Thermodynamic activities in silicon binary melts
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Abstract
The thermodynamic activities in the silicon binary melts with Al, Ca, Mg, Fe, Ti, Zn, Cu, Ag, Au, Sn, Pb, Bi, Sb, Ga, In, Pt, Ni, Mn and Rh are studied. The silicon activities along the liquidus are calculated through a quasi-regular solution model using the recently determined liquidus constants for the silicon binary systems. The silicon activities at its melting point are calculated considering regular solution approximation. The activities of the other melt component at the silicon melting point are also calculated through the graphical integration of the Gibbs–Duhem equation for the activity coefficient, which are further utilized to determine the corresponding activities along the liquidus. The calculated activities are presented graphically, and it is indicated that the results are consistent with the reported activity data in the literature. The activities in the dilute solutions are also calculated graphically. Moreover, the activities of particular dilute solute elements in silicon are calculated through a simple formula, which is a function of the liquidus constants.
Introduction
Silicon has a wide range of applications in metallurgical, semiconductor, photovoltaic (PV), electronic and chemical industries. Silicon transforms to its liquid state during its production and purification through metallurgical processes. In particular, molten silicon is treated through various metallurgical processes, such as slag refining, plasma refining, vacuum refining, and directional solidification to produce solar grade silicon. In these processes, the solute impurities in molten silicon are reduced to very low concentrations to fulfil the requirements of high silicon purity of 99.9999 wt% for PV applications. Speaking in general, the thermodynamic properties of the silicon melts are crucially important in any silicon-refining technique, and developing basic knowledge on the thermodynamics of silicon melts is needed.
The aim of the present study is to study the chemical activities in silicon binary melts through the application of a novel analytical method without the application of thermodynamic software and programming. The activities of the melt components are calculated, and the obtained results are evaluated with the experimentally and theoretically determined activities in the literature. Moreover, the activities of the silicon binary dilute solutions are determined, which are important regarding the production of solar grade silicon.
Activity calculation methodology
Activity of silicon
Activity of element Me
Liquidus constants and eutectic compositions for silicon binary systems [1]
System | a | b | T_{eut}^{a} (K) | X_{Si, eut} |
---|---|---|---|---|
Si–Al | −9789.7 | 3.7402 | 850.1 | 0.119 |
Si–Ca | −83,820 | −0.2271 | 1,296 | 0.694 |
Si–Mg | −73,857 | −32.612 | 1,217 | 0.527 |
Si–Fe | −99,915 | −47.18 | 1,479 | 0.724 |
Si–Ti | −32,8815 | −167.32 | 1,591 | 0.87 |
Si–Zn | 29,206 | 16.26 | 692.2 | 2.5 × 10^{−4} |
Si–Cu | −41,822 | −24.056 | 1,075 | 0.304 |
Si–Ag | −32,763.1 | −32.42 | 1,108 | 0.116 |
Si–Au | −49,248 | −24.058 | 633 | 0.183 |
Si–Pt | −197,194 | −93.921 | 1,220 | 0.656 |
Si–Sn | 31,162 | 4.0289 | 505 | 2 × 10^{−7} |
Si–Pb | 79,639 | 17.079 | 600 | 1.3 × 10^{−9b} |
Si–Bi | 61,648 | 8.3531 | 544 | 1.7 × 10^{−9b} |
Si–Sb | 14,273 | −5.99 | 902.7 | 3.2 × 10^{−3} |
Si–Ga | 14,844 | 4.75 | 302.8 | 3 × 10^{−10} |
Si–In | 46,903 | 13.97 | 429.6 | 2.5 × 10^{−10} |
Si–Pd | −73,424 | −28.385 | 1,165 | 0.52 |
Si–Ni | −89,383 | −36.856 | 1,266 | 0.591^{c} |
Si–Mn | −123,424 | −64.86 | 1,415 | 0.674 |
Si–Rh | −201,331 | −102.06 | 1,333 | 0.686 |
Results and discussion
Activities in the silicon melts
The activities calculated for the silicon binary melts are graphically presented. Moreover, they are evaluated and discussed with regard to the literature data.
Si–Al alloys
The behaviour of Al in low concentrations, which is important for solar grade silicon materials, can be seen in Fig. 2c. The activity curve for Miki et al. was calculated using their reported activity data: \( { \ln }\gamma_{\text{Si}}^{ \circ } \) = −3610/T + 0.452, and \( \varepsilon_{\text{Al}}^{\text{Al}} \) = 100,000/T − 40.1 [10] at 1,700 K (1,427 °C). There is a good correlation between the calculated curve of the present study and the calculated curve from their activity data for the dilute solutions with aluminium molar fractions smaller than X_{Al} = 0.05. The calculated activity curve fits also with the measured Al activities in silicon dilute solutions [9].
Si–Ca alloys
The activity data for Si–Ca melts in low calcium concentrations are relatively close as seen in Fig. 3b. The present calculated activities are consistent with the experimental measurements [10, 17]. In addition, the present calculation results are relatively close to the calculated curves obtained using the reported activity data: \( { \ln }\gamma_{\text{Ca}}^{ \circ } \) = −6.926, and \( \varepsilon_{Ca}^{Ca} \) = 10.858 [10, 21] at 1,723 K (1,450 °C).
Si–Mg alloys
Si–Fe alloys
Si–Ti alloys
Si–Zn alloys
Si–Cu alloys
Si–Ag alloys
Si–Au alloys
Si–Pt alloys
Si–Sn alloys
Si–Pb alloys
Si–Bi alloys
Si–Sb alloys
Si–Ga alloys
Si–In alloys
Si–Pd alloys
Si–Ni alloys
Si–Mn alloys
Si–Rh alloys
Evaluation of the calculated activities
The calculated activities of the present study and their comparison with the literature data indicate that the applied methodology can be used properly to determine the thermodynamic activities in silicon binary melts. The calculated silicon activities along the liquidus and at the silicon melting point for the silicon binary alloys with Al, Ag, Au, Sn, Pb and Ga are consistent with the reported data in literature. Moreover, the calculated activities for the other melt component through Gibbs–Duhem integration are also in agreement with literature data. All these binary alloys are simple eutectic systems containing no intermediate phases. This may accordingly indicate that the calculated activities for the similar silicon binary systems with Zn, Bi, Sb and In are reliable too.
The calculated silicon activities along the liquidus for the silicon binary systems containing intermediate phases such as Si–Ca, –Mg, –Fe, –Ti, –Cu, –Ni and –Mn systems are in agreement with regard to the literature data. Hence, the calculated silicon activities along the liquidus for Si–Pt, –Pd and –Rh melts are expected to be reliable. The silicon activities in the melts under isothermal conditions, i.e. 1,414 °C, are always in a small scale higher than what we obtain along the liquidus. The difference between these activities increases with decreasing silicon concentration due to the temperature decrease along the liquidus and its effect to the Eq. (4). A proper correlation between the calculated activities at 1,414 °C with the experimentally measured activities at different temperatures was observed for the binary melts which indicates that the activity of silicon in its binary systems is not significantly affected by temperature, while it is strongly dependent on the chemical composition. The calculated activities for the other melt component of the melts were consistent with the majority of the literature data, relying on the properly calculated silicon activities. The effect of the preliminary calculated silicon activity on the calculation accuracy of the other component activity is illustrated for the Si–Cu melts. This case indicates that the applied calculation method is very sensitive to the initially calculated silicon activities.
The behaviour of the silicon melts
A correlation between the thermodynamics behaviour of the silicon binary melts and the metallic properties of the other element is found in this study. The binary melts consisted of silicon and alkali earth metals (Ca, Mg) and transition metals (Ti, Fe, Mn, Ni, Cu, Rh, Pd, Au, Pt) show always negative deviation from the ideal solution. The phase diagram studies of these binary systems show that intermediate compounds exist between these elements and silicon. This may indicate that relatively strong forces exist between the components of their melts which causes the activity drop of each element because of the introduction of the other element; in other words, the negative behaviour compared to the ideal solution. This can be the reason for observing larger deviations for the silicon binary melts formed by the elements in higher period number and lower group number in the periodic table.
The binary melts consisting of silicon and poor metals (Zn, Al, Ga, Sn, Pb, Bi, In and Sb) show positive deviation from the ideal solution. It is seen that the phase diagrams of these binary systems contain no silicide, meaning the absence of significant forces between the solution atoms in the melts. Small positive deviation from the ideal solution is observed when the melt consisted of silicon and poor metals with close position to silicon in the periodic table such as Al, and Ga. In contrast 5, large positive deviation is seen when the metal is far from silicon, such as Pb and Bi. The presented argument here may be supported considering the behaviour of silver in Si–Ag melts, where it shows positive deviation and also negative deviation from the ideal solution. According to the argument here, silver is located in between the transition metals and poor metals in the periodic table and this behaviour of silver is expected.
Activity coefficients in the dilute solutions
Activity coefficients of the Si–Me dilute solutions at 1,414 °C
Si–Me melt | \( \gamma_{\text{Si}}^{ \circ } \) From activity curve | \( \gamma_{\text{Me}}^{ \circ } \) From activity curve | \( \gamma_{\text{Me}}^{ \circ } \) From Eq. (10) |
---|---|---|---|
Si–Al | 0.4184 | 0.370 | 0.3173 |
Si–Ca | 0.004 | 0.0032 | 0.0026 |
Si–Mg | 0.0015 | 0.0498 | – |
Si–Fe | 0.0017 | 0.014 | – |
Si–Zn | 3.0 | 1.4705 | 1.1349 |
Si–Cu | 0.0025 | 0.1865 | – |
Si–Ag | 0.275 | 2.703 | – |
Si–Au | 0.060 | 0.204 | – |
Si–Sn | 7.143 | 5.128 | 5.6813 |
Si–Pb | 78.919 | 37.481 | 37.4806 |
Si–Bi | 56.24 | 29.680 | 29.6853 |
Si–Sb | 4.170 | 4.879 | 5.6865 |
Si–Ga | 1.869 | 1.749 | 1.6275 |
Si–In | 8.0 | 5.598 | 5.2792 |
Si–Pd | 0.030 | 0.0286 | – |
Si–Ni | 0.00035 | 0.0205 | – |
Si–Mn | 0.00025 | 0.0030 | – |
The activity coefficients in the above mentioned silicon dilute solutions were calculated by Eq. (10), and the obtained results are presented in Table 2. It is seen that the calculated activity coefficients by this formula are consistent with those obtained graphically.
Applicability of the model
As indicated above, the calculated activities for the Si binary melts through the outlined model in this study are consistent with experimental measurements in literature. In the present study, 20 Si-binary melts were studied, and the model is applicable for studying other Si binary systems except those show significant solid solubility in silicon such as Si–B, Si–P and Si–As systems. The model may be applicable for studying other similar binary systems in which negligible solid solubility in one melt component is observed. And this element must show negligible changes in the heat capacity due to fusion, which was considered to derive the main Eq. (1). Authors believe that the model is applicable for the binary systems of the elements in the same group with silicon (Ge, In, Sn, Pb). Precise study must be dedicated to other binary melts to evaluate the model applicability for other systems.
Conclusions
The activities of 20 Si-binary systems were calculated considering the liquidus data on the silicon portion of the phase diagrams, where the activity coefficient of silicon along the liquidus can be calculated using the previously calculated liquidus constants. Silicon activities at the silicon melting point were further calculated through regular solution assumption, which were then used to calculate the other melt component activities by Gibbs–Duhem integration. It was found that the calculated activities of this study are consistent with the measured and calculated activities in the literature. A correlation between the behaviour of the Si-binary melts and the position of the other melt element in the periodic table was observed. In this case, Si-binary melts with alkali and transition metals show negative deviation from the ideal solution, whereas the melts consisting of poor metals show positive deviation from the ideal solution. The activities of the dilute solute elements in the melts were also determined graphically and tabulated. In addition, the activities of particular dilute solute elements in silicon were calculated through a derived simple formula and the results were found to be consistent with the graphically determined values.
Notes
Acknowledgements
The authors acknowledge the fund provided through the BASIC project (191285/V30) by the Norwegian Research Council.
Open Access
This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.
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