BULK AND GRAIN BOUNDARY DIFFUSION OF Cu IN NiAl
E. RABKIN
Department of Materials Engineering, Technion-Israel Institute of Technology, 32000 Haifa,
Israel
ABSTRACT The interdiffusion in Cu-(Ni-43 at.% Al) system was studied. It was shown that in
the bulk diffusion zone Cu atoms substitute mainly Al atoms and the diffusion problem can be
treated as a quasibinary. The concentration dependencies of interdiffusion coefficient were
determined in the temperature range of 923-1273 K. The rapid increase of interdiffusion
coefficient in the narrow region of Cu concentrations was interpreted in terms of percolation
model. Calculated percolation threshold for Cu atoms (6.06 at. %) agrees well with the
experimental data. The grain boundary interdiffusion was accompanied by nucleation and
growth of grain boundary pores. This porosity was discussed in terms of grain boundary
Kirkendall effect.
1. INTRODUCTION
Good mechanical properties, low density, high melting temperature and high oxidation resistance
of the ordered NiAl intermetallic compound with B2 structure have attracted scientific attention
to this material for more than three decades. For a better understanding of the high-temperature
mechanical properties of this compound, the knowledge about migration properties of atomic
defects is essential. From the fundamental point of view, understanding the diffusion
mechanisms in ordered intermetallic compounds with the B2 structure has remained elusive for
quite a long time until now [1]. The usual mechanism involving jump to the nearest neighbor
vacancy is improbable in B2 compounds as that migration may locally destroy the chemical
order. As a consequence, several alternative mechanisms were suggested, such as next nearest
neighbor jump mechanism, antistructural bridge mechanism and a group of mechanisms in
which the vacancy jumps between the nearest neighbors only are allowed, but chemical order is
restored after the jump cycle is completed. Recent data of Herzig and co-workers on Ni tracer
diffusion in NiAl support the triple defect mechanism of Ni diffusion [2]. According to this
1 - 84
mechanism, a correlated motion of the group of two Ni-sublattice vacancies and one Ni
atntistructural atom on the Al sublattice is responsible for Ni migration in the ordered structure.
For reaching a deeper understanding of the diffusion mechanisms in NiAl and attaining a greater
degree of confidence in favor of one mechanism or another, measurements of Al diffusivity in
the Al sublattice of this compound are ultimately needed. However, the radiotracer studies of Al
self-diffusion were not performed since the
26
Al radioactive isotope is hardly available. One
possibility to overcome this difficulty is to use a substitute for Al atoms that preferentially
occupies the substitutional sites on the Al sublattice. In the present study we have chosen Cu for
“probing” the diffusivity of Al sublattice in NiAl. Both experimental studies [3, 4] and
theoretical investigations [5, 6] show that Cu atoms preferentially occupy the Al sublattice in Nirich NiAl. The solubility of Cu in NiAl is reasonably high [7]. Therefore, we decided to
investigate the chemical interdiffusion in the Cu-NiAl system. The advantage of interdiffusion
experiment is that it allows determining the concentration dependence of interdiffusion
coefficient with the small increment in concentration from the single penetration profile.
The knowledge of interdiffusion parameters in the Cu-NiAl system is also important from the
applications’ point of view. Cu was suggested as a low-melting point interlayer in the process of
transient liquid phase (TLP) bonding of NiAl parts [8]. It was shown that the bonds between
NiAl parts exhibit the parent metal strength and fail in the bulk NiAl rather than at the interface.
Kinetics of TLP bonding can be predicted if the interdiffusion parameters in the Cu-NiAl system
are known.
In this work, we studied the bulk and grain boundary (GB) interdiffusion in the Cu-NiAl system
in the temperature range of 923-1273 K. The objective of this work was to determine the
parameters of interdiffusion and to discuss possible diffusion mechanisms in this system. More
details on these diffusion studies can be found in Refs [9, 10].
2. EXPERIMENTAL
The rod of NiAl intermetallic compound of a 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 casting in vacuum. The
as-cast alloy was then remelted and purified in a vacuum electron-beam floating zone melting
apparatus allowing two passes with a solidification velocity 6 mm/min to obtain a coarse-grained
polycrystalline ingot with an average grain size of 2 mm. 3 mm thick discs were cut from the ingot by
spark erosion. The surfaces of the discs were ground and polished to mirror finish using SiC papers
and diamond pastes down to 0.25 m particle size. The chemical composition of the discs (431 at.%
1 - 85
Al) was determined by the LINK-ISIS energy dispersive X-ray analysis (EDS) attached to the XL 30
Philips scanning electron microscope (SEM). The composition of the discs statistically varied in the
range 42-44 at. % Al with no systematic dependence of composition with radial distance from the
center. The decrease of Al content indicates that a part of it evaporated during the remelting process in
the vacuum electron-beam floating zone melting apparatus. The electrolytic deposition of Cu was
conducted with a solution containing 230 g of Cu2SO4*5H2O and 65 g of H2SO4 dissolved in 1000 ml
of distilled H2O with a current density of 0.1-0.2 A/cm2. After depositing about 100 m thick Cu
layer, the specimens were annealed in evacuated silica ampoules (10-5 Pa) at 7 different temperatures
in the range of 923-1273 K. At 1123 K, the samples were annealed for five different annealing times
of 5, 10, 20, 33 and 54 h. After annealing, the discs were cut in two halves and the cross-sections were
ground and polished. No Kirkendall porosity was observed in the bulk diffusion zone, though pores
did nucleate on some of the grain boundaries [10]. Careful etching with a solution containing 40 ml
HCl, 25 ml C2H5OH and 5 g CuCl2 in 30 ml of distilled H2O allowed us to identify the regions of
accelerated diffusion due to the GBs. The regions not affected by the grain boundary diffusion were
marked by microhardness indentations with an indentor attached to the microscope. Following
repolishing the same samples with indentation marks concentration profiles in the bulk interdiffusion
zone were determined by the EDS in the SEM. The EDS measurements were carried out using an
acceleration voltage of 20 kV, probe current of 0.1 nA, take off angle of X-ray radiation of 35°,
and acquisition time of 100 s per measurement, respectively. For all cases the standard deviation
of the measured intensity for a single measurement did not exceed 5 % relative. Pure elemental
standards (Cu, Ni and Al) and Kα analytical lines for all elements were used for the quantitative
analysis performed using conventional correction procedure included in LINK-ISIS software.
3. RESULTS
Figure 1 (a, b) show the concentration profiles of Cu, Ni and Al in the diffusion zone of the
samples annealed at 1123 and 1073 K, respectively. Qualitatively, the variation of Cu
concentration with penetration depth in the Ni-rich part of the diffusion zone resembles a
horizontal mirror reflection of that for variation of Al concentration with depth. This means that
the Cu atoms indeed substitute the Al atoms in the NiAl lattice. The slight decrease of Ni
concentration across the same interface suggests, however, that a minor fraction of Cu atoms
substitute Ni atoms in the NiAl lattice. All studied penetration profiles exhibited the behavior
similar to that shown in Fig. 1. The amplitude of variation of Ni concentration decreased with
decreasing temperature. The fact that the variation in Ni content is small when compared to that
1 - 86
C, at.%
80
1123 K, 54 h
60
Ni
40
Al
20
Cu
0 (a)
0
20
40
x, m
1073 K, 24 h
80
C, at. %
60
60
Ni
40
Al
20
Cu
0 (b)
0
5
10
x, m
15
20
Figure 1. Concentration profile of Al, Cu and Ni in the interdiffusion zone of Cu – NiAl couple
after annealing at 1123 K for 54 h (a) and at 1073 K for 24 h (b).
of Al and Cu allowed us to consider the diffusion problem as quasibinary and to apply the
standard Matano-Boltzmann procedure for extracting the concentration dependence of
interdiffusion coefficient from the concentration profile [1]. The corrections for differences in
partial molar volumes of Cu and Al in NiAl were not made since these data are not available in
1 - 87
the literature. Moreover, it has been demonstrated in a number of studies on metallic diffusion
couples that the application of exact Sauer-Freise analysis, that takes into account the difference
in partial molar volumes of diffusing components, does not significantly improve the results
obtained by original Matano-Boltzmann method without molar volume corrections [11].
To proceed with the Matano-Boltzmann processing of the concentration profiles we need to
prove that the characteristic dimensions of the diffusion zone increase with time according to the
parabolic law [1]. We defined the characteristic dimension, d, as a distance between the original
Cu/NiAl interface and the point of maximal slope in the steep region of the profile at low Cu
concentrations (see Fig. 1). The dependence of d on the annealing time t at 1123 K is shown in
Figure 2. The least square fit of the data by the exponential function d=aty, where a and y are the
fitting parameters, gives y=0.430.02. Therefore, one can conclude that the main condition for
the applicability of Matano-Boltzmann analysis is fulfilled with a reasonable degree of accuracy.
The method of determining the concentration dependence of interdiffusion coefficient is
illustrated in Figure 3. The right hand side of the concetration profile (to the right from original
Cu/NiAl interface) was fitted by the pseudo-Fermi type function
40
d=0.176 x t
1123 K
0.432
80
60
CCu, at.%
d, m
30
1273 K, 4 h
20
P1=33.53+0.4
P2=-0.15+0.005
P3=-131.69+0.5
P4=3.57+0.4
40
20
10
0
0
0
5
10
15
20
0
25
50
75
x, m
-4
t x 10 , s
Figure 2. Time dependence of distance between Figure 3. Iinterpolation of the penetration
the initial Cu/NiAl interface and the point of profile with the help of linear and pseudomaximal slope on the concentration profile.
Fermi [see equation (1)] functions.
C ( x) 
P1  xP2
 x  P3 
1  exp 

 P4 
1 - 88
(1)
where C and x are the concentration of Cu and diffusion depth, respectively, and P1, P2, P3 and
P4 are the fitting parameters. As can be seen in Fig. 3, this pseudo-Fermi function excellently fits
the experimentally determined concentration profile. Obviouslly, C()=0, which means that the
pseudo-Fermi function also satisfies the boundary condition of the diffusion problem. For the
low temperatures, the sum of two different pseudo-Fermi functions was used to fit the
experimental data. The Cu-rich part of the concentration profile (to the left from original
Cu/NiAl interface) was fitted by the straight line. The discontinuity region at the interface was
also fitted by the straight line, plotted through two adjacent points of the Cu-rich and Cu-poor
parts of the profile (this roughly corresponds to the spatial resolution limit of EDS in SEM). The
Matano interface was defined as the middle point between the intersections of the straight line in
the interfacial region with the straight line fitting the Cu-rich part of the profile and the pseudoFermi line fitting the Cu-poor part of the concentration profile.
The integration and differentiation needed to determine the interdiffusion coefficient by the
Matano Boltzmann method were performed using the fitting functions of the type given by
equation (1). The calculated concentration dependencies of quasibinary interdiffusion
~
coefficients, D , for all studied temperatures are presented in Figure 4. The curve for each
1273 K
-13
10
1223 K
1173K
-14
10
1123 K
2
D, m /s
-15
10
1073 K
-16
10
1023 K
-17
10
923 K
-18
10
0
5
10
15
20
CCu,at.%
Figure 4. Concentration dependence of the interdiffusion coefficient in the Cu-NiAl system for
the temperature range studied. The broken vertical line marks the percolation threshold for Cu
diffusion by the nn jumps mechanism.
1 - 89
temperature is an average calculated using 5-6 concentration profiles measured in different
~
places of the sample. It can be seen that D monotonously increases with increasing Cu
~
concentration in the temperature range of 1123-1273 K. The slow initial increase of D for Cu
concentrations below 6-8 at. % is followed by the rapid increase in the interval of 8-13 at. % Cu.
~
D increases by more than one order of magnitude in this concentration range. For Cu
~
concentrations above 13-14 at. %, D remains high but does not significantly increase with
~
increasing Cu concentration. The D (c ) dependencies at lower temperatures resemble the same
~
for D (c ) at high temperatures for low Cu concentrations. However, because of the switch of the
~
diffusion path in the direction of Ni3Al(Cu) phase at low temperatures, D begins to decrease
with increasing concentration, starting from a certain value of Cu concentration that is strongly
~
temperature-dependent. This apparent decrease of D should be treated with care. It is possible
that due to limitation in spatial resolution in the EDS analysis in SEM, the sharp interphase
boundary between the Ni3Al(Cu) and NiAl(Cu) phases is smeared out, which results in low, but
finite values of diffusivity obtained with the help of the processing scheme outlined above. At
the lowest temperature, precipitates in the diffusion zone were clearly visible in the optical
microscope and their face-centred cubic structure was identified with the help of electron
backscattering diffraction in SEM. However, the diffusion profile in the three-component system
may not experience a discontinuity at the interphase boundary. Obviously, the resolution of SEM
~
is insufficient to decide whether the decrease in D observed at low temperatures is an artefact
caused by the concentration discontinuity at the interphase boundary or the same reflects a
decreased atomic mobility in this region of concentrations.
Figure 5 (a, b) present the concentration dependence of the activation enthalpy, H, and pre~
exponential factor for interdiffusion, D 0 , respectively. These parameters were calculated using
the formal least square fit to the data obtained in the entire range of temperature (filled symbols)
and using the data only for the temperature interval 1073-1273 K (open symbols). For 0-9 at. %
Cu, the data for the entire temperature interval can be used, while for higher Cu concentrations
only the Arrhenius parameters for the high-temperature branch of data (open symbols) are
relevant. The minima for concentration dependence of activation enthalpy and pre-exponential
~
factor approximately correspond to the Cu concentration of the steepest slope on the D (c )
dependence (see Fig. 4).
1 - 90
10
400
923-1273 K
1073-1273 K
923-1273 K
1073-1273 K
10
D0, m /s
300
2
H, kJ/mol
350
250
200
5
2
10
-1
10
-4
(a)
0
5
10
15
10
20
cCu, at.%
-7
(b)
0
5
10
15
20
cCu, at.%
Figure 5. Concentration dependence of the activation enthalpy (a) and pre-exponential factor (b)
for interdiffusion in the Cu – NiAl system.
Cu
NiAl
(a)
40 m
(c)
(b)
(d)
Figure 6. The LM micrographs illustrating different morphologies of the GB pores formed as a result
of interdiffusion in Cu/NiAl couple at 850 C for 54 h.
1 - 91
In Fig. 6, the typical light microscopy (LM) images of Cu/NiAl diffusion zone in the vicinity of GBs
are presented. An appropriate etching regime allowed us to visualise the solid solution of Cu in NiAl
in the diffusion zone for the Cu concentration above approximately 5 at. %.
Slightly more than a half of all GB regions observed were of the type shown in Fig. 6a:

No porosity in the bulk and at the GB is observed;

Near-GB region enriched by Cu exhibits a characteristic wedge-like shape, which indicates
that the GB diffusion of Cu occurred in the B-regime of GB diffusion;

Some GB migration in the diffusion zone occurred (DIGM).
However, in some cases (Figs 6b-d) the pores at the GBs were observed (in Fig. 6 they are
marked by the white arrows). Three different pore morphologies can be distinguished:

Open elongated pore connected with the original Cu/NiAl interface (Fig. 6 b);

Isolated elongated pore which moved away from the original Cu/NiAl interface in
direction of NiAl (Fig. 6 c);

The chain of pores at the GB (Fig. 6d).
It should be noted that the largest pores in Figs 6 c-d are not completely convex, but exhibit a
kind of neck in the middle of the pore. Also in the cases in which the GB porosity was observed,
no pores were found in the adjacent regions of the bulk diffusion zone. Almost in all cases some
DIGM occurred.
4. DISCUSSION
4.1 Bulk interdiffusion
Our results clearly demonstrate that majority of the Cu atoms added to the Ni-43 at.% Al alloy
substitutes the Al atoms (see Fig. 1). The concentration profiles obtained by Gale and Guan [8]
during their studies of joining NiAl and Ni with a Cu interlayer at 1423 K are qualitatively
similar to the profiles obtained in the present work. This means that the entropy effects cannot
change the preferential occupation of Al sites by Cu atoms even at this relatively high
temperature. Therefore, Cu is a suitable substitute for Al for studying the atomic mobility in the
Al sublattice of the Ni-rich NiAl.
~
Initial increase of D and decrease of H with increasing concentration up to 6-8 at. % Cu can be
understood in terms of the relationship between the activation enthalpy for diffusion and solidus
temperature. For most pure metals, H  17 RTm , where Tm is the melting temperature [1]. This
1 - 92
relationship is also valid for dilute disordered alloys. No correlation of this type is available in
the literature for the ordered alloys with the high degree of long-range order. However, it is
reasonable to assume that the same proportionality between H and Tm is valid for ordered
alloys. Such a proportionality reflects a simple fact that the vacancy formation and migration
energies scale with the strength of the interatomic bonds. The solidus of ternary Al-Cu-Ni phase
diagram is known [7] and, indeed, for the constant Ni concentration the solidus temperature
decreases with increasing Cu concentration in the region of stability for the B2 phase.
~
For higher Cu concentrations the situation changes dramatically. D increases rapidly in the
interval 8-13 at. % Cu by more than one order of magnitude. The rate of this increase (per at. %
~
~
Cu) is much higher than that for increase in D in the initial part of D (c ) dependencies. The
location of the interval of concentrations in which this increase occurs is almost temperature
independent. This indicates that the increase of diffusivity is not associated with any phase
transition, since in that case the concentrations at which the diffusivity changes would be
temperature dependent as in the case of Fe-Si system [12]. One can argue that the order-disorder
transition may occur in the ternary Ni-Al-Cu system upon addition of Cu. As a rule, the diffusion
is faster in disordered phase than that in the ordered one [1]. However, the Cu concentration at
which the rapid growth of diffusivity begins in this case should increase with decreasing
temperature, since the stability domain of ordered phases always extends with decreasing
temperature. Obviously, our experimental data do not support this conclusion. Therefore, the
reason for this remarkable change of diffusivity observed in our work should be related to the
intrinsic properties of the B2-lattice.
According to the results of the present and previous [3-6] studies, Cu atoms substitute Al atoms
in the Al sublattice of NiAl. In these sites, all 8 nearest neighbors (nn) of Cu atom are Ni atoms.
Therefore, any jump of Cu atom to the nn site will lead to the situation in which the majority of
its nearest neighbors are Al atoms, which is energetically unfavorable. This is why the simple
direct Cu-vacancy exchanges and the diffusion of Cu through the nn sites should be forbidden.
This means that the complex correlated mechanisms (triple defect mechanism or 6-jumps cycle)
of diffusion should be involved in Cu transport, as that in the case for Ni tracer diffusion. The
correlation factor for these mechanisms is very low and, correspondingly, the diffusion
coefficients are also low if compared with the diffusivities in the disordered lattice. However, the
situation may change with the increase of deviation from the stoichiometry and Cu
concentration. It has been long established that the deviations from stoichiometry in Ni-rich NiAl
1 - 93
alloys are compensated for by the Ni antistructural atoms occupying the sites in the Al sublattice
[13]. Belova and Murch have shown that the percolation of Ni atoms through the nn sites
becomes possible and the antistructural bridge mechanism of diffusion through the nn sites
initiates as the atomic fraction of antistructural Ni atoms on Al sublattice of fully ordered NiAl
reaches the threshold value of 0.13 [14]. If we increase the deviation from stoichiometry and Cu
concentration in our ternary NiAl(Cu) alloy, the occasionally “trapped” Cu atom in the Ni
sublattice may increasingly find Ni or Cu atoms and not Al atoms as its nearest neighbors. Let us
assume that the sites in Ni sublattice that have more than half the 8 nn sites occupied by Ni or Cu
are now accessible for the nn jumps of Cu atoms substituting Al in the Al sublattice. According
to Belova and Murch [14], the percolation of Cu atoms through the ordered structure by the nn
jumps becomes possible if the concentration of such “permitted” sites on Ni sublattice exceeds
0.13. In the fully ordered Ni57Al43-xCux alloy, the probability, p, to find a Cu or a Ni atom on the
Al sublattice is
p
7 x
50
(2)
The probability, Ql, that l out of 8 nearest neighbors of Ni atom in the Ni sublattice are Al atoms
and the rest 8-l are Ni or Cu atoms is
Ql 
8!
l
p 8l 1  p 
l! 8  l !
(3)
Then the percolation condition for the Cu atoms under the assumption that at least half the
nearest neighbors of “permitted” site for Cu diffusion in the Ni sublattice is Cu or Ni atoms can
be written in the form:
p 8  8 p 7 1  p   28 p 6 1  p   56 p 5 1  p   70 p 4 1  p   0.13
2
3
4
(4)
This equation has only one real positive solution, p=0.2613. From equation (2) we find that the
concentration of Cu in the ternary alloy corresponding to this percolation threshold is x=6.06 at.
%. From Fig. 4 it is obvious that this is approximately the value of Cu concentration at which the
~
rapid growth of D with increasing concentration begins. It should be kept in mind that both the
effective cross-section of the infinite cluster and the contribution of the nn jumps to the overall
1 - 94
diffusion flux are small as long-range diffusion by nn jumps becomes feasible at the percolation
~
threshold. Therefore, no discontinuity or break in the concentration dependence of D should be
observed at the percolation threshold. For the concentrations above the threshold, the effective
cross-section of the infinite cluster increases rapidly with increasing Cu concentration, which
~
causes the corresponding increase in D . However, this increase is continuous, as is best
illustrated by Fig. 2 of Ref. [14], which exhibits the concentration dependence of the correlation
~
factor for diffusion very similar to our D (c ) dependencies in Fig. 4.
In the fully ordered system, the percolation threshold concentration should be temperature
independent, as indeed observed in our experiments. Therefore, one can conclude that our
percolation model for Cu diffusion provides a plausible explanation of the observed
~
concentration and temperature dependence of D .
4.2 Grain Boundary interdiffusion
In our opinion, this study represents a first direct experimental confirmation of the Kirkendall
effect during the GB diffusion. Indeed, the absence of any porosity in the bulk diffusion zone
indicates that either the bulk Kirkendall effect is weak or the climb of lattice dislocations is an
easy process which absorbs all excess vacancies formed in the process of interdiffusion. The
elongated morphology of the GB pores clearly indicates that the vacancy flux along the GBs
plays a decisive role in nucleation and development of these pores. Indeed, if one supposes that
the GB pores are the result of a simple heterogeneous nucleation of excess vacancies in the bulk
diffusion zone, the shape of the pore should be close to the spherical one, since the value of GB
energy, b, is only 30-40 % of that for the surface energy, s. Also the phenomenon of DIGM
which occurred in all GBs studied supports the idea of the importance of the GB vacancy flux in
diffusion process. In the next section, the quantitative theory of the growth of GB pores induced
by the vacancy flux along the GB will be developed.
Let us consider the following two-dimensional model of the growth of GB pore (see Fig. 7): the
flux of vacancies f0 caused by the different GB mobilities of diffusant (Cu) and matrix (Al)
atoms enters the pore at the triple line B. The vacancies then spread along the two internal
surfaces of the pore. For sufficiently small pores the divergence of the surface flux of vacancies,
fs, is the reason for the migration of the internal surface of the pore. At the opposite to B triple
line A no vacancies arrive (i.e. all of them were absorbed by the migrating internal surfaces of
the pore). Following Mullins [15], these two conditions can be written in the following form:
1 - 95
fs 
 s Ds  s k
kT

(5)
s
where s, Ds, k and s are the width of the surface diffusion layer, the surface diffusion coefficient
of vacancies, the surface curvature and the co-ordinate measured along the pore surface,
respectively. kT has its usual meaning. For the normal velocity, vn, of the pore internal surface
we have:
vn 
1 f s

 s
(6)
where  is the atomic volume. The Eqs. (5-6) should be completed by the following obvious
geometric conditions (see Fig. 7):
y
y
x

x
 cos  ;
 sin  ; k  
 cos   sin 
; vn 
s
s
s
t
t
(7)
where t stands for annealing time. The boundary conditions for the flux at the points A and B
read, respectively:
f s ( S  0)  0
(8a)
f s ( S  S max ) 
1
f0
2
(8b)
The condition of mechanical equilibrium at the triple lines reads
y
b
A
s


x
fs
V~t1/2
s
fs
f0~t-1/2
B
GB
Figure 7. To the calculation of the shape of GB pore growing by the mechanism of surface
diffusion.
1 - 96
2 s cos  0   b
(9)
The analysis of Eqs. (5-9) has shown that these equations have a time independent, self-similar
solution for the pore shape only in the case
f 0 ~ t 1 / 2
(10)
This condition is fulfilled in the case of GB diffusion in C-regime, in which the diffusion is
confined to the very GB core [16]. Indeed, in this case the total amount of Cu diffused into NiAl
is proportional to Db t 
1/ 2
, where Db is the GB diffusion coefficient, and the flux, which is the
time derivative of this amount, fulfils the Eq. (10). However, as can be seen from Fig. 6, our
experimental situation corresponds to the B-regime, in which the lateral bulk diffusion from the
GB cannot be neglected. Unfortunately, the f 0 t  dependence for the B-regime does not allow
the self-similar, time independent solution of Eqs. (5-8) to be obtained. Nevertheless, the Eq.
(10) captures the main feature of any diffusion process, namely, the decrease of flux with the
increasing time. Therefore, in what follows we will give a time-independent solution of Eqs. (59) with the vacancy flux given by Eq. (10), and will make only a qualitative comparison with the
experiment.
Following Mullins [15], we will introduce the following non-dimensional spatial variables:
s  2 Bt 
1/ 4
S
(11a)
X (S )
(11b)
Y (S )
(11c)
x  2 Bt 
1/ 4
y  2 Bt 
1/ 4
k
1
2 Bt 
1/ 4
K (S )
(11d)
and the non-dimensional vacancy flux, F:
f 
 s Ds  s
kT
1 - 97

1
2Bt 
1/ 2
F (S )
(11e)
where
B
 s Ds  s
kT
(12)
is the Mullins’ constant. Equation (11e) clarifies the reason for introduction of these new
variables: taking into account the Eq. (10) for the flux one can see that the new, non-dimensional
flux F is indeed time-independent. This justifies the separation of time and co-ordinate
dependencies in Eqs. (11a-d). From Eqs. (11) it follows that the volume of the pore, V, increases
with time according to the V~t1/2 law. In these new variables the Eqs. (5-8) can be rewritten in
the form:
F
 Y cos   X sin 
S
(13a)
X
 cos 
S
(13b)
Y
 sin 
S
(13c)
K
F
S
(13d)

 K
S
(13e)
Equations (13a-e) represent a system of five coupled differential equations for five unknown
functions X(S), Y(S), (S), K(S) and F(S). These equations should be completed by the boundary
conditions which follow from Eqs. (9) and (11):
for S=0: Y=0; =0; F=0
(14a)
for S=Smax: Y=0; = 0; F=F0
(14b)
kT
1/ 2
 f 0  Bt 
 s Ds  s
(14e)
where
F0 
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According to our main assumption expressed by Eq. (10) the non-dimensional initial vacancy
flux F0 at the triple line B (see Fig. 2) is a time-independent constant. Therefore, a remarkable
feature of the Eqs. (13) is that these equations describe the constant, time independent shape in
the properly defined non-dimensional co-ordinates given by Eqs. (11). It should be noted that
Eqs. (14) describe the evolution of the pore of an arbitrary complexity, contrary to the original
theory of Mullins [15], which was restricted by the small-slope, small curvature approximation.
We solved Eqs. (13) with the boundary conditions (14) by the Runge-Kutta method for 0=1.4
which is a typical value for the dihedral angle of GB grooves. The resulting pore shapes for
different values of F0 are shown in Fig. 8. For convenience, all four pores are shown together on
this Figure. This means that the symmetry axis (shown by the broken line) for each pore
represents the Y=0 line for this pore. The main features of the calculated pore shapes can be
summarized as follows:

All pores are entirely in the region X>0. Keeping in mind the definition of X by Eq. (11b)
one can conclude that the pore moves as a whole in the direction opposite to the vacancy
flux;

For small fluxes the shape of a pore is close to the lens, which represent the equilibrium
GB pore shape;

With the increasing flux the pore becomes longer and narrower. It is interesting that for
F0=4 the rear (with respect to the vacancy flux) part of the pore is clearly wider than the pore
front;
Neck
Pore migration
1
F =5
0
F =4
0
Y
F =2
Vacancies flux
0
F =1
0
0
2
4
X
6
8
Figure 8. Shapes of the GB pores (in the non-dimensional co-ordinates defined by Eqs. (7b, c))
calculated from Eqs. (9-10) for the different values of non-dimensional flux F0.
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
For high values of F0 a neck located approximately in the middle of the pore can be
observed. This means that for some critical value of F0 this neck is closing, and for higher F0
a self-similar solution for the pore shape does not exist. It can be speculated that for these
high values of F0 the process of pores reproduction will occur. After the nucleation of a small
pore at the GB it will grow, gradually changing it shape from the convex to the necked one.
At some moment the neck will be closed and the pore will split in two. Afterwards, the whole
process will be repeated for the small front pore. However, this is only a qualitative scenario
since for large F0 no self-similar solution of Eqs. (13-14) exists and the calculated shapes
(Fig. 8) are not applicable.
The calculated pore shapes reveal a striking similarity with the experimentally observed ones
(Fig. 6). The inward pore displacement from the original Cu/NiAl interface, the narrowing in the
middle of the pore and the results of pore reproduction all can be observed in Figs. 6c-d. Thus,
the model based on the GB vacancy flux and on the diffusion of vacancies along the internal
surface of the pore is confirmed experimentally.
Acknowledgements. This work is a result of author’s long-years cooperation with Drs Valery
Semenov, Leonid Klinger, Tatiana Izyumova and Ms Aliza Winkler.
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