# Contact Angles

In addition to the surface tension at a liquid-vapor ($\ell v$) interface discussed above, surface tensions can also exist at interfaces between solid-liquid interface ($s\ell$) and solid-vapor interface (sv); this can be demonstrated using a liquid-vapor-solid system in Fig. 1. The contact line is the locus of points where the three phases intersect. The contact angle, θ, is the angle through the liquid between the tangent to the liquid-vapor interface and the tangent to the solid surface. The contact angle is defined for the equilibrium condition. In 1805, Young published the basic equation for the contact angle on a smooth, insoluble, and homogeneous solid:

 $\cos \theta =\frac{{{\sigma }_{sv}}-{{\sigma }_{s\ell }}}{{{\sigma }_{\ell v}}}$ (1)
Figure 1 Drop of liquid on a planar surface.
Figure 2 Schematic of apparent contact angle θ: (a) stationary liquid, (b) liquid flows upward, (c) liquid flows downward.

which follows from a balance of the horizontal force components (Faghri, 1995), as shown in Fig. 1. When there is relative motion of a liquid drop over a solid surface, a different contact angle can be expected. When the relative motion stops, an angle different from the apparent (equilibrium) contact angle is seen; it depends upon the direction of the previous motion, i.e., whether it was a receding or advancing surface, as shown in Fig. 2. The minimum wetting contact angles for different solid-liquid combinations obtained by Stepanov et al. (1977) are reproduced in Table 5.1. All contact angle approaches are based on the following assumptions (Kwok and Neumann, 1999; Yang et al., 2003): (1) validity of Young’s equation (1), (2) pure liquid, (3) constant values of ${{\sigma }_{sv}}\text{, }{{\sigma }_{s\ell }}\text{, and }{{\sigma }_{\ell v}}$, (4) the value of liquid surface tension which should be higher than the anticipated solid surface tension, and (5) a value of σsv independent of the liquid used.

Table 5.1 Minimum wetting contact angle in arc degree (the upper and lower values are for advancing and receding liquid front, respectively; Stepanov et al. 1977)

 Acetone Water Ethanol R-113 Aluminum 73/34 Beryllium 25/11 63/7 0/0 Brass 82/35 18/8 Copper 84/33 15/7 Nickel 16/7 79/34 16/7 Silver 63/38 14/7 Steel 14/6 72/40 19/8 16/5 Titanium 73/40 18/8

For a rough surface, the contact angle θrough is related to the contact angle on a smooth surface θ by

 cosθrough = γcosθ (2)

where γ is the ratio of the rough surface area to the smooth surface area. Since γ is always greater than 1, cosθrough is greater than cosθ. Therefore, the contact angle θrough is less than the contact angle on a smooth surface, θ. Equilibrium contact angles can vary depending on the motion history of the contact line, particularly for rough surfaces. Equilibrium contact angles may be used in calculations for various heat transfer devices.

## References

Faghri, A., 1995, Heat Pipe Science and Technology, Taylor & Francis, Washington, DC.

Faghri, A., and Zhang, Y., 2006, Transport Phenomena in Multiphase Systems, Elsevier, Burlington, MA.

Stepanov, V.G., Volyak, L.D., and Tarlakov, Y.V., 1977, “Wetting Contact Angles for Some Systems,” Journal of Engineering Physics and Thermophysics, Vol. 32, No. 6, pp. 1000-1003.

Yang, J., Han, J., Isaacson, K., and Kwok, D. Y., 2003, “Effects of Surface Defects, Polycrystallinity, and Nanostructure of Self-Assembled Monolayers for Octadecanethiol Adsorbed on to Au on Wetting and its Surface Energetic Interpretation,” Langmuir, Vol. 19, pp. 9231-9238.

Yang, T.H., and Pan, C., 2005, “Molecular Dynamics Simulation of a Thin Water Layer Evaporation and Evaporation Coefficient,” International Journal of Heat and Mass Transfer, Vol. 48, pp. 3516-3526.

Kwok, D.Y., and Neumann, A. W., 1999, “Contact Angle Measurement and Contact Angle Interpretation,” Advances in Colloid and Interfaces Science, Vol. 81, pp. 167-249.