Sundar. V. Newnham. R.E. "Electrostriction The Electrical Engineering Handbook Ed. Richard C. Dorf Boca Raton CRC Press llc. 2000
Sundar, V., Newnham, R.E. “Electrostriction” The Electrical Engineering Handbook Ed. Richard C. Dorf Boca Raton: CRC Press LLC, 2000
50 Electrostriction 50.1 Introduction 0. 2 Defining Equations Piezoelectricity and Electrostriction. Electrostriction and V Sundar and Compliance Matrices. Magnitudes and Signs of Electrostrict R.E. Newnham Coefficients 50.3 PMN-PT-A Prototype Electrostrictive Material State Un The Pennsylvania 0.4 Applications 50.5 Summary 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 upling 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 producible, nonhysteretic, and tunable strain response to electric fields, which gives them an advantage over zoelectrics 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 his 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- trictive applications are also outlined. 50.2 Defining Equations Electrostriction is defined as the quadratic coupling between strain(x) and electric field (E), or between strain nd polarization(P). It is a fourth-rank tensor defined by the following relationship =MEE (50.1) where x; is the strain tensor, Em and En components of the electric field vector, and Mim the fourth-rank field- related electrostriction tensor. The M coefficients are defined in units of m/V2 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 electro triction coefficient Q mm,as c 2000 by CRC Press LLC
© 2000 by CRC Press LLC 50 Electrostriction 50.1 Introduction 50.2 Defining Equations Piezoelectricity and Electrostriction • Electrostriction and Compliance Matrices • Magnitudes and Signs of Electrostrictive Coefficients 50.3 PMN–PT — A Prototype Electrostrictive Material 50.4 Applications 50.5 Summary 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 polyvinylidene 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 electrostrictive materials are also compared. A few common criteria used to select relaxor ferroelectrics for electrostrictive applications are also outlined. 50.2 Defining 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: xij = Mijmn Em En (50.1) where xij is the strain tensor, Em and En components of the electric field vector, and Mijmn the fourth-rank fieldrelated electrostriction tensor. The M coefficients are defined in units of m2 /V2 . 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 electrostriction coefficient Qijmn, as V. Sundar and R.E. Newnham Intercollege Materials Research Laboratory, The Pennsylvania State University
QijinnPm Q coefficients are defined in units of m /C. The M and Q coefficients are equivalent. Conversions between the two coefficients are carried out using the field-polarization relationships Pm=nmEn, and En=X where nmm is the dielectric susceptibility tensor and %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 irect effect(dx, /dek dE Miu ). The second is the stress(Xy)dependence of the dielectric stiffness xmm,or the reciprocal dielectric susceptibility, called the first converse effect(d/dXy=Mmy). The third effect is the polarization dependence of the piezoelectric voltage coefficient gi, called the second converse effect(dgauldPi 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 Miur or Qiu 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 rroelectrics such as BaTiO, 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: and Pi, the strains arising from spontaneous polarizations, piezoelectricity, and electrostriction may be formulated xi=Qik PPi+ 2QikPP'+ Qir PpP (50.4) In the paraelectric state, we may express the strain as x=Qm Pp P, so that dx /dpi=8k=2QjuPr Converting to the commonly used d coefficients dijk=mk gim=2x mk Qijmn Pn 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, (50.6) but for electrostriction: Mil=Mial=Milk=Mi* Mkli-- MIki=Mali= Mikil c 2000 by CRC Press LLC
© 2000 by CRC Press LLC xij = QijmnPm Pn (50.2) Q coefficients are defined in units of m4 /C2 . The M and Q coefficients are equivalent. Conversions between the two coefficients are carried out using the field-polarization relationships: Pm = hmnEn , and En = cmn Pm (50.3) where hmn is the dielectric susceptibility tensor and cmn 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 (d2 xij/dEk dEl= Mijkl). The second is the stress (Xkl) dependence of the dielectric stiffness cmn, or the reciprocal dielectric susceptibility, called the first converse effect (dcmn/dXkl = Mmnkl). The third effect is the polarization dependence of the piezoelectric voltage coefficient gjkl , called the second converse effect (dgjkl/dPi = cmkcnlMijmn). 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 electrostrictive effects and obscure them. The electrostriction coefficients Mijkl or Qijkl 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 BaTiO3 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 xij = QijklPkPl, so that dxij/dPk = gijk = 2QijklPl. Converting to the commonly used dijk coefficients, dijk = cmk gijm = 2cmkQijmnPn (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. Compliance is a more symmetric fourth-rank tensor than is electrostriction. For compliance, in the most general case, sijkl = sjikl = sijlk = sjilk = sklij = slkij = sklji = slkij (50.6) but for electrostriction: Mijkl = Mjikl = Mijlk = Mjilk ¹ Mklij = Mlkij = Mklji = Mlkij (50.7) x Q P P Q P P Q P P ij ijkl k s l s ijkl k s l ijkl k l = + 2 ¢ + ¢ ¢
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 oo/mm are compared below. S1S2S300 S12S1S1300 S13S3S100 000 0 M3M3M3300 0000S 000 0 0 44 0 000002(S4-S2 0000(M1-M1 Compliance coefficients si3 and S3 are equal, but Mu3 and M3I 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 Magnitudes and Signs of Electrostrictive Coefficients The values of M coefficients range from about 10-4m/Vin low-permittivity materials to 10-6m/Vin 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 103m/C in relaxor ferroelectrics to greater than 1 m/C2 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 rel elaxor 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 depend 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 trics PMN(Pb(Mgir Nbx)O3)relaxor ferroelectric compounds were first synthesized more than erroelec- 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,, PT) and other materials are commonly added to PMn to shift Tmax 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. c 2000 by CRC Press LLC
© 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 s13 and s31 are equal, but M13 and M31 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 m2 /V2 in low-permittivity materials to 10–16 m2 /V2 in highpermittivity actuator materials made from relaxor ferroelectrics such as PMN–lead titanate (PMN–PT) compositions. 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 m4 /C2 in relaxor ferroelectrics to greater than 1 m4 /C2 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 ferroelectrics. PMN (Pb(Mg1/3Nb2/3)O3) 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 (PbTiO3, PT) and other materials are commonly added to PMN to shift Tmax 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 ferroelectrics, 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. S S S S S S S S S S S S S M M M M M M 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 000 0 0 0000 0 ( - ) È Î Í Í Í Í Í Í Í Í ˘ ˚ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ 00000 M11 M12 ( - ) È Î Í Í Í Í Í Í Í Í ˘ ˚ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙
Ta T (macro-micro) FIGURE 50. 1 Polarization and dielectric behavior of a relaxor ferroelectric as a function of temperature, showing the three temperature regimes. Transverse StrainⅨx,x10 P3(C2/m4 FIGURE 50.2 Transverse strain as a function of the square of the polarization in ceramic 0.9PMN-01PT, at RT. The quadratic (x= QP)nature of electrostriction is illustrated. Shaded circles indicate strain ed while polarization and unshaded circles indicate decreasing polarization 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 Ta, the depolarization temperature, the relaxor material is macropolar, exhibits a stable remanent polarization, and behaves as a piezoelectric. Tmax is the temperature at which the maximum dielectric constant is observed. Between T, and Tmax, 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 Tmax, 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. IPT. Figure 50.a and b show the longitudinal strain as a function of the applied electric ield 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 c 2000 by CRC Press LLC
© 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 Td , the depolarization temperature, the relaxor material is macropolar, exhibits a stable remanent polarization, and behaves as a piezoelectric. Tmax is the temperature at which the maximum dielectric constant is observed. Between Td and Tmax , the material possesses nanometer-scale microdomains that strongly influence the electromechanical behavior. Large dielectric permittivities and large electrostrictive strains arising from micro–macrodomain reorientation are observed. Above Tmax, 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 reproducibility 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 50.1 Polarization and dielectric behavior of a relaxor ferroelectric as a function of temperature, showing the three temperature regimes. FIGURE 50.2 Transverse strain as a function of the square of the polarization in ceramic 0.9PMN–0.1PT, at RT. The quadratic (x = QP2 ) nature of electrostriction is illustrated. Shaded circles indicate strain measured while increasing polarization and unshaded circles indicate decreasing polarization. Temperature (°C) Polarization Pa Dielectric constant K Td Tm III (macro-polar) II (macro-micro) I (electrostrictive)