# Gecko feet

This picture shows a crested gecko, Rhacodactylus ciliatus, climbing up the vertical side of a terrarium

The feet of geckos have a number of specializations. Their surfaces can adhere to any type of material with the exception of Teflon (PTFE). This phenomenon can be explained with three elements:

• Foot structure
• Structure of the material to which the foot adheres
• The ability to adhere to a surface and become a part of it

## Background

Geckos are members of the family Gekkonidae. They are reptiles that inhabit temperate and tropical regions. There are over 1,000 different species of geckos.[1] They can be a variety of colors. Geckos are carnivorous, feeding on a variety of foods, including insects and worms.[2] Most gecko species, including the crested gecko (Rhacodactylus ciliatus),[3] can climb walls and other surfaces.

## Structure

Close view of a gecko's foot
Micrometer- and nanometer-scale view of a gecko's toe[4]

### Chemical structure

The interactions between the gecko's feet and the climbing surface are stronger than simple surface area effects. On its feet, the gecko has many microscopic hairs, or setae (singular seta), that increase the Van der Waals forces between its feet and the surface. These setae are fibrous structural proteins that protrude from the epidermis, which is made of β-keratin,[5] the basic building block of human skin.

### Physical structure

The β-keratin bristles are approximately 5 μm in diameter. The end of each seta consists of approximately 1,000 spatulae that are shaped like an isosceles triangle. The spatulae are approximately 200 nm on one side and 10–30 nm on the other two sides.[6] The setae are aligned parallel to each other but not oriented normal to the toes. When the setae contact another surface, their load is supported by both lateral and vertical components. The lateral load component is limited by the peeling of the spatulae, and the vertical load component is limited by shear force.

## Van der Waals forces

### Hamaker surface interaction

The following equation can be used to quantitatively characterize the Van der Waals forces, by approximating the interaction as being between two flat surfaces:

${\displaystyle F=-{\frac {A_{\text{H}}}{12\pi D^{3}}}}$

where F is the force of interaction, AH is the Hamaker constant, and D is the distance between the two surfaces. Gecko setae are much more complicated than a flat surface, for each foot has roughly 14,000 setae that each have about 1,000 spatulae. These surface interactions help to smooth out the surface roughness of the wall, which helps improve the gecko to wall surface interaction.

• Surface roughness
• Adsorbed material, such as particles or moisture
• How the gecko's foot contacts the surface
• The material gradient properties (dependence of elastic modulus on the depth).[7]

## Interaction potential derivation

### Van der Waals interaction

Schematic diagram representing the Van der Waals interaction between a sphere and an infinite plane.

Using the combined dipole–dipole interaction potential between molecules A and B:

${\displaystyle W_{\mathrm {AB} }=-{\frac {C_{\mathrm {AB} }}{D^{6}}}}$

where WAB is the potential energy between the molecules (in joules), CAB is the combined interaction parameter between the molecules (in J m6), and D is the distance between the molecules [in meters]. The potential energy of one molecule at a perpendicular distance D from the planar surface of an infinitely extending material can then be approximated as:

${\displaystyle W_{\mathrm {A,Plane} }=-\iiint \limits _{\text{all space}}\,{\frac {C_{\mathrm {AB} }\ \rho _{\mathrm {B} }}{(D')^{6}}}dV}$

where D′ is the distance between molecule A and an infinitesimal volume of material B, and ρB is the molecular density of material B (in molecules/m3). This integral can then be written in cylindrical coordinates with x being the perpendicular distance measured from the surface of B to the infinitesimal volume, and r being the parallel distance:

{\displaystyle {\begin{aligned}W_{\mathrm {A,Plane} }&=-C_{\mathrm {AB} }\rho _{\mathrm {B} }\int _{0}^{\infty }\int _{0}^{\infty }{\frac {2\pi r}{\left((D+x)^{2}+r^{2}\right)^{3}}}\,dr\,dx\\&=-{\frac {\pi C_{\mathrm {AB} }\rho _{\mathrm {B} }}{2}}\int _{0}^{\infty }{\frac {1}{(D+x)^{4}}}\,dx\\&=-{\frac {\pi C_{\mathrm {AB} }\rho _{\mathrm {B} }}{6D^{3}}}\end{aligned}}}

### Modeling spatulae potential

Schematic diagram representing Van der Waals interaction between a cylinder and an infinite plane.

The gecko–wall interaction can be analyzed by approximating the gecko spatula as a long cylinder with radius rs. Then the interaction between a single spatula and a surface is:

${\displaystyle W_{\mathrm {seta,plane} }=-\iiint \limits _{\text{all space}}\,{\frac {\pi C_{\mathrm {AB} }\rho _{\mathrm {B} }\rho _{\mathrm {A} }}{6(D')^{6}}}\,dV}$

where D′ is the distance between the surface of B and an infinitesimal volume of material A and ρA is the molecular density of material A (in molecules/m3). Using cylindrical coordinates once again, we can find the potential between the gecko spatula and the material B then to be:

{\displaystyle {\begin{aligned}W_{\mathrm {s,p} }&=-{\frac {2\pi ^{2}C_{\mathrm {AB} }\rho _{\mathrm {A} }\rho _{\mathrm {B} }}{6}}\int _{0}^{\infty }\int _{0}^{r_{\mathrm {s} }}\,{\frac {r}{(D+x)^{3}}}\,dr\,dx\\&=-{\frac {\pi ^{2}C_{\mathrm {AB} }\rho _{\mathrm {A} }\rho _{\mathrm {B} }r_{\mathrm {s} }^{2}}{6}}\int _{0}^{\infty }\,{\frac {1}{(D+x)^{3}}}\,dx\\&=-{\frac {\pi ^{2}C_{\mathrm {AB} }\rho _{\mathrm {A} }\rho _{\mathrm {B} }r_{\mathrm {s} }^{2}}{12D^{2}}}\\&=-{\frac {A_{\mathrm {H} }r_{\mathrm {s} }^{2}}{12D^{2}}}\end{aligned}}}

where AH is the Hamaker constant for the materials A and B.

The Van der Waals force per spatula, Fs can then be calculated by differentiating with respect to D and we obtain:

${\displaystyle F_{\mathrm {s} }=-\left[{\frac {d}{dD}}(W_{\mathrm {s,p} })\right]=-{\frac {A_{\mathrm {H} }r_{\mathrm {s} }^{2}}{6D^{3}}}}$

We can then rearrange this equation to obtain rs as a function of AH:

{\displaystyle {\begin{aligned}r_{\mathrm {s} }&={\sqrt {\frac {6D^{3}F_{\mathrm {s} }}{A_{\mathrm {H} }}}}\approx {\sqrt {\frac {6(1.7\times 10^{-10}\ \mathrm {m} )^{3}(40\times 10^{-6}\ \mathrm {N} )}{A_{\mathrm {H} }}}}\\&=3.43\times 10^{-17}{\sqrt {\mathrm {N\ m^{3}} }}\times {\frac {1}{\sqrt {A_{\mathrm {H} }}}}\end{aligned}}}

where a typical interatomic distance of 1.7 Å was used for solids in contact and a Fs of 40 µN was used as per a study by Autumn et al.[5]

## Experimental verification

The equation for rs can then be used with calculated Hamaker constants[8] to determine an approximate seta radius. Hamaker constants through both a vacuum and a monolayer of water were used. For those with a monolayer of water, the distance was doubled to account for the water molecules.

Materials A/B AH (10−20 J) Calculated rs (µm)
Hydrocarbon/Hydrocarbon (vacuum) 2.6–6.0 0.21–0.14
Hydrocarbon/Hydrocarbon (water) 0.36–0.44 1.6–1.5
Hydrocarbon/Silica (vacuum) 4.1–4.4 0.17–0.16
Hydrocarbon/Silica (water) 0.25–0.82 1.9–1.1
Albumin/Silica (water) 0.7 1.2

These values are similar to the actual radius of the setae on a gecko's foot (approx. 2.5 μm).[5][9]

Stickybot, a climbing robot using synthetic setae[10]

Research attempts to simulate the gecko's adhesive attribute. Projects that have explored the subject include:

• Replicating the adhesive rigid polymers manufactured in microfibers that are approximately the same size as gecko setae.[11]
• Replicating the self-cleaning attribute that naturally occurs when gecko feet accumulate particles from an exterior surface between setae.[12]
• Carbon nanotube arrays transferred onto a polymer tape.[13] In 2015 commercial products inspired by this work were released.[14]

## References

1. ^ Skibinski, Brian. "All Species". Geckolist.com. Retrieved June 3, 2011.
2. ^
3. ^ "Crested Geckos". LLLReptile and Supply, Inc. 2006. Retrieved June 3, 2011.
4. ^ Autumn, K. (2006). "How gecko toes stick". American Scientist. 94 (2): 124–132. doi:10.1511/2006.58.124.
5. ^ a b c Autumn, K.; Setti, M.; Liang, Y. A.; Peattie, A. M.; Hansen, W. R.; Sponberg, S.; Kenny, T. W.; Fearing, R.; Israelachvili, J. N.; Full, R. J. (2002). "Evidence for Van Der Waals adhesion in gecko setae". PNAS. 99 (19): 12252–12256. doi:10.1073/pnas.192252799. PMC 129431. PMID 12198184.
6. ^ Prevenslik, T. (2009). "Electrostatic Gecko Mechanism". Tribology in Industry. 31 (1&2).
7. ^ Popov, Valentin L.; Pohrt, Roman; Li, Qiang (2017-09-01). "Strength of adhesive contacts: Influence of contact geometry and material gradients". Friction. 5 (3): 308–325. doi:10.1007/s40544-017-0177-3. ISSN 2223-7690.
8. ^ Butt, Hans-Jürgen; Graf, Karlheinz; Kappl, Michael (6 March 2006). Physics and Chemistry of Interfaces. John Wiley & Sons. ISBN 978-3-527-60640-5.
9. ^ Arzt, E.; Gorb, S.; Spolenak, R. (2003). "From micro to nano contacts in biological attachment devices". PNAS. 100 (19): 10603–10606. doi:10.1073/pnas.1534701100. PMC 196850. PMID 12960386.
10. ^ "Stickybot". Biomimetics and Dexterous Manipulation Laboratory, Stanford University.
11. ^ Majidi, C.; Groff, R. E.; Maeno, Y.; Schubert, B.; Baek, S.; Bush, B.; Maboudian, R.; Gravish, N.; Wilkinson, M.; Autumn, K.; Fearing, R. S. (18 August 2006). "High Friction from a Stiff Polymer using Micro-Fiber Arrays". Physical Review Letters. 97 (7): 076103. Bibcode:2006PhRvL..97g6103M. doi:10.1103/physrevlett.97.076103. PMID 17026251.
12. ^ Fearing, Ronald. "Self-Cleaning Synthetic Gecko Tape". University of California, Berkeley. Missing or empty |url= (help)
13. ^ Ge, Liehuie; Sethi, Sunny; Ci, Lijie; Ajayan, Pulickel M.; Dhinojwala, Ali (June 19, 2007). "Carbon nanotube-based synthetic gecko tapes". Proceedings of the National Academy of Sciences of the United States of America. 104 (26): 10792–10795. doi:10.1073/pnas.0703505104. PMC 1904109. PMID 17578915.
14. ^ Lavars, Nick (2015-12-22). "Gecko-inspired adhesive tape finally scales to market". www.gizmag.com. Retrieved 2015-12-23.