Diffusiophoresis of an Ellipsoid along the Axis of a Cylindrical Pore

Jun 2, 2010 - Laplace operator; nj, zj, Jj, and Dj are the number concentration, the valence, the number flux, and the diffusivity of ionic species j,...
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J. Phys. Chem. B 2010, 114, 8043–8055

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Diffusiophoresis of an Ellipsoid along the Axis of a Cylindrical Pore Jyh-Ping Hsu,* Xuan-Cuong Luu, and Wei-Lun Hsu Department of Chemical Engineering, National Taiwan UniVersity, Taipei, Taiwan 10617 ReceiVed: February 19, 2010

The influences of a boundary and the shape of a particle on its diffusiophoretic behavior are examined by considering the diffusiophoresis of a charged ellipsoid along the axis of an uncharged cylindrical pore filled with electrolyte solutions. The diffusiophoretic mobility of the particle under various conditions is evaluated through varying the particle aspect ratio, the size of the particle, the thickness of the double layer, the diffusivities of co-ions and counterions, and the level of the surface potential. Because the double-layer polarization, the electrophoresis effect, the boundary effect, and the electrical interaction between the particle and the co-ions outside the double layer all play a role, the diffusiophoretic behavior of the particle is complicated. For a fixed particle volume, the relative magnitude of the diffusiophoretic mobilities of particles of various shapes depends highly on the degree of the boundary effect. It is possible for a particle to migrate toward the low-concentration side, regardless of the level of the surface potential, which is not observed in the case of a sphere in a planar slit or in a spherical cavity. Introduction Diffusiophoresis, the migration of colloidal particles as a response to an applied concentration gradient,1–4 has many practical applications. The adhesion of high molecular weight polymers to metals,5 the ionic deposition and film coating,6,7and the scavenging of radioactive particles,8 for instance, all involve this phenomenon. Depending upon the nature of the dispersion medium, a particle can be driven by electrostatic force7,9–13 and by van der Walls and dipole forces.14,15 The former applies to electrolyte solutions and the latter to nonelectrolyte solutions. Derjaguin2,4 pioneered relevant theoretical analysis by considering the diffusiophoresis of a rigid, isolated nonconductive sphere in an infinite electrolyte solution having nonuniform ionic distribution. Assuming that the thickness of the double layer surrounding the particle is thin relative to its radius, the effect of double-layer polarization (DLP) was taken into account, and it was shown that the particle moves toward the highconcentration side. Prieve et al.12 showed that the diffusiophoretic velocity of an isolated sphere in an infinite solution containing symmetric electrolytes subject to an applied uniform concentration gradient comprises a chemiphoresis component and an electrophoresis component. The former comes from the nonuniform accumulation of the co-ions and counterions inside the double layer for the case where the diffusivities of these ions are the same, and the latter arises from the difference in ionic diffusivities. It was shown that if the double layer is much thinner than the local radius of curvature of the particle and the effect of DLP is negligible, then the diffusiophoretic velocity of the particle is independent of both its size and its shape. Applying the methodology of O’Brien and White16 for the analysis of a charged particle in an applied electric field, Prieve and Roman12 evaluated the diffusiophoretic velocity of a rigid, isolated sphere in symmetric electrolytes (KCl and NaCl) over a wide range of surface potential and double layer thickness. The presence of a boundary can have a profound influence on the diffusiophoretic behavior of a particle. Because the flow, * Corresponding author. Tel.: 886-2-23637448. Fax: 886-2-236223040. E-mail: [email protected].

the concentration, and the electric fields near a particle are all influenced by a neighboring boundary, so are the associated hydrodynamic and electric forces acting on the particle. Intuitively, the diffusiophoretic behavior of a particle when a boundary is present will be different both quantitatively and qualitatively from that when it is absent. This is justified theoretically by the result of the diffusiophoresis of a sphere normal to a plane in electrolyte solutions17,18 and by that of a charged sphere parallel to one and two planar walls;19 some interesting and complicated diffusiophoretic behaviors are predicted in these studies. For example, in the former, the diffusiophoretic velocity of a particle can have a local maximum as the particle-boundary distance varies, and both the sign and the magnitude of that velocity can vary as the particle gets closer to the boundary.17,18 In the latter, the particle can either be sped up or slowed down by the boundary.19 In a study of the diffusiophoresis of a charged sphere in an uncharged cavity, Hsu et al.20 proposed the presence of two types of DLPs. Together with the electrophoresis effect coming from the difference in the diffusivities of ionic species, they lead to complicated diffusiophoretic behavior. Several attempts have been made on the modeling of the diffusiophoresis of nonspherical particles such as cylindrical21,22 and elliptical particles.23 The results obtained reveal that the shape of a particle can play an important role. For example, the diffusiophoretic velocity of a particle decreases with a decrease in its maximum length in the direction of the concentration gradient,23 and the influence of a normal plane on the behavior of the cylindrical particle is stronger than that on a sphere.21 Because diffusiophoresis needs be conducted inevitably in a finite space,17–21,24–27 taking the boundary effect into account is highly desirable in the modeling stage from an apparatus design and/or data interpretation point of view. In addition, since the shape factor of a particle can be significant, considering a more generalized particle, the shape of which is closer to reality is necessary. To these ends, we consider the diffusiophoresis of an ellipsoidal particle, which has been considered previously in modeling electrokinetic problems,28,29 along the axis of a cylindrical pore in the present study. The shape of the particle

10.1021/jp1039157  2010 American Chemical Society Published on Web 06/02/2010

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Figure 1. Diffusiophoresis of an ellipsoid of semiaxes c and d in a cylindrical pore of radius b subject to an applied concentration gradient ∇n0 in the z-direction; r, θ, and z are the cylindrical coordinates with the origin at the center of the ellipsoid.

elementary charge, the Boltzmann constant, and the absolute temperature, respectively. Instead of solving eqs 1-4 directly, a perturbed problem is solved.16 In this approach, the dependent variables φ, v, p, and nj are first expressed as the sum of an equilibrium component, which is the value of the variable when ∇n0 is not applied, and a perturbed component coming from the application of ∇n0. That is, φ ) φe + δφ, v ) ve + δv, p ) pe + δp, and nj ) nje + δnj, where subscript e and prefix δ denote the equilibrium and the perturbed components, respectively. In addition, nj is expressed as18,20

[

nj ) nj0e exp chosen has the advantage that various kinds of particles can be simulated by varying its aspect ratio. The diffusiophoretic behavior of the particle under various conditions is discussed. In particular, the influences of the particle aspect ratio, the size of the particle, the thickness of the double layer, the diffusivities of co-ions and counterions, and the level of the surface potential on the diffusiophoretic mobility of the particle and the relevant forces acting on it are discussed in detail. Theory Let us consider the problem illustrated schematically in Figure 1, where an ellipsoidal particle with semiaxes d and c is placed on the axis of a cylindrical pore of radius b filled with an incompressible Newtonian fluid containing z1:z2 electrolytes with z1 and z2 being the valence of cations and that of anions, respectively. Let R ) -z2/z1. Through varying the aspect ratio (d/c), particles of various shapes can be simulated. For example, d/c > 1, d/c ) 1, and d/c < 1 represent prolates, spheres, and oblates, respectively. For convenience, the volume of a particle is fixed at (4/3)πa3 with a being the radius of the equivalent sphere. The cylindrical coordinates, r, θ, and z, are adopted with the origin located at the center of the particle, and only the (r, z) domain needs be considered because of the axial symmetric nature of the present problem. A uniform concentration gradient ∇n0 is applied in the z-direction, and the particle moves along the axis of the pore. Under the conditions of steady state, creeping flow, and constant physical properties, the governing equations of the present problem can be summarized as16,18,20

∇2φ ) -

[(

F ) ε

2

zjenj ε j)1

∑

(1)

) ]

(2)

-∇p + µ∇2v - F∇φ ) 0

(3)

∇·v)0

(4)

∇ · Dj ∇nj +

zje n ∇φ + njv ) 0 kBT j

In these expressions, φ is the electrical potential; ∇2 is the Laplace operator; nj, zj, Jj, and Dj are the number concentration, the valence, the number flux, and the diffusivity of ionic species j, respectively, j ) 1, 2; v and µ are the velocity and the viscosity of the liquid phase, respectively; F and p are the space charge density and the pressure, respectively; and e, kB, and T are the

]

zje (φ + δφ + gj) kBT e

(5)

where nj0e is the bulk concentration of ionic species j at equilibrium and gj, j ) 1,2, is a hypothetical scaled potential function used to modify the Poisson distribution of the ionic species. It is known that the spatial variation of the ionic species at equilibrium follows Boltzmann distribution. This distribution may no longer be valid when ∇n0 is applied, which yields asymmetric ionic distribution surrounding the particle.16,17,20 Suppose that a|∇n0| , n0e with n0e being the bulk concentration at equilibrium; that is, the concentration gradient coming from the application of ∇n0 is much smaller than that coming from the equilibrium concentration. Then, it can be shown that the governing equations for the perturbed problem become eqs A1-A6 in the Appendix.18,20 We assume the following: (a) The particle surface is maintained at a constant potential, and that of the pore is uncharged. (b) The electrical potential at a point far away from the particle is uninfluenced by its presence, and the macroscopic electric field arising from the difference in ionic diffusivities is proportional to β ) (D1 - D2)/(D1 + RD2). (c) Both the particle and the pore are nonconductive and impenetrable. (d) The net ionic flux vanishes at a point far away from the particle. (e) The ionic concentration reaches the value of (nj0e + z∇nj0) at a point far away from the particle. (f) Both the surface of the particle and that of the pore are nonslip. Therefore, the boundary conditions for the perturbed problem can be summarized in eqs A7-A17 in the Appendix. Note that because the flow field is in the creeping flow regime and the present problem is axisymmetric the rotation of the particle does not occur. Similar to the treatment of O’Brien and White16 in the analysis of a charged particle driven by an applied electrical field, the present problem is partitioned into two subproblems. In the first subproblem, the particle moves with constant velocity U in the absence of ∇n0, and in the second subproblem ∇n0 is applied but the particle remains still. To determine the diffusiophoretic velocity of a particle, both the electric force Fe and the hydrodynamic force Fd acting on the particle need to be calculated.25,30,31 Let Fe and Fdi be the z-components of Fe and Fd in subproblem i, respectively, F*ei ) Fei/ε(kBT/z1e)2 and F*di ) Fdi/ε(kBT/z1e)2 be the corresponding scaled forces, and Fi ) Fei + Fdi be the magnitude of the total force acting on the particle in the z-direction in subproblem i. Then F1 ) χ1U and F2 ) χ2∇n0 with χ1 and χ2 independent of U and ∇n0, respectively. The total force acting on the particle vanishes at steady state, yielding

U)-

χ2 ∇n χ1 0

(6)

Diffusiophoresis of an Ellipsoid

Figure 2. Variation of the scaled diffusiophoretic velocity U* as a function of the absolute value of the scaled surface potential potential |φr| for the case of an isolated sphere at R ) 1, β ) 0, d/c ) 1, and κa ) 1; solid curve, present result with λ ) 0.05; dotted curve, analytical result of Keh and Wei;33 dashed curve, numerical result of Prieve and Roman.12

Results and Discussion The governing equations for the perturbed problem, eqs A1-A6, and the associated boundary conditions, eqs A7-A17, are solved numerically by FlexPDE,32 a finite element method based software. Mesh is refined to ensure the convergence of the results. The applicability and the accuracy of the software adopted are also checked by solving the diffusiophoresis of an isolated rigid sphere in an infinite medium, the solution of which is available in the literature. For convenience, we define the scaled diffusiophoretic velocity of a particle U* as U* ) U/U0, where U0 ) εγ(kBT/z1e)2/aµ is a reference velocity with γ ) ∇*n*0 , n*0 ) n0/n0e, and ∇* ) a∇. Figure 2 shows the simulated variation of the scaled diffusiophoretic velocity U* as a function of the scaled potential based on the present method for the case of d/c ) 1. The corresponding results of Prieve and Roman12 and Keh33 are also presented for comparison; the latter is based on the conditions of low surface potential; that is, |φr| should not exceed about unity, where φr ) ξa(z1e/kBT) is the scaled surface potential of the particle with ξa being the corresponding surface potential. Figure 2 reveals that the performance of the software adopted in this study is satisfactory. The diffusiophoretic behavior of a particle under various conditions is examined through numerical simulation. For illustration, we let R ) 1. The parameter β is assumed to take one of the two values: 0 or -0.2. An example for the former is an aqueous KCl solution, and that for the latter is an aqueous NaCl solution.12 For the case where β ) 0, the particle is assumed to be positively charged and can be either positively or negatively charged for the case where β ) -0.2. Dispersion Medium with β ) 0. In this case, the effect of electrophoresis arising from the difference in the ionic diffusivities is absent. Because the applied concentration gradient ∇n0 directs from the left-hand side of a particle to its righthand side, the ionic concentration in the right-hand side region of the particle is higher than that in its left-hand side region. Therefore, the double layer on the right-hand side of the particle is thinner than that on its left-hand side. Since the particle is positively charged, the concentration of anions (counterions) inside the double layer on the right-hand side of the particle is higher than that on its left-hand side. The polarized double layer, defined as type I DLP by Hsu et al.,20 induces a local electric field driving the particle toward the high-concentration (righthand) side. This phenomenon was also observed by Pawar et al.34 and Dukhin et al.11 for the case of thin double layers. Type

J. Phys. Chem. B, Vol. 114, No. 24, 2010 8045 I DLP is illustrated in Figure 3, which shows the scaled perturbed concentration of ionic species, δn*j ) δnj/nj0e, j ) 1,2, along the axis of the pore. As can be seen in this figure, inside the double layer (1< |z*| < 2.8 in Figure 3a and 1 < |z*| < 2 in Figure 4b, where z* ) z/a) δn*2 > δn*1 in the right-hand side region and δn*1 > δn*2 in the left-hand side region. Figure 3b also indicates that outside the double layer (2 < |z*| < 5) δn*1 > δn*2 in the right-hand side region and δn*1 < δn*2 in the left-hand side region. This is defined as type II DLP by Hsu et al.,20 which drives the particle toward the low-concentration (lefthand) side. A comparison between Figure 3a and 3b reveals that both δn*2 and δn*1 increase with increasing λ ()c/b), implying that as the pore gets smaller the degree of DLP becomes more important. This is because as the space between the particle and the pore becomes small the double layer is compressed by the pore, and when a concentration gradient is applied, following Derjaguin’s analysis,2,4 those counterions must be redistributed along the direction of the applied concentration gradient, yielding a larger δn*2 . A similar result is also reported in the diffusiophoresis of a sphere normal to a plane.18 Moreover, as the space between the particle and the pore gets small, it is difficult for counterions to diffuse through that space from the high concentration side to the low concentration side, yielding an accumulation of counterions on the right-hand side of the particle, thereby increasing δn*2 . Due to electroneutrality, the δn*1 outside the double layer becomes larger accordingly. This is also why type II DLP appears clearly in Figure 3b (λ ) 0.8). Figure 3a and 3b also shows that δn*2 is much larger than δn*1 , implying that type I DLP dominates and δφ* ) δφ/ξa is positive on the low-concentration side and δφ* negative on the high-concentration side so that the particle is driven toward the high-concentration side. However, as seen in Figure 4, this is true only if λ is small (Figure 4a, where λ ) 0.1) and is not the case if λ is large (Figure 4b, where λ ) 0.8). This interesting observation arises mainly from the geometry considered. Different from that considered by Lou and Lee18 and Hsu et al.,20 the present geometry has the nature that as the co-ions outside the double layer diffuse from the high-concentration side of the particle to its low-concentration side they must pass through the gap between the double layer and the pore. If λ is large, both the electric repulsive force between the co-ions and the particle and the hydrodynamic force coming from the ionic diffusion become important. Figure 5 illustrates the variations of the scaled diffusiophoretic velocity U* as a function of κa () a[∑2j)1nj0e(zje)2/εkBT]2), which measures the relative thickness of the double layer, for various combinations of the absolute value of the scaled surface potential |φr| and the aspect ratio (d/c). As seen in this figure, |U*| has a local maximum as κa varies, and if κa is sufficiently large and |φr| is sufficiently low, then U* becomes positive; that is, the particle is driven to the high-concentration side. These behaviors can be explained by the variations of the scaled forces acting on the particle shown in Figure 6 and the flow field presented in Figure 7. For the present case, the electric forces come from type I DLP, type II DLP, and the electric interaction between the particle and the co-ions outside the double layer as they diffuse through the gap between the double layer and the pore. The first force drives the particle toward the high-concentration side, and the other two forces drive the particle toward the lowconcentration side. For a fixed λ, if κa is large, so is the gap between the double layer and the pore, and therefore, the electric repulsive force between the particle and the co-ions outside the double layer is small. A large κa also implies a great electric force coming from type I DLP,18,20 and therefore, the total

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Figure 3. Variation of the scaled perturbed concentration δn*j ) δnj/δnje0 (delta_n1 and delta_n2) along the axis of the pore at R ) 1, β ) 0, d/c ) 2, κa ) 1, and |φr| ) 1: (a) λ ) 0.1, (b) λ ) 0.8.

electric force is dominated by type I DLP, yielding U* > 0. On the other hand, if κa is small, then the electric repulsive force dominates, yielding a negative total electric force. The presence of the local maximum in |U*| as κa varies can be explained by that if κa is small, so is the surface charge density, and therefore, the electric force acting on the particle is small, yielding a small |U*|. As κa increases, |U*| increases accordingly. However, if κa is too large, so is the space between the double layer and the pore, leading to a small electric repulsive force, and therefore, |U*| becomes small. In the present case, both the chemiosmotic flow arising from the excess pressure on the highconcentration side of the particle and that from the fluid flow near the particle surface from the high-concentration side of the particle to its low-concentration side due to type I DLP drive the particle toward the low-concentration side. In addition, the larger the κa and the larger the amount of anions in the double layer, the fluid flow inside is faster, and therefore, the scaled hydrodynamic force F*d2 is greater,20 as can be inferred from

Figure 7. As |φr| increases, the electric repulsive force between the co-ions and the particle increases, as does the hydrodynamic force, and the particle tends to be driven to the low-concentration side. As seen in Figure 5, the qualitative behaviors of the U* of probate, sphere, and oblate are similar. Quantitatively, if κa is small, the values of |U*| of these particles rank as |U*|(prolate) > |U*|(sphere) > |U*|(oblate). As (d/c) varies from 2 (prolate) to 0.5 (oblate), because the particle volume is fixed, c increases, and with constant λ, b increases too, yielding a decrease in the electric repulsive force between the particle and the co-ions outside the double layer as they diffuse through the gap between the double layer and the pore. In addition, as (d/c) increases, d increases accordingly, implying that the thickness of the double layer on the right-hand side of the particle decreases and that on its left-hand side increases, yielding a more serious DLP. This is consistent with the result of Keh and Huang23 where the diffusiophoretic velocity of an ellipsoidal particle in an

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Figure 4. Contours of the scaled disturbed electric potential δφ* ) δφ/ξa on the half plane δφ* ) δφ/ξa for cases in Figure 3.

infinite medium is found to increase with an increase in its maximum length in the direction of the concentration gradient. Note that the variation in (d/c) influences not only the degree of DLP as mentioned above but also the degree of boundary effect. The behavior of |F*e2| shown in Figure 6 is affected by these two factors. As seen in this figure, if κa is small, then F*e2 < 0 and |F*e2|(prolate) > |F*e2|(sphere) > |F*e2|(oblate). The negative F*e2 arises from that the boundary effect is more important than the effect of DLP, and therefore, the electric force is dominated

by the electric repulsive force between the co-ions outside the double layer and the particle. The order of |F*e2| for various types of particles is due to that the degree of boundary effect increases with increasing (d/c). If κa is large, so is the space between the outer boundary of the double layer and the pore, and therefore, the negative electric force arising from the interaction between co-ions as they diffuse from the high concentration side to the low concentration side and the particle is small compared with the positive electric force coming from type I DLP, yielding a

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Figure 5. Variations of the scaled diffusiophoretic velocity U* as a function of κa for various combinations of |φr| and (d/c) at λ ) 0.5; solid line, d/c ) 1; dotted line, d/c ) 0.5; dashed line, d/c ) 2.

Figure 6. Variations of the scaled electric force F*e2 and F*d2 as a function of κa at various values of (d/c) for the case where |φr| ) 1 and λ ) 0.5; solid line, d/c ) 1; dotted line, d/c ) 0.5; dashed line, d/c ) 2.

positive F*e2. In this case, |F*e2|(prolate) > |F*e2|(sphere) > |F*e2|(oblate) because the larger the (d/c) the more significant the degree of DLP is. Note that an increase in the degree of DLP also leads to an increase in the |F*d2| arising from chemiosmosis. This is because as κa increases more ions are present in the double layer, making chemiophoresis more significant, as illustrated in Figure 7. If κa is large, because the double layer is thin, the scaled hydrodynamic force due to the chemiosmosis caused by DLP ranks as |F*d2|(prolate) > |F*d2|(sphere) > |F*d2|(oblate). This order is reversed at a small κa due to the influence of the perturbed electrical potential, which drives counterions (anions) to the direction opposite to that coming from DLP. As |φr| increases, the electric repulsive force between the co-ions outside the double layer and the particle and, therefore, |U*| increase. An increase in the amount of counterions inside the double layer also makes DLP more important, yielding both a greater electric driving force and a greater hydrodynamic retardation force. The variations in the scaled diffusiophoretic mobility of a particle U* as a function of its aspect ratio, (d/c), at various

Hsu et al. levels of |φr| are presented in Figure 8. Here, |U*| is seen to have a local maximum as (d/c) varies. Figure 9 reveals that the absolute value of the scaled viscous force acting on the particle, |F*d1| () |Fd1/ε(kBT/z1e)2|), has a local minimum as (d/c) varies. For a given λ, because the particle volume is fixed, a smaller (d/c) implies a larger c and, therefore, a larger pore radius and a less significant boundary effect. If the boundary effect is insignificant, the decrease in |F*d1| is mainly caused by the decrease in the particle surface area shown in Figure 10, which shows that if the particle volume is fixed then the sphere (d/c ) 1) has the minimum surface area. The increase of |F*d1| with increasing (d/c) after it passes the local minimum arises from the increase in the particle surface area, the increase in the projection area of the particle on a plane parallel to the z-axis, and a more important boundary effect. Figure 11 illustrates the variations of F*d2 and F*e2 for the case of Figure 8. The trend of F*e2 in this figure can be explained by that as (d/c) increases, because the particle volume is fixed, c must decrease, and because λ is constant, b also decreases. Therefore, the larger the (d/c) the closer is the double layer surrounding the particle to the pore. Since the space between the double layer and the pore becomes narrower for the co-ions to pass through, the more important the corresponding electric repulsive force between the co-ions and the particle. In addition, because the higher the |φr| the greater the repulsive force is, |F*e2| increases with increasing |φr|. Furthermore, since the particle volume is fixed, an increase in (d/c) implies that d increases; therefore, the double layer surrounding the particle becomes thinner, and the effect of DLP is enhanced, yielding an increase in the hydrodynamic force acting on the particle. However, because the gap between the double layer and the pore decreases with increasing (d/c), the chemiophoresis coming from the perturbed electrical potential decreases accordingly. The presence of the local maximum of |U*| in Figure 8 is the result of the competition between the forces Fd2 and Fe2, which both drive the particle toward the low-concentration side, and the hydrodynamic force Fd1 retarding the movement of the particle. The variations of the scaled diffusiophoretic velocity of a particle U* as a function of λ at various combinations of (d/c), |φr|, and κa are illustrated in Figure 12. This figure reveals that the sign and the relative magnitudes of U* for particles of various shapes depend highly upon the degree of the boundary effect measured by λ. If λ is small (λ ) 0.1), except for d/c ) 0.5 and |φr| ) 5, U* > 0 and U*(prolate) > U*(sphere) > U*(oblate); if λ takes a medium large value, then U* < 0 and |U*|(prolate) > |U*|(sphere) > |U*|(oblate); if λ is large, then U* < 0, and |U*|(prolate) > |U*|(sphere) < |U*|(oblate). These behaviors can be explained by that if λ is small the boundary effect is unimportant, and the behavior of the particle is governed by the effect of DLP. Since type I DLP dominates, the particle is driven toward the high-concentration side (U* > 0). For a fixed κa, because the degree of type I DLP increases with increasing (d/c), so is U*. As λ increases, U* declines to zero first, becomes negative, and then passes through a negative local maximum. This is because that for fixed κa the larger the λ, the closer the double layer is to the pore, thereby increasing the electric repulsive force between the co-ions outside the double layer and the particle. This has the effect of reducing the electric force coming from type I DLP, and if λ is sufficiently large, the electric force acting on the particle becomes negative (repulsive), as is justified in Figure 13, which shows that if λ is sufficiently large the absolute value of both the electric force and the hydrodynamic force acting on the particle is larger. A comparison between Figure 12a and 12b indicates that if λ is

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Figure 7. Contours of the scaled velocity on the plane v* ) v/U0 in the second subproblem for different values of κa at d/c ) 2, β ) 0, φr ) 1, and λ ) 0.5; (a) κa ) 1, (b) κa ) 5.

small then |U*| increases with increasing κa. This is because as κa increases the concentration of counterions inside the double layer increases, and the degree of type I DLP increases accordingly. A larger κa also implies a less significant boundary effect, a smaller electric repulsive force, and a smaller |U*|. A comparison between Figure 12a and 12c reveals that if λ is small ()0.1), because the electric force coming from type I DLP increases with increasing |φr| and the electric repulsive force between the co-ions outside the double layer and the particle is negligible, |U*| increases accordingly. Figure 12c shows that if |φr| is high, because both the hydrodynamic force acting on the particle and the projection area of the particle on a plane perpendicular to the z-direction are large, U*(d/c ) 0.5) is

negative at a small λ. If λ takes a medium large value, the electric repulsive force dominates, the magnitude of which increases with increasing significance in the boundary effect arising from the decrease in the ionic concentration, yielding |U*|(κa ) 1) > |U*|(κa ) 5) and |U*|(|φr| ) 1) < |U*|(|φr| ) 5). In addition, the magnitude of that electric repulsive force for various types of particles ranks as probate > sphere > oblate, yielding |U*|(probate) > |U*|(sphere) > |U*|(oblate). However, if λ is too large, because the increase in the electric repulsive force due to the increase in the significance of the boundary effect is less significant than that due to the increase in the ionic concentration (or thickness of the double layer), |U*|(κa ) 1) < |U*|(κa ) 5). Furthermore, if λ is sufficiently large, then the

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Figure 8. Variations of the scaled diffusiophoretic velocity U* as a function of the aspect ratio (d/c) for various levels of |φr| at λ ) 0.5, κa ) 1, and β ) 0.

Hsu et al.

Figure 10. Curve 1, variations of the ratio (surface area of a particle/ that of the equivalent sphere) as a function of the aspect ratio (d/c); curves 2 and 3, variations of the ratio (surface area of a particle projected on a plane perpendicular and parallel to the z-axis/that of the equivalent sphere).

Figure 9. Variations of the scaled electric force F*d1 as a function of the aspect ratio (d/c) for various levels of |φr| at λ ) 0.5, β ) 0, and κa ) 1.

hydrodynamic force due to the presence of the pore dominates. This force is related to the projection area of the particle on a plane parallel to the axis of the pore.28,29 As seen in Figure 10, this area for particles of various shapes ranks as probate > sphere > oblate, implying that |F*d1|(prolate) > |F*d1|(sphere) > |F*d1|(oblate), which yields |U*|(probate) < |U*|(sphere) < |U*|(oblate). The presence of the local maximum of |U*| as λ varies seen in Figure 12 is the result of the competition between the electric force and the hydrodynamic force acting on the particle. Both the increase in the electric repulsive force between the co-ions outside the double layer and the particle and that in the hydrodynamic force with increasing λ are due to a more significant boundary effect. As expected, the value of λ at which the local maximum occurs decreases with decreasing κa. Dispersion Medium with β * 0. Let us consider next the case where the diffusivity of cations is different from that of anions, implying that the effect of electrophoresis is present. According to eq A11, the induced electric field coming from this effect has the opposite direction as that of ∇n0. Therefore, if the particle is positively (negatively) charged, it is driven to the low- (high-) concentration side. Due to the presence of the electrophoresis effect, the |U*| in Figure 14a, where the particle is positively charged, is larger

Figure 11. Variations of the scaled forces in the second subproblem as a function of |φr| at λ ) 0.5, β ) 0, and κa ) 1. Solid curves, F*e2; dashed curves, F*d2.

than that in Figure 5. However, as seen in Figure 14b, where the particle is negatively charged, |U*| becomes smaller than that in Figure 5, and U* can be positive if |φr| is sufficiently low. This is because in this case the electric force coming from the electrophoresis effect tends to drive the particle to the highconcentration side. The direction of this force is the same as that of the electric force coming from type I DLP and is opposite to that of the electric repulsive force between the co-ions outside the double layer and the particle. The reduction in the electric force by the force coming from the electrophoresis effect yields a smaller |U*|. Figure 15 shows the variation of the scaled diffusiophoretic velocity U* of a positively charged particle as a function of λ at various combinations of (d/c), κa, and φr. A comparison between Figures 15 and 12 indicates that the more significant the boundary effect (larger λ) the less appreciable the influence of the electrophoresis effect on U*. For example, if λ ) 0.1, the values of U* are 0.00403, 0.00926, and 0.0138 in Figure 12a, where β ) 0, when (d/c) takes the values of 0.5, 1, and 2, respectively, and become -0.1099, -0.1246, and

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J. Phys. Chem. B, Vol. 114, No. 24, 2010 8051

Figure 12. Variations of U* as a function of λ for various combinations of (d/c), κa, and |φr| at β ) 0; (a) κa ) 1 and |φr| ) 1; (b) κa ) 5 and |φr| ) 1; (c) κa ) 1 and |φr| ) 5.

-0.1361 in Figure 15a, where β ) -0.2; if λ ) 0.6, the values of U* are -0.0243, -0.0268, and -0.0279 in Figures 12a when (d/c) takes the values of 0.5, 1, and 2, respectively, and become -0.0618, -0.06308, and -0.0624 in Figure 15a. This behavior is reasonable because the more significant the boundary effect the greater is the hydrodynamic drag acting on the particle. Figures 12 and 15 also indicate that the thinner the double layer (larger κa) and/or the higher the level of the surface potential, the larger the amount of variation in U* as β changes from 0 to -0.2, that is, the more significant the electrophoresis effect. This is because if the surface potential remains constant the larger the κa the thinner the double layer, the higher the surface

Figure 13. Variations of the scaled forces of the second subproblem as a function of λ for the cases where β ) 0; (a) κa ) 1 and |φr| ) 1; (b) κa ) 5 and |φr| ) 1; (c) κa ) 1 and |φr| ) 5; solid curves, d/c ) 1; dotted curves, d/c ) 0.5; dashed curves, d/c ) 2.

charge density, and, therefore, the greater the corresponding electric force. The effect of the level of the surface potential on the behavior of U* as β and λ vary can also be explained by similar reasoning. The behaviors of U* seen in Figure 15 are the net result of the competition of the electrophoresis effect, the effect of DLP, the electric interaction between the coins outside the double layer and the particle, and the hydrodynamic interaction between the particle and the pore.

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Figure 14. Variations of the scaled diffusiophoretic velocity U* as a function of κa for various combinations of φr and (d/c) at λ ) 0.5 and β ) -0.2; (a) φr > 0, (b) φr < 0; solid curves, d/c ) 1; dotted curves, d/c ) 0.5; dashed curves, d/c ) 2.

Figure 16 shows that for a negatively charged particle if λ is small, because the electric forces coming from type I DLP and the electrophoresis effect dominate, U* > 0 and is larger than the corresponding value of |U*| in Figure 15. Since these effects are more significant at a larger κa and/or |φr|, so is U*. As λ increases, the electrophoresis effect becomes less significant; the electric repulsive force between the co-ions outside the double layer and the particle increases; and the hydrodynamic drag increases. These all lead to a smaller U*. As seen in Figure 16c, if |φr| is sufficiently high, then U* < 0 and |U*| has a local maximum as λ varies. The latter is the result of the competition of the above three effects. Because the relative magnitude of U* for particles of various shapes depends highly upon both the level of λ and that of φr, that relative magnitude is complicated. Conclusions The diffusiophoresis of an ellipsoidal particle along the axis of a cylindrical pore is analyzed theoretically. Through the geometry considered, we are able to examine simultaneously the boundary effect and the influence of the shape of a particle on its diffusiophoretic behavior. We show that the factors key to the present problem are the polarization of the double layer surrounding the particle (DLP), the electrophoresis effect coming from the difference in the ionic diffusivities, and the electric

Figure 15. Variations of U* as a function of λ at various combinations of (d/c), κa, and φr at β ) -0.2; (a) κa ) 1, φr ) 1; (b) κa ) 5, φr ) 1; (c) κa ) 1, φr ) 5.

interaction between the particle and the coins outside the double layer as they diffuse across the gap between the double layer and the pore. These factors are related closely to the level of the surface potential, the thickness of the double layer, the relative magnitude of the particle (or pore), and the particle aspect ratio, leading to complicated diffusiophoretic behavior. Two types of DLPs are present: type I DLP refers to that the concentration of the counterions inside the double layer on the high-concentration side of the particle is higher than that on its low-concentration side; type II DLP refers to that the concentra-

Diffusiophoresis of an Ellipsoid

Figure 16. Variations of U* as a function of λ at various combinations of (d/c), κa, and φr at β ) -0.2; (a) κa ) 1, φr ) -1; (b) κa ) 5, φr ) -1; (c) κa ) 1, φr ) -5.

tion of the co-ions outside the double layer on the highconcentration side of the particle is higher than that on its lowconcentration side. Type I DLP drives the particle toward the high-concentration side, and type II DLP drives the particle toward the other side. If the electrophoresis effect is absent, we conclude the following: (a) If the boundary effect, which is measured by the parameter λ ()ratio of particle semiaxis/pore radius), is relatively unimportant, type I DLP dominates, and the smaller the pore the more important that effect is because more counterions are pushed into the double layer by the pore. On the other hand, if that effect is important, not only type II

J. Phys. Chem. B, Vol. 114, No. 24, 2010 8053 DLP but also the electric repulsive force between the particle and the co-ions outside the double layer and the hydrodynamic retardation due to the presence of the pore are important. (b) For a fixed λ, the absolute value of the scaled diffusiophoretic mobility of a particle, |U*|, has a local maximum as the thickness of double layer varies, and if the double layer is sufficiently thin and the scaled surface potential is sufficiently low, then the particle moves to the high-concentration side. (c) If the particle volume is fixed, the qualitative behaviors of U* for probate, sphere, and oblate are similar. Quantitatively, if the double layer is thick, then |U*|(prolate) > |U*|(sphere) > |U*|(oblate). This is because as the shape of a particle varies at a fixed volume both the degree of DLP and the electric interaction between the particle and the co-ions outside the double layer are influenced. If the double layer is thin, the boundary effect becomes unimportant compared with the effect of type I DLP. As the level of the surface potential increases, the electric repulsive force between the particle and the coions and, therefore, |U*| increases. (d) For fixed particle volume and λ, |U*| shows a local maximum as the ratio (semimajor axis/semiminor axis), (d/c), varies, which can be explained by the variations in the degree of the boundary effect, the particle surface area, and the projection area of the particle on a plane perpendicular to its direction of movement. (e) Due to the influences of the effect of DLP, the boundary effect, and the electric interaction between the particle and the co-ions outside the double layer, both the sign and the relative magnitudes of U* for particles of various shapes depend upon the level of λ. If the electrophoresis effect is present, then: (a) The electric field induced by the electrophoresis effect has the opposite direction as that of the imposed concentration gradient. Therefore, if the particle is positively (negatively) charged, it is driven to the low- (high-) concentration side. (b) The |U*| of a positively (negatively) charged particle is larger (smaller) than that in the case where the electrophoresis effect is absent. In addition, if the level of the surface potential is sufficiently low, the particle moves toward the high-concentration side. (c) The electrophoresis effect is more significant under the following conditions: the boundary effect is unimportant, the double layer is thin, and the surface potential is high. (d) If the boundary effect is unimportant, because the electric forces coming from type I DLP and the electrophoresis effect dominate, a negatively charged particle moves toward the high-concentration side. On the other hand, if the boundary effect is important, the electrophoresis effect becomes insignificant, and because the electric repulsive force between the particle and the co-ions outside the double layer and the hydrodynamic drag become significant, U* decreases accordingly. In this case, if the level of the surface potential is sufficiently high, then the particle moves toward the low-concentration side. Furthermore, |U*| has a local maximum as λ varies. (e) The relative magnitude of U* for particles of various shapes depends highly upon the level of the surface potential and the degree of boundary effect, making that relative magnitude complicated. Acknowledgment. This work is supported by the National Science Council of the Republic of China. Appendix Substituting φ ) φe + δφ, v ) ve + δv, p ) pe + δp, and nj ) nje + δnj into eqs 1-5 in the text, neglecting terms involving the product of two perturbed terms, and noting that ve ) 0 and, therefore, v ) δv, we obtain18,20

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∇*2φe ) -

Hsu et al.

(κa)2 [exp(-φrφ*e - exp(Rφrφ*)] e (1 + R)φr

(A1) ∇*2δφ* -

(κa)2 [exp(-φrφ*) e + (1 + R)φr

R exp(Rφrφ*)]δφ* ) e

(κa)2 [exp(-φrφ*)g e * 1 + (1 + R)φr (A2) R exp(Rφrφ*)g e *] 2

*1 + γPe1v* · ∇*φ*e ∇*2g*1 ) φr∇*φ*∇*g e

(A3)

*1 + γPe2v* · ∇*φ*e ∇*2g*2 ) Rφr∇*φ*∇*g e

(A4)

+ -φr2∇δp + γ∇*2v* + φ2r (∇*2φ*∇*δφ* e ∇*2δφ*∇*φ*) e ) 0 (A5) ∇* · v* ) 0

(A6)

Here, R ) -z2/z1,δp* ) δp/(εξ2a /a2), ∇* ) a∇, ∇*2 ) a2∇2, φr 2 ) ξa(z1e/kBT), κ ) [∑j)1 nj0e(zje)2/εkBT]2, Pej ) ε(kBT/z1e)2/µDj, v* ) v/U0, U0 ) εγ(kBT/z1e)2/aµ, γ ) ∇*n*0 , n*0 ) n0/n0e, φ*e ) φe/ξa, δφ* ) δφ/ξa, and g*j ) gj/ξa with ξa, U0,κ, and Pej being the surface potential of the particle, the reference velocity, the reciprocal Debye length, and the electric Peclet number of ionic species j, respectively. On the basis of the boundary conditions assumed in the text, the boundary conditions for the scaled variables can be summarized as follows18,20

φ*e ) 1 on the particle surface

(A7)

φ*e ) 0 on the pore surface

(A8)

1 (A9) λ n · ∇*δφ* ) 0 on the particle and the pore surfaces (A10) φ*e ) 0 as |z*| f ∞, r*