*Appendix A.2. Verification of Ability to Generate Chambers with Uniform Concentration Using Branch Channels*

To generate a concentration gradient based on the enclosure angle, the pressure must be uniformly supplied to each driving chamber. To verify this, an experiment was conducted using the flow path shown in Figure A2a. The driving chambers were arranged independently, so that they were not affected by the neighboring chambers, and the driving chambers were designed in a treelike pattern, so that there was no difference in flow length among the driving chambers. If concentrations in all the main chambers were equal (Figure A2a), concentration could be adjusted by the enclosure angle. The experimental results are shown in Figure A2b. Concentrations in the main chambers are not uniform, and it is difficult to adjust them on the basis of enclosure angle. When pneumatic pressure is used, the pressure may not be evenly transmitted if the flow path is narrow. Therefore, the application of air pressure to such a branch channel was not appropriate to generate a concentration gradient.

**Figure A2.** Verification of pressure uniformity. (**a**) To generate a concentration gradient using the enclosing angle, pressure must be uniform to generate a concentration gradient using the enclosing angle. (**b**) Experimental results. Concentrations were not uniform.

#### *Appendix A.3. Generation of Concentration Gradients Based on Serial-Type Flow Paths*

As mentioned in the previous section, it is difficult to evenly supply pressure to the driving chamber; therefore, we attempted to generate a concentration gradient based on pressure gradient. The flow path is shown in Figure A3a. The farther the distance from the air source, the more pressure loss occurs owing wall friction, and the more pressure gradient is generated in the driving chamber. The experimental results are shown in Figure A3b. No stirring occurred at all owing to the effect of the channel width, as described in the previous section.

Assuming that the channel width affects agitation, we designed the channel to generate the concentration gradient based on the pressure gradient by removing the effect of the channel width and conducted the experiment. The flow path shown in Figure A4a was designed in a way that the width of the flow path between the pneumatic source and each driving chamber was maximal. Experimental results are shown in Figure A4b. Although a concentration gradient was generated, the chambers located at both ends of the flow path exhibited a low concentration because of the pressure loss caused by wall friction.

120

**1st trial, 15s 2nd trial, 15s 3rd trial, 15s 500μm**

**Figure A3.** Use of pressure gradient. (**a**) Air was supplied from the left side of the figure, and the driving chamber was deformed by air pressure. The pressure gradient was generated by the pressure drop through the flow path, which generated the concentration gradient. (**b**) Experimental results of (**a**). Each chamber was driven for 15 s, but no stirring occurred.

(**b**)

**Figure A4.** Flow path without the effect of channel width. (**a**) Concentration gradient was generated by the pressure gradient after removing the effect of the channel width. (**b**) Experimental results. Concentration gradient was generated, but concentrations in the chambers at both ends were low.

The aforementioned results imply that it is difficult to supply pressure to the driving chamber as expected when the channel width of the driving chamber is narrow. Therefore, when air is used for pressure transmission and the frequency of expansion/contraction is high, it is better to increase the channel width of the driving chamber as much as possible.

#### *Appendix A.4. Influence of Deformation of Wall*

Another possible reason for the diluted concentration in the chambers at both ends is the difference in the deformation of the driving chamber. Figure A5 shows how the wall is deformed when the driving chamber is driven using a high-speed camera. There is a difference in the deformation of the driving chamber depending on the positional relationship between the air pressure source and driving chamber. It is not clear whether

this affects the concentration or not, but the parameters related to the concentration can be reduced by removing this effect.

**Figure A5.** Wall deformation in relation to the position of the pneumatic source. Deformation differeed depending on the position of the chamber.

#### **Appendix B. Raw Data of the Experiments**

Figure A7 shows raw data of Figure 7. In Figure A7, the graph exhibits rattling behavior owning to the recording frame rate. Figure A6 shows the n-th, N + 1st, and N + 2nd frames in the experiments. The main chamber repeatedly expanded and contracted, which caused the liquid to be gradually absorbed as it moved back and forth between the neck chambers, resulting in rattling luminance values during the driving chamber operation.

**Figure A6.** N-th to N + 2nd frame of an experiments. Main chamber repeatedly expanded and contracted, which caused rattling luminance values.

**Figure A7.** Time variation in mixing index until 9 s after the application of air pressure.

## **References**

