Bulk electroconvective instability at high Peclet numbers Brian D. Storey (Olin College) Boris Zaltzman & Isaak Rubinstein (Ben Gurion University of the Negev)

Physical setup x y Binary electrolyte (C+,C-) Equations Poisson-Nernst-Planck Incompressible Navier-Stokes Fixed potential Fixed concentration of C+ No flux of C- Solid surfaces are charge selective (electrode or ion exchange membrane).

Steady state (no flow) V=1 E, flux of C+ Bulk is electro-neutral, linear conc. profile Double layer, Debye =0.01 Typical dimensionless Debye = or less

Current-voltage relationship Resistor at low voltage

Different views on bulk stability Conflicting reports of bulk instability in present geometry Microfluidic observations of bulk instability with imposed concentration gradients Lin, Storey, Oddy, Chen & Santiago (2004) El Mochtar, Aubry, Batton (2003) Bulk instability. Grigin (1985, 1992) Bulk instability, but not sufficient for mixing. Bruinsma & Alexander (1990) Bulk instability. Rubinstein, Zaltzman, & Zaltzman (1995). No bulk instability. Buchanan & Saville (1999) No bulk instability. Highlighted problems with all earlier works reporting instability. Limited parameter space. Lerman, Zaltzman, Rubinstein (2005)

Bulk electroconvective (BE) model Convection/Diffusion of concentration Current continuity Navier-Stokes First 2 equations are derived from Poisson-Nernst-Planck, assuming electro-neutrality. Incompressibility

Parameters Peclet, approx. 1 for KCl in water Reynolds, approx.001 (so we disregard) Ratio of applied voltage to thermal voltage (25 mv) Ratio of diffusivity of ions 0

Hoburg-Melcher (HM) limit D=1, Pe=∞, low V analysis 0 0 Purely imaginary spectrum

Modified Hoburg-Melcher (MHM) Pe=∞, low V analysis 0 Summary D>1, Real, S 2 <0, Stable D 0, Unstable D=1, Imag, Oscillations

Finite voltage, Pe=∞ Unstable MHM model (Pe=∞), low V limit MHM model (Pe=∞) Stable

Bulk electroconvection (BE) model low V analysis Current, I max =4 unstable L=-68 k=4.74 Summary D>1, Real, Stable D<1, Real, Unstable (threshold) D=1, Stable

BE at finite voltage, D=0.1 Unstable Pe=9.9

BE at finite voltage D>1 MHM model (Pe=∞) Unstable

BE model, Pe=10000, V=4 Real Imag

Conclusions Bulk instability can exist, in theory. New bulk instability mechanism found when D+ < D-, that can occur at low V. Many previous studies only considered D+=D-, Pe ~ 1. Whether D+ > D- or vice versa can lead to different behaviors. Unresolved questions: –Are there cases where this instability could be experimentally observed? –How does bulk instability relate to instability in extended space charge region? (Zaltzman and Rubinstein, 2006). –Does asymmetry in electrolyte matter in microfluidic applications? (Oddy and Santiago, 2005). –Does this instability matter in concentration polarization flows observed in nanochannel applications? Kim, Wang, Lee, Jang, Han (2007)

Steady state (no flow) V=20 E, flux of C+ Bulk is electro-neutral, linear conc. profile Extended space charge Double layer, Debye =0.01

Finite voltage, Pe=10000 Unstable BE Unstable Stable MHM model (Pe=∞) BE, low V

Finite voltage, Pe=10000 Unstable BE, full Unstable V=4 Stable MHM model (Pe=∞)

Bulk electroconvection (BE) model low V, D=1 HM

Low voltage limit, Pe=10000 Unstable BE, low V limit Stable