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Diffusiophoresis of a sphere along the axis of a cylindrical pore
Journal of Colloid and Interface Science 342 (2010) 598–606
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Diffusiophoresis of a sphere along the axis of a cylindrical pore Jyh-Ping Hsu a,*, Wei-Lun Hsu a, Ming-Hong Ku a, Zheng-Syun Chen b, Shiojenn Tseng b a b
Department of Chemical Engineering, National Taiwan University, Taipei, Taiwan 10617, Taiwan, ROC Department of Mathematics, Tamkang University, Tamsui, Taipei, Taiwan 25137, Taiwan, ROC
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 21 August 2009 Accepted 20 October 2009 Available online 24 October 2009
The diffusiophoresis of a charged spherical particle along the axis of an uncharged cylindrical pore filled with an electrolyte solution is analyzed theoretically. The influence of chemiphoresis, which includes two types of double-layer polarization, and that of electrophoresis arising from the difference in the diffusivities of the ionic species on the diffusiophoretic behavior of the particle are discussed. We have underlined the important difference between two cases: the first is a possibility for a particle to migrate to the low concentration side at low surface potentials (25 mV) along the axis of a cylindrical pore, while the second is that this migration occurs at high surface potentials (150 mV) in the case of a sphere in a spherical cavity. Ó 2009 Elsevier Inc. All rights reserved.
Keywords: Diffusiophoresis Boundary effect Sphere in cylindrical pore Double-layer polarization
1. Introduction The spontaneous movement of a colloid particle in a solution having nonuniformly distributed solute is known as diffusiophoresis [1–4]. If the solute is electrolytes, the particle is driven by electrostatic force [5–7], and by van der Walls and dipole forces if it is non-electrolytes [8–10]. Diffusiophoresis plays an important role in many operations, including polymer-to-polymer adhesion [11], latex particle coating process [12], scavenging of radioactive particles [13], and lung deposition of aerosols [14], to name a few. For the case of an isolated particle, symmetric electrolytes, and uniform concentration gradient, Deryagin et al. [15] were able to derive the following expression for the diffusiophoretic velocity of the particle U:
U¼
ef kB T rn0 D2 D1 ln cosh f þ f l Ze n0 D2 þ D1
ð1Þ
where e, l, f, kB, T, e, Z, and n0 are the permittivity and the viscosity of the liquid phase, the surface potential of the particle, Boltzmann constant, the absolute temperature, the elementary charge, and the valence and the bulk concentration of ions, respectively. r is the gradient operator; Dj is the diffusivity of ionic species j (1 for cations and 2 for anions); f ¼ Zef=4kB T is the scaled surface potential. B T rn0 D2 D1 According to Eq. (1), U comprises two components, elf kZe n0 D2 þD1 ef kB T rn0 ln cosh f and l Ze n0 . The latter is contributed by chemiphoresis [16], f which is due to the nonuniform accumulation of the counterions and coions inside the double layer surrounding the particle. The former is contributed by electrophoresis [17], which comes from the * Corresponding author. Fax: +886 2 23623040. E-mail addresses: [email protected], [email protected] (J.-P. Hsu). 0021-9797/$ - see front matter Ó 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jcis.2009.10.043
difference in the diffusivities of counterions and coions. Chemiphoresis is sensitive to the presence of a boundary [18–25] because it affects not only the flow field [21] and the electric field [22] near a particle but also the osmotic flow inside the double layer [25]. Assuming a thin but polarized double layer, Keh and Jan [21] investigated the boundary effect on diffusiophoresis by considering a charged sphere moving normal to a plane in an electrolyte solution. Their analysis was extended by Lou and Lee [22] to the case of an arbitrarily thick double layer. It was shown that the compression of the double layer by the plane drives more counterions to the particle surface, yielding a more serious effect of double-layer polarization (DLP). In addition, if the particle is close to the plane, the diffusioosmotic flow, which is generated by the excess pressure on the high concentration side of the particle, drives the particle away from the plane. Hsu et al. [25] pointed out that the electric field arising from DLP and the difference between the diffusivity of cations and that of anions also contributes to that phenomenon. The diffusiophoresis of a charged sphere between two uncharged, planar parallel surfaces was studied by Chen and Keh [23] for the case where the particle moves parallel to the planes, and that for the case where the particle moves normal to the planes was considered by Chang and Keh [24]. It was shown that the boundary effect in the latter is more significant than that in the former. In a study of the diffusiophoresis of a charged sphere in an uncharged cavity, Hsu et al. [25] pointed out that two types of DLP are present and they can have a profound influence on the diffusiophoretic behavior of the sphere. The above discussions suggest that the boundary effect on diffusiophoresis is an important issue, and a thorough study on this effect is necessary. In particular, the geometry can play a key role is assessing the significance of that effect. In this study, we consider the diffusiophoresis conducted in a cylindrical pore. This
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extends previous analysis on the diffusiophoresis in a planar slit [23,24] and in a spherical cavity [25] to a geometry closer to reality. The effects of the factors key to the present problem, including the surface potential, the thickness of double layer, the radius of the pore, and the diffusivities of ionic species on the diffusiophoretic behavior of the particle are examined in detail.
where gj is a hypothetical perturbed function and nj0e is the equilibrium bulk concentration of ionic species j. We assume that the application of rn0 yields but a slight distortion of the equilibrium concentration distribution of each ionic species over the length scale a, that is, a|rn0| n0e with n0e being the bulk electrolyte concentration at equilibrium. Under these conditions, Eqs. (2)–(6) yield [27,28]
2. Theory
r2 /e ¼ The problem considered is illustrated in Fig. 1, where a charged spherical particle of radius a moves along the axis of a long cylindrical pore of radius b subject to an imposed uniform concentration gradient rn0. (r, h, z) are the cylindrical coordinates adopted with the origin at the center of the particle. rn0 is in the z-direction. Note that because the present system is h-symmetric, only the (r, z) coordinates and the domain 0 6 r 6 b and 1 < z < 1 need be considered. The pore is filled with an aqueous, incompressible Newtonian fluid containing z1:z2 electrolytes with z1 and z2 being the valence of cations and that of anions, respectively. The governing equations of the present problem can be summarized as following:
r2 / ¼
2 X q zj enj ¼ e e j¼1
r Dj rnj þ
zj e nj r/ þ nj v ¼ 0 kB T
ðjaÞ2 expð/r /e Þ expða/r /e Þ /r ð1 þ aÞ
ðjaÞ2 exp /r /e þ aexp a/r /e d/ 1þa ðjaÞ2 ¼ exp /r /e g 1 þ exp a/r /e ag 2 1þa
ð7Þ
r2 d/
r2 g 1 /r r /e r g 1 ¼ nPe1 v r /e
ð8Þ ð9Þ
ð2Þ
ð3Þ
rv ¼0
ð4Þ
rp þ lr2 v qr/ ¼ 0
ð5Þ
Here, /, v, p, q, and nj are the electrical potential, the relative velocity of the liquid phase to the particle, the pressure, the space charge density, and the number concentration of ionic species j, respectively. Adopting the treatment of O’Brien and White [26], where a charged particle is driven by an applied electric field, each dependent variable is partitioned into an equilibrium component and a perturbed component. The former is the value of a variable when rn0 is not applied, and the latter is that outside the particle when rn0 is applied. That is, / = /e + d/, v = ve + dv, p = pe + dp, and nj = nje + dnj, where the subscript e denotes the equilibrium value and the prefix d represents the perturbed value. Because the particle remains stagnant at equilibrium, ve = 0 and v = dv. To account for a possible distortion of the ionic cloud surrounding the particle, the concentration of ionic species j is expressed as [26]
zj eð/e þ d/ þ g j Þ nj ¼ nj0e exp kB T
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Fig. 2. Typical meshes used on the half plane h = 0 for the case where a = 1, b = 0, ja = 1, k = 0.5, /r = 3, n = 0.01.
ð6Þ
Fig. 1. Diffusiophoresis of a charged spherical particle of radius a along the axis of a cylindrical pore of radius b; (r, h, z) are the cylindrical coordinates adopted with the origin at the center of the particle; the applied concentration gradient rn0 is in the z-direction.
Fig. 3. Variation of the diffusiophoretic velocity U as a function of the absolute value of the scaled surface potential |/r|; solid curve, present results at a = 1, b = 0, ja = 1, k ? 0; dashed curve, numerical results of Prieve and Roman [39]; dotted curve, analytical results of Keh and Wei [40].
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Fig. 4. Contours of the scaled perturbed concentration dnj ¼ dnj =nj0e [delta_n1 N in (a) and delta_n2 N in (b)] on the half plane h = 0 at a = 1, b = 0, ja = 1, k = 0.8, /r = 3, n = 0.01; (a) i = 1 and (b) i = 2.
r2 g 2 þ a/r r /e r g 2 ¼ nPe2 v r /e
ð10Þ
r v ¼ 0
ð11Þ
/2r r dp þ nr2 v þ /2r r2 /e r d/ þ r2 d/ r /e ¼ 0
ð12Þ
In these expressions, a = z2/z1, /r = fz1e/kBT, r = ar, r2 = P a r2, dp = dp/(ef2/a2), j ¼ ½ 2j¼1 nj0e ðezj Þ2 =ekB T1=2 , /e ¼ /e =f, d/ = 2
d//f, g j ¼ g j =f, Pej = e(kBT/z1e)2/gDj, v = v/Ur, Ur = en(kBT/z1e)2/ag, n ¼ jr n0 j, and n0 ¼ n0 =n0e . Here, j, Pej, Ur, and n are the reciprocal Debye length, the electric Peclet number of ionic species j, the reference velocity, and the scaled concentration gradient, respectively.
We assume the following: (a) The surface of the particle is maintained at a constant potential and the cylindrical pore is uncharged. (b) Both the particle and the pore are non-conductive. (c) The macroscopic electric field coming from the difference in the diffusivity of counterions and that of coions is proportional to b [= (D1 D2)/(D1 + aD2)] [29]. (d) Both the surface of the particle and that of the pore are ion-impenetrable [30]. (e) The bulk concentration of ionic species j is (nj0e + zrnj0) [19,25]. (f) Both the surface of the particle and that of the pore are non-slip. (g) The liquid phase is stagnant at a point far away from the particle. These assumptions lead to the following boundary conditions:
/e ¼ 1 on the particle surface
ð13Þ
/e ¼ 0 on the pore surface
ð14Þ
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Fig. 5. Variation in the scaled perturbed concentration dnj ¼ dnj =nj0e (delta_n1 N and delta_n2 N) along the axis of the pore for the case of Fig. 4.
1 k
ð15Þ
independent of U and rn0, respectively. Because F1 + F2 = 0 at steady state,
n r d/ ¼ 0 on the particle surface
ð16Þ
U¼
n r d/ ¼ 0 on the pore surface
ð17Þ
/e ¼ 0;
jz j ! 1 and r <
1 r d/ ¼ bc ez ; /r
1 jz j ! 1 and r < k
ð18Þ ð19Þ
n r g j ¼ 0 on the pore surface
ð20Þ
1 z c d/ ; /r
1 z c d/ ; g 2 ¼ a/r
jz j ! 1 and r <
1 k
The governing equations, Eqs. (7)–(12), and the associated boundary conditions, Eqs. (13)–(25), are solved numerically by FlexPDE [34], a finite element based software. The end effect of the cylindrical pore can be neglected if its length exceeds ca. 12 times of the particle diameter [35–38]. Typically, using a total of 9033 nodes is sufficient to achieve grid independence. Fig. 2 shows an example of the mesh structure used.
ð21Þ 3.1. Negligible boundary effect (k ? 0)
1 jz j ! 1 and r < k
ð22Þ
v ¼ 0 on the particle surface
ð23Þ
v ¼ U ez
ð24Þ
v ¼ U ez ;
ð26Þ
3. Results and discussion
n r g j ¼ 0 on the particle surface
g 1 ¼
v2 rn v1 0
on the pore surface jz j ! 1 and r <
1 k
ð25Þ
In these expressions, z = z/a, r = r/a, k = a/b, U = U/Ur. n is the unit normal vector directed into the liquid phase, and ez is the unit vector in the z-direction. The present problem is mathematically partitioned into two hypothetical sub-problems [26]: rn0 is not applied but the particle moves at a constant velocity U, and rn0 is applied but the particle is held fixed in the space. In the present case, only the electrical force Fe and the hydrodynamic force Fd acting on the particle need be considered. Let Fei and Fdi be the z-components of Fe and Fd in sub-problem i, respectively [31–33], and Fi be the total force acting on the particle in the z-direction in sub-problem i with Fi being its magnitude. Then F1 = v1U and F2 = v2rn0, where v1 and v2 are
To justify that the numerical procedure adopted in the present study is appropriate, it is first applied to the limiting case of k ? 0, which was solved previously [39,40]. In this case, the presence of the pore, or the boundary effect, is insignificant, and the diffusiophoretic behavior of a particle is close to that of an isolated particle in an infinite solution having a uniform concentration gradient. Fig. 3 shows that for the range of the scaled surface potential considered, the present result is consistent with the numerical result of Prieve and Roman [39]. If |/r| is low, the present result coincides with the analytical result of Keh and Wei [40]; however, if |/r| exceeds about unity, the difference between the two becomes appreciable. This is expected because the analytical result is valid for low surface potentials only. Note that U has a local maximum near |/r| = 4; the analytical result is unable to predict this behavior. As explained by Pawar et al. [41], the presence of the local maximum arises from that only counterions can penetrate into the double layer significantly, and due to a strong electric repulsive force, coions accumulate in the region immediately outside the double layer. These phenomena were discussed by Hsu et al. [25] in detail, and they defined two types of DLP. Type I DLP is due to that the amount of the counterions inside the double
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layer on the high concentration side of the particle is larger than that on the low concentration side. Type II DLP comes from the accumulation of the coions immediately outside the double layer. Numerical simulations are conducted to examine the diffusiophoretic behavior of a particle under various conditions. Both the case of b = 0 and that of b – 0 are considered. 3.2. D1 = D2 If the diffusivity of cations is the same as that of anions, then b = 0. Typical example includes an aqueous KCl solution. Fig. 4 illustrates the distribution of the scaled perturbed concentration
Fig. 6. Variations of the scaled diffusiophoretic velocity U as a function of ja for various combinations of |/r| and k at a = 1, b = 0; (a) k = 0.2, (b) k = 0.5, and (c) k = 0.8.
of ions, dnj ¼ dnj =nj0e , j = 1, 2, on the half plane h = 0 for the case the particle is positively charged. As seen in Fig. 4a, because the electric repulsive force is appreciable, the perturbed concentration of coions (cations) near the particle surface is very low. This implies that most of the coions diffuse from the high concentration (right-hand) side to the low concentration (left-hand) side through the space between the outer boundary of the double layer and the pore. Fig. 4b shows the presence of type I DLP, where the amount of counterions (anions) inside the double layer on the high concentration side is larger than that on the low concentration side. This generates an electric field driving the particle to the high concentration side. The variation of dnj along the axis of the pore is plotted in Fig. 5, which shows that in the outer region of the double layer (2 < |z| < 5) dn1 > dn2 on the right-hand side, and dn2 > dn1 on the left-hand side. This is known as type II DLP, which induces another electric field with a direction opposite to that induced by type I DLP. In our case, because both the diffusiophoretic velocity U and the reference velocity Ur are proportional to the applied concentration gradient rn0, U (=U/Ur) becomes independent of rn0. This is consistent with the experimental observation of Ebel et al. [1]. Fig. 6 illustrates the variation of the scaled diffusiophoretic velocity of a particle, U, as a function of ja for various combinations of k and |/r|. Under the conditions of small k and thin double layer, the amount of coins penetrated into the double layer as they diffuse across the particle is small, yielding a small electric repulsive force between the coions and the particle. In this case, the diffusiophoresis of the particle is dominated by DLP and
Fig. 7. Variations of the scaled diffusiophoretic velocity U as a function of k for various combinations of ja and |/r| at a = 1, b = 0; (a) |/r| = 1 and (b) |/r| = 5.
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chemiosmotic flow. For a rigid particle, the electric field comes from type I DLP is usually stronger than that from type II DLP [25]. Therefore, as seen in Fig. 6a, if ja is large, except for |/r| = 5, the particle is driven toward the high concentration side (U > 0). Note that the chemiosmotic flow, which arises from the excess pressure on the high concentration side and type I DLP, drives the fluid near the particle surface from the high concentration side to the low concentration side. That is, its presence has the effect of driving the particle toward the low concentration side. For |/r| 6 4, because the electric force coming from DLP is greater than the hydrodynamic drag coming from chemiosmotic flow, the particle moves to the high concentration side. However, as |/r| gets higher (=5), because type II DLP becomes stronger the electric force contributed by DLP is reduced. In this case, the hydrodynamic drag coming from chemiosmotic flow becomes stronger than the electric force contributed by DLP, and the particle is driven toward the low concentration side. If ja is small (thick double layer), the coions drive the particle toward the low concentration side as they diffuse across its surface, yielding a smaller total electric force, which can be negative. Therefore, the particle migrates to the low concentration side, regardless of the level of |/r|. Fig. 6a shows that if |/r| is sufficiently high, U has a local minimum as ja varies. This is because if ja is small, the surface charge density, and, therefore, the electrical driving force acting on the particle is small. A small ja also implies a low concentration of coions, yielding a small repulsive force between the particle and the coions as they diffuse across the particle. On the other hand, if ja is large, then the double layer is thin and the electric repulsive force between the particle and the coions is also small. A comparison between Fig. 6a and b reveals that as k increases from 0.2 to 0.5 the value of ja at which the local minimum of U occurs shifts to a larger value. This is because the thickness of the double layer at k = 0.5 exceeds the pore radius when ja ffi 1, and therefore, the electric repulsive force between the particle and the coins is stronger than that at k = 0.2. In this case, that force becomes significant, and the U in Fig. 6b becomes more negative than that in Fig. 6a. That is, if the pore is small, so is the space between the outer boundary of the double layer and the pore, and the coions must penetrate into the double layer as they diffuse
603
across the particle, which has the effect of driving the particle toward the low concentration side. Because k = 0 in Fig. 6c, the particle is very close to the pore and the double layer occupies the entire space between the particle and the pore even at a large ja (=10), making it difficult for the coions to diffuse across the particle. In this case, the electric repulsive force between the particle and the coions dominates, driving the particle to the low concentration side (U < 0). Note that |U| increases with increasing ja, which is different from the behavior of U in Fig. 6a and b. This is because a larger ja implies a higher charge density on the particle surface and a higher concentration of coions in the liquid phase. Note that the value of |U| in Fig. 6c is smaller than that in Fig. 6b. This is because an increase in k not only makes it more difficult for the coions to diffuse across the particle but also results in a stronger hydrodynamic drag force acting on the particle. The boundary effect arising from the presence of the pore on the diffusiophoretic behavior of the particle is illustrated in Fig. 7. If k is small, the smaller the ja (thicker double layer) the stronger the electric repulsive force acting on the particle due to the diffusion of the coions, yielding a smaller U. On the other hand, if k is large, because the space between the particle and the pore is small, the electric repulsive force between the particle and the coions is strong, regardless of the level of ja. The magnitude of this force depends upon the surface charge density of the particle; the higher the surface charge density the stronger the repulsive force. Therefore, |U| increases with increasing ja. Note that U shows a local minimum as k varies. The presence of this local minimum is the result of the competition between the electric force and the hydrodynamic drag acting on the particle. The magnitudes of these forces increase with increasing k. The increase in the magnitude of the electric force is due to a stronger repulsive force between the particle and the coions, and that of the hydrodynamic drag is due to a more significant boundary effect. Note that U is negative in Fig. 7b (/r = 5) even if the boundary effect is unimportant (large ja and small k). This is because type II DLP is more important at higher surface potentials, making the electric force coming from DLP smaller than the hydrodynamic drag arising from chemiosmotic flow.
Fig. 8. Contours of the scaled perturbed electric potential d/ (delta_psiN) on the half plane h = 0 at a = 1, b = 0.2, ja = 1, k = 0.5, /r = 3, n = 0.01.
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3.3. D1 – D2 If the diffusivity of cations is different from that of anions, such as in the case of an aqueous NaCl solution, then b – 0. In this case, because the ionic flux of cations in the bulk solution must equal to that of anions, the diffusion of the ionic species yields an electric field, as is justified in Fig. 8. This electric field, known as the background electric field, is parallel to the direction of ionic diffusion. If the diffusivity of cations is larger than that of anions (b > 0), the background electric field has the same direction as that of the im-
Fig. 9. Variations of the scaled diffusiophoretic velocity U as a function of ja for various combinations of /r and k at a = 1, b = 0.2, /r > 0; (a) k = 0.2, (b), k = 0.5, and (c) k = 0.8.
posed concentration gradient rn0. On the other hand, if the diffusivity of cations is smaller than that of anions (b < 0), the background electric field has the opposite direction as that of rn0. Because b = 0.2 in Fig. 8, the background electric field drives a positively charged particle to the low concentration side and a negatively charged particle to the high concentration side. This phenomenon was observed experimentally [1], and called electrophoresis. The presence of the electrophoresis effect makes the value of U in Fig. 9 smaller than the corresponding value of U in Fig. 6. The electrophoresis effect is more appreciable when the boundary effect is less significant. For example, Figs. 6 and 9 show that for ja = 1 and /r = 5, the difference between the values of U in these two figures is ca. 0.55 at k = 0.2, and is ca. 0.02 at k = 0.8. This is because if k is large (boundary effect significant), so is the hydrodynamic drag acting on the particle. In this case, U becomes less sensitive to the electrophoresis effect, as can be seen more clearly in Fig. 10. A comparison between Figs. 10 and 7 reveals that the amount of decrease in U as b varies from 0 to 0.2 at a smaller k is much larger than that at a larger k. For instance, at ja = 5 and k = 0.8, U is 0.02 in Fig. 7a and 0.05 in Fig. 10a. On the other hand, at ja = 5 and k = 0.2, U is 0.025 in Fig. 7a and 0.125 in Fig. 10a. Similarly, at ja = 5 and k = 0.8, U is 0.22 in Fig. 7b and 0.31 in Fig. 10b, and at ja = 5 and k = 0.2, U is 0.0 in Fig. 7b and 0.4 in Fig. 10b. In addition, because the electrophoresis effect is more important at larger ja, the larger the ja the larger the amount of decrease in U as b decreases from 0 to 0.2. As seen in Fig. 10b, where |/r| is high, if k takes a medium large value,
Fig. 10. Variations of the scaled diffusiophoretic velocity U as a function of k for various combinations of ja and /r at a = 1, b = 0.2; (a) /r = 1 and (b) /r = 5.
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the behavior of U as ja varies is complicated. This is because that in this case the effect of electrophoresis is not that important as that in the case of small k, the effect of DLP becomes comparable to that of chemiosmotic flow when |/r| is high, and the hydrodynamic drag due to the boundary effect is not that important as that in the case of large k. That is, the magnitudes of all the relevant forces are on the same order, yielding complicated behavior in U. If the particle is negatively charged and b < 0, then the background electric field drives the particle to the high concentration side. Therefore, the value of U in Fig. 11 is larger than the corre-
Fig. 11. Variations of the scaled diffusiophoretic velocity U as a function of ja for various combinations of /r and k at a = 1, b = 0.2, /r < 0; (a) k = 0.2, (b) k = 0.5, and (c) k = 0.8.
605
sponding value of U in Fig. 6. These figures also reveal that the higher the |/r| the stronger the electrophoresis effect. As mentioned previously, this effect is more important at smaller values of k. In Fig. 11a, U is dominated by the electrophoresis effect and the electric repulsive force between the particle and the coions at small values of ja, and by the electrophoresis effect and DLP at large values of ja. For |/r| P 2, if ja < 0.4, U decreases with increasing ja because the increase in the surface charge density yields a stronger electric repulsive force between the particle and the coions. However, for /r = 1, U increases with increasing ja because an increase in the surface charge density leads to a stronger electrophoresis effect. Fig. 11a indicates that U increases with increasing ja for ja ranges from ca. 0.5 to 4. This is because the thinner the double layer the smaller the electric repulsive force between the particle and the coions. Because type II DLP is significant at high surface charge densities and the higher the surface charge density the more important it is, U decreases with increasing ja for ja exceeds ca. 4 and |/r| P 4. No general trend is observed for U in Fig. 11a as /r varies. This is because the higher the |/r| not only the more significant the electrophoresis effect, which drives the particle to the high concentration side, but also a stronger electric repulsive force between the particle and the coions, which drives the particle to the low concentration side. In addition, both type I and type II DLP are influenced by /r, as mentioned in the discussion of Fig. 3. Due to a more significant boundary effect, the electrophoresis effect in the case of Fig. 11b and c is less important than that in the case of Fig. 11a. The qualitative behaviors of U as /r, ja, and
Fig. 12. Variations of the scaled diffusiophoretic velocity U as a function of k for various combinations of ja and /r at a = 1, b = 0.2; (a) /r = 1 and (b) /r = 5.
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k vary in Fig. 11b and Fig. 11c are similar to those in Fig. 6b and Fig. 6c. The crosses of the curves near ja = 0.1 in Fig. 11b and Fig. 11c arise mainly from the electrophoresis effect. As mentioned previously, the higher the |/r| the larger is the amount of increase in U as b decreases from 0 to 0.2. Fig. 12 suggests that the electrophoresis effect is more appreciable at a smaller k and a larger ja, which is consistent with the results presented in Figs. 9 and 10. For instance, the difference between the values of U in Figs. 7a and 12a is ca. 0.15 at k = 0.2 and ja = 5, and is ca. 0.05 at k = 0.8 and ja = 5. 4. Summary The boundary effect on the diffusiophoretic behavior of a particle is analyzed by considering the diffusiophoresis of a charged spherical particle along the axis of an uncharged cylindrical pore filled with an electrolyte solution. The influences of the radius of the pore, the diffusivities of ionic species, the thickness of double layer, and the surface potential of the particle on the diffusiophoretic velocity of the particle are examined. We show that if the radius of the pore is large (boundary effect unimportant) and/or the double layer is thin, the diffusiophoretic behavior of the particle is dominated by two types of double-layer polarization (DLP). Because type I DLP is usually more important than type II DLP, most of the previous studies focused on the former, making it difficult to explain some observed phenomena. For example, under the conditions of high surface potential, thin double layer, and inappreciable boundary effect, where type II DLP dominates, a particle moves to the low concentration side, and the thinner the double layer the smaller is its diffusiophoretic velocity. Here, we show that if the double layer is thin and/or the surface potential is high, type II DLP can be more important than type I DLP. If the radius of the pore is small and/or the double layer is thick, then the diffusiophoretic behavior of the particle is dominated by the repulsive force between the particle and the coions outside the double layer as they diffuse across the particle. This force, which is not observed in the case of a sphere in a spherical cavity, drives the particle to the low concentration side. If the diffusivity of cations is different from that of anions, then the electrophoresis effect is present; the less significant the boundary effect the more important thus effect. Depending upon the difference between the diffusivity of cations and that of anions, and the sign of the surface charge of the particle, the electrophoresis effect is capable of driving the particle toward either the low concentration side or the high concentration side.
Acknowledgment This work is supported by the National Science Council of the Republic of China. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41]
J.P. Ebel, J.L. Anderson, D.C. Prieve, Langmuir 4 (1988) 396–406. H.J. Tu, H.J. Keh, J. Colloid Interface Sci. 231 (2000) 265–282. S.S. Dukhin, E.S. Malkin, A.S. Dukhin, Colloid J. USSR 40 (1978) 536–540. S.S. Dukhin, E.S. Malkin, A.S. Dukhin, Colloid J. USSR 41 (1979) 734–739. B. Abecassis, C. Cottin-Bizonne, C. Ybert, A. Ajdari, L. Bocquet, Nat. Mater. 7 (2008) 785–789. D.C. Prieve, J.L. Anderson, J.P. Ebel, M.E. Lowell, J. Fluid Mech. 148 (1984) 247– 269. P.O. Staffeld, J.A. Quinn, J. Colloid Interface Sci. 130 (1988) 69–87. H.J. Keh, Y.M. Lin, Langmuir 10 (1994) 3010–3017. J.L. Anderson, M.E. Lowell, D.C. Prieve, J. Fluid Mech. 117 (1982) 107–121. P.O. Staffeld, J.A. Quinn, J. Colloid Interface Sci. 130 (1988) 88–100. S.S. Voyutskii, V.L. Vakula, J. Appl. Polym. Sci. 7 (1963) 475–491. A.A. Korotkova, B.V. Deryagin, Colloid J. USSR 53 (1991) 719–722. P. Goldsmith, H.J. Delafield, L.C. Cox, Quar. J. Roy. Meteorol. Soc. 89 (1963) 43– 61. G.M. Hidy, J.R. Brock, Environ. Sci. Technol. 3 (1969) 563–567. B.V. Deryagin, Z.R. Ulberg, S.S. Dukhin, Colloid J. USSR 48 (1986) 570–573. S.S. Dukhin, B.V. Deryaguin, Surface and Colloid Science, Vol. 7, Wiley, New York, 1974. S. Misra, S. Varanasi, P.P. Varanasi, Macromolecules 23 (1990) 4258–4269. J. Lou, Y.Y. He, E. Lee, J. Colloid Interface Sci. 299 (2006) 443–451. J.P. Hsu, J. Lou, Y.Y. He, E. Lee, J. Phys. Chem. 111 (2007) 2533–2539. H.J. Keh, Y.L. Li, Langmuir 23 (2007) 1061–1072. H.J. Keh, J.S. Jan, J. Colloid Interface Sci. 183 (1996) 458–475. J. Lou, E. Lee, J. Phys. Chem. C 112 (2008) 2584–2592. P.Y. Chen, H.J. Keh, J. Colloid Interface Sci. 286 (2005) 774–791. C.Y. Chang, H.J. Keh, J. Colloid Interface Sci. 322 (2008) 634–653. J.P. Hsu, W.L. Hsu, Z.S. Chen, Langmuir 25 (2009) 1772–1784. R.W. O’Brien, L.R. White, J. Chem. Soc. Faraday Trans. 2 74 (1978) 1607– 1626. E. Lee, J.W. Chu, J.P. Hsu, J. Colloid Interface Sci. 196 (1997) 316–320. J.H. Masliyah, S. Bhattacharjee, Electrokinetic Transport Phenomena, fourth ed., Wiley, New York, 2006. Y.K. Wei, H.J. Keh, J. Colloid Interface Sci. 248 (2002) 76–87. E. Lee, J.W. Chu, J.P. Hsu, J. Colloid Interface Sci. 205 (1998) 65–76. J.P. Hsu, L.H. Yeh, M.H. Ku, J. Colloid Interface Sci. 305 (2007) 324–329. J.P. Hsu, L.H. Yeh, J. Chin. Inst. Chem. Eng. 37 (2006) 601–607. J. Happel, H. Brenner, Low-Reynolds Number Hydrodynamics, Nijhoff, Boston, 1983. FlexPDE Version 2.22, PDE Solutions Inc., USA, 2000. J.P. Hsu, M.H. Ku, C.Y. Kao, J. Colloid Interface Sci. 276 (2004) 248–254. J.P. Hsu, L.H. Yeh, M.H. Ku, Colloid Polym. Sci. 284 (2006) 886–892. J.P. Hsu, Z.S. Chen, Langmuir 23 (2007) 6198–6204. J.P. Hsu, S.H. Hung, C.Y. Kao, S. Tseng, Chem. Eng. Sci. 58 (2003) 5339– 5347. D.C. Prieve, R. Roman, J. Chem. Soc. Faraday Trans. 2 83 (1987) 1287–1306. H.J. Keh, Y.K. Wei, Langmuir 16 (2000) 5289–5294. Y. Pawar, Y.E. Solomentsev, J.L. Anderson, J. Colloid Interface Sci. 155 (1993) 488–498.
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Journal of Colloid and Interface Science www.elsevier.com/locate/jcis
Diffusiophoresis of a sphere along the axis of a cylindrical pore Jyh-Ping Hsu a,*, Wei-Lun Hsu a, Ming-Hong Ku a, Zheng-Syun Chen b, Shiojenn Tseng b a b
Department of Chemical Engineering, National Taiwan University, Taipei, Taiwan 10617, Taiwan, ROC Department of Mathematics, Tamkang University, Tamsui, Taipei, Taiwan 25137, Taiwan, ROC
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 21 August 2009 Accepted 20 October 2009 Available online 24 October 2009
The diffusiophoresis of a charged spherical particle along the axis of an uncharged cylindrical pore filled with an electrolyte solution is analyzed theoretically. The influence of chemiphoresis, which includes two types of double-layer polarization, and that of electrophoresis arising from the difference in the diffusivities of the ionic species on the diffusiophoretic behavior of the particle are discussed. We have underlined the important difference between two cases: the first is a possibility for a particle to migrate to the low concentration side at low surface potentials (25 mV) along the axis of a cylindrical pore, while the second is that this migration occurs at high surface potentials (150 mV) in the case of a sphere in a spherical cavity. Ó 2009 Elsevier Inc. All rights reserved.
Keywords: Diffusiophoresis Boundary effect Sphere in cylindrical pore Double-layer polarization
1. Introduction The spontaneous movement of a colloid particle in a solution having nonuniformly distributed solute is known as diffusiophoresis [1–4]. If the solute is electrolytes, the particle is driven by electrostatic force [5–7], and by van der Walls and dipole forces if it is non-electrolytes [8–10]. Diffusiophoresis plays an important role in many operations, including polymer-to-polymer adhesion [11], latex particle coating process [12], scavenging of radioactive particles [13], and lung deposition of aerosols [14], to name a few. For the case of an isolated particle, symmetric electrolytes, and uniform concentration gradient, Deryagin et al. [15] were able to derive the following expression for the diffusiophoretic velocity of the particle U:
U¼
ef kB T rn0 D2 D1 ln cosh f þ f l Ze n0 D2 þ D1
ð1Þ
where e, l, f, kB, T, e, Z, and n0 are the permittivity and the viscosity of the liquid phase, the surface potential of the particle, Boltzmann constant, the absolute temperature, the elementary charge, and the valence and the bulk concentration of ions, respectively. r is the gradient operator; Dj is the diffusivity of ionic species j (1 for cations and 2 for anions); f ¼ Zef=4kB T is the scaled surface potential. B T rn0 D2 D1 According to Eq. (1), U comprises two components, elf kZe n0 D2 þD1 ef kB T rn0 ln cosh f and l Ze n0 . The latter is contributed by chemiphoresis [16], f which is due to the nonuniform accumulation of the counterions and coions inside the double layer surrounding the particle. The former is contributed by electrophoresis [17], which comes from the * Corresponding author. Fax: +886 2 23623040. E-mail addresses: [email protected], [email protected] (J.-P. Hsu). 0021-9797/$ - see front matter Ó 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jcis.2009.10.043
difference in the diffusivities of counterions and coions. Chemiphoresis is sensitive to the presence of a boundary [18–25] because it affects not only the flow field [21] and the electric field [22] near a particle but also the osmotic flow inside the double layer [25]. Assuming a thin but polarized double layer, Keh and Jan [21] investigated the boundary effect on diffusiophoresis by considering a charged sphere moving normal to a plane in an electrolyte solution. Their analysis was extended by Lou and Lee [22] to the case of an arbitrarily thick double layer. It was shown that the compression of the double layer by the plane drives more counterions to the particle surface, yielding a more serious effect of double-layer polarization (DLP). In addition, if the particle is close to the plane, the diffusioosmotic flow, which is generated by the excess pressure on the high concentration side of the particle, drives the particle away from the plane. Hsu et al. [25] pointed out that the electric field arising from DLP and the difference between the diffusivity of cations and that of anions also contributes to that phenomenon. The diffusiophoresis of a charged sphere between two uncharged, planar parallel surfaces was studied by Chen and Keh [23] for the case where the particle moves parallel to the planes, and that for the case where the particle moves normal to the planes was considered by Chang and Keh [24]. It was shown that the boundary effect in the latter is more significant than that in the former. In a study of the diffusiophoresis of a charged sphere in an uncharged cavity, Hsu et al. [25] pointed out that two types of DLP are present and they can have a profound influence on the diffusiophoretic behavior of the sphere. The above discussions suggest that the boundary effect on diffusiophoresis is an important issue, and a thorough study on this effect is necessary. In particular, the geometry can play a key role is assessing the significance of that effect. In this study, we consider the diffusiophoresis conducted in a cylindrical pore. This
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extends previous analysis on the diffusiophoresis in a planar slit [23,24] and in a spherical cavity [25] to a geometry closer to reality. The effects of the factors key to the present problem, including the surface potential, the thickness of double layer, the radius of the pore, and the diffusivities of ionic species on the diffusiophoretic behavior of the particle are examined in detail.
where gj is a hypothetical perturbed function and nj0e is the equilibrium bulk concentration of ionic species j. We assume that the application of rn0 yields but a slight distortion of the equilibrium concentration distribution of each ionic species over the length scale a, that is, a|rn0| n0e with n0e being the bulk electrolyte concentration at equilibrium. Under these conditions, Eqs. (2)–(6) yield [27,28]
2. Theory
r2 /e ¼ The problem considered is illustrated in Fig. 1, where a charged spherical particle of radius a moves along the axis of a long cylindrical pore of radius b subject to an imposed uniform concentration gradient rn0. (r, h, z) are the cylindrical coordinates adopted with the origin at the center of the particle. rn0 is in the z-direction. Note that because the present system is h-symmetric, only the (r, z) coordinates and the domain 0 6 r 6 b and 1 < z < 1 need be considered. The pore is filled with an aqueous, incompressible Newtonian fluid containing z1:z2 electrolytes with z1 and z2 being the valence of cations and that of anions, respectively. The governing equations of the present problem can be summarized as following:
r2 / ¼
2 X q zj enj ¼ e e j¼1
r Dj rnj þ
zj e nj r/ þ nj v ¼ 0 kB T
ðjaÞ2 expð/r /e Þ expða/r /e Þ /r ð1 þ aÞ
ðjaÞ2 exp /r /e þ aexp a/r /e d/ 1þa ðjaÞ2 ¼ exp /r /e g 1 þ exp a/r /e ag 2 1þa
ð7Þ
r2 d/
r2 g 1 /r r /e r g 1 ¼ nPe1 v r /e
ð8Þ ð9Þ
ð2Þ
ð3Þ
rv ¼0
ð4Þ
rp þ lr2 v qr/ ¼ 0
ð5Þ
Here, /, v, p, q, and nj are the electrical potential, the relative velocity of the liquid phase to the particle, the pressure, the space charge density, and the number concentration of ionic species j, respectively. Adopting the treatment of O’Brien and White [26], where a charged particle is driven by an applied electric field, each dependent variable is partitioned into an equilibrium component and a perturbed component. The former is the value of a variable when rn0 is not applied, and the latter is that outside the particle when rn0 is applied. That is, / = /e + d/, v = ve + dv, p = pe + dp, and nj = nje + dnj, where the subscript e denotes the equilibrium value and the prefix d represents the perturbed value. Because the particle remains stagnant at equilibrium, ve = 0 and v = dv. To account for a possible distortion of the ionic cloud surrounding the particle, the concentration of ionic species j is expressed as [26]
zj eð/e þ d/ þ g j Þ nj ¼ nj0e exp kB T
599
Fig. 2. Typical meshes used on the half plane h = 0 for the case where a = 1, b = 0, ja = 1, k = 0.5, /r = 3, n = 0.01.
ð6Þ
Fig. 1. Diffusiophoresis of a charged spherical particle of radius a along the axis of a cylindrical pore of radius b; (r, h, z) are the cylindrical coordinates adopted with the origin at the center of the particle; the applied concentration gradient rn0 is in the z-direction.
Fig. 3. Variation of the diffusiophoretic velocity U as a function of the absolute value of the scaled surface potential |/r|; solid curve, present results at a = 1, b = 0, ja = 1, k ? 0; dashed curve, numerical results of Prieve and Roman [39]; dotted curve, analytical results of Keh and Wei [40].
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Fig. 4. Contours of the scaled perturbed concentration dnj ¼ dnj =nj0e [delta_n1 N in (a) and delta_n2 N in (b)] on the half plane h = 0 at a = 1, b = 0, ja = 1, k = 0.8, /r = 3, n = 0.01; (a) i = 1 and (b) i = 2.
r2 g 2 þ a/r r /e r g 2 ¼ nPe2 v r /e
ð10Þ
r v ¼ 0
ð11Þ
/2r r dp þ nr2 v þ /2r r2 /e r d/ þ r2 d/ r /e ¼ 0
ð12Þ
In these expressions, a = z2/z1, /r = fz1e/kBT, r = ar, r2 = P a r2, dp = dp/(ef2/a2), j ¼ ½ 2j¼1 nj0e ðezj Þ2 =ekB T1=2 , /e ¼ /e =f, d/ = 2
d//f, g j ¼ g j =f, Pej = e(kBT/z1e)2/gDj, v = v/Ur, Ur = en(kBT/z1e)2/ag, n ¼ jr n0 j, and n0 ¼ n0 =n0e . Here, j, Pej, Ur, and n are the reciprocal Debye length, the electric Peclet number of ionic species j, the reference velocity, and the scaled concentration gradient, respectively.
We assume the following: (a) The surface of the particle is maintained at a constant potential and the cylindrical pore is uncharged. (b) Both the particle and the pore are non-conductive. (c) The macroscopic electric field coming from the difference in the diffusivity of counterions and that of coions is proportional to b [= (D1 D2)/(D1 + aD2)] [29]. (d) Both the surface of the particle and that of the pore are ion-impenetrable [30]. (e) The bulk concentration of ionic species j is (nj0e + zrnj0) [19,25]. (f) Both the surface of the particle and that of the pore are non-slip. (g) The liquid phase is stagnant at a point far away from the particle. These assumptions lead to the following boundary conditions:
/e ¼ 1 on the particle surface
ð13Þ
/e ¼ 0 on the pore surface
ð14Þ
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601
Fig. 5. Variation in the scaled perturbed concentration dnj ¼ dnj =nj0e (delta_n1 N and delta_n2 N) along the axis of the pore for the case of Fig. 4.
1 k
ð15Þ
independent of U and rn0, respectively. Because F1 + F2 = 0 at steady state,
n r d/ ¼ 0 on the particle surface
ð16Þ
U¼
n r d/ ¼ 0 on the pore surface
ð17Þ
/e ¼ 0;
jz j ! 1 and r <
1 r d/ ¼ bc ez ; /r
1 jz j ! 1 and r < k
ð18Þ ð19Þ
n r g j ¼ 0 on the pore surface
ð20Þ
1 z c d/ ; /r
1 z c d/ ; g 2 ¼ a/r
jz j ! 1 and r <
1 k
The governing equations, Eqs. (7)–(12), and the associated boundary conditions, Eqs. (13)–(25), are solved numerically by FlexPDE [34], a finite element based software. The end effect of the cylindrical pore can be neglected if its length exceeds ca. 12 times of the particle diameter [35–38]. Typically, using a total of 9033 nodes is sufficient to achieve grid independence. Fig. 2 shows an example of the mesh structure used.
ð21Þ 3.1. Negligible boundary effect (k ? 0)
1 jz j ! 1 and r < k
ð22Þ
v ¼ 0 on the particle surface
ð23Þ
v ¼ U ez
ð24Þ
v ¼ U ez ;
ð26Þ
3. Results and discussion
n r g j ¼ 0 on the particle surface
g 1 ¼
v2 rn v1 0
on the pore surface jz j ! 1 and r <
1 k
ð25Þ
In these expressions, z = z/a, r = r/a, k = a/b, U = U/Ur. n is the unit normal vector directed into the liquid phase, and ez is the unit vector in the z-direction. The present problem is mathematically partitioned into two hypothetical sub-problems [26]: rn0 is not applied but the particle moves at a constant velocity U, and rn0 is applied but the particle is held fixed in the space. In the present case, only the electrical force Fe and the hydrodynamic force Fd acting on the particle need be considered. Let Fei and Fdi be the z-components of Fe and Fd in sub-problem i, respectively [31–33], and Fi be the total force acting on the particle in the z-direction in sub-problem i with Fi being its magnitude. Then F1 = v1U and F2 = v2rn0, where v1 and v2 are
To justify that the numerical procedure adopted in the present study is appropriate, it is first applied to the limiting case of k ? 0, which was solved previously [39,40]. In this case, the presence of the pore, or the boundary effect, is insignificant, and the diffusiophoretic behavior of a particle is close to that of an isolated particle in an infinite solution having a uniform concentration gradient. Fig. 3 shows that for the range of the scaled surface potential considered, the present result is consistent with the numerical result of Prieve and Roman [39]. If |/r| is low, the present result coincides with the analytical result of Keh and Wei [40]; however, if |/r| exceeds about unity, the difference between the two becomes appreciable. This is expected because the analytical result is valid for low surface potentials only. Note that U has a local maximum near |/r| = 4; the analytical result is unable to predict this behavior. As explained by Pawar et al. [41], the presence of the local maximum arises from that only counterions can penetrate into the double layer significantly, and due to a strong electric repulsive force, coions accumulate in the region immediately outside the double layer. These phenomena were discussed by Hsu et al. [25] in detail, and they defined two types of DLP. Type I DLP is due to that the amount of the counterions inside the double
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layer on the high concentration side of the particle is larger than that on the low concentration side. Type II DLP comes from the accumulation of the coions immediately outside the double layer. Numerical simulations are conducted to examine the diffusiophoretic behavior of a particle under various conditions. Both the case of b = 0 and that of b – 0 are considered. 3.2. D1 = D2 If the diffusivity of cations is the same as that of anions, then b = 0. Typical example includes an aqueous KCl solution. Fig. 4 illustrates the distribution of the scaled perturbed concentration
Fig. 6. Variations of the scaled diffusiophoretic velocity U as a function of ja for various combinations of |/r| and k at a = 1, b = 0; (a) k = 0.2, (b) k = 0.5, and (c) k = 0.8.
of ions, dnj ¼ dnj =nj0e , j = 1, 2, on the half plane h = 0 for the case the particle is positively charged. As seen in Fig. 4a, because the electric repulsive force is appreciable, the perturbed concentration of coions (cations) near the particle surface is very low. This implies that most of the coions diffuse from the high concentration (right-hand) side to the low concentration (left-hand) side through the space between the outer boundary of the double layer and the pore. Fig. 4b shows the presence of type I DLP, where the amount of counterions (anions) inside the double layer on the high concentration side is larger than that on the low concentration side. This generates an electric field driving the particle to the high concentration side. The variation of dnj along the axis of the pore is plotted in Fig. 5, which shows that in the outer region of the double layer (2 < |z| < 5) dn1 > dn2 on the right-hand side, and dn2 > dn1 on the left-hand side. This is known as type II DLP, which induces another electric field with a direction opposite to that induced by type I DLP. In our case, because both the diffusiophoretic velocity U and the reference velocity Ur are proportional to the applied concentration gradient rn0, U (=U/Ur) becomes independent of rn0. This is consistent with the experimental observation of Ebel et al. [1]. Fig. 6 illustrates the variation of the scaled diffusiophoretic velocity of a particle, U, as a function of ja for various combinations of k and |/r|. Under the conditions of small k and thin double layer, the amount of coins penetrated into the double layer as they diffuse across the particle is small, yielding a small electric repulsive force between the coions and the particle. In this case, the diffusiophoresis of the particle is dominated by DLP and
Fig. 7. Variations of the scaled diffusiophoretic velocity U as a function of k for various combinations of ja and |/r| at a = 1, b = 0; (a) |/r| = 1 and (b) |/r| = 5.
J.-P. Hsu et al. / Journal of Colloid and Interface Science 342 (2010) 598–606
chemiosmotic flow. For a rigid particle, the electric field comes from type I DLP is usually stronger than that from type II DLP [25]. Therefore, as seen in Fig. 6a, if ja is large, except for |/r| = 5, the particle is driven toward the high concentration side (U > 0). Note that the chemiosmotic flow, which arises from the excess pressure on the high concentration side and type I DLP, drives the fluid near the particle surface from the high concentration side to the low concentration side. That is, its presence has the effect of driving the particle toward the low concentration side. For |/r| 6 4, because the electric force coming from DLP is greater than the hydrodynamic drag coming from chemiosmotic flow, the particle moves to the high concentration side. However, as |/r| gets higher (=5), because type II DLP becomes stronger the electric force contributed by DLP is reduced. In this case, the hydrodynamic drag coming from chemiosmotic flow becomes stronger than the electric force contributed by DLP, and the particle is driven toward the low concentration side. If ja is small (thick double layer), the coions drive the particle toward the low concentration side as they diffuse across its surface, yielding a smaller total electric force, which can be negative. Therefore, the particle migrates to the low concentration side, regardless of the level of |/r|. Fig. 6a shows that if |/r| is sufficiently high, U has a local minimum as ja varies. This is because if ja is small, the surface charge density, and, therefore, the electrical driving force acting on the particle is small. A small ja also implies a low concentration of coions, yielding a small repulsive force between the particle and the coions as they diffuse across the particle. On the other hand, if ja is large, then the double layer is thin and the electric repulsive force between the particle and the coions is also small. A comparison between Fig. 6a and b reveals that as k increases from 0.2 to 0.5 the value of ja at which the local minimum of U occurs shifts to a larger value. This is because the thickness of the double layer at k = 0.5 exceeds the pore radius when ja ffi 1, and therefore, the electric repulsive force between the particle and the coins is stronger than that at k = 0.2. In this case, that force becomes significant, and the U in Fig. 6b becomes more negative than that in Fig. 6a. That is, if the pore is small, so is the space between the outer boundary of the double layer and the pore, and the coions must penetrate into the double layer as they diffuse
603
across the particle, which has the effect of driving the particle toward the low concentration side. Because k = 0 in Fig. 6c, the particle is very close to the pore and the double layer occupies the entire space between the particle and the pore even at a large ja (=10), making it difficult for the coions to diffuse across the particle. In this case, the electric repulsive force between the particle and the coions dominates, driving the particle to the low concentration side (U < 0). Note that |U| increases with increasing ja, which is different from the behavior of U in Fig. 6a and b. This is because a larger ja implies a higher charge density on the particle surface and a higher concentration of coions in the liquid phase. Note that the value of |U| in Fig. 6c is smaller than that in Fig. 6b. This is because an increase in k not only makes it more difficult for the coions to diffuse across the particle but also results in a stronger hydrodynamic drag force acting on the particle. The boundary effect arising from the presence of the pore on the diffusiophoretic behavior of the particle is illustrated in Fig. 7. If k is small, the smaller the ja (thicker double layer) the stronger the electric repulsive force acting on the particle due to the diffusion of the coions, yielding a smaller U. On the other hand, if k is large, because the space between the particle and the pore is small, the electric repulsive force between the particle and the coions is strong, regardless of the level of ja. The magnitude of this force depends upon the surface charge density of the particle; the higher the surface charge density the stronger the repulsive force. Therefore, |U| increases with increasing ja. Note that U shows a local minimum as k varies. The presence of this local minimum is the result of the competition between the electric force and the hydrodynamic drag acting on the particle. The magnitudes of these forces increase with increasing k. The increase in the magnitude of the electric force is due to a stronger repulsive force between the particle and the coions, and that of the hydrodynamic drag is due to a more significant boundary effect. Note that U is negative in Fig. 7b (/r = 5) even if the boundary effect is unimportant (large ja and small k). This is because type II DLP is more important at higher surface potentials, making the electric force coming from DLP smaller than the hydrodynamic drag arising from chemiosmotic flow.
Fig. 8. Contours of the scaled perturbed electric potential d/ (delta_psiN) on the half plane h = 0 at a = 1, b = 0.2, ja = 1, k = 0.5, /r = 3, n = 0.01.
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3.3. D1 – D2 If the diffusivity of cations is different from that of anions, such as in the case of an aqueous NaCl solution, then b – 0. In this case, because the ionic flux of cations in the bulk solution must equal to that of anions, the diffusion of the ionic species yields an electric field, as is justified in Fig. 8. This electric field, known as the background electric field, is parallel to the direction of ionic diffusion. If the diffusivity of cations is larger than that of anions (b > 0), the background electric field has the same direction as that of the im-
Fig. 9. Variations of the scaled diffusiophoretic velocity U as a function of ja for various combinations of /r and k at a = 1, b = 0.2, /r > 0; (a) k = 0.2, (b), k = 0.5, and (c) k = 0.8.
posed concentration gradient rn0. On the other hand, if the diffusivity of cations is smaller than that of anions (b < 0), the background electric field has the opposite direction as that of rn0. Because b = 0.2 in Fig. 8, the background electric field drives a positively charged particle to the low concentration side and a negatively charged particle to the high concentration side. This phenomenon was observed experimentally [1], and called electrophoresis. The presence of the electrophoresis effect makes the value of U in Fig. 9 smaller than the corresponding value of U in Fig. 6. The electrophoresis effect is more appreciable when the boundary effect is less significant. For example, Figs. 6 and 9 show that for ja = 1 and /r = 5, the difference between the values of U in these two figures is ca. 0.55 at k = 0.2, and is ca. 0.02 at k = 0.8. This is because if k is large (boundary effect significant), so is the hydrodynamic drag acting on the particle. In this case, U becomes less sensitive to the electrophoresis effect, as can be seen more clearly in Fig. 10. A comparison between Figs. 10 and 7 reveals that the amount of decrease in U as b varies from 0 to 0.2 at a smaller k is much larger than that at a larger k. For instance, at ja = 5 and k = 0.8, U is 0.02 in Fig. 7a and 0.05 in Fig. 10a. On the other hand, at ja = 5 and k = 0.2, U is 0.025 in Fig. 7a and 0.125 in Fig. 10a. Similarly, at ja = 5 and k = 0.8, U is 0.22 in Fig. 7b and 0.31 in Fig. 10b, and at ja = 5 and k = 0.2, U is 0.0 in Fig. 7b and 0.4 in Fig. 10b. In addition, because the electrophoresis effect is more important at larger ja, the larger the ja the larger the amount of decrease in U as b decreases from 0 to 0.2. As seen in Fig. 10b, where |/r| is high, if k takes a medium large value,
Fig. 10. Variations of the scaled diffusiophoretic velocity U as a function of k for various combinations of ja and /r at a = 1, b = 0.2; (a) /r = 1 and (b) /r = 5.
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the behavior of U as ja varies is complicated. This is because that in this case the effect of electrophoresis is not that important as that in the case of small k, the effect of DLP becomes comparable to that of chemiosmotic flow when |/r| is high, and the hydrodynamic drag due to the boundary effect is not that important as that in the case of large k. That is, the magnitudes of all the relevant forces are on the same order, yielding complicated behavior in U. If the particle is negatively charged and b < 0, then the background electric field drives the particle to the high concentration side. Therefore, the value of U in Fig. 11 is larger than the corre-
Fig. 11. Variations of the scaled diffusiophoretic velocity U as a function of ja for various combinations of /r and k at a = 1, b = 0.2, /r < 0; (a) k = 0.2, (b) k = 0.5, and (c) k = 0.8.
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sponding value of U in Fig. 6. These figures also reveal that the higher the |/r| the stronger the electrophoresis effect. As mentioned previously, this effect is more important at smaller values of k. In Fig. 11a, U is dominated by the electrophoresis effect and the electric repulsive force between the particle and the coions at small values of ja, and by the electrophoresis effect and DLP at large values of ja. For |/r| P 2, if ja < 0.4, U decreases with increasing ja because the increase in the surface charge density yields a stronger electric repulsive force between the particle and the coions. However, for /r = 1, U increases with increasing ja because an increase in the surface charge density leads to a stronger electrophoresis effect. Fig. 11a indicates that U increases with increasing ja for ja ranges from ca. 0.5 to 4. This is because the thinner the double layer the smaller the electric repulsive force between the particle and the coions. Because type II DLP is significant at high surface charge densities and the higher the surface charge density the more important it is, U decreases with increasing ja for ja exceeds ca. 4 and |/r| P 4. No general trend is observed for U in Fig. 11a as /r varies. This is because the higher the |/r| not only the more significant the electrophoresis effect, which drives the particle to the high concentration side, but also a stronger electric repulsive force between the particle and the coions, which drives the particle to the low concentration side. In addition, both type I and type II DLP are influenced by /r, as mentioned in the discussion of Fig. 3. Due to a more significant boundary effect, the electrophoresis effect in the case of Fig. 11b and c is less important than that in the case of Fig. 11a. The qualitative behaviors of U as /r, ja, and
Fig. 12. Variations of the scaled diffusiophoretic velocity U as a function of k for various combinations of ja and /r at a = 1, b = 0.2; (a) /r = 1 and (b) /r = 5.
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k vary in Fig. 11b and Fig. 11c are similar to those in Fig. 6b and Fig. 6c. The crosses of the curves near ja = 0.1 in Fig. 11b and Fig. 11c arise mainly from the electrophoresis effect. As mentioned previously, the higher the |/r| the larger is the amount of increase in U as b decreases from 0 to 0.2. Fig. 12 suggests that the electrophoresis effect is more appreciable at a smaller k and a larger ja, which is consistent with the results presented in Figs. 9 and 10. For instance, the difference between the values of U in Figs. 7a and 12a is ca. 0.15 at k = 0.2 and ja = 5, and is ca. 0.05 at k = 0.8 and ja = 5. 4. Summary The boundary effect on the diffusiophoretic behavior of a particle is analyzed by considering the diffusiophoresis of a charged spherical particle along the axis of an uncharged cylindrical pore filled with an electrolyte solution. The influences of the radius of the pore, the diffusivities of ionic species, the thickness of double layer, and the surface potential of the particle on the diffusiophoretic velocity of the particle are examined. We show that if the radius of the pore is large (boundary effect unimportant) and/or the double layer is thin, the diffusiophoretic behavior of the particle is dominated by two types of double-layer polarization (DLP). Because type I DLP is usually more important than type II DLP, most of the previous studies focused on the former, making it difficult to explain some observed phenomena. For example, under the conditions of high surface potential, thin double layer, and inappreciable boundary effect, where type II DLP dominates, a particle moves to the low concentration side, and the thinner the double layer the smaller is its diffusiophoretic velocity. Here, we show that if the double layer is thin and/or the surface potential is high, type II DLP can be more important than type I DLP. If the radius of the pore is small and/or the double layer is thick, then the diffusiophoretic behavior of the particle is dominated by the repulsive force between the particle and the coions outside the double layer as they diffuse across the particle. This force, which is not observed in the case of a sphere in a spherical cavity, drives the particle to the low concentration side. If the diffusivity of cations is different from that of anions, then the electrophoresis effect is present; the less significant the boundary effect the more important thus effect. Depending upon the difference between the diffusivity of cations and that of anions, and the sign of the surface charge of the particle, the electrophoresis effect is capable of driving the particle toward either the low concentration side or the high concentration side.
Acknowledgment This work is supported by the National Science Council of the Republic of China. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41]
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