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Microstructural simulation of phase decomposition in Cu–Ni alloys
Journal of Alloys and Compounds 460 (2008) 206–212
Microstructural simulation of phase decomposition in Cu–Ni alloys Erika O. Avila-Davila, Victor Lopez-Hirata ∗ , Maribel L. Saucedo-Mu˜noz, Jorge L. Gonzalez-Velazquez Instituto Politecnico Nacional, Metallurgy, Apartado Postal 118-556, 07051 Mexico, D.F., Mexico Received 29 March 2007; received in revised form 18 May 2007; accepted 18 May 2007 Available online 24 May 2007
Abstract The microstructure simulation of spinodal decomposition was carried out in the aged Cu–70 and 90 at.% Ni alloys, based on a solution of the non-linear Cahn–Hilliard partial differential equation by the finite difference method. The calculated concentration profiles were compared with the experimental ones determined by atom-probe field ion microscope analyses of the solution treated and aged Cu–70 at.% Ni alloy samples. Both the numerically simulated and experimental results showed a good agreement for the concentration profiles and microstructure evolution in the aged Cu–Ni alloys. A very slow growth kinetics of phase decomposition was observed to occur in this type of alloy. The morphology of decomposed phases consists of an irregular shape with no preferential alignment in any crystallographic direction. The wavelength of composition modulation was determined numerically to be about 2 nm and it remained constant after aging at 573 K for times as long as 8889 h. No phase decomposition was observed to occur for the numerical simulation of aging at temperatures lower than 523 K for a time as long as 1 year. © 2007 Elsevier B.V. All rights reserved. Keywords: Metals and alloys; Precipitation; Computer simulation
1. Introduction The miscibility gap in the Cu–Ni alloy system was predicted to appear at temperatures lower than 600 K based on thermodynamical data [1]. Several works have been carried out in order to confirm experimentally the spinodal decomposition in these alloys by different experimental techniques. Nevertheless, it has been difficult to verify its presence because of the low diffusivity at temperatures lower than 600 K [2]. Phase decomposition in this type of alloys has been studied using different experimental techniques such as, electrical resistivity [3], X-ray diffraction [4] and neutron scattering [5,6]. These works only reported the evidence of a clustering tendency in Cu–Ni alloys. The use of electron-irradiated enabled to enhance the diffusive process and to estimate the coherent spinodal temperature using small angle neutron scattering (SANS). However, the results were contrary to those expected from the spinodal decomposition theory proposed by Cahn–Hilliard [7]. This anomalous behavior was attributed
∗
Corresponding author. Tel.: +52 55 57296000; fax: +52 55 51191986. E-mail address: [email protected] (V. Lopez-Hirata).
0925-8388/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.jallcom.2007.05.070
to the more complicated phase decomposition kinetics in electron-irradiated samples because of the change in the excess defect concentration during aging [6]. Lopez-Hirata et al. [8] analyzed long-aged Cu–70 at.% Ni alloy samples using atomprobe field ion microscopy. They observed a slight increase in the composition modulation amplitude with time and thus, concluded that the phase decomposition took place spinodally. The phase continuum method has been applied successfully for the microstructure simulation of different phase transformations in different alloy systems [9]. For instance, Honjo and Saito [10] used a solution of the Cahn–Hilliard non-linear equation for the numerical simulation of the phase decomposition in Fe–Cr and Fe–Cr–Mo alloys. They reported a very good agreement between the numerically simulated results and the experimental ones. There is no report in the literature for the numerical microstructural simulation of phase decomposition in Cu–Ni alloys. Besides, the numerical analysis would be useful to analyze the kinetics of microstructural evolution in this alloy system because of its slow diffusive process. Thus, the purpose of this work was to carry out the numerical simulation of phase decomposition in the binary Cu–Ni alloy system and these results were compared with those determined
E.O. Avila-Davila et al. / Journal of Alloys and Compounds 460 (2008) 206–212
experimentally in the aged Cu–Ni alloys with an atom-probe field ion microscope (AP-FIM). 2. Numerical formulation The Cahn–Hilliard non-linear equation for a multicomponent system with a constant mobility can be reduced to the following equation [10]: ∂ci (x, t) 2 ∂fo (c) 2 (1) = Mi ∇ + −Ki ∇ ci ∂t ∂ci where ci (x, t) is the concentration as a function of distance x and time t, Mi the atomic mobility, fo the local free energy and Ki is the gradient energy coefficient. The local energy fo was defined using the regular solution model as follows [10]: fo = fCu cCu + fNi cNi + ΩCu−Ni cCu cNi +RT (cCu ln cCu + cNi ln cNi )
(2)
where R is the gas constant and T is the absolute temperature. fCu and fNi are the molar free energy of pure Cu and Ni, respectively, and ΩCu–Ni is the interaction parameter. The atomic mobility Mi is related to the interdiffusion coef−
207
The lattice, diffusion, thermodynamic and elastic constant for the microstructural simulation in the Cu–Ni alloys were taken from references [2,11–13] and they are shown in Table 1. The effect of coherency elastic strain energy was taken into account in the phase decomposition of Cu–Ni alloys in spite of the similar lattice parameters of copper and nickel [6], and thus a small strain energy would be expected; nevertheless, it might affect the growth kinetics of phase decomposition. This energy was introduced into Eq. (1), according to the simple definition proposed by Hilliard [7]: (7) fel = A η2 Y (c − c0 )2 dx where A is the cross-sectional area, Y the elastic constant defined by the elastic stiffness constants, and c11 , c12 and c44 for the Cuand Ni-rich phases. The parameter η is equal to d ln a/dc. In the case of fcc metals, the elastic energy will be a minimum for the 1 0 0 directions and thus, the Y value can be assumed similar to that corresponding to an isotropic material [7]: 2 c12 Y1 0 0 = c11 + c12 − 2 (8) c11 The elastic constants, cij , were defined as follows:
ficient Di as follows: 2 − ∂ fo D i = Mi ∂ci 2
(3) −
The interdiffusion coefficient Di was defined as follows [7]: −
Di = DNi cCu + (1 − cCu )DCu
(4)
The Cahn–Hilliard non-linear equation can be solved by the finite difference method [9]. Besides, the local free energy fo was defined by using the regular solution model. The gradient energy coefficient K was defined as proposed by Hilliard [7]: 2 K = 23 hM (5) 0.5 r0 where hM 0.5 is the heat of mixing per unit volume at c = 0.5 and ro is the nearest-neighbor distance. The heat of mixing hM was determined according to the following equation [10]: hM = cCu cNi ΩCu–Ni
(6)
cij = cijCu cCu + cijNi (1 − cCu )
(9)
Considering the elastic strain energy, fel , Eq. (1) was rewritten as follows: ∂fel ∂ci (x, t) 2 ∂fo (c) 2 (10) + − K i ∇ ci = Mi ∇ ∂t ∂ci ∂ci The microstructural simulation was carried out using the finite difference method with 101 × 101 points-square grit with a mesh size of 0.25 nm and a time-step size of 10 s. The simulations were performed for the Cu–70 and 90 at.% Ni alloys at temperatures between 523 and 595 K for different times. These compositions were selected because the maximum of the miscibility gap is located at about 70 at.% Ni and the other composition corresponds to an asymmetrical alloy, which can be used for comparison of the morphology and kinetics of the phase decomposition.
Table 1 Values of lattice, diffusion, thermodynamic and elastic constants Constant
Cu–Ni alloys
Lattice parameter (nm) Diffusion coefficient ΩCu–Ni (J mol−1 ) cij
(J m−3 )
η (nm)
Cu/Ni
(cm2 s−1 )
0.360 [11] Cu 1.5–2.3 exp(−230,000 − 260,000 J mol−1 )/RT [2] Ni 17–35 exp(−270,000 − 300,000 J mol−1 )/RT [2] (8366.0 + 2.802T) + (−4359.6 + 1.812T)(cCu– cNi ) [12] c11 = 16.84 × 1010 c12 = 12.14 × 1010 c44 = 7.54 × 1010 0.0016 [11]
c11 = 24.65 × 1010 [13] c12 = 14.73 × 1010 c44 = 12.47 × 1010
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3. Experimental procedure The Cu–70 and 90 at.% Ni alloys were prepared by vacuum-melting of pure elements in an alumina crucible. These alloys were homogenized at 1273 K for 98 h and then hot forged, swaged and drawn into wires of 0.3 mm diameter, using intermediate annealing treatments. Samples were solution treated at 1173 K for 1 h and subsequently quenched in water. The aging treatments were performed at temperatures between 573 and 595 K for times from 10 to 200 h. AP-FIM specimens were prepared by electropolishing in a 70 vol.% nitric acid solution with 12–15 V (dc). FIM observation of samples was performed at 20 and 115 K with Ne gas. The atom probe analysis of samples was only carried out for the Cu–70 at.% Ni alloy using an energy compensated time-of-flight AP-FIM at 40 K under a vacuum of about 4 × 10−8 Pa.
4. Results and discussion 4.1. Concentration profiles It is important to mention that in the case of simulated results, the amplitude of the composition modulation showed practically no increase in the case of the sample aged at 523 K for 8889 h (more than 1 year), compared with the concentration profile corresponding to the solution treated sample. Fig. 1 shows the numerically calculated plots of Cu concentration, cCu , versus distance, x, for the Cu–70 at.% Ni alloy solution treated (0 h), and aged at 573 K for different times. It can be seen that there is an increase in the modulation amplitude with aging time. The increase in amplitude is more evident in the sample aged at 595 K for 83 h, Fig. 2, than that corresponding to the sample aged at 573 K for 222 h. These increases in amplitude at 573 and 595 K confirmed that the phase decomposition took place spinodally in this alloy since it has been reported [7] that the increase in modulation amplitude with aging time is a basic characteristic of the spinodal decomposition mechanism. Besides, this result also verifies that the supersaturate solid solution decomposes into a mixture of Cu- and Ni-rich phases, as expected in the equilibrium Cu–Ni phase diagram [14]. The long simulated aging times also confirm that the kinetics of phase decomposition is very slow in this alloy system.
Fig. 1. Calculated plots of copper concentration, cCu , vs. distance, x, for the Cu–70 at.% Ni alloy aged at 573 K for 0, 222, 278, 361 and 444 h.
Fig. 2. Calculated plots of copper concentration, cCu , vs. distance, x, for the Cu–70 at.% Ni alloy aged at 595 K for 0, 83, 97, 111 and 125 h.
The numerically calculated concentration profiles of the Cu–90 at.% Ni alloy aged at 595 K for different times are shown in Fig. 3. It is evident the increase in amplitude for the sample aged at 595 K for 222 h. It was also detected than there was almost no increase in the modulation amplitude with the increase in aging time for the sample aged at 573 K for 8889 h. By comparison of the concentration profiles at 595 K for these two alloy compositions, it seems to be evident that the kinetics of phase decomposition of the Cu–70 at.% Ni alloy is faster than that corresponding to the Cu–90 at.% Ni alloy since, for instance, the increase in amplitude for the aging at 595 K for the Cu–70 at.% Ni alloy occurred after aging for 83 h and that for the Cu–90 at.% Ni alloy took place after aging for 222 h. The difference of the phase decomposition behavior for these two compositions can be analyzed using the theory of Cahn–Hilliard for spinodal decomposition [7]. A particular solution of the modified diffusion equation is: c − c0 = A(β, t) exp(iβx) dβ (11)
Fig. 3. Calculated plots of copper concentration, cCu , vs. distance, x, for the Cu–90 at.% Ni alloy aged at 595 K for 0, 222, 278, 361 and 444 h.
E.O. Avila-Davila et al. / Journal of Alloys and Compounds 460 (2008) 206–212
209
where c0 is the average composition and A(β, t) is the amplitude of the Fourier component of wave number β at time t (β = 2π/λ, where λ is the wavelength). This amplitude is defined in terms of the initial amplitude at time zero as follows: A(β, t) = A(β, 0) exp(R(β)t)
(12)
The amplification factor R(β) is given by: R(β) = −
M NV
∂2 f 2 2 β2 + 2η Y + 2Kβ ∂c2
(13)
where NV is the number of atoms per unit volume. The sign of R(β) is determined by that of the second partial derivative of f with respect to c, f . Inside the spinodal f < 0 and R(β) > 0 for all values of β. Thus, any composition modulation will grow. The magnitude of f will be the largest for an alloy composition located at the center of the spinodal and thus, the amplitude of composition modulation will be higher than that corresponding to an alloy composition close to the spinodal line. The first alloy has been named as symmetric alloy and the other one designated as asymmetric alloy [6]. In this work, the Cu–70 at.% Ni alloy is a symmetric alloy and thus the amplitude is higher than that of the asymmetric Cu–90 at.% Ni alloy. The increase in aging temperature also increases the mobility M and thus it promotes an increase in the modulation amplitude. In general, it could be stated that the kinetics of phase decomposition for the symmetric alloy would be faster than that of the asymmetric alloy since the R(β) is higher in the former case and it would cause a higher modulation amplitude for the same aging time, according to Eqs. (12) and (13).
Fig. 4. AP-FIM concentration profile for the Cu–70 at.% Ni alloy (a) solution treated and (b) aged at 573 K for 2000 h.
The AP-FIM experimental concentration profiles of the Cu–70 at.% Ni alloy, solution treated and aged at 573 K for 2000 h, are shown in Fig. 4a and b, respectively. It can be also seen a slight increase in the modulation amplitude with aging
Fig. 5. Simulated microstructure for the Cu–70 at.% Ni alloy aged at 573 K for (a) 0 h, (b) 222 h, (c) 278 h, (d) 361 h and (e) 444 h.
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time. This slight increase also confirmed the phase decomposition took place via spinodal decomposition in this alloy system. 4.2. Microstructural evolution The simulated microstructures of the Cu–70 at.% Ni alloy aged at 573 K for 0 (solution treated), 222, 278, 361 and 444 h are shown in Fig. 5a–e, respectively. The black and white regions correspond to the Cu- and Ni-rich phases, respectively. It can be seen that the morphology of the decomposed phases is irregular and interconnected. The volume fraction of the Ni-rich phase increased with aging time. A similar morphology of the decomposed phases was also observed for the simulated microstructure of the Cu–90 at.% Ni alloy after solution treating and aging at 595 K for 0, 222, 278, 361 and 444 h, as shown in Fig. 6a–e. The volume fraction of Ni-rich phase increased with aging time. The volume fraction of the Cu-rich phase (black zones) corresponding to the aged Cu–70 at.% Ni alloy was higher than that of the aged Cu–90 at.% Ni, as expected from the lever rule for these two alloy compositions. This type of microstructure has been named isotropic [6] and it has no preferential alignment in a specific crystallographic direction. In contrast, the microstructure of decomposed phases, which shows an alignment in the 1 0 0 crystallographic direction in order to reduce the coherency strain energy (2η2 Y) are known as anisotropic [6]. In the case of Cu–Ni alloys, the parameter η is about 0.0016 [11] and thus the contribution of the elastic coherent strain energy to the phase decomposition process is very small. That
is, the elastic strain energy affects in an isotropic manner to the decomposition process. Therefore, it has practically no effect on the morphology of the decomposed phases. The size of the decomposed phase is very small, about 2 nm, for the aging process of both alloy compositions and it seems to be reasonable for the slow diffusive process in this alloy system. The experimental Ne gas FIM images of the Cu–70 at.% Ni alloy aged at 573 K for 500 h and Cu–90 at.% Ni alloy aged at 595 K for 500 h are shown in Fig. 7a and b, respectively. The figures on the right side correspond to an enlargement of the decomposed phases. FIM observation was carried out at about 115 K to produce a larger difference in the field evaporation rate between the expected Cu- and Ni-rich phases [15]. This difference in evaporation rate is expected to image the Nirich phase brightly and the Cu-rich phase darkly. Fig. 7a and b resembles the FIM image obtained from spinodally decomposed Fe–Cr alloys [16]. In this type of alloys, there was no preferential alignment in a specific crystallographic direction due to the small coherency strain energy. That is, an isotropic spinodal decomposition was present during aging of these alloys. This type of phase decomposition is also expected to occur in the aging of Cu–Ni alloys because of the similar lattice parameters of copper and nickel. FIM observation of samples at 20 K showed only the regular concentric ring pattern in which no evidence of the phase decomposition could be observed. The experimental results of morphology and no preferential alignment for the decomposed phases showed, in general, a good agreement with the simulated microstructure for both the aged Cu–70 and 90 at.% Ni alloys.
Fig. 6. Simulated microstructure for the Cu–90 at.% Ni alloy aged at 595 K for (a) 0 h, (b) 222 h, (c) 278 h, (d) 361 h and (e) 444 h.
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211
Fig. 7. FIM images of the (a) Cu–70 at.% Ni alloy aged at 573 K for 500 h and (b) Cu–90 at.% Ni alloy aged at 595 K for 500 h.
4.3. Growth kinetics of phase decomposition The wavelength of the composition modulation can be determined by means of the analysis of calculated concentration profiles of Figs. 1–3 using the autocorrelation or structure function analyses [15,17]. Fig. 8 shows the variation of wavelength for the composition modulation as a function of aging time for the Cu–70 and 90 at.% Ni alloys aged at 595 K. It can be observed that the wavelength remains almost constant, 2.00 and 2.02 nm for the Cu–70 and 90 at.% Ni alloys, respectively, with aging time. This is a characteristic behavior observed during the early stages of aging in spinodally decomposed alloys [7,17]. Likewise, it can be seen that the wavelength for the aged Cu–90 at.% Ni alloy is slightly longer that that corresponding to the aged Cu–70 at.% Ni alloy. According to the Cahn–Hilliard theory of spinodal decomposition [7], the amplification factor R(β) is positive for 0 < β > βc with: 2 2π ∂ f/∂c2 βc = (14) = λc K The analysis of Eq. (14) suggests that the wavelength of composition modulation is shorter for a symmetric alloy and as the
aging temperature decreases, since f increases in these two latter cases. This fact is in good agreement with the results observed in Fig. 7. The autocorrelation analysis of the experimental AP-FIM concentration profiles of Fig. 4 indicated a value of wavelength
Fig. 8. Plots of composition modulation wavelength, λ, vs. aging time, t, at 595 K for the computer simulation of Cu–70 and 90 at.% Ni alloys.
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for composition modulation of about 2 nm, which is in agreement with the value order for wavelength determined from the numerical simulation, Fig. 8. These short wavelength values confirmed that the growth kinetics of phase decomposition is very slow in Cu–Ni alloys because of the low atomic diffusive process at temperatures lower than 600 K. The phase decomposition of Cu–Ni alloys was also simulated using the linearized Cahn–Hilliard equation [7] and the simulated results were practically the same that those obtained with the non-linear equation. This fact seems to suggest that the growth kinetics of phase decomposition is very slow and thus its evolution is in the early stages of phase decomposition in spite of the prolonged aging times. Besides, it was also observed that the elastic coherency-strain energy caused that the phase decomposition kinetics was slower compared with the simulated results determined with the non-linear Cahn–Hilliard equation without considering the elastic strain energy.
Acknowledgements The authors wish to thank the financial support from CGPI-COFAA-IPN and Fondo Sectorial para la EducacionCONACYT 47151. References [1] [2] [3] [4] [5] [6]
[7] [8] [9]
5. Conclusions The simulation of phase decomposition was carried out in the Cu–Ni alloy system using the solution of the non-linear Cahn–Hilliard partial differential equation by the finite difference method. The calculated results showed a good agreement with the experimental ones. The growth kinetics rate of phase decomposition was very slow in the aged Cu–Ni alloys. The phase decomposition kinetics of the symmetric alloy was faster than that corresponding to the asymmetric alloy. The morphology of the decomposed phases is similar to that reported in isotropically decomposed alloys.
[10] [11] [12] [13] [14] [15] [16]
[17]
M. Larrain, Can. Metall. Q. 18 (1979) 401–406. K. Monma, H. Suto, H. Oikawa, Bull. Jpn. Inst. Met. 4 (1984) 4–9. M.F. Ebel, Phys. Stat. Sol. A 5 (1971) 91–94. T. Tsakalakos, Scripta Metall. 15 (1981) 255–258. W. Wagner, R. Poerschke, H. Wollenberger, J. Phys. F Met. Phys. 12 (1982) 405–424. R. Poerschke, D. Schwan, in: P. Haasen (Ed.), Decomposition of alloys: the early stage, 2nd Acta-Scripta Met. Conf., Acta Scripta. USA, 1983, pp. 180-184. J.E. Hilliard, Phase Transformations, ASM International, Metals Park, OH, USA, 1970. V.M. Lopez-Hirata, T. Sakurai, K. Hirano, Scripta Metall. Mater. 26 (1992) 99–103. E.N. Kaufmann, Encyclopedia of Materials Characterization, vol. 1, Wiley Interscience, 2003. M. Honjo, Y. Saito, ISIJ Int. 40 (2000) 914–919. W.B. Pearson, A Handbook of Lattice Spacings and Structures of Metals and Alloys, ASM International, Metals Park, OH, USA, 1986. T. Ikoma, M. Kajihara, Mater. Sci. Eng. A 437 (2006) 293–300. G.E. Dieter, Mechanical Metallurgy, McGraw Hill, New York, NY, 2001. H. Baker, Alloy Phase Diagrams Handbook, vol. 3, ASM International, Materials Park, OH, USA, 1992. M.K. Miller, G.D.W. Smith, Atom Probe Analysis: Principles and Applications to Materials Problems, MRS, USA, 1989. S.S. Brenner, M.K. Miller, W.A. Soffa, Proceedings of the International Conference on Phase Transformations in Solids, Crete, Greece, 1983, pp. 309–316. G. Kostorz, Phase Transformations in Materials, Wiley-VCH, Germany, 2003.
Microstructural simulation of phase decomposition in Cu–Ni alloys Erika O. Avila-Davila, Victor Lopez-Hirata ∗ , Maribel L. Saucedo-Mu˜noz, Jorge L. Gonzalez-Velazquez Instituto Politecnico Nacional, Metallurgy, Apartado Postal 118-556, 07051 Mexico, D.F., Mexico Received 29 March 2007; received in revised form 18 May 2007; accepted 18 May 2007 Available online 24 May 2007
Abstract The microstructure simulation of spinodal decomposition was carried out in the aged Cu–70 and 90 at.% Ni alloys, based on a solution of the non-linear Cahn–Hilliard partial differential equation by the finite difference method. The calculated concentration profiles were compared with the experimental ones determined by atom-probe field ion microscope analyses of the solution treated and aged Cu–70 at.% Ni alloy samples. Both the numerically simulated and experimental results showed a good agreement for the concentration profiles and microstructure evolution in the aged Cu–Ni alloys. A very slow growth kinetics of phase decomposition was observed to occur in this type of alloy. The morphology of decomposed phases consists of an irregular shape with no preferential alignment in any crystallographic direction. The wavelength of composition modulation was determined numerically to be about 2 nm and it remained constant after aging at 573 K for times as long as 8889 h. No phase decomposition was observed to occur for the numerical simulation of aging at temperatures lower than 523 K for a time as long as 1 year. © 2007 Elsevier B.V. All rights reserved. Keywords: Metals and alloys; Precipitation; Computer simulation
1. Introduction The miscibility gap in the Cu–Ni alloy system was predicted to appear at temperatures lower than 600 K based on thermodynamical data [1]. Several works have been carried out in order to confirm experimentally the spinodal decomposition in these alloys by different experimental techniques. Nevertheless, it has been difficult to verify its presence because of the low diffusivity at temperatures lower than 600 K [2]. Phase decomposition in this type of alloys has been studied using different experimental techniques such as, electrical resistivity [3], X-ray diffraction [4] and neutron scattering [5,6]. These works only reported the evidence of a clustering tendency in Cu–Ni alloys. The use of electron-irradiated enabled to enhance the diffusive process and to estimate the coherent spinodal temperature using small angle neutron scattering (SANS). However, the results were contrary to those expected from the spinodal decomposition theory proposed by Cahn–Hilliard [7]. This anomalous behavior was attributed
∗
Corresponding author. Tel.: +52 55 57296000; fax: +52 55 51191986. E-mail address: [email protected] (V. Lopez-Hirata).
0925-8388/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.jallcom.2007.05.070
to the more complicated phase decomposition kinetics in electron-irradiated samples because of the change in the excess defect concentration during aging [6]. Lopez-Hirata et al. [8] analyzed long-aged Cu–70 at.% Ni alloy samples using atomprobe field ion microscopy. They observed a slight increase in the composition modulation amplitude with time and thus, concluded that the phase decomposition took place spinodally. The phase continuum method has been applied successfully for the microstructure simulation of different phase transformations in different alloy systems [9]. For instance, Honjo and Saito [10] used a solution of the Cahn–Hilliard non-linear equation for the numerical simulation of the phase decomposition in Fe–Cr and Fe–Cr–Mo alloys. They reported a very good agreement between the numerically simulated results and the experimental ones. There is no report in the literature for the numerical microstructural simulation of phase decomposition in Cu–Ni alloys. Besides, the numerical analysis would be useful to analyze the kinetics of microstructural evolution in this alloy system because of its slow diffusive process. Thus, the purpose of this work was to carry out the numerical simulation of phase decomposition in the binary Cu–Ni alloy system and these results were compared with those determined
E.O. Avila-Davila et al. / Journal of Alloys and Compounds 460 (2008) 206–212
experimentally in the aged Cu–Ni alloys with an atom-probe field ion microscope (AP-FIM). 2. Numerical formulation The Cahn–Hilliard non-linear equation for a multicomponent system with a constant mobility can be reduced to the following equation [10]: ∂ci (x, t) 2 ∂fo (c) 2 (1) = Mi ∇ + −Ki ∇ ci ∂t ∂ci where ci (x, t) is the concentration as a function of distance x and time t, Mi the atomic mobility, fo the local free energy and Ki is the gradient energy coefficient. The local energy fo was defined using the regular solution model as follows [10]: fo = fCu cCu + fNi cNi + ΩCu−Ni cCu cNi +RT (cCu ln cCu + cNi ln cNi )
(2)
where R is the gas constant and T is the absolute temperature. fCu and fNi are the molar free energy of pure Cu and Ni, respectively, and ΩCu–Ni is the interaction parameter. The atomic mobility Mi is related to the interdiffusion coef−
207
The lattice, diffusion, thermodynamic and elastic constant for the microstructural simulation in the Cu–Ni alloys were taken from references [2,11–13] and they are shown in Table 1. The effect of coherency elastic strain energy was taken into account in the phase decomposition of Cu–Ni alloys in spite of the similar lattice parameters of copper and nickel [6], and thus a small strain energy would be expected; nevertheless, it might affect the growth kinetics of phase decomposition. This energy was introduced into Eq. (1), according to the simple definition proposed by Hilliard [7]: (7) fel = A η2 Y (c − c0 )2 dx where A is the cross-sectional area, Y the elastic constant defined by the elastic stiffness constants, and c11 , c12 and c44 for the Cuand Ni-rich phases. The parameter η is equal to d ln a/dc. In the case of fcc metals, the elastic energy will be a minimum for the 1 0 0 directions and thus, the Y value can be assumed similar to that corresponding to an isotropic material [7]: 2 c12 Y1 0 0 = c11 + c12 − 2 (8) c11 The elastic constants, cij , were defined as follows:
ficient Di as follows: 2 − ∂ fo D i = Mi ∂ci 2
(3) −
The interdiffusion coefficient Di was defined as follows [7]: −
Di = DNi cCu + (1 − cCu )DCu
(4)
The Cahn–Hilliard non-linear equation can be solved by the finite difference method [9]. Besides, the local free energy fo was defined by using the regular solution model. The gradient energy coefficient K was defined as proposed by Hilliard [7]: 2 K = 23 hM (5) 0.5 r0 where hM 0.5 is the heat of mixing per unit volume at c = 0.5 and ro is the nearest-neighbor distance. The heat of mixing hM was determined according to the following equation [10]: hM = cCu cNi ΩCu–Ni
(6)
cij = cijCu cCu + cijNi (1 − cCu )
(9)
Considering the elastic strain energy, fel , Eq. (1) was rewritten as follows: ∂fel ∂ci (x, t) 2 ∂fo (c) 2 (10) + − K i ∇ ci = Mi ∇ ∂t ∂ci ∂ci The microstructural simulation was carried out using the finite difference method with 101 × 101 points-square grit with a mesh size of 0.25 nm and a time-step size of 10 s. The simulations were performed for the Cu–70 and 90 at.% Ni alloys at temperatures between 523 and 595 K for different times. These compositions were selected because the maximum of the miscibility gap is located at about 70 at.% Ni and the other composition corresponds to an asymmetrical alloy, which can be used for comparison of the morphology and kinetics of the phase decomposition.
Table 1 Values of lattice, diffusion, thermodynamic and elastic constants Constant
Cu–Ni alloys
Lattice parameter (nm) Diffusion coefficient ΩCu–Ni (J mol−1 ) cij
(J m−3 )
η (nm)
Cu/Ni
(cm2 s−1 )
0.360 [11] Cu 1.5–2.3 exp(−230,000 − 260,000 J mol−1 )/RT [2] Ni 17–35 exp(−270,000 − 300,000 J mol−1 )/RT [2] (8366.0 + 2.802T) + (−4359.6 + 1.812T)(cCu– cNi ) [12] c11 = 16.84 × 1010 c12 = 12.14 × 1010 c44 = 7.54 × 1010 0.0016 [11]
c11 = 24.65 × 1010 [13] c12 = 14.73 × 1010 c44 = 12.47 × 1010
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3. Experimental procedure The Cu–70 and 90 at.% Ni alloys were prepared by vacuum-melting of pure elements in an alumina crucible. These alloys were homogenized at 1273 K for 98 h and then hot forged, swaged and drawn into wires of 0.3 mm diameter, using intermediate annealing treatments. Samples were solution treated at 1173 K for 1 h and subsequently quenched in water. The aging treatments were performed at temperatures between 573 and 595 K for times from 10 to 200 h. AP-FIM specimens were prepared by electropolishing in a 70 vol.% nitric acid solution with 12–15 V (dc). FIM observation of samples was performed at 20 and 115 K with Ne gas. The atom probe analysis of samples was only carried out for the Cu–70 at.% Ni alloy using an energy compensated time-of-flight AP-FIM at 40 K under a vacuum of about 4 × 10−8 Pa.
4. Results and discussion 4.1. Concentration profiles It is important to mention that in the case of simulated results, the amplitude of the composition modulation showed practically no increase in the case of the sample aged at 523 K for 8889 h (more than 1 year), compared with the concentration profile corresponding to the solution treated sample. Fig. 1 shows the numerically calculated plots of Cu concentration, cCu , versus distance, x, for the Cu–70 at.% Ni alloy solution treated (0 h), and aged at 573 K for different times. It can be seen that there is an increase in the modulation amplitude with aging time. The increase in amplitude is more evident in the sample aged at 595 K for 83 h, Fig. 2, than that corresponding to the sample aged at 573 K for 222 h. These increases in amplitude at 573 and 595 K confirmed that the phase decomposition took place spinodally in this alloy since it has been reported [7] that the increase in modulation amplitude with aging time is a basic characteristic of the spinodal decomposition mechanism. Besides, this result also verifies that the supersaturate solid solution decomposes into a mixture of Cu- and Ni-rich phases, as expected in the equilibrium Cu–Ni phase diagram [14]. The long simulated aging times also confirm that the kinetics of phase decomposition is very slow in this alloy system.
Fig. 1. Calculated plots of copper concentration, cCu , vs. distance, x, for the Cu–70 at.% Ni alloy aged at 573 K for 0, 222, 278, 361 and 444 h.
Fig. 2. Calculated plots of copper concentration, cCu , vs. distance, x, for the Cu–70 at.% Ni alloy aged at 595 K for 0, 83, 97, 111 and 125 h.
The numerically calculated concentration profiles of the Cu–90 at.% Ni alloy aged at 595 K for different times are shown in Fig. 3. It is evident the increase in amplitude for the sample aged at 595 K for 222 h. It was also detected than there was almost no increase in the modulation amplitude with the increase in aging time for the sample aged at 573 K for 8889 h. By comparison of the concentration profiles at 595 K for these two alloy compositions, it seems to be evident that the kinetics of phase decomposition of the Cu–70 at.% Ni alloy is faster than that corresponding to the Cu–90 at.% Ni alloy since, for instance, the increase in amplitude for the aging at 595 K for the Cu–70 at.% Ni alloy occurred after aging for 83 h and that for the Cu–90 at.% Ni alloy took place after aging for 222 h. The difference of the phase decomposition behavior for these two compositions can be analyzed using the theory of Cahn–Hilliard for spinodal decomposition [7]. A particular solution of the modified diffusion equation is: c − c0 = A(β, t) exp(iβx) dβ (11)
Fig. 3. Calculated plots of copper concentration, cCu , vs. distance, x, for the Cu–90 at.% Ni alloy aged at 595 K for 0, 222, 278, 361 and 444 h.
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where c0 is the average composition and A(β, t) is the amplitude of the Fourier component of wave number β at time t (β = 2π/λ, where λ is the wavelength). This amplitude is defined in terms of the initial amplitude at time zero as follows: A(β, t) = A(β, 0) exp(R(β)t)
(12)
The amplification factor R(β) is given by: R(β) = −
M NV
∂2 f 2 2 β2 + 2η Y + 2Kβ ∂c2
(13)
where NV is the number of atoms per unit volume. The sign of R(β) is determined by that of the second partial derivative of f with respect to c, f . Inside the spinodal f < 0 and R(β) > 0 for all values of β. Thus, any composition modulation will grow. The magnitude of f will be the largest for an alloy composition located at the center of the spinodal and thus, the amplitude of composition modulation will be higher than that corresponding to an alloy composition close to the spinodal line. The first alloy has been named as symmetric alloy and the other one designated as asymmetric alloy [6]. In this work, the Cu–70 at.% Ni alloy is a symmetric alloy and thus the amplitude is higher than that of the asymmetric Cu–90 at.% Ni alloy. The increase in aging temperature also increases the mobility M and thus it promotes an increase in the modulation amplitude. In general, it could be stated that the kinetics of phase decomposition for the symmetric alloy would be faster than that of the asymmetric alloy since the R(β) is higher in the former case and it would cause a higher modulation amplitude for the same aging time, according to Eqs. (12) and (13).
Fig. 4. AP-FIM concentration profile for the Cu–70 at.% Ni alloy (a) solution treated and (b) aged at 573 K for 2000 h.
The AP-FIM experimental concentration profiles of the Cu–70 at.% Ni alloy, solution treated and aged at 573 K for 2000 h, are shown in Fig. 4a and b, respectively. It can be also seen a slight increase in the modulation amplitude with aging
Fig. 5. Simulated microstructure for the Cu–70 at.% Ni alloy aged at 573 K for (a) 0 h, (b) 222 h, (c) 278 h, (d) 361 h and (e) 444 h.
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time. This slight increase also confirmed the phase decomposition took place via spinodal decomposition in this alloy system. 4.2. Microstructural evolution The simulated microstructures of the Cu–70 at.% Ni alloy aged at 573 K for 0 (solution treated), 222, 278, 361 and 444 h are shown in Fig. 5a–e, respectively. The black and white regions correspond to the Cu- and Ni-rich phases, respectively. It can be seen that the morphology of the decomposed phases is irregular and interconnected. The volume fraction of the Ni-rich phase increased with aging time. A similar morphology of the decomposed phases was also observed for the simulated microstructure of the Cu–90 at.% Ni alloy after solution treating and aging at 595 K for 0, 222, 278, 361 and 444 h, as shown in Fig. 6a–e. The volume fraction of Ni-rich phase increased with aging time. The volume fraction of the Cu-rich phase (black zones) corresponding to the aged Cu–70 at.% Ni alloy was higher than that of the aged Cu–90 at.% Ni, as expected from the lever rule for these two alloy compositions. This type of microstructure has been named isotropic [6] and it has no preferential alignment in a specific crystallographic direction. In contrast, the microstructure of decomposed phases, which shows an alignment in the 1 0 0 crystallographic direction in order to reduce the coherency strain energy (2η2 Y) are known as anisotropic [6]. In the case of Cu–Ni alloys, the parameter η is about 0.0016 [11] and thus the contribution of the elastic coherent strain energy to the phase decomposition process is very small. That
is, the elastic strain energy affects in an isotropic manner to the decomposition process. Therefore, it has practically no effect on the morphology of the decomposed phases. The size of the decomposed phase is very small, about 2 nm, for the aging process of both alloy compositions and it seems to be reasonable for the slow diffusive process in this alloy system. The experimental Ne gas FIM images of the Cu–70 at.% Ni alloy aged at 573 K for 500 h and Cu–90 at.% Ni alloy aged at 595 K for 500 h are shown in Fig. 7a and b, respectively. The figures on the right side correspond to an enlargement of the decomposed phases. FIM observation was carried out at about 115 K to produce a larger difference in the field evaporation rate between the expected Cu- and Ni-rich phases [15]. This difference in evaporation rate is expected to image the Nirich phase brightly and the Cu-rich phase darkly. Fig. 7a and b resembles the FIM image obtained from spinodally decomposed Fe–Cr alloys [16]. In this type of alloys, there was no preferential alignment in a specific crystallographic direction due to the small coherency strain energy. That is, an isotropic spinodal decomposition was present during aging of these alloys. This type of phase decomposition is also expected to occur in the aging of Cu–Ni alloys because of the similar lattice parameters of copper and nickel. FIM observation of samples at 20 K showed only the regular concentric ring pattern in which no evidence of the phase decomposition could be observed. The experimental results of morphology and no preferential alignment for the decomposed phases showed, in general, a good agreement with the simulated microstructure for both the aged Cu–70 and 90 at.% Ni alloys.
Fig. 6. Simulated microstructure for the Cu–90 at.% Ni alloy aged at 595 K for (a) 0 h, (b) 222 h, (c) 278 h, (d) 361 h and (e) 444 h.
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Fig. 7. FIM images of the (a) Cu–70 at.% Ni alloy aged at 573 K for 500 h and (b) Cu–90 at.% Ni alloy aged at 595 K for 500 h.
4.3. Growth kinetics of phase decomposition The wavelength of the composition modulation can be determined by means of the analysis of calculated concentration profiles of Figs. 1–3 using the autocorrelation or structure function analyses [15,17]. Fig. 8 shows the variation of wavelength for the composition modulation as a function of aging time for the Cu–70 and 90 at.% Ni alloys aged at 595 K. It can be observed that the wavelength remains almost constant, 2.00 and 2.02 nm for the Cu–70 and 90 at.% Ni alloys, respectively, with aging time. This is a characteristic behavior observed during the early stages of aging in spinodally decomposed alloys [7,17]. Likewise, it can be seen that the wavelength for the aged Cu–90 at.% Ni alloy is slightly longer that that corresponding to the aged Cu–70 at.% Ni alloy. According to the Cahn–Hilliard theory of spinodal decomposition [7], the amplification factor R(β) is positive for 0 < β > βc with: 2 2π ∂ f/∂c2 βc = (14) = λc K The analysis of Eq. (14) suggests that the wavelength of composition modulation is shorter for a symmetric alloy and as the
aging temperature decreases, since f increases in these two latter cases. This fact is in good agreement with the results observed in Fig. 7. The autocorrelation analysis of the experimental AP-FIM concentration profiles of Fig. 4 indicated a value of wavelength
Fig. 8. Plots of composition modulation wavelength, λ, vs. aging time, t, at 595 K for the computer simulation of Cu–70 and 90 at.% Ni alloys.
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for composition modulation of about 2 nm, which is in agreement with the value order for wavelength determined from the numerical simulation, Fig. 8. These short wavelength values confirmed that the growth kinetics of phase decomposition is very slow in Cu–Ni alloys because of the low atomic diffusive process at temperatures lower than 600 K. The phase decomposition of Cu–Ni alloys was also simulated using the linearized Cahn–Hilliard equation [7] and the simulated results were practically the same that those obtained with the non-linear equation. This fact seems to suggest that the growth kinetics of phase decomposition is very slow and thus its evolution is in the early stages of phase decomposition in spite of the prolonged aging times. Besides, it was also observed that the elastic coherency-strain energy caused that the phase decomposition kinetics was slower compared with the simulated results determined with the non-linear Cahn–Hilliard equation without considering the elastic strain energy.
Acknowledgements The authors wish to thank the financial support from CGPI-COFAA-IPN and Fondo Sectorial para la EducacionCONACYT 47151. References [1] [2] [3] [4] [5] [6]
[7] [8] [9]
5. Conclusions The simulation of phase decomposition was carried out in the Cu–Ni alloy system using the solution of the non-linear Cahn–Hilliard partial differential equation by the finite difference method. The calculated results showed a good agreement with the experimental ones. The growth kinetics rate of phase decomposition was very slow in the aged Cu–Ni alloys. The phase decomposition kinetics of the symmetric alloy was faster than that corresponding to the asymmetric alloy. The morphology of the decomposed phases is similar to that reported in isotropically decomposed alloys.
[10] [11] [12] [13] [14] [15] [16]
[17]
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