THERMODYNAMICS OF GRAIN BOUNDARIES
IN Ni-RICH NiAl
E. Rabkin, L. Klinger, V.Semenov*
Department of Materials Engineering,
TECHNION-Israel Institute of Technology, 32000 Haifa, Israel
*Institute of Solid State Physics, Russian Academy of Sciences,
142432 Chernogolovka, Russia
Abstract
Thermal grooving at grain boundaries in Ni-rich NiAl was
studied by atomic force microscopy technique. The determined average
ratio of grain boundary to surface energy for large-angle grain
boundaries at 1400 C is 0.45, which is in a good agreement with the
results of computer simulations. It has been found that in most cases
thermal grooving at the grain boundaries is accompanied by relative
shift of the adjacent grains. This shift is associated with the grain
boundary sliding caused by the relaxation of internal substructure of the
specimen. A model of grain boundary grooving with the simultaneous
sliding is developed. The calculated grain boundary groove profiles are
in a good agreement with the experimentally measured ones.
1. Introduction
Good mechanical properties, low density, high melting
temperature and high oxidation resistance of the ordered NiAl
intermetallic compound with B2 structure attract attention to this
material for more than three decades. The main obstacle for structural
applications of NiAl is its severe intrinsic grain boundary (GB)
brittleness at low temperatures. Therefore, the knowledge of GB atomic
structure and energy is important for understanding the mechanism of
intergranular failure in NiAl. While the atomistic simulation studies
addressed both the structural and energetic aspects of GBs in NiAl [1-3],
only the atomic structure of some GBs in NiAl was studied
experimentally using high resolution electron microscopy [3, 4],
whereas the data on GB energies are virtually nonexistent. The GB
energy, b, is an important parameter directly related to the cohesive
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strength, G, of the polycrystal, which in the case of ideally brittle
intergranular fracture is given as
G   s1   s 2   b
(1)
where s1 and s2 - the energies of two surfaces formed as a result
of GB fracture.
Indirectly, the GB energy is also connected with GB diffusivity
and intergranular corrosion resistance. The main goal of the present
paper was experimental determination of GB energies in polycrystalline
NiAl. Simultaneously with the energy measurements, three of five
possible macroscopic geometrical degrees of freedom (DOFs) for the
same set of GBs, namely, those associated with misorientation
parameters of the adjacent grains, were identified. Finding the
geometrical DOFs of low-energy GBs serves as a starting point in the
concept of GB engineering [5].
Recently, a high potential of atomic force microscopy (AFM)
for determining relative GB energies in metals [6] and ceramics [7] has
been demonstrated. AFM combines the possibility to scan relatively
large surface areas with the atomic resolution in the vertical direction,
thus allowing determination of dihedral angle at the root of GB groove,
, with a very high accuracy unattainable by other methods [6]. Under
the assumption of surface isotropy, the relative GB energy is directly
connected with  by the relationship
b
 
 2 cos 
s
2
(2)
Relative GB energy values determined by eq. (2) contain
however a potential error caused by the neglection of the torque terms
which stem from the inclinational dependence of s. Nevertheless, the
values obtained in the study of GB grooving in ceramics [7] were
consistent with the results of previous studies. Since the anisotropy of
surface energy in metals and alloys is, in general, lower than in
ceramics, the torque terms will be neglected in the present study, too. An
additional important source of error in determining GB energy is GB
sliding that occurs in the process of annealing and leads to a difference
in the level of adjacent grains near the GB. The formation of GB steps
caused by such a sliding have been recently revealed in Fe(Si) alloy [8].
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The driving force for this process is the dependence of b on the
misorientation angle, whereas the selective absorption of lattice
dislocations by GB has been suggested as a possible mechanism of GB
sliding [8]. Earlier, a similar phenomenon of near-GB lattice rotations in
Ni has been revealed by tracking local lattice orientation across GB with
the aid of electron backscattering diffraction (EBSD) technique [9]. In
the present study we have also found that during high-temperature
annealing of NiAl polycrystal the GB grooving process occurs
simultaneously with the GB sliding, with the amount of sliding varying
from zero to approximately 1 m. If GB grooving is accompanied by
GB sliding, the relationship between the groove width and its depth
derived by Mullins [10] and used in [7] for determining  is no longer
valid. We suggested an algorithm for determining the GB energy value
which is not affected by the process of GB sliding and also modified
original Mullins’ theory of GB grooving by taking into account GB
sliding process.
2. Experimental
A rod of NiAl intermetallic compound with nominal
composition 48 at.% Ni + 52 at.% Al was produced from Ni of 99.95
at.% purity and Al of 99.99 at. % purity by vacuum casting. The as-cast
alloy was then remelted and purified by vacuum electron-beam floating
zone melting. After two passes of the melted zone through the rod at the
velocity 6 mm/min we obtained a polycrystal with averaged grain size 2 mm.
The discs of 3 mm in thickness cut from the rod by spark erosion were
annealed in vacuum of 10-5 Pa at 1400oC for 1 h to provide relaxation of
internal stresses and stabilize microstructure. Metallographic examination of
one of the discs in longitudinal section revealed columnar microstructure
with GBs running approximately along the rod axis. Another disc, ground
and polished to the mirror quality on SiC paper followed by diamond paste
down to 0.3 m particle size was annealed for the second time at the same
conditions. Its surface was then studied by the light microscopy (LM),
scanning electron microscopy (SEM), with the attachment for electron backscattering diffraction, and AFM. Chemical composition of the specimen after
the second annealing was determined by the energy dispersive X-ray
analysis using JSM840 (JEOL, Japan) scanning electron microscope
equipped with LINK ISIS (Oxford Instruments, England) energy
200
dispersive spectrometer (EDS). EDS measurements were carried out at
20 kV accelerating voltage and 1 nA probe current in Ni K  and Al K
radiation with 30° take-off angle using pure Ni and Al as standards. The
standard deviation of the measured intensity for a single measurement
with the acquisition time 100 s did not exceed 2 %. The result, averaged
over 10 measurements and normalized to 100 %, gives Al content as
43.60.5 at.%. The partial loss of Al in comparison with the nominal
composition (52 at.%) is due to its evaporation during the remelting process.
The AFM measurements were done using the Autoprobe CP AFM
of Park Scientific operated in the contact mode and the Si Ultralevers with
nominal tip radius of 10 nm. Both the topography signal and the feedback
loop error signal were collected. For the acquisition of AFM data, the AFM
cantilever was placed near the chosen GB using the on-axis optical
microscope of the AFM and then its tip was brought into contact with the
specimen surface. Using the AFM operating system, it was possible, by
observing the distance between the cantilever tip and GB, to adjust the slow
scan direction to be parallel to the GB groove with the accuracy of 2. After
the adjustment, AFM image containing 256 x 256 pixels was obtained by
scanning in the direction perpendicular to the GB groove, with the GB being
approximately in the middle of the scanned region.
The effect of the finite size of AFM tip on the measured topography
of GB grooves has been analysed in details in the previous publications [6,
7] but for the relatively large grooves (4-6 m wide) observed in the present
work this effect can be neglected. Saylor and Rohrer determined the dihedral
angle at the groove root by measuring the width (taken as the distance
between the two maxima) and depth of the GB groove [7], while
Schöllhammer et al. [6] used the direct fit of the observed topography by the
functions suggested by Mullins [10]. However, both methods cannot be
applied in the present work due to the observed grain sliding along GB. For
that reason, the known relationship between the groove width and depth
ceases to be valid and the solution of surface diffusion problem is to be
modified [10, 11]. Moreover, in the AFM used in the study, scanning is
performed by moving specimen mounted on the holder against the cantilever
tip. The movement, implemented by piezoelectric contraction/expansion of
the walls of the tube on which the specimen holder is mounted, displaces the
specimen surface along the axis of the piezoelectric tube. As a result, the
output AFM signal always contains some instrumental parabola-type
201
distortion which is added to the true specimen topography. This instrumental
distortion, which is a common feature of many AFMs, can be removed by
substracting the averaged parabola from the primary image. Some
“flattening” of the primary AFM image is also required if the specimen
surface is not exactly parallel to the scanning plane. Such a flattening may
cause an additional error in determining the dihedral angle of GB groove
from the groove width and depth.
In the present research a procedure for extracting the dihedral angle
of GB groove different from those employed in [6, 7] was developed. Each
of the 256 line scans of the flattened image perpendicular to the GB groove
was used in calculating dihedral angle. First, the point of absolute minimum
was determined for every scan. Then, a number of points for interpolation
was chosen along the scan line on both sides of the GB groove, with the
length of the interpolated regions being about 2/3 of the distance between the
minimum and each maximum of the profile. After parabolic interpolation,
performed separately for each side of the groove profile, the intersection
point of the two parabolas was determined by solving the corresponding
quadratic equation. The example of this procedure is given in Fig. 1.
200
100
y, nm
0
1 
-100
-200
-300
-400
Intersection: (8.9527; -333.43)
5
6
7
8
9 10 11 12
x,  m
Fig. 1. To the algorithm for determining the dihedral angles 1, 2 at the root of
GB groove. The parabolic interpolation is applied separately to the left and right
branches of the AFM profile.
202
The coordinates of the intersection point of two parabolas in Fig. 1
(8.9527 m; -333.43 nm) are slightly different from those of the measured
minimum point (8.9444 m; -330.60 nm). The difference is due to the finite
information density of the AFM image. Indeed, since the distance between
neighbouring pixels is 78 nm in the above example, the location of the
obtained minimum value does not necessarily corresponds to the root of GB
groove. After determining the intersection point, the dihedral angles 1 and
2 (Fig. 1) were calculated from the slopes of interpolating parabolas at the
intersection point. In the above example 1=82.015° and 2=74.41°. Based
on these values, the relative GB energy was calculated according to equation
b
 cos 1  cos  2
s
(3)
which is valid for the isotropic surface energy and immobile GB
perpendicular to the specimen surface. Metallographic examination on
the cross-sections showed that, as a rule, no GB migration was
associated with the grooving process (no GB energy determinations
were carried out for GBs which migrated on long distances due to some
other reasons) and, because of the columnar microstructure, GBs were
perpendicular to the specimen surface. In the above example,
b/s=0.408. The values of b/s averaged for all 256 line scans typically
show standard deviation as low as 0.01. The main advantage of the
parabolic interpolation used in the present work is its consistency with
the flattening procedure of primary AFM images, in which the linear or
parabolic functions are employed.
The measurements of the crystallographic orientations of the
individual grains were performed by EBSD method carried out with the
LINK OPAL (Oxford Instruments, England) equipment using the high
resolution field emission gun SEM LEO 982 (Zeiss - Leica). Analytical
conditions for EBSD measurements were as follows: accelerating
voltage - 20 kV, beam current - about 3 nA, working distance - 21 mm,
angle between the electron probe and the sample normal - 70°. Under
these conditions the probe diameter was about 40 nm. The image of the
analyzed area on the inclined surface was obtained employing the
conventional secondary electron detector mounted on the side wall of
the microscope chamber, or using the detector of forward scattered
203
electrons which is a part of the LINK OPAL assembly. After specific
grains of interest have been selected, the microscope was switched to the
spot mode for acquisition of EBSD patterns that were recorded on a
CCD screen. The identification of crystal planes and directions
appearing in the patterns (observed as bright bands and their
intersections) was performed using LINK OPAL software. For the
calibration of angular distances between bands on CCD screen we used
EBSD pattern obtained from (001) Si single crystal mounted on a
specimen holder close to the specimen. The error in the measurement of
angular distances did not exceed 1º. The orientation matrix was
determined for each grain of interest and Miller indices of the normal to
the grain surface were calculated. The misorientation between two
specific grains was expressed in terms of an axis common to the two
crystals and the angular rotation around this axis (the angle/axis pair).
3. Results
A representative AFM image of GB region (corresponding to
GB No. 1 in Table 1) is shown in Fig. 2. Good reproducibility of both
the groove shape and dihedral angle along the GB groove root should be
mentioned. At the same time, far from the groove root the level of the
left grain in Fig. 2 is visibly lower than the level of the right grain. Some
relative shift of the adjacent grains was observed for the majority of
studied GBs.
Fig. 2. AFM image of GB No. 1.
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Table 1
Orientations of adjacent grains, corresponding misorientation at their grain
boundaries (GBs), and amount of sliding at and relative energies of GBs
GB No.
Surface I
Surface II
GB
m
b/s
1
(3 –1 0)
(4 –1 0)
15.5<-33 –4 4>
0.13
0.390.01
2*
(3 –1 0)
(4 –1 0)
15.5<-33 –4 4>
0.02
0.400.01
3
(3 –1 0)
(010)
14.4<2 –1 3>
1.2
4
(013)
(010)
20.4<23 5 –24>
0.63
5
(810)
(010)
43.0<17 2 –2>
0.5
6
(810)
(701)
26.0<1 35 2>
0.14
7
(701)
(801)
3.0<12 –8 9>
8
(1 -7 -1)
(5 -1 0)
22.0<-17 0 1>
0.23
0.490.01
9
(5 -1 0)
(061)
21.5<0 35 1>
0.0
0.410.01
10
(910)
(5 -1 0)
19.2<34 8 -1>
0.15
0.490.02
11
(160)
(010)
12.8<11 21 26>
0.41
12
(160)
(010)
12.1<4 -5 -9>
0.32
13
(014)
(010)
16.5<011>
0.53
(010)
(010)
14.5<-1 -33 7>
0.12
14
0.590.02
0.110.01
0.260.01
0.330.01
*13 difference in in-plane inclination with respect to GB No.1.
Table 1 summarizes the amount of sliding and calculated
relative energy for all studied GBs. In the second and third columns of
Table 1 Miller indexes of the surfaces of grains forming the GB are
shown in such a way that the grain of higher level appears in the second
column. Fig. 3a shows LM image of GB, a portion of which migrated
slightly during annealing. AFM image of the area of GB migration
(Fig. 3b) indicates that the region swept by migrating GB exhibits the
intermediate level between those of the left and right grains. The root of
the GB groove at the original GB position is blunted thus indicating that
the GB left its original position in the middle of annealing process and
there was enough time for smoothing the sharp groove root by the
surface diffusion mechanism. The blunting of the groove root at the
205
original GB position is visualized by the line scan of the AFM image in
the direction perpendicular to the GB (Fig. 3c).
GB
100 m
(a)
0.6
(b)
0.4
OGB
y,  m
0.2
0.0
-0.2
-0.4
0
(c)
FGB
(c)
10
20
30
40
50
x,  m
Figure 3. LM (a) and AFM (b) images of migrating grain boundary (GB) and its
linear AFM profile (c) containing original (OGB) and final (FGB) GB
positions. Note the blunted groove root at OGB position.
206
One of the possible reasons for the relative shift of adjacent
grains might be the evaporation of Al from the specimen surface with
the rate dependent on the surface orientation. If crystallographic
orientations of adjacent grains are different, the difference in
evaporation rate would result in different contraction rate of the grains
leading to the formation of a step at their GB. To verify this possibility,
Al distribution across the GBs exhibiting the largest amount of sliding
was analyzed by EDS. LM image of the studied area is shown in Fig. 4a.
White triangles that cross the GB are pointed to the lower grains, with
the corresponding numbers indicating the height of the step between the
level of two adjacent grains. Fig.4b contains EBSD patterns obtained
from the four grains observed in Fig. 4a and used for indexing
crystallographic orientation of their surfaces. Results of EDS analysis of
Al distribution performed along lines aa’ and bb’ in Fig. 4a are
presented in Fig. 4c. Both grains show the same Al content and no
discontinuity of Al distribution across the GB is observed.
(410)
(010)
a
a’
b
b’
0.63
m
(013)
0.13
1.1
m
m
(310)
500 m
Fig. 4a. LM image of four grains, with their Miller indices indicated,
which form two GB junctions; triangular marks across the boundaries point to
the lower grain, with the amount of corresponding GB sliding shown near the
triangles
For the most right GB in Fig. 4a the AFM measurements of
GB grooving were performed at two locations differing by inplane inclinations visible on the sample surface (GBs No. 1 and
2 in Table 1). As can be seen from Table 1, the amount of GB
sliding in these two positions is very different. It can be
therefore
207
4. Discussion
4.1. The mechanism of GB sliding
The following reasons for the formation of steps at GBs during
annealing can be named:
- Residual shear stresses in the specimen produced during cooling after
the previous annealing. This possibility is excluded due to the fact
that certain amount of sliding occurs also at the final position of
migrated GB (Fig. 3). Indeed, the relaxation of residual stresses
would cause the GB sliding at the initial position, whereas, taking
into account high annealing temperature (1400 C), all internal
stresses should be fully relaxed before the GB arrives to its final
position.
- The presence of steps at GBs prior to annealing as a result of
specimen preparation (polishing). In this case, as well as in the
preceding one, no sliding is to be expected at the final position of
migrating GBs. In addition, no GB steps were revealed by AFM prior
to annealing, though LM allowed to distinguish separate grains
because of their different reflectivity.
- Selective evaporation of Al from the specimen surface during
annealing. This possibility is not confirmed by the Al concentration
profiles measured across GBs, which do not exhibit any
discontinuity. Moreover, interdiffusion coefficient in Ni-43 at.% Al
system extrapolated to 1400 C from the data of Shankar and Seigle
[12] is 4.310-12 m2/s. This corresponds to 125 m wide
Dt -law) after 1 h
interdiffusion zone (estimated according to
annealing. Therefore, if the shrinkage is caused by selective Al
evaporation, the change in the grain surface level should occur
gradually over a distanse exceeding 100 m because any
concentration discontinuity in this range would be smoothed due to
the bulk interdiffusion beneath the surface. Experimentally, however,
the difference in the levels of two adjacent grains is clearly visible on
distances lesser than 10 m across GBs (see Fig. 2 and 3c). In
addition, the grain with (010) surface exhibits the lowest level in Fig.
4a. The (010) plane is the second densely packed plane, after (011),
in the bcc lattice. Since the lower evaporation rate is expected for the
densely packed planes this observation contradicts the hypothesis of
208
selective evaporation. It can be therefore concluded that selective Al
evaporation cannot be the cause of the observed GB sliding.
As follows from the above arguments, the change in the level of
adjacent grains ought to be associated with some physical relaxation
process that occurs during annealing either within the GBs or in their
close vicinity. In the previous publication [8] the hypothesis of near-GB
lattice rotations being responsible for the step formation at GBs in an FeSi alloy was put forward. The process is illustrated schematically in
Fig. 5.
GB
y
x
Fig. 5. Illustration of the mechanism of near-GB lattice rotation. The driving
force for the rotation is the decrease of GB energy.
For simplicity, a low angle GB built of the array of edge dislocations
is considered. Its energy is an increasing function of the misorientation
angle and, therefore, there exists a thermodynamic driving force for the
grain rotation which decreases the misorientation angle. Rotation of
individual grains in the fine grained polycrystals is possible [13], while
only the lattice regions close to GBs can change their orientation in the
coarse grained structures. Such a local rotation of the lattice is made
possible due to the mechanism of selective absorption of lattice
dislocations at the GBs. Let us assume that, for some reason, the right
grain in Fig. 5 is more active than the left one in supplying dislocations
to the GB. There is an attraction force between the GB and the lattice
209
dislocations having the sign of their Burgers vector opposite to that of
GB dislocations. These lattice dislocations became trapped in the GB
and subsequently annihilate with the GB dislocations. As a result, the
linear density of GB dislocations decreases and, hence, the
misorientation angle between the adjacent grains decreases, too.
Moreover, as a result of this process the right grain is left with an excess
of dislocations having the same sign of Burgers vector as the GB
dislocations. This excess leads to the near-GB lattice rotation that is
consistent with the decrease of misorientation angle. The net
macroscopic result of the dislocation rearrangement of this type is the
formation of a step at the intersection of GB with the specimen surface
(Fig. 5). The suggested mechanism is consistent with our experimental
observations. Among the investigated GBs, GB No. 9 has the lowest
angular deviation (1.3 around <1 –9.4 5> axis) from the special
coincident site lattice (CSL) 13 boundary (26.62<010>), where 
stands for the reciprocal density of coincident sites in two misoriented
lattices, and also exhibits the lowest change of level among all studied
large angle GBs. This is consistent with the known fact that both the rate
of GB sliding under the action of the applied shear stress and the rate of
grain rotation due to the misorientation dependent GB energy tend to a
minimum for the singular CSL GBs [14, 15].
We do not consider here the origin of the bulk dislocations
responsible for the near-GB lattice rotation. This may be either growth
dislocation inherited from the re-melting process or dislocations
introduced into sub-surface layers in the process of mechanical
polishing, or both. As has been recently demonstrated [16], a significant
dislocations density is produced up to the depth of about 100 m below
the surface during the final polishing stage of NiAl single crystal using
the 0.3 m Al2O3.
An interesting observation was made by J.W. Cahn and coworkers during their molecular dynamics simulations of the migration of
a GB separating a small grain embedded in a larger one [17]. The
shrinkage of the small grain was accompanied by the increase of the GB
misorientation angle due to the increase in the density of GB
210
dislocations. This example clearly demonstrates that while the
microstructure evolution is driven by the decrease of the total interfacial
energy its actual direction can be determined by the decrease of either
GB energy or GB area. In our case, the potential for the latter is limited
because of the columnar microstructure, therefore, the former option is
realized. Moreover, the idealized computer simulations do not take into
account the defect structure of the bulk (dislocations).
4.2 GB energy
Among GBs listed in Table 1 there are three low-angle GBs
(No. 7, 12 and 14) with the misorintation angles of 3°, 12.1° and 14.5°
having the relative energies of 0.11, 0.26 and 0.33, respectively. The
averaged value of relative GB energy for the large angle GBs (those
with the misorientation angle above 15°) is 0.45. It can be, therefore,
concluded that the energy of low angle GBs increases with the
misorintation angle and is considerably lower than that of large angle
GBs. This result is in line with the current understanding of GB
energetics [14]. It should be also noted that the determined value of
relative GB energy in NiIAl (0.45) is closer to that in pure metals (0.30.4) [14] rather than in ceramics (1.0-1.5) [7].
As known from the computer simulation studies [1-2, 18], the
GB energy is sensitive to the microscopic position of GB plane with
respect to the two kinds of atomic sites. In the ordered B2 structure of
NiAl, even of stoichiometric composition, Ni-rich, Al-rich or
stoichiometric compositions are possible in the GB plane. For example,
the energy of Al-rich GB is considerably higher than that of Ni-rich GB
in Ni-rich alloy [18]. Based on these results, one would expect a high
scattering of the measured GB energies, since the microscopic positions
of the GB planes should be a random result of the specimen pre-history.
However, the experimental results of the present work do not support the
hypothesis of structural multiplicity of GBs in NiAl. Indeed, the value of
GB energy remained constant with a good accuracy along the whole GB
segment of AFM scans, the size of which varied from 12 to 30 m.
211
Moreover, two segments of the same GB separated by the distance of
approximately 100 m and having different in-plane inclinations
exhibited almost identical energies (GBs No. 1 and 2 in Table 1). It is,
therefore, very probable that in the studied structure the majority of GBs
belonged to the same class, presumably, of the Ni-rich GBs.
We took an advantage of the fact that the preparation procedure
of our sample resulted in a noticeable <001> texture and plotted the
value of relative GB energy as a function of misorientation angle for
GBs with their misorientation axis deviating from the <001> axis no
more than by 15° (see Fig. 6). (GB No. 9 is not included in Fig. 6
because, according to the low value of GB sliding associated with the
GB, it belongs to a special class of CSL boundaries characterized by a
sharp local cusp in GB energy vs. misorientation dependence.)
0.8
 b/ s
0.6
0.4
0.2
0.0
0
20
40
60
80
Misorientation around <001>, deg
Fig. 6. The dependence of measured relative GB energies on misorientation
angle for the GBs with the misorientation axis close to <001>. The solid line
represents the fit by equation (4).
D. Wolf demonstrated that formal substitution of the
misorientation angle  in the Read-Shockley formula for the energy of
low angle GBs by the sin(2) gives an empirical relationship which fits
very well the calculated energies of <001> tilt GBs in the whole range of
212
misorientations [19]. As can be seen in Fig. 6, there is a good fit between
our experimental data and an equation of this type
 b   0 sin 2 A  ln sin 2 
(4)
for 0=0.08515 and A=8.27435. According to equation (4) the
maximum relative energy in the family of <001> tilt GBs is about 0.7.
There are two sets of reliable computer simulation data (both for 0 K)
for comparison with our experimental results. Yan et al. [1] calculated
the energies of some symmetrical tilt <001> GBs in stoichiometric NiAl
using the semi-empirical embedded atom model (EAM) interatomic
potential that reproduces correctly different type of defects
accommodating the deviations of composition from the stoichiometry.
The calculated relative energies for the large angle GBs vary from 0.6 to
0.92. This is considerably higher than our experimental values. Mutasa
and Farkas calculated the energies of 5 tilt <001> GBs in NiAl alloys
of different composition [18]. The relative energy of non-singular Nirich (013)<001> GB in the Ni-rich alloy was approximately 0.5, which
is comparable with the our measurements. It can be therefore concluded
that the modified EAM potential for NiAl (that reproduces correctly not
only the type of defects but also the diffusion parameters of pure Ni and
Al), as developed by Mishin and Farkas [20] and used in Ref. [18], gives
the best description of GB energetics in Ni-rich NiAl.
5. The model
Let us assume that the GB stays planar during the annealing
process. We will try now to find a solution for the surface topography
evolution supposing that the surface diffusion mechanism is operative
and under the same set of simplifications as those in the original
Mullins’ paper [10]. Our basic assumption is that the change of the
vertical coordinate of the surface, y (see Fig. 7), is a result of two
independent processes, namely, the shape evolution by surface diffusion
and the GB sliding, i.e. independent movement of two adjacent grains in
the opposite directions by the distance h(t)
213
 y
 4 y  h( t )
B

t
t
x 4
(5)
where y+(x,t) and y-(x,t) describe the solutions for the right and
left grains, respectively, and
h(0)=0
(6a)
Far from the GB the curvature of the surface is zero and no
surface diffusion occurs. Therefore,
y , t   ht 
(6b)
In equation (5) B is the coefficient of Mullins:
B
 s Ds
kT
(7)
where , Ds and  are the thickness of surface diffusion layer,
surface diffusion coefficient and atomic volume, respectively, and kT
has its usual meaning.
S
 S

2h
b
y
x
GB
Fig. 7. To the model of GB grooving with simultaneous GB sliding.
214
It should be noted that equations (5)-(7) are derived in the
framework of original Mullins’ assumptions which are valid for the
single-component system [10]. It is very likely that the surface excess
energy in NiAl is very sensitive to the deviations of the local surface
composition from the equilibrium value, which is the case for the GB
energy [1,2, 18]. In this situation the equations (5)-(7) still can be used,
but the surface diffusion coefficient should be properly renormalized
[21]. The equations (5)-(7) should be completed by the following
boundary conditions at the root of GB groove x=0
y 0  y 0
y
x

x 0
 2 y
x 2
 3 y
x 3
y
x

x 0

x 0

x 0
 2 y
x 2
 3 y
x 3
b
s
(8a)
(8b)
(8c)
x 0
(8d)
x 0
Equation (8a) represents the obvious continuity condition;
equation (8b) is the small-slope approximation of the equilibrium
condition (3); equations (8c) and (8d) represent the conditions of
continuity of the chemical potential and the surface diffusion flux,
respectively. The system of equations (5) - (8) describes the GB
grooving process with simultaneous GB sliding, the amount of GB
sliding, 2h(t), being the function of time. The solution of equations (5)(6) is sensitive to the function h(t). Only in one case, namely,
ht   t1 / 4 , these equations allow the self-similar solution exhibiting
the time-independent shape of GB groove. For any other non-zero h(t)
the solution depends essentially on time and no “universal” solutions
like that suggested by Mullins [10] exist. As follows from the
215
experimental observations, the GB sliding occurs continuosly during the
annealing, however, the exact time dependence of the amount of GB
sliding is unknown. To get an idea about its time dependence, we
performed simple calculations (not shown here) based on the following
assumptions: (i) the elastic stress field of the GB is described by the
known expression for the low angle GBs; (ii) the velocity of lattice
dislocations is proportional to the elastic interaction force between an
individual dislocation and the GB; (iii) all lattice dislocations that have
approached the GB by the distance equal to the interdislocation spacing
in the GB are absorbed by this GB. The calculations resulted in the time
dependence of the type
h(t )  1  exp  t 
(9a)
with =const.
We will, nevertheless, assume a different type of time
dependence
ht    Bt 
1/ 4
(9b)
keeping in mind that this time dependence is chosen only for the
mathematical convenience and only the qualitative comparison of the
calculated and experimentally measured groove shapes is possible. Here
=const is the only new parameter in the Mullins-type model of the
phenomenon. It should be noted that dependence (9b) captures the main
features of more realistic dependence (9a), namely, the rapid GB sliding
at the beginning of annealing process and the decreasing sliding rate at
the later stages of annealing. Let us introduce the new dimensionless
coordinates u and g according to
x  u(Bt )1 / 4
y 
b
Bt 1 / 4 g  u    Bt 1 / 4
2 s
216
(10a)
(10b)
In the dimensionless coordinates the system of equations (5) – (8) is
transformed to
1
g  ug '
4
g  , t   0
g'''' 
g  0   g  0   
g ' 0  g ' 0  2
g ' ' 0  g ' ' 0
g ' ' ' 0  g ' ' ' 0
(11a)
(11b)
s
b
(11c)
(11d)
(11e)
(11f)
where the prime, double prime, etc. denote the derivatives
d
,
du
d2
etc., respectively. Note that for 0 (no GB sliding occurs)
du 2
g(u)=g(-u) and the problem (11) is reduced to the Mullins’ problem [10].
The general solution of equations (11) can be represented in the form
g  u   a1 g1 u   a0 g 0 u 
(12)
where a1 and a0 are the u-independent constants and g1, g0 are
two linearly independent solutions of equation (11a) for which
g   0 .
These functions satisfy the following boundary conditions in the
point u=0
g1 0   
1
2 5 / 4 
g0 0  1
g1 ' 0  1
217
 0.78012 ;
(13a)
5 / 4
 0.64092
2
1
g1 ' ' 0   
 0.61372
2 3 / 4 
g 0 ' 0  
(13b)
g0 ' ' 0   0
(13c)
g1 ' ' ' 0  0
5 / 4 
g 0 ' ' ' 0  
 0.361602
21 / 2 
(13d)
where  is the Euler’s gamma-function. Following Mullins [10],
the functions g(u) can be represented in the form of infinite power series.
The constants a1 and a0 are determined from the boundary
conditions (11c-f)
a1  a1  1
and
a0   a0  
2 s
b
(14)
Finally, with the definition (10b) we have
y u, t  


b
Bt 1 / 4  g1 u   2 s 1  g0 u 
2 s
b


(15)
The expression in the square brackets of the RHS of equation
(15) calculated for the different values of parameter  is shown in
Fig. 8a. In Fig. 8b, two experimentally measured line profiles of the GB
groove are shown for comparison. The calculated and measured profiles
are in a good qualitative agreement with each other. This agreement
proves that GB sliding process occurred continuously during the
specimen annealing. It should be noted that the process of GB sliding
decreases the width of GB groove, defined as a distance between the two
maxima on the groove profile. For example, the width of Mullins groove
(=0) is 4.6, while for /m=1 the value is decreased to 3.6 (see Fig. 8a).
218
1.5
1.0
y/m(Bt)1/4
Mullins' function
/m=0.5
0.5
0.0
/m=0.3
-0.5
-1.0
/m=1
(a)
-10
-5
0
x/(Bt)1/4
5
10
0
5
10
15
20
200
y, nm
0
(010)
(160)
(410)
(310)
-200
200
0
-200
(b)
0
5
10
15
20
x,  m
Figure 8. (a) Linear profiles of GB grooves calculated according to equation
(15); m denotes b/2s. (b) Experimentally measured profiles of GBs exhibiting
different amount of GB sliding.
5. Conclusions
From the results of the present study the following conclusions
can be drawn.
1. The topography of thermal GB grooves in the Ni-43.6 at.% Al
intermetallic after annealing at 1400 C for 1 h was studied by
AFM technique. It was found that in many cases the thermal GB
grooving is accompanied by the sliding at GBs, the amount of
sliding being below 1 m. The dependence of the relative GB
219
2.
3.
4.
5.
energy and of the amount of GB sliding on misorientational
degrees of freedom of the GBs was determined.
The ratio of GB-to-surface energy for the low angle GBs is low
(0.11 for the GB with the misorientation angle of 3) but
increases with the misorientation angle, reaching an average
value of 0.45 for the large angle GBs with the misorientation of
approximately 15.5. The values of relative GB energy are in a
good agreement with the results of computer simulation of
Mutasa and Farkas [18].
It is demonstrated that the amount of GB sliding is sensitive to
the in-plane inclination of the GB.
It is shown that the observed GB sliding can be reasonably
explained in terms of near-GB lattice rotation. The driving force
for the lattice rotation is the misorientation dependence of GB
energy, with the rotation itself occuring by the mechanism of
selective absorption of lattice dislocations by the GB.
The model of GB grooving with simultaneous GB sliding is
developed in the small-slope approximation of Mullins [10]. The
calculated topographies of the GB grooves are in a good
qualitative agreement with the observed shapes.
Acknowledgements
This research was supported by the Ministry of Science and
Culture of the German Federal State Niedersachsen, by the INTAS grant
No. 97-0118 and by the Centre for Absorption in Science, Ministry of
Immigrants Absorption, State of Israel. E.R. also wishes to thank the
support of the Fund for the Promotion of Research at the Technion.
Helpful discussions with Dr. Y. Fishman are heartily appreciated.
1.
2.
3.
4.
5.
References
Yan, M., Vitek, V., and Chen, S.P., Acta mater., 1996, 44, 4351.
Mishin, Y. and Farkas, D., Phil. Mag. A, 1998, 78, 29.
Fonda, R.W., Yan, M., and Luzzi, D.E., 1995, Phil. Mag. Lett.,
71, 221.
Nadarzinski, K. and Ernst, F., Phil. Mag. A, 1996, 74, 641.
Reitz, W.E., JOM, 1998, No. 2, 39.
220
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
Schöllhammer, J., Chang, L.-S., Rabkin, E., Baretzky, B., Gust,
W. and Mittemeijer, E.J., Z. Metallkd., 1999, 90, 687.
Saylor, D.M. and Rohrer, G.S., J. Amer. Cer. Soc., 1999, 82,
1529.
Rabkin, E., Semenov, V.N. and Bischoff, E., Z. Metallkd., 2000,
91, 165.
Thomson, C.B. and Randle, V., Acta mater., 1997, 45, 4909.
Mullins, W.W., J. Appl. Phys., 1957, 28, 333.
Robertson, W.M., J. Appl. Phys., 1971, 42, 463.
Shankar S. and Seigle, L.L., Metall. Trans. A, 1978, 9A, 1467.
Harris, K.E., Singh, V.V., and King, A.H., Acta mater., 1998, 46,
2623.
Sutton, A.P. and Balluffi, R.W., Interfaces in Crystalline
Materials, Clarendon Press, Oxford, 1995.
Martin, G., Phys. stat. sol. (b), 1992, 172, 121.
Simkin, B.A. and Crimp, M.A., Phil. Mag. Lett., 2000, 80, 395.
Cahn, J.W., Personal communication (1999).
Mutasa, B. and Farkas, D., Metall. Mater. Trans. A, 1998, 29A,
2655.
Wolf, D., Scripta metall., 1989, 23, 1713.
Mishin, Y. and Farkas, D., Phil. Mag. A, 1996, 75, 169.
Rabkin, E., Estrin, Y. and Gust, W., Mater. Sci. Engng. A, 1998,
249, 190.
221