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Adhesion and Friction Force Coupling of Gecko Setal Arrays: Implications for Structured Adhesive Surfaces† Boxin Zhao,‡ Noshir Pesika,‡ Kenny Rosenberg,‡ Yu Tian,‡,§ Hongbo Zeng,‡ Patricia McGuiggan,‡ Kellar Autumn,| and Jacob Israelachvili*,‡ Department of Chemical Engineering and Materials Research Laboratory, UniVersity of California at Santa Barbara, Santa Barbara, California 93106, State Key Lab of Tribology, Department of Precision Instruments, Tsinghua UniVersity, Beijing 100084, People’s Republic of China, and Department of Biology, Lewis & Clark College, Portland, Oregon 97219 ReceiVed July 16, 2007. In Final Form: October 12, 2007 The extraordinary climbing ability of geckos is partially attributed to the fine structure of their toe pads, which contain arrays consisting of thousands of micrometer-sized stalks (setae) that are in turn terminated by millions of fingerlike pads (spatulae) having nanoscale dimensions. Using a surface forces apparatus (SFA), we have investigated the dynamic sliding characteristics of setal arrays subjected to various loading, unloading, and shearing conditions at different angles. Setal arrays were glued onto silica substrates and, once installed into the SFA, brought toward a polymeric substrate surface and then sheared. Lateral shearing of the arrays was initiated along both the “gripping” and “releasing” directions of the setae on the foot pads. We find that the anisotropic microstructure of the setal arrays gives rise to quite different adhesive and tribological properties when sliding along these two directions, depending also on the angle that the setae subtend with respect to the surface. Thus, dragging the setal arrays along the gripping direction leads to strong adhesion and friction forces (as required during contact and attachment), whereas when shearing along the releasing direction, both forces fall to almost zero (as desired during rapid detachment). The results and analysis provide new insights into the biomechanics of adhesion and friction forces in animals, the coupling between these two forces, and the specialized structures that allow them to optimize these forces along different directions during movement. Our results also have practical implications and criteria for designing reversible and responsive adhesives and articulated robotic mechanisms.
Introduction The extraordinary climbing ability of geckos has stimulated extensive research because of interest in its own right and also because of technological implications in designing dry, responsive adhesive systems and robots. Much effort has been focused on understanding the high adhesion of geckos, which includes experimentally imaging and characterizing the fine structures of gecko foot pads1-4 and measuring the adhesive forces of single gecko foot hairs and even single spatulae5,6 and theoretically modeling the adhesion by applying continuum mechanics-based theories such as the Hertz and JKR theories.7-9 The superadhesive ability of geckos to stick to almost any surface, whether hydrophilic or hydrophobic, rough or smooth, has been attributed to the fine structure of its toe pads, which contain arrays consisting of thousands of micrometer-sized stalks (setae) that are terminated by millions of fingerlike pads (spatulae) having nanoscale dimensions. These millions of fingerlike pads allow for a large †
Part of the Molecular and Surface Forces special issue. * To whom correspondence should be addressed. E-mail: jacob@ engineering.ucsb.edu. Fax: 805-893-7870. Phone: 805-893-8407. ‡ University of California at Santa Barbara. § Tsinghua University. | Lewis & Clark College. (1) Ruibal, R.; Ernst, V. J. Morphol. 1965, 117, 271-294. (2) Hiller, U. Zoomorphologie 1976, 84, 211-221. (3) Rizzo, N. W.; Gardner, K. H.; Walls, D. J.; Keiper-Hrynko, N. M.; Ganzke, T. S.; Hallahan, D. L. J. R. Soc. Interface 2006, 3, 441-451. (4) Russell, A. P.; Bauer, A. M.; Laroiya, R. J. Zool. 1997, 241, 767-790. (5) Autumn, K.; Liang, Y. A.; Hsieh, S. T.; Zesch, W.; Chan, W. P.; Kenny, T. W.; Fearing, R.; Full, R. J. Nature 2000, 405, 681-685. (6) Huber, G.; Gorb, S. N.; Spolenak, R.; Arzt, E. Biol. Lett. 2005, 1, 2-4. (7) Arzt, E.; Gorb, S.; Spolenak, R. Proc. Natl. Acad. Sci. U.S.A. 2003, 100, 10603-10606. (8) Gao, H. J.; Wang, X.; Yao, H. M.; Gorb, S.; Arzt, E. Mech. Mater. 2005, 37, 275-285. (9) Persson, B. N. J. J. Chem. Phys. 2003, 118, 7614-7621.
“real” contact area to form so that the gecko foot pads can adhere to almost any surface via the weak but universal van der Waals force10,11 and other particular types of noncovalent forces such as capillary forces.12,13 The hierarchy of biological nano/micro/macrostructures and functions has been applied technologically in the design of manmade gecko mimics14-16 and the development of new methodologies to tune adhesion by fabricating fine microstructured surfaces.17,18 Thus, gecko foot pad analogues or “structured adhesives” are often straight pillars or fibers arrayed perpendicular to the substrate14 or inclined to the substrate,16,19 and are made of either stiff silicon materials15 or compliant polymers.14,16,20 Both experimental and theoretical studies9,19,21,22 have revealed that these fibrillar surfaces or structures are more deformable and have larger fracture zones so as to have stronger adhesion (10) Russell, A. P. J. Zool. 1975, 176, 437-476. (11) Autumn, K.; Sitti, M.; Liang, Y. C. A.; Peattie, A. M.; Hansen, W. R.; Sponberg, S.; Kenny, T. W.; Fearing, R.; Israelachvili, J. N.; Full, R. J. Proc. Natl. Acad. Sci. U.S.A. 2002, 99, 12252-12256. (12) Huber, G.; Mantz, H.; Spolenak, R.; Mecke, K.; Jacobs, K.; Gorb, S. N.; Arzt, E., Proc. Natl. Acad. Sci. U.S.A. 2005, 102, 16293-16296. (13) Sun, W. X.; Neuzil, P.; Kustandi, T. S.; Oh, S.; Samper, V. D. Biophys. J. 2005, 89, L14-L17. (14) Geim, A. K.; Dubonos, S. V.; Grigorieva, I. V.; Novoselov, K. S.; Zhukov, A. A.; Shapoval, S. Y. Nat. Mater. 2003, 2, 461-463. (15) Northen, M. T.; Turner, K. L. Nanotechnol. 2005, 16, 1159-1166. (16) Glassmaker, N. J.; Jagota, A.; Hui, C. Y.; Kim, J. J. R. Soc. Interface 2004, 1, 23-33. (17) Lamblet, M.; Verneuil, E.; Vilmin, T.; Buguin, A.; Silberzan, P.; Leger, L. Langmuir 2007, 23, 6966-6974. (18) Chan, E. P.; Greiner, C.; Arzt, E.; Crosby, A. J. MRS Bull. 2007, 32, 496-503. (19) Aksak, B.; Murphy, M. P.; Sitti, M. Langmuir 2007, 23, 3322-3332. (20) Kim, S.; Sitti, M. Appl. Phys. Lett. 2006, 89, 261911. (21) Hui, C. Y.; Glassmaker, N. J.; Tang, T.; Jagota, A. J. R. Soc. Interface 2004, 1, 35-48. (22) Jagota, A.; Bennison, S. Integr. Comp. Biol. 2002, 42, 1140-1145.
10.1021/la702126k CCC: $40.75 © 2008 American Chemical Society Published on Web 12/08/2007
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Figure 1. Hierarchical structures of a Tokay gecko (adapted from ref 5). (A) Optical image of a Tokay gecko; (B) scanning electron microscope (SEM) image of a setal array (i.e., rows of setae from a toe); (C) SEM image of a single seta; (D) SEM image of the fine terminal branches of a seta, called spatulae or spatular pads.
than bulk smooth materials. Current technological developments demand further understanding of the gecko adhesive system, which is motivated by its potential applications in designing robotic devices that can climb or run on walls and ceilings23,24 where rapid, strong adhesion and friction followed by equally rapid detachment is required. The hierarchical structures of Tokay geckos (Gekko geckos) are shown in Figure 1A-D: a body with 4 feet, each foot with 5 toes, each toe with about 20 rows of sticky setal arrays, with each setal array consisting of thousands of setal stalks amounting to approximately 200 000 setae per toe and each seta terminating with 100-1000 spatulae. Typically, gecko setae are ∼110 µm long and ∼5 µm wide and are inclined to the gecko foot skin at an angle of θ ≈ 45° (Figure 1C).25 The hierarchical structures of gecko foot pads are also highly anisotropic in shape. The setal stalks are almost straight at the base but curved and branched into fine hairy fingerlike spatulae at their ends. The stalks bend and deform during climbing, which is believed to facilitate the switching between attachment and detachment. Our current research efforts on the gecko adhesive system are focused on understanding the switching mechanism between the attachment and detachment forces and mechanisms in gecko climbing and on providing insights for designing responsive adhesive or robotic devices. On the basis of literature reports5,8,26 and our own studies of the gecko adhesive system,25,26 we hypothesize that the highly anisotropic and specialized microstructure of gecko setae gives rise to two types of quite different mechanisms during climbing: (23) Daltorio, K. A.; Gorb, S.; Peressadko, A.; Horchler, A. D.; Wei, T. E.; Ritzmann, R. E.; Quinn, R. D. MRS Bull. 2007, 32, 504-508. (24) Creton, C.; Gorb, S. MRS Bull. 2007, 32, 466-468. (25) Tian, Y.; Pesika, N.; Zeng, H. B.; Rosenberg, K.; Zhao, B. X.; McGuiggan, P.; Autumn, K.; Israelachvili, J. Proc. Natl. Acad. Sci. U.S.A. 2006, 103, 1932019325. (26) Autumn, K.; Dittmore, A.; Santos, D.; Spenko, M.; Cutkosky, M. J. Exp. Biol. 2006, 209, 3569-3579.
one, during contact and attachment, generates strong adhesion and friction forces, and the other, during detachment, reduces both of these forces to almost zero. Each of these steps takes about 10 ms. Autumn and co-workers26 proposed a “frictional adhesion” model in which gecko adhesion depends directly on the shear (friction, lateral) force in the gripping direction. More recently, we analyzed theoretically the gecko friction and adhesion behavior based on a tape model taking into account the nano- and microscopic geometry and the macroscopic action of gecko toes.25,27 It was shown that the lateral friction force and normal adhesion force of a single seta can be changed by more than 3 orders of magnitude during gecko attachment and detachment. In this article, we experimentally investigate the adhesion and friction force coupling of gecko setal arrays subjected to various loading, unloading, and shearing conditions at varied angles from the pure attachment to the pure detachment direction. The main purpose of this article is to measure and analyze the different motions and forces acting on setae, test the hypothesis, and provide criteria and implications for designing patterned adhesive surfaces. This was done by simulating the two motions (i.e., attachment and detachment) in a surface forces apparatus (SFA) in which setal arrays were pressed against and then slid across a substrate surface for various shearing distances. The results are analyzed and discussed in the frameworks of the classic AmontonsCoulomb friction law28 and JKR contact adhesion theory29 and compared with previous measurements on gecko setae and the general adhesion and friction behaviors of structured (rough or patterned) surfaces. (27) Pesika, N.; Tian, Y.; Zhao, B.; Rosenberg, K.; Zeng, H.; McGuiggan, P.; Autumn, K. J. Adhes. 2007, 83, 383-401. (28) Baumberger, T. In Physics of Sliding Friction; Persson, B. N. J., Tosatti, E., Ed.; Kluwer Academic Publishers: Boston, 1996; p 1. (29) Johnson, K. L.; Kendall, K.; Roberts, A. D. Proc. R. Soc. London, Ser. A 1971, 324, 301-320.
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Figure 2. Schematic illustrations and definitions of the displacements and adhesion and friction forces measured in the SFA experiments. (Left panels) Deformations of a setal array (A) or single setal stalk (B) under an applied normal preload L0, with no sliding (V ) 0). (Right panels) Deformations during steady-state sliding along the gripping (+V) and releasing (-V) directions.
Experimental Methods and Materials A surface force apparatus (SFA 2000) was used in this study. The SFA has proved to be a powerful tool for measuring both normal (e.g., adhesion) and lateral (e.g., friction, shear and lubrication) forces between surfaces.30,31 Forces are measured between two curved surfaces, usually two crossed-cylindrical surfaces that, according to the Derjaguin approximation, are locally equivalent to two spheres of radius 2R or a sphere of radius R near a flat surface. Details and operational principles of the SFA have been described.32,33 Briefly, the separation distance d between the two surfaces in an SFA can be controlled to 0.1 nm by use of a three-stage mechanism involving motor-driven micrometers, differential spring levers, and a piezoelectric crystal transducer, and surface separations can also be measured to 0.1 nm by employing an optical multiple beam interference technique.33,34 To measure the normal forces between (30) Claesson, P. M.; Ederth, T.; Bergeron, V.; Rutland, M. W. AdV. Colloid Interface Sci. 1996, 67, 119-183. (31) Israelachvili, J. N. Intermolecular and Surface Forces, 2nd ed.; Elsevier Academic Press: London, 1991. (32) Israelachvili, J. N.; McGuiggan, P. M. J. Mater. Res. 1990, 5, 22232231. (33) Israelachvili, J. N. J. Colloid Interface Sci. 1973, 44, 259-272. (34) Tolansky, S. Multiple-Beam Interferometry of Surfaces and Films; Dover Publications: New York, 1970; pp 8-24.
two surfaces, one of them, usually the lower, is mounted at the end of a double-cantilever normal-force-measuring spring of stiffness kN, as shown in Figure 2A. The deflection of this spring allows for both attractive and repulsive forces to be measured over a range of more than 6 orders of magnitude, all with a distance resolution of 0.1 nm. Equations 1 and 2 relate the normal forces or loads, L0 (the preload force) and L(v (the dynamic load during shearing at velocity V), to the displacements of the normal-force-measuring spring, (∆D - ∆d0) and (∆D - ∆d(v), that is, the piezo- or motor-induced displacements ∆D minus the deflections of the setal array ∆d0 (static) or ∆d(v (during shearing): L0 ) kN(∆D - ∆d0)
(1)
L(v ) kN(∆D - ∆d(v)
(2)
It is important to note that during sliding the preload L0 is kept constant but because the setal stalks deform during sliding the effective normal loads L(v experience changes from this initial preload. Lateral (or shear) movement of one of the surfaces was accomplished with a piezoelectric “bimorph slider” by applying a voltage across the electrode surfaces of the bimorph strips using a coaxial cable connected to a function generator outside the
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apparatus.35 The friction-force-measuring attachment or “friction device” (Figure 2A) is used together with the bimorph slider to measure the shear forces. Four semiconductor strain gauges are attached symmetrically to two oppositely bending surfaces of the vertical double-cantilever friction-force-measuring spring of stiffness kL, as shown in Figure 2A. These make up the four arms of a Wheatstone bridge, which is used to measure the lateral displacement produced on the upper surface by the motion of the lower surface (activated by the bimorph slider). Equations 3 and 4 relate the lateral or friction forces, F0 (static) and F(v (dynamic), to the displacements of the friction-force-measuring spring, (∆X - ∆x0) and (∆X(v ∆x(v); that is, the displacement of the driving stage ∆X minus the displacement of the setal array ∆x0 (static) or ∆x(v (during shearing): F0 ) kL(∆X0 - ∆x0)
(3)
F(v ) kL(∆X(v - ∆x(v)
(4)
The springs and strain gauges were calibrated independently using standard weights. The spring constant (stiffness) of the normal doublecantilever spring was kN ) 500 N/m, and that of the lateral doublecantilever spring (of the friction device) was kL ) 2000 N/m. Equipped with the bimorph slider and friction device, one can measure the static or dynamic friction (or shear) forces F at different normal loads L and sliding velocities V at the same time as the surfaces are moved both normally and (simultaneously, if desired) laterally relative to each other (i.e., with the angle of relative motion of the two surfaces varying from 0° (pure friction) to 90° (pure adhesion) to 180° (pure friction in the opposite direction)). The gecko setal arrays studied were from live nonmolting tokay geckos (G. gecko) using the methods modified from Autumn et al.11 The arrays were kept at ambient conditions (room temperature 24 °C and ∼50% relative humidity) before experiments. Prior to an experiment, a setal array (∼3 mm long and ∼0.5 mm wide) was glued onto the cylindrically curved surface of a silica disc with cyanoacrylate adhesive (Loctite 410; Henkel Loctite Corp., Rocky Hill, CT) and mounted as the bottom surface in the SFA (SFA 2000), as shown in Figure 2A. The “top” substrate surface was a polymeric solid surface (the nonsticky side of commercial singlesided Scotch tape) taped onto a silica glass disc. Loading and shearing measurements were conducted in ambient air at 24 °C.
Results, Analysis, and Discussion Prior studies5,26 have revealed that a preload may be necessary for gecko setae to acquire high adhesion and friction. Thus, our experiments were designed to involve two steps: an initial normal compression to a certain preload L0, followed by cyclical (toand-fro) sliding at sliding velocities of V ) 12.5-25 µm/s. Figure 3A shows reproductions of dynamic force versus time traces during a typical (single) shearing cycle, showing how the normal and lateral forces change during the cycle. It reveals two types of friction and normal force behavior: the adhesive-friction regime when shearing is along the gripping (+V) direction and the loaddependent friction regime when shearing is along the releasing (-V) direction. In the gripping regime, the normal forces are negative (tensile or adhesive), and the friction forces are high. In the releasing regime, the normal forces are positive (compressive or repulsive), and the friction forces are lower than the normal forces, L-V. Note that the classic Amontons’ friction law predicts a proportional dependence of the friction force on the normal load. Many previous studies in our labororatory and in others’ have reported symmetric friction traces,36 which are quite different from the asymmetric ones that we find here for gecko seta. In addition, the latter show no stick-slip friction, but most (35) Luengo, G.; Schmitt, F. J.; Hill, R.; Israelachvili, J. Macromolecules 1997, 30, 2482-2494. (36) Maeda, N.; Chen, N. H.; Tirrell, M.; Israelachvili, J. N. Science 2002, 297, 379-382.
Figure 3. Reproductions of the normal (loading) and lateral (friction) forces vs time traces for a gecko setal array pressed against a solid polymer surface in ambient air at 24 °C and ∼50% relative humidity. (A) Typical cycle, the fourth cycle in panel B, showing details of the main features of the changing normal forces, L, and friction forces, F, in the load-dependent friction regime and the adhesivefriction regime. L-V, L+V, F+V, and F-V are four characteristic forces (i.e., the maximum or steady-state forces during shearing at velocity (V). (B) Force traces of multiple shearing cycles showing an initial normal compression (preloading) and subsequent cyclical shearing commencing in the releasing direction. The shaded area is the cycle shown in panel A.
synthetic polymer surfaces do. Thus, overall, the friction of a gecko setal array is different from that of common materials that follow Amontons’ law except, to some extent, in the releasing regime. Details of the force changes during preloading and subsequent cyclical shearing are shown in Figure 3B, where Figure 3A is a blow up of the fourth cycle. In Figure 3B, preload L0 generates a lateral static friction force F0 (static because the surfaces are not being sheared) due to the compression of the tilted setal stalks (cf. Figure 2B). As the normal load was increased from 0 to L0 ) 2 mN and then kept at L0, the lateral force increased more gradually and then fluctuated with time, indicating possible rearrangements or stick-slip behavior of the setae. Lateral motion was initiated by starting the shearing of the setae along the releasing direction (i.e., the upward direction in Figure 3B). This enhanced both the initial compressive force and lateral (friction) forces. The shearing was continued for about 125 µm (0.125 mm) and then reversed for double this distance (peak-to-peak amplitude of ∼250 µm) before switching again to complete the first cycle and continuing the cycling at the same frequency of 0.1 Hz under the preload of L0 ) 2 mN. Note that the actual normal load L was not constant but varied with sliding. On changing the shearing direction from the releasing to the gripping direction (see also Figure 2B), the normal compressive force rapidly changed to a tensile or adhesive force; the resultant
Force Coupling of Gecko Setal Arrays
adhesion force exhibits a small overshoot but is basically constant throughout the gripping regime. On switching back to the releasing direction, the adhesion changes to a repulsion that exhibits a more pronounced overshoot followed by a gradual decrease. It is possible that this sharp overshoot is the point at which the feet of a running gecko detach. As for the corresponding lateral forces, these exhibited no overshoot but increased monotonically in the gripping direction while remaining roughly constant or decreasing slightly in the releasing direction. The increasing friction force during the gripping part of a cycle is likely due to the increasing adhesive contacts (binding sites) with time, as occurs in many other, especially polymer, systems.37,38 The very different forces and subtle changes occurring in the gripping and releasing regimes during a cycle must be due to the tilted initial (resting) configuration of the setal stalks, to the shapes of the their ends (cf. Figures 1 and 2), and to the rearrangements (changes in orientation, alignment, and bundling) that are quite different when shearing in the gripping and releasing directions. It has long been known that during climbing or running gecko toes grip (i.e., curl in) during attachment but uncurl (i.e., peel the tips of their toes away from surfaces) during detachment. Previous force measurements of pulling a single seta have shown that initial preload and lateral shearing are necessary to generate high adhesion and that the setal shear force also depends on whether the setal pads and spatulae point toward or away from the surface.5,26 Presumably, the setal spatulae project toward the surface in attachment, and the number of spatular contacts increases with the preload and lateral displacement (along the gripping direction). In detachment, the spatulae are pushed or sheared away from the surface, and the number of spatular contacts remains roughly constant with lateral displacement (along the releasing direction). We discuss this more later. In an attempt to fully quantify the trends observed, we define four force parameters that represent the peak lateral and normal forces in the gripping and releasing directions (cf. Figures 2B and 3): F+V and L+V are the maximum lateral and normal forces in the gripping direction (positive V) and F-V and L-V are those in the releasing direction (negative V). We investigated how these four forces were interrelated and how they depended on the preload, the various normal and lateral displacements of the surfaces, and the shearing conditions. Figure 4 shows how the static preload L0 and compression of the setal array ∆d0 vary as the base of the normal-force-measuring spring is displaced a distance ∆D (which was the way that the two surfaces were brought together or separated in the SFA, as illustrated in the two left panels of Figure 2). The preload force is given by eq 1 (L0 ) kN(∆D - ∆d0)) and is found to scale roughly as the square of the displacements or, more specifically, to follow the power law relationships L0 ∝ ∆d02.0 and L0 ∝ ∆D1.7. Because the stiffness kN is constant at low preloads ( ∆d0). The coupling of the friction and real loading forces is shown by plotting F(V versus L(V in Figure 5B. As already noted, the frictional behavior in the releasing direction follows Amontons’ law with a friction coefficient of µreal ) F-V/L-V ) 0.25, which is typical of many dry materials that have friction coefficients in the range of 0.2-0.3. A similar friction coefficient of 0.3 was reported for gecko setal arrays in previous measurements using a different technique.26 In contrast, as shown in the left panels of Figure 5A,B both the normal and lateral forces were highly nonlinear in the gripping direction. We may note in particular the maximum in the negative (adhesion) force L+V with increasing preload (Figure 5A), and the continually increasing friction force F+V as the normal force becomes more negative (adhesive) and then less negative (i.e., progressively more repulsive). This type of behavior is consistent with the previous conclusions regarding the increasing stiffness with increasing loading force L0 and, when considered together with the adhesion of the surfaces (in the gripping regime), suggests an analysis based on the Johnson-Kendall-Roberts (JKR) theory of contact mechanics coupled to current theories of adhesion-controlled friction.45,46 In this continuum mechanics-based approach, the friction force (45) Berman, A.; Drummond, C.; Israelachvili, J. Tribol. Lett. 1998, 4, 95101. (46) Homola, A. M.; Israelachvili, J. N.; McGuiggan, P. M.; Gee, M. L. Wear 1990, 136, 65-83.
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Figure 5. Dynamic forces during the shearing of a setal array along the gripping and releasing directions on a polymer surface in ambient air at various preloads and normal displacements. (A) Friction and real normal forces as functions of the preload L0. Solid points are measurements on loading; open points are measurements on unloading. (B) Friction forces vs the real normal forces; the dashed curve is a fit of the JKR friction model, eqs 6-8, using a contact fraction of φ ) A/A0 ) 0.0035. (C) Friction and normal forces as functions of the normal displacement. Positive normal forces are compressive, but negative normal forces are tensile or adhesive. Solid lines are the best fits to the loading data points. All of the forces shown in A-C, both on loading (solid points) and unloading (blank points), were reversible, with some hysteresis occurring around the zero-preload region.
is proportional to the real contact area A multiplied by the shear stress σ:
F ) Aσ
(5)
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The radius a of the contact junction (A ) πa2) and separation force Ls are given by the JKR theory as
R a3 ) [L + 6πRγ ( x12πRγL + (6πRγ)2] K
(6)
Ls ) -3πRγ
(7)
and
where R is the mean radius of the curvatures of the contacting surfaces (here R ≈ 20 mm), K is the elastic constant of the system, and γ is the surface or interface energy (half the cohesion energy for two similar materials) or half the adhesion energy for dissimilar materials and the minus sign in eq 6 applies only when the normal displacement is controlled instead of the load L+V. In load-controlled measurements, the two contacting surfaces separate spontaneously or jump apart at a finite area a0 when the load L+V ) Ls because of a mechanical instability, so only the positive sign applies in eq 6. In displacement-controlled measurements,47 there is no jumping apart: the surfaces are separated, and the load reaches a maximum negative value of Ls and then decreases to zero as the area A decreases monotonically to zero. In this case, both the + and - signs in eq 6 apply: the positive sign corresponds to a > a0, and the minus sign corresponds to a < a0. We note that in the JKR theory the contact area or its radius a varies with the load L in the way in which the friction force F+V varies with L+V in Figure 5B.47 On the basis of this analysis, the adhesion-controlled friction force, eq 5, can be expressed as a function of the normal load L49
F+V ) πa2σ )
(
)
R[L+V + 6πRγ ( x12πRγL+V + (6πRγ)2] π K
2/3
σ (8)
For a more quantitative analysis, we first note that in the case of the rough or patterned setal surfaces studied here the real contact area A is less than the projected area A0 assumed in the JKR equation, which lowers the effective surface energy γ*, effective elastic constant K*, and effective shear stress σ*. The contact area fraction φ ) A/A0 depends on both the intrinsic number density of the setal array and external loading conditions. We estimate the surface energy γ* by equating the maximum negative load L+V ) -1.8 mN in Figure 5B to the separation force Ls in eq 7; this gives an effective energy of γ* ≈ 10 mJ/m2, which is about 70% lower than the thermodynamic value γ ≈ 30 mJ/m2. The effective elastic modulus K* of the setal array is primarily determined by the number density of setae (∼104/ mm2), which has been theoretically estimated and experimentally measured to be K* ≈ 105 Pa,9,48 4 orders of magnitude lower than the modulus of bulk β-keratin of K ≈ (1-3) × 109 Pa.48 For a hydrocarbon or polymer sliding on a rigid, chemically inert surface where both surfaces are molecularly smooth (i.e., where there is no roughness and no entanglements), we expect σ ≈ 1 × 106 N/m2 if there is complete (100%) contact.46 For a patterned setal surface, to first approximation we assume the effective shear stress to be σ* ≈ σφ. (47) Johnson, K. L. Contact Mechanics; Cambridge University Press: Cambridge, U.K., 1985; p 128. (48) Autumn, K.; Majidi, C.; Groff, R. E.; Dittmore, A.; Fearing, R. J. Exp. Biol. 2006, 209, 3558-3568. (49) There is an additional “load-controlled” friction force term, µL, which for low loads is negligible compared to the “adhesion-controlled” contribution.
Inserting the above estimated γ*, σ*, and K* values and the contact fraction value φ ) 0.0035 into eq 8 yields the theoretical friction force versus load (dashed) curve shown in Figure 5B. The value used for the contact fraction of φ ) 0.0035 giving the best fit to the experimental data is only 0.35% of the projected or apparent area. It is surprising to see how well the JKR friction model fits the experimental data points for φ ) 0.0035. We suspect that shearing the compressed setal arrays enables an equivalent displacement-controlled separation process; otherwise, the surfaces would spontaneously separate at the maximum negative load because of the finite stiffness of the contacting surfaces. At the microscopic level, the likely changes in the configuration of the setal stalks under compression and sliding in the gripping regime (left) and releasing regime (right) are illustrated by the four schematics in Figure 5B: (i) at low L0, only a finite number of setae/spatulae contact the surface, with the number of contacts and/or contact density increasing with L0; (ii) the contact density reaches a maximum as the normal load reaches a minimum value; (iii) the setae are highly compressed as the normal load becomes progressively less negative; and (iv) in the releasing direction, the setae are highly compressed. The contact fractions are low independently of the preloads. Given the complexity of this type of system, a rigorous analysis might require the type of numerical modeling recently applied to rough and patterned Hertzian contacts by Robbins and coworkers44 but extended to adhesive JKR contacts, which is apparently a very difficult and still unsolved problem. However, using the JKR adhesion-controlled friction equation (eq 8) appears to be sufficiently satisfactory for describing the overall trends observed but is given in terms of effective parameters γ*, K*, and σ* that at the moment can only be estimated. Figure 5C plots the friction forces and loads as functions of the normal displacements. As described above, both the normal preload and lateral shearing affect the displacement (whether compressive or tensile) of the setal array, which in turn causes the variations observed in the dynamic forces F(V and real loads L(V. The F(V versus ∆d curves are approximately parabolic with minima at ∆d ) ∆dc and ∆d ≈ 0 and follow a similar trend as in Figure 4. However, in the gripping direction (blue curve in Figure 5C), the normal force L+V initially becomes more negative and after a certain displacement, ∆dc, less negative, eventually becoming positive (i.e., repulsive). This trend is analogous to a Lennard-Jones potential: as the surfaces come closer to each other, there is an increase and then a decrease in the attractive force (negative load), with an adhesion maximum and a potential energy minimum and then a purely repulsive regime (positive load) on further compression. Such a trend or force-distance curve is expected from a JKR analysis of load versus displacement of an elastic, adhesive contact junction: the long-range attraction comes from the surface adhesion forces, determined by γ; the short-range repulsion comes from the elastic compression of the materials, determined by K, which eventually dominates the interaction as the surfaces are compressed. The minimum (maximum adhesion) at L+V ) -2 mN occurred here at ∆d ) 35 µm. The apparent contact area at this point can be roughly estimated to be ∼3 mm2 from the equation A0 ) 2πR∆d, which is slightly larger than the size of the setal patch. Thus, the setal arrays should be fully compressed after this saturation point. We should note that for smooth surfaces the minimum JKR adhesive force or energy will occur at a much smaller distance or displacement (
