Calculations of the Energy of Subgrains in a Lattice-Parameter Gradient

The dislocation‐associated energy of subgrains in a lattice‐parameter (concentration) gradient is calculated using two‐dimensional isotropic elasticity and neglecting almost all subgrain‐subgrain interactions. The subgrains are rectangular and are bounded by regular arrays of pure edge dislocations with Burgers vector normal to the gradient. These boundaries are either lattice‐parameter step boundaries or tilt boundaries. The energy per unit volume normalized by μ/4π(1−ν) is $$R^{\text{−1}}\text{[2α−1+}\ln {\text{(b/2πρ}}_0\mathrm{)-\alpha \hspace{0.17em}}\ln \text{α−(1−α)}\ln \text{(1−α)−α\hspace{0.17em}}\ln \text{(QH/R)−(1−α)}\ln {\mathrm{(H/R)]+(H}}^2{\text{/12R}}^2\text{){8(1−α)}{\tan }^{\text{−1}}{\text{Q+(1−4α)Q+[(3−6α+3α}}^2{\text{)Q−(1−4α+3α}}^2{\mathrm{)Q}}^3]\ln {\text{(1+Q}}^{\text{−2}}{\text{)+[3α}}^2{\text{Q+(9α}}^2{\text{−6α)Q}}^{\text{−1}}]\ln {\text{(1+Q}}^2\mathrm{)\},}$$ where ρ₀ is the dislocation core radius, b is its Burgers vector, α is the fraction of dislocations in tilt boundaries, bH is the height of the subgrain (parallel to the concentration gradient), Q is its length to height ratio, bR is its relaxed radius of curvature due to its lattice parameter gradient, μ is the shear modulus, and ν is Poisson's ratio. Two low‐energy regimes are identified: first at low H, high Q, and α=0, the lattice‐step boundary regime; second at high H, low Q, and α=1, the tilt boundary regime. The latter appears to be slightly lower in energy.