adhesion design maps for fibrillar adhesives: the effect of shape

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Acta Biomaterialia 5 (2009) 597–606

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Adhesion design maps for fibrillar adhesives: The effect of shape

Christian Greiner a,*, Ralph Spolenak b, Eduard Arzt a,1

a Max Planck Institute for Metals Research, Heisenbergstrasse 3, 70569 Stuttgart, Germanyb Laboratory for Nanometallurgy, Department of Materials, ETH Zurich, 8093 Zurich, Switzerland

Received 30 June 2008; received in revised form 29 August 2008; accepted 3 September 2008Available online 25 September 2008

Abstract

The biomimetic reproduction of adhesion organs, as found in flies, beetles and geckoes, has become a topic of intense research overthe past years. Successes, however, have so far been limited. This is due to the vast range of parameters involved, including fibril size,elastic modulus, contact shape, surface roughness and ambient humidity. In previous studies, design and materials selection charts todetermine the optimum materials and design combination for dry adhesive systems have been established. The effect of shape on theadhesive properties of single fibers and fiber arrays has also been a research focus. In this paper both approaches are combined to providemore advanced guidelines for the design of optimal adhesive structures.� 2008 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

Keywords: Gecko adhesion; Contact mechanics; Surface patterning; Shapes; Materials selection

1. Introduction

The adhesion of fibrillar surfaces due to molecular (vander Waals) forces is subject to a strong size effect: the split-ting of large contacts into many smaller ones leads to animprovement in adhesion [1,2]. This suggests an explana-tion for the finding that bigger animals exhibit finer adhe-sive structures. It has recently become clear that van derWaals interactions are mostly responsible for the adhesionof geckos [3–6], while capillary forces may contribute to theadhesion [7,8], in some cases significantly. In a recent paper[9], these concepts were distilled into ‘‘adhesion designmaps” that can guide the implementation of artificial adhe-sion systems by suggesting optimal dimensions and mate-rial properties. In another paper, different scaling lawswere found for various contact shapes [10]. The presentpaper attempts to extend the concept of adhesion designmaps to non-spherical contact shapes, including those thatexhibit an axial asymmetry and thus allow easy detachment

1742-7061/$ - see front matter � 2008 Acta Materialia Inc. Published by Else

doi:10.1016/j.actbio.2008.09.006

* Corresponding author. Tel.: +49 7116893231.E-mail address: [email protected] (C. Greiner).

1 Present address: INM Leibniz Institute for New Materials, CampusD2 2, 66123 Saarbrucken, Germany.

in the appropriate pulling direction. The objective is toeventually develop predictive capabilities for reproducingbiomimetic adhesion systems (Table 1).

A typical adhesion design map proposed previously forspherical contacts [9] is shown in Fig. 1. It displays graph-ically the propensity of different mechanisms which limitthe adhesive strength (Fig. 1) and ultimately may guidethe design process: fiber fracture, ideal contact strength,adaptability, apparent contact strength and condensation.The axes chosen describe the elastic modulus of thefibers and their radius. The ‘‘fiber fracture” criterion (blue)acknowledges that, for very thin fibers, the adhesionstrength will exceed their theoretical fracture strength.The limit of ‘‘ideal contact strength” (red) is given by anideally fitting contact between the two surfaces, withoutnecessity for elastic deformation. As a certain elastic com-pliance is necessary to adapt to rough surfaces, the limit of‘‘adaptability” (green) was introduced. The apparent con-tact strength rapp (black), i.e. the pull-off force divided bythe apparent contact area Aapp, is shown as contours super-imposed on the maps. When the fibers tend to stick to eachother, rather than to the contact surfaces, the ‘‘condensa-tion” (or clumping, matting) limit (cyan) has been reached.As suggested by Spolenak et al. [9] we can define a conode

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Table 1List of symbols.

a Contact radius (m)Aapp Apparent contact area (m2)Ac Actual contact area (m2)Af Contact area of single fiber (m2)b Characteristic length of surface interactions (m)C Geometrical factor (-)E Young’s modulus (Pa)E* Reduced Young’s modulus (Pa)Eeff Effective Young’s modulus (Pa)Eopt Optimum Young’s modulus (Pa)f Pillar packing density (-)h Tape thickness (m)Pc Pull-off force (N)q Shape parameter (-)R Fiber radius (m)Ropt Optimal fiber radius (m)a Peel-off angle of elastic tape (�)c Work of adhesion (J/m2)c’ Work of adhesion between two fiber tips (J/m2)ceff Effective work of adhesion (J/m2)kopt Optimal fiber aspect ratio (–)m Poisson’s ratio (–)r* Interfacial strength (Pa)ropt

app Optimum apparent contact strength (Pa)rc Pull-off strength (Pa)rf Axial fiber stress (Pa)rm Tensile strength (Pa)rth Theoretical contact strength of van der Waals bonds (Pa)rf

th Theoretical fiber fracture strength (Pa)

598 C. Greiner et al. / Acta Biomaterialia 5 (2009) 597–606

(orange) in the maps. The conode links the loci of optimumapparent contact strength, while avoiding condensation

Fig. 1. Typical adhesion design map for spherical contact elements as developedcriterion of fiber fracture, the red line the ideal contact strength. The limit of fibones. The black lines are contours of equal apparent contact strength. In add

and ensuring adaptability. These loci are shown as blackdots at the intersections of the green and cyan lines for dif-ferent values of the fiber aspect ratio k. The optimum adhe-sive is found at the intersection of the conode with the fiberfracture limit, as indicated in the map (red circle). In thefollowing, new design maps based on this original principleare presented for a variety of shapes (Fig. 2).

2. Design map for flat tips

First we wish to formulate the equations for flat-tipended fibers. Lacking a simple analytical solution other-wise, we assume that the fibers are stiff in relation to thesubstrate and in perfect contact. The contours of apparentcontact strength rapp require an expression for the pull-offforce Pc of a flat punch [10–12]

P c ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi8pER3c

q; ð1Þ

where E is the Young’s modulus, R the fiber radius and cthe work of adhesion. With an apparent contact area ofAapp = R2p/f, where f is the area fraction of fibers, and set-ting rapp = Pc/Aapp the maximum fiber radius for a speci-fied value of rapp is defined as follows:

R 68Ecf 2

pr2app

: ð2Þ

Note that for flat tips the maximum radius scales line-arly with Young’s modulus, in contrast to spherical con-tacts, where it was independent of E ([9], see also Fig. 1).

in a previous publication by Spolenak et al. [9]. The blue line indicates theer condensation is indicated by the cyan lines and the adaptability by greenition, the conode and the optimum design parameters are indicated.

Fig. 2. Side and plan view of contact shapes considered for adhesion design maps: (a) hemisphere, (b) flat tip, (c) toroidal tip, (d) elastic band. In each case,the shaded area indicates contact (adapted from Ref. [10]).

1E-4 1E-3 0.01 0.1 1 10 100 10001E-3

0.01

0.1

1

10

100

λ =

300

adap

tabi

lity

condensation

aσ

σ

σ

pp

=0.

1M

Pa λ = 300

λ = 100

λ = 30

λ = 10

λ = 3

λ = 1

λ =

100

λ =

1

λ =

10

app

=10

MPaap

p

=1

MPa

fiber

radi

usR

(μm

)

Young's modulus E (GPa)

punch, f = 10 %, = 0.05 J/m2, b = 0.2 nm, Eeff

=1 MPa, S/F

fiber fracture

idea

l con

tact

stre

ngth

optimum

Fig. 3. Adhesion design map for flat tips with the same parameters as inFig. 1 (c = 0.05 J m�2, f = 10%, b = 0.2 nm and Eeff = 1 MPa). Thedashed line (orange) is the ‘‘conode”. Its intersection with the idealcontact strength criterion indicates optimum parameters and is highlightedwith a red circle.

C. Greiner et al. / Acta Biomaterialia 5 (2009) 597–606 599

Fiber fracture is avoided when the apparent contactstrength is smaller than f rf

th; here rfth is the theoretical fiber

fracture stress equated to rfth = E/10, as before [9], giving as

a non-fracture condition

R P800cpE

: ð3Þ

The limit of ideal contact strength is established follow-ing Spolenak et al. [9] as

R P8Eb2

pc: ð4Þ

where b is the characteristic length of surface interactions.As a condensation criterion, we assume the same equa-

tion to hold as for fibers with spherical contacts, i.e. [9]

R P8c0h1=2

f

Ek3; ð5Þ

where c0 denotes the work of adhesion between two fiberstips, and k is the aspect ratio L/2R, with L the fiber length.The function hf is defined as

1

hf¼ 1

hðf Þ ¼ ðffiffiffiffiffiffip

4f

r� 1Þ2: ð6Þ

The limit of adaptability is independent of shape, so thatthe condition derived in Ref. [9] is used throughout thispaper.

The resulting map is displayed in Fig. 3. Additional mapswere constructed applying the condensation limits devel-oped by Glassmaker et al. [13] and Persson [14], with onlyminor changes (see the Supporting Information).

Fig. 3 can be read as follows: the regions below the fiberfracture and the ideal contact strength limit should beavoided in the design of artificial systems. For Young’smoduli to the right of the adaptability criterion, the fiberstiffness is too high to account for rough surfaces; this cri-terion, as before, depends on aspect ratio k (different verti-cal green lines).

The conode (orange dashed line) is mathematically givenby:

R ¼ c

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffihðf ÞEC3f 3

E3effp

3

s: ð7Þ

Here, C is a dimensionless geometrical constant with avalue close to 10 and Eeff is the effective Young’s modulusof the fiber structure. In the log–log plot of Fig. 3, the con-ode has a slope of ½. Because the contours of apparent con-tact strength (black lines) have a slope of 1, larger radiiwould imply better adhesion. This is in contrast to sphericalcontacts, as shown in Fig. 1. The absolute optimum for flattips is now found at the intersection between the conode andthe ideal contact strength limit. The optimum values for theaspect ratio k and the fiber radius R can be derived in ananalogous way (see Table 2). The result for the optimumapparent contact strength ropt

app is given, as before, by [9]

Table 2Summary of the derived design criteria for the hemispherical, flat tip, torus and tape-like contact shape (both for a peel-off angle of a = 60� and for thegeneral case with Y = [(1�cos a)2 + (1�cos a)]).

Tip shape Ideal contact strength Apparent contact strength Fiber fracture

Sphere R P 8E2b3

3c2p2ð1�m2Þ2 R 6 3f c2rapp

R P 15cE

Flat tip R P 8Eb2

pc R 6 8Ecf 2

pr2app

R P 800cpE

Torus (self-similar) R > E2p2b3

211c2 R 6 ðf 3pEr3

appÞ1=2c R P

ffiffiffiffiffiffiffiffiffiffiffiffiffi1000pp

cE

Elastic tape (a = 60�) R P � 40c3E þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1600c2

9E2 þ b2q

R 6 � 40c3E þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1600c2

9E2 þ f 2c2

r2app

qR P 188c

E

Elastic tape R P�10c

E þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi100c2

E2 þYb2 sin2 a

qY R 6

�10cE þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi100c2

E2 þY sin2 ac2 f 2

r2app

qY R P

�10cE þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi100c2þY 2002 sin2 ac2

E2

qY


roptapp ¼

f cb: ð8Þ

Surface roughness will have a strong influence on theadhesion performance of flat tips; we refer to Appendix Bfor the construction of adhesion design maps for flat tipsagainst substrates with periodic roughness.

3. Design map for toroidal tips

We consider next a toroidal (doughnut-shaped) contactgeometry. As in Ref. [10], self-similar scaling is assumed,i.e. r = R/10, where r is the radius of the torus ring (seeFig. 1c). The pull-off force of a torus is given by [10]:

P c ¼ ðp4Ec2R4Þ1=3: ð9Þ

The resulting equations for the different limits arelisted in Table 2; the corresponding map is shown inFig. 4 (for further condensation criteria see the Support-

1E-4 1E-3 0.01 0.1 1 10 100 10001E-3

0.01

0.1

1

10

100

Emax λ=

300

optimum

idea

l con

tact

stre

ngthfiber fracture

λ=

300λ=

100λ=

30

λ=

10

λ=

3

λ=

1

λ=

100

λ=

1

λ=

10

σ app= 10 MPa

σ app= 1 MPa

fiber

radi

usR

(μm

)


Torus (self-similar), f = 10 %, γ = 0.05 J/m2, b = 0.2 nm, Eeff

=1 MPa, S/F

Emin

σ app= 0.1 MPa

Fig. 4. Adhesion design map for toroidal tips. The following parametersare assumed: c = 0.05 J m�2, f = 10%, b = 0.2 nm and Eeff = 1 MPa. Thedashed line (orange) is the ‘‘conode”. It has the same slope as the apparentcontact strength limit. The whole sector between fiber fracture and idealcontact strength limit—indicated by the solid red line—yields optimaladhesive properties.

ing Information). Note that the contours of apparentcontact strength now have a slope of ½, which liesbetween the cases for spherical contacts (slope 0) and flattips (slope 1). The conode equation derived in Section 2for flat tips (Eq. (7)) is still valid here. It is interestingthat the conode and the apparent contact strength crite-rion have the same slope. This implies that all points onthe conode correspond to the same apparent contactstrength. Therefore, a broad optimum range resultswhich lies along the appropriate conode, between itsintersection with the fiber fracture and ideal contactstrength criteria (marked in red). The modulus hencemust lie between

Emax ¼222=3Cf c2h1=3

f

p7=3b2Eeff

ð10Þ

and

Emin ¼10Eeffp4=3

Cf1

h1=3f

: ð11Þ

The other parameters such as the aspect ratio andthe radius can be deduced in an analogous way (seeTable 2).

4. Design map for elastic tapes

Next, adhesion design maps for elastic tapes are pre-sented. We make the following assumptions: the tape incontact with the substrate has a square area of dimensions2R � 2R; scaling of the tape thickness h is self-similar:h = R/10 (Fig. 1d). We further assume that the condensa-tion and adaptability criteria are the same as for sphericalcontacts. From Kendall [10,15] we get the pull-off force as afunction of peel angle a:

P c ¼ 2cRgða; kÞ ð12Þwhere:

gða; kÞ ¼ 2 sin affiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1�cos aÞ2þ2kþð1�cos aÞp

k ¼ 10cER

ð13Þ

For an (arbitrary) peel angle of a = 60�, this leads to thefollowing expression for Pc

1E-4 1E-3 0.01 0.1 1 10 100 10001E-3

0.01

0.1

1

10

100

λ =

100

σapp = 1 kPa

σapp = 10 kPa

fiber fracture

λ = 300λ = 100

λ = 30

λ = 10

λ = 3

λ = 1

λ =

300

λ =

1

λ =

10

σapp = 0.1 MPa

σapp = 1 MPa

fiber

rad

ius

R (

μm)


Tape (α = 60˚), f = 10 %, γ = 0.05 J/m2, b = 0.2 nm, Eeff

=1 MPa, S/F

optimum

Fig. 5. Adhesion design map for an elastic tape at a peel-off angle of 60�.The following parameters are assumed: c = 0.05 J m�2, f = 10%,b = 0.2 nm and Eeff = 1 MPa. The dashed line (orange) is the ‘‘conode”.Its intersection with the fiber fracture limit indicates the optimumparameters and is highlighted with a red circle.


P c ¼4cRffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ 80c

3ER

q : ð14Þ

From this, we determine the design limits listed in Table2 and the adhesion design map presented in Fig. 5. (See theSupporting Information for other condensation limits.)

Note that for the elastic tape, the contours of constantapparent contact strength are curved: its values are inde-pendent of Young’s modulus for large moduli, but scaleinversely with Young’s modulus for small moduli.

The conode has a slope of ½. The slope of the apparentcontact strength criterion is 0 at high moduli and �1 at low

1E-4 1E-3 0.01 0.1 11E-3

0.01

0.1

1

10

100

σ app

= 1

MPa

σ app

= 0

.1 M

Pa

fiber fracture

λ = 30

λ = 10λ = 3

λ = 1

λ =

1

λ =

10

fiber

rad

ius

R (

μm)

Young's modulus E

Punch, f = 10 %, γ = 0.05 J/m2, b = 0

Roughness: a =

σapp

= 6.8 kPa

idea

l con

ta

Fig. 6. Adhesion design map for a flat tip against a substrate with periodicaa = 500 nm is considered. The dashed, orange line is the ‘‘conode”. For R > 500For R > a, the apparent contact strength is constant over the entire map – contaby the fiber fracture limit.

moduli, so that the optimum can be found at the intersec-tion between the conode and the fiber fracture criterion.This results in the following optimum modulus (see Table2 for other optimum parameters)

Eopt ¼1882=3

h1=3f

Eeff pCf

: ð15Þ

This condition is presented as a function of a in Table 2.For arbitrary a, Eq. (15) becomes:

Eopt ¼102=3 �hf þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffih2

f þ 400Y sin2ðaÞh2f

qY 2

� �2=3

Eeffp

Y 2f Chf: ð16Þ

It is of interest to identify the peel-off angle a for whichthe pull-off force perpendicular to the surface is maximal.Brief calculation shows that this condition is reached fora � 60�. Adhesion design maps for pull-off angles between5� and 90� can be found in Ref. [16].

5. Discussion

5.1. Comparison of different shapes

The adhesion design map for flat tips (Fig. 3) suggestsoptimum adhesion at high modulus values. This is surpris-ing as we usually experience that softer materials ‘‘stick”

better than stiffer ones. The reason lies in the relationshipbetween pull-off force and Young’s modulus for a flatpunch (Eq. (1)): the pull-off force scales with the squareroot of the modulus, giving a higher adhesion force for astiffer punch.

On introducing roughness, these results change as isdemonstrated by the periodic model presented in Appendix

10 100 1000

λ =

300

Intermediate

Rough Surface

λ = 300λ = 100

λ =

100

(GPa)

.2 nm, Eeff

=1 MPa, S/F

500 nm

Smooth Surface

σ app

= 1

0 M

Pa

ct s

treng

th

l roughness. For the surface roughness, a radius of the small spheres ofnm, the presented model for flat tips against a rough surface is employed.ct splitting invariance – and the minimum Young’s modulus is determined


B. Even for small roughness, the adhesion performance ofthe punch is dramatically reduced (Fig. 6) as shown by Ful-ler and Tabor [17] and Johnson [18] in 1975. More recently,Persson investigated this problem in more depth bydescribing the surfaces with a roughness power spectrum[19,20]. Compared with these approaches for rough sur-faces, our model is simple but demonstrates the dramaticinfluence of surface roughness on adhesion performancevery well.

For the toroidal tips presented in Fig. 4, the conode hasthe same slope as the contours of constant apparent con-tact strength. This means that one can change Young’smodulus by over three orders of magnitude (from �0.6to 200 GPa) and still obtain the same adhesion perfor-mance. The optimum radius would change from about0.04 to 2 lm in this process. The reason for this behavioris a balance between contact splitting and the dependenceof the pull-off force on Young’s modulus. The latter scaleswith E1/3 and favors larger moduli, whereas contact split-ting shifts optimum values to smaller radii.

For the tapes, which have the same splitting efficiency asspheres [10], contact splitting is stronger than the depen-dence on Young’s modulus (Pc�E1/2): the optimum valuesare found at smaller moduli and radii. Both the contours ofconstant apparent contact strength and the limit of idealcontact strength are shifted to smaller radius values thanfor other shapes. The limit of ideal contact strength, forthe moduli considered, is found at radii smaller than theones included in the maps. Interestingly, the angle at whichgecko spatulae make contact with a substrate [21] is around

1E-4 1E-3 0.01 0.1 1 10 100 10001E-3

0.01

0.1

1

10

100

λ =

100

idea

l con

tact

stre

ngthfiber fracture

σapp = 1 MPa

sphere

σapp = 0.1 MPa

λ = 300

λ = 100

λ = 30

λ = 10

λ = 3λ = 1

λ =

300

λ =

1

λ =

10

σapp = 0.1 MPa

σapp = 1 MPa

fiber

rad

ius

R (

μ m)


f = 10 %, γ = 0.05 J/m2, b = 0.2 nm, Eeff

=1 MPa, S/F

Sphere and Tape (α = 15-90˚)

optimum

Fig. 7. Adhesion design map for hemispherical and band-like tip shape.The following parameters are assumed: c = 0.05 J m�2, f = 10%,b = 0.2 nm and Eeff = 1 MPa. For the elastic band, peel-off angles between15� and 90� are considered and the resulting areas for constant apparentcontact strength (gray) and for the fiber fracture limit (light blue) arehatched. The limits belonging to the spherical contact are indicated. Theintersections of the conode (orange line) with the fiber fracture limits yieldthe optimum parameters and are indicated with a red circle in the case ofthe sphere and by a solid red line for the tape contact.

55�, which is quite close to the optimum value of about 60�.This suggests that the maps may not only help in designingman-made adhesives but also in understanding biologicalstructures.

As spheres and tapes are so similar, we compare thedesign maps of both shapes in Fig. 7. The intervals for con-stant apparent contact strength and the fiber fracture limitfor peel angles between 15� and 90� are marked as hatchedareas. A first observation is the strong dependence of theapparent contact strength on Young’s modulus for the tapecontact. For small moduli it scales inversely with E,whereas for high E, it is independent of the modulus. Itreaches the value for the hemispherical contact for whichthe apparent contact strength is not a function of E. Aninteresting difference is the following: for the spheres, anoptimum Young’s modulus of around 30 MPa and an opti-mum fiber radius of �30 nm are found; in the case of theelastic tape, the corresponding values are 100–190 MPaand 50–70 nm. This shows that for finding the optimal fiberradius, the hemispherical approximation is appropriate,whereas for the Young’s modulus it yields values aboutone order of magnitude higher. It is of interest that themodulus effect is very small, and that the spherical contactcan be used as a first-order approximation for the elastictape.

5.2. What is the optimum shape?

Table A1 (Appendix) summarizes the results for opti-mum adhesives. As it is quite difficult to see trends withthese equations only, we insert the typical values for b, C,f, Eeff, a and c that were used to construct the maps in Figs.3–7 (b = 0.2 nm, C = 10, f = 0.1, Eeff = 1 MPa, a = 60�,c = 0.05 J m�2). The resulting numerical values for theoptimal design parameters are summarized in Table 3.

The flat tips result in the highest values for the optimumapparent contact strength, which reaches a value ofropt

app=25 MPa, well above that of the gecko (0.1–0.2 MPa[3,7]). However, the required values for the optimalYoung’s moduli (Eopt = 6 TPa) even surpass modulireached by the strongest materials (1 TPa). The value forthe pillar aspect ratio is very high (kopt � 730), far abovethose reached by current nano- and micro-lithographytechniques. On the other hand, carbon nanotubes exhibitaspect ratios of over 1000 and have, in recent experiments,demonstrated surprising adhesion capabilities [22–24].

Introducing surface roughness and using our simplemodel of periodic hemispherical bumps, the performanceof the punch drops dramatically and the apparent contactstrength for a roughness of Rz = 1 lm is no higher than�7 kPa (about five orders of magnitude smaller thanwithout roughness!). Such drastic reductions in adhesionperformance were modeled in the literature and have beenfound experimentally [17,19,20,25] as well. For the othertip shapes, the spherical contact shows the second highestadhesion performance (ropt

app = 2.8 MPa) followed by thetorus (0.7 MPa) and the elastic tape (74 kPa). The pre-

Table A1Summary of the optimum design parameters found by intersecting the conode with the appropriate limit. Besides the optimum Young’s modulus Eopt, theoptimal fiber radius Ropt, aspect ratio kopt, and the ultimate apparent contact strength ropt

app are presented.

Tip shape Eopt Ropt

Sphere Eopt ¼ 152=3pEeff

Cfh1=3f

Ropt ¼C151=3cfh1=3

f

pEeff

Flat tip Eopt ¼ hfC3f 3c4

64E3eff

b4pRopt ¼ hf C3f 3c3

8b2p2E3eff

Torus (self-similar) Emax ¼222=3Cf c2h1=3

f

p7=3b2EeffEmin ¼ 10Eeff p4=3

Cf1

h1=3f

Rmax ¼20481=3C2f 2c2h2=3

f

p8=3bE2eff

Rmin ¼10001=2cCfh1=3

f

10p5=6Eeff

Elastic tape (a = 60�) Eopt ¼ 1882=3

h1=3f

Eeff pCf Ropt ¼

1881=3ch1=3f Cf

pEeff

Tip shape kopt roptapp

Sphere kopt ¼ 151=3

2h1=6f

roptapp ¼

3pEeff

2C151=3h1=3f

Flat tip kopt ¼h1=2

f C2f 2c2

15E2eff b2p

roptapp ¼

cfb

Torus (self-similar) kmax ¼20481=3h1=6

f Cf c

2Eeff bp5=3 kmin ¼ ð52 Þ1=2ð phf

Þ1=6 rmax=minapps ¼ Eeff p4=3

h1=3f C

Elastic tape (a = 60�) kopt ¼ 1881=3

2h1=6f

roptapp �

0:16pEeff

h1=3f C

Table 3To ease the comparison between the different tip shapes and fibercondensation models, we insert typical values (Eeff = 1 MPa, C = 10,f = 0.1, c = 0.05 J m�2, b = 0.2 nm) into the equations for Eopt, Ropt, kopt

and the ultimate apparent contact strength roptapp, listed in Table A1, and

present the numerical results. For the spherical tips a Poisson’s ratio ofm = 0.3 is assumed.

Tip shape Eopt (Pa) Ropt (m) kopt (–) roptapp (Pa)

Sphere 2.83 � 107 2.65 � 10�8 1.50 2.83 � 106

Flat tip 5.98 � 1012 1.22 � 10�5 735.80 2.50 � 107

Torus (self-similar) 4.71 � 1011 3.42 � 10�6 193.60 6.81 � 105

6.81 � 107 4.11 � 10�8 2.30Elastic tape (a = 60�) 1.53 � 108 6.16 � 10�8 3.49 7.44 � 104


dicted optimal Young’s modulus and radius values areplausible and in the range possible in artificial systems.

The elastic tape not only promises reasonable adhesion,but due to its asymmetric nature allows easy detachment;this property is very desirable in technical systems. Wecan state that all optimal aspect ratio values, with theexception of the flat tip, are quite small and accessible withtoday’s micro-nanopatterning techniques.

In summary, for ideal surfaces without roughness, flattips are by far the ones with the best adhesion performancebut they are very sensitive to surface roughness. For theideal case, flat tips are followed by hemispheres and torii,whereas elastic tape has a comparatively low adhesionstrength. However, even tapes give maximum values forapparent contact strength that are well in the range ofgecko adhesion. All shapes require relatively high Young’smodulus values, which are above the MPa range. With theexception of the flat tip and the torus, optimum radii liebelow 1 lm. The optimum aspect ratios for the elastic tape,the torus and the sphere are mainly below 10, but evenexceed 700 for the flat tip.

5.3. Limitations and problems

The parameter space for adhesive structures is compli-cated. While the adhesion design maps presented in thispaper are quite complex, they are still based on severesimplifications. A central assumption of the concept isthe validity of the mechanistic equations over severalorders of magnitudes in size and Young’s modulus. Asimilar assumption is made in the celebrated ‘‘deforma-tion mechanism maps” introduced for bulk materialsby Ashby et al. [33]. Despite the same possible reserva-tion, this concept has become very successful in guidingthe application of existing materials and the design ofnew ones.

As usual in modeling, assumptions had to be made inderiving the mechanistic equations. All derivations arebased on continuum mechanics, which is applicable in thissize range. The fiber fracture limit relies on an approxima-tion for the theoretical fracture strength [27], which maynot be strictly applicable to polymers. For PDMS forexample, the fracture strength is about a factor of fourhigher than the Young’s modulus. For amorphous poly-mers a similar, elegant relationship between fracturestrength and modulus, as for crystalline materials, is notavailable. If a reusable adhesive structure is required, thecriterion of fiber fracture strength could be replaced byfiber yield strength, which is proportional the fiber elasticproperties for most polymeric materials except low modu-lus elastomers. In most cases, however, the limit of fiberfracture is not as critical as the criterion for fibercondensation.

For the flat punch contact, the equation for the pull-offforce (Eq. (1)) was originally derived assuming that thecontact radius is equal to the radius of the punch. This isonly valid for a punch with higher stiffness than that ofthe substrate it is in contact with. This will not be the case


for the entire modulus range of the design maps. Spolenaket al. showed that for a reduction of contact radius to 80%of the punch radius, the pull-off force is reduced by about10% [10]. We follow this argument and use Eq. (1) for allmoduli considered in the maps, bearing its limitation inmind.

A possibly unexpected result is the low optimum aspectratios that are predicted in this paper. One possible reasonlies in the neglect of surface roughness: animals whichadhere to surfaces with considerable roughness are gener-ally equipped with setae of larger aspect ratios (about 25for the gecko [28]). Further, there is experimental evidencethat the gecko adhesion benefits from humidity and thatcapillary effects might play a role [7]. One can imagineartificial adhesives combining capillary and van derWaals-type interactions [29]. In future, it would thereforebe interesting to include roughness and humidity in themaps. As far as further contact shapes are concerned futureefforts will include mushroom- and spatula-like structureswhich might also involve friction forces during pull-off, acontribution not accounted for so far.

6. Conclusions

In the present paper we extended the existing concept ofadhesion design maps for hemispherical contacts to threeother tip shapes: flat, toroidal and elastic band. Keepingin mind the assumptions made when constructing themaps, we draw the following main conclusions relevantto the design of an optimal adhesive system:

� For substrates without roughness, flat tips yield thestrongest adhesives. They are, however, sensitive to evensmall roughnesses. The second-best adhesion perfor-mance results from toroidal tips, followed by spheresand tapes. With regards to optimal values, it is interest-ing to observe that flat tips need to be hard and large,spherical tips have an optimum in the range of thegecko’s properties and toroidal shapes are within certainlimits scale independent.� If strong adhesion and easy release are required, the

elastic band still exhibits an adhesion performance inthe range of the adhesion of gecko. This geometry isof interest as the asymmetric tape allows for a controlof the adhesion via the pull-off angle. An optimal pull-off angle of 60� was determined that correlates well withpull-off angles found in nature.� In terms of contact optimization for adhesion in techni-

cal applications, a specific application will need a tuneddesign. Specifically, the roughness of the counter sur-faces needs to be known to choose the ideal shape. Withregards to future structure design this study showed thatthe condensation criterion strongly limits the maximumpossible adhesion for all shapes observed. Thus anattempt to reduce the affinity of fibers to each other bytopological or by chemical means would be highlydesirable.

Appendix A. Design maps for tapes: detailed derivation

Contours of constant apparent contact strength.

The apparent area fraction is given by:

Aapp ¼4R2

f: ðA1Þ

From this and with Eq. (12), we determine the apparentcontact strength [9]:

rapp ¼P c

Aapp

¼ f c

Rffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ 80c

3ER

q ; ðA2Þ

which results in a limit of:

r2appR2 þ r2

app

80c3E

R� f 2c2 ¼ 0 ðA3Þ

R1=2 ¼ �40c3E�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1600c2

9E2þ f 2c2

r2app

s: ðA4Þ

Since only positive values for R have a physical meaning wewrite:

R 6 � 40c3Eþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1600c2

9E2þ f 2c2

r2app

s: ðA5Þ

The limit of fiber fracture.

For the fiber fracture limit, we get [9],

rf ¼5P c

R26 rf

th �E10

ðA6Þ

and derive the following criterion:

R2 þ 80c3E

R� 40; 000c2

E2¼ 0: ðA7Þ

Since only positive R values have a physical meaning, thisyields:

RðpositiveÞ � 188cE

ðA8Þ

R P188c

E: ðA9Þ

The limit of ideal contact strength

Following Spolenak et al. [9],

rc ¼P c

4R26 rth �

cb

ðA10Þ

results in the following limit (again, only positive R valueshave physical meaning):

R2 þ 80c3E

R� b2 ¼ 0 ðA11Þ

RðpositiveÞ ¼ � 40c3E�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1600c2

9E2þ b2

rðA12Þ

R P � 40c3Eþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1600c2

9E2þ b2

r: ðA13Þ


Appendix B. Surface roughness in the adhesion design maps

for flat tips

Here we wish to show that the concept of adhesiondesign maps can be used to model surface roughness effectsin a simple manner. We model the surface as having peri-odic roughness and approximate its contour by hemispher-ical protrusions. In a two-dimensional representation, thesurface is a series of half-circles. We assume that thepunches themselves are absolutely flat (Fig. 6, right).

There are three cases imaginable:

(a) The periodicity of the roughness is of the same orderas the punch diameter. Then only one sphere toucheseach punch. This case leads to the adhesion designmap of spheres as previously published [9] (Fig. 1).

(b) The periodicity is so small that the substrate can beconsidered as a smooth surface (see Fig. 2b). Thisleads to the adhesion design map introduced in Sec-tion 2 (Fig. 3).

(c) The intermediate case will be analyzed in the follow-ing paragraph.

We assume that the hemispheres are in a closest two-dimensional packing of p=2

ffiffiffi3p

, or approximately 90.7%.From here on, we can calculate the number n of spheresin contact with a single flat tip of radius R. The surfaceroughness is described by the radius a of the hemispheres.According to surface roughness definitions [30], this radiuscorresponds to Rz/2 (Rz being the average peak to valleyheight). We find for the number of contacting spheres:

n ¼ pR2

4ffiffiffi3p

a2; ðB1Þ

which leads to a total JKR force of:

P c ¼ffiffiffi3p

p2cR2

8a: ðB2Þ

To construct a map for this case, the apparent contactstrength can be expressed as:

rapp ¼ f

ffiffiffi3p

pc8a

: ðB3Þ

Interestingly, this is independent of punch radius andYoung’s modulus, but depends on the surface roughness.Here we find a splitting effect: for smaller roughness (thus,smaller radius a), the apparent contact strength increases.For the limit of fiber fracture we use:

E ¼ 10ffiffiffi3p

pc8a

; ðB4Þ

which is also independent of R. Following Spolenak et al.[9], the contact radius ac at the instant of pull-off is definedas:

ac ¼9pca2

8E�

� �1=3

ðB5Þ

for one spherical contact, so that for n hemispheres in con-tact we get:

rc ¼P c

na2cp6 rth ðB6Þ

rc ¼ 6acE�2

81p2R4

� �1=3

: ðB7Þ

With this result, rth � c/b, E* = E/1�m2 and the assump-tion that m = 0.3 we find the following limit of ideal contactstrength:

R P 1:34ðabÞ3=4 Epc

� �1=2

ðB8Þ

The resulting adhesion design map for flat tips against asubstrate with periodic roughness is shown in Fig. 6. ForR � a, when the punch is of the same dimension as thehemispherical protrusions, the above-derived model canno longer be applied. When R � a/10, the case for a flatpunch against a smooth surface is reached and the mapis the same as in Fig. 3, with one exception: for the con-tours of constant apparent contact strength it has beentaken into account that some of the punches do not contactthe counter surface: Eq. (2) is divided by a factor of 10. Forthe intermediate part of a/10 6 R 6 a, we interpolated thelimits of fiber fracture and ideal contact strength accord-ingly. As the apparent contact strength is constant forR > a, neither small nor large radii are favored. A mini-mum value for the Young’s modulus is determined throughthe fiber fracture criterion. On the right of this vertical lineall radii give the same adhesion performance.

Appendix C. Supplementary data

Supplementary data associated with this article can befound, in the online version, at doi:10.1016/j.actbio.2008.09.006.

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adhesion design maps for fibrillar adhesives: the effect of shape

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