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

Abstract

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−1[2α−1+ln(b/2πρ0)−α lnα−(1−α)ln(1−α)−α ln(QH/R)−(1−α)ln(H/R)]+(H2/12R2){8(1−α)tan−1Q+(1−4α)Q+[(3−6α+3α2)Q−(1−4α+3α2)Q3]ln(1+Q−2)+[3α2Q+(9α2−6α)Q−1]ln(1+Q2)},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.