Nanotribology of Spiderman

Abstract

Spiders can produce cobwebs with high strength to density ratio and surprisingly display self-cleaning, strong and releasable adhesion (as geckos). Nanointerlocking, capillary and van der Waals forces, all potential adhesive mechanisms, were thus discussed, demonstrating the key role played by hierarchy to the design of super-hydrophobic, i.e. self-cleaning (by activating fakir drops as in lotus’ leaves) and super-adhesive materials. The reversibility of the strong attachment was quantified thanks to an improved nonlinear peeling model, for which the solution in closed form was provided. Thus, mimicking Nature, thanks to carbon nanotube-based technology, we have suggested [N. Pugno, J. Phys. Condens. Matter 19, 395001 (2007)] the feasibility of large invisible cables, as well as of self-cleaning, super-adhesive and releasable hierarchical smart materials. We have found that a man can be supported by a transparent cable with cross-section of 1 cm² and feasibly, with spider material gloves and boots, could remain attached even to a ceiling: a preliminary step towards a Spiderman suit.

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Introduction

The gecko’s ability to ‘run up and down a tree in any way, even with the head downwards’ was firstly observed by Aristotle, almost twenty-fifth centuries ago, in his Historia Animalium. A comparable ‘adhesive’ system is found in spiders, that in addition have the ability of producing fascinating cobwebs.

In general, when two solid (rough) surfaces are brought into contact with each other, physical/chemical/mechanical attractions occur (see [1]). The force developed that holds the two surfaces together is known as adhesion. Nanointerlocking (or friction), intermolecular forces, including capillary and van der Waals forces, suction, secretion of sticky fluids and electrostatic attraction are all potential adhesive mechanisms in biological attachment systems (see the review by [2]).

Suction cups operate under the principle of air evacuation, i.e., when they come into contact with a surface, air is forced out of the contact area, creating a pressure differential. The adhesive force generated is simply the pressure differential multiplied by the cup area. Thus, in our (sea level) atmosphere the achievable suction strength is $\sigma _s {\;\approx\;} 0.1\,{\textrm{MPa}}$, of the same order of magnitude of those observed in the other mentioned adhesive mechanisms or in spider/gecko adhesion. Even if suction can have an interesting role in producing synthetic adhesive materials, especially to be used in high-pressure environments, its mechanics is rather trivial. Moreover, although several insects and frogs rely on sticky fluids to adhere to surfaces, synthetic materials cannot evidently secrete these fluids without uncomfortable reservoirs. Furthermore, electrostatic attraction occurs only when two dissimilar heteropolar surfaces come in close contact. Accordingly, we will omit in our discussion these three mechanisms.

In geckos the main adhesive mechanisms are capillary [3] and van der Waals [4] forces, whereas in spiders [5], in addition to the main van der Waals adhesion, nanointerlocking could have a role (e.g., during cobweb gripping). Accordingly, in this paper, we focus our attention on these three adhesive mechanisms, with an eye to the role played by hierarchy and to reversibility.

Hierarchical miniaturized hairs without adhesive secretions are characteristic features of both spiders and geckos, see Fig. 1. In jumping spider evarcha arcuata [5], in addition to the tarsal claws (hooks with radius of ∼50 μm), a scopula (with surface area of 37,000 μm²) is found at the tip of the foot; the scopula is differentiated in setae, each of them covered with numerous setules (with an average density of ∼$2.1\,{\mu \textrm{m}}^{ - 2},$) terminating in a triangular contact (with surface area of ∼$0.17\,{\mu \textrm{m}}^2.$) The total number of setules per foot can be calculated at 78,000 and thus all 8 feet are provided with a total of ∼0.6 million points of contacts. The average adhesion force per setule was measured to be ∼41 nN, corresponding for the 8 feet or scopulae to $\sigma _{spider} {\;\approx\;} 0.24{\textrm{MPa}}$ and to a safety factor, that is the adhesive force over the body weight (∼15.1 mg), of $\lambda _{spider} {\;\approx\;} 173$.

Fig. 1

Spider and gecko feet showed by SEM. In the Tokay gecko (Fig. 1f) the attachment system is characterized by a hierarchical hairy structures, which starts with macroscopic lamellae (soft ridges ∼1 mm in length, Fig. 1h), branching in setae (30–130 μm in length and 5–10 μm in diameter, Fig. 1i,l; [6, 70–72]. Each seta consists of 100–1,000 substructures called spatulae [6, 70], the contact tips (0.1–0.2 μm wide and 15–20 nm thick, Fig. 1M; [5, 69] responsible for the gecko’s adhesion. Terminal claws are located at the top of each singular toe (Fig. 1 g). Van der Waals and capillary forces are responsible for the generated adhesive forces [37, 73] whereas claws guarantee an efficient attachment system on surfaces with very large roughness. Similarly, in spiders (e.g. Evarcha arcuata, [74]) an analogous ultrastructure is found. Thus, in addition to the tarsal claws, which are present on the tarsus of all spiders (Fig. 1c), adhesive hairs can be distinguished in many species (Figs. 1d, e). Like for insects, these adhesive hairs are specialised structures that are not restricted only to one particular area of the leg, but may be found either distributed over the entire tarsus, as for lycosid spiders, or concentrated on the pretarsus as a tuft (scopula) situated ventral to the claws (Fig. 1a, 1b), as in the jumping spider Evarcha arcuata [74]

Similarly, a tokay gecko (gecko gecko) foot consists of lamellae (soft ridges ∼1 mm in length), from which tiny curved setae (∼10 μm in diameter, density of ∼$0.014\,{\mu \textrm{m}}^{ - 2}$) extend, each of them composed by numerous spatulae (100–1,000 per seta, ∼0.1 μm in diameter) with terminal contact units (having surface area of ∼$0.1\,{\mu \textrm{m}}^2$) [6, 7]. The adhesive force of a single seta and even of a single spatula has recently been measured to be respectively ∼194 μN [8] or ∼11 nN [9]. This corresponds to an adhesive strength of $\sigma _{gecko} {\;\approx\;} 0.58\,{\textrm{MPa}}$ [8] and a safety factor of $\lambda _{gecko} {\;\approx\;} 102$ (to compute this value we have assumed a weight of ∼250 g and a single foot surface area of ∼110 mm²), comparable only with those of spiders (∼173 [5]), cocktail ants (>100, [10]) or knotgrass leaf beetles (∼50, [11]).

Note that such safety factors are ideal and thus are expected to be reduced by about one order of magnitude [12] as a consequence of the presence of ‘defects’ e.g. spurious particles, located at the contact interfacial zone. According to the previous values, we estimate for a gecko a total number of points of contacts of ∼3 billions, thus much larger than in spiders, as required by their larger mass (the number of contacts per unit area must scale as the mass to 2/3, see [13]). The total adhesive force could easily be overcome by subsequently detaching single setules and not the whole foot at once [14, 15].

Moreover several natural materials exhibit super-hydrophobicity, with contact angles between 150° and 165°; often a strategy for allowing a safe interaction with water. This is the case for the leaves of about 200 plants, including asphodelus, drosera, eucalyptus, euphorbia, ginkgo biloba, iris, tulipa and, perhaps the most famous, lotus [16, 17]. Similarly, animals can be super-hydrophobic, as for the case of water strider legs, butterfly wings, duck feathers and bugs [18–20]. These surfaces are generally composed of intrinsic hydrophobic material and N=2 hierarchical micro-sized levels [21].

Superhydrophobia is extremely important in micro/nano-fluidic devices for reducing the friction associated with the fluid flow, but also for self-cleaning: super-hydrophobic materials are often called self-cleaning materials since drops are efficiently removed taking with them the dirty particles which were deposit on them [17, 22]. This effect is extremely important in super-adhesive materials. Hansen and Autumn [23] have proved that gecko setae become cleaner with repeated use; this is probably a consequence of the hierarchical nature of the gecko foot, as we are going to demonstrate.

A replication of the characteristics of gecko [24] or spider feet would enable the development of a self-cleaning, as the lotus leaves (see the review by [21]), super-adhesive and releasable hierarchical material and, with the conjunction of large invisible cables [25], of a preliminary Spiderman suit, see Fig. 2 (Pugno, [26, 27].

Fig. 2

Spiderman (related inset from the web) must have large cobwebs and self-cleaning, super-adhesive and releasable gloves and boots. Invisible large cables [25] could be realized with nanotube bundles (related inset from [28]), whereas gloves and boots, mimicking spider (related inset from [5]) or gecko (related inset from [51]) materials, with hierarchically branched nanotubes (related inset from [65]) as suggested by our analysis. Note that nanotube forest is super-hydrophobic (water repellent) and thus self-cleaning (related inset from [75]) as lotus leaves (related inset: photo by the Author)

Invisible Macroscopic Spider Silk

In this section we present just an idea, no more no less, for realizing large invisible cables [25]; a discussion on their technological feasibility is also included. The strength, stiffness and density of the invisible cable are estimated, and the condition of invisibility is provided.

Consider a rectangular cable having width W, thickness T and length L; the cross-section being composed of n×m (multiwalled) carbon nanotubes with inner and outer diameter ${d_ {-}}$ and ${d_ {+}}$ respectively and length L. Let us assume that they are arranged in a square lattice with periodic spacing p=W/n=T/m (see Fig. 2, related inset). Then, the strength ${\sigma _{C}}$ of the bundle, defined as the failure tensile force divided by the nominal area W×T, is predicted as:

$$\sigma _C = \frac{\pi }{4}\frac{{d_ + ^2 - d_ - ^2 }}{{p^2 }}\sigma _{NT} ,\quad \sigma {\;\to\;} E,\rho $$

((1))

where ${\sigma _{NT}}$ denotes the strength of the single carbon nanotube. To derive Eq. 1 we have assumed a full transfer load between the nanotube shells, which seems to be plausible if intertube bridgings are present, otherwise ${\sigma _{NT}}$ would represent the nominal multiwalled nanotube strength. The same relationship is derived for the cable Young’s modulus ${E_{C}}$ considering in Eq. 1 the substitution $\sigma {\;\to\;} E$ and ${E_{NT}}$ as the Young’s modulus of the single carbon nanotube. Similarly, the cable density $\rho _C$, defined as the cable weight divided by the nominal volume W×T×L, is predicted according to Eq. 1 with the substitution $\sigma {\;\to\;} \rho$, where $\rho _{NT}$ would denote the carbon (nanotube) density. Thus, the same (failure) strain ${{\varepsilon _C = \sigma _C } \mathord{\left/ {\vphantom {{\varepsilon _C = \sigma _C } E}} \right. \kern-\nulldelimiterspace} E}_C {{ = \sigma _{NT} } \mathord{\left/ {\vphantom {{ = \sigma _{NT} } {E_{NT} }}} \right. \kern-\nulldelimiterspace} {E_{NT} }}$ and strength over density ratio ${{\sigma _C } \mathord{\left/ {\vphantom {{\sigma _C } \rho }} \right. \kern-\nulldelimiterspace} \rho }_C {{ = \sigma _{NT} } \mathord{\left/ {\vphantom {{ = \sigma _{NT} } {\rho _{NT} }}} \right. \kern-\nulldelimiterspace} {\rho _{NT} }}$ is expected for the bundle and for the single nanotube. This ratio is huge, at least theoretically, e.g., as required in the megacable of the space elevator [12]. Thus, Eq. 1 is a law to connect the nanoscale properties of the single nanotube with the macroscopic properties of the bundle.

On the other hand, indicating with λ the light wavelength, the condition for a nanotube to be invisible is:

$$d_ + < < \lambda $$

((2a))

whereas to have a globally invisible cable, we require in addition to not have interference between single nanotubes, i.e.:

$$p > > \lambda $$

((2b))

We do not consider here the less strict limitations imposed by the sensitivity of the human eye, that can distinguish two different objects only if their angular distance is larger than ~1^′. In other words, we want the cable to be intrinsically invisible.

Note that in the case of $p < < \lambda$ a transparence is still achievable considering a sufficient thin sheet, as suggested by classical aerosol mixtures (here not applicable).

Assuming ${{d_ + } \mathord{\left/ {\vphantom {{d_ + } \lambda }} \right. \kern-\nulldelimiterspace} \lambda } {\;\approx\;} {1 \mathord{\left/ {\vphantom {1 {10}}} \right. \kern-\nulldelimiterspace} {10}}$, ${p \mathord{\left/ {\vphantom {p \lambda }} \right. \kern-\nulldelimiterspace} \lambda } {\;\approx\;} 10$, from the theoretical strength, Young’s modulus and density of a single nanotube, we derive the following wavelength-independent invisible cable properties:

$$\sigma _C^{\left( {theo} \right)} {\;\approx\;} 10\,{\textrm{MPa,}}\quad E_C {\;\approx\;} 0.1\,{\textrm{GPa,}}\quad \rho _C {\;\approx\;} {\textrm{0}}{\textrm{.1}}\,{{{\textrm{kg}}} \mathord{\left/ {\vphantom {{{\textrm{kg}}} {{\textrm{m}}^{\textrm{3}} }}} \right. \kern-\nulldelimiterspace} {{\textrm{m}}^{\textrm{3}} }}$$

((3))

Meter-long multiwalled carbon nanotube cables can already be realized [28], suggesting that our proposal could became soon technologically feasible. For such a nanostructured macroscopic cable, a strength over density ratio of ${{\sigma _C } \mathord{\left/ {\vphantom {{\sigma _C } {\rho _C }}} \right. \kern-\nulldelimiterspace} {\rho _C }} {\;\approx\;} 120 - 144{{{\textrm{KPa}}} \mathord{\left/ {\vphantom {{{\textrm{KPa}}} {\left( {{{{\textrm{Kg}}} \mathord{\left/ {\vphantom {{{\textrm{Kg}}} {{\textrm{m}}^{\textrm{3}} }}} \right. \kern-\nulldelimiterspace} {{\textrm{m}}^{\textrm{3}} }}} \right)}}} \right. \kern-\nulldelimiterspace} {\left( {{{{\textrm{Kg}}} \mathord{\left/ {\vphantom {{{\textrm{Kg}}} {{\textrm{m}}^{\textrm{3}} }}} \right. \kern-\nulldelimiterspace} {{\textrm{m}}^{\textrm{3}} }}} \right)}}$ was measured, dividing the breaking tensile force by the mass per unit length of the cable (the cross-section geometry was not of clear identification). Thus, we estimate for the single nanotube contained in such a cable $\sigma _{NT} {\;\approx\;} 170{\textrm{MPa}}$ ($\rho _{NT} {\;\approx\;} {\textrm{1300\,K}}{{\textrm{g}} \mathord{\left/ {\vphantom {{\textrm{g}} {{\textrm{m}}^{\textrm{3}} }}} \right. \kern-\nulldelimiterspace} {{\textrm{m}}^{\textrm{3}} }}$), much lower than its theoretical or measured nanoscale strength [29]. This result was expected as a consequence of the larger probability to find critical defects in larger volumes [30]. Thus, defects limit the range of applicability of long bundles based on nanotubes, e.g. reducing their strength by about one order of magnitude [12]. However, the cable strength is expected to increase with the technological advancement. The cable density was estimated to be $\rho _C {\;\approx\;} 1.5{{{\textrm{Kg}}} \mathord{\left/ {\vphantom {{{\textrm{Kg}}} {{\textrm{m}}^{\textrm{3}} }}} \right. \kern-\nulldelimiterspace} {{\textrm{m}}^{\textrm{3}} }}$ [28], thus resulting in a cable strength of $\sigma _C {\;\approx\;} 200\,{\textrm{KPa}}$. Note that a densified cable with a larger value of ${{\sigma _C } \mathord{\left/ {\vphantom {{\sigma _C } {\rho _C }}} \right. \kern-\nulldelimiterspace} {\rho _C }} {\;\approx\;} 465\,{{{\textrm{KPa}}} \mathord{\left/ {\vphantom {{{\textrm{KPa}}} {\left( {{{{\textrm{Kg}}} \mathord{\left/ {\vphantom {{{\textrm{Kg}}} {{\textrm{m}}^{\textrm{3}} }}} \right. \kern-\nulldelimiterspace} {{\textrm{m}}^{\textrm{3}} }}} \right)}}} \right. \kern-\nulldelimiterspace} {\left( {{{{\textrm{Kg}}} \mathord{\left/ {\vphantom {{{\textrm{Kg}}} {{\textrm{m}}^{\textrm{3}} }}} \right. \kern-\nulldelimiterspace} {{\textrm{m}}^{\textrm{3}} }}} \right)}}$ was also realized [29], suggesting the possibility of a considerable advancement for this technology in the near future. For such cables a degree of transparency was observed, confirming that our proposal is realistic. Inverting Eq. 1 we deduce for them $p {\;\approx\;} 260\,{\textrm{nm}}$, in good agreement with the Scanning Electron Microscope (SEM) image analysis [28]. The nanotube characteristic diameter was $d_ + {\;\approx\;} 10\,{\textrm{nm}}$. Considering the visible spectrum, $\lambda {\;\approx\;} 400 - 600\,{\textrm{nm}}$, the condition (2a) was thus satisfied, whereas the condition (2b) was not satisfied. Thus, only a partial degree of transparency was to be expected (see Fig. 2, related inset).

Moreover, multiwalled carbon nanotubes with $d_ + {\;\approx\;} 50{\textrm{nm}}$ ($d_ - {\;\approx\;} 0{\textrm{nm}}$) spaced by $p {\;\approx\;} 5\,{\mu \textrm{m}}$ are expected to realize an invisible cable with the mechanical properties given in Eq. 3. For example, this would correspond to an invisible cable with a cross-section of 1 cm² and weight per unit length of only $0.01\,{{\textrm{g}} \mathord{\left/ {\vphantom {{\textrm{g}} {\textrm{m}}}} \right. \kern-\nulldelimiterspace} {\textrm{m}}}$, capable of supporting the weight of a man (1,000 N). However note that defects would decrease the cable strength, e.g., by one order of magnitude [12].

The nanotubes will remain parallel satisfying the condition (2b), if the cable will work under tension. A later force at the middle of the cable will tend to compact the nanotubes and at a strain of $\varepsilon {\;\approx\;} 8\left( {{W \mathord{\left/ {\vphantom {W L}} \right. \kern-\nulldelimiterspace} L}} \right)^2$ all of them will be in contact. Since for a cable ${W \mathord{\left/ {\vphantom {W L}} \right. \kern-\nulldelimiterspace} L} < < 1$ (e.g., $10^{ - 2}$), a strain of the order of $\varepsilon {\;\approx\;} 10^{ - 4}$, i.e. small if compared with that at failure $\varepsilon _{NT}^{\left( {theo} \right)} {\;\approx\;} {{\sigma _{NT}^{\left( {theo} \right)} } \mathord{\left/ {\vphantom {{\sigma _{NT}^{\left( {theo} \right)} } {E_{NT} {\;\approx\;} }}} \right. \kern-\nulldelimiterspace} {E_{NT} {\;\approx\;} }} 0.1$, will activate the nanotube interaction. In such a situation the cable would ‘appear’ near to the point of application of the lateral force, i.e. where the condition of Eq. 2b is not locally verified, to survive by activating the interaction; this behavior could help in visualizing the cable after having trapped a victim.

Obviously, reducing the requirement of invisibility to that of (a degree of) transparency would automatically lead to strongest macroscopic synthetic cobwebs. Graded cross-links are needed in order to improve the overall bundle strength [31].

Nanohooks

In this section an estimation of the elastic strength of hooks (Fig. 3a) with friction is summarized [32], treating them as elastic arcs (see [33]). We have quantified, as the intuition and Velcro® material suggest, that hooks allow reversible strong attachment, establishing elastic-plastic or hyper-elastic behaviours as dictated by the competition between friction and large displacements [32]. In addition, size-effects suggest that nano-contacts are safer. Thus, we describe here the main results of a ‘Velcro nonlinear mechanics’[32], that could have interesting applications also in different fields, as suggested by its recent observation in wood [34].

Fig. 3

Elastic hook with friction. Conditions of interlocking (a), ultimate “elastic” strength (b) and hauling (c)

The hook elastic critical force ${F_{h}}$ (Fig. 3b) can be estimated according to:

$$F_h {\;\approx\;} \frac{{\left( {{\pi \mathord{\left/ {\vphantom {\pi 2}} \right. \kern-\nulldelimiterspace} 2} + \varphi } \right)EI}}{{\pi R^2 }}$$

((4))

where ϕ is the friction coefficient between hook and substrate (or loop), E is the material Young modulus, I is the cross-sectional moment of inertia and R is the hook radius. Thus, if a number of hooks per unit area $\rho _h = {m \mathord{\left/ {\vphantom {m {\left[ {\pi \left( {2R} \right)^2 } \right]}}} \right. \kern-\nulldelimiterspace} {\left[ {\pi \left( {2R} \right)^2 } \right]}}$ is present, corresponding to an equivalent number m of hooks per clamp, the corresponding nominal strength will be:

$$\sigma _h = \rho _h F_h = \frac{{m\left( {{\pi \mathord{\left/ {\vphantom {\pi 2}} \right. \kern-\nulldelimiterspace} 2} + \varphi } \right)EI}}{{4\pi ^2 R^4 }} = \frac{{m\left( {{\pi \mathord{\left/ {\vphantom {\pi 2}} \right. \kern-\nulldelimiterspace} 2} + \varphi } \right)E}}{{16\pi \left( {{R \mathord{\left/ {\vphantom {R r}} \right. \kern-\nulldelimiterspace} r}} \right)^4 }}$$

((5))

where r is the equivalent radius (in terms of inertia) of the cross-section. For example, considering m=10, $\varphi = 0$, E=10 GPa (Young’s modulus for keratin material is E=1–20 GPa, see [35], and [36]) and ${R \mathord{\left/ {\vphantom {R r}} \right. \kern-\nulldelimiterspace} r} = 10$ corresponds to $\sigma _h {\;\approx\;} 0.3\,{\textrm{MPa}}$, comparable with the strength observed in evarcha arcuata spiders of ∼0.24 MPa ([5], note the two spider hooks in the related inset of Fig. 2).

On the other hand, the maximum force for hooking (Fig. 3c) is:

$$F_h^* = - \frac{{\varphi EI}}{{R^2 }}$$

((6))

Consequently the ratio:

$$\mu = \frac{{F_h }}{{\left| {F_h^* } \right|}} = \frac{1}{\pi } + \frac{1}{{2\varphi }}$$

((7))

is expected to be very large ($\mu \left( {\varphi {\;\to\;} 0} \right) {\;\to\;} \infty$), and thus strong and ‘reversible’ adhesion is expected in hooked materials. This can be easily verified in our own home making experiments on Velcro materials, directly measuring μ and thus deducing the related friction coefficient ϕ. For example for $\mu {\;\approx\;} 10 {\,-\,} 100$, $\varphi {\;\approx\;} 5 {\,\times\,} 10^{ - \left( {2 - 3} \right)}$.

If a contact area A supports the (e.g., animal body) weight W, the safety factor, i.e. the ratio between the attachment force and the weight $W = Mg = \rho Vg = \rho ALg$ (ρ is the density, g is the gravitational acceleration, V is the body volume and M its mass) is:

$$\lambda = \frac{{\sigma _h A}}{W} = \frac{1}{{\rho g}}\frac{{\sigma _h }}{L}$$

((8))

in which L=V/A is a characteristic size of the supported weight. Thus, smaller is safer. For example, since we expect $L \propto M^{{1 \mathord{\left/ {\vphantom {1 3}} \right. \kern-\nulldelimiterspace} 3}}$, assuming $\sigma _h {\;\approx\;} const$, the predicted scaling is $\lambda = kM^{ - {1 \mathord{\left/ {\vphantom {1 3}} \right. \kern-\nulldelimiterspace} 3}}$; noting that in the evarcha arcuata spiders [5] $\lambda _{spider} {\;\approx\;} 173$ and $M_{spider} {\;\approx\;} 15{\textrm{mg}}$ we deduce $k_{spider} {\;\approx\;} 43{\textrm{g}}^{1/3}$. Thus for a Spiderman ($M_{man} {\;\approx\;} 70\,{\textrm{kg}}$), defined as a man having gloves and shoes composed of spider material, we roughly (because self-similarity is assumed) expect $\lambda _{spiderman} {\;\approx\;} 1$. For gecko gloves, since for geckos $\lambda _{geckos} {\;\approx\;} 102$ and $M_{geckos} {\;\approx\;} 250\,{\textrm{g}}$, we would deduce $k_{geckos} {\;\approx\;} 643\,g^{{1 \mathord{\left/ {\vphantom {1 3}} \right. \kern-\nulldelimiterspace} 3}}$ and thus $\lambda _{spiderman} {\;\approx\;} 15$. Accordingly, such gloves are sufficient to support Spiderman even on a ceiling.

The force carried by one hook scales as $F_1 \equiv F_h \propto {{r^4 } \mathord{\left/ {\vphantom {{r^4 } {R^2 }}} \right. \kern-\nulldelimiterspace} {R^2 }}$, thus the bending, tensile, and nominal stresses in the hook must scale as $\sigma _b \propto {r \mathord{\left/ {\vphantom {r R}} \right. \kern-\nulldelimiterspace} R}$, $\sigma _t \propto \left( {{r \mathord{\left/ {\vphantom {r R}} \right. \kern-\nulldelimiterspace} R}} \right)^2$ and $\sigma _h \propto \left( {{r \mathord{\left/ {\vphantom {r R}} \right. \kern-\nulldelimiterspace} R}} \right)^4$ respectively. Accordingly, size-effects can be predicted. For example, splitting up the contact into n sub-contacts, i.e., $R {\;\to\;} {R \mathord{\left/ {\vphantom {R {\sqrt n }}} \right. \kern-\nulldelimiterspace} {\sqrt n }}$, would result in a force $F_n = n^\beta F_1$ with $\beta = 0$ if $r \propto R$ but $\beta = 2$ if $r = const$. Thus, for this last case, sub-contacts are found to be stronger, even if the hook will be higher stressed and its mechanical strength will impose a lower-bound to the radius of the smallest hook. This explains why Nature uses nano-sized bio-contacts, since usually $0 < \beta < 2$, as recently discussed on the basis of contact mechanics (for which $\beta = {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}$, see [13]). Pugno [26] has shown that this enhancement cannot continue ad infinitum. If the hook weight is a constant fraction of the body weight, the scaling of the safety factor is $\lambda \propto {{r^2 } \mathord{\left/ {\vphantom {{r^2 } {R^3 }}} \right. \kern-\nulldelimiterspace} {R^3 }}$, similarly to the prediction of Eq. 8.

Finally, the work of adhesion per unit area can be computed according to:

$$2\gamma _h = \rho _h \int\limits_0^{F_h } {\delta \left( F \right){\textrm{d}}F} = \left( {\frac{1}{2} + \kappa } \right)\sigma _h \delta \left( {F_h } \right)$$

((9))

where $\kappa = 0$ for linear systems. For example, considering $\kappa = 0$, $\sigma _h = 0.3\,{\textrm{MPa}}$ as previously computed, and $\delta \left( {F_h } \right) {\;\approx\;} R = 100\,{\textrm{nm}}$ we get $\gamma _h {\;\approx\;} 0.03{{\textrm{N}} \mathord{\left/ {\vphantom {{\textrm{N}} {\textrm{m}}}} \right. \kern-\nulldelimiterspace} {\textrm{m}}}$, comparable with the work of adhesion observed in geckos (0.05 N/m, see [37]).

Nanohooks could be useful, in addition to classical mechanisms, in order to reduce the dramatic role played by the surface roughness, that will imply contact ‘defects’ during especially macroscopic adhesion.

Self-Cleaning Super-Adhesive Materials

The contact angle (Fig. 4a) between a liquid drop and a solid surface was found by Young [38] according to $\cos \theta {\,=\,} {{\left( {\gamma _{SV} - \gamma _{SL} } \right)} \mathord{\left/ {\vphantom {{\left( {\gamma _{SV} - \gamma _{SL} } \right)} \gamma }} \right. \kern-\nulldelimiterspace} \gamma }_C$, where $\gamma _C {\;\equiv\;} \gamma _{LV}$ and the subscripts of the surface tensions describe the solid (S), liquid (L) and vapour (V) phases. Note that for ${{\left( {\gamma _{SV} - \gamma _{SL} } \right)} \mathord{\left/ {\vphantom {{\left( {\gamma _{SV} - \gamma _{SL} } \right)} \gamma }} \right. \kern-\nulldelimiterspace} \gamma }_C {\;>\;} 1$ the drop tends to spread completely on the surface and $\theta = 0^ \circ$, whereas for ${{\left( {\gamma _{SL} - \gamma _{SV} } \right)} \mathord{\left/ {\vphantom {{\left( {\gamma _{SL} - \gamma _{SV} } \right)} \gamma }} \right. \kern-\nulldelimiterspace} \gamma }_C {\;>\;} 1$ the drop is in a pure non-wetting state and $\theta = 180^ \circ$. According to the well-known Wenzel’s model [39, 40] the apparent contact angle ${\theta _{W}}$ is a function of the surface roughness w, defined as the ratio of rough to planar surface areas, namely, $\cos \theta _W = w\cos \theta$ (Fig. 4b). The apparent contact angle varies also with the heterogeneous composition of the solid surface, as shown by Cassie and Baxter [41]. Consider a heterogeneous surface made up of different materials characterized by their intrinsic contact angles $\theta _i$ and let ${\varphi _{i}}$ be the area fraction of each of the species; the individual areas are assumed to be much smaller than the drop size. Accordingly, the apparent contact angle $\theta _{CB}$ can be derived as $\cos \theta _{CB} = \sum\limits_i^{} {\varphi _i \cos \theta _i }$ [41].

Fig. 4

Contact angles for a drop on a flat surface (a) or on a rough surface in the Wenzel (b) or Fakir (c) state

A droplet can sit on a solid surface in two distinct configurations or states (Fig. 4b, c). It is said to be in Wenzel state (Fig. 4b) when it is conformal with the topography. The other state in which a droplet can rest on the surface is called the Fakir state, after Quéré [42], where it is not conformal with the topography and only touches the tops of the protrusions on the surface (Fig. 4c). The observed state should be the one of smaller contact angle, as can be evinced by energy minimization [43].

Let us consider a hierarchical surface (Fig. 5). The first level is composed by pillars in fraction ϕ (as in Fig. 4b, c). Each pillar is itself structured in n sub-pillars in a self-similar (fractal) manner, and so on. Thus, the pillar fraction at the hierarchical level N is $\varphi ^N$, whereas the related number of pillars at the level N is $n^N$. Applying the Cassie and Baxter law [41] for the described composite (solid/air) hierarchical surface (the contact angle in air is by definition equal to 180°) we find for the hierarchical fakir state:

$$\cos \theta _F^{\left( N \right)} = \varphi ^N \left( {\cos \theta + 1} \right) - 1$$

((10))

Fig. 5

A hierarchical surface with N=2 levels

Note that for $N = 0$ $\cos \theta _F^{\left( 0 \right)} = \cos \theta$ as it must be, whereas for N=1 $\cos \theta _F^{\left( 1 \right)} = \varphi \left( {\cos \theta + 1} \right) - 1$, as already deduced for the case described in Fig. 4c [44]. Equation 10 quantifies the crucial role of hierarchy and suggests that hierarchical surfaces are fundamental to realize super-hydrophobic materials (effective contact angle larger than $\theta _{SHpho} {\;\approx\;} 150^ \circ$), since we predict $\theta _F^{\left( \infty \right)} = 180^ \circ$. The minimum number of hierarchical levels necessary to achieve super-hydrophobia in the Fakir state is thus:

$$N_{SHpho}^{\left( F \right)} = \frac{{\log \left( {\frac{{1 + \cos \theta _{SHpho} }}{{1 + \cos \theta }}} \right)}}{{\log \varphi }}$$

((11))

and the logarithmic dependence suggests that just a few hierarchical levels are practically required.

By geometrical argument the roughness w of the introduced hierarchical surface Fig. 5 can be calculated in closed form. The roughness at the hierarchical level k is given by $w^{\left( k \right)} = 1 + {{S_L^{\left( k \right)} } \mathord{\left/ {\vphantom {{S_L^{\left( k \right)} } A}} \right. \kern-\nulldelimiterspace} A}$ in which A is the nominal contact area and $S_L^{\left( k \right)}$ is the total lateral surface area of the pillars. The pillar at the level k has an equivalent radius ${r_{k}}$ and a length ${l_{k}}$ and the pillar slenderness s, defined as the ratio between its lateral and base areas, is $s {\;=\;} {{2l_k } \mathord{\left/ {\vphantom {{2l_k } {r_k }}} \right. \kern-\nulldelimiterspace} {r_k }}$. The air surface area at the level k can be computed as $A\left( {1 - \varphi ^k } \right)$ or equivalently as $A {\;-\;} n^k \pi r_k^2$, thus we deduce $r_k = r_0 \left( {{\varphi \mathord{\left/ {\vphantom {\varphi n}} \right. \kern-\nulldelimiterspace} n}} \right)^{{k \mathord{\left/ {\vphantom {k 2}} \right. \kern-\nulldelimiterspace} 2}}$, with $A \equiv A_0 \equiv \pi r_0^2$. Consequently:

$$w^{\left( N \right)} = 1 + \frac{1}{{\pi r_0^2 }}\sum\limits_{k = 1}^N {2\pi r_k l_k n^k } = 1 + s\sum\limits_{k = 1}^N {\varphi ^k } = 1 + s\frac{{\varphi - \varphi ^{N + 1} }}{{1 - \varphi }}\quad \forall n$$

((12))

Note that the result becomes independent from n (and that $w^{\left( 0 \right)} \left( \varphi \right) = 1 = w^{\left( N \right)} \left( {\varphi = 0} \right)$, $w^{\left( 1 \right)} = 1 + s\varphi$, whereas $w^{\left( {N \ne 1} \right)} \left( {\varphi = 1} \right) = \infty$). Thus we find for the hierarchical Wenzel state:

$$\cos \theta _W^{\left( N \right)} = w^{\left( N \right)} \cos \theta = \left( {1 + s\frac{{\varphi - \varphi ^{N + 1} }}{{1 - \varphi }}} \right)\cos \theta $$

((13))

Equation 13 suggests that hierarchical surfaces can be interesting also to realize super-hydrophilic materials, since we predict $\cos \theta _W^{\left( \infty \right)} = w^{\left( \infty \right)} \cos \theta$ with $w_{}^{\left( \infty \right)} = 1 + {{s\varphi } \mathord{\left/ {\vphantom {{s\varphi } {\left( {1 - \varphi } \right)}}} \right. \kern-\nulldelimiterspace} {\left( {1 - \varphi } \right)}}$; thus if $\cos \theta > 0$ $\theta _W^{\left( \infty \right)} {\;\to\;} 0$, for $s {\;\to\;} \infty$ or $\varphi {\;\to\;} 1$. However note that for $\cos \theta < 0$, $\theta _W^{\left( \infty \right)} {\;\to\;} 180^ \circ$ ($s {\;\to\;} \infty$ or $\varphi {\;\to\;} 1$), and thus super-hydrophobia can take place also in the Wenzel state, without invoking fakir drops. The minimum number of hierarchical levels necessary to render the surface super-hydrophobic/hydrophilic in the Wenzel state is thus:

$$N_{SHpho,\,phi}^{\left( W \right)} = \frac{{\log \left( {1 + \frac{{\left( {1 - \varphi } \right)}}{{s\varphi }}\left( {1 - \frac{{\cos \theta _{SHpho,\,phi} }}{{\cos \theta }}} \right)} \right)}}{{\log \varphi }}$$

((14))

where effective contact angles smaller than $\theta _{SHphi}$ define super-hydrophily.

Comparing $\theta _W^{\left( N \right)}$ and $\theta _F^{\left( N \right)}$, we find that the Fakir state is activated at each hierarchical level for (we omit here second order problems, related to metastability, contact angle hysteresis and limit of the Wenzel’s approach, for which the reader should refer to the review by [21]):

$$\theta > \theta _{WF}^{} , \ \cos \theta _{WF}^{} = - \frac{{1 - \varphi ^N }}{{w^{\left( N \right)} - \varphi ^N }} = - \frac{1}{{w^{\left( \infty \right)} }} = - \frac{{1 - \varphi }}{{1 + \varphi \left( {s - 1} \right)}}\quad \forall N$$

((15))

Note that the result is independent from N and $\theta _{WF}^{} {\;\to\;} 90^ \circ$ for $s {\;\to\;} \infty$ or $\varphi {\;\to\;} 1$, and thus a hydrophobic\hydrophilic material composed by sufficiently slender or spaced pillars surely will/will not activate fakir drops and will become super-hydrophobic\hydrophilic for a large enough number of hierarchical levels. Thus hierarchy can enhance the intrinsic property of a material. The role of hierarchy is summarized in the phase diagram of Fig. 6.

Fig. 6

Effective contact angle $\theta ^* = \theta _{F,W}^{\left( N \right)}$ as a function of the intrinsic one θ by varying the number N of hierarchical levels. Thus, super-hydrophobic/hydrophilic surfaces can be obtained by an opportune design of the hierarchical architecture, according to this phase diagram and reported equations (note that metastable fakir drops could be observed also in the Wenzel region, dotted line, see [21])

For example, for plausibly values of $\varphi = 0.5$ and s=10 we find from Eq. 15 $\theta _{WF}^{} = 95.2^ \circ$; thus assuming $\theta = 95^\circ$ the Fakir state is not activated and the Wenzel state prevails (if the Fakir state still prevails it is metastable, see [21]). From Eq. 12 $w^{\left( 1 \right)} = 6$, $w^{\left( 2 \right)} = 8.5$, $w^{\left( 3 \right)} = 9.75$ and $w^{\left( 4 \right)} = 10.375$; accordingly from Eq. 13 $\theta _W^{\left( 1 \right)} {\;\approx\;} 122^\circ$, $\theta _W^{\left( 2 \right)} {\;\approx\;} 138^\circ$, $\theta _W^{\left( 3 \right)} {\;\approx\;} 148^\circ$ and $\theta _W^{\left( 4 \right)} {\;\approx\;} 155^\circ$, thus N=4 hierarchical levels are required for activating super-hydrophobia (from Eq. 14 $N_{SHpho}^{\left( W \right)} = 3.2$). On the other hand, assuming $\theta = 100^\circ$ fakir drops are activated and from Eq. 10 $\theta _F^{\left( 1 \right)} {\;\approx\;} 126^\circ$, $\theta _F^{\left( 2 \right)} {\;\approx\;} 143^\circ$ and $\theta _F^{\left( 3 \right)} {\;\approx\;} 154^\circ$, thus N=3 hierarchical levels are sufficient to achieve super-hydrophobia (from Eq. 11 $N_{SHpho}^{\left( F \right)} = 2.6$).

Some insects, such as the beetle Hemisphaerota cyanea, use capillary to stick to their substrate, generating a force close to 1 g (i.e. 60 times its body mass) for more than 2 min [45], allowing them to resist attacking ants; tokay geckos use the same principle (in addition to van der Waals forces) to generate their tremendous adhesion [3].

Between a spherical surface (contact angle θ) of radius ${r_{0}}$ and a flat plate (contact angle ${\theta _{P}}$), the capillary attractive or repulsive force is predicted to be $F_C = 2\pi r_0 \gamma _C \left( {\cos \theta + \cos \theta _P } \right)$ [46]. Thus, for a pillar of size ${r_{0}}$ composed by N hierarchical levels the force is $F_C^{\left( N \right)} = n^N 2\pi r_N \gamma _C \left( {\cos \theta + \cos \theta _P } \right)$ and the nominal strength $\sigma _C^{\left( N \right)} = {{F_C^{\left( N \right)} } \mathord{\left/ {\vphantom {{F_C^{\left( N \right)} } {\left( {\pi r_0^2 } \right)}}} \right. \kern-\nulldelimiterspace} {\left( {\pi r_0^2 } \right)}}$ becomes:

$$\sigma _C^{\left( N \right)} = \frac{{2\left( {\varphi n} \right)^{{N \mathord{\left/ {\vphantom {N 2}} \right. \kern-\nulldelimiterspace} 2}} }}{{r_0 }}\gamma _C \left( {\cos \theta _{W,\,F}^{\left( N \right)} + \cos \theta _P } \right)$$

((16))

Note that for N=0 such a capillary strength corresponds to the previously discussed law. For N=1 the strength scales as $\sqrt n$, in agreement with a recent discussion [14]: splitting up the contact into n sub-contacts would result in a stronger interaction (with a cut-off at the theoretical strength): smaller is stronger (see [30]). This explains the observed miniaturized size of biological contacts. Introducing the previously computed contact angle related to the hierarchical surface allows one to evaluate the hierarchical capillary force, with or without activation of the Fakir state. Super-attraction/repulsion can thus be achieved thanks to hierarchy, since $\sigma _C^{\left( N \right)} {\;\approx\;} \sigma _C^{\left( 0 \right)} \left( {\varphi n} \right)^{{N \mathord{\left/ {\vphantom {N 2}} \right. \kern-\nulldelimiterspace} 2}}$.

Thus, the analysis demonstrates and quantifies that super-hydrophobic/ hydrophilic and simultaneously super-attractive/repulsive surfaces can be realized, mimicking Nature thanks to hierarchical architectures. Assuming $\varphi = 0.5$, n=s=10 and $\theta {\;\approx\;} 120^\circ$ (as in lotus leaves), the analysis shows that fakir drops are activated and only two hierarchical levels are required to achieve super-hydrophobia ($\theta > \theta _{WF}^{} = 95.2^\circ$, $N_{SHpho}^{\left( F \right)} = 1.9$; $\theta _F^{\left( 1 \right)} {\;\approx\;} 139^\circ$, $\theta _F^{\left( 2 \right)} {\;\approx\;} 151^\circ$), in agreement with direct observations on super-hydrophobic plants (see our discussing in the Introduction). Simultaneously, we deduce $\sigma _C^{\left( 1 \right)} {\;\approx\;} 2.2\sigma _C^{\left( 0 \right)}$, $\sigma _C^{\left( 2 \right)} {\;\approx\;} 5.0\sigma _C^{\left( 0 \right)}$ and $\sigma _C^{\left( 3 \right)} {\;\approx\;} 11.2\sigma _C^{\left( 0 \right)}$, i.e. just three hierarchical levels (or even two, if $\varphi {\;\approx\;} 1$ and $n {\;\approx\;} 10$) are sufficient to enhance the capillary strength by one order of magnitude, generating super-attractive ($\sigma _C^{\left( 0 \right)} > 0$) or super-repulsive ($\sigma _C^{\left( 0 \right)} < 0$) surfaces. Thus, the analysis suggests the feasibility of innovative self-cleaning and simultaneously super-adhesive hierarchical materials, as observed in spiders and geckos.

Analogously, hierarchy simultaneously enhances the work of adhesion, and thus the corresponding force, per unit nominal area, due to the larger effective surface area. Accordingly, the maximum (assuming all the surfaces in contact) effective work of adhesion can be derived by the following energy equivalence:

$$\gamma _{\max }^{\left( {N_1,\, N_2 } \right)} {\;\approx\;} \gamma _C \left( {w_1^{\left( {N_1 } \right)} + w_2^{\left( {N_2 } \right)} } \right)$$

((17))

in which the subscripts 1 and 2 refer to the two surfaces in contact. For example, the adhesive force between two-hierarchical level surfaces, defined by $w_{1,2}^{\left( 2 \right)} = 1.75$ ($\varphi = 0.5$, s=1), is enhanced by hierarchy by a factor of 3.5 (with respect to the two corresponding flat surfaces). Note that for s=10 this factor becomes 18 and remains significantly larger than one (i.e. 10) even if one of the two surfaces becomes perfectly flat.

However we have to note that the Wenzel approach must loose its validity for large roughness w, for which an effective micro- (rather than macro-) roughness have to be considered.

Capillary and van der Waals Forces

The capillary force can also be derived according to the well-known Laplace’s law [47]. The attractive force between two flat plates of areas A, separated by a liquid of thickness t, with (liquid/vapour) surface tension $\gamma _C$ and (liquid/solid) contact angle θ is (see the review by [21]):

$$F_C = \frac{{2A\gamma _C \cos \theta }}{t}$$

((18))

Note that $\sigma _C = {{F_C } \mathord{\left/ {\vphantom {{F_C } A}} \right. \kern-\nulldelimiterspace} A}$ is a function of the liquid thickness but not of the size of the contact. Considering for example $\gamma _C = 0.05{{\textrm{N}} \mathord{\left/ {\vphantom {{\textrm{N}} {\textrm{m}}}} \right. \kern-\nulldelimiterspace} {\textrm{m}}}$, $\theta = 80^\circ$ and t=1 nm would yield $\sigma _C {\;\approx\;} 9\,{\textrm{MPa}}$. The force described by Eq. 17 is attractive for $\theta < 90^\circ$ (hydrophilic) or repulsive for $\theta > 90^\circ$ (hydrophobic). An additional viscous force can be generated $F_C^{\left( \eta \right)} \propto {\eta \mathord{\left/ {\vphantom {\eta \tau }} \right. \kern-\nulldelimiterspace} \tau }$, where η is the dynamic viscosity of the liquid and τ is the separation time interval.

Note the differences between the force predictions of Eq. 18 and that considered in the previous section [46], i.e.:

$$F_C = 2\pi r_0 \gamma _C \left( {\cos \theta + \cos \theta _P } \right)$$

((19))

in the limit of $r_0 {\;\to\;} \infty$, which suggest that we are far from a full understanding of the mechanism. In addition, both the approaches predict ${{\sigma _C = F_C } \mathord{\left/ {\vphantom {{\sigma _C = F_C } A}} \right. \kern-\nulldelimiterspace} A} = {{F_C } \mathord{\left/ {\vphantom {{F_C } {\left( {\pi r_0^2 } \right)}}} \right. \kern-\nulldelimiterspace} {\left( {\pi r_0^2 } \right)}} {\;\to\;} \infty$ for $t,r_0 {\;\to\;} 0$ in contrast to the common sense of a finite theoretical strength $\sigma _C^{\left( {th} \right)}$. This cut-off could be a consequence of a quantized (instead of a continuous) crack propagation, as discussed in the example reported by Pugno [26]. Thus, the following asymptotic matching can be straightforwardly proposed:

$$\sigma _C {\;\approx\;} \left( {\frac{2}{{r_0 + c}} + \frac{1}{{t + c}}} \right)\gamma _C \left( {\cos \theta + \cos \theta _P } \right), c {\;\approx\;} {{3\gamma _C \left( {\cos \theta + \cos \theta _P } \right)} \mathord{\left/ {\vphantom {{3\gamma _C \left( {\cos \theta + \cos \theta _P } \right)} {\sigma _C^{\left( {th} \right)} }}} \right. \kern-\nulldelimiterspace} {\sigma _C^{\left( {th} \right)} }}$$

((20))

Similarly, the van der Waals force between two parallel surfaces of area A is ([48]; see also [49]):

$$F_{vdW} = \frac{{HA}}{{6\pi t^3 }}$$

((21))

where H is the Hamaker’s constant, with a typical value around $10^{ - 20} {\textrm{J}}$ (as before, t<30 nm is the separation between the two surfaces). Note that $\sigma _{vdW} = {{F_{vdW} } \mathord{\left/ {\vphantom {{F_{vdW} } A}} \right. \kern-\nulldelimiterspace} A}$ is a function of the liquid thickness but not of the size of the contact. Considering for example t=1 nm would yield $\sigma _{vdW} {\;\approx\;} 0.5\,{\textrm{MPa}}$.

For the case of a spherical surface of radius ${r_{0}}$ and a flat plate, the contact force predicted according to the ‘JKR’ model of contact mechanics [50] is:

$$F_{vdW} = {3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-\nulldelimiterspace} 2}\pi \gamma _{vdW} r_0 $$

((22))

Thus also in this case, as formerly discussed by Arzt et al. [13], $F_n = \sqrt n F_1$. Moreover, since $F_{vdW} \propto r_0$ the results reported in the previous section are still applicable.

As for capillary, note the differences between the two approaches summarized in Eqs. 21 and 22, which suggest that we are far from a full understanding of the mechanism. In addition, Eqs. 21 and 22 predict ${{\sigma _{vdW} = F_{vdW} } \mathord{\left/ {\vphantom {{\sigma _{vdW} = F_{vdW} } A}} \right. \kern-\nulldelimiterspace} A} = {{F_{vdW} } \mathord{\left/ {\vphantom {{F_{vdW} } {\left( {\pi r_0^2 } \right)}}} \right. \kern-\nulldelimiterspace} {\left( {\pi r_0^2 } \right)}} {\;\to\;} \infty$ for $t,r_0 {\;\to\;} 0$ in contrast to the common sense of a finite theoretical strength $\sigma _{vdW}^{\left( {th} \right)}$ (and of a quantized crack propagation, see [26]). Thus, the following asymptotic matching can be straightforwardly proposed:

$$\sigma _{vdW} {\;\approx\;} {3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-\nulldelimiterspace} 2}\gamma _{vdW} \left( {\frac{1}{{r_0 + c}} + \frac{H}{{9\gamma _{vdW} \pi \left( {t + c} \right)^3 }}} \right), c = X\left( {6\pi \sigma _{vdW}^{\left( {th} \right)} \,; - 9\pi \gamma _{vdW} \,;0; - H} \right)$$

((23))

where $X\left( {a\,;b\,;c;d} \right)$ denotes the solution of the three-order polynomial equation $ax^3 + bx^2 + cx + d = 0$, derived imposing $c:\sigma _{vdW}^{} \left( {r_0 ,t {\;\to\;} 0} \right) = \sigma _{vdW}^{\left( {th} \right)}$ (one could also consider valid Eq. 23 with $c {\;\to\;} 0$, with a cut-off at $\sigma _{vdW}^{\left( {th} \right)}$). To have an idea of the theoretical strength note that $\sigma _{vdW}^{\left( {th} \right)} {\;\approx\;} 20\,{\textrm{MPa}}$ (see [51]).

The different force predictions for plausible values are of the same order of magnitude. Using Eq. 22, as done by Autumn et al. [4] and Arzt et al. [13], ($\gamma _{vdW} {\;\approx\;} 0.05\,{{\textrm{N}} \mathord{\left/ {\vphantom {{\textrm{N}} {\textrm{m}}}} \right. \kern-\nulldelimiterspace} {\textrm{m}}}$) for the gecko spatula ($r_0 {\;\approx\;} 0.05\,{\mu \textrm{m}}$) we get $F_{spatula} {\;\approx\;} 12\,{\textrm{nN}}$, comparable to the observed value of ∼11 nN [9]. Thus, for a seta composed by 1,000 spatulae, $F_{seta}^{} {\;\approx\;} 12\,{\mu \textrm{N}}$ [4, 8] measured values of ∼194 and ∼40 μN respectively); for a non hierarchical seta from Eq. 22 one would deduce ($r_0 {\;\approx\;} 5\,{\mu \textrm{m}}$) $F_{seta}^/ {\;\approx\;} 1.2\,{\mu \textrm{N}}$ and thus for a real, thus hierarchical, seta having 1,000 spatulae, $F_{seta}^{} {\;\approx\;} \sqrt {1000} \times 1.2\,{\mu \textrm{N}} {\;\approx\;} 38\,{\mu \textrm{N}}.$ Similarly for the setula of a spider $R_{eq} {\;\approx\;} \sqrt {{{0.17} \mathord{\left/ {\vphantom {{0.17} \pi }} \right. \kern-\nulldelimiterspace} \pi }} {\mu \textrm{m}} {\;\approx\;} 0.2\,{\mu \textrm{m}}$ (terminal surface area of ∼$0.17\,{\mu \textrm{m}}^2$) and $F_{setula}^{} {\;\approx\;} 47\,{\textrm{nN}}$ (observed value ∼41 nN, [5]).

Finally, we note that since different mechanisms could be simultaneously activated, the real adhesive force (or strength or fracture energy) could be computed as:

$$F = \sum\limits_i^{} {F_i\, f_i } $$

((24))

in which ${F_{i}}$ is the force activated by the i-th mechanism having weight ${f_{i}}$ $\left(\sum\limits_i^{} {f_i } = 1\right)$.

For example, for geckos a still partially unsolved question is to quantify the participation of capillary and van der Waals forces in their adhesion (nanohook and suction mechanisms in geckos have been ruled out, see the review by [2]). We note that Huber et al. [3] observed a humidity (U) dependence of the adhesion force in gecko spatulae, thus, from Eq. 24 we could write:

$$F = F_{dry}\, f_{dry} + F_{wet}\, f_{wet} {\;\approx\;} F_{vdW} \left( {1 - U} \right) + \left( {F_{vdW} + F_C } \right)U$$

((25))

By fitting their data we find $F_{vdW} {\;\approx\;} 7{\textrm{nN}}$ and $F_C {\;\approx\;} 5{\textrm{nN}}$, thus ${{F_{vdW} } \mathord{\left/ {\vphantom {{F_{vdW} } {F_C }}} \right. \kern-\nulldelimiterspace} {F_C }} {\;\approx\;} 1.4$, i.e., van der Waals are expected to be the main adhesive forces in geckos even if capillary ones play a significant role too.

Equations 20 and 23 can be straightforwardly extended to hierarchical surfaces according to our findings reported in the previous Section.

Smart Adhesion

Consider the detachment as the peeling of a thin film of (free-) length l, width b and thickness h, pulled at an angle ϑ by a force F Fig. 7. A non-linear stress-strain relationship $\sigma = E\left( \varepsilon \right)\varepsilon$ is considered. The total potential energy (elastic energy minus external work) of the film is $\Phi = bhl\int\limits_0^\varepsilon {E\left( \varepsilon \right)} \,\varepsilon \,{\textrm{d}}\varepsilon - Fl\left( {1 - \cos \vartheta + \varepsilon } \right)$. Thus, the energy release rate is:

$$2\Delta \gamma \equiv - \frac{1}{b}\frac{{{\textrm{d}}\Phi }}{{{\textrm{d}}l}} = - h\int\limits_0^\varepsilon {E\left( \varepsilon \right)} \,\varepsilon \;{\textrm{d}}\varepsilon + \frac{F}{b}\left( {1 - \cos \vartheta + \varepsilon } \right)$$

((26))

Fig. 7

Peeling of a thin film

The detachment will take place when $\Delta \gamma \equiv \gamma \equiv \gamma _1 + \gamma _2 - \gamma _{12}$, where $\gamma _{1,2}$ are the surface energies of the two materials in contact and $\gamma _{12}$ is that of the interface.

For quadratic nonlinearities, i.e., $E\left( \varepsilon \right) = E + E^/ \varepsilon$, a closed form solution is still reachable. Note that $E^/ < 0$ describes elastic-plastic materials (e.g., hooked surface with $\varphi > \varphi _C {\;\approx\;} {\pi \mathord{\left/ {\vphantom {\pi {39}}} \right. \kern-\nulldelimiterspace} {39}}$, see [26]), whereas $E^/ > 0$ hyper-elastic ones (e.g., hooked surface with $\varphi < \varphi _C$). The detachment force is found in the following form:

$$\begin{array}{ll}\varepsilon _C &= X\;\left( {4E^/ ;\;3E + 6E^/ \left( {1 - \cos \vartheta } \right)\;;\,\,6E\left( {1 - \cos \vartheta } \right)\;;\,\, - 12{\gamma \mathord{\left/ {\vphantom {\gamma h}} \right. \kern-\nulldelimiterspace} h}} \right), F_C\\ &= AE\left( {\varepsilon _C } \right)\varepsilon _C\end{array} $$

((27))

(as before $X\left( {a\,;b\,;c;d} \right)$ denotes the solution of the three-order polynomial equation $ax^3 + bx^2 + cx + d = 0$). For $E^/ {\;\to\;} 0$ the classical Kendall [52] prediction is recovered. Varying the pulling angle strong force variations are found, as can easily be evinced considering the simplest case in the limit of $E^{ - 1} ,E^/ {\;\to\;} 0$, deducing $F_C = {{2\gamma b} \mathord{\left/ {\vphantom {{2\gamma b} {\left( {1 - \cos \vartheta } \right)}}} \right. \kern-\nulldelimiterspace} {\left( {1 - \cos \vartheta } \right)}}$. Note that, also according to fracture mechanics, sub-contacts are safer ($b {\;\to\;} {b \mathord{\left/ {\vphantom {b {\sqrt n }}} \right. \kern-\nulldelimiterspace} {\sqrt n }}$,$F_n = \sqrt n F_1$; the paper by Pugno [26] shows that this cannot be ad infinitum).

Moreover the strongest attachment is achieved for $\vartheta = 0$ whereas the easier detachment for $\vartheta = \pi$. The ratio between the corresponding forces is:

$$\frac{{F_a }}{{F_d }} \equiv \frac{{F_C \left( {\vartheta = 0} \right)}}{{F_C \left( {\vartheta = \pi } \right)}} = g\left( {E^/ } \right)\frac{{1 + \sqrt {\chi + 1} }}{{\sqrt \chi }}, \chi \equiv {\gamma \mathord{\left/ {\vphantom {\gamma {\left( {hE} \right)}}} \right. \kern-\nulldelimiterspace} {\left( {hE} \right)}}$$

((28))

where $g\left( {E^/ } \right)$ is a known function describing the constitutive nonlinearity, which could have an important role for soft matter, and in particular $g\left( 0 \right) = 1$; in this case ${{F_a } \mathord{\left/ {\vphantom {{F_a } {F_d }}} \right. \kern-\nulldelimiterspace} {F_d }} {\;\to\;} 1,\infty$ for $\chi {\;\to\;} \infty ,0$. For example taking $\gamma = 0.05{{\textrm{N}} \mathord{\left/ {\vphantom {{\textrm{N}} {\textrm{m}}}} \right. \kern-\nulldelimiterspace} {\textrm{m}}}$ (of the order of the previously discussed $\gamma _{h,C,vdW}$ ), h=100 nm, E=10 GPa we find ${{F_a } \mathord{\left/ {\vphantom {{F_a } {F_d }}} \right. \kern-\nulldelimiterspace} {F_d }} {\;\approx\;} 283$. This value is of the same order of magnitude of the safety factor found in spiders [5], i.e., 173. Such a geometrical control can thus explain the smart safety factor reduction during detachment up to ∼1, needed for animal walk. Thus, this pulling angle control can represent the main mechanism to achieve reversible adhesion. For example, for a man with adhesive gloves capable of supporting 300 kg at $\vartheta {\;\approx\;} \pi$, only ∼1 kg must be applied at $\vartheta {\;\approx\;} 0$ to detach them. Probably the proper use of such hypothetical gloves would require opportune training, similarly to the use of a pair of skis, a paragliding or a wet suit.

The value of ${F_{C}}$ corresponds to a delamination (opening and/or sliding) and not necessarily to a detachment (opening prevails on sliding). To distinguish between these two different mechanisms we note that $2\gamma = {{\left( {K_{op}^2 + K_{sl}^2 } \right)} \mathord{\left/ {\vphantom {{\left( {K_{op}^2 + K_{sl}^2 } \right)} E}} \right. \kern-\nulldelimiterspace} E}$, where $K_{op,sl}^{}$ are the stress intensity factors at the tip of the interfacial crack for opening (mode I) or sliding (mode II) and $K_{op}^{} \propto F_ \bot ^{} \propto \sin \vartheta$ whereas $K_{sl}^{} \propto F_{//} \propto \cos \vartheta$; assuming as a first approximation a detachment for ${{K_{sl}^{} } \mathord{\left/ {\vphantom {{K_{sl}^{} } {K_{op} {\;\approx\;} }}} \right. \kern-\nulldelimiterspace} {K_{op} {\;\approx\;} }}\tan ^{ - 1} \vartheta {\,<\,} 1$ we derive a critical value of ∼$45^\circ$. A similar behaviour has recently been confirmed by numerical simulations on gecko seta [51]: for forces applied at an angle less than ∼$30^\circ$ the predominant failure mode was sliding, whereas larger angles correspond to detachment. Using the previous parameters we find ${{F_C \left( {\vartheta = 30^\circ } \right)} \mathord{\left/ {\vphantom {{F_C \left( {\vartheta = 30^\circ } \right)} {F_C \left( {\vartheta = 150^\circ } \right)}}} \right. \kern-\nulldelimiterspace} {F_C \left( {\vartheta = 150^\circ } \right)}} {\;\approx\;} 14$, which can still be sufficient to control adhesion of nonideal contacts, for which the strength is expected to be reduced by a factor of about one order of magnitude [12]. However note that in any case (i.e., also at $\vartheta {\;\approx\;} 0$) the total adhesive force could be overcome by subsequently detaching single points of contacts and not the whole surface at once [14], even if, when not in vivo, this mechanism could be hard to be activated. Note that the ratio predicted by Eq. 28 is compatible with homemade experiments that we have performed using adhesive tape. For larger thickness the behaviour would be that of a beam rather than of a film [26].

Observation on Living Geckos

In this section we summarize recent observations on adhesion of living geckos [53]. We report experimental observations on the times of adhesion of living Tokay geckos (Gekko geckos) on polymethylmethacrylate (PMMA) inverted surfaces. Two different geckos (male and female) and three surfaces with different root mean square (RMS) roughness (RMS=42, 618 and 931 nm) have been considered, for a total of 72 observations. The measured data are proved to be statistically significant, following the Weibull Statistics with coefficients of correlation between 0.781 and 0.955. The unexpected result is the observation of a maximal gecko adhesion on the surface with intermediate roughness of RMS=618 nm, that we note has waviness comparable to the seta size.

Surface roughness strongly influences the animal adhesion strength and ability. Its role was shown in different measurements on flies and beetles, walking on surfaces with well defined roughness [54], [55, 56], on the chrysomelid beetle Gastrophysa viridula [57], on the fly Musca domestica ([56]) as well as on the Tokay geckos [58]. Peressadko and Gorb [56] and Gorb [57] report a minimum of the adhesive/frictional force, spanning surface roughness from 0.3 to 3 μm. The experiments on the reptile Tokay gecko [58] showed a minimum in the adhesive force of a single spatula at an intermediate root mean square (RMS) surface roughness around 100–300 nm, and a monotonic increase of adhesion times of living geckos by increasing the RMS, from 90 to 3,000 nm. There are several observations and models in the literature, starting with the pioneer paper by Fuller and Tabor [59], in which roughness was seen to decrease adhesion monotonically. But there is also experimental evidence in the literature, starting with the pioneer paper by Briggs and Briscoe [60], which suggests that roughness need not always reduce adhesion. For example, Persson and Tosatti [61] and Persson [62], in the framework of a reversible model, have shown that for certain ranges of roughness parameters, it is possible for the effective surface energy to first increase with roughness amplitude and then eventually decreasing. Including irreversible processes, due to mechanical instabilities, Guduru [63] has demonstrated, under certain hypotheses, that the pull-out force must increase by increasing the surface wave amplitude. We have suggested [53] that roughness alone could not be sufficient to describe the three-dimensional topology of a complex surface and additional parameters have to be considered for formulating a well-posed problem.

Accordingly, we have machined and characterized three different Polymethylmethacrylate surfaces (PMMA 1,2,3; surface energy of ∼41 mN/m) with a full set of roughness parameters, as reported in Table 1: Sa represents the surface arithmetical average roughness; Sq=RMS is the classical mean square roughness; Sp and Sv are respectively the height of the highest peak and the deepness of the deepest valley (absolute value); Sz is the average distance between the five highest peaks and the five deepest valleys (detected in the analyzed area); Ssk indicates the surface skewness; Sdr is the effective surface area minus the nominal one and divided by the last one.

Table 1 Roughness parameters for the three different Polymethylmethacrylate (PMMA 1,2,3) surfaces

Two different Tokay gecko’s, female (G1, weight of ∼46 g) and male (G2, weight of ∼72 g), have been considered. The gecko is first placed in its natural position on the horizontal bottom of a box (50 × 50 × 50 cm³). Then, slowly, we rotated the box up to the gecko reaches a natural downwards position and, at that time, we start the measurement of the gecko time of adhesion. We excluded any trial in which the gecko walks on the inverted surface. The time measurement was stopped when gecko breaks loose from the inverted surface and falls on the bottom of the box (for G1) or at the first detachment movement of the gecko’s foot (for G2). The time between one measurement and the following, pertaining to the same set, is only that needed to rotate the box and placed the gecko again on the upper inverted surface (∼14 s). The experiments were performed at ambient temperature (∼22°C) and humidity (∼75%). The measured adhesion times are summarized in Table 2 and confirmed to be statistically significant by applying Weibull Statistics, see Fig. 8.

Table 2 Gecko adhesion times on PMMA 1,2,3 surfaces

Fig. 8

Weibull Statistics (F is the cumulative probability of detachment/failure and t_i are the measured adhesion times) applied to the measured adesion times on PMMA surfaces. PMMA 1 (green, for which we made the measurements in four different days and with both geckos G1 and G2), PMMA 2 (black, for which we made the measurements in two different days, one with gecko G1 and one with gecko G2) and PMMA 3 (red, for which we made the measurements in a single day with gecko G2)

We have observed a maximum in the gecko’s adhesion times on PMMA 2, having an intermediate roughness of RMS=618 nm. An oversimplified explanation could be the following. For PMMA 1 (Sq=42 nm, waviness of λ≈3–4 μm, h≈0.1 μm), the gecko’s seta (diameter of ~10 μm, represented in blue in Fig. 9, that must not be confused with the terminal nearly two dimensional spatualae) cannot penetrate in the characteristic smaller valleys and adhere on the side of each single one Fig. 9a, thus cannot optimally adapt to the surface roughness. For PMMA 2 (Sq=618 nm, λ≈7–8 μm, h≈1 μm) the gecko’s setae are able to adapt better to the roughness, adhering this time on the top of and on the side of a single asperity: in this way the effective number of setae in contact increases and, as a direct consequence, also the adhesion ability of the gecko increases Fig. 9b. On PMMA 3 (Sq=931 nm, λ≈10–12 μm and h≈2 μm) the waviness characterizing the roughness is larger than the seta’s size: as a consequence, a decrease in the number of setae in contact is expected Fig. 9c . As a result, on PMMA 2 an adhesion increment, of about 45%, is observed. According to Briggs and Briscoe [60] an increment of 40%, thus close to our observation, is expected for an adhesion parameter α equal to 1/3. Such a parameter was introduced as the key parameter in governing adhesion by Fuller and Tabor [59] as:

$$\alpha = \frac{{4\sigma }}{3}\left( {\frac{{4E}}{{3\pi \sqrt \beta \gamma }}} \right)^{{2 \mathord{\left/ {\vphantom {2 3}} \right. \kern-\nulldelimiterspace} 3}} $$

((29))

where σ is the standard deviation of the asperity height distribution (assumed to be Gaussian), β is the mean radius of curvature of the asperity, γ is the surface energy and E is the Young modulus of the soft solid (gecko foot). Even if the value of E of the entire foot cannot be simply defined, as a consequence of its non compact structure, we note that considering it to be of the order of 10 MPa (thus much smaller than that of the keratin material), with $\gamma {\;\approx\;} 0.05{{\textrm{N}} \mathord{\left/ {\vphantom {{\textrm{N}} {\textrm{m}}}} \right. \kern-\nulldelimiterspace} {\textrm{m}}}$ [8], $\sigma {\;\approx\;}$Sq, $\beta {\;\approx\;} \lambda$ would correspond to values of α close to 0.5.

Fig. 9

A simple interpretation of our experimental results on the adhesion tests of living geckos on PMMA surfaces having different roughness. (A) Setae cannot adapt well on PMMA 1; (B) on PMMA 2 the adhesion is enhanced thanks to the higher compatibility in size between setae and roughness; (C) on PMMA 3 only partial contact is achieved. On the right, we report the analyzed three-dimensional profiles of the roughness for all the three investigated surfaces (from the top: PMMA 1,2 and 3)

The reported maximal adhesion was not observed by Huber et al. [58]. Note that their tested polished surfaces were of five different types, with a nominal asperity size of 0.3, 1, 3, 9 and 12 μm, which correspond to RMS values of 90, 238, 1,157, 2,454 and 3,060 nm respectively. Huber et al. [58] have observed sliding of geckos on polishing paper with a RMS value of 90 nm for slopes larger than 135°. On a rougher substrate, with a RMS value of 238 nm, two individual geckos were able to cling to the ceiling for a while, but the foot-surface contact had to be continuously renewed because gecko toes slowly tend to slid off the substrate. Finally, on the remaining tested rougher substrates, animals were able to adhere stably to the ceiling for more than 5 min.

These different observations (assuming that the influences of claws and moult were minimized also by Huber et al. [58] suggest that the RMS parameter is not sufficient alone to describe all the aspects of the surface roughness. The use of a ‘complete’set of roughness parameters could help in better understanding the animal adhesion and thus in optimizing the Spiderman suit design.

Towards a Spiderman Suit

According to our analysis, a man (palm surfaces of ∼200 cm²) and gecko-material gloves ($\sigma _{gecko} {\;\approx\;} 0.58\,{\textrm{MPa}}$) could support a mass of ∼1,160 kg (safety factor ∼14), or with spider-material gloves ($\sigma _{spider} {\;\approx\;} 0.24\,{\textrm{MPa}}$) a mass of ∼480 kg (safety factor ∼6). We expect, due to non ideal contact, a safety factor reduction of about one order of magnitude [12], thus we still conclude that Spiderman suits could become feasible in the near future. Note that theoretical van der Waals gloves ($\sigma _{vdW}^{\left( {th} \right)} {\;\approx\;} 20\,{\textrm{MPa}}$) would allow one to support a mass of ∼40,000 kg (safety factor of ∼500). Carbon nanotubes could be one of the most promising candidates for our applications: on a small scale a carbon nanotube surface was able to achieve adhesive forces ∼200 times greater than those of gecko foot hairs [64], even if it could not replicate large scale gecko adhesion perhaps due to a lack of compliance and hierarchy. Thus, we propose the use of hierarchical branched long (to have the sufficient compliance) nanotubes [65] as good material for a Spiderman suit, with a number of hierarchical levels sufficient to activate self-cleaning, as quantifiable by our calculations. Their aspect ratio must not be too large, to avoid bunching [66, 67] and elastic self-collapse under their own weight, but sufficiently large to conform to a rough surface by buckling under the applied stress (see [2]), similar to the optimization done by Nature in spiders and geckos. In particular, following Glassmaker et al. [67] and Yao and Gao [68] for the bunching and introducing our result for the pillar radius at the level N ($r_N = r_0 \left( {{\varphi \mathord{\left/ {\vphantom {\varphi n}} \right. \kern-\nulldelimiterspace} n}} \right)^{{N \mathord{\left/ {\vphantom {N 2}} \right. \kern-\nulldelimiterspace} 2}}$) we find the anti-bunching and anti-self-collapse [69] conditions at the hierarchical level N in the following form:

$$\begin{array}{ll} & s < \min \left\{ 2\left( {\frac{{3^3 \pi ^4 }}{{2^5 \left( {1 - v^2 } \right)}}} \right)^{{1 \mathord{\left/ {\vphantom {1 {12}}} \right. \kern-\nulldelimiterspace} {12}}} \left( {\frac{{Er_0 }}{\gamma }} \right)^{{1 \mathord{\left/ {\vphantom {1 3}} \right. \kern-\nulldelimiterspace} 3}} \left( {\frac{\varphi }{n}} \right)^{{N \mathord{\left/ {\vphantom {N 6}} \right. \kern-\nulldelimiterspace} 6}} \left( {\sqrt {\left( {{{\varphi _{\max } } \mathord{\left/ {\vphantom {{\varphi _{\max } } {\varphi ^N }}} \right. \kern-\nulldelimiterspace} {\varphi ^N }}} \right)} - 1} \right)^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} ,\right.\\ & \left.\left( {\frac{{8\pi ^2 E}}{{r_0 \left( {{\varphi \mathord{\left/ {\vphantom {\varphi n}} \right. \kern-\nulldelimiterspace} n}} \right)^{{N \mathord{\left/ {\vphantom {N 2}} \right. \kern-\nulldelimiterspace} 2}} \rho g}}} \right)^{{1 \mathord{\left/ {\vphantom {1 3}} \right. \kern-\nulldelimiterspace} 3}} \right\},\end{array}$$

((29))

where v is the Poisson’s ratio, ρ is the material density and $\varphi _{\max } = {\pi \mathord{\left/ {\vphantom {\pi {2\sqrt 3 }}} \right. \kern-\nulldelimiterspace} {2\sqrt 3 }}$, $\varphi _{\max } = {\pi \mathord{\left/ {\vphantom {\pi 4}} \right. \kern-\nulldelimiterspace} 4}$ or $\varphi _{\max } = {\pi \mathord{\left/ {\vphantom {\pi {3\sqrt 3 }}} \right. \kern-\nulldelimiterspace} {3\sqrt 3 }}$ for triangular, square or hexagonal pillar lattices respectively. In order to have a uniform contact, the buckling [69] must be activated under the applied stress $\sigma _a$ (e.g., ~10 KPa, see [2]), thus imposing:

$$s > \pi \sqrt {{4 \mathord{\left/ {\vphantom {4 3}} \right. \kern-\nulldelimiterspace} 3}} \varphi ^{{N \mathord{\left/ {\vphantom {N 2}} \right. \kern-\nulldelimiterspace} 2}} \sqrt {\left( {{{\sigma _a } \mathord{\left/ {\vphantom {{\sigma _a } E}} \right. \kern-\nulldelimiterspace} E}} \right)} ,$$

((30))

Equations 29 and 30 can be used for an optimal design of hierarchical super-adhesive materials.

Accompanied by large transparent (if not fully invisible, to dispose of a higher strength) nanotube based cobwebs, a complete preliminary Spiderman suit could be realized, Fig. 2.

Conclusions

We have proposed [26] new laws to design futuristic self-cleaning, super-adhesive and releasable hierarchical smart materials, as well as large invisible cables, based on carbon nanotube technology.

Thus the formulas suggest the possibility of scaling up the amazing adhesion properties of a spider to the size of a man, thus the feasibility of a Spiderman suit. Strong attachment and self-cleaning are all properties that must be achieved simultaneously. Even if this seems to be impossible, since these mechanisms are in strong competitions, lotus leaves and gecko feet suggest the opposite. But one could deduce that this high nanotech project is unfeasible, since no adhesive-based animals larger than geckos exist in Nature. This is not fully right: Nature has often different scopes with respect to ours, for example animals are not interested in going into space, as we are. Consequently, rather than mimicking Nature we must be inspired by Nature: an airplane is not a big bird. Thus, the project could be feasible. We think that it is feasible, since for a Spiderman Suit we need an adhesive strength that is much lower (two order of magnitudes) than the theoretical (e.g. van der Waals) strength. This safety factor could allow us to produce a flaw-tolerant, a very important requirement since larger contact imperfections are expected at larger size scales, super-adhesive smart material.

The analysis thus represents a first step towards the feasibility of a Spiderman suit [26].