J. Phys. Chem. B 2009, 113, 3105–3111
3105
Ion Transport Models for Electroanalytical Simulation. 1. Theoretical Comparison S. Van Damme,* N. Smets, D. De Wilde, G. Weyns, and J. Deconinck Research Group Electrochemical and Surface Engineering, Vrije UniVersiteit Brussel (VUB), Pleinlaan 2, 1050 Brussel, Belgium ReceiVed: NoVember 11, 2008; ReVised Manuscript ReceiVed: January 7, 2009
Ion transport models are compared by computing the limiting current density of an electrodeposition on a rotating disk electrode for various hypothetical electrolytes. The first ion transport model is the pseudoideal solution model, on which many commercial electroanalytical simulation tools are built. The second, more rigorous model consists of the linear phenomenological equations for which the activity coefficients and Onsager coefficients are calculated locally with the mean spherical approximation (MSA). Introduction Ion transport in electrolyte solutions is well described by the linear phenomenological equations from irreversible thermodynamics.1,2 Unfortunately they are difficult to employ in numerical simulation as they contain a large number of coefficientssthe activity coefficients and the Onsager coefficientsswhich depend in a complicated manner on the local chemical composition. It takes considerable effort to determine them all experimentally at a sufficient number of compositions, particularly in multiion solutions. However, at infinite dilution the solution becomes ideal and all activity coefficients tend to unity and all offdiagonal Onsager coefficients tend to zero, so that the only unknown coefficients are the limiting diffusion constants, D0i . The pseudoideal solution model, employed in many commercial electroanalytical simulation tools, is based on this observation.3 To compensate for errors at finite concentrations, the limiting diffusion constants, D0i , are often replaced with effective diffusion constants, Di (also called integral diffusion constant). These effective diffusion constants should then be chosen to match the experimental bulk conductivity, κ. This can be accomplished easily by choosing
Di ) Di0
κ κ0
(1)
different results. Since the MSA involves more computations, it is important to know if it is worth the extra effort. In a second paper we will compare both models with experimental results for copper electrodeposition. General Theory According to irreversible thermodynamics the diffusive flux of a species depends linearly on the thermodynamic forces. Under isothermal-isobaric conditions and in the absence of magnetic fields these forces are related to variations in the chemical composition and the electrical state. The molar diffusive flux of a species i in the mass-fixed reference frame is then given by8
b Ji ) -
I
L
j)1
Lij ) Lji
I
F2 κ ) z 2c D0 RT i)1 i i i
∑
(2)
so that the potential drop over the electrolyte solution in the absence of concentration gradients is still correct. As an alternative to the pseudoideal solution model, the mean spherical approximation (MSA) provides analytical equations to estimate the activity coefficients and the Onsager coefficients, based on the ion diameters.4-7 In this first paper we compare the pseudoideal solution model with the MSA on a theoretical basis to determine under which conditions and to what extent the two models actually predict * To whom correspondence
[email protected].
should
be
addressed.
E-mail:
m
) (
+ zj -
) ]
mj bφ z F∇ m0 0 (i ) 1, 2, ..., I)
(3)
where µj, mj, and zj are respectively the molar chemical potential, the molar mass, and the charge number of species j, φ is the electrostatic potential, T is the absolute temperature, F is the Faraday constant, and Lij is the Onsager coefficient related to species i and j. The Onsager coefficients satisfy the Onsager reciprocal relations,
where κ0 is the ideal conductivity,
0
[(
∑ Tij ∇b µj - m0j µ0
(i, j ) 1, 2, ..., I)
(4)
The zeroth index is reserved for a reference species, usually a neutral solvent in which case z0 ) 0. Choosing the molar concentrations of the solutes, ci, and the electrostatic potential as the independent variables, the diffusive flux can be rewritten as
b Ji ) -
I
∑ Dij∇bcj - ziFuici∇bφ
(i ) 1, 2, ..., I)
(5)
j)1
where the diffusion coefficients are given by8 I
Dij )
(
mk ∂µ0 Lik ∂µk T ∂cj m0 ∂cj k)1
∑
)
10.1021/jp809947q CCC: $40.75 2009 American Chemical Society Published on Web 02/18/2009
(i, j ) 1, 2, ..., I) (6)
3106 J. Phys. Chem. B, Vol. 113, No. 10, 2009
Van Damme et al.
and the mobilities by
bc b J i ) -Di∇ i
ziFDi b c ∇φ RT i
(i ) 1, 2, ..., I) (15)
I
ui )
∑
1 zL ziciT j)1 j ij
(i ) 1, 2, ..., I)
(7)
Regardless of which model is chosen, the conductivity can always be expressed in terms of the mobilities as
With the Gibbs-Duhem equation,
I
κ ) F2 I
(8)
i)0
and the definition of the molar chemical potential,
(i ) 1, 2, ..., I)
(9)
where µ0i is the standard molar chemical potential and R is the ideal gas constant, equation 6 becomes
Dij ) R
[
(
)]
I I Lij mk ∂ ln yk ∂ ln yl +R Lik + 1+ cl cj ∂c m c ∂cj j 0 0 k)1 l)1 (i, j ) 1, 2, ..., I) (10)
∑
∑
The activity coefficients from theories where the solvent is regarded as a continuum are calculated at constant solvent chemical potential and should be converted to the atmospheric pressure condition. This so-called McMillan-Mayer to LewisRandall conversion is to a very good approximation given by9 I
ln yi ) ln yiMM - ViΦMM
∑ cj
(11)
Clearly, equation 15 cannot capture the full complexity of equation 5, not even with effective diffusion constants like eq 1, because (1) the coefficients depend on the local concentrations, (2) there is a diffusion term in each concentration gradient, and (3) there is no relation between the diffusion coefficients and the mobilities at finite concentrations. It is therefore worthwhile to investigate under which conditions and to what extent the pseudoideal solution model deviates from the complete linear phenomenological equations where the activity coefficients and the Onsager coefficients are calculated with the MSA. The Mean Spherical Approximation. In the MSA the ions are regarded as charged hard spheres with a certain diameter σi. The solvent is regarded as a continuum with relative permittivity εr and dynamic viscosity η. The central quantity of the MSA is the screening parameter which is defined by the implicit equation4,10
where Vi is the partial molar volume of species i and ΦMM is the McMillan-Mayer osmotic coefficient. Furthermore, the Onsager coefficients are calculated in the solvent-fixed reference frame and should be converted to the mass-fixed reference frame via8
ci I cj I cicj I mkLkjs mkLiks + 2 F k)1 F k)1 F k)1
∑
∑
l)1
Dij ) δijDi0
(i, j ) 1, 2, ..., I)
(13)
where δij is the Kronecker symbol, and the Nernst-Einstein relation holds for the mobilities,
ui )
Di0 RT
(i ) 1, 2, ..., I)
(14)
Replacing D0i with Di and plugging these results into eq 5 gives the flux in the pseudoideal solution model,
zi - Ψσi2 1 + Γσi
)
2
(17)
where
LB )
e2 4πε0εrkBT
(18)
I
Ψ)
ziciσi π 2∆Ω i)1 1 + Γσi
∑
Ω)1+
(12) At infinite dilution the diffusion coefficients simplify to
∑ ci i)1
I
∑ ∑ mkmlLkls
(
I
4Γ2 ) 4πLB
j)1
Lij ) Lijs -
(16)
i)1
∑ cidµi ) 0
µi ) µi0 + RT ln(yici)
∑ zi2ciui
I ciσi3 π 2∆ i)1 1 + Γσi
∑
∆ ) 1 - X3
(19)
(20)
(21)
I
Xp )
∑
π c σp 6 i)1 i i
(22)
The quantity 2Γ generalizes the Debye screening parameter κD. Because the MSA accounts for steric hindrance in the neutralizing ion cloud, 2Γ will be smaller than κD. The activity coefficients and the Onsager coefficients in the MSA have been reported in the literature recently.4-7 For the computation of the diffusion coefficients the derivatives of the activity coefficients are needed. After some tedious but straightforward algebra we find that
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J. Phys. Chem. B, Vol. 113, No. 10, 2009 3107
MM MM ∂ ln yMM ∂ ln yk,E ∂ ln yk,HS k ) + ∂cj ∂cj ∂cj
(23)
[]
[
where
[
MM ∂ ln yk,E RkRj ) -πLB + LB ∂cj ∆Ω
βkβj
(
B2 2Γ + LB 2A ∆Ω
)
]
-
A)π
(24) F3j )
[
∑ ciσi (1 + Γσ )3 i)1
6X22 X33
with
(zi - Ψσi2)2
]
3X22 /X32 6X22 /X3 3X1 - 3X22 /X32 + + ∆ ∆2 ∆3
[
[ ]
]
ln ∆
3X22 /X32 3X2 πσj 1 π + + + ∆ 6 ∆ ∆2 6
[]
3X1 -
I
[
6X2 6X2 /X3 πσj2 3 πσj F2j ) + 2 ln ∆ + + ∆ 6 6 ∆2 X3
3X22 /X32 2
+
∆
6X22 /X3 3
-
∆
6X22 X33
πσj3 6
(32)
]
πσj2 ln ∆ + 6
6X1X2 - 4X23 /X32 5X23 /X33 X0 + X23 /X33 + + + ∆ ∆2 ∆3 6X23 /X3
(25)
i
∆4
+
6X23 X34
]
ln ∆
πσj3 6
(33)
Results and Discussion I
B)π
∑
ciσi2
i)1
(
Ri ) σi
zi - Ψσi2
(26)
(1 + Γσi)2
Ψσi2 zi - Ψσi2 + 1 + Γσi 3
(
zi - Ψσi2 βi ) 1 + Γσi
)
2
-
)
BRi ∆Ω
(27)
(28)
The limiting current density reflects the rate of an electrochemical reaction which is dominated by the transport of the reacting species toward the electrode and is therefore a suitable quantity to compare the ion transport models. The theoretical comparison is based on the simulated limiting current densities of the electrodeposition of a hypothetical reacting species A from various binary (AC) and ternary (AC + BC) electrolyte solutions onto a rotating disk electrode. The rotating disk electrode is chosen, because there exists an analytical expression for the flow field b V in this setup and the concentration and electric potential variations are one-dimensional.3 The rotation speed is set arbitrarily to 1000 rpm. The stationary material balance equations,
b · (c b b ∇ i V + J i) ) 0
(34)
and the electroneutrality condition, and
I
∑ zici ) 0
(35)
i)1
MM ∂ ln yk,HS ) F0j + F1jσk + F2jσk2 + F3jσk3 ∂cj
(29)
with
F0j )
F1j )
3 1 πσj ∆ 6
[]
[ ]
2 3X2 πσj3 3 πσj + ∆ 6 ∆2 6
[]
(30)
(31)
are solved numerically for both ion transport models with the finite element method with upwind contributions for the convection term.11 The domain of calculation for the simulations is located within the first 100 µm from the electrode, as shown schematically in Figure 1. The grid is made up of 801 points, defining 800 elements exponentially contracting toward the electrode. The refinement factor is set to 1.011, which corresponds to a ratio of 1:10000 between the size of the first and the last element. At the limiting current density, the concentration of the reacting ion becomes zero on the electrode. On the solution side bulk concentrations and zero potential are imposed. The set of nonlinear equations is solved iteratively with Newton’s method. The diffusion coefficients and mobilities in the MSA are calculated locally at each node. The conversion of the activity coefficients to the Lewis-Randall reference frame is usually
3108 J. Phys. Chem. B, Vol. 113, No. 10, 2009
Van Damme et al.
Figure 1. Domain of calculation.
TABLE 1: Minimum and Maximum Values of the Parameters
minimum maximum a
ct (mol/m3)
R
Di0 (10-9 m2)/s)a
Di0 (10-9m2/s)b
σi (Å)a
σi (Å)b
mi (kg/mol)
10 1000
0.01 10
1 2
0.7 1.1
2.5 5.5
3.5 6.5
0.04 0.20
Figure 2. Histogram of the relative difference of the limiting current density ∆j for 1-1 electrolytes: ct ) 10 mol/m3 and DC0 ) 110-9 m2/s (black), ct ) 10 mol/m3 and DC0 ) 210-9 m2/s (white), ct ) 1000 mol/ m3 and DC0 ) 110-9 m2/s (dark-gray), ct ) 1000 mol/m3 and DC0 ) 210 × 10-9 m2/s (light-gray).
For monovalent ions. b For divalent ions.
not more than a few percent for total ion concentrations up to 1000 mol/m3 and will be omitted in this work.9 For the reference frame correction the density of pure water at 298.15 K is taken. The charge numbers of the ions are restricted to 1 or 2 so as to be representative for nonassociating electrolytes. This means that only 1-1, 1-2, and 2-1 binary electrolyte solutions and 1-1-1, 1-1-2, 1-2-1, and 2-1-1 ternary electrolyte solutions are considered. The reason for considering both binary and ternary electrolyte solutions is to vary the relative contribution of diffusion and migration to the flux of the reacting ion A. The physical or practical restrictions to the remaining parameters, that is, the bulk concentrations, the limiting diffusion constants, the ion diameters, and the molar masses, are summarized in Table 1. The extreme values of the total ion concentration ct are chosen so as to ensure deviation from ideal behavior and to stay well within the validity range of the MSA. The ratio R ) cAC/cAB is varied from an excess inert electrolyte at 0.01 to a maximum value of 10 imposed by numerical stability. At higher values, the rise in the concentration of B to compensate for the depletion of A becomes too excessive. The ranges of D0i , σi, and Mi are chosen to encompass common ions, like alkali metal ions, alkali earth metal ions, Ag+, Cu2+, Zn2+, halides, and oxyanions. The exceptionally mobile H+ and OH- are excluded. It should be remembered that in the MSA σi and mi correspond to the hydrated ion. To screen for important parameters the limiting current density is calculated with the pseudoideal solution model and the MSA for the full factorial design, that is, for each combination of the minimum and the maximum level of every parameter. The “experimental” bulk conductivity to compute the effective diffusion constants for the pseudoideal solution model is taken equal to the MSA conductivity. The binary electrolytes have seven parameters (ct, R, DA0 , D0C, σA, σC, mA, and mC), so 27 combinations are made, while the ternary electrolytes have 11 parameters (ct, R, DA0 , DB0 , D0C, σA, σB, σC, mA, mB, and mC), meaning 211 combinations. In reality D0i , σi, and mi are slightly correlated and can assume only distinct values. However, since the interest is in which parameters mostly affect the difference between the pseudoideal solution model and the MSA, the parameters are treated as independent and continuous. For each electrolyte a histogram of the relative
Figure 3. Histogram of the relative difference of the limiting current density ∆j for 1-2 electrolytes: ct ) 10 mol/m3 and DC0 ) 0.710-9 m2/s (black), ct ) 10 mol/m3 and DC0 ) 1.110 × 10-9 m2/s (white), ct ) 1000 mol/m3 and DC0 ) 0.710 × 10-9 m2/s (dark-gray), ct ) 1000 mol/m3 and DC0 ) 1.110 × 10-9 m2/s (light-gray).
deviation of the limiting current density predicted by the pseudoideal solution model with respect to the limiting current density predicted by the MSA,
∆j )
jI - jMSA jMSA
(36)
will provide an overall picture of the difference between both ion transport models. Next, an analysis of the variation of ∆j with each parameter will indicate the determining variables. The average absolute shift in ∆j will be computed for each parameter p,
∆p )
1 P-1
2
2P-1
min ∑ |∆jmax p,e - ∆jp,e |
(37)
e)1
where P is the number of parameters. There are 2P-1 electrolytes for which p is at its maximum level, and an equal amount of min electrolytes for which p is at its minimum level. ∆jmax p, e and ∆jp, e are the value of ∆j for electrolyte e, where p is at its maximum and minimum value, respectively. If ∆p is large it means that, on average over the remaining P - 1 variables, variable p has a strong influence on ∆j. Binary Electrolyte Solutions. The histograms of the binary electrolyte solutions, shown in Figures 2, 3, and 4, immediately reveal that the pseudoideal solution model always underestimates the limiting current density when compared to the MSA. The
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J. Phys. Chem. B, Vol. 113, No. 10, 2009 3109
Figure 4. Histogram of the relative difference of the limiting current density ∆j for 2-1 electrolytes: ct ) 10 mol/m3 and DC0 ) 110 × 10-9 m2/s (black), ct ) 10 mol/m3 and DC0 ) 210 × 10-9 m2/s (white), ct ) 1000 mol/m3 and DC0 ) 110 × 10-9 m2/s (dark-gray), ct ) 1000 mol/ m3 and DC0 ) 210 × 10-9 m2/s (light-gray).
TABLE 2: Average Absolute Shift for Binary Electrolytes p
1-1
1-2
2-1
ct DA0 DC0 σA σC mA mC
12.94 1.35 6.19 1.69 0.87 3.18 1.00
16.39 1.90 6.55 2.80 3.32 3.40 0.61
19.69 1.07 10.75 5.09 1.88 2.26 1.28
Because the solution adjacent to the electrode is very dilute at the limiting current density, an analogous result is obtained in the MSA,
(
b J AMSA ≈ - 1 -
underestimations reach up to 30% for the 1-1 electrolytes, 38% for the 1-2 electrolytes, and 46% for the 2-1 electrolytes. The average absolute shifts for the 7 parameters are summarized in Table 2. Not surprisingly, the total ion concentration ct is the most important parameter. In the limit of ct f 0 both ion transport models coincide, regardless of all other parameters. Only at high ct there is a difference. The second most important parameter is recognized as D0C. The remaining parameters vary in relative importance. The histograms demonstrate that ct and D0C alone largely determine ∆j. It is clear that for a concentrated binary electrolyte solution the pseudoideal solution model can significantly underestimate the limiting current density of the MSA, certainly if the counterion is slow. As the MSA was shown to reproduce the experimental properties of simple aqueous electrolyte solutions quite well,4,6 it can be inferred that the pseudoideal solution model will also underestimate the true limiting current density under these conditions. A comparison with experimental limiting current densities in aqueous CuSO4 solutions will be presented in a subsequent article. The importance of ct and D0C can be rationalized by examining the flux of the reacting ion A on the electrode, which is equivalent to the current density. The contribution of convection is zero, because the fluid is stagnant there. From the zero flux b φ and of the inert ion C a direct relation is obtained between ∇ b c C, ∇
b φ ) - RT ∇ bc ∇ zCFcC C
(38)
Substituting this expression into the flux equation of A and replacing cC by -(zAcA)/(cC) through the use of the electroneutrality condition gives in the pseudoideal solution model
(
b J AI ≈ - 1 -
)
zA b cI D ∇ zC A A
Figure 5. Concentration profile for a 2-1 electrolyte with ct ) 1000 mol/m3, DA0 ) 0.710 × 10-9 m2/s, DC0 ) 2.010 × 10-9 m2/s, σA ) 4.6 Å, σC ) 3.6 Å mA ) 0.06 kg/mol, and mC ) 0.04 kg/mol.
(39)
)
zA b cMSA D ∇ zC AA A
(40)
The values of the diffusion coefficients are
DA ) DA0
κ κ0
(41)
in the pseudoideal solution model and
DAA ≈ DA0
(42)
in the MSA. The conductivity of an electrolyte solution in the MSA can be written as the sum of an ideal part and a nonideal part due to the electrophoretic and the relaxation effect. The nonideal part always lowers the conductivity, that is, κ/κ0 < 1, explaining why the pseudoideal solution model always underestimates the limiting current density with respect to the MSA. The nonideal part becomes more important with increasing concentration, and the ideal part is directly proportional to the limiting diffusion constants, so that the deviation between the pseudoideal solution model and the MSA is indeed the greatest for high concentration and low limiting diffusion constants. There is an additional indirect effect of the parameters on ∆j via the concentration gradient of the reacting ion, which is responsible for the smaller influence of the limiting diffusion constant of the reacting ion. It can be seen in Figure 5 that the predicted concentration profiles are different for the two models. For the case of a 2-1 electrolyte (similar to CuCl2) the concentration of the active ion decreases more slowly toward the electrode in the pseudoideal model, such that at the electrode the concentration gradient is steeper. Ternary Electrolyte Solutions. The histograms of the ternary electrolyte solutions, shown in Figures 6 7, 8, and 9, are similar to those of the binary electrolytes. Again, all deviations are negative. The underestimations reach up to 29% for the 1-1-1 electrolytes, 41% for the 1-1-2 electrolytes, 41% for the
3110 J. Phys. Chem. B, Vol. 113, No. 10, 2009
Figure 6. Histogram of the relative difference of the limiting current density ∆j for 1-1-1 electrolytes: ct ) 10 mol/m3 and DC0 ) 110 × 10-9 m2/s (black), ct ) 10 mol/m3 and DC0 ) 210 × 10-9 m2/s (white), ct ) 1000 mol/m3 and DC0 ) 110 × 10-9 m2/s (dark-gray), ct ) 1000 mol/m3 and DC0 ) 210 × 10-9 m2/s (light-gray).
Van Damme et al.
Figure 9. Histogram of the relative difference of the limiting current density ∆JQ for 2-1-1 electrolytes: ct ) 10 mol/m3 and DC0 ) 110 × 10-9 m2/s (black), ct ) 10 mol/m3 and DC0 ) 210 × 10-9 m2/s (white), ct ) 1000 mol/m3 and DC0 ) 110 × 10-9 m2/s (dark-gray), ct ) 1000 mol/m3 and DC0 ) 210 × 10-9 m2/s (light-gray).
TABLE 3: Average Absolute Shift for Ternary Electrolytes
Figure 7. Histogram of the relative difference of the limiting current density ∆JQ for 1-1-2 electrolytes: ct ) 10 mol/m3 and DC0 ) 0.710 × 10-9 m2/s (black), ct ) 10 mol/m3 and DC0 ) 1.110 × 10-9 m2/s (white), ct ) 1000 mol/m3 and DC0 ) 0.710 × 10-9 m2/s (dark-gray), ct ) 1000 mol/m3 and DC0 ) 1.110 × 10-9 m2/s (light-gray).
Figure 8. Histogram of the relative difference of the limiting current density ∆JQ for 1-2-1 electrolytes: ct ) 10 mol/m3 and DC0 ) 110 × 10-9 m2/s (black), ct ) 10 mol/m3 and DC0 ) 210 × 10-9 m2/s (white), ct ) 1000 mol/m3 and DC0 ) 110 × 10-9 m2/s (dark-gray), ct ) 1000 mol/m3 and DC0 ) 210 × 10-9 m2/s (light-gray).
1-2-1 electrolytes, and 40% for the 2-1-1 electrolytes. The average absolute shift for the 11 parameters are summarized in Table 3. As expected, the total ion concentration ct is again the most important parameter. The second most important parameter is also again D0C. Two other parameters are also fairly important, namely R and DB0 , but the histograms demonstrate that ct and D0C alone explain a large portion of the variation in ∆j. Like for concentrated binary electrolyte solutions, the pseudoideal solution model may significantly underestimate the limiting current density for concentrated ternary electrolyte solutions, certainly if the counterion is slow.
p
1-1-1
1-1-2
1-2-1
2-1-1
ct R DA0 DB0 DC0 σA σB σC mA mB mC
13.83 1.92 1.08 2.10 4.00 0.48 1.22 1.11 1.11 0.06 0.36
19.32 3.57 1.45 2.72 4.28 0.98 2.49 1.70 1.31 0.05 0.24
19.03 3.25 1.42 1.98 4.89 0.42 1.10 2.63 0.84 0.04 0.28
13.20 5.22 1.73 1.75 5.23 1.52 0.76 1.04 0.88 0.05 0.49
An analogous reasoning as for the binary electrolytes explains the observed behavior of ∆j. The contribution of convection to the flux of A on the electrode is zero. At the limiting current density cA tends to zero, so the solution adjacent to the electrode is approximately a binary solution of BC. Even at R ) 10 it is still conductive enough to make the migration of A small compared to its diffusion. The additional diffusion terms b c and DAc∇ b c are small compared to DAA∇ b c , because the DAB∇ B C A cross diffusion coefficients of A are small if the concentration of A is low. Therefore,
b cI b J AI ≈ -DA∇ A
(43)
b cMSA b J AMSA ≈ -DAA∇ A
(44)
DAA is now no longer equal to the limiting diffusion constant, but it is only affected by the relaxation effect, so that it will still be greater than DA which suffers the full nonideality. This explains also, for the ternary electrolytes, why the pseudoideal solution model underestimates the limiting current density with respect to the MSA. If the bulk of the solution is already diluted with respect to the reacting ion and contains an excess of inert electrolyte, DAA will remain constant, so equation 44 is valid everywhere. Because the concentration variations are small, the potential will satisfy Laplace’s equation. In that case, it is clear that choosing DA ) DAA and reducing only the limiting diffusion constants of the inert ions to match the conductivity will be more accurate than applying 1 to all diffusion coefficients. Conclusion A comparison between the frequently used pseudoideal solution model and the MSA has pointed out that the former
Ion Transport Models tends to underestimate the limiting current density of electrodeposition processes, particularly if the concentrations are high and the inert ions have a low limiting diffusion constant. In such cases underestimations of 20% or more should be expected. Since many industrial processes are operated just below the limiting current to achieve maximum throughput while avoiding unwanted side reactions, it is essential to predict the limiting current accurately. It is to be expected that statistical mechanical theories, like the MSA will find applications in the simulation of electrochemical processes where the composition changes significantly inside the reactor such that the assumption of constant effective properties is inadequate. A special pseudoideal solution model is proposed for the case where the bulk concentration of the reacting ion is very low and the concentrations of the inert ions are high. A treatment with constant effective properties is then possible: only the diffusion coefficients of the inert ions should be adjusted to match the experimental conductivity, while the MSA diffusion coefficient DAA is used for the active ion.
J. Phys. Chem. B, Vol. 113, No. 10, 2009 3111 References and Notes (1) de Groot, S. R.; Mazur, P. Non-equilibrium Thermodynamics; Dover Publications, Inc.: New York, 1984. (2) Kreuzer, H. J. Nonequilibrium Thermodynamics and its Statistical Foundations; Oxford University Press: New York, 1981. (3) Newman, J. S. Electrochemical Systems; Prentice Hall, New York, 1991. (4) Dufreˆche, J.-F.; Bernard, O.; Durand-Vidal, S.; Turq, P. J. Phys. Chem. B 2005, 109, 9873–9884. (5) Simonin, J.-P.; Blum, L.; Turq, P. J. Phys. Chem. 1996, 100, 7704– 7709. (6) Van Damme, S.; Deconinck, J. J. Phys. Chem. B 2007, 111, 5308– 5315. (7) Dufreˆche, J.-F.; Bernard, O.; Turq, P. J. Chem. Phys. 2002, 116, 2085–2097. (8) Miller, D.; Vitagliano, V.; Sartorio, R. J. Phys. Chem. 1986, 90, 1509–1519. (9) Simonin, J.-P. J. Chem. Soc., Faraday Trans. 1 1996, 92, 3519– 3523. (10) Blum, L. Primitive Electrolytes in the Mean Spherical Approximation. In Theoretical Chemistry, AdVances and PerspectiVes; Eyring, H., Henderson, D., Eds.; Academic Press: New York, 1980; Vol. 5, pp 1-66. (11) Bortels, L.; Deconinck, J.; Van den Bossche, B. J. Electroanal. Chem. 1996, 404, 15–26.
JP809947Q
