Piezoelectric equations

Starting from the plane one-dimemional piezoelectric equations [102],... 

Electrostriction is related to the converse piezoelectric effect. At modest electric field strengths, the piezoelectric equations given previously are adequate and there is a linear relationship between strain and electric field. However, at higher electric field strengths, these equations need to be extended to include a further term quadratic with respect to the electric field. The strain is now given by... 

These are called the piezoelectric equations. The number of independent parameters for the lowest symmetry trigonal crystal are 21 for Sy, 18 for dmi... 

Piezoelectric effect is of fundamental importance for the piezoelectric measurements technology. General thermodynamic theory of a piezoelectric effect will be described more in details in this paragraph. Temperature or entropy a is held constant in the diagram according to Fig. 4.1. Hence the diagram is reduced just to the relationship between mechanical quantities or (stress or strain) and electrical quantities Ek or Dk (electric field or electric displacement). No special superscript for isothermal or adiabatic option is further used in linear piezoelectric equations of state. Omitting abovementioned thermal quantities, the system of 24 equations of state in Table 4.1 is reduced just to 8 equations - see Fig. 5.1 and Eqs. (5.1), (5.2), (5.3), (5.4), (5.5), (5.6), (5.7), and (5.8). 

P. Curie and J. Curie discovered the piezoelectric effect in 1880. It was found that, when a compressive or a tensile force was applied on some crystals along some special directions (for example, a quartz) electrical charges could be created on the corresponding surfaces of the crystal and the size of the created charge was proportional to the strength of the apphed force. This phenomenon is called the piezoelectric effect . All ferroelectric crystals show a piezoelectric effect. The piezoelectric effect can be described by piezoelectric equations. On the basis of thermodynamic principles, piezoelectric equations can be derived (e.g., see Xu, 1991). These equations describe linear relationships between the four variables stress tensor [T], strain tensor [S], electrical field vector E and electric displacement vector D. The piezoelectric equations can be expressed as four kinds of equations, depended on the variables. Selecting E and Tas variables, we have ... 

These are called the first piezoelectric equations, Sij are elastic compliance coefficients, are dielectric constants and dmi are piezoelectric coefficients (or piezoelectric moduli). Ifwe taking E and [S] as variables, the second piezoelectric equations will be as follows ... 

The boundary conditions for these piezoelectric equations are important (a) The condition mechanically free stipulates specifically that boundaries of a piezoelectric sample (e.g., a piezoelectric vibrator) can move, i.e., the vibrator vibrates with a variable strain and zero (or constant) stress. Under this condition, the coefficients in these equations carry a superscript T e.g., is the dielectric constant at constant stress, (b) The condition mechanically clamped stipulates specifically that the boundaries of a vibrator cannot move. This condition means that, when the frequency of the applied voltage is much higher than the resonance frequency of the vibrator, the strain is constant (or zero), while the stress varies. In this case, the coefficients in these equations carry a superscript S e.g., is the dielectric constant at constant strain, (c) The condition of electrical short circuit implies specifically that the electric field = 0 (or a constant), while the electric displacement D 0 inside the vibrator. This is the case when the two electrodes on the surface of the crystal sample are electrically connected (or the electric potential on the entire surface of the sample is constant). Under this condition, the coefficients in these equations carry a superscript E e.g., sfj (or c ) is the elastic compliance (or stiffness) coefficient at constant electric field, (d) The condition of electrical open circuit corresponds to the case when aU the free charges are kept on the electrodes of the sample (electrically insulated) and the internal electric field / 0, while = 0 in the sample, hi this case, the coefficients in these equations carry a superscript D e.g., sjj (or c ) is the elastic compliance (or stiffness) coefficient at constant polarization. 

Measurements from stress gauges, assuming equal accuracy and time resolution, are equivalent to measurements from particle velocity gauges in exploring a material s equation of state. Both piezoresistive and piezoelectric techniques have been used extensively in shock-compression science. 

A normal dielectric may be characterized by Eq. (4.1) with the piezoelectric terms deleted. For an isotropic dielectric subject to uniaxial strain and a collinear electric field this equation takes the form... 

Several polymers also are effective piezoelectric materials. The best known of these is PVDF (Equation 6.55), which is employed in loud speakers, fire and burglar alarm systems, earphones, and microphones. 

Nylon 11 (Equation 6.56) is also a piezoelectric material that can be aligned when placed in a strong electromagnetic field giving films used in infrared-sensitive cameras, underwater detection devices, and in electronic devices since it can be overlaid with printed circuits. 

The quartz crystal microbalance (QCM) is a piezoelectric device consisting of a thin (e.g.) quartz wafer sandwiched between two electrodes. A potential applied across the electrodes results in an oscillation of the quartz. The frequency of the oscillation, which can be measured accurately, is sensitive to mass loading. The relationship between frequency and mass loading is described by the Sauerbray equation ... 

A code has been written to enable the velocities of surface waves in multilayered anisotropic materials, at any orientation and propagation and including piezoelectric effects, to be calculated on a personal computer (Adler et al. 1990). The principle of the calculation is a matrix approach, somewhat along the lines of 10.2 but, because of the additional variables and boundary conditions, and because the wave velocities themselves are being found, it amounts to solving a first-order eight-dimensional vector-matrix equation. A... 

Scanning tunneling microscopy (STM), 787. 1157 bioelectrochemistry and, 1159 electrochemistry and. 1158 electrodeposition and. 1310 nanotechnology, 1345 piezoelectric crystal, 1158 tunneling current. 1157 underpotential deposition, 1313, 1315 Scavanger electrolysis, electrodeposition, 1343 Schlieren method, diffusion layer. 1235 Schmickler, 1495,1510 Schrodinger equation, 1456, 1490 Schultze 923,1497.1510 Screw dislocation, 1303, 1316, 1321, 1326 Secondary reference electrode, 815, 1109 Self-consumed electrode, 1040 Semiconductors... 

Equations (6) and (7) express these relationships. are the elastic compliance constants OC are the linear thermal expansion coefficients 4 and d jj,are the direct and converse piezoelectric strain coefficients, respectively Pk are the pyroelectric coefficients and X are the dielectric susceptibility constants. The superscript a on Pk, Pk, and %ki indicates that these quantities are defined under the conditions of constant stress. If is taken to be the independent variable, then O and are the dependent quantities ... 

Fundamental equations for piezoelectric effect due to uniform strain S are given in the form ... 

Naturally, the fixed composition phase transformations treated in this section can be accompanied by local fluctuations in the composition field. Because of the similarity of Fig. 17.3 to a binary eutectic phase diagram, it is apparent that composition plays a similar role to other order parameters, such as molar volume. Before treating the composition order parameter explicitly for a binary alloy, a preliminary distinction between types of order parameters can be obtained. Order parameters such as composition and molar volume are derived from extensive variables any kinetic equations that apply for them must account for any conservation principles that apply to the extensive variable. Order parameters such as the atomic displacement 77 in a piezoelectric transition, or spin in a magnetic transition, are not subject to any conservation principles. Fundamental differences between conserved and nonconserved order parameters are treated in Sections 17.2 and 18.3. 

Mercury binding leads to an increase of mass of the gold layer which can be detected by electro-acoustic transducers based on quartz microbalance (QMB the abbreviation QCM = quartz crystal microbalance is also widely used), surface acoustic waves (SAW)—devices [20] or microcantilevers [21,22], Adsorption of mercury vapour increases resonance frequency of shear vibrations of piezoelectric quartz crystals (Fig. 12.2). This process can be described by Sauerbrey equation [23]. For typical AT-cut quartz, this equation is... 

Abstract The aim of this contribution is to derive macroscopic equations describing flow of two-ionic species electrolytes through porous piezoelectric media with random, not necessarily ergodic, distribution of pores. Under assumption of ergodi-city the macroscopic equations simplify and are obtained by using the Birkhoff ergodic theorem. 

In this paper the problem of stationary flow of two-ionic species electrolyte through random piezoelectric porous media is studied, thus extending our earlier paper [14], where periodicity was assumed. To derive the macroscopic equations we use the method od stochastic two-scale convergence in the mean developed by [4], Solid phase was assumed to be piezoelectric since according to [9] wet bone reveals piezoelectric properties, cf. also [15], We recall that a strong conviction prevails that for electric effects in bone only streaming potentials are responsible. 

Equations of Flow of Electrolyte Through Piezoelectric Random Porous Medium... 

Telega, J.J. and Wojnar, R. (2000) Flow of Electrolyte through Porous Piezoelectric Medium Macroscopic Equations. C. R. Acad. Sci. Paris Serie lib 328, 225-230... 

Analogue to the dielectric case, the reversible contribution to the strain (see Equation 1.1) can be determined by the integration of the piezoelectric small signal coefficient d j over the applied bias field. 

The ieee Standard on Piezoelectricity [1] describes these measurements in considerable detail, and gives the necessary sample geometries for determination of a number of the piezoelectric constants. For example, the relations that enable determination of the other common piezoelectric moduli in bulk ceramics are given in Equation (2.7). 

As a result of the presence of the supporting substrate, the piezoelectric coefficients reported for films are usually effective numbers, rather than the free piezoelectric coefficients typically reported for bulk samples [23], The two coefficients most widely measured are the effective d j coefficient and the e j coefficients given in Equation (2.11). 

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