Enhancement of electrostrictive polymer efficiency for energy harvesting with cellular polypropylene electrets

Synthetic Metals 162 (2012) 1948–1953

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Enhancement of electrostrictive polymer efficiency for energy harvesting with cellular polypropylene electrets A. Eddiai a,b , M. Meddad b,c , D. Guyomar b , A. Hajjaji d , Y. Boughaleb a , K. Yuse b , S. Touhtouh d , B. Sahraoui e,∗ a

Département de Physique, Faculté des Sciences, Laboratoire de Physique de la Matière Condensée (LPMC), 24000 El Jadida, Morocco Laboratoire de Génie Electrique et Ferroélectricité (LGEF), INSA LYON, Bat. Gustave Ferrie, 69621 Villeurbanne Cedex, France c DAC HR Laboratory, Université Ferhat Abbas, 19000 Sétif, Algeria d Ecole Nationale des Sciences Appliquées d’El Jadida, Université d’el Jadida, EL Jadida, Morocco e LUNAM Université, Université d’Angers, CNRS UMR 6200, Laboratoire MOLTECH-Anjou, 2 bd Lavoisier, 49045 ANGERS cedex, France b

a r t i c l e

i n f o

Article history: Received 23 May 2012 Received in revised form 18 August 2012 Accepted 21 August 2012 Available online 13 October 2012 Keywords: Polypropylene electrets Electrostrictive polymer Energy harvesting efficiency Electromechanical conversion

a b s t r a c t The purpose of this paper is to propose new means for harvesting energy using electrostrictive polymers. The recent development of electrostrictive polymers has generated new opportunities for high-strain actuators. At the current time, the investigation of using electrostrictive polymer for energy harvesting, or mechanical-to-electrical energy conversion, is beginning to show its potential for this application. The objective of this work was to study the effect of cellular polypropylene electrets after high-voltage corona poling on an electrostrictive polyurethane composite filled with 1 vol.% carbon black at a low applied voltage in order to increase the efficiency of the electromechanical conversion with electrostrictive polymers. Theoretical analysis supported by experimental investigations showed that an energy harvesting with this structure rendered it possible to obtain harvested power up to 13.93 nW using a low electric field of 0.4 V/␮m and a transverse strain of 3% at a mechanical frequency of 15 Hz. This represents an efficiency of 78.14% at low frequency. This percentage is very significant compared to other structures. Finally, it was found that the use of polypropylene electrets with electrostrictive polymers was the best way to decrease the power of polarization in order to obtain a good efficiency of the electromechanical conversion for energy harvesting. © 2012 Elsevier B.V. All rights reserved.

1. Introduction With the broad advancements in the field of wireless sensor networks, certain applications require the sensor nodes to have a long lifetime. Using conventional batteries is not always advantageous since their replacement requires human intervention. Hence, acquiring the electrical power needed to operate these devices is a major concern, and an alternative energy source to conventional batteries must be considered. The electrical energy required to run these devices can be obtained by trapping the thermal, light, or mechanical energies available in the ambient environment. Such a process would help to provide unlimited energy for the lifespan of the electronic device. The process of extracting energy from the ambient environment and converting it into consumable electrical energy is known as energy harvesting. Typical ambient energies include sunlight, mechanical energy, thermal energy, and RF energy. These sources

∗ Corresponding author. E-mail address: [email protected] (B. Sahraoui). 0379-6779/$ – see front matter © 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.synthmet.2012.08.012

can be used to increase the lifetime and capability of the devices by either replacing or augmenting the battery usage [1–4]. Devices powered by energy harvesters can, when placed in inaccessible locations, be used to provide vital information on operational and structural circumstances [5]. There is an increasing volume of research carried out on energy harvesting [6]. Piezoelectric materials are long been used for mechanical-to-electrical energy harvesting [7–10]. There have been many investigations [11,12] into the behavior of piezoelectric materials and their application in the development of piezoelectric devices. Having reviewed several methods on energy harvesting, Roundy et al. [13,14] concluded that piezoelectric generators (energy converters) are very promising because of their high efficiency. However, these materials tend to be stiff and limited in mechanical strain abilities; so for many applications in which low-frequency and large-stroke mechanical excitations are available (such as human movement), direct coupling of the piezoelectric materials to the excitation source yields a very low input mechanical energy [15]. Electroactive polymers (EAPs), which change shape as an electric field is applied, are currently being studied by researchers as alternative materials in a large number of areas such as artificial

A. Eddiai et al. / Synthetic Metals 162 (2012) 1948–1953

muscles or vibration control [16,17]. The type of electroactive material known as electrostrictive polymer has shown considerable promise for a variety of actuator applications [18] and may be well suited for harvesting energy from vibration sources such as human motion [19–22]. With this background, the purpose of the present study was to provide a new structure based on the effect of cellular polypropylene electrets after high-voltage corona poling on the electrostrictive polyurethane composite filled with 1 vol% carbon black at a low applied voltage in order to increase the efficiency of the electromechanical conversion with electrostrictive polymers. Electromechanical properties of electrostrictive polymers and dielectric elastomers are intrinsically governed by the permittivity of the material. This parameter controls strain mode electromechanical actuator but also the power density recoverable in generator mode. An increase in permittivity causes a substantial increase in performance. The incorporation of black carbon in polyurethane matrices allows us to improve the performance of the energy harvesting of these polymers through a significant change in their characteristic low concentrations. Concerning energy harvesting, the major issue on the theme remains the energy balance between energy harvested and energy consumed by electronic circuits. Currently, micro-generators using electrostrictive polymer are not independent, a source of bias, produced by a battery or external power is required. Their expansion will be through the development using hybrid structure to empower them. This requires the introduction of active materials, piezoelectric, dedicated to the creation of potential needed to pump energy of vibration, in order to have a system to harvest energy from different sources, also improving their reliability. The objective of using electrets (PP) is to produce the static field required to capture energy through electrostrictive polymers to increase the efficiency of electromechanical conversion with electrostrictive polymers. This article thus discusses the development of a model that is able to predict the energy-harvesting capabilities of an electrostrictive polymer composite with cellular polypropylene electrets. The experimental data were found to be in good agreement with the theoretical predictions. Moreover, the final results indicated that the use of electrets is among the best solutions to enhance the electromechanical conversion of electrostrictive polymers.

2. Experimental procedure 2.1. Polymer materials and preparation Smart materials are a primary element in energy harvesting in that they represent a first stage in the conversion of ambient vibrations into electrical energy. Current technologies mainly focus on PZT materials [23,24]. However, because of brittleness and depolarization problems, PZT harvesters face challenges in long-term operations. Advances in the field of new smart electrostrictive polymers have suggested the investigation of their capabilities as energy converters [25,22]. Previous work on a polyurethane composite (PU 1%C) has demonstrated that the dielectric permittivity can be improved to increase the harvested power [19,20]. The composite was prepared in the laboratory by solution casting. Polyurethane (PU) granules were first dissolved in N,Ndimethylformamide (DMF) at 80 ◦ C for 30 min (10 wt%). Then carbon nanopowder (Aldrich, average particle size 30 nm) was added to the solution under stirring. The volume content of carbon in the composite was fixed to 1% in order to ensure a homogeneous distribution of the filler. The nanopowder was ultrasonically dispersed in the mixture PU (UP400S HIELSHER, probe H7; duty cycle 0.7, 70% power) and DMF during 12 min. The solution was

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Fig. 1. A schematic illustration of the tested films.

Table 1 Various properties of the polymers. Materials

Relative permittivity

Polyurethane (PU 1%C) 15 Polypropylene 2.1

Young’s modulus (MPa)

Thickness (␮m)

Surface (cm2 )

40 103

52 50

6.4 4.8

cooled with ice, after which the viscous mixture was poured onto a glass plate. It was cured at 60 ◦ C for 12 h and heated at 123 ◦ C during 3 h to remove most of the solvent. The obtained sample was rectangular (55 mm × 22 mm) and 52 ␮m thick. The dimensions of the sputtered gold electrode area on the obtained sample were 40 mm × 16 mm. The sample of cellular polypropylene, denoted PQ50, whose thickness and density were 50 ␮m and 600 kg/m3 , respectively, was supplied by Sodinor Company (France). In a next step, the two films were attached to one another with a conductive glue in order to obtain an electrical connection between the two samples. This is schematically illustrated in Fig. 1. Permittivity measurements were carried out using a Solartron 1255, 1296 interface and the Young modulus of the films was evaluated on a Newport table. The results of the measurement performed at 1 Hz are summarized in Table 1. 2.2. Experimental setup of the corona discharge method Fig. 2 presents the corona method that the samples were subjected to. With a corona triode, the electrical charges were injected for a duration of 10 min through the non-metalized surface of the samples which were placed under a stainless grid (the size of the mesh screen was 150 ␮m) in order to maintain a uniform surface potential. With the aim of investigating the charge storage performance and its influence on the electrostrictive characteristics of a fully charged sample, the corona voltages were kept at −30 kV and a grid voltage of −4 kV was employed. The setup is shown schematically in Fig. 2.

Fig. 2. The corona discharge setup.

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A. Eddiai et al. / Synthetic Metals 162 (2012) 1948–1953 45 Polyurethane 1%C 40

Polypropelene

35 30

εr

25 20 15 10 5 0 -1 10

0

10

10

1

2

10

10

3

4

10

5

10

6

10

Frequency(Hz) Fig. 3. Dielectric permittivity (εr ) as a function of the frequency for filled polyurethane and polypropylene.

2.3. Evaluation of the material properties Fig. 3 shows the permittivity versus frequency for a filled polyurethane (PU 1%C) and a polypropylene. The dielectric constants of the obtained samples were calculated from the capacitance at room and measurements by using an LCR meter (HP 4284A). The capacitance of these films was measured over the frequency range of 0.1 Hz to 1 MHz. A large decrease of the dielectric constant was observed around 10 Hz for the filled PU when the frequency increased. Such a behavior is known to be due to the loss of one of the polarization contributions (interfacial polarization, orientation polarization, electronic polarization, atomic polarization. . .) of the dielectric constant [26]. Considering the value of the frequency, this decrease could be unambiguously attributed to the loss of space charges inducing an interfacial polarization contribution. Moreover, the contribution of the space charge could be neglected for frequencies higher than some tens of Hertz. For the cellular polypropylene, the dielectric permittivity remained constant in the frequency range considered. Therefore, beyond a frequency of 1 Hz, it was possible to prevent a global permittivity for (PU1 + C% PP).The frequency dependence of the effective loss tangent for the filled polyurethane (PU 1%C) and polypropylene is shown in Fig. 4. For the PU 1%C, a minimum of

the effective loss was observed when the frequency was increased, starting from very high values at the lowest frequency (0.1 Hz) and reaching almost the same lowest value (0.003) at around 20 Hz. Furthermore, the value of the tangent loss for the PP was relatively low and constant over the considered frequency range. 3. Energy harvester 3.1. Principle of measurement of the harvested power Fig. 5 provides a schematic representation of the setup developed for characterizing the power harvested by the polymer film. The polymer film was mounted in a sample holder composed of two parts: one fixed and a second that could be moved in the 1-direction with the help of an XM550 ironless linear motor. As a consequence, the film was driven with a given strain profile and assumed to be strained along the 1-direction. The electrostrictive polymer was subjected to a dc bias electric field, produced by a function generator and amplified by a high-voltage power amplifier (Model 10/10, Trek Inc., Medina, NY). It was necessary to induce a polarization with a dc bias to obtain a pseudo-piezoelectric behavior in an otherwise non-piezoelectric material [19,28–29]. The sample was connected to an electrical load R, and the current was monitored by

0.18 polyurethane 1%C 0.16

Polypropelene

0.14

Tan(δ )

0.12 0.1 0.08 0.06 0.04 0.02 0 -1 10

10

0

10

1

2

10 10 Frequency(Hz)

3

10

4

10

5

Fig. 4. Losses as a function of the frequency for the filled polyurethane and the polypropylene.

10

6

A. Eddiai et al. / Synthetic Metals 162 (2012) 1948–1953

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Fig. 6. Mechanical and electric configuration of an electrostrictive polymer.

that the electric field is applied in the direction of the thickness (axis 3), (Eq. (2)) can, using the Voigt notation [30] (Fig. 6), be written as Fig. 5. A schematic illustration of the experimental setup for the energy-harvesting measurements.

a current amplifier (SR570, Stanford Research Systems Inc., Sunnyvale, CA). The RMS power harvested on the load was subsequently 2 , where I derived using P = R · Irms rms is the current measured by the current amplifier.

E T S1 = M31 E32 + s11 1

(2)

D3 = εT33 E3 + 2M31 E3 T1

The current induced by the transverse vibration can be measured as I = A(∂D3 /∂t), where A is the area of the electrostrictive polymer. The current produced by the polymer can thus be related to the strain and electric field by



3.2. Model of the harvester power on an electrical load I=A

∂E3 ∂t



εT33 +

2 E2 2M31 S1 − 6M31 3



+

E s11

2M31 (∂S1 /∂t)E3 E s11



(3)

The current modeling used in this study has been described in detail in a previous publication [20]. Nevertheless, its main features are briefly summarized in the following. Electrostriction is generally defined as a quadratic coupling between strain and electric field. Assuming a linear relationship between the polarization and the electric field, the strain Sij and electric flux density Dm are expressed as independent variables of the electric field intensity Em , En , and the stress Tij by the constitutive relations as [27]

E = Y ; Y is the Young’s modulus. Consequently, the where 1/s11 power harvested by the polymer can be expressed by

E T Sij = Mijkl Ek El + sijkl kl

PHarvested ∝ R · Ih2

Di =

εTijkl Ek

(1)

+ 2Mijkl El Tkl

because a dc electric field (Edc ) was applied to the sample so that ∂E3 /∂t = 0, the short circuit current can be expressed as



Ih = 2M31 YEdc A

∂S1 dA ∂t

(4)

(5)

where R is the electric load, and

E is the elastic compliance, M where sijkl ijkl is the electric-field-related

electrostriction coefficient, and εTijkl is the linear dielectric permittivity. One way of harvesting energy using electrostrictive polymers is to operate in the pseudo-piezoelectric mode. For this, the electrostrictive polymer was subjected to a dc-bias electric field. Assuming that the polymer is driven at a given strain in axis 1 and

  ∂S1

2 2  PHarvested ∝ R4M31 Y 2 Edc 

A

∂t

2 

dA .

4. Results and discussion The modeling of Eq. (4) demonstrated that there was a linear relationship between the electric field and the derivative of strain

Strain (%)

(a) 2 1 0 -1 -2 0

0.1

0.2

0.3

0.4

0.5 Time (s)

0.6

0.7

0.8

0.9

1

-8

Current (A)

(b) 2

x 10

1.5 1 0.5 0 0

0.05

0.1

(6)

0.15 Time (s)

0.2

0.25

Fig. 7. (a) The strain (S3 ) versus time; (b) the current (Ih ) function of time.

0.3

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A. Eddiai et al. / Synthetic Metals 162 (2012) 1948–1953

8

x 10

-8

2

-8

With PP Without PP

7

With PP Without PP

6

1.5

5

Current (A)

Current (A)

x 10

4 3 2

1

0.5

1 0.5

1

1.5

2 2.5 Edc (V/m)

3

3.5

4

0 0

4.5 6 x 10

Fig. 8. The short-circuit current I0 at 15 Hz as a function of the static electric field Edc for a constant strain of S1 = 3%.

versus time. To validate this, the current was measured as a function of the static electric field and derivative of strain, and the results are presented in Figs. 8 and 9, respectively. This was done to illustrate the theory presented in Section 3.2 and to demonstrate the effect of this structure with regard to the performance of electrostictive polymers for energy harvesting. Fig. 7(a and b) depicts the strain and the variation in harvested current versus time for a 52-␮m thick polyurethane (PU 1%C) film and a 50-␮m thick polypropylene (PP) film. The bias field Edc , strain S1 and mechanical frequency were respectively fixed at 0.4 V/␮m, 3% and 15 Hz. As expected, a linear relation between the current and electric fields or derivative  of strain was indeed observed. From the relation Ih = 2M31 YEdc A ∂S1 /∂t dA, the value of M31 , at a given strain, was determined from the slope of Ih versus Edc . Figs. 7 and 8 clearly demonstrated the linear behavior of the current harvested from the electric field bias and the derivative of strain. These curves show the comparison between the structure with and without the polypropylene film. We can see that through this new structure, the harvested current was increased without variation of the electric field bias (Edc ) and strain (S), thus rendering it possible to improve the efficiency of the electromechanical conversion of electrostrictive polymers. For example, for a bias field of 1 V/␮m, the current harvested with the use of PP was about 50 nA, and as a comparison, the value obtained from the current harvested without PP was 9.6 nA. The experimental application of the proposed structure (PU 1%C +PP) showed an outstanding performance, rendering it possible to harvest 5 times more currant than the standard method (PU 1%C). For the active energy harvesting system investigated within the scope of this study, electric energy was consumed by the polarization of the polymer. No direct measurement on the amount of consumed electric energy was performed. Based on the results of RP and the equivalent electric scheme of the setup, it was possible to estimate the efficiency of the conversion. Table 2 gives the power required for the polarization, as well as the harvested power (at 0.4 MV/m and 15 Hz). The last column of the table displays values

0.5

1

1.5

2

2.5 3 Strain(%)

3.5

4

4.5

5

5.5

Fig. 9. The short-circuit current I0 at 15 Hz as a function of the strain S1 for a static field Edc = 0.4 V/␮m.

of the efficiency of the energy conversion. The power required to induce polarization is given by Eq. (7)

 PPolarization = R

Vdc Rp + R

but RP  R so



PPolarization = R

Vdc Rp

2 (7)

2 (8)

Furthermore, the power harvested for this load could then be given by Eq. (5) PHarvested ∝ R · Ih2 . The harvested power is expressed in (Eq. (6)). Since the power required to induce polarization was proportional to the inverse of Rp , it would be interesting to employ a composite with high Rp values to obtain a decreased PPolarization . Under these conditions, one can clearly see that the use of polypropylene increased the performance of electrostrictive polymers up to 78.14% by a decrease in applied voltage (Fig. 9). The next experiment was performed to validate the increase in power. According to the theory presented in Section 3.2, the difference between the last experiments resided in an added electrical resistance for measuring the harvested power. Fig. 10 presents the harvested power versus the electric load for a given electric field (0.4 V/␮m), strain (3%) at 15 Hz and a resistor of various values. This data displays the existence of an optimal load resistance of 60 M and a maximum harvested power of 13.8 nW in the classical configuration. The magnification made it possible to see that the proposed structure clearly demonstrated an increase in output power, which

16

Harvested power ( nW)

0 0

PU 1%C with PP

14 12 10 8 6 4

Table 2 A comparison of the various power values of the materials at 15 Hz and with an electrical field of 0.4 MV/m. Material

PPolarization (nW)

PHarvested (nW)

Efficiency of the conversion (%)

PU 1%C + PP

3.84

13.93

78.14

2 0 5 10

10

6

10

7

Resistance (MΩ )

10

8

10

9

Fig. 10. The harvested power at 15 Hz for R = 60 M as a function of the load resistance R for a static electric field Edc = 0.4 V/␮m and a transverse strain S1 = 3%.

A. Eddiai et al. / Synthetic Metals 162 (2012) 1948–1953

gave rise to an improvement in the electromechanical conversion of electrostrictive polymers. In the present case, the efficiency of the conversion was positive (78.14%) due to the harvested power being higher than that consumed when working in the pseudo-piezoelectric mode. However, in the absence of the cellular polypropylene electrets and for the same values of static electric field (0.4 V/␮m), strain (3%) and mechanical frequency (15 Hz), the power consumed would be greater than the harvested power, causing the efficiency of the conversion to become negative. However, this value of static field was very low. In fact, the biggest advantage of electrostrictive polymers with polypropylene electrets might be the fact that, in addition to a good dynamic range at a low polarization power, they operate best at relatively long strokes and moderate forces. 5. Conclusion This paper has presented a new method to estimate the conversion efficiency in electroactive polymers and composites, reflecting the effect of cellular polypropylene on the electrostrictive polymers for energy harvesting. The setup is easy to use and can be operated with a high sensitivity. It is also possible to obtain a higher dynamic current by using PP charged by a low static electric field. According to the experimental results, the maximum output power on a resistive load with an ac-to-dc conversion could reach 14 nW for a transverse strain of 3% with a static electric field of 0.4 V/␮m at 15 Hz. This harvested power demonstrated the excellent potential of using the cellular polypropylene with electrostrictive polymers for mechanical energy harvesting in order to overcome the problem of applying a strong electric field and to increase the performance of these polymers for electromechanical conversion. Many applications appear feasible, but challenges remain. The studied material is believed to be most advantageous for applications requiring low frequencies, low cost, and large areas. Currently, piezoelectric materials are the most popular options for harvesting mechanical energy because of their compact configuration and compatibility with micro electromechanical systems (MEMS). However, their inherent limitations include aging, depolarization, and brittleness. Finally, it was found that electrostrictive polymers with cellular polypropylene electrets were promising candidates for replacing piezoelectric (PZT) materials. References [1] R. Amirtharajah, A.P. Chandrakasan, IEEE Symp VLSI Circuits, Dig. Tech. Papers, 1997, pp. 25–26.

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[2] A. Kansal, D. Potter, M.B. Srivastava, Proc. Measurement and Modeling of Computer Systems in Joint Int. Conf., 2004, pp. 223–234. [3] S. Meninger, J. Mur-Miranda, J. Lang, A. Chandrakasan, A. Slocum, M. Schmidt, R. Amirtharajah, IEEE Transactions on Very Large Scale Integration Systems 9 (1) (2001) 64–76. [4] M. Rahimi, H. Shah, G.S. Sukhatme, D. Estrin, IEEE Int. Conf. Robotics and Automation, 2003, pp. 19–24. [5] M. Lallart, T. Monnier, D. Guyomar, Structural Health Monitoring 9 (1) (2010) 87–98. [6] E. Lefeuvre, M. Lallart, C. Richard, D. Guyomar, Piezoelectric Ceramics, Sciyo, Rijeka, Croatia, 2010. [7] G.A. Lesieutre, G.K. Ottman, H.F. Hofmann, Journal of Sound and Vibration 269 (2004) 991–1001. [8] M. Umeda, K. Nakamura, S. Ueha, Japanese Journal of Applied Physics 36 (1997) 314–315. [9] M. Goldfarb, L.D. Jones, American Society of Mechanical Engineers Journal of Dynamic Systems, Measurement, and Control 121 (1999) 566–571. [10] H.A. Sodano, E.A. Magliula, G. Park, D.J. Inman, Conf. Adaptive Struct. Technol., 2002, pp. 153–157. [11] S. Kim, W.W. Clark, Q.M. Wang, Journal of Intelligent Material Systems and Structures 16 (10) (2005) 847–854. [12] E. Hong, S. Trolier-McKinstry, R. Smith, S.V. Krishnaswamy, C.B. Freidhoff, IEEE Transactions on Ultrasonics, Ferroelectrics and Frequency Control 53 (4) (2006) 697–706. [13] S. Roundy, P.K. Wright, J. Rabaey, Computer Communications 26 (11) (2003) 1131–1144. [14] S. Roundy, P.K. Wright, Smart Materials and Structures 13 (5) (2004) 1131–1142. [15] Y. Liu, K. Ren, H.F. Hofmann, Q.M. Zhang, Proceedings of SPIE, Int. Soc. Opt. Eng. 385 (1) (2004) 17–28. [16] R. Pelrine, R.D. Kornbluh, J. Eckerle, P. Jeuck, S. Oh, Q. Pei, S. Stanford, Proceedings of SPIE 4329 (2001) 148–156. [17] Y. Bar-Cohen, Electroactive Polymer (EAP) Actuator as Artificial Muscles (Reality, Potential, and Challenges), SPIE Press, Bellingham, WA, 2004. [18] L. Petit, B. Guiffard, L. Seveyrat, D. Guyomar, Sensors and Actuators A148 (2008) 105–110. [19] D. Guyomar, L. Lebrun, C. Putson, P.-J. Cottinet, B. Guiffard, S. Muensit, Journal of Applied Physics 106 (1) (2009), art. no. 014910. [20] L. Lebrun, D. Guyomar, B. Guiffard, P.J. Cottinet, C. Putson, Sensors and Actuators A 153 (2009) 251–257. [21] S. Boisseau, G. Despesse, A. Sylvestre, IOP Smart Materials and Structure (2010). [22] Y. Liu, K.L. Ren, H.F. Hofmann, Q. Zhang, Ferroelectrics and Frequency Control 52 (2005) 2411–2417. [23] Z. Wang, Y. Xu, Applied Physics Letters 90 (2007), art. no. 263512. [24] A. Hajjaji, D. Guyomar, S. Touhtouh, S. Pruvost, Y. Boughaleb, M. Rguiti, C. Courtois, A. Leriche, K. Benkhouja, Journal of Applied Physics 108 (2010) 064103. [25] P.-J. Cottinet, D. Guyomar, B. Guiffard, C. Putson, L. Lebrun, IEEE Transactions on Ultrasonics, Ferroelectrics and Frequency Control 57 (2010) 774. [26] B.S. Mitchell, An Introduction to Materials Engineering and Science for Chemical and Materials Engineers, Wiley-IEEE, 2004. [27] K. Ren, Y. Liu, H.F. Hofmann, Q.M. Zhang, Applied Physics Letters 91 (2007) 132910. [28] P.J. Cottinet, D. Guyomar, B. Guiffart, L. Lebrun, C. Putson, First Int. Conf. Sensor Device Technologies and Applications, Venice, Italy, Jul. 18–25, 2010, pp. 25–31. [29] M. Lallart, P.-J. Cottinet, L. Lebrun, B. Guiffard, D. Guyomar, Journal of Applied Physics 108 (2010), art. no. 034901. [30] S. Eury, R. Yimnirun, V. Sundar, P.-J. Moses, S.-J. Jang, R.E. Newnham, Materials Chemistry and Physics 61 (1) (1999) 18–23.