EXTENDED SPACE CHARGE EFFECTS IN CONCENTRATION POLARIZATION Isaak Rubinstein and Boris Zaltzman Blaustein Institutes for Desert Research Ben-Gurion University of the Negev Israel

Anomalous Rectification Copper deposition from 0.002N CuSO 4 solution 0.1V, 1MHz Dukhin’s Vortex E= 100V cm −1 Electrokinetic flow around a 1mm ion exchange granule S. Dukhin, N. Mischuk and P.Takhistov Coll. J. USSR89 Y. Ben and H.-C. Chang JFM02 I.R, Israel Rubinstein and E. Staude PCH85

S. J. Kim, Y.-Ch. Wang, J. H. Lee, H. Jang, and Jongyoon Han PRL 07 Windshield Wiper’s Effect

S.M. Rubinstein, G. Manukyan, A. Staicu, I. R., B. Zaltzman, R.G.H. Lammertink, F. Mugele, and M. Wessling PRL08 Nonequilibrium Electroosmotic Instability

Voltage-current curve of a C-membrane Current power spectra Overlimiting Conductance F. Maletzki, H.W. Rosler and E. Staude, JMS92

Electrodialysis applications J. Balster, M. Yildirim, R. Ibanez, R. Lammertink, D. Jordan, and M. Wessling, JPC B07 Top view 50 to 550 µm Cross section 50 µm 20µm

Classical picture of Concentration Polarization Stirred Bulk Cation-exchange membrane 0 Electric Double Layer x 1 1 C Diffusion layer,  I = V = 0 0 < I < 2 V  I=2 I

Tangential electric field, acting upon the space charge of the interfacial electric double layer, produces a tangential force whose action results in a slip-like flow known as electro-osmosis. Bulk Slip velocity C - (y)‏ C + (y)‏ Electric Double Layer - EDL Helmholtz (1879), Guoy-Chapman (1914), Stern (1924) Helmholtz-Smoluchowski 1879, 1903, 1921 HEURISTIC THEORY OF ELECTRO- OSMOTIC SLIP Assumptions: 1. Lateral hydrostatic pressure variation is negligible. 2. Electric field = superposition of the intrinsic field of EDL and a weak constant applied tangential field

ELECTROCONVECTION, STEADY STATE TWO TYPES OF ELECTROCONVECTION IN STRONG ELECTROLYTES “Bulk” electroconvection Classical quasiequilibrium electroosmosisNon-equilibrium electroosmosis

INNER SOLUTION: Boundary Conditions - Electroosmotic Slip, etc. OUTER SOLUTION:

OUTER SOLUTION: Locally Electroneutral “Bulk” Electroconvection EQUILIBRIUM ELECTROOSMOSIS Quasi-equilibrium Electric Double Layer Conduction stable: E. Zholkovskij, M. Vorotynsev, E. Staude J.Col.Int.Sc.96 Dukhin: 60s – 70s

Non-equilibrium Electric Double Layer I.R., L.Shtilman JCS Faraday Trans.79

Ionic concentration profiles ε=.001, 1 - V=0, 2 - V=7, 3 – V=15, 4 – V=25 Levich 1959, Grafov, Chernenko , Newman, Smyrl , Buck 1975, Listovnichy 1989, Nikonenko, Zabolotsky, Gnusin, 1989, Bruinsma, Alexander 1990, Chazalviel 1990, Mafe, Manzanares, Murphy, Reiss 1993, Urtenov 1999, Chu, Bazant 2005

Space charge density profiles ε=.001 O(ε 2/3 ) is the critical length scale, which dominates the EDL for the voltage range V=O(4/3|ln(ε)|), marking the transition from the quasi-equilibrium to non-equilibrium regimes of the double layer. For voltages larger than O(4/3|ln(ε)|), a whole range of scales appears for the extent of the space charge, anything from O(ε 2/3 ) to O(1). For such voltages, O(ε 2/3 ) is the length scale of the transition zone from the extended non-equilibrium space charge region to the quasi-electro-neutral bulk ε 2/3 ε

Basic Estimates

Toy Problem Stirred Bulk x 0 1 C I = V = 0 0 < I < 2 I Stirred Bulk 1 (1)‏ (2)‏

EIS of ESC

Anomalous Rectification

Limiting EOII flow problem, electroosmotic instability S. Dukhin 1989: Electrokinetic Phenomena of the Second Kind, Adv. Coll. Interf. Sc.,91 P. Takhistov 1989: Duhin’s vortex measurements A.V. Listovnichy 1989: Extreme asymptotic ESC, Sov. Electrochem.,89 I. R. and B.Z. 1999: Limiting EOII slip: Marginal stability curves 1 - D = 0.1, 2 - D = 1, 3 - D = 10

Mechanism of Non-equilibrium Electro-osmotic Instability Test vortex

BASIC 1D PROBLEM IN TERMS OF PAINLEVÉ EQUATION

Universal Electro-Osmotic Slip Formula Dukhin’s Formula for | ζ |=O(1)‏ |  |>>O(1), Extended Charge Electroosmosis B.Z., I.R. JFM07

Electro-neutral bulk FLOW DRIVEN BY NON-EQUILIBRIUM ELECTROOSMOSYS Universal Electro-Osmotic Formulation

Marginal stability curves for full electro-convective problem, D=1, 1- ε=1E-2, 2- ε=1E-3, 3- ε=3E-5

Comparison of Neutral-Stability Curves in the Full and Limiting Formulations

Voltage - Current Curves in the Limiting Electro-Osmotic Formulation ε = 0.001, ε = , ε =

Hysteresis Mechanism Stabilizing 1D conduction in EN Bulk and in the QE EDL Destabilizing 1D conduction in the Extended Space Charge Region Convective mixing Destruction of 1D CP Lowering the hampering effect of the bulk electric force

Voltage - Current Curves in the Limiting Formulation with and without the Bulk Force Term ε =

Gilad Yossifon and Hsueh-Chia Chang, PRL08

Laterally averaged concentration profiles for three voltages corresponding to the limiting and two overlimiting currents y 1 2 3

Laterally averaged concentration profiles for various values of voltage and ε

Laterally averaged concentration profiles for various values of voltage (Full Problem)‏

I V

CURRENT & Z 0 VERSUS VOLTAGE

SPACE CHARGE

SPACE CHARGE DENSITY

IONIC CONCENTRATIONS

CURRENT & TOTAL CHARGE VERSUS VOLTAGE

TOTAL CHARGE & ESC VERSUS VOLTAGE

Electrodialysis stack Ion Exchange Membranes

Voltage-current curve of a C-membrane Current power spectra Overlimiting Conductance through Ion Exchange Membranes F. Maletzki, H.W. Rosler and E. Staude, JMS92

Voltage-current characteristic for amalgamated copper cathode (A) and membrane C51 (B) with electrolyte immobilized by agar-agar Corresponding current-noise power spectrum of the membrane  =0.9V; working electrolyte 0.01M CuSO 4 23  C, theoretical limiting current: 126 mA Maletzki et al., 1992

Current-voltage curves of a C-membrane modified by a thin layer of cross- linked polyvinyl alcohol I [ mA/cm 2 ] U[V]

VISUALIZATION

Nonlinear Electro-convection ε = 0.01 Universal regular electro-osmotic formulation is needed

x y c

Concentration Level Lines and Streamlines (Electroosmotic Problem, ε = 0.001, V=35)‏

Overlimiting conductance

Numerical simulation of electroconvection in the limiting model for ε=10−6 showing hysteresis: black line – way up, blue line – way down. (a) Dimensionless current/voltage dependence; (b) flow streamlines’ pattern; (c) voltage dependence of the absolute value of the dimensionless linear flow velocity averaged over the diffusion layer; (d) current’s relaxation in the overlimiting regime.