Issue 
Eur. Phys. J. Appl. Phys.
Volume 89, Number 3, March 2020



Article Number  30901  
Number of page(s)  8  
Section  Physics of Energy Transfer, Conversion and Storage  
DOI  https://doi.org/10.1051/epjap/2020190102  
Published online  23 April 2020 
https://doi.org/10.1051/epjap/2020190102
Regular Article
PZT ceramic particles/polyurethane composites formalism for mechanical energy harvesting
^{1}
Université Chouaib Doukkali, Faculté des Sciences, LPMC,
El Jadida,
Morocco
^{2}
Université Chouaib Doukkali, ENSA, LabSIPE,
El Jadida,
Morocco
^{3}
Université de Lyon, INSA, LGEF,
Lyon,
France
^{4}
Université Hassan II, ENS, LBGIM,
Casablanca,
Morocco
^{*} email: rjafallah.a@ucd.ac.ma
Received:
27
March
2019
Received in final form:
12
October
2019
Accepted:
1
April
2020
Published online: 23 April 2020
More recently, the ferroelectric ceramic/polymer composites have been progressively replacing ferroelectric ceramics and polymers as they combine their interesting properties. Such as high compliance of polymers and high electromechanical coupling of ferroelectric ceramics those are required for piezoelectric transducer applications. At the same time, the ferroelectric ceramic/polymer composites formalism for predicting their energyconversion capabilities is of both academic and industrial interest. The novelty of this paper is that the electrical power harvested by the PZT/PU polarized composite has been expressed in terms of the effective longitudinal piezoelectric coefficient (d_{33}) of the composite via a parameter p related to the poling ratio. Besides, the parameter p, that is characterizing the PZT/PU composites with different longitudinal piezoelectric coefficients (d_{33}), was evaluated. The other parameters of the electrical power expression were calculated using the Yamada model for the dielectric, piezoelectric and elastic constants. Finally, a good agreement was found between experience and model.
© EDP Sciences, 2020
1 Introduction
The conversion of mechanical energy into electrical energy and vice versa is an exploited technique in an extensive range of applications, including transducers, sensors and actuators [1]. This energy conversion is done in coupled materials especially electroactive ones such as piezoelectric and electrostrictive materials [2]. Classical piezoelectric materials, like lead zirconate titanate ceramics known as PZT, were for a long time used for energy conversion due to their high electromechanical couplings. However, ferroelectric ceramics present higher rigidity and lower mechanical strain that make from them unsuitable for many applications where large flexibility and mechanical compliance are required. Recently, electroactive polymers (EAP) mainly electrostrictive polymers have demonstrated their energy conversion abilities [3]. As a result, they have been considered as an alternative to ferroelectric ceramics for transducers, sensors and actuators applications because of their large flexibility and mechanical compliance. More recently, ferroelectric ceramic/polymer composites have been examined as alternative to classical ferroelectric and electrostrictive materials for transducer, actuator, and sensor applications because they could combine the mechanical compliance and flexibility of polymers with the high piezoelectric and pyroelectric activities of ceramics [4,5].
Indeed, piezoelectricity was revealed in twophase systems consisting of polymeric medium and piezoelectric lead zirconate titanate powder [6–8]. Few different twophase systems, those consisting of a continuous polymeric medium and PZT dispersoid, have been investigated for mechanical energy harvesting [9,10]. In particular, the composite systems with a continuous polymeric medium and PZT dispersoid are very useful because of their plasticity and flexibility [11–13].
In fact, several theoretical studies have been done on twophase systems concerning the piezoelectric constant, the dielectric, and the elastic constant [13–26]. However, few theoretical models have been proposed for predicting their energyconversion capabilities [27].
In this regard, we have made, in our previous work [6], a detailed study of the electromechanical transduction, for the PZT/PU unpolarized and polarized composites submitted to sinusoidal mechanical strain with amplitude of 1.5% at very low frequencies (2 Hz and 4 Hz) and static electric field (10 V/μm) or without (0 V/μm). In addition, we have evaluated the contribution of both the electrostrictive and piezoelectric effects in electrical powers harvested by the PZT/PU polarized composites. In another previous work [8], we have extended the electromechanical characterization of the PZT/PU unpolarized and polarized composites. We have developed an analytical model predicting the energy harvested by the PZT/PU unpolarized and polarized composites from the electrical and mechanical properties of their constituent materials. This model is based on the approach of representing the experimental setup by an equivalent electrical circuit. Also, this model is able to predict the contribution of both the electrostrictive and piezoelectric effects in electrical powers harvested by the PZT/PU polarized composites. In our most recent published work [7], we have proposed a bridge transducer using the PZT/PU polarized composite in order to overcome the cracking problem in classic piezoelectric transducers. We have evaluated the mechanical tensile strain and electrical power output from the PZT/PU polarized composite based bridge transducer using the finite element method (FEM). Besides, we have compared the FEM results from PZT/PU and PZT5A based bridge transducers.
The novelty of the present paper with respect to the previous articles [6–8], is that the electrical power harvested by the PZT/PU polarized composite has been expressed in terms of the effective longitudinal piezoelectric coefficient (d_{33}) of the composite via a parameter p related to the poling ratio. Besides, the parameter p, that characterizes the PZT/PU composites with different longitudinal piezoelectric coefficients (d_{33}), was evaluated. The other parameters of the electrical power expression were calculated using the Yamada model for dielectric, piezoelectric and elastic constants. Finally, good agreements were found between the experience and model.
2 Theory
For a 03 ceramic/polymer composite, several models have been proposed in order to describe the behavior of the dielectric, piezoelectric, and elastic constants. The Yamada model has been developed to describe ellipsoidal particles in an isotropic polymer matrix [13]. This model is generally used to describe dielectric, piezoelectric, and elastic properties of random composites [28].
2.1 Dielectric constant
The dielectric constant (ε) of the twophase composite in the poling direction according to the Yamada model is given by (1)
where ε, ε_{1}, ε_{2} are the dielectric constants of the twophase composite, continuous medium, and ellipsoidal particles, respectively, n is the shape factor, and q is the volume fraction of the ellipsoidal particles.
2.2 Piezoelectric constant
Let us consider a twophase composite, as shown in Figure 1, which is composed of a dielectric continuous medium and piezoelectric particles. The piezoelectricity of the twophase composite is caused by the piezoelectric particles. The piezoelectric constant d according to the Yamada model is given by (2)
where, q is the volume fraction of the piezoelectric particles, α is the poling ratio, G is the local field coefficient, and d_{2} is the piezoelectric constant of the piezoelectric particles.
On the other hand, the local field coefficient G is defined by (3)
Putting equation (1) into equation (3), we obtain the local field coefficient, (4)
Therefore, equations (2) and (4) give the expression of the piezoelectric constant for the twophase composite.
Fig. 1 Twophase composite consisting of dielectric continuous medium and piezoelectric ellipsoidal particles [13]. 
2.3 Elastic constant
The Young’smodulus (Y) of the twophase composite according to the Yamada model is given by (5)
where Y, Y_{1}, Y_{2} are the Young’s modulus of the twophase composite, continuous medium, and ellipsoidal particles, respectively, q is the volume fraction of the particles, and n′ is a parameter attributed to the shape of the ellipsoidal particles that is given as follows. (6)
where, σ is the Poisson’s ratio for the continuous medium.
2.4 Open circuit output electrical power
The constitutive equations for a linear piezoelectric material under low mechanical stress (T) levels can be written as (7) (8)
where S_{μ} is the mechanical strain vector, D_{k} is the electric displacement vector, T_{v} is the mechanical stress vector, E_{i} is the electric field vector, is the mechanical compliance matrix under constant electric displacement, g_{iv} and are the direct and reverse piezoelectric coefficients where the superscript t means the transposed matrix, and is the dielectric susceptibility constant under constant mechanical stress. The subscripts i, k = 1, 2, 3 are the electrical subscripts and represent the different directions inside the Cartesian coordinate system of the piezoelectric material (i.e. x, y, and z). The subscripts μ, v = 1, 2, 3, 4, 5, 6 are the contracted mechanical subscripts corresponding to the three Cartesian coordinate axes and the rotational motion/shear motion around these three axes.
Considering the mechanical stress/strain in the xdirection and electrical field/displacement in the zdirection, the constitutive equations can be rewritten as (9) (10)
where d_{31} is the transverse piezoelectriccoefficient and is the dielectric constant.
Under a mechanical stress applied in the xdirection (T_{1}), the open circuit output voltage (V) of the piezoelectric material can be computed from equation (10) and is given as (12)
where t is the thickness of the piezoelectric material.
The charge (Q) generated on the piezoelectric material can be determined from equation (10) and is given as (13)
where, C is the capacitance of the piezoelectric material.
The above relationship shows that at low frequencies the piezoelectric plate can be assumed to behave like a parallel plate capacitor. Hence, electric power available under the cyclic excitation is given by equation (15) as follows (15)
The transverse piezoelectric coefficient is given by (17)
The longitudinal piezoelectric coefficient is also given by (18)
The 33 and 31 modes are the two modes in which piezoelectric material is generally exploited. The first number (3) means that the voltage is generated along the zaxis in both mode (i.e. the electrodes are attached to surfaces perpendicular to the zaxis) while the second number means the direction of the applied stress. In the 33mode, the stress is applied along the same axis as the voltage appears. In the 31mode, the stress is applied along the xaxis whereas the voltage appears in the zaxis.
where, B is the bulk modulus, Y is the Young’s modulus, and ν is the Poisson’s ratio.
In the case of an isotropic material, the bulk modulus (B) is related to the Young’s modulus (Y) and the Poisson’s ratio (ν) through the relation (20)
hence, the relationship between transverse and longitudinal coefficients is (21)
Putting equation (21) into equation (16), the new expression of the electric power reads as follows (22)
Alternatively, the electric power density can be expressed as follows (23)
Under certain experimental conditions, for a given material of fixed area and thickness, the electrical power is dependent on the square of d_{33}. Consequently, a material with a high d_{33} will generate high power.
2.5 Electrical power supplied to resistive load
Figure 2 shows the simplest method to provide electrical energy, produced by a piezoelectric material subjected to mechanical strain, to a resistive load (R). This method directly connects the resistive load (R) in series with the piezoelectric material. Moreover, it is easy to calculate the electrical power supplied to the given resistance using this method. The three different axes termed as 1, 2, and 3 (Fig. 2) are the three Cartesian coordinate axes x, y, and z used to define different directions in the piezoelectric material.
Figure 3 represents the equivalent electrical circuit of the configuration shown in Figure 2. In this circuit, the piezoelectric material was represented by a voltage source in series with a resistor (r).
By applying the voltage divider, the voltage across the resistance (R) is given as (24)
Hence, the electrical power provided to resistance (R) is as follows (25)
By using equations (11) and (12), the electrical power becomes (26)
Finally, by using equation (21), the final expression of the electrical power is given as (27)
Fig. 2 Easier configuration allowing powering the resistive load directly. 
Fig. 3 Equivalent electrical circuit of the piezoelectric material connected with a resistive load. 
3 Experimental procedure
3.1 Elaboration of the PZT/PU films
The PZT/PU composites were prepared in the Laboratory of Engineering Sciences for Energy (LabSIPE) at ENSAEl Jadida. PZT powder and PU pellets are two types of commercial materials that were used to synthesize the PZT/PU composites. PZT powder (PZTP189) was provided by the SaintGobain Quartz, whereas PU pellets (Estane 58888 NAT 021 (PU88A)) were supplied by Lubrizol. The synthesized PZT/PU composites are composites of polyurethanes loaded by a 50 volume percent of the PZT powder. This 50 volume percent gives flexible PZT/PU composites with moderate piezoelectric properties. In fact, the PZT/PU composites with a PZT fraction superior than 50% have suitable piezoelectric properties but lose their flexibility. Contrariwise, the PZT/PU composites with a PZT fraction less than 50% have a suitable flexibility but their piezoelectric properties are not significant.
First, the PU pellets were dissolved in the tetrahydrofuran solvent (THF) under magnetic stirring at a room temperature. Secondly, the PZT grains ranging in diameter from 1 to 10 μm were added to the solution under the magnetic stirring and then the ultrasonic agitation in order to homogeneously disperse the PZT grains in the solution. The solution casting method was then used to make the PU/PZT films with a thickness around 100 μm. Then, the PU/PZT films were dried at the room temperature for 12 h to evaporate the remaining solvent (see Fig. 4). Next, gold electrodes were deposited onto both sides of the PU/PZT films, with an area 1 cm × 4 cm, using cathodic sputtering (Cressington 208 HR) to make measurements later. The thickness of the gold layer is about 20 nm, which has been measured using the MTM 20 high resolution thickness controller that is available with the sputter coater 208 HR. Finally,the PU/PZT films were immersed in hot silicon oil at 80 °C and subjected to a bias electric field of 10 MV/m for 20 min for their poling. Indeed, 20 min was optimized and enough to pole the PZT/PU composites. Other tests with time higher than 20 min did not lead to better results.
Fig. 4 Elaboration process of the PU/PZT composites. 
3.2 Experimental setup
Figure 5 shows the schematic illustration of the experimental setup used for measuring the power harvested by PU50vol%PZT composites. The PU50vol%PZT composite was mounted on a test support consisting of two jaws: one immobile and the second that can be moved in one direction by means of a linear motor driven by a function generator via a controller. As a result, the composite film was strained with a given profile along this direction. Depending on the equipment, the tensile solicitation mode was used and the monitoring was done by fixing the applied strain and measuring the electric current produced by the composites. In our case, a harmonic strain was chosen for ease of implementation: S(L) = S_{M} cos(ω_{m}t) where L is the length of the composite film, ω_{m} = 2πf_{m} is the strain pulsation with f_{m} = 2 Hz is the strain frequency, and S_{M} = 1.5% is the strain amplitude. The composite was connected to an electrical resistance (R) for measuring the root mean squarepower (i.e. ) with I_{RMS} the current measured by a current amplifier (Keithley 617).
For making a meaningful experimental statistical analysis, the measurements were performed three times for each sample and repeated on more than 5 similar samples. Furthermore, these measurements were approximately the same for each specimen.
Fig. 5 Experimental setup for measuring the electrical power harvested by the composite in response to a mechanical excitation. 
4 Results and discussion
Figure 6 presents the electric current according to the frequency that were harvested by PU50vol%PZT composites of different piezoelectric constants (d_{33}). For a given frequency (f_{m}), it is remarkable to note that the parameter that had an influence on the harvested current is the piezoelectric constant (d_{33}). As expected, a material with a higher piezoelectric constant (d_{33}) has generated the higher electric current. In addition, a linear relationship between the shortcircuit current and the frequency was observed for a given piezoelectricconstant (d_{33}), surface of material (A), Young’s modulus (Y), and strain (S_{1}). These observations were validated by the piezoelectric current model in the framework of shortcircuit conditions and when a harmonic strain was applied (28)
where I_{ac} is the piezoelectric shortcircuitcurrent and ω_{m} = 2πf_{m} is the pulsation of the mechanical strain.
By using equation (21), the piezoelectric current model becomes (29)
Figure 7 shows the power density as a function of the resistance harvested by PU50vol%PZT composites of different piezoelectric constants (d_{33}). Again, it is remarkable to note, under the same working conditions, that the parameter that had an influence on the harvested power density is the piezoelectric constant (d_{33}). As mentioned above, a material with a higher piezoelectric constant (d_{33}) has generated a higher electric power density.
Fig. 6 Electric current as a function of the frequency, harvested by PU50vol%PZT composites with different d_{33}. 
Fig. 7 Power density as a function of the resistance harvested by PU50vol%PZT composites with different d_{33}. 
4.1 Model validation
In order to validate the theoretical model, equation (27) was used to draw the power density as a function of the resistance and for comparing it with the experimental curves. The parameters used in equation (27) were analytically calculated.
4.1.1 Parameters of model
First, the dielectric constant value () of the PU50vol%PZT composite was calculated using equation (1). By using the values of dielectric constants of both PU (ε_{1}) and PZTP189 (ε_{2}) from Table 1, the volume fraction for the PZTparticles (q = 0.5), and the parameter attributed to the shape of the PZTparticles (n = 8.5 [13]), the dielectric constant of the PU50vol%PZT composite () was found to be 43.
Second, the stress value (T_{1}) was calculated using the relationship . Equation (5) was used to determine the elastic constant (). Indeed, elastic constants values of PU and PZTP189 (Y_{1}, Y_{2} respectively) included in Table 1, the volume fraction for the PZTparticles (q = 0.5), and the parameter attributed to the shape of the PZTparticles (n′ = 0.6 calculated using Eq. (6) with σ = 0.3) gives an elastic constant equal to 106 MPa. Hence, with S_{1} = 1.5% (the strain value used in experimental tests), the T_{1} value equals 1.596 MPa. Moreover, bearing in mind that F_{1} = T_{1} × a where F_{1} is the applied force and a = t × l is the area on which the force (F_{1}) is applied, the F_{1} value equals 1.6 N. Table 2 summarize these calculated results.
Finally, the value of the optimal electrical resistance (r) was found by solving the following equation. (30)
This gives r = R_{opt} with R_{opt} the optimal resistance corresponded to the maximal power P_{Rmax}. Figure 7 shows that the maximal electrical powers corresponds to R_{opt} = 7.5 MΩ, thus the value of the resistance (r) equals 7.5 MΩ.
Dielectric and elastic properties of PU88A and PZTP189 materials.
Dielectric and elastic properties of PU50vol%PZT composites and some parameters used to calculate these properties.
4.1.2 Validity of the model
In this subsection, the comparison between experimental data and theoretical model was done. In fact, a new parameter (p) was introduced into the theoretical model so that the latter best fits the experimental data, as shown in Figure 8. Equation (27) becomes equation (31) after introducing this new parameter (p) as follows (31)
Figure 9 presents the value of the new parameter (p) corresponding to each d_{33} value. On the one hand, it was remarkable that the value of parameter (p) ranged from 0.64 to 0.74. On the other hand, it is possible to use equation (2) to explain the physical origin of this parameter (p). Since the volume fraction (q) and the piezoelectric constant (d_{2}) of the piezoelectric particles as well as the local field coefficient (G) are constant, the poling ratio (α), or a part of it, is probably the physical quantity represented by the parameter (p). Indeed, we don’t have, for the present, other clear explanations for this parameter (p). Finally, it can be noted that the theoretical model agreed well with experimental data.
Fig. 8 Comparison between the experimental results and theoretical model. 
Fig. 9 Values of the parameter (p) corresponding to the different d_{33}piezoelectric constants. 
5 Conclusion
The purpose of the current study is to propose a simple expression of the electrical power harvested by PZT/PU poled composites in terms of the effective longitudinal piezoelectric coefficient (d_{33}) of the composite via a parameter p related to the poling ratio.
This study has shown that PZT/PU poled composites are able to harvest the vibration energy. It has also shown that PZT/PU poled composites with different piezoelectric constants (d_{33}) generate different electrical powers. Moreover, an expression of the electrical power, delivered by the PZT/PU composites to a resistive load, as a function of the longitudinal piezoelectric coefficient (d_{33}) was proposed. Besides, the parameter p, that characterized the PZT/PU composites with different longitudinal piezoelectric coefficients (d_{33}), was evaluated. The other parameters of the electrical power expression were calculated using the Yamada model for the dielectric, piezoelectric and elastic constants. Finally, good agreements were found between experience and model.
Moreover, this theoretical model is able to predict the electrical power converted by PU/PZT composites with different longitudinal piezoelectric coefficients (d_{33}), and can be used to optimize the choice of materials for the composite.
These findings enhance our understanding of piezoelectric ceramic/polymer composites for energy harvesting, and will allow to increase their functionality, in view of enabling future devices fully autonomous.
Author contribution statement
All the authors were involved in the preparation of the manuscript. All the authors have read and approved the final manuscript.
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Cite this article as: Abdelkader Rjafallah, Abdelowahed Hajjaji, Fouad Belhora, Abdessamad El Ballouti, Samira Touhtouh, Daniel Guyomar, Yahia Boughaleb, PZT ceramic particles/polyurethane composites formalism for mechanical energy harvesting, Eur. Phys. J. Appl. Phys. 89, 30901 (2020)
All Tables
Dielectric and elastic properties of PU50vol%PZT composites and some parameters used to calculate these properties.
All Figures
Fig. 1 Twophase composite consisting of dielectric continuous medium and piezoelectric ellipsoidal particles [13]. 

In the text 
Fig. 2 Easier configuration allowing powering the resistive load directly. 

In the text 
Fig. 3 Equivalent electrical circuit of the piezoelectric material connected with a resistive load. 

In the text 
Fig. 4 Elaboration process of the PU/PZT composites. 

In the text 
Fig. 5 Experimental setup for measuring the electrical power harvested by the composite in response to a mechanical excitation. 

In the text 
Fig. 6 Electric current as a function of the frequency, harvested by PU50vol%PZT composites with different d_{33}. 

In the text 
Fig. 7 Power density as a function of the resistance harvested by PU50vol%PZT composites with different d_{33}. 

In the text 
Fig. 8 Comparison between the experimental results and theoretical model. 

In the text 
Fig. 9 Values of the parameter (p) corresponding to the different d_{33}piezoelectric constants. 

In the text 
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