Two-Dimensional to Three-Dimensional Transition in Soap Films Demonstrated by Microrheology

We follow the diffusive motion of colloidal particles of diameter $d$ in soap films of varying thickness $h$ with fluorescence microscopy. Diffusion constants are obtained both from one- and two-particle microrheological measurements of particle motion in these films. These diffusion constants are related to the surface viscosity of the interfaces comprising the soap films, by means of the Trapeznikov approximation and Saffman’s equation for diffusion in a 2D fluid. Unphysical values of the surface viscosity are found for thick soap films ($h/d>7\pm 3$), indicating a transition from 2D to 3D behavior.

Figure 1

Schematic of the Trapeznikov approximation, where the entire soap film is approximated as a single interface in contact with bulk air phases.

Figure 2

The solid symbols are the mean square displacements for five soap films with increasing $h/d$ ratios. See Table  for the viscosities ${\eta }_{\mathrm{bulk}}$ that correspond to these films. The open symbols represent data from the bulk (3D) solutions used to make the soap films for four cases (b, g, i/j).

Figure 3

Two-particle longitudinal correlation function $⟨D_{rr}/\tau ⟩$ for the five soap films described in Fig. 2, symbols being the same. An additional data set from 3 has been included (open hexagons). Solid lines are empirical fits to the data of the form $A\ln R+B$. The dashed line is the form of the correlation function expected for a 3D fluid with ${\eta }_{\mathrm{bulk}}=6.5\mathrm{mPa}·\mathrm{s}$ (squares).

Figure 4

(a) Two-particle diffusion constant $D_{s,2p}$ plotted against the one-particle $D_{s,1p}$ for the six soap films described in Figs. 2, 3. Four additional data sets have been included, details of which are given in Table . The straight line indicates equality between the diffusion constants. Errors in $D_{s,1p}$ and $D_{s,2p}$ are $\pm 5\%$, similar to that in ${\eta }_{\mathrm{bulk}}$, and are smaller than the size of the symbols in the figure. (b) Interfacial viscosity ${\eta }_{\mathrm{int}}$ as a function of $h/d$. Inset: Magnified view of ${\eta }_{\mathrm{int}}$ for $h/d<4$, with error bars included.