Calculating Van der Waals Bonding Force of Carbon Nanotubes

[SkepticJ]

I hope I've put this in the right forum, if not, please move it.

I'm wanting to calculate the van der Waals bonding force/area of 1 cm^2 of "setae" carbon nanotubes that are 0.4nm(4 angstroms) in diameter. Such nanotubes are the smallest physically possible so that's what I want to go with. I want to use the Johnson–Kendall–Roberts adhesion theory to calculate the force/area, but I don't understand it. Would someone explain this math to me so that I may go about doing calculations? Thank you.

Another problem is that these nanotubes don't have a hemisphere end, because carbon atoms aren't much smaller than this. The tubes do end in a dome-like cap, but it's only composed of a few atoms, so won't conform to simply making the ends of the tubes as perfect hemispheres in the calculations. What can I do?

 

[SkepticJ]

Hmmm, perhaps this was the wrong forum to post it in. Would one of the mods please move it to a better one? Thanks.

 

[tacman]

Hello,

Thank you for reaching out and providing the details of your question. Calculating the van der Waals bonding force of carbon nanotubes can be a complex task, but I will do my best to explain the process using the Johnson-Kendall-Roberts (JKR) adhesion theory.

First, let's briefly discuss the van der Waals force. This type of bonding force is a weak attraction between molecules or atoms that results from the fluctuations of their electron clouds. In the case of carbon nanotubes, the van der Waals force is due to the interaction between the carbon atoms in the nanotubes.

Now, the JKR theory is a widely accepted model for calculating the adhesive force between two surfaces. In this case, we will use it to calculate the van der Waals bonding force between your setae carbon nanotubes. The JKR theory takes into account the elastic properties of the materials, the contact area, and the surface energy.

To calculate the force/area, we will need to use the following equation:

F = (3/2) * (E * R * δ)^0.5

Where:
F - is the force per unit area (N/m^2)
E - is the elastic modulus of the material (Pa)
R - is the radius of the nanotube (m)
δ - is the surface energy (J/m^2)

To use this equation, we need to determine the values for each variable. For the elastic modulus, we can use the published value for carbon nanotubes, which is around 1 TPa (10^12 Pa). The radius of your nanotubes is 0.2 nm (2 angstroms), which is half of the given diameter. And for the surface energy, we can use a typical value for carbon nanotubes, which is around 0.2 J/m^2.

Substituting these values into the equation, we get:

F = (3/2) * (1 * 10^12 * 0.2 * 0.2 * 0.2)^0.5 = 1.7 * 10^-5 N/m^2

Now, to calculate the force per unit area for a 1 cm^2 area, we need to multiply this value by 10^4, which gives us:

F = 1.7 * 10^-1 N/m^2

This