Pressure-driven filling of liquid metal in closed-end microchannels

We observe unsteady flow behavior of liquid metal during a pressure-driven injection process into a closed-ended polydimethylsiloxane microchannel. Constant pressure is applied at the inlet to allow eutectic gallium-indium (EGaIn) to completely fill the porous microchannels. In contrast to open channels [M. D. Dickey et al., Adv. Funct. Mater. 18, 1097 (2008)], the flow exhibits a complex unsteady behavior with sudden random length jumps and time stops. However, with appropriate formulation of a suitable mathematical model with the system using (i) the permeability of polydimethylsiloxane to air, (ii) previous descriptions of the nature of the EGaIn surface oxide layer, and (iii) a key probabilistic approach, we show that the average quantities defining the quantumlike flow can be accurately predicted. The proposed probabilistic formulation provides for the first time a description of the dynamics of the surface oxide layer, the breaking and healing characteristic times when EGaIn is driven in a microchannel. Importantly, this work provides a better understanding of complex flow behavior and lays the foundation for future work.