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In this thesis, long-wave morphological instabilities during
directional solidification of a dilute binary alloy are
discusseed. The long-wave developments of the interfacial
disturbances can be described by a so-called evolution
equation, so the derivation of the evolution equation becomes
the most important step in studying the morphological
instability of a solidification system. In the dissertation,
three-dimensional cellular instabilities in non-equilibrium
directional solidifications are investigated. The infinitely
one-sided model and the frozen-temperature-approximation are
adopted in the analysis. An evolution equation was first
derived by an integral technique. Then the Segel-Stuart
equations. The equilibrium solutions associated with different
morphologies were evaluated and the stability analysis of them
to three-dimensional disturbances was studied. Finally, five
stability basins of bifurcation were addressed. The result
shows that the interface morphologies depend on which basin the
chosen critical condition is located in and how far the
operating point is exceeded from the critical condition. The
presence of disequilibrium can stabilize the steady cellular
mode, cause the wavenumber smaller and affect the location of
the chosen critical condition in the basin of bifurcation.
Meanwhile, during the directional solidification process, the
two-dimensional band-like cells and three-dimensional hexagonal
cells and nodes are likely to be observed. Whether these two-
and three-dimensioal microstructures are observable can be
predicted through the stability analysis of the equilibrium
solution of the interfacial disturbance. In the present
analysis, possibilities to observe these microstructures are
discussed for some particular range of control parameters,
respectively. It is concluded that two-dimensional band-like
cells are observable, while three-dimensiona hexagonal cells
and nodes are stable for the solvability constants in the
amplitude equation.
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