Thermodynamic driving force in ferroelectric crystals with a rank-2 laminated domain pattern, and a study of enhanced electrostriction

The thermodynamic driving force for domain growth in a rank-2 laminated ferroelectric crystal is derived in this article, and we used it, together with a homogenization theory, to study the issue of enhanced electrostrictive actuation recently reported by Burcsu et al. [2004. Large electrostrictive actuation of barium titanate single crystals. J. Mech. Phys. Solids 52, 823-846]. We derived this force from the reduction of Gibbs free energy with respect to the increase of domain concentration. It is shown that both the free energy and the thermodynamic force consist of three parts: the first arises from the difference in M₀ and M₁, the linear electromechanical compliances of the parent and product domains, respectively, at a given level of applied stress and electric field, the second stems from the electromechanical work associated with the change of spontaneous strain and spontaneous polarization during domain switch, and the third from the internal energy due to the distribution of polarizations strain and electric polarization inside the crystal. We prove that the first term is substantially lower than the second one, and the third one is identically zero with compatible domain pattern. The second one is, however, not exactly equal to the commonly written sum of the products of stress with strain, and electric field with polarization during switch, unless both domains have identical moduli in the common global axes. We also show that, with compatible domain patterns and when M₁=M₀, this driving force is identical to Eshelby's driving force acting on a flat interface due to the jump of energy-momentum tensor. Applications of the theory to a BaTiO₃ crystal subjected to a fixed axial compression and decreasing electric field from the [0 0 1] state reveal that the crystal undergoes a three-stage switching process: (i) the 0→90° switch to form a rank-1 laminate, (ii) the 0→180° switch inside the 0° domain to form a laminate I with a concurrent 90°→−90° switch inside the 90° domain to form laminate II, creating a rank-2-laminated domain pattern, and (iii) finally the 90→180° switch. It is the exchange of stability between the 0, 90°, and 180° domains under compression and electric field that is the origin of the enhanced actuation. We illustrate these intrinsic features by showing the evolution of these domains, and demonstrate how the reported large actuation strain can be attained with a rank-2 laminate.     

The influence of a compressive stress on the nonlinear response of ferroelectric crystals

The origin of nonlinearity in a ferroelectric crystal is domain reorientation, and such a process can be affected by the presence of a compressive stress. In this article we examine how a superimposed compression affects the evolution of new domain and how it changes the shape of the hysteresis loop. We start out by considering the thermodynamic driving force for domain reorientation, and then use a dual-phase homogenization theory to calculate the overall response. To uncover the influence of a compressive stress, the theory is used to calculate the hysteresis loop between the electric displacement D and the electric field E of a BaTiO₃ crystal, first without and then with a compression, using a two-consecutive 90° switch model (i.e. 0° → 90° → 180°). It is found that, from the initial 0° position, the compressive stress will increase the thermodynamic driving force and promote an earlier onset of the 90° domain, but its presence will cause a significant delay for the reorientation process to pass through the intermediate 90° state in route to its final 180° configuration. The D vs. E loop then exhibits a more round shape and a lesser steep slope near the coercive field. The delayed passage and more rounded shape are found to be consistent with a recent experimental observation [Burcsu et al., 2004. J. Mech. Phys. Solids 52, 823–846].     

A continuum analysis of charge induced ferroelectric domain wall motions

A continuum model is presented for the motion of a domain wall in a plane 90°-domain configuration subjected to an isolated extrinsic charge near the surface of a ferroelectric single crystal. Local pinning is postulated for the kinetic law. Before the appearance of the extrinsic charge, all polarization surface charges are taken to be neutralized by environmental charges. The domain wall motion after the appearance of the extrinsic charge is assumed to proceed sufficiently fast without any significant conductive currents on the surface or in the interior of the crystal such that new surface and interface polarization charges remain unscreened and contribute to the ferroelectric anisotropy energy. A non-admissible divergence of the electric field and consequently of the local thermodynamic driving force and of the domain wall velocity appears in the model if the domain wall charged by interface polarization charges intersects the crystal surface charged by surface polarization charges under an arbitrary angle. The physically possible domain wall angle is identified using the condition of a non-divergent driving force. The ferroelectric anisotropy energy and an intrinsic surface energy of the domain wall, however, do not provide stability of the domain wall trajectory against an unlimited increase of its curvature at the surface. The problem has been solved conceptually by proper account of the domain wall bending energy. Numerical and dimensional analysis explain also why domain walls driven by extrinsic charges remain almost straight in soft ferroelectrics.     

A rate-dependent three-dimensional free energy model for ferroelectric single crystals

The one-dimensional free energy model for ferroelectric materials developed by Smith et al. [Smith, R.C., Seelecke, S., Ounaies, Z., 2002. A free energy model for piezoceramic materials. In: 9th SPIE Conference on Smart Structures and Materials, San Diego, USA, pp. 17–22; Smith, R.C., Seelecke, S., Ounaies, Z., Smith, J., 2003. A free energy model for hysteresis in ferroelectric materials. J. Intell. Mater. Syst. Struct. 14, 719–739; Smith, R.C., Seelecke, S., Dapino, M.J., Ounaies, Z., 2005. A unified framework for modeling hysteresis in ferroic materials. J. Mech. Phys. Solids 54, 46–85] is generalized to three space dimensions including both polarization and strain. In the resulting nine-dimensional energy function, six free energy potentials representing the six distinct types of tetragonal variants of perovskite lattice structures are given as quadratic functions of polarization vector and strain tensor. Energy barrier expressions as functions of thermodynamic driving forces are obtained through a generalization of the one-dimensional equations derived from the model of Smith et al. This approach presents an alternative to the cumbersome determination of higher-dimensional saddle points and is attractive for a computationally efficient implementation. The energy barrier expressions are combined with evolution equations for the variant fractions based on the theory of thermally activated processes and thus allow for a natural treatment of rate-dependent effects. The predictions of the model are compared with recent measurements on BaTiO₃ single crystals by Burcsu et al. [Burcsu, E., Ravichandran, G., Bhattacharya, K., 2004. Large electrostrictive actuation of barium titanate single crystals. J. Mech. Phys. 52, 823–846]. The effects of applied stress and 90°- and 180°-switching processes are discussed in detail.     

Domain switching and non-linear coupling of ferroelectric composites

One of the most notable characteristics of ferroelectric materials is that they could undergo spontaneous polarization and spontaneous strain changes by applied fields. Reorientation of the spontaneous polarization and spontaneous polarization strain of ferroelectric inclusions in ferroelectric composites can change microstructures and affect effective electroelastic properties of ferroelectric composites. Based on orientation distribution function and its evolution as well as switching criterion, non-linear electromechanical coupling behaviour of ferroelectric composites is studied by application of micromechanics. A constitutive model of ferroelectric composites is developed. Comparison between analytical and experimental results shows that the model presented can describe many non-linear electromechanical coupling problems of ferroelectric composites such as polarization or depolarization, etc.     

Energy principle of ferroelectric ceramics and single domain mechanical model

Many physical experiments have shown that the domain switching in a ferroelectric material is a complicated evolution process of the domain wall with the variation of stress and electric field. According to this mechanism, the volume fraction of the domain switching is introduced in the constitutive law of ferroelectric ceramic and used to study the nonlinear constitutive behavior of ferroelectric body in this paper. The principle of stationary total energy is put forward in which the basic unknown quantities are the displacement u _i , electric displacement D _i and volume fraction ρ _I of the domain switching for the variant I. Mechanical field equation and a new domain switching criterion are obtained from the principle of stationary total energy. The domain switching criterion proposed in this paper is an expansion and development of the energy criterion. On the basis of the domain switching criterion, a set of linear algebraic equations for the volume fraction ρ _I of domain switching is obtained, in which the coefficients of the linear algebraic equations only contain the unknown strain and electric fields. Then a single domain mechanical model is proposed in this paper. The poled ferroelectric specimen is considered as a transversely isotropic single domain. By using the partial experimental results, the hardening relation between the driving force of domain switching and the volume fraction of domain switching can be calibrated. Then the electromechanical response can be calculated on the basis of the calibrated hardening relation. The results involve the electric butterfly shaped curves of axial strain versus axial electric field, the hysteresis loops of electric displacement versus electric filed and the evolution process of the domain switching in the ferroelectric specimens under uniaxial coupled stress and electric field loading. The present theoretic prediction agrees reasonably with the experimental results given by Lynch. The project supported by the National Natural Science Foundation of China (10572138).     

A micromechanics method to study the effect of domain switching on fracture behavior of polycrystalline ferroelectric ceramics

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On twinning and domain switching in ferroelectric Pb(Zr1−xTix)O3—part I: twins and domain walls

We study the electromechanical behavior of lead zirconate titanate ferroelectric ceramics (PZT), by means of a three-dimensional continuum model for deformable ferroelectric bodies in their polar phase characterized by spontaneous polarization and strain. Spontaneous polarization and strain organize into a domain structure which minimizes electrostatic and elastic energies and which can be modified by the application of electromechanical loads. Such process, which is called “domain switching”, is associated with electrical and mechanical hysteresis and can be studied as a minimization problem for a functional which reminds the micromagnetic energy of deformable ferromagnetics. In this paper, which is the first of two, we deal with the electromechanical model and related constitutive assumptions, as well as with the analysis of domain structure in PZT. In particular, following the discover of a new monoclinic phase in PZT carried by Noheda and co-workers, we analyze twinning between spontaneous strain at the various phase boundaries and show that both non-generic, non-conventional twins and finely-twinned laminates are possible, and also that the presence of a monoclinic phase may explain PZT exceptional properties.     

A rate-dependent two-dimensional free energy model for ferroelectric single crystals

The one-dimensional free energy model for ferroelectric materials developed by Smith et al. [29–31] is generalized to two dimensions. The two-dimensional free energy potential proposed in this paper consists of four energy wells that correspond to four variants of the material. The wells are separated by four saddle points, representing the barriers for 90°-switching processes, and a local maximum, across which 180°-switching processes take place. The free energy potential is combined with evolution equations for the variant fractions based on the theory of thermally activated processes. The model is compared to recent measurements on BaTiO₃ single crystals by Burcsu et al. [8], and predicitions are made concerning the response to the application of in-plane multi-axial electric fields at various frequencies and loading directions. The kinetics of the 90°- and 180°-switching processes are discussed in detail.     

Domain Engineered Relaxor Ferroelectric Single Crystals

Single crystal relaxor ferroelectric materials exhibit extraordinary electromechanical properties. They are being applied in high performance sensors, actuators, and transducers. Field induced polarization switching and phase transitions of these crystals lead to complex nonlinear behavior. In recent years experimental investigations have been conducted to characterize the polarization switching and phase transition behavior as a function of crystallographic orientation, temperature, electric field, and stress. The results give insight into the mechanism underlying the observed large field hysteretic behavior. This review article describes the observed behavior and presents results of multiscale modeling that predicts the macroscopic behavior from the single domain single crystal behavior and evolution of crystal variants at the microscale.     

Time dependence of effective properties of polycrystalline ferroelectric ceramics

In this paper, based on Merz[7] experimental results and classical nucleation theory, a micromechanics statistical model is proposed to describe the relation between the special microstructure-level evolution phenomena-domain switching and macro-response. The polycrystalline ferroelectric ceramics treated as a composition of switched domain and unswitched domain, the approaches of Eshelby's equivalent inclusion and Mori-Tanaka's mean field theory are used to analyze and predict its effective electroelastic properties. The model can incorporate the effects of time dependence of domain switching and shape of individual crystalline. To the BaTiO₃ polycrystalline ceramics, the analytical results are in good agreement with the experimental results.     

Continuum thermodynamics of ferroelectric domain evolution: Theory, finite element implementation, and application to domain wall pinning

A continuum thermodynamics framework is devised to model the evolution of ferroelectric domain structures. The theory falls into the class of phase-field or diffuse-interface modeling approaches. Here a set of micro-forces and governing balance laws are postulated and applied within the second law of thermodynamics to identify the appropriate material constitutive relationships. The approach is shown to yield the commonly accepted Ginzburg-Landau equation for the evolution of the polarization order parameter. Within the theory a form for the free energy is postulated that can be applied to fit the general elastic, piezoelectric and dielectric properties of a ferroelectric material near its spontaneously polarized state. Thereafter, a principle of virtual work is specified for the theory and is implemented to devise a finite element formulation. The theory and numerical methods are used to investigate the fields near straight 180° and 90° domain walls and to determine the electromechanical pinning strength of an array of line charges on 180° and 90° domain walls.     

Three dimensional phase field study on the thickness effect of ferroelectric polymer thin film

The electromechanical behavior of poly(vinylidene fluoride-trifluoroethylene) [P(VDF-TrFE)] ferroelectric thin film was investigated using the three dimensional (3D) phase-field method. Various energetic contributions, including elastic, electrostatic, and domain wall energy were taken into account in the variational functional of the phase field model. Evolution of the microscopic domain structures of P (VDF-TrFE) polymer film was simulated. Effects of the in-plane residual stress, the film thickness and externally applied electric bias field on the electromechanical properties of the film were explored. The obtained numerical results showed that the macroscopic responses of the electric hysteresis loops are sensitive to the residual stress and electric bias field. It was also found that thickness has a great effect on the electric hysteresis loops and remanent polarization.     

Micromechanics of ferroelectric domain switching behavior: Part II: Constitutive relations and hysteresis

A constitutive relation is developed to describe the nonlinear behavior of ferroelectric ceramics subjected to external stress and electric field. The theoretical development considers each domain as an inclusion. The Helmholtz and Gibbs free energy of the constituent element are derived by using a micromechanics approach. They are functionals of the orientation distribution function (ODF) that represents the domain distribution patterns. By applying the internal variable theory and expanding ODF in Fourier series, the yield condition, evolution of ODF, and constitutive relation are obtained. Theoretical results agree with experiments.     

A one-dimensional continuum model for ferroelectric ceramics

In this article, we introduce a one-dimensional continuum model for ferroelectric ceramics within a thermodynamical framework. The model consists of a free energy potential, a switching criterion, and a kinetic relation. The free energy potential is given as a function of polarization, strain, and two internal variables – remanent polarization and remanent strain. A polarization switching is described by evolutions of the two internal variables and evolution laws called kinetics are proposed based on the second law of thermodynamics. The predictions of the model are compared with experimental observations. It is suggested to model unpoled domains in the fully poled state for improved model responses.     

An I-integral method for crack-tip intensity factor variation due to domain switching in ferroelectric single-crystals

In the present study, an I-integral method is established for solving the crack-tip intensity factors of ferroelectric single-crystals. The I-integral combined with the phase field model is successfully used to investigate crack-tip intensity factor variations due to domain switching in ferroelectricity subjected to electromechanical loadings, which exhibits several advantages over previous methods based on small-scale switching. First, the shape of the switching zone around a crack tip is predicted by the time-dependent Ginzburg–Landau equation, which does not require preset energy-based switching criterion. Second, the I-integral can directly solve the crack-tip intensity factors and decouple the crack-tip intensity factors of different modes based on superimposing an auxiliary state onto an actual state. Third, the I-integral is area-independent, namely, the I-integral is not affected by the integral area size, the polarization distributions, or domain walls. This makes the I-integral applicable to large-scale domain switching. To this end, the electro-elastic field intensity factors of an impermeable crack in PbTiO₃ ferroelectric single crystals are evaluated under electrical, mechanical, and combined loading. The intensity factors obtained by the I-integral agree well with those obtained by the extrapolation technique. From numerical results, the following conclusions can be drawn with respect to fracture behavior of ferroelectrics under large-scale switching. Under displacement controlled mechanical loading, the stress intensity factors (SIFs) decrease monotonically due to the domain switching process, which means a crack tip shielding or effective switching-induced toughening occurs. If an external electric field is applied, the electric displacement intensity factor (EDIF) increases in all cases, i.e., the formed domain patterns enhance the electric crack tip loading. The energy release rate, expressed by the crack-tip J-integral, is reduced by the domain switching in all examples, which underlines the switching-induced-toughening effect. In contrast, under stress controlled load, the SIF evolves due to large-scale switching to a stable value, which is higher than the non-switching initial value, i.e., fracture is promoted in this case.     

Macroscopical non-linear material model for ferroelectric materials inside a hybrid finite element formulation

A new approach for modeling hysteretic non-linear ferroelectric ceramics is presented, based on a fully ferroelectric/ferroelastic coupled macroscopic material model. The material behavior is described by a set of yield functions and the history dependence is stored in internal state variables representing the remanent polarization and the remanent strain. For the solution of the electromechanical coupled boundary value problem, a hybrid finite element formulation is used. Inside this formulation the electric displacement is available as nodal quantity (i.e. degree of freedom) which is used instead of the electric field to determine the evolution of remanent polarization. This involves naturally the electromechanical coupling. A highly efficient integration technique of the constitutive equations, defining a system of ordinary differential equations, is obtained by a customized return mapping algorithm. Due to some simplifications of the algorithm, an analytical solution can be calculated. The automatic differentiation technique is used to obtain the consistent tangent operator. Altogether this has been implemented into the finite element code FEAP via a user element. Extensive verification tests are performed in this work to evaluate the behavior of the material model under pure electrical and mechanical as well as coupled and multi-axial loading conditions.     

Compatible domain arrangements and poling ability in oriented ferroelectric films

The microstructure in single crystal ferroelectric films is significantly affected by the substrate conditions and the crystallographic orientation of the film. Domain arrangements form to minimize the total energy producing a stable state corresponding to the external boundary conditions. In order to find low energy domain arrangements, this study uses exact compatibility conditions for periodic laminate structures which ensure that all the adjacent domains fit together compatibly. These conditions are applied to films with various orientations and crystal systems, such as the rhombohedral and tetragonal crystal systems. A systematic search is used to discover exactly compatible structures for given states of macroscopic strain and polarization in the film. The theory is applied to [001], [011], and [111] oriented rhombohedral and tetragonal films. The results indicate poling paths along which the microstructure can evolve continuously while maintaining compatibility, to get from a state of zero through-thickness polarization to the state with the greatest value of through-thickness polarization. The evolution of the domain arrangement along these poling paths is shown, and the poling ability, or the limit on the maximum polarization achieved, is discussed. The influence of a strain state imposed by the substrate on the microstructure and poling ability is studied. The use of the model is illustrated by developing poling maps for a tetragonal [001] oriented film to show the set of polarization states that can be achieved as a function of the imposed substrate strain.     

Interplay between polarization rotation and crack propagation in PMN-PT relaxor single crystals

Investigations on the interconnection between the polarization rotation and crack propagation are performed for [110]-oriented 74Pb(Mg_1/3Nb_2/3)O₃-26PbTiO₃ relaxor ferroelectric single crystal under electric loadings along [001] direction. The crystal is of predominantly monoclinic M_A phase with scatter distributed rhombohedral (R) phase under a moderate poling field of 900 V/mm in [001] direction. With magnitude of 800 V/mm, a through thickness crack is initiated near the electrode by electric cycling. Static electric loadings is then imposed to the single crystal. As the applied static electric field increases, domain switching in the monoclinic M_A phase and phase transition from M_A to R phase occur near the crack. The results indicate that the crack features a conducting one. Whether domain switching or phase transition occurs depends on the intensity of the electric field component that is perpendicular to the applied electric field.     

Utilizing mechanical loads and flexoelectricity to induce and control complicated evolution of domain patterns in ferroelectric nanofilms

We have conducted a systematical investigation to reveal the stability and evolution path of various ferroelectric domain patterns in nanofilms subjected to mechanical loads and related flexoelectric field. Within a rigorous framework of flexoelectricity, a phase-field approach has been established for simulating the domain structure of ferroelectric nanofilms. The electromechanical fields of the nanofilms are numerically solved by a fast Fourier transform technique (FFT) based on the combination of Khachaturyan's microscopic elastic theory and Stroh's formalism of anisotropic elasticity. Using this approach, we simulate eight types of domain patterns that can be stabilized in the nanofilms. It is further demonstrated that these domain patterns can be significantly affected by the mechanical loads and related flexoelectric field and exhibit fruitful evolution paths. To adapt the applied mechanical strain and strain gradient, the domain pattern may remain stable, evolve into another polydomain pattern, or become a monodomain state (an effect of domain erasing). The domain fraction, detailed domain morphology, average stresses in the nanofilms, average polarization and temporal evolution characteristics of the domain patterns under various mechanical loads and sources of flexoelectric field have been analyzed. This investigation should provide instructive information for the practical application of ferroelectric nanofilms under complex and changeable mechanical conditions.