Theory of Microstructural Development during Rapid Solidification

A variety of non-equilibrium growth processes are characterized by phase boundaries consisting of moving fingers, often with interesting secondary structures such as sidebranches. Familiar examples are dendrites, as seen in snowflake growth, and fluid fingers often formed in immiscible displacement. Such processes are characterized by a morphological instability which renders planar or circular shapes unstable, and by the competing stabilizing effect of surface tension. We survey recent theoretical work which elucidates how such systems arrive at their observed patterns. Emphasis is placed upon dendritic solidification, simple local models thereof, and the Saffman-Taylor finger in two-dimensional fluid flow, and relate these systems to their more complicated variants. We review the arguments that a general procedure for the analysis of such problems is to first find finger solutions of the governing equations without surface tension, then to incorporate surface tension in a non-perturbative manne...

Pattern selection in fingered growth phenomena

Solidification microstructures: recent developments, future directions

A numerical model of cellular and dendritic growth has been developed that can predict cellular and dendritic spacings, undercoolings, and the transition between structures. Fully self-consistent solutions are produced for axisymmetric interface shapes. An important feature of the model is that the spacing selection mechanism has been treated. A small, stable range of spacings is predicted for both cells and dendrites, and these agree well with experiment at both low and high velocities. By suitable nondimensionalization, relatively simple analytic expressions can be used to fit the numerical results. These expressions provide an insight into the cellular and dendritic growth processes and are useful for comparing theory with experiment.

Numerical modeling of cellular/dendritic array growth: spacing and structure predictions

The primary spacing data of Part I are compared to the existing theoretical models of Hunt and of Kurz and Fisher, and a significant disagreement is found. A theoretical model based on the Hunt model is developed, and it is found that the theory adequately explains the variation in primary spacing, λ1, with the growth rate,V. A maximum in λ1,vs V is predicted and the velocity at which the maximum occurs matches with the result obtained experimentally. It is shown that the maximum in λ1 corresponds to the dendrite-to-cell transition, and cellular structures are found to grow with much smaller spacings than dendritic structures under identical growth conditions.

Interdendritic Spacing: Part II. A Comparison of Theory and Experiment

Directional solidification experiments have been carried out in a succinonitrile-5.5 mol pct acetone system to characterize dendrite tip radius and interdendrite spacings as functions of growth rate and temperature gradient in the liquid. A maximum in primary dendrite spacing as a function of growth rate is observed, and this maximum is found to occur at the dendrite-cellular transition velocity. A scaling law is shown to exist between the secondary dendrite arm spacing, λ2, near the tip and the dendrite tip radius, p, which is λ2/ρ = 2.2 ± 0.3. Experimental results on ρ have been found to agree with the theoretical model based on the marginal stability criterion.

K. Somboonsuk

Papers

Interdendritic Spacing: Part I. Experimental Studies