Theory and computer simulation of tweed texture

Abstract

An analytical theory of the ordering interaction J(R _ij ) in structural phase transitions mediated by elastic relaxation in the material is outlined. The ordering process in cell i sets up a local stress field due to the sizes, shapes or displacements of atoms or atomic groups, which is propagated elastically to a distant cell j. The atomistic theory for ferro- and antiferro-elastic transitions takes into account two types of singularity, one due to elastic anisotropy and the other to the Zener interaction J _z of infinite range in ferroelastic transitions. The form of J _k in Fourier space is highly anisotropic with a few “soft” directions coinciding with the orientation of twin boundaries. The asymptoptic J(R) at large R is shown to be very anisotropic as well and decays as R ^?3 in ferroelastic and R ^?5 in antiferroelastic systems.

Computer simulations for a three-dimensional model of about 29,000 particles show a strong tendency to form tweed texture, as observed experimentally. Well above the structural phase transition temperature, the strain fluctuations show well-developed embryos of the tweed texture. On quenching to below the transition temperature, a pronounced micro-twinning appears which follows almost exactly the shape of the embryos and then develops towards a stripe texture. After a certain time needle-shaped domains are formed and a peculiar step-wise process of generating new stripes is observed.     

Hierarchical microstructure in structural phase transitions

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.     

Influence of transport current and thermal fluctuations on the resistive properties of HTSC+CuO composites

Measurements of the temperature dependence of the electrical resistance R(T) below the superconducting transition temperature have been performed at different values of the transport current in HTSC+CuO composites modeling a network of weak S-I-S Josephson junctions (S—superconductor, I—insulator). It has been shown experimentally that the temperature dependence R(T) at different values of the transport current is adequately described by means of the mechanism of thermally activated phase slippage developed by Ambegaokar and Halperin for tunnel structures. Within the framework of this model we have numerically calculated the temperature dependence of the critical current J _c(T) as defined by various criteria. Qualitative agreement obtains between the measured and calculated temperature dependences J _c(T). Fiz. Tverd. Tela (St. Petersburg) 41, 969–974 (June 1999)     

Non-planar spin texture driven dissipation in a ferromagnet-superconductor bilayer

We report the observation of a pronounced dip in the in-plane magnetic field (H_∥) dependence of the critical current density J_c(H) and a peak in resistance R(H) of a NbN-HoNi₅ bilayer at temperatures below the magnetic ordering temperature (T_Curie ≈ 3.5 K) of HoNi₅, which is lower than the onset temperature (≈9 K) of superconductivity in the NbN layer. The extrema in J_c(H) and R(H) appear at fields much below the upper critical field of NbN. We attribute these features to a coupling between localized out-of-plane moments present in the magnetic film and Pearl vortices of the superconducting layer. A spin re-orientation transition of the localized moments by H_∥ breaks this coupling, leading to the observed excess dissipation.     

Critical current of an inhomogeneous Josephson junction at an intergrain boundary with a random distribution of dislocations

The critical current J _c(θ) of an intergrain boundary is calculated as a function of the contact misorientation angle θ of the granules. It is assumed that the ordering parameter is suppressed in regions near boundaries with an enhanced mechanical stress induced by randomly distributed surface dislocations. The stress distribution function is determined using a probabilistic approach. Assuming that the weak coupling at the boundary is Josephson coupling, an analytic expression is found for the angular dependence J _c(θ) (for tilt and twist boundaries). The magnitude of the residual critical current of a boundary in a strong magnetic field is estimated. Fiz. Tverd. Tela (St. Petersburg) 40, 393–402 (March 1998)     

Mixed spin- and spin- Ising models with random nearest-neighbour interactions

Using the effective field theory with correlations, we study mixed spin?3/2 and spin?1/2 Ising models with random bonds and crystal-field interactions on the honeycomb lattice. The nearest-neighbour couplings J_ij are taken as random variables with distribution P(J_ij) = pδ(J_ij ? J)+(1 ? p)δ(J_ij ? αJ), where J > 0 and |α| ≤ 1. In a certain range of negative values of α, the phase diagrams exhibit re-entrant behaviour. In detail, we investigate separately two kinds of disorder: Bond dilution (α = 0) and random ±J interactions (α = ?1). In both cases, the influence of the an-isotropy on the phase diagrams shows some new outstanding features.     

Effects of random biquadratic couplings in a spin-1 spin-glass model

A spin-1 model, appropriated to study the competition between bilinear (J _ij S _i S _j ) and biquadratic (K _ij S _i ² S _j ²) random interactions, both of them with zero mean, is investigated. The interactions are infinite-ranged and the replica method is employed. Within the replica-symmetric assumption, the system presents two phases, namely, paramagnetic and spin-glass, separated by a continuous transition line. The stability analysis of the replica-symmetric solution yields, besides the usual instability associated with the spin-glass ordering, a new phase due to the random biquadratic couplings between the spins. Received 18 May 1999 and Received in final form 20 October 1999     

Reentrant transitions in the van Hemmen model of a spin glass

The van Hemmen model of a spin glass, which is an Ising model with random couplings J_ij between sites i and j equal to J₀ + J(ξ_iη_j + ξ_jη_i), where (ξ_i, η_i) are independent, identically distributed random variables, is studied in the pair approximation of the cluster variation method. For the family of probability distributions (1 − p)δ(ξ_i − a) + pδ(ξ_i) + (1 − p)δ(ξ_i + a), where p is varied, phase diagrams are constructed. They are qualitatively different from the mean-field phase diagrams and display a competition between tendencies towards spin-glass and towards ferromagnetic ordering, which results in reentrant transitions. It is argued that the observed effects are not accidental but are borne by the competition of bonds of the underlying lattice system.     

Application of epidemic models to phase transitions

The Susceptible-Infected-Recovered (SIR) and Susceptible-Exposed-Infected-Recovered (SEIR) models describe the spread of epidemics in a society. In the typical case, the ratio of the susceptible individuals fall from a value S ₀ close to 1 to a final value S_f , while the ratio of recovered individuals rise from 0 to R_f?=?1???S_f . The sharp passage from the level zero to the level R_f allows also the modeling of phase transitions by the number of “recovered” individuals R(t) of the SIR or SEIR model. In this article, we model the sol–gel transition for polyacrylamide–sodium alginate (SA) composite with different concentrations of SA as SIR and SEIR dynamical systems by solving the corresponding differential equations numerically and we show that the phase transitions of “classical” and “percolation” types are represented, respectively, by the SEIR and SIR models.     

The transverse spin-1 Ising model with random interactions

The phase diagrams of the transverse spin-1 Ising model with random interactions are investigated using a new technique in the effective field theory that employs a probability distribution within the framework of the single-site cluster theory based on the use of exact Ising spin identities. A model is adopted in which the nearest-neighbor exchange couplings are independent random variables distributed according to the law P(J_ij)=pδ(J_ij−J)+(1−p)δ(J_ij−αJ). General formulae, applicable to lattices with coordination number N, are given. Numerical results are presented for a simple cubic lattice. The possible reentrant phenomenon displayed by the system due to the competitive effects between exchange interactions occurs for the appropriate range of the parameter α.     

A relationship for the surface impedance at the skin effect foran arbitrary kernel of integral

For the case of the skin effect in metals and an arbitrary function J(R) entering into the nonlocal relation between the current density and the electric field, general distributions of the electric field and surface impedance are found. The results obtained are applicable to both normal and superconducting metals and are represented as the Fourier transform of the function J(R).     

The anisotropic 3D Ising model

An expression for the free energy of an (001) oriented domain wall of the anisotropic 3D Ising model is derived. The order--disorder transition takes place when the domain wall free energy vanishes. In the anisotropic limit, where two of the three exchange energies (e.g. J_x and J_y ) are small compared to the third exchange energy (J_z ), the following asymptotically exact equation for the critical temperature is derived, sinh(2J_z /k _B T _c)sinh(2(J_x ?+?J_y )/k _B T _c))?=?1. This expression is in perfect agreement with a mathematically rigorous result (k _B T _c/J_z ?=?2[ln(J_z /(J_x ?+?J_y ))?ln(ln(J_z /(J_x ?+?J_y ))?+?O(1)]^?1) derived earlier by Weng, Griffiths and Fisher (Phys. Rev. 162, 475 (1967)) using an approach that relies on a refinement of the Peierls argument. The constant that was left undetermined in the Weng et al. result is estimated to vary from ～0.84 at ((H_x ?+?H_y )/H_z )?=?10^?2 to ～0.76 at ((H_x ?+?H_y )/H_z )?=?10^?20.     

An efficient method for calculating the pair distribution functions of a ternary hard-sphere system in r space within the Percus-Yevick approximation

We present a direct method which allows an accurate and—at the same time—economical calculation of the pair distribution functions (PDFs) g_ij (r) of an additive ternary hard-sphere system within the Percus—Yevick approximation. The approach is based on the fact that for this approximation the Laplace transforms [?_ij (s)] of the PDFs are known analytically, so that the inversion of the ?_ij (s) into r space can be performed exactly. The expressions presented here allow the determination of the ?_ij (r) for r values up to 8R ₁, R ₁ being the diameter of the smallest species; this range in r space should be sufficient for applications in standard algorithms of liquid state theory, such as thermodynamic perturbation theories or integral-equation approaches.     

Approximate solution of the Percus-Yevick equation for a binary mixture of hard spheres with non-additive diameters

We present an approximate solution of the Percus-Yevick integral equation for a binary mixture of hard spheres with non-additive diameters. Defining R_ij the distance of closest approach between particles of species i and j by R ₁₂ = ½(R ₁₁ + R ₂₂) + α, we obtain a closed set of equations for the direct correlation functions c_ij (r) when 0 < α ? min [½(R ₂₂ - R ₁₁), ½R ₁₁]. Our expressions for c_ii (r), and for c ₁₂(r) in the range 0 < r ? ½[R ₂₂ - R ₁₁] - α, agree with those previously obtained by Lebowitz and Zomick.     

Role of thermal vibrations in magnetic phase transitions

This article addresses the contradiction that exists in studies on magnetic phase transitions between theoretical models that ignore the role of thermal vibrations and analysis of neutron diffraction data that always incorporates them. Ignoring thermal vibrations in both theoretical models and analysis of diffraction data leads to the latter giving different magnetic-order parameters for different reciprocal lattice lines. This leads to a unique consequence, the assumption to neglect a physical phenomenon turns a single-valued experimental observable into a multiple-valued one where all values are equally valid. This assumption is clearly unacceptable and must be rejected. Diffraction data constrain all theoretical models to incorporate thermal vibrations and represent the exchange interaction as temperature dependent, J_ij (T), instead of the current practice, J_ij (0).     

Sound propagation anomalies and phase diagram of chromium alloys

Calculations of the complex elastic moduli Ĉ(T) in dilute Cr alloys are compared to measurements of the velocity and damping of sound near T _N and at high temperatures T>T _N (T _N — Néel temperature). The thermodynamic calculation is based on the covalent bond model of 3d ions in a state with different numbers n of covalent electrons. The parameters A _ij ⁽ⁿ⁾ of indirect exchange between the ions of the i and j sublattices are expressed in terms of the covalent bond energy Γ _ij ⁽ⁿ⁾ . The stability of the charge and spin density waves (CDWs and SDWs) is found by a variational method and is determined by the dispersion of Γ _ij ⁽ⁿ⁾ and by the Coulomb parameters U _n. For a small structural vector Q the phase diagram contains a superantiferromagnetic phase (SAFM) at temperatures T _N<T≲2T _N. The peak of the defect |ΔE(T)| of the modulus and of the sound damping Δ_h(T _N) near the first-order SDW-SAFM transition is determined by the structure of the transitional domain. Measurements of the anomalous growth of E(T) at temperatures T>T _N make it possible to determine the magnetostriction constants λ(T) of Ce alloys in the SAFM phase on the basis of the SAFM theory. Fiz. Tverd. Tela (St. Petersburg) 41, 1467–1472 (August 1999)     

Low temperature far IR spectra of (CH34)N NiCl3, (CH3)4N NiBr3 and (CD3)4N NiBr3

The known phase transition in (CH₃)₄N NiCl₃, at 171 K, has been characterised by far IR spectroscopy. The transition is explained as due to a formation of weak labile C-H,…, Cl hydrogen bonds at low temperatures, restricting the “free” internal rotations of the methyl groups and perhaps at the same time ordering the orientations of the tetramethylammonium ions. No similar transition in (CH₃)₄N NiBr₃ was found.     

Spatial structure and stability based on random walks

A general expression for a recursion formula which describes a random walk with coupled modes is given. In this system, the random walker is specified by the jumping probabilities P⁺ and P^– which depend on the modes. The transition probability between the modes is expressed by a jumping probabilityR ^(ij) (orr _ij). With the aid of this recursion formula, spatial structures of the steady state of a coupled random walk are studied. By introducing a Liapunov function and entropy, it is shown that the stability condition for the present system can be expressed as the principle of the extremum entropy production.On leave of absence from Tohoku University, Department of Applied Science, Faculty of Engineering, Sendai, 980 Japan.     

Intergranular percolation in granular YBCO/BaTiO3 composites

Ferroelectrics and high temperature superconductors are two promising materials for future electronic devices. Both being perovskite ceramic structures with similar crystal chemistry a set of samples were prepared from the composite of (1−x)YBa₂Cu₃0_7−δ −(x)BaTiO₃ (YBCO/BT). These samples were investigated with temperature dependent resistance, FTIR, X-ray diffraction and SEM-EDX analysis. It has been found that the critical exponent in the T _c0 (R = 0) region is in agreement with the percolation theory. A long-range superconducting order results from thermally assisted percolation process through weak-links between the grains. The connectivity in the coherent transition region can be explained by a power law.