Microchannels with Self-Pumping Walls

When asymmetric Janus micromotors are immobilized on a surface, they act as chemically powered micropumps, turning chemical energy from the fluid into a bulk flow. However, such pumps have previously produced only localized recirculating flows, which cannot be used to pump fluid in one direction. Here, we demonstrate that an array of three-dimensional, photochemically active Au/TiO2 Janus pillars can pump water. Upon UV illumination, a water-splitting reaction rapidly creates a directional bulk flow above the active surface. By lining a 2D microchannel with such active surfaces, various flow profiles are created within the channels. Analytical and numerical models of a channel with active surfaces predict flow profiles that agree very well with the experimental results. The light-driven active surfaces provide a way to wirelessly pump fluids at small scales and could be used for real-time, localized flow control in complex microfluidic networks.

The calibration of the diffraction pattern using the defocusing method with a phase contrast microscope is conducted by immersing and immobilizing 1 µm PS spheres in a 3 % polyacrylamide (PAC) gel. 1 The reference stacks are shown in Figure S2a, and the corresponding diffraction pattern area vs. defocusing height is plotted in Figure S2c.

Supplementary Note 3. Fabrication of various topographical structures and comparison of pumping speeds
The fabrication of 3D Janus bars and 2D patches is achieved using photolithography and Glancing Angle Deposition. The 3D bars are fabricated similar to the 3D pillars described in the main text. The only difference is the photomask, which is patterned with micro rectangular structures to produce the bars. For 2D Janus patches, an array of micro holes with thickness of 1.7 µm was exposed at 0 • to an e-beam source with Au in a GLAD chamber for a 100 nm thick deposition, at 30 • to TiO 2 for a 50 nm thick deposition, followed by a lift-off process in acetone and air plasma and an annealing process at 450 • C for 2h in air.
The SEM images of the corresponding structures are shown in Figure S3a and Figure Figure S4: Illustration of the analytical model and parameters of a cross section of a channel. The sections L 1 are assumed to be passive (bare surfaces). The section containing pillars is indicated as L 0 . P 1 and P 2 are the pressure at the inlet and outlet of the active section. See text for details.
We assume that the inlet and outlet pressures are at a constant hydrostatic pressure, defined as P = 0. Because the active region is the sole source of fluid motion, it must generate a pressure gradient to drive the flow. Therefore, we define P 1 and P 2 as the pressures at the start and end of the active region.
The fluid flow within each region of the channel is described by the continuity equation Here p(x) is the (z-independent) pressure in the channel the channel, µ is the fluid viscosity, and u(z) is the fluid velocity in the x-direction.
Because the pressure depends only on x and the velocity only on z, the pressure must change linearly with x in each region of the channel. For convenience, we define the constant pressure gradient as dp/dx = −G. Then we can write the pressure gradient G across each section of channel as: To completely determine the flow profile in the 3-part channel, the unknown pressures P 1 and P 2 (and thus G 1 , G 2 , and G 0 ) must be specified in terms of the (controllable) wall slip velocities, u t w and u b w , within the active region. To relate these quantities, we leverage the incompressibility of the water, observing that the volume flow rate must be constant through all three sections of the channel. The volumetric flow rate (per unit depth of the channel into the page),Q, is calculated in each section (indexed here by i) by integrating the flow speed over the whole channel height: Enforcing continuity,Q 1 =Q 2 =Q 3 ≡Q, will provide the necessary constraints to completely determine the unknown pressures P 1 and P 2 , thereby fully describing the flow profile within the channel.
In the inert inlet and outlet segments, the flow profile will solve Eq. (1) with boundary and the flow rates areQ EnforcingQ 1 =Q 2 leads to the condition Assuming the flow is from left to right, P 2 > 0. The pressure head generated by the active walls is ∆P = 2P 2 , and the pressure gradient within the active region is G 0 = −2P 2 /L 0 .
Within the active region, the flow rate iṡ Equating Eq. (13) with Eq. (9), the pumping pressure P 2 can be expressed in terms of the wall velocities and the channel geometry: Finally, the flow velocity and flow rate can be rewritten in terms of the wall velocities and channel geometry: For each of the three channel configurations considered in the text (symmetric, antisymmetric, and skew), we now substitute the appropriate channel geometry and boundary conditions to calculate the flow profile, net flow rates, and pumping pressure using Eqs. (14)-(16).

Symmetric channel: Active top and bottom surfaces
Aligning the structures on top and bottom surfaces to pump in the same direction, the active surface provides a forward slip velocity on both walls, u b w = u t w = u w , so that and the associated flow rate is:Q The pumping pressure is: When the active pumping region is much longer than the side channels (L 0 L 1 ), Q = u w h, and all of the fluid is pumped forward in a plug flow (u 1 (z) = u w ). By contrast, when the pumping region is much smaller than the side channels (L 0 L 1 ), the flow rate is reduced by a factor L 0 2L 1 :Q = u w hL 0 /(2L 1 ). By adding longer inlets and outlets, flow resistance increases and the flow rate for a given slip velocity decreases.

Antisymmetric channel: Active top and bottom surfaces
When the active walls are assembled to pump in opposite directions, u b w = u w and u t w = −u w . Applying the boundary conditions, the flow within the active channel follows a linear profile: Because of the odd symmetry of this configuration, the antisymmetric channel generates no unidirectional pumping pressure and no net flow:

Skew channel: Active bottom surface
When only the bottom surface in a channel is active, u b w = u w and u t w = 0, and the flow profile within the active region is then a skew parabola: and the associated flow rate isQ The pumping pressure is By comparing to Eqs. (19) and (18), we see that when pumping with only one active wall, the generated pressure and flow rate are half of those generated by the channel with two active walls.

Additional comments
By assuming that the flow u is independent of position along the channel x, the analysis above inherently ignores inlet/outlet effects, including at the boundary between the active and inactive regions. Because of the sudden change from slip-to nonslip-boundary conditions, an additional pressure drop is expected to occur locally, and u will become dependent on x.
As a result, there must be some flow also in the z-direction locally around the boundary.
However, these effects will remain localized to the interface between regions, and can be neglected to leading order as long as the channel regions are much longer than they are wide. In this case, the core implications of the model above remain valid: the flow profile described by Eq. 5 develops in each section, the magnitude of the pumping pressure can be estimated from mass conservation, and the volumetric flow rate is accurately calculated from mass conservation.
The validity of the model is supported a posteriori from the agreement with experimental data in Fig. 5 (main text) and from the results of a numerical simulations described below.

Supplementary Note 5. Numerical simulation
Numerical studies were conducted using COMSOL 5.3 with creeping flow physics to crosscheck the validity of the approximations in the analytical model above. The simulation setup specific lengths L 1 and L 0 used for the numerical solution are those corresponding to the patterned surfaces and illumination area as used in the experiments (see Table 1 below).
The inlet and outlet pressures are P = 0 Pa. At inactive walls, non-slip boundary conditions are enforced. At each active wall, the prescribed slip velocity is applied. The value of this velocity is taken from experimental measurements, as described in Note 6.
To avoid a discontinuity in the wall slip speed, small transition sections are added between each inactive and active section. The length of the transition sections are 0.5 µm with a ramping velocity along each active wall 0 µm/s to u w (at the entrance to the active region) or u w to 0 µm/s (at the exit). Given that the numerical and analytical solutions agree, we find that these transition sections do not significantly affect the results.

Supplementary Note 6. Parameters used in theoretical and numerical calculations
As shown in Figure 3b of the main text, tracer particles close to the active surface undergo a zigzag motion due to the hydrodynamic influence of the pillars. Therefore, flow velocity measurements along the channel are underestimated when tracking such particles. For the analytical and numerical analysis of flow within the active channels (in Figure 4), we do not consider the speed measured directly at the wall, and instead use particle velocities measured at least 5 µm from the wall. To account for this difference from the experiments we model the channel with a smaller height, corresponding to the distance between the uppermost and lowermost measured particle velocities. We then treat these velocities as the slip condition within the channel. The channel sizes and wall velocities used in the models for Fig. 5 are listed in Table 1.