New lithium gas sorbents

Journal of Alloys and Compounds 456 (2008) 187–193

New lithium gas sorbents II. A mathematical model of the evaporation process K. Chuntonov a,∗ , A. Ivanov b , D. Permikin b a

Nanoshell Materials R&D GmbH, Primoschgasse 3, 9020 Klagenfurt, Austria b Ural State University, Lenin avenue, 51, 620083 Ekaterinburg, Russia

Received 31 January 2007; received in revised form 12 February 2007; accepted 13 February 2007 Available online 17 February 2007

Abstract A phenomenological theory of evaporation for a system where the vapor source is a filament made of a solid solution of a volatile metal in a nonvolatile metal has been created. Approximate solutions, which describe both the evaporation process and the kinetics of the deposit growth with sufficient accuracy, have been obtained. This theoretical analysis is part of a general program of the development of evaporable (EG) and nonevaporable (NEG) lithium getters. © 2007 Elsevier B.V. All rights reserved. Keywords: Lithium; Solid solutions; Evaporation rate; Film deposition; Sorption; Getters; Passivation

1. Introduction Evaporation processes from alloys containing a volatile component are the basis for a large number of technologies like refining of metals [1,2], deposition of metallic films [3–5], creation of an open pore alloy structure [6], etc. As vapor sources alloys often (for different reasons) appear to be a better choice than pure metals. But at this the evaporation and deposition procedures become more complicated due to an inevitable appearance of diffusion limitations. Besides, the decrease of the concentration of the volatile component during evaporation can lead to phase changes, which abruptly alter the evaporation rate. This kind of behavior is typical, e.g., for intermetallic compounds with a low-melting nonvolatile component [7]. In previous studies we considered the problem of evaporation of a volatile metal from an intermetallic compound with a fusible cover layer [8,9]. It has been shown that such a cover layer not only fulfills protective functions, but also stabilizes the evaporation flow in compliance with experimental data [7]. The goal of the present paper is the analysis of the evaporation process in cases, where a volatile metal is dissolved in a crystal

∗

Corresponding author. Tel.: +43 463 31 9329; fax: +43 463 31 9329x7. E-mail address: [email protected] (K. Chuntonov).

0925-8388/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.jallcom.2007.02.065

lattice of a nonvolatile metal. This project relates in particular to the need for a theoretical description of the evaporation of lithium from its solid solutions in transition metals. 2. The deposition principles The ability of lithium to form solid solutions with, e.g., Ag, Au, and Cu in a wide concentration range opens the way for using these alloys in getter applications. The advantages of these alloys over pure lithium are the following: • self passivation of solid solutions Li–Me (Me = Ag, Au, Cu) in the normal atmosphere, which makes the handling and the storage of the material easier; • a favorable combination of strength and ductility of these alloys, which allows applying mechanical methods for their shaping; • widening of the temperature range of the solid state, which facilitates the design of getter pumps. The high predictability of the changes in the properties of solid solutions with the change in relative concentrations should also be regarded as an advantage. As far as the production of getter films using Li–Me alloys is concerned, several methods of transferring the getter material onto a substrate through the vapor phase can be distinguished.

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Let us describe in short different types of films, which can be obtained with the help of these methods. 2.1. Dense lithium films Upon heating of a Li–Me alloy in a high vacuum chamber, lithium evaporates and forms dense films on cold surfaces. These films can bind many residual gases at room temperature. To increase the sorption effectiveness of the material, the deposition should be performed either continuously, with the rate close to the reaction rate between lithium and the gases, or periodi˚ cally to produce layers with a thickness not bigger than ∼120 A [10]. The sorption capacity of the deposited film can be increased by many times if instead of a smooth substrate surface a surface with a brush structure – a system of parallel microneedles elongated in the direction vertical to the surface – is used. This kind of structure can be (a) grown by crystallization from the vapor according to a vapor–liquid–solid mechanism [11–13], (b) obtained by directed crystallization of eutectics with subsequent vacuum or liquid etching of one of the phases [14–16], or (c) by electrolytic deposition [17]. A brush substrate of any nature is suitable for sorption at room temperature, including metals, ceramics, and even polymers. 2.2. Homogeneous Li–Me films Solid solution films of extremely high concentration of Li, e.g. Cu–20 at.% Li, Ag–50 at.% Li, etc. can be deposited by the methods of cathode sputtering on suitable parts of vacuum or gas devices for capturing residual gases or unwanted gas impurities. The maximum working temperature for getter coatings of this kind to be used in a vacuum according to the data on the thermal stability of Li–Me alloys is about 350 ◦ C [18], and for usage in gas purifiers according to the data on phase equilibria it is slightly higher than 400 ◦ C [19]. 2.3. Double-layer Li/Me films The vapor pressure of Li over Li–Me alloys is by many orders of magnitude higher than the vapor pressure of the second component. For this reason upon thermal evaporation of lithium in a vacuum practically pure Me remains from the alloy at the end of the process. If the temperature of such a residue is raised to values close to the melting point, then sufficiently intensive evaporation of Me will start and a layer of the second component Me will be deposited on the lithium film. A double-layer coating Li/Me will thus appear, the outside Me layer of which will be able to protect the lithium film from chemical damage under operational conditions during which a temporary contact between the film and the atmosphere is unavoidable. Two different kinds of Li/Me products can be manufactured depending on the ratio of the thickness of the lithium layer dLi and the layer of the second metal dMe . At dLi /dMe  1, the initial material after the first short annealing for homogenization turns into an active film, the composition and properties of which

differ very little from those of a pure Li film. Films of this type can be used for gas sorption at room temperature. If dLi /dMe ∼ 1, annealing for homogenization leads to the formation of the above mentioned Li–Me films which show acceptable sorption activity in the temperature interval of 150–400 ◦ C. Double-layer coatings Li/Me can also be used according to the conventional scheme known from the experience with nonevaporable getter (NEG) materials, i.e. by repeated cycles of activation and sorption. However, in the present case, the regeneration of a getter occurs due to the exit to the surface of a fresh portion of lithium atoms and not due to diffusion of gases into the bulk of the material. 3. The process model In order to limit the scope of the analysis, let us consider vapor sources in the form of filaments. The results thus obtained can be easily transformed for the case of point sources or of sources of a planar shape. 3.1. Model formulation A wire with the radius r0 is placed along an axis of an infinite tube, which has an inner radius L  r0 . The material of the wire is an ideal solid solution with concentration c0 of a volatile metal A in a nonvolatile metal B. Vacuum is maintained inside the tube. The wire is heated to the temperature T, at which the vapor pressure of A becomes sufficient for evaporation. The concentration of A in the surface layer of the wire decreases due to evaporation, and then a diffusion flux in a radial direction from the axis of the wire towards its surface appears. It is required to find the time law for evaporation of the component A and the rate of the film growth on the inner side of the tube, assuming that r0 = const. A mathematical model of these processes can be written down in the following way:   ∂c D ∂ ∂c (1) = r , t > 0, 0 < r < r0 ∂t r ∂r ∂r −D

∂c = hc, ∂r

c(r, t) = c0 ,

t > 0, r = r0 t=0

(2) (3)

where t is the time, r a radial coordinate, c(r,t) the concentration of metal A in the wire, D = D0 exp(−Q/RT) the coefficient of diffusion of atoms A in metal B, D0 a preexponential factor, Q the molar activation energy of diffusion, R the gas constant, h = (RT/2πM)1/2 exp(−E/RT) the evaporation coefficient, M the molar mass of a diffusant A, and E is the molar energy of evaporation of A. The flow j of the evaporating substance A in the frames of the problems (1)–(3) is determined in the following way (the amount of atoms evaporating in a unit of time from a unit of area of the lateral surface of the wire): j(t) = hc(r0 , t)

(4)

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3.2. Solution The solution of the system of Eqs. (1)–(3) leads to the following result for the evaporation flow (see Appendix I):   ∞  λ2k Dt (5) j(t) = hc0 Ck J0 (λk ) exp − 2 r0 k=1 Assuming that all atoms of A reaching the inner wall of the tube stick to it, we obtain for the deposition rate V(t) (the number of A atoms deposited on a unit of the inside area of the tube during a unit of time): jr0 L or for the linear growth of the film: V (t) (particles/cm2 s) =

v(t) (cm/s) =

jMr0 ρNL

(6)

(7)

where ρ is the density of metal A, and N is the Avogadro number. An exact solution (5) is rather substantial and requires special software. Besides, the series according to the Bessel function converge slowly at large values of the mass transfer parameter H (Appendix I). Thus simpler methods for calculating j(t), which were found with the help of a standard Laplace transformation technique, are also suggested (Appendix II). At H  1 (the kinetic regime) an expression independent of the diffusion coefficient D:   2ht j(t)  hc0 exp − , hr0  D (8) r0 follows from (23). If H  1 (in the diffusion regime), an expression independent of the evaporation coefficient h:   D D (9) − j(t) ≈ c0 πt 2r0

Fig. 1. Dependence of evaporation flow on time for the kinetic regime at different values of H: dots correspond to the exact solution (5), the solid line corresponds to solution (8); (1) H = 0.01, (2) H = 0.05, and (3) H = 0.1.

of the evaporation flow according to these formulas and for an exact solution (5) obtained by summation of 500 summands of a series are compared in Figs. 1–4. For greater generality the data is presented in a dimensionless form y(τ) and τ (Appendix I). The case of the kinetic regime (H  1) is presented in Fig. 1. It can be seen that a very simple formula (8) provides high accuracy for the value j(t) during the entire process at values of H up to ∼0.1 Another extreme case (the diffusion regime, H  1) is shown in Fig. 2: here a mathematically simple formula (9) provides results which coincide with the exact solutions starting from values τ ∼ 4/H2 . Since parameter H in this case is large, formula (9) can be used for calculations of j(t) everywhere except for the period of the very beginning of the process. Solution (10), which concerns the intermediate situation where H ∼ 1 (Fig. 3), precisely describes the evaporation process in the whole interval of times except for a small neighborhood

follows from (25). In the intermediate situation, when H ∼ 1, a generalization of formula (8) following from (23) can be recommended   8hDt 4Dhc0 , hr0 < D exp − (10) j(t) ≈ 4D + hr0 4D + hr0 And finally, for the initial stage of the process, (24) provides formula:  2   √  h t h t , t→0 (11) erfc √ j(t) ≈ hc0 exp D D which is valid for any H. 4. Discussion 4.1. Analysis of approximated solutions First of all, it is necessary to determine the applicability boundaries of the main approximations (8)–(11) and to estimate their accuracy. For this purpose the results of the calculation

Fig. 2. Dependence of evaporation flow on time for the diffusion regime at different values of H: dots correspond to the exact solution (5), the solid line corresponds to solution (9); (1) H = 5, (2) H = 10, and (3) H = 50.

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Fig. 3. Dependence of evaporation flow on time for the intermediate regime: dots correspond to the exact solution (5), the solid line corresponds to solution (10); (1) H = 0.5, (2) H = 1.0, and (3) H = 2.0.

of τ ∼ 0. However, as demonstrated in Fig. 4, this initial process stage can be described by the asymptotic formula (11) the results of which are indistinguishable from those of the exact solution (5). Thus, if the value of the parameter H is known, formula (5) can be substituted by a corresponding approximate expression, practically without any loss of accuracy, for the determination of the value of j(t). With H  1, it is enough to use (8) instead of (5). With H  1, and in the time interval from 0 to t ≈ D/h2 , it is possible to use (11); and at t > D/h2 formula (9) should be used. Finally, for the case of H ∼ 1 a sequential combination of (11) and (10) should be employed. 4.2. The two-stage character of the process For lithium alloys the estimation of values h and H gives 10−7 m/s ≤ h ≤ 10−6 m/s and 102 ≤ H ≤ 104 , respectively. Here

Fig. 4. Dependence of evaporation flow on time at the initial stage of the process: dots correspond to exact solution (5), the solid line corresponds to solution (11); (1) H = 0.1, (2) H = 1, and (3) H = 10.

the following data was used: 10−3 m ≤ r0 ≤ 10−4 m, 750 K ≤ T ≤ 900 K, 15 at.% Li ≤ c0 ≤ 30 at.% Li, E = 137 kJ/mol and D  10−13 to 10−14 m2 /s [20–22]. These results indicate that the evaporation of Li from solid solutions in nonvolatile metals is limited by diffusion, but this does not mean that evaporation is not possible in a kinetic regime. The value of H depends not only on the values of E, Q and T, but also on the size of the vapor source. In the strict sense, it is a function of time and it changes during the process of evaporation due to the variation in concentrations in the material. However, it is not reasonable to introduce the actual values of H(t) for consideration here, because in this case the simplicity of the concept (12)–(14) and also of the approximations (8)–(11) would be lost. Therefore it is more convenient to accept a two-stage model of the evaporation process and to retain the formalism (12)–(14). Experimental proof of the validity of the two-stage character of the evaporation process is given in [23]. Further speculative arguments in its favor are presented here. Evaporation from thin films due to the small thickness always takes place in a kinetic regime (in case of wires, e.g., H → 0 at r0 → 0). The initial stage of evaporation from chemically homogeneous, massive bodies does not differ practically from the evaporation from films: the exit of atoms of a volatile component into the vapor phase from a thin surface layer takes place in a diffusion-free way or with insignificant participation of diffusion. As shown by the evolution of the concentration profile, calculated for wire vapor sources according to Eq. (19) (see Appendix I), in the beginning a very thin layer is formed on the surface of the material which almost completely consists of atoms of the nonvolatile component B (Fig. 5). At evaporation temperatures, at which the general mobility of atoms is sufficiently increased, the surface layer B consolidates into a dense film of B, which starts slowing down the evaporation of A. This initial stage I (Fig. 6) with duration t  D/ h2 corresponds to the part of the curve j(t), where the exact solution (5)

Fig. 5. Radial distribution of concentration u(x) at different times τ at H = 1000 (according to the results of a summation of 5000 summands of a series (15)): (1) τ = 0, (2) τ = 0.0001, (3) τ = 0.0005, and (4) τ = 0.001.

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4.3. Aspects of applications

Fig. 6. Two stages of evaporation for H  1: I, the field of the initial evaporation stage; II, the field of the diffusion stage; the solid line corresponds to formula (5), dots correspond to formula (8), crosses to formula (9).

coincides with the kinetic approximation (8), but differs from the diffusion approximation (9). Therefore we identify stage I with the kinetic regime of evaporation. After the formation of cover layer B, the evaporation process enters a diffusion stage II, which is characterized by a slower evaporation rate compared with the kinetic stage. The thickness of the surface layer of the material depleted of component A continues to grow with time (Fig. 5) and the evaporation flow gradually decreases with the decrease of the radial gradient of concentrations in the wire. It is important to note that the division considered here for the evaporation process into two stages, a short kinetic stage and a long-duration diffusion stage, is correct only for a system A–B representing a solution of a volatile metal A in a nonvolatile metal B, i.e., for systems with H  1. This is obvious from Fig. 7, which shows the concentration distribution of the evaporating component inside the wire for different times, when H = 1. In spite of a certain depletion of the periphery of the source by the volatile component, in general the concentration gradient is not large and a shell of a nonvolatile component is not formed on the surface.

Turning back to the aim of the present analysis, it appears that the phenomenological description of the evaporation process in the variants (5)–(7) is quite satisfactory. Though formula (5) is not a simple one, it accurately defines the value j(t) for the case of initially homogeneous wires of composition c0 . Combining (5) with (6) or (7), we form a mathematical basis for the programmed control of the growth of a film deposit of metal A. That is, Eqs. (5)–(7) can help to solve most of the problems concerning evaporation and deposition of lithium from homogeneous vapor sources. At the same time, the concept of a two-stage character of the evaporation process introduced here allows an efficient use of approximations (8) and (9). The interpretation of the initial stage of evaporation as a kinetic stage, which creates a film of component B on the surface of the wire, changes the approach of formula (8). The formation of a coating of a chemically inert metal B on the surface of a wire containing a chemically active metal A means that a qualitatively new, corrosion-resistant product has appeared. Thus, formula (8) becomes an instrument of controlling a thermovacuum procedure producing a protective coating B on the surface of a vapor source. In fact, integrating (8) it is easy to find for each composition c0 the time t, which is necessary for the formation of a film of component B with a set thickness on the surface of a wire. The transient nature of the kinetic stage makes it especially attractive as a technological alternative to other methods of chemical passivation of Li–Me alloys. Together with (8), the approximation (9) also acquires significance in a technological niche. As evaporation processes in alloys with a cover layer take place in the diffusion regime, formula (5) becomes unnecessary and can be substituted without any risk by the simplified expression (9). Substitution of (9) into (6) or (7) provides in this case a simpler mathematical basis for regulating the thickness of the deposit of A growing on a substrate. Finally, it should be observed that the general group of volatile metals includes a series of 22 metallic elements of particular relevance in this context such as As, Ba, Ca, Cd, Cs, Eu, Na, Sm, Yb, etc. [24]. For these metals, the values of parameters h and H are close to those of Li. Therefore evaporation processes from alloys containing these metals will follow the same laws as Li–Me alloys. In as much as all these volatile metals are either toxic or chemically active, creation of a protective coating on the surface of these alloys appears to be an important option. This constitutes the main potential for application of Eqs. (8) and (9). 5. Summary and conclusions

Fig. 7. Radial distribution of concentration u(x) at different times τ for H = 1: (1) τ = 0, (2) τ = 0.01, (3) τ = 0.05, and (4) τ = 0.1.

1. A mathematical description of the evaporation of a volatile metal A from its solid solution in a nonvolatile metal B, e.g. evaporation of Li (A) from Li–Me alloys, where Me (B) = Ag, Cu, etc., has been presented. 2. The evaporation process from the considered materials consists of two sequential stages, the first one being kinetically, the second and main one diffusion controlled.

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3. The initial stage of evaporation can be used for a fast method of creating a protective layer of component B on the surface of an alloy A–B. 4. The deposition of films of component A from chemically homogeneous vapor sources A–B can be controlled with the help of the exact solution (5) or with the help of approximations with formulae containing only exponential or power functions. 5. The deposition of films of component A from solid solution sources A–B with a cover layer of component B can be controlled with the help of a simple diffusion approximation (9). Acknowledgements

Appendix A The problems (1)–(3) can be solved precisely by the method of separation of variables. Let us substitute dimensional variables and functions for dimensionless values: x = r/r0 , τ = Dt/r02 , u(τ,x) = c(t,r)/c0 . Then the system of equations transforms in the following way:   ∂u 1 ∂ ∂u x , τ > 0, 0 < x < 1 (12) = x ∂x ∂x ∂τ ∂u = Hu, ∂x

u(τ, x) = 1,

τ > 0, x = 1

(13)

τ=0

(14)

Here H = r0 h/D is a dimensionless mass transfer parameter, which is equal to the ratio of two characteristic times tD /th , where tD = r02 /D is the characteristic time of diffusion and th = r0 /h is the characteristic time of evaporation. When H  1, we have th  tD , and in this case the process is limited by evaporation kinetics. In the opposite situation, i.e. at H  1, we have th  tD , which corresponds to the diffusion-limited evaporation regime. The solution of the problems (12)–(14) can be presented in the form of an infinite series: u(τ, x) =

∞  k=1

Ck J0 (λk x) exp(−λ2k τ)

(15)

where J0 (z) is the Bessel zero order function and λk are eigenvalues, which are found from a transcendental equation: λk J1 (λk ) = HJ0 (λk ),

y(τ) = u(τ, x = 1) =

k = 1, 2, 3 . . .

(16)

for which only positive roots are taken into account. The symbol J1 (z) in (16) stands for the Bessel first order function. Expansion coefficients Ck have the form: 1 J0 (λk x)x dx 2H Ck =  10 = (17) 2 J0 (λk )(λ2k + H 2 ) 0 J0 (λk x) x dx

∞  k=1

= 2H

Ck J0 (λk ) exp(−λ2k τ)

∞  exp(−λ2 τ) k

k=1

(18)

λ2k + H 2

Returning to the dimensional values, we arrive at the final solution:     ∞  λ2k Dt λk r c(t, r) = c0 Ck J0 , (19) exp − 2 r0 r0 k=1 ∞ 

The authors would like to thank Prof. H. Schmidbaur, TU M¨unchen, for his support. The research was partly supported by RFBR Grant No. 04-01-96008.

−

wherefrom, for a dimensionless flow y(τ) = j(t)/hc0 , we obtain



λ2 Dt j(t) = hc0 Ck J0 (λk ) exp − k 2 r0 k=1



Appendix B For a solution of the problems (12)–(14) a Laplace integral transformation according to the time variable τ can be employed. Upon introducing a Laplace image according to ∞ U(x; s) ≡ u(τ, x) exp(−sτ) dτ (20) 0

we obtain a solution in the form √ √ √ √ sI1 ( s) + H[I0 ( s) − I0 (x s)] √ √ √ U(x; s) = s[ sI1 ( s) + HI0 ( s)]

(21)

where s is the parameter of the Laplace transformation, and I0 (z) and I1 (z) are modified Bessel zero and first order functions. The value of U(1;s) of this image at the point x = 1 designates the surface concentration u(τ, x = 1) and, consequently, a dimensionless flow y(τ): √ I1 ( s) √ √ U(1; s) = √ √ (22) s[ sI1 ( s) + HI0 ( s)] Using the known properties of the integral Laplace transformations [25], we arrive at ⎧ 4 ⎪ ⎪ s < 1 or H < 1 ⎪ ⎪ ⎪ (4 + H)s + 8H ⎪ ⎪ ⎨ 1 √ s1 U(1; s) ≈ √ s(H + s) ⎪ ⎪ ⎪ ⎪ H 1 ⎪ ⎪ ⎪ √ 2 H  1, s > 1 ⎩ √s(H + √s) − √ 2 s(H + s) or after inverse transformations at   4 8H y(τ) ≈ exp − τ , τ  1 or H < 1 4+H 4+H √ y(τ) ≈ exp(H 2 τ) erfc(H τ), τ → 0

(23) (24)

√ y(τ) ≈ exp(H 2 τ) erfc(H τ)

√ 1 − exp(H 2 τ/4) erfc(H τ/2) − , 2H

H 1

(25)

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where erfc(z) is an additional probability integral: ∞ 2 exp(−z2 ) dz erfc(z) = √ π z References [1] W.D. Lawson, S. Nielsen, Preparation of Single Crystals, Butterworths, London, 1958. [2] G.G. Devjatyh, J.E. Elliev, High Purification of Substances, Vysshaja shkola, Moskva, 1974. [3] A.H. Sommer, Photoemissive Materials, John Wiley, NY, 1968. [4] J.P. Gimigliano, US Patent 3,637,421 (January 25, 1972). [5] D.M. Mattox, Vacuum Deposition, Reactive Evaporation, and Gas Evaporation, vol. 5, ASM Handbook, Materials Park, OH, 1996, pp. 556–572. [6] K. Chuntonov, EP 1,600,232 (November 30, 2005). [7] C.Y. Lou, G.A. Somorjai, J. Chem. Phys. 55 (1971) 4554. [8] K.A. Chuntonov, V.V. Mansurov, Vacuum 47 (1996) 463. [9] K.A. Chuntonov, V.G. Postovalov, A.G. Kesarev, Vacuum 55 (1999) 101. [10] K. Chuntonov, J. Setina, J. Alloys Compd. 455 (2008) 489–496. [11] R.S. Wagner, W.C. Ellis, Appl. Phys. Lett. 4 (1964) 89.

193

[12] E.I. Givargizov, Crystallization from Vapor (Gas) Phase, Modern Crystallization, vol. 3, Nauka, Moskva, 1980, pp. 233–279. [13] X.D. Wang, J.H. Song, P. Li, J.H. Ryou, R.D. Dupuis, C.J. Summers, Z.L. Wang, J. Am. Chem. Soc. 127 (2005) 7920. [14] R. Elliot, Eutectic Solidification Processing: Crystalline and Glassy Alloys, Butterworths, London, 1983. [15] H.E. Cline, J. Appl. Phys. 41 (1970) 76. [16] K. Chuntonov, H. Ipser, K. Richter, EP 06002762.0 (February 10, 2006). [17] C.L. Rinne, J.J. Hren, P.S. Fedkiw, J. Electrochem. Soc. 149 (2002) C150. [18] J.A. Rodriges, J. Hrbek, J. Phys. Chem. 98 (1994) 4061. [19] H. Okamoto, Phase Diagrams for Binary Alloys, ASM International, Material Park, OH, 2000. [20] A. Raho, M. Kadi-Hanifi, Ann. Chem. Sci. Mater. 29 (2004) 45. [21] J. Vacik, U. K¨oster, V. Hnatowicz, J. Cervena, G. Pasold, Defect Diffus. Forum 237–240 (2005) 485. [22] J.M. Titman, B.M. Moores, J. Phys. F: Met. Phys. 2 (1972) 592. [23] K. Chuntonov, J. Setina, A. Ivanov, D. Permikin, J. Alloys Compd., in preparation. [24] A. Brown, J.H. Westbrook, Formation Techniques, Intermetallic Compounds, John Wiley and Sons, NY, 1967, pp. 303–336. [25] G.A. Korn, T.M. Korn, Mathematical Handbook for Scientists and Engineers, McGraw-Hill, New York, 1961.