Controlling Flow Rates in Paper Networks
An essential component of our paper-fluidics toolbox is the ability to control the flow rates of fluids in paper networks. Two classes of controls that we have investigated are geometric controls and the use of dissolvable barriers.
We can investigate flow in the simplest 2D structures to create some basic design rules for transport in paper networks. For example, what happens to the fluid front in regions of expanding or contracting geometry?
The image of Figure 1 shows strips that contain a simple expansion at different locations downstream, and a constant width strip for comparison. Flow initially follows the Washburn relation in all the strips. Transition to a greater width results in a deviation from the Washburn relation and a greater degree of slowing of the fluid front. The plot of Figure 1 shows the distance vs. the square root of time for the two left most strips. Here you can see the initial Washburn flow and then a further slowing that starts at the point of the expansion. The image in Figure 2 shows that for a greater width expansion, there is a greater degree of slowing of the fluid front. Thus, simple control parameters for slowing down the transport time of the fluid front are the downstream location of the expansion and the width of the expansion.
Figure 1. Transport of the fluid front in simple expansion geometries.
Figure 2. Transport of the fluid front for different width expansions.
For the case of a contraction geometry, a transition to a smaller width, as shown in Figure 3, the flow starts out Washburn, increases transiently at the constriction, and then resumes Washburn flow as the larger width section serves as a non-limiting source for the smaller width section. The result is that the transport time of the fluid front, i.e. the time that the fluid front takes to travel the length of the strip, is decreased relative to a constant-width strip. The downstream location of the constriction can be used to control the transport time of the fluid front, and this time is minimized when the lengths of the two sections of different widths are equal.
Figure 3. Transport of the fluid front in the case of a contraction geometry.
In the case of fully-wetted flow (i.e. when the fluid front has reached the wicking pad), we can use the electrical circuit analogy to Darcy’s Law for fluidic circuits. The pressure difference across the circuit is analogous to potential difference, the volumetric flow rate is analogous to current, and the fluidic resistance is equal to the physical parameters of the system including geometric factors of the paper circuit. Fluidic resistances in series are summed, while fluidic resistances in parallel can be added in reciprocals.
Figure 4. Electrical circuit analogy to fluid flow.
Using this electrical circuit analogy for resistances in series, we can calculate the relative resistances of simple structures, such as those shown in Figure 5. Since resistance is proportional to length over cross-sectional area, the resistance of A is the greatest and the resistance of B is the smallest. For a uniform pressure difference across all three structures, the volumetric flow rate in A is the smallest and in B the greatest. The transport time for flow through a strip with a multi-segment geometry can be calculated from t = V / Q, where V is the volume of the geometry, and Q is the volumetric flow rate. Assuming that permeability and viscosity are constant, differences in the transport times of fluids in two strips will be solely due to geometric factors. The prediction is that the transport time will be the fastest in the contant width strip A, and for the strips of varying widths, the transport time should be faster in strip C than in strip B. Figure 5 shows a time series comparison of experimental and simulation results (Comsol Multiphysics) for flow in the strips of different geometries, while Figure 6 shows a plot of position vs. time for the marked band in structures B and C. The transport times show good quantitative agreement, demonstrating that we have the ability to predict and control flow rates for simple changes in geometry.
Figure 5. Comparison of experimental and model results for Darcy’s Law flow shows good agreement in both the shapes and the locations of the bands.
Figure 6. Comparison of experimental and model results shows good quantitative agreement in the transport times of the marked bands through the multi-segment geometries.
We have been investigating the use of sugar solutions to create delays in the transport of the fluid within a subcircuit. The two image series of Figure 7 demonstrate how the extent of the barrier can be used to vary the delay time. In this case, the barrier has been created by applying a trehalose solution across the channel and allowing it to dry. In the top series, the presence of a barrier in the right leg creates a delay of the fluid front in that leg. In the lower series, barriers were created in both legs, and the longer sugar barrier in the right leg results in a greater delay in the fluid front in the right leg compared to the left leg.