The shift of Curie temperature and evolution of ferroelectric domain in ferroelectric crystals

A micromechanics-based thermodynamic model for the phase transition of ferroelectric crystals is developed and, with it, the shift of Curie temperature and evolution of ferroelectric phase upon cooling are examined. This approach differs from the classical phenomenological one in that the evolution of new domain concentration can be predicted. We start out by formulating the Gibbs free energy of a generic, two-phase crystal consisting of the parent paraelectric phase and the transformed ferroelectric phase, at a given level of temperature, stress, and electric field. The thermodynamic driving force for domain growth is then derived and, together with the resistance force, a kinetic equation is established. The derived driving force is found to arise from three different sources of Gibbs free energy: (i) the interaction energy due to the heterogeneity of electromechanical moduli of the parent and product phases, (ii) the energy dissipation due to spontaneous polarization, and (iii) the self-energy of the dual-phase system due to the existence of polarization strain and electric polarization. For a BaTiO₃ crystal the electromechanical heterogeneity is found to play a rather significant role that seems not to have been recognized before. The derived shift recovers to the Clausius-Clapeyron relation if such heterogeneity disappears. We have examined in detail several factors that affect the shift of Curie temperature, and calculated the evolution of overall polarization and dielectric constant of a BaTiO₃ crystal. The results are found to be consistent with available test data.     

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.     

多孔极化PZT95/5铁电陶瓷单轴压缩力学响应与放电特性

采用添加造孔剂的方法制备了四种不同孔隙率PZT95/5铁电陶瓷,对其进行电场极化,随后开展了准静态单轴压缩实验,讨论了畴变、相变以及孔隙率对极化PZT95/5铁电陶瓷的力学响应与放电特性的影响. 研究结果表明：(1)多孔极化PZT95/5铁电陶瓷非线性力学响应行为主要归因于畴变和相变的共同作用,与内部孔洞变形和坍塌基本无关;(2) 在准静态单轴压缩下极化PZT95/5铁电陶瓷的去极化机制是畴变和相变的共同作用;(3) 孔隙率对极化PZT95/5铁电陶瓷的弹性模量、压缩强度有明显的影响,而对断裂应变的影响较小;(4)极化PZT95/5铁电陶瓷畴变和相变开始的临界应力都随着孔隙率的增大而线性衰减,但相变开始的临界体积应变却不依赖孔隙率;(5)极化PZT95/5铁电陶瓷电荷饱和释放量随着孔隙率呈线性减小,但孔隙率对电荷释放速率基本没有影响。     

Two-scale micromechanics-based probabilistic modeling of domain switching in ferroelectric ceramics

A two-scale micromechanics model is developed in this paper to analyze domain switching in ferroelectric ceramics, using a probabilistic domain switching criterion based on energetic analysis. The microstructure of ferroelectric ceramics at two distinct length scales, domains and grains, has been carefully analyzed. The interaction at domain level is accounted for by energy minimization theory, while the fluctuation at grain level is analyzed using ellipsoidal two-point correlation function. The model has been implemented by Monte Carlo method, and applied to simulate the electric poling and mechanical depoling of Pb(Zr_xTi_1-x)O₃ (PZT) ceramics across morphotropic phase boundary (MPB). The drastically different switching characteristics of PZT ceramics across MPB has been captured, and good agreement with experiments has been observed. The effects of the transformation strains and spontaneous polarizations are highlighted, confirming the proposition of Li et al. [2005. Domain switching in polycrystalline ferroelectric ceramics. Nature Materials 4, 776–781] that the strain compatibility plays a dominant role in domain switching in ferroelectric ceramics.     

铁电陶瓷PZT53复杂力电耦合行为的实验研究

通过实验研究了平行和垂直于极化方向的正应力对铁电陶瓷锆钛酸铅(PZT53)的电滞回线(E3-P3)和电致应变曲线(E-ε)的影响. 实验发现平行于极化方向的压应力对PZT53陶瓷的电滞回线、电致应变曲线形状以及矫顽场大小都有明显的影响,但是垂直于极化方向的拉、压应力只对PZT53陶瓷的电致应变曲线形状有明显的影响,但对电滞回线形状和矫顽场大小都没有显著影响. 采用畴翻转的模型详细解释了观察到的实验现象,所得结果为建立铁电陶瓷的多轴力、电耦合本构模型,提供了物理基础.     

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.     

Numerical simulation of domain switching in multilayer ferroelectric actuators

Micromechanical finite element methods are developed based on a nonlinear constitutive model of ferroelectric polycrystals. Electromechanical behaviors ahead of an internal electrode tip are numerically simulated in multilayer ferroelectric actuators. Around the electrode edge, the nonuniform electric field generates a concentration of stress due to the incompatible strain as well as spontaneous strain. The preferred domain switching enhances the concentration of residual stress and may cause the actuators to crack. An electrically permeable crack emanating from an internal electrode is analyzed. A large scale domain switching zone is found in the vicinity of crack tips. The larger the actuating strain and electric field are, the larger the switching zone will be. The size of switching zone even reaches the scale of crack length with increasing electromechanical loading.     

Micromechanical modelling of remanent properties of morphotropic PZT

The present paper investigates the capability of micromechanical material models to predict the ferroelectric behaviour of morphotropic PZT ceramics in a rate-independent approximation based on realistic microscopic material parameters. Starting point is a three-dimensional tetragonal model, which builds on the model of Pathak and McMeeking [2008. Three-dimensional finite element simulations of ferroelectric polycrystals under electrical and mechanical loading. Journal of the Mechanics and Physics of Solids 56, 663-683]. Volume fractions of the crystallographic variants represent the domain structure inside the grains. Interactions between the grains are taken into account by means of a representative volume element of the grain compound. A simplified set of realistic microscopic material parameters of the lattice in terms of Young's modulus, Poisson's ratio, dielectric constant, and spontaneous strain and polarisation is derived from experimental data and theoretical results given in the literature. The simulation of the macroscopic remanent polarisation and strain response due to two load cases shows explicitly that the tetragonal model is not capable to reproduce the behaviour of morphotropic PZT. Therefore, the model is extended by the rhombohedral phase, allowing a mixture of both phases with varying quantities inside the grains. A comparison of our results with experimental data shows a remarkably good agreement, revealing the capability of the extended model.     

高应变率下多孔未极化PZT95/5铁电陶瓷的非线性力学行为

采用添加造孔剂的方法制备了4种不同孔隙率的未极化PZT95/5铁电陶瓷。采用基于超高速相机与数字图像相关性方法的试样全场应变测量技术以及分离式霍普金森压杆（SHPB）技术,对多孔未极化PZT95/5铁电陶瓷进行高应变率单轴压缩实验研究。全场应变测量结果显示:轴向应变仅在试样中部分布较均匀,将该区域的平均应变作为应力-应变关系中的试样应变测量值较为合理,而由SHPB原理计算的试样应变值明显偏大,需要摒弃或修正传统的SHPB数据处理方法。通过波形整形技术实现了恒应变率加载,弱化了径向惯性效应的影响,揭示出多孔未极化PZT95/5铁电陶瓷的压缩强度具有显著的应变率效应。通过分析试样轴向应变和径向应变随着加载应力的变化,阐明多孔未极化PZT95/5铁电陶瓷的非线性变形行为的物理机制是畴变和相变共同作用,并发现畴变临界应力和相变临界应力都随着应变率升高而增大。保持加载应变率不变,讨论了孔隙率对多孔未极化PZT95/5铁电陶瓷动态力学行为的影响,发现随着孔隙率的升高,动态压缩强度呈非线性衰减,而畴变临界应力和相变临界应力则基本呈线性衰减。     

A ferroelectric and ferroelastic 3D hexahedral curvilinear finite element

An isoparametric 3D electromechanical hexahedral finite element integrating a 3D phenomenological ferroelectric and ferroelastic constitutive law for domain switching effects is proposed. The model presents two internal variables which are the ferroelectric polarization (related to the electric field) and the ferroelastic strain (related to the mechanical stress). An implicit integration technique of the constitutive equations based on the return-mapping algorithm is developed. The mechanical strain tensor and the electric field vector are expressed in a curvilinear coordinate system in order to handle the transverse isotropy behavior of ferroelectric ceramics. The hexahedral finite element is implemented into the commercial finite element code Abaqus® via the subroutine user element. Some linear (piezoelectric) and non linear (ferroelectric and ferroelastic) benchmarks are considered as validation tests.     

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.     

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).     

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.     

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.     

Time dependent behavior of ferroelectric materials undergoing changes in their material properties with electric field and temperature

This paper presents a time dependent polarization constitutive model suitable for predicting nonlinear polarization and electro-mechanical strain responses of ferroelectric materials subject to various histories of electric fields. The constitutive model is derived based on a single integral form with nonlinear (electric field and temperature dependent) integrand. The total polarization consists of the time-dependent and residual components. The residual component of the polarization is due to polarization switching in the ferroelectric materials. We use an ‘internal clock’ concept to incorporate the effect of electric field on the rate of polarization. The corresponding strain response is determined through the use of third order piezoelectric constant and/or fourth order electrostrictive constant that vary with polarization stage. It is assumed that in absence of polarization, both piezoelectric and electrostrictive constants are zero. To incorporate the effect of temperature on the overall polarization behavior all material parameters in the constitutive model are allowed to change with the ambient temperature. We present numerical studies on the effect of time, temperature, and electric field on the response of ferroelectric material followed by verification of the constitutive model. Experimental data on lead zirconate titanate (PZT) materials available in the literature are used to verify the model.     

铁电陶瓷微观电畴的成核率模型

大量的实验已经证实电畴翻转是铁电材料非线性和迟滞性本构曲线的根本原因。研究者已经对铁电陶瓷的微观电畴翻转行为进行了大量详细的研究。针对描述电畴成核的物理实验结果和经典的成核率实验数据,为了建立电畴翻转体积分数的演化方程提出了反应微观电畴翻转的成核率模型。针对铁电试样电畴随机分布的情况,应用该模型对铁电陶瓷的非线性本构行为进行了研究。理论结果与实验数据的比较表明,模型能较好的描述铁电材料的非线性本构行为。同时模型所揭示的微观反转的物理本质可进一步的指导宏观唯象模型的改进。     

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.     

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.     

Three-dimensional finite element simulations of ferroelectric polycrystals under electrical and mechanical loading

Complex, non-linear, irreversible, hysteretic behavior of polycrystalline ferroelectric materials under a combined electro-mechanical loading is a result of domain wall motion, causing simultaneous expansion and contraction of unlike domains, grain sub-divisions that have distinct spontaneous polarization and strain. In this paper, a 3-dimensional finite element method is used to simulate such a polycrystalline ferroelectric under electrical and mechanical loading. A constitutive law due to Huber et al. [1999. A constitutive model for ferroelectric polycrystals. J. Mech. Phys. Solids 47, 1663-1697] for switching by domain wall motion in multidomain ferroelectric single crystals is employed in our model to represent each grain, and the finite element method is used to solve the governing conditions of mechanical equilibrium and Gauss's law. The results provide the average behavior for the polycrystalline ceramic. We compare the outcomes predicted by this model with the available experimental data for various electromechanical loading conditions. The qualitative features of ferroelectric switching are predicted well, including hysteresis and butterfly loops, the effect on them of mechanical compression, and the response of the polycrystal to non-proportional electrical loading.     

A Continuum Theory of Deformable,Semiconducting Ferroelectrics

Ferroelectric solids, especially ferroelectric perovskites, are widely used as sensors, actuators, filters, memory devices, and optical components. While these have traditionally been treated as insulators, they are in reality wide-band-gap semiconductors. This semiconducting behavior affects the microstructures or domain patterns of the ferroelectric material and the interaction of ferroelectrics with electrodes, and is affected significantly by defects and dopants. In this paper, we develop a continuum theory of deformable, semiconducting ferroelectrics. A key idea is to introduce space charges and dopant density as field (state) variables in addition to polarization and deformation. We demonstrate the theory by studying oxygen vacancies in barium titanate. We find the formation of depletion layers, regions of depleted electrons, and a large electric field at the ferroelectric–electrode boundary. We also find the formation of a charge double layer and a large electric field across 90° domain walls but not across 180° domain walls. We show that these internal electric fields can give rise to a redistribution or forced diffusion of oxygen vacancies, which provides a mechanism for aging of ferroelectric materials.