Introduction

Sn–Pb solders were widely used for many years by electrical and electronic equipment (EEE) manufacturers as a joining material because of low cost, good wetting, easy melting and superior mechanical properties despite the well-known fact of toxic effect of Pb on human health and severe environment pollution [1, 2]. However, rapid advances in microelectronic design and technology in recent decades led to multiple accelerations of Pb-containing electronic waste (i.e., the used EEE) accumulation and made Pb pollution a major issue [3]. Beginning with the middle of the noughties, a number of legislations restricting the use and disposal of Pb–Sn solders (e.g., EU Directive 2002/95/EC) had been implemented in the different parts of the world [4]. Since that time, the process of gradual substitution of traditional soldering alloys with nonpolluting alternative materials was launched and a wide variety of Pb-free soldering compositions was developed [1, 4,5,6,7,8,9,10,11,12].

A binary and ternary alloys of Sn with Ag and/or Cu are recognized [6, 9, 13,14,15] as the most promising materials in this line, but some of their functional properties are inferior to those of conventional Pb-containing solders [2, 3, 13, 16, 17]. Ag–Sn (Ag-Cu–Sn) alloys are potential candidates for using as individual lead-free solders as well as the base of more complicated compositions. An improvement in these lead-free alloys may be achieved by adding a fourth alloying element (Mn, Ce, Ni, Ti, Co, Al, Fe, Bi, Sb, In or Au) [13, 16, 18, 19] or by reinforcing the soldering material with carbon nanoparticles [20, 21]. In any case, the knowledge of the base alloys mixing enthalpies is essential for the further development of new high-performance solders by time- and labor-saving way.

Thermodynamics of the considered Ag–Sn system formation was studied earlier, and the data on mixing enthalpy ∆H mix of liquid alloys Ag–Sn were reported by different authors [22,23,24]. The published results are in good qualitative agreement with each other. In all the cases, a distinct exothermic interaction of liquid components was observed within compositional range between 0 and 65–80 at.% Sn with the extremum − 2.7 ÷ − 3.3 kJ mol−1 at about 20 at.% Sn. For the rest compositions, a weak (less than 0.5 kJ mol−1) endothermic interaction is typical. Meanwhile, the data presented by different authors have a quantitative discrepancy up to 0.5 kJ mol−1, and it may be attributed to the lack of identical experimental conditions, temperatures in particular.

The present work is a part of a large-scale investigation of alloy formation in the ternary system Ag–Cu–Sn including boundary binary solutions Ag–Sn, Cu–Sn, Ag–Cu. To obtain the accurate results, it is critically important to measure all the involved compositions under the identical conditions. This is why we undertook one more independent attempt of Ag–Sn ∆H mix calorimetric determination using the fixed local measuring parameters. Earlier the mixing enthalpies of liquid alloys Ag–Cu–Sn were measured by Luef et al. [5] at the temperatures not higher 900 °C, i.e., in the limited compositional range. We carry out experiments at the elevated temperature 1150 °C above meting points of the pure components and hope much to get new information on Ag–Cu–Sn alloys mixing enthalpies, which will be a supplementary contribution to the existing thermodynamic databases such as the COST-Action 531 of the European Union.

Materials

The considered alloys were prepared from tin (99.999% purity) and silver (999.99% purity). All measurements were taken in a protective dynamic atmosphere of argon 5.0 (purity 99.999) with a flow rate of this gas of 10 ml min−1. The instrument sensitivity calibration was taken using metal components of the alloy and standard samples of synthetic leucosapphire. The masses of metal and sapphire calibrating samples were 55–130 and 50–54 mg, respectively.

Experimental

The components were mixed by drop calorimetry in the experimental chamber of a SETARAM MHTC thermal analyzer. The mixing enthalpy was measured as follows: First, pure metals were cut into the samples of approximately equiaxial shape and overall size not more than 4 mm and a mass within 43–152 mg. The latter were weighed on an analytical balance Sartorius CPA225D with an accuracy of ± 0.01 mg. Then, the metal pieces were loaded into the programmable introducing device for their subsequent feeding into the hot experimental chamber every 15 min. The main parameters of the experiment (initial and final temperatures, the duration and temperature of the isothermal exposures, the sequence of the gas circuit electromagnetic valves switching, etc.) were set with the use of Calisto 1.088 software.

After evacuating the air from the calorimeter down to the air pressure 2.5 × 10−2 torr and creating protective inert atmosphere in it, the prespecified sequence of the temperature stages of the experiment was started. The analytical signal of the heat flux, the sample temperature and the current time were recorded using the Calisto program with the formation of a data file on the computer’s hard disk. Afterward, the recorded data were processed to determine the values of the enthalpy change following the introduction of each sample into a measuring detector heated to a predetermined temperature. A detailed premise of the procedure for determining the integral mixing enthalpy ∆H mix by drop calorimetry is given in [25].

The reference values on pure metal component enthalpies that are necessary for the estimation of the calibration factor k were taken from [22, 26]. It was found out that the data on Ag in [22] and [26] at 1150 °C are practically the same. But this relationship is quite another matter concerning pure Sn. The difference between enthalpy values of Sn given by Dinsdale in [26] and by Kubaschewski and Alcock in [22] at a temperature 1150 °C is as much as 10%. Therefore, we had performed an independent study of Sn enthalpy increment [27, 28] and came to the conclusion that the data [22] are the most suitable one for making calorimeter’s calibration. Comparison between various reference data on ∆H of Sn at high temperatures [22, 26, 29, 30] and our results [27, 28] is considered in Table 1. One can see that the values reported by Veryatin et al. [30] are evidently overestimated. The values of ∆H taken from [26] and [29] are almost identical. Our experimental points are in a best agreement with the temperature dependence of ∆H [22] being extrapolated up to 1200 °C.

Table 1 Experimental values of Sn enthalpy increment in comparison with the reference data
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As it is well known (see, e.g., [25]), the correctness of the calorimeter’s calibration is of critical importance for the accurate and reliable calorimetric results. In our case, the determination of the sensitivity calibration factor k of the instrument consisted of two parts. Initially, the samples of the first component of the alloy were successively introduced into the measuring detector for the gradual accumulation of the first liquid metal in analytical crucible. Taking into account the all-round pattern of heat flux recording in a 3D-type measuring detector, the value of k was postulated to be constant and independent of the number of introduced samples. It was determined as the arithmetic mean for all the drops. After completion of the multiple adding of the second component with the formation of an alloy of approximately equimolar composition, several pieces of a reference sapphire were additionally fed to the liquid metal mirror and the value of k was again calculated. Finally, to measure the mixing enthalpy, we have not used k = const, but assumed a linear change in k with the mass of the melt during the dilution of the alloy with the second component. The application of factor k correction by sapphire samples adding to mixing enthalpy measurement was discussed in detail earlier in [31]. The error of enthalpy determination taking into account possible inaccuracies of experiments and reference data was about ± 5%.

Results and discussion

Experimental results on heat effects of the binary liquid Ag–Sn alloy formation at 1150 °C are represented in Fig. 1. The experimental data cover the entire compositional range from pure silver to pure tin and lie completely in the negative region of the graph indicating the exothermic nature of interparticle interactions.

Fig. 1
figure 1

Compositional dependency of Ag–Sn alloys mixing enthalpy at 1150 °C. 1 Experimental data after the primary calibration using pure components; 2 the same data after additional correction of calibration by leucosapphire samples; 3 approximation of experimental points by Eq. (1); 4 approximation of experimental points by Eq. (3)

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The dots show the compositional dependence of the mixing enthalpy ∆H mix at 1150 °C, represented by two opposite branches. The position of the asterisks on the graph was found taking into account the instrument’s sensitivity calibration by the first pure components only. It can be seen that such calibration does not give a smooth transition of one branch into another. The squares represent the same results after applying the supplementary calibration by sapphire samples.

Both series of experimental points contain smoothed stepwise segment in the range 0.47 < N Sn < 0.53. The nature of this phenomenon is unclear. To our regret, drop calorimetry alone is not able to explain the mentioned anomaly. An extra study is required. However, it is important to note that the tabulated raw data on Ag–Sn mixing enthalpy at 1250 °C reported in [23] demonstrate the similar behavior around 55–60 at.% Sn being plotted versus composition.

Correction of k by adding sapphire samples significantly improves the convergence of the counter branches of the ∆H mix isotherm, but it does not solve the problem completely. Therefore, two branches of square dots were finally approximated by a common analytic function of the following kind:

$$\Delta H_{\text{mix}} = N_{\text{Ag}} \cdot N_{\text{Sn}} \cdot \left( {a + b \cdot N_{\text{Sn}} + c \cdot N_{\text{Sn}}^{2} + d \cdot N_{\text{Sn}}^{3} + \cdots } \right),$$

where a, b, c, d… are const, and N Ag, N Sn are molar fractions of the components.

For this purpose, we reduced the measured points to ∆H mix·N −1Ag ·N −1Sn and described experimental data in Fig. 2 with a power polynomial using the least square method to determine the best degree and coefficients of polynomial. It was found out that the power polynomial should be at least of the fifth order and has the following constant parameters for good approximation of ∆H mix·N −1Ag ·N −1Sn with R-squared coefficient of determination R 2 = 0.999:

$$\Delta H_{\text{mix}} = N_{\text{Ag}} \cdot N_{\text{Sn}} \cdot \left( {{-}\,22.068{-}60.141 \cdot N_{\text{Sn}} + 577.07 \cdot N_{\text{Sn}}^{2} {-}1300.6 \cdot N_{\text{Sn}}^{3} + 1233.7 \cdot N_{\text{Sn}}^{4} {-}429.49 \cdot N_{\text{Sn}}^{5} } \right),{\text{kJ}}\,{\text{mol}}^{ - 1}.$$
(1)
Fig. 2
figure 2

Approximation of the experimental data on ∆H mix/(N Ag·N Sn) (circles) using Eq. (1) (solid line) and Eq. (3) (dash-and-dot line)

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Results of the experimental data fitting with the use of Eq. (1) are represented with a solid line in Figs. 1 and 2. The above-stated approximation is accurate, but it is purely mathematical procedure and has no certain physical meaning.

That is why we performed one more approximation using the theory of solutions. Results of the latter fitting are represented in Figs. 1 and 2 with a dash-and-dot line. Owing to the complex shape of the ∆H mix isotherm, a quasi-chemical model of the subregular solutions considering pair and ternary interparticle interactions in the melt was selected to approximate the experimental points. In general, the mixing heat isotherm in this model is described by the equation [32]:

$$\Delta H_{\text{mix}} = N_{\text{Ag}} \cdot N_{\text{Sn}} \cdot (\alpha_{1} \cdot N_{\text{Ag}} + \alpha_{2} \cdot N_{\text{Sn}} {-}\alpha_{3} \cdot N_{\text{Ag}} \cdot N_{\text{Sn}} ),$$
(2)

where α 1, α 2, α 3 are const.

In order to fit the experimental points by Eq. (2), the values of N Ag in brackets were replaced with (1 − N Sn) giving finally the following equation:

$$\Delta H_{\text{mix}} /(N_{\text{Ag}} \cdot N_{\text{Sn}} ) = A + B \cdot N_{\text{Sn}} + C \cdot N_{\text{Sn}}^{2} ,$$

where A, B, C are const and are equal to α 1, − α 1 + α 2 − α 3 and α 3, respectively.

The experimental results taken as ∆H mix/(N Ag·N Sn) were described by a parabolic function using the least square method with the determination of A, B and C values (Fig. 2). The R-squared coefficient of determination was R 2 = 0.983. This procedure allowed us to estimate the values of the α coefficients in Eq. (2), which finally acquired the form:

$$\Delta H_{\text{mix}} = N_{\text{Ag}} \cdot N_{\text{Sn}} \cdot \left( {{-}\,27.7075 \cdot N_{\text{Ag}} {-}0.9977 \cdot N_{\text{Sn}} + 27.7079 \cdot N_{\text{Ag}} \cdot N_{\text{Sn}} } \right),\;{\text{kJ}}\,{\text{mol}}^{ - 1} .$$
(3)

Both the experimental points and the approximating curve in Fig. 1 have pronounced asymmetry, while the strongest exothermic interparticle interaction being observed at a tin content is about 23%.

In Fig. 3 the integral ∆H mix of Ag–Sn system is shown together with the partial molar mixing enthalpies Δ\(\bar{H}_{\text{mix}}\) of components. The partial molar mixing enthalpy of tin is determined using differentiation of Eq. (3) with respect to N Sn, and Δ\(\bar{H}_{\text{mix}}\) of silver was found from the Gibbs–Duhem equation.

Fig. 3
figure 3

Enthalpies of components mixing in the Ag–Sn system at 1150 °C. Solid line—integral mixing enthalpy; dot line and dash-and-dot line—partial molar mixing enthalpies

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Our results are compared with the available information on Ag–Sn ∆H mix [22,23,24] in Fig. 4. In general, our results do not contradict the known data concerning value and shape of the ∆H mix(N Sn) curve. But at a high tin contents our isotherm remains in the negative region, whereas the data of the other authors are slightly positive.

Fig. 4
figure 4

Ag–Sn mixing enthalpy isotherms by different authors. 1 Data [5] at 1250 °C, 2 data [22] at 1150 °C, 3 data [24] at 1127 °C, 4 our data at 1150 °C

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As mentioned above, the reference data on enthalpy increment of pure Sn by Dinsdale [26] seem to be less suitable for heat calibration of the calorimeter in comparison with [22]. Nevertheless, we made an attempt to recalculate experimental data and reproduce Fig. 1 using data [26] for both pure Sn and Ag. Results are given in Fig. 5.

Fig. 5
figure 5

Reconstruction of Fig. 1 by the application of data [26] on the enthalpy of both pure components to sensitivity calibration of the calorimeter. Circles—experimental data after the primary calibration using pure components; rhomb dots—the same data after additional correction of calibration by leucosapphire samples; dash line—approximation of experimental points by quasi-chemical model of the subregular solutions

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One can see from Fig. 5 that the sensitivity calibration of the calorimeter using the substitution of data [22] on Sn enthalpy increment at 1150 °C with that of [26] makes the shape of ∆H mix curve and its sign closer to the previous data of the other authors. But it does not provide satisfactory convergence of the opposite branches of the isotherm.

Conclusions

It was shown by the experimental study that the obtained ΔH mix isotherm in general does not disagree with the results of other authors, but our curve does not change its sign and lies entirely in the region of exothermic thermal effects. The detected difference may be explained by the fact that previous researchers had used less accurate high-temperature literature data on the enthalpy of tin for calorimeter calibration.