Separation of Bulk, Surface, and Topological Contributions to the

Article pubs.acs.org/JPCC

Separation of Bulk, Surface, and Topological Contributions to the Conductivity of Suspensions of Porous Particles R. Bouchet, D. Devaux, V. Wernert, and R. Denoyel* MADIREL, CNRS/Aix-Marseille University, Centre de St Jérôme, 13397 Marseille Cedex 20, France ABSTRACT: Electromigration of ions through porous silica particles dispersed in an electrolyte is studied by conductivity measurements. By determining the suspension conductivity at infinite dilution of particles where the Maxwell equation is applicable, the conductivity of the particles is determined. At high ionic strength, this allows calculation of the tortuosity of the particles. The tortuosity is then used to extract the pore conductivity from the particle conductivity under low ionic strength conditions where the surface conductivity is not negligible. Evolution of pore conductivity, which appears to be related to pore size, is not monotonous when ionic strength increases, showing first a decrease at very low ionic strengths, i.e., in conditions of double layers overlap in the pores, followed by an increase to trend toward the bulk conductivity at high ionic strength. This unexpected behavior can be explained by the fact that the initial surface conductivity in pores is mainly due to the protons, provided by spontaneous dissociation of surface silanol sites in water, which are subsequently exchanged by sodium.

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INTRODUCTION A better understanding of ion transport in complex porous media is needed in various fields such as cement chemistry,1 ion exchange chromatography,2 membrane science,3 electrochemistry,4 or geophysics.5 In many of these domains a large range of pore sizes is involved in the transport, from nanometer to millimeter scales. The transport of any molecule in a pore structure depends on structural parameters, fluid properties, and interaction between fluid and interfaces. The generally considered structural parameters are surface area, pore size distribution, and pore topology. The latter may be quantified by a parameter such as tortuosity.6 The incidence of fluid properties on transport may be quantified by the various diffusion coefficients of molecules. The interaction between the fluid and the solid surface leads to adsorption and friction effects that have a strong incidence on the kinetics of transfer. In the case of ion transport, another effect has to be taken into account: formation of the surface charge that leads to establishment of the so-called double/triple and diffuse layers (DL).7 Consequently, the transport of ions in porous materials will be affected by a number of potential gradients: the potential gradient in the DL and the membrane potential due to a different composition of fluid inside and outside pores. It can itself be decomposed in a diffusion component due to the difference of mobility between anion and cation and an exclusion component due to the influence of the surface charge of pore walls that modifies the ionic composition of the fluid filling the pores. Despite this complexity, the study of ion transport can take benefit of the fact that ionic diffusivity, electrical conductivity, and membrane potentials are interrelated and so to use © 2012 American Chemical Society

impedance measurements to characterize this transport as well as the material itself.9−11 Determination of the conductivity of a porous plug has been used for a long time to determine the so-called formation factor12 and tortuosity.13 In that aim, the porous media is filled with a highly concentrated electrolyte in order to be under conditions where the surface conduction is negligible which often corresponds to a diffuse layer (DL) thickness that is negligible as compared to the pore size. The tortuosity is simply given by τ=

εκ°

(1) κeff where κ° is the bulk conductivity of the solution that fills the pores, κeff that of the porous plug, and ε the plug porosity. The formation factor is simply the ratio κ°/κeff. Equation 1 can also be applied to membranes or suspensions of particles. The tortuosity obtained by conductivity is generally in good agreement with values obtained by other methods based on efficient diffusion coefficient determination.6,14,15 When the electrolyte concentration, i.e., the ionic strength, is decreased, the condition of negligible DL thickness is no longer valid and the surface conductivity must be taken into account. Surface conduction is the excess conduction that takes place in dispersed systems due to the presence of electric double layers. Because this is an excess quantity, a subtraction has to be made to measure it: conduction without double layer has to be Received: November 4, 2011 Revised: January 25, 2012 Published: January 26, 2012 5090

dx.doi.org/10.1021/jp210614h | J. Phys. Chem. C 2012, 116, 5090−5096

The Journal of Physical Chemistry C

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subtracted, but it is more or less straightforward. 16 Experimental determination of conductivity in pores in relation with surface conduction has been done mainly in two situations: (i) homogeneously dispersed beds or suspensions17,18 and (ii) porous plugs10 or porous membranes.19 These situations generally correspond to only one pore scale, which simplifies analysis. Several models have been used to model the conductivity in a pore as a function of pore size, surface charge, ionic strength, and ζ potential.7 They often use the Dukhin’s number, which represents the ratio between surface conductivity and bulk electrical conductivity.7,20 Clearly, there is interest in determining the conduction properties of porous materials because they could bring various types of information about the material: topological (tortuosity), electrical (ζ potential21), and transport (conductivity and diffusivity are related either by the Nernst−Einstein equation or the Nernst−Hartley equation depending on the presence of concentration gradients22). To our knowledge, only two papers consider the possibility of several pore size scales in the case of porous particles plugs: one with zeolites23 and the other with Stober silica.24 In both cases, the size of the internal pores of the particles was so small (lower than 2 nm, which is in the micropore range of the IUPAC classification25) that the contribution of the particle conductivity to the bed conductivity was considered as negligible. The conductivity of porous particles suspensions can be used to determine the conductivity of the particle when the measurements are carried out at various solid/liquid ratios.26 By this method, called the infinite dilution method in the following, it was possible to determine the tortuosity of the particles by making these measurements at high ionic strength using eq 1 with the particle conductivity and particle porosity. The objective of the present paper is to apply this method but decreasing the ionic strength in order to show the effect of pore size and surface conduction. In that aim, the conductivity of various porous particle suspensions is determined at various ionic strengths and solid/liquid ratios. Several materials differing mainly by their pore sizes but with similar surface chemistry are analyzed. A method is proposed to determine the conductivity inside pores that can be then compared to that calculated by existing models.

pore size. All porous materials selected for this work are made of spherical particles. Nonporous silica (C600) is also used for comparison. Surface areas, pore sizes, and particle porosities measurements were carried out either by mercury porosimetry (Autopore 9220 from Micromeritics) or by nitrogen adsorption at 77 K (ASAP 2010, Micromeritics) using well-known methods.27,28 Samples were heated several hours at 150 °C under a residual pressure lower than 1 Pa before the gas adsorption experiment. The main characteristics of the samples obtained by these methods are given in Table 1 (specific surface area, pore volume, and mean pore size). Complementary data about the pore structure characteristics or electrical properties, such as the ζ potential, of Spherosil-type silica can be found in the literature.29−31 SEM images of sample MB2000 are given in Figure 1 to give an idea of the structure of such porous materials (Philips XL30 Environmental Scanning Electronic Microscopy). Conductivity measurements were carried out in a cell whose temperature was controlled within 0.1 K thanks to a circulating thermostatted bath. Electrical resistance measurements were carried out by impedance spectroscopy (from 0.01 Hz to 1 MHz) using a standard two-electrode conductivity cell integrated in the dispersed bed. Impedance spectroscopy is a very precise tool for measuring electrical properties of materials and fluids.32,33 The amplitude of the exciting signal was fixed to a small value of 30 mV. A Solartron 1260 frequency analyzer controlled by a personal computer using Z-Plot software (Scribner) was used. Z-view software (Scribner) was used to visualize the impedance diagrams and separate suspension resistance from electrode effects. The resolution on the measured impedance is 3 per 10 000. Measurements were done in the whole frequency domain: the resistance of the medium was deduced by fitting the experimental data using an equivalent circuit based on a resistance in parallel with a constant phase element (CPE) the whole in series with a CPE for the electrode/electrolyte blocking interface. The conductivity of the solution was measured on the supernatant after powder sedimentation, whereas the suspension conductivity was measured inside the suspension where the particles were homogeneously dispersed using a magnetic stirrer. Before the experiment, the particles were washed with deionized water (Millipore Milli-Q) until the resistivity of the supernatant is higher than 0.5 MΩ cm−1 and then dried under vacuum. The variation of the total porosity of the dispersed bed at constant ionic strength was obtained by adding powders by 0.5 g increments to 25 cm3 of electrolyte. At each step, the cumulative volume Vp of particles added to the solution was calculated by dividing the cumulative weight of dried material, mp (in g), by the density ρ (2.2 g cm−3) of the material mp Vp = ρ (2)

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MATERIALS AND METHODS The list of the porous particles used in this study is given in Table 1 together with their suppliers, their chemical nature, and their main structural properties: specific surface area and mean Table 1. Main Characteristics of the Samplesa sample name/ manufacturer C600/Siffraco Spherosil MB2000 Biosepra Spherosil XOB030 Biosepra Spherosil XOA200 Biosepra

surface area/m2 g−1

mean pore size/nm

particle size/μm

particle pore volume/cm3 g−1

4.3 18

200

0.6 50−100

1

The total porosity ε was then deduced by

ε= 47

60

50−100

1

199

20

50−100

1

Vi Vi + Vp

(3)

where Vi is the initial volume of electrolyte. Variation of ionic strength at “nearly” constant total porosity was obtained by injecting either a concentrated solution of sodium chloride or pure solid sodium chloride. The pH was controlled all along the experiment with a Quick pH meter and a combined microelectrode calibrated with standard buffers

a

The mean pore size and pore volume of the particles are determined by mercury porosimetry or gas adsorption. 5091



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Figure 1. SEM images of sample MB2000 showing the spherical porous particles (left) and a zoom of their surface (right).

before and after experiments. The cell was maintained under argon to avoid CO2 contamination. Particle porosity and tortuosity values were reproducible within 5%.

where εp denotes the porosity of the particles. The following relationship holds between the various porosities

RESULTS We will first briefly discuss the principle of our “infinite dilution method” allowing determination of the conductivity and tortuosity of porous particles in the case of high ionic strength.26 A porous particle made of nonconducting solid is conducting because the pores are filled by the electrolyte. The method is based on the effective medium theory of Maxwell34 proposed to predict the effective conductivity of a continuous phase containing a homogeneously dispersed phase. This theory, which is supposed to be valid for diluted suspensions where there is no interaction between particles, leads to the socalled Maxwell equation, in which the effective conductivity κeff of the suspension is related to the conductivities of spherical particles (κp) and electrolyte (κ0) by

The first application of this method is then to give the tortuosity of porous particles, which is an important parameter of transport properties. Interestingly, the conductivity of the particle is also obtained by this calculation. What is the information brought by this method when applied at a lower ionic strength where the surface conductivity is no longer negligible? An example of a measurement of the tortuosity of the suspension as a function of its total porosity calculated by eq 1 is given in Figure 2 at two ionic strengths. For both ionic strengths the data can be well represented by eq 5 in the whole porosity range explored, indicating that it is possible to deduce in both cases the conductivity of the particle. The main difference between the two curves is that the value of the tortuosity is higher than 1 in the case of high ionic strength, whereas it may be smaller than 1 under low ionic strength

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2κ0 + κ p − 2(1 − εext)(κ0 − κ p) = κ0 2κ0 + κ p + (1 − εext)(κ0 − κ p)

ε = εext + ε p(1 − εext)

(8)

κ eff

(4)

where εext denotes the external porosity. The tortuosity of the dispersed bed is obtained by combining eqs 1 and 4 κp κp ⎞ ⎛ 2 + 0 + (1 − εext)⎜1 − 0 ⎟ ⎝ κ κ ⎠ τ=ε κp κp ⎞ ⎛ 2 + 0 − 2(1 − εext)⎜1 − 0 ⎟ ⎝ κ κ ⎠

(5)

In the case where the particles are nonporous, they are nonconducting and the external porosity and total porosity are equal, i.e., κp = 0 and εext = ε, eq 7 simplifies to τ = 1 + 0.5(1 − ε) =

3−ε 2

Figure 2. Apparent tortuosity versus total suspension porosity in the case of a suspension of porous particles. Sample: Spherosil XOB030. Pore size 60 nm. Squares: 10−4 mol L−1 NaCl. Disks: 1 mol L−1 NaCl. Triangles correspond to an experiment where the total porosity was maintained nearly constant whereas NaCl concentration was varied between 10−4 and 1 mol L−1. Lines correspond to eq 5.

(6)

The expression of the tortuosity for a suspension obeying Maxwell theory is then rather simple when particles are nonconducting. It is in agreement with experiment only at high porosity.26 In the case of porous particles, the tortuosity of the suspension is measured as a function of total porosity by eq 1 and calculated theoretically by eq 5 where the ratio κp/κ° is varied until experimental and calculated data are tangent and tend toward 1 at high total porosity (infinite dilution of particles). The tortuosity τp of the particles may then be calculated classically by τp =

conditions. Also, the data obtained at almost constant total porosity (Figure 2, triangles) but increasing ionic strength are added. Data points hit between the two lines showing that results are independent of introduction order of the constituents. A tortuosity, defined as a topological parameter, lower than one has no meaning. It simply shows that the conductivity of the particles is higher than that of the bulk electrolyte. This is due to the fact that the surface conductivity is no longer negligible: the conductivity of the fluid inside pores is higher than the bulk conductivity. For these reasons the Y axis of

κ0ε p κp

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determined by the infinite dilution method at high ionic strength. κp is determined at any other ionic strength by the same method (Figure 2), which allows one to plot the average pore conductivity κ00 as a function of ionic strength for the three porous samples studied here (Figure 4). This type of result has already been obtained in the case of membranes or porous plugs for which there is only one scale of porosity. In that case, the correction for porous material tortuosity is much simpler.37 At high ionic strength, the pore conductivity tends toward that of the bulk, as expected. When the ionic strength decreases, the pore conductivity becomes much higher than that of the bulk electrolyte in equilibrium because of the increasing contribution of surface conductivity. The difference increases when pore size decreases, which is expected because the surface/volume ratio of the pore

Figure 2 is called apparent tortuosity, as done by other authors when the measured tortuosity value depends on the probe or on the mechanism of transport.35 The right-hand side of eq 1 is a real topological tortuosity only when the ionic strength is high enough to be able to neglect surface effects, and then τ becomes constant and higher than 1. This is illustrated in Figure 3, where the apparent tortuosity is plotted as a function of ionic strength at constant total porosity (0.975) for both the nonporous silica and the three porous silicas differing by their pore size. At high ionic strength a plateau above 1 is reached, which highlights this range where the measured tortuosity is representative of the topology of the material. At lower ionic strength, the apparent tortuosity decreases due to the

Figure 3. Apparent tortuosity versus NaCl concentration for the three porous silica and the nonporous one. Pore diameters are indicated in the figure. Total porosity is 0.975. Lines are guides for the eyes.

Figure 4. Evolution of pore conductivity with ionic strength for the studied porous silica samples. Pore sizes: (circles) 200, (triangles) 60, and (stars) 20 nm. Squares show the corresponding pH variation in the case of the 20 nm sample (right-hand scale). Solid lines correspond to eq 10, applied to each sample. Dotted line is the conductivity of the supernatant.

contribution of surface conductivity. Interestingly, for the set of silicas tested here, where the pH variations are small along the curves and similar from a sample to the other, there is a clear separation between the curves corresponding to the various samples. The smaller the pore size the larger the variation of apparent tortuosity and the higher the ionic strength at which the effect of surface conductivity appears. In the case of the nonporous sample the effect of surface conductivity becomes only significant at a very low ionic strength (below 5 × 10−5 mol L−1). This could lead to a method of characterization of porous materials based on conductivity measurements, as proposed in the case of membranes.19 Nevertheless, a preliminary modeling of the phenomena is needed to relate the observed data to the structural and surface charge properties of the material. Indeed, this method allows determination of the conductivity of the porous particles as a function of ionic strength. To relate this measured property to the pore size or to the surface area it is needed to know the conductivity of the fluid in the pore because only this data can be modeled as a function of surface charge and pore size. If the conductivity of the fluid in the pores was the same for all pore sizes it would be simply related to the particle conductivity by the equation defining tortuosity τpκ p κ o° = εp (9)

increases. A new feature is observed here when compared with prior results from the literature:22,16,8,38 there is a minimum in the curve representing pore conductivity versus ionic strength. We observe that the data presented here are obtained in a very large range of salt concentrations including concentrations much lower than that used in the literature. Some recent data obtained with nanoslits39,40 were also determined in the very low salt concentration range, but this minimum was not experimentally observed. Nevertheless, this behavior is predicted by the model proposed by Balderrassi,40,41 whatever the hypothesis about the surface charge: constant zeta potential, constant surface charge density, or charge regulation. This author starts from the classical theory of electrical doublelayer based on the Poisson equation and Boltzman distribution but then considers a nanochannel in equilibrium with end channel wells and includes the effect of local ionic strength on ion mobility. The minimum is much less pronounced in the case of charge regulation, which reproduces well his data that do not present the minimum. It is not surprising to us since the pH variation is large in his experiments, whereas it is small in those presented here (see Figure 4). Because the surface conductivity is considered as an excess quantity, the conductivity of the pore can be considered as the sum of the contribution of the bulk conductivity assumed to be constant all along the pore section and of the contribution of the surface conductivity. In fact, because the conductances are

With this definition κ00 denotes the average conductivity in pores. The narrower the pore size distribution, the closer to the actual pore conductivity is that calculated by eq 9. Introduction of the effect of the pore size distribution in eq 9 is a difficult task36 that is out of the scope of this paper. The tortuosity τp is 5093



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number of counterions in the pore due to surface group dissociation may be very high as compared with that due to bulk concentration defined by pH. Considering the same pore volume (10 nm diameter by 100 nm length) and a surface charge density of 1 μC cm−2,42 the number of counterions (H+) in the pore due to silanol dissociation would be higher than 100 (this number is obtained by multiplying the surface density of dissociated silanols, derived from surface charge density, by the surface area of the pore). These numbers show that conductivity is ensured mainly by the counterions (their transport number is very close to one). If now another counterion Na+ is introduced in the system with a new co-ion Cl−, the blocking effect for the co-ion is maintained, but counterions (H+ and Na+) can be exchanged without electrical energy demand between the pore and the bulk. Because the mobility of H+ is much higher than that of Na+, the absolute conductivity in the pore decreases. Simultaneously the pH decreases (Figure 4). The decrease of conductivity with salinity continues until the electrical blocking effect is no more predominant with respect to self-diffusion of co-ions. This probably occurs when the potential at the center of the pore is null, i.e., when the double layers no longer overlap. This is why the position of the minimum depends on pore size. The position of the minimum corresponds to NaCl concentrations around 8 × 10−6, 5 × 10−5, and 9 × 10−5 mol L−1 for samples with pore sizes 200, 60, and 20 nm, respectively. This is also the range of the double-layer thickness for these concentrations (150, 60, and 45 nm, respectively). To model this behavior the simple approach proposed by Schoch and Renaud39 can be used. The conductivity in the pore can be considered as the sum of the conductivity of the ions at the concentration of the bulk plus the conductivity due to the counterions produced by the surface charge formation

additives, the following equation can be derived for the conductivity of a cylindrical pore7

κ o° = κ o + 2κs/R p

(10)

where κ denotes the surface conductivity and Rp the pore radius. This simple equation is able to fit the data (Figure 4) from high salt concentration to the position of the minimum. The values of κs that give the best fit are of the same order for the three porous samples (3 × 10−10, 5 × 10−10, and 3 × 10−10 S for MB2000, XOB030, and XOA200, respectively), which can be expected for a set of samples with a similar surface chemistry. If the surface conductivity is decomposed in its inner layer (stagnant layer) and diffuse layer s

κs = κ i + κ D

(11) 16

one may suppose, following Lyklema et al., that most of the surface conductivity is due to its inner layer component, because the inner layer may accommodate until 80% of the countercharge. Revil also underlined, in the case of clays, that for salinities above 10−3 mol·L−1 the surface conductivity appears independent of salinity.22 Models that are more complex have been used to model the pore conductivity versus bulk conductivity. For example, Szymczyk et al.38 applied a space charge model that takes into account the surface potential. However, their data can be also fitted by the simple eq 10. Use of conductivity to determine the zeta potential is then questionable since the relative contributions of κi and κD are unknown in eq 11. Only κD is relatively easy to model in the frame of the classical diffuse layer theory. Finally, the last point to discuss is the behavior at low ionic strength. As seen in Figure 3, when starting from the clean solid immerged in pure water, the pore conductivity starts to decrease when a salt is added to the medium. A minimum is then reached before the expected increase of pore conductivity with ionic strength is observed. The position of the minimum depends on the pore size. The narrower the pore is the higher the ionic strength corresponding to the minimum. This is perhaps a reason why such effect has not yet been shown because most experiments found in the literature were done with larger pores and higher ionic strengths. A possible interpretation is the following. Without salt, dissociation of surface groups leads to formation of surface charge yielding counterions in the solution; protons are produced in the case of silica at a pH higher than its point of zero charge, which is in the pH range 2−3, following

κ 00 = (uNa+c Na+ + u H+c H+ + uCl−cCl− + uOH−cOH−)L e + 2(u +α + u +(1 − α))Q */Dp Na

H

S

(12)

where ci is the bulk concentration of ion i, ui its bulk mobility, L the Avogadro constant, e the electron charge, and QS* the surface charge density. The main difference with Schoch and Renaud’s approach is introduction of the parameter α to take into account that the counterions in the pore may be either proton or sodium ions. α is the fraction of counterion in pores that are sodium. Because, as proposed above, there is no energetic barrier to the exchange between sodium and proton in the system, α can be equal to the total cationic fraction of sodium in the system, i.e., this fraction is constant through the system. It can be derived from the composition of the external liquid by the following equation

Si−OH = Si−O− + H+

Because the porous particle must stay electrically neutral as a whole, these ions stay in the pores (the external surface area is negligible). This is why the initial pH is close to 7 in these experiments (see Figure 4). The only co-ions in the initial states are OH− anions (if silica solubility is neglected). The co-ions stay outside the pores because if they enter a pore the bulk solution as well as the pore would no longer be neutral as expected if a counterion accompanies it. The driving force for the latter process, i.e., introduction of both a cation and an anion, is not electrical but diffusive. Without salt, at a pH close to 7, the number of proton and hydroxyls which would be in a volume equal to that of a pore with diameter 10 nm and length 100 nm is a fraction of a molecule (lower than 0.001, this number is obtained by multiplying the ion concentration deduced from pH by the pore volume). On the other hand, the

0 + c 0+ − c H+) (c NaCl H α= 0 + c 0+ c NaCl H

(13)

0 where cNaCl denotes the total concentration of sodium chloride in the system and cH+ that of protons outside the pores, calculated from the pH measurement and cH0 + the initial concentration of protons in the suspension before salt addition. This is an approximation since the external concentration of sodium chloride is not exactly equal to its total concentration in the system. Co-ions (chloride) are indeed excluded from the double layer, leading to an increase of their concentration outside the pores and, as a consequence, to an increase of the

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the results presented here, this is shown by the continuous decrease of pH with salt concentration, which is even steeper after the minimum of the conductivity curves in Figure 4. Nevertheless, the variation of surface charge that can be calculated from this pH variation is only a fraction of the total surface charge Q* accounting for our results (less than 1%). The systems have been studied here under conditions where the surface charge variation is very small. Equation 12 is able to reproduce our conductivity data; nevertheless, this is mainly a qualitative model since α = 1 inside pores would lead to a large displacement of proton toward the supernatant: using the surface charge to calculate the expected pH variation leads to values below 4, which is much smaller than measured values. Other parameters probably have to be taken into account for a complete description such as, for example, solubility of the silica or variation of mobility with respect to the distance to the surface.

overall external concentration of electrolyte (Donnan effect). The error, an underestimation of NaCl external concentration in eq 13, is at maximum equal to the ratio pore volume to total volume of liquid, which is always lower than 5% in our experiments. The mobilities used in eq 12 are those of the bulk even for the ions in the pores. This is a reasonable approximation for the present purpose: Lyklema et al.16 arrived at the conclusion that the mobility, even in the stagnant layer, may be at least 50% of the bulk mobility. The results given by eq 12 are given in Figure 5. The value used for Q* (0.7, 1.3, and 0.8 μC cm2 for XOA200, XOB030, and MB2000, respectively) are in agreement with recent surface charge titration.42 The minimum is well described by this equation. It was also supported by the model of Balderrassi41 but not in agreement with his experimental data for the reasons already described. This minimum is however not present among the experimental results described in the paper by Schoch and Renaud.39 The common point to these papers is the use of nanochannels (or nanoslits) carved on wafers in contact with reservoirs

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CONCLUSION This paper presents a method allowing separation of the topological and electrical effect in ion transport through porous particles. Using a measurement of the conductivity at various solid/liquid ratios, the infinite dilution method, the effective conductivity of the particles is determined allowing their internal tortuosity to be determined at high ionic strength. It can then be used to extract the pore conductivity in low ionic strength conditions where surface conductivity is not negligible. This pore conductivity can then be modeled versus pore size using eq 12, which takes into account bulk and surface conductivity. The surface charge obtained by the fitting of conductivity data is in agreement with surface charge titrations given in the literature. Assuming that the initial surface conductivity is only due to protons, the pore conductivity curves versus ionic strength show a minimum that can be explained by the progressive exchange between proton and sodium in the pores in conditions of double-layer overlap.

Figure 5. Evolution of pore conductivity with ionic strength. Pore sizes: (circles) 200, (triangles) 60, and (stars) 20 nm. Full lines correspond to eq 12.

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Corresponding Author

containing the electrodes. In that case the surface/volume ratio is very small as compared to our procedure which uses high surface area porous solids. Consequently, the amount of protons produced by surface charge formation is very small as compared to the amount of protons and cations already present in solution. The effect of Na+/H+ exchange on conductivity cannot be observed. If one looks in more detail to the evolution of the two main terms of the right-hand side of eq 12 it can be observed that the surface conductivity term no longer changes above a salinity of 10−4 mol L−1 (where α is very close to 1), which is a smaller value than that quoted above.5 This behavior is unexpected if one considers the chemistry of surface charge formation in the presence of an electrolyte. It is indeed generally assumed that introduction of counterions enhances the surface charge density thanks to a binding reaction, which can be written in the case of silica and sodium chloride as Si−OH + Na+ = (Si−O−Na+) + H+

AUTHOR INFORMATION

*E-mail: renaud.denoyel@univ-provence. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We are thankul for support from the CNRS-ATILH Contrat de Programme de Recherche: Résistance, Porosité et Transport des Matériaux Cimentaires.

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REFERENCES

(1) Schwartz, L. M.; Garboczi, E. J.; Bentz, D. P. J. Appl. Phys. 1995, 78, 5898. (2) Liapis, A. I.; Grimes, B. A. J. Sep. Sci. 2005, 28, 1909. (3) Quarterone, E.; Mustarelli, P.; Magistris, A. J. Phys. Chem. B 2002, 106, 10828. (4) Thorat, I. V.; Stephenson, D. E.; Zacharias, N. A.; Zaghib, K.; Harb, J. N.; Wheeler, D. R. J. Power Sources 2009, 188, 592. (5) Revil, A.; Darot, M.; Pezart, P. A. Geophys. Res. Lett. 1996, 23, 1989. (6) Ben Clennell, M. In Developments in Petrophysics; Lowell, M. A., Harvey, P. K., Eds.; Geological Society Special Publication, 1997; Vol. 122, p 299. (7) Lyklema, J. Fundamentals of Interface and Colloid Science; Academic Press: New York, 1995; Vol. II.

(a)

The so-called “site binding models” are often used to quantify the equilibrium constant of this reaction. It shows that by increasing salt concentration the number of SiO− is increased and consequently the surface charge as well, which is clearly proven by surface charge titration experiments.7 In the case of 5095



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Separation of Bulk, Surface, and Topological Contributions to the

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