Sintering in practice is the control of both densification and
grain growth. Densification is the act of reducing porosity in a sample, thereby making it denser. Grain growth is the process of grain boundary motion and
Ostwald ripening to increase the average grain size. Many properties (
mechanical strength, electrical breakdown strength, etc.) benefit from both a high relative
density and a small grain size. Therefore, being able to control these properties during processing is of high technical importance. Since densification of powders requires high temperatures, grain growth naturally occurs during sintering. Reduction of this process is key for many engineering ceramics. Under certain conditions of chemistry and orientation, some grains may grow rapidly at the expense of their neighbours during sintering. This phenomenon, known as
abnormal grain growth (AGG), results in a bimodal grain size distribution that has consequences for the mechanical, dielectric and thermal performance of the sintered material. For densification to occur at a quick pace it is essential to have (1) an amount of liquid phase that is large in size, (2) a near complete solubility of the solid in the liquid, and (3) wetting of the solid by the liquid. The power behind the densification is derived from the capillary pressure of the liquid phase located between the fine solid particles. When the liquid phase wets the solid particles, each space between the particles becomes a capillary in which a substantial capillary pressure is developed. For submicrometre particle sizes, capillaries with diameters in the range of 0.1 to 1 micrometres develop pressures in the range of to for silicate liquids and in the range of to for a metal such as liquid cobalt. Striving to minimize its energy leads to the coarsening of the
microstructure to reach a metastable state within the specimen. This involves minimizing its GB area and changing its
topological structure to minimize its energy. This grain growth can either be
normal or abnormal, a normal grain growth is characterized by the uniform growth and size of all the grains in the specimen.
Abnormal grain growth is when a few grains grow much larger than the remaining majority.
Grain boundary energy/tension The atoms in the GB are normally in a higher energy state than their equivalent in the bulk material. This is due to their more stretched bonds, which gives rise to a GB tension \sigma_{GB}. This extra energy that the atoms possess is called the grain boundary energy, \gamma_{GB}. The grain will want to minimize this extra energy, thus striving to make the grain boundary area smaller and this change requires energy. For a solid, one can derive an expression for the change in Gibbs free energy, dG, upon the change of GB area, dA. dG is given by \sigma_{GB} dA \text{ (work done)} = dG \text{ (energy change)} = \gamma_{GB} dA + A d\gamma_{GB}\,\! which gives \sigma_{GB} = \gamma_{GB} + \frac{Ad\gamma_{GB}}{dA}\,\! \sigma_{GB} is normally expressed in units of \frac{N}{m} while \gamma_{GB} is normally expressed in units of \frac{J}{m^2} (J = Nm) since they are different physical properties. Grains strive to minimize their energy, and a curved boundary has a higher energy than a straight boundary. This means that the grain boundary will migrate towards the curvature. The consequence of this is that grains with less than 6 sides will decrease in size while grains with more than 6 sides will increase in size. Grain growth occurs due to motion of atoms across a grain boundary. Convex surfaces have a higher chemical potential than concave surfaces, therefore grain boundaries will move toward their center of curvature. As smaller particles tend to have a higher radius of curvature and this results in smaller grains losing atoms to larger grains and shrinking. This is a process called Ostwald ripening. Large grains grow at the expense of small grains. Grain growth in a simple model is found to follow: G^m= G_0^m+Kt Here
G is final average grain size,
G0 is the initial average grain size,
t is time,
m is a factor between 2 and 4, and
K is a factor given by: K= K_0 e^{\frac{-Q}{RT}} Here
Q is the molar activation energy,
R is the ideal gas constant,
T is absolute temperature, and
K0 is a material dependent factor. In most materials the sintered grain size is proportional to the inverse square root of the fractional porosity, implying that pores are the most effective retardant for grain growth during sintering.
Reducing grain growth Solute ions If a
dopant is added to the material (example: Nd in BaTiO3) the impurity will tend to stick to the grain boundaries. As the grain boundary tries to move (as atoms jump from the convex to concave surface) the change in concentration of the dopant at the grain boundary will impose a drag on the boundary. The original concentration of solute around the grain boundary will be asymmetrical in most cases. As the grain boundary tries to move, the concentration on the side opposite of motion will have a higher concentration and therefore have a higher chemical potential. This increased chemical potential will act as a backforce to the original chemical potential gradient that is the reason for grain boundary movement. This decrease in net chemical potential will decrease the grain boundary velocity and therefore grain growth.
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 F = \pi r \lambda \sin (2\theta)\,\! 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 f = \frac{4}{3} \pi r^3 N\,\! 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: n = \frac{3f}{2 \pi r^2}\,\! Now, assuming that the grains only grow due to the influence of curvature, the driving force of growth is \frac{2 \lambda}{R} 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: n F_{max} = \frac{2 \lambda}{D_{crit}}\,\! This can be reduced to D_{crit} = \frac{4r}{3f} \,\! 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. ==Sintering of catalysts==