Dynamical scaling for underdamped strain order parameters quenched below first-order phase transitions

In the conceptual framework of phase ordering after temperature quenches below transition, we consider the underdamped Bales-Gooding-type 'momentum conserving' dynamics of a 2D martensitic structural transition from a square-to-rectangle unit cell. The one-component or $N_{\rm OP} =1$ order parameter is one of the physical strains, and the Landau free energy has a triple well, describing a first-order transition. We numerically study the evolution of the strain-strain correlation, and find that it exhibits dynamical scaling, with a coarsening length $L(t) \sim t^{\alpha}$. We find at intermediate and long times that the coarsening exponent sequentially takes on respective values close to $\alpha=2/3$ and $\alpha=1/2$. For deep quenches, the coarsening can be arrested at long times, with $\alpha \simeq 0$. These exponents are also found in 3D. To understand such behaviour, we insert a dynamical-scaling ansatz into the correlation function dynamics to give, at a dominant scaled separation, a nonlinear kinetics of the curvature $g (t) \equiv 1/ L(t)$. The curvature solutions have time windows of power-law decays $g \sim 1/t^\alpha$, with exponent values $\alpha$ matching simulations, and manifestly independent of spatial dimension. Applying this curvature-kinetics method to mass-conserving Cahn-Hilliard dynamics for a double-well Landau potential in a scalar $N_{\rm OP}=1$ order parameter yields exponents $\alpha = 1/4$ and $1/3$ for intermediate and long times. For vector order parameters with $N_{\rm OP} \geq 2$, the exponents are $\alpha = 1/4$ only, consistent with previous work. The curvature kinetics method could be useful in extracting coarsening exponents for other phase-ordering dynamics.