Non-Newtonian droplet-based microfluidics logic gates

Droplet-based microfluidic logic gates have many applications in diagnostic assays and biosciences due to their automation and the ability to be cascaded. In spite of many bio-fluids, such as blood exhibit non-Newtonian characteristics, all the previous studies have been concerned with the Newtonian fluids. Moreover, none of the previous studies has investigated the operating regions of the logic gates. In this research, we consider a typical AND/OR logic gate with a power-law fluid. We study the effects of important parameters such as the power-law index, the droplet length, the capillary number, and the geometrical parameters of the microfluidic system on the operating regions of the system. The results indicate that AND/OR states mechanism function in opposite directions. By increasing the droplet length, the capillary number and the power-law index, the operating region of AND state increases while the operating region of OR state reduces. Increasing the channel width will decrease the operating region of AND state while it increases the operating region of OR state. For proper operation of the logic gate, it should work in both AND/OR states appropriately. By combining the operating regions of these two states, the overall operating region of the logic gate is achieved.


Results and discussion
The effects of the power-law index and the bubble/droplet length Figure 1 shows the snapshots of droplet breakup process in times before and after breakup for two different droplet lengths l/w1=1.8 and l/w1=2.4. The time evolutions of velocity and pressure fields for AND state are shown for the non-breakup (l/w1=1.8) and breakup (l/w1=2.4). When the droplet with length l/w1=1.8 enters the bifurcation completely, the pressure and shear force are small compared to the interfacial tension. Thus, the deformation rate is not adequate for breaking the droplet. At t= 92 ms, most of the continuous fluid flows between the right corner of the bifurcation and the droplet. With the passage of time, the flow rate increases and this pushes the droplet to pass through the left branch without breakup.
For the droplet with l/w1=2.4 the continuous phase fluid squeezes the middle part of droplet. The neck thickness decreased with time. Then, the droplet breaks up into two droplets. It shows that at t=92 ms the droplet starts to breakup.  Figure 2 shows the snapshots of droplet breakup process for different times before and after breakup. The time evolutions of the velocity and pressure fields for OR state are shown for the non-breakup (n=1) and breakup (n=1.15). At time t=64 ms the merged droplet in the middle channel completely enters the outlet channels. When the droplet with n=1 enters the bifurcation completely, the pressure behind the droplet tends to squeeze the droplet but the pressure and shear force are small compared to the interfacial tension. Thus, the deformation rate is not adequate for breaking the droplet. At t= 66 ms, most of the continuous fluid flows between the right corner of the bifurcation and the droplet. The flow rate gradually increases and this pushes the droplet to pass through the left branch without breakup. Therefore, the OR state is working.
For the droplet with n=1.15 the continuous phase fluid squeezes the middle section of droplet. The neck thickness decreases with time. Then, the droplet breaks up into two droplets. It shows that at t=70 ms the droplet starts to breakup. Therefore, the OR state is not working.  The neck thickness decreases with time. Then, the droplet breaks up into two droplets. It shows that at t=144 ms droplet starts to breakup. Therefore, the OR state is not working. For the droplet with w4/w1=1 the continuous phase fluid squeezes the middle section of droplet.
The neck thickness decreased as the time goes on. Then the droplet breaks up into two droplets. It shows that at t=89 ms droplet starts to breakup. Therefore, the AND state is working.  Figure 4. Droplet behavior in AND state in non-breakup (point A of Fig. 12) and breakup (point B of Fig. 12) for l/w1=1.4, n=0.7 and Ca=0.015, (a) evolution of velocity distribution for w4/w1=1.2 (non-breakup) and w4/w1=1 (breakup), and (b) evolution of pressure distribution for w4/w1=1.2 (non-breakup) and w4/w1=1 (breakup).

Effect of physical parameters on the breakup
According to investigation of Bedram et al. the most important parameter which affect the droplet size, is the channel width ratio (w4/w1) 1 . As the droplet is non-Newtoian, we consider the effects of fluid power index (n), too. The results are as below:  decreases and more flow rate passes through channel A+B. Thus, more volume of the droplet tends to go to the channel A+B and less to the channel A.B. Consequently, when the droplet breakup occurs, the breakup percentage decreases.
Figures 5 and 6 also show that increasing the fluid power index (n), increases the breakup percentage. Because by increasing n, the viscosity increases and this causes the velocity gradient to be smaller in the droplet. This reduces vortex production and lowers the resistance to deformation and, therefore, the droplet breaks up faster. It means that when the droplet enters the exit branches, the deformation rate is enough to start the breakup and there is less time for the droplet to go to channel A+B. Thus, the droplet volume ratio increases.