Spin-glass nature of tweed precursors in martensitic transformations

Abstract

Many displacive solid-solid transformations, despite being first order, show pronounced precursor effects, such as the mesoscopic, micron-scale, "tweed'' pattern seen in shape-memory alloys. We model this tweed theoretically using a nonlinear, nonlocal elastic free energy, and argue that quenched concentration inhomogeneities drive the local tweed modulations. We report (1) the construction of a model for {l l}/(lT) shear transformations in square systems, (2) a simulation including concentration inhomogeneities, and (3) a mapping of the disordered 2D martensite onto an infinite-range spin model, identifying tweed with the spin-glass phase.PACS numbers: 8I.30.Kf, 61.70.Wp, 75.10.Nr In textbook first-order phase transitions critical fluctuations are negligible: Water shows no hint of incipient solidity above 0°C. First-order solid-solid displacive transformations are strikingly different; precursors are common for tens through hundreds of degrees away from the transitions. We study here the tweed pattern [l] observed in electron micrographs well above the transition temperature in the so-called medium and weak martensites, including the shape-memory alloys FePd [2] and NiAl [3], the superconducting ^-15's [4], and high-T^ Y-Ba-Cu-Co-O, and Y-Ba-Cu-Al-O [5,6]. The tweed appears as a characteristic cross-hatched pattern. X-rayand electron-dilTraction measurements [2,3,7-10] indicate that tweed is a local mixture of undeformed and deformed regions, and that it arises from a local {llO}/ <lTO) shear [1,3].This paper considers the two-dimensional square to rectangular transition as a simple model for the threedimensional cubic to tetragonal transition in metallic systems. Near the transition, these materials show a marked softening [11] of the elastic constant C' = (Cii -Ci2)/2 [the resistance to rectangular (deviatoric) shears], and have large elastic anisotropy [12]; hence C44/C and (Cii+2Ci2)/C' are both large. We thus consider the formal limit where this elastic anisotropy is infinite; i.e., the only allowed deformations are rotations, translations, and volume-preserving rectangular stretches (deviatoric strain). We subsequently relax these constraints.Our theoretical model formally supports the proposal that the existence of some disorder is of central importance in the tweed precursor [13]. Suggestively, in pure zirconium and titanium, central-peak precursors do not appear above the w phase transition, but do appear in samples alloyed with as little as 1% oxygen or nitrogen [14]. We are further motivated by the drastic dependence of the martensitic transition temperature on concentration. For example, in Fei -^^^Pd^^, [2], the martensitic transformation temperature changes from 0 to 300 K as the concentration 770 varies from 32% to 29%. Within a local region in any alloy, the local concentration vjix.y) will exhibit static, quenched-in, statistical variation about the nominal concentration r/o; we consider the proposal that these inhomogeneities drive the local deformations in the material, forming tweed.Our or...

Complexity at Mesoscale

“…Figure 6 shows formation of low temperature T < T 0 parallel-domain structures with only bulk compatibility included. Note that the structures have the proper 45 • orientation (as in [17]) but are not true twins: there is no well defined width scale. In Fig.…”

“…For (Fourier-expandable) strains ǫ(r) in the bulk, one obtains via the Lagrangian multiplier formalism [17] that e 1 ( k), e 2 ( k) are proportional to ǫ( k), with k dependent coefficients. An OP strain-strain potential…”

“…The OP is one or more components of a strain tensor ε µν (µ, ν = x, y, z); not a true scalar. For simplicity, we consider here a 2D square-to-rectangle transformation [16,17,18,19], with a rectangular or deviatoric strain as the OP. The non-OP strain components are implicit functions of the OP, through a compatibility (differential) equation [20].…”

“…The compression-shear (CS) strains e 1 (r), e 2 (r) are written in terms of the order parameter ǫ(r) by solving the elastic compatibility (differential) equation. The analysis with six strain tensor components in 3D can be carried out [22], but for simplicity, we confine our discussion to 2D with a compatibility constraint, satisfied at all times [17,23]:…”

Mechanochemical spinodal decomposition: a phenomenological theory of phase transformations in multi-component, crystalline solids

“…We extend this gradient dependence to the strain measures ∇e, as did Barsch and Krumhansl 8 following Toupin. 9 (Also see Karatha et al, 13,14 for treatments using infinitesimal and finite strain, respectively.) These gradient dependences appear in a non-uniform free-energy density Gðc; e; ∇c; ∇eÞ.…”