Calculation of Surface Tension

Surface tension is a thermophysical property closely related to the bulk thermodynamic properties of the material. It is important to understand the surface tension properties as it can have a big influence on the way in which alloys solidify. In particular it affects the solerability solderability. In conventional solders lead had a particularly desirable property of reducing the surface tension. Understanding how the addition of impurities affect the surface tension of lead free solders is important in attempts to remove the use of lead from electronic interconnections. Measurements of surface tension has been carried out systematically within the COST531 Action resulting in the Surdat database developed by Moser and colleagues. The Figure below shows a key of measurements carried out on the surface tension of pure Sn [W Gasior, Z Moser, J Pstrus, J. Phase Equil. 2001, 221(1), 20-25].

Surface tension of Sn from the work of Gasior et al.

While surface tension is not a thermodynamic property of the bulk material it is neverthless a reflection of the thermodynamic properties of the atoms near the surface. For a pure material, of course, the surface has the same composition as that of the bulk material. However when it comes to an alloy the different component elements have different tendencies to be surface active causing the composition of the surface to be quite different from that of the bulk material.

Tanaka [99Tan] has attempted to model the surface tension of a wide range of materials, both ionic and metallic melts, by using an equation developed by Butler [32But] which seeks to express the surface tension in terms of an equilibrium between the bulk material and a surface layer. Tanaka expressed the Butler equation as:

Butler equation for surface tension expressed by Tanaka

which expresses the fact the chemical potential of the components is equal in the bulk and surface layer or alternatively that there is chemical equilibrium between the bulk and the surface layer. The right hand side of the above equations relate to the bulk thermodynamic properties while on the left hand side of the equations relate to the surface. The thermodynamic properties of the bulk are, of course, well represented by standard thermodynamic models and data as stored in comprehensive thermodynamic databases such as the SOLDERS database. The surface thermodynamic properties are given by:

relative to the pure elements in the liquid phase. The surface area contributions, A₁ and A₂, to the surface properties could be related to the volumes of the pure elements in then the liquid:

where N₀ is the Avogadro number and V₁ the molar volume of the first element. Tanaka calculated the surface tension for a number of metallic systems and found an empirical relationship relating G^mix , the excess Gibbs energy of mixing, for the surface and G^mix for the bulk:

It is now possible to calculate the equilibrium between the bulk material and the surface layer. The unknown surface tension, σ, of the liquid for a particular composition is calculated to be the value which just brings the surface into equilibrium with the bulk. This could be seen as analogous to a procedure for calculating the pressure required to bring a new phase into equilibrium with a chosen phase.

While the empirical relationship above derived by Tanaka is useful in the absence of any other information it may be more useful to derive the excess properties for the surface layer from any experimental surface tension data available. The way in which this problem is now formulated means that the approach can easily be generalised to any number of components. Indeed the most practical way to introduce a capability for calculating surface tension is to create a database for the thermodynamic properties of the surface. The figure below shows the calculated surface tension of Bi-Sn liquids with experimental points superimposed. 

The composition of the surface can also be calculated at the same time as shown in the next figure and this shows how Bi is markedly more surface active than Sn.

The next figure shows the calculated surface tension of Ag-Sn alloys at 1000ºC.  

While the actual value of the surface tension is important, it is often more important to know how the surface tension changes with temperature. Computational thermochemistry provides this capability as shown in the next figure where the temperature dependence of the surface tension is predicted to change sign.