Capillary instabilities in solid thin films: Lines

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Abstract
The linear morphological instability of a line of film on a substrate has been examined for contact angles between 0 and π. The base state of the line is an infinitely long cylinder with cross‐sectional shape a segment of a circle. We assume that mass flows by diffusion along the film surface and that local equilibrium holds. We find that for non‐zero contact angles there is a finite range of perturbation wavenumbers in the axial direction which correspond to instability and will potentially lead to agglomeration of the line of film. All unstable perturbations are of the varicose (sausage) type. The presence of the substrate is stabilizing; the range of unstable wavelengths is always less than that of a freely‐suspended circular cylinder with the same volume and decreases to zero width at zero contact angle. The maximum growth rate of the instability varies strongly with the contact angle and approaches zero as the contact angle approaches zero. Our results agree qualitatively with the experimentally observed wavelength and growth rates of the instability.