WWhen are rough surfaces sticky?
1, 2 and Mark O. Robbins Johns Hopkins University, Department of Physics and Astronomy,3400 North Charles Street, Baltimore, MD 21218, USA Fraunhofer IWM, MikroTribologie Centrum µ TC,W¨ohlerstraße 11, 79108 Freiburg, Germany
At the molecular scale there are strong attractive interactions between surfaces, yet few macroscopicsurfaces are sticky. Extensive simulations of contact by adhesive surfaces with roughness on nanometerto micrometer scales are used to determine how roughness reduces the area where atoms contact and thusweakens adhesion. The material properties, adhesive strength and roughness parameters are varied byorders of magnitude. In all cases the area of atomic contact rises linearly with load, and the prefactor riseslinearly with adhesive strength for weak interactions. Above a threshold adhesive strength, the prefactorchanges 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 ordersof magnitude in load. It relates the threshold strength to roughness and material properties, explainingwhy most macroscopic surfaces do not stick. The numerical results are qualitatively and quantitativelyinconsistent with classical theories based on the Greenwood-Williamson approach that neglect the range ofadhesion and do not include asperity interactions. a r X i v : . [ c ond - m a t . m t r l - s c i ] N ov urfaces are adhesive or “sticky” if breaking contact requires a ﬁnite force. Few of the surfaceswe encounter are sticky even though almost all are pulled together by van der Waals interactions atatomic scales. Gecko setae and engineered adhesives use this ubiquitous attraction to achievepull off forces per unit area that are orders of magnitude larger than atmospheric pressure, and ourworld 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 para-dox. Experiments show that a key factor underlying this paradox is surface roughness, whichreduces the fraction of surface atoms that are close enough to adhere.
Quantitative calcula-tions of this reduction are extremely challenging because of the complex topography of typicalsurfaces, which have bumps on top of bumps on a wide range of scales.
In many cases theycan be described as self-afﬁne fractals from a lower wavelength λ s of order nanometers to an up-per wavelength λ L in the micrometer to millimeter range. Here, we use an efﬁcient Green’sfunction approach to calculate adhesive contact of surfaces with roughness from subnanometer tomicrometer scales. Numerical results for a wide range of surfaces, adhesive interactions and ma-terial properties are presented and used to develop a simple, parameter-free equation that predictsthe effect of adhesion on contact.The traditional Greenwood-Williamson (GW) approach for calculating contact of rough sur-faces approximates their complex topography by a set of spherical asperities of radius R whoseheight distribution is determined from self-afﬁne scaling. The long-range elastic interactions be-tween different asperities are neglected. This approach is analytically tractable and provided asimple explanation for the observation that the area of contact between nonadhesive elastic sur-faces is proportional to the normal force or load pushing them together. Later generalizations considered the effect of adhesion between surfaces and found that the key parameter was theratio of the root mean squared (rms) height variation h rms to the normal displacement δ c of a sin-gle asperity due to adhesion. If the work of adhesion gained per unit area of contact is w , then δ c = (3 / R ( πw/E ∗ ) with contact modulus E ∗ = E/ (1 − ν ) for an isotropic material withYoung’s modulus E and Poisson ratio ν . GW based adhesion models predict that the forceneeded to separate surfaces drops rapidly as h rms /δ c increases and is negligible for h rms /δ c > .In the last decade, Persson has developed a scaling theory that includes an approximate treat-ment of asperity interactions. At the same time, large scale numerical calculations of contactbetween rough surfaces have become feasible.
Both approaches reveal limitations in the GWtreatment of nonadhesive surfaces. For example, the deﬁnition of R is ambiguous, the predicted2ange of linear scaling between area and load is orders of magnitude too small, and predictionsfor the geomety of individual contacts and the spatial correlations between them are qualitativelywrong. 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 rangeof surface separations over which attractive interactions are signiﬁcant, ∆ r , is zero. He hasapplied this theory to speciﬁc cases and found a reduction in adhesion with increasing h rms , butthis 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 surfaceswith roughness from subnanometer to micrometer scales. The numerical results are clearly incon-sistent with expressions based on the GW approximation. In particular, the relevant length scaledescribing the roughness is not h rms and the range of adhesive interactions determines a charac-teristic adhesive pressure w/ ∆ r that plays a critical role. Numerical results for a wide range ofsurfaces, adhesive interactions and material properties are presented and used to develop a simple,parameter-free equation that predicts the effect of adhesion on contact. RESULTS
Figure 1(a) shows the geometry of the simulations. There is a rigid upper surface with self-afﬁne roughness. The change in height h over a lateral distance x increases as x H where the Hurstexponent H is between 0 and 1. The elastic substrate is the (100) surface of an fcc crystal withatomic spacing a , and behaves like a continuous medium in the limit of large λ s /a . We identifyregions where atoms feel a repulsive force with the contact area A rep (see methods).Fig. 1(b) shows the complex spatial distribution of A rep for nonadhesive interactions. BothGW and more recent approaches predict that A rep is much smaller than the total area A and riseslinearly with the load N pushing the surfaces together. By dimensional analysis, thesurface geometry can only enter through the dimensionless rms surface slope, h (cid:48) rms = (cid:112) (cid:104)|∇ h | (cid:105) (see Fig. 1g). Steeper and stiffer surfaces are harder to bring into contact, so that NA rep = h (cid:48) rms E ∗ κ rep ≡ p rep , (1)where numerical solutions, such as the grey line in Fig. 2, ﬁnd the dimensionless constant κ rep isclose to 2 while GW and Persson give κ rep ≈ . and 1.6, respectively. Note that the ratio of loadto area represents the mean repulsive pressure p rep in contacting regions, which depends only on h (cid:48) rms and E ∗ .Figures 1(b)-(e) and 2 show how adding adhesion affects the distribution of contacting regions3nd the relation between load and A rep . There is no need to separately consider the effect of E ∗ and w because they always enter as a ratio with dimensions of length: (cid:96) a ≡ w/E ∗ . As discussedbelow, (cid:96) a /a is typically much less than unity and we use it to quantify the relative strength ofadhesion.Our ﬁrst ﬁnding is that there is always a linear relation between the total load and the area inintimate repulsive contact at low N (Fig. 2). This can be described by Eq. (1) with κ rep replacedby a renormalized constant κ . As the strength of adhesion increases, κ and the ratio of A rep toload rise. Eventually the ratio diverges and the surfaces become sticky when κ changes sign. Anegative κ leads to an elastic instability that pulls surfaces into contact and a pulloff force equal tothe 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 affectscontact geometry. Figures 1(c-e) show contacting regions (orange) that interact with repulsiveforces and attracted regions (black) that are close enough to feel adhesive forces (Fig. 1(f,g)). Thestrength of adhesion is varied at constant total repulsive area A rep . We ﬁnd that the correspondingrepulsive load N rep and mean pressure p rep also remain constant (see upper inset in Fig. 3) and thatthere are only minor morphological changes in the shape of A rep . The main change is that the totalarea feeling an attractive force, A att , spreads around the periphery of A rep as the range of adhesiveforces, ∆ 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 thenanometer and micrometer range. 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 constantmean adhesive pressure w/ ∆ r in the attractive region (see Fig. 3) and that the ratio of repulsiveand attractive areas is independent of load at low loads. The ﬁrst two observations allow us towrite the total load as N = p rep A rep − ( w/ ∆ r ) A att . From Eq. (1) we immediately ﬁnd /κ = 1 /κ rep − /κ att (2)with κ rep ≈ and κ att = h (cid:48) rms ∆ r(cid:96) a A rep A att . (3)The remaining unknown is the ratio of repulsive to attractive area.If A att (cid:28) 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 reaches4he interaction range ∆ r (Fig. 1(g)). For a general area, A rep = P d rep /π where d rep is the meandiameter (see Fig. 1(f) and Suppl. S-II). For ordinary two dimensional objects like a circle, theperimeter and diameter are proportional and increase as the square root of the area. This behavioris assumed in conventional theories of contact between rough surfaces that ignore long-range elas-tic interactions between individual contacting asperities, such as the previously discussed GW and related adhesive models. Including elastic interactions leads to qualitative changes incontact geometry. The contact area becomes a fractal with the same fractal dimension as theperimeter (a true “monster” ). We ﬁnd that d rep is then nearly independent of contact area, loadand 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 surfacesalways rises as x / . We use the standard prefactor for a cylinder with contact radius d rep / andequate d att to the lateral distance where the separation equals ∆ r (Suppl. S-I). Combined with ourexpression for P , this gives the constant ratio between attractive and repulsive areas A rep A att = d rep πd att = (cid:20) π (cid:21) / (cid:20) h (cid:48) rms d rep π ∆ r (cid:21) / . (4)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 descrip-tion of A rep /A att and κ rep /κ att for a wide range of surface geometries. Deviations are only largerthan the numerical uncertainty when the attractive area has grown too large to be approximated asa 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 / ).Eqs. (3) and (4) provide a simple and quantitative explanation for the changes in Fig. 2. Asthe adhesion energy (and therefore (cid:96) a ) increases, there is a proportional increase in /κ att . At ﬁrstadhesion merely produces a small change in the ratio of area to load. The sign of the ratio changeswhen /κ att becomes bigger than /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 theundeformed surface. As above and in Suppl. S-I, we approximate the contacting part of asperitiesby a cylinder with radius R , which is calculated from the rms curvature of the rough surface /R = h (cid:48)(cid:48) rms / (cid:112) (cid:104) ( ∇ h ) (cid:105) / . If the contact has diameter d rep and slope h (cid:48) rms at the contact edge,then d rep = 4 h (cid:48) rms /h (cid:48)(cid:48) rms . Following the same reasoning, the length in the numerator of Eq. (4) is5roportional to the height change δh from the contact edge to center: δh = [ h (cid:48) rms ] /h (cid:48)(cid:48) rms = h (cid:48) rms d rep / (5)The values of h (cid:48) rms and h (cid:48)(cid:48) rms can readily be evaluated from the statistical properties of the roughsurface in real or reciprocal space. The lower inset of Fig. 3 shows that P and A rep are proportionaland that the proportionality constant is always within a factor of two of h (cid:48) rms /h (cid:48)(cid:48) rms .The contact area is not directly accessible to experiment, but changes with load in the meanseparation between surfaces u can be measured with sufﬁciently stiff mechanical devices. Thenormal contact stiffness deﬁned as k N = dN/du is typically found to rise linearly with load fornonadhesive surfaces. In the regime where surfaces are not sticky we ﬁnd that the relationbetween 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 factorof κ rep /κ , the normal stiffness is reduced by the same factor. This is a small correction unlessthe surfaces are close to becoming sticky, and nonadhesive predictions are likely to be withinexperimental uncertainties. DISCUSSION
Surfaces are sticky when the total adhesive force, which is adhesive pressure times attractivearea, exceeds the total repulsive force p rep A rep . This corresponds to /κ att > /κ rep , and ournumerical results show stickiness if and only if this condition is met. It can be recast as a conditionon the ratios of pressures and areas, ( w/ ∆ r ) /p rep > A rep /A att , or using our analytic expressions: h (cid:48) rms ∆ rκ rep (cid:96) a (cid:20) δh ∆ r (cid:21) / < π (cid:20) (cid:21) / ≈ (6)where the ﬁrst factor on the left reﬂects the pressure ratio and the second comes from the arearatio. As noted above and in the supplementary material the effective range of the potential istypically less than but of order of the atomic bond-distance a . Any height change δh is possiblein continuum theory, but there is a natural lower bound of order a in atomic systems. For example,roughness on crystalline surfaces occurs in the form of terraces with height ∼ a . Inserting thisbound in Eq. (4) one ﬁnds a necessary but not sufﬁcient criterion for adhesion: (cid:96) a /a > ∼ . h (cid:48) rms .Note that the above prediction for the onset of adhesion is qualitatively different than previousmodels 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 beenidentiﬁed before. Continuum theories have considered the limiting cases of ∆ r equal to zero
6r inﬁnity and concluded ∆ r had little effect, while we ﬁnd more adhesion at small ∆ r becausethe adhesive pressure is increased. Finally, our relations only include quantities that are determinedby short wavelength roughness – the rms surface slope and curvature. The rms roughness is thekey surface property in past GW theories, and rises with the upper wavelength of roughness as λ HL .The numerical results with different symbol size in Fig. 4 have values of λ HL that vary by morethan 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 dataand traditional GW theories.To determine the implications of Eq. (6), we ﬁrst consider the extreme case where w is theadhesive energy for joining crystals of the same material. Then for atomistic solids the sameinteractions determine both E ∗ and w . The value of (cid:96) a is of order of the relative displacementneeded to change the elastic energy by the binding energy, and (cid:96) a /a (cid:28) . For example, diamondhas E ∗ ≈ Pa and w ≈
10 J / m , yielding (cid:96) a /a ≈ . with a the carbon bond length. Thesimple Lennard-Jones potential has (cid:96) a /a ≈ . . For these typical values of (cid:96) a , adhesion shouldoccur for h (cid:48) rms of order 0.1 and below. The stickiest cases considered in Fig. 4 (closed red symbols)are indeed for the case (cid:96) a /a = 0 . , h (cid:48) rms = 0 . , and small d rep . Increasing h (cid:48) rms to 0.3 suppressesadhesion.Exposing surfaces to the environment typically reduces the adhesive forces to van der Waalsinteractions with w ∼ mJ/m . The value of (cid:96) a and the root mean square slope needed toeliminate adhesion are reduced by two to three orders of magnitude. Only exceptionally smoothsurfaces like atomically ﬂat mica and the silicon surfaces used in wafer bonding have slopeslow enough to stick ( < ∼ − ). For most surfaces κ ≈ κ rep . This provides an explanation for thesuccess of models for friction that ignore van der Waals adhesion. Most of the sticky surfaces we are familiar with break the connection between w and E ∗ toincrease (cid:96) a /a . Geckos and recently manufactured mimics break the solid up into a hierarchicalseries of separate rods with pads at the ends. This allows adjacent pads to contact the surface atdifferent heights without a large elastic energy, leading to a small effective E ∗ even though thecomponents are stiff. Tape, rubber, and elastomers adhere via van der Waals interactions, but havesmall elastic moduli associated with the entropy required to stretch polymer segments betweenchemical crosslinks. Eq. (6) implies that surfaces with w = 50 mJ/m , h (cid:48) rms ∼ , ∆ r ∼ . and d rep ∼
10 nm will be sticky if E ∗ < ∼
10 MPa , which is common for soft rubbers and elastomerswhile paper is much stiffer ( > Taking h (cid:48) rms ∼ and ∆ r ∼ . , one ﬁnds adhesionfor d rep < ∼ µ m . Adhesives of this type can stick even to structured surfaces with macroscopicgrooves. Once an adhesive bond is formed, the viscoelastic properties of the adhesive that can beused to greatly increase the force needed to break the adhesive bond. In summary, we have presented numerical simulations of adhesive contact between rough sur-faces for a wide range of adhesion strength, surface geometries and material properties. In allcases the area in intimate repulsive contact rises linearly with the applied load at low loads. Theratio of area to load increases with adhesion and changes sign when surfaces begin to stick. Thistransition only occurs in the limits of smooth surfaces, high surface energy and low stiffness. Theresults are qualitatively inconsistent with traditional GW theories, but in quantitative agreementwith a simple parameter-free theory based on observed changes in contact geometry. This theorymakes speciﬁc predictions for experimental systems and may aid in the design of adhesives, andin engineering surface roughness to enhance or eliminate adhesion. It also provides a simple ex-planation for our everyday experience with macroscopic adhesion. For most materials the internalcohesive interactions that determine elastic stiffness are stronger than adhesive interactions andsurfaces will only stick when they are extremely smooth. Tape, geckos and other adhesives stickbecause the effect of internal bonds is diminished to make them anomalously compliant.8 ethods —
Calculations were performed for a rigid rough surface contacting a ﬂat elasticsubstrate. In continuum theory this is equivalent to contact between two rough elastic surfacesand the mapping remains approximately valid at atomic scales. Self-afﬁne rough surfaces withthe desired Hurst exponent H , h (cid:48) rms , λ s and λ L are generated using a Fourier-ﬁltering algorithmdescribed previously. Fourier components for each wavevector (cid:126)q have a random phase and anormally distributed amplitude that depends on the magnitude q . The amplitude is zero for q > π/λ s , proportional to q − − H for π/λ s > q > π/λ L , and rolls over to a constant for q < π/λ L .Periodic boundary conditions with period L are applied in the plane of the surface to preventedge effects. The elastic response is determined using a Fourier-transform technique with alinearised surface Green’s function corresponding to Poisson ratio ν = 1 / . Results are shown forperiod L = 2 λ L = 4096 a with rigid boundary conditions applied at depth L below the surface.Systematic studies were performed with L and λ L from 512 a to 8192 a to ensure that ﬁnite-sizeeffects are small.Atoms on the elastic substrate interact with the rigid rough surface through a potential thatonly depends on the height difference z . We use a - Lennard-Jones potential that represents theintegral over a half space of the usual - Lennard-Jones potential between atoms. The potentialis truncated smoothly using a cubic spline from the potential minimum at z = a to the cutoff at a + ∆ r . The potential and its ﬁrst two derivatives are continuous and vanish at the cutoff. In ourcalculations we ﬁx the stiffness k of the potential at a value that is consistent with the stiffnessof interactions within the substrate: k = E ∗ a / . Consistent results were obtained with otherpotentials, including an untruncated - Lennard-Jones potential.As is common for atomic-scale calculations, the contact area A rep is deﬁned as the areacovered by atoms that feel a repulsive force ( z < a ). Similarly, the attractive area A att is the areacovered by atoms that feel an attractive force ( a < z < a + ∆ r ). We only show results for A rep /λ s > ∼ , so that there is a statistical number of contacting asperities. Numerical values of κ , κ rep and κ att are computed at contact area from the ratios of load and area.9 cknowledgements — This work was supported by the Air Force Ofﬁce of Scientiﬁc Re-search (grant FA9550-0910232), the U.S. National Science Foundation (grant OCI-108849, DMR-1006805, CMMI-0923018), the Simons Foundation (M.O.R.) and the European Commission(Marie-Curie IOF 272619 for L.P.). Computations were carried out at Johns Hopkins Universityand the J¨ulich Supercomputing Center. 10
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Phys Rev B a)(c) (d) (e) ℓ a / a = 0 ℓ a / a = 0.01 ℓ a / a = 0.02 d rep d att Δ r (g)(f) d rep d att d att (b) | ∇ h | FIG. 1:
Contact geometry. (a)
The rigid top surface is a self-afﬁne fractal with Hurst exponent H = 0 . ,root mean square slope h (cid:48) rms = 0 . , lower wavelength cutoff λ s = 32 a and upper wavelength cutoff λ L = 2048 a . The elastic substrate is initially ﬂat with atoms spaced by a . Substrate deformationsproduced by a typical adhesive contact are shown. All height variations are magniﬁed to show on the scaleof the ﬁgure. (b-e) Atoms that feel repulsive (orange) and attractive (black) forces at a ﬁxed A rep ∼ . A .Nonadhesive results for the entire system are shown in (b) and expanded views of the region indicated by asquare are shown in (c-e) for the indicated w/E ∗ a = (cid:96) a /a . (f) Magniﬁed view of an a wide regionof (e). The mean diameter d rep is obtained by averaging the distance across A rep . (g) Vertical slice througha contact patch showing the rigid rough (gray) surface and the deformed elastic substrate. The root meansquare slope, h (cid:48) rms , is the rms average of the local slope, |∇ h | , as indicated on the right. The attractivelength d att is the distance from the repulsive patch edge at which the gap equals the interaction range ∆ r .The surface separation rises as a 3/2 power law for nonadhesive surfaces, leading to the 2/3 exponent inEq. (4). .01 0.00 0.01 0.02 0.03 N/h rms E ∗ A A r e p / A sticky < n o n a d h e s i v e rep ‘ a / a = . ‘ a / a = . ‘ a / a = . ‘ a / a = . FIG. 2:
Contact area as a function of normal load.
For nonadhesive contact, the contact area A rep rises approximately linearly with load N with dimensionless prefactor κ rep ≈ (Eq. 1). As the strengthof adhesion (cid:96) a = w/E ∗ increases, the area rises more rapidly with load. The initial ratio of area to loadcorresponds to a renormalized κ that diverges at the onset of stickiness (green line). The red line showsa sticky case where κ < and the pull-off force is nonzero. Results shown are for a surface with Hurstexponent H = 0 . , root mean square slope h (cid:48) rms = 0 . , and lower wavelength cutoff λ s = 32 a . -6 -5 -4 -3 -2 -1 A att /A -6 -5 -4 -3 -2 -1 ∆ r N a tt / ( w A ) -4 -3 -2 -1 A rep /A -4 -3 -2 -1 N r e p / h r m s E ∗ -4 -3 -2 -1 A rep /A -4 -3 -2 -1 h r m s h r m s P π A FIG. 3:
Test of predictions for N rep and N att . The attractive load N att is equal to the attractive area A att times the mean adhesive pressure w/ ∆ r . Upper inset: The repulsive load N rep is equal to the repulsivearea A rep times the mean repulsive pressure p rep ≡ h (cid:48) rms E ∗ /κ rep where κ rep remains close to 2 even forthe adhesive case with ﬁnite (cid:96) a . Lower inset: The length of the perimeter P of the repulsive area A rep isproportional to the area itself. Plot is normalized to show that d rep ≡ πA rep /P is nearly independent of A rep and d rep ≈ h (cid:48) rms /h (cid:48)(cid:48) rms . Deviations by up to a factor of 2 from this expression for d rep are responsible forthe spread in the ﬁgure. For a given system, changes in d rep with A rep are less than 25% over 2-3 decades in A rep . All plots show multiple contact areas for each realization of a surface. Results are shown for H = 0 . (triangles), 0.5 (squares) and 0.8 (circles) and (cid:96) a /a = 0 . (black), 0.005 (blue) and 0.05 (red). Closedand open symbols are for h (cid:48) rms = 0 . and 0.3, respectively. The symbol size increases as λ s /a increasesfrom 4 to 64 in powers of 2. -3 -2 -1 ‘ a h rms ∆ r h π i / h π ∆ rh rms d rep i / -3 -2 -1 / a tt sticky -1 h π i / h h rms d rep π ∆ r i / A r e p / A a tt FIG. 4:
Test of predictions for κ att and A att . The attractive /κ att is plotted against the prediction inEq. (3). The solid line has unit slope. Results are shown for Hurst exponents H = 0 . (triangles), 0.5(squares) and 0.8 (circles) and adhesive strengths (cid:96) a /a = 0 . (black), 0.005 (blue) and 0.05 (red).Closed and open symbols are for root mean square slopes h (cid:48) rms = 0 . and 0.3, respectively. The symbol sizeincreases as the lower wavelength cutoff λ s /a increases from 4 to 64 in powers of 2. Adhesion is observedif and only if /κ att > /κ rep . The shaded region shows the prediction for adhesion using the value κ rep = 2 found in the continuum limit (Suppl. S-2). Inset: Plot of the ratio of repulsive to attractive area A rep /A att against the prediction of Eq. (4). upplementary Material for“When are rough surfaces sticky?” Lars Pastewka , and Mark O. Robbins Johns Hopkins University, Dept. Physics and Astronomy, Baltimore, MD 21218, USA Fraunhofer IWM, MikroTribologie Centrum µ TC, 79108 Freiburg, Germany
S-I. SURFACE SEPARATION, d att , AND THE EFFECTIVE RANGE ∆ r FOR ARBITRARY IN-TERACTION POTENTIALS
As discussed in the main text, we use standard results from continuum theory for nonadhesivecontact between smooth surfaces. If x is the lateral distance from the edge of the contact, thenfor small x the separation ∆( x ) between surfaces always rises as x / outside the contact. Theprefactor rises with the surface slope at the edge of the contact which we take to be h (cid:48) rms . Forsimple geometries like spheres, cones or cylinders, the only length scale that enters is the radius ofthe contact area. Since the contacting region in our numerical simulations has a constant averagediameter d rep , we use the standard prefactor for a cylinder: x ) d rep = √ h (cid:48) rms (cid:18) xd rep (cid:19) / (S-1)Our numerical data for the average surface separation at a given distance from the perimeter areconsistent with this relation without adjustable parameters. To ﬁnd d att we just equate ∆( d att ) to ∆ r in Eq. (S-1), yielding d att /d rep = (cid:18)
34 ∆ rh (cid:48) rms d rep (cid:19) / . (S-2)An effective range of interaction ∆ r can be deﬁned for arbitrary forms of the interaction po-tential using Eq. (S-1). We deﬁne p (∆) as the attractive pressure between surfaces separated by ∆ > . Then w = ∞ (cid:82) d∆ p (∆) where w is the work of adhesion. As in the calculation of A att in themain document, we assume that the perimeter changes direction slowly enough that we can writethe total load as the perimeter times a contribution per unit length. With ∆( x ) being the separation18t distance x from the contact edge, this yields: N att = P ∞ (cid:90) d x p (∆( x )) . (S-3)This is equivalent to the expressions in the main text for truncated potentials with d att / ∆ r = ∞ (cid:90) d x p (∆( x )) /w . (S-4)From Eqs. (S-1), (S-2) and (S-4) one ﬁnds an expression for ∆ r (∆ r ) − / = 23 ∞ (cid:90) d∆ ∆ − / p (∆) (cid:44) ∞ (cid:90) d∆ p (∆) . (S-5)With this value of ∆ r , all of the relations in the main text for truncated potentials carry over to anarbitrary potential. Note that the integrals are well deﬁned because p represents the total force andgoes 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 that minimizes the energy. For example, if the 9-3 Lennard-Jones potential is used to inﬁnitedistances, one ﬁnds ∆ r = 1 . a . For the 12-6 Lennard-Jones potential, ∆ r = 0 . a . Note thatthese ranges are referenced to the potential minimum at a and that most of the binding energycomes from these short scales ( > %).The spline potential used in the calculations reported in the main text has p ( z ) = ka a (cid:104)(cid:0) a z (cid:1) − (cid:0) a z (cid:1) (cid:105) if z ≤ a − ( z − a ) + r ( z − a ) − r ( z − a ) if a < z < a + ∆ r if z > a + ∆ r (S-6)where a is the minimum of the potential and k = E ∗ a / is the stiffness at z = a . The adhesionenergy was varied by changing ∆ r at ﬁxed k . The range given by Eq. (S-5) for this potential isessentially the same as the actual range (within 6%). For a general truncated potential, quantitativeagreement 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 . Maugis has found that the DMTapproximation is accurate for spheres when a dimensionless ratio λ Maugis is small: λ Maugis ≡ (cid:20) R(cid:96) a π (∆ r ) (cid:21) / . (S-7)19s above and in the main text, we determine R = d rep / h (cid:48) rms by assuming a locally cylindricalgeometry with contact diameter d rep and slope h (cid:48) rms at the edge of the contact. We ﬁnd that theapproximations used in the main text are accurate even when λ Maugis exceeds unity. Using typicalvalues of (cid:96) a ∼ · − for atomistic solids exposed to the environment, h (cid:48) rms = 0 . and ∆ r/a = 1 ,this corresponds to d rep of close to a millimeter. For elastomers and other compliant systems,surface deformation becomes important at smaller scales, but the surfaces are usually sticky andother corrections are also required. For example, A att is typically not small compared to A rep .Previous analytic studies of rough surface adhesion have considered the Johnson-Kendall-Roberts(JKR) limit where the area outside the contact does not contribute signiﬁcantly to the adhesion. This corresponds to λ Maugis > where the contact radius becomes of macroscopic dimensions.Finally, many of the above relations assume that the surface slope is relatively small so thatthe total and projected areas are nearly equal and the potential interaction depends only on thevertical separation. The strains induced by contact are of order the surface slope and plasticityalso becomes important for very steep surfaces. S-II. DETERMINING CONTACT PATCH GEOMETRIES
For continuous curves, the perimeter and area are related through d rep , the mean length ofcontiguous segments in horizontal (or vertical) slices through A rep (see Fig. 1(f) in main text).Suppose the slices are made at a spacing dz that can be made arbitrarily small. The total area canbe approximated by the sum over all contiguous segments of the segment length times dz . If thetotal number of segments is N tot , then A rep = N tot d rep dz . Each end of a segment contributes dz tothe projection of the perimeter perpendicular to the slice. The projected perimeter length is then N tot dz . The total perimeter length is πN tot dz if one assumes that all orientations are sampledequally – as for a circle. This gives the relation A rep = P d rep /π cited in the text.Figure S-1 illustrates how area and perimeter are deﬁned on the discrete geometry used in oursimulations. Atoms on the substrate surface form a square grid with spacing a . The surface isdivided into a grid of square cells centered on each atom. A grid cell contributes a to A rep if thecorresponding atom feels a repulsive force and to A att if the force is attractive.Grid cells are deﬁned to be neighbors if they share an edge (Fig. S-1a). The correspondingatoms are then nearest neighbors. The repulsive area is divided into connected patches like thatshown in Fig. S-1b. Grid cells (atoms) that belong to a patch but have less than four neighbors in20he patch are part of the perimeter. The perimeter length P is calculated as: P = βa N P where N P is the number of perimeter cells and β corrects for the discreteness of the lattice. Consider a line oflength L at an angle θ to the horizontal axis. The perimeter cells will form a stepped approximationto this line. It is easy to show that the number of perimeter cells is equal to L/ ( a cos θ ) for θ between − π/ and π/ . Counting these cells and multiplying by a underestimates P by a factorof / cos θ . We ﬁnd β = 4 sinh − (1) /π ≈ . by assuming isotropy and averaging over angles.Figure S-2 tests the above relations by plotting the predicted ratio of perimeter to area as afunction of d rep for the full range of H , λ s and w discussed in the main text. Agreement is excel-lent in the continuum limit. For d rep /a > results are within the numerical uncertainty. Thedeviation at the greatest d rep is reduced if λ L /d rep is increased to remove ﬁnite-size effects. As d rep /a decreases below 10, there is an accelerating drop in the plotted ratio. The above relationsassumed lines were straight, and curvature can not be ignored when the radius of curvature of theperimeter, ∼ d rep / , is comparable to the grid spacing, a . S-III. COMPARISON TO THEORIES BASED ON THE GREENWOOD-WILLIAMSON AP-PROXIMATION
As noted in the main text, traditional theories for the effect of adhesion start from theGreenwood-Williamson approximation. Fuller and Tabor found the pulloff force needed to sep-arate the surfaces in the JKR limit of short range potentials ( ∆ r → ) and Maugis found similarresults for the opposite DMT limit of long range potentials ( ∆ r → ∞ ). In both cases, the pulloffforce is a function of h rms /δ c , where δ c is the normal displacement of a single asperity due to ad-hesion δ c = (3 / R ( πw/E ∗ ) . The pulloff force drops rapidly for h rms /δ c > and is extremelysmall for h rms /δ c > . In contrast to our results, no clear transition to nonadhesive behavior witharea 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 = 3 πwR/ the pulloff force for each inthe JKR limit. From statistical studies of rough surfaces, N ∼ . A /Rh rms . This gives N P c ∼ wA / h rms , with no explicit dependence on R .Figure S-3 shows our results for the adhesive force as a function of h rms /δ c and the predictionof Fuller and Tabor, which is very close to the expressions obtained by Maugis. Note that we plotthe maximum force N max obtained as surfaces are brought together, because the pulloff force is21ot unique and depends on the peak loading pressure. However the pulloff force is always largerthan 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 withnumerical solutions of the model they were derived for. The predicted pulloff force falls orders ofmagnitude below the numerical results, and systems with the same value of h rms /δ c have pulloffforces that vary by almost three orders of magnitude. As noted in the main text, the numericalresults depend only on the rms slope and curvature which are predominantly determined by smallwavelength roughness. The numerical results do not vary with the long wavelength cutoff ofroughness λ L , while h rms ∼ λ HL changes more than an order of magnitude.Fuller and Tabor did not actually use their model to ﬁt their data. They noted that the numberof asperities should vary as /Rh rms but then dropped this dependence. Instead, they normalizedby the pulloff force for smooth surfaces of the same chemistry. The data were then collapsed byﬁtting δ c to ﬁnd an effective radius rather than obtaining R from the actual surface. The resultingradius was about 50 µ m, which is much larger than the smallest asperities on typical surfaces. Thisapproach of rescaling both axes to match the theoretical prediction has been typical of subsequentwork and masks quantitative errors in the theory. Note that normalizing our numerical data bythe smooth surface result, wA / ∆ r , introduces a parameter that is not present in past theories anddoes not improve the correlation between pulloff force and h rms /δ c . Johnson, K. L.
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Tribol. Int. , 213 (1974). a) (b) FIG. S-1:
Identifying the contact patch perimeter. (a)
Grid cells are neighbors when they share an edge.Gray squares are neighbors of the black square in the middle. (b)
A patch showing perimeter cells with lessthan four neighbors in gray and interior cells with four neighbors in black. d rep /a P d r e p / π A r e p FIG. S-2:
Test of relationship between perimeter P , mean diameter d rep and area A rep . The ratio of
P d rep /π to repulsive area as a function of the mean width of contacting regions d rep . Results are shownfor H = 0 . (triangles), 0.5 (squares) and 0.8 (circles) and (cid:96) a /a = 0 . (black), 0.005 (blue) and 0.05(red). Closed and open symbols are for h (cid:48) rms = 0 . and 0.3, respectively. The symbol size increases as λ s /a increases from 4 to 64 in powers of 2. h rms /δ c -3 -2 -1 h r m s N m a x / ( w A ) FIG. S-3:
Comparison with Greenwood-Williamson-type theories.
Maximum force N max upon ap-proach of the two surfaces plotted against the adhesion parameter h rms /δ c with δ c = 3 / π w R/ ( E ∗ ) ) / identiﬁed by Fuller & Tabor. The force is plotted in units of
N P c where P c = 3 πwR/ is the Johnson-Kendall-Roberts pulloff force for a single sphere and N is the number of asperities. We use therms curvature to express /R = h (cid:48)(cid:48) rms / and additionally make use the frequently quoted relationship Rh rms N/A = 0 . . Clearly, there is little correlation between adhesion parameter and maximum force.Results are shown for H = 0 . (triangles), 0.5 (squares) and 0.8 (circles) and (cid:96) a /a = 0 . (blue) and0.05 (red). Closed and open symbols are for h (cid:48) rms = 0 . and 0.3, respectively. The symbol size increases as λ s /a increases from 4 to 64 in powers of 2.increases from 4 to 64 in powers of 2.