Piezoelectric displacement

Furthermore, for the voltage regimes used here, no evidence was found that pfm measurements should lack from mobile charge deposition upon switching. In contrast, we prove that the pfm signal purely reflects the measured piezoelectric displacement which hence may be compared to the local polarization distribution. 

Since acceleration is the second derivative of displacement, a piezoelectric accelerometer sensor with an integrator becomes a velocity transducer. This arrangement is gradually superseding the self-generating mo ing-coil velocity sensor (where a coil of wire moves relative to a magnetic field). 

Depth-sensing nanoindentation is one of the primary tools for nanomechanical mechanical properties measurements. Major advantages to this technique over AFM include (1) simultaneous measurement of force and displacement (2) perpendicular tip-sample approach and (3) well-modeled mechanics for dynamic measurements. Also, the ability to quantitatively infer contact area during force-displacement measurements provides a very useful approach to explore adhesion mechanics and models. Disadvantages relative to AFM include lower force resolution, as well as far lower spatial resolution, both from the larger tip radii employed and a lack of sample positioning and imaging capabilities provided by piezoelectric scanners. 

For mechanical wave measurements, notice should be taken of the advances in technology. It is particularly notable that the major advances in materials description have not resulted so much from improved resolution in measurement of displacement and/or time, but in direct measurements of the derivative functions of acceleration, stress rate, and density rate as called for in the theory of structured wave propagation. Future developments, such as can be anticipated with piezoelectric polymers, in which direct measurements are made of rate-of-change of stress or particle velocity should lead to the observation of recognized mechanical effects in more detail, and perhaps the identification of new mechanical phenomena. 

Piezoelectric solids are characterized by constitutive relations among the stress t, strain rj, entropy s, electric field E, and electric displacement D. When uncoupled solutions are sought, it is convenient to express t and D as functions of t], E, and s. The formulation of nonlinear piezoelectric constitutive relations has been considered by numerous authors (see the list cited in [77G06]), but there is no generally accepted form or notation. With some modification in notation, we adopt the definitions of thermodynamic potentials developed by Thurston [74T01]. This leads to the following constitutive relations ... 

From a constitutive relation of the form t = t(D, ri) (here t is stress not time) it can be readily shown that, since there is no change in electric displacement in an open-circuit, thick-sample configuration, there are no secondary stresses due to electromechanical coupling. Nevertheless, the wavespeed is that of a piezoelectrically stiffened wave. 

Fig. 4.6. Piezoelectric pulse diagrams can be used to obtain explicit representations of the time dependent electric fields in piezoelectric substances. The magnitudes and orientations of these electric fields are critical to development of shock-induced conduction. As an example, the diagram on the left shows the polarization and displacement relations for a location at the input electrode. The same functions for a location within the crystal is shown on the right (after Davison and Graham [79D01]).

FIGURE 9.4 The direct force measurement apparatus shown here ean measure the forees between two eurved molecularly smooth surfaces in liquids. Mica surfaces, either raw or eoated, are the primary surfaees used in this apparatus. The separation between the surfaces is measured by optieal teehniques to better than 10 nm. The distance between the two surfaces is controlled by a three-stage meehanism that ineludes a voltage-driven piezoelectric crystal tube supporting the upper mica surface this crystal tube can be displaced less than 10 nm in a controlled fashion. A force-measuring spring is attached to the lower mica surface and its stiffness can be varied by a factor of 1,000 by shifting the position of a movable clamp. Reprinted with permission from Proc. Natl. Acad Sci. USA, 84, July 1987, 4722. 

Fig. 9.2. The inverse piezoelectric effect. A thin and long quartz plate, QQ, is sandwiched between two tin foils. By applying a voltage to the tin foils, the quartz plate elongates or contracts according to the polarity of the applied voltage. To measure the very small displacement. Curie (1889a) used a lever ABD with a small piece of glass V attached at its end, the displacement of which is then measured with an optical microscope. (After Curie, 1889a.)...

A few months later, Lippman (1881) predicted the existence of the inverse piezoelectric effect By applying a voltage on the quartz plate, a deformation should be observed. This effect was soon confirmed by the Curie brothers (Curie and Curie, 1882), who designed a clever experiment to measure the tiny displacement, as shown in Fig. 9.2. Here, a light-weight lever with an arm of about 1 100 amplifies the displacement by two orders of magnitude. An optical microscope further amplifies it by two orders of magnitude. The displacement is then measured by an eyepiece with a scale. 

In the early years of STM instrumentation, tripod piezoelectric scanners were the predominant choice, as shown in Fig. 9.6. The displacements along the X, y, and z directions are actuated by three independent PZT transducers. Each of them is made of a rectangular piece of PZT, metallized on two sides. Those three PZT transducers are often called x piezo, y piezo, and z piezo, respectively. By applying a voltage on the two metallized surfaces of a piezo, for example, the x piezo, the displacement is... 

If larger displacements are required, an arrangement as shown in Fig. 9.7, the bimorph, can be applied. The principle is similar to the bimetal thermometer. Two thin plates of piezoelectric material are glued together. By applying a voltage, one plate expands and the other one contracts. The composite flexes. 

Consider a simple case a piezoelectric tube with one end fixed to a base plate, and another end fixed to an end block. The two ends, however, are allowed to experience relative displacements freely. Thus, the strain patterns at different cross sections are identical. 

Angular displacement sensed by piezoelectric accelerometers frequency from vibrator input G M G M... 

As has been well established, piezoelectricity in a non-polar crystal is brought about by the internal strain in the crystal. The internal strain means the displacement of atoms which is not affine to the deformation of crystal lattice. In the case of a polymer film which is not electrically conductive and where the charges are possibly embedded, a description of piezoelectricity can be reached by considering not only the internal strain in the lattice but also the displacement of these charges which is not affine to the average deformation of the whole system. 

When an electric field E is applied, Eq changes by an amount proportional to E, as usually expressed by (a — 1) E/4n, where e is the dielectric constant. When a strain is applied to the film, changes in and hence changes in polarization are divided into two components 1. displacement equal to the macroscopic displacement and 2. residual displacement. The latter is the internal strain and causes the intrinsic piezoelectricity. The effect of internal strain on P is expressed by eu, where e is the intrinsic piezoelectric constant and u is the elongational strain along the x-axis. The electric displacement D can therefore be written... 

Order parameters may also refer to underlying atomic structure or symmetry. For example, a piezoelectric material cannot have a symmetry that includes an inversion center. To model piezoelectric phase transitions, an order parameter, r], could be associated with the displacement of an atom in a fixed direction away from a crystalline inversion center. Below the transition temperature Tc, the molar Gibbs free energy of a crystal can be modeled as a Landau expansion in even powers of r (because negative and positive displacements, 77, must have the same contribution to molar energy) with coefficients that are functions of fixed temperature and pressure,... 

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. 

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