Peculiarities of Fe and Ni Diffusion in Polycrystalline Cu
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Article
Peculiarities of Fe and Ni Diffusion in Polycrystalline Cu
Alexey Rodin * and Ainur Khairullin
Department of Physical Chemistry, National University of Science and Technology “MISiS”, NUST “MISiS”, 4,
Leninsky pr-t, Moscow 119049, Russia
* Correspondence: rodin@misis.ru
Abstract: The peculiarities of Ni and Fe diffusion in polycrystalline Cu and Cu-Fe alloys at a
650–750 ◦ C temperature range were studied. It was shown that the bulk diffusion coefficients
were in a good agreement with the available literature data obtained for different temperature con-
centration ranges. The obtained values of the Ni GB diffusion triple product could be described
−165 kJmol −1
as sδDb = 8.8 × 10 − 13 × exp m3 s−1 . The obtained values were an order of magnitude
RT
less than the values obtained by the radiotracer method. This fact was explained by the negative
segregation of Ni at the Cu GB. A stronger negative segregation of Fe in the copper GB reduced
the penetration depth of iron along the GB because of an additional (negative) “driving” force; this
explained why there was no advanced grain boundary diffusion. Another peculiarity of the Fe
diffusion that was confirmed was the significant supersaturation of the Cu-based solid solution near
the Fe/Cu interface.
Keywords: diffusion; grain boundary; copper; segregation
1. Introduction
Diffusion in grain boundaries (GBs) has been of great interest for the last 70 years due
to a significant (several orders of magnitude) difference between the GB and bulk diffusion
coefficients. This effect is especially pronounced at moderate and low temperatures (below
0.7 Tm , where Tm is the melting point); the mass transport in polycrystals associated with
the flux of the diffusant in the GB becomes dominant over the direct flow through the grain
Citation: Rodin, A.; Khairullin, A. volume. Over these years of research, various methods to study grain boundary diffusion
Peculiarities of Fe and Ni Diffusion in and approaches to their accurate mathematical treatment have been developed, so that
Polycrystalline Cu. Materials 2023, 16, we can now postulate that a foundation for predicting the behavior of various elements in
922. https://doi.org/10.3390/ polycrystalline systems has already been constructed.
ma16030922 The attempt to enlarge the approaches—developed for bulk diffusion to GB diffusion,
Academic Editor: Frank Czerwinski
but taking into account the peculiarities of this object—looks natural. For example, the
basic method for the experimental determination of diffusion parameters is the radiotracer
Received: 14 November 2022 method (see e.g., [1]), which can be described as a diffusion of the tracer in its dilute solution
Revised: 11 January 2023 in a matrix. A matrix can be a one-component system or a multicomponent system and
Accepted: 12 January 2023 the concentration dependence of the diffusion coefficient can be obtained by measuring
Published: 18 January 2023 parameters at different concentrations of the alloying elements. A more complex approach,
both in creating a physical model and in the mathematical description, is based on the study
of the mutual diffusion of elements in a wide concentration range because the diffusion
of two or more components (generally speaking, not independent) is accompanied by a
Copyright: © 2023 by the authors.
possible phase formation, pore formation, shift of interface and other effects that must be
Licensee MDPI, Basel, Switzerland.
This article is an open access article
taken into account. The situation with grain boundaries is even more complicated because
distributed under the terms and
they are not the same, even within one sample. Their properties are very sensitive to
conditions of the Creative Commons the angle of grain misorientation, the presence of impurities, the possibility of boundary
Attribution (CC BY) license (https:// migration and other effects [2]. In this sense, a comparison of the grain boundary diffusion
creativecommons.org/licenses/by/ data obtained by different authors requires a deep analysis, especially if different methods
4.0/). for the determination of parameters are used. It is necessary to underline that in the
Materials 2023, 16, 922. https://doi.org/10.3390/ma16030922 https://www.mdpi.com/journal/materials
Materials 2023, 16, 922 2 of 11
practically important temperature regime of moderate temperatures (close to 0.5 Tm ), the
grain boundary diffusion can be described by a so-called triple product P = sδDb , where
s = ccb x=±δ/2 is the GB enrichment factor (ratio of the GB and bulk concentrations), δ is
the grain boundary width and Db is the GB diffusion coefficient. This is because the bulk
diffusion determines not only the direct flux from the surface into the grain, but also the
flux from the GB to the grain [1,2]. It is evident that the greater the GB segregation factor,
the greater the triple product and the deeper the penetration of the diffusant along the GB.
In this paper, we present a comparative analysis of the results of the study of the
bulk and GB diffusion obtained by different methods (the methods of radiotracer isotopes
and a microprobe analysis (EPMA)) for iron and nickel in copper. As additional data to
the discussion, the results of the computer modeling for these systems as well as for a
cobalt–copper system were used.
The choice of the system was based on scientific interest of a system with negative GB
segregation (segregation factor s < 1) [3,4], but Cu-Fe alloys are of a practical interest as a
soft magnetic material [5] as well as being conductive materials with enhanced mechanical
and wear-resistant properties [6]. Fe-based coatings allow the improvement of wear
resistance for several tools [7,8]. The stability of the microstructure and properties at
elevated temperatures are connected to the diffusion properties of polycrystalline alloys.
The bulk and GB diffusion of the chosen systems have been studied in a number
of works. One can observe a good agreement between the bulk diffusion data both for
Ni [9–14] and Fe [9,15–18] diffusion in Cu. Regarding the GB diffusion, the difference in
the results is significant.
Triple product values for Ni in Cu obtained by autoradiography [19–21] demonstrate
a difference of two orders of magnitude. The most reliable data, obtained by the radio-
tracer technique [22], provided intermediate values; however, the data obtained by an
electron probe microanalysis (EPMA) [23] obtained an order of magnitude with higher
values than [22]. This is surprising, because radiotracer methods typically provide the
characteristics for the fastest GB [24]. n o
According to [25], the triple product of Fe diffusion is P = 9.1 × 10−12 exp −121kJ/mol
RT ,
but this could only be determined in a temperature range of 949–1247 K, which is more than
0.7 Tm . Our results [26–28] demonstrated the absence of a concentration excess near the GB
in comparison with the bulk at high (above 1073 K) and low (below 873 K) temperatures;
thus, the GB diffusion parameters could not be determined. On the other hand, the triple
product recalculated from the growth rate of cobalt–iron particles at the GBs of a copper
alloy at 823 and 873 K showed the absence of any anomalies [29], providing a reasonable
value for the GB triple product close to Cu self-diffusion.
In order to ascertain the possible reasons of these contradictions, a diffusion study with
an enlarged time–temperature range was conducted for an Fe-Cu system. Special attention
was paid to the correct treatment of the results; thus, the experiment was conducted both
for the cross-section and the foils at the same temperatures. In order to study the possible
effect of Fe on the diffusion characteristics of the grain boundaries in Cu, the effect of the
alloying of Cu by Fe was studied by using Fe and Ni as diffusants.
2. Materials and Methods
An energy dispersive X-ray microanalysis (EDX) was used to determine the concen-
tration of the diffusing elements near the grain boundary and far from it. It allowed us to
obtain the data by a direct comparison of the penetration depths of the elements. If there
was a significant difference in the penetration depths, there was the possibility of separately
determining the parameters of the GB and bulk diffusion. The accuracy of this method
does not exceed 0.1%; therefore, only data above 0.2–0.3% were taken into account for the
analysis.Materials 2023, 15, x FOR PEER REVIEW 3 of 11
Materials 2023, 16, 922
method does not exceed 0.1%; therefore, only data above 0.2–0.3% were taken into ac- 3 of 11
count for the analysis.
For the research, copper of 99.995% purity, iron of 99.99% purity, chemically pure
(better than 99%) nickel and iron sulfates (NiSO4·7H2O and FeSO4·7H2O) were used.
For the research, copper of 99.995% purity, iron of 99.99% purity, chemically pure
The Cu-Fe alloys were prepared by dissolving the master alloys (containing 1.25%
(better than 99%) nickel and iron sulfates (NiSO4 ·7H2 O and FeSO4 ·7H2 O) were used.
iron) in liquid copper at a temperature of 1200 °C in Ar with the small addition of a H2
The Cu-Fe alloys were prepared by dissolving the master alloys (containing 1.25%
atmosphere. The melts were kept for 4 h at this temperature and cooled down by re-
iron) in liquid copper at a temperature of 1200 ◦ C in Ar with the small addition of a H2
moving the quartz reactor from the furnace under the conditions of a weak Ar flux.
atmosphere. The melts were kept for 4 h at this temperature and cooled down by removing
The
theingots
quartz(diameter
reactor from of about 20 mm)under
the furnace were the
keptconditions
at a temperature
of a weak of 1050 °C in an Ar
Ar flux.
atmosphereThe foringots
20 h. (diameter
They wereofcut about 20 mm) were kept at a temperature of 1050 ◦these
into discs of approximately 3 mm thickness; C in an Ar
were carefully
atmosphere cleaned,
for 20 h.thinned by deformation
They were cut into discs(70%) and polished.
of approximately 3 mmAfter that, recrys-
thickness; these were
tallization annealing
carefully cleaned, was carriedby
thinned out in a hydrogen
deformation (70%)atmosphere
and polished. at 1000
After°Cthat,
for 1recrystallization
h.
The iron content in the samples was determined by
annealing was carried out in a hydrogen atmosphere at 1000 C for 1 h. atomic emission
◦ spectroscopy
with inductively
The iron coupled
contentplasma. Two types
in the samples of samples with
was determined 0.4 at%
by atomic and 0.3
emission at% were with
spectroscopy
obtained.
inductively coupled plasma. Two types of samples with 0.4 at% and 0.3 at% were obtained.
An electrochemical
An electrochemical deposition fromfrom
deposition sulfate aqueous
sulfate aqueous electrolytes
electrolyteswaswasusedused to to
pro-
produce
duce the sample for the diffusion study. The electrolyte for the nickel
the sample for the diffusion study. The electrolyte for the nickel deposition comprised deposition com-
prisednickel
nickelsulfate
sulfate(250(250g/L),
g/L),boric
boricacid
acid(30(30 g/L)and
g/L) andsodium
sodium chloride
chloride (10(10 g/L)
g/L) asas
wellwell
as as
distilled
distilled
water. The Fe deposition comprised iron sulfate (250 g/L), potassium sulfate(150
water. The Fe deposition comprised iron sulfate (250 g/L), potassium sulfate (150 g/L)
g/L) and
andoxalic
oxalicacid
acid(4(4g/L).
g/L).TheThetime
time was
was selected
selected inin
order
order toto
obtain
obtain a a2020μm-thick
µm-thicklayer.layer. The
The samples
sampleswerewereannealed
annealedininquartz
quartzampoules
ampoulesand andevacuated
evacuatedtotoa a1010 mm
−3 − 3 mm HgHg vacuum.
vacuum.
Another type of
Another typesample was was
of sample an 18anμm-thick
18 µm-thick CuCu foilfoil
with
withFeFe onononeoneside,
side,as aspro-
produced
ducedinin[23,24].
[23,24].Typical
Typical views
views of of
thethe diffusion
diffusion couple
couple cross-section
cross-section (points
(points 1–5 corre- with
1–5 correspond
spondthe with
bulkthe bulk diffusion
diffusion measurementsmeasurements
and pointsand 6–14points
are the6–14 are the measurements)
GB diffusion GB diffusion and
measurements) and the reverse side of the annealed foil
the reverse side of the annealed foil are presented in Figure 1. are presented in Figure 1.
(a)
Figure 1. Cont.Materials 2023, 15, x FOR PEER REVIEW 4 of 11
Materials 2023,16,
Materials2023, 15,922
x FOR PEER REVIEW 4 of1111
4 of
(b) (b)
Figure 1. SEM 1.
Figure
Figure images:
1. SEM (a) surface
SEM images:
images: (a) of the cross-section
surface of Cu-Ni
of the cross-section
cross-section ofdiffusion
of Cu-Ni couple;couple;
Cu-Ni diffusion
diffusion (b) the (b)
couple; foil after
(b)the
thefoil
foilafter
after
annealing (opposite
annealing to Fe layer
(opposite to side).
Fe layer side).
annealing (opposite to Fe layer side).
3.3.Results
3. Results Results
3.1.
3.1.Ni
NiDiffusion
3.1. Ni Diffusion
Diffusion
The ◦ C;
Thediffusion
The diffusion diffusionofofnickel
of nickel into into
pure
nickel into pure
purecopper
copper was
wasstudied
was studied
copper atat
temperatures
at temperatures
studied ofof
of 650
temperatures 650
and and
650750 750750
and
with the alloys containing ◦ C.
°C; with°C;
thewith thecontaining
alloys alloys 0.3 0.3
andand
containing 0.3
0.4%0.4%
and iron,
0.4%
iron, the
iron,
the temperatures
the temperatures
temperatures were
were 650,
were
650, 700
650,
700 and and
700 750750
and
750
Typical
°C. Typical concentration
°C. Typical
concentration profiles
concentration
profiles at the
profiles
at the at GB
GB andand
the GB in the
and
in the in bulk
bulk theareare
bulk presented in Figure
are presented
presented 2. 2. 2.
in Figure
in Figure
120
120
100 this work
this work
100
80 calculation interface Ni-
80 (0.4Fe+Cu)
С, аt.%
60 Wipple
С, аt.%
interface Ni-
('0.4Fe+Cu) 60 calculation
40
40
20
20
0
0 10 20 30 40
0
0 20 40 60
х, µm х, µm
(a) (b)
Figure 2. Concentration profiles for bulk (a) and GB (b) diffusion of Ni in Сu + 0.4 Fe at% at T = 700
Figure
°С for 72Figure
h. 2.2.Concentration
Concentrationprofiles
profilesfor
forbulk
bulk(a)
(a)and
andGB
GB(b)
(b)diffusion
diffusionofof
NiNi
inin
CuСu + 0.4
+ 0.4 FeFe
at%at% at=T700
at T = 700
◦C
°С for
for 72 h.72 h.Materials 2023, 16, 922 5 of 11
The data on bulk diffusion were described by the erf-like equation (solution of the
diffusion equation with the initial condition corresponding with a layer of finite thickness a
and the diffusion coefficient D being constant):
x−a
C0 x+a
C(x, t) = (erf √ + erf √ ) (1)
2 2 Dt 2 Dt
For the data on the grain boundary diffusion, a modified Whipple solution [30] was
used:
Cb = C ( x, t) + Cb1 ( x, y, t)
∆
0.5 δ √
C0 ∗( x − a) R 1 ( x − a )2 1 Db − D y− 2(σ −1) D Dt (2)
Cb1 ( x, y, t) = √
σ3/2
∗ exp 4σDt ∗ erfc 2 D −σD ∗ √ 2 + sδDb dσ
4 πDt b Dt
1
This was essential because the Whipple solution is valid for the case of the boundary
condition corresponding with a constant concentration on the surface. In our case, this
condition was not fulfilled.
The obtained values of the bulk and grain boundary diffusion (averaged from at least
5 different profiles) parameters are presented in Table 1.
Table 1. Ni bulk diffusion coefficient (D) and GB diffusion triple product (P = sδDb ).
GB Diffusion Triple Product
Bulk Diffusion Coefficient D × 1016 , m2 /s
◦C P × 1021 , m3 /s
T,
Pure Cu Cu + 0.3% Fe Cu + 0.4% Fe Pure Cu Cu + 0.3% Fe Cu + 0.4% Fe
750 1.5 2.5 2.4 2 5 2.5
700 - 0.41 0.4 - 0.84 1.2
650 0.13 0.18 0.12 0.25 0.23 0.36
The mean square displacement could be estimated as 10–20% for each D obtained
from the given profile and around 30% for the averaged value at a given temperature. Thus,
we observed only a small difference in the values of the GB diffusion for the pure Cu and
the Cu-Fe alloys. The temperature dependences for the Ni bulk diffusion coefficient and the
P-value are presented in Figure 3 in comparison with the data from [9–14,19–22]. It could
be seen that the values of DNi slightly differed from the other data; PNi was significantly
smaller than the data obtained in [22,23]. Notably, the Wipple approach [30] was used
directly whereas in [23], the data were recalculated from the values obtained with the use
of the Fisher solution for a quasi-stationary approximation [31].
The temperature dependence of the GB diffusion triple product (assuming the absence
of a difference between the data for pure Cu and Cu-Fe alloys) could be expressed as
follows:
−165 kJmol −1
−13
sδDb = 8.8 × 10 × exp m 3 s −1 (3)
RT
3.2. Results of Bulk and Grain Boundary Diffusion of Iron into Copper and Copper-Based Alloys
The diffusion of iron into pure copper was studied at three temperatures: 650 (138 h),
700 (24, 96, 192 and 312 h) and 750 ◦ C (30 h). The annealing time was varied at one
temperature to estimate the concentration change near the iron/copper interface. For the
copper–iron alloys, the annealing times were 138 and 190 h (650 ◦ C), 49 and 72 h (700 ◦ C)
and 30 h (750 ◦ C). The typical concentration profiles obtained from the EDX analysis are
shown in Figure 4.Materials
Materials 2023,
2023, 16,15,
922x FOR PEER REVIEW 66of
of 11
11
Figure 3. Arrhenius plots for bulk [9–14] (a) and GB [19–22] (b) diffusion of Ni in Cu, according to
different authors. Points correspond with the data obtained in the present work. Averaged value
for Cu-Fe alloys are presented as triangles.
3.2. Results of Bulk and Grain Boundary Diffusion of Iron into Copper and Copper-Based Alloys
The diffusion of iron into pure copper was studied at three temperatures: 650 (138
h), 700 (24, 96, 192 and 312 h) and 750 °C (30 h). The annealing time was varied at one
temperature to estimate the concentration change near the iron/copper interface. For the
Figure3. 3. Arrheniusplots
Arrhenius plotsfor
forbulk
bulk[9–14]
[9–14](a)
(a)and
andGB
GB [19–22](b)
(b) diffusion of
of Ni
Ni in
in Cu, according
according to
Figure
copper–iron alloys, the annealing times were 138[19–22]
and 190 hdiffusion
(650 °C), 49 andCu,72 h (700 °C)to
different
different authors. Points correspond with the data obtained in the present work. Averaged value
and
for 30 hauthors.
Cu-Fe (750 Points
°C).
alloys areThe
correspond with the dataprofiles
typicalasconcentration
presented triangles.
obtainedobtained
in the present
fromwork. Averaged
the EDX value
analysis for
are
Cu-Fe alloys are
shown in Figure 4. presented as triangles.
3.2. Results of Bulk and Grain Boundary Diffusion of Iron into Copper and Copper-Based Alloys
The diffusion of iron into pure copper was studied at three temperatures: 650 (138
h), 700 (24, 96, 192 and 312 h) and 750 °C (30 h). The annealing time was varied at one
temperature to estimate the concentration change near the iron/copper interface. For the
copper–iron alloys, the annealing times were 138 and 190 h (650 °C), 49 and 72 h (700 °C)
and 30 h (750 °C). The typical concentration profiles obtained from the EDX analysis are
shown in Figure 4.
Figure 4.
Figure 4. Typical
Typical concentration
concentration profiles
profiles corresponding
correspondingwith
withmeasurements
measurementsfar
faraway
away(a)
(a)and
andnear
neartoto
the GB
the GB (b)
(b) for
for Fe
Fe diffusion
diffusion of
of Сu
Cu ++ 0.3
0.3 Fe
Fe at.%
at.%(T
(T==700
700°С;
◦ C;t t==4949h).
h).
The concentration was
The was expressed
expressed as as the
thedifference
differencebetween
betweenthethemeasured
measuredvalue
valueandand
the concentration
the concentration in the matrix
matrix (0,
(0, 0.3
0.3 and
and0.40.4at%
at%for
forthe
thedifferent
differentsamples).
samples).Vertical
Verticallines
lines
showed the distance corresponding with the foil thickness (18 µm), taking into account the
typical depth of the analysis (3 µm).
The maximum concentration (concentration at the Fe/Cu interface) was obtained by
Figure 4. Typical concentration profiles corresponding with measurements far away (a) and near to
approximating the diffusion profile to a zero depth, as posited in [26,27]. To determine the
the GB (b) for Fe diffusion of Сu + 0.3 Fe at.% (T = 700 °С; t = 49 h).
DFe and Cs, the erfc-like equation was used:
The concentration was expressed as the difference
x − a between the measured value and
the concentration in the matrix C ((0, t) =
x, 0.3 CS0.4
and erfc √ the different samples). Vertical lines
at% for (4)
2 DtMaterials 2023, 16, 922 7 of 11
It is important to note that CS was almost the same for all times of annealing at a given
temperature (see Table 2); thus, Equation 4 could be used. On the other hand, it decreased
with an increase in the temperature (Table 3). This indicated that if one could usually use
Cs as a solubility at this temperature (C0 ), here, it was only a value corresponding with a
stationary regime.
Table 2. Results of determination of Cs obtained by the extrapolation of Equation (4) on x = a at
T = 700 ◦ C.
Time 24 h 48 h 72 h 96 h 192 h 312 h
Cs, at.% 2.9 3.1 3.2 3.1 2.7 3.4
Table 3. Bulk diffusion coefficient for Fe-Cu, Fe-(Cu + Fe) and Fe-(Cu + S).
Bulk Diffusion Coefficient D × 1016 , m2 /s Max. Conc., Solubility,
T, ◦ C
Pure Cu Cu + 0.3% Fe Cu + 0.4% Fe Cu + 0.002% S [27] [9] Cs C0 [32]
750 12 11 18 10.2 12.7 3.1 0.21
700 4.8 5.5 5.4 - 3.4 3.0 0.3
650 1.7 1.2 1.7 1 0.82 2.2 0.46
Note the two features of the obtained profiles:
• The measured values of the concentration profiles were significantly above the solubil-
ity limit at the temperatures of annealing (the solubility at different temperatures is
given in Table 3).
• The profiles near and far from the grain boundary were practically the same and a
matrix concentration was reached at the same depth.
The level of iron supersaturation in the copper near the surface was determined as the
ratio CS /C0 . At a temperature of 650 ◦ C, the supersaturation was 15; at 700 ◦ C, it was 9
and at 750 ◦ C, it was 5. The value of supersaturation decreased not only due to an increase
in solubility, but also due to a decrease in Cs.
As it was impossible to determine the parameters of the grain boundary diffusion in
the absence of a difference in the profiles of the bulk and GB diffusion, only the average
values of the bulk diffusion coefficients were determined, which are given in Table 3 and
Figure 5. Additionally, data obtained earlier [27] are presented. We noted that the results
were in a good agreement with the previous data.
In order to compare the results with the measured values of the concentration on
the opposite side of the copper foils annealed for different times, the concentrations of Fe
corresponding with 18 µm were calculated (Table 4). The Cs values were also taken from
our measurements and, as one can see, the calculated results were in a good agreement
with the experimental ones for a short time of annealing.
Table 4. Calculated and measured concentration (at.%) far from the GB and the measured value near
to the GB opposite to the diffusant layer side of the foil.
Grain Boundary, Calculation Bulk,
Fe-Cu Average Bulk, at.%
at.% at.%
650 ◦ C, 65 h 0.4 0.3 0.3
650 ◦ C, 120 h 0.4 0.3 0.4
700 ◦ C, 24 h 0.3 0.3 0.3
700 ◦ C, 45 h 0.4 0.4 0.5
750 ◦ C, 15 h 0.4 0.4 0.3
750 ◦ C, 27 h 0.4 0.3 0.4
750 ◦ C, 70 h 0.5 0.4 1.0Materials
Materials 2023,
2023, 15,
16, x922
FOR PEER REVIEW 8 8ofof 11
11
Figure Arrheniusplot
Figure 5. Arrhenius plotfor
forbulk
bulk diffusion
diffusion of in
of Fe FeCu
in according
Cu according to different
to different authors
authors [9,15–18].
[9,15–18]. Points
Points correspond with the data obtained in the present
correspond with the data obtained in the present work. work.
4. Discussion
In order to compare the results with the measured values of the concentration on the
opposite side oftothe
According thecopper foilsresults,
presented annealed for different
the used times, the
method obtained concentrations
a reasonable of Fe
agreement
for the bulk diffusion
corresponding with 18data
μmforwereboth systems.(Table
calculated For the4).case
TheofCs
Nivalues
diffusion,
werethe concentration
also taken from
profiles
our were quite symmetric,
measurements and, as onewith canthesee,interface corresponding
the calculated with its
results were in initial
a good position. The
agreement
slightthe
with non-symmetry
experimentalcould
ones beforexplained
a short time by of
theannealing.
concentration dependence of the diffusion
coefficient. One could expect a notable difference in the diffusion coefficients at different
concentration
Table ranges
4. Calculated andbecause
measured ofconcentration
the significant difference
(at.%) far frominthetheGB
melting
and thetemperatures
measured value of
near to the
copper and GBnickel
opposite ◦ C diffusant
to the
(1084 and 1455layer ◦ C, correspondingly).
side of the foil. However, as shown in [33], the
concentration dependence of the diffusion coefficient was significant at a high temperature
Calculation Bulk,
only. For
Fe-Cuthe temperature range 900–1000
Grain Boundary, at.% ◦ C,Average
the difference in the mutual diffusivity for
Bulk, at.%
at.%
different Ni concentrations was almost 100 times; at 850 ◦ C, it was 10 times and at 730 ◦ C,
650 °С,
it was only653htimes and could 0.4 be observed at a narrow 0.3 concentration range 0.3 near 80%
650 °С, 120 h 0.4 0.3
Ni. For other concentration ranges, the diffusion coefficient was of the same value. The 0.4
700 °C, 24 hprofiles obtained
concentration 0.3 in the present work confirmed 0.3 this fact. The0.3Fe diffusion
700 °C, was
coefficient 45 h also close to the0.4literature data. In both0.4 cases, the bulk diffusion 0.5coefficients
did750
not°C, 15 h on the preliminary
depend 0.4 alloying of the copper 0.4 with iron. 0.3
Regarding
750 °C, 27 h the Ni GB diffusion,
0.4 a significant difference
0.3 could be seen, with 0.4the experi-
mental data70measured
750 °C, h in a similar
0.5 way [23]. First, we noted 0.4 that it was necessary 1.0 to modify
the Whipple equation for the calculation because the conditions did not fully correspond
with
4. the diffusion from a source, with a constant concentration for both the bulk and GB
Discussion
diffusion. A significant difference from the radiotracer results [22] was reasonable because
According to the presented results, the used method obtained a reasonable agree-
the treatment of the tails of the concentration profiles provided the diffusion coefficient for
ment for the bulk diffusion data for both systems. For the case of Ni diffusion, the con-
the fastest GB but not the average [24]. Taking into account that the scattering of the GB
centration profiles were quite symmetric, with the interface corresponding with its initial
diffusion parameters can be 1–2 orders of magnitude [34], the large scatter of our results and
position. The slight non-symmetry could be explained by the concentration dependence
the difference with the data, as mentioned above, appeared to be natural. We also observed
of the diffusion coefficient. One could expect a notable difference in the diffusion coeffi-
a significant difference in the GB diffusion effective activation energy. The calculated value
cients at different concentration ranges because of the significant difference in the melt-
(160 kJ/mol) was significantly higher than the typical value of 12 E (with half the activation
ing temperatures of copper and nickel (1084 °C and 1455 °C, correspondingly). However,
energy of the bulk diffusion being around 100 kJ/mol) and the values obtained in other
as shown in [33], the concentration dependence of the diffusion coefficient was signifi-
works (e.g., 91 and 74 kJ/mol in [13] for Cu samples of different purity). The effective
cant at a high
activation temperature
energy only. For
of GB diffusion the temperature
consists of two termsrange 900–1000
connected °C, activation
with the the difference in
energy
the mutual diffusivity for different Ni concentrations was almost 100 times;
of diffusion and the enthalpy of segregation. This is the reason why, typically, the activation at 850 °C, it
was 10 times
energy and atGB
of chemical 730diffusion
°C, it was onlythan
is less 3 times and could be (the
for self-diffusion observed
enthalpyat aof
narrow con-
adsorption
centration range near 80% Ni. For other concentration ranges, the diffusion coefficientMaterials 2023, 16, 922 9 of 11
is negative). However, the results of a number of studies on equilibrium adsorption in
copper–nickel systems allowed us to conclude that copper positively segregated in the
entire concentration range; thus, the segregation of Ni was negative [3]. It opposes the
calculation from the diffusion results, which provided a positive segregation [22]. The
difference in the results may have been due to the fact that the radiotracer method requires
relatively short times. The determined enrichment coefficient of the GB did not correspond
with an equilibrium state or even a quasi-stationary regime. The diffusion process can be
described as the movement of the diffusing element along the GB and atoms partly enter
the volume from the grain boundary. Thus, on the GB, the concentration is constantly
greater than in the adjacent bulk. Studies with non-radioactive isotopes require significantly
longer times and the established ratio is closer (although it may still be different) to that of
the equilibrium.
The absence of the effect of pre-alloying was obtained both in this work and in [22]. It is
clear that diffusion, especially isotopic diffusion, is sensitive to changes in the GB structure;
thus, it is natural to conclude that the structure of the GB and its kinetic characteristics
do not change under the influence of such alloying. In [25], it was concluded that iron
negatively segregated and exited from the grain boundaries (or the grain boundary moved
from the line of Fe atoms). The same conclusion can be reached by analyzing the results
of the computer modeling performed in [35]. Despite the different models taken for the
description of the interaction between atoms, the general effect is the same. The close effect
for the diffusion of Co in Cu provides the same conclusion that a negative segregation
significantly slows down GB diffusion; thus, it cannot be ascertained if the concentration in
the bulk is relatively high or not.
There are no direct measurements of iron (as well as cobalt) segregation on the surface
and at grain boundaries. However, the isotherms of the surface and grain boundary energy
of cobalt and iron in copper obtained in [4,36] directly indicated the growth of surface
energy at the initial section of the curve, which, according to the Gibbs adsorption equation,
indicates a negative adsorption. The enrichment coefficient estimated from this data was
approximately equal to 0.1–0.3. As was shown in a modified model of GB diffusion, taking
into account the surface energy gradient as an additional driving force [37], the additional
term was only important in the case of negative segregation and only in the case of a
relatively high concentration level. As was shown, the concentration of iron reached
significantly higher values than the solubility limit. The reason for the formation of a
supersaturated solid solution was the absence (or almost an absence) of Cu diffusion in
iron. In contact with a copper-based solid solution, we obtained almost pure iron instead
of an iron-based solid solution. A superimposed solution can be stable for a long time
due to the same reasons; for its decay, the formation of a solid solution of copper in iron is
necessary.
Summarizing the results of the present work and our earlier studies [26–28], we
concluded that there was an absence of advanced grain boundary diffusion of iron in
copper in undiluted copper-based solutions at a wide temperature range (above 550 ◦ C).
This could be explained by the negative adsorption of Fe in Cu, which decreased the p-value
and provoked an additional braking force.
5. Conclusions
The grain boundary diffusion of two elements with negative segregation (Ni and Fe)
in Cu was studied. The results of the determination of the bulk diffusion coefficients for
Ni and Fe demonstrated the sufficient accuracy of the methods used for the GB diffusion
studies. The weak concentration dependence of the bulk diffusion coefficient allowed us
to use, in this case, the Wipple solution for the GB diffusion problem, with modifications
to the initial and boundary conditions. In contradiction to the literary data, it was shown
that Ni GB diffusion at a high concentration was significantly slower than in a dilute
solution and could be characterized by a high value of effective activation energy of 160
kJ/mol, which could be explained by the negative adsorption of nickel on the Cu GB.Materials 2023, 16, 922 10 of 11
There was also no concentration dependence for the iron diffusion, although the studies
that were carried out corresponded with supersaturated solid solutions with a maximum
concentration 10–15 times higher than the solubility at a given temperature. Regarding
the grain boundary diffusion of iron, a significant negative adsorption reduced the depth
of the penetration of iron along the GB and explained why there was no advanced grain
boundary diffusion.
Author Contributions: Conceptualization, A.R.; Formal analysis, A.K.; Investigation, A.K.; Data
curation, A.K.; Writing—review & editing, A.R.; Supervision, A.R. All authors have read and agreed
to the published version of the manuscript.
Funding: The work was made with the financial support of the NUST “MISiS” program “Priority-
2030” Agreement # 075-15-2021-1306, dated 30 September 2021.
Institutional Review Board Statement: Not applicable.
Informed Consent Statement: Not applicable.
Conflicts of Interest: The authors declare no conflict of interest.
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