 |
Grain boundaries may contain special
defects that only exist in grain
boundaries; the most prominent ones are grain boundary dislocations. Grain
boundary dislocations are linear defects with all the characteristics of
lattice dislocations, but with very specific Burgers vectors that can only occur in grain boundaries. |
|
 |
To construct grain boundary dislocations, we can
use the universal
Volterra
definition.We start with a "low S" boundary and make a cut in the habit plane of
the boundary. The cut line, as before, will define the dislocation line vector
l which by definition will be contained in the boundary. |
|
|
Now we displace one grain with respect to the
other grain by the Burgers vector b so as to preserve the
structure of the boundary everywhere except
around the dislocation line. In other words: the structure of the boundary
after the shift looks exactly as before the shift. |
|
What does that mean? What is the
"structure of the boundary" and how do we preserve it? |
|
|
Well, we have a CSL on both sides of the
boundary. We certainly will preserve the structure of the boundary if we shift
by a translation vector of the CSL, i.e. by a rather large Burgers
vector. We than would preserve the coincidence site lattice - which is fine, but far too limited. We
already preserve the structure of the boundary if we simply preserve the coincidence! |
|
|
It is best to illustrate what this means with a
simple animation: Two superimposed
lattices form a CSL marked in blue.
The red lattice moves to the left, and at first there are no more coincidences
of lattice points - the CSL has disappeared and we have a different
structure. However, after a short distance of shifting - far smaller than a
lattice vector of the CSL, coincidence
points appears and we have a CSL again - but with the
coincidence points now in different
positions. |
|
|
|
|
|
|
|
|
|
|
|
We found a displacement vector that preserved the
structure of the boundary - sort of experimentally. There are others, too, and
the possible displacements vectors that conserve the CSL obviously are
not limited to vectors of the crystal lattice; they
can be much smaller. This we can generalize: |
|
The set of all
possible displacement vectors which preserve the CSL defines a new kind of lattice, the so-called
DSC-lattice. The abbreviation "DSC"
stands for "Displacement Shift
Complete", not the best of possible names, but time-honored by
now. |
|
|
A better way of thinking about it would be to
interprete the abbreviation as "Displacements
which are Symmetry Conserving". Displacing one grain of a grain
boundary with a CSL by a vector of the corresponding DSC lattice thus
preserves the structure of the boundary because it preserves the symmetries of
the CSL. We now conclude: |
|
|
Translation vectors of
the DSC lattice are possible Burgers vectors bGB for
grain boundary dislocations. As for lattice dislocations, only the
smallest possible values will be encountered for energetic reasons. |
|
|
Grain boundary dislocations
constructed in this way by (Volterra)
definition, have most of the properties of real dislocations - just
with the added restriction that they are confined to the boundary. Strain- and
stress field, line energy, interactions, forming of networks - everything
follows the same equations and rules that we found for lattice
dislocations. |
|
|
It remains to be seen how the DSC-lattice
can be constructed. From the illustration it is clear that every vector that
moves a lattice site of grain 1 to a lattice site of grain 2 is a
DSC-lattice vector. This leads to a simple "working" definition: |
|
The DSC-lattice is the coarsest
sub-lattice of the CSL that has all atoms of both
lattices on its lattice points. Most lattice points of the
DSC-lattice, however, will be empty |
|
|
|
|
|
|
This is the DSC-lattice for the animation above. Its
easy enough to obtain, but: |
|
A formal and general definition of the
DSC lattice (including near CSL orientations) is one of the most
difficult undertakings in grain boundary theory. If you love tough nuts, turn
to chapter 7.3 and proceed. |
|
|
|
|
|
|
|
Any translation of one of the two crystals along a
vector of the orange DSC-lattice will keep the CSL, but will generally
shift its origin. Only if a DSC vector is chosen that is also a vector
of the CSL, will the origin of the CSL remain in place. |
|
|
Looking
back at the S5 boundary from before, we
now can enact the cut and the displacement procedure and generate a picture of
the dislocations that must result. The result contains a
little surprise and is shown in cross-section below: |
|
|
|
|
|
|
|
|
|
|
|
The cut was made from the right. The top crystal
(red lattice points) was shifted by a unit vector of the DSC lattice,
which is a 1/5<210> vector in both crystal lattices in this case.
The second crystal (green lattice points) was left completely unchanged. The
coincidence points are blue. We observe two somewhat surprising effects: |
|
|
The boundary plane (as indicate by the pink line)
after the shift is not identical with the plane of the cut |
|
|
The CSL has an interruption in both grains
- it doesn't fit anymore. Disturbing - but totally
unimportant. The CSL, after all, is totally meaningless for real crystals - the
(mathematical) coincidence points in the
grains have no significance for the grains. The only significance of a coincidence orientation is
that it provides an especially good fit of the two grains at a boundary, i.e.
it allows for a particularly favorable boundary structure. And the structure of the grain boundary is unchanged by the
introduction of the grain boundary dislocation, except around its core region.
This is indicated by the characteristic diamond shapes (yellow) in the picture
above that can be taken as the hallmark of this S5 structure. |
|
|
Think about it! Finding the yellow diamonds is the
practical way of finding the position of the boundary. However you define the
position - you will find the preserved structure as expressed in the yellow
diamonds here. |
|
|
Introducing the grain boundary
dislocation thus had the unexpected additional effect of introducing a
step in the grain boundary. Some
atoms had to be changed from green to red
to obtain the structure, but that again is an artifact
of the representation. Real atoms are all the same; they do not come
in green and red and do not care to which crystal they belong. |
|
We see that the recipe works:
Dislocations in the DSC lattice preserve the structure of the boundary;
they leave the coincidence relation unchanged. However, they also may introduce steps in the plane of the boundary -we
cannot yet be sure that this always the case. |
|
|
Note that is not directly obvious how
the step relates to the dislocation, i.e. how the vector describing the step
can be deduced from the DSC lattice vector used as Burgers vector. (If
you see an obvious relationship - please tell me. I'm not aware of a simple
formula applicable in all cases). |
|
|
Note also: While many (if not all)
grain boundary dislocations are linked with a step, the reverse is not true:
There are many possible steps in a boundary that do not have any dislocation character. More to that in
chapter 8.3 |
|
The extension to three dimensions is obvious, but also a bit
mind-boggling. Still, some general rules can be given |
|
|
The larger the elementary cell of the CSL,
the smaller the elementary cell of the DSC-lattice! |
|
|
If you suspected it by now: The DSC lattice
indeed can be seen as the reciprocal lattice (in space) of the
CSL. |
|
|
The volumes of CSL, crystal lattice and
DSC lattice relate as S : 1 :
S –1 for cubic
crystals. |
|
What are all these lattices good for?
The main import is: |
|
|
A grain boundary between two
grains that is close to, but not exactly at
a low-energy (= low S) orientation may
decrease its energy if grain boundary dislocations with a Burgers vector of the
DSC lattice belonging to the low-S
orientation are introduced so that the dislocation free parts are now in the
precise CSL orientation and all the
misalignment is taken up by the grain boundary dislocations. |
|
|
We will see how this works in the next
sub-chapter. |
|
|
|
© H. Föll
