- Email: [email protected]
The interdiffusion in copper-nickel alloys
Journal of Alloys and Compounds 687 (2016) 104e108
Contents lists available at ScienceDirect
Journal of Alloys and Compounds journal homepage: http://www.elsevier.com/locate/jalcom
The interdiffusion in copper-nickel alloys ski b Bartek Wierzba a, *, Wojciech Skibin a b
co w Warszawy 12, 35-959 Rzeszo w, Poland Rzeszow University of Technology, Faculty of Mechanical Engineering and Aeronautics, al. Powstan w, Poland AGH University of Science and Technology, al. Mickiewicza 30, 30-059 Krako
a r t i c l e i n f o
a b s t r a c t
Article history: Received 21 March 2016 Received in revised form 2 June 2016 Accepted 9 June 2016 Available online 11 June 2016
In this paper the interdiffusion process and Frenkel effect (Kirkendall porosity) are studied in Ni-Cu and NiAl-Cu systems at 1273 K. The generalized Boltzmann-Matano method is used for calculation of the intrinsic diffusion coefficients as a polynomial function in the whole composition range. The calculated interdiffusion coefficient in Cu-Ni diffusion couple decrease with increasing Ni content. The total decrease of interdiffusion coefficient exceeds two orders of magnitude. The equation for the void growth rate is obtained. The experimental results of diffusion and voids formation are analyzed. The voids radii is estimated and compared with experiments. © 2016 Elsevier B.V. All rights reserved.
Keywords: Voids Frenkel effect Diffusion Boltzmann-Matano
1. Introduction High oxidation resistance, mechanical properties, low density and high melting temperature of the NiAl intermetallic compound have attracted scientific attention to this material [1]. The B2 type offers potential as a highservice-temperature material, for example, in aero gas turbine engines [2,3]. However, many application of NiAl would require access to suitable joining technologies [3]. The pure Cu can be used for interlayer during diffusional bonding process [2]. The copper shows extensive solid solubility in both NiAl and nickel [4]. The essential is the knowledge about high temperature diffusion for better understanding of the mechanical properties. The scientists nowadays debate about the diffusion mechanism in B2 compounds. From Rabkin works, we know, that the usual mechanism involving jump to the nearest neighbor vacancy is improbable in B2 compounds as that migration may locally destroy the chemical order [1]. Recent data of Herzig and co-workers on Ni tracer diffusion in NiAl support the triple defect mechanism of Ni diffusion [5]. The main kinetic parameter describing the diffusion in multicomponent system is the intrinsic diffusion coefficient. In binary systems the Boltzmann-Matano (B-M) analysis [6] can be used for determination of this parameter. However in ternary and higher
* Corresponding author. E-mail address: [email protected] (B. Wierzba). http://dx.doi.org/10.1016/j.jallcom.2016.06.085 0925-8388/© 2016 Elsevier B.V. All rights reserved.
order systems the B-M method is invalid. In our previous paper we have shown the generalization of B-M analysis [7,8]. The key idea is the knowledge about the position of Kirkendall plane. This additional data allows for determination of the intrinsic diffusion coefficient at the position of the markers. Then, assuming, that the logarithm of diffusivities are determined by polynomial function (at least of order 2) the diffusion coefficients for each component in multicomponent diffusion couple can be approximated [7]. During the interdiffusion process, due to the differences in diffusion coefficients numerous phenomena occur: the lattice shift, the stress generation and relaxation, the nonequilibrium distribution of vacancies and voiding [9e13]. Voiding is a result of relaxation of pure material or alloy supersaturated with vacancies. The relaxation of vacancy subsystem can proceed by joining of vacancies into voids [14,15]. In this work the interdiffusion in Cu-Ni and Cu-NiAl at the temperature 1273 K will be investigated. Thus, the objective of this work is to determine the kinetic parameters - intrinsic diffusion coefficients. Moreover the voids radii - the Frenkel and Kirkendall effects will be investigated in both experimental and numerical analysis.
2. Voids growth and its radius The void radius depends on the concentration of the vacancies and its diffusion coefficient. The analysis bases on the expression derived by Gusak and Storozhuk [16]:
ski / Journal of Alloys and Compounds 687 (2016) 104e108 B. Wierzba, W. Skibin
dR 1 eq 1 ¼ DV NV NV þ dt LV R
(1)
where: LV denote the mean free path of vacancies, NV and NVeq are the vacancy molar fraction and equilibrium vacancy molar fraction ðNVeq ¼ 0:0002Þ, respectively. The generalized vacancy diffusion Pr I I Di Dj
j;i¼1
coefficient DV ¼
NV
Pjr> i i;j¼1
DIi Nj
where DIi denotes the intrinsic
isj
diffusion coefficient of the i-th component. The equation for the time evolution of the vacancy concentration is defined as: eq
vNv Nv Nv þ divjv þ ¼0 vt tv
(2)
where Nv is the vacancy molar fraction, jv is the vacancy flux. The Nveq and tv denote the vacancy equilibrium molar fraction and relaxation time, respectively. The strength of sinks can be characterized by the mean free path of vacancies LV z DVtV [16]. The vacancy flux is defined as a sum of the diffusion and drift parts:
jv ¼ jdv þ Nv ydrift
In Equation (3) term NVy is small (NVy z 0), since the vacancy ratio is small (NV ≪ Ni) and can be neglected. Already from Darken analysis, we know that the overall sum of the fluxes should equal zero. Thus the vacancy flux is a sum of the fluxes of the components, mainly: r X
drift
DIi grad Ni
(4)
i¼1
where Ni is the molar ratio and DIi is intrinsic diffusion coefficient of the component. Moreover, voids growth during multi-component diffusion process will lay on the generalized Darken approach. The core of the model is the mass conservation law for each component:
vNi þ div vt
DIi VNi þ Ni
r X
! DIi VNi
¼0
(5)
i¼1
2.1. Solution - numerical method The numerical treatment of the method comes down to solving
DIi
resulting from the space discretization was used [17]. The uniform grid, contained 100 mesh points, was used and the concentrations were defined at points xk. The space derivatives in the equations were approximated by two point (first derivative) and three point (second derivative) uniform finite differences: kþ1 F k1 F vF ðxk ; tÞz vx xkþ1 xk1
(6)
and kþ1 Fðx x k1 Fðx k v2 F k k1 Þ þ kþ1 xk Þ Fðxkþ1 xk1 Þ ðx ; tÞz k 0:5ðxkþ1 xk1 Þðxkþ1 xk Þðxk xk1 Þ vx2
(7) Both were calculated from Taylor expansion. The time integrator, used in the present computations to solve the ODEs, was based on the adaptive step size Runge-KuttaFehlberg method. The six evaluations of the functions from the fifth-order Runge-Kutta algorithm were used to make the another combinations implemented in the fourth-order Runge-Kutta method. A difference between these two estimates served as an estimate of the truncation error. Hence, the step size was adjusted [18].
(3) drift
jv ¼
105
2.2. The initial and boundary conditions The remaining initial data necessary in the calculations include: terminal composition of the diffusion couple (representing the terminal points of the diffusion path), diffusion coefficients of the components in all phases, processing time and temperature - results section. The boundary conditions in presented model should be applied for the fluxes of the components in each phase. Mainly, the closed system is assumed, thus the diffusion fluxes at the boundary equals zero:
jdi ¼ DIi VNi ¼ 0 at the boundary
the two-phase zone formation is not introduced into the model. The above set of equations allows calculating the voids and concentration time evolution in the system where diffusion coefficients are known. In this paper the intrinsic diffusion coefficients were determined from the Ni-Cu experimental results with Generalized Boltzmann-Matano analysis. This method allows determination of the diffusivities in multicomponent systems where the Kirkendall position is known. In this method the intrinsic diffusion coefficient can be estimated at the Kirkendall plane position as [8]:
2 3 ZxK x 1 vx K 5 * * * ∞ i ∞ * ∞ 4 Ni ¼ Ni ðt; xK Þ ¼ N Ni dx þ Ni Ni x N Ni ; i ¼ 1; 2; …; r 2t vNi N* K i 2t 1=2 i
(8)
(9)
∞
a set of ordinary differential equations dy=dt ¼ f ðy; tÞ. Thus, the explicit Euler method can be used - the passage from time tk to tkþ1 ¼ tkþDt was carried out by one-step evaluation. Moreover, the advancement tk / tkþ1 was executed in which ykþ1 ¼ yk þ Dt$f(yk,tk) with f(y,t) evaluated for t ¼ tk and y ¼ yk. The method of lines to solve numerically the ODEs system
where xK denotes the Kirkendall plane position, hence position for which intrinsic diffusivity is calculated. Thus, the unique logarithm of intrinsic diffusivity as a polynomial (parabolic) function of molar ratio can be approximated. The comparison of the original BM and GBM method in Ni-Cu system is presented in results section.
106
ski / Journal of Alloys and Compounds 687 (2016) 104e108 B. Wierzba, W. Skibin
Fig. 1. The experimental result of Kirkendall porosity (Frenkel effect) in Cu-Ni diffusion couple; T ¼ 1273 K, t ¼ 200 h.
Fig. 3. The experimental result of Kirkendall porosity (Frenkel effect) in Cu-NiAl diffusion couple; T ¼ 1273 K, t ¼ 150 h.
3. Results The highly pure nickel (99.99%) and copper (99.99%) foils (0.5 mm thickness) provided by Sigma Aldrich were used as starting materials. The diffusion couple was constructed from pure copper and nickel and from pure copper and b-NiAl. Moreover, small particles ~0.3 mm of thorium dioxide (ThO2) were used as fiducial (Kirkendall) markers to find the Kirkendall shift in Ni-Cu couple. Diffusion couple was held in furnace in temperature 1273 K for several times. Protective atmosphere (argon) was used as well as external load to provide good adherence. After annealing and standard metallographic preparation, the diffusion couple was examined using scanning electron microscopy (SEM) coupled with EDX detector. The concentration profiles were measured using EDX-analysis. Moreover, the position of the inert ThO2 markers after interdiffusion process was studied. The experimental results of cross-sections describing the voids in Ni-Cu after annealing for 200 h and bNiAl-Cu system after annealing for 150 h at 1273 K are shown on Figs. 1e3. The experimentally determined Kirkendall plane position, xK relative to the Matano plane position xM (xM ¼ 0) in Ni-Cu system at 1273 K for 200 h, analyzed in the present work, is shown in Table 1. Basing on experimental concentration profile the BoltzmannMatano analysis was used. Intrinsic diffusion coefficients of nickel
and copper in Ni-Cu system were determined using Generalized Boltzmann-Matano method [19]. The method assumes the parabolic dependence of the logarithm of intrinsic diffusion coefficients versus molar ratio of element in the diffusion couple. To check the validity of GBM approach in Ni-Cu system concentration profiles were calculated by the generalized Darken method [20] and compared with experimental results.
3.1. Diffusion in the Ni-Cu system The concentration dependent intrinsic diffusion coefficients of nickel and copper at 1273 K in Ni-Cu system were calculated using Generalized Boltzmann-Matano approach from the experimental concentration profile [19]. The intrinsic diffusion coefficients of Ni and Cu were approximated in the whole range of concentration in Ni-Cu alloy, Figs. 4 and 5. The results of diffusion coefficients are described by polynomials:
DNi ¼ 1012:794:94NNi þ2:37NNi and DCu 2
¼ 1019:37þ1:38NCu þ1:19NCu cm2 s1 2
The polynomials were constructed from approximation method of self diffusion coefficients and intrinsic diffusion coefficient at Kirkendall plane position (points at Figs. 4 and 5). The intrinsic diffusion coefficients of components was than used to calculate the ~ ¼ N DI þ N DI , and compare with interdiffusion coefficient, D Cu Ni Ni Cu the one obtained by original Boltzmann-Matano method, Fig. 6. The interdiffusion coefficient decrease with increasing Ni content, thus the self diffusion coefficient of pure Cu is grater than self diffusion coefficient of Ni. Fig. 7 presents the comparison between experimental and calculated concentration profiles of Cu and Ni over diffusion couple cross-section in analyzed system.
Table 1 The experimental Kirkendall plane position. Ni-Cu system at 1273 K
Fig. 2. The experimental result of Kirkendall porosity (Frenkel effect) in Cu-NiAl diffusion couple; T ¼ 1273 K, t ¼ 150 h.
Annealing time [h] Kirkendall plane position [mm] Measurement error [mm]
200 84.9 2.3
ski / Journal of Alloys and Compounds 687 (2016) 104e108 B. Wierzba, W. Skibin
107
Fig. 4. Intrinsic diffusion coefficient of nickel in the Ni-Cu system at 1273 K by Generalized Boltzmann-Matano method.
Fig. 7. Concentration profile of Cu and Ni in the interdiffusion zone of Cu-Ni binary couple after annealing at 1273 K for 200 h.
Fig. 5. Intrinsic diffusion coefficient of copper in the Ni-Cu system at 1273 K by Generalized Boltzmann-Matano method.
3.2. Diffusion in the bNiAl-Cu system The diffusion couple will be treated as half-infinite, thus the initial concentration profile will be constructed from two Heaviside-functions (Fig. 8). The interdiffusion coefficients in bNiAl phase exhibits strong dependence on the concentration. In this
Fig. 6. The Comparison between Ni-Cu interdiffusion coefficient by BoltzmannMatano method and Generalized Boltzmann-Matano at 1273 K, 200 h.
work the Campbell approximation [21,22] was used. The integrated diffusion coefficients in b-NiAl are DNi¼DA l¼ 9.15$1010; in Cureach diffusion couple side are DNi ¼ 5.29$1014 [23], DAl ¼ 2.91$1013 [24] and the intrinsic diffusion coefficient of Cu (for the whole range) will be used from previous calculations e 2 DCu ¼ 1019:37þ1:38NCu þ1:19NCu cm2 s1 . The results of the calculated concentration profile over the distance compared with the experiments are presented in Fig. 8. The 0 jump’ over the concentration profile, Fig. 8, characterizes the discontinuity (boundary) between the b-NiAl and Cu-rich phases. Moreover, the experimental diffusion path of the process is shown in Fig. 9. The diffusion path is a mapping of the stationary concentrations onto the isothermal section of the equilibrium phase diagram. Thus, the diffusion path connects initial compositions of the diffusion couple and can go across the single-,two- and three-phase fields. It starts at the composition of one alloy and ends at the other. The ’jumps; presented on Figs. 8 and 9 corresponds to the nonisothermal section of the equilibrium phase diagram. Thus, no two-phase zone was presented. The homogeneity region of individual phases (isothermal section of the equilibrium phase diagram) is marked by gray filling.
Fig. 8. Concentration profile of Al, Cu and Ni in the interdiffusion zone of Cu-NiAl couple after annealing at 1273 K for 150 h.
108
ski / Journal of Alloys and Compounds 687 (2016) 104e108 B. Wierzba, W. Skibin
3 The calculated interdiffusion coefficient in Cu-Ni diffusion couple decrease with increasing Ni content. The total decrease of interdiffusion coefficient exceeds two orders of magnitude. 4 It was demonstrated, that the voids radii evolution can be estimated in Cu-NiAl system. The rate of void growth is deter eq 1 þ 1 . The mean mined by equation dR ¼ D ðN N Þ V V LV R V dt migration length for vacancy base on the theory of mean free path (average distance traveled by a moving particle between successive impacts). Thus, the mean migration length for vacancy in Cu-NiAl system is approximated to LV ¼ 106m. The estimated void radii was 8.9 mm, while the measured was equal to 9.1 mm. Acknowledgements This work has been supported by the National Science Centre (NCN) in Poland, decision number 2013/09/B/ST8/00150. References Fig. 9. Diffusion path in the ternary Al-Ni-Cu system at 1273 K; gray filling - the homogeneity region.
The experimental voids radii in Cu-NiAl system during diffusion process was estimated and compared with the theoretical one. Assuming the mean migration length for vacancy LV ¼ 106 m and using Equation (1) the theoretical void radius after 150 h is equal to 8.9 mm, while the measured mean void radii was equal to 9.1 mm. 4. Conclusions In this paper the interdiffusion processes in Cu-Ni and Cu-NiAl at 1273 K were studied. From the results the following conclusions can be formulated: 1 It was found, that at 1273 K the diffusion path in the ternary phase diagram starts in the region of Cu-based ternary solid solution and ends in the homogeneity region of b-NiAl(Cu) solid solution. Moreover, the diffusion path follows the tie line - no two-phase zone is formed. 2 It was demonstrated, that the generalized Boltzmann-Matano method is an effective tool for determining the intrinsic diffusion coefficient in multicomponent systems where the position of Kirkendall plane is determined. The Kirkendall plane position is an additional factor for diffusivities determination.
[1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24]
E. Rabkin, V.N. Semenov, A. Winkler, Acta Mater. 50 (2002) 3227. W.F. Gale, Y. Guan, MMTA 27A (1996) 3621. D.M. Dimiduk, D.B. Miracle, C.H. Ward, Mater. Sci. Technol. 8 (1992) 367. T.B. Massalski (Ed.), Binary Alloy Phase Diagrams, ASM, Metals Park, OH, 1986. St Frank, S.V. Divinski, U. Soderval, Chr Herzig, Acta Mater. 49 (2001) 1399. C. Matano, Jpn. J. Phys. 8 (1933) 109. ski, Phys. A 392 (2013) 4316. B. Wierzba, W. Skibin ski, Physica A 440 (2015) 100. B. Wierzba, W. Skibin A.D. Smigelskas, E.O. Kirkendall, Trans. AIME 171 (1947) 130. L.S. Darken, Trans. AIME 174 (1948) 184. ti, A.A. Kodentsov, F.J.J. van Loo, Phys. Rev. M.J.H. van Dal, A.M. Gusak, C. Cserha Lett. 86 (2001) 3352. B. Wierzba, Competition between Kirkendall and Frenkel effects during multicomponent interdiffusion process, Phys. A () (in print). A. Poczekajlo, Z. Grzesik, S. Mrowec, High Temp, Mater. Proc (2013), http:// dx.doi.org/10.1515/htmp-2013-0055. K.N. Tu, Solder Joint Technology, Springer, New York, 2007. K.N. Tu, Electronic Thin_Films Reliability, Cambridge University, New York, 2011. A.M. Gusak, N.V. Storozhuk, Phys. Metals Metall 114 (2013) 197. W.E. Schiesser, The Numerical Method of Lines, Academic Press, San Diego, 1991. W.H. Press, B.P. Flannery, S.A. Teukolsky, W.T. Vetterling, Numerical Recipes in C: the Art of Scientific Computing, Cambridge University Press, 1992. ski, Phys. A 392 (2013) 4316. B. Wierzba, W. Skibin M. Danielewski, B. Wierzba, Acta Mater. 58 (2010) 6717. C.E. Campbell, Acta Mater. 56 (2008) 4277. B. Wierzba, K. Tkacz-Smiech, A. Nowotnik, K. Dychton Chem, Vap. Depos. 20 (2014) 1. C.A. Mackliet, Phys. Rev. 109 (1958) 1964. R.L. Fogelson, Ya A. Ugay, A.V. Pokoyev, Izv Vyssh, Uchebn Zaved, Tsvetn. Metall. 3 (1973) 143.
Contents lists available at ScienceDirect
Journal of Alloys and Compounds journal homepage: http://www.elsevier.com/locate/jalcom
The interdiffusion in copper-nickel alloys ski b Bartek Wierzba a, *, Wojciech Skibin a b
co w Warszawy 12, 35-959 Rzeszo w, Poland Rzeszow University of Technology, Faculty of Mechanical Engineering and Aeronautics, al. Powstan w, Poland AGH University of Science and Technology, al. Mickiewicza 30, 30-059 Krako
a r t i c l e i n f o
a b s t r a c t
Article history: Received 21 March 2016 Received in revised form 2 June 2016 Accepted 9 June 2016 Available online 11 June 2016
In this paper the interdiffusion process and Frenkel effect (Kirkendall porosity) are studied in Ni-Cu and NiAl-Cu systems at 1273 K. The generalized Boltzmann-Matano method is used for calculation of the intrinsic diffusion coefficients as a polynomial function in the whole composition range. The calculated interdiffusion coefficient in Cu-Ni diffusion couple decrease with increasing Ni content. The total decrease of interdiffusion coefficient exceeds two orders of magnitude. The equation for the void growth rate is obtained. The experimental results of diffusion and voids formation are analyzed. The voids radii is estimated and compared with experiments. © 2016 Elsevier B.V. All rights reserved.
Keywords: Voids Frenkel effect Diffusion Boltzmann-Matano
1. Introduction High oxidation resistance, mechanical properties, low density and high melting temperature of the NiAl intermetallic compound have attracted scientific attention to this material [1]. The B2 type offers potential as a highservice-temperature material, for example, in aero gas turbine engines [2,3]. However, many application of NiAl would require access to suitable joining technologies [3]. The pure Cu can be used for interlayer during diffusional bonding process [2]. The copper shows extensive solid solubility in both NiAl and nickel [4]. The essential is the knowledge about high temperature diffusion for better understanding of the mechanical properties. The scientists nowadays debate about the diffusion mechanism in B2 compounds. From Rabkin works, we know, that the usual mechanism involving jump to the nearest neighbor vacancy is improbable in B2 compounds as that migration may locally destroy the chemical order [1]. Recent data of Herzig and co-workers on Ni tracer diffusion in NiAl support the triple defect mechanism of Ni diffusion [5]. The main kinetic parameter describing the diffusion in multicomponent system is the intrinsic diffusion coefficient. In binary systems the Boltzmann-Matano (B-M) analysis [6] can be used for determination of this parameter. However in ternary and higher
* Corresponding author. E-mail address: [email protected] (B. Wierzba). http://dx.doi.org/10.1016/j.jallcom.2016.06.085 0925-8388/© 2016 Elsevier B.V. All rights reserved.
order systems the B-M method is invalid. In our previous paper we have shown the generalization of B-M analysis [7,8]. The key idea is the knowledge about the position of Kirkendall plane. This additional data allows for determination of the intrinsic diffusion coefficient at the position of the markers. Then, assuming, that the logarithm of diffusivities are determined by polynomial function (at least of order 2) the diffusion coefficients for each component in multicomponent diffusion couple can be approximated [7]. During the interdiffusion process, due to the differences in diffusion coefficients numerous phenomena occur: the lattice shift, the stress generation and relaxation, the nonequilibrium distribution of vacancies and voiding [9e13]. Voiding is a result of relaxation of pure material or alloy supersaturated with vacancies. The relaxation of vacancy subsystem can proceed by joining of vacancies into voids [14,15]. In this work the interdiffusion in Cu-Ni and Cu-NiAl at the temperature 1273 K will be investigated. Thus, the objective of this work is to determine the kinetic parameters - intrinsic diffusion coefficients. Moreover the voids radii - the Frenkel and Kirkendall effects will be investigated in both experimental and numerical analysis.
2. Voids growth and its radius The void radius depends on the concentration of the vacancies and its diffusion coefficient. The analysis bases on the expression derived by Gusak and Storozhuk [16]:
ski / Journal of Alloys and Compounds 687 (2016) 104e108 B. Wierzba, W. Skibin
dR 1 eq 1 ¼ DV NV NV þ dt LV R
(1)
where: LV denote the mean free path of vacancies, NV and NVeq are the vacancy molar fraction and equilibrium vacancy molar fraction ðNVeq ¼ 0:0002Þ, respectively. The generalized vacancy diffusion Pr I I Di Dj
j;i¼1
coefficient DV ¼
NV
Pjr> i i;j¼1
DIi Nj
where DIi denotes the intrinsic
isj
diffusion coefficient of the i-th component. The equation for the time evolution of the vacancy concentration is defined as: eq
vNv Nv Nv þ divjv þ ¼0 vt tv
(2)
where Nv is the vacancy molar fraction, jv is the vacancy flux. The Nveq and tv denote the vacancy equilibrium molar fraction and relaxation time, respectively. The strength of sinks can be characterized by the mean free path of vacancies LV z DVtV [16]. The vacancy flux is defined as a sum of the diffusion and drift parts:
jv ¼ jdv þ Nv ydrift
In Equation (3) term NVy is small (NVy z 0), since the vacancy ratio is small (NV ≪ Ni) and can be neglected. Already from Darken analysis, we know that the overall sum of the fluxes should equal zero. Thus the vacancy flux is a sum of the fluxes of the components, mainly: r X
drift
DIi grad Ni
(4)
i¼1
where Ni is the molar ratio and DIi is intrinsic diffusion coefficient of the component. Moreover, voids growth during multi-component diffusion process will lay on the generalized Darken approach. The core of the model is the mass conservation law for each component:
vNi þ div vt
DIi VNi þ Ni
r X
! DIi VNi
¼0
(5)
i¼1
2.1. Solution - numerical method The numerical treatment of the method comes down to solving
DIi
resulting from the space discretization was used [17]. The uniform grid, contained 100 mesh points, was used and the concentrations were defined at points xk. The space derivatives in the equations were approximated by two point (first derivative) and three point (second derivative) uniform finite differences: kþ1 F k1 F vF ðxk ; tÞz vx xkþ1 xk1
(6)
and kþ1 Fðx x k1 Fðx k v2 F k k1 Þ þ kþ1 xk Þ Fðxkþ1 xk1 Þ ðx ; tÞz k 0:5ðxkþ1 xk1 Þðxkþ1 xk Þðxk xk1 Þ vx2
(7) Both were calculated from Taylor expansion. The time integrator, used in the present computations to solve the ODEs, was based on the adaptive step size Runge-KuttaFehlberg method. The six evaluations of the functions from the fifth-order Runge-Kutta algorithm were used to make the another combinations implemented in the fourth-order Runge-Kutta method. A difference between these two estimates served as an estimate of the truncation error. Hence, the step size was adjusted [18].
(3) drift
jv ¼
105
2.2. The initial and boundary conditions The remaining initial data necessary in the calculations include: terminal composition of the diffusion couple (representing the terminal points of the diffusion path), diffusion coefficients of the components in all phases, processing time and temperature - results section. The boundary conditions in presented model should be applied for the fluxes of the components in each phase. Mainly, the closed system is assumed, thus the diffusion fluxes at the boundary equals zero:
jdi ¼ DIi VNi ¼ 0 at the boundary
the two-phase zone formation is not introduced into the model. The above set of equations allows calculating the voids and concentration time evolution in the system where diffusion coefficients are known. In this paper the intrinsic diffusion coefficients were determined from the Ni-Cu experimental results with Generalized Boltzmann-Matano analysis. This method allows determination of the diffusivities in multicomponent systems where the Kirkendall position is known. In this method the intrinsic diffusion coefficient can be estimated at the Kirkendall plane position as [8]:
2 3 ZxK x 1 vx K 5 * * * ∞ i ∞ * ∞ 4 Ni ¼ Ni ðt; xK Þ ¼ N Ni dx þ Ni Ni x N Ni ; i ¼ 1; 2; …; r 2t vNi N* K i 2t 1=2 i
(8)
(9)
∞
a set of ordinary differential equations dy=dt ¼ f ðy; tÞ. Thus, the explicit Euler method can be used - the passage from time tk to tkþ1 ¼ tkþDt was carried out by one-step evaluation. Moreover, the advancement tk / tkþ1 was executed in which ykþ1 ¼ yk þ Dt$f(yk,tk) with f(y,t) evaluated for t ¼ tk and y ¼ yk. The method of lines to solve numerically the ODEs system
where xK denotes the Kirkendall plane position, hence position for which intrinsic diffusivity is calculated. Thus, the unique logarithm of intrinsic diffusivity as a polynomial (parabolic) function of molar ratio can be approximated. The comparison of the original BM and GBM method in Ni-Cu system is presented in results section.
106
ski / Journal of Alloys and Compounds 687 (2016) 104e108 B. Wierzba, W. Skibin
Fig. 1. The experimental result of Kirkendall porosity (Frenkel effect) in Cu-Ni diffusion couple; T ¼ 1273 K, t ¼ 200 h.
Fig. 3. The experimental result of Kirkendall porosity (Frenkel effect) in Cu-NiAl diffusion couple; T ¼ 1273 K, t ¼ 150 h.
3. Results The highly pure nickel (99.99%) and copper (99.99%) foils (0.5 mm thickness) provided by Sigma Aldrich were used as starting materials. The diffusion couple was constructed from pure copper and nickel and from pure copper and b-NiAl. Moreover, small particles ~0.3 mm of thorium dioxide (ThO2) were used as fiducial (Kirkendall) markers to find the Kirkendall shift in Ni-Cu couple. Diffusion couple was held in furnace in temperature 1273 K for several times. Protective atmosphere (argon) was used as well as external load to provide good adherence. After annealing and standard metallographic preparation, the diffusion couple was examined using scanning electron microscopy (SEM) coupled with EDX detector. The concentration profiles were measured using EDX-analysis. Moreover, the position of the inert ThO2 markers after interdiffusion process was studied. The experimental results of cross-sections describing the voids in Ni-Cu after annealing for 200 h and bNiAl-Cu system after annealing for 150 h at 1273 K are shown on Figs. 1e3. The experimentally determined Kirkendall plane position, xK relative to the Matano plane position xM (xM ¼ 0) in Ni-Cu system at 1273 K for 200 h, analyzed in the present work, is shown in Table 1. Basing on experimental concentration profile the BoltzmannMatano analysis was used. Intrinsic diffusion coefficients of nickel
and copper in Ni-Cu system were determined using Generalized Boltzmann-Matano method [19]. The method assumes the parabolic dependence of the logarithm of intrinsic diffusion coefficients versus molar ratio of element in the diffusion couple. To check the validity of GBM approach in Ni-Cu system concentration profiles were calculated by the generalized Darken method [20] and compared with experimental results.
3.1. Diffusion in the Ni-Cu system The concentration dependent intrinsic diffusion coefficients of nickel and copper at 1273 K in Ni-Cu system were calculated using Generalized Boltzmann-Matano approach from the experimental concentration profile [19]. The intrinsic diffusion coefficients of Ni and Cu were approximated in the whole range of concentration in Ni-Cu alloy, Figs. 4 and 5. The results of diffusion coefficients are described by polynomials:
DNi ¼ 1012:794:94NNi þ2:37NNi and DCu 2
¼ 1019:37þ1:38NCu þ1:19NCu cm2 s1 2
The polynomials were constructed from approximation method of self diffusion coefficients and intrinsic diffusion coefficient at Kirkendall plane position (points at Figs. 4 and 5). The intrinsic diffusion coefficients of components was than used to calculate the ~ ¼ N DI þ N DI , and compare with interdiffusion coefficient, D Cu Ni Ni Cu the one obtained by original Boltzmann-Matano method, Fig. 6. The interdiffusion coefficient decrease with increasing Ni content, thus the self diffusion coefficient of pure Cu is grater than self diffusion coefficient of Ni. Fig. 7 presents the comparison between experimental and calculated concentration profiles of Cu and Ni over diffusion couple cross-section in analyzed system.
Table 1 The experimental Kirkendall plane position. Ni-Cu system at 1273 K
Fig. 2. The experimental result of Kirkendall porosity (Frenkel effect) in Cu-NiAl diffusion couple; T ¼ 1273 K, t ¼ 150 h.
Annealing time [h] Kirkendall plane position [mm] Measurement error [mm]
200 84.9 2.3
ski / Journal of Alloys and Compounds 687 (2016) 104e108 B. Wierzba, W. Skibin
107
Fig. 4. Intrinsic diffusion coefficient of nickel in the Ni-Cu system at 1273 K by Generalized Boltzmann-Matano method.
Fig. 7. Concentration profile of Cu and Ni in the interdiffusion zone of Cu-Ni binary couple after annealing at 1273 K for 200 h.
Fig. 5. Intrinsic diffusion coefficient of copper in the Ni-Cu system at 1273 K by Generalized Boltzmann-Matano method.
3.2. Diffusion in the bNiAl-Cu system The diffusion couple will be treated as half-infinite, thus the initial concentration profile will be constructed from two Heaviside-functions (Fig. 8). The interdiffusion coefficients in bNiAl phase exhibits strong dependence on the concentration. In this
Fig. 6. The Comparison between Ni-Cu interdiffusion coefficient by BoltzmannMatano method and Generalized Boltzmann-Matano at 1273 K, 200 h.
work the Campbell approximation [21,22] was used. The integrated diffusion coefficients in b-NiAl are DNi¼DA l¼ 9.15$1010; in Cureach diffusion couple side are DNi ¼ 5.29$1014 [23], DAl ¼ 2.91$1013 [24] and the intrinsic diffusion coefficient of Cu (for the whole range) will be used from previous calculations e 2 DCu ¼ 1019:37þ1:38NCu þ1:19NCu cm2 s1 . The results of the calculated concentration profile over the distance compared with the experiments are presented in Fig. 8. The 0 jump’ over the concentration profile, Fig. 8, characterizes the discontinuity (boundary) between the b-NiAl and Cu-rich phases. Moreover, the experimental diffusion path of the process is shown in Fig. 9. The diffusion path is a mapping of the stationary concentrations onto the isothermal section of the equilibrium phase diagram. Thus, the diffusion path connects initial compositions of the diffusion couple and can go across the single-,two- and three-phase fields. It starts at the composition of one alloy and ends at the other. The ’jumps; presented on Figs. 8 and 9 corresponds to the nonisothermal section of the equilibrium phase diagram. Thus, no two-phase zone was presented. The homogeneity region of individual phases (isothermal section of the equilibrium phase diagram) is marked by gray filling.
Fig. 8. Concentration profile of Al, Cu and Ni in the interdiffusion zone of Cu-NiAl couple after annealing at 1273 K for 150 h.
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3 The calculated interdiffusion coefficient in Cu-Ni diffusion couple decrease with increasing Ni content. The total decrease of interdiffusion coefficient exceeds two orders of magnitude. 4 It was demonstrated, that the voids radii evolution can be estimated in Cu-NiAl system. The rate of void growth is deter eq 1 þ 1 . The mean mined by equation dR ¼ D ðN N Þ V V LV R V dt migration length for vacancy base on the theory of mean free path (average distance traveled by a moving particle between successive impacts). Thus, the mean migration length for vacancy in Cu-NiAl system is approximated to LV ¼ 106m. The estimated void radii was 8.9 mm, while the measured was equal to 9.1 mm. Acknowledgements This work has been supported by the National Science Centre (NCN) in Poland, decision number 2013/09/B/ST8/00150. References Fig. 9. Diffusion path in the ternary Al-Ni-Cu system at 1273 K; gray filling - the homogeneity region.
The experimental voids radii in Cu-NiAl system during diffusion process was estimated and compared with the theoretical one. Assuming the mean migration length for vacancy LV ¼ 106 m and using Equation (1) the theoretical void radius after 150 h is equal to 8.9 mm, while the measured mean void radii was equal to 9.1 mm. 4. Conclusions In this paper the interdiffusion processes in Cu-Ni and Cu-NiAl at 1273 K were studied. From the results the following conclusions can be formulated: 1 It was found, that at 1273 K the diffusion path in the ternary phase diagram starts in the region of Cu-based ternary solid solution and ends in the homogeneity region of b-NiAl(Cu) solid solution. Moreover, the diffusion path follows the tie line - no two-phase zone is formed. 2 It was demonstrated, that the generalized Boltzmann-Matano method is an effective tool for determining the intrinsic diffusion coefficient in multicomponent systems where the position of Kirkendall plane is determined. The Kirkendall plane position is an additional factor for diffusivities determination.
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