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

Source: [Dynamical scaling for underdamped strain order parameters quenched below first-order phase transitions](<https://www.sota2.com/research/paper/68fa34cf1e209131ce89c56f>)

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Data updated: 2025-10-24T20:53:51.966Z

Authors: N. Shankaraiah, Awadhesh K. Dubey, Sanjay Puri, Subodh R. Shenoy

Published: 2016-12-06T10:29:57.000Z

## Abstract

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.

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## Original sources and implementation

[Original paper](<https://arxiv.org/abs/1612.01737>)

[PDF](<https://arxiv.org/pdf/1612.01737.pdf>)