To keep a nanoparticle dispersion stable, it helps if the particles are, on average, far apart. But for a given wt% of particle, the interparticle distance decreases with decreasing radius rparticle. You can choose between whether the particles can end up loose-packed or close-packed at their limit (when d=0). The text boxes give you the distances at your chosen value of Wt%.
To convert wt% to volume fraction for the calculation, we need the densities, ρ, of the particle and liquid.
As particles get smaller they have a greater tendency to clump because their surface area to volume ratio is higher. Things are made worse because the distance between particles also decreases, so they are more likely to bump into each other. And they move faster because their diffusion coefficient is larger (1/r).
The calculation from Riman1 depend on volume fraction of particles, φ (up to the packing limit φm), but we tend to work in weight percent and often it is the weight percentage that is quoted by manufacturers. So enter the densities of the solid and liquid as well as Wt% solids and particle diameter. Two interparticle distances, d, are given because the value depends on when you consider the particles to be packed:
d=2r((φm/φ)0.333-1 ) φm = 0.59 Loose Packed, 0.63 Close Packed
The graph shows the two values for your given particle size over a range of Wt %, starting at 4% simply to give a reasonable Y-axis scale.
An earlier version used a formula from Woodcock, but the reference and justification was difficult to trace so I've discontinued it.
1Tian Hao, Richard E. Riman, Calculation of interparticle spacing in colloidal systems, Journal of Colloid and Interface Science 297 (2006) 374–377