pubs.acs.org/Langmuir © 2010 American Chemical Society
Effect of Multiple Ionic Species on the Electrophoretic Behavior of a Charge-Regulated Particle Jyh-Ping Hsu* and Yi-Hsuan Tai Department of Chemical Engineering, National Taiwan University, Taipei, Taiwan 10617 Received July 27, 2010. Revised Manuscript Received September 15, 2010 It is often assumed in the conventional electrophoresis analysis that the liquid phase contains only one kind of each cation and anion. That analysis is extended to the case where the liquid phase contains multiple ionic species in this study so that the conditions considered are closer to reality. Using a dispersion of SiO2 particles, which is of a charge-regulated nature, as an example, where the dispersion pH is adjusted by HCl and NaOH, numerical simulation is conducted to examine the electrophoretic behaviors of the particle under various conditions. We show that the presence of multiple ionic species is capable of yielding profound and interesting electrophoretic behaviors, which are justified by the experimental data in the literature. In addition, we show that two types of double-layer polarization (DLP) are present that have not been reported previously in the electrophoresis analyses. Type I DLP, which reduces the mobility of a particle, occurs inside the double layer, and type II DLP, which raises that mobility, occurs immediately outside the double layer.
Introduction Electrophoresis, driven by a charged colloidal particle in an applied electric field,1 has been applied in many areas of both scientific and industrial significance for a long period of time. This operation provides an efficient way to characterize the physicochemical properties of a particle such as the density of the dissociable functional groups on its surface.2 DNA identification, for example, often involves an electrophoresis method. Under simplified conditions, Smoluchowski3 was able to show that the electrophoretic mobility of a particle, defined as the electrophoretic velocity of the particle per unit strength of the applied electric field, correlates linearly with its surface (zeta) potential. Many attempts have been made afterwards with respect to the extension of that analysis to account for conditions of more practical significance. Considering the operation in practice, several key factors of electrophoresis deserve detailed study. These include the geometry of a system, the charged conditions of the particle, the concentrations and types of ionic species in the liquid phase, the effect of the particle concentration, and the effect of the boundary. The charged conditions of a particle are capable of influencing its electrophoretic behavior both quantitatively and qualitatively. Often, it is assumed that the surface of a particle is maintained either at a constant potential4-7 or at a constant charge density.8,9 These are the limiting cases of a more general model, a charge*Corresponding author. Tel: 886-2-23637448. Fax: 886-2-23623040. E-mail:
[email protected]. (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) Hsu, J. P.; Huang, S. W.; Hsieh, T. S.; Young, T. H.; Hu, W. W. Electrophoresis 2002, 23, 2001. (3) von Smoluchowski, M. Z. Phys. Chem. 1918, 92, 129. (4) Hsu, J. P.; Chen, Z. S. Langmuir 2007, 23, 6198. (5) Chih, M. H.; Lee, E.; Hsu, J. P. J. Colloid Interface Sci. 2002, 248, 383. (6) Ennis, J; Anderson, J. L. J. Colloid Interface Sci. 1997, 185, 497. (7) Zydney, A. L. J. Colloid Interface Sci. 1995, 169, 476. (8) Hsu, J. P.; Hung, S. H. Langmuir 2003, 19, 7469. (9) Qian, S.; Wang, A.; Afonien, J. K. J. Colloid Interface Sci. 2006, 303, 579. (10) Tang, Y. P.; Chih, M. H.; Lee, E.; Hsu, J. P. J. Colloid Interface Sci. 2002, 242, 121. (11) Hsu, J. P.; Chen, C. Y.; Lee, D. J.; Tseng, S.; Su, A. J. Colloid Interface Sci. 2008, 325, 516.
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regulated surface,10,11 which is widely adopted to simulate biocolloids and several inorganic colloids. In this model, dissociation/ association reactions are assumed to occur on the particle surface, yielding fixed charge, the density of which depends upon the degree of those reactions. The level of the surface potential of a particle also plays a key role in electrophoresis modeling. If it is lower than ca. 25 mV, the corresponding theoretical treatment can be simplified drastically. However, the resultant model is unsuitable for simulating factors such double-layer polarization, which can lead to interesting and significant electrophoretic behaviors. The modeling of electrophoresis is based on the electrical and hydrodynamic interactions between the particle and the fluid in the double layer surrounding it. Under general conditions, solving the equations governing an electrophoresis problem analytically is almost impossible because they are coupled, highly nonlinear partial differential equations. To make the relevant analysis simpler, earlier theoretical studies are based mainly on the limiting cases where the double layer is either very thin or very thick12,13 and/or on the assumption that the surface potential is low. One of the possible approaches to circumvent that difficulty is to adopt a perturbation method where the applied electric field needs to be much weaker than the electric field established by the particle.14 Recent advances in computation technologies and facilities also make it possible to solve complicated electrophoresis problem under conditions closer to reality. Regarding the influence of the types of ionic species in the liquid phase on the electrophoretic behavior of a particle, previous theoretical studies almost always assume that the liquid phase contains only one kind of each cation and anion. Available results for the case where multiple ionic species are present are very limited.15 That assumption, although it simplifies the mathematical analysis, is unrealistic in practice because other ionic species are usually present. For instance, in addition to the background ionic species, Hþ and OH- are always present in an aqueous (12) Hunter, R. J. Foundations of Colloid Science, 2nd ed.; Oxford University Press: New York, 2001. (13) Keh, H. J.; Anderson, J. L. J. Fluid Mech. 1985, 153, 417. (14) O’Brien, R. W.; White, L. R. J. Chem. Soc., Faraday Trans. 2 1978, 74, 1607. (15) Hsu, J. P.; Hsieh, T. S.; Young, T. H.; Tseng, S. Electrophoresis 2003, 24, 1338.
Published on Web 10/05/2010
DOI: 10.1021/la102968u
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dispersion and can play a role if the level of pH is either very low or very high. Therefore, extending the conventional analyses to take the presence of multiple ionic species into account is very desirable. This is conducted in the present study by considering the electrophoresis of a charge-regulated sphere under the conditions of an arbitrary level of surface potential and double-layer thickness.
Theory As illustrated in Figure 1, we consider the electrophoresis of a spherical particle of radius a subject to an applied electric field E along the axis of a cylindrical computation domain of radius b with b.a, that is, the boundary effect, if there is any, is neglected in our analysis. The cylindrical coordinates are adopted with the origin at the center of the particle, and because of the symmetric nature of the present problem, only the (r, z) domain needs be considered. Let ΩP, ΩW, and ΩE be the surface of the particle, that of the cylindrical computational domain, and the end surfaces of the computational domain, respectively. Suppose that the liquid phase is an incompressible Newtonian fluid containing N kinds of ionic species and that the system is in a pseudosteady state. Then, the present problem can be described by the following set of equations, which include those for the electrical potential, φ, the number concentration of ionic species j, nj, and the velocity of the fluid relative to that of the particle, v: r2 φ ¼ -
N X F zj enj ¼ ε ε j¼1
ð1Þ
zj e nj rφ þ nj v Jj ¼ - Dj rnj þ kB T
ð2Þ
r 3 Jj ¼ 0
ð3Þ
- rp þ ηr2 v - Frφ ¼ 0
ð4Þ
r3v ¼ 0
ð5Þ
In these expressions, r and r2 are the gradient operator and the Laplace operator, respectively; ε is the permittivity of the liquid phase; -Frφ is the electric body force acting on the fluid with F being the space charge density; e is the elementary charge; zj, Jj, and Dj are respectively the valence, the flux, and the diffusivity of ionic species j, with j = 1, 2,..., N; kB and T are the Boltzmann constant and the absolute temperature, respectively; η is the fluid viscosity; and p is the hydrodynamic pressure. Because of their coupled and nonlinear nature, analytically solving eqs 1-5 is nontrivial, if not impossible. However, if the strength of E is weak compared with that of the electric field established by the particle and/or the pore, then this difficulty can be alleviated by adopting a perturbation approach14 where each dependent variable is partitioned into an equilibrium component and a perturbed component. The former is the value of a variable in the absence of E, and the latter represents its value resulting from the application of E. For example, φ = φe þ δφ, where φe and δφ are the equilibrium potential and the perturbed potential, respectively. In subsequent discussions, the subscript e and the prefix δ denote the equilibrium and the perturbed properties, respectively. Because both the level of surface potential and the thickness of the double layer surrounding a particle are arbitrary in the present study, the phenomenon of double-layer polarization (DLP) can be significant. In this case, the double layer surrounding the 16858 DOI: 10.1021/la102968u
Figure 1. Electrophoresis of a charge-regulated particle of radius a along the axis of a cylindrical computational domain of radius b (b.a) subject to an applied electric field E in the z direction; r, θ, z are the cylindrical coordinates adopted with the origin at the center of the pore; ΩP, ΩW, and ΩE are the surface of the particle, that of the computational domain, and the end surfaces of the computational domain, respectively.
particle may no longer remain spherical because of the convective motion of the ionic species, which is expressed in the last term on the right-hand side of eq 2. To take the effect of DLP into consideration, nj is expressed as14 zj eðφe þ δφ þ gj Þ ð6Þ nj ¼ nj0 exp kB T where gj is a hypothetical potential function simulating a polarized double layer and nj0 is the bulk ionic concentration. Assuming that the strength of E is weak compared with that of the electric field established by the particle, eqs 1-3 and 6 yield
N ðKaÞ2 X R11 exp½- R10 φe N P R21 j ¼ 1
ð7Þ
N ðKaÞ2 X R21 ðδφ þ gj Þ exp½- R10 φe N P R21 j ¼ 1
ð8Þ
r 2 φe ¼
j¼1
r 2 δφ ¼
j¼1
r 2 gj ¼ R10 r φe 3 r gj þ Pej v 3 rφe
nj ¼ expð- R10 φe Þ½1 - R10 ðδφ þ gj Þ
- r p þ r 2 v þ r 2 φe r δφ þ r 2 δφr φe ¼ 0 r 3 v ¼ 0
ð9Þ ð10Þ ð11Þ ð12Þ
Here, r*=ar and r*2 =a2r2 are the scaled gradient P operator and the scaled Laplace operator, respectively; κ = [ 2j=1nj0(ezj)2/ εkBT]1/2 is the reciprocal Debye screening length; nj* = nj/nj0, φ*=φ e e/ζa, δφ*=δφ/ζa, and g*=g j j/ζa with ζa=kBT/z1e being the thermal potential; and Pej = ε(kBT/z1e)2/ηDj is the electric Peclet Langmuir 2010, 26(22), 16857–16864
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number of ionic species j, with j = 1-4. v* = v/U 0 with U 0 = ε(kBT/z1e)2/aη being a reference velocity; p*= p/pref with pref = εζ2a/a2 being a reference pressure; and Rβγ = zβj nγj0/zβ1nγ10, where subscript 1 denotes a reference ionic species, which is one of the ionic species in the system. Note that because the particle is stagnant at equilibrium, ve =0 and therefore v = δv. Using the SiO2 particle as an example,16 we assume that the surface of a particle is capable of carrying both positive and negative fixed charges through the following two main reactions: AOH SAO - þ Hþ
ð13Þ
AOH þ Hþ S AOH2 þ
ð14Þ
If we let Ka and Kb be the equilibrium constants of these reactions, then Ka ¼
NAO - ½Hþ NAOH
ð15Þ
Kb ¼
NAOH2 þ NAOH ½Hþ
ð16Þ
where NAO-, NAOH2þ, and NAOH are the surface densities of AO-, AOH2þ, and AOH, respectively. According to eqs 13 and 14, if the concentration of Hþ is high (low pH), then the particle is positively charged and it becomes negatively charged if the concentration of Hþ is low (high pH). If we let Ntotal be the total number of AOH molecules on the particle surface, then Ntotal ¼ NAO - þ NAOH þ NAOH2 þ
ð17Þ
Combining eqs 15-17 and assuming a Boltzmann distribution for Hþ at equilibrium, we obtain σ surface ¼ Ka
- FNtotal
þ
2
þ
Kb ð½H 0 expð- φe e=kB TÞÞ þ ½H 0 expð- φe e=kB TÞ þ Ka
-
Kb ð½Hþ 0 expð- φe e=kB TÞÞ2 Ka þ ½Hþ 0 expð- φe e=kB TÞ þ Kb ð½Hþ 0 expð- φe e=kB TÞÞ2
ð18Þ
Here, σsurface is the surface charge density, F is the Faraday constant, and [Hþ]0 is the bulk concentration of Hþ. The surface potential can be obtained from eq 18 by applying Gauss’s law. We assume that the following boundary conditions apply to the electric and the concentration fields:
n 3 r φe ¼ aeFNtotal Ka 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 on ΩP
ð19Þ
n 3 r δφ ¼ 0 on ΩP
n 3 r gj ¼ 0 on ΩP (16) Binner, J.; Zhang, Y. J. Mater. Sci. Lett. 2001, 20, 123.
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ð20Þ ð21Þ
φe ¼ 0 on ΩW
n 3 r δφ ¼ 0 on ΩW
n 3 r gj ¼ 0 on ΩW
φe ¼ 0 on ΩE
n 3 r δφ ¼ - Ez on ΩE
gj ¼ - δφ on ΩE
ð22Þ ð23Þ ð24Þ ð25Þ ð26Þ ð27Þ
In these expressions, σsurface* = σsurfaceae/εkBT is the scaled surface charge density; Ez* = Ezea/kBT; and n is the unit outer normal vector on a surface. Equations 20 and 23 imply that both the surface of the particle and that of the pore are nonconductive, and eqs 21 and 24 indicate that they are impenetrable to ionic species. As stated in eqs 22 and 25, because the pore is uncharged the equilibrium potential vanishes both on the pore surface and at a point far away from the particle. Equation 26 indicates that the electric field on the pore surface comes solely from the applied electric field. Equation 27 states that the ionic concentration at a point far away from the particle reaches the bulk value, as it should. We assume that the surface of the particle and that of the pore are nonslip and the flow field at a point far away from the particle is not influenced by its presence. These yield the following boundary conditions for the flow field v ¼ 0 on ΩP
ð28Þ
v ¼ - ðU=U 0 Þez on ΩW
ð29Þ
v ¼ - ðU=U 0 Þez on ΩE
ð30Þ
where ez is the unit vector in the z direction and U is the particle velocity. The approach of O’Brien and White14 is adopted to evaluate the mobility of the particle. In this approach, the original problem is partitioned into two hypothetical subproblems. In the first subproblem, the particle moves with a constant velocity U without the application of E, and in the second subproblem, E is applied but the particle is held fixed in space. If we let Fi be the total force acting on the particle in the z direction in subproblem i and Fi be the corresponding magnitude, then F1 = χU and F2 = βE. Note that χ and β are independent of U and E, respectively. At steady state, F1 þ F2 = 0, yielding U ¼ -
β E χ
ð31Þ
Therefore, the mobility of the particle, μE, can be expressed as μE ¼ -
β F2 U ¼ χ F1 E
ð32Þ
This expression suggests that μE can be evaluated by the following steps. (i) Assume an arbitrary electrophoretic velocity and solve the governing equations for the flow field in the first subproblem. (ii) Substitute the result obtained in the previous step into the governing equations for the electric field in the second subproblem, and solve the resultant expressions. (iii) Evaluate the electric force and the hydrodynamic force acting on the particle. (iv) DOI: 10.1021/la102968u
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Calculate μE with eq 32. (v) Determine if the total force acting on the particle in the z direction vanishes. If this is true, then the assumed electrophoretic velocity is correct and the solution procedure is finished. Otherwise, assume another electrophoretic velocity and go back to step i. In the present case, the forces acting on the particle include mainly the electrical force Fe and the hydrodynamic force Fd. Let Fei and Fdi be the z components of Fe and Fd in subproblem i, respectively, and let Fei* = Fei/εζa2 and Fdi* = Fdi/εζa2 be the corresponding scaled quantities. Then, Fei and Fdi can be evaluated respectively by17 0 ! 1 Z Dφ Dδφ Dφ Dδφ e @ e nz A dΩP Fei ¼ ð33Þ Dn Dz Dt Dt ΩP and Fdi
Z ¼
ΩP
ðσ H 3 nÞ 3 ez dΩP
ð34Þ
ΩP*
Here, is the dimensionless surface area of the particle scaled by a2; n and t are the magnitudes of the unit normal vector n and that of the unit tangential vector t, respectively; nz is the z component of n; and σ H*=σ H/(εζa2/a2) is the scaled shear stress tensor with σ H being the corresponding shear stress tensor.
Figure 2. Variation of the scaled electrophoretic mobility μE* =
μE(ηe/εkBT) as a function of κa for a rigid particle of constant potential in an uncharged cylindrical pore for the case where λ = 0.1 and φe* = 1 on the particle surface. (-) Present results and (2) the analytical solution of Shugai and Carnie.20
Results and Discussion The relevant governing equations and the associated boundary conditions are solved numerically by FlexPDE.18 This finite element method-based software is found to be efficient and sufficiently accurate for solving similar boundary-value problems.19 To justify its applicability, the problem considered by Shugai and Carnie,20 where the electrophoresis of a rigid particle of constant surface potential is solved analytically, is reanalyzed by the present method. Note that κa needs to be small enough (ca. 0.8) that the surface potential of the present model is roughly constant. Figure 2 illustrates the value of the scaled mobility μE*= μE(ηe/εkBT) evaluated by the software adopted and the corresponding analytical result of Shugai and Carnie.20 Note that μE* in the former is slightly smaller than in the latter, which is expected because the effect of DLP is not considered in the latter. In general, the performance of the software adopted is satisfactory. The size of the cylindrical computation domain needs to be defined appropriately so that the electrophoretic behavior of a particle is not influenced by its presence. Figure 3 shows the variation of the critical ratio of λ (= a/b), λc, as a function of the molar concentration of NaCl, CNaCl. Here, λc is defined as the value of λ below which the mobility of a particle is roughly constant as λ decreases and the shear force at ΩW vanishes at the same time. Figure 3 reveals that λc increases with increasing CNaCl. This is expected because the lower the CNaCl, the thicker the double layer and therefore the larger the b value that needs be assumed to ensure that the presence of the boundary of the computational domain is unimportant. Considering the conditions of subsequent discussions, a/b = 0.1 is assumed. To justify the applicability of the present electrophoresis model, it is used to describe the experiment data of the SiO2 particle (AEROSIL OX 50).21 Figure 4 show the variation of the (17) (18) (19) (20) (21) 215.
Hsu, J. P.; Yeh, L. H.; Ku, M. H. J. Colloid Interface Sci. 2007, 305, 324. FlexPDE, version 2.22; PDE Solutions Inc.: Spokane Valley, WA, 2000. Hsu, J. P.; Hung, S. H.; Yu, H. Y. J. Colloid Interface Sci. 2004, 280, 256. Shugai, A. A.; Carnie, S. L. J. Colloid Interface Sci. 1999, 213, 298. Sonnefeld, J.; L€obbus, M.; Vogelsberger, W. Colloids Surf., A 2001, 195,
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Figure 3. Variation of the critical ratio λc as a function of CNaCl at a=20 nm, pH 5, Ka=10-7, Kb=10-2, and Ntotal=5 10-5 mol/m2.
electrophoretic mobility μE as a function of pH. Both the experimental data and the results based on the present model are presented. The results predicted by the present theoretical model agree reasonably well with the experiment data. At the concentration of 0.001 M NaCl, the fitted values of the adjustable parameters are a=20 nm, pKb =2, pKa =7.4, and Ntotal =5 10-5 mol/m2. The estimated value of Ntotal is different with that estimated by Sonnefeld et al.,21 8 10-6 mol/m2. This might be caused by the fact that the actual surface reactions are far more complicated than those described by eqs 13 and 14. Note that because there are eight adjustable parameters the electrophoresis model of Sonnefeld et al.21,22 is complicated and fitting it to the experimental data is challenging. In addition, because only Naþ and Cl- are included in that model, it can be unrealistic when the pH is either too low or too high, where the presence of Hþ or OH- should not be ignored. In contrast, because the particle size is measurable and (Ka þ Kb) can be estimated directly from the isoelectric point, there are only two adjustable parameters in our model, namely, Ka (22) O’Brien, R. W.; Canon, D. W.; Rowlands, W. N. J. Colloid Interface Sci. 1995, 173, 406.
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Figure 4. Variation of the electrophoretic mobility μE as a function of pH for the case where the liquid phase contains particles of radius 20 nm in 0.001 M NaCl. (-) Present result with pKa = 7.4, pKb = 2, and Ntotal = 5 10-5 mol/m2 and (b) experiment data of Sonnefeld et al.21
Figure 6. (a) Variations of the surface charge density σsurface and the equilibrium surface potential as a function of pH at a = 20 nm, pKa = 7, pKb = 2, CNaCl = 0.001 M, and Ntotal = 5 10-6 mol/m2. (b) Variations of the electrostatic force Fe2, curve 1, and the hydrodynamic force Fd2, curve 2, as a function of pH for the case in plot a.
Figure 5. Variation of the electrophoretic mobility μE as a function of pH for the case where the liquid phase contains particles of radius 20 nm in 0.001 M NaCl at pKa = 7, pKb = 2, and Ntotal = 5 10-6 mol/m2. In curve 1, only Naþ and Cl- are considered, but in curve 2, Naþ, Cl-, OH-, and Hþ are all considered.
(or Kb) and Ntotal. Furthermore, the present model is applicable to the entire pH range. Figure 5 illustrates the influence of the types of ionic species present in the liquid phase on the electrophoretic behavior of a particle, where the variations in the mobility μE as a function of pH for different combinations of the types of ionic species are presented. For the present case, the point of zero charge is at pH ∼2.5. As can be seen in Figure 5, the mobility curves corresponding to different combinations of ionic species coincide for pH ranging from ca. 2 to 4.5, that is, close to the point of zero charge but become different both quantitatively and qualitatively for pH 4.5, especially at higher pH. This is because at a low (high) level of pH, the Hþ (OH-) coming from the dissociation of H2O and HCl (NaOH) becomes significant and should not be ignored. Note that if the presence of these ionic species is neglected then |μE| will be overestimated. It is interesting to note that if Hþ, OH-, Naþ, and Cl- are considered in the electrophoresis model, then μE shows a local minimum at pH ∼8.8. This local minimum disappears, however, Langmuir 2010, 26(22), 16857–16864
if Hþ and OH- are not included in the model, as is usually done in the literature. These behaviors can be explained by the variation of the surface charge density and the relevant forces acting on the particle as the pH varies as illustrated in Figure 6. Figure 6a shows that if the pH exceeds ca. 7 then the surface charge density σsurface (C/m2) remains roughly constant, implying that the electrical driving force coming directly from the surface charge should behave similarly. However, Figure 6b reveals that the scaled electric force acting on the particle, Fe2, dominates in that pH range and has a negative local minimum. This implies that the electric force is influenced by some other factor. Figure 7 shows the contours of the scaled perturbed ion distribution (delta_n_N) = [(nanion - nanion,e) - (ncation - ncation,e)]/(nsodium,0) at pKa = 7, pKb = 2, Ntotal = 5 10-6 mol/m2, and pH 9 on the plane θ = π/2, where nanion and ncation are the number concentrations of anions and cations, respectively, and nanion,e and ncation,e are the equilibrium number concentrations of anions and cations, respectively. nsodium,0 is the bulk concentration of Naþ. This Figure reveals that the application of E yields an appreciable difference between (delta_n_N) on the top of the particle and that on the bottom. That is, more (fewer) anions (cations) accumulated near the top of the particle than that near the bottom. This is known as doublelayer polarization (DLP),23 which induces an internal electric field, the direction of which is opposite to that of E, thereby (23) Lee, E; Chu, J. W.; Hsu, J. P. J. Colloid Interface Sci. 1998, 205, 65.
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Figure 7. Contours of the scaled perturbed ion distribution (delta_n_N) = [(nanion - nanion,e) - (nanion - nanion,e)]/(nsodium,0) at a = 20 nm, pKa = 7, pKb = 2, Ntotal = 5 10-6 mol/m2, CNaCl = 0.001 M, and pH 9 on the plane θ = π/2.
reducing the electric driving force coming from E. Note that the effect of DLP is most significant if the thickness of the double layer is comparable to the size of a particle and its surface potential is sufficiently high.24 In our case, if the pH exceeds 7.5, then the absolute value of the surface potential exceeds 80 mV, which can be considered high. The influences of the pH value of the bulk phase and the surface charge density on the degree of DLP are illustrated in Figure 8, where the variation of (delta_n_N) along the z direction for the case of Figure 5 at two levels of pH are presented. As seen in Figure 8a, at pH 6 the scaled perturbed concentration of anions on the top of the particle is higher than that on the bottom. It is interesting to observe that if the pH is raised to 9 then two types of DLP are present: type I DLP, which is seen in Figure 8b, occurs inside the double layer, and type II DLP occurs immediately outside the double layer. In contrast to the case of type I DLP, the scaled perturbed concentration of anions on the top of the particle is lower than that on the bottom, thereby inducing an internal electric field, the direction of which is the same as that of E. That is, the presence of type I DLP reduces the absolute value of the mobility of the particle, but the presence of type II DLP raises that value. As seen in Figure 8b, however, the electric field induced by type II DLP is much weaker than that induced by type I DLP. Note that the extent of type I DLP also increases with increasing pH. For example, the maximum (delta_n_N) is ca. 1.3 10-2 at pH 6 and becomes 2.8 10-2 at pH 9. The presence of type II DLP was proposed by Hsu et al.25 in a study of the diffusiophoresis of a charged particle in an electrolyte solution but has never been reported in previous electrophoresis studies. Figure 9 shows the influence of the concentration of NaCl, CNaCl, on the mobility of a particle μE at various pH levels. In the present case, the magnitude of the mobility increases with increasing thickness of the double layer. As seen in Figure 10, this is because the rate of increase in the hydrodynamic retardation force with decreasing double-layer thickness is faster than that in the electric driving force.11 Figure 9 reveals that if the pH is low (e6) then |μE| decreases monotonically with increasing CNaCl. It is interesting that at pH 7, μE has both a negative local maximum and a negative local minimum as CNaCl varies. As the pH increases, the presence of those local extrema becomes more apparent and (24) Hsu, J. P.; Chen, Z. S. J. Phys. Chem. B 2008, 112, 11270. (25) Hsu, J. P.; Hsu, W. L.; Ku, M. H.; Tseng, S. J. Colloid Interface Sci. 2010, 342, 598.
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Figure 8. Contours of the scaled perturbed ion distribution (delta_n_N) along the z direction for the case of Figure 5 at (a) pH 6 and (b) pH 9, where the boundary of the double layer is marked by dashed lines.
Figure 9. Variations of the electrophoretic mobility μE as a function of CNaCl for various levels of pH at a = 20 nm, pKa = 7, pKb = 2, and Ntotal = 5 10-6 mol/m2.
the values of CNaCl at which they appear also become larger. These behaviors result from the effect of DLP because in the present case the equilibrium surface potential of the particle exceeds ca. 55 mV and κa is on the order of unity. As illustrated in Figure 10, at pH 7 the variation of the slope of the electrical force acting on the particle in the second subproblem, Fe2, varies appreciably for CNaCl ranging from 0.004 to 0.01 M and from Langmuir 2010, 26(22), 16857–16864
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Figure 10. Variations of the electrostatic force Fe2 (curves 1, 3, and 5) and the hydrodynamic force Fd2 (curves 2, 4, and 6) acting on a particle as a function of CNaCl for various levels of pH at a = 20 nm, pKa = 7, pKb = 2, and Ntotal = 5 10-6 mol/m2. Curves 1 and 2, pH 6; curves 3 and 4, pH 7; and curves 5 and 6, pH = 8.
Figure 11. Contours of the scaled perturbed ion distribution (delta_n_N) on the plane θ = π/2 at CNaCl = 0.005 M and pH 7 (a) and CNaCl = 0.01 M and pH 8 (b) for the case where a = 20 nm, pKa = 7, pKb = 2, and Ntotal = 5 10-6 mol/m2.
0.008 to 0.04 M at pH 8. The value of CNaCl at which the local maximum (minimum) of μE occurs is in the first (second) range where DLP occurs, as illustrated in Figure 11. In an experimental study of the electrophoresis of amorphous silica dioxide, Ute26 (26) Pyell, U. Electrophoresis 2008, 29, 576.
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Figure 12. Variations of the electrophoretic mobility μE as a function of pH at various values of a for the case where CNaCl = 0.001 M, pKa = 7, pKb = 2, and Ntotal = 7 10-7 mol/m2.
also suggested that the presence of the local extrema in μE could be attributed to the effect of DLP. The influence of the radius of a particle on its mobility is illustrated in Figure 12. Note that because the surface area of the particle increases with its radius, so does the amount of surface charge and therefore the electric force acting on the particle. However, a larger surface area also yields a greater viscous force acting on the particle. As seen in Figure 12, if the pH is lower than ca. 8, then |μE*| increases with increasing particle size, implying that the electric force dominates. This is because in the absence of DLP the scaled surface charge density (σsurface*) is proportional to the particle radius. However, if the pH exceeds ca. 8, then because the effect of DLP is important, the dependence of |μE*| on the particle size becomes complicated. Figure 12 also indicates that the effect of DLP becomes unimportant when the particle is either too small or too large because |μE*| does not show a local minimum as the pH varies. A similar phenomenon was also observed in the electrophoresis of silica dioxide particles27 and fumed oxide particles.28 This is consistent with the result of Hsu and Chen,24 where DLP is found to be significant when κa is on the order of unity and becomes unimportant if it is either too small or too large.
Conclusions We extend the conventional electrophoresis analyses, where the liquid phase is usually assumed to contain only one kind of each cation and anion to the case where multiple ionic species are present. The applicability of the derived electrophoresis model is justified by fitting it to the experimental data of SiO2 particles reported in the literature, where the dispersion pH is adjusted by HCl and NaOH. Numerical simulations are conducted to investigate the electrophoretic behavior of a particle under various conditions, and the obtained results can be summarized as follows. (i) To describe the specific behavior of the mobility of a chargeregulated particle at high pH, the conventional electrophoresis model includes too many adjustable parameters, making the datafitting procedure nontrivial and unrealistic. In contrast, through incorporating the ionic species actually present in the liquid phase, the present model is much more concise and closer to reality. (ii) If the dispersion pH is close to the point of zero charge, then (27) Rodriguez-Santiago, V.; Fedkin, M. V.; Lvov, S. N. Rev. Sci. Instrum. 2008, 79, 093302. (28) Gun’ko, V. M.; Zarko, V. I.; Leboda, R.; Chibowski, E. Adv. Colloid Interface Sci. 2001, 91, 1.
DOI: 10.1021/la102968u
16863
Article
incorporating Hþ and OH- into the electrophoresis model is unnecessary; otherwise, neglecting those ionic species can yield an electrophoretic mobility that deviates both quantitatively and qualitatively from the true value. For example, because of the effect of double-layer polarization, if both Hþ and OH- are included in the electrophoresis model then the mobility shows a local minimum as the pH varies, which is consistent with the experimental observation. However, that local minimum disappears if Hþ and OH- are not included in the electrophoresis model. (iii) Two types of double-layer polarization (DLP) are observed, which has not been reported previously in electrophoresis analyses. Type I DLP, which reduces the absolute value of the mobility of a particle, occurs inside the double layer, and type II DLP, which raises that value, occurs immediately outside the
16864 DOI: 10.1021/la102968u
Hsu and Tai
double layer. Under typical conditions, the effect of type II DLP is much less significant than that of type I DLP. (iv) If the pH is low (e6), then the absolute value of the mobility decreases monotonically with increasing background concentration of NaCl, CNaCl; if the pH takes a medium large value (= 7), then the mobility has both a negative local maximum and a negative local minimum as CNaCl varies and the higher the pH, the more apparent the presence of those local extrema and the higher the CNaCl at which they appear. (v) If the pH is lower than ca. 8, then the absolute value of the mobility increases with increasing particle size. However, if the pH exceeds that value, then because the effect of DLP is important, the dependence of the mobility on the particle size becomes complicated. That effect is unimportant if the particle is either sufficiently small or sufficiently large.
Langmuir 2010, 26(22), 16857–16864
