Realizing an infinite microfluidic stage using spatial transformation and conformal mapping

Microscale phenomena are typically observed under microscopes within confined geometries in traditional laboratory settings. Such confinement includes cover slips, air-water interfaces, or microfluidic channels, introducing either intentional or undesirable boundaries. Here, we demonstrate that microflows can be generated uniformly in all directions within a six-channel microfluidic junction, with uniformity strictly maintained by symmetry-group-based control. A dynamically controlled flow velocity is thus integrated into the 3D movement of an effectively infinite microfluidic stage, without any boundary acting on the specimen. Additionally, any imperfections in the movement of the "stage" can be corrected by spatially deforming a test movement pattern to align with the desired one. We show that this restored symmetry can be extended to any movement pattern through conformal mapping, which remains differentiable along the entire path. These conformally mapped movements can thus be used to displace specimens smoothly and rapidly, achieving accuracies as high as submicron scales. To demonstrate its capabilities, we use this infinite microfluidic stage to study, for long periods of time in the absence of any stresses, the interactions of Brownian particles and freely swimming microorganisms.