## Abstract

Motivated by the observation that the spontaneous polarization process of a ferroelectric polycrystal under the influence of a superimposed stress and/or electric field involves heterogeneous evolution of the ferroelectric phase among its constituent grains, a self-consistent electromechanical model is developed to determine the effective behaviour of the polycrystalline ceramic from such a heterogeneous electromechanical state. We start out from consideration of a micromechanics-based thermodynamic process to establish the kinetic equation of the crystallite and use it to evaluate the evolution of its ferroelectric domain. Then together with the Curie–Weiss law for the dielectric constants of the tetragonal phase, a dual-phase mixture theory is adopted to determine the change of its electromechanical moduli as temperature cools down below its Curie point. The overall property of the polycrystal is subsequently calculated by the self-consistent model through orientational average over its constituent grains. This two-level micromechanics model is applied to examine the shift of Curie temperature and evolution of the effective electromechanical moduli of a BaTiO_{3} ceramic under cooling. The calculated results show that its Curie temperature decreases with increasing hydrostatic pressure, but increases with a superimposed axial compression or a biased electric field. The predicted temperature shift and change of the dielectric constants are found to be consistent with experimental observations.

## 1. Introduction

A ferroelectric ceramic consists of many ferroelectric crystallites whose orientations change from one to another across the grain boundaries. When such a ceramic is cooled down from above its Curie point *T*_{c}, spontaneous polarization takes place simultaneously in its grains. In the absence of an external electric/mechanical load, the ferroelectric domains evolve equally and the polycrystal as a whole remains isotropic. Under the influence of a biased electromechanical load—such as an axial electric field or a mechanical tension or compression—the evolution of a new domain in all crystallites is not uniform, and the overall electromechanical properties of the polycrystal is transversely isotropic. Such a transversely isotropic characteristic is marked by five effective elastic compliances, three piezoelectric moduli and two dielectric constants.

A schematic of the polycrystal model during the cubic→tetragonal phase transition is depicted in figure 1*a*–*c*, using BaTiO_{3} for illustration. The paraelectric state of the cubic crystallites above *T*_{c} is shown in figure 1*a*, and a generic polarization state in which the parent cubic phase and the transformed tetragonal phase coexist at *T*<*T*_{c} is shown in figure 1*b*. In the latter case, each crystallite is in essence a two-phase composite and, due to the change of new domain concentration from grain to grain, the electromechanical moduli of the constituent grains are orientation-dependent. The overall properties of the polycrystalline ceramic at the generic state then depend on the volume concentration of the tetragonal domain in each grain, and are the volume-averaged values over those of its crystallites. In addition, at lower *T* when temperature approaches the boundary of tetragonal→orthorhombic phase transition, the dielectric constants of the tetragonal phase may follow the Curie–Weiss law to rise, making these constants temperature-dependent. Finally, as polarization reaches the saturation state, each grain turns into a tetragonal crystallite, as shown in figure 1*c*.

With this simple sketch, it is evident that the issue of spontaneous polarization of a ferroelectric ceramic can perhaps be most logically addressed first from the spontaneous polarization of its crystallites, and then by an orientational average, to calculate their collective response. If in each crystallite we refer the cubic phase as phase 0 and the tetragonal phase as phase 1, and denote the volume concentration of phase r as *c*_{r} (i.e. *c*_{1}+*c*_{0}=1), then the volume concentration *c*_{1} will continue to evolve from 0 to 1 as temperature traverses down from *T*_{c} to the saturation state. In this paper we will start from Gibbs free energy of the cubic–tetragonal dual-phase system to derive the thermodynamic driving force and the kinetic equation for the cubic→tetragonal phase transition in the crystallites, and make use of a homogenization theory to determine their effective electromechanical response. The volume-average process will be addressed by a self-consistent scheme. The issues of Curie-temperature shift and change of the effective electromechanical moduli of the polycrystal without and with an external stress or electric field will be examined using this two-level micromechanical model.

This approach differs from the classic Landau–Ginsburg–Devonshire theories in several significant ways. The classical theories are phenomenological, in which the Gibbs or Helmholtz free energy of the system is based on functional representation often in the form of polynomials of the electromechanical field. These theories are simple to use, and have proven valuable in interpreting and predicting many observed phenomena, but, unlike the present theory, they are not micromechanics-based and could not deliver the evolution of new domain concentration *c*_{1}, account for the difference Δ*L* in the electromechanical moduli between the parent and product phases, and address the heterogeneous state of the constituent grains. For a review on these classic theories and other pertinent phenomena, one may refer to Jona & Shirane (1993), Fatuzzo & Merz (1967), Jaffe *et al*. (1971), Lines & Glass (1977), Ikeda (1996) and Strukov & Levanyuk (1998), for instance.

More pertinent to the present work are the micromechanical studies that we have seen in recent years. Most notable among these include Hwang *et al*. (1995), Arlt (1996), Chen *et al*. (1997), Hwang *et al*. (1998) and Chen & Lynch (1998). In addition, a polycrystal model patterned after that of the rate-independent crystal plasticity was developed by Huber *et al*. (1999), and a rate-dependent model by Huber & Fleck (2001). Finite-element analysis has also been carried out by Hwang & Waser (2000) and Fotinich & Carman (2000). Furthermore, a micromechanics-based thermodynamic model making use of an effective domain to represent the collective effects of all active domains was developed by Li & Weng (1999, 2002, 2004) and Li (2003). This body of work has provided an important micromechanical basis for ferroelectric response.

These micromechanical studies, however, are all concerned with domain switch, not with spontaneous polarization that is the focus of this investigation. Among these cited works, Huber *et al*. (1999) and Huber & Fleck (2001) have also adopted a self-consistent approach. As such, it is perhaps necessary to clarify the differences between their formulation and the present one. First, their formulation is *incremental* involving the tangent moduli, while the present one is in a *total* form involving the secant moduli. The incremental formulation has the merit of wider applicability for it can address issues of non-radial loading, but the total one is substantially simpler for calculations under proportional loading, and this is the case here. Second on the grain level, they have included the contributions of all potential variants (six in tetragonal crystallites) but have not considered the domain morphology, while we will adopt the lamellar morphology, which has been uncovered by Merz (1952, 1954) and Hooton & Merz (1955) in BaTiO_{3} crystals and by DeVries & Burke (1957) in BaTiO_{3} ceramic (see figure 2*a*,*b*). In addition, their formulation takes the electromechanical moduli of the cubic and tetragonal phases to be isotropic and equal, while the present one considers their full anisotropy and makes distinction between the differences in their electromechanical moduli. As a consequence, their model does not provide the change of the effective electromechanical moduli of the polycrystal in the course of domain switch or spontaneous polarization. Finally, their transformation criterion was established similar to the yield criterion for the operation of slip systems in crystal plasticity, while we follow a micromechanics-based thermodynamic approach to derive the Gibbs free energy and the thermodynamic driving force for phase transition. This driving force is exact when the domain morphology takes the lamellar shape, which also represents the minimum energy configuration during the martensitic, diffusionless-type processes (Khachaturyan 1983; Wayman 1992).

While our adoption of piezoelectric laminate in each grain was motivated by both the reported morphology and its minimum energy configuration, our selection of the active variant from the six potential ones will be based on the criterion of maximum thermodynamic driving force at a given state of stress and electric field. This procedure differs from the approach of Shu & Bhattacharya (2001), Li & Liu (2004) and Li *et al*. (2005), who started from consideration of the geometrical and electrical compatibility of variants to find the correspondent variants that gives rise to a minimum energy state. Our approach also satisfies the conditions of geometrical and electric compatibility, but in the sense of Eshelby (1957)—not in the sense of crystallography in the unloaded state—with the cubic phase serving as the matrix and the tetragonal phase with an additional eigenstrain and eigen-polarization serving as inclusions. Their approach has shed great insight into the crystallography of variant formation. At this stage, however, it is mainly kinematic; no kinetics is involved. As a result it cannot address the issues of domain evolution, moduli change, heterogeneity of the electromechanical state, shift of Curie temperature, etc. that are the concerns of this study. Their approach has its root in the classical theory of invariant plane strains during martensitic transformation, and it is also a valuable tool for domain engineering.

## 2. Electromechanical constitutive equations of the cubic and tetragonal phases in the constituent grains

The linear electromechanical constitutive relations are generally written with the stress *σ*_{ij} and electric displacement *D*_{m} as a pair, and strain *ϵ*_{kl} and electric field *E*_{n} as another, as (Ikeda 1996)(2.1)in tensorial notations, where ^{(E)} and ^{(ϵ)} are the elastic modulus and dielectric permittivity tensors measured at constant electric field and constant strain, respectively, and is the piezoelectric tensor (these quantities when written without indices are all tensorial). The superscript ‘T’ stands for transpose.

Such a pairing has the advantage that, in the absence of body force and free-charge density, both *σ* and *D* satisfy the divergence theorem(2.2)whereas *ϵ* and *E* are derivable from the displacement *u*_{i} and electric potential *ϕ* as(2.3)

To examine the spontaneous polarization under a thermo-electro-mechanical load, however, it is more suitable to cast the constitutive relations with (*ϵ*_{ij}, *D*_{m}) as a pair and (*σ*_{kl}, *E*_{n}) as another in the form(2.4)where ^{(E)} and ^{(σ)} are the elastic compliance and dielectric permittivity tensors measured at constant electric field and constant stress, respectively. Tensor is the piezoelectric compliance tensor, but it has also been called the piezoelectric modulus tensor as it represents the direct piezoelectric effect of a crystal under the application of stress.

Owing to the change of unit cell from cubic to tetragonal symmetry, the ferroelectric phase further possesses an eigenstrain (*ϵ*^{sp}) and eigen-polarization (*P*^{sp}), so that its complete constitutive relations are(2.5)where the superscript ‘sp’ stands for spontaneous polarization. Then in view of figure 3 and taking direction 3 to be the poling axis, these eigenfields are given by(2.6)in Nye's (1979) contracted notations, where *a*_{0}, *a* and *c* are the lattice constants of the cubic and tetragonal unit cells, and *P*_{s} the saturation polarization.

The coupled relations can be put together in a unified set of notations as(2.7)with *X* serving as the load and *Y* as the response, i.e.(2.8)Tensor is then the electromechanical compliance tensor. The electromechanical moduli tensor will be denoted by , so that =^{−1}.

During the cubic→tetragonal phase transition in each grain, we will denote the compliances of the cubic phase with a subscript 0, and those of the tetragonal phase with a subscript 1. The tetragonal phase carries the 4*mm* symmetry, but in measurement it has been frequently approximated with the 6*mm* symmetry, such that(2.9)and(2.10)It follows that(2.11)Both _{0} and Δ will play an important role in the thermodynamic driving force of the cubic→tetragonal phase transition.

## 3. The self-consistent model linking the constituent grains to the polycrystal

When the polycrystalline ceramic is under an externally applied electromechanical load , the average internal load of a crystallite, , differs from it (an overbar signifies that it is a volume-averaged quantity). As all the constituent grains exist on equal geometrical footing, such a problem can be most conveniently addressed by the self-consistent scheme in which an oriented grain is embedded in the homogeneous effective medium with a property that represents the orientational average of all constituent grains. The embedding procedure is carried out for each grain, and its average load is then evaluated self-consistently in terms of its average response . Such a relation was first derived by Hill (1965) in an incremental form to study the plasticity of a polycrystal, and later modified to a total form by Berveiller & Zaoui (1979) under the special case of proportional loading. The calculated stress–strain relations of a face-centred-cubic polycrystal by Hutchinson (1970) and by Weng (1982) using the incremental and the total forms, respectively, were found to be sufficiently close. In the thermoelastic context, such a relation was further developed by Lu & Weng (1998) to study the martensitic transformation of polycrystalline shape-memory alloys. We now extend it to examine the spontaneous polarization of a ferroelectric ceramic under a thermo-electro-mechanical load.

Using the unified notations of (2.7) and (2.8) with *X* representing the electromechanical load and *Y* the response, the incremental self-consistent relation between the average fields of a crystallite (with a subscript ‘c’) and those of the polycrystal can be cast into(3.1)where ^{*} is the electromechanical constraint tensor of the polycrystalline matrix, with a value that is dependent upon its *tangent* electromechanical moduli. Under proportionally increasing load this incremental relation can be extended to the total form in terms of the *secant* moduli (with a subscript ‘s’), as(3.2)where the secant constraint tensor is related to the secant moduli _{s} of the polycrystal and Eshelby (1957)-type -tensor of a spherical inclusion when the effective medium is characterized by its, yet-unknown, secant electromechanical moduli tensor _{s}. More specifically, it is given by(3.3)

When the ceramic is under a biased electric field or uniaxial tension or compression, the effective medium is transversely isotropic, and the corresponding -tensor for a spherical inclusion can be found in Huang & Yu (1994) and Dunn & Wienecke (1997). The components of the -tensor derived in both papers were given using (*ϵ*^{*}, *E*^{*}) as the pair of eigenfields, and so it must be converted to the (*ϵ*^{*}, *D*^{*}) pair, using, for instance, the procedure of Li & Dunn (2001).

Since the total response consists of a linear and a nonlinear term, i.e.(3.4)where _{c}—the inverse of _{c}—is the *linear* effective electromechanical compliance tensor of the grain (crystallite) and that of the overall polycrystal. Since _{s} also satisfies the relation(3.5)we have(3.6)where matrix *I* is the 9×9 identity matrix.

When the external load is at , it is this and temperature *T* that control the level of the tetragonal domain *c*_{1} in each grain. We shall return to this issue in §4.

Several quantities are involved here: the linear effective electromechanical compliances of a crystallite _{c} and its effective polarization at a given concentration of tetragonal phase *c*_{1}, and the linear effective electromechanical compliances of the polycrystal , its secant constraint tensor and its effective polarization . We shall leave the determination of the crystallite properties until §4, but for now assuming that they are known for each grain, then the overall polarization and the overall effective electromechanical moduli of the polycrystal can be evaluated from the orientational averages over all constituent grains.

For the determination of overall spontaneous strain and electric polarization of the polycrystal , care must be exercised that it does not follow directly from the orientational average of those of the grains, i.e. , where (*θ*, *ϕ*, *ψ*) are the Euler angles defining the orientation of a grain. Instead, it is the total response and the total load that satisfy such a direct orientational average, i.e.(3.7)and then the nonlinear term can be evaluated from . The relation holds only when _{c}= for all crystallites, i.e. when all crystallites are isotropic with identical moduli. Hill (1967) first identified such a characteristic as one of the ‘essential features’ of constitutive law in the elastoplastic deformation of metal composites and polycrystals. He proved rigorously that, because of the elastic heterogeneity, the elastic and plastic components of the overall strain are *not* the direct means of their microscopic counterparts. More recently, it has also been recognized in the evaluation of effective magnetostriction of polycrystals and composites (Chen *et al*. 2003).

The linear effective electromechanical moduli can be calculated from the elastic relation of Hill (1965) and Walpole (1969) type, but written in the electromechanical context, as(3.8)Here _{c} is the electromechanical response concentration tensor of an oriented crystallite in the linear context. This relation is sufficiently general even for the evaluation of grain shape effect on (Qiu & Weng 1991). Once _{c} is known for all crystallites, it can be used to calculate the evolution of the effective elastic compliances, piezoelectric constants and dielectric constants of the polycrystal in the course of spontaneous polarization without or with a superimposed stress or electric field.

We now proceed to determine the effective spontaneous polarization of a crystallite and its linear effective electromechanical compliances _{c} that are needed in this self-consistent scheme.

## 4. Determination of the crystallite properties

To calculate the overall properties of the polycrystal self-consistently from those of its constituent grains, both and _{c} must be determined. Since a crystallite at its generic state is a dual-phase material consisting of the parent cubic phase and the transformed tetragonal phase, these two quantities will depend on the volume concentration *c*_{1} of the tetragonal one. It is thus imperative that this *c*_{1} be determined first at a given level of (). This calls for the establishment of the kinetic equation of phase transition. We now address this issue from the standpoint of a micromechanics-based thermodynamic approach.

### (a) Tetragonal domain concentration *c*_{1}

The kinetic equation provides the evolution of the tetragonal domain concentration *c*_{1} as the thermo-electro-mechanical load continues to increase. This may be derived from consideration of the thermodynamic driving force that results from the reduction of Gibbs free energy and the resistance force. The change of Gibbs free energy at a given consists of the chemical free energy and the electromechanical potential energy as(4.1)

The chemical part depends on the entropy change Δ*s*(*T*_{c}) and specific-heat change Δ*c*_{p} from the parent (cubic) to the product (tetragonal) phase. By noting that the change of chemical free energy of a phase is given by d*G*(*T*)=−*s*(*T*)d*T*, with ∂*s*/∂*T*|_{p}=*c*_{p}/*T* under the isobaric condition, and integrating it from *T*_{c} to *T* for both phases, it is straightforward to arrive at(4.2)where *c*_{1} is the volume concentration of tetragonal phase at the current temperature *T*.

The electromechanical potential energy of a crystallite, on the other hand, can be derived from (Stratton 1941; Eshelby 1957)(4.3)where are the mechanical traction and electric potential on the boundary surface *S* of a grain with an outward normal *n*_{j} at position *x*_{i}, to give rise to a uniform stress and electric field . The potential energy of the grain under the same electromechanical load but without phase transition is given by(4.4)and it is the difference that gives rise to required in (4.1).

To evaluate this quantity explicitly, we have been guided by the observed lamellar morphology such as shown in figure 2*a*,*b*. This has prompted us to invoke the Eshelby (1957) and Mori & Tanaka (1973) approach for a two-phase composite containing aligned ellipsoidal inclusions. This approach is similar to the Maxwell-Garnett approximation in transport phenomena (Nan 1993) and coincides with Willis' (1977) bounds in the context of composite elasticity (Weng 1992). It has the further virtue of being exact when the inclusions take the lamellar structure (Walpole 1969; Weng 1990). Application of this approach leads to(4.5)where *Y*^{*}=(*ϵ*^{*}, *P*^{*}) is Eshelby's equivalent electromechanical eigenfield introduced into the inclusion regions (the tetragonal phase) so that _{1} can be replaced by _{0} to yield the same electromechanical field in it. Here all quantities are tensorial and the subscript 1 refers to phase 1, the tetragonal phase. In addition,(4.6)where(4.7)noting that *c*_{0}=1−*c*_{1}. In the thermoelastic context, this set of relations reduces to that derived in Lu & Weng (1997) for martensitic transformation in shape-memory alloys.

Now following the principle of irreversible thermodynamics (Rice 1975), the conjugate thermodynamic driving force for the evolution of new domain is given by(4.8)which, after some heavy algebra on the Δ*G*_{p} term, can be cast into(4.9)with(4.10)and .

During spontaneous polarization, this driving force is countered by a resistance force that arises primarily from the energy dissipation associated with the domain wall growth. In the study of martensitic transformation in shape-memory alloys, Sun & Hwang (1993*a*,*b*) suggested a linear function to represent the energy dissipation, but we have found that the resistance force is stronger than linear. This finding is based on the observation that, under pure cooling, the driving force in (4.9) is mainly a linear function of *T*, but phase transition is very rapid when temperature initially passes through *T*_{c}, then slows down, and eventually approaches an asymptotic state at saturation. This phenomenon can be easily visualized during the first-order transition, and remains so for the second-order transition. In order to model the entire range of transformation, a nonlinear function that exhibits such a characteristic is essential, and for this reason one may choose an exponential or a tangent function. Here we chose *G*_{d}=*h* tan(*bc*_{1}), leading to the expression for the resistance force(4.11)

Then the condition of phase transition is reached when the driving force is sufficient to overcome the resistance force, and this results in the kinetic equation(4.12)This equation allows one to determine the volume concentration, *c*_{1}, of the tetragonal domain of a crystallite at the given level of electromechanical field and temperature *T*. (In computations it is easier to use the converse effect by treating *c*_{1} as the independent variable and let it increase from 0 or some initial value to 1, to compute the corresponding variation of and *T*.)

This equation must satisfy the initial condition that, under a pure cooling process (i.e. ), onset of phase transition occurs at *T*=*T*_{c}, at which the volume concentration of the tetragonal domain jumps from 0 to some initial value *c*_{1(i)} for the first-order transition, and is exactly 0 for the second-order transition (i.e. *c*_{1(i)}=0). At this point(4.13)where *C*_{0} is the value of *C* at *c*_{1}=*c*_{1(i)}. Equation. (4.13) provides the connection among the three parameters *h*, *b* and *c*_{1(i)}. Since *c*_{1(i)} is easily identifiable from the polarization or the dielectric constant curve, this leaves *b* to be the sole parameter to be specified.

### (b) Effective spontaneous strain and polarization , and effective electromechanical compliances _{c}

Once the tetragonal domain concentration *c*_{1} has been determined from the kinetic equation, the overall electromechanical response of the dual-phase crystallite can be calculated from (Li & Weng 1999)(4.14)where *Y*^{*} is the equivalent polarization field of the tetragonal phase as given in (4.6). In view of the first of (3.4), the effective spontaneous polarization field is then given by(4.15)As with for the polycrystal, for the crystallite, and is so only when _{1}=_{0}, under which *Y*^{*}=0 and _{c}=_{0}.

The overall effective electromechanical compliances of the crystallite _{c} can be calculated from (see Weng 1990 in the context of linear elasticity)(4.16)where the compliances of the cubic and tetragonal phases are specified in (2.9) and (2.10).

## 5. The shift of Curie temperature of a BaTiO_{3} ceramic and evolution of its effective electromechanical moduli upon continuous cooling

In order to place the developed self-consistent model in perspective, we now apply it to study the shift of Curie temperature and evolution of the effective electromechanical moduli of a BaTiO_{3} polycrystal as the temperature cools down below its Curie point. This will be carried out without and with a superimposed stress or electric field. At a given stage of thermo-electro-mechanical load, the polycrystal behaviour is calculated from the orientational average of its constituent grains as outlined in §3, and the crystallite properties are in turn determined from the cubic–tetragonal, dual-phase theory of §4. The polycrystal is taken to consist of 150 differently oriented grains, generated from a base cubic crystal through uniform increments of rotation characterized by the three Euler angles. Owing to the cubic symmetry of the parent phase we varied them within the standard triangle of the stereographic projection as in Weng (1983) in the study of polycrystal plasticity; i.e. in Goldstein's (1950) notations, the first angle *ϕ* increases from 0 to 45° with a uniform increment of 7.5°, the second one *θ* increases also by 7.5° but always stays inside or on the [100]–[111] symmetric line, resulting in a total of 25 orientations. Then for each orientation we again rotated it by a uniform increment of 30° for the third angle *ψ* from 0 to 180°, creating a total of 150 orientations. We have found that this produces a fairly isotropic polycrystal.

To make use of the lamellar morphology of the tetragonal phase in each crystallite, we have also derived the non-vanishing components of the -tensor of a lamellar inclusion in a cubic matrix; these components are listed in appendix A. There are six potential tetragonal variants in each crystallite. Under a biased stress or electric field, the orientation of the active variant is chosen according to the criterion of maximum thermodynamic driving force. This criterion reduces to the maximum energy dissipation under homogeneous electromechanical compliances (_{1}=_{0}), and is similar to Bishop & Hill's (1951*a*,*b*) maximum plastic work in crystal plasticity. Under a non-biased field such as hydrostatic pressure, the orientations are chosen randomly so that the polycrystal as a whole remains isotropic at a generic state of phase transition.

The material constants of the BaTiO_{3} crystallites are taken as follows. The Curie point was *T*_{c}=120 °C (Merz 1949), entropy change was Δ*s*(*T*_{c})=−1.25×10^{4} J m^{−3} °C^{−1} (Megaw 1957) and specific heat Δ*c*_{p}=12.4×10^{4} J m^{−3} °C^{−1} (Merz 1950). In addition, the components of polarization strain and electric polarization during the cubic→tetragonal phase transition are (Merz 1949; Jona & Shirane 1993)(5.1)which indicates a negative volume change of the unit cell during the phase transition (e.g. ) and a saturation polarization of *P*_{s}=0.16 C m^{−2}. The elastic compliances, piezoelectric and dielectric constants of the cubic and tetragonal phases at *T*_{c} are listed in table 1. It is well known that, as temperature decreases from above *T*_{c} towards it, the dielectric constant of the cubic phase tends to follow the Curie–Weiss law and rise, bringing it to a very high value (10 000*k*_{0}). Likewise, as temperature further decreases towards the tetragonal–orthorhombic transition temperature (−5 °C), the dielectric constants of the tetragonal phase can rise according to the Curie–Weiss law. Merz's experiment (1949) for a BaTiO_{3} crystal between the temperature range of 120 and −5 °C showed that, within the low temperature range where the cubic→tetragonal transition is almost complete, the *k*_{33} (*c*-axis) of the crystal remains relatively flat (see figure 4), but *k*_{11} (*a*-axis) exhibits a pronounced rise (see figure 5). These features indicated that while *k*_{33} of the tetragonal phase can be taken as constant, its *k*_{11} is temperature-dependent, represented by the Curie–Weiss relation(5.2)where a relatively small temperature-independent term has been neglected. Then by using Merz's data between 20 and −5 °C, i.e. with *k*_{a}*/k*_{0}=4.74×10^{3} at 20 °C and 6.92×10^{3} at −5 °C, we determined *C*=374×10^{3} °C and *T*_{0}=−59 °C. This also results in *k*_{a}*/k*_{0}=2.09×10^{3} at 120 °C listed in table 1. Finally, the cubic→tetragonal phase transition of the BaTiO_{3} crystal is known to be of the first-order type (Jaffe *et al*. 1971; Strukov & Levanyuk 1998). In order to reflect the property of a BaTiO_{3} crystal tested by Merz (1949), we took *c*_{1(i)}=0.60 and *b*=1.355.

Based on these material constants and the dual-phase theory outlined in §4, we first calculated the evolution of the dielectric constants *k*_{33} and *k*_{11} of the dual-phase crystal under a pure cooling process when temperature decreases from 120 °C; the results are shown in figures 4 and 5, respectively. The good agreement with Merz's test data in both cases suggests that this dual-phase theory with a Curie–Weiss representation for *k*_{11} of the tetragonal phase could reasonably represent the ferroelectric properties of this crystal.

### (a) The shift of Curie temperature of polycrystalline BaTiO_{3}

We first examine how the transition temperature of a BaTiO_{3} polycrystal can be shifted under the influence of a superimposed electromechanical load. Since Curie temperature is a fundamental property of ferroelectric ceramics, and its up or down shift can have a significant effect on the design of ferroelectric devices, it is important that the nature of the shift be accurately predicted. This onset condition occurs when the kinetic equation (4.12) in the most favourably oriented grain is satisfied under the given level of applied load. That is, one seeks the value of temperature *T* at the corresponding level of when *c*_{1}=*c*_{1(i)}.

#### (i) Influence of a superimposed hydrostatic pressure

Many devices are used under the environment of hydrostatic pressure and so it is of practical interest to see how the transition temperature changes with it. Figure 6 shows the shift of Curie temperature as the applied pressure increases from 0 to 400 MPa. Here Δ*T*_{c}≡*T*−*T*_{c} is the difference between the transition temperature with a superimposed stress or/and electric field, and the original Curie point *T*_{c} without it. The computational result for the polycrystalline BaTiO_{3} is plotted in solid line, while the dashed lines are Merz's (1950) experimental data for a BaTiO_{3} crystal. Merz attributed the scattered data at higher pressure to the late pressure he applied in those tests, and suggested the linear one to be the best result. The additional dot–dashed line corresponds to the experimental result of a barium–strontium titanate ceramic (Ba75–Sr25)TiO_{3} (Shirane & Sato 1951). Although the Curie temperature of (Ba75–Sr25)TiO_{3} is lower than that of BaTiO_{3}, it has been known that the dielectric properties of the solid solution Ba–Sr titanate are essentially similar to those of pure barium titanate (Shirane & Sato 1951), and that the nature of its Curie-temperature shift has often served as a general reference to that of BaTiO_{3} ceramic (Jaffe *et al*. 1971). These results indicate that the Curie temperature drops almost linearly as the applied hydrostatic pressure increases. The close agreement between the calculated polycrystal shift and Merz's single crystal shift is consistent to the fact that single crystals and polycrystals with cubic structure have essentially the same response under a pure hydrostatic pressure.

#### (ii) Effect of a superimposed axial pressure

Under an axial pressure our calculation finds that the Curie temperature of the BaTiO_{3} ceramic increases. The calculated result is shown by the solid line in figure 7. The slope of the increase is about 5.0×10^{−2} °C MPa^{−1}. In a series of tests on a BaTiO_{3} ceramic, Takagi *et al*. (1948) also found that the transition temperature increased with axial pressure with a slope of about 5.4×10^{−2} °C MPa^{−1} (the dashed line). In addition, Shirane & Sato's (1951) have also measured the shift on the same (Ba75–Sr25)TiO_{3} discussed in figure 6, and found a slope of about 4.1×10^{−2} °C MPa^{−1} (the dot–dashed line). Thus, contrary to the effect of hydrostatic pressure, the Curie temperature increases with increasing axial pressure. This opposite trend is well predicted by the theory.

#### (iii) The shift of *T*_{c} under a biased electric field

Instead of a mechanical load, a biased electric field was then superimposed on the polycrystal. The calculated shift of the Curie temperature is shown in figure 8, showing an increase of the transition temperature. Also plotted here is the slope of Baerwald & Berlincourt's (1953) measurement on a BaTiO_{3} ceramic that showed an increase of about 4 °C with a field of 0.5 MV m^{−1}. The predicted shift and the measured slope are seen to be sufficiently close.

### (b) Evolution of the effective elastic, piezoelectric and dielectric constants during spontaneous polarization

As temperature traverses down from *T*_{c}, the tetragonal domains grow at the expense of the cubic phase in the crystallites. As a consequence, the overall electromechanical properties of the polycrystal also continue to evolve, until the saturation state is reached in all constituent grains. Under a pure cooling process or with an additional superimposed hydrostatic pressure, each crystallite transforms identically and the polycrystal as a whole is isotropic. But when it is further subjected to a biased mechanical stress or electric field during the cooling process, the evolution of tetragonal domains in the crystallites is non-uniform and the polycrystal as a whole is anisotropic. In order to uncover the nature of the change in its effective electromechanical properties, we consider here four loading conditions: (i) pure cooling, (ii) with a superimposed axial compression , (iii) with a superimposed axial tension and (iv) with a biased electric field .

#### (i) Pure cooling process

The corresponding changes of the isotropic bulk and shear moduli of the polycrystal are shown in figure 9*a*,*b*, respectively, whereas that of the overall dielectric constant are shown in figure 9*c*. These effective properties have been calculated through the orientational average of (3.8), in which the effective anisotropic electromechanical compliances of each crystallite were evaluated from (4.16) at a given value of *c*_{1}. These properties were initially those of the isotropic BaTiO_{3} with cubic grains (*c*_{1}=0), then jumped to those of the cubic–tetragonal mixture at *c*_{1}=0.6, and then evolved steadily to the saturation state at *c*_{1}=1, corresponding to the values of the polycrystal containing randomly oriented tetragonal grains. It is found that the bulk modulus increases with increasing polarization, but the shear modulus decreases with it. The dielectric constant also drops sharply from the initial 10×10^{3}*k*_{0} of the cubic phase to about 3×10^{3}*k*_{0} of the cubic–tetragonal mixture at *T*_{c}, and then decreases further due to an increase of the tetragonal phase. It eventually rises again when the Curie–Weiss characteristic of *k*_{11} of the tetragonal phase becomes a dominant feature. The agreement with the measurement of Jaffe *et al*. (1971) is also excellent, apart from the later departure during which the test data shows another drop (this could be a result of an early tetragonal→orthorhombic transition).

#### (ii) Effect of a superimposed axial pressure

When the polycrystal is superimposed with a compression of during the cooling process, the evolution of tetragonal domain differs from grain to grain and the overall polycrystal is transversely isotropic. By taking the orientational average over the 150 grain orientations, the changes of the transversely isotropic electromechanical compliances are shown in figure 10. Unlike the preceding case where all constituent grains experience the onset of phase transition simultaneously, the onset condition under an axial compression (and an axial tension or a biased electric field next) is satisfied sequentially from the most favourably oriented grain to the least favourably oriented one. As a consequence, there is no sudden jump in the overall properties at *T*_{c}. There is also an early transition by about 5 °C now. All five elastic compliances (*s*_{11}, *s*_{33}, −*s*_{12}, −*s*_{13}, *s*_{44}) increase with decreasing temperature, and so do the three piezoelectric constants (*d*_{15}, *d*_{33}, −*d*_{31}). The two dielectric constants (*k*_{3}, *k*_{1}), however, decrease first and then increase again with decreasing temperature.

#### (iii) Effect of a superimposed axial tension

With a superimposed tensile stress , the evolution of the transversely isotropic electromechanical compliances is shown in figure 11. Contrary to the preceding case, the *s*_{33} is now higher than *s*_{11} and −*s*_{13} is also higher than −*s*_{12}. The order of (*d*_{15}, *d*_{33}, −*d*_{31}) remains unchanged, but that of (*k*_{3}, *k*_{1}) has also been reversed.

#### (iv) With a biased electric field

When the polycrystal is superimposed with a biased field during the cooling process, the change of the transversely isotropic electromechanical compliances is depicted in figure 12. The increase of the five elastic and three piezoelectric constants, and the decrease and then increase of the two dielectric constants, are similar to those under axial tension, but at this level of electric field they are slightly higher. In contrast to the case of axial compression, both and tensile give rise to higher *s*_{33} over *s*_{11}, and lower *k*_{3} over *k*_{1}, for the reason that the *s*_{33} value of the tetragonal phase is greater than *s*_{11}, and its *k*_{3} is less than *k*_{1}. Both and tensile tend to promote the tetragonal domain in each grain to align toward the direction 3, but the compressive would inhibit it, resulting in these two opposite trends.

## 6. Concluding remarks

In this study we have developed a two-level micromechanical model to calculate the spontaneous polarization of a ferroelectric polycrystal. The first level involves the phase transition in each grain, which at its generic state is taken to consist of the cubic and tetragonal phases with lamellar structure. The volume concentration of the tetragonal phase at a given level of thermo-electro-mechanical load is determined from a thermodynamic-based kinetic equation, and then with a Curie–Weiss temperature-dependent *k*_{11} of the tetragonal phase, a dual-phase homogenization theory is used to determine the effective electromechanical moduli of the crystallites. The second level involves the transition of the field quantities from the grain to the polycrystal level through a self-consistent scheme, and with it, the overall property of the polycrystal is calculated from the orientational average of its constituent grains. This model can account for the anisotropy and heterogeneity of the electromechanical moduli of the cubic and tetragonal phases in each grain, and their heterogeneous stress and electric field state. It can also account for the heterogeneous evolution of the tetragonal phase among the constituent grains. We have applied the developed theory to study the shift of Curie temperature of a BaTiO_{3} ceramic without and with a superimposed electromechanical load, and to calculate the evolution of its elastic, piezoelectric and dielectric constants in the course of spontaneous polarization. The calculated shift in *T*_{c} with a superimposed hydrostatic pressure, uniaxial compression, and a biased electric field and the predicted overall dielectric constant, are found to be consistent with experimentally measured data.

## Acknowledgments

This work was supported by the National Science Foundation, under grants CMS-0114801 and 0510409.

## Footnotes

- Received November 8, 2004.
- Accepted November 22, 2005.

- © 2006 The Royal Society