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.

Figure 1

Experimental setup: (a) microchannels; (b) image of the filling process; (c) three-dimensional schematic of the PDMS channel.

Figure 2

Experimental measurements of the nondimensional length of the EGaIn-filled part of each channel in the system as a function of time, under an imposed inlet pressure of (a) 1200 mb and (b) 1600 mb. Each color represents the filling process in one of the nine microchannels considered. Plots illustrate the complexity of the global evolution due to the strong stochastic component present in the evolution of each individual microchannel.

Figure 3

Illustration of the random sudden compression and slow degassing processes taking place during the penetration of EGaIn in the PDMS channel.

Figure 4

Statistical cumulative distributions of both $X_i$ (blue line) and $X_i^{\prime }$ (gray line). The dashed black line is a lognormal cumulative distribution function, $F\left(X\right)$, with mean $\mu =-4$ and standard deviation $\xi =2$.

Figure 5

Simulation of the EGaIn filling process under an imposed inlet pressure of 1600 mb; 15 simulated runs (or independent channels).

Figure 6

Average filling length of liquid metal in all captured fingers under the imposed inlet pressure of 1200 mb (left) and 1600 mb (right), respectively. TOpen circles represent experimental data; solid lines, numerical results.