A note on the effect of surface topography on adhesion of hard elastic rough bodies with low surface energy

Abstract Adhesion between bodies is strongly influenced by surface roughness. In this note, we try to clarify how the statistical properties of the contacting surfaces affect the adhesion under the assumption of long-range adhesive interactions. Specifically, we show that the adhesive interactions are influenced only by the roughness amplitude hrms, while the rms surface gradient h0rmsonly affects the non-adhesive contact force. This is a remarkable result if one takes into account the intrinsic difficulty in defining h r m s ′ . $h_{\mathrm{rms}}^{^{\prime }}.$ Results are also corroborated by a comparison with self-consistent numerical calculations.


Introduction
Adhesion of surfaces is a widely investigated problem and is of fundamental importance in many elds of science, like biology [1], medicine [2,3], and engineering [4]. The adhesion between rough surfaces is of practical interest both for elastic [5] and viscoelastic bodies [6,7], as roughness alters the e ective surface energy of contacting bodies.
For this reason, several researchers investigated the problem from both the experimental and theoretical point of views. Fuller and Tabor (FT), for example, elaborated the rst model aimed at explaining the e ect of rough-*Corresponding Author: Luciano A errante: Department of Mechanics, Mathematics and Management, Politecnico of Bari, V.le Japigia, 182, 70126, Bari, Italy, E-mail: luciano.a errante@poliba.it Guido Violano, Giuseppe Demelio: Department of Mechanics, Mathematics and Management, Politecnico of Bari, V.le Japigia, 182, 70126, Bari, Italy ness on adhesion [8]. They extended the Greenwood-Williamson (GW) asperity model [9] to the case of adhesion described with the Johnson, Kendall and Robert (JKR) theory [10]. They found a surprising agreement with experiments, which showed roughness destroys adhesion reducing pull-o force. Moreover, they found that the pull-o force depends on a single parameter, "which may be regarded as representing the statistically averaged competition between the compressive forces exerted by the higher asperities trying to prise the surfaces apart and the adhesive forces between the lower asperities trying to hold the surfaces together" (Ref. [11]). However, this picture of adhesion is valid only when roughness has a single length scale, but roughness usually occurs on many di erent length scales. Moreover, the formalism used by Fuller and Tabor is ne if the area of real contact (and the adhesion force) is very small. For this reason, the FT theory received several criticisms [12,13]. In particular, Pastewka and Robbins [13], comparing their fully numerical predictions of pull-o force with the Fuller and Tabor ones, found pullo data very far from the FT predictions. However, such large deviation was partly due to e ects of truncation in the tails of the heights distribution of their surfaces [14].
Persson with a completely di erent methodology based on a multiscale approach [15], found that small quantities of roughness can induce an increase in the effective interfacial energy as a result of the increase in the surface area. However, such e ect is not observed when the adhesive contact of hard solids is investigated [16]. In this case and at su ciently small wavelength, the JKR approach may become inappropriate, since the amplitude g of the sinusoid is comparable with the length scale ε of the Lennard-Jones force law, and attractive tractions in the separation regions will then have a signi cant e ect [17,18]. In this limit, a DMT-type solution [19] may be preferred to a JKR one, as remarked already in Ref. [16,20,21].
The above picture witnesses the existence of an open debate in the scienti c community on which e ects roughness induces on adhesion of elastic surfaces.
In the present note, with the aid of an advanced multiasperity model [20,22], we try to shed light on this problem investigating the in uence that some important roughness parameters have on the adhesion and, in particular, on the pull-o force. Our results do not give a de nitive response to the initial question, but they put the attention on the effect that the mean square roughness amplitude hrms and gradient h rms have on adhesion in a precise limit: the contact of hard solids with long-range adhesion interactions, where DMT-type models are known works quite well.

On the e ect of mean square roughness amplitude h rms and gradient h rms on pull-o force
Results are obtained with the Interacting and Coalescing Hertzian Asperities (ICHA) model [22,23] where adhesion is modeled as suggested in Ref. [20]. For details of the formulation the reader is referred to these works. Here, we brie y recall that solution is obtained solving rst the adhesiveless contact problem under the action of the force F . Then, according to the DMT hypothesis for which adhesive interactions do not alter the deformation of the bodies and act only outside the contact area, the e ective contact force F N producing the contact area A is calculated as difference between the non-adhesive (or repulsive) force F and the adhesive one F ad where Anc is the non-contact area and pa(u) is the adhesive force per unit area, whose value depends on the separation u between bodies, according to the equation (see Ref. [16,19]) where w is the work of adhesion and dc is the range of attractive forces of the order of the interatomic distance. Notice, the adhesive force F ad = Anc d x pa [u (x)] can be alternatively calculated as (see Ref. [16]) where P (u) denotes the interfacial gap probability distribution, which is calculated by the solution of the adhesiveless contact problem, and A is the nominal contact area.
Calculations are performed on self-a ne fractal surfaces with power spectral density (PSD) assumed in a power law relation with the wave vector q = (qx , qy), with a constant value in a low wavenumber roll-o region and zero otherwise. Surfaces are numerically generated using the spectral methodology developed in Ref. [24,25], and using q L = . · m − , q = q L , and q = Nq , being N the number of scales. Two sets of simulations are considered. In the rst one, surfaces have been generated by keeping constant the root mean square roughness amplitude hrms; in the second one, instead, we xed the root mean square gradient h rms . Adhesion energy w, composite elastic modulus E * and interatomic bond distance dc have been assumed equal to . J/m , . · GPa and nm, respectively. Fig. 1a shows the normalized contact area A/A as a function of the dimensionless pressuresF N = F N /(A E * ), F = F /(A E * ), andF ad = F ad /(A E * ) for di erent values of the rms gradient h rms and xed hrms = . nm. Fig.  1b shows, instead, the same type of plot for xed h rms = .
and various values of hrms. As expected, the dependence of the contact area on the repulsive load F is practically linear and it is affected by h rms according to the known relation A/A F / A E * h rms , i.e., at xed F , the relative contact area decreases as h rms increases. On the contrary, the surface rms gradient has no e ect on the relation between contact area and adhesive force F ad . Indeed, in such case, all curves collapse in a single one. As a result, the dependence of the curves on h rms observed in Fig. 1a is exclusively due to the contribution of the repulsive interactions.
For the same reasons, Fig. 1b shows that the contribution of F is not a ected by hrms and just a little increase in rms roughness amplitude is enough to strongly reduce the adhesion force.
Medina and Dini [26], investigating the adhesive contact between a rough elastic sphere and a rigid half-space, found that a very modest contact hysteresis appears for small roughness in the range where the DMT approach is widely believed to be valid. For this reason, in the framework of our model, it is reasonable neglecting hysteresis loss and assuming no change in the area-load curves during the loading and unloading phases. In micro-and nano-devices, for example, stickiness of contacting surfaces may represent an important problem and it can occur even when adhesive hysteresis is missing.
Under such hypothesis, the pull-o force, i.e., the force required to completely detach the surfaces, is the , . , . , .
lowest negative force value of the area-load curve; its dependence on the surface statistical parameters is shown in Fig. 2a. Data show a negligible in uence of h rms on the sticky behavior of the surface. On the contrary, the pullo force is strongly a ected by the average roughness in agreement with results of Fig. 1b, where one can observe that variations in hrms directly a ect the attractive forces. For hrms = . nm, we have also plotted data of fully numerical GFMD calculations given in Ref. [16] for a validation of our results. The agreement is quite good, and Fig.  Figure 19 by Ref. [16].
2b further shows that such agreement concerns the whole curve relating the contact area and the applied load.
There is an intrinsic di culty in de ning the rms gradient, both on theoretical and practical levels. Indeed, the rms gradient is related to short wavelength components of the PSD spectrum. In particular, the value of h rms depends on the high cut-o frequency at which the truncation of the PSD spectrum is xed. Real surfaces present very broad spectra, and roughness is characterized by several wavelengths from nano to micro scales. The choice of the cut-o frequency represents a critical step in modelling rough contacts. However, Solhjoo and Vakis [27] suggested that the limit of the high cut-o frequency can be identi ed via the PSD of relaxed atomic structures. They found q ≈ . nm − . Lorenz et al. [28] suggested that the truncation should occur where the rms gradient reaches h rms (q ) = . , although there are no data available to interpret the generality of this recommendation. Other authors [29] suggested many factors could be associated to the truncation cut-o , including small dirt particles or rubber wear particles. From an experimental point of view, in practical cases, measurements of surface local gradient are closely related to instruments sensitivity [30]. Therefore, in view of this di culty, the above results, showing independence of the pull-o force from the short wavelengths, are remarkable and are also in agreement with recent ndings of Joe, Thouless and Barber [31] that showed the adhesive behavior of self-a ne fractal surfaces is not in uenced by the high frequency cut-o (and hence by the smallest roughness structures).

Conclusions
In this work, we have shown that adhesive forces are inuenced only by the rms roughness amplitude hrms, while repulsive interactions depend on the rms surface gradient h rms . As a result, we have also found that the pull-o force is almost independent of h rms in agreement with recent ndings of other works of the literature. The present results apply in the limit of hard solids with long-range adhesive interactions and, consequently, the debate on which geometrical parameters a ect the stickiness of randomly rough surfaces remains open.