The # of particle embedded should increase proportionally to the area, thickness or volume of the foil?

Answer 1

I suppose you're talking about collecting particles with varying mass, velocity or both on a foil placed in the path of a spatially uniform particle stream. Keeping the thickness constant means the same capture rate per unit area, so proportional to area would be right. Keeping the volume constant allows different area/thickness ratios, and the capture rate is very unlikely to be proportional to thickness*, so both thickness and volume are losers.
*For instance, if mass and velocity are constant, a certain thickness will stop all the particles.