Verification of a model for the d33-coefficientof ferroelectrets

J. Hillenbrand, G. M. Sessler, and X. ZhangInstitute for Communications Technology

Darmstadt University of Technology64283 Darmstadt, Germany

Abstract- An existing model for the piezoelectric d33-coefflcientsof charged cellular polymers (ferroelectrets) is tested with ex-perimental data obtained from two different cellular polypropyl-ene (PP) materials. The model assumes the cellular film to consistof plane parallel layers of solid and gaseous material with thesurfaces of the solid layers charged in a specifilc way. Films ofcharged cellular PP are expanded by a pressure treatment. Sub-sequently, due to viscoelastic relaxation, the thickness of the filmsdecreases, thus causing a change of their Young's modulus Y.Values of Y are obtained from interferometric measurements ofthe thickness resonance frequency. Together with the measuredthickness of the solid layers and air layers in the material, the d33-coefficients can be determined from the model. These values arecompared with experimental results for d33 also obtained inter-ferometrically by means of the inverse piezoelectric effect. Verygood agreement between the calculated and measured d33-coefficients and their change with film thickness and time isobtained.

I. INTRODUCTION

Ferroelectrets based on cellular polymers have been of con-siderable recent interest because of their large piezoelectricd33-coefficients [1]. Most studies on this group of materialswere conducted on cellular polypropylene (PP). After suitablecharging, a charge distribution consisting of charge of onepolarity on all "upper" surfaces of the voids and charge of theopposite polarity on the "lower" surfaces is obtained.Such piezoelectric films of cellular PP show d33-coefficients

of about 200 pC/N. The coefficients can be increased by acontrolled pressure-expansion process before charging [2-6]and can be further enhanced by repeating the pressure-expansion after charging [7]. Such experiments have resultedin d33-coefficients of up to 2000 pC/N measured quasistati-cally and in values of about 600 pC/N at audio frequencies [7].Expansion experiments are always followed by a viscoelastic

relaxation and thus a thickness decrease of the films. In thecourse of such studies it was observed that, as a function offilm thickness, the d33-coefficient first rises and then falls,assuming a pronounced maximum in between [3]. This findingwas later confirmed in somewhat different experiments [5,7]and attributed to the inverse behavior of Young's modulus Ywhich showed a distinct minimum at intermediate densities. Atheoretical model of the piezoelectric activity of cellular mate-rials, which is based on charged plane parallel solid and gase-ous layers, actually predicts a proportionality of d33 and 1/Y,apart from dependencies on charge density and thickness of air

and solid layers [8-10]. The expected dependencies of d33 onall of these parameters have, however, not yet been tested.A comparison of the measured and calculated dependence of

the d33-coefficient on Young's modulus and on sample thick-ness is made in this paper. The experimental values of d33were obtained interferometrically on samples, which undergothe above-described relaxation of the film thickness after apressure-expansion. For the model calculations of d33, inde-pendently determined values of Young's modulus and of filmthickness were measured and constant charge densities on theintemal surfaces are assumed. Comparison of the calculatedwith the measured values will indicate whether the theoreticalmodel with its simplifications is suitable for describing thed33-coefficients of cellular materials.

II. SAMPLES AND MEASURING METHODS

The measurements were performed on two different types ofcellular PP. The first type is VHD50 film from Treofan(Neunkirchen, Germany), originally 50 Mm thick with a den-sity of 0.029 kg/M2. The voids in the film are generated duringstretching due to calcium carbonate filler particles embeddedin the polymer matrix. The films are uncharged as receivedfrom the manufacturer. The second material is type HSOI filmfrom VTT (Tampere, Finland), originally 70 pm thick with adensity of 0.0231 kg/M2. These films were charged by themanufacturer. Both film types have been used before in ex-periments conceming the piezoelectric behavior of cellularmaterials [ 1-1 1].The VHD50 samples were subjected to a double expansion

process, as described before [7]. The first expansion isachieved by a pressure treatment consisting of the exposure ofthe film to nitrogen at a pressure of 20 bar at room tempera-ture for a period of 3 h. Thereafter, the samples were annealedat a temperature of 80 °C for 2 h, followed by a reduction ofthe pressure to atmosphere and subsequent slow cooling. Thehigh pressure inside the samples then inflates the cellularstructure. Slow-relaxation of the polymer results in a gradualdecrease of the thickness. This is followed by corona chargingfor 60 s with a 32 kV corona setup without grid. Metallizationof the samples reduces the sample thickness further [7]. Forachieving high d33-coefficients, a second expansion process isadvisable [7]. To avoid charge dissipation, this expansion hasto be performed at a relatively low temperature. For theVHD50 films it was carried out at 8 bar at room temperaturefor a period of 2 h and thereafter for another 2 h at 45 'C. Fastrelease of the pressure completes the process. Following this

149

second expansion, or additional expansions, the thicknessdecreases again by viscoelastic relaxation.The HSO1 samples were corona charged and thereafter met-

allized on both sides by the manufacturer. Stored at roomtemperature, these films show quasistatically measured d33-coefficients of about 200 pC/N [8,9] even years after charging.Prior to the measurement of Y and d33, an expansion processwas also carried out on these films. This pressure treatmentconsisted of the exposure of the films to nitrogen at 15 bar atroom temperature for a period of 2 days. Thereafter, the sam-ples were annealed at a temperature of 55 °C for 2 h, followedby a fast reduction of the pressure to atmosphere and subse-quent slow cooling. Again, the films are inflated and relaxthereafter, resulting in a gradual decrease of the thickness.The measurement of the d33-coefficient on such inflated

films was performed interferometrically [12]. The sample isexcited to vibrations by application of an ac-voltage V chosento be 100 Veff at the desired frequency. Interferometric detec-tion of the vibration amplitude A allows one to determine d33by means of

d33=A/V. (1)

For the determination of Young's modulus, the frequencyresponse of A was measured. With our equipment this is pos-sible up to frequencies of 500 kHz. For most of the cellularfilms this range includes the thickness extension resonanceand thus allows one to determine the corresponding resonancefrequency fr. This frequency is related to Young's modulus Yby means of the relation [6]

Y =13fr pS2 (2)

where p is the density of the cellular film and s its thickness.Since the frequency response of d33 and thus f, were measuredas a function of film thickness, Y can be determined as a func-tion of film thickness by using Eq. (2).

III. EXPERIMENTAL RESULTS AND DISCUSSIONA. Frequency responses of the d33-coefficientInterferometrically measured frequency responses of the vi-

bration amplitude of a typical VHD50 sample upon ac-excitation, as well as the d33-coefficient determined from it bymeans of Eq. (1), are shown in Fig. 1. Responses were taken atvarious times after the second expansion. The curves shown inthe figure are a typical selection of more data that has beenobtained on a twice expanded VHD50 film.All curves show the pronounced thickness resonance of the

film, found already several years ago in first interferometric ordielectric measurements of such materials [13] and studiedagain more recently [4,6,7]. Resonance frequencies fr firstdecrease (curves for 58, 70, and 111 min) and then increaseagain (270 and 8835 min). From the data,fr can be determinedquite accurately; an exception are the results for 270 minwhich show a double peak. Such double peaks appeared occa-sionally in the frequency responses. For the evaluation of Y,the first and larger peak was used.

E 600

cD 500:2

m 400EC 3000L- 200>0

100

0100 150

Frequency [kHz]

-oN0CD0

C)00CD

0(D:2o.

W1-

Fig. 1. Interferometrically determined frequency responses of the vibrationamplitude and the d33-coefficient of the double-expanded VHD5O film meas-

ured at five different times after the second expansion.

B. Young 's modulusYoung's modulus of two VHD50 films and a HSO1 film

were determined from the resonance frequencies by means ofEq. (2). One of the VHD50 films and the HSO1 film wereinvestigated directly after the low-temperature expansion andthe other VHD50 film was measured after a third expansion.While the thickness of all three films and Young's modulus

of the HSOI film decrease monotonically after the last expan-sion, Young's modulus of both VHD50 films first decreasesbut starts to increase again after 100 min.During each interferometric measurement of Young's

modulus Y (see sect. II) the film thickness s was measuredsimultaneously. The relationships between Y and s are re-quired for the model calculations and are shown for all threefilms in Fig. 2.The curves in the figure for the VHD50 films, which are

similar to those obtained recently in other studies [5,7], show alocal minimum of Y. This behavior of the VHD50 films ispossibly due to the fact that, for an ellipsoidal gas bubble withsolid walls, Y is expected to decrease continuously as theheight of the ellipsoid decreases due to the shrinkage of thefilm. Eventually, as some of the voids become very flat (gaslayer smaller than vibration amplitude due to the applied ac-voltage) the material assumes the much larger stiffness of thesolid ("densification" in the stress-strain relationship [14]).The continuous decrease of Y of the HSO1 films with decreas-ing thickness can be explained if one assumes that densifica-tion is of minor importance in the investigated thicknessrange.

C. Test of the modelfor d33If d33 and Y, together with the thicknesses of the solid and

gaseous layers, are known one can test the relationship

SI1S2iQiY 52 (SI + £52 )2

(3)

150

10

co

0-cn

-c0)

0

0 .150 60 70 80 90 100 110

Film thickness [pm]Fig. 2. Young's modulus of the three cellular PP films as a function of film

thickness.

obtained from a model [8-10] of the piezoelectricity in cellularmaterials. In Eq. (3) E is the permittivity of the solid material,s, and s2 are the total thicknesses of the solid and gaseouslayers, respectively, with s = s1 + s2 the total film thickness, s2iis the thickness of the i-th gaseous layer with I s2i = S2, andis the charge density on the surface of the i-th layer. As men-tioned above, the model assumes the films to consist of a

number of plane parallel solid and gaseous layers with thesolid layers all charged to one polarity on their "top" surfaceand to the other polarity on their "bottom" surface.Since the individual c' s are not known, the model is further

simplified by assuming that all co -values are identical andequal to a. Then the sum in Eq. (3) equals s2aand one obtains

1+ 2

dEC Si

33y (I+ES22j2

Si

(4)

According to Eq. (4), d33 can be calculated from sl, S2, Y, anda. The quantities si and S2 follow directly from the area den-sity of the samples, which is given by the manufacturer, andfrom the measured film thickness. Young's modulus Y isdetermined as described above. Finally, the charge density acan be estimated from scanning-electron microscopic (SEM)data [15], but an accurate value can not be given. It is there-fore treated as an adjustable parameter which is kept constantfor a sample while its thickness changes. The fact that thecharge density does not change during a low-temperatureexpansion and shrinkage process was shown before [7], andwill again be shown below.If d33 thus obtained from Eq. (4) is to be compared with ex-

perimental values, those values must be obtained at the same

frequency as Y which is determined by means of Eq. (2) at theresonance frequency. However, the resonance value of d33, as

shown in Fig. 1, can not be directly used since Eq. (4) does notconsider the effect of mechanical resonances of the cellularmaterial. Including these resonance phenomena in a general-ised theory is presently not feasible since details about theseresonances are not known too well and since they may beoverlapping in some cases. Thus, d33 must be determined for

the resonance frequency but without the effect of the reso-nance on its value.This is approximately possible by using data for d33 far be-

low the first resonance and extrapolating this data to the reso-nance frequency. This procedure is justified because d33 belowresonance is a very weak function of frequency generallygiven by a power law, i.e. represented by a straight line in adouble-logarithmic plot [6,7].Using such extrapolated values one obtains for the VHD

sample expanded twice the measured d33-coefficients plottedin Fig. 3 as function of film thickness. The data shows a peakat a film thickness of about 58 pm, corresponding to a relativedensity (relative to solid PP) of 0.55. A similar peak in the d33-coefficient has been observed before [5]. The measured d33-values for this sample are now compared in the figure with thecoefficients calculated by substituting the known data for sl,S2, and Y and the numerically fitted value o= 0.41 mC/m2 intoEq. (4).

C'CY)

I._c

a)

a)0

a)0

N

a)a.

0 Measured d33

600 n A Calculated d

00 - 33

400

OA

200

50 55 60 65 70 75 80Film thickness [pm]

Fig. 3. Comparison of measured and calculated d33-coefficients as a functionof film thickness for the double expanded VHD50 film.

A similar comparison between experimental and theoreticalvalues of d33 for the second VHD50 sample, which was ex-

panded three times, is made in Fig. 4. For this sample, whichwas charged exactly like the VHD50 film discussed in Fig. 3,a best fit is obtained for cr= 0.36 mC/mi2. Finally, a compari-son of experimental and theoretical d33-values for the HSOIfilm is shown in Fig. 5 with a fitted charge density ofv= 0.43 mC/m2.The comparison of experimental and calculated data in Figs.

3 to 5, particularly the dependence on film thickness, is excel-lent considering the fact that the above-discussed simplifica-tions have been made in the theory. It must be borne in mindthat the change of d33 caused by the variation of Y and of s

over the range of thickness values amounts, for a single sam-

ple, to factors of up to 2.5 and 1.5, respectively. If one com-

pares the three samples investigated, Young's modulus varieseven by a factor of 30 (see Fig. 2) while the thickness changesby a factor of 2.2. In spite of these relatively strong variations,the deviations between theory and measurement are small over

the modulus and thickness ranges investigated. Particularlynoteworthy is the correspondence of the thicknesses at which

151

-0- VHD50 after second expansion'-0- VHD50 after third expansion-A- HS01 after first expansion

1

the calculated an(maximum value in

- 800Q0~

0-

V' 600

.acJ

0

qaz 400

a)O0

50

Fig. 4. Comparison of rof film thick

z [0

d measured d33-coefficients reach their thickness dependence obtained from a theoretical model agreeFigs. 3 and 4. surprisingly well over the entire range of film thicknesses. The

calculated values are obtained by substituting independentlydetermined values of Young's modulus and thickness of air

0 Measured d layers and by using a value for he charge density in each film

0 ~~~~~~~~~~33fi/ Calculated Ud33 that remains constant during its hickness change. It can there-

fore be concluded that the model, despite of the simplifyingassumptions made in its derivation, is a good description ofthe piezoelectric effect in piezoelectric cellular materials.

A The model can also be used to determine indirectly an aver-age value of the charge density on the surfaces of the voids ofthe cellular films. A comparison with an existing SEM-methodfor studying charge distributions in such films indicates thefollowing: The SEM technique yields the spatial distribution

. in the thickness direction with pm resolution but delivers only55 60 65 70 75 80 approximate values for the charge density. As opposed to this,

Film thickness [pm] the new procedure based on Eq. (4) allows one to determinemeasured and calculated d33-coefficients asafunction the mean charge density quite accurately but provides noness for the triple expanded VHD50 film. information on the distribution, neither in the lateral nor in the

thickness direction. The two methods thus supplement eachother with respect to their capabilities.

a%-I0 Measured d40 33co 40 > A Calculated d

.1 300

O 20

g Aia)100N

a)CL n

Fig. 5. Comp

The chargclose to vafilm type Idischarge cafter twocharge peran initial ci

with the a]nature of tifor all threeis actuallythickness oand corona

All of thesness, moduaccuracy.

For severamanufacturof the meas

ACKNOWLEDGEMENTS

The authors are indebted to Treofan and to VTT for provid-ing the PP films and to the Deutsche Forschungsgemeinschaft(DFG) and the Volkswagen Foundation for financial support.

REFERENCES

[1] R. Gerhard-Multhaupt, IEEE Trans. Diel. and Electr. Ins. vol. 9, 850,2002.

)70 to 90 100 110 120 [2] M. Paajanen, M. Wegener, and R. Gerhard-Multhaupt, J. Phys. D: Appl.~~~~~~~~~~~~Phys.,vol. 34, 2482. 2001.

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,e density of 0.43 mC/m2 found for the HSOl film is New York, 2003.lues determined by a SEM method for the same [5] M. Wegener, W. Wirges, R. Gerhard-Multhaupt, M. Dansachmiiller, R.Schwodiauer, S. Bauer-Gogonea, S. Bauer, M. Paajanen, H. Minkkinen,[15]. In these SEM studies, an almost complete and J. Raukola, Appl. Phys. Lett., vol. 84, 392, 2004.

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SEM exposure periods. Since the amount of lost 1226,2004.exposure period was estimated to be 0.25 MC/iM2, [7] X. Zhang, J. Hillenbrand, and G. M. Sessler, J. Phys. D: Appl. Phys.,harge density of 0.5 MC/M2 follows. The agreement vol. 37, 2146, 2004.

[8] G. M. Sessler and J. Hillenbrand, Appl. Phys. Lett., vol 75, 3405, 1999.,bove value is good considering the approximate [9] J. Hillenbrand and G. M. Sessler, IEEE Trans. DieL and Electr. Ins.,he SEM method. It should also be emphasized that vol. 7, 537, 2000.- samples similar charge densities were found. This [10] M. Paajanen, H. Valimiiki, and J. Lekkala, 10th International Sympo-

expected since the same polymer (PP) is used, the sium on Electrets, Delphi, pp. 735-738, IEEE, New York, 1999.If the voids in the cellular films is about the same, [11] J. Lekkala, R. Poramo, K. Nyholm, and T. Kaikkonen, Med. Biol. Eng.charging at breakdown was used in all cases. Comput., vol. 34, 67, 1996.se facts indicate that Eq. (4) is describing the thick- [12] J. Hillenbrand and G. M. Sessler, IEEE Trans. Diel. and Electr. Ins.,

-Vol. 1, 72 2004

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Gogonea, S. Bauer, J. Hillenbrand, R. Kressmann, G. M. Sessler, M.Paajanen, and J. Lekkala, Appl. Phys. Lett., vol. 77, 3827, 2000.

IV. CONCLUSIONS [14] L. J. Gibson and M. F. Ashby, Cellular solids, Cambridge UniversityPress, Cambridge, 2001.

il samples of cellular PP, supplied by two different [15] J. Hillenbrand and G. M. Sessler, Conference on Electrical Insulationers and produced in different ways, the dependence and Dielectric Phenomena, Victoria, pp. 161-165, IEEE, New York,sured d,a1-coefficients on csqmnlte. thirkni-Pc snrl tlip 2000.

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