## Abstract

We determined how preload and work of adhesion control the force required to pull a circular cylindrical indenter off a microfibre array. Five regimes, with different contact behaviours, are identified for the unloading phase of indentation. These regimes are governed by two dimensionless parameters. Above a critical preload, the pull-off force and the pull-off stress reach a plateau value. The critical preload, as well as the plateau pull-off force (stress), is found to depend on a single dimensionless parameter *q*, which can be interpreted as a normalized work of adhesion.

## 1. Introduction

Recent interest in bio-inspired adhesives has motivated many researchers to fabricate microfibre arrays and to measure their adhesion (Sitti & Fearing 2003; Peressadko & Gorb 2004; Kim & Sitti 2006; Northen & Turner 2006; Aksak *et al*. 2007; del Campo *et al*. 2007; Gorb *et al*. 2007; Greiner *et al*. 2007; Murphy *et al*. 2007; Noderer *et al*. 2007; Yao *et al*. 2007). Most of these bio-inspired adhesives are made of soft elastomers to promote good contact. However, some arrays using carbon nanotubes as fibres are found to exhibit excellent adhesion, as demonstrated by the recent works of Tong *et al*. (2004), Jin *et al*. (2005), Zhao *et al*. (2006) and Sethi *et al*. (2008).

Many of the theoretical works in this area have focused on contact mechanics and adhesion of microfibre arrays (Jagota & Bennison 2002; Persson & Gorb 2003; Tang *et al*. 2005; Bhushan *et al*. 2006; Schargott *et al*. 2006; Tian *et al*. 2006; Yao & Gao 2006; Bhushan 2007; Glassmaker *et al*. 2007; Persson 2007; Yao & Gao 2007). These works give important insights into design principles of fibrillar interfaces. The present work concentrates on the interpretation of experiments that characterize the adhesive properties of these arrays. Indentation experiments based on the theory of Johnson Kendall and Roberts (JKR; Johnson *et al*. 1971) have been used extensively to characterize the adhesion of soft materials. An excellent review can be found in Shull *et al*. (1998). Application of this technique to study the adhesion of microfibre arrays is relatively recent. In these experiments, a compressive preload is applied to bring a rigid smooth indenter into intimate contact with a microfibre array. The microfibre array is usually part of a backing layer made of the same material. Once contact is established, the indenter is retracted. As unloading begins, the contact line is usually pinned (contact area fixed). Eventually, the contact line is unpinned and the contact area starts to shrink stably until a critical tensile force is achieved, which is known as the pull-off force.

To put our work into perspective, we briefly review works on measuring adhesion of microfibre arrays. Sitti & Fearing (2003) characterized adhesion of their array to silicon by measuring the force required to pull a silicon atomic force microscope probe off it. Peressadko & Gorb (2004) and Gorb *et al*. (2007) used the force needed to pull a flat glass off a microfibre array to characterize adhesion. Northen & Turner (2006) measured the force required to pull a circular ‘flat punch’ (FP) off a microfibre array and defined the adhesive strength as the pull-off force divided by the cross-sectional area of the punch. To avoid alignment problems associated with a flat indenter, Kim & Sitti (2006) used a spherical glass indenter to measure the pull-off force *F*_{c}. They also computed the energy dissipated in a load cycle *Γ* (hysteresis) and suggested that *F*_{c}/*A*_{max} or *Γ*/*A*_{max} can be used to characterize adhesion, where *A*_{max} is the maximum contact area. They interpreted *Γ*/*A*_{max} as the work of adhesion, but did not study the connection between *F*_{c}/*A*_{max} and *Γ*/*A*_{max}. The connection between hysteresis and work of adhesion is investigated in more detail by Noderer *et al*. (2007) using a film terminated microfibre array. Aksak *et al*. (2007) and Murphy *et al*. (2007) used exactly the same methodology employed by Kim & Sitti (2006) to measure the adhesion of their fibrillar arrays. In Yao *et al*. (2007), the pull-off force was used to characterize the adhesion of microfibre arrays with different fibre orientations. del Campo *et al*. (2007) and Greiner *et al*. (2007) also used spherical indenters to measure the pull-off forces of their microfibre arrays.

The survey above indicates that the pull-off force is a widely accepted measure of the adhesion. However, its usage becomes ambiguous if the pull-off force depends on the preload. This dependence was demonstrated by Greiner *et al*. (2007). They explained their observation using a theory developed by Schargott *et al*. (2006). In this theory, the fibrils are modelled as a spring foundation, while the indenter and the backing layer are assumed to be rigid. This assumption is reasonable for the indenter, which is typically very stiff in comparison with the highly deformable microfibres. However, the deformation of the backing layer may not be small, since its compliance can be comparable to the fibre array. Indeed, recent experiments of Kim *et al*. (2007) and a theory by Long *et al*. (2008) have demonstrated that backing layer thickness can significantly affect the pull-off force. It should be noted that the analysis of Long *et al*. (2008) cannot be used to study the effect of preload on the pull-off force, since the indenter was assumed to be flat. The aim of this work is to study how the pull-off force depends on the preload, the geometry (e.g. the indenter radius) and material properties (e.g. stiffness of backing layer). In the following, we will show that, above a critical preload, the pull-off force is independent of preload. Also, for the special case of a rigid backing layer, our results agree with those proposed by Schargott *et al*. (2006).

The plan of this paper is as follows. The governing equations are derived in §2. Numerical and analytical results are presented in §3. Section 4 presents the dependence of the pull-off force and the pull-off stress on preload and geometry. The summary and discussion are in §5. A summary of notations is given in the electronic supplementary material.

## 2. Formulation

We consider the problem of a circular cylindrical punch, despite the fact that most tests are carried out with a spherical indenter. However, cylindrical indenters have been used to measure adhesion (Chaudhury *et al*. 1996; She & Chaudhury 2000) and in some situations it is more desirable. For example, Shen *et al*. (submitted) have recently used a cylindrical indenter to study normal and shear contact (friction); this specimen has the advantage that the history of a line of fibres can be followed during sliding. In steady sliding, this line of fibres can be taken to be representative of the entire contact region.

The geometry is shown in figure 1*a*. The indenter has radius *R* and is very long in the out-of-plane direction. The deformation of the backing layer is in-plane strain. Since the backing layer thickness is usually much thicker than the fibre height, we model the backing layer as an elastic half space with Young's modulus *E* and Poisson's ratio *v*. To account for finite thickness of the backing layer, we allow the elastic half space to have a different elastic modulus to the microfibres. This approximation can be justified by the recent work of Long *et al*. (2008). They showed that the compliance of the backing layer can be modified by either decreasing its thickness or by changing its modulus. For example, to study the response of very thin backing layers, the modulus of the half space can be taken as infinite.

The geometry and the coordinate system are shown in figure 1*b*. The *x*-axis coincides with the top surface of the elastic backing layer. The fibre array lies between the indenter and the *x*-axis. The circular indenter has radius *R* and the contact region is a long strip occupying . The vertical displacement of the backing layer at *y*=0 is denoted by *v*.

Since the spacing and the diameter of the fibres are very small in comparison with the substrate thickness and punch radius, the fibre array is treated as an elastic foundation that behaves as(2.1)where *σ* is the contact pressure on the foundation; *k* is its effective stiffness; and *δ* is the *difference* in normal displacement between the surface of the indenter that is in contact with the fibre array and the surface of the backing layer. To justify the usage of the foundation model, we have carried out simulations of normal contact where the fibres are discrete. We found that the foundation model produces essentially the same result as the discrete model, even for a very small number of fibres (greater than four). Finally, it should be noted that our theory does not take into account fibril buckling, which will occur at sufficiently large preloads. When a fibre buckles, it loses contact and reduces adhesion (Hui *et al*. 2007). Fibre buckling can also dramatically increase the compliance of the fibre array. Fortunately, such a significant change in compliance can usually be observed from the loading data.

In this paper, we use the standard notation in contact mechanics, which is a positive *σ* is compressive. The effective stiffness *k* is related to the stiffness of a fibre *K*_{f} by(2.2)where *ρ* is the number of fibres per unit area. For example, if the fibres are bars with height *L* and cross-sectional area *A*, then the stiffness is(2.3)where *Y* is Young's modulus of the fibrils. The backing layer is assumed to be linear elastic with Young's modulus *E* and Poisson's ratio *v*. Fibrils are assumed to have identical pull-off strength, i.e. a fibril will be detached when the force acting on it reaches *K*_{f}*δ*_{c}, where *δ*_{c} is the critical stretch a fibril can withstand. Translating this to the foundation model, the interface will fail at a critical stress *kδ*_{c} and the effective work of adhesion of the interface *W*_{eff} is(2.4)This model for the behaviour of the microfibril array is similar to that proposed by Schargott *et al*. (2006). In their calculation, they neglect the deformation of the backing layer. If the deformation of the backing layer is taken into account, then the contact condition is (following Johnson 1985)(2.5)where *σ*(*x*) denotes the contact pressure in |*x*|<*a* and a prime denotes differentiation with respect to *x*. The displacement gradient *v*′ along the *x*-axis is related to the normal contact stress by (Johnson 1985)(2.6)where . Equation (2.6) is valid for all *x*. For , the integral is interpreted as a principal integral. Combining equations (2.5) and (2.6) gives(2.7)The contact stress is related to the applied normal indenter load *F* (*F*>0 compression) by(2.8)It should be noted that all the forces in this work are actually force per unit out-of-plane thickness. Additional constraints must be imposed on (2.7) to solve for the contact width. This constraint depends on whether the contact area is increasing (crack healing) or decreasing (crack growth).

### (a) Preload

During preload, the indenter is under compression and the contact line moves outwards. A useful way to think about the mechanics of indentation is to view the contact line as the front of an external crack that occupies the air gap between the indenter and the substrate. In the preload phase, this external crack heals. Since the goal of using microfibrils is to increase the adhesion, most arrays exhibit large hysteresis, i.e. adhesion is usually small in the preload phase in comparison with the retraction phase. Therefore, we assume no adhesion during preload to reduce the number of parameters in our analysis. The absence of adhesion (Hertz contact) implies that the fibres at the contact edge cannot bear tension. Continuity of traction requires the normal stress to vanish at the edge, i.e.(2.9a)Introduce the normalized variables(2.9b)where *F*_{p} is the applied preload and *a* is the contact width. Equations (2.7) and (2.8) become(2.10a)(2.10b)(2.10c)where *α* is the normalized contact width and 1/*c* is the normalized preload, i.e.(2.10d)Thus, *c* is known for a given preload. The normalized contact pressure *Σ* and the normalized contact width *α* are determined by solving (2.10*a*) and (2.10*b*), together with the normalized Hertz condition(2.10e)

### (b) Unloading

Denote the contact width at the start of unloading by *a*_{0}. For a given preload *F*_{p}, *a*_{0} or its normalized form *α*_{0} is determined by solving (2.10*a*), (2.10*b*) and (2.10*e*). During unloading, the contact width is *non*-increasing, *a*≤*a*_{0}. In many experiments, the contact line does not move until the indenter is under tension. This contact or crack pinning is due to the fact that the work required to advance the crack by a given amount is typically much greater than the energy released when the crack closes by the same amount. In our case, contact is adhesion-less in the preload phase, so the contact line will always be pinned at the beginning of unloading, until a critical force *F*_{u} is achieved. We will call *F*_{u} the unpinning force. Denote the force required to advance the contact line at a given contact width *a*(*a*≤*a*_{0}) by *F*. Physically, *F* and *F*_{u} should increase with *δ*_{c} or, equivalently, with the effective work of adhesion *W*_{eff} defined by equation (2.4).

The governing equations during unloading are still given by (2.7) and (2.8), except (2.9*a*) is replaced by the condition of decreasing contact or crack advance, i.e.(2.11)

In the unloading phase, we normalize the contact stress using(2.12)With this new renormalization, (2.7), (2.8) and (2.11) become(2.13a)(2.13b)where . The dimensionless parameter *β* in (2.13*a*) is defined by(2.13c)The condition for crack advance (2.11) becomes(2.13d)The retraction force *F* is normalized in the same way as the preload, i.e.(2.13e)

## 3. Results

Numerical results are presented first for the preload phase, followed by the unloading phase.

### (a) Results for the preload phase

The solution is expected to lie between two limits. The first corresponds to the case of stiff springs or Hertz contact, i.e. *α*≫1. By setting *c*=2/*α*^{2} and using *α*≫1, the deformation due to springs can be neglected in (2.10*a*). Equation (2.10*a*) simplifies to(3.1)which is the integral equation governing the indentation of a circular cylinder on a flat half-space (no microfibre array; Johnson 1985). The solution of (3.1) is(3.2a)(3.2b)The other limit corresponds to *α*→0, i.e. when the spring is very soft in comparison with the substrate. This is essentially the case considered by Schargott *et al*. (2006). In this case, and the integral term in (2.13*a*) can be neglected. Equation (2.13*a*) reduces to(3.3)where we have used to determine the constant of integration. To find *c*, we use (2.10*b*), this gives(3.4)In summary, the relations between preload and contact width for soft and hard springs are given by(3.5a)and(3.5b)respectively. The intermediate cases are obtained by solving (2.10*a*) and (2.10*b*) numerically. The normalized force versus normalized contact width is shown in figure 2. This figure shows that the indenter force versus contact width can be determined very accurately using the following expression:(3.6)where is the normalized indenting force during preload (2.10*d*). Equation (3.6) is the Hertz curve for the preloading of a microfibre array.

### (b) Results for unloading *a*≤*a*_{0}

Unloading takes place in two stages: in the first, the contact line is pinned (*a*=*a*_{0}) and in the second, the contact line unpins and propagates inwards (crack growth). The unpinning of the contact line will eventually lead to indenter pull-off. In a load control test, the pull-off force *F*_{c} (*F*_{c}<0) is defined to be the minimum of the retraction force *F*. The minimum can occur at *a*=*a*_{0} or *a*<*a*_{0}. In the first case, pull-off occurs at *a*=*a*_{0}, so the unpinning force is the same as the pull-off force, i.e. . In the second case, . We determine the load *F* to advance the contact line using the following procedure. The contact width *a*_{0} at the start of unloading is known from the preload calculation. Note that *β* in (2.13*a*) can be expressed in terms of , i.e.(3.7)Note that *α* and *β* are known once *a* is given. For a given *a*≤*a*_{0}, we solve (2.13*a*) for the normalized stress *ϕ*, then evaluate (2.13*e*) to obtain the normalized retraction force . The force required to unpin the crack, *F*_{u}, is obtained by setting *a*=*a*_{0} in *F*.

Five regimes can be identified during unloading. Each regime corresponds to different values of and . These regimes are discussed below.

#### (i) Regime I α≪1, *β*≪1

This regime corresponds to very compliant fibres and a flat indenter. Equation (2.13*a*) becomes approximately(3.8)The solution of (3.8), using the boundary condition , is(3.9)Equation (3.9) states that the contact pressure is uniformly distributed. For this reason, this regime is called ‘equal load sharing’ (ELS). The retraction force is(3.10)where *q* is defined in (3.7) and is independent of the contact width. Since *F* decreases with *a*, retraction in a force control test is unstable once the contact line is unpinned. The pull-off force *F*_{c} is the same as the unpinning force *F*_{u} in this regime. The relation between the pull-off force and the effective work of adhesion *W*_{eff} is obtained using (3.10) and (2.4), i.e.(3.11)Since *α*≪1, the pull-off force can be expressed in terms of the preload *F*_{p} using (3.5*a*), i.e.(3.12)A typical plot of normalized load versus normalized contact width in this regime is shown in figure 3*a*. Similar plots are generated for all the other four regimes. To summarize, in the ELS regime, the pull-off force increases with the preload, and the pull-off occurs as the contact line unpins.

#### (ii) Regime II *α*≪1, *β*≈O(1)

This regime corresponds to soft springs, stiff backing layer or small punch radius. Since we treated the fibrils as continuous spring foundation, this regime is called the ‘spring dominated’ (SD) limit. As mentioned in the introduction, Schargott *et al*. (2006) developed a model to explain how the pull-off force depends on the preload. In their model, both the indenter and the substrate were assumed to be rigid. In regime II, (2.13*a*) can be approximated by(3.13)The solution of equation (3.13) subject to the boundary condition (2.13*d*) is(3.14)Using (2.13*e*), the retraction force *F* is(3.15)The question of whether the unpinning force is the same as the pull-off force can be addressed by noting that *F* versus *a* has a unique minimum at . This implies that retraction is stable for and unstable for in a force control test. Therefore, if . The pull-off force is(3.16a)(3.16b)Using (2.4) and (3.5*a*), the pull-off force can be expressed in terms of the work of adhesion and preload *F*_{p}, i.e.(3.17a)(3.17b)The condition of stability in terms of the preload is(3.18)Equations (3.17*a*) and (3.17*b*) state that the pull-off force depends on the preload: for small preloads, the magnitude of the pull-off force increases with preload and is given by (3.17*a*); as the preload increases beyond (see equation (3.18) above), it reaches a plateau value that is independent of preload, given by (3.17*b*). The existence of a plateau pull-off force was observed by Greiner *et al*. (2007).

A typical normalized load versus normalized contact width in this regime is plotted in figure 3*b*. The magnitude of the pull-off force increases with the preload and reaches a maximum given by (3.17*b*). For preloads satisfying (3.18) or , instability will occur when the contact width shrinks to . For small preloads or , instability will occur right after the crack unpins. Note also that *β*=1 and at , consistent with our assumption that . In this regime, the indenting displacement *Δ* is well defined since the substrate is rigid. It can be shown that at instability.

#### (iii) Regime III *α*≫1, *β*≫1

This regime corresponds to an indenter with very large radius. Inside the contact region, the curvature of the indenter can be neglected and the circular indenter can be approximated by a FP. The condition implies that the backing layer is much more compliant than the fibrils. Therefore, this regime corresponds to a FP indenting on a elastic half-space, and is called the FP limit, where the governing equation (2.13*a*) can be approximated by(3.19)The solution of (3.19) satisfying (3.8) is given by the classical punch solution (Johnson 1985)(3.20)where is the normalized retraction force. Note that the normalization condition (2.13*e*) is satisfied. Equation (3.20) shows that the contact pressure goes to infinity as the edge of the contact region is approached; therefore, the boundary condition (2.13*d*) cannot be satisfied by this solution. However, (3.20) is valid except close to the contact edge, which can be treated as the tip of an external crack. The stress intensity factor of this crack is(3.21)The force required to move the contact line is determined by equating the energy release rate, , to the effective work of adhesion, . This argument results in(3.22)Equation (3.22) implies that the unloading process is unstable once the contact line is unpinned. Therefore, the pull-off force . The pull-off force can be expressed in terms of the work of adhesion by(3.23a)Using (3.5*b*), since in this regime, the pull-off force is(3.23b)Equation (3.23*b*) shows how the pull-off force depends on preload. A schematic normalized load versus normalized contact width curve in this regime is shown in figure 4*a*.

#### (iv) Regime IV *α*≈O(1), *β*≪1

In this regime, the circular indenter can be approximated as a FP, but the stiffness of the fibrils is comparable to the backing layer. We call this regime ‘FP with springs’ (FPS). The governing equation (2.13*a*) can be approximated by(3.24)(3.25)There is no simple analytic solution for (3.24) and (3.25). However, (3.24) and (3.25) can be solved numerically. Equations (3.24), (3.25) and (2.13*b*) imply that depend only on *α*. The normalized retraction force is related to by(3.26)Figure 5 shows how changes with *α*. Note in the ELS limit, which is approached as *α*→0. It is also possible to generate a higher-order asymptotic solution (see appendix A). Here, we state the result(3.27)As *α*→∞, the FP limit is approached and (3.22) shows that , which implies(3.28)Figure 5 shows that(3.29)gives an excellent fit to our numerical results.

To determine the pull-off force *F*_{c}, we examine how *F* varies with *a*. Using (3.25) and (3.29), we found(3.30)Equation (3.29) implies that for all *α*, so the retraction force is always tensile. Furthermore, (3.30) shows that *F* is a decreasing function of *a*. This indicates that unloading is unstable once the crack unpins, i.e. *F*_{c}=*F*_{u}. We can also express the pull-off force in terms of effective work of adhesion *W*_{eff}(3.31)where . We can use (3.31) and (3.6) to determine the dependence of the pull-off force *F*_{c} on the preload *F*_{p}. A typical plot of the normalized load versus normalized contact width is given in figure 4*b*.

#### (v) *Regime V**α*≫1, *β*≫1

This regime corresponds to the classical JKR limit (Johnson *et al*. 1971), where the fibrils are very stiff in comparison with the substrate. Equation (2.13*a*) can be approximated by(3.32)which is the governing equation for a circular cylindrical indenter in normal contact with an elastic half space. The analytical solution of (3.32) is found to be (Johnson 1985)(3.33)Similar to the FP limit, (3.33) cannot satisfy the boundary condition (2.13*d*). The force after the contact line unpins is determined using energy balance, similar to the analysis of the FP limit. The stress intensity factor is found to be(3.34)Equating the energy release rate, , to the effective work of adhesion, we obtain(3.35)Equation (3.35) implies that *F* achieves its minimum(3.36)at(3.37)Note that at , consistent with *α*≫1 and *β*≫1. As in the SD limit, when , retraction is unstable once the contact line unpins, i.e.. On the other hand, if the preload is sufficiently large so that , retraction is stable until the contact line shrinks to , then pull-off occurs. In this case, the pull-off force *F*_{c} reaches the plateau value *F*_{min} and is independent of preload. To summarize, the pull-off force is(3.38a)(3.38b)Since in this regime, the pull-off force can be expressed as a function of preload force using (3.5*b*); this results in(3.39a)(3.39b)The stability condition can be expressed in terms of the preload as(3.40)A schematic load versus contact width plot in this regime is given in figure 6. We summarize our results of the five regimes in table 1.

Since both the parameters depend on the contact width, different regimes can be observed in a test. Specifically, both *α* and *β* decrease during unloading, so it is possible that, at the start of unloading, the test is conducted in the JKR regime, but pull-off can occur in the other regimes. Of course, it is impossible to start unloading at a regime with a smaller *α* or *β*, and end up in a regime with a larger *α* or *β*. For example, it is impossible to go from the SD regime to the FPS regime. An exception is the ELS regime, this regime if it occurs, is the only one possible. This is because implies . ELS is possible if and only if the condition is satisfied. In terms of the preload, these conditions are(3.41)More than one regime may be valid for certain values of *α* and *β*. For example, the FPS regime collapses to the ELS regime for small *α*, the SD regime also reduces to the ELS regime for small *β*. Other examples are the JKR regime reduces to the FP regime for small *β* and the FPS regime coincides with the FP regime for large *α*. It should be noted that figures 3*a*,*b*, 4*a*,*b* and 6 were plotted assuming that one regime dominates, ignoring the fact that this may not be always possible.

## 4. Effect of preload on the pull-off force and stress

### (a) Pull-off force

To illustrate how the pull-off force in a typical test depends on the preload, we numerically evaluate the pull-off force for different preloads, assuming typical values for the parameters *R*, *k*, *W*_{eff} and *E*^{*}. Once these quantities are fixed, *β* is completely determined by *α* through *β*=*qα*^{2}, where *q* is defined by (3.7). Typical values of *q* in the literature are between 0.01 and 10. The solution procedure is as follows. For each *α*_{0}, we solve (2.13*a*), (2.13*b*) and (2.13*d*) to obtain the pull-off force. The relation between preload and *α*_{0} is obtained using (3.6). Figure 7 plots the pull-off force versus the preload for four different values of *q*. Predictions of different regimes are also marked in these figures. The pull-off forces *F*_{c} in the ELS, SD, FP, FPS and JKR regimes are given by (3.12), (3.17*a*), (3.17*b*), (3.23*b*), (3.31), (3.39*a*) and (3.39*b*), respectively. As expected, the magnitude of the pull-off force increases with the preload for small preloads. For sufficiently large preloads, the pull-off force approaches a plateau value that depends only on *q*. The existence of the plateau corresponds to the existence of a *smooth* minimum in the force versus contact width curve. It should be noted that the five regimes indicated above do not cover the entire parameter space (*α*, *β*). Thus, the JKR and SD regimes are not the only regimes with a plateau pull-off force.

Let us consider these figures in more detail. Figure 7*a* can be divided into four regions. In each region, the numerical result can be well approximated by analytical expressions of some regimes, which are indicated on the plot. Note that monotonically increases with *α* (see (3.6) or figure 2). In the first region, *α*_{0} and are both small; this means that pull-off is in the ELS, SD and FPS regimes. In the second region, *α*_{0} is of order 1, but is still small, so pull-off is in the FPS regime. Since *q*=0.01, there is a small region where *α*_{0} is large, but *β*_{0} remains small. In this narrow region, the FP, FPS and JKR regimes are all valid. Finally, *α*_{0} becomes so large that only the JKR regime is valid. For larger *q*, e.g. *q*=1 and 10, the second region where FPS is valid disappears. In the plateau region, neither the SD nor the JKR regime is approached. Of particular interest is the case of large *q* (figure 7*d*). For this case, the pull-off force is well approximated by (3.17*a*) and (3.17*b*) (SD limit) for practically all preloads.

Denote the plateau pull-off force by *F*_{∞} and its normalization by . The normalized plateau pull-off force depends only on the dimensionless parameter *q* (see (3.7)), which is inversely proportional to the square root of the effective work of adhesion. The dependence of on *q* is shown in figure 8. Note that a large *q* implies small indenter radius, small effective work of adhesion, soft fibrils or stiff backing layer. Of the five regimes, only the SD and JKR regimes predict a plateau pull-off force. in these regimes can be obtained using (3.17*a*), (3.17*b*) and (3.39*a*)(4.1a)(4.1b)Figure 8 shows that is a monotonically decreasing function of *q*. In addition, for large and small *q*, is governed by (4.1*a*) and (4.1*b*), respectively. We found that the numerical results for can be well approximated by(4.2)Denote the minimum preload where the plateau pull-off force is first reached by . We shall call the critical preload. Just as , the normalized critical preload depends only on *q*. Using (3.18) and (3.20), in the SD and JKR regimes are(4.3a)(4.3b)An expression that agrees well with these numerical results and is consistent with (4.3*a*) and (4.3*b*) is given by(4.4)

### (b) Pull-off stress

Another measure of adhesion is the pull-off stress, which is the pull-off force divided by the contact area at pull-off. The pull-off stress is harder to measure, since it may not be possible to measure the contact area directly. These two characterizations of adhesion are very different. For example, the pull-off force can be very small in the ELS regime (figure 9), even though the theoretical upper bound for the pull-off stress is achieved in the ELS regime. The existence of a plateau pull-off force implies that the pull-off stress should also approach a plateau. Denote the plateau pull-off stress by *σ*_{c}. Equations (2.13*e*) and (4.1*a*) and (4.1*b*) imply that(4.5)Since *kδ*_{c} is the maximum stress the interface can withstand, we normalize the plateau pull-off stress by *kδ*_{c}, i.e.(4.6)Since *α*_{c} depends only on *q*, equation (4.5) implies that depends only on *q*. As shown in figure 8, large *q* favours the SD, whereas small *q* favours the JKR regime. The behaviours of in these two limits are(4.7a)(4.7b)Figure 9 plots versus *q*. The asymptotic behaviour of predicted by (4.7*a*) and (4.7*b*) is also plotted for comparison. The expression(4.8)is consistent with (4.7*a*) and (4.7*b*) and provides an excellent fit to our numerical results.

## 5. Summary

We studied the effect of preload on the force needed to pull a circular cylindrical indenter off a microfibril array. The fibril array was modelled as an elastic foundation and the backing layer was assumed to be infinitely thick. In the preload phase, we obtained an approximate expression of normalized preload as a function of a dimensionless parameter, *α*, which can be interpreted as a normalized contact width. In the unloading phase, the force required to unpin and to move the contact line is determined by two dimensionless parameters *α* and *β*. We found five regimes that are characterized by different values of *α* and *β*. These regimes are chosen because the dependence of the pull-off force on preload can be determined by simple analytical expressions. We also show that there exists a critical preload above which the pull-off force reaches a plateau value. This critical preload, as well as the plateau pull-off force, depends on a single dimensionless parameter *q* defined by (3.7). Simple analytical expressions are also obtained for the plateau pull-off force and the critical preload. In the design of fibrillar adhesives, one is often interested in the pull-off stress instead of the pull-off force. Equation (4.8) relates the plateau pull-off stress to geometry and material properties through the dimensionless parameter *q*.

Our analysis suggests several ideas that can lead to more accurate characterization of adhesion of microfibre arrays. First, it is important to conduct tests in the plateau region, where the pull-off force is independent of the preload. One can reach the plateau region by applying preload greater than the critical preload *F*_{p}, which is given in (4.4) for all *q*. Second, the normalized plateau pull-off force is also obtained as a function of *q*. Since *q* is inversely proportional to the square root of the effective work of adhesion (3.7), we are able to calculate the effective work of adhesion by simply measuring the plateau pull-off force. Third, it is useful to recognize that the pull-off force depends on both the indenter radius (similar to the JKR theory) and the backing layer properties. For example, the effect of a thin backing layer is to increase *E*, which increases *q*. This will increase the pull-off load. Finally, while it is reasonable to characterize adhesion using *Γ*/*A*_{max} (see §1, third paragraph), it is not useful to characterize adhesion using *F*_{c}/*A*_{max}. Unlike the concept of the pull-off stress, *F*_{c}/*A*_{max} approaches zero for large preload since *F*_{c} approaches a constant, whereas *A*_{max} increases with preload. This characterization of adhesion may mislead one to believe that adhesion is small, while in reality, it is not.

## Acknowledgments

This work is supported by a grant from the Department of Energy (DE-FG02-07ER46463) and by a grant from the National Science Foundation (CMS-0527785). C.Y.H. appreciates discussions with Anand Jagota.

## Footnotes

- Received September 6, 2008.
- Accepted November 7, 2008.

- © 2008 The Royal Society