Editing Sintering (section)

==== Mechanical equilibrium ====
In a two-dimensional [[isotropic material]] the grain boundary tension would be the same for the grains. This would give angle of 120° at GB junction where three grains meet. This would give the structure a [[hexagonal]] pattern which is the [[metastable]] state (or [[mechanical equilibrium]]) of the 2D specimen. A consequence of this is that, to keep trying to be as close to the equilibrium as possible, grains with fewer sides than six will bend the GB to try keep the 120° angle between each other. This results in a curved boundary with its [[curvature]] towards itself. A grain with six sides will, as mentioned, have straight boundaries, while a grain with more than six sides will have curved boundaries with its curvature away from itself. A grain with six boundaries (i.e. hexagonal structure) is in a metastable state (i.e. local equilibrium) within the 2D structure.<ref name="Fundamentals of Materials Science" /> In three dimensions structural details are similar but much more complex and the [[metastable]] structure for a grain is a non-regular 14-sided [[polyhedra]] with doubly curved faces. In practice all arrays of grains are always unstable and thus always grow until prevented by a counterforce.<ref name="Physical Metallurgy ch 28">{{cite book|author=Cahn, Robert W. and Haasen, Peter |title=Physical Metallurgy|year=1996|isbn=978-0-444-89875-3|pages=2399–2500|publisher=Elsevier Science |edition=Fourth}}</ref>

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 <!--clarify--> curvature.{{clarify|date=September 2012|reason="the <!--clarify--> the curvature" is wrong, but I'm not sure how to fix it}} 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.<ref name="Ceramic materials ch sintering">{{cite book|last1=Carter|first1=C. Barry|last2=Norton|first2=M. Grant|title=Ceramic Materials: Science and Engineering|url=https://archive.org/details/ceramicmaterials00cart|url-access=limited|year=2007|publisher=Springer Science+Business Media, LLC.|isbn=978-0-387-46270-7|pages=[https://archive.org/details/ceramicmaterials00cart/page/n425 427]–443}}</ref>

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:
<math display="block">G^m= G_0^m+Kt</math>

Here ''G'' is final average grain size, ''G<sub>0</sub>'' is the initial average grain size, ''t'' is time, ''m'' is a factor between 2 and 4, and ''K'' is a factor given by:
<math display="block">K= K_0 e^{\frac{-Q}{RT}}</math>

Here ''Q'' is the molar activation energy, ''R'' is the ideal gas constant, ''T'' is absolute temperature, and ''K<sub>0</sub>'' 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.