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When are rough surfaces sticky?

At the molecular scale there are strong attractive interactions between surfaces, yet few macroscopic surfaces are sticky. Extensive simulations of contact by adhesive surfaces with roughness on nanometer to micrometer scales are used to determine how roughness reduces the area where atoms contact and thus weakens adhesion. The material properties, adhesive strength and roughness parameters are varied by orders of magnitude. In all cases the area of atomic contact rises linearly with load, and the prefactor rises linearly with adhesive strength for weak interactions. Above a threshold adhesive strength, the prefactor changes sign, the surfaces become sticky and a finite force is required to separate them. A parameter-free analytic theory is presented that describes changes in these numerical results over up to five orders of magnitude in load. It relates the threshold strength to roughness and material properties, explaining why most macroscopic surfaces do not stick. The numerical results are qualitatively and quantitatively inconsistent with classical theories based on the Greenwood-Williamson approach that neglect the range of adhesion and do not include asperity interactions.

Surfaces are adhesive or "sticky" if breaking contact requires a finite force. Few of the surfaces we encounter are sticky even though almost all are pulled together by van der Waals interactions at atomic scales. 1 Gecko setae 2,3 and engineered adhesives 4 use this ubiquitous attraction to achieve pull off forces per unit area that are orders of magnitude larger than atmospheric pressure, and our world would come to a halt if these pressures operated on most macroscopic surfaces.
The discrepancy between atomic and macroscopic forces has been dubbed the adhesion paradox. 5 Experiments show that a key factor underlying this paradox is surface roughness, which reduces the fraction of surface atoms that are close enough to adhere. [5][6][7][8] Quantitative calculations of this reduction are extremely challenging because of the complex topography of typical surfaces, which have bumps on top of bumps on a wide range of scales. 9,10 In many cases they can be described as self-affine fractals from a lower wavelength λ s of order nanometers to an upper wavelength λ L in the micrometer to millimeter range. 7,11 Here, we use an efficient Green's function approach to calculate adhesive contact of surfaces with roughness from subnanometer to micrometer scales. Numerical results for a wide range of surfaces, adhesive interactions and material properties are presented and used to develop a simple, parameter-free equation that predicts the effect of adhesion on contact.
The traditional Greenwood-Williamson (GW) 12 approach for calculating contact of rough surfaces approximates their complex topography by a set of spherical asperities of radius R whose height distribution is determined from self-affine scaling. The long-range elastic interactions between different asperities are neglected. This approach is analytically tractable and provided a simple explanation for the observation that the area of contact between nonadhesive elastic surfaces is proportional to the normal force or load pushing them together. Later generalizations 6,13 considered the effect of adhesion between surfaces and found that the key parameter was the ratio of the root mean squared (rms) height variation h rms to the normal displacement δ c of a single asperity due to adhesion. If the work of adhesion gained per unit area of contact is w, then δ 3 c = (3/4) 3 R(πw/E * ) 2 with contact modulus 14 E * = E/(1 − ν 2 ) for an isotropic material with Young's modulus E and Poisson ratio ν. GW based adhesion models 6,13 predict that the force needed to separate surfaces drops rapidly as h rms /δ c increases and is negligible for h rms /δ c > 3.
In the last decade, Persson has developed a scaling theory that includes an approximate treatment of asperity interactions. 15,16 At the same time, large scale numerical calculations of contact between rough surfaces have become feasible. [17][18][19][20] Both approaches reveal limitations in the GW treatment of nonadhesive surfaces. For example, the definition of R is ambiguous, 22 the predicted range of linear scaling between area and load is orders of magnitude too small, 23 and predictions for the geomety of individual contacts and the spatial correlations between them are qualitatively wrong. 16,17 As shown below, these geometrical features determine the effect of adhesion.
In recent work, Persson has extended his theory to include adhesion in the limit where the range of surface separations over which attractive interactions are significant, ∆r, is zero. 15, [24][25][26] He has applied this theory to specific cases and found a reduction in adhesion with increasing h rms , but this powerful approach has not yet led to simple analytic predictions for general surfaces.
Here, we use an efficient Green's function approach to calculate adhesive contact of surfaces with roughness from subnanometer to micrometer scales. The numerical results are clearly inconsistent with expressions based on the GW approximation. In particular, the relevant length scale describing the roughness is not h rms and the range of adhesive interactions determines a characteristic adhesive pressure w/∆r that plays a critical role. Numerical results for a wide range of surfaces, adhesive interactions and material properties are presented and used to develop a simple, parameter-free equation that predicts the effect of adhesion on contact. Figure 1(a) shows the geometry of the simulations. There is a rigid upper surface with selfaffine roughness. The change in height h over a lateral distance x increases as x H where the Hurst exponent 10 H is between 0 and 1. The elastic substrate is the (100) surface of an fcc crystal with atomic spacing a 0 , and behaves like a continuous medium in the limit of large λ s /a 0 . We identify regions where atoms feel a repulsive force with the contact area A rep (see methods). GW and more recent approaches predict that A rep is much smaller than the total area A 0 and rises linearly with the load N pushing the surfaces together. 12,15,16,18-20 By dimensional analysis, the surface geometry can only enter through the dimensionless rms surface slope, h rms = |∇h| 2 (see Fig. 1g). Steeper and stiffer surfaces are harder to bring into contact, so that 12,15,16,18-21

RESULTS
where numerical solutions, such as the grey line in Fig. 2, find the dimensionless constant κ rep is close to 2 while GW and Persson give κ rep ≈ 2.6 and 1.6, respectively. Note that the ratio of load to area represents the mean repulsive pressure p rep in contacting regions, which depends only on h rms and E * . There is no need to separately consider the effect of E * and w because they always enter as a ratio with dimensions of length: a ≡ w/E * . As discussed below, a /a 0 is typically much less than unity and we use it to quantify the relative strength of adhesion.
Our first finding is that there is always a linear relation between the total load and the area in intimate repulsive contact at low N (Fig. 2). This can be described by Eq. (1) with κ rep replaced by a renormalized constant κ. As the strength of adhesion increases, κ and the ratio of A rep to load rise. Eventually the ratio diverges and the surfaces become sticky when κ changes sign. A negative κ leads to an elastic instability that pulls surfaces into contact and a pulloff force equal to the magnitude of the most negative load (see Fig. 2) is needed to separate them.
A quantitative model for the changes in κ can be derived by analyzing how adhesion affects contact geometry. Figures 1(c-e) show contacting regions (orange) that interact with repulsive forces and attracted regions (black) that are close enough to feel adhesive forces ( Fig. 1(f,g)). The strength of adhesion is varied at constant total repulsive area A rep . We find that the corresponding repulsive load N rep and mean pressure p rep also remain constant (see upper inset in Fig. 3) and that there are only minor morphological changes in the shape of A rep . The main change is that the total area feeling an attractive force, A att , spreads around the periphery of A rep as the range of adhesive forces, ∆r, increases ( Fig. 1(e,f)). This type of behavior is assumed in the Derjaguin-Muller-Toporov (DMT) approximation for adhesion which is typically valid for spherical asperities in the nanometer and micrometer range. 27,28 Different behavior might be observed if λ s was much larger (Suppl. S-I).
The key observations needed to calculate κ are that p rep remains constant, that there is a constant mean adhesive pressure w/∆r in the attractive region (see Fig. 3) and that the ratio of repulsive and attractive areas is independent of load at low loads. The first two observations allow us to write the total load as N = p rep A rep − (w/∆r)A att . From Eq. (1) we immediately find with κ rep ≈ 2 and The remaining unknown is the ratio of repulsive to attractive area.
If A att A rep , it can be approximated by A att ≈ P d att where P is the length of the perimeter of A rep and d att the average lateral distance from the perimeter where the surface separation reaches the interaction range ∆r ( Fig. 1(g)). For a general area, A rep = P d rep /π where d rep is the mean diameter (see Fig. 1(f) and Suppl. S-II). For ordinary two dimensional objects like a circle, the perimeter and diameter are proportional and increase as the square root of the area. This behavior is assumed in conventional theories of contact between rough surfaces that ignore long-range elastic interactions between individual contacting asperities, such as the previously discussed GW 12 and related adhesive 6,13,29 models. Including elastic interactions leads to qualitative changes in contact geometry. [16][17][18] The contact area becomes a fractal with the same fractal dimension as the perimeter 18 (a true "monster" 10 ). We find that d rep is then nearly independent of contact area, load and adhesive strength and present an analytic expression for it below.
We calculate d att using continuum theory for nonadhesive contact of locally smooth surfaces. If x is the lateral distance from the edge of a contact, then for small x the separation between surfaces always 14 rises as x 3/2 . We use the standard prefactor for a cylinder with contact radius d rep /2 and equate d att to the lateral distance where the separation equals ∆r (Suppl. S-I). Combined with our expression for P , this gives the constant ratio between attractive and repulsive areas Inserting this result in Eq. (3), gives the prediction for κ att .
As shown in Fig. 4, our simple analytic expressions provide a quantitatively accurate description of A rep /A att and κ rep /κ att for a wide range of surface geometries. Deviations are only larger than the numerical uncertainty when the attractive area has grown too large to be approximated as a thin rim around the repulsive region (i.e. when A att > A rep ), which is well into the sticky regime.
The continuum expression for d att also fails in this limit (d att > d rep /2).
Eqs. (3) and (4) provide a simple and quantitative explanation for the changes in Fig. 2. As the adhesion energy (and therefore a ) increases, there is a proportional increase in 1/κ att . At first adhesion merely produces a small change in the ratio of area to load. The sign of the ratio changes when 1/κ att becomes bigger than 1/κ rep and the surface becomes sticky.
The length d rep is always of order λ s and has a simple relation to statistical properties of the undeformed surface. As above and in Suppl. S-I, we approximate the contacting part of asperities by a cylinder with radius R, which is calculated from the rms curvature of the rough surface If the contact has diameter d rep and slope h rms at the contact edge, then d rep = 4h rms /h rms . Following the same reasoning, the length in the numerator of Eq. (4) is proportional to the height change δh from the contact edge to center: The values of h rms and h rms can readily be evaluated from the statistical properties of the rough surface in real or reciprocal space. The lower inset of Fig. 3 shows that P and A rep are proportional and that the proportionality constant is always within a factor of two of 4h rms /h rms .
The contact area is not directly accessible to experiment, but changes with load in the mean separation between surfaces u can be measured with sufficiently stiff mechanical devices. 30,31 The normal contact stiffness defined as k N = dN/du is typically found to rise linearly with load for nonadhesive surfaces. 32,33 In the regime where surfaces are not sticky we find that the relation between surface separation and N rep is nearly unchanged, just as the relation between N rep and A rep is nearly the same ( Fig. 3 upper inset). Since adhesion reduces the total load N by a factor of κ rep /κ, the normal stiffness is reduced by the same factor. This is a small correction unless the surfaces are close to becoming sticky, and nonadhesive predictions are likely to be within experimental uncertainties.

DISCUSSION
Surfaces are sticky when the total adhesive force, which is adhesive pressure times attractive area, exceeds the total repulsive force p rep A rep . This corresponds to 1/κ att > 1/κ rep , and our numerical results show stickiness if and only if this condition is met. It can be recast as a condition on the ratios of pressures and areas, (w/∆r)/p rep > A rep /A att , or using our analytic expressions: where the first factor on the left reflects the pressure ratio and the second comes from the area ratio. As noted above and in the supplementary material the effective range of the potential is typically less than but of order of the atomic bond-distance a 0 . Any height change δh is possible in continuum theory, but there is a natural lower bound of order a 0 in atomic systems. For example, roughness on crystalline surfaces occurs in the form of terraces with height ∼ a 0 . Inserting this bound in Eq. (4) one finds a necessary but not sufficient criterion for adhesion: a /a 0 > ∼ 0.5h rms .
Note that the above prediction for the onset of adhesion is qualitatively different than previous models for rough surface adhesion, which do not include two of the key lengths in Eqs. (3)- (6).
The characteristic width of contacting regions d rep reflects their fractal nature and has not been identified before. Continuum theories have considered the limiting cases of ∆r equal to zero 6,15,29 or infinity 13 and concluded ∆r had little effect, 13 while we find more adhesion at small ∆r because the adhesive pressure is increased. Finally, our relations only include quantities that are determined by short wavelength roughness -the rms surface slope and curvature. The rms roughness is the key surface property in past GW theories, and rises with the upper wavelength of roughness as λ H L . The numerical results with different symbol size in Fig. 4 have values of λ H L that vary by more than an order of magnitude but collapse onto the universal curve predicted by Eqs. (3) and (4).
Supplemental section S-III presents plots that show qualitative discrepancies between these data and traditional GW theories.
To determine the implications of Eq. Exposing surfaces to the environment typically reduces the adhesive forces to van der Waals interactions with w ∼ 50mJ/m 2 . The value of a and the root mean square slope needed to eliminate adhesion are reduced by two to three orders of magnitude. Only exceptionally smooth surfaces like atomically flat mica 1 and the silicon surfaces used in wafer bonding 34 have slopes low enough to stick ( < ∼ 10 −3 ). For most surfaces κ ≈ κ rep . This provides an explanation for the success of models for friction that ignore van der Waals adhesion. 12,14, 28,35 Most of the sticky surfaces we are familiar with break the connection between w and E * to increase a /a 0 . Geckos 2,3 and recently manufactured mimics 4 break the solid up into a hierarchical series of separate rods with pads at the ends. This allows adjacent pads to contact the surface at different heights without a large elastic energy, leading to a small effective E * even though the components are stiff. Tape, rubber, and elastomers adhere via van der Waals interactions, but have small elastic moduli associated with the entropy required to stretch polymer segments between chemical crosslinks. Eq. (6) implies that surfaces with w = 50mJ/m 2 , h rms ∼ 1, ∆r ∼ 0.5 nm and d rep ∼ 10 nm will be sticky if E * < ∼ 10 MPa, which is common for soft rubbers and elastomers while paper is much stiffer (>1GPa). Tapes are normally designed to have moduli below 0.1MPa, which is known as the Dahlquist criterion. 36 Taking h rms ∼ 1 and ∆r ∼ 0.5 nm, one finds adhesion for d rep < ∼ 100 µm. Adhesives of this type can stick even to structured surfaces with macroscopic grooves. Once an adhesive bond is formed, the viscoelastic properties of the adhesive that can be used to greatly increase the force needed to break the adhesive bond. 5 In summary, we have presented numerical simulations of adhesive contact between rough sur- Methods -Calculations were performed for a rigid rough surface contacting a flat elastic substrate. In continuum theory this is equivalent to contact between two rough elastic surfaces and the mapping remains approximately valid at atomic scales. 37 Self-affine rough surfaces with the desired Hurst exponent H, h rms , λ s and λ L are generated using a Fourier-filtering algorithm described previously. 38 Fourier components for each wavevector q have a random phase and a normally distributed amplitude that depends on the magnitude q. The amplitude is zero for q > 2π/λ s , proportional to q −1−H for 2π/λ s > q > 2π/λ L , and rolls over to a constant for q < 2π/λ L . Systematic studies were performed with L and λ L from 512a 0 to 8192a 0 to ensure that finite-size effects are small.
Atoms on the elastic substrate interact with the rigid rough surface through a potential that only depends on the height difference z. We use a 9-3 Lennard-Jones potential that represents the integral over a half space of the usual 12-6 Lennard-Jones potential between atoms. The potential is truncated smoothly using a cubic spline from the potential minimum at z = a 0 to the cutoff at a 0 + ∆r. The potential and its first two derivatives are continuous and vanish at the cutoff. In our calculations we fix the stiffness k of the potential at a value that is consistent with the stiffness of interactions within the substrate: k = E * a 0 /2. Consistent results were obtained with other potentials, including an untruncated 12-6 Lennard-Jones potential.
As is common for atomic-scale calculations, 37,41,42 the contact area A rep is defined as the area covered by atoms that feel a repulsive force (z < a 0 ). Similarly, the attractive area A att is the area covered by atoms that feel an attractive force (a 0 < z < a 0 + ∆r). We only show results for A rep /λ 2 s > ∼ 10, so that there is a statistical number of contacting asperities. 18 Numerical values of κ, κ rep and κ att are computed at 1% contact area from the ratios of load and area.   As discussed in the main text, we use standard results from continuum theory for nonadhesive contact between smooth surfaces. If x is the lateral distance from the edge of the contact, then for small x the separation ∆(x) between surfaces always rises as x 3/2 outside the contact. 1 The prefactor rises with the surface slope at the edge of the contact which we take to be h rms . For simple geometries like spheres, cones or cylinders, the only length scale that enters is the radius of the contact area. Since the contacting region in our numerical simulations has a constant average diameter d rep , we use the standard prefactor for a cylinder: 2,3 Our numerical data for the average surface separation at a given distance from the perimeter are consistent with this relation without adjustable parameters. To find d att we just equate ∆(d att ) to ∆r in Eq. (S-1), yielding An effective range of interaction ∆r can be defined for arbitrary forms of the interaction potential using Eq. (S-1). We define p(∆) as the attractive pressure between surfaces separated by where w is the work of adhesion. As in the calculation of A att in the main document, we assume that the perimeter changes direction slowly enough that we can write the total load as the perimeter times a contribution per unit length. With ∆(x) being the separation at distance x from the contact edge, this yields: This is equivalent to the expressions in the main text for truncated potentials with From Eqs. (S-1), (S-2) and (S-4) one finds an expression for ∆r With this value of ∆r, all of the relations in the main text for truncated potentials carry over to an arbitrary potential. Note that the integrals are well defined because p represents the total force and goes linearly to zero at the equilibrium surface separation ∆ = 0.
In general we find that ∆r from Eq. (S-5) is comparable to or smaller than the atomic spacing a 0 that minimizes the energy. For example, if the 9-3 Lennard-Jones potential is used to infinite distances, one finds ∆r = 1.15a 0 . For the 12-6 Lennard-Jones potential, ∆r = 0.62a 0 . Note that these ranges are referenced to the potential minimum at a 0 and that most of the binding energy comes from these short scales (> 85%).
The spline potential used in the calculations reported in the main text has where a 0 is the minimum of the potential and k = E * a 0 /2 is the stiffness at z = a 0 . The adhesion energy was varied by changing ∆r at fixed k. The range given by Eq. (S-5) for this potential is essentially the same as the actual range (within 6%). For a general truncated potential, quantitative agreement with results for the relation between area and load are better when Eq. (S-5) is used.
In calculating the load we have assumed that the variation of the surface separation ∆ with x is not affected by adhesion. A similar approximation is made in the Derjaguin-Muller-Topov (DMT) theory for contact of a spherical asperity of radius R. 4 Maugis has found that the DMT approximation is accurate for spheres when a dimensionless ratio λ Maugis is small: 5 .

(S-7)
As above and in the main text, we determine R = d rep /2h rms by assuming a locally cylindrical geometry with contact diameter d rep and slope h rms at the edge of the contact. We find that the approximations used in the main text are accurate even when λ Maugis exceeds unity. Using typical values of a ∼ 5·10 −4 for atomistic solids exposed to the environment, h rms = 0.1 and ∆r/a 0 = 1,  Grid cells are defined to be neighbors if they share an edge (Fig. S-1a). The corresponding atoms are then nearest neighbors. The repulsive area is divided into connected patches like that shown in Fig. S-1b. Grid cells (atoms) that belong to a patch but have less than four neighbors in the patch are part of the perimeter. The perimeter length P is calculated as: P = βa 0 N P where N P is the number of perimeter cells and β corrects for the discreteness of the lattice. Consider a line of length L at an angle θ to the horizontal axis. The perimeter cells will form a stepped approximation to this line. It is easy to show that the number of perimeter cells is equal to L/(a 0 cos θ) for θ between −π/4 and π/4. Counting these cells and multiplying by a 0 underestimates P by a factor of 1/ cos θ. We find β = 4 sinh −1 (1)/π ≈ 1.1222 by assuming isotropy and averaging over angles. As noted in the main text, traditional theories for the effect of adhesion start from the Greenwood-Williamson approximation. Fuller and Tabor 7 found the pulloff force needed to separate the surfaces in the JKR limit of short range potentials (∆r → 0) and Maugis 8 found similar results for the opposite DMT limit of long range potentials (∆r → ∞). In both cases, the pulloff force is a function of h rms /δ c , where δ c is the normal displacement of a single asperity due to ad- The pulloff force drops rapidly for h rms /δ c > 1 and is extremely small for h rms /δ c > 3. In contrast to our results, no clear transition to nonadhesive behavior with area proportional to load was discussed.
These traditional theories expressed the pulloff force as a ratio to the maximum force N P c where N is the number of spherical asperities and P c = 3πwR/2 the pulloff force for each in the JKR limit. From statistical studies of rough surfaces, 8-10 N ∼ 0.05A 0 /Rh rms . This gives N P c ∼ wA 0 /4h rms , with no explicit dependence on R. the maximum force N max obtained as surfaces are brought together, because the pulloff force is not unique and depends on the peak loading pressure. However the pulloff force is always larger than N max which would move the numerical data even farther from the theoretical prediction.
It is clear that traditional theories are both quantitatively and qualitatively inconsistent with numerical solutions of the model they were derived for. The predicted pulloff force falls orders of magnitude below the numerical results, and systems with the same value of h rms /δ c have pulloff forces that vary by almost three orders of magnitude. As noted in the main text, the numerical results depend only on the rms slope and curvature which are predominantly determined by small wavelength roughness. The numerical results do not vary with the long wavelength cutoff of roughness λ L , while h rms ∼ λ H L changes more than an order of magnitude. Fuller and Tabor did not actually use their model to fit their data. They noted that the number of asperities should vary as 1/Rh rms but then dropped this dependence. Instead, they normalized by the pulloff force for smooth surfaces of the same chemistry. The data were then collapsed by fitting δ c to find an effective radius rather than obtaining R from the actual surface. The resulting radius was about 50µm, which is much larger than the smallest asperities on typical surfaces. This approach of rescaling both axes to match the theoretical prediction has been typical of subsequent work and masks quantitative errors in the theory. Note that normalizing our numerical data by the smooth surface result, wA 0 /∆r, introduces a parameter that is not present in past theories and does not improve the correlation between pulloff force and h rms /δ c .