1 Introduction

Microfibre arrays are bio-inspired adhesion structures fabricated to mimic the attachment devices of climbing animals such as geckos and lizards ^[1-5]. Most of these structures are made of elastomers with soft or hard backings. The compliant fibrils result in softness of the array, allowing it to adapt to rough surfaces under compressive preloads so as to promote good contact ^[6-8]. Yet, indentation tests demonstrated that the pull-off force of a sphere indenter from a microfibre array depends on the preload ^[9-11]. This leads to ambiguity in characterizing the adhesive strength. To understand the phenomenon, Greiner et al. ^[10] analyzed a microfibre array with rigid backing by the model proposed by Schargott et al. ^[12] for the pads of the great green bushcricket. The case of deformable backing layer was also examined by Long and Hui ^[13] in two dimensions. A common conclusion is that the pull-off force increases with the increasing preload and then is saturated after the preload exceeds a critical value.

Evidently, the preload-dependence of pull-off force arises from the non-planar profile of the indenter. A preload increases the number of fibrils adhered to the indenter, and thus enhances the adhesive force when the indenter is retracted. Similar situation is also encountered in practical applications. Since the surface that a microfibre array contacts is often rough and irregular, a compressive preload plays an equal role in affecting the adhesion behavior. Nonetheless, detailed analysis of this problem has not been seen.

In this work, we formulate a theoretical model to quantify the preload-responsive adhesion between a microfibre array and a rough surface. The microfibre array consists of identical elastic fibrils clamped perpendicularly to a rigid backing layer with regular in-plane arrangement, and the free end of the fibrils may be of any shape. The rough surface is assumed rigid as well, and its profile is represented by randomly located steps with normally distributed height. Analytical expression of the adhesion force is derived for both the approach and retraction processes. The result reflects the preload-dependence of the pull-off force as observed in the experiments, and thus provides a theoretical guidance for adhesion control in technical applications.

2 Adhesive performance of single fibril

Figure 1 is a typical case of the microfibre array in contact with the rough surface. As already mentioned, the fibrils are elastic rods of the length $l$, the radius $r$, and Young's modulus $E$, clamped perpendicularly on a hard backing with regular array (e.g., square or hexagonal). The rough surface is treated as rigid and represented by randomly located steps with the normally distributed height $z$. When the distance from the backing layer to the average flat plane (dashed line) is $d$, some fibrils may be in contact with the rough surface while the others are not. In this case, a force $P$ per unit area needs to be applied on the backing layer to maintain mechanical equilibrium. The sign of $P$ may be positive (upward) or negative (downward), and the positive maximum ${P_{\max }}$ is the pull-off force. We will derive explicit expression of $P$.

We first focus on the adhesion between a single fibril and a typical step of height $z$. The variation of the axial force $F(z)$ in the fibril is illustrated schematically in Fig. 2 ^[14]. In the process approaching the surface, the fibril is initially free of force. Adhesion occurs when the free end of the fibril is sufficiently close to the surface so that van der Waals force works, like pull-in instability in nano-electro-mechanical systems ^[15]. Then, the fibril becomes linearly compressed. Once the compressive force reaches the critical value, that is, ${F_\mathrm b} = - 5.05EA{(r/l)^2}$ for a cantilevered rod ^[16], the fibril buckles. If the preload is not very large, the buckled fibril exhibits reversible adhesion, and the axial force essentially remains unchanged ^{[11, 17]}. Accordingly, we can write

(1)

with ${d^*} = d - z$, $\varepsilon = ({d^*} - l)/l$, $A = \pi {r^2}$, and ${l_\mathrm b} = l + {F_\mathrm b}l/(EA)$. In the retraction process, the buckled fibril firstly restores its straight configuration of length ${l_\mathrm b}$ under the constant force $ - {F_\mathrm b}$, and then experiences a gradual change from compressed to tensioned. So long as the tensile force attains the adhesion force ${F_\mathrm a}$, the fibril is pulled off, and the axial force $F(z)$ suddenly vanishes. Therefore, if the backing layer is retracted so that its distance to the average flat plane is increased from ${d_0}$ to $d$, the axial force in the fibril is given by

(2)

where ${l_\mathrm a} = l + {F_\mathrm a}l/(EA)$ and $H(x)$ denotes the Heaviside step function which equals 1 for $x \ge 0$ and vanishes for $x ＜ 0$. Note that the adhesion force ${F_\mathrm a}$ depends on the shape of the fibril end. For example, if the fibril has a hemispherical end, it is known from the Johnson-Kendall-Roberts (JKR) theory ^[18] that ${F_\mathrm a} = 3\pi rW/2$, where $W$ is the work of adhesion between the surfaces.

3 Adhesion force of microfibre array

The expression of $F(z)$ enables us to obtain the force $P$ applied to the backing layer by averaging the stresses within all the fibrils,

(3)

Here, $f$ denotes the area friction of the fibril cross-section in the array, and $\phi (z, \beta )$ is the probability density function of the normally distributed step height, given by

(4)

in which $\beta $ is the standard deviation reflecting the roughness of the surface. For convenience, the following normalized quantities are defined:

(5)

With these notations, the applied force in the approach process can be obtained as

(6)

while that in the retraction process is

(7)

in which ${\rm{erfc}}(x)$ stands for the complementary error function.

Equations (6) and (7) characterize the adhesion behavior of the microfibre array in the approach and retraction processes, respectively. If the array is brought by a preload ${p_0}$ to the position of a distance ${x_0}$ and then retracted by a force $p$ to the distance $x$ from the average flat plane of the rough surface, the preload ${p_0}$ is determined by setting $x = {x_0}$ in Eq.(6) and the force $p$ is given by Eq.(7). Because both $p$ and ${p_0}$ are related to ${x_0}$, it can be judged that $p$ depends on ${p_0}$. The reason is clear, as the fibrils which are not in contact with the rough surface at the distance ${x_0}$ do not contribute to the force $p$ in the subsequent retraction process.

4 Preload-dependence of pull-off force

To gain quantitative insights into the preload-dependence of the adhesion force, we now consider a squared array of fibrils with hemispherical ends. The material and geometric parameters are given by ^[10] $E = 1.07\;{\mathop{\rm MPa}\nolimits} $, $W = 0.068\;{{\mathop{\rm J\cdot m}\nolimits} ^{ - 2}}$, $r = 2.5\;\mu $m, $l = 20\;\mu {\mathop{\rm m}\nolimits} $, and $f = 0.23$. Consequently, we have ${l_\mathrm a} = 20.76\;\mu {\mathop{\rm m}\nolimits} $ and ${l_\mathrm b} = 18.42\;\mu {\mathop{\rm m}\nolimits} $. These data allow us to obtain numerical results for the adhesion forces with the help of Eqs.(6) and (7) in the approach and retraction processes.

Figure 3 is the variation of the applied force $p$ with the distance $x$, where the normalized standard deviation of the step height is taken as $\rho=0.01$. In the approach process, the force $p$ is initially zero and then becomes compressive at the distance of about $x=1.01$ where some fibrils begin to contact the rough surface. With the decreasing distance, the number of contacting fibrils continuously increases, and so does the compression. At the position near $x = 0.98$, almost all the fibrils are in contact with the rough surface, and the compressive force then increases linearly, until some fibrils buckle at about $x=0.93$. After all the fibrils are buckled around $x=0.91$, the compressive force is saturated. If the microfibre array is retracted from ${x_0}=0.90$, as shown by the red line, the fibrils restore their straight configurations reversely, and then the applied force $p$ varies linearly from compressive at $x = 0.93$ to tensile at $x = 1.02$. Thereafter, some fibrils start to be pulled off so that the increase of the tensile force slows down. After attaining the maximum ${p_{\max }} = 0.050$ (the normalized pull-off force) at $x = 1.028$, the force decreases and finally vanishes. If the microfibre array is retracted from ${x_0} = 1.0$, the situation changes because only a part of the fibrils are adhered to the rough surface at that position. A remarkable feature, as revealed by the blue line, is that the force $p$ increases along a different path with the increasing distance $x$, and the pull-off force ${p_{\max }} = 0.032$ at $x = 1.038$ becomes significantly smaller. The result clearly reflects the preload-dependence of the pull-off force.

The relationship between the pull-off force and the preload is demonstrated in Fig. 4, where the results for different surface roughness are also provided for comparison. Each curve experiences an initial growth followed by a plateau. The reason is attributed to the increase and saturation of the number of adhered fibrils caused by the increasing preload. It is seen that the rougher the surface, the slower the growth and the lower the plateau. This is due to the fact that less number of fibrils in the array can make contact with a rougher surface at the same time.

5 Conclusions

Based on the adhesive performance of a single fibril, we formulate a model for the adhesion between a microfibre array and a rough surface. Analytical expressions of the adhesion force are obtained for both the approach and retraction processes. Consistent with the experimental observations, with the increasing preload, the predicted pull-off force increases at first and then attains the plateau value. The result is of significance in practical applications as it may be used in adhesion control. It is noted that we have only considered the case of hard backing layer. If the backing layer is soft, the fibrils interact with each other and the scenario becomes much more complicated. This will be explored in our future work.