Stability and Elasticity of Soap Films

The relative stability of soap films is partially due to their elasticity. To see this, we consider a film with an equilibrium state represented by the surface excess T and the dissolved surfactant concentration c. A local stretching disturbs these parameters. If the disturbance occurs on a short time scale, the soap molecules do not have time to diffuse out from the inner fluid to the surface, so the same number of soap molecules remains both inside and at the surface. As the area increases, Tg decreases, which results in an increase of the surface tension according to Eq. (2.11), thus opposing the stretching. This process is called the Marangoni elasticity and explains the film stability to rapid disturbances. If the time scale of the stretching is 

In principle, the limit between the Marangoni and the Gibbs elasticity is given by the time scale % = h /D, which characterizes the time needed to diffuse away by the thickness h. Since the diffusion coefficienf is D 4 x 10-1° mVs, for films of 1 pm thick is in the order of 0.01 s. However, it is arguedi that impurities make the equilibrium more difficult to reach, so that Marangoni elasticity can be observed up to tiie order of a second. 

For small concentrations n = TgRT (see (2.11)), i.e., the elastic modulus simplifies to  

In thick films, there is no direct interaction between the two surface layers, so that the film thickness can have any value. Therefore, we can assume that the concentration of the soap is uniform and can be described as  

To calculate the Gibbs elastic constant Eq, we allow the variation of c and /], i.e., from (2.15) we write  

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