J. Phys. Chem. B 2002, 106, 2117-2122
2117
Effect of Ionic Sizes on the Electrophoretic Mobility of a Particle with a Charge-Regulated Membrane in a General Electrolyte Solution Shih-Wei Huang and Jyh-Ping Hsu* Department of Chemical Engineering, National Taiwan UniVersity, Taipei, Taiwan 10617, R.O.C.
Yung-Chih Kuo Department of Chemical Engineering, National Chung Cheng UniVersity, Chia-Yi, Taiwan 62102, R.O.C.
Shiojenn Tseng Department of Mathematics, Tamkang UniVersity, Tamsui, Taipei, Taiwan 25137, R.O.C. ReceiVed: May 1, 2001; In Final Form: NoVember 1, 2001
The electrophoretic mobility of a planar particle covered by a charge-regulated membrane in a mixed (a:b) + (c:b) electrolyte solution, taking the effect of the sizes of charged species into account, is analyzed theoretically. The influences of the key parameters, which include the frictional coefficient of the membrane layer, pH, membrane thickness, ionic strength, valence of counterions, fixed charge density, fraction of multivalent counterions, dissociation equilibrium constant of the functional groups, and sizes of cations and anions, are discussed. We demonstrate the importance of the ionic steric effects by showing a substantial difference between the classic point-charge model and the present model.
I. Introduction The surface properties of charged entities are often estimated by electrophoresis in which an electric field is applied to an electrolyte dispersion of the entity. The sign of the surface charge and the level of the surface potential of biocolloids such as cells and microorganisms, for example, are usually determined by this technique. The classic model of Smoluchowski1 provides fundamental theory for the estimation of the electrophoretic mobility of a particle. The original derivation of this model is based on rigid entities in which the charge carried by an entity is distributed over its surface. Therefore, it needs to be modified when nonrigid particles are considered. A typical example includes biocolloids and particles covered by an artificial membrane. Human erythrocytes, leukocytes, and most neurons belong to the former. In these cases, an entity comprises a rigid core and an ion-penetrable layer, which carries fixed charges due to the dissociation of the functional groups it bears.2,3 Ohshima and Kondo4 modeled the electrophoretic behavior of a particle coated with an ion-penetrable membrane layer by assuming a uniformly distributed fixed charge. The case of linear or exponential distributed fixed charge was considered by Hsu et al.5 Ohshima6 was able to derive a general expression for the electrophoretic mobility of a rigid spherical particle covered with a polyelectrolyte layer. The analysis of Ohshima and Kondo4 and that of Hsu et al.5 was extended by Hsu and Fan7 to take into account the effect of the spatial variations on both the fixed charge and the dielectric constant in the membrane phase. Hsu et al.8 estimated the electrophoretic mobility of a planar particle covered by an ion-penetrable membrane in an asymmetric electrolyte solution by assuming a uniform fixed charge distribution. A more detailed analysis * To whom correspondence should be addressed. Fax: 886-2-23623040. E-mail:
[email protected].
was conducted by Tseng et al.9 to take into account the effect of the degree of dissociation of the functional groups in the membrane layer. The results discussed above are based on the classic GouyChapman theory1 in which ionic species are treated as point charges; that is, the effect of their sizes on the behavior of the system under consideration has not been considered. It was concluded that double layer behavior is influenced significantly by the effective radii of the electrolyte ions.10-13 The effect of ionic sizes on the ionic concentration of an infinite planar charged surface in a strong electrolyte solution was examined by Torrie and Valleau14 through a Monte Carlo simulation. Grand canonical ensemble Monte Carlo simulations of ions with different sizes and charges in a charged spherical cavity have also been conducted,15-17 and similar conclusions have been drawn. Lozada-Cassou et al.18 examined the nonlinear effects of the electrophoresis of a spherical colloidal particle by incorporating ionic size effects. The effect of the sizes of charged species on the electrical potential and the properties of double layer for the case of an ion-penetrable charged membrane was investigated by Hsu and Kuo19 and by Kuo and Hsu.20 The electrical interaction between two particles covered by an ionpenetrable charged membrane in an electrolyte solution was estimated by Kuo and Hsu21 taking the effect of the sizes of charged species into account. The ionic sizes are found to have a significant influence on the electrical interaction force between two particles, which is consistent with the result for the shortrange repulsive forces between silica surfaces.22 Employing a microelectrophoresis technique, Mironov and Dolgaya23 studied the surface charge of the outer membrane of rat dorsal root ganglion neurons. Their experimental observations revealed that the magnitude of electrophoretic mobility of rat dorsal root ganglion neurons when Ca2+ is present is smaller that when divalent organic cations, dimethonium or hexa-
10.1021/jp011644b CCC: $22.00 © 2002 American Chemical Society Published on Web 02/02/2002
2118 J. Phys. Chem. B, Vol. 106, No. 8, 2002
Huang et al. For convenience, the system is divided into five regions: region I, X < Xca, which comprises the charge-free region (-∞ < X < 0) and the inner uncharged membrane (0 < X < Xca), X being the scaled distance; region II, Xca < X < Xan, which contains cations only; region III, Xan < X < Xi, which contains both cations and anions; region IV, Xi < X < Xo, which contains all charged species; region V, Xo < X, which includes the outer uncharged membrane (Xo < X < d) and the liquid phase (d < X < ∞). Xi and Xo are, respectively, the locations of the inner plane of fixed charge and the outer plane of fixed charge, and Xca and Xan represent respectively the most interior locations of positive and negative charges. Xi is the effective scaled radius of the fixed charged group. The scaled symbols are defined by X ) κr, Xca ) κσca, Xan ) κσan, Xi ) κσf, and Xo ) d - κσf, κ and r being, respectively, the reciprocal Debye length and the distance from the rigid core.
Figure 1. Schematic representation of the system under consideration. Xca, Xan, Xi, Xo, and d are respectively scaled locations of the effective most interior locations of positive and negative charges, the locations of inner plane of the fixed charge and outer plane of fixed charge, and the scaled thickness of the membrane. The scaled size of fixed charged group is Xi. Regions I-V denote respectively the charge-free region, the region in which only cations are present, the region in which both cations and anions are present, the region in which all charged species are present, and the region which comprises the outer uncharged membrane and the diffuse double layer.
methonium, are present. That is, the larger the size of counterions, the greater the mobility. Their results also show that the mobility increases with the increase in pH and decreases with the increase in electrolyte concentration. Note that two or more kinds of cations are present in the liquid phase and the membrane layer may carry one or more types of functional groups in their experiments. This implies that the classic theory for electrophoresis needs to be modified. In the present study, the electrophoretic behavior of a planar particle coated with a charge-regulated membrane layer in a general electrolyte solution is investigated taking the sizes of charged species into account. The model of Hsu and Kuo20,24 is extended to the case in which both acidic and basic functional groups are present in the membrane layer and a mixed (a:b) + (c:b) electrolyte solution is considered. On the basis of the latter, the effect of the presence of multivalent counterions on the electrophoretic mobility of a particle can be examined. II. Analysis The problem under consideration is shown in Figure 1, where we consider a planar particle comprised of a rigid, uncharged core and an ion-penetrable membrane layer of scaled thickness d. The particle moves with constant velocity U0 in a mixed (a:b) + (c:b) electrolyte solution, where “a” and “c” are the valences of cations and “b” is the valence of anions. The continuum solvent (primitive model) nature is assumed. An electric field E parallel to the surface of the particle is applied. The membrane layer contains uniformly distributed functional groups, and the dissociation of these functional groups yields fixed charge. Let σf, σan, and σca be the effective radii of fixed charged groups, anions, and cations, respectively. Without loss of generality, we assume that σf > σan > σca. The fixed charged groups are arranged so that the margin of the leftmost one coincides with the core-membrane interface, and that of the rightmost one coincides with the membrane-liquid interface.
1. Electrical Field. Suppose that the spatial variation in the scaled electrical potential in the system under consideration can be described by the scaled Poisson-Boltzmann equation
d2ψ ) 0,region I dX2 d2ψ -[(1 - ξ) exp(-aψ) + ξ exp(-cψ)] ) , region II (a + b) + (c - a)ξ dX2
(1)
(2)
d2ψ exp(bψ) - [(1 - ξ) exp(-aψ) + ξ exp(-cψ)] ) , (a + b) + (c - a)ξ dX2 region III (3) d2ψ exp(bψ) - [(1 - ξ) exp(-aψ) + ξ exp(-cψ)] + N , ) (a + b) + (c - a)ξ dX2 region IV (4) d2ψ exp(bψ) - [(1 - ξ) exp(-aψ) + ξ exp(-cψ)] ) , (a + b) + (c - a)ξ dX2 region V (5) where ψ ) eφ/kBT, ξ ) cnc0/bnb0, κ2 ) e2[a(a + b)na0 + c(b + c)nc0]/0rkBT, bnb0 ) ana0 + cnc0, and N ) NA(N-i)1 - N+i)1)/ bnb0. Here, φ is the electrical potential; ξ is the fraction of cations of valence c in the bulk liquid phase; na0, nb0, and nc0 are, respectively, the number concentrations of ionic species of valences a, -b, and c in the bulk liquid phase; e and NA are, respectively, the elementary charge and the Avogadro number; r and 0 are the relative permittivity of the electrolyte solution and the permittivity of a vacuum, respectively; kB and T denote, respectively, the Boltzmann constant and the absolute temperature; N-i)1 and N+i)1 are, respectively, the densities of the negative and the positive fixed charges in membrane layer. Note that the Laplace equation (1) in the charge-free region I implies a constant electric field and hence a linear mean electrostatic potential. We assume the following: (i) the electric field is absent in the rigid core of the particle, (ii) the gradient of the electrical potential vanishes at a point far from the particle surface, and (iii) both the electrical and its gradient are continuous at the intersection of two adjacent regions. Therefore
Electrophoretic Mobility of a Particle
J. Phys. Chem. B, Vol. 106, No. 8, 2002 2119
we have
dψ f 0as X f 0 and X f ∞ dX
(dψ dX ) (dψ dX )
) X)Xca+
) X)Xan+
(dψ dX )
(dψ dX )
X)Xca-
(dψ dX )
X)Xan-
(dψ dX )
)
X)Xi+
and ψ(Xca+) ) ψ(Xca-)
(7)
d2U -λ2U ) 0,0 < X < Xca dX2
and ψ(Xan+) ) ψ(Xan-)
(8)
and ψ(Xi+) ) ψ(Xi-)
(9)
-[(1 - ξ) exp(-aψ) + ξ exp(-cψ)] d 2U , - λ 2U ) L 2 (a + b) + (c - a)ξ dX Xca < X < Xan (19)
X)Xi-
dψ dψ ) and ψ(Xo+) ) ψ(Xo-) dX X)Xo+ dX X)Xo-
( )
(6)
3. Flow Field. The relative velocity between the particle and the liquid phase is in the direction parallel to the particle surface. Suppose that the flow field can be described by the scaled Navier-Stokes equation
( )
(10)
Equation 6 implies that the system is at electroneutrality, and the electrical field vanishes at the membrane-core interface. The former means that the fixed charge carried by the membrane layer is balanced by the mobile ions inside and outside the membrane layer, and the latter is due to the fact that the rigid core is free of charge. It should be pointed out that although the electric field vanishes at the membrane-core interface, the electrical potential inside the rigid core has a nonzero constant value. 2. Dissociation of Functional Groups. Suppose that both acidic and basic functional groups are present in the membrane layer. The dissociation of these functional groups can be expressed by
AHZa-(n-1)(n-1)- S AHZa-nn- + H+, Ka,n, n ) 1, 2, ..., Za
d 2U - λ 2U ) dX2 exp(bψ) - [(1 - ξ) exp(-aψ) + ξ exp(-cψ)] , L (a + b) + (c - a)ξ Xan < X < d (20) exp(bψ) - [(1 - ξ) exp(-aψ) + ξ exp(-cψ)] d 2U )L , 2 (a + b) + (c - a)ξ dX d < X < ∞ (21) where U(X) is the scaled velocity, η is the viscosity of the electrolyte solution, and f is the frictional coefficient of the membrane layer. The scaled quantities are defined by U ) u/U0, λ2 ) f/ηκ2, and L ) 0rkBTE/ηeU0. For convenience, the particle remains fixed and the fluid moves in the inverse direction from that of U. The boundary conditions associated with (18)-(21) are assumed as
U ) 0 as X f 0
(11) BHZb-(n-1)[Zb-(n-1)]+ S BHZb-n(Zb-n)+ + H+, Kb,n, n ) 1, 2, ..., Zb (12) In these expressions Za is the number of dissociable protons of the acidic group AHZa, Zb is the number of absorbable protons of the basic group B, and Ka,n, n ) 1, 2, ..., Za, and Kb,n, n ) 1, 2, ..., Zb, are the corresponding equilibrium constants. It can be shown that the concentration of negative fixed charges, N-i)1, and that of positive fixed charges, N+i)1, are24 i)1
N-i)1 )
Na Qa 1 + Pa
(
(13)
)
(14)
, V ) a, b
(15)
N+i)1 ) Nbi)1 Zb -
Qb 1 + Pb
In these expressions
[ ] ∑[ ∏ ] ZV
PV )
m
∑∏
m)1 j)1 CH+ ZV
QV )
KV,j
n
n
n)1
KV,j
j)1 CH+
, V ) a, b
CH+0 is the bulk concentration of H+.
(dU dX ) (dU dX )
) X)Xca+
)
(dU dX )
(dU dX )
and U(Xca+) ) U(Xca-)
(23)
(dU dX )
and U(Xan+) ) U(Xan-)
(24)
and U(d+) ) U(d-)
(25)
X)Xca-
X)Xan+
X)d+
(22)
X)Xan-
)
(dU dX )
X)d-
U f -1 as X f ∞
(26)
Equation 22 implies that the slipping plane is chosen at the core-membrane interface. Equations 23-25 state that both the velocity and the tangential component of the stress tensor are continuous at the closest approach of cations Xca, at the closest approach of anions Xan, and at the membrane-liquid interface d. Solving (1)-(5) and (18)-(21) simultaneously subject to the boundary conditions (6)-(10) and (22)-(26) provides the spatial variation of electrical potential and that of fluid velocity. The electrophoretic mobility, µ, defined by µ ) U0/E, can then be evaluated based on these results. III. Results and Discussion
(16)
where the concentration of H+, CH+, can be evaluated by
CH+ ) CH+0 exp(-ψ)
(18)
(17)
The behavior of the system under consideration is investigated through numerical simulation. The governing equations are solved numerically by PDEase,25 which is based on Galerkin’s finite element method of weighted residuals with quadratic basis. For illustration, we assume that the fixed charge in the membrane layer is negative with Za ) Zb ) 1. Also, we assume r ) 78.5 and T ) 298 K. Figure 2 shows the simulated variation in the scaled electrical potential ψ as a function of the scaled
2120 J. Phys. Chem. B, Vol. 106, No. 8, 2002
Figure 2. Variation of scaled potential ψ as a function of scaled distance X. Solid curve, present model; dashed curve, PCM. Key: T ) 298 K, r ) 78.5, ionic strength ) 0.1 M, η ) 8.91 × 10-4 N s/m2, f ) 1 × 1011 N s/m4; d ) 1, a ) b ) 1, c ) 2, ξ ) 0, Za ) Zb ) 1, pH ) 7, pKa,1 ) 3, pKb,1 ) 5, Nai)1 ) Nbi)1 ) 1 M, E ) 800 V/m, Xca ) 0.05, Xan ) 0.1, and Xi ) 0.2.
Figure 3. Variation of mobility µ as a function of pH for various f. Solid curves, present model; dashed curves, PCM. Curves 1, f ) 1 × 1010 N s/m4; curves 2, f ) 1 × 1011 N s/m4; curves 3, f ) 1 × 1012 N s/m4. Key: same as Figure 2.
distance X. For comparison, the corresponding result based on the classic point charge model (PCM) is also represented in this figure. Figure 2 reveals that |ψ| exhibits a local maximum as X varies which is not observed for the case of PCM in which |ψ| decreases monotonically with X. In general, the latter will overestimate |ψ|. This is because the fixed charge is present in region IV only in the present model, but it is present in the whole membrane layer in the corresponding PCM. Figure 3 illustrates the variation of the mobility of a particle µ as a function of the pH of the bulk liquid phase for various friction coefficients f. The results for the corresponding PCM are also presented for comparison. Figure 3 indicates that, for a fixed pH, |µ| decreases with the increase in f. This is expected since the larger the f the greater the resistance of the membrane layer for liquid flow. As can be seen in Figure 3, |µ| increases with the increase in pH for pH < 7, and becomes about constant for pH > 7, which is consistent with the experimental results of Mironov and Dolgaya.23 A similar relation between |µ| and pH was also observed experimentally by Hayashi et al.26 This is because if pH is low, the degree of dissociation of acidic functional groups in membrane layer becomes small and, at the same time, it is easy for H+ to bind to the basic groups in the
Huang et al.
Figure 4. Variation of mobility µ as a function of scaled membrane thickness d at various pH values. Solid curves, present model; dashed curves, PCM. Curves 1, pH ) 6; curves 2, pH ) 7; curves 3, pH ) 8. Key: same as Figure 2.
membrane layer. These lead to a low concentration of negative fixed charge in the membrane layer and, therefore, a small mobility. On the other hand, if pH is sufficiently high, the dissociation of acidic functional groups is essentially complete, and the concentration of negative fixed charge remains constant, as does the mobility. According to the result of Mironov and Dolgaya,23 as pH varies from 5 to 9, |µ| ranges from 1 to 1.4 µm‚s-1‚V-1‚cm in a 0.0001 M 1:1 electrolyte solution, and from 0.1 to 3.2 µm‚s-1‚V-1‚cm in 0.02 M KNO3.26 It will be shown later that a higher ionic strength leads to a lower |µ|. The ionic strength used in Figure 3 is 0.0001 M, and the range of |µ| obtained is reasonable. This figure also reveals that the corresponding PCM will overestimate the absolute mobility, and the deviation increases with the increase in pH. The variation of the mobility of a particle µ as a function of the scaled membrane thickness d at various pH is illustrated in Figure 4. The results based on the corresponding PCM are also shown for comparison. Here, the total amount of functional groups is held constant. Figure 4 reveals that for a fixed pH |µ| exhibits a local maximum as the thickness of membrane varies, which is not observed in the corresponding PCM under the conditions assumed. The existence of the local maximum can be explained as follows. If d is small, the fixed charge is more concentrated near the plane X ) 0 on which the fluid velocity vanishes. This implies that the contribution to |µ| by fixed charge is lowered.4 Therefore, if d is small, |µ| increases with the increase in d. On the other hand, since the friction coefficient of the membrane layer is constant, the thicker the membrane, the greater its total resistance, and therefore, |µ| decreases with the increase in d. Figure 4 also reveals that PCM will overestimate |µ|, and the deviation increases with the decrease in membrane thickness. As in the case of Figure 3, for a fixed membrane thickness, |µ| increases with the increase in pH. Figure 5 shows the variation of the mobility of a particle µ as a function of ionic strength I at various pH values. Here the total number of dissociable functional groups is fixed. For comparison, the results based on the corresponding PCM are also presented. As can be seen in Figure 5, for a fixed pH, the higher the ionic strength I (or the concentration of electrolyte solution), the smaller the |µ|, which is consistent with the theoretical prediction of Lozada-Cassou18 and the experimental observation of Mironov and Dolgaya.23 This is because the increase in I has the effect of increasing the degree of screening the surface charge by counterions. The neutralization of surface
Electrophoretic Mobility of a Particle
Figure 5. Variation of mobility µ as a function of ionic strength I at various pH values. Solid Curves, present model; dashed curves, PCM. Curves 1, pH ) 6; curves 2, pH ) 7; curves 3, pH ) 8. Key: same as Figure 2.
Figure 6. Variation of mobility µ as a function of pH for various valances of cations. Curve 1, a ) 1, b ) 1; curve 2, a ) 2, b ) 1; curve 3, a ) 3, b ) 1. Key: same as Figure 2.
charge through counterion binding is also improved. Figure 5 also suggests that PCM will overestimate the absolute mobility, and the deviation decreases with the increase in ionic strength. As in the case of Figures 3 and 4, for a fixed I, |µ| increases with pH. Figures 3-5 imply that the sizes of the charged species should be taken into account for the case of thin membrane layer, high pH, and low ionic strength (or electrolyte concentration). Makino et al.27 found that the |µ| for free rat basophilic leukemia cells decreases with an increase in ionic strength. They pointed out that |µ| changes from -1.85 to -1.05 µm‚s-1‚V-1‚cm in solutions as the ionic strength varies from 0.01 to 0.154 M. Hayashi et al.26 also concluded that |µ| for various strains of bacterial cells decreases with an increase in ionic strength. For Escherichia coli and Pseudomonas putida, |µ| ranges from -2 to -1 µm‚s-1‚V-1‚cm in electrolyte solutions as the ionic strength varies from 0.05 to 0.14 M. The results presented in Figure 5 are consistent with these two experimental observations. The variation of the mobility of a particle µ as a function of pH for various valances of cations is shown in Figure 6. Here we assume that the cations with various valences have the same sizes. This figure reveals that the higher the valence of cations, the smaller the |µ|. This is because the membrane is negatively charged, and therefore, the higher the valence of cations, the greater its shielding effect, which leads to a lower absolute potential and a smaller |µ|.
J. Phys. Chem. B, Vol. 106, No. 8, 2002 2121
Figure 7. Variation of mobility µ as a function of N*. Curve 1 uses X-Y1 axes; curve 2 uses X-Y2 axes. Curve 1, acidic functional groups; curve 2, basic functional groups. Key: same as Figure 2.
Figure 8. Variation of mobility µ as a function of ξ at various pKa,1 values. Curve 1, pKa,1 ) 3; curve 2, pKa,1 ) 5; curve 3, pKa,1 ) 6. Key: same as Figure 2.
Figure 7 shows the variation of the mobility of a particle µ as a function of N*. N* ) ZaNai)1/bCb0 for acidic functional groups, and N* ) ZbNbi)1/bCb0 for basic functional groups, where Cb0 is the bulk molar concentration of anions. Figure 7 reveals that, for the case of acidic functional groups, the larger the N*, the greater the |µ|. This is because the membrane layer is assumed to carry negative charges, and therefore, a larger N* yields a higher density of negative charges, which leads to a greater |µ|. In contrast, for the case of basic functional groups, the larger the N*, the smaller the |µ|, as can be seen in Figure 7. The variation of the mobility of a particle µ as a function of the fraction of multivalent cations ξ at various equilibrium dissociation constants for acidic functional groups Ka,1 is illustrated in Figure 8. This figure reveals that the larger the ξ, the lower the |µ|. Since a ) b ) 1 and c ) 2, this implies that the presence of multivalent cations in the liquid phase has the effect of raising the shielding effect and reducing the absolute mobility. Figure 8 also shows that the smaller the Ka,1 (or larger pKa,1), the smaller the |µ|. This is expected since a smaller the Ka,1 implies a lesser degree of dissociation of acidic functional groups, which leads to a lower density of fixed charge in the membrane layer. The variation of the mobility of a particle µ as a function of the scaled effective radius of cations Xca for a fixed size of anions at various pH values is illustrated in Figure 9. The
2122 J. Phys. Chem. B, Vol. 106, No. 8, 2002
Huang et al. the result for the case when both co-ions and fixed charged groups have the same size; that is, region III vanishes. Figure 10 reveals that |µ| decreases with the increase in Xan. The change of |µ| as Xan varies, however, is negligible. As in the case of Figure 8, Figure 10 also suggests that |µ| decreases with the increase in ξ. IV. Conclusions
Figure 9. Variation of mobility µ as a function of scaled effective radius of cations Xca at various pH values for the case Xan ) 0.1 and Xi ) 0.2. Curve 1, pH ) 6; curve 2, pH ) 7; curve 3, pH ) 8. Key: same as Figure 2.
In summary, the electrophoretic mobility of a planar particle covered by a charged-regulated membrane layer immersed in a mixed (a:b) + (c:b) electrolyte solution is analyzed. Previous theory is extended in that a general charged condition in the membrane layer, which mimics biological cells, is considered, and the sizes of the charged species are taken into account. The results of numerical simulation reveal that for particles carrying net negative fixed charges the classic point charge model will overestimate the mobility; the deviation increases with the increase in pH, decrease in the thickness of membrane layer, and decrease in the ionic strength. If the number of functional groups in the membrane layer is constant, then the higher the pH and/or the lower the ionic strength, the greater the mobility. If the sizes of counterions of various valences are the same, the following conditions for counterions lead to a greater absolute mobility: larger equilibrium dissociation constant, lower valence, and larger ionic size. The last finding is consistent with experimental observation. The effect of the size of co-ions on the mobility is found to be negligible. Acknowledgment. This work is supported by the National Science Council of the Republic of China. References and Notes
Figure 10. Variation of mobility µ as a function of scaled effective radius of anions Xan at various ξ values for the case Xca ) 0.05 and Xi ) 0.2. Curves 1-3 use X-Y1 axes; curve 4 uses X-Y2 axes. Curves 1 and 4, ξ ) 0; curve 2, ξ ) 0.5; curve 3, ξ ) 1. Key: same as Figure 2.
leftmost point of each curve represents the result for the case when counterions are treated as point charges, and the rightmost point of each curve is the result for the case when both counterions and co-ions have the same size; that is, region II vanishes. Figure 9 suggests that the larger the cations the higher the |µ|. This is because the smaller the cations the easier for them to bond to the negative fixed charge in membrane layer, which has the effect of lowering the absolute electrical potential. This result is consistent with the experimental observation of Mironov and Dolgaya.22 As in the case of Figure 6, Figure 9 also shows that |µ| increases with the increase in pH. The variation of the mobility of a particle µ as a function of the scaled effective radius of anions Xan for a fixed size of cations at various ξ values is presented in Figure 10. The leftmost point of each curve represents the result for the case when both counterions and co-ions have the same size; that is, region II vanishes. The rightmost point of each curve represents
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