We present a critical review that encompasses the fundamentals and state-of-the-art knowledge of barium titanate-based piezoelectrics. First, the essential crystallography, thermodynamic relations, and concepts necessary to understand piezoelectricity and ferroelectricity in barium titanate are discussed. Strategies to optimize piezoelectric properties through microstructure control and chemical modification are also introduced. Thereafter, we systematically review the synthesis, microstructure, and phase diagrams of barium titanate-based piezoelectrics and provide a detailed compilation of their functional and mechanical properties. The most salient materials treated include the (Ba,Ca)(Zr,Ti)O3, (Ba,Ca)(Sn,Ti)O3, and (Ba,Ca)(Hf,Ti)O3 solid solution systems. The technological relevance of barium titanate-based piezoelectrics is also discussed and some potential market indicators are outlined. Finally, perspectives on productive lines of future research and promising areas for the applications of these ma...

BaTiO3-based piezoelectrics: Fundamentals, current status, and perspectives

P>We complete the development and description of a thermodynamic method for the computation of phase equilibria and physical properties of multiphase mantle assemblages. Our previous paper focused on the computation of physical properties. In this paper, our focus shifts to the phase equilibria. We further develop our theory to specify the ideal and excess contributions to solution properties and derive properties of multiphase assemblages. We discuss our global inversion strategy for determining the values of the free parameters in our theory and compare inverted parameter values with expectations based on scaling arguments. Comparisons between our method and experimental phase equilibria data encompass the pressure-temperature regime of Earth's mantle. Finally, we present applications of our method that illustrate how it may be used to explore the origins of mantle structure and mantle dynamics. Continuing rapid advances in experimental and theoretical petrology and mineral physics have motivated an expansion of the scope of our model via the addition of several new phases, and of the soda component: an appendix lists all parameters in our model and references to the experimental and theoretical studies that constrain them. Our algorithm for global minimization of the Gibbs free energy is embodied in a code called HeFESTo, and is detailed in a second appendix.

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Thermodynamics of mantle minerals - II. Phase equilibria

Foreword Acknowledgements Definitions of symbols used 1. Some fundamentals 2. The harmonic approximation and lattice dynamics of very simple systems 3. Dynamics of diatomic crystals: general principles 4. How far do the atoms move? 5. Lattice dynamics and thermodynamics 6. Formal description 7. Acoustic modes and macroscopic elasticity 8. Anharmonic effects and phase transitions 9. Neutron scattering 10. Infrared and Raman spectroscopy 11. Formal quantum mechanical description of lattice vibrations 12. Molecular dynamics simulations Appendices Problems Bibliography Index.

Introduction to Lattice Dynamics

A lattice-dynamical treatment of displacive phase transitions leads naturally to the softmode model, in which the phase-transition mechanism involves a phonon frequency that falls to zero at the transition temperature. The basic ideas of this approach are reviewed in relation to displacive phase transitions in silicates. A simple free-energy model is used to demonstrate that Landau theory gives a good approximation to the free energy of the transition, provided that the entropy is primarily produced by the phonons rather than any configurational disorder. The "rigid unit mode" model provides a physical link between the theory and the chemical bonds in silicates and this allows us to understand the origin of the transition temperature and also validates the application of the soft-mode model. The model is also used to reappraise the nature of the structures of high-temperature phases. Several issues that remain open, such as the origin of first-order phase transitions and the thermodynamics of pressure-induced phase transitions, are discussed.

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Theory of displacive phase transitions in minerals

A new thermodynamic model for the plagioclase-liquid exchange reaction between the albite (NaAlSi 3 O 8 ) and anorthite (CaAl 2 Si 2 O 8 ) components is presented, which can be used as a plagioclaseliquid hygrometer or thermometer. The model incorporates calorimetric and volumetric data for the pure liquid and crystalline components, which permits the effect of temperature and pressure on the exchange reaction to be calculated independently from the effect of composition. This allows a more accurate assessment of the effect of melt composition (including dissolved water concentration) on the exchange reaction from plagioclase-liquid equilibrium experiments. Activity-composition relations for the plagioclase solid solution are taken from Holland and Powell (1992). The new hygrometer is calibrated on 71 plagioclase-liquid experiments, of which 45 are hydrous and 26 are anhydrous. Three filters were applied to the phase-equilibrium data: (1) crystallanities <30%; (2) pure H 2 O fluidsaturated; and (3) compositional totals (including H 2 O component) of 97‐101% for hydrous quenched glasses. The final data set spans a wide range of liquid compositions (46‐74 wt% SiO 2 ), plagioclase compositions (An 93 -An 37 ), temperatures (825‐1230 °C), pressures (0‐300 MPa), and dissolved melt water concentrations (0‐7 wt% H 2 O). The standard error of estimate (SEE) for the model is ±0.32 wt% H 2 O, and all liquid compositions are fitted equally well. When the model is used as a thermometer, all measured temperatures are recovered equally well within ±14 °C on average. The model is only recommended for applications that fall within the compositional bounds of the calibration data set (i.e., metaluminous basalts through rhyolites in equilibrium with An 95 -An 35 ). It is not yet calibrated for rhyolites crystallizing plagioclase more sodic than An 30 , owing to an absence of phase-equilibrium experiments on rhyolites that pass the required filters. The new plagioclase-liquid hygrometer/thermometer is available as a Visual Basic program that runs on Excel 2004.

http://www.minsocam.org/MSA/AmMin/TOC/Abstracts/2009_Abstracts/Apr09_Abstracts/Lange_p494_09.pdf

A thermodynamic model for the plagioclase-liquid hygrometer/thermometer

The temperature evolution of the structural order parameter in displacive phase transitions with high transition temperatures (e.g. quartz, As2O5, LaAlO3, CaCO3, NaNO3, Pb3(PO4)2 etc.) follows a Landau-type behaviour over large temperature intevals. Below a saturation temperatureT

 s
 , however, the order parameter tends to become temperature independent. A quantitative analysis shows the correlationk

 B
 
T

 s
 =ħΩ/4, where Ω is the frequency of the relevant local excitation atT

 s
 . A general Landau-type expression for the Gibbs energy is given which includes the effects of order-parameter saturation.

Order-parameter saturation and low-temperature extension of Landau theory

Spontaneous strains for the a ↔ b transition in quartz were determined from lattice parameter data collected by X-ray powder diffraction and neutron powder diffraction over the temperature range ;5‐1340 K. These appear to be compatible with previous determinations of the order parameter variation in a quartz only if there is a non-linear relationship between the individual strains and the square of the order parameter. An expanded form of the 2-4-6 Landau potential usually used to describe the phase transition was developed to account for these strains and to permit calculation of the elastic constant variations. Calibration of the renormalized coefficients of the basic 2-4-6 potential, using published heat capacity data, provides a quantitative description of the excess free energy, enthalpy, entropy, and heat capacity. Values of the unrenormalized coefficients in the Landau expansion that include all the strain-order parameter coupling coefficients were used to calculate variations of the elastic constants. Values of the bare elastic constants were extracted from published elasticity data for b quartz. Calculated variations of C11 and C12 match their observed variations closely, implying that the extended Landau expansion provides a good representation of macroscopic changes within the (001) plane of quartz. Agreement was not as close for C33, suggesting that other factors may influence the strain parallel to [001]. The geometrical mechanism for the transition involves both rotations and shearing of SiO4 tetrahedra, with each coupled differently to the driving order parameter. Only the shearing part of the macroscopic distortions appears to show the same temperature dependence as other properties that scale with Q 2 . Coupling between the strain and the order parameter provides the predominant stabilization energy for a quartz and is also responsible for the first-order character of the transition.

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Calibration of excess thermodynamic properties and elastic constant variations associated with the alpha beta phase transition in quartz

The phonon spectra ofIR-active vibrations in alkali feldspars have been systematically investigated in the spectral range 50-2000 cm-I at temperatures between 20 and 900 K. Samples with high degrees of Al-Si order display IR absorption line profiles with spectral line widths that are systematically smaller than those for samples with low degrees of AlSi order. Phonon frequencies depend in a nonlinear manner on the Na-K content, indicating nonideal mixing properties. The mode Griineisen parameters are largest for lowfrequency lattice modes and Si-O stretching modes; some negative Griineisen parameters were found at phonon energies below 430 cm-1. The temperature evolution of the phonon spectra shows clear indications of the C2/mCT transition in Al-Si disordered Na-rich samples. An anomaly for highly Al-Si ordered microcline at 270 K is confirmed.

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Phonon spectra of alkali feldspars; phase transitions and solid solutions

The structural phase transition in titanite near 500 K (averaged symmetries A2/a→P2
1/a) and a second anomaly around 900 K have been studied using infrared spectroscopy on single crystals aqnd powder samples, measurements of the dielectric properties and the specific heat. The same synthetic single crystal was used in all experiments. The phase transition near 500 K is associated with a break in the temperature evolution of phonon frequencies and absorption intensities. Some phonon signals decrease rapidly under further heating and their extrapolated intensities disappear at ca. 850 K. The most dominant temperature effect relates to Ti-O phonons with amplitudes along the crystallographic a axis. These phonons show large LO-To splitting and continue to soften under heating even at temperatures above the transitions point (ca. 500 K). The softening of these modes correlates directly with the increase of the real part of the dielectric constant with a well-pronounced anomaly at 500 K. The dielectric losses also increse with increasing temperature. Measurements under strong field do not show antiferroelectricity. The transition at 500 K generates a small but sharp λ-anomaly in the excess specific heat. A second, weaker anomaly was found near 850 K. The results are discussed in terms of thermodynamic models.

Phase transition(s) in titanite CaTiSiO 5 : An infrared spectroscopic, dielectric response and heat capacity study

The effective thickness of ferroelastic twin-walls (W-walls) in Pb3(PO4)2 at room temperature ( 0.7 T c ) was determined by X-ray diffraction. The diffraction profiles were analysed using a wall profile e=eotanh x/w where W is the effective wall thickness including the effects of wall bending and surface relaxations. The experimental value W is around 10 unit cells which is larger than expected from standard renormalization arguments. The limitations of such experimental studies are discussed.

B. Wruck

Papers

Order-parameter saturation and low-temperature extension of Landau theory

Calibration of excess thermodynamic properties and elastic constant variations associated with the alpha beta phase transition in quartz

Phonon spectra of alkali feldspars; phase transitions and solid solutions

Phase transition(s) in titanite CaTiSiO 5 : An infrared spectroscopic, dielectric response and heat capacity study

On the thickness of ferroelastic twin walls in lead phosphate Pb3(PO4)2 an X-ray diffraction study