Electric Double Layer and Concentration Polarization

Boris ZaltzmanBen-Gurion University of the Negev

Israel

Isaak Rubinstein

Conduction from an electrolyte into a charge-selective solid (ion exchangemembrane or metal electrode)

------------------C-(y)

C+(y)

ψ(y)

ψ0

y

Electric double layer

Charged Surface of Cation-Selective Membrane

Diffusion Layer

Electro-neutral bulk Stirred bulk

Membrane and Micro-Nano-Channel Device

Voltage-current curve of a C-membrane

+

_

CuSO4

MEMBRANE

Cu

Cu

2δ

Prototypical experiment, I. Rubinstein 70-th

MEMBRANE

C0

y

C0

δ

δ

Measuring Voltage-Current Dependence

Current power spectra

Classical picture of concentration polarizationmembrane: y=0 outer edge of diffusion layer: y=1

( )2 ,0)0(

12,ln,1)1(2

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

,0

.10 ,

lim =→=∞→

−==+−=

−===

=+=−

<<==

−

−+

IIcV

eI c yIc

V c

Icc cc

yccc

V

yyyy

ϕ

ϕϕϕϕ

Voltage, V

Cur

rent

den

sity

, I

I

II

III

I lim

?

Electrolyte

C-MEMBRANE

y

Prototypical experiment, I. Rubinstein 70-th

Voltage-current characteristic for amalgamated copper

cathode (A) and membrane (B) with electrolyte

immobilized by agar-agar, F. Maletzki et al 1992

Convective mixing+

_

CuSO4

MEMBRANE

Cu

Cu

2δ

MEMBRANE

C0

y

C0

δ

δ

Electro-Convection

No Gravitational Convection

100Ra10 ,200100 ,1.001.0 <<<<<< μδNC

52122/1

0

2/1

20

1

2

1010 ,)(2

)(

,Sc ,5.04

Pe

0 , , ,0 :00

Sc1

)( Pe ,10)( Pe

−−

−

+

−−−

−−+

+−

−−−−

++++

<<=

==≈

==

=−−=====∇

∇−∇Δ+Δ=

−=Δ∇−∇∇=∇+

∞<<∞−<<∇+∇∇=∇+

επ

ε

νπη

ϕϕ

ϕϕ

ϕεϕϕ

cFdRT

DDD

DDd

FRT

DLv

ccVpcvyv

pvv

cccccvc

xyccDcvc

y

t

y

t

t

Bulk

Slip velocity

ε<<1 OUTER SOLUTION: BULK ELECTRO-CONVECTION

INNER SOLUTION: ELECTRO-OSMOTIC SLIP

TWO TYPES OF ELECTRO-CONVECTION IN STRONG ELECTROLYTES

Bulk electro-convection Electro-osmosis

0 . 0 0

1 . 0 0

2 . 0 0

3 . 0 0

4 . 0 0

5 . 0 0I [ m A / c m ]2

0 . 0 0 . 5 1 . 0 1 . 5 2 . 0

U [ V ]

I [ m A / c m ] / 0 . 1 m2 μI [ m A / c m ] / 0 . 2 m2 μI [ m A / c m ] / 0 . 3 m2 μI [ m A / c m ] / 0 . 4 m2 μI [ m A / c m ] / 1 . 0 m2 μI [ m A / c m ] / 1 . 0 m2 μI [ m A / c m ] / 2 . 0 m2 μI [ m A / c m ] / 2 . 0 m2 μI [ m A / c m ] / o r i g i n a l2

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

U[V]

Rubinstein, Zaltzman, Pretz, Linder

For low-molecular electrolytes: Bulk Electric force - stabilizes like gravitation for stable stratification.

+

_

CuSO4

MEMBRANE

Cu

Cu

2δ

Overlimiting conductance is surface driven!

Electric Double Layer study

Helmholtz (1879), Guoy-Chapman (1914), Debye & Hückel (1917)Stern (1924)

Electric Double Layer study

Modeling Stern Layer (Boundary condition for Diffuse Layer Model)P. Delahay, R. Macdonald, L. A. Geddes, E. M. Itskovich, A. A. Kornyshev,

M. A. Vorotyntsev, A. J. Bard…Historical Review in M. Z. Bazant, K.T. Chu, B.J.Bayly, SIAM J. Appl. Math

65, 1463-1484 (2005), K.T. Chu, M.Z. Bazant PHYSICAL REVIEW E 74, 011501 2006, Theoretical Modeling of Stern layer

Frumkin-Butler-Volmer boundary condition for Stern/Diffuse layer interface

Electrical circuit approach

+ Bulk as a resistor

( )exp exp( )

| , oxidation, reduction,

kinetic constants, transfer coefficients, dimensionless Stern layer thickness

F R O R s O R O s

s S

J k C f k C fd O Rdx

k

α ϕ α ϕϕϕ δ

αδ

+ +± = − Δ − Δ

±Δ = − −

− −−

Modeling Diffuse layer (Gouy – Chapman)

2

2 ( )dd c cdy

ϕ + −= − −

Poisson equation describes variation of potential in a spatial charge distribution)

Nernst-Planck equations describe ionic transport

0

0, 1,

d dc F dD cdy dy RT dy

d dc F dD c z zdy dy RT dy

ϕ

ϕ

++

+

−−

− + −

+ =

− = = =

Scaling1/2

20 0

, , , = , = dd

rF c y dRTc x rRT c l l c Fϕϕ ε

±±

= = =

22

20, 0, d dc d d dc d dc c c cdx dx dx dx dx dx dx

ϕ ϕ ϕε+ −

+ − − + + = − = = −

Modeling Diffuse layer (Gouy – Chapman)

ln const. const.

ln const. const.

dc d dj c c c c edz dz dz

dc d dj c c c c edx dx dx

ϕ

ϕ

ϕε μ μ ϕ

ϕε μ μ ϕ

++ + + + + + + −

−− − − − − − −

= + = = + = =

= − = = − = =

Asymptotic analysis

, 0 , 0 - charged solid/solution interface, - EDL outer edge EN bulk xz z z zε

= < < ∞ = = ∞ →

Boundary conditions at EDL outer edge

( )2

2

2

2

( ) ( ) , ( ),

= =2 sinh( )

=2 ( ) - Debye-Huckel approximation | | 1

c c cc ce c ced c e e cdzd cdz

ϕ ϕ ϕ ϕ

ϕ ϕ ϕ ϕ

ϕ ϕ

ϕ ϕ ϕ

ϕ ϕ ϕ ϕ ϕ

+ −

+ − + −

− −

∞ = ∞ = ∞ == =

− −

− − <<

Poisson-Boltzmann Equation

cosh( / 4) sinh( / 4)exp( 2 )2lncosh( / 4) sinh( / 4)exp( 2 )

(0)

z cz c

ς ςϕ ϕς ς

ς ϕ ϕ

+ −= +− −

= −

Simplest Model Problem (Ideally-Selective membrane)Electrolyte

C-MEMBRANE

x

1 10 : (0) , fixed charge density (0) , overall potential drop across Electrolyte Layer

0, impermability for co-ions (anions)

x c p pV V

dc dcdx dx

ϕϕ

+

−−

= = −= − −

− =

1

1

ln const. ln ln

ln , 0x x

c c p VcV c cp

μ ϕ ϕ

ς ϕ ϕ

+ += + = + = −

= − − = − =

potential drop across diffuse EDL

Membrane/solution Interface

EDL/EN Bulk “Outer” Boundary

The Electric Double Layer and Electrokinetic Phenomena

Electrokinetic phenomena result from the differential movement of two phases where the interface is an electrical double layer. The region containing the double layered is sheared at some distance from the solid surface creating a thin film associated with the solid. The electrical potential at the shearing plane is the zeta potential.

Electrophoresis – a suspended, charged particle moves as a result of an applied electrical fieldElectrosmosis – a liquid flows along a charged surface when an electric field is applied parallel to the surfaceSedimentation potential – an electrical potential created by the movement of charged particles through a liquid by gravityStreaming potential – an electric potential created when a liquid is forced to move along a charged surface

The Electric Double Layer and Electrokinetic Phenomena

------------------C-(y)

C+(y)

ψ(y)

ψ0

y

Electric double layer

Assumptions: 1. Lateral hydrostatic pressure variation is negligible.

2. Electric field = superposition of the intrinsic field of EDL and weak constant applied tangential field

Bulkace and ENthe interfeen drop betwpotential

EEu

Eu

x

yyyy

−−=−=

=+

Σ

ςϕς

ϕ ,|

0y

Helmholtz-Smoluchowski 1879, 1903, 1921HEURISTIC THEORY OF ELECTRO-OSMOTIC

SLIP

Matched asymptotic expansions (Dukhin 60s – 70s)

( ) ) :S-(H 21

2110 ,

),( ),,( ,

2

222

xzzzzxzzxzzz

zzzzz

uu

ppwjwiuv

zxzxcyz

ϕϕϕϕϕ

ϕε

ϕϕε

ϕε

−=−=

===+=

= ±

( ) ( )( )

( ) ( )2

1lnln4ln),()( )(

1111ln2)(),(

)0,(0

)0,(0

2/

22/2/

22/2/

)0,(),(

))0,(),((

ς

ςς

ςςϕϕϕϕ

ϕϕ

ϕϕ

ϕςϕς

ϕϕϕ

ϕϕ

eccxuxuxV

eeeeeexzxeeccc

excccc

excccc

xx

zc

zc

zz

xzxzz

xzxzz

+−+=∞=−−=

−−+−+++=−=−=

==−

==+

−

−−−+−

−−−−

−−+++

Quasi-Equilibrium Electro-Osmotic Slip.

1. Impermeable Charged Surface

2. Charge-Selective Solid (Cation-Selective Membrane)

xuconstc ϕς== ,

( ) ( )

lim

/2

1

(0) 0, 21ln ln 4 ln

24ln 2 4ln 2 ln( )

x

x x

Concentration PolarizationV c I I

ec p V const u

V u c

ς

ϕ ϕ

ς φ

→ ∞ = → =++ = − = =

→ ∞ → −∞ = − =

1D Conduction stable: Zholkovskij,Vorotynsev, Staude (1996)

( ) ( )2

1lnln4ln),()( 2/ς

ϕς eccxuxu xx+−+=∞=

v

Stability of 1D Quasi-Equilibrium Conduction Electro-osmotic Instability ( )( ,0) 4ln 2 ln( )xu x c=

Vortex Fluctuation

c

Valid for , fails at the limiting current.

Breakdown of Quasi-Equilibrium at the Limiting Current

)0,(),())0,(),(( )0,( ,)0,( xzxxzx exccexcc ϕϕϕϕ −−−−+ ==

∞<> |)0,(| ,0)0,( xxc ϕ

Rubinstein, Shtilman 1979

2

1

( ) 0, ( ) 0, , 0 1

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

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

x x

x

x x x x xx

x

c c c c c c x

Depleted membrane surface :V c p c c

Stirred Bulk :c c

ϕ ϕ ε ϕ

ϕ ϕ

ϕ

+ + − − − +

+ − −

+ −

+ = − = = − < <

= − = − =

= = =C-membrane

Electrolytex

0.0

0.4

0.8

1.2

1.6

C- ,C+

Y0 0.2 0.4 0.6 0.8 1

V=0

V=4

V=20

V=100

______ C+

_ _ _ _ _C-

V=2

Grafov, Chernenko 1962-1964, Newman, Smyrl 1965-1967, Buck 1975, Rubinstein, Shtilman 1979,Listovnichy 1989 , Nikonenko, Zabolotsky, Gnusin, 1989, Bruinsma, Alexander 1990, Chazalviel 1990, Mafe, Manzanares, Murphy, Reiss 1993, Urtenov 1999, Chu, Bazant 2005

Transition from Quasi-Equilibrium to Non-Equilibrium Regime

O(ε2/3) - the critical length scale for V=O(4/3|ln(ε)|) -the transition from QE-EDL to NE-EDL. For V> 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

Dukhin (1989) : NE-EDL Electrokinetic Phenomena of the Second Kind

Ionic concentration profilesε=.01, V=0, V=2, V=4, V=20, V=100

Levich 1959

C-(x)

C+(x)

------------------

C+(x)= C-(x)=C(x)

Electric Double layer

Electro-neutral part of Diffusion Layer

Stirred Bulk

y

C-(x)

C+(x)

------------------

C+(x)= C-(x)=C(x)

Quasi-equilibrium portion of EDL

Electro-neutral part of Diffusion Layer

Stirred Bulk

y

Extended Space Charge

C-(x)=0,

C+(x)>0

Breakdown of Quasi-Equilibrium at the Limiting Current

constc

constc

=−=

=+=−

−

++

ϕμϕμ

ln

ln

0.001 0.01 0.1x

1E-008

0.0001

1

c+ -c _

0.001 0.01 0.1 1x

1E-005

0.1

c+ -c _

1. Threshold

scale and Extended Space Charge Typical Length scale

( )2/3

(1)

( ) ( )

j O

O x Oε ϕ

+ =

=

2/31 ( ) ln | | ( )O O xϕ ε ε< ≤ =

2/3ε

2

0, (1)

x x

xx

c c Oc c j

c c

ϕε ϕ

− +

+ + +

− +

= <<+ = −

= −

2xx x jε ϕ ϕ +=

3. Extreme NE Regime

2. Transitional Regime 2/3(1 )( ) , 0 1 ( )O O xα αϕ ε α ε− −= < < =

1( ) ( ) 1O O xϕε

= =

Electric Migration Domination Appearance of ESC

BASIC 1D PROBLEM IN TERMS OF PAINLEVÉ EQUATION

2/3 1/3

1/32/3

30

2 2/3 2/31 0

0

,

1 ( ) 1, 21 2 , 2

( ) 0

x

z

F I F Electric fieldxz I - Inner variable

F F z z F

F F I p z

F

ε ϕ

ε

ε

−

− −

=

= − −

=

′′ = + − +

′ + = +

∞ =

Chargeded Space - Extenzsition) - TranO(z

- QE-EDL,-zζ)on of V (ng functi increasi z

0

0

0

1 ,1

10

>>=<<

−−

-0.4 0 0.4 0.8x

0

0.2

0.4

0.6

0.8

1

c

x0 x0

x02 dcIdx

=

( )2

3 ,)(2

|)ln|( 1

3/2

00

34

0

VzVFdzzz0zzF

ncerge Domina Space Cha, Extended - NE-EDLVz

0 ≈−≈=<<−−≈

>>>>

ζ

ε

IIVzxuxu x

z 8),(lim)0,(

2

−==∞→

Extreme Non-Equilibrium Electro-osmosis

Instability of Quiescent Conduction+− === ccc ,0ε

.0 ,8

,0 :0

0 ,

,1

2 Pe

,10

2Sc1

=−===

=∇∇−Δ=

Δ+

=∇+

∞<<∞−<<

wccVucy

vpvv

cD

Dcvc

xy

y

yx

t

t

0 ,0 == vyc Quiescent Conduction (Concentration Polarization at the Limiting Current)

yc

v

Mechanism of Non-equilibrium Electro-osmotic Instability

( )2 2

00

( ,0) ln8 8

yxx y

y y

cV Vu x cc =

=

= − ≈ −

Vortex Fluctuation

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

+

_

CuSO4

MEMBRANE

Cu

Cu

2δ

VISUALIZATION

Gilad Yossifon and Hsueh-Chia Chang, PRL08

Figure indicates that the CPL thickness is selected by the instability. The cross-plot of the imaged diffusion layer thickness against the vortex pair wavelength in clearly shows that it obeys the linear scaling predicted by Rubinstein and Zaltzman with an empirical coefficient of 1.25, another confirmation that it is the same instability (the complex process of wave length selection by small vortices breaks up through fusion and transformation into still larger vortices).

S. J. Kim, Y.-Ch. Wang, J. H. Lee, H. Jang, and Jongyoon Han PRL 07

EDL Charge Dynamics under current

lim1

1

2Charge= ( (0)) for 2p c I Ip

ε − < =

lim1Charge 2 for 2p I Iε= ≈ =

lim1 0Charge 2 2 for 2p x I I Iε= + ≥ =

12

0

Charge ( ) (0)xc c dx ε ϕ+ −= − =

10 100 1000 10000V

0

1

2

3

4

5

j+

4

4.4

4.8

5.2

Σ tota

l /ε

(1)

(2)

EDL Charge Dynamics under current

0.00001 0.00010 0.00100 0.01000 0.10000x

0.0001

0.001

0.01

0.1

1

10

c+ -c -

(1)(2)

(3) (4) (5)

Mechanism of EDL Charge Dynamics under current

0.0

0.4

0.8

1.2

1.6

C- ,C+

Y0 0.2 0.4 0.6 0.8 1

V=0

V=4

V=20

V=100

______ C+

_ _ _ _ _C-

V=2

min

min

2min

0

12

min

ChargeQEEDL ( ) ( (0) ( ))

ChargeESC ( ) ( )

x

x x

xx

c c dx x

c c dx x

ε ϕ ϕ

ε ϕ

+ −

+ −

= − = −

= − =

0 0

min 0

2min 0

2

min2

0

, ESC ( )

, ESC ( )

( ) ( ) in ESC

( ) in ESC

1 in ESC 2 ( )

ChargeESC 2

x

xx

x x

x x

x

c I x x x

c x x x

x c c dx c dx

x

d I I x xdx

Ix

ϕε ϕ

σ

σϕεσσ σ ε

ε

ε

+

+

+ − +

= < <

= − < <

= − =

≈

≈ ≈ −

=

min min: ( ) 0dcx xdx

+

=Extended Space Charge

lim1 0Charge 2 2 for 2p x I I Iε= + ≥ =

1 0ChargeQEEDL 2 2p Ixε ε= −

EDL Charge Dynamics under current

0.001 0.01 0.1x

0.0001

0.001

0.01

0.1

c+ , c -

(1)

(2)

(3)(4)

0 0.5 1 1.5 2j+

0

0.0004

0.0008

0.0012

0.0016

M + Q

E-E

DL ,

M _ Q

E-E

DL

(1)

(2)

Ionic Mass decreases Charge increases due tofaster decrease of anions’ concentration No anions –charge saturation

Voltage Increase

Decrease of the total Ions’ number in the EDL and Increase of the charge due to the faster decrease of the anions’ mass

Limiting current regime: Moderate Voltage (Membrane Devices) EDL Charge Saturation (No anions in the EDL) , Appearance of the ESC = ε1/3 EDLCharge at ε2/3 distance from the membrane (~10ηm)High Voltage (Micro Nano channel Devices) “Overlimiting” Current, Further Increase of the ESC at up to O(1) distance from the Membrane (~1-3μm)Hydrodynamic Instability?

EDL Charge Dynamics under current

0ESCharge 2Ixε=

1 0 0ChargeQEEDL 2 2 2p x I Ixε ε= + −

1 0TotalCharge 2 2p x Iε= +

Mechanism of EDL Charge Dynamics under current2 2

2 2 2 21

1 1(0) ( ) [ (0) ( ) ( )]2 2

xx xx x x x x x

x x

c c c c c c I

x p c c x c x Ix

ε ϕ ε ϕ ϕ ϕ ϕ

ε ϕ ε ϕ

− + − + − +

− + −

= − = − = + −

− − + − − =

Drop of Maxwell Pressure Drop of Osmotic Pressure Total Friction of Water Balance of forces acting on the ions

QE-EDL2 2 2 2

1

2 2 2 2lim 1

21

( ) :1 1(0) ( ) [ (0) ( ) ( )] 02 2

1 1 (0) ( ) 02 2

2QEEDLCharge=TotalCharge+ESCharge

x x

x x

x O

x p c c x c x

I I x p

p

ε

ε ϕ ε ϕ

ε ϕ ε ϕ

ε

− + −

=

− − + − − =

≥ − − =

Conclusions

0.001 0.01 0.1x

0.0001

0.001

0.01

0.1

c+ , c -

(1)

(2)

(3)(4)

Current passes through perm-selective media

Charge saturation in QE-EDL, reduction of the charge carriers number

Limiting current regime and formation of the depleted region

Redistribution of the charge carriers

Extended space charge generationcharge is located at the depleted region outer edge

2 x x xxIc Ic

ϕ ϕ ε ϕ ρ≈ ≈ = −

0.00001 0.00010 0.00100 0.01000 0.10000x

0.0001

0.001

0.01

0.1

1

10

c+ -c -

(b)

(1)(2)

(3) (4) (5)

Extended space charge generationcharge is located at some distance from the wall and has much stronger affect than“immobile” QE-EDL

“Classical” Membrane Devices 10-100ηm

Micro-Nanochannels 1-5μm

Three-layer setup

Electrolytelayer

Electrolyte layer

Model

2

2

const., const.

(1 2) : , 0 fixed charge density

(0 1, 2 3):

/ Continuity of potential and concentrat

x x x x

xx

xx

j c c j c c

Membrane x N c c N

Solution x x c c

Membrane Solution Interface

ϕ ϕ

ε ϕε ϕ

+ + + − − −

− +

− +

= + = = − =

< < = + − > −

< < < < = −

ion (solubility=1)

Modeling Three-layer setup

2

:

1 '

1' /

(1 2) : , 0 fixed charge density

(0 1, 2 3)xx

Membrane EN Part c c NxN Membrane s EDL Scale zN

Membranes EDL Thickness Electrolyte EDL Thickness ON

Membrane x N c c N

Solution x x

ε ϕ

+ −

− +

= +

>> =

=

< < = + − > −

< < < < 2: xx c cε ϕ − += −

x=1

Membrane EDL Electrolyte Layer EDL

Modeling Three-layer setupForce balance across the EDL

2

2

const., const.

(1 2) : , 0 fixed charge density

(0 1, 2 3):

/ Continuity of potential and concentrat

x x x x

xx

xx

j c c j c c

Membrane x N c c N

Solution x x c c

Membrane Solution Interface

ϕ ϕ

ε ϕε ϕ

+ + + − − −

− +

− +

= + = = − =

< < = + − > −

< < < < = −

ion (solubility=1)

22 2

2

22 2

2

( (2) (2 )) (2) (2) 2 (2 ) ( )

(2) (2) 2 (2 ) ( )

x

x x

x

x x

N c c c J O

c c c J O

ε ϕ ϕ ϕ ε

ε ϕ ε

= + −

= −

= + + −

=

= − − + + − − + ⋅

= + − + + ⋅

Force Balance Across the EDL

Maxwell tension drop

Total El. Force Generated by Fixed Charge

Osmotic pressure drop Total Friction Force

Modeling Three-layer setupForce balance across the EDL

22 (2 ) 2 (2 )2 12(2) (2 )

4 2

(2 ) 0 (2)

c cN NN Nc c e

NN or c ce

+ + − + +

+

= + + +

→ ∞ + = =

(2 ) 0

(2) (2 ) 1

N or c

ϕ ϕ

→ ∞ + →

− − =

Maximum potential drop across Membrane EDL is 1 (thermal electric potential),Membrane’s EDL is always QE

2

( (2) (2 )) 2 (2 ) 2 (2 )

2 (2 ) (2 )(2) (2 ) 1 2

N c c

c cN N

ϕ ϕ

ϕ ϕ

− − = − − +

+ + − − = + −

Probing the Quasi-Equilibrium EDL by harmonic disturbances (Linearized problem – Impedance Spectroscopy, Next order

term – Rectification affect)

2, ;

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

t x xx

i t

c j c c

j t c t Nt V e t

c t c t

ω

ε ϕ

ϕ α ϕ

± ± − +

− +

+ −

= = −

= = >>= − + == =

( ) ( )( ) ( )

( ) ( )

2 2 20 22 22 20

2 2 222 22 20

2 2 222 22 20

;

;

;

i t i t i t i t

i t i t i t i t

i t i t i t i t

c c C e C e C e C e C

e e e e

I I Ie Ie I e I e I

ω ω ω ω

ω ω ω ω

ω ω ω ω

α α

ϕ ϕ α α

α α

± ± ± ± − ± ± − ±

− −

− −

= + + + + +

= + Φ + Φ + Φ + Φ + Φ

= + + + + +

Probing the Quasi-Equilibrium EDL by harmonic disturbances, Anomalous rectification effect

Voltage, V

Cur

rent

den

sity

, I

I

II

III

I lim

4/3 4/3 2/3[ ] [ ]t xω ε ε ε−= = =

Senda M, 1955; Sparnaay MJ, 1957; Tachi I, 1955;I.Rubinstein, I.Rubinstein, E.Staude, 1985

Probing the Quasi-Equilibrium EDL by harmonic disturbances

( )20 20 0 20 20 0 2Rex x x xI C c C Cϕ+ + + += + Φ + + ΦQE-EDL Increase of the counter-ion concentration

yields decrease of the field ( decreases)00

|xx

c ++ =

=

Extended Space Charge regime, no co-ions, increase of the outer counter-ion concentration increases the charge and yields positive fields response