Critical Role of Crystalline Anisotropy in the Stability of Cellular Array Structures in Directional Solidification

Abstract

We calculate numerically the full Floquet-Bloch stability spectrum of cellular array structures in a symmetric model of directional solidification. Our results demonstrate that crystalline anisotropy critically influences the stability of these structures. Without anisotropy, the stability balloon of cells in the plane of wave number and velocity closes near the onset of morphological instability. With a finite, but even small, amount of anisotropy this balloon remains open and a band of stable solutions persists for higher velocities into a deep cell regime. The width of the balloon depends critically on the anisotropy strength. [S0031-9007(96)01426-3] PACS numbers: 81.10.Aj, 05.70.Ln, 81.30.FbDirectional solidification has been an important focus of research in the pattern formation and materials science communities for many years [1]. In a generic experiment, a binary liquid mixture contained between two narrowly spaced glass plates is pulled at constant velocity through an externally imposed temperature gradient. The morphology of the solid-liquid interface depends sensitively on the pulling velocity. Below a critical velocity the planar interface is stable, and above this critical velocity it undergoes the well-known Mullins-Sekerka (MS) instability [2] which gives rise to cellular array structures that can exist for a range of spacings.Close to the onset of instability, cells are typically of small amplitude, while in an intermediate range of higher velocities, they develop a larger amplitude. The latter structures are marked by deep liquid grooves that determine the scale of the final microsegregation profile inside the solidified alloy, and hence its physical properties. Therefore much effort has been devoted to the search for the instabilities which limit the range of possible stable spacings. However, despite more than three decades since the MS analysis, there is still no coherent picture of cell stability in the low velocity regime of directional solidification that has been traditionally studied experimentally. In addition, it is not known how stability is influenced by crystalline anisotropy that is known to play a crucial role in interfacial pattern formation [3].This lack of understanding is largely due to the technical difficulty of calculating the stability of nonlinear structures in nonlocal models of diffusion controlled growth, and to the fact that spatially extended arrays can be subject to a broad range of instabilities. These can be either oscillatory or nonoscillatory (steady) and can range in wavelength from the cell spacing l to very long wavelengths. For these reasons, studies to date have been restricted to analyzing specific secondary instabilities [4-10], or to analyzing all possible instabilities but in a high velocity regime where the diffusive growth problem reduces to local equations [11].A host of instabilities have so far been identified. These include, in a low-velocity regime, the classic long wavelength Eckhaus instability [4,5], which is generically present in one-dimensio...

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