If the grain size of a material varies, what does happen? Are the permeability and resistivity of the material changed with the variation of that grain size? If they do, how they do?
$\begingroup$ If you mean grain material as small spherical balls then finer the grain lower the resistance. You basically fill more space with the conductive stuff. Opposite for permeability to things much smaller than the grain size. In this permeability case capillaries might interfere with my pictorial reasoning. Moreover if the things passing through have a not spherical geometry, they could go through faster as bigger they are. Unless they cannot go through at all. $\endgroup$– AlchimistaFeb 3, 2018 at 12:58
Resistivity increases as grain size drops because you have more grains crossed per unit length which gives more grain boundary scattering events. This effect is nonlinear because there are other factors affecting resistivity (thermal motion, point defects, or dopants for a degenerately doped semiconductor--a sort of metal). Once the grains get big enough, these other factors may dominate the resistivity.
I disagree with the comment above that smaller grains fill space better. All else equal (same microstructure and grain shape), space filling will be the same fraction regardless of particle size. Consider bowling balls, marbles, ball bearings.