Studies of electrochemical properties of compacted clays by concentration potential method

Journal of Colloid and Interface Science 309 (2007) 262–271 www.elsevier.com/locate/jcis

Studies of electrochemical properties of compacted clays by concentration potential method Andriy Yaroshchuk ∗,1 , Martin A. Glaus, Luc R. Van Loon Laboratory for Waste Management, Paul-Scherrer Institute, 5232 Villigen PSI, Switzerland Received 1 December 2006; accepted 9 February 2007 Available online 16 February 2007

Abstract The development of concentration (membrane) potential upon step-wise change in salt concentration has been studied for diaphragms made of various strongly compacted clays (montmorillonite, illite, kaolinite) equilibrated with 0.1 M NaCl solution. Porous ceramic filters were used to confine the clays mechanically to be able to achieve high extent of compaction (dry density ∼2000 kg/m3 ). A theoretical analysis has revealed that the relaxation pattern is primarily controlled by the properties of porous filters and only slightly depends on the clay properties. At the same time, quasi-stationary values of concentration potential are directly related to the electrochemical perm-selectivity of clay. This property has revealed considerable differences in the electrochemical behaviour of various clays used in this study. This has been attributed to the differences in the micro-structure of clays, in particular to the existence or nonexistence of the so-called interlayer water where cations may retain some mobility. It has also been shown that in clays with high electrochemical perm-selectivity, one can expect a strong increase in the diffusivity of cationic radio-tracers with decreasing ionic strength of equilibrium electrolyte solution. At the same time, low electrochemical perm-selectivity means no noticeable dependence of this kind. The correctness of this observation has been corroborated by the comparison of our findings with the literature data on the diffusion of cationic radio-tracers through compacted montmorillonite (high perm-selectivity) and kaolinite (low perm-selectivity). To check the self-consistency of our approach, we have also carried out sample measurements of diffusion of cationic and anionic radio-tracers through compacted illite. It has been found that the measured effective diffusion coefficients were in excellent agreement with the electrochemical perm-selectivity estimated for this clay from the measurements of concentration potential. © 2007 Elsevier Inc. All rights reserved. Keywords: Compacted clay; Concentration potential; Ion transport number; Transient; Radio-tracers; Diffusion

1. Introduction It has been found experimentally that in compacted clays, the effective diffusion coefficients of cationic and anionic radiotracers are quite different [1,2]. This phenomenon has been sometimes interpreted in terms of the so-called surface diffusion of cations [3,4] and of reduced accessible pore volume for anions [5]. It is logical to assume that this asymmetry is characteristic not only of trace ions but also of cations and anions of dominant salt(s). If that is true, the diffusion of dominant salt * Corresponding author. Fax: +34 93 4015814.

E-mail address: [email protected] (A. Yaroshchuk). 1 Current address: Departament d’Enginyeria Química (EQ), Universitat

Politècnica de Catalunya, Av. Diagonal, 647, Edifici H, 4a planta, 08028 Barcelona, Spain. 0021-9797/$ – see front matter © 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.jcis.2007.02.030

has to be accompanied by the appearance of the so-called concentration (or membrane, or diffusion) potential. The concentration potential in clays has been studied experimentally in noncompacted K-montmorillonite [6,7], in slightly compacted bentonite [8,9] as well as in consolidated samples of natural Callovo-Oxfordian argillite [10,11]. To our knowledge, no studies have been performed of concentration potential in strongly compacted mono-mineral clays. Besides that, correlations between the concentration potential and the diffusivities of cationic and anionic radio-tracers have also not been investigated. Finally, no comparative studies of electrochemical phenomena in various clays (compacted or noncompacted) have been carried out. The purpose of this study is to measure and interpret concentration potential with three different compacted clays (montmo-

A. Yaroshchuk et al. / Journal of Colloid and Interface Science 309 (2007) 262–271

rillonite, illite, and kaolinite) previously equilibrated with 0.1 M NaCl solution. The concentration potential will be measured at various concentration ratios, which will enable us to obtain information on the dependence of clay electrochemical properties on the salt concentration. The results of electrochemical measurements will be compared with those of sample measurements of diffusion of cationic and anionic radio-tracers through the same clay samples. This will enable us to check the selfconsistency of our approach. 2. Theory In the studies of compacted clays, the clays have to be mechanically confined. For that purpose, we use porous filter plates. Due to that, even a step-wise salt concentration change in the inlet reservoir does not give rise to an immediate concentration change at the inlet filter/clay interface. At the same time, it can be shown that in our experimental setup, the diffusion front practically never reaches the interface between the clay and the outlet reservoir. Due to that, the clay can be considered semi-infinite, and we can limit ourselves to considering only two media, namely a porous filter and semi-infinite clay. Each medium is characterized by two properties, namely, the effective diffusion coefficient, Di , and the specific sorption capacity per unit volume, αi . The equation of nonstationary diffusion in each of the media has this classical form ∂ci (x, t) ∂ 2 ci (x, t) αi (1) , = Di ∂t ∂x 2 where the index “i” denotes either the filter (“f”) or the clay (“c”). The boundary conditions follow from the continuity of salt concentration and flux at the interface between the filter and the clay cf (h, t) = cc (h, t),   ∂cf  ∂cc  Df = Dc , ∂x x=h ∂x x=h

(2) (3)

where h is the thickness of the filter. The initial condition is the step-wise change in concentration at the external filter boundary at time t = 0, that is cf (0, t) = cf0 + cfs H (t),

(4)

where H (t) is the unit-step function, cf0 is the initial concentration, and cfs is the step magnitude. If the latter is not too large compared to the initial concentration, one can assume the coefficients in Eq. (1) to be independent of salt concentration. In this case Eq. (1) is linear and can be solved by means of Fourier transform. In particular, for the time-dependent salt concentration at the interface between the filter and the clay, one can obtain this cfs cf (h, t) = cf0 + 2    ∞ 1 dω exp(−iωt/t0 ) × 1 − Im , √ √ π ω cosh( −iω) + r sinh( −iω) −∞

(5)

where the characteristic time, t0 , is defined in this way αf 2 h t0 ≡ Df and r≡

263

(6)

 αc Dc . αf Df

(7)

Now let us show that in homogeneous media, the concentration potential is independent of the shape of concentration profile and is controlled by the values of concentration at the medium boundaries, alone. If the contribution of convection to the ion transfer can be neglected,2 the local ion fluxes can be represented in this way [12] (e)

gt± dμ± j± = − , (F Z± )2 dx

(8)

where g is the electric conductivity per unit area, t± are the socalled ion transport numbers [13], F is the Faraday constant, (e) Z± are the ion charges, μ± are the electrochemical potentials. Under conditions of conventional diffusion, electric current is equal to zero, that is Z+ j+ + Z− j− = 0.

(9)

By using this definition of electrochemical potential (e)

μ± ≡ RT ln(as ) + F Z± ϕ,

(10)

where as is the salt activity, R is the universal gas constant, T is the absolute temperature, and ϕ is the electrostatic potential. By substituting Eqs. (8) and (10) into Eq. (9), we obtain this   t− d ln(a) RT t+ dϕ + =− , (11) dx F Z+ Z− dx where we have taken into account that t+ + t− ≡ 1. In a macroscopically homogeneous medium where the ion transport numbers can be considered to be not explicitly dependent on the coordinate, Eq. (11) can be integrated to yield RT ϕ −ϕ = F 



a  a 

 t− t+ + d ln(a), Z+ Z−

(12)

where the prime and double prime denote the quantities at the left and right boundaries of the medium, respectively. From Eq. (12), it is seen that in a macroscopically homogeneous medium, the concentration potential is fully determined by the values of salt concentration at the medium boundaries and, thus, is independent of the shape of concentration profile. In the particular case of (1:1) salt used in this study, Eq. (12) can be rewritten this way RT ϕ ≡ ϕ  − ϕ  = F

a 



2t+ (a) − 1 d ln(a),

(13)

a 

2 In the case of compacted clays, this assumption is justified due to their very low hydraulic permeability.

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where we have stressed the fact that the ion transport numbers may be dependent on the salt concentration. In this study, we deal with two macroscopically homogeneous media, namely, the inlet porous filter and the clay.3 Accordingly, the total electric potential difference between two reservoirs can be represented as a sum of two components both having the structure of right-hand side of Eq. (13) but distinct ion transport numbers. Since the ion transport numbers within the coarse-porous filter can be reasonably considered independent of salt concentration, the corresponding integral can be taken, and we obtain  

Fϕ(t) (f) af0 + afs = 2t+ − 1 ln RT af (h, t) a f (h,t)



+

(c) 2t+ − 1 d ln(a),

(14)

3. Experiment 3.1. Materials Reagents of highest purity were obtained from Fluka (Buchs, Switzerland) or Merck (Dietikon, Switzerland). De-ionised water was used throughout (Milli-Q water) for preparation of solutions. The montmorillonite was from Milos (Greece). Illite originated from Le Puy (France). These clays were equilibrated three times in series with 1 M NaCl (solid-to-liquid ratio ∼25 g/dm3 ) to remove all soluble salts and/or sparingly soluble minerals such as calcite and to convert the clay into the homo-ionic Na-form. After washing out NaCl by dialysis of the suspensions against water, the clay was freeze-dried. Kaolinite (KGa-2) was obtained from the Source Clay Mineral Repository (University of Missouri, Columbia) and was used as received.

af0 (f)

(c)

where a denotes the salt activity and t+ , t+ are the cation transport numbers in the filter and in clay, respectively. Immediately after the concentration step, the salt activity at the filter/clay interface is equal to the initial activity, af0 . Accordingly, the second term in Eq. (14) is equal to zero, and the initial value of concentration potential is controlled by the cation transport number in the filter. At sufficiently long times, the diffusion front penetrates ever deeper into the clay, and the salt concentration at the filter/clay interface becomes practically equal to that in the inlet reservoir. Accordingly, the first term in Eq. (8) vanishes, and the concentration potential is controlled by the cation transport number in the clay. The total variation of concentration potential from the moment of concentration step until sufficiently long times is

F ϕ(∞) − ϕ(0) RT af0   +afs (c)

(f)

afs = 2t+ − 1 d ln(a) − 2t+ − 1 ln 1 + . (15) af0 af0

Equation (15) can be rewritten in this symmetrical form 2RT ϕ(∞) − ϕ(0) = F

af0 +afs



(c) (f)

t+ − t+ d ln(a).

(16)

af0

If because of any reason the difference of ion transport numbers between the clay and the filter can be considered independent of salt concentration, Eq. (16) reduces to   2RT (c) afs (f)

ϕ(∞) − ϕ(0) = (17) t+ − t+ ln 1 + . F af0 3 We assume that the reservoirs are perfectly stirred. Therefore, the salt concentration is constant up to the surfaces of the filters, and, accordingly, there are no concentration potential contributions due to the reservoirs. Besides that, it can be shown that at the time scale of our experiments, the concentration front never reaches the outlet filter. Accordingly, there is no concentration gradient and no electric potential difference across it.

3.2. Equipment The concentration potential measurements were carried out with 1 cm thick confined cylindrical clay plugs in the test cell shown in Fig. 1. Steel parts of the cell were covered with PEEK encasements in order to prevent electrical shortcuts. The porous filter plates were made from Al2 O3 ceramics (Metoxit, Switzerland). Their thickness was 1.5 mm, the porosity (estimated from the water uptake) was ∼30%, and the pore size ∼5 µm (manufacturer’s data). Dry clay samples were compacted directly in the inner ring of the measuring cell to a target dry bulk density of 2000 kg m−3 . After covering each side of the clay sample with a porous ceramic filter disc and screwing together the measuring cell, the clay was allowed to equilibrate with 0.1 M NaCl solution for ∼4 weeks in the continuous flow system shown in Fig. 2. Initially (during ∼3 days), the solution was circulated at one side of the cell only (to let air escape). Subsequently, circulation was set up at both sides of the cell to accelerate the equilibration process. A peristaltic pump (Ismatec) was used to circulate the solutions. The volume of reservoirs in the circulation circuits was ca. 80 ml. The indicator electrodes were Ag/AgCl from Metrohm (with Argenthal silver ion trap) filled with 3 M KCl. The filling solutions were separated from the working solutions by porous ceramic plugs. Their thickness was 2 mm and the diameter was ca. 1 mm. Proceeding from the estimated diffusion coefficient of KCl of 2 × 10−10 m2 /s (about one order of magnitude lower than in the bulk due to the finite porosity and the pore tortuosity) we could estimate that due to the diffusion through the plug the concentration of KCl in the working solutions could be about 10−4 kmol/m3 by the end of a 10 h experiment, which is rather insignificant as compared with the concentrations of our working solutions (0.1–0.5 M). The filling solutions of the electrodes were replaced at regular intervals of ∼2 weeks. 3.3. Procedure Experience has shown that reproducible results for the concentration potential were only obtained, if the solutions in the

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265

Fig. 1. Test cell.

Fig. 2. Equipment: (1) clay sample; (2) porous filter discs; (3) test cell; (4) liquid supply (peristaltic pump); (5) backflow; (6) Ag/AgCl electrode; (7) potentiometer.

continuous flow system were not in a too long contact with the clay. For this reason, the solutions were replaced with fresh equilibrium solution (0.1 M NaCl) in both circulation circuits prior to the measurements. After this step, the asymmetry potential was measured. The measurement of concentration potential was subsequently started upon rapidly (within ∼1 min) replacing the electrolyte in the inlet circuit by purging the circulation tubings with a syringe. Subsequently, the circulation rate was set at 0.1 ml/min. In our experimental setup, the solutions were delivered to the external surfaces of the ceramic filters through snail-shape channels engraved in the surfaces of the plates used to compress the system mechanically. The channel length was ca. 40 cm long, and about 1 mm broad and deep. With these channel dimensions and circulation rate, the characteristic time of solution residence in the channel was about 250 s. At the same time, it can be shown that the characteristic time needed for the diffusion through the ceramic filter (see below for the estimates of its diffusion permeability) to change the salt concentration in the channel by 10% was larger than

1000 s. Therefore, the salt concentration at the external surface of the filter could be considered practically constant along the channel and equal to the concentration in the reservoir. The signal of the potentiometer was recorded by a plotter. In the case of montmorillonite, we observed noticeable asymmetry potentials (sometimes up to 10 mV). To correct for them, the measured values were shifted in such a way that the concentration potential immediately after the concentration step would be equal to the theoretical value of diffusion potential in the coarse-porous filter. Under assumption that the asymmetry potential did not vary in the course of the measurements, this procedure did not have effect on the estimates of the total variation of concentration potential from the moment of concentration step until sufficiently long times. 4. Results and discussion 4.1. Interpretation of time transients Fig. 3 shows the results of sample calculations of timeevolution of deviation of salt concentration at the interface between the filter and the clay from the initial value. It is seen that the plots calculated for relatively small values of parameter r run pretty close to each other and to the limiting case of totally impermeable clay (r = 0). Thus, at small r, the time evolution of concentration potential is only slightly dependent on the clay properties. The only qualitative difference in behaviour between the cases of slightly permeable and absolutely impermeable clay is the slow evolution at longer times in the former case while with absolutely impermeable clay the concentration potential does not change anymore starting from t  3t0 . At the same time, the pattern essentially depends on the properties of the filter (through the characteristic time t0 ). During a relatively short period of time, the interfacial concentration remains practically at the initial level. This period roughly corresponds to the time needed for the concentration profile to become ap-

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Fig. 3. Time dependence of normalized salt concentration at the inlet filter/clay interface: r = 0; 0.02; 0.05; 0.1; 0.25; 0.5 (from top to the bottom).

proximately linear within the filter. Thereafter, the concentration at the filter/clay interface increases in a monotone way and reaches about 90% of the new value within the characteristic time. After that, there is the aforementioned slow evolution. At larger values of parameter r, the picture is roughly similar but the separation between those two zones of rapid and slow time evolution is not that clear cut. For the ceramic filter used in this study, the effective diffusion coefficient4 of traces of radio-active sodium has been estimated from independent measurements at 1.1 × 10−10 m2 /s. By assuming that in the coarse-porous filter, the ratio of mobilities of sodium and chloride ions is the same as in the corresponding electrolyte solution, for the NaCl salt we obtain Df = 1.33 × 10−10 m2 /s. It can be shown that in coarse-porous media, the sorption capacity is equal to the porosity. By using the estimated value of salt diffusion coefficient, the porosity of 30% and the filter thickness of 1.5 mm, for the characteristic time, we obtain t0 = 5075 s ≈ 85 min. To estimate the value of parameter r, we need to know the effective salt diffusion coefficient and the sorption capacity of the clay with respect to the salt. Such data are not directly available for the clays used in this study. However, they can be substituted with data obtained for anionic radio-tracers. In the case of montmorillonite compacted to the same dry density and equilibrated with the solution of the same ionic strength as used in the concentration potential experiments, by the classical through-diffusion method for chloride radio-tracer for the effective diffusion coefficient and sorption capacity, we have estimated DCl ≈ (1.0 ± 0.5) × 10−13 m2 /s, 4 Defined as the proportionality coefficient between the diffusion flux of salt and the gradient of salt concentration.

αCl ≈ 0.01 ± 0.005. It can be shown that in ion-exchange media, both the salt diffusion coefficient and the sorption capacity with respect to a salt are roughly two times larger than the corresponding values for the co-ions (anions in our case) [14]. By taking this into account and by using the corresponding values for the filter, parameter r has been estimated at 0.01 ± 0.004. Fig. 4 shows an attempt to fit experimental time transients of concentration potential obtained for compacted montmorillonite by using Eq. (5) and the values of t0 and r estimated above. It is seen that the fit quality is amazingly good especially if one keeps in mind that the nonideality of the solution was neglected while deriving Eq. (5). It should be noted that the time evolution of salt concentration at the filter/clay interface has been calculated in the approximation of ideal solution. Nevertheless, the concentration potential was calculated with Eq. (14) where the nonideality is accounted for. Besides that, we assumed that the ion transport numbers were independent of salt concentration. From Fig. 4, it is seen that the measured concentration potential practically reached plateau values after ca. 250 min. From Eq. (15), it is seen that these quasi-stationary values can be estimated without making the assumption of solution ideality and of ion transport numbers independent of salt concentration. Just those values are plotted in Fig. 6 below. In ion-exchange media the salt diffusivity and sorption capacity depend on the salt concentration. This also had to be neglected while deriving Eq. (5) because otherwise one would obtain a nonlinear equation and could not apply the method of Fourier transform to solve the problem in quadratures. Nevertheless, as discussed above, strongly compacted montmorillonite may be considered almost impermeable as compared to the filter. In the approximation of impermeable clay, the equation of nonstationary diffusion inside it does not need to be solved, and the fact that this equation is nonlinear does not matter.

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267

Fig. 4. Time dependence of concentration potential measured with Na-montmorillonite: the points are experimental data and the lines are theoretical fits; the concentration ratio is equal to 2; 3; 5 (from bottom to the top).

Fig. 5. Time dependence of concentration potential measured with Na-illite: the points are experimental data and the lines are theoretical fits; the concentration ratio is equal to 1.5; 2; 3; 5 (from bottom to the top).

It is known that illite swells essentially less than montmorillonite. Due to that, at the same dry density, in saturated state this clay may be essentially less dense and may have higher salt diffusivity and sorption capacity. Indeed, from independent measurements of through-diffusion of chloride radio-tracer through compacted illite, we obtained these values DCl = (2.8 ± 0.6) × 10−11 m2 /s, αCl = 0.1 ± 0.05. By taking into account that in cation-exchange media the corresponding values for the salts are approximately two times larger than for anions, for the parameter r, we obtain r = 0.5 ± 0.15.

This value (and the same value of characteristic relaxation time as above) was used to fit the time transients of concentration potential obtained with compacted illite and shown in Fig. 5. It is seen that, in agreement with our theoretical analysis, the approach of quasi-stationary values at long times in this case is noticeably slower than in the case of montmorillonite. Accordingly, longer measurements had to be performed. Nevertheless, the really stationary values could not be reached, and a theoretical extrapolation to very long times had to be used to estimate them. Just these extrapolated values are shown in Fig. 6 below. Although the accuracy of this extrapolation procedure is, probably, not very high (first of all, due to the assumption of solution

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Fig. 6. Quasi-stationary concentration potential as a function of logarithm of ratio of activities.

ideality and to the neglect of dependence of clay properties on the salt concentration), it should be stressed that the difference between the extrapolated and the last directly measured values did not exceed 1–2 mV. From the discussion above, it is clear that the relaxation pattern and the characteristic relaxation time are primarily controlled by the properties of the filter. It should be stressed that, in our system a time evolution of concentration potential as such occurred due to the fact that the system was inherently heterogeneous and consisted of a porous filter and clay with distinct electrochemical perm-selectivities. A time evolution of concentration potential upon a concentration step has also been observed in [10] with samples of consolidated CallovoOxfordian argillite. In these experiments, no porous filters were used. Accordingly, if the clay sample were macroscopically homogeneous no time evolution of concentration potential would have to occur, and a quasi-stationary value would have to establish immediately after the concentration step. However, a rather slow time evolution (with the characteristic time of ca. 24 h) has been actually observed. The authors of [10] interpreted their findings in terms of characteristic time of linearization of salt concentration profile within the sample. From our analysis, it follows that the profile nonlinearity alone is not sufficient to give rise to a time evolution of concentration potential. Time evolution of electric potential difference arising across slightly compacted bentonite due to a salt concentration difference has also been observed in [8,9]. In these experiments, the salt concentration in the reservoirs was not kept constant (because of rather small volumes), and the time evolution of concentration potential occurred due to the dissipation of concentration difference across the clay sample.

4.2. Dependence of quasi-stationary concentration potential on the ratio of salt activities From Eq. (16), it is seen that from the local slope of dependence of variation of concentration potential on the logarithm of ratio of salt activities after and before the concentration step, one can determine the difference of cation transport numbers between the clay and the filter as a function of salt activity in the inlet reservoir after the concentration step. Since the cation transport number in the filter is considered known, from this relationship, one can determine the sought-for cation transport number in the clay. Fig. 6 shows the variation of concentration potential from the moment of concentration step until sufficiently long times presented in the corresponding coordinates. In the case of montmorillonite, the plot is only slightly sublinear. This means, that the cation transport number is practically independent of salt concentration within the studied concentration range. In the case of illite, the plot is more noticeably sublinear, i.e., the cation transport number somewhat decreases with the salt concentration. Finally, in the case of kaolinite, the slope is too small to reliably differentiate between a linear and a nonlinear pattern within the accuracy of our experimental data. The table shows the sodium transport number in the clays obtained from the initial slope of plots shown in Fig. 6. To estimate the cation transport number at the base concentration, polynomial approximations of experimental data were used and the initial slopes at very small activity ratios were obtained through extrapolation. The table also shows the so-called electrochemical perm-selectivity defined in this way

A. Yaroshchuk et al. / Journal of Colloid and Interface Science 309 (2007) 262–271 (c)

P≡

(b)

t+ − t+

(b) 1 − t+ (b)

,

(18)

where t+ is the cation transport number in the bulk electrolyte solution. It is seen that montmorillonite has the highest perm-selectivity for cations while kaolinite has the smallest one. Illite is somewhere in between, but closer to montmorillonite. This is in qualitative agreement with the micro-structure of these clays. Indeed, it is known that montmorillonite has a lot of the socalled inter-layer water. Due to its nano-metric dimensions and high cation-exchange capacity (CEC ≈ 1 eq/kg of dry montmorillonite), anions are excluded from the inter-layer water. At the same time, cations appear to retain some mobility there [15]. At the high compactions used in this study, the inter-layer water makes up almost the whole water content of montmorillonite, which gives rise to the high cationic perm-selectivity practically independent of salt concentration (no penetration of anions at any salt concentration within the studied concentration range). Kaolinite is known to have no permanent charges and therefore also no interlayer water. The inter-particle pore water of kaolinite has mainly bulk properties. The micro-structure of illite is somewhere between that of montmorillonite and kaolinite. Illite has permanent charges that are compensated by potassium. The interlayer is collapsed and therefore, no interlayer water is present. However, the dimensions of the inter-particle pores in this clay are such that they are noticeably overlapped by the diffuse parts of double layer, which are ca. 1 nm thick at the base ionic strength of 0.1 M used in this study. Accordingly, an important part of the inter-particle water has properties similar to that of interlayer water. Therefore, in the case of illite, a noticeable perm-selectivity can be observed. At the same time, the diffuse parts of double electric layers within the inter-particle pores may well be not perfectly overlapped, and the extent of their overlap may depend on the salt concentration. Due to that, the cation transport number is somewhat smaller than unity and dependent on the salt concentration. Quasi-stationary concentration potential as a function of concentration ratio and of clay compaction has been studied in [11] with Callovo-Oxfordian argillite. It was found that the concentration potential was approximately a linear function of logarithm of concentration ratio, in agreement with our findings. It was also found that the clay electrochemical permselectivity decreased with increasing ratio of characteristic heterogeneity scale (∼average pore size) to the Debye screening length. The findings made here are also in qualitative agreement with observations made for tracer diffusion of cations in compacted montmorillonite [15]. In that work it was postulated that the diffusion of tracer cations occurs predominantly through the interlayer porosity and is driven by the gradient of sorbed tracer ions. The latter gradient was calculated based on selectivity coefficients using a cation-exchange mechanism for the competitive binding of the tracer cation and the electrolyte cation to the negatively charged surface of the clay mineral. Effective diffusion coefficients calculated based on the cation tracer gradient in the external aqueous phase became thus dependent on the

269

concentration of the background electrolyte. Below, it is shown that these observations are consistent with the observation of high values for electrochemical perm-selectivity. As a first step by using an electrostatic Donnan-equilibrium approach it is demonstrated in what way the distribution coefficient of tracer cations depends on the ratio of fixed negative charges of the clay and the concentration of background electrolyte. In the following step a relation between the cation transport numbers and the tracer distribution between the clay and external water is established. 4.3. Equilibrium distribution of ions between clay and bulk solution within the scope of Donnan model The clay and equilibrium solution are considered homogeneous phases, and it is assumed that there is a local thermodynamic equilibrium across the interface between them. Therefore, the (electro)chemical potentials of all solutes must be equal at both sides of the interface. From this condition, we obtain   F Zi φD = a exp − , a¯ i (19) i RT where R is the universal gas constant, T is the absolute temperature, ai is the solute activity, Zi is the solute charge, F is the Faraday constant, φD is the so-called Donnan potential defined as the difference of electric potentials between an ion-exchange phase and the equilibrium electrolyte solution, and the horizontal bar denotes the properties in the clay. The condition of local thermodynamic equilibrium should be complemented by the conditions of electric neutrality of both phases. In binary (1:1) electrolyte solutions, these conditions read this way c+ = c − ≡ c s ,

(20)

c¯+ − c¯− = X,

(21)

where cs is the salt concentration in the equilibrium bulk solution, and X is the fixed charge concentration per unit pore volume (including interlayer where appropriate). By substituting Eq. (19) into Eq. (21) and by using Eq. (20), we obtain γ− γ+ exp(−F φD /RT ) − exp(F φD /RT ) = X/cs . (22) γ¯+ γ¯− To obtain explicit expressions for the ion distribution coefficients as functions of salt concentration, we need to specify the activity coefficients. Though this is still a point of debate, it is often assumed that the activity coefficients of cations and anions in bulk electrolyte solutions are the same. By following this approach, we obtain 1 1 exp(−F φD /RT ) − exp(F φD /RT ) = X/as , γ¯+ γ¯−

(23)

where as is the salt activity in the bulk solution, which is experimentally measurable and extensively tabulated as a function of concentration for a large number of salts. The activity coefficients in the clay are a more involved issue since it is pretty difficult to determine them directly from experiment. However, from the physics of the phenomena, it follows that when the salt

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concentration is much lower than the fixed charge concentration, the anions are almost completely excluded from the clay, and the solution composition in it is determined by the condition of electric neutrality. In this limiting case, it can be assumed that the cation activity coefficient in the clay is just a constant independent of salt concentration in the bulk. Besides that, the second term in the left-hand side of Eq. (23) can be neglected to yield: X γ¯+ . (24) as Finally, from Eqs. (19) and (24), it follows that the distribution coefficient of trace ions is given by   γi γ¯+ X Zi . Γi = (25) γ¯i as

Table 1 Electrochemical properties of compacted clays equilibrated with 0.1 M NaCl solution Clay

Sodium transport number

Electrochemical perm-selectivity

Ratio of effective diffusion coefficients of cations and anions

Montmorillonite Illite Kaolinite

0.98 0.92 0.49

0.97 0.87 0.15

50 11.5 1.0

exp(−F φD /RT ) =

In the particular case of single-charge trace ions, it can be assumed that the activity coefficients of trace ions both in the bulk and in the clay are approximately equal to those of dominant ions. In this approximation X (26) . cs Evidently, the same relationship can be obtained for the distribution coefficient of dominant cations. Equation (22) can also be resolved with respect to exp(−ϕD / RT ) by assuming that the activity coefficients in the clay and in the bulk are approximately the same. In this approximation, the activity coefficients in the left-hand side of Eq. (22) cancel out. It is also consistent to assume that the difference of activity coefficients in clay and in the bulk can also be neglected in the definition of distribution coefficients of Eq. (19). In this way, for the single-charge cations (dominant ones as well as traces), we obtain    X X 2 Γ+ = (27) +1+ . 2cs 2cs

Γi =

For the anions, we obtain    X X 2 +1− . Γ− = 2cs 2cs

(28)

The assumption of equal activity coefficients is evidently valid in the limiting case of high salt concentration as compared to the fixed charge concentration. Interestingly, in the opposite limiting case of relatively low salt concentration, Eq. (27) reduces to Eq. (26), which was obtained above without making assumption that the activity coefficients are equal in clay and in the bulk. Thus, it may well be that Eq. (27) is pretty accurate at intermediate salt concentrations, too. On one hand, this is a clear advantage over the nonelectrostatic approach proposed in [15], where the dependence of tracer diffusivities on the concentration of background electrolyte is the same across the whole range of salt concentrations. On the other hand, the approach described above cannot account for the different chemical behavior of individual cations of the same charge, such as Na+ and Cs+ , contrary to the approach based on the selectivity coefficients.

4.4. Comparison with through-diffusion experiments In [16], it is shown that the effective ion diffusion coefficients5 in ion-exchange media can be represented in this way  (c) (c) = D± Γ± , D¯ ± (29) where Γ+ , Γ− are the distribution coefficients of cations and (c) anions between the clay and the equilibrium salt solution, D+ , (c) D− are the local diffusion coefficients and the brackets mean the averaging over the pore volume (including the interlayer where appropriate). For (1:1) binary electrolytes, the cation transport number in clay can be represented in this way [16] (c)

t+ =

(c) D¯ +

(c) (c) D¯ + + D¯ −

(30)

.

If the cation transport number is known, from Eq. (30), we can obtain the ratio of effective diffusion coefficients of cations and anions shown in Table 1. It is seen that, for example, the contribution of cations to the current transfer in compacted montmorillonite is ca. 50 times larger than that of anions.6 By using Eqs. (27), (28), and (30), in the limiting case cs X, and in the homogeneous approximation used to derive Eqs. (27) and (28), one can obtain this simple relationship (c) 

(c)

t+ ≈ 1 − 4

D−

(c)

D+

cs X

4 .

(31)

Equation (31) shows that with decreasing salt concentration or increasing fixed charge density, the cation transport number rapidly approaches unity. At the same time, from Eqs. (20) and (28), it is seen that in this limiting case, the effective diffusion coefficient of single-charge cationic radio-tracers becomes inversely proportional to the salt concentration. This yields a slope of one if the corresponding experimental data are plotted as a function of salt concentration (ionic strength) in equilibrium electrolyte solution in log–log coordinates. Exactly this kind of dependence has been observed in [15] in throughdiffusion experiments with the same Na-montmorillonite compacted to the same extent as in this study. Moreover, from 5 Defined with the concentration difference in the equilibrium electrolyte solution as the driving force. 6 It has to be stressed that when the cation transport number approaches unity, the estimate of the ratio of partial conductivities becomes very sensitive to the exact value of transport number. Due to that, the estimate for montmorillonite can only be very rough.

A. Yaroshchuk et al. / Journal of Colloid and Interface Science 309 (2007) 262–271

Eq. (25), it is seen that in the limiting case cs X, the effective diffusion coefficient of double-charge cationic radio-tracers becomes approximately inversely proportional to the square of salt concentration. This means the slope of two in log–log coordinates. Again, exactly this kind of dependence was observed in [15] for the diffusion of traces of radioactive 85 Sr through compacted Na-montmorillonite. On the other hand, from Eqs. (27), (28), and (30), it is seen that in the opposite limiting case cs X, the cation transport number approaches its value in the bulk electrolyte solution, and the effective diffusion coefficients cease to depend on the salt concentration. In this study, we have found that in compacted kaolinite in equilibrium with 0.1 M NaCl solution, the cation transport number is quite close to the bulk value (b) (t+ = 0.4). In agreement with this, practically no dependence of effective diffusion coefficient of 85 Sr radio-tracer on the salt concentration has been observed in through-diffusion experiments performed with kaolinite in [15]. From Eq. (30), it is seen that if one has determined the effective diffusion coefficient of one of the ions as well as the ion transport numbers, one can estimate the effective diffusion coefficient of another ion. In principle, this approach is modelindependent and applicable to any clay. However, if the cation transport number is very close to unity, this estimate becomes very sensitive to the precise value of transport number. Therefore, such a check would not be very informative in the case of montmorillonite. In the case of illite, the cation transport number is not that close to unity, and the aforementioned check may be more informative. Therefore, we have carried out sample measurements of diffusion of 22 Na radio-tracer through a diaphragm of compacted Na-illite confined by two porous filters by using the same measurement protocol as in [15]. The estimated values of effective diffusion coefficient and specific sorption capacity in the case of clay equilibrated with 0.1 M electrolyte solution are (c) D¯ Na = (3.1 ± 0.5) × 10−10 m2 /s,

αNa = 4.5 ± 1.5. On the other hand, from the measurements of diffusion of chloride radio-tracer through the same “sandwich,” the effective diffusion coefficients of chloride in compacted Na-illite equilibrated with 0.1 M electrolyte solution was estimated at (2.8 ± 0.6) × 10−11 m2 /s. By using this value and the cation transport number from Table 1, for the effective diffusion coefficient of sodium, we obtain (3.2 ± 0.6) × 10−10 m2 /s. This is in excellent agreement with the value estimated directly from the through-diffusion experiment. 5. Conclusions In the studies of strongly compacted clays, porous filter plates have to be used to confine the swelling clay. Due to that,

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even a step-wise salt concentration change in the inlet reservoir does not give rise to an immediate concentration change at the inlet filter/clay interface. However, the relaxation pattern of transient concentration potential is primarily controlled by the diffusion properties of the filter, which can be studied in separate experiments. At the same time, quasi-stationary values of concentration potential achieved after relaxation are directly related to the electrochemical perm-selectivity of clay. This property reveals considerable differences in the electrochemical behaviour of various clays used in this study. This can be attributed to the differences in their micro-structure, in particular to the existence or non-existence of the so-called interlayer water where cations may retain some mobility. We have also predicted theoretically and confirmed experimentally that there is a close correlation between the electrochemical perm-selectivity, on one hand, and the ionic-strength dependence of effective diffusion coefficients of cationic radiotracers, on the other. The measurements of concentration potential are more rapid than those of through diffusion. Therefore, one can benefit from this correlation for the extrapolation of reference through-diffusion measurements to different ionic strengths. Acknowledgments The partial financial support of NAGRA (Wettingen, Switzerland) is gratefully acknowledged. We are also grateful to J. Ledermann for the help in the design and construction of test cell and to R. Rosse for the technical assistance with the measurements. References [1] A. Muurinen, P. Penttilä-Hiltunen, K. Uusheimo, Mater. Res. Soc. Proc. 127 (1989) 743–748. [2] A. Muurinen, J. Rantanen, P. Penttilä-Hiltunen, Mater. Res. Soc. Proc. 50 (1985) 617–624. [3] J.C. van Schaik, W.D. Kemper, S.R. Olsen, Soil Sci. Soc. Am. Proc. 30 (1966) 17–22. [4] D.W. Oscarson, Clays Clay Miner. 42 (1994) 534–543. [5] G.H. Bolt, F.A.M. de Haan, Anion exclusion in soil, in: G.H. Bolt (Ed.), Soil Chemistry. B. Physico-Chemical Models, Elsevier, Amsterdam, 1982. [6] M. Tschapek, C. Wasowski, S. Fallasca, Colloid Polym. Sci. 269 (1991) 1190–1195. [7] M. Tschapek, Langmuir 8 (1992) 334–535. [8] K. Heister, P.J. Kleingeld, J.P.G. Loch, J. Colloid Interface Sci. 286 (2005) 294–302. [9] S. Bader, K. Heister, J. Colloid Interface Sci. 297 (2006) 329–340. [10] A. Revil, P. Leroy, K. Titov, J. Geophys. Res. 110 (2005) B06202. [11] M. Rosanne, M. Paszkuta, P.M. Adler, J. Colloid Interface Sci. 297 (2006) 353–364. [12] A.E. Yaroshchuk, Sep. Purif. Technol. 22–23 (2001) 143–158. [13] R.A. Robinson, R.H. Stokes, Electrolyte Solutions, second ed., Dover, New York, 1993, p. 590. [14] A.E. Yaroshchuk, V. Ribitsch, Chem. Eng. J. 80 (2000) 203–214. [15] M.A. Glaus, B. Baeyens, M.H. Bradbury, A. Jakob, L.R. Van Loon, A. Yaroshchuk, Environ. Sci. Technol. 41 (2007) 478–485. [16] A.E. Yaroshchuk, Adv. Colloid Interface Sci. 60 (1995) 1–93.