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Calculating the critical thickness of a layer with
lattice constant a1 on top of a substrate with lattice
constant a2 can become rather involved, if all
components contributing to the elastic energy are taken into account. |
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In particular, you may want to consider the anisotropy of the
situation, the effect of a finite thickness of the top layer, the real geometry
with respect to the dislocations (their line energy depends on this and that,
and they may be split into partial dislocations). |
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Then, after arriving at a formula, you may chose to make all
kinds of approximations. |
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In the backbone part of the script we had a simple formula
(taken from a paper of the very well known scientist Sir Peter
Hirsch) which you can
find in the
link
(together with some comments): |
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| dcrit = |
= |
b
8 · p · f ·
(1 + n) |
· ln |
e · dcrit
r0 |
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With b = Burgers
vector of the dislocations; usually somewhat smaller than a lattice
constant, f = misfit parameter = (a1 -
a2)/a1, n = Poisson ration » 0,4, e = really e = base of
natural logarithms r0 = inner core radius of the
dislocation; again in the order of lattice constant. |
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Lets look at some other approaches |
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A formula taking into account most everything going back to J.
W. Matthews and
A.E. Blakeslee
(1974) , who pioneered this field of research, is |
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| dcrit |
= |
b · (1 – n) ·
cos2Q
8 · p · (1 +
n) · f · cosl |
· |
æ
ç
è |
ln |
æ
è |
dcrit
b |
ö
ø |
+ 1 |
ö
÷
ø |
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with Q = angle between the
dislocation line and its Burgers vector, l =
angle between the slip direction and that line in the interface plane that is
normal to the line of intersection between the slip plane and the
interface. |
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For simple systems (Q =
90o and l = 0o), we
have |
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| dcrit |
= |
b
8 · p · f ·
(1 + n) |
· |
æ
ç
è |
ln |
æ
è |
dcrit
r0 |
ö
ø |
+ 1 |
ö
÷
ø |
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And that is Sir Peters equation if you insert ln(e) for
the 1 in the ln term. |
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While Sir Peter used the simple approximation |
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the comparison with the (computer-generated) correct
functional dependence suggests |
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which is a bit different! |
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A plot of the full formula and the approximation
looks like this: |
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Similar curves are contained in the books of
Mayer and
Lau or Tu,
Mayer and Feldmann; they supposedly use the same equation but show rather different results. |
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Well, somewhere should be a mistake (maybe I made one?). In
any case, it nicely demonstrates the point made in the backbone section:
Do not blindly believe a
theory. In case of doubt, try it out. |
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© H. Föll
To index
8.1.1 Phase Boundaries