Flow fields in soap films: Relating viscosity and film thickness

We follow the diffusive motion of colloidal particles in soap films with varying $h/d$, where $h$ is the thickness of the film and $d$ is the diameter of the particles. The hydrodynamics of these films are determined by looking at the correlated motion of pairs of particles as a function of separation $R$. The Trapeznikov approximation [A. A. Trapeznikov, Proceedings of the 2nd International Congress on Surface Activity (Butterworths, London, 1957), p. 242] is used to model soap films as an effective two-dimensional (2D) fluid in contact with bulk air phases. The flow fields determined from correlated particle motions show excellent agreement with what is expected for the theory of 2D fluids for all our films where $0.6\le h/d\le 14.3$, with the 2D shear viscosity matching that predicted by Trapeznikov. However, the parameters of these flow fields change markedly for thick films $\left(h/d>7\pm 3\right)$. Our results indicate that three-dimensional effects become important for these thicker films, despite the flow fields still having a 2D character.

Figure 1

Absorption spectrum of a soap film (50/50 water/glycerol mixture, 2% Dawn by weight) as a function of normally incident wavelength of light. From the peaks in this spectrum and Eq. (2), the thickness of the film is inferred to be $h=510\text{nm}$. Inset: the time dependence of $h$ for two different soap films prepared with 60:40 water/glycerol ratio and 2% concentration of Dawn.

Figure 2

(Color online) Schematic of the Trapeznikov approximation, where the entire soap film is approximated as a single interface in contact with bulk air phases. Reproduced and modified with permission from [16].

Figure 3

(Color online) Coupled motions of particles embedded in the soap films. There are four possible eigenmodes associated with this coupling, correlated and anticorrelated motion parallel $\left(D_{x\pm }\right)$ and perpendicular $\left(D_{y\pm }\right)$ to the lines joining the centers of the particles.

Figure 4

MSD for a soap film, sample (f) in Table (solid circles), compared to MSD in a bulk solution that comprises the fluid layer of the soap film (open circles). Dashed line shows a slope of 1, indicating free diffusion. The effective 2D shear viscosity of the film can be evaluated from Eq. (4) and is given by ${\eta }_{SD}=8.03\text{nPa}\text{s}$.

Figure 5

Correlation functions for the same soap film as in Fig. 4. Symbols are $⟨D_{x+}/\tau ⟩$, solid circles; $⟨D_{x-}/\tau ⟩$, open circles; $⟨D_{y+}/\tau ⟩$, solid squares; $⟨D_{y-}/\tau ⟩$, open squares. Solid lines are fits to the data from Eq. (8), with $B=0.54\mu {\text{m}}^2/\text{s}$ (horizontal dashed line), $C=0.13$, and $L=64\mu \text{m}$ (vertical dashed line).

Figure 6

Scaled (a) longitudinal $\left(⟨D_{x\pm }/\tau ⟩/B\right)$ and (b) transverse $\left(⟨D_{y\pm }/\tau ⟩/B\right)$ correlation functions for five soap films plotted against the scaled separation $R/d$. Solid symbols represent correlated motion while open symbols represent anticorrelated motion. Refer to Table for details about the soap films. Symbols are triangles, sample (a); hexagons, sample (c); squares, sample (d); circles, sample (f); stars, sample (i). Solid lines are of the form $1\pm C\text{ln}\left(L/R\right)$ and $1\pm C\left[\text{ln}\left(L/R\right)-1\right]$, respectively, based on the values of $C=0.13$ and $L=325d$ that roughly fit the data for the films with $h/d<5$. In particular, sample (i) (stars) is a thicker film with $h/d=14.3$ and those data deviate the most from these straight lines.

Figure 7

Fit parameters for nine soap films, including the five shown in Fig. 6, as a function of $h/d$. (a) $C$; the dashed line indicates a constant value of 0.13 for all soap films with $h/d<7\pm 3$. (b) $L/d$; the dashed line represents a value of 325, from the scaling of the correlation functions shown in Fig. 6. (c) $B$ (solid circles) compared to the one-particle diffusion constant $D_{s,1p}$ (open circles). The inset shows the ratio $B/D_{s,1p}$ to be nearly constant with a value of $1.9\pm 0.1$.

Figure 8

The scaled quantity $k_{B}T/\left(2\pi BC\right)$ compared to the one-particle effective 2D shear viscosity ${\eta }_{SD}$ (solid circles) and ${\eta }_B$ (open circles). The dashed line indicates equality between the two quantities.