aa r X i v : . [ c ond - m a t . m t r l - s c i ] F e b On roughness-induced adhesion enhancement
M. Ciavarella
Politecnico di BARI. Center of Excellence in Computational Mechanics. Viale Gentile182, 70126 Bari. [email protected]
Abstract
While adhesion reduction due to roughness is not surprising, roughnessinduced adhesion remained a puzzle until recently Guduru and coworkershave shown a very convincing mechanism to explain both the increase ofstrength and of toughness in a sphere with concentric single scale of wavi-ness. Kesari and coworkers have later shown some very elegant convenientasymptotic expansion of Guduru’s solution. This enhancement is very highand indeed, using Kesari’s solution, it is here shown to depend uniquely on aJohnson parameter for adhesion of a sinusoidal contact. However, counterin-tuitively , it leads to unbounded enhancement for conditions of large roughnessfor which Johnson parameter is very low. Guduru postulated that this en-hancement should occur after sufficiently large pressure has been applied toany spherical contact. Also, that although the enhancement is limited to theJKR regime of large soft materials with high adhesion, the DMT limit forthe smooth sphere is found otherwise. However, for hard materials, even theDMT limit for the smooth solids is very hard to observe, which suggest thatalso adhesion reduction is yet not well understood.The limitations of the assumption of simply connected area is here furtherdiscussed, and a well known model for hard particles in contact with roughplanes due to Rumpf is used to show that in the range where unboundedincrease is predicted, orders of magnitude reduction is instead expected forrigid solids. We suggest that Guduru’s model may be close to an upperbound to adhesion of rough bodies, while the Rumpf-Rabinovich model maybe close to a lower bound.
Keywords:
Roughness, Adhesion, Guduru’s theory, Fuller and Tabor’s theory
Preprint submitted to journal September 21, 2018 . Introduction
Fuller and Tabor (1975) were perhaps the first to measure adhesion oflow modulus materials like smooth rubber lenses against roughened surfaces,and clearly showed very small amounts of roughness amplitude (few microns)were sufficient to destroy adhesion almost completely. They then proceededto develop an asperity model, which is commonly believed not to permit en-hancement of adhesion, but instead very rapid extinction . In nature, variousmechanisms have been shown to lead to adhesion enhancements in insectswhich use adhesion for their locomotion including varying the shape of eachcontact as well as increasing the number of them, so as to obtain a benefitfrom contact splitting (Hui et al ., 2004, Kamperman et al , 2010, Gao & Yao,2004, Yao & Gao, 2006). Adhesion enhancement was measured by Briggs& Briscoe (1977), and Fuller & Roberts (1981), who were unable to explainthe data, particularly the increase of energy dissipation. Persson (2002) pos-tulated that an increase of adhesion may occur for the increase of surfacearea induced by roughness, but later on this has been shown to be not thereason for the enhancement clearly explained by Guduru and collaborators(Guduru (2007), Guduru & Bull (2007), Waters et al (2009)). Guduru’s so-lution is a very classical contact mechanics exact solution assuming a simplyconnected contact area develops in a spherical contact having a concentricaxisymmetric waviness. Clearly, as it was apparent also to Guduru (2007),there are some limitations for this solution to hold, as one expects the contactto occur only on the crests of the waviness, for ”sufficiently” large amplituderoughness and this would limit the enhancement shown in the simply con-nected contact area model. The separated contact solution is unfortunatelynot possible in closed form, and therefore it was not studied by Guduru, whonoticed however the is a large set of conditions for which we could assumeit holds. Guduru’s theory is very brilliantly described and corroborated byexperiments, and serves the main purpose of showing when the enhancementcan occur. However, we move here from a opposite perspective, trying tounderstand why the large enhancement is not commonly observed. We willtherefore discuss two main aspects of the solution: the assumption of simply In reality, reduction is assumed from the outset in the model, as Fuller and Taborpostulated that the smooth sphere case corresponded to the aligned asperity case in thenominally plane model. et al (2009) develop aMaugis-Dugdale solution with an annulus of uniform tension at the edge ofthe contact area, showing enhancement is limited to the JKR regime. How-ever, they seem to imply, at least with the limited set of parameters theystudy, the limit should be that of a DMT smooth sphere, which in the caseof a rough sphere, is in fact, although not an enhancement, still very high.We will therefore consider a limit case, that of rigid bodies in contact, show-ing that the limit is more complex than that, and permitting a very largereduction indeed. Much of the debate on adhesion of spherical bodies hasconcentrated on the transition from the DMT theory (see Greewood, 2007for a detailed treatment about this controversial theory) to the JKR limit.However, the two theories differ, for pull-off, only on a small prefactor (3/2in JKR, and 2 in DMT), whereas the more we understand about the ef-fect of roughness on adhesion, the more we are confused between orders ofmagnitude reduction (as it is commonly observed) and 1 order of magnitudeenhancement (or higher?), as Guduru’s theory shows. This note thereforeattempts to compare various results in order to hopefully arrive at a bettercomprehension between these extreme limits.In some of the developments, we use the recently developed asymptoticexpansion of Guduru’s solution developed by Kesari et al (2010,2011), findingsome reduced parameter dependence in the solution, which in fact is exactly aJKR solution, different in the loading and the unloading phase. We examinetherefore where this solution holds, therefore providing some hopefully insightin further understanding of the more general complex problem of adhesion inrough surfaces which, in view of the competition between enhancement andreduction mechanisms, is still not well understood.
2. Guduru-Kesari theory
Waters et al (2009) give a good summary of Guduru’s theory and experi-ments. They have a surface defined as f ( r ) = r R + A (cid:0) − cos πrλ (cid:1) , where R
3s the sphere radius, λ is wavelength of roughness, h is its amplitude, and A can be both positive in the case of a central convex asperity, and negative, fora central concave trough. The solution of the contact problem with adhesionis possible assuming a simply connected contact area, as a function of twoparameters α G = ARλ , β G = λ E ∗ πwR (1)where w is the surface energy, and E ∗ the plane strain elastic modulus. Theresults are given in terms of a dimensionless load b P = P/ (cid:0) πwR (cid:1) so that b P = 1 corresponds to the smooth sphere, and also for dimensionless approach b h = hR/λ , and contact radius b a = a/λ , as b P G ( b a ) = 83 β (cid:20) b a + α (cid:18) π b a π b a H (2 π b a ) − π b a H (2 π b a ) (cid:19)(cid:21) − p β b a (2) b h G ( b a ) = b a + απ b aH (2 π b a ) − pb a/β (3)Kesari et al (2010,2011) have developed a very elegant ”envelope” solutionof the Guduru problem. The envelop is obtained by joining in an asymptoticsolution for small roughness sizes, in particular λ << a : said otherwise, theremuch be enough wavelengths of roughness in the contact area — a conditionquite close to what was used in the Archard cascade process of redistributionof loads in the adhesionless contact problem (Ciavarella et al. (2000)), andwhich can be checked a posteriori. They notice that the solution has somesimplified behaviour but do not further discuss this aspect. The envelopsolution in terms of load and indentation depth is P K ( a ) = 43 R E ∗ a − a / (cid:18) √ πwE ∗ ± πE ∗ A √ λ (cid:19) (4) h K ( a ) = a R − a / r πwE ∗ ± π A √ λ ! (5)where we have grouped the term with the same power in contact radius a —the theory obviously corresponds to the known JKR theory for A √ λ = 0. An obvious remark about the equations (4,5) is that they are exactly those of the JKR theory also in the case of roughness, but with a corrected4 enhanced or reduced , respectively for unloading or loading) surface energy P ( a ) = 43 R E ∗ a − a / √ πwE ∗ (cid:18) ± √ πα KLJ (cid:19) (6) h ( a ) = a R − a / r πwE ∗ (cid:18) ± √ πα KLJ (cid:19) (7)where α KLJ = r wλπ E ∗ A (8)is the parameter Johnson (1995) introduced for the JKR adhesion of a nomi-nally flat contact having a single scale sinusoidal waviness of amplitude A andwavelength λ . We can also recast the JKR ”envelope” equations of Kesari etal (2011) in terms of only one of the Guduru dimensionless parameters, β G together with α KLJ b P K ( b a ) = 83 β G b a − pb a β G (cid:18) ± √ πα KLJ (cid:19) (9) b h K ( b a ) = b a − (cid:18) b aβ G (cid:19) / (cid:18) ± √ πα KLJ (cid:19) (10)It is therefore very interesting that the increase of adhesion should scalewith the same parameter of sinusoidal contact. It is also very interesting andunexpected a priori that an important value for this parameter in eqt. (9,10) is α KLJ = 1 √ π ≃ .
56 (11)which is exactly the value for which the sinusoid self-flattens to full contactunder no applied load. In fact, the behaviour of a sinusoidal contact shouldbe explained in few words, following Fig.1, where α KLJ is indicated as α because in general, for a multiscale roughness, this parameter can only bedefined appropriately to a single sinusoid, and in particular tends to increasefor low fractal dimensions, which is the common case observed in practice(see Afferrante et al, = Α = Α = Α = Α = - - Π a (cid:144) Λ p (cid:144) p * Fig.1. Behaviour of a nominally flat single sinusoidal contactSuppose we follow the curve corresponding to α = 0 .
3. Under zeroload, since the curve is decreasing (and hence unstable), the contact will”jump” to point B, similarly to any JKR solution. From this point on, acompressive load p , moves the system along the curve BC, until at pointC, again instability occurs and complete contact occurs. The maximummean pressure needed to establish full contact is a fraction κ of p ∗ , where p ∗ = πE ∗ A/λ is the pressure for full contact without adhesion, and reduceswith α . For α > α cr ≃ .
56, partial contact does not occur and the surfacesimmediately snap together until full contact occurs. At that point, whencontact is established, it can be maintained also for negative arbitrarily high(tensile) mean pressures, provided ¯ p ≥ ηp ∗ , where η is a negative parameter,function of α .Notice there is a discussion in Waters et al (2009), based on qualita-tive energy balance, which essentially repeats the same process in Johnson’s6inusoid α KLJ > .
56, which for Guduru’s parameter becomes gives α G < √ πβ G (12)Waters et al (2009) remark, correctly, that this condition only approximately estimates for which parameters the roughness is flattened and simply con-nected area is established spontaneously — that is, within the macroscopiccontact area.Returning to the actual Guduru model of a spherical contact with si-nusoidal waviness, it wasn’t necessary that the same parameter relative tothe waviness in Guduru’s solution appeared, and indeed, it did not appear in Guduru’s solution, but only in its envelope from Kesari’s solution. As-suming a simply connected area, the condition α KLJ > .
56 seems sufficientto guarantee that this solution is appropriate, but as we shall see, enhance-ment is this region is relatively low, and instead, surprisingly, the largestenhancement occurs for low α KLJ . We can estimate the actual ”radius ofspontaneous contact” therefore precisely from the full Guduru solution, andindeed also approximately from Kesari’s solution.It would seem that α KLJ > .
56 shows an important transition too:the loading curve envelope becomes Hertzian for this value, and ”less thanHertzian” for lower values α KLJ < .
56, which in contrast would be the val-ues where the enhancement of pull-off load would be even more than 4. Inparticular, an effective energy on loading and unloading, respectively, can bedefined as √ w eff,loading = w (cid:18) − √ πα KLJ (cid:19) ; w eff,unloading = w (cid:18) √ πα KLJ (cid:19) (13)where we have left the square root for w eff,loading to take into account thatfor α KLJ < / √ π , √ w eff,loading should be negative.Notice that for the Kesari expansion to be valid, one needs b a >> . Therefore, for the condition at pull-off to be reasonably evaluated from thisanalysis, one needs in general low values of β G <<
1. In this limit only, onecould use pull-off from the unloading curve, so the size of contact area atpull-off is b a c,lowβ G = 9 π R λ wE ∗ (cid:18) √ πα KLJ (cid:19) = 98 β G (cid:18) √ πα KLJ (cid:19) (14)7nd the actual value of pull-off is b P min ,lowβ G = − (cid:18) √ πα KLJ (cid:19) (15)Instead, the size of the contact area when the load is zero, is obtainedfrom the loading curve (only for α KLJ > / √ π ) as b a eq,lowβ G = 92 β G (cid:18) − √ πα KLJ (cid:19) (16)and this suggests an alternative map from Fig.5 of Waters et al (2009) wheredependence on α G , β G is shown. Indeed, in view of the convenience of writingKesari’s equation in terms of α KLJ , perhaps a clearer notation is to rewritethe Guduru equations in terms of β G , α KLJ instead of their original α G , β G using α G = π / α KLJ √ β G and obtaining b P G ( b a ) = 83 β G b a + (cid:20) √ β G π / α KLJ (cid:18) π b a π b a H (2 π b a ) − π b a H (2 π b a ) (cid:19)(cid:21) − p β G b a (17) b h G ( b a ) = b a + (cid:20) π / b aH (2 π b a ) α KLJ √ β G (cid:21) − s b aβ G (18)where the terms under square parentheses cause the fluctuation in Guduru’sequations, and are substituted in the envelop by the factors (cid:16) ± √ πα KLJ (cid:17) . In fig.2,3 we show some examples of load-approach, approach-area andload-area, in 2 interesting cases with low β G : where Kesari’s envelope (rep-resented also here as blue and red curves) works well, and with low andwith high α KLJ . At low α KLJ , the enhancement in pull-off is greater andthe loading curves actually fold on each other in the load-approach diagram(Fig.1a). As the Kesari’s equation predict, the spontaneous jump into con-tact practically does not exist, and therefore the large enhancement can onlybe obtained with sufficiently large pressure during the loading stage, andKesari et al (2010,2011) give also additional results on how to compute theloading and unloading curves, as well as the integral of the envelop curve tocompute energy dissipation. 8 h `- - - P ` ( a ) a ` h ` ( b ) a ` P ` ( c )Fig.2 - Load-approach (a), approach-area (b) and load-area (c), for low9 KLJ = 0 .
25 and low β G = 0 .
15. Blue and red lines are the Kesarienvelopes for unloading and loading, respectively. Notice that in this caseof α KLJ = 0 .
25 the curves on loading fold on each other.10 h `- P ` ( a ) a ` h ` ( b ) a ` P ` ( c )11ig.3 - Load-approach (a), approach-area (b) and load-area (c), for high α KLJ = 1 and low β G = 0 . α KLJ , instead, we have more noticeable spontaneous jump intocontact and in this particular case, unloading from this point already seemsto lead to a value of pull-off close to the Kesari envelope. Notice that Guduru(2007) had attempted an empirical fit for the pull-off enhancement of the type b P min = − (cid:18) Cw β (cid:19) (19)where β was observed to vary between 0.5 and 0.9. The equation above (15)seems to justify this, considering that α KLJ ∼ √ w , but does not permit abetter fit including the deviations from Kesari’s expansion at large β G .The enhancement of pull-off defined as (cid:12)(cid:12)(cid:12) b P min (cid:12)(cid:12)(cid:12) is shown in Fig.4 in theasymptotic limit at low β G i.e. from eqt.(15). It is clear that enhancementsare small when α KLJ >
10 (about 10% increase, and less), significant in α KLJ <
5, has a value of 4 at the critical value α KLJ = 0 .
56 and continues toincrease with lower α KLJ . This is significant as the factor α KLJ measures thehow sticky is the waviness in itself, and all the equations are saying is that avery sticky small amplitude waviness doesn’t add nor destroy the stickinessof the macroscopic sphere. The biggest enhancement would seem to occur,surprisingly perhaps, at low values of α KLJ , when the value of pull-off indimensional terms would seem to be P min , lim → − πwR (cid:18) √ πα KLJ (cid:19) = − π E ∗ R (cid:18) A √ λ (cid:19) (20)which no longer depends on surface energy and seems instead to be related toelastic modulus alone and geometrical quantities . This limit does not have Let us check this equation for example with atomic roughness, of both amplitude andwavelength a . As wE ∗ = l a , and for Lennard-Jones potential, l a /a = 0 . P min , lim → − π E ∗ Ra = − π wR = − wR (21)which in fact corresponds to an enhancement with respect to the smooth sphere (?) of afactor . = 100. But one could take wavelength of size a and higher amplitude, and thiswould grow even further with A without any apparent limit. Α KLJ P ` min ¤ Fig.4 - Asymptotic dependence of pull-off enhancement on α KLJ = q wλπ E ∗ A for low values of β G << α KLJ = 0 .
3. Limits on the enhancement
There are various reasons why unbounded enhancement doesn’t occur.Even those aspects which were already discussed by Guduru and coworkersare given here in more details, in the new notation permitted by Kesari’sequation, and new comparisons are added.13 .1. Deviations at large β G As we have discussed in the previous paragraph, the Kesari expansion isvalid in the limit b a >>
1, which implies in general low values of β G << . As clear in the Fig.5 which compare the actual pull-off values from the fullGuduru model with those of the asymptotic expansion, the enhancement ismuch reduced at large β G . In particular, following the various curves in Fig.5at increasing β G , the enhancement is much reduced for intermediate values of α KLJ although the low α KLJ paradoxical behaviour seems preserved. Noticewe can actually switch to reduction instead of enhancement for large β G , butthis is due, as noticed by Guduru (2007), to the fact that in the limit ofvery large wavelength (which is also large β G ), we have essentially only onecontact at separation, that obtained the sphere with just the central crest ofthe wavy surface, having a reduced equivalent radius1 R eff = 1 R + 4 π Aλ (22)If we consider the pull-off value with this reduced radius, this leads to b P min ,largeβ G = − R eff R = −
11 + 4 π α G = − π / α KLJ √ β G π / α KLJ √ β G + 4 π (23)which goes to 1 both for large β G and large α KLJ as it is clear from Fig.5.Combinations of large β G and small α KLJ shows this reduction is not toolarge. Guduru suggested there is an optimal wavelength where the enhance-ment is highest; however, this must necessarily be just a different rewordingof the dependences on the parameters. At low β G , there is so far no reallimit on the enhancement, if it wasn’t for the conditions on the followingparagraphs. 14 .1 0.2 0.5 1.0 2.0 5.0 Α KLJ P ` min ¤ Α KLJ = Β G = Β G = Β G = - K + Fig.5 - Enhancement of pull-off. Black thick line is obtained with Kesariequation on unloading, blue line with Kesari equation on loading, andred,green,cyan are obtained with the full Guduru solution with β G = 10 , , The enhancement predicted from the Kesari envelope curve on unloadingbecomes arbitrarily large at low α KLJ and indeed also Guduru (2007) inFig.11, shows values of the order of 30-40 factor of increase, with valueshigher than 15 being actually measured in his experimental validation paper(Guduru & Bull, 2007). Guduru (2007) has a very preliminary discussionabout the validity of the assumption of simply connected contact area, as forthe case of non-adhesive contact, it can be cast very clearly in terms of theparameter α G = ARλ . In particular, to have the gap function monotonicallyincreasing, it is enough to have α G = ARλ = 1 π / α KLJ √ β G < . .
12 (24)or β G > β lim = (cid:18) . α KLJ (cid:19) (25)Hence, at low α KLJ enhancement in the Guduru-Kesari model can onlyhold if β G is large, as otherwise separation in the contact may well occur.As we are interested in the range α KLJ < .
56, the restriction on the shaperequires β G > . et al (2009), where it is clear that for α G < .
12 weare generally below (cid:12)(cid:12)(cid:12) b P min (cid:12)(cid:12)(cid:12) = 4 . This is because at large β G , we are lookingat the problem of a single asperity detaching, as we have seen in point 1) ofthe discussion above, which reduced adhesion, instead of enhancing it.In Fig.6, we plot the boundary defined by β G = β lim with a red curve:notice that this corresponds to increasingly high values of β G the more wemove towards the left. Only points below this curve are ”certainly” satisfyingthe monotonicity of the punch and hence of the simply connected contact.It is clear that within this region, the enhancement is lower than 4 which isalso the highest amplification with Johnson parameter α KLJ > . Α KLJ P ` min ¤ Α KLJ = Β lim K - K + Fig.6 - The boundary of enhancement of pull-off. Black thick line isobtained with Kesari equation on unloading, in blue curve Kesari equationon loading, and red curve is β G = β lim ( α KLJ )Guduru (2007) notices that the condition α G < .
12 is too restrictive ontwo grounds: one, because of the effect of adhesion, which can only make thelikelihood of contact greater: indeed, even from the single sinusoid solutionthat we described in the first paragraph, we know that the pressure to reachfull contact decreases from the purely mechanical one p ∗ = πE ∗ A/λ andcould be defined as p ∗ ( α KLJ ), see Afferrante et al (2015). This is importantas we know from that analysis, that once in full contact, there is little chanceto return to separated contact (Johnson (1995) has to postulate a flaw atthe interface, or else there is theoretical strength as the only possible limit),16ther than from the contact edge. Second, because the monotonicity of thepunch profile is reobtained for large compressions, i.e. loading the contactsubstantially before starting the unloading. In particular, from the functionof the profile, the condition is immediately found as14 π α G > − sin (2 πr/λ )(2 πr/λ ) (26)which has a maximum giving the absolute α G < . r > r c , where one choice is r c λ = 2 πA Rλ = 2 πα G >> π · .
12 = 0 . > − sin (cid:0) π A Rλ (cid:1) . This is not further discussed in Guduru(2007), but here we will make more considerations. First of all, we shouldnotice that to have the radius of contact less than, say, 1/4 of the radius ofthe sphere, we need 2 π Aλ R < R , which gives Aλ < π = 0 .
04 (27)This together with α G = ARλ > .
12 gives Rλ > . A/λ = 3 or higher. Withrespect to α KLJ , the condition on Aλ < π leads to α KLJ > q l a πA , where l a = w/E ∗ . Since we want α KLJ < .
56, for example α KLJ = 0 .
1, thenwe need l a A < π (cid:0) . (cid:1) or A > l a . But A < λ π so that λ > l a , and therefore R >
38 454 l a , so we need big enough spheres with respect toadhesion characteristic length.Let us further estimate the pressure distribution (from Guduru (2007),under adhesionless contact only) with respect to p ∗ = πE ∗ A/λ, noting thatwith Aλ = 0 .
04 this corresponds to a quite high compressive stress, just 1/10 ofthe elastic modulus, with the warning that finite strains and other deviationsmay occur. We could convince ourselves that the waviness has been squashedout mechanically if the local pressure is positive. Now, rearranging Guduru’sequation in the form pp ∗ (cid:16) ra (cid:17) = 2 (cid:18) π α G + 2 (cid:19) aλ r − (cid:16) ra (cid:17) + Z r/a H (cid:0) π xa aλ (cid:1)q(cid:0) xa (cid:1) − (cid:0) ra (cid:1) d xa − π (cid:16) aλ (cid:17) Z r/a xa H (cid:0) π xa aλ (cid:1)q(cid:0) xa (cid:1) − (cid:0) ra (cid:1) d xa (28)17e can plot it for aλ = r c λ = 2 πα G (Fig.7a) and α G = 0 . , . , ,
5, as well asfor even higher contact radius aλ = 3 r c λ = 6 πα G .(Fig.7b). The results showthat at the very low α G = 0 .
075 (thick black line) the pressure is indeedalways compressive and this should occur for any value of the contact area.However, some tension appears for α G = 0 .
15 (thick blue line) or higher val-ues, which is where the ”local” monotonicity condition postulated by Guduru(2007) should also suggest compression always for any value of the contactarea. Only with triple radius of contact, as in Fig.7b, we do find alwayscompressive stresses. In the case of adhesion, since we are interested in thecase α KLJ < .
56, there is no guarantee therefore that the ”local” conditionof Guduru (2007) justifies the simple connected area assumption. This, to-gether with the fact that the pressure may be impractically large, suggeststhat we may not be able to observe the actual enhancement predicted by thesimply connected contact area solution.18 .2 0.4 0.6 0.8 1.0 r (cid:144) a123p (cid:144) p * Α G = Α G = Α G = Α G = Α G = ( a ) (cid:144) a12345p (cid:144) p * Α G = Α G = Α G = Α G = Α G = ( b )Fig.7 - Pressure distributions with pure mechanical contact, for α G = 0 . , . , . , . , . aλ = r c λ = 2 πα G or(b) aλ = 3 r c λ = 3 × πα G
4. Deviation from the JKR regime
The transition in adhesion from Bradley-rigid behaviour to fully elasticJKR behaviour is well known for the sphere. It depends on the well known It is more precise to say from rigid behaviour to JKR regime, as the DMT solution,commonly referred to as the limit for low Tabor parameter, has indeed various forms, andmost of them not exact. µ = σ th E ∗ (cid:18) RE ∗ πw (cid:19) / (29)where σ th is the theoretical strength and we have defined the parameter withreference to the sphere, but we could also define a value appropriate for theroughness.Anyway, the phenomenon of enhancement of toughness and strength be-cause of surface waviness has been shown to be restricted primarily to theJKR adhesion regime in Waters et al (2009), or µ >
1. This is impor-tant in view of application of the Guduru model to rough surfaces. Guduruand Kesari have considered very soft materials with very small amounts ofroughness. If we consider the more standard situation of macroscopic con-tacts with many asperities, the local behaviour of asperities will not show anyenhancement. However, in intermediate situations the situation is unclear.However, a further remark is that the limit of ”rigid” roughness is notnecessarily that of the sphere without roughness, as it seems suggested inWaters et al (2009). Indeed, first of all, we have the limit at large β G whichshould reduce the pull-off to the case of contact between the sphere andthe first crest of the waviness. Secondly, we have at our disposal a solutionfor rigid adhesion between a spherical particle and a rough plane (simplifiedwith a single small asperity, and otherwise smooth plane), due to Rumpf(1990), later modified by Rabinovich et al (2000) which is obtained applyingDerjaguin’s approximation and contains two terms: a first term representsthe interaction sphere/asperity (which increases with r ), and “noncontact”particle/flat separated by the height (radius) of the asperity (decreasing with r ) P ad P smooth = 11 + R/r + 1(1 + r/H ) (30)where H is some atomic size length scale. This equation has been used verymuch in the area of particle adhesion and powder technology, as well as drugdelivery, semiconductor fabrication, xerographic processes, and paint formu-lation or aerosol formation, amongst others) and shown to be reasonably in Rabinovicz et al (2000) modified this to take into account a one-scale roughness butthis simply changes the radius of the asperity into r = 1 . rms , but the behaviour isthe same. In the nanoscale roughness regime, a dramatic decrease in adhesion force ispredicted for this size adhering particle. r/R →
0, or very largeradius of asperity r (when the contact is essentially between the sphere anda single asperity which in this case has grown in curvature so that it is aflat plane itself), this model leads to the value for the particle alone on theflat surface P smooth which is in this case the Bradley result for the sphere,2 πRw . This is the value Waters et al (2009) seem to obtain for µ → . Themodel shows therefore with increasing r/R first a decrease of adhesion due toroughness and then, after reaching a minimum, an increase towards a lineartrend in R of the Bradley equation.In Guduru’s notation r = λ A (2 π ) and rR = 1(2 π ) α G (31)and therefore the Rumpf-Rabinovich model gives P ad P smooth = 11 + (2 π ) α G + 1 (cid:16) λ AH (2 π ) (cid:17) (32)This formula has been shown to work pretty well with nano and atomicsize roughness, with hard solids. In our case, as we are uncertain especially ofthe case α G > .
12, corresponding to rR < .
21, it is clear that the first termis small (interaction sphere asperity), of 0 .
174 and less. On the other hand,the second term giving the interaction with the smooth plane separated bythe hemiasperity is also extremely small too when r >> H (atomic size) asit is common. Suppose R = 10 H , and the Rumpf-Rabinovich model gives P ad P smooth = 11 + (2 π ) α G + 1 (cid:16) (2 π ) α G (cid:17) (33)We explore the range r/H = 1 , , i.e. α G = π ) , (2 π ) in Fig.8 below,which clearly shows that reduction of pull-off of various orders of magnitudefor α G increasing in the range where we cannot assume full contact.21
100 10 Α G - - P ad (cid:144) P smooth ( a )Fig.8 - The Rumpf-Rabinovich model as applied to the Guduru problemIt is clear that, by assuming a simply connected contact area, Waters et al (2009) found only a very limited range of possible decay of adhesion,although they should have found at least the regime at high β G where thecontact is expected between the sphere and the first crest of the sinusoidalwaviness, therefore reduced with respect to the Bradley result for the sphere.
5. Conclusions
We have revisited the Guduru model, and discussed the possible reasonsfor the limitation of the very high enhancement of adhesion found in thatmodel. In particular, we have observed a reduced dependence on the param-eters, we have identified the Johnson parameter for single sinusoidal contactto govern also the amplification of adhesion, at least in an asymptotic regimewhere the envelope solution by Kesari et al holds.Finally, we have shown that, as very large amplification is expected fromthe Guduru model, even greater reduction is expected in the separated con-tact regime, as estimated from a rigid model adhesion equation by Rumpfand Rabinovich. The latter gives perhaps the lower bound of the adhesion,as Guduru gives the upper bound.
6. References6. References