## Abstract

We present studies of a bio-inspired film-terminated fibrillar surface that has significantly enhanced adhesion and contact compliance compared with a flat control. We show that adhesion hysteresis arises from the architecture of the interfacial region. Measured in cyclic indentation experiments, it can be nearly five times the absolute work of adhesion for a flat control. Using a two-dimensional model, we propose that hysteresis develops as a result of crack trapping by fibril edges. The crack propagation mode is consistent with this model, as is the observation that with increasing hysteresis crack opening adhesion energy increases whereas crack closing adhesion energy decreases compared with a flat control. Contact compliance of the fibrillar structure is up to seven times more than that of a flat control. We present a model for the contact compliance of such structures which is in good agreement with measurements.

## 1. Introduction

Fibrillar surfaces are commonly found on the contact surfaces of the feet of many lizards and insects (e.g. Ruibal & Ernst 1965; Hiller 1968, 1976; Autumn *et al*. 2000; Eisner & Aneshansley 2000; Scherge & Gorb 2001; Rizzo *et al*. 2006). Since these creatures rely on clinging and climbing abilities for survival, a plausible supposition is that they have evolved fibrillar surface architectures in a way which enhances their chance of survival (Scherge & Gorb 2001). Drawing this hypothesis a bit further, one deduces that fibrillated surfaces must provide desirable adhesion and friction properties. In particular, past studies have shown that fibrillar surfaces are more compliant than flat surfaces of the same material, which allows better adhesion against rough surfaces (Persson 2003; Persson & Gorb 2003; Hui *et al*. 2005). Moreover, owing to the small size and compliant nature of the contacting tips or ‘spatulas’ of biological setae, fibrillar surfaces are able to attain stronger adhesion than flat surfaces of the same material (Autumn *et al*. 2000; Arzt *et al*. 2003).

With such biological systems as motivation, several groups have recently attempted to mimic the biological architecture in order to attain enhanced adhesion (Sitti & Fearing 2003; Glassmaker *et al*. 2004; Hui *et al*. 2004; Peressadko & Gorb 2004; Yurdumakan *et al*. 2005; Gorb *et al*. 2006; Kim & Sitti 2006; Majidi *et al*. 2006; Aksak *et al*. 2007; Greiner *et al*. 2007). Simple single-level structures generally fail to achieve theoretically predicted enhancement in strength and toughness due to loss of contact area, lateral collapse and buckling of fibrils (Glassmaker *et al*. 2004; Hui *et al*. 2004). Fibrillar structures with ‘mushroom’ ends have been shown to enhance adhesion significantly (Gorb *et al*. 2006; Kim & Sitti 2006; Aksak *et al*. 2007; Greiner *et al*. 2007). (See Jagota *et al*. (2007) and other articles from the same issue for reviews on this subject.)

A two-level structure which consists of a simple array of micro-posts connected at the terminal end by a thin flexible film has recently been shown to significantly improve adhesion compared with a flat unstructured control (figure 1; Glassmaker *et al*. 2006, 2007). While inspired by biological setal adhesion, this architecture is distinct from any that we are aware of in nature, although the geometry of the setal system found in the grasshopper *Tettigonia viridissima* is quite similar (Gorb & Scherge 2000). As shown in this paper, adhesion enhancement is due to the spatial variation of energy available to drive a crack (for a monotonically changing remote load) as it moves from fibril to fibril. With this undulation in available energy, the crack is forced to propagate unstably and requires a larger load than would be necessary when the fibrillar region is not present.

In addition to enhancing adhesion, the structure shown in figure 1 is advantageous for several other reasons. Specifically, by its design, it avoids two undesirable phenomena observed for large aspect ratio, free-standing posts: lateral collapse and buckling (Hui *et al*. 2002; Glassmaker *et al*. 2004; Sharp *et al*. 2004). Lateral collapse and buckling reduce contact area and adhesion. The fibrillar geometry and the terminal film of the structure in figure 1 also provide increased compliance precisely where it is needed, i.e. at the contact interface. This allows the surface to achieve initial contact more easily and, as just mentioned, it results in crack tip pinning.

In this paper, we present results of a study of the film-terminated fibrillar structure via indentation experiments. We intend to characterize quantitatively the compliance and adhesion properties of this new structure. In addition to distinguishing the differences between fibrillar and non-fibrillar flat control samples, we aim to understand the effects of fibril spacing and height on adhesion and compliance. By these studies, we hope to give insight into why the structure enhances compliance and adhesion, and how to design the geometry to tailor its properties to specified adhesion or compliance performance goals.

## 2. Experimental methods

### (a) Fabrication

The fabrication method for the structure shown in figure 1 has been discussed elsewhere (Glassmaker *et al*. 2006, 2007). Briefly, a fibrillar array is created by moulding a polymer precursor (in this case, poly(dimethylsiloxane) (PDMS)) using a negative topographic master. The master consists of an array of square holes in Si, created by photolithography and a deep reactive ion etching process. Then, the liquid polymer precursor is flowed into the holes, cured and subsequently peeled out of the master. An array of polymer posts results. The cross-sectional geometry and spacing of the posts is identical to that of the array of holes on the master, and the post height is equal to the depth of the hole.

To attach the terminal film to the ends of the posts, a polymer precursor (again PDMS, here) is first spin-coated on a hydrophobic substrate. Then, the array of posts is placed on the film while the film is still liquid. The liquid film partially wicks up the fibrils and is cured in place. The entire sample is removed from the substrate manually after a glass cover-slip is attached as a cantilever to the backing of the sample. All samples have the same terminal film thickness. Our analysis predicts that thinner films would increase adhesion. However, thinner films more readily tear and are therefore more difficult to manufacture. A typical final fibrillar structure is shown in figure 1. We usually fabricate three fibrillar samples with different structures on a single 10 cm Si wafer. On each wafer, we also fabricate a flat control. The thin film is adhered onto the flat control in the same fashion as to the fibrillar sample.

Nine fibrillar samples are discussed in this paper. All of the samples have a nominal film thickness of 4 μm and have fibrils with a square cross-section of 14 μm per side arranged in a hexagonal pattern on a 650 μm thick backing. The fibril height and fibril spacing were varied to analyse the effects of geometry. Each sample had a fibril height of 53, 60 or 67 μm and a fibrillar spacing of 38, 62 or 87 μm. For purposes of clarity and brevity, samples are referred to simply by their fibrillar spacing and fibrillar height. For example, a sample with a fibrillar spacing of 87 μm and a fibrillar height of 60 μm is referred to as ‘S87H60’.

### (b) Indentation experiments

Indentation experiments were carried out in a custom apparatus built on an inverted optical microscope, shown schematically in figure 2. It consists of a precision vertical stage attached to a rigid load train containing a strain gauge type load cell in line with a spherical glass indenter with a 4 mm radius. The surface of the glass indenter was treated with an organo-silane monolayer in order to minimize hysteresis and time-dependent effects in the control samples.1 The organo-silane monolayer was stable throughout the experiments as indicated by the reproducible results obtained from the control samples. The stage lowered the indenter into contact with the sample of interest, which was supported on the microscope platform. As this happened, the load cell voltage and stage displacement were recorded by a computer data acquisition system, and the contact area between the indenter and sample was viewed through the microscope. The computer used for data acquisition also recorded a direct digital streaming video of the contact evolution.

Typical experimental force and displacement data from a fibrillar surface (S87H60) and a flat control surface are shown in figure 3. In either case, the specimen was indented to a depth of 30 μm in the first cycle, the indenter was retracted to different depths, then cycled to the maximum depth 10 times and finally retracted completely out of contact. Note the markedly different behaviour of a fibrillar surface and flat control surface. First, the fibrillar surface is much more compliant than the flat surface; the latter requires much greater force for the same indenter depth. Second, the fibrillar surface requires a greater tensile force to separate the indenter from the sample. Third, the fibrillar surface requires a greater amount of work to separate the indenter from the sample. Fourth, the fibrillar sample shows greater hysteresis in an indentation cycle. Micrographs of the contact surface are shown in figure 3 for when the indenter is at a depth of 0 μm (a,e), 15 μm (b,d) and 30 μm (c) for both indentation and retraction. The fibrillar surface has a greater contact area than the control at all indentation depths. This is in large measure due to the deformation of the thin surface layer (fibril+film), which changes the kinematic relationship between indenter depth and contact area (Johnson 1985). Figure 3 shows that the evolution of contact of the flat control surface is approximately symmetric about maximum indentation; note the similarity between pairs (a,e) and (b,d). This is not the case for the fibrillar samples; note the difference between micrographs (a,e) and (b,d). The contact area on the fibrillar surface remains pinned when the indenter retracts from 30 to 15 μm. This fact is of central importance in our explanation of the energy dissipation process. It also turns out to be useful in determining the sample contact compliance and the work of adhesion. Particular attention should be directed towards the shape of the contact areas. As expected, the spherical indenter produces circular contact areas on the flat control surface. However, the same spherical indenter produces irregular contact areas that approach a hexagonal shape, especially when the contact area is small.

The behaviour of the control corresponds to adhesive indentation, fitting the Johnson–Kendall–Roberts (Johnson *et al*. 1971) model, which implies that(2.1)where *P* is the load; *a* is the radius of contact; *R* is the radius of the indenter; *E*^{*} is the plane strain elastic modulus; and *γ* is the surface energy. Plotting the left-hand side of equation (2.1) as a function of *a*^{3/2}, a JKR plot, as shown in figure 4, was used to estimate the surface energy and elastic modulus of the flat controls in this experiment. The flat control samples fabricated from PDMS had an average work of adhesion (on contact growth) and plane strain modulus with a 95% confidence interval of 0.0692±0.002 J m^{−2} and 3.81×10^{6}±4×10^{4} N m^{−2}, respectively.

The JKR model (equation (2.1)) cannot be applied to the fibrillar surfaces because the samples are neither isotropic nor homogeneous. Therefore, two separate experiments were devised to extract adhesion hysteresis, the work of adhesion and compliance of the fibrillar surfaces.

The goal of the first set of experiments was to measure the sample compliance. These experiments use the fact that the contact area is pinned when the indenter begins to retract from the surface, as seen in figure 3 (points c, d). For a given fixed contact area, the compliance is the change in indenter displacement per unit force, d*δ*/d*P*. A suitable normalization for the fibrillar compliance is by the flat punch (Boussinesq) value for a half space at the same contact area (Johnson 1985), i.e.(2.2)where is the contact radius and *A* is the measured contact area.

The compliance experiments were carried out by starting the indenter away from the sample. The indenter was lowered at a displacement rate of 1 μm s^{−1} to a specified depth and then completely retracted from the sample. The process was repeated for a series of depths, and a linear fit was used to find the slope of the force versus displacement curve immediately after the indenter reached its specified depth, as shown in figure 5. For the parts of the curves used to find compliance, the contact line was observed to be pinned between two rows of fibrils, and the contact area was nearly fixed.

The interfacial hysteresis experiment is similar to the compliance experiment except that the glass sphere was indented to a depth of 30 μm and then retracted to a specified depth. The indenter was then cycled 10 times between the maximum and minimum indentation depths. The process was repeated for a series of varying minimum indentation depths, as shown in figure 6. The net area under the force–displacement curve during a cycle was measured along with the contact area at the maximum and minimum indentation depths. Interfacial hysteresis was then calculated by(2.3)where Δ*A* is the difference between the maximum and minimum contact area in a loading cycle. The interfacial hysteresis was normalized by the work of adhesion of the flat control surface, *W*_{ad}=2*γ* (equation (2.1)). This process was repeated for several (6–12) minimum indentation depths. Figure 6 shows an example with three different minimum depths. There is little hysteresis for the smallest cycle (light grey curve); it is observed that, over this range of retraction, the contact remains pinned. By contrast, once the contact area changes significantly over the course of a cycle (dark grey and black curves), the adhesion hysteresis also increases significantly. This clearly indicates that hysteresis does not arise from the bulk material properties but is due to the process of separation. Since the hysteresis is large compared with that measured in a flat control, we conclude that hysteresis arises due to the fibrillar structure.

The analyses of energy loss during cyclic indentation and compliance during unloading are attractive because they allow extraction of these properties of the interface in a model-independent manner. Unfortunately, this does not extend to the extraction of absolute work of separating the interface. However, for the samples where hysteresis dominates, we show how it is possible to estimate this value with one reasonable assumption.

Consider figure 7 below in which the line represents a typical force–displacement measurement for a fibrillar sample. Suppose we are at a point A in the unloading cycle at which the displacement is *δ* and we know the contact area as well. To calculate the work of separating the interface, we attempt to compute the difference between the stored strain energy in the fibrillar sample and the external work done on it in reaching this point. Since the bulk sample is elastic, the strain energy depends only on the geometry of the final state. To calculate the strain energy stored in the sample at that point, we follow a specific path (Shull 2002). First, we assume that the adhesive forces are turned off as in an adhesion-less ‘Hertz’ contact and we indent the sample until we reach a point B, which has the same contact radius as A. Let us call this indentation depth *δ*_{H}. The strain energy of the system at the end of this first step is given by the area under the load–displacement curve until that point (*δ*_{H}).(2.4)Then, we fix the contact area and retract the indenter until we reach a displacement of *δ*. Because the contact is fixed in the second step, and we know the contact compliance, it is simple to compute the change in strain energy in the second step.(2.5)The total strain energy is(2.6)The problem in computing *U*_{E} is that, without using a model, we do not know the Hertz curve for our material. However, for samples dominated by hysteresis, we can neglect adhesion during the loading phase in comparison with its value during unloading so that the loading part of the force–displacement curve is taken as the Hertz curve. Then, the total strain energy at point A is shown in figure 7 by the area shaded light grey. The net external work done on the system is simply the area under the force–indentation curve; the difference between the two, a strain energy *deficit*, is shaded dark grey in figure 7. We identify this deficit as the work required to separate the interface.

To implement this strategy, we pick a series of points corresponding to ‘B’ in figure 7 where the compliance is already known, as described earlier. Using the known value of compliance, we extend a straight line from points B until they intersect the experimental force–displacement curve on unloading (figure 8, points ‘A’).

These intersections now represent points where we know how to calculate strain energy and external work, and hence the deficit in energy. A typical plot of the deficit versus the area is shown in figure 9; its slope is the estimate of absolute work of separation during unloading. As expected, this method works well and produces values consistent with the hysteresis experiment in cases where the hysteresis dominates. We also extract an effective work of adhesion during loading from the same experiments by making a JKR plot as in figure 4.

The result in figure 9 supports the following models of interfacial hysteresis. In the control samples, there is little bulk dissipation, so the change in strain energy in a cycle is small. This means that the measured hysteresis energy approximately equals the work done on the interface over a cycle. If we assume that the work done in separating the interface takes a single constant value on opening, *W*^{+}, and a *different* and also constant value on healing, *W*^{−}, we have(2.7)Equation (2.7) implies that the hysteresis per unit area vanishes if the work of opening the interface equals the work released on its closing. If the work of opening and the work of closing are different, hysteresis is directly proportional to Δ*A*. The data in figure 9 support the fact that *W*^{+} is a constant; the slope of the line is −*W*^{+}. Note that, if *W*^{+}≫*W*^{−}, then the hysteresis per unit area is approximated well by the work of separating the interface. In §3*a*, we will develop a model that is consistent with this assumption.

## 3. Results and discussion

Before presenting and discussing experimental results, we develop a theoretical picture to understand how microstructural parameters enhance adhesion and contact compliance. We present a semi-quantitative model for crack trapping based on the idea that the energy release rate available to a crack tip in a structured interface varies periodically. We show how this leads to enhanced energy dissipation. We also develop a model for the contact compliance of a fibrillar interface that is shown to be in good quantitative agreement with experimental data.

### (a) Crack trapping mechanism

To understand the origin of hysteresis in our experiments, we have analysed a two-dimensional plane stress model in which the fibrillar structure consists of a single row of pillars, as shown in figure 10. Fibrils have width *b* and are spaced with a period *w*. The terminal film has thickness *t*. The material is infinite in extent in the *x*-direction, and the interfacial crack between the strip and the substrate is assumed to be semi-infinite. A uniform vertical displacement, *δ*, is applied on the upper surface of the strip. We wish to analyse how the energy release rate available to propagate the interfacial crack, *G*, varies with the spatial position of its tip. This problem has translational symmetry in the sense that relative to the crack tip all fields repeat with a period equal to fibril spacing, *w*. We thus need only to consider a unit cell starting at an arbitrary location, *c*, with *x* in the range *c*<*x*<*c*+*w*.

Dimensional analysis and linearity imply that the local energy release rate, *G*, can be written as(3.1)where *ϕ* is a dimensionless function of crack tip position, geometry and Poisson's ratio, *ν*. We compute *G* as a function of crack tip position in the periodic cell using a finite element method (details will be reported elsewhere). Its typical variation with crack position within the unit cell is shown in figure 11 (*w/b*=13/3, *h/b*=20/3, *t/b*=2/3). The vertical dotted lines represent pillar edges. We have normalized *G* by(3.2)which is the energy release rate of a flat control. The horizontal lines in figure 11 represent the normalized work of adhesion of the interface, *W*_{ad}/*G*_{0}. Since *G*_{0} increases with *δ*, this horizontal line shifts downwards monotonically with increasing applied displacement.

When the crack is between fibrils, the energy available for crack growth comes primarily from the thin film. Thus, the energy release rate is expected to be low and a decreasing function of crack length until the crack reaches the next fibril whereupon it starts to increase. This behaviour is apparent in figure 11; indeed, the minimum in *ϕ* occurs just to the left of a fibril and the maximum at its right edge. Consider a situation in which the crack tip is initially located at some position to the left of the fibril as shown in figure 11. We study the growth of this crack as the remote applied displacement increases. The condition for stable crack growth is(3.3)i.e. to keep the crack in stable equilibrium, the energy release rate at the crack tip must equal the work of adhesion and it should be a decreasing function of crack tip location.

Equations (3.1) and (3.2) imply that with increasing applied displacement the *G/G*_{0} curve, the connected points, does not change while the normalized work of adhesion line moves downwards in figure 11. The *G/G*_{0} curve represents the dimensionless function *ϕ*. The crack will remain at its initial position until the applied displacement increases sufficiently so that the intersection of the horizontal line and the function *ϕ* occurs at the crack tip location. With further increase in applied displacement, the crack tip moves stably to the right until it reaches the point where *ϕ* is at a minimum, in which circumstance the applied displacement is *δ*_{max}. Any slight increase in applied displacement will result in unstable crack growth since d*G*/d*x*>0 after the minimum. Thus, the crack will propagate at an applied displacement of *δ*_{max}. In contrast, for a flat control, crack propagation occurs when the normalized work of adhesion equals unity, . Since(3.4)the ratio of applied displacement needed to propagate the crack in a fibrillar sample versus a flat control is(3.5)An external observer unaware of the microstructure will find that it takes greater applied displacement to separate the interface. Since the fibril height is small compared with the layer thickness, *h*≪*H*, and the material modulus is the same, this observer will conclude by applying equation (3.5) that the effective work of adhesion of the interface has increased by a factor of 1/*ϕ*_{min}. Moreover, the arrested crack would always be found at the position where *ϕ* is a minimum, *x*_{min}, i.e. just to the left of a fibril.

Now, with the crack tip at *x*_{min}, consider what happens as the loading is reversed. As applied displacement is decreased, the crack will close stably until the crack tip reaches the position, *x*_{max}, where *ϕ* achieves its maximum *ϕ*_{max} and *δ*=*δ*_{min}. Any slight decrease in applied displacement results in unstable crack closure. The equivalent statement to equation (3.5) now becomes(3.6)and the external observer will conclude that the effective work of adhesion during crack closure is 1/*ϕ*_{max}. The works of adhesion for crack opening and healing, *W*^{+}, *W*^{−}, respectively, introduced above can now be related to *ϕ*_{min} and *ϕ*_{max} as(3.7)

During crack opening, our argument implies that the external loading apparatus will have to release energy at a rate in excess of that required by the interface itself. Where is this extra energy expended, one might ask, in a purely elastic system? In particular, if the crack had traversed the entire periodic cell stably, we must insist that the mean energy release rate should have equalled the intrinsic work of adhesion. This has been confirmed by our numerical simulations. During part of its traversal, some of the remotely supplied energy is stored by the fibrils. During the remaining part, this stored energy would be released. As we have shown, for realistic loading (i.e. monotonically increasing remotely), the part of the cycle where the energy is stored corresponds to stable crack growth. However, the part where the stored energy is released is unstable and the energy is not recoverable.

Figure 12 shows a sequence of images in which the crack closes (indentation) and then opens (retraction of indenter) over the same set of fibrils. These images are separated by equal time intervals of 1 s. Dark regions represent the contact; light regions are the crack. The arrow on the first indentation panel (figure 12*a*) points out that, on closure, the crack is trapped on the right-hand side of the fibril, consistent with our model. Indentation panels (figure 12*d*,*e*) show the unstable crack jumping across the trapping fibril. The first retraction panel (figure 12*f*) shows trapping of the crack front by the left edge of the fibril. In figure 12*i*,*j*, the crack jumps across this fibril. It is apparent that the crack trapping geometry in the experiments is three dimensional, and not all observations can be captured by our two-dimensional model. For example, our two-dimensional model would predict that the crack location relative to the trapping fibril is the same before and after the jump. In the experiments, the crack front is wavy and its location is evidently determined both by the next trapping fibril and by the position of the rest of the front.

### (b) Contact compliance

In this section, a contact mechanics model is developed to study the compliance of the fibrillar array. We assume that the contact compliance is dominated by the behaviour of the fibrils themselves and of their connection with the backing material. That is, the role of the terminal thin film on the compliance of the fibrillar structure is assumed to be negligible. Prior to buckling, the fibrils are assumed to deform as bars and, therefore, bending and buckling effects are ignored.

We assume that fibrils are identical, have circular cross-sections, and are arranged in a hexagonal array. The radius and height of the fibrils are denoted by *a*_{f} and *L*, respectively. Given the radius of contact *R*, the number of fibrils in contact, *N*, can be obtained by simple geometry,(3.8)where , in which *w* is the spacing between the centre lines of the nearest fibrils.

The displacement of the *k*th fibrils *k*=1,2, …, *N*, relatively to the substrate, denoted by is(3.9)where *F*_{k} is the force acting on the *k*th fibril and *E* is the modulus of the fibril.

Let the average vertical displacement of the area directly underneath the *k*th fibril be denoted by . It can be estimated by assuming that the pressure is uniform in the circular region common to the *k*th fibril and the substrate. Another possible approach is to assume uniform displacement in the circular region. These two methods give nearly the same prediction; we will present only the former. According to Johnson (1985), the vertical displacement due to a uniform pressure applied on a circular region with its centre at the origin is(3.10)(3.11)where ** r** is a vector from the centre of the fibril to any point on the interface. The functions and are the complete elliptic integrals of the first and second kind with character , respectively. Using (3.13), we can calculate the average displacement of the region underneath the fibril, i.e. , to be(3.12)Here, we have assumed that the fibril layer is thin in comparison with the backing (PDMS), since the solution above assumes that the substrate is a half space.

Because the material is linearly elastic, the displacement at any point is given by superposition of that due to the force applied at that very point and that due to forces at all other points(3.13)where *F*_{i} is the force acting on the *i*th fibril and is the normalized distance between the centre of the *i*th and *k*th fibrils, which can be expressed as .

We further add to this the displacement due to the fibril itself, given by (3.9), to find the total displacement of the *k*th fibril, *V*_{k}. Next, consider the unloading process of a rigid indenter, which is initially compressed into the fibrillar array and, subsequently, has its contact area pinned. The incremental vertical displacement, *Δ*, on the fibrils is uniform. Therefore, for each fibril *k*, we have(3.14)where(3.15)

Equation (3.14) applies for each of the *N* fibrils. Together, this set of linear equations is solved numerically to determine the *N* unknown fibrillar forces, *F*_{j}, for a given *Δ*. The compliance is then computed as , where *F* is the total force, .

Predictions of this model are compared with experimental measurements of compliance in figure 13 for a sample with fibril height of 53 μm and three different fibril spacings. (Data and predictions for other samples are provided in the electronic supplementary material.) Fibrils are square in cross-section and have a width of 14 μm. Actual dimensions are somewhat higher and vary along the fibril height. For these reasons, we allow the fibril radius in the model to vary as a fitting parameter. Results shown in figure 13 have a width of 17 μm, from which we calculate an effective radius for use in the model by equating the cross-sectional area so that *a*_{f}=9.6 μm. Compliance as a function of contact area is found by varying the number of fibrils *N* according to equation (3.11).

For moderate contact area, there is good agreement between the model and the experiment (see also the data in the electronic supplementary material). However, at a certain area, there is a marked departure of the experimental trend from the theory. It is observed experimentally that this change occurs when compressive loads are sufficient to cause buckling of fibrils, as discussed later. Buckling significantly increases the contact compliance. A comparison of the fibril and substrate displacement components (given by equations (3.9) and (3.13), respectively) reveals that, in most of the experimental cases examined here, compliance is dominated by the fibrillar layer. Thus, it is not surprising that buckling of the fibrils strongly influences the overall compliance. Additionally, the effects of buckling are most pronounced on shorter fibrils (53 μm). Intuitively, one would expect longer fibrils to buckle more readily, but the opposite is true in this experiment because the tests are run with displacement control, as opposed to force control. Apparently, the higher load experienced by shorter fibres, for a given displacement, more than compensates for their higher buckling load.

Direct evidence of buckling and its effect on the material response are shown in figure 14. We observe an inflection in the load–displacement curve between points ‘*a*’ and ‘*b*’. The micrographs corresponding to these points clearly show that this feature corresponds to buckling of compressively loaded fibrils. It is interesting to note that this buckling is not detrimental to the adhesion properties, as seen in previous studies of simple fibrillar structures without a terminating film (Glassmaker *et al*. 2004; Sharp *et al*. 2004). The 10 indentation cycles shown in figure 14 demonstrate the structure's ability to buckle and then to unbuckle repeatedly.

It is instructive to examine two limiting cases in which fibrils are very short, *h*→0, or very long. In the first case, compliance is dominated by the substrate and one expects it to scale with the contact area according to the Boussinesq flat punch solution for a circular region on the surface of an elastic half space with uniform applied vertical displacement, i.e. as the inverse of the square root of the contact area (equation (2.2)). This is shown in figure 13 by the line with a slope of −1/2. When the fibrils are long, they dominate the compliance and, since their number increases linearly with the contact area, the compliance in this limit should scale inversely with the contact area. This is shown in figure 13 by a slope of −1. Our model and the experiments fall between these two limits, but somewhat closer to the latter.

Figure 15 shows data for normalized compliance as a function of contact area for samples with a fibril length of 53 μm (see the electronic supplementary material for data on samples with longer fibrils). Recall that the measured fibrillar compliance is normalized by the Boussinesq compliance, given in equation (2.2). Normalized compliance decreases towards an asymptotic value with increasing contact area. The increase in normalized compliance beyond an area of approximately 4×10^{5} μm^{2} is due to buckling of fibrils. Data from all nine samples at a fixed value of contact area 4×10^{5} μm^{2} are given in the electronic supplementary material. They confirm that there is a systematic increase in compliance with spacing between fibrils, as expected. The effect of fibril height is less clear due to the narrow range tested experimentally.

### (c) Hysteresis and works of adhesion and separation

We return now to the examination of experimental results for hysteresis and works of adhesion. Figure 16 shows measured adhesion hysteresis normalized by the work of adhesion of the flat control for samples with a fibril height of 67 μm and three different fibril spacings. Beyond a certain contact area, the normalized hysteresis approaches an asymptotic value, consistent with our picture of a well-defined work of adhesion for opening and another one for closing of the crack. Note that the hysteresis is greatest for the intermediate spacing of 62 μm. Figure 17 shows the aggregated data for normalized interfacial hysteresis for all nine samples. Note that some fibrillar samples have a value nearly five times the work of adhesion of a flat control. Interestingly, it appears that maximum hysteresis is achieved at an intermediate spacing and an intermediate fibril height. Increasing the inter-fibril distance along the direction of crack propagation decreases *ϕ*_{min} and hence promotes the crack trapping mechanism. However, because our specimens are isotropic in the plane, this separation also reduces the number of trapping sites per unit length of the crack front. Therefore, one anticipates that hysteresis will be small in the limit of both very high and low density of fibrils, and that it will be maximum at some optimal spacing. While the two-dimensional model presented earlier is sufficient to explain the crack trapping mechanism, a three-dimensional model is required to address this phenomenon. Further details on analysis of crack propagation through this structure will be presented elsewhere. Figure 18 shows maximum pull-off force data from all nine samples, normalized by its measured value for the flat control. It presents a picture consistent with that painted by the hysteresis data.

In table 1, for the set of samples with a fibril height of 62 μm and varying spacing, we compare the absolute works of adhesion measured during loading and unloading with that for a control. Note that an increase in the unloading work of adhesion is accompanied by a decrease in the loading work of adhesion, as predicted by the crack trapping model.

## 4. Summary

We have fabricated and studied a new type of synthetic fibrillar adhesion surface, inspired by biological setal systems, which consists of an array of fibrils terminated by a continuous film. The film-terminated fibrillar interface studied in this paper has demonstrated enhanced contact compliance and adhesion as measured by hysteresis, pull-off force and absolute works of adhesion. Adhesion and hysteresis enhancement in our system is due to a crack trapping mechanism, which we have explored theoretically using a two-dimensional model. The crack is trapped under the thin film between fibrils. Because the film is thin, pillars near the advancing crack front alternately absorb and release elastic energy. The absorption process is stable whereas the release is unstable, resulting in energy loss. The crack trapping mechanism predicts that the work of adhesion will be reduced and the work of separation increased relative to a flat control, which we have shown to occur in the experiments. For a crack propagating to the right, consistent with observations, the model also predicts that on opening the crack will be trapped at the left edge of a fibril, whereas on closing (retraction to the left) the crack will be trapped at the right fibril edge. It hops dynamically from one trapped location to the next.

While the two-dimensional model captures several features of interfacial separation and healing, there are important three-dimensional effects that render it quantitatively inaccurate. For example, for similar dimensions, the two-dimensional predicted adhesion enhancement factor 1/*ϕ*_{min} is considerably larger than measured experimentally. In addition, it increases without bound in the two-dimensional model as fibril separation is increased with other geometrical parameters held constant. In contrast, our experiments indicate that there is an optimal separation (and fibril height) corresponding to maximum adhesion enhancement. This is a three-dimensional effect that results from a competition between two tendencies: increased separation (i) increases the energy stored in each fibril at the point of instability and (ii) decreases the number of fibrils available per unit area. This mechanism for energy enhancement is somewhat different from the one proposed previously for fibrillar interfaces (Jagota & Bennison 2002; Glassmaker *et al*. 2005) and similar to the lattice trapping argument for crystalline solids (Thomson *et al*. 1971; Rice 1978).

The samples we have discussed here were made entirely of PDMS and were indented with a sphere having a hydrophobic coating. While this has allowed us to study the adhesion enhancement process relatively cleanly, it also means that absolute values of adhesion energy remain modest. It is our expectation that the adhesion enhancement can be increased by coupling it to stronger and possibly dissipative adhesive processes. The process we have used could be applied to other materials. For materials that cannot be moulded, direct etching techniques can be used to produce arrays of posts (Glassmaker *et al*. 2004).

## Acknowledgments

This work was supported by a grant from the DuPont company, by start-up funds for A.J. from Lehigh University and by the National Science Foundation under grant CMS-0527785. In addition, this work was performed in part at the Cornell NanoScale Facility, a member of the National Nanotechnology Infrastructure Network, which is supported by the National Science Foundation (grant ECS 03-35765). We have benefited from discussions with Prof. Manoj K. Chaudhury of Lehigh University, particularly from his suggestion on how to attach a film to the fibrillar structure.

## Footnotes

Electronic supplementary material is available at http://dx.doi.org/10.1098/rspa.2007.1891 or via http://www.journals.royalsoc.ac.uk.

↵Our purpose in treating the indenter with a hydrophobic self-assembled monolayer is to establish a control with low hysteresis and with adhesion dominated by van der Waals forces. In the absence of such a treatment, it is well-known that rate-dependent and hysteretic interfacial processes occur that (Ghatak

*et al*. 2000), in our experiments, would cloud the effects of specimen geometry on the hysteresis that we wish to establish. Hexadecyltrichlorosilane was evaporated for 1 hour onto the glass after precleaning it with 70% H_{2}SO_{4}, 30% H_{2}O_{2}for 30 min and then low-energy oxygen plasma for 1 min.- Received February 27, 2007.
- Accepted June 14, 2007.

- © 2007 The Royal Society