Abstract
The effect of aspect ratio on the pulloff stress and stiffness of soft elastic fibres is studied using elasticity and numerical analysis. The adhesive interface between a soft fibre and a smooth rigid surface is modelled using the Dugdale–Barenblatt model. Numerical simulations show that, while pulloff stress increases with decreasing aspect ratio, fibres get stiffer. Also, for sufficiently low aspect ratio fibres, failure occurs via the growth of internal cracks and pulloff stress approaches the intrinsic adhesive strength. Experiments carried out with various aspect ratio polyurethane elastomer fibres are consistent with the numerical simulations.
1. Introduction
Fibrillar adhesion has attracted much attention in recent years. The remarkable attachment ability of some of the species that bear fibrillar footpads leads researchers to be inspired from and mimic these fibrillar adhesive pads [1,2]. In conjunction with fabrication attempts, many works were conducted to analyse the mechanism behind this unusual attachment ability [1,3,4]. Experimentally, it has been found that singlelevel structures consisting of long fibres with nominally flat ends generally fail to achieve theoretical predicted enhancement in strength owing to loss of contact area, lateral collapse and buckling of fibres [5–7]. In practice, enhanced adhesion is usually accomplished by fabricating an additional terminal element at the end of a fibre [8–13]. How the shape of these terminal elements affects adhesion has been studied theoretically by Gao & Yao [14], Greiner et al. [15], Spolenak et al. [16] and Soto et al. [17].
The mechanics of contact and adhesion of fibrillar surfaces can be studied using a cohesive zone model that describes the adhesive interaction between two contacting surfaces. The simplest of such models is the Dugdale–Barenblatt (DB) model [18,19]. In this model, the interface separates when the normal interfacial stress reaches the theoretical strength of the interface, denoted by σ_{o}. The interface continues to separate at this stress until the separation reaches a critical distance δ_{c}, after which the interface can no longer support any stress, resulting in a crack initiation. The region where the separation of interface occurs is referred to as the cohesive zone. In this model, the work of adhesion of the surfaces in contact is given by . Based on this model, Tang et al. [20] as well as Gao & Yao [14] have established that a reduction in the characteristic lateral dimension of fibres forces the entire interface of an individual fibre to fail simultaneously rather than via propagation of the crack. This phenomenon increases adhesion strength and compliments contact splitting [21], explaining why larger animals have finer diameter fibrils. Works by Arzt et al. [21] and Tang et al. [20] include detailed analysis on fibrillar interfaces and how it could enhance adhesion compared with flat unstructured surfaces.
While animals such as geckos favour long fibrils, some animals such as bush crickets use short pillars as adhesive pads [22]. Long fibres are advantageous over short ones in rough surface applications where fibre array compliance is crucial for roughness adaptation and high adhesion strength. On the other hand, smooth surface applications may not require long fibres. Short microfibre arrays are often used as adhesive footpads for climbing/walking robots for smooth surfaces [23–26]. In addition to short microfibre arrays, thin unstructured polymer footpads are also used as adhesive footpads in climbing robots [27]. The effect of aspect ratio (i.e. the ratio of the length of the fibre to its effective lateral dimension) on adhesion strength and compliance is crucial to be understood in order to design effective adhesives for these climbing robots. Previous analyses were carried out on high aspect ratio fibres,^{1} that is, the fibres are long in comparison with their diameter. The contact mechanics of short fibres are quite different from those of long ones. Indeed, because fibres are made of incompressible materials, the compliance of the system decreases dramatically with increasing fibre aspect ratio. The geometry for the adhesion of short fibres is practically identical to the socalled ‘pokerchip test’ [28], where a thin layer of elastomer is sandwiched between two rigid plates, which are pulled apart under tension. Another closely related test is the ‘probetack test’ [29]. In a typical ‘tack’ test, a rigid flat cylindrical punch is brought into intimate contact with a uniform thin layer of elastomer (a pressuresensitive adhesive); the probe is then withdrawn and the force required for pulloff as well as the energy expended for the pulloff force is measured by finding the area under the load versus the displacement curve. When the interface between the punch and the elastic layer is sufficiently weak, failure occurs by the propagation of an interfacial edge crack. However, for highstrength adhesives, the layer fails by the initiation and growth of the internal interfacial cracks. These internal interfacial cracks are pinned along their fronts; as a result, they grow into large cavities with the long axis in the loading direction. The elastomeric material between these cavities acts as long fibrils to support the applied load.
In this work, the effect of aspect ratio on pulloff stress and stiffness of soft elastomeric fibres in contact with a rigid smooth surface is studied. Since the Poisson ratio of elastomers is very close to 0.5, fibres are assumed to be incompressible in this work. In §2, we present the elasticity formulation for the pulloff stress and the stiffness of a soft elastic fibre followed by numerical simulations in §3. Experiments carried out with various aspect ratio fibres and comparison with the simulation results are shown in §4. Discussions and conclusions are given in §5.
2. Formulation
2.1. Elastic fibre problem
In this section, we study how the fibril aspect ratio affects the force required to pull a fibre off a rigid substrate as shown in figure 1. Let us assume that a soft incompressible elastic fibre of length h and radius a is perfectly bonded to a rigid flat substrate and a uniform displacement Δ is applied to the free end of the fibre. Let P be the force exerted on the fibre owing to the displacement at the free end. For very long fibres, the applied displacement will cause a uniaxial stress state P/πa^{2} in the fibre except in a region very close to the fibre edge. For these fibres, P/πa^{2} is directly proportional to the strain ɛ = Δ/h, 2.1where E is the Young modulus of the fibre. As the fibre becomes shorter relative to its radius, deviation from uniaxial behaviour becomes significant; as a result, equation (2.1) must be modified to reflect this deviation. Dimensional analysis and the linearity of the governing equation imply that 2.2where p is a dimensionless function of a single parameter a/h. According to equation (2.2), p(a/h) can be interpreted as the stiffness ratio of a fibre of aspect ratio h/a to one that has aspect ratio of infinity (i.e. very long fibre) and p(a/h → 0) → 1.
For an elastic fibre in contact with a rigid substrate, the normal stress σ_{zz} near the fibre edge has the form 2.3where 0 < α < 1 is a numerical factor that depends on the Poisson ratio, e.g. α ≈ 0.4 for ν = 0.5 [30]. Equation (2.3) predicts that the normal stress becomes infinite as the edge is approached. The factor A in equation (2.3) represents the amplitude of the singular stress field. Dimensional analysis and the linearity of the governing equations imply that the amplitude A is given by 2.4where g is a dimensionless function that depends only on a/h. We do not know g in general; however, in the limit of a/h → 0, 2.5In this limit, g approaches g(0), which is a numerical constant.
2.2. Pulloff stress
It is assumed that the interfacial failure initiates when the amplitude of the singular field A reaches a critical value A_{c}, which is assumed to be a system constant independent of geometry, 2.6Here, we assume that the initiation of interfacial failure corresponds to pulloff. Whether pulloff (peak load) occurs at the instant of crack initiation or at a later time will be addressed in §3. Using equations (2.2), (2.4) and (2.6), one can determine the pulloff load as 2.7The value of A_{c} could be extracted from the previous work by Tang et al. [20], in which they estimated the pulloff force of a soft elastic long cylindrical fibre in contact with a rigid flat surface, whose height is much larger than its radius. In their study, they used the DB model to model the interfacial adhesion between the tip of the fibre and the adhering substrate. Define σ_{s} as the average stress the fibre bears at pulloff, . They found that normalized pulloff stress depends on a single dimensionless parameter χ such that where 2.9
The dimensionless parameter is the ratio of two length scales: a and . The former is the radius of the fibre and the latter the size of dominance of the interfacial adhesive forces. Here, for incompressible materials and B is a numerical constant of order 1 and was determined numerically to be B = 0.83.
Our assumption that the interface fails when A = A_{c} is valid in the regime where , called the ‘flawsensitive’ regime. In this regime, the cohesive zone is very small in comparison with the fibre's lateral dimension; its breakdown results in failure via the unstable growth of an interface crack at the edge of the fibre. In this case, . In the ‘flawinsensitive’ regime, where χ < 1, the cohesive zone occupies the entire interface and, therefore, . In the flawsensitive regime, for a/h → 0, A_{c} can be obtained from equating (2.7) and (2.8b), 2.10As pointed out earlier, A_{c} is a system constant and is independent of geometry. Using equation (2.10), equation (2.7) can be rewritten to obtain the pulloff force of the elastic fibril and the pulloff stress (adhesive strength) in the flawsensitive regime, respectively, as 2.11aand 2.11b
3. Numerical simulations
Finiteelement method (FEM) simulations were carried out using COMSOL MultiPhysics 3.4 to simulate the stiffness and pulloff stress. The DB cohesive zone model is used in the pulloff stress calculations. Simulation details can be found in appendix A. Unless stated otherwise, the simulation parameters are a = 1 µm, E = 2 MPa and σ_{o} = 100 kPa. The height of the fibre h is varied from 0.1 to 10 µm. χ is varied by changing δ_{c} from 0.2 to 3 nm for fixed a, E, ν and σ_{o}. Corresponding χ and W_{ad} range from 2 to 30 and from 20 to 300 µJ m^{−2}, respectively.
3.1. Crack initiation and pulloff
For high aspect ratio fibres, because the length of the fibre is much larger than its radius, crack growth occurs under load control and is expected to be unstable. Therefore, pulloff is expected to occur once a crack initiates at the edge of the fibre. As the length of the fibre decreases, the detachment occurs under displacement control. In this case, crack growth can be stable, which could lead to pulloff after the crack initiation (i.e. during crack propagation). FEM simulations were carried out beyond the instant of crack initiation to determine whether pulloff occurs either at the instant of crack initiation or later during crack propagation. Figure 2 plots the simulation results for the average stress the fibre bears, , as a function of the applied displacement Δ (figure 2a) and normal interfacial displacement δ_{m} (figure 2b). Both plots show that the peak load is reached either at the instant of or soon after the crack initiation. These results indicate that the load the fibre bears at the instant of crack initiation is a good approximation for the maximum attainable tensile load before interfacial failure and will be used as the pulloff load in this study.
3.2. Fibre stiffness and pulloff stress
The stiffness (i.e. spring constant) of the fibre is defined by rearranging equation (2.1) as 3.1
The function p(a/h) in equation (3.1) can be interpreted as the stiffness of a fibre normalized by the stiffness of a fibre with very high aspect ratio. It is calculated by fixing a and changing h in our FEM simulations. Figure 3a shows the FEM results for p(a/h). Results for k_{f} are plotted along with the stiffness predictions for uniform strain across the cross section (i.e. p(a/h) = 1) in figure 3b. As expected, for small a/h values, the actual stiffness deviates very little from the uniformstrain approximation. However, the difference continues to grow for larger a/h, reaching close to an order of magnitude for a/h = 10.
Figure 4a shows the simulation results for the a/h dependence of pulloff stress for varying χ values (see equation (2.11b)). Pulloff stress increases for increasing a/h for all χ and approaches the intrinsic adhesive strength σ_{o} for sufficiently large a/h. As expected for larger χ, pulloff stress approaches the intrinsic adhesive strength at smaller a/h. For each χ value, normalized pulloff stress increases initially with increasing a/h, reaches an inflection point and then keeps increasing until it reaches the intrinsic adhesive strength (figure 4a). The trend observed in figure 4a is consistent with the theoretical expectations. Let us rearrange equation (2.11b) to obtain the normalized pulloff stress as 3.2Numerical simulations indicate that, as a/h increases, the value of H(a/h) also increases, which according to equation (3.2) results in a higher pulloff stress. However, the maximum possible value of pulloff stress is the intrinsic adhesive strength, which implies that equation (3.2) has an upper bound of 1. Recall that equation (3.2) is valid only in the flawsensitive regime (that is, the cohesive zone is small in comparison with the fibre dimensions); therefore, despite the fact that equation (3.2) could theoretically be greater than 1 for sufficiently large a/h, the value of cannot exceed 1, as required by the DB model. Simulation results plotted in figure 4a confirm this result.
Crack initiation behaviour is affected by the fact that cannot exceed 1. In figure 4a, the inflection point marks the a/h value where the crack initiation switches from the edge of the fibre to the centre of the fibre. For a given χ that is larger than 1, the critical separation δ_{c} is reached at the edge of the fibre that initiates a crack from the edge. Beyond a critical a/h, the critical separation δ_{c} is reached at the centre of the fibre first, which leads to the crack initiation at the centre of the fibre.
The function H(a/h)/H(0) can be calculated by rearranging equation (3.2) as 3.3where normalized pulloff stress is calculated using FEM simulations, χ is calculated using equation (2.9), B = 0.83 and α = 0.4. The equality in equation (3.3) holds with the given parameters only when the crack is initiated from the edge of the fibre. Results for equation (3.3) are included in figure 4b. As in figure 4a, inflection points in figure 4b also represent the transition of crack initiation from the edge to the centre of the fibre. Note that in the regime before the inflection points where the crack initiates from the edge, equation (3.3) holds and all the curves coincide and correspond to H(a/h)/H(0). After the inflection point, crack initiation transitions to the centre of the fibre and, in this regime, equations (3.2) and (3.3) are no longer valid. Therefore, the regime after the inflection point in figure 4b is not equal to H(a/h)/H(0) and thus all the curves in this regime follow different paths for increasing a/h values. The transition of the crack initiation from the edge to the centre of the fibre is shown in figure 5, which plots the normal separation of the tip of the fibre from the opposing rigid surface for χ = 12 and varying a/h. For small a/h, critical separation is reached at the edge of the fibre (r/a = 1). As a/h increases, crack initiation transitions from the edge to the centre of the fibre (r/a = 0).
4. Experiments
4.1. Material and methods
Experiments were performed with polyurethane elastomer (ST1060; BJB Enterprises) cylinders, which were fabricated using a moulding technique. Fabrication details can be found in appendix B. Fabricated cylinders are 0.75 mm in radius and the length ranges from 0.155 to 3 mm. A glass slide (Microscope Slides; Fisher Scientific) was used as the adhering surface. Prior to testing, the glass slide was cleaned with acetone, ethyl alcohol and diionized water followed by nitrogen treatment for 2 min.
A custom adhesion measurement setup was used to conduct the experiments (figure 6a). This system consists of an inverted optical microscope (Eclipse TE200; Nikon) with an automated highprecision stage (MFACC; Newport) that holds a highresolution load cell (GSO50; Transducer Techniques Inc.). In order to ensure proper alignment, an acrylic disc attached to a cylinder was first placed on the adhering surface (glass slide). Then, the acrylic disc–polyurethane cylinder assembly was glued to the load cell by lowering the load cell in contact with the acrylic disc while the polyurethane cylinder was in intimate contact with the adhering surface to ensure alignment between the adhering surface of the cylinder and the glass slide. Alignment was also checked using the optical microscope and necessary corrections were made with a twoaxis rotational stage (GON 40L and GON 40U; Newport, UK). The automated stage, which was controlled by custom realtime software, lowered the cylinder into contact with the glass at a fixed velocity of 1 µm s^{−1} until a preload of 100 mN was reached. The cylinder was then retracted at a speed of 1 µm s^{−1} until it detached from the sample. The measurement system continuously captured the force data, which was used to produce the force–distance plot, as seen in figure 6b. Effective stiffness k_{e} was estimated from the slope of the retract curve away from the pulloff region. Pulloff force is the minimum force (maximum tensile force) on the force–distance plot. Measurements were repeated three times for each sample.
4.2. Experimental results
Experiments were performed to measure the pulloff force and the stiffness of the soft fibres. Figure 7a plots the experimental results for the stiffness of soft fibres along with simulations. The stiffness value in figure 7a is the effective spring constant k_{e} from the serial connection of the soft fibre and the measurement system. The effective spring constant for our system is 4.1where k_{d} is the stiffness of the measuring device and is measured to be k_{d} = 3.48 kN m^{−1}. The FEM data in figure 7a are obtained by calculating k_{f} from FEM simulations and using equation (4.1) to estimate the effective stiffness k_{e}.
The cohesive zone simulations presented in this study assume that the fibre is free of defects and in intimate contact with the smooth substrate. In reality, it is very challenging to fabricate fibres with perfectly vertical, defectfree edges. In order to determine the effect of a potential defect at the edge of the fibreadhering substrate interface, the defect is modelled as a fixed length circumferential crack. Simulation results for an edge crack are presented as a reference along with cohesive zone simulations and experimental results for pulloff force in figure 7b. Parameters used in the cohesive zone simulations are: W_{ad} = 93 mJ m^{−2} [6], σ_{o} = 10 MPa (fitting parameter), E = 2 MPa [31], a = 0.75 mm and h is measured using a calibrated optical microscope. While an intrinsic adhesive strength of 10 MPa might seem high, it is attainable considering that a typical van der Waals interaction could reach much higher adhesive stresses [32]. Some explanation of why van der Waals forces are much weaker than the theory predicts can be found in Hui et al. [33] and Tang et al. [34]. Using these values, χ is calculated from equation (2.9) to be 48 130. For fixed length circumferential crack simulations, the length of the crack measured from the edge of the fibre is 1 µm. Please refer to appendix A for simulation details.
5. Discussions and conclusions
The measured stiffness and simulation results in figure 7a exhibit a close match. It is, however, important to emphasize the limitations of the stiffness measurement performed in this work. In general, the stiffness of a structure is measured using a measuring device with a spring constant that is much larger than that of the structure whose spring constant is being measured. While, for small a/h values, stiffness can be measured accurately with the measuring device used for experiments in this work, it is prone to large errors for larger a/h values. The measured effective stiffness in figure 7a is very close to the stiffness of the measuring device for a/h values larger than 2, which implies that the fabricated fibres with a/h > 2 are much stiffer than the measuring device.
It was found that the results from the fixed length circumferential crack simulations were very close to those of cohesive zone simulations, neither of which demonstrated an exact fit to the experimental data, as shown in figure 7b. The discrepancy between the experimental data and the simulations suggests larger edge cracks with increasing a/h, as opposed to a fixed length edge crack for all a/h. It is possible that the size of the crack at the edge increases with thinner samples owing to variations in the fabrication technique, which could explain the plateau for the experimental data at higher a/h values. Nonetheless, both experimental and simulated pulloff stress increase with increasing a/h. It is important to note that the inflection point in figure 4a is not present in the FEM results plotted in figure 7b. The absence of an inflection point indicates that the crack initiates from the edge of the fibre for all a/h used in the experiments. χ = 48 130 is much larger than 1 for our experimental system. Even our relatively large a/h cannot force the system to be in the flawinsensitive regime (i.e. ). Thus, pulloff occurs by crack initiation and growth from the edge for all the experimental a/h values.
The interfacial strength σ_{o} used in the FEM simulations is larger than the Young modulus of the fibre material, which implies that, at pulloff, the region around the cohesive zone experiences nonlinear deformations. In addition, polyurethanes are viscoelastic materials. Although experiments were performed at 1 µm s^{−1}, ratedependent behaviour could still be present. In our simulations, fibre material is assumed linearly elastic and incompressible.
In the light of the simulations and the experimental results, the effect of aspect ratio becomes evident particularly when the length of the fibre gets smaller than the radius of the fibre (i.e. a/h > 1). Existing literature suggests that, from a theoretical standpoint, an ideal design for fibrillar adhesives on the footpad of a climbing robot would be composed of long fibres with sufficiently small diameters to ensure flawinsensitive adhesion [14,20]. However, for most elastomers, a flawinsensitive regime could only be achieved with fibre diameters smaller than 1 µm, which might be very challenging to fabricate. Our study suggests that similar enhancement is possible with larger diameter, smaller aspect ratio fibres. The loss in compliance owing to lowering the aspect ratio could be compensated by adding a compliant element to the backing of the fibres, such as foam tapes, soft polymers [25], or by incorporating hierarchical fibre designs [35–38]. It is worth noting that this work is concerned with smooth surface adhesion and proposes enhancement in adhesive strength via lowering the aspect ratio of fibres. For climbing robot applications, the design of the footpads cannot solely be based on the adhesive strength. There are several other vital design aspects such as shear and peel strength of adhesive footpads as well as their ability to conform to roughness. An effective design should consider the effect of lowering the aspect ratio on all of the design parameters.
In summary, we investigated the effect of aspect ratio on the adhesion and stiffness of an elastic fibre in contact with a rigid smooth surface. Functions affecting pulloff stress and stiffness which only depend on the aspect ratio were determined using finiteelement simulations. Results suggest that lower aspect ratios stiffen the fibre drastically and well beyond the stiffness calculated from beam equations. FEM analyses showed that pulloff stress increases with decreasing aspect ratio. In addition, it was found that, for fibres with sufficiently low aspect ratios, the crack initiates at the centre of the fibre and lowering the aspect ratio further results in flawinsensitive adhesion as expected. Experiments performed with various aspect ratio polyurethane fibres showed an increase in adhesive strength and stiffness for lower aspect ratio fibres, which are consistent with numerical simulations.
Acknowledgements
This work was funded by the National Science Foundation (CMMI0800408).
Appendix A. Comsol implementation details
A.1. Cohesive zone simulations
In the simulations, χ is varied by changing δ_{c} from 0.2 to 3 nm for fixed a, E, ν and σ_{o}. Aspect ratio is varied by fixing a and varying h. In COMSOL, Lagrange quadratic elements are used in twodimensional axial symmetry mode to mesh elastic fibre with a = 1 µm. Height h is varied from 0.1 to 10 µm to adjust the aspect ratio. The elastic fibre's Young modulus is 2 MPa. For stiffness simulations, the fibre tip is assumed to have full friction contact with the rigid backing and the substrate (i.e. the fibre is perfectly bonded to the rigid backing and the substrate). While the rigid substrate is fixed, a displacement of is applied to the rigid backing of the fibre for each h to calculate the stiffness of the fibre. Load p is calculated by integrating the normal stress distribution at the midsection of the fibre over the crosssectional area. Effective stiffness k_{e} is determined using equation (4.1).
Pulloff simulations are performed by implementing the DB cohesive zone model at the tip of the fibre (i.e. interface between the fibre and the rigid substrate). Displacement is constrained to have no nodal displacement in the radial direction. The DB cohesive zone model is modified to incorporate into the FEM software. In the DB cohesive zone model, the interface is not allowed to separate until the stress reaches σ_{o} (i.e. infinite stiffness interfacial interaction). When the stress reaches σ_{o}, the interface is allowed to separate and pulloff takes place when normal separation reaches the critical separation distance δ_{c} (figure 8a). In the modified DB model (figure 8b), the infinite stiffness interface is modified to a high stiffness interface where the interface is allowed to separate a small amount (δ_{1}) compared with δ_{c} until interfacial stress reaches σ_{o}. The switch from σ_{o} to 0 for δ larger than δ_{c} is approximated using the sigmoid function. The equation for the modified DB model is
Such modifications were found necessary in order to ensure a convergent solution and avoid oscillations owing to the step functions involved in the DB model. δ_{1} = 0.01 nm for all simulations. The deviation of work of adhesion from the DB model owing to the modification is negligible provided that the sigmoid function has a steep decline and .
A.2. Circumferential crack simulations
The energy release rate for a circumferential crack of length c (measured from the edge of the fibre) at the interface between a long cylindrical fibre of radius a and a smooth rigid adhering surface is [39] A 1for c ≪ a where p is the applied load and E *= E/(1−ν^{2}). Linearity and dimensional analysis imply that, for varying a/h, G becomes A 2where f(a/h) depends only on a/h and the boundary conditions at the interface. Pulloff occurs when the energy release rate G is equal to the critical energy release rate for fracture G_{c}. Rearranging equation (A 2), pulloff load can be calculated from A 3
The FEM simulations were performed for a fixed circumferential crack length of c = 1 µm and a = 750 µm. The Young modulus E = 2 MPa and fibre material is assumed incompressible. The interface from the centre of the fibre to the tip of the crack is assumed to be perfectly bonded to the substrate. Displacement Δ is applied incrementally for a/h ≪1 until a desired load is reached. At this load, which is taken to be the pulloff load at a/h ≪1, G_{c} is found using a Jintegral routine around the crack tip [40]. Then, pulloff load for the rest of the a/h values was found by increasing Δ until G = G_{c}. Equation (A 3) implies that, for constant G_{c}/c, pulloff load depends only on f(a/h). Therefore, these simulations effectively give us f(a/h), which can be multiplied by an appropriate constant to fit the experimental pulloff data.
Appendix B. Fibre fabrication
The fabrication method is designed to form a replica of a smooth cylindrical steel rod from polyurethane elastomer. The steel rod is first press fitted into an acrylic plate with spacers placed on two sides. Liquid silicone rubber (HS II, Dow Corning) is poured around the steel rod onto an acrylic plate on the bottom (figure 9a). A second acrylic plate with a circular through hole, the same size as the diameter of the steel rod, is placed through the rod and pressed onto the spacers (figure 9b). The silicone rubber is allowed to cure for 24 h at room temperature and removed from the rod to form the mould for fibre fabrication. Then, this mould is placed onto a smooth flat silicone rubber layer. This way, the mould adheres to the substrate leaving no opening in between. Polyurethane (ST1060; BJB enterprises) is poured into the mould and an acrylic cap is placed on top of it (figure 9c). After curing at room temperature for 24 h, the polyurethane is peeled off from the mould to obtain a soft cylinder with a rigid acrylic backing (figure 9d). Scotch tape is used as a spacer to fabricate thin moulds for very low aspect ratio fibres. Fabricated fibres are 0.75 mm in diameter with lengths ranging from 0.155 to 3 mm.
Footnotes

↵1 The term ‘fibre’ usually refers to structures with very high aspect ratios. In the text, the term fibre is used for both low and high aspect ratio structures for consistency throughout the paper.
 Received October 21, 2010.
 Accepted December 21, 2010.
 This Journal is © 2011 The Royal Society