The elastic contribution to ledge growth on coherent interfaces in the system of a cube-shaped γ′ precipitate in nickel alloys. Part I: The elastic fields

In the framework of linear elasticity we found a solution to the problem of a ledge on a coherent boundary of a second phase particle with an arbitrary misfit. The solution is obtained by the superposition of elementary precipitates: a cube-shaped precipitate and a plate-shaped one. It is applied to the study of the elastic fields associated with a ledge on the face of a cube-shaped γ′ precipitate in a nickel-base alloy and to follow the evolution of the fields as the ledge grows. The solution makes it evident that the perturbations to the elastic fields given rise are by the presence of a ledge are just the fields generated by the plate-shaped precipitate. Although the transformation strains of γ′ are pure dilation, the perturbation to the elastic fields is anisotropic and its source is traced back to the intrinsic material anisotropy and to geometric anisotropy, associated with the shape of the plate (defined by the ratio of the riser height, c, to the terrace width, a). Hence the contribution of the ledge to the elastic fields evolves with the lateral growth of the ledge, unlike the case of ledges on free surfaces. The trace of the stress tensor in the matrix in front of the ledge determines the interaction of the ledge with solute atoms. In the case of dilatational transformation the non-vanishing of the trace of the stress tensor is due solely to the material anisotropy. Its range and magnitude are, however, determined by the shape of the precipitate. In the matrix adjacent to a simple cube-shaped precipitate the trace is compressive, implying repulsive interaction with large solute atoms; its absolute value decreases near the edges of the cube. A ledge makes a tensile contribution to the trace ahead of its riser, forming a deep trap for large solute atoms and elasically preferred routes for their diffusion toward the ledge. The tensile trace contribution is maximal when c/a ∼ 0.04 and at that state the rate of ledge growth is expected to be maximal.