Editing Sintering (section)

=== Grain growth ===
{{main|Grain growth}}

A [[grain boundary]] (GB) is the transition area or interface between adjacent [[crystallites]] (or grains) of the same chemical and [[crystal lattice|lattice]] composition, not to be confused with a [[phase boundary]]. The adjacent grains do not have the same orientation of the lattice, thus giving the atoms in GB shifted positions relative to the lattice in the [[crystal]]s. Due to the shifted positioning of the atoms in the GB they have a higher energy state when compared with the atoms in the crystal lattice of the grains. It is this imperfection that makes it possible to selectively etch the GBs when one wants the microstructure to be visible.<ref name=Smallman>{{cite book|last=Smallman R. E.|first=Bishop, Ray J|title=Modern physical metallurgy and materials engineering: science, process, applications|year=1999|publisher=Oxford : Butterworth-Heinemann|isbn=978-0-7506-4564-5}}</ref>

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 [[Grain growth#Normal vs abnormal|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.<ref name="Fundamentals of Materials Science">{{cite book|last=Mittemeijer|first=Eric J.|title=Fundamentals of Materials Science The Microstructure–Property Relationship Using Metals as Model Systems|url=https://archive.org/details/fundamentalsmate00mitt_322|url-access=limited|year=2010|publisher=Springer Heidelberg Dordrecht London New York|isbn=978-3-642-10499-2|pages=[https://archive.org/details/fundamentalsmate00mitt_322/page/n479 463]–496}}</ref>

==== 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 <math>\sigma_{GB}</math>. This extra energy that the atoms possess is called the grain boundary energy, <math>\gamma_{GB}</math>. The grain will want to minimize this extra energy, thus striving to make the grain boundary area smaller and this change requires energy.<ref name="Fundamentals of Materials Science" />

"Or, in other words, a force has to be applied, in the plane of the grain boundary and acting along a line in the grain-boundary area, in order to extend the grain-boundary area in the direction of the force. The force per unit length, i.e. tension/stress, along the line mentioned is σGB. On the basis of this reasoning it would follow that:

<math display="block">\sigma_{GB} dA \text{ (work done)} = \gamma_{GB} dA \text{ (energy change)}\,\!</math>

with dA as the increase of grain-boundary area per unit length along the line in the grain-boundary area considered."<ref name="Fundamentals of Materials Science" /><sup>[pg 478]</sup>

The GB tension can also be thought of as the attractive forces between the atoms at the surface and the tension between these atoms is due to the fact that there is a larger interatomic distance between them at the surface compared to the bulk (i.e. [[surface tension]]). When the surface area becomes bigger the bonds stretch more and the GB tension increases. To counteract this increase in tension there must be a transport of atoms to the surface keeping the GB tension constant. This diffusion of atoms accounts for the constant surface tension in liquids. Then the argument,

<math display="block">\sigma_{GB} dA \text{ (work done)} = \gamma_{GB} dA \text{ (energy change)}\,\!</math>

holds true. For solids, on the other hand, diffusion of atoms to the surface might not be sufficient and the surface tension can vary with an increase in surface area.<ref name=Sintering>{{cite book|last=Kang|first=Suk-Joong L.|title=Sintering: Densification, Grain Growth, and Microstructure|url=https://archive.org/details/sinteringdensifi00kang_089|url-access=limited|year=2005|publisher=Elsevier Ltd.|isbn=978-0-7506-6385-4|pages=[https://archive.org/details/sinteringdensifi00kang_089/page/n21 9]–18}}</ref>

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
<math display="block">\sigma_{GB} dA \text{ (work done)} = dG \text{ (energy change)} = \gamma_{GB} dA + A d\gamma_{GB}\,\!</math>

which gives
<math display="block">\sigma_{GB} = \gamma_{GB} + \frac{Ad\gamma_{GB}}{dA}\,\!</math>

<math>\sigma_{GB}</math> is normally expressed in units of <math>\frac{N}{m}</math> while <math>\gamma_{GB}</math> is normally expressed in units of <math>\frac{J}{m^2}</math> <math>(J = Nm)</math> since they are different physical properties.<ref name="Fundamentals of Materials Science" />

==== 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.