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The relation between the spacing of the
dislocations and the tilt- or twist angle in the special cases given was simple
enough - but what about arbitrary small angle grain boundaries with twist and
tilt components? What kind of dislocation structure and what geometry should be
expected? |
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As we have seen, the detailed structure of the network can be
quite complicated and depends on materials parameters like stacking fault
energies. We can not expect to have a simple formula giving us the
answers. |
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The relation giving the distance between dislocations in a boundary and the
orientation relationship for arbitrary low-angle orientations (meaning that the
two rotation angles needed for a general description are both small, lets say
£ 10o - 15o) was
first given by Frank.
It is Franks formula referred to
before. |
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Franks formula is derived in the advanced
section, here we only give the result. The low-angle grain boundary shall be
described by: |
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Its dislocation network consisting of dislocations with
Burgers vectors b. |
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An arbitrary vector r contained in the
plane of the boundary. |
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A (small) angle α around an
arbitrary axis described by the (unit) vector l (then one
angle is enough) that describes the orientation relationship between the
grains. We may then represent the rotation by a polar vector R =
α · l |
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Franks formula then is: |
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with B = sum of all the specific Burgers
vectors bi cut by r ; i.e.
B = Σi
bi. |
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Since the formula is formally applicable to any boundary, but
does not make much sense for large angles α
(can you see why?) we only consider low-angle boundaries. Then we can replace
sin α/2 approximately by α/2 and obtain the simplified version |
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Let's illustrate this: |
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Shown are two vectors r1 and
r2 contained in a boundary plane with an arbitrarily
chosen dislocation network consisting of two types of dislocations having
Burgers vectors b1 and
b2. |
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Franks formula ascertains that (r ×
l) · α equals the sum of the Burgers
vectors encountered by r, i.e. B =
2b2 + 3b1 for
r1, and B =
3b1 for r1 in the
picture above. |
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This is a major achievement, but not overly helpful when you
try to find out the geometry of the network for some arbitrary boundary,
because their is no simple and unique way of decomposing a sum of Burgers
vectors into its individual parts. |
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This "simple" formula, however, contains
the special cases that we have considered before, and leaves enough room for
complications. It does not, however, say anything about preferred planes or network geometries. For this one
needs the full power of Bollmanns
O-lattice theory. |
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Franks formula is not applicable
to large angle grain boundaries because the distance between the dislocations
would become so small as to be meaningless. |
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In this case the grain boundary may be viewed as the
(dislocation free) "low Σ" boundary
closest to the actual orientation with a superimposed low-angle grain boundary
formed by dislocations in the corresponding DSC
lattice. |
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Franks formula then can be used for the low angle part and
will give the correct over-all Burgers vector count. |
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The precise geometry of the network, however, can
become hopelessly complicated because all the additional features, e.g.
extrinsic dislocations and steps, are not only still present but become more
complicated, too. |
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In addition, some special perversions may evolve, e.g. the
splitting of DSC lattice
dislocations into partial dislocations in the DSC lattice, producing
stacking faults in the DSC lattice! |
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And we are implicitly talking grain boundaries between cubic
crystals! In less symmetric crystals everything is even more complicated - it
is time then to study O-lattice theory! |
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One last feature should be mentioned that now can
be understood: The reactions of lattice dislocations that move into grain
boundaries. So
far, a grain boundary was just seen as an internal surface on which a
dislocation can somehow end. |
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Now we know better: It simply decomposes into the intrinsic
(DSC-lattice) grain boundary dislocations present. This is quite
satisfying because the logical problems encountered when thinking more in more
detail about how a dislocation "just ends" on a grain boundary. It is
also what one sees if looking closely, an example is
shown in the
illustration. |
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© H. Föll