Editing Sintering (section)

==== 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" />