1998 Annual Report
Basic Energy Sciences
Modeling of Microstructural Pattern Formation in Directional Solidification
A. Karma and M. Plapp, Northeastern University
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Research Objectives
Our research aims at understanding the complicated spontaneous
pattern formation processes occuring during the growth of solids.
Examples for such phenomena are the dendritic or eutectic microstructures
formed when an alloy is directionally solidified, or the growth
spirals observed during epitaxial thin-film growth. The particular
difficulty common to all these diffusion-limited processes is
the presence of moving phase boundaries of complex and changing
shape. We use the phase-field method to investigate these notoriously
difficult free-boundary problems. Computational Approach
The traditional formulation of free-boundary problems in terms
of mathematically sharp interfaces between different thermodynamic
phases is difficult to implement because of front tracking. The
phase-field method introduces indicator fields (or phase fields)
and allows for a finite interface thickness. The equations of
motion become nonlinear partial differential equations that can
be solved using standard methods. In this setting, the problem
can be very easily parallelized. We simulated systems of considerable
complexity on the NERSC T3E. Accomplishments
We have investigated the influence of the surface tension anisotropy
on the dynamics of pattern selection during the directional solidification
of a binary alloy. Whereas the importance of this anisotropy is
well documented for free dendritic growth, its role in directional
solidification was recognized only recently. The possibility of
simulating large arrays of solidification cells allowed us to
monitor the complete dynamics of spacing selection, starting from
a slightly perturbed interface (as is usually the case in experiments).
In particular, it was shown that recently discovered multiple
cells can be dynamically selected if the initial perturbation
has a fixed periodicity. These results are in excellent agreement
with recent experiments where a perturbation of fixed wavelength
is imposed at the beginning of the solidification process, using
a modulated laser beam. |
We have also developed a phase-field model for the directional
solidification of eutectic alloys with an additional ternary impurity.
Simulations show the development of large two-phase eutectic cells,
in agreement with experiments. The dynamics of eutectic cell ("colony")
formation are at present ill understood and can be further investigated
with our model.
As an extension of our main project, we have recently adapted
the phase-field method to model step-flow growth on faceted crystal
surfaces. The step dynamics in epitaxial thin-film growth are
governed by the interplay of the flux of deposited atoms, the
surface diffusion, and the desorption from the surface. In particular,
we have investigated the dynamics of spiral ridges around screw
dislocations. Whereas these spirals are well understood in the
regime where desorption is dominant, our method enables us to
investigate the opposite case, relevant for molecular beam epitaxy,
where surface diffusion is dominant. Significance
Both directional solidification and epitaxial growth are important
industrial methods for advanced materials fabrication. Many properties
of the final material are largely influenced by the microstructure
of the material, which in turn is determined by the dynamics of
the growth front. Understanding the influence of various control
parameters on the growth dynamics is therefore of great practical
importance. Publications
A. Karma and W.-J. Rappel, "Quantitative phase-field modeling
of dendritic growth in two and three dimensions," Phys. Rev.
E 57, 4323-4349 (1998).
G. W. Losert, D. A. Stillman, H. Z. Cummins, P. Kopczynski, W.-J.
Rappel, and A. Karma, "Selection of doublet cellular patterns
in directional solidification through spatially periodic perturbations,"
Phys. Rev. E (in press, 1998).
A. Karma and M. Plapp, "Spiral surface growth without desorption,"
Phys. Rev. Lett. (in press, 1998). |
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