Introduction

The best soldering materials in the electronics sector are lead–tin alloys because of their exceptional mechanical, chemical, and physical properties. They are very reliable and easily manufactured and have dominated the soldering industry for many years without much improvement. However, finding a new alternative solder alloy to replace lead has been difficult due to its excellent properties except for its environmental and health hazards. Due to the development of advanced technology, most of the electronic items, including mobile phones, etc., have been miniaturized. This technology has grown and advanced quite quickly. Electronic products lose their effectiveness after a short span of time, so, as a result, a lot of electronic garbage is produced.1,2,3,4,5,6,7,8 The majority of old gadgets are thrown away in landfills, where the lead eventually seeps into the soil and contaminates the groundwater. The European Union is making great efforts to enforce a lead ban on devices. In June 2000, the EU approved the Waste of Electrical and Electronic Equipment Directive and the Regulation on the Restrictions of the Use of Certain Hazardous Substances.9,10 There is a big motivation in the electronic industry to shift from lead–tin to lead-free solders. In recent years, electronic assembly manufacturing has undergone considerable modifications. To enable lead-free solder in electronics, some manufacturers have already begun to adapt the design. Accurate phase diagram determination and thermodynamic property measurements are the initial stages in the creation of novel solder alloys (lead-free). Many experts in this field have suggested the creation of a thermodynamic database for solder systems that are free of lead, which is why it is very important to carry out experiments on this type of alloy system to generate a reliable database of their thermodynamic properties.

Literature Survey

Bi-Sn System

Several researchers have conducted experimental investigations on the phase equilibria and thermodynamic characteristics of this system, but there is a significant discrepancy in some of the findings. The specific heat was measured by Nagasaki and Fujita11 and the heat of mixing was calculated by Oelsen and Golucke.12 Ohtani and Ishida,13 who tested the eutectic temperature using differential scanning calorimetry, more recently released liquidus data. These data generally agree well with one another. Numerous writers have also looked at the mixing enthalpies of liquid Bi-Sn alloys, including.14,15,16,17 As noted by Ohtani and Ishida,13 research done prior to the 1940s14 exhibits significant scatter, but later data15,16,17 exhibit rather good agreement. Electromotive force (EMF) techniques have been used by numerous authors to measure the activities of Sn in the liquid phase; Asryan and Mikula's contribution18 is one of the most recent efforts in this area. Bi in the liquid bath of Sn was also measured by Yazawa and Koike19 at 1100 °C, while Rickert et al.20 did so between 750 °C and 1050 °C. The activities were measured by the several authors and there was good agreement; however, the calculated enthalpies revealed significant disparities between them. Only Seltz and Dunkerley's models21 had good agreement with the data that were directly measured.15,16,17 Enthalpies of formation are also different from Oelsen and Golucke's calorimetric observations.12 The study by Lee et al.22 contains a list of other important sources.

Sb-Sn Binary System

The literature has several calorimetric analyses of the enthalpy of mixing of liquid Sb-Sn alloys. Kawakami23 was the first to record it, in 1930, at 800 °C. Later, Kleppa24 did so at 450 °C and 700 °C. All these facts have been included in Hultgren et al compilation's.25 Later, Sommer et al.26 and Azzaoui et al.27 calculated the enthalpy of mixing of liquid alloys in the temperature range of 510–835 °C, and at 619 °C (0.5 ≤ xSn ≤ 1) and 640 °C (0 ≤ xSn ≤ 0.5), respectively, to come to the ultimate conclusion that all the experimental results were satisfactory. Sommer et al.26 proposed a minor but considerable temperature dependence of Hmix and the formation of two compounds, SbSn and SbSn2. Frantik and Mcdonald28 and Vassiliev et al.29 used EMF methods to calculate the relevant Hmix values. Recent research by Chen et al.30 utilizing a regular solution model, and the assumption of temperature-independent heat of mixing, defined the liquid phase in the Sb-Sn system. The computed values and the experimental results published by Wittig et al.,15 Sommer et al.,26 and Azzaoui et al.27 are in good agreement.

Bi-Sb Binary System

The binary system Bi-Sb, which features an equilibrium between solid and liquid solutions, is only briefly discussed in a few papers. Hansen and Anderko31,32,33 established a phase diagram, with a liquidus and a solidus that were somewhat less precisely known. The well-known works of Hultgren et al.25 and Smithells et al.34 both cite these findings. For alloys with a composition of xSb = 0.21–0.77 and at a temperature of 1073 K, Kawakami et al.4,23 calculated the enthalpy of formation through the direct reaction method. At a temperature of 298 K, Kubaschewski and Seith35 found the mixing temperatures of the constituents in the solid state. Geguzin and Pines36 explored solid solutions in the binary Bi-Sb system. For the alloy with xSb = 0.63 in the temperature range of 623–823 K, they calculated the specific heat.

Sn-Bi-Sb Ternary System

Katayama et al.37 employed the fused salt EMF method to obtain the activity of tin between 700 K and 1000 K along pseudo-binary sections of Sn(xSb/xBi = 1/3, 1/1 and 3/1) and in binary alloys of Sn-Sb and Sn-Bi systems. Based on these findings, Darken's equation was used to determine the concentration dependence of the excess Gibbs energy of mixing. Manasijevic et al.38 examined phase equilibria in the Sn-Bi-Sb ternary system using phase diagram calculations as well as experimental results.

Awe et al.39 studied the thermodynamic activity of Sn in the ternary lead-free solder system (Sn-Bi-Sb) at 900 K using a molecular interaction model and predicted the activities of all the components in Sn-Bi-Sb along three cross-sections from the corner of each metal with a molar ratio of the other two components (1:3, 1:1, and 3:1) in the temperature range of 500–1000 K.

Zhang et al.40 studied the effect of Sb content on the properties of Sn-Bi solders and found that the shear strength of Sn-Bi-Sb solders increases as the Sb content increases. This alloy has a wider melting range because it is highly sensitive to Sb content. Also as the Bi content decreases and the Sb content increases, the alloy showed more ductility. So, the literature review suggests that the thermodynamic properties like activities and Gibbs free energy of Sn-Bi-Sb were predicted by using different computational modeling and EMF measurement techniques, but no enthalpy data are available for the entire composition range for the given Sn-Bi-Sb system.

Therefore, calorimetric measurements of the Sn-Bi-Sb system have been carried out along five of the cross-sections (see Fig. 1) in the temperature range of 923–1023 K using a calorimeter (96 Line Evo; Setaram Instruments, France), as set out in the "Calorimetric Measurements" section, below. Partial and integral thermodynamic properties were determined from the calorimetric data. Integral mixing enthalpy plots were used to generate iso-enthalpy curves. The substitutional solution Redlich–Kister–Muggianu (RKM) model was used to derive the interaction parameter based on ternary enthalpy values, and, to obtain these parameters, a least square fitting model was used. Comparisons are made between the estimated and measured values.

Fig. 1
figure 1

Measured cross-sections (intersections a to e indicated; see Table V, below) and alloy compositions in the Sn-Bi-Sb ternary system.

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Experimental

Materials

The binary and ternary alloys formed in this study were made from pure metals (Sn, Bi, and Sb). α-Al2O3 needles from the National Institute of Standards & Technology (NIST; Gaithersburg, MD, USA), were used as the calibration reference. The metals were cleaned with n-hexane in a supersonic bath and then vacuum-dried in a glove box antechamber to eliminate any remaining solvent. They were then divided into small bits and precisely weighed to the accuracy of 10−4 g. Table I lists the specifications for the pure metals and argon (protective gas).

Table I The materials used for this study
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Calorimetric Measurements

In this investigation, the drop calorimeter was used to obtain the enthalpy values of the given system. It is equipped with a furnace of resistance-type made up of a graphite tube, and having an arrangement for the flow of water throughout the outer section of the furnace for continuous cooling and a thermopile of 56 pairs of S-type thermocouples. The furnace could operate at a maximum temperature of 1593 K. It has a motorized dropping apparatus that helps in dropping the metals automatically into the crucible, which had an outer diameter of 12 mm and length of 60 mm. For the calibration of the calorimeter, four pieces of NIST SRM 720 α-Al2O3 needles were dropped at the end of each sample series. The these needles were certified by NIST as adhering to its standards. CALISTO Data Processing software was used to monitor the calorimeter. A specified amount of base metal was added to the crucible before the trials began. The CALISTO software managed the flow of gas and the water-cooling system. Prior to the start of the trials, everything needs to be programmed, such as the rate of heating and cooling, how time the temperature needed to be held for, and the rate of gas flow. To create a completely inert atmosphere, all the air was flushed out of the crucible and the protective tube (the space between outer and inner walls of the furnace) by using a vacuum pumping system and then argon gas was passed through the crucible. Throughout the studies, a steady supply of argon (about 30 mL/min) kept the alloy inert from oxidizing. Argon also took the fumes arising from the heating process from the furnace, and so behaved as a carrier gas as well as a protective gas. There was a 30-min gap between two successive drops and then the heat signals were recorded. At 923 K, 973 K, and 1023 K, calorimetric analyses of the Sn-Bi-Sb system were conducted along five of the cross-sections (see Fig. 1). Pieces of pure tin were dropped into molten Sb0.25Bi0.75, Sb0.50Bi0.50, Sb0.75Bi0.25 alloys. bismuth into Sb0.50Sn0.50 , and antimony into Bi0.50Sn0.50. Prior to introducing the sample material, the system underwent a roughly 16-h thermal equilibration process to establish a stable baseline. The entire mass of the crucible and the samples were weighed before and after the measurements to confirm that there was only a little weight loss due to evaporation. Each experiment was repeated twice while holding the parameters constant. The CALISTO software was then used to integrate the recorded heat signals and estimate the enthalpy values.

Results and Discussion

With the aid of the CALISTO software, the heat flow curves' peaks were integrated. The heat signal value associated with the α-Al2O3 needle drops served as the basis for determining the calibration constant (K). After being multiplied by the calibration constant, the integral values of each heat signal were then converted to enthalpy, which is known as the heat effect. Here, the reaction enthalpy (\({\Delta H}_{\mathrm{Reaction},\mathrm{X},\mathrm{i}}\) ) which is a function of heat effect (\(\Delta {H}_{\mathrm{Signal},\mathrm{X},\mathrm{i}}\cdot K\)) and the change in enthalpy (\({\Delta H}_{\mathrm{X},\mathrm{i}}^{{T}_{\mathrm{D}} \to {T}_{\mathrm{M}}}\)), is calculated by Eq. 1 when species X is dropped from the drop temperature (TD) to the bath temperature (TM):

$$\Delta H_{{{\text{Reaction}},{\text{X}},{\text{i}}}} = \left( {\Delta H_{{{\text{Signal}},{\text{X}},{\text{i}}}} \cdot K} \right) - \left( {\Delta H_{{{\text{X}},{\text{i}}}}^{{T_{{\text{D}}} \to T_{{\text{M}}} }} \cdot n_{{{\text{X}},{\text{i}}}} } \right)$$
(1)

where \({n}_{\mathrm{X},\mathrm{i}}\) represents the amount of species X (no. of moles) which was dropped by the help of automatic dropping device into the liquid bath and, corresponding to every peak, there is a \(\Delta {H}_{\mathrm{Signal},\mathrm{X},\mathrm{i}}\) value which is the integrated area (µV/s) after integrating the peaks corresponding to each drop by using the baseline integration tool. \({\Delta H}_{\mathrm{X},\mathrm{i}}^{{T}_{\mathrm{D}} \to {T}_{\mathrm{M}}}\) represents the change in the molar enthalpy of species X and this change can be calculated by the help of enthalpy data available in the literature for each species corresponding to a given temperature range.

Equation (2) can be used to calculate the partial enthalpy \(\Delta {\overline{H} }_{\mathrm{X},\mathrm{i}}\) as a very small amount of species X was added to liquid metal in the crucible:

$$ \Delta \overline{H}_{{\text{X,i}}} \approx \Delta H_{{\text{Reaction,X,i}}} /n_{{\text{X,i}}} $$
(2)

Each measurement involved adding specific amounts of one of the three metals to the crucible, then dropping the remaining two metals at the specified temperature. Thus, Bi was placed into the crucible and Sb was then dropped, and then Sn was dropped into the binary Bi-Sb; next, Sb was placed into the crucible and Sn was then dropped and then Bi was dropped into the binary Sb-Sn; and finally Sn was placed into the crucible and Bi was then dropped and then Sb was dropped into the binary Bi-Sn. According to Eq. 3, the integral molar mixing enthalpy \(\Delta {H}_{\mathrm{mix}}\) is computed as:

$$\Delta {H}_{\mathrm{mix}}= \frac{\sum \Delta {H}_{\mathrm{Reaction}}}{({n}_{\mathrm{crucible} }+ \sum {n}_{\mathrm{i}})}$$
(3)

The Bi, Sn, and Sb enthalpy increments from room temperature to the drop temperature were acquired from the literature25,41 and used in Eq. 1. Tables II, III, and IV provide the molar integral enthalpies which were derived by the help of Eq. 3 for the given ternary system along five sections considering various compositions. They additionally include partial mixing enthalpies values with the variation in the composition of the different species.

Table II Integral and partial mixing enthalpies of Sn-Bi-Sb alloys when Sn dropped
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Table III Partial and integral mixing enthalpies of Sn-Bi-Sb alloys when Bi dropped
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Table IV Partial and integral mixing enthalpies of Sn-Bi-Sb alloys when Sb dropped
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Errors in calorimetric measurements can come from a number of different places, including the calorimeter's construction type, calibration, heat flow curve baseline integration, degree of solute solubility in the solvent, and impurity concentration. The calorimeter's overall experimental uncertainty ranges from 10 to 12%. It was determined that the calibration error caused by the dropping α-Al2O3 needles was less than ± 1.5%.

The enthalpies of mixing of the five cross-sections depicted in Fig. 1 in the Sn-Bi-Sb ternary system were determined at 923 K, 973 K, and 1023 K. Pieces of pure tin were dropped into molten Sb0.25Bi0.75, Sb0.50Bi0.50, Sb0.75Bi0.25 alloys; bismuth into Sb0.50Sn0.50, and antimony into Bi0.50Sn0.50 alloys. The closeness of readings near the intersection points on the five cross-sections indicates the quality of our experimental data (see Table V; Fig. 1). In each attempt, the variation was significantly reduced and hardly noticeable. However, systemic errors, such as those resulting from insufficient mixing reactions, cannot be completely ruled out.

Table V Integral enthalpy of mixing values at the sites of intersection a, b, c, d, and e; see also Fig. 1
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Figure 2 illustrates the change in mixing enthalpies for three different isopleths and two binaries. For the three isopleths and the Sb-Sn binary, the mixing enthalpies are exothermic across the Sn composition, but endothermic at the low-Sn composition end. For these three cross-sections, it was found that the mixing enthalpy’s minima was exothermic in nature which makes the alloy more stable.

Fig. 2
figure 2

The enthalpies of mixing of the given ternary alloy system corresponding to the cross-section of (Sb0.25Bi0.75)1–xSnx, circle; (Sb0.50Bi0.50)1–xSnx, triangle; and (Sb0.75Bi0.25)1–xSnx, star; at 973 K, Sb-Sn binary data from the literature,38 pentagon; and Bi-Sn binary data from literature, square.38

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Mixing enthalpy’s minima corresponding to the cross-sections (Sb0.25Bi0.75)1–xSnx is close to xSn ~ 0.62 and for the cross-section (Sb0.50Bi0.50)1–xSnx is approximately equal to xSn ~ 0.55 and. if we consider the cross-section (Sb0.75Bi0.25)1–xSnx , it is almost symmetric about the tin composition.

The enthalpy of mixing falls with increasing tin content and reaches a minimum at the tin composition indicated above. As the xSb/xBi ratio decreases, it can also be seen in Fig. 2 that the lowest value (minima) in the curve representing mixing enthalpy becomes less exothermic in nature, and reaches its peak at equi-atomic composition after which it decreases at the extremes. The Bi, Sb, and Sn atoms' interatomic interaction may have diminished as a result.

Regardless of the mixing entropy’s sign, the mixing enthalpy adds adversely to the Gibbs energy for the majority of the tin composition. The measured mixing enthalpy is found to be negative which suggests that the atoms in this system have finite amounts of interatomic interaction energy. This indicates that the heat is produced by a mixing process making it more exothermic.

Figures 3, 4, and 5 illustrate the nature of mixing enthalpies with the variations in the composition of Sn. Additionally, it can be seen that the enthalpy curves at 923 K, 973 K, and 1023 K are quite close to each other. This makes the mixing enthalpy of Sn-Bi-Sb system's independent of temperature. Temperature has very little effect on how the Bi, Sn, and Sb atoms in this system interact with one another in the above temperature range. The degree of mixing is better when the curve is more symmetric. In the temperature range of 923–1023 K, the (Sb0.75Bi0.25)1–xSnx alloy mixes better than the other two alloys.

Fig. 3
figure 3

The enthalpies of mixing of the given ternary alloy system for the cross-section of (Sb0.25Bi0.75)1–xSnx.

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Fig. 4
figure 4

The enthalpies of mixing of the given ternary alloy system for the cross-section of (Sb0.50Bi0.50)1–xSnx.

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Fig. 5
figure 5

The enthalpies of mixing of the given ternary alloy system for the cross-section of (Sb0.75Bi0.25)1–xSnx.

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Figure 6 illustrates the nature of mixing enthalpies with the variations in the composition of Bi. In the given cross-section, the mixing enthalpy maxima are endothermic in nature. The asymmetrical nature of the enthalpy of mixing curve and the large value of the enthalpy of mixing makes the alloy less stable. The amount of heat needed to produce these alloys would depend on their composition. The enthalpy of mixing rises with Bi concentration, reaching a peak around xBi ~ 0.72.

Fig. 6
figure 6

The enthalpies of mixing of the given ternary alloy system for the cross-section of (Sb0.50Sn0.50)1–xBix at 973 K, star; Bi-Sb binary data from the literature,38 circle; and Bi-Sn binary data from literature, square38.

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Figure 7 further shows the enthalpies of mixing plots for a given ternary system at 923 K, 973 K, and 1023 K as a function of composition. These plots are quite close to each other. This might mean that, in the Sn-Bi-Sb system, mixing enthalpy is almost independent of temperature and mixing behavior does not significantly alter with the variation of temperature.

Fig. 7
figure 7

The enthalpies of mixing of the given ternary alloy system for the cross-section of (Sb0.50Sn0.50)1–xBix.

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Figure 8 illustrates the relationship between mixing enthalpies with variation in Sb composition of the given ternary alloy system. In the given cross-section, the minima of the mixing enthalpy is exothermic in nature which makes the alloy more stable. In the cross-section (Bi0.50Sn0.50)1–xSbx, the minima of enthalpy of mixing is almost symmetric with regard to antimony composition.

Fig. 8
figure 8

The enthalpies of mixing of the given ternary alloy system for the cross-section of (Bi0.50Sn0.50)1–xSbx at 973 K, star; Bi-Sb binary data from literature,38 square; and Sb-Sn binary data from literature, triangle.38

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The enthalpy of mixing falls as antimony concentration rises, reaching minima at xSb ~ 0.50. It reaches its peak at equi-atomic percent and then decreases at the end.

The Bi, Sb, and Sn atoms' interatomic attraction may have diminished as a result. Regardless of the mixing entropy’s sign, the mixing enthalpy adds adversely to the Gibbs energy for the majority of the Sn composition. The measured mixing enthalpy was found to be negative, which suggests that the atoms in this system have finite amounts of interatomic interaction energy. Therefore, heat produced by the mixing process is exothermic and is released during the mixing process.

Figure 9 further shows that the plots at 923 K, 973 K, and 1023 K are quite near to each other. This suggests that the Sn-Bi-Sb system's enthalpy of mixing is nearly temperature-independent.

Fig. 9
figure 9

The enthalpies of mixing of the given ternary alloy system for the cross-section of (Bi0.50Sn0.50)1–xSbx.

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The plots between molar enthalpy of mixing and the composition of dropping species was used to create the above curves. It is shown in Fig. 10 for nine iso-enthalpies, i.e., 0.2 kJ/mol, 0.4 kJ/mol, − 0.2 kJ/mol, − 0.4 kJ/mol, − 0.6 kJ/mol, − 0.8 kJ/mol, − 1.0 kJ/mol, − 1.2 kJ/mol, and − 1.4 kJ/mol. The mixing enthalpy is seen to be more negative when the alloy’s composition is nearer to the Sb-Sn binary system and positive when it is nearer to the Bi-Sb binary system. This is because the Sb-Sn binary system has the largest negative mixing enthalpy values among the three binaries, whereas the Bi-Sb alloys have the highest positive enthalpy of mixing values. The binary Sb-Sn affects the majority of the iso-enthalpy curves. This suggests that the Sb and Sn atoms in the given ternary systems' sub-lattices may have a greater influence on the orientation of the enthalpy of mixing curves.

Fig. 10
figure 10

Different iso-enthalpy curves of the given liquid ternary alloy at 973 K; values are in kJ/mol.

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The thermodynamic characteristics of various type of systems may be ternary or a higher order may be deduced from the data of different binaries using a variety of theoretical models. The theoretical model put forward by Ansara and Dupin is significant.42 The RKM polynomial has also been employed by Luef et al.43 for substitutional solutions. We have used the least squares fitting for the given dataset as in:

$$ \Delta H_{{{\text{mix}}}} = \mathop \sum \limits_{{\text{i}}} \mathop \sum \limits_{{{\text{j}} > {\text{i}}}} \left[ {x_{{\text{i}}} x_{{\text{j}}} \mathop \sum \limits_{{\upupsilon }} L_{{{\text{i }}:{\text{ j}}}}^{\left( \upsilon \right)} \left( {x_{{\text{i}}} - x_{{\text{j}}} } \right)^{\upsilon } } \right] + x_{{\text{i}}} x_{{\text{j}}} x_{{\text{k}}} \left( {L_{{{\text{i}}:{\text{j}}:{\text{k}}}}^{0} x_{{{\text{Sn}}}} + L_{{{\text{i}}:{\text{j}}:{\text{k}}}}^{1} x_{{{\text{Bi}}}} + L_{{{\text{i}}:{\text{j}}:{\text{k}}}}^{2} x_{{{\text{Sb}}}} } \right) $$
(4)

where \({L}_{\mathrm{i }:\mathrm{ j}}^{(\upsilon )}\)(ν = 0, 1, 2, …) are the interaction parameters (binary) of the binary systems and \({L}_{\mathrm{i}:\mathrm{j}:\mathrm{k}}^{\nu }\) (ν = 0, 1, 2,…) are the interaction parameters (ternary). The mixing enthalpy obtained experimentally is compared with the calculated values considering the binary contributions only. The difference gives the ternary interaction contribution in this system, which is further optimized using Eq. 4. The ternary mixing enthalpy for Sn-Bi-Sb system for the cross-sections of (Sb0.25Bi0.75)1–xSnx, (Sb0.50Bi0.50)1–xSnx, (Sb0.75Bi0.25)1–xSnx, (Sb0.50Sn0.50)1–xBix, (Bi0.50Sn0.50)1-xSbx is represented in Fig. 11, along with the RKM modeled data fitted curve at 973 K as per Eq. 4.

Fig. 11
figure 11

Comparison of mixing enthalpies between RKM-modeled data, bold line, and experimental data of liquid Sn-Sb-Bi alloys along the cross-sections of (Sb0.25Bi0.75)1–xSnx, circle; (Sb0.50Bi0.50)1–xSnx, triangle; and (Sb0.75Bi0.25)1–xSnx, star; at 973 K.

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The binary interaction parameters and findings from this study have been used to derive the ternary interaction parameters using the least square fit. The binary interaction parameters are taken from data in the literature38 that was readily available. Table VI lists the interaction parameters for binary and ternary systems. The comparison between the experimental data with the values obtained from the RKM model considering the ternary interaction are shown in Fig. 11. It can be seen that both the values for the mixing enthalpies were in good agreement except for a few compositions. Therefore, the RKM substitutional model is very well followed by this system in the temperature range of 923–1023 K.

Table VI Binary and ternary interaction parameters
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In Fig. 12, integral enthalpies of mixing for one of the five cross-sections (Bi0.50Sn0.50)1–xSbx at 923 K, 973 K and 1023 K are plotted along with the literature data, 38which were calculated using ternary interaction parameters of the boundary binaries of the ternary Bi-Sn-Sb system based on the RKM polynomial. It is inferred from Fig. 12 that the experimental results of the present investigations are in good agreement with the data reported in the literature.

Fig. 12
figure 12

Comparison of experimental results of this study with data available in the literature38 along the cross-sections of (Bi0.50Sn0.50)1–xSbx at 923 K, 973 K and 1023 K.

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Summary and Conclusions

Integral and partial mixing enthalpies of the Sn-Bi-Sb ternary system were investigated using a drop calorimeter in the temperature range of 923–1023 K along five of the cross-sections: (Sb0.25Bi0.75)1–xSnx, (Sb0.50Bi0.50)1–xSnx, (Sb0.75Bi0.25)1–xSnx, (Sb0.50Sn0.50)1–xBix, (Bi0.50Sn0.50)1–xSbx by drop methods. Pure Sb, Sn, and Bi were dropped into the crucible depending on the cross-sections. It has been found that the enthalpies of mixing are nearly temperature-independent.

The curves representing iso-enthalpy were plotted using the mixing enthalpy values for the all five cross-sections. Most of these curves were oriented towards the Sb-Sn binary system. This could be due to the stronger interatomic interaction between Sb and Sn atoms in the given sub-lattice of the given ternary system. Ternary interaction parameters were determined by using the RKM model, followed by least square optimization. The agreement between the experimental results and theoretical model was very good.