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Absolute interfacial energies of [001] tilt and twist grain boundaries in copper
ABSOLUTE INTERFACIAL ENERGIES OF [OOl] TILT TWIST GRAIN BOUNDARIES IN COPPER* N, A. GJOSTEINI_
AND
and F. N. RHINESS
The absolute interfacial energies of [OOl] tilt and twist boundaries in copper have been determined, by using a Zeiss Interference Microscope to measnre the dihedral angles which form during the thermal grooving process. An analysis of the results obtained for small angle, lineage boundaries, in terms of the appropriate dislocation model, shows that the Read-Shockley equation predicts the energy up to misorientations of 5 to 6”, but that for larger misorient&ions, it gives much too small an energy. It is shown also that van der Merwe’s treatment, which avoids some of the limitation8 present in the Read-Shockley derivation, is capable of predicting the energy up to misorientations of 8 to 9”. Moreover, from the analysis presented herein, it is possible to offer an explanation for the apparent agreement that was obtained, at large misoriantations (25 to 30”), between the previous measurements of relative grain boundary energies and the Read-Shockley equation. The large angle grain boundary is characterized by a broad maximum in energy that shows no energy cusps, for either tilt or twist boundaries. Both types of boundaries have nearly the same energy for misorientations less than la’, but for the range of misorientations, over which there is a maximum in energy, twist boundaries have a lower energy by about 130 ergs/cmr. An attempt to explain this difference from the results obtained from a calculation made on an atomic basis is discussed.
ENERGIES
ABSOLUES
d’INTERFACE DES JOINTS FLEXION [OOl] DU CUIVRE
DE TORSION
ET DE
Les enorgies absolues d’interface des joints de torsion et de flexion [OOI] du cuivre sont determinees au microscope interferentiel Zeiss, par la mesure de.8 angles di&dres qui apparaissent au tours du meoanisme d’entaille thermique. L’analyse des resultats obtenus pour des joints iafaible angle, eompte tenu du modele de dislocation approprie, montre que l’bquation de Read-Shockley prevoit correctement l’energie jusqu’a des divergences d’orientation de 5-6”, mais qu’au-dela elle conduit a une bnergie trop faible. Cette analyse montre Bgalement que la methode de Van der Merwe, qui Qvite certaines limitations de l’equation de Read-Shockley, permet de prevoir l’energie jusqu’a des divergences d’orientation de 8-9”. En outre, la discussion des resultats obtenus iei, permet d’avancer une explication de l’aocord apparent obtenu pour de grandes divergences (W-30”), entre les mesures obtenues pr&edemment et l’equation de Read-Shoekley, dans le calcul des Energiesrelatives des joints. Le joint it grand angle, qu’il soit de flexion ou de torsion, est earact&id par un large maximum Bnerg&ique. Les deux types de joint ont pratiquement la mdme Bnergiepour des divergences d’orientation inferieures it 18”, mais pour les angles auxquels correspond un maximum energ&ique, les joints de torsion ont une Bnergie moindre de 130 ergs/cm2 environ. Les auteurs discutent alors, sur la base de leurs resultats, d’une tentative d’explication de cette difference, par un calcul qui s’appuie sur une base atomique. ABSOLUTE
G~ENZFLACH~~ENERGIEN VERDREHUNGSKORNGRENZEN
DER [loo] NEIGUNGS-
UND
BON KUPFER
Die absoluten Grenzflachenenergien der [loo] Neigungs- und Verdrehungskorngrenzen van Kupfer wurden beatimmt unter Verwendung eines Zeiss Interferenz-Mikroskops zur Messung der Diederwinkel, die sich beim thermischen Furchungsvorgang ausbilden. Eine Analyse der Ergebnisse an KleinwinkelVerzweigungsstrukturgrenzen auf Grund des angemessenen Versetzungmodells zeigt, dass die ReadShockle~-Gleichung die Energie bis zu Orier~tierun~un~~chieden von 5 bis 6” voraussagt, sie gibt jedoch fur griissere Oreinti~~ngsunterschiede eine zu kleine Energie. Zudem wird gezeigt, dam die Behandlung van der Merwes, die einige Beschrankungen der Read-Shockleyschen Ableitung vermeidet, die Energie bis su Orientierungsunterschieden von 8 bis 9” voraussagen kann. Weiterhin ist es duroh die hior gegebene Analyse moglich eine Erkliirung fur die anscheinende Ubereinstimmung zu bieten, die grossen Orientierungsunterschieden(25 bis 30”) zwisohen friiheren Messungen der relativen Korngrenzenergien und der Read-Shockley-Gleiehung erhalten wurde. Die Grosswinkelhorngrenze wird dureh ein breites Energiemaximum charakterisiert, das weder fur Neigungs- noch fur Verdrehun~ko~~enzen ~nergiespitzen aufweist. Beide Typen van Korngrenzen haben fur Orientie~gsuntersohiede unterhalb 18” beinahe dieselbe Energie, doch im Bereich der Orientierungsunterschiede, in dem ein Maximum der Energie auftritt, haben dio Verdrehungskorngrenzen oine urn etwa 130 erg/cma geringere Energie. Es wird ein Versuch erortert diese Unterschiede aus den Ergebnissen einer atomistischen Bereohnung zu erklaren. * Abstracted from a thesis presented by the Aluminum Company of America Fellow, N. A. Gjostein, to the Faculty of the School of Engineering and Science, Carnegie Institute of Technology, in partial fulfillment of the requirements for the Degree of Doctor of philosophy in ~et~l~gieal Engineering. Received July 24, 1958. t Research and Development Laboratory, Thompson Ramo Wooldridge, Inc., Cleveland, Ohio. $ Department of Metallurgical Engineering, Carnegie Institute of Technology, Pittsburgh, Penna. ACTA METALLURGICA,
VOL. 7, MAY
1959
319
ACTA
320
METALLURGICA,
1. INTRODUCTION
From measurements of the absolute interfacial energies of small angle boundaries, it is possible to provide a rigorous test of the dislocation model of the grain boundary, for in this case the experimental energy-angle relationship may be compared quantitatively with the predicted relationships, which have been derived by Read and Shockleyd) and van der Merwe.c2) It is expected that the Read-Shockley treatment, which is based on linear elasticity theory, will be valid only when the spacing of the dislocations D is considerably larger than DL, the diameter of a cylindrical surface, surrounding the dislocation line, which encloses material experiencing non-Hookean strains (DL might be defined also as a dislocation width). Subject to this limitation Read and Shockley found that the interfacial energy E of an array of dislocations varies with the relative rotation 0 between the crystals, according to the expression E = E&A
-
In 6)
(1)
where E, is a parameter which can be evaluated independently from known constants of the material and A is related directly to the core energy of the dislocation, i.e. the energy residing within the cylinder of diameter D,. When measurements of relative grain boundary energiesf3-‘) were compared with the Read-Shockley equation, it was found that equation (1) appeared to fit (qualitatively) the form of experimental E vs. 8 curves up to large values of e (25 to 30”), indicating that the dislocation widths in these materials were unusually small, i.e. DL s b, where b is the magnitude of the Burger’s vector b. Concerning this anomaly, Lomer and Nye (lo) have commented that it seems unreasonable for the dislocations to be so narrow, and Readc8) and Brooks@) have indicated that the agreement at large angles is probably due to a fortuitous cancellation of errors in the derivation. Further insight into this problem can be obtained from the following considerations. To compare the relative energy measurements with equation (1) it was necessary to fit the calculated curve to the large angle data; the result of this was that only the form of the energy-angle relationship was tested. It was not possible to determine whether the Read-Shockley equation gave the correct absolute magnitude of the grain boundary energy, for example, by obtaining an experimental value for E, at small angles, and comparing it with the proper theoretical value. It seems possible, therefore, that the values of the parameters used to fit the calculated curve to the large angle relative energy values may not have any theoretical
VOL.
7, 1959
significance. An ambiguity of this sort can be avoided when absolute energies are measured, since in this case E,, which is obtained from a plot of E/B vs. In 8, can be compared with the value derived from the appropriate dislocation model. Absolute energies of tilt boundaries in silver(ll) and zinc’12) have been measured but in neither case was it possible to determine E, at small angles. With the recent advanceso3) in interference microscopy, it was felt that it would be feasible to determine the absolute energies of [OOl] tilt and twist boundaries in copper from measurements of dihedral angles formed by the thermal grooving process.(11~14)It was considered important to extend the measurements over as wide a range of 8 as possible, with particular emphasis on small values of 0, and other discrete large values of 0, where energy cusps might be expected to occur. In addition, considerable attention was paid to the effect of boundary orientation on its energy. From these measurements, it was thought that it would be possible to draw conclusions, concerning the structures of both small and large angle boundaries, the latter type being particularly interesting in view of the possibility of calculating their energies on an atomic basis.(15y16) 2. EXPERIMENTAL
PROCEDURE
2.1 Preparation of specimens Cathode sheet copper, having a purity of 99.98 per cent Cu, was used to prepare from the melU5) bicrystals, containing [OOl] tilt and twist boundaries. Bot,h types of bicrystals were prepared from seeds having a cube direction within 0.5’ of their longitudinal axes. Thus, each pair of crystals had in common a cube direction, about which they were rotated with respect to each other by the angle 8. Tilt boundaries in the range 5” < 8 < 78” were symmetrical, i.e. the plane of the boundary made an angle 812 with the [OlO] direction of either crystal. It was possible to extend the energy measurements to 8EO.6” by making use of the lineage boundaries formed during the growth of most of the bicrystals. The lineage boundaries were asymmetric tilt boundaries, whose twist component rarely exceeded 0.2”. The asymmetry angle $, i.e. the angle between the mean of [OIO] directions of the crystals and boundary plane varied from 0 to 28”. In order to insure that the axis of relative rotation was within 1.0” of the [OOl] direction, and that the magnitude of the rotation 8 could be predetermined with the same accuracy, the necessary seeding operations were done mechanically by means of a special type of goniometer.07) For the lineage
GJOSTEIN
AND
RHINES:
TILT
AND
TWIST
GRAIN
BOUNDARIES
IN
COPPER
321
FIG. la. Interferogram showing the perpendicular-type fringe pattern of a thermal groove which formed at a 32” symmetrical [OOl] tilt boundary; the plane of the surface shown is parallel to the common [OOl] direction and normal to the gram boundary; contour interval, 0.27 ,u; x 762.
FIR. lb. Interferogram showing the parallel-type fringe pattern of a thermal groove formed at a 37” symmetrical [OOI] tilt boundary; the plane of the surface shown is parallel to the common [OOl] direction and normal to the gram boundary; contour interval, 0.27,~; x 762.
boundaries,
izes the Linnik interference
8 was
measured
to
fO.2”
by
super-
system,og)
and is parti-
imposing the Laue spots from each crystal on a single
cularly well suited for large wedge angle interference.
film.
A discussion
With
some
modificationo7)
used to measure the value obtained
in this way was usually
1 .O” of the predetermined The section
this method
was
0 for the large angle boundaries; within
misorientation.
axis, and were deeply etched.
dihedral angles of thermal
grooves has been given by Mykura.os) It is convenient
bicrystals, which had a rectangular cross (A in. x 1 in.), were sectioned in a plane
normal to the growth
of the Linnik system, and its limitations
with respect to measuring
to determine
from either perpendicular Figs.
la
and
lb,
the dihedral
respectively.
shown that, for the perpendicular
Amelinckx(20)
has
fringe pattern,
the
Both the as-cast (top) surface and the cut surface were
angle u, that one side of the groove
mechanically
vertical,
prepare
polished,
them
and then electropolished(ls)
for the thermal
In order to avoid the formation
grooving
to
treatment.
Deoxo
hydrogen unit,
and
atmosphere over
During this treatment, in recrystallized
copper
of facets on the sur-
(passed turnings
through
a
at 650°C).
the specimens were supported
alumina
crucibles,
surrounded
by
copper foil. The specimens were annealed for 37 hr at 1065°C; the shape of the thermal groove profile after annealing under these conditions
makes with the
is related to the corresponding
fringe angle q
by the relation
face, and to obtain reproducible dihedral angles, it was found necessary to anneal the specimens in a purified
angle
or parallel fringe patterns,
is shown in Fig. la.
2.2 .Mea.surement of dihedral angles Dihedral angles were measured by the use of a Zeiss Interference Microscope. This instrument util-
2L
(2)
tan cc = ;Im tan *II where L is the fringe spacing, il the wavelength
of the
light source, and m is the linear magnification. The errors involved in measuring 7 from enlarged photographs and the related errors in u have been discussed by Gjostein.(l’) This method is versatile, and can be used for either high or low energy boundaries. To give some idea of the magnitudes of a and q, it may be said that as u varied from 89”30’ to 80”0’, r,~varied from 50 to IO”. Hilliard(21) has suggested a method
of determining
cc by plotting a profile from the parallel fringe pattern,
ACTA
322
METALLURGICA,
Fig. lb. In this investigation it was found that this method could be applied only to grooves having a depth of greater than 0.5 ,u, i.e. for high energy boundaries. Therefore, the majority of the measurements were made by using the first method, but, for some high energy boundaries, both techniques were used. Both methods give an over-estimate of u due to the obliquity effect, reported by Tolmon and Wood, for interference microscopes using convergent illumination. This results in an apparent decrease of about 10 per cent in the grain boundary energy.(r7) Accordingly, all energy values reported herein have been increased by 10 per cent to compensate for this effect.
TABLE
a+B y(J = 2y, cos ~ 2
(3)
‘7, 1959
1. Dihedral angles and ratio ya/ys for tilt boundaries at 1065°C.
tf 0%) 0.65 o”::, 0.85 0.9 1.3 1.3 1.5 K 3.0 5.5 :*: rz:o 16.0 19.0 22.0 26.0 28.0 32.0 37.0 40.0 45.0 53.0 65.0 78.0
3. RESULTS
During the course of this investigation, a variety of groove profiles was observed. It was found, however, that only the normal-type profile, having the characteristics shown in Fig. la, gave reproducible values of a. In Appendix I, evidence is presented which indicates that it is permissible to neglect the orientation dependent terms of Herring’s general equation for interfacial equilibrium,(23~24)provided that the measurements of a are taken from normal profiles. Under these conditions, the grain boundary surface tension* ya may be calculated from the equation
VOL.
a-l-B
---.* Corrected for obliquity effect reported by Tolmon and Wood.‘=’ 4. ANALYSIS
OF RESULTS
Equation (1) can be used to represent the energy data by placing it in the form Ry = R&A
where a and B are the angles that each side of the groove makes with the vertical? and ys is the average surface tension of the free surface; the ratio yo/ys has been calculated from equation (2) and corrected by 10 per cent as indicated previously. Tables 1 and 2 give the values of a + /3 obtained for [OOI] tilt and twist boundaries at 1065C~ from perpendicular fringe patterns, along with corrected ratios of yG/ys. Each value of a + /l represents an average taken from 15 to 20 fringes. In many cases, the specimen was repolished and completely retreated to produce a new groove which was measured using 15 to 20 fringes. It was found that yG/ys was reproducible to within &lo per cent for most boundaries.
* The definition of “surface tension” used here is that $msn)y Herring’2*’ and should not be taken to mean “surface t The plane of the grain boundary was always within 2’ of being normal to the free surface. $ There is a measurable temperature coefficient for ye in the case of large angle boundaries, but this effect does not influence the conclusions stated herein.
0.038 0.044 0.047 0.048 0.051 0.064 0.052 0.066 0.084 0.099 0.094 0.135 0.144 0.202 0.222 0.294 0.300 0.328 0.339 0.338 0.356 0.350 0.374 0.362 0.353 0.322 0.217
178O2’ 177”42’ 177’36’ 17v32 177y22’ 176’40 177”18’ 176”36 17v40 174O54’ 175’8 173”4 172’36’ 169’36’ 168’32’ 164O54 164’32’ 163”s’ 162O28’ 162’34’ 161”38’ 160”48’ 160’49’ 161°1;f 161’56’ 163’20’ 168’48
-
In @)
where R, = E/ys = ya/ys, and R, = Eo/ys- Plots of RY/Ovs. In 6 for [OOl] tilt and twist boundaries are shown in Figs. 2 and 3, respectively. It can be seen TABLE
2. Dihedral angles and ratio ye/jjs for twist boundaries
at 1065°C
3 (d+s)
-.
Yell/s’
a+B -_-_
--
E 5:o ii:::
10.0 13.0 15.0 17.0 23.0 30.0 35.0 37.0 40.0 43.0 45.0
176”44’ 175’28 174”6 173O2’ 170”44 170’12’ 168’48 168OO’ 167”36’ 166”36’ 164”52’ 164’8 165”24 164’4 165”56 164”52’
0.063 0.088 0.115 0.136 0.180 0.190 0.218 0.234 0.240 0.261 0.293 0.308 0.282 0.307 0.295 0.293
* Corrected for the obliquity effect reported by Tolmon and Wood.‘=’
GJOSTEIN
0.5 4.01 ’
I I
AND
RHINES:
8 I Degrees1 5 IO
i!
I
TILT
AND
TWIST
GRAIN
TABLE
20
BOUNDARIES
3. Pammeters
I
%
COPPER
323
R, and A 8s determined from Ry/% vs. In 0 plots
50
I
IN
Range of @
A
[OOl] Tilt series-1065°C 0.91 & 0.05 -0.72 0.49 f 0.01 $0.71 [OOl] Twist series-1065°C 0.46 0.54
10 43
0.6 < f3 < 12 3<8<53
36
5<8<45 ___.--
* Calculated from A = 1 + In &lax. (8)
i
I
s.0
-4.0
t -3.0
-2.0 In 0
-1.0
0
Fra. 2. Plot of Ry/i3 VS. In 19for [OOl] tilt boundaries at 1065°C.
that the data for tilt boundaries, Fig. 2, cannot be represented by one straight line, and that the slope (--R,) increases (negatively) as 8 decreases. For purposes of a.nalysis, the results were divided into two overlapping ranges, 0.6” < 8 < 12” and 3” < 8 < 45’. Then, using the method of least squares,(25) linear regression lines were determined for each range of 8. Prom the slopes and the intercepts of the regression lines, it was possible to determine the small and large angle values for R, and A; these are given in Table 3. In view of the results obtained for tilt bo~daries, it was expected that a plot of R,/6 vs. In r9for twist boundaries also would show a change in slope at small
1.6
8 (Degrees) IQ 2,O
’
59 I
0’
I
-3.0
I
I
I
I -1.0
-2.0
II 0
In 8 FIG. 3. Hot of Rv/@ vs. ln 8 for [OOI] twist boundaries at 1065°C.
values of 8. Unfortunately, it was not possible to obtain any energy data for 0 < 3” in this series and therefore it is not clear whether twist boundaries present the same effect, Fig. 3. The large angle values of R, and A for twist boundaries are given in Table 3. Energy vs. 0 curves, which were calculated from equation (4), using the large angle parameters, are shown in Figs. 4 and 5 with the experimental point superimposed. These curves have no theoretical significance, as will be seen in the next section, but they can be used as an empirical representation of the energy data over a wide range of 0. On these plots, in addition to the dimensionless ratio ya/ys, absolute values of yG are reported. For this purpose, yS was taken to be 1670 ergslclnz. According to Fisher and Dunnt2@, the most reliable value of ys for copper is 1430 ergs/cm2 in the range 950°C to 105O”C, as determined by Udin et c&(2’) It has been pointed out,(2s*29) however, that Udin et al. neglected the effect of transverse grain boundaries in computing ys. Taking the number of grains per unit length along the wire to be 80(2s) and yG/ys = 0.37, it can be shown that a more probable average value for ys is 1670 ergs/cm2. The calculated curve for the [OOl] tilt series has a maximum at 8 = 43’. Beyond @ = 43’ the calculated curve would begin to decrease; therefore, in this region the curve has been drawn to fit the points, the dashed line indicating uncertainty in the data. The calculated curve which best fits the twist boundary series has a maximum at 6 := 36”, but the experimental points could as well fit a curve with a maximum at 0 = 45”. There is no need to extend the measurements beyond 45’ because the energyangle relationship for an [OOI] twist boundary series must be symmetrical about this value. The difference in the energy between tilt and twist boundaries is not very large until 8 exceeds about 18’, where the tilt series has a higher energy, by some 130 ergs/cm2. It is important to mention here that no abnormal scatter, i.e. greater than fl0 per cent, was observed
ACTA
324
METALLURGICA,
VOL.
7, 1959
0.40
0.32
0.24 a” \ P
0.16
8 (Degrees) FIU. 4. Dependence of grain boundary energy on misorientation for [OOI] tilt boundaries at 1065°C. Solid line represents the curve c&~&&d from equation (I), using the lwge a~& paraw&ers. Although the curve has no theoretical significance it can be used as an empirical representation of the energy data over the range 5’ < 0 < 43”. Beyond 43”, the ourve has been drawn to fit the experimental points.
at 8 = 37” (310) and 8 = 53’ @IO), the presumed major energy cusp positions of the [OOl] tilt series. Since for each critical angle three different bicrystals were measured, the probability that the cusp has escaped detection is small. Moreover, no energy cusp was detected at 3’7“ in the [OOl] twist series. No effect of the orientation of the boundary upon its energy was found for large angle boundaries, including those near energy euap positions. 5. DISCUSSION
where $’ = # rJz45” refers to the axes [llO] and [IiO], the sign of 45” being chosen so that $’ and # he between 0 and 90”, and cI1, etc. are the elastio constants of the material. In the second model, four sets of of ~slocations are needed; each set has equal edge and screw components. For copper, however, E,, calculated from this model, is higher than for the first one (31) and therefore, this model need not be considered further, since in a welLannealed specimen, the lowest energy configuration should obtain. Table 4 gives the comparison between experimental and theoretical values of E,. The theoretical E, was computed from equation (5), using the room
To provide a rigorous test for the Read-Shockley equation, it is necessary to compare the experimental values of the parameters E, and A with those derived from dislocation theory. Read and Shockley(30) have made use of Frank’s formula(31) to determine the dislocation content of an [OOl] tilt boundary in a face centered cubic lattice, and they find that there are two possible dislocation models. Model one consists of two sets of edge dislocations, having Burger’s vectors
212
&=,b[llO]
and
1/i
b,=~b[llO].
_
Bor this model E, is given by E 0- -
cos q +
~-__
sin#
47r X--
e
b (Q,, + C,z)
- Cl,) -______ J~llf%O’Dell (@ + Cl, + 2C**)
(Degrees)
Fro. 5. Dependence of gmin boundary energy on misorientation for [OOl] twist boundaries at 1065°C. The curve was calculated from equation (l), using the large ungle parameters, and therefore it has no theoretical signiflcanoe, but it can be used to represent the experimental points.
GJOSTEIN TABLE
AND
RHINES:
TILT
4. Comparison of theoretical and for [OOl] tilt boundaries
E,(ergs/cm2)
AND
experimental
TWIST E,
GRaIN
BOUNDARIES
IN
COPPER
325
where e is the Naperian base and e is Poisson’s ratio. Foreman,
Jaswon
and Wood(35)
have
considered
more realistic force laws for the center of a dislocation
Method of determination
from which Lomer(33) has obtained the energy per unit 1550 1480 1520 & 100 820 * 15
* Computed 1670 ergs/cu?.
temperature 15 per
R,
from appropriate
elastic constants
cent
for
measured between the [loo]
=
directions)
determined
The averaging measurements
of E,
Examination
and the other
from the relation
d4’ = I.35 E,(O)
having
selecting
a
of the values listed in Table 4 reveals
considering
the
the proper
uncertainties
temperature
well with
involved
coefficient
-
[
1
-1s -sL,-
.
0
Read-Shockley
equation is valid at large 0; therefore,
neither of these methods Table
in
for the
will be discussed.
will be considered
obtained
from equation
from equations
(8) and (9).
Experimental
values of A, are listed in addition. the theoretical
treatment
for A, is not. as
rigorous as that for E,, the fact that the small angle A, lies between the two best theoretical determinat,ions is considered
to be significant.
From the discussion
that has preceded,
to the treatment
it is clear
boundary,
of Read and Shockley,
according agrees with
angle value for E, agrees more closely
with theory
the measured energy only when 0 is small.
than does the large angle one, which
differs
can be demonstrated
by a
with dislocation
data with the theoretical
E vs. 0 curves, Fig. 6.
theory.
are shown. One was calculated from equation 1, using the small angle parameters (i.e. essentially the
Read
and Shockleyo)
have
in terms of a non-
energy E, which resides inside a circle of
radius rL = D,/2, and an elastic energy ElII, which depends on the unknown boundary conditions at rL.
They show that the unknowns
can be represented
rL, E, and E’II,
by a single parameter
r. < rL, in
theoretical parameters),
Two Read-Shockley
while the other one represents
theoretical
+ln&
5” to 6”. At larger angles it begins to deviate and shows much too small an energy at the theoretical maximum.
density
7r 0 where A, refers to c$’ = 0 or 90”.
points
out
that
the
relationship
To evaluate
from t’he core radius r. to a distance
TABLE
5. Comparison
for T < rL. Nabarro(33), of Peierls(34), has
is cut off at a radius
b ro = ~~~~e(1 - a)
of the
r,/b,
concerning
determined the energy associated with each edge dislocation of an interacting pair, and he shows that the same energy can be obtained from elasticity theory, if the integration
Read-Shockley
the elastic energy
order b/28, where the stress fields of the adjacent
it is necessary to make some assumption, using the sinusoidal approximation
The
curve fits the data very well up to about
formula was derived by integrating
A,=1
curves
an empirical curve fitted to the large angle data.
Nabarro(33)
which case
the stress-strain
This fact
by comparing
values for A also may be compared
shown that A can be expressed Hookean
rather effectively
the small angle experimental
factor of about 1.8.
These
(7) using values of r,/b
ys.
point to note that the small
Van der
below.
values of A,.
5 lists the theoretical
were derived
that the energy of a dislocation
Experimental
(9)
for ro. Koehler’s method, however, contains errors(33) and Brooks has assumed that the
elastic constants and in selecting the correct value for It is an important
a
l-
as
Koehler(36), Brooks(g), and van der Merwec2) also have
Although
from 45” to 17”.
that the small angle E, agrees reasonably theory,
(6)
for 0 < 5”
on boundaries
four for
given formulae
theoretical
was done because
were made
range of asymmetry
In r,/b = In 2 l---o
Merwe’s treatment
s
in terms of a parameter
of approximately
In this case, In r,/b can be expressed
3, using 78 =
Two
has the value
copper.
for copper,(32) reduced
“do, 1,” 45” Eo@‘)
o.15 ~
a, which
one refers to +’ = 45” (6 was
is an average value z.
0
in Table
temperature.(30)
values of E, are given;
3
length of an edge dislocation
Equation (5), with 9’ = 45” Equation (6) Experimental value;* small angle boundaries Experimental value;* large angle boundaries
of theory parameter
with A.
experiment
for
the
Method of determination
-0.23 -1.50 -0.57 to -0.76 0.86
Theoretical; equations (7) and (8) Theoretical; equations (7) and (9) Experimental;* small angle value Experimental; large angle value
_____ * Since A varies with 4 ’ ,(8’ it was necessary to report * range of values for A,,, corresponding to 4’ = 45” to 17’.
ACTA
326
METALLURGICA,
VOL.
7,
1959
able to express A,, as A
0
=
1
(’ - a)po
In
+
(1 -
where ,u is the elastic
shear modulus
constant
order
of
determines
the
same
the amplitude
(10)
2o)Trp
of
and p.
is a
magnitude
which
of the sinusoidal
stress-
strain relation.
Taking A, = -0.72, Table 3, (neglecting the variation of A with c#‘) it was found that ,u/,uo must be about 3.65, in order to make van der Merwe’s E vs. 8 formulat2) agree with Read-Shockley’s at small 8.*
Based on these conditions,
a theoretical
curve was calculated from van der Merwe’st2) formula. It can be seen, Fig. 4, that it does not deviate appreciably 9”.
from the experimental
As 8 increases
beyond
data until &’= 8 to
this point,
the van der
Merwe curve does not drop off rapidly, remains considerably
although
it
below the actual energy curve.
While this method seems to give better agreement for larger I3 (Degrees)
angles
pointed
FIQ. 6. Comparison of the energies of small angle, lineage boundaries with those predicted by the formulae of Read and Shockley”] and van der Merwe.‘al
used
than
Read-Shockley’s,
out that the sine function,
by van
der Merwe,
approximation
not
to the force law for copper.
This inequality
finding that p/p0 > 1.
strain relation for copper has a much smaller maximum stress than that given by the sine function
strain energy resides in regions near the dislocation
amplitude
cores, where the stress-strain relationship is non-linear.
city
An estimate of rL can be obtained from the treatment
agreement
of dislocation
widths
given
by
Hooke’s
Foreman
law becomes
distance
approximates
comparable
Assuming
curve,
i.e.
D/2 = 5b. As 8 increases
theory
for
small
strains.
between theory
Undoubtedly,
and experiment
the
could be
repeated,
using the force law of Foreman
et uZ.(~~)
that this
rL, it can be seen that it is
to the point at which the Read-Shockley
curve begins to deviate appreciably mental
when its
is fixed so as to agree with classical elasti-
et uZ.(~~) extended to larger values of 8, if, as Lomer and Nye(lO) suggest, van der Merwe’s calculations were invalid
at a distance (along the slip plane) of about 4.7b (see Figs. 2 and 3 in reference (35)).
a good This can
reflects the fact that the true stress-
b/28 = r, since at this point the greater part of the
For a,, = 4 (copper),
be
which has been
is probably
be seen from the experimental dislocations have equal importance. It is expected that this approximation will become invalid when
it should
from the experi-
at 0 = 5 to 6” or equivalently
5.2. Comparison
of results with previous
energy dais
for tilt boundaries In the past, measurements boundaries
as a function
of the energies of tilt
of misorientation,
with two
exceptions,(11912) have been made on a relative basis, the
higher
order
elastic stress field become important.@)
terms of the Brooks(g) has
and therefore it was not possible to obtain E, from a plot of E/B vs. In 8.
Instead
summed these terms, and he finds that they add an
energy-angle
infinite series in powers of t3 to A-ln 0, the first term being b,02, where b, = 1. This factor would raise
fitting the Read-Shockley
the theoretical E vs. 0 curve slightly, but not enough to account for the discrepancy between the ReadShockley formula and experiment, for 8 > 5 to 6”. Van der Merwe(2) avoids some of the difficulties inherent in the Read-Shockley method by assuming that the stress-strain relation at the interface is sinusoidal. Since van der Merwe accounts for the non-linear forces at the core of the dislocation, he is
relationship
the shape of relative
was tested
essentially
by
equation to the large angle
energy data; it was concluded(197-s) that the equation described the data very well over the entire range of 8. This
investigation
has
demonstrated
that
such
a
* For small 8, van der Merwe’s expression’2’ gives the same functional dependence of E on 0 as the Read-Shockley formula. In order to make the magnitudes of each expression equivalent, however, the factor ,427r(l - u) of van der Merwe’s equation must be replaced by the appropriate one from anisotropic elasticity thoery, in addition to requiring that p/p0 = 3.65.
CJOSTEIN
AI+D RHINES:
TILT
AND
TWIST
procedure is not valid in the case of copper, since it results in values for E, and A, that are not in agreement with theory. The same situation may be true for the previous relative energy measurements. It will be recalled that Brooks’ calculation of the experimental E,(s) for Fe-Si boundaries from the relationship E, = ~~~~~e~~~(*)gave a value 50 per cent lower than the theoretical one. This might be expected, since large angle values for E,,, and 8,,, were used. Further support for this suggestion may be obtained from a consideration of the previous determinations of A. Unlike Es, the value of this constant can be determined from either absolute or relative energy measurements. This investigation has verified the idea put forth by Brooks@) and Parker et a1.02) that A is a function of 8. The small angle A, for copper is comparable to the values, -0.82 to -1.62, obtained by Parker et oZ.02) for small angle tilt boundaries in zinc. This fact, coupled with the knowledge that the theoretical A,, value for copper probably lies in the range -0.23 to -1.50 (the larger negative value is probably more nearly correct), indicates that the positive values (0.20 to 0.55, with tin being an exception) obtained for other metals may be too large. It should be noted also that the large angle A, = +0.86; for copper* is similar to those obtained by previous investigator, indicat~g that they did not extend their measurements to small enough angles to detect a change in slope in the E/Ovs. In 0 plot, and a consequent decrease toward negative values of A,. 5.3. Dislocation model and energies of [OOl] twist bou~ar~e~
GRAIN
BOUNDARIES
ZN COPPER
327
From equation (11), E, was found to be 1450 ergs/cm2, a value essentially the same as that calculated for tilt boundaries having variable asymmetry. For sufficiently small angles, such that --In 0 > A, it would be impossible with the present experimental techniques to measure the difference in their energies. There is some evidence, however, which indieates that A for small angle [OOl] twist boundaries is more negative than the corresponding value for tilt boundaries. This was shown in the folloltig manner. The experimental energy for 3 to 9’ was plotted on an expanded scale and the curve that fitted the points was extrapolated to the origin. Taking E, to be 1450 ergs/cm2, E vs. 0 curves were calculated for various values of A. The appropriate theoretical curve was taken to be the one which best fitted the extrapolated portion of the experimental curve over the largest range of 8. With this method, the best fit was obtained for A = -1.5 in which case the two curves began to deviate near 2”, and the theoretical maximum occurred at approximately 5”. Assuming A = -1.5, van der Merwe’s treatmentc2) of the twist boundary gives ~l~o = 1.95, confirming the suggestion put forth earlier to the effect that the sine function is not a good approximation to the force law for copper. Using j,~/fi~= 1.95, an energy curve was calculated from van der IEerwe’s equation, which agreed well with the experimental curve up to 4” before falling below it. The analysis given above, suggests that screw dislocations begin to interact at larger spacings than edge dislocations, indicating that their widths are larger on the (001) plane. It is clear, however, that the lack of data for 8 < 3” makes it impossible to draw any conclusions eoncern~g this latter point.
An exact evaluation of the energies of twist boundaries in terms of the appropriate dislocation model 5.4. Structure and energy of the large angle grain is difficult because of t.he lack of data at very small bo~~~~ar~ angles. Since the small angle E, could not be deterIt has been shown in the preceding sections that mined from an E/O vs. In 8 plot, any analysis of the there appears to be no advantage in considering the data must be carried out using the theoretical E,. large angle grain boundary in copper to be composed Such an assumption is probably reasonable in view of dislo~atiolls, in order to compute theoretieall~ its of the good agreement between experiment and interfacial energy. It was decided, therefore, to theory for the case of tilt boundaries. attempt an explanation of such experimental facts as A twist boundary in the (001) plane consists of in (1) the occurrence of a maximum in the energy-angle crossed grid of screw dislocations, having Burger’s relationship beginning at 18 to 22O for [OOl] tilt and vectors twist boundaries, and (2) the difference in maximum 1/2 %G b,=2-b[llO] and ba= Yb[llO.] * It will be noted that A, = f0.86 for copper is somew&t larger than the correspondingvalues for high energy boundaries in other materials. This results from the fact that in deter. Es may be computed from the formula(s). mining A, from an E/O VS. In 6 plot a more positive value will &?
0
=1:
_
b
27r J
C44(Cl,
-
2
(712)
(11)
be obtained if more large angle data are included in the least squares analysis. In this investigation results up to 0 =: 53” were included, whereas previous investigators rarely included measurements beyond @ = 30 to 35”
ACTA
328
energies for tilt and twist boundaries the grain boundary The reduced
complexity when
METALLURGICA,
by calculating
energy on an atomic basis. of this problem
it is noted
that
rational
7, 1959
no quantitative
comparison
the calculated
is considerably
for
VOL.
grain
boundaries the geometry of the lattices requires that the atoms be arranged in repetitive patterns along the The coincidence plots of Kronberg and interface.
could be made between
energies for twist and tilt boundaries,
the calculation
did give indications
angle twist boundary
that
the large
should have the lower energy.
This is primarily due to the fact that on twisting two (001) planes about their common
[OOI] direction
Wilson(37) illustrate such patterns for [OOI] and [ill]
be less on the average than for a corresponding
twist boundaries
rotation. These ca.lculations have demonstrated
and similar patterns may be drawn
for tilt boundaries. such a pattern,
It is possible to find the energy of
by assuming that the atoms interact
according to central forces, provided that (1) the number and (2) the positions of the atoms within the unit are known. (2) by assuming
Friedel et uZ.(16)determined (1) and that for the most part the atoms
of atoms within the repetitive
unit on the basis of a and thus it seems that
in order to predict boundaries
from the pattern to avoid
having
with large compressional
section. As a final point,
Vassamillet’s(r7)
(1) and (2) from
approach
structure
was to determine
of boundaries
found
in
soap bubble rafts.
from
the conditions
concerning
(1) and
(2) but
rather
to
these
given earlier in this
it is interesting
energy cusps of appreciable
of large angle
to determine
to note that no
depth,
i.e. greater than
&IO per cent, were found in either series. well mean that (1) the comparatively
In principle it should not be necessary to make some assumption
the characteristics
it will be necessary
some atoms were removed strains.
that it is not
trial and error type calculation,
quantities
bonds
tilt
feasible to select the proper number and arrangement
within the unit remained at their normal lattice sites; interatomic
the
change in bond lengths across the interface appear to
This may
ordered arrange-
ment of atoms needed to produce an energy cusp has been replaced
by a more disordered
one at the high
determine them by requiring that the Helmholtz free energy of the pattern be stationary with respect to
temperature of measurement (1065”C), due to lowering of interfacial free energy that could occur from its
any virtual displacement
higher entropy contribution, the plane of the boundary
of the pattern.
atoms are assumed to interact
When the
according
to central
forces, this is equivalent to determining their positions from the condition
If there are N atoms
in the pattern,
results in a set of 3N
If multiple
to observe the effect.
that the resultant force on every
atom in the pattern vanishes. simultaneous
well-aligned
or (2) the crystals and/or were never sufficiently
this condition
equations solutions
in 3N position
co-ordinates.
for these equations
correct ones are found
by minimizing
energy of the pattern.
Attempts
arise, the
the total free
were made to solve
6. CONCLUSIONS
(1) It accurate angle,
has
been
values
lineage
possible
to
obtain
for the absolute boundaries
reasonably
energies
in copper.
An
of these results in terms of the appropriate model shows that the Read-Shockley
of small analysis
dislocation
equation
pre-
these equations for a very simple arrangement of atoms, but even in this case a prohibitive amount of
dicts the energy up to 0 = 5 to 6”, where the dislocations are about 10 atom planes apart. For angular
work was involved. An alternative procedure
theory
misorientations also was attempted.
In
larger than this value, linear elasticity
no longer gives the correct
energy, probably
feasible
because most of the strain energy resides in regions
of atoms were selected for
the pattern, and for each pattern an energy (not a free
where Hooke’s law is not obeyed. For 0 > 5 to 6” van der Merwe’s equation is a better approximation to
energy) was calculated.
the actual energy-angle
this
case,
for
a given
boundary,
numbers and arrangements
able difficulty
several
This method led to consider-
for primarily
two
reasons:
(1) The
by Read and Shockley.
than that given
It was shown also that the
interfacial energy was found to be very sensitive to the
Peierls’
particular array of atoms used in the calculation,
van der Merwe, is not a very good representation
thus
making it difficult to detect a trend toward a minimum energy configuration, and (2) this, in turn, made it impossible to compare the energies of two boundaries having different misorientations and consequently it was not possible to predict the point at which the energy maximum should begin. Although,
for the reasons mentioned
in (1) and (2),
sinusoidal
relationship
stress-strain
relationship,
used by to
the actual force law for copper. (2) The energy data for twist boundaries did not extend to small enough values of 8 to provide a rigorous test of the dislocation model. An analysis of these results, however, suggests the deviations from the Read-Shockley equation occurs at smaller values of 0, i.e. 2 to 3”, than for the tilt boundaries.
GJOSTEIN
AND
RHINES:
TILT
AND
TWIST
(3) At large angles, both [OOl] tilt and twist boundaries, show a rather broad energy maximum beginning near 18 to 22’, with twist boundaries having a considerably lower maximum energy than tilt boundaries. It is shown that these effects cannot be predicted by assuming that the atoms interact according to central forces until the positions of atoms along the interface are known more accurately. (4) For large angle boundaries (tilt or twist) no appreciable effect of boundary orientation on its energy was found, nor were any energy cusps observed. APPENDIX
I
Herring(23,z4) has demonstrated that pure metals at low temperatures should have a ys-plot composed of maxima and sharply-~usped minima (co~espond~g to low index planes). If this is the case, a smoothly curved surface should be replaced by a hill and valley Structure,@3 i.e. faceting should occur. When faceting is present the local equilibrium condition at the root of a thermal groove can be represented by(23’
where M.has been taken equal to ,6and ys is taken to be the same for both sides of the groove. If the y8 plot ?K9 exhibits sharply-cusped minima, the term __ may au have an appreciable magnitude, and Yc: as calculated from equation 3 (text) will be subject to error. %s * Several factors indicate that aa 1s not large for oopper at high temperatures (1065%) under an oxygen-free atmosphere (purified hydrogen). (I) The groove profile was smoothly curved down to the root; no large scale facets were observed with this type of profile. (2) The ratios yQ/yAswere r~sonably repr~u~ible (&IO%).
If, as Brooks estimates,@‘)
magnitude 0.01 for copper, it would be expected that Ycr/Ys would fluctuate by 50.01. For large angle boundaries this amount would be within the experimental error, but for lineage boundaries YG 1y8 z 0.04, hence a variation of fO.01 should have been detected easily. (3) The shape of the normal profile agrees(21J7) reasonably well with that predicted by Mulling, who in his treatment, neglected the variation of surface properties with orientation. (4) If faceting is permitted to occur, the profile becomes badly distorted, and the dihedral angle fluctuates by large amo~ts.(l’) 3
GRAIN
BOUNDARIES
IN
COPPER
329
The effect discovered by Mykura,‘3g) i.e. for a given pair of twin boundaries in niokel, one dihedral angle is less than 180’ and the other is greater 180’ by nearly the same amount, shows that in this case _2 8Ys -._ may be as large as 0.05. Mykura gives the ys, 1 aa I impression that all nickel twins exhibit this effect, but some recent observations’40) indicate that there are exceptions. Mykura also believes that the same is true for copper. During this investigation, twenty pairs of twin bo~daries were examined, and the Mykura effect was found in only one case. Some pairs, however, showed decidedly unequal dihedral angles, both being less than 180”. This latter phenomenon was found to occur in some instances in grains which have a slightly roughened surface. Although much information has to be learned yet about the orientation dependence of ys, it is believed that the observations given above indicate that for copper surfaces at 1065 C in a hydrogen atmosphere aY, in computing yalys t,he neglect of the term -2 -ys I aa I probably does not result in a large systematic error in the energy values reported herein. ACKNOWLEDGMENTS
The authors wish to thank the Aluminum Company of America for supporting this research in the form of a fellowship grant to one of US (N. A. G.). Thanks are due also to Dr. P. Shewmon for helpful discussions and for the preparation of many copper single crystals. The authors are indebted to the following laboratories for the generous use of their Interference Microscopes : (1) Westinghouse Research Laboratories, Westinghouse Electric Corp. (East Pittsburgh, Penna.) (2) Research Laboratories, Westinghouse Atomic Power Division (Bettis Plant). (3) Central Research Laboratories, Pittsburgh Plate Glass Company, (Creighton, Penna.). REFERENCES &XL 78,276 (1950). 1. W. T. READ and W. SEOCKLEY,Phys. 2. J. H. VAN DER MERWE, Proo. Phys. Sm. A83, 616 (1950). 3. C. G. DUNN and F. LIONETTI, Trans. Amer. Inst. Min. (M&U.) Eragws. 185, 125 (1949). 4. C. G. DUNN, F. W. DANIELS and M. J. BOLTON, Tram. A?ner. Is&. iTfin. (~~~~ZZ.) Bngm. 128, 1245 (1950). 5. K. T. Aus~ and B. CHALMERS, Pmt. Roy. Sm. A 201, 210 6. !??)~uYT (1950). 7. ~~~io~~
and B CKALMERS,Proc. Roy. Soe. A 204, 359 Tram. Amw.
In&.
Min.
(~~e~~.)
.Enpx.
8. W. k’. READ, Di,rlocationuin Cm@ds. McGraw-Hill (1953). 9. H. BROOKS, Metal Interfaces, p. 20. American Society for Metals Monograph (1951). 10. W. M. LOMERand J. F. NY& PTOC. Roy. Sot. A 212, 576 (1952).
330
ACTA
METALLURGICA,
11. A. P. GREENOU~H and R. KING, J. Inst. Met. 79, 415 (1951). 12. R. B. SHAW, T. L. JOHNSTON,R. J. STOILES,J. WASHBURN and E. R. PARKER, Mine& Research Laboratory Report, Ser. 27, Issue 14. University of California (May; 1958). 13. H. MYPURA, Proc. Phys. Sot. B 67, 281 (1954). 14. B. CHALMERS, R. KING and R. SHUTTLEWORTH,PTOC. Roy. Sot. A 193, 465 (1948). 15. J. FRIEDEL, B. D. CULLITY and C. CRUSSARD,Acta Met. 1, 79 (1953). 16. L. VASSAMILLET, Term Paper, Carnegie Institute of Technoloev 11954). 17. N. A. GJOSTEIN; PhD Thesis, Carnegie Institute of Technology (February, 1958). 18. P. JACQUET, Trans. Electrochem. Sot. 69, 629 (1936). 19. W. LINNIK. Dokl. Akad. Nauk SSSR 21 (1933). 20. S. AMELIN~KX, Physica 19, 1175 (1953). 21. J. E. HILLIARD, to be published. 22. J. R. TOLMON and J. G. WOOD, J. Sci. In&rum. 33, 236 (1956). 23. C. HERRING, The Physics of Powder Metallurgy, p. 143. McGraw-Hill (1951). 24. C. HERRING, Structure and Properties of Solid Surfaces, p. 5. University of Chicago Press (1952). 25. P. 0. HOEL, Introduction to Mathematical Statistics, p. 76. Wiley, New York (1951).
VOL.
7, 1959
26. J. C. FISHER and C. G. DUNN. Imverfections in Nearlu Perfect Crystals, p. 317. Wiley, ‘NewaY&k (1950). 27. H. UDIN, A. J. SHALER and J. WULFF, J. Metals N.Y. 1, 186 (1949). 28. R. SHUTTLEWORTH,Discussion to reference 26. 29. H. UDIN, Trans. Amer. Inet. Min. (Met&l.) Engr. 189, 63 (1951). 30. W. T. READ and W. SHOCKLEY, Imperfections in Nearly Perfect Crystals, p. 352. Wiley, New York (1952). 31. F. C. FRANK, Carnegie Institute of Technology Symposiuwa on the Plastic DeformationSolids. Office of Naval Research (1950). 32. E. SCHMID and W. BOAS, Plasticity of Crystals. Hughes, London (1950). 33. F. R. N. NABARRO, Advanc. Phys. 1, 269 (1952). R. E. PEIERLS, PTOC. Phvs. Sot. 52. 34 (1940). ::: A. J. FOREMAN, M. A. ~ASWON aid J.‘ K. &OOD, PTOC. Phys. Sot. A 64, 156 (1951). J. S. KOEHLER, Phys. Rev. 60, 397 (1941). M. L. KRONBERO and F. H. WILSON, Trans. Ame?. In&. Min. (Met&Z.) Engrs. 185, 501 (1949). 38. W. W. MULLINS, J. App. Phys. 28, (3), 333 (1957). H. MYKURA, Acta Met. 5, 346 (1957). ::: P. SKEWMON, private communication.
AND
and F. N. RHINESS
The absolute interfacial energies of [OOl] tilt and twist boundaries in copper have been determined, by using a Zeiss Interference Microscope to measnre the dihedral angles which form during the thermal grooving process. An analysis of the results obtained for small angle, lineage boundaries, in terms of the appropriate dislocation model, shows that the Read-Shockley equation predicts the energy up to misorientations of 5 to 6”, but that for larger misorient&ions, it gives much too small an energy. It is shown also that van der Merwe’s treatment, which avoids some of the limitation8 present in the Read-Shockley derivation, is capable of predicting the energy up to misorientations of 8 to 9”. Moreover, from the analysis presented herein, it is possible to offer an explanation for the apparent agreement that was obtained, at large misoriantations (25 to 30”), between the previous measurements of relative grain boundary energies and the Read-Shockley equation. The large angle grain boundary is characterized by a broad maximum in energy that shows no energy cusps, for either tilt or twist boundaries. Both types of boundaries have nearly the same energy for misorientations less than la’, but for the range of misorientations, over which there is a maximum in energy, twist boundaries have a lower energy by about 130 ergs/cmr. An attempt to explain this difference from the results obtained from a calculation made on an atomic basis is discussed.
ENERGIES
ABSOLUES
d’INTERFACE DES JOINTS FLEXION [OOl] DU CUIVRE
DE TORSION
ET DE
Les enorgies absolues d’interface des joints de torsion et de flexion [OOI] du cuivre sont determinees au microscope interferentiel Zeiss, par la mesure de.8 angles di&dres qui apparaissent au tours du meoanisme d’entaille thermique. L’analyse des resultats obtenus pour des joints iafaible angle, eompte tenu du modele de dislocation approprie, montre que l’bquation de Read-Shockley prevoit correctement l’energie jusqu’a des divergences d’orientation de 5-6”, mais qu’au-dela elle conduit a une bnergie trop faible. Cette analyse montre Bgalement que la methode de Van der Merwe, qui Qvite certaines limitations de l’equation de Read-Shockley, permet de prevoir l’energie jusqu’a des divergences d’orientation de 8-9”. En outre, la discussion des resultats obtenus iei, permet d’avancer une explication de l’aocord apparent obtenu pour de grandes divergences (W-30”), entre les mesures obtenues pr&edemment et l’equation de Read-Shoekley, dans le calcul des Energiesrelatives des joints. Le joint it grand angle, qu’il soit de flexion ou de torsion, est earact&id par un large maximum Bnerg&ique. Les deux types de joint ont pratiquement la mdme Bnergiepour des divergences d’orientation inferieures it 18”, mais pour les angles auxquels correspond un maximum energ&ique, les joints de torsion ont une Bnergie moindre de 130 ergs/cm2 environ. Les auteurs discutent alors, sur la base de leurs resultats, d’une tentative d’explication de cette difference, par un calcul qui s’appuie sur une base atomique. ABSOLUTE
G~ENZFLACH~~ENERGIEN VERDREHUNGSKORNGRENZEN
DER [loo] NEIGUNGS-
UND
BON KUPFER
Die absoluten Grenzflachenenergien der [loo] Neigungs- und Verdrehungskorngrenzen van Kupfer wurden beatimmt unter Verwendung eines Zeiss Interferenz-Mikroskops zur Messung der Diederwinkel, die sich beim thermischen Furchungsvorgang ausbilden. Eine Analyse der Ergebnisse an KleinwinkelVerzweigungsstrukturgrenzen auf Grund des angemessenen Versetzungmodells zeigt, dass die ReadShockle~-Gleichung die Energie bis zu Orier~tierun~un~~chieden von 5 bis 6” voraussagt, sie gibt jedoch fur griissere Oreinti~~ngsunterschiede eine zu kleine Energie. Zudem wird gezeigt, dam die Behandlung van der Merwes, die einige Beschrankungen der Read-Shockleyschen Ableitung vermeidet, die Energie bis su Orientierungsunterschieden von 8 bis 9” voraussagen kann. Weiterhin ist es duroh die hior gegebene Analyse moglich eine Erkliirung fur die anscheinende Ubereinstimmung zu bieten, die grossen Orientierungsunterschieden(25 bis 30”) zwisohen friiheren Messungen der relativen Korngrenzenergien und der Read-Shockley-Gleiehung erhalten wurde. Die Grosswinkelhorngrenze wird dureh ein breites Energiemaximum charakterisiert, das weder fur Neigungs- noch fur Verdrehun~ko~~enzen ~nergiespitzen aufweist. Beide Typen van Korngrenzen haben fur Orientie~gsuntersohiede unterhalb 18” beinahe dieselbe Energie, doch im Bereich der Orientierungsunterschiede, in dem ein Maximum der Energie auftritt, haben dio Verdrehungskorngrenzen oine urn etwa 130 erg/cma geringere Energie. Es wird ein Versuch erortert diese Unterschiede aus den Ergebnissen einer atomistischen Bereohnung zu erklaren. * Abstracted from a thesis presented by the Aluminum Company of America Fellow, N. A. Gjostein, to the Faculty of the School of Engineering and Science, Carnegie Institute of Technology, in partial fulfillment of the requirements for the Degree of Doctor of philosophy in ~et~l~gieal Engineering. Received July 24, 1958. t Research and Development Laboratory, Thompson Ramo Wooldridge, Inc., Cleveland, Ohio. $ Department of Metallurgical Engineering, Carnegie Institute of Technology, Pittsburgh, Penna. ACTA METALLURGICA,
VOL. 7, MAY
1959
319
ACTA
320
METALLURGICA,
1. INTRODUCTION
From measurements of the absolute interfacial energies of small angle boundaries, it is possible to provide a rigorous test of the dislocation model of the grain boundary, for in this case the experimental energy-angle relationship may be compared quantitatively with the predicted relationships, which have been derived by Read and Shockleyd) and van der Merwe.c2) It is expected that the Read-Shockley treatment, which is based on linear elasticity theory, will be valid only when the spacing of the dislocations D is considerably larger than DL, the diameter of a cylindrical surface, surrounding the dislocation line, which encloses material experiencing non-Hookean strains (DL might be defined also as a dislocation width). Subject to this limitation Read and Shockley found that the interfacial energy E of an array of dislocations varies with the relative rotation 0 between the crystals, according to the expression E = E&A
-
In 6)
(1)
where E, is a parameter which can be evaluated independently from known constants of the material and A is related directly to the core energy of the dislocation, i.e. the energy residing within the cylinder of diameter D,. When measurements of relative grain boundary energiesf3-‘) were compared with the Read-Shockley equation, it was found that equation (1) appeared to fit (qualitatively) the form of experimental E vs. 8 curves up to large values of e (25 to 30”), indicating that the dislocation widths in these materials were unusually small, i.e. DL s b, where b is the magnitude of the Burger’s vector b. Concerning this anomaly, Lomer and Nye (lo) have commented that it seems unreasonable for the dislocations to be so narrow, and Readc8) and Brooks@) have indicated that the agreement at large angles is probably due to a fortuitous cancellation of errors in the derivation. Further insight into this problem can be obtained from the following considerations. To compare the relative energy measurements with equation (1) it was necessary to fit the calculated curve to the large angle data; the result of this was that only the form of the energy-angle relationship was tested. It was not possible to determine whether the Read-Shockley equation gave the correct absolute magnitude of the grain boundary energy, for example, by obtaining an experimental value for E, at small angles, and comparing it with the proper theoretical value. It seems possible, therefore, that the values of the parameters used to fit the calculated curve to the large angle relative energy values may not have any theoretical
VOL.
7, 1959
significance. An ambiguity of this sort can be avoided when absolute energies are measured, since in this case E,, which is obtained from a plot of E/B vs. In 8, can be compared with the value derived from the appropriate dislocation model. Absolute energies of tilt boundaries in silver(ll) and zinc’12) have been measured but in neither case was it possible to determine E, at small angles. With the recent advanceso3) in interference microscopy, it was felt that it would be feasible to determine the absolute energies of [OOl] tilt and twist boundaries in copper from measurements of dihedral angles formed by the thermal grooving process.(11~14)It was considered important to extend the measurements over as wide a range of 8 as possible, with particular emphasis on small values of 0, and other discrete large values of 0, where energy cusps might be expected to occur. In addition, considerable attention was paid to the effect of boundary orientation on its energy. From these measurements, it was thought that it would be possible to draw conclusions, concerning the structures of both small and large angle boundaries, the latter type being particularly interesting in view of the possibility of calculating their energies on an atomic basis.(15y16) 2. EXPERIMENTAL
PROCEDURE
2.1 Preparation of specimens Cathode sheet copper, having a purity of 99.98 per cent Cu, was used to prepare from the melU5) bicrystals, containing [OOl] tilt and twist boundaries. Bot,h types of bicrystals were prepared from seeds having a cube direction within 0.5’ of their longitudinal axes. Thus, each pair of crystals had in common a cube direction, about which they were rotated with respect to each other by the angle 8. Tilt boundaries in the range 5” < 8 < 78” were symmetrical, i.e. the plane of the boundary made an angle 812 with the [OlO] direction of either crystal. It was possible to extend the energy measurements to 8EO.6” by making use of the lineage boundaries formed during the growth of most of the bicrystals. The lineage boundaries were asymmetric tilt boundaries, whose twist component rarely exceeded 0.2”. The asymmetry angle $, i.e. the angle between the mean of [OIO] directions of the crystals and boundary plane varied from 0 to 28”. In order to insure that the axis of relative rotation was within 1.0” of the [OOl] direction, and that the magnitude of the rotation 8 could be predetermined with the same accuracy, the necessary seeding operations were done mechanically by means of a special type of goniometer.07) For the lineage
GJOSTEIN
AND
RHINES:
TILT
AND
TWIST
GRAIN
BOUNDARIES
IN
COPPER
321
FIG. la. Interferogram showing the perpendicular-type fringe pattern of a thermal groove which formed at a 32” symmetrical [OOl] tilt boundary; the plane of the surface shown is parallel to the common [OOl] direction and normal to the gram boundary; contour interval, 0.27 ,u; x 762.
FIR. lb. Interferogram showing the parallel-type fringe pattern of a thermal groove formed at a 37” symmetrical [OOI] tilt boundary; the plane of the surface shown is parallel to the common [OOl] direction and normal to the gram boundary; contour interval, 0.27,~; x 762.
boundaries,
izes the Linnik interference
8 was
measured
to
fO.2”
by
super-
system,og)
and is parti-
imposing the Laue spots from each crystal on a single
cularly well suited for large wedge angle interference.
film.
A discussion
With
some
modificationo7)
used to measure the value obtained
in this way was usually
1 .O” of the predetermined The section
this method
was
0 for the large angle boundaries; within
misorientation.
axis, and were deeply etched.
dihedral angles of thermal
grooves has been given by Mykura.os) It is convenient
bicrystals, which had a rectangular cross (A in. x 1 in.), were sectioned in a plane
normal to the growth
of the Linnik system, and its limitations
with respect to measuring
to determine
from either perpendicular Figs.
la
and
lb,
the dihedral
respectively.
shown that, for the perpendicular
Amelinckx(20)
has
fringe pattern,
the
Both the as-cast (top) surface and the cut surface were
angle u, that one side of the groove
mechanically
vertical,
prepare
polished,
them
and then electropolished(ls)
for the thermal
In order to avoid the formation
grooving
to
treatment.
Deoxo
hydrogen unit,
and
atmosphere over
During this treatment, in recrystallized
copper
of facets on the sur-
(passed turnings
through
a
at 650°C).
the specimens were supported
alumina
crucibles,
surrounded
by
copper foil. The specimens were annealed for 37 hr at 1065°C; the shape of the thermal groove profile after annealing under these conditions
makes with the
is related to the corresponding
fringe angle q
by the relation
face, and to obtain reproducible dihedral angles, it was found necessary to anneal the specimens in a purified
angle
or parallel fringe patterns,
is shown in Fig. la.
2.2 .Mea.surement of dihedral angles Dihedral angles were measured by the use of a Zeiss Interference Microscope. This instrument util-
2L
(2)
tan cc = ;Im tan *II where L is the fringe spacing, il the wavelength
of the
light source, and m is the linear magnification. The errors involved in measuring 7 from enlarged photographs and the related errors in u have been discussed by Gjostein.(l’) This method is versatile, and can be used for either high or low energy boundaries. To give some idea of the magnitudes of a and q, it may be said that as u varied from 89”30’ to 80”0’, r,~varied from 50 to IO”. Hilliard(21) has suggested a method
of determining
cc by plotting a profile from the parallel fringe pattern,
ACTA
322
METALLURGICA,
Fig. lb. In this investigation it was found that this method could be applied only to grooves having a depth of greater than 0.5 ,u, i.e. for high energy boundaries. Therefore, the majority of the measurements were made by using the first method, but, for some high energy boundaries, both techniques were used. Both methods give an over-estimate of u due to the obliquity effect, reported by Tolmon and Wood, for interference microscopes using convergent illumination. This results in an apparent decrease of about 10 per cent in the grain boundary energy.(r7) Accordingly, all energy values reported herein have been increased by 10 per cent to compensate for this effect.
TABLE
a+B y(J = 2y, cos ~ 2
(3)
‘7, 1959
1. Dihedral angles and ratio ya/ys for tilt boundaries at 1065°C.
tf 0%) 0.65 o”::, 0.85 0.9 1.3 1.3 1.5 K 3.0 5.5 :*: rz:o 16.0 19.0 22.0 26.0 28.0 32.0 37.0 40.0 45.0 53.0 65.0 78.0
3. RESULTS
During the course of this investigation, a variety of groove profiles was observed. It was found, however, that only the normal-type profile, having the characteristics shown in Fig. la, gave reproducible values of a. In Appendix I, evidence is presented which indicates that it is permissible to neglect the orientation dependent terms of Herring’s general equation for interfacial equilibrium,(23~24)provided that the measurements of a are taken from normal profiles. Under these conditions, the grain boundary surface tension* ya may be calculated from the equation
VOL.
a-l-B
---.* Corrected for obliquity effect reported by Tolmon and Wood.‘=’ 4. ANALYSIS
OF RESULTS
Equation (1) can be used to represent the energy data by placing it in the form Ry = R&A
where a and B are the angles that each side of the groove makes with the vertical? and ys is the average surface tension of the free surface; the ratio yo/ys has been calculated from equation (2) and corrected by 10 per cent as indicated previously. Tables 1 and 2 give the values of a + /3 obtained for [OOI] tilt and twist boundaries at 1065C~ from perpendicular fringe patterns, along with corrected ratios of yG/ys. Each value of a + /l represents an average taken from 15 to 20 fringes. In many cases, the specimen was repolished and completely retreated to produce a new groove which was measured using 15 to 20 fringes. It was found that yG/ys was reproducible to within &lo per cent for most boundaries.
* The definition of “surface tension” used here is that $msn)y Herring’2*’ and should not be taken to mean “surface t The plane of the grain boundary was always within 2’ of being normal to the free surface. $ There is a measurable temperature coefficient for ye in the case of large angle boundaries, but this effect does not influence the conclusions stated herein.
0.038 0.044 0.047 0.048 0.051 0.064 0.052 0.066 0.084 0.099 0.094 0.135 0.144 0.202 0.222 0.294 0.300 0.328 0.339 0.338 0.356 0.350 0.374 0.362 0.353 0.322 0.217
178O2’ 177”42’ 177’36’ 17v32 177y22’ 176’40 177”18’ 176”36 17v40 174O54’ 175’8 173”4 172’36’ 169’36’ 168’32’ 164O54 164’32’ 163”s’ 162O28’ 162’34’ 161”38’ 160”48’ 160’49’ 161°1;f 161’56’ 163’20’ 168’48
-
In @)
where R, = E/ys = ya/ys, and R, = Eo/ys- Plots of RY/Ovs. In 6 for [OOl] tilt and twist boundaries are shown in Figs. 2 and 3, respectively. It can be seen TABLE
2. Dihedral angles and ratio ye/jjs for twist boundaries
at 1065°C
3 (d+s)
-.
Yell/s’
a+B -_-_
--
E 5:o ii:::
10.0 13.0 15.0 17.0 23.0 30.0 35.0 37.0 40.0 43.0 45.0
176”44’ 175’28 174”6 173O2’ 170”44 170’12’ 168’48 168OO’ 167”36’ 166”36’ 164”52’ 164’8 165”24 164’4 165”56 164”52’
0.063 0.088 0.115 0.136 0.180 0.190 0.218 0.234 0.240 0.261 0.293 0.308 0.282 0.307 0.295 0.293
* Corrected for the obliquity effect reported by Tolmon and Wood.‘=’
GJOSTEIN
0.5 4.01 ’
I I
AND
RHINES:
8 I Degrees1 5 IO
i!
I
TILT
AND
TWIST
GRAIN
TABLE
20
BOUNDARIES
3. Pammeters
I
%
COPPER
323
R, and A 8s determined from Ry/% vs. In 0 plots
50
I
IN
Range of @
A
[OOl] Tilt series-1065°C 0.91 & 0.05 -0.72 0.49 f 0.01 $0.71 [OOl] Twist series-1065°C 0.46 0.54
10 43
0.6 < f3 < 12 3<8<53
36
5<8<45 ___.--
* Calculated from A = 1 + In &lax. (8)
i
I
s.0
-4.0
t -3.0
-2.0 In 0
-1.0
0
Fra. 2. Plot of Ry/i3 VS. In 19for [OOl] tilt boundaries at 1065°C.
that the data for tilt boundaries, Fig. 2, cannot be represented by one straight line, and that the slope (--R,) increases (negatively) as 8 decreases. For purposes of a.nalysis, the results were divided into two overlapping ranges, 0.6” < 8 < 12” and 3” < 8 < 45’. Then, using the method of least squares,(25) linear regression lines were determined for each range of 8. Prom the slopes and the intercepts of the regression lines, it was possible to determine the small and large angle values for R, and A; these are given in Table 3. In view of the results obtained for tilt bo~daries, it was expected that a plot of R,/6 vs. In r9for twist boundaries also would show a change in slope at small
1.6
8 (Degrees) IQ 2,O
’
59 I
0’
I
-3.0
I
I
I
I -1.0
-2.0
II 0
In 8 FIG. 3. Hot of Rv/@ vs. ln 8 for [OOI] twist boundaries at 1065°C.
values of 8. Unfortunately, it was not possible to obtain any energy data for 0 < 3” in this series and therefore it is not clear whether twist boundaries present the same effect, Fig. 3. The large angle values of R, and A for twist boundaries are given in Table 3. Energy vs. 0 curves, which were calculated from equation (4), using the large angle parameters, are shown in Figs. 4 and 5 with the experimental point superimposed. These curves have no theoretical significance, as will be seen in the next section, but they can be used as an empirical representation of the energy data over a wide range of 0. On these plots, in addition to the dimensionless ratio ya/ys, absolute values of yG are reported. For this purpose, yS was taken to be 1670 ergslclnz. According to Fisher and Dunnt2@, the most reliable value of ys for copper is 1430 ergs/cm2 in the range 950°C to 105O”C, as determined by Udin et c&(2’) It has been pointed out,(2s*29) however, that Udin et al. neglected the effect of transverse grain boundaries in computing ys. Taking the number of grains per unit length along the wire to be 80(2s) and yG/ys = 0.37, it can be shown that a more probable average value for ys is 1670 ergs/cm2. The calculated curve for the [OOl] tilt series has a maximum at 8 = 43’. Beyond @ = 43’ the calculated curve would begin to decrease; therefore, in this region the curve has been drawn to fit the points, the dashed line indicating uncertainty in the data. The calculated curve which best fits the twist boundary series has a maximum at 6 := 36”, but the experimental points could as well fit a curve with a maximum at 0 = 45”. There is no need to extend the measurements beyond 45’ because the energyangle relationship for an [OOI] twist boundary series must be symmetrical about this value. The difference in the energy between tilt and twist boundaries is not very large until 8 exceeds about 18’, where the tilt series has a higher energy, by some 130 ergs/cm2. It is important to mention here that no abnormal scatter, i.e. greater than fl0 per cent, was observed
ACTA
324
METALLURGICA,
VOL.
7, 1959
0.40
0.32
0.24 a” \ P
0.16
8 (Degrees) FIU. 4. Dependence of grain boundary energy on misorientation for [OOI] tilt boundaries at 1065°C. Solid line represents the curve c&~&&d from equation (I), using the lwge a~& paraw&ers. Although the curve has no theoretical significance it can be used as an empirical representation of the energy data over the range 5’ < 0 < 43”. Beyond 43”, the ourve has been drawn to fit the experimental points.
at 8 = 37” (310) and 8 = 53’ @IO), the presumed major energy cusp positions of the [OOl] tilt series. Since for each critical angle three different bicrystals were measured, the probability that the cusp has escaped detection is small. Moreover, no energy cusp was detected at 3’7“ in the [OOl] twist series. No effect of the orientation of the boundary upon its energy was found for large angle boundaries, including those near energy euap positions. 5. DISCUSSION
where $’ = # rJz45” refers to the axes [llO] and [IiO], the sign of 45” being chosen so that $’ and # he between 0 and 90”, and cI1, etc. are the elastio constants of the material. In the second model, four sets of of ~slocations are needed; each set has equal edge and screw components. For copper, however, E,, calculated from this model, is higher than for the first one (31) and therefore, this model need not be considered further, since in a welLannealed specimen, the lowest energy configuration should obtain. Table 4 gives the comparison between experimental and theoretical values of E,. The theoretical E, was computed from equation (5), using the room
To provide a rigorous test for the Read-Shockley equation, it is necessary to compare the experimental values of the parameters E, and A with those derived from dislocation theory. Read and Shockley(30) have made use of Frank’s formula(31) to determine the dislocation content of an [OOl] tilt boundary in a face centered cubic lattice, and they find that there are two possible dislocation models. Model one consists of two sets of edge dislocations, having Burger’s vectors
212
&=,b[llO]
and
1/i
b,=~b[llO].
_
Bor this model E, is given by E 0- -
cos q +
~-__
sin#
47r X--
e
b (Q,, + C,z)
- Cl,) -______ J~llf%O’Dell (@ + Cl, + 2C**)
(Degrees)
Fro. 5. Dependence of gmin boundary energy on misorientation for [OOl] twist boundaries at 1065°C. The curve was calculated from equation (l), using the large ungle parameters, and therefore it has no theoretical signiflcanoe, but it can be used to represent the experimental points.
GJOSTEIN TABLE
AND
RHINES:
TILT
4. Comparison of theoretical and for [OOl] tilt boundaries
E,(ergs/cm2)
AND
experimental
TWIST E,
GRaIN
BOUNDARIES
IN
COPPER
325
where e is the Naperian base and e is Poisson’s ratio. Foreman,
Jaswon
and Wood(35)
have
considered
more realistic force laws for the center of a dislocation
Method of determination
from which Lomer(33) has obtained the energy per unit 1550 1480 1520 & 100 820 * 15
* Computed 1670 ergs/cu?.
temperature 15 per
R,
from appropriate
elastic constants
cent
for
measured between the [loo]
=
directions)
determined
The averaging measurements
of E,
Examination
and the other
from the relation
d4’ = I.35 E,(O)
having
selecting
a
of the values listed in Table 4 reveals
considering
the
the proper
uncertainties
temperature
well with
involved
coefficient
-
[
1
-1s -sL,-
.
0
Read-Shockley
equation is valid at large 0; therefore,
neither of these methods Table
in
for the
will be discussed.
will be considered
obtained
from equation
from equations
(8) and (9).
Experimental
values of A, are listed in addition. the theoretical
treatment
for A, is not. as
rigorous as that for E,, the fact that the small angle A, lies between the two best theoretical determinat,ions is considered
to be significant.
From the discussion
that has preceded,
to the treatment
it is clear
boundary,
of Read and Shockley,
according agrees with
angle value for E, agrees more closely
with theory
the measured energy only when 0 is small.
than does the large angle one, which
differs
can be demonstrated
by a
with dislocation
data with the theoretical
E vs. 0 curves, Fig. 6.
theory.
are shown. One was calculated from equation 1, using the small angle parameters (i.e. essentially the
Read
and Shockleyo)
have
in terms of a non-
energy E, which resides inside a circle of
radius rL = D,/2, and an elastic energy ElII, which depends on the unknown boundary conditions at rL.
They show that the unknowns
can be represented
rL, E, and E’II,
by a single parameter
r. < rL, in
theoretical parameters),
Two Read-Shockley
while the other one represents
theoretical
+ln&
5” to 6”. At larger angles it begins to deviate and shows much too small an energy at the theoretical maximum.
density
7r 0 where A, refers to c$’ = 0 or 90”.
points
out
that
the
relationship
To evaluate
from t’he core radius r. to a distance
TABLE
5. Comparison
for T < rL. Nabarro(33), of Peierls(34), has
is cut off at a radius
b ro = ~~~~e(1 - a)
of the
r,/b,
concerning
determined the energy associated with each edge dislocation of an interacting pair, and he shows that the same energy can be obtained from elasticity theory, if the integration
Read-Shockley
the elastic energy
order b/28, where the stress fields of the adjacent
it is necessary to make some assumption, using the sinusoidal approximation
The
curve fits the data very well up to about
formula was derived by integrating
A,=1
curves
an empirical curve fitted to the large angle data.
Nabarro(33)
which case
the stress-strain
This fact
by comparing
values for A also may be compared
shown that A can be expressed Hookean
rather effectively
the small angle experimental
factor of about 1.8.
These
(7) using values of r,/b
ys.
point to note that the small
Van der
below.
values of A,.
5 lists the theoretical
were derived
that the energy of a dislocation
Experimental
(9)
for ro. Koehler’s method, however, contains errors(33) and Brooks has assumed that the
elastic constants and in selecting the correct value for It is an important
a
l-
as
Koehler(36), Brooks(g), and van der Merwec2) also have
Although
from 45” to 17”.
that the small angle E, agrees reasonably theory,
(6)
for 0 < 5”
on boundaries
four for
given formulae
theoretical
was done because
were made
range of asymmetry
In r,/b = In 2 l---o
Merwe’s treatment
s
in terms of a parameter
of approximately
In this case, In r,/b can be expressed
3, using 78 =
Two
has the value
copper.
for copper,(32) reduced
“do, 1,” 45” Eo@‘)
o.15 ~
a, which
one refers to +’ = 45” (6 was
is an average value z.
0
in Table
temperature.(30)
values of E, are given;
3
length of an edge dislocation
Equation (5), with 9’ = 45” Equation (6) Experimental value;* small angle boundaries Experimental value;* large angle boundaries
of theory parameter
with A.
experiment
for
the
Method of determination
-0.23 -1.50 -0.57 to -0.76 0.86
Theoretical; equations (7) and (8) Theoretical; equations (7) and (9) Experimental;* small angle value Experimental; large angle value
_____ * Since A varies with 4 ’ ,(8’ it was necessary to report * range of values for A,,, corresponding to 4’ = 45” to 17’.
ACTA
326
METALLURGICA,
VOL.
7,
1959
able to express A,, as A
0
=
1
(’ - a)po
In
+
(1 -
where ,u is the elastic
shear modulus
constant
order
of
determines
the
same
the amplitude
(10)
2o)Trp
of
and p.
is a
magnitude
which
of the sinusoidal
stress-
strain relation.
Taking A, = -0.72, Table 3, (neglecting the variation of A with c#‘) it was found that ,u/,uo must be about 3.65, in order to make van der Merwe’s E vs. 8 formulat2) agree with Read-Shockley’s at small 8.*
Based on these conditions,
a theoretical
curve was calculated from van der Merwe’st2) formula. It can be seen, Fig. 4, that it does not deviate appreciably 9”.
from the experimental
As 8 increases
beyond
data until &’= 8 to
this point,
the van der
Merwe curve does not drop off rapidly, remains considerably
although
it
below the actual energy curve.
While this method seems to give better agreement for larger I3 (Degrees)
angles
pointed
FIQ. 6. Comparison of the energies of small angle, lineage boundaries with those predicted by the formulae of Read and Shockley”] and van der Merwe.‘al
used
than
Read-Shockley’s,
out that the sine function,
by van
der Merwe,
approximation
not
to the force law for copper.
This inequality
finding that p/p0 > 1.
strain relation for copper has a much smaller maximum stress than that given by the sine function
strain energy resides in regions near the dislocation
amplitude
cores, where the stress-strain relationship is non-linear.
city
An estimate of rL can be obtained from the treatment
agreement
of dislocation
widths
given
by
Hooke’s
Foreman
law becomes
distance
approximates
comparable
Assuming
curve,
i.e.
D/2 = 5b. As 8 increases
theory
for
small
strains.
between theory
Undoubtedly,
and experiment
the
could be
repeated,
using the force law of Foreman
et uZ.(~~)
that this
rL, it can be seen that it is
to the point at which the Read-Shockley
curve begins to deviate appreciably mental
when its
is fixed so as to agree with classical elasti-
et uZ.(~~) extended to larger values of 8, if, as Lomer and Nye(lO) suggest, van der Merwe’s calculations were invalid
at a distance (along the slip plane) of about 4.7b (see Figs. 2 and 3 in reference (35)).
a good This can
reflects the fact that the true stress-
b/28 = r, since at this point the greater part of the
For a,, = 4 (copper),
be
which has been
is probably
be seen from the experimental dislocations have equal importance. It is expected that this approximation will become invalid when
it should
from the experi-
at 0 = 5 to 6” or equivalently
5.2. Comparison
of results with previous
energy dais
for tilt boundaries In the past, measurements boundaries
as a function
of the energies of tilt
of misorientation,
with two
exceptions,(11912) have been made on a relative basis, the
higher
order
elastic stress field become important.@)
terms of the Brooks(g) has
and therefore it was not possible to obtain E, from a plot of E/B vs. In 8.
Instead
summed these terms, and he finds that they add an
energy-angle
infinite series in powers of t3 to A-ln 0, the first term being b,02, where b, = 1. This factor would raise
fitting the Read-Shockley
the theoretical E vs. 0 curve slightly, but not enough to account for the discrepancy between the ReadShockley formula and experiment, for 8 > 5 to 6”. Van der Merwe(2) avoids some of the difficulties inherent in the Read-Shockley method by assuming that the stress-strain relation at the interface is sinusoidal. Since van der Merwe accounts for the non-linear forces at the core of the dislocation, he is
relationship
the shape of relative
was tested
essentially
by
equation to the large angle
energy data; it was concluded(197-s) that the equation described the data very well over the entire range of 8. This
investigation
has
demonstrated
that
such
a
* For small 8, van der Merwe’s expression’2’ gives the same functional dependence of E on 0 as the Read-Shockley formula. In order to make the magnitudes of each expression equivalent, however, the factor ,427r(l - u) of van der Merwe’s equation must be replaced by the appropriate one from anisotropic elasticity thoery, in addition to requiring that p/p0 = 3.65.
CJOSTEIN
AI+D RHINES:
TILT
AND
TWIST
procedure is not valid in the case of copper, since it results in values for E, and A, that are not in agreement with theory. The same situation may be true for the previous relative energy measurements. It will be recalled that Brooks’ calculation of the experimental E,(s) for Fe-Si boundaries from the relationship E, = ~~~~~e~~~(*)gave a value 50 per cent lower than the theoretical one. This might be expected, since large angle values for E,,, and 8,,, were used. Further support for this suggestion may be obtained from a consideration of the previous determinations of A. Unlike Es, the value of this constant can be determined from either absolute or relative energy measurements. This investigation has verified the idea put forth by Brooks@) and Parker et a1.02) that A is a function of 8. The small angle A, for copper is comparable to the values, -0.82 to -1.62, obtained by Parker et oZ.02) for small angle tilt boundaries in zinc. This fact, coupled with the knowledge that the theoretical A,, value for copper probably lies in the range -0.23 to -1.50 (the larger negative value is probably more nearly correct), indicates that the positive values (0.20 to 0.55, with tin being an exception) obtained for other metals may be too large. It should be noted also that the large angle A, = +0.86; for copper* is similar to those obtained by previous investigator, indicat~g that they did not extend their measurements to small enough angles to detect a change in slope in the E/Ovs. In 0 plot, and a consequent decrease toward negative values of A,. 5.3. Dislocation model and energies of [OOl] twist bou~ar~e~
GRAIN
BOUNDARIES
ZN COPPER
327
From equation (11), E, was found to be 1450 ergs/cm2, a value essentially the same as that calculated for tilt boundaries having variable asymmetry. For sufficiently small angles, such that --In 0 > A, it would be impossible with the present experimental techniques to measure the difference in their energies. There is some evidence, however, which indieates that A for small angle [OOl] twist boundaries is more negative than the corresponding value for tilt boundaries. This was shown in the folloltig manner. The experimental energy for 3 to 9’ was plotted on an expanded scale and the curve that fitted the points was extrapolated to the origin. Taking E, to be 1450 ergs/cm2, E vs. 0 curves were calculated for various values of A. The appropriate theoretical curve was taken to be the one which best fitted the extrapolated portion of the experimental curve over the largest range of 8. With this method, the best fit was obtained for A = -1.5 in which case the two curves began to deviate near 2”, and the theoretical maximum occurred at approximately 5”. Assuming A = -1.5, van der Merwe’s treatmentc2) of the twist boundary gives ~l~o = 1.95, confirming the suggestion put forth earlier to the effect that the sine function is not a good approximation to the force law for copper. Using j,~/fi~= 1.95, an energy curve was calculated from van der IEerwe’s equation, which agreed well with the experimental curve up to 4” before falling below it. The analysis given above, suggests that screw dislocations begin to interact at larger spacings than edge dislocations, indicating that their widths are larger on the (001) plane. It is clear, however, that the lack of data for 8 < 3” makes it impossible to draw any conclusions eoncern~g this latter point.
An exact evaluation of the energies of twist boundaries in terms of the appropriate dislocation model 5.4. Structure and energy of the large angle grain is difficult because of t.he lack of data at very small bo~~~~ar~ angles. Since the small angle E, could not be deterIt has been shown in the preceding sections that mined from an E/O vs. In 8 plot, any analysis of the there appears to be no advantage in considering the data must be carried out using the theoretical E,. large angle grain boundary in copper to be composed Such an assumption is probably reasonable in view of dislo~atiolls, in order to compute theoretieall~ its of the good agreement between experiment and interfacial energy. It was decided, therefore, to theory for the case of tilt boundaries. attempt an explanation of such experimental facts as A twist boundary in the (001) plane consists of in (1) the occurrence of a maximum in the energy-angle crossed grid of screw dislocations, having Burger’s relationship beginning at 18 to 22O for [OOl] tilt and vectors twist boundaries, and (2) the difference in maximum 1/2 %G b,=2-b[llO] and ba= Yb[llO.] * It will be noted that A, = f0.86 for copper is somew&t larger than the correspondingvalues for high energy boundaries in other materials. This results from the fact that in deter. Es may be computed from the formula(s). mining A, from an E/O VS. In 6 plot a more positive value will &?
0
=1:
_
b
27r J
C44(Cl,
-
2
(712)
(11)
be obtained if more large angle data are included in the least squares analysis. In this investigation results up to 0 =: 53” were included, whereas previous investigators rarely included measurements beyond @ = 30 to 35”
ACTA
328
energies for tilt and twist boundaries the grain boundary The reduced
complexity when
METALLURGICA,
by calculating
energy on an atomic basis. of this problem
it is noted
that
rational
7, 1959
no quantitative
comparison
the calculated
is considerably
for
VOL.
grain
boundaries the geometry of the lattices requires that the atoms be arranged in repetitive patterns along the The coincidence plots of Kronberg and interface.
could be made between
energies for twist and tilt boundaries,
the calculation
did give indications
angle twist boundary
that
the large
should have the lower energy.
This is primarily due to the fact that on twisting two (001) planes about their common
[OOI] direction
Wilson(37) illustrate such patterns for [OOI] and [ill]
be less on the average than for a corresponding
twist boundaries
rotation. These ca.lculations have demonstrated
and similar patterns may be drawn
for tilt boundaries. such a pattern,
It is possible to find the energy of
by assuming that the atoms interact
according to central forces, provided that (1) the number and (2) the positions of the atoms within the unit are known. (2) by assuming
Friedel et uZ.(16)determined (1) and that for the most part the atoms
of atoms within the repetitive
unit on the basis of a and thus it seems that
in order to predict boundaries
from the pattern to avoid
having
with large compressional
section. As a final point,
Vassamillet’s(r7)
(1) and (2) from
approach
structure
was to determine
of boundaries
found
in
soap bubble rafts.
from
the conditions
concerning
(1) and
(2) but
rather
to
these
given earlier in this
it is interesting
energy cusps of appreciable
of large angle
to determine
to note that no
depth,
i.e. greater than
&IO per cent, were found in either series. well mean that (1) the comparatively
In principle it should not be necessary to make some assumption
the characteristics
it will be necessary
some atoms were removed strains.
that it is not
trial and error type calculation,
quantities
bonds
tilt
feasible to select the proper number and arrangement
within the unit remained at their normal lattice sites; interatomic
the
change in bond lengths across the interface appear to
This may
ordered arrange-
ment of atoms needed to produce an energy cusp has been replaced
by a more disordered
one at the high
determine them by requiring that the Helmholtz free energy of the pattern be stationary with respect to
temperature of measurement (1065”C), due to lowering of interfacial free energy that could occur from its
any virtual displacement
higher entropy contribution, the plane of the boundary
of the pattern.
atoms are assumed to interact
When the
according
to central
forces, this is equivalent to determining their positions from the condition
If there are N atoms
in the pattern,
results in a set of 3N
If multiple
to observe the effect.
that the resultant force on every
atom in the pattern vanishes. simultaneous
well-aligned
or (2) the crystals and/or were never sufficiently
this condition
equations solutions
in 3N position
co-ordinates.
for these equations
correct ones are found
by minimizing
energy of the pattern.
Attempts
arise, the
the total free
were made to solve
6. CONCLUSIONS
(1) It accurate angle,
has
been
values
lineage
possible
to
obtain
for the absolute boundaries
reasonably
energies
in copper.
An
of these results in terms of the appropriate model shows that the Read-Shockley
of small analysis
dislocation
equation
pre-
these equations for a very simple arrangement of atoms, but even in this case a prohibitive amount of
dicts the energy up to 0 = 5 to 6”, where the dislocations are about 10 atom planes apart. For angular
work was involved. An alternative procedure
theory
misorientations also was attempted.
In
larger than this value, linear elasticity
no longer gives the correct
energy, probably
feasible
because most of the strain energy resides in regions
of atoms were selected for
the pattern, and for each pattern an energy (not a free
where Hooke’s law is not obeyed. For 0 > 5 to 6” van der Merwe’s equation is a better approximation to
energy) was calculated.
the actual energy-angle
this
case,
for
a given
boundary,
numbers and arrangements
able difficulty
several
This method led to consider-
for primarily
two
reasons:
(1) The
by Read and Shockley.
than that given
It was shown also that the
interfacial energy was found to be very sensitive to the
Peierls’
particular array of atoms used in the calculation,
van der Merwe, is not a very good representation
thus
making it difficult to detect a trend toward a minimum energy configuration, and (2) this, in turn, made it impossible to compare the energies of two boundaries having different misorientations and consequently it was not possible to predict the point at which the energy maximum should begin. Although,
for the reasons mentioned
in (1) and (2),
sinusoidal
relationship
stress-strain
relationship,
used by to
the actual force law for copper. (2) The energy data for twist boundaries did not extend to small enough values of 8 to provide a rigorous test of the dislocation model. An analysis of these results, however, suggests the deviations from the Read-Shockley equation occurs at smaller values of 0, i.e. 2 to 3”, than for the tilt boundaries.
GJOSTEIN
AND
RHINES:
TILT
AND
TWIST
(3) At large angles, both [OOl] tilt and twist boundaries, show a rather broad energy maximum beginning near 18 to 22’, with twist boundaries having a considerably lower maximum energy than tilt boundaries. It is shown that these effects cannot be predicted by assuming that the atoms interact according to central forces until the positions of atoms along the interface are known more accurately. (4) For large angle boundaries (tilt or twist) no appreciable effect of boundary orientation on its energy was found, nor were any energy cusps observed. APPENDIX
I
Herring(23,z4) has demonstrated that pure metals at low temperatures should have a ys-plot composed of maxima and sharply-~usped minima (co~espond~g to low index planes). If this is the case, a smoothly curved surface should be replaced by a hill and valley Structure,@3 i.e. faceting should occur. When faceting is present the local equilibrium condition at the root of a thermal groove can be represented by(23’
where M.has been taken equal to ,6and ys is taken to be the same for both sides of the groove. If the y8 plot ?K9 exhibits sharply-cusped minima, the term __ may au have an appreciable magnitude, and Yc: as calculated from equation 3 (text) will be subject to error. %s * Several factors indicate that aa 1s not large for oopper at high temperatures (1065%) under an oxygen-free atmosphere (purified hydrogen). (I) The groove profile was smoothly curved down to the root; no large scale facets were observed with this type of profile. (2) The ratios yQ/yAswere r~sonably repr~u~ible (&IO%).
If, as Brooks estimates,@‘)
magnitude 0.01 for copper, it would be expected that Ycr/Ys would fluctuate by 50.01. For large angle boundaries this amount would be within the experimental error, but for lineage boundaries YG 1y8 z 0.04, hence a variation of fO.01 should have been detected easily. (3) The shape of the normal profile agrees(21J7) reasonably well with that predicted by Mulling, who in his treatment, neglected the variation of surface properties with orientation. (4) If faceting is permitted to occur, the profile becomes badly distorted, and the dihedral angle fluctuates by large amo~ts.(l’) 3
GRAIN
BOUNDARIES
IN
COPPER
329
The effect discovered by Mykura,‘3g) i.e. for a given pair of twin boundaries in niokel, one dihedral angle is less than 180’ and the other is greater 180’ by nearly the same amount, shows that in this case _2 8Ys -._ may be as large as 0.05. Mykura gives the ys, 1 aa I impression that all nickel twins exhibit this effect, but some recent observations’40) indicate that there are exceptions. Mykura also believes that the same is true for copper. During this investigation, twenty pairs of twin bo~daries were examined, and the Mykura effect was found in only one case. Some pairs, however, showed decidedly unequal dihedral angles, both being less than 180”. This latter phenomenon was found to occur in some instances in grains which have a slightly roughened surface. Although much information has to be learned yet about the orientation dependence of ys, it is believed that the observations given above indicate that for copper surfaces at 1065 C in a hydrogen atmosphere aY, in computing yalys t,he neglect of the term -2 -ys I aa I probably does not result in a large systematic error in the energy values reported herein. ACKNOWLEDGMENTS
The authors wish to thank the Aluminum Company of America for supporting this research in the form of a fellowship grant to one of US (N. A. G.). Thanks are due also to Dr. P. Shewmon for helpful discussions and for the preparation of many copper single crystals. The authors are indebted to the following laboratories for the generous use of their Interference Microscopes : (1) Westinghouse Research Laboratories, Westinghouse Electric Corp. (East Pittsburgh, Penna.) (2) Research Laboratories, Westinghouse Atomic Power Division (Bettis Plant). (3) Central Research Laboratories, Pittsburgh Plate Glass Company, (Creighton, Penna.). REFERENCES &XL 78,276 (1950). 1. W. T. READ and W. SEOCKLEY,Phys. 2. J. H. VAN DER MERWE, Proo. Phys. Sm. A83, 616 (1950). 3. C. G. DUNN and F. LIONETTI, Trans. Amer. Inst. Min. (M&U.) Eragws. 185, 125 (1949). 4. C. G. DUNN, F. W. DANIELS and M. J. BOLTON, Tram. A?ner. Is&. iTfin. (~~~~ZZ.) Bngm. 128, 1245 (1950). 5. K. T. Aus~ and B. CHALMERS, Pmt. Roy. Sm. A 201, 210 6. !??)~uYT (1950). 7. ~~~io~~
and B CKALMERS,Proc. Roy. Soe. A 204, 359 Tram. Amw.
In&.
Min.
(~~e~~.)
.Enpx.
8. W. k’. READ, Di,rlocationuin Cm@ds. McGraw-Hill (1953). 9. H. BROOKS, Metal Interfaces, p. 20. American Society for Metals Monograph (1951). 10. W. M. LOMERand J. F. NY& PTOC. Roy. Sot. A 212, 576 (1952).
330
ACTA
METALLURGICA,
11. A. P. GREENOU~H and R. KING, J. Inst. Met. 79, 415 (1951). 12. R. B. SHAW, T. L. JOHNSTON,R. J. STOILES,J. WASHBURN and E. R. PARKER, Mine& Research Laboratory Report, Ser. 27, Issue 14. University of California (May; 1958). 13. H. MYPURA, Proc. Phys. Sot. B 67, 281 (1954). 14. B. CHALMERS, R. KING and R. SHUTTLEWORTH,PTOC. Roy. Sot. A 193, 465 (1948). 15. J. FRIEDEL, B. D. CULLITY and C. CRUSSARD,Acta Met. 1, 79 (1953). 16. L. VASSAMILLET, Term Paper, Carnegie Institute of Technoloev 11954). 17. N. A. GJOSTEIN; PhD Thesis, Carnegie Institute of Technology (February, 1958). 18. P. JACQUET, Trans. Electrochem. Sot. 69, 629 (1936). 19. W. LINNIK. Dokl. Akad. Nauk SSSR 21 (1933). 20. S. AMELIN~KX, Physica 19, 1175 (1953). 21. J. E. HILLIARD, to be published. 22. J. R. TOLMON and J. G. WOOD, J. Sci. In&rum. 33, 236 (1956). 23. C. HERRING, The Physics of Powder Metallurgy, p. 143. McGraw-Hill (1951). 24. C. HERRING, Structure and Properties of Solid Surfaces, p. 5. University of Chicago Press (1952). 25. P. 0. HOEL, Introduction to Mathematical Statistics, p. 76. Wiley, New York (1951).
VOL.
7, 1959
26. J. C. FISHER and C. G. DUNN. Imverfections in Nearlu Perfect Crystals, p. 317. Wiley, ‘NewaY&k (1950). 27. H. UDIN, A. J. SHALER and J. WULFF, J. Metals N.Y. 1, 186 (1949). 28. R. SHUTTLEWORTH,Discussion to reference 26. 29. H. UDIN, Trans. Amer. Inet. Min. (Met&l.) Engr. 189, 63 (1951). 30. W. T. READ and W. SHOCKLEY, Imperfections in Nearly Perfect Crystals, p. 352. Wiley, New York (1952). 31. F. C. FRANK, Carnegie Institute of Technology Symposiuwa on the Plastic DeformationSolids. Office of Naval Research (1950). 32. E. SCHMID and W. BOAS, Plasticity of Crystals. Hughes, London (1950). 33. F. R. N. NABARRO, Advanc. Phys. 1, 269 (1952). R. E. PEIERLS, PTOC. Phvs. Sot. 52. 34 (1940). ::: A. J. FOREMAN, M. A. ~ASWON aid J.‘ K. &OOD, PTOC. Phys. Sot. A 64, 156 (1951). J. S. KOEHLER, Phys. Rev. 60, 397 (1941). M. L. KRONBERO and F. H. WILSON, Trans. Ame?. In&. Min. (Met&Z.) Engrs. 185, 501 (1949). 38. W. W. MULLINS, J. App. Phys. 28, (3), 333 (1957). H. MYKURA, Acta Met. 5, 346 (1957). ::: P. SKEWMON, private communication.