EXTENDED SPACE CHARGE EFFECTS IN CONCENTRATION POLARIZATION

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

2 Anomalous Rectification Copper deposition from 0.002N CuSO4 solution 0.1V, 1MHz I.R, Israel Rubinstein and E. Staude PCH85 Dukhin s Vortex S. Dukhin, N. Mischuk and P.Takhistov Coll. J. USSR89 Y. Ben and H.-C. Chang JFM02 E= 100V cm 1 Electrokinetic flow around a 1mm ion exchange granule

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

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

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

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

7 Classical picture of Concentration Polarization Diffusion layer, δ c0, D D (cx c x ) x 0 Stirred Bulk (cx c x ) x 0 x 0 c(0) 1 (0) 0 Cation-exchange membrane x 1 (cx c x ) x 1 I I=V=0 1 C V (cx c x ) x 1 0 0<I<2 I=2 Electric Double Layer 1 0 I x

8 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. Electric Double Layer - EDL Helmholtz (1879), Guoy-Chapman (1914), Stern (1924) -- Bulk Slip velocity C+(y) yy E u xx 0, Helmholtz-Smoluchowski 1879, 1903, 1921 C-(y) HEURISTIC THEORY OF ELECTROOSMOTIC 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 E u E, (0) ( ) potential drop between the interface and the EN Bulk

9 ELECTROCONVECTION, STEADY STATE TWO TYPES OF ELECTROCONVECTION IN STRONG ELECTROLYTES Bulk electroconvection Classical quasiequilibrium electroosmosis Non-equilibrium electroosmosis

10 OUTER SOLUTION: Pe( v )c Pe( v )c p 0 ( c c ) ( c c ) v 0 v 0 v? INNER SOLUTION: Boundary Conditions - Electroosmotic Slip, etc. membrane v ui wj x y solution z, c, ( x, z ), ( x, z ) y u x wz 0 w 0 pz 2 z zz p( x, z ) 2 z p ( x, 0) px x zz u zz 0 z x zz u zz 0 x 2

11 EQUILIBRIUM ELECTROOSMOSIS Quasi-equilibrium Electric Double Layer 2 c c c z c z 0 c ( x, z ) c ( x,0)e ( x,0) ( x, z ) c z c z 0 c ( x, z ) c ( x,0)e ( x, z ) ( x,0) zz c c zz c ( x,0) e ( x, z ) ( x,0) e ( x,0) ( x, z ) ( x, z ) ( x,0) 2ln c ( x,0) c ( x,0) e 1 (e 1)e z e 1 (e 1)e z cx cx u ( x,0) x 4ln 2 4ln e / 2 1, ( x) ( x,0) ( x,0) c c Dukhin: 60s 70s cx ln c const x u (4ln 2) x c Conduction stable: E. Zholkovskij, M. Vorotynsev, E. Staude J.Col.Int.Sc.96

12 Non-equilibrium Electric Double Layer I.R., L.Shtilman JCS Faraday Trans.79 c c y c y 0 y c y 0 y y 2 yy c c c y c 2 c y y 0,2 p1 c dy 2 0 y 0 V y 2 0 y 0,2 0

13 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

14 Space charge density profiles ε=.001 ε2/3 ε2/3 ε O(ε2/3) 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(ε2/3) to O(1). For such voltages, O(ε2/3) is the length scale of the transition zone from the extended non-equilibrium space charge region to the quasi-electro-neutral bulk

15 Basic Estimates j (c y c y ) jdiff c y c, yy c c QE EDL & ESC : c 2 jmigr c y 0, yy c QE EDL : jdiff jmigr, c O(1) y, c y 2 ESC extended counterions ' concentration min: 2 jdiff jmigr I c, 2 c I, O(1) I 1/ 3 2 / / / 3 2/3 2/3 1/ 3 4 / 3 c I I, q I ESC 2 c c

16 Toy Problem Stirred Bulk (1) x 1 c( 1) 1 ( 1) 0 I c1 ( x) 1 (1 x) 2 1 C 0 0 : c x c x I, c x c x 0 2 xx c c c x 0 x 0 x x 0 0 c(1) 1 (1) V I=V=0 I c2 ( x) 1 (1 x) 2 0 cx c x I cx c x 0 % x 0 ln c x 0 0 x 1 0<I<2 0<I<2-1 Stirred Bulk (2) I 1 1 I / 2 e V / 2 1 I / 2 x

17 EIS of ESC

18 Anomalous Rectification

19 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: 1 2 I ' 1 2 cn u V V 8 I 8 cn Limiting EOII flow problem, electroosmotic instability Marginal stability curves 1 - D = 0.1, 2 - D = 1, 3 - D = 10 Pe( v )c c p v 0 v 0 c 0 v 1 2 V 8 2 c n c n vn 0 CP: x, 0 y 2 c0 ( y ) y, v 0

20 Mechanism of Non-equilibrium Electro-osmotic Instability u ( x, 0) E V 2 c yx u ( x,0) = 8 cy y =0 Test vortex cy v

21 BASIC 1D PROBLEM IN TERMS OF PAINLEVÉ EQUATION

22 Universal Electro-Osmotic Slip Formula B.Z., I.R. JFM07 ζ >>O(1), Extended Charge Electroosmosis Dukhin s Formula for ζ =O(1) ζ 2/8 U I cxy U u U x U x I ln I x, I 2c y 3 cy 3 ln c ln p1 2 max( z0, 0) V 1, 3I 1/ 2 c I 2/3 3/ 2

23 FLOW DRIVEN BY NON-EQUILIBRIUM ELECTROOSMOSYS Universal Electro-Osmotic Formulation 2/3 z 0 >>1, c x,0, t ε c y x,0,t Electro-neutral bulk Pe v c=d c+c ϕ, 0<y<1, <x< Pe v c= c c ϕ 3/2 2max z,0 0 ln c x, 0, t +ϕ x, 0, t =ln p1 V 1/2, Δ v = Δϕ ϕ+ p 3I v =0 c xy u x, 0, t = U ϕ [ V ϕ x,0, t ] ϕ x x, 0, t +U I c y ln c x,1,t +ϕ x,1,t =ln p 1,c y x,1,t c x,1,t ϕ y x,1,t =0, w x,1,t =0, u x,1,t = 4ln2ϕ x,1,t, c y x,0,t c x,1,t ϕ y x,0,t =0, w x,0,t =0. z0 z0 ( ), V ( x, 0, t ), F ( z, z0 )dz 0

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

25 Comparison of Neutral-Stability Curves in the Full and Limiting Formulations ε= D a s h e d lin e V = - 4 /3 ln ε + c o n s t V V D a s h e d lin e k = - 1 /3 ln ε + c o n s t 3.4 D=1 20 kc k ε

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

27 Hysteresis Mechanism r r 1 Sc v t v p, Stabilizing 1D conduction in EN Bulk and in the QE EDL r v 0 Destabilizing 1D conduction in the Extended Space Charge Region Convective mixing Destruction of 1D CP hampering effect of the bulk electric force Lowering the

28 Voltage - Current Curves in the Limiting Formulation with and without the Bulk Force Term ε = r 0 v p U I cxy u U x 3 cy r 0 v p

29 Gilad Yossifon and Hsueh-Chia Chang, PRL08

30 Laterally averaged concentration profiles for three voltages corresponding to the limiting and two overlimiting currents <C> y

31 Laterally averaged concentration profiles for various values of voltage and ε

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

33 I V

36 CURRENT & Z0 VERSUS VOLTAGE

39 SPACE CHARGE

40 SPACE CHARGE DENSITY

41 IONIC CONCENTRATIONS

42 CURRENT & TOTAL CHARGE VERSUS VOLTAGE

43 TOTAL CHARGE & ESC VERSUS VOLTAGE

44 Ion Exchange Membranes Electrodialysis stack

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

47 Voltage-current characteristic for amalgamated copper cathode (A) and membrane C51 (B) with electrolyte immobilized by agar-agar Maletzki et al., 1992 Corresponding current-noise power spectrum of the membrane φ=0.9v; working electrolyte 0.01M CuSO4 23οC, theoretical limiting current: 126 ma

48 [ma/cm2]] I I[mA/cm I [ma/cm2] / 0.1 µm I [ma/cm2] / 0.2 µm 4.00 I [ma/cm2] / 0.3 µm I [ma/cm2] / 0.4 µm I [ma/cm2] / 1.0 µm I [ma/cm2] / 1.0 µm 3.00 I [ma/cm2] / 2.0 µm I [ma/cm2] / 2.0 µm I [ma/cm2] / original U [V] U[V] Current-voltage curves of a C-membrane modified by a thin layer of crosslinked polyvinyl alcohol

49 VISUALIZATION

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

51 c x y 3.5

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

53 Overlimiting conductance

54 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. ε

Ben-Gurion University of the Negev The Jacob Blaustein Institutes for Desert Research The Albert Katz International School for Desert Studies

Ben-Gurion University of the Negev The Jacob Blaustein Institutes for Desert Research The Albert Katz International School for Desert Studies Analysis of the Electrical Double Layer With and Without Added