Sedimentation of a pH-Regulated Nanoparticle in a Generalized

Article Cite This: J. Phys. Chem. C XXXX, XXX, XXX-XXX

pubs.acs.org/JPCC

Sedimentation of a pH-Regulated Nanoparticle in a Generalized Gravitational Field Jyh-Ping Hsu,§,# Yu-You Chu,† and Shiojenn Tseng*,‡ §

Department of Chemical Engineering, National Taiwan University of Science and Technology, Taipei, Taiwan 10607 Department of Chemical Engineering, National Taiwan University, Taipei 10617, Taiwan ‡ Department of Mathematics, Tamkang University, New Taipei City 25137, Taiwan †

ABSTRACT: Taking account of the effect of double-layer polarization, we modeled the sedimentation of a pH-regulated nanoparticle in a generalized field. The influences of the radius and the density of the functional groups of the particle, the pH, and the bulk salt concentration of the liquid phase and the Reynolds number Re on the sedimentation behavior of the particle are examined in detail. We found that as Re increases, because more counterions are dragged away from the particle surface, its averaged charge density decreases accordingly. The smaller the particle, the thicker the double layer so that double-layer polarization is more significant and the distribution of the surface charge density is more nonuniform. Interestingly, the smaller the particle, the higher the averaged surface charge density, but the smaller the electric force acting on it. If the particle is sufficiently large (300 nm in radius), its averaged surface charge density is insensitive to Re as it varies from 0.0025 to 0.04 but can have an appreciable difference (ca. 10%) if it is small (75 nm).

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particle as it settles with or without an applied electric field. In the case where an electric field is applied,8 they found that the more seriously the counterions are dragged away from a particle the more significant the influence of the electric force acting on its behavior. In a study of the electrokinetic behavior of a charged particle Sonnefeld et al.12 found that its surface charge density increases with increasing pH for pH ranging from 4 to 8. Applying this result, Atalay et al.13 analyzed the interaction between a charged nanoparticle and a charged plane. It was found that if the particle is sufficiently close to the plane the overlapping of their double layers yields an asymmetric distribution in H+, resulting in a significant reduction in the surface charge density in the interaction region. Keller et al.14 investigated the sedimentation of a rigid sphere having a constant surface potential in a salt solution subject to a generalized gravitational field. It was found that the electric force acting on the particle has a local maximum as Reynolds number varies. This was attributed to the asymmetric distribution of counterions near the particle. Based on a perturbation approach, Keh and Ding15 derived the dependence of the sedimentation velocity of a rigid sphere having a constant surface charge density in an aqueous, Newtonian salt solution on the thickness of double layer. Taking account of the effect of double-layer relaxation, Yeh et al.16 modeled the settling of a deformable polyelectrolyte in an unbounded salt solution subject to an applied centrifugal field. The sedimentation behavior of the polyelectrolyte was explained by its effective charge density and the local electric field induced near it. They showed that the

INTRODUCTION The sedimentation of colloidal particles in an applied field (e.g., gravitational and centrifugal) has been applied to various areas of practical significance, including, for instance, wastewater treatment,1 space propellant recognition, and winemaking. Applying a molecular dynamics approach, Whitmer et al.2 studied the sedimentation crystallization of colloidal particles; the influences of the gravity, the attractive interaction between aggregating colloids, and their Brownian motion were discussed. They concluded that a moderate level of attractive interaction can enhance the rate of crystallization. Buzzaccaro et al.3 analyzed the sedimentation kinetics of polymeric colloids. Through a novel optical method, they measured the sedimentation profile of dispersions having a very low turbidity. Based on a Stokes approximation and diffusion along the negative gradient of concentration, Song et al.4 modeled the sedimentation of particles and colloidal aggregates taking account of both streaming and seepage. The results obtained explained why a suspension having a high particle concentration is more unstable. The sedimentation behavior of a charged colloidal particle is more complicated than that of an uncharged particle. In the former, the double layer surrounding a particle deforms as it moves, yielding a local electric field which tends to retard its movement. If the particle surface is charge-regulated, its charged conditions are influenced by several factors including, for example, the solution pH and the bulk salt concentration.5,6 The distribution of the surface charge of a particle might also be affected by its movement.7 Taking account of counterion condensation and double-layer polarization, Bhattacharyya et al.8−11 evaluated the electric and the hydrodynamic forces acting on a charged, rigid, or porous © XXXX American Chemical Society

Received: July 13, 2017 Revised: August 24, 2017

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DOI: 10.1021/acs.jpcc.7b06909 J. Phys. Chem. C XXXX, XXX, XXX−XXX

Article

The Journal of Physical Chemistry C

Figure 1. Sedimentation of a rigid, pH-regulated sphere of radius Rp and surface Ωp subject to a generalized gravitational field Cg with g being the gravitational acceleration. For numerical solution purposes, a large cylindrical computation domain of radius b, length l, and surfaces Ωt, Ωl, and Ωb is defined, and the cylindrical coordinates (r, θ, z) are chosen with the origin at the particle center.

that the sedimentation and diffusion lead to differences in the degree of association of particles and cells. A decrease in the particle size minimizes the effect of sedimentation on cellular dose or the association of particles and cells. Caruso et al.20 investigated the influence of sedimentation and particle shape on the interaction between cells and particles in a dynamic flow. They showed that this flow minimizes the influence of sedimentation. In this study, the sedimentation of a rigid, pH-regulated sphere in an applied general gravitation field is modeled focusing on the dependence of its velocity on the solution pH, the bulk salt concentration, and the associated Reynolds number. In addition, the variation of the charged conditions of the particle with its sedimentation velocity is examined for the first time. This extends previous studies, where a constant potential/ charge density or a uniform charge distribution is almost always assumed.

Figure 2. Variation of the surface charge density of an isolated, spherical silica particle of radius 58 nm in an aqueous KCl solution with pH at Cbulk = 100 mM, Ntotal = 2.1 sites/nm2, pKA = 6.38, and pKB = 1.87. Discrete symbols: experimental data of Sonnefeld et al.;12 curve: present numerical result.

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THEORY Let us consider the sedimentation of a pH-regulated, isolated spherical particle of radius a and surface Ωp in an aqueous salt solution subject to an applied field Cg shown in Figure 1, where g denotes the gravitational acceleration. For numerical solution purposes, we defined a large cylindrical computation domain of radius b and length l. The cylindrical coordinates (r, θ, z) are adopted with its origin at the particle center. Ωt, Ωb, and Ωl denote the top, bottom, and lateral surfaces of the computation domain, respectively. Suppose that the liquid phase is an incompressible fluid and the system is at a pseudosteady state. If we let ϕ, u, and p be the

sedimentation behavior of a loosely structured polyelectrolyte is dominated by fluid convection. In addition, the shape of a polyelectrolyte is capable of influencing its behavior through affecting the amount of counterions attracted into its interior. Adopting a cell model, Lee et al.17 analyzed the sedimentation of a concentrated dispersion of charge-regulated colloidal particles focusing on the effects of the density of the dissociable functional groups on the particle surface, the degree of their dissociation, and the volume fraction of the particle. In an experimental study of the uptake of gold nanoparticles Cho et al.18 showed that the amount of uptake depends upon the rate of sedimentation of those particles. Cui et al.19 demonstrated

Figure 3. Variation of the scaled electric force (Fe/Fe,ref) acting on a rigid TiO2 particle of radius 300 nm, zeta potential 76.2 mV, and density ρp = 4000 kg/m3 in an aqueous KCl solution with Cbulk = 1.04 × 10−3 mM (a) and the ratio of the forces (electric force/hydrodynamic force) = (Fe/Fh) (b) with the reference Reynolds number Re. Discrete symbols: numerical result of Keller et al.;14 curve: present numerical results. B


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Figure 4. Variation of the averaged surface charge density σP (a) and the scaled electric force acting on the particle Fe* (b) with the bulk salt concentration Cbulk for various values of Ntotal at Re = 0.01 and pH 8. Curve 1: Ntotal = 5 × 10−7 mol/m2; 2: Ntotal = 7.5 × 10−7 mol/m2; 3: Ntotal = 10−6 mol/m2; 4: Ntotal = 1.25 × 10−6 mol/m2; 5: Ntotal = 1.5 × 10−6 mol/m2; 6: Ntotal = 2 × 10−6 mol/m2. N

μ∇2 u − ∇p − (∑ FzjCj)∇ϕ = 0 (2)

j=1

(3)

∇·u = 0 j = 1, 2, 3, 4

(4)

⎛ ⎞ FzjCj ∇·Nj = ∇·⎜Cj u − Dj∇Cj − Dj ∇ ϕ⎟ RT ⎝ ⎠

(5)

∇·Nj = 0,

Here, ∇, ∇ , j, ρe, e, ε, μ, F, R, and T are the gradient operator, Laplace operator, the number of ionic species, the space density of mobile ions, the elementary charge, the fluid permittivity, the fluid viscosity, Faraday constant, the gas constant, and the absolute temperature, respectively. zj, Cj, Nj, and Dj are the valence, the molar concentration, the flux, and the diffusivity of ionic species j, respectively. For convenience, the Reynolds number is defined as Re = 2RPρuref/μ, where uref = 2RP2(ρp − ρ)Cg/9μ is the terminal velocity based on Stokes’ law with ρp and g being the particle density and the magnitude of g, respectively. The particle surface bears functional groups MOH of surface densities Ntotal capable of undergoing the following dissociation/ association reactions: 2

Figure 5. Contours of the concentration difference (CK+ − CK+,bulk) (mol/m3) on the half plane θ = 0 at Cbulk = 0.1 mol/m3, Re = 0.01, pH 8, Ntotal = 10−6 mol/m2, and Rp = 300 nm.

electric potential, the fluid velocity, and the pressure, respectively, then the present problem can be described by 2

−∇ ϕ =

ρe ε

N

=

∑ j=1

FzjCj ε

MOH ⇔ MO− + H+

(6)

MOH 2+ ⇔ MOH + H+

(7)

The corresponding equilibrium constants are KA = NMO− [H+]S/ NMOH and KB = NMOH [H+]S/NMOH2+, where [H+] (mol/m2) is the

(1)

Figure 6. Variation of the averaged surface charge density of a particle σP (a) and the magnitude of the scaled electric force acting on it |FE*| (b) with the bulk salt concentration Cbulk for various levels of pH at Re = 0.01 and Ntotal = 10−6 mol/m2. Curve 1: pH 5; 2: pH 5.5; 3: pH 6; 4: pH 7; 5: pH 7.5; 6: pH 8. C


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Figure 7. Variation of the averaged surface charge density σP (a) and the scaled electric force acting on the particle Fe* (b) with the Reynolds number Re for various values of Ntotal at Cbulk = 0.1 mol/m3 and pH 8. Curve 1: Ntotal = 5 × 10−7 mol/m2; 2: Ntotal = 7.5 × 10−7 mol/m2; 3: Ntotal = 10−6 mol/m2; 4: Ntotal = 1.25 × 10−6 mol/m2; 5: Ntotal = 1.5 × 10−6 mol/m2; 6: Ntotal = 2 × 10−6 mol/m2.

Figure 8. Variation of the averaged surface charge density of a particle σP (a) and the magnitude of the scaled electric force acting on it |FE*| (b) with the bulk salt concentration Cbulk for various levels of pH at Cbulk = 0.1 mol/m3 and Ntotal = 10−6 mol/m2. Curve 1: pH 5; 2: pH 5.5; 3: pH 6; 4: pH 7; 5: pH 7.5; 6: pH 8.

surface molar concentration of H+. Because Ntotal = NMO− + NMOH + NMOH+2 , the surface charge density of the particle, σP (C/m2), is 2 ⎛ ⎞ KA − [H+]S /KB ⎟⎟ σP = FNtotal ⎜⎜ + + ⎝ KA + [H ]S + [H ]S /KB ⎠

are reliable and sufficiently accurate. Usually, using ca. 200,000 mesh elements is appropriate.

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CODE VALIDATION To verify the applicability of the solution procedure adopted, we first simulate the variation of the surface charge density of an isolated, spherical silica of Rp = 58 nm with pH. The result shown in Figure 2 indicates that the surface charge density increases with increasing pH, and the quantitative values agree well with the experimental data reported by Sonnefeld et al.12 The second example is the sedimentation of an isolated rigid TiO2 particle in an aqueous KCl solution solved numerically by Keller et al. Figure 3 summarizes the variation of the scaled electric force (Fe/Fe,ref) acting on the particle and the ratio of the forces (electric force/hydrodynamic force) = (Fe/Fh) with the reference Reynolds number Re. As can be seen in this figure, our model is capable of describing the behavior of both (Fe/Fe,ref) and (Fe/Fh).

(8)

For the case where the solution pH is adjusted by HCl and KOH the following conditions must satisfy: [H+]0 = 10−pH, [OH−]0 = 10−(pKw−pH), [K+]0 = CKCl, and [Cl−]0 = CKCl + 10−pH − 10−(pKw−pH) for pH < pKw/2; [H+]0 = 10−pH, [OH−]0 = 10−(pKw−pH), [K+]0 = CKCl + 10−(pKw−pH) − 10−pH, and [Cl−]0 = CKCl for pH ≥ pKw/2. The square brackets denote the molar concentration of a species and the subscripts the bulk value. Suppose that the electric potential and the pressure at a point sufficiently far away from the particle are uninfluenced by the particle. Therefore σp n ·∇ϕ = − onΩ p (9) ε n ·Nj = 0 on Ω p and Ω l

(10)

ϕ = 0 on Ω b and Ω t

(11)

Cj = Cj0 on Ω l and Ω b and Ω t

(12)

u = −uref ez on Ω l

(13)

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NUMERICAL SIMULATION To examine the behavior of the particle under various conditions, a numerical simulation is conducted by varying its functional group density, the solution pH, the bulk salt concentration, and the Reynolds number, which is regulated by the applied external field. For illustration, we assume b = 7500 nm and l = 15 000 nm. We consider a TiO2 particle with Rp = 300 nm, ρp = 4000 kg/m3, pKA = 7.8, pKB = −4.95, and T = 300 K. We assume that the liquid phase is an aqueous KCl solution so that DH+ = 9.3 × 10−9 m2/s, DOH− = 5.3 × 10−9 m2/s, DK+ ≅ DCl− ≅ 2 × 10−9 m2/s, ε = 6.9505 × 10−12 (F/m), μ = 8.91 × 10−4 kg/(ms), and F = 96485 C/mol. For convenience, we define the scaled electric

n is the unit outer normal vector of Ωp, Cj the ionic concentration, and ez the unit vector in the z direction. COMSOL MulitiPhysics (version 4.3a, www.comsol.com) is adopted to solve the present problem numerically. Mesh independence is checked to ensure that the results obtained D


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Figure 9. Variations in the surface charge density of a particle σP with Re at Ntotal = 10−6 mol/m2, Rp = 300 nm, and pH 8. The levels of Cbulk are 10, 1, 0.1, and 0.01 mol/m3 in (a), (b), (c), and (d), respectively. (e) Variation of the percentage difference in the averaged surface charge density

D=

σP(Re = 0.0025) − σP(Re = 1) σP(Re = 0.0025)

× 100%.

force acting on the particle Fe* = Fe/Ee,ref, where Fe,ref = εϕref is a reference electric force.

charged, a decrease in [H+] yields a higher negative charge density. On the other hand, if the particle is positively charged, [Cl−] increases with increasing Cbulk and [H+] increases accordingly. Equation 7 indicates that its averaged surface charge density also increases. We conclude that regardless of the sign of σP its magnitude always increases with increasing Cbulk. The behavior of the curves in Figure 6(b) is similar to that in Figure 4(b), except that the local maximum of |FE*| occurs at Cbulk ≅ 0.1 mol/m3 for all the levels of pH examined. As expected, the more the pH deviates from the isoelectric point, the greater the |FE*|. Influence of Reynolds Number. Figure 7 demonstrates that the Reynolds number has an impact on the averaged surface charge density and the scaled electric force. As can be seen in Figure 6(a), the averaged surface charge density has only a slight increase responding to the increase of Re, and this phenomenon will be discussed later. Being different from the averaged surface charge density, the scaled electric force makes an obvious increase with respect to the enlarging Re. The larger |FE*| results from two factors: the large electric potential gradient and the large space which contains the steep potential gradient. In this case, the increasing Re contributes to the higher amount of the counterions that is dragged away, causing the potential gradient to get larger, which results in a larger |FE*|.

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RESULTS AND DISCUSSION Influence of Bulk Salt Concentration. The influence of the density of the functional group of the particle, Ntotal, on its averaged surface charge density, σP, and that on the scaled electric force acting on it, Fe*, at pH 8 are presented in Figure 4. Since the isoelectric point of the point is 6.375, the particle is negatively charged in this case. Figure 4 reveals that the larger the Ntotal the higher the σP, which is expected because the larger the Ntotal the larger amount of ions can be dissociated. Note that Fe* has a local maximum as Cbulk varies. For Ntotal = 2 × 10−6 mol/m2 this local maximum occurs at Cbulk ≅ 1 mol/m3, while it occurs at Cbulk ≅ 0.1 mol/m3 for other values of Ntotal. This arises from the effect of double-layer polarization, which induces a local electric field retarding the particle movement. The asymmetric ionic distribution shown in Figure 5 illustrates the polarized double layer. Figure 6 reveals that the averaged surface charge density of the particle σP is influenced by the bulk salt concentration Cbulk. This can be explained by that the higher the Cbulk the higher the [K+], so that [H+] is lower. The reaction expressed in eq 6 tends to shift toward its right-hand side. Therefore, if the particle is negatively E


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Figure 10. Variations of the surface charge density of a particle σP with Re at Ntotal = 10−6 mol/m2, Rp = 300 nm, and Cbulk = 0.1 mol/m3. The levels of pH are 5, 6, 7, and 8 in (a), (b), (c), and (d), respectively. (e) Variation of the percentage difference in the averaged surface charge density

D=

σP(Re = 0.0025) − σP(Re = 1) σP(Re = 0.0025)

× 100%.

Figure 11. Variation of the surface charge density of a particle σP (a) and the scaled electric force acting on it FE* (b) with the Reynolds number Re for various values of particle radius Rp at Cbulk = 0.1 mol/m3, Ntotal = 10−6 mol/m2, and pH 8. Curve 1: Rp = 75 nm; 2: Rp = 100 nm; 3: Rp = 150 nm; 4: Rp = 200 nm; 5: Rp = 300 nm.

present the variation of the surface charge density with pH at Re = 0.01 here. This is because the value of the averaged surface charge density is almost the same in this range. It should be noticed that when we expand the upper limit to Re = 1 the results will be completely different. The influence of the Reynolds number on the averaged surface charge density of a particle σP at various levels of the bulk salt

Figure 8(b) shows the variations of the magnitude of the scaled electric force, |FE*|, with Re for various values of pH, which reveals that the more the pH deviates from the isoelectric point the greater the |FE*|, which has been discussed in Figure 6(b). The behavior of |FE*| corresponding to Re is similar to that of Figure 7(b). We choose to present the variation of the averaged surface charge density for different values of Ntotal, while we only F


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Figure 12. Contours of the concentration difference (CK+ − CK+,bulk) (mol/m3) on the half plane θ = 0 at Cbulk = 0.1 mol/m3, Re = 0.01, pH 8, and Ntotal = 10−6 mol/m2. (a) Rp = 75 nm and (b) Rp = 300 nm.

Figure 13. Contours of the electric potential ϕ (V) on the half plane θ = 0 at Cbulk = 0.1 mol/m3, Re = 0.01, pH 8, and Ntotal = 10−6 mol/m2. (a) Rp = 75 nm and (b) Rp = 300 nm.

Figure 14. Distributions of the surface charge density σP along the particle surface from its south pole (s = 0) to its north pole (s = π) for various levels of Rp at Cbulk = 0.1 mol/m3, Ntotal = 10−6 mol/m2, and pH 8. Curve 1: Rp = 75 nm; 2: Rp = 100 nm; 3: Rp = 150 nm; 4: Rp = 200 nm; 5: Rp = 300 nm.

Figure 15. Distributions of the surface charge density σP along the particle surface from its south pole (s = 0) to its north pole (s = π) for various levels of Re at Cbulk = 0.1 mol/m3, Ntotal = 10−6 mol/m2, Rp = 75 nm, and pH 8. G


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Figure 16. Variations of the averaged surface charge density of a particle σP (a) and the magnitude of the scaled electric force acting on it Fe* (b) with the density of the functional groups Ntotal for various levels of Re at pH 8 and Cbulk = 0.1 mol/m3. Curve 1: Re = 0.01; 2: Re = 0.02; 3: Re = 0.03; 4: Re = 0.04.

Figure 17. Variations of the averaged surface charge density of a particle σP (a) and the scaled electric force acting on it Fe* (b) with the density of the functional groups Ntotal for various values of the bulk salt concentration Cbulk at Re = 0.01 and pH 8. Curve 1: Cbulk = 0.1 mol/m3; 2: Cbulk = 1 mol/m3; 3: Cbulk = 5 mol/m3; 4: Cbulk = 10 mol/m3.

Figure 10(e). Since the isoelectric point of the particle is 6.375, it is positively charged in Figure 10(a) and (b). In this case, the lower the pH, the larger the D is. In contrast, the particle is negatively charged in Figure 10(c) and (d), where the higher the pH the smaller the D. As can be seen in Figure 10(e), D decreases with increasing pH with the largest D of 11.20% occurring at pH 5. Influence of Particle Radius. The results illustrated in Figure 11 suggest that the smaller the particle the higher its averaged surface charge density. This can be explained by the difference between the concentration of K+ near the particle and the corresponding bulk concentration of K+ (CK+ − CK+,bulk) shown in Figure 12 for two particle radii. This figure reveals that the distribution of (CK+ − CK+,bulk) for a smaller particle (Rp = 75 nm) is more nonuniform than that for a larger particle (Rp = 300 nm). In addition the value of (CK+ − CK+,bulk) at Rp = 75 nm is larger than that at Rp = 300 nm, implying that the [H+] of the former is lower than that of the latter. Therefore, according to eq 6, the surface charge density at Rp = 75 nm is higher than that at Rp = 300 nm, which is seen in Figure 11(a). The percentage difference in the averaged surface charge density as Re varies

Figure 18. Variations of the magnitude of the scaled electric force acting on a particle |Fe*| with pH for various levels of Re at Ntotal = 10−6 mol/m2 and Cbulk = 0.1 mol/m3. Curve 1: Re = 0.01; 2: Re = 0.02; 3: Re = 0.03; 4: Re = 0.04.

concentration Cbulk is summarized in Figure 9(a)−(d). As mentioned previously, the degree of drag away of the counterions near a particle increases with increasing Re, yielding a lower σP. This phenomenon is more significant at lower bulk salt concentration. Figure 9(e) shows the variation of the percentage difference in the averaged surface charge density D =

σP(Re = 0.0025) − σP(Re = 1) σP(Re = 0.0025)

from 0.0025 to 0.04, D′ =

σP(Re = 0.0025) − σP(Re = 1) σP(Re = 0.0025)

× 100%,

for Rp = 75 nm is 8.50% and is 0.96% for Rp = 300 nm. The qualitative behavior of the scale electric force acting on a particle shown in Figure 11(b) is different from that of σP seen in Figure 11(a). The former reveals that the smaller the particle the smaller the electric force acting on it. As shown in Figure 13, this arises because the smaller the particle, the lower the surface potential. The distributions of the surface charge density σP along the particle surface from its south pole (s = 0) to its north pole (s = π)

× 100%.

This figure reveals that the lower the Cbulk the larger the D. At Cbulk = 0.01 mol/m3, D = 22.96%. The variations of the Reynolds number on the averaged surface charge density of a particle σP at various levels of pH are presented in Figure 10(a)−(d), and the variation in the percentage difference in the surface charge density D=

σP(Re = 0.0025) − σP(Re = 0.04) σP(Re = 0.0025)

× 100% with pH is shown in H


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Figure 19. Variations of the averaged surface charge density of a particle σP (a) and the magnitude of the scaled electric force acting on it |Fe*| (b) with pH for various values of the bulk salt concentration Cbulk at Re = 0.01 and Ntotal = 10−6 mol/m2. Curve 1: Cbulk = 0.01 mol/m3; 2: Cbulk = 0.1 mol/m3; 3: Cbulk = 1 mol/m3; 4: Cbulk = 5 mol/m3; 5 Cbulk = 10 mol/m3.

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CONCLUSIONS Taking account of the effect of double-layer polarization, the sedimentation of a pH-regulated nanoparticle subject to a generalized gravitation field is modeled theoretically. The radius and the density of the functional groups of the particle, the pH, and the bulk salt concentration Cbulk of the liquid phase and the Reynolds number Re are examined for their influences on the sedimentation behavior of the particle. We show that as the bulk salt concentration varies the sedimentation velocity of the particle has a local maximum occurring at Cbulk ≅ 0.1 mol/m3. This arises from the dependence of the concentration of H+ and, therefore, the charge density of the particle, with Cbulk. As Re increases, more counterions are dragged away from the particle, yielding a greater electric force acting on it, but its averaged surface charge density becomes smaller.

for various values of particle size Rp are presented in Figure 14, and those for various values of the Reynolds number Re are presented in Figure 15. Figure 14 reveals that the smaller the particle, the more nonuniform the surface charge distribution. For example, the percentage differences in the surface charge density, D″ =

σP(s = 0) − σP(s = π ) σP(s = 0)

× 100%, for Rp = 75, 100, 150,

200, and 300 nm are 3.07, 2.33, 1.83, 1.25, and 0.95%, respectively. This can be explained by the smaller the particle, the more significant the degree of double-layer polarization. The amount of counterions dragged away from the double layer of the particle increases with increasing Re, resulting in a larger D″, as seen in Figure 15. Influence of Density of Functional Groups. The influences of the density of the functional groups of a particle Ntotal on its averaged surface charge density σP and the magnitude of the scaled electric force acting on it Fe* are presented in Figure 16(a), indicating that the larger the Ntotal the higher the σP, which is expected since the larger the Ntotal the greater the number of functional groups available for dissociation. The trend of Fe* as Re increases seen in Figure 16(b) can be explained by the larger the Re the greater the amount of ions that is dragged away from the particle. Figure 17(a) reveals that the averaged surface charge density of a particle σP increases with increasing bulk salt concentration Cbulk. The rationale behind this is mentioned in the discussion of Figure 6. Note that in Figure 17(b) the curve of Fe* at Cbulk = 0.1 mol/m3 intersects with that at Cbulk = 1 mol/m3. This can be explained by Figure 4(b), where the Cbulk at which the local maximum of Fe* occurs shifts from ≅0.1 mol/m3 to ≅1 mol/m3 as Ntotal increases from 5 × 10−7 mol/m2 to 2 × 10−6 mol/m2. Influence of pH. The behavior of the magnitude of the scaled electric force acting on a particle as pH varies shown in Figure 18 is similar to that of the averaged surface charge density seen in Figure 8(a). This is expected because the more the pH deviates from the isoelectric point of the particle the higher its charge density and, therefore, the greater the electric force acting on it. As the Reynolds number Re is raised from 0.01 to 0.04, the amount of the counterions dragged away from the particle increases accordingly, resulting in a stronger potential gradient and, therefore, a stronger electric force. The results illustrated in Figure 19 are consistent with those shown in Figure 18; that is, the more the pH deviates from the isoelectric point of a particle the higher its charge density and the greater the electric force acting on it. As observed in Figure 6(b), |Fe*| shows a local maximum at Cbulk ≅ 0.1 mol/m3.

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AUTHOR INFORMATION

Corresponding Authors

*Tel.: 886-2-26215656 ext. 2508. E-mail: [email protected]. edu.tw. ORCID

Jyh-Ping Hsu: 0000-0002-4162-1394 Shiojenn Tseng: 0000-0001-6679-6337 Present Address #

Department of Chemical Engineering, National Taiwan University, Taipei, Taiwan 10617.

Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work is supported by the Ministry of Science and Technology, Republic of China. REFERENCES

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Sedimentation of a pH-Regulated Nanoparticle in a Generalized

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