Superlattices and Microstructures, Vol. 3, No. 6, 7987 599 zyxwvu INTERFA CIA L PATTERN FORMATION FAR FROM EQUILIBRIUM E. Ben-JacoblIZ, P. Gariki” . and D. Grier’ ‘Department of Physics Randall Laboratory University of Michigan Ann Arbor, Michigan 48109 ‘Department of Condensed Matter Physics School of Physics and Astronomy Tel Aviv University 69978 Tel Aviv Israel (Received 19 May 1987) Over the past few years diffusion-controlled systems have been shown to share a common set of interfacial morphologies. The singular nature of the microscopic dynamics of surface tension and kinetic growth far from equilibrium are critical to morphology selection, with special importance attributed to the anisotropy of these effects. The morphologies which develop can be organized via a morphol- ogy diagram according to the driving force and the effective anisotropy. We focus on the properties of the dense-branching morphology (DBM) which appears for sufficiently weak effective anisotropy, and the nature of morphology transitions between the DBM and dendritic growth stabilized by either surface tension or ki- netic effects. The DBM is studied in the Hele-Shaw cell, and its structure analyzed by linear stability analysis. A comparison is made between the power spectrum of the structure and the stability analysis. We then provide a detailed account of the morphology diagram and morphology transitions in an anisotropic Hele-Shaw cell. Theoretically the question of morphology transitions is addressed within the boundary-layer model by computing selected velocities as a function of the un- dercooling for different values of the surface tension and the kinetic term. We argue that the fastest growing morphology is selected whether it is the DBM. surface tension dendrites, or kinetic dendrites. A comparison is made with our experimental results in electrochemical deposition for the correspondence between growth velocities and morphology transitions. Over the past several years unifying principles governing the development of interfacial patterns have been arrived at after intensive study of theoretical models and experimental systems. It is now recog- nized that structures whose growth is diffusion-con- trolled share a common set of possible interfacial pat- terns and that a correspondence can be made between the control parameters which determine the selected morphology in these systems. Although simple scal- ing between systems may not be realizable, qualita- tive rules of morphology correspondence have been established by insightful identification of these con- trol parameters. The macroscopic controls are self- evident, e.g., undercooling in solidification, supersat- uration in precipitation, voltage drop in electrochem- ical deposition, or pressure differential in viscous fin-- gering. The critical advance in understanding has been the discovery of the mathematically singular na- ture of the microscopic dynamics of crystalline aniso- tropy, growth kinetics, and surface tensionlm3. Thus, it is now possible to speak broadly and say that when noise dominates the dynamics of the interface, the pattern which evolves has structure on many length scales, is likely to be fractal, and to resemble the com- puter generated diffusion-limited aggregation (DLA) morphology 4.5; in the presence of surface tension and zyxwvutsrqpo 0749-6036/87/060599+ zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA 17 $02.00/O 0 1987 Academtc Press Limited