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Evaluation of macroscopic polarization and actuation abilities of electrostrictive dipolar

polymers using the microscopic Debye/Langevin formalism

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2012 J. Phys. D: Appl. Phys. 45 205401

(http://iopscience.iop.org/0022-3727/45/20/205401)

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IOP PUBLISHING JOURNAL OF PHYSICS D: APPLIED PHYSICS

J. Phys. D: Appl. Phys. 45 (2012) 205401 (9pp) doi:10.1088/0022-3727/45/20/205401

Evaluation of macroscopic polarizationand actuation abilities of electrostrictivedipolar polymers using the microscopicDebye/Langevin formalismJean-Fabien Capsal, Mickael Lallart, Jeremy Galineau, Pierre-JeanCottinet, Gael Sebald and Daniel Guyomar

Laboratoire de Genie Electrique et Ferroelectricite (LGEF), Universite de Lyon, INSA-Lyon, EA 682, 8rue de la physique, F-69621, Villeurbanne, France

E-mail: jean-fabien.capsal@insa-lyon.fr (J-F Capsal)

Received 1 December 2011, in final form 13 March 2012Published 1 May 2012Online at stacks.iop.org/JPhysD/45/205401

AbstractElectrostrictive polymers, as an important category of electroactive polymers, are known tohave non-linear response in terms of actuation that strongly affects their dynamic performanceand limits their applications. Very few models exist in the literature, and even fewer arecapable of making reliable predictions under an electric field. In this paper, electrostrictivestrain of dipolar polymeric systems is discussed through constitutive equations derived fromthe Boltzmann statistics and Debye/Langevin formalism. Macroscopic polarization isexpressed as a function of the inherent microscopic parameters of the dielectric material.Electrostrictive strain, polarization and dielectric permittivity are described well by the modelin terms of dipole moment and saturation of dipole orientation, allowing the physical definitionof the electrostrictive coefficient Q. Maxwell forces generated by dipolar orientation inducingsurface charges are also used to explain the electrostrictive strain of polymers. The assessmentof this analysis through a comparison with experimental data shows good agreement betweenreported values and theoretical predictions. These materials are generally used inlow-frequency applications, thus the interfacial phenomena that are responsible for lowsaturation electric field should not be omitted so as not to underestimate or overestimate thelow electric field response of the electrostrictive strain.

1. Introduction

Over the last decade, electrostrictive organic materials haveattracted considerable interest due to their large strain inducedunder an electric field, low weight and good mechanical prop-erties [1–4]. These electroactive polymers show promisingpotential in various applications where electromechanical con-version is required, such as low-frequency energy harvestingor actuation (for instance micropumps, long stroke actuatorsor artificial muscles) [5–11]. Despite the numerous studiespublished, electrostrictive strain of polymers is still a subjectof controversy [12, 13]. On the one hand, electrostriction in di-electric polymers is defined as the strain induced by attractiveforces exerted by an electrostatic field [14] and, on the otherhand, it is supposed that a thermodynamic analysis describes

well the constitutive behaviour of the material [15]. Thus, themicroscopic origin of electrostrictive strain is a subject of inter-est to model and predict the strain induced by an electric field.

It is experimentally shown that the electrostrictive strain,which is a compressive strain, deviates from the expectedquadratic relation S(E) [16, 17]. However, recent studiesdemonstrate a saturation of the electric field induced strainwhich occurs at a few per cent of deformation [3–18]. Earlyanalyses tend to regard the material as hyperelastic withdifferent models such as Mooney–Rivlin, Ogden and Yeoh togive a physical explanation to this phenomenon [15, 19, 20].While these approaches may explain some experimentallyobserved phenomena, their physical origin remains unclearand the predictions are far from being accurate. In thisstudy we propose to use the Boltzmann statistics of dipole

0022-3727/12/205401+09$33.00 1 © 2012 IOP Publishing Ltd Printed in the UK & the USA

J. Phys. D: Appl. Phys. 45 (2012) 205401 J-F Capsal et al

orientation and the Debye/Langevin formalism to describethe macroscopic polarization of a dielectric material as afunction of the inherent microscopic parameters of a dipolarpolymer. Although classical, never applied to electrostriction,the proposed model describes well the experimental data interms of dipole moment and saturation of dipole orientation.Thus, it is possible to extract or predict the strain under a highelectric field and the molecular origin of the electrostrictivestrain of a dielectric material.

This paper is organized as follows. Section 2 describes thesample elaboration and material characterization. Sections 3and 4 introduce the analysis of microscopic polarization andits implication in terms of macroscopic behaviour using theDebye/Langevin formalism. Section 5 exposes the comparisonof the theoretical predictions obtained through this study withexperimental data and discusses the validity of the proposedapproach. Section 6 briefly concludes the paper, recalling themain findings of this study.

2. Experimental

2.1. Sample elaboration

Three types of dipolar polymers were used in this study.

• Bi-axially stretched poly(propylene) (PP) films with athickness of 12 µm were used as-received (AVX).

• Poly(vinylidene fluoride–trifluoroethylene–chlorofluoro-ethylene) (P(VDF–TrFE–CFE)) terpolymer powder(Piezotech) was dissolved in methyl ethyl ketone (MEK).The solution was deposited onto glass substrates duringthe evaporation of the solvent. Then the as-preparedfluorinated terpolymer films with a thickness of 60 µmwere annealed at 110 ◦C for 3 h in order to remove theresidual solvent and optimize the crystalline degree of thepolymer.

• Another class of polymer studied was a polyurethane (PU)elastomer (Estane). Films were prepared by dissolutionof polyurethane granules in N ,N -dimethyl formamide(DMF). The solution was then deposited on a glasssubstrate to ensure good homogeneity of the film anda reproducible thickness of 100 µm. The wet filmwas subsequently dried in an oven in order to correctlyeliminate the DMF solvent.

For each type of polymer a piece of diameter 2 cm was cut fromthe films and a 50 nm gold layer electrode was then sputteredon both sides.

2.2. Material characterization

Electrical field induced strains were measured using a double-beam laser interferometer (Agilent 10889B) with a precisionon the order of 10 nm. In all cases, the thickness strain wascompressive due to the negative electrostriction coefficient.The film samples were placed on horizontal stainless steel discs(20 mm in diameter) in order to avoid measuring a parasiticflexure motion, and a second brass disc positioned on topof the film rendered it possible to apply a bipolar electricfield. A function generator (Agilent 33250A) amplified by a

factor of 1000 through a high-voltage amplifier (Treck 10/10B)delivered the required bipolar voltage at a low frequency of100 mHz.

The ground current between the sample holder and theground was measured using a current amplifier (Keithley 428).As a sinusoidal shape of the electric field with a period of 10 swas applied on the sample film, the current on the externalcircuit is the summation of two currents; the displacementcurrent (Id) based on the dielectric displacement and theconduction (IR) current according to equation (1):

I = Id + IR = Id +E

R(1)

where I is the poling current, Id is the displacement current, IR

is the conduction current, E is the applied electric field and R isthe electrical resistance. Thus, the dielectric displacement wascalculated from the time integration of the current calculatedafter the subtraction of the electrical resistance of the sample.

3. Theoretical dielectric behaviour of dipolarpolymers

A complete description of the Langevin formalism appliedto dipolar orientation is given in the appendix as sucha development is quite classical. It is shown from theBoltzmann/Langevin formalism that the dipole moment µ isrelated to the dipole moment of a single dipolar entity of amolecule or a particle (crystal phase for example or nano-fillers) within the dielectric material. As most polymersare semi-crystalline and heterogeneous systems by nature,the total macroscopic polarization can be expressed as thesummation of the dipolar orientation of the dipoles in theamorphous phase, the dipolar orientation of the dipoles withinthe crystalline structure and interfacial polarization. If thecrystalline structure is non-polar only the amorphous phaseis involved in the macroscopic polarization and the dipolemoment is low. However, in some cases and under highelectric fields, parts of the crystalline structure are polar whichleads to a high dipole moment compared with the amorphousphase. Very high dipole moments are also generated byinterfacial effects such as the Maxwell Wagner Sillars (MWS)polarization. According to the definition of Esat, a higherdipole moment leads to a lower saturation electric field. Thus,the higher the interaction energy between the dipole and theelectric field, the lower the electric field required to compensatethe thermal energy.

Figure 1(a) presents the macroscopic polarization versusthe applied electric field simulated from equation (A6) fora dielectric material with a dielectric susceptibility in thelinear regime of χ = 47. Three different dipole moments(µ = 244 D, 122 D and 61 D) were reported corresponding tothree different values of Esat. It can be seen from figure 1(a)that Esat represents the electric field required to reach the non-linear zone.

From equation (A8), one can extract Esat as a function ofthe thermal energy, χ and the density of dipoles N as follows:

E2sat = NkBT

3χ. (2)

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J. Phys. D: Appl. Phys. 45 (2012) 205401 J-F Capsal et al

Figure 1. Effect of the saturation electric field (Esat) and appliedelectric field on the simulated macroscopic polarization P (a) andrelative dielectric permittivity ε∗ (b) of a dielectric material with afixed low electric field relative permittivity of ε = 48.

At constant dielectric permittivity, the increase in thepolarization at saturation with the increase in Esat is relatedto a higher dipole density N of the dielectric material. Theevolution of the dielectric susceptibility versus the appliedelectric field is depicted in figure 1(b). As the electric field isincreased, the dielectric susceptibility is decreased indicatingthat most of the dipoles are oriented in the direction parallelto the applied electric field and cannot be tilted any further.Dielectric materials with a high dipole moment, therefore, havea more pronounced decrease in dielectric susceptibility with anapplied electric field.

4. Theoretical electrostrictive strain of dielectricpolymers

Longitudinal electrostriction in dielectric materials can beexpressed as the quadratic dependence of the electrostrictivestrain S33 with the macroscopic polarization P [16, 17]:

S33 = Q33P3(E)2 = Q33P(E)2 (3)

where S3 is the longitudinal electrostrictive strain, Q33 is theelectrostrictive coefficient (<0 for polymers), and P(E) is thenon-linear macroscopic polarization.

The strain induced by the Maxwell forces (SMaxwell) isexpressed as a function of the non-linear dielectric permittivity(ε∗) and the Young modulus (Y ) of the polymer:

SMaxwell = ε∗E2

Y= χ∗E2

Y+

ε0E2

Y= dP(E)

dE

E2

Y+

ε0E2

Y.

(4)

For high dielectric permittivity (ε∗ > ε0), the electrostrictivestrain induced by the Maxwell forces can be expressed by

SMaxwell = ε∗E2

Y≈ χ∗E2

Y= dP(E)

dE

E2

Y. (5)

For the sake of simplicity, it is assumed that for E � Esat, theLangevin function can be simplified as

�(x) ∼= tanh(x

3

). (6)

Hence, macroscopic polarization can be identified withthe commonly used empirical expression of macroscopicpolarization, allowing a physical explanation of thisphenomenon:

P(E) = 3Esatε tanh

(E

3Esat

). (7)

AnddP(E)

dE= ε∗ = ε

cosh2(E/3Esat)(8)

where ε is the low electric field permittivity (linear regime)and ε0 is the vacuum permittivity.

Assuming that electrostriction is mainly generated by theattractive interaction between the charged surfaces [14], theelectrostrictive coefficient can be defined as

Q33 = 1

Y

E2(dP(E)/dE)

P 2(E)= 1

9Yε

E2

E2sat sinh2(E/3Esat)

= 1

9Yε

x2

sinh2(x/3). (9)

And for x < 1, (i.e. E � Esat—linear regime)

f (x) = x2

sinh2(x/3)∼= 9. (10)

Therefore, considering low electric fields (linear regime), theelectrostrictive coefficient can be related to the mechanicaland dielectric properties of the material according to anelectromechanical coupling constant Q33:

Q33 = 1

Yε. (11)

The molecular origin of the electrostrictive strain in polymericsystems is still a subject of controversy. As electrostrictionis a function of the macroscopic polarization, it is possibleto extract the microscopic origin of the electrostrictive strainof a dipolar material from measurements of P(E) and ε(E).Verifying the validity of equation (11) in the linear regime(below the saturation electric field) confirms that the Maxwellstress is responsible for the electrostrictive strain of dipolarpolymers.

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J. Phys. D: Appl. Phys. 45 (2012) 205401 J-F Capsal et al

Figure 2. Polarization versus applied electric field (measurementand modelling from equation (A9)) for bi-axially stretchedpolypropylene (f = 0.1 Hz).

5. Discussion

The main purpose of this study is to give a physicalinterpretation of electrostriction as well as the saturation ofthe electrostrictive strain in terms of dipole orientation. To doso, three different dielectric polymers were chosen for theirintrinsic dielectric properties:

• Poly(propylene) (PP) is a low dielectric polymer withlow dielectric loss. Thus, the low-frequency interfacialresponse can be neglected (high Esat).

• P(VDF–TrFE–CFE) is a highly polar polymer with lowdielectric loss (low Esat).

• Poly(urethane) (PU) is a highly heterogeneous polymerwith high loss above the glass transition temperature.Thus, interfacial polarization mainly governs the dielectricresponse of the polymer at a low frequency (very low Esat).

From these polymers, one can extract the influence of eachkind of polarization phenomena on the electrostrictive strainresponse and saturation effect.

5.1. Dielectric response under electric field

Figure 2 presents the macroscopic polarization P(E) as afunction of the electric field for 12 µm thick, bi-axiallystretched polypropylene (PP). Polypropylene is a weakly polarsemi-crystalline polymer commonly used in organic capacitorsfor its high breakdown electric field, good energy density(∼2 J cm−3 at 640 V µm−1) and high mechanical properties.The dependence of P(E) is linear on a wide range of electricfields and the macroscopic polarization reaches 8 mC m−2 foran applied electric field of 230 V µm−1. The low hysteresisfound in the P(E) measurement is related to the dielectriclosses of polypropylene at low frequencies. The modelling(grey curve) of experimental P(E) from equation (A9) allowsone to extract the relative dielectric permittivity at low electricfield (ε) and saturation field (Esat). The relative low fielddielectric permittivity was found to be εr = 4 This valueis consistent with low-frequency dielectric measurements

Figure 3. Polarization versus applied electric field (measurementand modelling from equation (A9)) for (a) relaxor ferroelectricP(VDF–TrFE–CFE) and (b) associated relative dielectricpermittivity calculated from equation (A10) (f = 0.1 Hz).

reported in the literature [21]. A high saturation electric fieldwas found (Esat = 22 kV µm−1) which is well above thebreakdown electric field of the polymer (which therefore isalways linear in real applications) and corresponds, accordingto the definition of Esat, to a low dipole moment of µ =0.051D. In a discussion about the origin of dielectric loss innon-polar or weakly polar polymers, Work et al [22] estimatedthe effective dipole moment of polypropylene to be of the orderof µ = 0.054D. Thus, the value extracted from the Debyeformalism model is in very good agreement with the expectedvalue of µ. The values of Esat and ε clearly indicate that theamorphous phase mainly influences the polarization behaviourof the polymer under high electric fields. The large saturationelectric field, which is well above the breakdown electric fieldof the PP polymer, induces a constant dielectric permittivityacross the entire electric field range.

Another example of P(E) measurement is given infigure 3(a). In this case, highly polar P(VDF–TrFE–CFE)terpolymer was chosen. This terpolymer is known for itshigh dielectric constant, high electrostrictive and electrocaloricbehaviour [23, 24]. These properties are dependent on themacroscopic polarization of the material, and it is of interest toknow the molecular origin of polarization in order to improve

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J. Phys. D: Appl. Phys. 45 (2012) 205401 J-F Capsal et al

the electroactive properties. The measured P(E) cycle forthis terpolymer presents a low remanent polarization, whichis consistent with its relaxor ferroelectric behaviour. Fromequation (A9), the relative dielectric permittivity under lowelectric field was found to be εr = 50. This value is in verygood agreement with the value reported by Zhang et al [25]and Bauer et al [26]. A very high dipole moment of µ = 24 Dwas extracted from the theoretical derivation. This value islarger than the dipole moment of the single VDF dipole of themacromolecule (µ = 2.2 D) [27]. Zhang et al [28] and Chuet al [29] supposed that the introduction of CFE moleculesexpands the interchain lattice that lowers the energy of the transgauche trans gauche’ (TGTG’) conformation and achievesa reversible change from non-polar TGTG’ conformation topolar (all trans) conformation within the crystalline phase. Thehigh value of µ extracted from the model is in good agreementwith this conclusion and indicates that for these kinds ofpolymers, the polarization behaviour lies in the crystallinephase, as the crystallites dispersed in the amorphous phasetend to act as particles with a very large dipole moment.

The evolution of the relative dielectric permittivity (ε∗)under a high electric field extracted from the Langevinmodel (equation (A10)) for P(VDF–TrFE–CFE) terpolymeris reported in figure 3(b). The experimental data points are ingood agreement with the work of Chu et al [29]. The dynamicdielectric permittivity of the terpolymer decreases with theincrease in the applied electric field. For an electric field ofE = 120 V µm−1, the dielectric permittivity of the fluorinatedterpolymer decreases to ε = 18. This decrease is attributed tothe saturation effect caused by the large dipole moments of thecrystalline phase of the terpolymer which is described well bythe Debye/Langevin approach.

Electrostrictive polymers such as P(VDF–TrFE–CFE) aresemi-crystalline and therefore a biphasic model is neededin order to assess the behaviour of the crystalline andamorphous phases at very high electric fields (up to E =400 V µm−1). Moreover, to demonstrate the validity ofthe proposed approach, the theoretical permittivity obtainedthrough the Debye/Langevin formalism is compared withhigh electric field permittivity measurements reported in theliterature [28] (figure 4).

In a biphasic system such as P(VDF–TrFE–CFE)terpolymers, the macroscopic polarization and dielectricsusceptibility can be expressed as the summation of thepolarization of the amorphous phase (Esat1, χ1) and thepolarization of the crystalline phase (Esat2, χ2) according tothe equation

P(E) = P(E)1 + P(E)2, (12)

anddP(E)

dE= ε∗ − ε0 = dP1(E)

dE+

dP2(E)

dE. (13)

The dipole moment of the VDF molecule is µ = 2.2 D whichleads to a very high saturation field Esat = 476 V µm−1 anda dielectric permittivity of ε = 12 in the case of P(VDF).Introducing these values into equation (13), it is then possibleto adjust the experimental data with the contribution of eachphase of the heterogeneous system. Figure 4 shows perfectagreement between the experimental points and the biphasic

Figure 4. Evolution of the dielectric permittivity under an electricfield for P(VDF–TrFE–CFE) (58.3/34.2/7.5%). (�) experimentaldata from Chu et al [28], (· · · · · ·) monophasic theoretical modelfrom equation (A10) and (——) biphasic theoretical model fromequation (13).

model. The dipole moment of the crystalline phase extractedfrom the biphasic model equals µ = 18 D which is closeto the dipole moment deduced from the P(E) measurement(µ = 24 D). This clearly shows that for low electric fields, thedielectric behaviour of the polymer is mainly governed by thecrystalline phase. For high electric fields, all dipoles of thecrystalline phase are oriented and the dielectric permittivity isthen governed by the amorphous phase.

Figure 5(a) reports the low-frequency unipolar P(E)

cycle measurement of the PU elastomer. A hysteresis wasobserved that was attributed to the high dielectric loss of thePU elastomer. The low-frequency polarization is high andthe saturation electric field is low. The relative dielectricpermittivity and dipole moment under a low electric field,calculated from the Langevin formalism, were found to beεr = 116 and µ = 257 D (Esat = 5 V µm−1). These valuesare consistent with a large interfacial polarization (MWS)effect caused by charges trapped at the interfaces betweencrystalline and amorphous regions of the polymer [32]. Thus,heterogeneities within the system highly influence the low-frequency dielectric response of the PU polymer and lead to anearly saturation of the polarization. The electric field influenceon the dielectric permittivity of the PU polymer extracted fromthe Langevin formalism is depicted in figure 5(b). Largedipoles induced by interfacial phenomena lead to a highdielectric permittivity under a low electric field and an earlysaturation of the dipoles’ orientation. Thus, a decrease in thepermittivity from ε = 116 to ε = 12 in an electric field rangeof E = 25 V µm−1 is reported. According to equation (5), theelectrostrictive performance of the polymer should be modifiedby the low electric field evolution of ε∗.

5.2. Electrostrictive strain under an electric field

The value of the electrostrictive strain constant ofpolypropylene using the fitting parameters of equation (A9)is Q33 = 15 m4 C−2. This value is consistent with the

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J. Phys. D: Appl. Phys. 45 (2012) 205401 J-F Capsal et al

Figure 5. Polarization versus applied electric field (measurementand modelling from equation (A9)) for (a) polyurethane and (b)associated relative dielectric permittivity calculated fromequation (A10) (f = 0.1 Hz).

experimental value of Q33 = 16 m4 C−2 reported by Euryet al [30], denoting that in this class of polymer the Maxwellforces induced by dipole orientation within the amorphousphase are responsible for the electrostrictive strain.

Figure 6 presents the longitudinal electrostrictive strain ofP(VDF–TrFE–CFE) versus the applied electric field, measuredat a frequency of f = 0.1 Hz. It is shown that the longitudinalelectrostrictive strain is a compressive strain that does not havea quadratic dependence on the electric field. The maximummeasured strain is nearly 2.5% for an applied electric fieldof 100 V µm−1. Experimental data were adjusted usingequation (5) with a Langevin model. The Maxwell straincalculated with a constant low electric field relative permittivityof ε = 50 is also depicted. Using the fitting parametersdeduced from the ε∗(E) measurements (figure 3(b)) and thetensile mechanical modulus measured using the tensile test(Young’s modulus Y = 70 MPa) good agreement betweenexperimental data and theoretical predictions was found. Thiscan be explained by the fact that the measurements areperformed for electric fields below 200 V µm−1, where thedielectric response of the crystalline phase dominates. Thegood fitting indicates that the electrostrictive response at

Figure 6. Longitudinal strain (at f = 0.1 Hz) under a high electricfield for P(VDF–TrFE–CFE), (——) modelling from equation (5)and the dielectric permittivity extracted from P(E) measurement,and (- - - -) Maxwell strain calculated with a constant low electricfield dielectric permittivity ε = 50.

Figure 7. Prediction of the longitudinal strain under high electricfield for P(VDF–TrFE–CFE), (——) biphasic theoretical model and(— · · —) monophasic theoretical model.

electrical fields below E = 100 V µm−1 is mainly due toattractive forces between opposite surfaces of the polymer(Maxwell force) induced by dipole orientation within thecrystalline phase of the fluorinated terpolymer. Experimentaldata adjusted by Maxwell strain calculated with a constantlow electric field permittivity overestimate the electric fieldresponse of the electrostrictive strain. Thus, identifying andpredicting the saturation effect of the intrinsic dipolar processis of prior interest to model the electrostrictive strain responseof dipolar polymers and their application as actuators.

As shown previously, above an applied electric field of100 V µm−1, both amorphous and crystalline phases must betaken into account in the modelling of the dielectric properties.Thus, it is very interesting to pay particular attention to thepredicted electrostrictive strain of the P(VDF–TrFE–CFE)terpolymer using both models for E > 200 V µm−1 (figure 7).If only one phase is taken into account, the electrostrictive

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J. Phys. D: Appl. Phys. 45 (2012) 205401 J-F Capsal et al

Figure 8. Longitudinal strain (at f = 0.1 Hz) under a high electricfield for polyurethane, (——) modelling from equation (5) and thedielectric permittivity extracted from P(E) measurement, (- - - -)Maxwell strain calculated with a constant low electric fielddielectric permittivity (ε = 116) and (— · · —) Maxwell straincalculated with a constant high frequency (ε = 6).

strain saturates above 200 V µm−1. This can be seen as thesaturation of the dipoles of the crystalline phase that leads toconstant electrical charges on both surfaces. In the case of abiphasic system, over the saturation of the dipolar crystallinephase, the charges on the surface of the polymer still increasedue to the non-saturation of the dipoles of the amorphousphase. Chen et al showed that the dielectric breakdown ofa terpolymer with thermally evaporated aluminium electrodesis nearly Ebreakdown = 400 V µm−1 [31]. Thus, it might bepossible to reach an electrostrictive strain of nearly 10% underE = 350 V µm−1, leading to high mechanical energy.

The longitudinal electrostrictive strain of PU polymermeasured at a frequency of f = 0.1 Hz is presented infigure 8. Below an electric field of E = 7 V µm−1, theelectrostrictive strain has a quadratic dependence on theelectric field. Up to E = 10 V µm−1, the longitudinalelectrostrictive strain reaches a constant value of S33 =−0.87%. The mechanical Young modulus of PU polymermeasured using the tensile test was found to be Y = 7 MPa. Infigure 8 the Maxwell strains modelled from equation (5) with adielectric permittivity extracted from the Langevin formalism(figure 5(b)) and Maxwell strains calculated from a constantdielectric permittivity of ε = 116 (f = 0.1 Hz and lowelectric field permittivity) and ε = 6 (dielectric permittivityat f = 1 kHz) are also reported. Very good agreement of theexperimental data and the Langevin formalism model is found.This reflects the necessity of taking into account the saturationof the polarization induced by the strong dipole momentsunder the penalty of significantly overestimating the value ofachievable strain. Polymers being heterogeneous materials,it is necessary to carefully consider the different polarizationphenomena that are highly sensitive to the excitation frequencyunder the penalty of underestimating the real electrostrictivestrain under low electric fields. At low electric fields,electrostrictive strain of PU polymers is attributed mainly to

attractive forces between opposite surfaces of the polymer(Maxwell force) induced by charges trapped at the interfacebetween crystalline and amorphous regions that lead to a largeinterfacial polarization.

In both cases, electrostrictive strain response andsaturation effect are described well by the Langevinformalism indicating that pure Maxwell effects govern theelectromechanical response of dipolar polymers. Thus, it isnecessary to pay particular attention to the physical origin ofthe dipolar processes involved in the electrical response. Thesematerials are generally used in low-frequency applications,thus the interfacial phenomena should not be omitted so asnot to underestimate or overestimate the low electric fieldresponse of the electrostrictive strain. Understanding thephysical meaning of electrostriction allows one to design high-performance electroactive polymer actuators (EAP).

6. Conclusion

The physical origin of electric field induced electrostrictivestrain of dielectric polymers is a controversial subject, and thephysical interpretation of the saturation effect is usually basedon empirical laws. The purpose of this present work is toinvestigate the microscopic origin of the electrostrictive strainof dipolar polymers. The Debye/Langevin formalism wasused to describe the evolution of the macroscopic polarizationand the dielectric properties of polymers under high electricfields. Saturation of dipole orientation is responsible for thenon-quadratic dependence of the electrostrictive strain on theapplied electric field when the latter features high values.For the three classes of polymers studied, the electrostrictivestrain is described well by the Maxwell forces induced byopposite charges on the surface of the polymer generatedby dipole orientation of amorphous or crystalline phases andinterfacial effects. It is shown that the Debye/Langevinformalism is a useful technique to predict and improve theelectroactive performance of polymeric dipolar polymers, andis validated by comparing experimental data with theoreticalvalues obtained with this model.

Appendix. Microscopic polarization in theDebye/Langevin formalism

The macroscopic polarization of a dielectric material can berelated to the microscopic dipole moment µi of the moleculesor particles in volume V according to the equation [32, 33]:

�P = 1

V

∑i

�µi + �P∞ = N〈 �µ〉 + �P∞ (A1)

where P is the macroscopic polarization, µi is the dipolemoment of the molecule or particle i, P∞ is the polarizationrelated to non-dipolar orientation phenomena (Maxwell–Wagner–Sillars or electrode polarization), N is the dipoledensity and 〈 �µ〉 is the mean dipole moment.

Assuming that only dipolar orientation of molecular orparticle dipoles significantly contributes to the polarization,P∞ can be neglected. The simplest approach is to consider

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J. Phys. D: Appl. Phys. 45 (2012) 205401 J-F Capsal et al

no interaction between dipoles and a uniform distribution ofthe electric field so that the local electric field (Eloc) acrossthe dipoles is equal to the external electric field applied to thesample (E):

�Eloc = �E. (A2)

According to the Boltzmann statistics, the mean value 〈 �µ〉 ofthe dipole moment �µ is given by the counterbalance of theinteraction energy W (W = −µ · E) of the dipole with theelectric field and the thermal energy W ′ (W ′ = kBT , where kB

is the Boltzmann constant and T is the temperature):

〈 �µ〉 =∫

4π�µ exp(W/W ′) d�∫

4πexp(W/W ′) d�

=∫

4π�µ exp( �µ · �E/kBT ) d�∫

4πexp( �µ · �E/kBT ) d�

(A3)

In spherical coordinates, the mean value of the dipole momentthrough the thickness direction can be expressed as

〈µ〉 =∫ π

0 µ cos θ exp(µE cos θ/kBT ) 12 sin θ dθ∫ π

0 exp(µE cos θ/kBT ) 12 sin θ dθ

= µ〈cos θ〉

(A4)

where θ is the angle between the orientation of the dipolemoment and the electric field and µ is the value of the momentof the dipolar chain.

The right-hand side term can be expressed using theLangevin function [32]:

〈cos θ〉 = �

(µE

kBT

)= exp(µE/kBT ) + exp(−µE/kBT )

exp(µE/kBT ) − exp(−µE/kBT )

− kBT

µE(A5)

Using equations (A1) and (A5), the macroscopic polarizationcan be related to the microscopic parameters of the dielectricmaterial as

P(E) = Nµ

[coth

(µE

kBT

)− kBT

µE

]

= Nµ

[coth

(E

Esat

)− Esat

E

](A6)

where Esat = kBT /µ is the electric field that compensatesthe temperature depolarization of the dipole. For low electricfields (E � Esat), the Langevin function is linear according tothe equation

�

(µE

kBT

)∼= µE

3kBT. (A7)

And then,

P(E) = Nµ2

3kBTE = (ε − ε0)E (A8)

where ε is the dielectric permittivity at low electric fields (linearregime) and ε0 is the vacuum permittivity.

From equations (A6)–(A8) the macroscopic polarizationfor high electric fields can be written as a function of the linearregime parameter χ and Esat:

P(E) = 3Esat(ε − ε0)

[coth

(E

Esat

)− Esat

E

]. (A9)

Hence, for high electric fields, the dielectric susceptibility isno longer constant but its dynamic value can be defined fromequation (A9) as

dP(E)

dE= (ε∗ − ε0) = 3(ε − ε0)

[E2

sat

E2− 1

sinh2(E/Esat)

].

(A10)

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