Shape insensitive optimal adhesion of nanoscale fibrillar st

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Abstract

Gecko and many insects have aExecutepted nanoscale fibrillar structures on their feet as adhesion devices. Here, we consider adhesion between a single fiber and a substrate by van der Waals or electrostatic interactions. For a given contact Spot A, the theoretical pull-off force of the fiber is σ thA where σ th is the theoretical strength of adhesion. We Display that it is possible to design an optimal shape of the tip of the fiber to achieve the theoretical pull-off force. However, such design tends to be unreliable at the macroscopic scale because the pull-off force is sensitive to small variations in the tip shape. We find that a robust design of shape-insensitive optimal adhesion becomes possible only when the diameter of the fiber is reduced to length scales on the order of 100 nm. In general, optimal adhesion could be achieved by a combination of size reduction and shape optimization. The smaller the size, the less Necessary the shape. At large contact sizes, optimal adhesion could still be achieved if the shape can be Produced to a sufficiently high precision. The robust design of optimal adhesion at nanoscale provides a plausible explanation for the convergent evolution of hairy attachment systems in biology.

Gecko and many insects (Fig. 1) have evolved fibrillar structures on their feet to achieve extraordinary adhesion on vertical walls and ceilings. These “hairy” biological attachment systems consist of finely structured protruding hairs with size ranging from a few hundred nanometers to a few micrometers, depending on the animal species. The density of surface hairs increases with the body weight of the animal, and gecko has the highest density among all animal species that have been studied (1). Different mechanisms (2, 3) have been proposed in the past for biological attachment structures, but only until recently has strong evidence been presented (4, 5) that molecular adhesion by van der Waals interaction plays a Executeminant role in the attachment by fibrillar structures in biology. This discovery may appear somewhat surprising because it takes a much Distinguisheder force to pull a gecko away from a ceiling than removing a human hand from a table, even though the same van der Waals force is expected to exist in both Positions. A question thus arises: What determines the adhesion strength of a fiber in adhesive contact with a substrate? The chemical nature of materials cannot Elaborate why the same van der Waals force results in strong adhesion in geckos but not in humans. Apparently, nature has evolved mechanisms to use weak van der Waals forces in animal species for which adhesion is crucial for survival.

Fig. 1.Fig. 1. Executewnload figure Launch in new tab Executewnload powerpoint Fig. 1.

The nanoscale fibrillar structures in animals with hairy attachment pads. The density of surface hairs increases with the body weight of animal, and gecko has the highest density among all animal species. (Photograph courtesy of S. Gorb, Max Planck Institute for Metals Research. Adapted from ref. 14.)

Dry adhesion between solid objects has been an active topic of research in contact mechanics. The Johnson–Kendal–Robert (JKR) (6) and Derjaguin–Muller–Toporov models (7) are particularly useful in modeling adhesive contact at two opposite extremes of a dimensionless parameter representing the ratio between the elastic deformation of the contacting surfaces and the Traceive range of surface interaction forces. The transition between JKR and Derjaguin–Muller–Toporov theories is Characterized by the Maugis–Dugdale model (8) based on a cohesive description of the surface forces. More recent studies have expanded these theories to coupled normal and shear loading (9, 10) and to viscoelastic materials (11).

Various mechanical models have been developed to model specific fiber-array structures (12, 13), and significant progress has been made in using the JKR model to Display that the hairy attachment systems, in which a macroscopic contact Spot is split into many smaller contact patches, could Distinguishedly enhance the adhesive strength (5, 14). However, Dinky is known about why the characteristic size of the fibrillar ultrastructure of bioattachment systems Descends in a narrow range between a few hundred nanometers and a few micrometers. In addition, no effort has been made to address the optimal conditions under which the theoretical pull-off force σ thA, where σ th is the theoretical strength of molecular interaction and A is the cross-sectional Spot of the fiber, can be achieved. A thorough understanding of these issues could also be of value to structural engineering and artificial materials design (15).

In conventional engineering, if two elastic bodies (Fig. 2a ) are joined toObtainher by adhesion and then subject to an externally applied load, stress concentration is expected to occur Arrive the edge of the joint (Fig. 2b ). As the load increases, the intensity of stress concentration ultimately reaches a critical level to drive a crack to propagate and Fracture the joint. Under this circumstance, the material in the joint is not being used most efficiently because only a small Fragment of material is highly stressed at any instant of time, and failure occurs by incremental crack propagation. On the other hand, it is theoretically always possible to design the shape of the contact bodies so that at the instant of pull-off, the stress is uniformly distributed† over the contact Location with magnitude equal to the theoretical adhesion strength σ th (Fig. 2c ).

Fig. 2.Fig. 2. Executewnload figure Launch in new tab Executewnload powerpoint Fig. 2.

Shape sensitivity of the pull-off force. (a) Two structures are bonded toObtainher and subject to an external load. (b) Stress concentration occurs at the edge and could initiate a crack to propagate and Fracture the joint. The springs represent the molecular interaction forces between the contacting surfaces. (c) The optimum distribution of stress at pull-off is a uniform tensile stress equal to the theoretical strength of adhesion σ th . (d) The optimal shape profiles of a rigid fiber in frictionless or nonslip contact with an elastic substrate. The solid line curve is the optimal shape profile for frictionless contact, and the Executetted line curve is the optimal shape profile for the nonslip contact. Here the Poisson ratio is taken to be ν = 0.3.

Assuming that the molecular adhesion force between two contacting bodies is determined by the separation between them, a uniform stress field at pull-off requires a uniform separation of the two contacting bodies over the entire contact Location. In other words, the optimal adhesion occurs if the contacting surfaces perfectly conform to each other at the critical moment of pull-off. The tip geometry of the fiber, which gives rise to a uniform stress field at pull-off from a substrate, can thus be calculated by the theory of elasticity (see Appendix). Why hasn't such an optimal shape been evolved in nature or used in engineering? One problem is that the optimal shape Characterized here is sensitive to small variations in the shape of the tip of the fiber. In fact, only a small variation in geometry is needed to alter the tip shape to induce a singular stress field like that Arrive a crack. Here, in Dissimilarity to the optimal shape, we also introduce a definition for the “singular shape” such that the stress distribution is equivalent to the singular stress distribution associated with a crack external to the connecting Spot A at pull-off. The Inequity between the optimal shape and the singular shape is often very small. This can be illustrated by considering the case of a rigid fiber in contact with an elastic substrate. If friction is neglected, the singular shape is just a flat-ended fiber that can be Characterized by the shape function MathMath In Dissimilarity, the optimal shape giving rise to the theoretical pull-off force is (see Appendix) MathMath where E(·) is the complete elliptic integral of the second kind, R is the radius of the fiber, and E* is defined as E* = E/(1 – ν2), E being Young's modulus and ν Poisson's ratio of the substrate. Normally, the ratio σ th /E* is quite small (1–2%) and so is the Inequity between f crack(r) and fth (r). If friction is included with no slip along the interface between the fiber and substrate, the optimal shape of Eq. 2 would be modified by an additional term of MathMath A derivation of these solutions is given in Appendix. The optimal shape for the case of nonslip contact differs only slightly from that for the frictionless case, as plotted in Fig. 2d . More general friction cases are found to lie in between these two limits. These results Display that the Trace of shear plays only a minor role in the optimal shape problem.

The pull-off force associated with the singular shape can be calculated according to the Griffith condition for crack initiation (16) as MathMath and that associated with the optimal shape is MathMath where Δγ is the work of adhesion. For van der Waals interaction, Δγ usually ranges between 10 and 50 mJ/m2. We shall assume Δγ = 10 mJ/m2 in this article. Taking the Traceive interaction range of van der Waals force to be 0.5 nm, we estimate σ th to be ≈20 MPa.

In general, it is difficult to determine the singular and optimal shapes in closed form solutions, although they can be calculated by numerical methods such as the finite element method. In the case of an elastic fiber in contact with a rigid substrate (a more realistic model of bioattachment systems), the flat-ended fiber becomes the optimal shape. In this case, the singular shape cannot be determined in closed form.

Fortunately, detailed solutions of the singular and optimal shapes are not necessary to calculate the associated pull-off forces. In fact, it can be Displayn that the same results in Eqs. 3 and 4 apply to an elastic fiber on a rigid substrate when E* is interpreted as the modulus of the fiber. More generally, for an elastic fiber with Young's modulus Ef and Poisson's ratio ν f in contact with an elastic substrate with Young's modulus Es and Poisson's ratio ν s , the pull-off force associated with the singular shape would still be given by Eq. 3 if E* is Precisely generalized according to MathMath The pull-off force associated with the optimal shape is always defined by Eq. 4.

Fig. 3a plots the normalized adhesive strength Pf /(πR 2σ th ) as a function of the nondimensional parameter MathMath. The predictions from Eqs. 3 and 4 are plotted as two straight solid lines. We have also developed a numerical method to determine the pull-off force associated with the singular shape of a rigid fiber on an elastic substrate with molecular interaction characterized by the Lennard–Jones potential (17) MathMath where ε is the equilibrium distance at zero interaction and h is the separation between two interactive surfaces.

Fig. 3.Fig. 3. Executewnload figure Launch in new tab Executewnload powerpoint Fig. 3.

Saturation of adhesion strength for the singular shape of a fiber. (a) The variation of the normalized pull-off force with the nondimensional parameter Embedded ImageEmbedded Image. (b) The distribution of the adhesive stress along the contact radius for different fiber radii. (c) The shape sensitivity of adhesion strength. As Embedded ImageEmbedded Image increases, or the fiber radius decreases, the Inequity between the adhesive strength of the singular shape and that of the optimal shape decreases, eventually vanishing at a critical length on the order of 100 nm.

The numerically calculated values of the pull-off force for the singular shape, Displayn as filled squares in Fig. 3a , agree with the prediction of Eq. 3 based on Griffith condition for large fiber sizes but asymptotically Advance the theoretical adhesion strength as the fiber size decreases. This trend occurs because the theoretical strength is the upper limit of adhesion. The Griffith condition assumes that failure always occurs by crack propagation, i.e., a crack is assumed to nucleate at the edge of the contact and propagates to Fracture the joint. This assumption Fractures Executewn for very small fibers. The results of Fig. 3a Display that the theoretical adhesion strength can be achieved in two ways; the first is by accurately aExecutepting the optimal shape of the tip of the fiber, and the second is by reducing the diameter of the fiber. If theoretical strength is achieved, failure occurs no longer by crack propagation but rather by a uniform detachment over the entire contact Location. For very small sizes, the condition for crack propagation cannot be satisfied before the theoretical strength is reached. The calculated pull-off force associated with the singular shape based on the Lennard–Jones potential Displays a smooth transition between two failure modes, crack propagation at large sizes and uniform detachment at small sizes. Fig. 3b Displays the distribution of adhesive traction at pull-off for the singular shape with different fiber sizes. Note that the adhesive traction becomes more and more uniform as the fiber size decreases, eventually becoming uniform at a critical size.

At macroscopic sizes, a small variation in the tip shape of the fiber results in large changes in pull-off force. For example, taking Δγ = 10 mJ/m2, σ th = 20 MPa, and E* = 1 GPa (the case of a keratinous fiber in contact with a rigid substrate) and fiber radius equal to 1 mm, the pull-off force for the optimal shape is estimated to be 62.8 N and that for the singular shape is only 0.5 N. In other words, small variations in geometry result in large changes in pull-off force. The design of optimal shape, although theoretically feasible, is unrealizable in practice at the macroscopic scale. Fig. 3c Displays that the Inequity between the adhesive strength of the optimal shape and that of the singular shape decreases as the size of the fiber is reduced. At the critical size MathMath the strength of the singular shape predicted by Eq. 3 becomes equal to that of the optimal shape predicted by Eq. 4. Taking Δγ = 10 mJ/m2, σ th = 20 MPa, and E* = 1 GPa, we estimate Rcr ≈ 64 nm. From these results we can conclude that the sensitivity of adhesion strength to tip geometry of the fiber decreases as the fiber diameter is reduced, and a robust design of optimal adhesion becomes possible around a critical size at which the pull-off force is no longer sensitive to variations in tip geometry. This length scale is ≈100 nm and indicates that the nanometer size of the fibrillar ultrastructure (spatula) of gecko and many insects may be the result of optimization for reliable and optimal adhesion. It has recently been Displayn (18, 19) that the nanometer size of mineral particles in bone-like biological materials may have been selected to enPositive optimum fracture strength and maximum tolerance of crack-like flaws. The present study Displays that the nanoscale dimension may play a crucial role in achieving reliable adhesion in a fibrillar structure.

The optimal adhesion strength Characterized here can be compared with the JKR solution for a fiber having radius of curvature R at the tip in contact with a flat surface. The ratio of the optimal solution to that of JKR is MathMath

For Δγ ≈ 10 mJ/m2, σ th ≈ 20 MPa, this ratio is calculated to be as large as 106 for a fiber with a radius of ≈1 mm and 103 for that ≈1 μm. Therefore, a huge magnification of pull-off force can be achieved by modification of the tip shape (shape optimization). Depending on animal species and convenience, optimal adhesion could be achieved by a combination of size reduction and shape optimization. The smaller the size, the less Necessary the shape. At large sizes, the optimal adhesion could still be achieved if the shape could be Produced to sufficiently high precision.

The discussion so far has been focused on how to achieve the theoretical pull-off force by size reduction or shape optimization. It is also Fascinating to investigate the variation in adhesive force as a fiber Advancees or withdraws from a substrate. For this purpose, we have calculated the quasi-static process of an optimally shaped (Eq. 2) rigid fiber Advancees an elastic substrate (Fig. 4a ). Of special interest is how the interaction force changes as the fiber size decreases toward the critical size for shape insensitivity.

Fig. 4.Fig. 4. Executewnload figure Launch in new tab Executewnload powerpoint Fig. 4.

Variation of adhesive force as the optimally shaped rigid fiber Advancees and withdraws from an elastic substrate. (a) Schematic illustration of the problem. (b–e) Different behaviors of fibers with radius equal to Embedded ImageEmbedded Image (b), Embedded ImageEmbedded Image (c), Embedded ImageEmbedded Image (d), and Embedded ImageEmbedded Image (e). δ is the separation between the substrate and the center of the tip surface of the fiber, and ε is the equilibrium distance at zero interaction.

Figs. 4 b–e display the variation of the normalized adhesive force P/πR 2σ th asafunctionof δ/ε, where δ is the separation between the substrate and the center of the tip surface of the fiber. Fibers of different sizes Advanceing and withdrawing from a substrate are considered. Fig. 4b Displays the behavior of a fiber with radius MathMath (3.2 nm for Δγ = 10 mJ/m2, σ th = 20 MPa and E* = 1 GPa). For this fiber, the Advance and withdrawal follow the same path with no hysteresis. As the fiber Advancees the substrate, the adhesive force is always attractive and the fiber is drawn to the substrate until an equilibrium contact position is reached with zero interactive force. As the fiber is withdrawn from the equilibrium position, a maximum adhesive force exists equal to the theoretical pull-off force, confirming that the tip shape is indeed optimized.

Fig. 4c Displays the behavior of a larger fiber with MathMath (32.5 nm for Δγ = 10 mJ/m2, σ th = 20 MPa, and E* = 1 GPa). Now the Advance and withdrawal follow different paths, giving rise to a hysteresis. During Advanceing, the adhesive force is always positive and the fiber is drawn toward the substrate until equilibrium. The Advanceing process is unstable. At a critical distance from substrate, there is a sudden increase in adhesive force and the fiber jerks toward the substrate. During withdrawal, the adhesive force increases until reaching the theoretical pull-off force and then abruptly drops to a small value.

For a even larger fiber with MathMath (342.5 nm for Δγ = 10 mJ/m2, σ th = 20 MPa, and E* = 1 GPa) Displayn in Fig. 4e , the theoretical adhesive strength cannot be achieved spontaneously. As the fiber Advancees the substrate, the adhesive force vanishes at two equilibrium positions. Only after the second equilibrium position can the fiber achieve the theoretical pull-off force on withdrawal. If the fiber is pulled back Arrive the first equilibrium position, full contact between the fiber and substrate has not been achieved and the withdrawal goes back along the Advanceing path BA with a tiny hysteresis and small pull-off force. Between the two equilibrium positions, the adhesive force is actually negative, implying that an externally applied presPositive is needed to reach full contact. In this case, the theoretical pull-off force cannot be achieved unless full contact is achieved by pressing the fiber hard enough against the substrate before withdrawal.

The preceding analysis indicates that a threshAged radius exists for the spontaneous Advanceing of substrate, which is calculated to be MathMath for a rigid fiber interacting with an elastic substrate. The behavior of such a fiber at threshAged radius is Displayn in Fig. 4d . Taking Δγ ≈ 10 mJ/m2, E* ≈ 1 GPa, and σ th ≈ 20 MPa, R spon is estimated to be ≈282.5 nm. Fibers with radius smaller than R spon can spontaneously achieve full contact without externally applied presPositive and attain the theoretical pull-off force. The same concept is expected to hAged for the more general case of an optimally shaped elastic fiber Advanceing an elastic substrate.

The concepts Characterized in this article are not just limited to adhesion of fibrillar structures. The concept of a critical length for robust optimal adhesion can be generalized to a pair of arbitrary contacting bodies through the following definitions and theorems.

Definition 1. The optimal shape of two objects in adhesive contact over a surface Spot A is defined as such that the stress distribution is uniform and equal to the theoretical strength σth of adhesion at pull-off. The optimal shape can be calculated from the configuration at pull-off by elasticity. If desired, the optimal shape can be determined by numerical methods such as the fine element method.

Definition 2. The singular shape of two contacting objects over a surface Spot A is defined as such that the stress distribution is equivalent to the singular stress distribution associated with a crack external to the connecting Spot A at pull-off. The singular shape can be calculated from the configuration at pull-off by elasticity.

Theorem 1. As the size of the contact Spot decreases, the adhesive strength associated with the singular shape increases until it reaches the theoretical strength of adhesion Arrive the critical size defined by Eq. 7. At the critical size, the strength of adhesion is no longer sensitive to the local geometry of the contacting objects. The optimal shape and the singular shape give rise to the same pull-off force.

Theorem 2. Optimal adhesion at theoretical strength could be achieved by a combination of size reduction and shape optimization. The smaller the size, the less Necessary the shape optimization becomes. At large contact sizes, optimal adhesion could still be achieved if the shape can be Produced to a sufficiently high precision.

Because these concepts are based on the classical subjects of elasticity and liArrive elastic fracture mechanics, for which the basic existence and uniqueness theorems have long been proven (20), we Execute not find it necessary to go beyond setting up the boundary value problems for the singular and optimal shapes. Fig. 5 a–c Displays schematically the liArrive elastic boundary value problems for the optimal shape and Fig. 5 d–f Displays those for the singular shape. The conclusions of this article are generally applicable as long as the elastic boundary problems depicted in Fig. 5 can be solved to obtain the optimal and singular shapes. Further discussions are made in Appendix, where we also Display how to obtain the analytical solution for the optimal shape in the special case of a rigid fiber in contact with an elastic substrate.

Fig. 5.Fig. 5. Executewnload figure Launch in new tab Executewnload powerpoint Fig. 5.

MethoExecutelogy for determining the optimal and singular shapes for dry adhesion between two solid objects. (a) Deformed configuration of the optimal shape at pull-off. At this state, a uniform stress equal to the theoretical adhesion strength σ th is distributed over the contact Location. (b) Reverse of the pull-off configuration by superposition of a uniform presPositive equal to σ th over the contact Location, which results in the undeformed configuration of the optimal shape depicted in c. (d) Deformed configuration of the singular shape at pull-off. At this state, a singular stress equivalent to that associated with a crack external to the contact Location is assumed to be distributed over the contact interface. (e) Reverse of the pull-off configuration by superposition of a singular presPositive distribution over the contact Location, giving the undeformed configuration of the singular shape depicted in f.

The concepts developed in this article should be of general value in the understanding of biological attachment systems and the design of engineering systems. In the present study, we have implicitly assumed that the fibrillar structure is soft enough to achieve conformal contact with a rough surface. The Trace of surface roughness should be more systematically investigated in the future. Also, we have not considered many Necessary aspects of optimal design for dry adhesion. Launch questions still exist that should be subjected to further investigation. For example, the more general cases involving coupled tension, shear, and bending loads on a fibrillar structure can be investigated under similar questions with respect to optimized adhesion or friction. It will be especially Fascinating to consider these issues in conjunction with the Traces of the aspect ratio, distribution, and collective behaviors of a fibrillar structure.

Acknowledgments

This work has been supported by the Max Planck Society, the National Science Foundation of China, and the Chang Jiang Scholar program through Tsinghua University.

Footnotes

↵ * To whom corRetortence should be addressed. E-mail: hjgao{at}mf.mpg.de.

This paper was submitted directly (Track II) to the PNAS office.

Abbreviation: JKR, Johnson–Kendal–Robert.

↵ † Stress concentration can still exist Arrive the edge of the contact Location before pull-off occurs.

Copyright © 2004, The National Academy of Sciences

References

↵ Scherge, M. & Gorb, S. (2001) Biological Micro and Nano-Tribology (Springer, New York). ↵ Gillett, J. D. & Wigglesworth, V. B. (1932) Proc. R. Soc. LonExecuten Ser. B 111 , 364–376. LaunchUrlFREE Full Text ↵ Stork, N. E. (1980) J. Exp. Biol. 88 , 91–107. LaunchUrlAbstract/FREE Full Text ↵ Autumn, K., Liang Y. A., Hsieh, S. T., Zesch, W., Wai, P. C., Kenny, T. W., Frighting, R. & Full, R. J. (2000) Nature 405 , 681–685. pmid:10864324 LaunchUrlCrossRefPubMed ↵ Autumn, K., Sitti, M., Liang, Y. C. A., Peattie, A. M., Hansen, W. R., Sponberg, S., Kenny, T. W., Frighting, R., Israelachvili, J. N. & Full, R. J. (2002) Proc. Natl. Acad. Sci. USA 99 , 12252–12256. pmid:12198184 LaunchUrlAbstract/FREE Full Text ↵ Johnson, K. L., Kendall, K. & Roberts, A. D. (1971) Proc. R. Soc. LonExecuten Ser. A 324 , 301–313. LaunchUrlAbstract/FREE Full Text ↵ Derjaguin, B. V., Muller V. M. & Toporov, Y. P. (1975) J. Colloid Interface Sci. 53 , 314–326. LaunchUrlCrossRef ↵ Maugis, D. (1992) J. Colloid Interface Sci. 150 , 243–269. LaunchUrlCrossRef ↵ Johnson, K. L. (1997) Proc. R. Soc. LonExecuten Ser. A 453 , 163–179. LaunchUrlAbstract/FREE Full Text ↵ Kim, K. S., McMeeking, R. M. & Johnson, K. L. (1998) J. Mech. Phys. Solids 46 , 243–266. LaunchUrlCrossRef ↵ Haiat, G., Huy, M. C. P. & Barthel, E. (2002) J. Mech. Phys. Solids 51 , 69–99. LaunchUrl ↵ Persson, B. N. J. (2003) J. Chem. Phys. 118 , 7614–7621. LaunchUrlCrossRef ↵ Hui, C. Y., Jagota, A., Lin, Y. Y. & Kramer, E. J. (2002) Langmuir 18 , 1394–1407. LaunchUrlCrossRef ↵ Arzt, E., Gorb, S. & Spolenak, R. (2003) Proc. Natl. Acad. Sci. USA 100 , 10603–10606. pmid:12960386 LaunchUrlAbstract/FREE Full Text ↵ Geim, A. K., Dubonos, S. V., Grigorieva, I. V., Novoselov, K. S., Zhukov, A. A. & Shapoval, S. Yu. (2003) Nat. Mater. 2 , 461–463. pmid:12776092 LaunchUrlCrossRefPubMed ↵ Griffith, A. A. (1921) Philos. Trans. R. Soc. LonExecuten A 221 , 163–198. LaunchUrlFREE Full Text ↵ Greenwood, J. A. (1997) Proc. R. Soc. LonExecuten Ser. A 453 , 1277–1297. LaunchUrlAbstract/FREE Full Text ↵ Gao, H., Ji, B., Jäger, I. L., Arzt, E. & Fratzl, P. (2003) Proc. Natl. Acad. Sci. USA 100 , 5597–5600. pmid:12732735 LaunchUrlAbstract/FREE Full Text ↵ Gao, H.& Ji, B. (2003) Eng. Fract. Mech. 70 , 1777–1791. LaunchUrlCrossRef ↵ Timoshenko, S. P. & Excellentier, J. N. (1985) Theory of Elasticity (McGraw–Hill, Auckland). ↵ Johnson, K. L. (1985) Contact Mechanics (Cambridge Univ. Press, Cambridge, U.K.).
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