A constrained domain-switching model for polycrystalline ferroelectric ceramics. Part II: Combined switching and application to rhombohedral materials

A combined switching assumption (CSA) is incorporated into the constrained domain-switching model presented in part I to address the nonsymmetric deformations of ferroelectric ceramics under tension and compression. Using the CSA, the domain-switching process in rhombohedral ferroelectric/ferroelastic ceramics is analyzed in detail. The results show that in rhombohedral lead titanate zirconate (PZT) ceramics, most 109° switching can be accomplished with a minor fraction of 71° switching during electric poling, which is quite different from BaTiO₃ and tetragonal PZT ceramics in which only a few per cent of 90° switching can occur. Domain switching under combined uniaxial electric and mechanical tension/compression is also studied. Uniaxial tension is found to enhance electric poling while uniaxial compression inhibits it.     

Nonlinear electric–mechanical behavior and micromechanics modelling of ferroelectric domain evolution

Domains exist in ferroelectric ceramics. External loads, such as electric field and stress, can cause domain switching. Domain switching always results in nonlinear ferroelectricity and ferroelasticity of ferroelectric ceramics. In this investigation, nonlinear electric–mechanical behavior related to ferroelectric and ferroelastic domain switching is experimentally and theoretically studied. In the experimental work, the electric–mechanical response of a soft PZT ferroelectric ceramic subjected to combined electric–mechanical loads was observed. The effect of different compressive stress levels on the electromechanical response was examined. In the theoretical modelling, the orientation of each domain is defined by its local coordinate relative to a fixed global coordinate. Orientation distribution function (ODF) is used to describe the domain pattern. For mathematical simplicity, the Reuss average is used in the modelling. According to the proposed theory, a domain has different Gibbs' energy at different orientation states and the energy difference forms the domain switching driving force. The domain pattern and its evolution are determined by the joint action of the domain switching driving force and the dissipation during domain switching. In ferroelectricity and ferroelasticity, 90° and 180° domain switchings play different roles and have different switching dissipations associated with them. A criterion considering the difference between the 90° switching and the 180° switching is established by the thermodynamic approach. There is an agreement between theoretical and experimental results. It should be pointed out that the micromechanical model proposed in this paper is restricted to ferroelectric materials exhibiting transformation from cubic to tetragonal only.     

A temperature-dependent two-step domain-switching model for ferroelectric materials

Piezoelectric stack actuators are promising candidates for use in fuel injection technology. Experimental results for stack actuators have shown that a significant amount of heat is generated when they are driven under high electric-field magnitudes and/or high frequency, both of which occur in fuel injectors. They also exhibit hysteretic nonlinear behavior when driven under high electric-field magnitudes. In this paper, a new domain-switching model for PZT materials is developed. The model is based on changes in potential energy, and accounts for the temperature effect on domain switching. It also accounts for full thermo-electro-mechanical coupling. Additionally, different energy levels are assumed for different domain-switching types. It is assumed that 180° switching is a two-step process caused by two 90° switchings. A finite element implementation of a thermopiezoelectric continuum based on the proposed switching model is presented. The model shows good agreement with experimental results at different temperatures and loading conditions.     

Study of electrical and mechanical contribution to switching in ferroelectric/ferroelastic polycrystals

Polarization switching in a polycrystalline ferroelectric/ferroelastic ceramic is simulated with a finite element model. It is assumed that a crystallite switches if the reduction in potential energy of the polycrystal exceeds a critical energy barrier per unit volume of switching material. Each crystallite, represented by a cubic element in a finite element mesh, is a single domain that switches completely without a simulated domain wall motion. The possible dipole directions of each crystallite are assigned randomly subject to crystallographic constraints. The model accounts for electric field induced (i.e. ferroelectric) switching and stress induced (i.e. ferroelastic) switching without piezoelectric interaction. Different weights for the mechanical and electrical contribution to switching are selected phenomenologically to simulate electric displacement vs electric field and strain vs electric field of a ceramic lead lanthanum zirconate titanate (PLZT). Although the critical energy barriers for 90° and 180° switching are assumed to be the same, 90° switching is favored when the electrical contribution to switching (i.e. electrical energy) is dominant, but 180° switching is favored when the mechanical contribution to switching (i.e. elastic strain energy) is dominant. With increasing mechanical contribution and decreasing electrical contribution, the simulated electric displacement deviates from the Rayleigh law under a low applied electric field, and the shape of a switching region (or a process zone) changes from a prolonged ellipsoid to a sphere.     

Study of effect of 90° domain switching on ferroelectric ceramics fracture using the moiré interferometry

Fracture behavior of ferroelectric ceramics during in-plane and out-of-plane 90° domain switching was studied using the moiré interferometry technique. The specimens used in the experiment were three-point-bending beams, each with a single through-notch, which were subjected to a mechanical load, an electrical load and a combined electrical and mechanical load, respectively. The main subject of interest is to investigate the influence of 90° domain switching induced by the electrical load on fracture toughness and material brittleness. In the experiment, compared with out-of-plane 90° domain switching, in-plane 90° domain switching occurs in the region of approximately a 45° band. It causes larger in-plane tensile strain ε_xx in almost all regions of a specimen, especially in the 45° band. In-plane 90° domain switching greatly decreases fracture toughness and weakens the material brittleness of ferroelectric ceramics. The out-of-plane 90° domain switching does not exhibit such a great influence on fracture toughness and material brittleness as the in-plane 90° domain switching does.     

Frequency effects on fatigue crack growth and crack tip domain-switching behavior in a lead zirconate titanate ceramic

The microstructural origins of the effect of frequency on the electrical fatigue behavior of pre-cracked soft ferroelectric Pb(Zr_0.48Ti_0.52)O₃ is investigated by means of a high spatial resolution hard X-ray synchrotron source. It is found that there is a strong link between the frequency of the applied bipolar field, domain-switching behavior in terms of ferroelastic reorientation of the domains around the crack tip and the resultant crack growth. The crack growth is accentuated under increased ferroelastic switching and, in particular, found to be more pronounced under low-frequency loading. The concept of domain wall viscoelasticity is applied to explain why lower frequencies accelerate crack growth under a bipolar electric field.     

Direct observation of asymmetric domain wall motion in a ferroelectric capacitor

We report in situ transmission electron microscopy observations of the 180° polarization switching process of a PbZr_0.2Ti_0.8O₃ (PZT) capacitor. The preferential, but asymmetric, nucleation and forward growth of switched c-domains were observed at the PZT/electrode interfaces, arising due to the built-in electric field induced at each interface. The subsequent sideways growth of the switched domains was inhibited by the depolarization field due to the imperfect charge compensation at the counter-electrode and also at the boundaries with preexisting a-domains, which contributed further to the asymmetric switching behavior. It was found that the preexisting a-domains split into fine a- and c-domains constituting a 90° stripe domain pattern during the 180° polarization switching process, revealing that these domains also actively participated in the out-of-plane polarization switching. The real-time observations uncovered the origin of the switching asymmetry and further clarified the importance of charged domain walls and the interfaces with electrodes in the ferroelectric switching processes.     

Lateral size and thickness dependence in ferroelectric nanostructures formed by localized domain switching

Ferroelectric nanostructures can be formed by local switching of domains using techniques such as piezo-force microscopy (PFM). Understanding the dependence of the switching behavior on the lateral size of the electrode is important to determine the minimum feature size for writing ferroelectric nanostructures. To understand these lateral size effects, we use the time-dependent Ginzburg–Landau equations in a two-dimensional square to rectangle ferroelectric transition to simulate localized switching of domains for PFM-type and parallel-plate capacitor configurations. Our investigations indicate that fringing electric fields lead to switching via intermediate 90° domains even in the absence of substrate or clamping effects for films of sufficient thicknesses, and via 180° rotations at smaller thicknesses. The voltage required to switch the domain increases by decreasing the lateral size, and at very small lateral sizes the coercive voltage becomes so large that it becomes virtually impossible to switch the domain.     

Conducting crack propagation driven by electric fields in ferroelectric ceramics

Ferroelectric ceramics are susceptible to fracture under high electric fields, which are commonly generated in the vicinity of electrodes or conducting layers. In the present work, we extend a phase-field model of fracture in ferroelectric single crystals to the simulation of the propagation of conducting cracks under purely electrical loading. This is done by introducing the electrical enthalpy of a diffuse conducting layer into the phase-field formulation. Simulation results show oblique crack propagation and crack branching from a conducting notch, forming a tree-like crack pattern in a ferroelectric sample under positive and negative electric fields. Microstructure evolution indicates the formation of tail-to-tail and head-to-head 90° domains, which results in charge accumulation around the crack. The charge accumulation, in turn, induces a high electric field and hence a high electrostatic energy, further driving the conducting crack. Salient features of the results are compared with experiments.     

Nonlinear constitutive behavior of soft and hard PZT: experiments and modeling

This work characterizes and models the nonlinear inelastic behavior of lead zirconate–titanate (PZT) ceramics. A detailed investigation of the hysteresis loop of the stress–strain curve shows that the process of ferroelastic non-180° polarization switching and of switching saturation are, respectively, a smooth softening process and a gradual hardening process. The inflection point in the curve is found to be a boundary point that distinguishes the transition from a softening process to a hardening one. Based on underlying physical mechanisms, this nonlinear behavior is simulated by a mechanical model, consisting of several parallel-arranged Maxwell chains with each chain taking a main role in a different, successive stage of the switching process. Each chain includes a nonlinear spring and a frictional slider. The generalized displacement of the slider, indicating an internal variable, is introduced to describe the local process of the cooperative switching of the corresponding grain group. The hardening and softening processes are controlled by variations of stiffness of the nonlinear springs. Furthermore, the observed similarities between the hysteresis loop of the stress–strain curve and that of stress–electric displacement curve are used to develop an equivalent stress concept and to formulate constitutive laws for PZT ceramics. Comparisons between the experimental hysteresis loops and those calculated by the proposed model show satisfactory agreement for both hard PZT and C5800 and soft C5500.     

外场作用下BaTiO_3单晶裂纹扩展和畴变

通过偏振光显微镜对连续应力和恒电场作用下的BaTiO_3单晶中的裂纹扩展和畴变过程进行原位观察.结果表明.在连续应力的作用下,裂尖处应力集中导致出现畴变带,裂纹与畴变带垂直相交,畴带和裂纹一起向前移动,畴变在先,裂纹扩展在后;在恒电场作用下,畴变引起的不协调应变导致电致裂纹扩展,畴变始终发生在裂纹扩展之前.裂纹向前扩展时不断的切过前方的畴带,直到不再有畴变发生时裂纹停止扩展.     

THREE DIMENSIONS PHASE FIELD SIMULATION OF MECHATRONIC COUPLE FOR FERROELECTRIC MATERIALS

利用三维相场理论模拟了铁电材料的自发极化、电滞回线以及力电耦合对畴变的影响. 结果表明, 外场下的畴变是通过新畴形核及畴壁移动所引起的长大实现的. 逐渐改变外场时, 在矫顽场附近各类电畴均发生90°畴变, 从而导致极化强度突变. 沿垂直电场方向加 拉、压应变能阻止或促进沿电场方向的畴变.     

R-curve behavior of BaTiO3- and PZT ceramics under the influence of an electric field applied parallel to the crack front

The fracture resistance curves (R-curves) of BaTiO₃ and commercial PZT–PIC 151 were measured with compact tension specimens under the influence of an electric field applied parallel to the crack front. A strong influence of the electric field on the starting and plateau value was found as well as on the length of the R-curve. Generally a toughness increase was detected with increasing electric field. The toughening effect is estimated from the change in crack tip stress intensity induced by ferroelastic domain switching near the crack tip using the weight function formalism developed for stress-induced transformation toughening of zirconia ceramics. In order to obtain a quantitative prediction of toughening, ferroelastic and ferroelectric properties were measured.     

Influences of temperature and electric field on the bending strength of lead zirconate titanate ceramics

The present work studies the effects of temperature and a d.c. electric field on the bending strength of PZT-841 ceramics using three-point bending measurement. In the temperature range from 30°C to the Curie point T_c (≈272°C), the bending strength as a function of temperature exhibits a valley shape and the valley floors at a temperature around 225°C, revealing a 25% reduction in comparison with the bending strength at room temperature. Meanwhile, elastic compliance and damping factor exhibit peaks, respectively, at 225°C and 220°C, implying a strong correlation between bending strength and compliance. A positive or negative electric field larger than 3 kV/cm reduces the bending strength of PZT-841 ceramics significantly. For example, the bending strength under a positive field of 20 kV/cm is only one half of that without application of any electric field. The electric field is able to fracture mechanically sustained samples. Under a constant load of 70 MPa, the mean value of the critical electric field at fracture is 11.0 kV/cm. A 90°-domain wall kinetic model is herein proposed to understand the observed phenomena.     

Dielectric properties and relaxor ferroelectric behavior of (1–y)Ba(Zr0.1Ti0.9)O3–yBa(Zn1/3Nb2/3)O3 ceramics

The microstructure, dielectric and ferroelectric properties of (1–y)Ba(Zr_0.1Ti_0.9)O₃–yBa(Zn_1/3Nb_2/3)O₃ (y=0–0.05) ceramics prepared by traditional solid state method were investigated by X-ray diffractometer, scanning electron microscope, electric parameter testing system and ferroelectric tester. It is found that the barium zirconate titanate based ceramics are single-phase perovskites as y increases up to 0.05 and their average grain size decreases with the increase of y. The permittivity maximum ε_r,max is suppressed from 8948 to 1611 at 1 kHz with increasing y, and the ferroelectric–paraelectric phase transition temperature T_m decreases from 93 to –89 °C at 1 kHz as y increases. The composition-induced diffuse phase transition is enhanced with increasing y. The relaxor-like ferroelectric behavior with a strong frequency dispersion of T_m and permittivity at T<T_m accompanied by a strong diffuse phase transition is found for the system with high y value. The remnant polarization decreases with increasing y, while the coercive field decreases remarkably and then increases with the increase of y.     

Gamma Ray Irradiation Effects on the Ferroelectric and Piezoelectric Properties of Barium Titanate Ceramics

The effect of heavy dose gamma ray irradiation on the ferroelectric and piezoelectric properties of barium titanate (BaTiO₃) ceramics has been investigated. It is found that on irradiation the ferroelectric property decreases and polarization behavior shows double loop hysteresis. The piezoelectric properties including piezoelectric charge constant (d ₃₃), electromechanical coupling coefficient (K _p), and electrostrictive strain also decreases. The most probable reason for decreased ferroelectric and piezoelectric properties may be the occurrence of random local strain upon irradiation. The phase transition temperature from ferroelectric to paraelectric decreases and degree of diffuseness increases on irradiation. The thermoluminescence (TL) glow curve showed a peak at 226 °C showing that irradiated BaTiO₃ has TL properties. Presence of TL clearly indicates that gamma ray irradiation causes trapped holes and electrons and these trapped charges are released at temperature higher than 226 °C. The creation of trapped holes and electrons effected the microstrain of BaTiO₃ ceramic leading to change in the ferroelectric and piezoelectric properties of BaTiO₃ ceramic.     

Toughening of conducting cracks due to domain switching

Recent investigations [Heyer et al., Acta mater. 46 (1998) 6615] of conducting cracks under combined electrical and mechanical loading in piezoelectric ceramics revealed that the fracture load (K_I) changes with applied electric field (K_E). The experimental observations contradict theoretical predictions based on linear piezoelectricity and strip-saturation models. In this paper, interaction between a semi-infinite conducting crack and switching induced strains and polarizations is studied to examine the above discrepancy. New fundamental solutions for the electroelastic field intensity factors induced by transformation strains and polarizations are derived and used to develop a model for variation of fracture toughness due to domain switching. A work energy based criterion for domain switching and a Reuss-type approximation for poly-domain piezoelectrics reported previously are used in the modeling. The influence of electromechanical loading and poling direction on the switching-induced stress intensity factors (toughness variation) is studied for PZT-5H. Under combined electrical and mechanical loading, a positive electric field is found to increase apparent fracture toughness of a crack parallel to the poling direction, while a negative one tends to reduce it. This prediction is consistent with experimental observations.     

Reinterpretation of Type II Hot Corrosion of Co-Base Alloys Incorporating Synergistic Fluxing

The components of gas-turbine engines operating in marine environments are highly susceptible to hot corrosion, which is typically classified as Type II (650–750 °C) and Type I (900–950 °C) hot-corrosion attack. Even though hot-corrosion has been widely investigated in the last 50 years, several critical questions remain unanswered and new ones have emerged based on recent observations that, in part, are associated with the increasing complexity of the alloy systems and the sulfate-deposit chemistries. The present work is focused on the Type II hot-corrosion mechanism for Co-base alloys. Observations for a CoCrAlY model alloy (isothermally exposed at 700 and 800 °C under different atmospheres, including: air and O₂ with 100 and 1000 ppm SO₂) suggest the rapid dissolution of Co (as Co-oxide) is not the controlling factor in the degradation mechanism, as was proposed by Luthra, since the γ-phase which is richer in Co, is not attacked as significantly as the Al-rich β-phase. To the contrary, it is suggested that Al (and Cr) is (are) the element(s) which is (are) removed first. A modified interpretation of the Type II hot-corrosion mechanism is proposed, which is based on the synergistic fluxing model developed by Hwang and Rapp.     

A modified Rietveld method to model highly anisotropic ceramics

High energy X-ray diffraction was employed to probe the complex constitutive behavior of a polycrystalline ferroelectric material in various sample orientations. Pb(Zn,Nb)O₃-Pb(Zr,Ti)O₃ (PZN-PZT) ceramics were subjected to a cyclic bipolar electric field while diffraction patterns were taken. Using transmission geometry and a two-dimensional detector, lattice strain and texture evolution (domain switching) were measured in multiple sample directions simultaneously. In addition, texture analysis suggests that non-180° domain switching is coupled with lattice strain evolution during uniaxial electrical loading. As a result of this material’s high strain anisotropy, the full-pattern Rietveld method was inadequate to analyze the diffraction data. Instead, a modified Rietveld method, which includes an elastic anisotropy term, yielded significant improvements in the data analysis results.     

Collective dynamics in nanostructured polycrystalline ferroelectric thin films using local time-resolved measurements and switching spectroscopy

Grain-to-grain long-range interactions and the ensuing collective dynamics in the domain behavior of nanostructured polycrystalline Pb(Zr,Ti)O₃ ferroelectric thin films have been investigated. To identify the key factors and interactions controlling local polarization dynamics we utilize a synergistic approach based on focused ion beam (FIB) milled damage-free nanostructures to isolate single grains and grain clusters, time-resolved piezoresponse force microscopy and switching spectroscopy PFM (SSPFM) (PFM) to address polarization dynamics within individual grains, and finite-element simulations to quantify the local ferroelectric interactions and hence assess the weight of several possible switching mechanisms. The experiments find that of the three possible switching mechanisms, namely direct electromechanical coupling, local built-in electric field and strain, and grain boundary electrostatic charges, the last one is the dominant mechanism. Although finite-element simulations find that direct electromechanical coupling and local built-in field-induced switching are possible, calculations confirm that for the utilized material properties, the aforementioned mechanisms are energetically unfavored.