Mechanics of Materials 36 (2004) 949–958 www.elsevier.com/locate/mechmat The effective pyroelectric and thermal expansion coefficients of ferroelectric ceramics JiangYu Li * Department of Engineering Mechanics, University of Nebraska-Lincoln, W317.5 Nebraska Hall, Lincoln, NE 68588-0526, USA Received 5 November 2002; received in revised form 13 May 2003 Abstract We present a micromechanical analysis on the effective pyroelectric and thermal expansion coefficients of ferro- electric ceramics in terms of their microstructural information. The overall behaviors of ferroelectric ceramics are profoundly influenced by the microstructural phenomena, where the macroscopic pyroelectric effect can be induced in an originally isotropic, thus non-pyroelectric ceramic composed of randomly oriented ferroelectric grains through poling, during which the polar axes of grains are switched by the applied electric field or mechanical stress. To analyze these complicated phenomena, we will first establish an exact connection between the effective thermal moduli and the effective electroelastic moduli of ferroelectric ceramics, and then combine the exact connection with the effective medium approximation to provide an estimate on the effective pyroelectric and thermal expansion coefficients of fer- roelectric ceramics in terms of the orientation distribution of grains and poling conditions, where the texture evolution as a result of domain switching during poling has been taken into account. Numerical results are presented and good agreements with known theoretical results and some experimental data are observed. Ó 2003 Elsevier Ltd. All rights reserved. Keywords: Pyroelectricity; Ferroelectricity; Piezoelectricity; Ceramics; Effective moduli 1. Introduction recent years, and have been utilized for various application such as infrared detection, imaging The crystals having a spontaneous polarization systems, and thermal-medical diagnostics (Cross, along a unique polar axis are called pyroelectric 1993). (Jona and Shirane, 1993), where the spontaneous A ceramic made of pyroelectric grains does not polarization is dependent on the temperature, and necessarily possess overall pyroelectric effect, be- electric charges can be induced on the crystal cause the center-symmetry of randomly oriented surface by temperature change. Pyroelectric de- ceramics prevents pyroelectric effect at macro- vices relying on the temperature sensitivity of scopic scale. If the grains are also ferroelectric, i.e., polarization have received increasing interest in if the polar axis can be switched by the applied electric field or mechanical stress, then macro- scopic pyroelectric effect can be induced in the * Tel.: +1-402-4721631; fax: +1-402-4728292. ceramics through poling, where a high electric E-mail address: [email protected] (J.Y. Li). field is applied to the ceramic at an elevated 0167-6636/$ - see front matter Ó 2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechmat.2003.05.005 950 J.Y. Li / Mechanics of Materials 36 (2004) 949–958 temperature. The microstructural evolution during governing the effective constitutive moduli in Sec- poling is very complex and involves polarization tion 3. Micromechanics approximation will be switching, domain wall movement, intergranular introduced in Section 4, and microstructural fea- constraint, and microstress generation. The com- tures such as texture and orientation distribution plicated microstructural phenomena obviously function (ODF) under various poling conditions have profound influence on the macroscopic are discussed in Section 5. We will then present pyroelectricity of ceramics, and this work will some numerical results and discussion on the study the effects of microstructural phenomena at effective pyroelectric and thermal expansion coef- domain and grain levels, including orientation ficients of ferroelectric ceramics in Section 6. It is distribution of domains and grains, on the effective demonstrated that the microstructure has impor- pyroelectric and thermal expansion coefficients of tant effect on the overall pyroelectric and thermal ferroelectric ceramics. The texture evolution in expansion coefficients of ceramics, and good ferroelectric ceramics during poling will be taken agreements between numerical results and known into account. theoretical results and some experimental data are Although much effort has been made to under- observed. stand the effective piezoelectric moduli of ferroelec- tric ceramics (Dunn, 1995; Nan and Clarke, 1996; Rodel€ and Kreher, 1999; Li, 2000a), little work has 2. Basic equations and notation been done to predict the effective thermal moduli of ferroelectric ceramics in terms of their microstruc- We consider the piezoelectric analog of the ture. Instead, quite a few micromechanical analysis uncoupled theory of thermoelasticity, where the on the effective thermal moduli of piezoelectric constitutive equations are given by composites have been developed (Dunn, 1993a,b; E E Benveniste, 1994; Chen, 1994a; Qin et al., 1998; Li rij ¼ L ekl þ eijkðÀEkÞÀk h; ijkl ij ð1Þ and Dunn, 1998; Levin et al., 1999; Aboudi, 2001). e e Di ¼ eiklekl À j ðÀEkÞÀp h This is probably because the microstructural phe- ik i nomena in ferroelectric ceramics are more in- or its inverse volved. In this work, we extend the self-consistent D D approach originally developed for electroelastic eij ¼ Mijklrkl þ gijkDk þ aij h; ð2Þ moduli (Li, 2000a) to predict the effective pyro- r r À Ei ¼ giklrkl À tikDk þ ci h: electric and thermal expansion coefficients of fer- roelectric ceramics. The analysis shares the same In the equations, rij and ekl are stress and strain spirit as many works in elastic polycrystals, where tensors, respectively; Di and Ek are electric dis- an exact connection between the effective thermal placement and electric field, respectively; h is the moduli and the effective electroelastic moduli is temperature change with respect to a reference E established first (Levin, 1967; Rosen and Hashin, temperature. The meanings of elastic moduli Lijkl D 1970; Hashin, 1984; Schulgasser, 1987; Kreher, and Mijkl, piezoelectric coefficients eijk and gijk, e r 1988; Li, 2000b), and the effective medium ap- dielectric constants jik and tik, thermoelastic con- E D e proximation is then applied to analyze the local stants kij and aij , and thermoelectric constants pi r electromechanical fields in ceramics (Walpole, and ci are clear from the constitutive equations. 1969; Willis, 1977; Qiu and Weng, 1991). The re- We notice that ÀEi rather than Ei is used as cent results on the equilibrium domain configura- dependent variable, because it gives us the sym- tion in a saturated ferroelectric ceramics have been metric moduli matrix which proves to be conve- employed to estimate the effective texture of poled nient. We also notice that by choosing different ferroelectric ceramics (Bhattacharya and Li, 2001; sets of independent variables, other sets of con- Li and Bhattacharya, 2002). stitutive equations can be obtained. For example, We introduce the basic equation and notation it is easier to measure thermal expansion coeffi- o in Section 2, followed by the exact relationships cients aE eij and pyroelectric coefficients ij ¼ oh jrmn;Em J.Y. Li / Mechanics of Materials 36 (2004) 949–958 951 r oDi p ¼ j , and they are related to current À1 i oh rmn;En H ¼ G ; C ¼ H K : ð8Þ moduli by JilK lKJi Ji JilK lK Finally, we notice the differential constraints on aD ¼ aE À g pr; cr ¼ tr pr: ð3Þ ij ij ijk k i ij j the field variables in terms of the equilibrium r Notice that ci is the figure of merit used by a de- equation signer to assess the performance of a pyroelectric RiJ;i ¼ 0; ð9Þ material for a typical device (Newnham et al., 1978). and gradient equation To proceed, we adopt the notation introduced ZJi ¼ UJ;i; ð10Þ by Barnett and Lothe (1975) that treats the elastic and electric variables on an equal footing. It is with similar to the conventional indicial notation with uj; J ¼ 1; 2; 3; the exception that both lowercase and uppercase UJ ¼ ð11Þ /; J ¼ 4; subscripts are used as indices, where lowercase ones take on the range 1–3, and uppercase ones where ui and / are displacement and electric po- take on the range 1–4, and repeated uppercase tential, respectively. subscripts are summed over 1–4. With this nota- tion, the field variables are expressed as 3. The effective moduli rij; J ¼ 1; 2; 3; RiJ ¼ Di; J ¼ 4; ð4Þ Let us now consider a ferroelectric polycrys- ekl; K ¼ 1; 2; 3; talline ceramic subjected to a uniform temperature ZKl ¼ ÀEl; K ¼ 4; change h, with linear displacement and electric 0 potential ZKlxl, or uniform traction and electric the electroelastic moduli are expressed as 0 8 displacement RkLnk applied at the boundary, where > LE ; J; K ¼ 1; 2; 3; x is the position vector and n is the outward > ijkl l k < normal. The polycrystal is composed of grains eijl; J ¼ 1; 2; 3; K ¼ 4; GiJKl ¼ with certain orientation distribution, each having > e ; J ¼ 4; K ¼ 1; 2; 3; :> ikl distinct constitutive moduli in the global coordi- Àje ; J; K ¼ 4; nate system due to the variation in grainsÕ orien- 8 il ð5Þ D tation and grain anisotropy. The constitutive > Mijkl; J; K ¼ 1; 2; 3; <> moduli of polycrystalline aggregate, as a result, is gijl; J ¼ 1; 2; 3; K ¼ 4; HJilK ¼ heterogenous at microscopic level. This leads to a > g ; J 4; K 1; 2; 3; :> ikl ¼ ¼ heterogeneous local electromechanical field in r Àtil; J; K ¼ 4 polycrystal at grain level, which depends on X, the orientation of individual grains. However, at the and the thermal moduli are expressed as ( macroscopic level the effective constitutive moduli E kij; J ¼ 1; 2; 3; can be defined through the effective constitutive KiJ ¼ e equation if the polycrystal is statistically homo- pi ; J ¼ 4; ( ð6Þ geneous (Nemat-Nasser and Hori, 1999), aD; J ¼ 1; 2; 3; C ij à à Ji ¼ r hRiJ ðXÞi ¼ GiJKlhZKlðXÞi À KiJ h; ci ; J ¼ 4: ð12Þ Ã Ã hZKlðXÞi ¼ HKliJ hRiJ ðXÞi þ CKlh; As a result, the constitutive equations (1) and (2) can be rewritten as where hÁi is used to denote the orientation-aver- aged quantities in polycrystal, weighted by the R ¼ G Z À K h; Z ¼ H R þ C h; ð7Þ iJ iJKl Kl iJ Ji JilK lK Ji orientation distribution function (ODF), which where describes the probability of a grain falling into 952 J.Y.