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Diffusiophoresis of a Nonuniformly Charged Sphere in a Narrow Cylindrical Pore Jyh-Ping Hsu* and Xuan-Cuong Luu Department of Chemical Engineering, National Taiwan University, Taipei, Taiwan 10617
Shiojenn Tseng Department of Mathematics, Tamkang University Tamsui, Taipei, Taiwan 25137 ABSTRACT: The influences of the boundary effect and the charged conditions of a particle on its diffusiophoretic behavior are studied by considering the diffusiophoresis of a nonuniformly charged sphere along the axis of a narrow cylindrical pore. We show that the surface charge distribution of the particle is capable of yielding the complicated phenomenon of double-layer polarization and, together with the presence of the pore, leads to profound and interesting diffusiophoretic behaviors. For example, if the electrophoresis effect coming from the difference in the ionic diffusivities is absent, then the diffusiophoretic velocity of a particle with the sign of the surface charge on its middle part different from that on its left- and right-hand parts has both a local maximum and a local minimum as the fraction of the middle varies; that velocity has only a local minimum if the whole particle is charged with the same sign. If the electrophoresis effect is significant, then the diffusiophoretic behavior of the particle depends mainly on its averaged surface charge density.
’ INTRODUCTION In the past few decades, both the fundamental theory and the application of micronanotechnology have made considerable advancement in fields such as microfluidic and nanofluidic,1 drug delivery,2 and autonomous motion,3,4 to name a few. When the length scale of a process reduces to micro- or nanoscales, the interfacial electrokinetic phenomena come into play and the associated mechanisms need be taken into account.5 These include, for instance, electrophoresis,6 electroosmosis,7 diffusioosmosis,8 self-electrophoresis,9 and diffusiophoresis.1012 The last mechanism, refering to the spontaneous movement of a charged particle driven by an applied concentration gradient, plays a key role in processes such as the separation of microsized particles from air,13 microfluidic applications,14,15 and polymeric coatings.16 It is known that the application of a uniform concentration gradient to a dispersion comprising charged particles and electrolyte solution makes the double layer surrounding a particle polarized.10,12,17 Together with the back ground electric field arising from the difference in ionic diffusivities, the local electric field resulting from the polarized double layer explains the diffusiophoretic movement of the particle.12,15 As proposed by Hsu et al.,10 two types of double-layer polarization (DLP) are present. Type I DLP comes from the nonuniform distribution of counterions inside the double layer, driving the particle toward the high-concentration side. Type II DLP arises from the accumulation of co-ions immediately outside the double layer r 2011 American Chemical Society
due to the requirement of electroneutrality, driving the particle toward the low-concentration side. The mechanisms involved in diffusiophoresis imply that the diffusiophoretic behavior of a particle is closely related to the charged conditions on its surface. This was justified by Hsu et al.11 in a recent study of the diffusiophoresis of a nonuniformly charged particle in an infinite electrolyte medium, where they showed that the interaction between the polarized double layers of the neighboring parts of the particle yields complicated and interesting diffusiophoretic behavior. Note that in practice particles may acquire nonuniformly distributed surface charge, which arises from, for example, the heterogeneity in chemical structure.1820 It is known that the presence of a boundary is capable of influencing significantly the diffusiophoretic behavior of a particle through deforming its double layer, thereby affecting both the electric and hydrodynamic forces acting on the particle. For example, the diffusiophoretic velocity of a uniformly charged sphere moving perpendicular to a plane has a local maximum,21 and if the sphere is sufficiently close to the plane, then both the sign and the magnitude of that velocity might vary. For the case where the sphere moves parallel to one or two planes,22 its diffusiophoretic velocity can be either raised or reduced by the Received: January 19, 2011 Revised: May 2, 2011 Published: June 07, 2011 12592
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by the following equations:10,12,21,27,28 r2 φ ¼
Figure 1. Diffusiophoresis of a nonuniformly charged spherical particle of radius a immersed in an electrolyte solution subject to an applied uniform concentration gradient rn0 in the z direction along the axis of a narrow cylindrical pore of radius b; r, θ, z are the cylindrical coordinates with the origin at the center of the particle; the particle is partitioned into the right, middle, and left parts by dividing the azimuthal angle j into three intervals: 0 < jR < 90 j, 90 j < jM < 90 þ j, and 90 þ j < jL 1, since the amount of positive charge on the middle part of the particle increases with increasing j, so does the negative electric force acting on that part of the particle, and |U*| increases accordingly. Note that because rn0 is in the z direction, the amount of ionic species accumulated near the poles of a particle, and therefore, the corresponding type I DLP, is sensitive to the charged conditions near those points. 12598
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The Journal of Physical Chemistry C If the azimuthal angle j is sufficiently large, then the degree of the type I DLP of the middle part of the particle increases rapidly with increasing j, and therefore, the negative electric force acting on that part of the particle reduces accordingly, so is |U*|. Figure 6 indicates that |U*|(R = 3) > |U*|(R = 2) > |U*|(R = 1), which is expected because in this case the amount of positive charge on the middle part of the particle increases with increasing R, so does the negative electric force acting on that part of the particle. The variations of the scaled diffusiophoretic velocity for the case where R < 0 can also be explained by the variations of the scaled electric force presented in Figure 7. This figure shows that * , which comprises the scaled electric force if R < 0 then Fe2M coming from the interaction between the co-ions outside the double and the middle part of the particle and the type I DLP of its right- and the left-hand parts, is always negative. This makes * small and U* negative for the range of the azimuthal angle j Fe2 examined. If j is small, because the amount of negative charge on the middle part of the particle increases with increasing j, the electrical repulsion makes the amount of anions accumulate on the right-hand side of the particle increases too, and the degree of the type I DLP of its right- and left-hand parts increases accordingly. In this case, the decrease in |U*| as j increases results from the contribution of the positive electric force acting * . on the right- and the left-hand parts of the particle, Fe2RL However, an increase in j also enhances the negative electric force acting on the middle part of the particle, thereby raising |U*|. The competition between the positive scaled electric force * and the negative scaled electric force Fe2M * , both increase Fe2RL with increasing j, yields a local minimum in |U*|. The presence of the local maximum of |U*| can be explained by the same reasoning as that employed in the discussion of the case where R > 1. Figure 6 also indicates that if j is sufficiently small, then |U*(R = 3)| < |U*(R = 2)| < |U*(R = 1)|, and this order is reversed if j is sufficiently large. If j is small, since the larger the |R| the more significant the type I DLP of the right- and the left* , yielding |U*(R = hand parts of the particle, the larger the Fe2RL 3)| < |U*(R = 2)| < |U*(R = 1)|. However, if j is sufficiently large, then F*e2M becomes important, and the larger * |, yielding the reverse order in |U*|. the |R| the larger the |Fe2M Figure 6 also reveals that for a fixed value of |R| if j is small then |U*(R < 0)| < |U*(R > 0)| and |U*(R < 0)| > |U*(R > 0)| if j is large. As illustrated in Figure 7, the latter results from the fact that * (R < 0)| > |Fe2 * (R > 0)|. if j is large then |Fe2 Case 2. β = 0.2. In this case, because β 6¼ 0 the effect of electrophoresis arising from the difference in the ionic diffusivities needs be taken into account.10,12,15,21,26,27 According to eq 17, since β < 0, the background electric field coming from that effect drives a positively (negatively), which is determined by the sign of σh, charged particle toward the low- (high-) concentration side. Figure 8 shows the influence of parameter R on the scaled diffusiophoretic velocity U* at various combinations of j and ka at β = 0.2. In Figure 8a, where the double layer is thick (ka = 1), if R > 1, then U* |U*|(j = 45) > |U*|(j = 30). Note that, because σh > 0 in this case and the boundary effect is significant, both the electric force coming from electrophoresis effect and that from the interaction between the particle and the co-ions outside the double layer are negative and so is U*. Since σ h increases with increasing R so is the negative electric force acting on the particle, and therefore, |U*| increases accordingly. The behavior of |U*| as
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Figure 8. Variations of the scaled diffusiophoretic velocity U* as a function of R for various combinations of j and ka at β = 0.2, σR = σL = 4, σM = RσR, and λ = 0.5; (a) ka = 1 and (b) ka = 5.
the azimuthal angle j varies results from the fact that σ h increases with increasing j. If R < 0 and |R| is sufficiently large, although the electric force coming from electrophoresis effect is positive, it is smaller than the negative electric force arising from the interaction between the particle and the co-ions (anions) outside the double layer, yielding a negative U*. The increase of |U*| with increasing R and |U*|(j = 60) > |U*|(j = 45) > |U*|(j = 30) can be explained by the same reasoning as that employed in the case where R is positive and sufficiently large. The double layer in Figure 8b is thin (ka = 5), the boundary effect is insignificant, and the electrophoresis effect becomes significant. If |R| is small, then, because the negative electric force acting on the particle is small, so is |U*|. In addition, since |R| is small, the amount of charge carried on the middle part of the particle decreases with increasing j, the negative electric force decreases accordingly, so is |U*|. A comparison between panels b and a in Figure 8 shows that the U* in the former is greater than the corresponding U* in the later, and U* can be positive if R is sufficiently small. This is because the negative electric force acting on the particle is insignificant due to the fact that the influence of the pore is unimportant, and the electrophoresis effect, which drives a 12599
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Figure 9. Variations of the scaled diffusiophoretic velocity U* as a function of λ for various combinations of ka and R at β = 0, σR = σL = 4, σM = RσR, and j = 30; (a) R = 1, (b) R = 3, (c) R = 1, and (d) R = 3.
particle having negative (positive) σ h toward the high- (low-) concentration side, is significant. In addition, if |R| > 1, then U*(j = 60) < U*(j = 45) < U*(j = 30), and that trend in reversed if R < 1. The former results from the fact that, if R > 1, then σ h increases with increasing azimuthal angle j, and the latter from the fact that, if R < 1, then σ h decreases with increasing j. Effect of Pore Size λ. Case 1. β = 0 Figure 9a indicates that for a uniformly charged particle (R = 1), if ka is small and/or λ is sufficiently large, that is, the boundary effect is important, then the particle is driven by the applied concentration gradient toward the low-concentration side (U* < 0). This is because, in this case, the negative (repulsive) force coming from the interaction between the co-ions outside the double layer and the particle is important. Otherwise, the electric force coming from type I DLP dominates, driving the particle toward the highconcentration side (U* > 0). For negative values of U*, |U*| is seen to increase with decreasing ka, which results from the increase of the negative electric force with increasing degree of boundary effect. |U*| is seen to have a local maximum in Figure 9a as λ varies. This arises from the competition between the negative electric force coming from the interaction between
the co-ions outside the double layer and the particle with the hydrodynamic retardation force arising from the presence of the pore; both of these forces increase with increasing degree of boundary effect. The qualitative behavior of U* shown in Figure 9b, where R is raised to 3, is similar to that in Figure 9a. As expected, because all of the relevant electric forces increase with increasing R, the |U*| in Figure 9b is larger than the corresponding |U*| in Figure 9a. In Figure 9c, the middle part of the particle is negatively charged and its other two parts are positively charged. Therefore, the particle experiences both negative and positive electric forces, and both of these forces increase with increasing λ. However, U* is always negative for the ranges of the parameters considered. Note that the negative electric force acting on the middle part of the particle includes that which comes from the interaction between the co-ions (anions) outside the double layer and the middle part of the particle and that from the type I DLP of its right- and left-hand parts. Because the middle part of the particle is negatively charged, if λ is large, then anions tend to accumulate near its right-hand side, thereby enhancing the degree of type I DLP of its other two parts. In Figure 9c, the charge density of the 12600
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Figure 10. Variations of the scaled diffusiophoretic velocity U* as a function of λ for various values of ka at β = 0, σR = σL = 4, and j = 0.
Figure 11. Variations of the scaled diffusiophoretic U* as a function of λ for various values of R at β = 0.2, σR = σL = 4, σM = RσR, j = 30, and ka = 1.
middle part of the particle is the same as that of its other two parts (R = 1). In this case, if ka is small, then |U*| has a local maximum as λ varies, and it has both a local minimum and a local maximum if ka is large. The former is because, if ka is small (boundary effect is significant), then the negative electric force dominates, and therefore, |U*| increases with increasing λ. However, if λ is sufficiently large, then the hydrodynamic retardation arising from the presence of the pore becomes significant, and |U*| decreases accordingly. The latter arises if ka is large, because the boundary effect is unimportant, the negative electric force becomes small, and it is further reduced by the positive electric force coming from the type I DLP of each part of the particle. If λ is small, because the rate of increase in the positive electric force coming from type I DLP with increasing λ is faster than that in the negative electric repulsive force, |U*| decreases accordingly. However, if λ is sufficiently large, then because the boundary effect is important and the negative electric force dominates, |U*| increases with increasing λ. These lead to a local minimum in |U*| as λ varies. The presence of the local maximum in |U*| at a large λ can be explained by the same reasoning as that employed in the discussion of Figure 9a. In Figure 9d, the middle part of the particle is negatively charged with a density higher than that of its other two parts. Due to that high charge density, the negative force acting on the middle part of the particle dominates. As in the case of Figure 9d, U* is seen to be negative for the ranges of the parameters considered. Figure 9d indicates that |U*| has a local maximum as λ varies, and its presence can be explained by the same reasoning as that employed in the discussion of Figure 9a. The effect of the positive electric force coming from the type I DLP of the right- and the left-hand parts of the particle becomes appreciable only if λ is sufficiently large. In this case, |U*| increases with increasing ka. For example, if λ exceeds ca. 0.5, then |U*(ka = 0.5)| < |U*(ka = 1)|. Consider next a dipole-like particle comprising a positively charged right-hand half, which is on the high-concentration side, and a negatively charged left-hand half, which is on the lowconcentration side, with σR = σL = σ > 0, where σR and σL are the scaled surface charge density of the right-hand half of the particle and that of the left-hand half, respectively. In this case, if
the boundary effect is unimportant, the application of rn0 yields a negative scaled perturbed potential on both sides of the particle,11 and the hydrodynamic drag coming from the circular flows near the particle dominates, which drives the particle toward the low-concentration (left-hand) side, making U* negative. On the other hand, if the boundary effect is serious, then the diffusion of cations (anions) from the high-concentration (righthand) side of the particle toward the low-concentration (lefthand) side is hindered by the positive (negative) surface charge on the right- (left-) hand part of the particle. Let us consider next the case where a particle comprises a positively charged right-hand half, which is on the high-concentration side, and a negatively charged left-hand half, which is on the low-concentration side. Figure 10 reveals that under the conditions assumed, U* is always negative. In addition, if ka is small, then |U*| has a local maximum as λ varies, and it has both a local minimum and a local maximum as ka gets large. This is because if ka is small, the amount of cations and that of anions accumulated on the right-hand side of the particle increase with increasing λ. This enhances both the excess pressure and the circular flows, thereby raising the hydrodynamic drag, and therefore, |U*|. However, the hydrodynamic retarded force acting on the particle due to the presence of the pore also increases with increasing λ, making |U*| decreases. As λ increases, the competition of these two factors yields the presence of the local maximum in |U*|. If ka is sufficiently large, |U*| decreases first with increasing λ due to the increase in the electric force coming from the type I DLP of both parts of the particle resulting from the compression of the double layer by the pore. The presence of the local maximum in |U*| can be explained by the same reasoning as that employed in the case where ka is small. Figure 10 also reveals that if λ is fixed then |U*| increases with increasing ka when ka is small but decreases with increasing ka when ka is large. The former results from the increase in the amount of counterions inside the double layer due to the increase in the bulk ionic concentration, and the hydrodynamic drag acting on the particle increases accordingly. The latter results from the decrease in the amount of counterions inside the double layer, which arises from that both the surface potential and the degree of boundary effect the decrease with increasing ka. 12601
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Case 2. β = 0.2. The influence of the boundary effect, measured by λ (=a/b), on the scaled diffusiophoretic velocity U* at various values of R is illustrated in Figure 11. Note that if λ is small, the boundary effect becomes insignificant, and the electrophoresis effect is significant, which drives a positively (negatively) charged particle toward the low- (high-) concentration side, yielding a negative (positive) U*. At small values of λ, if R < 0 and |R| is sufficiently large, then U* is positive and decreases with increasing R. The former is because σh is negative, and the later is because σ h increases with increasing R. If λ is large, the boundary effect becomes important and the diffusiophoretic behavior of the particle is complicated. If R = 0, U* increases monotonically with increasing λ, and if R > 0, then U* shows a negative local minimum as λ varies. The former results from the fact that, if the middle part of the particle is uncharged (R = 0), then the negative electric force acting on that part vanishes, and since the hydrodynamic retardation force acting on the particle increases with increasing λ, |U*| decreases. The latter is because, if R > 0, the electric repulsive force between the particle and the co-ions outside the double layer increases with increasing λ and |U*| increasing accordingly. However, an increase in λ also raises the hydrodynamic retardation force acting on the particle, which reduces |U*|. The competition between these two factors yields a negative local minimum in U*. If R < 0, then as λ increases, U* decrease from a positive value to become negative and has a negative local minimum. The decrease in U* results from the increase in the negative electric force acting on the particle arising from the interaction between co-ions (anions) and the negatively charged middle part of the particle and the influence of type I DLP on the right- and the left-hand parts of the particle on its middle part with increasing λ. The presence of the negative local minimum in U* as λ varies can be explained by the same reasoning as that employed in the case when R > 0. Figure 11 also reveals that if λ is sufficiently large, then, because the boundary effect in significant, |U*| increases with increasing |R|. This is because of the negative electric force acting on the middle part of the particle, and therefore, on the particle, coming both from the interaction between co-ions (anions) and the negatively charged middle part of the particle and from the influence of the type I DLP of the right- and the left-hand parts of the particle on its middle part decrease with decreasing |R|.
(low-concentration side), where the charged conditions of the middle part can be different from that of the other two parts. Then the degree of the DLP of each part of the particle is affected by that of the other two parts. If the signs of the surface charge of all the three parts of the particle are the same, then the electric force acting on the particle comprises three terms, namely, that which comes from the type I DLP of each part of the particle, that which comes from the type II DLP due to the accumulation of the co-ions on the right-hand side of the particle, and that which comes from the 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. In contrast, if the sign of the surface charge of the middle part of a particle is different from that of its right- and left-hand parts, then the type II DLP of those two parts vanishes and their type I DLP is enhanced. (b) Suppose that the boundary effect, which is measured by λ (=particle radius a/ pore radius b) is significant. Then, if the signs of the surface charge of all three parts of the particle are the same but the charge density of the middle part is higher than that of the rest two parts, then the scaled diffuophoretic velocity has a negative local minimum as the fraction of the middle part varies. On the other hand, if the sign of the surface charge of the middle part of the particle is different from that of its right- and left-hand parts, then the scaled diffusiophoretic velocity has both a negative local maximum and a negative local minimum. (c) If a particle comprises a positively charged right-hand half (on the high-concentration side) and a negatively charged left-hand half (on the low-concentration side), then, regardless of the level of the boundary effect, the scaled diffusiophoretic velocity is always negative. In addition, if the double layer is thick, then the scaled diffusiophoretic velocity has a negative local minimum as λ varies, and it has both a negative local maximum and a negative local minimum if the double layer is thin. If the electrophoresis effect is present, then we conclude the following. (a) The background electric field induced by the electrophoresis effect drives a particle of positive (negative) nature toward the low- (high-) concentration side. (b) If the pore is sufficiently large and/or the double layer is sufficiently thin, then the boundary effect is unimportant and the electrophoresis effect becomes significant. (c) If the boundary effect is important, then the diffusiophoretic behavior of the particle depends upon both the electrophoresis effect and the boundary effect.
’ CONCLUSIONS We modeled the diffusiophoretic behavior of a nonuniformly charged particle under the conditions where the boundary effect can be significant by considering the diffusiophoresis of a sphere along the axis of a long cylindrical pore. We show that the double layer surrounding the particle is distorted by the pore and, together with the nonuniformly charged conditions on the particle surface, lead to complicated and interesting diffusiophoretic behaviors. The mechanisms involved in the present problem include type I double-layer polarization (DLP) reflecting the asymmetric distribution of the counterions inside the double layer, type II DLP reflecting that of the co-ions immediately outside the double layer, the electrophoresis effect coming from the difference in the ionic diffusivities, and the 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. If the electrophoresis effect is absent, we conclude the following. (a) Suppose that a particle comprises a right-hand part (high-concentration side), a middle part, and a left-hand part
’ AUTHOR INFORMATION Corresponding Author
*Tel: 886-2-23637448. Fax: 886-2-236223040. E-mail: jphsu@ ntu.edu.tw.
’ ACKNOWLEDGMENT This work is supported by the National Science Council of the Republic of China. ’ NOTATIONS a radius of particle (m) b radius of cylindrical pore (m) diffusivity of ionic species j (m2/s) Dj e elementary charge (C) unit vector in the z direction (-) ez Fe, Fd electric and hydrodynamic forces acting on the particle (N) 12602
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The Journal of Physical Chemistry C Fei, Fdi Fi F*ei * Fdi F2* gj gj* Jj kB n nj nj0e n0e n0* p Pej (r,θ,z) r r* R T U U0 U* v ve v* z z* zj
z components of Fe and Fd, respectively, in subproblem i (N) =Fei þ Fdi magnitude of the total force acting on the particle in the z direction in subproblem i (N) =Fei/ε(kBT/z1e)2, scaled force (-) =Fdi/ε(kBT/z1e)2, scaled force (-) =Fe2 * þ Fd2 * (-) hypothetical potential simulating the deformation of double layer j = 1,2 (V) =gj/ξa (-) number flux of ionic species j (1/m2/s) Boltzmann constant (J/K) unit normal vector (-) number concentration of ionic species j (1/m3) bulk number concentration of ionic species j at equilibrium (1/m3) bulk number concentration at equilibrium (1/m3) =n0/n0e (-) pressure (Pa) =ε(kBT/z1e)2/μDj, electric Peclet number of ionic species j (-) cylindrical coordinates radial coordinate (m) =r/a (-) =σM/σL(-) absolute temperature (K) diffusiophoretic velocity (m/s) =εγ(kBT/z1e)2/aμ, reference velocity (m/s) =U/U0, scaled diffusiophoretic velocity (-) velocity of the liquid phase (m/s) equilibrium velocity (m/s) =v/U0 (-) axial coordinate (m) =z/a (-) valence of ionic species j, j = 1,2 (-)
’ GREEK LETTERS R =z2/z1 (-) β =(D1 D2)/(D1 þ RD2) (-) scaled charge density on the particle surface (-) σp σR, σM, σL scaled charge densities on the right, middle, and left particle surface, respectively γ = r*n0*, scaled concentration gradient (-) =δnj/nj0e, scaled perturbed concentration of ionic δnj* species j (-) δv perturbed velocity (m/s) δφ perturbed potential (V) δφ* =δφ/ξa (-) θ angular coordinate (radian) j azimuthal angle (degree) ε permittivity of water (C/V/m) k =[∑2j=1nj0e(zje)2/εkBT]2, reciprocal Debye length (1/m) λ =a/b (-) μ viscosity of the liquid phase (kg/m/s) surface potential of particle (V) ξa F space charge density (C/m3) φ potential of the surface with applied concentration gradient (V) equilibrium potential (V) φe =φe/ξa (-) φ*e =ξa(z1e/kBT), scaled surface potential of particle (-) φr
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r r* rn0 r2 r*2 χ1, χ2
gradient operator (1/m) =ar (-) concentration gradient (1/m4) Laplace operator (1/m2) =a2r2 (-) proportional constant (-)
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