1. Calculation of the Pressure Drop Before Trapping of Beads

Supporting information

Contents 1. Calculation of the pressure drop before trapping of beads 2. Calculation of the pressure drop after trapping of beads (containing Fig. S1) 3. Fig. S2. Microscopic image and histogram of prepared agarose hydrogel microbeads. 4. Supplementary Movie 1. Clustering triple hetero microbeads. 5. Supplementary Movie 2. Adjusting the contacting modes of triple hetero microbeads.

1. Calculation of the pressure drop before trapping of beads Details of the calculation of pressure drop are shown below. The calculation was handled with Mathematica (Wolfram Research, USA). By applying the Darcy-Weisbach equation to a rectangular channel, we obtain the expression: where  is the fluid viscosity; L is the length of the channel; Q is the volumetric flow rate; A and R are the cross-sectional area and perimeter of channel, respectively; and C() denotes a constant that is a function of the aspect ratio . The widths of the parts are W1–W6, the lengths along the x axis are X1–X5, the lengths along the y axis are Y1–Y3, and all parts are of identical height H. As a result, A, R, and C() of each part are represented as A = W H, R = 2 (W + H), and C() = |W / H|, respectively, and L of each part is represented by using X1–X5 or Y1–Y5. A1– A12, R1– R12, C1– C12(), and L1– L12 are represented using X1–X5, Y1–Y5, W1–W5, and H, as follows, respectively:

A1 = W3 H, R1 = 2 (W3 + H), C1() = |W3 / H|, L1 = X1

A2 = W4 H, R2 = 2 (W4 + H), C2() = |W4 / H|, L1 = X2

A3 = W3 H, R3 = 2 (W3 + H), C3() = |W3 / H|, L3 = X3

A4 = W3 H, R4 = 2 (W3 + H), C4() = |W3 / H|, L4 = X4

A5 = W3 H, R5 = 2 (W3 + H), C5() = |W3 / H|, L5 = X5

A6 = W1 H, R6 = 2 (W1 + H), C6() = |W1 / H|, L1 = Y1

A7 = W5 H, R7 = 2 (W5 + H), C7() = |W5 / H|, L7 = X1 + X2 + X3 + Y1

A8 = W1 H, R8 = 2 (W1 + H), C8() = |W1 / H|, L8 = X1 + X2 + X3 + Y1 + Y2

A10 = W1 H, R10 = 2 (W1 + H), C10() = |W1 / H|, L10 = Y3

A11 = W2 H, R11 = 2 (W2 + H), C11() = |W2 / H|, L11 = X1 + X2 + X3 + X4 + X5

A12 = W1 H, R12 = 2 (W1 + H), C12() = |W1 / H|, L12 = Y1 + Y2 + Y3

By using A, R, C() and Q1, Q2, Q3, Q4, the pressure drops of each part of the channel are represented as follows:

Here, (i = 1, 2, , 12) From Eqs. (1)–(3), the values of , , and were calculated as follows:

As a result, under the hypothesis of Q4 = 1, the flow rates of Q1, Q2, and Q3 are represented as follows: 2. Calculation of the pressure drop after trapping of the beads The above-identified Q1, Q2, Q3, and Q4 are the values when the trapping spot is empty. After the beads are trapped, the flow rate changes. When forward flow is applied, the value of after trapping one/two bead(s) in the trapping spot for three beads is significant for trapping of the following beads. Therefore, the values of Q1, Q2, Q3, and Q4 after trapping the beads is recalculated by evaluating the plugging by the trapped bead under some hypothesis such that: (i) after bead trapping the cross-sectional area of the part of the channel including the bead becomes equal to the cross-sectional area of the channel minus that of the bead; (ii) the assumed cross section of the part is homothetic to the cross section of the part before the trapping of beads. Figure S1 shows the cross section of the trapped bead in the part of the channel. In the figure, W is the width of the part of the channel and Db is the diameter of the trapped bead. are not functions of W and Db. In contrast, R and A are functions of W and Db. Eq. (1) is represented as follows:

Here, the ratio between the cross-sectional area of the part of the channel and that of the trapped bead is defined as kc. kc is represented using W, H, and Db as follows:

Here, the pressure drop after trapping the bead is defined as : the cross-sectional area and perimeter of the channel after trapping the bead are defined as A’ and R’, respectively. is represented as follows:

Here, and . From Eq. (4), the ratio of is represented by using as follows: (5) By using Eq. (5), the pressure drop is recalculated as follows. After trapping the single bead in the trapping spot for three beads, the trapped bead changes the pressure drop of parts p2 and p3. As for p3, the trapped bead is involved with part p3. Therefore, W3 is greater than Db. In such cases (), is represented as follows: . By using Eq. (5), the value of is identified. As for p2, the bead is stuck to the entrance of part p2. Therefore, Db is greater than W4. In such cases, is represented as follows: . By using Eq. (5), the value of is identified. From these values ( and ) and other parameters, the value of after trapping a single bead in the trapping spot for three beads is identified. should be greater than 1.3 so that the following bead will be trapped in the trapping spot. After trapping two beads in the trapping spot for three beads, the secondly trapped bead changes the pressure drop of part p4 in addition to parts p2 and p3. As for p2 and p3, and have already been calculated above, respectively. As for p4, is represented as follows: . By using Eq. (5), the value of is identified. From these values (, , and ) and other parameters, the value of after trapping two beads in the trapping spot for three beads is identified. should be greater than 1.3 so that the following bead will be trapped in the trapping spot, resulting in the trapping of three beads. In this study, to satisfy until the trapping spot is fully occupied and to trap 100-m-sized microbeads, the following dimensions were employed: W1 = 115 (m), W2 = 115, W3 = 115, W4 = 35, W5 = 38, W6 = 43, X1 = 100, X2 = 30, X3 = 100, X4 = 100, X5 = 100, Y1 = 150, Y2 = 100, Y3 = 3000, and H = 120. With these dimensions, the values for in each condition are identified as follows: (i) with forward flow, the value of before trapping beads in the trapping spot for three beads is identified as 9.02; the value of is identified as 5.39; the value of is identified as 1.68; and the value of is identified as 5.39. From these values (, , and ) and other dimensions, the value of after trapping a single bead in the trapping spot for three beads is identified as 5.87; the value of after trapping two beads in the trapping spot for three beads is identified as 4.74. is greater than 1.3 both after trapping the single bead and after trapping two beads in the trapping spot for three beads. Therefore, the subsequent bead will be trapped in the trapping spot until the spot is fully occupied. As a result, the device with these parameters can trap three 100-m-sized microbeads with forward flow. On the other hand, (ii) with forward flow, the value of before trapping beads in the trapping spot for a single bead is identified as 2.57. before trapping beads in the trapping spot for a single bead is greater than 1.3. Therefore, the subsequent bead will be trapped in the trapping spot until the spot is fully occupied. As a result, the device with these parameters can trap single 100-m- sized microbead with backward flow. To conclude, our device can trap three 100-m-sized microbeads in the trapping spot for three beads with forward flow and a single 100-m-sized microbead in the trapping spot for a single bead with backward flow. With the device designed above and a flow rate of 30 l/min, the preliminary success rate of rearrangement was around 80% when the rearrangement process was performed 20 times. This rate should be raised by more a precise design of the microfluidic device or a higher flow rate.

Fig. S1 Schematic diagram of trapping spot B with trapped microbeads. Red area shows the cross- sectional area of the trapped bead at lines A-A’, B-B’, and C-C’. After trapping a single bead, parts p2 and p3 are plugged by the trapped bead. After trapping two beads, in addition to parts p2 and p3, part p4 is plugged by the secondly trapped bead. White areas within the solid-lined square work as a channel after trapping the beads.

3. Fig. S2. Microscopic image and histogram of prepared agarose hydrogel microbeads By using axisymmetric flow-focusing device (AFFD), we obtained monodisperse agarose hydrogel beads as shown in Fig. S2a. The monodispersity was calculated as the coefficient of variation (CV) = 4.5%.

Fig. S2 (a) Image of monodisperse agarose hydrogel beads prepared by using AFFD. (b) Histogram of prepared hydrogel beads shown in Fig. 3a. The mean diameter of the hydrogel beads was 94.7 m, and the CV was 4.8%.