 So, today we have the last session, it will be mostly the first part will be lively and the second one will be online. I hope that everybody will stay throughout. I want to announce that at the end of today's meeting, at the end of the conference after the last talk by Zhenquan Xu, the last speaker who will speak on remote after that, there will be a session where the three best posters will be awarded the prize and the committee will run that part of the meeting and then we will go for lunch after that. So with this, we begin with the first talk by last part of energy dissipation in soft adhesion from a deepening perspective. Thanks Aerio for the introduction and also thanks to you and Ernst and Andrea and Quintuuy for giving me the opportunity to speak here and to talk about the latest and greatest on adhesion of soft solids on rough surfaces or something like what you see on the bottom of this slide. So, let me first point out that I'm sort of located on the meso-scale end of this meeting. So, I'm not talking about atoms today, I'm talking about contact line and deepening of contact lines and the work was done mainly by my PhD student Antoine. And I'm a theoretician, I do computer simulations, but I will show you a few experiments that were done in essentially in Pittsburgh, in the group of Tavis Jacobs and in Akron, in the group of Ali Ginovala. So, since this meeting is dedicated to Mark Robbins, let me say a couple of words. Maybe let me tell you why I'm in adhesion at all. So I started doing a post-doc with Mark, something like 11 years ago, and I had written this beautiful proposal, a Marie Curie proposal that got funded on how surfaces run in, how surface topography changes during sliding. And then I arrived in Baltimore and Mark told me, well, but adhesion is also interesting. And I think he said this three times and then we started working on adhesion. I never did what was in the original proposal, so if anyone or any of the referees is here, I'm sorry that we never implemented this. And his way of thinking has greatly shaped the way I do research, and what I'll show you today is also, of course, inspired from many interactions with him, and he also worked on deep pinning of contact lines, and we talked about this and part of why I'm here talking about this adhesion story is also due to him. Okay, so let me set the stage about what I will talk about today. So I will talk about adhesion of soft solids on rough surfaces, and the canonical experiment is a soft swir, in this case a PDMS swir that's contacted with some solid body. In this case, this is a diamond surface, and then you measure the force as a function of displacement. And in this case, Ali at Acron, they also measured the contact area through an optical technique by just looking into the contact. So in this experiment, the substrate was actually varied, sort of to do the experiment on substrates of different roughness, and this is what these acronyms are down here. So this is ultrananocrystalline diamond, microcrystalline diamond, nanocrystalline diamond, diamond films with different crystal sizes and different roughness. So what does this look like? So this shows the contact radius as a function of force. So if you move into contact during loading, you typically find something that looks like this, and if you want, you can fit it with a JKR model, and you will get a relatively nice fit to JKR. We've heard a lot about JKR here already, so it's a contact model that assumes an interface energy for high interface energy and soft solids. So JKR is sort of a model that assumes thermodynamic equilibrium, so when you move back, you should move back along the same line with the except of the point of pull-off. But what you typically find, and this is not the only experiment there's lots of report in the literature, is that you actually move back a different way. So there's a hysteresis in these contacts. And if you want, you can even fit a JKR model at least to part of this pull-off curve, and you would get a different work of adhesion, of course, if you extracted the effective work of adhesion from these solids. So it gets worse, so now we can do the same experiment on a different substrate and you get different curves. So here the approach looks similar, but the moving out of contact, the hysteresis is larger. There's even sort of a region where we are at constant area or at constant radius when we move out of contact. And just to show you a little more data, so these are the other experiments, so they all look different. They somehow seem to depend on the topography of these films. So this means there's some sort of dissipation process going on. This may not be super obvious from this plot, but of course you also see the hysteresis in the normal force versus displacement plot. And if you integrate that, that's just the work you do on the system, and sort of you do work on the system in these contact cycles. But the typical explanation for this is something is going on either in the solid or at the interface. So there may be contact aging, what's now called the quality of the contact may change. So there may be interatomic bonds that form because you hold the contact. This is taken from Jacob Isayelach-Willis' book on sort of contact of silicon or silicon oxide surfaces that of course form covalent bonds, or there may be viscoelastic damping going on, viscoelasticity also of course leads to time dependence and also to dissipation. So these experiments that were done by Ali, he also did control experiments on flat silicon wafers that were OTS coated, and there's no hysteresis on these silicon wafers. They were done at very low velocity. So we are reasonably sure that viscoelasticity can be ruled out. There may still be of course viscoelasticity, but the question that I want to talk about today is whether we can get adhesion hysteresis from purely elastic mechanisms, and of course the answer to that is yes, and the key is the pinning of the contact line. Okay, so let me take a step back and show you something that you have seen already maybe ten times in this meeting, the Prandtl-Tomlinson model. For me it's actually the first time that I have it on the slide, but many of you will be familiar with this. I'll tell you in a second how this relates to adhesion. So the Prandtl-Tomlinson model is essentially, what we have is a slider that is pulled by a spring of stiffness K over a corrugated substrate that's typically, or corrugated energy landscape that's typically modeled by a sign, what you see here is actually the force balance expression of the energy. Now in the limit where sort of the amplitude of the force corrugation is much larger or is larger than the stiffness or K times A, the slider gets stuck in the minima of the corrugated energy landscapes, and this leads to energy dissipation. So let me just show you a little video of what this looks like. So for floppy driving force, for floppy spring, what you find is that your slider gets stuck in this minima and then there is jumps into the next minimum position. And sort of the idea here is we jump from here to here, and then the energy difference, this energy, is dissipated simply because you decouple the velocity of the jump is decoupled from the velocity of the external motion. So you even get dissipation at zero velocity. This is an elastic instability, and this is a key mechanism, Martin Moussa alluded to this already in one of his comments at the beginning of the week. This is a key mechanism for energy dissipation in the quasi-static limit or at very low velocities. So again, I want to point out that sort of this is a force balance expression, and here this is a driving force, and the driving force is linear in the extension of the spring in this case. So what does this have to do with adhesion? So let me show you a little tiny toy model. So I'm now considering the contact of a sphere on a deformable substrate. So what you see up here, this is our sphere. You see that the contact neck has deformed because of course it's snapping into contact because of the adhesive interaction, but the work of adhesion is not a constant. I'm considering these concentric rings of work of adhesion that you see down here. So as we move from the center of the contact to the contact edge, we have a sinusoidal oscillation of the work of adhesion. Let me show you what happens. If you consider this model, think about it maybe as a simulation. So what happens is that the contact line actually gets pinned at the boundary between the high and the low work of adhesion region and then jumps into the next minimum. I'll show it to you in a second. So it's pinned between these two regions. The reason is of course in a high work of adhesion region, it tries to increase the contact radius, and then it hits the low work of adhesion region where the contact radius should be smaller, so there's an equilibrium between the two. So we have unstable jumps between these two situations. Let me make this a bit more quantitative and sort of connect back to the Pantel-Tomlinson model. So what you see here is the energy release rate plotted as a function of the contact radius. What is the energy release rate? The energy release rate is the energy that is given to the system from the elastic deformation or release from the elastic deformation per area swept by the contact line. So it's something like the derivative of an energy is essentially a force. Think about it as a force in the Pantel-Tomlinson model. So this is our driving force. So this is this black curve here. This is essentially the spring. In the Pantel-Tomlinson, the spring is linear. Here this is the JKR expression. It's a non-linear expression that tells us what the driving force looks like as a function of the contact radius. This needs to be balanced by the energy gain of the surface that you get from the surface for making contact. And this here is the work of adhesion. This is this gray curve here that is denoted by W of A. So essentially we are sort of scanning across the contact here and looking at the oscillation in the work of adhesion. And the equilibrium between the two. So this here is the point where the system finds a stable equilibrium. So if I run this model, then you see that there's unstable and it jumps very similar to what you see in the Pantel-Tomlinson model just for this adhesive case. Let's move back out of contact. Let's take a step. Let's move back out of contact. So if we move back out of contact, what happens is that we jump at a different location and this is the reason we get hysteresis. Here we get pinned essentially at a different point on this work of adhesion field. What you see up here is a plot of the penetration versus the radius. This is essentially the force balance in the whole system. And there's the regions where the energy release rate is larger than the work of adhesion, so the energy gained from the surface. And the other way around are marked in different colors. And the equilibrium position is this one here. And you see that there's discrete jumps between these positions. Now what I'm doing is let me make these amplitudes much, much smaller than what you see here. So this is now a much, much smaller amplitude. What you end up having is you end up sampling only the peaks of the work of adhesion field. So you get a difference, a different effective or apparent work of adhesion during approach, which corresponds to the minimum on the work of adhesion field. Then from retraction, which is the maximum. And the difference, the hysteresis between them is actually proportional to the root mean square fluctuations on that surface. And we can also look here at the penetration versus area. And you see that we get nice JKR, so these curves again are JKR curves with these effective values of the work of adhesion. So this toy model essentially is a gist of what I want to tell you about. So if you want, you can fall asleep now, or you can listen to me telling you what happens if we now go from these concentric rings to random work of adhesion, and then how to go from a random work of adhesion to surface roughness. So what I will, you mean, what do you mean? This one down here, the gray curve. So this is just what we stuck into the system. This is our work of, no, this is per surface area. So the W values, this NG is per surface area. It's an energy per surface area. Okay, so what happens on heterogeneous surfaces? So we have to do a little more thinking, a little more complication in order to do this. So what I will show you now is two types of simulation. One is a boundary element method calculation that is fairly standard in the community. Mark and myself have used it to study adhesion. What you do is you solve for the elastic deformation in the substrate. And we do this on a sort of work of adhesion field that is random. And then I will also show you a model that is a correct front model. Essentially, it's a model for just the contact line. So we are reducing the problem to a 1D problem, which allows us to do extremely large calculations. It's a factor of 100 to 1000 faster than the BAM model. And we can do realistic roughness and spend all the way to macroscopic scales. This is something that we recently published in JNPS this year. So what is the idea behind the correct front model? It's exactly what I've told you before on these concentric rings, is we have force equilibrium. So the energy release rate needs to equal the work of adhesion. But now locally, so A of theta and theta is a point on our work of adhesion map. So a point along the contact radius here, yeah? So for each point, we require this force equilibrium. And then this here is the fractal mechanics model for the contact line. It's a perturbation around the JKR solution. So let me just point out what this looks like. This here, this delta to the power of one half is a fractional aplasian of the contact line. Think about it as something like a curvature. It's not really a curvature, but it's something like a curvature. So it prevents the contact line from going out, from making a curvature. So oscillations of the contact line cost energy. And sort of it's pulled back. The contact line behaves like an elastic spring, essentially. Let me show you what this looks like. Maybe first look at the right image. So this will show you in a second the motion of the contact line. You see that there's, again, jumps. But these jumps are now localized along the rim of the contact line. So we have individual instabilities on the heterogeneous work of adhesion. You also see that the BAM and the crack-front model, essentially, gives the same answer. Let me run this again and point out that sort of here we have these, of course, these jumps, these instabilities, also correspond to jumps in the external driver, in the penetration versus normal force. You can see them here because this is not really a very finely resolved graph. So this is something that is still a very coarse calculation. So what type of insights can we gain from this model? Let me discuss a little bit what's happening in these calculations. So we have a line moving in a random field. So there's different limits that we can consider here. The first limit is the line is extremely rigid. If the line is very rigid, what we are doing is we are just sampling the work of adhesion, which is just taking the average of the work of adhesion along the line. It can deform. And in that limit, there's no hysteresis. The work of adhesion for retraction and for approach is the same. The opposite limit is the limit of an extremely soft line, in which case, essentially, in this discretized or pixelized representation here, we can regard each of these columns individually. And each of these columns is something like this concentric ring model that I've shown you before. So it will sample simply the peaks, the maxima, and minima along these lines. So what you get is the difference between the work of adhesion during retraction and approach is something as proportional to the root means square fluctuations on that surface. Now the limit that you typically have is something that's intermediate, which is called the collective pinning limit, in which case the line has some stiffness. So we think about this as we can't deform the line over a characteristic length scale that's called the elastic coherence length or the larkin length. And what happens here is that, essentially, we then need to average the work of adhesion over this elastic coherence length. And this reduces this expression from the soft limit by a square root of n term that comes from sort of randomly averaging random terms. This is actually something that's very well described in literature. The original larkin argument was done for flux lines and superconductors. Mark Robbins and Dournier have used it for the wetting fronts, and it's also been used in the fracture community. So just one another expression. So this larkin length can be related to the properties of the work of adhesion field as given by the mean square divided by the rms square. So at least it's not the length, and it's sort of the number of this value and that we have here. Let me just show you two simulations of what this looks like. So this is a relatively large elastic coherence length. These yellow flashes are the individual instabilities that we jump over. So you see that the contact line has jumps over this characteristic length scales. And this simulation down here has a smaller elastic coherence length. OK, so how do we now get to rough surfaces? So this is not yet roughness. This is the variant locally varying surface energies, locally varying work of adhesion. So what I will show you now is that you can, that there's a mapping that maps heights onto these work of adhesion fields. And in order to sort of discuss this mapping, let me start with what does contact of geometrically rough surfaces look like. So the typical picture that you have, think about non-adhesive contact is we have a rough surface contacting an elastic solid. Of course, at the contacting spot, the elastic solid deforms and you get this contact area and then you can also ask the question, what is the pressure that you see? Pressure is a function or functional of the displacement and if we use space, it's diagonal and this is an expression that you find in any good contact mechanics textbooks. So if you move further into contact, of course, at some point, we will arrive at a limit where we are conforming. Where the two bodies are perfectly conforming. Now what does adhesion do? Well, the elastic deformation costs us energy. We need to pay energy to elasticly deform the solid and from adhesive interaction, we gain energy. We gain the surface energy during deformation. And in the following, I will discuss only this limit. So this is the limit of soft solids where we actually have conforming surfaces. So this is an assumption that we stick in here. And for this limit, there's a famous result by Voperson and Eriotossati, both of whom I had this conference, that says the following. If I'm in this conforming limit, then the apparent work of adhesion that I should see is given by what I call here the intrinsic work of adhesion. So this is what your surface interaction fundamental forces also give you. So this is the intrinsic work of adhesion. And then I have to subtract the energy now per surface area that I need to pay to conform elasticly. And the energy to conform elasticly can also be written down in a very simple expression. So this is, again, an expression in Fourier space. This is the elastic energy for conforming, can essentially derive from the pressure expression that I've given you here. What you see is an integral over q, and then here, you just have the displacement of the surface squared. And if you are conforming, then essentially the displacement here becomes h. And so I want to point out that this elastic is essentially a geometric parameter of the topography. So we only have the modulus e here in front. If we disregard this, there's no, it doesn't say anything about the elastic properties are sort of factored out. And if you look at this, this is d squared q, then you have this q here. If you have q to the power of zero, you have the rms amplitude, q to the power of two is the rms slope, and this q is sort of an intermediate in between them. This is what determines the elastic energy. It's a roughness parameter, essentially, something that you can compute from the topography. Okay, so we, there's, as I said, there's this mapping. I don't, I won't show you the expression for the integral transformation from heights to w, but I want to discuss it. So Antoine had derived it, and then we found out that it's actually already in the literature. Anders and Rice had derived it in the context of a dislocation near a correct tip, where you also have a displacement field of the dislocation and the correct tip, and it's essentially exactly the same expression that we, that we are using here. Let me just discuss the properties of this work of adhesion field that we get when we apply, when we transform from heights to work of adhesion. So first of all, the mean value is exactly the Passon-Tosati limit. That is the expression that I discussed before. It's the intrinsic work of adhesion minus the elastic energy. And then with a little bit of additional equations, so essentially you need to assume, or you need to, I mean, you need to realize that the stress intensity factor is linear in the heights, and then you use Irving's expression for the relationship between stress intensity factor and energy release rate. You can also derive an approximate expression for the fluctuations, which also again contains the intrinsic work of adhesion here and the elastic energy. So this is the amplitude of fluctuations of this, on this work of adhesion field. So again, if we do calculations of this, we get JKR, so this here is JKR, JKR on the way out, JKR on the way in with two different works of adhesion for approach and retraction. And we can write down an analytic expression on approximation for these works of adhesion coming into contact and moving out of contact, which is given by the mean value here. So this is Passon-Tosati plus minus and this can be derived from this lacking argument that I've shown you on the slide before. If you put all the equations together that I've given you, you will see that essentially the hysteresis has to be proportional to this RMS square divided by the mean value. And this then becomes Passon-Tosati plus minus a pre-factor times the elastic energy and from numerical calculations, I show them in a second, we get something like K equal three. So Passon-Tosati is the mean field limit plus minus, you get the K times the elastic energy contribution, you get the hysteresis. So let me just show you this applied. So this is a synthetic rough surface with a specific power spectral density. It's a self-defined surface where you have a power spectral density of the height. So this is essentially the amplitude of the height fluctuations in Fourier space that is a power law with the first exponent H here. So if we apply the integral transform, then the PSD transforms approximately like this. Let me just point out that this is the expression for W-RMS that I've shown you on the slide before. So we have W int times, this here is the elastic energy or the contribution to the elastic energy of that specific wave factor. So this is essentially the generalization of this expression that I've shown you before. And then you get a work of adhesion with a power spectrum that looks like this. It's essentially a product with Q. So all the exponents decrease by one. So now we did these crack front calculations. These are the dots. And what you see as the dashed line here is now the difference between the retraction and approach work of adhesion is six times E elastic. And this is exactly this K equal three Ks. So it's plus minus K elastic, which was three. And our calculations match this at least for a larger elastic reasonably well. We did the parametric study with different realizations, different Hearst exponents. So they all seem to collapse on a master curve. So we can probably get better than just this linear relationship between the hysteresis and the elastic energy. Okay, yeah? Yeah, just one more slide and then I'm done here. So let's connect this back to the experiment. So as I said in the beginning, Ali, you know, while I had done these experiments on these diamond films, and this is one example that you see here of a force contact radius curve. And now in order to apply our theory, we need the roughness of these films. And these diamond films were characterized down to the atom in the group of Tevis Jacobs. So essentially what he did is he combined stylus, a profilometry, AFM and cross-sectional TEM technique to piece together the PSD, in this case, over seven orders of magnitude in length scale. So for all of these samples, we know the PSD fully down to the atomic scale. So they all have been characterized with TEM. So now we can stick this in our model in this crack-front type simulation, and this is what we get. Let me point out that this intrinsic work of attrition is fitted to the approach data. But with this fit, we do reasonably well, I would say, sort of predicting what happens when we move out of contact. Let me just show you this data in a slightly different presentation. So what I'm now doing is I'm not showing the normal force, but I'm showing the energy release rate. Think about this as fitting a JKR model to each of these points and extracting an effective work of attrition from these JKR points. Then on approach, here we get a constant work of attrition. And our model would predict actually a constant work of attrition doing retraction. That's what I've talked about today. And they see in the experiments a slight increase here, which is something that we're still looking at. Of course, there may still be viscoelastic effects or contact-aging effects or other things on top of sort of this elastic pinning. But we are reasonably sure that elastic pinning is one of the main contributions to the attrition histories this year. So I'm essentially done. Thank you for your attention. Antoine did all of the work, as I said. I mean, he did a great job. He's one of the students where I have to catch up. So he does things that I then need to try to understand. So he's really excellent. And I think this is, it's the next step sort of in predicting really pull-off forces from measurements of rough surfaces. Something else that I just want to point out is a bit of advertisement. We're running an online service contact engineering that allows you to upload measurements of rough surfaces and do contact mechanics calculations and analyze the surfaces. You can compute PSDs. And since a couple of weeks ago, you can also publish your surfaces. You get the DOI. So if you need to have a paper and you want to publish topography data, you can go there, click the publish button. You get the DOI for citation in your papers. With that, I'm done. Thank you for your attention. And I'm happy to take questions. Thank you very much. Very good work. Questions? Should I do it? I can click on here, right? Martin, Müser, Roberto Gerner. Well, if nobody else asks a question, I can always ask one. I mean, now you're sometimes used to the contact line and a great talk by the way, and then one has done. So are there ranges where you don't get the hysteresis? And did you look into the question, how that correlates with the stickiness criteria? I guess you're familiar with the work by Pascal Robbins and, of course, with the work by Bob Hatt. Martin, the god of the internet. Did you get into the question if this existence of hysteresis correlates with the breaking up the god of the internet doesn't want you to ask your question, it seems. OK. So I want to point out that what I've shown you today is the limit of soft surfaces and conforming contact. So the paper with Mark, the PNAS, with the stickiness criterion is the opposite limit, where we are sort of in a DMT-like limit of hard surfaces and we are not conforming. And I think something that's very important and that's often overlooked is that these limits lead to different physical behavior, and you have to treat them differently. I don't think there's a theory that sort of can span from sort of non-conforming contact. Maybe both theory can do it, but I'm not sure if you are fully there yet so that can span from non-conforming contact all the way to the very soft limit. So as soon as you have partial contact and also Roberto's question, you ask how much trapped air could limit the full contact goes in that direction. Of course, you will have internal contact lines and maybe this crack-front thing still works, but you would need to consider the internal contact as well. So all bets are off in that case. I may even increase adhesion because you have additional contact lines that you need to get rid of. And with regards to the experiment and to the trapped air, I think that PEM is actually permeable for you, but the experimentalists can probably answer that better. So I don't think that would be a hot candidate while you get extra apparent adhesion on the retraction line that you get partial contact perhaps. Perhaps something like this, yeah. OK, interesting. All questions or comments? OK, in that case, we thank Lars again. Thank you very much. Move on to the next talk.