Sundar, V., Newnham, R.E. “Electrostriction” The Electrical Engineering Handbook Ed. Richard C. Dorf Boca Raton: CRC Press LLC, 2000 50 Electrostriction 50.1Introduction 50.2Defining Equations Piezoelectricity and Electrostriction?Electrostriction and Compliance Matrices?Magnitudes and Signs of Electrostrictive Coefficients 50.3PMN–PT — A Prototype Electrostrictive Material 50.4Applications 50.5Summary 50.1 Introduction Electrostriction is the basic electromechanical coupling mechanism in centric crystals and amorphous solids. It has been recognized as the primary electromechanical coupling in centric materials since early in the 20th century [Cady, 1929]. Electrostriction is the quadratic coupling between the strain developed in a material and the electric field applied, and it exists in all insulating materials. Piezoelectricity is a better-known linear coupling mechanism that exists only in materials without a center of symmetry. Electrostriction is a second-order property that is tunable and nonlinear. Electrostrictive materials exhibit a reproducible, nonhysteretic, and tunable strain response to electric fields, which gives them an advantage over piezoelectrics in micropositioning applications. While most electrostrictive actuator materials are perovskite ceramics, there has been much interest in large electrostriction effects in such polymer materials as poly- vinylidene fluoride (PVDF) copolymers recently. This chapter discusses the three electrostrictive effects and their applications. A discussion of the sizes of these effects and typical electrostrictive coefficients is followed by an examination of lead magnesium niobate (PMN) as a prototype electrostrictive material. The electromechanical properties of some common electro- strictive materials are also compared. A few common criteria used to select relaxor ferroelectrics for electro- strictive applications are also outlined. 50.2Defining Equations Electrostriction is defined as the quadratic coupling between strain (x) and electric field (E), or between strain and polarization (P). It is a fourth-rank tensor defined by the following relationship: x ij = M ijmn E m E n (50.1) where x ij is the strain tensor, E m and E n components of the electric field vector, and M ijmn the fourth-rank field- related electrostriction tensor. The M coefficients are defined in units of m 2 /V 2 . Ferroelectrics and related materials often exhibit nonlinear dielectric properties with changing electric fields. To better express the quadratic nature of electrostriction, it is useful to define a polarization-related electros- triction coefficient Q ijmn , as V. Sundar and R.E. Newnham Intercollege Materials Research Laboratory, The Pennsylvania State University ? 2000 by CRC Press LLC x ij = Q ijmn P m P n (50.2) Q coefficients are defined in units of m 4 /C 2 . The M and Q coefficients are equivalent. Conversions between the two coefficients are carried out using the field-polarization relationships: P m = h mn E n , and E n = c mn P m (50.3) where h mn is the dielectric susceptibility tensor and c mn is the inverse dielectric susceptibility tensor. Electrostriction is not a simple phenomenon but manifests itself as three thermodynamically related effects [Sundar and Newnham, 1992]. The first is the well-known variation of strain with polarization, called the direct effect (d 2 x ij /dE k dE l = M ijkl ). The second is the stress (X kl ) dependence of the dielectric stiffness c mn , or the reciprocal dielectric susceptibility, called the first converse effect (dc mn /dX kl = M mnkl ). The third effect is the polarization dependence of the piezoelectric voltage coefficient g jkl , called the second converse effect (dg jkl /dP i = c mk c nl M ijmn ). Piezoelectricity and Electrostriction Piezoelectricity is a third-rank tensor property found only in acentric materials and is absent in most materials. The noncentrosymmetric point groups generally exhibit piezoelectric effects that are larger than the electro- strictive effects and obscure them. The electrostriction coefficients M ijkl or Q ijkl constitute fourth-rank tensors which, like the elastic constants, are found in all insulating materials, regardless of symmetry. Electrostriction is the origin of piezoelectricity in ferroelectric materials, in both conventional ceramic ferroelectrics such as BaTiO 3 as well as in organic polymer ferroelectrics such as PVDF copolymers [Furukawa and Seo, 1990]. In a ferroelectric material, that exhibits both spontaneous and induced polarizations, P s i and P￠ i , the strains arising from spontaneous polarizations, piezoelectricity, and electrostriction may be formulated as (50.4) In the paraelectric state, we may express the strain as x ij = Q ijkl P k P l , so that dx ij /dP k = g ijk = 2Q ijkl P l . Converting to the commonly used d ijk coefficients, d ijk = c mk g ijm = 2c mk Q ijmn P n (50.5) This origin of piezoelectricity in electrostriction provides us an avenue into nonlinearity. In this case, it is the ability to tune the piezoelectric coefficient and the dielectric behavior of a transducer. The piezoelectric coefficient varies with the polarization induced in the material, and may be controlled by an applied electric field. The electrostrictive element may be tuned from an inactive to a highly active state. The electrical impedance of the element may be tuned by exploiting the dependence of permittivity on the biasing field for these materials, and the saturation of polarization under high fields [Newnham, 1990]. Electrostriction and Compliance Matrices The fourth-rank electrostriction tensor is similar to the elastic compliance tensor, but is not identical. Com- pliance is a more symmetric fourth-rank tensor than is electrostriction. For compliance, in the most general case, s ijkl = s jikl = s ijlk = s jilk = s klij = s lkij = s klji = s lkij (50.6) but for electrostriction: M ijkl = M jikl = M ijlk = M jilk 1 M klij = M lkij = M klji = M lkij (50.7) xQPPQPPQP ij ijklk s l s ijklk s l ijklkl =+￠+ ￠￠2 ? 2000 by CRC Press LLC This means that for most point groups the number of independent electrostriction coefficients exceeds those for elasticity. M and Q coefficients may also be defined in a matrix (Voigt) notation. The electrostriction and elastic compliance matrices for point groups 6/mmm and ￥/mm are compared below. Compliance coefficients s 13 and s 31 are equal, but M 13 and M 31 are not. The difference arises from an energy argument which requires the elastic constant matrix to be symmetric. It is possible to define sixth-rank and higher-order electrostriction coupling coefficients. The electrostriction tensor can also be treated as a complex quantity, similar to the dielectric and the piezoelectric tensors. The imaginary part of the electrostriction is also a fourth-rank tensor. Our discussion is confined to the real part of the quadratic electrostriction tensor. Magnitudes and Signs of Electrostrictive Coefficients The values of M coefficients range from about 10 –24 m 2 /V 2 in low-permittivity materials to 10 –16 m 2 /V 2 in high- permittivity actuator materials made from relaxor ferroelectrics such as PMN–lead titanate (PMN–PT) com- positions. Large strains of the order of strains in ferroelectric piezoelectric materials such as lead zirconate titanate (PZT) may be induced in these materials. Q values vary in an opposite way to M values. Q ranges from 10 –3 m 4 /C 2 in relaxor ferroelectrics to greater than 1 m 4 /C 2 in low-permittivity materials. Since the strain is directly proportional to the square of the induced polarization, it is also proportional to the square of the dielectric permittivity. This implies that materials with large dielectric permittivities, like relaxor ferroelectrics, can produce large strains despite having small Q coefficients. As a consequence of the quadratic nature of the electrostriction effect, the sign of the strain produced in the material is independent of the polarity of the field. This is in contrast with linear piezoelectricity where reversing the direction of the field causes a change in the sign of the strain. The sign of the electrostrictive strain depends only on the sign of the electrostriction coefficient. In most oxide ceramics, the longitudinal electrostriction coefficients are positive. The transverse coefficients are negative as expected from Poisson ratio effects. Another consequence is that electrostrictive strain occurs at twice the frequency of an applied ac field. In acentric materials, where both piezoelectric and electrostrictive strains may be observed, this fact is very useful in separating the strains arising from piezoelectricity and from electrostriction. 50.3 PMN–PT — A Prototype Electrostrictive Material Most commercial applications of electrostriction involve high-permittivity materials such as relaxor ferroelec- trics. PMN (Pb(Mg 1/3 Nb 2/3 )O 3 ) relaxor ferroelectric compounds were first synthesized more than 30 years ago. Since then, the PMN system has been well characterized in both single-crystal and ceramic forms, and may be considered the prototype ferroelectric electrostrictor [Jang et al., 1980]. Lead titanate (PbTiO 3 , PT) and other materials are commonly added to PMN to shift T max or increase the maximum dielectric constant. The addition of PT to PMN gives rise to a range of compositions, the PMN–PT system, that have a higher Curie range and superior electromechanical coupling coefficients. The addition of other oxide compounds, mostly other ferro- electrics, is a widely used method to tailor the electromechanical properties of electrostrictors [Voss et al., 1983]. Some properties of the PMN–PT system are listed here. SSS SSS SSS S S SS MMM MMM MMM M M 11 12 13 12 11 13 13 13 11 44 44 44 12 11 12 13 12 11 13 31 31 33 44 44 00 0 00 0 00 0 000 0 0 0000 0 000002 00 0 00 0 00 0 000 0 0 0000 0 - ( ) é ? ê ê ê ê ê ê ê ê ù ? ú ú ú ú ú ú ú ú 000000 11 12 MM- ( ) é ? ê ê ê ê ê ê ê ê ù ? ú ú ú ú ú ú ú ú ? 2000 by CRC Press LLC Based on dielectric constant vs. temperature plots, the electromechanical behavior of a relaxor ferroelectric may divided into three regimes (Fig. 50.1). At temperatures less than T d , the depolarization temperature, the relaxor material is macropolar, exhibits a stable remanent polarization, and behaves as a piezoelectric. T max is the temperature at which the maximum dielectric constant is observed. Between T d and T max , the material possesses nanometer-scale microdomains that strongly influence the electromechanical behavior. Large dielec- tric permittivities and large electrostrictive strains arising from micro–macrodomain reorientation are observed. Above T max , the material is a “true electrostrictor” in that it is paraelectric and exhibits nonhysteretic, quadratic strain-field behavior. Since macroscale domains are absent, no remanent strain is observed. Improved repro- ducibility in strain and low-loss behavior are achieved. Figure 50.2 illustrates the quadratic dependence of the transverse strain on the induced polarization for ceramic 0.9PMN–0.1PT. Figure 50.3a and b show the longitudinal strain as a function of the applied electric field for the same composition. The strain-field plots are not quadratic, and illustrate essentially anhysteretic nature of electrostrictive strain. The transverse strain is negative, as expected. FIGURE 50.1Polarization and dielectric behavior of a relaxor ferroelectric as a function of temperature, showing the three temperature regimes. FIGURE 50.2Transverse strain as a function of the square of the polarization in ceramic 0.9PMN–0.1PT, at RT. The quadratic (x = QP 2 ) nature of electrostriction is illustrated. Shaded circles indicate strain measured while increasing polarization and unshaded circles indicate decreasing polarization. Temperature (°C) Polarization P a Dielectric constant K T d T m III (macro-polar) II (macro-micro) I (electrostrictive) ? 2000 by CRC Press LLC The averaged longitudinal and transverse electrostriction coefficients have been measured for poled ceramic PMN to be Q 33 ~ 2.3 ′ 10 –2 m 4 /C 2 , Q 13 ~ –0.64 ′ 10 –2 m 4 /C 2 . The corresponding field-related coefficients are M 33 ~ 1.50 ′ 10 –16 m 2 /V 2 and M 13 ~ –4.19 ′ 10 –17 m 2 /V 2 . Induced strains of the order of 10 –4 may be achieved with moderate electric fields of ~40 kV/cm. These strains are much larger than thermal expansion strains, and are in fact equivalent to thermal expansion strains induced by a temperature change of ~1000°C. M 33 values for some other common ferroelectrics and a PVDF copolymer are listed in Table 50.1. The mechanical quality factor Q M for PMN is 8100 (at a field of ~200 kV/m) compared with 300 for poled barium titanate or 75 for poled PZT 5-A [Nomura and Uchino, 1981]. The induced piezoelectric coefficients d 33 and d 31 can vary with field (Fig. 50.4). The maxima in the induced piezoelectric coefficients for PMN as a function of biasing electric field are at E ~ 1.2 MV/m, with d 33 = 240 pC/N and –d 31 = 72 pC/N. Pb(Mg 0.3 Nb 0.6 Ti 0.1 )O 3 is a very active composition, with a maximum d 33 = 1300 pC/N at a biasing field of 3.7 kV/cm. FIGURE 50.3Longitudinal (a) and transverse (b) strains as a function of applied electric field in 0.9PMN–0.1PT, at RT. x is not quadratic with E except at low fields. ? 2000 by CRC Press LLC 50.4 Applications The advantages that electrostrictors have over other actuator materials include low hysteresis of the strain-field response, no remanent strain (walk off), reduced aging and creep effects, a high response speed (<10 ms), and strain values (>0.03%) achievable at realizable electric fields. Displacement ranges of several tens of microns may be achieved with ±0.01m reproducibility. Most actuator applications of electrostrictors as servotransducers and micropositioning devices take advantage of these characteristics. Mechanical applications range from stacked actuators through inchworms, microangle adjusting devices, and oil pressure servovalves. Multilayer actuators produce large displacements and high forces at low drive voltages. The linear change in capacitance with applied stress of an electrostrictor can be used as a capacitive stress gauge [Sundar and Newnham, 1992]. Electrostrictors may also be used as used in field-tunable piezo- electric transducers. Recently, electrostrictive materials have been integrated into ultrasonic motors and novel flextensional transducers. Electrostrictors have also been integrated into “smart” optical systems such as bistable optical devices, interferometric dilatometers, and deformable mirrors. Electrostrictive correction of optical aberrations is a significant tool in active optics. Electrostrictors also find applications in “very smart” systems such as sen- sor–actuator active vibration-suppression elements. A shape memory effect arising from inverse hysteretic behavior and electrostriction in PZT family antiferroelectrics is also of interest. A recent survey [Uchino, 1993] predicts that the market share of piezoelectric and electrostrictive transducers is expected to increase to more than $10 billion by 1998. TABLE 50.1Electrostrictive and Dielectric Data for Some Common Actuator Materials a Composition M 33 ′ 10 –17 m 2 /V 2 Dielectric Constant K Ref. Pb(Mg 1/3 Nb 2/3 )O 3 (PMN) 15.04 9140 Nomura and Uchino, 1983 (Pb 1-x La 2x/3 )(Zr 1-y Ti y )O 3 (PLZT 11/65/35) 1.52 5250 Landolt-Bornstein BaTiO 3 (poled) 1.41 1900 Nomura and Uchino, 1983 PbTiO 3 1.65 1960 Landolt-Bornstein SrTiO 3 5.61 ′ 10 –2 247 Landolt-Bornstein PVDF/TrFE copolymer 43 12 Elhami et al., 1995 a At room temperature, low frequency (<100 Hz) and low magnitude electric fields (<0.1 MV/m). FIGURE 50.4Induced piezoelectric coefficients d 33 and –d 31 as a function of applied biasing field for ceramic PMN, 18°C. ? 2000 by CRC Press LLC In selecting electrostrictive relaxor ferroelectrics for actuator and sensor applications, the following criteria are commonly used. A large dielectric constant and field stability in the K vs. E relations are useful in achieving large electrostrictive strains. These criteria also lead to large induced polarizations and large induced piezoelec- tric coefficients through the second converse effect. Broad dielectric transitions allow for a large operating temperature range. In the case of relaxors, this implies a large difference between T max and T d . Minimal E–P hysteresis and no remanent polarization are useful in achieving a low-loss material that is not susceptible to joule heating effects. These factors are listed in Table 50.2. 50.5 Summary Electrostriction is a fundamental electromechanical coupling effect. In ceramics with large dielectric constants and in some polymers, large electrostrictive strains may be induced that are comparable in magnitude with piezoelectric strains in actuator materials such as PZT. The converse electrostrictive effect, which is the change in dielectric susceptibility with applied stress, facilitates the use of the electrostrictor as a stress gauge. The second converse effect may be used to tune the piezoelectric coefficients of the material as a function of the applied field. Electrostrictive materials offer tunable nonlinear properties that are suitable for application in very smart systems. Defining Terms Elastic compliance: A fourth-rank tensor (s ijkl ) relating the stress (X) applied on a material and the strain (x) developed in it, x ij = s ijkl X kl . Its inverse is the elastic stiffness tensor (c ijkl ). Electrostriction:The quadratic coupling between strain and applied field or induced polarization. Conversely, it is the linear coupling between dielectric susceptibility and applied stress. It is present in all insulating materials. Ferroelectricity:The phenomenon by which a material exhibits a permanent spontaneous polarization that can be reoriented (switched) between two or more equilibrium positions by the application of a realistic electric field (i.e., less than the breakdown field of the material). Perovskite: A crystal structure with the formula ABO 3 , with A atoms at the corners of a cubic unit cell, B atoms at the body-center position, and O atoms at the centers of the faces. Many oxide perovskites are used as transducers, capacitors, and thermistors. Piezoelectricity:The linear coupling between applied electric field and induced strain in acentric materials. The converse effect is the induction of polarization when stress is applied. Relaxor ferroelectric:Relaxor ferroelectric materials exhibit a diffuse phase transition between paraelectric and ferroelectric phases, and a frequency dependence of the dielectric properties. Smart and very smart systems:A system that can sense a change in its environment, and tune its response suitably to the stimulus. A system that is only smart can sense a change in its environment and react to it. Related Topic 58.5 State-of-the-Art Smart Materials TABLE 50.2Selection Critera for Relaxor Ferroelectrics for Electrostrictive Devices Desirable Properties Material Behavior ? Large strain, induced polarization, and induced piezoelectricity ? Large dielectric constants ? Large operating temperature range ? T max – T d is large ? Broad dielectric transition ? Low-loss, low-joule heating, minimal hysteresis, no remanent polarization ? Operation in paraelectric regime (T > T max ) ? 2000 by CRC Press LLC References W. G. Cady, International Critical Tables, vol. 6, p. 207, 1929. K. Elhami, B. Gauthier-Manuel, J. F. Manceau, and F. Bastien, J. Appl. Phys., vol. 77, p. 3987, 1995. T. Furukawa and N. Seo, Jpn. J. Appl. Phys., vol. 29, p. 675, 1990. S. J. Jang, K. Uchino, S. Nomura, and L. E. Cross, Ferroelectrics, vol. 27, p. 31, 1980. Landolt-Bornstein, Numerical Data and Functional Relationships in Science and Technology, New Series, Gruppe III, vols. 11 and 18, Berlin: Springer-Verlag, 1979, 1984. R. E. Newnham, Chemistry of Electronic Ceramic Materials, in Proc. Intl. Conf., Jackson, Wyo., 1990; NIST Special Publication 804, 39, 1991. S. Nomura and K. Uchino, Ferroelectrics, vol. 50, p. 197, 1983. V. Sundar and R. E. Newnham, Ferroelectrics, vol. 135, p. 431, 1992. K. Uchino, MRS Bull., vol. 18, pp. 42, 1993. D. J. Voss, S. L. Swartz, and T. R. Shrout, Ferroelectrics, vol. 50, p. 1245, 1983. Further Information IEEE Proceedings of the International Symposium on the Applications of Ferroelectrics (ISAF) IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control (UFFC) American Institute of Physics Handbook, 3rd ed., New York: McGraw-Hill, 1972 M. E. Lines and A. M. Glass, Principles and Applications of Ferroelectric Materials, Oxford: Clarendon Press, 1977 ? 2000 by CRC Press LLC