Grain boundary
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A grain boundary is the interface between two grains in a polycrystalline material. Grain boundaries disrupt the motion of dislocations through a material; reducing crystallite size is therefore a common way to improve strength, as described by the Hall-Petch relationship. Since grain boundaries are defects in the crystal structure they tend to decrease the electrical and thermal conductivity of the material. The high interfacial energy and relatively weak bonding in grain boundaries makes them preferred sites for the onset of corrosion and for the precipitation of new phases from the solid. They are also important to many of the mechanisms of creep.
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[edit] Types of boundary
It is convenient to separate grain boundaries by the extent of the misorientation between the two grains: high angle grain boundaries (HAGB) are those with a misorientation greater than some value while low angle grain boundaries (LAGB) are those below this figure. Generally speaking a low-angle boundary is composed of an array of dislocations and its properties and structure are a function of the misoriention. In contrast a high-angle boundary's properties are normally found to be independent of the misorientation (with notable exceptions). The transition between low- and high-angle is typically quoted as 10-15° but this is necessarily somewhat arbitrary.
Consider the simplest case of a tilt boundary where a single, contiguous crystallite or grain is gradually bent by some external force. The energy associated with the elastic bending of the lattice can be reduced by inserting a dislocation, which is essentially a half-plane of atoms that act like a wedge. As the grain is bent further, more and more dislocations must be introduced to accommodate the deformation resulting in a growing wall of dislocations - a low-angle boundary. The grain can now be considered to have split into two sub-grains of related crystallography but notably different orientations. Eventually, as the density of dislocations increases, the cores of the dislocations will begin to overlap and the ordered nature of the boundary will begin to breakdown. At this point the boundary can be considered to be high-angle and the original grain to have separated into two entirely separate grains.
An alternative is a twist boundary where the misorientation occurs around and axis that is perpendicular to the boundary. This type of boundary incorporates two sets of screw dislocations. If the Burgers vectors of the dislocations are orthogonal then the dislocations do not strongly interact and form a square network. In other cases the dislocations may interact to form a more complex hexagonal structure.
[edit] High angle boundary structure
High angle boundaries were originally understood to be some form of amorphous layer but it is now accepted that a boundary consists of regions of good and bad fit between the two grains. This concept is called the coincidence site lattice. The degree of fit (Σ) between the two grains is described by the reciprocal of the ratio of CSL sites to the total number of sites. Thus a boundary with high Σ is more random than one with low Σ. Low-angle boundaries, where the distortion is entirely accommodated by dislocations, are Σ1. Some other special bounadries are known and these may have a lower than expected Σ or other special properties. Examples include coherent twin boundaries (Σ3) and high-mobility boundaries in FCC materials (Σ7). Deviations from the ideal CSL may be accommodated by local atomic relaxation or the inclusion of dislocations into the boundary.
[edit] Describing a boundary
A boundary can be described by the orientation of the boundary with respect to the two grains and the 3-D rotation required to bring the grains into coincidence. Thus a boundary has 5 macroscopic degrees of freedom. However, it is common to describe a boundary only in terms of the orientation relationship of the neighbouring grains. Generally, the convenience of ignoring the boundary plane orientation, which is very difficult to determine, outweighs the reduced information. The relative orientation of the two grains is described using the rotation matrix:
Using this system the rotation angle θ is:
while the direction [uvw] of the rotation axis is:
![[(a_{32}-a_{23}),(a_{13}-a_{31}),(a_{21}-a_{12})] \,\!](../../../math/7/8/b/78bc63456ef7eee0ec9be191c5a8343e.png)
The nature of the crystallography involved limits the misorientation of the boundary. A completely random polycrystal, with no texture, thus has a characteristic distribution of boundary misorientations (see figure). However, such cases are rare and most materials will deviate from this ideal to a greater or lesser degree.
[edit] Boundary energy
The energy of a low-angle boundary is dependent on the degree of misorientation between the neighbouring grains up to the transition to high-angle status. In the case of simple tilt boundaries the energy of a boundary made up of dislocations with Burgers vector b and spacing h is predicted by the Read-Shockley equation:
where θ = b/h, γ0 = Gb/4 π(1-ν), A = 1 + ln(b/2 πr0), G is the shear modulus, ν is possions ratio, and r0 is the radius of the dislocation core. It can be seen that as the energy of the boundary increases the energy per dislocation decreases. Thus there is a driving force to produce fewer, more misoriented boundaries (i.e grain growth).
The situation in high-angle bounadries is more complex. Although theory predicts that the energy will be a minimum for ideal CSL configurations, with deviations requiring dislocations and other energetic features, empirical measurements suggest the relationship is more complicated. Some predicted troughs in energy are found as expected while others missing or substantially reduced. Surveys of the available experimental data have indicated that simple relationships such as low Σ are misleading [1]:
It is concluded that no general and useful criterion for low energy can be enshrined in a simple geometric framework. Any understanding of the variations of interfacial energy must take account of the atomic structure and the details of the bonding at the interface. : - Sutton and Balluffi
[edit] Boundary migration
The movement of grain boundaries (HAGB) has implications for recrystallization and grain growth while subgrain boundary (LAGB) movement strongly influences recovery and the nucleation of recrystallization.
A boundary moves due to a pressure acting on it. It is generally assumed that the velocity is directly proportional to the pressure with the constant of proportionality being the mobility of the boundary. The mobility is strongly temperature dependent and often follows an Arrhenius type relationship:
The apparent activation energy (Q) may be related to the thermally activated atomistic processes that occur during boundary movement. However, there are several proposed mechanisms where the mobility will depend on the driving pressure and the assumed proportionality may break down.
It is generally accepted that the mobility of low-angle boundaries is much lower than that of high-angle boundaries. The following observations appear to hold true over a range of conditions:
- The mobility of low-angle boundaries is proportional to the pressure acting on it.
- The rate controlling process is that of bulk diffusion
- The boundary mobility increases with misorientation.
Since low-angle boundaries are composed of arrays of dislocations and their movement may be related to dislocation theory. The most likely mechanism, given the experimental data, is that of dislocation climb, rate limited by the diffusion of solute in the bulk [2].
The movement of high-angle boundaries occurs by the transfer of atoms between the neighbouring grains. The ease with which this can occur will depend on the structure of the boundary, itself dependent on the crystallography of the grains involved, impurity atoms and the temperature. It is possible that some form of diffusionless mechanism (akin to diffusionless phase transformations such as martensite) may operate in certain conditions. Some defects in the boundary, such as steps and ledges, may also offer alternative mechanisms for atomic transfer.
Since a high-angle boundary is imperfectly packed compared to the normal lattice it has some amount of free space or free volume where solute atoms may possess a lower energy. As a result a bounadry may be associated with a solute atmosphere that will retard its movement. Only at higher velocities will the boundary be able to break free of its atmosphere and resume normal motion.
Both low- and high-angle boundaries are retarded by the presence of partilces via the so-called zener pinning effect. This effect is often exploited in commercial alloys to minimise or prevent recrystallization or grain growth during heat-treatment.
[edit] References
- AP Sutton, RW Balluffi (1987). "Overview no. 61: On geometric criteria for low interfacial energy". Acta Metallurgica 35 (9): 2177–2201.
- FJ Humphreys, M Hatherly (2004). Recrystallisation and related anealing phenomena. Elsevier.
- RD Doherty; DA Hughes; FJ Humphreys; JJ Jonas; D Juul Jenson; ME Kassner; WE King; TR McNelley; HJ McQueen; AD Rollett (1997). "Current Issues In Recrystallisation: A Review". Materials Science and Engineering A238: 219–274.
