Self-similar evolution of a precipitate in inhomogeneous elastic media

In this paper, we present linear theory and nonlinear simulations to study the self-similar growth and shrinkage of a precipitate in a 2D elastic media. This work is motivated by a series of studies by Li et al., where the existence and morphological stability of self-similarly evolving crystals were demonstrated in a diffusional field. Here, we extend the theory and simulations to account for solid-state phase transformations where elasticity plays an important role in regularizing the evolution of the precipitate. For given applied strain or stress boundary conditions, we show that depending on the mass flux entering/exiting the system, there exist critical scalings of flux and elasticity at which compact self-similar growth/shrinkage occurs in the linear regime. We then develop a spectrally accurate boundary integral method combined with a rescaling scheme to investigate the effects of nonlinearity and the morphological stability of these linear self-similar precipitates. Our numerical results reveal that at long times there exists nonlinear stabilization that leads the precipitate to evolve to compact universal limiting shapes selected by the applied stress and mass diffusion flux. These theoretical and numerical results suggest that the classical Mullins–Sekerka instability, which drives precipitates to acquire dendritic or dense-branching morphologies, can be controlled. This could potentially provide a new route to controlling material microstructures.

► Develop a self-similar theory for a solid/solid diffusion problem with elasticity. ► Discover stable, nonlinear self-similar shrinking in addition to self-similar growth. ► Provide a new route to controlling material microstructures.