electro-mechanical response of elastomer membranes coated with ultra-thin metal elecgtrodes

8/7/2019 Electro-mechanical Response of Elastomer Membranes Coated With Ultra-thin Metal Elecgtrodes

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Journal of the Mechanics and Physics of Solids53 (2005) 25572578

The electro-mechanical response of elastomermembranes coated with ultra-thin

metal electrodes

Matthew R. Begley a, , Hilary Bart-Smith b , Orion N. Scott b ,Michael H. Jones c, Michael L. Reed c

a Structural and Solid Mechanics Program, Department of Civil Engineering & Department of MaterialsScience and Engineering, University of Virginia, Charlottesville, VA 22904, USA

b Department of Mechanical and Aerospace Engineering, University of Virginia, Charlottesville,VA 22904, USA

cDepartment of Electrical and Computer Engineering, University of Virginia, Charlottesville,VA 22904, USA

Received 19 October 2004; accepted 5 May 2005

Abstract

This paper presents experimental and theoretical analyses of the electro-mechanicalresponse of metal/elastomer multilayers. A novel test has been devised to determine therelationship between the mechanical response of clamped elastomer membranes, coated onboth sides with metal electrodes, and an applied electric eld. The load-deection response of the multilayers subjected to different voltages was measured using an instrumented spherical

indenter having dimensions comparable to the freestanding span. The measurements are usedwith closed-form solutions for membrane deection to determine the effective plane-strainmodulus of cracked multilayers and electrically induced in-plane strains. The experimentsdemonstrate that: (i) electrically induced strains vary with the square of the electric eld, asexpected from electrostatic models of parallel plate capacitors, (ii) the transverse stiffness of membranes can be controlled using applied electric elds, (iii) analytical models accuratelypredict the relationship between electrode crack spacing, layer properties and effective moduli.Finally, we estimate the toughness of the sub-micron metal electrodes, using cracking models

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www.elsevier.com/locate/jmps

0022-5096/$- see front matter r 2005 Published by Elsevier Ltd.doi:10.1016/j.jmps.2005.05.002

Corresponding author. Tel.: +1 4342438728; fax: +1 434982 2951.E-mail address: [email protected] (M.R. Begley).
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that relate crack spacing, imposed strain and the energy release rate governing channel crackformation.r 2005 Published by Elsevier Ltd.

Keywords: Electro-mechanical; Membrane; Elastomer; Cracking; Thin lm toughness

1. Introduction

The use of electro-mechanical devices based on polymers coated with ultra-thin,conductive electrodes is of growing interest (e.g. Pelrine et al., 1998, 2000 ; Kofod,2001 ; Bar-Cohen, 2004 ; Yang et al., 2003 ; Pope et al., 2004 ; Goulborne et al., 2004 ;Wagner et al., 2004 ). The key concept of these systems is the exploitation of theinherent exibility of the dielectric and electrodes to create devices capable of largedeformation. The multilayer must be sufciently compliant such that relatively weakelectrostatic pressures can induce signicant deformation in the through-thicknessdirection. For nearly incompressible dielectrics, such as silicone-based elastomers,the through-thickness deformation generates signicant in-plane deformation, asshown schematically in Fig. 1 . The phenomenon illustrated in Fig. 1 has led to theuse of the term dielectric elastomers to describe this type of multilayered actuator(Pelrine et al., 1998, 2000 ; Kofod, 2001 ; Bar-Cohen, 2004 ; Yang et al., 2003 ; Pope etal., 2004 ; Goulborne et al., 2004 ).

Previous work has illustrated the function of such devices, which are often referredto as electrostrictive because the deformation is proportional to the square of theelectric eld (e.g. Pelrine et al., 1998, 2000 ; Kofod, 2001 ; Bar-Cohen, 2004 ). Thisquadratic relationship can arise from two different mechanisms: (i) an electrostrictivedielectric material that exhibits electrically induced strains (arising from inter-molecular forces), without any mechanical coupling with the electrodes, and (ii)attractive electrostatic forces between the electrodes that generate strain viamechanical coupling with the dielectric. In this work, we make no attempt todirectly quantify the relative contributions of these two mechanisms, 1 and use theterm electrostrictive to refer to the behavior of the entire multilayer.

Previous efforts have focused primarily on the coupled electro-mechanicalresponse of specimens loaded or constrained in the direction parallel to the planeof the lm ( Pelrine et al., 1998, 2000 ; Kofod, 2001 ; Bar-Cohen, 2004 ; Yang et al.,2003 . However, the in-plane strains generated by the application of the electric eldcan also be exploited to control out-of-plane behavior; this has been demonstratedby Frecker and Mockenstrum, who describe the function of elastomer diaphragmpumps ( Pope et al., 2004 ; Goulborne et al., 2004 ). Here, we describe the performanceof a compliant metal /elastomer system in coupling electrical input and out-of-planebehavior, which differs signicantly from previous efforts that have focused solely on

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1However, we illustrate that a completely consistent electro-mechanical model can be developed withoutincluding electrostrictive effects of the dielectric itself, indicating that the second mechanism probablydominates.

M.R. Begley et al. / J. Mech. Phys. Solids 53 (2005) 255725782558


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in-plane behavior ( Pelrine et al., 1998, 2000 ; Kofod, 2001 ; Bar-Cohen, 2004 ; Yanget al., 2003 ) or studied the tranverse loading of multilayers with conductive carbongrease electrodes ( Pope et al., 2004 ; Goulborne et al., 2004 ).

The performance of metal/elastomer multilayers is strongly inuenced by thecompliance and conductivity of the electrodes, and the presence of defects that act tolocalize electrical charge and initiate breakdown of the dielectric ( Pelrine et al., 1998,2000 ; Kofod, 2001 ; Begley and Bart-Smith, 2005 ). This paper describes a completeframework that can be used to predict multilayer behavior in terms of individuallayer properties, electrode cracking, and applied electric elds. This framework iscomprised of three signicant and original contributions: rst, it provides

experimental validation of constitutive theories for cracked laminates comprised of layers whose elastic modulus and thickness mismatch span several orders of magnitude ( Begley and Bart-Smith, 2005 ). Second, it provides experimentalvalidation of closed-form membrane load-deection solutions that involve pre-strain ( Begley and Mackin, 2004 ). Lastly, it provides the complete mechanicsframework needed to extract toughness values for nanoscale lms from membranestretching experiments. The combination of these three contributions illustrate thatelectrical functionality is preserved despite electrode cracking, and enablequantitative predictions of actuator performance.

We consider a novel arrangement, wherein a circular sandwich (comprised of an

elastomer with electrodes coated on both sides) is clamped along the outer edge andmechanically loaded transverse to the plane of the lm, as shown in Fig. 2a . Theelectric eld generates electrostatic attractive forces in the through thicknessdirection that compress the lm and increase the effective length of the clampedsection. This phenomenon is identical to that used in pre-stretched diaphragmpumps described elsewhere ( Pope et al., 2004 ; Goulborne et al., 2004 ). The load-deection response of the lm is measured using an instrumented indenter shaftequipped with a large Teon sphere. A relatively large sphere (compared with themembrane span) is chosen to avoid highly localized deformation near the indenterand increase the force for a given indenter displacement. We consider large

deections relative to the lm thickness, such that the bending stiffness of themultilayer is negligible compared to that arising from in-plane stretching; hence,membrane solutions are appropriate ( Begley and Mackin, 2004 ; Scott et al., 2004 ).

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Fig. 1. Schematic diagram of compliant multilayers that generate in-plane strains via electrostaticattraction of the electrodes: unconstrained in-plane deformation due to transverse electric eld.

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The chosen geometry has important advantages. First, the mechanical andelectrical boundary conditions are well established, since the electrodes cover theentire specimen. For the geometry shown in Fig. 2 , there is no pathway for dielectricbreakdown except through the specimen (i.e. charge cannot conduct around theedges of the dielectric). Secondly, relatively thin specimens can be easily fabricated:this reduces the voltage required to induce measurable actuation strain. Finally, thegeometry involves forces and displacements easily measured using off-the-shelf load

cells and positioning stages; this is a critical concern when dealing with ultra-compliant specimens with low modulus ( $ 12 MPa) and thickness ( $ 100 mm).Although an instrumented indenter is used here, pressure loading can be used inconjunction with a non-contacting displacement measurement, such as white lightinterferometry.

With regards to characterization of cracking behavior, out-of-plane deection isless desirable in some respects than a simple axial experiment involving multilayers.However, axial congurations are often difcult to grip and align, they require ultra-low load resolution (owing to the thin lm dimensions needed to limit appliedvoltages) and they may involve un-coated regions near the edges that complicate

boundary conditions.As will be demonstrated, the nonlinear load-deection response of membrane

deection can be accurately described using closed-form solutions that relate

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Fig. 2. (a) Side view of the membrane deection specimen, (b) Oblique cut-away view of the membranedeection specimen.



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indenter size, membrane properties, load, deection, and in-plane strain induced bythe electric eld. The response at zero applied electric eld is used to determine theeffective modulus of the multilayers, which is a strong function of the spacing of thecracks that develop in the electrodes during multilayer deection. Theoreticalrelationships between crack spacing, lm properties and effective modulus areveried using these experiments. Once the effective modulus of the crackedmultilayer is established, the closed form membrane solutions can be used todetermine the in-plane strain induced by the applied electric eld, which can be inturn related to the electrostatic pressure ( Stratton, 1941 ). It should be emphasizedthat the in-plane strain discussed here does not arise from an initial tension imposedprior to activation. Rather, the electrically induced strain acts to slacken the lm,and arises due to electro-mechanical coupling.

2. Experimental approach

The samples consist of a single piece of silicone-based elastomer coated on bothsides with two thin copper layers, as shown schematically in Fig. 2 . The presentexperiments focus on copper electrodes, as they exhibit strong adhesion with silicone.Preliminary experiments with gold illustrated that a titanium adhesion layer isrequired to inhibit electrode delamination; this is undesirable as the titaniumdecreases compliance without improving electrical performance. An uncured

poly(dimethyl siloxane)-based resin (Dow Sylgard 186) is cast into a mold to createthe sample geometry as shown in Fig. 2b (Scott et al., 2004 ). This technique andresulting geometry are used to eliminate slip at the grips that occurs when a uniformthickness membrane (e.g. created via spinning or dipping) is loaded transversely(Scott et al., 2004 ). This mold is then placed in a vacuum to remove entrained air.The PDMS-based resin is cured by baking at 60 1C for 5 h. The thickness of thePDMS dielectric in the gauge section, i.e. the center span that deects under loading,is nominally controlled by adjusting the unlled gap size of the molds. The PDMSthickness of all of the samples considered here is less than 500 mm to limit the appliedvoltage magnitude needed to cause measurable deformation. The present thickness

and operating voltages generate electrostatic pressures comparable to previousstudies ( Pelrine et al., 1998, 2000 ).

As shown in Fig. 3 , the sub-micron metal electrodes are thin enough to transmitlight through the specimen; the logo is printed on the paper resting beneath thesample. Despite having thickness in the 15300 nm range, the electrodes mechanicalproperties strongly inuence the response of the multilayer, owing to the fact thattheir elastic modulus is $ 105 times greater than the substrate. This effect is mitigatedby the fact that the electrodes develop cracks with relatively small spacing, as shownin Fig. 3 . The characteristics of the mold surfaces play a critical role in sampleperformance, since any surface roughness is ultimately transferred to the electrodes.

Imperfections promote electrode cracking and reduce the effective modulus of themultilayer. As will be illustrated, this can reduce the modulus of the electrodeswithout compromising electrical performance.

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experiments at different applied voltages that were held xed throughout the loadingand unloading.

After a complete set of tests, the lm thickness of the PDMS gauge section wasdetermined by cutting samples from the gauge section and viewing a cross sectionwith an optical microscope. To characterize the spatial variation in membranethickness, nine sections were cut from each sample. In contrast, the electrodethickness was determined a priori by placing a sacricial silicon wafer next to thePDMS layer during deposition. After deposition, the copper was etched to create astep whose height corresponded to the electrode thickness. The step height betweenthe electrode and surface of the silicon wafer was then measured with a Tencorsurface prolometer. This procedure was used on the thicker samples to establish therelationship between deposition time and thickness; the thickness of the two thinnestsamples is extrapolated using this relationship.

2.2. Fabrication of elastomer substrates

Two groups of specimens were fabricated using different molds; both moldsgenerate the specimen geometry shown in Fig. 2 . Group 1 was fabricated using analuminum mold, with two circular aluminum pucks that are xed into position witha small gap, whose dimension is controlled using metal shims as spacers. Completedetails of the mold and this fabrication procedure are given in Scott et al. (2004) . Theshims are removed prior to lling, and after curing the entire mold is separated from

the specimen in two halves. To reduce electrode imperfections resulting from theelastomer copy of the mold surfaces, the aluminum disks that comprise the upperand lower surfaces of the gauge section were mechanically polished.

Unfortunately, polishing of such large mold surfaces ( $ 5 cm diameter) led toslight misalignments during mold assembly, which resulted in signicant thicknessvariations both within a single gauge section and from sample to sample. It isinteresting to note that the electro- mechanical response for Group 1 is still welldescribed using the models presented here, provided one uses the effective modulusdetermined from the experiments. For this reason, we include the results for Group1, despite the signicant variations in sample thickness.

To reduce the thickness variation, a second mold was fabricated from acrylic usinga CNC precision milling machine. This mold consisted of two identical pieces with acircular step, and a at plate with a hole placed between these pieces. The nominalgauge thickness is equal to the thickness of the spacer minus the sum of the stepheights. Specimens fabricated from this mold, referred to as Group 2, hadsignicantly less spatial variation in thickness, and led to greatly improvedagreement with theoretical models for the multilayer mechanical response. Table 1gives the pertinent geometric details for each sample.

2.3. Electrode deposition and wiring

After curing and removal from the mold, both sides of the elastomeric specimenswere coated with a sub-micron thick copper layer via sputtering. The copper lms

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(99.999% pure) were deposited in a magnetron sputter deposition system. Thesamples were loaded above the deposition guns and rotated at 3 rpm while undervacuum to ensure uniform lm thickness. Varying the deposition time pro-duced different electrode thicknesses. The sputtering process creates a relativelybrittle layer of copper (with failure strains on the order of 1%); this is furtherdiscussed in Section 5.

Wire leads were attached to the edge of either side of the sandwich structure using

a conductive epoxy commonly used in micro-electronics (CW2400 ConductiveEpoxy, Chemtronics). Residual stresses generated during curing lead to highlylocalized lm cracking around the junction, as shown in Fig. 3 .

2.4. Load-deection measurements

The coated elastomer was placed between two insulating acrylic plates, with thewire leads running between the specimen and insulating plates, as shown in Fig. 2 .These three layers were then clamped into the membrane-testing rig shown in Fig. 4(Scott et al., 2004 ). Deection tests were conducted using a Teon sphere (with

radius $ 19 mm) xed to an indenter shaft supported by an air bearing. A load cellplaced between the indenter shaft and positioning stage measured the load on thesample; displacement was measured using the positioning stage itself. The specimenswere subjected to a maximum deection of $ 38 mm, or $ 1540% of the membranespan. While larger deections are easy to achieve, the resulting load-deectionresponse is not amenable to closed-form solutions ( Begley and Mackin, 2004 ; Scottet al., 2004 ). The load cell resolution is $ 5 mN and the displacement resolution is$ 20 mm. Comprehensive studies of the test rig dynamics (including measurementdrift, sample rate-dependence, air bearing performance, etc.) are described in Scottet al. (2004) . These studies revealed that rate-dependence and friction in the air

bearing are negligible over the range of indentation speeds used here (typically$ 20 mm/s). Time-dependent material response is not observed for such indentationspeeds ( Scott et al., 2004 ). Voltage was applied across the electro-mechanical

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Table 1Membrane sample properties

Sample Deposition time (min) Membrane thickness ( mm) Electrode thickness (nm)

Group 1 1A 1 162 (stdv 92) 161B 1.5 300 (stdv 92) 241C 3 219 (stdv 108) 501D 9 182 (stdv 78) 1501E 18 144 (stdv 70) 285

Group 2 2A 1 93.3 (stdv 17.7) 162B 3 94.6 (stdv 15.2) 482C 6 89 (stdv 27.2) 962D 12 160.8 (stdv 30.1) 192



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specimen using a high-voltage source (Glassman High Voltage, Inc., Model#FC15P8). The membrane deection experiments were conducted several secondsafter the voltage was applied to minimize the presence of any time-dependentcharging effects introduced by the voltage source. Experiments involving time-xedmembrane displacement and voltage showed negligible changes in load over the timescale corresponding to the deection tests, i.e. several seconds.

3. Theoretical models

In Section 3.1, we rst describe the mechanical response of cracked multilayers, toidentify the effective modulus of the multilayer, E , in terms of layer properties andcrack density. Section 3.2 describes the load-deection behavior of the membrane interms of this parameter. Once the effective modulus of the cracked multilayer isdetermined from cases without electrically induced strain, the deection models canbe used to relate the induced in-plane strain to the applied voltage, as described in

Section 3.3.For the scenarios considered here, strains are less than several percent. As such, a

linear elastic constitutive description is a good approximation to the nonlinearbehavior typically observed for elastomers at larger strains, particularly with regardsto the load-deection results ( Begley and Mackin, 2004 ). We assume the membraneis nearly incompressible (with v 12), such that the effective elastic modulus of themultilayer, E , completely describes the multilayers mechanical response.

3.1. Mechanical response of cracked multilayers

Begley and Bart-Smith have developed quasi-analytical models for the modulus of cracked multilayers comprised of ultra-thin stiff lms on highly compliant substrates(Begley and Bart-Smith, 2005 ). The models describe the relationship between the

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Fig. 4. Schematic diagram of the experimental test-rig used to clamp and displace the membrane while

measuring indenter loads; further details can be found in Goulborne et al. (2004) .



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average macroscopic stress s and macroscopic strain of the multilayer ( ), the crackspacing, L , crack opening displacement, and the steady-state energy release rate forchanneling crack formation, G ss . Key dimensionless parameters are identied basedon a shear lag analysis (similar to that used for brous composites, e.g. Hutchinsonand Jensen (1990) ) of a periodically cracked section subjected to plane-straindeformation ( Begley and Bart-Smith, 2005 ). The agreement between nite elementmodels and those presented below is excellent ( Begley and Bart-Smith, 2005 ). Asummary of the variables, measurements and predictions for each sample tested isgiven in Table 2 .

The plane-strain modulus of the intact multilayer (i.e. with no cracks), E o , isdened as

s 2^

hf E f ^

hs E s E o , (1)

where^

hf hf =2hf hs and^

hs hs=2hf hs, where hf and hs are the lm(electrode) and substrate (dielectric) thickness, respectively. The plane-strain moduliof the lm and substrate are dened as E f E f =1 v2f and E s E s=1 v

2s ,

respectively, where E f and vf are the elastic modulus and Poissons ratio of the lm,and E s and vs are the elastic modulus and Poissons ratio of the substrate. Theeffective Poissons ratio of the multilayer is assumed to be dominated by the thickerPDMS dielectric, such that for the entire multilayer , v$ 12 and E

43 E .The effective modulus of the cracked layer, E L, relates the average multilayer

stress, s , and macroscopic plane strain imposed on the multilayer, , according to

s E L . The effective modulus of the cracked multilayer is naturally lower than E o , and decreases with increasing crack density (or decreasing crack spacing) ( Begleyand Bart-Smith, 2005 ):

E E o

1 0:2hf E f L E s

^

hf E f ^

hs E o !" #1

, (2)

where the pre-factor 0.2 has been determined from nite element models. For sub-micron metal electrodes and elastomer substrates with thickness $ 0.1 mm, the term

^

hf E f =^

hs E o ffi 1 (see Table 2 ).

Begley and Bart-Smith also present closed-form results for the crack driving forcesfor periodic cracking in the electrodes. Their results are extended here to consider thescenario of sequential cracking of the multilayer, rather than the spontaneousformation of an array of cracks as considered in Begley and Bart-Smith (2005) .Given an initial crack spacing of Lo , we are interested in determining the crackdriving force for an array of cracks with the nal spacing L f . Assuming a uniformcrack opening displacement through the thickness of the lm, the steady-state energyrelease rate for the formation of channel cracking, G ss , is given by

G ss s f L o d L f , (3)

where s f L o is the stress in the intact portion of the electrode prior to the formationof the nal set of cracks, and d L f is the crack opening displacement after theformation of the nal set of cracks.

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Table 2Properties, measurements and predictions for the mechanical response of the multilayers (zero voltage)

E f 125lGPa, E s 1:3MPa

Specimen hs(mm)

hf (nm)

hf E f =hs E o E o(MPa)

L (mm) hf E f =L E s PredictedE c (MPa)

MeasuredE c (MPa)

Measuredd max (mm)

1A 162 16 0.90 27 18 84 1.3 1.6 6.24 1B 300 24 0.88 23 36 63 1.4 2.5 5.27 1C 219 50 0.96 60 38 125 1.8 2.9 5.54 01D 182 150 0.99 210 126 113 6.7 3.8 5.41 01E 144 285 1.00 499 373 72 24.2 15.6 3.66 0

2A 93.3 16 0.94 46 26 58 2.9 2.9 6.49 02B 94.6 48 0.98 130 40 114 4.2 4.8 6.17 02C 89 96 0.99 273 66 157 7.3 6.6 5.38 02D 160.8 192 0.99 302 106 171 6.5 6.4 4.76 0


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Begley and Bart-Smith illustrate that the nal crack opening displacement is wellapproximated as

d 0:2hf E f E s

E L f E o , (4)

where is the macroscopic strain imposed on the multilayer. The stress in the lm prior to cracking is approximated as s f L o E f E f s = E o , since the lm isintact. Noting that s E Lo , the energy release is given by

G ss % 0:2 E Lo

E o

E L f E o

E f 2hf . (5)

If one assumes that crack densities are large enough such that hf E f =L E sb 1 (as

supported by Table 2 ), the second term in Eq. (2) dominates, and Eq. (5) can besimplied considerably. In experiments, it is easier to observe crack saturation thanthe onset of cracking. Assume that the crack spacing prior to the nal cracking eventis exactly twice that of the nal cracks spacing, i.e. Lo ! 2L f . In this case, Eq. (5)simplies to

G ss % 10 E sL f 2

hf

E s E o Lhf

2 s 2hf E o

. (6)

This is an interesting result that differs signicantly from the case of a lm on asemi-innite substrate, and reects a number of interactions between the lm, nitethickness substrates and periodic crack spacing. In most channel cracking models,the lm makes a negligible contribution to the overall stiffness of the multilayer. Inthis limit, the crack openings scale with the lm thickness because changes in lmthickness do not affect the overall macroscopic strain (at xed stress). However, inthe scenario considered here, decreasing the lm thickness will appreciably increasethe macroscopic strain (at xed stress), leading to larger crack opening displace-ments. Alternatively, at xed strain, decreasing the lm thickness reduces theconstraint imposed by the lm at the crack, allowing for larger crack openings.

The key implication of Eq. (6) is that for stiff lms on relatively compliantsubstrates, crack spacing should increase with increasing lm thickness. This is clearupon setting G ss G c and solving for the crack spacing. This result is contrary to thetrend expected for scenarios where the lm makes a negligible contribution to theoverall stiffness of the multilayer.

3.2. Deection of membranes with in-plane strain using spherical indenters

We use results for membrane deection outlined in Begley and Mackin (2004) ,wherein closed-form solutions relate the indenter load, displacement, and strain.While these solutions invoke several approximations, they have been shown to be

accurate when compared with full numerical solutions and experiments performedon identical elastomer specimens to those considered here ( Begley and Mackin, 2004 ;Scott et al., 2004 ). The solutions are expressed in terms of a uniform in-plane strain,

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o , which in this case is a result of the transverse pressure generated by the electriceld.

The contact radius as a function of applied load is given by ( Begley and Mackin,2004 ):

rc ffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffi32p P EhR 6p 2o s 3 ov uut, (7)

where rc rcR is the contact radius, P is the indenter load, h is the total multilayerthickness, E is the effective modulus of the multilayer, and R is the indenter radius.The maximum strain arising from deection is dictated by uniform stretching of themembrane in the contact region ( Begley and Mackin, 2004 ). Inside the contact, the

strain is approximately equal to

i ffi16

r2c 16 ffiffiffiffiffiffiffiffiffiffiffi32p P EhR 6p 2o s 12 o . (8)

The load-deection response of the membrane is given by

D

R

23p

aR

3=4 ffiffiffiffiffiffiffiffiffiffiffi9p 4P =EhR 2 64p 6 3oq 3p P =EhR 2=3 4p 2 o

ffiffiffiffiffiffiffiffiffiffi9p 4P =EhR 2 64p 6 3o

q 3p P =EhR

1=3

8>>>>>:

9>>>=>>>;

, (9)

where D is the deection of the center of the membrane.In the limit of zero electrically induced strain (i.e. o ! 0), Eq. (9) simplies to

d

R

aR

3=4 169p

P EhR

1=3

. (10)

Eq. (10) can be used with deection experiments at zero applied voltage to extractthe effective modulus of the multilayer, as discussed further in Section 5. Eq. (10)bears close resemblance to Schwerin-type point-load solutions. However, the

dependence on indenter radius is quite different, as expected for scenarios wherethe contact radius is a signicant fraction of the freestanding span ( Begley andMackin, 2004 ; Scott et al., 2004 ).

While no distinction has yet been made between positive-(tensile) or negative(compressive)-induced strain, it must be assumed that applied voltage causes in-plane strains that are smaller than those induced by lm deection. Otherwise, thestress resultants in the lm are compressive, and the analysis is not appropriate.Thus, we neglect the possibility of buckling or wrinkling of the multilayer. Thesolutions outlined above are only applicable when the strain (or contact radius) ispositive, as indicated by Eq. (7). In the present context, these approximate solutions

will not be applicable for combinations of relatively low loads and relatively highapplied voltages, which lead to compressive strains that do not support transverseloads.

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3.3. Electrostatic models to relate applied electric eld to in-plane strain

The pressure generated by the electrostatic interaction of the charged electrodescan be estimated using a parallel plate capacitor model ( Stratton, 1941) . Theelectrostatic energy stored in the capacitor can be written as

U k o k

2V h

2

Aoho , (11)

where k o 8:85 1012 C2=N m 2is the permittivity of free space, k is the dielectricconstant of the elastomer, V is the applied voltage (N m/C), Ao is the initial surfacearea of the membrane and ho is the initial thickness of the dielectric. Assuming thatthe dielectric is incompressible, the volume Ao ho is constant, such that theelectrostatic pressure is ( Pelrine et al., 1998 )

pz 1A

dU dh

k o kV h

2

. (12)

Assuming uni-axial compression, vpz E o , such that the uniform in-plane straininduced by the electric eld is given by

o F V 2, (13)

where

F

k o k

2Eh 2 , (14)where once again, the effective Poissons ratio of the multilayer is taken as one-half.

4. Results

4.1. Mechanical response during deection experiments with zero electric eld

Experimental load vs. displacement measurements are shown in Fig. 5 for zero

applied voltage, along with theoretical predictions (i.e. Eq. (10)) adopting E as thetting parameter. The membrane deection is limited to distances for which Eq. (10)has been shown to agree with numerical solutions which include large strainelastomer response ( Begley and Mackin, 2004 ). The maximum imposed strains areless than 3%, as indicated by Eq. (8). Table 2 summarizes both the experimentallydetermined effective modulus, and that predicted via the models in Section 3. Thedata is truncated for loads below 5 mN, which represents the resolution limit of theload cell. This also eliminates the inuence of bending effects at small displacements.

A comparison of the measured moduli of the cracked multilayers is shown inFig. 6 as a function of crack density. We have chosen to present the results in terms

of dimensionless quantities because the normalization accounts for variations insubstrate thickness from one sample to the next. The error bars represent the effectof using the minimum and maximum substrate thickness (i.e. h Dh) in the

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normalization. Absolute values for the predicted modulus and crack spacing areprovided in Table 2 . It is clear that the second sample set (with more carefullycontrolled substrate thickness) yields excellent agreement with the analytical models.The variability of the rst sample set is most likely a result of the spatial variation insubstrate thickness, which leads to less uniform crack spacing.

The present experiments support the crack spacing/lm thickness relationshipimplied by the cracking model, i.e. Eq. (6). Table 2 reveals that crack spacingincreases systematically with lm thickness. An estimate for the lms fracture

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40 60 80 100 120 140 160 180 2 00

Crack Density,

0.15

0.1

0.05

0

E f f e c

t i v e

M o

d u

l u s ,

E c / E

oEquation (2)

(1B)

(2B)

(2A)(1A)

(1E)

(1D)(2C)

(1C)

(2D)

Fig. 6. Comparison of the predicted and measured modulus of the cracked multilayers.

Fig. 5. Experimental load-displacement measurements of all samples after three load cycles, at zeroapplied voltage. The lines represent Eq. (10), with E used as a tting parameter.



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toughness and a quantitative evaluation of Eq. (6) can be made from the presentexperiments as follows. The maximum strain imposed on the multilayer duringdeection can be estimated by combining Eqs. (8) and (10)

i 18

aR

3=8 3d

2a 1=2

. (15)

This is strictly true only for the strain in the contact region; however, thetheoretical and numerical results of Begley and Mackin indicate that for a=R o 2, thestrain at the outer edge of the membrane is only $ 20% lower than that predicted byEq. (15) ( Begley and Mackin, 2004 ). Hence, the strain in the membrane does nothave signicant spatial variations and Eq. (15) is a reasonable estimate for themacroscopic strain imposed on the multilayer.

We have chosen to focus on Group 2, since the modulus predictions andmeasurements are in much better agreement. However, the results of applying thefracture model for both groups are provided in Table 2 . Fig. 7 depicts crack spacingas a function of ffiffiffiffiffiffiffiffiffiffiffihf = 2i p , and illustrates that the present experiments are consistentwith the above predictions. The error bars reect a 20% uncertainty in imposedstrain, and 10% uncertainty in crack spacing. A least squares t of the form given byEq. (7) implies a toughness value for the lms of G c % 152J =m 2. If one were todetermine the toughness for each case using Eq. (7) directly, the results range fromG c % 1402 240J =m 2. The reasonableness of these results is discussed in Section 5.1.

4.2. Mechanical response as a function of applied voltage

Once the effective modulus has been identied using the results for zero appliedvoltage, the relationship between applied voltage and induced strain can be

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Fig. 7. Crack spacing as a function of lm thickness.



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determined by conducting deection experiments at different voltages. Fig. 8 showsthe effect of applied voltage on the measured load-displacement response for Sample 2A, corresponding to the membrane with the thinnest electrodes (and lowest effectivemodulus) in Group 2. The presence of an electric eld has a dramatic effect on the

membrane displacement, indicating that signicant in-plane strains have beengenerated due to the attractive force between the electrodes. Note that for largeapplied voltages, the displacement at zero load is not zero; this is because inducedstrains cause the samples to bulge outward.

The measured response of the membrane is tted with the Begley-Mackin model(Eq. (9)), using the zero-voltage experimental result for E and adopting the inducedstrain o as a tting parameter. The in-plane strain can then be determined as afunction of applied voltage by tting a sequence of curves such as that shown inFig. 8 . The results of this procedure are shown in Fig. 9 , which illustrates therelationship between induced strain and the applied voltage for all samples in both

groups. There is a clear indication that the induced in-plane strain varies as with thesquare of the voltage, as implied by Eq. (13). Furthermore, the effect of the appliedvoltage decreases with increasing electrode thickness and increasing crack spacing, asexpected.

Eqs. (13) and (14), which relate electrically induced deformation, specimendimensions, and the modulus of the multilayer can be directly evaluated using theseresults. The experimental coefcient F which relates the induced strain and appliedvoltage is shown in Fig. 10 as a function of the parameter Eh 2s . These coefcientswere determined by a least squares t to the results shown in Fig. 9 ; the modulus istaken as the experimentally determined effective modulus listed in Table 2 . The error

bars represent the standard deviation arising from thickness variation. Fig. 10illustrates excellent agreement between the experiments and the theoretical relation-ship between induced strain and the effective modulus of the multilayer.

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Fig. 8. Load vs. deection for Sample 2A showing variation of compliance of membrane due to presencesof electric eld. A comparison between the measured response and Eq. (9) assuming effective modulus, E ,and o as tting parameter is shown.



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5. Discussion

5.1. Crack spacing and inferred lm toughness

The toughness values inferred from the present experiments raise two interestingquestions. First, why is the average toughness so much lower than typical values for

bulk metals? Second, what is the rationale for the systematic trend seen in Group 2(though not in Group 1), wherein toughness increases with decreasing lm thickness?

First, we address the issue of measured average toughness values that arecharacteristic of brittle materials. While the models involve several approximations,it is unlikely that any of them lead to dramatically lower energy release rates thatyield signicantly higher inferred toughness values. The most signicant approxima-tions are that the elastomer substrate is assumed to be linear elastic, and largedeformations are neglected. However, the tests involve macroscopic strains less than3% and for the elastomers used here, the tangent modulus does not changeappreciably in this range ( Scott et al., 2004 ). Hence, the approximations are most

questionable in a highly localized region near the crack tip.It is difcult to say without more detailed calculations if these effects are

important. On the one hand, large deformation formulations generally predict

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Fig. 9. Electrically induced strain varies as with the square of the voltage, as implied by Eq. (13).



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smaller crack openings: on the other, the decrease in tangent modulus observed inelastomers (for $ 0:1 1) would allow for greater crack openings. Given thesecompeting mechanisms, it seems highly unlikely that the net effect is to change crackopening predictions by more than 100%. This is strongly supported by the fact thatthe predicted modulus, which is derived directly from predicted crack openingdisplacements ( Begley and Bart-Smith, 2005 ), agrees closely with experimental

measurements. Thus, it seems very unlikely that the toughness of these metal lms isgreater than $ 500 J/m 2 .It is reasonable that one would obtain such low toughness values when one

considers the nanoscale thickness of the lms and recent studies of dislocationbehavior in small volumes (e.g. Espinosa et al., 2004 ; Hsia et al., 1994 ). Thesputtering of sub-micron lms generates sub-micron grains ( Espinosa et al., 2004 ),whose boundaries act as dislocation barriers, which inhibit the plastic deformationthat would normally lead to higher toughness values. Furthermore, theoreticalmodels used to rationalize brittle fracture in the presence of plasticity havepostulated the presence of a dislocation-free zone ahead of the crack tip that elevates

stress and promotes cleavage ( Hsia et al., 1994 ). These models predict that the upperlimit for the size of this zone is $ 300500 nm. Hence, it seems likely that thethickness of the lms and their ne microstructure signicantly inhibit plasticdeformation, and dramatically reduce the toughness over micron-scale lms.

Despite the overall low toughness values, there is a clear systematic decrease intoughness with increasing lm thickness for Group 2. (The highly variable substratethickness for Group 1 decreases the homogeneity of lm cracking and eliminates thistrend.) At rst glance, the trend in toughness values for Group 2 is at odds with theprevious discussion of dislocation behavior; one might expect that the thicker lmswould yield larger toughness values. However, even the thickest lms in the present

samples have sub-micron thickness, implying that the lms are not thick enough toapproach bulk behavior. That is, even the thickest lms have sub-micron grains thatsignicantly limit plastic deformation and keep the toughness relatively low.

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Fig. 10. The coefcient relating in-plane strain with applied voltage as a function of Eh 2s .



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The increasing lm toughness with decreasing thickness can be rationalized in thecontext of recent theoretical work regarding the necking and fracture of thin metalliclms on polymer substrates. This work indicates that debonding of the lm near thecrack has a strong inuence on lm cracking ( Li et al., 2005 ; Begley et al., 2005 ). Thefailure strain of lms adhered to substrates may be larger than freestanding lms dueto the constraint against necking imposed by the substrate. The present experimentsindicate failure strains on the order of $ 1%, which is consistent with freestandingcopper lms ( Espinosa et al., 2004 ). It is difcult to determine if debonding near thecracks occurs in the present experiments, since this behavior is highly localized (i.e.probably occurs at length scales near or beneath one micron). The present lms areadherent in the sense that the electrodes do not visibly ake off, even between cracksseparated by only a few microns. The consistent inverse size-dependence of thetoughness, as seen in Table 2 for Group 2, can be explained using analytical modelsof highly localized necking instabilities ( Begley et al., 2005 ). These models reveal thatthicker lms are less affected by substrate effects, such that the necking instability isnot suppressed.

5.2. Electro-mechanical coupling

It has been shown that the application of an electric eld between two parallelelectrodes, separated by a compliant dielectric, will produce strains perpendicular tothe eld. The experimental relationship between in-plane strain and applied voltage

shows that the model accurately predicts the electro-active response of themembranes. Using the results in Fig. 10 , the effective dielectric constant of themultilayer, k , can be calculated using Eq. (14). This yields 3.1 and 1.7 for Group 1and Group 2, respectively. These results are close to standard reference values forsilicone-based elastomers, as one would expect based on the parallel plate capacitormodel described in Section 3.3. The difference between the two groups (with respectto dielectric constant) is most likely due to differences in cracking patterns caused bydifferences in substrate variability (i.e. both thickness and mold surface topology). Itis difcult to envision that one would obtain values identical to the substrate materialitself, given the likelihood of uneven charge distribution in the presence of cracking.

In the present experiments, the amount of electrically induced strain is limited bydielectric breakdown across the electrodes. The present experiments achievebreakdown elds which are 4050% of those reported elsewhere ( Pelrine et al.,1998, 2000 ). The lower breakdown eld reported here is most likely due to defects inthe substrate introduced during casting (i.e. spun-cast samples are much freer of defects), and charge localization near cracks in the electrodes. Further work isneeded to develop laminates that (i) exhibit higher breakdown voltages (i.e. are freeof defects) and (ii) are amenable to membrane deection experiments.

Indeed, it may even be surprising that the multilayers exhibit any signicantelectro-mechanical coupling, given that electrode cracking would intuitively lead to a

loss of electrical performance. The cracking models described in Section 3.1 can beused to explain why electrical functionality is retained in the presence of electrodecracking. Begley and Bart-Smith point out that the charge distribution in cracked

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electrodes will depend on the increment in voltage applied to the wire lead and thecrack spacing between adjacent islands of electrode ( Begley and Bart-Smith, 2005 ).If the electric eld generated by a charge differential between adjacent islands islarger than the breakdown eld in air, the cracked layers will continue to distributecharge. Thus, the condition for charge distribution is that DV 4 E b d , where DV is thevoltage increment applied to the electrical lead and E b $ 10MV =m is the breakdowneld over the cracks.

Table 2 lists the anticipated crack opening for each of the samples based onEq. (4), which incorporates the effective crack spacing. It is clear that heavily crackedlms lead to small crack opening displacements, such that DV $ 10 V. Even if thecrack openings are an order of magnitude larger than the predicted values, 3 theapplied voltages needed to induce signicant mechanical strains are much higherthan this, such that electrode cracking does not play a signicant role in chargedistribution. Put another way, the crack openings are much smaller than thesubstrate thickness, such that electric eld generated between adjacent islands ismuch larger than the eld required for breakdown from one electrode to the other.

The experimental and theoretical relationships between crack spacing and electro-mechanical performance clearly indicate that crack spacing improves electricalperformance, for two reasons. (i) High crack densities reduce the effective in-planemodulus of the multilayer, increasing actuation strain. (ii) High crack densitiesreduce crack openings, promoting breakdown across adjacent islands of electrodematerial and leading to charge distribution in the cracked electrodes. It is interesting

to note that Group 1, which has signicantly greater thickness variation from sampleto sample, yield noticeably better electro-mechanical coupling (on average Fig. 10 ).This may be due to uneven yet favorable crack distributions that lower the effectivemodulus and improve charge distribution in localized regions. While this leads tobetter average performance, the response varies a great deal, making comparisonswith theoretical models problematic.

6. Concluding remarks

This novel experimental procedure can be used to characterize both themechanical and electrical performance of metal/elastomer membranes. The Begley-Mackin membrane deection model enables extraction of both the effective modulusof the cracked multilayer, and electrically induced strain. These experiments haveillustrated that the models are capable of predicting the coupled electro-mechanicalresponse of a metal/elastomer membrane subject to an electric eld. A keyimplication of the present experiments and theoretical models is that electrodecracking should be promoted, as it reduces the effective modulus of the multilayerand improves charge distribution because crack openings are smaller. The mechanics

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3This is highly unlikely; the agreement between the experimental and theoretical effective modulus isexcellent. Since the theoretical models are based on the crack opening contribution to macroscopic strain,it is unlikely the predicted crack openings are that far off.



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framework presented here also enables the determination of fracture toughness forsub-micron lms.

Acknowledgements

MRB and MHJ gratefully acknowledge support through the National ScienceFoundation, award #9984517 and an accompanying Research Experience forUndergraduates (REU) grant. HBS and ONS gratefully acknowledge the support of the National Science Foundation through Award #0348448 and the David andLucile Packard Foundation through the Packard Fellowship for Science andEngineering.

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Begley, M.R., Mackin, T.J., 2004. Spherical indentation of freestanding circular thin lms in themembrane regime. J. Mech. Phys. Solids 52, 20052023.

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electro-mechanical response of elastomer membranes coated with ultra-thin metal elecgtrodes

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