Some difficulties associated with interface theory
Consider two crystals A and B, both of cubic structure, related by a rigid body rotation.
Any boundary containing the axis of rotation is a tilt boundary; for the special case of the
symmetrical tilt boundary, lattice B can be generated from A by reflection across the boundary
plane. By substituting the rigid body rotation for (A S A) in equation 25, the dislocation
structure of the symmetrical tilt boundary may be deduced (example 22) to consist of a single
array of dislocations with line vectors parallel to the tilt axis.
Symmetry considerations imply that the rigid body rotation has up to 23 further axis–angle
representations. If we impose the condition that the physically most significant representation
is that which minimizes the Burgers vector content of the interface, then the choice reduces to
the axis–angle pair involving the smallest angle of rotation.
On the other hand, (A S A) can also be a lattice–invariant twinning shear on the symmetrical
tilt boundary plane
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; crystal B would then be related to A by reflection across the twin plane so
that the resulting bicrystal would be equivalent to the case considered above. The dislocation
content of the interface then reduces to zero since the invariant–plane of the twinning shear is
fully coherent.
This ambiguity in the choice of (A S A) is a major difficulty in interface theory
5
. The problem
is compounded by the fact that interface theory is phenomenological - i.e. the transformation
strain (A S A) may be real or notional as far as interface theory is concerned. If it is real
then we expect to observe a change in the shape of the transformed region, and this may help
in choosing the most reasonable deformation (A S A). For example, in the case of mechanical
twinning in FCC crystals, the surface relief observed can be used to deduce that (A S A) is a
twinning shear rather than a rigid body rotation. In the case of an FCC annealing twin, which
grows from the matrix by a diffusional mechanism (during grain boundary migration), the same
twinning shear (A S A) may be used to deduce the interface structure, but the deformation is
now notional, since the formation of annealing twins is not accompanied by any surface relief
effects. In these circumstances, we cannot be certain that the deduced interface structure for
the annealing twin is correct.
The second major problem follows from the fact that the mathematical Burgers vector content
b
t
given by equation 25 has to be factorised into arrays of physically realistic dislocations with
Burgers vectors which are vectors of the DSC lattice. There is an infinite number of ways in
which this can be done, particularly since the interface dislocations do not necessarily have
Burgers vectors which minimise their elastic strain energy.
Secondary dislocations are referred to an exact CSL as the reference lattice. Atomistic calcu-
lations suggest that boundaries in crystals orientated at exact coincidence contain arrays of
primary dislocations (whose cores are called structural units)
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. The nature of the structural
units varies with the Σ3 value, but some favoured CSL’s have boundaries with just one type
of structural unit, so that the stress field of the boundary is very uniform. Other CSL’s have
boundaries consisting of a mixture of structural units from various favoured CSL’s. It has
been suggested that it is the favoured CSL’s which should be used as the reference lattices in
the calculation of secondary dislocation structure, but the situation is unsatisfactory because
the same calculations suggest that the favoured/unfavoured status of a boundary also depends
of the boundary orientation itself. (We note that the term “favoured” does not imply a low
interface energy).
Referring now to the rigid body translations which exist in materials with “hard” atoms, it
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