Langmuir 2000, 16, 9941-9943
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Modeling of the Salt Permeability in Fixed Charge Multilayer Membranes Konstantin Lebedev,†,‡ Patricio Ramı´rez,§ Salvador Mafe´,*,† and Julio Pellicer† Departament de Termodina` mica, Facultat de Fı´sica, Universitat de Vale` ncia, E-46100 Burjassot, Spain, and Departament de Cie` ncies Experimentals, Universitat Jaume I, Apdo. 224, E-12080 Castello´ , Spain Received May 3, 2000. In Final Form: September 20, 2000
Introduction Recently, much interest has been devoted to fixed charge multilayer membranes with alternating layers of cationic and anionic polyelectrolytes.1-3 These membranes could be considered as a series array of many bipolar membranes. Bipolar membranes are composed of one cation and one anion ion-exchange layers joined together in series.4-9 The highly selective ion transport that results from the rejection of the salt mobile ions by the fixed charge membrane groups (Donnan exclusion)3,7,10 constitutes one of the most interesting properties of the membranes. We have modeled here the salt permeability of a fixed charge multilayer membrane taking into account simultaneously many effects: the charge numbers of the salt ions, the diffusion boundary layers (DBLs) flanking the membrane, and the different values of the ion diffusion coefficients in these layers and the membrane. The calculations correspond to a number of layer pairs ranging from N ) 1 to N ) 100 and show clearly the effect of the ratio between the salt solution concentration and the membrane fixed charge concentration on the membrane permeability for different ionic charge numbers. Previous theoretical models for ion transport in fixed charge bipolar4-9,14,15 and multilayer membranes11-13 have not considered the problem under the above general conditions. We neglect the osmotic transport, the changes in the swelling of the multilayer structure with the salt concentration, and the effects of the adsorption of some polyelectrolyte material on the membrane surfaces. The theoretical approach is based on the Nernst-Planck flux equations16 together with the Donnan theory10 for the ionic equilibria at the membrane boundaries. We solve numerically the system of equations without invoking * To whom correspondence should be addressed. E-mail: smafe@ UV.es. † Universitat de Vale ` ncia. ‡ On leave from Kuban State University, Krasnodar, Russia. § Universitat Jaume I. (1) Decher, G. Science 1997, 277, 1232. (2) Krasemann, L.; Tieke, B. J. Membr. Sci. 1998, 150, 23. (3) Krasemann, L.; Tieke, B. Langmuir 2000, 16, 287. (4) Simons, R. J. Membr. Sci. 1993, 78, 13. (5) Strathmann, H.; Rapp, H. J.; Bauer, B.; Bell, C. M. Desalination 1993, 90, 303. (6) Zabolotskii, V. I.; Shel’deshov, N. V.; Gnusin, N. P. Russ. Chem. Rev. 1988, 57, 801. (7) Urairi, M.; Tsuru, T.; Nakao, S.; Kimura, S. J. Membr. Sci. 1992, 70, 153. (8) Higa, M.; Tanioka, A. J. Phys. Chem. 1997, 101, 2321. (9) Mafe´, S.; Ramı´rez, P. Acta Polym. 1997 48, 234. (10) Helfferich, F. Ion Exchange; McGraw-Hill: New York, 1962. (11) Kedem, O.; Katchalsky, A. Trans. Faraday Soc. 1963, 59, 1941. (12) Sonin, A. A.; Grossman, G. J. Phys. Chem. 1972, 76, 3996. (13) Higuchi, A.; Nakagawa, T. J. Membr. Sci. 1987, 32, 267. (14) Higa, M.; Kira, K. J. Phys. Chem. 1995, 99, 5089. (15) Tsuru, T.; Nakao, S.; Kimura, S. J. Membr. Sci. 1995, 108, 269. (16) Buck, R. P. J. Membr. Sci. 1984, 17, 1.
Figure 1. Sketch of the fixed charge multilayer membrane and the two diffusion boundary layers flanking it. The zone extending from x ) 0 to x ) dT corresponds to the membrane, and it is not to scale for the sake of clarity. Xi is the fixed charge concentration of layer i. The external salt solution concentrations are cLS ≡ c ) 10-4 mol/cm3 and cRS ) 0 mol/cm3.
simplifying approximations such as the Goldman constant field assumption10 and obtain the membrane permeability for typical values of the parameters characteristic of the problem. The theoretical results reproduce qualitatively some of the experimental trends observed concerning the charge number of the ions, the number of polyelectrolyte layers in the membrane, and the membrane fixed charge concentration.3 Theory and Numerical Solution Consider the multilayer membrane shown in Figure 1. The membrane stack separating two salt solutions of concentration cLS (left) and cRS (right) has a total thickness dT and is composed of N pairs of alternating anion- and cation-exchange layers. Layer i has a thickness di and a fixed charge concentration Xi (i ) 1, ..., 2N). We assume that the left salt solution is in front of an anion-exchange layer, so that Xi > 0 for odd i and Xi < 0 for even i. To make the notation more compact, the subscripts i ) -1, 0, 2N + 1, and 2N + 2 will denote, respectively, the left salt solution, the left DBL, the right DBL, and the right salt solution. The DBLs flanking the membrane can arise because of inefficient stirring10and extend from x ) -δL to x ) 0 (left solution) and from x ) dT to x ) dT + δR (right solution). Dk,i and ck,i are the diffusion coefficient and the local concentration of ion k (k ) 1 for cations and k ) 2 for anions) in layer i (i ) 0, ..., 2N + 1), respectively. cjk,i (k ) 1, 2; i ) -1, ..., 2N + 2) denotes the concentration of ion k at the left (j ) L) or right (j ) R) border of layer i. Therefore, the boundary conditions for the concentraR ) cRS . tions are cL1,0 ) cLS and c1,2N+1 The equations describing the transport of ions through the multilayer membrane are the Nernst-Planck equations
[
Jk,i ) -Dk,i
]
dck,i F dφi + zkck,i , dx RT dx k ) 1, 2; i ) 0, ..., 2N + 1 (1)
the electroneutrality conditions 2
∑ zkck,i + Xi ) 0, k)1
i ) 0, ..., 2N + 1
(2)
where X0 ) X2N+1 ) 0, the Donnan equilibrium conditions10
10.1021/la0006420 CCC: $19.00 © 2000 American Chemical Society Published on Web 11/15/2000
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Notes
L R R ck,i ) ck,i-1 exp[-zkF(φLi - φi-1 )/RT], k ) 1, 2; i ) 1, ..., 2N + 1 (3)
the condition of zero electric current 2
∑
k)1
2
zkJk,i )
∑ zkJk ) 0,
i ) 0, ..., 2N + 1
(4)
k)1
the continuity equations
Jk,i-1 ) Jk,i ≡ Jk, k ) 1, 2; i ) 1, ..., 2N + 1 (5) In eqs 1-5, zk (k ) 1, 2) is the charge number of ion k, φi (i ) 0, ..., 2N + 1) is the local electric potential in layer i, φji (i ) 0, ..., 2N + 1) is the electric potential value at the left (j ) L) and right (j ) R) border in layer i (see Figure 1), Jk,i (k ) 1, 2; i ) 0, 2N + 1) is the flux of ion k in layer i, and F, R, and T have their usual meanings.10 The unknowns in eqs 1-5 are the ion fluxes Jk, the mobile ion concentrations, and the electric potential through the membrane. To solve the problem, we define a 2N + 1-dimensional function F(Z) with F ) {f1, ..., f2N+1} and Z ) {Z1, ..., Z2N+1}: R R L L -z1/z2 (c2,i-1 )-z1/z2 - c1,i (c2,i ) fi ) c1,i-1 R , i ) 1, 3, ..., 2N - 1; Zi ) c2,i i ) 2, 4, ..., 2N;
(6)
L Zi ) c1,i , Z2N+1 ) J1 (7)
L R and c1,i (k ) 1, 2; i ) 1, The other boundary values ck,i R L (k ) 1, 2; i ) 0, 3, ..., 2N + 1) together with ck,i and c2,i 2, ..., 2N) are found by integrating numerically the Nernst-Planck equations (in dimensionless form) using a fourth-order Runge-Kutta method. Each membrane layer is divided into 100 segments for integration. The resulting system of nonlinear equations
F(Z) ) 0
(8)
is solved by Newton’s method according to the numerical algorithm previously described.17 Once J1 has been determined, the permeability of the membrane defined as
P≡
J1 cL1,0
-
Figure 2. The membrane permeability P versus the number of membrane layer pairs N for the values of the cation charge number indicated on the curves. The salt solution and membrane fixed charge concentrations are cLS ≡ c ) 10-4 mol/cm3 and cRS ) 0 mol/cm3, and X2i-1 ) -X2i ≡ X ) 10-3 mol/cm3 (i ) 1, ..., N), respectively. The membrane layer and DBL thicknesses are di ≡ d ) 10-5 cm (i ) 1, ..., 2N) and δL ) δR ≡ δ ) 0 cm, respectively. The ion diffusion coefficients in the membrane are10 D1m ) 0.5D2m ) 10-7 cm2/s.
R c1,2N+1
(9)
can be readily obtained. Results and Discussion The results of Figures 2-4 are obtained according to the above numerical procedure for the following values of the parameters unless otherwise specified. The salt solution and membrane fixed charge concentrations are cLS ≡ c ) 0.1 M ) 10-4 mol/cm3 and cRS ) 0 M, and X2i-1 ) -X2i ≡ X ) 1 M ) 10-3 mol/cm3 (i ) 1, ..., N), respectively. The membrane layer thicknesses are di ≡ d ) 10-5 cm (i ) 1, ..., 2N), and the total membrane thickness is dT ) 2Nd. The diffusion coefficient of ion k is assumed to be the same in every layer, Dk,i ≡ Dkm (k ) 1, 2; i ) 1, ..., 2N). In the two DBLs we assume Dk,0 ) Dk,2N+1 ≡ Dks (k ) 1, 2). We have chosen the values10 D1m ) 0.5D2m ) 10-7 cm2/s in the membrane and D1s ) 0.5D2s ) 10-5 cm2/s in the (17) Lebedev, K. A.; Kovalev, I. V. Elektrokhimiya 1999, 35, 224 (in Russian).
Figure 3. Decimal logarithm of membrane permeability, log(P), versus the decimal logarithm of the ratio between the salt solution and membrane fixed charge concentrations, log(c/X), for the values of the cation charge number indicated on the curves. All parameters are as in Figure 2 except for N ) 60 (which gives a total membrane thickness dT ) 2Nd ) 1.2 × 10-3 cm) with X changing according to the values of (c/X) in the figure.
DBLs. The anion charge number is z2 ) -1, and the cation charge number takes the values z1 ) 1, 2, and 3. Figure 2 shows the membrane permeability P versus the number of membrane layer pairs N in the absence of diffusion boundary layers (δL ) δR ≡ δ ) 0) for the values of the cation charge number indicated on the curves. Note the significant decrease of P with N as well as the changes in the P values obtained for the different charge numbers of the cations. The theoretical results obtained reproduce qualitatively the experimental trend observed: the Donnan exclusion is more effective the higher the charge number of the cation.3 However, the Donnan exclusion considered here cannot be the only relevant effect, since membrane layer thicknesses considerably higher than the reported3 one (d ∼ 10-7 cm) were assumed in Figure 2 to obtain theoretical permeabilities of the order of the experimental ones. We may tentatively ascribe the above discrepancy to the swelling of the multilayer structure during the diffusion experiments. This swelling would lead to an increase of the individual layer thicknesses. Alternatively, we could obtain similar results to those in Figure 2 with significantly lower membrane layer thicknesses by decreasing the effective ionic diffusion coef-
Notes
Figure 4. Membrane permeability P versus the DBL thickness δL ) δR ≡ δ for the values of the cation charge number indicated on the curves. Here N ) 60, and the total membrane thickness is dT ) 1.2 × 10-3 cm. The ion diffusion coefficients are D1m ) 0.5D2m ) 10-7 cm2/s in the membrane and D1s ) 0.5D2s ) 10-5 cm2/s in the diffusion boundary layers. The membrane fixed charge and salt solution concentrations are those of Figure 2.
ficients (due to the limited water content in the membrane). Other effects suggested by one of the reviewers may be the adsorption of some polyelectrolyte material inside the pores as well as on the membrane surfaces. Finally, although we ignored the osmotic transport, its effect is probably not so large to account for the discrepancy between the theory and the experimental data.3 Figure 3 shows the decimal logarithm of the membrane permeability, log(P), versus the logarithm of the ratio between the left salt solution and the membrane fixed charge concentrations, log(c/X), for the values of the cation charge number indicated on the curves. The parameters are those of Figure 2 except for N ) 60 (which gives a total membrane thickness dT ) 2Nd ) 1.2 × 10-3 cm) with X changing according to the values of (c/X) in the figure. As expected, the Donnan exclusion is more effective (and then the permeability is lower) the higher the charge number of the cation and the lower the ratio between the salt solution concentration and the membrane fixed charge concentration. This effect should not then be exclusive of fixed charge multilayer membranes: experimental data for anions of different charge numbers in monolayer negatively charged membranes show also that the higher the absolute value of the charge number the higher the salt rejection.18 Figure 3 shows that the membrane permeability decreases significantly when the fixed charge concentration of the polyelectrolyte layers is increased (at constant salt solution concentration). This is in qualitative agreement with the reported experimental data.3 However, there is a reversal in the dependence of the permeabilities on the cation charge number z1 for high enough salt solution concentrations. This reversal can readily be explained: the numerical solution of the transport equa(18) Tsuru, T.; Urairi, M.; Nakao, S.; Kimura, S. J. Chem. Eng. Jpn. 1991, 24, 518.
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tions shows that the cation concentration within the first positively charged membrane layer (see Figure 1) increases with the charge number z1 for high values of (c/X) while it decreases with z1 for low values of (c/X). Figure 4 shows the membrane permeability P versus the DBL thickness δL ) δR ≡ δ for the values of the cation charge number indicated on the curves. Here N ) 60, with a total membrane thickness dT ) 1.2 × 10-3 cm. Typical experimental values10 for δ are within the range 10-3 to 10-1 cm, although higher values than these are also included in Figure 4 to show the limiting behavior of P with δ. It is clear from Figure 4 that the permeability does not change with δ until the DBL thickness reaches values higher than the membrane thickness dT ) 1.2 × 10-3 cm. Therefore, the multilayer membrane controls completely the transport until δ ) 10-2 cm, approximately. For higher δ values the effects of the DBLs become significant and cannot be ignored. These effects are more important the lower the salt solution concentration.10,19 In summary, the theoretical results presented reproduce qualitatively some of the experimental trends observed concerning the charge number of the ions, the number of polyelectrolyte layers in the membrane, and the membrane fixed charge concentration.3 The Donnan exclusion phenomenon must then play a role in the observed membrane selectivity,7,15 although we would obtain theoretical permeabilities considerably higher than the experimental ones if we were to introduce in our theoretical model the relatively thin experimental layer thicknesses reported.3 Therefore, other physicochemical effects not taken into account here explicitly (the swelling of the multilayer structure during the diffusion experiments, the dependence of the ionic partition equilibrium at the membrane/ solution interface and the ionic diffusion through the membrane with the membrane water content, the change of the membrane fixed charge concentration with the density and molecular structure of the polyelectrolyte chains, etc.) should also affect significantly the membrane permeability.3 The above effects could be included if additional experimental data concerning the ionic partition coefficients were introduced in the Donnan equilibrium equations. Therefore, the present theoretical estimations are potentially useful for the understanding of the transport phenomena through fixed charge multilayer membranes and constitute a first approximation for more elaborate models to come. Acknowledgment. Fundacio´ Caixa-Castello´ (project P1B98-12), the Ministry of Education of Spain (project PB98-0419), and the University of Valencia (Visiting Professorship to K.L.) are gratefully acknowledged. Thanks are also given to Dr. Jose´ A. Manzanares as well as to one of the reviewers for valuable comments. LA0006420 (19) Mafe´, S.; Aguilella, V. M.; Pellicer, J. J. Membr. Sci. 1988, 36, 497.
