## Abstract

Viscosity modeling for pure metals and alloys is widely studied, and many solutions for dependence of viscosity on temperature can be found in the literature for pure metals and alloys. Many of these depend on experimental data for pre-exponential and exponential coefficients. Two key models include: (i) Kaptay model for pure metals, which is completely independent of experimental data and depends only on general constants A and B for a large set of pure metals with few exceptions and (ii) Kaptay viscosity model for liquid alloys derived on the same principles, a temperature-dependent viscosity only as a function of thermophysical properties of the alloy components. In the case of surface tension, the main available models are divided into four groups: Butler formulation-based models, density-functional models, semi-empirical models, and thermodynamic geometric models. Considering the absence of adequate models for surface tension, in this work, two equations relating surface tension and viscosity for pure metals are analyzed as a function of temperature. Regarding the Egry surface tension-viscosity relation for pure metals, a new relation equation for multicomponent alloys is proposed. By applying the proposed equation, the surface tension is calculated and plotted as a function of temperature for ternary and quaternary aluminum alloys.

### Keywords

- viscosity
- surface tension
- thermophysical properties
- multicomponent alloys
- surface tension-viscosity correlation equation

## 1. Introduction

Surface tension is a phenomenon that occurs whenever a liquid is in contact with other liquids or even gases; then, an interface is established and it acts like a stretched elastic membrane. A surface is called wet if the contact angle is less than 90° and nonwet otherwise. A substrate, like dust or pollution, can contribute energetically to the membrane, decreasing its contact angle. Another feature is the magnitude of the surface tension

Many solutions can be found in the literature concerning the viscosity dependence on temperature for pure metals [1, 2, 3, 4, 5, 6, 7] and for alloys [8, 9, 10, 11, 12, 13, 14, 15]. Budai et al. have reviewed the existing models used to predict dependence of viscosity on temperature of alloys, for cases where the viscosities of pure components are already known [10, 15, 16, 17, 18] and those that are independent of experimental data [19]. Budai et al. extended the Kaptay unified equation [10] for the viscosity of alloys, which has been named BBK model [20]. The BKK model was shown to fail in the prediction of viscosity for alloy systems with components that melt congruently [21].

Solutions for surface tension as a function of temperature are generally based on: Butler formulation [22, 23, 24, 25]; statistical thermodynamics surface density-functional theory [26, 27, 28]; semi-empirical thermodynamic model [29]; and thermodynamic models [30, 31]. All these models are normally specific for certain binary or ternary alloy systems, or they are general but considerably difficult to apply.

In 1992, Egry [32] derived a relation between surface tension and viscosity deduced from statistical mechanics for the melting temperature, based upon the expressions of Fowler [26] for surface tension and Born and Green [33] for viscosity. Both expressions are expressed as integrals over the product of interatomic forces and the pair distribution functions. The author extended this relation to a finite temperature range by using data available in the literature [34].

In this work, by using a straightforward solution for viscosity for molten pure metals [7] and alloys [10], a comparison between numerical simulations and experimental data for the surface tension and viscosity of pure liquid metals and liquid alloys is provided, in order to validate Egry’s relation for pure molten metals [34]. An extension of this relation is derived for multicomponent alloys. The surface tension is calculated and plotted against temperature for ternary and quaternary aluminum alloys.

## 2. Modeling

The modeling section is divided into models dealing with the viscosity of pure metals and multicomponent alloys, and with the surface tension-viscosity relation equations for pure liquid metals and alloys.

### 2.1 Model for the viscosity of pure liquid metals

Kaptay [7] derived a unified equation for the viscosity of pure liquid metals as a function of temperature, which encompasses the activation energy and the free volume concept. Based on the Andrade’s equation [35] as a starting approach, the activation energy concept has been incorporated. By combining again with Andrade’s formulation with free volume concept, an equation for the dependence of viscosity of pure metals on temperature has been derived. As this equation obeys both concepts, the authors named it as a unified equation for the viscosity of pure metals. The derived equation for the viscosity of pure metals as a function of temperature is given by,

where * i*, A and B are temperature-independent semi-empirical parameters being approximately identical for all liquid metals,

*(*i

### 2.2 Model for the viscosity of multicomponent alloys

An Arrhenius-type viscosity equation can be extended to deal with viscosity of multicomponent alloy, by applying Redlich-Kister polynomial to excess viscosity,

where

where

then, we have,

where

Kaptay [10], based on the Seetharaman-Du Sichen equation, regarding the theoretical relationship between the cohesion energy of the alloy and the activation of viscous flow, proposed the following equation:

where

### 2.3 Relation between surface tension and viscosity for pure liquid metals

Egry [32] derived a relation between surface tension and viscosity deduced from statistical mechanics for a finite temperature range, based upon the expressions of Fowler [26] for surface tension (

where

In a very similar way, Born and Green [33] derived an expression for the viscosity (

where

where

In 2005, Kaptay [7] derived a surface tension-viscosity relation, as

where

## 3. A surface tension and viscosity relation model for multicomponent alloys

Based on the formula originally proposed by Egry [34], a new formula is derived for the relation between surface tension and viscosity for multicomponent alloys, that is,

where

Substituting Eq. (12) into Eq. (6), we get,

## 4. Results and discussion

Figure 1 presents the viscosity of pure aluminum as a function of temperature, simulated by Kaptay [7] theoretical model compared to experimental data [37]. The model fits the experimental scatter well, but a correction in the melting temperature for pure aluminum was carried out according to the model description provided in [7], although, for the case of Al, no correction was applied by the author [7]. Figure 2 shows the evolution of viscosity of molten Cu as a function of temperature, where the theoretical model is compared to experimental results [37], where a deviation is noticed for temperatures close to the melting temperature.

In Figure 3, the viscosity of pure molten silicon is plotted against temperature. As mentioned, the melting temperature of Si has been assumed as

In Figure 4, it can be seen that Kaptay’s model [7] well the experimental data well for the viscosity of molten magnesium as a function of temperature [38, 39, 40].

In Figures 5, 6, 7, 8, the surface tension for pure Al, Cu, Si, and Mg is depicted as a function of temperature, calculated from surface tension-viscosity relations according to Egry [34] and Kaptay [7] formulations. For all cases, except for Si in Figure 7, the surface tension exhibits a trend to decrease as the temperature increases. It also can be noticed that for all cases, they diverge close to the melting point. For high temperatures, both models yield very close results. The best agreement observed between the two models is that of pure Mg, as shown in Figure 8.

Figures 9, 10, 11, 12 show the evolution of viscosity as a function of temperature for all examined alloys, where it can be seen that the two models generally exhibit similar results. The greatest deviation between the models can be observed for the Al-6wt%Cu-3wt%Si alloy, Figure 10; but even in this case, a relatively good agreement can be considered. This may be related to the non-ideal term of viscosity of the Redlich-Kister equation, whose coefficients do not depend on the alloy composition but depend on the temperature [36].

Figure 13 shows the application of the derived surface tension relation equation for alloys, Eq. (13), to ternary Al-Cu-Si and quaternary Al-Cu-Si-Mg alloys. As can be noticed, the increase in the alloy Si content decreases the surface tension. The lowest surface tension profile is associated with the alloy having the highest Si content, that is, for the Al-12wt%Si-1wt%Cu-1wt%Mg quaternary alloy.

## 5. Conclusion

A solution for viscosity of alloys has been derived based on an extension of Egry’s surface tension-viscosity relation equation for pure metals [34] and Kaptay’s [10] unified solution for viscosity of multicomponent alloys. Kaptay and Egry’s surface tension-viscosity relations were plotted against temperature. It was shown that for the pure metals analyzed in the present study, for lower temperatures, both relations deviated from each other. However, for higher temperatures, a better agreement has been achieved. The derived solution for surface tension was plotted against temperature for ternary Al-Cu-Si and Al-Cu-Si-Mg alloys. It was shown that with increasing Si alloy content, the surface tension of Al-based alloys decreases, which is in agreement with the casting practice of Aluminum-based alloys.

## Acknowledgments

The authors acknowledge the financial support provided by FAPERJ (The Scientific Research Foundation of the State of Rio de Janeiro), CAPES, and CNPq (National Council for Scientific and Technological Development).

## References

- 1.
Dinsdale AT, Quested PN. The viscosity of aluminium and its alloys—A review of data and models. Journal of Materials Science. 2004; 39 :7221-7228 - 2.
Brooks RF, Dinsdale AT, Quested PN. The measurement of viscosity of alloys—A review of methods, data and models. Measurement Science and Technology. 2005; 16 :354-362 - 3.
Iida T, Shiraishi Y. In: Kawai Y, Shirashi Y, editors. Chapter 4: Viscosity Handbook of Physico-Chemical Properties at High Temperatures. Japan: Iron and Steel Institute; 1988 - 4.
Hildebrand JH. Viscosity and Diffusivity: A Predictive Treatment. New York: John Wiley & Sons Inc; 1977 - 5.
Walther C. The evaluation of viscosity data. Erdöl und Teer. 1931; 7 :382-384 - 6.
Chhabra RP, Tripathi A. A correlation for the viscosity of liquid metals. High Temperatures - High Pressures. 1993; 25 :713-718 - 7.
Kaptay G. A unified equation for the viscosity of pure liquid metals. Zeitschrift fuer MetaIlkunde. 2005; 96 :1-8 - 8.
Zivkovic D, Manasijevic D. An optimal method to calculate the viscosity of simple liquid ternary alloys from the measured binary data. Calphad. 2005; 29 :312-316 - 9.
Kucharski M. The viscosity of multicomponent systems. Zeitschrift fuer MetaIlkunde. 1986; 77 :393-396 - 10.
Kaptay G. A new equation to estimate the concentration dependence of the viscosity of liquid metallic alloys from the heat of mixing data. In: Proceedings of microCAD 2003 Conference. Section Metallurgy, University of Miskolc; 2003. pp. 23-28 - 11.
Zhong XM, Liu YH, Chou KC, Lu XG, Zivkovic D, Zivkovic Z. Estimating ternary viscosity using the thermodynamic geometric model. Journal of Phase Equilibria. 2003; 24 :7-11 - 12.
Singh RN, Nadan R, Sommer F. Viscosity of liquid alloys: Generalization of Andrade’s equation. Monatshefte für Chemie. 2012; 143 :1235-1242 - 13.
Jeyakumar M, Hamed M, Shankar S. Rheology of liquid metals and alloys. Journal of Non-Newtonian Fluid Mechanics. 2011; 166 :831-838 - 14.
Solek K, Korolczuk-Hejnak M, Slezak W. Viscosity measurement for modeling of continuous steel casting. Archives of Metallurgy and Materials. 2012; 57 :333-338 - 15.
Moelwyn-Hughes EA. Physikalische Chemie. Stuttgart: Thieme; 1970 - 16.
Iida T, Ueda M, Morita Z. On the excess viscosity of liquid alloys and the atomic interaction of their constituents. Tetsu to Hagane. 1976; 62 :1169-1178 - 17.
Kozlov LY, Romanov LM, Pod’yachev AP, Petrov NN, Kostygova OV. Calculation of the viscosity of multicomponent iron- and nickel-base melts. Izvestiya Vysshikh Uchebnykh Zavedenii, Chernaya Metallurgiya. 1982; 5 :1-4 - 18.
Seetharaman S, Du SC. Estimation of the viscosities of binary metallic melts using Gibbs energies of mixing. Metallurgical and Materials Transactions B: Process Metallurgy and Materials Processing Science. 1994; 25B :589-595 - 19.
Hirai M. Estimation of viscosities of liquid alloys. ISIJ. 1993; 33 (2):281-285 - 20.
Budai I, Benkö MZ, Kaptay G. Comparison of different theoretical models to experimental data on viscosity of binary liquid alloys. Materials Science Forum. 2007; 537–538 :489-496 - 21.
Cheng J, Gröbner J, Hort N, Schmid-Fetzer R. Measurement and calculation of the viscosity of metals—A review of the current status and developing trends. Measurement Science and Technology. 2014; 25 :062001. 9pp - 22.
Butler JAV. The thermodynamics of the surfaces of solutions. Proceedings of the Royal Society of London A. 1932; 135 :348-375 - 23.
Tanaka T, Lida T. Application of a thermodynamic database to the calculation of surface. Tension for iron-base liquid alloys. Steel Research International. 1994; 65 :21-28 - 24.
Hoar TP, Melford DA. The surface tension of binary liquid mixtures: Lead + tin and lead + indium alloys. Transactions of the Faraday Society. 1957; 53 :315-326 - 25.
Tanaka T, Hack K, Lida T, Hara S. Application of thermodynamic databases to the evaluation of surface tension of molten alloys, salt mixture and oxide mixtures. Zeitschrift fuer MetaIlkunde. 1996; 87 :380-389 - 26.
Fowler RH. A tentative statistical theory of Macleod’s equation for surface tension, and the parachor. Proceedings of the Royal Society of London A. 1937; 159 :229-246 - 27.
Shih W-H, Stroud D. Theory of the surface tension of liquid metal alloys. Physical Review B. 1985; 32 :804-811 - 28.
Aqra F, Ayyad A, Takrori F. Model calculation of the surface tension of liquid Ga-Bi alloy. Applied Surface Science. 2011; 257 :3577-3580 - 29.
Goicoechea J, Garcia-Cordovilla C, Louis E, Pamies A. Surface tension of binary and ternary aluminium alloys of the systems Al-Si-Mg and AI-Zn-Mg. Journal of Materials Science. 1992; 27 :5247-5252 - 30.
Egry I. Surface tension of compound forming liquid binary alloys: A simple model. Journal of Materials Science. 2004; 39 :6365-6366 - 31.
Amore S, Brillo J, Egry I, Novakovic R. Surface tension of liquid Cu–Ti binary alloys measured by electromagnetic levitation and thermodynamic modelling. Applied Surface Science. 2011; 257 :7739-7745 - 32.
Egry I. On a universal relation between surface tension and viscosity. Scripta Metallugica et Materialia. 1992; 26 :1349-1352 - 33.
Born M, Green HS. A general kinetic theory of liquids III. Dynamical properties. Proceedings of the Royal Society of London A. 1947; 190 :455-474 - 34.
Egry I. On the relation between surface tension and viscosity for liquid metals. Scripta Metallugica et Materialia. 1993; 28 :1273-1276 - 35.
Andrade EN da C. The theory of the viscosity of liquids: I. Philosophical Magazine. 1934; 17 :497-511 - 36.
Zhang F, Du Y, Liu S, Jie W. Modeling of the viscosity in the Al-Cu-Mg-Si system: Database construction. Calphad. 2015; 49 :79-86 - 37.
Kobatake H, Schmitz J, Brillo J. Density and viscosity of ternary Al-Cu-Si alloys. Journal of Materials Science. 2014; 49 :3541-3549 - 38.
Lihl F, Nachtigall E, Schwaiger A. Viskositätsmessungen an binären Legierungen des Aluminiums mit Silizium, Zink, Kupfer und Magnesium. Zeitschrift fuer MetaIlkunde. 1968; 59 :213-219 - 39.
Culpin MF. The viscosity of liquid magnesium and liquid calcium. Proceedings of the Physical Society. 1957; 70 :1079-1086 - 40.
Gebhardt E, Becker M, Dorner S. Effect of alloying additions on the viscosity of aluminum. Aluminum. 1955; 31 :315-321