Editing Sintering (section)

== Densification, vitrification and grain growth ==
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 {{convert|175|psi}} to {{convert|1750|psi}} for silicate liquids and in the range of {{convert|975|psi}} to {{convert|9750|psi}} for a metal such as liquid cobalt.<ref name=Kingery/>

Densification requires constant [[capillary pressure]] where just solution-precipitation material transfer would not produce densification. For further densification, additional particle movement while the particle undergoes grain-growth and grain-shape changes occurs. Shrinkage would result when the liquid slips between particles and increases pressure at points of contact causing the material to move away from the contact areas, forcing particle centers to draw near each other.<ref name=Kingery/>

The sintering of liquid-phase materials involves a fine-grained solid phase to create the needed capillary pressures proportional to its diameter, and the liquid concentration must also create the required capillary pressure within range, else the process ceases. The vitrification rate is dependent upon the pore size, the viscosity and amount of liquid phase present leading to the viscosity of the overall composition, and the surface tension. Temperature dependence for densification controls the process because at higher temperatures viscosity decreases and increases liquid content. Therefore, when changes to the composition and processing are made, it will affect the vitrification process.<ref name=Kingery/>

=== Sintering mechanisms ===
Sintering occurs by diffusion of atoms through the microstructure. This diffusion is caused by a gradient of [[chemical potential]] – atoms move from an area of higher chemical potential to an area of lower chemical potential. The different paths the atoms take to get from one spot to another are the "sintering mechanisms" or "matter transport mechanisms".

In solid state sintering, the six common mechanisms are:<ref name="Kingery" />

# surface diffusion – diffusion of atoms along the surface of a particle
# vapor transport – evaporation of atoms which condense on a different surface
# lattice diffusion from surface – atoms from surface diffuse through lattice
# lattice diffusion from grain boundary – atom from grain boundary diffuses through lattice
# grain boundary diffusion – atoms diffuse along grain boundary
# plastic deformation – dislocation motion causes flow of matter.

Mechanisms 1–3 above are non-densifying (i.e. do not cause the pores and the overall ceramic body to shrink) but can still increase the area of the bond or "neck" between grains; they take atoms from the surface and rearrange them onto another surface or part of the same surface. Mechanisms 4–6 are densifying – atoms are moved from the bulk material or the grain boundaries to the surface of pores, thereby eliminating porosity and increasing the density of the sample.

=== 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.

=== Reducing grain growth ===

==== Solute ions ====
If a [[dopant]] is added to the material (example: Nd in BaTiO<sub>3</sub>) 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

<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>