Primary and secondary pyroelectric effects of ferroelectric 0 3 composites

K.-H. Chew, F. G. Shin, B. Ploss, H. L. W. Chan, and C. L. Choy

Citation: J. Appl. Phys. 94, 1134 (2003); doi: 10.1063/1.1583154

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Primary and secondary pyroelectric effects of ferroelectric 0-3 composites

K.-H. Chew,a)F. G. Shin, B. Ploss, H. L. W. Chan, and C. L. Choy

Department of Applied Physics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong

共Received 11 December 2002; accepted 28 April 2003兲

Simple and tractable analytical expressions for determining the pyroelectricity in ferroelectric 0-3 composites have been developed. For the dilute suspension limit, expressions for the effective pyroelectric and other thermal electromechanical properties are derived within the framework of the Maxwell–Wagner approach. Then, an effective-medium theory is employed to examine the first and second pyroelectric coefficients in the concentrated suspension regime. The effective-medium approach as compared to the Maxwell–Wagner approach results in a better agreement with known experimental data up to higher volume fraction of inclusions. The pyroelectricity of 0-3 composites of ceramic inclusions embedded in the P共VDF–TrFE兲 copolymer matrix and of P共VDF–TrFE兲 inclusions embedded in a ceramic matrix are analyzed numerically under different polarization configurations. The theoretical predictions show that the secondary pyroelectric effects in composite systems with ceramic as the matrix are stronger than those with the copolymer as the matrix and can sometimes dominate the pyroelectricity for certain compositions. © 2003 American Institute of Physics. 关DOI: 10.1063/1.1583154兴

I. INTRODUCTION

Pyroelectric materials are extensively used for thermal infrared detectors1due to advantages such as good sensitiv-ity, room temperature operation, and low cost.2 In recent years, the research in the field of pyroelectricity has been concentrated on discovering materials with higher figures of merit. One way to attain this is to dilute the ferroelectric material in order to reduce the dielectric constant as observed in sol–gel processed (Pb,Ca)TiO3.3Composite materials of

ferroelectric ceramics particles embedded in polymer matri-ces with different connectivities4 aroused great interest be-cause of their mechanical flexibility, low cost, and excellent pyroelectricity.5 Another advantage is that their dielectric, thermal, and pyroelectric properties can be changed by vary-ing the volume fraction of ceramic particles or polvary-ing procedure.6

Many experimental works have been done on the pyro-electric properties of ferropyro-electric composites.7Theoretically, Bhalla et al. investigated the primary and secondary pyro-electric effects and derived expressions for the effective py-roelectric coefficients based on simple series and parallel connections.5Yamazaki et al. derived the pyroelectric coef-ficient of 0-3 composites with identical spherical ferroelec-tric particles.8 Based on a modified Clausius–Mossotti relation,9 Wang et al.10 formulated the pyroelectric coeffi-cient of ferroelectric composites by taking into account the depolarization coefficient. Nan11developed a formulation to describe the pyroelectricity of ferroelectric composite mate-rials. The theoretical framework is developed in terms of a Green’s function method and perturbation theory which is more rigorous but complicated. Recently, Levin and Luchaninov have derived expressions for the effective pyro-electric constants of composite materials with spherical

inclusion12 by means of effective field theory. While exten-sive theoretical studies8 –12have been made on the pyroelec-tricity of 0-3 composites, the works are mainly concerned with the effective pyroelectric effect of ferroelectric compos-ites and not explicitly on the secondary effect.

This work concerns deriving explicit analytical expres-sions for the primary and secondary pyroelectric effects of ferroelectric 0-3 composites. We employ two methods in suc-cession in this study: the Maxwell–Wagner共MW兲approach and then effective-medium 共EM兲 theory; the former gives expressions for the case of a dilute suspension of inclusion particles while the latter gives results for a wider range of particle concentration. Section II contains the general expres-sions for the primary and secondary effects of ferroelectric 0-3 composites. The definition of pyroelectricity is given, followed by a detailed derivation of primary and secondary pyroelectric coefficients within the framework of the two methods mentioned earlier. In Sec. III, the pyroelectricity of composite systems consisting of ferroelectric ceramic and polymer phases is examined. Comparison with experimental results is performed to verify the two theoretical models. Some conclusions are given in Sec. IV.

II. THEORY

Pyroelectricity is the electrical response of a polar mate-rial as a result of a change in temperature. The pyroelectric effects at constant stress ␴and electric field E are defined as13confining to scalar notations,

冉

⳵D

⳵T

冊

_E,␴⫽

冉

⳵D

⳵T

冊

_E,e⫹

冉

⳵D

⳵e

冊

_E,T

冉

⳵e

⳵T

冊

_E,␴, 共1兲

where D, e, and T represent, respectively, the electric dis-placement, strain, and temperature of the material. The term (⳵D/⳵T)E,e of Eq.共1兲is called the primary pyroelectric

re-sponse. The primary pyroelectric effect corresponds to the a兲_{Electronic mail: khchew2@hotmail.com}

1134

charges produced owing to the change in polarization with temperature when the dimensions of the pyroelectric material are fixed. For a material under constant stress, its dimensions change with temperature due to thermal expansion resulting in an additional contribution of piezoelectrically induced charge. This leads to the secondary pyroelectric effect which is defined as the final term of Eq.共1兲.

For ferroelectric materials, the electric displacement can be related as

D⫽P⫹d␴, 共2兲

with the polarization term P contributed by the ‘‘switchable’’ polarization P_S and electric field E given by

P⫽PS⫹␧E. 共3兲

In Eqs.共2兲and共3兲, d is the piezoelectric coefficient and␧the permittivity of the material.

From Eqs.共1兲to共3兲, the primary pyroelectric coefficient is expressed as

␳1⬅

冉

⳵D

⳵T

冊

E,␴

⫽

冉

⳵_⳵TP

冊

E,␴

⫽⳵PS ⳵T ⫹E

⳵␧

⳵T, 共4兲

where the first term ⳵PS/⳵T represent ␳1 at zero applied

electric field at a temperature below the transition tempera-ture TC. The second term exploits the ␳1 near the Curie

temperature where⳵␧/⳵T can be quite large. The secondary pyroelectric coefficient is written as

␳2⬅

冉

⳵D

⳵e

冊

_E,T

冉

⳵e

⳵T

冊

_E,

␴

⫽

冉

⳵D

⳵␴

冊

E,T

冉

⳵␴

⳵e

冊

_E,T

冉

⳵e

⳵T

冊

_E,

␴

⫽dc␣, 共5兲 which is the contribution induced by the coupled effects of thermal deformation and piezoelectricity. c and ␣ are the elastic stiffness and thermal expansion coefficients of the py-roelectric material. In the present study, explicit expressions for both the primary and secondary pyroelectric responses of ferroelectric 0-3 composites are derived with examples given for T sufficiently far below TCof the two constituents, where

the pyroelectric contribution from the dielectric permittivity is weak and can be neglected.

A. Single spherical inclusion in a homogeneous pyroelectric matrix

Consider the single inclusion problem of a spherical in-clusion embedded in an infinite matrix, in which both mate-rials are ferroelectric as well as dielectrically and elastically isotropic and homogeneous, as shown in Fig. 1. We assume that both the constituents may be under hydrostatic stresses as may be created by thermal expansion. Under the action of a weak electric field Emin the matrix region far away from

the inclusion, the electric displacements and electric fields in the two materials are related by the following expression 共Appendix A兲:

具

Di

典

⫹2␧m„

具

Ei

典

⫺

具

Em

典

…⫽

具

Dm

典

, 共6兲

with

具

D_i

典

⫽

具

P_i

典

⫹␧_i

具

E_i

典

⫹d_hi

具

␴_i

典

, 共7兲 and

具

Dm

典

⫽

具

Pm

典

⫹␧m

具

Em

典

⫹dhm

具

␴m

典

, 共8兲

where

具

␨

典

⫽1/V兰␨dV denotes the volumetric averaging of the quantity ␨. Subscripts i and m refer to inclusion and matrix. P, ␧, dh, and␴represent the polarization, dielectric

permittivity, piezoelectric coefficient, and hydrostatic stress, respectively.

The hydrostatic stresses are related by static equilibrium conditions in a similar way as can be obtained by manipu-lating with standard results such as given by Christensen14 共Appendix B兲, thus

具

␴m

典

⫽

具

␴i

典

⫹

4

3␮m„

具

ei

典

⫺

具

em

典

…, 共9兲

where e is volumetric strain and␮is shear modulus. For deformation involving a change of temperature⌬T, the relationship between the hydrostatic stress, volumetric strain, and temperature of the inclusion is15

具

␴i

典

⫽ki„

具

ei

典

⫺␣i⌬T…, 共10兲

where k and ␣ represent the bulk modulus and the thermal expansion coefficient of the material, respectively. Here, ⌬T⬅T⫺Trefis the temperature difference from a reference

temperature Tref at which thermal stress does not arise or

taken as zero. A similar equation may be written for the matrix material.

B. Maxwell–Wagner„MW…approach for deriving the effective constants of pyroelectric 0-3

composites in the dilute suspension limit

[image:3.612.342.535.52.221.2]

Consider now a composite material consisting of a well-dispersed, dilute suspension of spherical pyroelectric inclu-sions in a pyroactive matrix phase, the properties of which are to be determined. Imagine a single sphere of such a com-posite material is embedded in an infinite matrix of the origi-nal matrix phase 共Fig. 2兲. This constitutes again the single inclusion problem discussed above. This is so because the sphere of the composite material may be considered to be a

FIG. 1. Spherical inclusion embedded in matrix phase of permittivity␧m.

homogenous sphere of an equivalent material having the ef-fective properties of the composite. Thus, for the present problem, the analog of Eq.共6兲is

具

D

典

⫹2␧_m„

具

E

典

⫺

具

E_m

典

…⫽

具

D_m

典

, 共11兲

where quantities without subscripts relate to the composite. Similarly, a result analogous to Eq.共9兲on hydrostatic stress is

具

␴

典

⫹4

3␮m„

具

e

典

⫺

具

em

典

…⫽

具

␴m

典

. 共12兲

The dielectric permittivity␧ and hydrostatic piezoelec-tric coefficient dh of the composite are related to the electric

displacement

具

D

典

, polarization

具

P

典

, applied field

具

E

典

, and stress 具␴典as

具

D

典

⫽

具

P

典

⫹␧

具

E

典

⫹dh

具

␴

典

, 共13兲

and the deformation to a change in temperature⌬T and hy-drostatic stress as15

具

␴

典

⫽k„

具

e

典

⫺␣⌬T…, 共14兲

where k and ␣ represent the bulk modulus and thermal ex-pansivity of the composite material, respectively.

From Eqs. 共9兲 and 共12兲, we obtain another form of stress–strain relationship as

具

␴

典

⫹4

3␮m

具

e

典

⫽

具

␴i

典

⫹

4

3␮m

具

ei

典

. 共15兲

The electric field in the inclusion can be obtained by solving Eqs. 共6兲,共7兲, and 共11兲, thus

具

E_i

典

⫽

具

D

典

⫹2␧m

具

E

典

⫺

具

Pi

典

⫺dhi

具

␴i

典

␧i⫹2␧m

. 共16兲

Manipulations of Eqs. 共10兲, 共14兲, and 共15兲 provide the expression for stress acting on the inclusion as

具

␴i

典

⫽

具

␴

典

冉

1⫹4␮m 3k

1⫹4␮m 3ki

冊

⫹共␣⫺␣i兲⌬T

冉

4␮m

3

1⫹4␮m 3ki

冊

,

共17兲

as well as the relationship between the strain of the inclusion

具

ei

典

and temperature change ⌬T as

具

ei

典

⫽

共k⫹43␮m兲

具

e

典

⫺共k␣⫺ki␣i兲⌬T

共ki⫹

4 3␮m兲

. 共18兲

We are now ready to solve the pyroelectric coefficients of the pyroelectric composite material. From Eq. 共C6兲 共 Ap-pendix C兲together with Eqs.共10兲and共14兲, we have

„k⫺k_m…

具

e

典

⫺„k␣⫺k_m␣_m…⌬T

⫽␾†„ki⫺km…

具

ei

典

⫺„ki␣i⫺km␣m…⌬T‡. 共19兲

Substitution of

具

e_i

典

into Eq.共19兲and by equating terms involving

具

e

典

and⌬T give

共k⫺km兲⫽␾共ki⫺km兲

冉

k⫹4 3␮m

ki⫹43␮m

冊

, 共20兲

and

共k␣⫺km␣m兲⫽␾共ki⫺km兲

冉

k␣⫺ki␣i

ki⫹

4 3␮m

冊

⫹␾共ki␣i⫺km␣m兲. 共21兲

After some mathematical manipulation, Eq. 共20兲 gives the bulk modulus of the composite which is similar to Hashin’s16as

1

k⫽ 1 k_i

冉

␾

1⫹4␮_m/3k_i

冊

⫹ 1 k_m

冉

1⫺␾ 1⫹4␮_m/3k_m

冊

冉

␾

1⫹4␮m/3ki

冊

⫹

冉

1⫺␾

1⫹4␮m/3km

冊

, 共22兲

and Eq.共21兲together with Eq.共22兲give Levin’s17expansion coefficient formula in the form

␣⫺␣i⫽„␣m⫺␣i…

冉

1/ki⫺1/k

1/ki⫺1/km

冊

. 共23兲

Using Eq. 共C3兲 together with Eqs. 共7兲, 共16兲, and 共17兲 gives

冉

1⫺␾ ␧i⫺␧m ␧i⫹2␧m

冊

具

D

典

⫽␾

冉

3␧m

␧i⫹2␧m

冊

具

Pi

典

⫹„1⫺␾…

具

Pm

典

⫹␾

冉冉

3␧m

␧i⫹2␧m

冊

[image:4.612.76.273.52.222.2]

d_hi⫺d_hm

冊

⫻„␣⫺␣i…⌬T

冉

4␮m

3

1⫹4␮m 3ki

冊

⫹

冉

2␧m␾

冉

␧i⫺␧m

␧i⫹2␧m

冊

⫹␧m

冊

具

E

典

⫹

再

dhm⫹␾

冉冉

3␧m

␧i⫹2␧m

冊

dhi⫺dhm

冊

⫻

冉

1⫹4␮m 3k

1⫹4␮m 3ki

冊冎

具

␴

典

. 共24兲

Comparing Eq. 共24兲 with Eq. 共13兲, we obtain the total polarization

具

P

典

, effective permittivity ␧, and effective pi-ezoelectric coefficient dh of the composite within the MW

共Ref. 18兲framework as

具

P

典

⫽␾

冉

3␧m

␧i⫹2␧m

1⫺␾ ␧i⫺␧m ␧i⫹2␧m

冊

具

Pi

典

⫹„1⫺␾…

冉

1

1⫺␾ ␧i⫺␧m ␧i⫹2␧m

冊

具

Pm

典

⫹␾

冉

3␧_m ␧i⫹2␧m

dhi⫺dhm

冊

„␣⫺␣i…

冉

4␮m

3

1⫹4␮m 3ki

冊

1⫺␾ ␧i⫺␧m ␧i⫹2␧m

⌬T,

共25兲

␧⫽

冉

␧m⫹␾2␧m

␧i⫺␧m

␧i⫹2␧m

1⫺␾ ␧i⫺␧m ␧i⫹2␧m

冊

, 共26兲

and

d_h⫽

dhm⫹␾

冉

3␧m

␧i⫹2␧m

dhi⫺dhm

冊

冉

1⫹4␮m 3k

1⫹4␮m 3ki

冊

1⫺␾ ␧i⫺␧m ␧i⫹2␧m

. 共27兲

By using Eqs.共4兲and共25兲, the primary pyroelectric co-efficient of the composite is expressed as

␳1⫽␾

冉

3␧m

␧i⫹2␧m

1⫺␾ ␧i⫺␧m ␧i⫹2␧m

冊

␳i

⫹„1⫺␾…

冉

1

1⫺␾ ␧i⫺␧m ␧i⫹2␧m

冊

␳m, 共28兲

where␳_i and␳_m denote the pyroelectric response of the in-clusion and matrix, respectively. The pyroelectric contribu-tion arising from the temperature dependence of permittivity is ignored in the present study as we consider measurement temperatures far below the Curie temperature TC of the two

phases. However, for temperatures in the vicinity of a tran-sition point of the phases, the temperature dependence of material properties might have significant effects on the py-roelectric activity.19,20The secondary pyroelectricity is writ-ten as

␳2⫽␾

冉

3␧_m ␧i⫹2␧m

dhi⫺dhm

冊

共␣⫺␣i兲

冉

4␮m

3

1⫹4␮m 3ki

冊

1⫺␾ ␧i⫺␧m ␧i⫹2␧m

,

共29兲 which occurs as a result of the deformation mismatch be-tween the two phases. Summations of Eqs. 共28兲 and 共29兲 give the total pyroelectric coefficient for the composite. Equations共25兲–共29兲are derived based on the MW approach which is believed to give expressions applicable to the dilute suspension regime.

C. Effective-medium „EM…approach for deriving the effective constants of 0-3 pyroelectric

composites in the concentrated suspension limit

This section illustrates the derivation of expressions for the effective constants of ferroelectric 0-3 composites in the concentrated suspension regime based on an effective me-dium approach.16From here onward, we skip the

具 典

notation in the interest of simplicity and convenience when no confu-sion can be caused共see Fig. 3兲.

Equation 共D15兲 of Appendix D provides the EM equa-tions as

⫺ d␾

兵1⫺␾其⫽ d Pm

再

⳵P

⳵␾␾⫽0

冎

⫽ d␧m

再

⳵␧

⳵␾␾⫽0

冎

⫽ d␮m

再

⳵␮ ⳵␾␾⫽0

冎

⫽ dkm

再

⳵k

⳵␾␾⫽0

冎

⫽ d␣m

再

⳵␣ ⳵␾␾⫽0

冎

⫽ ddhm

再

⳵dh ⳵␾␾⫽0

冎

which is a set of six simultaneous equations.

The equality linking k_m and ␣_m can be integrated to yield a first integral

␺1⫽

␣i⫺␣m

1 k_i⫺

1 k_m

, 共31兲

which can be further manipulated to obtain a relation be-tween k and␣ of the composite as

␣i⫺␣

1 k_i⫺

1 k

⫽␣i⫺␣m

1 k_i⫺

1 k_m

, 共32兲

which is a result identical to Eq.共23兲.

The equalities in Eq.共30兲that link P_m, d_hm, and␧_mcan be combined by making use of the first integral␺₁ to give

d Pm⫺␺1⌬Tddhm

再冉

⳵P

⳵␾_␾⫽0

冊

⫺␺1⌬T

冉

⳵dh ⳵␾_␾⫽0

冊冎

⫽ d␧m

再

3␧m

␧i⫺␧m

␧i⫹2␧m

冎

,

共33兲 where, from Appendix D,

⳵P

⳵␾_␾⫽0⫽共Pi⫺Pm兲

冉

3␧m

␧i⫹2␧m

冊

⫹

冉

4␮_m 3ki ⫺

4␮_m 3km

1⫹4␮m 3ki

冊

⫻␺1⌬T

冉

dhm⫺dhi

3␧_m ␧i⫹2␧m

冊

,

and

⳵d_h

⳵␾_␾⫽0⫽dhm

冉

␧i⫺␧m

␧i⫹2␧m⫺

1⫹4␮m 3k_m

1⫹4␮m 3ki

冊

⫹dhi

冉

3␧_m ␧i⫹2␧m

1⫹4␮m 3k_m

1⫹4␮m 3ki

冊

.

Manipulation of Eq.共33兲yields

d„Pm⫺␺1⌬Tdhm…

Pi⫺Pm⫺␺1⌬T共dhi⫺dhm兲

⫽ d␧m

␧i⫺␧m

,

which may be integrated to get another first integral as

␺2⫽

Pi⫺Pm⫺␺1⌬T共dhi⫺dhm兲

␧i⫺␧m

, 共34兲

which then yields

P_i⫺P⫺␺₁⌬T共d_hi⫺d_h兲

␧i⫺␧ ⫽

P_i⫺P_m⫺␺₁⌬T共d_hi⫺d_hm兲 ␧i⫺␧m

,

共35兲

and thus the total polarization may be solved in the form

P⫽Pi

冉

␧⫺␧m

␧i⫺␧m

冊

⫹Pm

冉

␧i⫺␧

␧i⫺␧m

冊

⫹

冉

␣i⫺␣m

1 k_i⫺

1 k_m

冊

⫻⌬T

冉

dh⫺dhi

␧⫺␧m

␧i⫺␧m

⫺dhm

␧i⫺␧

␧i⫺␧m

冊

. 共36兲

Hence, the pyroelectric coefficients can be obtained from Eq. 共36兲to give the primary pyroelectric coefficient

␳1⫽␳i

冉

␧⫺␧m

␧i⫺␧m

冊

⫹␳m

冉

␧i⫺␧

␧i⫺␧m

冊

, 共37兲

and the secondary pyroelectric coefficient

␳2⫽

冉

␣i⫺␣m

1

ki

⫺ 1

km

冊

冉

d_h⫺d_hi␧⫺␧m ␧i⫺␧m

⫺d_hm ␧i⫺␧ ␧i⫺␧m

冊

. 共38兲

Here, again, we have neglected the temperature dependence of the material parameters for the reason indicated in the earlier section. The effective dielectric permittivity␧in Eqs. 共37兲and共38兲can be obtained by considering the equality of Eq. 共30兲that relates␧mand␾:

⫺_兵1d_⫺␾_␾其⫽ d␧m

再

3␧_m ␧i⫺␧m ␧i⫹2␧m

冎

, 共39兲

which can be easily integrated to give

冉

␧i⫺␧

␧1/3

冊

⫽共1⫺␾兲

冉

␧i⫺␧m

␧m

1/3

冊

, 共40兲

a result identical to the well-known Bruggeman21 formula. An exact expression for the effective piezoelectric coef-ficient dh is not easily obtainable from Eq.共30兲. However,

[image:6.612.80.272.52.153.2]

Wong et al.22have derived an explicit formula for dh as

FIG. 3. Illustration of the effective-medium scheme for a composite system. Spherical inclusions are divided into two groups of ‘‘dark’’ and ‘‘bright’’ color embedded in matrix phase of effective permittivity␧m. The darker

spheres occupy a fraction␾2by volume in the composites, surrounded by a

‘‘matrix’’ material which by itself is a composites material with a fraction of

dh⫽

1

␾

冉

␧⫺␧m

␧i⫺␧m

冊

冉

1

k⫺ 1

km

1

ki⫺

1

km

冊

dhi⫹

1 共1⫺␾兲

冉

␧i⫺␧

␧i⫺␧m

冊

⫻

冉

1

ki

⫺1

k 1

ki⫺

1

km

冊

dhm, 共41兲

where k is the bulk modulus as expressed in Eq. 共22兲. Substituting Eq.共41兲and Eq.共22兲into Eq.共38兲, the sec-ondary pyroelectric coefficient becomes

␳2⫽

冉

␣i⫺␣m

3

4␮m⫹ ␾

km⫹

1⫺␾ ki

冊

⫻

冉

␾ ␧i⫺␧

␧i⫺␧m

dhm⫺共1⫺␾兲

␧⫺␧m

␧i⫺␧m

dhi

冊

, 共42兲

where the dielectric permittivity ␧is described by Eq.共40兲.

III. RESULTS AND DISCUSSION

To illustrate the pyroelectricity of ferroelectric 0-3 com-posites, the biphasic system comprising ferroelectric ceramic and poly共vinylidene fluoride–trifluoroethylene兲 copolymer 关P共VDF–TrFE兲兴is chosen. Lead zirconate titanate共PZT兲and lead titanate 共PT兲 are the ceramic phases considered. PT is chosen as another ceramic in the present study with the pur-pose of comparison with PZT because the piezo and me-chanical properties of PT such as piezoelectric coefficient, bulk modulus, and shear modulus are larger than those of PZT ceramic. The pyroelectricity of the 0-3 composites are investigated theoretically under various polarization configu-rations, particularly for the secondary pyroelectric effect. The investigations were performed theoretically for two types of composites:共i兲copolymer-matrix composites共Figs. 4 and 5兲 and共ii兲 ceramic-matrix composites共Figs. 6 and 7兲. In prac-tice, a composite23system prepared by making use of

‘‘cor-als’’ as a template to produce a ceramic replica and then filling its pores with polymer might be a way of realizing a ceramic-matrix composite with polymer inclusions. The ceramic-matrix composite might be an interesting system to study, particularly for its secondary pyroelectric effect.

In order to illustrate and differentiate the dilute suspen-sion 共MW兲 and concentrated suspension 共EM兲 models, the experimental results of PZT/P共VDF–TrFE兲 共Ref. 24兲 关 ma-trix: P共VDF–TrFE兲; inclusion: PZT兴up to a high loading of 0.6 volume fraction of PZT inclusions are compared with our theoretical predictions. The physical parameters adopted in the calculations are listed in Table I.25The Young’s modulus Y and Poisson’s ratio ␯ of PT are 126.7 GPa and 0.22, re-spectively, as given by Ref. 21. Following the relation k

⫽Y /3(1⫺2␯) and␮⫽Y /2(1⫹␯) for the bulk modulus and shear modulus, the values for k and␮ are found to be 75.4 and 51.9 GPa, respectively. By using equation dh⫽d33 ⫹2d31 关d33⫽94.0 pCN⫺1 and d31⫽⫺9.5 pCN⫺1 共Ref.

22兲兴, the hydrostatic piezoelectric coefficient d_h for PT is calculated as 75.0 pCN⫺1. The poling ratios for the constitu-ents are for simplicity assumed to be 0.65 for both ceramic and polymer. For the composite system PZT/P共VDF–TrFE兲, the poling ratios are also taken as 0.65 for the convenience of discussion and comparison.

[image:7.612.99.518.555.703.2]

Figures 4 and 5 show the calculated total pyroelectric and secondary pyroelectric coefficient, respectively, for com-posite systems of PZT and PT dispersed in the P共VDF– TrFE兲copolymer matrix. It is found that generally the mag-nitudes associated with the secondary pyroelectric activity are only a small fraction of the total activity. The comparison of experimental results with our theory is given in Fig. 4共a兲. The EM prediction shows a good agreement with the experi-mental data up to a higher volume fraction of 0.6共with most data points falling within the EM calculated lines兲while the MW approach also shows a good accuracy in prediction for volume fraction up to 0.3. This comparison shows that the effective pyroelectric coefficient derived based on the EM approach results in a better agreement with known experi-mental results than the MW method.

From Fig. 4, the共negative兲pyroelectric coefficient of the composites increases with increasing volume fraction␾, in-dicating the contribution of the polarized ceramic phase in all the three different polarization configurations considered:共i兲 the constituent phases polarized in parallel 共‘‘parallel’’ con-figuration兲,共ii兲the constituent phases polarized in antiparal-lel directions共‘‘antiparallel’’ configuration兲, and共iii兲only the ceramic phase polarized 共‘‘ceramic-only’’ configuration兲. Composites polarized in the ‘‘parallel’’ configuration possess the highest pyroelectricity for all ceramic volume fraction. For PZT/P共VDF–TrFE兲 composites, the EM prediction shows that for the three configurations the total⫺␳ increases gradually up to volume fraction of 0.6 and thereafter in-creases significantly with increasing␾. However, the calcu-lations from the MW method indicate a rapid enhancement of⫺␳ only beyond␾⫽0.8. For the PT/P共VDF–TrFE兲 com-posites shown in Fig. 4共b兲, both models also predict a mono-tonic rise in the total⫺␳value throughout the whole ceramic volume fraction range. An interesting feature that can be ob-served in both PZT and PT composite systems for the ‘‘an-tiparallel’’ configuration is that the magnitude of the pyro-electric activity ␳ is suppressed gradually with increasing volume fraction of ceramic inclusions. Until the pyroactivity

totally vanishes at a ‘‘critical’’ nonzero volume fraction 关␾C⬇0.5共PZT兲and␾C⬇0.4共PT兲for EM;␾C⬇0.6共PZT兲

and ␾C⬇0.5 共PT兲 for MW兴 before enhancing with further

increase in ␾. It is noted that PZT/P共VDF–TrFE兲 loses its pyroelectricity at higher ceramic concentration ␾ than PT/ P共VDF–TrFE兲 composites, while the hand-waiving one would be tempted to believe the opposite because it would take more PT particles共of lower pyroelectric coefficient than PZT兲to counteract the effect of the copolymer matrix.

In general, Fig. 5 shows that the contributions of piezo-electrically induced charges in the two composite systems are not significant compared to the charges produced due to variation of polarization with temperature. The secondary pyroelectric activity in the PT/P共VDF–TrFE兲 composites is slightly larger than that of the PZT/P共VDF–TrFE兲system for all three polarization configurations. For the two systems, the secondary pyroelectric response reaches its peak at ceramic volume fraction greater than 0.6. Among the three polariza-tion configurapolariza-tions, the secondary responses of the ‘‘paral-lel’’ and ‘‘ceramic-only’’ configurations for the two compos-ite systems show a similar trend.

Figures 6 and 7 show the pyroelectric responses of com-posite systems of P共VDF–TrFE兲inclusions in the

ferroelec-FIG. 5. Calculated secondary pyroelectric coefficient␳2of ferroelectric 0-3 composites with P共VDF–TrFE兲copolymer as the matrix. Dependence of␳2on

[image:8.612.102.512.52.196.2]

the volume fraction␾of共a兲PZT and共b兲PT with P共VDF–TrFE兲as the matrix phase.共i兲Both the matrix and inclusion constituent phases polarized in parallel; 共ii兲the constituent phases polarized in antiparallel directions; and共iii兲only the ceramic phase is polarized. The dashed curves represent the MW approach predictions and solid curves are based on the EM method.

[image:8.612.98.515.574.713.2]

tric ceramic matrix. The predictions show that the pyroelec-tric response decreases monotonically with increasing ␾, indicating a weakening of pyroelectric activity as the con-centration of copolymer inclusions is higher, which is ex-pected as the pyroelectricity of the copolymer is weaker than that of the two ceramics. An interesting feature is shown in Fig. 6 by the ‘‘antiparallel’’ polarization configuration in which the pyroelectric response declines with increasing co-polymer volume fraction and vanishes at some critical con-centration␾C. The elimination of pyroelectricity in the

com-posite systems near ␾C means the strengths of the primary

and secondary effects are similar, but with opposite signs. However, a higher volume fraction of copolymer inclusions is required to cancel the pyroelectricity of the PZT-matrix compared to PT-matrix composite system. This is because the pyroelectric response of PZT is much stronger than that of PT. Upon further increasing the volume fraction of co-polymer inclusions, the pyroelectric activity of the PT-matrix composite is increased in an opposite direction and

ap-proaches a minimum around␾⫽0.85关Fig. 6共b兲兴before turn-ing back. A similar trend is also observed for the ceramic-only polarization configuration, as shown in Fig. 6共b兲.

Compared to the copolymer-matrix composites in Fig. 5共b兲, the secondary pyroelectric response for ceramic-matrix composite systems exhibits negative values in all the three polarization configurations. The secondary pyroelectric ac-tivity is stronger than that of the conventional copolymer-matrix systems. A feature noted in Fig. 7 is that the second-ary pyroelectric activity of the PT-matrix composite system dominates at a certain range of copolymer volume fraction. This clearly demonstrates that the charge induced due to pi-ezoelectricity resulting from thermal deformation is stronger than that arising from polarization changes at that particular concentration region.

IV. CONCLUSION

Analytical expressions for the pyroelectric activities of ferroelectric 0-3 composites have been developed using two different methods: the MW approach and the EM method. 0-3 composites consisting of ferroelectric ceramics and P共VDF–TrFE兲are adopted for the present study. For the ce-ramic phases, PZT and PT are considered in this investiga-tion owing to their difference in piezo and mechanical prop-erties, which are the important factors for secondary pyroelectric effects. The experimental results for the copolymer–matrix composites of PZT/P共VDF–TrFE兲 共Ref. 24兲are used to compare with the calculation results from the two theoretical approaches. The predictions from EM show a better agreement with experimental data up to higher volume fraction of inclusion than that from the MW model.

[image:9.612.109.507.52.195.2]

Two different kinds of 0-3 composites have been exam-ined theoretically: 共a兲copolymer-matrix composites and共b兲 ceramic-matrix composites. The copolymer-matrix compos-ite is the conventional composcompos-ite system comprising copoly-mer and ceramic as matrix and inclusion, respectively. Ceramic-matrix composites are a special type of composites consisting of ceramics as matrix phase and copolymer as inclusions. For copolymer-matrix composites, the results show that PZT-inclusion composites 共i.e., composites with

FIG. 7. Calculated secondary pyroelectric coefficient␳2of ferroelectric 0-3 composites with copolymer as the matrix. Dependence of␳2 on the volume

[image:9.612.52.298.554.692.2]

fraction␾of P共VDF–TrFE兲with共a兲PZT and共b兲PT as the matrix phase.共i兲Both the matrix and inclusion constituent phases polarized in parallel;共ii兲the constituent phases polarized in antiparallel directions; and共iii兲only the ceramic phase is polarized. The dash curves represent the MW predictions and solid curves are based on the EM method.

TABLE I. Physical parameters for P共VDF–TrFE兲, PZT, and PT phases adopted in the numerical calculations for the composites as shown in Figs. 4 –7.

P共DVF–TrFE兲a _PZTa _PTb,c,d

␳e

(␮Cm⫺2K⫺1)

⫺16.71 ⫺346.45 ⫺104.00b

k

共GPa兲 3.42 62.30 75.42

c

␮

共GPa兲

0.73 27.10 51.93c

␣f

(⫻10⫺6_K⫺1₎

74.50 3.90 3.57d

dh

(pCN⫺1₎ ⫺

9.30 34.00 75.00c

␧ 9.50g _{1116.00 140.00}b

a_{Reference 24.} b_{Reference 19.} c

Reference 22.

d_{Reference 25.}

e_{The values are adopted by assuming a poling ratio of 0.65.} f_{Linear thermal expansion coefficient.}

PZT inclusions兲 exhibit a higher pyroelectric effect than copolymer-matrix composites with PT as inclusions. Com-pared to PZT-inclusion composites, the secondary pyroelec-tric coefficient for PT-inclusion composites is larger. This is expected as the piezo and mechanical properties of PT are higher than those of PZT ceramic. In general, the results show that the secondary effect is only a small fraction of the total pyroelectric effect in the copolymer-matrix composites. This indicates that the pyroelectric activities in copolymer-matrix composites are dominated by the primary pyroactivi-ties. In ceramic-matrix composites, the results show that the secondary effect is larger than the copolymer-matrix com-posites. Although the secondary pyroelectric coefficient of the composites can be enhanced in ceramic-matrix geometry, the composites 共the ceramic-matric composites兲still exhibit a dominant primary pyroactivity, similar to the copolymer-matrix composites.

More interesting results can be found by examining the pyroelectric responses of the ferroelectric 0-3 composites un-der three different polarization configurations, particularly with the constituent phases poled in antiparallel directions. For the ‘‘antiparallel’’ configuration, the piezoelectricities of the two constituent phases are enhanced whereas the pyro-electric activities are partially eliminated. For instance, the contributions of the two constituent phases共ceramic and co-polymer兲 to the effective pyroelectric coefficient for the copolymer-matrix composite are neutralized at a ‘‘critical’’ nonzero volume fraction of ␾_C⬇0.4 for PT or␾_C⬇0.5 for PZT 共based on the predictions from the EM method兲. Pyroelectric-compensated piezoelectric composite materials are predicted to occur at a lower volume fraction of inclu-sions for copolymer-matrix composites than for ceramic-matrix composites.

In this study, we are concerned only with the pyroelec-tric response at a temperature well below the Curie tempera-ture of the two constituents. At temperatempera-tures near the transi-tion temperature of P共VDF–TrFE兲,19,20 its dielectric permittivity exhibits a strong dependence on temperature.19,20A detailed theoretical study on the pyroelec-tric effect by taking into account the temperature dependence of dielectric permittivity will be performed in a later publication.

ACKNOWLEDGMENT

This work was supported by the Centre for Smart Mate-rials and an internal research grant of The Hong Kong Poly-technic University.

APPENDIX A

Consider a system comprising a single dielectric sphere of radius R in an infinite ferroelectric matrix under an ap-plied field in the z direction. We assume that there are no surface charges on the sphere and that hydrostatic stresses may be present in the constituents. We write the electric dis-placement D of the inclusion (i) and matrix (m) as Di⫽Pi

⫹␧iEi⫹dhi␴i and Dm⫽Pm⫹␧mEm⫹dhm␴m where the

po-larizations P and stresses␴are assumed uniform, which may be denoted by their average values

具 典

. E, ␧, and dh denote

the electric field, permittivity, and hydrostatic piezoelectric coefficients. Under these simplifying assumptions, we need to solve the Laplace’s equation

ⵜ2_␸

l共r,␪兲⫽0, l⫽i,m, 共A1兲

where ␸i(r,␪) is the electric potential for r⬍R inside the

spherical inclusion and ␸m(r,␪) is the potential for r⬎R

outside of the inclusion. Coordinates r and ␪ represent the distance from the center of the sphere and azimuthal angle, respectively, ␪⫽0 being the z axis.

The Laplace’s equations are subject to the following conditions:

共i兲 at r⫽0, ␸i(0,␪) is finite; 共A2兲

共ii兲 by assuming the field in the matrix sufficiently far away from the inclusion is uniform and denoted by Em_⬁ at

r→⬁, ␸m共r→⬁,␪兲⫽⫺Em⬁r cos␪; 共A3兲

共iii兲 at r⫽R, the boundary conditions for the continuity of the electric potential is

␸i共R,␪兲⫽␸m共R,␪兲, 共A4兲

and for the normal component of the electric displace-ment is

⫺␧i ⳵␸i

⳵r⫹

具

Pi

典

cos␪⫹dhi

具

␴i

典

cos␪

⫽⫺␧m

⳵␸m

⳵r ⫹

具

Pm

典

cos␪⫹dhm

具

␴m

典

cos␪. 共A5兲

From Eq.共A1兲together with the conditions of Eqs.共A2兲 and 共A3兲, the general solutions for the spherical inclusion and matrix are

␸i共r,␪兲⫽

兺

n_⫽0 ⬁

Anrn␳n共cos␪兲, 共A6兲

and

␸m共r,␪兲⫽⫺Em⬁r cos␪⫹

兺

n⫽0

⬁ _C

n

rn⫹1␳n共cos␪兲, 共A7兲

where␳n(cos␪) are Legendre polynomials.

Using Eqs. 共A6兲 and共A7兲together with boundary con-ditions共A4兲and共A5兲, we obtain

A1⫽⫺

冉

3␧m

␧i⫹2␧m

冊

Em⬁

⫹

具

Pi

典

⫺

具

Pm

典

⫹dhi

具

␴i

典

⫺dhm

具

␴m

典

␧i⫹2␧m

, 共A8兲

C1⫽R3

再冉

␧i⫺␧m

␧i⫹2␧m

冊

Em⬁

⫹

具

Pi

典

⫺

具

Pm

典

⫹dhi

具

␴i

典

⫺dhm

具

␴m

典

␧i⫹2␧m

冎

, 共A9兲

An⫽Cn⫽0 for n⫽1. 共A10兲

Substituting Eqs.共A8兲–共A10兲into Eq.共A6兲, the electric field in the inclusion is found to be independent of (r,␪),

thus

具

E_i

典

⫽E_i. E_m⬁ is likewise obtained from Eq. 共A7兲, which readily verifies that

具

E_m

典

⫽E_m⬁ by averaging over a spherical matrix volume r⬎R outside the inclusion. The ex-pression for the field in the inclusion is thus

具

E_i

典

⫽„␧m

具

Em

典

⫹

具

Pm

典

⫹dhm

具

␴m

典

…⫺„

具

Pi

典

⫹dhi

具

␴i

典

…⫹2␧m

具

Em

典

␧i⫹2␧m

, 共A11兲

which can be rewritten in terms of the average dielectric displacement and electric field as

具

D_i

典

⫹2␧_m„

具

E_i

典

⫺

具

E_m

典

…⫽

具

D_m

典

, 共A12兲 where

具

Di

典

⫽␧i

具

Ei

典

⫹

具

Pi

典

⫹dhi

具

␴i

典

,

and

具

Dm

典

⫽␧m

具

Em

典

⫹

具

Pm

典

⫹dhm

具

␴m

典

.

APPENDIX B

Consider a spherical inclusion of radius r⫽a embedded in a continuous matrix. The center of the sphere is chosen as the origin of spherical coordinates (r,␪,␾). Let us consider the system subjected to hydrostatic stresses. Hydrostatic stresses are independent of␪and␾, thus shear stresses van-ish and␴_␪␪⫽␴_␾␾. The equilibrium equation is

⳵␴rr ⳵r ⫹

2共␴rr⫺␴␪␪兲

r ⫽0, 共B1兲

and the displacement equation corresponding to stress 共B1兲 is in the form

⳵2_u

rr ⳵r2 ⫹

2

r

⳵u_rr

⳵r ⫺ 2

r2urr⫽0. 共B2兲

Rearranging the above equation results in

⳵ ⳵r

冋

⳵urr ⳵r ⫹

2

rurr

册

⫽0,

which can be solved by successive integration to give

urr⫽Ar⫹

B

r2, 共B3兲

where A and B are integration constants.

Equation共B3兲gives the general solutions for the inclu-sion and matrix as

urri⫽Air, 共B4兲

and

urrm⫽Amr⫹

Bm

r2 , 共B5兲

where subscripts i and m represent inclusion and matrix, respectively. The coefficient Bi for the inclusion vanishes

because urri(0,␪,␾) is finite at r⫽0.

The strains of the inclusion and matrix can be obtained from the displacement Eqs.共B4兲and共B5兲as

erri⫽Ai, 共B6兲

and

errm⫽Am⫹

B_m

r3 . 共B7兲

The corresponding radial stresses for the inclusion and ma-trix are found to be

␴rri⫽3kiAi, 共B8兲

and

␴rrm⫽3kmAm⫺4␮m

Bm

r3 , 共B9兲

where k and ␮ are bulk modulus and shear modulus, respectively.

The constants of integration are subject to boundary con-ditions as below:

at r⫽a, uri⫽urm,

and 共B10兲

␴rri⫽␴rrm.

Equations 共B4兲and共B5兲together with boundary conditions 共B10兲give

Ai⫽Am⫹

Bm

a3, 共B11兲

and Eqs. 共B8兲 and 共B9兲 together with boundary conditions 共B10兲give

3kiAi⫽3kmAm⫺4␮m

B_m

a3 . 共B12兲

Solving Eqs.共B11兲and共B12兲yields

3kmAm⫽3kiAi⫺4␮m共Ai⫺Am兲. 共B13兲

The stress–strain relationship can be obtained from Eqs. 共B6兲to共B9兲as

具

␴m

典

⫽

具

␴i

典

⫹

4

3␮m共

具

ei

典

⫺

具

em

典

兲, 共B14兲

where the volumetric strains for matrix and inclusion are

APPENDIX C

Consider the following manipulations in which

具

D

典

,

具

E

典

, and 具␴典 are the volume averaged dielectric displace-ment, electric field, and hydrostatic stress, respectively; sym-bols with subscripts i and m are associated with inclusion and matrix, respectively, and those without usually refer to the composite. We first consider the volume-averaged quantity

I1⬅

具

D⫺Pm⫺␧mE⫺dhm

典

⫽

具

D

典

⫺

具

P_m

典

⫺␧_m

具

E

典

⫺d_hm

具

␴

典

. 共C1兲

Since P_mhas been assumed uniform, it is the same as

具

P_m

典

. I₁can also be expressed in terms of the volume of the com-posite V and of inclusion V_ias

I1⫽

1

V

冕

V„

D⫺Pm⫺␧mE⫺dhm␴…dV,

⫽_V1

冕

V_i„

D⫺Pm⫺␧mE⫺dhm␴…dV, 共C2兲

⫽Vi

V„

具

Di

典

⫺

具

Pm

典

⫺␧m

具

Ei

典

⫺dhm

具

␴i

典

…,

where it is noted that the integral over the matrix volume is zero because

具

Dm

典

⫽

具

Pm

典

⫹␧m

具

Em

典

⫹dhm

具

␴m

典

.

Equation共C1兲then yields

具

D

典

⫺

具

Pm

典

⫺␧m

具

E

典

⫺dhm

具

␴

典

⫽␾„

具

Di

典

⫺

具

Pm

典

⫺␧m

具

Ei

典

⫺dhm

具

␴i

典

…, 共C3兲

where ␾⬅Vi/V is the volume fraction of the inclusion

phase.

Similarly, the volume-averaged quantity

I2⬅

具

␴⫺kme⫺km␣m⌬T

典

⫽

具

␴

典

⫺km

具

e

典

⫹km␣m⌬T,

共C4兲 may be expressed as

I2⫽

1

V

冕

V„␴⫺

kme⫹km␣m⌬T…dV,

⫽1

V

冕

V_i„␴⫺

k_me⫹k_m␣_m⌬T…dV, 共C5兲

⫽Vi

V„

具

␴i

典

⫺km

具

ei

典

⫹km␣m⌬T…,

where, again, it is noted that the integration over the matrix volume gives zero as

具

␴m

典

⫽km

具

e

典

⫹km␣m⌬T. Then, Eqs.

共C4兲and共C5兲yield

具

␴

典

⫺km

具

e

典

⫹km␣m⌬T⫽␾„

具

␴i

典

⫺km

具

ei

典

⫹km␣m⌬T….

共C6兲

APPENDIX D

We now consider a composite system consisting of in-clusions with total volume fraction ␾ in a matrix material. For the convenience of illustration, the inclusions are divided into two groups, which are represented in Fig. 3 as bright and dark spheres. The dark spheres occupy共say兲a fraction␾2of

the composite and they are dispersed in a ‘‘matrix’’ environ-ment, which is itself a composite material made up of bright spheres of 共say兲 fraction ␾₁ in pure matrix materials. The total volume fraction␾of the inclusions expressed in terms of ␾1 and␾2 is

␾⫽共1⫺␾2兲␾1⫹␾2. 共D1兲

Let the formulas for calculating the overall or effective polarization, dielectric permittivity, piezoelectric coefficient, bulk modulus, shear modulus, and thermal expansion coeffi-cient of the composite be represented by functions of the constituent properties and the inclusion fraction as below:

P⫽P共Pi, Pm,␧i,␧m,ki,km,␮i,␮m,␣i,␣m,dhi,dhm,␾兲,

共D2兲

␧⫽␧共␧i,␧m,␾兲, 共D3兲

dh⫽dh共␧i,␧m,ki,km,␮i,␮m,dhi,dhm,␾兲, 共D4兲

k⫽k共ki,km,␮i,␮m,␾兲, 共D5兲

␮⫽␮共ki,km,␮i,␮m,␾兲, 共D6兲

and

␣⫽␣共ki,km,␣i,␣m,␾兲, 共D7兲

where the functions P, ␧, dh, k, ␮, and␣ are to be

deter-mined by the EM approach. In writing down these equations, the relevant independent variables are decided by referring to Eqs.共22兲,共23兲,共25兲,共26兲, and共27兲for the dilute suspension case. We refer, e.g., to the Hashin16 shear modulus

␮⫽␮m

再

1

⫹_7⫺5␯ 15共1⫺␯m兲共␮i/␮m⫺1兲␾

m⫹2共4⫺5␯m兲关␮i/␮m⫺共␮i/␮m⫺1兲␾兴

冎

,

where␯mis Poisson’s ratio of the matrix phase, to obtain the

functional dependence as expressed in Eq. 共D6兲. Thus, e.g.,

␣of the composite is calculated from the knowledge of k and

␣ of the constituents.

First, we wish to simplify the rather clumpsy notation in Eqs. 共D1兲–共D7兲. Let ⌰ represent an array of properties for the composite material system and ⌰n its individual ele-ments. The symbol⌰n_{for n⫽1, 2, 3, 4, 5, and 6 correspond}

to P, ␧, d_h, k, ␮, and ␣, respectively. When⌰ is written with a subscriptᐉas⌰_ᐉ, thenᐉlabels the specific system or material in question. Thus, we denote by ⌰_i and ⌰_m the arrays of properties of the inclusion material and matrix ma-terial, respectively. Thus, Eqs.共D2兲–共D7兲can be abridged as

⌰n_⫽⌰n_共⌰

i,⌰m,␾兲 for n⫽1,2,...,6, 共D8兲

or simply as ⌰⫽⌰(⌰i,⌰m,␾), which can be written in

terms of␾1 and␾2 by using Eq.共D1兲as

⌰⫽⌰关⌰i,⌰m,共1⫺␾2兲␾1⫹␾2兴. 共D9兲

Now back to the main discussion. An effective-medium scheme can be developed by considering that the dark inclu-sions of volume fraction ␾2 in the composite 共Fig. 3兲 are

k₁,␮₁, and␣₁. According to the notation above, we denote the array of properties of the latter composite by⌰₁and that of the former composite by ⌰₂. Thus, the effective proper-ties of the original composite can be calculated as

⌰2⫽⌰共⌰i,⌰1,␾2兲. 共D10兲

Now

⌰1⫽⌰共⌰i,⌰m,␾1兲. 共D11兲

By utilizing Eq.共D11兲, Eq.共D10兲can be reexpressed as

⌰2⫽⌰关⌰i,⌰共⌰i,⌰m,␾1兲,␾2兴. 共D12兲

However, all the time we have been looking at the same composite system we started with共Fig. 3兲, only from differ-ent considerations. The assertion that the differdiffer-ent consider-ations should give the same results is the statement on which the EM scheme of calculations is based. Thus, we have

⌰关⌰i,⌰m,共1⫺␾2兲␾1⫹␾2兴

⫽⌰关⌰i,⌰共⌰i,⌰m,␾1兲,␾2兴. 共D13兲

Differentiation of Eqs.共D13兲with respect to␾1and then

evaluating the results at␾1⫽0 yield six equations of similar

structure, the first one (⌰1 _{or P) of which is listed below:}

⫺⳵⌰

1

⳵␾ 兵1⫺␾其⫹

兺

n ⳵⌰1

⳵⌰m n

⳵⌰n

⳵␾_␾⫽0⫽0, 共D14兲

where use has been made of the equalities

⌰共⌰i,⌰m,0兲⫽⌰m,

describing the evident fact that the properties of a ‘‘compos-ite’’ with zero fraction of inclusion material are just the prop-erties of the matrix material.

Written in a more usual way, Eq.共D14兲 is a first-order partial differential equation of the form

⫺兵1⫺␾其⳵P

⳵␾⫹ ⳵P

⳵␾_␾⫽0 ⳵P

⳵Pm

⫹_⳵␾⳵␧

␾⫽0

⳵P

⳵␧m

⫹_⳵␾⳵k

␾⫽0

⳵P

⳵km

⫹⳵␮_⳵␾ ␾⫽0 ⳵P ⳵␮m ⫹⳵␣_⳵␾ ␾⫽0 ⳵P ⳵␣m

⫹⳵dh ⳵␾_␾⫽0

⳵P

⳵d_hm⫽0, 共D15兲

where P is the unknown function to be solved whereas the derivatives at ␾⫽0 are known functions pertaining to the dilute limit, which may be calculated from Eqs. 共22兲, 共23兲, 共25兲,共26兲, and共27兲and the adopted Hashin’s16shear modu-lus 共as mentioned earlier兲. Thus,

⳵P

⳵␾_␾⫽0⫽

再

共Pi⫺Pm兲

冉

3␧m

␧i⫹2␧m

冊

⫹

冉

4␮m/3

1⫹4␮m/3ki

冊

⫻共␣i⫺␣m兲⌬T

冉

dhm⫺dhi

3␧m

␧i⫹2␧m

冊冎

,

⳵␧

⳵␾_␾⫽0⫽

再

3␧m

␧i⫺␧m

␧i⫹2␧m

冎

,

⳵k

⳵␾_␾⫽0⫽

再

km

ki共

k_i⫺k_m兲

冉

1⫹4␮m/3km 1⫹4␮m/3ki

冊冎

,

⳵␮ ⳵␾_␾⫽0⫽

再

15共1⫺␯_m兲共␮_i⫺␮_m兲␮_m 共7⫺5␯m兲␮m⫹2共4⫺5␯m兲␮i

冎

,

⳵␣

⳵␾_␾⫽0⫽

再

共␣i⫺␣m兲

冉

1⫹4␮m/3km

1⫹4␮_m/3k_i

冊冎

,

and

⳵d_h

⳵␾_␾⫽0⫽

再

dhm

冉

⫺

1⫹4␮_m/3k_m 1⫹4␮m/3ki⫹

␧i⫺␧m

␧i⫹2␧m

冊

⫹d_hi

冉

3␧m ␧i⫹2␧m

冊冉

1⫹4␮m/3km

1⫹4␮m/3ki

冊冎

.

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