Encoding and Controlling of Two Droplet Trains in a Microfluidic Network with
the Loop-like Structure
Generally, the total flow rates 𝑄1 for the upper T-junction and 𝑄2 for the lower
T-junction are different, these two T-junctions will produce two different droplet trains.
The viscosity of the two droplet train may also be different in the model. These
differences are expressed by the droplets intervals 𝜆1 𝐼 and 𝜆2 𝐼 , the droplets flow
resistances 𝑅𝑑1 and 𝑅𝑑2 etc. 𝑅𝑑1 and 𝑅𝑑2 depend on the parameters such as
droplet viscosity, flow rate, surface tension, droplet size. The schematic of the
microfluidic network is shown below. The two T-junctions produce two trains of
droplets which are denoted by 𝜉 for the upper train and 𝜁 for the lower train.
Figure 1: Schematic of the microfluidic network. The network contains two T junctions which
generate two droplet trains respectively, two dilution or concentration modules, a loop-like
structure and one outlet channel. After generation, droplets undergo four stages of I, II, III and IV;
finally, they leave the network from the outlet channel with different patterns. Droplets are
represented with black hollow and solid circles. Related parameters are marked in the schematic.
1
After the system become stable, we begin to mark the droplets and derive the
model when two droplet arrive at nodes a and b, respectively, and both of them are
marked with 0 as shown in the figure 1 above. But it seems not general. So before the
two droplets arrive at nodes c and d, we should consider the situation about the
droplets arriving at nodes a and b. Generally, there is a distance delay about the lower
train because of the arbitrary two droplet trains.
We assume when number 𝜉 droplet of train 1 arrives at node a, number 𝜁
droplet of train 2 will arrive at the location that has a distance of 𝑧 away from node b
as shown in the following figure 2. 𝜉1 and 𝜁1 should be introduced. 𝜉1 is an
arbitrary droplet following droplet 𝜉 of train 1 and 𝜁1 is an arbitrary droplet
following droplet 𝜁 of train 2.
Figure 2: Schematic of the general situation that droplets arrive at nodes a and b.
z can be easily got by analysis:
𝑧 = 𝜁𝜆2 𝐼 − 𝑉2 𝐼
𝜉𝜆1 𝐼
𝑉1 𝐼
(1)
where 0 ≤ 𝑧 < 𝜆2 𝐼 .
In general, when number 𝜉1 droplet of train 1 arrives at node c, number 𝜁1
droplet of train 2 will arrive at the location that has a distance of 𝑥 away from node d.
2
Due to the symmetrical nature of the network structure, 𝑥 satisfies the condition:
0 ≤ 𝑥 < 𝜆2 𝐼𝐼 . The total time for droplet 𝜉1 of train 1 to arrive at node c is ∆𝑡1,𝜉1 =
∆𝑡1 𝐼𝐼 + (𝜉1 − 𝜉)𝜆1 𝐼 /𝑉1 𝐼 (𝜉 ≥ 0), where ∆𝑡1 𝐼𝐼 = 𝑋/𝑉1 𝐼𝐼 ; similarly, the total time for
droplet 𝜁1 of train 2 to arrive at the location that has an 𝑥 distance away from node
d is ∆𝑡2,𝜁1 = ∆𝑡2 𝐼𝐼 + [(𝜁1 − 𝜁)𝜆2 𝐼 + 𝑧]/𝑉2 𝐼 (𝜁 ≥ 0) , where ∆𝑡2 𝐼𝐼 = (𝑋 − 𝑥)/𝑉2 𝐼𝐼 .
Considering ∆𝑡1,𝜉1 = ∆𝑡2,𝜁1 , we obtain:
𝑋/𝑉1 𝐼𝐼 + (𝜉1 − 𝜉)𝜆1 𝐼 /𝑉1 𝐼 = (𝑋 − 𝑥)/𝑉2 𝐼𝐼 + [(𝜁1 − 𝜁)𝜆2 𝐼 + 𝑧]/𝑉2 𝐼
(2)
Substituting equation (3) into equation (2) yields:
𝑋/𝑉1 𝐼𝐼 + 𝜉1 𝜆1 𝐼 /𝑉1 𝐼 = (𝑋 − 𝑥)/𝑉2 𝐼𝐼 + 𝜁1 𝜆2 𝐼 /𝑉2 𝐼
(3)
From equation (3) we can see the controlling result of 𝑥 is independent on z, so
we don’t need to introduce z when deriving the model to make it looks more general.
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