Electrical conductivity enhanced dielectric and piezoelectric properties of ferroelectric 0-3 composites

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1 JOURNAL OF APPLIED PHYSICS 97, Electrical conductivity enhanced dielectric and piezoelectric properties of ferroelectric 0-3 coposites C. K. Wong a Departent of Applied Physics, he Hong Kong Polytechnic University, Hong Kong, China F. G. Shin Departent of Applied Physics, Materials Research Center and Center for Sart Materials, he Hong Kong Polytechnic University, Hong Kong, China Received 8 Septeber 2004; accepted 29 Deceber 2004; published online 14 March 2005 We have investigated the effects of electrical conductivity of the constituents on the dielectric and piezoelectric properties of ferroelectric 0-3 coposites. he tie-dependent internal electric fields are first derived, which can be induced by an applied ac field in dielectric easureent or stress in piezoelectric easureent. Our previously developed odel C. K. Wong, Y. M. Poon, and F. G. Shin, J. Appl. Phys. 90, has been extended to include the additional contribution fro the electrical conductivities and the frequency of easureent, which can be significant for ceraic/polyer coposites possessing high conductivity in the atrix phase. he odel provides an explanation to the surprisingly high piezoelectric d 33 values reported by, e.g., Chen et al. Sens. Actuators, A 65, Explicit expressions for the transient and steady-state responses are given and the effective perittivity, d 33, d 31, and d h coefficients have been derived Aerican Institute of Physics. DOI: / I. INRODUCION a Electronic ail: wongck.a@polyu.edu.hk One of the priary goals of ebedding piezoelectric ceraic particles within a polyer atrix i.e., 0-3 coposite is to cobine the better properties of ceraic and polyer. Ferroelectric ceraics have high piezoelectric properties, but their poor echanical properties and the large difference in acoustic ipedance with water restrict their usage for certain applications. Ferroelectric polyers, on the other hand, have good echanical flexibility, but their piezoelectric d values are low. he use of ferroelectric coposites sees capable of overcoing these deficiencies and their properties can be tailored for specific applications. One ay in principle design a 0-3 coposite with high ceraic volue fraction for applications necessitating high piezoelectric d values. However, it is difficult to fabricate a 0-3 coposite saple of high ceraic concentration and the high ceraic content will also lower the flexibility of the coposite. o iprove the piezoelectric as well as pyroelectric properties of 0-3 coposites and also for facilitating poling, Sakaoto et al. attepted to fabricate graphite doped lead zirconate titanate/polyurethane PZ/PU 0-3 coposites. 1 By adding about 1% by volue of graphite to increase the electrical conductivity of PU, both the pyroelectric and piezoelectric d 33 coefficients of the coposite increased. Indeed, the effect of higher conductivity in the polyer atrix is not liited only to enhancing the pyroelectric and piezoelectric properties. Recently we find that, in the poling process of ferroelectric coposites, it shortens the tie developent of the electric field acting on the ceraic to reach saturation, therefore, poling of the ceraic phase in such coposite systes can be ore efficient, 2,3 which is of practical significance since the pyroelectric and piezoelectric activities of the constituents are generally proportional to their degree of poling. 3,4 In another report, Chen et al. investigated the influence of solvent treatent on the piezoelectric and dielectric properties of a ferroelectric 0-3 coposite. 5 hey iersed the coposite saple of lead zirconate titanate/polyvinylidene fluoride PZ/PVDF in a certain solvent e.g., N-diethylacetaide DMA for a period of tie. he piezoelectric and dielectric properties were easured after the saple was reoved fro the solvent environent. hey observed that both the perittivity and d 33 coefficient were one to five ties larger than those of the virgin specien. Such high values are quite likely to be out of reach of the existing odel predictions for noral dielectric and piezoelectric 0-3 coposites. Chen et al. suggested that the polyer was swollen by the soaking solvent and the swollen coposite would then transfor fro 0-3 to 1-3 type, hence the enhanced d 33 value. 5 We believe that these enhanceents could very well be related to the increent of electrical conductivity in the atrix aterial brought about by the solvent. Actually, the use of different solvent treatents with varying concentration has been a popular technique to increase the conductivity of polyer electrolytes, in which a considerable aount of research work has been done due to the potential application in solid-state batteries, etc his work attepts to investigate the effect of electrical conductivity on the dielectric and piezoelectric properties of a ferroelectric coposite of a dispersion of spherical inclusions in a continuous atrix, assuing the constituents possess finite conductivity. he saple is excited by an ac electric field dielectric easureent or ac stress piezoelectric easureent of sall aplitude. An analytical odel is /2005/976/064111/9/$ , Aerican Institute of Physics Downloaded 25 Mar 2011 to Redistribution subject to AIP license or copyright; see

2 C. K. Wong and F. G. Shin J. Appl. Phys. 97, proposed and explicit expressions are derived for the dynaic behavior of the electric fields in the constituents. he steady-state solutions are then used to obtain explicit expressions for the effective perittivity, d 31, d 33, and d h coefficients. Copared to the Maxwell Wagner forula 11 for perittivity and our previously derived analytical expressions for d 31, d 33, and d h, 12 which assue perfectly insulating constituents, the set of expressions contains a factor describing the coupled effects of perittivity, conductivity, and the easuring frequency. heoretical evaluations are based on the constituent paraeters of PZ/PVDF. he surprisingly large d 33 values reported by Chen et al. can be qualitatively siulated by assuing an increase in the conductivity of PVDF after soaking in solvent. 5 We will deonstrate that this enhanceent will naturally be significant for saples with a ildly conductive atrix phase. II. HEORY o find the effective perittivity and piezoelectric d coefficients of a 0-3 coposite of two ferroelectric phases, we will first obtain the tie developent of the internal electric fields within the individual phases, given the external sinusoidal electric field and stress. hen the steady-state solution of the in-phase coponent of internal fields will be used to obtain expressions for the perittivity, d 31, d 33, and d h coefficients. A. A forulation for calculating the dynaic behavior of internal electric fields Suppose the coposite is initially polarized in the z direction, in which case we only need to be concerned with the electric field and polarization in the 3 direction. We first write the voluetric average electric displaceent D and conduction current density j for the ferroelectric constituent aterials in the coposite as 13 D 3i = i E 3i + P 3i 1 D 3 = E 3 + P 3, j 3i = i E 3i 2 j 3 = E 3, the angular brackets denote volue-averaged fields enclosed. P is polarization, and denote perittivity and electrical conductivity, respectively, and E is electric field. Subscripts i and denote inclusion and atrix, respectively. Consider the single inclusion proble of a ferroelectric sphere surrounded by a ferroelectric atrix ediu with a unifor electric field applied along the z direction far away fro the inclusion. he boundary-value proble gives the following equations: 13 D 3i +2 E 3i E 3 = D 3 q 0, 3 j 3i +2 E 3i E 3 = j 3 + q 0 /t. In Eqs. 3 and 4, we have assued that both constituent aterials are uniforly polarized and the hoogeneously polarized sphere is covered with surface charge of density q 0 at the pole along the polarizing direction =0 with a distribution given by q 0 cos. For a coposite coprising a dilute suspension of spherical particles uniforly distributed in the atrix aterial, the voluetric averages of the electric fields satisfy 12 E 3 = E 3i + 1 E 3, 5 is the volue fraction of the inclusion phase. We obtain fro Eqs E 3i t + E 3i = 3 E 3 + E 3 /t + 1 P 3 P 3i /t, i +2 = i i E 3i t + E 3i = / and = L E E 3 t + E 3, 8 B. Effective perittivity of a ferroelectric 0-3 coposite For dielectric easureent, assue E 3 =E 0 sin t =2f and f is the frequency of the applied field. We further assue that the aplitude of E 3 i.e., E 0 is sall such that the contribution fro the hysteresis behavior ay be neglected. hus, P 3 /t=p 3i /t=0 in Eq. 6 which becoes L E = i +2. Assuing E 3i is initially zero, the solution fro solving Eq. 8 is 9 E 3i = E 0/ L E e t/ sin t cos t. 10 Downloaded 25 Mar 2011 to Redistribution subject to AIP license or copyright; see

3 C. K. Wong and F. G. Shin J. Appl. Phys. 97, In a dielectric easureent, the electric current flowing in the coposite is used to deterine the perittivity value. For a coposite coprising a dilute suspension of spherical particles uniforly distributed in the atrix aterial, the voluetric average of the total current density ay be written as J 3 = j 3 + D 3 /t, j 3 = j 3i + 1 j 3 D 3 = D 3i + 1 D At sufficiently long tie fro the start of the dielectric easureent i.e., at steady state, the ter exp t/ in Eq. 10 ay be oitted. he coponents of E 3i in phase and 90 out of phase with the applied field are P3i t P 3 t = d i t = d t, 20 d i =P 3i /, d =P 3 /. When the stress is applied along the x direction, i.e., = xx, then d i =d i and d =d, d i = L + L d 31i + L d 33i, 21 and d = + d 31 + d 33, L = I 1 1 3I J 1 1 3J, E 3i in phase = E 0/ L E 1+ 2 sin t, E 3i out of phase = E 0/ L E cos t. 13 L = I 1 1 3I + 2J 1 1 3J, In this work, we only focus on the in-phase coponent of the dielectric property. he perittivity is J 3 dt in phase =. 15 E 3 Substituting Eqs into Eq. 15 and aking use of Eqs. 1, 2, and 5, the effective perittivity is obtained as = + L E i, 16 = neg 1 / , 17 neg = i, i or in a ore failiar for, 18 i i =. 19 i +2 i Note that when i = =0, =1 and Eq. 19 reduces to the well-known Maxwell Wagner forula. 11 C. Effective piezoelectric coefficients of a ferroelectric 0-3 coposite In the piezoelectric easureent, the polarizations P 3i and P 3 vary with the applied stress piezoelectricity. Suppose the coposite is subjected to an external tensile stress. he rates of change of polarizations in Eq. 6 can be related to the external stress due to piezoelectric effect. hus see Appendix, = L 1 3 = I J, I = 1 3 = 1 L 1 3 = I J, k i k 3k +4 3k i +4, J = 5 3k +4 i k +2 i + 9k +8 Here d 31 and d 33 are the piezoelectric coefficients. k and denote bulk odulus and shear odulus, respectively. When the stress acts along the z direction, i.e., = zz, then d i =d i and d =d, d i = L d 33i +2L d 31i, 29 d = d d In short-circuit condition i.e., E 3 =0, Eq. 6 becoes E 3i t + E 3i 1 d d i = i +2 t. 31 Assuing = 0 sin t with 0 being the aplitude of applied stress and E 3i is initially zero, the solution fro solving Eq. 31 is E 3i = d d i i +2 sin t + cos t e t/, 32 =2f and f is the frequency of the applied stress. Fro Eq. 5, E 3 = E 3i /1, since E 3 =0 in the piezoelectric easureent. Downloaded 25 Mar 2011 to Redistribution subject to AIP license or copyright; see

4 C. K. Wong and F. G. Shin J. Appl. Phys. 97, For a sufficiently long tie in the piezoelectric easureent i.e., at steady state, the ter exp t/ in Eq. 32 ay be oitted. he coponents of E 3i in phase and 90 out of phase with the applied stress are E 3i in phase = d d i sin t i +2, 33 E 3i out of phase = d d i cos t i Again, we only focus on the in-phase coponent of the piezoelectric responses. he piezoelectric d coefficient of the coposite obtained fro the easureent is J 3 dt in phase d =. 35 Substituting Eq. 33 into Eqs. 1, 2, and 12, we obtain fro Eq. 35, d = d i + 1 d + 1 d L E L Ed i d, 36 d = / neg , 37 L E = 1 L E 1 = i i Further, using Eqs. 21, 22, 29, and 30, the effective d 31 and d 33 coefficients can be reexpressed as d 31 = 1 d 1 L E L + L d 31i + L d 33i d 1 L E + d 31 + d 33, d 33 = 1 d 1 L E L d 33i +2L d 31i d 1 L E d d he effective hydrostatic piezoelectric d h coefficient is defined by d h =d 33 +2d 31 and siilarly for inclusion and atrix, thus, d h = 1 d 1 L E L h d hi d 1 L E h d h, 41 Note that when i = =0, d =1 and Eqs reduce to the expressions of our previous odel. 12 D. Effective piezoelectric coefficients of a coposite in ters of effective, k, and he foregoing Eqs are calculated for the dilute suspension regie. One can reexpress the L E s and L s Eqs. 9, 37, 23 26, 42, and 43 in ters of the effective dielectric and elastic properties of the coposite as in Ref. 12. L E ay be directly reexpressed fro Eq. 16 and relabeled E, E = i A siilar derivation can also be ade for reexpressing L E. Using Eqs. 1 5 and again assuing P 3 /t =P 3i /t=0, we can write E 3 t + E 3 = L E E t + E pos, 45 pos = i +2 / i +2. he solution of Eq. 45 for the sae sinusoidal input is E 3 = E 0/ pos L E pos e t/ pos sin t pos cos t. 46 We then follow the sae footsteps as before in deriving Eq. 16 and finally obtain = 1 L E i, = pos neg 1 / pos and L E is as given by Eq. 38. After solving for L E by rearranging Eq. 47 and relabeling by E, we obtain 1 i E = i Concerning L s, we can adopt the sae transfored expressions called F s as in Ref. 12. his technique has been deonstrated there to give results which are applicable to higher, provided that better estiates of effective properties are available. his schee provides siple and tractable explicit expressions for higher. hus Eqs are transfored to L h =2L 3k +4 k i + L =, 3k +4 k i +41 k h =2 + = 1 L h 1 = 3k i +4 k 3k +4 k i +41 k d 31 = 1 d 1 E F + F d 31i + F d 33i d 1 EF + F d 31 + F d 33, d 33 = 1 d 1 E F d 33i +2F d 31i d 1 EF d 33 +2F d 31, 51 Downloaded 25 Mar 2011 to Redistribution subject to AIP license or copyright; see

5 C. K. Wong and F. G. Shin J. Appl. Phys. 97, d h = 1 d 1 E F h d hi d 1 EF h d h, F = F k 1 1 k k 1 1 i k i 1, = 1 1 k 1 1 k 3 k 1 1 i k i 1, F = F 1 = k 1 i k 1 3 k 1 1 i k 1 1 i i 1, F = 1 F 1 = 1 F h = 1 k 1 1 k k 1 1 i k, k 1 i k 1 3 k 1 1 i k i i 1, 56 F h = 1 F h 1 = 1 k 1 i k 1 1 k 1 1 i k In the above, we have assued that the value of d is unchanged. III. RESULS AND DISCUSSION A previous article 5 reported that the piezoelectric d 33 coefficient and perittivity of a PZ/PVDF 0-3 coposite would be significantly increased by solvent treatent. he saples were iersed in a certain solvent e.g., DMA for a period of tie and the piezoelectric and dielectric properties were easured after the saples were reoved fro the solvent environent. Chen et al. found that the dielectric constant increased fro 60 to 120 and the d 33 constant fro 27 to 130 pc/n. However, the elastic constant of the coposite was not changed by the solvent treatent. his suggests that the anoaly is not directly related to the structural configurational changes in the treated coposite which ay be induced by the diffusion of solvent as the authors postulated. he phenoenon is ost likely the effect of the draatic increent of electrical conductivity. Since the conductivity of solvent cobinations such as ethylene carbonate EC, propylene carbonate PC, diethylacetaide DMA, diethylforaide DMF, and diethylsulphoxide DMSO are about c 1, 9 the effective conductivity of PVDF is thought to be draatically increased, especially when the coposite saple is just taken out fro the solvent environent. On the other hand, Chen et al. also observed that, after several hours, both the anoalous dielectric and piezoelectric constants gradually decreased to their virgin values again. his is consistent with the thinking that the conductivity in the atrix phase diinishes as the solvent evaporates away. ABLE I. Properties of constituents for PZ/PVDF 0-3 coposites. / 0 a 1 c 1 Y a d 33 GPa b pc/n A. Effect of and easuring frequency on and d 33 coefficients of 0-3 coposites d 31 pc/n PZ c a 140 PVDF 12 varied a Reference 5. b Reference 12. c Reference 14. Here, we will deonstrate that high plays a significant role in the effective and d 33. he adopted properties of constituents for the calculations are listed in able I. Y and ay be transfored to k and by using k=y /3 6 and =Y /2+2. Chen et al. used a dc field of 10 MV/ to polarize their coposite. Since the coercive field of PVDF is about 45 MV/, 15 which is uch larger than the poling field adopted by Chen et al., the electric field acting in the atrix phase will be insufficient to give a high degree of reanent polarization after poling, hence the piezoelectric activities will also be liited. In another report for the piezoelectric properties of PZ/PVDF coposites, Furukawa et al. reported d 31 =0.3 pc/n for PVDF, which is negligibly sall. 16 Hence, we adopted d 33 =d 31 =0 in able I. he calculated results of Eq. 19 and d 33 Eq. 40 for PZ/PVDF 0-3 coposites of sall volue fraction are shown in Fig. 1. he easuring frequency for dielectric constant is 1 khz. 5 For coparison purpose, we assue the sae easuring frequency the highest achievable in coercial d 33 eters for d 33 constant. In Fig. 1, the solid lines are based on our previous odel, 12 which corresponds to =1 in Eq. 19 and d =1 in Eq. 40 of the present odel. A typical value of conductivity for PVDF at roo teperature is about c 1. he prediction based on this value overlaps with our previous odel. hus, the effect of electrical conductivity on the effective and d 33 of untreated saples can norally be neglected at roo teperature. When is increased, no notable increent of and d 33 is observed until = c 1. More enhanceent of and d 33 is observed at higher ceraic volue fraction. Further increent of in order of agnitude will draatically increase the value of and d 33 for a given. In Fig. 1, we deonstrate the significant effect when c 1. It shows that both and d 33 are very sensitive to a sall change of in this region. According to Chen et al., the loading of the PZ in their coposite is approxiately 55%. 5 At high ceraic concentration, the calculated values shown in Fig. 1 do not fit well with their easured values because Eqs. 19 and 40 are derived fro dilute suspension considerations, although it is already observed that can significantly enhance and d 33. Later on, we will use Eq. 51 higher volue fraction calculation for d 33 to predict the d 33 values of the experiental data of Chen et al.. In ost of the calculations shown in Fig. 1, i is taken as c 1. In Fig. 1, the predictions for i = = c 1 are also shown. It clearly reveals that the calculated and d 33 values are only slightly different Downloaded 25 Mar 2011 to Redistribution subject to AIP license or copyright; see

6 C. K. Wong and F. G. Shin J. Appl. Phys. 97, FIG. 2. he variation of Eq. 7 of PZ/PVDF coposites with the agnitude of conductivity in the atrix phase. he easuring frequency is 1kHz. FIG. 1. he variation of a perittivity Eq. 19 and b d 33 constant Eq. 40 of PZ/PVDF coposites with the agnitude of conductivity. Equations 19 and 40 with s=1 solid lines represent that the constituents are perfectly insulating. constants. Suing up, with electrical conductivity considered, the odel suggested in this article yields significant effects for coposite saples with high conductivity in the polyer atrix, but only inial effects for noral coposites possessing relatively low conductivity. hus the factors Eqs. 17 and 37 do not affect in a noticeable way the goodness of fit already obtained by any existing odels for ordinary coposite saples. For saples possessing high, the effect of the factors has to be included. fro the case of i = c 1 with the sae = c 1. Actually, we generally find that and d 33 are uch less affected by i than by. In addition, the enhanceent of and d 33 shown in Fig. 1 will actually continue for increasing until c 1. When c 1, a steady state is reached and no further notable increent of and d 33 is observed. his apparent liit will be saller for a lower easuring frequency. In Eqs. 17 and 37, is the governing factor which depends on the constituent perittivities, conductivities, and, rather than just i or. Moreover, one can also appreciate fro Eq. 7 that the i is of lesser significance than. Figure 2 shows the ceraic volue fraction dependence of with different adopted. he figure shows that decreases onotonically with for all s. hus, and d 33 increase with ceraic volue fraction, as noted in Fig. 1. Fro Fig. 2, it is found that a higher generally displaces the whole curve to lower values. Fro Eqs. 17 and 37, the s should also be sensitive to changes in the easuring frequency. Figure 3 shows the variation of and d 33 with the easuring frequency f, assuing = c 1. At frequencies above 1 Hz, no notable enhanceent is found in and d 33. Further reduction in f will have an effect siilar to increasing, resulting in significant enhanceent of and d 33. he above analyses for piezoelectricity have been confined to the d 33 constant. Actually, the enhanceent by conductivity also applies to the d 31 Eq. 29 and d h Eq. 34 FIG. 3. he variation of a perittivity Eq. 19 and b d 33 constant Eq. 40 of PZ/PVDF coposites with the easuring frequency. Equations 19 and 40 with s=1 represent that the constituents are perfectly insulating. Other lines with s1 assues that the constituents possess sall but finite conductivity. Downloaded 25 Mar 2011 to Redistribution subject to AIP license or copyright; see

7 C. K. Wong and F. G. Shin J. Appl. Phys. 97, FIG. 4. heoretical prediction by Eq. 51 is copared with the experiental data of Chen et al. see Ref. 5 for the d 33 vs of a PZ/PVDF coposite. have been assued to decay linearly with tie fro c 1 to c 1 and the easuring frequency is 200 Hz. B. Coparison with experiental data Here we attept to apply the present odel to investigate the anoalously high d 33 values reported by Chen et al. 5 hey easured the perittivity and d 33 constant after the PZ/PVDF saples were reoved fro the solvent environent and obtained very high and d 33 values which decayed with tie. heir experiental data deonstrated that the percentage increent of d 33 is significantly larger than the perittivity. he variation of d 33 with can be obtained fro the experiental data of Chen et al. and is shown in Fig. 4. We can ake use of Eq. 51 to predict this d 33 relationship, using the experientally easured and elastic constant =810 9 N 2. 5 In the absence of rigorous theory to odel the decay of with tie due to solvent evaporation, we assue that decays linearly with tie. he value of when the saple is just reoved fro the solvent environent ay be roughly estiated by Eq. 19. We use Eq. 19 to calculate the s fro the experiental given by Chen et al. and the results suggest that increases onotonically fro c 1 to c 1 when varies fro 66 0 to Note that the variation of is sall which is due to the use of Eq. 19 for the dilute suspension case. In the regie of close to 66 0 i.e., the saple has been reoved fro the solvent environent for a long tie and the effect of conductivity is negligible, the calculated will generally be significantly larger than the true value =66 0 = c 1 versus a typical value of c 1 for untreated saples. In the regie of close to i.e., the saple is just taken out fro the solvent environent and the effect of conductivity is significant, the calculated by Eq. 19 ay also be larger than the true value, but the discrepancy should not be as large as in the sall regie. It is because the value of should be high at the large regie. he calculated high value is thought to be not so affected by a high or low calculation when the effect of electrical conductivity is effectively doinating in Eq. 19. Hence, ay be considered to have a value equal to or slightly less than c 1 for the largest obtained by Chen et al.. FIG. 5. a Matheatical fit to the experiental data of Chen et al. see Ref. 5 for the d 33 vs relation of a PZ/PVDF coposite. b Calculated vs tie using Eq. 51 shows an apparent two-stage behavior: quite constant during the first four hours and linearly decays thereafter. Assue decays linearly with tie fro c 1 to c 1 during the 12-h easureent perfored by Chen et al. and their d 33 eter has a typical easuring frequency of 200 Hz. Moreover, Eq. 51 requires two elastic properties. Apart fro the elastic constant easured by Chen et al., we have also adopted 12,17 k i k k = k k i k /k +4 /3 59 to calculate the effective bulk odulus. heoretical prediction of Eq. 51 with the above considerations is shown in Fig. 4. We notice that, apart fro the rightost regie of extreely large when the saple is just taken out fro the solvent environent, the general trend of d 33 vs has been reproduced by the above siple assuption for. In Fig. 4, the dashed line is based on our previous odel, 12 which corresponds to d = = =1 in Eq. 51 of the present odel. he prediction is very far away fro the experiental data. he solid line present odel shows a draatic iproveent. It sees that the consideration of high electrical conductivity in the atrix phase is essential to understand this interesting phenoenon. he discrepancy in the large regie suggests that ay not decay linearly at the initial tie. o investigate this, we first fit the experiental data atheatically to give a sooth trend for d 33 vs see Fig. 5a. Using the fitted curve, the true behavior of versus tie is calculated by Downloaded 25 Mar 2011 to Redistribution subject to AIP license or copyright; see

8 C. K. Wong and F. G. Shin J. Appl. Phys. 97, coposites with nonpyroelectric atrix. 19 Note that when d =1 which corresponds to i = =0 and Eq. 61 reduces to the expression for priary pyroelectric coefficient suggested by Chew et al. 20 IV. CONCLUSIONS FIG. 6. heoretical prediction by Eq. 51 is copared with the experiental data of Chen et al. see Ref. 5 for the d 33 vs relation of a PZ/PVDF coposite. have been assued to decay linearly with tie fro c 1 to c 1 and the easuring frequency is 100 Hz. Eq. 51 and the result is shown in Fig. 5b. he result shows an apparent two-stage behavior: changes only slowly for the first four hours, followed by a linear decay behavior siilar to our assuption in the calculation of Fig. 4. his suggests that the initial evaporation of solvent, allegedly fro the surface regions, ay not have affected the properties in the bulk substantially because solvent diffusion fro the bulk takes longer tie. Further investigation is required with a ore sophisticated odel for solvent evaporation. Since Chen et al. have not reported the easuring frequency for their piezoelectric easureent, we have assued f = 200 Hz in the above analysis. If f = 100 Hz is adopted for our calculations, an equally good fit is obtained solid line shown in Fig. 6. For the reduced easuring frequency, a slightly lesser value at =134 0 = c 1 is adopted. his is because reducing f is effectively reducing, as we have deonstrated in Fig. 1. he present forulation neglects the loss coponents of the elastic and piezoelectric properties in the constituents. In the case that the loss coponents are also considered, all constituent paraeters involved should take on coplex values, and the final expressions would becoe uch ore coplicated. 18 A variation of the present odel is also suitable for the study of the pyroelectric coefficient of 0-3 coposites. We ay write P 3i /t= p i /t and P 3 /t= p /t p denotes pyroelectric coefficient and is the applied sinusoidal teperature. 3 he effective p can be directly rewritten fro Eq. 36 as hus p = p i + 1 p + 1 d L E L Ep i p. 60 p = 1 d 1 L E p i d 1 L Ep. 61 his equation is a generalization of the expression suggested by La et al. who investigated the pyroelectric properties of In conclusion, we have included the effect of electrical conductivity fro the constituents in the prediction of dielectric and piezoelectric properties. Explicit expressions have been derived for, d 33, d 31, and d h coefficients. High conductivity in the atrix phase can significantly enhance the perittivity and piezoelectric coefficients of ferroelectric 0-3 coposites. A high value ay be induced by solvent treatent of a coposite. he enhanced and piezoelectric d constants have saturation values which reain unchanged beyond a certain. Coparison of our theoretical results with the experiental data of Chen et al. 5 shows fairly good agreeent, when is assued to decay roughly linearly with tie due to solvent evaporation. he electrical conductivity effect is quite likely responsible for the reported anoalously high d 33 values in the work of Chen et al. ACKNOWLEDGMEN his work was partially supported by the Center for Sart Materials of he Hong Kong Polytechnic University. APPENDIX Suppose the coposite is subjected to tensile stresses in the x, y, and z directions siultaneously. he contribution in polarization due to the stresses in the constituents can be written as P 3i = d 31i xxi + d 32i yyi + d 33i zzi A1 P 3 = d 31 xx + d 32 yy + d 33 zz, d 31, d 32, and d 33 are the piezoelectric coefficients and xx represents the volue-averaged stress in the x direction within the atrix aterial, and so forth. In a previous article, we have considered the elasticity proble of a coposite with spherical inclusions subjected to external stresses xx, yy, and zz and obtained 12 xxi =L yyi L zzi xx =L yy zz L L L L L L xx L yy zz, xx yy zz, A2 A3 L, L,, and are given by Eqs Substituting Eqs. A2 and A3 into Eq. A1 gives Downloaded 25 Mar 2011 to Redistribution subject to AIP license or copyright; see

9 C. K. Wong and F. G. Shin J. Appl. Phys. 97, P 3i = L d 31i + L d 32i + d 33i xx + L d 32i + L d 31i + d 33i yy + L d 33i + L d 31i + d 32i zz. P 3 = d 31 + d 32 + d 33 xx + d 32 + d 31 + d 33 yy + d 33 + d 31 + d 32 zz A4 We assue that both constituents are piezoelectrically transversely isotropic. hus, d 31i =d 32i and d 31 =d 32 and we can write P3i t P 3 t = d i xx t = d xx t A5 for xx 0 and yy = zz =0, d i =L +L d 31i +L d 33i and d = + d 31 + d 33. For zz 0 and xx = yy =0, P3i zz = d t i, t A6 P 3 zz = d t t d i =L d 33i +2L d 31i and d = d d W. K. Sakaoto, P. Marin-Franch, and D. K. Das-Gupta, Sens. Actuators, A 100, Y.. Or, C. K. Wong, B. Ploss, and F. G. Shin, J. Appl. Phys. 93, Y.. Or, C. K. Wong, B. Ploss, and F. G. Shin, J. Appl. Phys. 94, Furukawa, IEEE rans. Electr. Insul. 24, X. D. Chen, D. B. Yang, Y. D. Jiang, Z. M. Wu, D. Li, F. J. Gou, and J. D. Yang, Sens. Actuators, A 65, G. Żukowska, M. Rogowska, A. Wojda, E. Zygadlo-Monikowska, Z. Florjańczyk, and W. Wieczorek, Solid State Ionics , S. S. Sekhon and H. P. Singh, Solid State Ionics , A. Magistris, E. Quartarone, P. Mustarelli, Y. Saito, and H. Kataoka, Solid State Ionics , S. S. Sekhon, Bull. Mater. Sci. 26, S. S. Sekhon, N. Arora, and H. P. Singh, Solid State Ionics 160, J. C. Maxwell, A reatise on Electricity and Magnetis, reprinted Dover, New York, 1954, Vol C. K. Wong, Y. M. Poon, and F. G. Shin, J. Appl. Phys. 90, C. K. Wong, Y. W. Wong, and F. G. Shin, J. Appl. Phys. 92, H. L. W. Chan, Y. Chen, and C. L. Choy, Integr. Ferroelectr. 9, Ikeda, Fundaentals of Piezoelectricity Oxford, New York, 1996, p Furukawa, K. Ishida, and E. Fukada, J. Appl. Phys. 50, Z. Hashin, J. Appl. Mech. 29, C. K. Wong, Y. M. Poon, and F. G. Shin, J. Appl. Phys. 92, K. S. La, Y. W. Wong, L. S. ai, Y. M. Poon, and F. G. Shin, J. Appl. Phys. 96, K.-H. Chew, F. G. Shin, B. Ploss, H. L. W. Chan, and C. L. Choy, J. Appl. Phys. 94, Downloaded 25 Mar 2011 to Redistribution subject to AIP license or copyright; see