Hysteretic behavior in three-dimensional soap film rearrangements

We report experiments on soap film configurations in a triangular prism for which the shape factor can be changed continuously. Two stable configurations can be observed for a range of the shape factor $h$, being the prism-height/edge-length ratio. A hysteretic behavior is found, due to the occurence of another local minima in the free energy. Contrary to a common belief, soap films can be trapped in a particular configuration being different from a global surface minimization. This metastability can be evidenced from a geometrical model based on idealized structures. Depending on the configuration, the transition is either first or second order, providing clues on the structural relaxations taking place into three-dimensional foams, such as T1 rearrangements.

Figure 1

Three soap film configurations are considered in the present paper. Characteristic lengths $A$, $H$, $L,$ and $B$ are denoted. Configuration ${\text{I}}_{\text{curved}}$ with curved edges is observed for small $h=H/A$ ratios. In order to perform calculations on the latter structure, we consider a similar structure, called ${\text{I}}_{\text{flat}}$, with flat surfaces only. The double tetrahedra configuration II is observed when the ratio $h$ is high.

Figure 2

Pictures of the soap film configurations studied in this paper. The closeup near vertices reveal the curvature of the ${\mathrm{I}}_{\mathrm{curved}}$ configuration (top) while the soap films remain flat for configuration II (bottom).

Figure 3

The total surface area $s=S/A^2$ of soap films in the triangular prism as given by the Surface Evolver algorithm (data adapted from [6]). Both configurations ${\mathrm{I}}_{\mathrm{curved}}$ and II are represented. The coexistence of both configurations is possible for $0.41\lesssim h\lesssim 0.49$. Dots corresponds to instability points. The data for ${\mathrm{I}}_{\mathrm{flat}}$ are also given.

Figure 4

Illustration of the triangular framework used for the experiments with relevant parameters. The framework consists of three vertical rigid wires attached to a horizontal and regular triangle. The length of a triangular edge is $A=10$ cm. A mobile triangle, parallel to the former, can be placed at different vertical positions $H$ along the vertical framework by using a thin wire (dashed line).

Figure 5

Sketch of the experimental setup. The frame is immerged into a soap and water mixture. After removing the frame from the solution, thin soap films are created. A motor can increase continuously or decrease $h$ by using to a thin wire. Above the framework, a high definition CCD camera records top and lateral images of the soap films with the help of an inclined mirror.

Figure 6

Measurements of the geometrical characteristics for configurations I and II as a function of the dimensionless parameter $h$. Error bars are indicated. (top) The area $\sigma =\Delta /A^2$ of the horizontal face for configuration I. (bottom) The vertical length $\ell =L/A$ of the Plateau border in configuration II. In both plots, the experimental data are represented by either up or down triangles for respectively increasing $h$ and decreasing $h$ situations. The trend of the data are emphasized by gray lines, as guides for the eye. The hysteretic loop is emphasized by a filled polygon. Numbered arrows indicate the experimental sequence (see text).

Figure 7

Energy landscape of both idealized configurations ${\mathrm{I}}_{\mathrm{flat}}$ (through parameter $b$) and II (through parameter $\ell$) for different $h/a$ values. The dots indicates the global and local minima for each curve. The hysteresis loop is emphasized by thick gray curves and arrows numbered according to Fig. 6.