Post on 19-Mar-2020
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