Nonlinear Constitutive Behavior Of Ferroelectric Materials

The ferroelectric specimen is considered as an aggregation of many randomly oriented domains. According to this mechanism, a multi-domain mechanical model is developed in this paper. Each domain is represented by one element. The applied stress and electric field are taken to be the stress and electric field in the formula of the driving force of domain switching for each element in the specimen. It means that the macroscopic switching criterion is used for calculating the volume fraction of domain switching for each element. By using the hardening relation between the driving force of domain switching and the volume fraction of domain switching calibrated, the volume fraction of domain switching for each element is calculated. Substituting the stress and electric field and the volume fraction of domain switching into the constitutive equation of ferroelectric material, one can easily get the strain and electric displacement for each element. The macroscopic behavior of the ferroelectric specimen is then directly calculated by volume averaging. Meanwhile, the nonlinear finite element analysis for the ferroelectric specimen is carried out. In the finite element simulation, the volume fraction of domain switching for each element is calculated by using the same method mentioned above. The interaction between different elements is taken into account in the finite element simulation and the local stress and electric field for each element is obtained. The macroscopic behavior of the specimen is then calculated by volume averaging. The computation results involve the electric butterfly shaped curves of axial strain versus the axial electric field and the hysteresis loops of electric displacement versus the electric field for ferroelectric specimens under the uniaxial coupled stress and electric field loading. The present theoretical prediction agrees reasonably with the experimental results.

Energy Principle And Nonlinear Electric-Mechanical Behavior Of Ferroelectric Ceramics

Many experimental observations have shown that a single domain in a ferroelectric material switches by progressive movement of domain walls, driven by a combination of electric field and stress. The mechanism of the domain switch involves the following steps: initially, the domain has a uniform spontaneous polarization; new domains with the reverse polarization direction nucleate, mainly at the surface, and grow though the crystal thickness; the new domain expands sideways as a new domain continues to form; finally, the domain switch coalesces to complete the polarization reversal. According to this mechanism, the volume fraction of the domain switching is introduced in the constitutive law of the ferroelectric material and used to study the nonlinear constitutive behavior of a ferroelectric body in this paper. The principle of stationary total potential energy is put forward in which the basic unknown quantities are the displacement u(i), electric displacement D-i and volume fraction rho(I) of the domain switching for the variant I. The mechanical field equation and a new domain switching criterion are obtained from the principle of stationary total potential energy. The domain switching criterion proposed in this paper is an expansion and development of the energy criterion established by Hwang et al. [ 1]. Based on the domain switching criterion, a set of linear algebraic equations for determining the volume fraction rho(I) of domain switching is obtained, in which the coefficients of the linear algebraic equations only contain the unknown strain and electric fields. If the volume fraction rho(I) of domain switching for each domain is prescribed, the unknown displacement and electric potential can be obtained based on the conventional finite element procedure. It is assumed that a domain switches if the reduction in potential energy exceeds a critical energy barrier. According to the experimental results, the energy barrier will strengthen when the volume fraction of the domain switching increases. The external mechanical and electric loads are increased step by step. The volume fraction rho(I) of domain switching for each element obtained from the last loading step is used as input to the constitutive equations. Then the strain and electric fields are calculated based on the conventional finite element procedure. The finite element analysis is carried out on the specimens subjected to uniaxial coupling stress and electric field. Numerical results and available experimental data are compared and discussed. The present theoretic prediction agrees reasonably with the experimental results.

Electromechanical influence of crack velocity at bifurcation for poled ferroelectric materials

The piezoelastodynamic field equations are solved to determine the crack velocity at bifurcation for poled ferroelectric materials where the applied electrical field and mechanical stress can be varied. The underlying physical mechanism, however, may not correspond to that assumed in the analytical model. Bifurcation has been related to the occurrence of a pair of maximum circumferential stress oriented symmetrically about the moving crack path. The velocity at which this behavior prevails has been referred to as the limiting crack speed. Unlike the classical approach, bifurcation will be identified with finite distances ahead of a moving crack. Nucleation of microcracks can thus be modelled in a single formulation. This can be accomplished by using the energy density function where fracture initiation is identified with dominance of dilatation in relation to distortion. Poled ferroelectric materials are selected for this study because the microstructure effects for this class of materials can be readily reflected by the elastic, piezoelectic and dielectric permittivity constants at the macroscopic scale. Existing test data could also shed light on the trend of the analytical predictions. Numerical results are thus computed for PZT-4 and compared with those for PZT-6B in an effort to show whether the branching behavior would be affected by the difference in the material microstructures. A range of crack bifurcation speed upsilon(b) is found for different r/a and E/sigma ratios. Here, r and a stand for the radial distance and half crack length, respectively, while E and a for the electric field and mechanical stress. For PZT-6B with upsilon(b) in the range 100-1700 m/s, the bifurcation angles varied from +/-6degrees to +/-39degrees. This corresponds to E/sigma of -0.072 to 0.024 V m/N. At the same distance r/a = 0.1, PZT-4 gives upsilon(b) values of 1100-2100 m/s; bifurcation angles of +/-15degrees to +/-49degrees; and E/sigma of -0.056 to 0.059 V m/N. In general, the bifurcation angles +/-theta(0) are found to decrease with decreasing crack velocity as the distance r/a is increased. Relatively speaking, the speed upsilon(b) and angles +/-theta(0) for PZT-4 are much greater than those for PZT-6B. This may be attributed to the high electromechanical coupling effect of PZT-4. Using upsilon(b)(0) as a base reference, an equality relation upsilon(b)(-) < upsilon(b)(0) < upsilon(b)(+) can be established. The superscripts -, 0 and + refer, respectively, to negative, zero and positive electric field. This is reminiscent of the enhancement and retardation of crack growth behavior due to change in poling direction. Bifurcation characteristics are found to be somewhat erratic when r/a approaches the range 10(-2)-10(-1) where the kinetic energy densities would fluctuate and then rise as the distance from the moving crack is increased. This is an artifact introduced by the far away condition of non-vanishing particle velocity. A finite kinetic energy density prevails at infinity unless it is made to vanish in the boundary value problem. Future works are recommended to further clarify the physical mechanism(s) associated with bifurcation by means of analysis and experiment. Damage at the microscopic level needs to be addressed since it has been known to affect the macrocrack speeds and bifurcation characteristics. (C) 2002 Published by Elsevier Science Ltd.

A field model interpretation of crack initiation and growth behavior in ferroelectric ceramics: change of poling direction and boundary condition

Strain energy density expressions are obtained from a field model that can qualitatively exhibit how the electrical and mechanical disturbances would affect the crack growth behavior in ferroelectric ceramics. Simplification is achieved by considering only three material constants to account for elastic, piezoelectric and dielectric effects. Cross interaction of electric field (or displacement) with mechanical stress (or strain) is identified with the piezoelectric effect; it occurs only when the pole is aligned normal to the crack. Switching of the pole axis by 90degrees and 180degrees is examined for possible connection with domain switching. Opposing crack growth behavior can be obtained when the specification of mechanical stress sigma(infinity) and electric field E-infinity or (sigma(infinity), E-infinity) is replaced by strain e and electric displacement D-infinity or (epsilon(infinity), D-infinity). Mixed conditions (sigma(infinity),D-infinity) and (epsilon(infinity),E-infinity) are also considered. In general, crack growth is found to be larger when compared to that without the application of electric disturbances. This includes both the electric field and displacement. For the eight possible boundary conditions, crack growth retardation is identified only with (E-y(infinity),sigma(y)(infinity)) for negative E-y(infinity) and (D-y(infinity), epsilon(y)(infinity)) for positive D-y(infinity) while the mechanical conditions sigma(y)(infinity) or epsilon(y)infinity are not changed. Suitable combinations of the elastic, piezoelectric and dielectric material constants could also be made to suppress crack growth. (C) 2002 Published by Elsevier Science Ltd.

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 rho (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 rho (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.

Displacive To Order-Disorder Two-Step Phase Transition Model For Para-Ferroelectric Transition

Two-step phase transition model, displacive to order-disorder, is proposed. The driving forces for these two transitions are fundamentally different. The displacive phase transition is one type of the structural phase transitions. We clearly define the structural phase transition as the symmetry broking of the unit cell and the electric dipole starts to form in the unit cell. Then the dipole-dipole interaction takes place as soon as the dipoles in unit cells are formed. We believe that the dipole-dipole interaction may cause an order-disorder phase transition following the displacive phase transition. Both structural and order-disorder phase transition can be first-order or second-order or in between. We found that the structural transition temperatures can be lower or equal or higher than the order-disorder transition temperature. The para-ferroelectric phase transition is the combination of the displacive and order-disorder phase transitions. It generates a variety of transition configurations along with confusions. In this paper, we discuss all these configurations using our displacive to order-disorder two-step phase transition model and clarified all the confusions.

Nematic ordering pattern formation in the process of self-organization of microtubules in a gravitational field

Papaseit et al. (Proc. Nati. Acad. Sci. U.S.A. 97, 8364, 2000) showed the decisive role of gravity in the formation of patterns by assemblies of microtubules in vitro. By virtue of a functional scaling, the free energy for MT systems in a gravitational field was constructed. The influence of the gravitational field on MT's self-organization process, that can lead to the isotropic to nematic phase transition, is the focus of this paper. A coupling of a concentration gradient with orientational order characteristic of nernatic ordering pattern formation is the new feature emerging in the presence of gravity. The concentration range corresponding to a phase coexistence region increases with increasing g or NIT concentration. Gravity facilitates the isotropic to nernatic phase transition leading to a significantly broader transition region. The phase transition represents the interplay between the growth in the isotropic phase and the precipitation into the nematic phase. We also present and discuss the numerical results obtained for local NIT concentration change with the height of the vessel, order parameter and phase transition properties.

Phase unwrapping based on branch cut placing and reliability ordering

A new 2-D quality-guided phase-unwrapping algorithm, based on the placement of the branch cuts, is presented. Its framework consists of branch cut placing guided by an original quality map and reliability ordering performed on a final quality map. To improve the noise immunity of the new algorithm, a new quality map, which is used as the original quality map to guide the placement of the branch cuts, is proposed. After a complete description of the algorithm and the quality map, several wrapped images are used to examine the effectiveness of the algorithm. Computer simulation and experimental results make it clear that the proposed algorithm works effectively even when a wrapped phase map contains error sources, such as phase discontinuities, noise, and undersampling. (c) 2005 Society of Photo-Optical Instrumentation Engineers.

Electrochromism accompanying ferroelectric domain inversion in congruent RuO2 : LiNbO3 crystal

Electrochromic phenomena accompanying the ferroelectric domain inversion in congruent RuO2-doped z-cut LiNbO3 crystals at room temperature are observed in experiments. During the electric poling process, the electrochromism accompanies the ferroelectric domain inversion simultaneously in the same poled area. The electrochromism is completely reversible when the domain is inverted from the reverse direction. The influences of electric field and annealing conditions on domain inversion and electrochromism are also discussed. We propose the reasonable assumption that charge redistribution within the crystal structure caused by domain inversion is the source for electrochemically oxidation and reduction of Ru ion to produce the electrochromic effect. (c) 2005 Optical Society of America.

Phase unwrapping based on branch cut placing and reliability ordering

A new 2-D quality-guided phase-unwrapping algorithm, based on the placement of the branch cuts, is presented. Its framework consists of branch cut placing guided by an original quality map and reliability ordering performed on a final quality map. To improve the noise immunity of the new algorithm, a new quality map, which is used as the original quality map to guide the placement of the branch cuts, is proposed. After a complete description of the algorithm and the quality map, several wrapped images are used to examine the effectiveness of the algorithm. Computer simulation and experimental results make it clear that the proposed algorithm works effectively even when a wrapped phase map contains error sources, such as phase discontinuities, noise, and undersampling. (c) 2005 Society of Photo-Optical Instrumentation Engineers.

Digital holographic interferometry and the angular spectrum method applied to measuring the domain inversion in ferroelectric crystal

The application of digital holographic interferometry on the quantitative measurement of the domain inversion in a RuO2: LiNbO3 crystal wafer is presented. The recorded holograms are reconstructed by the angular spectrum method. From the reconstructed phase distribution we can clearly observe the boundary between the inverted and un-inverted domain regions. Comparisons with the results reconstructed by use of the Fresnel transform method are given. Factors that influence the measurement include the spectrum filter size and the spectrum movement are discussed. The spectrum filter size has an effect on the measurement of the details. Although the spectrum movement affects every single reconstructed image, it has no influence on the final measurement.

Distinct two dimensional lateral ordering of self-assembled quantum dots

We report a quantum dot (QD) ensemble structure in which the in-plane arrangements of the dots are in a hexagonal way while the dots are also vertically aligned. Such a distinct lateral ordering of QDs is achieved on a planar GaAs(l 0 0) rather than on a prepatterned substrate by strain-mediated multilayer vertical stacking of the QDs. The analysis indicates that the strain energy of the lateral island-island interaction is minimum for arrangement of the hexagonal ordering. The ordered dots demonstrate strong photoluminescence (PL) emission at room temperature (RT) and the full width at half maximum of PL peak at RT is only 50 meV. (C) 2007 Elsevier B.V. All rights reserved.

Abnormal ferroelectric response and enhancement of piezoelectricity of PbTiO3 under uniaxial compression

The ferroelectricity of rhombohedral PbTiO3 under uniaxial compression is investigated from first-principles study. Upon compression, the ferroelectricity decreases until a critical stress of -29 GPa and then increases with a further increase of the magnitude of the uniaxial compressive stress. We also find that uniaxial compression could enhance piezoelectricity and that the maximum piezoelectric coefficient d(33) occurs at sigma(33)=-49 GPa, which supports the experimentally observed piezoelectric behavior in rhombohedral Pb(Mg1/3Nb2/3O3)-0.32PbTiO(3) [Q. Wan, C. Chen, and Y. P. Shen, J. Appl. Phys. 98, 024103 (2005)]. Our calculated results show that the Pb, Ti, and O atoms have different contributions to the total polarization with increasing the magnitude of uniaxial compressive stress, and that when -sigma(33)>55 GPa, the Ti atoms no longer have contributions to the polarization, which leads to the changes of ferroelectricity and piezoelectricity. (C) 2008 American Institute of Physics.

Magnetic ordering and irreversible magnetization between ZFC and FC states in RCo5Ga7 compounds

The magnetic properties of RCo5Ga7 (R = Y, Tb, Dy, Ho and Er) compounds which crystallize in the ScFe6Ga6-type structure have been studied. The compounds with R, Y, Tb, Dy, Ho and Er display behaviour similar to semiconductors. The Co transition metal sublattice is ferrimagnetic with a very low spontaneous magnetization. The ferrimagnetic ordering observed for R = Y, Tb, Dy, Ho and Er is due to the transition metal sublattice with transition temperatures at about 295 K. At low temperatures, the magnetic ordering for R Tb, Dy, Ho and Er is due to the rare-earth sublattice, which is ferromagnetic with a Curie temperature below 5 K. By fitting the linear part of the inverse magnetization, the effective magnetic moment of the R ion is found to be close to its expected theoretical value, with paramagnetic Curie temperatures below 5 K. Due to the paramagnetic nature of the R sublattice above 60 K, the ferrimagnetic ordering temperature of the Co sublattice does not vary with the type of rare-earth ion. The irreversibility of the magnetization of YCo5Ga7, as measured in zero-field cooled (ZFC) and field cooled (FC) states, is attributed to movement of domain walls. Application of a large enough applied field completes the movement of the domain wall from the low-temperature to the high-temperature one at 5 K. With a very low magnetic field 100 Oe, the difference between the ZFC and the FC shrinks. (C) 2004 Elsevier B.V. All rights reserved.