J. Phys. Chem. 1996, 100, 12503-12508
12503
Transport of Ions through Cylindrical Ion-Selective Membranes Jyh-Ping Hsu* and Kun-Lin Yang Department of Chemical Engineering, National Taiwan UniVersity, Taipei, Taiwan 10617, Republic of China ReceiVed: January 2, 1996; In Final Form: May 10, 1996X
The rate of transport of ions through an ion-penetrable cylindrical membrane bearing negative fixed charges is investigated theoretically. For the same available space, the present novel device is capable of providing more surface area for ion transport than the conventional planar membranes through an array of hollow fibers type of apparatus. We consider a general linear distribution for the fixed charges in the membrane phase. A perturbation method is adopted for the resolution of the governing Nernst-Planck equation. The result of numerical simulation reveals that if the fixed charges are uniformly distributed, the current efficiency η is independent of both the inner radius (or the curvature) of the membrane and the current density. We show that if the parameters of a membrane are chosen adequately, the value of η can be increased by increasing the degree of nonuniformity of the fixed charge distribution under a constant amount of fixed charges.
If a membrane containing water-dissociable cations is immersed in an aqueous solution containing electrolytes, it becomes negatively charged. In this case, the cations in the liquid phase tend to migrate toward the membrane due to the electrostatic interaction, and the anions tend to move away from the membrane. A membrane having this characteristic is called a cation-selective membrane. In practice, if an electric field is applied, it can be used as a device to separate the ions in the liquid phase. Apparently, the nature of a membrane, especially the distribution of its fixed charges, plays a significant role in the assessment of its performance.1-6 Reiss and Bassignana7 proposed that a sandwich-type membrane comprises three sections, each with a different uniform fixed charge distribution (i.e., position independent). It was shown that the performance as an ion-separation device of this membrane is better than that if the fixed charges were distributed uniformly throughout the membrane. The behavior in which the performance of a membrane can be enhanced by varying the distribution of fixed charges is termed the superselectivity.7 Often, Henderson’s assumption8 is made, which leads to a reasonable result for uniformly distributed fixed charges. For nonuniformly distributed fixed charges, however, its performance is unsatisfactory5,9,10 and may yield results that violate thermodynamic laws.4 A more realistic approach is based on the Nernst-Planck equation coupled with the Poisson equation. This was adopted by Manzanares et al.4 for the case of a membrane having linearly distributed fixed charges. The analysis was also extended to trignometric functions.1 Selvey and Reiss1 suggested that the nonhomogeneity of fixed charges was due to the existence of some ionic groups in the membrane phase, and the distribution of fixed charges should close to trignometric functions. These distributions have the effect of enhancing the current efficiency. The level of the enhancement, however, is limited. Manzanares et al.2 found that the current efficiency of a membrane carrying nonhomogeneous fixed charges is not necessary greater than that of a membrane bearing homogeneously distributed fixed charges. Furthermore, depending upon the type of fixed charge distribution, increasing the current density of the system can have a positive or negative effect on current efficiency. Sokirko et al.3 discussed the effect of concentration polarization; both linear and exponential fixed charge distributions were consid* To whom correspondence should be addressed. X Abstract published in AdVance ACS Abstracts, July 1, 1996.
S0022-3654(96)00005-6 CCC: $12.00
ered. It was shown that the current efficiency is affected by the distribution of fixed charges. For some level of current density, the greater the degree of nonuniformity of fixed charge distribution, the higher the current efficiency. However, variation in the current density may have the effect of reducing the degree of enhancement of current efficiency. A key assumption to these analyses is local electroneutrality,2-4 which is satisfied if the thickness of a membrane is much greater than that of the electrical double layer in the liquid phase.1,4 The problem becomes complicated if this assumption is violated, and a numerical procedure is inevitable for the resolution of the governing equations. Selvey and Reiss1 suggested that a perturbation approach may be applicable for the case in which the local electroneutrality is not satisfied. No attempt was made to the resolution of the governing equations. For more complex situations, e.g., various types of ions with different valences11 or unsteady-state operations,12-14 the governing equations need to be solved numerically. Previous efforts are almost always based on a planar geometry. In practice, due to the limitation of space, a planar membrane is capable of providing a limited area for the transfer of ions. In the present study, an attempt is made to extend the analysis on a planar geometry to the case of a cylindrical geometry. A separation device similar to the hollow fiber reactors15 adopted widely in bioengineering can be designed to provide a large surface area in a finite volume, a highly desirable property. Modeling By referring to Figure 1, we consider a cylindrical membrane immersed in a symmetric electrolyte solution. For convenience, the solutions on both sides of the membrane have the same composition. As pointed out by Sokirko et al.,3 for planar membranes, the effect of boundary layer can be much more significant than that of the membrane inhomogenity. Here, we do not consider this effect, for simplicity, and the analysis is focused on the curvature effect of the membrane structure. Suppose that the diffusivities of cation and anion are the same. In this case, the transfer of ions are governed by the NernstPlanck equation. For a 1:1 electrolyte solution, we have1-4,11-14
[
Jj dCj q dψ ) -D + (-1)j+1 C r dr kT dr j © 1996 American Chemical Society
]
(1)
12504 J. Phys. Chem., Vol. 100, No. 30, 1996
Hsu and Yang ξ ) r/w, ξi ) ri/w, and
h)
J1w DriCi
(7a)
g)
J2w DriCi
(7b)
hc )
hri w
(7c)
gc )
gri w
(7d)
[ ]
4πq2Ci L)w κkT Figure 1. Schematic representation of the system under consideration. Here J1 and J2 are the fluxes of cation and anion, respectively, Ci and Co are the concentrations of cation in the core region and outside membrane, respectively, C1 and C2 are the concentrations of cation and anion in the membrane phase, respectively, C3 is the concentration of fixed charges, ri and ro are the inner and outer radii of the membrane, and I is the current density.
where ψ denotes the electrical potential, r is the radial distance, j is a species index (j ) 1 for cation, j ) 2 for anion), D is the diffusivity of ion, Cj is the concentration of ion species j, q represents the elementary charge, and k and T are the Boltzmann constant and the absolute temperature, respectively. Here the flux density of ion species j is written as Jj/r so that Jj is a constant proportional to the flux of this species, which is independent of the position at steady state. The electrostatic potential distribution is described by the Poisson equation
d2ψ 1 dψ -4πq + ) (C1 - C2 - C3) r dr k dr2
(2)
where C3 is the concentration of fixed charges in the membrane. Without loss of generality, we consider a cation-selective membrane; i.e., the fixed charges are negative. The current density, I, can be expressed as
I ) F∑zjJj
(3)
1/2
(7e)
In these expressions, w denotes the width of membrane, ri is the inner radius of membrane, and Ci and Co are, respectively, the concentrations of ions at r ) ri and at r ) ro, ro being the outer radius of membrane. (3) and (7a)-(7d) imply that i ) I/DCiF. If the condition of local electroneutrality applies,2-4 (6) reduces to
p - n - F(ξ) ) 0
(8)
If p is known, n can be calculated from this expression. Define the current efficiency, η, as7
η)
|h| |h| + |g|
(9)
This definition leads to 0.5 e η e 1. If half of the current is due to the transport of anion from the outer boundary of a membrane to its inner boundary, and half of that is due to the transport of cation in the inverse direction, η is at its lowest possible value of 0.5. In this case, ion separation does not occur. On the other hand, if the current is due to the transport of cation only, η has its highest possible value of unity, and the membrane is an idealized one. We assume a general linear fixed charge distribution
F(ξ) ) R + βξ
(10)
where
j
where F is the Faraday constant and zj is the valence of ion species j. Note that J1 is positive and J2 is negative. For 1:1 electrolytes, (1)-(3) can be rewritten in the following dimensionless forms:
-
dφ hc dp -p ) dξ dξ ξ
(4)
-
dn dφ gc +n ) dξ dξ ξ
(5)
d2φ 1 dφ + ) -L2[p - n - F(ξ)] dξ2 ξ dξ
(6)
i ) hc - gc
(7)
Here, φ ) qψ/kT, p ) C1/Ci, n ) C2/Ci, F ) C3/Ci, θ ) Co/Ci,
R)
3γ(ξ2o - ξi2) 2δ(ξ3o - ξi3) + 3(ξ2o - ξi2) β ) Rδ
(10a) (10b)
δ is an arbitrary parameter. It can be shown that the average dimensionless concentration of fixed charges is γ, which is independent of both R and β.16 On the basis of (4), (5), and (8), we have
gc - hc - βξ dφ ) dξ ξ(2p - R - βξ)
(11)
Combining this expression with (4) yields
dp p(hc + gc - βξ) - hc(R + βξ) ) dξ ξ(R + βξ - 2p)
(12)
This expression needs to be solved subject to the following
Transport of Ions
J. Phys. Chem., Vol. 100, No. 30, 1996 12505
Figure 2. Variation of current efficiency as a function of the difference between the dimensionless concentration of cation at the outer boundary and that at the inner boundary of a membrane for various dimensionless inner radii of the membrane. Parameters used are i ) 1 and γ ) 1. The slope β in (10) is positive. The dashed line denotes the result for a planar membrane (ξi f ∞).
boundary conditions
p ) pi, ξ ) ξi
(13)
p ) po, ξ ) ξo
(14)
The values of pi and po can be determined by Donnan equilibrium7 1/2 1 1 pi ) F(ξi) + F2(ξi) + 1 2 4
[ ] 1 1 p ) F(ξ ) + [ F (ξ ) + θ ] 2 4 o
2
o
2
o
1/2
(15) (16)
Solving (12) analytically is nontrivial, if not impossible. However, its numerical solution can be obtained without too much difficulty. If a membrane is highly charged, an approximate perturbation solution for (12) can be derived (Appendix). By referring to (7c) and (7d), the solution of (12) contains two unknowns, h and g. There is one more boundary condition that needs to be satisfied, (14). The current efficiency η can be estimated through an iterative method. For a fixed dimensionless current density i, an arbitrary η is assumed, and (7) and (9) are used to calculate h and g. The concentration of cations at the outer boundary of the membrane is then evaluated. The result obtained is compared with that based on (14). If the difference between these two values is reasonable, the assumed value for η is acceptable; otherwise, a new η is assumed, and the procedure is repeated.
Figure 3. Same as in Figure 2, except that the slope β in (10) is negative.
C3i are, respectively, the concentration of fixed charges at the outer boundary and that at the inner boundary of a membrane. Note that (Fo - Fi) is a measure of the degree of uniformity of the distribution of fixed charges in the membrane phase. The smaller its value, the more uniform the distribution. If the fixed charges in membrane are uniformly distributed, i.e., (Fo - Fi) ) 0, η is independent of ξi. For a fixed (Fo - Fi), η increases with ξi. This implies that the smaller the curvature of membrane surface, the greater the η. Note that if ξi is sufficiently large, η increases with (Fo - Fi). However, if ξi is small, the reverse is true. This suggests that a membrane having a nonuniform fixed charge distribution may not have a higher current efficiency than a membrane having a uniform fixed charge distribution. The result for the case of negative β is illustrated in Figure 3. This figure reveals that the general trend in Figure 2 can be reversed by changing the sign of β. This suggests that increasing η can always be achieved by choosing an appropriately designed membrane. For example, if ξi ) 2.5, a negative β should be chosen. Figures 2 and 3 suggest that if the distribution of fixed charges is uniform, (Fo - Fi) ) 0, the current efficiency is independent of both the inner radius of the membrane and the current density. This can be deduced from the analytical solution. Letting h ) ηi and g ) (η - 1)i in (A9), we obtain
[
exp
[
]
In the case of homogeneously distributed fixed charges, p0 ) pi. Since [(2η - 1)2iξ/R] never vanishes, (17) suggests that
p0(2η - 1) - ηR ) pi(2η - 1) - ηR ) 0
Discussion The performance of a cylindrical membrane is investigated through numerical simulation. The results presented in the following discussions are based on the exact numerical solution of (12). Figure 2 shows the variation of current efficiency, η, as a function of the difference between the dimensionless concentration of fixed charges at the outer boundary and that at the inner boundary of a membrane, (Fo - Fi), for various dimensionless inner radii of the membrane, ξi, with a positive β in (10). (Fo - Fi) is defined as (C3o - C3i)/Ci, where C3o and
]
[p0(2η - 1) - ηR] 2(2η - 1) ) (p0 - pi) i R [p (2η - 1) - ηR] (2η - 1)2iξ (17) exp R
(18)
or
η)
pi 2pi - R
(19)
In other words, η is independent of both ξ and i; the former is a function of ri. Note that, if the curvature of a membrane vanishes (ξi approaches infinity), it reduces to a planar membrane. This can be inferred from (A2) and is illustrated
12506 J. Phys. Chem., Vol. 100, No. 30, 1996
Hsu and Yang
Figure 4. Variation of current efficiency as a function of the difference between the dimensionless concentration of cation at the outer boundary and that at the inner boundary of a membrane for various dimensionless current densities. Parameters used are ξi ) 10 and γ ) 1. The slope β in (10) is positive.
Figure 6. Variation of current efficiency as a function of the difference between the dimensionless concentration of cation at the outer boundary and that at the inner boundary of a membrane for various average dimensionless fixed charge concentrations in the membrane. Parameters used are ξi ) 10 and i ) 1. The slope β in (10) is positive.
Figure 5. Same as in Figure 4, except that the slope β in (10) is negative.
Figure 7. Same as in Figure 6, except that the slope β in (10) is negative.
in Figures 2 and 3, where the distribution of η approaches a finite value as ξi increases to infinity. The variation of η as a function of (Fo - Fi) for various dimensionless current densities i is presented in Figure 4. The slope β in (10) is positive. Figure 4 shows that if the fixed charges in the membrane are uniformly distributed, η is independent of i. For a fixed (Fo - Fi), η increases with i. If i is sufficiently large, η increases with (Fo - Fi), and the reverse is true if i is small. The result for the case of negative β is pictured in Figure 5. Again, to increase η by increasing the degree of nonuniformity of fixed charge distribution can be achieved by adjusting the sign of β. Figure 6 shows the variation of η as a function of (Fo - Fi) for various average dimensionless concentrations of the fixed charges in the membrane, γ. The slope β in (10) is positive. The result for the case of negative β is illustrated in Figure 7. Figure 6 reveals that for a fixed (Fo - Fi), η increases with γ, as expected. If γ is small, η increases with (Fo - Fi), and the reverse is true if γ is large. A comparison of Figures 6 and 7 shows that if γ is sufficiently small (je.g., 0.5), η increases with (Fo - Fi), regardless of the sign of β. If γ is close to unity, η is insensitive to (Fo - Fi).
It should be pointed out that when the effect of ξi on η is discussed, the thickness of membrane and its volume cannot be fixed simultaneously. Similarly, the average concentration of fixed charges and the total amount of fixed charges cannot be fixed at the same time. In the numerical simulation, the thickness of membrane and the average density of fixed charges are fixed. In the mathematical treatment, the diffusivity of cation, D+, and that of anion, D-, are assumed to be the same. The derivation can be extended to D+ * D- without too much difficulty. In this case, the D in (7a) and (7b) needs to be replaced by D- and h replaced by (hD-/D+). The rest of the expressions remain the same. Here we show an example to illustrate the typical value of η for parameters used in practical applications. According to Mulder,18 cylindrical membranes can be classified, on the basis of their inner radii ri, into three categories: hollow fiber (ri < 0.25 mm), capillary membrane (0.25 mm < ri < 2.5 mm), and tubular membrane (ri > 2.5 mm). The thickness of membrane used for electrodialysis ranges from 100 to 500 µm, and the diffusivity of ions in the membrane is on the order of 10-14 to
Transport of Ions
J. Phys. Chem., Vol. 100, No. 30, 1996 12507
10-10 m2/s. If ri ) 0.25 mm, w ) 100 µm, i ) 1, D ) 10-10 m2/s, and Ci ) Co ) 0.1 M, then ξi ) 5, I ) 0.96 mA/m2, and η ) 0.7228.
TABLE 1: Deviation in the Current Efficiency Estimated on the Basis of the Coion-Exclusion Assumption from the Corresponding Exact Value Based on (12) for the Case i ) 0.5, ξi ) 5, and (Go - Gi) ) 0.5a
Appendix
γ
exact η
approximate η
% deviation
Suppose that the thickness of a membrane is thin compared with its inner radius, i.e., (ξo - ξi) , ξi. If we define the perturbation parameter as ) 1/ξi, then17
1 2 3 4 5
0.7273 0.8556 0.9171 0.9478 0.9646
0.6684 0.8160 0.8919 0.9321 0.9531
8.0984 4.6283 2.7478 1.6565 1.1922
ξ ) ξi[1 + (ξ - ξi)]
(A1)
Expanding p into a power series of yields
p ) p0 + p1 + 2p2 + ‚‚‚
(A2)
Substituting this expression into (12) gives
d(p0 + p1 + 2p2 + ‚‚‚)/dξ ) {[p0(h + g - β) - h(R + βξ)] + [-p0β(ξ - ξi) + p1(h + g - β)] + [p2(h + g β) - p1β(ξ - ξi)]}/{[R + βξ - 2p0] + [(ξ - ξi)(R + 2
βξ - 2p0) - 2p1] + 2[-2p1(ξ - ξi) - 2p2]} (A3) It can be shown that
1 1 y y2 - xz 2 ) + + ‚‚‚ x + y + 2z x x2 x3
a γ is the concentration of fixed charges; γ ) 1 corresponds to the case that the concentration of average fixed charges is the same as the bulk concentration of ions.
transformation z ) ξ + R/β in (A5) and solving the resultant expression subject to (A7a) and (A7b), we obtain
(
)
V20 + mV0 + n
1/2
× V2 + mV + n (2V + m - ∆)[2V0 + m - ∆]
{
(2V + m + ∆)[2V + m + ∆]
}
m/2∆
)
βz (A8) R + βξi
where V0 denotes the value of V at ξ ) 0, V ) [(p/z) - (β/2)] and
n)
(A4)
Expanding (A3) and collecting terms of the same order in , we obtain 0:
-2hβ + (h + g - β)β 4
∆)
(A8a)
x
(h + g)2 + 2hβ 4
(A8b)
If β ) 0, solving (A5) subject to (A7a) and (A7b) leads to
dp0 p0(h + g - β) - h(R + βξ) ) dξ R + βξ - 2p0
(A5) exp
1:
dp1/dξ ) {[-p0β(ξ - ξi) + p1(h + g - β)]}/{R + βξ 2p0} - {[p0(h + g - β) - h(R + βξ)][(R + βξ - 2p0)(ξ ξi) - 2p1]}/{(R + βξ - 2p0)2} (A6) 2:
{
}
[p0(h + g) - hR] 2(h + g) ) (p - pi) i R(h - g + β) 0 [p (h + g) - hR]
[
exp
]
(h + g)2ξ (A9) R(g - h)
If a membrane is highly charged, p0 can be approximated by (R + βξ), the so-called coion-exclusion assumption. In this case (A6) reduces to
dp1 h - g + β + p + (2h + g)(ξ - ξi) ) 0 dξ R + βξ 1
dp2/dξ ) {p2(h + g - β) - p1β(ξ - ξi)}/{R + βξ - 2p0} {[(ξ - ξi)(R + βξ - 2p0) - 2p1][-p0(ξ - ξi) +
(A10)
p1(h + g - β)]}/{(R + βξ - 2p0)2} + {[(ξ - ξi)(R +
Solving this equation yields
βξ - 2p0) - 2p1]2 - (R + βξ - 2p0)[-2p1(ξ - ξi) -
p1 ) {(R + βξ)[Rg + 2Rh + gξi(3β - g) + hξi(6β - g + 2) - gξ(2β - g) - hξ(4β + g - 2)]}/{(2β - g + h)(3β -
2p2]}{[p0(h + g - β) - h(R + βξ)]/(R + βξ - 2p0) } (A7) 3
g + h)} + k(R + βξ)(-β+g-h)/β (A11)
l where The associate boundary conditions are
p0 ) pi, ξ ) ξi
(A7a)
p0 ) po, ξ ) ξo
(A7b)
pm ) 0, ξ ) ξi, m ) 1, 2, ...
(A7c)
pm ) 0, ξ ) ξo, m ) 1, 2, ...
(A7d)
Here pi and po are the dimensionless concentrations of cation at the inner and the outer boundaries of the membrane, respectively, defined in (15) and (16). If β * 0, making the
k)-
R[Rg + 2Rh + gξi(3β - g) + hξi(6β - g + 2)] (2β - g + h)(3β - g + h)R(-β+g-h)/β
(A11a)
Substituting p0 and p1 into (A2) gives the first-order perturbation solution for (12). If vanishes (or ξi approaches infinity), the membrane becomes planar. In this case, (A2) yields p ) p0, which is governed by (A5). In other words, the zeroth-order perturbation solution of (12) corresponds to that of a planar membrane. The coion-exclusion assumption is satisfied if a membrane is highly charged. This is usually the case in practice. Table
12508 J. Phys. Chem., Vol. 100, No. 30, 1996 1 illustrates the deviation of the current efficiency estimated on the basis of the coion-exclusion assumption from the corresponding exact value. As can be seen from this table, the percent deviation is about 10% if the average concentration of fixed charges in membrane is the same as the bulk concentration of ions and is about 1% if the former is five times of the latter. References and Notes (1) Selvey, C.; Reiss, H. J. Membr. Sci. 1985, 23, 11. (2) Manzanares, J. A.; Mafe, S.; Pellicer, J. J. Chem. Soc., Faraday Trans. 1992, 88, 2355. (3) Sokirko, V. A.; Manzanares, J. A.; Pellicer, J. J. Colloid Interface Sci. 1994, 168, 32. (4) Manzanares, J. A.; Mafe, S.; Pellicer, J. J. Phys. Chem. 1991, 95, 5620. (5) Takagi, R.; Nakagaki, M. J. Membr. Sci. 1986, 27, 285. (6) Later, R. Chem. ReV. 1990, 90, 355.
Hsu and Yang (7) Reiss, H.; Bassignana, I. C. J. Membr. Sci. 1982, 11, 219. (8) Lakshminarayanaiah, N. Equations of Membrane Biophysics; Academic Press: New York, 1984. (9) Nakagaki, M.; Takagi, R. J. Membr. Sci. 1985, 23, 29. (10) Takagai, R.; Nakagaki, M. J. Membr. Sci. 1990, 53, 19. (11) Brumleve, T. R.; Buck, R. P. J. Electroanal. Chem. 1978, 90, 1. (12) Garrido, J.; Mafe, S.; Pellicer, J. J. Membr. Sci. 1985, 24, 7. (13) Manzanares, J. A.; Murphy, W. D.; Mafe, S.; Reiss, H. J. Phys. Chem. 1993, 97, 8524. (14) Murphy, W. D.; Manzanares, J. A.; Mafe, S.; Reiss, H. J. Phys. Chem. 1992, 96, 9983. (15) Broek, A. P.; Teunis, H. A.; Bargeman, D.; Sprengers, E. D.; Smolders, C. A. J. Membr. Sci. 1992, 73, 143. (16) Kuo, Y. C.; Hsu, J. P. J. Chem. Phys. 1995, 102, 22. (17) Hsu, J. P.; Kuo, Y. C. J. Chem. Soc., Faraday Trans. 1995, 91, 1223. (18) Mulder, M. Basic Principles of Membrane Technology; Kluwer Academic: Boston, MA, 1951.
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