# Simultaneous Measurement of Isotope-Free Tracer Diffusion Coefficients and Interdiffusion Coefficients in the Cu-Ni System

## Abstract

A new formalism recently developed by Belova et al., based on linear response theory combined with the Boltzmann–Matano method, allows determination of tracer and interdiffusion coefficients simultaneously from a single, isotope-free, traditional diffusion couple experiment. An experimental methodology with an analytical approach based on the new formalism has been carried out using the model Cu-Ni system to effectively determine tracer diffusion coefficients from an isotope-free diffusion couple experiment. Cu thin films were deposited in between several binary diffusion couples with varying terminal alloy compositions (Cu-25Ni, Cu-50Ni, Cu-75Ni, Ni). Diffusion couples were annealed at 800, 900 and 1000 °C, and the superimposed concentration profiles of thin film and interdiffusion were analyzed for the simultaneous determination of tracer and interdiffusion coefficients. Processed concentration profiles obtained from the diffusion experiments were also fitted with simple Gaussian distribution function. Results were compared to existing literature data obtained independently by radiotracer experiments, and an excellent agreement has been observed.

## Keywords

experimental techniques interdiffusion tracer diffusivity## 1 Introduction

Diffusion coefficients are important material property parameters that help the understanding of phase transformations and microstructural development. For metallic alloys, investigation of diffusion typically involves measurement of interdiffusion coefficients, intrinsic diffusion coefficients and/or tracer diffusion coefficients. Both interdiffusion and intrinsic diffusion coefficients are ‘thermo-kinetic’ parameters since they include both thermodynamic and kinetic terms based on the Darken model. Tracer diffusion experiments provide the most fundamental information about the number of jumps (per unit time) an atom makes between the lattice sites.[1] The tracer diffusion coefficient is a kinetic parameter that is typically employed for constructing mobility diffusion databases, and can be coupled with thermodynamic information to constitute interdiffusion and intrinsic diffusion coefficients.

Traditional techniques[2,3] for experimentally determining the tracer diffusion coefficient in alloys can be challenging and costly due to the use of isotopes and the number of experiments required to assess composition and temperature dependence. Therefore, an efficient way of obtaining tracer diffusion coefficients is highly sought after. Furthermore, simultaneous measurement of tracer and interdiffusion coefficients would provide an efficient and consistent way to distinguish the kinetic (e.g., mobility) and thermodynamic (e.g., thermodynamic factor, Φ) influences on the overall diffusion process in metallic alloys, for example, through the Darken model.

A new formalism[4,5] for simultaneous measurement of isotope tracers and interdiffusion coefficients in multicomponent alloys has been recently reported. This formalism utilized Onsager’s phenomenological relationships[6] combined with the Boltzmann–Matano transformation. In this study, we implement an analytical method based on this new formalism to simultaneously determine interdiffusion coefficients, \(\tilde{D}\) and tracer (\(D_{\text{Cu}}^{*}\)) diffusion coefficients in the Cu-Ni system at 800, 900 and 1000 °C. The Cu-Ni system was chosen for this investigation to demonstrate the validity of the new analytical framework, because there are ample data on both tracer diffusion and interdiffusion in the literature.

## 2 Analytical Framework

*A*

_{x}

*B*

_{(1−x)}−

*A*

_{y}

*B*

_{(1−y)}. Within the couple, consider that atoms of the same type (

*A*) have fraction of them that is situated at a given (known) location in the couple and has a different relative composition from that at the remaining locations. In other words, the

*AB*couple constructed in such a way that there is a fraction of

*A*atoms that can be separated from the rest of the

*A*atoms based on its location. Therefore, it is possible to distinguish them and

*A*atoms can be split into two groups: atoms

*A*1 and

*A*2. Then, due to the difference in their specific locations and compositions, the respective thermodynamic forces (\(X_{A1}\) and \(X_{A2}\)) of these atoms will not be equal. Furthermore, the flux of

*A*atoms, \(J_{A}\), can be split into two fluxes, \(J_{A1}\) and \(J_{A2}\) accordingly. Given this information, the Onsager expressions for the diffusion fluxes of the atomic components

*A*and

*B*in the lattice coordinate system can be written as:

*L*

_{ij}’s are the Onsager phenomenological transport coefficients, the

*X*

_{j}’s are the forces acting on atoms, \(J_{A} = J_{A1} + J_{A2}\) and \(X_{A1} \ne X_{A2}\).

*A*component, and for simplicity only a unidirectional flow in the

*x*direction is considered. The two types of

*A*atoms constitute the total composition of

*A*atoms, \(c_{A}\):

*N*is the number of lattice sites per unit volume, \(\tilde{J}_{i}\) is the flux of type

*i*atoms in the laboratory coordinate system and \(J_{i}\) is the flux of type

*i*atoms in the lattice coordinate system. Then

In Eq 7, there are two terms still in the lattice reference frame, which need to be further replaced with experimentally measurable terms.

*A*atoms, \(F_{B}\) is a correlation function between the movements of two different

*B*atoms, and \(F_{AB}\) is a cross-correlation function between the movements of

*A*and

*B*atoms.

*A*that includes the partial concentration profile of

*A*1. Now, assuming the diffusion couple was annealed for a time

*t*, and

*N*(the number of lattice sites per unit volume) is not changing with composition, the diffusion equations for components

*A*,

*A*1 and

*B*can be written as:

*f*, \(\tilde{D}\) and \(D_{A}^{ *}\) are approximated as functions of \(\lambda\) only. After integrating Eq 19 with respect to the variable \(\lambda\) we have that:

*f*yields:

*t*, will always be a constant, Eq 22 can be rewritten as:

*a*is also a constant. The role of this constant in the equation is to control the sign of the numerator and denominator. Thus, determining the value of

*a*should be done in a way that the resulting tracer coefficient will be positive. Finally, the most convenient form of the equation can be written as:

## 3 Experimental Procedure

Stacking sequence, anneal temperatures and anneal times of diffusion couples

Couple | Stacking sequence | Temp., °C | Time, s | |||
---|---|---|---|---|---|---|

Metal 1 | Metal 2 | Thin film | Metal 3 | |||

1 | Cu50Ni | Cu25Ni | Cu | Cu50Ni | 1000 | 900 |

2 | Cu50Ni | Cu25Ni | Cu | Cu50Ni | 900 | 900 |

3 | Cu50Ni | Cu25Ni | Cu | Cu50Ni | 800 | 5400 |

4 | Cu75Ni | Cu50Ni | Cu | Cu75Ni | 1000 | 7200 |

5 | Cu75Ni | Cu50Ni | Cu | Cu75Ni | 900 | 18,000 |

6 | Cu75Ni | Cu50Ni | Cu | Cu75Ni | 800 | 75,600 |

7 | Cu75Ni | Ni | Cu | Cu75Ni | 1000 | 3600 |

8 | Cu75Ni | Ni | Cu | Cu75Ni | 900 | 3600 |

9 | Cu75Ni | Ni | Cu | Cu75Ni | 800 | 10,800 |

For diffusion couple assembly, alloy surfaces were ground and metallographically polished down to a 1 μm surface finish. Samples were then ultrasonically cleaned with high purity ethanol. On selected alloys, electron beam physical vapor deposition (EB-PVD) technique was employed for Cu deposition. Cu deposition was carried out with the main chamber pressure of 1.2 × 10^{−7} Torr, power of 10 kV, current of 320 mA, and a deposition rate of 1.0 ~ 1.4 Å/s. Deposition rate was monitored by using an oscillating crystal. Based on the target thickness of 3 µm, the Cu deposition period lasted 80 minutes. The prepared alloys were assembled into diffusion couples as listed in Table 1. They were held together by a stainless steel jig with alumina (Al_{2}O_{3}) spacers to avoid any reaction between the stainless steel and the diffusion couple. The assembled diffusion couples were encapsulated in quartz tubes using an oxy-propylene torch. Prior to sealing, the capsules were repeatedly evacuated and flushed with Ar and H_{2}. Quartz tubes were sealed when a vacuum of 8 × 10^{−6} Torr or better was achieved.

^{TM}Ultra-55 field emission scanning electron microscopy (FE-SEM) equipped with x-ray energy dispersive spectroscopy (XEDS). Standardless quantification via XEDS was employed to obtain concentration profiles across both diffusion zones: one with interdiffusion only and the other with the imposed thin film (e.g., tracer and interdiffusion). Concentration profiles extracted from XEDS data were iteratively fitted using OriginPro™ 8.5 software using the expression:[11]

*R*

^{2}values approached 1.

## 4 Interdiffusion

Trønsdal and Sørum[14] quantitatively determined the temperature and concentration dependence of interdiffusion coefficients at medium concentrations for the temperature range 700-1000 °C. The interdiffusion coefficients reported by Trønsdal and Sørum[14] at 900 and 800 °C were significantly higher than the interdiffusion coefficients determined by this study as presented in Fig. 2(b) and (c). Hayashi et al.[15] also examined the Cu-Ni interdiffusion for the temperature range 765 and 906 °C. While the composition-dependence was similar, the magnitude of the interdiffusion coefficients was lower than those determined in this study. Impurity diffusion coefficients reported by Askil,[16] also plotted in Fig. 2(b) demonstrate that the magnitude and the trend in composition-dependence are more consistent with interdiffusion coefficients determined in this study.

Zhao et al.[17] investigated interdiffusion coefficient of Cu and Ni in the temperature range of 650 and 850 °C as presented in Fig. 2(c). Interdiffusion coefficients at 800 °C for selected compositions were also reported by Trønsdal and Sørum[14] as represented in Fig. 2(c). Results obtained from this study demonstrate that the composition-dependence is consistent among all studies but the magnitude determined by this study falls between those determined by Zhao et al.[17] and Trønsdal and Sørum.[14]

## 5 Tracer Diffusion Coefficient Determination from Thin Film Diffusion

As was explained above, the diffusion couples assembly consisted of a part with just interdiffusion and the other part with a thin film sandwiched in-between as shown in Fig. 1. Annealing of the couple created a local ‘spike’ in the composition of one of the alloy components as schematically illustrated in Fig. 1 (i.e., pure Cu thin film), which include contributions from both, interdiffusion and thin film diffusion. The analysis of the thin film diffusion profile (spike) gave the total composition of Cu and then the analysis of the standard interdiffusion profile allow for separation of the fraction (i.e., c_{Cu2} fraction) of total Cu atoms (Eq 3 with A = Cu) and corresponding atomic fluxes as expressed by Eq 11a and 11b. This approach of distinguishing thin film and interdiffusion profiles, i.e., Cu and Cu_{2} profiles, is based on the assumption that the thickness of pure Cu layer is negligibly small.

_{Cu1}profile, the interdiffusion contribution is subtracted from the total c

_{Cu}profile. This was achieved by taking the mirror image of the interdiffusion profile and subtracting \(c_{{{\text{Cu}}_{2} }}\) profile from total

*c*

_{Cu}profile as demonstrated in Fig. 3. Once the concentration profiles were obtained, Eq 27 was used to determine the tracer diffusion coefficient.

_{Cu1}profile and plotted against the corresponding spike composition. Table 2 summarizes the tracer diffusion coefficient, \(D_{\text{Cu}}^{*}\) obtained by application of the Belova et al. formalism as a function of Cu composition (at.%) and the tracer diffusion coefficient calculated using the Gaussian distribution function. When the existing literature tracer diffusion data obtained by radioactive isotopes studies[2] compared with the data obtained by this study utilizing Gaussian distribution function, good agreement was found, as presented in Fig. 5.

Tracer diffusion coefficient derived from the Belova et al. formalism and Gaussian distribution function

Cu, at.% | \(D_{\text{Cu}}^{*}\) Belova et al. formalism, cm | \(D_{\text{Cu}}^{*}\) Gaussian distribution function, cm |
---|---|---|

47 | 8.6 × 10 | 1.3 × 10 |

48 | 8.6 × 10 | |

49 | 8.6 × 10 | |

50 | 8.6 × 10 |

Summary of tracer diffusion coefficients obtained from each couple annealed

Couple | Cu composition, at.% | Temp., °C | Time, s | \(D_{\text{Cu}}^{*}\), cm | Average \(D_{\text{Cu}}^{*}\), cm | Standard deviation |
---|---|---|---|---|---|---|

1 | 73 | 1000 | 900 | 5.4 × 10 | … | … |

2-A | 67 | 900 | 900 | 5.7 × 10 | 4.3 × 10 | 7 × 10 |

2-B | 67 | 900 | 900 | 4.6 × 10 | ||

3 | 67 | 800 | 5400 | 8.0 × 10 | … | … |

4-A | 47 | 1000 | 7200 | 1.8 × 10 | 1.4 × 10 | 6 × 10 |

4-B | 47 | 1000 | 7200 | 9.0 × 10 | ||

5 | 48 | 900 | 18,000 | 1.6 × 10 | … | … |

6 | 47 | 800 | 75,600 | 2.5 × 10 | … | … |

7-A | 21 | 1000 | 3600 | 4.1 × 10 | 4.5 × 10 | 4 × 10 |

7-B | 21 | 1000 | 3600 | 4.9 × 10 | ||

8 | 22 | 900 | 3600 | 4.2 × 10 | … | … |

9-A | 22 | 800 | 10,800 | 9.9 × 10 | 8.7 × 10 | 2 × 10 |

9-B | 22 | 800 | 10,800 | 7.5 × 10 |

## 6 Conclusions

The tracer diffusion coefficient, \(D_{\text{Cu}}^{*}\) and interdiffusion coefficients have been simultaneously determined using the experimental methodology based on the new formalism and Gaussian distribution function fitting. This was performed using standard diffusion couple experiments without any radioactive or stable isotopes. Results produced were in excellent agreement with previously reported values determined independently by radiotracer and interdiffusion experiments, when Gaussian solution of thin-film concentration profile was utilized. This investigation demonstrated that the new formalism can be successfully applied to binary systems without using radiotracers and self/tracer diffusivities can be obtained from traditional diffusion couple experiments.

## Notes

### Acknowledgments

IVB and GEM would like to acknowledge support of the Australian Research Council through its Discovery Project Grants Scheme (DP170101812).

## References

- 1.H. Mehrer,
*Diffusion in Solids: Fundamentals, Methods, Materials, Diffusion-Controlled Processes*, Springer, Berlin, 2007, p 212-214CrossRefGoogle Scholar - 2.A. Benninghoven, F.G. Rudenauer, and H.W. Werner,
*Secondary Ion Mass Spectrometry: Basic Concepts, Instrumental Aspects, Applications and Trends*, John Wiley & Sons, New York, 1987, p 257–258Google Scholar - 3.A. Paul, T. Laurila, V. Vuorinen, and S.V. Divinski,
*Thermodynamics, Diffusion & The Kirkendall Effect in Solids*, Springer, Berlin, 2014, p 167Google Scholar - 4.I.V. Belova, Y.H. Sohn, and G.E. Murch, Measurement of Tracer Diffusion Coefficients in an Interdiffusion Context for Multicomponent Alloys,
*Philos. Mag. Lett.*, 2015,**95**(8), p 416-424ADSCrossRefGoogle Scholar - 5.I. Belova, N.S. Kulkarni, Y.H. Sohn, and G. Murch, Simultaneous Measurement of Tracer and Interdiffusion Coefficients: An Isotopic Phenomenological Diffusion Formalism for the Binary Alloy,
*Philos. Mag.*, 2013,**93**(26), p 3515-3526ADSCrossRefGoogle Scholar - 6.L. Onsager, Reciprocal Relations in Irreversible Processes, I,
*Phys. Rev.*, 1931,**37**(4), p 405ADSCrossRefGoogle Scholar - 7.J.G. Kirkwood, R.L. Baldwin, P.J. Dunlop, L.J. Gosting, and G. Kegeles, Flow Equations and Frames of Reference for Isothermal Diffusion in Liquids,
*J. Chem. Phys.*, 1960,**33**(5), p 1505ADSMathSciNetCrossRefGoogle Scholar - 8.L.S. Darken, Diffusion, Mobility and Their Interrelation Through Free Energy in Binary Metallic Systems,
*Trans. AIME*, 1948,**175**, p 184-194Google Scholar - 9.J.R. Manning, Correlation Factors for Diffusion in Nondilute Alloys,
*Phys. Rev. B*, 1971,**4**(4), p 1111-1121ADSCrossRefGoogle Scholar - 10.A.R. Allnatt and A.B. Lidiard,
*Atomic Transport in Solids*, Cambridge University Press, Cambridge, 1993, p 161-180Google Scholar - 11.D. Liu, L. Zhang, Y. Du, H. Xu, and Z. Jin, Ternary Diffusion in Cu-Rich fcc Cu-Al-Si Alloys at 1073 K,
*J. Alloys. Compd.*, 2013,**566**, p 156-163CrossRefGoogle Scholar - 12.T. Heumann and K. Grundhoff, Diffusion and Kirkendall Effect in the Cu-Ni System,
*Z. Metallkunde*, 1972,**63**(4), p 173-180, in GermanGoogle Scholar - 13.Y. Iijima, K. Hirano, and M. Kikuchi, Determination of Intrinsic Diffusion Coefficients in a Wide Concentration Range of a Cu-Ni Couple by the Multiple Markers Method,
*Trans. Jpn. Inst. Metals*, 1982,**23**(1), p 19-23CrossRefGoogle Scholar - 14.G.O. Trønsdal and H. Sørum, Interdiffusion in Cu-Ni, Co-Ni, and Co-Cu,
*Phys. Status Solidi (B)*, 1964,**4**(3), p 493-498ADSCrossRefGoogle Scholar - 15.E. Hayashi, Y. Kurokawa, and Y. Fukai, Hydrogen-Induced Enhancement of Interdiffusion in Cu-Ni Diffusion Couples,
*Phys. Rev. Lett.*, 1998,**80**(25), p 5588-5590ADSCrossRefGoogle Scholar - 16.J. Askill,
*Tracer Diffusion Data for Metals, Alloys and Simple Oxides*, Springer, Berlin, 1970, p 31-80CrossRefGoogle Scholar - 17.J. Zhao, J.E. Garay, U. Anselmi-Tamburini, and Z.A. Munir, Directional Electromigration Enhanced Interdiffususion in the Cu-Ni System,
*J. Appl. Phys.*, 2007,**102**(11), p 114902ADSCrossRefGoogle Scholar - 18.K. Monma, H. Suto, and H. Oikawa, Diffusion of Ni 63 and cu 64 in Nickel-Copper Alloys (On the Relation Between High Temperature Creep and Diffusion in Nickel Base Solid Solutions),
*J. Jpn. Inst. Metals*, 1964,**28**(188), p 192-196, in JapaneseCrossRefGoogle Scholar - 19.E.A. Smirnov, L.I. Ivanov, and E.A. Abranyan, Experimental study of self-diffusion in copper,
*Izv. Akad. Nauk SSSR Metal.*, 1967,**168**Google Scholar - 20.M.S. Anand, S.P. Murarka, and R.P. Agarwala, Diffusion of Copper in Nickel and Aluminum,
*J. Appl. Phys.*, 1965,**36**(12), p 3860-3862ADSCrossRefGoogle Scholar