# Soap film sudden transition

Soap films under gravity have a gradual thickness gradient, as evident in the smooth interference fringes.

But why is there an abrupt transition from "white" to transparent 2/3 of the way up? This transparent region grows over time as the film thins.

• The transparent region is a film that is too thin to produce optical interference in the visible region of the spectrum. As gravity causes liquid to drain down over time, the very thin region expands, and the transparent region increases in size until it is too thin to support the surface tension of the film ... at which time, the film collapses. – David White Jan 18 '18 at 2:41
• The constructive interference would gradually fade away as thickness reduces and the waves fall out of phase. There must be a sudden transition in thickness. Also it's possible to maintain small fully transparent films for hours. The key is to enclose it in a container so it doesn't dry out. – Kevin Kostlan Jan 18 '18 at 5:05

Short answer: the smooth upwards-decreasing thickness profile turns into a constant function when the minimum, stable thickness for the film is reached.

This transition point moves downward with time:

Source: Ropars et al., Dynamics of gravity-induced gradients in soap film thicknesses (e-print).

## Soap film interference

The thin film of soap solution offers two surfaces of reflection: the frontal one, a transition from air to film, and the back one, from film to air. The bands are caused by the interference between the reflections at the two interfaces, their color reflecting the difference in optical path length, which in turn depends on the film thickness ($d$ in the scheme below).

The reflection at a "hard interface" (from a region with lower refraction index such as air to one of a higher refraction index such as soap film) leads to an inversion in the wave phase (a change in $\pi=180^{\circ}$), while a reflection at a "soft interface" (from higher to lower refraction indexes) leads to no change in phase. That means that if the film is too thin relative to the wavelength, then destructive interference will prevent the light from being reflected: that's the dark band on the top, which is the thinest portion of the film (between about $10$ nm or $25$ nm and $50$ nm), do to the effect of gravity.

The color of the soap film under white light, as a function of thickness is shown in this figure:

The question is about why the transition from bright to dark is abrupt. So first it's relevant to point out that it doesn't have to be abrupt, but can be smoother for an away-from-equilibrium film:

## Abrupt transition

As for the abrupt transition, it's located at the transition line between the non-stationary$^1$ region where the solution is still flowing down and the stationary region where gravity is counterbalanced by, e.g., capillary forces (the capillary length for the film is about $4$ meters) and the Marangoni effect. The films walls are kept apart by repulsion forces, which are mainly (Wikipedia, and also this answer): "steric (the surfactants can not interlace) and electrostatic (if surfactants are charged)".

Source: Gaulon et al., Sound and vision: visualization of music with a soap film.

$^1$: I'm using stationary instead of equilibrium, because besides Marangoni flows, the stable state may include dynamic phenomena such as marginal regeneration, which consists in convection flows along the frame of the soap film (see, e.g., this paper, e-print) which seems necessary to prevent rupture, according to this paper (this need could conceivably explain the relatively short life time of films on square frames (e-print).

• There must be a phase-transition going on between the "white" and "black" regions. The slope of the thickness curves in the bottom plot is steeper across the transition region than just below it. Also, your "smooth" example has a region of "droplets" of each phase near the top (it also has smaller bubbles attached to the ring, but that's besides the point). – Kevin Kostlan Jan 19 '18 at 6:24
• @KevinKostlan, I also think we might associate phases to the different regions. Notice, though, that the last plot (Gaulon) has very low resolution in the transition region (a single point in the black film) so I think we should not read too much from it (that's why I emphasized the 1st, Roupars' plot). As for the 'smooth' example, I called it so because of the (literally) gray region above the full white one - the transition to the fully black one is indeed more complicated. – stafusa Jan 19 '18 at 8:12
• @stufusa yes I would wish they made a more detailed plot. The black curve that reaches down through the white region in the question's photo is more evidence of phase-behavior. – Kevin Kostlan Jan 30 '18 at 1:55