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  • Electric Double Layer and Concentration Polarization

    Boris Zaltzman Ben-Gurion University of the Negev

    Israel

    Isaak Rubinstein

  • Conduction from an electrolyte into a charge-selective solid (ion exchange membrane 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

    Prototypical experiment, I. Rubinstein 70-th

    MEMBRANE

    C0

    y

    C 0

    δ

    δ

    Measuring Voltage-Current Dependence

    Current power spectra

  • Classical picture of concentration polarization membrane: 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 =→=∞→

    −==+−=

    −===

    =+=−

  • 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

    MEMBRANE

    C0

    y

    C 0

    δ

    δ

    Electro-Convection

    No Gravitational Convection

    100Ra10 ,200100 ,1.001.0

  • 5212 2/1

    0

    2/1

    2 0

    1

    2

    1010 , )(2

    )(

    ,Sc ,5.0 4

    Pe

    0 , , ,0 :0 0

    Sc 1

    )( Pe ,10)( Pe

    −−

    +

    −−−

    −−+

    +−

    −−−−

    ++++

  • 0 . 0 0

    1 . 0 0

    2 . 0 0

    3 . 0 0

    4 . 0 0

    5 . 0 0 I [ 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 μ I [ m A / c m ] / 0 . 2 m2 μ I [ m A / c m ] / 0 . 3 m2 μ I [ m A / c m ] / 0 . 4 m2 μ I [ m A / c m ] / 1 . 0 m2 μ I [ m A / c m ] / 1 . 0 m2 μ I [ m A / c m ] / 2 . 0 m2 μ I [ m A / c m ] / 2 . 0 m2 μ I [ 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

    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 f d O R dx

    k

    α ϕ α ϕ ϕϕ δ

    α δ

    + +± = − Δ − Δ

    ±Δ = − −

    − − −

  • Modeling Diffuse layer (Gouy – Chapman)

    2

    2 ( ) dd c c dy

    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 c dy dy RT dy

    d dc F dD c z z dy dy RT dy

    ϕ

    ϕ

    + +

    +

    − −

    − + −

      + = 

       

    − = = =   

     

     

    Scaling 1/2

    2 0 0

    , , , = , = d d rF c y dRTc x r

    RT c l l c F ϕϕ ε

    ± ±  = = =  

     

     

    2 2

    20, 0, d dc d d dc d dc c c c dx 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 e dz dz dz

    dc d dj c c c c e dx 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 z ε

    = < < ∞ = = ∞ →

    Boundary conditions at EDL outer edge

    ( ) 2

    2

    2

    2

    ( ) ( ) , ( ) ,

    = =2 sinh( )

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

    c c c c ce c ce d c e e c dz d c dz

    ϕ ϕ ϕ ϕ

    ϕ ϕ ϕ ϕ

    ϕ ϕ

    ϕ ϕ ϕ

    ϕ ϕ ϕ 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 p V V

    dc dc dx dx

    ϕ ϕ

    +

    − −

    = = − = − −

    − =

    1

    1

    ln const. ln ln

    ln , 0x x

    c c p V cV c c p

    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 field Electrosmosis – a liquid flows along a charged surface when an electric field is applied parallel to the surface Sedimentation potential – an electrical potential created by the movement of charged particles through a liquid by gravity Streaming 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 interf een drop betwpotential

    EEu

    Eu

    x

    yyyy

    − −=−=

    =+

    Σ

    ς ϕς

    ϕ ,|

    0 y

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

    SLIP

  • Matched asymptotic expansions (Dukhin 60s – 70s)

    ( ) ) :S-(H 2 1

    2 110 ,

    ),( ),,( ,

    2

    2 22

    xzzzzxzzxzzz

    zzzzz

    uu

    ppwjwiuv

    zxzxcyz

    ϕϕϕϕϕ

    ϕ ε

    ϕϕ ε

    ϕ ε

    −=−=

    ===+=

    = ±

    

    ( ) ( )( ) ( ) ( )

    2 1lnln4ln),()( )(

    11 11ln2)(),(

    )0,(0

    )0,(0

    2/

    22/2/

    22/2/

    )0,(),(

    ))0,(),((

    ς

    ςς

    ςς ϕϕϕϕ

    ϕϕ

    ϕϕ

    ϕςϕς

    ϕϕϕ

    ϕ ϕ

    eccxuxuxV

    eee eeexzxeeccc

    excccc

    excccc

    xx

    zc

    zc

    zz

    xzx zz

    xzx zz

    +−+=∞=−−=

    −−+ −+++=−=−=

    ==−

    ==+

    − −−+−

    −−−−

    −−+++

  • Quasi-Equilibrium Electro-Osmotic Slip.

    1. Impermeable Charged Surface

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

    xuconstc ϕς== ,

    ( ) ( )

    lim

    /2

    1

    (0) 0, 2 1ln ln 4 ln

    2 4ln 2 4ln 2 ln( )

    x

    x x

    Concentration Polarization V c I I

    ec p V const u

    V u c

    ς

    ϕ ϕ

    ς φ

    → ∞  = → = ++ = − =  =

    → ∞  → −∞  = − =

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

    ( ) ( ) 2