Interaction Coefficients of Mo, B, Ni, Ti and Nb with Sn in Molten Fe–18mass%Cr Alloy

Abstract

Increasing the utilization of steel scrap is strongly required for reducing CO₂ emission in iron- and steel-making processes. In steel scrap recycling, the content of tramp elements in steel (such as copper and tin) inevitably increases. Accordingly, it is important to understand the thermodynamic characteristics of relevance to the accumulation of tramp elements in molten steel. The values of the interaction coefficients of Mo, B, Ni, Ti, and Nb with Sn in molten iron were reported previously. However, little is known about the interaction coefficients of alloying elements with tramp elements in molten high-chromium steel. In this work, the interaction coefficients of Mo, B, Ni, Ti, and Nb with Sn in the molten Fe–18mass%Cr alloy were measured at 1873 K by a chemical equilibration technique that uses the liquid immiscibility of the Fe–18mass%Cr alloy and Ag, yielding the following results:

The results show that the values of the interaction coefficients of M with Sn in the Fe-18mass%Cr alloy are smaller than those for molten iron, which were measured in the previous work, except for titanium. The interaction coefficients of M with Sn in Fe and Fe–18mass%Cr alloy were estimated based on a regular solution model. The estimated interaction coefficients of B, Ni, and Ti with Sn in molten iron and Ni and Ti with Sn in the molten Fe–18mass%Cr alloy reasonably agree with the measured values.

1. Introduction

An increase in the use of steel scrap is strongly required for reducing CO₂ emission in iron- (Fe-) and steel-making processes.^1,2,3,4) However, urban scrap often contains several tramp elements that are difficult to remove after their dissolution in molten iron. Accordingly, the amount of tramp elements in recycled steel products increases with increasing recycling age. Thus, it is important to investigate the influence of tramp elements on the steelmaking process and steel properties. For example, copper (Cu) and tin (Sn) have been reported to cause hot shortness during solidification and hot rolling processes.^{5,6,7,8,9,10,11)}

Up to date, many studies have addressed the removal of Cu and Sn.^{12,13,14,15,16,17,18,19,20,21,22,23,24,25, 26, 27, 28,29,30)} However, a proper removal process has not yet been established for practical applications. Kuzuhara et al.³¹⁾ investigated the content of impurities in steel during the refining in an electric furnace. They reported that the Cu content was 0.37 mass% and was higher than any other impurities. Daigo et al.³²⁾ investigated the influence of fifteen impurity elements mixed in recycled steels on the tensile strength, elongation, yield point or proof stress, soundness in the welding area, and fracture toughness and reported that properties related to strength and stress were found to be enhanced by the presence of almost any impurity, while elongation and welding soundness were often compromised. Consequently, the amount of tramp elements in recycled steel scrap has to be tightly monitored.

On the other hand, steel products utilizing tramp elements have been introduced, such as antibacterial ferritic stainless steel, which contains more than 1 mass% Cu^33,34,35) and Sn-added stainless steel with low chromium (Cr) and nickel (Ni) contents.³⁶⁾ These successes indicate that high-grade steel products can be produced from steel scraps if the effects of tramp elements are known and their content is controlled. Transition metals are usually added to high-grade steels.³⁷⁾ The activity data are useful for controlling the content of such alloying elements. The tramp elements in molten Fe are considered to affect the activities of the coexisting alloying elements, owing to the thermodynamic interactions between them.

The thermodynamic interaction between two elements in molten Fe is captured by the interaction coefficient. The values of the interaction coefficients for various elements that were found in molten Fe have been reported. Recently, the interaction coefficients of alloying elements with tramp elements in molten Fe were reported.^{38,39,40,41,42)} However, data on the interaction coefficients of transition elements with tramp elements in molten alloy steels have been very limited. In our previous work, the interaction coefficients of boron (B), cobalt (Co), and Ni with Cu⁴¹⁾ and those of Mo, B, Ni, titanium (Ti), and niobium (Nb) with Sn in molten Fe⁴²⁾ were reported. Cr is known to be an essential element in the production of high-grade steels such as stainless steels, heat-resistant steels, high-tensile-strength steels, and tool steels. To produce such special steels, it is important to know the interaction coefficients between the alloying elements and tramp elements in high-Cr steels. Accordingly, in this work, the interaction coefficients of Mo, B, Ni, Ti, and Nb with Sn in molten Fe–18mass%Cr alloys were measured using a chemical equilibration technique at 1873 K.

2. Method for Determining the Interaction Coefficients of M (M: Mo, B, Ni, Ti, and Nb) with Sn

In this work, the interaction coefficients of M (M: Mo, B, Ni, Ti, and Nb) with Sn were determined by investigating the effect of M on the partition ratio of Sn between the Fe–18mass%Cr (Fe–18Cr) and silver (Ag) phases, using the liquid immiscibility of each phase at 1873 K.⁴³⁾ When equilibrium is attained, the partial molar Gibbs energy of Sn in the molten Fe–18Cr phase (${\overline{G}}_{\text{Sn(in}\mathrm{\hspace{0.17em} } \text{Fe}-18\text{Cr)}}$) is equal to that in the molten Ag phase (${\overline{G}}_{\text{Sn(in}\mathrm{\hspace{0.17em} } \text{Ag)}}$). This relationship is expressed by Eq. (1):   

$${\overline{G}}_{\text{Sn}\left(\text{in}\mathrm{\hspace{0.17em} } \text{Fe}-18\text{Cr}\right)}={\overline{G}}_{\text{Sn}\left(\text{in}\mathrm{\hspace{0.17em} } \text{Ag}\right)}$$(1)

This can be rewritten as Eq. (2) when the standard state of the activity of Sn in each phase is taken as an equally pure liquid:   

$$\text{ln}{a}_{\text{Sn}\left(\text{in}\mathrm{\hspace{0.17em} } \text{Fe}-18\text{Cr}\right)}=\text{ln}{a}_{\text{Sn}\left(\text{in}\mathrm{\hspace{0.17em} } \text{Ag}\right)}$$(2)

where a_{Sn(in Fe–18Cr)} and a_{Sn(in Ag)} denote the activity of Sn in Fe–18Cr and Ag, respectively. In the (Fe–18Cr)–Ag–Sn system, the activities of Sn in the molten Fe–18Cr and molten Ag phases are expressed by Eqs. (3) and (4), respectively, when Sn is diluted in each phase:   

$$\begin{array}{l}\text{ln}{a}_{\text{Sn}\left(\text{in}\mathrm{\hspace{0.17em} } \text{Fe}-18\text{Cr}\right)}=\text{ln}{x}_{\text{Sn}\left(\text{in}\mathrm{\hspace{0.17em} } \text{Fe}-18\text{Cr}\right)}+\text{ln}{\gamma }_{\text{Sn}\left(\text{in}\mathrm{\hspace{0.17em} } \text{Fe}-18\text{Cr}\right)}^0\\ +x_{\text{Sn}\left(\text{in}\mathrm{\hspace{0.17em} } \text{Fe}-18\text{Cr}\right)}{\epsilon }_{\text{Sn}\left(\text{in}\mathrm{\hspace{0.17em} } \text{Fe}-18\text{Cr}\right)}^{\text{Sn}}+x_{\text{Ag}\left(\text{in}\mathrm{\hspace{0.17em} } \text{Fe}-18\text{Cr}\right)}{\epsilon }_{\text{Sn}\left(\text{in}\mathrm{\hspace{0.17em} } \text{Fe}-18\text{Cr}\right)}^{\text{Ag}}\end{array}$$(3)

  

$$\begin{array}{l}\text{ln}{a}_{\text{Sn}\left(\text{in}\mathrm{\hspace{0.17em} } \text{Ag}\right)}=\text{ln}{x}_{\text{Sn}\left(\text{in}\mathrm{\hspace{0.17em} } \text{Ag}\right)}+\text{ln}{\gamma }_{\text{Sn}\left(\text{in}\mathrm{\hspace{0.17em} } \text{Ag}\right)}^0+x_{\text{Sn}\left(\text{in}\mathrm{\hspace{0.17em} } \text{Ag}\right)}{\epsilon }_{\text{Sn}\left(\text{in}\mathrm{\hspace{0.17em} } \text{Ag}\right)}^{\text{Sn}}\\ +x_{\text{Cr}\left(\text{in}\mathrm{\hspace{0.17em} } \text{Ag}\right)}{\epsilon }_{\text{Sn}\left(\text{in}\mathrm{\hspace{0.17em} } \text{Ag}\right)}^{\text{Cr}}+x_{\text{Fe}\left(\text{in}\mathrm{\hspace{0.17em} } \text{Ag}\right)}{\epsilon }_{\text{Sn}\left(\text{in}\mathrm{\hspace{0.17em} } \text{Ag}\right)}^{\text{Fe}}\end{array}$$(4)

where x_{i(in α)}, ${\gamma }_{i(\text{in}\mathrm{\hspace{0.17em} } \alpha )}^0$, ${\epsilon }_{i(\text{in}\mathrm{\hspace{0.17em} } \alpha )}^j$ are the molar fraction of element i in phase α in the (Fe–18Cr)–Ag–Sn system (α: Fe–18Cr or Ag), the activity coefficient of element i in phase α at infinite dilution, and the interaction coefficient of element j on i in phase α, respectively. In the (Fe–18Cr)–Ag–Sn–M system, the activities of Sn are similarly expressed by Eqs. (5) and (6), respectively:   

$$\begin{array}{l}\text{ln}{a}_{\text{Sn}\left(\text{in}\mathrm{\hspace{0.17em} } \text{Fe}-18\text{Cr}\right)}=\text{ln}{X}_{\text{Sn}\left(\text{in}\mathrm{\hspace{0.17em} } \text{Fe}-18\text{Cr}\right)}+\text{ln}{\gamma }_{\text{Sn}\left(\text{in}\mathrm{\hspace{0.17em} } \text{Fe}-18\text{Cr}\right)}^0\\ +X_{\text{Sn}\left(\text{in}\mathrm{\hspace{0.17em} } \text{Fe}-18\text{Cr}\right)}{\epsilon }_{\text{Sn}\left(\text{in}\mathrm{\hspace{0.17em} } \text{Fe}-18\text{Cr}\right)}^{\text{Sn}}+X_{\text{Ag}\left(\text{in}\mathrm{\hspace{0.17em} } \text{Fe}-18\text{Cr}\right)}{\epsilon }_{\text{Sn}\left(\text{in}\mathrm{\hspace{0.17em} } \text{Fe}-18\text{Cr}\right)}^{\text{Ag}}\\ +X_{\text{M}\left(\text{in}\mathrm{\hspace{0.17em} } \text{Fe}-18\text{Cr}\right)}{\epsilon }_{\text{Sn}\left(\text{in}\mathrm{\hspace{0.17em} } \text{Fe}-18\text{Cr}\right)}^{\text{M}}\end{array}$$(5)

  

$$\begin{array}{l}\text{ln}{a}_{\text{Sn}\left(\text{in}\mathrm{\hspace{0.17em} } \text{Ag}\right)}=\text{ln}{X}_{\text{Sn}\left(\text{in}\mathrm{\hspace{0.17em} } \text{Ag}\right)}+\text{ln}{\gamma }_{\text{Sn}\left(\text{in}\mathrm{\hspace{0.17em} } \text{Ag}\right)}^0+X_{\text{Sn}\left(\text{in}\mathrm{\hspace{0.17em} } \text{Ag}\right)}{\epsilon }_{\text{Sn}\left(\text{in}\mathrm{\hspace{0.17em} } \text{Ag}\right)}^{\text{Sn}}\\ +X_{\text{Cr}\left(\text{in}\mathrm{\hspace{0.17em} } \text{Ag}\right)}{\epsilon }_{\text{Sn}\left(\text{in}\mathrm{\hspace{0.17em} } \text{Ag}\right)}^{\text{Cr}}+X_{\text{Fe}\left(\text{in}\mathrm{\hspace{0.17em} } \text{Ag}\right)}{\epsilon }_{\text{Sn}\left(\text{in}\mathrm{\hspace{0.17em} } \text{Ag}\right)}^{\text{Fe}}+X_{\text{M}\left(\text{in}\mathrm{\hspace{0.17em} } \text{Ag}\right)}{\epsilon }_{\text{Sn}\left(\text{in}\mathrm{\hspace{0.17em} } \text{Ag}\right)}^{\text{M}}\end{array}$$(6)

where X_i(inα) is the molar fraction of element i in phase α in the (Fe–18Cr)–Ag–Sn–M system. From Eqs. (2), (3), (4) and Eqs. (2), (5), and (6), Eqs. (7) and (8) can be derived as follows.   

$$\begin{array}{l}0=\text{ln}\left\{\frac{x_{\text{Sn}\left(\text{in}\mathrm{\hspace{0.17em} } \text{Ag}\right)}}{x_{\text{Sn}\left(\text{in}\mathrm{\hspace{0.17em} } \text{Fe}-18\text{Cr}\right)}\hspace{1em}}\right\}+\text{ln}\left\{\frac{{\gamma }_{\text{Sn}\left(\text{in}\mathrm{\hspace{0.17em} } \text{Ag}\right)}^0}{{\gamma }_{\text{Sn}\left(\text{in}\mathrm{\hspace{0.17em} } \text{Fe}-18\text{Cr}\right)}^0\hspace{1em}}\right\}+x_{\text{Sn}\left(\text{in}\mathrm{\hspace{0.17em} } \text{Ag}\right)}{\epsilon }_{\text{Sn}\left(\text{in}\mathrm{\hspace{0.17em} } \text{Ag}\right)}^{\text{Sn}}\\ -x_{\text{Sn}\left(\text{in}\mathrm{\hspace{0.17em} } \text{Fe}-18\text{Cr}\right)}{\epsilon }_{\text{Sn}\left(\text{in}\mathrm{\hspace{0.17em} } \text{Fe}-18\text{Cr}\right)}^{\text{Sn}}+x_{\text{Cr}\left(\text{in}\mathrm{\hspace{0.17em} } \text{Ag}\right)}{\epsilon }_{\text{Sn}\left(\text{in}\mathrm{\hspace{0.17em} } \text{Ag}\right)}^{\text{Cr}}\\ +\left\{x_{\text{Fe}\left(\text{in}\mathrm{\hspace{0.17em} } \text{Ag}\right)}{\epsilon }_{\text{Sn}\left(\text{in}\mathrm{\hspace{0.17em} } \text{Ag}\right)}^{\text{Fe}}-x_{\text{Ag}\left(\text{in}\mathrm{\hspace{0.17em} } \text{Fe}-18\text{Cr}\right)}{\epsilon }_{\text{Sn}\left(\text{in}\mathrm{\hspace{0.17em} } \text{Fe}-18\text{Cr}\right)}^{\text{Ag}}\right\}\end{array}$$(7)

  

$$\begin{array}{l}{X}_{\text{M}\left(\text{in}\mathrm{\hspace{0.17em} } \text{Fe}-18\text{Cr}\right)}{\epsilon }_{\text{Sn}\left(\text{in}\mathrm{\hspace{0.17em} } \text{Fe}-18\text{Cr}\right)}^{\text{M}}=\\ \text{ln}\left\{\frac{X_{\text{Sn}\left(\text{in}\mathrm{\hspace{0.17em} } \text{Ag}\right)}}{X_{\text{Sn}\left(\text{in}\mathrm{\hspace{0.17em} } \text{Fe}-18\text{Cr}\right)}\hspace{1em}}\right\}+\text{ln}\left\{\frac{{\gamma }_{\text{Sn}\left(\text{in}\mathrm{\hspace{0.17em} } \text{Ag}\right)}^0}{{\gamma }_{\text{Sn}\left(\text{in}\mathrm{\hspace{0.17em} } \text{Fe}-18\text{Cr}\right)}^0\hspace{1em}}\right\}\\ +\left\{X_{\text{Fe}\left(\text{in}\mathrm{\hspace{0.17em} } \text{Ag}\right)}{\epsilon }_{\text{Sn}\left(\text{in}\mathrm{\hspace{0.17em} } \text{Ag}\right)}^{\text{Fe}}-X_{\text{Ag}\left(\text{in}\mathrm{\hspace{0.17em} } \text{Fe}-18\text{Cr}\right)}{\epsilon }_{\text{Sn}\left(\text{in}\mathrm{\hspace{0.17em} } \text{Fe}-18\text{Cr}\right)}^{\text{Ag}}\right\}\\ +X_{\text{Sn}\left(\text{in}\mathrm{\hspace{0.17em} } \text{Ag}\right)}{\epsilon }_{\text{Sn}\left(\text{in}\mathrm{\hspace{0.17em} } \text{Ag}\right)}^{\text{Sn}}-X_{\text{Sn}\left(\text{in}\mathrm{\hspace{0.17em} } \text{Fe}-18\text{Cr}\right)}{\epsilon }_{\text{Sn}\left(\text{in}\mathrm{\hspace{0.17em} } \text{Fe}-18\text{Cr}\right)}^{\text{Sn}}\\ +X_{\text{Cr}\left(\text{in}\mathrm{\hspace{0.17em} } \text{Ag}\right)}{\epsilon }_{\text{Sn}\left(\text{in}\mathrm{\hspace{0.17em} } \text{Ag}\right)}^{\text{Cr}}+X_{\text{M}\left(\text{in}\mathrm{\hspace{0.17em} } \text{Ag}\right)}{\epsilon }_{\text{Sn}\left(\text{in}\mathrm{\hspace{0.17em} } \text{Ag}\right)}^{\text{M}}\end{array}$$(8)

The mutual solubilities in Fe–Ag binary system are reported to be x_{Ag(in Fe)} 0.002 to 0.004 and x_{Fe(in Ag)} 0.006 to 0.017.⁴⁴⁾ These values are the saturated solubility of Fe in Ag phase and Ag in Fe phase, respectively, and do not change significantly with a small amount of added element M. Moreover, the solubility of Ag in Fe–18Cr is also considered to be small because Cr and Ag are also immiscible at 1873 K.⁴⁵⁾ Accordingly, when it is assumed that the effect of M on the mutual solubility of Fe and Ag is negligibly small, the interaction coefficient of M with Sn in Fe–18Cr, ${\epsilon }_{\text{Sn(in}\mathrm{\hspace{0.17em} } \text{Fe}-18\text{Cr)}}^{\text{M}}$, is derived from Eqs. (7) and (8) as follows:   

$$\begin{array}{l}{\epsilon }_{\text{Sn}\left(\text{in}\mathrm{\hspace{0.17em} } \text{Fe}-18\text{Cr}\right)}^{\text{M}}=\\ [ln\left\{\frac{X_{\mathrm{S}\mathrm{n}\left(\mathrm{i}\mathrm{n}\mathrm{\hspace{0.17em} } \mathrm{A}\mathrm{g}\right)}}{X_{\mathrm{S}\mathrm{n}\left(\mathrm{i}\mathrm{n}\mathrm{\hspace{0.17em} } \mathrm{F}\mathrm{e}-18\mathrm{C}\mathrm{r}\right)}\hspace{1em}}\right\}-ln\left\{\frac{x_{\mathrm{S}\mathrm{n}\left(\mathrm{i}\mathrm{n}\mathrm{\hspace{0.17em} } \mathrm{A}\mathrm{g}\right)}}{x_{\mathrm{S}\mathrm{n}\left(\mathrm{i}\mathrm{n}\mathrm{\hspace{0.17em} } \mathrm{F}\mathrm{e}-18\mathrm{C}\mathrm{r}\right)}\hspace{1em}}\right\}+{\epsilon }_{\mathrm{S}\mathrm{n}\left(\mathrm{i}\mathrm{n}\mathrm{\hspace{0.17em} } \mathrm{A}\mathrm{g}\right)}^{\mathrm{S}\mathrm{n}}\left\{X_{\mathrm{S}\mathrm{n}\left(\mathrm{i}\mathrm{n}\mathrm{\hspace{0.17em} } \mathrm{A}\mathrm{g}\right)}-x_{\mathrm{S}\mathrm{n}\left(\mathrm{i}\mathrm{n}\mathrm{\hspace{0.17em} } \mathrm{A}\mathrm{g}\right)}\right\}\\ -{\epsilon }_{\mathrm{S}\mathrm{n}\left(\mathrm{i}\mathrm{n}\mathrm{\hspace{0.17em} } \mathrm{F}\mathrm{e}-18\mathrm{C}\mathrm{r}\right)}^{\mathrm{S}\mathrm{n}}\left\{X_{\mathrm{S}\mathrm{n}\left(\mathrm{i}\mathrm{n}\mathrm{\hspace{0.17em} } \mathrm{F}\mathrm{e}-18\mathrm{C}\mathrm{r}\right)}-x_{\mathrm{S}\mathrm{n}\left(\mathrm{i}\mathrm{n}\mathrm{\hspace{0.17em} } \mathrm{F}\mathrm{e}-18\mathrm{C}\mathrm{r}\right)}\right\}+{\epsilon }_{\mathrm{S}\mathrm{n}\left(\mathrm{i}\mathrm{n}\mathrm{\hspace{0.17em} } \mathrm{A}\mathrm{g}\right)}^{\mathrm{C}\mathrm{r}}\left\{X_{\mathrm{C}\mathrm{r}\left(\mathrm{i}\mathrm{n}\mathrm{\hspace{0.17em} } \mathrm{A}\mathrm{g}\right)}-x_{\mathrm{C}\mathrm{r}\left(\mathrm{i}\mathrm{n}\mathrm{\hspace{0.17em} } \mathrm{A}\mathrm{g}\right)}\right\}]/\\ X_{\mathrm{M}\left(\mathrm{i}\mathrm{n}\mathrm{\hspace{0.17em} } \mathrm{F}\mathrm{e}-18\mathrm{C}\mathrm{r}\right)}+L_{\mathrm{M}}^{\prime }{\epsilon }_{\mathrm{S}\mathrm{n}\left(\mathrm{i}\mathrm{n}\mathrm{\hspace{0.17em} } \mathrm{A}\mathrm{g}\right)}^{\mathrm{M}}\end{array}$$(9)

where $L_{\text{M}}^{\prime }$ is the partition ratio of M, which is defined as $L_{\mathrm{M}}^{\prime }=\frac{X_{\text{M(in}\mathrm{\hspace{0.17em} } \text{Ag)}}}{X_{\text{M(in}\mathrm{\hspace{0.17em} } \text{Fe}-18\text{Cr)}}\hspace{1em}}$.

In this study, the interaction coefficients were derived using Eq. (9) from the experimental results for the partition ratios of Sn and M between the Fe–18mass%Cr (Fe–18Cr) and Ag phases. In the case of M = Ti or Nb, Al was added, as explained in Section 3. Therefore, the interaction coefficient of Al with Sn was taken into account as follows:   

$$\begin{array}{l}{\epsilon }_{\text{Sn}\left(\text{in}\mathrm{\hspace{0.17em} } \text{Fe}-18\text{Cr}\right)}^{\text{M}}=\\ [ln\left\{\frac{X_{\mathrm{S}\mathrm{n}\left(\mathrm{i}\mathrm{n}\mathrm{\hspace{0.17em} } \mathrm{A}\mathrm{g}\right)}}{X_{\mathrm{S}\mathrm{n}\left(\mathrm{i}\mathrm{n}\mathrm{\hspace{0.17em} } \mathrm{F}\mathrm{e}-18\mathrm{C}\mathrm{r}\right)}\hspace{1em}}\right\}-ln\left\{\frac{x_{\mathrm{S}\mathrm{n}\left(\mathrm{i}\mathrm{n}\mathrm{\hspace{0.17em} } \mathrm{A}\mathrm{g}\right)}}{x_{\mathrm{S}\mathrm{n}\left(\mathrm{i}\mathrm{n}\mathrm{\hspace{0.17em} } \mathrm{F}\mathrm{e}-18\mathrm{C}\mathrm{r}\right)}\hspace{1em}}\right\}+{\epsilon }_{\mathrm{S}\mathrm{n}\left(\mathrm{i}\mathrm{n}\mathrm{\hspace{0.17em} } \mathrm{A}\mathrm{g}\right)}^{\mathrm{S}\mathrm{n}}\left\{X_{\mathrm{S}\mathrm{n}\left(\mathrm{i}\mathrm{n}\mathrm{\hspace{0.17em} } \mathrm{A}\mathrm{g}\right)}-x_{\mathrm{S}\mathrm{n}\left(\mathrm{i}\mathrm{n}\mathrm{\hspace{0.17em} } \mathrm{A}\mathrm{g}\right)}\right\}\\ -{\epsilon }_{\mathrm{S}\mathrm{n}\left(\mathrm{i}\mathrm{n}\mathrm{\hspace{0.17em} } \mathrm{F}\mathrm{e}-18\mathrm{C}\mathrm{r}\right)}^{\mathrm{S}\mathrm{n}}\left\{X_{\mathrm{S}\mathrm{n}\left(\mathrm{i}\mathrm{n}\mathrm{\hspace{0.17em} } \mathrm{F}\mathrm{e}-18\mathrm{C}\mathrm{r}\right)}-x_{\mathrm{S}\mathrm{n}\left(\mathrm{i}\mathrm{n}\mathrm{\hspace{0.17em} } \mathrm{F}\mathrm{e}-18\mathrm{C}\mathrm{r}\right)}\right\}+{\epsilon }_{\mathrm{S}\mathrm{n}\left(\mathrm{i}\mathrm{n}\mathrm{\hspace{0.17em} } \mathrm{A}\mathrm{g}\right)}^{\mathrm{C}\mathrm{r}}\left\{X_{\mathrm{C}\mathrm{r}\left(\mathrm{i}\mathrm{n}\mathrm{\hspace{0.17em} } \mathrm{A}\mathrm{g}\right)}-x_{\mathrm{C}\mathrm{r}\left(\mathrm{i}\mathrm{n}\mathrm{\hspace{0.17em} } \mathrm{A}\mathrm{g}\right)}\right\}\\ -X_{\mathrm{A}\mathrm{l}\left(\mathrm{i}\mathrm{n}\mathrm{\hspace{0.17em} } \mathrm{F}\mathrm{e}-18\mathrm{C}\mathrm{r}\right)}{\epsilon }_{\mathrm{S}\mathrm{n}\left(\mathrm{i}\mathrm{n}\mathrm{\hspace{0.17em} } \mathrm{F}\mathrm{e}-18\mathrm{C}\mathrm{r}\right)}^{\mathrm{A}\mathrm{l}}]/X_{\mathrm{M}\left(\mathrm{i}\mathrm{n}\mathrm{\hspace{0.17em} } \mathrm{F}\mathrm{e}-18\mathrm{C}\mathrm{r}\right)}+L_{\mathrm{M}}^{\prime }{\epsilon }_{\mathrm{S}\mathrm{n}\left(\mathrm{i}\mathrm{n}\mathrm{\hspace{0.17em} } \mathrm{A}\mathrm{g}\right)}^{\mathrm{M}}\end{array}$$(10)

The interaction coefficients in the mole fraction coordinates (${\epsilon }_{\text{Sn(in}\mathrm{\hspace{0.17em} } \text{Fe}-18\text{Cr)}}^{\text{M}}$) can be transformed to those in the mass percent coordinates ($e_{\text{Sn(in}\mathrm{\hspace{0.17em} } \text{Fe}-18\text{Cr)}}^{\text{M}}$), using the following equation:   

$${\epsilon }_{\text{Sn}\left(\text{in}\mathrm{\hspace{0.17em} } \text{Fe}-18\text{Cr}\right)}^{\text{M}}=230\frac{M_{\text{M}}}{M_{\text{Fe}-18\text{Cr}}}{e}_{\text{Sn}\left(\text{in}\mathrm{\hspace{0.17em} } \text{Fe}-18\text{Cr}\right)}^{\text{M}}+\left(1-\frac{M_{\text{M}}}{M_{\text{Fe}-18\text{Cr}}}\right)$$(11)

where M_M and M_Fe–18Cr are the atomic mass of element M and the average molar mass of Fe–18Cr, respectively.

3. Experimental

The effects of M (M: Mo, B, Ni, Ti, and Nb) on the partition ratio of Sn between the molten Fe–18Cr alloy and molten Ag were investigated at 1873 K. The experimental apparatus and arrangement are shown in Fig. 1. Three samples, contained in three Al₂O₃ crucibles (OD: 15 mm, ID: 12 mm, H: 100 mm), were heated to 1873 K in a mullite tube (OD: 70 mm, ID: 60 mm, H: 1000 mm) in an electric resistance furnace under argon. A larger Al₂O₃ crucible (OD: 45 mm, ID: 38 mm, H: 100 mm) was used as the outer crucible. In the case of M = Ti or Nb, a Fe–15mass%Al alloy, which was prepared in advance, was added to prevent the Al₂O₃ crucible from being reduced by Ti or Nb, and the Ti or Nb contents from decreasing. The initial masses of Fe, Ag, Cr, Sn, M (M: Mo, B, Ni, Ti, and Nb), and of the Fe–15mass%Al alloy are listed in Table 1, for each sample. After the temperature reached 1873 K, the samples were held for 5 h to attain equilibrium. Subsequently, the samples were withdrawn from the furnace and quenched in cooled water. The metal samples were carefully cut for chemical analyses. The Cr, Sn, M, and Al contents of the Fe–18Cr and Ag phases were analyzed using inductively coupled plasma atomic emission spectroscopy (ICP-AES).

Fig. 1.

Schematic of the experimental apparatus and arrangement of materials. (Online version in color.)

Table 1. Initial masses and experimental results. (a): Mo, B, and Ni, (b): Ti and Nb.

(a)
No.	M	Initial mass (g)					Experiment results (mass%)						Calculated mass (g)			Molar ratio (-)
							Fe phase			Ag phase
		Fe	Ag	Cr	Sn	M	Cr	Sn	M	Cr	Sn	M	m (Cr)	m (Sn)	m (M)	$L_{\text{Sn}}^{\prime }$	$L_{\text{M}}^{\prime }$
1	–	13.6	7.01	3.40	0.0521	–	18.5	0.051	–	0.0509	0.540	–	3.12	0.05	–	25.4	–
2		13.6	7.00	3.40	0.103	–	18.6	0.106	–	0.0582	1.09	–	3.14	0.10	–	24.6	–
3		13.6	6.99	3.40	0.201	–	18.9	0.196	–	0.0455	2.07	–	3.21	0.18	–	25.1	–
4		13.7	7.00	3.26	0.052	–	18.5	0.038	–	0.0639	0.60	–	2.99	0.05	–	27.9	–
5		13.7	7.00	3.26	0.102	–	18.7	0.104	–	0.0640	1.20	–	3.03	0.10	–	22.7	–
6		13.7	7.00	3.26	0.151	–	18.4	0.124	–	0.0588	1.79	–	2.97	0.15	–	25.7	–
7	Mo	13.7	7.00	3.26	0.100	0.100	18.1	0.101	0.589	0.0531	1.13	0.0035	3.06	0.097	0.10	21.8	0.0116
8		13.7	7.00	3.26	0.102	0.200	14.7	0.094	0.895	0.0544	1.14	0.0075	2.40	0.097	0.15	23.6	0.0163
9		13.7	7.00	3.26	0.102	0.230	17.7	0.100	1.304	0.0648	1.19	0.0072	3.01	0.101	0.23	23.9	0.0108
10		13.7	7.00	3.26	0.099	0.244	18.2	0.085	1.397	0.0564	1.12	0.0063	3.11	0.098	0.23	24.7	0.0087
11		13.6	7.00	3.26	0.101	0.275	17.8	0.099	1.510	0.0603	1.22	0.0067	3.06	0.098	0.26	23.5	0.0086
12		13.7	7.00	3.26	0.102	0.304	18.7	0.105	1.761	0.0549	1.11	0.0082	3.25	0.098	0.31	21.0	0.0091
13		13.7	7.00	3.26	0.102	0.346	18.0	0.100	2.068	0.0672	1.17	0.0052	3.11	0.101	0.35	22.8	0.0049
14	B	13.6	7.01	3.36	0.109	0.027	19.6	0.118	0.101	0.0480	1.12	0.0003	3.42	0.109	0.02	23.1	0.0057
15		13.7	7.01	3.26	0.103	0.036	17.9	0.092	0.108	0.0561	1.13	0.0081	2.98	0.098	0.02	24.1	0.1472
16		13.6	7.00	3.36	0.105	0.0512	18.3	0.094	0.187	0.0585	1.10	<0.0001	3.09	0.101	0.03	28.2	–
17		13.7	7.00	3.26	0.102	0.063	17.8	0.097	0.236	0.0455	1.04	<0.0001	2.91	0.087	0.04	25.7	–
18		13.7	7.00	3.27	0.100	0.075	17.9	0.087	0.231	0.0634	1.13	0.0069	2.99	0.094	0.04	25.5	0.0588
19		13.7	7.03	3.27	0.100	0.078	17.9	0.098	0.293	0.0456	1.02	<0.0001	3.00	0.089	0.05	25.1	–
20		13.7	7.00	3.26	0.102	0.100	18.3	0.084	0.299	0.0578	1.16	0.0067	3.07	0.098	0.05	27.6	0.0441
21		13.7	7.00	3.26	0.100	0.125	19.3	0.076	0.568	0.0592	1.14	0.0031	3.29	0.094	0.10	29.2	0.0108
22		13.7	7.00	3.26	0.100	0.150	16.6	0.076	0.656	0.0550	1.13	0.0032	2.75	0.093	0.11	28.9	0.0097
23		13.7	7.00	3.26	0.100	0.175	17.8	0.065	0.687	0.0601	1.14	0.0020	2.99	0.092	0.12	35.4	0.0060
24		13.7	7.00	3.26	0.100	0.200	18.1	0.058	0.811	0.0517	1.19	0.0016	3.07	0.094	0.14	41.9	0.0041
25		13.7	7.00	3.26	0.100	0.225	16.8	0.065	0.796	0.0611	1.20	0.0026	2.79	0.096	0.13	37.6	0.0067
26	Ni	13.7	7.00	3.26	0.100	0.100	17.9	0.103	0.530	0.0532	1.19	0.0042	3.01	0.101	0.09	23.0	0.0154
27		13.7	7.00	3.26	0.100	0.150	17.1	0.098	0.835	0.0623	1.20	0.0057	2.86	0.102	0.14	24.8	0.0134
28		13.7	7.00	3.26	0.100	0.200	16.4	0.116	1.254	0.0548	1.09	0.0075	2.73	0.097	0.21	19.4	0.0117
29		13.7	7.00	3.26	0.100	0.250	17.1	0.107	1.360	0.0595	1.17	0.0097	2.87	0.101	0.23	21.4	0.0139
30		13.7	7.00	3.26	0.100	0.300	17.8	0.101	1.419	0.0555	1.11	0.0086	3.01	0.096	0.24	21.5	0.0118
31		13.7	7.00	3.26	0.100	0.350	17.8	0.110	1.911	0.0614	1.16	0.0108	3.03	0.101	0.33	20.6	0.0111

(b)
No.	M	Initial mass (g)						Experiment results (mass%)								Calculated mass (g)				Molar ratio (-)
								Fe phase				Ag phase
		Fe	Ag	Cr	Sn	M	Fe-15Al	Cr	Sn	M	Al	Cr	Sn	M	Al	m (Cr)	m (Sn)	m (M)	m (Al)	$L_{\text{Sn}}^{\prime }$	$L_{\text{M}}^{\prime }$
32	Ti	13.7	7.00	3.26	0.100	0.054	0.485	19.5	0.077	0.301	0.324	0.0518	1.10	0.0033	0.0007	3.44	0.091	0.053	0.008	27.9	0.0214
33		13.7	7.00	3.26	0.103	0.104	0.483	17.7	0.082	0.557	0.086	0.0516	1.20	0.0020	0.0661	3.06	0.099	0.096	0.003	28.8	0.0069
34		13.7	7.00	3.26	0.099	0.151	0.483	17.6	0.098	0.800	0.073	0.0576	1.16	0.0004	0.0403	3.07	0.100	0.139	0.002	23.2	0.0009
35		13.7	7.00	3.26	0.100	0.204	0.486	19.4	0.088	0.894	0.027	0.0559	1.33	0.00	0.0217	3.46	0.110	0.159	0.004	29.7	0.0000
36		13.7	7.00	3.26	0.101	0.253	0.486	17.2	0.082	1.121	0.037	0.0553	1.21	0.00	0.0058	2.98	0.100	0.194	0.005	29.1	0.0000
37		13.7	7.00	3.27	0.098	0.312	0.489	21.5	0.094	1.160	0.043	0.0578	1.09	0.00	0.0222	3.94	0.094	0.212	0.006	22.7	0.0014
38	Nb	13.7	7.00	3.26	0.100	0.046	0.484	17.8	0.088	0.244	0.157	0.0585	1.20	0.00	0.0358	3.08	0.101	0.042	0.004	26.9	0.0000
39		13.7	7.00	3.26	0.099	0.095	0.483	19.0	0.075	0.550	0.223	0.0555	1.13	0.00	0.0364	3.34	0.094	0.097	0.006	29.4	0.0000
40		13.7	7.00	3.26	0.102	0.154	0.486	17.5	0.098	0.245	0.119	0.0532	1.16	0.00	0.0652	3.02	0.099	0.042	0.004	23.2	0.0000
41		13.7	7.00	3.26	0.102	0.205	0.487	17.5	0.085	1.044	0.112	0.0572	1.16	0.00	0.0545	3.05	0.097	0.181	0.014	26.5	0.0000
42		13.7	7.00	3.26	0.102	0.248	0.484	16.3	0.095	1.189	0.089	0.0581	1.19	0.00	0.0576	2.79	0.101	0.204	0.011	24.3	0.0000
43		13.7	7.00	3.25	0.101	0.301	0.485	17.1	0.094	1.114	0.111	0.0684	1.17	0.00	0.0658	2.97	0.099	0.193	0.014	24.3	0.0000
44		13.7	6.99	3.26	0.100	0.323	0.484	16.2	0.081	0.899	0.125	0.0674	1.12	0.00	0.0572	2.76	0.093	0.153	0.015	26.9	0.0000
45		13.7	7.00	3.26	0.101	0.349	0.488	17.7	0.068	0.995	0.087	0.0634	1.13	0.00	0.0382	3.08	0.092	0.173	0.011	32.4	0.0000

4. Results and Discussion

4.1. Self-interaction Coefficient of Sn in the Ag Phase

The Cr, Sn, M, and Al contents of the Fe–18Cr and Ag phases after the experiments are listed in Table 1. It was confirmed that the Cr content of Fe–18Cr was maintained at approximately 18 mass% in all experiments. The molar partition ratio of Sn, defined as $L_{\text{Sn}}^{\prime }=\frac{x_{\text{Sn(in}\mathrm{\hspace{0.17em} } \text{Ag)}}}{x_{\text{Sn(in}\mathrm{\hspace{0.17em} } \text{Fe}-18\text{Cr)}}\hspace{1em}}$, was calculated and is listed in Table 1. It was found that the Sn content of the Fe–18Cr phase was low, generally less than 0.1 mass%, and the value of $L_{\text{Sn}}^{\prime }$ varied in the 19.4–41.9 range. It was also found that the M content of the Ag phase was very low.

In the experiments of the (Fe–18Cr)–Ag–Sn system (Nos.1 to 6), it was found that x_{Sn(in Ag)} was much larger than x_{Sn(in Fe–18Cr)} and x_{Cr(in Ag)}, and the terms of $x_{\text{Sn(in}\mathrm{\hspace{0.17em} } \text{Fe}-18\text{Cr)}}{\epsilon }_{\text{Sn(in}\mathrm{\hspace{0.17em} } \text{Fe}-18\text{Cr)}}^{\text{Sn}}$ and $x_{\text{Cr(in}\mathrm{\hspace{0.17em} } \text{Ag)}}{\epsilon }_{\text{Sn(in}\mathrm{\hspace{0.17em} } \text{Ag)}}^{\text{Cr}}$ in Eq. (7) were considered negligible. Moreover, assuming that the mutual solubility of Ag and Fe in each phase does not depend on the Sn content, Eq. (7) can be rewritten as follows:   

$$\text{ln}{L}_{\text{Sn}}^{\prime }=-x_{\text{Sn}\left(\text{in}\mathrm{\hspace{0.17em} } \text{Ag}\right)}{\epsilon }_{\text{Sn}\left(\text{in}\mathrm{\hspace{0.17em} } \text{Ag}\right)}^{\text{Sn}}+\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}.$$(12)

According to Eq. (12), the experimental results of ln$L_{\text{Sn}}^{\prime }$ are plotted against x_{Sn(in Ag)} in Fig. 2. From Fig. 2, a linear relationship is observed, and ln$L_{\text{Sn}}^{\prime }$ is seen to slightly decrease with increasing the Sn content of the Ag phase. From the slope of the solid line in Fig. 2, which is a linear regression of the experimental plots, the value of the self-interaction coefficient of Sn in the Ag phase, ${\epsilon }_{\text{Sn(in}\mathrm{\hspace{0.17em} } \text{Ag)}}^{\text{Sn}}$, was determined as 3.80.

Fig. 2.

Concentration dependence of the partition ratio of Sn, for the (Fe–18Cr)–Ag Sn system.

4.2. Effect of M (Mo, B, Ni, Ti, or Nb) on the Activity of Sn in the Fe–18Cr Alloy

The dependence of $L_{\text{Sn}}^{\prime }$ on X_{M(in Fe–18Cr)} in the (Fe–18Cr)–Ag–Sn–M system is shown in Fig. 3. As shown in Fig. 3, the molar partition ratio of Sn increased with an increase in the B, Ti, or Nb contents, and decreased with an increase in the Mo or Ni contents. The results show that the activity of Sn in Fe–18Cr increased owing to B, Ti, and Nb, while it decreased owing to Mo and Ni. The values of $L_{\text{M}}^{\prime }=\frac{X_{\text{M(in}\mathrm{\hspace{0.17em} } \text{Ag)}}}{X_{\text{M(in}\mathrm{\hspace{0.17em} } \text{Fe}-18\text{Cr)}}\hspace{1em}}$ were in the 0.00–0.06 range, as shown in Table 1, which can be judged to be negligibly small. Furthermore, the absolute values of the X_{Sn(in Fe–18Cr)} – x_{Sn(in Fe–18Cr)} and X_{Cr(in Ag)} – x_{Cr(in Ag)} terms were less than 0.002 and 0.003, respectively, which are also negligible. Thus, the $L_{\text{M}}^{\prime }{\epsilon }_{\text{Sn(in}\mathrm{\hspace{0.17em} } \text{Ag)}}^{\text{M}}$, X_{Sn(in Fe–18Cr)} – x_{Sn(in Fe–18Cr)} and X_{Cr(in Ag)} – x_{Cr(in Ag)} terms in Eqs. (9) and (10) were ignored in the calculation. Substituting the derived value of 3.80 into ${\epsilon }_{\text{Sn(in}\mathrm{\hspace{0.17em} } \text{Ag)}}^{\text{Sn}}$ in Eq. (9), the following equation was obtained:   

$$\begin{array}{l}{\epsilon }_{\text{Sn}\left(\text{in}\mathrm{\hspace{0.17em} } \text{Fe}-18\text{Cr}\right)}^{\text{M}}=\\ \frac{\text{ln}\left\{\frac{X_{\text{Sn}\left(\text{in}\mathrm{\hspace{0.17em} } \text{Ag}\right)}}{X_{\text{Sn}\left(\text{in}\mathrm{\hspace{0.17em} } \text{Fe}-18\text{Cr}\right)}\hspace{1em}}\right\}-\text{ln}\left\{\frac{x_{\text{Sn}\left(\text{in}\mathrm{\hspace{0.17em} } \text{Ag}\right)}}{x_{\text{Sn}\left(\text{in}\mathrm{\hspace{0.17em} } \text{Fe}-18\text{Cr}\right)}\hspace{1em}}\right\}+3.80\left\{X_{\text{Sn}\left(\text{in}\mathrm{\hspace{0.17em} } \text{Ag}\right)}-x_{\text{Sn}\left(\text{in}\mathrm{\hspace{0.17em} } \text{Ag}\right)}\hspace{0.5em}\right\}}{X_{\text{M}\left(\text{in}\mathrm{\hspace{0.17em} } \text{Fe}-18\text{Cr}\right)}}\end{array}$$(13)

Rearranging Eq. (13),   

$$\begin{array}{l}\text{ln}{\gamma }_{\text{Sn}}^{\text{M}}={\epsilon }_{\text{Sn}\left(\text{in}\mathrm{\hspace{0.17em} } \text{Fe}-18\text{Cr}\right)}^{\text{M}}{X}_{\text{M}\left(\text{in}\mathrm{\hspace{0.17em} } \text{Fe}-18\text{Cr}\right)}\\ \hspace{1em}\hspace{1em} =\text{ln}\left\{\frac{X_{\text{Sn}\left(\text{in}\mathrm{\hspace{0.17em} } \text{Ag}\right)}}{X_{\text{Sn}\left(\text{in}\mathrm{\hspace{0.17em} } \text{Fe}-18\text{Cr}\right)}\hspace{1em}}\right\}-\text{ln}\left\{\frac{x_{\text{Sn}\left(\text{in}\mathrm{\hspace{0.17em} } \text{Ag}\right)}}{x_{\text{Sn}\left(\text{in}\mathrm{\hspace{0.17em} } \text{Fe}-18\text{Cr}\right)}\hspace{1em}}\right\}\\ \hspace{1em}\hspace{1em}\hspace{0.5em}+3.80\left\{X_{\text{Sn}\left(\text{in}\mathrm{\hspace{0.17em} } \text{Ag}\right)}-x_{\text{Sn}\left(\text{in}\mathrm{\hspace{0.17em} } \text{Ag}\right)}\right\}\end{array}$$(14)

where $\text{ln}{\gamma }_{\text{Sn}}^{\text{M}}$ represents the effect of M on the activity coefficient of Sn in Fe–18Cr. From Eq. (14), in a linear relationship between $\text{ln}{\gamma }_{\text{Sn}}^{\text{M}}$ and X_{M(in Fe–18Cr)}, the slope corresponds to the interaction coefficient of M with Sn in Fe–18Cr and ${\epsilon }_{\text{Sn(in}\mathrm{\hspace{0.17em} } \text{Fe}-18\text{Cr)}}^{\text{M}}$. This equation was applied to M=Mo, B, or Ni. In the case of M = Ti or Nb, the interaction coefficient of Al with Sn in molten Fe, ${\epsilon }_{\text{Sn(in}\mathrm{\hspace{0.17em} } \text{Fe)}}^{\text{Al}}$ = 3.10³⁷⁾ was used, and the following equations were derived from Eq. (10):   

$$\begin{array}{l}{\epsilon }_{\text{Sn}\left(\text{in}\mathrm{\hspace{0.17em} } \text{Fe}-18\text{Cr}\right)}^{\text{M}}=\\ \frac{\text{ln}\left\{\frac{X_{\text{Sn}\left(\text{in}\mathrm{\hspace{0.17em} } \text{Ag}\right)}}{X_{\text{Sn}\left(\text{in}\mathrm{\hspace{0.17em} } \text{Fe}-18\text{Cr}\right)}\hspace{1em}}\right\}-\text{ln}\left\{\frac{x_{\text{Sn}\left(\text{in}\mathrm{\hspace{0.17em} } \text{Ag}\right)}}{x_{\text{Sn}\left(\text{in}\mathrm{\hspace{0.17em} } \text{Fe}-18\text{Cr}\right)}\hspace{1em}}\right\}+3.80\left\{X_{\text{Sn}\left(\text{in}\mathrm{\hspace{0.17em} } \text{Ag}\right)}-x_{\text{Sn}\left(\text{in}\mathrm{\hspace{0.17em} } \text{Ag}\right)}\hspace{0.5em}\right\}}{X_{\text{M}\left(\text{in}\mathrm{\hspace{0.17em} } \text{Fe}-18\text{Cr}\right)}}\\ -3.10\frac{X_{\text{Al}\left(\text{in}\mathrm{\hspace{0.17em} } \text{Fe}-18\text{Cr}\right)}}{X_{\text{M}\left(\text{in}\mathrm{\hspace{0.17em} } \text{Fe}-18\text{Cr}\right)}}\end{array}$$(15)

  

$$\begin{array}{l}\text{ln}{\gamma }_{\text{Sn}}^{\text{M}}={\epsilon }_{\text{Sn}\left(\text{in}\mathrm{\hspace{0.17em} } \text{Fe}-18\text{Cr}\right)}^{\text{M}}{X}_{\text{M}\left(\text{in}\mathrm{\hspace{0.17em} } \text{Fe}-18\text{Cr}\right)}\\ =\text{ln}\left\{\frac{X_{\text{Sn}\left(\text{in}\mathrm{\hspace{0.17em} } \text{Ag}\right)}}{X_{\text{Sn}\left(\text{in}\mathrm{\hspace{0.17em} } \text{Fe}-18\text{Cr}\right)}\hspace{1em}}\right\}-\text{ln}\left\{\frac{x_{\text{Sn}\left(\text{in}\mathrm{\hspace{0.17em} } \text{Ag}\right)}}{x_{\text{Sn}\left(\text{in}\mathrm{\hspace{0.17em} } \text{Fe}-18\text{Cr}\right)}\hspace{1em}}\right\}+3.80\left\{X_{\text{Sn}\left(\text{in}\mathrm{\hspace{0.17em} } \text{Ag}\right)}-x_{\text{Sn}\left(\text{in}\mathrm{\hspace{0.17em} } \text{Ag}\right)}\right\}\\ -3.10X_{\text{Al}\left(\text{in}\mathrm{\hspace{0.17em} } \text{Fe}-18\text{Cr}\right)}\end{array}$$(16)

According to Eqs. (14) and (16), the experimental results for the (Fe–18Cr)–Ag–Sn–M (M=Mo, B, Ni, Ti, or Nb) systems are plotted in Figs. 4, 5, 6, 7, 8, respectively. The results for the Fe–Ag–Sn–M system that were reported in our previous work³⁶⁾ are also shown in Figs. 4, 5, 6, 7, 8. The solid and dashed lines correspond to the linear regression approximations to the (Fe–18Cr)–Ag–Sn–M and Fe–Ag–Sn–M systems, respectively. That is, the slopes of the solid and dashed lines indicate ${\epsilon }_{\text{Sn(in}\mathrm{\hspace{0.17em} } \text{Fe}-18\text{Cr)}}^{\text{M}}$ and ${\epsilon }_{\text{Sn(in}\mathrm{\hspace{0.17em} } \text{Fe)}}^{\text{M}}$, respectively. The values of ${\epsilon }_{\text{Sn(in}\mathrm{\hspace{0.17em} } \text{Fe}-18\text{Cr)}}^{\text{M}}$ were determined from the slopes of the solid lines in Figs. 4, 5, 6, 7, 8. The values of the interaction coefficients in mass percent composition coordinates ($e_{\text{Sn(in}\mathrm{\hspace{0.17em} } \text{Fe}-18\text{Cr)}}^{\text{M}}$) were also calculated using Eq. (11). The calculated values of ${\epsilon }_{\text{Sn(in}\mathrm{\hspace{0.17em} } \text{Fe}-18\text{Cr)}}^{\text{M}}$ and $e_{\text{Sn(in}\mathrm{\hspace{0.17em} } \text{Fe}-18\text{Cr)}}^{\text{M}}$ as well as ${\epsilon }_{\text{Sn(in}\mathrm{\hspace{0.17em} } \text{Fe)}}^{\text{M}}$ that was measured in the previous work³⁶⁾ are summarized in Table 2, where the values in parentheses show the corresponding standard deviations, which were calculated from the standard deviations of $\text{ln}{\gamma }_{\text{Sn}}^{\text{M}}$ and arithmetic means of X_{M(in Fe−18Cr)} in Figs. 4, 5, 6, 7, 8, assuming that the $\text{ln}{\gamma }_{\text{Sn}}^{\text{M}}$ values obey the Gaussian distribution and the deviation of X_{M(in Fe−18Cr)} can be ignored. It was found that the values of the interaction coefficients of M with Sn in Fe–18Cr are smaller than those in molten Fe, except for Ti. It was also found that the sign of the interaction coefficient of Mo with Sn changes from positive to negative with an increase in the Cr content of 18 mass%.

Fig. 3.

Dependence of $L_{\text{Sn}}^{\prime }$ on X_{M(in Fe–18Cr)}, for the (Fe–18Cr)–Ag Sn–M system. (Online version in color.)

Fig. 4.

Effect of Mo on the activity coefficient of Sn, for Fe–18Cr and Fe,⁴²⁾ at 1873 K. (Online version in color.)

Fig. 5.

Effect of B on the activity coefficient of Sn, for Fe–18Cr and Fe,⁴²⁾ at 1873 K. (Online version in color.)

Fig. 6.

Effect of Ni on the activity coefficient of Sn, for Fe–18Cr and Fe,⁴²⁾ at 1873 K. (Online version in color.)

Fig. 7.

Effect of Ti on the activity coefficient of Sn, for Fe–18Cr and Fe,⁴²⁾ at 1873 K. (Online version in color.)

Fig. 8.

Effect of Nb on the activity coefficient of Sn, for Fe–18Cr and Fe,⁴²⁾ at 1873 K. (Online version in color.)

Table 2. Interaction parameter of M with Sn, for the molten Fe–18mass%Cr, ${\epsilon }_{\text{Sn(in}\mathrm{\hspace{0.17em} } \text{Fe}-18\text{Cr)}}^{\text{M}}$, $e_{\text{Sn(in}\mathrm{\hspace{0.17em} } \text{Fe}-18\text{Cr)}}^{\text{M}}$ and those for Fe,⁴²⁾ ${\epsilon }_{\text{Sn(in}\mathrm{\hspace{0.17em} } \text{Fe},\mathrm{\hspace{0.17em} } 1\mathrm{\hspace{0.17em} }873\mathrm{\hspace{0.17em} } \text{K)}}^{\text{M}}$ at 1873 K.

M	${\epsilon }_{\text{Sn(in}\mathrm{\hspace{0.17em} } \text{Fe}-18\text{Cr)}}^{\text{M}}$	$e_{\text{Sn(in}\mathrm{\hspace{0.17em} } \text{Fe}-18\text{Cr)}}^{\text{M}}$	${\epsilon }_{\text{Sn(in}\mathrm{\hspace{0.17em} } \text{Fe)}}^{\text{M}}$36)
Mo	–2.9 (±3.2)	–0.0054 (±0.0099)	11.7 (±4.7)
B	9.6 (±1.9)	0.19 (±0.02)	9.9 (±1.2)
Ni	–8.9 (±2.6)	–0.036 (±0.011)	–6.8 (±1.7)
Ti	10.4 (±5.7)	0.051 (±0.028)	6.0 (±1.2)
Nb	20.5 (±8.0)	0.055 (±0.022)	31.7 (±1.8)

4.3. Interaction Coefficients of M (Mo, B, Ni, Ti, or Nb) with Sn in Fe–18Cr

The activity coefficient of M in the Fe–Sn–M ternary system, lnγ_{M(in Fe–Sn–M)}, can be estimated based on Toop’s equation⁴⁶⁾ by considering the bonds between the nearest atoms, as described in Eq. (17):   

$$\begin{array}{l}\text{ln}{\gamma }_{\text{M}\left(\text{in}\mathrm{\hspace{0.17em} } \text{Fe}-\text{Sn}-\text{M}\right)}=\text{ln}{\gamma }_{\text{M}\left(\text{in}\mathrm{\hspace{0.17em} } \text{Fe}-\text{M}\right)}\frac{x_{\text{Fe}}}{x_{\text{Fe}}+x_{\text{Sn}}}+\text{ln}{\gamma }_{\text{M}\left(\text{in}\mathrm{\hspace{0.17em} } \text{Sn}-\text{M}\right)}\frac{x_{\text{Sn}}}{x_{\text{Fe}}+x_{\text{Sn}}}\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{0.5em}-{\left(1-x_{\text{M}}\right)}^2\frac{\mathrm{\Delta }{G}_{\text{Fe}-\text{Sn}}^{Ex}}{RT}\end{array}$$(17)

where lnγ_{M(in Fe–M)} and lnγ_{M(in Sn–M)} are the activity coefficients of M in the Fe–M and Sn–M binary systems, and $\mathrm{\Delta }{G}_{\text{Fe}-\text{Sn}}^{Ex}$ is the excess Gibbs free energy of mixing in the Fe–Sn binary system, respectively. $\mathrm{\Delta }{G}_{\text{Fe}-\text{Sn}}^{Ex}$ is expressed by Eq. (18) based on the regular solution model:   

$$\mathrm{\Delta }{G}_{\text{Fe}-\text{Sn}}^{Ex}=RT\left\{x_{\text{Fe}}\text{ln}{\gamma }_{\text{Fe}\left(\text{in}\mathrm{\hspace{0.17em} } \text{Fe}-\text{Sn}\right)}+x_{\text{Sn}}\text{ln}{\gamma }_{\text{Sn}\left(\text{in}\mathrm{\hspace{0.17em} } \text{Fe}-\text{Sn}\right)}\right\}$$(18)

When x_M is much smaller than 1, 1−x_M and x_Fe + x_Sn become approximately unity, and γ_{M(in Fe–M)} and γ_{M(in Sn–M)} can be replaced by ${\gamma }_{\text{M(in}\mathrm{\hspace{0.17em} } \text{Fe)}}^0$ and ${\gamma }_{\text{M(in}\mathrm{\hspace{0.17em} } \text{Sn)}}^0$, respectively. Thus, Eq. (19) can be derived from Eqs. (17) and (18) as follows:   

$$\begin{array}{l}\text{ln}{\gamma }_{\text{M}\left(\text{in}\mathrm{\hspace{0.17em} } \text{Fe}-\text{Sn}-\text{M}\right)}=x_{\text{Fe}}\text{ln}{\gamma }_{\text{M}\left(\text{in}\mathrm{\hspace{0.17em} } \text{Fe}\right)}^0+x_{\text{Sn}}\text{ln}{\gamma }_{\text{M}\left(\text{in}\mathrm{\hspace{0.17em} } \text{Sn}\right)}^0\\ -\left(x_{\text{Fe}}\text{ln}{\gamma }_{\text{Fe}\left(\text{in}\mathrm{\hspace{0.17em} } \text{Fe}-\text{Sn}\right)}+x_{\text{Sn}}\text{ln}{\gamma }_{\text{Sn}\left(\text{in}\mathrm{\hspace{0.17em} } \text{Fe}-\text{Sn}\right)}\right)\\ =x_{\text{Fe}}\text{ln}{\gamma }_{\text{M}\left(\text{in}\mathrm{\hspace{0.17em} } \text{Fe}\right)}^0+\left(1-x_{\text{Fe}}\right)\text{ln}{\gamma }_{\text{M}\left(\text{in}\mathrm{\hspace{0.17em} } \text{Sn}\right)}^0-\left(1-x_{\text{Fe}}\right)\text{ln}{\gamma }_{\text{Sn}\left(\text{in}\mathrm{\hspace{0.17em} } \text{Fe}\right)}^0\end{array}$$(19)

On the other hand, the activity coefficient of M in the Fe–Sn–M ternary system can be expressed as $\text{ln}{\gamma }_{\text{M(in}\mathrm{\hspace{0.17em} } \text{Fe}-\text{Sn}-\text{M)}}$ = $\text{ln}{\gamma }_{\text{M(in}\mathrm{\hspace{0.17em} } \text{Fe)}}^0$ + ${\epsilon }_{\text{M(in}\mathrm{\hspace{0.17em} } \text{Fe)}}^{\text{Sn}}(1-x_{\text{Fe}})$. Because the right-hand side of this expression and Eq. (19) are equal, the following equation is obtained:   

$${\epsilon }_{\text{M}\left(\text{in}\mathrm{\hspace{0.17em} } \text{Fe}\right)}^{\text{Sn}}=\text{ln}{\gamma }_{\text{M}\left(\text{in}\mathrm{\hspace{0.17em} } \text{Sn}\right)}^0-\text{ln}{\gamma }_{\text{Sn}\left(\text{in}\mathrm{\hspace{0.17em} } \text{Fe}\right)}^0-\text{ln}{\gamma }_{\text{M}\left(\text{in}\mathrm{\hspace{0.17em} } \text{Fe}\right)}^0$$(20)

Based on Eq. (20), the interaction coefficient of M with Sn in molten Fe can be calculated from the activity coefficient of each binary system. The following equation was obtained by replacing Fe with Fe–Cr in Eq. (20):   

$${\epsilon }_{\text{M}\left(\text{in}\mathrm{\hspace{0.17em} } \text{Fe}-\text{Cr}\right)}^{\text{Sn}}=\text{ln}{\gamma }_{\text{M}\left(\text{in}\mathrm{\hspace{0.17em} } \text{Sn}\right)}^0-\text{ln}{\gamma }_{\text{Sn}\left(\text{in}\mathrm{\hspace{0.17em} } \text{Fe}-\text{Cr}\right)}^0-\text{ln}{\gamma }_{\text{M}\left(\text{in}\mathrm{\hspace{0.17em} } \text{Fe}-\text{Cr}\right)}^0$$(21)

Assuming that Cr in molten Fe obeys Henry’s law, the activity coefficients of Sn and M are expressed as $\text{ln}{\gamma }_{\text{Sn(in}\mathrm{\hspace{0.17em} } \text{Fe}-\text{Cr)}}^0$ = $\text{ln}{\gamma }_{\text{Sn(in}\mathrm{\hspace{0.17em} } \text{Fe)}}^0$ + ${\epsilon }_{\text{Sn(in}\mathrm{\hspace{0.17em} } \text{Fe)}}^{\text{Cr}}{x}_{\text{Cr}}$ and $\text{ln}{\gamma }_{\text{M(in}\mathrm{\hspace{0.17em} } \text{Fe}-\text{Cr)}}^0$ = $\text{ln}{\gamma }_{\text{M(in}\mathrm{\hspace{0.17em} } \text{Fe)}}^0$ + ${\epsilon }_{\text{M(in}\mathrm{\hspace{0.17em} } \text{Fe)}}^{\text{Cr}}{x}_{\text{Cr}}$, respectively. Thus,   

$$\begin{array}{c}{\epsilon }_{\text{M}\left(\text{in}\mathrm{\hspace{0.17em} } \text{Fe}-\text{Cr}\right)}^{\text{Sn}}=\text{ln}{\gamma }_{\text{M}\left(\text{in}\mathrm{\hspace{0.17em} } \text{Sn}\right)}^0-\left(\text{ln}{\gamma }_{\text{Sn}\left(\text{in}\mathrm{\hspace{0.17em} } \text{Fe}\right)}^0+{\epsilon }_{\text{Sn}\left(\text{in}\mathrm{\hspace{0.17em} } \text{Fe}\right)}^{\text{Cr}}{x}_{\text{Cr}\left(\text{in}\mathrm{\hspace{0.17em} } \text{Fe}-\text{Cr}\right)}\right)\\ -\left(\text{ln}{\gamma }_{\text{M}\left(\text{in}\mathrm{\hspace{0.17em} } \text{Fe}\right)}^0+{\epsilon }_{\text{M}\left(\text{in}\mathrm{\hspace{0.17em} } \text{Fe}\right)}^{\text{Cr}}{x}_{\text{Cr}\left(\text{in}\mathrm{\hspace{0.17em} } \text{Fe}-\text{Cr}\right)}\right)\end{array}$$(22)

Subtracting Eq. (20) from Eq. (22), the following equation is derived:   

$${\epsilon }_{\text{Sn}\left(\text{in}\mathrm{\hspace{0.17em} } \text{Fe}-\text{Cr}\right)}^{\text{M}}={\epsilon }_{\text{Sn}\left(\text{in}\mathrm{\hspace{0.17em} } \text{Fe}\right)}^{\text{M}}-\left({\epsilon }_{\text{Sn}\left(\text{in}\mathrm{\hspace{0.17em} } \text{Fe}\right)}^{\text{Cr}}+{\epsilon }_{\text{M}\left(\text{in}\mathrm{\hspace{0.17em} } \text{Fe}\right)}^{\text{Cr}}\right)x_{\text{Cr}\left(\text{in}\mathrm{\hspace{0.17em} } \text{Fe}-\text{Cr}\right)}$$(23)

The interaction coefficients of M with Sn in molten Fe and Fe–18Cr were estimated using Eqs. (20) and (23) using the values in Tables 3 and 4, where x_{Cr(in Fe–Cr)} was calculated as 0.191 for Fe–18Cr. The estimated values are shown in Table 5 and are compared with the measured values in Fig. 9. From Fig. 9, it was found that the estimated interaction coefficients of B, Ni, and Ti with Sn in molten Fe and Ni and Ti with Sn in molten Fe–18mass%Cr based on the regular solution model reasonably agree with the measured values. On the other hand, the difference between the measured and estimated values of the interaction coefficient of Mo with Sn in molten Fe–18mass%Cr is comparatively larger, and it cannot be estimated using the regular solution model. It is important to measure the interaction coefficients experimentally to determine the activity of the solute element precisely.

Table 3. The values of ${\gamma }_{\text{M(in}\mathrm{\hspace{0.17em} } \text{Sn)}}^0$, ${\gamma }_{\text{M(in}\mathrm{\hspace{0.17em} } \text{Fe)}}^0$, and ${\gamma }_{\text{M(in}\mathrm{\hspace{0.17em} } \text{Fe)}}^0$, from the literature.^47,48,49)

M	${\gamma }_{\text{M(in}\mathrm{\hspace{0.17em} } \text{Sn)}}^0$	${\gamma }_{\text{M(in}\mathrm{\hspace{0.17em} } \text{Fe)}}^0$47)	${\gamma }_{\text{M(in}\mathrm{\hspace{0.17em} } \text{Fe)}}^0$47)
Mo	–	2.58	1
B	20448)		0.022
Ni	0.0023848)		0.66
Ti	0.65149)		0.004
Nb	12.448)		0.2

Table 4. The values of ${\epsilon }_{\text{Sn(in}\mathrm{\hspace{0.17em} } \text{Fe)}}^{\text{M}}$, ${\epsilon }_{\text{Sn(in}\mathrm{\hspace{0.17em} } \text{Fe)}}^{\text{Cr}}$, and ${\epsilon }_{\text{M(in}\mathrm{\hspace{0.17em} } \text{Fe)}}^{\text{Cr}}$, from the literature.^42,47)

M	${\epsilon }_{\text{Sn(in}\mathrm{\hspace{0.17em} } \text{Fe)}}^{\text{M}}$42)	${\epsilon }_{\text{Sn(in}\mathrm{\hspace{0.17em} } \text{Fe)}}^{\text{Cr}}$47)	${\epsilon }_{\text{M(in}\mathrm{\hspace{0.17em} } \text{Fe)}}^{\text{Cr}}$47)
Mo	11.7	3.27	–0.0068
B	9.9		–
Ni	–6.8		–0.0027
Ti	6.0		3.5
Nb	31.7		–

Table 5. The estimated values of the interaction coefficients of M (M: Mo, B, Ni, Ti, and Nb) with Sn, for the molten Fe–18Cr and for the molten Fe, at 1873 K, derived from the regular solution model.

M	${\epsilon }_{\text{Sn(in}\mathrm{\hspace{0.17em} } \text{Fe)}}^{\text{M}}$	${\epsilon }_{\text{Sn(in}\mathrm{\hspace{0.17em} } \text{Fe}-18\text{Cr)}}^{\text{M}}$
Mo	–	11.1
B	8.2	–
Ni	–6.6	–7.4
Ti	4.1	4.7
Nb	3.2	–

Fig. 9.

Relationship between the measured ${\epsilon }_{\text{Sn(in}\mathrm{\hspace{0.17em} } \text{Fe)}}^{\text{M}}$, ${\epsilon }_{\text{Sn(in}\mathrm{\hspace{0.17em} } \text{Fe}-18\text{Cr)}}^{\text{M}}$ values and those derived from the regular solution model.

5. Conclusions

The interaction coefficients of Mo, B, Ni, Ti, and Nb with Sn in molten Fe–18mass%Cr alloy at 1873 K were measured using the liquid immiscibility of Fe–18 mass%Cr and Ag. The conclusions are summarized as follows:

(1) The interaction coefficients of Mo, B, Ni, Ti, and Nb with Sn in the molten Fe–18mass%Cr alloy at 1873 K were determined as follows:   

$${\epsilon }_{\mathrm{S}\mathrm{n}\left(\mathrm{i}\mathrm{n}\mathrm{\hspace{0.17em} } \mathrm{F}\mathrm{e}-18\mathrm{C}\mathrm{r}\right)}^{\mathrm{M}\mathrm{o}}=-2.9\left(\pm 3.2\right),\mathrm{\hspace{0.17em} }{e}_{\mathrm{S}\mathrm{n}\left(\mathrm{i}\mathrm{n}\mathrm{\hspace{0.17em} } \mathrm{F}\mathrm{e}-18\mathrm{C}\mathrm{r}\right)}^{\mathrm{M}\mathrm{o}}=-0.0054\left(\pm 0.0099\right)$$

  

$${\epsilon }_{\mathrm{S}\mathrm{n}\left(\mathrm{i}\mathrm{n}\mathrm{\hspace{0.17em} } \mathrm{F}\mathrm{e}-18\mathrm{C}\mathrm{r}\right)}^{\mathrm{B}}=9.6\left(\pm 1.9\right),\mathrm{\hspace{0.17em} } e_{\mathrm{S}\mathrm{n}\left(\mathrm{i}\mathrm{n}\mathrm{\hspace{0.17em} } \mathrm{F}\mathrm{e}-18\mathrm{C}\mathrm{r}\right)}^{\mathrm{B}}=0.19\left(\pm 0.02\right)$$

  

$${\epsilon }_{\mathrm{S}\mathrm{n}\left(\mathrm{i}\mathrm{n}\mathrm{\hspace{0.17em} } \mathrm{F}\mathrm{e}-18\mathrm{C}\mathrm{r}\right)}^{\mathrm{N}\mathrm{i}}=-8.9\left(\pm 2.6\right),\mathrm{\hspace{0.17em} } e_{\mathrm{S}\mathrm{n}\left(\mathrm{i}\mathrm{n}\mathrm{\hspace{0.17em} } \mathrm{F}\mathrm{e}-18\mathrm{C}\mathrm{r}\right)}^{\mathrm{N}\mathrm{i}}=-0.036\left(\pm 0.011\right)$$

  

$${\epsilon }_{\mathrm{S}\mathrm{n}\left(\mathrm{i}\mathrm{n}\mathrm{\hspace{0.17em} } \mathrm{F}\mathrm{e}-18\mathrm{C}\mathrm{r}\right)}^{\mathrm{T}\mathrm{i}}=10.4\left(\pm 5.7\right),\mathrm{\hspace{0.17em} } e_{\mathrm{S}\mathrm{n}\left(\mathrm{i}\mathrm{n}\mathrm{\hspace{0.17em} } \mathrm{F}\mathrm{e}-18\mathrm{C}\mathrm{r}\right)}^{\mathrm{T}\mathrm{i}}=0.051\left(\pm 0.028\right)$$

  

$${\epsilon }_{\mathrm{S}\mathrm{n}\left(\mathrm{i}\mathrm{n}\mathrm{\hspace{0.17em} } \mathrm{F}\mathrm{e}-18\mathrm{C}\mathrm{r}\right)}^{\mathrm{N}\mathrm{b}}=20.5\left(\pm 8.0\right),\mathrm{\hspace{0.17em} } e_{\mathrm{S}\mathrm{n}\left(\mathrm{i}\mathrm{n}\mathrm{\hspace{0.17em} } \mathrm{F}\mathrm{e}-18\mathrm{C}\mathrm{r}\right)}^{\mathrm{N}\mathrm{b}}=0.055\left(\pm 0.022\right)$$

(2) The interaction coefficients of Mo, B, Ni, Ti, and Nb with Sn in molten Fe and in the molten Fe–18mass%Cr alloy were estimated based on the regular solution model. The estimated interaction coefficients of B, Ni and Ti with Sn in molten Fe and Ni and Ti with Sn in the molten Fe–18mass%Cr alloy reasonably agree with the measured values.

(3) The estimated interaction coefficient of Mo with Sn in the molten Fe–18mass%Cr alloy is larger than the measured value, and the sign of the value is not in accord. It is important to measure the interaction coefficients experimentally to determine the activity of the solute element precisely.

Acknowledgements

This work was supported by a Grant-in-Aid for Scientific Research (B) (Number 18H03405) from the Japan Society for the Promotion of Science (JSPS). We would like to thank Editage (www.editage.com) for English language editing.

References

• 1)   H.  Katayama and  Y.  Mizukami: Materia Jpn., 35 (1996), 1283 (in Japanese). https://doi.org/10.2320/materia.35.1283
• 2)   Y.  Igarashi,  I.  Daigo,  Y.  Matsuno and  Y.  Adachi: ISIJ Int., 47 (2007), 753. https://doi.org/10.2355/isijinternational.47.753
• 3)   H.  Hatayama,  I.  Daigo,  Y.  Matsuno and  Y.  Adachi: Environ. Sci. Technol., 44 (2010), 6457. https://pubs.acs.org/doi/pdf/10.1021/es100044n
• 4)   S.  Hisashige,  T.  Nakagaki and  T.  Yamamoto: ISIJ Int., 59 (2019), 598. https://doi.org/10.2355/isijinternational.ISIJINT-2018-355
• 5)   K.  Shibata,  S. J.  Seo,  M.  Kaga,  H.  Uchino,  A.  Sasanuma,  K.  Asakura and  C.  Nagasaki: Mater. Trans., 43 (2002), 292. https://doi.org/10.2320/matertrans.43.292
• 6)   M.  Hatano,  K.  Kunishige and  Y.  Komizo: Tetsu-to-Hagané, 88 (2002), 142 (in Japanese). https://doi.org/10.2355/tetsutohagane1955.88.3_142
• 7)   K.  Yamamoto,  H.  Shibata,  K.  Nakajima and  S.  Mizoguchi: Tetsu-to-Hagané, 90 (2004), 788 (in Japanese). https://doi.org/10.2355/tetsutohagane1955.90.10_788
• 8)   Y.  Kondo and  H.  Tanei: ISIJ Int., 55 (2015), 1044. https://doi.org/10.2355/isijinternational.55.1044
• 9)   A.  Takemura and  Y.  Tanaka: Tetsu-to-Hagané, 102 (2016), 501 (in Japanese). https://doi.org/10.2355/tetsutohagane.TETSU-2015-109
• 10)   M.  Hatano and  K.  Kunishige: Tetsu-to-Hagané, 89 (2003), 659 (in Japanese). https://doi.org/10.2355/tetsutohagane1955.89.6_659
• 11)   T.  Fukagawa: Ph.D. thesis, Osaka University, (1997), 10, https://doi.org/10.11501/3161879, (accessed 2020-12-18) (in Japanese).
• 12)   H.  Yamauchi,  M.  Miwa and  H.  Kobayashi: Bull. Iron Steel Inst. Jpn., 4 (1999), 378.
• 13)  Committee for Removing Residual Elements from Steel Scrap: Fundamental Studies on Separation of Residual Elements from Steel Scrap, The Iron and Steel Institute of Japan, Tokyo, (1996), 3 (in Japanese).
• 14)   M.  Nakamoto,  Y.  Okumura,  T.  Tanaka and  T.  Yamamoto: Tetsu-to-Hagané, 100 (2014), 761 (in Japanese). https://doi.org/10.2355/tetsutohagane.100.761
• 15)   N.  Sano: Bull. Jpn. Inst. Met., 30 (1991), 905 (in Japanese). https://doi.org/10.2320/materia1962.30.905
• 16)   T.  Imai and  N.  Sano: Tetsu-to-Hagané, 74 (1988), 640 (in Japanese). https://doi.org/10.2355/tetsutohagane1955.74.4_640
• 17)   C.  Wang,  J.  Hirama,  T.  Nagasaka and  S.  Ban-ya: Tetsu-to-Hagané, 77 (1991), 353 (in Japanese). https://doi.org/10.2355/tetsutohagane1955.77.3_353
• 18)   C.  Wang,  T.  Nagasaka,  M.  Hino and  S.  Ban-ya: Tetsu-to-Hagané, 77 (1991), 504 (in Japanese). https://doi.org/10.2355/tetsutohagane1955.77.4_504
• 19)   C.  Wang,  T.  Nagasaka,  M.  Hino and  S.  Ban-ya: Tetsu-to-Hagané, 77 (1991), 644 (in Japanese). https://doi.org/10.2355/tetsutohagane1955.77.5_644
• 20)   H.  Horie,  M.  Nakamura,  S.  Hiratsuka,  T.  Kowata and  Y.  Ishikawa: Imono (J. Jpn. Foundarymen’s Soc.), 64 (1992), 245 (in Japanese). https://doi.org/10.11279/imono.64.4_245
• 21)   A.  Matsui,  Y.  Uchida and  Y.  Takahashi: JFE Tech. Rep., 38 (2016), 58 (in Japanese).
• 22)   X.  Chen,  N.  Ito,  K.  Nakashima and  K.  Mori: Tetsu-to-Hagané, 81 (1995), 959 (in Japanese). https://doi.org/10.2355/tetsutohagane1955.81.10_959
• 23)   T.  Matsuo: Tetsu-to-Hagané, 75 (1989), 82 (in Japanese). https://doi.org/10.2355/tetsutohagane1955.75.1_82
• 24)   T.  Hidani,  K.  Takemura,  R. O.  Suzuki and  K.  Ono: Tetsu-to-Hagané, 82 (1996), 135 (in Japanese). https://doi.org/10.2355/tetsutohagane1955.82.2_135
• 25)   M.  Sasabe,  E.  Harada and  S.  Yamashita: Tetsu-to-Hagané, 82 (1996), 129 (in Japanese). https://doi.org/10.2355/tetsutohagane1955.82.2_129
• 26)   K.  Yamaguchi and  H.  Ono: ISIJ Int., 52 (2012), 18. https://doi.org/10.2355/isijinternational.52.18
• 27)   K.  Yamaguchi,  H.  Ono and  T.  Usui: Tetsu-to-Hagané, 96 (2010), 531 (in Japanese). https://doi.org/10.2355/tetsutohagane.96.531
• 28)   K.  Kato and  H.  Ono: J. Jpn. Soc. Exp. Mech., 18 (2018), 258 (in Japanese). https://doi.org/10.11395/jjsem.18.258
• 29)   K.  Yamaguchi,  H.  Ono and  E.  Takeuchi: Tetsu-to-Hagané, 101 (2015), 636 (in Japanese). https://doi.org/10.2355/tetsutohagane.TETSU-2015-039
• 30)   H.  Ono and  J.  Aboshi: ISIJ Int., 59 (2019), 703. https://doi.org/10.2355/isijinternational.ISIJINT-2018-420
• 31)   S.  Kuzuhara,  K.  Kudo,  H.  Hayashi and  I.  Daigo: ISIJ Int., 59 (2019), 2120. https://doi.org/10.2355/isijinternational.ISIJINT-2019-235
• 32)   I.  Daigo,  K.  Tajima,  H.  Hayashi,  D.  Panasiuk,  K.  Takeyama,  H.  Ono,  Y.  Kobayashi,  K.  Nakajima and  T.  Hoshino: ISIJ Int., 61 (2021), 498. https://doi.org/10.2355/isijinternational.ISIJINT-2020-377
• 33)   S.  Nakamura,  N.  Okubo,  K.  Miyakusu,  M.  Hasegawa and  Y.  Munesue: Nisshin Steel Tech. Rep., 76 (1997), 48 (in Japanese).
• 34)   S.  Nakamura,  S.  Suzuki,  N.  Okubo,  M.  Hasegawa and  K.  Miyakusu: CAMP-ISIJ, 11 (1998), 1147 (in Japanese).
• 35)   S.  Nakamura,  S.  Suzuki,  N.  Okubo,  M.  Hasegawa and  K.  Miyakusu: CAMP-ISIJ, 12 (1999), 517 (in Japanese).
• 36)   M.  Hatano,  H.  Matsuyama,  Y.  Tamura and  T.  Morita: Nippon Steel Tech. Rep., 416 (2020), 40 (in Japanese).
• 37)   H.  Ono and  M.  Nakamoto: Bull. Iron Steel Inst. Jpn., 23 (2018), 253 (in Japanese).
• 38)   N.  Lv,  X.  Gao,  S.  Ueda and  S.  Kitamura: Tetsu-to-Hagané, 105 (2019), 378 (in Japanese). https://doi.org/10.2355/tetsutohagane.TETSU-2018-102
• 39)   K.  Sugiyama,  S.  Ueda,  X.  Gao,  S.  Kim and  S.  Kitamura: ISIJ Int., 57 (2017), 625. https://doi.org/10.2355/isijinternational.ISIJINT-2016-676
• 40)   K.  Sugiyama,  S.  Ueda,  X.  Gao,  S.  Kim and  S.  Kitamura: ISIJ Int., 60 (2020), 2147. https://doi.org/10.2355/isijinternational.ISIJINT-2019-550
• 41)   H.  Ono,  T.  Maeda and  H.  Konishi: Tetsu-to-Hagané, 104 (2018), 121 (in Japanese). https://doi.org/10.2355/tetsutohagane.TETSU-2017-068
• 42)   H.  Ono, H. O and S. Fujiwara: Tetsu-to-Hagané, 105 (2019), 363 (in Japanese). https://doi.org/10.2355/tetsutohagane.TETSU-2018-119
• 43)   H.  Ono,  T.  Miki and  M.  Nakamoto: Tetsu-to-Hagané, 105 (2019), 344 (in Japanese). https://doi.org/10.2355/tetsutohagane.TETSU-2018-148
• 44)   L. J.  Swartzendruber: Bull. Alloy Phase Diagr., 5 (1984), 560. https://doi.org/10.1007/BF02868316
• 45)   M.  Venkatraman and  J. P.  Neumann: Bull. Alloy Phase Diagr., 11 (1990), 263. https://doi.org/10.1007/BF03029296
• 46)   H.  Ono-Nakazato,  D.  Hirai and  T.  Usui: J. High Temp. Soc., 26 (2000), 139.
• 47)  The Japan Society for the Promotion of Science, The 19th Committee on Steelmaking: Steelmaking Data Sourcebook, Gordon and Breach Science Publishers, New York, (1988), 278.
• 48)   T. B.  Massalski: Binary Alloy Phase Diagrams, 2nd ed., Plus Updates, ASM International, Materials Park, OH, (1996), CD-ROM.
• 49)   S.  Itoh and  Y.  Inoue: Mater. Trans., 52 (2011), 704. https://doi.org/10.2320/matertrans.M-M2011801