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Large superelastic deformation in ferroelastic-ferroelectrics (FMs) is a complex phenomenon involving multiple mechanisms operating simultaneously. Understanding how these mechanisms contribute corporately is critical to apply this useful property to the intrinsically brittle FMs, which can therefore display both excellent functional and mechanical performance. Here, we have directly observed and quantitatively analyzed in situ in a transmission electron microscope the three main mechanisms of twinning domain, phase transformation and mobile point defect contributing to extremely large superelastic deformation in single-crystal BaTiO3 (5.0% strain) and Pb(Mg1/3Nb2/3)O3-PbTiO3 (10.1% strain). Our results reveal the hierarchical origin of large recoverable strain in “brittle” FMs.

A phase field method is proposed to determine the equilibrium fields of a magnetoelectroelastic multiferroic with arbitrarily distributed constitutive constants under applied loadings. This method is based on a developed generalized Eshelby’s equivalency principle, in which the elastic strain, electrostatic, and magnetostatic fields at the equilibrium in the original heterogeneous system are exactly the same as those in an equivalent homogeneous magnetoelectroelastic coupled or uncoupled system with properly chosen distributed effective eigenstrain, polarization, and magnetization fields. Finding these effective fields fully solves the equilibrium elasticity, electrostatics, and magnetostatics in the original heterogeneous multiferroic. The paper formulates a variational principle proving that the effective fields are minimizers of appropriate close-form energy functional. The proposed phase field approach produces the energy minimizing effective fields (and thus solving the general multiferroic problem) as a result of artificial relaxation process described by the Ginzburg–Landau–Khalatnikov kinetic equations. [ABSTRACT FROM AUTHOR]

By using coarse graining model, the dependence of magnetoelectric (ME) coupling on the mutual orientations of magnetic and electric fields with respect to the orientation of layers in polycrystalline multiferroic laminates is investigated. It is shown that the ME coefficient, described by polarization change in response to the applied magnetic field, is proportional to the trace of effective piezomagnetic strain tensor projected onto laminate interfaces. The piezomagnetic strain significantly depends on the orientation of applied magnetic fields. The results obtained here demonstrate that the magnetoelectric effect in layered composites can be significantly improved by optimizing the configuration of applied electric/magnetic fields. [ABSTRACT FROM AUTHOR]

Trilayer composites consisting of 0.9Pb(Zr0.52Ti0.48)O3–0.1Pb(Zn1/3Nb2/3)O3 (0.9 PZT-0.1 PZN) and Ni0.6Cu0.2Zn0.2Fe2O4 (NCZF) in the configuration NCZF-(0.9 PZT-0.1 PZN)-NCZF were synthesized using pressure assisted sintering. Composites with optimized magnetostrictive to piezoelectric thickness ratio showed a high magnetoelectric (ME) coefficient of 525 mV/cm Oe. Further enhancement in the magnitude of ME coefficient was obtained (595 mV/cm Oe) when the angle of applied dc magnetic field was changed to 45°. Changing the intermediate piezoelectric layer from single to trilayer stack geometry configuration leads to the realization of giant ME response of 782 mV/cm Oe in sintered composites. [ABSTRACT FROM AUTHOR]

The generic case of a ferroelectric solid solution is considered wherein different symmetry phases located at opposing ends of the diffusionless phase diagram are separated by a morphotropic boundary (MB). It is shown that the Landau theory of weak first-order phase transformations automatically predicts vanishing of the anisotropy of polarization, continuity of thermodynamic properties, and a drastic decrease in domain wall energy near the MB line that results in the formation of adaptive ferroelectric nanodomain states. Low-resolution diffraction from these adaptive states acquired at the coherence lengths of elastic x-ray or neutron scattering probes will produce the same diffraction pattern as attributed to monoclinic (MA,MB,MC) phases. It is further shown that the electric- or stress-field-induced reconfiguration of these adaptive nanodomain states results in a softening of the piezoelectric, elastic, and dielectric properties near the MB line. In addition, the spherical degeneration of the polarization direction, reflecting the decoupling of the polarization from the crystal lattice at the MB, also predicts the formation of a polar glass state whose properties should be similar to the special properties of amorphous ferromagnets. In particular, the vanishing of the polarization anisotropy at the MB should result in ferroelectric domains with irregular shapes exhibiting high configurational sensitivity to external forces. The theory further predicts that two tricritical points will occur on the line of paraelectric→ferroelectric transitions and it is shown that all two-phase equilibrium lines of the diffusionless phase diagram—including the MB line—must be replaced by two-phase fields. Within these two-phase fields, the adjacent ferroelectric-ferroelectric and paraelectric-ferroelectric phases coexist. Possible topologies of the equilibrium MB phase diagram illustrating these two-phase equilibrium fields are computed and discussed. [ABSTRACT FROM AUTHOR]

The strain-mediated magnetoelectric coupling in piezoelectric-piezomagnetic composites with arbitrary spatial arrangement of particles of constituent phases is investigated. The free energy of such a composite is formulated as a functional of a phase field that is a shape function describing a spatial distribution of the phases. The free energy is obtained by a minimizing of the extended Landau free energy complemented by the electrostatic and strain-induced interactions caused by piezoelectric and piezomagnetic strains. The magnetoelectric coupling tensor derived from this functional also depends on a composite microstructure. A minimization of this functional with respect to configurational variables gives the optimal composite architecture. It is shown that a platelike configuration of constituent phases provides the largest strength of magnetoelectric coupling. The phase field computer modeling confirms this conclusion. [ABSTRACT FROM AUTHOR]

The atomic density function kinetic theory of a spontaneous atomic rearrangement at the atomic scale is developed. In this theory, the evolution of the system is described not by the individual movement of each atom in the same system but by the evolution of the atomic density functions, which are averages over the time-dependent ensemble. The temporal evolution of atomic densities is driven by the free energy relaxation. The thermodynamic aspects of the instability of the homogeneous liquid state and the formation of crystalline and amorphous state are discussed. Examples of self-assembling of atoms in the formation of a three-dimensional nanocrystal, and the two-dimensional diffusionless transformation in the crystalline and amorphous states are considered. The modeling reveals the special features of displacive phase transformations in the crystalline state through a transient stress-accommodating state. It also demonstrates that the thermodynamic relaxation of the liquid below its critical instability point produces a relaxed amorphous state that is practically a metastable phase. [ABSTRACT FROM AUTHOR]

Ferroelectric and ferroelastic phases with very low domain wall energies have been shown to form miniaturized microdomain structures. A theory of an adaptive ferroelectric phase has been developed to predict the microdomain-averaged crystal lattice parameters of this structurally inhomogeneous state. The theory is an extension of conventional martensite theory, applied to ferroelectric systems with very low domain wall energies. The case of ferroelectric microdomains of tetragonal symmetry is considered. It is shown for such a case that a nanoscale coherent mixture of microdomains can be interpreted as an adaptive ferroelectric phase, whose microdomain-averaged crystal lattice is monoclinic. The crystal lattice parameters of this monoclinic phase are self-adjusting parameters, which minimize the transformation stress. Self-adjustment is achieved by application of the invariant plane strain to the parent cubic lattice, and the value of the self-adjusted parameters is a linear superposition of the lattice constants of the parent and product phases. Experimental investigations of Pb(Mg[sub 1/3]Nb[sub 2/3])O[sub 3]–PbTiO[sub 3] and Pb(Zn[sub 1/3]Nb[sub 2/3])O[sub 3]–PbTiO[sub 3] single crystals confirm many of the predictions of this theory. © 2003 American Institute of Physics. [ABSTRACT FROM AUTHOR]

Three-dimensional computer simulations are used to examine the development of magnetic domains and migration of domain walls in hard ferromagnetic thin films of polytwinned microstructure. The polytwinned microstructure, consisting of three different orientation variants of the L1[sub 0] ordered phase with orthogonal c-axes, is characterized by a multi-component long-range order (LRO) parameter. The coupling between the magnetic domains and the polytwinned microstructure is described by the coupling between magnetization and the LRO parameter in the anisotropy free energy of the system. The magnetodynamics is described by Landau-Lifshitz equations that are solved numerically in the reciprocal space using a parallel Fast Fourier Transform algorithm. It is found that the coupling results in a multilevel hierarchy in the magnetic domain structure, with intracolony boundaries (across multiple variants within a single polytwinned colony) and intercolony boundaries (across multiple colonies). In thin films of high magnetic anisotropy we find the intracolony boundary to be of {100} type and the intercolony one to be on average of {110} type. The migration of the magnetic domain boundaries in external magnetic fields is studied in a number of simple cases in which the fields are applied along different directions. The simulation results are analyzed and compared with existing experimental observations. [ABSTRACT FROM AUTHOR]

Three-dimensional micromagnetic simulations are used to study the effect of crystallographic textures on the magnetic properties ofuniaxial nanocrystalline films of hard magnetic materials with arbitrary grain shapes and size distributions. The correlation lengths (effective ferromagnetic exchange interaction radius and domain wall width) are assumed to be smaller than the typical grain size. The Landau-Lifshitz equations of magnetization dynamics are employed to describe the distribution of magnetization in ferromagnetic domains, domain evolution during magnetization switching, and the hysteresis curve. The equations are solved numerically in reciprocal space using the fast Fourier transform technique. Simulations are performed for films of different grain textures. The results show that magnetic coupling between grains in thin films significantly affects the morphology of the magnetic domains and their response to the magnetic field applied. The greater the deviation of the uniaxial directions of the grains from the film normal, the smaller the coercivity and the remanence magnetization. It is also shown that the remanence magnetization and coercivity respond differently to the variations in film texture and that the magnetic reversal process is a collective process involving groups of grains. In particular, it is found that the crystallographic texture of the grains has a more complicated effect on the coercivity than on the remanence magnetization and on the character of topological changes during the magnetization reversal process. [ABSTRACT FROM AUTHOR]

The phase field microelasticity theory of a three-dimensional elastically anisotropic solid of arbitrarily inhomogeneous modulus also containing arbitrary structural inhomogeneities is proposed. The theory is based on the equation for the strain energy of the elastically and structurally inhomogeneous system presented as a functional of the phase field, which is the effective stress-free strain of the "equivalent" homogeneous modulus system. It is proved that the stress-free strain minimizing this functional fully determines the exact elastic equilibrium in the elastically and structurally inhomogeneous solid. The stress-free strain minimizer is obtained as a steady state solution of the time-dependent Ginzburg-Landau equation. The long-range strain-induced interaction due to the elastic and structural inhomogeneities is explicitly taken into account. Systems with voids and cracks are the special cases covered by this theory since voids and cracks are elastic inhomogeneities that have zero modulus. Other misfitting defects, such as dislocations and coherent precipitates, are also integrated into this theory. Examples of elastic equilibrium of elastically inhomogeneous solid under applied stress are considered. [ABSTRACT FROM AUTHOR]

The phase field microelasticity theory of a three-dimensional elastically anisotropic single crystal with multiple voids and cracks is developed. It is extended to the case of elastically isotropic polycrystal. The theory is based on the exact equation for the strain energy of the “equivalent” continuous elastically homogeneous body presented as a functional of the phase field. This field is the equivalent stress-free strain. It is proved that the equivalent stress-free strain minimizing the strain energy of the elastically homogeneous body fully determines the elastic strain and displacement of the body with voids/cracks. The geometry and evolution of multiple voids and cracks are described by the phase field, which is a solution of the stochastic time-dependent Ginzburg–Landau equation. Other stress-generating defects, such as dislocations and precipitates, are trivially integrated into this theory. The proposed model does not impose a priori constraints on the configuration of multiple voids and cracks or on possible configurations and evolutions of cracks along their propagation paths. Examples of computations of the elastic equilibrium of systems with voids and/or cracks, the evolution of cracks in single crystals and polycrystals, and material toughening due to second-phase obstacles are considered. © 2002 American Institute of Physics. [ABSTRACT FROM AUTHOR]

A Ginzburg-Landau type theory of interaction of randomly distributed local dipoles in a paraelectric crystal is developed. The interaction is caused by the polarization of the host lattice generated by these dipoles. The obtained effective Hamiltonian of the dipole-dipole interaction is employed for the Monte Carlo simulation of ferroelectric properties of a system with off-center dopant ions producing local dipoles. The computer simulation shows that at low dopant ion concentration the paraelectric state transforms into a macroscopically paraelectric state consisting of randomly oriented polar clusters. These clusters amplify the effective dipole moment and dramatically increase the dielectric constant. The interaction between the clusters results in a spectrum of relaxation time and transition to the relaxor state. The real and imaginary parts of the susceptibility of this state are calculated. At intermediate dopant concentration, the material undergoes a diffuse phase transition into a ferroelectric state smeared within a temperature range. A further increase in the dopant concentration makes the transition sharper and closer to the conventional ferroelectric transition. The results obtained are compared with the behavior of the K[sub 1 -x]Li[sub x]TaO[sub 3] relaxor ferroelectric. [ABSTRACT FROM AUTHOR]

Coarse graining model was used to examine the dependence of magnetoelectric (ME) coupling on the mutual orientations of magnetic and electric fields with respect to the orientation of layers in polycrystalline multiferroic laminates. The results obtained demonstrated that the magnetoelectric effect in layered composites could be significantly improved by optimizing the configuration of applied electric/magnetic fields.

The atomic density function kinetic theory of a spontaneous atomic rearrangement at the atomic scale is developed and the evolution of the system is described by the evolution of the atomic density functions, which are averages over the time-dependent ensemble. The modeling has revealed the special features of displacive phase transformations in the crystalline state through a transient stress-accommodating state and has shown that the thermodynamic relaxation of the liquid below its critical instability point produces a relaxed amorphous state.

The magnetic domain structure of Fe-55 at %Pd alloy at different stages of atomic ordering was analyzed using Lorentz microscopy imaging technique. The stripe magnetic domain where the magnetization direction or the c axis of the L1(sub 0) grain is perpendicular to the surface orientation and the plate-like magnetic domain where the magnetization direction of the L1(sub 0) grain is parallel to the specimen surface orientation are the two types of magnetic domains that were found in these coarsened grains.

The phase field microelasticity theory of a three-dimensional elastically anisotropic solid of arbitrary inhomogeneous modulus containing arbitrary structural inhomogeneities is presented. The stress-free strain minimizing this functional fully determines the exact elastic equilibrium in the elastically and structurally inhomogeneous solid.