Modeling guided design of dielectric elastomer generators and actuators

Tiefeng Lia,b,* , Shaoxing Qua , Christoph Keplingerb,c, Rainer Kaltseisc, Richard Baumgartnerc,

Siegfried Bauerc, Zhigang Suob and Wei Yanga

aInstitute of Applied Mechanics, Zhejiang University, 38 Zheda Road, Hangzhou 310027, China bSchool of Engineering and Applied Sciences, Harvard University, Cambridge, MA 02138, USA

cSoft Matter Physics, Johannes Kepler University, Altenbergerstrasse 69, A-4040 Linz, Austria

ABSTRACT

Mechanical energy and electrical energy can be converted to each other by using a dielectric elastomer transducer. Large voltage-induced deformation has been a major challenge in the practical applications. The voltage-induced deformation of dielectric elastomer is restricted by electromechanical instability (EMI) and electric breakdown. We study the loading path effect of dielectric elastomer and introduce various methods to achieve giant deformation in dielectric elastomer and demonstrate the principles of operation in experiments. We use a computational model to analyze the operation of DE generators and actuators to guide the experiment. In actuator mode, we get three designing parameters to vary the actuation response of the device, and realize giant deformation with appropriate parameter group. In the generator mode, energy flows in a device with inhomogeneous deformation is demonstrated. Keywords: Dielectric elastomer, electromechanical stability, large deformation, energy harvesting

1. INTRODUCTION Dielectric elastomer (DE) is one of the most promising electroactive polymer material for stretchable transducers.

DE is capable of larger deformation, high energy density, light weight and rust free. When an electric field is applied through the thickness direction, a DE membrane reduces its thickness and expands its area. This process has been widely explored in applications including actuators, sensors and generators [1-15]. Electromechanical instability, electric breakdown and rupture have significantly influenced the performances of DE actuators and generators and will lead to the failure of the whole system. Recently, a lot of research has been focused on investigating the failure mechanism of DE membrane in different operation modes, designing new material and structures to achieve desirable performances of DE actuators and generators and averting failures during operation [13-36]. DE generators (DEGs) operate from various sources have also been widely studied. Recently, experiments and theoretical investigations monitor electrical and mechanical energy flows and show the cycle of energy conversion in work-conjugate planes [31,37]. Figure 1 shows mechanisms of the two different modes of DE transducers: the actuator mode and the generator mode. This paper present an analytical model with nonlinear theory of DE with inhomogeneous deformation, with particular configuration referred to the DE membrane inflation system in both the actuator mode and the generator mode. Theoretical studies and numerical simulations guide the design of experiments. Large voltage-induced deformation and energy harvesting with two capacitors are realized. The work is presented as follows. Section 2 derives the governing equations with consideration of the nonlinear theory of soft dielectrics. Section 3 studies the large voltage-induced deformation in the actuator mode. Section 4 studies the energy harvesting in the generator mode.

Electroactive Polymer Actuators and Devices (EAPAD) 2012, edited by Yoseph Bar-Cohen, Proc. of SPIE Vol. 8340, 83401X · © 2012 SPIE · CCC code: 0277-786X/12/$18 · doi: 10.1117/12.915424

Proc. of SPIE Vol. 8340 83401X-1

Figure 1. A membrane of a dielectric elastomer is sandwiched between two compliant electrodes. (a) In the actuator mode, electrical energy is transferred to mechanical energy. When a voltage is applied, the DE membrane expands in area and reduces in thickness. (b) In the generator mode, mechanical energy is transferred to electrical energy.

2. GOVERNING EQUATIONS

Governing equations for DE membranes inflation are well established, as shown in Figure 2, ( )Rr and ( )Rz specify the deformation of the membrane. Consider a ring element Rd of the membrane with radius R in the reference state. In a deformed state, the ring element deforms to radius ( )Rr and ( )Rz . The deformation causes the latitudinal

stretch Rr /2 =λ . In the reference state, the material particles between the two circles form an annulus of width Rd .

In the deformed state, these material particles form a circular band of width Rd1λ , where 1λ is the longitudinal stretch.

Let ( )Rθ be the slope of a membrane at material particle R . The geometry dictates that θλ sindd 1=Rz ,

θλ cosdd 1=Rr . With the thickness H and radius A in the reference state, in the deformed state. Because the

DE membrane is incompressible, the thickness at of the membrane is 21λλHh = .

Figure 2. A membrane of a dielectric elastomer is sandwiched between two compliant electrodes.(a) DE membrane inflated slightly by pressure (b) DE membrane undergoes large deformation when certain voltage and pressure is applied (c) The stress-free state of the DE membrane. (d) The axisymmetric stressed state of the ring element during inflation.

When the membranes are inflated, the stretches are inhomogeneous in the membranes, and are described by functions ( )R1λ and ( )R2λ . In a deformed state, the membrane is subject to longitudinal stress ( )R1σ and latitudinal stress ( )R2σ . Adopting the previous studies on DE membrane inflation actuators [38,39], one may write the mechanical

Proc. of SPIE Vol. 8340 83401X-2

equilibrium of the membrane as

( ) 0dsind 1 =− rprhr θσ . (1)

( ) 0sin

cosd 21 =−+θ

σθσ dzhprdzhr . (2)

Following the equation of state for soft dielectric membrane characterized by Gent material model[40], one obtains the explicit expressions for the stresses 1σ and 2σ [39]:

( ) 2

221

2

22

21

22

21lim

22

21

21lim

1 3λλε

λλλλλλλμσ ⎟

⎠⎞

⎜⎝⎛ Φ−

−−−+−

= −−

−−

HJJ

(3)

( ) 2

22

1

2

22

21

22

21lim

21

22

22lim

2 3λλε

λλλλλλλμσ ⎟

⎠⎞

⎜⎝⎛ Φ−

−−−+−

= −−

−−

HJJ

(4)

As suggested in the experiments of the DE elastomers[41], we choose the permittivity F/m1016.4 11−×=ε , the shear modulus kPa45=μ and the stretch limit 270lim =J . The following dimensionless variables are used: the

voltage HμεΦ=Φ~ , the pressure ( )HpAp μ=~ , the volume 3~ AVV = and the charge

( )εμ2~ AQQ = . We solve equations (1)-(4) under the boundary conditions for axi-symmetric membrane inflation to give the states of equilibrium during DE membrane deformation in both actuator mode and generator mode.

3. LARGE VOLTAGE-INDUCED DEFORMATION In the actuator mode, voltage is applied to the DE membrane inflation system to actuate the DE membrane. We

demonstrate the large voltage-induced deformation in the DE membrane inflation system. Several aspects enable the voltage to trigger giant deformation. First, the DE membrane is pressurized to the state near the verge of instability. Second, voltage is applied to trigger the instability. Third, the deformation of the DE membrane is restrained after snaps to a stable state without electric breakdown. These aspects are examined in this section by using numerical simulation. Instead of pushing the membrane very close to the verge of instability, we may restrain the excessive deformation by bending the snap-through path. These operation principles can be accomplished by the volume of the chamber, CV (as shown in Figure 2 (a) and (b)). When a membrane is mounted on the chamber, the membrane is flat, and the pressure in the chamber is the same as the atmosphere pressure, atmp . Then air is pumped into the chamber through a valve, the

membrane is inflated into a balloon of volume 0V , and the pressure in the chamber becomes atm0 pp + . The valve is subsequently closed, fixing the amount of air enclosed by the chamber and the membrane. When the voltage Φ is applied, the membrane is further inflated with the volume under the membrane V , and the pressure in the chamber becomes atmpp + . Because the amount of air enclosed by the chamber and the balloon remains unchanged, the ideal-gas law requires that

( )( ) ( )( )C0atm0Catm VVppVVpp ++=++ . (5) As the volume of the balloon increases, the pressure p drops. For a chamber of a given volume, the pressure-volume curve of the air (5) is a curve on the pressure-volume plane, representing the snap-through path. We fix the dimensionless value ( ) 50//atm =AHp μ and the initial volume of the balloon 42.2/ 3

0 =AV for a chamber of volume

1600/ 3C =AV . We solve equations (1)-(5) to give the states of equilibrium. The pressure-volume curve of the air

(the dashed line) intersects with the pressure-volume curves of the membrane at several points, each defining a state of equilibrium (Figure 3(c)). The balloon is in state A before the voltage is applied. When the voltage is turned on and ramped up, the balloon expands further, and the pressure-volume curves of the balloon lowers. When the voltage reaches a certain level, the pressure-volume curve of the air is tangential to the pressure-volume curve of the balloon at B, and the balloon snaps to state C. As the voltage increases, the balloon expands further, until it suffers electric breakdown. If

Proc. of SPIE Vol. 8340 83401X-3

the voltage is ramped down from state C, the balloon will shrink back to D when voltage drops to zero. These states of equilibrium are also shown in the voltage-volume plane (Figure. 3(d)). The chamber restrains the excessive deformation after the balloon snaps. From state A to C, the voltage-triggered expansion of area of the DE membrane is around 1692%, large voltage-induced deformation is achieved.

Figure 3. Experimental results [41]: (a) balloon shapes correspond to the states during actuation. (b) Numerical results for a membrane mounted on a chamber of volume [39]. (c) Pressure-volume curves of the membrane at several values of voltage. The dashed line is the pressure-volume curve of the air. (d) Voltage-volume curve.

4. ENERGY HARVESTING

Dielectric elastomer generators convert mechanical work into electrical energy. An experiment to monitor the electrical and mechanical energy flows is demonstrated using the DE membrane inflation system[37]. The experiment determines the specific electrical energy generated per cycle, and the mechanical to electrical energy conversion efficiency. The generator is operated between two charge reservoirs of different voltages, and describes the cycle of energy conversion in work-conjugate planes. To characterize and investigate the harvesting process, an analytical model for the membrane inflation DEG with inhomogeneous deformation is presented [42]. Attention is focused on the harvesting cycle in the experiment. As shown in Figure 4 (a) and (b), the harvesting cycles of DEG are plotted in the pressure-volume and voltage-charge work-conjugate planes.

Figure 4. (a) and (b) show the harvesting process of the DEG in the work conjugate planes. The yellow area depicts the mechanical energy consumed and that generates electrical energy. Green area shows the mechanical energy lost during unloading. (c) inhomogeneous fields on the DE membrane of four states 1-4 during the harvesting process [42].

The area enclosed by the contour represents the consumed mechanical energy. Ignoring the dissipative effects

such as viscoelasticity or current leakage, one may use the yellow area in Figure 4(b) to quantify the generated electrical energy, which quantitatively equals to the energy of the yellow area in Figure 4(a). Accordingly, for volume controlled loading/unloading cycle, the theoretical conversion efficiency may reach 100%. However, during the inflation stage of experiments, the pressure keeps dropping. The unloading path fails to follow the

Proc. of SPIE Vol. 8340 83401X-4

N-shaped equation of state (the blue curve linking state 3 and state 4). Snap-back would take place during the deflation, as depicted by the black arrow. The deflation from state 3 to state 4 first follows the black arrow and then the blue curve after the intersection. The harvesting cycle is redefined by this snap-back unloading path. The consumed mechanical energy is the yellow area plus the green area. The latter stands for the mechanical energy loss that transfers to the kinetic of the air and heat loss, instead of contributes to the output electrical energy. The predicted energy conversion efficiency decreases to 67%. This result indicates that instabilities degrade the efficiency of energy conversion and the specific electrical energy generated per cycle. Figure 4(c) shows the inhomogeneous deformation of the DE membrane in different states during harvesting. The stretch monotonically increases from the minimum at the outer end to the maximum at the apex of the membrane. The DE membrane undergoes the highest stretches in state 2. Care is needed to restrict the harvesting cycle in a safe operation range to avert electric breakdown and rupture of the DE membrane. The analysis may guide the design of dielectric elastomer generators.

5. CONCLUSIONS

Dielectric elastomers are susceptible to electromechanical instability and electric breakdown. The instability can be harnessed to achieve giant voltage-induced deformation. We demonstrate large voltage induced actuation and energy harvesting with a commonly used experimental setup—a dielectric membrane inflation system. The membrane is inflated by pressurizing the chamber, and by applying voltage through the thickness of the membrane. Numerical simulation captures inhomogeneous deformation, the bifurcation diagrams and the energy harvesting cycle. The theoretical prediction has guided us to perform the system in both the actuator mode and the generator mode with suitable values of the parameters and loading paths. High voltage-induced deformation and energy harvesting cycle are achieved in different modes by designing certain loading paths. The contour plots of fields may guide the design and the operation of DEGs to avert failures.

REFERENCES

[1] Yoseph Bar-Cohen, “Electroactive Polymer (EAP) Actuators as Artificial Muscle: Reality, Potential and Challenges”, SPIE Press monograph, 482 (2001).

[2] R. Pelrine, R. Kornbluh, Q. B. Pei, and J. Joseph, “High-speed electrically actuated elastomers with strain greater than 100%”, Science, 287, 836 (2000).

[3] F. Carpi and D. De Rossi, “Contractile dielectric elastomer actuator with folded shape”, IEEE Trans. Dielectr. Electr. Insul. 12, 835 (2005).

[4] G. Kofod, M. Paajanen, and S. Bauer, “Self-organized minimum-energy structures for dielectric elastomer actuators”, Applied Physics A: Materials Science & Processing, 85,141 (2006).

[5] F. Carpi, G. Gallone, F. Galantini, and D. Rossi, “Silicone–poly(hexylthiophene) blends as elastomers with enhanced electromechanical transduction properties”, Adv. Funct. Mater., 18, 235–241 (2008).

[6] F. Carpi, G. Frediani S. Tarantino, and D. Rossi, “Millimetre-scale bubble-like dielectric elastomer actuators”, Polym Int., 59, 407–414 (2010).

[7] L. Liu, J. Fan, Z. Zhang, L. Shi, Y. Liu and J. Leng, “Analysis of the novel strain responsive actuators of silicone dielectric elastomer”, Adv. Mater. Res. 47-50, 298 (2008).

[8] R. E. Pelrine, R. D. Kornbluh, and J. P. Joseph, “Electrostriction of polymer dielectrics with compliant electrodes as a means of actuation”, Sensors & Actuators: A. Physical, 64, 77 (1998).

[9] J. S. Plante and S. Dubowsky, “On the properties of dielectric elastomer actuators and their design implications”, Smart Materials and Structures, 16, S227 (2007).

[10] M. Wissler and E. Mazza, “Mechanical behavior of an acrylic elastomer used in dielectric elastomer actuators”, Sensors & Actuators: A. Physical, 120, 185 (2007).

[11] T. B. Xu and J. Su, “heoretical modeling of electroactive polymer-ceramic hybrid actuation systems”, J. Appl. Phys., 97, 034908 (2005).

Proc. of SPIE Vol. 8340 83401X-5

[12] E. M. Arruda and M. C. Boyce, “On the dynamic electromechanical loading of dielectric elastomer membranes”, J. Mech. Phys. Solids, 41, 389 (1993).

[13] J. S. Plante and S. Dubowsky, “Large-scale failure modes of dielectric elastomer actuators”, Int. J. Solids Struct., 43, 7727 (2006).

[14] N. C. Goulbourne, E. M. Mockensturm, and M. I. Frecker, “Electro-elastomers: large deformation analysis of silicone membranes”, Int. J. Solids Struct. 44, 2609 (2007).

[15] L. Patrick, K. Gabor, and M. Silvain, “Characterization of dielectric elastomer actuators based on a hyperelastic film model”, Sens. Actuators, A 135,748 (2007).

[16] E. Corona and S. Kyriakides, “On the collapse of inelastic tubes under combined bending and pressure”, Int. J. Solids Struct., 24, 505 (1998).

[17] G.Kofoda, W. Wirges, M. Paajanen and S. Bauer, “Energy minimization for self-organized structure formation and actuation“, Appl. Phys. Lett., 90, 081916 (2007).

[18] E. M. Mockensturm and N. Goulbourne, “Dynamic response of dielectric elastomers”, Int. J. Non-Linear Mech. 41, 388 (2006).

[19] X. Zhao and Z. Suo, “Method to analyze electromechanical stability of dielectric elastomers”, Appl. Phys. Lett., 91, 061921 (2007).

[20] X. Zhao, W. Hong, and Z. Suo, “Electromechanical coexistent states and hysteresis in dielectric elastomers “, Physical Review B, 76, 134113 (2007).

[21] Z. Suo, X. Zhao, and W. H. Greene, “A nonlinear field theory of deformable dielectrics”, J. Mech. Phys. Solids, 56, 476 (2008).

[22] J. Zhou, W. Hong , X. Zhao , Z. Zhang and Z. Suo , “Propagation of instability in dielectric elastomersJ. Solids Struct”, I. J. Solids Struct, 45, 3739 (2008).

[23] X. Zhao and Z. Suo, “Electrostriction in elastic dielectrics undergoing large deformation”, J. Appl. Phys., 101, 123530 (2008).

[24] Y. Liu, L.Liu, Z. Zhang, L.Shi, and J. Leng, “Comment on “Method to analyze electromechanical stability of dielectric elastomers”, Appl. Phys. Lett. 93, 106101 (2008).

[25] A. N. Norrisa, “Comment on “Method to analyze electromechanical stability of dielectric elastomers””, Appl. Phys. Lett. 92, 026101 (2007).

[26] R. Díaz-Calleja, E. Riande, and M. J. Sanchis, “On electromechanical stability of dielectric elastomers”, Appl. Phys. Lett., 93, 101902 (2008).

[27] G. Kofod, P. Sommer-Larsen, R. Kronbluh, and R. Pelrine, “Actuation response of polyacrylate dielectric elastomers”, J. Intell. Mater.Syst. Struct , 14, 787(2000).

[28] Y. Liu, L. Liu, S. Sun, L. Shi, and J. Leng, “Comment “On electromechanical stability of dielectric elastomers””, Appl. Phys. Lett., 94, 096101 (2009).

[29] J. Leng, L. Liu, Y. Liu, K Yu, and S Sun, “Electromechanical Stability of Dielectric Elastomer”, Appl. Phys. Lett., 94, 211901 (2009).

[30] Y. Liu, L. Liu, S. Sun, and J. Leng, Stability analysis of dielectric elastomer film actuator. Science China Series E, 52(9): 2715-2723 (2009).

[31] S. Adrian Koh, Xu. Zhao, and Z. Suo “Maximal energy that can be converted by a dielectric elastomer generator”, Applied Physics Letters 94, 262902 (2009).

[32] B. Li, L. Liu and Z. Suo, “Extension limit, polarization saturation, and snap-through instability of dielectric elastomers”, International Journal of Smart and Nano Materials, 2, 59-67 (2011).

[33] Z. G. Suo, Theory of dielectric elastomers. Acta Mechanica Solida Sinica, 23, 6, 549-578, (2010). [34] L. W. Liu, Y. J. Liu, Z. Zhang, B. Li, J. S. Leng, “Electromechanical stability of electro-active silicone filled with

high permittivity particles undergoing large deformation”, Smart Materials ＆ Structures, 19, 115025 (2010). [35] Y. J. Liu, L. W. Liu, Z. Zhang, Y. Jiao, S. H. Sun, J. S. Leng, “Analysis and manufacture of an energy harvester

based on a Mooney-Rivlin–type dielectric elastomer”, Europhysics Letters, 90, 36004 (2010). [36] Keplinger C., Kaltenbrunner M., Arnold N., and Bauer S. Röntgen’s, “electrode-free elastomer actuators without

electromechanical pull-in instability”, PNAS, 107, 4505 (2010). [37] R. Kaltseis, C. Keplinger, R. Baumgartner, M. Kaltenbrunner, T.F. Li, P. Machler, R. Schwodiauer, Z.G. Suo, S.

Bauer, “Method for measuring energy generation and efficiency of dielectric elastomer generators”, Appl. Phys. Lett., 99, 162904 (2011).

Proc. of SPIE Vol. 8340 83401X-6

[38] J. Zhu, S.Q. Cai, Z.G. Suo, “Resonant behavior of a membrane of a dielectric elastomer”, Int. J. Solids Struct., 47, 3254-3262 (2010).

[39] T.F Li, C. Keplinger, R. Baumgartner, S. Bauer, W. Yang, Z.G. Suo, “Giant voltage-induced deformation in dielectric elastomers near the verge of snap-through instability”, submitted for publication.

[40] A.N. Gent, 1996. A new constitutive relation for rubber. Rubber Chem Technol 69, 59-61(1996). [41] C. Keplinger, T.F Li, R. Baumgartner, Z.G. Suo, S. Bauer, “Harnessing snap-through instability in soft dielectrics

to achieve giant voltage-triggered deformation”, Soft Matter 8, 285-288 (2012). [42] T.F Li, S.Q. Qu, W. Yang, “Energy harvesting of dielectric elastomer generators concerning inhomogeneous

fields and viscoelastic deformation”, in preparation.

Proc. of SPIE Vol. 8340 83401X-7