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On the density of coincidence sites in grain boundaries
Scripta
METALLURGICA
V o l . 8, Printed
pp. 1 1 9 7 - 1 2 0 0 , in t h e U n i t e d
ON THE DENSITY OF COINCIDENCE
1974 States
Pergamon
Inc
Press,
SITES IN GRAIN BOUNDARIES
D.A. Smith Department of Metallurgy
and Science of Materials
University of Oxford, U.K.
(Received
An implicit relationship coincidence
August
26,
1974)
is often assumed between the vol,~me reciprocal density of
sites Z and the energy of high angle grain boundaries.
some support from the results of Chaudhari orientations
and Matthews
This view receives
(i) who found that coincidence
occurred frequently in twist boundaries of MgO and CdO and that, with
one exception when like charges approached too closely, of any particular
orientation was inversely proportional
the internal energy of symmetrical
the frequency of the occurrence to E.
Computer calculations
tilt high angle grain boundaries
of
(2) i.e. twins in
f.c.c, materials have shown marked energy minima at the E = 3 and Z = 11 coincidence orientations
but not for ~ = 5.
Since boundaries
are planar defects insofar as their
energy does depend on some function of the density of shared sites or, more generally but numerically
equivalently,
to the size of some repeating unit, it seems more plausible
to
use an area measure of the size of the repeating unit, o, rather than a volume measure such as Z; ~ is the area per coincidence point or of the repeating unit in the boundary plane
(3).
Z and ~ are not simply related owing to the variation of the symmetry and
axial ratios of the csl cell for different axes of rotation (except ~ = 39, 50.13"/<123>) direction
[hkl] so that Z
(4).
All cels for Z < 50
(5) can be regarded as twins, i.e. rotations of ~ about a
~[h 2 + k 2 + 12 ] where a = 1 if h 2 + k 2 + 12 is odd end
= 2 if h 2 + k 2 + 12 is even.
All atoms in the (hkl) plane are coincident.
This means
that in these cases the csl cell has as base a rectangular mesh which is the unit mesh in (hkl). b.c.c,
The area per coincidence
and 4 for f.c.c,
atom is thus
(h2 + k 2 + 12)%/B where B is 2 for
lattices.
Fig 1 shows the relationship between o and ~ for f.c.c, lowest o value for any given Z has been used in constructing is not monotonically with E reflecting
related to Z in general,
the geometrical
lattices and Z < 50; the figs 1 and 2.
Although O
for any particular rotation axis o increases
similarity of the csl cells for any given rotation axis
e.g. for <001> axes the csl cells are tetragonal
1197
and for axes orthorhcmbic.
1198
DENSITY
OF COINCIDENCE
SITES
Vol.
8,
No.
Fig 2 shows the relatior~ship between o end Z for b.c.c, lattices end Z < 50. is a monotonic relation between o end ~.
I0
There
The importance of indexing correctly the atom
planes is exemplified by the strong difference between o and Z for the f.c.c, and b.c.c. cases, and the absence of a monotonic relation between o end Z in the f.c.c, case in which a43 < u 7 for example. Csl geometry is evidently not the sole determinent of grain boundary energy (cf (a) the variation of stacking fault energy from one f.c.c, metal to another even though the geoNetry is identical end there exists an excellent "good fit" structure or (b) the facetting of twins into low density csl planes)
(6).
It is interesting to note the
correlation between fig 1 and (i) the calculated and measured internal energy of coincidence boundaries
(2) end (b) the results of the ball experiment of Hermann et al,
(7).
In
this last experiment single crystal spheres of copper are annealed on a flat oriented single crystal copper surface end a texture is observed to develop from an originally rendom set of orientations.
Again the Z = ii orientation occurs frequently.
On the basis
of the calculation underlying fig I it would be expected that p(Z = 3) • p(Z = ii) > p(Z = 19) • p(Z = 5) • p(Z = 27) etc where p(Z = J) means the probability of occurrence of
theZ=
J orientation.
In addition at 16°6'/110, 17°52'/110 end 20°2'/110 there exist
respectively the Z = 33, 83 and 51 csl orientations end Hermann et al, observed that p(Z = 83) • p(Z = 33) end p(Z = 51) and again this may be understood since 083 is the smallest in this set.
The whole of the above discussion rests on the hypothesis that the
boundary plane between each copper ball end the substrate had rotated into the twin orientation.
It might be expected on the basis of f i g 2 that the ball experiment would
show quite different results for a b.c.c, metal end that p(ZJ) would be proportional to Z (neglecting degeneracy).
Acknowledgments The author is grateful to Professor J.W. Christian and to Dr R.C. Pond for ~Iseful discussions, to the Armourers end Brasiers' Company for a research fellowship and to Professor P.B. Hirsch, FRS for inspiration and facilities.
References i.
P. Chaudhari end J.W. Matthews, J.App1.Phys. 42, 3063 (1971).
2.
G. Hasson, J.¥. Boos, I. Herbeuval, M. Biscondi end C. Goux , Surf.Sol. 3~I, 115 (1972).
3.
B. Chalmers and H. Gleiter, Prog. Materials Science 16 (1972).
4.
R.C. Pondt Cen.Met.Quarterly,
in press (1974).
5.
M.A. Fortes, Proc. Universidade de Louren~o Marques, 7
6.
R.C. Pond and D.A. Smith, Can. Met, Quarterly, in press (1974~.
7.
G. Hermann, G. Baro end H. Gleiter, Fourth Bolton Lending Conference, Grain Boundaries in Engineering Materials
(1974).
(1972).
Vol.
8, No.
i0
DENSITY
,
3
OF C O I N C I D E N C E
SITES
1199
I
11 13 15 17 19 21 23 2S 27 29 31 33 35 37 39 1,1 1,3 /,5 1,7
Fig 1 is a plot of the minimum area in the twin plane per coincidence site versus Z with Z < 50 for the f.c.c, lattice (taking a lattice parameter of unity and with the ordinate in units of ~2)
BCC
0
3 S 7 9
13 'IS 17 19 21 23 25 27 ~ 31 33 3S 37 3S ~1 J,1 1,5 1,7 I,!
T. Fig 2 Is a plot
o f t h e minimum a r e a i n t h e t w i n p l a n e p e r c o i n c i d e n c e
site
versus ~ with ~ <_ 50 for the b.c.c, lattice (taking a lattice parameter of unity and with the ordinate in units of ~2)
METALLURGICA
V o l . 8, Printed
pp. 1 1 9 7 - 1 2 0 0 , in t h e U n i t e d
ON THE DENSITY OF COINCIDENCE
1974 States
Pergamon
Inc
Press,
SITES IN GRAIN BOUNDARIES
D.A. Smith Department of Metallurgy
and Science of Materials
University of Oxford, U.K.
(Received
An implicit relationship coincidence
August
26,
1974)
is often assumed between the vol,~me reciprocal density of
sites Z and the energy of high angle grain boundaries.
some support from the results of Chaudhari orientations
and Matthews
This view receives
(i) who found that coincidence
occurred frequently in
one exception when like charges approached too closely, of any particular
orientation was inversely proportional
the internal energy of symmetrical
the frequency of the occurrence to E.
Computer calculations
tilt high angle grain boundaries
of
(2) i.e. twins in
f.c.c, materials have shown marked energy minima at the E = 3 and Z = 11 coincidence orientations
but not for ~ = 5.
Since boundaries
are planar defects insofar as their
energy does depend on some function of the density of shared sites or, more generally but numerically
equivalently,
to the size of some repeating unit, it seems more plausible
to
use an area measure of the size of the repeating unit, o, rather than a volume measure such as Z; ~ is the area per coincidence point or of the repeating unit in the boundary plane
(3).
Z and ~ are not simply related owing to the variation of the symmetry and
axial ratios of the csl cell for different axes of rotation (except ~ = 39, 50.13"/<123>) direction
[hkl] so that Z
(4).
All cels for Z < 50
(5) can be regarded as twins, i.e. rotations of ~ about a
~[h 2 + k 2 + 12 ] where a = 1 if h 2 + k 2 + 12 is odd end
= 2 if h 2 + k 2 + 12 is even.
All atoms in the (hkl) plane are coincident.
This means
that in these cases the csl cell has as base a rectangular mesh which is the unit mesh in (hkl). b.c.c,
The area per coincidence
and 4 for f.c.c,
atom is thus
(h2 + k 2 + 12)%/B where B is 2 for
lattices.
Fig 1 shows the relationship between o and ~ for f.c.c, lowest o value for any given Z has been used in constructing is not monotonically with E reflecting
related to Z in general,
the geometrical
lattices and Z < 50; the figs 1 and 2.
Although O
for any particular rotation axis o increases
similarity of the csl cells for any given rotation axis
e.g. for <001> axes the csl cells are tetragonal
1197
and for
1198
DENSITY
OF COINCIDENCE
SITES
Vol.
8,
No.
Fig 2 shows the relatior~ship between o end Z for b.c.c, lattices end Z < 50. is a monotonic relation between o end ~.
I0
There
The importance of indexing correctly the atom
planes is exemplified by the strong difference between o and Z for the f.c.c, and b.c.c. cases, and the absence of a monotonic relation between o end Z in the f.c.c, case in which a43 < u 7 for example. Csl geometry is evidently not the sole determinent of grain boundary energy (cf (a) the variation of stacking fault energy from one f.c.c, metal to another even though the geoNetry is identical end there exists an excellent "good fit" structure or (b) the facetting of twins into low density csl planes)
(6).
It is interesting to note the
correlation between fig 1 and (i) the calculated and measured internal energy of coincidence boundaries
(2) end (b) the results of the ball experiment of Hermann et al,
(7).
In
this last experiment single crystal spheres of copper are annealed on a flat oriented single crystal copper surface end a texture is observed to develop from an originally rendom set of orientations.
Again the Z = ii orientation occurs frequently.
On the basis
of the calculation underlying fig I it would be expected that p(Z = 3) • p(Z = ii) > p(Z = 19) • p(Z = 5) • p(Z = 27) etc where p(Z = J) means the probability of occurrence of
theZ=
J orientation.
In addition at 16°6'/110, 17°52'/110 end 20°2'/110 there exist
respectively the Z = 33, 83 and 51 csl orientations end Hermann et al, observed that p(Z = 83) • p(Z = 33) end p(Z = 51) and again this may be understood since 083 is the smallest in this set.
The whole of the above discussion rests on the hypothesis that the
boundary plane between each copper ball end the substrate had rotated into the twin orientation.
It might be expected on the basis of f i g 2 that the ball experiment would
show quite different results for a b.c.c, metal end that p(ZJ) would be proportional to Z (neglecting degeneracy).
Acknowledgments The author is grateful to Professor J.W. Christian and to Dr R.C. Pond for ~Iseful discussions, to the Armourers end Brasiers' Company for a research fellowship and to Professor P.B. Hirsch, FRS for inspiration and facilities.
References i.
P. Chaudhari end J.W. Matthews, J.App1.Phys. 42, 3063 (1971).
2.
G. Hasson, J.¥. Boos, I. Herbeuval, M. Biscondi end C. Goux , Surf.Sol. 3~I, 115 (1972).
3.
B. Chalmers and H. Gleiter, Prog. Materials Science 16 (1972).
4.
R.C. Pondt Cen.Met.Quarterly,
in press (1974).
5.
M.A. Fortes, Proc. Universidade de Louren~o Marques, 7
6.
R.C. Pond and D.A. Smith, Can. Met, Quarterly, in press (1974~.
7.
G. Hermann, G. Baro end H. Gleiter, Fourth Bolton Lending Conference, Grain Boundaries in Engineering Materials
(1974).
(1972).
Vol.
8, No.
i0
DENSITY
,
3
OF C O I N C I D E N C E
SITES
1199
I
11 13 15 17 19 21 23 2S 27 29 31 33 35 37 39 1,1 1,3 /,5 1,7
Fig 1 is a plot of the minimum area in the twin plane per coincidence site versus Z with Z < 50 for the f.c.c, lattice (taking a lattice parameter of unity and with the ordinate in units of ~2)
BCC
0
3 S 7 9
13 'IS 17 19 21 23 25 27 ~ 31 33 3S 37 3S ~1 J,1 1,5 1,7 I,!
T. Fig 2 Is a plot
o f t h e minimum a r e a i n t h e t w i n p l a n e p e r c o i n c i d e n c e
site
versus ~ with ~ <_ 50 for the b.c.c, lattice (taking a lattice parameter of unity and with the ordinate in units of ~2)