Article pubs.acs.org/Langmuir
Electrophoresis of a pH-Regulated Zwitterionic Nanoparticle in a pH-Regulated Zwitterionic Capillary Nan Wang,†,‡ Chien-Pai Yee,† Yu-Yen Chen,† Jyh-Ping Hsu,*,† and Shiojenn Tseng*,§ †
Department of Chemical Engineering, National Taiwan University, Taipei, Taiwan 10617 College of Chemistry and Chemical Engineering, Huazhong University of Science and Technology, Wuhan 430074, PR China § Department of Mathematics, Tamkang University, Tamsui, Taipei 25137, Taiwan ‡
ABSTRACT: We consider the electrophoresis of a rigid sphere along the axis of a narrow cylindrical capillary; both are pH-regulated and zwitterionic. This extends available analyses in the literature to a more general and realistic case. Adopting a titanium oxide (TiO2) particle in a silicon dioxide (SiO2) capillary as an example, we examine the capillary radius, the solution pH, and the electrolyte concentration (or double-layer thickness) for their influences on the electrophoretic behavior of a particle. Because the pH solution is adjusted by HCl and NaOH, the presence of four kinds of ionic species, namely, H+, OH−, Na+, and Cl−, should be considered if NaCl is the background electrolyte. This also extends conventional electrophoresis analyses to the case of multiple ionic species. The interactions of the electroosmotic flow, the properties of the particle and the solution, and the capillary wall yield complicated electrophoretic behavior that can be regulated by the solution pH and the background electrolyte concentration. The results gathered are necessary for the future design of nanopore-based electrophoresis devices.
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INTRODUCTION Modern electrophoresis applications usually involve boundary effects. In nanopore-based DNA sensing techniques, for example, the length scale of a device can be close to that of target entities, implying that the presence of the device wall might play a role. Therefore, classical analyses such as those for an isolated particle1 and a monodisperse suspension2,3 need modification. In fact, theoretical and experimental evidence revealed that the boundary effect can influence the electrophoretic behavior of a particle both quantitatively4,5 and qualitatively.6−15 In general, a boundary is capable of influencing the behavior of a particle during its electrophoresis both electrically and hydrodynamically, and that influence can be profound if the boundary is charged. Taking account of the presence of multiple ionic species in the liquid phase, Hsu et al.16 modeled the electrophoresis a particle, the surface of which is maintained at a constant charge density, in a chargeregulated, zwitterionic cylindrical pore. Because of the presence of the electroosmotic flow coming from the charged pore, several interesting and significant behaviors were observed. In many modern applications of electrophoresis, such as DNA identification,17,18 separation technology,19 biological analysis,20 particle surface characterization,21 and nanofabrication technology,22,23 the charged conditions of a particle are usually complicated and cannot be described by those in conventional analysis, namely, constant surface potential or constant surface charge density. In this case, a charge-regulated surface model might be applicable, where the charged conditions on the particle surface depend upon the properties of the surrounding medium such as the pH and bulk ionic concentration. A charge-regulated surface is a general model © XXXX American Chemical Society
that covers both constant surface potential and constant surface charge density models as limiting cases and should be adopted, in general. Inorganic entities such as metal oxide particles in an aqueous environment, for example, usually react with H+ and/or OH− so that its surface is charge-regulated. Similarly, because of the dissociation of (association with) H+, organic particles such as proteins and microorganisms also have that nature. Considering various modern applications of electrophoresis, we investigate, for the first time, the electrophoretic behavior of a pH-regulated, zwitterionic particle in a capillary of similar nature. This model is closer to reality in many potential applications and is capable of simulating essentially all types of colloidal particles. When we take a titanium dioxide (TiO2) particle in a silicon oxide (SiO2) capillary as an example, the influences of solution pH, bulk ionic concentration, and capillary radius on the electrophoretic behavior of the particle are discussed in detail.
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THEORY As shown in Figure 1, we consider the electrophoresis of a rigid spherical particle of radius a in a cylindrical capillary of radius b subject to an applied uniform electric field E of strength E in the z direction. In a study of electrophoresis, Ai and Qian24 concluded that particles tend to align along the center line of a nanopore before entering it. Therefore, we assume that the particle is moving along the axis of the capillary. The influence of the capillary wall is measured by the Received: March 13, 2013 Revised: May 11, 2013
A
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N
∑j=1
∇*2 gj* =
zj z1
⎛ zj ⎞ (δϕ* + gj*)exp⎜ − ϕe*⎟ ⎝ z1 ⎠
zj 2nj0
N
(κa)2
∇*2 δϕ* =
∑
z 2n j = 1 1 10
2
zj nj0 z12n10
(6)
∇*ϕe*·∇*gj* + Pej v *·∇ϕe* = 0,
j = 1, 2, ..., N Figure 1. Electrophoresis of a pH-regulated, zwitterionic particle of radius a along the axis of charge-regulated, zwitterionic cylindrical capillary of radius b subject to an applied uniform electrical field E in the z direction; r and z are the radial and axial distances of the cylindrical coordinates adopted with the origin at the center of the particle. Ωp, ΩW, and ΩE are the surface of the particle, the lateral surface of the capillary, and the surface of the inlet and the outlet, respectively.
−∇*p* + ∇*2 v* + ∇*2 ϕe*∇*δϕ* + ∇*2 δϕ*∇*ϕe* = 0 (8)
⎤ ⎛ zj ⎞⎡ zj nj* = exp⎜ − ϕe*⎟⎢1 − (δϕ* + gj*)⎥, z1 ⎦ ⎝ z1 ⎠⎣
(10)
A kOH ⇔ A kO− + H+,
(1)
(2)
−∇p + η∇ v − ρ∇ϕ = 0
(3)
∇·v = 0
(4)
Kak =
∇ , ∇, ρ, e, kB, T, ε, and η are the Laplace operator, the gradient operator, the space charge density, the elementary charge, Boltzmann constant, the absolute temperature, the permittivity of the liquid, and the viscosity of the liquid, respectively. zj, Jj, and Dj are the valence, the flux, and the diffusivity of ionic species j, j = 1, 2,..., N, respectively. −ρ∇ϕ denotes the electric body force acting on the fluid. Under the conditions of a weak applied electric field, E ≪ 25 kV/m, a perturbation approach25−27 can be applied that considerably simplifies the solution procedure. In this case, dependent variables v, ϕ, p, nj, and ρ are partitioned into an equilibrium component, symbol with a subscript e, and a perturbed component, symbol with a prefix δ, namely, v = ve + δv, ϕ = ϕe + δϕ, p = pe + δp, nj = nje + δnj, and ρ = ρe + δρ. The equilibrium component comes from the presence of the particle and the capillary in the absence of E, and the perturbed component comes from the application of E. Substituting the above partitioned expressions into eqs 1−4, neglecting terms that involve a product of two perturbed terms, and expressing the resultant expressions in scaled symbols yield the following equations: (κa)2 N
∑j=1
2
zj nj0 z12n10
N
∑ j=1
⎛ zj ⎞ exp⎜ − ϕe*⎟ z1n10 ⎝ z1 ⎠
(11)
k = 1, 2
(12)
The equilibrium constants of these reactions can be expressed as
2
∇*2 ϕe* = −
k = 1, 2
A kOH + H+ ⇔ A kOH 2+,
j = 1, 2, ..., N
2
j = 1, 2, ..., N
Here, ∇*2 = a2∇2, ∇* = a∇, κ = [∑j=1Nnj0(ezj)2/εkBT]1/2, nj* = nj/nj0, ϕe* = ϕe/ζa, δϕ* = δϕ/ζa, gj* = gj/ζa, ζa = kBT/z1e, Pej = εζa2/ηDj, v* = v/U0, U0 = εζa2/aη, p* = p/pref, and pref = εζa2/a2. nj0 is the bulk ionic concentration, gj is a hypothetical potential function simulating a polarized double layer, ζa is the thermal potential, κ is the reciprocal Debye length, Pej is the electric Peclet number of ionic species j, U0 is a reference velocity, and pref is a reference pressure. Because the particle is stagnant at equilibrium, ve = 0, yielding v = δv. Suppose that the surfaces of both the particle and the capillary are zwitterionic, such as in the cases of SiO2 and TiO2,28 capable of undergoing dissociation/association reactions:
2
⎤ ⎡ ⎛ ⎞ zje nj∇ϕ⎟ + nj v⎥ = 0, ∇·⎢ − Dj⎜∇nj + ⎢⎣ kBT ⎝ ⎠ ⎦⎥
(9)
∇*·v* = 0
parameter Λ = b/a. Let r and z be the radial and axial distances of the cylindrical coordinates adopted, the origin of which is at the particle center. We assume that the capillary is sufficiently long so that its end effect is negligible. A cylindrical computational domain confined by inlet and outlet surfaces is defined for convenience. Let Ωp, ΩW, and ΩE be the surfaces of the particle, the capillary, and the inlet and outlet, respectively. The capillary is filled with an incompressible Newtonian fluid with constant physical properties containing N kinds of ionic species. Suppose that the system under consideration is at pseudosteady state. It can then be described by the following set of equations describing the electrical potential, ϕ, the pressure, p, the number concentration of ionic species j, nj, and the fluid velocity, v: N z en ρ j j ∇ ϕ = − = −∑ ε ε j=1
(7)
Kbk =
NAkO−[H+]sk NAkOH
k = 1, 2
,
(13)
NAkOH2+ NAkOH[H+]sk
k = 1, 2
,
(14)
where the subscript k is 1 (2) for the particle (capillary) surface. Here, NAkO−, NAkOH+2 , and NAkOH are the surface densities of AkO−, AkOH+2 , and AkOH, respectively, and [H+]sk is the concentration of H+ on the particle (capillary) surface. If we let Ntotal k be the total number density of the functional groups on the particle (capillary) surface, then Ntotal k = NAkO− + NAkOH + NAkOH2+
(15)
It can be shown that the charge density on the particle (capillary) surface, σk, is29 ⎛ ⎜ σk = − FNtotal k ⎜ ⎜ ⎛ + ⎜ Kbk ⎜[H ]0 exp ⎝ ⎝
Kak −ϕee
⎞2 −ϕee + ⎟ + [H ] exp − 0 kBT ⎠
( ) kBT
2
zjnj0
−
Kbk([H+]0 exp(− ϕee/kBT )) Kak + [H+]0 exp
(5)
−ϕee
( ) kBT
⎛ + Kbk ⎜[H+]0 exp ⎝
−ϕee
(
)+K
⎞ ⎟ ⎟, 2⎟ ⎞ ⎟ ⎟ ⎠ ⎠
k = 1, 2
ak
( ) kBT
(16) B
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F*2 = βE*, with χ and β being proportionally constant and F*1 = Fi/εζa2. Because F*1 + F*2 = 0 at steady state, the scaled electrophoretic mobility, μE*, can be expressed as
where F is the Faraday constant and [H+]0 is the bulk concentration of H+. We assume that the particle surface is charge-regulated, nonconductive, and impermeable to ionic species. These yield the following boundary conditions: n ·∇*ϕe* = −σ1*
ΩP
on
F *U * β μE* = − = − 2 χ F1*E*
(17)
n ·∇*δϕ* = 0
on
ΩP
n ·∇*gj∗ = 0,
j = 1, 2, ..., N
If we let Fei and Fdi be the values of Fe and Fd in subproblem i, respectively, then16
(18)
on
ΩP
(19)
Fei* =
Suppose that the capillary surface is nonconductive, remains charge-regulated, and is impermeable to ionic species. Then the following boundary conditions apply: n ·∇*ϕe* = −σ2*
ΩW
on on
ΩW
n ·∇*gj∗ = 0,
j = 1, 2, ..., N
ΩW
Fdi* =
on
ΩE
n ·∇*δϕ* = −Ez*
on
gj∗ = −δϕ*, j = 1, 2, ..., N
(22)
ΩE
i = 1, 2
(32)
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RESULTS AND DISCUSSION The parameters that are key to the present problem, including the capillary size (or boundary effect), the solution pH, and the solution electrolyte concentration (or double-layer thickness), are examined for their influence on the electrophoretic behavior of a particle. For illustration, unless otherwise mentioned, we consider a titanium oxide (TiO2, pKa1 = 7.8, pKb1 = −4.95) particle in a silicon dioxide (SiO2, pKa2 = 7.5, pKb2 = 1.8) capillary
(24)
on
i = 1, 2
Here, nz is the z component of n, and ∂/∂n and ∂/∂t denote the derivative along the direction of n and that along the direction of the unit tangential vector t, respectively. F*ei = Fei/εζa2, F*di = Fdi/εζa2, Ω*P = ΩP/a2, and σH* = σH/(εζa2/a2) with σH being the corresponding hydrodynamic stress tensor.
(23)
ΩE
∫Ω * (σ H*·n)·ez dΩP*, P
We assume that on ΩE the electric potential is uninfluenced by the particle, the electric field comes solely from the applied electric field, and the ionic concentration reaches the corresponding bulk equilibrium value. These yield the boundary conditions below: n ·∇*ϕe* = 0
⎤ ∂δϕ* ⎛ ∂ϕe* ∂δϕ* ⎞ ⎥ ⎟⎟nz dΩ P*, − ⎜⎜ ⎝ ∂t ∂t ⎠ ⎥⎦ ⎣ ∂n ∂z e
(31)
(21)
on
⎡ ∂ϕ *
∫Ω * ⎢⎢ P
(20)
n ·∇*δϕ* = 0
(30)
(25)
Here, σk* = σkae/εkBT, k = 1, 2, and Ez* = Ezea/kBT. n is the unit normal vector directed into the liquid phase on ΩP and ΩW and upward on ΩE. Because the origin is at the center of the particle, the fluid moves with a relative velocity of −(U/U0)ez on the capillary surface with U and ez being the particle velocity and the unit vector in the z direction, respectively. On the inlet and outlet surfaces, the flow field is uninfluenced by the particle. Therefore, v* = 0
on
ΩP
(26)
v* = −(U /U 0)ez
on
n ·∇*vz∗ = 0
on
ΩE
vr ∗ = 0
ΩE
on
ΩW
(27) (28) Figure 2. Variation of the scaled electrophoretic mobility μ*E with pH for various levels of CNaCl at a = 20 nm and Λ = 10.
(29)
The governing equations and the associated boundary conditions are solved numerically by FlexPDE (PDE Solutions, Spokane Valley, WA), a finite element method-based software. Because the particle velocity U, which is present in the boundary conditions, is unknown, the solution procedure involves trial and error, in general. This difficulty can be avoided by adopting the approach proposed by O’Brien and White,26 where the original problem is partitioned into two subproblems. In the first subproblem, the particle moves with a velocity U in the absence of E, and in the second subproblem, E is applied but the particle is motionless. Let Fi be the force acting on the particle in the z direction in subproblem i with magnitude Fi. The forces involved in electrophoresis include the electrical force Fe and the hydrodynamic force Fd with magnitudes Fe and Fd, respectively. Therefore, the total forces acting on the particle in the z direction in those two subproblems are F1* = χU* and
Figure 3. Variation of the scaled equilibrium surface potential ϕe* with pH for the case of Figure 2. ϕ*e of a particle, solid line; that of a capillary, dashed line. C
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Figure 4. Contours of the scaled fluid velocity v* near a positively charged particle in subproblem two at various combinations of CNaCl and pH for the case of Figure 2. (a) CNaCl = 10−4 M and pH 3, (b) CNaCl = 10−3 M and pH 3, (c) CNaCl = 10−2 M and pH 3, (d) CNaCl = 10−4 M and pH 4, (e) CNaCl = 10−3 M and pH 4, and (f) CNaCl = 10−2 M and pH 4.
be neglected, which is satisfied if the capillary length exceeds ca. 30 times the particle radius.31 The point of zero charge (PZC) of the particle is 6.375, and that of the capillary is 2.850. Therefore, the capillary is negatively charged for the range of pH considered, 3 to 8.5, and the particle can be either positively or negatively charged. The former implies that the EOF associated with the capillary can play a significant role. Influence of pH. The influence of the solution pH on the electrophoretic behavior of a particle is summarized in Figure 2 for various levels of bulk electrolyte concentration CNaCl. It is interesting to see in this figure that the behavior of the scaled electrophoretic mobility μ*E as the pH varies depends greatly upon CNaCl. For the pH range from 3 to 4, μ*E variation is complicated by CNaCl. As illustrated in Figure 3, this is because the variation in the surface potential of the particle resulting from the variation in CNaCl (or double-layer thickness) is different from that of the capillary. In that pH range, the μE* at CNaCl = 10−4 M is seen to
as an example. The total number density of the surface functional groups of the particle and that of the capillary are 7.65 × 10−6 and 7.50 × 10−6 mol/m2, respectively.13,30 The background electrolyte is NaCl, and the solution pH is adjusted with HCl and NaOH, implying that four major ionic species need to be considered; namely, Na+, Cl−, H+, and OH− should be considered, and z1 = 1. If we let Kw be the dissociation constant of water, then the following relationship applies: [H+]0 = 10−pH, [Na+]0 = CNaCl, [Cl−]0 = CNaCl + 10−pH − 10−(pKw− pH), and [OH−]0 = 10−(pKw − pH) for pH ≤ Kw/2; [H+]0 = 10−pH, [Na+]0 = CNaCl + 10−(pKw − pH) − 10−pH, [Cl−]0 = CNaCl, and [OH−]0 = 10−(pKw − pH) for pH ≥Kw/2. In addition, the following values are adopted: a = 20 nm, T = 298 K, kB = 1.38 × 10−23 J/K, ε = 7 × 10−10 CV−1 m−1, η = 10−3 kg m−1 s−1, e = 1.6 × 10−19 C. The diffusivities of H+, Na+, Cl−, and OH− are 9.38 × 10−9, 1.33 × 10−9, 2.00 × 10−9, and 5.29 × 10−9 m2 s−1, respectively; therefore, their electric Peclet numbers are 0.05, 0.235, 0.353, and 0.08868, respectively. The capillary is long enough that its end effects can D
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EOF. Therefore, as CNaCl increases from 10−4 to 10−2 M, μ*E decreases continuously because of a weaker EOF and a stronger retardation flow surrounding the particle. Note that at CNaCl = 10−4 and 10−3 M, μ*E remains roughly constant for the pH range from 6.375 (PZC of TiO2) to 7. This is because the increase in the |μE*| of the negatively charged particle with increasing pH is offset by the increase in the EOF velocity. The slight decrease in μE* at CNaCl = 10−2 M in that pH range arises from the increase in the EOF because the pH increase is not enough to offset the increase in the |μE*| of the particle. If the pH exceeds ca. 7, then the EOF is significant enough to alter the direction of the velocity of the negatively charged particle such that μ*E increases rapidly with increasing pH at all levels of CNaCl. Note that although the particle is negatively charged its μ*E is dominated by the EOF induced by the charged capillary with a high surface potential.
increase with increasing pH. This is because the strength of the retardation flow near the particle at this level of CNaCl decays more rapidly than that at the other two levels of CNaCl shown in Figure 4. In addition, the EOF at CNaCl = 10−4 M has a more appreciable influence than that at the other two levels of CNaCl as a result of the difference in the surface potential of the capillary. For the pH range from 4 to 6.375 (PZC of the capillary), the result of competition between the decrease in μE* of the positively charged particle and the increase in the EOF velocity depends upon the level of CNaCl. At CNaCl = 10−4 M, μE* is seen to increase gradually with increasing pH, which arises from the increase in the EOF velocity due to the increase in the level of the capillary surface potential. The level of the capillary surface potential decreases with increasing CNaCl, as does the strength of the corresponding
Figure 5. Variation of the scaled electrophoretic mobility μE* with pH for various values of Λ. (a) CNaCl = 10−2 M, (b) CNaCl = 10−3 M, and (c) CNaCl = 10−4 M.
Figure 6. Contours of the scaled z-direction velocity at the outlet of the capillary for the case of Figure 4c at (a) Λ = 5 and pH 6, (b) Λ = 10 and pH 6, (c) Λ = 5 and pH 7, and (d) Λ = 10 and pH 7. E
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capillary. The EOF velocity depends upon κb, and its value at the capillary center is roughly constant when κb exceeds ca. 7.31 In this case, because the particle is almost uncharged near pH 6.375, its mobility is dominated by EOF; therefore, μ*E is almost independent of Λ in Figure 5a,b. However, if pH exceeds this value, then because the surface charge of the particle changes from negative to positive, μE*(Λ = 10) < μE*(Λ = 5). Note that for a negatively charged particle, the larger the Λ, the larger the |μE*|; therefore, the curves in Figure 5c, where CNaCl = 10−4 M and the particle is negatively charged, do not intersect with each other. This arises from the fact that κb is small in the present case, the boundary effect is significant, and therefore the EOF velocity is slow, as shown in Figure 6, where the scaled radial velocities under various conditions are presented. Note that for a fixed pH the larger the b/a the more significant the influence of EOF on the mobility. Influence of Capillary Nature. The charged conditions of the capillary surface depend upon its properties such as the total number of functional groups and the associated equilibrium constants. The influence of this effect on μ*E is simulated in Figure 7. As seen for the two sets of equilibrium constants assumed for the capillary (pKa2 = 6.8, pKb2 = 1.8 andpKa2 = 7.5, pKb2 = 1.8), the influence of Λ (or the boundary) on μE* is inappreciable. However, a slight change in pKa2 from 6.8 to 7.5 is capable of yielding different behavior in μ*E for pH μE*(pKa2 = 7.5, pKb2 = 1.8). Influence of Bulk Salt Concentration. Figure 8 shows the influence of the bulk salt concentration CNaCl on the scaled particle mobility μ*E for various levels of pH. This figure reveals
Figure 3 shows that if the pH exceeds ca. 7, because it is far from the PZC of the capillary, its surface potential at each level of CNaCl is high enough to yield an appreciable EOF. Influence of Capillary. The influence of the capillary (boundary), measured by parameter Λ = b/a, on the electrophoretic behavior of a particle is summarized in Figure 5. As seen in Figure 5a,b, if the pH is lower than ca. 6.375 (PZC of TiO2), then μE*(Λ = 10) > μE*(Λ = 5). This is expected because the smaller the Λ, the more significant the boundary effect and the greater the hydrodynamic drag acting on the particle. However, this effect becomes insignificant as the pH gets close to the PZC of the particle, where its surface potential almost vanishes and therefore its mobility is governed essentially by the EOF as a result of the charged
Figure 7. Variation of the scaled electrophoretic mobility μE* with pH for various capillary properties at CNaCl = 10−3 M. Black curves: pKa2 = 6.8 and pKb2 = 1.8. Blue curves: pKa2 = 7.5 and pKb2 = 1.8. Solid curve: Λ = 10. Dashed curve: Λ = 5.
Figure 8. Variation of the scaled electrophoretic mobility μE* with CNaCl for various levels of pH at Λ = 10.
Figure 9. Variation of the scaled surface charge density σ*particle (a) and scaled equilibrium surface potential ϕ*e (b) of the particle with CNaCl for the case of Figure 8. F
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that the behavior of μ*E depends strongly on pH. At pH 3, μ*E shows a local maximum occurring at CNaCl ≅ 4 × 10−2 M. As shown in Figure 9, the increase in μE* for CNaCl < 4 × 10−2 M is due to the increase in the surface charge density of the particle, as is the electric driving force acting on it. The decrease in μ*E as CNaCl exceeds ca. 4 × 10−2 M arises from the fact that the electric double layer becomes so thin that the hydrodynamic drag acting on the particle becomes strong. For the case of pH 5, μ*E shows a local minimum, where it decreases monotonically with increasing CNaCl for CNaCl up to ca. 5 × 10−3 M and then increases with a further increase in CNaCl. As illustrated in Figure 9, the surface potential of the capillary is sufficiently high so that the effect of EOF dominates. In this case, ϕ*e decreases with increasing CNaCl, yielding the initial decrease in μ*E . However, if CNaCl is sufficiently high (>5 × 10−3 M), then the effect of the EOF becomes less significant than that of the increase in the electric driving force acting on the particle resulting from the increase in σ*particle with increasing CNaCl (Figure 9a). In this case, μ*E increases with increasing CNaCl, yielding a local minimum. If the pH is raised to 7, because ϕe* is sufficiently high (Figure 9b), and σparticle * is insensitive to the variation in CNaCl (Figure 9a), then μ*E is dominated by the EOF. In this case, because |ϕ*e | decreases monotonically with increasing CNaCl, so does μE*, as observed in Figure 8.
can decrease monotonically. This arises from the competition between the electric driving force acting on the particle and the EOF of the capillary.
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AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected] (J.P.H.);
[email protected] (S.T.). Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS This work is supported by the National Science Council of the Republic of China. REFERENCES
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CONCLUSIONS Extending previous analyses to conditions closer to reality, we study the electrophoresis of a rigid, zwitterionic pH-regulated particle in a zwitterionic, pH-regulated, cylindrical capillary filled with multiple ionic species. Taking a TiO2 (pKa1 = 7.8 and pKb1 = −4.95) in an SiO2 (pKa2 = 7.5 and pKb2 = 1.8) capillary filled with H+, OH−, Na+, and Cl− as an example, we simulated the particle mobility under various conditions. We conclude the following: (i) The conventional electrophoresis model based on binary ionic species is inappropriate to describe the electrophoresis behavior of a zwitterionic, charge-regulated particle as the ionic strength varies, especially at higher or lower pH values. (ii) Because of the variations in the particle and capillary surface potentials, the influence of the solution pH on the electrophoretic behavior of a particle depends strongly upon the bulk salt concentration. (iii) If the thickness of the capillary double layer, measured by the product (capillary radius/Debye length), exceeds ca. 7, then the associated electroosmotic flow (EOF) is essentially independent of the bulk salt concentration. (iv) The influence of the capillary on the electrophoretic behavior of a particle depends upon its point of zero charge (PZC). If the pH is lower than the PZC, because of the boundary effect, then the smaller the capillary, the smaller the particle mobility. However, this effect becomes insignificant as pH gets close to the PZC, where the mobility is governed essentially by the EOF. In this case, either the mobility can be independent of the capillary radius, or the smaller the capillary, the larger the particle mobility. (v) For the range of pH considered (3 to 8.5), a slight change in the PZC of the capillary can appreciably influence, both quantitatively and qualitatively, the particle mobility. (vi) If the capillary surface potential is low, then the particle mobility is mainly influenced by its surface charge density. However, if the double layer is sufficiently thin (bulk salt concentration sufficiently high), because of a strong EOF and the associated hydrodynamic drag, then the particle mobility is influenced significantly by the EOF at high pH. (vii) Depending upon the level of pH, the curve of particle mobility against bulk salt concentration can have a local maximum or a local minimum or G
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