text: If particles of a second phase which are insoluble in the matrix phase are added to the powder in the form of a much finer powder, then this will decrease grain boundary movement. When the grain boundary tries to move past the inclusion diffusion of atoms from one grain to the other, it will be hindered by the insoluble particle. This is because it is beneficial for particles to reside in the grain boundaries and they exert a force in opposite direction compared to grain boundary migration. This effect is called the Zener effect after the man who estimated this drag force to
  where r is the radius of the particle and λ the interfacial energy of the boundary if there are N particles per unit volume their volume fraction f is
  assuming they are randomly distributed. A boundary of unit area will intersect all particles within a volume of 2r which is 2Nr particles. So the number of particles n intersecting a unit area of grain boundary is:
  Now, assuming that the grains only grow due to the influence of curvature, the driving force of growth is  where (for homogeneous grain structure) R approximates to the mean diameter of the grains. With this the critical diameter that has to be reached before the grains ceases to grow:
  This can be reduced to
  so the critical diameter of the grains is  dependent on the size and volume fraction of the particles at the grain boundaries.
  It has also been shown that small bubbles or cavities can act as inclusion
  More complicated interactions which slow grain boundary motion include interactions of the surface energies of the two grains and the inclusion and are discussed in detail by C.S. Smith.