Grain boundary diffusion and segregation of 57Co in high-purity copper: Radiotracer measurements in B- and C-type diffusion regimes

Accepted Manuscript 57 Grain boundary diffusion and segregation of Co in high-purity copper: Radiotracer measurements in B- and C-type diffusion regimes Daniel Gaertner, Gerhard Wilde, Sergiy V. Divinski PII:

S1359-6454(17)30057-5

DOI:

10.1016/j.actamat.2017.01.045

Reference:

AM 13507

To appear in:

Acta Materialia

Received Date: 11 November 2016 Revised Date:

18 January 2017

Accepted Date: 22 January 2017

Please cite this article as: D. Gaertner, G. Wilde, S.V. Divinski, Grain boundary diffusion and segregation 57 of Co in high-purity copper: Radiotracer measurements in B- and C-type diffusion regimes, Acta Materialia (2017), doi: 10.1016/j.actamat.2017.01.045. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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Grain boundary diffusion and segregation of 57Co in high-purity copper: Radiotracer measurements in B- and C-type diffusion regimes Daniel Gaertner*, Gerhard Wilde, and Sergiy V. Divinski† Institute of Materials Physics, University of Münster

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Wilhelm-Klemm-Str. 10, 48149 Münster, Germany

Abstract

Grain boundary diffusion of 57 Co in high-purity polycrystalline copper is investigated using the radiotracer technique in Harrison’s B- (850 − 1150 K) and C-type (550 − 950 K) kinetic regimes. The triple product P = s · δ · Dgb (s is the segregation factor and

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δ the grain boundary width) and the grain boundary diffusion coefficient Dgb of Co in Cu are determined to obey the Arrhenius laws with the activation enthalpies of Qgb = 66.2 kJ/mol and Hgb = 100.9 kJ/mol, respectively. Using the experimental estimate of δ, δ∼ = 0.5 nm, Co is found to segregate strongly at Cu grain boundaries and the corresponding

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segregation factor follows an Arrhenius dependence with the segregation enthalpy of Hs =

−34.7 kJ/mol . Co-diffusion experiments with the

57 Co

and

110m Ag

isotopes support a

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’sub-interface’-type of grain boundary segregation of Co in Cu.

Keywords: Copper; Grain boundary diffusion; Grain boundary segregation; Cobalt

* Corresponding author: † Corresponding author:

[email protected] [email protected]

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1 Introduction Radiotracer measurements of solute grain boundary (GB) diffusion in a polycrystalline solid are typically performed in true dilute limit conditions and, as a result, equilibrium solute segregation

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can be determined [1]. The key idea is to combine the GB diffusion experiments with a solute in both Harrison’s B and C type conditions [2] for the same polycrystalline material. At low temperatures and/or relatively short annealing times (Harrison’s C regime) the tracer diffuses dominantly along GBs, there is practically no bulk diffusion, and the GB diffusion coefficient,

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Dgb , can directly be measured. At relatively high temperatures and/or longer annealing times

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(Harrison’s B regime) the tracer diffuses fast along GBs with a subsequent leakage into the √ adjacent grain interiors which is characterized by the corresponding bulk diffusion length Dv t (Dv and t are the bulk diffusion coefficient and the diffusion time, respectively). The latter has to be significantly larger than the GB width δ remaining smaller than the grain size d. As an exact solution of such diffusion measurement, the so-called triple product

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P = s · δ · Dgb

(1)

can be evaluated [3]. Here the pertinent solute segregation factor s is determined as the ratio

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between the solute concentrations in the GB, cgb , and in the adjacent bulk, cv cgb s= cv x=±δ/2

(2)

and δ is the diffusional GB width, respectively. Here x is the coordinate perpendicular to the

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grain boundary which in Fisher’s model [4] is considered as a homogeneous slab of the thickness δ.

Combining the B- and C-type regime measurements, the product of the solute segregation factor, s, and the diffusional GB width, δ, can be determined as:

s·δ =

P Dgb

(3)

The combination of the B- and C-type regime measurements for GB self-diffusion, when the solute segregation factor s = 1, allows an experimental determination of the diffusional GB 2

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width δ. The GB self-diffusion measurements over very large temperature intervals in NiO [5], pure Ag [6, 7], Ni [8, 9], Fe [10], α-Ti [11] and in a nanocrystalline γ-FeNi-alloy [12, 13, 14] provided a temperature independent value of about 0.5 nm (within the limits of experimental uncertainties) for the diffusional GB width . Using this value of δ, the solute segregation factor

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s can be determined from Eq. (3), assuming that the GB width negligibly depends on the type of tracer atoms [15]. A pre-requisite of the application of the approach, Eqs. (1) – (3), is the stability of the same GB structure and an absence of any structure transition in the whole temperature interval of the GB diffusion measurements [15]. In the case of a GB structure

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transition, a kink in the corresponding Arrhenius dependencies can be observed [17, 18].

So far, the GB diffusion of different solutes – Au [19], Ag [20, 21] , Se [22], Ge [23], Bi [24],

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Ni [25, 26] , Fe [27, 28] and Cu self-diffusion [29] – in the same high-purity copper (99.9998 wt.%) has been measured and the corresponding segregation factors have been determined. The investigation of Ni grain-boundary diffusion in 5N Cu depending on the sulfur content and preannealing treatments by Tôkei et. al. [26] verifies the importance of using the same high-purity matrix in order to determine the intrinsic diffusion and segregation properties. Note that GB

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diffusion of Ag in Cu was investigated using Auger electron spectroscopy and applying the Hwang-Balluffi method [21] and the results are similar to those determined by the radiotracer technique [20].

It seems that presently Cu is the most intensively investigated metal with respect to grain

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boundary diffusion and segregation. A further example is pure Al of nominally the same purity but different origin in which GB diffusion of different solutes – i.e. Fe [30], Zn [31, 32, 33], and

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Ga [34] – has been investigated using the radiotracer technique or electron probe micro-analysis (predominantly in the B-type kinetic regime). Comparing all solutes studied, the ferromagnetic impurities like nickel and iron represent a special case of solutes with relatively small atomic radii and high surface energies with respect to those of pure copper. However, nickel and iron differ significantly concerning their solubility in Cu – nickel is completely miscible (a miscibility gap is though suspected at low temperatures) and iron is almost immiscible. While for Ni in Cu a moderate but distinct segregation has been established [25], the case of Fe diffusion and segregation in Cu turned out to be quite intricate and even formal C-type profiles were measured at high temperatures in the B regime [27, 28]. 3

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It was proposed that at higher temperatures GB diffusion-induced Fe coverage of a GB core in Cu provokes grain boundary instability with respect to lateral shifts by several atomic planes that induces Fe-rich layers in the adjacent grains [28]. As a consequence the Cu matrix is over-

measurements under B-type kinetic conditions [28].

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saturated locally and this fact changes the kinetics of the GB penetration which complicates

The present paper aims to investigate GB diffusion and segregation of another ferromagnetic impurity, namely of cobalt, in the same high-purity copper. Cobalt is similar to Ni and Fe in many respects, it has a smaller atomic radius and a higher surface energy as those of Cu and

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is almost immiscible with it [35]. Actually, investigations of Co segregation in Cu-Co-alloys were performed [36], but the segregation behavior of Co in GBs of high-purity polycrystalline

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copper remains unknown. Rodin et al. [37] already tried to investigate GB diffusion of Co in Cu using an electron probe micro analysis, however, no Co GB diffusion flux could be detected. In view of the similarity of the Fe–Cu and Co–Cu systems, similar segregation behaviors may potentially be expected. However, a fundamental difference is found with a strong temperature dependence of Co GB segregation, whereas Fe segregation in the same Cu material is practically

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temperature-independent.

2 Experimental procedure

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2.1 Sample preparation

Copper of the nominal purity 5N8 was used (which corresponds to the material A used for the

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copper self-diffusion investigation in Ref. [29]). The preparation procedure of the cylindrical samples of about 10 mm in diameter and 3 mm in thickness was equivalent to the procedure described in Ref. [25]. One face of the specimen was polished by a standard metallographic procedure to a mirror-like quality. In order to recover the defects introduced by the preparation procedure, the specimen was sealed in a silica tube under a purified (5N) Ar atmosphere and annealed at 1023 K for 24 hours. After this pre-diffusion annealing each sample was further annealed at the temperature of the intended diffusion treatment for at least the double duration (in order to achieve equilibrium GB segregation of all spurious impurity elements inherent in

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the material and to minimize grain growth during the subsequent solute tracer diffusion experiment). After each step of the preparation procedure the samples were etched carefully with nitric acid. The average grain size was about 200 µm, which is appropriate for GB diffusion measure-

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ments in the B-type regime. For the C-type regime measurements, the grain size was reduced to about 60 − 80 µm applying mechanical deformation to about 20% before the above described thermal treatment.

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2.2 Radiotracer experiments

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The radiotracer 57 Co (half-life 271.7 d) was available as a HCl solution. To achieve the required specific activity of the tracer material, the solution was highly diluted with double-distilled water. The tracer solution (with the total activity of about 12 kBq) was applied on the polished sample surface and dried. Under a purified Ar atmosphere the samples were sealed into silica tubes and subjected to the diffusion annealing in a temperatue range of 550 − 1150 K for the chosen durations. The temperatures were measured and controlled with a Ni-NiCr thermocou-

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ple to an accuracy of ±1 K. After the diffusion annealing, the samples were reduced by about 2 mm in diameter in order to remove the effects of lateral and surface diffusion. The penetration profiles were determined by parallel mechanical sectioning using a microtome and weighting the sections using a microbalance.

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The isotope 57 Co decays after capturing an electron into an excited state of the isotope 57 Fe and after emitting γ-radiation the isotope 57 Fe turns into a stable state [38]. In order to measure

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the activity of each section a germanium detector was used. Additionally, a Co/Ag co-diffusion experiment has been performed. A 110m Ag tracer solution (half-life 249.8 d) with an activity of approximately a half of that of the used. The

110m Ag

57 Co

tracer was

isotopes decay with emission of mainly 658 and 885 keV γ-quanta [50],

which can easily be distinguished from the

57 Co

decays, γ-peaks at 122 and 136 keV, by an

available germanium detector with a 16K multi-channel analyzer.

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3 Experimental results The way to analyze the GB penetration profiles depends crucially on the kinetic regime in which the measurements are performed. The key parameter is the value of the Le Claire parameter α

sδ α= √ 2 Dv t

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[2] (which is generally unknown for solute diffusion):

(4)

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This parameter relates the effective GB width, s · δ, with the corresponding diffusion length √ in the grain volume, 2 Dv t. According to the present knowledge [15], α > 1 corresponds to Harrison’s C-type regime (lower temperatures and/or shorter diffusion times) and α < 0.1

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corresponds to the B-type regime (higher temperatures and/or longer diffusion times), while the interval 0.1 < α < 1 represents the transition regime BC (for a theoretical estimate of the limits of diffusion regimes see also Ref. [16]). In the case of self-diffusion (s = 1), the GB diffusion regimes can directly be established ’on demand’ by a proper choice of the annealing time t at the given temperature T. However, in the case of solute diffusion the segregation factor is a

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priori unknown and this fact complicates a rigorous analysis of the GB diffusion profiles. At lower temperatures, the conditions of the C-type regime can reliably be satisfied without √ the knowledge of the segregation factor, if the condition α/s = δ/2 Dv t > 1 is satisfied (if s > 1 can safely be assumed, this condition corresponds definitely to the C-type regime) and as

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a result, the GB diffusion coefficient Dgb can directly be measured. With increasing temperature, the bulk diffusivity and the GB diffusivity increase exponentially, but with different rates,

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so that α becomes smaller than 1 at a given temperature or for the given diffusion time. In this case, the C-type type diffusion conditions will not be satisfied any more. An analysis of the penetration profiles as measured in the C-type regime will result in an underestimation of the Dgb values [39]. Subsequent calculations of s are necessery to determine rigorously the limits of both C- and B-type regimes in a self-consistent manner (see below). This approach is used in the present paper. In order to analyze the GB diffusion kinetics and the penetration profiles in the B type regime, the volume diffusion coefficient Dv of Co in Cu must be known. In the present work, the assessment of Neumann and Tölle [40] of the original experimental data from Refs. [41, 42] 6

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was used:

=7.4 × 10

−5

7.36 × 10

217.1 kJ mol−1 · exp − RT

−2

!

312.6 kJ mol−1 · exp − RT

!

(5)

2 −1

m s

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Here R is the gas constant.

+

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DvCo

3.1 C-type diffusion regime

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The radiotracer experiments under intended C-type regime conditions were performed at 5 different temperatures between 550 K and 950 K. As an example, Figure 1a presents the concentration profile measured at 550 K. After an abrupt near-surface decrease of the tracer concentration (which is proportional to the GB density or inversely proportional to the grain size), the profile follows first the Gaussian solution of the diffusion problem (the logarithm of the tracer con-

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centration decreases linearly with the depth squared). However, at a depth of about 60 to 100 µm that corresponds to the grain size there is a strong change of the slope with large irregularities between individual points, red dashed line in Figure 1a. The corresponding activities are about 10−3 Bq/mg which only slightly exceed the background level. Since only few representa-

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tive points for such a branch of the penetration profiles could be measured applying reasonable acquisition times and since the tentatively derived diffusion coefficients are characterized by

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more than 100% uncertainties, we will not focus on these contributions in the present study. The main parts of the penetration profiles measured in the temperature interval of 550 K to 750 K are shown in Figure 1b as a function of the depth squared. After an abrupt near-surface decrease of the tracer concentration, the profiles follow the Gaussian solution of the diffusion problem (solid lines) and the GB diffusion coefficients, Dgb , can be determined as:

Dgb

1 =− 4t



∂ ln c¯ ∂y2

 −1

(6)

Here c¯ and y are the relative specific activity of the layer, which is proportional to the solute

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concentration and the penetration depth, respectively.

∗

t [s] 7.16 × 106 1.21 × 106 6.05 × 105 8.64 × 104 1.73 × 105 1.08 × 104 1.12 × 106 3.60 × 103 1.80 × 103

√

Dv t [m] 1.12 × 10−9 1.77 × 10−8 1.83 × 10−7 6.91 × 10−8 7.58 × 10−7 9.55 × 10−7 9.74 × 10−6 2.04 × 10−6 4.26 × 10−6

Dgb [m2 s−1 ] 1.26 × 10−17 3.99 × 10−16 3.87 × 10−15 4.50 × 10−15 6.37 × 10−15 2.51 × 10−14 – – –

P [m3 s−1 ] – – – – 5.74 × 10−21 1.83 × 10−20 1.54 × 10−20 4.10 × 10−20 9.87 × 10−20

s 2.95 × 103 2.19 × 103 1.76 × 103 1.76 × 103 1.49 × 103 1.31 × 103 1.31 × 103 1.18 × 103 1.08 × 103

α

β

Kinetic regime

658.88 30.88 2.41 6.38 0.49 0.34 0.03 0.14 0.06

– – – – 1140 113 9.38 8.64 1.15

C C C C Transition BC Transition BC B B B

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T [K] 550 650 750∗ 750 850 950 950 1050 1150

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Table 1: Diffusion parameters of the radiotracer experiment (the parameters α and β are defined by Eqs. (4) and (8), respectively). The uncertainty of the Dgb values is typically below 20%, if not explicitly specified.

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The uncertainty of the determined value of Dgb is large, about +270% and -40% that is affected by a relatively low number of the corresponding expimetal points.

The concentration profiles measured in the C-type regime were followed to penetration depths of about 150 µm. The relevant parameters of the diffusion experiments and the determined diffusion coefficients are summarized in Table 1. Table 1 confirms that α > 1 for the investigated temperatures from 550 K to 750 K, according to the subsequent determination of

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the segregation factor s (see below).

3.2 B-type diffusion regime

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The diffusion experiments under intended type B regime conditions were performed at 4 different temperatures in a range of 850 − 1150 K. As mentioned above, the lower limit of the cor-

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responding temperature interval cannot be determined a priori. The penetration profiles which turned out to be unambiguously measured in the B-type regime are shown in Fig. 2. The other 2 profiles turned out to belong to the transition BC regime and they are presented in Fig. 3. A pronounced near-surface part of the penetration profile corresponding to the diffusion experiment at 950 K for 13 days (green triangles) is caused by direct bulk diffusion and the tentatively derived value of the bulk diffusion coefficient agrees well with that predicted by Eq. (5). According to the Le Claire analysis [43] of Suzuoka’s exact solution [44], the GB-related tail of the penetration profiles (in the coordinates of ln c¯ against y1.2 ) has to be linear. The Co diffusion penetration profiles (see Fig. 2) show such a dependence. The pertinent slope, 8

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y [µm]

a)

10

3

0 20

40

ky [µm]

b)

60

80

0

100

150

200

750 K (7 d) 750 K (1 d) 650 K

2

10

10

10

-1

-2

k=2 k=2

-3

k = 2.5

-4

0

10

20

30

y

2

[10

40 -10

50

60

70

0

1

2

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10

0

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10

1

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10

550 K

k=1

relative specific activity [arb. units]

-1

relative specific activity [Bq mg ]

10

2 2

ky

2

m ]

[10

3

-8

4

5

2

m ]

Figure 1: Penetration profile measured at 550 K (a) and the main branches of the concnetration profiles measured in the C-type kinetic regime (b). y is the penetration depth and k is a scaling factor. In (a) the black and red lines indicate two branches of the concentration profiles.

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1.2 , allows the determination of the value of the triple product P = s · δ · D [3], ¯ ∂ ln c/∂y gb

P = 1.146



Dv1.44 t1.52

1/2.96 

∂ ln c¯ − 1.2 ∂y

−5/2.96

(7)

β=

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β,

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This, somewhat cumbersome solution is applied if the value of the second Le Claire parameter,

P √

2Dv Dv t

(8)

is within the interval 102 < β < 104 [3]. In the case of smaller β values, the numerical exponents in Eq. (7) have to be properly modified [3] and in the case of β < 100 the following expression was used,

P = 2.0



Dv t

1/2 

0.70β

0.018

5/3  ∂ ln c¯ −5/3 − 1.2 ∂y

(9)

The diffusion parameters and the determined values of the triple products P are summarized

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y [µm] 0

100

200

950 K (13 d) 1050 K

0

1

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relative specific activity [arb. units]

1150 K

2

y

1.2

[10

3

-5

m

1.2

4

]

Figure 2: Penetration profiles measured under type B diffusion regime conditions. y is the penetration depth.

in Table 1.

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The determined values of the GB diffusion coefficients, Dgb , and of the triple products, P, are presented in Fig. 3 as functions of the inverse temperature. For a convenient comparison, the Dgb values are multiplied by the GB width δ = 0.5 nm. For the two temperatures of 850 K and

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950 K (the shorter annealing time in the latter case), both P and Dgb values were determined (by processing the same penetration profiles according to the B-, Eq. (7), or C-type, Eq. (6), kinetic regimes, respectively) and they are presented in Fig. 3 by open symbols.

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The triple product P seems to follow an Arrhenius dependence within the experimental temperature interval from 850 K to 1150 K (red dashed line) and an Arrhenius dependence may be proposed for the temperature dependence of Dgb , too (blue dashed line). The scatter of the determined P-values is relatively small, suggesting the reliability of the Arrhenius presentation, whereas the scatter of the Dgb values is larger. Figure 3 substantiates that the measured P values are systematically larger than the product δ · Dgb and Co does segregate to Cu grain boundaries, s 6= 1. By re-grouping terms in Eq. (3), one obtains

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T [K] 1200

10

10

600

500

Type B

-20

-21

-22

10

-23

s

P;

D

gb

[m

3

-1

s ]

10

-19

800

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10

1000

Corr

s

10

-25

-26

Type C

8

10

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10

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10

-24

12

14

T

-1

[10

-4

16

18

20

-1

K ]

Figure 3: Arrhenius diagram for GB diffusion of Co in Cu in type B (red circles) and type C kinetics

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(blue squares). The values measured in the transition BC regime are presented by open symbols. Dashed lines represent the original temperature dependencies without a proper analysis of the segregation factor. The corrected temperature dependencies (solid lines) and the corrected P values for the transition BC regime (filled red triangles) are shown. Note, that two values at 750 K (see Table 1) were measured, but the data points are overlapping. The values are distinguished by the error bars. The determination of the segregation factor s is illustrated. For the correction procedure see text.

see Fig. 3.

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s=

P δ · Dgb

(10)

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Having determined the value of the segregation factor, Le Claire’s parameters α and β can be estimated, Table 1. An inspection of Table 1 suggests that the experiments at 1150 K and 950 K (for diffusion time of 13 d) were unambiguously performed under type B conditions. The value of α at 1050 K is only slightly above 0.1 (α = 0.14). The experiments performed at 850 K and 950 K (short annealing) have to be re-analyzed to provide reliable data, since they fall into the transition BC regime, 0.1 < α < 1.

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3.3 Transition BC regime At temperatures of 850 K and 950 K (short annealing) the penetration profiles are significantly curved when plotted as ln c¯ against y2 (see Fig. 4, top penetration profiles). If GB diffusion

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is measured in the transition BC regime, the determined Dgb values will be underestimated [39]. Thus, a downward deviation from the low-temperature Arrhenius dependence would be expected and it may be anticipated from Fig. 3. However, in view of large uncertainties of the GB diffusion coefficients determined at 650 K and especially at 750 K, a common Arrhenius-

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type dependence may initially be suggested (blue dashed line).

The same layer activities plotted against y1.2 (Fig. 4, bottom penetration profiles) demon-

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strate less curved profiles, thus fulfilling the predicted dependencies for the B-type kinetics. y

0

2

[10

-8

1

2

m ] 2

3

850 K

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relative specific activity [arb. units]

950 K (3 h)

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0

1

2

y

1.2

[10

-5

m

3 1.2

]

Figure 4: Penetration profiles determined in the transition regime and plotted as measured in the B-type regime (bottom x-axis) or in the C-type regime (top x-axis). y is the penetration depth.

In the present paper we will follow a (slightly modified) approach of Szabo and co-workers [39] to process these penetration profiles as it was outlined in Ref. [15]. The main steps of the correction procedure are as follows. • The penetration profiles are replotted as a function of the reduced depth αw4/5 , where w 12

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is determined by y w=√ P



4Dv t

1/4

y =p sδDgb



4Dv t

1/4

(11)

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In order to have a first approximation of w, the profiles are analyzed as measured in the Bregime, because α < 0.5 [15]. The values of P will be determined from the corresponding slope of the profile and the reduced depths w and accordingly the product αw4/5 will be

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estimated [15].

• The correction factor for the triple-product P can be determined applying the graphical dependencies plotted in [39] or using pure numerical fitting [15], which reproduce the

est of 0.01 ≤ αw4/5 ≤ 30.

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correction factors calculated in [39] to an accuracy better than 1% in the interval of inter-

• Having determined the correction factor for the apparent triple product G1 the corresponding correction factor for the apparent GB diffusion coefficient G2 , which is defined as

exp

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G2 =

Dgb

theor Dgb

(12)

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can be determined using the relation

G2 = (1 − G1 )(1 + G2 /4)

(13)

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• Having corrected the triple products P and the GB diffusion coefficients Dgb , the segregation factor can be re-evaluated by using Eq. (10).

• These steps were repeated several times until convergence was reached, first for Pexp (950 K) and then for Pexp (850 K).

In Table 2 the re-evaluated values of the triple products P, the segregation factor s and the parameters α, β for the transition regime are summarized. The newly determined values of α and β for all others temperatures are listed, too.

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Table 2: Diffusion parameters of the radiotracer experiment after introducing correction factors (the parameters α and β are defined by Eqs. (4) and (8), respectively). P [m3 s−1 ] – – – – 6.52 × 10−21 2.13 × 10−20 1.54 × 10−20 4.10 × 10−20 9.87 × 10−20

The uncertainty of Dgb at 750 K after

s

α

β

5.92 × 103 1324.91 – 1.84 × 103 25.98 – 7.83 × 102 1.07 – 7.83 × 102 2.83 – 4.07 × 102 0.13 1137 2.43 × 102 0.06 113 2.43 × 102 0.01 9 1.60 × 102 0.02 8 1.13 × 102 0.01 1 5 6.05 × 10 seconds is about +270% and

-40%.

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∗

Dgb [m2 s−1 ] 1.26 × 10−17 3.99 × 10−16 3.87 × 10−15 4.50 × 10−15 – – – – –

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T [K] 550 650 750∗ 750 850 950 950 1050 1150

The iterative analysis proves self-consistently that the diffusion experiment at 950 K (short

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diffusion time of 3 h) was performed under type B conditions. The value of the parameter α is smaller than 0.1. The experiment at 850 K was also performed under type B conditions, however the α value is formally 0.13 and no further convergence can be obtained if the correction procedure is continued. Since the numerical corrections remain small and since this particular α value is only slightly larger than unity, we stopped the analysis here.

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The finally determined triple product P and GB diffusion coefficient Dgb of Co in Cu are described by the following Arrhenius equations, solid lines in Fig. 3, 

(66.2 ± 6.0) kJ mol−1 × 10−17 · exp − RT

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8.7 PCorr = 7.4+ −4.0





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Corr 6.2 Dgb = 4.9+ −2.7



!

m3 s − 1

(14)

!

m2 s − 1

(15)

(100.9 ± 4.1) kJ mol−1 −8 × 10 · exp − RT

3.4 Determination of the segregation factor s After measuring and re-evaluating Dgb and P = s · δ · Dgb , the segregation factor s can finally be determined, see Eq. (10). In a temperature range from 550 K to 750 K the grain boundary segregation of Co in Cu increases with decreasing temperature, following a linear Arrhenius dependence, 

4.3 sCorr = 3.0+ −2.3



(−34.7 ± 7.3) kJ mol−1 · exp − RT 14

!

(16)

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4 Discussion Figure 5 compares the determined values of Co GB diffusion coefficients, Dgb , and triple products, P, with the existing data for other ferromagnetic solutes, i.e. Ni [25] and Fe [28]. As a

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reference, the Cu self-diffusion data [29] are shown, too. Since the same copper material was used in all these investigations, a reliable comparison of the diffusivities is possible.

Co of Co GB diffusion in Figure 5 substantiates that the triple product PCo = sCo · δ · Dgb

Cu for Cu self-diffusion. Such Cu is lower than the value of the double product PCu = δ · Dgb

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behaviour was already observed for Ni [25]. In general, with increasing group number of the solute element an enhancement of Psolute is expected, which is very likely to result from the

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decrease of solid solubility in Cu with increasing group number. Thus, an enhancement of segregation and thus larger values of the triple-product P are expected [19, 20, 22, 23, 24, 27, 28]. Additionally, the triple product of Co has a similar magnitude as the triple product of Ni and the activation enthalpy is only slightly smaller than the activation enthalpy of Ni GB diffusion in Cu. Such similarity to Ni is unexpected for the reason that Co (like Fe) is almost immiscible with Cu in contrast to Ni. Since a strong segregation of Co is expected, the GB

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diffusivity of Co in Cu has to be relatively low.

The directly measured GB diffusivity of Co in Cu follows a trend generally observed for solutes in polycrystalline Cu, namely that the GB diffusivity of a solute is about the same, or

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less by a few orders of magnitude than the GB self-diffusivity under type C conditions. Existing results in the low-temperature range of the C-type experiments for different solutes in pure Cu and Ag [1] and also for Ag in an Fe-Ni alloy [45] support this assessment.

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The rate of Co GB diffusion in Cu is lower than that of GB self-diffusion by about three orders of magnitude in the low-temperature range. Another remarkable feature is that the activation enthalpy of Co GB diffusion is only slightly larger than that of Cu GB self-diffusion indicating large entropic effects and/or repulsive vacancy – Co atom interactions in Cu grain boundaries. The results presented in Fig. 5 reveal a Ni-like behaviour of Co GB diffusion in Cu under both, type B conditions and type C conditions. Ni and Co are similar with respect to atom radius and surface energy, but are distinguished by their solubility in Cu. How could this Ni-

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T [K] 10

10

10

10

10

Type C

Cu

-12

Fe

10

-13

10

Ni -14

10

-15

10

Co -16

10

-13

-14

-15

-16

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10

10

10

10

10

10

D

-19

s

Co

-20

-21

10 Co

10

10

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Type

B

Ni

10

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-24

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8

-18

10

-19

-20

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10

Cu

-21

-22

-23

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10

-17

2

10

[m

-17

3

10

P

[m

-12

-1

Fe

-11

gb

-1

s ]

10

600

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10

800

s ]

1200 1000

-11

12

14

T

-1

16

[10

-4

18

20

-24

22

-1

K ]

Figure 5: Comparison of the determined triple-products P (left ordinate) of Co GB diffusion (green

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triangles, present work) and the Co GB diffusion coefficients, right ordinate, Dgb , (green triangles with black markings, present work) with the literature data on Ni (straight blue lines) [25] and Fe (straight red lines, red squares for high-temperature measurements) [28] in Cu. The Cu self-diffusion data [29] are shown by black lines. The way to determine the product sCo · δCo for Co in Cu is exemplified. Co and like behaviour and the large difference between the corresponding GB diffusivities, Dgb Cu , be understood? In case of Ni, it was suggested [25] that Ni atoms do not segregate to Dgb

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a GB core, but to deeper atomic planes, which are adjacent to the GB core (the case of ’subinterface’ segregationin polycrystalline copper). Such behavior was observed for the case of Ni

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GB diffusion in pure Ag [46] and in Ag-Ni alloys [47]. Fe ’sub-interface’ segregation in the same high-purity Cu was also assumed, with the distinction of a higher Fe segregation level [28]. The present results of Co GB diffusion indicate a ’sub-interface’ type of segregation, too. The temperature dependencies of the segregation factor s, determined from the original data (Table 1, dashed green line), and sCorr , which correspond to the corrected ones (Table 2, green line), are presented in Fig. 6 and compared with the results obtained on other solutes (Ag [20], Bi [24], Ni [25] and Fe [27, 28]). If we tentatively use the direct experimental data, Table 1, the segregation factor of Co would be in a range of 103 to 3 × 103 and the corresponding segrega16

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tion enthalpy, HsCo , would be almost zero, HsCo = −8.8 kJ mol−1. However, the requirement of self-consistency forces a re-analysis of the data measured in the transition BC regime and a strong temperature dependence of Co segregation is found. The determined segregation enthalpy, HsCorr = −34.7 kJ mol−1, is almost equal to the segregation enthalpy of Ag in Cu [20].

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The absolute values of the Co segregation factor are by more than one order of magnitude larger than those of the Ag segregation factor in the same high-purity polycrystalline Cu. Both, Co [35] and Ag [48] are immiscible with Cu, whereas the atomic radius of Ag is significantly larger than that of Cu and Co. The affinity to segregation should be stronger for Ag due to the elastic

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effects, which provide a dominant contribution according to McLean [49], in clear contradiction with the present experimental results. Such a contradiction between Ni segregation and Ag

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segregation in the Ag–Ni system has already been discussed in Ref. [25]. T [K]

1000 4

800

10

600

Fe

Co

Bi

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Segregation factor,

s

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10

Ag

2

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10

Ni

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1

10

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12

14

16 -1

T

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-4

18

20

22

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Figure 6: Temperature dependence of the segregation factor s for different solutes in high-purity polycrystalline Cu: Co (present work, dashed and straight green line, see text), Ag [20] (orange line), Bi [24] (purple line), Ni [25] (blue line) and Fe [28] (red line).

The larger segregation factor of Co with respect to that of Ag supports the assumption that Co atoms segregate next to the GB core, i.e. sub-interface segregation occurs. Diffusion experiments with both Co and Ag solutes have to clarify the ’sub-interface’-type of Co segregation in high-purity polycrystalline Cu. 17

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4.1 A 57Co/110m Ag co-diffusion experiment A Co/Ag co-diffusion experiment was prepared in a similar way as described in the experimental part.

100

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y [µm] 0

200 57

Co

110m

Ag

-1

relative specific activity [Bq mg ]

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2

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-2

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[10

-5

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m

1.2

4

]

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y

Figure 7: Penetration profiles of Co (blue filled circles) and Ag (red filled squares) under type B regime conditions. y is the penetration depth.

The diffusion experiment was performed at a temperature of 950 K under type B conditions

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and the diffusion time of Co was set to tCo = 1123200 s like in the already performed Co

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diffusion experiment in this work. of Ag was chosen to approach the same q The diffusion time p Ag bulk diffusion lengths for Ag, Dv tAg , and Co, DvCo tCo , in Cu. Note that similar relations

were chosen, e.g., in the experiments by Bernardini with co-workers on Zn and Fe GB diffusion in Cu [51]. Since at 950 K the bulk diffusion length of Co in Cu is 9.74 × 10−6 m for the

diffusion time of 1123200 s, the diffusion time for Ag in Cu was set at tAg = 75989 s. The data

of Barreau et al. [52] on bulk diffusion of Ag in Cu were used, Ag Dv

= 0.61 × 10

−4

194.4 kJ mol−1 · exp − RT

!

m2 s − 1

(17)

Correspondingly, the Co tracer was first applied to a Cu specimen and it was annealed at 18

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950 K for 1047211 s. Then Ag tracer was additionally applied and the specimen with both radioisotopes was further annealed at 950 K for 75989 s. T [K] 10

1200

-16

1100

1000

900

110m

57

10

Co (present work)

-17

co-diffusion experiment: 110m

Ag

57

10

10

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gb

P = s D

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-18

[m

3

-1

s ]

Ag

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-21

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Co

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Ag (Divinski et al. [20])

Cu self-diffusion (Surholt et al. [27])

9

10

T

-1

[10

-4

11

12

-1

K ]

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Figure 8: The determined triple-products P of Co (blue unfilled circle) and Ag (red filled square) compared with the Arrhenius dependence of the triple-product of Co (straight blue line and blue filled circles - present work) and Ag (straight red line [20]). The straight black line represents the Cu GB self-diffusion under type B regime conditions [29]. Figure 7 shows the measured penetration profiles for Co (blue filled circles) and Ag (red

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filled squares) in the same polycrystalline Cu sample. It is remarkable that Co atoms penetrate deeper than Ag atoms. In the given coordinates, the GB diffusion-related tails of each

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penetration profile show a linear dependence in accordance with the Le Claire analysis [43] of Suzuoka’s exact solution for B-type kinetcs [44]. The pertinent slopes differ by about 10 % and the triple-products were determined by Eq. (9), because the corresponding parameter β turned out to be small. As a result, the values of PCo (950 K) = 1.38 × 10−20 m3 s−1 and PAg (950 K) = 1.75 × 10−19 m3 s−1 were determined. These values of the triple products

are shown in Fig. 8 in comparison to the data on Co (present work, Eq. (14)) and Ag ([20], Eq. (18)). In Ref. [20], the following temperature dependence of Ag GB diffusion in pure Cu was established,

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0.6 PAg = 1.4+ −0.4



(69.1 ± 2.5) kJ mol−1 × 10−15 · exp − RT

!

m3 s − 1

(18)

Figure 8 reveals that the estimated values of PCo (unfilled blue circle) and PAg (red filled

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square) are slightly below the corresponding Arrhenius plots; though considering experimental uncertainties, the agreement between the results of the co-diffusion experiment with the independently measured data for single solutes is acceptable. In fact, the values are similar within an uncertainty range of 20% that is typical for the diffusion measurements.

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These results verify unambiguously that the diffused Co atoms have only negligible influence on the diffusion process of the Ag atoms in the same GBs. Thus, Co atoms do not segregate

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in the GB core, but reside at sites belonging to ’sub-interfaces’. This type of segregation does not influence diffusion of Ag atoms which are believed [53] to segregate at the GB core.

5 Summary

In this paper GB diffusion of Co in high-purity polycrystalline Cu is measured in an extended

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temperature interval. At low temperatures (the C-type kinetic regime), Co atoms diffuse relatively slow (by three orders of magnitude slower than the Cu self-diffusion rate) along Cu GBs with an activation enthalpy of about 101 kJ/mol, which is larger than the activation enthalpy

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for Cu self-diffusion (72 kJ/mol [29]). Co is found to segregate strongly at Cu GBs with the segregation enthalpy of −34.7 kJ/mol and the equilibrium segregation factor increases from

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approximately 7 × 102 to 6 × 103 while the temperature decreases from 750 K to 550 K.

A co-diffusion experiment with Co and Ag atoms diffusing into the same Cu GBs proves the sub-interface type of segregation of Co in Cu GBs.

Acknowledgement Financial support from the Deutsche Forschungsgemeinschaft (DFG) within the priority program SPP-1713 chemomechanics (research project DI 1419/7-1) is gratefully acknowledged.

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