Samson Et Al-1999-International Journal for Numerical Methods in Engineering

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING

Int. J. Numer. Meth. Engng. 46, 20432060 (1999)

MODELLING ION DIFFUSION MECHANISMSIN POROUS MEDIA

E. SAMSON1; 2, J. MARCHAND1; 2;, J.-L. ROBERT1 AND J.-P. BOURNAZEL3

1Universite Laval; Quebec; Canada G1K 7P42SIMCO Technologies Inc.; 1400; boul. du Parc Technologique; Quebec; Canada G1P 4R7

3Laboratoire dEtudes et de Recherches sur les Materiaux 23; rue de la Madeleine-BP 136 13631 Arles Cedex; France

SUMMARY

The main features of a numerical model aiming at predicting the drift of ions in electrolytic solutions arepresented. The mechanisms of ionic diusion are described by solving the NernstPlanck system of equations.The electrical coupling between the various ionic uxes is accounted for by the Poisson equation. Twoalgorithms using the nite element method for spatial discretization are compared for simple test cases. Oneis based on the Picard iteration method while the other is based on the NewtonRaphson scheme. Test resultsclearly indicate that the range of application is broader for the algorithm based on the NewtonRaphsonmethod. Selected examples of the application of the algorithm to more complex 1-D and 2-D cases aregiven. Copyright ? 1999 John Wiley & Sons, Ltd.

KEY WORDS: ionic diusion; NernstPlanck; Poisson; electrical coupling

1. INTRODUCTION

In many engineering problems, the behaviour of porous materials is directly aected by the trans-port of ions under a concentration gradient. For instance, it has been shown that the swellingof clays is predominantly controlled by the penetration of ions by diusion in their interlayerspaces [1]. Given their inuence on various phenomena such as the ltration by ion exchangemembranes and the transport of pollutants in soils, the mechanisms of ionic diusion in porousmedia has also received a great deal of attention from chemical and geological engineers [2; 3].The process of ionic diusion remains of primary importance in many civil engineering problemssince the long-term durability of many building materials, such as concrete, is directly aected bythe transport of chemical species [4].Over the years, it has been established that the mechanisms of ionic diusion can be adequately

modelled by the NernstPlanck=Poisson set of equations [2; 4]. These equations take into accountthe electrical coupling between the dierent ions present in an ideal solution (i.e. no chemicalactivity eects are considered). According to this model, the drift of an ionic species stronglyinuences that of all other ions dissolved in the electrolytic solution.

Correspondence to: J. Marchand, CRIB-Department of Civil Engineering, Laval University, Quebec, Canada G1K 7P4

CCC 0029-5981/99/36204318$17.50 Received 26 January 1999Copyright ? 1999 John Wiley & Sons, Ltd. Revised 6 April 1999

2044 E. SAMSON ET AL.

Although the electrical coupling between the various ionic uxes is well known to electro-chemists and engineers, most existing models aiming at describing the mechanisms of ionicdiusion tend to neglect this phenomenon [3; 5]. Furthermore, a comprehensive bibliographicalreview has recently shown that the proposed analytical or numerical ionic transport models areunsatisfactory, all of them being limited to unidimensional and steady-state cases. A summary ofthis literature survey is given in the following section.In order to extend the application of the NernstPlanck=Poisson set of equations, two algorithms

were tested. The rst one is based on the Picard iteration method. The second algorithm uses aNewtonRaphson scheme. Both rely on the nite element method for spatial discretization.After this comparison, selected examples of calculations are presented to illustrate the applica-

tion of the second algorithm to the treatment of steady-state and transient problems involving animportant number of multivalent ionic species. An example of application of the algorithm to theresolution of 2-D cases (axisymmetrical geometries) is also given.

2. MATHEMATICAL MODEL

The NernstPlanck model, which describes the ux of an ionic species i in solution, is given by

ji= [Di](grad (ci) +

ziFRT

ci grad (V ))

(1)

where ji stands for the ux of the species i, [Di] is the diusion coecient tensor of the species, ciis the ionic concentration of the species, zi is the valence number of the species, F is the Faradayconstant, R is the perfect gas constant, T is the temperature, and V is the electrical potential thatis locally induced in the electrolytic solution by the movement of all ionic species.It should be emphasized that the presence of this electrical potential is probably the most

important feature that distinguishes ionic diusion from molecular diusion. In an ionic solution,the local electroneutrality shall be preserved at any point. The conservation of electroneutralityrequires that the transport of all diusing species should be coupled. During the diusion process,all ions are not drifting at the same speed. Some ions tend to diuse at a higher rate. However,any excess charge transferred by the faster ions builds up a local electric eld (Valso called thediusion potential) which slows down the faster ions, and reciprocally accelerates the slower ionicparticles. The diusion potential has to be accounted for even in cases where an external electricaleld is applied to the system. In that case, the diusion potential is superimposed to the externaleld.It should also be underlined that equation (1) does not consider any chemical activity eects

or the transport of ions by convection of the liquid phase in the pore system. The inuence ofactivity phenomena on the mathematical treatment of the diusion problem is discussed elsewhere[6]. A detailed discussion on the limits of equation (1) is given by Helerich [2].For each of the ionic species present in solution, the mass conservation law is given by

@ci@t+ div( ji)= 0 (2)

This equation does not account for any chemical or physical interactions that can develop betweenthe solid and the various ionic species in solution. A comprehensive discussion of the inuence

Copyright ? 1999 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng. 46, 20432060 (1999)

MODELLING ION DIFFUSION MECHANISMS IN POROUS MEDIA 2045

of chemical reactions and physical interaction phenomena on the mathematical treatment of ionicdiusion problems is given in [7; 8].By replacing equation (1) in equation (2), one nds the complete NernstPlanck equation:

@ci@t

div([Di]

(grad (ci) +

ziFRT

ci grad (V )))

(3)

In order to complete this system of equations, one has to dene one last relationship that willcouple the transport of all ionic species to one another. Over the past decades, numerous authorshave chosen to simplify the problem by assuming that the electroneutrality of the solution ispreserved at any points:

Ni=1

zici + w=0 (4)

where N is the number of species and w is a xed charge density over the domain. For mostporous materials, the xed charge density is not a relevant parameter. It is, however, often usedin the description of biological systems like thin membranes.In many cases, it is also assumed that the global ow of all ions across the membrane yields a

nil current:

Ni=1

zi ji=0 (5)

However, as will be discussed in the following section, these assumptions are not always valid.A more rigorous way to treat the problem [2] is to dene the variation of the electric potentialaccording to the spatial distribution of the electric charges. This relationship is given by the Poissonequation

2V + =0 (6)

where is the electrical charge density and is the dielectric constant of the surrounding medium.The electrical charge density is a function of the concentration of the various ions in solution andcan be calculated using the following equation:

=F(

zici + w) (7)

3. NUMERICAL MODELS FOR SOLVING THE NERNSTPLANCK=POISSONSYSTEM OF EQUATIONS

3.1. Bibliographical review

Over the past decades, numerous authors have tried to develop solutions for the NernstPlanckequation. The rst solution was obtained in 1890 by Planck himself [9]. He studied the case ofsteady-state unidimensional ionic diusion of two monovalent species (+1 and 1) through amembrane. In the problem considered by Planck, xed concentration and potential were imposed



on both sides of the domain. The solution was obtained using the electroneutrality and nil currentassumptions. The geometry and boundary conditions of this problem, although simple and veryidealized, still remain about the only ones for which an analytical solution is possible.Later on, Schlogl [10], Helerich [2] and Teorell [11] developed other solutions for the treatment

of unidimensional steady-state problem. Schlogl [10] derived an analytical solution using the sameboundary conditions as Planck. Schlogls solution was developed in such a way that it couldaccount for any number of ions, whatever their valence number. Once again, the electroneutralityand nil current assumptions were at the basis of the development of his solution.The application of all previous analytical solutions was limited to steady-state cases. Always

using the electroneutrality and nil current assumptions, Conti and Eisenman [12] proposed a dif-ferent solution that could be applied to the treatment of unsteady-state problems. They developedan expression for the variation of the electrical potential through a membrane but could not obtainneither the concentration nor the electrical potential proles. Furthermore, the validity of theirsolution was solely restricted to electrolytic solutions made of monovalent ions.It should also be emphasized that all the previous analytical solutions were developed on the

basis of the electroneutrality and nil current assumptions. In 1943, Goldman [13] presented asolution for the steady-state case assuming a constant electric eld across the membrane (i.e. alinear variation of the electrical potential across the system). This simplication of the problemallowed the author to integrate directly the NernstPlanck equation.The considerable diculty of developing analytical solutions for this system of equations has

led researchers to use numerical methods. In 1965, Cohen and Cooley [14] presented an algorithmthat allows solving the NernstPlanck equation for transient cases. Their solution was obtainedusing a predictor-corrector scheme. But the predictor step uses Plancks analytical solution, thuslimiting the application of the algorithm to very simple cases.More recently, Hwang and Helerich [15] developed an algorithm that can be used to solve the

NernstPlanck equation for any number of ions, for any valence number, and for transient problems,with a nite-dierence discretization. The discretization is not directly performed on equation (3).The system of equations rst has to be transformed according to the electroneutrality and nil currenthypotheses. This transformation introduces two new terms that are used as iteration coecientsin the algorithm. However, after being transformed, the equations become very complex, whichmakes the conversion in two or three dimensions extremely dicult. Moreover, the transformationof the equations complicates the treatment of the boundary conditions. Patzay [16] modied theapproach by using three iteration coecients instead of two. In this case, no modication in thetreatment of the boundary conditions is required.Harden and Viovy [17], who worked on membranes subjected to a current of variable inten-

sity, have directly discretized the conservation equation (3) using the nite-dierence method. Anexplicit Euler scheme is used for time discretization. For a case with N ionic species, the con-centration proles of N 1 ions are calculated using the concentration and potential calculated atthe preceding time step. For the remaining species, the concentration is calculated using the elec-troneutrality condition. The new electric eld is determined knowing that the current introducedin the membrane must be equal to the internal current. The advantage of this method, comparedto all those previously described, is that it can be easily transposed in two or three dimensions.This solution, however, still relies on the electroneutrality assumption.All the previous numerical models are based on the assumption that the coupling of the Nernst

Planck equation with either the electroneutrality condition or the constant eld relationship yieldsa reliable description of the ionic diusion mechanisms. Even if these hypotheses may constitute



a good choice in some practical cases, Helerich [2] mentions that the Poisson equation should beused in a more rigorous approach of the problem. MacGillivray and Hare [18; 19] have demon-strated that the electroneutrality and constant eld hypotheses are, in fact, nothing but particularapplications of the Poisson equation. The electroneutrality assumption is applicable only when theconcentrations are high while the constant eld hypothesis is rather valid for low concentrations.Some researchers have tried to couple the NernstPlanck equation to the Poisson equation.

For instance, James et al. [20] coupled both equations to the Stokes equation to model the owof a liquid containing charged particles in a cylinder. Their analysis was limited to steady-statecases. The conservation equations were discretized using the nite element method and a Galerkinresidual weighting. The algorithm consists in solving the equations one after the other startingfrom an initial concentration prole (Picard iterations), the values obtained being used as startingpoints for the following iterations, until convergence is reached.Kato [20] has proposed a numerical method to solve the NernstPlanck=Poisson system of

equations for unidimensional steady-state cases. Knowing that, once the steady state is reached,the uxes are constant, a rst analytical integration of the NernstPlanck equation can be per-formed. The solution obtained has to be discretized afterwards by the nite-dierence method. Thenumerical scheme used is similar to that of James et al. [20], the main dierence being that thestarting point is a potential prole, which is subsequently used to calculate the concentrations.As can be seen, the development of numerical solutions clearly appears to be the most promising

approach for the treatment of the NernstPlanck=Poisson system of equations. It should howeverbe emphasized that transient problems with any number of species for 1-D or 2-D cases with anite element discretization have never been investigated. As previously discussed, all attempts todevelop numerical solutions for this set of equations were limited to simpler cases. The object ofthis paper is to compare the ability of two numerical algorithms to treat complex ionic diusionproblems using the NernstPlanck=Poisson equations.

3.2. First algorithm: uncoupled equations

The rst algorithm proposed is based on the Picard iteration technique, also known as the succes-sive substitution technique [21]. It is a rst-order method, thus having a low rate of convergence.The algorithm is similar to the numerical resolution proposed by James et al. [22] and Kato [20],in which the equations are solved one after another.The equations are discretized separately following the standard nite element method. The results

are thus presented without much detail. The weighted residual form of the NernstPlanck equation(3) over the domain is given by

W = [@ci@t

div([Di]

(grad (ci) +

ziFRT

ci grad (V )))]

d=0 (8)

where is the weighting function. From that point on, an axisymmetrical case with an orthotropicmaterial is considered. Performing an integration by parts on W yields the following weak form:

W = @ci@t

r dr dz + ; r ; z

[[Di]

{ci; rci; z

}+

ziFRT

ci

{V; rV; z

}]r dr dz (9)

The boundary terms are omitted, since all the simulations will be performed considering Dirichletconditions. Equation (9) is discretized according to the Galerkin method. The unknown ci is



interpolated at the nodes, as is , according to

ci = N {cin} (10) = N { n} (11)

where N are the shape functions and the subscript n indicates node values. The elementarymatrices are written as

[Kei ] =e[B]T[Di] [B] + [B]T[Ei] [N N ]r dr dz (12)

[M ei ] =e{N}N r dr dz (13)

where [Ei] is the matrix coupling the concentration of each species to the electrical potential. Itis written as

[Ei] =

DriziFRT

@V@r

0

0DziziFRT

@V@z

(14)

The matrices [B] and [N N ] are dened as

[B] =

@@r

N @@z

N

(15)

[N N ] =[N N

](16)

The various integrals are calculated using a Gaussian quadrature method. The terms @V=@r and@V=@z in the matrix [Ei] are evaluated at the integration points.The Poisson equation is discretized using the same technique. The elementary matrices are

[Ke] =e[B]T[B]r dr dz (17)

{Fe}=e{N}

zici

r dr dz (18)

{Fes }=e{N}w

r dr dz (19)

where {Fes } is the solicitation vector coming from the xed charge density w. The concentrationsin equation (18) are calculated at the integration points.The resolution steps for a steady-state problem are presented. For a transient problem, a standard

method [21] could be used.

1. An initial concentration is assumed for each ionic species.2. The Poisson equation is solved using the concentration of the previous iteration level.3. A loop is performed on all the ionic species:

(a) The gradient potential is calculated on the integration points from the numerical solutionof the Poisson equation. The results are used to build the matrix [Ei].



(b) The solution of the NernstPlanck equation is obtained for the species i using the ele-mentary matrices (12) and (13).

4. The L2 norm of the vector u= c1 c2 : : : V kc1 c2 : : : V k1 is calculated where k standsfor the iteration level.

5. If the norm is higher than a tolerance threshold , go back to step 2 by using the concentrationjust calculated. The loops are performed until convergence is reached.

The main advantage of this algorithm is that the same calculation code can be used, whateverthe number of ionic species. Only the number of loops (step 3) is dierent.

3.3. Second algorithm: coupled equations

In the second algorithm, the NernstPlanck and Poisson equations are coupled and solved simul-taneously. A classical NewtonRaphson method [21; 23] is used to solve the non-linear set ofequations. As it is a second-order scheme, convergence is expected to be faster than what isobtained for Picard iteration method. Once again, the discretization follows the standard niteelement procedure.The weighted residual form is written as [23]

W =1 2 : : :

R1R2...

d=0 (20)

where the Ris are the residuals associated to each of the equations and the is are the corre-sponding weighting functions. For each of the ionic species, the residual is

Ri=@ci@t

div([Di]

(grad(ci) +

ziFRT

ci grad(V )))

(21)

and for the Poisson equation,

RV =2V + F (

zici + w) (22)

For a case limited to ionic species, the integration by parts leads to the following weak form:

W =c1 c2 V

c1c20

r dr dz

+c1; r c1; z c2; r c2; z V; r V; z

Dr1c1; r +Dr1z1FRT

c1V; r

Dz1c1; z +Dz1z1FRT

c1V; z

Dr2c2; r +Dr2z2FRT

c2V; r

Dz2c2; z +Dz2z2FRT

c2V; z

V; r

V; z

r dr dz



+c1 c2 V

F

z1c1

Fz2c20

r dr dz

+c1 c2 V

00

Fw

r dr dz (23)

The weak form is discretized using a Galerkin weighting. The vector of the unknown variablesis written as

c1c2...V

= [N ]{Un} (24)

[N ] =

N1 N2 NG

N1 N2 NG. . .

. . . . . .N1 N2 NG

(25)

Un= c11 c21 : : : V1 c12 c22 : : : V2 c1G c2G : : : VG (26)where G is the number of nodes in the element. The small dots (: : :) indicate the terms to addwhen considering more ionic species, whereas the big dots () stand for the missing shapefunction terms. The subscripts i and j designate the species i and the node j. The elementarymatrices can thus be expressed as

[K e] =e

[B]T[D1][B]

[KeD]

+ [N ]T[D2][N ] [KeP]

+ [B]T[D3][N N ] [KeNP

]

r dr dz (27)

[M e] =e[N ]T[D4][N ]r dr dz (28)

{Fe}=e[N ]T

00...

Fw

r dr dz (29)

The matrix [Ke] is divided into three parts. The rst part,[KeD

], includes all the diusion terms. The

two other parts are related to the coupling between the concentration and the electrical potential.The matrix

[KeP

]comes from the discretization of the Poisson equation (the third line in equation

(23)) and[KeNP

]is related to the coupling between the concentration and the electrical potential



gradient in the NernstPlanck equation. The [N N ] matrix is given by

[N N ] =

N1 N2 NGN1 N2 NG

N1 N2 NGN1 N2 NG

. . .. . . . . .

N1 N2 NGN1 N2 NG

(30)

For an axisymmetrical case, the matrix [B] corresponds to

[B] =

N1; x NG;xN1;y NG;y

N1; x NG;xN1;y NG;y

. . . . . .N1; x NG;xN1;y NG;y

(31)

The matrices [D1][D4] are given by

[D1]=

Dr1Dz1

Dr2Dz2

. . .1

(32)

[D2]=

00...

Fz1

Fz2

0

(33)

[D3]=

Dr1z1FRT

@V@r

Dz1z1FRT

@V@z

Dr2z2FRT

@V@r

Dz2z2FRT

@V@z

. . .00

(34)



[D4]=

11

. . .0

(35)

The tangent matrix is obtained by calculating the rst variation of the weak form W (Equation(20)) [23]:

W =1 2 : : :

R1R2...

d=0 (36)

For a transient case solved with an implicit Euler scheme (=1), the elementary tangent matrixis given by

[KeT]= [Me] + t[Ke] + t

e[BT][D5][B] r dr dz (37)

where

[D5]=

0 0 0 0 : : :Dr1z1FRT

c1 0

0Dz1z1FRT

c1

Dr2z2FRT

c2 0

0Dz2z2FRT

c2

......

0 00 0

(38)

In contrast to the preceding numerical scheme, the characteristics of the various matrices varywith the number of ionic species accounted for. Furthermore, larger matrices have to be stored.

4. COMPARISON OF THE TWO ALGORITHMS

In order to compare both algorithms, an example rst presented by Kato [20] will be used. Itis an unidimensional steady-state problem involving two ionic species. The development of theelementary matrices for 1-D cases is not presented but is straightforward starting from the matricesof the axisymmetrical case. The equations are rst rewritten in a non-dimensional form by stating

V =RTF

V (39)

x= Lx (40)



Di = DiD0 (41)

ji =c0D0L

ji (42)

ci = c0ci (43)

where the subscript 0 designates a reference value and L is the length of the domain. Equa-tions (1) and (6) are rewritten as

ji = (

DiD0

)dcidx

zici(

DiD0

)dVdx

(44)

d2V

dx2+

2= 0 (45)

with

2 =RT

c0F2L2(46)

=

zici + (w=c0) (47)

in the domain 06x61. The value of the parameter 2 is determined mainly by the scale ofthe problem under consideration. For a medium kept at 25C (298K) and saturated with anionic solution having a concentration of 1mmol=l and a permittivity of 70832 1010 C=V m,2' 2 1016=L2. For biological membranes that have thickness around 1m, 2 ' 00002. Forporous construction materials like concrete, the characteristic length is of the order of about 1 cmand thus 2 ' 2 1012. For two problems having the same characteristic length L, a variationin 2 corresponds to a variation in the level of concentration involved.For this specic example, it is assumed that two monovalent species are present in solution

(z1 =+1 and z2 =1). For both species, Di=1 and c0 =w=1. It is also assumed that bothspecies have a dimensionless concentration of 10 at x=0 and 15 at x=1. The dimensionlesspotential is set at 00 at x=0 and 10 at x=1.To solve this problem, both algorithms were used with a linear two-node element.

4.1. Results obtained with the rst algorithm

To begin the calculations, initial concentrations were considered to vary linearly across thesystem. Figure 1 shows the solution obtained with 2 = 05 and computed with 20 elements. Twelveiterations were needed to perform the calculations. The dierence with the solution obtained byKato [20] does not exceed 2 per cent.The next set of simulations consists in varying the value of 2 while keeping the same boundary

conditions. It appeared that under 2 = 025, no convergence could be obtained. Figure 2 clearlyshows the variation of the convergence rate as a function of 2. All simulations were made with 20elements. In order to rectify this situation, dierent solutions were tested: a variation in the numbersof elements, an increase in the order of the interpolation polynomials, and a modication in theinitial solution introduced to the algorithm. Results revealed that all this work was done in vain.A modication of the algorithm was also made, by updating the solution after a calculation step



Figure 1. First algorithm: solution for 2 = 05

Figure 2. First algorithm: norm vs. number of iterations for dierent values of 2

according to the following procedure:

{u}k1 = {u}k1 + {u}k2

2(48)

This modication allowed us to break the 025 limit, but only to nd the same problem for avalue of about 015. For such values of 2, only thin membrane problems can be solved. This iswhat justied the development of the coupled equation algorithm, in an attempt to broaden theeld of application of the NernstPlanck=Poisson set of equations.

4.2. Results obtained with the second algorithm

The same tests were carried out for the second algorithm, and proved to provide much betterresults. For the second algorithm, there is no evidence of a critical value of 2 for which divergenceoccurs. The number of required iterations is lower and limited to approximately four. Furthermore,this number varies very little according to 2.



Figure 3. Second algorithm: solution for 2 = 00001

To illustrate the superior behaviour of the second algorithm, the same problem was solved,but this time with 2 = 00001. The results are shown in Figure 3, and were obtained with 100elements. More elements are needed to avoid oscillations near the boundaries, due to high gradientvalues. Comparisons with the results of Kato [20] are impossible since the author did not performany tests for low values of 2.Following these results, it is clear that the second algorithm is much more robust. It was used

to obtain the results that will be presented afterwards.Various other cases were considered to test the second algorithm. The rst example was inspired

from Helerich [2]. It consists in a 1-D steady-state problem where four species are present:Mg2+, SO24 , Na

+ and K+. Dimensionless variables were used in the treatment of the problem.The diusion coecients were 10 for SO24 , Mg2+ and Na+, and 167 for K+. At x=0, theboundary conditions were 10 for SO24 , 05 for Mg2+ and zero for the other species. At x=1,the conditions were 08345 for SO24 , 00 for Mg2+, 0169 for Na+ and 05 for K+. The potentialwas established at 00 at x=0 and 50 at x=1. The xed charge density in the membrane, w=c0,was equal to 10. The only remaining unknown variable was the value of 2.The solution obtained analytically by Helerich [2] for that same problem was developed on

the basis of the electroneutrality hypothesis. Considering that MacGillivray and Hare [18; 19]have demonstrated that the validity of this assumption is restricted to very low values of 2, thecalculations were performed with 2 = 1 1010.The solution, computed with 50 elements, is shown in Figure 4. The dierence with the analytical

solution presented by Helerich [2] is about 3 per cent, which conrms the results of MacGillivrayand Hare [18; 19] concerning the value of 2.The other example concerns a transient diusion problem. It was selected to evaluate the in-

uence of the 2 parameter for a xed length domain. The calculations were performed usingan implicit Euler scheme. The dimensionless variables were once again used with the additionof t=D0t=L2 as a time variable. Two species were considered, with z1 = + 1 and D1 = 3 for therst one, and z2 = 1, D2 = 1 for the second. Initially, the concentrations were set equal to zeroover the entire domain, which consisted of a bar of length 1. At t=0, a unit concentration wasimposed at x=0 for both species. The concentrations were set at zero at x=1.



Figure 4. Concentration proles for the Helerich problem

Figure 5. Concentration proles at t=001; 2 = 001

Two simulations were performed for values of 2 equal to 1 102 and 1 104 respectively.This implies that, for a given domain length, the concentrations involved in the second case werea hundred times higher than in the rst one. Results are given in Figures 5 and 6. These resultswere obtained at t=001 and reached after 200 steps of 5105. All calculations were performedwith 50 elements. As a comparative basis, the solutions computed without taking into account anyelectrostatic eect, thus derived on the sole basis of Ficks law, are also given in Figures 5 and 6.In Figure 5, the inuence of the electrical coupling between the species is clearly noticeable.

On the one hand, results indicate that the progression of the fastest ion is slowed down by theelectrical potential. On the other, the slowest ion, with a diusion coecient being three timeslower, is accelerated. In the case of higher concentration (Figure 6), this phenomenon is evenmore obvious. Both concentration proles are so close that they became superposed.These results conrm, once more, the conclusion of MacGillivray and Hare [18; 19], who sug-

gested that the electroneutrality is gradually approached as the value of 2 decreases. Figure 7shows that the electrical charge density tends toward zero as 2 is reduced.



Figure 6. Concentration proles at t=001; 2 = 00001

Figure 7. Charge density for the two transient cases

The last example shows the use of this numerical scheme for a transient problem with anaxisymmetrical geometry. The simulations were performed to investigate the eect of the presenceof electrical charges on the inner surface of a pore. In this case, ions are diusing in and out ofthe pore space. The pore, presented in Figure 8, has a radius of 1 m and a length of 10 m.Five ions were considered in the simulations: OH, Na+, K+, SO24 , and Cl

. Their respectivediusion coecients (in m2=s) are: 5273 109, 1334 109, 1957 109, 1065 109, and2032 109.The initial concentrations in the pore were xed at: 690mmol= l of OH, 286mmol= l of Na+,

500mmol= l of K+, 48mmol= l of SO24 , and 0mmol= l of Cl.

The boundary conditions, at the entrance of the pore, were xed at 500mmol= l of Cl,800mmol= l of Na+, and 300mmol= l of OH. For the electrical (diusion) potential, the initialvalue was set at 0.Two cases were investigated. In the rst case, it was assumed that no electrical charges were

present on the inner surface of the pore. In this case, the electrical potential was set at 0 at theentrance of the pore, since a reference value has to be xed at some point. In the second case,



Figure 8. The axisymmetrical problem

Figure 9. Potential prole in a pore

the electrical potential on the inner surface of the pore was set equal to 10 mV as shown inFigure 8. The system was considered to remain in isothermal conditions (i.e. at 22C ) overthe entire duration of the process. The dielectric constant of the system was xed at 70823 1010 C=Vm. The two simulations were performed in ve time steps of 00002 s. For the spa-tial discretization, a regular mesh of 10 50 linear three-node triangles was used, as shown inFigure 8.According to the numerical simulations, the potential on the inner surface of the pore has no

signicant eect on the concentration proles. Its inuence is limited to a small region at thevicinity of the surface. Hence, the ux of ions in the pore is the same for both simulations.For the electrical potential, the situation is however dierent. The potential on the inner surface

tends to lower the dierence of potential between both ends of the pore. For the case without anyelectrical charges on the pore wall, this dierence has a value of 5mV. In the other case, thisvalue drops to 31mV.



5. CONCLUSION

The comparison between a numerical scheme based on the Picard iteration technique and anotherscheme based on the NewtonRaphson method has clearly showed that the former, even though ithas already been used in some recent papers, cannot be used to solve the NernstPlanck=Poissonsystem of equations for all cases.By coupling all the equations and using the NewtonRaphson method, the NernstPlanck=Poisson

system of equations could be solved for problems with a high number of ionic species with dier-ent valence number, either for steady state or transient cases. Furthermore, the numerical schemeworks for 1-D geometry as well as 2-D and axisymmetrical ones, and could easily be extendedfor 3-D cases.This work opens the way for the treatment of complex multiionic species models of transport

in porous media involving chemical reactions between the species and the solid matrix, whileconsidering the electrical coupling.

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Samson Et Al-1999-International Journal for Numerical Methods in Engineering