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Ion-Partitioning between Charged Capillaries and Bulk Electrolyte Solution: An Example of Negative “Rejection” V. Vlachy Faculty of Chemistry and Chemical Technology, University of Ljubljana, 1000 Ljubljana, P.O.B. 537, Slovenia Received June 15, 2000. In Final Form: August 31, 2000 The equilibrium partitioning of a primitive model +3:-3 electrolyte between a charged cylindrical capillary and the bulk solution was studied using the grand canonical Monte Carlo method. The ions were considered to be charged hard spheres embedded in a dielectric continuum with the fixed charge smeared on the hard inner walls of the capillary. The electrolyte in a micropore was assumed to be in equilibrium with the external bulk electrolyte of the same chemical composition. Simulation results are presented for several charge densities of the fixed charge and for different concentrations of the external electrolyte. Under certain conditions, that is, for moderate to high charge densities on the inner walls and at low enough concentration of the external electrolyte, we obtained an increase of the electrolyte concentration within the microcapillary, and not a decrease as expected on the basis of the traditional electrical double-layer theory. This “negative rejection” seems to be a consequence of the strong correlation between multivalent ions.
Introduction Material containing charged micropores surrounded by a bulk electrolyte often contains a lower concentration of ions than the surrounding equilibrium solution (for review see refs 1 and 2). The ability of porous material to exclude electrolyte forms the basis of important technological processes for desalination of water (see, for example, refs 1, 3, 4, and 5). The equilibrium distribution of ions between the capillary and bulk solution is important also for understanding electrokinetic phenomena6,7 and also for some engineering applications where the mean activity coefficient of an electrolyte contained in a charged micropore is needed.8,9 Normally, the concentration of an invading aqueous electrolyte is lower in the porous phase, hence the term rejection of electrolyte. The phenomenon can be explained on the basis of the interaction between the charge fixed on the inner surfaces and ions of the simple electrolyte in the micropore. Due to charged groups on the internal walls, there is an excess of counterions (ions of the opposite charge sign to those fixed on the walls), while the co-ions are partly excluded from this region. As a result, the electrolyte concentration (co-ion concentration) within the capillary is reduced below the bulk value. Among the parameters affecting electrolyte exclusion, the most important are (i) the radius of the microcapillary, (ii) the concentration of invading electrolyte, and (iii) the capacity (proportional to surface charge density) of the micropore. Experimental data indicate that greater rejection is obtained for dilute solutions and that rejection of (1) McKelvey, J. G.; Spiegler, K. S.; Wyllie, M. R. J. Chem. Eng. Prog. Symp. Ser. 1959, 55, 199. (2) Vlachy, V.; Haymet, A. D. J. Aust. J. Chem. 1990, 43, 1961. (3) Dressner, L.; Kraus, K. A. J. Phys. Chem. 1963, 67, 990. (4) Marcinkowsky, A. E.; Kraus, K. A.; Phillips, H. O.; Johnson, J. S., Jr.; Shor, A. J. J. Am. Chem. Soc. 1966, 88, 5744. (5) Jacazio, G.; Probstein, R. F.; Sonin, A. A.; Yung, D. J. Phys. Chem. 1972, 76, 4015. (6) Westermann-Clark, G. B.; Anderson, J. L. J. Electrochem. Soc. 1983, 130, 839. (7) Christoforou, C. C.; Westermann-Clark, G. B.; Anderson, J. L. J. Colloid Interface Sci. 1985, 106, 1. (8) Dressner, L. J. Phys. Chem. 1963, 67, 2333. (9) Dolar, D.; Vlachy, V. Vest. Slov. Kem. Drus. 1981, 23, 327.
a +1:-1 electrolyte increases on increasing the capacity of the microporous material. Further, an electrolyte solution containing divalent co-ions is rejected more strongly than a solution with monovalent co-ions.4 These results can be understood in terms of the classical theory of the electrical double-layer.2 Of course, there are also exceptions, and they seem to happen when electrolyte solutions with multivalent counterions are forced through the porous material. Kraus et al.,4,10 for example, report that normal rejection is destroyed in the presence of dior trivalent counterions. Surprisingly, they found the downstream electrolyte solution to be more concentrated than the upstream (feed) solution. In other words the “rejection” coefficient was found to be negative. As suggested by the authors,10 there must be other interactions which overshadow the simple rejection due to the surface charge. Negative “rejection” with divalent counterions was reported also in the work of Staude and coworkers.11 Westermann-Clark and Anderson6 measured the streaming potential for various electrolyte solutions in charged cylindrical capillaries. The authors reported good agreement of the electrostatic space charge theory, based on the classical electrical double-layer theory of Gouy and Chapman12 and experimental data for a +1:-1 electrolyte. No such agreement was obtained for a system containing divalent counterions. In the latter case, the streaming potential was found to change its sign. All these results indicate that the simple picture of electrostatic exclusion of co-ions by the charged walls of the capillary needs to be revised when multivalent counterions are present. The phenomena mentioned above have most often been ascribed to specific (non-Coulombic) interactions, which are not included in the electrostatic model. In this paper, however, we wish to show that negative “rejection”, or reversal of the sign of the streaming potential, may be a consequence of the strong interionic correlation in the system. This correlation is neglected by the classical (10) Kraus, K. A.; Marcinkowsky, A. E.; Johnson, J. S.; Shor, A. J. Science 1966, 151, 194. (11) Staude, E.; Hinde, E.; Malejka, F. Colloids Surf. 1989, 42, 356. (12) Kortu¨m, G. Treatise on Electrochemistry; Elsevier: Amsterdam, 1965; pp 389-394.
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electrostatic theory (for reviews, see refs 13 and 14 and the references therein) but is included in the more rigorous approach applied below. Model and Method In the present work, as in several previous papers,2,15-23 the porous material is pictured as an array of very long cylindrical micropores with the fixed charge distributed on the inner surfaces. The average micropore is assumed to be of infinite length and electroneutral, mimicking a macroscopic portion of the porous material. The equilibrium properties may then be calculated for this average micropore: The electrolyte in a micropore is assumed to be in equilibrium with a bulk electrolyte of the same chemical composition. In this respect the situation is analogous to the classical Donnan equilibrium.24 The calculation is based on the primitive model of an electrolyte solution; the ions are presented as charged hard spheres of equal diameter. The system as a whole is treated as a continuous dielectric with dielectric constant 0r. In this model the interaction potential between two ions of species i and j reads
{
2 1 zizj e0 if r g a , ij ij uij(rij) ) 4π0r rij if rij < aij ∞
(1)
where ai denotes the diameter of a particle of type i and rij is the distance between the two particles. Further, zi is the valency of an ion of type i, and e0 is the proton charge. Only charge and size symmetric electrolytes are considered here; therefore, ai ) aj ) a and zi ) -zj ) 3. An important parameter of the model is the surface charge density σ, defined as
σ)
e 2πRch
(2)
where Rc ) R + a/2 is the radius and h the length of the capillary, while e denotes the fixed charge. For all the calculations presented in this paper T ) 298 K, r ) 78.54, Rc ) 4.20 nm, and a ) 0.75 nm. To obtain the physicochemical properties of the confined electrolyte, the spatial distributions of ions in a micropore must first be calculated. Traditionally, these distributions were obtained via the Poisson-Boltzmann (PB) approximation (see, for example, refs 2 and 9), but in recent years other theoretical approaches have been successfully applied.2,17-23,25 Among these methods, the grand canonical Monte Carlo (GCMC) method proved to be very useful.2,16-18,20 In the open ensemble simulation the chemical potential of the solution is fixed by the external (13) Kjelander, L. Ber. Bunsen-Ges. Phys. Chem. 1996, 100, 894. (14) Vlachy, V. Annu. Rev. Phys. Chem. 1999, 50, 145. (15) Olivares, W.; Croxton, T. L.; McQuarrie, D. A. J. Phys. Chem. 1980, 84, 867. (16) Vlachy, V.; McQuarrie, D. A. J. Phys. Chem. 1986, 90, 3248. (17) Vlachy, V.; Haymet, A. D. J. J. Am. Chem. Soc. 1989, 111, 477. (18) Jamnik, B.; Vlachy, V. J. Am. Chem. Soc. 1993, 115, 660. (19) Yeomans, L.; Feller, S. E.; Sanchez, E.; Lozada-Cassou, M. J. Chem. Phys. 1993, 98, 1463. (20) Jamnik, B.; Vlachy, V. J. Am. Chem. Soc. 1995, 117, 8010. (21) Lo, W. Y.; Chan, K. Y. Mol. Phys. 1995, 86, 745. (22) Lo, W. Y.; Chanm, K. Y.; Lee, M.; Mok, K. L. J. Electroanal. Chem. 1998, 450, 265. (23) Lozada-Cassou, M. In Fundamentals of Inhomogeneous Fluids; Henderson, D., Ed.; Marcel Dekker: New York, 1992; pp 303-361. (24) Guggenheim, E. A. Thermodynamics, 5th ed.; North-Holland: Amsterdam, 1967; pp 305-307. (25) Lozada-Cassou, M.; Olivares, W.; Sulbarian, B. Phys. Rev. E. 1996, 53, 522.
Table 1. Exclusion Coefficient G for Various Charge Densities of the Microporea cs/M
-ln(γ()
0.0281 0.0998 0.1982
2.669 3.324 3.606
σ(1)
σ(2)
σ(4)
+0.005 ( 0.008 -0.263 ( 0.005 -0.810 ( 0.008 +0.109 ( 0.005 -0.022 ( 0.010 -0.231 ( 0.008 +0.141 ( 0.005 +0.078 ( 0.005 -0.028 ( 0.005
a σ(1), σ(2), and σ(4) apply to σ ) 0.035 665, 0.071 33, and 0.142 66 A s/m2, respectively.
reservoir of electrolyte with concentration cs at temperature T and mean activity csγ(, while the number of ions in the capillary fluctuates.26 The main advantage of the GCMC method is that, by sampling at constant chemical potential, the relevant bulk electrolyte phase is defined unambiguously. The open ensemble simulation method, used in this study, requires advance knowledge of the excess chemical potential of the bulk electrolyte phase at concentration cs. The data for +3:-3 size symmetric electrolytes with ionic diameter a ) 0.75 nm are, to our knowledge, not available in the literature. The mean activity coefficients needed as input for the GCMC calculation of the electrolyte in the micropore may be obtained either by separate grand canonical simulations of the bulk electrolyte27 or by some more approximate theoretical method.28 In this calculation the GCMC method was used to obtain the mean activity coefficient of the equilibrium bulk electrolyte. The computer simulations were performed as described before27,29 with 500-600 ions in the system; the results for the logarithm of γ( are given in column 2 of Table 1. The numerical uncertainties in cs and ln(γ() were estimated to be below 0.3%. In parallel we also solved the hypernetted-chain (HNC) integral equation for this case.28 We note in passing that the HNC mean activity coefficients, evaluated by the approximate expression,30,31 were in fair agreement with the new GCMC results from simulations of isotropic electrolyte solution. Results The GCMC simulation procedures used in this work have been described elsewhere,2,17,18,20,26,27 so only the details important for the present calculation will be mentioned. The model assumes the capillary to be of infinite length; the actual length of the capillary in the calculation was h ) 204 nm. From earlier experience for this length, the end effects are negligible and simple periodic boundary conditions in the direction of the z-axis were used. In performing the statistics, the first 20-25 million configurations were discarded and not included in the averaging. The following 60-80 million configurations were utilized in the averaging process. We have to stress that the electroneutrality condition in the micropore was satisfied at all times during the simulation.2,17,18,20 This is consistent with the model of the average (infinite) capillary representing the microporous material, as described in the previous section. In this respect our simulation differs from some other calculations,22,25,32 where other boundary conditions were investigated. Due to the charged groups on the inner surfaces, the co-ions are partly excluded from the capillary. As a result, (26) Torrie, G. M.; Valleau, J. P. J. Chem. Phys. 1980, 73, 5807. (27) Valleau, J. P.; Cohen, L. K. J. Chem. Phys. 1980, 72, 5935. (28) Rasaiah, J. C.; Friedman, H. L. J. Chem. Phys. 1968, 48, 2742. (29) Vlachy, V.; Ichiye, T.; Haymet, A. D. J. J. Am. Chem. Soc. 1991, 113, 1077. (30) Belloni, L. Chem. Phys. 1985, 99, 43. (31) Vlachy, V.; Prausnitz, J. M. J. Phys. Chem. 1992, 96, 6465. (32) Sørensen, T. S.; Sloth, P. J. Chem. Soc., Faraday Trans. 1992, 88, 571.
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the co-ion concentration in the capillary is expected to be lower than that in the equilibrium bulk electrolyte of concentration cs. The effect is conveniently measured by the so-called exclusion coefficient, Γ, defined as
Γ)
cs - 〈cco-ion〉 cs
(3)
where 〈cco-ion〉 is the average concentration of co-ions in the micropore. This quantity has been studied before: Vlachy and Haymet2,17,33 reported the exclusion coefficients for size and charge symmetric +1:-1 and +2:-2 electrolytes at various concentrations and the charge densities of the inner walls. In all calculations presented so far, rejection of the model electrolyte from the charged capillary was observed: in other words, in all calculations, positive values of Γ were obtained. The same conclusion holds true for micropores containing charge asymmetric electrolytes18 or mixtures.20 The results for the rejection coefficient, at several charge densities and for three different concentrations of the model +3-:3 electrolyte, are presented in Table 1. At σ ) 0.142 66 A s/m2 (the results denoted by σ(4) in Table 1) we found the average concentration of co-ions in the micropore 〈cco-ion〉 to be larger than the concentration of the equilibrium bulk solution cs, for all the values of cs studied here. The rejection coefficient Γ is therefore negative! For the lowest value of σ presented in Table 1, that is for σ ) 0.003 566 5 A s/m2, the rejection coefficient is positive for the concentration range from 0.0281 to 0.1982 M. At intermediate charge density (results denoted by σ(2) in Table 1) we obtained negative values of the exclusion coefficient at cs ) 0.0281 M and cs ) 0.0998 M, which turned into small positive values (i.e. 〈cco-ion〉 is smaller than cs) at higher concentration. The exclusion coefficient Γ therefore increases with increasing concentration cs. Note that this result differs qualitatively from the corresponding dependence for mono- or divalent counterions in the capillary.18 Further, the exclusion coefficient decreases with increasing charge density on the inner surface. Note that, despite the long GCMC runs, the numerical uncertainty in some results (for Γ close to zero) is large, due to the small differences between the two equilibrium concentrations in eq 3. The exclusion coefficient can also be obtained from the solution of the Poisson-Boltzmann equation (see, for example, ref 2). We solved the PB equation for this case, but since the results for Γ were not very accurate, we choose not to present them here. The PB calculations for a +3:-3 electrolyte yield positive values for Γ at all the charge densities and concentrations presented in Table 1. For the lowest charge density σ ) 0.003 566 5 A s/m2, the PB theory and GCMC calculations agree at least in the sign of the exclusion coefficient, but the PB values are considerably too high. For example, at cs ) 0.0998 M the GCMC result for the exclusion coefficient Γ is 0.106 ( 0.005, while the PB prediction is 0.192 ( 0.001. This finding is in agreement with our previous calculations for +2:-2 electrolytes,17,33 where the average concentration of an electrolyte in a micropore obtained from simulations was always considerably higher than that predicted by the Poisson-Boltzmann approach. Also, the PoissonBoltzmann theory predicts an increase in the exclusion coefficient (stronger rejection) with increasing surface charge density, and a decrease of Γ with increasing (33) Vlachy, V.; Haymet, A. D. J. J. Electroanal. Chem. 1990, 283, 77.
Figure 1. Local concentrations of trivalent counterions (+) and co-ions (b) inside a capillary as obtained by the GCMC simulation and the Poisson-Boltzmann equation (counterions, continuous line; co-ions, dashed line). The concentration of the equilibrium bulk electrolyte cs was 0.0281 M, and the surface charge density σ was 0.071 33 C/m2.
Figure 2. Reduced mean electrostatic potential y(r) as obtained by the GCMC simulation (+) and the Poisson-Boltzmann equation (line); data as for Figure 1.
concentration cs, while the GCMC results are just the opposite of that. To show the source of the discrepancy between the predictions of the classical theory and the computer simulations, the spatial distributions of ions in the capillary are shown in Figure 1. The concentration of the equilibrium bulk electrolyte cs was 0.0281 M, and the surface charge density was 0.071 33 C/m2 in this example. As we can see from this figure, the Poisson-Boltzmann prediction for the distribution of co-ions (lower line) differs qualitatively and quantitatively from the corresponding GCMC result. More precisely, the GCMC distribution of co-ions has a notable peak at r/a ) 3.93, that is, at about 1.25 nm from the charged surface. This peak is a consequence of the strong correlation between the co-ions and counterions. The latter, namely, form a dense layer next to the inner surface of the micropore (electrical doublelayer), and in response to that, there is an increased concentration of co-ions in the adjacent region. This kind of crossover behavior of the distribution functions has been noticed before,34-36 but here the consequences seem to be more dramatic; the result is a negative value of the exclusion coefficient. The GCMC and PB results for Γ in this example are -0.263 ( 0.005 and +0.500 ( 0.001, respectively. The potential distribution y(r) ) e0ψ(r)/kBT (e0 is the proton charge and kB is Boltzmann’s constant), (34) Torrie, G. M.; Valleau, J. P. J. Phys. Chem. 1982, 86, 3251. (35) Gonzales-Tovar, E.; Lozada-Cassou, M.; Henderson, D. J. Chem. Phys. 1985, 83, 361. (36) Vlachy, V.; Haymet, A. D. J. J. Chem. Phys. 1986, 84, 5874.
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concentration of ions present in the capillary. Note that the rejection coefficient as predicted by the GCMC simulation is small in this case; it is Γ ) -0.028 ( 0.005. Concluding Remarks
Figure 3. Same as for Figure 1: cs was 0.0998 M, and the surface charge density σ was 0.035 665 C/m2.
Figure 4. Same as for Figure 1: cs was 0.1982 M, and the surface charge density σ was 0.142 66 C/m2.
calculated as described in refs and 36 and 37, is for this case shown in Figure 2. The spatial distributions of ions in the capillary are also shown in Figures 3 and 4. In Figure 3 the GCMC and PB concentration profiles are compared for the case where the exclusion coefficient assumes a positive value: cs ) 0.0998 M and σ ) 0.035 665 C/m2 in this example. The results shown in next figure (Figure 4) apply to the highest concentration and charge density studied here, where cs ) 0.1982 M, and the charge density σ is 0.142 66 C/m2. The nonmonotonic profiles shown in this figure seem to be a consequence of the interplay of electrostatic and exclusion volume effects, reflecting a relatively high (37) Murthy, C. S.; Bacquet, R. J.; Rossky, P. J. J. Phys. Chem. 1985, 89, 701.
In this paper we showed that for a +3:-3 electrolyte solution in equilibrium with a charged micropore it is possible to obtain an increase of the average concentration in the system and therefore a negative “rejection” coefficient. This unexpected result cannot be explained by the classical electrical double-layer theory based on the solution of the Poisson-Boltzmann equation. The latter approach, namely, always predicts a positive value of Γ as a result. The simulation results presented in this paper are not inconsistent with experimental data. Negative “rejection”, that is, a higher concentration of the co-ions in the micropore than in the equilibrium bulk solution, was observed in cases where di- or trivalent counterions were present in solution.4,10,11 Of course, these experimental findings can also be attributed to the specific interactions between ions and the charged surface. Our computer simulations demonstrate, however, that the role of the ion-ion correlations in these systems should not be underestimated. This conclusion is consistent with recent theoretical studies proving that strong correlations between trivalent counterions yield clustering of highly charged macroions in micellar solutions.38,39 The results presented in this paper are based on the primitive model, where the solvent is treated as a dielectric continuum. Equilibrium concentrations of electrolyte in the capillary have recently been investigated by GCMC simulation of the solvent primitive model (SPM).40 In this model (SPM) the solvent molecules are described as neutral hard spheres. The conclusion is that addition of a neutral component yields concentration profiles which are more structured than those in the absence of the extra species. On the other hand, the volume-averaged quantities are much less model dependent. This gives some confidence in the thermodynamic results obtained for the primitive model. Acknowledgment. V.V. acknowledges the support of the Slovene Ministry of Science and Technology. LA000826E (38) Lobaskin, V.; Linse, P. Phys. Rev. Lett. 1999, 83, 4208. (39) Hribar, B.; Vlachy, V. Biophys. J. 2000, 78, 694. (40) Lee, M.; Chan, K.-Y.; Nicholson, D.; Zara, S. Chem. Phys. Lett. 1999, 307, 89.
