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Relative energies of grain boundaries near a coincidence orientation relationship in high-purity lead
RELATIVE
ENERGIES ORIENTATION
OF
GRAIN
BOUNDARIES
RELATIONSHIP G.
DIMOUt
and
IN K.
NEAR
A
HIGH-PURITY
COINCIDENCE LEAD*
‘I’. AUST:
The effect of deviations from ideal coincidence orientation relationship on the relative energy of a coincidence grain boundary was studiecl in tricrystals of high purity lead. Loner relative energies were with a maximum energv decrease observed within &2’ of the ideal 36.9” (100) comcidence relationship. of about 30 per cent. The present results. combined with previous energy data for different high-density coincidence boundaries in high purity lead, are shown to be related to the calculated number of shared These results provicle esperimental etom sites per unit area (in lattice units) of the grain bounclary. support for the boundary coincidence and “relaxed” coincidence moclels. ESERGIES
RELATIVES DES JOISTS DE GR.iISS PROCHES D’USE ORIESTATIOS DE COIXCIDESCE D-ASS LE PLOJIB DE HAUTE PURETE L’influence des &carts par rapport 8. la relation d’orientation cle coincidence id&ale sur l’energie relative dun joint de grains de coincidence a 6th BtudiCe dans cles tricristaus de plomb de haute purete. Des energies relatives plus faibles ont et& observeea B mains de $2’ de la relation de coincidence id&de de 36,9> (100) avec une diminution maximale d’bnergie de 30°0 environ, Lee auteurs montrent que ces reeultats, combines WY valeurs antbrieures de l’energie pour differents joints cle forte clenaite de coincid. ence dans le plomb de haute purete, sont relies au nombre cle sites atomiques commune par unite de surface du joint cle grains. Ces resultats fournissent uu support experimental pour les mocleles de coi’ncidence et de coincidence “relax&e.” RELATIVE
ESERGIES VOS KORSGRESZES IS DER S.iHE EINER KOISZIDESZ. ORIESTIERCSGSBEZIEHCSG IS HOCHREISEM BLEI An hochreinen Bleitrikristallen murde der Eintlul3 einer Abweichung von der idealen KoinzidenzOrientierungsbeziehung auf die relative Energie einer Koinzidenzkorngranze untersucht. Innerhalb 52” der idealen 36.9” (lOO)-Koinzidenzbsziehun g wurden niedrigero relative Energien bsobachtet mit man die vorliegenden Ergebnisse mit einer masimalen Energieabnahme von etwva 30“&. Kombiniert friiheren Energiemessungen fur verschiedene Koinzidenzgrenzen hoher Dichte in hochreinem Blei. so findet man eine _1bhi%ngigkeit von der berechneten Zahl geteilter ;\tomlagen pro Fliicheneinheit (in Diese Ergebnisse sind eine eaperimentelle Bestiitigung fur Gitterparsmtereinheiten) der Korngrenze. das Korngrenzenkoinziclenzmoclsll und clas Model1 .relasierter” Iioinziclenz.
from the ideal orientation would eliminate lattice coincidence. Bishop and Chalmers(g) suggested a boundary coincidence model in which atoms common to both lat.tices are at the boundary, e.g. Fig. 1 (b). Therefore, although in the Kronberg-Wilson model deviations from the ideal coincidence angles would result in t,he destruction of lattice coincidence, in the BishopChalmers model the common boundary atoms could still be maintained at the expense of a certain amount of cryst,al distortion. Chalmers and Gleitero”) have recently presented a modified or “relaxed” coincidence modelwhich appears to be less stringently dependent on the orientation relationship. This model can best be illustrated by the use of Fig. 1 (c). In Fig. 1 (b), the boundary coincidence model as proposed by Bishop and Chalmers and having a periodicity p is shown. Figure l(c) shows the same coincidence orientation relationship but with a relative displacement of the two crystals along the boundary plane. In this latter case, therearenoshared atoms and yet, according to computer calculations of Weins et c~Z.,(~nthe free energy- of the grain boundary is lowered. The important criterion for the existence of coincidence boundaries according to Chalmers and Gleiter, therfore, is not the sharing of atoms but the presence of a skuctural periodicit- whose structure
The energy of a grain boundary has been shown to be dependent on the angle of misorientation (0) between the two crystal lattices when the angle 6 is small.(i) For large-angle boundaries, the energy is independent of misorientation except for certain specific misorientations at which a drop or cusp in the energy versus 8 curve has been observed.@+ The large-angle boundaries having a lower energy correspond to interfaces with a coincidence relationship, e.g. coherent and incoherent twin boundaries, and certain types of grain boundaries, e.g. with 6 = 38” (111) in zone refined lead.‘“) A lattice coincidence model was first proposed by Kronberg and V’ilson(s) in 1949. This model suggests that with certain misorientations, atoms are common to the latiice of both grains adjacent to the boundary. For example, a 36.9” (100) coincidence grain boundary shovvn in Fig. l(a) has one atom in five common to both lattices. Such a model implies that any special properties resulting from the presence of coincidence should be found only at. precise and narrow orientation relationships since even the slightest departure * Received June 4, 1953. t G. Dimou, presently with Camron Ltd., R. & D. Resdale, Ontario. $ K. T. Aust, Department of Metallurgy and Materials Science, Materials Research Centre, University of Toronto Canada. ACTA
JIETALLURGICA,
VOL.
22, J_ASX_7ARY
1974
1i
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7 per cent less energy for 31’ or IS” (111) boundaries in zone-refined lead. Hasson et aZ.ci) have measured the energy of boundaries in aluminum of 99.998 per cent purity and find local minima at some of the angles predicted by the coincidence model. Chaudhari and Xatthews(5*6) have shown that twist boundaries formed on (100) planes between smoke particles of XgO and Cd0 show minima at angles predicted by the coincidence model. The present no& was carried out to obtain further experiment’al data concerning the energy of a coincidence grain boundary and the effect of deviations from ideal coincidence relationship on this energy. Such data should then provide a critical test of the various coincidence or relaxed coincidence models(8-10i proposed for grain boundary structures. EXPERIMENTAL
FIG. 1. Lattice coincidence model of & 36.9(100) tilt (Fig. la). A 36.9” (100) tilt boundary showing shared Iattice sites with a periodioity of p (Fig. lb). The same tilt boundary in which the two lattices have been displaced to remove coincidence; the structural periodicity is again p (Fig. lc). boundery
is equivalent to the structure present when the atoms are being shared. The period for any given coincidence orient,at.ion relationship would be the same for both the model with shared atom sites and the one without. In the example given in Fig. 1 for a 36.9” (100) tilt boundary, the period for both models is p. Due to the absence of coincidence atoms in the model of Fig. (lc), the extent of coincidence cannot be specified as the density of shared sites but rather as the number of shared sites that can exist, in the plane of the boundary per unit area of that boundary. It follows from this modified or relaxed coincidence model that deviations from ideal coincidence relationships would not be as cat,astrophic to the special properties attributed to the lattice coincidence model as first thought. Aside from energy studies of coherent and noncoherent kin boundaries, very few studies have provided direct experimental evidence for a drop in orientation relationships. energy at coincidence Rutter and _&ust(2)noted a 15 per cent lower energy for a 38” (111) grain boundary and possibly about
Single crystals and tricrystal specimens were prepared from the melt using lead of 99.9999% purity, with and without solute additions of 2 atom ppm of Sn, Ca or Au. The single-crystals were grown in a horizontal furnace under purified argon. Each single crystal, having dimensions of 3f16 in. x 3/16 in. x 6 in. was grown at 0.5 in/hr. at approximately330”C from the same seed crystal having a cube orientation. Each crystal was acid cut into three equal lengths for subsequent use in the growth of a t,ricrystal. The crystals were chemically polished in a solution of 50 % acetic acid and 50 ‘A hydrogen peroxide, washed with water and rinsed with alcohol before drying. The method described by Gleiter@i for growing tricrystals was used here, in which the junction of three properly oriented seed crystals is melted and then slowly cooled into a tricrystal \vith controlled The seed crystals were secured in the orientations. three single crystal holders of the tricrystalgrowing apparatus. Each crystal wasX-rayedand,ifneceseary,readjusted before positioning in t,he tricrystal apparatus. The single crystal holders were designed to be removable from the rest of the apparatus and capable of allowing the X-raying of an already positioned single crystal. The complete tricrystal growing apparatus was capable of adjusting each crysta1 individually by rotating wit,h respect to its major axis, tilting along a vertical axis, moving each crystal forward and back along its major axis and changing the orientation between crystals on the horizontal growing plane. Each tricrystal was grown in a vacuum of lob6 torr by melting the junction of t.he three seed crystals and t,hen allowing t,he melt to cool slowly. Due to the geometry of the tricrgstal graphite boat and the relatively large mass of the single crystal holders,
DIJiOTZ
directional
ACST:
ATD
ESERGIE8
cooling n-as easily achieved
of the Jingle crystals. place at about directions,
OF
GR_1IIS
along the ases
Alsol cooling of the melt took
an equal rate along each of the three
thus forming
a tricrystal
of equally
cr,vstala and having a grain boundary
junction
large at the
centre of the sample.
The rate of growth
crystals was controlled
by slowly decreasing the current
through
the furnace
one minute
during
an interval
to more than ten minutes
of the triof less than
from the time
of complete melting until complete tricryetalformation. In order to avoid the growth the bottom slightly
surface
superheat
thickness The
the
crystallographic in Fig.
were removed acetic
acid
surfaces
the
The
pinnedt wing
50%
t\vo
as shown
a spark
to prevent subsequent an
annealing.
ture
of
Each
ahoxed
grain boundary After alcohol sample
parbially
ghost motion
annealing, etched
and the polishing mounted
bath
holder
and
the
the
and scanning
junction
were
electron
microscopy
cations up to about 4000 x . d typical equilibrium
grain
boundary
lead is given in Fig.
3.
three using
at magnifi-
example
angles
Variations
of
measured
for high
of the purity
in the boundary
angle measurements
were as much as 4’ which gave
rise to a maximum
error
in the relative
boundary
energy of &S per cent. RESULTS
AND
DISCUSSION
The relative energies of t,he boundaries were determined according to the following relationship:“’ ^JAB 7
sin x where
yan,
ysc
=-
*/lx
sin p
=-
YBD
and yBD are the
unit area of the boundaries
AB,
(1)
sin b free energies
and 1, B and 6 are the equilibrium
holes
as shown in Fig.
cidence
boundary
non-coincidence
and
2. BC
boundaries,
Taking and
XB
BD
boundary as the coin-
as high-angle
yBc = yBrJ = constant
during glass
Ilost a result of of
cycle. was polished
immersions
and
in ethyl
The sample was
electron
orientation
microscope
of each
of the
100
110 C
L
ii0
w
001
100
010
100
FIG. 2. Crvstallogaphic
orientation
of
the
tricrystals.
per
BC and BD respec-
furnace. as
solution.
on a scanning
at
annealing
of the
The dihedral
angles
and there was no evidence
alternate
optical
8, was + 1’.
angles
after
analysis
determination
n-as sealed
evacuated
boundaries
each tricrystal by
boundary
boundaries
back-reflection
the
150 hr at a tempera-
beyond the 150 hr annealing
.slightl;v then
and
for at least
motion
in
39
tively
was necessary
tricrystal
by S-rav error
of misorientation,
grain
COISCIDESCE
were
of the grain boundaries
323 _C 1°C in a salt
samples
1 mm dia.
pinning
by grain boundary
argon-flushed
motion
The
angle
Both
boundaries
2, with
determined
cutting
smoothed
SEdR
maximum
of 50 ‘A glacial
peroxide.
high-angle
the disappearance
and annealed
by acid
grains The
the
the seed crystals
were chemically
machine.
tube
maintain
(100) plane and chemically
in Fig.
from the specimen
to
of the tricrystals
growth,
hydrogen
to the horizontal
polished.
also
tricrystals
of each tricrystal
parallel
and
strip in a solution
and
from
at less than about 4 mm.
After
from
crystals
it was necessary
orientation
2.
with a polyethylene
in
boat
of the tricrystal
is shown
of stray
of the boat
BOUSDXRIES
FIG. 3. Equilibrium grain boundary angles in highpurity lead. 0 = 3i.5’. _,_yR-= 0.71. Magnification: X hbW.
Or
13 = 6 = (180 -
z/2).
eyuation
(1)
becomes:
(‘1 The validity of ecluation (1) is based on the assumption that the torque terms of the boundaries AB, BC and BD are negligible. The torque forces on the coincidence boundary AB are zero because the boundary is symmetric; the energy of the noncoincidence large-angle boundaries BC and BD is The assumptions are independent of inclination. threfore justifiable and the relative energy yu of the coincidence bounclary in its final fortn is; sin a
where yuc was arbitrarly set equal to unity. The results of the experiments are summarized in Table 1. and in a plot of in vs the deviation from the ideal coincidence value, 36.9” (loo), shon-n in Fig. 4. It is apparent that the lowering of energy of the coincidence boundary is present within 2’ of the ideal co~ne~den~~ m~sorientation relationship. In t,he vicinity of t,he coincidence boundary, the relative energy- shows a consistent drop of BS much as 25-30 per cent Unfortunately, no data for the ideal 36.9” (100) misorientation lvere obtained and, therefore, the-exact extent of the energy decrease could not be determined. However, since the 19 (100) values closest to the ideal value were within & 0.6” (Fig. 4) and the error inroIvec1 in the 8 determination is as high as &l’, the observed 25-30 per cent drop in ~a may represent the actual extent of the energy cllop for the ideal coincidence relationship. It should be noted in Table 1 and Fig. 4 that the relative energy values for specimens having solute additions of tin, go!d or calcium were not very different from those of pure lead. It woulcl appear that the
FIG. 4. Relat~ive grain boundary energy (;J~) vs. deviation (&l) from ideal coincidence angle of 36.9’ t 1~1).
solute additions (2 atom ppm) were not sufficient to cause any measurable differences, either for a solute with relatively high solid solubi1it.v (Sk1 in Pb) or for solutes with low solid solubility (-Au or Ca in Pb). Gleiter(*) observed a 32 per cent lower energy for a high density coincidence high angle boundary in high-purity lead at 3%.5”C. However. in lead containing about SO atom ppm of copper this same coincidence boundary gave only a 12 per cent lower energy than the non-coincidence boundary. A summary of previous and present data for relative energies of different coincidence boundaries in highpurity zone-refined lead is shown in Table 1. It is evident, that the present results are con&tent n-ith the previous studies for different high den&y coincidence boundaries. The data in Table 2 are plotted in Fig. 5 and indicate a good correlation between relative energy of coincidence boundaries and the calculated number of shared atom sites per unit area [in lattice
DIJIOU T.LBLE
hXD
1.
.AUST:
ESERGIES
OF
GRAIS
BOUSDARIES
SEdR
COISCIDESCE
31
The relative energy of coincidence boundaries and their density of shared atom sites in high purity lead
Coincidence
Density of shared atom sites
Relative energy* 7s
60’ (I 11) (coherent twin boundary) 70.5’ (110) (incoherent twin boundary) 36.9” (100)
9*3o-’ (101 l.lj,-2 (13, 0.91a-* ‘lo’
35” (111) 2s” or 34” (111)
0*33a-2 113) O,*To-’ 113,
0.05 f O.OlA’i?’ 0.65 + 0.05c’j 0.71 f 0.06 (present work) 0.55 + 0.05’Z 0.93 + O.O5l?’
* Specimens
boundary orientation relationship
equilibrated
in temperature
range of 31%316.5”C.
‘.
‘. \
Fm. 5. Relative
energies of coincidence boundaries atom sites in high purity lead.
units where “a” is the lattice constant) of the boundary.oOJs) A similar good correlation is obtained between relative boundary energy and size of the periodic unit, in terms of the dimension of the unit lattice cell, for these same boundaries, as suggested by the “relaxed” coincidence mode1.(1c*r4) These results are believed to provide support for the boundary coincidence and “relaxed” coincidence models advanced by Bishop and Chalmers(s) and Chalmers and Gleiter,“O) respectively. Gleite?) found that the energy of a 3’7” (100) tilt boundary in zone-refined lead is 10 per cent lower than the energy of a 18.6’ (100) tilt boundary. Utilizing the results of the present work for the relative energy of a 37” (100) boundary, the energy of the 18.5 (100) boundary is about 20 per cent lower than the energy of a high-angle non-coincidence boundary. This result is consistent with t.he plot of Fig. 5, since the 18.5” (100) boundary corresponds closely to an ideal coincidence relationship (19’ (100)) with a calculated densit.y of shared atom sites of 0.33 a--2.(10) The correlat,ion given in Fig. 5 may differ in nonmetallic materials with more complicated bonding.(r5)
Rolling
and ‘1 Winqard (12) ‘6
vs. density
of shared
For example, energy calculations of twist boundaries in MgOo5) indicate that a 28’ (100) coincidence boundary has a positive contribution to energy from an electrostatic term and therefore has a shallow energy minima. Also, it should be emphasized that the relationship found in Fig. 6 for zone-refined lead would change drastically if strongly segregating impurities are present. For example, Gleiter@) has found that the relative energy value of 0.68 for the 70.5’ (110) coincidence boundary increases to approximately 0.88 with the addition of about 50 ppm by weight of copper to the lead. This is attributed to a stronger segregation of copper atoms at non-coincidence boundaries than at coincidence boundaries, thereby decreasing the energy of a non-coincidence boundary (yno) more rapidly than the energy of a coincidence boundary (yas) i.e. increasing the ratio of YABIYBC. CONCLUSIONS
1. Lower relative energy values were detected within f 2’ of the ideal 36.9” (100) coincidence boundary relationship in high-purity lead. The extent of this decrease in energy corresponds to
32
dCT_1
METILLCRGIC_~,
about 30 per cent that for the energies of non-coincidence grain boundaries. 2. Solute additions of 2 atom ppm of gold, calcium or tin appeared to hare little effect on relative boundary energies in high-purity lead. 3. A good correlation is found between the relative energy of coincidence grain boundaries and the density of shared atom sites per unit. area of the boundary in high-purity lead. 4. The results of this study are in agreement with the boundary coincidence model of Bishop and Chalmers and t.he “relaxed” coincidence model of Chalmers and Gleiter. ACKNOWLEDGElMENTS
The authors would like to acknowledge the very helpful assistance given by Dr. H. Gleiter through both informal and written discussions. The authors would also like to thank Mr. H. Sang for useful suggestions and Mr. D. Abdula for the construction of the crystal growing apparatus. Financial support n-as provided by the National Research Council of Canada and the Defence Research Board of Canada.
VOL.
22,
1974 REFERENCES
1. Ii. T. _%USTand B. CH_%LMERS, Xelal Inle+ces, p. 153. -Am. Sot. Metals. Cleveland. (19?iP). 2. J. W. RUTTER and I<. T. XT%, refeked to by K. T. Aust in Surfaces and Interfaces, I Chemical and Physical Chmactetitics, edited bv BLXKE, REID and WEISS. . D. . 435. Syracuse University P&s, (1967). 3. K. T. .~CST and B. CH_UYERS, Xet. Trans. 1, 1095 (19iO). 4. H. G~EITER, &ta Met. 18, 117 (1970). 3. P. CH.~~DHARIand J. W. X%TTHEWS, Appl. Phya. Lett. 17, 115 (1970). 6. P. C!E_~~DH~RI and J. W. X~TTHE~S, J. appl. Phys. 42, 3063 (1971). 7. G. Hassos, J. Y. Boos, I. HERBEXX_~L,>I. BISCOXDIand C. GOIX, Surface Sci. 31, 115 (1972). S. JI. L. KROYBERG and F. H. WILSOX, Trans. .-I.Z.X.E., 185, 501 (1949). 9. G. H. BISHOP and B. CH~~UERS, Scripta -Ilet. 2, 133 (1968). 10. B. CHALMERS and H. GLEITER, Phil. Xag. 23, 121 (1971). 11. 31. J. WEISS, H. GLEITER and B. CHALMERS,J. appl. Phys. 42, 2639 (Igil). 17. G. F. BOLLIXG and W. C. WISECARD, J. Inst. Net& 86, 492 (1958). 13. H. GLEITER, private communication (1972). 14. H. GLEITER, unpublished; quoted in H. GLEITER and B. CH~LJIERS,High-angle g&in boundaries, Progrew in Materials Science, Pergamon Press Vol. 16, pp. 24, S (1972). 15. P. CHAUDH~RIand H. CHARBXAV, Surface Sci. 31, 10-L (1972).
ENERGIES ORIENTATION
OF
GRAIN
BOUNDARIES
RELATIONSHIP G.
DIMOUt
and
IN K.
NEAR
A
HIGH-PURITY
COINCIDENCE LEAD*
‘I’. AUST:
The effect of deviations from ideal coincidence orientation relationship on the relative energy of a coincidence grain boundary was studiecl in tricrystals of high purity lead. Loner relative energies were with a maximum energv decrease observed within &2’ of the ideal 36.9” (100) comcidence relationship. of about 30 per cent. The present results. combined with previous energy data for different high-density coincidence boundaries in high purity lead, are shown to be related to the calculated number of shared These results provicle esperimental etom sites per unit area (in lattice units) of the grain bounclary. support for the boundary coincidence and “relaxed” coincidence moclels. ESERGIES
RELATIVES DES JOISTS DE GR.iISS PROCHES D’USE ORIESTATIOS DE COIXCIDESCE D-ASS LE PLOJIB DE HAUTE PURETE L’influence des &carts par rapport 8. la relation d’orientation cle coincidence id&ale sur l’energie relative dun joint de grains de coincidence a 6th BtudiCe dans cles tricristaus de plomb de haute purete. Des energies relatives plus faibles ont et& observeea B mains de $2’ de la relation de coincidence id&de de 36,9> (100) avec une diminution maximale d’bnergie de 30°0 environ, Lee auteurs montrent que ces reeultats, combines WY valeurs antbrieures de l’energie pour differents joints cle forte clenaite de coincid. ence dans le plomb de haute purete, sont relies au nombre cle sites atomiques commune par unite de surface du joint cle grains. Ces resultats fournissent uu support experimental pour les mocleles de coi’ncidence et de coincidence “relax&e.” RELATIVE
ESERGIES VOS KORSGRESZES IS DER S.iHE EINER KOISZIDESZ. ORIESTIERCSGSBEZIEHCSG IS HOCHREISEM BLEI An hochreinen Bleitrikristallen murde der Eintlul3 einer Abweichung von der idealen KoinzidenzOrientierungsbeziehung auf die relative Energie einer Koinzidenzkorngranze untersucht. Innerhalb 52” der idealen 36.9” (lOO)-Koinzidenzbsziehun g wurden niedrigero relative Energien bsobachtet mit man die vorliegenden Ergebnisse mit einer masimalen Energieabnahme von etwva 30“&. Kombiniert friiheren Energiemessungen fur verschiedene Koinzidenzgrenzen hoher Dichte in hochreinem Blei. so findet man eine _1bhi%ngigkeit von der berechneten Zahl geteilter ;\tomlagen pro Fliicheneinheit (in Diese Ergebnisse sind eine eaperimentelle Bestiitigung fur Gitterparsmtereinheiten) der Korngrenze. das Korngrenzenkoinziclenzmoclsll und clas Model1 .relasierter” Iioinziclenz.
from the ideal orientation would eliminate lattice coincidence. Bishop and Chalmers(g) suggested a boundary coincidence model in which atoms common to both lat.tices are at the boundary, e.g. Fig. 1 (b). Therefore, although in the Kronberg-Wilson model deviations from the ideal coincidence angles would result in t,he destruction of lattice coincidence, in the BishopChalmers model the common boundary atoms could still be maintained at the expense of a certain amount of cryst,al distortion. Chalmers and Gleitero”) have recently presented a modified or “relaxed” coincidence modelwhich appears to be less stringently dependent on the orientation relationship. This model can best be illustrated by the use of Fig. 1 (c). In Fig. 1 (b), the boundary coincidence model as proposed by Bishop and Chalmers and having a periodicity p is shown. Figure l(c) shows the same coincidence orientation relationship but with a relative displacement of the two crystals along the boundary plane. In this latter case, therearenoshared atoms and yet, according to computer calculations of Weins et c~Z.,(~nthe free energy- of the grain boundary is lowered. The important criterion for the existence of coincidence boundaries according to Chalmers and Gleiter, therfore, is not the sharing of atoms but the presence of a skuctural periodicit- whose structure
The energy of a grain boundary has been shown to be dependent on the angle of misorientation (0) between the two crystal lattices when the angle 6 is small.(i) For large-angle boundaries, the energy is independent of misorientation except for certain specific misorientations at which a drop or cusp in the energy versus 8 curve has been observed.@+ The large-angle boundaries having a lower energy correspond to interfaces with a coincidence relationship, e.g. coherent and incoherent twin boundaries, and certain types of grain boundaries, e.g. with 6 = 38” (111) in zone refined lead.‘“) A lattice coincidence model was first proposed by Kronberg and V’ilson(s) in 1949. This model suggests that with certain misorientations, atoms are common to the latiice of both grains adjacent to the boundary. For example, a 36.9” (100) coincidence grain boundary shovvn in Fig. l(a) has one atom in five common to both lattices. Such a model implies that any special properties resulting from the presence of coincidence should be found only at. precise and narrow orientation relationships since even the slightest departure * Received June 4, 1953. t G. Dimou, presently with Camron Ltd., R. & D. Resdale, Ontario. $ K. T. Aust, Department of Metallurgy and Materials Science, Materials Research Centre, University of Toronto Canada. ACTA
JIETALLURGICA,
VOL.
22, J_ASX_7ARY
1974
1i
o. Q)o*o*
“8 Q
00*(-J
000 0 (a)
00
*
0
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0
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*,o”
0
* * OoO 0” “0” 00 0 o&f
00
8
0”
0
0”
..P,o;. 0
34.9‘3
0 00
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$
00
1
ooOoQOOO *o*o 0008*0*
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es
(b)
7 per cent less energy for 31’ or IS” (111) boundaries in zone-refined lead. Hasson et aZ.ci) have measured the energy of boundaries in aluminum of 99.998 per cent purity and find local minima at some of the angles predicted by the coincidence model. Chaudhari and Xatthews(5*6) have shown that twist boundaries formed on (100) planes between smoke particles of XgO and Cd0 show minima at angles predicted by the coincidence model. The present no& was carried out to obtain further experiment’al data concerning the energy of a coincidence grain boundary and the effect of deviations from ideal coincidence relationship on this energy. Such data should then provide a critical test of the various coincidence or relaxed coincidence models(8-10i proposed for grain boundary structures. EXPERIMENTAL
FIG. 1. Lattice coincidence model of & 36.9(100) tilt (Fig. la). A 36.9” (100) tilt boundary showing shared Iattice sites with a periodioity of p (Fig. lb). The same tilt boundary in which the two lattices have been displaced to remove coincidence; the structural periodicity is again p (Fig. lc). boundery
is equivalent to the structure present when the atoms are being shared. The period for any given coincidence orient,at.ion relationship would be the same for both the model with shared atom sites and the one without. In the example given in Fig. 1 for a 36.9” (100) tilt boundary, the period for both models is p. Due to the absence of coincidence atoms in the model of Fig. (lc), the extent of coincidence cannot be specified as the density of shared sites but rather as the number of shared sites that can exist, in the plane of the boundary per unit area of that boundary. It follows from this modified or relaxed coincidence model that deviations from ideal coincidence relationships would not be as cat,astrophic to the special properties attributed to the lattice coincidence model as first thought. Aside from energy studies of coherent and noncoherent kin boundaries, very few studies have provided direct experimental evidence for a drop in orientation relationships. energy at coincidence Rutter and _&ust(2)noted a 15 per cent lower energy for a 38” (111) grain boundary and possibly about
Single crystals and tricrystal specimens were prepared from the melt using lead of 99.9999% purity, with and without solute additions of 2 atom ppm of Sn, Ca or Au. The single-crystals were grown in a horizontal furnace under purified argon. Each single crystal, having dimensions of 3f16 in. x 3/16 in. x 6 in. was grown at 0.5 in/hr. at approximately330”C from the same seed crystal having a cube orientation. Each crystal was acid cut into three equal lengths for subsequent use in the growth of a t,ricrystal. The crystals were chemically polished in a solution of 50 % acetic acid and 50 ‘A hydrogen peroxide, washed with water and rinsed with alcohol before drying. The method described by Gleiter@i for growing tricrystals was used here, in which the junction of three properly oriented seed crystals is melted and then slowly cooled into a tricrystal \vith controlled The seed crystals were secured in the orientations. three single crystal holders of the tricrystalgrowing apparatus. Each crystal wasX-rayedand,ifneceseary,readjusted before positioning in t,he tricrystal apparatus. The single crystal holders were designed to be removable from the rest of the apparatus and capable of allowing the X-raying of an already positioned single crystal. The complete tricrystal growing apparatus was capable of adjusting each crysta1 individually by rotating wit,h respect to its major axis, tilting along a vertical axis, moving each crystal forward and back along its major axis and changing the orientation between crystals on the horizontal growing plane. Each tricrystal was grown in a vacuum of lob6 torr by melting the junction of t.he three seed crystals and t,hen allowing t,he melt to cool slowly. Due to the geometry of the tricrgstal graphite boat and the relatively large mass of the single crystal holders,
DIJiOTZ
directional
ACST:
ATD
ESERGIE8
cooling n-as easily achieved
of the Jingle crystals. place at about directions,
OF
GR_1IIS
along the ases
Alsol cooling of the melt took
an equal rate along each of the three
thus forming
a tricrystal
of equally
cr,vstala and having a grain boundary
junction
large at the
centre of the sample.
The rate of growth
crystals was controlled
by slowly decreasing the current
through
the furnace
one minute
during
an interval
to more than ten minutes
of the triof less than
from the time
of complete melting until complete tricryetalformation. In order to avoid the growth the bottom slightly
surface
superheat
thickness The
the
crystallographic in Fig.
were removed acetic
acid
surfaces
the
The
pinnedt wing
50%
t\vo
as shown
a spark
to prevent subsequent an
annealing.
ture
of
Each
ahoxed
grain boundary After alcohol sample
parbially
ghost motion
annealing, etched
and the polishing mounted
bath
holder
and
the
the
and scanning
junction
were
electron
microscopy
cations up to about 4000 x . d typical equilibrium
grain
boundary
lead is given in Fig.
3.
three using
at magnifi-
example
angles
Variations
of
measured
for high
of the purity
in the boundary
angle measurements
were as much as 4’ which gave
rise to a maximum
error
in the relative
boundary
energy of &S per cent. RESULTS
AND
DISCUSSION
The relative energies of t,he boundaries were determined according to the following relationship:“’ ^JAB 7
sin x where
yan,
ysc
=-
*/lx
sin p
=-
YBD
and yBD are the
unit area of the boundaries
AB,
(1)
sin b free energies
and 1, B and 6 are the equilibrium
holes
as shown in Fig.
cidence
boundary
non-coincidence
and
2. BC
boundaries,
Taking and
XB
BD
boundary as the coin-
as high-angle
yBc = yBrJ = constant
during glass
Ilost a result of of
cycle. was polished
immersions
and
in ethyl
The sample was
electron
orientation
microscope
of each
of the
100
110 C
L
ii0
w
001
100
010
100
FIG. 2. Crvstallogaphic
orientation
of
the
tricrystals.
per
BC and BD respec-
furnace. as
solution.
on a scanning
at
annealing
of the
The dihedral
angles
and there was no evidence
alternate
optical
8, was + 1’.
angles
after
analysis
determination
n-as sealed
evacuated
boundaries
each tricrystal by
boundary
boundaries
back-reflection
the
150 hr at a tempera-
beyond the 150 hr annealing
.slightl;v then
and
for at least
motion
in
39
tively
was necessary
tricrystal
by S-rav error
of misorientation,
grain
COISCIDESCE
were
of the grain boundaries
323 _C 1°C in a salt
samples
1 mm dia.
pinning
by grain boundary
argon-flushed
motion
The
angle
Both
boundaries
2, with
determined
cutting
smoothed
SEdR
maximum
of 50 ‘A glacial
peroxide.
high-angle
the disappearance
and annealed
by acid
grains The
the
the seed crystals
were chemically
machine.
tube
maintain
(100) plane and chemically
in Fig.
from the specimen
to
of the tricrystals
growth,
hydrogen
to the horizontal
polished.
also
tricrystals
of each tricrystal
parallel
and
strip in a solution
and
from
at less than about 4 mm.
After
from
crystals
it was necessary
orientation
2.
with a polyethylene
in
boat
of the tricrystal
is shown
of stray
of the boat
BOUSDXRIES
FIG. 3. Equilibrium grain boundary angles in highpurity lead. 0 = 3i.5’. _,_yR-= 0.71. Magnification: X hbW.
Or
13 = 6 = (180 -
z/2).
eyuation
(1)
becomes:
(‘1 The validity of ecluation (1) is based on the assumption that the torque terms of the boundaries AB, BC and BD are negligible. The torque forces on the coincidence boundary AB are zero because the boundary is symmetric; the energy of the noncoincidence large-angle boundaries BC and BD is The assumptions are independent of inclination. threfore justifiable and the relative energy yu of the coincidence bounclary in its final fortn is; sin a
where yuc was arbitrarly set equal to unity. The results of the experiments are summarized in Table 1. and in a plot of in vs the deviation from the ideal coincidence value, 36.9” (loo), shon-n in Fig. 4. It is apparent that the lowering of energy of the coincidence boundary is present within 2’ of the ideal co~ne~den~~ m~sorientation relationship. In t,he vicinity of t,he coincidence boundary, the relative energy- shows a consistent drop of BS much as 25-30 per cent Unfortunately, no data for the ideal 36.9” (100) misorientation lvere obtained and, therefore, the-exact extent of the energy decrease could not be determined. However, since the 19 (100) values closest to the ideal value were within & 0.6” (Fig. 4) and the error inroIvec1 in the 8 determination is as high as &l’, the observed 25-30 per cent drop in ~a may represent the actual extent of the energy cllop for the ideal coincidence relationship. It should be noted in Table 1 and Fig. 4 that the relative energy values for specimens having solute additions of tin, go!d or calcium were not very different from those of pure lead. It woulcl appear that the
FIG. 4. Relat~ive grain boundary energy (;J~) vs. deviation (&l) from ideal coincidence angle of 36.9’ t 1~1).
solute additions (2 atom ppm) were not sufficient to cause any measurable differences, either for a solute with relatively high solid solubi1it.v (Sk1 in Pb) or for solutes with low solid solubility (-Au or Ca in Pb). Gleiter(*) observed a 32 per cent lower energy for a high density coincidence high angle boundary in high-purity lead at 3%.5”C. However. in lead containing about SO atom ppm of copper this same coincidence boundary gave only a 12 per cent lower energy than the non-coincidence boundary. A summary of previous and present data for relative energies of different coincidence boundaries in highpurity zone-refined lead is shown in Table 1. It is evident, that the present results are con&tent n-ith the previous studies for different high den&y coincidence boundaries. The data in Table 2 are plotted in Fig. 5 and indicate a good correlation between relative energy of coincidence boundaries and the calculated number of shared atom sites per unit area [in lattice
DIJIOU T.LBLE
hXD
1.
.AUST:
ESERGIES
OF
GRAIS
BOUSDARIES
SEdR
COISCIDESCE
31
The relative energy of coincidence boundaries and their density of shared atom sites in high purity lead
Coincidence
Density of shared atom sites
Relative energy* 7s
60’ (I 11) (coherent twin boundary) 70.5’ (110) (incoherent twin boundary) 36.9” (100)
9*3o-’ (101 l.lj,-2 (13, 0.91a-* ‘lo’
35” (111) 2s” or 34” (111)
0*33a-2 113) O,*To-’ 113,
0.05 f O.OlA’i?’ 0.65 + 0.05c’j 0.71 f 0.06 (present work) 0.55 + 0.05’Z 0.93 + O.O5l?’
* Specimens
boundary orientation relationship
equilibrated
in temperature
range of 31%316.5”C.
‘.
‘. \
Fm. 5. Relative
energies of coincidence boundaries atom sites in high purity lead.
units where “a” is the lattice constant) of the boundary.oOJs) A similar good correlation is obtained between relative boundary energy and size of the periodic unit, in terms of the dimension of the unit lattice cell, for these same boundaries, as suggested by the “relaxed” coincidence mode1.(1c*r4) These results are believed to provide support for the boundary coincidence and “relaxed” coincidence models advanced by Bishop and Chalmers(s) and Chalmers and Gleiter,“O) respectively. Gleite?) found that the energy of a 3’7” (100) tilt boundary in zone-refined lead is 10 per cent lower than the energy of a 18.6’ (100) tilt boundary. Utilizing the results of the present work for the relative energy of a 37” (100) boundary, the energy of the 18.5 (100) boundary is about 20 per cent lower than the energy of a high-angle non-coincidence boundary. This result is consistent with t.he plot of Fig. 5, since the 18.5” (100) boundary corresponds closely to an ideal coincidence relationship (19’ (100)) with a calculated densit.y of shared atom sites of 0.33 a--2.(10) The correlat,ion given in Fig. 5 may differ in nonmetallic materials with more complicated bonding.(r5)
Rolling
and ‘1 Winqard (12) ‘6
vs. density
of shared
For example, energy calculations of twist boundaries in MgOo5) indicate that a 28’ (100) coincidence boundary has a positive contribution to energy from an electrostatic term and therefore has a shallow energy minima. Also, it should be emphasized that the relationship found in Fig. 6 for zone-refined lead would change drastically if strongly segregating impurities are present. For example, Gleiter@) has found that the relative energy value of 0.68 for the 70.5’ (110) coincidence boundary increases to approximately 0.88 with the addition of about 50 ppm by weight of copper to the lead. This is attributed to a stronger segregation of copper atoms at non-coincidence boundaries than at coincidence boundaries, thereby decreasing the energy of a non-coincidence boundary (yno) more rapidly than the energy of a coincidence boundary (yas) i.e. increasing the ratio of YABIYBC. CONCLUSIONS
1. Lower relative energy values were detected within f 2’ of the ideal 36.9” (100) coincidence boundary relationship in high-purity lead. The extent of this decrease in energy corresponds to
32
dCT_1
METILLCRGIC_~,
about 30 per cent that for the energies of non-coincidence grain boundaries. 2. Solute additions of 2 atom ppm of gold, calcium or tin appeared to hare little effect on relative boundary energies in high-purity lead. 3. A good correlation is found between the relative energy of coincidence grain boundaries and the density of shared atom sites per unit. area of the boundary in high-purity lead. 4. The results of this study are in agreement with the boundary coincidence model of Bishop and Chalmers and t.he “relaxed” coincidence model of Chalmers and Gleiter. ACKNOWLEDGElMENTS
The authors would like to acknowledge the very helpful assistance given by Dr. H. Gleiter through both informal and written discussions. The authors would also like to thank Mr. H. Sang for useful suggestions and Mr. D. Abdula for the construction of the crystal growing apparatus. Financial support n-as provided by the National Research Council of Canada and the Defence Research Board of Canada.
VOL.
22,
1974 REFERENCES
1. Ii. T. _%USTand B. CH_%LMERS, Xelal Inle+ces, p. 153. -Am. Sot. Metals. Cleveland. (19?iP). 2. J. W. RUTTER and I<. T. XT%, refeked to by K. T. Aust in Surfaces and Interfaces, I Chemical and Physical Chmactetitics, edited bv BLXKE, REID and WEISS. . D. . 435. Syracuse University P&s, (1967). 3. K. T. .~CST and B. CH_UYERS, Xet. Trans. 1, 1095 (19iO). 4. H. G~EITER, &ta Met. 18, 117 (1970). 3. P. CH.~~DHARIand J. W. X%TTHEWS, Appl. Phya. Lett. 17, 115 (1970). 6. P. C!E_~~DH~RI and J. W. X~TTHE~S, J. appl. Phys. 42, 3063 (1971). 7. G. Hassos, J. Y. Boos, I. HERBEXX_~L,>I. BISCOXDIand C. GOIX, Surface Sci. 31, 115 (1972). S. JI. L. KROYBERG and F. H. WILSOX, Trans. .-I.Z.X.E., 185, 501 (1949). 9. G. H. BISHOP and B. CH~~UERS, Scripta -Ilet. 2, 133 (1968). 10. B. CHALMERS and H. GLEITER, Phil. Xag. 23, 121 (1971). 11. 31. J. WEISS, H. GLEITER and B. CHALMERS,J. appl. Phys. 42, 2639 (Igil). 17. G. F. BOLLIXG and W. C. WISECARD, J. Inst. Net& 86, 492 (1958). 13. H. GLEITER, private communication (1972). 14. H. GLEITER, unpublished; quoted in H. GLEITER and B. CH~LJIERS,High-angle g&in boundaries, Progrew in Materials Science, Pergamon Press Vol. 16, pp. 24, S (1972). 15. P. CHAUDH~RIand H. CHARBXAV, Surface Sci. 31, 10-L (1972).