Density, Surface Tension, and Viscosity of Liquid Pb–Sb Alloys

Article Cite This: J. Chem. Eng. Data XXXX, XXX, XXX−XXX

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Density, Surface Tension, and Viscosity of Liquid Pb−Sb Alloys Tomasz Gancarz* and Wladyslaw Gasior Institute of Metallurgy and Materials Science, Polish Academy of Sciences, 30-059 Krakow, Poland S Supporting Information *

ABSTRACT: With the use of the discharge crucible method, the density, surface tension, and viscosity of Pb−Sb liquid alloys were determined. The experiments were performed in the temperature range between 573 and 1050 K. The measurements were conducted in a glovebox filled with high-purity argon, which was continuously circulated between the glovebox and the purification system. The measurements were conducted for four alloys of the composition 16.75, 40, 60, and 90 atom % Sb. The experimental data on the surface tension and viscosity were compared to those from modeling. The new experimental data and the corresponding model-predicted values exhibit good agreement, and only small differences in linear changes between the surface tensions of Pb and Sb can be observed. This can be attributed to small negative deviations of the thermodynamic properties from Raoult’s law and a relatively small difference between the surface tensions of the pure components. In the case of viscosity, the measured data were in excellent agreement with those calculated from one of the models and those measured earlier. The obtained results for the densities of Pb−Sb liquid alloys were only a little higher than the densities of ideal solutions.

1. INTRODUCTION

mechanical properties and increase the porosity of the cast alloys in the foundry process.8 The thermophysical properties of liquid Pb and Sb were taken from previous studies.9−11 The density, surface tension, and viscosity of Sb9 were obtained by the same method used in this study. The properties of Pb were measured using the dilatometric technique10 for density, the maximum bubble pressure technique10 for surface tension, and the capillary method11 for viscosity. The literature contains limited data for Pb−Sb alloys. Sato and Munakata12 studied density, Ma et al.13 examined surface tension, and Sato and Munakata12 and Gebhardt and Kostlin14 investigated viscosity. In addition, Gou et al.15 suggested that the higher temperature of liquid Pb20Sb80 alloy caused the formation and stabilization of Sb clusters and a Peierls barrier16 in the liquid, which caused a significant increase in the activation energy of the viscous fluid. Because of the high viscosity values obtained for all of the tested alloys and the lack of structural study, further work is needed to expand upon the theories.15

Advances in solar energy require the development of battery technology for energy storage. The liquid Pb−Sb−Li system proposed by the Sadoway group1 shows the possibility of using these alloys as liquid batteries. However, binary alloys should be investigated fully before ternary Pb−Sb−Li alloys can be further explored. The Li−Sb−Pb alloys, which could be used for batteries with liquid lithium negative electrodes, working at higher temperature with a molten salt electrolyte, and a liquid antimony−lead alloy positive electrode, allow self-segregation by density into three distinct layers due to the immiscibility of the contiguous salt and metal phases. The all-liquid construction confers the advantages of higher current density, longer life cycle, and simpler manufacturing of large-scale storage systems.2 The addition of Sb to Pb caused improved mechanical properties and better castability, allowing complicated shapes to be created.3 According to the Pb−Sb phase diagram,4 with the eutectic point at 524.5 K for 16.75 atom % Sb, Pb−Sb alloys with 1−10 atom % Sb can be used in continuous casting and gravity casting applications, most commonly for lead electrodes for batteries.5−7 Higher Sb content in Pb−Sb alloys could lead to dendritic microstructure and segregation, which would reduce corrosion resistance and © XXXX American Chemical Society

Received: December 1, 2017 Accepted: April 12, 2018

A

DOI: 10.1021/acs.jced.7b01049 J. Chem. Eng. Data XXXX, XXX, XXX−XXX

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In order to verify the thesis presented by Guo et al.15 that the structure of liquid Pb−Sb alloys with higher Sb content changes and that clusters of Sb are created in the liquid, measurements of four Pb−Sb alloys were proposed. The discharge crucible (DC) method was used to obtain the density, surface tension, and viscosity of liquid Pb−Sb alloys. The obtained experimental results were compared with the models.

pure Sb were also obtained using the DC method, as for the Pb−Sb alloys, but these data were used to verify the DC method using pure Sb, Sn, and Zn, as presented in a previous paper.9

3. RESULTS AND DISCUSSION 3.1. Thermophysical Properties of Liquid Pb and Sb. During the study of thermophysical properties of liquid Pb−Sb alloys, the corresponding properties of the pure elements Pb and Sb were also evaluated from literature data.9−56 The thermophysical properties for pure Pb were chosen from literature data on density from Iida et al.,28 data on surface tension from Ma et al.,13 and data on viscosity from Rothwell.44 The thermophysical properties for pure Sb were determined previously by Gancarz et al.9 using the DC method (as for Pb− Sb alloys). The experimental data describing temperature dependence are collected in Tables 1−3 and presented in Figures 1−6 together with the literature data. 3.1.1. Density. Figure 1 presents the density of pure Pb, obtained from the literature data over the temperature range of

2. EXPERIMENTAL SECTION Pure metal Pb (99.999%) and Sb (99.99%) were used for the preparation of the alloys. The chemical compositions of the Pb−Sb alloys were 16.75, 40, 60, and 90 atom % Sb. The alloys were prepared directly in the measured crucible. After melting and stabilization of the temperature, the liquid alloys were mixed, and the experiment began. The measurements were performed in a glovebox with a high-purity Ar (6N) atmosphere provided with a purification system (catalytic Cu, molecular sieve, high-temperature reactor with shavings of Ti, working at a temperature of 1123 K). The levels of O2 and H2O were below 0.001% as measured by solid-state analyzers for oxygen and moisture. The DC method, described in detail in previous work,17−20 was used to obtain the thermophysical properties of Pb−Sb alloys. The DC method leads to the resolution of eq 1, which is based on the measured mass changes over time and describes the correlation of the change in the height of the liquid in the crucible, thus allowing the density, surface tension, and viscosity of the investigated liquid to be determined: he = f (Ve) ⎧ ⎪ 1⎪ ⎨ = 2g ⎪ ⎡ ⎪ ρ⎢⎣a4 ⎩ σ + ρgr0

Ve 3

( ) 2r0Ve η

2

( )

+ a3

2r0Ve η

+ a2

( ) 2r0Ve η

⎫2 ⎪ ⎪ ⎬ ⎤⎪ + a1⎥ ⎪ ⎦⎭ (1)

Figure 1. Temperature dependence of the density of pure Pb from literature data: black ▼, Strauss et al.;21 blue ★, Kirshenbaum et al.;22 red ◆, McAlister;23 green ●, Thresh et al.;24 orange +, Lucas;25 brown ○, Ruppersberg and Spiecher;26 purple ▲, Khairulin and Stankus;27 cyan ×, Iida et al.28

where ρ is the density of the liquid (kg/m ), g is the gravitational acceleration (m/s2), r0 is the orifice radius in the bottom of crucible (m), Ve is the free flow rate (m3/s), η and σ are the viscosity (mPa·s) and surface tension (mN/m), respectively, and a1, a2, a3, and a4 are constants in the polynomial describing the dependence of the discharge coefficient (Cd) on the Reynolds number (Re).18 The thermophysical properties were calculated using a numerical program based on the Hooke−Jeeves method.18−20 The obtained results for density, surface tension, and viscosity for the Pb−Sb alloys are collected in Tables 1−3, respectively. The results for density and surface tension as functions of temperature were elaborated by a linear equation and those for viscosity by an Arrhenius equation. The results are presented in Tables 1−3 along with the equation parameters, estimated errors, and values calculated at a temperature of 923 K. The experiment was conducted for four chemical compositions of Pb−Sb alloys with Sb contents of 16.75, 40, 60, and 90 atom %, and the experimental data are collected in Tables S1− S3. Because of the large amount of literature data9−56 on the physicochemical properties of pure Pb and Sb, the experiment was not repeated for the pure elements because these properties were obtained using several methods. The data for 3

605−1950 K using the Archimedes technique21,22,25,27 and dilatometer,23 pycnometer,24 and maximum bubble pressure (MBP)26 methods. The recommended parameters of the linear temperature dependence of the density of pure Pb are presented in Table 1 and correspond to the values from Iida et al.28 Figure 2 presents the density data for pure Sb available in the literature. The values obtained by Gancarz et al.,9 Sato and Munakata,12 and Iida et al.28 show very good agreement, with differences lower than 0.5%. The results reported by Greenway29 and Fisher and Phillips30 are slightly higher in comparison with those cited above, although the observed differences do not exceed 1%. 3.1.2. Surface Tension. The literature data on the surface tension of pure Pb are presented in Figure 3. The differences among the literature data for the surface tension of pure Pb from Gasior et al.,10 Ma et al.,13 White,32 Hoar and Melford,33 Lang,34 Matuyama,35 Plevachuk et al.,36 Bircumshaw,37 and Adachi et al.,31 are in the range below 5%. Gasior et al.10 used B

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Figure 2. Temperature dependence of the density of pure Sb from the literature: brown dashed line, Gancarz et al.;9 orange dot-dot-dashed line, Sato and Munakata;12 blue dot-dashed line, Iida et al.;28 blue dashed line, Greenway;29 black solid line, Fisher and Phillips.30

Figure 4. Temperature dependence of the surface tension of pure Sb from the literature: red ◇, Gancarz et al.;9 green solid line and black dashed line, Keene;39 purple ▽, Gasior et al.;40 orange □, Somol and Berenek;41 blue ○, Lauerman and Sauerwald;42 light blue △, Lazarev.43

suggested by Keene39 for the value of the surface tension of Sb, only those works where the surface tension was higher than the average results were taken into account. The results obtained by Lazarev43 show a slightly lower surface tension value compared with the others. 3.1.3. Viscosity. The viscosity data for pure Pb are presented in Figure 5. The literature data were taken from Gasior et al.,11

Figure 3. Temperature dependence of the surface tension of pure Pb from literature data: purple □, Gasior et al.;10 black ○, Ma et al.;13 red dashed line, Adachi et al.;31 blue +, White;32 blue □, Hoar and Melford;33 green ☆, Lang;34 black dot-dashed line, Matuyama;35 green △, Plevachuk et al.;36 orange dashed line, Bircumshaw;37 brown ▽, Abdel-Aziz and Kirshah.38

the MBP method to obtain the surface tension, but the obtained data show the lowest values. However, on the basis of the very good agreement of the surface tension data for Pb−Sb alloys with those from Ma et al.13 (see Table 2), the surface tension values from Ma et al.13 for pure Pb were recommended and used in modeling. Figure 4 presents the literature data on the surface tension of pure Sb. The elaborated temperature dependence (green solid line) based on the experimental data (as the average of all literature data), suggested in the review by Keene,39 shows very good agreement with the surface tension values measured by Gancarz et al.,9 Gasior et al.,40 Somol and Beranek,41 and Lauerman and Sauerwald.42 However, the equation for the surface tension of Sb proposed by Keene39 (black dashed line) is about 5% higher compared with the experimental results. Keene39 assumed that the purer metal has a higher surface tension, which leads to a more negative temperature gradient of the surface tension, dσ/dT. To determine the equation

Figure 5. Temperature dependence of the viscosity of pure Pb from literature data: red long-dashed line, Gasior et al.;11 black short-dashed line, Gebhardt and Kostlin;14 blue □, Iida et al.;28 green +, Rothwell;44 orange ☼, Sauerwald;45 blue ☆, Esser et al.;46 purple ▽, Gering and Sauerwald;47 magenta △, Yao and Kondic;48 blue ○, Johns and Davis;49 pink dot-dashed line, Piertuk and Pietruk;50 brown ◆, Thresh and Crawley.51

Gebhardt and Kostlin,14 Iida et al.,28 Rothwell,44 Sauerwald,45 Esser et al.,46 Gering and Sauerwald,47 Yao and Kondic,48 Johns and Davies,49 Piertuk and Pietruk,50 and Thresh and Crawley.51 The highest values were obtained by Gasior et al.11 using the capillary method, and the values obtained by other authors are lower by less than 10%. The curves of the slopes obtained by Gasior et al.,11 Iida et al.,28 Gebhardt and Kostlin,14 Rothwell,44 and Thresh and Crawley51 are very similar, although the C

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observed maximal differences are as cited above. The recommended viscosity data collected in Table 3 that were used for modeling were taken from Rothwell.44 Figure 6 presents the viscosity data for pure Sb. The literature data of Gancarz et al.,9 Sato and Munakata,12

Figure 7. Temperature dependences of the densities of Pb−Sb alloys. Experimental data from this work: black ★, Pb16.75Sb; blue ◆, Pb40Sb; red ▲, Pb60Sb; green ●, Pb90Sb. Literature data for the pure elements: orange dashed curve, Gancarz et al.9 for Sb; purple solid line, Iida et al.28 for Pb. Literature data for Pb−Sb alloys from Sato and Munakata:12 green □, Pb20.2Sb; pink △, Pb36.7Sb; blue ○, Pb57.5Sb; brown ◇, Pb80Sb.

Figure 6. Temperature dependence of the viscosity of pure Sb from literature data: red ☆, Gancarz et al.;9 orange ☼, Sato and Munakata;12 green ○, Gebhardt and Kostlin;14 purple □, Iida et al.;28 black ◇, Fisher and Phillips;30 purple ×, Herwing and Wobst;52 blue ▽, Postolov et al.;53 pink △, Bienias and Sauerwald;54 olive +, Nakajima.55

3.2.3. Viscosity. The temperature dependences of the viscosities for the Pb−Sb alloys are presented in Figure 9. The experimental results were described by the Arrhenius equation for each alloy. The obtained parameters (activation energy and pre-exponential factor) are gathered in Table 3. The viscosities of the studied alloys, as in the case of density and surface tension, diminish with increasing Sb content. In Figure 9, the experimental results are compared with the literature data from Sato and Munakata12 and Gebhardt and Kostlin.14 The viscosity data obtained by Sato and Munakata12 for alloys of the composition Pb42.5Sb and Pb80Sb are slightly higher than the experimental results for Pb40Sb and Pb90Sb obtained here. For the lower Sb content in the Pb19.5Sb alloy, the viscosity data of Sato and Munakata12 at lower temperature are similar to those for the Pb16.75Sb alloy investigated in this study. However, above around 750 K the viscosity values are close to those for Pb40Sb. The viscosity data obtained by Gebhardt and Kostlin14 for the Pb15.9Sb and Pb36.3 alloys show better agreement with the experiment results for Pb16.75Sb and Pb40Sb obtained here. The data obtained by measurements carried out by Gou et al.15 are much higher, over 50% in fact, than our experimental data for viscosity. To aid the clarity of Figure 9, they are not included. 3.3. Modeling. 3.3.1. Modeling of Density. As proposed in the previous work,56,57 the ideal density, ρideal, for the Pb−Sb system was calculated using eq 2: 1 ρideal = XPbmPb XSbmSb XPbmPb + XSbmSb XPbmPb + XSbmSb + ρ ρ

Gebhardt and Kostlin,14 Iida et al.,28 Fisher and Phillips,30 Herwing and Wobst,52 Postolov et al.,53 Bienias and Sauerwald,54 and Nakajima55 show good agreement for the viscosity of Sb. The highest viscosity value was obtained by Fisher and Phillips,30 although the difference did not exceed 7%. Taking into account that the thermophysical properties (density, surface tension, and viscosity) of pure Sb were obtained using the same DC method and bearing in mind the very good agreement of the obtained results with those in the literature, the results of Gancarz et al.,9 the parameters of which describing the temperature dependence are collected in Tables 1−3, were recommended for modeling. 3.2. Thermophysical Properties of Pb−Sb Alloys. 3.2.1. Density. The thermophysical properties of four Pb−Sb alloys with Sb concentrations of 16.75, 40, 60, and 90 atom % (denoted as Pb16.75Sb, Pb40Sb, Pb60Sb, and Pb90Sb, respectively) were measured over the temperature range from 573 to 923 K. The determined temperature dependences of the densities for the Pb−Sb alloys are presented in Figure 7 together with literature data for Pb28 and Sb.9 The parameters of the linear equations describing the obtained experimental results are collected in Table 1. As shown in Figure 7, the density of the Pb−Sb liquid alloy decreases as the Sb concentration in the alloy increases. This is in very good agreement with the data of Sato and Munakata.12 3.2.2. Surface Tension. Figure 8 shows the temperature dependences of the surface tensions of Pb−Sb alloys. The experimental results for each alloy were described by a linear equation, and their parameters are collected in Table 2. The surface tension decreases in a manner similar to the density as the Sb content in the Pb−Sb alloy increases. The experimental data obtained in this study for Pb−Sb alloys show values close to those determined by Ma et al.13

Pb

Sb

(2)

where XPb and XSb are the atomic concentrations of Pb and Sb, respectively, and mPb and mSb are the atomic weights and ρPb and ρSb the densities of the pure components Pb and Sb, as taken from Iida et al.28 and Gancarz et al.,9 respectively. The determined experimental density values for Pb−Sb liquid alloys were interpreted using the model proposed by Brillo and Egry,58,59 which is expressed by eq 3: D

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Table 1. Coefficients of the Linear Temperature Dependence of the Density (ρ = A + BT) and Their Uncertainties,a Density Values Calculated at 923 K, and Standard Deviations of the Fitsb for Pb−Sb Alloysc atom % Sb

A (g·cm−3)

u(A)

B (g·cm−3·K−1)

u(B)

ρ(923 K) (g·cm−3)

s(ρ)

11.471 10.84 9.98 8.99 7.691 6.982

− 0.02 0.02 0.05 0.025 −

−0.00132 −0.00113 −0.00108 −0.00097 −0.00082 −0.00075

− 0.00003 0.00002 0.00006 0.00003 −

10.254 9.979 8.983 8.095 6.936 6.485

− 0.017 0.008 0.003 0.002 −

28

0 16.75 40 60 90 1009 a

Estimated from linear regression error as implemented in the Grapher software package. bEstimated from computed differences between 1 N experimental values (ρexp) and values calculated from the linear density fit (ρfit): s(ρ) = N ∑i = 1 |ρexp − ρ fit |, where N is the number of experimental points. cThe standard uncertainties u are u(T) = 1 K, u(p) = 0.1 kPa at atmospheric pressure (p = 0.1 MPa), and u(X) = 0.0001 for Pb and Sb.

Figure 8. Temperature dependences of the surface tensions of Pb−Sb alloys. Experimental data from this work: black ★, Pb16.75Sb; blue ◆, Pb40Sb; red ▲, Pb60Sb; green ○, Pb90Sb. Literature data for the pure elements: black dashed line, Ma et al.13 for Pb; orange solid line, Gancarz et al.9 for Sb. Literature data for Pb−Sb alloys from Ma et al.:13 green ◇, Pb17.5Sb; pink □, Pb40Sb; blue ○, Pb60Sb; brown △, Pb80Sb.

ρ=

XPbmPb + XSbmSb m m XPb ρPb + XSb ρSb + VE Pb

Figure 9. Temperature dependences of the viscosities of the Pb−Sb alloys. Experimental data from this work: black ★, Pb16.75Sb; blue ◆, Pb40Sb; red ▲, Pb60Sb; green ○, Pb90Sb. Literature data for the pure elements: black dashed line, Gebhardt and Kostlin14 for Pb; orange solid line, Gancarz et al.9 for Sb. Literature data for Pb−Sb alloys from Sato and Munakata:12 purple □, Pb42.5Sb; green ×, Pb80Sb. Literature data for Pb−Sb alloys from Gebhardt and Kostlin:14 brown ○, Pb15.9Sb; pink ▽, Pb36.3Sb; blue △, Pb47.1Sb.

VE = XPbXSbVX

(3)

Sb

(4)

in which VX is a type of interaction parameter. In our case, the value of VX obtained from such a fit is equal to approximately −1.29 cm3·mol−1. The isothermal section of density at 923 K, presented in Figure 10, shows very good agreement with the experimental and model data.59 The experimental results for the density of

where XPb and XSb are the atomic concentrations of Pb and Sb, respectively, mPb and mSb are the atomic weights of Pb and Sb, respectively, and VE is the excess volume, which depends on the concentrations according to the following equation:

Table 2. Coefficients of the Linear Temperature Dependence of the Surface Tension (σ = A + BT) and Their Uncertainities,a Surface Tension Values Calculated at 923 K, and Standard Deviations of the Fitsb for Pb−Sb Alloysc atom % Sb 13

0 16.75 40 60 90 1009

A (mN·m−1)

u(A)

B (N·m−1·K−1)

u(B)

σ(923 K) (mN·m−1)

s(σ)

533.0 478.6 467.0 450.8 445.2 439.5

− 0.3 0.9 0.6 1.8 −

−0.112 −0.072 −0.072 −0.065 −0.079 −0.075

− 0.001 0.001 0.001 0.002 −

429.6 412.0 400.4 390.7 372.4 370.3

− −0.1 −0.2 0.1 0.2 −

a

Estimated from linear regression error as implemented in the Grapher software package. bEstimated from computed differences between 1 N experimental values (σexp) and values calculated from the linear surface tension fit (σfit): s(σ ) = N ∑i = 1 |σ exp − σ fit|, where N is the number of c experimental points. The standard uncertainties u are u(T) = 1 K, u(p) = 0.1 kPa at atmospheric pressure (p = 0.1 MPa), and u(X) = 0.0001 for Pb and Sb. E

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Table 3. Coefficients from the Arrhenius Relation for the Temperature Dependence of the Viscosity (η = A exp(−Ea/RT)) and their Uncertainties,a Viscosity Values Calculated at 923 K, and Standard Deviations of the Fitsb for Pb−Sb Alloysc atom % Sb 44

0 16.75 40 60 90 1009

A (mPa·s)

u(A)

Ea (J·mol−1)

u(Ea)

η(923 K) (mPa·s)

s(η)

0.543 0.530 0.455 0.342 0.167 0.160

− 0.002 0.002 0.001 0.001 −

7889.6 7495.0 7960.9 9814.1 16090.8 16938.1

− 19.3 23.7 31.6 68.8 −

1.603 1.407 1.283 1.227 1.361 1.512

− 0.018 0.017 0.002 0.004 −

a

Estimated from linear regression error as implemented in the Grapher software package. bEstimated from computed differences between 1 N experimental values (ηexp) and values calculated from the Arrhenius viscosity fit (ηfit): s(η) = N ∑i = 1 |ηexp − η fit|, where N is the number of experimental points. cThe standard uncertainties u are u(T) = 1 K, u(p) = 0.1 kPa at atmospheric pressure (p = 0.1 MPa), and u(X) = 0.0001 for Pb and Sb.

the bulk and surface phases, which were given by Tanaka et al.62 and Gasior.63,64 The new equations in Gasior63,64 for the calculation of the monatomic surface layer and the relation between the excess Gibbs free energies in the bulk and surface phases allow the excess Gibbs free energy of the bulk phase to be calculated on the basis of the surface tension data with much higher accuracy. The modeled isotherms of the surface tension are shown in Figure 11 together with the values for ideal

Figure 10. Density vs XSb (atom %) isotherms at 923 K for Pb−Sb alloys: ◆, experimental values; blue dashed line, Brillo model;59 red solid line, ideal changes between the densities of Pb and Sb.

the Pb−Sb system show a slight positive deviation from linear changes, similar to those calculated for the densities of ideal solutions. According to the thermodynamic properties of the Pb−Sb liquid phase,60 the activities of both components show a very small negative deviation from Raoult’s law, and the mixing enthalpy of the liquid phase varies between −40 and −70 J/mol for composition XSb = 0.5. The density of Pb−Sb alloys should show a positive deviation, as confirmed by the obtained experimental density data, in line with values from the Brillo model.59 3.3.2. Modeling of Surface Tension. To model the surface tension of Pb−Sb alloys, the Butler model61 was used, which in general form can be written as follows: σ = σi +

aiB

aS RT ln iB , Ai ai

Figure 11. Surface tension vs XSb (atom %) isotherms at 923 K for Pb−Sb alloys: ●, experimental values; red dashed line, Butler model for ideal solutions; blue dot-dashed line, Butler model using the relation of the monatomic surface layer and surface excess Gibbs energy given by Tanaka et al.;62 black solid line, Butler model with the polarized atoms model given by Gasior.63,64

solutions (excess Gibbs energy equals 0). Instead of the low differences between the surface tensions of Pb and Sb and the small negative deviations of the excess Gibbs free energy from the ideal solution values, the differences among all of the isotherms are small. In addition, all of the isotherms possess values slightly higher than the experimental values. 3.3.3. Modeling of Viscosity. As presented in a previous work57,58 the viscosity models that allow the viscosities of liquid alloys to be calculated use thermodynamic and physical properties as well as density, molar volume, atomic mass, and atomic radius. In this study, the models of Kozlov et al.,65 Kaptay,66 Morita et al.,67 and Gasior68 were used for viscosity calculations. The viscosities of Pb and Sb were taken from Rothwell44 and Gancarz et al.,9 respectively. Figure 12 presents the viscosity isotherm section at 923 K as determined in this

i = Pb, Sb (5)

aiS

where and are the activities of component i in the bulk (B) and monatomic surface phases (S) and Ai is the surface area of the molar monomeric layer of component i. The greatest advantage of the Butler model61 is that the surface tension for a multicomponent system can be calculated using only the surface tensions and densities of the pure elements in the alloy and the partial Gibbs free energies of the components in the liquid phase. The calculations were conducted for two different models of the monatomic surface layer and the relation between the excess Gibbs free energies in F

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the measurement of physical properties of the Pb−Sb−Li ternary system.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jced.7b01049. Experimental data (Tables S1−S3) (PDF)

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

Tomasz Gancarz: 0000-0001-7737-2193 Funding

This work was financed by the Ministry of Science and Higher Education of Poland as the Research Program of IMMS PAS Krakow, Task No. Z-9 in 2016−2019.

Figure 12. Viscosity vs XSb (atom %) isotherm at 923 K for Pb−Sb alloys: black ◆, experimental values from this work; green dot-dotdashed line, model of Kozlov et al.;65 red short-dashed line, Kaptay model;66 purple long-dashed line, model of Morita et al.;67 solid orange line, Gasior model;68 blue ▲, literature data from Gebhardt and Kostlin.14

Notes

The authors declare no competing financial interest.

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REFERENCES

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study and compares it with values obtained from the model65−68 and experimental data.14 The experimental results obtained in this study and by Gebhardt and Kostlin14 show good agreement with the values calculated by applying the Gasior model.68 The experimental data for the viscosities of Pb−Sb alloys show negative deviations from the ideal solution model, which is confirmed by the models of Kozlov et al.65 and Gasior.68 The values for the pure elements play an important role in calculating viscosity in the modeling. The data calculated using the Gasior model68 show the best correlation with those obtained experimentally.

4. CONCLUSIONS In this work, new experimental data on the density, surface tension, and viscosity of four Pb−Sb alloys were obtained as functions of temperature and composition. The measurements were performed for temperatures ranging between 573 and 1050 K by the DC method. The obtained experimental data for density and surface tension show very good agreement with literature data of Sato and Munakata12 and Ma et al.,13 respectively. The viscosity values obtained in this study and reported in the literature by Gebhardt and Kostlin14 for Pb−Sb alloys up to 50 atom % Sb were observed to be similar. However, the significantly increased viscosity for the Pb90Sb alloy suggested by Guo et al.15 was not observed, and neither was this confirmed by modeled values. The slopes of both density and surface tension with respect to temperature showed negative values, as did the linear temperature dependence in the investigated temperature regions. The calculated activation energy in the Arrhenius equation increased with increasing Sb concentration in the studied Pb−Sb alloys. The opposite tendency was observed for the pre-exponential parameter. Very good agreement was observed between the experimental density data and the values obtained using the Brillo model.59 Similar agreement was observed for the measured and modeled surface tensions determined in this work and by Gebhardt and Kostlin14 and for the viscosity modeled using the Gasior model.68 Results from this study will be used for future work in G

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