“Marginal pinching” in soap films

Abstract

Abstract. -We discuss the behaviour of a thin soap film facing a frame element: the pressure in the Plateau border around the frame is lower than the film pressure, and the film thins out over a certain distance λ(t), due to the formation of a well-localized pinched region of thickness h(t) and extension w(t). We construct a hydrodynamic theory for this thinning process, assuming a constant surface tension: Marangoni effects are probably important only at late stages, where instabilitites set in. We find λ(t) ∼ t 1/4 , and for the pinch dimensions, h(t) ∼ t −1/2 and w(t) ∼ t −1/4 . These results may play a useful role for the discussion of later instabilitites leading to a global film thinning and drainage, as first discussed by K. Mysels under the name "marginal regeneration".Early experiments by K. Mysels and coworkers [1] showed that a vertical soap film, suspended on a frame, (and made with "mobile" surfactant) thins out by nucleation and growth of black, thin spots near the Plateau borders. They called this process "marginal regeneration".There are in fact (at least) two steps in marginal regeneration: a) The pressure in the Plateau border is lower than the pressure in the film. This thins out the film near the border, and leads to a "pinch". b) The pinched state must have an intrinsic instability leading to the black spots. The two steps are very different: the pinch can occur at constant surface tension-i.e. without any Marangoni effect. On the other hand, the later instabilities are probably triggered by Marangoni flows, as pointed out by a number of authors [2][3][4][5].Our aim in the present note is restricted to the first step, i.e. the description of the pinched state with its dynamics. Pinching has already been studied in connection with the elimination of dimples in the coalescence of drops [6,7] or in the drainage of thin films [8,9]. However, in these problems, the extension of the dimpled zone is prescribed and fixes one of the spatial scales involved. Our case is different: we consider a semi-infinite film (of initial thickness e 0 ) facing a straight Plateau border. At t = 0 the film begins to pinch, and is perturbed over a certain distance λ(t) increasing with time (as we shall see λ(t) ∼ t 1/4 ). Also the width w(T ) of the small pinched region decreases with time (w(t) ∼ t −1/4 ) and so does the thickness of the pinch h(t) ∼ t −1/2 . It seems important to know these scales in detail before embarking into the second step: all data on dimples suggest that instabilities can only occur after a long

On the absence of marginal pinching in thin free films

“…If one assumes, as in [1], that the surfaces of the film are immobile, then the drainage timescale is found to be…”

“…Generic solutions of the so-called lubrication equation (4), subject to the boundary conditions (7), were shown by [1] to exhibit "marginal pinching". At large times, the film thickness approaches a non-monotonic profile, attaining its continually decreasing minimum value in the margin between the lamella and the Plateau border.…”

“…With a view to understanding the initial stages of this process, Aradian, Raphaël and de Gennes [1] studied "marginal pinching", which refers to the development of a pinched region in the neighbourhood of the Plateau border at the edge of a soap film. Under the assumption that the fluid velocity is zero at the surfactantcovered boundaries of the film, the film thickness was found to satisfy a nonlinear partial differential equation familiar from lubrication theory (e.g.…”

“…The evolution equation analysed by [1] is a limiting case, in which the free surfaces are so loaded with surfactant, and the stresses due to even small surfactant gradients so large, that the free surfaces are effectively tangentially immobile. molecular weight surfactants or polymers or from mixtures of surfactants such as SDS and dodecanol (e.g.…”

Evolution of nonconformal Landau-Levich-Bretherton films of partially wetting liquids

“…While significant progress has been made in understanding the evolution of such films in various limiting cases [8], a comprehensive analysis that encompasses all the stages of film evolution, and which ultimately predicts the rupture time, is still lacking. Briefly, the early stages of thinning have been studied in the context of marginal soap pinching and ophthalmology [6,9], while the main features of the final collapse are also understood [10]. We analyze the full dynamic evolution of such thin films, from deposition to thinning to rupture, with theoretical rupture times that can be compared to experiment.…”