EXTENDED SPACE CHARGE EFFECTS IN CONCENTRATION POLARIZATION Isaak Rubinstein and Boris Zaltzman...
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Transcript of EXTENDED SPACE CHARGE EFFECTS IN CONCENTRATION POLARIZATION Isaak Rubinstein and Boris Zaltzman...
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.