## Abstract

We study the adhesion of a surface with a ‘dimple’ which shows a mechanism for a bi-stable adhesive system in surfaces with spaced patterns of depressions, leading to adhesion enhancement, high dissipation and hysteresis. Recent studies were limited mainly to the very short range of adhesion (the so-called JKR regime), while we generalize the study to a Maugis cohesive model. A ‘generalized Tabor parameter’, given by the ratio of theoretical strength to elastic modulus, multiplied by the ratio of dimple width to depth has been found. It is shown that bistability disappears for generalized Tabor parameter less than about 2. Introduction of the theoretical strength is needed to have significant results when the system has gone in full contact, unless one postulates alternative limits to full contact, such as air entrapment, contaminants or fine scale roughness. Simple equations are obtained for the pull-off and for the full contact pressure in the entire set of the two governing dimensionless parameters. A qualitative comparison with results of recent experiments with nanopatterned bioinspired dry adhesives is attempted in light of the present model.

## 1. Introduction

In the recent explosion of interest for insect adhesion and bioinspired adhesives, the development of patterned surfaces follows various avenues and methods of constructions, one with pillars of various shapes and tips, and another with depressions, or ‘dimples’. In general, the understanding of mechanisms of adhesion on nano- and micro-rough surfaces is critical, but is far from complete at present. Elastic very compliant materials (Young's modulus less than 1 MPa) can adapt well to the roughness of hard surfaces, although there is then a problem of releasing adhesion once it is realized. Geckos recur to more complex mechanisms, namely their hierarchical fibrillar structure, because they show relatively rigid modulus material (approx. 1 GPa), but even this system shows less ability to adhere for some specific range of roughness [1,2], showing that a truly efficient system for multiscale arbitrary roughness is extremely difficult to achieve.

Akerboom *et al.* [3] describe a method to build nanopatterned bioinspired dry adhesives using colloidal lithography, which results in elastomers patterned with nanodimples with a wavelength about 10 times the size of the dimple (qualitatively described in figure 1*a*). Adhesion and friction properties of nanopatterned surfaces with varying dimple depth against a spherical probe counter-surface showed enhancement of adhesion, attributed to reaching full contact and an energy-dissipating mechanism during pull-off. For this kind of nanopatterned surface, one of most elegant models proposed is the ‘dimple’ model of McMeeking *et al.* [4] (McMeeking–Ma–Arzt (MMA) model in the following), which essentially consists in a single depression in one of the surfaces, as seen in figure 1*b*. This leads to a mechanism for a bi-stable adhesive system, which they analysed mainly within the assumption of JKR model [5] which corresponds to very short range adhesion where adhesive forces are all within the contact area.

Zhou *et al.* [6] apply the MMA model to describe the contact between smooth insect pads and a substrate with cylindrical pillars, making several simplifying assumptions. In particular, that full contact with the bottom of the pillar array is made once the pad is able to fill out a ‘dimple’ within the square element formed by four pillars. Both the pads of cockroaches (*Nauphoeta cinerea*) and hairy pads of dock beetles (*Gastrophysa viridula*) showed adhesion between pillars for larger spacings, but not on the densest arrays.

Cañas *et al.* [7] used this model to fit qualitatively some results of adhesion of biomimetic polydimethylsiloxane (PDMS) pillar arrays with mushroom-shaped tips to nano- and micro-rough surfaces, and the pull-off strength for patterned PDMS was apparently controlled by the estimated deepest dimple-like feature on the rough surface. The fit was very qualitative and will be further discussed in the Discussion section.

Therefore, it is important to make the MMA dimple model more realistic by introducing various features. Here we consider the role of cohesive stresses, leading to a more general model of adhesion, like that introduced by Maugis in the context of spherical contact, for the first time in 1992 [8].

## 2. Preliminary remarks

The dimple model has some features in common with the periodical contact problem of wavy surfaces with adhesion, and it is very useful to make connections to the parameters ruling the latter problem. For a single sinusoid of wavelength and amplitude λ, *h*, Johnson [9] discusses the JKR regime [5] is governed by a single parameter, *α*, defined as
2.1where *l*_{a} = *w*/*E** is a characteristic length of adhesion, *w* the work of adhesion, *E** the composite elastic modulus of the two materials in plane strain. Also, *α* represents the square of the ratio of the surface energy in one wavelength to the elastic strain energy when the wave is flattened. Solving the problem, it turns out that for *α* > 0.57, there is a spontaneous snap into full contact, from which detachment should occur only at values of stress close to theoretical strength. This ‘paradoxical’ behaviour led Johnson to postulate that (especially for two-dimensional roughness) there would be a limit due to entrapment of air in the valleys, or contaminants, or short scale roughness. Postulating a defect at the interface, a JKR analysis can find a finite pull-off value again. The full solution for sinusoidal roughness is possible analytically only for one-dimensional profile, and not for two-dimensional roughness, where we lose axisymmetry. In the dimple case, axisymmetry permits one to obtain a complete solution also in the more general cohesive regime, as we shall see.

Various authors have recently generalized Johnson's JKR one-dimensional sinusoidal problem to the cohesive regime, using the Maugis simple law of potential [10,11], which gives a constant value of attraction between zero and a finite range of attraction distance, or with a double-Westergaard solution [12], or finally fully numerically with a Lennard-Jones potential [13], showing very similar results. The solution and the pull-off strength depend crucially on another parameter, the so-called *Tabor parameter* that governs also the transition for the sphere problem [8,14] between the rigid behaviour and the JKR behaviour [15]
2.2where *σ*_{0} is theoretical strength, and *R* is the radius of the tip of the sinusoid. Depending on the combination of *α, μ*, a quite complicated pattern of different transition processes between partial and full contact during loading/unloading stages emerges, which features one or more jump instabilities. It is important to remark that, given the Tabor parameter increases with *R*, rougher surfaces enhance adhesion both by increasing the magnitude of the pull-off force and by inducing more energy loss due to adhesion hysteresis—in these respects, this corresponds to what has been shown clearly also by Guduru [16] for a spherical contact with an axisymmetric waviness. Waters *et al.* [17] remark that this enhancement of pull-off and of hysteresis pertains mostly to the JKR regime when Tabor parameter is *μ* > 1, and this point is important and further discussed in the Discussion section.

The dimple model has two geometrical length scales as the periodic sinusoid (an amplitude *δ*_{0} and a radius *b*), and therefore we shall try, before generalizing the results of the dimple model to the cohesive one, to show more precise parallelism between the two problems introducing similar dimensionless parameters.

## 3. MMA solution in JKR regime

We summarize MMA findings, in a more concise, dimensionless notation. MMA solve entirely the JKR regime of the axisymmetric dimple, and make a start in solving a cohesive model, which however they only attempt qualitatively in plane strain therefore without a full comparison with the JKR predictions. For an axisymmetric dimple of amplitude *δ*_{0} and radius *b* the shape of the dimple is (figure 1*b*)
3.1where *ɛ*(*r*/*b*) is the complete integral of the second kind and *κ*(*b*/*r*) is the complete elliptic integral of the first kind.

In the absence of adhesion, a ‘full contact state’ is reached when the external mean pressure *σ*_{A} is large enough (in compression) to realize inside the dimple the full contact pressure
3.2

This full contact pressure *T* depends then on the geometry (the ‘slope’) of the dimple, and can be comparable to theoretical strength *σ*_{0} for not too low a ‘slope’ *δ*_{0}/*b*. To compare with the sinusoid, remark that the full contact pressure for a sinusoid is *p** = *π**E***g*/λ, and therefore this is very similar to *T* when ‘slope’ is measured as *g*/λ.

In the full adhered state, the combination of the localized stress *T* in *r* < *b*, and the remote tension *σ*_{A} in the entire plane, lead to a very simple crack problem, which we can solve using standard results for axisymmetric cracks [14] under internal pressure *p*(*r*). For a crack of radius *c*, we derive an auxiliary function [14, 3.114a]
3.3and
3.4Then, [14, 3.117] gives the stress intensity factor as Hence, the Griffith (JKR) case is obtained equating *K*_{I} = *K*_{Ic}, the ‘toughness’ of the material pair, or the energy release rate *G* = *w*, where 2*E***w* = *K*_{Ic}^{2}, leading to eqn 6(a,b) of McMeeking *et al.* [4]. However, we can define the ‘Johnson parameter for the dimple’, analogous with (2.1), as proportional to the ratio between the work of adhesion, and the elastic energy to flatten the dimple:
3.5Using (3.5), we can restate the JKR curves normalizing the stresses by *T* as and all length scales by *b* as obtaining^{1}
3.6and
3.7

These curves are plotted in figure 2*a* for various *α*_{d} and for *α*_{d} = 0.8 in figure 2*b*. In particular, there is a minimum at which is (indicated as point A). If a dimple is loaded in compression, at point A there is an unstable jump into full contact. Moreover, the intersection with is indicated as point O, and this is the equilibrium state at zero external load. When *α*_{d} > 1, since point O moves in the tensile region, there is spontaneous full contact. This condition (*α*_{d} > 1) corresponds to the condition *α* > 0.57 for Johnson's parameter in the sinusoidal contact.

If there is no previous jump into full contact the pull-off corresponds to the unstable jump at point B to complete opening of the interface, which corresponds to *σ*_{A}|_{max}. This can be found obviously evaluating which results in
3.8The resulting expression for the pull-off is complicated as we would need to insert into (3.7), but an extremely good fit we found is
3.9

Expression (3.9) has only sense for *α*_{d} > 1, because otherwise there is spontaneous full contact. From the full contact state, the JKR analysis cannot predict any finite to separate the contact, as in Johnson's sinusoid.

## 4. Cohesive model

For a cohesive model, we need to impose for continuity of stresses at the crack tip (no singularity), adding the ‘internal’ cohesive stresses *σ*_{0} in an annular region *d* < *r* < *c* which adds a contribution which tends to close the crack. Two equations result from the two conditions: that there is no singularity of stresses at the crack tip, and that the crack opening displacement (COD) at the end of the cohesive zone is equal to a limit value . With the calculations obtained in appendix A, we obtain that the equilibrium is given by the solution of two equations
4.1where and
4.2Note that the closest to a ‘Tabor parameter’ for the dimple is
4.3which contains no cubic root as the original Tabor parameter. As we shall see, it is for of the order of 1 where the transition from ‘rigid’ to JKR behaviour occurs.^{2}

### 4.1. Full cohesion

A simpler case is when (full cohesion) and we have to assume that when separation starts, the entire region of the original dimple separates, as the internal stress is equal to *σ*_{0}. Hence, we impose a standard contact mechanics condition on the absence of singularity of the stresses *K*_{I} + *K*_{i} = 0 ((4.2) with *d* = 0) which gives
4.4and this particular equation was in fact also obtained as eqn 10 in [4]. So, when the stress rises above , the crack opens in a continuous manner until the critical condition is reached on the COD. In equation (4.1), the integral can be easily obtained and (4.1) with leads to
4.5Substituting for *ĉ* (4.4) and solving
4.6which correspond to For , i.e. we have that a full cohesive solution exists in the entire dimple, and from (4.4)
4.7so the pull-off tends to the theoretical strength in the rigid limit when . This is a quite expected result, considering we have in this limit a very shallow depression completely immersed in the cohesive stresses.

## 5. Results

To solve the system of the two equations (4.2) and (4.1), for a given *α*_{d} and *ĉ*, we used a Newton–Raphson scheme associated with a pseudo-arc-length continuation, obviously unless we were in the full cohesive branch, where a fully analytical solution has been obtained.

Figure 3 shows the curves load versus radius of the crack (or ‘non-contact’ area) *ĉ* with a low *α*_{d} = 0.5, which requires compression to jump into full contact. In the following figures, solid and dashed lines, respectively, represent stable/unstable solutions. Each subplot shows a value of , increasing from (*a*) to (*d*) as respectively. For there is very low hysteresis and only one possible pull-off (point F) which is practically coincident with the JKR prediction, in fact, there is an unstable jump from point C to the EF JKR branch. We can call the segments BC as ‘full cohesive branches’, and segments ‘EF’ as ‘JKR branches’. The major difference at low is that the pull-off in F can be easily reached even starting from a full-contact state, while JKR would need an infinite stress for separation. This has important implications, as it removes the bi-stable behaviour between a low and a high adhered state. When , the unloading path is virtually already JKR, except that there is a cohesive branch ABC at that at leads to complete separation (point C). The loading branch is instead practically coincident with JKR. In these respects, obviously, there is a large hysteresis that was expected but not fully explicit in the JKR model.

In figure 4, the case with *α*_{d} = 1 is shown. Note that this is the highest value of *α*_{d} that permits a finite pull-off in the JKR case, as the minimum is exactly at With regard to the cohesive solution, for a compressive force is required to allow the dimple to jump into full contact, contrary to the JKR prediction. The case *α*_{d} = 1.5 (figure 5) is more interesting as the JKR solution would show always complete adhesion (without specific information on theoretical strength), and instead a cohesive model shows more clearly that the behaviour is related to the theoretical strength, particularly for low . Note that in figure 5*a* and thus all the branch is full-cohesive and the pull-off coincides with the theoretical strength at .

Figure 6 shows results for the size of the cohesive radius versus *ĉ* for *α*_{d} = 1 and (*a*) (*b*) (*c*) (*d*) . Cohesive zones are obviously large when is low, whereas the JKR limit is approached for high The jumps in the solutions are also indicated with arrows.

In figure 7, physical values of pull-off are plotted for the usual three values of *α*_{d} = 0.5 − 1 − 1.5 (symbol size increases with *α*_{d}). Three regimes can be identified as the dimple ‘Tabor parameter’ is increased. First, the rigid regime for , where (equation (4.7) with squares). We recall that this regime is full cohesive for all *ĉ* and that the pull-off is reached when the half-space completely detaches. Then, in the second regime (circles), the pull-off happens in partial contact (point F, figures 3–5). This value of is insensitive to is almost coincident with the JKR value (3.9) and exists if the branch E–F crosses as this ensures that the branch can be physically reached. At a third branch appears (triangles) that is almost linear with (see equation (4.6) for high ), and indeed is close to the theoretical strength of the material which is the dashed limit, closer the higher is *α*_{d}. This branch corresponds to the case when pull-off is reached after full contact (point C, figures 3–5). For high *α*_{d}, only pull-off at point C exists, as the contact naturally jumps into full contact at zero applied tension.

Figure 8 summarizes some results for the pressure to reach full contact . The pressure needed for full contact is Hence, for we have , otherwise For *α*_{d} > 1, increases up to 0 and then remains constant as there is a spontaneous jump into full contact.

## 6. Discussion

Akerboom *et al.* [3] build nanopatterned bioinspired dry adhesives using colloidal lithography, and use MMA model for estimating the critical dimple depth below which full contact occurs. The adhesion and friction properties of nanopatterned surfaces where measured with varying dimple depth against a *spherical* probe of radius *R* = 2.38 mm. Enhancement of adhesion, and large friction was attributed to reaching full contact which is qualitatively in agreement with the simple MMA model (JKR regime). However, considering an average dimple depth of 100 nm and a dimple radius of 500 nm, we note that the height-to-radius ratio of the dimple was quite high, of the order of 1/5, and therefore our ‘dimple Tabor’ parameter is likely to be quite low, of the order of , which probably is smaller than 1. According to our results, at this level of dimple Tabor parameter, there is no bistability and enhancement—the behaviour is quite ‘rigid’. In these respects therefore, their adhesion enhancement is much less clearly explained by the dimple model than what they suggest. But the problem is more complicated, as we really have a spherical contact with spaced dimples, and no exact analysis can be made of this problem. However, Waters *et al.* [17] studied the detachment of a rigid sphere from an elastic axisymmetric wavy surface. They demonstrated that the JKR–rigid transition seems to occur depending on the original Tabor parameter (2.2) for the sphere, rather than the ‘local’ one owing to waviness, in their case. If we estimate the sphere Tabor parameter in Akerboom *et al.*'s case *w* = 0.163 Jm^{−2}, *E** = 1.5 MPa, we obtain *μ* ≃ 28*σ*_{0}/*E** which is quite likely now larger than 1 and therefore justifies some enhancement, also in accord to the figures reported in Waters *et al.* [17] for the rough sphere. In the literature, there have been attempts to define a ‘scale-dependent’ Tabor parameter [18], so that at high magnification, where nanoscale roughness is observed involving asperities which may have radius of curvature in the *nm* range, Tabor parameter tends to decrease: but the Persson & Scaraggi [18] definition is not of immediate use, and the experiments of Akerboom *et al.* [3] seem to suggest a useful information about the way Tabor parameter should be defined for nanopatterned surfaces against spherical probes.

Moving to the experiments of Zhou *et al.* [6], the transition from partial to full contact occurred on the arrays with tall (1.4 μm) pillars and 4 μ m spacing, which produces again a ‘dimple Tabor’ parameter that is likely to be quite low, similar to that of Akerboom *et al.* [3] and of the order of . Again, it is difficult in these conditions to use the original MMA model, and the present analysis may be more appropriate. However, one problem is that the pad's effective elastic modulus was estimated from the MMA model prediction to be approximately 270 kPa, rather than measured, so their apparent good fit with the original MMA model may be more qualitative than quantitative, although it turns out consistent with order of magnitude of smooth pads of bushcrickets and stick insects. Less tall pillars produce full contact already according to the MMA model, but this may result also from the fact that the dimple Tabor parameter tends to increase, and therefore the JKR regime may be more correct. It is difficult without having the actual data, to be more precise.

Finally, Cañas *et al.* [7] used a dimple model for a much more complex problem involving a nanopatterned surface against random rough surfaces, assuming for simplicity that the dimple represents a depression in the surface with random roughness, so that *δ*_{0} represents the root mean square roughness. Nanopatterned surfaces (biomimetic PDMS pillar arrays with mushroom-shaped tips) showed always superior performances with respect to non-patterned surfaces when adhering to nano- and micro-rough surfaces, and Cañas *et al.* [7] attempted a fit of the results of the patterned surfaces using the MMA model. In doing that, they consider that the equivalent ‘dimple’ had equal height and radius , which would immediately break the JKR assumption as the dimple Tabor parameter would result in this case . However, as they extract from least-squares fit the elastic modulus and the work of adhesion, they obtain a very good fit, except that, as they indeed note, the best-fit elastic modulus is about 100 times lower and the work of adhesion is 30 times lower than expected. As they used the equation for pull-off from partial contact (3.9), we can rewrite it, putting
6.1Had they used another (more realistic) value for , which is quite arbitrary at this stage, they would have obtained
6.2Hence, equating from the original and the new equation, we do not need to repeat the fit: they would find , so to get closer to the expected modulus as well as the work of adhesion , they would need to use
Naturally, this value of is unrealistic. This shows that it is optimistic to use a simple dimple model for a problem with a patterned surface against a random rough surface.

## 7. Conclusion

The adhesion of a surface with a ‘dimple’ has been studied with a cohesive model, extending the original JKR study of McMeeking *et al.* [4]. The model shows a mechanism for a bi-stable adhesion in the JKR regime, but as expected, tends to behave less hysteretically in the range of low ‘Tabor’ parameter. In particular, the Tabor parameter is here defined as , the ratio between the theoretical strength of the material to the dimple full contact pressure *T*. The bi-stable behaviour can be restricted more specifically to the range of low *α*_{d} and high The JKR model of McMeeking *et al.* [4] is valid when and a full study and simplified equations have been obtained, permitting pull-off to be obtained with simple equations. For low values of Tabor parameter and the dimple does not show spontaneous jump into full contact and has a pull-off coincident with . From the engineering point of view, these results could trace a baseline solution for engineered patterned surfaces that use dimples to enhance adhesion or to obtain selective low and high adhered states. Obviously, the system remains still highly idealized, in that no air entrapment or contaminant, or an additional scale of roughness, has been considered, nor has possible interaction between dimples been considered.

## Competing interests

We declare we have no competing interests.

## Funding

A.P. was supported by DFG (German Research Foundation), project HO 3852/11-1.

## Appendix A. Details of the calculations

Starting from where we left off with the JKR calculation, for a cohesive model, we need to impose for continuity of stresses at the crack tip (no singularity), and a COD condition at the mouth of the crack. These two conditions are obtained from the crack under internal pressure, where we add to the localized stress *T* and the uniform stress the ‘internal’ cohesive stresses *σ*_{0} in a annular region *d* < *r* < *c*: this adds a contribution which tends to close the crack and corresponds to the auxiliary function
A 1Collecting the results from (3.3), (3.4), (A 1), requires
A 2which permits one to find the size of the cohesive end zone *d*. In particular, we would get for *c*/*b* < 1 but we do not need this, because with a cohesive model the crack is either closed or separated with *c*/*b* > 1, in which case we obtain
A 3Solving, we obtain equation (4.2).

Then, we need to impose that the COD at the end of the cohesive zone, , where *u*_{y} is displacement of the crack faces. The equation for COD at *d* is [14, 3.112]
A 4with .

When *d*/*b* < 1, from (3.3), (A 1)
A 5which gives easily^{3} equation (4.1), where the similar case *d*/*b* > 1 has been also included.

## Footnotes

- Received December 9, 2016.
- Accepted January 18, 2017.

- © 2017 The Author(s)

Published by the Royal Society. All rights reserved.