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# On Pastewka and Robbins' Criterion for Macroscopic Adhesion of Rough Surfaces

On Pastewka and Robbins' Criterion for Macroscopic Adhesion of Rough Surfaces

M. Ciavarella

[+] Author and Article Information

J. Tribol 139(3), 031404 (Nov 30, 2016) (5 pages)

Paper No: TRIB-16-1057; doi: 10.1115/1.4034530

History: Received February 15, 2016; Revised July 18, 2016

ARTICLE

REFERENCES

Abstract

Abstract | Introduction | A Simple Asperity Model | Pull-Off | Discussion | Conclusion | References

Pastewka and Robbins (2014, “Contact Between Rough Surfaces and a Criterion for Macroscopic Adhesion,” Proc. Natl. Acad. Sci., 111(9), pp. 3298–3303) recently have proposed a criterion to distinguish when two surfaces will stick together or not and suggested that it shows quantitative and qualitative large conflicts with asperity theories. However, a comparison with asperity theories is not really attempted, except in pull-off data which show finite pull-off values in cases where both their own criterion and an asperity based one seem to suggest nonstickiness, and the results are in these respects inconclusive. Here, we find that their criterion corresponds very closely to an asperity model one (provided we use their very simplified form of the Derjaguin–Muller–Toporov (DMT) adhesion regime which introduces a dependence on the range of attractive forces) when bandwidth α is small, but otherwise involves a root-mean-square (RMS) amplitude of roughness reduced by a factor α−−√α. Therefore, it implies that the stickiness of any rough surface is the same as that of the surface where practically all the wavelength components of roughness are removed except the very fine ones.

http://tribology.asmedigitalcollection.asme.org/article.aspx?articleid=2...

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## Comments

## the story continues with some papers I have submitted

so if anyone is interested, I can discuss privately the developments.........

mike

## here for example some comparison with experiments

Despite these days it is impossible to talk of GW models because the "fractal" community suggests interaction and multiscale effects are completely wrong in GW, here are some GW models with adhesion which seem to work against experiments, showing the main effect of rms roughness, contrary to what the PR model says in the rush to forget all GW, that stickiness depends only on slopes and curvature https://www.researchgate.net/profile/Attilio_Frangi/publication/28805652...

## the deviation from GW model cannot be all due to scatter

In this new paper accepted in Trib Int we show that the deviation from PR to GW cannot be all attributed to scatter due to imperfect tails of Gaussian surfaces as easily provoqued by low fractal dimension with insufficient size of the domain

https://www.researchgate.net/publication/312153253_Adhesion_between_self...

## Pure coincidence DMT approx work in Pastewka-Robbins model?

This paper may help shed some ligth why the very numerous crude approximations in PR model may have coincidentally led to reasonable agreement in their set of parameters. But not for any extrapolation fundamental purpose! The paper is submitted but probably siib accepted, as reviewers were positive.

1. arXiv:1701.04300 [pdf, ps, other]

On the use of DMT approximations in adhesive contacts, with remarks on random rough contactsMichele CiavarellaComments: 11 pages, 5 figuresSubjects: Materials Science (cond-mat.mtrl-sci)

## Spectral Moments of self-affine fractals

Sir, when we are generating a self-affine fractal rough surface using the Power Spectral Density, does the Longuet-Higgins theory that "by counting the zeros and the extrema we can estimate the moments m2 and m4" still apply? Thank you.

## question on PSD

Sir, I don't understand your question. If you know PSD, you can integrate it to get the moments. Longuet-Higgins is a theory to study the maxima, minima, zero-crossings, no need to use that to compute m2 and m4. Please be precise.

## Sir, Thank you so much for

Sir, Thank you so much for the prompt reply.

I meant let's say,

a) we generate an isotropic rough surface using PSD and get the moments m2 and m4 by integrating the PSD.

b) Then after generating the rough surface, if we take an arbitrary profile of the surface and count the number of zeros and extrema and using the method mentioned in Longuet-Higgins' paper on isotropic rough surfaces caclulate the moments m2 and m4.

Will the values of m2 and m4 obtained by methods a) and b) match? Thank you.

## this is indeed a more subtle point

The relation between profile and surface is not that obvious. I suspect many papers around make some mistake (possibly even Pastewka-Robbins one on which this post started!). In short, Nayak paper of 1971 explains it all, although it may not be the easiest. There is no problem with m0, but there is with m2 --- the slope has a factor 2 difference, because slope in x direction and slòpe in y-direction are independent uncorrelated processes see

Some observations on Persson's diffusion theory of elastic contact

## Also a query on insistence on Gaussianity of height distribution

Thank you for the insight and the link. I was always confused by the interchanging use of Surface PSD and Profile PSD in many of the papers. This mostly clears up that aspect. Nayak's paper is indeed a bit tough to understand.

Why do most of the papers insist on the Gaussianity of height distribution? Like in this paper by Yastrebov et al. they have a detailed discussion on how the lower and upper cut-off wavenumbers affect the Gaussianity of the surface. Is it because all the major asperity contact theories and Persson's theory consider the Gaussian rough surfaces or is their a practical aspect to this as well? Thank you.

## Gaussianity

The reason to insist on gaussianity is that the maths is much simpler. Most people are use to measure roughness assuming it is gaussian, and do not know much statistics.

In general, gaussian is the very reason of success of Greenwood-Williamson model: quite a few models before that assumed unrealistic asperity distributions, including a russian one assuming uniform distribution of which I forget the name. GW paper itself doesn't show a very good evidence of gaussianity of experimental surfaces, except perhaps the top of the distribution. This is discussed further in a recent paper of mine

On the effect of wear on

asperityheight distributions, and the corresponding effect in the mechanical responseOn the significance ofasperitymodels predictions of rough contact with respect to recent alternative theoriesWhen the random processes appeared on the scene, the need to assume gaussianity became even more a condition for much easier mathematical development of Nayak and Longuet-Higgins. Again, this is vaguely based, as any gaussian distribution, on the Central Limit Theorem, which applies with a large number of independent process

of about the same variance.This last condition is not really easy to obtain from a fractal. In a typical fractal, you have essentially a Fourier series whose terms have different "size", and therefore CLT does not apply. Persson has made his entire career on the gaussian assumption because his model strongly assumes gaussianity. Notice that the original derivation of Persson's theory is very involute and takes 50 pages of physics journals, whereas I obtain it in two steps from a more mechanical procedure in

Rough contacts near full contact with a very simple

asperitymodelYastrebov is right to show that to obtain gaussian fractals, you have to be very careful. Persson and co. have suggested "roll-off" component of PSD to increase the number of "nearly equal" component and get closer to CLT. But this is really a distraction and very difficult to understand if real surfaces really have a roll-off or not. Too much roll-off, and you have no longer a fractal! So you do need a very large window rather than roll-off, in your random surface, i.e. much larger than lower cutoff in wavenumber, and you should have a good gaussian surface.

But the question remains: are real surfaces gaussian? Some people now are starting to general Weibull fractals using RMD. I will explain later how to do that. This is mainly because of my papers above, especially in adhesion, which question this fuss about gaussianity.

Final remark: the fact that you have a power-law PSD does not imply you have a fractal. It may simply be a square-form signal whose Fourier decomposition gives a invese cubic power-law! You would beed to check the phases between different components, or higher-order autocorrelation functions.

Anyway, the sad part of this huge literature, it that it is really academic. The main point of GW is essentially showing the linearity of real contact area and load due to the fact that the number of contact spots increases with load in a way that both area and load grow proportionally. What Persson's theory found was generally quite academic improvement.

Is it a secret to explain what you are planning to do?

## Gaussianity

Thank you sir, this addresses many of my doubts and as it happens my doubt on gaussianity, partly, was a result of thoughts on the first two papers you mentioned here. Correct me if I'm wrong, but from what I've seen from most of the papers that they address only the gaussianity because they are more concerned about the "qualitative" behaviour of the load-separtion & load-area relationships.

As I am new to this I am currently starting with a 1D roughness which is equivalent to the 2D isotropic rough as in papers by Popov and also in this 2013 paper by Scaraggi et al and analyse it's contact with a plane surface using FEM. So, the questions on finding m2 and gaussianity. Thank you.

## FEM is not very efficient, unless you want to deal with plastic

FEM is not the best way forward for rough contact. There are full 2D codes available public domain, including surface generators by Lars Pastewka in fact.

the contact mechanics calculator is here:

http://contact.engineering/

You can find a rough surface generator here:

https://gist.github.com/pastewka/72ab48e6570c72792a3cd0ff85d0e653

So you need to define a more interesting project!

## Very helpful material.

Wow! I searched a lot about this type of material/codes/notes that could help me kick start but I couldn't find much. These are hugely helpful for me. Are there any other forums/repositories/discussion boards where computational contact is discussed?

As I am fairly new to not just rough surface contact but to contact itself and already a bit familiar with FE we thought we'll start with something small just to get a feel of rough surface generation and contact. We (i.e me and my guide) haven't yet finalized on the directon. Thank you so much.

## The contact mechanics challenge: Problem definition

on numerical methods for rough contacts, almost everything has been said, you may look atThe

contactmechanicschallenge: Problem definitionThat effort is going to appear as a paper soon in Trib. Letters. We discussed this in a recent Lorentz worshop. It has taken almost a decade after Persson's paper, to converge on some conclusions. Mainly because Persson never clearly said if his solution were exact or not: a very good trick to attract attention and citations! His solution in the end did result to be approximate, and not much better than asperity one.

His solution for load-separation is even worse, and in fact there is a little work to be done there, which perhaps I will do.

There is a lot more open problems in adhesion of course, as this entire discussion shows.

## a mathematica code for surface generation

anyway, there is already one code in public domain mathematica http://demonstrations.wolfram.com/TwoDimensionalFractionalBrownianMotion/

## Amodified form of Pastewka-Robbins criterion for adhesion

Amodified form of

Pastewka-Robbins criterion for adhesionMCiavarella, APapangelo- The Journal of Adhesion, 2017 - Taylor & Francis AbstractRecent numerical investigation on self-affine Gaussian surfaces by Pastewka & Robbins have led to a criterion for “stickiness” based on when the slope of the (repulsive) area-load relationship appears to become vertical in numerical simulations at a ratio of contact area to nominal one (rather arbitrarily) fixed to 1%. Since pull-off and slope of the area-load are two faces of the same medal, a simple check of the results in terms of pull-off shows that Pastewka & Robbins have many more data which fail their criterion than the ones who satisfy it, and this is evident even in their own Figures. As a small improvement, a proposal to modify the criterion to better fit their own data is put forward. However, the pull-off decay seems rather exponential so that it is unclear if their slope criterion really corresponds to a “thermodynamic” limit, and consequently their conclusion that stickiness should depend only on slopes and curvature may be an artefact of their assumption of defining a secant at 1% contact area ratio, rather than a true important property of rough contact. Both the PR criterion and the present modified one imply that for fractal dimension D<2.4, stickiness should increase with resolution, so the problem of truncation of the spectrum seems ill-defined: in fact, PR define rigid self-affine surfaces with rather smooth and well defined slopes, and not a realistic atomic roughness as first studied by Luan and Robbins.

## Effect of rms amplitude on adhesive behavior of surfaces

There is no qualitative contrast between classical asperity models and Persson models or any numerical recent calculation about the slope of area-load curve: the only geometrical parameter entering the area-load slope is the rms slope of the surface.

The question arises with adhesion. Pastewka-Robbins suggested that the slope in this case becomes dependent additionally on rms curvature, and not on rms amplitude, whereas asperity models (Fuller-Tabor is the only one, the proper rough random surface one is not in the Literature but we are about to publish it) involve also rms amplitude.

However, also Pastewka-Robbins do find that pull-off depends on rms amplitude. So in their case there is a curious threshold: for non-sticky surfaces, they say there is no dependence on rms amplitude, whereas for sticky one there is, as they also find.

Can you beleive this?

## Some arguments on why Pastewka-Robbins is not general

First of all, PR use a "truncated potentials", a convenient numerical representation of the Lennard-Jones potential, but certainly a numeric artefact. Their potential is truncated at a short distance, a0+delta_r where delta_r is of the order of a0 itself (the atomic distance), and therefore, there is an artificial effect there: after 2 atomic steps, there is no longer adhesive force. This perhaps hides some effects of the rms amplitude?

In a true situation, Lennard-Jones extends to infinity, and it is clear that this fact already lowers the significance of their finding. Even at very long distances, in theory one should always have some attractive force, and this cannot be zero, and this means the area-load bends in the tensile quadrant

. Their criterion therefore has a lot to do with their arbitrary definition of the truncated potential.always## What is the pull-off for theoretically flat surfaces?

An interesting problem is the following: what happens for a truly flat surface? Even with a truncated potential, it is easy to show that the decaying adhesive force induces an instability --- even if the surface were flat. I have done the calculation for this instability, and it turns out, for the Lennard-Jones potential, that this instability occurs if we have a periodic wavelenght of the order of 50 a0. Therefore, not very large at all. Two flat surfaces would be in equilibrium at distance a0 and show the theoretical Lennard Jones pull-off force (theoretical strength) only if they were constrained not to assume wavy configurations.

This in fact could be an interesting problem to study, which however requires a numerical solution. Anyone interested?

## sorry, there was a mistake here

Obviously despite there is an instability, the surfaces will jump into contact uniformly much before this instability appears. Sorry.

## Another study which seems in contrast to Pastewka and Robbins

I see this just accepted paper on http://www.sciencedirect.com/science/article/pii/S0301679X17301081

Tribology International

Available online 3 March 2017

In Press, Accepted Manuscript — Note to users

Effect of fine-scale roughness on the tractions between contacting bodies

Persson’s theory for the elastic contact of rough surfaces is modified to include the compliance associated with an interface force law such as the Lennard-Jones law. We determine the effect of adding a small packet of waves on the probability distribution function [PDF] of the local interfacial gap (including the effect of elastic deformation). This procedure is then used iteratively to develop an algorithm for determining the PDF for a rough surface with a prescribed power spectral density. The results show that for untruncated quasi-fractal surfaces, the PDF then converges at large wavenumber, in contrast to the result when only elastic deformation is taken into account. If the roughness is restricted to wavenumbers greater than a certain critical value, the algorithm predicts a converged relation between nominal traction and mean gap that can be regarded as a modified interfacial force law describing the influence of just the fine-scale roughness on the contact. In particular, the area under the resulting curve can be interpreted as a measure of interface energy as reduced by this roughness. The remaining macroscopic features of the surface can then be described using the JKR methodology in combination with this modified interface energy.

This also seems in contrast to PR --- since the procedure converges when fine details are added, and hence the stickiness and any other property of the solution CANNOT depend on slopes and curvatures! Rather, this study also shows the main dependence on the rms amplitude.

## An asperity model for adhesion between rough surfaces

Articlein Journal of Adhesion Science and Technology · March 2017 DOI: 10.1080/01694243.2017.1304856A simple asperity model using random process theory is developed in the presence of adhesion, using the Derjaguin, Muller and Toporov model for each individual asperity.

A new adhesion parameter is found, which perhaps improves the previous parameter proposed by Fuller and Tabor which assumed identical asperities– the model in all his variants for the radius always gives a finite pull-off force, as in Fuller and Tabor, and contrary to the exponential asperity height distribution, where the force is either always compressive, or always tensile. It is shown that a model with spheres having a radius only dependent on height is a reasonable approximation with respect to models having also a distribution of radius curvatures – the three models differ considerably, as opposed to the adhesionless case where these details did not matter.The important surface parameters in the theory determining the pull-off force are the three moments m0, m2, m4. The asymptotic form of the model at large separation is solved in closed form. As the theoretical pull-off of aligned asperities having the same radius (the average value) increases with the square root of the Nayak bandwidth of the roughness, and as asperity models are known to describe less well the surface at large bandwidth parameters, the limit behavior at large bandwidths remains uncertain.Link to the paper: http://www.tandfonline.com/eprint/2tGWbVTiTXB4KJRVQEzi/full

## the problem is still unsolved!

While the asperity model shows qualitatively similar results to Pastewka and Robbins for the pull-off value (and therefore main dependence on m0, as suggested also by the recent very elegant Persson's extended solution by Junki Joe Scaraggi and Barber), the area-load slope (not necessarily a very interesting quantity) predicted by PR is in contrast to asperity models. Anyone joining the effort to solve the puzzle?

## I wonder how Pastewka & Robbins would justify Fig11 experiments

PR argue that their findings suggests stickiness in strong contrast to the classical findings of FT, whose emphasis in on rms amplitude of roughness, whereas their findings are independent on it.

I wonder how this can be reconciled with the pull-off experiments in Fig.11 from the literature on our recent paper

http://www.tandfonline.com/eprint/aNhs6YfWn7P23MPZa6Dh/full

## Besides, even Persson's study are against Pastewka & Robbins

Mulakaluri, N., Persson, B.: Adhesion between elastic solids with randomly rough surfaces: comparison of analytical theory with molecular-dynamics simulations. Europhys. Lett. 96, 66003 (2011) [PDF] arxiv.org

In all the figures, from fig.2 to fig.5, the MAIN parameter ruling pull-off is RMS roughness.... against PR criterion. The puzzle continues...