Straight to zigzag transition of foam pseudo-Plateau borders on textured surfaces

The structure of liquid foams follows simple geometric rules formulated by Plateau 150 years ago. By placing such foam on a microtextured hydrophilic surface, we show that the bubble footprint exhibits a morphological transition. This transition concerns the liquid channels, also called pseudo-Plateau borders, which are straight between vertices on a smooth surface. We demonstrate experimentally that for a sufficiently large roughness size compared to the width of the liquid channels, the footprint adopts a zigzag shape. This transition is associated with the absence of a wetting film between the pillars caused by capillary suction of the foam, observed by confocal microscopy. We rationalize the number of zigzag segments by a geometric distribution describing the observations made with the footprint perimeter and the mesh size of the asperities.

Figure 1

Example of footprint zigzag transition of a monodisperse foam $R=1000\mu \mathrm{m}$ on a textured surface $a=60\mu \mathrm{m}$ (High resolution with segment labeling is provided in the SM [24]). The image (a) is for a liquid fraction $\phi =0.28$% (i.e., $a/r_{\mathrm{pb}}=0.65$) where straight edge footprints are observed. In (b), for $\phi =0.03\%$ (i.e., $a/r_{\mathrm{pb}}=2.0$), edges have a zigzag morphology. The yellow line shows the recorded path of the PPBs constituted of segments. Scale bars in white represents 0.6 mm. Histograms are the PDF of the footprint perimeters $\ell$ in the (c) straight and (d) zigzag regimes, respectively. The vertical dashed lines are the mean values, which are represented in (e) as a function of $R$. The black line of equation $\langle \ell \rangle =7R$ is a guide for the eye.

Figure 2

Average number of segments $\langle n\rangle$ as a function of the dimensionless size of asperities $a/r_{\mathrm{pb}}$. Horizontal lines are mean values for $a/r_{\mathrm{pb}}<1$ and $a/r_{\mathrm{pb}}>1$, and are equal to 6.2 and 14.6, respectively. The markers encode the visual classification made in Fig. 1 with $\circ$ for straight PPBs and $+$ for the zigzag morphology. Images illustrating the PPBs in the zigzag regime are provided in the SM [24]. The inset is the PDF of the number of segments for footprints satisfying $a/r_{\mathrm{pb}}<1$.

Figure 3

(a), (b) Images obtained by confocal microscopy where the textured surface is represented in yellow and the liquid is in blue for $a=100\mu \mathrm{m}$. (a) for $r_{\mathrm{pb}}=180\mu \mathrm{m}$, the surface is filled by a liquid film of the thickness of the pillars, and (b) the surface pores are empty ($r_{\mathrm{pb}}=75\mu \mathrm{m}$). The width of the imaged surface is 1.2 mm. (c) Phase diagram of the PPB morphology representing $a$ versus $r_{\mathrm{pb}}$. Large blue circles are for surfaces covered by a liquid film, illustrated in (a) and large yellow triangles are for dry surfaces as shown in (b). The black line is the equality between axes.

Figure 4

Statistics in the zigzag regime. (a) The mean number of segments $\langle n\rangle$ is plotted against $\phi$ for $a=130\mu \mathrm{m}$ in the regime $a/r_{\mathrm{pb}}>1$. The dashed lines are the mean values for each bubble radius $R$. Vertical error bars represent the standard deviation, and horizontal ones the interval. (b) The main plot shows $\mathcal{E}\langle n\rangle$ as a function of the dimensionless parameter $\langle \ell \rangle /2a$ for the entire dataset in the zigzag regime. The solid line represents equality between axes. The inset is the PDF $\mathcal{P}$ of the lengths ${\ell }_i/2a$ for five different sizes $a$. On both plots, the vertical error bar represents the standard deviation and the horizontal one is the range over which the data are regrouped.