## Abstract

The mechanics of reversible adhesion of the gecko is investigated in terms of the attachment and detachment mechanisms of the hierarchical microstructures on its toe. At the bottom of the hierarchy, we show that a spatula pad of tiny thickness can be well absorbed onto a substrate with a large surface area and a highly constrained decohesion process zone, both of which are beneficial for robust attachment. With different peeling angles, the peeling strength of a spatula pad for attachment can be 10 times larger than that for detachment. At the intermediate level of hierarchy, we show that a seta can achieve a stress level similar to that in the spatula pad by uniformly distributing adhesion forces; as a consequence, the 10 times difference in the peel-off force of a single spatula pad for attachment and detachment is magnified up to a 100 times difference in adhesion energy at the level of seta. At the top of the hierarchy, the attachment process of a gecko toe is modelled as a pad under displacement-controlled pulling, leading to an adhesive force much larger than the gecko's body weight, while the associated detachment process is modelled as a pad under peeling, resulting in a negligible peel-off force. The present work reveals, in a more systematic way than previous studies in the literature, that the hierarchical microstructures on the gecko's toe can indeed provide the gecko with robust adhesion for attachment and reversible adhesion for easy detachment at the same time.

## 1. Introduction

Experiments have shown that the gecko's amazing ability to climb on walls and ceilings with bare feet is mainly due to the van der Waals interaction between the gecko's toes and a surface (Autumn *et al*. 2000, 2002), with capillary effects playing a somewhat minor role (Huber *et al*. 2005*a*,*b*). The underlying adhesion mechanisms of how the gecko can adhere robustly but also elegantly release adhesion has stirred a lot of interest in recent years (Persson & Gorb 2003; Gao & Yao 2004; Hui *et al*. 2004; Gao *et al*. 2005; Autumn *et al*. 2006*a*,*b*; Tian *et al*. 2006; Yao & Gao 2006; Kim & Bhushan 2007; Peattie & Full 2007; Peattie *et al*. 2007; Gravish *et al*. 2008).

Different microscopic techniques have been employed to investigate the tissue on the gecko's toes, and fine structures have been recognized all the way down to the nanoscale. For example, a detailed description of these microstructures for *Gekko gecko* was provided in Tian *et al*. (2006). As shown in figure 1, the toe of a *Gekko gecko* has a lamella structure that contains millions of hairs called setae. Each seta is approximately 120 μm long and has a cross-sectional diameter approximately 4.2 μm. The seta further branches into hundreds of spatulas through several shaft levels. At the very end of the branches, the spatula shaft holds a spatula pad that is 0.3 μm in length, 0.2 μm in width and 5 nm in thickness.

Various adhesion mechanisms for attachment and detachment of a gecko toe have been proposed. On robust attachment, Yao & Gao (2006) demonstrated theoretically that a self-similar, multilevel fibrillar microstructure could be designed from the nanoscale and up to achieve flaw-tolerant adhesion at multiple length scales. For releasable adhesion, Yao & Gao (2006) modelled the hairy microstructure as a strongly anisotropic material and showed that the adhesion strength varies with the pulling direction. Based on the force balance in an ‘adhesion-controlled’ friction model, Tian *et al*. (2006) obtained a peel-off force on the spatula pad, which varies strongly as a function of the peel-off angle. Assuming that the number of spatulas in contact with the surface is much larger during attachment than detachment, Tian *et al*. (2006) provided a different explanation for the large difference in adhesion strength between these two states. Kim & Bhushan (2007) developed a three-level hierarchical spring model to investigate capillary effects on gecko adhesion.

While the previous studies have provided valuable insights into the adhesion mechanisms of gecko, it appears that the existing models are rather fragmented and incomplete. For example, the self-similar hierarchical model of Yao & Gao (2006) could explain the role of structural hierarchy in robust attachment, but the role of hierarchy was not considered in their anisotropic material model for detachment. On the other hand, the model by Tian *et al*. (2006) could explain the large difference in adhesion strength for attachment and detachment, but failed to account for the specific geometry of the hierarchical microstructures selected by nature, such as the diameter and area density of setae which appear quite critical for the gecko to obtain robust adhesion. The shortcomings in these models have motivated us to set up a more coherent and self-consistent modelling framework to examine the hierarchical microstructures of the gecko, in the light of mounting experimental observations on this subject.

Our hierarchical modelling framework is aimed to provide a more comprehensive description of the adhesion mechanisms of the gecko from spatula pads at the bottom of the hierarchy all the way up to the movements of the gecko's toes during attachment and detachment. We will attempt to explain how the gecko secures robust attachment at each level of the hierarchy. We are also interested in finding out how the same microstructure leads to negligible detaching force when it is used in a different way by the gecko. The forces and geometric parameters predicted at different scales will be compared with corresponding experimental observations.

## 2. Adhesion at the bottom of the hierarchy: a single spatula pad

### (a) The spatula pad can be easily absorbed on to a surface owing to its tiny thickness

When a spatula pad approaches a surface, the van der Waals interaction will absorb it onto the surface. At this scale, a simplified two-dimensional elastic-beam model of the spatula pad can be used to explain that the tiny thickness of the spatula pad would enable it to have a relatively large area of contact with the surface so that the whole microstructure under the gecko's toe would be well grounded on a surface.

Figure 2*a* shows a beam of unit width with elastic modulus *E* and thickness *H* adhering to a rigid surface. The beam interacts with the surface with van der Waals adhesion energy *γ*. Let the beam be subjected to a deflection *Δ* and a rotation *θ* at one end. The total free energy in the system consists of the elastic energy stored in the beam and the interfacial adhesion energy *γl*_{d}. It has been shown (Chen *et al*. 2003) that the elastic energy in such a configuration is , where is the bending stiffness of the beam with unit width. The competition between the elastic and adhesion energies determines a stable equilibrium length *l*_{d} of the detached portion of the beam. From , we find(2.1)For simplicity, assuming in equation (2.1) yields(2.2)

Taking *E*=2.6 GPa for the elastic modulus of the spatula pads (Tian *et al*. 2006) and *γ*=0.011 J m^{−2} (Yao & Gao 2006), the dependence of *l*_{d} on the beam thickness *H* is plotted in figure 2*b* for *θ*=*π*/6 and *π*/2. When *θ*=*π*/6 and *H*=5 nm, the detaching length is *l*_{d}=39 nm which is well below the length of 300 nm for the whole spatula pad. Therefore, the spatula pad would be well attached to the surface over a total length of 261 nm. When *θ*=*π*/2 and *H*=5 nm, the detaching length is *l*_{d}=116 nm with an attaching length of 184 nm. In this way, the tiny thickness results in a large portion of the spatula pad being firmly absorbed onto the surface.

### (b) The spatula pad can achieve maximal adhesion strength even in discontinuous contact with a surface

Owing to the presence of surface roughness (Persson & Gorb 2003) or contaminants, the contact between the spatula pad and a surface may not always be smooth or continuous. Chen *et al*. (2008) have recently studied the adhesion strength of a thin film in discrete contact with a surface. It was discovered that when the film thickness *H* is much smaller than a characteristic length scale , the size of the decohesion process zone along the interface, analogous to the notion of the cohesive zone in fracture mechanics, is(2.3)for 90° peeling, where *σ*_{c} denotes the theoretical adhesion strength and is the reference cohesive zone size for a semi-infinite crack. According to equation (2.3), the decohesion process zone along the interface between a very thin film and a substrate can be constrained by the film thickness. The decohesion process zone size is an important length scale in small-scale mechanics (Gao & Ji 2003; Gao & Yao 2004; Hui *et al*. 2004). Considering a film in contact with a substrate via periodically distributed adhesion patches, Chen *et al*. (2008) showed that the adhesion strength of the film varies with its thickness. For large film thickness with a large cohesive zone size compared with the spacing between the adhesion patches, the apparent adhesion energy is given by the average adhesion energy of the interface, as expected. On the other hand, for very thin films, as the cohesive zone size is highly confined compared to the characteristic length scale of the adhesion patches, the apparent adhesion energy of the interface can be given by the peak value of adhesion energy along the interface, thus becoming independent of the discreteness of contact.

If we take *σ*_{c}=20 MPa and *γ*=0.01 J m^{−2} (Yao & Gao 2006), is estimated to be 28 nm and the cohesive zone size is *l*_{cn}∼10 nm according to equation (2.3). Therefore, the apparent adhesion energy could be maintained at the maximum adhesion energy of the interface as long as a spatula pad can continuously attach to the surface for a length comparable to its thickness; thus, the spatula pad can gain robust adhesion even in the presence of discrete contact between the pad and the surface.

### (c) With different peeling angles, the peeling strength for attachment and detachment can be 10 times different for a single spatula pad

Since the cohesive zone size is much smaller than the expected attaching length of a spatula pad on a surface, we can use energy balance to estimate the peeling strength, i.e. the critical peel-off force, of a spatula pad at different peeling angles. Kendall (1975) calculated the critical force required to peel an elastic thin film off a rigid surface as(2.4)where *θ* is the peel-off angle and *W* is the width of the film. If the Kendall model is adopted to describe the peeling of a spatula pad off a surface, equation (2.4) clearly shows that the peel-off force varies with the peeling angle *θ*. The variation of *P*_{cr} with *θ* at different values of *γ*/*EH* is plotted in figure 3. Although the Kendall model has been employed in some of the previous studies to explain the difference between attachment and detachment of geckos and insects (Gao *et al*. 2005; Huber *et al*. 2005*a*,*b*; Spolenak *et al*. 2005), it has not been well integrated in the analysis of a hierarchical structure.

Using the observed parameters for the spatula pad, we estimate . Figure 3 shows that the critical peel-off force is *γW* when *θ*=*π*/2, and can be further reduced for peel-off angles larger than *π*/2, according to equation (2.4). On the other hand, it can increase up to one order of magnitude higher than *γW* as *θ* is decreased from *π*/2. For example, it becomes 10 *γW* when *θ* is slightly smaller than *π*/6. The peel-off force on the spatula pad is estimated to be approximately 2 nN when *θ*=*π*/2 and 20 nN when *θ* is slightly smaller than *π*/6. These results are in good range compared with the experiments by Huber *et al*. (2005*a*,*b*) and Sun *et al*. (2005), which showed a peel-off force on the order of 10 nN, although the capillary effect is not considered in our work.

It has been observed (Tian *et al*. 2006) that the gecko rolls in its toes to grip a surface during attachment, resulting in a small peeling angle at the level of spatula pads; on the other hand, the gecko rolls out the toes during detachment, leading to a spatula peeling angle approximately *π*/2. The orientational dependence of the peeling-off force predicted from the Kendall model is consistent with the observation that the gecko tends to adopt lower peel-off angles on the spatula, resulting in larger peel-off forces, during attachment and higher peel-off angles, resulting in smaller peel-off forces, during detachment. Note that the Kendall model does not provide a linear relation between normal and shear forces as suggested by Autumn *et al*. (2006*a*,*b*).

Tian *et al*. (2006) accommodated the findings of Autumn *et al*. (2006*a*,*b*) with a theory of adhesion-controlled friction and predicted a peel-off force of 16 nN at 90° for detachment and 400 nN at approximately 10° for attachment. We note that the predicted difference in peel-off forces between attachment and detachment is approximately 10 times in both the Kendall model and the Tian *et al*. (2006) model. In other words, both of these models indicate that, by changing the peeling angle of a spatula pad, the gecko can achieve an attachment force one order of magnitude higher than the detachment force at the level of a single spatula pad. However, the predicted peel-off force for the attachment of a spatula pad in Tian *et al*. (2006) appears to be too large. To be conservative without losing generality, we take 2 nN as the peel-off force for detachment, corresponding to a strain level of 0.08% at the end of a spatula pad, and 20 nN as the peel-off force for attachment, corresponding to a strain level of 0.8% in the spatula, in the present analysis.

## 3. Adhesion of a single seta

To understand how adhesion works at the level of the seta, let us first imagine that the spatula pads are grown on an array of long fibrils, with one spatula pad per fibril, on the gecko's toe. The stress level and the corresponding strain level in the fibrils will be the same as those at the end of the spatula pad. The adhesion energy for this array of extended adhesive bonds consists of the energy dissipated along the interface and the elastic energy stored in the fibril volume,(3.1)where *L* is the length of the fibrils and *φ* is the area fraction of the fibril array. The inclusion of the stored elastic energy into the adhesion energy of a fibrillar structure can also be found in Gao *et al*. (2004), Hui *et al*. (2004) and Yao & Gao (2006).

According to equation (3.1), the larger the length of the fibrils, the higher will be the adhesion energy. Taking *L*=120 μm as the length of a seta, we can find *γ*_{e}≫*γ*. For *φ*≈1, when the peeling angle of the spatula pad is *θ*=*π*/2, we have *γ*_{e}=10*γ*≡0.1 J; and when the peeling angle of the spatula pad is slightly smaller than *θ*=*π*/6, we find *γ*_{e}=924*γ*≡9.24 J. Therefore, extending the spatula pads to the length of the seta gives rise to a structural unit with much higher adhesion energy than the van der Waals interaction energy. The increase in adhesion energy is tremendous. At the same time, the effect of orientation-dependent adhesion energy for peeling at different angles is magnified through the term in equation (3.1). Thus, at the scale of the seta, the adhesion energy for attachment is almost two orders of magnitude higher than that for detachment.

However, if an adhesive layer consisting of millions of such extended adhesive fibrils is directly placed on the gecko's toe, there will be a serious problem with self-bunching among them because their aspect ratio would be too large. The bunching would cause the spatula pads to be entangled so that their peeling angle cannot be controlled by the toe movement of the gecko. In reality, the gecko orderly adopts a hierarchical structure to bundle up these adhesive bonds so that the spatula pads would be extended in an organized way to large lengths and their peeling angle can be controlled by the motion of the toe. It has been shown that the spacing between neighbouring setae could prevent self-bunching (Glassmaker *et al*. 2005; Yao & Gao 2006). In addition, according to equation (2.1), the angle between neighbouring fibres could also be used to prevent self-bunching, such as the branching angles of neighbouring spatula shafts shown in figure 1.

But, adopting a hierarchical structure generates a new problem. As it is schematically shown in figure 4, the seta branches into hundreds of spatula shafts that then adhere to a surface. When a tensile force is applied to the seta, small perturbations or contact flaws would cause stress to be non-uniformly distributed among the shafts. According to the theory of fracture mechanics, crack-like flaws due to surface roughness or contaminants could result in a huge reduction of the stress level in the seta from that in the spatulas. On the other hand, the stress level in the seta would approach the stress level in the spatula shafts if the diameter of the seta is limited below a critical value, *d*_{s} (Gao *et al*. 2004, 2005; Hui *et al*. 2004; Yao & Gao 2006). Under this condition, equation (3.1) would remain valid in the presence of a hierarchy, with *d*_{s} given by (Gao & Ji 2003; Gao & Yao 2004), where is the adhesion strength and *v*_{s} is the Poisson ratio of seta. Neglecting the van der Waals interaction energy in equation (3.1), the critical diameter of the seta is,(3.2)where *l*_{b} is the length of the branch. It is interesting that the critical diameter of a seta does not depend on *P*_{cr} and is on the order of the length of its branches according to equation (3.2). The length of the spatula shafts is estimated from figure 1 to be approximately 1.5 μm, which gives the critical diameter approximately 4.5 μm. This value agrees well with the reported value of 4.2 μm for the diameter of a seta. Therefore, with size confinement, the stress in the seta can approach that in the spatula.

Because the stress in the seta approaches that in the spatula pads and there are 100–1000 spatulas in a seta, the attachment force of a single seta predicted from the current model will be 2–20 μN for peeling spatula pads at an angle slightly smaller than *θ*=*π*/6, in contrast to a detachment force of only 0.2–2 μN for peeling around *θ*=*π*/2. Autumn *et al*. (2000) obtained a maximal force of approximately 200 μN for a single seta from their experiments. In their case, the force on a single spatula pad is 0.2–2 μN. With the cross-sectional area of a spatula pad approximately 10^{3} nm×nm, the stress level in the spatula in their experiment would be of the order of 0.2–2 GPa, which would correspond to a very large strain level in the range of 8–77%.

## 4. Adhesion of a toe

### (a) Adhesion under displacement-controlled pulling

Figure 5 shows an elastic pad partially adhering on a rigid surface. The pad has elastic modulus *E* and Poisson's ratio *v*. Its thickness *H* is assumed to be much smaller than its length. The pad has adhesion strength *σ*_{c} and adhesion energy *γ* with the substrate.

When a displacement is uniformly imposed on the entire upper surface of the pad, the strain field on the far left end is considered to be uniform. Instead of using force-balance criteria, the maximum stress in the bonded portion of the pad should be derived based on the Griffith energy criterion as(4.1)where *ϕ* is the pulling angle of the reaction force (Tada *et al*. 2000). According to equation (4.1), smaller *H* results in larger *σ*. Generally, *σ* will be smaller than the adhesion strength except for very small *H* where the energy criteria should be replaced with the force-balance criteria (Gao & Ji 2003; Buehler *et al*. 2006).

### (b) Adhesion force for toe attachment and detachment

It has been observed that the gecko rolls in and grips its toes inward for attachment and rolls them out for detachment (Tian *et al*. 2006). To understand the mechanics of such movements, an adhesive pad under displacement-controlled pulling is used to model the ‘toe rolling-in and gripping’ motion, while a thin film under *π*/2 peeling is used to model the ‘toe rolling-out’ motion.

Suppose that a gecko stays on a vertical wall with *ϕ*≈0 and *v*=0.3 during attachment. Using equation (4.1), the attaching stress when the gecko rolls in and grips its toe is given by(4.2)By contrast, as the gecko rolls out its toe and peels off the surface, according to equation (2.4), the detaching stress in the toe under 90° peeling is close to(4.3)where the same adhesion energy has been used for both attachment and detachment. These relations suggest(4.4)where . The smaller the parameter *ϵ*_{d}, the larger the difference between *σ*_{a} and *σ*_{d}. If a relatively large value of *ϵ*_{d}, say *ϵ*_{d}=0.01, is taken, the stress in the toe for attachment would be approximately nine times larger than that for detachment, according to equation (4.4). The attachment force for the toe will be *σ*_{a} multiplied by a large fraction of the lower surface area of the toe, while the detachment force will be *σ*_{d} multiplied by the cross-sectional area of the toe. If *γ*=0.1 J is used as the adhesion energy for detachment, the total detaching force of a foot will be 0.001 N when its width is taken to be 10 mm. It will be very easy for a gecko to peel off its toes with such a small force.

Irschick *et al*. (1996) reported from experiments that the lower attaching area of the two front feet of a *Gekko gecko* is 227 mm^{2} and its body mass is 43.4 g. Since the exact elastic modulus of a toe is not available, a parameter study is carried out for the attaching force on a vertical wall. We use *γ*=9.24 J for the adhesion energy of a seta and take *H*=2 mm to be a rough estimate for the thickness of the gecko's toe. According to equation (4.1), when *E*=10 MPa, the force on the two front feet will be 43 N; when *E*=1 MPa, it will be 13 N; when *E*=100 kPa, it will be 4 N; when *E*=10 kPa, it will be 1.3 N. These predicted values are all much higher than the body weight of 0.4 N for *Gekko gecko*. Meanwhile, the maximal adhesion force on the two front feet was measured by Irschick *et al*. (1996) to be approximately 20 N *in vivo*.

## 5. Role of hierarchy in gecko toe adhesion

With the hierarchical modelling described previously, what possibly happens when a gecko climbs on a wall can be described as follows.

At each step, the gecko's toes take turns approaching a surface, rolling in and gripping for attachment, then rolling out for detachment. As a toe approaches a surface, the tiny thickness of spatula pads would cause a large area of the pads to be absorbed onto the surface. Then, the toe rolls in and grips for further attachment enhancement. The spatula pads are now at an angle smaller than *π*/6 to the surface. This low angle peeling results in a large adhesion force and large adhesion energy at the level of the seta. At the level of the toe, the ‘rolling-in and gripping’ movement results in displacement-controlled pulling that further enhances the adhesion force for robust attachment. As the toe rolls out for detachment, the toe is under peeling. At this stage, the peeling angle of a spatula pad is approximately *π*/2, leading to a much reduced adhesion force. The reduced adhesion force in the spatula also leads to much reduced adhesion energy of the seta, resulting in easy detachment of the toe.

## 6. Discussion

### (a) The effect of seta curvature on adhesion energy

We have treated the seta as a straight elastic fibre so that the stored elastic energy in equation (3.1) comes mostly from its axial elongation. In studying the orientation dependence of the pull-off force of a single seta, Gao *et al*. (2005) took a curved geometry into account. The natural curvature of the seta may also store extra elastic energy due to bending. To understand this issue, we have conducted a finite-element calculation of a curved elastic beam with Young's modulus *E*=2.6 GPa, Poisson's ratio *v*=0.3, cross-sectional diameter *D*=4.2 μm and beam arc length *L*=120 μm. The beam is fixed at one end and stretched at the other end. Two different radii of curvature, *R*=120 and 360 μm, are studied. Results from the simulation show that the pulling force tends to first straighten the beam as schematically given in the inset of figure 6. At the beginning of stretching, the beam behaves much more softly due to bending compared with pure axial elongation (Persson & Gorb 2003; Gravish *et al*. 2008; Kim & Bhushan 2007). The stored elastic energy associated with bending is found to be comparable with that of stretching when *R*=120 μm, but much smaller than that of stretching when *R*=360 μm, as shown by the areas under the curves in figure 6. We can estimate the stored elastic energy *U*_{B} associated with bending as(6.1)where is the moment of inertia of a beam of circular cross-section while the stored elastic energy *U*_{T} due to axial elongation is , *ϵ* being the axial elongation strain. It can be shown that(6.2)

According to equation (6.2), the ratio of *U*_{B}/*U*_{T} decreases as the radius of curvature or the axial strain increases. When *ϵ*=0.8%, corresponding to the strain level for attachment, we estimate *U*_{B}/*U*_{T}=1.2 for *R*=120 μm and *U*_{B}/*U*_{T}=0.1 for *R*=360 μm. This is consistent with the finite-element results estimated from the area ratio under the curves for different stages of strain in figure 6. Since the radius of curvature of the seta appears to be much larger than the seta length 120 μm (figure 1), it seems reasonable to consider only the elastic stored energy from the axial elongation of the seta during attachment. On the other hand, for detachment, the strain level in the fibrillar structure is taken to be approximately 0.08% in our model. In this case, the stored elastic energy due to bending can be comparable or even larger than that due to axial elongation, according to equation (6.2). Note that the relative contribution from bending tends to decrease as the strain in the seta increases.

### (b) Limitations of the present model

We have treated the spatula pad as a uniform thickness of 5 nm (Tian *et al*. 2006) while it was taken as 5 nm at the tip and 20 nm at the end of the pad by Persson & Gorb (2003). With uniform thickness, the stress level at the end of the spatula pad would be higher than that at the tip. Although the enhanced thickness at the end of the pad should not change the overall prediction of our model, it can beneficially lower the stress level at the end of the spatula pad to prevent, for example, plastic damage.

In our model, seta, the seta branch and spatula pad are all treated as linear elastic isotropic materials with the same Young's modulus. In reality, these structures, mostly composed of β-keratin, are anisotropic and their modulus may also vary with location as well as temperature and relative humidity.

In addition, we note that the peeling arm in Kendall's thin-film model is assumed to be very long, which may not be the case at the level of either the spatula pad or the toe. Although these simplifications or approximations have allowed us to explain the physics involved in gecko toe adhesion rather concisely, it should be interesting to address these issues in more depth. Furthermore, since the strain level from the experiments by Autumn *et al*. (2000) appeared to be very high, it may be interesting to incorporate a full stress–strain relation, including yielding and post-yielding behaviours, of a spatula pad into Kendall's model.

## 7. Conclusion

We have studied the mechanics of reversible adhesion of the gecko in terms of the attachment and detachment mechanisms of the hierarchical microstructures on its toe.

At the bottom level of the hierarchy, the spatula pads have a tiny thickness of approximately 5 nm, allowing them to be easily absorbed on to a solid surface. The decohesion process zone size of the spatula pad at the interface is small so that a small contact area would enable the spatula pad to make full use of the van der Waals interaction energy. Taking a peeling angle slightly smaller than *π*/6 for adhesion and *π*/2 for release results in a 10 times higher peel-off force on the spatula pad during attachment than that during detachment.

A slender hairy structure can not only provide a much higher adhesion energy than the intrinsic energy associated with van der Waals interaction but also highly magnifies the difference in adhesion energy between attachment and detachment. Limiting the diameter of the seta below a critical value ensures uniform stress distribution in the structure.

By rolling in and pulling on the toe for attachment while rolling out and peeling the toe for detachment, the difference in adhesion forces between the two states has been further magnified at the scale of the toe. In this way, the gecko attains an adhesion force much higher than its body weight with displacement-controlled pulling and a detachment force much lower than its body weight with peeling at a large angle.

## Acknowledgments

B.C. acknowledges work is supported by the Natural Sciences and Engineering Research Council of Canada (NSERC).

## Footnotes

- Received December 3, 2007.
- Accepted February 18, 2008.

- © 2008 The Royal Society