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The stability of a free, thin liquid film against small, spontaneous thickness fluctuations is explored. The film is unstable with respect to fluctuations with wavelengths larger than a critical wavelength Λc=–2π2γ/(d2V/dh2)½, where γ is the interfacial tension and V(h) the free energy of interaction as a function of the film thickness h. V(h) may include van der Waals attraction and double-layer repulsion. The kinetics of the growing fluctuations is obtained by assuming a laminar liquid flow between rigid film surfaces at a constant viscosity. There are stable fluctuation-modes, which grow exponentially with time, each with a characteristic time constant τ, and modes with certain wavelengths grow faster than all others (τ=τm). If the van der Waals forces predominate Λc and τm are given by eqn. (4.2) and (4.3) respectively. For A= 10–14-10–12erg, γ= 30 dyne/cm and h= 100–1000 Å, Λc ranges from 0.6–600 µ and τm from a fraction of a second to several hours. The life-time and critical thickness hc of an unstable film are also calculated; they depend on the time constant τm and on the time of draining. The critical thickness is calculated for microscopic, circular films and compared with measurements of Scheludko and Exerowa. For water and aniline films, the calculated hc are 410 and 750 Å respectively, whereas the experimental values are 270 and 410 Å.
A. Vrij (Sat,) studied this question.