Appendix A: Future Directions
A.7.3 Example for the Interaction of Structural, Aerodynamic,
9.1 Smart Ferroelectric Ceramics for Transducer Applications
9.1.2 Piezoelectric and Electrostrictive Effects in Ceramic Materials
Piezoelectricity, discovered in Rochelle salt by Jacques and Pierre Curie, is the term used to describe the ability of certain
9
Piezoelectric and Electrostrictive Ceramics Transducers and Actuators
9.1 Smart Ferroelectric Ceramics for Transducer Applications ... 9-1 Introduction • Piezoelectric and Electrostrictive Eff ects in Ceramic
Materials • Measurements of Piezoelectric and Electrostrictive Eff ects • Common Piezoelectric and Electrostrictive Materials • Piezoelectric Composites • Applications of Piezoelectric and Electrostrictive Ceramics • Current Research and Future Trends
References ... 9-11 9.2 Smart Ceramics: Transducers, Sensors, and Actuators ... 9-12
Introduction • Piezoelectricity • Piezoelectric Materials • Applications of Piezoelectricity
References ... 9-27 9.3 Noncontact Ultrasonic Testing and Analysis of Materials ... 9-27
Introduction • NCU Transducers • NCU System and Signal Processing • NCU Techniques and Applications • Perusal of NCU
References ... 9-40
materials to develop an electric charge that is directly propor- tional to an applied mechanical stress (Figure 9.1a). The piezoelectric materials also show the converse eff ect, i.e., they deform (strain) proportionally to an applied electric fi eld (Figure 9.1b). To exhibit piezoelectricity, the crystal should belong to one of the 20 noncentrosymmetric crystallographic classes. An important subgroup of piezoelectric crystals is fer- roelectrics, which possess the averaged dipole moment per unit cell (spontaneous polarization) that can be reversed by an application of the external electric fi eld. Above the certain temperature (Curie point), most ferroelectrics lose their ferro- electric and piezoelectric properties and become paraelectrics, i.e., materials having centrosymmetric crystallographic struc- ture with no spontaneous polarization. Electrostriction is a second order eff ect, which refers to the ability of all materials to deform under the application of an electric fi eld. Th e phenom- enological master equation describing the deformations of an insulating crystal subjected to both elastic stress and electric fi eld is given by
= + m+ m n,
ij ijkl kl mij mnij
x s X d E M E E (9.1)
where
xij are the components of elastic strain sijkl is the elastic compliance tensor Xkl are the stress components
dmij are the piezoelectric tensor components Mmnij is the electrostrictive tensor
Em and En are the components of an external electric fi eld In this equation, the Einstein summation rule is used for repeat- ing indices. Typically, the electrostriction term (μEmEn) is more than an order of magnitude smaller than the piezoelectric term
in Equation 9.1, i.e., the electrostrictive deformations are much smaller than piezoelectric strains. In this case, under zero stress, Equation 9.1 simply transforms to
≈ m.
ij mij
x d E (9.2)
Equation 9.2 describes the converse piezoelectric eff ect where the electric fi eld changes dimensions of the sample (Figure 9.1b).
In centrosymmetric materials, the piezoelectric eff ect is absent and the elastic strain is only due to the electrostriction. In single domain ferroelectrics having centrosymmetric paraelectric phase, the piezoelectric and electrostriction coeffi cients can be described in terms of their polarization and dielectric constant.
For example, longitudinal coeffi cients (both electric fi eld and deformation are along tetragonal axis, symmetry 4 mm) can be described as follows:
=
33 2 11 0 33 3,
d Q e e P (9.3a)
= 2
11 11( 0 33) ,
M Q e e (9.3b)
where e33 and P3 are the dielectric constant and polarization along the polar direction, e0 = 8.854 × 10−12 F/m is the permittivity of vacuum, and Q11 is the polarization electrostriction coeffi - cient, which couples longitudinal strain and polarization due to general electrostriction equation:
= 2
3 11 3.
S Q P (9.4)
Th e mathematical defi nition of the direct piezoelectric eff ect where applied elastic stress causes charge on the major surfaces of the piezoelectric crystal is given by
P
(a)
(b) X
Y P
P
Force X− ΔX
Voltage
− +
Charge
− +
Y + ΔY
P X
Y
FIGURE 9.1 Schematic representations of the direct and converse piezoelectric eff ect: (a) an electric fi eld applied to the material changes its shape and (b) a mechanical force on the material yields an electric fi eld across it.
= ,
m mi i
P d X (9.5)
where Pm is the component of electrical polarization. In the case of electrostriction (centrosymmetric crystals), no charge appears on the surface of the crystal upon stressing, and the converse electrostriction eff ect is simply a change of the inverse dielectric constant under mechanical stress:
Δ( /(l ee0)) 2= Q X11 3. (9.6) It should be noted that the reduced matrix notation [2] for the piezoelectric and electrostriction coeffi cients, and stress tensor is used in Equations 9.3 through 9.6.
Th e piezoelectric and electrostrictive eff ects were described for the case of single domain crystals in which the spontaneous polarization is constant everywhere. A technologically impor- tant class of materials is piezoelectric and electrostrictive ceram- ics, which consist of randomly oriented grains, separated by grain boundaries. Ceramics are much less expensive in process- ing than single crystal and typically off er comparable piezoelec- tric and electrostrictive properties. Apparently, in nonferroelectric ceramics, piezoelectric eff ect of individual grains is canceled out by averaging over the entire sample and the whole structure will have a macroscopic center of symmetry and negligible piezo- electric properties. Only electrostriction can be observed in such
ceramics. Sintered ferroelectric materials (single crystals or ceramics) consist of regions with diff erent orientations of spon- taneous polarization, the so-called ferroelectric domains.
Domains appear when the material is cooled down through the Curie point to minimize the electrostatic and elastic energy of the system. Domain boundaries or domain walls are movable under applied electric fi eld, so the ferroelectric can be poled, i.e., domains become oriented in the crystallographic direction clos- est to the direction of applied electric fi eld (Figure 9.2). Typically, poling is performed under high electric fi eld at elevated temper- ature to facilitate domain alignment. As a result, initially centro- symmetric ceramic sample loses the inversion center and becomes piezoelectric (symmetry ∞∞m). Th ere are three inde- pendent piezoelectric coeffi cients: d33, d31, and d15, which relate longitudinal, transverse, and shear deformations, respectively, to the applied electric fi eld (see Figures 9.1 and 9.3).
Another set of materials constants that is frequently used to characterize the piezoelectric properties of ceramics are the piezo- electric voltage coeffi cients, gij defi ned in a matrix notation as
= ,
i ij j
E g X (9.7)
where Ei are the components of electric fi eld arising due to the external stress Xj. Th e charge coeffi cients dij and voltage coeffi - cients are related by the following equation:
= /( 0 33).
ij ij
g d e e (9.8)
An important property of piezoelectric and electrostrictive transducers is their electromechanical coupling coeffi cient, k, defi ned as
=
=
2
2
resulting mechanical energy/input electrical energy, or
resulting electrical energy/input mechanical energy.
k
k
(9.9)
Th e coupling coeffi cient represents the effi ciency of the piezo- electric in converting electrical energy into mechanical energy and vice versa. Since the energy conversion is never complete, the coupling coeffi cient is always less than unity.
Unpoled
Ep
(a) (b)
Poled
FIGURE 9.2 Schematic of the poling process in piezoelectric ceram- ics: (a) in the absence of an electric fi eld, the domains have random orientation of polarization with zero piezoelectric activity; (b) the polarization within the domains and grains are aligned in the direction of the electric fi eld.
FIGURE 9.3 Schematic of the longitudinal (a), transverse (a), and shear deformations (b) of the piezoelectric ceramic material under applied electric fi eld.
(a)
V P V P
(b)
9.1.3 Measurements of Piezoelectric