THE THEORY OF LIMITING DIFFUSION-MIGRATION CURRENTS IN PARTIALLY DISSOCIATED ELECTROLYTES

           Yu.I.Kharkats, A.V.Sokirko
           The A.N.Frumkin Institute of Electrochemistry,
           Academy of Sciences of the USSR,
          117071, Moscow V-71, Leninsky Prospekt, 31, USSR
          The processes of diffusion-migration  transport  of  ions  in
      solutions of completely dissociated electrolytes have  been  well
      studied [1]. Of interest is to analyze the peculiarities of these
      processes proceeding under the conditions, where the  electrolyte
      is only partially dissociated according to the reaction equation
                                           k
                       z +         |z |-    1
                        1            2
                   n  A     +  n  B       "=='   A  B               (1)
                    1           2                 n  n
                                           k       1  2
                                            2
            z +      |z |-
             1         2
      Here A    and B      are cation and anion, z  and  z  are  charge
                                                  1       2
      numbers, A  B   is neutral molecule, n  and n  are stoichiometric
                n  n                        1      2
                 1  2
      coefficients, k  and k  are  association  and  dissociation  rate
                     1      2
      constants. The stoichiometric coefficients n  and n  coinside with
                                                  1      2
      |z | and z in the  case  when  |z |  and  z are  mutually  simple
        2       1                      2         1
      numbers ( do  not have  any  common divisors ).
          The diffusion transport of  ions  in  systems  with  chemical
      equilibria in the absence of electromigration was analyzed in [2-4].
          The dependence of a limiting diffusion-migration  current  of
                                      1
      cation discharge on the equilibrium constant of partially  disso-
      ciated electrolyte was studied in [5] under an  assumption,  that
      dissociation and recombination rate constants are rather high, so
      that in the whole diffusion layer concentrations of  cations  C ,
                                                                     1
      anions C  and non-dissociated neutral molecules C  are related by
              2                                        3
      the equilibrium condition
                              n    n
                               1    2
                           b C  3 C   = C                          (3)
                              1    2     3
      wrere b  =  k /k   is  equilibrium  constant.  One  of  the  most
                   1  2
      interesting results of [5] is the fact, that for sufficienly high
      values of non-dissociated  substance  diffusion  coefficient  the
      limiting current can be several times higher that the diffusion -
      migration current in fully dissociated electolyte.
          This paper presents the calculation of a limiting current  of
      cation discharge in partially  dissociated electrolyte in a  more
      rigorous formulation without a supposition of equilibrium  condi-
      tion (2). Presented analysis is based on Nernst  diffusion  layer
      model [1], which is widely used in electrochemical  macrokinetics
      and accounts in implicit form the convective transfer of ions. De-
      veloped theory can also be applied to the systems  with  membrane
      covered electrodes.
      1. Statement of the problem and the general solution.
                                                   z +
                                                    1
          Let us consider discharge  of  cations  A     reducing  in  a
      steady  state  conditions  to  neutral  species  which  does  not
                                      2
      interact with any other substances in the solution. We shall also
                           z -
                            2
      suppose that anions B    are  electrochemically  inert  and  that
      direct discharge of molecules A  B   is  impossible  in  all  the
                                     n  n
                                      1  2
      potential region.
          The system of electrodiffusion equations describing distribu-
      tion of component concentrations C , C , C  and electric potenci-
                                        1   2   3
      al in the diffusion layer close to the electrode can be written as
                 dC           dCd            dJ      i
                   1            3
                -----        -----          ----   -----
             D        + n D        + z D C       =      ,          (3)
              1          1 3          1 1 1
                 dx           dx             dx     z F
                                                     1
                 dC           dCd            dJ
                   2            3
                -----        -----          ----
             D        + n D        + z D C       = 0,              (4)
              2          2 3          2 2 2
                 dx           dx             dx
                  2
                 d C
                    1                n    n
                -----                 1    2
             D        = k  ( C  - b C  3 C   ),                    (5)
              3    2     2    3      1    2
                 dx
             z  C  = |z | C .                                      (6)
              1  1     2   2
      Here D , D ,  D   are  diffusion  coefficients  of  corresponding
            1   2    3
      components, J=Ff/RT is the dimensionless potential, i is the  ca-
      tion discharge current density, x is  coordinate,  the  remaining
      designations are generally accepted.
          The first and the  third  terms  in  equations  (3)  and  (4)
      describe diffusion and migration fluxes of  cations  and  anions.
      The second terms in these equations correspond to the transfer of
                                      3
                 z +      |z |-
                  1         2
      substaces A    and B      due to diffusion  of  A  B  .  Equation
                                                       n  n
                                                        1  2
      (5) descri- bes diffusion  transport  of  A  B    molecules  with
                                                 n  n
                                                  1  2
      account of reac- tion (1). Finally the equation (6) presents  the
      condition of local electroneutrality in the diffusiuon layer.
          The system of equations (3)-(5)  should  be  supplemented  by
      the boundary coudition
                             dC   |
                               3  |
                            ----- |
                                  |      = 0,                       (7)
                                  |
                             dx   | x=0
      corresponding to the electrochemical inertness of molecules A  B
                                                                   n  n
                                                                    1  2
      and the condition
                            C (0) = 0,                              (8)
                             1
      corresponding to the limiting current of cation discharge. At the
      diffusion layer boundary x = L concentrations C , C , C  are equal
                                                     1   2   3
      to their equilibrium values:
                                 Ü
                       C  (L) = C  (b),    i = 1,2,3.               (9)
                        i        i
                                                   Ü   Ü      Ü
          The values of equilibrium concentration C , C  and C  can  be
                                                   1   2      3
                                           Ü
      related with the total concenration C  of substance A  B    in  a
                                                           n  n
                                                            1  2
      solution and with the equilibrium constant b by equations
                           Ü n     Ü n     Ü
                              1       2
                       b (C )  3 (C )   = C ,                      (10)
                           1       2       3
                           Ü         Ü
                       z  C  = |z | C ,                            (11)
                        1  1     2   2
                        Ü       Ü       Ü
                       C  + n  C  = n  C .                         (12)
                        1    1  3    1
      Combining equations (10)-(12),  one  obtains  the  equation  that
                                      4
                      Ü
      determines the C  (b) dependence
                      1
                        Ü         Ü m            n       Ü
                                                  2
                       C  + n b (C ) ( z  /|z | )   = n C ,         (13)
                        1    1    1     1    2         1
      where m = n  + n   is  the  formal  order  of  the  recombination
                 1    2
      reaction. Substituting the solution of equation  (13)  into  (11)
      and (10), one determines the equilibrium concentrations  appeared
      in (9).
          The calculations in [5], based on the solution of a system of
      equations (2)-(4), (6) with boundary conditions (7)-(9),  corres-
      pond to the  limiting  case,  when  the  dimensionless  parameter
                2
      d = D /k L  tends  to  zero,  so  that  equation  (5)  should  be
           3  2
      replaced by (2) for any 0 < x < L.
          It follows from the local electroneutrality condition (6) and
      from equations (3), (4), that
           dC          z1 *     D n    D n  *  dCd        i
             1                   3 1    3 2      3
          ----- |     ----|  |  ----   ---- | -----     -----
                 1 +       +         +               =        ,   (14)
                |         |  |              |
           dx   7     |z |8  7   D      D   8  dx       z FD
                        2         1      2               1  1
          After passing to dimensionless variables:
                                                 Ü
                  x = x / L,          c  = C  / C ,                (15)
                                       i    i
      equations (5), (14) and boundary conditions (7) - (9) are  writen
      in the form:
                       dc      dc
                         1       3
                      -----   -----
                    g       +       = j,                           (16)
                       dx      dx
                        2
                       d c
                          3            m
                      ------        _
                    d        = c  - b3c ,                          (17)
                         2      3      1
                       dx
                      Ü    Ü                  Ü    Ü
             c (1) = C  / C  = k,    c (1) = C  / C  = l,          (18)
              1       1               3       3
                                      5
                                      dc   |
                                        3  |
                                     ----- |
             c (0) = 0,                    |     = 0,              (19)
              1                            |
                                      dx   | x=0
      wrehe the following designations for combinations  of  parameters
      are introduced:
                          z1  *       D n    D n  *
                                 /     3 1    3 2
                   |     ---- |   |    ----   --- |
               g =  1 +         /           +      ,
                   |          |   |               |
                   7     |z | 8/  7    D      D   8
                           2            1      2
                                 D n      D n  *
                     iL      /    3 1      3 2
                   --------   | ------   ------|
               g =          /          +        ,
                         0    |                |
                   z FD C  /  7   D        D   8
                    1  1           1        2
                         m-1            n
               _       Ü                 2
               b = b (C )    ( z /|z | )  .
                                1   2
      Integrating (16), one obtains
                        g c  + c  = jx + b.                       (20)
                           1    3
      Using conditions (19), one concludes that c (0) =  b,  and  using
                                                 3
      conditions (18), one has
                                        Ü
                               j + b = j ,                        (21)
      where
                                Ü
                               j  = gk + l.
                Ü
      Quantity j  is the expression for a dimensionless current in  the
      case d = 0, i.e. under the conditions of equilibrium of the  dis-
      sociation-recombination reaction. Indeed, letting d = 0  in  (17)
                                 Ü
      and (19), one obtains j = j . Quantity b can be  treated  in  two
      ways simultaneously: as a  dimensionless  concentration  of  non-
      dissociated substance near the electrode and as a  correction  to
                             Ü
                                                                    -
      dimensionless current j  for small d. As it was shown above, b ~0
                                      6
           -
      at d  ~ 0.
          The system of equations (16) - (17) with  boundary conditions
      (18)-(19) is nonlinear ( c  enters (17) in power of m > 2 )  and,
                                1
      hence, it does not have any general  analytical  solution.  Below
      will be given analytical solutions for the limiting cases of  re-
      combination reaction d = 0, as well as for the cases of high  and
      low recombination-dissociation reaction rates ( d , 1 and d . 1).
      The results of numerical solution of a system will be  given  for
      the intermediate region of d values of the order of unity.
      2. The case of the recombination-dissociation reaction equilibrium
                                        Ü
          As mentioned above, quantity j  is the expression for  dimen-
      sionless limiting current in case of the  dissociation-recombina-
      tion reaction equilibrium d = 0, that was analyzed in paper  [5].
      The corresponding limiting current in dimensionless units can  be
      written as:
                                                            Ü
           z FD           z           D n    D n           Cd
            1  1  #        1  * Ü      3 1    3 2 *   Ü     1  *$
          -------        ----         ----   ----         ----
      i =         ? |1 +      |C  + |       +     || C  -      |?.
                                1
             L    3 7    |z | 8     7  D      D   87       n   84   (22)
                           2            1      2            1
                                                     _             Ü _
      In the expression of (22) current i depends on b  via  the  C (b)
                                                                   1
      dependence only.
                 _
          The i (b) dependence, determined by formulae (13),  (22),  is
                                                           _
      shown in Fig.1. For a high dissociation degree, when b ,  1,  the
                                                  Ü
      dimensionless concentration of cations is  c   ~  |z |,  and  the
                                                  1       2
      current tends to the value
                                      7
                                                Ü
                       i = ( z  + |z | ) z F D C  / L,
                              1     2     1   1
      that coincides with the value of i in a binary solution of  fully
                                                                   _
      dissociated electrolyte. For a low dissociation degree, when b.1,
                                                                Ü
      the dimensionless concentration of electroactive ions is c  ,  1,
                                                                1
      and the limiting current tends to the value
                                   Ü
                                                   *
                          z F D D C     n      n
                         --1---1 3--- |- 1      2  |
                     i =                     +     |.
                               ___    | ____   ____
                              L       7  D      D  8
                                          1      2
          One should mention some important fact.  Namely,  though  the
      concentration of dischaging cations in a solution  decreses  with
      lowering the dissociation degree, the limiting current  tends  to
      the asymptotic constant value,  that  depends  on  the  ratio  of
      diffusion coefficient D , D , D  of components and parameters  n
                             1   2   3                                1
                                              _ -
      and n . In this case the value of i for b  ~ 7 can be either hig-
           2
                                       _ -
      her, or lower than its value for b  ~ 0.
          In the simpliest case, when all  diffusion  coefficients  are
      equal ( D  = D  = D  ), dimensionless limiting current  is  equal
               1    2    3
                                  _
      to 1 and does not depend on b.
                                                            _
          A physical explanation for such a behavior  of  i(b)  is  the
      fact, that electroactive cations are transferred in  a  diffusion
      layer both via their diffusion and migration (  the  distribution
                                                 _
      of concentrations c (x) for some values of b is shown in Fig. 2 )
                         1
      and due to diffusion transfer to the  electrode  with  subsequent
      dissociation of neutral molecules A  B  . ( The  distribution  of
                                         n  n
                                          1  2
      concentration c (x) is shown in Fig.3.) The rate  of  the  latter
                     3
                                      8
      mechanism is proportional to  the  diffusion  coefficient  D   of
                                                                  3
      neutral molecules, and the corresponding  contribution  into  the
      limiting current is given by the second term in formula (22).
          In case of D  = D  = D  the decrease of the  contribution  of
                      1    2    3
                                       z +
                                        1
      diffusion-migration transfer of A    cations, caused by  decreas-
           Ü              _
      ing c  with growing b, is fully compensated by  diffusion  supply
           1
      of dissociating neutral A  B   molecules to the electrode,  which
                               n  n
                                1  2
                                                            _
      just provides the independence of limiting current on b.
      3. Analytical solution for the case of high reaction rates d , 1.
          Since equation (17) contains a small parameter  at  a  higher
      order derivative and is nonlinear, we shall replace the dependent
      and independent variables in this equation in such a manner, that
      all terms of equation (17) be of  the  same  order  of  magnitude
                       #_+                                #_+
      [6-8]. Let y = x/ed . As follows from (20), for x   ed   the  sum
                                                         #_+
      of concentrations gc  + c  is also of the order of ed . We  shall
                          1    3
      seek the solution of equation (17) in the form of
                    m/2                 1/2
              c  = d    W(y),     c  = d    U(y),                 (23)
               3                   1
      where W(y) and U(y) are functions of the order of unity. It  fol-
                                                    m/2
      lows from conditions (19) and (20), that b   d   . Neglecting  in
                                                 m/2
      relation (20) the terms of the order  of  d     one  obtains  the
      approximate expression for function U:
                           U(y) = j y / g.                        (24)
      Substituting (23) and (24) into (17), one  obtains  the  equation
                                      9
      for function W :
                        2                  m
                       d W            j y *
                      -----       _  -----
                            = W - b |     |                       (25)
                         2
                       dy           7  g  8
      with boundary conditions
                       dW  |                          -m/2
                      ----
                           |   =0,         W (0) = b d    .       (26)
                       dy  |y=0
      The general solution W(y) of homogeneous equation (25) is
                     W = s  exp (-y) + s  exp (y).                (27)
                          1             2
      The particular solution W of a non-homogeneous  equation  can  be
      found by the method of constants variation. Summing up the  gene-
      ral solution of homogeneous equation with the particular  soluti-
      on of a non-homogeneous equation, one obtains the general soluti-
      on of a non-homogeneous equation, that satisfies the  W'(0)  =  0
      condition, in the form
                                7                y
                           #  y       -y m    -y       y m      $
              _                 i                i
      W(y) = (b/2) (j/g) m ? e    dt e  t  + e     dt e t  + m! ?, (28)
                                j                j
                           3                                    4
                                y                0
      where m! is the factorial function. Using (28) and the second co-
                                                          m
                                               _    #_+
      ndition in (26), one finds the value b = b  (jed /g) m!.  Substi-
      tuting the latter one into (21), one obtains the equation for j:
                      Ü    m/2          m
                               _
                 j = j  - d    b ( j/g )  m!                      (29)
                                                             Ü
      One may neglect the small difference between  j  and  j   in  the
      right-hand side of (28) and write down the approximate  exression
      for dimensionless current in the form:
                      Ü    m/2      Ü    m
                               _
                 j = j  - d    b ( j /g )  m!                     (30)
                                     10
                                            #_+
          Thus, for low values of parameter ed , i.e. for high dissoci-
      ation rates the limiting  diffusion-migration  current  decreases
                         (n +n )
                           1  2
      proportionally to d       .
      4.  The case of low reaction rates d . 1
          In this case the solution can be sought in the form of expan-
                                         -1
      sion in powers of small parameter d  :
                                     -1
                           c  = X + d  Y,                         (31)
                            1
      where X, Y are functions of the order of utity. Substituting this
      expansion into (17), taking into account (16)  and  equating  the
      terms at d, one gets
                               2
                              d X
                             -----
                                   = 0.
                                2                                 (32)
                              dx
      After satisfying the boundary conditions (18), (19)  one  obtains
      from the last equation the major part of solution for c :
                                                             1
                              X = k x.                            (33)
      To find Y, we equate the terms not containing  d  and,  substitu-
      ting (33), we obtain
            2
           d Y         m
          ----   _
        g      = b (kx)  - ( jx + k +l - j - gkx ).              (34)
           dx
      Relations (20) and (21) were also taken into account in  deriving
      (34). Function Y satisfies homogeneous boundary conditions:
                     Y (0) = 0,      Y (1) = 0.                   (35)
      Integrating (34) with using (35), one obtains:
                                     11
                  m  m+2               3               2    $
             # _
           1   b k (x   - x)   (gk-j)(x -x)   (j-k-l)(x -x) ?
          -- | ------------- - ------------   -------------
      Y =    ?                              +               ?.
                                                                  (36)
           g ? ( m+2 )( m+1 )       6              2        |
             3                                              4
      This expression and the j= dc /dx|    condition give rise to  the
                                   1   |x=0
      expression for a flux in case of low dissociation rates ( d . 1):
                                                         m
                                                        _
                     1  #    1     g *     1         b k      $
                    ---     --- - ---     ---   -------------
            j = k +     ? |          |k -     +               ?.  (37)
                    d g 3 7  2     6 8     2     (m+2) (m+1)  4
                _
          The j(b) dependence, given by (37), is mainly  determined  by
      the first term and represents a monotoneously decreasing function.
      5. The numerical solution.
          The system of equations (16)-(19) has also been solved  nume-
      rically for some intermediate d values by using  the  Runge-Kutta
      method and the optimization procedure  of  searching  for  the  j
      value satisfying the boundary conditions.
                                 _
          Fig. 4 shows the j (lg b) depedences, calculated by numerical
      solution of the problem for some values of parameter d.  As  fol-
      lows  from  numerical  calculations  and  from  the  results   of
      approximate analytical solution of the problem, the limitig  cur-
      rent of reduction of cations decreaes as parameter d grows.
      6.  Conclusion.
          The above investigation shows  that  limiting  current  in  a
      partially dissociated binary electrolyte depends, first,  on  the
      electrolyte dissociation rate constant and, second, on the  equi-
      librium constant. Analytical expressions (30) and (37) for  limi-
                                     12
      ting current, obtained for the cases of high and low  (  d  ,  1,
      d . 1 ) electrolyte dissociation rate constans, allow to determine
      equilibrium constatn b from experimental values of i and k .  In
                                                                2
      case of intermediate d values constant b  can  be  determined  by
                                            _
      using the family  of  curves  j  (lg  b)  obtained  by  numerical
                                        -
      solution of the problem. In the d  ~ 0 limit the calculated j (b)
      dependence transforms into formula for j obtained in [5]. For low
      values of dissociation rate constant ( d.1 ) the value  of  limi-
      ting diffusion-migration current  is  mainly  determined  by  the
      value of equilibrium concentration of  electroactive  cations  in
      the solution.
                                                                  Ü
          Note in conclusion, that by changing the concentration C  in
                                                          _
      the solution, one can vary the value  of  parametr  b,  which  is
                        Ü m-1
      proportional to (C )   , whereas the value of parametr d does not
                 Ü
      depend on C . This allows, in principle, to find the dissociation
      rate constant k  and the reverse  recombinanation  reaction  rate
                     1
      constant k  from the comparison of the experimental dependence of
                2
                                          Ü                         _
      limiting currrent on concentration C   and  calculated  j  (lnb)
      curves for different values of d.
                                     13
                       References.
      1.  Newman J. Electrochemical systems. New Jersey; Prentice-Hall,
          1973.
                                                       "
      2.  Fetter K. Elektrochemische Kinetik.  Berlin-Gottingen-Heidel-
          berg: Springer-Verlag, 1961.
      3.  Galvell J.R. J.Electrochem. Soc., 1976, V.123, P.464.
      4.  Galvell J.R. Corros. Sci., 1981, V.21, P.551.
      5.  Kharkats Yu.I. Elektrokhimiya, 1988, V.24, P.539.
      6.  Nayfeh A.H. Intoduction to Perturbation Techniques.-
          New York: Wiley Interscience, 1981.
      7.  Schlichting  H.   Grenzschicht-Theorie.   Karlsruhe:   Verlaq
          G.Braun, 1965.
      8.  Vorotyntsev M.A. Elektrokhimiya, 1988, V.24, P.1239.
                                     14
                               Figure Captions
                   _
      Fig.1. The i(b) dependence for z  = 2,  |z |  =  1  and  for  the
                                      1         2
             values of parameter D z /D  + D |z |/D :
                                  3 1  2    3  2   1
             1 - 1; 2 - 2; 3 - 3; 4 - 4; 5 - 5.
      Fig.2. The c (x) dependence for z  = 2, |z | = 1 and D  = D  = D
                  1                    1        2           1    2    3
                               _
             and for values of b:
             1 - 0.0343; 2 - 0.2187; 3 - 1; 4 - 6.481; 5 - 225.
      Fig.3. The c (x) dependence for the values of  parameters  listed
                  3
             in caption to Fig.2.
                                                                     _
      Fig.4. The dependence of cation flux to the electrode on  lg  (b)
             for different values of d:
             1 - d = 0.02; 2 - d = 0.1; 3 - d = 1; 4 - d = 10.
                                     15
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