Role of fluctuations in viscous fingering and dendritic crystal growth: a noise-driven model with non-periodic sidebranching and no threshold for onset

Abstract

A noise-driven model is developed to describe the role of fluctuations in side- branch phenomena in growth patterns for the fluid displacement problem and for dendritic crystal growth. Simulation results are compared with recent experiments on NH,Br den- drites. It is found that the RMS sidebranch amplitude is an exponential function of distance from the tip, with no apparent onset threshold. Moreover, the sidebranches are non-periodic (at all distances from the tip) with apparently random variations in amplitude. What is the physical mechanism whereby sidebranches 'spontaneorrsly' appear a short distance behind the growing tip of a dendritic form? For generations, this question has fascinated scientists in a variety of fields, ranging from metallurgy and crystal growth on the one hand to botany and embryology on the other. Recently interest has focused on extremely simple systems that spontaneously develop sidebranches. It has been found that, when a low-viscosity fluid displaces a high-viscosity anisotropic fluid under pressure, a sidebranch pattern develops that resembles dendritic crystal growth. For example, Buka et a1 (l) use air to displace a viscous solution of a nematic liquid crystal. The anisotropy can also be in the medium itself. Horvath er a1 (2) have shown that a single scratch in one wall of the confining Hele-Shaw cell is sufficient to produce a dendritic pattern. Similarly, Ben-Jacob et a1 (3) find dendritic fluid patterns when they scratch a triangular lattice onto the cell. Most surprising, perhaps, is the observa- tion of Couder et af (4) that dendritic growth patterns can occur when the anisotropy is provided by a simple bubble of air on the tip of the growing viscous finger. Can these similarities between diverse systems be understood in terms of underlying physical principles common to all? Here we tentatively suggest a physical model that seems to account for such sidebranch phenomena. Although fluid displacement phenomena are striking, a larger number of quantitative results is known for dendritic crystal growth. Hence we shall focus attention on the latter. In paiticular, Dougherty et a1 (5) have recently made a detailed analysis of photographs of growing NH4Br dendrites, taken at 20s intervals. They have found three surprising results: (i) sidebranch positions are non-periodic at any distance from the tip, with almost random variations in both phase and amplitude, (ii) sidebranches on opposite sides of the dendrite are essentially uncorrelated in position and length and (iii) the sidebranch amplitude is an exponential function of distance from the tip, with no apparent onset threshold distance.