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rspa.royalsocietypublishing.org
Role of different factors
affecting interdiffusion in
Cu(Ga) and Cu(Si) solid
solutions
Sangeeta Santra1 , Hongqun Dong2 , Tomi Laurila2
and Aloke Paul1
Research
Cite this article: Santra S, Dong H, Laurila T,
Paul A. 2014 Role of different factors affecting
interdiffusion in Cu(Ga) and Cu(Si) solid
solutions. Proc. R. Soc. A 470: 20130464.
http://dx.doi.org/10.1098/rspa.2013.0464
Received: 13 July 2013
Accepted: 21 October 2013
Subject Areas:
materials science
Keywords:
diffusion, solid solution, driving force,
indentation, vacancy wind effect
Author for correspondence:
Aloke Paul
e-mail: [email protected]
1 Department of Materials Engineering, Indian Institute of Science,
Bangalore, Karnataka 560012, India
2 Microsystem Technology, Department of Electronics, School of
Electrical Engineering, Aalto University, Espoo, Finland
A detailed diffusion study was carried out on Cu(Ga)
and Cu(Si) solid solutions in order to assess the
role of different factors in the behaviour of the
diffusing components. The faster diffusing species in
the two systems, interdiffusion, intrinsic and impurity
diffusion coefficients, are determined to facilitate the
discussion. It was found that Cu was more mobile in
the Cu–Si system, whereas Ga was the faster diffusing
species in the Cu–Ga system. In both systems, the
interdiffusion coefficients increased with increasing
amount of solute (e.g. Si or Ga) in the matrix (Cu).
Impurity diffusion coefficients for Si and Ga in Cu,
found out by extrapolating interdiffusion coefficient
data to zero composition of the solute, were both
higher than the Cu tracer diffusion coefficient. These
observed trends in diffusion behaviour could be
rationalized by considering: (i) formation energies
and concentration of vacancies, (ii) elastic moduli
(indicating bond strengths) of the elements and
(iii) the interaction parameters and the related
thermodynamic factors. In summary, we have shown
here that all the factors introduced in this paper
should be considered simultaneously to understand
interdiffusion in solid solutions. Otherwise, some of
the aspects may look unusual or even impossible to
explain.
Electronic supplementary material is available
at http://dx.doi.org/10.1098/rspa.2013.0464 or
via http://rspa.royalsocietypublishing.org.
2013 The Author(s) Published by the Royal Society. All rights reserved.
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1. Introduction
...................................................
rspa.royalsocietypublishing.org Proc. R. Soc. A 470: 20130464
Over the years, many studies on various important dilute solid solutions have been conducted
to discern the diffusion mechanism by taking into consideration some of the responsible factors,
such as the (i) vacancy formation energy, (ii) driving force, (iii) vacancy–impurity bonding and
(iv) vacancy wind effect. Birchenall [1] and Le Claire [2] qualitatively established a correlation
between the melting point of the alloy and the interdiffusion coefficients. It states that the
interdiffusion coefficients increase with a decrease in the melting point of the alloy and vice
versa. Heumann [3] developed relations for the vacancy–impurity binding energy considering the
correlation factor of the impurity and the jump–frequency relationships in dilute substitutional
binary solid solutions. Howard & Manning [4] characterized the diffusion behaviour of the
impurity and matrix atoms using a five-frequency model. Furthermore, it is expected that the
molar volume and the elastic modulus of the matrix will play a substantial role in the diffusion
behaviour. Finally, the thermodynamics of the system must be taken into consideration to
rationalize the trends in the diffusion data as a function of concentration. It should be noted
that more than one factor is typically important in a given system. Unfortunately, the exact
contribution from different factors simultaneously has not been examined earlier in detail, which
may be owing to the lack of important data required for the analyses.
Cu–Si and Cu–Ga systems provide an interesting case, as here the solute Si has a higher melting
point compared with Cu in Cu(Si), and Ga has a lower melting point in Cu(Ga). The required
data for the diffusion analysis are also readily available in the literature. Furthermore, both the
Cu-rich solid solutions crystallize as face-centred cubic (FCC). Consequently, the structure factor
required for the calculation of the vacancy wind effect is also available. Moreover, the variations
in activation energies for the vacancy formation with composition are available, which can be
used to interpret the trends in interdiffusion coefficients. Thermodynamic assessments of both
systems can also be found in the literature. Both systems are also important in many technological
applications. For example, both alloys are used in the manufacturing of superconducting wires
using the bronze technique where pure V rods are inserted in the Cu(Ga) or Cu(Si)–bronze ingots
to obtain V3 Ga and V3 Si layers on annealing, respectively [5,6]. Diffusion of Ga or Si out of the
respective bronzes influences the microstructural evolution and kinetics of the product layer that,
in turn, affects the intrinsic superconducting properties, such as critical current density (Jc ) and
critical temperature (Tc ) [7].
A few experimental studies are available in the literature on Cu–Ga and Cu–Si systems where
the focus is on understanding the diffusion behaviour of Cu(Ga)– and Cu(Si)–bronze alloys. An
interdiffusion study on Cu–Ga solid solution was performed by Wilhelm [8] at temperatures
ranging from 500 to 700◦ C. Later on, Smithells [9] determined the interdiffusion coefficient D̃ over
a narrow range of concentrations up to 3 at.% Ga. Takahashi et al. [10] evaluated D̃ over a range
of temperatures 730–930◦ C and in wide concentration ranges (up to 13.4 at.% Ga). However, the
intrinsic or tracer diffusion coefficients were not estimated.
Aaronson et al. [11] determined D̃ in FCC Cu 5.82 wt.% Si using the moving interphase
boundary (MIB) method at temperatures between 655 and 775◦ C. However, as pointed out by
Minamino et al. [12], the MIB method assumes D̃ to be independent of composition. Rhines &
Mehl [13] calculated D̃ by a slicing spectroanalysis method only at 1000◦ C where the solubility
of Si in Cu is around 5 at.% Si [14]. Minamino et al. [12] investigated interdiffusion on Cu–Si
over wide temperature 725–900◦ C and composition ranges (up to 9.8 at.% Si). They measured
the impurity diffusion coefficients of Si in pure Cu. However, the intrinsic diffusion coefficients
were not estimated. An extensive diffusion study was conducted by Iijima et al. [15] at T = 627–
877◦ C using the Kirkendall markers that report Cu to be the faster diffusing species at all the
compositions at 857◦ C. The tracer diffusion coefficient of 67 Cu in pure Cu and alloys up to
1.8 at.% Si at 877◦ C was measured by applying the radiotracer serial sectioning method. These
were compared with the intrinsic diffusivity values of Cu and Si calculated from bulk diffusion
experiments, suggesting a weak interaction between a vacancy and an Si atom in the Cu matrix.
Gierlotka & Haque [16] and Cao et al. [17] have thermodynamically assessed the atomic mobilities
2
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(a) Diffusion couple technique
Required amounts of pure Cu (99.999 wt.%), pure Si (99.9999 wt.%) and pure Ga (99.9999 wt.% in)
were measured and Cu(15 at.% Ga) and Cu(8 at.% Si) alloys were melted in an arc melting unit
that attains a vacuum of 10−4 Pa in an Ar-purged atmosphere. The composition for the two alloys
was chosen according to the equilibrium solid solubility limit to FCC Cu solid solution. The alloys
were melted thrice and flipped every time in order to improve homogeneity. These ingot samples
were homogenized at 775◦ C for 50 h. The homogeneity of the alloys was checked in an electron
microprobe analyser (EPMA) by measuring the composition randomly at various spots. The
deviation was found to be within ±0.2 at.% from the average composition. The alloys were then
electro discharge machining cut into pieces of 1 mm thickness. After a standard metallographic
preparation for the diffusion couples, TiO2 markers of 1 µm particle size, suspended in acetone,
were introduced on one of the diffusion couple members. It was then clamped with the second
one in a stainless steel fixture with the smallest possible pressure for a proper bonding. The
molybdenum foil was used as the barrier between the fixture and the end-members. The Cu(15
at.% Ga)/Cu and Cu(8 at.% Si)/Cu diffusion couples were subjected to heat treatments at 650,
700, 750 and 775◦ C for 16 and 25 h respectively, in a calibrated tube furnace. These were selected
because the heat treatments during the bronze technique are usually carried out in this range.
The furnace has a temperature control of ±5◦ C, and the vacuum is typically maintained at
10−4 Pa. After the heat treatment, the diffusion couples were cross-sectioned and prepared
metallographically for further examination. The composition profiles across the interdiffusion
zone were measured in the EPMA, and the Kirkendall marker plane position was determined
using the X-ray peaks originated from Ti.
(b) Indentation
Indentation studies were performed on a Cu(Ga)/Cu diffusion couple and Cu(Si) alloys to
estimate the variation of the modulus with composition. A Vickers indenter was used, and a
maximum load of 1000 mN was applied. A loading and unloading rate of 2000 mN s−1 with a
pause of 5 s at the peak load was used. The distance between each of the indents was maintained
as a minimum of five times the average diagonal length of the indents, so that the strain fields
do not interact. The interdiffusion zone of the Cu(Si)/Cu diffusion couple is narrow, so the
...................................................
2. Experimental details
3
rspa.royalsocietypublishing.org Proc. R. Soc. A 470: 20130464
of the elements in the FCC Cu(Si) solid solution independently. Cao et al. have also calculated
the temperature dependence of the self-diffusion coefficient of Si considering their atomic
mobility parameters.
The need for an optimized matrix composition in manufacturing of superconductors provides
a strong motivation to carry out detailed diffusion studies on Cu-rich Cu(Ga) and Cu(Si) solid
solutions in order to understand the detailed diffusion mechanisms. In this study, the markerbased diffusion experiments have been conducted to evaluate the intrinsic and tracer diffusion
coefficients. Furthermore, the experimental results obtained in this study were rationalized by
using the following factors available in the literature or determined in this work: (i) changes in
the vacancy concentration as a function of the composition, (ii) changes in the thermodynamic
factor, (iii) changes in the lattice parameters of the involved phases, (iv) melting points of the
elements, (v) changes in the impurity vacancy binding energy, (vi) variation of elastic modulus
with composition, (vii) the vacancy wind effect and (viii) interaction parameters between the
diffusing element and the matrix. Even though all the above-defined factors play a definite role
in diffusion behaviour of a given system, typically, the weight of the factors changes from system
to system. Therefore, here we try to identify the important factors influencing interdiffusion of
components in these two systems.
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(a)
(b)
Cu
Cu
(15 at.% Ga)
Cu
Cu
(8 at.% Si)
100 µm
(c)
100 µm
(d ) 9
14
8
7
6
5
4
3
2
1
0
–1
XK
12
XK
Si at.%
10
Ga at.%
...................................................
rspa.royalsocietypublishing.org Proc. R. Soc. A 470: 20130464
XK
XK
8
6
4
2
Ga measured
Ga smoothed
0
–2
0
100
200 300 400 500
displacement (µm)
600
Si measured
Si smoothed
0
50
100
150 200
displacement (µm)
4
250
Figure 1. (a) Backscattered electron images of the interdiffusion zone developed in Cu(15 at.% Ga)/Cu and (b) Cu(8 at.%
Si)/Cu diffusion couples annealed at 775◦ C (1048 K) at 16 and 25 h, respectively. (c) Corresponding measured and smoothed
composition profiles of Cu(15 at.% Ga)/Cu and (d) Cu(8 at.% Si)/Cu couples. (Online version in colour.)
indentations were performed on different Cu–Si alloyed pieces having different Si percentages
to achieve the modulus value at every Si at.%. Before that, these alloys were also homogenized
at 775◦ C for 50 h in the same calibrated furnace, and the same approach was followed to ensure
their homogeneity. On the other hand, a wide range of Ga at.% was covered in the measurements
because the diffusion zone thickness was already large after a single diffusion couple experiment.
The modulus values were calculated using the standard Olivers and Pharr method at each indent.
Ga concentration corresponding to the indentation was measured using EPMA.
(c) X-ray diffraction measurements
The lattice parameters of pure Cu, Cu(15 at.% Ga) and Cu(8 at.% Si) were determined using the
X-ray diffraction (XRD) technique. Measurements were performed for two alloys to compare the
results with existing data in these two systems. Diffraction peaks were determined using CuKα1
radiation (λ = 1.540593 Å), and the strongest four diffraction peaks were used for the analysis.
The lattice parameter values were calculated using UNITCELL software. The 2θ values, along
with their corresponding intensities, were fed as the input parameters. Accuracy of the obtained
lattice parameter was within ±0.00056, ±0.0016 and ±0.0005 nm for pure Cu, Cu(15 at.% Ga) and
Cu(8 at.% Si), respectively. After validating the results, the variations of the lattice parameters
with composition were used to estimate the size misfit for both the systems.
3. Results
The typical interdiffusion zones developed on annealing Cu(15 at.% Ga)/Cu and Cu(8 at.%
Si)/Cu at 775◦ C for 16 and 25 h are displayed in figure 1a and b, respectively. The corresponding
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(b) 7.5
3.66
Vm ( ×10–6 m3 mol–1)
3.64
3.63
3.62
3.61
0
4
2
(c)
6
8 10
at.%
1.6
Y/Vm ( ×105 mol m–3)
xo= 142.9 µm
xo
1.2
0.8
0.4
12
14
7.3
7.2
7.1
7.0
16
0
(d) 1.6
1.4
1.2
1.0
0.8
0.6
xK
Y/Vm ( ×105 mol m–3)
3.60
7.4
2
4
6
8 10
at.%
12
14
16
xo = 89.4 µm
xK
xo
0.4
0.2
0
–0.2
0
0
100 200 300 400 500 600
displacement (µm)
0
50 100 150 200
displacement (µm)
250
Figure 2. Variation of (a) lattice parameter and (b) molar volume with composition in Cu(Ga) and Cu(Si) solid solution regions.
(c) Location of the initial contact plane xo in the Y/Vm versus x profiles of Cu(15 at.% Ga)/Cu and (d) Cu(8 at.% Si)/Cu diffusion
couples. (Online version in colour.)
composition profiles across the interdiffusion zone are presented in figure 1c,d, and the location
of the Kirkendall marker plane is marked as xK . The analytic approach undertaken here for
calculating the interdiffusion coefficients at different compositions is based on Wagner’s relation
[18] taking the smoothed composition profile into consideration,
x∗
x+∞
∗
Y
(1 − Y)
Vm
∗
∗
∗
dx + Y
dx ,
(3.1)
D̃(Y ) = ∗ (1 − Y )
Vm
x−∞ Vm
x∗
2t dY
dx
where Y = (Ni − Ni− )/(Ni+ − Ni− ) is the composition normalizing variable, Ni is the mole fraction
of the component i and Ni− and Ni+ are the end-member compositions on the left- and righthand sides of the diffusion couple, respectively. Vm (m3 mol−1 ) is the molar volume, t (seconds)
is the annealing time, x (metres) is the position parameter. The terms x−∞ and x+∞ are the
position parameters at the unaffected parts of the diffusion couple (left- and right-hand sides,
respectively). The asterisk represents the position of interest.
The variation of molar volume with composition is an important prerequisite for D̃ estimation.
It is estimated for both the systems considering the lattice parameter data as available in the
literature [19,20] and is presented in figure 2b. The interdiffusion coefficients, D̃ for Cu(Ga)
and Cu(Si) systems at different compositions and temperatures are shown in figure 3a and b,
respectively. It is observed that D̃ increases linearly with the Ga and Si concentration in the
respective solid solution regions, which is in accordance with the trend reported by Iijima et al.
[15] for the Cu(Si) system presented in figure 3b, whereas for the Cu(Ga) system, the trend closely
follows the one reported by Takahashi et al. [10].
...................................................
3.65
5
Cu–Ga system
Cu–Si system
Cu–Ga system
Cu–Si system
rspa.royalsocietypublishing.org Proc. R. Soc. A 470: 20130464
lattice parameter (Å)
(a)
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(a)
(b)
10–12
6
10–13
~
D (m2 s–1)
10–15
650°C
700°C
750°C
775°C
10–15
10–16
10–16
2
(c)
10–12
~
D (m2 s–1)
10–13
650°C
700°C
750°C
775°C
4
6
Ga at.%
8
10
1
2
3
4
5
Si at.%
6
7
(d )
2 at.%
4 at.%
6 at.%
8 at.%
10 at.%
10–13
2 at.%
4 at.%
6 at.%
10–14
10–14
10–15
10–16
9.4 9.6 9.8 10.0 10.2 10.4 10.6 10.8 11.0
1/T ( ×10–4 K–1)
10–15
10–16
9.4 9.6 9.8 10.0 10.2 10.4 10.6 10.8 11.0
1/T ( ×10–4 K–1)
Figure 3. Variation of interdiffusion coefficients at 650, 700, 750 and 775◦ C (923, 973, 1023 and 1048 K) with composition in
(a) Cu(15 at.% Ga)/Cu and (b) Cu(8 at.% Si)/Cu systems. Arrhenius plots of interdiffusion coefficients for selected (c) Ga and
(d) Si composition in respective systems. (Online version in colour.)
The activation energies for D̃ at different compositions are calculated using the Arrhenius
equation expressed as
Q
,
(3.2)
D̃ = D̃o exp −
RT
where D̃o (m2 s−1 ) is the pre-exponential factor, Q (J mol−1 ) is the activation energy, R (J mol−1 K)
is the gas constant and T is the temperature in K. The Arrhenius plots for D̃ in the FCC Cu(Ga)
and Cu(Si) solid solutions are displayed in figure 3c,d, from which Q and D̃o are estimated at
each of these compositions. The variation of these two parameters against the respective solutes
is shown in figure 4a,b. The activation energy is almost invariable in the case of Cu(Si), whereas
figure 4a illustrates a slightly increasing trend of activation energy with respect to Ga at.% beyond
6 at.% Ga. Rhines & Mehl [13] have reported the activation energy for interdiffusion in the Cu(Si)
system to follow an increasing trend, whereas the results of both Minamino et al. [12] and Iijima
et al. [15] show a decreasing trend of activation energy with Si content. However, Aaronson et al.
[11] found the trend to be insensitive to the Si concentration, similar to the one observed in this
study. Takahashi et al. [10] found no such particular trend with regards to the activation energy for
interdiffusion in a Cu(Ga) solid solution, whereas studies by Wilhelm [8] show a trend consistent
with the findings in this study.
According to Takahashi et al. [10], D̃o for Cu (13.4 at.% Ga) increases by an order of magnitude,
whereas Wilhelm [8] reports the increment of the same in Cu (10.3 at.% Ga) by three orders of
magnitude. Iijima et al. [15] reports D̃o for Cu(8 at.% Si) to increase with the Si content, similar
to the trend noted in this study. The values reported by Minamino et al. [12] exhibit no such
distinct behaviour for the pre-exponential factor. However, as D̃o increases for both the systems,
...................................................
10–14
10–14
rspa.royalsocietypublishing.org Proc. R. Soc. A 470: 20130464
10–13
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(b)
Q (kJ mol–1)
~
log D o (m2 s–1)
~
log D o (m2 s–1)
activation energy
pre-exponential factor
–6
–7
–8
–9
7
160
140
120
100
80
60
activation energy
pre-exponential factor
–7
–8
2
4
6
Ga at.%
8
10
1
2
3
4
Si at.%
5
6
Figure 4. Variation of activation energies and pre-exponential factors at selected (a) Ga and (b) Si composition in their
respective systems. (Online version in colour.)
it explains the increasing behaviour of the interdiffusion coefficients with the solute content. It
should be noted that the interdiffusion coefficients represent a kind of averaged diffusivity of
the elements, as explained below (see equation (3.4)). Thus, knowledge of the intrinsic diffusion
coefficients is indispensable to have an idea about the relative mobility of the species.
The ratio of intrinsic diffusivities of components i and j can be estimated at the Kirkendall
marker plane using the relation developed by van Loo [21], expressed as
+ xK
− x+∞ 1−Y
Y
Di Vi Ni x−∞ Vm dx − Ni xK Vm dx
=
(3.3)
x+∞ 1−Y ,
Y
K
Dj Vj −N+ x−∞
dx + N−
dx
j
x
Vm
j
xK
Vm
where Vi is the partial molar volume of component i. It can be observed in the Y/Vm versus x
profile in figure 2c,d that the marker plane (xK ) moves to the Cu(Ga) region from the initial contact
plane (xo ) and towards the Cu-rich side in the case of the Cu–Si system. The location of xo could
be found by equalizing the areas in Y/Vm versus x and (1 − Y)/(Vm ) versus x plots. As the molar
volume varies linearly, the location found by both the plots should be the same. It is indeed found
to be more or less the same, and an average of these two is taken as the location of this plane. The
procedure is explained in detail in the electronic supplementary material. The locations of the
initial contact and the Kirkendall marker planes indicate that Ga is the faster diffusing species in
the Cu(Ga) solid solution, whereas Cu is the faster diffusing species in the Cu(Si) system.
The compositions at the marker plane at different annealing temperatures are listed in tables 1a
and 2a. The ratio of intrinsic diffusivities, DGa /DCu increases, and DCu /DSi decreases with the
increase in temperature. The absolute values of the intrinsic diffusivities of the species, as listed
in tables 1a and 2a, are calculated from the known values of the interdiffusion coefficients and the
ratios of the diffusivities using the relation expressed as [22]
D̃ = Vi Ci Dj + Vj Cj Di .
(3.4a)
By neglecting the molar volume (which was actually derived by Darken [23]), it can be written as
D̃ = Ni Dj + Nj Di ,
(3.4b)
where Ci is the concentration of element i. The partial molar volumes of Cu, Ga and Si calculated
from the molar volume variations are 7.08 × 10−6 , 8.74 × 10−6 and 7.43 × 10−6 m3 mol−1 ,
respectively. As we did not find much error when the variation of molar volume was neglected,
we estimated the values using equation (3.4b).
The impurity diffusion coefficients of Ga and Si in pure Cu can be computed by extrapolating
the interdiffusion coefficient data to 0 at. % of Ga and Si, respectively [24], and the values are
...................................................
160
140
120
100
80
60
rspa.royalsocietypublishing.org Proc. R. Soc. A 470: 20130464
Q (kJ mol–1)
(a)
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Table 1. Interdiffusion, intrinsic, tracer diffusion coefficients and vacancy wind effects of Cu and Ga determined in the Cu(Ga)
system.
NCu
D̃ (m2 s−1 )
DGa (m2 s−1 )
DCu (m2 s−1 )
DGa /DCu
..........................................................................................................................................................................................................
650
0.089
0.911
8.8 × 10−15
9.0 × 10−15
7.5 × 10−15
1.2
−14
−14
−15
..........................................................................................................................................................................................................
700
0.091
0.909
1.5 × 10
1.6 × 10
8.3 × 10
1.9
750
0.0885
0.912
4.0 × 10−14
4.2 × 10−14
1.9 × 10−14
2.2
0.908
Φ
−14
−14
−14
..........................................................................................................................................................................................................
..........................................................................................................................................................................................................
775
T (◦ C)
0.092
NGa
7.7 × 10
D∗Ga(v) (m2 s−1 )
8.2 × 10
D∗Cu(v) (m2 s−1 )
3.4 × 10
(1 + WGa )
2.4
(1 − WCu )
(b)
..........................................................................................................................................................................................................
650
0.089
4.3
2.1 × 10−15
1.8 × 10−15
1.003
0.97
700
0.091
4.2
3.8 × 10−15
2.3 × 10−15
1.016
0.86
3.9
−14
−15
1.019
0.79
1.025
0.77
..........................................................................................................................................................................................................
..........................................................................................................................................................................................................
750
0.0885
1.1 × 10
6.1 × 10
..........................................................................................................................................................................................................
775
0.092
4
−14
2.0 × 10
−14
1.1 × 10
..........................................................................................................................................................................................................
Table 2. Interdiffusion, intrinsic, tracer diffusion coefficients and vacancy wind effects of Cu and Si determined in the Cu(Si)
system.
T (◦ C)
(a)
NCu
NSi
D̃ (m2 s−1 )
DCu (m2 s−1 )
DSi (m2 s−1 )
DCu /DSi
..........................................................................................................................................................................................................
650
0.9517
0.0483
1.8 × 10−15
1.1 × 10−14
1.9 × 10−15
5.6
−15
−14
−15
..........................................................................................................................................................................................................
700
0.955
0.045
4.0 × 10
2.3 × 10
3.6 × 10
6.3
750
0.9538
0.0462
7.7 × 10−15
4.2 × 10−14
8.4 × 10−15
5.0
−14
−14
−14
..........................................................................................................................................................................................................
..........................................................................................................................................................................................................
775
T (◦ C)
0.954
NSi
0.046
Φ
1.1 × 10
D∗Cu(v) (m2 s−1 )
5.0 × 10
D∗Si(v) (m2 s−1 )
1.6 × 10
(1 + WCu )
3.1
(1 − WSi )
(b)
..........................................................................................................................................................................................................
650
0.0483
3.4
2.6 × 10−15
2.5 × 10−16
1.22
0.951
700
0.045
3.2
5.8 × 10−15
1.1 × 10−15
1.24
0.956
−14
−15
1.20
0.959
1.15
0.968
..........................................................................................................................................................................................................
..........................................................................................................................................................................................................
750
0.0462
3.18
1.1 × 10
2.6 × 10
..........................................................................................................................................................................................................
775
0.046
3.1
−14
1.4 × 10
−15
5.2 × 10
..........................................................................................................................................................................................................
shown in figure 5. The activation energies for the impurity diffusion coefficients of Ga in Cu
(DGa(Cu) ) and Si in Cu (DSi (Cu )) are found to be 99 ± 8 and 138 ± 24 kJ mol−1 , respectively. The
results show that the impurity diffusion coefficients of both Ga and Si are higher than the Cu tracer
diffusion coefficients [25]. A similar trend was reported by both Iijima et al. [15] and Minamino
et al. [12].
To further understand the atomic mechanism of the diffusing species, it is essential to compute
the tracer diffusion coefficients because they eliminate the effect of the chemical driving force on
the elements. The diffusion couple technique is an indirect but reliable technique to calculate these
values. It is also important to estimate the contribution from the vacancy wind effect. In general,
this is evaluated from the experimentally estimated intrinsic and tracer diffusion coefficients.
However, this may lead to a large error in estimation because of associated errors in both the
techniques. As explained below, the vacancy wind effect can also be estimated from one of the
diffusion coefficients, as explained for the Cu(Sn) system [26]. This helps to reduce the error in
evaluation of the vacancy wind effect. In a binary system, the intrinsic diffusion coefficients, Di
...................................................
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T (◦ C)
(a)
8
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10–14
9
DGa(Cu) Q = 99 ± 8 kJ mol–1 (this study)
DSi(Cu) Q = 138 ± 24 kJ mol–1 (this study)
Cu tracer diffusion coefficient [25]
Q = 200 ± 17 kJ mol–1
D (m2 s–1)
10–17
9.2
9.6
10.0
10.4
1/T ( ×10–4 K–1)
10.8
11.2
Figure 5. Impurity diffusion coefficients of Ga and Si in pure Cu and Cu tracer diffusion coefficient. Adapted from [25]. Copyright
c 1969 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim. (Online version in colour.)
and Dj , of individual elements are related to the tracer diffusion coefficients, D∗i and D∗j , following
the random alloy model developed by Manning [27],
and
D∗i (1 + Wi ) Di
=
,
D∗j (1 − Wj ) Dj
(3.5a)
Di = D∗i Φ(1 + Wi )
(3.5b)
Dj = D∗j Φ(1 − Wj ),
(3.5c)
where Φ = d ln ai /d ln Ni = d ln aj /d ln Nj is the thermodynamic factor using the Gibbs–Duhem
relationship for a binary system. Note here that we have neglected the contribution of molar
volume change with composition because it is expected to contribute only a negligible error.
Relations considering molar volumes can be found in [22]. In some systems, it can be very
important to consider the variation of molar volume, as was found in the Cu(Sn) solid solution
[26]. In our calculations, we have considered the faster diffusing component as i.
Shin et al. [28] have thermodynamically assessed for FCC Cu(Si) in the Cu–Si system by
taking into account also the enthalpy of mixing calculated from first principles. The drawbacks
of the previous thermodynamic modellings of the Cu–Si system lie mainly on the fact that the
thermochemical data available for the FCC region have not been used while evaluating the
thermodynamic parameters for the phase in question. Thus, the thermodynamic assessments
carried out by Shin et al. [28] are considered for the determination of the thermodynamic factors
in the Cu(Si) system. The thermodynamic assessment conducted by Li et al. [29] is used for the
Cu(Ga) system. The Φ values for Cu(Ga) and Cu(Si) extracted using THERMOCALC software are
plotted in figure 6a and b, respectively. It should be noted that the thermodynamic factor for both
of the elements is the same in the case of a binary system.
Manning introduced a correction factor called the vacancy wind effect that describes the effect
of vacancy flux on the intrinsic diffusivity of an element and can be expressed [27,30] as
Wi =
and
Wj =
2Ni (D∗i − D∗j )
(3.5d)
Mo (Ni D∗i + Nj D∗j )
2Nj (D∗i − D∗j )
Mo (Ni D∗i + Nj D∗j )
,
(3.5e)
...................................................
10–16
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10–15
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10
5
4
3
2
1
0
2
4
6
8 10
Ga at.%
12
14
16
18
thermodynamic factor (F)
(b) 5
650ºC
700ºC
750ºC
775ºC
4
3
2
1
0
2
4
Si at.%
6
8
Figure 6. Variation of driving force with (a) Ga in Cu(Ga) and (b) Si content in Cu(Si) systems at 923, 973, 1023 and 1048 K (650,
700, 750 and 775◦ C). (Online version in colour.)
where Wi and Wj are the vacancy wind factors of the species i and j, respectively. We use i for
the faster and j for the slower diffusion component in the two different systems. Mo is a constant
that depends on the crystal structure and for the FCC crystals, the value is 7.15 [27]. Therefore,
after substituting equations (3.5d) and (3.5e) in equations (3.5b) and (3.5c), we can first estimate the
tracer diffusion coefficients from known intrinsic diffusion coefficients. Then, we can estimate the
vacancy wind effects on Ga, Cu and Si using equations (3.5d) and (3.5e). Accordingly, it is clear that
from the knowledge of intrinsic diffusion coefficients themselves we can estimate both the tracer
diffusion coefficients and the vacancy wind effect. This will introduce smaller errors compared
with considering two experiments. Consequently, if we estimate the vacancy wind effect from the
experimentally determined values of tracer and intrinsic diffusion coefficients separately from
experiments carried out by radioisotopes and diffusion couples, the resulting error should be
higher because of the error associated in both the techniques. To calculate the ratio of the tracer
diffusion coefficients after neglecting the vacancy wind effect, Wi should be considered as zero
in equations (3.5a–c). The estimated tracer diffusion coefficients considering the vacancy wind
effects are summarized in tables 1b and 2b, along with the intrinsic and interdiffusion coefficients.
Furthermore, we can estimate the correlation factors of the components fi and fj from the
known overall correlation factor in an FCC crystal, fo , which is 0.7815. Based on the sum rule,
...................................................
650ºC
700ºC
750ºC
775ºC
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thermodynamic factor (F)
(a) 6
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Table 3. Atom–vacancy exchange frequency ratios and correlation factors of (a) Cu and Ga in the Cu(Ga) system and (b) Cu and
Si in the Cu(Si) system.
wGa /wCu
fGa
fCu
..........................................................................................................................................................................................................
650
0.089
1.2
0.75
0.784
..........................................................................................................................................................................................................
700
0.091
1.9
0.67
0.793
750
0.0885
2.2
0.64
0.795
775
T (◦ C)
0.092
NSi
2.40
wCu /wSi
0.62
fCu
0.798
fSi
..........................................................................................................................................................................................................
..........................................................................................................................................................................................................
(b)
..........................................................................................................................................................................................................
650
0.0483
5.8
0.773
0.952
700
0.045
6.4
0.773
0.956
750
0.0462
5.0
0.774
0.945
775
0.046
3.1
0.775
0.914
..........................................................................................................................................................................................................
..........................................................................................................................................................................................................
..........................................................................................................................................................................................................
..........................................................................................................................................................................................................
which relates the phenomenological coefficients and the ratios of the exchange frequencies, w,
Belova & Murch [31,32] have shown that the ratio of intrinsic diffusion coefficients is equal to the
vacancy–atom exchange frequencies, such that
Di wi
= .
Dj wj
Furthermore, these are related to the correlation factors by [33]
1−fi
wi
fi
= 1−f j
wj
(3.6a)
(3.6b)
fj
and
fo = Ni fi + Nj fj .
(3.6c)
The estimated correlation factors in the two different systems are listed in table 3a,b. It can be
seen that with increasing temperature, the ratio of exchange frequencies, wGa /wCu , increases.
In the Cu–Si system, wCu /wSi decreases with temperature, which means that wSi /wCu increases.
Therefore, in both systems, the ratio of minor to major component vacancy exchange increases
with temperature. On the other hand, the correlation factor for the minor component decreases,
whereas it increases for the major component with an increase in temperature.
4. Discussion
There are several factors that can simultaneously contribute to the experimentally observed
diffusion behaviour in the two systems under consideration. These include (i) changes in the
vacancy concentration as a function of the composition, (ii) changes in the thermodynamic
factor, (iii) changes in the lattice parameters of the involved phases, (iv) melting points of the
elements, (v) changes in the impurity vacancy binding energy, (vi) variation in elastic modulus
with composition (vi) the vacancy wind effect and vacancy–atom exchange frequency and
(vii) interaction parameters between the diffusing element and the matrix. Here, we discuss
each one of the above-mentioned plausible factors individually, and then at the end, we draw
everything together and point out the most important factors in order to find the most reasonable
explanations for the observed behaviour.
...................................................
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T (◦ C)
(a)
11
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(a)
4
12
cv ( ×10–3)
2
1
0
500
(b)
600
700
800
T (°C)
900
1000
1100
1.15
Cu(Ga)
Cu(Si)
1.10
Ef (eV)
1.05
1.00
0.95
0.90
0.85
0
4
8
at.%
12
16
Figure 7. (a) Absolute vacancy concentration in pure Cu (adapted from [34], copyright 1992 by the American Physical Society)
and Cu(3.04 at.% Si) (adapted from [35], copyright 1997 by Trans Tech Publications). (b) Variation of vacancy formation energies
in Cu(Ga) and Cu(Si) solid solutions with solute content (adapted from [36], copyright 1984 by American Physical Society). (Online
version in colour.)
(a) Changes in the vacancy formation energy
Hehenkamp et al. [34] and Mosig et al. [35] determined absolute vacancy concentrations in pure
Cu and Cu(3.04 at.% Si) using a differential dilatometer. Data from both the sources are plotted
in figure 7a for comparison. It can be observed from the figure that at a particular temperature,
the vacancy concentration is significantly higher in the alloy with 3.05 at.% Si than in the case
of pure Cu. Fukushima & Doyama [37] measured the apparent vacancy formation energy of a
single vacancy for Ga and Si bronzes using the positron annihilation method. The outcome of
from
the investigations supports the above-mentioned results. They observed a decrease in Evac
f
1.13 ± 0.02 to 0.92 ± 0.02 eV when the Si content in Cu was 3.9 at.% and 7.8 at.%, respectively.
being lowered down to 0.88 ± 0.02 from 1.10 ± 0.02 eV when the Ga
Similarly, they noted the Evac
f
at.% increased to 15 at.% from the original 4.6 at.%. The theoretically estimated effective formation
energy by Kim [36] for disordered Cu(Ga) and Cu(Si) alloys presented in figure 7b illustrates the
decrease in the formation energy as a function of the increase in Ga or Si content in the two
bronze alloys. Although the vacancy concentration in the Cu(Ga) solid solution has not yet been
...................................................
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Cu(3.04 at.% Si)
pure Cu
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The driving force (considered with respect to the thermodynamic factor, Φ) for diffusion could
have a strong effect on D̃. The thermodynamic factor, as displayed in figure 6a,b, increases as a
function of the solute content in both systems. A slightly stronger dependence of the interdiffusion
coefficient on the thermodynamic factor can be noted in the Cu(Si) alloy in comparison with that
in Cu(Ga). For example, at a solute content of 8 at.% at 650◦ C, the thermodynamic factor in the
Cu–Si system is about 15% larger than that in the Cu–Ga system. The behaviour of the thermodynamic
factor explains the increasing trend of D̃ as a function of Ga and Si content in the respective solid solutions
shown in figure 3a,b. As discussed later on, the larger thermodynamic factor in the Cu–Si system
in comparison with the Cu–Ga system also indicated higher attractive interaction between the
elements in the previous system. Previous studies on systems such as Ni(Mo) [38], Co(Mo) [39],
Cu(Sn) [26], Fe–Pt [40] and U–Zr [41] have illustrated how the changes in the Φ values are
reflected in the D̃ curves.
(c) Changes in the lattice parameter
The lattice parameters increase with the addition of solutes in both Ga and Si bronzes [19,20],
suggesting an increase in molar volume, as shown in figure 2a,b. However, as can be seen from
the same figure, the increase in the molar volume in the Cu(Ga) solid solution is considerably
more than that of Cu(Si). This is also reflected in the behaviour of the pre-exponential factor D̃o in the two
systems, as shown in figure 4a,b. The pre-exponential term depends on the vibrational frequency, the
geometry factor, the correlation factor and the square of the lattice parameter, among other things.
It can be seen from figure 4 that D̃o in Cu–Si is always larger than that in the Cu–Ga system, and
it also increases a bit more in the previous case. However, based solely on the lattice parameter
changes determined experimentally, the opposite would be expected, as the lattice parameter
change is much larger in the Cu–Ga system than in the Cu–Si system. This issue will be discussed
further later on in §4e.
(d) Melting points of the elements
As shown above, Ga and Cu are the faster diffusing elements in Cu(Ga) and Cu(Si), respectively.
This could be attributed to the melting point of the diffusing elements (Ga, 29.7◦ C; Cu, 1084.6◦ C
and Si, 1414◦ C) [42]. At a particular temperature, the element having the lower melting point should
possess higher thermal vibration of the atoms, and thus a higher diffusion rate. Intrinsic diffusion studies
on many systems, for example, Cu–Ag [43], Cu–Au [43], Cu–Zn [44] and U–Mo [45] show the
movement of the Kirkendall marker plane towards the element with the lower melting point.
This could be considered to be one of the reasons for finding higher impurity diffusion coefficients
of Ga in Cu than in the Cu tracer diffusion coefficients. The case of Si is not as straightforward
because Si has a higher melting point than Cu. Therefore, other factors than merely the melting
point must contribute to the observed behaviour (see §4f ).
Another interesting outcome is noted on comparing the impurity diffusion coefficients of the
respective solutes, as displayed in figure 5. The values of the impurity diffusion coefficients of
the solute elements, i.e. Ga and Si, fall according to the melting points of those elements. Ga,
whose Tm is the lowest, has the highest impurity diffusion coefficient in Cu, followed by DSi(Cu) .
Thus, the melting point of a solute atom could have a significant effect on the impurity diffusion
coefficient of that element in a matrix.
...................................................
(b) Thermodynamic factor
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determined, the decreasing Evac
trend suggests a distinct increase in the vacancy concentration
f
with Ga addition in Cu. Thus, the interdiffusion coefficient should increase as the solute content in the
bronze alloy becomes higher in both Cu (Ga) and Cu(Si) systems. There are no significant differences
between the two systems in this respect.
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(e) Impurity vacancy binding energy
(f) Variation of elastic modulus with composition
For further reasoning, the effect of modulus on the atomic mobility in Cu(Ga) and Cu(Si) alloys
is quantitatively examined using the Vickers indenter test. The corresponding measured elastic
modulus values as a function of solute content are presented in figure 8a,b. An average modulus of
112 GPa for pure Cu is obtained, which agrees fairly well with the reported literature value [49]. A
decreasing trend in the modulus is noted with the addition of Si in the Cu matrix. In order to get
the indents at a large number of Ga concentrations in the interdiffusion zone, the indentations
were not performed in a single row. It could be noted in the composition profile (figure 1c)
that the change on Ga at.% from 1 to approximately 9 at.% is within 150 µm. Thus, not many
indents could be performed in this region. Nevertheless, a clear decreasing trend in the elastic
modulus with addition of Ga is observed in the Cu(Ga) solid solution. As modulus is related to
the bonding of components, it should become somewhat easier for the solute atoms to migrate in
the matrix with a lower bulk modulus. Furthermore, Cu–Si and Cu–Ga bonds are strongly mixed
in their character, and the charge distribution in the Cu matrix around Cu atoms must be changed
accordingly. As a consequence, it is expected that the bonding is weakened with respect to the
‘pure’ metallic Cu matrix. This could be one of the reasons for finding higher impurity diffusion
coefficients for both the systems compared with the Cu tracer diffusion coefficient because bond
strengths of Cu–Ga and Cu–Si are lower compared with Cu–Cu. It should further be noted that
the decrease in the modulus in the Cu–Ga system (as well as the size misfit) is larger than that in
...................................................
where aCu and a are the lattice parameter values of pure Cu and Cu–X alloy, respectively. | da
dc |
is the absolute value of the slope obtained on fitting the lattice constant values and the solute
concentration in Cu. The aCu value considered for misfit calculation is 0.362054 ± 0.00056 nm as
obtained in this study by XRD measurement, which corresponds well to the reported value in the
literature [48]. The obtained values for Cu(Ga) and Cu(Si) are 0.366796 ± 0.0016 and 0.362562 ±
0.0005 nm, respectively, which agrees fairly well with the reported ones. As the accuracy is within
±0.0016 and ±0.0005 nm for Cu(Ga) and Cu(Si) respectively, | da
dc | is determined by considering
the lattice parameter values reported in the literature given with the 1 at.% increment of solute
content in Cu, as presented in figure 2a. The same data are also considered for the calculation of
other diffusion parameters. The size misfit for Cu(Ga) and Cu(Si) calculated using equation (4.1)
are 7.73 and 1.82%, respectively. It is noted that the activation energy for interdiffusion of Cu(Ga)
is slightly lower in comparison with that of Cu(Si). This follows the reported trend of decrease
in activation energy with increase in the misfit, as found for the interdiffusion study of Pt group
metals in Ni [46]. The solute–vacancy binding energy is expected to be large when the size misfit
is large. However, it is to be noted that there is a contradiction between the magnitudes of the
solute–vacancy binding as determined by Mosig et al. and the size misfit in the two systems. In the
Cu–Ga system, the size misfit is larger than that in the Cu–Si system, but still in [35], the vacancy
binding energy is reported to be larger in Cu–Si than in the Cu–Ga system. With respect to this,
the tracer diffusion studies carried out in [15], as discussed in §1, on the other hand, indicated
that there is only a weak interaction between a vacancy and an Si atom in the Cu matrix.
rspa.royalsocietypublishing.org Proc. R. Soc. A 470: 20130464
The impurity vacancy bonding should also play a significant role in the diffusion behaviour.
Mosig et al. [35] measured the vacancy–impurity binding energies in dilute Cu(Ga) and Cu(Si)
alloys to be 0.14 and 0.19 eV, respectively, using a differential dilatometry technique. Karunaratne
& Reeds [46] explained qualitatively the correlation between the solute–vacancy binding energy
and the size misfit. Then, they rationalized the possible considerations for the activation energy
for interdiffusion. The size misfit parameter of a solute element, ε, that considers the lattice
parameter variation with the solute addition, is expressed as [47]
1 da × 100,
(4.1)
ε=
aCu dc 14
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(a)
15
160
E (GPa)
100
80
60
40
0
2
4
6
8
10
Ga at.%
12
14
16
(b) 160
140
E (GPa)
120
100
80
60
40
0
2
4
Si at.%
6
8
Figure 8. Variation of elastic moduli with (a) Ga in Cu(Ga) and (b) Si content in Cu(Si) systems.
the Cu–Si system, which is reflected in the higher impurity diffusion coefficient of the Ga in Cu
when compared with Si, according to the arguments presented above. In addition, the decrease
in elastic modulus is also related to the decrease in the stiffness, and thus should be shown as
the increase in vibrational amplitude. Thus, when the pre-exponential factors are considered in
the two systems, based on the increase in lattice parameter alone, it would be expected that the
pre-exponential term in Cu–Ga should increase more strongly than that in the Cu–Si system.
However, based on the discussion above, it may be possible that the decrease in the vibrational
term could act to hinder this effect. Only at 10 at.% of Ga in the FCC Cu(Ga) solid solution can
one see a marked increase in D̃o .
As discussed above, the higher diffusion rate of Cu compared with Si was related to the
lower melting point of Cu. However, the faster diffusion of Cu in Cu(Si) may, in addition, be
related to a preferable interaction between Cu and the vacancies. This point was considered
theoretically by Iijima et al. [15] using the impurity and the tracer diffusion coefficient knowledge.
The basic conclusion in [15] was that there is, in fact, a preferable interaction between Cu
and vacancies. Following the above discussion on the Cu(Si) case, there could be a preferable
interaction between Ga atoms and the vacancies also in the Cu–Ga case. This would allow
the vacancies to exchange positions with the neighbouring Ga atoms rather than with the Cu
atoms. Nevertheless, a quantitative estimation of jump frequencies is required to deduce a
detailed reasoning.
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(g) Vacancy wind effect and component vacancy–atom frequency
Based on the assessed thermodynamic data on the Cu–Ga and Cu–Si systems, the interaction
parameters (L0 ) are −38799.9 + 17.8403 × T and −41305 + 26.591 × T for the Cu–Ga and Cu–Si
systems, respectively. Thus, there is a slightly larger attractive interaction between Cu and Si
than that between Cu and Ga, as reflected also in the values of the thermodynamic parameters
(figure 6). Consequently, it is expected that the interdiffusion coefficient is smaller in the Cu–Si
system than that in the Cu–Ga system as the interaction between the diffusing element and matrix
is stronger in the Cu–Si system. Support for this argument can be found in figure 3a,b, where
the interdiffusion coefficient is smaller throughout the measured concentration range in the Cu–
Si systems compared with that in the Cu–Ga system. The stronger interaction between Cu and
Si when compared with that between Ga and Cu is further exhibited in the impurity diffusion
coefficient values, as discussed in the context of the melting point of elements above. Accordingly,
the impurity diffusion coefficient of Ga is higher than that of Si in the Cu matrix. Typically,
when attractive interaction between the matrix and atom diffusing in the matrix is increased,
the diffusion rate should decrease as the matrix induces higher ‘lattice friction’ to the diffusing
element. This is shown, for example, in Au–Sn, Cu–Sn and Ni–Sn systems, where diffusion of Au
in Sn is fastest, Cu in Sn the next and Ni in Sn is by far the slowest. Interaction between the metal
in question and Sn increases in the same direction [50,51].
5. Conclusion
It has been long known that many factors influence the diffusion of components at the same time.
However, these have not been discussed together in the context of any real system before. Owing
to the reasons outlined in §1, the Cu–Ga and Cu–Si systems provide a unique opportunity to carry
out this type of analysis for the first time.
Kirkendall marker experiments indicated that Ga is the faster diffusing component in
Cu(Ga), whereas Cu is the faster diffusing component in Cu(Si). This can be easily understood
firstly, based on the melting points of the components. In both the systems, components with
lower melting points have higher diffusion rates because of higher thermal vibration. Another
contributing factor is considered to be the Ga–vacancy interaction in the Cu(Ga) alloy, which
is higher than the Cu–vacancy interaction in the Cu(Si) alloy. This difference in the impurity–
vacancy interaction was validated by Iijima et al. [15] for Cu(Si). We could not compare the
vacancy wind effects or the vacancy–atom exchange frequencies between these two systems
because these are determined at different compositions.
It was found that the impurity diffusion coefficients of Si and Ga in Cu are higher compared
with the tracer diffusion rate of Cu in Cu. In the case of Si, this cannot be explained by the melting
point of the elements. However, the observed decrease of the elastic modulus with the increasing
content of the solute indicated that the Cu–Cu bonding strength must be higher than that of
...................................................
(h) Interaction parameters
rspa.royalsocietypublishing.org Proc. R. Soc. A 470: 20130464
Tracer diffusion coefficients and the vacancy wind effects were determined from the estimated
values of intrinsic diffusion coefficients. It can be noted that the vacancy wind effect is higher
in Cu(Si) compared with Cu(Ga) solid solution because of the difference in the ratios of the
intrinsic diffusivities in these systems. The higher the ratio of intrinsic diffusivities, i.e. the higher
the difference in the diffusivities of the components, the higher the vacancy wind effect. This
can be understood from the (1 ± Wi ) values, as listed in tables 1b and 2b. Furthermore, the
major component, i.e. Cu, is affected more compared with the minor components, which can be
understood from equations (3.5d) and (3.5e), where mol fraction Ni dictates how composition in
terms of mol fraction influences the vacancy wind effect. The estimation of the intrinsic diffusion
coefficients indicates that the ratio for minor to major vacancy–atom exchange frequency increases
with temperature.
16
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India, which enabled us to carry out this research.
References
1. Birchenall CE. 1951 Atom movements, pp. 112–128. Metals Park, OH: ASM.
2. Le Claire AD. 1949 Diffusion of metals in metals. Prog. Metal. Phys. 1, 306–379. (doi:10.1016/
0502-8205(49)90009-X)
3. Heumann T. 1979 Diffusion, correlation effects and jump frequency ratios in binary
substitutional solid solutions. J. Phys. F Met. Phys. 9, 1997–2010. (doi:10.1088/0305-4608/
9/10/010)
4. Howard RE, Manning JR. 1967 Kinetics of solute-enhanced diffusion in dilute face-centeredcubic alloys. Phys. Rev. 154, 561–568. (doi:10.1103/PhysRev.154.561)
...................................................
Acknowledgements. We acknowledge the financial support from the Department of Science and Technology,
17
rspa.royalsocietypublishing.org Proc. R. Soc. A 470: 20130464
Cu–Ga and Cu–Si. Because of the higher bond strength of Cu–Cu, the diffusion rate of Cu is
lower compared with the solutes. Furthermore, the size misfit, which directly relates to the void
size, was higher in Cu–Ga in comparison with Cu–Si. Accordingly, the activation energy for Ga
impurity diffusion was lower when compared with that of Si. As expected, the highest activation
energy for the impurity diffusion was that of Cu in Cu with zero size misfit.
The interdiffusion coefficient in the Cu–Si system is smaller than that in the Cu–Ga system,
as expected based on the stronger interaction between Cu and Si in comparison with that
between Cu and Ga. Furthermore, in both the systems, the interdiffusion coefficient increases
with the increase in solute content. The increase in thermodynamic parameters with the increase
in solute content supports this finding. Decrease in the vacancy formation energy, i.e. increase
in vacancy concentration with increasing solute content, further supports the observed trends in
the interdiffusion coefficient. In addition, the increase in the lattice parameter as a function of
the solute content points to the same conclusion. Interestingly, it can be noted that the increase in
interdiffusion coefficient is mainly attributed to the increase in the pre-exponential factor because,
in both systems, the activation energy is more or less constant at different compositions.
The reasons for the changes in the pre-exponential factor include changes in lattice parameters
and in the elastic moduli. It was shown that consideration of only one factor (lattice parameter
change) may lead to confusing results between the two systems. As the vacancy formation energy
decreases with increasing solute content, we would expect to see a decrease also in the activation
energy for interdiffusion. As this is not observed, it may well be possible that the energy required
for migration also increases as a function of the amount of solute. This is supported by the
noted increase in the thermodynamic factor as a function of concentration of Si and Ga in Cu.
In addition, the vacancy–impurity binding energy decreases as a function of increasing solute
content.
The estimation of the vacancy wind effect indicates that the major component Cu is affected
more in both systems, which is expected [26,52]. It is to be noted that the faster diffusing
component becomes even faster, and the slower diffusing component becomes slower because of
this effect. As Cu is the slower diffusing component in Cu(Ga) and the faster diffusing component
in Cu(Si), it is affected differently in these two systems.
As discussed in §1, this kind of overall analysis about the role of several different factors
in diffusion behaviour in a given system has not been carried out before. Even though it was
found that all the different factors have a role in the diffusion behaviour, they are strongly
interrelated, and the weights are different. For example, vacancy formation energies, changes
in lattice parameter, bond strengths and the melting point of elements are related. In this case,
it can be stated that the three most important factors that can be used to rationalize most of
the experimental data are (i) number and formation energies of vacancies, (ii) elastic modulii
(indicating bond strengths) of the elements and (iii) the interaction parameters and related
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