High-throughput determination of the composition-dependent interdiffusivities in Cu-rich fcc Cu-Ag-Sn alloys at 1073 K

Article abstract of DOI:10.1016/j.jallcom.2015.05.030

Abstract Based on the recently developed pragmatic numerical inverse method for determining the composition-dependent interdiffusivities in a ternary system by using a single diffusion couple, high-throughput determination of the interdiffusivities in Cu-rich fcc Cu-Ag-Sn alloys at 1073 K was performed in the present work. The composition-dependent interdiffusivity matrices along the entire diffusion paths of five fcc Cu-Ag-Sn diffusion couples were obtained. The reliability of the interdiffusivities determined by the pragmatic numerical inverse method was first validated by Fick's second law applied to numerical simulation of composition profiles and interdiffusion fluxes for each diffusion couple. The excellent agreement between the simulated results and the experimental data was obtained. In order to further validate their reliability, the traditional Matano-Kirkaldy method was employed to evaluate the interdiffusivities at the intersection point of the diffusion paths for every two diffusion couples. The good agreement between the interdiffusivities determined by the pragmatic numerical inverse method and those by the traditional Matano-Kirkaldy method was also observed. These facts indicate that the interdiffusivities determined by the pragmatic numerical inverse method are reliable.

Full text of DOI:10.1016/j.jallcom.2015.05.030

Journal of Alloys and Compounds 644 (2015) 687–693
Contents lists available at ScienceDirect
Journal of Alloys and Compounds
journal homepage: www.elsevier.com/locate/jalcom
High-throughput determination of the composition-dependent
interdiffusivities in Cu-rich fcc Cu–Ag–Sn alloys at 1073 K
Huixia Xu , Weimin Chen , Lijun Zhang , Yong Du , Chengying Tang b
a
a
a,
⇑
a
a
State Key Laboratory of Powder Metallurgy, Central South University, Changsha, Hunan 410083, PR China
Guangxi Key Laboratory of Information Materials, Guilin University of Electronic Technology, Guilin 541004, PR China
b
a r t i c l e i n f o
a b s t r a c t
Article history:
Based on the recently developed pragmatic numerical inverse method for determining the
composition-dependent interdiffusivities in a ternary system by using a single diffusion couple,
high-throughput determination of the interdiffusivities in Cu-rich fcc Cu–Ag–Sn alloys at 1073 K was per-
formed in the present work. The composition-dependent interdiffusivity matrices along the entire diffu-
sion paths of five fcc Cu–Ag–Sn diffusion couples were obtained. The reliability of the interdiffusivities
determined by the pragmatic numerical inverse method was first validated by Fick’s second law applied
to numerical simulation of composition profiles and interdiffusion fluxes for each diffusion couple. The
excellent agreement between the simulated results and the experimental data was obtained. In order
to further validate their reliability, the traditional Matano–Kirkaldy method was employed to evaluate
the interdiffusivities at the intersection point of the diffusion paths for every two diffusion couples.
The good agreement between the interdiffusivities determined by the pragmatic numerical inverse
method and those by the traditional Matano–Kirkaldy method was also observed. These facts indicate
that the interdiffusivities determined by the pragmatic numerical inverse method are reliable.
Ó 2015 Elsevier B.V. All rights reserved.
Received 27 February 2015
Received in revised form 3 May 2015
Accepted 6 May 2015
Available online 12 May 2015
Keywords:
Fcc Ag–Cu–Sn alloys
Diffusion
Interdiffusivity
Diffusion couple
Numerical inverse method
Matano–Kirkaldy method
1
. Introduction
Due to its low melting point, high wetting property and excel-
phases, i.e., face centered cubic (fcc) and intermetallic phases. The
major obstacle for development of atomic mobility in solid solder
alloys is the lack of accurate diffusivity information. Taking the
ternary Cu–Ag–Sn system for example, there is no any experimen-
tal diffusivity information for fcc phase available in the literature.
Therefore, there is a need to remedy this situation.
lent comprehensive performance, the Ag–Cu–Sn solder is recog-
nized to be the most widely used lead-free solder alloy [1–6]. In
order to fully understand the microstructure evolution during the
welding and the later service processes, for instance, the interfacial
reaction and interdiffusion behavior between the solder and sub-
strate, the accurate diffusivity information in the Ag–Cu–Sn system
is the prerequisite. Moreover, a project aiming at development of
atomic mobility database in the Sn–Ag–Bi–Cu–In–Pb solder alloy
via an integration of experimental measurement, DICTRA
The single-phase diffusion couple technique is frequently
employed to determine the interdiffusivities of the target phase.
For a ternary system, the well-known Matano–Kirkaldy method is
the most widely used one in materials community [13–15]. With
the Matano–Kirkaldy method, the four independent interdiffusivi-
ties at the intersection point along the diffusion paths of two diffu-
sion couples can be obtained for a ternary system. Though the
Matano–Kirkaldy method can give reasonable interdiffusivities,
its efficiency is very low. With such a low efficiency, a large number
of diffusion couple experiments should be performed if the
composition-dependent interdiffusivities needs to be evaluated
for a high-quality atomic mobility database. Moreover, its low effi-
ciency cannot meet the requirement of abundant experimental data
in the MGI (Materials Genome Initiative) [16,17] and/or ICME
(Integrated Computational Materials Engineering) [18] projects
nowadays. In order to solve this problem, a pragmatic numerical
inverse method was developed and realized in a home-made code
by Chen et al. [19] from our research group very recently. With such
(
DIffusion-Controlled TRAnsformation) simulation and atomistic
simulation is currently carried out in our research group. Up to
now, the atomic mobility database for liquid phase in Sn–Ag–Bi–
Cu–In–Pb solder has been established by Chen et al. [7–9] using
a modified Sutherland equation [10] and the phenomenological
treatment of diffusion in liquid [11,12]. The established liquid
atomic mobility database was also successfully applied to simulate
the dissolution of Ag and Cu substrates into liquid solder alloys
during the reflow process [9]. However, this is not the case for solid
⇑
Corresponding author. Tel.: +86 731 88877963; fax: +86 731 88710855.
E-mail addresses: xueyun168@gmail.com, lijun.zhang@csu.edu.cn (L. Zhang).
http://dx.doi.org/10.1016/j.jallcom.2015.05.030
925-8388/Ó 2015 Elsevier B.V. All rights reserved.
0

6
88
H. Xu et al. / Journal of Alloys and Compounds 644 (2015) 687–693
a numerical inverse method, the composition-dependent interdif-
fusivities in ternary systems can be effectively determined by using
a single diffusion couple. Thus, the efficiency increases dramatically
in comparison with the traditional Matano–Kirkaldy method.
Here, ~J
concentrations of solutes 1 and 2, x is the diffusion distance, and t is
i
1 2
is the interdiffusion flux of component i, c and c are the
the time. D~ and D~ are the two main interdiffusion coefficients,
3
3
1
1
22
while D~ and D~ 3 are the two cross interdiffusion coefficients in
3
1
2
21
Consequently,
high-throughput determination
of
the
the ternary system. In general, the interdiffusion coefficients are
composition-dependent at a constant temperature (and pressure).
As stated in Section 1, there are two methods to determine the
composition-dependent interdiffusion coefficients in ternary sys-
tems, i.e., the pragmatic numerical inverse method and the
Matano–Kirkaldy method, as briefly described in the following.
composition-dependent interdiffusivities in fcc Cu–Ag–Sn alloys
is chosen as the target in the present work. The large amount of
interdiffusivity information obtained in the present work will serve
as the important basis for the establishment of atomic mobility
database of fcc Cu–Ag–Sn system. The major objectives of the pre-
sent work are (i) to determine the composition-dependent interdif-
fusivities in Cu-rich fcc Cu–Ag–Sn alloys at 1073 K by using the
pragmatic numerical inverse method together with 5 groups of
fcc single-phase Cu–Ag–Sn diffusion couples, and (ii) to validate
the obtained composition-dependent interdiffusivities by compre-
hensively comparing with the results due to the Matano–Kirkaldy
method as well as by Fick’s second law applied to numerical sim-
ulation of composition profiles for each diffusion couple.
3.1. Pragmatic numerical inverse method
The pragmatic numerical inverse method was recently
proposed by Chen et al. [19] to determine the composition-
dependent interdiffusivities in ternary systems by using a single
diffusion couple. According to Manning’s random alloy model
~
3
ij
[
20], the interdiffusivities D (i, j = 1 or 2) and the mobility M
i
are
related by:
2
. Experimental procedure
h
ꢂ
ꢃi
P
~
3
ij
3
3
3
3
The terminal compositions of the five Cu–Ag–Sn diffusion couples are listed in
D ¼ RT M
i
/ ꢁ c
i
M
1
/
þ M
2
/
þ M
3
/
ij
1j
2j
3j
Table 1. In order to make up the diffusion couples, binary/ternary alloys with termi-
nal compositions need to be prepared first. Pure copper (purity: 99.99%), silver
"
#
3
2c
i
RT
ðM
m
/
Þ
m
mj
þ s ðM
i
ꢁ c
1
M
1
ꢁ c
2
M
2
ꢁ c
3
M
3
Þ
P
ð3Þ
(
purity: 99.99%), and tin (purity: 99.99%) were used as starting materials.
A
0
ðc
m
M
m
Þ
m
Different amounts of pure metal elements Cu, Ag, and Sn corresponding to the
terminal compositions were encapsulated in the separate vacuum quartz tubes.
All the quartz tubes were then placed in a high-temperature furnace (GSL1700X,
Hefei kejing materials technology Co., Ltd., Hefei, China) at 1473 K for 4 days to
ensure that all the elements were molten. After cooling to room temperature, all
the samples were then re-melted by arc melting under a high-purity argon atmo-
sphere using a non-consumable tungsten electrode (WKDHL-1, Opto-electronics
Co., Ltd., Beijing, China) for four times to improve their homogeneities.
3
where R is the gas constant and T is the temperature. /ij is the ther-
modynamic factor, and can be expressed as:
ꢀ
ꢁ
c
RT
i
@
li
@
li
3
ij
/
¼
ꢂ
ꢁ
@c
3
ð4Þ
@c
j
Subsequently, the samples were linearly cut into blocks of approximate dimensions
where li is the chemical potential of component i, and can be
obtained from the corresponding thermodynamic descriptions,
which are usually available for most alloy systems nowadays. The
second part of Eq. (3) denotes the vacancy-wind effect, which con-
siders the contribution of the vacancy flux. When s equals to 1, the
vacancy-wind effect is considered. While s equals to 0, the
3
5
ꢀ 5 ꢀ 10 mm after mechanically removing the surface material, and then sealed
into an evacuated quartz tubes, and homogenized at 1073 ± 5 K for 20 days in an
L4514-type diffusion furnace (Qingdao Instrument & Equipment Co., Ltd., China),
followed by quenching in water. The polished and cleaned blocks were bound
together by molybdenum wires to form five diffusion couples according to the
assembly listed in Table 1. These couples were then sealed into quartz tubes under
vacuum atmosphere, and annealed at 1073 K for 36 h in the L4514-type diffusion
furnace. After that, the couples were quenched into cold water. The annealed
couples were then metallographic polished along the planes parallel to diffusion
direction. The solute concentration profiles of all the five diffusion couples were
measured by means of EPMA technique (JXA-8230, JEOL, Japan) on the polished
section.
0
vacancy-wind effect is not considered. The parameter A is a factor
depending only on crystal structure, and here equals to 7.15 for the
face centered cubic (fcc) crystals [20]. The function developed by
Andersson and Ågren [11] and incorporated in DICTRA software
[
21] was directly employed in the present work to express the
atomic mobility for element i:
3
. Methods for evaluating ternary interdiffusion coefficients
ꢀ
ꢁ
1
RT
DG
i
RT
M
i
¼
exp
ð5Þ
In a hypothetical 1–2–3 ternary system (assuming 1 and 2 are
solutes while 3 is chosen as the solvent. Here, the solvent is Cu,
and the solutes are Ag and Sn), Fick’s first and second law read
as respectively
where
¼ c
þ c
Here,
i
DG can be expanded by the Redlich–Kister polynomial:
1
i
2
i
3
i
1;2
1;3
1 3
c D
Gi
D
G
i
1
D
G þ c
2
D
;3
G þ c
3
D
G þ c
1
c
2
D
Gi þ c
2
1;2;3
2
c
3
D
Gi þ c
1
c
2
c
3
D
Gi
ð6Þ
@
c
1
@c
2
~
¼ ꢁ D~ 3
~
3
i2
J
i
ꢁ D
ð1Þ
ð2Þ
1
i
2
i
3
i
i1
D
G ,
D
G and
D
G are the end-members for diffusion of
@
x
@x
1
;2
1;3
2;3
element i in elements 1, 2 and 3, while
D
Gi
,
D
Gi
,
D
Gi
and
ꢀ
ꢁ
@
c
t
i
@
@c
1
@c
2
1;2;3
~
3
i1
~
3
DGi
are the interaction parameters for the mobility of element
¼
D
þ D
i2
@
@x
@x @x
i in the 1–2, 1–3, 2–3 and 1–2–3 systems, respectively. For
Table 1
List of terminal compositions of the five diffusion couples in the present work.
Couple name
Composition (at.%)
Temperature (K)
Diffusion time (hours)
C1
C2
C3
C4
C5
Cu–1.8Sn/Cu–1.9Ag
Cu/Cu–1.7Sn–2.6Ag
Cu/Cu–3.8Sn–1.0Ag
Cu–2.9Sn/Cu–2.0Ag
Cu–1.98Sn/Cu–3.5Ag
1073
1073
1073
1073
1073
36
36
36
36
36

H. Xu et al. / Journal of Alloys and Compounds 644 (2015) 687–693
689
simplicity, the end-members corresponding to self and impurity
diffusivities are fixed by using the experimental data,
first-principles calculations, molecular dynamic simulation or
some semi-empirical relations. The other coefficients such as
all the 5 diffusion couples measured by EPMA in the present work
are presented in Fig. 2. Based on Eqs. (8) and (9), the corresponding
interdiffusion fluxes in these diffusion couples can be also calcu-
lated and presented in Fig. 2. It should be noted that all the exper-
imental data are denoted in symbols.
1
;2
1;3
2;3
1;2;3
DGi
,
DGi
,
DGi and
DGi
are treated as adjustable parameters.
Based on the measured composition profiles, the
composition-dependent interdiffusivities of fcc Cu–Ag–Sn alloys
at 1073 K along the entire diffusion path of each diffusion couple
were then effectively determined by using the pragmatic numerical
inverse method. During the evaluation of the interdiffusivities, the
thermodynamic descriptions for fcc phase in the Ag–Cu–Sn system
were directly taken from Kattner [23], from which the thermody-
namic factor in Eq. (4) can be provided. While the end-members
Moreover, only one or two of them are needed to be evaluated in
most cases for one diffusion couple. Therefore, by combining Eqs.
(
2)–(6), one can evaluate the composition-dependent interdiffusiv-
ities based on the measured concentration profiles together with
the available thermodynamic description.
During the evaluation of interdiffusivities, the Fick’s second law
was applied to simulate the concentration profiles based on the
computed interdiffusivities, an optimal set of adjustable parame-
1
2
3
1
;2
1;3
2;3
1;2;3
for the three binary systems,
DG , DG and DG , were taken from
ters such as
D
Gi
,
D
Gi
,
D
Gi and/or
DGi
were carefully chosen
i i i
Refs. [24–26], and fixed all the time. Moreover, the vacancy-wind
effect was considered in the present work. One or two of the adjus-
by iteratively fitting until the minimization of the error between
the measured and the simulated concentration profiles is achieved:
1
;2
1;3
2;3
1;2;3
*
+
table parameters,
D
Gi
,
D
Gi
,
D
Gi and
D
Gi , were tried to get the
N
ꢂꢄ
ꢄꢃ
ꢄ
XX
1
N
ꢄ
cal
exp
best fit to the experimental composition profiles for each diffusion
couple. With the final optimal set of adjustable parameters for each
diffusion couple, the ternary interdiffusivities of composition
dependence can be directly calculated by using Eqs. (2)–(6). All
the calculated interdiffusivities were subject to the examination
of the following thermodynamically stable constraints [27],
min < error > ¼ min
ꢄc ꢁ cij
ꢄ
ð7Þ
ij
i¼1;2 j¼1
where c ic jal and cij are the calculated and the experimental concen-
exp
trations of component i at the jth point, respectively, and N is the
total number of the experimental data. With the optimal set of
the fitting parameters, the concentration-dependent interdiffusivi-
ties in the target ternary system can be computed via Eq. (3).
~
Cu
~
Cu
D
þ D
> 0
ð12Þ
ð13Þ
SnSn
AgAg
3.2. Matano–Kirkaldy method
~
Cu
~
Cu
AgAg
~
Cu
~
Cu
AgSn
D
ꢂ D
ꢁ D
ꢂ D
P 0
SnSn
SnAg
Based on the Fick’s first law in Eq. (1), Kirkaldy successfully
extended the Boltzmann–Matano method into ternary and even
higher-order systems [13,14].
Assuming that each component has the same molar volume, the
interdiffusion flux of each component can be determined directly
from the concentration profiles without using the interdiffusion
coefficients [22]. The interdiffusion flux of component i can be
expressed as:
ꢂ
ꢃ
2
~
Cu
~
Cu
AgAg
þ 4 ꢂ D~ Cu ꢂ D
~
Cu
AgSn
D
ꢁ D
P 0
ð14Þ
SnSn
SnAg
Only those interdiffusivities fulfill the above constraints can be
output.
The finally obtained ternary interdiffusivities, D~ Cu , D
and D~
Fig. 3 as a three-dimensional (3-D) illustration. For readers’ conve-
nient usage, all the original experimental data are also provided as
the electronic Supplementary Materials. The errors of the interdif-
fusivities determined by the pragmatic numerical inverse method
were evaluated according to the scientific method proposed by
Lechelle et al. [28], who considered the error propagation via the
following function,
~
Cu
SnAg
~
Cu
AgSn
, D
SnSn
Cu
for the 5 fcc Cu–Ag–Sn diffusion couples are presented in
AgAg
Z
c
i
1
t
~
J
i
¼ 2
ðx ꢁ x
þ1
orci
0
Þdc
i
ð8Þ
ꢁ1
c
i
where t is the diffusion time, x
0
is the position of Matano plane
[
13,14] and can be obtained from the following relation:
Z
þ1
c
i
ðx ꢁ x
0
Þdc ¼ 0
ð9Þ
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
u
ꢀ
ꢁ
ꢁ
1
2
c
u X
i
@f
@a
2
t
uðfðA; B . . .ÞÞ ¼
ðuð
a
ÞÞ
ð15Þ
ꢁ
1
þ1
and ci and ci (i = 1, 2) are the terminal compositions at the left
and right sides of the diffusion couple. The initial and boundary
conditions for the semi-infinite diffusion couples are:
a
¼A;B...
Here, A and B. . . are the correlation quantities of function f like
Eqs. (3) and (5), while u(
able like concentration. During the evaluation of the errors, the
differences between the experimental concentration profiles and
the predicted ones propagate to calculation of atomic mobility in
Eq. (5) and then to calculation of diffusivities in Eq. (3). In order
to eliminate the effect of the absolute value, the relative error
a) (a = A, B. . .) is the uncertainty of vari-
c
i
ðꢃx; 0Þ ¼ c
i
ðꢃ1; tÞ ¼ cꢃi 1
ð10Þ
a
Combining Eqs. (1) and (8), one can obtain:
Z
ꢀ
ꢁ
c
i
dc
dx
1
dc
2
dx
~
3
i1
~
3
i2
ðx ꢁ x
0
Þdc
i
¼ ꢁ2t D
þ D
ð11Þ
ꢁ
1
c
ci
i
(
i.e., equals to the uncertainty divided by the absolute value of
With four equations similar to Eq. (11) from two diffusion cou-
ples, the four main and cross interdiffusivities in Eq. (1) of the
intersection point can be then determined.
the interdiffusivity) rather than the uncertainty itself was utilized
in the present work. Moreover, considering that one relative error
can be determined for one diffusivity, numerous errors should be
evaluated, displayed and stored. For a clear display of the
concentration-dependent diffusivities along the entire diffusion
paths, as well as to save the space, only the average error of the dif-
fusivities was provided here for simplification. The average relative
error of the interdiffusivities obtained by this numerical inverse
method was finally evaluated to be 8%. It can be clearly seen from
Fig. 3 that the ternary interdiffusivities for all the diffusion couples
vary apparently along with the composition of Sn and Ag. All the
4
. Results and discussion
Considering that all the 5 diffusion couples are in the same fcc
single-phase region, one typical microstructure of the diffusion
zone is given in Fig. 1, which shows the backscattered electron
image (BEI) of C5 (Cu–1.98Sn/Cu–3.5Ag) diffusion couple annealed
at 1073 K for 36 h. The concentration profiles of each component in

6
90
H. Xu et al. / Journal of Alloys and Compounds 644 (2015) 687–693
Fig. 1. Backscattered electron image of the microstructure of the Cu–1.98Sn/Cu–3.5Ag diffusion couple annealed at 1073 K for 36 h.
main interdiffusivities determined by the pragmatic numerical
inverse method are positive, while all the cross ones are negative.
According to the thermodynamic analysis by Liu et al. [29], the
negative sign of the cross interdiffusvities indicates a negative gra-
dient of diffusion potentials versus composition. Moreover, the
between the calculated interdiffusion fluxes and the experimental
data is fairly good. Both facts in Fig. 2 indicate that the presently
obtained interdiffusivities by using the numerical inverse method
are reasonable.
In order to further validate the reliability of the obtained inter-
diffusivities, the Matano–Kirkaldy method was utilized to calculate
the interdiffusivities at the specific intersection points of the diffu-
sion paths for all the 5 diffusion couples. Based on Fig. 3, there are
totally 7 intersection points of the diffusion paths. But the intersec-
tion points of C1/C3 and C3/C5 diffusion couples, as well as those of
C2/C4, C2/C5 and C4/C5 diffusion couples, are close to each other,
these intersection points are approximately supposed to be one
point. Thus, the interdiffusivities for the total 4 intersection points
were then determined. Considering that all the measured concen-
tration profiles of Sn and Ag are almost symmetric, the standard
Boltzmann function,
main interdiffusivity D~ Cu
is larger than the other main one
SnSn
~
Cu
D
, which indicates that the diffusion rate of Sn in the fcc Cu–
AgAg
Ag–Sn alloys at 1073 K is faster than that of Ag. In addition, one
more nice feature of such 3-D plot in Fig. 3 is that the projection
of the interdiffusivities for each diffusion couple on the composi-
tion plane is exactly the diffusion path of the respective diffusion
couple. In order to further provide a quantitative description of
how the diffusivities vary with concentrations and also give an idea
of the scatter in the diffusivity data, the contour maps for the
obtained interdiffusivities are plotted and displayed in Fig. 4. As
can be seen in the figure, the variation of the main interdiffusivities
~
Cu
SnSn
and D~
Cu
AgAg
D
, and the absolute values of cross interdiffusivity
A
1 þ e
1
ꢁ A
2
cðxÞ ¼
þ A
2
ð16Þ
ðxꢁx Þꢂdx
0
~
Cu
SnAg
D
depends heavily on Sn content, while that of the absolute val-
ues of cross interdiffusivity D~ Cu depends heavily on Ag content.
AgSn
was used to fit the experimental profiles of Sn and Ag and generate
the smooth concentration profiles c (x) relative to the distance.
i
Moreover, the main interdiffusivities D~ Cu and D~
Cu
, and the abso-
SnSn
AgAg
Here, A
1
, A
2
and dx are the fitting parameters for each concentration
here should be an
lute values of cross interdiffusivity D~ Cu increase as Sn content
SnAg
profile. Moreover, the position of Matano plane x
0
increases. However, the absolute values of cross interdiffusivity
average one of Sn and Ag in one diffusion couple. And the fitted con-
centration profiles of Cu were obtained based on the relationship
~
Cu
AgSn
D
decrease as Ag content increases.
Based on the obtained composition-dependent interdiffusivities
xCu = 1 ꢁ x ꢁ xSn. Based on the fitted Boltzmann functions together
Ag
along the entire diffusion path, the composition profiles for each
diffusion couple can be simulated by using Fick’s second law (i.e.,
Eq. (2)). The simulated composition profiles of Cu, Ag and Sn for
each diffusion couple annealed at 1073 K for 36 h are compared
with the corresponding experimental data, as shown in Fig. 2. As
can be seen from the figure, the simulated concentration profiles
are in excellent agreement with the experimental data.
Moreover, on the basis of the simulated composition profiles for
each diffusion couple together with either Eqs. (1) or (8), the corre-
sponding interdiffusion fluxes of each elements can be directly
computed, as also presented in Fig. 2. Again, the agreement
with the above-mentioned Matano–Kirkaldy method, the four
interdiffusivities at the interaction point of the diffusion path for
the two diffusion couples can be then obtained. All the finally
obtained interdiffusivities by the Matano–Kirkaldy method are
listed in Table 2. Moreover, the uncertainty of the interdiffusivities
was also evaluated according to the scientific method by Lechelle
et al. [28]. During the evaluation of uncertainties of the interdiffu-
sivities by the Matano–Kirkaldy method, the uncertainties of the
elemental concentrations due to different sources, like the experi-
mental measurement and the Boltzmann function fitting, firstly
propagated to calculation of the interdiffusion fluxes (e.g., Eq. (8))

H. Xu et al. / Journal of Alloys and Compounds 644 (2015) 687–693
691
Fig. 2. Comparison between the experimental and the simulated concentration profiles/interdiffusion fluxes of the diffusion couples annealed at 1073 K for 36 h based on the
ternary interdiffusion coefficients obtained in the present work; (a) C1: Cu–1.8Sn/Cu–1.9Ag, (b) C2: Cu/Cu–1.7Sn–2.6Ag, (c) C3: Cu/Cu–3.8Sn–1.0Ag, (d) C4: Cu–2.9Sn/Cu–
2
.0Ag, and (e) C5: Cu–1.98Sn/Cu–3.5Ag. Symbols are due to the experimental measurement.
and then to calculation of the interdiffusivities (e.g., Eq. (11)). Again,
the relative error (i.e., equals to the uncertainty divided by the abso-
lute value of the interdiffusivity) rather than the uncertainty itself
was utilized here. The finally relative errors for the interdiffusivities
evaluated by the Matano–Kirkaldy method are also listed in Table 2.
In order for a direct comparison, the interdiffusivities at the same
compositions calculated by using pragmatic numerical inverse
method were also listed in Table 2. It can be seen from the table that
(i) there exists differences (lower than 50% in general) among the
interdiffusivities measured at nearly the same composition by

6
92
H. Xu et al. / Journal of Alloys and Compounds 644 (2015) 687–693
~
Cu
SnSn
Fig. 3. 3-D illustration of the obtained ternary interdiffusivities along the entire diffusion paths based on the numerical inverse method in the present work: (a) D
,
b) ꢁD~
Cu
; (c) ꢁD~
Cu
; and (d) D
~
Cu
.
(
SnAg
AgSn
AgAg
~
Cu
~
Cu
~
Cu
; and (d) D~ Cu . The
Fig. 4. Contour maps for the obtained ternary interdiffusivities based on the numerical inverse method in the present work: (a) D
, (b) ꢁD
; (c) ꢁD
SnSn
SnAg
AgSn AgAg
points denote the interdiffusivities evaluated by the numerical inverse method in the present work. The dashed iso-diffusivity lines are drawn by hand based on the points.

H. Xu et al. / Journal of Alloys and Compounds 644 (2015) 687–693
693
Table 2
Diffusion coefficients in Cu-rich fcc Cu–Ag–Sn alloys obtained in this work.
ꢁ
14
m2
s )
ꢁ1
Compositions
at.%)
Diffusion coefficients (10
Matano–Kirkaldy method
D~ Cu D~ Cu
(
Numerical inverse method
D~ Cu D~ Cu
Sn
Ag
D~ Cu
AgSn
D~ Cu
AgAg
(RE)a
D~ Cu
AgSn
(RE: 8%)b
D~ Cu
AgAg
(RE: 8%)b
Couple name
SnSn
SnAg
SnSn
SnAg
(
RE)a
12.54
2.93%)
(RE)a
(RE)a
(RE: 8%)b
(RE: 8%)b
1
.012
1.619
ꢁ1.25
ꢁ1.89
4.01
(2.74%)
10.25
9.94
ꢁ0.18
ꢁ0.15
ꢁ0.17
ꢁ0.43
ꢁ0.21
ꢁ0.35
ꢁ0.33
ꢁ0.50
ꢁ0.16
ꢁ0.13
ꢁ5.68
ꢁ6.65
ꢁ9.26
ꢁ12.86
ꢁ5.19
ꢁ11.68
ꢁ5.74
ꢁ9.08
ꢁ8.66
ꢁ5.18
4.92
3.51
3.12
5.36
4.34
4.37
5.49
6.45
3.49
4.28
C2
C4
C5
C1
C3
C5
C3
C4
C1
C2
(
(14.95%)
(7.28%)
12.15
1
.337
0.323
10.97
2.77%)
ꢁ1.38
ꢁ0.29
5.55
(6.27%)
17.75
9.31
(
(26.80%)
(26.72%)
15.7
1
.999
.810
0.594
1.103
11.14
2.72%)
11.17
2.63%)
ꢁ1.65
ꢁ0.15
5.34
(4.02%)
3.94
10.86
14.96
11.92
9.22
(
(25.62%)
ꢁ0.74
(47.57%)
ꢁ2.37
0
(
(17.3%)
(9.17%)
(2.56%)
a
Here, ‘RE’ means the relative error of each interdiffusivity obtained by the Matano–Kirkaldy method.
Here, ‘RE: 8%’ means the average error of the interdiffusivities obtained by the numerical inverse method is 8%.
b
different sets of diffusion couples via the pragmatic numerical
inverse method. That is because in such inverse method the concen-
tration gradient has effect on the evaluated interdiffusivities
acknowledges the support from Shenghua Scholar Program of
Central South University, China.
[
(
30,31]. For the same composition, the compositions gradients
and also interdiffusion flux gradients) in different diffusion couples
Appendix A. Supplementary material
are completely different (see Fig. 2). However, a further analysis of
the data indicate that such effect can be eliminated if the
model-predicted concentration profiles can reproduce the experi-
mental data exactly. For instance, the model-predicted concentra-
tion profiles of couple 3 fit better to the experimental data than
those of couples 1 and 5. The diffusivities obtained from couple 3
are closer to the diffusivities obtained by the Matano–Kirkaldy
method than those from couples 1 and 5; (ii) the main interdiffusiv-
ities at the intersection points by these two methods are almost the
same, while the cross interdiffusivities show some differences.
Actually, such a difference is normal because it is well-known that
the cross interdiffusivities cannot be precisely determined by the
traditional Matano–Kirkaldy method. Therefore, it can be concluded
that the presently obtained interdiffusivities by using the numerical
inverse method are reliable.
Supplementary data associated with this article can be found, in
the online version, at http://dx.doi.org/10.1016/j.jallcom.2015.05.
0
30.
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The financial support from the National Natural Science
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