Importance of Multiple Ionic Species on the Diffusiophoresis of a Rigid

Article pubs.acs.org/JPCC

Importance of Multiple Ionic Species on the Diffusiophoresis of a Rigid, Charged-Regulated, Zwitterionic Sphere Jyh-Ping Hsu* and Yu-Kui Fu Department of Chemical Engineering, National Taiwan University, Taipei, Taiwan 10617 S Supporting Information *

ABSTRACT: The importance of the presence of multiple ionic species in the liquid phase on the diffusiophoresis of a chargedregulated, zwitterionic, rigid sphere is investigated. This analysis extends previous ones, where only two kinds of ionic species were assumed to present and a particle is maintained at either constant surface potential or constant surface charge, to a more general and realistic case. We show that the diffusiophoretic behavior of the particle is influenced significantly by the solution pH, the thickness of double layer, the strength of the applied salt concentration, and both the number and the type of ionic species. In particular, because both H+ and OH− are polarized by the electrical field arising from the applied salt concentration gradient, the strength of that electrical field is reduced accordingly. This effect is especially important when pH is either very high or very low. In addition, assuming the presence of only two kinds of ionic species can yield appreciable deviation, especially when the salt concentration is low.

established local electric field points to the direction opposite to that of the applied concentration gradient. Depending on the sign of the charge of a particle, the electrophoresis effect is capable of driving it toward opposite directions.20 The diffusiophoretic velocity of a particle depends on the result of competition between chemiphoresis and electrophoresis. These effects were studied experimentally by Ebel et al.16 The diffusiophoretic behavior of a particle depends highly on its charged conditions. Previous diffusiophoresis analyses usually assumed that the particle surface is maintained at either constant charge density or constant potential.16,21,22 Although this makes mathematical treatment simpler, it can be unrealistic because the actual surface is usually somewhere between those two idealized models. SiO2 and TiO2 particles, for instance, have both acidic and basic functional groups, and their surface charge depends on the degree of dissociation/association of these functional groups, which in turn depends on the solution pH. In this case, assuming either constant surface potential or constant surface charge is inappropriate, and a charge-regulated surface,23 a more general model, which covers these two idealized models as limiting cases, needs be adopted. A considerable amount of effort has been made in the literature on the modeling of diffusiophoresis under various conditions. However, almost all of them assumed that the liquid phase contains only two kinds of ionic species. Considering the applications of diffusiophoresis, where the presence of multiple ionic species is not

1. INTRODUCTION Remarkable advances have been made in the past decade in nanotechnology and fabrication techniques, making studies and applications relevant to electrokinetic phenomena grow vigorously. These include, for example, deposition1−4 and adhesion5 of particles, biosensor, drug delivery, and DNA sequencing,6−9 to name a few. When the length scale comes to micro- or nanometer, electrokinetic phenomena such as electrophoresis, electroosmosis, and diffusiophoresis play the key role. Among these, diffusiophoresis, where particles are driven by an applied concentration gradient of electrolyte or nonelectrolyte,10−12 is interesting because the mechanisms involved can be profound. Compared with electrophoresis, diffusiophoresis has the merit that Joule heat effect is absent13 and therefore is suitable to characterize/separate temperature-sensitive entities such as biocolloids. For the case where a salt concentration gradient is applied to a charged particle, two main effects are involved in diffusiophoresis, namely, chemiphoresis and electrophoresis.14−16 Chemiphoresis comes from the nonuniform ionic distribution near the particle. Depending on the conditions assumed, two types of double-layer polarization (DLP) can be significant. Type I (II) DLP drives the particle toward the same (opposite) direction as that of the applied concentration gradient.17−19 An electrophoresis effect is present when the diffusivities of the ionic species are different. In this case, a local electric field is established and, depending on the relative magnitudes of the diffusivity of cations and that of anions, a particle can be driven toward either the direction of the applied concentration gradient or the opposite direction. For example, in an aqueous NaCl solution, because the diffusivity of Na+ is smaller than that of Cl−, the © 2012 American Chemical Society

Received: April 11, 2012 Revised: June 18, 2012 Published: June 20, 2012 15126

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(4)

∇·v = 0

uncommon, available results in the literature can be insufficient, and a more general analysis is necessary. For example, the buffer solution used to maintain the solution pH usually provides extra ionic species. Another typical example is acidic or alkalic solutions, where the presence of H+ or OH− can be significant compared with background ionic species. In this study, the diffusiophoresis of a rigid, charged-regulated, zwitterionic sphere is investigated, focusing on the importance of the presence of multiple ionic species in the liquid phase, which is the first attempt in the literature.

−∇p + η∇2 v − ρ∇ϕ = 0

(5)

In these expressions, ▽ and ▽ are the Laplace operator and the gradient operator, respectively; ρ, ε, e, kB, and T are the space charge density of mobile ions, the permittivity of the liquid phase, the elementary charge, Boltzmann constant, and the absolute temperature, respectively; zj, Dj, and nj are the valence, the diffusivity, and the number concentration of ion species j, respectively. Suppose that the concentration field surrounding the particle is only slightly disturbed by ▽n0, that is, a|▽n0| ≪ n0e.25 Then, because the electric field induced by ▽n0 is weak compared with that of the equilibrium electric field established by the particle, each dependent variable can be partitioned into an equilibrium component, the value of that variable in the absence of ▽n0, and a perturbed component coming from ▽n0.17,24 If a dependent variable with a subscript e (prefix δ) denotes the equilibrium (perturbed) component of that variable, then ϕ = ϕe + δϕ, nj = nje + δnj, p = pe + δp, and v = ve + δ v. Because the particle is stationary in the absence of ▽n0, ve = 0, yielding v = δv. To take account of the effect of DLP, nj is expressed as 2

2. THEORY The problem under consideration is illustrated in Figure 1, where we consider the diffusiophoresis of a rigid, nonconductive particle of

⎡ zje(ϕ + δϕ + g ) ⎤ e j ⎥ nj = nj0 exp⎢ − ⎢⎣ ⎥⎦ kBT

(6)

where gj is an electrochemical potential term simulating the possible deformation of the double layer surrounding a particle.26 Equations 1−6 yield27−29 ∇*2 ϕe = −

∇*2 δϕ* =

(κa)2 N

∑ j = 1 α210 (κa)2 N

∑ j = 1 α210

N

∑ α110 exp(−α100ϕ*e ) (7)

j=1 N

∑ α210(δϕ* + gj*) exp[−α100ϕe*] j=1

(8) 2

∇* g * j − α100∇*ϕe*·∇*g * j − Pej v*·∇ϕ*e = 0

(9) (10)

∇*·v* = 0

−∇*δp* + ∇*2 v* + ∇*2 ϕ*e ·∇*δϕ* + ∇*2 δϕ*·∇ϕ*e

Here ▽* = a▽, ▽*2 = a2▽2, κ = [ΣNj=1 nj0(ezj)2/εkBT]1/2, ϕ*e = ϕe/(kBT/z1e), δϕ* = δϕ/(kBT/z1e), g*j = gj(kBT/z1e), δp* = δp/pref, pref = ε(kBT/z1e)2/a2, Pej = ε(kBT/z1e)2/ηDj, U0 = εξ(kBT/z1e)2/aη, ξ = ▽*n*0, and v* = v/U0. Pej and κ are the electric Peclet number of ionic species j and the reciprocal Debye length, respectively. U0 and ξ are a reference velocity and the scaled concentration field, respectively. αimn = zljnmj0Dnj /zl1nm10Dn1, where subscript 1 denotes a reference ionic species, which can be any of the ionic species in the system. Without loss of generality, we assume that the particle surface carries zwitterionic groups AOH capable of undergoing the following dissociation/association reactions

radius a and surface Ωp subject to an applied uniform concentration field ▽n0. r, φ, θ are the spherical coordinates adopted with the origin at the center of the particle. ▽n0 is in the z direction. Let ϕ, Jj, v, and p be the electric potential, the flux of ionic species j, the fluid velocity, and the pressure, respectively. Then, the present problem can be described by17,24 N

∇2 ϕ = −

ρ = −∑ zjenj ε j=1

(1)

⎞ ⎛ zjenj Jj = nj v − Dj⎜∇nj + ∇ ϕ⎟ kBT ⎠ ⎝

(2)

∇·Jj = 0

(3)

(11)

=0

Figure 1. Diffusiophoresis of a charge-regulated zwitterionic particle of radius a and surface Ωp subject to an applied uniform concentration field ▽n0. r, φ, and θ are the spherical coordinates adopted with the origin at the center of the particle. ▽n0 is in the z direction.

AOH ↔ AO− + H+

(12)

AOH + H+ ↔ AOH 2+

(13)

Let Ka and kB be the equilibrium constants of these reactions, with Ka = NAO−[H+]s/NAOH and kB = NAOH2+/NAOH[H+]s, where NAO−, NAOH2+, and NAOH are the surface densities of AO−, AOH2+, 15127

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shear stress tensor with σH being the corresponding nonscaled value. FlexPDE, a finite-element method based software, is applied to solve the governing equations and the associated boundary conditions. Grid independence is checked during the numerical calculation to ensure the convergence of the results obtained. Typically, a total of ca. 4000 and 8000 nodes are necessary for the resolution of the flow field and the electric field, respectively. The results obtained are substituted into eqs 21 and 22 to evaluate F*ei and F*di, respectively, and χ1 and χ2 are then calculated, followed by using eq 20 to evaluate U.

and AOH, respectively, and the subscript s denotes the surface property. Equations 12 and 13 imply that depending on the level of pH the particle surface can be positively or negatively charged. If we let Ntotal be the total surface density of AOH, then Ntotal = NAOH + NAO− + NAOH2+. The spatial variation in the concentration of H+ at equilibrium is assumed to follow Boltzmann distribution, that is, [H+]s = [H+]0 exp( −ϕ*e ) +

(14)

+ 0

where [H ]s and [H ] are the surface density and the bulk H+ concentration, respectively. The surface charge density of the particle, σs, can be derived from eqs 11−14 as ⎞ ⎛ K a − Kb[H+]s2 ⎟ σs = −FNtotal ⎜ + 2 + ⎝ Kb[H ]s + [H ]s + K a ⎠

3. RESULTS AND DISCUSSION The software adopted and the solution procedure are first verified by solving the diffusiophoresis of an isolated, rigid sphere of constant surface potential, which was previously solved, both analytically (low surface potential)34 and numerically (arbitrary surface potential).19 The present result and corresponding literature results are summarized in the Supporting Information.

(15)

where F is Faraday constant.30 Note that the effects of ionic size and the associated steric restriction problem are ignored in our analysis for simplicity.31,32 Suppose that the particle is nonconductive, nonslip, and ionimpermeable, then the boundary conditions associated with eqs 7−11 on Ωp can be summarized as below n ·∇*ϕ*e = −σ *s

(16)

n ·∇*δϕ* = 0

(17)

n ·∇*g * j = 0

(18)

v* = 0

(19)

The scaled equilibrium potential ϕ*e can be obtained from eq 7. Substituting this value into eqs 8−11 yields a set of linear equations, suggesting that they can be solved indirectly by partitioned the original problem into two subproblems:17,24 (i) the particle moves with a constant velocity U in the absence of ▽n0 and (ii) ▽n0 is applied but the particle remains fixed. To evaluate the diffusiophoretic velocity of the particle, we need to evaluate the forces acting on it. These include the electrical force, Fe, and the hydrodynamic force, Fd. Note that only the z components of these forces need be considered. If we let Fi be the total force acting on the particle in the z direction in subproblem i, i = 1, 2, and Fi is its magnitude, then F1 = χ1U and F2 = χ2▽n0, where χ1 and χ2 are proportional constants. Assuming that the system is at a pseudo-steady state, we obtain F1 + F2 = 0, yielding χ U = 2 ∇n 0 χ1 (20)

Figure 2. Variation of the scaled surface potential ϕ*s as a function of pH at pKa = 7.8, pKb = 4.9, CKCl = 10−3 M, and Ntotal = 5 × 10−6 mol/m2.

If we let Fei and Fdi be the electric force and the hydrodynamic force acting on the particle in the z direction in subproblem, i, respectively, with F*ei = Fei/ε(kBT/ez1)2 and F*di = Fdi/ε(kBT/ ez1)2 being the corresponding scaled values, respectively, then33 F *ei =

F *di =

∬S

⎛⎡ ∂ϕ* ∂δϕ* ∂δϕ* ∂φe* ⎤ ⎜⎢ e + ⎥ ∂n ∂z* ⎦ * ⎝⎣ ∂n ∂z*

⎡ ∂ϕ* ∂δϕ* ∂ϕ* ∂δϕ* ⎤ ⎞ −⎢ e + e ⎥nz⎟ dS* ⎣ ∂n ∂n ∂t ∂t ⎦ ⎠

(21)

∫S* (σ *H ·n)·ez dS*

(22) Figure 3. Variations of the scaled diffusiophoretic velocity U* as a function of pH for various levels of the background salt concentrations CKCl at pKa = 7.8, pKb = 4.9, and Ntotal = 5 × 10−6 mol/m2. Solid curves: K+, Cl−, H+, and OH− are all considered; dashed curves: only K+ and Cl− are considered.

2

where S* = S/a is the scaled surface area of the particle with S being the particle surface area. ∂/∂n and ∂/∂t are the rate of change with distance along the unit normal vector n and the unit tangential vector t, respectively. σ*H = σH/(εζ2/a2) is the scaled 15128

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Figure 4. Distribution of the net scaled perturbed ionic concentration (nK+ + nH+ − nCl− − nOH−) (delta_nD) on the plane θ = π/2 along the z direction for two levels of pH at pKa = 7.8, pKb = 4.9, CKCl = 10−4 M, and Ntotal = 5 × 10−6 mol/m2. (a) pH 6 and (b) pH 3. Red lines: outer boundary of double layer.

As can be seen, the performance of the software adopted is satisfactory, and the solution procedure is appropriate. The diffusiophoretic behaviors of a particle under various conditions are examined in detail through numerical simulation by varying parameters key to the present problem. These include the pH, the thickness of double layer, the strength of the applied salt concentration gradient, and both the number and the type of ionic species. For illustration, we consider the case where TiO2

particles are dispersed in an aqueous KCl solution; that is, the liquid phase contains K+, Cl−, H+, and OH−. Assuming room temperature, the electric Peclet numbers of these ions are 0.235, 0.235, 0.050, and 0.088, respectively. For TiO2 particles, pKa = 7.8 and pKb= −4.9; therefore, the point of zero potential is pHzpc = (7.8 + 4.9) = 6.35.35 Dependence of Surface Potential on pH. Let us examine first the influence of the solution pH on the particle surface 15129

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Figure 5. Contours of the scaled disturbed electric potential δϕ* (Psi2N) for the case of Figure 4.

charge. The decrease in |ϕ*s| in the ranges pH 10 can be explained by the fact that the potential of a surface with constant charge density decreases with increasing ionic strength.30 Influence of pH. The influence of the solution pH on the diffusiophoretic behavior of a particle is shown in Figure 3, where

potential. As seen in Figure 2, the scaled surface potential ϕ*s > 0 for pH 6.35, which is expected. Note that ϕ*s has a positive local maximum at pH ∼2, and a negative local minimum at pH ∼12. This is because the reactions expressed in eqs 12 and 13 are complete at these pH values; therefore, the particle surface behaves like one with constant 15130

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Figure 6. Contours of the scaled disturbed electric potential δϕ* Z (Psi2N) on the plane θ = π/2 at two levels of pH for the case where the liquid phase contains K+ and Cl− only with pKa = 7.8, pKb = 4.9, CKCl = 10−4 M, and Ntotal = 5 × 10−6 mol/m2. (a) pH 6, (b) pH 3.

the scaled diffusiophoretic velocity U* is plotted against pH at various salt concentrations. Both the result for the case where only K+ and Cl− are considered and that where all K+, Cl−, H+, and OH− are considered are presented. Note that U* is

symmetric about pH 6.35 (pHzpc) in the former but becomes asymmetric in the latter. That is, if we consider K+ and Cl− only, then the magnitude of the particle mobility is the same, regardless of the sign of its surface potential, as long as the absolute value of 15131

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that potential is the same. However, this is not the case when K+, Cl−, H+, and OH− are all considered. As seen in Figure 3, the behavior of U* in this case is more complicated and interesting when the background electrolyte concentration of KCl is low (CKCl = 10−4 M). Here U* shows two different local maxima as pH varies, one at pH 4.8 and the other at 8.8. The presence of these local maxima comes from the competition of type I and type II DLP. In our case, the salt concentration near the top region of the particle is higher than that near its bottom region; therefore, the double layer in the former is thinner than that in the latter. Note that the amount of counterions inside the double layer on the high-concentration side is greater than that on lowconcentration side, defined as type -I DLP.20 This induces a local electric field driving the particle toward the high-concentration side, regardless of the sign of its surface charge. Figure 4 shows the contours of the scaled net perturbed ionic concentration δn = (nK+ + nH+ − nCl− − nOH−) on the plane θ = π/2 along the z direction at two levels of pH. In Figure 4a, pH 6, where the particle is positively charged, δn is negative (positive) on the high- (low-) concentration side, exemplifying the presence of type I DLP. Note that if the particle surface potential is sufficiently high, then DLP II occurs, where the amount of coions immediately outside the double layer near the highconcentration side of the particle is larger than that near its lowconcentration side. This induces a local electric field, the direction of which is opposite of that comes from type I DLP, driving the particle to the low-concentration side. Figure 4b illustrates the presence of type II DLP, where pH 3 and the particle surface potential are higher than that in Figure 4a. As seen in Figure 4b, δn has local maximum (minimum) at Z ≅ 3 (−3), the outer boundary of the double layer. The contours of the scaled disturbed electric potential δϕ* for the case of Figure 4 are presented in Figure 5. As can be seen, the |δϕ*| at pH 6 (Figure 5a) is lower than that at pH 3 (Figure 5b), implying that the effect of type II DLP in the former is less significant than that in the latter. Influence of Multiple Ionic Species. Figure 3 shows that at a low background salt concentration (CKCl = 10−4 M), if only K+ and Cl− are considered, then the scaled mobility U* increases monotonically with pH for pH >6.35. This is because if the presence of H+ and OH− is neglected, then the double-layer thickness does not vary with pH, and the κa is too small to observe type II DLP, as illustrated in Figure 6a, where pH 6; therefore, δϕ* is negative (positive) on the high- (low-) concentration side. It is interesting to note in Figure 6b that at pH 3, even the surface potential is sufficiently high; as suggested in Figure 5b, the qualitative behavior of δϕ* remains the same as that in Figure 6a, implying that the presence of type II DLP is insignificant. This indicates that if the background salt concentration is low, then the presence of H+ and OH− becomes significant. Because the qualitative behavior of a particle can be influenced, the presence of those ionic species should be considered. As seen in Figure 3, if K+, Cl−, H+, and OH− are all considered, then U* shows a local maximum as pH varies for pH >6.35. Note that the curve corresponding to CKCl = 10−4 M where K+, Cl−, H+, and OH− are all considered is asymmetric to pHzpc (6.35). This is because if K+, Cl−, H+, and OH− are all considered, the thickness of double layer (measured by κa) varies with pH, and the lower the background salt concentration the more significant the variation. As illustrated in Figure 3, if K+, Cl−, H+, and OH− are all considered, then the local maximum of U* for CKCl = 10−4 M occurs at pH ≅ 5 and 9. The ionic strength (or the level of κa) at these pH values is the same. The difference

between the levels of those two local maximum comes from the difference in the corresponding surface potentials. The absolute value of the surface potential at pH 9 (142 mV), is higher than that at pH 5 (76 mV), yielding a larger local maximum in U*. Figure 3 indicates that if only K+ and Cl− are considered, then as CKCl varies from 10−4 to 10−2 M, the U* at CKCl = 10−3 M is the smallest, but this is not the case when K+, Cl−, H+, and OH− are considered, where κa varies with pH. The former is because at that level of CKCl (κa ≅ 1) the increase in the viscous force, which reduces particle mobility, with increasing κa is more significant than that in the electric force. In general, the U* for the case where only K+ and Cl− are considered is larger than that where K+, Cl−, H+, and OH− are all considered. This is because the polarization of H+ and OH− by the electric field induced from type I DLP reduces that effect. However, if CKCl takes a medium large value and pH is either very low or very high, then the U* for the case where only K+ and Cl− are considered becomes smaller than that where K+, Cl−, H+, and OH− are all considered. This is because as pH deviates appreciably from 7, κa deviates from the value at which the viscous force acting on the particle is the most significant (where κa ≅ 1). That is, the change in double-layer thickness at CKCl = 10−3 M is crucial in the case where all four ionic species are considered. Note that in this case the mobility is larger than that in the case where only two kind of ionic species are considered, which arises from the competition between the electric force and the viscous force acting on the particle. In our study, KCl is adopted as background salt so that the effect of electrophoresis coming from the difference in ionic diffusivities is negligible and we can focus solely on the effect of the presence of multiple ionic species. If another electrolyte, NaCl for instance, is adopted, then the electrophoresis effect is important. In this case, the diffusiophoretic behavior of the particle might be influenced both quantitatively and qualitatively.20 For simplicity, the possible interactions of the background ionic species with the surface zwitterionic groups are neglected in our analysis. In a study of the influence of the ionic species on the charged properties of silica surfaces, Wang and Revil36 concluded that if the salinity exceeds 1 mM, then that influence needs be considered. In our study, TiO2 particles are considered, and, according to the results shown in Figure 3, the effect of multiple ionic species is most important when the background salt concentration is on the order of 10−4 M. If TiO2 behaves similarly as SiO2, then this implies that the interactions of the ionic species with the surface zwitterionic groups may be ignored. Nevertheless, a more rigorous analysis should take account of those interactions.

4. CONCLUSIONS The importance of the presence of multiple ionic species in the liquid phase on the diffusiophoresis of a particle is investigated by considering a charge-regulated, rigid, zwitterionic spherical particle. This extends previous analyses, where only two kinds of ionic species (cation and anion, one kind each) are considered and a particle is maintained at either constant surface potential or constant surface charge, to a more general and realistic case. Adopting TiO2 particles in an aqueous KCl solution as an example (i.e., K+, Cl−, H+, and OH− are present), we conclude the following. (1) The particle surface potential varies with pH and has both a local maximum and a local minimum at pH ∼2 and pH ∼12, respectively, where the dissociation/association reactions of the acidic/basic functional groups on the particle surface are complete. (2) The polarization of H+ and OH− by the local electric fields established by types I and II DLP reduces the influence of these two types of DLP. (3) The presence of H+ and 15132

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OH− should not be ignored, for example, when the background salt concentration is low (e.g., 10−4 M). In this case, the diffusiophoretic behavior of a particle when K+, Cl−, H+, and OH− are considered is different both quantitatively and qualitatively from that when only K+ and Cl− are considered. This is because the double-layer thickness is independent of pH in the latter and type II DLP is insignificant. (4) If the background salt concentration is low (e.g., 10−4 M), the doublelayer thickness depends highly on pH for both pH 10. The results gathered in this study provide valuable information for both interpretations of experimental data and design of diffusiophoresis devices.

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(20) Hsu, J. P.; Hsu, W. L.; Chen, Z. S. Langmuir 2009, 25, 1772−1784. (21) Ennis, J.; Anderson, J. L. J. Colloid Interface Sci. 1997, 185, 497− 514. (22) Chih, M. H.; Lee, E.; Hsu, J. P. J. Colloid Interface Sci. 2002, 248, 383−388. (23) Hsu, J. P.; Chen, Z. S.; Lee, D. J.; Tseng, S.; Su, A. Chem. Eng. Sci. 2008, 63, 4561−4569. (24) Hsu, J. P.; Lou, J.; He, Y. Y.; Lee, E. J. Phys. Chem. B 2007, 111, 2533−2539. (25) O’Brien, R. W.; White, L. R. J. Chem. Soc., Faraday Trans. II 1978, 74, 1607−1626. (26) Hunter, R. J. Foundations of Colloid Science, 2nd ed.; Oxford University Press: New York, 2001. (27) Hsu, J. P.; Liu, K. L.; Hsu, W. L.; Yeh, L. H.; Tseng, S. Langmuir 2010, 26, 16037−16047. (28) Hsu, J. P.; Hsu, W. L.; Liu, K. L. Langmuir 2010, 26, 8648−8658. (29) Hsu, J. P.; Liu, K. L.; Hsu, W. L.; Yeh, L. H.; Tseng, S. J. Phys. Chem. B 2010, 114, 2766−2778. (30) Hsu, J. P.; Tai, Y. H. Langmuir 2010, 26, 16857−16864. (31) Miklavic, S. J.; Ninham, B. W. J. Colloid Interface Sci. 1990, 134, 305−311. (32) Espinosa-Marzal, R. M.; Drobek, T.; Balmer, T.; Heuberger, M. P. Phys. Chem. Chem. Phys. 2012, 14, 6085−6093. (33) Hsu, J. P.; Chen, Z. S.; Ku, M. H.; Yeh, L. H. J. Colloid Interface Sci. 2007, 314, 256−263. (34) Keh, H. J.; Wei, Y. K. Langmuir 2000, 16, 5289−5294. (35) Fox, M. A.; Dulay, M. T. J. Photochem. Photobiol, A: Chem. 1996, 98, 91−101. (36) Wang, M.; Revil, A. J. Colloid Interface Sci. 2010, 343, 381−386.

ASSOCIATED CONTENT

* Supporting Information S

Variation of the scaled diffusiophoretic velocity U* as a function of the scaled surface potential ϕs*. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*Tel: 886-2-23637448. Fax: 886-2-23623040. E-mail: jphsu@ ntu.edu.tw. 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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dx.doi.org/10.1021/jp303499d | J. Phys. Chem. C 2012, 116, 15126−15133