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Models for electrokinetic phenomena in montmorillonite
Colloids and Surfaces A: Physicochemical and Engineering Aspects 195 (2001) 171– 180 www.elsevier.com/locate/colsurfa
Models for electrokinetic phenomena in montmorillonite J.-F. Dufreˆche *, V. Marry, O. Bernard, P. Turq ANDRA and Laboratoire Liquides Ioniques et Interfaces Charge´es, case courrier 51, Uni6ersite´ P. et M. Curie, 4 place Jussieu, F-75252 Paris Cedex 05, France
Abstract Clays present remarkable electrokinetic features since they exist from very dilute colloidal state to nanoporous materials, depending on the water/clay ratio. The case of low volume fraction Vwater/Vtot which corresponds to compact systems is examined. The ionic distributions have been evaluated by Poisson– Boltzmann like models and compared to discrete solvent simulations. Several electrokinetic properties (electroosmosis and conductance) have been calculated, in the framework of the mean spherical approximation introduced in the Fuoss– Onsager transport theory. It is found that Onsager’s limiting laws in terms of external concentration are not valid on the grounds of the Donnan effect. © 2001 Elsevier Science B.V. All rights reserved. Keywords: Electrokinetic phenomena; Montmorillonite; Clays
1. Introduction Clay minerals as montmorillonite have a significant role in the storage of radionuclear wastes: when compacted, they are used as a barrier towards radionuclides. This is one of the reasons why the study of transport properties in these materials is essential. Clays are formed by large platelets composed of stacks of elementary sheets. In the case of montmorillonite, a sheet is made up of a layer of octahedral oxides (Al3 + , Mg2 + , etc.) between two layers of tetrahedral oxides (Si4 + , Al3 + , etc.). As some of the cations are substituted by other cations of lower valency, clay sheets are negatively charged. Counterions which are located between sheets are partly responsible for the swelling be* Corresponding author.
havior of montmorillonite in the presence of water. Many factors such as the general structure of the clay and its level of hydration seem to be involved in the retention of elements. When they are not adsorbed on the surface of a clay particle, ions or radionuclides can move either between platelets or between the sheets inside a platelet. In this article, the motion study will be reduced to the transport between two parallel sheets of clay. Nevertheless, this approximation seems to be acceptable for a compacted clay whose platelets are inclined to stick to each other. Monte-Carlo and molecular dynamics simulations have already been undertaken to describe the equilibrium and dynamic properties of this system: several kinds of counterions such as Li+ [1,2], Na+ [1–6], K+ [1,7,8] and Cs+ [9,10] were studied. These simulations use microscopic descriptions of the system: water and clay sheets are
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considered to be discrete sets of atoms. Unfortunately, this method is expensive as it is time consuming, and gives dynamic quantities as diffusion coefficients only at very short times (about a nanosecond). In this article, we develop a quite simple mesoscopic model to describe ions’ motions in confined water. Clay sheets are considered as uniformly charged planes and water as a continuum. Such a model, based on the Poisson– Boltzmann equation, was already used to calculate equilibrium quantities such as osmotic pressure [11]. Ionic distributions between two clay sheets were compared with those obtained by continuous solvent Monte-Carlo simulations [12]. It was shown that the Poisson– Boltzmann equation was satisfactory for medium and large interlayer spacings. Donnan effects deduced from Poisson –Boltzmann for didodecyldimethylammonium bromide, which is a similar swollen lamellar medium, were in good agreement with experiments of atomic adsorption [13]. In this article, ionic distributions from Poisson – Boltzmann are compared to those obtained by discrete solvent Monte-Carlo simulations. Then they are used to determine transport properties for several salt concentrations in the interlayer spacing (electroosmosis and conductivities). As a continuous solvent model, approximations are made towards the microscopic description. However, it allows one to get dynamic quantities for times comparable with experiments. The calculation that will be described next is based on the mean spherical approximation (MSA) transport theory. A similar theory was previously used for Nafion membranes and succeeded in reproducing experimental conductivity data [14].
2. Equilibrium properties: microscopic and mesoscopic approaches
2.1. Ionic distributions In the adopted model, clay sheets are considered as parallel infinite planes. They are uniformly charged and the surface charge density is |= 0.0161/2e A, − 2. The water inside the interlayer spacing is a continuum. Its viscosity and dielectric
constant are bulk water’s ones: at 298 K, p= 0.89× 10 − 3 Pa s and mr = 78.3, respectively. Counterions are Na+ and the added salt NaCl. The electrolyte is simply treated by taking an average radius.
2.1.1. Poisson– Boltzmann treatment If ci (r) is the concentration of the ion i at a distance r from a central particle we can write, according to Boltzmann statistics:
ci (r)= Mi exp −
Vi (r) kBT
(1)
where Vi (r) can be identified with the electrostatic energy Vi (r)= ei(r). ei = Zi e is the charge of i and (r) the electrostatic potential. In the case of two horizontal charged planes separated by a distance L, and ci only depend on z. Mi is obtained by integrating ci (z):
&
+ L/2
ci (z) dz= Mi
&
− L/2
+ L/2
exp −
− L/2
=Lc
ei(z) dz kBT
0 i
(2)
0 i
where c is the average concentration of i in the pore. By replacing in the Poisson equation D = − i ei ci (r)/m0mr, we get: D = − % i
ei Mi e exp − i kBT m0mr
(3)
Setting = e/kBT, we get: D = − 4yLB% Zi Mi e − Zi
(4)
i
where LB = e 2/4ym0mrkBT is the Bjerrum length. The symmetry of the system implies d/dz (z= 0)= 0 as one of the boundary conditions. Thus, is defined within an additive constant and the Mi depend on the choice of this constant. Indeed, if is changed into % with = %+ 0 the concentrations become − Zi % ci = Mi e − Zi = Mi e − Zi 0e − Zi % = M %e i
(5)
Thus we can choose either the constant or one of the Mi. When there is no added salt in the interlayer spacing, the differential equation can be analytically solved. The counterionic concentration is given by:
J.-F. Dufreˆ che et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 195 (2001) 171–180
h 1 2yZ 2LB cos2(hz)
173
2
c(z)=
(6)
with h tan(hL/2) =2yZLB|/e. In the presence of added salt, (z) and ci (z) are numerically determined. The ionic distributions of this system calculated by the Poisson– Boltzmann equation have already been validated by Monte-Carlo simulations in which water is considered as a continuum [12]. At low distances L ( B10 A, ), discrete solvent simulations showed that distributions presented oscillations, indeed even peaks, which are characteristic of the discrete nature of ions and water molecules. The latter organize themselves in layers and the system is more stable for distances L corresponding to mono-, bi- or trilayers of water molecules. As we assumed that this feature would die down as L increases, we decided to compare Poisson– Boltzmann distributions with those obtained with a discrete solvent Monte-Carlo program for larger values of the interlayer spacing (\ 20 A, ). As the model used in the simulation takes the fine structure of the montmorillonite and the solvent into account, it allows us to test the accuracy of the simple mesoscopic model where the clay is described by the oversimplified Poisson– Boltzmann equation.
2.1.2. Discrete sol6ent simulations The structure of the montmorillonite used in the microscopic simulation was taken from X-ray diffraction studies given in the literature [15,16]. Its formula is Na0.75[Si7.75Al0.25](Si7.75Al0.25) O20(OH)4. The simulation box is formed by two half-sheets containing eight clay units. The charges of the sheets’ atoms are Skipper’s ones [17]. Van der Waals interactions are reduced to Lennard-Jones potentials, whose parameters are given by Smith [10]. The water is described by the well-known SPC/E model [18]. This potential has been chosen because its dielectric constant m SPC/ r E= 819 5 [19] is very close to the experimental value (mr = 78.3). Indeed, in the Poisson–Boltzmann model, the solvent is only described by this physical quantity, so that this value is essential for the comparison. Monte-Carlo equilibration has been performed in the (N, P, T) ensemble. The interlayer spacing
varies in order to get a pressure P= 1 bar. The number of water molecules in the box was fixed and equal to 300. The average height of the box evaluated after equilibration was 35.5 A, . By subtracting the thickness of the clay sheet, we get the distance between the two charged planes of the mesoscopic model. Poisson–Boltzmann ionic distributions are then calculated by taking an average radius of about 2.1 A, for counterions Na+. This is close to the Pauling radius (2.0 A, ).
2.1.3. Comparison of ionic distributions The counterionic distributions and their integrals obtained by the Poisson–Boltzmann equation and discrete solvent simulation when no salt is added are given in Figs. 1 and 2. We notice the presence of oscillation in the discrete solvent calculation curve. The two sharp peaks close to the planes represent ions that are strongly associated to the surface. Their mobility is very low: they have to be described in terms of surface conductivity. The Poisson–Boltzmann equation underestimates their intensity. The discrete solvent curves oscillate around the Poisson–Boltzmann values. These oscillations do not come from the ionic correlations (due to the size of the ions) that are
Fig. 1. Counterionic distributions by discrete solvent simulation and Poisson – Boltzmann continuous solvent calculation. The distributions are symmetrical between 0 and 12.3 A, . This corresponds to the half-size of the simulation box minus the thickness of the clay sheet (6.45 A, /2) and the radius of Na+ (2.1 A, ).
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Fig. 2. Integrated values of the counterionic distributions. The distributions are symmetrical between 0 and 12.3 A, . This corresponds to the half-size of the simulation box minus the thickness of the clay sheet (6.45 A, /2) and the radius of Na+ (2.1 A, ).
not taken into account in the Poisson– Boltzmann equation. They come from the discrete description of water. Indeed, the period of the oscillations is close to the mean distance between two oxygen atoms in bulk water (2.7 A, ). Furthermore, in continuous solvent Monte-Carlo simulations [12] the oscillations are very different; their intensity is much less substantial. Thus, the Poisson–Boltzmann approximation averages the oscillations. The two distributions are in agreement, particularly in the middle of the pore. The transport properties of electrolytes that are measured are generally linked to the integral of the concentration (see, e.g. Eq. (29)). In this case (Fig. 2), the agreement is excellent. Likewise, the thermodynamic quantities obtained from the integration of the pair correlations functions in the Debye –Hu¨ ckel electrolyte theory can be used for dilute solutions whereas the pair correlations functions are themselves much less valid. This justifies a continuous solvent model of transport properties in montmorillonite. This mesoscopic model will be more acceptable for rather large and medium interlayer spacings (L \ 20 A, ) even if in this article transport property calculations have been made until L = 10 A, .
In the microscopic model, the sheets have a thickness and they are periodically reproduced in all directions. In contrast, in the mesoscopic model, there are only two sheets that have no thickness. In fact, this difference between the boundary conditions does not explain the differences between the results. Indeed, on the grounds of the Gauss theorem, if the sheets are uniformly charged, the electric fields created by the other planes are canceled. In the same way, the thickness of the sheets does not modify the electric field in the interlayer spacing. Consequently, the Poisson–Boltzmann solution calculated with the boundary conditions of the mesoscopic model is the same when compared to the solution calculated with the boundary conditions of the microscopic model. To conclude, even if the model is all the more accurate since the interlayer spacing is large, that is to say not completely appropriate for a compacted clay, the accuracy between the curves allows to assume that the Poisson–Boltzmann model is a good approximation for ionic distributions although it is simple. It had been previously pointed out that in the case of clays, the Poisson– Boltzmann equation was in agreement with continuous solvent simulations [12]. This study shows that in addition, this model is consistent with discrete solvent simulations, at least if no salt is added.
2.2. Donnan effect When a salt, NaCl for example, is added in the water in contact with the clay, some salt penetrates between the sheets. It is well known that cint " cext where cint and cext are, respectively, the concentrations of salt inside and outside the pore: this is the Donnan effect. Between clay sheets, the average concentration of the cation (which is the counterion) c 01 and the average concentration of the anion c 02 are given by: c 01 =
2| + cint LZe
c 02 = cint
(7) (8)
J.-F. Dufreˆ che et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 195 (2001) 171–180
Experimentally, it is useful to evaluate the Donnan effect because the internal concentration cint is often unknown. The relation between cint and cext is found by equating the chemical potentials of the salt inside and outside the pore: v sext =v sint. The expression of the chemical potential of the ion i in the interlayer spacing that is compatible with the Poisson –Boltzmann approximation is vi =v 0i + kBT ln ci (z) +Zi e(z)
(9)
Using Eq. (1) and Z1 = − Z2 =Z, we get the chemical potential of the salt inside the clay: v sint =v 01 +v 02 +kBT ln M1M2
(10)
In the aqueous solution: v sext =v 01 + v 02 +2kBT ln cext
(11)
M1M2 =c 2ext
(12)
As is defined within an additive constant, we can choose [13] M1 =cext. Then, M2 =cext too. This choice allows one to simplify the Poisson– Boltzmann equation: d2 s 2ext = sinh(Z) dz 2 Z
In the case of high salinities or interlayer spacings (sextL 1), the effects of both charged planes can be decoupled, so that a superposition approximation can be used: (z)= 0(z− L/2)+ 0(z+ L/2)
(13)
(15)
where 0(z) is the Poisson–Boltzmann potential for a simple charged surface at z=0: tanh
Z0 = ke − sextz 4
(16)
k is an integration factor, given by k= 1 +1/ ZE − 1/ZE with E=2yLB|/sexte. Assuming that c1(0)= c2(0)= cext and = 0 at z= 0, i.e. the charged planes are far enough from each other for the electrolyte to be homogeneous in the middle of the pore, we get: cint =
Thus, the chemical equilibrium implies
175
1 L
&
L/2
c2(z) dz=
− L/2
cext L
&
L/2
e − Z(z) dz
(17)
− L/2
which gives, according to Eqs. (15) and (16): cint k 8 = 1− cext 1+k sextL
(18)
In the whole concentration range, the Donnan ratio cint/cext can be approximated by a Pade´ expansion:
where sext = 8yLBZ 2cext is the Debye screening constant. By resolving the Poisson– Boltzmann equation, we are able to calculate cint as a function of cext. The Donnan ratio cint/cext as a function of cext is given in Fig. 3 for several interlayer spacings. In the case of low salinities or interlayer spacings (sextL 1), is not disrupted by the presence of salt. For a highly charged clay: F = yZLLB|/ e] 1, a development of the analytical solution of the Poisson – Boltzmann equation without salt leads to [13]
cint s 2extL 2 3 = 1+ 2 F cext 8y
(14)
For the montmorillonite described in Section 1 and NaCl as an added salt, we get |/e: 0.008 A, − 2. As LB :7 A, for an aqueous solution at ambient temperature, we notice that F ] 1 as soon as L exceeds 5 A, . It is the case here.
Fig. 3. Ratio of internal over external concentration as a function of the external concentration. At low interlayer spacings, cint/cext tends rapidly to a linear function when cext tends to zero, exhibiting the limiting behavior cint 8c 2ext. For large interlayer spacings, cint/cext tends to a step function as expected.
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cint A(sextL)2 : cext 1+ABsextL + A(sextL)2
(19)
where A=(1 +3/F)/(8y 2) and B =8k/(1 +k). This Pade´ approximation gives both limiting cases. In the intermediate range, this analytical expansion is consistent with the Poisson–Boltzmann numerical solution with 5– 10% error. It has been already verified that the Donnan effect calculated by the Poisson– Boltzmann equation was in quantitative agreement with Monte-Carlo simulations in the grand-canonical ensemble and with experimental data [13]. If the external concentration is known, we are then able to calculate the internal concentrations from which ionic distributions can be determined. Then, the study of transport properties follows.
3. Transport in interlayer spacing: electroosmosis and conductivity
3.1. Electroosmosis In this section, we take an adaptation of the classical Smoluchowski method for electroosmosis to porous systems. The electroosmotic velocity (or the solvent velocity) vs verifies the Stokes equation:
Fig. 4. Electroosmotic velocity profiles for different interlayer spacings without added salt.
vs(z)=
e [(z)− (L/2)]Eey 4ypLB
(23)
In the case of no added salt, (z)=(2/ Z) ln cos hz, so that we get: vs(z)=
e cos hz ln Ee 2ypZLB cos hL/2 y
(24)
with 9·vs = 0 if the solvent is incompressible. p is the solvent shear viscosity, 9p is the pressure gradient and F6 is the body force per unit volume. If the electric field is directed along the y-axis, the induced electroosmotic velocity verifies vs = 6s(z)ey, where ey is the unit vector along y. The projection along the y-axis gives:
The electroosmotic mobility profiles without added salt, for several interlayer spacings, are represented in Fig. 4. In the case of an added salt, electroosmotic velocities are calculated numerically. The variation of the electroosmotic velocity profile is represented for an interlayer distance of 40 A, , for various concentrations (Fig. 5). For high added salt concentrations, the parabolic profile is replaced by a plateau pattern corresponding to the rapidly decreasing double layer thickness. As expected, a higher salt concentration increases electrostatic screening and reduces electroosmosis.
p6¦s (z)+zelE = 0
3.2. Conducti6ity
pDvs +F6 − 9p = 0
(20)
(21)
The charge density can be obtained from the Poisson equation ¦(z) = −zel/m0mr. Thus p6¦s (z)−m0mr¦(z)E =0
(22)
By setting =e/kBT, with the boundary conditions vs(z)=0 for z = 9 L/2, we get after integration
3.2.1. Specific conducti6ity The conductivity has several contributions whose order of magnitude varies with increasing concentration. Without added salt the conductivity is only due to the intrinsic contribution of counterions and to the electroosmotic flow.
J.-F. Dufreˆ che et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 195 (2001) 171–180
= int + e.o.
(25)
The ionic atmosphere around cations is not distorted by the opposite movement of the possible anions. In our model, the intrinsic contribution of counterions does not involve any relaxation nor electrophoretic effect, and follows strictly the Nernst– Einstein relation D/u= RT/ZF
(26)
where D is the diffusion coefficient of the counterion and u its intrinsic mobility. It could be determined by measuring the self-diffusion coefficient of those counterions by NMR or by tracers. Experiments are planned for montmorillonite. This intrinsic conductivity of counterions will be our reference value, with respect to which any other effect will be evaluated. It depends on each particular experimental case and cannot be theoretically predicted except by computer simulation. Indeed, it contains the surface conductance in the double layer. The electroosmotic conductivity has different values according to the interlayer spacing. In the case of no added salt, the electroosmotic conductivity can be directly evaluated by averaging the electroosmotic velocity over the interlayer spacing:
Fig. 5. Electroosmotic velocity profiles for different added salt concentrations. The interlayer spacing is 40 A, . For high added salt concentrations, the velocity pattern tends rapidly to a plateau value.
e.o. =
tan(hL/2) e h −1 hL/2 4y 2Z 2L 2Bp 2
2
n
177
(27)
In the case of added salt a new contribution appears, which has to be added to the intrinsic and electroosmotic ones: the salt conductivity. However, the intrinsic and electroosmotic contributions are modified by the presence of the added electrolyte and have to be evaluated according to this new situation. In this paper we neglect any modification of the intrinsic conductivity. The salt contribution can be estimated noticing that it results from the extension of the intrinsic mobility (case of no added salt) and from the presence of supplementary co- and counterions corresponding to the added salt. The increasing ionic strength leads to non-ideal contributions of the same nature as for ordinary electrolytes in solution, which can therefore be analyzed in the same way, by introducing non-ideality MSA factors for electrolytic conductance [14]. We have then: = int + salt + e.o.
(28)
In the presence of an added electrolyte the electroosmotic contribution is given by: e.o. =
1 L
& L/2
− L/2
% Zi eci (z) us(z) dz
(29)
i
where us(z)= 6s(z)/E. As we have seen in Section 2, the relevant variable is not the internal concentration of added salt, but the external salt concentration; we use it as a variable in the estimation of the electroosmotic specific conductivity, for different interlayer spacings, as represented in Fig. 6. As expected, the electroosmotic specific conductivity is higher for thinner interlayer spacing and decreases slightly with increasing external salt concentration. The MSA treatment of the specific conductivity of the salt gives salt by using the Fuoss–Onsager equations for the non-equilibrium pair distribution functions. We used an analytical theory that gives extended laws for the variations with concentration of transport coefficients of strong and associated electrolytes [20–22]. In the case of bulk solutions, this theory was found to be in agreement with experimental data up to molar concentrations. The basic ingredients are the MSA
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Fig. 6. Electroosmotic conductivity versus external concentration for different interlayer spacings.
equilibrium pair distribution functions, whereas the hydrodynamic interactions are treated by the mean of Oseen’s tensor approximation. The results for a NaCl model with a mean ionic diameter d=3.3 A, are presented in Fig. 7. The specific MSA conductivity varies almost linearly with the external concentration except for small interlayer spacing and small concentrations, on the grounds of the Donnan effect. The total specific conductivity (taking as zero value the intrinsic contribution to the counterion conductivity) can then be represented as a func-
Fig. 8. Total specific conductivity (taking the intrinsic conductivity as zero reference value) as a function of the external concentration.
tion of the external concentration. The small interlayer spacing has a relatively high conductivity without external salt, because of the dominant electroosmotic current. The results are presented in Fig. 8. There are clearly two domains in the curves. For low concentrations, electroosmosis is the predominant effect. The conductivity is all the more important since the interlayer spacing is small. In contrast, for high concentrations, the salt conductivity overcomes the electroosmosis: the total quantity is more important for large interlayer spacings. Between these two domains, the intersection point is obtained for cext =0.5 mol l − 1.
3.2.2. Equi6alent conducti6ity The linearity between the specific conductivity and the external concentration simply expresses the fact that the number of ions is proportional to the external concentration. As with the bulk theory, the deviation from this ideal behavior can be expressed in terms of equivalent conductivity \=
Fig. 7. Salt conductivity versus external concentration for different interlayer spacings.
Zcext
(30)
The MSA –Fuoss –Onsager theory is very interesting here because it is able to describe the conductivity of electrolyte solutions up to molar concentrations. In Figs. 9 and 10 are presented
J.-F. Dufreˆ che et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 195 (2001) 171–180
Fig. 9. Equivalent electroosmotic conductivity versus external concentration for different interlayer spacings.
the electroosmotic and MSA equivalent conductivities. This last one starts from zero without added salt, as required by the electrostatic repulsion between the charged sites of montmorillonite and the coions of the added salt. In the presence of an excess of added salt, it tends to a nearby plateau value depending strongly on the interlayer spacing. Larger interlayer spacings exhibit a broad maximum at low ionic strength. The curves are close to the bulk conductivity only for large interlayer spacings and important external concentrations.
179
Fig. 11. Total equivalent conductivity (taken from the total specific conductivity without the intrinsic contribution) versus external concentration for different interlayer spacings.
In any case (large or small interlayer spacing), the total equivalent conductance represented in Fig. 11 exhibits a rapid decrease with increasing external concentration. The divergence of the electroosmosis ant the total equivalent conductivities simply comes from the electroosmosis of the counterion which is not zero for cext = 0. Thus, this aspect is not very significant.
3.2.3. Limiting laws It should be noted that Onsager’s limiting laws for the relation between equivalent conductance and external concentration are not valid. Indeed, it can be showed that \e.o. and \salt do not vary linearly as a function of the square root of the external concentration cext at low salinities, on the grounds of the Donnan effect. The internal concentration is given by Eq. (14) which is equivalent to cint =
L 2Z 2LB 3 1+ c 2ext = xc 2ext y F
(31)
Then, by using the analytical solution of the Poisson–Boltzmann equation without added salt, we obtain the limiting law for the electroosmotic conductivity: Fig. 10. Equivalent salt conductivity versus external concentration for different interlayer spacings.
e.o. = 0e.o. − Gc 2ext
(32)
J.-F. Dufreˆ che et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 195 (2001) 171–180
180
where 0e.o. is given by Eq. (24) and G is a positive constant. Thus, the added salt decreases the electroosmotic conductivity. At low salinities, the salt conductivity can be obtained from the Onsager limiting law of the bulk electrolyte: salt = A−B cint =A − B xcext Zcint
(33)
where A and B are known positive constants. The specific conductivity defined with the external concentration is then: \salt =
salt =Axcext −Bx 3/2c 2ext Zcext
2. Moreover, microscopic equivalent to Smoluchowski velocity could be derived giving Kubo like relations for the transport of solvent and added salt. This would permit to compare the transport given by molecular dynamics and continuous solvent models.
(34)
Consequently, the total equivalent conductivity varies in cext at low salinities. 4. Conclusions The present study shows the consistency of the two approaches, discrete solvent and continuous solvent models, with regard to the ionic distributions between the sheets in the absence of added salt. The main difference is the presence of oscillations that are not due to the size of the ions: they express the molecular description of the solvent. The tractability of continuous solvent description gives the possibility to incorporate the salt effect in equilibrium and transport properties in clays. The electroosmosis and the conductivity have been calculated. They strongly depend on the interlayer spacing. The bulk limiting laws are not valid with the external concentration which is the experimental parameter. Experiments are planned to determine the transport coefficients of ions and water in montmorillonite, in order to check the validity of the above-presented model. Those experiments, except for local dynamics, will concern mainly the case of added salt. Other models are obviously possible: 1. DFT related models for continuous solvent constitute a good alternative way to improve the Poisson – Boltzmann method which is the simplest DFT.
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Models for electrokinetic phenomena in montmorillonite J.-F. Dufreˆche *, V. Marry, O. Bernard, P. Turq ANDRA and Laboratoire Liquides Ioniques et Interfaces Charge´es, case courrier 51, Uni6ersite´ P. et M. Curie, 4 place Jussieu, F-75252 Paris Cedex 05, France
Abstract Clays present remarkable electrokinetic features since they exist from very dilute colloidal state to nanoporous materials, depending on the water/clay ratio. The case of low volume fraction Vwater/Vtot which corresponds to compact systems is examined. The ionic distributions have been evaluated by Poisson– Boltzmann like models and compared to discrete solvent simulations. Several electrokinetic properties (electroosmosis and conductance) have been calculated, in the framework of the mean spherical approximation introduced in the Fuoss– Onsager transport theory. It is found that Onsager’s limiting laws in terms of external concentration are not valid on the grounds of the Donnan effect. © 2001 Elsevier Science B.V. All rights reserved. Keywords: Electrokinetic phenomena; Montmorillonite; Clays
1. Introduction Clay minerals as montmorillonite have a significant role in the storage of radionuclear wastes: when compacted, they are used as a barrier towards radionuclides. This is one of the reasons why the study of transport properties in these materials is essential. Clays are formed by large platelets composed of stacks of elementary sheets. In the case of montmorillonite, a sheet is made up of a layer of octahedral oxides (Al3 + , Mg2 + , etc.) between two layers of tetrahedral oxides (Si4 + , Al3 + , etc.). As some of the cations are substituted by other cations of lower valency, clay sheets are negatively charged. Counterions which are located between sheets are partly responsible for the swelling be* Corresponding author.
havior of montmorillonite in the presence of water. Many factors such as the general structure of the clay and its level of hydration seem to be involved in the retention of elements. When they are not adsorbed on the surface of a clay particle, ions or radionuclides can move either between platelets or between the sheets inside a platelet. In this article, the motion study will be reduced to the transport between two parallel sheets of clay. Nevertheless, this approximation seems to be acceptable for a compacted clay whose platelets are inclined to stick to each other. Monte-Carlo and molecular dynamics simulations have already been undertaken to describe the equilibrium and dynamic properties of this system: several kinds of counterions such as Li+ [1,2], Na+ [1–6], K+ [1,7,8] and Cs+ [9,10] were studied. These simulations use microscopic descriptions of the system: water and clay sheets are
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considered to be discrete sets of atoms. Unfortunately, this method is expensive as it is time consuming, and gives dynamic quantities as diffusion coefficients only at very short times (about a nanosecond). In this article, we develop a quite simple mesoscopic model to describe ions’ motions in confined water. Clay sheets are considered as uniformly charged planes and water as a continuum. Such a model, based on the Poisson– Boltzmann equation, was already used to calculate equilibrium quantities such as osmotic pressure [11]. Ionic distributions between two clay sheets were compared with those obtained by continuous solvent Monte-Carlo simulations [12]. It was shown that the Poisson– Boltzmann equation was satisfactory for medium and large interlayer spacings. Donnan effects deduced from Poisson –Boltzmann for didodecyldimethylammonium bromide, which is a similar swollen lamellar medium, were in good agreement with experiments of atomic adsorption [13]. In this article, ionic distributions from Poisson – Boltzmann are compared to those obtained by discrete solvent Monte-Carlo simulations. Then they are used to determine transport properties for several salt concentrations in the interlayer spacing (electroosmosis and conductivities). As a continuous solvent model, approximations are made towards the microscopic description. However, it allows one to get dynamic quantities for times comparable with experiments. The calculation that will be described next is based on the mean spherical approximation (MSA) transport theory. A similar theory was previously used for Nafion membranes and succeeded in reproducing experimental conductivity data [14].
2. Equilibrium properties: microscopic and mesoscopic approaches
2.1. Ionic distributions In the adopted model, clay sheets are considered as parallel infinite planes. They are uniformly charged and the surface charge density is |= 0.0161/2e A, − 2. The water inside the interlayer spacing is a continuum. Its viscosity and dielectric
constant are bulk water’s ones: at 298 K, p= 0.89× 10 − 3 Pa s and mr = 78.3, respectively. Counterions are Na+ and the added salt NaCl. The electrolyte is simply treated by taking an average radius.
2.1.1. Poisson– Boltzmann treatment If ci (r) is the concentration of the ion i at a distance r from a central particle we can write, according to Boltzmann statistics:
ci (r)= Mi exp −
Vi (r) kBT
(1)
where Vi (r) can be identified with the electrostatic energy Vi (r)= ei(r). ei = Zi e is the charge of i and (r) the electrostatic potential. In the case of two horizontal charged planes separated by a distance L, and ci only depend on z. Mi is obtained by integrating ci (z):
&
+ L/2
ci (z) dz= Mi
&
− L/2
+ L/2
exp −
− L/2
=Lc
ei(z) dz kBT
0 i
(2)
0 i
where c is the average concentration of i in the pore. By replacing in the Poisson equation D = − i ei ci (r)/m0mr, we get: D = − % i
ei Mi e exp − i kBT m0mr
(3)
Setting = e/kBT, we get: D = − 4yLB% Zi Mi e − Zi
(4)
i
where LB = e 2/4ym0mrkBT is the Bjerrum length. The symmetry of the system implies d/dz (z= 0)= 0 as one of the boundary conditions. Thus, is defined within an additive constant and the Mi depend on the choice of this constant. Indeed, if is changed into % with = %+ 0 the concentrations become − Zi % ci = Mi e − Zi = Mi e − Zi 0e − Zi % = M %e i
(5)
Thus we can choose either the constant or one of the Mi. When there is no added salt in the interlayer spacing, the differential equation can be analytically solved. The counterionic concentration is given by:
J.-F. Dufreˆ che et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 195 (2001) 171–180
h 1 2yZ 2LB cos2(hz)
173
2
c(z)=
(6)
with h tan(hL/2) =2yZLB|/e. In the presence of added salt, (z) and ci (z) are numerically determined. The ionic distributions of this system calculated by the Poisson– Boltzmann equation have already been validated by Monte-Carlo simulations in which water is considered as a continuum [12]. At low distances L ( B10 A, ), discrete solvent simulations showed that distributions presented oscillations, indeed even peaks, which are characteristic of the discrete nature of ions and water molecules. The latter organize themselves in layers and the system is more stable for distances L corresponding to mono-, bi- or trilayers of water molecules. As we assumed that this feature would die down as L increases, we decided to compare Poisson– Boltzmann distributions with those obtained with a discrete solvent Monte-Carlo program for larger values of the interlayer spacing (\ 20 A, ). As the model used in the simulation takes the fine structure of the montmorillonite and the solvent into account, it allows us to test the accuracy of the simple mesoscopic model where the clay is described by the oversimplified Poisson– Boltzmann equation.
2.1.2. Discrete sol6ent simulations The structure of the montmorillonite used in the microscopic simulation was taken from X-ray diffraction studies given in the literature [15,16]. Its formula is Na0.75[Si7.75Al0.25](Si7.75Al0.25) O20(OH)4. The simulation box is formed by two half-sheets containing eight clay units. The charges of the sheets’ atoms are Skipper’s ones [17]. Van der Waals interactions are reduced to Lennard-Jones potentials, whose parameters are given by Smith [10]. The water is described by the well-known SPC/E model [18]. This potential has been chosen because its dielectric constant m SPC/ r E= 819 5 [19] is very close to the experimental value (mr = 78.3). Indeed, in the Poisson–Boltzmann model, the solvent is only described by this physical quantity, so that this value is essential for the comparison. Monte-Carlo equilibration has been performed in the (N, P, T) ensemble. The interlayer spacing
varies in order to get a pressure P= 1 bar. The number of water molecules in the box was fixed and equal to 300. The average height of the box evaluated after equilibration was 35.5 A, . By subtracting the thickness of the clay sheet, we get the distance between the two charged planes of the mesoscopic model. Poisson–Boltzmann ionic distributions are then calculated by taking an average radius of about 2.1 A, for counterions Na+. This is close to the Pauling radius (2.0 A, ).
2.1.3. Comparison of ionic distributions The counterionic distributions and their integrals obtained by the Poisson–Boltzmann equation and discrete solvent simulation when no salt is added are given in Figs. 1 and 2. We notice the presence of oscillation in the discrete solvent calculation curve. The two sharp peaks close to the planes represent ions that are strongly associated to the surface. Their mobility is very low: they have to be described in terms of surface conductivity. The Poisson–Boltzmann equation underestimates their intensity. The discrete solvent curves oscillate around the Poisson–Boltzmann values. These oscillations do not come from the ionic correlations (due to the size of the ions) that are
Fig. 1. Counterionic distributions by discrete solvent simulation and Poisson – Boltzmann continuous solvent calculation. The distributions are symmetrical between 0 and 12.3 A, . This corresponds to the half-size of the simulation box minus the thickness of the clay sheet (6.45 A, /2) and the radius of Na+ (2.1 A, ).
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Fig. 2. Integrated values of the counterionic distributions. The distributions are symmetrical between 0 and 12.3 A, . This corresponds to the half-size of the simulation box minus the thickness of the clay sheet (6.45 A, /2) and the radius of Na+ (2.1 A, ).
not taken into account in the Poisson– Boltzmann equation. They come from the discrete description of water. Indeed, the period of the oscillations is close to the mean distance between two oxygen atoms in bulk water (2.7 A, ). Furthermore, in continuous solvent Monte-Carlo simulations [12] the oscillations are very different; their intensity is much less substantial. Thus, the Poisson–Boltzmann approximation averages the oscillations. The two distributions are in agreement, particularly in the middle of the pore. The transport properties of electrolytes that are measured are generally linked to the integral of the concentration (see, e.g. Eq. (29)). In this case (Fig. 2), the agreement is excellent. Likewise, the thermodynamic quantities obtained from the integration of the pair correlations functions in the Debye –Hu¨ ckel electrolyte theory can be used for dilute solutions whereas the pair correlations functions are themselves much less valid. This justifies a continuous solvent model of transport properties in montmorillonite. This mesoscopic model will be more acceptable for rather large and medium interlayer spacings (L \ 20 A, ) even if in this article transport property calculations have been made until L = 10 A, .
In the microscopic model, the sheets have a thickness and they are periodically reproduced in all directions. In contrast, in the mesoscopic model, there are only two sheets that have no thickness. In fact, this difference between the boundary conditions does not explain the differences between the results. Indeed, on the grounds of the Gauss theorem, if the sheets are uniformly charged, the electric fields created by the other planes are canceled. In the same way, the thickness of the sheets does not modify the electric field in the interlayer spacing. Consequently, the Poisson–Boltzmann solution calculated with the boundary conditions of the mesoscopic model is the same when compared to the solution calculated with the boundary conditions of the microscopic model. To conclude, even if the model is all the more accurate since the interlayer spacing is large, that is to say not completely appropriate for a compacted clay, the accuracy between the curves allows to assume that the Poisson–Boltzmann model is a good approximation for ionic distributions although it is simple. It had been previously pointed out that in the case of clays, the Poisson– Boltzmann equation was in agreement with continuous solvent simulations [12]. This study shows that in addition, this model is consistent with discrete solvent simulations, at least if no salt is added.
2.2. Donnan effect When a salt, NaCl for example, is added in the water in contact with the clay, some salt penetrates between the sheets. It is well known that cint " cext where cint and cext are, respectively, the concentrations of salt inside and outside the pore: this is the Donnan effect. Between clay sheets, the average concentration of the cation (which is the counterion) c 01 and the average concentration of the anion c 02 are given by: c 01 =
2| + cint LZe
c 02 = cint
(7) (8)
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Experimentally, it is useful to evaluate the Donnan effect because the internal concentration cint is often unknown. The relation between cint and cext is found by equating the chemical potentials of the salt inside and outside the pore: v sext =v sint. The expression of the chemical potential of the ion i in the interlayer spacing that is compatible with the Poisson –Boltzmann approximation is vi =v 0i + kBT ln ci (z) +Zi e(z)
(9)
Using Eq. (1) and Z1 = − Z2 =Z, we get the chemical potential of the salt inside the clay: v sint =v 01 +v 02 +kBT ln M1M2
(10)
In the aqueous solution: v sext =v 01 + v 02 +2kBT ln cext
(11)
M1M2 =c 2ext
(12)
As is defined within an additive constant, we can choose [13] M1 =cext. Then, M2 =cext too. This choice allows one to simplify the Poisson– Boltzmann equation: d2 s 2ext = sinh(Z) dz 2 Z
In the case of high salinities or interlayer spacings (sextL 1), the effects of both charged planes can be decoupled, so that a superposition approximation can be used: (z)= 0(z− L/2)+ 0(z+ L/2)
(13)
(15)
where 0(z) is the Poisson–Boltzmann potential for a simple charged surface at z=0: tanh
Z0 = ke − sextz 4
(16)
k is an integration factor, given by k= 1 +1/ ZE − 1/ZE with E=2yLB|/sexte. Assuming that c1(0)= c2(0)= cext and = 0 at z= 0, i.e. the charged planes are far enough from each other for the electrolyte to be homogeneous in the middle of the pore, we get: cint =
Thus, the chemical equilibrium implies
175
1 L
&
L/2
c2(z) dz=
− L/2
cext L
&
L/2
e − Z(z) dz
(17)
− L/2
which gives, according to Eqs. (15) and (16): cint k 8 = 1− cext 1+k sextL
(18)
In the whole concentration range, the Donnan ratio cint/cext can be approximated by a Pade´ expansion:
where sext = 8yLBZ 2cext is the Debye screening constant. By resolving the Poisson– Boltzmann equation, we are able to calculate cint as a function of cext. The Donnan ratio cint/cext as a function of cext is given in Fig. 3 for several interlayer spacings. In the case of low salinities or interlayer spacings (sextL 1), is not disrupted by the presence of salt. For a highly charged clay: F = yZLLB|/ e] 1, a development of the analytical solution of the Poisson – Boltzmann equation without salt leads to [13]
cint s 2extL 2 3 = 1+ 2 F cext 8y
(14)
For the montmorillonite described in Section 1 and NaCl as an added salt, we get |/e: 0.008 A, − 2. As LB :7 A, for an aqueous solution at ambient temperature, we notice that F ] 1 as soon as L exceeds 5 A, . It is the case here.
Fig. 3. Ratio of internal over external concentration as a function of the external concentration. At low interlayer spacings, cint/cext tends rapidly to a linear function when cext tends to zero, exhibiting the limiting behavior cint 8c 2ext. For large interlayer spacings, cint/cext tends to a step function as expected.
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cint A(sextL)2 : cext 1+ABsextL + A(sextL)2
(19)
where A=(1 +3/F)/(8y 2) and B =8k/(1 +k). This Pade´ approximation gives both limiting cases. In the intermediate range, this analytical expansion is consistent with the Poisson–Boltzmann numerical solution with 5– 10% error. It has been already verified that the Donnan effect calculated by the Poisson– Boltzmann equation was in quantitative agreement with Monte-Carlo simulations in the grand-canonical ensemble and with experimental data [13]. If the external concentration is known, we are then able to calculate the internal concentrations from which ionic distributions can be determined. Then, the study of transport properties follows.
3. Transport in interlayer spacing: electroosmosis and conductivity
3.1. Electroosmosis In this section, we take an adaptation of the classical Smoluchowski method for electroosmosis to porous systems. The electroosmotic velocity (or the solvent velocity) vs verifies the Stokes equation:
Fig. 4. Electroosmotic velocity profiles for different interlayer spacings without added salt.
vs(z)=
e [(z)− (L/2)]Eey 4ypLB
(23)
In the case of no added salt, (z)=(2/ Z) ln cos hz, so that we get: vs(z)=
e cos hz ln Ee 2ypZLB cos hL/2 y
(24)
with 9·vs = 0 if the solvent is incompressible. p is the solvent shear viscosity, 9p is the pressure gradient and F6 is the body force per unit volume. If the electric field is directed along the y-axis, the induced electroosmotic velocity verifies vs = 6s(z)ey, where ey is the unit vector along y. The projection along the y-axis gives:
The electroosmotic mobility profiles without added salt, for several interlayer spacings, are represented in Fig. 4. In the case of an added salt, electroosmotic velocities are calculated numerically. The variation of the electroosmotic velocity profile is represented for an interlayer distance of 40 A, , for various concentrations (Fig. 5). For high added salt concentrations, the parabolic profile is replaced by a plateau pattern corresponding to the rapidly decreasing double layer thickness. As expected, a higher salt concentration increases electrostatic screening and reduces electroosmosis.
p6¦s (z)+zelE = 0
3.2. Conducti6ity
pDvs +F6 − 9p = 0
(20)
(21)
The charge density can be obtained from the Poisson equation ¦(z) = −zel/m0mr. Thus p6¦s (z)−m0mr¦(z)E =0
(22)
By setting =e/kBT, with the boundary conditions vs(z)=0 for z = 9 L/2, we get after integration
3.2.1. Specific conducti6ity The conductivity has several contributions whose order of magnitude varies with increasing concentration. Without added salt the conductivity is only due to the intrinsic contribution of counterions and to the electroosmotic flow.
J.-F. Dufreˆ che et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 195 (2001) 171–180
= int + e.o.
(25)
The ionic atmosphere around cations is not distorted by the opposite movement of the possible anions. In our model, the intrinsic contribution of counterions does not involve any relaxation nor electrophoretic effect, and follows strictly the Nernst– Einstein relation D/u= RT/ZF
(26)
where D is the diffusion coefficient of the counterion and u its intrinsic mobility. It could be determined by measuring the self-diffusion coefficient of those counterions by NMR or by tracers. Experiments are planned for montmorillonite. This intrinsic conductivity of counterions will be our reference value, with respect to which any other effect will be evaluated. It depends on each particular experimental case and cannot be theoretically predicted except by computer simulation. Indeed, it contains the surface conductance in the double layer. The electroosmotic conductivity has different values according to the interlayer spacing. In the case of no added salt, the electroosmotic conductivity can be directly evaluated by averaging the electroosmotic velocity over the interlayer spacing:
Fig. 5. Electroosmotic velocity profiles for different added salt concentrations. The interlayer spacing is 40 A, . For high added salt concentrations, the velocity pattern tends rapidly to a plateau value.
e.o. =
tan(hL/2) e h −1 hL/2 4y 2Z 2L 2Bp 2
2
n
177
(27)
In the case of added salt a new contribution appears, which has to be added to the intrinsic and electroosmotic ones: the salt conductivity. However, the intrinsic and electroosmotic contributions are modified by the presence of the added electrolyte and have to be evaluated according to this new situation. In this paper we neglect any modification of the intrinsic conductivity. The salt contribution can be estimated noticing that it results from the extension of the intrinsic mobility (case of no added salt) and from the presence of supplementary co- and counterions corresponding to the added salt. The increasing ionic strength leads to non-ideal contributions of the same nature as for ordinary electrolytes in solution, which can therefore be analyzed in the same way, by introducing non-ideality MSA factors for electrolytic conductance [14]. We have then: = int + salt + e.o.
(28)
In the presence of an added electrolyte the electroosmotic contribution is given by: e.o. =
1 L
& L/2
− L/2
% Zi eci (z) us(z) dz
(29)
i
where us(z)= 6s(z)/E. As we have seen in Section 2, the relevant variable is not the internal concentration of added salt, but the external salt concentration; we use it as a variable in the estimation of the electroosmotic specific conductivity, for different interlayer spacings, as represented in Fig. 6. As expected, the electroosmotic specific conductivity is higher for thinner interlayer spacing and decreases slightly with increasing external salt concentration. The MSA treatment of the specific conductivity of the salt gives salt by using the Fuoss–Onsager equations for the non-equilibrium pair distribution functions. We used an analytical theory that gives extended laws for the variations with concentration of transport coefficients of strong and associated electrolytes [20–22]. In the case of bulk solutions, this theory was found to be in agreement with experimental data up to molar concentrations. The basic ingredients are the MSA
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Fig. 6. Electroosmotic conductivity versus external concentration for different interlayer spacings.
equilibrium pair distribution functions, whereas the hydrodynamic interactions are treated by the mean of Oseen’s tensor approximation. The results for a NaCl model with a mean ionic diameter d=3.3 A, are presented in Fig. 7. The specific MSA conductivity varies almost linearly with the external concentration except for small interlayer spacing and small concentrations, on the grounds of the Donnan effect. The total specific conductivity (taking as zero value the intrinsic contribution to the counterion conductivity) can then be represented as a func-
Fig. 8. Total specific conductivity (taking the intrinsic conductivity as zero reference value) as a function of the external concentration.
tion of the external concentration. The small interlayer spacing has a relatively high conductivity without external salt, because of the dominant electroosmotic current. The results are presented in Fig. 8. There are clearly two domains in the curves. For low concentrations, electroosmosis is the predominant effect. The conductivity is all the more important since the interlayer spacing is small. In contrast, for high concentrations, the salt conductivity overcomes the electroosmosis: the total quantity is more important for large interlayer spacings. Between these two domains, the intersection point is obtained for cext =0.5 mol l − 1.
3.2.2. Equi6alent conducti6ity The linearity between the specific conductivity and the external concentration simply expresses the fact that the number of ions is proportional to the external concentration. As with the bulk theory, the deviation from this ideal behavior can be expressed in terms of equivalent conductivity \=
Fig. 7. Salt conductivity versus external concentration for different interlayer spacings.
Zcext
(30)
The MSA –Fuoss –Onsager theory is very interesting here because it is able to describe the conductivity of electrolyte solutions up to molar concentrations. In Figs. 9 and 10 are presented
J.-F. Dufreˆ che et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 195 (2001) 171–180
Fig. 9. Equivalent electroosmotic conductivity versus external concentration for different interlayer spacings.
the electroosmotic and MSA equivalent conductivities. This last one starts from zero without added salt, as required by the electrostatic repulsion between the charged sites of montmorillonite and the coions of the added salt. In the presence of an excess of added salt, it tends to a nearby plateau value depending strongly on the interlayer spacing. Larger interlayer spacings exhibit a broad maximum at low ionic strength. The curves are close to the bulk conductivity only for large interlayer spacings and important external concentrations.
179
Fig. 11. Total equivalent conductivity (taken from the total specific conductivity without the intrinsic contribution) versus external concentration for different interlayer spacings.
In any case (large or small interlayer spacing), the total equivalent conductance represented in Fig. 11 exhibits a rapid decrease with increasing external concentration. The divergence of the electroosmosis ant the total equivalent conductivities simply comes from the electroosmosis of the counterion which is not zero for cext = 0. Thus, this aspect is not very significant.
3.2.3. Limiting laws It should be noted that Onsager’s limiting laws for the relation between equivalent conductance and external concentration are not valid. Indeed, it can be showed that \e.o. and \salt do not vary linearly as a function of the square root of the external concentration cext at low salinities, on the grounds of the Donnan effect. The internal concentration is given by Eq. (14) which is equivalent to cint =
L 2Z 2LB 3 1+ c 2ext = xc 2ext y F
(31)
Then, by using the analytical solution of the Poisson–Boltzmann equation without added salt, we obtain the limiting law for the electroosmotic conductivity: Fig. 10. Equivalent salt conductivity versus external concentration for different interlayer spacings.
e.o. = 0e.o. − Gc 2ext
(32)
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180
where 0e.o. is given by Eq. (24) and G is a positive constant. Thus, the added salt decreases the electroosmotic conductivity. At low salinities, the salt conductivity can be obtained from the Onsager limiting law of the bulk electrolyte: salt = A−B cint =A − B xcext Zcint
(33)
where A and B are known positive constants. The specific conductivity defined with the external concentration is then: \salt =
salt =Axcext −Bx 3/2c 2ext Zcext
2. Moreover, microscopic equivalent to Smoluchowski velocity could be derived giving Kubo like relations for the transport of solvent and added salt. This would permit to compare the transport given by molecular dynamics and continuous solvent models.
(34)
Consequently, the total equivalent conductivity varies in cext at low salinities. 4. Conclusions The present study shows the consistency of the two approaches, discrete solvent and continuous solvent models, with regard to the ionic distributions between the sheets in the absence of added salt. The main difference is the presence of oscillations that are not due to the size of the ions: they express the molecular description of the solvent. The tractability of continuous solvent description gives the possibility to incorporate the salt effect in equilibrium and transport properties in clays. The electroosmosis and the conductivity have been calculated. They strongly depend on the interlayer spacing. The bulk limiting laws are not valid with the external concentration which is the experimental parameter. Experiments are planned to determine the transport coefficients of ions and water in montmorillonite, in order to check the validity of the above-presented model. Those experiments, except for local dynamics, will concern mainly the case of added salt. Other models are obviously possible: 1. DFT related models for continuous solvent constitute a good alternative way to improve the Poisson – Boltzmann method which is the simplest DFT.
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