Supplementary Materials
Microfluidic Network Based Combinatorial Dilution DeviceFor High ThroughputScreening and Optimization
Kangsun Lee, a Choong Kim, b Geunhui Jung, b Tae Song Kim, b Ji Yoon Kang, b and Kwang W. Oh a
aSMALL (nanobio Sensors and MicroActuators Learning Laboratory), Department of Electrical Engineering, University at Buffalo, The State University of New York (SUNY at Buffalo), 332 Bonner Hall, Buffalo, NY, 14260, USA
b Nano-Bioresearch Center, Korea Institute of Science and Technology (KIST),39-1 Hawolgok Dong, Songbuk Gu, 136-791, Seoul, Korea.
In these supplementary materials, mathematical modeling using an equivalent electrical circuit as shown in Fig. 2, the result of the electrical circuit analysis using P-Spice, and design parameters of the proof-of-concept device are detailed.
1. Detailed mathematical modeling
From a Kirchhoff’s current law (KCL), the following set of flow rates for the additive and the buffer channels is satisfied.
(1)
Eq. 1 represents the total input flow rates for the additive solutions and thebuffer solution. The total input flows ofthe additive solutions are divided into 4 sub-flows at4 cross junctions (, , , and for the additive solution A), and the total input flow of the buffer solution is divided into 7 sub-flows for the 7 buffer channels as shown in Fig. 2. Thus, the total output flow ratesin each output port ()are a sum of the flows joining into the each buffer channel in the 1st layer.
(2)
The concentration values associated with each additive in the output ports are a function of the mixing ratio of the additive volume and the total volume.Eq. 3 represents a concentration set of a ternary blend (ABC/D): = {, , } in the 1st output port (e.g., the case of C (n, k) = 3C3).
(3)
Eq. 4 represents concentration sets of binary blends (AB/D, BC/D, and AC/D): = {, , 0} in the 2nd output port, = {0, , } in the 3rd output port, and = {, 0, } in the 4th output port, respectively (e.g., the case of C (n, k) = 3C2).
(4a)
(4b)
(4c)
Eq. 5 represents concentration sets of pure additives (A/D, B/D, and C/D): = {, 0, 0} in the 5th output port, = {0, , 0} in the 6th output port, and = {0, 0, } in the 7th output port, respectively (e.g., the case of C (n, k) = 3C1).
== (5a)
== (5b)
== (5c)
The output flow rates () and the output concentrations () can be independently variable in each output port. Using Eq. 3, Eq. 4, and Eq. 5, the sub-flow rates supplied from each additive channel can be calculated as functions of and .
(6a)
(6b)
(6c)
The sub-flow rates in the buffer channels can be also expressed from Eq. 2 and Eq. 6.
(7)
In this study, to simplify the analysis and design of the circuit, the following set of initial design rules, notated with a symbol of ‘*’ in Fig. 2 was applied.
=2 unit, =2 unit, = 2 unit, = 2unit,= 10 unit
= 10 unit, = 15 unit(8)
Where, , , , and are resistances of channel segments associated with the additive A, the additive B, the additive C, and the buffer solution D, respectively.are resistances of channel segments for mixing between the additive and the buffer solutions, which is based on diffusion times at the given output flow rates. are resistances of channel segments for final mixtures in the output ports. The parameters can be chosen any values theoretically, but theywere set to be multiples of unity.
As mentioned in section 2.1, since microfluidic circuits are similar to electrical circuits, linear equations can be derived by Ohm’s and Kirchhoff’s lawsto determine proper channel resistances in the microfluidic circuits. With the given conditions for the desired output flow rates () and concentrations (), flow resistances of each channel can be determined by solving the linear equations. Eq. 9representsflow resistances of channel segments for the additive A, the additive B, and the additive C.The channel resistances can be expressed by the ratios of pressure drops (Vyi-Vy(i+1)) and flow rates () between neighboring cross junctions. We assume that ,, and.
(9)
Eq. 10representsflow resistances of channel segments for the output ports.We assume that .
(10)
In order to satisfy all the conditions, ,,, and , the resistances of channel segments for distribution of the buffer solution were chosen as the following set, where the values can be optionally selected depending on designs and layouts of the device.
=8 unit, = 15 unit, = 12 unit,= 9 unit
= 10 unit, = 10 unit(11)
In this design, 1 unit = 8 mm. Consequently, if the cross-sections of all channels in the circuit are identical, the only variable parameter to control the fluidic resistance is the channel length, which shows relatively simple design and fabrication. With the given conditions for the desired output flow rates (), concentrations (), and the initial design rules, the channel lengths of each segment can be determined by solving the linear equations.
2. P-Spice simulation
An electrical circuit used for the P-Spice simulation is shown in Fig. S1. In the analysis, all flow rates were treated as electrical currents and all flow resistances (or the channel lengths) as electrical resistances. The input flow ratesof the buffer solution and the additive solutions were treated as constant current sources, output ports were treated as ground (GND). Also, we assumed that resistances of the via holes were negligible because of the relatively large dimension of the via holes. With the parameters listed in Table 1, we investigated electric currents of the output ports. When the input currents were 4 A for the buffer solution and 1 A for the additive solutions, the output currents were exactly 1 A. The current values in each segment were identical to those from the mathematical modeling.
Fig. S1 The result of the electrical circuit analysis using a P-Spice software tool.
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3. Detailed design parameters
All design parameters including the channel lengths and their associated concentrations and flow rates were summarized in Table 1.
Table 1. The design parameterscalculated by numerical analysis (modeling and simulations), including the channel length (resistances), concentrations, and flow rates for each channel segment.
Channel segments / D0 / D1 / D2 / D3 / D4 / D5 / D6 / D7 / Mi,j / A0 / A1 / A2 / A3 / B0Concentration / CA / 0 / 0 / 0 / 0 / 0 / 0 / 0 / 0 / - / 1 / 1 / 1 / 0 / 0
CB / 0 / 0 / 0 / 0 / 0 / 0 / 0 / 0 / - / 0 / 0 / 0 / 1 / 1
CC / 0 / 0 / 0 / 0 / 0 / 0 / 0 / 0 / - / 0 / 0 / 0 / 0 / 0
Length (mm) / 8 / 80 / 64 / 120 / 96 / 72 / 80 / 80 / 80 / 8 / 16 / 32 / 24 / 8
Flow rate (µL/min) / 12 / 0.75 / 1.5 / 1.5 / 1.5 / 2.25 / 2.25 / 2.25 / - / 3 / 2.25 / 1.5 / 0.75 / 3
Channel segments / B1 / B2 / B3 / C0 / C1 / C2 / C3 / O1 / O2 / O3 / O4 / O5 / O6 / O7
Concentration / CA / 0 / 0 / 0 / 0 / 0 / 0 / 1/4 / 1/4 / 0 / 1/4 / 1/4 / 0 / 0 / 0
CB / 1 / 1 / 0 / 0 / 0 / 0 / 1/4 / 1/4 / 1/4 / 0 / 0 / 1/4 / 0 / 1/4
CC / 0 / 0 / 1 / 1 / 1 / 1 / 1/4 / 0 / 1/4 / 1/4 / 0 / 0 / 1/4 / 0
Length (mm) / 42.6 / 16 / 80 / 8 / 53.4 / 16 / 48 / 120 / 68.5 / 80 / 72 / 26 / 40 / 60
Flow rate (µL/min) / 2.25 / 1.5 / 0.75 / 3 / 2.25 / 1.5 / 0.75 / 3 / 3 / 3 / 3 / 3 / 3 / 3
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