Editing Sintering (section)

==== Fine second phase particles ====
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

<math display="block"> F = \pi r \lambda \sin (2\theta)\,\!</math>
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
<math display="block"> f = \frac{4}{3} \pi r^3 N\,\!</math>

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:

<math display="block">n = \frac{3f}{2 \pi r^2}\,\!</math>

Now, assuming that the grains only grow due to the influence of curvature, the driving force of growth is <math>\frac{2 \lambda}{R} </math> 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:

<math display="block">n F_{max} = \frac{2 \lambda}{D_{crit}}\,\!</math>
This can be reduced to
<math display="block">D_{crit} = \frac{4r}{3f} \,\!</math>

so the critical diameter of the grains is dependent on the size and volume fraction of the particles at the grain boundaries.<ref name="Physical Metallurgy">{{cite book|author=Cahn, Robert W. and Haasen, Peter |title=Physical Metallurgy|year=1996|publisher=Elsevier Science |isbn=978-0-444-89875-3|edition=Fourth}}</ref>

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.<ref name="C. S. Smith">{{cite journal|last=Smith|first=Cyril S.|title=Introduction to Grains, Phases and Interphases: an Introduction to Microstructure|date=February 1948}}</ref>