Gecko feet

Gecko feet are the four feet of the Gecko. Their surfaces can adhere to any type of material with the exception of Teflon (which is the trade name for PTFE). This phenomenon can be explained three elements:

This picture shows a Crested Gecko, Rhacodactylus Ciliatus, climbing up the vertical side of a terrarium

• Foot structure
• Structure of the material to which the foot adheres
• The type of force interaction between the two

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 insects and worms. Most gecko species, including the crested gecko (Rhacodactylus Ciliatus),^[2] can climb walls and other surfaces.

Structure of the Gecko’s Feet

Close view of a gecko's foot

Micro and nano view of gecko's toe^[3]

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 seta (pl. setae), 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,^[4] the basic building block of human skin.

Physical Structure

The β-keratin bristles are approximately 5 microns in diameter. The end of each seta consists of approximately 1,000 spatulae that are shaped like an isoceles triangle. The spatulae are approximately 200 nanometers on one side and 10–30 nanometers on the other two sides.^[5] 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.

Type of Interaction - 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 between two flat surfaces:

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

where F is the force of interaction, A_H 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 seta that each have about 1000 spatulae. These surface interactions help to smooth out the surface roughness of the wall, which helps improve the gecko to wall surface interaction.

Factors affecting adhesion

Many factors affect adhesion, including:

• Surface roughness
• Adsorbed material, such as particles or moisture
• How the gecko's foot contacts the surface

Interaction potential derivation

Van der Waals Interaction

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

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

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

Where $W_{AB}$ is the potential energy between the molecules [J], $C_{AB}$ is the combined interaction parameter between the molecules [J m$^6$], and D is the distance between the molecules [m]. 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:

$W_{A,Plane}=-\iiint\limits_{3D} \, \frac{C_{AB}\ \rho_B}{(D')^6}dV$

where D' is the distance between molecule A and an infinitesimal volume of material B, and $\rho_B$ is the molecular density of material B [molecules/m$^3$]. 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:

$W_{A,Plane} = -C_{AB}\ \rho_B\int_{0}^{\infty}\int_{0}^{\infty}\frac{2\pi r}{[(D+x)^2+r^2]^3}\, dr\,dx$

$= -\frac{\pi C_{AB}\ \rho_B}{2} \int_{0}^{\infty} \frac{1}{(D+x)^4}\, dx$

$W_{A,Plane} = -\frac{\pi C_{AB}\ \rho_B}{6D^3}$

Modeling Seta 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 setae as a long cylinder with radius $r_s$. Then the interaction between a single seta and a surface is:

$W_{seta,plane} = -\iiint\limits_{3D}\, \frac{\pi C_{AB}\ \rho_B\ \rho_A}{6(D')^6}\, dV$

where D' is the distance between the surface of B and an infinitesimal volume of material A and $\rho_A$ is the molecular density of material A [molecules/m$^3$]. Using cylindrical coordinates once again, we can find the potential between the gecko seta and the material B then to be:

$W_{s,p} = -\frac{2\pi^2C_{AB}\ \rho_A\ \rho_B}{6}\int_{0}^{\infty}\int_{0}^{r_s}\,\frac{r}{(D+x)^3}\, dr\,dx$

$= -\frac{\pi^2 C_{AB}\ \rho_A\ \rho_B\ r_s^2}{6}\int_{0}^{\infty}\,\frac{1}{(D+x)^3}\, dx$

$= -\frac{\pi^2 C_{AB}\ \rho_A\ \rho_B\ r_s^2}{12D^2}$

$W_{s,p} = -\frac{A_H\ r_s^2}{12D^2}$

where $A_H$ is the Hamaker constant for the materials A and B.

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

$F_s = -\left [\frac{\operatorname{d}}{\operatorname{d}D} (W_{s,p})\right ] = -\frac{A_H\ r_s^2}{6D^3}$

We can then rearrange this equation to obtain $r_s$ as a function of $A_H$:

$r_s = \sqrt{\frac{6 D^3 F_s}{A_H}} \approx \sqrt{\frac{6(1.7 \times 10^{-10}\ \mathrm{m})^3(40 \times 10^{-6}\ \mathrm{N})}{A_H}}$

$= 3.43 \times 10^{-17} \sqrt{\mathrm{Nm^3}} \times \frac{1}{\sqrt{A_H}}$

where a typical interatomic distance of 1.7 Å was used for solids in contact and a $F_s$ of 40 µN was used as per a study by Autumn et. al.^[4]

Experimental verification

The equation for $r_s$ can then be used with calculated Hamaker constants^[6] to determine an approximate seta radius. Hamaker constants through both a vacuum and a monolayer of water was used. For those with a monolayer of water, the distance was doubled to account for the water molecules.

Materials A/B	$A_H$ ($10^{-20}$J)	Calculated $r_s$ (µm)
Hydrocarbon/Hydrocarbon (vacuum)	2.6–6.0	0.21–0.14
Hydrocarbon/Hydrocarbon (water)	0.36–0.44	1.6–1.5
Hydrocarbon/SiO$_2$ (vacuum)	4.1–4.4	0.17–0.16
Hydrocarbon/SiO$_2$ (water)	0.25–0.82	1.9–1.1
Albumin/SiO$_2$ (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).^[4][7]

Synthetic adhesives

Stickybot, a climbing robot using synthetic setae^[8]

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

• Replicate the adhesiveness igid polymers manufactured in microfibers that are approximately the same size as gecko setae.^[9]
• Replicate the self cleaning attribute that naturally occurs when gecko feet accumulate particles from an exterior surface between setae.^[10]
• Carbon nanotube arrays transferred onto a polymer tape.^[11] In 2015 commercial products inspired by this work were released.^[12]

References

1. Skibinski, Brian. "All Species". Geckolist.com. Retrieved June 3, 2011.
2. "Crested Geckos". LLLReptile and Supply, Inc. 2006. Retrieved June 3, 2011.
3. Autumn, K. (2006). How gecko toes stick. American Scientist 94, 124–132.
4. 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:10.1038/news020826-2.
5. Prevenslik, T. (2009). "Electrostatic Gecko Mechanism". Tribology in Industry 31 (1&2).
6. Butt, Hans-Jürgen; Graf, Karlheinz; Kappl, Michael (6 March 2006). Physics and Chemistry of Interfaces. John Wiley & Sons. ISBN 978-3-527-60640-5.
7. Arzt, E.; Gorb, S.; Spolenak, R. (2003). "From micro to nano contacts in biological attachment devices". PNAS 100 (19): 10603–10606.
8. Biomimetics and Dexterous Manipulation Labtory,Stanford University
9. 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.
10. Fearing, Ronald. "Self-Cleaning Synthetic Gecko Tape". University of California, Berkeley.
11. 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. doi:10.1073/pnas.0703505104.
12. Lavars, Nick (2015-12-22). "Gecko-inspired adhesive tape finally scales to market". www.gizmag.com. Retrieved 2015-12-23.