Thermodynamic assessments of six binary systems of alkali metals

CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 35 (2011) 446–454

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Thermodynamic assessments of six binary systems of alkali metals Xin Ren, Changrong Li ∗ , Zhenmin Du, Cuiping Guo School of Materials Science and Engineering, University of Science and Technology, Beijing, No.3 Xueyuan Road, Beijing 100083, China

article

info

Article history: Received 12 April 2011 Received in revised form 11 June 2011 Accepted 16 June 2011 Available online 13 July 2011 Keywords: K–Na, K–Cs, K–Rb, Cs–Na, Cs–Rb and Na–Rb binaries Alkali metal systems Thermodynamic assessment CALPHAD technique

abstract Among the phase diagrams of six binary systems composed of alkali metals, Sodium (Na), Potassium (K), Rubidium (Rb) and Cesium (Cs), three of them, those of the Cs–K, Cs–Rb and K–Rb systems, indicate a complete range of solid solubility, and the other three, those of the Cs–Na, K–Na and Na–Rb systems, have the eutectic type characteristic. The computational thermodynamic descriptions of the six binary systems were modeled by the CALPHAD technique. The solution phases, liquid and body-centered cubic, were described using a substitution model, with the excess Gibbs energies formulated by the Redlich–Kister expression. The intermediate phases with no solubility ranges, KNa2 and CsNa2 , were treated as stoichiometric compounds. The thermodynamic properties and the phase diagrams of the six binary systems were predicted. The consistency between the calculated results and the reported experimental measurements for the phase equilibria and the thermochemical properties has been obtained for all of the six binary systems. © 2011 Elsevier Ltd. All rights reserved.

1. Introduction Among the metallic elements, the alkali metals have a particularly simple electronic structure in possessing a single valence electron outside a closed rare-gas shell. The alkali alloys show excellent electric transport performance and the liquid metals and metallic alloys can be used as non-crystalline conductors [1]. In addition, the liquid K–Na alloy plays an important role in nuclear reactors as coolants [2]. The alloys formed between Rb and the three elements Cs, K and Na can be used to remove the residual gas in a high vacuum system. The Rb and Na–Rb alloys also serve as a working liquid to carry a nuclear power system [3]. Owing to their superior performance, the alkali metal alloys are given more and more attention. So there is a large number of theoretical and experimental studies on the thermodynamic properties of the alkali metal alloys. However, the phase diagrams of the binary systems formed among the alkali metal elements are at present inadequately studied. In order to better understand the alkali metal alloys, it is necessary to investigate the phase diagrams and the thermodynamic properties of the related binary systems. For the six binary systems, K–Na, K–Cs, K–Rb, Cs–Na, Cs–Rb and Na–Rb, Bale and Pelton [4–9] reviewed in detail and calculated their phase diagrams using numerical expressions. For five of the six binaries besides the K–Na system, the calculated phase diagrams can be found on the MTDATA website

∗

Corresponding author. Tel.: +86 10 82377789; fax: +86 10 62333772. E-mail address: [email protected] (C. Li).

0364-5916/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.calphad.2011.06.005

(http://mtdatasoftware.tech.officelive.com/default.htm) without published thermodynamic parameters [10]. Recently, Zhang et al. [11] optimized the K–Na system by a combined firstprinciples and CALPHAD (CALculation of Phase Diagram) method. In the present work, the thermodynamic descriptions of all the phases in the six binary systems (Cs–K, Cs–Na, Cs–Rb, K–Na, K–Rb and Na–Rb) are modeled integrally by means of the CALPHAD technique, among which those of the K–Na system are mainly based on Ref. [11]. The thermodynamic properties and the phase diagrams of the six binary systems are assessed according to all the available experimental data of both phase equilibria and thermodynamic properties. 2. Experimental details The experimental thermodynamic properties and phase equilibria of the Cs–K, Cs–Na, Cs–Rb, K–Na, K–Rb and Na–Rb binary systems are available in literatures. In the three binaries of Cs–K, Cs–Rb and K–Rb, the phase diagrams display complete ranges of solubility for liquid and solid [12,13]. Tanigawa and Doyama [14] and Soma et al. [15] predicted that in these three systems, the miscibility gaps may exist at low temperature based on the Helmholtz free energy curves obtained by the pseudo-alloy atom model. For the Cs–K binary system, Steinberg et al. [16] observed a phase separation below 185 K, which is taken as the critical temperature for the present assessment. And for the Cs–Rb and the K–Rb binaries, there were no experimental observations for the existence of the miscibility gaps, and the predicted critical temperatures, about

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Table 1 Condensed phases in the six binary systems. Binary system

Phase

Cs–K

Liquid Bcc_ A2

Cs–Na

Cs–Rb K–Na

K–Rb Na–Rb

Prototype

Composition(X2)

Formula

Reference

cl2

W

0–1.0 0–1.0

(Cs, K) (Cs, K)

[12]

cl2 cl2

W W

0–1.0 0 1.0 0.667

(Cs, Na) (Cs) (Na) (Cs)1/3 (Na)2/3

[29]

Liquid Bcc_A2

cl2

W

0–1.0 0–1.0

(Cs, Rb) (Cs, Rb)

[13]

Liquid Bcc_A2

cl2

W

(K, Na) K, Na

[37]

KNa2

hP12

MgZn2

0–1.0 0–0.046 0.992–1.0 0.667

(K)1/3 (Na)2/3

[18]

(K, Rb) (K, Rb)

[13]

(Na, Rb) (Na) (Rb)

[19]

Liquid Bcc_Cs Bcc_Na CsNa2

Pearson symbol

Liquid Bcc_A2

cl2

W

0–1.0 0–1.0

Liquid Bcc_Na Bcc_Rb

cl2 cl2

W W

0–1.0 0 1.0

50 K for both of the systems, by Soma et al. [15] are used in the present assessment, since their prediction is close to the experiment in the Cs–K system. The binaries of Cs–Na and K–Na have only one intermediate compound in each system, CsNa2 and KNa2 with the hexagonal (C14) structure [17,18]. The binary Na–Rb is a simple eutectic system [19]. The elements K and Na, Na and Rb have no solubility in each other in the K–Na and Na–Rb binary systems respectively [18,19]. The crystallographic data for the six binary systems are listed in Table 1. 2.1. Equilibrium information 2.1.1. Cs–K The Cs–K system was studied by Goria [20] using thermal analyses, Rinck [21] using both thermal analyses and electrical conductivities, and Bohm and Klemm [22] using both magnetic susceptibilities and X-ray studies. Later, Goates et al. [12] carried out a detailed thermal analysis for the liquid–solid phase equilibria, presenting both the liquidus and the solidus curves. In addition, several authors investigated the low-temperature phase transitions in solid solutions [23–27]. By means of thermal and electrical measurements, Shmueli et al. [27] reported the existence of the intermediate phase CsK2 below −88 °C and the compound may have a hexagonal structure. A second intermediate phase Cs6 K7 reported by Simon et al. [24] has not yet been adequately investigated. In the present work, only two stable condensed phases (liquid and Bcc) are involved. 2.1.2. Cs–Na Rinck [28], Goria [20] and Ott et al. [29] investigated the Cs–Na binary system by thermal analysis, while Bohm and Klemm [22] studied this system using both thermal analysis and magnetic susceptibility measurements. According to Villars [17], the single intermediate phase CsNa2 in the Cs–Na system has the same C14 structure as KNa2 . The size factor rule proposes that when the atomic radii differ by more than 15%, the solid solution solubility is restricted. On the basis of this rule, the negligible solid solubility is indicated in the Cs–Na system due to the large atomic radius difference between Cs and Na. Evidence suggests there may be a small region of liquid–liquid immiscibility, but this has not been verified. In the present work, the liquidus immiscibility is not taken into account and the two terminal phases are modeled as pure elements. The equilibrium phase diagram was taken from the results obtained by Ott et al. [29].

2.1.3. Cs–Rb Hansen [30] evaluated the Cs–Rb binary system according to the studies by several authors using different methods, including Goria [20] using thermal analyses, Rinck [31] using both thermal and conductivity measurements, and Bohm and Klemm [22] using both magnetic susceptibility and X-ray measurements. The phase diagram of the Cs–Rb system based on the detailed thermal analysis of liquid–solid phase equilibria by Goates et al. [13] is in good agreement with much earlier work of Rinck [31], particularly on the Cs side. 2.1.4. K–Na The K–Na binary system was investigated using thermal analysis, freezing point and resistivity measurements by different authors [32–36]. The phase diagram was constructed by Ott et al. [37] based on their detailed thermal analysis. According to Laves and Wallbaum [18], the intermediate solid phase KNa2 in this system has the hexagonal structure isotype with MgZn2 , as shown in Table 1. Krier et al. [36] measured both the eutectic point and the incongruent melting point of KNa2 using the calorimetry method. MacDonald et al. [35] reported the existence of the Bcc solid solution using electrical resistance measurement, which was confirmed by the results of Rimai and Bloembergen [38] using nuclear magnetic resonance. Combining the first-principles approach and CALPHAD technique, Zhang et al. [11] investigated the K–Na system, using the ionic model to describe the liquid phase, and reproduced the phase diagram and the thermodynamic properties well. In the present work, for the general consideration in the six binary systems composed of alkali metals, the model for the liquid phase of the K–Na system is changed into the substitutional solution. The parameters are mainly based on Ref. [11] with a little modification. 2.1.5. K–Rb Kurnakov and Nikitinskii [39] investigated the K–Rb binary system by both conductivity and flow pressure measurements, and Goria [40] and Rinck [41] using both thermal analyses and microscopic observation. The phase diagram of the K–Rb system based on the detailed thermal analysis by Goates et al. [13] is in good agreement with the compilation by Hansen [30] as well as the much earlier work of Rinck [41]. According to Goates et al. [13], the accuracy of liquidus data was within ±0.1 K and the uncertainty of the solidus data was ±1 K except in the central compositions where the uncertainty was only ±0.1 K. So the liquidus and solidus data measured by Goates et al. [13] are mainly referred in the present work.

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Table 2 Thermodynamic data used for optimized of the six binary systems. Thermodynamic parameter

Temperature (K)

Type of investigation

Method of investigation

Phase

Binary system

References

Enthalpy of mixing

298 384±2 400

Calculation Experiment Calculation

First-principles calculation Reaction calorimetry Heat of mixing

Bcc

[11] [43] [44]

773–1200

Calculation

Pressure of saturated vapors

K–Na All systems Cs–K Cs–Na Cs–K Cs–Na Cs–Rb

384 400 520 384–1073 384 384 384

Experiment Experiment Experiment Calculation Calculation Calculation Calculation

Spectrophotometry Effusion Emf Heat of mixing Heat of mixing ab initio calculation Heat of mixing

Activity

Free energy of mixing Entropy of mixing

2.1.6. Na–Rb Using thermal analyses, Rinck [42], Goria [40], Bohm and Klemm [22] and Goates et al. [19] investigated the Na–Rb binary system. In Ref. [19], the liquidus data were judged to be accurate to ±0.15 K and the maximum solid solubilities were 1.1 at.% Na in Rb and 0.5 at.% Rb in Na, which were obtained by varying the compositions of the terminal solutions until an eutectic was observed. Since these values were so small, it is possible that the values exceeded the limit of detection of the small amount of eutectic available. In this instance, the solubilities would be even smaller. So the solubilities of Na and Rb in each other are ignored and both of the terminal phases are treated as pure elements in the present work.

Liquid

[45]

[50] [44] [51] [47] [52] [53] [52]

K–Na Na–Rb K–Na K–Rb K–Na

The K–Rb alloys are the least investigated among the six binary systems of alkali metals. The free energy and the enthalpy of mixing of the K–Rb binary system at 384 K were obtained by Singh [53] using ab initio calculations and their computed values of the heat of mixing were in close agreement with the experimental observation. For the Na–Rb binary system, the activities of Na and Rb at different temperatures, 384, 473, 773 and 1073 K, were obtained by Lokshin [47] on the basis of the enthalpy of mixing. The above mentioned thermodynamic properties are listed in Table 2. 3. Thermodynamic modeling 3.1. Pure elements

2.2. Thermodynamic information The enthalpies of mixing of liquid at 384±2 K in the six binary systems (Cs–K, Cs–Na, Cs–Rb, K–Na, K–Rb and Na–Rb) have been measured by Yokokawa and Kleppa [43] using reaction calorimetry, which are used in the present work. Among the six binary systems, only the Cs–Rb shows the negative enthalpy of mixing. This is due to the relative electro-negativity factor which correlates with the negative enthalpy contribution plays a dominate role in the measurement of enthalpy of mixing. For the Cs–K binary system, Kagan [44] investigated the activities of Cs and K at 400 K on the basis of the heat of mixing measured by Yokokawa and Kleppa [43]. At high temperatures, 773, 900, 1000, 1073 and 1200 K, Shpil’rain et al. [45] studied the activities of Cs and K based on the experimental data on the pressures of the saturated vapors over liquid measured by themselves [46]. Lokshin [47] determined the activities of both elements Cs and Na in the Cs–Na binary system at temperatures 384, 473, 773 and 1073 K based on the heat of mixing studied by Yokokawa and Kleppa [43]. In the Cs–Rb binary system, at high temperatures, 773, 900, 1000, 1073 and 1200 K, Shpil’rain et al. [48] also studied the activities of Cs and Rb based on the experiment data on the pressures of the saturated vapors over liquid measured by Pokrasin and Roshchupkin [49]. For the K–Na binary system, the experimental and thermodynamic properties data are rich in literature. Cafasso et al. [50] measured the activities of Na and K in liquid at 384 K using spectrophotometry and Kagan [44] using an effusion method at 400 K. Lokshin [51] investigated the activities of K at 520 K by electromotive force (EMF). Hultgren et al. [52] compiled the free energy and entropy of mixing of liquid at 384 K, which were not considered in the work by Zhang et al. [11]. In addition, Zhang et al. [11] also calculated the enthalpy of mixing of Bcc at 298 K using the firstprinciples calculation.

The Gibbs energy function for the element i in the phase ϕ is described by an equation of the following form: 0

ϕ

ϕ

Gi (T ) = Gi (T ) − HiSER (T )(298.15 K)

(1)

where HiSER (T )(298.15 K ) is the molar enthalpy of the element i at 298.15 K in its standard element reference (SER) state, Bcc for Cs, K, Na and Rb. The Gibbs energy of the element i, in its SER state, is denoted by GHSERi as follows: Bcc Bcc GHSERi = 0 GBcc i (T ) = Gi (T ) − Hi (298.15 K).

(2)

The Gibbs energy functions of pure elements are taken from the SGTE compilation by Dinsdale [54] in the present work. 3.2. Solution phases The liquid phase in the six binary systems (Cs–K, Cs–Na, Cs–Rb, K–Na, K–Rb, Na–Rb) and the Bcc phase in the four binary systems (Cs–K, Cs–Rb, K–Na and K–Rb) are described using a substitution model, (M, N), with M, N = Cs, K, Na, Rb and M ̸= N. The Gibbs free energy function for the phase ϕ (ϕ = liquid and Bcc) is given as: ϕ

ϕ

Gϕm = xM 0 GM + xN 0 GN + RT (xM ln xM + xN ln xN ) + E Gϕm 0

ϕ

0

(3)

ϕ

where xM and xN are the molar fractions, GM and GN the molar Gibbs energy of the pure element M and N respectively; R is the ϕ gas constant, T the absolute temperature, and E Gm the excess molar Gibbs energy expressed by the Redlich–Kister Polynomial [55]: E

Gϕm = xM xN

−

i ϕ L M ,N

(xM − xN )i

(4)

i=0

ϕ

where i LM ,N is the interaction coefficient between the elements M and N in the phase ϕ and takes the following form: i ϕ L M ,N

= A + BT

(5)

where the constants A and B are to be optimized in the present work.

X. Ren et al. / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 35 (2011) 446–454

a

b

c

d

e

f

449

Fig. 1. Calculated enthalpy of mixing of liquid at 384 ± 2 K in the six binary systems and comparison with the experimental data [43]: (a) Cs–K; (b) Cs–Na; (c) Cs–Rb; (d) K–Na; (e) K–Rb; (f) Na–Rb.

3.3. Intermetallic compounds The intermetallic compounds CsNa2 and KNa2 in the Cs–Na and the K–Na binary systems respectively are described as MN2 (M=Cs

or K, and N=Na) and modeled as stoichiometric compounds by a two-lattice model, (M)1/3 (N)2/3 . The parentheses separate species in different sublattices and the constants present the mole fraction of the different sublattices. The Gibbs energy for one mole atoms

450

X. Ren et al. / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 35 (2011) 446–454

a

b

c

d

e

Fig. 2. Calculated activities of elements in different systems at different temperatures and comparison with experimental data [44,45,47,48,50,51]: (a) Cs–K; (b) Cs–Na; (c) Cs–Rb; (d) K–Na; (e) Na–Rb.

of MN2 is given by the following expression: 2 GMN m

= 1/3

0

GBcc M

+ 2/3

0

GBcc N

+ A + BT

3.4. Unary phase (6)

where the parameters A and B are to be optimized for each of the compounds CsNa2 and KNa2 in the present work.

No noticeable solubilities of K in Na, Na in Rb and vice versa in the K–Na and the Na–Rb binary systems have been reported. Thus, the terminal phases in the K–Na and the Na–Rb systems

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a

451

b

c

Fig. 3. Calculated thermodynamic properties of the K–Na binary: (a) and (b) Gibbs free energy and entropy of mixing of liquid at 384 K, compared with the calculated results using the Gibbs–Duhem relation by Ref. [52] and also the calculated values by Zhang et al. [11]; (c) Enthalpy of mixing of Bcc phase at 298 K, compared with the optimization results and the predicted values using the first-principles calculation by Zhang et al. [11].

were treated as the pure substances. The Gibbs energy function of the unary phase is the same as that of the corresponding pure element. 4. Results and discussion On the basis of the thermodynamic data of the pure elements Cs, K, Na and Rb compiled by Dinsdale [54] and the experimental data of the phase equilibria and the thermochemical properties, the thermodynamic parameters of the six binary systems, Cs–K, Cs–Na, Cs–Rb, K–Na, K–Rb and Na–Rb, are optimized using the Parrot module of Thermo-Calc software package [56]. It works by minimizing an error of sum with each piece of selected information given a certain weight. The weight was given and adjusted based upon the original data uncertainties and upon the personal evaluation. For checking of the results, the systems were also calculated using Pandat software [57]. The thermodynamic parameters obtained in the present work are shown in Table 3. 4.1. Evaluation of thermodynamic properties Fig. 4. Calculated Gibbs free energy of liquid at 384 K and comparison with the values calculated using ab initio calculations [53].

Fig. 1 presents the calculated enthalpy of mixing of liquid at 384 K for six binary systems, which are compared with

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Table 3 Thermodynamic parameters of the six binary systems. Binary system Cs–K

Phase Liquid

Bcc_ A2

Cs–Na

Cs–Rb

Liquid

= 426.310 − 0.289T

1 Liq LCs,K

= −173.221 + 1.348T

2 Liq LCs,K

= 177.307 − 1.339T

(Cs, K)

(Cs, Na)

0 Bcc_A2 LCs,K

= 4120.709 − 6.131T

1 Bcc_A2 LCs,K

= −365.975 − 0.036T

0 Liq LCs,Na

= 3603.202 + 0.299T

1 Liq LCs,Na

= −2274.182 + 0.150T

2 Liq LCs,Na

= 1463.096 − 1.509T

GBcc_Cs = 0 GBcc Cs Cs

Bcc_ Na

(Na)

GBcc_Na = 0 GBcc Na Na

CsNa2

(Cs)1/3 (Na)2/3

0 Bcc GCs:Na2 = 1/30 GBcc Cs + 2/3 GNa − 478.531 + 1.222T

(Cs, Rb)

0 Liq LCs,Rb

= −531.166 + 0.616T

1 Liq LCs,Rb

= 22.921 − 0.293T

2 Liq LCs,Rb

= 191.480 − 0.207T

Liquid

Liquid

CsNa

0 Bcc_A2 LCs,Rb

(Cs, Rb)

= 165.587 − 1.062T

2 Bcc_A2 LCs,Rb

= −290.015 + 1.615T

0 Liq LK,Na

(K, Na)

(K, Na)

= 726.052 − 1.386T

1 Bcc_A2 LCs,Rb

= 2938.543 + 0.413T

1 Liq LK,Na

= −748.741 + 1.432T

2 Liq LK,Na

= 240.989 − 1.304T

0 Bcc_A2 LK,Na

= 11799.512 − 10.004T

1 Bcc_A2 LK,Na

= −1298.161 − 1.000T

KNa

KNa2

(K)1/3 (Na)2/3

0 Bcc GK,Na2 = 1/30 GBcc K + 2/3 GNa + 10.126 − 0.921T

Liquid

(K, Rb)

0 Liq LK,Rb

Bcc_ A2

Na–Rb

(Cs, K)

(Cs)

Bcc_ A2

K–Rb

Thermodynamic parameter 0 Liq LCs,K

Bcc_ Cs

Bcc_ A2

K–Na

Model

Liquid

(K, Rb)

(Na, Rb)

= 501.200 − 0.831T

1 Liq LK,Rb

= 9.789 − 0.728T

2 Liq LK,Rb

= 56.082 − 0.160T

0 Bcc_A2 LK,Rb

= 917.318 − 0.764T

1 Bcc_A2 LK,Rb

= −53.585 − 0.349T

0 Liq LNa,Rb

= 4948.676 + 0.341T

1 Liq LNa,Rb

= 1743.399 − 1.417T

2 Liq LNa,Rb

= 251.419 + 1.887T

Bcc_ Na

(Na)

GBcc_Na = 0 GBcc Na Na

Bcc_ Rb

(Rb)

GBcc_Rb = 0 GBcc Rb Rb

the available experimental data [43]. The calculated values agree well with the experimental measurements. Among the six binary systems, only the Cs–Rb binary shows a negative enthalpy of mixing of liquid, Fig. 1(c). For most cases, as two elements become more separated in the periodic table, the positive enthalpy of mixing tends to increase. However, the comparison between the enthalpies of mixing in the Cs–Na and in the Na–Rb binaries is an exception, as in Fig. 1(b) and (f). Fig. 2 illustrates the activities of the related elements in the Cs–K, the Cs–Na, the Cs–Rb, the K–Na and the Na–Rb binary systems at different temperatures with the experimental data from

literatures [44,45,47,48,50,51]. The activities of the elements in the Cs–Na, the K–Na and the Na–Rb binaries vary distinctly with temperatures, while those in the Cs–K and the Cs–Rb binaries vary slightly. Fig. 3(a) and (b) show, respectively, the Gibbs energies and the entropies of mixing of liquid at 384 K for the K–Na binary system. Fig. 3(c) shows the enthalpies of mixing in the Bcc phase at 298 K. In Fig. 3, the calculation results based on both the present optimization and the previous optimization [11] are compared with the Gibbs–Duhem relation [52] in (a) and (b), and with the first-principles calculation [11] in (c). The present

X. Ren et al. / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 35 (2011) 446–454

a

b

c

d

e

f

453

Fig. 5. Calculated phase diagrams of the six binary systems and comparison with the experimental data [12,13,18,19,29,32–34,37]: (a) Cs–K; (b) Cs–Na; (c) Cs–Rb; (d) K–Na; (e) K–Rb; (f) Na–Rb.

optimization illustrates some improvement compared with that of Ref. [11]. Fig. 4 demonstrates the calculated Gibbs energies of liquid at 384 K in the K–Rb binary system and the comparison with the ab initio calculations [53].

4.2. Evaluation of phase equilibria The calculated phase diagrams on the basis of the present thermodynamic descriptions of the six binary systems are shown in Fig. 5 and are compared with the experimental data. In the

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X. Ren et al. / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 35 (2011) 446–454

Table 4 Invariant reactions in the six binary systems. Binary system

Reaction

Cs–K

Liquid ←→ Bcc_A2

Cs–Na

Liquid ←→ Bcc_Cs + CsNa2 Liquid+Bcc_Na ←→ CsNa2

Cs–Rb

Liquid ←→ Bcc_A2

K–Na

Liquid ←→ Bcc_A2 +KNa2 Liquid+Bcc_A2 ←→ KNa2

K–Rb

Liquid ←→ Bcc_A2

Na–Rb

Liquid ←→ Bcc_Na+Bcc_Rb

Compositions (X2 )

0.209 0.209 0.296 0.309

0.319 0.300 0.598 0.592

0.821 0.821

three systems of the Cs–K, the Cs–Rb and the K–Rb, the solid immiscibilities at low temperatures are shown in Fig. 5(a), (c) and (e), which are consistent with the result obtained by Steinberg et al. [16] and the prediction of the existence of phase mixtures according to Soma et al. [15] respectively. For all systems, the calculated temperatures and compositions of the invariant reactions of the six binary systems are listed in Table 4 and are compared with the experimental measurements obtained by different authors. 5. Conclusion The thermodynamic descriptions of the six binary systems, Cs–K, Cs–Na, Cs–Rb, K–Na, K–Rb and Na–Rb, are assessed based on the experimental data of the phase equilibria and the thermodynamic properties in the present work. The calculated phase diagrams and thermodynamic properties are compared favorably with the experimental measurements from literatures. A set of self-consistent thermodynamic parameters was obtained for each of the systems. The calculated results can reproduce the experimental data well. Acknowledgments The authors would like to acknowledge National Natural Science Foundation of China (No. 50731002 and No. 50671009) for their financial support of this project. Thanks to Royal Institute of Technology and CompuTherm LLC for supplying the Thermo-Calc and Pandat Software packages respectively. Appendix. Supplementary data Supplementary material related to this article can be found online at doi:10.1016/j.calphad.2011.06.005. References [1] M. Aditya, Commun. Theor. Phys. 54 (2010) 159–166. [2] Y.A. Odusote, Physica B 403 (2008) 2877–2883. [3] Hans Ulrich Borgstedt, Cherian K. Mathews, Applied Chemistry of the Alkali Metals, Plenum Press, 1987. [4] C.W. Bale, A.D. Pelton, Bull. Alloy Phase Diagr. 4 (1983) 379–382. [5] C.W. Bale, Bull. Alloy Phase Diagr. 3 (1982) 310–313. [6] C.W. Bale, A.D. Pelton, Bull. Alloy Phase Diagr. 4 (1983) 382–384. [7] C.W. Bale, Bull. Alloy Phase Diagr. 3 (1982) 313–318. [8] C.W. Bale, A.D. Pelton, Bull. Alloy Phase Diagr. 4 (1983) 385–387. [9] C.W. Bale, Bull. Alloy Phase Diagr. 3 (1982) 318–321. [10] R.H. Davies, A.T. Dinsdale, S.M. Hodson, J.A. Gisby, N.J. Pugh, T.I. Barry, T.G. Chart, MTDATA — The NPL databank for metallurgical thermochemistry in User aspects of phase diagrams, in: Proceedings of the international conference held at the Joint Research Centre, Petten, the Netherlands, 25–27th June 1990.

0.505 0.505 0 0 1 1 0.470 0.470 0.046 0.030 0.992 0.991 0.667 0.667 0 0

0.667 0.667 0.667 0.667

0.667 0.667 0.667 0.667

1 1

T (K)

Reaction type

Reference

235.0 235.1 241.3 241.5 265.2 264.9 282.7 282.4 260.5 261.9 280.1 281.0 307.0 307.1 268.7 267.7

Congruent Congruent Eutectic Eutectic Perutectic Peritectic Congruent Congruent Eutectic Eutectic Peritectic Peritectic Congruent Congruent Eutectic Eutectic

[12] This work [29] This work [29] This work [13] This work [37] This work [37] This work [13] This work [19] This work

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