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Intrinsic diffusion in Ni3Al system
Intermetallics 11 (2003) 291–297 www.elsevier.com/locate/intermet
Intrinsic diffusion in Ni3Al system C. Cserha´tia,*, A. Paulb, A.A. Kodentsovb, M.J.H. van Dalc, F.J.J. van Loob a
Department of Solid State Physics, University of Debrecen, PO Box 2, 4010 Debrecen, Hungary b Laboratory of Solid State and Materials Chemistry, Eindhoven University of Technology, PO Box 513, 5600 MB Eindhoven, The Netherlands c Philips Research Leuven, Kapeldreef 75 B-3001 Leuven, Belgium
Received 24 September 2002; received in revised form 7 November 2002; accepted 7 November 2002
Abstract Interdiffusion experiments were performed between different Ni–Al two-phase alloys. A method is given to estimate the tracer diffusion coefficient of one of the constituents in a binary alloy knowing the tracer diffusion coefficient of the other species. The procedure is based on the so-called Darken-Manning analysis. The tracer diffusion coefficient of Al was determined in this way in Ni3Al: 243 16 KJ=mol 2 þ2:28106 m =s DAl ¼ 5:05 107 1:11710 exp 7 RT Our results are compared with others available in the literature. The values for the self diffusivity of Al calculated from interdiffusion experiments give consistent result. All these measurements support the theory that diffusion of the minor element occurs through a vacancy mechanism in Ni3Al. # 2003 Elsevier Science Ltd. All rights reserved. Keywords: B. Diffusion
1. Introduction Nickel-based intermetallic compounds are still in the focus of interest. Extensive investigations have been carried out on mechanical properties of these materials, with special attention on Ni3Al which is the g0 phase in the binary Ni–Al system. Its high mechanical strength at high temperatures explains the persistent intention for applications. Several processes are driven by diffusion like recrystallization, grain growth and solid state reactions and therefore, knowledge of diffusion is essential for the fabrication of such materials and also for practical use at high temperatures. Diffusion data in Ni3Al, studied by radiotracer and interdiffusion experiments, are summarized in [1] and recently in [2,3]. Tracer measurements concentrated
mainly on the determination of Ni diffusivity because of the lack of a proper Al isotope. Larikov [4] is the only one who reported the tracer coefficient of Al in Ni3Al as almost equal to the Ni diffusivity. This measurement is, however, doubtful since the Ni tracer diffusion data they measured are significantly larger than in other recent reports [1]. The goal of the present study was to determine the self diffusion coefficient of Al in Ni3Al from interdiffusion experiments by applying the Darken–Manning analysis [5,6]. To this end interdiffusion experiments were designed between incremental couples of different Ni–Al two-phase alloys in which Ni3Al formed from its saturated adjacent phases by diffusion.
2. Experimental procedure * Corresponding author. Tel.: +36-52-316073; fax: +36-52316073. E-mail address: cserhati@delfin.klte.hu (C. Cserha´ti).
Pure Ni (99.995%) and Al (99.999%) provided by Goodfellow (UK) were used as starting materials. Four different binary alloys were arc-melted to form two
0966-9795/03/$ - see front matter # 2003 Elsevier Science Ltd. All rights reserved. PII: S0966-9795(02)00235-2
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292
types of diffusion couples: A: Ni65Al35/Ni85Al15 and B: Ni72Al28/Ni78Al22. The alloy ingots were re-melted three times in order to improve their homogeneity. Recrystallization and grain growth were accomplished by annealing in vacuum at 1000 C for 48 h. The couple halves were cut and polished down to 0.25 mm. The parts were cleaned ultrasonically in ethanol. Prior to the annealing small ThO2 particles ( 0.3mm) were introduced at the initial interface as Kirkendall markers. The diffusion couples were heat treated in a vacuum furnace (5107 mbar) under an external load of about 5MPa for different times at various temperatures. During the annealing the temperature was controlled within 2 C. The details of the annealings are given in Table 1. After standard metallographic preparation the heat treated samples were investigated by optical microscope, Scanning Electron Microscope (SEM) and Electron Probe Microanalysis (EPMA).
3. Experimental findings A typical morphology of an annealed diffusion couple can be seen in Fig. 1. One can clearly observe the twophase initial materials, in which the precipitates of the final reaction product were present. The Ni rich side contains Ni rich g0 phase precipitated out from the Ni solid-solution, while the other side is full with Al rich g0 phase precipitates dispersed in the NiAl parent phase. The interface between the end members and the formed Ni3Al is wavy which can be explained by the heterogeneous structure of the initial two phase material. The moving interface of the growing Ni3Al incorporates the precipitates on both sides of the new phase. In the course of this process the interface becomes irregular. This nature of the interface makes it difficult to determine the thickness of the phase and the exact position of the Kirkendall plane although the straight row of white particles in the middle of the grown phase Table 1 Measuring details as well as the composition at the Kirkendall plane is given [NNi(K)] Couple
Temp (K)
Time (h)
NNi(K) (at.%)
(DNiVAl)/(DAlVNi)a
A A A A A A A B B B B
1173 1223 1248 1273 1273 1323 1323 1373 1373 1373 1473
400 196 196 196 196 392 98 196 196 196 196
74.6 75.0 75.0 74.5 74.5 74.9 74.8 74.9 75.3 73.9 76.0
2.3 4.4 5.5 4.0 5.4 6.9 4.2 6.6 2.4 2.4 3.4
a
In the last column the calculated ratio of the intrinsic diffusion coefficients is written.
Fig. 1. Back-scattered electron image of the diffusion zone after annealing at 1000 C for 196 h. The Kirkendall markers (ThO2 particles) are located in the formed Ni3Al phase.
marks it nicely. One clearly sees the different crystal morphology of Ni3Al on both sides of the Kirkendall plane, caused by the different nucleation sites [7]. In order to reduce this trouble the SEM images were processed by computer. The contour of the interfaces were fitted with a polinom. The areas of the reaction product on both sides of the original welding plane were calculated. Summing the two areas gives the average thickness of the reaction layer, while by taking the ratio the position of the Kirkendall plane was determined. In Fig. 2. a typical concentration profile is shown where the Kirkendall plane is indicated as well. As Fig. 2 displays the ThO2 particles are located at the stoichiometric Ni3Al composition within the experimental error (1–2 at.%). The thickness of the reaction product and the position of the Kirkendall plane changes along the sample due to the wavy nature of the interface. To perform the Boltzmann-Matano analysis the concentration profile was measured to determine the magnitude of the concentration step at the interfaces on both sides of the g0 phase. Then the concentration profile was scaled to the calculated average phase thickness (mentioned above). In this way a stepwise linear concentration profile was constructed (see Fig. 2.). Time dependence of the phase growth was measured at two different temperatures (1173 K, 1323 K). Parabolic growth was found which demonstrates the diffusion control of the process. It was reported by Janssen [8] that below 1000 C grain boundary diffusion becomes the dominant process. In order to determine the average grain size in the g0 phase the sample was etched to make the grain structure visible. The average grain size in the new phase was comparable with the thickness of the whole Ni3Al layer. Moreover, the protruding of the interface corresponds nowhere to the grain boundaries. This points to the fact that the importance of grain boundary diffusion was small compared to volume diffusion.
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293
Fig. 2. A typical concentration profile is shown (1000 C, 196 h). It was scaled to the average thickness of the newly formed phase (see the text for the details). The Kirkendall plane (K) is indicated.
The usual Boltzmann-Matano analysis was applied to calculate the interdiffusion coefficients at the Kirkendall plane. At one temperature (1373 K, see Table 1) three experimental runs were performed which enabled us to estimate the error of the mutual diffusion coefficients as the maximal deviation from the average value. The interdiffusion coefficients found at the stoichiometric composition are plotted in Fig. 3 together with data obtained from the literature [8,9]. The authors in these experiments performed multi-phase diffusion measurement, but their starting materials were different. Whilst Janssen [8] used NiAl(55 at.% Ni)-pure Ni diffusion couples, Watanabe [9] measured interdiffusivity in NiAl(75 at.% Ni)-pure Ni system by AEM. The circumstances in our experiments were closer to Janssen’s since the composition of the end members we designed were different from the composition of the reaction product. This might explain that our results are in better agreement with his data. Looking at the measured concentration profile (Fig. 2) one can realize that the composition at the Ni rich and the Al rich side of the Ni3Al phase is different from the concentration of the equilibrium phase diagram taken from the well known book edited by Massalski [10]. To check what the real equilibrium compositions are, another set of experiments was developed in which two phase alloys of known compositions had been equilibrated at the same temperature range where the diffusion annealings were performed. It was found that the composition on both sides of the g-phase we obtained from the diffusion
couples correspond well with these measurements. It means that our result differ with the phase diagram printed in [10], but are in beautiful accordance with the data book of Hansen and Anderko [11] (see Appendix).
4. Determining the diffusivity of Al Although the interdiffusion coefficient is a perfect measure of the redistribution of the components during the diffusion process it gives no information on the relative diffusivities of the species. A more fundamental quantity is the intrinsic diffusivity (Di) which is directly related to the atomic fluxes (Ji) with respect to the lattice planes (Kirkendall frame of reference) via Fick’s first law [12]: JA ¼ DA
@CA DA VB @NA ¼ ; @x V2m @x
ð1Þ
where Ni and Vi is the mole fraction and the partial molar volume of element i respectively, Vm is the molar volume, Ci the composition in moles of component i per m3 and x is the diffusion direction. Since inert ThO2 particles were used to mark the initial welding plane, the intrinsic diffusion coefficient at this position can be calculated. For our purposes it was more interesting to compute the ratio of the intrinsic diffusion coefficients at the Kirkendall plane at position xk [7]:
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294
Fig. 3. Interdiffusion coefficients are plotted on an Arrhenius-plot together with data obtained from the literature. The interdiffusion coefficients were calculated at the Kirkendall plane, i.e. at the stoichiometric composition (see Fig. 2).
DA VA ¼ DB VB 2 þ 3 Ð Ð NA N þ xk A 1 NA NA dx NA xk dx 7 6 NA 1 Vm Vm 6 7 þ 7 6 Ð Ð 4 5 N N N N A xk A 1 A A dx þ N dx Nþ B 1 B xk Vm Vm ð2Þ N+ i
N i
and indicate the nominal composition of the starting materials, i.e. the composition at the place where no diffusion has occurred yet (see also Fig. 2).The molar volume Vm is considered to be constant in the near-stochiometric compound Ni3Al. In this manner we simplified the problem to area calculations. The ratio VA/VB is difficult to determine in an intermetallic compound and, in fact, one measures the ratio (DAVB)/ (DBVA) which can be found in Table 1. The intrinsic diffusion coefficient moreover is associated to the tracer diffusivities Di* via the Darken– Manning formula [6,12]: Vm @lnaA DA ¼ DA ð 1 þ WA Þ ; ð3Þ @lnNA VB where ai is the chemical activity of component i (the term lnai/lnNi is called the thermodynamic factor)
and Wi is the so called vacancy wind factor. Taking the ratio of the intrinsic diffusion fluxes of both species the advantage of this method becomes clear. In this way we can avoid the difficulties provoked by thermodynamic factors since in binary alloys the thermodynamic factors of the two components are equal. The measured ratio (DAVB)/(DBVA) (see Table 1) can be related to the tracer-diffusion coefficients through: DA VB DA ð1 þ WA Þ : ¼ DB VA DB ð1 WB Þ
ð4Þ
The vacancy wind factor Wi [6] formulates how the net vacancy flux influences the mobility of the diffusing species (i.e. it takes the cross terms into account [12,13]): 2NA DA DB ; WA ¼ Mo NA DA þ NB DB ð5Þ 2NB DA DB WB ¼ Mo NA DA þ NB DB In this equation the correlation effects are considered through Mo. Although Manning’s theory was developed
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for a random solid solution, it has been suggested that it might be applied for L12 compounds where the diffusion of the minor element occurs by ordinary vacancy mechanism over the sublattice of the other species. Employing the statements of Ikeda et al. [2] just as Numakura et al. [3] this assumption seems to be adoptable in our case as well. Employing the proposed mechanism Mo=4.43 was taken [2,14]. Since the tracer coefficient of Ni in Ni3Al is well documented in the literature, the self diffusivity of Al can be computed. Applying Eqs. (4) and (5) and the measured Ni tracer data a second order equation was obtained from which the tracer diffusion coefficient of Al in Ni3Al at the position of the Kirkendall plane (i.e. at 75 at.% Ni, see Table 1) was calculated. We note here that by using the value Mo=7.15 for a random fcc lattice as proposed by Manning [iii], instead of Mo=4.43 suggested by [2], the (DAVB)/(DBVA) values would decrease by about 7–10% which is within the experimental scatter of the measurements. The obtained self diffusion coefficients can be represented by an Arrhenius plot (Fig. 4) together with the employed Ni tracer diffusion data measured in [15]. The activation energy of Al diffusion in Ni3Al has been calculated using the linear-least square fitting method: þ2:28106 DAl ¼5:05 107 1:11710 7 ð6Þ 243 16 KJ=mol 2 exp m =s RT For comparison, the tracer diffusion coefficients for Ni given in the literature are [15,16]: DNi ¼1:592 104 exp 291 9 KJ=mol 2 m =s RT 303 KJ=mol 2 DNi ¼ 3:59 104 exp m =s RT 5. Discussion Our results are compared with literature data in Fig. 4. The triangle representing the result of Ikeda et al. [2], measured in single phase interdiffusion experiment, fits well to our result although in that work besides the slightly different circumstances the correct value of the thermodynamic factor was also needed. Fujiwara et al. [17] performed intrinsic diffusion measurement using NiAl/Ni diffusion couples. Since volume change has occurred during the reaction they applied the Sauer-Freise method to calculate the interdiffusion coefficients at the Kirkendall plane. They were also able to estimate the self diffusion coefficient of Al in Ni3Al using the Darken-Manning expressions. Their
295
results are more close to the Ni self diffusion coefficients, however their initial materials were different [Ni3Al(38 at.% Al)-pure Ni]. The difference of the starting material can be essential since the measured diffusion profiles might be inaccurate. NiAl exists in a large composition range and Al can dissolve into the Ni solid solution up to about 16 at.% at 1200 C. Measuring the long diffusion tail in both end phases with the desired accuracy is technically difficult. If this is the case, determining the position of the Kirkendall plane in the diffusion zone becomes dangerous. One of the advantages of our method is to avoid this element of uncertainty. Moreover in our calculations, using two-phase initial materials and only the ratio of the intrinsic diffusion coefficients, we did not have to consider either the thermodynamic factor nor volume changes during the diffusion process. It was not necessary to cope with the partial molar volumes as well since these quantities drop out during the calculations applying Eqs. (2) and (4). Our results are in accordance with the theory that in Ni3Al the diffusion of the minor element occurs via the sublattice of the major constituent. The structure of L12 (Cu3Au-type) alloys can be derived from the fcc lattice. Atoms of the major element (Ni) occupy face-centre sites (a sublatice) while those of the minor one (Al) inhabit cube-corner sites (b sublattice). Comparing Ni self-diffusion in Ni3Al and in pure Ni on the normalized temperature scale, the diffusion data coincide with each other almost exactly [18]. This—as well as other experimental evidences [18]—suggests that Ni diffusion occurs via nearest neighbour jumps on its own a sublattice mediated by thermal vacancies. Since next nearest neighbour jump would be improbable for Al a straightforward assumption is supposing the minor element uses the a sublattice as well for diffusion. This idea is supported by theoretical as well as experimental work [3,19]. In thermal equilibrium a small fraction of Al atoms occupy antisite positions decreasing the degree of order in the L12 structure. On the other hand, an exchange of an antisite atom with a vacancy on the nearest neighbour a site would not make further disorder. In this respect the Al diffuses like an impurity atom on the a sublattice i.e. the five jump-frequency model can be employed in which the diffusion coefficient can be written as [19]: 0 2 0 ! 0 DAl ¼ a02 CV 40 !02 fi p : 3 !3
ð7Þ
Note that, according to literature agreement, the prime in the expression refers to data in Ni3Al. Here 2/3 0 is the geometrical factor, a 0 is the lattice parameter, CV is the probability that an a site is vacant, !0 2 is the exchange frequency of a vacancy on the a sublattice with an antisite (in our case Al) atom, !0 3 and !0 4 are the frequencies of the dissociative and associative jumps
296
C. Cserha´ti et al. / Intermetallics 11 (2003) 291–297
Fig. 4. The Al tracer diffusion coefficients are represented in an Arrhenius-plot. For comparison the Ni tracer diffusion data are represented as well. The error of the tracer diffusion coefficients were calculated with the Gauss’s equation for error propagation using the uncertainties of the experimental measurements.
of a vacancy antisite atom pair, f 0 i is the correlation factor for impurity diffusion (Al) in the fcc lattice and pa the occupation probability of a sites. This last factor decreases the diffusivity of the minor element since this factor denotes the fraction of antisite (Al) atoms on the a (Ni) sublattice. The experimentally found activation energy for Al self-diffusion is comparable with that of Ni i.e. the diffusion is almost as easy for Al as for Ni. Even so, Al diffuses 2–5 times slower since the preexponential factor is several orders of magnitude smaller compared to Ni. The small value of Do originates from the low value of pa [see Eq. (7)]. Since Ni3Al is highly ordered up to its melting point (Tm=1668 K) the probability for an Al atom to occupy an antisite position is low. As a consequence the pa occupation probability is small in thermal equilibrium at the temperature range in which the experiments were performed.
6. Conclusions The self diffusion coefficient of Al has been deduced from Kirkendall experiments applying incremental diffusion couples in the temperature range of 900–1200 C.
A comparison with other experimental findings has been made. We illustrated that the present work together with literature data leads to consistent values of the tracer diffusivity of Al. It was demonstrated that these measurements support the theory that in Ni3Al diffusion of the minor element (Al) occurs through a sublattice vacancy mechanism using the Ni sublattice. We have shown that the occupation probability (pa), describing the fraction of antisite atoms, is accountable for the small value of Do. Appendix In order to check the equilibrium composition of the different phases in the given temperature and composition range two different binary alloy has been prepared. The same pure Ni (99.995%) and Al (99.999%) provided by Goodfellow (UK) were used as starting materials as for the diffusion couples. The binary alloys were arc-melted to form Ni0.7225Al0.2775 and Ni0.7896Al0.2104. The alloys were annealed for 200 h at 1100, 1050, 1000 C and 50 h at 1250 C. After standard metallographic preparation the composition of the different Ni–Al phases was measured by EPMA (WDS) using pure element standards. For
C. Cserha´ti et al. / Intermetallics 11 (2003) 291–297
each temperature and phase we measured the composition at least at 20 different places to obtain Table A1 in which the numbers indicate the concentration of Ni in atomic fraction. Table A1 Part of the remeasured Ni–Al phase diagrama Temperature ( C) NiAl (b) Ni3Al (g0 )
NiSolid solution
Al-rich side Ni-rich side 1250 C
62:52þ0:6 0:3
þ0:5 73:460:6
þ0:7 76:890:3
85:22þ0:2 0:4
1100 C
þ0:4 63:19þ0:5 0:7 72:430:3
þ0:3 77:240:4
84:24þ0:5 0:5
1050 C
þ0:5 63:02þ0:3 0:3 72:490:4
þ0:5 77:380:3
84:86þ0:6 0:5
1000 C
þ0:4 62:48þ0:4 0:4 72:570:4
þ0:6 77:190:3
85:16þ0:5 0:8
a The numbers are the Ni concentrations in atomic percent. The numbers are in nice agreement with the diffusion couple measurements and with the phase diagram in [10] as well.
References [1] Mehrer H, editor. Diffusion in solids metals and alloys. Landolt Bo¨rnstein, vol. III/26; Springer; 1990.
297
[2] Ikeda T, Almazouzi A, Numakura H, Koiwa M, Sprengel W, Nakajima H. Acta Mater 1998;46:5396. [3] Numakura H, Ikeda T, Koiwa M, Almazouzi A. Phil Mag A 1998;77:887. [4] Larikov LN, Geichenko VV, Falchenko VM. Diffusion processes in ordered alloys [trans. from Russian by NBS and NSF] New Delhi: Amerind; 1981. p. 117. [5] Darken LS. In: Hollomon JH, editor. Atom movements. Cleveland (OH): ASM; 1951. p. 1. [6] Manning JR. Acta Met 1967;15:817. [7] van Loo FJJ. Prog Solid St Chem 1990;20:47. [8] Janssen MMP. Met Trans 1973;4:1623. [9] Watanabe M, Horita Z, Nemoto M. Defect and Diffusion Forum 1997;143–147:345. [10] Massalski TB, editor. Binary alloy phase diagrams. ASM International; 1992. [11] Hansen M, Anderko K. Constitution of binary alloys. McGrawHill; 1958. [12] Philibert J. Atom movements and mass transport in solids. Les Editions de Physique; 1991. [13] Onsager L. Phys Rev 1931;38:2265. [14] Koiwa M, Ishioka S. Phil Mag A 1983;48:1. [15] Cserha´ti C, Szabo´ IA, Erde´lyi G, Ma´rton Zs. Intermetallics 2002; 10:887. [16] Frank St, So¨dervall U, Herzig Ch. PhysSta Sol (b) 1995;191:45. [17] Fujiwara K, Horita Z. Defect and Diffusion Forum 2001;194199:565. [18] Mehrer H. Mat Trans JIM 1996;37:1259. [19] Numakura H, Yamada T, Koiwa M, Szabo´ IA, Hono K, Sakurai T. Defect and Diffusion Forum 1993;95-98:869.
Intrinsic diffusion in Ni3Al system C. Cserha´tia,*, A. Paulb, A.A. Kodentsovb, M.J.H. van Dalc, F.J.J. van Loob a
Department of Solid State Physics, University of Debrecen, PO Box 2, 4010 Debrecen, Hungary b Laboratory of Solid State and Materials Chemistry, Eindhoven University of Technology, PO Box 513, 5600 MB Eindhoven, The Netherlands c Philips Research Leuven, Kapeldreef 75 B-3001 Leuven, Belgium
Received 24 September 2002; received in revised form 7 November 2002; accepted 7 November 2002
Abstract Interdiffusion experiments were performed between different Ni–Al two-phase alloys. A method is given to estimate the tracer diffusion coefficient of one of the constituents in a binary alloy knowing the tracer diffusion coefficient of the other species. The procedure is based on the so-called Darken-Manning analysis. The tracer diffusion coefficient of Al was determined in this way in Ni3Al: 243 16 KJ=mol 2 þ2:28106 m =s DAl ¼ 5:05 107 1:11710 exp 7 RT Our results are compared with others available in the literature. The values for the self diffusivity of Al calculated from interdiffusion experiments give consistent result. All these measurements support the theory that diffusion of the minor element occurs through a vacancy mechanism in Ni3Al. # 2003 Elsevier Science Ltd. All rights reserved. Keywords: B. Diffusion
1. Introduction Nickel-based intermetallic compounds are still in the focus of interest. Extensive investigations have been carried out on mechanical properties of these materials, with special attention on Ni3Al which is the g0 phase in the binary Ni–Al system. Its high mechanical strength at high temperatures explains the persistent intention for applications. Several processes are driven by diffusion like recrystallization, grain growth and solid state reactions and therefore, knowledge of diffusion is essential for the fabrication of such materials and also for practical use at high temperatures. Diffusion data in Ni3Al, studied by radiotracer and interdiffusion experiments, are summarized in [1] and recently in [2,3]. Tracer measurements concentrated
mainly on the determination of Ni diffusivity because of the lack of a proper Al isotope. Larikov [4] is the only one who reported the tracer coefficient of Al in Ni3Al as almost equal to the Ni diffusivity. This measurement is, however, doubtful since the Ni tracer diffusion data they measured are significantly larger than in other recent reports [1]. The goal of the present study was to determine the self diffusion coefficient of Al in Ni3Al from interdiffusion experiments by applying the Darken–Manning analysis [5,6]. To this end interdiffusion experiments were designed between incremental couples of different Ni–Al two-phase alloys in which Ni3Al formed from its saturated adjacent phases by diffusion.
2. Experimental procedure * Corresponding author. Tel.: +36-52-316073; fax: +36-52316073. E-mail address: cserhati@delfin.klte.hu (C. Cserha´ti).
Pure Ni (99.995%) and Al (99.999%) provided by Goodfellow (UK) were used as starting materials. Four different binary alloys were arc-melted to form two
0966-9795/03/$ - see front matter # 2003 Elsevier Science Ltd. All rights reserved. PII: S0966-9795(02)00235-2
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292
types of diffusion couples: A: Ni65Al35/Ni85Al15 and B: Ni72Al28/Ni78Al22. The alloy ingots were re-melted three times in order to improve their homogeneity. Recrystallization and grain growth were accomplished by annealing in vacuum at 1000 C for 48 h. The couple halves were cut and polished down to 0.25 mm. The parts were cleaned ultrasonically in ethanol. Prior to the annealing small ThO2 particles ( 0.3mm) were introduced at the initial interface as Kirkendall markers. The diffusion couples were heat treated in a vacuum furnace (5107 mbar) under an external load of about 5MPa for different times at various temperatures. During the annealing the temperature was controlled within 2 C. The details of the annealings are given in Table 1. After standard metallographic preparation the heat treated samples were investigated by optical microscope, Scanning Electron Microscope (SEM) and Electron Probe Microanalysis (EPMA).
3. Experimental findings A typical morphology of an annealed diffusion couple can be seen in Fig. 1. One can clearly observe the twophase initial materials, in which the precipitates of the final reaction product were present. The Ni rich side contains Ni rich g0 phase precipitated out from the Ni solid-solution, while the other side is full with Al rich g0 phase precipitates dispersed in the NiAl parent phase. The interface between the end members and the formed Ni3Al is wavy which can be explained by the heterogeneous structure of the initial two phase material. The moving interface of the growing Ni3Al incorporates the precipitates on both sides of the new phase. In the course of this process the interface becomes irregular. This nature of the interface makes it difficult to determine the thickness of the phase and the exact position of the Kirkendall plane although the straight row of white particles in the middle of the grown phase Table 1 Measuring details as well as the composition at the Kirkendall plane is given [NNi(K)] Couple
Temp (K)
Time (h)
NNi(K) (at.%)
(DNiVAl)/(DAlVNi)a
A A A A A A A B B B B
1173 1223 1248 1273 1273 1323 1323 1373 1373 1373 1473
400 196 196 196 196 392 98 196 196 196 196
74.6 75.0 75.0 74.5 74.5 74.9 74.8 74.9 75.3 73.9 76.0
2.3 4.4 5.5 4.0 5.4 6.9 4.2 6.6 2.4 2.4 3.4
a
In the last column the calculated ratio of the intrinsic diffusion coefficients is written.
Fig. 1. Back-scattered electron image of the diffusion zone after annealing at 1000 C for 196 h. The Kirkendall markers (ThO2 particles) are located in the formed Ni3Al phase.
marks it nicely. One clearly sees the different crystal morphology of Ni3Al on both sides of the Kirkendall plane, caused by the different nucleation sites [7]. In order to reduce this trouble the SEM images were processed by computer. The contour of the interfaces were fitted with a polinom. The areas of the reaction product on both sides of the original welding plane were calculated. Summing the two areas gives the average thickness of the reaction layer, while by taking the ratio the position of the Kirkendall plane was determined. In Fig. 2. a typical concentration profile is shown where the Kirkendall plane is indicated as well. As Fig. 2 displays the ThO2 particles are located at the stoichiometric Ni3Al composition within the experimental error (1–2 at.%). The thickness of the reaction product and the position of the Kirkendall plane changes along the sample due to the wavy nature of the interface. To perform the Boltzmann-Matano analysis the concentration profile was measured to determine the magnitude of the concentration step at the interfaces on both sides of the g0 phase. Then the concentration profile was scaled to the calculated average phase thickness (mentioned above). In this way a stepwise linear concentration profile was constructed (see Fig. 2.). Time dependence of the phase growth was measured at two different temperatures (1173 K, 1323 K). Parabolic growth was found which demonstrates the diffusion control of the process. It was reported by Janssen [8] that below 1000 C grain boundary diffusion becomes the dominant process. In order to determine the average grain size in the g0 phase the sample was etched to make the grain structure visible. The average grain size in the new phase was comparable with the thickness of the whole Ni3Al layer. Moreover, the protruding of the interface corresponds nowhere to the grain boundaries. This points to the fact that the importance of grain boundary diffusion was small compared to volume diffusion.
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293
Fig. 2. A typical concentration profile is shown (1000 C, 196 h). It was scaled to the average thickness of the newly formed phase (see the text for the details). The Kirkendall plane (K) is indicated.
The usual Boltzmann-Matano analysis was applied to calculate the interdiffusion coefficients at the Kirkendall plane. At one temperature (1373 K, see Table 1) three experimental runs were performed which enabled us to estimate the error of the mutual diffusion coefficients as the maximal deviation from the average value. The interdiffusion coefficients found at the stoichiometric composition are plotted in Fig. 3 together with data obtained from the literature [8,9]. The authors in these experiments performed multi-phase diffusion measurement, but their starting materials were different. Whilst Janssen [8] used NiAl(55 at.% Ni)-pure Ni diffusion couples, Watanabe [9] measured interdiffusivity in NiAl(75 at.% Ni)-pure Ni system by AEM. The circumstances in our experiments were closer to Janssen’s since the composition of the end members we designed were different from the composition of the reaction product. This might explain that our results are in better agreement with his data. Looking at the measured concentration profile (Fig. 2) one can realize that the composition at the Ni rich and the Al rich side of the Ni3Al phase is different from the concentration of the equilibrium phase diagram taken from the well known book edited by Massalski [10]. To check what the real equilibrium compositions are, another set of experiments was developed in which two phase alloys of known compositions had been equilibrated at the same temperature range where the diffusion annealings were performed. It was found that the composition on both sides of the g-phase we obtained from the diffusion
couples correspond well with these measurements. It means that our result differ with the phase diagram printed in [10], but are in beautiful accordance with the data book of Hansen and Anderko [11] (see Appendix).
4. Determining the diffusivity of Al Although the interdiffusion coefficient is a perfect measure of the redistribution of the components during the diffusion process it gives no information on the relative diffusivities of the species. A more fundamental quantity is the intrinsic diffusivity (Di) which is directly related to the atomic fluxes (Ji) with respect to the lattice planes (Kirkendall frame of reference) via Fick’s first law [12]: JA ¼ DA
@CA DA VB @NA ¼ ; @x V2m @x
ð1Þ
where Ni and Vi is the mole fraction and the partial molar volume of element i respectively, Vm is the molar volume, Ci the composition in moles of component i per m3 and x is the diffusion direction. Since inert ThO2 particles were used to mark the initial welding plane, the intrinsic diffusion coefficient at this position can be calculated. For our purposes it was more interesting to compute the ratio of the intrinsic diffusion coefficients at the Kirkendall plane at position xk [7]:
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Fig. 3. Interdiffusion coefficients are plotted on an Arrhenius-plot together with data obtained from the literature. The interdiffusion coefficients were calculated at the Kirkendall plane, i.e. at the stoichiometric composition (see Fig. 2).
DA VA ¼ DB VB 2 þ 3 Ð Ð NA N þ xk A 1 NA NA dx NA xk dx 7 6 NA 1 Vm Vm 6 7 þ 7 6 Ð Ð 4 5 N N N N A xk A 1 A A dx þ N dx Nþ B 1 B xk Vm Vm ð2Þ N+ i
N i
and indicate the nominal composition of the starting materials, i.e. the composition at the place where no diffusion has occurred yet (see also Fig. 2).The molar volume Vm is considered to be constant in the near-stochiometric compound Ni3Al. In this manner we simplified the problem to area calculations. The ratio VA/VB is difficult to determine in an intermetallic compound and, in fact, one measures the ratio (DAVB)/ (DBVA) which can be found in Table 1. The intrinsic diffusion coefficient moreover is associated to the tracer diffusivities Di* via the Darken– Manning formula [6,12]: Vm @lnaA DA ¼ DA ð 1 þ WA Þ ; ð3Þ @lnNA VB where ai is the chemical activity of component i (the term lnai/lnNi is called the thermodynamic factor)
and Wi is the so called vacancy wind factor. Taking the ratio of the intrinsic diffusion fluxes of both species the advantage of this method becomes clear. In this way we can avoid the difficulties provoked by thermodynamic factors since in binary alloys the thermodynamic factors of the two components are equal. The measured ratio (DAVB)/(DBVA) (see Table 1) can be related to the tracer-diffusion coefficients through: DA VB DA ð1 þ WA Þ : ¼ DB VA DB ð1 WB Þ
ð4Þ
The vacancy wind factor Wi [6] formulates how the net vacancy flux influences the mobility of the diffusing species (i.e. it takes the cross terms into account [12,13]): 2NA DA DB ; WA ¼ Mo NA DA þ NB DB ð5Þ 2NB DA DB WB ¼ Mo NA DA þ NB DB In this equation the correlation effects are considered through Mo. Although Manning’s theory was developed
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for a random solid solution, it has been suggested that it might be applied for L12 compounds where the diffusion of the minor element occurs by ordinary vacancy mechanism over the sublattice of the other species. Employing the statements of Ikeda et al. [2] just as Numakura et al. [3] this assumption seems to be adoptable in our case as well. Employing the proposed mechanism Mo=4.43 was taken [2,14]. Since the tracer coefficient of Ni in Ni3Al is well documented in the literature, the self diffusivity of Al can be computed. Applying Eqs. (4) and (5) and the measured Ni tracer data a second order equation was obtained from which the tracer diffusion coefficient of Al in Ni3Al at the position of the Kirkendall plane (i.e. at 75 at.% Ni, see Table 1) was calculated. We note here that by using the value Mo=7.15 for a random fcc lattice as proposed by Manning [iii], instead of Mo=4.43 suggested by [2], the (DAVB)/(DBVA) values would decrease by about 7–10% which is within the experimental scatter of the measurements. The obtained self diffusion coefficients can be represented by an Arrhenius plot (Fig. 4) together with the employed Ni tracer diffusion data measured in [15]. The activation energy of Al diffusion in Ni3Al has been calculated using the linear-least square fitting method: þ2:28106 DAl ¼5:05 107 1:11710 7 ð6Þ 243 16 KJ=mol 2 exp m =s RT For comparison, the tracer diffusion coefficients for Ni given in the literature are [15,16]: DNi ¼1:592 104 exp 291 9 KJ=mol 2 m =s RT 303 KJ=mol 2 DNi ¼ 3:59 104 exp m =s RT 5. Discussion Our results are compared with literature data in Fig. 4. The triangle representing the result of Ikeda et al. [2], measured in single phase interdiffusion experiment, fits well to our result although in that work besides the slightly different circumstances the correct value of the thermodynamic factor was also needed. Fujiwara et al. [17] performed intrinsic diffusion measurement using NiAl/Ni diffusion couples. Since volume change has occurred during the reaction they applied the Sauer-Freise method to calculate the interdiffusion coefficients at the Kirkendall plane. They were also able to estimate the self diffusion coefficient of Al in Ni3Al using the Darken-Manning expressions. Their
295
results are more close to the Ni self diffusion coefficients, however their initial materials were different [Ni3Al(38 at.% Al)-pure Ni]. The difference of the starting material can be essential since the measured diffusion profiles might be inaccurate. NiAl exists in a large composition range and Al can dissolve into the Ni solid solution up to about 16 at.% at 1200 C. Measuring the long diffusion tail in both end phases with the desired accuracy is technically difficult. If this is the case, determining the position of the Kirkendall plane in the diffusion zone becomes dangerous. One of the advantages of our method is to avoid this element of uncertainty. Moreover in our calculations, using two-phase initial materials and only the ratio of the intrinsic diffusion coefficients, we did not have to consider either the thermodynamic factor nor volume changes during the diffusion process. It was not necessary to cope with the partial molar volumes as well since these quantities drop out during the calculations applying Eqs. (2) and (4). Our results are in accordance with the theory that in Ni3Al the diffusion of the minor element occurs via the sublattice of the major constituent. The structure of L12 (Cu3Au-type) alloys can be derived from the fcc lattice. Atoms of the major element (Ni) occupy face-centre sites (a sublatice) while those of the minor one (Al) inhabit cube-corner sites (b sublattice). Comparing Ni self-diffusion in Ni3Al and in pure Ni on the normalized temperature scale, the diffusion data coincide with each other almost exactly [18]. This—as well as other experimental evidences [18]—suggests that Ni diffusion occurs via nearest neighbour jumps on its own a sublattice mediated by thermal vacancies. Since next nearest neighbour jump would be improbable for Al a straightforward assumption is supposing the minor element uses the a sublattice as well for diffusion. This idea is supported by theoretical as well as experimental work [3,19]. In thermal equilibrium a small fraction of Al atoms occupy antisite positions decreasing the degree of order in the L12 structure. On the other hand, an exchange of an antisite atom with a vacancy on the nearest neighbour a site would not make further disorder. In this respect the Al diffuses like an impurity atom on the a sublattice i.e. the five jump-frequency model can be employed in which the diffusion coefficient can be written as [19]: 0 2 0 ! 0 DAl ¼ a02 CV 40 !02 fi p : 3 !3
ð7Þ
Note that, according to literature agreement, the prime in the expression refers to data in Ni3Al. Here 2/3 0 is the geometrical factor, a 0 is the lattice parameter, CV is the probability that an a site is vacant, !0 2 is the exchange frequency of a vacancy on the a sublattice with an antisite (in our case Al) atom, !0 3 and !0 4 are the frequencies of the dissociative and associative jumps
296
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Fig. 4. The Al tracer diffusion coefficients are represented in an Arrhenius-plot. For comparison the Ni tracer diffusion data are represented as well. The error of the tracer diffusion coefficients were calculated with the Gauss’s equation for error propagation using the uncertainties of the experimental measurements.
of a vacancy antisite atom pair, f 0 i is the correlation factor for impurity diffusion (Al) in the fcc lattice and pa the occupation probability of a sites. This last factor decreases the diffusivity of the minor element since this factor denotes the fraction of antisite (Al) atoms on the a (Ni) sublattice. The experimentally found activation energy for Al self-diffusion is comparable with that of Ni i.e. the diffusion is almost as easy for Al as for Ni. Even so, Al diffuses 2–5 times slower since the preexponential factor is several orders of magnitude smaller compared to Ni. The small value of Do originates from the low value of pa [see Eq. (7)]. Since Ni3Al is highly ordered up to its melting point (Tm=1668 K) the probability for an Al atom to occupy an antisite position is low. As a consequence the pa occupation probability is small in thermal equilibrium at the temperature range in which the experiments were performed.
6. Conclusions The self diffusion coefficient of Al has been deduced from Kirkendall experiments applying incremental diffusion couples in the temperature range of 900–1200 C.
A comparison with other experimental findings has been made. We illustrated that the present work together with literature data leads to consistent values of the tracer diffusivity of Al. It was demonstrated that these measurements support the theory that in Ni3Al diffusion of the minor element (Al) occurs through a sublattice vacancy mechanism using the Ni sublattice. We have shown that the occupation probability (pa), describing the fraction of antisite atoms, is accountable for the small value of Do. Appendix In order to check the equilibrium composition of the different phases in the given temperature and composition range two different binary alloy has been prepared. The same pure Ni (99.995%) and Al (99.999%) provided by Goodfellow (UK) were used as starting materials as for the diffusion couples. The binary alloys were arc-melted to form Ni0.7225Al0.2775 and Ni0.7896Al0.2104. The alloys were annealed for 200 h at 1100, 1050, 1000 C and 50 h at 1250 C. After standard metallographic preparation the composition of the different Ni–Al phases was measured by EPMA (WDS) using pure element standards. For
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each temperature and phase we measured the composition at least at 20 different places to obtain Table A1 in which the numbers indicate the concentration of Ni in atomic fraction. Table A1 Part of the remeasured Ni–Al phase diagrama Temperature ( C) NiAl (b) Ni3Al (g0 )
NiSolid solution
Al-rich side Ni-rich side 1250 C
62:52þ0:6 0:3
þ0:5 73:460:6
þ0:7 76:890:3
85:22þ0:2 0:4
1100 C
þ0:4 63:19þ0:5 0:7 72:430:3
þ0:3 77:240:4
84:24þ0:5 0:5
1050 C
þ0:5 63:02þ0:3 0:3 72:490:4
þ0:5 77:380:3
84:86þ0:6 0:5
1000 C
þ0:4 62:48þ0:4 0:4 72:570:4
þ0:6 77:190:3
85:16þ0:5 0:8
a The numbers are the Ni concentrations in atomic percent. The numbers are in nice agreement with the diffusion couple measurements and with the phase diagram in [10] as well.
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