Capillary Instabilities of Interfaces

This simple calculation is valid in the limit of small viscosities finite viscosity gives rise to a finite damping of these capillary modes. Experimental measurements of this dispersion relation using light scattering yields an estimate of the interfacial tension y.  

In the case of capillary waves, the flat surface or interface is basically stable and the waves are the result of thermal fluctuations. However, there are interface geometries where a simple interface shape — such as a cylindrical tube of one phase in other or the surface of a cylinder of fluid or solid — is intrinsically unstable because there are other geometries of lower surface areas and hence lower interfacial free energies. A simple energetic/thermodynamic argument (see the next example) can be used to show that the free energy of an undulated cylinder with undulations whose wavelength exceeds a critical value proportional to the cylinder radius, is lower than that of the perfect cylinder. These undulations eventually lead to the breakup of the cylinder into 

While the Rayleigh instability of a cylindrical surface is driven by the surface energy, the flat surface of a thin film can become unstable under the influence of van der Waals forces (see Chapter 5). In this case, the surface tension provides a restoring force which stabilizes short-wavelength undulations. Again, the critical wavelength is determined by a balance of the free energies (see the problems at the end of this chapter). 

Show that a planar interface, governed only by its surface tension is stable with respect to smaii unduiations of the interface, whiie a cyiindricai interface is unstabie to iong-waveiength undulations. 

To determine the stabiiity, we consider the interfaciai free energy of Eq. (3.1) and determine whether a smaii perturbation of the interface raises or iowers the free energy, if the free energy is increased by the perturbation, the surface is stabie — aithough the entropy gained by unduiations results in a finite ampiitude to these undulations (see Fig. 3.1 and Fig. 3.2) via thermal fluctuations. If the free energy is lowered by the perturbation, the 

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