Temperature dependences of surface tension, density and viscosity study of Sn-Ag-Cu with Bi additions using theoretical models

In this work, the kohler, Muggianu,Toop and Hillert geometric models were used to calculate the surface tension, molar volume and density of the liquid Sn-Ag-Cu-Bi quaternary alloys along three selected sections xSn:xAg:xCu = 1:1:1, 1:1:2 and 1:2:1 in the temperature range of 923 K–1423 K. The choice of this temperature range was made on the basis of the calculation results of the liquidus line of the alloys belonging to the three sections. The same properties have been estimated for five selected Sn2.7Ag0.86Cu3.86Bi, Sn3.13Ag0.48Cu4.02Bi, Sn2.95Ag0.53Cu6.81Bi, Sn2.68Ag1.01Cu6.62Bi and Sn3.24Ag0.75Cu1.76Bi quaternary alloys between 623 K and 1123 K for comparison with the available experimental data. Moreover, the surface tension and density of these five alloys have also been calculated on the basis of Guggenheim and theoretical equation, respectively. In addition, the Seetharaman-sichen and Kaptay equations were extended to estimate the viscosity of SAC + Bi alloys. We also discussed the influence of Bismuth addition in liquid Sn-Ag-Cu-Bi. Estimated values show that Bi increases molar volume and density but decreases the surface tension and viscosity. On the other hand, the surface tensions diminish with the temperature for the all studied models, with the exception of some concentration of Bismuth; an inverse tendency is observed (dσ/dT) > 0. While, the density diminishes with increasing temperature for all alloys (dσ/dT) < 0. These models have been shown to be a great alternative for calculating the thermo-physical properties of quaternary systems.

In recent years, researchers have been very interested in identifying alternatives for lead solders because they are harmful to the environment. This trend has been encouraged in Europe by the RoHS and WEEE European Directive. Based on the analysis of several multi-component systems, two sets of alloys are potential substitutes for lead soldering. The first groups are the Sn-Ag-Cu (SAC) alloys with the addition of different metals (Bi, Zn, In and Sb) [1][2][3][4][5][6] . Previous studies have recommended the Sn-Ag-Cu-Bi quaternary system as a promising candidate for lead-free solders, as the biggest advantage of SnAgCuBi over welds SnAgCu is the lower melting temperature. In SnAgCuBi, bismuth plays an important role in lowering the melting temperature of the alloy. It is possible to reduce the melting temperature of the alloy from 217 °C to 208 °C, by balancing the amount of bismuth as well as the amounts of silver and copper in the alloys. However, the addition of Bi to Sn-Ag-Cu alloys increases the mechanical properties of solders 7 .
Since surface properties play a key role in the development of welding, the development of new welds requires data on thermophysical properties such as surface tension that can be obtained experimentally or by numerical modeling if reliable data exist for subsystems and pure components. In addition, theoretical model is sometimes preferred because of the increasing complexity and costs of experimentation, especially for multi-component systems (quaternary, quinary…). The analysis of the influence of the amount of bismuth in the Sn-Ag-Cu-Bi quaternary welds on various properties was carried out by Hwang 8 and Takao et al. 9 . However, little previous work on thermo-physical and electrical properties has been reported in the literature 10 . Indeed, Moser et al. 11 , in 2006, have measured the thermo-physical and mechanical proprieties of Sn-Ag-Cu-eutectic alloy with different additions of Bismuth using different techniques. They showed a linear dependence of surface tension as a function of

Method of calculation
Numerous studies have been carried out for the creation of adequate databases of thermo-physical properties (surface tension, density and viscosity) of Sn, Ag, Cu and Bi under the framework of developing lead-free solders. In this work, we used the pure constituents of Sn, Ag, Cu and Bi inside the temperature range (523 K-1473 K) of our prediction. Moreover, among these numerous studies, we have chosen the values for pure elements that led to values of quaternary system Sn-Ag-Cu-Bi closest to the experimental ones.
Modeling surface tension of liquid Sn-Ag-Cu-Bi system from geometric models. The information of the properties of binary alloys is indispensable for the prediction of thermo-physical properties of quaternary alloys for the purpose of comparing with experimental data. The models proposed by various researchers are exposed in the following sections. The models of Kohler 15 and Muggianu et al. 16 were used in this work as simple models based on binary data, while the Toop 17 and Hillert 18 models are asymmetric.
Kohler model. The Kohler and Muggianu models have been used to estimate the surface tension of Sn-Ag-Cu-Bi quaternary alloys. Consequently, the properties of any quaternary system can be calculated from the knowledge of the corresponding properties of the boundary binaries.
Indeed, for the two symmetric models, the surface tension for a four-component system has been expressed by the following equations: Muggianu model. Toop model. On the other hand, the asymmetric models of Toop and Hillert are respectively expressed by the two equations: Hillert model.
In all equations σ E 234 and σ − − − E 1 2 3 4 correspond to the excess surface tension for ternary and quaternary system, while x 1 , x 2 , x 3 and x 4 represent the mole fractions of components in investigated system.
The excess surface tension of six binary systems. The excess surface tension σ E for binary system is a composition dependency of surface tension of mixture that can be defined as follows: with σ and σ i represent the surface tension of binary alloys and the surface tension of the ideal alloys, respectively. σ i can be defined as following: with X i and σ i represent the molar fraction of a constituent i and the surface tension of the pure constituent i, respectively. The surface tension, as a function of temperature, of the pure constituent i is presented in Table 1.
The excess surface tension values of the six binary alloys are taken from previous work Sn-Ag 23 , Sn-Cu 24 , Sn-Bi 23 , Ag-Cu 25 and Ag-Bi 23 . The values of Cu-Bi sub-binary alloys are predicted based on Butler's equation 13 .
Using of guggenheim equation. Based on the model of regular solutions, the Guggenheim 19 equation relies on statistical approximations, in which the constituents of the binary are distributed randomly within the quasi-crystalline liquid. Moreover, this equation has been developed for quaternary alloys, which can be written in the form: where σ M is the surface tension of the quaternary system, σ 1 , σ 2 , σ 3 and σ 4 are surface tensions of the individual components of the alloy, and A is the molar surface area which is defined by:  Table 2). N A is the Avogadro number and f is the atomic arrangement factor for the liquid surface. www.nature.com/scientificreports www.nature.com/scientificreports/ The excess molar volume of six binary systems. In this research, we used Kohler 15 , Muggianu 16 , Toop 17 and Hillert 18 to calculate molar volume of Sn-Ag-Cu-Bi alloys. The excess molar volume, V E for binary system is a composition dependency of molar volume of mixture that can be defined as follows: E i V and V i represent the molar volume of binary alloys and the molar volume of the ideal alloys, respectively. V i can be defined as follow: X i and V i represent the molar fraction of a constituent i and the molar volume of the pure constituent i, respectively ( Table 3).
The excess molar volume values of the sub-binary systems are taken from previous work Sn-Ag 23 , Sn-Cu 24 , Sn-Bi 23 , Ag-Cu 25 and Ag-Bi 23 .
The molar volume of Cu-Bi binary system was estimated using: Calculation of molar volume of liquid phase in the Sn-Ag-Cu-Bi system. The same equations previously used for the calculation of the excess surface tension σ E will be used for the calculation of the excess molar volume V E .
Density calculation of liquid the Sn-Ag-Cu-Bi system. The relationship between density and molar volume is an example is expressed as: where ρ is the density of mixture, while x i and M i are the molar fraction and molar mass of component i. V M is a molar volume of a mixture, calculated using Kohler 15 , Muggianu 16 , Toop 17 and Hillert 18 models. The mathematical prediction of density is based on semi-empirical equations as well as on theoretical equations 20 , ever since the atomic volumes of most liquid binary alloys is in fact a linear equation of composition, the calculation of alloy density can be approximated by an addition. A similar procedure can be used for higher order alloys, as indicated by: where x 1 , x 2 , x 3 , x 4 are the atomic fractions of the constituents of the alloys, ρ 1 , ρ 2 ρ 3 and ρ 4 are the densities of the pure components (Table 2).

Viscosity in the liquid Sn-Ag-Cu-Sn quaternary alloys.
Viscosity is one of the key properties of alloys that influence the performance of pyrometallurgical processes in several ways. As such, the researcher has made considerable efforts in numerous experimental studies to quantify the dependence on viscosity composition and temperature of many simple and complex alloy systems. However, because of the problems and difficulties inherent in high temperature measurements, the available experimental results only cover a limited range of compositions and temperatures and do not fully meet the needs of the industry. In addition, the accuracy or reliability   www.nature.com/scientificreports www.nature.com/scientificreports/ of some published data has been found to be unsatisfactory. The discrepancy between some of the data is significantly large 26 .
As a result, in recent decades, various models have been allocated to calculate the viscosity of metal alloys. These models are based on one of the theoretical equations used to estimate the viscosity of single liquids, such as Eyring's equation 27 , the Bockris Bockris' equation 28 , Weymann's equation 29 and Frenkel's equation 30 . There have also been some studies to correlate viscosity with self-defined parameters to find some consistency 26 . All modeling and correlation studies were useful and provided a reasonably good description of viscosity over a range of temperatures and compositions.
Seetharaman-Sichen equation. The Seetharaman-Du Sichen equation 21 is a mathematical equation for estimation the viscosity used and developed for quaternary Sn-Ag-Cu-Bi alloys:  This excess Gibbs energy was calculated using the Kohler geometric 15 .  E represent the excess Gibbs energies of the boundary binary systems taken along the quasi-binary sections X i /X j = x i /x j . x i and Xi are respectively the molar fraction of a constituent i in the quaternary and binary system.
The contribution of the six binary systems for the Sn-Ag-Cu-Bi quaternary alloy is described by polynomial Redlich-Kister polynomial 31 :  www.nature.com/scientificreports www.nature.com/scientificreports/   www.nature.com/scientificreports www.nature.com/scientificreports/ where ΔV E is the molar volume of excess for liquid quaternary alloys (m 3 /mol) and ΔH mix represent the integral enthalpy of the mixture, α is a semi-empirical parameter of the model. It is worth (0.155 ± 0.015) and can be neglected when experimental data are not available 32 .
Most thermodynamic data of ternary and multicomponent systems will come from a theoretical calculation rather than direct experiences because of their difficulties especially for metallurgical systems. The geometric model has been applied to estimate the integral enthalpy of mixing. Chou 33 has presented a general geometric model for calculating the thermodynamic properties of ternary and multi-component systems from binary data 32 . www.nature.com/scientificreports www.nature.com/scientificreports/  www.nature.com/scientificreports www.nature.com/scientificreports/

Results and Discussion
Surface tension in the liquid Sn-Ag-Cu-Bi quaternary alloys. The phase equilibria in the Sn-Ag-Bi-Cu quaternary system have been studied theoretically using thermodynamic calculations for three sections x Sn :x Ag :x Cu = 1:1:1, 1:1:2 and 1:2:1 and 1:2:1. The Gibbs energy values and the interaction parameters of all the phases of this system have been taken from NIST solder database 34 . The calculations were performed using Open Calphad software 35 .
As shown in Fig. 1, the results show that above 923 K, all the quaternary alloys of the three studied sections are in the liquid phase. www.nature.com/scientificreports www.nature.com/scientificreports/ One of the objectives of this article is to show that the geometric models and the Guggenheim equation work well for poly-constituent systems, especially for the quaternaries. Our calculations were done along three ternary sections Sn:Ag:Cu = 1:1:1,1:1:2 and 1:2:1. In addition, (Sn-Ag-Cu) eutc + Bi have been done using different predicting methods such as Kohler 15   www.nature.com/scientificreports www.nature.com/scientificreports/  www.nature.com/scientificreports www.nature.com/scientificreports/ bismuth addition in amount 0.1-0.2 mole an opposite tendency is observed   Table 7. Temperature dependence of the viscosity (mPa/s) of Sn-Ag-Cu-Bi quaternary alloys at various temperatures using Seetharaman-Sichen equation.
lowest surface tension of the four alloy metals (σBi < σSn < σAg < σCu), the system seems to reduce its energy by separating the component with the small surface tension at the surface 36 . Indeed, the bismuth component plays an important role not only in reducing the melting temperature of the alloy but also in reducing the surface tension of the SnAgCuBi alloys. It is important to note that the same behavior was observed for the other models (Muggianu and Hillert) and the Guggenheim equation.
Our calculated surface tensions of five quaternary Sn-Ag-Cu-Bi alloys are illustrated in Fig. 4 together with the measured ones 11,14 at 623 K-1123 K. It can be seen that a good agreement between the theoretical and measured values is obtained. In addition, for the comparison check, we calculated the mean square deviation corresponding to the experimental results for each traditional model and the Guggenheim equation.  where σ th,i and σ exp,i represent the surface tension of Sn-Ag-Cu-Bi alloys a permanent composition i for a theoretical models and an experimental values, respectively, while N is the total amount of investigate alloys. The calculations of square deviation are collected in Table 5. www.nature.com/scientificreports www.nature.com/scientificreports/ Generally, the surface tensions calculated with Guggenheim equations are in good accord with those obtained experimentally 11,14 . The calculated values from geometric models are slighly superior than the experimental ones. With the exception of certain temperatures, geometric models are in good accord with experimental data. Molar volume in the liquid Sn-Ag-Cu-Bi quaternary alloys. The molar volume of Sn-Ag-Cu-Bi in the liquid phase over a wide temperature range (from 923 to 1423 K) was calculated using geometric models (Kohler, Muggianu, Toop and Hillert).
We give only the calculated results of the molar volume using the Kohler model (Fig. 5). www.nature.com/scientificreports www.nature.com/scientificreports/ It can be noted in Fig. 5 that the molar volume of Sn-Ag-Cu-Bi augments linearly with the rise in temperature. Besides, the rise in the quantity of bismuth has an effect on the molar volume of Sn-Ag-Cu-Bi quaternary system. The similar behavior was observed for other models (Muggianu, Toop and Hillert).
Density in the liquid Sn-Ag-Cu-Bi quaternary alloys. The density of liquid Sn-Ag-Cu-Bi alloys as a function of temperature along the three sections x Sn :x Ag :x Cu = 1:1:1, 1:1:2 and 1:2:1 was calculated from the molar volume.
As an example, we present only the results of Kohler. The values are illustrated in the Fig. 6. The results obtained clearly show that the density of the quaternary Sn-Ag-Cu-Bi system decreases linearly with temperature < ρ ( ) 0 d dT and increases with the concentration of Bismuth at a given temperature (Fig. 6). The increase in density with addition of Bi was observed. The same effect has been observed by other authors 11 . This effect can be interpreted as bismuth having the most robust density as tin (ρSn <ρBi).
It can be noted simply from Fig. 7, our calculated density results in this study using the theoretical equation is in good accord with the experimental data. However, the geometric models are higher than experimental one.
It may be concluded that the estimated density values for the quaternary Sn-Ag-Cu-Bi using theoretical equation are characteristics to ideal solution at different temperatures (623 K-1123 K).
Standard deviations were determined for all models and for theoretical equation as: where ρ th,i and ρ exp,i represent the density of Sn-Ag-Cu-Bi alloys a permanent composition i for a theoretical models and an experimental values, respectively, while N is the totality amount of investigate alloys. The calculations of square deviation are collected Table 6. It can be seen from S that the calculated values of the density using the (Toop and Hillert) models give results close to experimental values 11 , in particular for temperature range at 823-1123 K. While for temperature range 623 K-1123 K, the results calculated using the theoretical equation give the values closest to the experiment data 14 .

Viscosity in the liquid Sn-Ag-Cu-Bi quaternary alloys. Based on Seetharaman-Sichen statistic and
Kaptay equations for Sn-Ag-Cu-Bi alloys, viscosities were calculated. The theoretical results for the viscosity of Sn-Ag-Cu-Bi alloys obtained using Seetharaman-Sichen and Kaptay equations are presented in Tables 7 and 8, respectively as an Arrhenius equation at various temperatures and given away in Figs 8 and 9.
As seen from Fig. 8, the viscosity of these alloys decreases curvilinearly with increasing temperature and bismuth. This decrease of viscosity by adding bismuth can be explained by its low viscosity in comparison of the other metals (ŋBi < ŋSn < ŋAg < ŋCu).
The results show a similar behavior of the viscosity in function of the temperature than those obtained using Seetharaman-Sichen equation. The viscosity decreases curvilinear with increasing temperature, and decreases with function of temperature and Bi content for all sections (Fig. 9). The predicted results are compared to each other and to the experimental ones (Fig. 10).
As seen in Fig. 10, the viscosity values obtained by Seetharamn sichen and Kaptay equations are slightly lower than those measured by Gancarz et al. 14   www.nature.com/scientificreports www.nature.com/scientificreports/ conclusions In this work the some thermo-physical properties (Surface tension, molar volume, density and viscosity) of the quaternary system Sn-Ag-Cu-Bi have been predicted at different temperatures. Therefore, we reformulated models, which permit one to calculate the surface tension, molar volume and density. This study is carried out using traditional geometric models and theoretical equation, such as Kohler, Muggianu, Toop and Hillert as symmetric and asymmetric models and Guggenheim, Seetharaman-sichen and Kaptay equations. Some important results of our predictions reveal the following conclusions: • For all compositions of Bismuth (except for the composition range with Bismuth molar content lower than x Bi = 0.2) in the quaternary system Sn-Ag-Cu-Bi, the surface tension decreases with increasing temperature. Indeed, the surface tension diminishes with addition of Bismuth concentration. • Addition of Bismuth to ternary Sn-Ag-Cu increases the molar volume and density but diminishes the viscosity. • We can conclude that among all the traditional predictive models and the theoretical equation, Guggnheim (for surface tension) and the theoretical equation (for viscosity) give the best agreement with the experimental one. • The viscosity results of the present work obtained by Seetharamn sichen and Kaptay equations are slightly lower than those measured by Gancarz et al. 14 .