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. They can be a variety of colors. Geckos are carnivorous, feeding on a variety of foods, including insects and worms. Most gecko species, including the crested gecko (Rhacodactylus ciliatus), can climb walls and other surfaces.

Structure

Close view of a gecko's foot

Micrometer- and nanometer-scale view of a gecko's toe

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, 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. 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:

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

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 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.

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
• The material gradient properties (dependence of elastic modulus on the depth).

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:

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

where W_AB is the potential energy between the molecules (in joules), C_AB 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:

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

where D′ is the distance between molecule A and an infinitesimal volume of material B, and ρ_B 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:

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

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 r_s. Then the interaction between a single spatula and a surface is:

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

where D′ is the distance between the surface of B and an infinitesimal volume of material A and ρ_A 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:

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

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

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

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

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

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

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

Experimental verification

The equation for r_s can then be used with calculated Hamaker constants 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.

Calculated seta radii

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).

Synthetic adhesives

Stickybot, a climbing robot using synthetic setae

Main article: Synthetic setae

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.
• Replicating the self-cleaning attribute that naturally occurs when gecko feet accumulate particles from an exterior surface between setae.
• Carbon nanotube arrays transferred onto a polymer tape. In 2015 commercial products inspired by this work were released.