J Adhesion 2011 elastic on patterned substrate

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Contact Instability of a Soft Elastic Film Bonded to a Patterned SubstrateJayati Sarkara; Hemalatha Annepua; Ashutosh Sharmab

a Department of Chemical Engineering, Indian Institute of Technology Delhi, New Delhi, India b

Department of Chemical Engineering, Indian Institute of Technology Kanpur, Kanpur, India

Online publication date: 11 March 2011

To cite this Article Sarkar, Jayati , Annepu, Hemalatha and Sharma, Ashutosh(2011) 'Contact Instability of a Soft ElasticFilm Bonded to a Patterned Substrate', The Journal of Adhesion, 87: 3, 214 — 234To link to this Article: DOI: 10.1080/00218464.2011.557332URL: http://dx.doi.org/10.1080/00218464.2011.557332

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Contact Instability of a Soft Elastic Film Bonded to aPatterned Substrate

Jayati Sarkar1, Hemalatha Annepu1, andAshutosh Sharma2

1Department of Chemical Engineering, Indian Institute ofTechnology Delhi, New Delhi, India2Department of Chemical Engineering, Indian Institute ofTechnology Kanpur, Kanpur, India

A linear stability analysis is presented for the contact instability of a soft thin elas-tic film which is rigidly bonded to a physically patterned substrate, and isin adhesive contact with a smooth rigid contactor. Increasing roughness byenhancing the substrate-amplitude produces increasingly smaller instabilitylength-scales. The smallest wavelengths obtainable are 0.3�h, an order of magni-tude smaller than that observed with films on flat substrates (3�h). Instabilitylength-scales are found to be largely independent of substrate length-scales. Forvan der Waals interaction, increase in substrate roughness increases the energypenalty and, consequently, requires smaller gap distances (<1nm for stiff films)to engender instabilities. When an externally controllable long-range electric fieldis employed instead, instabilities can be initiated at very low critical voltages(�32V) even in relatively stiff films, making it a more suitable route to produceminiaturized instability patterns.

Keywords: Contact instability; Elastic film; Linear stability analysis; Pattern forma-tion; Patterned substrate; Soft adhesion

1. INTRODUCTION

Instabilities in both liquid and solid thin films have been extensivelystudied over the last few decades because of their potential for wide

Received 2 August 2010; in final form 3 December 2010.

One of a Collection of papers honoring Chung-Yuen Hui, the recipient in February2011, of The Adhesion Society Award for Excellence in Adhesion Science, Sponsored by3M.

Address correspondence to Jayati Sarkar, Department of Chemical Engineering,Indian Institute of Technology Delhi, New Delhi 110016, India. E-mail: [email protected]

The Journal of Adhesion, 87:214–234, 2011

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ISSN: 0021-8464 print=1545-5823 online

DOI: 10.1080/00218464.2011.557332

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applications in fabrication of opto-electronic devices, lab-on-chip smallsensors [1,2], MEMS [3], functional coatings [1], pressure sensitiveadhesives [4], etc., and in understanding the physics underlying theseinstabilities. For example, the instabilities initiated in the viscousliquid wetting films of thicknesses (h) less than 100nm are governedby a competition between the destabilizing intermolecular interactionsbetween the film and the substrate and the surface tension=energywhich has a restoring effect. The wavelength of the long-wave insta-bility in this case depends on the nature and magnitude of the desta-bilizing force and scales non-linearly with the film thickness (forexample, it varies as h2 for the van der Waals interactions) [5–10].

Instabilities have been observed in elastic films in two different con-figurations. In peel geometry, the film is rigidly bonded to a substrateand a glass slide rests on the film in cantilever configuration to initiatea crack. When a normal force is applied at the end of the glass slide,well defined 1-D finger-like patterns with a wavelength �3�h–4�happear at the crack opening [11–14]. In the peel experiments, theinitiated crack propagates laterally throughout the length of the film[13] and, if the film has incisions in it or has a fibrillar structure[15–19], the crack needs to reinitiate at each of these structures. Sothe pull-off forces required to debond and the work of adhesion arefound to be higher for the patterned structures [15–19] giving someinsight into the working of a gecko’s feet.

In the adhesive-contact geometry considered here, the free surfaceof the film in contact proximity with an external contactor sponta-neously roughens to form a 2-D labyrinthic pattern when the stiffnessof the attractive forces (for example, the van der Waals or electrostaticforces) exceeds the elastic stiffness of the film. However, the wave-length of instability in this case is independent of the nature of theinteractions and the elastic modulus and the wavelength of the surfaceundulations scales as �3�h [20–32]. This wavelength produces theleast elastic energy penalty for surface deformations. In elastic filmsdiscussed here, the instabilities are, thus, short-wave and arise dueto a conflict between the attractive forces that destabilize the filmand the elastic energy that opposes it [21,22]. These instabilities arequite distinct from the instabilities in viscous wetting films, as theyare uninfluenced by the nature of the interactions present betweenthe film and the contactor and are short-waved as they scale linearlywith the film thickness. The independence of the instability wave-lengths with respect to the nature of the interactions has been verifiedby introducing an electric field between the contactor and the filmsurface [28–30]. The instabilities formed still scale as �3�h but thepatterns were well-ordered [29] compared with the labyrinthic

Instability of a Soft Film on a Patterned Substrate 215


patterns obtained from VDW interactions. At the onset of instability,the film surface forms columnar structures. On further debonding,the intervening cavities deepen and the columns peel off from thesides, reducing the contact area, until they finally snap-off. Thepull-off forces required to debond are orders of magnitude less thanthat required to debond two flat surfaces in adhesive contact, as alsofound in the peel experiments. In simulations, the pathways ofdebonding were found to depend on factors like step-size of debonding,noise present in the system, etc., which reduce the snap-off distanceand the pull-off force [25,26]. Noise introduced in the form of rough-ness of the patterned contactor is also found to influence the adhesion-debonding cycle [27]. In such a spatially non-uniform system, surfaceinstabilities initiate around the protruded parts of the contactor, butthe periodicity of the contactor-pattern had little effect on the surfacemorphologies [27,31,32]. The peak debonding force and the work ofadhesion, however, were found to reduce tremendously because ofthe presence of roughness in the contactor [27].

More recently, instability of an elastic bilayer was analyzed toexplore if surface length scales smaller than 3�h can be achieved, asminiaturization of instability patterns by this self-organization routecan be a potential technique for micro-patterning of soft functionalfilms. PDMS-metal bilayers were still found to give a wavelength�3�h [33], whereas viscous-viscous [34], visco-elastic [35] and elastic-elastic bilayers [36,37] showed lower length scales, with the smallestbeing �0.5�h for the elastic bilayers.

In order to eliminate the technical difficulties involved in castingbilayers, in the present paper we investigate an alternative approachof pattern miniaturization by depositing a smooth elastic film on atopographically patterned substrate. Since the film thickness now var-ies along the length of the film, we address the question of which thick-ness governs the instability-wavelength: the local minimum thickness,hmin, or the local maximum thickness, hmax, or the mean thickness, h?It may be anticipated that a new length scale depending on the sub-strate topography may arise because the total energy now becomes afunction of the substrate pattern parameters. An experimental studyof the contact instability on a periodic substrate [38] has focused onthe conditions for the alignment of the instability pattern with thesubstrate pattern. The effect of substrate topography (amplitude andperiodicity) was not investigated under a wide range of parameters.The linear stability analysis presented here shows that the instabilitylength scale can be modulated substantially by the substrate topogra-phy, which can guide the design and interpretation of future experi-ments in this area.

216 J. Sarkar et al.


In this paper, we focus on the linear stability analysis to understandthe effect of parameters of the patterned substrate on the instabilitylength scales. The linear stability analysis covers a wide range ofpattern parameters (substrate pattern amplitude, bp, and number ofsubstrate pattern wave modes n2) to uncover the new length scalesthat are formed at the film surface. The linear stability analysis per-formed here gives information about the magnitude of the thresholdinteraction force required for the inception of surface instabilitiesand the dominant wavelength of these instabilities at the criticalforce=separation distance. The contactor and the film surface can inter-act via either a short-range (van derWaals, VDW) or a long-range (elec-tric field) interaction. Both cases are considered in the present study.

2. MATHEMATICAL FORMULATION

Figure 1 illustrates the schematic of a soft, thin, incompressible,initially stress-free elastic film of length, L, thickness, h, andshear-modulus, l (<10MPa), deposited on a step-patterned substrate.The topography of the patterned substrate can be described by itsamplitude, bp, and wavelength, kp. The wavelength, kp, is L=n2 where,n2 is the number of wave modes of the patterned substrate. The para-meter n2 can be considered as the number of elevations (or depres-sions) in the step-profiled substrate. It is the repetitive units of thedyads (formed by one elevation and one depression of width L1 each)

FIGURE 1 Schematic diagram of a soft, thin, incompressible elastic film oflength L, thickness, h, and shear-modulus, l, bonded to a rigid patterned sub-strate in the form of a step-profile with amplitude, bp, width of elevations=depressions, L1 and situated at a gap distance, d, away from a contactor.



present in the patterned substrate. Thus, the parameters bp and n2,which dictate the substrate landscape, will be our controllingparameters. The top surface of the film is considered to be smooth(i.e. free of any defects) and it is situated at a gap distance, d, awayfrom an external surface (contactor). The contactor and the film sur-face can interact via either a short-range (van der Waals) or along-range (electric field) interaction. Both cases are considered inthe present study.

When the contactor is at a very close gap distance (�10nm) theVDW force becomes effective and the film surface undulates as shownby the broken line in Fig. 1. The surface interaction potential for thiscase, consisting of an attractive VDW component, is a function of thegap distance d�u .n only and can be represented by:

U u � nð Þ ¼ � A

12pðd� u � nÞ2ð1Þ

where u is the displacement vector, n represents the surface normaland A is the Hamaker constant (�10�20 J).

Long-range interactions in the form of an electric field can bepresent between the film and the contactor if an external voltage isapplied between the contactor and the substrate. The surface undu-lations shown in Fig. 1 can also appear in this case if the appliedvoltage exceeds a critical value. The electric potential of thiscapacitor-geometry can be represented by [28,29]:

U u � nð Þ ¼ � e0ef V2

2 ef � 1� �

ðd� u � nÞ þ dþ h� bp� � ð2Þ

where, V is the applied voltage, ef is the dielectric constant (2.5 forPDMS films), e0 is the free-space permittivity (8.85� 10�12C2=J-m)and h� bp denotes the local thickness of the elastic film (h� bp atelevations and hþ bp at depressions of the substrate).

A short-range Born repulsion term, which is active only atfilm-contactor gap distances less than �0.1368 nm, is also present inboth the cases of interaction potentials considered above. Since inthe linear stability analysis we are looking at only the onset of insta-bility, which occurs at a gap distance of few nanometers in the case ofVDW interactions and even higher values (up to the micron range) inthe case of a long-ranged electric field, this term is neglected. Therange of the film thicknesses of importance considered in the experi-ments as well as in the simulation studies are generally of the orderof few microns [11–14, 20–38]. Hence, any destabilizing interactions



of the film surface with the substrate are of little consequence and donot contribute to the interaction energy.

For linear stability analysis, a linearized form of the interactionforce is used. The interaction potential is, thus, expanded in its Taylorseries with the undeformed film as the reference state and terms to thequadratic order are retained as given by:

PU ¼ZS

U u � nð ÞdS ¼ZS

U0 þ F u � nð Þ þ 1

2Y u � nð Þ2

� �dS; ð3Þ

where,

U0 ¼ U 0ð Þ; F ¼ U0 0ð Þ; and Y ¼ U00 0ð Þ: ð4Þ

Here Y is called the interaction stiffness and plays an important rolein determining the unstable modes. The form of interaction stiffness,Y, varies according to the interactions under consideration. WhenVDW interactions are considered, the form of this parameter is,

Y ¼ A

2pd4ð5Þ

For electrostatic interactions, the interaction energy depends on thesubstrate topology as well as undulations of wavemodes on the filmsurface and the interaction stiffness has the form:

Y ¼ef � 1� �2

e0ef V2

dþ h� bp� �3 ð6Þ

The stored elastic energy will try to resist any deformation caused bythe interaction potential in an attempt to restore the film to its orig-inal configuration and this total elastic energy of the film is:

PE ¼ZV

l2rijeijdV ð7Þ

where

r ¼ pI þ l ruþruT� �

: ð8Þ

The parameter r represents the stress in the film, p is the pressurefield across the film, e is the strain tensor and is defined as the sym-metric part of ru.

In the linear stability analysis, a 2-D configuration will be con-sidered as shown in Fig. 1 and, thus, the total energy per unit lengthand depth of the film (P) composed of the two antagonistic energies



given by Eqs. 3 and 7 is:

P¼PUþPE ¼ 1

L

Z L

0

U0þF u �nð Þþ1

2Y u �nð Þ2

� �dx1þ

Z L

0

l2rijeijdx1

� �

ð9ÞThe step-patterned substrate, consisting of blocks of elevations and

depressions of width L1 each, is assumed to be piecewise-continuous.So in the following formulation, all the governing equations are takenseparately over every block where the thickness of the film is uniform(i.e. h� bp at the elevations and hþ bp at the depressions) and thenassembled (as given later) to describe the entire problem.

The equilibrium displacement that minimizes the total energysatisfies the stress equilibrium condition in the bulk:

lui;jj þ p;i ¼ 0; ð10Þ

and the incompressibility condition:

ui;i ¼ 0: ð11Þ

The boundary conditions at the film-contactor interface are given bythe displacement condition and the shear-free condition:

u2 x1; 0ð Þ ¼ acos k1x1ð Þ and r12 x1; 0ð Þ ¼ 0: ð12Þ

The rigid boundary conditions at the film-substrate boundary aregiven by

u1 x1;�h� bp� �

¼ 0 and u2 x1;�h� bp� �

¼ 0 ð13Þ

The perturbed displacement and the pressure fields are considered tobe of the form,

u1 x1; x2ð Þ ¼ ~uu1 x2ð Þsinðk1x1Þ

u2 x1; x2ð Þ ¼ ~uu2 x2ð Þcosðk1x1Þ

p x1; x2ð Þ ¼ ~pp x2ð Þcosðk1x1Þ ð14Þ

The afore mentioned stress and displacement equilibrium conditionslead to the following fourth-order differential equation in the displace-ment field, valid over every elevation and depression of the patternedsubstrate:

~uuiv2 x2ð Þ � 2k21 ~uu

002 x2ð Þ þ k41 ~uu2ðx2Þ ¼ 0 ð15Þ



The general solution of this differential equation has the followingform:

u1 x1; x2ð Þ ¼ � ak1

Qþ k1ðPþQx2Þð Þek1x2 þ S� k1ðRþ Sx2Þð Þe�k1x2� �

� sinðk1x1Þ;u2 x1; x2ð Þ ¼ a ðPþQx2Þek1x2 þ ðRþ Sx2Þe�k1x2

� �cosðk1x1Þ; and

p x1; x2ð Þ ¼ �2la Qek1x2 þ Se�k1x2� �

cosðk1x1Þ: ð16Þ

The unknowns P, Q, R, and S depend on the boundary conditions(Eqs. 12 and 13) which are functions of the substrate morphology.These coefficients evaluated are given in Appendix A. Thus, the dis-placement and the pressure fields involving these coefficients, andgiven by the solutions of Eq. 16, become functions of bp and n2.As a result, the total energy (given by Eq. 9) composed of the elasticenergy and the interaction energy will be a function of the substrateparameters. Thus, the morphology obtained by minimizing this totalenergy (as can be shown by non-linear analysis) also becomes afunction of the substrate parameters (amplitude bp and wave modesn2). If the form of Eq. 16 is considered, the rigid boundary con-ditions, given by Eq. 13, are exactly satisfied at the elevationsand the depressions. However, a slip is present at the vertical wallsbetween the elevations and the depressions. The magnitude of theslip can be evaluated from the values of the displacements u1(x1,x2) and u2(x1, x2) obtained from Eq. 16 at any point, x2, on the ver-tical wall boundary (where, (�hþ bp)< x2< (�h� bp)). (Theunknowns P, Q, R and S of Eq. 16 now become a function of�(h� x2) rather than �bp as can be found out from Eq. A1). The dis-placements thus calculated at the vertical walls are extremely neg-ligible in magnitude (�10�17a, where a<<h), and, thus, slip at thevertical walls of the patterned substrate is safely neglected in thepresent study.

2.1. Linear Stability Analysis

For performing the linear stability analysis, non-dimensionalisation ofthe length scales is done as described below:

X1 ¼ x1=hð Þ; X2 ¼ x2=hð Þ; b ¼ bp=h� �

; L0 ¼ L=hð Þ; d0 ¼ d=hð Þð17Þ



The total energy (of Eq. 9) in the reduced co-ordinates has the form:

Pðb;n2;n1Þ ¼1

L0

�Z L0

0

lð2P� 1Þ�2pn1

L0

�a2 cos2

�2pn1

L0�X1

�dX1

þZ L0

0

1

2hYa2 cos2

�2pn1X1

L0

�dX1

� ð18Þ

Minimization of the above total energy with respect to a needs the sat-isfaction of the following equation:

Z L0

0

2ð2P� 1Þ�2pn1

L0

�cos2

�2pn1

L0�X1

�dX1

�Z L0

0

��hY

l

�cos2

�2pn1X1

L0

�dX1 ¼ 0

ð19Þ

The parameter 2P� 1ð Þ of Eq. 19 is now equal to Sstep (b, n2, n1)and its exact form is given in Appendix A (Eq. A3). For VDW inter-actions, Y in the second term of Eq. 19 is a function of the gapdistance only and does not vary with X1 (from Eq. 5). The non-dimensional parameter (�hY=l) is a measure of the interaction stiff-ness (�Y) to the elastic stiffness (l=h) of the film and determinesthe net strength of the destabilizing influence exerted by the contac-tor on the film. This ratio is called the interaction stiffness ratio andis a constant for VDW interactions. Consequently, Eq. 19 for VDWinteractions becomes:

4

L02pn1

L0

� �Sstep b;n2;n1ð Þ ¼ �hY

lð20Þ

The main task of the linear stability analysis for the VDW case is tofind the critical or the minimum value of the interaction stiffnessratio (�hYc=l) at which the instabilities initiate. Mathematically,this means to obtain a minimum value of the expression in the righthand side of Eq. 20 for which non-trivial values of n1exist andsatisfy the Eq. 19. This value of critical interaction stiffness thusfound suggests the critical gap distance (dcrit) between the surfacesat which the surface starts to roughen.

An electric field, instead of VDW interactions, is often used toinduce instabilities in soft thin elastic films to get easily controllableand desired aspect ratio features which are also highly ordered [28–30].There is often a minimum gap-distance dictated by constraints ofmicro-fabrication. It is, thus, crucial to find the minimum externalvoltage to be applied at a fixed gap-distance for the onset of instability.



Thus, for long range e-field induced instabilities, Eq. 19 can be recastin the following form:

4 2pn1

L0

� �Sstep b;n2;n1ð Þ

R L0

0

cos22pn1L0 �X1

� �h3 d0þ 1�Bð Þ½ �3 dX1

¼ �2h ef � 1

� �2e0ef V2

lð21Þ

The value of B in the above equation is given in Eq. A2. The objectiveof the linear stability analysis in the present case is to find theminimum=critical value of the voltage (Vcrit) as given by the right handside in the above expression for which non-trivial values of n1crit, exist.This determines the number of wavemodes of instability triggered atthe critical voltage.

If the boundary conditions in Eq. 13 are taken with bp¼ 0, then thewhole problem can be reformulated to obtain the asymptotic case of aflat substrate.

3. RESULTS

This section discusses the results of the linear stability analysis thathas been carried out for a smooth, incompressible elastic film cast ona patterned substrate. The ranges of the parameter values of the sub-strate patterns taken in the present study are 1�n2� 150 and0� b� 0.9. In order to span over a wider range, the maximum numberof wavemodes in the substrate pattern is set to a high value of 15 timesthe natural wavemode of the elastic film. Substrate amplitude values(b) were restricted to a maximum of 0.9.

Figure 2 shows the contour plot of n1crit (number of instability wave-modes at which the critical interaction stiffness is attained) [Eq. 20] inthe n2-b plane for the VDW case. These results indicate that n1crit

takes on a higher value when the substrate amplitude increases andthe smallest length scales obtainable are when the substrate ampli-tude, b, is high (�0.9). For this case of a VDW force between the filmand the contactor, the smallest possible length scale obtained isk� 0.34�h (corresponding to n1crit¼ 95), as is seen from the darkestregions in the figure. It is observed that in the domain whereb< 0.15, if either n2 or b is varied, the critical wavelength of instabilitydoes not vary substantially. An earlier experimental work, involvingfilms on patterned substrates [38], indeed observed that decreasingthe lateral length scales of the substrate pattern or increasing sub-strate pattern amplitude had no effect on the final instability lengthscales formed for this range of parameters considered. The substrateused was the polycarbonate part of the commercial data storage disks.



Two amplitudes of the substrates viz. 35 nm (for substrate obtainedfrom DVD) and 65nm (for substrate obtained from CD) were con-sidered. The minimum thickness of the elastic films was 530nm. Thus,the maximum value of b chosen in their case was restricted to 0.12which falls in the zone of b< 0.15. Thus, the present study revealsthe reason for the observed instability length scales to be independentof substrate pattern length scales and amplitudes.

Figure 3 illustrates the same results as in Fig. 2, but now for thecase of an electric field induced force acting between the film andthe contactor instead of the VDW force. The material parametersconsidered are: ef¼ 2.5 (for PDMS films), e0¼ 8.85� 10�12C2=J-m,m¼ 10MPa, h¼ 0.1 mm, h=d¼ 30. The results for this case are qualitat-ively the same as in the case of VDW interaction, i.e., the instabilitylength scale decreases when the substrate amplitude increases. Thelength scales obtained here are, however, substantially smaller thanthose obtained for the short-range forces. For example, the minimumwavelengths obtained with the electric field are up to 0.3�h. Besidesyielding smaller–length scales, this route also offers the advantageof tuning the external force field, which is not the case for VDW

FIGURE 2 Contours of n1crit in n2-b plane when VDW interactions are acting.As b increases, n1crit also increases and maximum value attained by n1crit is 95(k� 0.34�h) when b� 0.9.



interactions. The latter depend only on the material properties of thefilm and the contactor and, hence, cannot be controlled.

Interestingly, for b> 0.15, there exist some pockets for both theVDW and electric fields where higher values of n1crit penetrate intoregions of the lower values of n1crit. These overlap zones are more pro-found in the case of electric field–induced instability where the pocketscan also be visualized for b< 0.15. This feature is clear from the Figs. 2and 3 by observing the darker shades of grey penetrating into thelighter grey shades. It can be seen that, in these zones, the substratepattern wave mode n2 affects the final number of instability wave-modes n1crit, i.e., within these pockets, lower substrate patterns(higher n2) engender smaller wavelengths at the film surface. Thus,for even a small value of the substrate amplitude, b, much shorterinstabilities can be obtained if the number of substrate wavemodesis chosen along the darkest boundaries of the pockets.

It is evident from Fig. 4 that, in the case of VDW interaction, theinteraction stiffness ratio defined as, �hY=l¼Ah=(2lpd4), increaseswith the amplitude, b. In the pockets where higher values of n1crit

FIGURE 3 Contours of n1crit in n2-b plane when electrostatic interactions arepresent for ef¼ 2.5, e0¼ 8.85� 10�12 C2=J-m, m¼ 10MPa, h¼ 0.1 mm, h=d¼ 30.As b increases, n1crit increases and highest value attained by n1crit is 106 (k �0.3�h) when b � 0.9.



penetrate into zones of lower n1crit (as seen in Figs. 2 and 3), theenergy penalty required also decreases. Thus, these sets of parametricvalues of n2 and b provide the most optimal substrate design para-meters for minimization of wavelength, thus obviating the need toemploy high amplitude substrates such as the case of b¼ 0.9 discussedpreviously. This can be seen more clearly in the Fig. 4 which is the con-tour map of the hYc=l in n2-b plane. It shows that the interaction stiff-ness is indeed lower near the pocket boundaries where higher valuesof n1crit exist. In the rest of the zone, n2 has a negligible effect onthe interaction stiffness and the instability length scales are predomi-nantly controlled by the substrate amplitudes only. Thus, it is seenthat roughness of the substrate pattern in the form of high amplitude(high b) requires a high critical interaction stiffness ratio for the onsetof instabilities. The increase in the interaction stiffness ratio due toincrease in the roughness can be attributed to the effective reductionin film thickness brought about by roughness in the system. From ear-lier studies [21, 22] the critical interaction stiffness required to bringin instabilities for a flat-substrate configuration for a single film isfound to be 6:22 l

h, where l=h represents the elastic stiffness of the sin-gle film. The patterned substrate configuration with laterally varying

FIGURE 4 Critical interaction stiffness ratio (–hYc=l) for step-profiled sub-strate in n2-b plane for VDW interactions. Interaction stiffness ratio increasesas b increases and is almost independent of n2.



film thickness can be considered as a spring assembly of alternatingelastic springs of stiffness K1 and K2 where the film thickness ish(1� b) and h(1þ b), respectively. This spring assembly under theinfluence of interaction stiffness can be considered to be as a springsystem in parallel and the effective elastic spring stiffness, Keff, canbe written as:

1

Keff¼ 1

K1þ 1

K2¼ wh 1� bð Þ

lþ 1�wð Þh 1þ bð Þ

lð22Þ

where w and (1�w) are the weightings of the contribution to the elas-tic stiffness from films of thickness h(1� b) and h(1þ b), respectively.Thus, for the present case of the spring assembly system, the interac-tion required to being in instability will be 6.22Keff. Comparing thisvalue with the critical interaction stiffness, Yc, obtained in the presentstudy it is evident that:

w 1� bð Þ þ 1�wð Þ 1þ bð Þ ¼ 6:22

� hYc

l

ð23Þ

On substituting the interaction stiffness values we obtained for thepatterned substrate configuration (refer to Fig. 4), we find that whenb is low (say 0.1), �hYc=l � 6:27 making w ! 0.5 indicating that thecontribution of both the thicknesses towards the effective stiffnessare the same. However, as the value of b increases, the weighted con-tribution from the term h(1� b) is very high compared with that fromthe term h(1þ b). For example, for b¼ 0.9, � hYc

l ¼ 47:43 and, hence,w¼ 0.983, suggesting that only the thin parts of the film are effectivein determining the stiffness. Thus, increased roughness decreases theeffective thickness of the film, subsequently increasing the effectivestiffness (for example, Keff h=l 1 for b=0.1 and Keff h=l¼ 7.625 forb¼ 0.9) which now requires a higher interaction stiffness to bring ininstability.

In the case of VDW interaction, increase in the interaction stiffnessratio means that a smaller inter-surface gap distance is requiredfor triggering instabilities. As suggested from Fig. 4, the gap distance,dcritical, required for surface roughening should decrease with anincrease in the value of the substrate amplitude (b). However, Fig. 5shows that change in the critical inter-surface distance is not sub-stantial with a variation in b (for example, dcritical 22nm for b¼ 0and dcritical 16nm for b¼ 0.9 for a film of A¼ 10�19 J, h¼ 10mm,m¼ 0.1MPa and of compliance DGh=ld2

e � 4�108). However, it is seen



that dcritical decreases drastically for less compliant or stiff films, wherecompliance of the film is defined asDGh=mle

2. HereDG¼A=12p le2 is the

energy of adhesion, h is the film thickness, m is shear modulus of theelastic film and de is the cut-off distance, which is�0.158nm in the caseof VDW interactions. Compliance is a measure of the ratio of theadhesive stiffness to the film stiffness and a higher value of this ratiocorresponds to a more compliant film. It can be seen that for very stifffilms where instability length scales are the smallest (A¼ 10�21 J,h¼ 0.1 mm, m¼ 10MPa and compliance DGh=mde

2� 4� 102), the valueof the dcritical required at b¼ 0.9 is 0.5 nm (<< 1). This is practicallyimpossible to achieve for surfaces other than atomically smooth ones.

The limitation in terms of unachievable gap distances for stifferfilms with VDW interaction can be overcome if an electric field isemployed. For the latter case, a realistic gap-distance in conjunctionwith relatively high substrate amplitudes (b! 0.9) is sufficient to trig-ger the instability even in a stiff film. The minimum voltage requiredis also realistic as can be seen from Fig. 6 (where the lighter greyshades denote lower values of Vcrit) which represents the results fora stiff film (m¼ 10MPa, h¼ 0.1 mm). This is a direct consequence of

FIGURE 5 Variation of critical distance (dcritical) with b, for films of differentcompliances for VDW interactions. For stiff substrates the dcritical value is lessthan 1nm. As b increases the dcritical value decreases.



the fact that the interaction stiffness for this case is also a function ofthe local film thickness (Eq. 6), unlike in the VDW case where it is afunction of the gap distance alone (Eq. 5). This means that at gap dis-tances, d, much smaller than film thickness, h (as is usually the case),the critical voltage that triggers instabilities is proportional to thelocal thickness of the film, h� bp. Thus, as b increases, the local filmthickness decreases and the critical voltage that initiates instabilitiesalso decreases. Thus, smaller length scale patterns can be formed,even in stiff films, if an external electric field is applied at practicalgap distances instead of making the VDW force operational by reduc-ing the gap distance to impractical values. At b¼ 0.9 where the smal-lest length scales are formed, the critical voltage required is as smallas 32V. Thus, the electric field route gives smaller length scales evenin stiff films for smaller critical voltages. The critical voltage (Fig. 6)decreases in the overlap zones of Fig. 3 where higher values of n1crit

penetrate into regions of lower values of n1crit. Thus, these zones areoptimal for the substrate design effective in reducing the pattern-wavelength. The twin advantages of engendering smaller instabilitiescompared with VDW interaction, and the ease to form patterns even in

FIGURE 6 Contour map in n2-b plane showing the critical voltage (Vcrit)required to trigger instabilities when electrostatic interactions are present.ef¼ 2.5, e0¼ 8.85� 10�12C2=J-m, m¼ 10MPa, h¼ 0.1mm, h=d¼ 30. As bincreases, Vcrit decreases to very low values like 32V when b� 0.9.



very stiff films by providing lower critical voltages, makes the electro-static force field a far more favorable route.

3. CONCLUSIONS

Roughness of the contacting surface in the form of a patterned contac-tor or noise was incorporated in a previous study [27] to understand itseffect on the mechanisms of adhesion-debonding at soft interfaces. Thepresent study focuses on the influence of the substrate roughness onthe onset and wavelength of surface instability in a soft, thin, incom-pressible elastic film cast on a step-patterned substrate. The insta-bility is considered to be triggered either by the attractiveshort-range VDW force or a longer-ranged electric field. A linear stab-ility analysis is performed with special attention towards finding theconditions for reduction in pattern size. A comparative study of theroles of different substrate pattern amplitudes and force fields pro-duces the following conclusions.

1. The substrate patterning reduces the length scales of instabilitycompared with a flat substrate where the smallest length scalesobserved are �3�h [20–29,31,32]. Increased substrate amplitude, b,engenders progressively smaller wave lengths which can be as lowas 0.34�h for the VDW force and 0.3�h when the electric field is appliedat b� 0.9. These pattern length scales are similar to the minimumwavelength obtained in elastic bilayers, �0.5�h [36,37].

2. At regions where b< 0.15, the instability length scale is nearlyindependent of both the substrate amplitude and the substrate lengthscale, as has also been observed in experiments [38]. Elsewhere, theinstability length scale declines significantly with the increase in pat-tern amplitude. When b> 0.15, n1crit is influenced by the substrateperiodicity, n2, within some selected pockets of the parameters. Inthese zones, increasing substrate roughness by considering smallersubstrate patterns (higher n2), indeed gives shorter instability wave-lengths. Thus, for even small substrate amplitudes, much smallerinstabilities can be obtained if the number of substrate wavemodesis chosen appropriately.

3. Although shorter wavelengths are formed with increase in b;increase in roughness effectively increases the elastic stiffness of thefilm which brings in an escalation in the elastic energy. Thus, strongerdestabilizing forces are now required to bring in instability in thesepatterned substrate systems.

4. For the VDW case, increase in the interaction stiffness ratioimplies reduction in critical gap distance. Thus, VDW force cannot



engender instability in highly stiff films that require unrealisticallysmall inter-surface distances of less than 1nm. Thus, patterning withthe VDW force works only for compliant films.

5. The difficulty encountered in destabilizing stiff films in the VDWcase can be overcome by employing an electric field. Unlike the VDWcase, increasing the substrate amplitude now does not pose any dif-ficulty because the critical voltage, in fact, decreases as the substrateamplitude increases (32V when b� 0.9). In addition to this, in thepockets where high values of n1crit are achieved, the critical voltagesrequired to bring in instability are found to be very low, making thecorresponding values of n2 and b the most suitable design parametersfor the substrate profile.

These results clearly reveal the potential of patterned substrates inpatterning of soft elastic films. If an elastic film is cast on a patternedsubstrate and is under the influence of an external electric field, thecritical voltages and the instability length scales can both be dimin-ished by an order of magnitude compared with those obtained insmooth substrates (Vcrit 32V, k� 0.3�h for b¼ 0.9 and Vcrit 118V,k� 3�h for b¼ 0). Also, the electric field can be externally controlledunlike the VDW force field.

ACKNOWLEDGMENTS

The authors acknowledge the support of the Department of Science &technology, New Delhi, to JS and AS as an Intensification of Researchin high Priority Areas grant. Many fruiftful discussions withVijay Shenoy over the years are gratefully acknowledged. It is a spe-cial pleasure to be a part of the Festschrift honoring Professor HerbertHui whose work in contact mechanics has been a source of inspirationto us.

REFERENCES

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APPENDIX A

The values of the constants of Eq. 16 that are obtained after solvingEq. 15 using the boundary conditions of Eqns. 12 and 13 innon-dimensional form are:

PðB;n2;n1Þ ¼1þ exp

�4pn1

L0 ð1þBÞ�� 4pn1

L0

�1þB

��1� 2pn1

L0 ð1þBÞ�

2 sinh

�4pn1


L0 ð1þBÞ

QðB;n2;n1Þ ¼ �2pn1

L0

�1þ exp

�4pn1

L0

�1þB

�� 4pn1

L0 ð1þBÞ�

2 sinh

�4pn1


L0 ð1þBÞ

8>><>>:

9>>=>>;

RðB;n2;n1Þ ¼ 1� PðB;n2;n1Þ; and SðB;n2;n1Þ ¼2pn1

L0 þQðB;n2;n1Þ

ðA1Þ

The discretized domain of the substrate and the value of the constantB of Eq. A1 over each of these discrete substrate blocks is given below:

B ¼ �b for

0 < X1 <L01

2

2i� 12

� �L01 < X1 < 2iþ 1

2

� �L01

2n2 � 12

� �L01 < X1 < 2n2L

0

8><>:

where i ¼ 1; 2; . . . ::ðn2 � 1Þ

B ¼ b for

�2jþ 1

2

� �L01 < X1 < 2jþ 3

2

� �L01

where j ¼ 0;1; 2; . . . ::ðn2 � 1Þ

ðA2Þ



The dimensionless function Sstep(b, n2, n1) of Eqns. 20 and 21 is givenby:

Sstepðb;n2;n1Þ ¼ S1ð�b;n2;n1Þ�Z L0=4n2

0

cos2�2pn1

L0 X1

�dX1

þZ L0

L0ð1�1=4n2Þcos2

�2pn1

L0 X1

�dX1

þXn2�1

i¼1

�Z L0n2ðiþ1=4Þ

L0n2ði�1=4Þ

cos2�2pn1

L0 X1

�dX1

��

þ S1ðb;n2;n1ÞXn2�1

i¼0

�Z L0n2ðiþ3=4Þ

L0n2ðiþ1=4Þ

cos2�2pn1

L0 X1

�dX1

�

ðA3Þ

where,

S1 b;n2;n1ð Þ ¼1þ cosh 4pn1

L0 1þ bð Þ� �

þ 2 2pn1

L0 1þ bð Þ �2

sinh 4pn1

L0 1þ bð Þ� �

� 4pn1

L0 1þ bð ÞðA4Þ



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