Statistical properties of defect-dependent detachment strength in bioinspired dry adhesives

Abstract

Dry adhesives using surface microstructures inspired by climbing animals have been recognized for their potentially novel capabilities, with relevance to a range of applications including pick-and-place handling. Past work has suggested that performance may be strongly dependent on variability in the critical defect size among fibrillar sub-contacts. However, it has not been directly verified that the resulting adhesive strength distribution is well described by the statistical theory of fracture used. Using in situ contact visualization, we characterize adhesive strength on a fibril-by-fibril basis for a synthetic fibrillar adhesive. Two distinct detachment mechanisms are observed. The fundamental, design-dependent mechanism involves defect propagation from within the contact. The secondary mechanism involves defect propagation from fabrication imperfections at the perimeter. The existence of two defect populations complicates characterization of the statistical properties. This is addressed by using the mean order ranking method to isolate the fundamental mechanism. The statistical properties obtained are subsequently used within a bimodal framework, allowing description of the secondary mechanism. Implications for performance are discussed, including the improvement of strength associated with elimination of fabrication imperfections. This statistical analysis of defect-dependent detachment represents a more complete approach to the characterization of fibrillar adhesives, offering new insight for design and fabrication.

1. Introduction

Inspired by biological systems (e.g. Tokay gecko [1]), a fibrillar surface microstructure has been identified as a potential alternative to intrinsically soft pressure sensitive adhesives (PSAs) in temporary bonding applications, particularly where ease of detachment (reversibility) and repeat attachment (reusability) are required. The ability to exploit the physical principles governing the functionality of this microstructure could allow for improved performance in a diverse range of applications, from skin adhesives to industrial pick-and-place handling [2].

There have been numerous efforts to fabricate dry adhesives comprised of synthetic fibrillar structures. In this work, we limit our attention to fibrils designed to contact the substrate via optimized tip structures (e.g. [3–7]), rather than those intended for shear-actuated side contact (e.g. [8–10]). At the interface of the fibril tip and the substrate, we anticipate regions where the separation of the two surfaces exceeds the range of the intermolecular interaction between them. These regions, referred to as cracks or defects, may result from surface roughness, fabrication imperfections or contaminant particles. The characteristics of the defects may also be dependent on the elastic properties and the preload applied when making contact. Defects are known to be stress raisers, with the severity of the stress concentration being proportional to the lateral extent of the defect. Defects exceeding a critical size will propagate at loads much lower than predicted in the absence of flaws, given the underlying strength of the interaction. For a typical elastomer system adhering by van der Waals forces to a much stiffer substrate, linear elastic fracture mechanics predicts sensitivity of the fibril adhesive strength (applied load on or elongation of a fibril at detachment) to defects with characteristic dimensions exceeding tens-of-nanometres. Since the characteristic dimensions of typical synthetic fibrils range from several micrometres to hundreds of micrometres, it is anticipated detachment will indeed occur via the propagation of defects at the interface of the fibril tip and substrate. This is supported by various experimental studies (e.g. [11–14]).

Figure 1 is a schematic of the factors influencing the adhesive strength of an individual fibril. Defect propagation is typically localized to the region of the fibril-tip interface where tensile stresses are highest. The location of this region can be controlled by the geometric features [15] or the material properties [16]. For both punch-like and mushroom-tipped fibrils, the region of high tensile stress is typically at or near the perimeter of the stalk. For punch-like fibrils, this region is the contact edge while for mushroom-tipped fibrils, on account of the presence of the flange, it is within the contact area (as shown in figure 1). Within the high-stress region, intense stress magnitudes will be further localized at the tip of individual defects. The severity of this stress concentration or intensification is controlled by the characteristic defect size. In combination, these effects dictate that the fibril adhesive strength is dependent on both the defect size and the geometric properties of the fibril. For a compliant fibril adhered to a much stiffer substrate, the detachment force has the general form

$$f_{\mathrm{c}}=\beta {\left(\frac{EW}{\pi a}\right)}^{1/2},$$ 1.1

where E is the elastic modulus of the fibril, W is the work of adhesion, a is the characteristic size of the critical defect and $\beta$ is a shape factor which is a function of the geometric properties of the fibril and has units of length squared. In general, the shape factor is dependent on the position of the defect within the interface, and on the defect size (thus changing the power law dependence on a, e.g. [17]). Only for a very small defect located in a region of nominal interface stress and far from any singular stress field is the shape factor independent of defect size. For an array of identical fibrils, it is expected that one detachment mechanism will dominate, as dictated by the geometry, such that the functional form of $\beta$ (in addition to the properties E and $W$) will be unchanged across the array. However, statistical variation in the critical defect size, a, from fibril to fibril in the array is expected as a result of surface roughness, inhomogeneities due to fabrication or contaminant particles. This will yield a distribution in the adhesive strength of fibrils in the array.

Figure 1.

Schematic of the effects controlling adhesive strength of an individual mushroom-tipped fibril. The tensile stress distribution shown is hypothesized to arise under the assumption of a perfect contact without roughness. The size of surface asperities and interfacial defects are exaggerated. (Online version in colour.)

For an adhesive patch consisting of an array of fibrillar microstructures, the analysis of the global strength of the contact bears striking similarity to the failure of fibril bundles which have been studied extensively in the context of composite materials. Classical work on this topic showed that, under conditions of equal load sharing, the strength of a bundle decays as the variability in strength of the component fibrils increases [18]. Significant effort in modelling of these systems has since been put forth (as reviewed in [19]), guiding the experimental characterization of the statistical properties of fibre strength (e.g. [20–22]).

An equivalent theoretical approach to the study of fibrillar adhesives was first adopted in [23], with a theoretical investigation involving Monte Carlo simulations having been performed assuming power-law distributed strength across a fibril array. It was confirmed that the strength of the array decayed as the variability in fibril strength increased. It was later hypothesized in [24], that defect-dependent detachment of fibrils should follow the statistical theory of fracture [25]. Under the assumption of validity of the empirical defect density function of Weibull, it was shown that the scaling of adhesive strength with contact perimeter was dependent on the distribution of defect size, potentially reconciling discrepancies across experimental studies [24].

While the role of statistical variation is suggested by these scaling irregularities, it has not been directly verified that the statistical properties of fibril adhesive strength are well described by the Weibull distribution. In this work, we seek to address this by testing the adhesive performance of a fibrillar surface microstructure using a platform which permits in situ contact visualization, allowing for the determination of the local strength of individual fibril contacts and assessment of the defect character. The results are subject to analysis based on the statistical theory of fracture. Implications for the performance of fibrillar adhesive systems are explored.

2. Overview of experiment

Figure 2 shows a schematic of the adhesion test performed using an array of mushroom-tipped PDMS fibrils on a backing layer of the same material. Details of the adhesive fabrication process and the experimental platform are given in the Material and methods section. The geometric parameters are given in the figure caption. The adhesive surface and the glass substrate were brought into contact via normal approach, which was halted when the specified compressive preload, P, was reached. They were then separated via normal retraction. The total load, F, and the displacement, u, measured from the position at which F = 0, were recorded. In situ contact visualization was performed, with high contrast between contacting and non-contacting regions obtained by frustrated total internal reflection (FTIR). Details of the experiment and its results have been published elsewhere [14].

Figure 2.

Schematic of the adhesion test of an array of mushroom-tipped PDMS fibrils, on a backing layer of the same material, contacting a glass substrate. Fibrils have radius a, and height h. They are arranged in a square packing configuration for which the centre-to-centre distance is d. The backing layer thickness is H. For the experimental system examined, $a=200\hspace{0.17em}\mu \mathrm{m}$, $h=1600\hspace{0.17em}\mu \mathrm{m}$,$\hspace{0.17em}d=800\hspace{0.17em}\mu \mathrm{m}$ and $H=5000\hspace{0.17em}\mu \mathrm{m}$. SEM of the fibril is also shown, with the mushroom tip outlined to highlight a typical fabrication imperfection. The intended array geometry is exactly as depicted, with $N=241$, however, fibrils lost in fabrication resulted in $N=237$ [14]. (Online version in colour.)

In order to correlate the time of detachment of a fibril from the in situ video with the local load on the fibril at that instant, thus determining the fibril adhesive strength, it is necessary to ensure that the load per fibril is uniform across the array. This is achieved when the backing layer is thin, the array dimensions are small, fibrils are compliant and they exhibit high strain at detachment [26]. Load sharing was assessed by ensuring that there is neither a preference for detachment of fibrils close to the array perimeter, nor any correlation between the detachment of one fibril and subsequent detachment of a neighbour. Verification of a uniform load distribution is detailed in appendix A. Furthermore, to obtain the adhesive strength of each fibril accurately the load cell should be sufficiently stiff so as not to trigger unstable detachment. This is verified by examination of the load–displacement data.

Figure 3a shows a plot of the tensile load, F, versus displacement, u. We observe that as the displacement is increased, separating the surfaces, the load increases. Progressive detachment is evidenced by the reduction in stiffness with increasing load, as well as by in situ contact visualization shown in the inset (video of the test is provided in the electronic supplementary material). The reduction in stiffness eventually leads to a load maximum, $F_{\max }=3.92\hspace{0.17em}\mathrm{N}$. If the assumption of equal load sharing holds, progressive detachment is evidence of a distribution in the adhesive strength of individual fibrils.

Figure 3.

(a) Tensile load, F, versus displacement, u, for the experiment described in §2. The compressive preload phase, $P=1\hspace{0.17em}\mathrm{N}$, is not shown. Zero displacement is defined at the point of zero load during retraction, such that the fibrils are approximately undeformed. The insets show the contact at two points during retraction, demonstrating progressive detachment of fibrils. (b) Exemplary detachments due to both a centre defect and an edge defect.

The in situ contact visualization provides sufficient spatial resolution to determine the character of defect propagation for individual fibrils. We observe two distinct mechanisms of detachment, exemplified in figure 3b. As hypothesized in figure 1, we observe defects nucleating from within the contact, under the edge of the stalk. They propagate outward to the contact edge. These are henceforth referred to as centre defects. They account for detachment in 159 of 237 fibrils (67%). A second mechanism of detachment is also observed, with defects propagating from the perimeter across the contact. These are referred to as edge defects. They account for detachment in the remaining 78 fibrils (33%).

Figure 4 shows a plot of the detached fibril fraction, $N_{\mathrm{d}}/N$, versus displacement, u. Each data point corresponds to the detachment of an individual fibril, as determined by in situ observation. The displacement for each data point, therefore, corresponds to the elongation at detachment for that fibril, a convenient measure of the adhesive strength. We seek to combine these data with knowledge of the detachment mechanism, thus giving a more complete picture of the strength distribution.

Figure 4.

Detached fibril fraction, $N_{\mathrm{d}}/N$, versus displacement,$\hspace{0.17em}u$. Each data point corresponds to the detachment of an individual fibril. Edge defect detachments and centre defect detachments are represented by orange diamonds and blue circles, respectively. (Online version in colour.)

Figure 5 shows the histogram of fibril elongation at detachment. The mean ${\overline{u}}_{\mathrm{c}}=0.225\hspace{0.17em}\mathrm{mm}$, and the standard deviation $\sigma =0.080\hspace{0.17em}\mathrm{mm}$. Determining the statistical properties of each defect population independently is a nuanced problem which is explored in §3. For now, we simply note that the edge defect propagation generally yields lower fibril adhesive strength. Edge defects, therefore, dominate early in the detachment process.

Figure 5.

Histogram of elongation at detachment, $u_{\mathrm{c}}$, for the fibril array. The frequency is in the form of the fibril fraction, $\mathrm{\Delta }N/N$. The bin size $\mathrm{\Delta }{u}_{\mathrm{c}}=0.04\hspace{0.17em}\mathrm{mm}$. (Online version in colour.)

To summarize, in the majority of fibrils, we observe centre defect propagation controlled by the characteristic interfacial stress distribution for the mushroom tip geometry. These fibrils operate as anticipated, with the flange having sufficiently reduced strain energy at the contact edge to prevent defect nucleation in this region. However, a significant number of fibrils do detach via the propagation of defects from the perimeter across the contact. It is hypothesized that they are the result of missing sections of flange, due to unintended damage during demoulding. An example of this damage is highlighted in the SEM inset of figure 2. The missing section of flange directly exposes the region at the edge of the stalk, rendering the stress state at the tip of this defect similar to that at the edge of a punch-like fibril [17]. It is, therefore, unsurprising that the strength of fibrils which detach due to propagation of these defects is greatly reduced when compared to the fundamental mechanism. Given the associated reduction in strength, we anticipate that the existence of this fabrication-imperfection-controlled mechanism is damaging to overall performance. If possible these defects should be eliminated by improving the yield of undamaged flanges in fabrication. With this in mind, as we move to statistically analyse fibril adhesive strength we seek to characterize each distribution independently. If this can be achieved, the properties obtained for the fundamental design-dependent mechanism represent the optimal performance metrics for that specific fibril, substrate, and set of environmental conditions. Furthermore, the properties of the secondary mechanism will allow us to quantify how detrimental it is to overall adhesive performance.

3. Theory and analysis

3.1. Unimodal statistical theory

We first consider each defect population independently. For a single population (a single mechanism of detachment), it is assumed that all defects propagate from the same region of the fibril-tip–substrate interface, and that this region is symmetric about the fibril centre with circumference S. Note that this is equally valid for both defect populations observed in the experiment. For edge defects, this region would be the fibril perimeter, while for centre defects it would be the axisymmetric region within the contact highlighted by the dashed line in figure 1. The statistical theory of fracture assumes that defects are highly dispersed within these regions. In this case, the number of critical defects in non-overlapping sections are independent, the probability of a critical defect existing within a small increment of the region of interest is proportional to its size, $\mathrm{d}S$, and the probability of multiple critical defects existing within this region is negligible. Known as the Poisson postulates, these are the basis for deriving a governing differential equation for the detachment probability (the full details of which can be found in many introductory statistics texts, e.g. [27]). Upon solution of this equation we obtain

$$\mathrm{\Phi }(\hspace{0.17em}f,S)=1-\exp \left(-{\int }_f{\int }_{S}g(\hspace{0.17em}f,S)\mathrm{d}S\mathrm{d}f\right),$$ 3.1

where $\mathrm{\Phi }$ is the probability of detachment, f is the local tensile load applied to the fibril and $g(\hspace{0.17em}f,S)$ is the number of defects per unit length of perimeter which yield fibril adhesive strength between f and $f+\mathrm{d}f$. The functional dependence of g on S accounts for the possibility of a non-uniform stress state along the high-stress region. In the system examined, the contact is axisymmetric and thus the stress state does not vary along the perimeter. The probability of detachment becomes

$$\mathrm{\Phi }(\hspace{0.17em}f,S)=1-\exp \left(-S{\int }_{f}g(f)\mathrm{d}f\right).$$ 3.2

The function $g(f)$ is representative of the distribution of size of the critical defect (from fibril to fibril) through its dependence on the load. If the relationship between detachment load and defect size can be deduced from fracture mechanics, i.e. if the parameters in equation (1.1) are known, then it is possible to obtain this distribution directly from the functional form of $g(f)$.

Having obtained the fundamental law of the statistical theory of fracture, the relevant task becomes characterizing $g(f)$ for a given material system. The most common approach has been to assume an empirical form and test its suitability by fitting to experimental data. On account of its simplicity and versatility, a power-law form of $g(f)$ was proposed by Weibull [25]

$$g(f)=\frac{m}{S_0{f}_0}{\left(\frac{\hspace{0.17em}f}{\hspace{0.17em}{f}_0}\right)}^{m-1},$$ 3.3

where $S_0$ is the reference fibril contact perimeter, $f_0$ is the reference value for the fibril adhesive strength and m is the Weibull modulus. This yields the detachment probability

$$\mathrm{\Phi }=1-\exp \left[-\frac{S}{\hspace{0.17em}{S}_0}{\left(\frac{\hspace{0.17em}f}{\hspace{0.17em}{f}_0}\right)}^m\right].$$ 3.4

The dependence on S yields a monotonic increase in the detachment probability with increasing fibril dimensions (assuming self-similar scaling). This is reflective of the increased likelihood of encountering a critical defect as the size of the high-stress region is increased. Moving forward we will consider only an array of identical fibrils, for which $S=S_0$ and the detachment probability simplifies to

$$\mathrm{\Phi }=1-\exp \left[-{\left(\frac{\hspace{0.17em}f}{\hspace{0.17em}{f}_0}\right)}^m\right].$$ 3.5

At this point, it is convenient to note the equivalence of the local tensile load, f, and the displacement/elongation, u, in describing the adhesive strength of a fibril. The critical force identified in equation (1.1) could be alternatively stated in terms of the elongation at detachment by recognizing that

$$u_{\mathrm{c}}=\frac{\hspace{0.17em}{f}_{\mathrm{c}}}{k},$$ 3.6

where k is the axial stiffness of a fibril. The detachment probability can, therefore, be written as

$$\mathrm{\Phi }=1-\exp \left[-{\left(\frac{u}{u_0}\right)}^m\right],$$ 3.7

where $u_0$ is the reference value for the elongation at detachment. Since elongation at detachment is the most experimentally convenient measure of fibril adhesive strength, we proceed with this definition. We use the term strength when referring to either the maximum tensile load supported by an individual fibril, or the elongation of the fibril at detachment.

The parameters $u_0$ and m are reflective of the statistical properties of the fibril strength for a particular substrate and set of environmental conditions. The reference strength, $u_0$, is related to the arithmetic mean, ${\overline{u}}_{\mathrm{c}}$, via

$${\overline{u}}_{\mathrm{c}}=u_0\mathrm{\Gamma }\left(\frac{1}{m}+1\right),$$ 3.8

where $\mathrm{\Gamma }$ is the gamma function. Over the entire physical range of m, the value of $\mathrm{\Gamma }$ varies between 0.88 and 1, meaning that $u_0$ can be viewed as primarily representative of the average fibril adhesive strength (and therefore dependent upon the average interfacial defect size). The Weibull modulus is a measure of the variability in this strength, with $m=1$ representing the stochastic limit, and $m=\mathrm{\infty }$ representing the deterministic limit in which the strength is uniquely $u_0$. The Weibull modulus is related to the standard deviation, $\sigma$, as

$$\sigma =u_0{\left[\mathrm{\Gamma }\left(\frac{2}{m}+1\right)-\mathrm{\Gamma }{\left(\frac{1}{m}+1\right)}^2\right]}^{1/2}.$$ 3.9

This is a monotonically decreasing function which in the limit $m=1$ yields $\sigma =u_0$, and in the limit $m=\mathrm{\infty }$ yields $\sigma =0$. We observe that in an absolute sense, the standard deviation is also dependent on the reference strength $u_0$. The probability density distribution for strength is given by

$$\psi =m{\left(\frac{u_c}{u_0}\right)}^{m-1}\exp \left[-{\left(\frac{u_c}{u_0}\right)}^m\right].$$ 3.10

The typical approach in the study of fracture is to test many samples independently, assign a probability to each based on ranking the strength observed. For the purpose of characterization, conditions of equal load sharing dictate that the experiment described in §2 effectively obtains the adhesive strength data for N fibrils simultaneously. The detached fibril fraction, $N_{\mathrm{d}}/N,$ is exactly the normalized rank in strength from lowest to highest, and is therefore equivalent to the detachment probability

$$\mathrm{\Phi }=\frac{N_{\mathrm{d}}}{N},$$ 3.11

and, for a single defect population, equation (3.7) can be fit to these experimental data. If the assumed functional form of the defect density distribution is appropriate, leading to a good fit, then the Weibull statistical properties $u_0$ and m are obtained.

3.2. Bimodal statistical theory

If two defect populations exist concurrently at the interface of the fibril tip and substrate then the situation is complicated significantly. Consider that a fibril detaching due to an edge defect, necessarily contains a centre defect which would have resulted in higher adhesive strength. The information about this defect is lost, and the emerging statistical properties of the centre defect distribution are distorted.

One possible approach to this problem is to develop a statistical framework which accounts for the possibility of multiple defect populations. Consider that the probability of detachment must be the product of the probabilities of detachment due to each population individually. If the first population exists for all fibrils, while the secondary population exists only among a fraction of all fibrils, $\alpha$, (known as partial concurrency [28]) then this is given by

$$\mathrm{\Phi }=1-(1-\alpha )\exp \left[-{\left(\frac{u}{u_{01}}\right)}^{m_1}\right]-\alpha \exp \left[-{\left(\frac{u}{u_{01}}\right)}^{m_1}-{\left(\frac{u}{u_{02}}\right)}^{m_2}\right],$$ 3.12

where $u_{0i}$ and $m_i$ are the statistical properties of individual populations. Note that in the limit $\alpha =0$ we obtain the result for a single population, while in the limit $\alpha =1$ we have full concurrency of the two populations among all fibrils. The issue is that in this form, absent additional information, equation (3.12) lacks utility for fitting to experimental data. The parameter space is extremely large and the probability itself is not unique for all combinations of parameters.

The other approach, afforded by knowledge of the detachment mechanism on a fibril-by-fibril basis, is to try to decouple the populations and use the unimodal framework of §3.1 to characterize their statistical properties individually. The simplest approach is to isolate the centre defects and re-rank them within the interval [1,N], relaxing the constraint on integer ranking and evenly spacing the data points. However, this approach neglects the influence of coupling of the populations. An improved method is mean order ranking [29], which considers the position of edge defects within the sequence. The more edge defects that are encountered, the lower the rank assigned to the next centre defect relative to the simple re-ranking approach. This reflects the increased probability that the disguised centre defect strengths would have exceeded subsequent data in the sequence. Mathematically, this is achieved by considering each data point in the sequence in order. When an edge defect is encountered in the sequence, the increment in the rank for the next centre defect is recalculated as

$$\mathrm{\Delta }=\frac{N+1-j}{1+(N-i)},$$ 3.13

where j is the re-rank of the previous centre defect in the sequence. Upon re-ranking, the associated detachment probability is calculated as

$$\mathrm{\Phi }=\frac{\hspace{0.17em}j}{N}.$$ 3.14

Figure 6 shows the detachment probability, $\mathrm{\Phi }$, versus displacement,$\hspace{0.17em}u$ for both the mean order ranked centre defect data, and the original bimodal data. The shift in centre defect data achieved for both simple re-ranking (not shown) and mean order re-ranking, reflects the higher strength of centre defects. However, mean order re-ranking accounts for the dominance of edge defects at low displacement by reducing the detachment probability in this region, relative to simple re-ranking. With the mean order rank of centre defect detachments, we are able to fit the unimodal statistical framework of equation (3.7). A nonlinear least-squares fitting method is preferred, as described in appendix B. This yields a reference strength $u_0=0.285\hspace{0.17em}\mathrm{mm}$ and Weibull modulus $m=4.08$. The corresponding average elongation at detachment, obtained from equation (3.8), is ${\overline{u}}_{\mathrm{c}}=0.259\hspace{0.17em}\mathrm{mm}$. The standard deviation, obtained from equation (3.9), is $\sigma =0.091\hspace{0.17em}\mathrm{mm}$.

Figure 6.

Detachment probability, $\mathrm{\Phi }$, versus displacement, $u$. Raw experimental data, containing both edge defect detachments (orange diamonds) and centre defect detachments (blue circles), is shown alongside mean order ranked data for centre defects (blue crosses). Results of fitting equation (3.7) to each dataset are shown as solid lines. For the mean order ranked data the associated parameters are $u_0=0.285\hspace{0.17em}\mathrm{mm}$ and $m=4.08$. The root mean square error of the fit is $\mathrm{\Sigma }=0.036$. For the raw data, the associated parameters are $u_0=0.250\hspace{0.17em}\mathrm{mm}$ and $m=3.05$. The root mean square error for the fit is $\mathrm{\Sigma }=0.020$. (Online version in colour.)

If the experimental platform is not able to determine the defect character for each fibril, the data have to be treated as if there is only one defect family that controls the adhesive strength. The raw experimental data would then be fitted to a single Weibull model. While this does yield a reasonable fit, the extracted statistical parameters (given in the caption of figure 6) do not accurately characterize either defect population. They represent an underestimation of the reference strength and Weibull modulus of the fundamental detachment mechanism by 12% and 25%, respectively. Furthermore, this mischaracterization, on account of a failure to recognize bimodality, would lead to a loss of insight when considering the performance of fibrillar microstructured samples. This will be discussed in detail in §3.3.

For now, we seek to characterize the statistical properties of the edge defect population. Mean order ranking could be applied to this population, but this would assume that edge defects exist in all fibrils. Given the hypothesis that edge defects are primarily associated with damaged mushroom tips, this is unlikely. Accordingly, we adopt an alternative approach. As we now have an estimate of the statistical properties of the centre defect population, the parameter space of equation (3.12) can be greatly reduced. With $u_{01}$ and $m_1$ taken from the preceding analysis, we perform a nonlinear least-squares fit of this equation to the raw experimental data to obtain the three remaining unknown parameters, $u_{02}$, $m_2$ and $\alpha$, all associated with the edge defect population.

Figure 7 shows the fit when mean order ranked statistical properties of the centre defect population are combined with the bimodal framework of equation (3.12). We prescribe $u_{01}=0.285\hspace{0.17em}\mathrm{mm}$ and $m_1=4.08,$ and obtain the statistical properties of the edge defect population as $u_{02}=0.201\hspace{0.17em}\mathrm{mm}$ and $m_2=3.35$. The fraction of fibrils exhibiting the edge defect population is determined to be $\alpha =0.390$ or 39%. This is physically meaningful, given that 33% of detachments were due to edge defects in the experiment. It is expected that this value should be less than $\alpha$ given that some fibrils with edge defects will still detach due to large, and therefore critical, centre defects. In figure 7, we also show the probability functions associated with each population individually (dashed curves), to highlight the limits between which the coupled behaviour lies.

Figure 7.

Detachment probability, $\mathrm{\Phi }$, versus displacement, $u$, showing the resulting fit when combining the statistical properties of centre defects obtained by mean order ranking, $u_{01}=0.285\hspace{0.17em}\mathrm{mm}$ and $m_1=4.08$, with the bimodal detachment probability of equation (3.12). The three unspecified parameters obtained by fitting are $u_{02}=0.201\hspace{0.17em}\mathrm{mm}$, $m_2=3.35$ and $\alpha =0.390$. The root mean square error for the fit is $\mathrm{\Sigma }=0.019$. The associated unimodal detachment probabilities are shown for each defect population. (Online version in colour.)

Given the large parameter space of the bimodal framework, we seek to confirm the validity of the statistical parameters obtained. To this end, we perform a Monte Carlo simulation which takes as its input the statistical parameters $u_{01}$, $m_1$, $u_{02}$, $m_2$ and $\alpha$, and generates a discrete bimodal distribution in which the statistical origins of each data point can be identified. This permits the generation of a histogram of fibril adhesive strength, decomposed by defect type. Differences in these histograms are observed, even where combinations of parameters lead to very similar attachment probability distributions. This is described in detail in Appendix C. Histograms generated are compared qualitatively to the experimental result of figure 5. Excellent agreement is observed for the parameters obtained by combining the method of mean order ranking of the primary population with nonlinear least-squares fitting of the bimodal framework for the secondary population. This gives confidence that each defect distribution independently has been well characterized.

3.3. Performance of fibril arrays exhibiting Weibull distributed strength

Fibril adhesive strength, in the form of elongation at detachment, appears to be well described by Weibull's statistical theory of fracture. We, therefore, seek to examine the influence of this statistical variation on the adhesive performance of a fibrillar microstructured surface. For the fibrillar array (shown in figure 2), the load developed during contact

$$F=\sum _{i=1}^{N_{\mathrm{a}}}{f}_i,$$ 3.15

where $N_{\mathrm{a}}$ is the number of attached fibrils and $f_i$ is the local load experienced by fibril i. Under conditions of equal load sharing, the total load is

$$F=N_{\mathrm{a}}f,$$ 3.16

where f is the local load experienced by all attached fibrils. This can be stated in terms of the displacement as

$$F=N_{\mathrm{a}}ku.$$ 3.17

The probability of individual fibril detachment is equivalent to the fraction of fibrils within the array which have detached, yielding

$$N_{\mathrm{a}}=N-N_{\mathrm{d}}=(1-\mathrm{\Phi })N.$$ 3.18

For an array exhibiting two partially concurrent defect populations, the load is given by

$$F=Nku\left[(1-\alpha )\exp \left(-{\left(\frac{u}{u_{01}}\right)}^{m_1}\right)+\alpha \exp \left(-{\left(\frac{u}{u_{01}}\right)}^{m_1}-{\left(\frac{u}{u_{02}}\right)}^{m_2}\right)\right].$$ 3.19

In the limit of a single population, $\alpha =0$, an analytical expression for the maximum load can be obtained by determining the point of zero gradient, as

$$u{|}_{\frac{\mathrm{d}F}{\mathrm{d}u}=0}=u_0{\left(\frac{1}{m}\right)}^{1/m},$$ 3.20

yielding

$$F_{\max }=Nk{u}_0{\left(\frac{1}{m}\right)}^{1/m}\exp \left(-\frac{1}{m}\right),$$ 3.21

which is an upper bound on the adhesive strength of a fibril array exhibiting the statistical parameters $u_0$, and m. As expected, increase in the average fibril adhesive strength, or equivalently reduction in the average defect size, results in increase of the array adhesive strength. Of equal significance in determining the array strength is the variability in strength or defect size, as reflected by the dependence on m. Terms in m yield a reduction in the strength by a factor of 0.368 in the stochastic limit, $m=1$, as compared to the deterministic limit, $m=\mathrm{\infty }$. This is a consequence of weak fibrils in the distribution, which cannot be compensated for by fibrils with higher than average adhesive strength. Early detachments are more damaging as they increase the share of load on fibrils which remain in contact and lead to a load maximum at lower displacement.

The preceding results allow for the assessment of the influence of the weaker edge defect population on the performance of the fibril array tested. We obtain the stiffness of the fibril array from the experimental load–displacement data as $Nk=41.6\hspace{0.17em}\mathrm{N}\hspace{0.17em}\mathrm{m}{\mathrm{m}}^{-1}$. In combination with the statistical parameters of the fundamental mechanism, $u_0=0.285\hspace{0.17em}\mathrm{mm}$ and $m=4.08$ (i.e. obtained from centre defect mean order ranking), equation (3.21) provides an estimate of the upper bound on load, $F_{\max }=6.58\hspace{0.17em}\mathrm{N}$. The maximum load observed in the experiment was just $F_{\max }=3.92\hspace{0.17em}\mathrm{N}$, suggesting that the impact of the edge defect population is a reduction in the adhesive strength of the fibril array of the order of 40%. This is of the order of the percentage of fibrils which possess these fabrication imperfections, and highlights the adhesive strength that may be gained by improving the yield of undamaged fibril tips assuming similar substrate properties and environmental conditions.

4. Discussion and conclusion

Equation (3.21) demonstrates that the variability in local adhesive strength from fibril to fibril can play an important role in determining the global adhesive strength of a microstructured sample. This emphasizes the significance of characterizing the statistical properties of fibril adhesive strength in a systematic way. By in situ observation of the contact, we have demonstrated that it is possible to determine elongation at detachment on a fibril-by-fibril basis across the array. In addition to observing defect propagation from within the contact below the stalk edge, defect propagation was also observed at the contact edge. This extraneous detachment mechanism is most likely associated with mushroom tips damaged during fabrication. While the statistical theory of fracture suggests a framework for characterization of the statistical properties of fibril adhesive strength, the coupling of these two defect populations complicates this process. Detachments due to one mechanism disguise statistical information about the other. The expectation that edge defects exist among only a fraction of all fibrils further increases the parameter space. These challenges are addressed by first decoupling the statistical properties of the fundamental centre defect population, before combining the statistical properties which emerge from this method with a bimodal probability framework. On the basis of this analysis, it is observed that, individually, the populations appear to be well characterized by the defect density function of Weibull. The capability of the model to predict an upper bound on array adhesive strength subsequently proved useful in determining the influence of the secondary, fabrication-dependent mechanism. We observe that the percentage reduction in strength is of the same order as the percentage of fibrils with fabrication imperfections, approximately 40%, and thus large increases in strength may be possible on smooth surfaces by increasing the yield of undamaged mushroom tips. It is anticipated that when making improvements to the fabrication process, repetition of this statistical analysis would be a valuable tool in the assessment of progress.

In considering how the results may change for different fibril geometries, we return to the discussion of §1 and the fibril adhesive strength given in equation (1.1). It is noted that each defect type will be associated with a different form of the shape factor $\beta$. For edge defects, as compared to centre defects, it is expected that this will lead to lower strength for the same characteristic defect size. Unfortunately, to the authors knowledge, no analysis of these specific defect types has been performed. Analyses of mushroom-tipped fibrils have typically focused on comparison to punch-like fibrils for defects at the contact edge (e.g. [30]), and have not analysed the strength due to defects within the contact. Consequently, exact solutions for the shape factors as a function of geometric features (fibril length, stalk diameter, flange diameter, flange thickness) are not available. Intuitively, we expect that for centre defects nucleating below where the stalk meets the flange, reduction in the thickness or reduction of the diameter of the flange will give lower fibril adhesive strength. Increase in the diameter of the stalk should also lead to lower strength for both centre defects and for edge defects associated with missing sections of flange. These effects are expected to change the average strength, with limited impact on the defect size distribution and thus the statistical variation in fibril adhesive strength. The dependence of the adhesive strength of a fibrillar sub-contact on the fibril length is expected to be weak, although fibril length can play an important role on system performance in other ways, for example governing the contribution to the toughness at large length scales [31–34] or controlling the tendency for fibril matting [34–39].

Changes in the stalk diameter have further significance in that, unlike other geometric properties discussed, they are expected to directly influence the statistical aspects of failure for the centre defect population. Increasing the stalk diameter increases the extent of the high-stress region highlighted in figure 1. The probability of sampling a critical defect thus increases, as is reflected in equation (3.4). One option for comparison of fibrils with different stalk diameters is to use equation (3.5), which contains no explicit size effect, for fitting purposes. In this case, the changes in geometry will be reflected implicitly in the values of $f_0$ and m obtained. Alternatively, equation (3.4) can be used directly to verify that the scaling of strength with stalk diameter is as predicted by this statistical framework.

The study of fibrils with reduced characteristic dimensions is an important technical challenge to be addressed in future work, and will require consideration of both the resolution of in situ visualization capabilities and the ability to maintain equal load sharing conditions. In regard to the latter, backing layer effects are expected to become more pronounced as the fibril length is reduced, and as the array size is increased [26]. Contact height differences due to inhomogeneities in the backing layer or roughness at the fibril scale [23,32,40], as well as loading imperfections [41,42], will become more pronounced with respect to the fibril length. Statistical models may, therefore, have to account for non-equal load sharing, as has been required in the study of the failure of fibres in composite materials [43].

Moving forward, it is our hope that the statistical characterization of fibril strength will prove to be a useful tool for assessment of performance in a variety of conditions. Of particular, interest is the examination of adhesion on substrates with more severe surface roughness [44–46]. Where roughness exists on much smaller scales than the fibrils themselves (as in the present study) we anticipate that the tip geometry will control the interfacial stress distribution and thus the region from which defect propagation occurs. Roughness will control the size of nucleation points within this region. As the length scale of roughness increases, the dominant role of the tip geometry may be precluded. Defects may propagate from multiple regions across the tip–substrate interface. In this case, it will be necessary to consider the non-uniform load distribution at the interface within the statistical framework if the defect size distribution is to be characterized, as has been done in the context of brittle fracture [47,48]. We also note that the influence of surface roughness may change in the presence of fluid at the interface, with the formation of capillary bridges effectively extending the range of the surface interaction [49].

Other topics of interest include systematic assessment of the role of surface contaminants [50], where introduction of a known size distribution of particles to the substrate may provide insight as to how these give rise to defects and in turn control fibril strength. Intentional introduction of fabrication imperfections [51] could also provide fundamental insight into the mechanics governing fibril adhesive strength. Furthermore, statistical characterization at various stages of cyclic loading can shed light on the durability of fibrils, where damage may accumulate over time [2]. The presence of shear at the interface [52] will also change the characteristics of the interfacial stress distribution and thus likely the statistical properties of fibril adhesive strength.

5. Material and methods

The microstructured sample was made from polydimethylsiloxane (PDMS, Sylgard 184, Dow Corning, Midland, MI, USA) via replica moulding as described in detail in ref. [14]. Briefly, for replica moulding, an aluminium mould with milled microscopic holes (negative of the mushroom structure) was used as template. The bottom of the mould was sealed using a polyethylene terephthalate (PET) film, Sigma (SIG GmbH, Düsseldorf, Germany). The surface roughness of the PET film was transferred to tips of the mushroom-shaped microstructures. This has been characterized by atomic force microscopy and is determined to have a Hurst exponent close to unity, and a roll of wavenumber of 2.5 µm⁻¹. Qualitatively the roughness is on the scale of tens of nanometres, with an RMS height difference of 37 nm and an RMS gradient of 35.

Adhesion tests were performed with a tensile tester (Inspekt table BLUE, Hegewald&Peschke, Nossen, Germany) equipped with a 50 N load cell. We corrected the measured displacement by accounting for the machine compliance, $C_{\mathrm{s}}=7.43\hspace{0.17em}\mu \mathrm{m}\hspace{0.17em}{\mathrm{N}}^{-1}$. The tensile tester was modified to perform adhesion tests on a smooth and nominally flat glass substrate. A θ-ϕ-goniometer (MOGO, Owis, Staufen im Breisgau, Germany) was used in order to align the substrate with the microstructured sample. A mirror and a camera were mounted below the transparent glass substrate. The contact of each pillar with the substrate was visualized in situ by the principle of FTIR, as described in detail in ref. [14]. Videos of contact formation and detachment were recorded and, subsequently, correlated with force and displacement data.

In the adhesion measurements, specimen and substrate were brought together until a compressive preload, $P=1\hspace{0.17em}\mathrm{N}$ was reached. The velocity of approach and retraction was $v=1\hspace{0.17em}\mathrm{mm}\hspace{0.17em}\mathrm{mi}{\mathrm{n}}^{-1}$. After reaching the compressive preload, the specimen was immediately withdrawn until it detached from the substrate. Measurements were performed using one adhesive specimen, repeated at three different positions on the substrate. There were no significant differences between the tests, hence a representative result is shown.

Force–displacement data were correlated with image sequences as follows. Image sequences were binarized by threshold using Fiji [53] such that contact (white) and non-contact (black) areas of mushroom pillars were identified. The position of each contact, together with the time of attachment and detachment, were determined using the ‘Analyze Particle’ tool in Fiji. Position and time data were imported into a Matlab routine (MathWorks, MA, USA) and correlated with force, time and displacement data from the adhesion test. For synchronization, the image showing the detachment of the last pillar was attributed to the time when the tensile force relaxed to zero. Crack types were determined manually for each fibril in the array.

Appendix A. Verification of uniform load distribution

Figure 8 is a histogram showing the number of neighbours in contact at detachment of each fibril in the array. The experimental data are compared to three simulated cases using the real areal geometry of the array. The first simulated case corresponds to random detachment, as would be expected if the fibrils detachments were uncoupled and the load distribution were uniform. One representative test is shown. The three remaining cases are designed to mimic detachment modes which may result from non-uniform load distributions. The array edge detachment is characteristic of backing layer compliance [26]. The sequence is based upon the position relative to the array centre, from furthest to closest. The peel-like detachments are characteristic of misalignment [41,42], occurring according to position along a single axis. Two such detachment sequences are shown, corresponding to alignment with the axis along which the fibril separation is d and the axis along which it is $\sqrt{2}d$, respectively.

Figure 8.

Histogram of nearest neighbours in contact at detachment for all fibrils in the array. The experimental data are compared to three simulated cases, a random detachment process, an array edge detachment and a peel-like detachment from one side of the array to the other. (Online version in colour.)

Fibrils in the bulk of a square array have four nearest neighbours. If the detachment is random then there is a steady reduction in the number of neighbours in contact when a fibril detaches. The distribution is fairly uniform, with a slight bias towards lower numbers of neighbours in contact. Conversely, all simulated sources of correlation in the detachment sequence lead to a clear peak of two attached neighbours. The similarity of the random simulated data and the experimental data are clear, leading to the conclusion that the load distribution is uniform and the detachment sequence is controlled by the distribution in fibril adhesive strength.

Appendix B. Fitting method for detachment probability

It is possible to linearize equation (3.7) for the purpose of least-squares fitting. However, this transformation is found to cause significant bias when small deviations occur at low strength (e.g. [54–56]). In general, maximum-likelihood [57] or nonlinear least-squares methods [58] are preferred. We proceed with the latter on the basis of general observation of lower root mean square error. A constrained minimization is performed using a sequential quadratic programming method in the Matlab subroutine ‘fmincon’ [59]. The constraints imposed are $0\le u_0\le \mathrm{\infty }$, $1\le m\le \mathrm{\infty }$, and, where the third parameter is involved, $0.32\le \propto \le 1$. Error estimates on the statistical properties are obtained by performing Monte Carlo simulations, based on randomly resampling N fibril adhesive strengths from the resulting distribution and refitting [57]. In each case, the standard deviation is very small, less than 0.01%, and so is not reported on a case by case basis.

Appendix C. Monte Carlo simulation of bimodal distribution

A Monte Carlo simulation is performed to generate a discrete bimodal probability distribution. Two strengths, one from each distribution, are randomly sampled. This is repeated for n samples, with only $\alpha n$ being assigned a strength from the secondary mode. For each fibril, the minimum of the two strengths persists in the resulting discrete bimodal distribution. The distribution from which the lower strength is obtained is stored along with the strength itself. This permits the generation of a histogram, decomposed by defect type. Such a histogram can be compared qualitatively to the experimental result of figure 5.

Figure 9 shows histograms of fibril adhesive strength for two combinations of bimodal statistical parameters. The first combination, shown in (a), are those obtained in §3.2 by combining fitting of equation (3.7) to mean order ranked data for the centre defect population with fitting of equation (3.12) to raw data to obtain the remaining three parameters. The other combination, shown in (b), is obtained by fitting raw data to equation (3.12) for all five parameters, with the constraints $0\le u_{0i}\le \mathrm{\infty }$, $1\le m_i\le \mathrm{\infty }$ and $0.32\le \propto \le 1$. This also results in a high-quality fit, $\mathrm{\Sigma }=0.0102$, and yields almost identical behaviour when considering the cumulative strength distribution. However, when decomposed by defect type we observe that the behaviour is very different. Only in (a) do we observe qualitative similarity to the experimental result of figure 5. This highlights the issue fitting of equation (3.12) without fibril-by-fibril knowledge of the detachment mechanism, and gives confidence in the result obtained by using the mean order ranking method to reduce the parameter space before fitting.

Figure 9.

Histogram of elongation at detachment, $u_{\mathrm{c}}$, produced by Monte Carlo simulation for two partially concurrent defect populations exhibiting the statistical parameters given in the insets of (a) and (b). Centre defects are represented in blue and edge defects are represented in orange. To avoid discretization error, the number of samples $n=100\hspace{0.17em}000$. The bin size $\mathrm{\Delta }{u}_{\mathrm{c}}=0.04\hspace{0.17em}\mathrm{mm}$. (Online version in colour.)

Data accessibility

The datasets supporting this article have been uploaded as part of the electronic supplementary material.

Authors' contributions

All authors were involved in conception of the work. J.A.B. performed the theoretical modelling. V.T. performed the experiments. J.A.B. and V.T. analysed the data. J.A.B., R.H. and R.M.M. interpreted the results. All authors were involved in preparation of the manuscript.

Competing interests

We declare we have no competing interests.

Funding

The work of R.M.M. was partially supported by the MRSEC Program of the National Science Foundation under award no. DMR 1720256. R.M.M. also acknowledges support from a Humboldt Alumni Award. The work of V.T., R.H. and E.A. was funded by the European Research Council (ERC) under the European Union's Seventh Framework Program (FP/2007–2013)/ERC Advanced grant no. 340929.