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Theoreticalmodelanddesignofelectroadhesivepadwithinterdigitatedelectrodes

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DOI:10.1016/j.matdes.2015.09.162

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Materials and Design 89 (2016) 485–491

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Materials and Design

j ourna l homepage: www.e lsev ie r .com/ locate / jmad

Theoretical model and design of electroadhesive pad withinterdigitated electrodes

Changyong Cao a,b,c,⁎, Xiaoyu Sun c,d, Yuhui Fang a,b, Qing-Hua Qin a, Aibing Yu e, Xi-Qiao Feng c

a Research School of Engineering, Australian National University, Canberra, ACT 0200, Australiab Department of Mechanical Engineering and Materials Science, Duke University, Durham, NC 27708, USAc Institute of Biomechanics and Medical Engineering, Department of Engineering Mechanics, Tsinghua University, Beijing 100084, Chinad Department of Engineering Mechanics, School of Civil Engineering, Wuhan University, Wuhan 430072, Chinae Department of Chemical Engineering, Monash University, Clayton, VIC 3800, Australia

⁎ Corresponding author at: Department of MechaniScience, Duke University, Durham, NC 27708, USA.

E-mail address: changyong.cao@duke.edu (C. Cao).

http://dx.doi.org/10.1016/j.matdes.2015.09.1620264-1275/© 2015 Elsevier Ltd. All rights reserved.

a b s t r a c t

a r t i c l e i n f o

Article history:Received 24 May 2015Received in revised form 28 September 2015Accepted 29 September 2015Available online xxxx

Keywords:Theoretical modelElectroadhesive padInterdigitated electrodesElectroadhesionDry adhesionSoft robotics

Electric fields alter adhesive forces betweenmaterials. Electroadhesive forces have been utilized in diverse appli-cations ranging from climbing robots, electrostatic levitation to electro-sticky boards. However, the design ofelectroadhesive devices still largely relies on empirical or “trial-and-error” approaches. In this work, a theoreticalmodel is presented to analyze the electrostatic field between the supporting wall and the electroadhesive devicewith periodic coplanar electrodes. The air-gap between the surface of electroadhesive device and the dielectricwall is explicitly taken into account in the model to consider its significant impact on electroadhesive forces.On the basis of this model, the electroadhesive force is calculated by using the Maxwell stress tensor. The effectsof key design parameters and working environments on the electroadhesion behavior are further investigated.This study not only provides a tool to reveal the underlyingmechanisms of electroadhesion but also suggests po-tential strategies to optimize novel electroadhesive devices for engineering applications.

© 2015 Elsevier Ltd. All rights reserved.

1. Introduction

Electric fields can alter adhesive forces between materials, whichallow surfaces to reversibly attach to each other without the use of me-chanical grippers, suckors, fasteners, or chemical adhesives [1–6].Electroadhesive devices have been developed by patterning electrodesin air or matrices of insulating dielectrics. Once an electric voltage is ap-plied, the devices can invoke the adhesive forces to a wide variety ofsubstrate materials (e.g., silicon, silicon dioxide, wood, drywall, glass,concrete, steel, and plastics) with various shapes, sizes, and roughness[2,5–7]. Compared to mechanical and chemical adhesions, theelectroadhesion has distinct advantages, e.g., fast response (responsetime b 10 ms), quiet operation, low cost, easy control, low energy con-sumption, and dust tolerance [2]. Due to these many benefits,electroadhesive devices have been widely used in various applications.Examples include electrostatic levitation used in liquid-crystal-displaymanufacturing [4,8–12], electroadhesive grippers for handling

cal Engineering and Materials

microcomponents of semiconductors [13,14], and electroadhesivepads for wall climbing robots [2,5,6,15,16] and sticky boards [17].

An electroadhesive device uses the electrostatic force between thesupporting material (e.g., wall surface) and the electroadhesive pad.For Coulomb-type electrostatic pads (ESP), the electrostatic force is gen-erated by the dielectric polarization due to the electric potential differ-ence. Based on the charge (or electrode) configuration, ESPs can beclassified into two types:mono-polar (plate-plate-capacitors) and bipo-lar (interdigitated electrodes) [1]. Mono-polar type ESPs generate adhe-sive forces in accordance with the principle that the two plates in aparallel capacitor attract each other if there is a voltage difference be-tween them. Therefore, the target structure (e.g., a concrete wall) is re-quired to be conductive such that a capacitor is formed between theelectrode and the supporting wall [18]. A bipolar type electroadhesivedevice is typically constituted by interlacing the fingers of twoconducting combs, as shown in Fig. 1. The space between the electrodefingers is usually filled with an electrical insulator. The fingers and thefiller are insulated from the substrate by a dielectric layer (i.e. thecover in Fig. 1). When alternating positive and negative charges are in-duced on the adjacent electrodes and the device is placed in contactwith a wall, the electric fields set up opposite charges on the wall andthus cause electrostatic adhesion between the electrodes and the in-duced charges on the supporting wall. This force may be explained bytwo mechanisms, namely, the gradient force and the Johnsen–Rahbeckforce [19].

Fig. 1. Schematic of an electroadhesive pad with embedded interdigitated electrodes: (a)3D view and (b) side view.

486 C. Cao et al. / Materials and Design 89 (2016) 485–491

Despite these technologically important applications ofelectroadhesion, the design of electroadhesive devices has largely reliedon trial-and-error approaches and the electroadhesive stresses and en-ergy are usually estimated by empirical equations [2,7,10,13,15].Prahlad and coworkers experimentally investigated the compliantelectroadhesion technique and developed a variety of wall climbing ro-bots such as tracked and legged robots [2]. They tested the attractiveforce of the robots climbing on distinct surfaces of materials and dem-onstrated the validity and superior performance of electroadhesion inclimbing robots. Yamamoto et al. [15] designed a wall climbing robotwith flexible electrodes that were fabricated by a plastic film and a con-ductive foil. They tested the attaching performance of the flexible elec-trode panel against both conductive and non-conductive walls, inwhich the adhesive forces is proportional to the square of voltage.Other innovative designs, e.g. gecko inspired electrostatic-chuck (ESC)[1], were also proposed to improve the poor performance of conven-tional ESCs on relatively rough surfaces. However, most of the studieswere mainly experimental and there is still a lack of theoretical modelsto reveal the relationship between the adhesive force and the key designparameters. Optimal design principles are still unavailable for amajorityof electroadhesive devices, in which the electrodes are actuallyembedded in insulating dielectrics. Only recently, Woo and Higuchi[9] developed a numerical model of electrostatic-levitation to calculatethe forces generated by electrodes and discussed the optimization ofthe electric charging rate with respect to the geometrical parametersof electrodes.

In this paper, we attempt to theoretically address the basic designprinciples of electroadhesion. A semi-analytical method is presentedto calculate the electric field in a four-layer or five-layer model. TheMaxwell stress tensor is used to calculate the desired electroadhesive

force. We demonstrate that various key factors, including the dielectricconstants of constituent materials, the thicknesses of cover layer, theair-gap, and the supporting walls, as well as the applied voltage andthe dielectric constant of walls, have significant effect on the perfor-mance of electroadhesive devices. This work not only helps in the com-prehension of the adhesionmechanisms in electrostatic devices but alsosupplies guidance for their optimal design.

2. Model of electroadhesive forces

Consider an electroadhesive pad consisting of periodic coplanarelectrodes sandwiched between a dielectric cover and a substrate, asshown in Fig. 1. Such electroadhesive pads can be made in a variety ofmethods. To enhance the compliance of the pad, the interdigitated elec-trodes can be deposited by flexiblematerials such as carbonmixedwitha polymer binder [2] or crumpled metal film [20]. The cover and sub-strate can bemade of elastomers, polymers or plastics with different di-electric constants while the supporting wall can be any commonmaterial, e.g. concrete cement or wood.

A four-layer model is here established, including the dielectric wall,the air gap, the cover, and the dielectric substrate. To investigate theelectroadhesion, we first need to calculate the electric field distributiongenerated by the electrodes embedded in the electroadhesive device. Itiswell known that the electricfield E in the solution space can be readilycalculated from the electric potential ϕ by

E ¼ −∇ϕ; ð1Þ

and that the potential in a dielectric medium satisfies the Laplaceequation

∇2ϕ ¼ 0: ð2Þ

When the solution domain of the problem is filled with a homoge-nous material, the electrostatic field can be analytically solved by theconformal mapping technique [21,22]. However, for a structure madeof several different materials as shown in Fig. 2(a), the conformal map-ping technique fails to solve the problem. Therefore, a point matchingmethod is used here to calculate the electric fields [9,23]. In thismethod,the potential function in each layer is defined by a series expansion interms of the solutions of the Laplace equation, which are periodic inthe direction parallel to the electrodes. For different layers, the coeffi-cients in the series are related to each other and to the electric potentialsapplied on the electrodes. The boundary conditions in the electrodeplane are satisfied at N discrete points, with N being the number ofterms in the series expansion [23]. The details of the calculationmethodare given in Supplemental information (SI) [24].

As shown in Fig. 2(a), the electroadhesive device consists of a copla-nar array of electrodes whose uniform pitch and width are assumed tobe L = 2b and 2a, respectively. The electrodes are assumed to have anegligible thickness and to be charged alternatingly by the electric po-tentials Φ and −Φ. Due to the periodic feature of the electrodes, onlyone period of the structure is analyzed. More details of the model andthe solution method are given in SI [24]. As shown in Fig. 2(b)–(d),using this four-layermodel, we calculate the distributions of electric po-tential ϕ and electric field components Ex and Ey for one period of thestructure with the assumed dielectric and geometric parameters εw=10ε0, εc=2ε0, a/b=0.5, and t/b=0.02, where ε0 denotes the dielectricconstant of air, εwis the dielectric constant of the wall (supporting sub-strate), εcthe dielectric constant of the cover, and t is the air gap thick-ness between the electroadhesion pad and the wall. It can be seen thatEy is the dominating component and its intensity in the air gap ismuch larger than those in the dielectric wall and the cover, yielding arelative larger attractive force between the pad and the wall.

The electrostatic attractive force exerted on the dielectric plate arisesfrom the interaction between the applied non-uniform electric field and

Fig. 2. (a) A four-layermodel for an electroadhesive padwith periodic coplanar electrodes sandwiched between a dielectric cover and a substrate. Contour plots of the calculated (b) elec-tric potential ϕ, (c) electric field component Ex, and (d) electric field componentEy for a period of the model in (a).

487C. Cao et al. / Materials and Design 89 (2016) 485–491

the induced conduction and polarization charge [9]. Once we have ob-tained the electric field, the attractive force can be calculated from theMaxwell stress tensor Tij, which is defined as

Tij ¼ ε EiE j−12δij Ej j2

� �; ð3Þ

where ε denotes the electric permittivity and δij the Kronecker delta.The attraction force FN along the normal direction of the dielectric sur-face is obtained by integrating the Maxwell stress tensor Tij over theenclosing surfaces of the dielectric medium. Assuming the pad surfaceis parallel to the wall (Fig. 2(a)), the electroadhesive force acting onthe wall of unit length and width L can be calculated by

FN ¼ ∮V f ydv ¼ ∮STyyds ¼ 12ε0Z L

0Ey

2−Ex2

� �dl; ð4Þ

where Ex and Ey are the electric field components in the air-gap be-tween the electroadhesive device and the dielectric wall.

Because the wall is regarded to be infinite in length along the y-direction, the attraction force acting on the upper surface of thewall is negligible. The electric field is uniform in the z-directionwhen the marginal part of the pad is neglected. Then theelectroadhesive force acting on the whole dielectric wall of a periodlength L is calculated by

FN ¼ 12ε0Z L

0εw=ε0ð Þ2E1y2−E1x

2h i

dl����y¼hþ2

; ð5Þ

where E1x and E1y are the components of the electric field on the sur-face of the dielectricwall. Thus, the average adhesion stress can be givenby

σad ¼ 14b

ε0Z 2b

0

εwε0

� �2

E1y2−E1x

2

" #dx: ð6Þ

With the assumption of the electric potential ϕ1 for the infinite wall,

as shown in SI [24], one can prove ∫L

0E1y

2dxjy¼hþ2

¼ ∫L

0E1x

2dxjy¼hþ2

. Thus,

Eq. (6) becomes

σad ¼ 12ε0

εwε0

� �2

−1

" #C

ab;h1b;tb;εcεw

� �Φ2b

� �2

; ð7Þ

where Cðab ; h1b ; tb ;εcεwÞ is defined as a dimensionless function of geo-

metric parameters ab,

h1b and t

b, and material parameter εcεw. ðΦ2bÞ

2 is the nor-malized electric field generated by the applied voltage. The factor12 ε0½ðεwε0 Þ

2−1� is a constant depending only on the dielectric constantsof the air and the wall. It is seen from Eq. (7) that the electroadhesiveforce is proportional to the square of the applied voltage [15] when

the function Cðab ; h1b ; tb ;εwεcÞ and the dielectric constant of the wall εw are

fixed. In the following section, we will use Eq. (7) to examine the effectof geometric andmaterial parameters on the electroadhesive force gen-erated by the electroadhesive pad.

3. Results and discussions

3.1. Electroadhesion without air-gap at the interface

We first examine the special case, in which the electroadhesion padcontacts the supportingwall without the air-gap, i.e., there exists a zero-thickness air-gap between the contact surface of electroadhesion padand the wall. The dielectric wall and the cover of the pad usually havedifferent dielectric constants, i.e., εc≠εw. Since no analytical solutioncan be obtained for such an inhomogeneous medium, the pointmatching method is employed to calculate the electric field with thefour-layer model presented in Section 2. With the zero-thickness air-gap and εc≠εw, Eq. (7) can be re-expressed as

σad ¼ 12ε0

εwε0

� �2

−1

" #C0 a

b;h1b;εcεw

� �Φ2b

� �2

; ð8Þ

Fig. 3. Variation of C′(a/b,h1/b,εc/εw) as a function of dielectric constant ratio εc/εw (a) forvarious normalized cover thickness h1/bwhen a/b=0.8 and εw=5ε0, and (b) for variousnormalized electrodewidth a/bwhen h1/b=0.02 and εw=5ε0. (c) Comparison of the pre-dicted electroadhesion with the experimental data from literature.

488 C. Cao et al. / Materials and Design 89 (2016) 485–491

where C′(a/b,h1/b,εc/εw) is a newly defined function of geometricparameter a/b, h1/b and material parameter εc/εw.

Fig. 3(a) and (b) plot the variation of C′(a/b,h1/b,εc/εw) with respectto the ratio of dielectric constants εc/εw for various h1/b and a/b, respec-tively. It can be seen from Fig. 3 that C′(a/b,h1/b,εc/εw) increases withthe increase in εc/εw for a fixed value of h1/b. However, the influenceof εc/εw becomes less significant when εc/εwN5, and thus each curve

has a plateau. As a result, from the viewpoint of fabrication, it is desiredto select materials with higher dielectric constants as the cover. This in-spires us to select proper devices in various working environments [2].It is also found that the dielectric constant of the substrate material hasno influence on the electroadhesive performance of the device. In addi-tion, Fig. 3 shows that both the normalized electrode width a/b and thecover thickness h1/b have significant effects on electroadhesion, andthat a smaller h1/b or a larger a/b results in a larger electrostatic attrac-tive force on the supporting wall. These results indicate that it is betterto reduce the cover thickness and to increase the electrode width inorder to enhance the adhesion force of the electroadhesive device. How-ever, it should be noticed that the electrodesmust not reach the value a/b=1, at which the positive and negative electrodes would touch eachother to form a short circuit. In the above discussions, we have ignoredthe restrictions of dielectric materials and allow the maximum voltageto be increased as required. However, the applied voltage cannot be in-creased infinitely due to the breakdown voltage of dielectric materials.For electroadhesion, this limit is determined by the breakdown electricstrength of air EBDair, which is about 3 × 106 V/m. Therefore, when the ap-plied voltage reaches the threshold of breakdown, there is an optimalvalue for a/b to maximize the electroadhesive force (see Supplementalinformation SI.3 [24]). To validate the above theoretical model, we cal-culate the electroadhesion force generated by an electroadhesive paddescribed in the literature [25], which has the same structural configu-ration as shown in Fig. 1(b). In this design, the cover layer of the padis 123 μm in thickness, made of epoxy resin (dielectric constant εc =7.8), and the substrate is made of glass-epoxy resin. The copper elec-trodes are 35 μm thick and 1mmwide. Tomeasure the electroadhesiongenerated by the pad, a 4-inch silicon wafer was used as the supportingwall. As shown in Fig. 3(c), the theoretical results agree with the exper-imental data. The little discrepancy may result from the fact that in theideal model the electrodes are considered to have zero thickness andthat there existed fringe effect in the testing device.

3.2. Electroadhesion with an air gap at the interface

In the former section, we discussed the ideal case with perfect con-tact between the electroadhesive device and the supporting wall. Inpractice, however, a small air-gapwith a thickness of just a few hundrednanometers exists between the electroadhesive pad and the wall [18].Thus, it is necessary to investigate the effect of the air-gap thickness

on the electroadhesive force. Fig. 4(a) shows the variation ofCðab ; h1b ; tb ;εcεw

Þ as a function of normalized air-gap t/b when h1/b=0.02, εc=7ε0 andεw=5ε0 for different values of the electrode width a/b. It can be seenthat the air-gap indeed has a significant effect on electroadhesion, anda very small air-gap (e.g., 0.05b) can drastically reduce about by 70%of its original forcewhen a/b=0.8. It is also noted that the degenerationof the attractive force is relatively faster for a smaller thickness of thecover. For some representative values of the air-gap t/b in the range

from 0 to 0.2, Fig. 4(b) shows the variation of Cðab ; h1b ; tb ;εcεwÞ as a function

of the electrode width a/b when h1/b=0.02, εc=7ε0 and εw=5ε0. It isfound that the attractive force increases with the increase in the elec-trode width a/b, but the influence of a/b sharply reduces when the air-gap t/b is large. Thus it is concluded that the reducing air-gap thicknessis important to increase the electroadhesive force.

3.3. Electroadhesion on a wall with finite thickness

In the above analyses, for the sake of simplicity, we assumed that thesupporting wall is of infinite thickness compared with theelectroadhesive pad. However, in such practical applications as electro-static-levitation, we often encounter objects with finite thickness, e.g.,thin LCD panels. Therefore, it is essential to understand the influenceof the thickness of targeting substrates (i.e., attachments, walls orpanels). In what follows, we employ a five-layer model (For details of

Fig. 4. (a) Variation of C(a/b,h1/b,t/b,εc/εw) as a function of dielectric constants ratio t/b forvarious normalized electrode width a/bwhen h1/b=0.02, εc=7ε0, and εw=5ε0. (b) Var-iation of C(a/b,h1/b,t/b,εc/εw) as a function of dielectric constants ratio a/b for various nor-malized thickness of air-gap t/bwhen h1/b=0.02, εc=7ε0, and εw=5ε0.

Fig. 5. (a) Variation of Cða=b;h1=b; t=b; εc=εwÞ as a function of normalized wall thicknessH/b for different dielectric constants ratio εc/εw when a/b=0.8, h1/b=0.02, and t = 0.(b) Variation of Cða=b;h1=b; t=b; εc=εwÞ as a function of normalized wall thickness H/bfor different air-gap t/b when a/b=0.8, h1/b=0.02, εc=5ε0 and εw=7.5ε0.

489C. Cao et al. / Materials and Design 89 (2016) 485–491

themodel, see Supplemental information SI.2 [24]) to study the effect ofthe thickness of supporting structure or attachment (e.g., wall). Theelectroadhesive force acting on the whole dielectric wall of a periodlength L is calculated by

FN ¼ 12ε0Z L

0E3y

2−E3x2

� �− E1y

2−E1x2

� �h idl; ð9Þ

where E1x and E1y are the electric field components above the dielec-tric wall, and E3x and E3y are the electric field components below thewall (see Fig. S1 in Supplemental information [24]). The average adhe-sion stress is written as

σad ¼ ε0Cab;h1b;tb;εcεw

� �Φ2b

� �2

; ð10Þ

where Cðab ; h1b ; tb ;εcεwÞ is a defined dimensionless factor. Fig. 5(a) de-

scribes the variation of Cðab ; h1b ; tb ;εcεwÞwith the normalized wall thickness

H/b. For each distinct εw/εc, we fix the electrode width as a/b=0.8 andh1/b=0.02, and assume there is no air-gap at the contact interface. Itcan be found from Fig. 5 that finite wall thickness has a significant effecton electroadhesion. For a given value of εw/εc, the electroadhesion stressincreases with wall thickness, and the stress approaches a limit valuewhen the wall thickness exceeds 2b. For different air-gaps, Fig. 5(b)

shows the variation of Cðab ; h1b ; tb ;εcεwÞ with respect to the normalized

wall thickness H/b, where we take the parameters a/b=0.8, h1/b=0.02, εc=5, and εw=7.5. It can be seen that the wall thickness hasmore significant impact when the air-gap is smaller.

3.4. Effect of surface roughness

In Section 3.2, we have shown that the electro-attractive force is sig-nificantly affected by the air-gap between pad and wall. Thus, theelectroadhesive force of a rigid padwill be sensitive to its surface rough-ness. Therefore, one possible approach to alleviate the negative influ-ence of roughness is to develop a compliant pad, e.g., depositingcompliant electrodes into elastomeric insulators, which is able to adap-tively adhere to a wide range of roughness [15]. Another promisingmethod is to design a compliant electrostatic pad by mimicking thestructure of gecko feet [26–28]. Such a biomimetic structure enablesthe electroadhesive pad to maintain a much larger electroadhesiveforce on rough surfaces. To design an adaptive electroadhesion padlike gecko feet, one can add a layer consisting of short hairs on top ofthe cover layer of the pad to enhance its adaptability on rough surfaces,as shown in Fig. 6(a). To analyze the performance of the biomimeticcompliant pad using the present model, the added layer can be treatedas a composite with an effective dielectric constant εm, which can beevaluated by a parallel mixing rule [29] as εm=Vaεc+Vaε0, where Va

and Vc are the volume fractions of the dielectric phase and air-gapphase (Va+Vc=1), respectively. For different values of the hair length,

Fig. 6. (a) Schematic design of a compliant electroadhesive pad with added hair layer by

mimicking Gecko feet. (b) Variation of Cða=b;h1=b; t=b; εc=εw; εm=εw;hm=bÞ as a functionof dielectric constants ratio εm/εw for different hair length hm/b when a/b=0.9, h1/b=0.01, and t = 0, where εc=7ε0 and εw=5ε0.

490 C. Cao et al. / Materials and Design 89 (2016) 485–491

Fig. 6(b) shows the variation of Cðab ; h1b ; tb ;εcεw; εmεw ;

hmb Þ as a function of di-

electric constants ratio εm/εw where we take a/b=0.9, h1/b=0.01, t/b=0, εw=5ε0, and εc=7ε0. It is obvious that the electroadhesion re-duces as the length of hair layer increases and will reach a peak valuefor each hm/b when the mixed dielectric constant εm is larger than εw.It is expected that such a design has another attractive merit, i.e., com-bining the electroadhesive force together with the van der Waals forceto generate stronger adhesion. It is known that when the dimension ofgecko-feet-like structures goes down to the micro or even nano-scale,the van der Waals force generated by the hair layer becomes muchstronger [26–28,30,31]. Therefore, the hair layer not only improves theadaptability of a pad on rough surfaces but also enhances the attractionforce. It will be promising to fabricate a novel electroadhesive device orto develop a new micro-transfer technique by superimposing bothelectroadhesive and van der Waals forces.

4. Conclusions

In summary, we have theoretically analyzed the electroadhesiveforces of an interdigitated type electroadhesive device by employing asemi-analytical model and discussed the optimal design principles ofelectroadhesive devices for a wide variety of applications. It is foundthat, to enhance the electroadhesive force, the cover layer should em-ploy dielectric materials with larger dielectric constants, and the elec-trodes should be wider while the cover should be thinner. However,for the extreme case where the electric potential is equal to the

breakdown voltage, the maximum electroadhesive force will occurwith a very small ratio of a/b. We have also proposed a novel route toimprove the performance of electroadhesive pads by mimicking thestructure of gecko feet, which enables the electrodes to maintain agood proximity with rough surfaces and to combine bothelectroadhesion and van derWaals forces together to increase the over-all adhesive force. The present theoretical model is helpful to designnew electroadhesive devices and to develop electroadhesion-basedmicro-transfer techniques.

Acknowledgments

The work was financially supported by the fellowships from Austra-lian National University, Duke University and NSF's Research TriangleMaterials Research Science and Engineering Center (DMR-1121107).C.C. would like to thank Prof. Xuanhe Zhao fromMIT for helpful discus-sion and support. Q.Q. thanks the financial support from the AustralianResearch Council. X.F. acknowledges the 973 Program of MOST(2012CB934001).

Appendix A. Supplementary data

Supplementary data to this article can be found online at http://dx.doi.org/10.1016/j.matdes.2015.09.162.

References

[1] J. Berengueres, M. Urago, S. Saito, K. Tadakuma, H. Meguro, Gecko Inspired Electro-static Chuck, Robotics and Biomimetics, 2006 ROBIO '06 IEEE International Confer-ence on 2006, pp. 1018–1023.

[2] H. Prahlad, R. Pelrine, S. Stanford, J. Marlow, R. Kornbluh, Electroadhesive Robots—Wall Climbing Robots Enabled by a Novel, Robust, and Electrically Controllable Ad-hesion Technology, Robotics and Automation, 2008 ICRA 2008 IEEE InternationalConference on 2008, pp. 3028–3033.

[3] A.B. Vankov, P. Huie, M.S. Blumenkranz, D.V. Palanker, Electroadhesive forceps fortissue manipulation. Proc SPIE 5314, Ophthalmic Technologies XIV 5314 (2004)270–274.

[4] J.U. Jeon, T. Higuchi, Electrostatic suspension of dielectrics, IEEE Trans. Ind. Electron.45 (1998) 938–946.

[5] MurphyMP, Kute C, Mengüç Y, Sitti M.Waalbot II: Adhesion recovery and improvedperformance of a climbing robot using fibrillar adhesives. Int. J. Robot. Res. 2011;30:118–33.

[6] E. Noohi, S.S. Mahdavi, A. Baghani, M.N. Ahmadabadi, Wheel-based climbing robot:modeling and control, Adv. Robot. 24 (2010) 1313–1343.

[7] D. Longo, G. Muscato, G. Tarantello, Performance Evaluation of Electrostatic Adhe-sion for Climbing Robots, Proceedings of the 13th International Conference onClimbing and Walking Robots and the Support Technologies for Mobile Machines2010, pp. 1029–1036.

[8] W. Shao Ju, J.-U. Jeon, T. Higuchi, J. Ju, Electrostatic Force Analysis of ElectrostaticLevitation System, SICE '95 Proceedings of the 34th SICE Annual Conference Interna-tional Session Papers 1995, pp. 1347–1352.

[9] Woo S, Higuchi T. Electric field and force modeling for electrostatic levitation oflossy dielectric plates. J. Appl. Phys. 2010;108:104906–10.

[10] L. Jin, T. Higuchi, M. Kanemoto, Electrostatic levitator for hard disk media, IEEETrans. Ind. Electron. 42 (1995) 467–473.

[11] K. Asano, F. Hatakeyama, K. Yatsuzuka, Fundamental study of an electrostatic chuckfor silicon wafer handling, Industry Applications, IEEE Transactions on 38 (2002)840–845.

[12] S. Kanno, T. Usui, Generationmechanism of residual clamping force in a bipolar elec-trostatic chuck, Journal of Vacuum Science & Technology B: Microelectronics andNanometer Structures 21 (2003) 2371–2377.

[13] G.J. Monkman, Robot grippers for use with fibrous materials, IInternational Journalof Robotics Research 14 (1995) 144–151.

[14] Monkman G. Electroadhesive microgrippers. Industrial Robot: An InternationalJournal 2003;30:326–30.

[15] A. Yamamoto, T. Nakashima, T. Higuchi, Wall Climbing Mechanisms Using Electro-static Attraction Generated by Flexible Electrodes, Micro-NanoMechatronics andHuman Science, 2007 MHS '07 International Symposium on 2007, pp. 389–394.

[16] M.P. Murphy, M. Sitti, Waalbot: Agile Climbing with Synthetic Fibrillar Dry Adhe-sives, Proceedings of the 2009 IEEE international conference on Robotics and Auto-mation, IEEE Press, Kobe, Japan 2009, pp. 2637–2638.

[17] Ltd. JI. http://www.justick.com/Products.aspx. 2006.[18] J. Yoo, J.S. Choi, S.J. Hong, T.H. Kim, S.J. Lee, Finite Element Analysis of the Attractive

Force on a Coulomb Type Electrostatic Chuck, International Conference on ElectricalMachines and Systems 2007, pp. 1371–1375.

[19] S. Kanno, T. Usui, Generationmechanism of residual clamping force in a bipolar elec-trostatic chuck, J. Vac. Sci. Technol. B 21 (2003) 2371–2377.

491C. Cao et al. / Materials and Design 89 (2016) 485–491

[20] C. Cao, H.F. Chan, J. Zang, K.W. Leong, X. Zhao, Harnessing localized ridges for high-aspect-ratio hierarchical patterns with dynamic tunability and multifunctionality,Adv. Mater. 26 (2014) 1763–1770.

[21] Y.C. Lim, R.A. Moore, Properties of alternately charged coplanar parallel strips byconformal mappings, IEEE Transactions on Electron Devices 15 (1968) 173–180.

[22] H.A. Wheeler, Transmission-line properties of parallel wide strips by a conformal-mapping approximation, Microwave Theory and Techniques, IEEE Transactions on12 (1964) 280–289.

[23] D. Marcuse, Electrostatic field of coplanar lines computed with the point matchingmethod, IEEE J. Quantum Electron. 25 (1989) 939–947.

[24] See Supplemental information for the detailed formulations of four-layer model andfive-layer model 2014.

[25] K. Yatsuzuka, F. Hatakeyama, K. Asano, S. Aonuma, Fundamental characteristics ofelectrostatic wafer chuck with insulating sealant, Industry Applications, IEEE Trans-actions on 36 (2000) 510–516.

[26] H. Gao, X. Wang, H. Yao, S. Gorb, E. Arzt, Mechanics of hierarchical adhesion struc-tures of geckos, Mech. Mater. 37 (2005) 275–285.

[27] H. Yao, H. Gao, Mechanics of robust and releasable adhesion in biology: bottom–updesigned hierarchical structures of gecko, Journal of the Mechanics and Physics ofSolids 54 (2006) 1120–1146.

[28] K. Autumn, M. Sitti, Y.A. Liang, A.M. Peattie, W.R. Hansen, S. Sponberg, et al., Evi-dence for van der Waals adhesion in gecko setae, Proc. Natl. Acad. Sci. 99 (2002)12252–12256.

[29] Y. Wu, X. Zhao, F. Li, Z. Fan, Evaluation of mixing rules for dielectric constants ofcomposite dielectrics by MC-FEM calculation on 3D cubic lattice, J. Electroceram.11 (2003) 227–239.

[30] H. Lee, B.P. Lee, P.B. Messersmith, A reversible wet/dry adhesive inspired bymusselsand geckos, Nature 448 (2007) 338–341.

[31] K. Autumn, N. Gravish, Gecko adhesion: evolutionary nanotechnology, Philos. Trans.R. Soc. A Math. Phys. Eng. Sci. 366 (2008) 1575–1590.