Convective instability of electrokinetic flows in a cross-shaped microchannel

We present a parametric experimental study of convective electrokinetic instability (EKI) in an isotropically etched, cross-shaped microchannel using quantitative epifluorescence imaging. The base state is a three-inlet, one-outlet electrokinetic focusing flow configuration where the centre sample stream and sheath flows have mismatched ionic conductivities. Electrokinetic flows with conductivity gradients become unstable when the electroviscous stretching and folding of conductivity interfaces grows faster than the dissipative effect of molecular diffusion. Scalar images, critical applied fields required for instability, and temporal and spatial scalar energy are presented for flows with a wide range of applied d.c. electric field and centre-to-sheath conductivity ratios. These parameters impose variations of the electric Rayleigh number across four orders of magnitude. We introduce a scaling for charge density in the bulk fluid as a function of local maximum conductivity gradients in the flow. This scaling shows that the flow becomes unstable at a critical electric Rayleigh number ($Ra_{e, \ell} \,{=}\, 205$) and applies to a wide range of applied field and centre-to-sheath conductivity ratios. This work is relevant to on-chip electrokinetic flows with conductivity gradients such as field amplified sample stacking, flow at the intersections of multi-dimensional assays, electrokinetic control and separation of sample streams with poorly specified chemistry, and low-Reynolds number micromixing.