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Peculiarities of the Dynamics of Ion Exchange in Supersaturated Solutions and Colloid Systems Dmitri Muraviev,*,† Ruslan Kh. Khamizov,‡ and Nikolai A. Tikhonov§ Department of Chemistry, Autonomous University of Barcelona, E-08193 Bellaterra, Barcelona, Spain, Vernadsky Institute of Geological and Analytical Chemistry, Kosygin Str. 19, 117975 Moscow, Russia, and Department of Mathematics, Physical Faculty, Lomonosov Moscow State University, 119899 Moscow, Russia Received May 27, 2003. In Final Form: September 30, 2003 This paper reports the results obtained by experimental and theoretical study of the dynamics of ion exchange accompanied by the ion-exchange isothermal supersaturation (IXISS) effect. This effect is observed for a number of ion-exchange systems where the frontal or the reverse frontal separation is accompanied by the formation of extremely stable supersaturated solutions of low solubility substances in the interstitial space of ion-exchange columns. After leaving the column, a supersaturated solution crystallizes spontaneously, which allows for designing a practically ideal ion-exchange process where a crystalline product is obtained directly after the ion-exchange treatment cycle. The paper comprises results on the experimental investigation of IXISS of magnesium carbonate on carboxylic resins, which is observed in desorption of Mg2+ from the resin in the Mg form with Na2CO3 solutions or with solutions of Na2CO3NaHCO3 mixtures. The physical and mathematical models of the kinetics and the dynamics of the ionexchange processes proceeding in a colloid system comprising a supersaturated solution and an ion exchanger, which surface is partially blocked by adsorbed precrystalline aggregated (micelles), are proposed, and their validity is experimentally confirmed.
Introduction Ion exchange has been widely applied in different fields of science and technology since the middle of the last century as a powerful separation and purification technique,1 which is mainly used for the selective removal (in purification) or fractionation (in separation) of ionic species. Under certain conditions, the ion-exchange purification process is accompanied by the concentration of one of the components of the mixture under separation. For example, this effect is observed when using the frontal1-3 and the reverse frontal1,2,4,5 separation techniques. The component under purification can also interact with the counterion of the initial ionic form of the resin (in frontal separation) or with the co-ion of the displacing agent (in reverse frontal separation). In both cases, this interaction may result in the formation of low solubility substances, whose concentration exceeds their solubility at a given temperature. Moreover, this supersaturated solution may remain stable within the column interstitial space for a long period (see the following). This phenomenon known as “ion-exchange isothermal supersaturation” (IXISS) was discovered in 1979 by Muraviev.6 The practical application of the IXISS effect was shown to be mainly dealing with a possibility to design highly * Corresponding author. Tel.: 34-93-5801853. Fax: 34-935805729. E-mail:
[email protected]. † Autonomous University of Barcelona. ‡ Vernadsky Institute of Geological and Analytical Chemistry. § Lomonosov Moscow State University. (1) Gorshkov, V. I.; Muraviev, D.; Warshawsky, A. In Ion Exchange. Highlights of Russian Science; Muraviev, D., Gorshkov, V. I., Warshawsky, A., Eds.; Marcel Dekker: New York, 2000; Vol. 1, p 1. (2) Gorshkov, V. I.; Muraviev, D.; Warshawsky, A. Solvent Extr. Ion Exch. 1998, 16 (1), 1. (3) Bobleter, O.; Bonn, G. In Ion Exchangers; Dorfner, K., Ed.; Walter de Gruyter: Berlin, 1991; p 1208. (4) Muraviev, D.; Chanov, A. V.; Denisov, A. M.; Omarova, F. M.; Tuikina, S. R. React. Funct. Polym. 1992, 17, 29. (5) Muraviev, D. In Integrated Analytical Systems; Alegret, S., Ed.; Elsevier: Amsterdam, 2003; p 37. (6) Muraviev, D. Zh. Fiz. Khim. 1979, 53 (2), 438 (Russian).
effective and ecologically clean (“green”) ion-exchange technologies.7-12 This effect appears also to be useful in the modeling of some geochemical processes such as percolation of supersaturated natural mineralized solutions through the porous media followed by their spontaneous crystallization.13 The main interest in the theory of IXISS deals with an adequate explanation of the abnormal stability of supersaturated solutions in the bed of granulated ion exchangers14,15 and some effects accompanying the IXISS phenomenon such as, for example, the unusual shape of the breakthrough curves obtained in some IXISS systems.16-18 Unlike conventional ionexchange systems, these curves are characterized by the clearly pronounced plateau. The maximum concentration corresponding to the plateau range is far lower than that which can be expected from the conventional material balance. The principle possibility of supersaturation in ion exchange deals with the structural features of ion (7) Muraviev, D. In Encyclopedia of Separation Science; Wilson, I. D., Poole, C. F., Adlard, T. R., Cooke, M., Eds.; Academic Press: London, 2000; p 2644. (8) Muraviev, D.; Khamizov, R. Kh.; Tikhonov, N. A. In Advances in Ion Exchange for Industry and Research; Williams, P. A., Ed.; SCI: Cambridge, 1999; p 20. (9) Khamizov, R. Kh.; Mironova, L. I.; Tikhonov, N. A.; Bychkov, A. V.; Poezd, A. D. Sep. Sci. Technol. 1995, 31, 1. (10) Muraviev, D.; Khamizov, R. Kh.; Tikhonov, N. A.; Krachak, A. N.; Zhiguleva, T. I.; Fokina, O. V. Ind. Eng. Chem. Res. 1998, 37, 1950. (11) Khamizov, R. Kh.; Muraviev, D.; Tikhonov, N. A.; Krachak, A. N. Zhiguleva, T. I.; Fokina, O. V. Ind. Eng. Chem. Res. 1998, 37, 2496. (12) Muraviev, D.; Noguerol, J.; Gaona, J.; Valiente, M. Ind. Eng. Chem. Res. 1999, 38, 4409. (13) Putnis, A.; Prieto, M.; Fernandez Diaz, L. Geol. Mag. 1995, 132 (1), 1. (14) Muraviev, D.; Drozdova, N. V.; Dolgina, N. B.; Karabutov, A. A. Langmuir 1998, 14, 1822. (15) Muraviev, D. Langmuir 1998, 14, 4169. (16) Muraviev, D.; Khamizov, R. Kh.; Tikhonov, N. A. Solvent Extr. Ion Exch. 1998, 16 (1), 151. (17) Muraviev, D.; Khamizov, R. Kh.; Tikhonov, N. A. In Ion Exchange. Highlights of Russian Science; Muraviev, D., Gorshkov, V. I., Warshawsky, A., Eds.; Marcel Dekker: New York, 2000; p 151. (18) Muraviev, D.; Khamizov, R. Kh.; Tikhonov, N. A.; Kirshin, V. V. Langmuir 1997, 13, 7186.
10.1021/la030216p CCC: $25.00 © 2003 American Chemical Society Published on Web 11/22/2003
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exchangers and the mechanisms of the ion-exchange reaction. An ion-exchange interaction at the initial stage suggests the formation of a molecular distribution of components, which reproduces the distribution of functional groups of the ion exchanger. For an ion exchanger in the initial ionic form A interacting with the external solution of compound BC of concentration C0 with the formation of compound AC, the condition for formation of a supersaturated interbead solution of AC (IXISS effect) can be written in the simplest case of equally charged ions (z ) zA ) zB ) zC) as follows:19
(x
)
R λC0 g S 1-R
(1)
Y R ) [KBA(Y)]2 1-Y
(2)
where
and S is the molar solubility; KBA is the equilibrium constant of A-B exchange; Y is the molar fraction of B in the resin phase, and λ is the distribution coefficient of co-ions C between the resin and the solution phases. The validity of relationship (1) in a wide range of parameters was shown by Ferapontov et al.20 The problem of long-term stabilization of supersaturated solutions of inorganic substances in the bed of the ion exchanger still remains unsolved. Nevertheless, some recently obtained results permit the suggestion of possible mechanisms of this stabilization. For example, the results of electromigration experiments carried out with supersaturated MgCO3 solutions obtained by using the IXISS technique showed that the sign of the magnesium carbonate micelles (precrystalline aggregates) appeared to be positive.19 These supersaturated MgCO3 solutions with the supersaturation degree (γ) of ≈5 coexisted inside the column with the granulated resin bed for more than 72 h. To explain these experimental facts it was suggested that the stabilization of the supersaturated solution in this system is due to the sorption of MgCO3 micelles on the surface of the resin beads similar to the mechanism of stabilization of zwitterlyte solutions reported by Muraviev.15 The conclusion about the formation of a surface layer of adsorbed colloid particle on the beads of ion exchangers, which decreases the rate of the exchange reaction and, hence, strongly influences the dynamics of the ion exchange, has been also reported by Putnis et al.13 and Khamizov et al.21 Similar suggestions were made indirectly by Muraviev et al.18 in the phenomenological description of the dynamics of ion exchange in supersaturated solutions. A further development of this hypothesis within the new physical and mathematical models of the kinetics and dynamics of ion exchange in supersaturated solutions and colloid systems is reported in this paper. The novelty of these models (in comparison with those previously proposed)18 consists of a deeper insight of the physical processes accompanying the IXISS effect and substantially enhances the quality of the mathematical modeling. A new parameter, R, characterizing the blockage of the surface of the ion exchanger by colloid particles (having a clear physical sense) and the equation connecting the R values with the concentrations of all the system components were proposed for the first time. Unlike those (19) Khamizov, R. Kh.; Myasoedov, B. F.; Tikhonov, N. A.; Rudenko, B. A. Dokl. Ross. Akad. Nauk 1997, 356 (2), 216. (20) Ferapontov, N. B.; Gorshkov, V. I.; Trobov, Kh. T.; Parbuzina, L. R. Zh. Fiz. Khim. 1994, 68, 1109.
previously reported,18 the new mathematical model permitted a quantitative description of all the experimental data obtained in a wide range of eluate concentrations (seven breakthrough curves) by using one set of the model parameters. Results and Discussion All materials, ion exchangers, and analytical and experimental techniques used in this work were identical to those described elsewhere.18 Physical and Kinetic Models. Let us consider an ionexchange column loaded with a carboxylic ion exchanger in the Mg form, through which a solution of the Na2CO3NaHCO3 mixture is passed. Desorbed from the resin, MgCO3 forms a supersaturated solution, which remains stable in the resin bed but crystallizes simultaneously following its removal from the column. This solution can be considered as a mixture of Mg2+, Na+, CO32-, H+, and OH- ions; MgCO3 molecules; and MgCO3 micelles (colloid particles) where the total MgCO3 concentration equals C. These micelles contain a nucleus and surrounding shell, which includes the potential-forming Mg2+ ions and (most probably) the Stern layer of adsorbed ions. Let us denote the surface charge density of the shell by σ and suppose that micelles are sorbed by the resin as a result of the ion-exchange interaction between potential-forming Mg2+ ions and Na+ counterions fixed on the functional groups of the resin. The nuclei of the micelles are a bit removed from the surface of the resin. The total charge of the layer of adsorbed particles (LAP) prevents the formation of the second layer as a result of the repulsion of micelles approaching the LAP from the solution phase. Hence, as follows from this model, the formation of the LAP on the surface of the resin essentially results in the change of its effective charge to the opposite, which provides stabilization of the supersaturated solution. In other words, the stabilization mechanism in this case (micelles bearing a charge opposite to that of the functional groups) appears to be reduced to that described by Muraviev et al. for amino acid (zwitterlyte) solutions.6,15,22-24 As follows from the results reported recently by Tikhonov et al.,25 if the density of LAP equals n, the total MgCO3 concentration in the solution phase equals C and the potential barrier for removal of the micelles from the LAP is A, then n ∼ CA and A ∼ σ2. It has been also shown by using Gouy-Chapman theory that σ ∼ N1/2, and n ∼ CN. In a more general case, n ∼ C(eNR - 1), where R is a coefficient and N is the concentration of the charges in the solution.26 The probability of exchange of the Mg2+ counterion for two Na+ ions in the stripping solution WMg-2Na is proportional in the first approximation to CNa2. Hence, WMg-2Na ) R1CNa2, where R1 is a coefficient. As the maximum WMg-2Na value corresponds to the initial stage of the process (21) Khamizov, R. Kh.; Novitsky, E. G.; Mironova, L. I.; Fokina, O. V.; Zhiguleva, T. I.; Krachak, A. N. Tekhn. Machin. 1996, 4, 112 (Russian). (22) Muraviev, D.; Saurin, A. D. Zh. Fiz. Khim. 1980, 54, 1271. (23) Muraviev, D.; Gorshkov, V. I.; Fesenko, S. A. Zh. Fiz. Khim. 1982, 56, 1567. (24) Muraviev, D.; Obrezkov, O. N. Zh. Fiz. Khim. 1986, 60 (2), 396. (25) Tikhonov, N. A.; Khamizov, R. Kh.; Kirshin, V. V. Zh. Fiz. Khim. 2000, 74 (2), 309. (26) Parameter N in the Gouy-Chapman theory denotes the equivalent concentration of all positively (N+) or negatively (N-) charged ions in some micro- or macrovolume of bulk solution sufficiently removed from the charged surface of the colloid particle so that the electric field of the latter does not influence the distribution of these ions. The main criteria of the absence of such influence is the electroneutrality condition, that is, N ) N+ ) N-. When approaching the colloid particle (in the diffusion part of electrical layer), this condition is not fulfilled.
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Figure 2. β(R) dependence for R ) 0.1. Figure 1. Schematic diagram of the surface of ion exchanger with adsorbed colloid particles and area D of the Nernst film (see text).
when CNa ) CNa0 (the concentration of the stripping solution entering the column), let us use R1(CNa0)2 as a normalization unit. The probability of the complete exclusion of the Mg2+ counterion from the ion-exchange process (Wbl) due to its blocking by a micelle from the LAP with density n is proportional to n, that is, Wbl ) R2n (R2 is a coefficient). From the previous discussion, it seems obvious that the degree of exclusion of Mg2+ counterions from exchange is characterized by the ratio R2Wbl/R1WMg-2Na. After substitution by using the relation n ∼ CN and rearrangement, one obtains the dimensionless parameter R ) hCN/(CNa0)2 (where h is a dimensionless coefficient), which can be considered as a factor characterizing the blockage of ion exchange by colloid particles (LAP micelles). In the further mathematical modeling, the dimensionless coefficient h is chosen so that at R ) 1 the ion exchange does not proceed. The kinetics of ion exchange in the system under consideration can be described by the following equation:
∂ai/∂t ) βi(R)(Ci - φi)
(3)
where ai and Ci are the concentration of component i in the ion exchanger and in the bulk solution, respectively, and φi is the concentration of the solution in equilibrium with ai. By their physical sense, the R and 1 - R values determine the fractions of open surface area and that blocked by colloid particles of the resin beads. At R ) 0, the LAP is absent and β achieves its maximum β(0) ) β0. At R ) 1, the LAP completely blocks the ion-exchange process and β(1) ) 0. Let us consider the process of stationary (in time) diffusion of one of the components through the Nernst film surrounding the resin bead and having under the given conditions thickness L, as it is shown schematically in Figure 1. As near the surface of the resin phase the charged LAP can affect the equilibrium concentration in the solution phase, the φi value in this area can vary. To disregard the influence of the variable (in space) potential of the electrical field on the diffusion process, let us consider for the two-dimensional case the function U(x, y) describing the deviation of the component concentration in the solution phase from the equilibrium value. Let x ) 0 and x ) L correspond to the surface of the resin and the outer boundary of the Nernst film, respectively. Let us also imagine the LAP as a sort of periodical structure of regions with widths of 2Rδ and 2(1 - R)δ, corresponding respectively to the blocked (diffusion does not proceed) and open (diffusion proceeds) surface areas of the resin. The 2δ value (δ , L) determines the distance between the centers of the blocked regions.
Let us consider the diffusion through the rectangular area D shown in Figure 1. The position of this area is chosen so that the blocked regions of the resin surface are located symmetrically toward D sides parallel to the x axis. Therefore, the substance flux through these D sides is absent. The process is described by the following relationships:
∆U ) 0 U(0, Rδ < y < δ) ) 0, U(L, y) ) C - φ
(4)
∂U ∂U ∂U (x, 0) ) (x, δ) ) (0, 0 < y < Rδ) ) 0 ∂y ∂y ∂x An average flux density J of the component (from the solution to the sorbent) depends on R and is determined by the following equation:
J)
(L, y) dy ∫0δd∂U ∂x
1 δ
(5)
To find J, let us use the transformation of the area D by considering that it belongs to the plane of the complex variable z ) x + iy (see Appendix). The ∂a/∂t value (see eq 3) is proportional to J, and at the same time at fixed C and φ, it is proportional β. From here, by using eq 5 and (eq A4, see Appendix) one obtains:
β(R) 0
β
)
J(R) πR ) 1 - R ln cos 2 J(0)
(
)
-1
(6)
where R ) 2δ/πL. Figure 2 shows the β(R) dependence for R ) 0.1. Mathematical Model of the Dynamics of Ion Exchange. Let us formulate now the mathematical problem taking into account the previously mentioned reasonings. In addition to the the used notations just mentioned, let us introduce the following parameters of the process: the longitude coordinate along the column ) z, the porosity of the sorbent bed ) , the solution flux through the unit of the column cross-sectional area ) q, the ion-exchange constants ) Ki (i ) Na, Mg, H), the dissociation constants ) ki, and the exchange capacity of the ion exchanger ) aΣ. The mathematical model will include the following relationships:
material balance in column ∂CNa ∂aNa ∂CNa +q + (1 - ) )0 ∂t ∂z ∂t
∂ ∂ (CMg + CMgCO3) + q (CMg + CMgCO3) + ∂t ∂z ∂aMg )0 (1 - ) ∂t
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∂aH ∂ ∂ )0 (CH + CHCO3) + q (CH + CHCO3) + (1 - ) ∂t ∂z ∂t material balance in solution ∂ (CCO3 + CMgCO3 + CHCO3) + ∂t ∂ q (CCO3 + CMgCO3 + CHCO3) ) 0 ∂z kinetics of ion exchange βi0 ∂ai ) βi(Ci - φi), βi ) , where ∂t 1 - R ln cos(πR/2) CMgCO3(2CCO3 + CHCO3) R)h (CNa0)2 equilibrium of ion exchange
( )
KMg
φMg aMg
1/2
)
φNa φH ) KH aNa aH
mass balance in resin phase 2(aH + aNa) + aMg ) aΣ dissociation constants kMg )
CMgCO3
, kH )
CMgCCO3
CHCO3 CHCCO3
initial and boundary conditions at t ) 0: CCO3 ) CH ) 0, CNa ) CNainit, CMg ) CMginit, Ci ) φi at z ) 0: CMg ) 0, CNa ) CNa0, CCO3 ) CCO30, CH ) CH0 In these relationships, the dissociation constants ki are known from the literature, and , aΣ, Ki, and βi0 values have been found in independent experiments.27 The previous system of equations was solved by using the finite differences technique with precision of the first order in respect to the coordinate and the time step values by using the computation procedure described elsewhere.28 Figure 3 shows the results of dynamic experiments carried out (points) and computations made (curves). The curves were computed by using the following values of the model parameters: R ) 0.05; h ) 8.8 ( 0.4; aΣ ) 3.5 ( 0.2 equiv/dm3; and βi0 ) 0.55 ( 0.1 min-1. It should be emphasized that, unlike the previously reported model,18 the model described in this paper permits the description of all the experimental curves (seven curves) obtained in a wide range of the system parameters, such as the concentration of magnesium displacing agent (Na2CO3NaHCO3 mixtures), by using just one set of the model parameters. It seems also useful to emphasize that the new mathematical model based on a deeper physical insight of the process substantially improves the precision of the computation technique (up to e7% versus ∼15% of that previously reported).18 Hence, one can characterize (27) Tikhonov, N. A.; Khamizov, R. Kh. Sorption and Chromatographic Processes, 2001, 1 (6), 968 (Russian). (28) Tikhonov, N. A. Comput. Methods Math Phys. 1995, 35 (3), 467.
Figure 3. Experimental (points) and computed (lines) concentration-volume histories obtained in the desorption of Mg2+ from KB-4 resin in Mg form with Na2CO3 + NaHCO3 mixtures of different compositions: 1.5 M + 0.6 M (1); 0.57 M + 0.87 M (2); 0.57 M + 0.37 M (3); 0.5 M + 0.0 M (4); 1.13 M + 0.73 M (5); 1.5 M + 0.0 M (6); and 1.0 M + 0.0 M (7). V ) volume passed through a unit of the cross-sectional area of the column (mL).
the proposed mathematical model as permitting to move from a semiquantitative to a quantitative description of the experimental concentration-volume histories. A good agreement between the experimental and the computed results testifies to the validity of the mathematical model proposed and confirms the correctness of the physical model of the process just described. In conclusion, this is the first time that the model of the dynamics of ion exchange in supersaturated solutions and colloid systems has been proposed and applied successfully for the quantitative description of the IXISS-based process. Acknowledgment. The authors would like to sincerely acknowledge all their co-workers mentioned in different references and throughout the text for making this publication possible. A part of this work was supported by Research Grants ND-2000 and ND-2300 from the International Science Foundation, from the Science and Technology Programs of Russian Federation “Global Ocean” (Grant 02.08.1), and from Russian Foundation for Basic Research (Grant 02-03-33144). The Autonomous University of Barcelona and Institute for Material Science of Barcelona are acknowledged with thanks for the financial support of D.M. during the preparation of this paper.
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{
xtg πR2 + 1 - 4e } ) πR e arcth{ 1 - 2 cos }≈ x 2
πR πw ≈ arcth cos 2δ 2
-zπ/δ
2
-zπ/δ
2
(
)
Figure 4. Conformal transformation of area D (see Figure 1). Conditions ∂U/∂n ) 0, U ) 0, and U ) C - φ are valid in the boundary regions denoted by points 1, 2, and 4; 5; and 3, respectively (see Appendix).
(
)
πR πR 1 ) - ln e-zπ/δ cos2 ) 2 2 2 πR yπ 1 (A2) +i - ln e-xπ/δ cos2 2 2 2δ
arctg 1 - 2e-zπ/δ cos2
(
)
From here, within the limits of the indicated precision in the vicinity of boundary x ) L one has a unique conformity
ξ)x-
πR 2δ ln cos π 2
(
)
and y ) η
Appendix It can be demonstrated29 that conformal transformation
w ) f(z) )
{
xth 2δπz + tg πR2 }(A1)
πR 2δ arcth cos π 2
2
One obtains that area G is a rectangle (0 < ξ < ξ0 ) L - (2δ/π) ln[cos(πR/2)], 0 < η < δ), where one has the following problem:
2
transfers area D into area G on the plane of the complex variable w ) ξ + iη, as it is shown schematically in Figure 4. As follows from eq A1, area G has straight regions of its boundaries 1, 2, 4, and 5 and a curvilinear boundary region 3. The same numbers denote the corresponding boundary regions of areas D and G in Figure 4. By taking into account that δ , L, let us disregard in the further rearrangements the e-2πL/δ value in comparison with e-πL/δ. The analysis shows that a possible error of such an approximation does not exceed 3%. Then, in the vicinity of the boundary x ) L, one has thπz/2δ ≈ 1 - 2e-zπ/δ. Hence,
πz th2 ≈ 1 - 4e-zπ/δ 2δ and (29) Lavrentiev, M. A.; Shabat, B. V. Methods of theory of complex variable functions; Nauka: Moscow, 1951 (Russian).
∆U ) 0 (A3) U(0, η) ) 0, U(ξ0, η) ) C - φ, ∂U ∂U (ξ, 0) ) (ξ, δ) ) 0 ∂η ∂η whose solution seems sufficiently evident: U ) [(C - φ)/ ξ0]ξ. From here, one obtains
C-φ ∂U (ξ , η) ) ∂ξ 0 ξ0 After moving back to area D, one obtains
|
| |
∂U ∂U ∂ξ C-φ ) ) 1) ∂x x)L ∂ξ ξ)ξ0∂x x)L ξ0 πR C-φ 2δ 1/ 1 ln cos L πL 2
[
LA030216P
]
-1
(A4)
