Self-diffusion Measurements of Liquid Sn Using the Shear Cell Technique and Stable Density Layering

Abstract

By utilizing the shear cell technique and achieving stable density layering with the addition of an alloying element Bi, the self-diffusion coefficients of liquid Sn were measured on the ground at 573 K, 773 K, and 973 K (300 °C, 500 °C, and 700 °C). Moreover, the impurity diffusion coefficients of Bi in the liquid Sn were simultaneously measured, to confirm the suppression of natural convection. From the experimental results, natural convection was confirmed to be suppressed, given that the impurity diffusion coefficients of Bi were in good agreement with the microgravity reference data. Upon changing the amounts of added Bi within 5 at. pct Bi, the self-diffusion coefficients of liquid Sn did not vary significantly. Moreover, the SnBi system within 5 at. pct Bi can be regarded as a dilute solution by calculating the activity of Sn in the SnBi system beforehand. The self-diffusion coefficients of the liquid Sn were in good agreement with the power law of temperature dependence from the microgravity reference data. After confirming the suppression of natural convection and that the SnBi system is a dilute solution, the self-diffusion coefficient of liquid Sn was determined as 5.00 ± 0.16 × 10⁻⁹ m² s⁻¹ at 773 K (500 °C). The effectiveness of the experimental method used in this study for the measurement of the self-diffusion coefficients was confirmed, under the condition that the liquid system could be regarded as a dilute solution.

Appendices

Appendix A

The quasi-chemical theory is a theoretical model that assumes the existence of chemical complexes in a liquid binary alloy. In this study, the SnBi system was assumed to be the regular solution. Thus, the excess free energy of mixing $G_{\text{m}}^{\text{ex}}$ in the SnBi system, using the quasi-chemical theory, can be expressed as

$$ \begin{array}{*{20}c} {\frac{{G_{\text{m}}^{\text{ex}} }}{RT} = N_{\text{Sn}} \ln \left( {\gamma_{\text{Sn}} } \right) + (1 - N_{\text{Sn}} )\ln \left( {\gamma_{\text{Bi}} } \right)} \\ \end{array} , $$

(A1)

where $\gamma_{\text{Sn}}$ is given by the following expressions:

$$ \begin{array}{*{20}c} {\gamma_{\text{Sn}} = \left\{ {\frac{{\beta - 1 + 2N_{\text{Sn}} }}{{N_{\text{Sn}} (\beta + 1)}}} \right\}^{z/2} ,} \\ \end{array} $$

(A2)

$$ \begin{array}{*{20}c} {\beta = \left[ {1 + 4N_{\text{Sn}} (1 - N_{\text{Sn}} )(\eta^{2} - 1)} \right]^{1/2} } \\ \end{array} , $$

(A3)

$$ \begin{array}{*{20}c} {\eta^{2} = \exp \left( {\frac{2\omega }{{zk_{\text{B}} T}}} \right)} \\ \end{array} . $$

(A4)

Therefore, using Eq. [6], $\varPhi_{\text{Sn}}$ is given by the following expressions:

$$ \begin{array}{*{20}c} {\varPhi_{\text{Sn}} = 1 + \frac{{\partial \ln \gamma_{\text{Sn}} }}{{\partial \ln N_{\text{Sn}} }} = 1 + \frac{z}{2}N_{\text{Sn}} \left( {\frac{{\beta ^{\prime} + 2}}{{\beta - 1 + 2N_{\text{Sn}} }} - \frac{{\beta ^{\prime}}}{1 + \beta } - \frac{1}{{N_{\text{Sn}} }}} \right)} \\ \end{array} , $$

(A5)

$$ \begin{array}{*{20}c} {\beta ^{\prime} = 2(\eta^{2} - 1)(1 - 2N_{\text{Sn}} )\left[ {1 + 4N_{\text{Sn}} (1 - N_{\text{Sn}} )(\eta^{2} - 1)} \right]^{ - 1/2} } \\ \end{array} , $$

(A6)

where the value of z was set as 10.2 at 773 K (500 °C) in liquid Sn.[25] The value of ω/k_B was calculated from the reference of the activity at 600 K (327 °C)[26] and was set as 126.6.

Appendix B

In diffusion experiments, it is important to obtain a concentration profile with a small variation. An increase in the concentration step (c₂ − c₁) of the diffusion couple effectively reduces the variation in the concentration. However, the use of the ¹²⁴Sn sample in the self-diffusion experiments in this study was limited, due to its low abundance. Moreover, it was necessary to accurately determine the minimum amount of ¹²⁴Sn for the self-diffusion experiments. The concentration profiles were therefore simulated to obtain the conditions for suitable fitting with Eq. [1]. From the simulation, the value of σ in the concentration step with self-diffusion was calculated.

Figure B1 and Table BI present the simulation of the concentration profile and the simulation properties of the self-diffusion experiment, respectively. In the simulation, the only variable parameter was c₂. First, an ideal concentration profile was created at a certain concentration step. Second, each ideal concentration was changed within the measurement error of ICP-MS used for the concentration measurement in this study, using a random function. The simulated concentration profile was obtained as shown in Figure B1. The measurement error of ICP-MS was 1.95 pct, which was obtained from the preliminary experiment.

Fig. B1

Simulation of concentration profile. The concentration step is 20 at. pct

Table BI Simulation Conditions for Self-diffusion Experiments

In preliminary experiments, five solutions of ¹²⁴Sn in nitric acid were prepared for each concentration level. For each solution, the measurements using ICP-MS were repeated four times. For each concentration level, a total of 20 measurements were performed, and the average and standard deviation (relative error) were calculated, as shown in Figure B2. The average of the relative error in the range of 6 at. pct to 25 at. pct of ¹²⁴Sn was 1.95 pct.

Fig. B2

Results of preliminary experiments

Finally, the simulated concentration profile was fitted again with Eq. [1], and X²_sim was obtained. From X²_sim and X²_meas, the standard deviation was calculated for each concentration step, and the error σ was calculated. For each concentration step, the simulation was repeated 10 times. Figure B3 presents the results of the simulation. In this study, the concentration step of ¹²⁴Sn was set as 20 at. pct. Therefore, the value of σ was 3.14 pct, obtained from Figure B3.

Fig. B3

Simulation results