# Electrokinetic Instability in Microflows With Conductivity Gradients

## Motivation

As electrokinetic microfluidic systems increase in complexity, robustness to heterogeneous sample streams becomes a major concern. Heterogeneities can arise from a variety of factors including temperature gradients, differences in ionic strength, or unintentional differences in sample preparation. One important heterogeneity is differences and/or gradient in ionic conductivity. Electric fields can couple with conductivity gradients in the field to generate electric body forces and electrokinetic instabilities (EKIs).

## Scaling Laws

- Applied electric fields can couple with conductivity gradients to generate net body forces in electric fields. For binary electrolytes (one positive, one negative species) we can derive the following simple expression for this effect [1]:

where is permittivity and is net charge.

- The natural velocity scale in microscale EKI is typically the
*electroviscous velocity*that results from a balance between the*viscous*and*electric stresses*in the liquid,

, and has the form

Here is a characteristic electric field, is the characteristic scale for viscous stresses, is the characteristic scale for conductivity gradients, and is viscosity.

- The onset of EKI can be described in terms of an electric Rayleigh number, , which is the ratio of time scale corresponding to the stretching of conductivity gradients due to electroviscous velocity to the time scale corresponding to molecular diffusion:

- Convective versus the absolute nature of electrokinetic instabilities is controlled by the ratio of electroviscous to electroosmotic velocity, . Simply put, convective instabilities transition to absolute instabilities when disturbance strengthen enough to travel upstream.

## Experiments

- We have studied electrokinetic flow instabilities that occur in DC-power-driven microflows when electric field is applied perpendicular to the conductivity gradient.

*Flow configurations*:(i) two parallel streams with different conductivity [1], and

(ii) hydrodynamically focused flow stream between a sheath flow with mismatched

conductivity [2].

- We have performed a detailed parametric experimental study by varying applied DC electric field and the conductivity ratio, i.e., varying the electric Rayleigh number.

- Depending on the conditions, our experiments show both convective and absolute

instabilities [1].

**Figure 1: Experimental visualization flow instability induced by a conductivity gradient. In these experiments, we electrokinetically inject two streams of electrolytes into a common mixing channel. The bottom stream is dyed with a neutral fluorescent dye. When both streams have the same conductivity we see a stable, diffusive interface between the two streams. Whereas, in the case of mismatched conductivity, a disturbance originates in the intersection region and is convected downstream.**

** Video 1: Experimental visualization of electrokinetic instability for varying electric fields and conductivity ratios. See Posner et al. [2] for more details.**

## Analysis

We have also analyzed electrokinetic instabilities using a variety of methods including

- Spectral analysis of experimental data (see Figure 2)

- Linear stability analysis [1,3,4],

- Detailed two and three-dimensional non-linear numerical simulations [3,4] (see Figure 3)

**Figure 2: Spectral analysis of EKI experimental data at four different electric Rayleigh numbers. Plots on the left show measured instantaneous scalar concentration fields in unstable electrokinetic flow. The dyed center stream flows from the left to the right and has higher conductivity than the background electrolyte sheath streams. The sheath steams enter from the top and bottom forming a pinched throat at the intersection. Plots on the right show power spectra measured at a downstream location on the mid-plane of the channel. At low electric field (low electric Rayleigh number, Rae) power spectra shows a single peak corresponding to the dominant unstable mode. At higher electric fields, we observe multiple modes which result due to bifurcation and period doubling. See Posner et al. [2] for more details.**

**Figure 3: Comparison of experimental visualization with detailed two-dimensional numerical simulation of electrokinetic instability induced by orthogonal conductivity gradient and electric field. Plotted is the intensity neutral fluorescent scalar which is initially mixed in the top stream. For both experiment and simulation, the electric field and bulk flow directions were from left to right. Simulation results show qualitative agreement with experimental visualizations and captures essential features of the instability, including the time when perturbations become visible and the subsequent distortion of conductivity field. See Lin et al. [3-4] for more details. **

Reference

1. Chen, C., Lin, H., Lele, S. K., and Santiago, J. G., "Convective and absolute electrokinetic instability with conductivity gradients",* J. Fluid. Mech.* 2005, *524*, 263-303. (pdf)

2. Posner, J. D., and Santiago, J. G., "Convective instability of electrokinetic flows in a cross-shaped microchannel", *J. Fluid. Mech. *2006, *555*, 1-42. (pdf)

3. Lin, H., Storey, B. D., Oddy, M. H., Chen, C., and Santiago, J. G., "Instability of electrokinetic microchannel flows with conductivity gradients", *Phys. Fluids* 2004, *16*, 2004. (pdf)

4. Lin H., Storey, B. D., and Santiago, J. G., "A depth-averaged electrokinetic flow model for shallow microchannels", *J. Fluid. Mech. *2008, *608*, 43-70. (pdf)