A grain boundary is the interface between two grains, or crystallites, in
a polycrystalline material. Grain boundaries are defects in the crystal
structure, and tend to decrease the electrical and thermal conductivity of
the material. Most grain boundaries are 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. On the other hand, grain
boundaries disrupt the motion of dislocations through a material, so
reducing crystallite size is a common way to improve strength, as described by
the Hall–Petch relationship. High and low angle boundaries
It is convenient to separate grain boundaries by the extent of the
mis-orientation between the two grains. Low angle grain boundaries or subgrain
boundaries are those with a misorientation less than about 15
degrees. Generally speaking they are composed of an array of dislocations and
their properties and structure are a function of the misorientation. In
contrast the properties of high angle grain boundaries, whose misorientation
is greater than about 11 degrees, are normally found to be independent of the
misorientation. However, there are 'special boundaries' at particular
orientations whose interfacial energies are notably lower than those of general
HAGBs. The most simple boundary is that of a
tilt boundary where the rotation axis is parallel to the boundary plane. This
boundary can be conceived as forming from a single, contiguous crystallite or
grain which 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, that creates a permanent misorientation between the two sides. 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.
An alternative is a twist boundary where the mis-orientation occurs around an
axis that is perpendicular to the boundary plane. 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.
These concepts of tilt and twist boundaries represent somewhat idealized
cases. The majority of boundaries are of a mixed type, containing dislocations of
different types and Burgers vectors, in order to create the best fit between the
neighboring grains. If the dislocations in the boundary
remain isolated and distinct, the boundary can be considered to be
low-angle. If deformation continues, the density of dislocations will increase
and so reduce the spacing between neighboring dislocations. Eventually,
the cores of the dislocations will begin to overlap and the ordered nature of the
boundary will begin to break down. At this point the boundary can be
considered to be high-angle and the original grain to have separated into
two entirely separate grains. In comparison to LAGBs, high-angle
boundaries are considerably more disordered, with large areas of poor fit
and a comparatively open structure. Indeed, they were originally thought to
be some form of amorphous or even liquid layer between the grains. However, this
model could not explain the observed strength of grain boundaries and, after
the invention of electron microscopy, direct evidence of the grain structure
meant the hypothesis had to be discarded. It is now accepted that a
boundary consists of structural units which depend on both the misorientation
of the two grains and the plane of the interface. The types of structural unit
that exist can be related to the concept of the coincidence site lattice, in
which repeated units are formed from points where the two misoriented
lattices happen to coincide. In coincident site lattice theory, the
degree of fit between the structures of the two grains is described by the
reciprocal of the ratio of coincidence sites to the total number of sites. Thus
a boundary with high Σ might be expected to have a higher energy than one with
low Σ. Low-angle boundaries, where the distortion is entirely accommodated by
dislocations, are Σ1. Some other low-Σ boundaries have special properties,
especially when the boundary plane is one that contains a high density of
coincident sites. Examples include coherent twin boundaries and
high-mobility boundaries in FCC materials. Deviations from the ideal CSL
orientation may be accommodated by local atomic relaxation or the inclusion of
dislocations into the boundary. Describing a boundary
A boundary can be described by the orientation of the boundary 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 as 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: 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. However, such cases are rare and most materials will deviate
from this ideal to a greater or lesser degree.
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: with is the shear modulus, is
Poisson's ratio, and 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. The situation in high-angle boundaries
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 are
misleading: 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. Boundary migration
The movement of grain boundaries has implications for recrystallization and
grain growth while subgrain boundary 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 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.
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 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 boundary 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 particles
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. See also
Crystal Crystallite
Crystallography Segregation in materials
References = Bibliography =
FJ Humphreys, M Hatherly. Recrystallisation and related anealing
phenomena. Elsevier. AP Sutton, RW Balluffi. "Overview no.
61: On geometric criteria for low interfacial energy". Acta Metallurgica
35: 2177–2201. doi:10.1016/0001-6160(87)90067-8.
Further reading RD Doherty; DA Hughes; FJ Humphreys; JJ
Jonas; D Juul Jenson; et al.. "Current Issues In Recrystallisation: A Review".
Materials Science and Engineering A 238: 219–274.
doi:10.1016/S0921-5093(97)00424-3. G Gottstein, LS Shvindlerman. Grain
Boundary Migration in Metals: Thermodynamics, Kinetics, Applications,
2nd Edition. CRC Press.