pubs.acs.org/Langmuir © 2010 American Chemical Society
Diffusiophoresis of a Charge-Regulated Sphere along the Axis of an Uncharged Cylindrical Pore Jyh-Ping Hsu,* Wei-Lun Hsu, and Kuan-Liang Liu Department of Chemical Engineering, National Taiwan University, Taipei, Taiwan 10617 Received December 16, 2009. Revised Manuscript Received February 8, 2010 The diffusiophoresis of a charge-regulated spherical particle along the axis of a cylindrical pore is modeled. The problem considered allows us to examine simultaneously the effects of the presence of a boundary, the charge conditions on the particle surface, the thickness of the double layer, and the nature of a dispersion medium including its pH value and the diffusivities of the ionic species with respect to the diffusiophoretic behavior of a particle. We show that, in addition to the factors of double-layer polarization, electrophoresis, and the osmotic flow of solvent, the diffusiophoretic behavior of the particle can also be affected significantly by the electrical repulsive interaction between the particle and the co-ions immediately outside the double layer as they diffuse through the gap between the double layer and the pore. The last effect, which can influence the diffusiophoretic behavior of a particle both quantitatively and qualitatively, is absent in the diffusiophresis of a sphere in a spherical cavity. The competition of those effects yields many interesting results that are of practical significance in the design of a diffusiophoretic apparatus and/or the interpretation of experimental data.
Introduction A particle of colloidal size suspended in a solution can be driven by nonuniformly distributed solute, which is known as diffusiophoresis.1-4 In addition to many potential applications, such as particle coating5,6 and aerosols deposition,7,8 diffusiophoresis may also play a role in the exchange of information between living cells and organisms.9 For the case in which the solute is composed of electrolytes and the particle is charged, the diffusiophoretic behavior of the particle is complicated by the presence of various types of forces. For example, the application of a concentration gradient makes the double layer near the highconcentration of the particle thinner than that on the lowconcentration side. This deformation in the double layer, known as type I double-layer polarization (DLP),10 drives the particle to the high-concentration side. Similarly, because the concentration of co-ions immediately outside the double layer on the highconcentration side is higher than that on the low-concentration side, a local electric field is induced, which is known as type II DLP.10 The direction of the electric field coming from type II DLP is opposite to that coming from type I DLP; the former drives the particle toward the low-concentration side. In general, the electric field coming from type I DLP is stronger than that from type II DLP. The difference in the diffusivity of counterions *Corresponding author. Tel: 886-2-23637448. Fax: 886-2-23623040. E-mail:
[email protected]. (1) Deryagin, B. V.; Dukhin, S. S.; Korotkova, A. A. Kolloidn. Zh. 1961, 23, 409. (2) Dukhin, S. S.; Deryagin, B. V. Surface and Colloid Science; Wiley: New York, 1974; Vol. 7. (3) Prieve, D. C.; Anderson, J. L.; Ebel, J. P.; Lowell, M. E. J. Fluid Mech. 1984, 148, 247. (4) Staffeld, P. O.; Quinn, J. A. J. Colloid Interface Sci. 1988, 130, 69. (5) Korotkova, A. A.; Deryagin, B. V. Colloid J. USSR 1991, 53, 719. (6) Yamenko, A. B.; Sednev, D. V. Colloid J. 1995, 57, 396. (7) Hidy, G. M.; Brock, J. R. Environ. Sci. Technol. 1969, 3, 563. (8) Kousaka, Y.; Endo, Y. J. Aerosol Sci. 1993, 24, 611. (9) Ibele, M.; Malouk, T. E.; Sen, A. Angew. Chem., Int. Ed. 2009, 48, 3308. (10) Zhang, X.; Hsu, W. L.; Hsu, J. P.; Tseng, S. J. Phys. Chem. B 2009, 113, 8646. (11) Misra, S.; Varanasi, S.; Varanasi, P. P. Macromolecules 1990, 23, 4258. (12) Hsu, J. P.; Lou, J.; He, Y. Y.; Lee, E. J. Phys. Chem. B 2007, 111, 2533. (13) Keh, H. J.; Li, Y. L. Langmuir 2007, 23, 1061.
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and that of co-ions also induces an electric field,11-14 and the corresponding effect is known as the electrophoresis effect.1-4 For instance, because the diffusivity of Naþ (1.33 10-9 [m2/s]) in an aqueous NaCl solution is smaller than that of Cl- (2.03 10-9 [m2/s]),15 an electric field directed toward the low-concentration side is established. This electric field comes from the electric force between counterions and co-ions: the ions with larger diffusivity speed up the ions with smaller diffusivity. Depending upon the charged conditions on the particle surface and the diffusivities of ionic species, the electrophoresis effect is capable of driving the particle toward either the low- or highconcentration side. If the diffusivity of cations is larger than that of anions, then the direction of the electric field coming from the electrophoresis effect is in the same direction as that of the applied concentration gradient and it is in the opposite direction to that of the applied concentration gradient if the diffusivity of cations is smaller than that of anions. In diffusiophoresis, the movement of a particle driven by the electrophoresis effect is usually called electrophoresis, but it is called chemiphoresis if the particle is driven solely by a nonuniformly distributed solute. The osmotic solvent flow16,17 inside the double layer surrounding a particle due to the excess pressure on the high-concentration side and the electric field induced by the solute concentration gradient also play important roles in the diffusiophoresis of the particle. The direction of the osmotic solvent flow is affected by the pressure and the electric fields coming from type I DLP, type II DLP, and the electrophoresis effect. The osmotic solvent flow in diffusiophoresis, known as diffusioosmotic flow, comprises two components: that caused by the difference in the diffusivities of ionic species, called electroosmotic flow, and that caused solely by the nonuniformly distributed solute, called chemiosmotic flow. Because the above-mentioned driving forces including those coming from the double-layer polarization, the electrophoresis effect, and the solvent flow across a particle are not independent (14) (15) (16) (17)
Ebel, J. P.; Anderson, J. L.; Prieve, D. C. Langmuir 1988, 4, 396. Chun, B.; Ladd, A. J. C. J. Colloid Interface Sci. 2004, 274, 687. Keh, H. J.; Hsu, L. Y. Microfluid. Nanofluid. 2008, 5, 347. Keh, H. J.; Ma, H. C. J. Power Sources 2008, 180, 711.
Published on Web 02/25/2010
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Figure 1. Problem considered: the diffusiophoresis of a spherical particle of radius a along the axis of a cylindrical pore of radius b; (r, θ, z) represents the cylindrical coordinates adopted with the origin at the center of the particle; 3n0 is the applied concentration gradient in the z direction.
of each other, the diffusiophoretic behavior of the particle becomes complicated and interesting. Previous analyses of diffusiophoresis include that conducted in nonelectrolyte solutions18-20 and in electrolyte solutions.3,4,14,21 Both theoretical3,18,19 and experimental results4,14,20 are available in the literature. Several attempts were made with respect to the diffusiophoresis of a particle for the case where a boundary is present.10,22-27 It was found that the influence of a boundary on the diffusiophoretic behavior of a particle can be profound. Because the presence of the boundary effect is not uncommon in practice, a detailed understanding of the role it plays in various possible geometries is very desirable. In this study, the diffusiophoresis of a charge-regulated28-31 spherical particle along the axis of an uncharged cylindrical pore filled with electrolyte solutions is analyzed. Through the geometry considered, the boundary effect on the diffusiophoretic behavior of a particle can be investigated. The charge-regulated nature of the particle implies that its surface may not be maintained at a constant potential or constant charge density, as is usually assumed in relevant studies. This simulates more realistically the charge conditions of biocolloids such as biological cells.32-34 Numerical simulations are conducted to examine the influences of the thickness of the double layer, the relative size of the cylindrical pore, the diffusivities of the ionic species, the pH of the bulk liquid phase, and the charge conditions on the particle surface on the diffusiophoretic behavior of a particle.
Theory The problem under consideration is illustrated in Figure 1, where a spherical particle of radius a is driven by an applied uniform concentration gradient 3n0, with n0 being the bulk solute concentration, along the axis of a cylindrical pore of radius b. The surface of the particle is charge-regulated, and the pore is filled with an aqueous, incompressible Newtonian fluid containing z1/z2 electrolytes, with z1 and z2 being the valences of cations (18) Keh, H. J.; Lin, Y. M. Langmuir 1994, 10, 3010. (19) Anderson, J. L.; Lowell, M. E.; Prieve, D. C. J. Fluid Mech. 1982, 117, 107. (20) Staffeld, P. O.; Quinn, J. A. J. Colloid Interface Sci. 1988, 130, 88. (21) Abecassis, B.; Cottin-Bizonne, C.; Ybert, C.; Ajdari, A.; Bocquet, L. Nat. Mater. 2008, 7, 785. (22) Keh, H. J.; Jan, J. S. J. Colloid Interface Sci. 1996, 183, 458. (23) Lou, J.; Lee, E. J. Phys. Chem. C 2008, 112, 2584. (24) Chen, P. Y.; Keh, H. J. J. Colloid Interface Sci. 2005, 286, 774. (25) Chang, C. Y.; Keh, H. J. J. Colloid Interface Sci. 2008, 322, 634. (26) Hsu, J. P.; Hsu, W. L.; Chen, Z. S. Langmuir 2009, 25, 1772. (27) Lou, J.; Shih, C. Y.; Lee, E. J. Colloid Interface Sci. 2005, 286, 774. (28) Ninham, B. W.; Parsegian, V. A. J. Theor. Biol. 1971, 31, 405. (29) Chan, D.; Perram, J. W.; White, L. R.; Healy, Y. W. J. Chem. Soc., Faraday Trans. 1 1975, 71, 1046. (30) Hsu, J. P.; Hung, S. H. J. Colloid Interface Sci. 2003, 264, 121. (31) Lou, J.; Shih, C. Y.; Lee, E. Langmuir 2010, 26, 47. (32) Ji, J. H.; Bae, G. N.; Yun, S. H.; Jung, J. H.; Noh, H. S.; Kim, S. S. Aerosol Sci. Technol. 2007, 41, 786. (33) Rodriguez, V. V.; Busscher, H. J.; Norde, W.; van der Mei, H. C. Electrophoresis 2002, 23, 2007. (34) Leitch, E. C.; Willcox, M. D. P. Curr. Eye Res. 1999, 19, 12.
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and anions, respectively. (r, θ, z) denotes the cylindrical coordinates with the origin at the center of the particle; 3n0 is in the z direction. The present problem is 2D, where only the (r, z) coordinates need to be considered, with 0 e r e b and - ¥ < z < ¥. To describe the diffusiophoretic behavior of the particle, the following equations, which include those for the electric, the concentration, and the flow fields, need to be solved simultaneously:10,12,23,27 r2 φ ¼ "
2 X F zj enj ¼ ε ε j ¼1
# zj e nj rφ þ nj v ¼ 0 r 3 - Dj rnj þ kB T
ð1Þ
ð2Þ
r3v ¼ 0
ð3Þ
- rp þ ηr2 v - Frφ ¼ 0
ð4Þ
In these expressions, φ, ε, F, e, kB, T, nj, Dj, v, η, and p are the electrical potential, the permittivity of the liquid phase, the space charge density, the elementary charge, the Boltzmann constant, the absolute temperature, the diffusivity of ionic species j, the velocity, the viscosity, and the pressure of the liquid phase, respectively. Instead of solving eqs 1-4 directly for dependent variables φ, nj, v, and p, each of them is partitioned into an equilibrium component, which is the value of the dependent variable when 3n0 is not applied, and a perturbed component, which comes from the application of 3n0.10,12,23,27 That is, we let φ = φe þ δφ, v = ve þ δv, p = pe þ δp, and nj = njeþδnj, where the subscript e and the prefix δ denote the equilibrium component and the perturbed component, respectively. Note that ve = 0 because the particle is stagnant when 3n0 is not applied. The partitioning of the original problem into an equilibrium problem and a perturbed problem has the advantage that the set of nonlinear equations (eqs 1-4) reduces to a set of linearized equations. For convenience, nj is expressed as zj eðφe þ δφ þ gj Þ nj ¼ nj0e exp kB T
ð5Þ
where gj is a perturbed function and nj0e is the bulk ionic concentration of ionic species j at equilibrium. Similar to the case where an external electric field is applied,30 eq 5 is capable of describing the deformation of the double layer surrounding a charged particle when 3n0 is applied. Under the condition that a|3n0| , n0e, with n0e being the bulk solute concentration when 3n0 is not applied, eqs 1-5 lead to10,26
r 2 φe ¼
r 2 δφ -
ðKaÞ2 ½expð- φe Þ - expðRφe Þ 1þR
ð6Þ
ðKaÞ2 ½expð- φe Þ þ R expðRφe Þδφ 1þR
ðKaÞ2 ½expð- φe Þg1 þ expðRφe ÞRg2 1þR
ð7Þ
r 2 g1 - r φe 3 r g1 ¼ ξPe1 v 3 r φe
ð8Þ
r 2 g2 þ Rr φe 3 r g2 ¼ ξPe2 v 3 r φe
ð9Þ
¼
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Hsu et al. r 3 v ¼ 0
ð10Þ
- r δp þ ξr 2 v þ r 2 φe r δφ þ r 2 δφr φe ¼ 0 ð11Þ In these expressions, R = -z2/z1, 3* = a3, 3*2 = a232, κ = [Σ2j=1nj0e(ezj)2/εkBT]1/2, φe* = φez1e/kBT, δφ* = δφz1e/kBT, v*=v/[εξ(kBT/z1e)2/aη], ξ = 3*n0*, and δp* = δp/[ε(kBT/ z1e)2/a2]. gj* = gjz1e/kBT and Pej = ε(kBT/z1e)2/ηDj, j = 1, 2, with Pej being the electric Peclet number of ionic species j. For the present case, the charge conditions on the particle surface depend upon the degree of dissociation of the function groups it carries. Assume that the particle surface contains multiprotonic functional groups AHZa with dissociation constant Ka,n, n = 1, 2,..., Za. Then ðn - 1Þ AHZa - ðn - 1Þ
Ka, n ¼
S
AHnZ-a - n þ Hþ ,
½AHnZ-a - n s ½Hþ s ðn - 1Þ -
½AHZa - ðn - 1Þ s
n ¼ 1, 2, :::, Za
, n ¼ 1, 2, :::, Za
ð12Þ
ð13Þ
Here, Za is the number of dissociable protons in AHZa, [AHZa-nn-]s and [AHZa-(n-1)(n-1)-]s are the numbers of AHZa-nn- and AHZa-(n-1)(n-1)- per unit of particle surface area, respectively, and [Hþ]s is the concentration of Hþ on the particle surface. If we let Ns be the number of functional groups per unit particle surface area, including both dissociated and nondissociated groups, that is, Ns ¼
ðn - 1Þ -
½AHZa - ðn - 1Þ
ð14Þ
then eqs 13 and 14 yield n Q
Ka, k Ns þ ½H k ¼1 " #, n ¼ 1, 2, :::, Za ½AHnZ-a - n s ¼ Z m Pa Q Ka, k 1þ þ m ¼1 k ¼1 ½H
ð15Þ
Assume that the spatial variation of the concentration of Hþ follows the Boltzmann distribution eφ ½Hþ ¼ ½Hþ 0 exp - e kB T
ð16Þ
with [Hþ]0 being the bulk concentration of Hþ. Then the charge density on the particle surface, σ = -eΣZan=1n[AHn-Za-n], can be expressed by30 2 3 - eNs
Za 6 m P 6 Q 6m 4 k ¼1 m ¼1
2
σ ¼ 1þ
Za 6 m P 6Q 6 4 m ¼1 k ¼1
Ns Qa
n 3 r φe ¼
7 Ka, k 7 7 5 eζ ½Hþ 0 exp kB T 3
ð17Þ
7 Ka, k 7 7 eζ 5 ½Hþ 0 exp kB T
where ζ is the surface potential. We assume the following: (a) Both the particle and the pore are nonconductive. (b) The net charge flux across the plane z = 8650 DOI: 10.1021/la904726k
Pa þ ½Hþ 0 expðφe Þ
on the particle surface
φe ¼ 0 on the pore surface
ð18Þ ð19Þ
φe ¼ 0, jzj f ¥, r < b
ð20Þ
n 3 r/ δφ/ ¼ 0 on the particle surface
ð21Þ
n 3 r/ δφ/ ¼ 0 on the pore surface
ð22Þ
r δφ ¼ - βγez , jzj f ¥, r < b
n 3 r gj ¼ 0 on the particle surface
ð23Þ ð24Þ
n 3 r gj ¼ 0 on the pore surface
ð25Þ
ð26Þ
g1 ¼ - zγ - δφ, jzj f ¥, r < b
ZX a þ1 n ¼1
constant vanishes; that is, (z1J1 þ z2J2) 3 z = 0, with Jj being the ion flux of ionic species j.11-14 (c) Both the surface of the particle and that of the pore are ion-impermeable and nonslip. (d) The concentration of ionic species in the bulk liquid phase reaches the value (nj0e þ z3nj0).35,36 (e) The liquid phase is stagnant at a point far away from the particle. These assumptions lead to the following boundary conditions:
g2 ¼
zγ - δφ, jzj f ¥, r < b R
ð27Þ
v ¼ 0 on the particle surface
ð28Þ
v ¼ - U ez on the pore surface
ð29Þ
v ¼ - U ez , jzj f ¥, r < b
ð30Þ
In these expressions, z* = z/a, Ns* = e2a2Ns/εkBT, [Hþ]0* = [Hþ]0/Ka,1, Qa = ΣZam=1[mΠmk=1(Ka,k/[Hþ]]/Ka,1/[Hþ], Pa = ΣZam=1[Πmk=1 (Ka,k/[Hþ])]/Ka,1/[Hþ], β = (D1 - D2)/(D1 þ RD2), and U* = U/[εξ(kBT/z1e)2/aη], where U is the diffusiophoretic velocity of the particle. n is the unit normal vector directed into the liquid phase, and ez is the unit vector in the z direction. Let F be the total force acting on the particle in the z direction and let F be its magnitude. Two types of forces are considered, namely, the electrical force Fe with magnitude Fe and the hydrodynamic force Fd with magnitude Fd; that is, F = Fe þ Fd and F = Fe þ Fi. Because we assume that the system is in a pseudosteady state, the diffusiophoretic velocity U, which is unknown in the present stage, should make the total force acting on the particle vanish. In general, U needs to be determined through a trial-anderror procedure. This tedious procedure can be avoided by partitioning the original problem into two subproblems:10,12,23,27 (a) the particle moves at a constant velocity U in the absence of 3n0 and (b) 3n0 is applied but the particle remains fixed. Mathematically, it can be shown that these two subproblems are equivalent to the original problem, both in the governing equations and in the boundary conditions. Physically, it has the advantage that we are able to examine separately the influence of each relevant force on the diffusiophoretic behavior of a particle. For example, the (35) Wei, Y. K.; Keh, H. J. Langmuir 2001, 17, 1437. (36) Keh, H. J.; Huang, T. Y. Colloid Polym. Sci. 1994, 272, 855.
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electric force in the second problem comes solely from the applied concentration gradient. On the basis of these, F is partitioned into F1, the total force acting on the particle in the z direction in the first subproblem, and F2, that in the second subproblem. If we let F1 and F2 be the magnitudes of F1 and F2, respectively, then F1 = χ1U and F2 = χ23n0, where χ1 is independent of U and χ2 is independent of 3n0. At steady state, F1 þ F2 = 0, yielding U ¼ -
χ2 rn0 χ1
ð31Þ
This velocity needs to be substituted into the original problem to determine if the total force acting on the particle vanishes. If we let Fei and Fdi be the values of Fe and Fd in subproblem i, with i = 1, 2, and Fei and Fdi be the corresponding magnitudes, then10,12,23,27,37-39 ! Z Dφe Dδφ Dφe Dδφ nz dS, i ¼ 1, 2 ð32Þ Fei ¼ Dz Dt Dt S Dn
Fdi ¼
Z
S
ðσ H 3 nÞ 3 e z dS , i ¼ 1, 2
ð33Þ
Here, Fdi* = Fdi/ε(kBT/z1e)2, Fei* = Fei/ε(kBT/z1e)2, and S* = S/ a2 with S being the particle surface; n and t are the magnitudes of the unit normal vector and the unit tangential vector, respectively; nz is the z component of n. σH* = σH/[ε(kBT/z1e)2/a2] with σH being the shear stress tensor.
Results and Discussion The governing equations and the associated boundary conditions are solved numerically by a finite element method using FlexPDE.40 To ensure that the end effect can be neglected, the length of a pore is chosen to be 15 times the diameter of a particle.41,42 The applicability of the software adopted is examined by considering first the case of an isolated particle with constant surface potential, where numerical43 and analytical44 results are available in the literature. To this end, we define λ = a/b, which measures the significance of the boundary effect; the smaller the λ value, the less significant the boundary effect. The limiting case of λ f 0 corresponds to the boundary-free case. Figure 2a indicates that the absolute value of the scaled equilibrium surface potential ζ* (= ζz1e/kBT), |ζ*|, increases with increasing Ns*, which is expected because the larger the Ns* value, the higher the concentration of the dissociable functional groups on the particle surface, yielding a higher surface charge density. On the basis of the relationship between ζ* and Ns* shown in Figure 2a, the surface conditions necessary for the construction of Figure 2b,c can be determined. The latter reveals that if |ζ*| is low, the present result is essentially the same as of that of Prieve and Roman43 and Keh and Wei.44 However, as |ζ*| becomes high, the result of Keh and Wei44 deviates from that of Prieve and Roman43 or the present result. This is expected because the result of Keh and Wei is valid for low surface potentials only. Note that the (37) Hsu, J. P.; Yeh, L. H.; Ku, M. H. J. Colloid Interface Sci. 2007, 305, 324. (38) Hsu, J. P.; Yeh, L. H. J. Chin. Inst. Chem. Eng. 2006, 37, 601. (39) Happel, J.; Brenner, H. Low-Reynolds Number Hydrodynamics; Nijhoff: Boston, 1983. (40) FlexPDE, version 2.22; PDE Solutions Inc.: Spokane Valley, WA, 2001. (41) Hsu, J. P.; Chen, Z. S. Langmuir 2007, 23, 6198. (42) Hsu, J. P.; Hung, S. H.; Kao, C. Y.; Tseng, S. Chem. Eng. Sci. 2003, 58, 5339. (43) Prieve, D. C.; Roman, R. J. Chem. Soc., Faraday Trans. 2 1987, 83, 1287. (44) Keh, H. J.; Wei, Y. K. Langmuir 2000, 16, 5289.
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Figure 2. (a) Variation of the scaled surface potential ζ* as a function of the scaled total number of acidic functional groups per unit particle surface area Ns*. (b, c) Variation of the diffusiophoretic velocity U* as a function of the scaled surface potential ζ* at R = 1, Za = 1, pKa,1(= -log Ka,1) = 5, pH(= -log[Hþ]0) = 7, λ = 0.05. β = (b) 0 and (c) -0.2; (-) present results; (---) numerical results of Prieve and Roman;43 and ( 3 3 3 ) analytical results of Keh and Wei.44.
value of U* for the case of β = -0.2 (aqueous NaCl solution) is much larger than the corresponding value of U* for the case of β = 0 (aqueous KCl solution). This arises from the effect of electrophoresis, which drives the negatively charged particle toward the high-concentration side.
Spatial Variations of δφ* and v* Note that whereas the solution of the first subproblem provides the magnitudes of the hydrodynamic drag and the electric force acting on a particle,26 which affect the magnitude of the DOI: 10.1021/la904726k
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Figure 3. (a) Contours of the scaled perturbed electric potential δφ* (delta_psiN) on the θ = 0 half-plane in the second subproblem and (b) the corresponding variation of the scaled perturbed electric potential δφ* (delta_psiN) along the axis of the pore for the case where R = 1, Za = 1, pKa,1 = 5, pH = 7, ξ = 0.01, and λ = 0.05 at β = 0; those at β = -0.2 are shown in plots c and d.
diffusiophoretic velocity of the particle, the direction of diffusiophoresis needs to be determined from the solution of the second subproblem, in particular, the sign of the sum Fe2*þFd2*. In a recent study of the diffusiophoresis of a sphere of constant surface potential in a cylindrical pore, Hsu et al.45 pointed out that the electrical interaction between the particle and the co-ions as they diffuse across the particle can play a role. This is justified by Figure 3, where the contours of the scaled perturbed electric potential δφ* on the θ = 0 half-plane in the second subproblem and the variation of δφ* along the axis of the pore are plotted. Figure 3a reveals that a local electric field directed toward the high-concentration side is present, which drives the charged particle toward the low-concentration side. As can be explained by the variation in the scaled perturbed concentration δnj*=δnj/ nj0e (delta_n1N and delta_n2N) along the axis of the pore shown in Figure 4a, this local electric field does not come from type II DLP. This Figure indicates that because type I DLP is more important than type II DLP the net electric force coming from DLP should drive the particle toward the high-concentration side. However, as can be inferred from Figure 3, the net electric force drives the particle toward the low-concentration side, implying that another electric force might be present and it can be more important than the electric force coming from DLP. Hsu et al.45 suggested that the extra electrical force may arise from the diffusion of the co-ions across the particle surface. The diffusiophoretic behavior of a particle for the case where the electrophoresis effect is present is more complicated than that for the case where it is absent. Figure 3c,d reveals that in the inner region of the double layer (0 < |z| < 1.2), δφ* > 0 on the high(45) Hsu, J. P.; Hsu, W. L.; Ku, M. H.; Chen, Z. S. J. Colloid Interface Sci. 2010, 342, 598.
8652 DOI: 10.1021/la904726k
concentration side but δφ* < 0 on the low-concentration side. That is, the direction of the induced electric field coming from type I DLP is opposite to that of the concentration gradient and that the induced electric field drives the particle toward the highconcentration side. In the outer region of the double layer (|z|>1.2), δφ* < 0 on the high-concentration side and δφ* > 0 on the low-concentration side, which arises from two effects. The first effect comes from type II DLP, where more co-ions accumulate on the high-concentration side than on the low-concentration side. The second effect arises from co-ions driving the particle toward the low-concentration side as they diffuse across its surface. Outside the double layer (ca. |z| > 5), a uniform electric field is observed, where δφ* > 0 on the high-concentration side and δφ* < 0 on the low-concentration side. This yields an electric field with a direction opposite to that of the applied concentration gradient. The presence of this electric field comes mainly from the difference in the ionic diffusivities. A comparison between parts a and b Figure 4 reveals that, because they are close to each other both quantitatively and qualitatively, the degree of DLP is insensitive to the value of β. This suggests that DLP depends mainly on the imposed concentration gradient; it is insensitive to the difference in the ionic diffusivities. The contours of the scaled velocity field in the second subproblem on the θ = 0 half-plane for different values of β are shown in Figure 5. Figure 5a suggests that the solvent in the inner part of the double layer is driven by the excess pressure on the highconcentration side and the induced electric field coming from type I DLP. This yields an osmotic flow of solvent from the highconcentration side to the low-concentration side along the particle surface. This flow drives the particle toward the low-concentration side. Figure 5a,b indicates that the osmotic flow of solvent at β = -0.2 is more appreciable compared with that at β = 0. This is Langmuir 2010, 26(11), 8648–8658
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Figure 4. (a) Variation in the scaled perturbed concentration δnj*=δnj/nj0e (delta_n1N and delta_n2N) along the axis of the pore for the case of Figure 3a and (b) that for the case of Figure 3c.
because in the present case the electric field induced by electrophoresis is in the same direction as that induced by type I DLP.
Effect of Double-Layer Thickness κa The influence of the thickness of the double layer on the diffusiophoretic behavior of a particle is illustrated in Figure 6, where the scaled velocity U* is plotted against κa at various values of Ns* for β = 0 and -0.2. The corresponding variations of the scaled electric force, Fe2*, and the scaled hydrodynamic force, Fd2*, acting on the particle in the second subproblem are presented in Figures 7 and 8, respectively. As seen in Figure 6a, where the effect of electrophoresis is absent (β = 0) unless κa exceeds ca. 10, the scaled diffusiophoretic velocity U* is negative. This implies that the particle tends to move to the low-concentration side unless the double layer is sufficiently thin. The behavior of U* in Figure 6a can be explained by the variations of Fe2* and Fd2* shown in Figures 7a and 8a, respectively. Figure 7a indicates that if κa is smaller than ca. 1.5 then Fe2* is negative, and it becomes positive when κa exceeds ca. 1.5. If κa is small, then the movement of the particle is dominant by the repulsive force Langmuir 2010, 26(11), 8648–8658
between the co-ions immediately outside the double layer and the particle. These co-ions penetrate into the double layer and drive the particle toward the low-concentration side as they diffuse from the high-concentration side to the low-concentration side. As seen in Figure 7a, Fe2* has a minimum in the interval 0.4 < κa < 0.5. This is because a larger κa implies a higher ionic concentration and therefore a greater repulsive force between the co-ions immediately outside the double layer and the particle. However, a larger κa also implies a thinner double layer and therefore a larger space between the outer boundary of the double layer, and the pore increases. That is, it is easier for the co-ions immediately outside the boundary layer to diffuse across the particle from the high-concentration side to the low-concentration side, resulting in a smaller repulsive force between the co-ions immediately outside the double layer and the particle. Therefore, if κa is sufficiently large, then the movement of the particle is no longer dominated by the repulsive force between the co-ions immediately outside the double layer and the particle but is dominated by the effect of DLP. As stated previously, two types of DLP are present, where type I DLP drives the particle toward the high-concentration side and type II drives the particle toward DOI: 10.1021/la904726k
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Figure 5. Contours of the scaled velocity field v* (vN) on the θ = 0 half-plane in the second subproblem at R = 1, Za = 1, pKa,1 = 5, pH = 7, ξ = 0.01, and λ = 0.05; β = (a) 0 and (b) -0.2.
the low-concentration side. As seen in Figure 7a, Fe2* is positive at large values of κa, implying that the electric field coming from type I DLP is stronger than that coming from type II DLP. In fact, the electric field coming from type I DLP is much stronger than that coming from type II DLP in most cases. This is because the concentration of the counterions inside the double layer is much higher than that of the counterions immediately outside the double layer and therefore the induced electric field arising from the former is much stronger than that arising from the latter. However, the strength of the electric field coming from type II DLP can be comparable to that coming from type I DLP. This occurs, for instance, if the ionic concentration and/or the surface potential of the particle is sufficiently high, as is justified by Figure 7a, where Fe2* has a local maximum in the interval 5 < κa < 6. This is because an increase in κa yields both a stronger type I DLP electric field and a stronger type II DLP, especially when κa is large. If the rate of increase in the strength of the type I DLP electric field as κa increases is slower than that of the type II DLP electric field, then Fe2* decreases (6 < κa < 10). That is, if κa is sufficiently large, then the diffusiophoretic velocity of a particle 8654 DOI: 10.1021/la904726k
may decrease with increasing κa. This phenomenon is observed both theoretically26 and experimentally.14 The behavior of U* for the case where the effect of electrophoresis is present (β = -0.2), as shown in Figure 6b, is more complicated than that for the case where the effect of electrophoresis is absent, as shown in Figure 6a. In the former, most values of U* are positive; that is, the particle tends to move to the high-concentration side. This is because the dissociation of Hþ from the functional groups on the particle surface makes it negatively charged and the difference in diffusivity between cations and anions yields an induced electric field, which is directed toward the low-concentration side. Therefore, the values of Fe2* in Figure 7b, most of which are positive, are larger than the values of |Fe2*| in Figure 7a, almost half of which are negative. We conclude that a negatively charged particle in a solution with β < 0 (e.g., an aqueous NaCl solution) is more likely to move to the high-concentration side compared to that in a solution with β = 0 (e.g., an aqueous KCl solution). Figure 6 also reveals that if β = 0, then the larger the Ns*, the larger the |U*|, which may not be the case if β = -0.2. The latter arises from the competition between Langmuir 2010, 26(11), 8648–8658
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Figure 6. Variations of the scaled diffusiophoretic velocity U* as a
function of κa for various levels of Ns* at R = 1, λ = 0.5, Za = 1, pKa,1 = 5, and pH = 7; β = (a) 0 and (b) -0.2.
chemiphoresis and electrophoresis; the former drives the particle toward the low-concentration side, but the latter drives the particle toward the high-concentration side. The relative significance of these two effects depends upon the level of κa. Referring to eq 4, the electric field and the nonuniform pressure distribution yield solvent flow inside the double layer. The scaled hydrodynamic drag acting on the particle in the second subproblem, Fd2*, correlates with that flow, especially for the solvent close to the particle surface. Figure 8a (β = 0) shows that Fd2* is negative, implying that the solvent in the inner part of the double layer flows from the high-concentration side to the low-concentration side, driving the particle toward the low-concentration side. In addition, |Fd2*| has a local maximum. This is because for small to medium-large values of κa both the excess pressure on the high-concentration side and the electric field coming from type I DLP drive the flow of solvent inside the double layer to the lowconcentration side. In this case, because the strength of the electric field coming from type I DLP increases with increasing κa, so does |Fd2*|. The increase in κa also increases the strength of the electric field coming from type II DLP, which drives the flow of solvent outside the double layer to the high-concentration side. Therefore, if κa is sufficiently large, then |Fd2*| decreases with increasing κa. A comparison between parts a and b of Figure 8 (β = -0.2) reveals that the presence of the electrophoresis effect arising from the difference in the diffusivities of cations and anions makes not only a larger |Fd2*| but also causes the disappearance of the local maximum in |Fd2*|. This is because the electrophoresis effect drives the solvent inside the double layer to the low-concentration side, known as electroosmotic flow, causing |Fd2*| to increase with κa even at larger values of κa. That is, the sum of the rate of increase of electroosmotic flow and that of type I chemiosmotic Langmuir 2010, 26(11), 8648–8658
Figure 7. Variations of the scaled electric force in the second subproblem, Fe2*, as a function of κa for the case of Figure 6 with ξ = 0.01.
flow is larger than the rate of increase of type II chemiosmotic flow. However, as justified by Figure 7b, type II chemiosmotic flow still plays an important role, especially when κa is sufficiently large. For instance, the absolute value of the slope of the curve corresponding to Ns* = 5 starts to decline as κa exceeds ca. 1.
Effect of Pore Size λ The variations of the scaled diffusiophoretic velocity U* as a function of λ at various combinations of κa and β are shown in Figure 9. In Figure 9a, where the electrophoresis effect is absent (β = 0), |U*| shows a local maximum as λ varies. The presence of the local maximum is the result of the competition between the electric force and the hydrodynamic force acting on the particle. The magnitudes of both of these forces increase with increasing λ. The increase in the magnitude of the electric force is due to a stronger repulsive force between the co-ions immediately outside the double layer and the particle at a larger λ. This is because if the pore is small then so is the space between the outer boundary of the double layer and the pore. Therefore, the co-ions must penetrate into the double layer as they diffuse from the highconcentration side to the low-concentration side, driving the particle in the same direction. The increase in the magnitude of the hydrodynamic force is due to a more significant boundary effect. A comparison between parts a and b of Figure 9 (β = -0.2) reveals that the presence of the electrophoresis effect yields a more positive U* and the increase in U* is more significant at smaller λ. For instance, the increase in U* is ca. 0.35 at λ = 0.1 and κa = 0.5 (U* = 0.02 in Figure 9a and 0.37 in Figure 9b), and it is ca. 0.02 at λ = 0.8 and κa = 0.5 (U* = -0.02 in Figure 9a and 0 in Figure 9b), implying that the less important the boundary effect DOI: 10.1021/la904726k
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Figure 8. Variations of the scaled hydrodynamic force in the second subproblem, Fd2*, as a function of κa for the case of Figure 6 with ξ = 0.01.
the more significant the effect of electrophoresis. This is expected because the more important the boundary effect the greater the hydrodynamic drag acting on the particle, making the increase in U* due to the electrophoresis effect less appreciable.
Effect of pH In our case, the degree of dissociation of the functional groups on the particle surface is closely related to the bulk concentration of Hþ, or the pH. Figure 10 illustrates the variations of the scaled diffusiophoretic velocity U* as a function of κa at various combinations of pH, β, and Ns*. In general, |U*| is seen to increase with increasing pH, which is expected because the higher the pH the lower the [Hþ]0 and therefore the easier it is for the acidic functional groups on the particle surface to dissociate, yielding a higher surface charge density, σ. Note that, however, the level of σ is limited by the number of acidic functional groups and the equilibrium constant Ka,n; as [Hþ]0 f 0, σ f σlim, where σlim is the limiting surface charge density. For example, as seen in Figure 10a,c, U*(pH = 7) = U*(pH = 8), suggesting that under the conditions assumed, σ(pH = 0.8) = σlim. Note that the difference between U*(pH = 7) and U*(pH = 8) in Figure 10b,d is larger than the corresponding difference in Figure 10a,c. This is because a larger Ns* implies a larger σ and therefore U* becomes more sensitive to the variation in pH. The variations of the scaled diffusiophoretic velocity U* as a function of pH at various combinations of Za, β, and Ns* are shown in Figure 11. As stated above, because a larger Ns* yields a 8656 DOI: 10.1021/la904726k
Figure 9. Variations of the scaled diffusiophoretic velocity U* as a
function of λ for various values of κa at R = 1, Za =1, pKa,1 = 5, and pH = 7; β = (a) 0 and (b) -0.2.
larger σ, the |U*| in Figure 11b,d is larger than that in Figure 11a, c. As seen in Figure 11a, |U*| increases with increasing pH and approaches a constant value as the pH becomes high. This is expected because the higher the pH the more complete the degree of dissociation of the acidic functional groups. If the pH is sufficiently high, then the dissociation of those functional groups becomes complete. As expected, the larger the Za, the higher the pH necessary to achieve complete dissociation of the acidic functional groups and the larger the corresponding |U*|. A comparison between Figure 11a,b indicates that the greater the number of acidic functional groups on the particle surface the higher the pH necessary to achieve complete dissociation. Note that because of the presence of the electrophoresis effect, U* becomes positive in Figure 11c. Intuitively, the higher the surface charge density σ (comes from higher pH and/or larger Za), the stronger the electrophoresis effect and therefore the larger the U*. However, Figure 11c reveals that this is not the case at Za = 3, where U* shows a local maximum near pH = 7. In addition, as the pH exceeds ca. 8, U* decreases with increasing Za. This can be explained by the fact that a higher σ also yields a greater electrical repulsive force between co-ions and particle as they diffuse through the gap between the particle and the pore. The presence of the local maximum of U* at Za = 3 is the result of the competition between the effect of electrophoresis and that of repulsive force. Figure 11d suggests that if the number of acidic functional groups is sufficiently large then the repulsive force between the co-ions and the particle dominates and U* becomes negative. Langmuir 2010, 26(11), 8648–8658
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Figure 10. Variations of the scaled diffusiophoretic velocity U* as a function of κa for various levels of pH at R = 1, λ = 0.5, Za = 1, and pKa,1 = 5. (a) Ns* = 1, β = 0; (b) Ns* = 5, β = 0; (c) Ns* = 1, β = -0.2; and (d) Ns* = 5, β = -0.2.
Figure 11. Variations of the scaled diffusiophoretic velocity U* as a function of κa for various values of n at R = 1, λ = 0.5, pKa,1 = 5, and pKa,n = pKa,n-1 þ 1. (a) Ns* = 1, β = 0; (b) Ns* = 5, β = 0; (c) Ns* = 1, β = -0.2; and (d) Ns* = 5, β = -0.2. Langmuir 2010, 26(11), 8648–8658
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Conclusions In summary, we analyzed the diffusiophoresis of a chargeregulated spherical particle along the axis of a cylindrical pore under the conditions of an arbitrary level of surface potential. The problem under consideration allows us to examine simultaneously the following effects on the diffusiophoretic behavior of a particle: the presence of a boundary, the charge conditions on the particle surface, the thickness of double layer, and the nature of a dispersion medium including its pH value and the diffusivities of the ionic species. We show that for the present case the diffusiophoretic behavior is governed by two types of doublelayer polarization, the electrical repulsive force between the coions and the particle arising from the diffusion of the former through the gap between the latter and the pore, the effect of electrophoresis coming from the difference in the ionic diffusivities, and the osmotic solvent flow induced by the pressure difference and the perturbed electric field. The competition of
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these effects yields many interesting results. For instance, depending upon the conditions assumed, the particle is capable of migrating either to the high- or to the low-concentration side and the diffusiophoretic mobility of the particle can have both a local maximum and a local minimum as the thickness of the double layer varies. These observations are of practical significance in the design of a diffusiophoretic apparatus and/or the interpretation of experimental data. Regarding the influence of the nature of the dispersion medium, we show that higher surface concentrations of acidic functional groups, higher numbers of protons carried by each functional group, and/or higher pH leads to larger diffusiophoretic mobility velocity when the electrophoresis effect is unimportant. However, this might not be the case when the electrophoresis effect is important. Acknowledgment. This work is supported by the National Science Council of the Republic of China.
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