Diffusiophoresis of Polyelectrolytes in Nanodevices: Importance of

Article pubs.acs.org/JPCC

Diffusiophoresis of Polyelectrolytes in Nanodevices: Importance of Boundary Jyh-Ping Hsu,† Kuan-Liang Liu,† and Shiojenn Tseng*,‡ †

Department of Chemical Engineering, National Taiwan University, Taipei, Taiwan 10617 Department of Mathematics, Tamkang University, Tamsui, Taipei, Taiwan 25137

‡

S Supporting Information *

ABSTRACT: Considering the potential applications of diffusiophoresis conducted in nanodevices, a thorough theoretical analysis is performed for the first time on the diffusiophoresis of a polyelectrolyte (PE) in the presence of two representative types of boundaries: the direction of diffusiophoresis is either normal (type I) or parallel (type II) to a boundary, using two large parallel disks and a cylindrical pore as an example, respectively. It is interesting to observe that due to the effects of double-layer polarization, counterion condensation, polarization of condensed counterions, and diffusion of co-ions across a PE, its diffusiophoretic behavior can be influenced both quantitatively (magnitude of mobility) and qualitatively (direction of diffusiophoresis) by a boundary. In general, type I (II) boundary raises the diffusiophoretic mobility of a PE toward the high (low) salt concentration side. The results gathered provide necessary information for applications in, for example, designing catalytic swimmers and nanopore-based biosensor devices.

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INTRODUCTION Diffusiophoresis,1−3 the motion of colloid particles driven by an externally applied solute concentration gradient, is applied widely in processes such as surface adhesion and coating,4−7 separation, and purification.8−12 It is also a significant mechanism in biological environment, known as chemotaxis, the migration of cells and microorganisms as a response to a local concentration gradient of nutrients, toxic species, or other signal materials.13−16 Owing to recent remarkable advances in nanotechnology, diffusiophoresis also finds potential applications in various fields. For example, experimental evidence revealed that an applied solute concentration gradient is capable of driving catalytic swimmer (nanomotors),17−19 and regulating novel electrophoresis based biosensor devices.20,21 Many attempts, both theoretical22−25 and experimental,2,26−28 were made on the analysis of the diffusiophoresis of charged particles in an electrolyte medium subject to an applied salt concentration gradient. In this case, several mechanisms are involved. First, the electrostatic interaction between the applied salt concentration gradient and the double layer surrounding a particle yields a local electric field EDLP, known as double-layer polarization (DLP).22,26,29 EDLP consists of EDLPI and EDLPII, coming respectively from type I DLP and type II DLP. EDLPI (EDLPII) occurs mainly inside (outside) the double layer, driving the particle toward the high (low) salt concentration side. This mechanism of DLP is most important when the thickness of double layer is comparable to the particle size. The net effect of EDLP depends upon the physical properties of the particle, the bulk salt concentration, and the presence of a boundary.22,23,29,30 Second, the difference in the ionic diffusivities induces a background electric field, Ediffusivity.26,29−31 The direction of Ediffusivity can be either toward © 2013 American Chemical Society

the high or low salt concentration side, depending upon the electrolyte. Third, for the case where diffusiophoresis is conducted along a narrow space the diffusion of co-ions from the high salt concentration side to the low salt concentration side might induce a local electric field ECO, which drives a particle toward the low salt concentration side.32−34 To mimic the electrophoretic behavior of a polyelectrolyte (PE) such as polypeptides, DNA, and DNA-complex, Hermans and Fujita35,36 proposed using an entirely porous particle. Although this kind of particle seems to be a limiting case of a soft particle, which comprises a rigid core and a porous membrane layer,23,25,33,37,38 the electrokinetic behavior between the two can be different both quantitatively and qualitatively. This is because factors such as counterion condensation, electroosmotic flow, and electrostatic relaxation for an entirely porous particle are much more significant than those for a soft particle.29,39−42 Compared to the efforts made on the analysis of PE electrophoresis,42−44 those of PE diffusiophoresis are very limited. Assuming low fixed charge density, Wei and Keh45 studied the diffusiophoresis of a PE molecule. Liu et al.29 extended their analysis to the case of arbitrary charge density. For the first time, they demonstrated that, in addition to the conventional effects coming from EDLP and Ediffusivity, the polarization of the condensed counterions inside a PE and the relaxation effect might also be significant if its fixed charge density is high. However, none of these analyses considered the Received: December 24, 2012 Revised: April 6, 2013 Published: April 10, 2013 9469

dx.doi.org/10.1021/jp312702r | J. Phys. Chem. C 2013, 117, 9469−9476

The Journal of Physical Chemistry C

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the dielectric constant ε and the viscosity η of the liquid phase inside the PE are the same as those outside it.48−50 Because the present problem is θ symmetric, only (r,z) domain needs be considered. Suppose that the concentration field inside and near the PE is only slightly distorted by ▽n0, that is, a|▽n0| ≪ n0e,26 with n0e being the bulk salt concentration in the absence of ▽n0. Then, the electric potential, ψ(r,z), the number concentration of ionic species j, nj(r,z), the fluid velocity v(r,z), and the pressure, p(r,z) after ▽n0 is applied can all be partitioned into an equilibrium component, the value of that variable in the absence of ▽n0, and a perturbed component arises from ▽n0.22−24 Let subscript e and prefix δ denote the equilibrium and the perturbed components, respectively, then ψ = ψe + δψ, nj = nje + δnj, v = ve + δv, and p = pe + δp. These independent variables can be described by a continuum-based model comprising of a set of Poisson−Boltzmann-Planck equations and modified Stokes equations. Poisson-Nernst−Planck Equations. The equilibrium concentration of ionic species j can be described by the Boltzmann distribution,51 nje = nj0e exp(−zjeψe/kBT) with nj0e, e, kB, and T being the equilibrium bulk concentration of ionic species j, the elementary charge, Boltzmann constant, and the absolute temperature, respectively. When ▽n0 is applied, nj no longer follows that distribution. To reflect this, we let nj = nj0e exp[−zje(ψe + δψ + gj)/kBT], where gj is a hypothetical potential function simulating the possible deformation of double layer.29,33,52 In addition, instead of solving δnj directly, gj is solved.29,44 Once δψ and gj are obtained, δnj can be evaluated by δnj = nj0e exp[−zje(δψ + gj)/kBT], which measures the degrees of type I DLP, type II DLP, electric relaxation, and polarization of condensed counterions.29 It can be shown that in our case the electric field is described by29,53

influence of boundary, which is significant when diffusiophoresis is conducted in a narrow space. In practice, the influence of a boundary on the diffusiophoretic behavior of a particle can be classified roughly into two categories: the direction of diffusiophoresis (or applied salt concentration gradient) is either normal or parallel to a boundary. In the study of Sen et al.,17 for example, nanomotors (Pt−Au rods) are driven by a hydrogen peroxide (H2O2) concentration gradient established by H2O2-rich gel. In this case, the gel surface can be viewed as a boundary normal to the concentration gradient. Another typical example for the presence of a normal boundary is in surface adhesion/coating application of diffusiophoresis. It was proven that applying a salt concentration gradient is capable of enhancing both the capture efficiency and current signal recognition20,21 in electrophoresis based biosensor devices, where the ionic current blockade can be measured when a DNA molecule passes through a nanopore.46,47 In this application of diffusiophoresis, the boundary (nanopore wall) is parallel to the direction of diffusiophoresis. The above discussions suggest that a thorough analysis on the diffusiophoresis of a PE in the presence of a boundary is necessary. In particular, both a normal boundary and a parallel one should be considered. This is done in the present study, where the dependence of the diffusiophoretic behavior of a PE and the associated mechanisms on factors such as the PE charge density, the double layer thickness (or bulk salt concentration), and the PE−boundary distance are examined in detail.

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THEORETICAL MODEL Let us consider the steady migration of a charged spherical PE of radius a and outer boundary Ωp driven by an applied salt concentration gradient ▽n0. Two typical cases are modeled: the diffusiophoresis of a PE is either normal or parallel to a boundary. As illustrated in Figure 1a, ▽n0 is normal to two

(κa)2 * [exp(−ψe*) − exp(αψe*)] − iρfix Δ*ψe* = − 1+α

(1)

(κa)2 [exp(−ψe*)(δψ * + g1*) + α exp(αψe*) Δ*δψ * = − 1+α (δψ * + g *)] (2) 2

Δ*g1* = ∇*ψe*·∇*g1* + γm1 v *·∇*ψe*

(3)

Δ*g2* = −α∇*ψe*·∇*g2* + γm2 v *·∇*ψe*

(4)

Here, α = −z2/z1, ψe* = z1eψe/kBT, δψ* = z1eδψ/kBT, and gj* = z1egj/kBT. Δ* = a2Δ and ▽* = a▽ are the scaled Laplace and gradient operators, respectively. i is a region index: i = 1 and 0 denote the region inside and outside the PE, respectively. κ = {∑j2= 1[nj0e(ezj)2/εkBT]}1/2 is the reciprocal Debye length. ρfix *= ρfixa2z1e/εkBT is the scaled fixed charge density of the PE. γ = ∇*n*0 is the scaled applied salt gradient. mj = ε(kBT/ez1)2/ηDj is a dimensionless number measuring the relative significance of ion transports due to electro-migration and osmosis.54 n0* = n0/ n0e, and Dj is the diffusivity of ionic species j. Modified Stokes Equations. Suppose that as the liquid flows through the PE, it experiences an extra friction force characterized by a uniform friction coefficient τfix. Then, the reciprocal shielding length λ can be expressed as λ = (τfix/ η)1/2.55 Because ve = 0, v = δv. Let pref = ε(kBT/z1e)2/a2 and Uref = ε▽n0(kBT/z1e)2/nj0eη be a reference pressure and a reference velocity, respectively, and δp* = δp/pref, v* = v/Uref.

Figure 1. Diffusiophoresis of a spherical polyelectrolyte (PE) of radius a and outer boundary Ωp subject to an applied salt gradient ▽n0 normal to two large parallel disks of half separation distance bd, (a), and along the axis of a cylindrical pore of radius bp, (b). (r,θ,z) are the cylindrical coordinates adopted with the origin at the center of the PE; ▽n0 is in the z direction.

large, parallel disks of half distance bd in the first case, and parallel to the axis of a long cylindrical pore of radius bp in the second case, shown in Figure 1b. The cylindrical coordinates (r,θ,z) are adopted with the origin at the PE center, and ▽n0 is in the z-direction. The PE carries a fixed charge of density ρfix dispersed in an incompressible, Newtonian solution containing z1/z2 electrolytes with z1 and z2 being the valences of cations and anions, respectively. For simplicity, we assume that both 9470



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Then the flow field can be described by the equation of continuity and modified Stokes equation as23,29,42

g2* =

v * = −U *ez ,

* )∇*δψ * −∇*δp* + γ Δ*v* + (Δ*ψe* + iρfix 2

+ Δ*δψ *∇*ψe* − iγ(λa) v* = 0

(6)

er ·v* = 0,

Boundary Conditions-Two Large Disks. In this case, we assume that the salt concentration gradient is established by two uncharged parallel disks. Let U and U* = U/Uref be the diffusiophoretic velocity of the PE and the corresponding scaled quantity, respectively. If these disks are nonslip, then they move with a velocity −U*ez relative to the PE. Therefore, the following boundary conditions apply n ·∇*ψe* = 0,

bd a

Z=±

g1* = −Zγ − δψ *,

Zγ − δψ *, α

v* = −U *ez ,

bd a

(9)

as R → ∞

n ·∇*δψ * = 0,

Z=±

Z=±

(10)

bd a

bd a

or as R → ∞

or as R → ∞

bd a

Z=±

(11)

(12)

FE* =

(13)

n ·∇*(er ·v*) = n ·∇*(ez ·v*) = 0,

as R → ±∞

n ·∇*ψe* = 0,

R=

(14)

FD* =

(15)

as Z → ±∞

(16)

n·∇*δψ * = 0,

R=

n ·∇*δψ * = ∓βγ ,

n ·∇*g1* = n ·∇*g2* = 0, g1* = −Zγ − δψ *,

(17)

as Z → ±∞

R=

(18)

bp a

(19)

as Z → ±∞

(20)

as Z → ±∞

as Z → ±∞

(22) (23) (24)

(25)

∬Ω* (σH*·n)·ez dΩ*p

(26)

where FE* = FE(z1e/kBT)2/ε, FD* = FD(z1e/kBT)2/ε, and ∫ ∫ Ω p*(•)dΩp* is the integration of the function (•) over the scaled PE surface Ω*p = Ωp/a2. The scaled equilibrium electric potential ψe* can be solved independently from eq 1 subject to either eqs 7 and 8 or eqs 15 and 16. Substituting the ψ*e obtained into eqs 2−6 yields a set of linear equations, which can be solved subject to either eqs 9−14 or eqs 17−24. Let U and U* = U/Uref be the diffusiophoretic velocity of the PE and the corresponding scaled quantity, respectively. Because U* (eqs 13 and 22) is unknown, the strategy of O’Brien and White52 for solving an electrophoresis problem is adopted to avoid a trial-and-error procedure. Here, the origin problem is partitioned into two subproblems: the PE migrates with a constant velocity in the absence of ▽n0 in the first subproblem, and ▽n0 is applied but it is held fixed in the second subproblem. More details about the evaluation of U* and the interpretation of relevant mechanisms can be found in Liu et al.29 It has been shown that the continuum model is capable of capturing the physics within a narrow space if its half-linear size is larger than 3 nm.58,59 The present approach and the solution procedure adopted have been validated22,23,29 by comparing the diffusiophoresis results under limiting conditions with the analytical solutions,45,60 numerical results,30 and experimental data26 in the literature.

bp a

a

∬Ω* (σE*·n)·ez dΩ*p p

bp a

bp

p

Here, Z = z/a, R = r/a, and n is the unit normal vector directed into the liquid phase; er and ez are the unit vectors in the r and z directions, respectively. β = (D1 − D2)/(D1 + αD2) measures the difference between the diffusivity of cations and that of anions, and therefore, the degree of Ediffusivity. Equation 9 implies the net charge flux vanishes on the disk surface. Boundary Conditions-Cylindrical Pore. In this case, we assume that the pore is nonconductive and uncharged, the ionic concentration reaches essentially its bulk value as Z→ ± ∞, and the net charge flux vanishes there. These yield the following boundary conditions

ψe* = 0,

(21)

The assumptions of equal ε and η inside and outside the PE imply that the following quantities are continuous across Ωp: ψe*, n·∇ψe*, δψ*, n·∇δψ*, gj*, n·∇gj*, and v*. Solution Procedure. A finite-element method based software, FlexPDE (Version 4.24, PDE Solutions, Spokane Valley, WA) is adopted to solve the scaled dependent variables ψe*(R,Z), δψ*(R,Z), gj*(R,Z), v*(R,Z), and δp*(R,Z). The diffusiophoretic velocity of a PE can be evaluated based on the condition that the total force acting on it vanishes at steady state, which is supported by experimental observation.28 Because the present problem is θ symmetric, only the zcomponents of the relevant forces need be considered. These include the electrical force FEez and the hydrodynamic force FDez, where FE and FD can be evaluated by integrating respectively the Maxwell stress tensor σE = εEE − (1/2)εE2I and the shear stress tensor σH = −δpI + 2ηΛ over the outer boundary of the PE, Ωp. Here, E = ▽ψ, I, Λ=[▽v + (▽v)T]/ 2, and T denote the local electric filed, the unit tensor, the rate of deformation tensor, and matrix transpose, respectively. As proposed by Hsu et al.56 and Happel and Brenner,57

(8)

Z=±

R=

n ·∇*(ez ·v*) = 0,

(7)

as R → ∞

n ·∇*δψ * = ∓ βγ ,

g2* =

as Z → ±∞

(5)

∇*·v* = 0

ψe* = 0,

Zγ − δψ *, α

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that if κa exceeds ca. 6, U* decreases with increasing κa. This arises from the shielding effect, which reduces the effective charge density of a PE due to the accumulation of counterions in its interior.40,61 Figure 2 reveals that the smaller the bd/a the larger the value of κa at which the first local maximal of U* occurs; however, the value of κa at which the second local maximal of U* occurs is almost uninfluenced. As expected, these indicate that the thicker the double layer (smaller κa) the more significant the boundary effect is. A schematic representation of the mechanisms for the presence of asymmetric ionic distribution near the PE for the case of two large disks is shown in Figure 3a, and the corresponding contours of the scaled perturbed potential δϕ* * = −5. When ▽n0 is applied, the in Figure 3b, where ρfix concentration of counterions (K+) near or inside the upper part of the PE is higher than that near of inside its lower part. That is, the double layer is polarized by the applied salt gradient, known as type I DLP, exerting an electric force on the PE toward the high salt concentration side. As can be inferred from Figure 3b, this yields a local electric field of strength EDLPI near the PE. Note that another local electric field of strength EDLPII having the direction opposite to that of the former one is also established outside the double layer. This is known as type II DLP, resulting from the diffusion of co-ions (Cl−). Because this diffusion is impeded by the negative fixed charge of the PE, the perturbed concentration of the co-ions outside the upper (lower) part of the double layer is higher (lower) than that of the counterions. A decrease in bd/a implies a decrease in the PE-boundary distance, so is the available space for type II DLP to take place. In other words, the shorter that distance is the thinner the double layer necessary for type II DLP to become significant, and therefore the larger the value of κa at which the first local maximum of U* occurs. Note that the boundary (disks) might also compress the double layer, resulting in a more significant type I DLP. The enhancement in EDLPI and the reduction in EDLPII due to the presence of the boundary is illustrated by Figure 4, where E*DLP = −∂δψ*/∂Z is the z-component of the induced local electric field coming from the applied salt * , instead of EDLP * , is plotted gradient. For convenience, −EDLP because the PE is negatively charged. Therefore, a positive −E*DLP implies that the local induced electric field drives the PE

RESULTS AND DISCUSSION For illustration, we assume that the background liquid phase is an aqueous KCl solution and a KCl concentration gradient is applied. Because DK+ ≅ DCl− ≅ 2 × 10−9 m2/s, Ediffusivity is negligible.22,26 Therefore, for the case of PE diffusiophoresis along the axis of a cylindrical pore its behavior is governed mainly by EDLP and/or ECO. In addition, we assume γ = 10−3, λa = 2, and T = 298 K, yielding m1 = m2 = 0.235. Diffusiophoresis Normal to Two Large Disks. The variations of the scaled velocity U* as a function of bulk KCl concentration, measured by κa, at various levels of bd/a for the case of two large disks are presented in Figure 2. As seen in this

Figure 2. Variations of the scaled diffusiophoretic velocity U* for the case of two large disks as a function of κa for various values of bd/a at * = −50. ρfix

figure, for the range of κa considered, U* has two local maxima and one local minimum. Note that the result for bd/a = 10 (blue curve) in Figure 2 is similar to that of Liu et al.,29 where an isolated PE is considered. They concluded that the first local maximum of U* arises from the competition between types I and double-layer polarization (DLP). As the bulk ionic concentration (or κa) increases, because type II DLP becomes more significant, U* decreases accordingly. However, if κa is sufficiently large, the diffusioosmotic flow becomes significant, driving the PE toward the high salt concentration side.29 Note

Figure 3. (a) Mechanisms for the presence of asymmetric ionic distribution near the PE, and therefore, EDLPI and EDLPII, for the case of two large * = −5, and bd/a = 6. disks. (b) Contours of δϕ* on the plane θ = π/2 at κa = 1, ρfix 9472



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attracted into its interior. If that density exceeds a certain level, counterion condensation occurs.62 In this case, the counterions accumulated inside the PE might be polarized, thereby reducing the strength of the electric field inside the PE, as shown in Figure 4b. Figure 5a shows schematically the mechanisms for the presence of asymmetric ionic distribution near the PE, EDLPI, EDLPII, and the polarization of the condensed counterions. The corresponding contours of the scaled perturbed potential δϕ* at ρ*fix = −50 are presented in Figure 5b. As seen in Figure 5a, the strong EDLPI drives the counterions condensed inside the PE toward its lower part. A comparison between Figures 3b and 5b reveals that this has the effect of diminishing the net local electric field inside and near the PE. Figure 4b also suggests that the shorter the PE−disk distance, the more significant the polarization of the condensed counterions. This is because the shorter that distance the stronger the type I DLP is. We conclude that the thicker the double layer and/or the shorter the PE−disk distance the faster the PE diffusiophoretic velocity toward the high salt concentration side. A regression analysis on the results shown in Figure 2 yields eq S1 in the Supporting Information. Diffusiophoresis along the Axis of a Cylindrical Pore. In this case, the boundary is parallel to the applied salt concentration gradient. Figure 6 illustrates the variations of the scaled diffusiophoretic velocity U* as a function of κa at various values of bp/a for the case of a cylindrical pore. As mentioned previously, the boundary effect is negligible at bp/a = 10. As can be seen in Figure 6, the shorter the PE−pore distance, the faster the U* toward the low salt concentration side, and this phenomenon is pronounced at low bulk salt concentration (small κa). The influence of a parallel boundary on the diffusiophoresis of a PE can be explained by Figure 7, where the mechanisms involved and the contours of δϕ* at ρ*fix = −50 are presented. As illustrated in Figure 7a, if the PE is close to the pore, or its double layer occupies the entire cross section of the pore, then it is difficult for co-ions (Cl−) to diffuse from the high salt concentration side to the low salt concentration side. In this case, they are forced to diffuse through the double layer, or even the interior of the negatively charged PE, yielding a

Figure 4. Variations of the scaled electric field, −EDLP * (=∂δψ*/∂Z), for various values of bd/a at R = 0 as a function of Z in the second * = −5, (a), and ρfix * = −50, (b) The gray subproblem at κa = 0.5 and ρfix region (|Z| ≤ 1) denotes the interior of PE.

toward the high salt concentration side (due to type I DLP), and a negative −EDLP * implies that it drives the PE toward the low salt concentration side (due to type II DLP). As seen in Figure 4a, if the boundary effect is insignificant (large bd/a), −EDLP * is positive the near the PE boundary and is negative outside that region. This is because type I DLP occurs mainly inside the PE and type II outside the double layer, as can be inferred from Figure 3b. As bd/a decreases, type I DLP is enhanced and type II DLP depressed. We conclude that the presence of the boundary (disks) has the effect of raising the diffusiophoretic velocity toward the high salt concentration side, in general. As the fixed charge density of the PE increases, so are the significance of type I DLP and the amount of counterions

Figure 5. (a) Mechanism for the presence of asymmetric ionic concentration, and therefore, EDLPI, EDLPII, and polarization of condensed * = −50 and bd/a = 6. counterions; (b) contours of δϕ* on the plane θ = π/2 at κa = 1, ρfix 9473



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Figure 6. Variations of the scaled diffusiophoretic velocity U* as a * = −50 for the case of a function of κa for various values of bp/a at ρfix cylindrical pore.

repulsive electric field, ECO driving it toward the low salt concentration side. The influence of the above-mentioned diffusion of coins (or ECO) on the PE diffusiophoresis is similar to that of type II DLP (or EDLPII); the diffusion of co-ions as a response to the applied salt concentration gradient is impeded in both cases. However, for the case of a parallel boundary, the ECO is usually much stronger than other induced local electric fields. Figure 8 depicts the variations of the sum of the scaled electric field −(EDLP * + ECO * ) as a function of Z at R = 0 for varies values of the PE−pore distance. As seen in Figure 8a, where ρfix * = −5, type II DLP is enhanced, but type I DLP is depressed. As mentioned previously, the diffusion of co-ions across a PE from the high salt concentration side to the low salt concentration side resulting in a strong electric repulsive force acting on the PE toward the low salt concentration side. If the co-ions are forced to penetrate through the PE, that repulsive force also diminishes the effect of type I DLP. The shorter the PE-pore distance the more significant the ECO, and therefore, the U* toward the low salt concentration side increase, as seen in Figure 6. Figure 8b suggests that the shorter the PE−pore distance, the less significant the polarization of the condensed counterions.

Figure 8. Variations of the scaled electric field, −(EDLP * + ECO * ) (=∂δψ*/∂Z), for various bp/a at R = 0 as a function of Z in the second * = −5, (A), and ρfix * = −50, (B). The subproblem at κa = 0.5 and ρfix gray region (|Z| ≤ 1) denotes the interior of PE.

As that distance decreases, because type I DLP is offset by the diffusion of co-ions the polarization of the condensed counterions becomes unimportant. We conclude that the presence of the parallel boundary has the effect of raising the PE velocity toward the low salt concentration side. In addition, the thicker the double layer and/or the shorter the PE−pore distance, the more significant the ECO is. A regression analysis on the results illustrated in Figure 6 gives eq S2 in the Supporting Information.

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CONCLUSIONS The influence of a boundary on the diffusiophoretic behavior of a polyelectrolyte is analyzed by considering two typical cases: the direction of diffusiophoresis is normal to a boundary (two

Figure 7. (a) Mechanism for the presence of ECO for the case of a cylindrical pore. (b) Contours of δϕ* on the plane θ = π/2 at κa = 0.5, ρ*fix = −50, and bp/a = 3. 9474



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large parallel disks, case I) and parallel to a boundary (along the axis of cylindrical pore, case II). We show that these two types of boundary might have opposite influences: the PE velocity is enhanced toward the high salt concentration side in case I but toward the low salt concentration side in case II. The presence of a boundary is capable of yielding complicated and interesting diffusiophoretic behaviors; in general, the closer (narrower) the boundary and/or the thicker the double layer the more complicated the boundary effects are. The results gathered provide necessary theoretical background for further studies, information for design of diffusiophoresis relevant devices such as catalytic nanomotors and nanopore based biosensor devices, and tools for interpreting experimental observations.

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ASSOCIATED CONTENT

S Supporting Information *

Additional information and equations. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*(S.T.) Tel: 886-2-26215656 ext. 2508. E-mail: topology@mail. tku.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.

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REFERENCES

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