1. DSC Lattice, Grain Boundary Dislocations - Basics

2. Our current understanding of the structure of high angle grain boundaries has been derived from geometrical concepts, based on dislocation models of low angle grain boundaries.
A fundamental reason for the failure of the lattice dislocation concept for larger angles of rotation is the requirement of a strictly periodic dislocation arrangement to minimize grain boundary energy.
The spacing of dislocations, however, changes discretely, namely at least by one atomic distance. As a consequence the angle of rotation  = b / d also changes in steps rather than continually.
For larger rotations, the orientation difference between two consecutive periodic dislocation arrangements becomes substantial. If, for instance, there is an arrangement with a dislocation spacing d = 4b, then  = 14.3°. Changing the dislocation spacing to 3 interatomic distances requires an angle of rotation of  = 19.2°.
The problem arises of what is the grain boundary structure for 14.3° <  < 19.2°, or, in general, between rotations that represent a periodic arrangement of (primary) crystal dislocations?

3. Coincidence site lattice (CSL) and structure of a 36
Coincidence site lattice (CSL) and structure of a 36.9° (100) ( = 5) grain boundary in a cubic crystal lattice. Right side of figure: grain boundary plane
II plane of the paper (twist boundary); left side of figure: grain boundary plane  plane of the paper (tilt boundary).

4. If we want to apply this crystallographic concept to real grain boundaries, there is the fundamental problem that the coincidence site lattice occurs only for very special rotations, and  does not change monotonically with the angle of rotation, as seen above

5. This problem is entirely complementary to the requirement of a discrete change of the angle of rotation in order to obtain a periodic crystal dislocation arrangement
In essence, a strictly periodic arrangement of crystal dislocations is absolutely identical with the relaxed structure of a CSL grain boundary
However, even for tiny deviations from the exact rotation relationship the long range coincidence is lost.
Just as a crystal tries to compensate a small misorientation by a periodic arrangement of crystal dislocations, we expect that the bicrystal will try to maintain its ideal fit and to compensate deviations from this perfect fit by localized perturbations, i.e. dislocations
These dislocations must have a Burgers vector that conserves the CSL just as lattice dislocations conserve the crystal lattice when forming a low angle grain boundary

6. Relationship between the coincidence site lattice and the primary dislocation structure at a grain boundary. If two identical, interlocking lattices (a) are turned symmetrically towards each other about an axis perpendicular to the plane of the page (b), a coincidence site lattice forms. The coincidence points are marked by overlapping circles and squares. The associated configuration of the resulting double dislocation is relaxed along the boundary (c), and the structure of a symmetrical low-angle tilt boundary forms (d).

7. The determination of the precise dislocation structure needed to transform a near-coincidence boundary into a true coincidence boundary with some superimposed grain boundary dislocation network can be exceedingly difficult
Obviously, we are somewhat off the Σ1 position. Introducing grain boundary dislocations now will establish the exact Σ1 relation between the dislocations (and something heavily disturbed at the dislocation cores).
The DSC-lattice as well the CSL are identical with the crystal lattice in this case, so the grain-boundary dislocations are simple lattice dislocations.

8. Introducing a sequence of edge dislocations in the tilt case and a network (not necessarily square) of screw dislocations in the twist case will do the necessary transformation
Between the dislocation lines we now have a perfect Σ1 relation (apart from some elastic bending)
All of the misfit relative to a perfect Σ orientation is concentrated in the grain boundary dislocations
The grain boundary energy is lowered in the area between the dislocations and raised it along the dislocations - there is the possibility of optimizing the grain boundary energy

9. 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 Voltera definitions. We start with a "low Σ" 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.

10. Two superimposed lattices form a CSL marked in blue
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.

11. As the most trivial example, the CSL will not be changed, if dislocations are introduced for which the Burgers vectors are lattice vectors of the CSL.
It is possible, for the Burgers vector to be a vector of the crystal lattice.
However, the elastic energy of dislocations increases with the square of the Burgers vector
The energy of the grain boundary would increase dramatically if dislocations with a very large Burgers vector were incorporated in the grain boundary.
The energy of a grain boundary is only determined by the density of coincidence sites, not their location.
We can relax the requirement that the location of the coincidence sites has to be conserved.

12. There are very small vectors which conserve the size of the CSL if the locations of the coincidence sites are allowed to change. The displacement vectors which satisfy this condition define the so-called Displacement Shift Complete (DSC) lattice
The DSC lattice is the coarsest grid that contains all lattice points of both crystal lattices
Coincidence site lattice (CSL) and DSC lattice at 36.9° (100) rotation in a cubic lattice.

13. All translation vectors of the CSL and the crystal lattices are also vectors of the DSC lattice, but the elementary vectors of the DSC lattice are much smaller.
Since the dislocation energy increases with the square of the Burgers vector, only base vectors of the DSC lattice qualify for Burgers vectors of the so-called secondary grain boundary dislocations (SGBDs).
Dislocations with DSC Burgers vectors are referred to as SGBDs, in contrast to primary grain boundary dislocations, which are crystal lattice dislocations, the periodic arrangement of which generates the CSL.
SGBDs are confined to grain boundaries, since their Burgers vectors are not translation vectors of the crystal lattice and their introduction into the crystal lattice would cause a local disruption of the crystal structure.

14. With regard to their geometry and correspondingly, to their elastic properties, secondary grain boundary dislocations can be treated like primary dislocations.
Much as primary dislocations can compensate a misorientation in a perfect crystal by a low angle grain boundary, secondary grain boundary dislocations can compensate an orientation difference to a CSL relationship while conserving the CSL.
Since SGBDs also have an elastic strain field as does any dislocation, they can be imaged in a TEM

15. A better way of thinking about it would be to interpret 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.
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.
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

16. Introducing the grain boundary dislocation thus had the unexpected additional effect of introducing a step in the grain boundary.
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.
While many (if not all) grain boundary dislocations are linked with a step, the reverse is not true.
A grain boundary between two grains that is close to, but not exactly at a low-energy (= low Σ) orientation may decrease its energy if grain boundary dislocations with a Burgers vector of the DSC lattice belonging to the low-Σ 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.

17. The larger the elementary cell of the CSL, the smaller the elementary cell of the DSC-lattice!
The DSC lattice indeed can be seen as the reciprocal lattice (in space) of the CSL.
The determination of the precise dislocation structure needed to transform a near-coincidence boundary into a true coincidence boundary with some superimposed grain boundary dislocation network can be exceedingly difficult.
Grain boundaries containing grain boundary dislocations which account for small misfits relative to a preferred (low) Σ orientation, are in general preferable to dislocation-free boundaries.
The Burgers vectors of the grain boundary dislocations could be translation vectors of one of the crystals, but that is energetically not favorable because the Burgers vectors are large and the energy of a dislocation scales with Gb2 and there is a much better alternative
The dislocations accounting for small deviations from a low Σ orientation are dislocations in the DSC lattice belonging to the CSL lattice that the grain boundary Σ endeavors to assume. Why should that be so? There are several reasons:

18. Dislocations in the DSC lattice belong to both crystals since the DSC lattice is defined in both crystals.
Burgers vectors of the DSC lattice are smaller than Burgers vectors of the crystal lattice, the energy of several DSC lattice dislocations with a Burgers vector sum equal to that of a crystal lattice dislocations thus is always much smaller.
With Σibi(DSC) = b(Lattice), we always have Σibi2(DSC) << b2(Lattice). This is exactly the same as in the case of lattice dislocations split into partial dislocations.
We can always imagine a low angle boundary of crystal lattice dislocations that produces exactly the small misorientation needed to turn an arbitrary boundary to the nearest low Σ position and superimpose it on this boundary.
Next, we decompose the crystal lattice dislocations into dislocations of the DSC lattice belonging to the low Σ orientation. This will be the dislocation network that we are going to find in the real boundary!

19. d = spacing of the DSC lattice dislocations and b = Burgers vector of the DSC lattice dislocations

20. A special property of the SGBDs is that at the location of the dislocation core, the grain boundary usually has a step
This step is a consequence of the fact that the CSL is displaced when an SGBD is introduced.
If an SGBD moves along the grain boundary, the step moves along with the dislocation and thus, the grain boundary is displaced perpendicular to its plane, i.e. the grain boundary will migrate by the distance of the step height.

21. (a) Grain boundary dislocations in a tilt boundary in stainless steel
(a) Grain boundary dislocations in a tilt boundary in stainless steel . (b) The pattern of the generation of a grain boundary edge dislocation. () Positions of atoms (small dots), coincidence sites (big dots) and the DSC-Iattice. () Orientation of grain boundary and positions of atoms at the grain boundary. () Rearrangement of material makes the grain boundary shift in sectors. () Partially shifted grain boundary. () Generation of a grain boundary edge dislocation by moving atoms along the grain boundary

22. Atomic configuration of a grain boundary edge dislocation at a  = 5 grain boundary in an fcc lattice. (a) Burgers vector parallel to the grain boundary; (b) Burgers vector inclined to the grain boundary.

23. A fundamental property of all dislocations is that they cause a shear deformation upon their motion. Therefore, the motion of a grain boundary dislocation always will cause a combination of grain boundary migration and grain boundary sliding.
In the case that the Burgers vector of the SGBD is parallel to the grain boundary plane, the dislocations need only to glide to cause the grain boundary to be displaced
If the Burgers vector is inclined to the grain boundary plane, the dislocation can only move by a combination of glide and climb, which requires the diffusion of vacancies, and is, therefore, a thermally activated process
The concept introduced is based on geometrical arguments only. Such a consideration cannot make predictions about the force equilibrium of the atomic arrangements considered.
This problem can only be solved by computer simulations, which allow us to calculate the position of the atoms at an equilibrium of interatomic forces, i.e. by relaxation of the geometrical arrangement.
In contrast to the basis of the geometrical considerations, coincidence is almost always lost upon relaxation, but the periodicity remains and that means the conceptual frame work is still correct.

24. Structure of a symmetrical 36
Structure of a symmetrical 36.9° (100) ( = 5) tilt boundary in aluminum, calculated by computer simulation. (a) Configuration after rigid rotation of the crystallites. (b) and (c) Structure of grain boundaries after relaxation. The
staggered vertical lines at the grain boundary indicate the shift of the crystallites. Hence, for a given misorientation there can be more than one structure

25. Some solutions for fcc crystals are given in the table:

26. https://www. tf. uni-kiel. de/matwis/amat/def_en/kap_7/backbone/r7_2_1
Dingley and Pond, Acta Met. 27, 667, 1979
A network of grain boundary dislocations with Burgers vectors b = a/82 <41,5,4> and an average distance of 20 nm is visible. The two sets of dislocations run parallel to the lines indicated by H and J