MICROSTRUCTURAL SCALING RELATIONSHIPS IN CELLULAR PATTERNS

摘要

In a cellular array, a strong coupling of solute field exists between neighboring cells so that a significant scatter in primary spacing and in cell shape is present in a given array. Through experimental studies and phase-field modeling, we have found that the apparent disorder in cells follows some definite scaling laws. When only cells are present in an array, all apparently differing cell shapes collapse (within optical resolution) onto a single shape when they are scaled with the local spacing. This shape interpolates between two known limits of this problem: the Scheil equation and the 2-d Saffman-Taylor (ST) equation which describes the Laplacian limit of directional growth. The former predicts the asymptotic shape of the interface far from the tip while the latter is found here to predict reasonably well the cell tip of experimental and numerical cell shapes. The predicted tip undercooling based on the ST shape is also in good agreement with the 2-d phase-field results. In addition, experimental results are presented showing that the cells in the presence of intercellular eutectic follow the solution of the 3-d ST problem. Finally, the phase-field results are strikingly different for 2-d shapes and 3-d axisymmetric shapes: steady-state cells exist in 2-d over a very wide range of spacing, but they only exist in 3-d up to a maximum spacing that is significantly smaller than the lower limit of the experimentally observed range of spacing. The existence of this band gap in 3-d cell solutions is consistent with previous numerical solutions of the 3-d ST problem.