Kinematic Scaling and Crossover to Scale Invariance in Martensite Growth

Abstract

Scale invariance in acicular martensites emerges as a consequence of a competition between the front velocity $v$ and the rate of nucleation $I$ of the martensite grains. A renormalization group for the stationary probability distribution of the martensite grain size, $P(l)$, has two fixed points---an unstable fixed point at $v/I\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0$, associated with a gamma distribution, and a stable (critical) fixed point at $v/I\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\ensuremath{\infty}$ with a L\'evy distribution, $P(l)\ensuremath{\sim}{l}^{\ensuremath{-}\ensuremath{\alpha}}$. Crossover functions and exponents governed by the unstable fixed point describe $P(l)$ at intermediate $v/I$. Acicular martensites thus represent a novel example of a flow towards self-organized criticality. Our results provide a method to compute dynamic quantities ( $v$ or $I$) from static optical micrographs of martensite grains.