EXTENDED SPACE CHARGE EFFECTS IN CONCENTRATION POLARIZATION

Isaak Rubinstein and Boris ZaltzmanBlaustein Institutes for Desert Research

Ben-Gurion University of the Negev Israel

Anomalous RectificationCopper deposition from 0.002N CuSO4 solution

0.1V, 1MHz

Dukhin’s Vortex

E= 100V cm−1

Electrokinetic flow around a 1mmion exchange granule

S. Dukhin, N. Mischuk and P.TakhistovColl. J. USSR89Y. Ben and H.-C. Chang JFM02

I.R, Israel Rubinstein and E. StaudePCH85

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 section50 µm

20µm

Classical picture of Concentration Polarization

0 ,c D D

Stirred Bulk

0

(0) 1

(0) 0

x

c

Cation-exchange membrane

0

1

1

1

( )

( ) 0

x x x

x x x

x

c c I

c c

Electric Double Layer

x1

1

C

Diffusion layer,

( ) 0

( ) 0x x x

x x x

c c

c c

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

0,

| , (0) ( )

yy xxE u E

u E potential drop between the interface and the EN Bulk

ELECTROCONVECTION, STEADY STATE

TWO TYPES OF ELECTROCONVECTION IN STRONG ELECTROLYTES

“Bulk” electroconvection

Classical quasiequilibrium electroosmosis Non-equilibrium electroosmosis

0

( ) ( )

( ) ( )

0

0

?

Pe c c c

Pe c c c

p

v

v

v

v

v

INNER SOLUTION: Boundary Conditions - Electroosmotic Slip, etc.

,

22 2

2

, ( , ), ( , )

1 1 10 0 ( , ) ( ,0)

21

0 02

x z z z zz z

x x zz zz z x zz zzx

u w

yz c x z x z

u w w p p x z p x

p u u

v i j

OUTER SOLUTION:

me

mb

ran

e

y

x

solution

2

( ,0) ( , )

( , ) ( ,0)

( , ) ( ,0) ( ,0) ( , )

2 ( ,0)2 2

22 2

0 ( , ) ( ,0)

0 ( , ) ( ,0)

( ,0)

1 ( 1)( , ) ( ,0) 2 ln

1 ( 1)

x x zz z

x z xz z

x z x x x zzz zz

z c x

z

c c

c c c x z c x e

c c c x z c x e

c c c x e e

e e ex z x

e e e

( ,0)

/ 2( ,0) 4 ln 2 4ln 1 , ( ) ( ,0) ( ,0)

ln (4 ln 2)

c x

x xx

xx x

c cu x e x x x

c c

cc const u

c

OUTER SOLUTION: Locally Electroneutral “Bulk” Electroconvection

EQUILIBRIUM ELECTROOSMOSISQuasi-equilibrium Electric Double Layer

Conduction stable: E. Zholkovskij, M. Vorotynsev, E. Staude J.Col.Int.Sc.96

Dukhin: 60s – 70s

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

2

0,2

10,2

2

0

0

2

0

0

0

2

0

y y y

y y y

yy

y yy

y

y

y

c c

c c

c c

c c

c p

c dy

V

Ionic concentration profiles ε=.001, 1 - V=0, 2 - V=7, 3 – V=15, 4 – V=25

Levich 1959, Grafov, Chernenko 1962-1964, Newman, Smyrl 1965-1967, 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 profilesε=.001

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

ε2/3

ε2/3

ε

Basic Estimates

2 2 22

2

22

( ) ,

& : 0,

: , (1) ,

' min:

,

y y diff y migr y

yy yy

diff migr y y

diff migr

cj c c j c j c c

c c QE EDL ESC c c

cQE EDL j j c O c

ESC extended counterions concentration

j j I c c

2

2 1/3 2/33

22/3 2 4/3 2/3 2/3 1/3 4/3

2

, (1)

, ESC

I O I

c I I q I

Toy ProblemStirred Bulk

1

( 1) 1

( 1) 0

x

c

-1 x0

1C

0 0

0

0

ln 0

x x

x x

x x

c c I

c c

c

I = V = 0

0 < I < 2

I

0 < I < 2

Stirred Bulk

1

1

(1) 1

(1)

x

c

V

(1) (2)

1( ) 1 (1 )2

Ic x x

2 ( ) 1 (1 )2

Ic x x / 21 / 2

1 / 2VI

eI

2

0 0 0

0 : , 0

0

x x x x

xx

xx x x

c c I c c

c c

c

EIS of ESC

Anomalous Rectification

Limiting EOII flow problem, electroosmotic instability

S. Dukhin 1989: Electrokinetic Phenomena of the Second Kind, Adv. Coll. Interf. Sc.,91P. Takhistov 1989: Duhin’s vortex measurementsA.V. Listovnichy 1989: Extreme asymptotic ESC, Sov. Electrochem.,89

I. R. and B.Z. 1999: Limiting EOII slip:

'2 21 1

8 8n

n

I cu V V

I c

22

0

( )

0

0

0

1

8

0

CP: , 0 2

( ) , 0

n

Pe c c

p

c

c nv V

c n

v

x y

c y y

v

v

v

v

Marginal stability curves 1 - D = 0.1, 2 - D = 1, 3 - D = 10

yc

v

Mechanism of Non-equilibrium

Electro-osmotic Instability0

2

8)0,(

yy

yx

c

cVxu

Test vortex

( ,0)u x E

BASIC 1D PROBLEM IN TERMS OF PAINLEVÉ EQUATION

Universal Electro-Osmotic Slip Formula

3/ 2

01 1/ 2

ln , 23 3

2max( ,0)ln ln

3

xyI Ix x yx

y

cU Uu U U I I c

c

zc p V

I

Dukhin’s Formula for | ζ |=O(1) ||>>O(1), Extended Charge Electroosmosis

2/3| | 1, c I

8/2

B.Z., I.R. JFM07

Electro-neutral bulk

0

)( Pe

,10 ),( Pe

v

pv

cccv

xyccDcv

y

xyIx c

cUtxtxVUtxu

I

zVptxtxc

),0,( )],0,([),0,(

,3

)0,max(2ln),0,(),0,(ln

2/1

2/30

1

.0),0,( ,0),0,(),1,(),0,(

),,1,(2ln4),1,( ,0),1,(

,0),1,(),1,(),1,( ,ln),1,(),1,(ln 1

txwtxtxctxc

txtxutxw

txtxctxcptxtxc

yy

yy

FLOW DRIVEN BY NON-EQUILIBRIUM ELECTROOSMOSYS

Universal Electro-Osmotic Formulation

0 0 0

0

( ), ( ,0, ), ( , )z z V x t F z z dz

),0,(),0,( ,1 3/20 txctxcz y

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

22.5

25

27.5

V

0 0.0025 0.005 0.0075 0.01

1.8

2.6

3.4

k c

D ashed lineV=-4/3 ln+const

D ashed line k=-1/3 lnconst

5103

0 2 4 6 8

k

10

20

30

40

50

V

D =1

Voltage - Current Curves in the Limiting Electro-Osmotic Formulation ε = 0.001, ε = 0.0001, ε = 0.00001

Hysteresis Mechanism

1Sc , 0tv v p v

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 = 0.00001

0 v p

0 v p

3

xyIx

y

cUu U

c

Gilad Yossifon and Hsueh-Chia Chang, PRL08

Laterally averaged concentration profiles for three voltages corresponding to the limiting and two overlimiting currents

y

<C>

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 & Z0 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 CuSO4

23C, theoretical limiting current: 126 mA

Maletzki et al., 1992

0.00

1.00

2.00

3.00

4.00

5.00

I [mA/cm ]2

0.0 0.5 1.0 1.5 2.0

U [V]

I [mA/cm ] / 0.1 m2 I [mA/cm ] / 0.2 m2 I [mA/cm ] / 0.3 m2 I [mA/cm ] / 0.4 m2 I [mA/cm ] / 1.0 m2 I [mA/cm ] / 1.0 m2 I [mA/cm ] / 2.0 m2 I [mA/cm ] / 2.0 m2 I [mA/cm ] / original2

Current-voltage curves of a C-membrane modified by a thin layer of cross-linked polyvinyl alcohol

I[mA/cm2]

U[V]

VISUALIZATION

Nonlinear Electro-convection ε = 0.01

Universal regular electro-osmotic formulation is needed

0.5 1 1.5 2 2.5 3 3.5

0.5

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.