant role.
- To help understand the observed behavior, we have developed a simple
model that takes into account effects of buoyancy-driven convection.
The model assumes that because of natural convection the fluid is
well-mixed and isothermal outside of a boundary layer or
stagnant film near the surface of the dendrite, and assumes that
the heat transport within the stagnant film takes place by diffusion alone.
A closed-form solution to the
thermal problem can be found that depends on the assumed thickness of
the stagnant film. The macroscopic flow outside the stagnant film is given by a
large Raleigh number approximate solution in which the region
consisting of the network of growing dendrites is approximated by an
isothermal sphere of radius R. The stagnant film
thickness is then determined self-consistently by a balance of
convective and diffusive heat transfer at the edge of the stagnant film.
- The predictions of the resulting theory are shown as the top curves
in the above figure. The theory depends on a single adjustable parameter,
representing the ratio of the gravitational acceleration, g, and the radius, R.
The top three curves in the figure correspond to the terrestrial value g_e
for g and radii of 0.5, 1.0, and 2.0 cm, which are
typical of the geometries in the experiments. The bottom three curves
correspond to R = 1 cm and g = 0.0001 g_e, g = 0.00001 g_e, and
g = 0. The theory does a good job of
predicting the values of undercooling for which the effects of convection
become important.
- This work has been described in a paper entitled ``Stagnant Film
Model of the Effect of Natural Convection on the Dendrite Operating State,''
by R. F. Sekerka, S. R. Coriell and G. B. McFadden, that has
been submitted for publication in the Journal of Crystal Growth.