Monte Carlo simulations of strain pseudospins: Athermal martensites, incubation times, and entropy barriers

Abstract

We study martensitic transition kinetics through temperature-quench Monte Carlo simulations for a square/rectangle ferroelastic transition, described by a Hamiltonian of three-state pseudospins S, without extrinsic disorder. Here S = 0 for high-temperature austenite, and S = ±1 for the two martensite variants. The temperaturedependent pseudospin Hamiltonian comes from the total scaled free energy functional, evaluated at the three minima of Landau polynomials in order-parameter strains. It includes power-law anisotropic interactions from the St. Venant compatibility constraint, which orient the elastic domain walls in a symmetry-breaking diagonal direction. We find that temperature-time-transformation (TTT) plots for domain-wall phase evolution have phase crossover temperatures, which are understood through an effective-droplet energy parametrization. For temperature cycling through the phases, there are hysteretic peaks in physical quantities. For temperature quenches, a "vapor" of martensitic droplets converts at a time t m (T ) to a vibrating "liquid" of bidiagonal domain walls, which then orient at a time t C (T ) to a static "crystal" of single-diagonal martensitic twins, which can have bound residual austenite. Focusing on the conversion time t m , we find a material-parameter phase diagram, which has regions of nonactivated "athermal" and activated "isothermal" martensites. In an athermal, nonactivated regime, there are explosive austenite-martensite conversions at temperatures below a residual-austenite spinodal in the TTT diagram, while above it, there are conversion tails, as in experiment. We find t m (T ) has a quasi-universal Vogel-Fulcher divergence at transition, with a log-normal conversion-rate distribution. The incubation times t m , t C are attributed to entropy barriers, with signatures of flat energies, during pathway searches for finite-scale transition textures, which are explicitly identified through textural and internal-stress snapshots. In a glasslike energy landscape picture, these entropic pathways to many locked-twin states dominate those to a single-variant martensite state, of almost the same energy density. Other transitions in 2D and 3D can be similarly studied. More generally, the models could be used to explore conceptual issues of how systems equilibrate after a deep quench.

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“…In the temperature-timetransformation (TTT) diagram with temperature on Xaxis and time on Y -axis 17,20,21 : (i) athermal martensites have rapid conversions below a transition temperature and delays above it as in experiments, with Vogel-Fulcher divergences that are insensitive to Hamiltonian energy scales, understood from the presence of non-activated entropy barriers; and (ii) isothermal ones have U-shaped conversion curves, as expected to arise from activated energy barriers. The shape of TTT curves transform from rapid to slow (or athermal to isothermal) at low temperatures on changing the material elastic stiffness constant 17,20,21 .…”

“…A subset of physical strain components (N OP ) are the order parameters (OP) and the remaining non-OP strains are minimized subject to a no-defect Saint-Venant compatibility constraint that induces scale-free, power-law, anisotropic interactions which orients the domain walls in preferred crystallographic directions 16 . Martensites can have exponentially large number of multivariant twinned states or nonuniform metastable local minima competiting with a single uniform global minimum 17 . Martensites are classified based on conversion times 18,19 : (i) athermal, which are expected to have rapid austenite-to-martensite conversions in milli-seconds on quenching below a transition temperature and no conversions above it; and (ii) isothermal can have slow conversions in minutes or hours.…”

“…Systematic temperature-quench Monte Carlo (MC) simulations are performed on strain-pseudospin clockzero model Hamiltonians in 2-spatial dimensions for scalar-OP (N OP = 1) square-rectangle (SR, N v + 1 = 3) transition 17,20 , and vector-OP (N OP = 2) trianglecentered rectangle (T CR, N v + 1 = 4), square-oblique (SO, N v + 1 = 5), and triangle-oblique (T O, N v + 1 = 7) transitions 21 and found both isothermal and athermal martensite parameter regimes. The pseudospin strain textures obtained from MC simulations and local meanfield 17,[20][21][22] are in very good agreement with experiments [23][24][25] .…”

“…The VogelFulcher conversion-delays that are insensitive to Hamiltonian energy scales are found to have Log-normal distributions that are signatures of rare events 33 . The number of successful conversions, that are also insensitive to energy scales, vanishes where the entropy barriers diverges 12,17,20 . In the incubating state, the crossing of entropy barrier is identified in energy occupancy distributions, MC acceptance fractions and heat and work releases as the structure factor enters into the Brillouin zone (BZ) golf-course through searches for rare energylowering pathways and elastic photocopying 34,35 .…”

“…We define conversion time t = t m when n m (t m ) = 0.5 or 50% 17,20,21,26 . On quenching to different temperatures, we find the conversion success-fraction 17 φ m that is the number of successful conversion pathways to martensite out of N runs .…”