Three-dimensional crystal growth—I: linear analysis and self-similar evolution

Abstract

In this paper, Part I of our study, we revisit the linear analysis (J. Appl. Phys. 34 (1963) 323; J. Appl. Phys. 36 (1965) 632; in: H.S. Peiser (Ed.), Crystal Growth, Pergamon, Oxford, 1967, p. 703) of the quasi-steady diffusional evolution of growing crystals in 3-D. We focus on a perturbed spherical solid crystal growing in an undercooled liquid with isotropic surface tension and interface kinetics. We investigate the relation between the far field flux of temperature and undercooling in the far field. In 3-D, the flux scales with the undercooling and with the instantaneous size of the crystal; this behavior is qualitatively different from 2-D, where there is no dependence on the size. As a consequence of this peculiarity, we demonstrate using linear analysis that in 3-D there exist critical conditions of flux at which self-similar evolution occurs. This leads to nonspherical, shape-invariant growing crystals. Rather than that using the concept of critical radius (Mullins and Sekerka, 1963; Coriell and Parker, 1965), we repose the problem in terms of a critical flux. The critical flux increases with increasing wave number of the perturbation, and separates regimes of stable and unstable growth, where stable growth implies that the perturbation decays with respect to the underlying sphere. The interfacial kinetics have a strong stabilizing effect (Coriell and Parker, 1965, 1967), which is explored in detail here. These results demonstrate that the classical Mullins–Sekerka (Mullins and Sekerka, 1963) instability, that arises in the presence of constant undercooling, can be suppressed by maintaining near-critical flux conditions (our formulation reduces to that in Mullins and Sekerka (1963) under constant undercooling). Correspondingly, there is little creation of unstable modes during growth and unstable growth is very constrained or completely eliminated. Near-critical flux conditions can be achieved by appropriately varying the undercooling in time; thus this work has important implications for shape control in processing applications. Experiments are currently being designed (by Stefano Guido and coworkers (Personal communication) at the University of Naples) to test this possibility. Moreover, in Part II (Cristini and Lowengrub, in preparation) of our study, we will investigate the nonlinear evolution using adaptive boundary-integral simulations.