Direct Measurement of Dendritic Array Stability

Abstract

Stability of a uniformly spaced dendritic array against a spatial period doubling instability was studied through UV thermal perturbations of every other dendrite tip. We observed that above a critical pulling speed the array is stable against these perturbations and we measured decreasing decay rates of the perturbed mode as we approached the critical speed. In the linear regime, our measurements are qualitatively consistent with the Warren-Langer linear stability analysis for a dendritic array [J. A. Warren and J. S. Langer, Phys. Rev. A 42, 3518 (1990); Phys. Rev. E 47, 2702], while in the nonlinear regime fitting to a third order amplitude equation shows that the transition is subcritical.[S0031-9007(98)06563-6] PACS numbers: 81.10. Aj, 68.70.+ w, 82.65.Dp In directional solidification experiments a dilute binary alloy with solute concentration C`is put in a temperature gradient G and a flat solid-liquid interface is established. The sample is then pulled with pulling speed V P toward the cold side; i.e., the alloy is solidified. Above a critical pulling speed the flat interface becomes cellular and at higher speeds a dendritic pattern is established. The dendrite coarsening process, sidebranching, and the interdendritic spacing l 1 are critical problems in pattern formation, and much work has been devoted to studying the time evolution, the structure, and the stability of the observed patterns.Detailed studies by Trivedi and co-workers, who started crystal growth from a planar front with constant pulling speed V P , indicated that in steady state the interdendritic spacing l 1 is reproducibly determined by the growth conditions and changes smoothly with pulling speed V P [1] or solute concentration C` [2]. Such observations suggested the existence of a dynamical selection mechanism for l 1 . Warren and Langer (WL) carried out a linear stability analysis of an array of dendrites [3,4] using microscopic solvability theory [5,6] to describe individual tip shapes and calculating the interaction between tips through the solute diffusion field. WL found that no dynamical selection mechanism should exist, but for a given V P a range of interdendritic spacings is allowed. Their calculations predict that for given growth conditions there is a range of pulling speeds in which an array of spacing l 1 is stable with a lower limit value V PD ͑l 1 ͒. Below V PD ͑l 1 ͒ the array is unstable and a period-doubling bifurcation should be observed, with every other dendrite continuing to grow.Subsequently, Huang et al. [7] observed that if directional solidification is initiated at some V 0 P until a steady state array develops and the pulling speed is then changed from V 0 P to V P , different values of l 1 are found for the same final V P , depending on the choice of V 0 P . They found that, for a given V P , there is a range of l 1 values whose lower limit lies close to the WL prediction. In a more recent experiment Han and Trivedi [8] observed that once a dendritic array has been established, V P can be increased consid...

Phase-field simulation of tilted growth of dendritic arrays during directional solidification

“…Hence, investigations of the primary spacing evolution from the initial planar instability during directional solidification are very important for the determination of the final mean primary spacing. Experiments [7,8] and simulations [28][29][30] for the planar instability and subsequent spacing adjustment have been carried out. However, few quantitative investigations have been devoted to the tilted growth of dendritic arrays.…”

“…Previous theoretical [4,5], experimental [6][7][8], and numerical [9,10] investigations have shown that a wide range of primary spacing can be selected for the same control parameters in directional solidification. Beyond this range of primary spacing, there is an abrupt transition to a new stable array due to the spacing adjustment.…”

Local control of the dendritic microstructure through perturbation

“…The necessary perturbation strength can be quantified from the response to small perturbations [11]. Here a 0 and a 1 are linear and third order growth coefficients, respectively.…”

Transient convective instabilities in directional solidification

“…6,7 In the current understanding of the problem, this transient determines the result of the selection process. 4,5 Indeed, calculations performed by using the instability of the steady state are known to yield wrong results on the selected wavelength when comparing to experiments. 8,9 While thermal and concentration gradients can induce convection in the melt, this effect has received little attention in time-dependent solidification problems, but it has nevertheless been studied by using different approximations in some solidification setups.…”

“…For instance, in dendritic growth, solutions with different wavelengths are possible for a given growth condition, 3 and the final one is dynamically selected during the initial transient states of the solidification process. 4,5 The fact is that after setting the final value of the pulling velocity, there is a transient in which a layer of solute is built up ahead of the advancing front, during which the instability can occur. 6,7 In the current understanding of the problem, this transient determines the result of the selection process.…”