ASYMPTOTIC AND EXACT SELF-SIMILAR EVOLUTION OF A GROWING DENDRITE

In this paper, we investigate numerically the long-time dynamics of a two-dimensional dendritic precipitate. We focus our study on the self-similar scaling behavior of the primary dendritic arm with profile x ∼ t^α1 and y ∼ t^α2, and explore the dependence of parameters α₁ and α₂ on applied driving forces of the system (e.g. applied far-field flux or strain). We consider two dendrite forming mechanisms: the dendritic growth driven by (i) an anisotropic surface tension and (ii) an applied strain at the far-field of the elastic matrix. We perform simulations using a spectrally accurate boundary integral method, together with a rescaling scheme to speed up the intrinsically slow evolution of the precipitate. The method enables us to accurately compute the dynamics far longer times than could previously be accomplished. Comparing with the original work on the scaling behavior α₁ = 0.6 and α₂ = 0.4 [Phys. Rev. Lett. 71(21) (1993) 3461–3464], where a constant flux was used in a diffusion only problem, we found at long times this scaling still serves a good estimation of the dynamics though it deviates from the asymptotic predictions due to slow retreats of the dendrite tip at later times. In particular, we find numerically that the tip grows self-similarly with α₁ = 1/3 and α₂ = 1/3 if the driving flux J ∼ 1/R(t) where R(t) is the equivalent size of the evolving precipitate. In the diffusive growth of precipitates in an elastic media, we examine the tip of the precipitate under elastic stress, under both isotropic and anisotropic surface tension, and find that the tip also follows a scaling law.