EXTENDED CHARGE ELECTRO-OSMOSIS AND ELECTRO-CONVECTIVE INSTABILITY

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

Voltage-current curve of a C-membrane

Current power spectra

Ilim

F. Maletzki et al 1992

Electrolyte

C-MEMBRANE

y

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

ϕϕ

ϕϕ

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

No Free Surface

No Marangoni Convection

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

−−

−

+

−−−

−−+

+−

−−−−

++++

<<=

==≈⎟⎠⎞

⎜⎝⎛==

=−−====

=∇

∇−∇∆+∆=

−=∆∇−∇∇=∇+

∞<<∞−<<∇+∇∇=∇+

επ

ε

νπη

ϕϕ

ϕϕ

ϕεϕϕ

cFdRT

DDD

DDd

FRT

DLv

ccVpcvyv

pvv

cccccvc

xyccDcvc

y

t

y

t

t

r

r

rr

r

r

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

“BULK” ELECTRO-CONVECTION (NO SLIP)

0 ,0 , :1,00

Sc1

)( Pe)( Pe

10 ,

==−=+==∇

∇∆+∇−∆=

∇−∇∇=∇+∇+∇∇=∇+

<<== +−

vccIccyv

pvv

cccvcccDcvc

yccc

yyyy

t

t

t

r

r

rr

r

r

ϕϕ

ϕϕ

ϕϕ

For low-molecular electrolytes:

1. Conduction - stable,

2. Electric force - stabilizes like gravitation for stable stratification.

0 ,ln ,1)1(2

)( 0000 ==+−= vcyIyc rϕ

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 µ

I [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 ig in a l2

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

I[mA/cm2]

U[V]

Rubinstein, Zaltzman, Pretz, Linder

Theory of Electric Double Layer and Electro-Osmotic Slip

Helmholtz (1879), Guoy-Chapman (1914), Stern (1924)

------------------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)

( ) xzzxzzz

xzzzz

u

ppwjwiuv

zxzxcyz

ϕϕϕ

ϕε

ϕϕε

ϕε

−=

=⇒=⇒=+=

= ±

2

222

21

2110 ,

),( ),,( ,

rrr

( ) ( )( )

( ) ( )

field applied the on depends osmosis charge Induced constant material approach Classical

eccxuxuV

eeeeeexzxeeccc

excccc

excccc

xx

zc

zc

zz

xzxzz

xzxzz

ςς

ϕςϕς

ϕϕϕ

ϕ

ϕ

ς

ςς

ςςϕϕϕϕ

ϕϕ

ϕϕ

−−−

+−+=∞=−−=

−−+−++

+=⇒−=−=

=⇒=−

=⇒=+

−

−−−+−

−−−−

−−+++

;2

1lnln4ln),()( )0(

1111ln2)(),(

)0,(0

)0,(0

2/

22/2/

22/2/

)0,(),(

))0,(),((

Quasi-Equilibrium Electro-Osmotic Slip.

1. Impermable Charged Surface

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

xuconstc ϕς== ,

( ) x

x

uV

euconstVpc

IIcVonPolarizati ionConcentrat

ϕς

ϕϕς

2ln42

1ln4lnln

2 ,0)0( 2/

1

lim

−=⇒−∞→⇒∞→

+=⇒=−=+

=→=⇒∞→

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

( ) ( )2

1lnln4ln),()( 2/ς

ϕς eccxuxu xx+

−+=∞=

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

. 0)2()2( )2( ,(2) ,0)2(

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

20 , ,0)( ,0)(

1

1

2

=−==

=−=−=

<<−==−=+

−−+

−−+

+−−−++

y

y

yyyyyy

ccpc surface:membrane Enriched

ccpcVurface:membrane s Depleted

ycccccc

y

y

yy

ϕϕ

ϕϕ

ϕεϕϕ

C-membrane

Electrolytey

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

BASIC 1D PROBLEM IN TERMS OF PAINLEVÉ EQUATION

0)(

, 221

,1)(21

,

013/23/2

0

2

03

3/13/2

3/13/2

=∞

+=⎟⎠⎞

⎜⎝⎛ +′

+−+=′′

=

−−=

−−

=

−

F

zpIFF

FzzFF

variable - InnerIyz

field Electric FIF

z

y

ε

ε

ϕε

e Chargended Spac1 - ExtezransitionO(1) - Tz

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

0

0

0

>>=<<

−−

, -10

( ) 0lim ,0)0(

0 ,21 2

==

∞<<−=

∞→ zz

xzzxzzz

uu

zu ϕϕϕ

IIVzxuxu x

z 8),(lim)0,(

2

−==∞→

( )2

3

,)(2

|)ln|( 1

3/2

0

0

34

0

VzVFdz

zz0zzF

ncerge Domina Space Cha, Extended - NE-EDLVz

0

≈⇒−≈=

<<−−≈

>>>>

∫ ζ

ε

Extreme Non-Equilibrium Electro-osmosis

Instability of Quiescent Conduction

0

)( Pe)( Pe

,0

Sc1

=∇∇−∇∆+∆=∇−∇∇=∇+∇+∇∇=∇+

=== +−

vpvv

cccvcccDcvc

ccc

t

t

t

r

rr

r

r

ϕϕϕϕ

ε

.0 ,8

,0 :0

0 ,

,1

2 Pe

,10

2Sc1

=−===

=∇∇−∆=

∆+

=∇+

∞<<∞−<<

wccVucy

vpvv

cD

Dcvc

xy

y

yx

t

t

rrr

r

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

yc

vr

Mechanism of Non-equilibrium Electro-osmotic Instability

0

2

8)0,(

=

−=yy

yx

ccVxu

Vortex Fluctuation

Overlimiting conductance

Neutral Stability Curve

)1(Pe8

coth2coshsinhcoshsinh4

21Pe

81

22

+=

−−

=+

DDV

kkkkkkkV

D)(D

0 5 10 15 20 25k

4

6

8

10

12

V

Short-wave singularity: ∞→⇒→ kVV

Full Nonlinear Electro-convection Numerical Solution for ε = 0.01

We need a universal (valid for all regimes), regular, limiting electro-osmotic formulation

0)( ,0|2

,121

),(2)(2)(

00

3

3/20

13/13/1

=∞=⎟⎟⎠

⎞⎜⎜⎝

⎛−

+

++=″

++−++

−=

= szzs

sss

s

Fzz

F

zFFF

Oz

zzFpIz

zF

current limiting the Near

εε

Basic Singular Painleve Solution

),( ,),( 000

0 εζζ zzdzzzF ==∫∞

( )

( )2/1

2/30

1 3)0,max(2ln),0,(),0,(ln

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

IzVptxtxc

cc

UtxtxVUtxuy

xyIx

−−=+

+−−+=

ϕ

ϕϕϕ

Dukhin’s Formula for |ζ|=O(1) - for |ζ|>>O(1)8/2ζ−

[ ] 3/2),0,(),0,( ,1|| txctxc yες ≈>>

Electro-neutral bulk

0 ,

)( Pe

,10 ),( Pe

Sc1 =∇∇−∇∆+∆=

∇−∇∇=∇+

∞<<∞−<<∇+∇∇=∇+

vpvv

cccvc

xyccDcvc

t

t

t

ϕϕ

ϕ

ϕ

( )

( )y

xyIx c

cUtxtxVUtxu

IzVptxtxc

+−−+=

−−=+

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

,3

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

2/30

1

ϕϕ

ϕ

ϕ

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

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

==−−==

=−=+

txwtxtxctxctxtxutxw

txtxctxcptxtxc

yy

yy

ϕϕ

ϕϕ

FLOW DRIVEN BY NON-EQUILIBRIUM ELECTROOSMOSYSUniversal Electro-Osmotic Flow Formulation

∫∞

=−−==0

000 ),,( ),,0,( ),,( ζεϕζεζ dzzzFtxVzz

Comparison of Neutral-Stability Curves in Full and Limiting Formulations

22.5

25

27.5

V

0 0.0025 0.005 0.0075 0.01

ε

1.8

2.6

3.4

kc

Dashed lineV=-4/3lnε+const

Dashed line k=-1/3lnε+const

5103 −⋅=ε

0 2 4 6 8

k

10

20

30

40

50

V

D=1

0.5 1 1.5 2 2.5 3 3.5

0.5

Full Nonlinear Electro-Osmotic Convection Numerical Solution for ε = 10-4