Electrophoresis of a Charge-Regulated Sphere in a Narrow

ARTICLE pubs.acs.org/JPCB

Electrophoresis of a Charge-Regulated Sphere in a Narrow Cylindrical Pore Filled with Multiple Ionic Species Jyh-Ping Hsu,* Yi-Hsuan Tai, and Li-Hsien Yeh Department of Chemical Engineering, National Taiwan University, Taipei, Taiwan 10617

Shiojenn Tseng Department of Mathematics, Tamkang University, Tamsui, Taipei, Taiwan 25137

bS Supporting Information ABSTRACT: The electrophoresis of a charge-regulated particle is modeled under the conditions where the dispersion medium contains multiple ionic species and the influence of a boundary can be significant. The conditions assumed in this study are much closer to reality than those of the available analyses in the literature, and, therefore, the results gathered provide valuable information to both theoreticians and experimentalists. For illustration, we consider the electrophoresis of a sphere in a cylindrical pore filled with four kinds of ionic species, namely, Hþ, OH, Naþ, and Cl; these ions are usually present in experiments where the bulk pH is adjusted by HCl and NaOH. We show that the presence of the pore has the effect of inhibiting the polarization of the double layer near a particle, thereby influencing both quantitatively and qualitatively the electrophoretic behavior of the particle. If the pore is charged, then the effect of electroosmotic flow needs be considered, yielding interesting phenomena that might play a key role in practice.

’ INTRODUCTION Electrophoresis has been applied widely as an analytical technique in fundamental study and as a separation tool in industrial production.1,2 This electrokinetic phenomenon has been studied extensively since the first experimental observation of Reuss.3 The history of theoretical analysis on electrophoresis dates back more than 100 years to that conducted by Helmholtz in 1879.4 The application of the result obtained is limited, however, because the dielectric effect of the liquid medium was neglected. Assuming low surface potential and infinitely thin double layer, Smoluchowski extended Helmholtz’s analysis to arrive at the following expression, which relates the mobility of a rigid particle μE and its surface (zeta) potential ζ:5 U εζ μE ¼ ¼ E η

ð1Þ

where ε and η are the dielectric constant and the viscosity of the fluid, respectively, and U and E are the particle velocity and the strength of the applied electric field, respectively. The r 2011 American Chemical Society

other extreme case of infinitely thick double layer was considered by H€uckel;6 he obtained μE ¼

2 εζ 3 η

ð2Þ

In practice, because the double layer surrounding a particle has a finite thickness, knowing the relationship between the mobility and the thickness double layer is highly desirable. This problem was considered by Henry,7 and the following expression was derived: μE ¼

2 εζ f ðkaÞ 3 η

ð3Þ

The derivation of this expression is based on the assumption of low surface potential, that is, the surface potential of a particle is fairly lower than a thermal voltage at 298 K, ca. 25 mV.8 Since violation of this condition is not uncommon in Received: December 15, 2010 Revised: February 27, 2011 Published: March 18, 2011 3972

dx.doi.org/10.1021/jp111889m | J. Phys. Chem. B 2011, 115, 3972–3980

The Journal of Physical Chemistry B

ARTICLE

the experiment,9 eq 3 needs be extended to take this factor into account. Often, the liquid phase in electrophoresis operations contains several kinds of cations and anions. For example, if HCl and NaOH are used to adjust the pH of a dispersion system, as is usually done in practice, the presence of Hþ, Naþ, OH, and Cl need be considered inevitably when the pH deviates appreciably from 7. For dispersions involving biocolloids, the presence of other ionic species such as Ca2þ, Mg2þ, and PO43 are not uncommon, making the liquid composition even more complicated. Previous theoretical analyses, however, almost always assume that the liquid phase contains only one kind of cation and one kind of anion even in a charge-regulated system.1012 This assumption, although it makes the mathematical treatment easier, can be unrealistic in practice. Recently, Hsu and Tai13 extended the traditional electrophoresis analyses to the case where the liquid phase might contain multiple ionic species through adopting a SiO2 dispersion as a model system. As verified by the experimental data of Sonnefeld et al.,14 they showed that the presence of those ionic species is capable of yielding interesting electrophoretic behaviors, and neglecting them can yield electrophoretic mobility that deviates both quantitatively and qualitatively from the experimental result. Modern fabrication technology makes the implement of electrophoresis in submicrometer-scaled devices feasible.15,16 Various electrophoresis experiments have also been conducted in nanoporous support media.17,18 In these cases, because the linear size of a particle can be comparable to that of a device, the boundary effect comes from the device wall should not be ignored. According to Hsu and Tai,13 the distribution of the ionic species surrounding a particle plays an important role in the description of its electrophoretic behavior. This implies that the presence of a boundary can influence significantly that behavior. Considering the rapid growth in the demand of performing electrophoresis in a narrow space, extending the analysis of Hsu and Tai13 to take the boundary effect into account is highly desirable. This is done in the present study by considering the electrophoresis of a charge-regulated sphere in a narrow cylindrical pore filled with an aqueous electrolyte solution containing multiple ionic species at an arbitrary surface potential.

’ THEORY Let us consider the electrophoresis of a rigid sphere of radius a along the axis of a cylindrical pore of radius b subject to an applied uniform electric field E illustrated in Figure S1 of Supporting Information. Let λ = (a/b), which measures the degree of boundary effect. The present problem is two-dimensional, and therefore, the cylindrical coordinates (r, z) are adopted with the origin at the center of the particle. Let ΩP, ΩE, and ΩW be the surface of the particle, the surface of the two ends of the pore, and the lateral surface of the pore, respectively. Suppose that the surface of the particle has functional group HA, which is capable of undergoing the following dissociation/ association reactions: AH S A þ Hþ

ð4Þ

AH þ Hþ S AHþ 2

ð5Þ

where NA, NAHþ2 , and NAH are the surface densities of A, AHþ 2, and AH, respectively, and [Hþ] is the concentration of Hþ. The total number of the functional groups on the particle surface can be expressed as Ntotal = NA þ NAH þ NAHþ2 . We assume that the system is at a pseudosteady state, the liquid phase is an incompressible Newtonian fluid containing N kinds of ionic species, and the flow field is in the creeping flow regime, both the surface of the particle and that of the pore are nonslip, nonconducting, and impenetrable to ionic species, the surface of the pore is maintained at a constant potential of ζw, and the flow field at a point far away from the particle is uninfluenced by its presence. Similar to the perturbation treatments of O’Brien and White19 and Ohshima,20,21 which are based on the condition that E is much weaker than the electric field established by the particle and the pore, each dependent variable is partitioned into an equilibrium component and a perturbed component; the former is the value of that variable when E is not applied and the latter denotes that coming from the application of E. Then the equations governing the present problem and the associated boundary conditions can be summarized as following:

ðkaÞ2

r 2 φe ¼

N

∑ R21

N

∑ R11 exp½ R10 φe j¼1

ð6Þ

j¼1

N

ðkaÞ2

r 2 δφ ¼

N

∑ R21

∑ R21 ðδφ þ gj Þ exp½ R10 φe j¼1

ð7Þ

j¼1

r 2 gj ¼ R10 r φe 3 r gj þ Pej v 3 rφe

ð8Þ

r p þ r 2 v þ r 2 φe r δφ þ r 2 δφ r φe ¼ 0 ð9Þ

r 3v ¼ 0

ð10Þ

nj ¼ expð R10 φe Þ½1 R10 ðδφ þ gj Þ, j ¼ 1, 2, :::, N

ð11Þ

n 3 r φe

( aeFNtotal Ka ¼ 2 þ εkB T Kb ð½H 0 expð φe ÞÞ þ ½Hþ 0 expð φe Þ þ Ka ) Kb ð½Hþ 0 expð φe ÞÞ2 Ka þ ½Hþ 0 expð φe Þ þ Kb ð½Hþ 0 expð φe ÞÞ2

¼ σsurface

n 3 r δφ ¼ 0

n 3 r gj ¼ 0

φ e ¼ ζw

Then the equilibrium constants of these reactions, Ka and Kb, can be expressed as Ka = NA[Hþ]/NAH and Kb = NAHþ2 /NAH[Hþ],

Ωp

ð12Þ

on

Ωp

ð13Þ

Ωp

on on

n 3 r δφ ¼ 0 3973

on

ΩW on

ΩW

ð14Þ ð15Þ ð16Þ

dx.doi.org/10.1021/jp111889m |J. Phys. Chem. B 2011, 115, 3972–3980


n 3 r gj ¼ 0

ΩW

on

I0 ðkrÞ

φ e ¼ ζw

n 3 r δφ ¼ Ez

gj ¼ δφ

v ¼0

on

on

ð18Þ

ΩE

ð19Þ

ΩE

on

ΩP

v ¼ ðU=U 0 Þez

on

ð17Þ

ΩE

on

I0 ðkbÞ

ARTICLE

ð20Þ

ΩW

ð22Þ on

ΩE ð23Þ

In these expressions, r* = ar, r* = a r εkBT]1/2, nj* = nj/nj0, φe* = φe/ζa, δφ* = δφ/ζa, gj* = gj/ζa, ζa = kBT/z1e, ζw* = ζw/ζa, Pej = εζa2/ηDj, v* = v/U0, U0 = εζa2/aη, p* = p/pref, pref = εζa2/a2, E*z = Ezea/kBT, and σ*surface = σsurfaceae/ εkBT. I0 is zero-order modified Bessel function of the first kind; ez is the unit vector in the z-direction; n is the unit normal vector directed into the liquid phase on ΩP and ΩW, and upward on ΩE. Rβγ = zβj nγj0/zβ1 nγ10, where subscript 1 denotes a reference ionic species, one of the ionic species in the system. The subscript e and the prefix δ denote the equilibrium and the perturbed properties, respectively. r2 is Laplace operator; ε is the permittivity of the liquid phase; F is the space charge density; e is the elementary charge; zj and Dj are the valence and the diffusivity of ionic species j, j = 1, 2, ..., N; kB and T are Boltzmann constant and absolute temperature, respectively; η is the fluid viscosity; φ is the electrical potential; nj is the number concentration of ionic species j; p is the pressure; v is the fluid velocity relative to the particle; gj is a hypothetical potential function simulating a polarized double layer; nj0 is the bulk ionic concentration; ζw is the surface potential of the pore; σsurface is the surface charge density; F is the Faraday constant; [Hþ]0 is the bulk concentration of Hþ. Note that because the particle is stagnant at equilibrium, ve = 0, and therefore, v = δv. Equation 18 describes the relationship between the scaled equilibrium potential and the surface potential of the pore on ΩE22 According to eq 19, the electric field on the pore surface comes solely from the applied electric field. Equations 20 implies that the ionic concentration at a point far away from the particle obeys Boltzmann distribution. As indicated by eq 23, if the pore is charged, then an electroosmotic flow is present.23 As in the previous analysis,13 the present problem is partitioned into two subproblems: the particle moves with a constant velocity U in the absence of E in the first subproblem, and E is applied but the particle is held fixed in the space in the second subproblem. Then, under a pseudosteady condition, the mobility of the particle μE can be evaluated by13 2

χ F2 U μE ¼ 2 ¼ F1 E χ1

2

2 , κ = [∑N j=1nj0(ezj) /

ΩP

ðσ H 3 nÞ 3 ez dΩP

ð26Þ

where n and nz are the magnitude and the z-component of n, respectively; t is the magnitude of the unit tangential vector t; Ω P* = ΩP/a2 is the scaled surface area of the particle; and σH* = σH/(εζa2/a 2) is the scaled shear stress tensor. A trialand-error procedure can be used to estimate μE.13 For convenience, we define the scaled mobility μE* as μE* = μE(ηe/εkBT) = F2*/F 1*.

’ RESULTS AND DISCUSSION Equations 611 are solved numerically subject to the boundary conditions specified in eqs 1223 by FlexPDE,25 a finiteelement method based commercial software, which is found to be sufficiently efficient and accurate to the present boundary-value problem.26 Its applicability was verified previously13 by solving numerically the electrophoresis of a rigid particle of constant surface potential, which was also solved analytically by Shugai and Carnie.27 The present theoretical model is first fitted to the experimental data of Sonnefeld et al.,14 where the electrophoresis of SiO228 particles of radius 20 nm was conducted. In this case four kinds of ionic species (i.e., N = 4) need be considered, including Naþ, Cl, Hþ, and OH, let [Naþ]0, [Cl]0, [Hþ]0, and [OH]0 be the bulk molar concentrations of these ionic species, respectively, and CNaCl be the background molar concentration of NaCl. If Kw denotes the dissociation constant of water, then for pH e pKw/2, the following relations apply: ½Hþ 0 ¼ 10pH

ð27Þ

½Naþ 0 ¼ CNaCl

ð28Þ

½Cl 0 ¼ CNaCl þ 10pH 10ðpK w pHÞ

ð29Þ

½OH 0 ¼ 10ðpK w pHÞ

ð30Þ

and for pH g pKw/2, the following relations apply:

ð24Þ

Here, F1 = χ1U and F2 = χ2Ε, Fi is the magnitude of the force acting on the particle in the z-direction in subproblem i, Fi, and χ1 and χ2 are proportional constants. The forces involved in the present problem include mainly the electrical force Fe and the

Z

Fdi ¼

ð21Þ

I0 ðkrÞ 0 v ¼ ζw 1 E þ ðU=U Þ ez I0 ðkbÞ z 2

hydrodynamic force Fd. The z-components of the scaled magni* = Fdi/ tudes of these forces in subproblem i, Fei* = Fei/εζa2 and Fdi εζa2, where Fei and Fdi are the corresponding z-components of Fe and Fd in subproblem i, can be calculated by integrating the Maxwell stress tensor σE and the shear stress tensor σH over ΩP as24 ! Z Dφe Dδφ Dφe Dδφ nz dΩP Fei ¼ ð25Þ Dz Dt Dn Dt ΩP

½Hþ 0 ¼ 10pH

ð31Þ

½Naþ 0 ¼ CNaCl þ 10ðpK w pHÞ 10pH

ð32Þ

½Cl 0 ¼ CNaCl

ð33Þ

½OH 0 ¼ 10ðpK w pHÞ

ð34Þ

Figure 1 shows both the experimental data and the corresponding values predicted by the present theoretical model based on 3974



ARTICLE

Figure 1. Variations of the electrophoretic mobility μE as a function of pH for the case where a = 20 nm. Discrete symbols are the experimental data of Sonnefeld et al.,14 and curves represent the values predicted by the present theoretical model at λ = 0.1, pKa = 7.4, pKb = 2, and Ntotal = 5 105 mol/m2; blue curve, CNaCl = 0.01 M, black curve, CNaCl = 0.001 M.

Figure 2. Variations of the scaled electrophoretic mobility μ*E as a function of pH for various values of λ at ζw* = 0 and Ntotal = 7 107 mol/m2.

the estimated values of pKa = 7.4, pKb = 2, and Ntotal = 5 105 mol/m2. Here, λ is assumed a small value of 0.1 so that the presence of the pore can be neglected. Figure 1 reveals that the present theoretical model predicts successfully the experimental data. μE is seen to have a negative local minimum as pH varies, which can be explained by the effect of double-layer polarization (DLP).13 Note that the higher the background concentration of NaCl (thinner double layer), the less appreciable that effect, and therefore, the presence of the local minimum is. In subsequent discussions, the electrophoretic behaviors of a particle under various conditions are examined through numerical simulation. For illustration, we assume that a = 20 nm, pKa = 7, and pKb = 2, and the liquid phase contains mainly Hþ, Naþ, Cl, and OH. Suppose that T = 298 K. Then, because kB = 1.38 1023 J/K, z1 = þ1, ε = 7 1010 CV1m1, η = 103 kg m1 s1, e = 1.6 1019 C, the diffusivities of Hþ, Naþ, Cl, and OH are 9.38 109 m2 s1, 1.33 109 m2 s1, 2 109 m2 s1, and

Figure 3. Contours of the scaled perturbed ion distribution δn* = [(nanion nanion,e) (nanion nanion,e)]/(nsodium,0) at ζw* = 0, Ntotal = 5 106 mol/m2, CNaCl = 0.001 M, and pH = 9 on the plane θ = π/2. (a) λ = 0.1, (b) λ = 0.5, (c) λ = 0.7.

5.29 109 m2 s1,29 respectively, and the electric Peclet numbers of Hþ, Naþ, Cl, and OH are 0.05, 0.353, 0.235, and 0.089, respectively. The dimensionless length of the pore, scaled by the particle radius, is assumed a sufficiently large value of 30 so that the end effects can be neglected. Regarding the charged conditions on 3975



ARTICLE

Figure 4. Variations of the scaled electrophoretic mobility μ*E as a function of CNaCl for various values of λ (= a/b) at ζw* = 0 and Ntotal = 5 106 mol/m2.

the surface of the pore, two possible cases are considered; namely, it is either charged or uncharged. Case 1 Uncharged Pore. Let us consider first the case where the cylindrical pore is uncharged (ζw* = 0). Figure 2 illustrates the variations of the scaled electrophoretic mobility μ*E as a function of pH for various values of λ. As seen in this figure, if λ is sufficiently large, then in the range of pH considered |μE*| increases monotonically with increasing pH. This is because the negative surface charge density increases with increasing pH. Note that if λ is sufficiently small, that is, the boundary effect is relatively insignificant, then |μE*| has a local maximum near pH = 8, which can be explained by the effect of DLP.13 In this case, because more anions (cations) accumulate near the bottom (top) region of the particle, inducing a local electric field, the direction is opposite to that of the applied electric field. As λ gets large, that is, the boundary effect is significant, that local maximum in |μ*| E disappears, implying the effect of DLP is inhibited by the boundary. This is verified in Figure 3, where the contours of the scaled net perturbed anions concentration, δn* = [(nanion nanion,e) (ncation ncation,e)]/(nsodium,0), at various values of λ are presented, where nanion and ncation are the number concentrations of anions and cations, respectively, and nanion,e and ncation,e are the corresponding equilibrium values; nsodium,0 is the bulk number concentration of Naþ. As can be seen in this figure, at λ = 0.1, the application of E yields an accumulation of a net amount of anions (cations) near the bottom (top) region of the particle, and the direction of the induced electric field points downward. As λ is increased to 0.5, the phenomenon of DLP is alleviated, with some of the perturbed ions accumulating near the surface of the pore. If λ is increased to 0.7, then almost all of the perturbed ions appear near the pore surface and the effect of DLP becomes become unimportant. In this case, the local maximum in |μ*| E disappears. If λ is further increased to 0.9, then because hydrodynamic retardation force dominates, |μE*| is close to 0. Figure 4 shows the variations of the scaled mobility μE* as a function of the background concentration of NaCl, CNaCl, at various values of λ. Under the conditions assumed, μE* is always * < 0 and Fd2 * > 0 with |Fe2 * | > |Fd2 * |, as negative because Fe2 illustrated in Figure 5a. Figure 4 also shows that, for a fixed level of CNaCl, |μ*| E decreases with increasing λ. This arises mainly

Figure 5. Variations of the scaled forces in the second subproblem, F*e2 and Fd2 * , as a function of CNaCl at various levels of λ for the case of Figure 4 (without λ = 0.9) (a) and that of the corresponding scaled total force F2* (= Fe2 * þ Fd2 * ) at λ = 0.1, 0.3, and 0.5 (b).

from the increase in the scaled z component of the hydrodynamic retardation force acting on the particle in the first subproblem, F1*, with increasing degree of boundary effect. It is interesting to see that the qualitative behavior of μ*E as CNaCl (or double layer thickness) varies depending upon the level of λ: if λ exceeds ca. 0.5, |μE*| increases monotonically with increasing CNaCl (or decreasing double layer thickness); if λ = 0.3, |μ*| E increases first with increasing CNaCl up to CNaCl = 0.05 M and then decreases with a further increase in CNaCl; if λ = 0.1, |μE*| shows both a local minimum and a local maximum as CNaCl varies. This can be explained by the results presented in Figure 5. Figure 5a indicates that the qualitative behavior of the scaled hydrodynamic force * is sensitive to the variation in λ. In general, the smaller the Fd2 * . Note that if λ and/or the higher the CNaCl the larger the Fd2 CNaCl is low, then F*d2 is negative, and if CNaCl is sufficiently high, then F*d2 becomes positive. The scaled electric force F*e2 is always negative, and its qualitative behavior is relatively insensitive to the variation in λ. According to Figure 5a, for the case where λ = 0.1 and CNaCl is either higher than ca. 0.015 M or lower than ca. 0.005 M and for the case where λ = 0.3 and CNaCl is higher than ca. 0.05 M, the scaled z component of the total force acting on the * . This particle in the second subproblem, F2*, is dominated by Fd2 is because the rate of increase of F*d2 with increasing CNaCl is 3976



Figure 6. Variations of the scaled electrophoretic mobility μE* as a function of pH for various values of λ at ζ*w = 1, CNaCl = 0.001 M, and Ntotal = 7 107 mol/m2.

slightly faster than that of |Fe2 * |, and therefore, |F2*| decreases with increasing CNaCl. In these cases, |μ*| E decreases with increasing CNaCl (decreasing double layer thickness). However, if λ = 0.1 and 0.005 M < CNaCl < 0.015 M, then due to the effect of DLP, * | and, since |Fe2 * | increases with increas|F2*| is dominated by |Fe2 ing CNaCl, so is |F*|. 2 Therefore, in this region of CNaCl, |μ*| E increases with increasing CNaCl, yielding both a local minimum and a local maximum in |μE*|. On the other hand, if λ is smaller than ca. 0.3 and CNaCl is sufficiently low, then F*2 is dominated by F*e2. That is, if the boundary effect is insignificant but the double layer is sufficiently thick, then |μE*| increases with decreasing double layer thickness, as is seen in Figure 4 for the case where λ = 0.3 and CNaCl < 0.05 M. Figure 5 shows that if λ g 0.5, then, regardless of the level of CNaCl (or double layer thickness), because the rate of change of F2* as CNaCl varies dominated by that * , |μE*| increases monotonically with increasing CNaCl of Fe2 (decreasing double layer thickness), as seen in Figure 4. Note that the general behavior of μ*E in Figure 4 is similar to that of F*2 in Figure 5b, implying that under the conditions assumed the electrophoretic behavior of the particle is dominated by the total force acting on it in the second subproblem. The results illustrated in Figure S2 of Supporting Information indicate that the effect of DLP is important only if the boundary effect is unimportant and the thickness of the double layer surrounding a particle takes a medium large value, as in Figure S2c, where λ = 0.1 and CNaCl = 0.005 M (κa = 3.4). If CNaCl is too high (Figure S2a,b), then because the double layer is too thin, its polarization becomes unimportant. On the other hand, if CNaCl is too low (Figure S2e,f), then although DLP is significant, its influence on the electrophoretic behavior of the particle is unimportant because the cations (anions) accumulated near the top (bottom) region of the particle are too far from the particle, making the induced electric field weak. These also explain the presence of the negative local maximum in the curve corresponding to λ = 0.1 in Figure 4. Case 2 Charged Pore. Let us consider next the case where the surface of the pore is charged, and for illustration, we assume that ζw*=1. This type of surface condition can be established, for example, by applying gate electrodes on the pore surface.30 Figure 6 shows the variations of the scaled electrophoretic mobility μ*E as a function of pH for various values of λ, the

ARTICLE

Figure 7. Variations of κa and the equilibrium surface potential as a function of pH for the case of Figure 6.

corresponding variations of κa and the equilibrium surface potential are presented in Figure 7, and Figure 8 illustrates the variations of the scaled forces in the z-direction in the second subproblem, F*, 2 F* e2, F* d2, F* d2(p), and F* d2(v) at λ = 0.1. As seen in Figure 6, because the positively charged pore induces a downward electroosmotic flow, the mobility of a particle is more negative than that presented in Figure 2, where the pore is uncharged. Due to the inhibition of DLP by the positive charged pore, the local maximum of |μE*| in Figure 2 is not seen in Figure 6. Note that if λ is sufficiently large (ca. 0.5), then μE* has a negative local maximum at a low pH (ca. 4.5), which will be discussed later. Figure 7 indicates that the larger the λ (particle closer to pore) the less negative the surface potential of the particle, and this potential can become positive if pH is sufficiently low. The variation in the surface potential of the particle is due to its electrical interaction with the charged pore. As expected, the value of pH at which the surface potential of the particle vanishes increases with increasing λ. Note that the curve of the scaled surface potential at λ = 0.1 essentially coincides with that at λ = 0.3, implying that the presence of the pore can be neglected when λ < 0.3. κa is seen to decrease with increasing pH and approaches a constant value (ca. 2) when pH is sufficiently high (ca. 4.5). This is because the concentration of Hþ decreases with increasing pH and is dominant by CNaCl when pH is sufficiently high. Because λ is small (= 0.1), the boundary effect is relatively unimportant in Figure 8. Note that the qualitative behavior of μ*E in Figure 6 is similar to that of F2* in Figure 8a, implying that the former is dominated by the latter. As can be seen in Figure 8a, |F*e2| goes through a local maximum at pH = 8 and then decreases due to the effect of DLP. However, the extent of decrease in |F*e2| is not as severe as that in the case where the electroosmotic flow is absent.13 This is why a local maximum in |μE*| is not seen in Figure 6. It is interesting to note in Figure 8a that F*d2 also has a positive local maximum at pH = 7.5. To explain this behavior, * is decomposed into a pressure component, Fd2(p) * , and a Fd2 * . As can be seen in Figure 8b, Fd2 * is viscous component, Fd2(v) dominated by F*d2(v) for pH ranges from 3 to ca. 7. As illustrated in Figure 7, this is because the surface potential of the particle is low in that pH range, and therefore, the electroosmotic flow that comes from the charged pore dominates F*d2. Note that if pH is lower than ca. 4.5, then κa (or κb) becomes important to 3977



Figure 8. Variations of the scaled forces in the z-direction in the second subproblem, F2*, Fe2 * , and Fd2 * (a) and Fd2(p) * and Fd2(v) * (b) as a function of pH for the case of Figure 6 at λ = 0.1.

Fd2(v) * because the speed of the electroosmotic flow is sensitive to the ionic concentration, and the smaller the κa (or κb) the slower that speed.31 Figure 7 reveals that if pH exceeds ca. 4.5, then although κa is roughly constant, the surface potential of the particle increases appreciably with increasing pH. If pH is sufficiently high, then the electroosmotic flow coming from the particle might compete with that from the pore. As illustrated in Figure S3a of Supporting Information, where pH = 5, the speed of the electroosmotic flow coming from the particle is small due to its low surface potential. Therefore, although that flow is upward because the particle is negatively charged, the net fluid flow is downward, and the corresponding viscous force is negative. If pH = 9, then, as shown in Figure S3b, the electroosmotic flow coming from the particle is significant. In this case, the net fluid flow near the particle becomes upward, and the corresponding viscous force is positive. Note that, as can be seen in Figure 8b, * , Fd2(p) * , also increases with the pressure component of Fd2 increasing pH. As seen in Figure S4 of Supporting Information, the difference between the scaled pressure near the top of the particle and that near its bottom at pH = 9 is about 10 times than that at pH = 6. In addition, the contours in Figure S4b are more compact than that in Figure S4a, where the surface potential in the former is more

ARTICLE

Figure 9. Variations of the scaled forces in the z-direction in the second subproblem, F*, 2 F* e2, and F* d2 (a) and F* d2(p) and F* d2(v) (b) as a function of pH for the case of Figure 6 at λ = 0.7.

negative than that in the latter. This is because a surface with a higher potential is capable of attracting more counterions, yielding a more significant electroosmotic flow, and the corresponding pressure, which drives the particle downward, is higher. The presence of the local maximum of F*d2 in Figure 8a arises from the * and Fd2(p) * . competition of Fd2(v) Figure 9 illustrates the variations of the scaled forces in the z-direction in the second subproblem, F*, 2 F* e2, F* d2, F* d2(p), and F*d2(v), for the case of Figure 6 at λ = 0.7. As mentioned previously, the presence of the pore has the effect of inhibiting the DLP surrounding the particle. This can be seen in Figure 9a, where the decrease of F*e2 with increasing pH becomes unimportant as pH exceeds ca. 7. The behavior of μ*E in Figure 6 at λ = *. 0.7 is similar to that of F2* in Figure 9a, and F2* is dominated by Fd2 It is interesting to observe that at this level of λ the behavior of F*d2 is uninfluenced by F*d2(v). This is because the direction of fluid flow surrounding the particle is dominated by the electroosmotic flow coming from the pore, and the variation of this electroosmotic flow with pH is not as sensitive as that of the osmotic pressure. The trend of F*d2(p) seen in Figure 9b is consistent with that of the scaled surface potential presented in Figure 7. Recall that the μE*pH curve in Figure 6 has a negative local maximum at a low level of pH when λ is sufficiently large. This is because under these conditions, the overlapping between the double layer 3978



Figure 10. Variations of the scaled electrophoretic mobility μE* as a function of CNaCl for various values of λ at ζ*=1 and Ntotal = 5 106 w 2 mol/m .

surrounding the particle and the pore is significant, leading to a decrease in |F*d2(p)|, as seen in Figure 9b. As illustrated in Figure 7, if pH is low, then the surface potential of the particle remains roughly the same, but the thickness of the double layer increases appreciably with increasing pH. In this case, if λ is sufficiently large, then because the particle and the pore are oppositely charged, the overlapping of their double layers yields a decrease in the * | decreases, accordingly. On electroosmotic pressure, and |Fd2(p) the contrary, as shown in Figure 8b, because the overlapping between the double layer and the pore is insignificant at λ = 0.1, the pressure does not vary appreciably in the range where pH is lower than ca. 4.5. However, as shown in Figure 7, if pH exceeds this value, then the surface potential increases rapidly with * |, as seen in Figure 8b. increasing pH, and so does the |Fd2(p) Therefore, we conclude that the presence of the negative local maximum of μ*E in Figure 6 for λ g 0.5 arises from the overlapping of the double layer surrounding the particle with the pore and the increase in the particle surface potential. Figure 10 illustrates the variations of the scaled electrophoretic mobility μ*E as a function of the background NaCl concentration, CNaCl, for various values of λ. The general behavior of μ*E in this figure is similar to that in Figure 4, where the pore is uncharged, except that the local maximum and the local minimum of μE* at λ = 0.1 disappears. This implies that the effect of DLP is inhibited by the electroosmotic flow coming from the charged pore.

’ CONCLUSIONS The electrophoretic behavior of a charged-regulated particle is investigated under the conditions where the dispersion medium contains multiple ionic species and the influence of a boundary can be significant. These conditions are much closer to reality than those considered in relevant analyses in the literature and, therefore, the results obtained provide valuable information to both theoreticians and experimentalists. Using the case of a SiO2 particle in a cylindrical pore filled with Hþ, OH, Naþ, and Cl as an example, the mobility of the particle under various combinations of pH and background concentration of NaCl, CNaCl, is discussed in detail, and several interesting phenomena are observed. For the case where the pore is uncharged, the presence of an uncharged pore is capable of inhibiting the effect of double

ARTICLE

layer polarization (DLP), especially when the ratio λ = (particle radius/pore radius) exceeds 0.5. The absolute value of the mobility of a particle decreases with increasing CNaCl, in general, when the boundary effect is important. However, if the boundary effect is insignificant, then the mobility increases with increasing CNaCl, and, due to the effect of DLP, the mobility has a negative local minimum, which occurs at a medium large CNaCl. The mobility has a negative local minimum as pH varies. This local minimum disappears, however, if the boundary effect is important. On the other hand, if the pore is positively charged, an electroosmotic flow is present. In this case, due to the competition between the electroosomtic flow induced by the particle and that by the pore, the direction of the flow field near the particle can vary with pH. If the boundary effect is important, then due to the overlapping of the double layer surrounding the particle and the pore, instead of having a negative local minimum as pH varies, the mobility has a negative local maximum. Because the particle is negatively charged under the conditions assumed (point of zero charge is 2.5 and pH g 3), the variation of its electrophoretic behavior as CNaCl varies for the case where the pore is positive charged is similar to that for the case where the pore is uncharged, except that the effect DLP might become insignificant.

’ ASSOCIATED CONTENT

bS

Supporting Information. Figures S1S4 as described in the text. This material is available free of charge via the Internet at http://pubs.acs.org.

’ AUTHOR INFORMATION Corresponding Author

*Tel: 886-2-23637448. Fax: 886-2-23623040. E-mail: jphsu@ ntu.edu.tw.

’ ACKNOWLEDGMENT This work is supported by the National Science Council of the Republic of China. ’ NOTATION a radius of particle (m) b radius of cylindrical pore (m) diffusivity of ionic species j (m2/s) Dj e elementary charge (C) unit vector in z direction () ez E magnitude of applied electric field (V/m) strength of applied electric field in z direction (V/m) Ez = Ezea/kBT () E*z E applied electric field (V/m) F Faraday constant (C/mol) z component of Fe in subproblem i (N) Fei electrical force acting on particle (N) Fe z component of Fd in subproblem i (N) Fdi hydrodynamic force acting on particle (N) Fd * = Fdi(v) * þ Fdi(p) * Fdi rescaled form of the viscous component of Fdi () F*di(v) rescaled form of the pressure component of Fdi () F*di(p) (= Fei þ Fdi) magnitude of Fi (N) Fi total force acting on particle in z direction in subproFi blem i (N) 3979


The Journal of Physical Chemistry B gj gj* I0 kB n nz nj nj0 n*j n p pref p* Pej r t tz t T u U U0 v v* z zj

hypothetical perturbed potential associated with ionic species j (V) =gj/ζa () zero-order modified Bessel function () Boltzmann constant (J/K) magnitude of n () z component of n () number concentration of ionic species j (no./m3) bulk number concentration of ionic species j (no./m3) = nj/nj0 () unit normal vector () pressure of fluid phase (Pa) = εζa2/a2 (Pa) =p/pref () (= ε(ζa)2/ηDj) electric Peclet number of ionic species j () radial coordinate (m) magnitude of t () z component of t () unit tangential vector on particle surface () absolute temperature (K) velocity of particle (m/s) z component of particle velocity (m/s) = ε(ζa)2/ηa (m/s) fluid velocity relative to particle (m/s) = v/U0 () axial coordinate (m) valence of ionic species j ()

Greek Letters

rβγ δφ δφ* ε ζ ζa ζw η κ

κa κb λ μE μ*E σH σsurface σ*surface φe φ*e ΩE χ1 χ2 ΩP ΩW r r* r2 r*

= zβj nγj0/zβ1 nγ10 () perturbed potential arising from E (V) = δφ/ζa () permittivity of fluid phase (C2/(N m2)) zeta potential (V) = kBT/ez1 (V) surface potential of pore (V) viscosity of fluid phase (kg/(m s)) 2 1/2 ) reciprocal Debye length (= [∑N j=1nj0(ezj) /εkBT] 1 (m ) scaled double layer thickness of particle () scaled double layer thickness of pore () =a/b () electrophoretic mobility (m2/(V s)) = μE(ηe/εkBT) () hydrodynamic stress tensor (N/m2) charge density on particle surface (C/m2) = σsurfaceae/εkBT () equilibrium potential (V) = φe/ζa () end surface (area) of pore (m2) proportional constant (kg/s) proportional constant (kg m2/s2) surface (area) of particle (m2) lateral surface (area) of pore (m2) gradient operator (1/m) =ar () Laplace operator (1/m2) =a2r ()

ARTICLE

Subscript

e i j

equilibrium property index of subproblem index of ionic species

Prefix

δ

perturbed property

’ REFERENCES (1) Masliyah, J. H. Electrokinetic Transport Phenomena; AOSTRA Technical Publication Series, no. 12; Alberta Oil Sands Technology and Research Authority: Edmonton, Alberta, Canada, 1994. (2) Ennis, J.; Zhang, H.; Stevens, G.; Perera, J.; Scales, P.; Carnie, S. J. Membr. Sci. 1996, 119, 47. (3) Reuss, F. F. Mem. Soc. Imperiale Naturalistes de Moscow 1809, 2, 327. (4) Helmholtz, H. Ann. Phys. Chem. 1879, 7, 337. (5) von Smoluchowski, M. Z. Phys. Chem. 1918, 92, 129. (6) H€uckel, E. Phys. Z. 1924, 25, 204. (7) Henry, D. C. Proc. R. Soc. London 1931, 133, 106. (8) Ennis, J.; Anderson, J. L. J. Colloid Interface Sci. 1997, 185, 497. (9) Pyell, U. Electrophoresis 2008, 29, 576. (10) Yu, H. Y.; Hing, S. H.; Hsu, J. P. Colloid Polym. Sci. 2004, 283, 10. (11) Tang, Y. P.; Chih, M. H.; Lee, E.; Hsu, J. P. J. Colloid Interface Sci. 2001, 242, 121. (12) Hsu, J. P.; Lee, E.; Yen, F. Y. J. Chem. Phys. 2000, 112, 6404. (13) Hsu, J. P.; Tai, Y. H. Langmuir 2010, 26, 16857. (14) Sonnefeld, J.; L€obbus, M.; Vogelsberger, W. Colloids Surf., A 2001, 195, 215. (15) Liang, L. T.; Ai, Y.; Zhu, J.; Qian, S.; Xuan, X. J. Colloid Interface Sci. 2010, 347, 142. (16) Yalcin., S. E.; Lee, S. Y.; Joo, S. W.; Baysal, O.; Qian, S. J. Phys. Chem. B 2010, 114, 4082. (17) Hatch, A. V.; Herr, A. E.; Throckmorton, D. J.; Brennan, J. S.; Singh, A. K. Anal. Chem. 2006, 78, 4976. (18) Long, Z. C.; Liu, D. Y.; Ye, N. N.; Qin, J. H.; Lin, B. C. Electrophoresis 2006, 27, 4927. (19) O’Brien, R. W.; White, L. R. J. Chem. Soc., Faraday Trans. 2 1978, 74, 1607. (20) Ohshima, H. J. Colloid Interface Sci. 1995, 62, 189. (21) Ohshima, H. Electrophoresis 2006, 27, 526. (22) Rice, C. L.; Whitehead, R. J. Phys. Chem. 1965, 69, 4017. (23) Hsu, J. P.; Ku, M. H.; Kao, C. Y. J. Colloid Interface Sci. 2004, 276, 248. (24) Hsu, J. P.; Yeh, L. H.; Ku, M. H. J. Colloid Interface Sci. 2007, 305, 324. (25) FlexPDE, version 2.22; PDE Solutions Inc.: Spokane Valley, WA, 2000. (26) Hsu, J. P.; Hung, S. H.; Yu, H. Y. J. Colloid Interface Sci. 2004, 280, 256. (27) Shugai, A. A.; Carnie, S. L. J. Colloid Interface Sci. 1999, 213, 298. (28) Binner, J.; Zhang, Y. J. Mater. Sci. Lett. 2001, 20, 123. (29) Poling, B. E.; Prausnitz, J. M.; O’Connell, J. P. The Properties of Gases and Liquids, 5th ed.; McGraw-Hill: New York, 2001. (30) Ai, Y.; Liu, J.; Zhang, B. K.; Qian, S. Anal. Chem. 2010, 82, 8217. (31) Hsu, J. P.; Yeh, L. H. J. Phys. Chem. B 2007, 111, 2579.

Superscript

*

scaled property 3980


Electrophoresis of a Charge-Regulated Sphere in a Narrow

Recommend Documents