A continuum analysis of the driving force of ferroelectric/ferroelastic domain wall motions

First, Eshelby's driving force for the motion of a sharp interface is rederived from general thermodynamic principles. Ferroelectric and ferroelastic domain walls represent a special class of such interfaces in mechanically stressed crystals subject also to an electric field. The corresponding bulk contributions to the driving force are caused by variations of the ferroelectric/ferroelastic anisotropy energy, whereas the interface contributions arise from variations of the intrinsic surface energy of the domain wall and domain wall bending. The general expressions for the local driving force per unit area of such domain walls are specialized then to domain walls in piezoelectric crystals.     

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.     

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.     

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.     

Unconventional domain band structure at the crack tip in ferroelectric ceramics

Domain polarization switch near the tip of a crack or an electrode plays a critical role in the fracture or toughening of ferroelectric ceramics. The intensive electric field near a crack tip stimulates local domain switching. Experiment indicates that the domain band structure in front of an indentation crack under lateral electric loading is unconventional, attributed to the highly localizing crack tip electric field. The partially switched ferroelectric grain resembles a banded Eshelby inclusion embedded in a polycrystalline ferroelectric matrix. The domain wall energy for unconventional domain structures is estimated via arrays of misfit dislocations. Mesomechanics analysis quantifies the unconventional domain band structures. The predicted parameters include the volume fraction, the thickness, and the orientation of switched domain bands.     

微液滴在PDMS软基体表面的动态摩擦行为研究

通过固液界面摩擦力测试装置研究了微液滴在PDMS软基体表面运动时的动态摩擦学行为,并对微液滴体积、滑动速度及软基体力学性能对固液界面动态摩擦行为的影响进行了分析. 结果表明:微液滴在软基体表面运动时表现出最大静摩擦力和动态摩擦力. 最大静摩擦力与微液滴黏度和速度梯度呈正比,动态摩擦力与微液滴体积、滑动速度和基体力学性能有关. 随着微液滴体积的增加,三相接触线长度增加,动态摩擦力增加;随着相对滑动速度增加,三相接触线长度及接触角滞后增加,动态摩擦力增加;随着软基体弹性模量降低,固液界面黏附力增加,固液界面运动能量耗散增加,动态摩擦力增加. 研究结果可为PDMS软基体表面微液滴的精确驱动和运动参数优化提供理论指导,也可进一步丰富固液界面摩擦理论.      

Mechanics and thermodynamics of surface growth viewed as moving discontinuities

Surface growth is presently described as the motion of a moving interface of vanishing thickness, physically representing the generating cells, separating a zone not yet affected by growth from a domain in which growth has occurred. The jump conditions of density, velocity, momentum, energy, and entropy over the moving front are expressed from the general balance laws of open systems in both physical and material format. The writing of the jump of the internal entropy production in material format allows the identification of a driving force for surface growth, thermodynamically conjugated to the material velocity of the moving front.     

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.     

Cracks with surface charge in poled ferroelectrics

A solution is developed for a prismatic elliptical cavity inserted into an infinite, poled, ferroelectric body subjected to uniform mechanical and electrical loading at infinity consistent with planar fields. The surface of the cavity is also permitted to attract free charge in such a way that the electric field in the cavity remains uniform. The solution is used to develop the mechanical and electrical intensity factors at the tip of the crack obtained by allowing the minor axis of the ellipse to become very small. The energy release rate upon propagation of the crack is also derived. It is found that both the polarization of the ferroelectric and the free charge on the surface of the crack can have a very substantial effect on the magnitudes of both the intensity factors and the energy release rate. The reality of remanent polarization in common ferroelectrics means that it cannot be neglected in the fracture mechanics of such materials. Furthermore, it is likely that remanent polarization leads to free charge being attracted to crack surfaces, so consideration of the latter would also seem to be of importance in the fracture mechanics of ferroelectrics.     

A SIMPLIFIED MODEL FOR DOMAIN SWITCHING OF FERROELECTRIC CRYSTAL

I. INTRODUCTION Domain switching is the main source of nonlinear characteristics of ferroelectric materials. The trans-formation performance of domain is the basis for ferroelectric constitutive research. In many literatures[1??10], complete switching models were adopted, in which ferroelectric materials are considered tobe consist of several basic sorts of domains, which are independent of each other. Under the actionof su?ciently strong electric ?elds or mechanical stress, the orienta…     

An analysis of 2D piezoelectric half-plane with a fixed conductor surface

Solutions to a piezoelectric half-plane with a fixed conductor surface electrode subjected to two generalized singularities (line dislocation and/or line force and free charge) are presented. Coulomb forces acting on the singularities due to the boundary polarization charges of medium and the induction charges of conductor electrode are analyzed in detail. The interaction between the two singularities is also analyzed numerically. Results show that Coulomb forces will become important as the free charge approaches the boundary or two singularities move closely. Project supported by the National Science Foundation of China (No. 10172036).     

Moving inelastic discontinuities and applications to martensitic phase transition

Summary In this work, equations of the kinetics and kinematics are developed for heterogeneous materials containing inelastic discontinuities with moving boundaries. From the derived free energy and the power of external forces one obtains the driving force acting on the moving boundary. Introducing the interface operators and some hypothesis on inelastic fields, one gets the driving force for the formation of an ellipsoidal domain. The theoretical model is illustrated by the derivation of nucleation and growth conditions of a martensitic plate inside an inhomogeneous plastic strain field. The obtained results are combined with a study of the kinetics and kinematics to derive the constitutive equation of an austenitic single crystal, from which the overall behavior of polycrystalline TRIP steels is deduced using the self-consistent scale-transition method. Comparison with experimental data shows a good agreement. Received 7 May 1999; accepted for publication 14 June 1999     

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).     

Shape gradient and classical gradient of curvatures: driving forces on micro/nano curved surfaces

Recent experiments and molecule dynamics simulations have shown that adhesion droplets on conical surfaces may move spontaneously and directionally. Besides, this spontaneous and directional motion is independent of the hydrophilicity and hydrophobicity of the conical surfaces. Aimed at this important phenomenon, a general theoretical explanation is provided from the viewpoint of the geometrization of micro/nano mechanics on curved surfaces. In the extrinsic mechanics on micro/nano soft curved surfaces, we disclose that the curvatures and their extrinsic gradients form the driving forces on the curved spaces. This paper focuses on the intrinsic mechanics on micro/nano hard curved surfaces and the experiment on the spontaneous and directional motion. Based on the pair potentials of particles, the interactions between an isolated particle and a micro/nano hard curved surface are studied, and the geometric foundation for the interactions between the particle and the hard curved surface is analyzed. The following results are derived: (a) Whatever the exponents in the pair potentials may be, the potential of the particle/hard curved surface is always of the unified curvature form, i.e., the potential is always a unified function of the mean curvature and the Gaussian curvature of the curved surface. (b) On the basis of the curvature-based potential, the geometrization of the micro/nano mechanics on hard curved surfaces may be realized. (c) Similar to the extrinsic mechanics on micro/nano soft curved surfaces, in the intrinsic mechanics on micro/nano hard curved surfaces, the curvatures and their intrinsic gradients form the driving forces on the curved spaces. In other words, either on soft curved surfaces or hard curved surfaces and either in the extrinsic mechanics or the intrinsic mechanics, the curvatures and their gradients are all essential factors for the driving forces on the curved spaces. (d) The direction of the driving force induced by the hard curved surface is independent of the hydrophilicity and hydrophobicity of the curved surface, explaining the experimental phenomenon of the spontaneous and directional motion.     

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.     

Micromechanical analysis of volumetric growth in the context of open systems thermodynamics and configurational mechanics. Application to tumor growth

We adopt in this paper the physically and micromechanically motivated point of view that growth (resp. resorption) occurs as the expansion (resp. contraction) of initially small tissue elements distributed within a host surrounding matrix, due to the interfacial motion of their boundary. The interface motion is controlled by the availability of nutrients and mechanical driving forces resulting from the internal stresses that built in during the growth. A general extremum principle of the zero potential for open systems witnessing a change of their mass due to the diffusion of nutrients is constructed, considering the framework of open systems thermodynamics. We postulate that the shape of the tissue element evolves in such a way as to minimize the zero potential among all possible admissible shapes of the growing tissue elements. The resulting driving force for the motion of the interface sets a surface growth models at the scale of the growing tissue elements, and is conjugated to a driving force identified as the interfacial jump of the normal component of an energy momentum tensor, in line with Hadamard’s structure theorem. The balance laws associated with volumetric growth at the mesoscopic level result as the averaging of surface growth mechanisms occurring at the microscopic scale of the growing tissue elements. The average kinematics has been formulated in terms of the effective growth velocity gradient and elastic rate of deformation tensor, both functions of time. This formalism is exemplified by the simulation of the avascular growth of multicell spheroids in the presence of diffusion of nutrients, showing the respective influence of mechanical and chemical driving forces in relation to generation of internal stresses.     

A phenomenological constitutive model for ferroelastic and ferroelectric hysteresis effects in ferroelectric ceramics

This paper is concerned with a macroscopic constitutive law for domain switching effects, which occur in ferroelectric ceramics. The three-dimensional model is thermodynamically consistent and is determined by two scalar valued functions: the Helmholtz free energy and a switching surface. In a kinematic hardening process the movement of the center of the switching surface is controlled by internal variables. In common usage, the remanent polarization and the irreversible strain are employed as internal variables. The novel aspect of the present work is to introduce an irreversible electric field, which serves instead of the remanent polarization as internal variable. The irreversible electric field has only theoretical meaning, but it makes the formulation very suitable for a finite element implementation, where displacements and the electric potential are the nodal degrees of freedom. The paper presents an appropriate implementation into a hexahedral finite brick element. The uni-axial constitutive model successfully reproduces the ferroelastic and the ferroelectric hysteresis as well as the butterfly hysteresis for ferroelectric ceramics. Furthermore it accounts for the mechanical depolarization effect, which occurs if the polarized ferroelectric ceramic is subjected to a compression stress.     

GIANT ANHYSTERETIC RESPONSE OF FERROELECTRIC SOLID SOLUTIONS WITH MORPHOTROPIC BOUNDARIES: THE ROLE OF POLAR ANISOTROPY

Computer modeling and simulation for the Pb(Zr 1 x Ti x )O 3 (PZT) system reveal the role of polar anisotropy on the giant anhysteretic response and structural properties of morphotropic phase boundary (MPB) ferroelectrics. It is shown that a drastic reduction of the composition- dependent polar anisotropy near the MPB flattens energy surfaces and thus facilitates reversible polarization rotation. It is further shown that the polar anisotropy favors formation of polar domains, promotes phase decomposition and results in a two-phase multidomain state, which response to applied electric field is anhysteretic when the polar domain reorientation is only caused by polarization rotation other than polar domain wall movement. This is the case for the decomposing ferroelectrics under a poling electric field with the formation of a two-phase multidomain microstructure, wherein most domain walls are pinned at the two-phase boundaries. Indication of the microstructure dependence of the anhysteretic strain response opens new avenues to improve the piezoelectric properties of these materials through the microstructure engineering.     

The Motion Behavior of a Group of Charged Droplets in Liquid-Liquid Electrostatic Spray

The droplet sizes and electrical charges under different applied electrical voltages are experimentally measured for a liquid-liquid electrostatic spray system. Considering droplet size and charge distributions, the two-dimensional motion for a group of charged droplets in a liquid-liquid electrostatic atomization system is simulated. From measured droplet size and charge distributions, the simulation can obtain the velocities and positions in a two-dimensional domain for all simulated droplets at different times. The various forces acting on droplet as well as their effects on droplet velocity and trajectory are analyzed and the liquid-liquid electrostatic atomization characteristics are revealed. In addition, for one-dimensional motion trajectory of larger droplet, the comparison between simulation and experiment is also conducted and a general agreement can be obtained.