Droplet Traffic in Microfluidic Networks: A Simple Model for Understanding and Designing

We propose a simple model to analyze the traffic of droplets in microfluidic “dual networks.” Such functional networks which consist of two types of channels, namely, those accessible or forbidden to droplets, often display a complex behavior characteristic of dynamical systems. By focusing on three recently proposed configurations, we offer an explanation for their remarkable behavior. Additionally, the model allows us to predict the behavior in different parameter regimes. A verification will clarify fundamental issues, such as the network symmetry, the role of the driving conditions, and of the occurrence of reversible behavior. The model lends itself to a fast numerical implementation, thus can help designing devices, identifying parameter windows where the behavior is sufficiently robust for a device to be practically useful, and exploring new functionalities.

Figure 1

Graph representations of three recently proposed dual networks: (a) a loop [7]; (b) a ladder [8]; (c) a bypassed $\mathsf{T}$ junction [12]. Full and dotted lines correspond to transport channels (accessible to droplets) and bypass channels (forbidden to droplets). Droplets are indicated by circles.

Figure 2

Final distances $\Delta x_n$ between 70 droplets after a return journey through the loop of the device in Fig. 1a, as a function of the initial period $\lambda$. Reversibility is observed where all 69 symbols collapse on the diagonal, see also the zoomed panel b. The results differ for a finite droplet train (panels c and d), where also the number of droplets matters. The parameters $L_u/L_{\ell }=1.112$, $R_d/{\overline{R}}_{\ell }=1.5$, are compatible with data from Ref. 8 [15].

Figure 3

Change of droplet misalignment in the ladder device. Only the boundary condition (a) leads to a contraction; fixed pressures on both ends (b) do not affect the offset, while it grows in (c). $R_d/{\overline{R}}_p=20$, $R_{\mathrm{by}}/{\overline{R}}_p=5$, $L_r/L_p=5$.

Figure 4

Fraction of wrong (not strictly alternating) decisions in $\mathsf{T}$ junctions with and without bypass [Fig. 1c]. The interval without errors around the average droplet distance $⟨\Delta x_{\mathrm{in}}⟩\approx 1.5L_s$ corresponds to the experimental findings in Fig. 3 of Ref. 12. The robustness of the behavior is challenged by an asymmetry ($L_u/L_{\ell }=1.01$) and a random perturbation of the incoming distances (standard deviation is 5% of the average value). $R_d/{\overline{R}}_s=0.3$.