Influence of surface tension-driven network parameters on backflow strength

Surface tension-driven flow is widely used, owing to its spontaneous motion, in microfluidic devices with single channel structures. However, when multiple channels are used, unwanted backflow often occurs. This prevents precise and sophisticated solution flow, but has been rarely characterized. We hypothesize that, with an analytical model, the parameters that influence backflow can be systematically characterized to minimize the backflow. In a microfluidic network, inlet menisci and channels are modeled as variable pressure sources and fluidic conductors, respectively. Through the model and experiment, the influence of each network element on the backflow strength is studied. Backflow strength is affected by the interplay of multiple inlet-channel elements. With the decrease (increase) of the fluidic channel conductance (inlet size), the backflow pressure of the corresponding inlet decreases. On the other hand, backflow volume reaches its peak value during the radius change of the corresponding inlet. In networks consisting of five inlet-channel elements, backflow pressure decreases with increasing step number. Our results provide the foundations for microfluidic networks driven by the Laplace pressure of inlet menisci.


Modeling pressures of inlet meniscus and channel junction.
By Young-Laplace equation, the meniscus pressure (P i ) of inlet i is given as a function of the radius of curvature (R i ) and the surface tension (); see Figure S1a: If the meniscus at inlet i does not exceed the edge of inlet i, by the Pythagorean theorem, R i is a function of the height (h i ) of the meniscus and the radius (r i ) of inlet i: Equation (S2) is obtained by the condition where inlet meniscus is pinned at the inlet rim. By inserting eq S2 into eq S1, we can rewrite P i as 2 2 4 ( ) ( ( )) Then, the sum of flow rates passing through the channel junction is obtained by the analogy of Kirchhoff's current law (see Figure S1b): where C i is the fluidic conductance of channel i, which is the inverse of fluidic resistance; and P JCT is the channel junction pressure. Herein, channel fluidic resistance, where µ j is dynamic viscosity of each solution that passes the microchannel, and w i L i , and h are the width, length, and height of the microchannels, respectively.

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Then, the junction pressure P JCT is

Modeling the relation between meniscus height and inlet pressure.
Owing to the similarity between fluidic resistance and electrical resistance, the flow rate (Q i ) from the channel junction to inlet i via channel i can be expressed as (see Figure S1b) In step k, the volume of the meniscus (V i ,) at inlet i is obtained by The change rate of V i is thus expressed as 2 2 π( ( ( )) ) ( ) 2 Becuase dV i /dt is equal to Q i , by applying eq S6 into eq S8, we obtain

Meniscus pinning condition at the inlet rim
We experimentally confirmed the meniscus pinning at the inlet rim at the condition of (1) heightto-radius ratio h i /r i < 0.5 and (2) solution contact angle >49°. A stereo microscope was used to observe the side view of menisci at the inlet rims. To have different values of the water contact angles, the PDMS surface was plasma treated. By adjusting the plasma power and treatment duration, we obtained equilibrium contact angles of 49, 70, and 105° on the PDMS surface. By the constant flow input (0.5 L/s) from a syringe pump, the inlet volume was increased and the inlet meniscus became more convex (Fig. S2a). The inlet meniscus remains pinned at the inlet rim up to 42 s. Then, the contact line of the meniscus moves over the rim. As such, we measured the maximum meniscus height (h max ) that maintains meniscus pinning for inlet radii (r) of 1 and 2 mm and contact angles of 49, 70, and 105°. Fig. S2b shows that the value of h max / r increases with increasing contact angle. Here, symbols of rectangle and circle correspond to r = 1 and 2 mm, respectively.

S4
We analyzed the effect of the size of channel 2 on V B2 , under constant condition of C 2 . Fig. S3 shows that V B2 decreases with the increase in the width (W) and height (h) of channel 2. When W and h increase, the length (L) of channel 2 increases to keep C 2 constant. Because V Ch2 = WhL, V Ch2 increases. However, V 2 , which is the inlet meniscus volume, does not change because the other conditions do not change. Thus, V B2 decreases because V B2 = V 2 /V Ch2 .