We study the growth (‘‘coarsening’’) of domains following a quench from infinite temperature to a temperature T below the ordering transition. The model we consider is an Ising ferromagnet on a square or cubic lattice with weak next-nearest-neighbor antiferromagnetic (AFM) bonds and single-spin-flip dynamics. The AFM bonds introduce free-energy barriers to coarsening and thus greatly slow the dynamics. In two dimensions, the barriers are independent of the characteristic length scale L(t), and therefore the long-time (t→∞) growth of L(t) still obeys the standard ${\mathit{t}}^{1/2}$ law. However, in three dimensions, a simple physical argument suggests that for quenches below the corner-rounding transition temperature, ${\mathit{T}}_{\mathrm{CR}}$, the barriers are proportional to L(t) and thus grow as the system coarsens. Quenches to T<${\mathit{T}}_{\mathrm{CR}}$ should, therefore, lead to L(t)∼ln(t) at long times. Our argument for logarithmic growth rests on the assertion that the mechanism by which the system coarsens involves the creation of a step across a flat interface, which below ${\mathit{T}}_{\mathrm{CR}}$ costs a free energy proportional to its length. We test this assertion numerically in two ways: First, we perform Monte Carlo simulations of the shrinking of a cubic domain of up spins in a larger sea of down spins. These simulations show that, below ${\mathit{T}}_{\mathrm{CR}}$, the time to shrink the domain grows exponentially with the domain size L.

This confirms that the free-energy barrier, ${\mathit{F}}_{\mathit{B}}$(L,T), to shrinking the domain is indeed proportional to L. We find excellent agreement between our numerical data and an approximate analytic expression for ${\mathit{F}}_{\mathit{B}}$(L,T). Second, to be sure that the coarsening system cannot somehow find paths around these barriers, we perform Monte Carlo simulations of the coarsening process itself and find strong support for L(t)∼ln(t) at long times. Above ${\mathit{T}}_{\mathrm{CR}}$ the step free energy vanishes and coarsening proceeds via the standard ${\mathit{t}}^{1/2}$ law. Thus, the corner-rounding transition marks the boundary between different growth laws for coarsening in much the same way that the roughening transition separates different regimes of crystal growth. We also find logarithmic coarsening following a quench in a two-dimensional ‘‘tiling’’ system, which models the corner-rounding transition of a [111] interface in our three-dimensional model. However, if instead of quenching, we cool the system slowly at a constant rate Γ, we find the final length scale L to have a power-law dependence on 1/Γ, i.e., L∼${\mathrm{\Gamma }}^{\mathrm{-}1/4}$, in accordance with a theoretical argument. The predictions concerning the dynamics of the tiling model should, in principle, be experimentally testable for a [111] interface of sodium chloride.

• Received 15 April 1992

DOI:https://doi.org/10.1103/PhysRevB.46.11376