Electrostriction in dielectric elastomer: Effect on. electromechanical actuation

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1 Electrostriction in dielectric elastomer: Effect on electromechanical actuation Bo Li 1,, Liwu Liu,, Jiuhui Wu 1, Zicai Zhu 1, Hualing Chen 1, 1. School of Mechanical Engineering, Xi an Jiatong University, Xi an, Shannxi, , P.R.China. School of Engineering and Applied Sciences, Harvard University, Cambridge, MA 018, United States. Department of Astronautical science and mechanics, Harbin Institute of Technology (H.I.T), P.O.Box 01, No.9, West Dazhi Street, Harbin, , P.R.China ABSTRACT In this paper, we study the electro-stress in dielectric elastomer (DE) undergoing large deformation subjected to a high voltage. The electrostriction is investigated and evaluated by the free-energy model when the dielectric permittivity does not remain constant in actuation. We investigate the nominal and true electric fields as the DE stretched with the electrostriction involved or not, and the stable domain for safe actuation is provided. Keywords: Dielectric elastomer, electrostriction, instability, actuator 1. INTRODUCTION Dielectric elastomer (DE) is a kind of long-chain soft polymers, capable of large deformation up to 00% [1]. Subjected to a high voltage, DE deforms and converts electrical into mechanical energy. In recent years, DE actuators of different design configurations, have been designed and fabricated for applications including robotics, prosthetic devices, medical implants, pumps, and valves [-10][]. Under a constant voltage, thinner elastomer would result in a higher electric field which may thin down the film further and cause the instability in DE actuation.this electromechanical instability in DE actuation may hence suppress further exploration of DE. Figure 1 shows a typical DE actuator. The elastic membrane is sandwiched between two complaint electrodes. When a voltage applied, the DE membrane reduces its thickness and expands its area as a result of electro-stress. Generally, in the non-polar dielectric without electric field, the electro-stress is blancolee@gmail.com * Author to whom correspondence should be addressed, Tel/Fax: , hlchen@mail.xjtu.edu.cn Electroactive Polymer Actuators and Devices (EAPAD) 010, edited by Yoseph Bar-Cohen, Proc. of SPIE Vol. 764, SPIE CCC code: X/10/$18 doi: / Proc. of SPIE Vol Downloaded From: on 08//014 Terms of Use:

2 composed of both Maxwell stress and electrostriction; the later is related to the change of material dielectric constant. In previous study of DE instability, some researchers only involved Maxwell stress accounting for the electro-stress as they assumed the dielectric constant was unaffected by the actuation. Nevertheless, in recent experimental reports, the dielectric constant of DE is strongly dependent of both deformation and high electric field, and the variation of dielectric constant value ranges from to 6 [11-1], which implies the presence of electrostriction. Some researcher have noticed this issue and tried to solve it theoretically. Zhao and Suo [14] studied the electrostriction in the large DE deformation state, and pointed out the electrostriction would become pronounced when the stretch ratio went above 5. In the light of Suo s study, Liu et al. [15] proposed a nonlinear expression of dielectric constant as a function of stretch ratio and obtained the electrostrictive parameters by curve fitting. However, as values of dielectric constant differ from experiment to experiment, a thoughtful comparison of these experimental data is expected before fitting the electrostrictive coefficients. λ L P λ1l 1 Φ λl X X P 1 P Figure 1 the free energy model of DE electromechanical coupling system In this paper, we present a physical study about the impact of both deformation and external electric field on the electrostriction in DE actuation, which might help to select the desirable experimental data. By using the free energy model, we study the critical stretch ratio in the actuation. In the pages that follow, a free energy model, involving both Maxwell stress and electrostriction, is proposed in section. Section will discuss the electrostriction physically to evaluate the effect on the actuation and instability, and Section 4 & 5 will summarize the results and conclude with discussion. Proc. of SPIE Vol Downloaded From: on 08//014 Terms of Use:

3 . FREE ENERGY MODEL INVOLVING BOTH MAXWELL STRESS AND ELECTROSTRICTION For DE polymers (M, VHB series) which are non-polar in the absence of voltage, the electro-stress is mainly referred to Maxwell stress or electrostriction or both. Maxwell stress, also known as electrostatic pressure, is induced by the charges accumulated on both electrode surfaces. The unlike charges on the opposing electrodes attract each other and form a compressive stress in the vertical direction on the film, as shown in Figure 1. However, electrostriction is strongly dependent of atomic aligning; charged particles inside the dielectric tend to displace relative to one another on the present of voltage, and this process is companied by the elongation of the material in the vertical direction. Therefore, Maxwell stress tends to reduce the film thickness, while electrostriction would thicken the film. Actuation under the two stress origins have not been fully investigated, and the misunderstandings might lead to the imprecise research of DE actuation and instability. The free-energy model is employed to analyze the transformation between electric and mechanical energy in DE actuation [16-18, ]. Similarly, we will study the electro-stress involving both Maxwell stress and electrostriction to study the instability with a proper physical explanation, and we obtain the free-energy model as D 1 1 W ( λ1, λ, λ, D% ) = WS + % λ1 λ λ ε (1) where W is the elastic stain energy density function, only dependent on stretch ratio λ at each of the S i three directions, ε = εε 0 r is the DE permittivity, ε ε is the dielectric constant of DE, D= Q/ ( LL ) r and D Q/ ( λlλ L ) 1 =. F/m is permittivity of vacuum, % is the nominal electric displacement = 1 1 is the true electric displacement. Define % as the true electric field. field, and E =Φ/ ( λ L ) 1 E % =Φ/ L as the nominal electric In this paper, we will investigateε when it is strongly dependent of both deformation and electric r field. In electrostriction theory [19], the relation ofε r and strain ξ ij is expressed as ε = εδ + aξ + a ξ δ rij r ij 1 ij kk ij () Proc. of SPIE Vol Downloaded From: on 08//014 Terms of Use:

4 where ε r is the dielectric constant of the un-deformed state, a1 and a is electrostrictive coefficients usually obtained from experimental data, δij is the kronecker symbol. If the elastic polymer is not electrostrictive, a1 and a vanishes, indicating the dielectric constant is independent of deformation and only Maxwell stress exists. In Eq.(), by introducing a 1 and a the electrostriction is considered as well as Maxwell stress. Supposing our DE is linear dielectric, the electric field relates to the electric displacement as [19] D = ε E α αβ β β () In our DE electromechanical system, the electric field and displacement exist only on the x direction, thus Eq.() reduces to Forξ λ 1 i =, we obtain i ( + ) ε = ε0εr = ε0 εr + a1ξ + a ξ11 + ξ ξ (4) ( 1) ( ) ε = ε0 εr + a1 λ + a λ1+ λ + λ (5) and W W s D% 1 D% σ λ 1 λ λ a ε 0 λ 1 λ % = = λ λ1 λ1 ε ε W W s D% 1 D% σ 1 1 λ 1 λ λ a ε 0 λ 1 λ % = = λ λ λ ε ε W W s D% 1 1 D% 1 1 % σ = = + λ1 λ ε 0( a1+ a) λ1 λ λ λ λ ε ε W D% 1 1 E% = = λ1 λ λ D% ε (6) (7) (8) (9) where % σ, i = 1,, are the nominal stresses. Recall the definition of nominal value and true value, Eq.(6-9) become i Proc. of SPIE Vol Downloaded From: on 08//014 Terms of Use:

5 σ σ λ λ % λ λ % ε λ λ P 1 W s 1 1 D D a = = % = 0 1 λlλl λλ λ1 ε ε σ σ λ λ % λ λ % ε λ λ P 1 W s 1 1 D D a = = % = 0 1 λ1l1λl λλ 1 λ ε ε σ σ λ λ % λ λ % ε λ λ λ P 1 W s 1 1 D D = = % = ( a1+ a) 1 λ1l1λl λλ 1 λ ε ε (10) (11) (1) Φ E% D% 1 1 D E = = = λ1 λ = λ L λ ε ε Eq.(1) reveals the relation between the true electric field and displacement, and it fully satisfies the assumption in Eq.(). Therefore, we have developed a free energy model involving both Maxwell stress and electrostrictive effect. In the next section, we will calculate the values of each electro-stress and critical values in DE actuation and instability. (1). ELECTROSTRICTION IN DE ACTUATION.1 Electrostrictive coefficients in isotropic DE polymer Consider a case when P1=P=P, P=0 and DE is isotropic, the stretch ratios in the planar direction are equal, which can be denoted as λ = λ1 = λ.thus, Eq.(1) becomes % = ε λλ 1 = ε λ. D E E The Poisson ratio of VHB is 0.49 [], which allow us to view the elastomer as incompressible. Therefore we have λ 1/ σ. = λ and ε = ε0 εr + aλ+ ( a1+ a) λ a1a The true stresses in Eq.(10) and (11) become Ws 0= λ εε ( ( ) 0 re + ε0e a1+ 6a a1+ a λ 5aλ) λ (14) In Eq.(14), the electro-stress in the thickness direction is composed of two origins; Maxwell stress () 1 = εεe, which is used to represent as the electro-stress in previous study, and 0 electrostriction ( ) ( 6 ( ) 5 ) 0E a1 a a1 a a σ = ε + + λ λ. In the next section, we will present the research of electrostrictive coefficients and estimate the value of electrostriction in large deformation. Proc. of SPIE Vol Downloaded From: on 08//014 Terms of Use:

6 A lot of experiments have been conducted to study the dielectric constant on different pre-stretch ratios, but these groups of data differ greatly from one to another [11-1]. We compare their research and investigate the reason in their discrepancy. Wissler et al. [1] and Kofod et al. [1] constructed the DE actuator as a planer capacitor and measured dielectric constant under four pre-stretch levels, ranging from 100% to 400%. This is a conventional method for dielectric constant measurement and the effect of frequency is inevitably involved in the results. Note that our research would mainly limit on the actuation under DC voltage, the AC supply and its effect of frequency on instability should be eliminated. Moreover, the dielectric constant varies as a result of polarization. For DE polymers, the aligning of dipoles on each molecular chain is not only dependent of deformation when it reaches the expansion limit, but strongly related to the external electric field as well. Considering our DE actuator undergoes both deformation and high electric field, the dielectric constant should be measured under both conditions, while the data obtained in Ref [1-1] were resulted with voltage as low as 1-V. Figure the variation of dielectric constant at different pre-stretch levels [0] Figure illustrates the result of VHB dielectric constant when the area was stretched [0]. The dash dot line shows the dielectric constant values when a high voltage (kv) was applied, and figure of merits of the solid line and dash line were obtained from equivalent capacitor with golden and carbon grease, respectively. Recall that DE actuator works under a relative high voltage (>kv), in the study of the change of DE dielectric constant, we should consider the dependence of both high voltage and deformation. In our research presented, we adopt the experiment data of the dash dot line and obtain a 1 = and a = by curve fitting.. Effect of electrostriction on DE deformation Using Neo-Hookean strain energy function Proc. of SPIE Vol Downloaded From: on 08//014 Terms of Use:

7 where 4 ( λ1 λ λ ) ( λ λ ) WS = μ + + = μ + 6 μ = N m [14] is the shear modulus, and we express Eq.(14) as 10 / 5 ( ) 0 r 0 ( 1 ( 1 ) ) o= μ λ λ ε ε E + ε E a + 6a a + a λ 5a λ (15) (16) Let En = E/ μ / ε 0 represents the normalized true electric field, Eq.(16) becomes E n = 4 ( λ λ ) ( a1 a ( a1 a) a ) ε λ 5 λ r (17) Since E % = Eλ = Eλ, the normalized nominal electric field can be written as E% n = λ 4 ( λ λ ) ( a1 a ( a1 a) a ) ε λ 5 λ r (18) Figure plots normalized nominal electric field as stretch ratio increasing from 1 to 5. The peak maximum values of the two curves in Figure indicate the maximum nominal electric fields that DE undergoes. In Figure the peak value of dash line is greater than that of the solid line, which indicates the electrostriction help to reduce the energy consumed by the DE in its actuation. 0.5 Nominal electric field E ~ /(μ/ε 0 ) 1/ Maxwell stress + electrostriction Only Maxwell stress Stretch ratio λ Figure With/without electrostriction, normalized nominal electric field vs. stretch ratio Proc. of SPIE Vol Downloaded From: on 08//014 Terms of Use:

8 Maximizing E % in Eq.(5), we obtain the critical stretch ratios λ = 1. 7 and λ = 1. 59, which c c correspond the % 8 max V/m and E = % 8 max V/m with and without E = electrostriction, respectively. When the electrostrictive effect is involved, the maximum deformation before eletromechanical instability is improved from 1.6 to 1.7 which suggests that electrostriction may help DE actuators undergoing large deformation...5 Maxwell stress + electrostriction Only Maxwell stress True electric field E/(μ/ε 0 ) 1/ Stretch ratio λ Figure 4 With/without electrostriction, normalized true electric field vs. stretch ratio Figure 4 shows the normalized true electric field in DE deformation. The curves in Fig 5 reveal three phenomenons, listed as follows: The true electric field is a monotonic function of the stretch ratio, whether electrostriction involved or not. This is due to the positive feedback produced at constant voltage, and would eventually leads to the electromechanical instability. For stretch under 1.5, it is reasonable to neglect the electrostrictive effect and take the dielectric constant as a fixed value of 4.7. However, if the dielectric elastomer undergoes deformation larger than, the electrostriction become pronounced, and requires to be considered as part of the eletro-stress. At a fixed stretch ratio value, higher electric field is required in previous study where dielectric constant of 4.7 was employed. This indicated that when involving the electrostriction, lower voltage is required in DE actuators application. Proc. of SPIE Vol Downloaded From: on 08//014 Terms of Use:

9 . Electromechanical instability In order to study the electromechanical instability, We express W W W = = λ λ λ 1 W W W = = λ, 1 D % λ D% λ D% as W 10D% 4D% D% ( ) λ ε ε ε = μ 1+ 5λ + λ + λ M + λ M W W D % D % = = λ λ M λ D% D% λ ε ε (19) (0) W λ = ε 4 D % (1) M = ε a a + a Where ( ) and we might rewrite H as λ H W W λ λ D% W W D% λ D% = () The stable actuation requires W > 0 λ, W > 0 D % () W W W > λ D% λ D % 8 Substituting D % = ελ E % into Eq. (4) and (4), we obtain λ E% μ( 1+ 5λ ) + 10ελ E% + 10ελ ME% + M > 0 ε (4) (5) λ ε 4 > 0 (6) Proc. of SPIE Vol Downloaded From: on 08//014 Terms of Use:

10 17 μ λ M ( λ + 5λ ) λ E% E% > 0 ε ε (7) as the stable actuation requirement, which is plotted in Figure 5. x Nominal electric field E ~ /(μ/ε 0 ) 1/ Stable domain Stretch ratio λ Figure 5 Stable domain for DE actuation The area below the curves in Figure 5 shows the stable domain in DE electromechanical coupling actuation, which may serve as criteria for the actuator design and safe working. 4. CONCLUSION The change of dielectric constant is affected not only by the deformation but also related to the external voltage that applied on the DE surfaces. In this case, we fit the electrostrictive coefficients with selected experimental data, which were measured as the elastomer at different stretch level and subjected to kv DC voltage. Electrostriction, generated in the actuation, upgrades the critical stretch ratio, which could help improve the stability in DE actuation. The DE actuation and stability is closely related to the electro-stress in its deformation. The electrostriction, as a result of change in material dielectric constant, is originated by the polarization and alignment of dipoles, and is affected by various factors as temperature and voltage frequency. In this DE film we investigated in the paper, there is no permanent dipole at the absence of electric field, but situations may become complicated when the polymer is heated and the dielectric relaxation is unavoidable at high frequency. Further study, both theoretically and experimentally, has been carried out about the electrostriction at complicated excitation. Proc. of SPIE Vol Downloaded From: on 08//014 Terms of Use:

11 5. ACKNOWLEDGEMENT This research is supported by the National Natural Science Foundation of China (Grant No ). Bo Li acknowledges China Scholarship Council for supporting his 1-month-visting at Harvard University, and thanks for group members in Suo s group for their constructive comments of this paper. REFERENCE 1. Ron Pelrine, et al., High-Speed Electrically Actuated Elastomers with Strain Greater Than 100%, 87 Science, 86 (000).. Kwangmok Jung, et al., Artificial annelid robot driven by soft actuators, Bioinsp. Biomim. (007) S4 S49.. Elaine Biddiss, Tom Chau, Dielectric elastomers as actuators for upper limb prosthetics: Challenges and opportunities, Medical Engineering & Physics 0 (008) Gabor Kovacs, Patrick Lochmatter, Michael Wissler, An arm wrestling robot driven by dielectric elastomer actuators, Smart Mater. Struct. 16 (007) S06 S Federico Carpi, Claudio Salaris, Danilo De Rossi, Folded dielectric elastomer actuators, Smart Mater. Struct. 16 (007) S00 S05 6. Andreas Wingert, Matthew D. Lichter, Steven Dubowsky, On the Design of Large Degree-of-Freedom Digital Mechatronic Devices Based on Bistable Dielectric Elastomer Actuators, IEEE/ASME TRANSACTIONS ON MECHATRONICS, VOL. 11, NO. 4, AUGUST 006, Arun Rajamani, Michael D. Grissom, Christopher D. Rahn, Qiming Zhang, Wound Roll Dielectric Elastomer Actuators: Fabrication, Analysis, and Experiments, IEEE/ASME TRANSACTIONS ON MECHATRONICS, VOL. 1, NO. 1, FEBRUARY 008, Stéphanie P. Lacour, Harsha Prahlad b, Ronald Pelrine b, Sigurd Wagner, Mechatronic system of dielectric elastomer actuators addressed by thin film photoconductors on plastic, Sensors and Actuators A 111 (004) Federico Carpi and Danilo De Rossi, Biomimetic Dielectric Elastomer Actuators, The First IEEE/RAS-EMBS International Conference on Biomedical Robotics and Biomechatronics, 006, 10. Xuanming Pang, Bo Li, Dongmei Xia, Sufang Jing, Application of Dielectric Elastomer Planar Actuators in a Micropump Chip, 4th IEEE Conference on Industrial Electronics and Applications, 009. ICIEA H. R. Choi, K. Jung, N. H. Chuc, M. Jung, I. Koo, J. Koo, J. Lee, J. Lee, J. Nam, M. Cho, and Y. Lee, in Effects of prestrain on behavior of dielectric elastomer actuator, San Diego, CA, USA, 005 (SPIE), p M. Wissler and E. Mazza, Electromechanical coupling in dielectric elastomer actuators,sensors and Actuators: A. Physical 18, 84-9 (007). Proc. of SPIE Vol Downloaded From: on 08//014 Terms of Use:

12 1. G. Kofod, P. Sommer-Larsen, R. Kornbluh, and R. Pelrine, Actuation response of polyacrylate dielectric elastomers,journal of Intelligent Material Systems and Structures 14, (00) 14. Xuanhe Zhao and Zhigang Suo, Electrostriction in elastic dielectrics undergoing large deformation, JOURNAL OF APPLIED PHYSICS 104, 150 (008) 15. Jinsong Leng, Liwu Liu, Yanju Liu, Kai Yu, and Shouhua Sun, Electromechanical stability of dielectric elastomer, APPLIED PHYSICS LETTERS 94, 11901(009) 16. Xuanhe Zhao and Zhigang Suo. Method to analyze electromechanical stability of dielectric elastomers. APPLIED PHYSICS LETTERS 91, (007) 17. R. Díaz-Calleja,E. Riande, and M. J. Sanchis. On electromechanical stability of dielectric elastomers. APPLIED PHYSICS LETTERS 9, (008) 18. Xuanhe Zhao, Wei Hong, and Zhigang Suo. Electromechanical hysteresis and coexistent states in dielectric elastomers. PHYSICAL REVIEW B 76, 1411 (007) 19. John David Jackson. Classic Electrodynamic, Springer Netherlands (Print) (Online)Volume 9, Number 1 (1999 ) 0. T. G. McKay, E. Calius, and I. Anderson, The Dielectric Constant of M VHB: a Parameter in Dispute, Proc. of SPIE Vol. 787, 7870P 1. Jean-Sebastien Plante, Steven Dubowsky, Large-scale failure modes of dielectric elastomer actuators, International Journal of Solids and Structures 4 (006) Y. Liu, L. Liu, S Sun and J. Leng, An investigation on electromechanical stability of dielectric elastomer undergoing large deformation, Smart Materials & Structures. 18, (009). Design and fabrication of a microfluidic chip driven and adhered by dielectric elastomers. Bo Li, et.al,. [C] Proc. of SPIE 009 Vol S(1-9) Proc. of SPIE Vol Downloaded From: on 08//014 Terms of Use: