Subodh R. Shenoy

Three-Dimensional Elastic Compatibility and Varieties of Twins in Martensites

We model a cubic-to-tetragonal martensitic transition by a Ginzburg-Landau free energy in the symmetric strain tensor. We show in three dimensions (3D) that solving the St. Venant compatibility relations for strain, treated as independent field equations, generates three anisotropic long-range potentials between the two order parameter components. These potentials encode 3D discrete symmetries, express the energetics of lattice integrity, and determine 3D textures. Simulation predictions include twins with temperature-varying orientation, helical twins, competing metastable states, and compatibility-induced elastic frustration. Our approach also applies to improper ferroelastics.

Hierarchical pattern formation in elastic materials

Abstract We study hierarchical structures such as branched twins in elastic materials based on a model of martensitic materials in which hierarchical twinning near the habit plane (austenite-martensite interface) is a new and crucial ingredient. The model includes (1) a triple-well potential ( θ 6 model) in local strain, (2) strain gradient terms up to second order in strain and fourth-order in gradient, and (3) all symmetry-allowed compositional fluctuation-induced strain gradient terms which favor hierarchical structures and enable communication between macroscopic (cm) and microscopic (A˚) regions essential for shape memory. Below the transition temperature ( T 0 ) we obtain the conditions under which branching of twins is energetically favorable. This hierarchy of length scales provides a related hierarchy of time scales and thus the possibility of non-exponential decay. Results based on 2D simulations of the time-dependent Ginzburg-Landau (TDGL) equation are shown for twins, tweed and hierarchy formation. We also apply stability analysis to study the formation of modulated structures at early time and obtain an approximate phase diagram for the model.

Hysteresis as rate competition: a Landau model example

An understanding of hysteresis is obtained as a competition between the field sweep rate h, and the inverse mean first-passage time (FPT), of jump rate, using the numerical Langevin dynamics of a Landau 4m model. Both mean jump values h,(h), and jump distributions p(hJ,h), agree quantitatively with previous FPT-based theoretical results. Rate competition and jump distribution ideas could have relevance for laser bistability and glass transition and jump distribution ideas could have relevance for laser bistability and glass transition problems.

Langevin Dynamic Simulation of Hysteresis in a Field-Swept Landau Potential

Numerical simulations are done of Langevin dynamics for a uniform-orderparameter, field-swept Landau model,Φ= −|a/2|m2+|b/4|m4−mh(t) , to study hysteresis effects. The field is swept at a constant rateh(t)=h(0)+ht. The stochastic jump values of the field {hJ from an initially prepared metastable minimumm(0) are recorded, on passage to a global minimum m(τ). The results are: (a) The mean jump¯hJ(h) increases (hysteresis loop widens) with h, confirming a previous theoretical criterion based on rate competition between field-sweep and inverse mean first-passage time 〈τ〉 (FPT); (b) The broad jump distributionρ(hJ,h) is related to intrinsically large FPT fluctuations (〈τ2〉−〈τ〉2)/〈τ2〉 ∼ O(1), and can be quantitatively understood. Possible experimental tests of the ideas are indicated.

Re-equilibration after quenches in athermal martensites: conversion delays for vapor-to-liquid domain-wall phases

Entropy barriers and aging states appear in martensitic structural-transition models, slowly re-equilibrating after temperature quenches, under Monte Carlo dynamics. Concepts from protein folding and aging harmonic oscillators turn out to be useful in understanding these nonequilibrium evolutions. We show how the athermal, nonactivated delay time for seeded parent-phase austenite to convert to product-phase martensite arises from an identified entropy barrier in Fourier space. In an aging state of low Monte Carlo acceptances, the strain structure factor makes constant-energy searches for rare pathways to enter a Brillouin zone “golf hole” enclosing negative-energy states, and to suddenly release entropically trapped stresses. In this context, a stress-dependent effective temperature can be defined, that re-equilibrates to the quenched bath temperature.

Critical Temperature, Critical Current, and Vortex Loops in 3D Anisotropic Josephson Junction Arrays

Monte Carlo simulations of a 3D anisotropic Josephson junction array are performed, to determine the critical temperature T c ( I ,α) for the onset of phase incoherence. Here I is the external current drive at opposite surfaces, and α≡ E J ⊥ / E J // is the inter-/intra-layer Josephson coupling ratio. T c ( I ,α) is equivalent to a critical current, I c ( T ,α) for the breakdown of global phase coherence. The temperature and anisotropy dependence of I c ( T ,α) supports a physical picture of two vortex populations, quasi-2D and 3D in nature at small and large scales respectively, separated by a length scale \(r_{\perp}\sim 1/\sqrt{\alpha}\).

Glassy relaxation in hierarchically frustrated Josephson arrays

Abstract Josephson arrays with hierarchical patterns of frustration may provide a physical prototype for glasses. Flux/disorder patterns {ƒI} with ƒ I =±ƒ , Σ I ƒ I =0 that have a self-similar character on coarse-graining, can be con structed iteratively, composed of quasi-neutral clusters. A nonequilibrium population of topological excitations {mI=±1}, ΣImI=0, trapped on such a hierarchical background, should annihilate sequentially ∼t −T T a due to the hierarchy of time scales. (Here T T a is a scaled temperature.) Arrays with a hierarchical pattern of flux penetration areas/intergrain couplings could show glassy behaviour, on rapid cooling or rapid reduction of magnetic fields H. This might be relevant for understanding magnetic moment relaxation M(t)∼−H3Tln t seen in high-Tc glassy superconductors.

Athermal martensites, Temperature-Time-Transformation diagrams and thermal hysteresis: Monte Carlo simulations of strain pseudospins

We explore the kinetics of a three-state strain pseudospin model for a square/rectangle ferroelastic transition, described by a temperature dependent hamiltonian without quenched disorder, using temperature quench Monte Carlo simulations. The model hamiltonian includes power law anisotropic long range interactions, which lock the domain walls in a symmetry breaking diagonal direction. In athermal parameter regime, there are fast conversions at the athermal transition temperature, but with delay tails above it, as in experiment. The conversion delay tails have a Vogel-Fulcher divergence at transition to austenite. The incubation delays and their insensitivity to elastic energy scales are attributed to entropy barriers. Temperature cycling shows hysteretic behavior in physical quantities.

Tc Dependence vs. Spacing d by Josephson-Junction between CuO2 Layers

We argue that the Josephson Junction coupling energy by Cooper pair tunneling between CuO2 layers enhances the critical temperature Tc for small spacing range (d=3 A − 4 A), while the charge coupling energy for the capacitance between CuO2 layers depresses the critical temperature Tc in the case of large spacings (d>4 A). Vortex loop scaling analysis for the 3-dimensional XY model (superfluid He or 3d-Josephson Junction Array model) is speculated on. These results are consistent with the Tc dependence versus spacing between CuO2 layers for layered high Tc oxides materials obtained by image processing analysis of electron microscope photographs.

Hierarchical microstructure in structural phase transitions

Abstract We consider a model in the context of martensitic materials in which hierarchical twinning near the habit plane (austenite-martensite interface) is a new and crucial ingredient. The model includes (1) a triple-well potential in local deviatoric (rectangular) strain, (2) strain gradient terms up to second order in strain and fourth order in gradient, and (3) all symmetry allowed compositional fluctuation-induced strain gradient terms. The last term favors branching of domain walls which enables communication between macroscopic and microscopic regions essential for shape memory. Below the transition temperature (T0) we obtain the conditions under which branching of twins is energetically favorable. Above T0 a hierarchy of branched domain walls also stabilizes tweed formation (criss-cross patterns of twins). External stress or pressure modulates (“patterns”) the spacing of domain walls. Results based on 2D time-dependent Ginzburg-Landau simulations are shown for twins, tweed and hierarchy formation.