Electrostrictive conversion enhancement of polymer composites using a nonlinear approach

Physics Letters A 375 (2011) 260–264

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Physics Letters A www.elsevier.com/locate/pla

Electrostrictive conversion enhancement of polymer composites using a nonlinear approach Daniel Guyomar, Mickaël Lallart ∗ , Pierre-Jean Cottinet LGEF, INSA-Lyon, 8 rue de la Physique, F-69621, Lyon, France

a r t i c l e

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Article history: Received 11 June 2010 Received in revised form 6 December 2010 Accepted 10 December 2010 Available online 17 December 2010 Communicated by A.R. Bishop Keywords: Electrostrictive Polymer Nonlinear Energy harvesting

a b s t r a c t Recent trends in electromechanical conversion demonstrated the advantages of using electrostrictive polymers for actuation or energy harvesting. However, their conversion abilities are lower than usual electroactive materials, such as piezoelectrics. The purpose of this Letter is to propose a solution for artificially increasing the coupling factor of electrostrictive materials. Based on an intermittent switching on an electrical circuit that leads to a voltage increase as well as a reduced phase between voltage and velocity, it is shown that such a process allows increasing the converted energy by 1200% and a gain up to 4 in terms of transferred electrical energy. © 2010 Elsevier B.V. All rights reserved.

1. Introduction Thanks to their flexibility, processability and high productivity, electrostrictive polymer materials have been of particular interest over the last few years in order to replace piezoelectric elements as actuators and transducers [1–7]. However, the conversion abilities of electrostrictive polymer composites are much less than those of piezoelectric materials [6]. Therefore, the proliferation of such materials for actuation or energy harvesting applications requires enhancing the electromechanical conversion capabilities of the composites. In order to achieve this purpose, many studies on the material itself have been devoted to increasing the permittivity of the material or including conductive nano-particles for decreasing the required bias electric field [1,6,8,9], leading however to costly or unstable composites. Other studies, such as the works performed by Liu et al. [10] and Ren et al. [11], focused on the optimization of the energy transfer process through the use of proper interfaces. However, in this latter case, external energy has to be supplied to the material, preventing the design of stand-alone systems. In addition, the use of such active energy harvesting schemes requires driving continuously the polymer parameters such as stress or voltage, which may not be practicable, and possibly leading to a negative energy balance because of the losses in the system. Electrostrictive materials may also be used in pseudo-piezoelectric mode by applying a bias voltage to the polymer sam-

*

Corresponding author. E-mail address: [email protected] (M. Lallart).

0375-9601/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2010.12.026

ples [12]. In such a configuration, the dynamic voltage is varying almost linearly with the strain in the material, assuming a high bias electric field compared to the electrical field generated by the mechanical solicitation. Hence, using such an approach, it is possible to apply techniques used with piezoelectric materials in order to increase the conversion abilities of electrostrictive polymers. In particular, it has been shown that a simple nonlinear treatment can artificially increase the coupling coefficient of piezoelectric elements [13]. Hence, the purpose of this Letter is to expose the application of such an approach for increasing the conversion abilities of electrostrictive materials. However, because of the quadratic dependence of the strain with the electric field and the need of applying a bias voltage for working in pseudo-piezoelectric coupling, it will be shown that the application of the nonlinear technique is not as straightforward it may be thought. 2. Modeling and conversion enhancement Contrary to piezoelectric element, electrostrictive polymers have to be submitted to a bias voltage (V bias ) in order to exhibit a pseudo-piezoelectric behavior. Under these conditions and assuming a linear behavior of the sample, it can be shown that the current I flowing out of the polymer may be expressed as:

I = α u˙ − C 0 V˙ ,

(1)

with V the polymer AC voltage, u the displacement, α the force factor (which depends on the bias voltage), and C 0 the clamped capacitance of the material. Such a relationship between the current, voltage and displacement is very similar than in the piezoelectric

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Fig. 1. (a) Schematics of the nonlinear conversion enhancement; (b) typical waveforms and (c) energy cycle.

case [13], except that the voltage is varying around a bias value for electrostrictive materials, hence requiring a modified approach for allowing the nonlinear treatment. From the energy analysis of the mechanical equation of the polymer, it can be shown that the part of mechanical energy converted into electrical energy is given by:

 E conv = α

V u˙ dt .

(2)

Hence, in order to increase the converted energy, it can be stated from Eq. (2) that:

• The dynamic voltage V has to be increased. • The time delay (or phase shift under sinusoidal excitation) between the AC voltage V and the velocity u˙ has to be decreased. Another constraint for the true conversion enhancement is that no external electrical power should be supplied to the system; therefore the conversion increase has to be stand-alone. The achievement of such an improvement may be obtained by switching the electrostrictive element on a electrical circuit as depicted in Fig. 1. The principles of operation of the device consist in inverting the dynamic polymer voltage (i.e., inverting the voltage with respect to V bias ) when the displacement is either maximum or minimum. The voltage inversion is done by connecting the electrostrictive element on an inductance, shaping a resonant electrical network with the polymer capacitance. Therefore the voltage oscillates around V bias , and is inverted (with respect to its bias value) if the switching time period equals half the oscillating pseudo-period. However, because of the resistive losses, the inversion is not perfect and characterized by the inversion coefficient γ (0  γ  1). Hence, such a nonlinear treatment generates an additional piecewise voltage component that has the same sign than the speed (as the inversion process occurs on minimum and maximum displacements), along with a voltage magnitude increase thanks to a cumulative effect offered by the dielectric behavior of the polymer. Therefore, the proposed concept permits both a voltage increase and a reduction of the phase shift between the voltage and the speed, denoting the conversion enhancement. It can be noted that such an approach allows an improvement of the conversion without requiring driving the polymer itself; the nonlinear treatment allowing an optimized energy cycle automatically. It should also be noted that such a treatment does not require any power; the energy given by the voltage source during a switching cycle corresponding to a positive slope is compensated by the energy restored to the source during the next switching cycle, corresponding to a negative slope. Fig. 2(a) depicts the obtained AC voltage waveforms obtained across the polymer under classical operations (in this case the digital switch is OFF, and presents a high impedance which however allows the polymer polarization) and when using the nonlinear approach, using the experimental set-up depicted in Fig. 3. The used

Fig. 2. Experimental polymer voltage waveforms with a 100 Hz vibration.

polymer features an electroded area of 4 × 1.6 cm2 and a thickness of 60 μm. The sample was subjected to a bias voltage of 400 V, leading to an electric field of 6.7 V μm−1 , which is actually much less than those used in active conversion [11]. The sample was then stretched using a shaker, and the total displacement has been set to 20 μm. The switching device, made of MOS transistors, was controlled using an external generator, although being possibly self-powered [14]. Preliminary measurements have been done in order to obtain model parameters under working conditions, yielding α = 84 μN V−1 , C 0 = 7 nF and γ = 0.72. Fig. 2(a) clearly demonstrates the ability of the proposed approach for increasing the voltage levels and decreasing the phase shift between the voltage and velocity. Hence, the voltage magnitude is 3.45 times greater in the nonlinear case, meaning that the maximal available electrostatic energy is 12 times higher than in the classical case. 3. Electrical energy generation enhancement Furthermore, from Eq. (1), it can be also shown that the converted energy may be decomposed as:

E conv =

1 2



2

C0 V +

V I dt ,

(3)

where the first term of the right side member corresponds to the dynamic electrostatic energy on the sample and the second term to the energy exchanged with the electrical interface and losses in the

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Fig. 3. Experimental set-up.

Fig. 4. Energy transfer: (a) classical approach; (b) nonlinear approach.

inductance during the inversion process. The energy transferred to the electrical interface may be evaluated by connecting a resistance R in parallel with the polymer sample (Fig. 4). In addition, a capacitor is used in order to filter the DC component introduced by the bias voltage (only the AC part intervenes in the energy conversion), and a high value resistance R b is used in order to let the voltage across the sample varying (this resistance has to be large enough so that almost all the current generated by the electromechanical coupling of the sample flows to the load R rather than to the bias voltage source – [15]). Therefore, the expression of the current turns to I = V / R. Hence, in this case, it can be shown from Eq. (1) that the converted power when using a classical approach may be expressed in the frequency domain by [13]:

P stand =

1

R α2

2 1 + ( RC 0 ω)2

ω2 u M 2 ,

(4)

where ω and u M denote the angular frequency and displacement magnitude. When including the switching device into the system (in this case, R b is obtained using the OFF resistance of the switch), solving in the time domain Eq. (1) with I = V / R over half an oscillation period (between two switching time instants) yields the expression of the power [13]:

P sw =

R α2

1

Experimental waveforms and powers using such a configuration are depicted in Fig. 2(b) (optimal load in the standard configuration case), Fig. 2(c) (optimal load in the nonlinear configuration case) and Fig. 5. In the latter chart, theoretical expected powers on the load obtained using Eqs. (4) and (5) are also depicted, showing a good agreement with the experimental results. These results demonstrate that the use of the proposed nonlinear treatment allows a significant increase of the transferred power by a factor of 4 whatever the frequency is, thanks to the conversion enhancement offered by the voltage inversion. This therefore allows an artificial increase of the material figure of merit (energy density per squared bias voltage magnitude and squared strain magnitude – [12]) from 6.6 × 10−10 J m−1 V−2 cycle−1 in the classical case to 2.9 × 10−9 J m−1 V−2 cycle−1 when using the nonlinear process. The optimal load is also increased when using the exposed concept, as the reduced phase shift between the dynamic voltage and velocity decreases the capacitive effect of the polymer. 4. Performance discussion Assuming low losses, pure electrostrictive material and small solicitations so that the polarization may be considered proportional to the electric field, the variation in the energy density dW available in the material is given by:

dW = T dS + MY E 2 dS ,

2 1 + ( RC 0 ω)2 2π  ( RC 0 ω)3 (1 + γ ) (e RC 0 ω − 1) × + 1 ω2 u M 2 . π π 1 + ( RC 0 ω)2 (e RC 0 ω − γ )2



(5)

(6)

where T , S and E refer to the stress, strain and electric field. Y is the Young modulus and M is given as the electrostrictive coefficient. d is given as the differentiation operator. In Eq. (6), the first term of the right side member is the mechanical energy density, while the second term refers to the converted energy density.

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Fig. 5. Experimental and theoretical power transferred to the load.

Fig. 6. Comparison of conversion cycles for several schemes.

Assuming that the average electric field E 0 (bias electric field in the case of pseudo-piezoelectric behavior) is much higher than the electric field E ac generated by the mechanical solicitation, the converted energy density may be approximated by:

MY E 2 dS ≈ MY E 0 2 dS + MY E 0 E ac dS .

(7)

The first term having a null time-domain integral as the strain is periodic. Hence, the total energy density dW conv converted by the electrostrictive polymer yields:

dW conv ≈ MY E 0 E ac dS .

(8)

Hence, the conversion abilities of electrostrictive materials may be obtained from the area of the electric field-strain relationship in the ( E ac , MY E 0 S ) domain. The comparison of the converted energy density between active scheme [11], standard pseudo-piezoelectric mode with optimal load and pseudo-piezoelectric mode using the switching device with optimal load is shown in Fig. 6. In order to make these charts independent from the system parameters, the value of the electric field has been normalized with respect to the bias or initial electric field, and the y-axis to the maximal value of the parameter MY E 0 S max , with S max the maximal strain value. In the case of the nonlinear treatment, the value of the inversion coefficient has been set to γ = 0.8. This figure shows that, under the same condition, the active and classical pseudo-piezoelectric approaches are almost equivalent, although the active conversion scheme performs slightly better. However, thanks to the increase in the electrostrictive coupling, the nonlinear approach outperforms the two previously techniques, by a typical factor of 8.

5. Conclusion This Letter proposed the application of a nonlinear approach for increasing the conversion abilities of electrostrictive polymers. Based on an intermittent voltage inversion around the bias voltage that allows both an increase of the dynamic voltage as well as a reduction of the phase shift between this voltage and the velocity, such a treatment therefore permits a significant increase in terms of converted and transferred power, while being simple enough to be easily implemented (e.g., no external driving is necessary). Results indicated that a typical gain of 12 in terms of converted power and 4 in terms of transferred power to a load is easily achievable using off-the-shelf components. Acknowledgements The authors wish to gratefully acknowledge the DGA (Délégation Générale pour l’Armement) that partly funded this work. References [1] [2] [3] [4] [5] [6] [7] [8] [9]

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