Convection effect on the diffusion-limited-aggregation fractal: Renormalization-group approach

A nonlinear effect of convection on the diffusion-limited-aggregation (DLA) fractal is investigated by use of the real-space renormalization-group method. The renormalization-group equations are derived for two parameters: the diffusion constant on the interface and the Péclet number P^ representing a ratio of the convection effect to the diffusion effect. It is found that the correlation length ξ (limiting the validity of the self-similarity) diverges as ξ∼‖P^${\Vert }^{\mathrm{-}\nu }$, where ${\nu }^{\mathrm{-}1}$=ln(∂P^ ’/∂P^${)}_{\mathrm{P}\mathrm{^}=0\mathrm{}}$/lnb (b is a scale factor). The correlation-length exponent ν is approximately given by ν=d${\mathrm{}\left(0\right)\mathrm{}}^{\mathrm{-}1}$, where d(0) is the surface fractal dimension of the DLA. The aggregate is the DLA fractal on smaller length scales than the correlation length, but becomes a nonfractal structure on large length scales. The renormalization-group method is applied to the formation of solidification patterns in two cases: one is the case of an undercooled melt and the other the case in which the aggregate acts as a heat sink. In the undercooled-melt case the convection acts as a destabilizing force on the interface, but, for the aggregate with a heat sink, the convection acts as a stabilizing force. It is shown that the DLA fractal crosses over to one of the two distinct nonfractal structures depending on the direction of the heat flux.