Partial Equilibration Scenario in 3D athermal martensites
quenched below first-order transition temperatures

The re-equilibration of a system after a quench is a long-standing problem in non-equilibrium statistical mechanics, and a generic Partial Equilibration Scenario (PES) has been proposed by Ritort and colleagues R1 ; R2 ; R3 ; R4 After a quench, the system spreads rapidly over an energy shell $E$ in configuration space. The system lowers its energy by intermittent energy releases $\delta E<0$ to the heat bath at $T$, and then again spreads ergodically over the next energy shell $E^{\prime}=E+\delta E<E$. The iteration of the fast/ slow steps ratchets the system down to a canonical equilibrium at $T$.The distribution of energy changes $\{\delta E\}$ has a signature exponential tail R1 ; R2 ; R3 ; R4 in the heat release distribution $P_{0}(\delta E;t_{w})\sim e^{\delta E/2T_{eff}(t_{w})}$, whose slope at the origin defines the (inverse) effective temperature $\beta_{eff}\equiv 1/T_{eff}$, dependent on the post-quench waiting time $t_{w}$. The tail for negative energies is part of a shifted gaussian that peaks at positive energies. The PES has been confirmed through analytic Monte Carlo (MC) calculations of relaxing independent harmonic oscillators R1 ; R2 ; by simulations of spin-glass models and Lennard-Jones binary mixturesR3 ; R4 , and through voltage noise intermittency R3 . The MC updates of ageing harmonic oscillators R2 have to hit the ever-shrinking target of as yet unrelaxed oscillators, and these rising entropy barriers induce slow decreases in energies $1/\ln t$ and acceptances $\sim 1/t\ln t$.

It is natural to test PES ideas in interacting systems with slow relaxations. The structural glass transition R5 is an attempted equilibration that is arrested at a glassy freezing temperature $T_{G}$, pre-empting crystallization R5 ; R6 . The $T$-dependent effective viscosity increases above a Vogel-Fulcher-Tammann (VFT) singularity R7 ; R8 $e^{1/|T-T_{G}|}$, that has been studied for a century, but is not yet fully understood. Further, decays around the glass temperature have non-exponential time dependence; and non-Debye frequency responses R9 . We consider other systems with structural transitions and equilibration delays. Martensitic steel alloys, when quenched from high temperature parent austenite to low temperature ‘martensite’ R10 ; R11 , show strain domain-wallR12 patterns R13 ; R14 ; R15 .They can exhibit puzzling delays in conversion to martensite R16 ; R17 ; R18 ; R19 ; R20 , that increase rapidly with temperature: raising $T$ to nearer transition by a few percent, can raise delays from $1$ sec to $10^{4}$ seconds R18 .

We do Monte Carlo (MC) simulations of quenches in $T$, of martensitic discretized-strainR21 ; R22 ; R23 models in 3D. The model hamiltonians describe the elastic Domain Walls (DW) or mobile twin boundaries, of four 3D structural transitions, each with characteristic anisotropic Compatibility interactions between order-parameter strainsR15 . The four transitions R21 can occur in martensite-related functional materials, where OP strains will be coupled to the functionality variables. The transitions are tetragonal-orthorhombic (YBCO, superconductivity); cubic-tetragonal (FePd, shape memory); cubic-orthorhombic (BaTiO, ferroelectrics); cubic trigonal (LaSrMnO, colossal magnetoresistance).

Our 3D simulations yield post-quench evolutions as in R24 ; R25 ; R26 2D, passing through three Domain Wall (DW) states. At first there is a majority-austenite DW Vapour state of a martensite droplet in an austenite background. This converts to majority-martensite DW Liquid, of randomly wandering walls. Finally the DW Liquid orders to a DW Crystal microstructure, with the walls along preferred directions R13 , as in parallel ‘twins’.

We focus on the conversion delays of the first evolution of DW Vapour $\rightarrow$ DW Liquid, that corresponds to austenite to martensite conversions, or a rise of the martensite fraction $n_{m}(t)$ from zero to unity. As in the earlier 2D case R24 ; R25 ; R26 , a phase diagram in material parameters is obtained. In the ‘athermal’ martensite regime, there are curious ‘incubations’, or no apparent changes after a quench, terminated by sudden avalanche conversions R16 ; R17 ; R18 ; R19 ; R20 . In this regime, we find three characteristic temperatures, with $T_{1}<T_{d}<T_{0}$. Here $T_{0}$ is the meanfield transition to uniform ordering.

Avalanche conversions in a single MC time step $t_{m}=1$,occur for $T<T_{1}$, identified as the martensite start temperatureR10 ; R11 $M_{s}=T_{1}$. Quenches into $T_{1}<T<T_{d}$ show (postponed) avalanches R27 or incubation behaviour: the conversion fraction $n_{m}(t)$ remains flat at zero, until there is a jump up to unity a time $t=t_{m}(T)$. The incubation delay time $t_{m}(T)$ extends rapidly, on approaching a divergence temperature $T_{d}$.The physical picture for delays is of Vapour-droplet Fourier profile attempting entry to a negative-energy region of effective Hamiltonian spectra $\epsilon(\vec{k},T)<0$. The profile has to pass through a zero-energy $\vec{k}$-space contour at a bottleneck, like a Golf Hole edge R28 ; R29 ; R30 ; R31 . This transit passage delay differs from the familiar critical slowing down from a divergent Order Parameter length. The $T$-dependent, anisotropic bottleneck shrinks on warming, with a topological shape change at $T_{d}$, that blocks entry, so entropy barriers diverge. The precursorR14 ; R15 region $T_{d}<T<T_{0}$, where 2D results suggest dynamic tweed R27 , may be studied elsewhere.

We use generic equilibration scenario of Ritort and colleagues to analyze the statistics R1 ; R2 ; R3 ; R4 ; R31 , of the set of energy changes $\{\delta E\}$ from each MC step (usually used but not retained R32 ). These heat releases are recorded only up to an aging time $t_{w}=t_{m}(T)$, so the effective temperature depends on the quench temperature, $T_{eff}(t_{w})\rightarrow T_{eff}(T)$: non-stationary distributions become time independent. We confirm the signature PES exponential tail for all four transitions. We find the search temperature vanishes linearly $T_{eff}(T)\sim T_{d}-T$, driving an entropy barrier divergence at $T_{d}$. The martensite-conversion times are thus predicted to show glass-like VFT behaviour, $\ e^{1/T_{eff}(T)}\sim e^{1/(T_{d}-T)}$ here understood as an arrest of PES cooling. Such VFT behaviour, is extracted from simulation and experimental R16 ; R17 ; R18 data.

The plan of the paper is as follows. In Section II, we discuss the generic Partial Equilibration Scenario and our specific case of quenching across a phase transition. Section III describes for the four transitions in 3D, the discrete-strain clock-like spins and their $T$-dependent Hamiltonians. In Section IV we describe the MC simulations, with delay results from the phase space bottlenecks in Section V. Section VI shows that PES signatures are seen in all four transitions. Section VII shows that both 3D simulations and metallic alloy experiments exhibit VFT behaviour. Finally Section VIII is a summary.

Appendix A illustrates how a continuum double-well Landau free energy induces a T-dependent, Ising effective Hamiltonian. Appendix B obtains the athermal phase diagram for four transitions. Movies of post-quench DW evolutions are in Supplementary Material Videos R27 .

How do systems re-equilibrate, after a temperature quench? Ritort and coworkers have suggested that if an equilibrium canonical ensemble in thermal contact with a heat bath suddenly has its bath quenched to a lower temperature $T$, then the system goes into an ageing ensembleR1 ; R2 ; R3 ; R4 , that has a quasi- microcanonical description of states of the system. There are sequential passages through decreasing-energy configurational shells, and intervening entropy barriers in the system-search for the new equilibrium. While delays from energy barriers are from attempts through activated jumps, to cross mountains, delays from entropy barriers are from attempts through constant-energy searches, to find rare channels going through or around, the mountain R31 .

In this Section we i) outline (our understanding of) the generic Partial Equilibration Ensemble R1 ; R2 ; R3 ; R4 or PES, and ii) state how this ageing scenario is applied to our specific case, that has quenches across phase transitions, and order parameter emergence from zero.

i) Generic PES for ageing after quenches:

The PES for the equilibration process considers a system of energy $E$ in contact with a (larger) heat bath. A familiar textbook derivation R33 of the canonical ensemble, applies the microcanonical ensemble to the system plus heat bath, of constant total energy $E_{tot}=E+E_{bath}$. The total number of states is the product of the $\Omega(E)$ states of the system, and of the $\Omega_{bath}(E_{tot}-E)$ states of the bath, summed over all allowed system energies $0<E<E_{tot}$:

The system configurational entropy is $S(E)=\ln\Omega(E)$, and the inverse effective temperature is $\beta_{eff}\equiv 1/T_{eff}=dS(E)/dE$. Similarly, the inverse bath temperature is $\beta_{bath}\equiv 1/T_{bath}=dS_{bath}/dE_{bath}$. The change in total entropy depends on the system energy change $dE$ as

with the equality at equilibrium, when the system and bath temperatures are equal $\beta_{eff}=\beta_{bath}$. The Second Law inequality $dS_{tot}>0$ must hold, for the irreversible cases. After a bath quench from initial to a final $T_{bath}=T$, the system is left hotter, $T_{eff}>T$, or $\beta_{eff}-\beta<0$. The energy changes are negative $dE=dQ<0$, with heat released by the system to the bath.

At equilibrium, the terms in the sum of Equ (1) are dominated by clusters of energy shells selected R33 by a sharp peak, arising from the product of a rising number of system states $\Omega(E)$, and a falling bath factor $e^{-E/T_{bath}}$. The peak width is the energy fluctuations from stochastic system-bath exchanges. These equilibrium ideas describe states, not processes.

The Partial Equilibration Scenario postulates a plausible post-quench non-equilibrium process for the system to evolve between the initial and the final equilibrium state. A sudden change in bath temperature or quench, will induce a shifted peak, around a different equilibrium state. The post-quench system, initially stranded in non-optimum states, is visualized as moving through the sequentially lower energy shells of Equ (1) in its search for the shifted peak, tracked by an ageing time $t_{w}$.

The Scenario postulates that the system i) rapidly spreads ergodically through all states of a shell of energy E, and ii) slowly dribbles out energy $\{\delta E\}$ to the ever-present energy bath. Since the system is partially equilibrated in the quasi-microcanonical shell, the equilibrium definitions can be retained, of the shell entropy $S(E)$ and its energy derivative $1/T_{eff}(t_{w})$. The back-and-forth energy exchanges to rapidly surmount internal energy barriers and explore all shell configurations, are summoned by the system from the bath (‘stimulated’). The slow changes on passages to a lower-energy shell, are releases by the system to the bath (’spontaneous’).

Fig 1 is a schematic of the Partial Equilibration Scenario. The successive shells have lower energy $E^{\prime}<E$ and hence lower number of configurations $\Omega(E^{\prime})<\Omega(E)$ and entropies $S(E^{\prime})<S(E)$. There is a generic entropy barrier $S_{B}\equiv-\Delta S=-\ln[\Omega(E^{\prime})/\Omega(E)]>0$ to finding the rarer states. Key seeks lock: most attempts fail.

From an ageing Fluctuation Relation R3 the nonequilibrium energy-change probability R1 ; R2 ; R3 ; R4 is a peak at the origin, times an exponential tail for negative changes. This generic PES signature tail depends on the ageing time through the the effective temperature $T_{eff}(t_{w})$, that scales the heat releases:

with an even prefactor ${P^{(+)}}_{0}(\delta E;t_{w})$.

In an important result, other effective temperatures, from the Fluctuation-Dissipation Theorem; and from non-equilibrium fluctuations of system variables, are shown to be equal to the PES effective temperature R1 : there is only one $T_{eff}$.

ii) Specific PES from quenching across a transition.

For our case of quenching to $T$ across a first-order transition, the Order Parameter (OP) has to rise from zero, and so the wait times $t_{w}$ to reach OP marker events will depend on the quench temperature, $t_{w}=t_{m}(T)$.The effective temperature and PES distribution will thus also depend on the quench temperature, $T_{eff}(t_{w})\rightarrow T_{eff}(T)$ and $P_{0}(\delta E;t_{w})\rightarrow P_{0}(\delta E,T)$. The even prefactor $P^{(+)}_{0}(\delta E,T)$ can be exponentiated and expanded to quadratic order, Equ (3) is then a gaussian peaked at the origin, times an exponential falloff. Completing the square yields a PES signature of a shifted gaussian, peaked at positive mean changes $M(T)=<\delta E>$, and scaled in $T_{eff}$:

For small heat releases $\delta E=-|\delta E|<0$, the PES distribution takes a Boltzmann-like form $P_{0}\simeq e^{-\frac{1}{2}\beta_{eff}(T)|\delta E|}$. This gives a physical meaning to the effective temperature: it is a search range for accessible energy shells. If $\beta_{eff}\rightarrow 0$, entropy barriers collapse, and passages are immediate. If $T_{eff}\rightarrow 0$, then entropy barriers diverge, and passage-searches freeze.

We postulate that the OP-related bottlenecks can be of two types, depending on the depth of the quench. a) The DW Vapour $\rightarrow$ Liquid delays are attributed to phase space bottlenecks R27 , suggested by concepts in protein folding R28 ; R29 ; R30 . Hamiltonian energy-spectrum contours in Fourier space of zero energy are like a $T$-dependent Golf Hole (GH) edge, with a negative-energy Funnel region inside it leading to the final state R28 . The entropic delays are from finding and entering the bottleneck.
b) For deeper quenches, the DW Liquid $\rightarrow$ Crystal delays could be conceptually related to spin facilitation models R27 ; R34 ; R35 , through the $T$-dependent sparseness of austenitic-hotspot dynamical catalysts, or other facilitating fields R20 ; R27 .

The transition-specific, derived effective Hamiltonians have been presented in detailR15 ; R21 ; R22 ; R23 ; R24 ; R25 ; R26 , and we just outline as conceptual background: A) Strains and Compatibility constraints. B) Reduction of continuum strains to discrete-strain ‘pseudo-spins’. C) Reduction of continuum strain free energies to effective ‘spin’ Hamiltonians.

It is useful to define $N_{OP}$ the number of components of the OP strains; and $N_{V}$ the number of Landau ‘variant’ minima at nonzero OP strain values. In terms of discretized strains, $N_{OP}$ is the number of vector spin components, that can point in $N_{V}$ variant directions. A double-well Landau free energy for a scalar magnetization maps onto (Appendix A) an Ising model with $N_{OP}=1$ and $N_{V}=2$. We consider four first-order transitions R21 with $N_{OP}=1,2,2,3$. The nonzero, unit-magnitude variant vectors point respectively to corners of symmetry-dictated polyhedra with $N_{V}=2,3,6,4$ corners, inscribed in a unit circle or unit sphere: a geometrically pleasing sequence of line, triangle, hexagon, and tetrahedron. These transitions are respectively, tetragonal-orthorhombic, cubic-tetragonal, cubic-orthorhombic, and cubic-trigonal.

Simulations were done on models for four transitions in 3D. The initial state $t=0$ is $2\%$ randomly and dilutely seeded martensite cells, in an austenite sea of $\vec{S}=0$. Evolutions proceed at quench temperatures $T<T_{0}$. Typical parameters are $T_{0}=1$; $\xi_{0}^{2}=1$; $T_{c}=0.81$ to $0.97$; $A_{1}=1$ to $85$; $E_{0}=3,4,5,6$; $N=L^{3}=16^{3}$; $N_{runs}=100$; and holding times $t_{h}=10^{4}$ MC sweeps, each over $N$ sites.

The Compatibility kernels arise from the non-OP harmonic terms, with $(6-N_{OP})$ elastic constants. For all transitions, we specify the fixed ratio of other non-OP elastic constants, to $A_{1}$. For the tetragonal-orthorhombic transition, with $N_{OP}=1$ and $S_{2}$ as the OP spin, the non-OP elastic constants are the other deviatoric constant $A_{3}$, and the three shear constants $A_{4},A_{5},A_{6}$, set to be $A_{3}=A_{4}=A_{5}=A_{6}=A_{1}/2$. Similarly, for the cubic-tetragonal and cubic-orthorhombic cases with $N_{OP}=2$ and $(S_{3},S_{2})$ as the OP spins, the non-OP constants are set to be $A_{4}=A_{5}=A_{6}=A_{1}/2$. Finally, for the cubic-trigonal case with $N_{OP}=3$ and the shears $(S_{4},S_{5},S_{6})$ as the OP, the constants are set as $A_{2}=A_{3}=A_{1}/2$.

The standard MC procedure R32 is followed, with an extra data retention R1 ; R2 ; R3 ; R4 of energy changes.
0. Take $N$ lattice sites, each with a vector spin of $N_{OP}$ components, in one of $N_{V}+1$ possible values (including zero) at MC time $t$. Each $\{{\vec{S}}(\vec{r})\}$ set is a ‘configuration’. With $n_{m}=0$ or $1$ for uniform austenite or martensite, the average martensite fraction is $n_{m}(t)\leq 1$. The conversion time $t_{m}$ is defined as when R24 $n_{m}(t_{m})=1/2$.
1. Randomly pick one of $N$ sites, and randomly flip the spin on it to a new direction/value, and find the (positive or negative) energy change $\delta E$.
2. If $\delta E\leq 0$, then accept the flip. If $\delta E>0$, then accept flip with probability $e^{-\delta E/T}$. Record this spin-flip $\delta E$, that is usually not retained after use.
3. Repeat steps 1 and 2. Stop after $N$ such spin-flips. This is the $t+1$ configuration with $n_{m}(t+1)$.
4. We collect all recorded $\{\delta E\}$ over each MC sweep of every run while tracking $n_{m}(t)$. The collection is only up to a waiting time equal to the martensite conversion time. $t=t_{w}\leq t_{m}(T)\leq t_{h}.$ We do six quenches, from $T=T_{1}$ upwards to $T_{d}$.

A single-variant athermal martensite droplet or ‘embryo’ R11 can form anywhere, and after a local conversion at a waiting time $t_{w}=t_{m}$, can propagate rapidly to the rest of the system R11 ; R19 . Hence it is the mean rates ${\bar{r}}_{m}$ (or inverse times), that are averaged over $N_{run}$ runs, analogous to total resistors in parallel determined by the smallest resistance. The mean times ${\bar{t}}_{m}$ are defined as the inverse mean ratesR24 .

For an un-successful $n$-th run that does not convert in a holding time $t_{h}$, the rate $r_{m}(n)$ could be assigned a value of either $1/t_{h}$ (conversion right after cutoff) or $0$ (conversion never occurs). We choose the $1/t_{h}$ cutoff, that shows up as a flattening of the mean rate near $T_{d}$. An extrapolation of the linear part of $r_{m}(T)$ to the temperature axis yields a valueR24 for $T_{d}$.

For the four transitions, we consider i) Delays from sluggish Domain Walls; ii) Bottlenecks in Fourier space.