## Shell Calculations

### Quick Start

When you buy a nanoparticle, what percent of it is "particle" rather than "dispersant"? The answer is that simple geometry tells us that it can be a surprisingly small amount once the particle gets smaller.

For a given particle radius rparticle, the graph shows the rapidly decreasing particle % as the shell size gets larger. If you need the mass % then you need to enter the respective densities, ρ.

### Shell Calculations

ρparticle g/cc
ρshell g/cc
rparticle nm

We often have to provide a stabilising shell around a nanoparticle. This sounds normal, and the shell is often "only a few nm" so we give it no tought. It is rather important, therefore, to know what volume percentage of your "nanoparticle" is particle of radius r and what volume percentage is shell of thickness δ.

The problem is that the volume of the shell depends on (r+δ)³ so it can rapidly become a large fraction of the whole.

"% Particle" = 100(4/3πr^3)/(4/3π(r+δ)^3) = 100r^3/(r+δ)^3

Depending on the relative densities of the particle and the shell, the Mass ratio might not be so bad as the Volume ratio.