Contact between rough surfaces and a criterion formacroscopic adhesionLars Pastewkaa,b and Mark O. Robbinsa,1

aDepartment of Physics and Astronomy, Johns Hopkins University, Baltimore, MD 21218; and bMikroTribologie Centrum μTC, Fraunhofer-Institut fürWerkstoffmechanik (IWM), D-79108 Freiburg, Germany

Edited* by Jacob N. Israelachvili, University of California, Santa Barbara, CA, and approved January 23, 2014 (received for review November 5, 2013)

At the molecular scale, there are strong attractive interactionsbetween surfaces, yet few macroscopic surfaces are sticky. Exten-sive simulations of contact by adhesive surfaces with roughnesson nanometer to micrometer scales are used to determine howroughness reduces the area where atoms contact and thus weak-ens adhesion. The material properties, adhesive strength, androughness parameters are varied by orders of magnitude. In all cases,the area of atomic contact is initially proportional to the load. Theprefactor rises linearly with adhesive strength for weak attractions.Above a threshold adhesive strength, the prefactor changes sign,the surfaces become sticky, and a finite force is required to separatethem. A parameter-free analytic theory is presented that describeschanges in these numerical results over up to five orders ofmagnitude in load. It relates the threshold adhesive strength toroughness and material properties, explaining why most macro-scopic surfaces do not stick. The numerical results are qualitativelyand quantitatively inconsistent with classical theories based on theGreenwood−Williamson approach that neglect the range of adhe-sion and do not include asperity interactions.

surface roughness | contact mechanics

Surfaces are adhesive or “sticky” if breaking contact requiresa finite force. At the atomic scale, surfaces are pulled to-

gether by van der Waals interactions that produce forces per unitarea that are orders of magnitude larger than atmosphericpressure (1). This leads to strong adhesion of small objects, such asGecko setae (2, 3) [capillary forces may also contribute to Geckoadhesion in humid environments (4, 5)] and engineered mimics (6),and unwanted adhesion is the main failure mechanism in micro-electromechanical systems with moving parts (7). Although tapeand gecko feet maintain this strong adhesion at macroscopic scales,few of the objects we encounter are sticky. Indeed, our world wouldcome to a halt if macroscopic objects adhered with an averagepressure equal to that from van der Waals interactions.The discrepancy between atomic and macroscopic forces has

been dubbed the adhesion paradox (8). Experiments show thata key factor underlying this paradox is surface roughness, whichreduces the fraction of surface atoms that are close enough to ad-here (8–11). Quantitative calculations of this reduction are ex-tremely challenging because of the complex topography of typicalsurfaces, which have bumps on top of bumps on a wide range ofscales (12, 13). In many cases, they can be described as self-affinefractals from a lower wavelength λs of order nanometers to anupper wavelength λL in the micrometer to millimeter range (10, 14).The traditional Greenwood−Williamson (GW) (15) approach

for calculating nonadhesive contact of rough surfaces approx-imates their complex topography by a set of spherical asperities ofradius R. The distribution of asperity heights is assumed to beeither exponential or Gaussian, and the long-range elastic inter-actions between different asperities are neglected. This approachis analytically tractable and provided a simple explanation for theobservation that the area of contact between nonadhesive elasticsurfaces is proportional to the normal force or external loadpushing them together. Later generalizations (9, 16) consideredthe effect of adhesion between surfaces. These theories predict

that adhesion depends on the ratio of the root-mean-squared(rms) variation in surface separation hrms to a length δc thatcharacterizes the strength of adhesion. The calculated force neededto separate surfaces drops rapidly as hrms=δc increases.In the last decade, there has been great progress in studies of

nonadhesive contact between self-affine rough surfaces. Persson hasdeveloped a scaling theory that includes an approximate treatmentof asperity interactions (17, 18). At the same time, large-scale nu-merical calculations of contact between rough surfaces have becomefeasible (19–23). Both approaches reveal limitations in the GWtreatment of nonadhesive surfaces. For example, the definition of Ris ambiguous (24), the predicted range of linear scaling betweenarea and load is orders of magnitude too small (25), and predictionsfor the geometry of individual contacts and the spatial correlationsbetween them are qualitatively wrong (18, 19). As shown below,these geometrical features determine the effect of adhesion.In recent work, Persson has extended his theory to include ad-

hesion and presented numerical solutions of his equations for sev-eral cases (26–30). The results indicate a general trend with in-creasing adhesion (29). At first, adhesion leads to an increase in theratio of contact area to load. Above a threshold adhesive strength,there is a finite contact area at zero load, and the surfaces becomesticky. This powerful approach has not yet led to simple analyticpredictions for general surfaces or been tested against extensivenumerical simulations of adhesive contact.Here, we use an efficient Green’s function approach to cal-

culate adhesive contact of surfaces with roughness from sub-nanometer to micrometer scales for a wide range of surfaceroughness, adhesive interactions, and material properties. Thenumerical results are clearly inconsistent with expressions basedon the GW approximation. They also indicate the importance of

Significance

Macroscopic objects rarely stick together, yet the van derWaals interactions between surface atoms produce attractivepressures that are orders of magnitude larger than atmo-spheric pressure. This “adhesion paradox” has been linked tosurface roughness, which reduces the area of intimate atomiccontact to summits on the rough landscape. This paper pres-ents a parameter-free theory that captures the interplay be-tween elasticity, interatomic attraction, and surface roughness.It predicts how adhesion changes contact area and when sur-faces are sticky. The results offer a simple explanation for whytape sticks to our desktops but a sheet of paper does not, andmay aid in the design of adhesives and in engineering surfaceroughness to enhance or eliminate adhesion.

Author contributions: L.P. and M.O.R. designed research; L.P. and M.O.R. performed re-search; L.P. and M.O.R. analyzed data; L.P. and M.O.R. wrote the paper; and L.P. wrote thecontact mechanics code and carried out simulations.

The authors declare no conflict of interest.

*This Direct Submission article had a prearranged editor.1To whom correspondence should be addressed. E-mail: mr@jhu.edu.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1320846111/-/DCSupplemental.

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parameters that have not yet been explicitly included in Persson’stheory. We use insights obtained from these calculations to developa simple, parameter-free equation that predicts the effect of adhe-sion on contact.

ResultsIn continuum theory, contact between two rough, elastic surfacescan be mapped to contact between a flat elastic substrate anda rigid rough surface (31). The combined compliance of the sur-faces determines the elastic modulus of the substrate, and theheight profile hðx; yÞ of the rigid surface matches their changingsurface separation. Our simulations take advantage of this map-ping to reduce the parameter space.Fig. 1A shows the geometry of the simulations. There is a rigid

upper surface with self-affine roughness between lower and upperwavelengths λs and λL (see Methods). This implies that the rmschange in surface height over a lateral distance r scales as rH forλs < r < λL. H is called the Hurst exponent and is between 0 and1 (13). One additional parameter is needed to characterize howrough the surface is. A convenient choice is the rms gradient orsurface slope (Fig. 1G) at small scales ð<λsÞ,

h′rms ≡ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiDj∇hj2

Er : [1]

The elastic substrate is the (100) surface of an fcc crystal withatomic spacing a0. The interactions within the substrate arechosen so that it responds like an isotropic solid with Young’smodulus E and Poisson ratio v (seeMethods). The effective modulusthat controls the response to contact forces is Ep ≡ E=ð1− ν2Þ.

The interactions between surfaces determine the work ofadhesion, w, which is defined as the free energy gained per unitcontact area. There is no need to separately consider the effectof w and Ep because they always enter as a ratio with dimen-sions of length: ℓa ≡w=Ep. As described in Discussion, ℓa=a0 istypically much less than unity and we use it to quantify therelative strength of adhesion. The value of ℓa=a0 is increased byincreasing both the range Δr over which attractive forces aresignificant and the characteristic attractive pressure pulling sur-faces together, w=Δr. In the following discussion, we assume thatattractive forces vanish at separations greater than Δr, but gen-eral equations for long-range potentials are given in SI Text,section S-I. In general, Δr is of the order of the equilibriumspacing between surfaces z0 ∼ a0. The latter length was used toestimate the range of adhesion in Tabor’s early theory of adhesionby a single spherical asperity (32).The surfaces are brought together under an external load N.

We identify regions where atoms feel a repulsive force (Fig. 1)with the true contact area Arep (see Methods). The total force onthese atoms is Nrep. The corresponding area and force associ-ated with regions where there is an attractive interaction are Aattand −Natt. The external load N =Nrep −Natt.Fig. 1B shows the complex spatial distribution of Arep for

nonadhesive interactions ðNatt = 0Þ. Analytic and numerical studiesfind that Arep is much smaller than the nominal area A0 and riseslinearly with the external load N pushing the surfaces togetherfor small Arep (15, 17, 18, 20–23). This implies a constant meanrepulsive pressure prep in contacting regions. Steeper and stiffersurfaces are harder to bring into contact, and both numerical andanalytic calculations find (15, 17, 18, 20–23)

A B

C D E F

G

Fig. 1. Geometry of surfaces and contacting regions. (A) A rigid surface with a self-affine fractal height profile hðx,yÞ is pushed down on an elastic substrateby an external load N. The substrate is initially flat with atoms spaced by a0. Substrate deformations produced by a typical adhesive contact are shown withheight variations magnified to show on the scale of the figure. (B−E) Surface map of atoms that feel repulsive (orange) and attractive (black) forces at a fixedcontact area Arep ∼ 0:02A0. Nonadhesive results for the entire system are shown in B, and expanded views of the region indicated by a square in B are shownin C−E for the indicated ℓa=a0 =w=E*a0. (F) Magnified view of an 80a0 wide region of E. The mean diameter drep is obtained by averaging the distance acrossArep. (G) Vertical slice through a contact patch showing the rigid rough (gray) surface and the deformed elastic substrate. The root-mean-square slope, h′rms, isthe rms average of the local height gradient, j∇hj, as indicated on the right. The attractive length datt is the distance from the contact perimeter at which thesurface separation equals the interaction range Δr. The separation rises as a 3/2 power of the distance from the perimeter for nonadhesive surfaces, leadingto the 2/3 exponent in Eq. 6.

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prep ≡Nrep

Arep=h′rmsE p

κrep ; [2]

where κrep is a dimensionless constant that is close to 2. Thisbehavior is illustrated by the gray line in Fig. 2.Figs. 1 B−E and 2 show how adding adhesion affects the

distribution of contacting and attracting regions and the relationbetween load and true contact area Arep. Our first finding is thatthere is always a linear relation between the external load N andthe area in intimate repulsive contact at low Arep (Fig. 2):

NArep

=h′rmsE p

κ ; [3]

where κ is renormalized from its nonadhesive value κrep. As thestrength of adhesion increases, κ and the ratio of Arep to loadrise. Eventually, the ratio diverges, and the surfaces becomesticky when κ changes sign. A negative κ leads to an elasticinstability that pulls surfaces into contact. A pull-off force equalto the magnitude of the most negative load (Fig. 2) is thenneeded to separate them.A quantitative model for the changes in κ can be derived by

analyzing how adhesion affects the geometry of interacting areasand the associated forces. Fig. 1 C−E illustrates how the geometrychanges with increasing adhesion strength ℓa. The external load isadjusted to hold the total contact area Arep constant, and thefigures demonstrate that attractive forces have little effect on thedetailed morphology of Arep. The corresponding repulsive forceNrep and mean pressure prep =Nrep=Arep are also nearly un-changed, which means that Nrep and Arep continue to be related byEq. 2. Fig. 3 Upper Inset shows that this relation holds for allsystems and contact areas studied. The observation that adhesion

produces little change in the surface deformation indicates that weare in the Derjaguin−Muller−Toporov (DMT) limit (33). Asdiscussed in SI Text, section S-I, this is consistent with directevaluation of the Tabor (32) and Maugis (16) parameters.The main change from Fig. 1C to Fig. 1E is that the total area

feeling an attractive force, Aatt, spreads around the periphery of Arepas ℓa is increased by extending the range of adhesive forces, Δr (Fig.1 F and G). As shown in Fig. 3, the total force from the attractiveregion Natt is just Aatt times the mean attractive pressure w=Δr,

Natt = ðw=ΔrÞAatt: [4]

The fact that the mean pressures in the repulsive and attrac-tive regions are constant (Fig. 3) allows us to write the exter-nal load as N = prep Arep − ðw=ΔrÞAatt. From Eqs. 2−4, we find1=κ= 1=κrep − 1=κatt with κrep ≈ 2 and

κatt = h′rmsΔrℓa

Arep

Aatt : [5]

The remaining unknown is the ratio of repulsive to attractivearea, which we now show is independent of load.If Aatt � Arep, it can be approximated by Aatt ≈Pdatt where P is

the length of the perimeter of Arep and datt is the average lateraldistance from the perimeter where the surface separation rea-ches the interaction range Δr (Fig. 1G). Using a general relation

Fig. 2. True contact area Arep as a function of external normal load N fordifferent adhesive interactions. For nonadhesive contact, Arep rises approx-imately linearly with N with dimensionless prefactor κrep ≈ 2 (Eq. 2). As thestrength of adhesion ℓa =w=E* increases, the area rises more rapidly withload. Straight lines show linear fits at small Arep whose slopes give therenormalized values of κ (Eq. 3). The value of κ initially rises with adhesion(blue lines) and diverges at the onset of stickiness (green line). The red lineshows a sticky case where κ< 0 and the pull-off force is nonzero. Resultsshown are for a surface with Hurst exponent H= 0:8, root-mean-squareslope h′rms = 0:1, and lower wavelength cutoff λs =32a0.

Fig. 3. Plots showing the linear relationships between repulsive and at-tractive areas and forces, and between perimeter and area for the widerange of parameters indicated by the symbols defined below. Over fiveorders of magnitude in both quantities, the ratio between Natt and Aatt isequal to the mean adhesive pressure w=Δr (Eq. 4). (Upper Inset) For thesame parameters, Nrep and the contact area Arep follow the linear relationpredicted by Eq. 2, showing that adhesion does not change the mean re-pulsive pressure prep ≡h′rmsE*=κrep in the contact. (Lower Inset) Plot showingthat the contact area Arep is proportional to the length of its perimeterP. The plot is normalized to show that the proportionality constantdrep=π ≈ 4h′rms=πh″rms (Eq. 9). Deviations by up to a factor of 2 from this ex-pression for drep are responsible for the spread in the figure. For a givensystem, changes in drep with Arep are less than 25% over 2–3 decades in Arep.All plots show multiple contact areas for each realization of a surface.Results are shown for H= 0:3 (triangles), 0.5 (squares), and 0.8 (circles) andℓa=a0 = 0:0005 (black), 0.005 (blue), and 0.05 (red). Closed and open symbolsare for h′rms = 0:1 and 0.3, respectively. The symbol size increases as λs=a0increases from 4 to 64 in powers of 2.

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between area and perimeter, we can write Arep =Pdrep=π, wheredrep is the mean diameter (see Fig. 1F and SI Text, section S-II).For ordinary 2D objects like a circle, the perimeter and diameterare proportional and increase as the square root of the area. Thisbehavior is assumed in conventional theories of contact betweenrough surfaces that ignore long-range elastic interactions betweenindividual contacting asperities, such as the previously discussedGW (15) and related adhesive (9, 16, 34) models. Including elasticinteractions leads to qualitative changes in contact geometry(18–20). The contact area becomes a fractal with the samefractal dimension as the perimeter (20) [a true “monster” (13)].We find that drep is then nearly independent of contact area,load, and adhesive strength and present an analytic expressionfor it below.We calculate datt using continuum theory for nonadhesive

contact of locally smooth surfaces. If x is the lateral distance fromthe edge of a contact, then for small x, the separation betweensurfaces always (31) rises as x3=2. We use the standard prefactorfor a cylinder with contact radius drep=2 and equate datt to thelateral distance where the separation equals Δr (Fig. 1G and SIText, section S-I). Combined with our expression for P, this givesthe constant ratio between attractive and repulsive areas

Arep

Aatt=

drepπdatt

=�169π

�1=3�h′rmsdrepπΔr

�2=3: [6]

Inserting this result in Eq. 5, gives the prediction

κatt =h′rmsΔr

ℓa

�169π

�1=3�h′rmsdrepπΔr

�2=3: [7]

As shown in Fig. 4, our simple analytic expressions providea quantitatively accurate description of Arep=Aatt and κatt for awide range of surface geometries. Deviations are only larger thanthe numerical uncertainty when the attractive area has grown toolarge to be approximated as a thin rim around the repulsive re-gion (i.e., when Aatt >Arep), which is well into the sticky re-gime. The continuum expression for datt also fails in this limitðdatt > drep=2Þ.Eq. 7 provides a simple and quantitative explanation for the

changes in Fig. 2. As the work of adhesion (and therefore ℓa)increases, there is a proportional increase in 1=κatt. At first, ad-hesion merely produces a small change in the ratio of repulsivecontact area to external load. The sign of the ratio changes when1=κatt becomes bigger than 1=κrep and the surface becomes sticky.We have also examined how adhesion changes the variation in

the mean surface separation u with load. The normal contactstiffness defined as kN = dN=du is typically found to rise linearlywith external load for nonadhesive surfaces (35, 36). In the regimewhere surfaces are not sticky, we find that the relation betweensurface separation and Nrep is nearly unchanged, just as the re-lation between Nrep and Arep is nearly the same (Fig. 3 UpperInset). Because adhesion reduces the external load N by a factor ofκrep=κ, the normal stiffness is reduced by the same factor. This isa small correction unless the surfaces are close to becoming sticky.The length drep is always of order λs and has a simple relation

to statistical properties of the undeformed surface. As above andin SI Text, section S-I, we approximate the contacting part ofasperities by a cylinder with radius R, where R is calculated fromthe rms curvature of the rough surface at small scales ð<λsÞ,

1=R= h″rms=2=ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiD�

∇2h�2Er

=2: [8]

If the contact has diameter drep and slope h′rms at the contactedge, then drep = 4h′rms=h″rms. Following the same reasoning, thelength in the numerator of Eqs. 6 and 7 is proportional to theheight change δh from the contact edge to center:

δh= ½h′rms�2=h″rms = h′rmsdrep=4: [9]

Fig. 3 Lower Inset shows that Arep and P are proportional andthat the proportionality constant drep=π is always within a factorof 2 of 4h′rms=πh″rms.

DiscussionSurfaces are sticky when the total adhesive force, which is at-tractive pressure times attractive area, exceeds the total repulsiveforce prep Arep. This corresponds to 1=κatt > 1=κrep ≈ 1=2, and ournumerical results show stickiness if and only if this condition ismet. It can be recast as a condition on the ratios of pressures andareas, ðw=ΔrÞ=prep >Arep=Aatt, or using our analytic expressions:

h′rmsΔrκrepℓa

�h′rmsdrep4Δr

�2=3< π

�316

�2=3≈ 1; [10]

where the first factor on the left reflects the pressure ratio andthe second comes from the area ratio.A simpler necessary condition for adhesion can be obtained

from Eq. 10. As noted above and in SI Text, section S-I, theeffective range of the potential is typically less than but ofthe order of the atomic bond distance a0. Any height changeδh is possible in continuum theory, but there is a natural lowerbound of order a0 in atomic systems. For example, roughness

Fig. 4. Values of κatt and Arep=Aatt from simulations with the wide rangeof parameters indicated below are plotted against the predictions fromthe model, Eqs. 6 and 7. For weak adhesion, deviations from the pre-dictions (solid lines) are comparable to statistical errors that are of theorder of the largest symbol size. The geometrical assumptions underlyingEqs. 6 and 7 break down for strong adhesion where Aatt becomes com-parable to or larger than the contact area Arep. Adhesion is observed ifand only if 1=κatt > 1=κrep. The shaded region shows the prediction foradhesion using the value κrep = 2. Results are shown for Hurst exponentsH= 0:3 (triangles), 0.5 (squares), and 0.8 (circles) and adhesive strengthsℓa=a0 = 0:0005 (black), 0.005 (blue), and 0.05 (red). Closed and open sym-bols are for root-mean-square slopes h′rms = 0:1 and 0.3, respectively. Thesymbol size increases as the lower wavelength cutoff λs=a0 increases from4 to 64 in powers of 2.

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on crystalline surfaces occurs in the form of terraces withheight ∼ a0. Using Eq. 9 and inserting the estimates Δr≈ a0 ≤ δhin Eq. 7, one finds a necessary but not sufficient criterion foradhesion: ℓa=a0 ≳ 0:5h′rms.Note that the prediction for the onset of adhesion in Eq. 10 is

qualitatively different from previous models for rough surfaceadhesion (9, 16, 27, 34), which do not include two of the keylengths in Eqs. 5–10. The characteristic width of contactingregions drep reflects their fractal nature and has not been iden-tified before. Continuum theories have considered the limitingcases of Δr equal to zero (9, 27, 34) or infinity (16) and con-cluded Δr had little effect (16), whereas we find more adhesionat small Δr because the adhesive pressure is increased. Finally,our relations only include quantities that are determined byroughness at short wavelengths (10, 37)—drep and the rms surfaceslope h′rms and curvature h″rms. In contrast, the rms height variationhrms ∝ h′rmsλsðλL=λsÞH is the key surface property in past GW the-ories. Our numerical results for different λs, h″rms, and H in Fig. 4collapse onto a universal curve even though the value of hrms variesmore than an order of magnitude for the same predicted valueof κatt. SI Text, section S-III presents plots that show qualitativediscrepancies between these data and traditional GW theories.Persson’s theory starts at large length scales and gradually

introduces finer and finer structure (26–30). The value of Δr isnot explicitly included, but a scale-dependent work of adhesionis introduced that may capture some of the effects of the in-teraction range. The effect of adhesion has only been determinedfor a limited range of parameters, and we hope that our pre-dictions will be compared with future results from this model.Persson has also presented a theory of capillary adhesion (38)that includes a calculation of the distribution of surface separa-tions and may be able to predict Aatt.Direct comparisons to past experiments are difficult. Past

studies reported hrms because that was believed to be the controlparameter. Values of h′rms are not generally reported. They can inprinciple be obtained from the surface height profile hðx; yÞ, butthis can be challenging in practice. Because h′rms is sensitive toroughness at small length scales, hðx; yÞ must be measured withhigh spatial resolution (10, 37).We can reach general conclusions about the experimental

implications of Eq. 10 using typical values of ℓa=a0. We firstconsider the extreme case where w is the work of adhesion forjoining crystals of the same material. Then, for atomistic solids,the same interactions determine both Ep and w. The value ofℓa=a0 is of the order of the fractional change in bond lengthneeded to change the elastic energy by the binding energy, andℓa=a0 � 1. For example, diamond has Ep ≈ 1012Pa and w≈10 J=m2, yielding ℓa=a0 ≈ 0:06 with a0 the carbon bond length.The simple Lennard−Jones potential has ℓa=a0 ≈ 0:05. For thesetypical values of ℓa, adhesion should occur for h′rms of order 0.1and below. The stickiest cases considered in Fig. 4 (closed redsymbols) are indeed for the case ℓa=a0 ≈ 0:05, h′rms = 0:1, andsmall drep. Increasing h′rms to 0.3 suppresses adhesion.Exposing surfaces to the environment typically reduces the

adhesive forces to weaker van der Waals interactions withw∼ 50mJ=m2. The value of ℓa and the root-mean-square slopeneeded to eliminate adhesion are reduced by two to three ordersof magnitude. Only exceptionally smooth surfaces like atomicallyflat mica (1) and the silicon surfaces used in wafer bonding (39)have slopes low enough to stick ð. 10−3Þ. For most surfaces, Eq.7 predicts κatt � 1 and hence κ≈ κrep. This provides an expla-nation for the success of models for friction that ignore van derWaals adhesion (15, 31, 40).Most of the sticky surfaces we are familiar with break the con-

nection between w and Ep to increase ℓa=a0. Tape, rubber, andelastomers adhere via van der Waals interactions over the entiresurface, but have small elastic moduli associated with the entropyrequired to stretch polymer segments between chemical crosslinks.

Eq. 10 implies that surfaces with w= 50mJ=m2, h′rms ∼ 1, Δr∼0:5  nm, and drep ∼ 10  nm will be sticky if Ep. 10 MPa, which iscommon for soft rubbers and elastomers, whereas paper is muchstiffer (> 1GPa). Tapes are normally designed to have modulibelow 0.1 MPa, which is known as the Dahlquist criterion (41).Taking h′rms ∼ 1 and Δr∼ 0:5  nm, one finds adhesion for drep .100 μm. Adhesives of this type can stick even to structured sur-faces with macroscopic grooves. Once an adhesive bond is formed,the viscoelastic properties of the adhesive can be used to greatlyincrease the force needed to break the adhesive bond (8).Animals use two different mechanisms to increase ℓa=a0 that

have evolved independently in a number of species (42). Some usesmooth pads with a large surface that maximizes w and a sparsefibrous internal structure that minimizes Ep (43). Geckos (2, 3)and recently manufactured mimics (6) break the solid up intoa hierarchical series of separate rods with pads at the ends. Thisallows adjacent pads to contact the surface at different heightswithout a large elastic energy, leading to a small effective Ep eventhough the components are stiff.In summary, we have presented numerical simulations of ad-

hesive contact between rough surfaces for a wide range of ad-hesion strength, surface geometries, and material properties. Inall cases, the area in intimate repulsive contact is proportional tothe external load at low-contact areas. The ratio of area to ex-ternal load increases with adhesion and changes sign when sur-faces begin to stick. This transition only occurs in the limits ofsmooth surfaces, high work of adhesion, and low stiffness. Theresults are qualitatively inconsistent with traditional GW theoriesbut in quantitative agreement with a simple parameter-free theorybased on observed changes in contact geometry. This theory makesspecific predictions for experimental systems and may aid in thedesign of adhesives, and in engineering surface roughness to en-hance or eliminate adhesion. It also provides a simple explanationfor our everyday experience with macroscopic adhesion. For mostmaterials, the internal cohesive interactions that determine elasticstiffness are stronger than adhesive interactions, and surfaces willonly stick when they are extremely smooth. Tape, geckos, and otheradhesives stick because the effect of internal bonds is diminished tomake them anomalously compliant.

MethodsSelf-affine rough surfaces with the desired Hurst exponent H, h′rms, λs, and λLare generated using a Fourier-filtering algorithm described previously (44).Fourier components for each wavevector q→ have a random phase and a nor-mally distributed amplitude that depends on the magnitude q. The amplitudeis zero for q> 2π=λs, proportional to q−1−H for 2π=λs >q> 2π=λL, and rolls overto a constant for q< 2π=λL. Periodic boundary conditions with period L areapplied in the plane of the surface to prevent edge effects. The elastic re-sponse is determined using a Fourier-transform technique (45, 46) with a lin-earized surface Green’s function corresponding to Poisson ratio ν= 1=2. Resultsare shown for period L= 2λL = 4096a0 with rigid boundary conditions appliedat depth L below the surface. Systematic studies were performed with L and λLfrom 512a0 to 8192a0 to ensure that finite-size effects are small.

Atoms on the elastic substrate interact with the rigid rough surfacethrough a potential that only depends on the height difference z. We usea 9-3 Lennard−Jones potential that represents the integral over a half spaceof the usual 12-6 Lennard−Jones potential between atoms. The potentialis truncated smoothly using a quartic function from the potential minimumat z= a0 to the cutoff at a0 +Δr. The potential and its first two derivativesare continuous and vanish at the cutoff. In our calculations, we fix thestiffness k of the potential at a value that is consistent with the stiffness ofinteractions within the substrate: k= E*a0=2. Consistent results were obtainedwith other potentials, including an untruncated 12-6 Lennard−Jones potential.

As is common for atomic-scale calculations (47−49), the contact areaArep isdefined as the area covered by atoms that feel a repulsive force ðz<a0Þ.Similarly, the attractive area Aatt is the area covered by atoms that feel anattractive force ða0 < z<a0 +ΔrÞ. We only show results for Arep=λ

2s &10, so

that there is a statistical number of contacting asperities (20). Numericalvalues of κ, κrep, and κatt are computed at 1% true contact area from theratios of the relevant forces and areas.

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ACKNOWLEDGMENTS. This work was supported by the Air Force Office ofScientific Research (Grant FA9550-0910232), the U.S. National Science Foun-dation (Grants OCI-108849, DMR-1006805, and CMMI-0923018), the Simons

Foundation (M.O.R.), and the European Commission (Marie-Curie IOF 272619,to L.P.). Computations were carried out at Johns Hopkins University and theJülich Supercomputing Center.

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