Dielectric Displacement Research Articles

Multi-field variational formulations and related finite elements for piezoelectricshells

Smart structures technology characterized by structurally integrated sensors and actuatorshas recently expanded significantly especially as regards lightweight constructions inaeronautics and robotics, e.g. to allow vibration suppression and noise attenuation. In orderto be capable of solving these complex issues the finite element method as a wellestablished design tool has to be extended. This paper focuses on shallow sandwichcomposite shell structures with thin piezoelectric patches bonded to the surfaces. For theproper design of plate and shell structures with integrated piezoelectric materials, variousvariational formulations and corresponding finite elements are presented. The starting pointis the well known two-field variational formulation where the linear piezoelectric effect istaken into account so that the displacements and the electric potential serve asindependent variables. Here, the mostly assumed linear variation of the electricpotential through the thickness is assumed. Next, it is shown that a quadraticvariation of the electric potential through the thickness can be deduced directlyfrom the charge conservation condition. This quadratic variation of the electricpotential in the thickness direction is compared with the linear gradient of the firsttwo-field variational formulation. Moreover, in order to allow the implementation ofalternative formulations of the constitutive equations by switching of the independentvariables and nonlinear material behaviour, a three-field variational formulation ispresented in analogy to the Hu–Washizu principle. Adopting this variationalprinciple a hybrid finite element is derived where the dielectric displacement isformulated as an additional degree of freedom. This independent variable canbe condensed on the element level and does not enter the system of equations.For the first time all these different variational formulations are developed for aReissner–Mindlin shallow shell element formulation and compared with each other.

Electrostatic energy in fields provides a route to dielectric permittivity of proteins

The expressions of electrostatic energy of a system of charges and dielectric describe in terms of electric fields and dielectric displacements provides a route to the calculation of dielectric permittivity of a protein. We derived and showed that once an electric field is generated from sources, the electrostatic energy of interaction is calculated directly from field by careful handling of the self energy terms. Calculations are done in vacuum and results are in good agreement with the experimentally determined dielectric constant of protein powders in the presence of a weak electric fields.

On the dynamics of piezoelectric cylindrical shells

In the present paper new equations of motion are derived for the vibration of piezo-ceramic thin-walled cylindrical shells, generalizing Flügge's shell theory for this type of material. These new equations differ from the ones known from the literature in that here the electric field is not assumed as constant over the thickness but is obtained by solving an additional differential equation in the thickness direction. The shells are polarized in the radial direction and the electrodes are in the form of identical sectors. Such shells are used e.g. as stators in some piezoelectric ultrasonic travelling wave motors and it is therefore important to study their free and forced vibrations. The momentum and moment of momentum balance used in Flügge's shell theory are of course unchanged. The constitutive relations used in the theory of elastic shells are replaced by those of a linear piezoelectric material, so that additional field variables are introduced. These are subject to Maxwell's laws, which in particular have to be fulfilled by the electric field inside the shell. For a thin radially polarized shell, only dielectric displacements in the radial direction are taken into account. Due to the absence of free electric charges in dielectric media such as PZT, the divergence of the dielectric displacement vanishes. This condition leads to an ordinary differential equation of the Euler type in the radial electric field, which can be solved in closed form. Together with the thin shell assumptions and with the piezoceramic constitutive equations, this results in the equations of motion for thin piezoceramic shells with non-constant electric field over the thickness.

Computing the electrostatic free-energy of complex molecules: The variational Coulomb field approximation

We introduce a novel approximate electrostatic method yielding the electrostatic fields around a molecule of complex shape embedded in a continuum dielectric solvent and the electrostatic solvation free-energies. This method extends the widely used Coulomb field approximation by supposing that the dielectric displacement can be written as the Coulomb field created by a set of fictitious “image” charges placed on the solute atomic sites. The electrostatic problem is solved by minimizing a polarization density functional with respect to the image charges. The method presents computational advantages which are reminiscent to those of the Coulomb field approximation; in particular, the solvation free-energy can be cast into a form which requires only the evaluation of space integrals limited to the interior of the solute. Its accuracy is demonstrated for simple solutes in water, ion pairs, the Tanford–Kirkwood globular protein model, and small polypeptides. It is shown also that our approach provides a systematic correction beyond the Coulomb field approximation which is able to improve the estimation of the atomic self-energies and associated Born radii in the generalized Born method.

Nematic Liquid Crystals Dispersed in a Ferroelectric Copolymer of Vinylidene Fluoride and Trifluoroethylene

We constructed polymer-dispersed liquid crystals which consisted of nematic liquid crystal 4'-n-pentil-4-cyanobiphenyl (5CB) and a ferroelectric copolymer of vinylidene fluoride and trifluoroethylene [P(VDF/TrFE)] by using a solvent-induced phase separation method. The hysteresis between the dielectric displacement and the external electric field was observed simultaneously with the electrooptical behavior in the poling process of the sample. The memory of the applied electric field appeared as a peak shift in the electrooptical dependence. This memory decayed on the order of 100 s. We consider that this decay was due to the screening of the spontaneous polarization of P(VDF/TrFE) by the redistribution of conductive charges in the crystal region of P(VDF/TrFE).

Non-local dielectric functions on the nanoscale: electronic polarization and fluctuations

In this work, we use a non-local dielectric theory to describe the electronic polarization induced by perturbing fields, and to characterize spontaneous, quantum mechanical fluctuations in the polarization. The theory holds on the intramolecular length scale, within the Born-Oppenheimer approximation. A non-local dielectric function ε d ( r , r ′) acts as an integral kernel that gives the dielectric displacement D( r) at point r within a molecule, in terms of the electric field E( r′) acting at other points r ′. Due to the inhomogeneity of the intramolecular environment, ε d ( r , r ′) differs from the non-local dielectric function ε v ( r , r ′) that characterizes the screening of applied potentials; however, the longitudinal component of the spatial Fourier transform ε d ( k , k ′) is completely determined by k, k′, and ε v ( k , k ′). For a molecule in equilibrium with a radiation bath at a fixed temperature, the fluctuation-dissipation theorem relates correlations of the spontaneous polarization fluctuations to the imaginary part of the non-local polarizability density. From this result, we derive a relation between the correlations of the fluctuating electronic polarization and the dielectric function ε d ( r , r ′;iω).

The electrical potential difference across cracks in PZT measured by Kelvin Probe Microscopy and the implications for fracture

An indentation crack in a poled PZT ceramic subjected to an electric field is investigated using AFM and KFM to determine the crack opening displacement and the electrical potential difference across the crack. The experimental results are used to calculate the crack tip stress and dielectric displacement intensity factors and the crack tip energy release rate. From the applied electric field and the measured field interior to the crack, the dielectric constant of the crack interior is determined to be 40. The consequences of this permittivity on the crack tip energy release rate are illustrated for a Griffith crack. The theoretically predicted effect of an applied electric field in retarding crack growth decreases significantly with increasing permittivity. In practical situations in terms of crack length, applied load and electric field level, the retardation of crack growth is negligible when the dielectric constant of the crack interior is higher than 20.

Novel Feature of the Phase Transitions in BaTiO 3 Revealed by "mK-Stabilized Cell"

Phase transitions in BaTiO 3 single crystals were investigated by "mK-stabilized cell" having fine temperature stability. It enables us to make various kind of measurements not only on heating and but also on cooling at a very small rate of temperature change. Heat flow measurements have revealed that both the uppermost and the lowermost phase transitions are accompanied by complicated thermal anomaly, whereas the intermediate temperature transition shows a single peak. Taking into account, these results together with those of dielectric constant, displacement current and X-ray diffraction measurements, we suggest the paraelectric-ferroelectric transition in BaTiO 3 takes place in multi-step, leading to a complicated structure in the lower temperature region.

Carga em Presença de Folha Dielétrica e a Atração entre Elas

Resolve-se o problema de uma carga pontual em presença de uma folha dielétrica fina, com o uso de imagens de cargas e dipolos cujos campos satisfazem as condições de continuidade do potencial e da componente normal do deslocamento elétrico nas superfícies da folha. Calcula-se, então, a força de atração resultante.

The use of harmonic analysis of the strain response in Pb(Mg(1/3)Nb(2/3)) O3-based ceramics to calculate electrostrictive coefficients.

The electromechanical response of ceramics has long been described with Landau Devonshire phenomenology, wherein the strain response is linked to a polynomial expansion in electric field or dielectric displacement. Consequently, the electromechanical response has been modeled with a variety of basis functions. However, these models have failed to accommodate hysteresis and the harmonic response that arises with saturation phenomena. In addition, no quantitative criterion has been used to truncate the expansion. By implementing a discrete Fourier transform in conjunction with Devonshire phenomenology, these three problems can be overcome as demonstrated with a dielectrically aged, lead magnesium niobate relaxor ferroelectric well above its Tmax, i.e., operating in the electrostrictive regime.

Ferroelectric behavior in solvent cast poly(vinylidene fluoride/hexafluoropropylene) copolymer films

The ferroelectric behavior of thin films of vinylidene fluoride (VDF) and hexafluoropropylene (HFP) random copolymers with two different HFP molar contents was investigated. Films were prepared by solvent casting using dimethyl formamide (DMF) as the solvent. The HFP molar content of the solvent cast films were 5 and 15%. The observed remanent polarization (Pr) obtained from dielectric displacement versus electric field (D–E) data for the 5 and 15% HFP copolymer samples were 80 and 50mC/m2, respectively. The Pr of these films is discussed in terms of the HFP molar content.

Measurement of Young's Moduli for Lead Zirconate Titanate (PZT) Ceramics

Abstract Knowledge of the Young's moduli is necessary for piezoelectric ceramics in order to separate the total deformation into an elastic and a plastic contribution. The methods for the determination of the Young's moduli, well known from conventional ceramics, are much more complicated due to the special electrical boundary conditions. The main aim of the report is to discuss the question in which mechanical tests the conditions of a constant electrical field are fulfilled and in which type of test constant dielectric displacements occur. It will be illustrated that the results from tests with homogeneous stress states are much more trustworthy than results from bending tests.

Inclusion and Inhomogeneity Problems in the Hexagonal Electroelastic Medium

The elastic Eshelby tensor plays a fundamental role for elastic inclusion problems [1]. For electroelastic media a generalization exists [2, 3], namely the electroelastic analogue of Eshelby tensor (EAET), a linear operator consisting of four tensors of fourth, third and second ranks. In order to derive explicit closed-form expressions for the EAET we assume a linear material law connecting stress and dielectric displacements with strain and electric field. The governing field equations for stress and dielectric displacements are determined by the elastic equilibrium and the conservation of free electric charges. In the modelling of the properties of such linear piezoelectric material Systems the absence of explicit Green's functions for electroelastic media has hindered progress for a long time. Using Michelitsch's recently derived explicit electroelastic 4 4 Green's function1† of the hexagonal (transversely isotropic) infinite medium [4] we calculated the EAET in explicit compact form for a spheroidal inclusion with the same ten hexagonal electroelastic moduli and orientation as the sourrounding medium [5]. The inclusion is assumed to be rotational symmetric with respect to the hexagonal c-axis with semi-axes (a1 ˆ a2 ˆ a, a3), a3 being parallel to the c-axis (x3-axis). The EAET generally is an operator with the structure [2, 3, 5]

Determination of nonlinear electromechanical energy conversion and the field-dependent elastic modulus of Pb(Mg/sub 1/3/Nb/sub 2/3/)O/sub 3/-based electrostrictors

Energy conversion efficiency is a critical parameter for all electromechanical materials. Although excellent techniques are available for linear materials, nonlinearities complicate the determination of conversion efficiency in electrostrictive Pb(Mg(1/3)Nb(2/3 ))O(3) (PMN). The field dependence of the elastic modulus presents additional problems. A technique combining pulse-echo ultrasound has been developed to approximate the change in the Young's modulus with applied field. A 30% decrease was observed at 1 MV/m (from ~100 to ~70 GPa). The boundary condition for these measurements was a constant electric field as opposed to constant dielectric displacement. In combination with the results from harmonic analysis, the modulus data may be used to determine the electromechanical conversion efficiency. This has been accomplished using an energy balance criterion for a PMN-based composition (with and without dc bias). The resulting longitudinal coupling values are >0.5 for practical electric fields (<1 MV/m peak).

Cross-polarization characteristics of a probe-fed hemispherical dielectric resonator antenna

The cross-polarization characteristics of a probe-fed hemispherical dielectric resonator antenna are studied theoretically. The mode-matching method was used to find the exact antenna Green's functions rigorously. The effects of the dielectric constant, probe length, and probe displacement on the cross-polarization level are examined.

Fundamental issues in modeling of mixed ionic-electronic conductors (MIECs)

The reaction at an MIEC/gas interface is electrochemical in nature when one of its charge-transfer steps influences the rate of the overall reaction. Changes in electrical states of the surfaces of an MIEC may not only enhance or hinder the rate of the reaction but also reverse the direction of the reaction (from anodic to cathodic or vice versa). Expressions similar to the Butler–Volmer equation are the most proper phenomenological description of the kinetics at an MIEC electrode, although the effect of the electrical state can be much smaller than that of the chemical state under certain conditions. Unless the kinetics are infinitely fast, the instantaneous relationship among concentrations of the species involved in a reaction is determined more by kinetics than thermodynamics. If an MIEC is assumed to be metallic in which bulk dielectric displacement cannot occur, both electric field and polarization in the bulk phase must vanish. If a huge capacitive response of an MIEC is indeed caused by space-dependent bulk polarization, displacement current density in the MIEC cannot be ignored in impedance analysis and the validity of local electroneutrality must be carefully evaluated. If local electroneutrality prevails, dielectric displacement vanishes, and the Fermi level is uniform in an MIEC, then the concentration of all defects must be uniform as must be the composition of the MIEC. It is important to adhere to these fundamental principles in order to obtain proper relations for the ionic, electronic, and oxygen fluxes and the correct influence of the interfaces on the charge-transfer and chemical reactions.

Contributions to the nonlinear dielectric and piezoelectric response of ferroelectric thin films and ceramics

The dielectric and piezoelectric response of ferroelectric thin films and ceramics was investigated as a function of the driving field amplitude. Detailed analysis of the amplitude and phase angle of the third harmonic of the dielectric displacement and piezoelectrically induced charge shows that the usual phenomenological approach is inadequate to describe experimentally observed field dependence of the dielectric and piezoelectric response. Qualitatively better agreement with experimental data is obtained using Rayleigh relations, suggesting that the observed field dependence of the polarization and piezoelectric charge may be associated with domain-wall displacement in a medium with a random distribution of pinning centers. The results indicate that the mechanism which controls the electromechanical response is very complex, and that it depends on history of samples, preparation conditions, material microstructure and on driving field amplitude.

The dielectric modulus: relaxation versus retardation

Experiments which access the quantity ε*( ω) are usually termed dielectric relaxation methods, although ε*( ω) and ε( t) actually refer to dielectric retardation. The true dielectric relaxation, the modulus M( t), corresponds to the decay of the electric field under the conditions of a constant dielectric displacement. We have measured the polarization in terms of a real dielectric relaxation technique by studying the decay of the electric field E( t) ∝ M( t) for times 10 −3 s≤ t≤10 6 s under constant charge conditions. This conceptually straightforward method bears the advantage of the ability to measure extremely large retardation times. Using the equivalent thermally stimulated technique we also investigate the equilibrium and non-equilibrium dynamic response of amorphous condensed matter well below its caloric glass-transition.

Thermally stimulated modulus relaxation in polymers: method and interpretation

We have measured the thermally stimulated decay of the dielectric modulus M(T, t) for poly(vinylacetate) by monitoring the electric field E(t) under the condition of a constant dielectric displacement for t > 0. This thermally stimulated polarization experiment is realized by cooling the sample at zero field to well below its glass transition temperature, applying a certain amount of charge, and then recording the voltage of the sample capacitor U(T, t) while ramping the temperature. The result of a temperature invariant distribution of relaxation times observed previously in isothermal experiments allows a direct translation of M(T, t) data into the variation of a characteristic relaxation time τ(T) or of a fictive temperature Tf(T) as a function of the actual temperature.

Dielectric relaxation of the electric field in poly(vinyl acetate): a time domain study in the range 10 −3–10 6 s

We have measured the polarization of poly(vinyl acetate) in terms of a real dielectric relaxation technique by studying the decay of the electric field E( t) for times 10 −3 s ≤ t ≤ 10 6 s under the condition of a constant dielectric displacement for t > 0. This represents a direct measurements of the dielectric modulus M( t), opposed to the conventional method of measuring the dielectric retardation ϵ( t), or its frequency domain analogue ϵ ∗(ω) . The advantages of accessing M( t) are a simple experimental set-up and for polar materials experimental times which are shorter by at least a factor of ϵ s/ ϵ ∞ relative to ϵ( t) decays. The latter effect is based on the well-known relation τ L = ϵ ∞/ ϵ s τ D between transversal and longitudinal dielectric time constants. For the polymer under study, we find that no significant change of the relaxation time distribution occurs within the experimental time window, i.e. from well above the glass transition at T g (if defined via τ g = τ( T g) = 100 s) to temperatures below T g where the average retardation times 〈τ〉 attain values of up to 10 7 s.