Local details versus effective medium approximation: A study of diffusion in microfluidic random networks made from Voronoi tessellations

We measured the effective diffusion coefficient in regions of microfluidic networks of controlled geometry using the fluorescence recovery after photobleaching (FRAP) technique. The geometry of the networks was based on Voronoi tessellations, and had varying characteristic length scale and porosity. For a fixed network, FRAP experiments were performed in regions of increasing size. Our results indicate that the boundary of the bleached region, and in particular the cumulative area of the channels that connect the bleached region to the rest of the network, are important in the measured value of the effective diffusion coefficient. We found that the statistical geometrical variations between different regions of the network decrease with the size of the bleached region as a power law, meaning that the statistical error of effective medium approximations decrease with the size of the studied medium with no characteristic length scale.

Figure 1

Left: Photography of an assembled microfluidic channel. Inlet and outlet tubes are visible. Right: Fluorescence micrography of a region of one microfluidic network. White areas correspond to channels, filled with the fluorescent solution.

Figure 2

Micrographies of networks with different porosities (networks B–E from Table 1). As the width of the channels increases, the reticular aspect of the network is lost. Color is inverted, so dark regions represent fluorescent-filled channels.

Figure 3

Color-inverted micrographies of bleached areas with different $\gamma$. Images correspond to network E in Table 1.

Figure 4

Procedure for obtaining the bleached and complementary masks. (a) Image before photobleaching. (b) First image after photobleaching. (c) and (d) Bleached and complementary masks.

Figure 5

(a) Evolution of the average fluorescence of the bleached and complementary regions. (b) Recovery curve $F\left(t\right)$ and the corresponding fit. The curves corresponds to a FRAP experiment in network F with $\gamma =13.2$.

Figure 6

(a)–(d) Images of FRAP experiments with varying number of feeding channels. All cases correspond to network A in Table 1. Dashed red circles indicate the bleached region. (e) Normalized effective diffusion coefficient, $D_{\text{eff}}/D_0$ for different number of feeding channels, $N$.

Figure 7

(a) Normalized effective diffusion coefficient, $D_{\text{eff}}/D_0$ as a function of $\gamma$ for network F. (b) Standard deviation of $D_{\text{eff}}$, normalized by $D_{\text{eff}}$, for ten realizations in arbitrary locations of the same network, at fixed $\gamma$. The dashed line marks the 5% accuracy limit of our measurements.

Figure 8

Normalized effective diffusion coefficient, $D_{\text{eff}}/D_0$ as a function of the porosity $\varphi$ for fixed $\gamma =3.99$.

Figure 9

(a) Perimeter fraction of feeding channels as a function of $\gamma$. (b) Fractional error of the perimeter fraction of feeding channels as a function of $\gamma$. The solid line represents the fit of Eq. (16).

Figure 10

Examples of observation and bleaching profiles. The dashed lines show $R=198\mu \mathrm{m}$ and $1/e$.

Figure 11

(a) Recovery curve (black line) and best fit (red line). The dashed and dot-dashed lines represent the model curves with $G$ decreased and increased in 10%, respectively. In (b), the same curves are normalized between 0 and 1. The curves with $G$ decreased and increased in 10% are virtually indistinguishable from the best fit.

Figure 12

Recovery curve based only on the bleached region, from the same data of Fig. 5.