pubs.acs.org/Langmuir © 2009 American Chemical Society
Diffusiophoresis of Concentrated Suspensions of Spherical Particles with Charge-regulated Surface: Polarization Effect with Nonlinear Poisson-Boltzmann Equation James Lou, Chun-Yu Shih, and Eric Lee* Department of Chemical Engineering, Institute of Polymer Science and Engineering, National Taiwan University, Taipei, Taiwan 10617 Received June 12, 2009. Revised Manuscript Received July 23, 2009 Diffusiophoresis in concentrated suspensions of spherical colloids with charge-regulated surface is investigated theoretically. The charge-regulated surface considered here is the generalization of conventional constant surface potential and constant surface charge density situations. Kuwabara’s unit cell model is adopted to describe the system and a pseudospectral method based on Chebyshev polynomial is employed to solve the governing general electrokinetic equations. Excellent agreements with experimental data available in literature were obtained for the limiting case of constant surface potential and very dilute suspension. It is found, among other things, that in general the larger the number of dissociated functional groups on particle surface is, the higher the particle surface potential, hence the larger the magnitude of the particle mobility. The electric potential on particle surface depends on both the concentration of dissociated hydrogen ions and the concentration of electrolyte in the solution. The electric potential on particle surface turns out to be the dominant factor in the determination of the eventual particle diffusiophoretic mobility. Local maximum of diffusiophoretic mobility as a function of double layer thickness is observed. Its reason and influence is discussed. Corresponding behavior for the constant potential situation, however, may yield a monotonously increasing profile.
Introduction Diffusiophoresis, the motion of a charged particle in an electrolyte solution due to the concentration gradient of the electrolytes, is an important and interesting fundamental electrokinetic phenomenon with potential in practical industrial applications, such as the deposition of colloidal paints in the traditional car industry.1,2 Moreover, in recent years, diffusiophoresis has found abundant novel applications in various fields involving manipulation of colloidal particles, which has been triggered in particular by the huge development of lab-on-a-chip technologies in the context of biological and chemical analysis.3 On the other hand, in the fundamental study of biological transport, the diffusiophoretic phenomenon shares similarities with chemotaxis, the ability of organisms such as bacteria or cells to move toward higher or lower concentrations of chemicals, nutrients, or poisons.4-7 Although chemotaxis motion is known to be turned actively by chemosensors detecting chemical gradient, diffusiophoresis appears to be the subsequent physiological driving force, at least in consistent with it. As these salts or chemicals are everywhere within a living biological system, corresponding diffusiophoresis of a suspending colloid is universal as well, although it may be coupled with other phoretic transport phenomena simultaneously. *To whom correspondence should be addressed. Telephone: 886-223622530. Fax: 886-2-23622530. E-mail:
[email protected].
(1) Smith, R. E.; Prieve, D. C. Chem. Eng. Sci. 1982, 37, 1213–1223. (2) Dukhin, S. S.; Ulberg, Z. R.; Dvornichenko, G. L.; Deryagin, B. V. Bull. Russ. Acad. Sci., Div. Chem. Sci. 1982, 31, 1535–1544. (3) Abecassis, B.; Cottin-Bizonne, C.; Ybert, C.; Ajdari, A.; Bocquet, L. Nat. Mater. 2008, 7, 785–789. (4) Parent, C. A.; Devreotes, P. N. Science 1999, 284, 765–770. (5) Dekker, L. V.; Segal, A. W. Science 2000, 287, 982. (6) Paxton, W. F.; Sundararajan, S.; Mallouk, T. E.; Sen, A. Angew. Chem., Int. Edit. 2006, 45, 5420–5429. (7) Prieve, D. C. Nat. Mater. 2008, 7, 769–770.
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The term “Diffusiophoresis” was first introduced by Deryagin and co-workers.8,9 They indicated that diffusiophoresis was caused by the polarization of the double layer under the influence of a bulk concentration gradient. The results of their theoretical analysis were verified both theoretically9 and experimentally10 later. Meanwhile, Anderson et al.11,12 conducted a theoretical investigation of the diffusiophoresis of a spherical particle immersed in both the electrolyte and nonelectrolyte solutions. Prieve and his co-workers1,13 studied a very dilute latex system, used in the car industry, both experimentally and theoretically. As for the concentrated colloidal dispersions, Lee and his coworkers14,15 considered recently the diffusiophoresis of concentrated spherical particles with arbitrary double layer thickness and zeta potential suspended in electrolyte solutions. The effect of double layer overlapping was considered. They showed, among other things, that the diffusiophoretic velocity exhibits a local maximum as well as a local minimum with varying zeta potential or double layer thickness due to the double layer polarization effect. In the study of suspensions of liquid drops,16 they observed that invicid liquid drops have a magnitude about three times greater in magnitude as compared with the corresponding rigid particles, while about two times as compared with liquid drops with similar viscosity of the suspending medium. Moreover, Lee (8) Deryagin, B. V.; Dukhin, S. S.; Korotkova, A. A. Colloid J. USSR 1978, 40, 531–536. (9) Dukhin, S. S.; Deryagin, B. V. Surface and Colloid Science; Wiley: New York, 1974; Vol. 7. (10) Ulberg, Z. R.; Dukhin, A. S. Prog. Org. Coat. 1990, 18, 1–41. (11) Anderson, J. L.; Lowell, M. E.; Prieve, D. C. J. Fluid Mech. 1982, 117, 107–121. (12) Prieve, D. C.; Anderson, J. L.; Ebel, J. P.; Lowell, M. E. J. Fluid Mech. 1984, 148, 247–269. (13) Prieve, D. C.; Roman, R. J. Chem. Soc., Faraday Trans. 2 1987, 83, 1287–1306. (14) Lou, J.; He, Y. Y.; Lee, E. J. Colloid Interface Sci. 2006, 299, 443–451. (15) Hsu, J. P.; Lou, J.; He, Y. Y.; Lee, E. J. Phys. Chem. B. 2007, 111, 2533– 2539. (16) Lou, J.; Lee, E. J. Phys. Chem. C 2008, 112, 12455–12462.
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and his co-workers17,18 extended their approach to study the diffusiophoresis of a spherical particle normal to a planar surface to examine the boundary effect. They concluded that the planar boundary poses not only as a conventional hydrodynamic retarding force, but distorts the shape of double layer greatly hence alters the particle velocity. This additional electrostatic influence of the planar surface also has profound impact on the motion of the particle when the particle is close to the plane. Hsu et al.19 observed similar behavior in a corresponding study of a particle in a spherical cavity. The previous analyses for the diffusiophoresis of charged particles in either dilute or concentrated suspensions were all based on the assumption of constant surface electric situation such as constant surface potential or constant surface charge density. While this assumption may be convincing under certain conditions, it only leads to idealized results for special cases and may be impractical for some particles. As pointed out by Ninham and Parsegian,20 these are idealized models representing limiting cases, and can be unsatisfactory sometimes in practice. The surface of biocolloids, such as bacterial cell surfaces,21,22 is capable of adjusting its charged conditions as a response to the variation in the nearby environment. The actual surface charge (or surface potential) for these particles is usually determined by the dissociation of ionizable surface groups and/or adsorption (or sitebinding) of specific ions. The degree of these dissociation and adsorption reactions will be a function of the local concentrations of specific ions at the particle surfaces. Krozel and Saville23 observed that the dissociation/adsorption of ions leads to a surface which has electric potential between that of a constant surface potential and that of constant surface charge density in practice. Carnie and Chan24-27 conducted a series of theoretical investigations about electrostatic interaction between two spherical particles. They observed that the electrostatic interaction or interaction free energy predicted by the charge-regulation model is always intermediate between that predicted by the constant-charged model and the constant-potential model. Menon and Zydney28 then extended it further to consider a spherical particle in a cylindrical pore. Kuo et al.29 evaluated the deposition rate of charge-regulated biocolloids on a charged surface. In recent years, Hong and Brown22 studied the electrostatic behavior of the chargeregulated bacterial surface, which plays an important role in bacterial interactions with other surfaces and in bacterial adhesion. All of the studies reviewed so far have focused on particle-particle or particle-interface electrostatic interaction. As for the electrokinetics involving particle motion, Lee and his coworkers30-32 considered the electrophoresis of charge-regulated particles. They showed that the number of the functional groups on particle surface and the pH of the liquid phase have profound impact on the electrophoretic mobility of the particles. The assumptions of constant surface charge density and constant surface potential were served as two limiting cases for the (17) Lou, J.; Lee, E. J. Phys. Chem. C 2008, 112, 2584–2592. (18) Lou, J.; Shih, C. Y.; Lee, E. J. Colloid Interface Sci. 2009, 331, 227–235. (19) Hsu, J. P.; Hsu, W. L.; Chen, Z. S. Langmuir 2009, 25, 1772–1784. (20) Ninham, B. W.; Parsegian, V. A. J. Theor. Biol. 1971, 31, 405–412. (21) Hong, Y.; Brown, D. G. Colloids Surf. B 2006, 50, 112–119. (22) Hong, Y.; Brown, D. G. Langmuir 2008, 24, 5003–5009. (23) Krozel, J. W.; Saville, D. A. J. Colloid Interface Sci. 1992, 150, 365–373. (24) Carnie, S. L.; Chan, D. Y. C. J. Colloid Interface Sci. 1993, 155, 297–312. (25) Carnie, S. L.; Chan, D. Y. C. J. Colloid Interface Sci. 1993, 161, 260–264. (26) Carnie, S. L.; Chan, D. Y. C. Langmuir 1994, 10, 2993–3009. (27) Carnie, S. L.; Chan, D. Y. C. J. Colloid Interface Sci. 1994, 165, 116–128. (28) Menon, M. K.; Zydney, A. L. Anal. Chem. 2000, 72, 5714–5717. (29) Kuo, Y. C.; Hsieh, M. Y.; Hsu, J. P. J. Phys. Chem. B 2002, 106, 4255–4260. (30) Lee, E.; Yen, F. Y.; Hsu, J. P. Electrophoresis 2000, 21, 475–480. (31) Hsu, J. P.; Lee, E.; Yen, F. Y. J. Chem. Phys. 2000, 112, 6404–6410. (32) Tang, Y. P.; Chih, M. H.; Lee, E.; Hsu, J. P. J. Colloid Interface Sci. 2001, 242, 121–126.
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Figure 1. Schematic representation of the system under consideration where a is the radius of a particle and b is that of a liquid cell. The applied concentration gradient rn0 is in the z-direction.
combined electrostatic and hydrodynamic interaction effects on the charge-regulated surfaces. Under essentially the same assumptions, Hsu and his co-workers33-35 further extended this approach to study several different kinds of boundary effects in electrophoresis. Keh and his co-workers36-38 conducted a series of research in the field of electrophoresis and diffusiophoresis, focusing on the limiting cases of low surface potential or charge density. Linearized Poisson-Boltzmann equation is adopted as a result and approximate analytical expressions were obtained. In this work, the diffusiophoretic behavior of a concentrated spherical colloidal dispersion is analyzed. A general charged condition on particle surface is considered; the classic constant surface potential model can be recovered as special cases of the present model. Compared with the results in the literature before, our study of the diffusiophoresis of concentrated colloidal dispersion takes into account the effects of arbitrary level of surface dissociation, as well as finite double layer thickness, and extensive volume fractions of the particles. Polarization effect of double layer is taken into account properly by solving the original nonlinear Poisson-Boltzmann equation. Kuwabara’s unit cell model39 is adopted to describe the concentrated suspension colloidal systems under study. In order to solve the resulted general electrokinetic equations, which are highly nonlinear, the powerful pseudospectral method40 is employed in this study. Results are analyzed and presented in the following sections.
Theory Let us consider the diffusiophoresis of concentrated rigid spherical particles of radius a in an aqueous solution of z1: z2 electrolytes, z1 and z2 are respectively the valences of cations and anions. R is defined as - z2/z1, for example, R = 1 for KCl. The electroneutrality in the bulk liquid phase requires that n20 = n10/ R, n10 and n20 being respectively the bulk concentrations of cations and anions, and R = -z2/z1as mentioned above. The dispersion is simulated by the unit cell model of Kuwabara.39 Referring to Figure 1, the dispersion is modeled by a representative particle of (33) Hsu, J. P.; Ku, M. H.; Kuo, C. C. Langmuir 2005, 21, 7588–7597. (34) Hsu, J. P.; Ku, M. H.; Kuo, C. C. Electrophoresis 2008, 29, 348–357. (35) Hsu, J. P.; Chen, C. Y.; Lee, D. J.; Tseng, S.; Su, A. J. Colloid Interface Sci. 2008, 325, 516–525. (36) Keh, H. J.; Ding, J. M. J. Colloid Interface Sci. 2003, 263, 645–660. (37) Ding, J. M.; Keh, H. J. Langmuir 2003, 19, 7226–7239. (38) Keh, H. J.; Li, Y. L. Langmuir 2007, 23, 1061–1072. (39) Kuwabara, S. J. Phys. Soc. Jpn. 1959, 14, 527–539. (40) Chu, J. W.; Lin, W. H.; Lee, E.; Hsu, J. P. Langmuir 2001, 17, 6289–6297.
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radius a surrounded by a concentric spherical liquid shell of radius b. Let j = (a/b)3, which is a measure of the volume fraction of the present dispersion. Suppose that a uniform concentration gradient rn0 is applied to the system in the z-direction, and as a response, the particle moves in the z-direction with a constant velocity U. The spherical coordinates (r,θ,j) with origin located at the center of the representative particle are adopted. For the present problem, we have to solve simultaneously the governing equations for the electric, the flow, and the concentration fields. These equations can be summarized as below:
surface potential is much larger than 25.6 mV at 25 C. The inverse Debye length κ is defined by K ¼½
2 X
nj0 ðezj Þ2 =εkB T1=2
ð1Þ
It should be noted that the conventional standard electrokinetic model is adopted in this study in that no surface conductivity is considered here. The slipping plane coincides with the particle surface. Let us consider the case where the surface of a particle contains a functional group AH, which is capable of undergoing the dissociation reaction
r3v ¼ 0
ð2Þ
AHSA - þHþ
μr2 v -rp -Frφ ¼ 0
ð3Þ
r2 φ ¼ -
2 X F zj enj ¼ ε ε j ¼1
r 3 f j ¼ 0,
j ¼ 1, 2
ð4Þ
f j ¼ -Dj rnj þ
n j zj e rφ þnj v, kT
j ¼ 1, 2
ð5Þ
In these expressions, φ is the electrical potential, v is the liquid velocity, F and ε are respectively the space charge density and the permittivity of the liquid phase, and e is the elementary charge. nj, fj, and Dj are respectively the number concentration, the concentration flux, and the diffusion coefficient of ionic species j. p and μ are respectively the pressure and the viscosity of the liquid phase. Furthermore, to account for possible concentration polarization arising from the movement of particle in the present problem, we assume a modified Boltzmann distribution of the form:14 nj ¼ nj0 exp -
zj e ðφ þδφþgj Þ kB T e
ðKaÞ2 ½expð -φe Þ -expðRφe Þ ¼ ð1þRÞ
where φ*e = φe/(kBT/e). It should be noted that eq 7 can not be further linearized when the scale of φe* is large; that is, the particle
ð10Þ
½Hþ s ¼ ½Hþ 0 expð -φe Þ þ
ð11Þ
þ
where [H ]0 is the equilibrium value of [H ]s. If we let Ns be the concentration of the functional groups on particle surface, then Ns ¼ ½AHþ½A -
ð12Þ
Equations 10-12 lead to
ð6Þ
ð7Þ
½A - ½Hþ s ½AH
where [•] denotes concentration and [Hþ]s is the value of [Hþ] at the particle surface. Suppose that the spatial distribution of [Hþ] follows the Boltzmann distribution, that is
½A - s ¼
That is, the electrical potential is decomposed into φe, δφ, and gj, representing respectively the equilibrium electric potential in the corresponding static problem, the induced electric potential arising from the movement of the liquid phase, and an equivalent perturbed potential arising from the convection of the electrolyte ions. It is known that in a static environment nj follows Boltzmann distribution at equilibrium, if the electrolyte concentrations and particle’s potentials are not too high, which is the case under study here. The two factors δφ and gj in our system are sometimes several orders of magnitude smaller than φe where significant deviations from Boltzmann distribution were reported experimentally. Note that diffusiophoresis can be characterized by an electrophoresis driven by the local gradient of electric potential coming from the bulk concentration gradient. This implies that the equations governing a diffusiophoresis problem can be deduced directly from those of the corresponding electrophoresis problem. It can be shown that, the governing equations of the present problem, in dimensionless forms, are as follows.14,31 (A) Equilibrium system. The governing equations for the equilibrium electric potential φe can be described by the Poisson-Boltzmann equation
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ð9Þ
Let Ka be the dissociation constant of this reaction. Then Ka ¼
r2 φe
ð8Þ
j ¼1
½Hþ 1þ Ka 0
Ns exp - kBeT φe
ð13Þ
The charge density on the surface of the representative particle, σ, is given by σ ¼ -e½A -
ð14Þ
Substituting eq 14 into eq 13 yields σ ¼
½H þ 1þ Ka 0
-eNs exp - kBeT φe
ð15Þ
If the relative permittivity of the representative particle is much smaller than that of the liquid phase, then Dφe σ ¼ -ε Dr r ¼a
ð16Þ
Combining eqs 15 and 16, we obtain Dφe 1 eNS , ¼ ε 1þ ½Hþ 0 exp - e φ Dr Ka kB T e
at r ¼ a
ð17Þ
For convenience, scaled quantities are used in the discussion hereafter. Equation 17 can be further reduced to
Dφe A ¼ , Dr 1þB expð -φe Þ
at r ¼ 1
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where r* = r/a, A = ae2Ns/εkBT, and B = [H þ]0/Ka. This expression describes the boundary condition at the particle surface. If the surface potential is low, eq 15 can be further simplified by: dσ -eNs σ ¼ σjφe ¼0 þ φ ¼0 ðφe -0Þ ¼ dφe e f1þ½Hþ 0 =Ka g -
ðe2 Ns =kB TÞf½Hþ 0 =Ka g f1þ½Hþ 0 =Ka g2
φe
the scaled stream function, and E*4 is the operator of E*2E*2, which is defined as: E
ð19Þ
ðz1 f 1 þz2 f 2 Þ 3 rB ¼ 0, ð20Þ
Note that eq 20 is a simple linear combination of the constant potential and constant charge situations. For example, it reduces to the constant charge case with ∂φe*/∂r*= A if B is set to zero, and constant potential case with φe* = (1 þ B)/B if A is very large. It should be noted that although there is no such simple mathematical expression in terms of a linear combination as the surface potential gets high, the charge-regulation model employed here remains as the general expression for the actual physical situation on the colloid surface. At unit cell surface, we suppose that the unit cell as a whole is electrically neutral, thus there is no electric current between adjacent cells, as shown in eq 21. at r ¼
b a
ðKaÞ2 ½expð -φe Þ -expðRφe Þ ð1þRÞ ð22Þ
The conservation equation of ions, eq 4, is converted into dimensionless form by substituting eq 5 and eq 6 into it
r 2 gj -r φe 3 r gj -Pej v 3 ðr φe þr δφþr gj Þ ð23Þ -ðr δφþr gj Þ 3 r gj ¼ 0, j ¼ 1, 2 Taking curl on eq 3 and introducing the stream function in the spherical coordinates, we get rid of the continuity equation and obtain " # 2 ðKaÞ Dφ Dg1 Dg2 f n1 þn2 ðRn2 Þ E ψ ¼ Dθ 1þR Dr Dr " # Dφ Dg Dg g sin θ - n1 1 þn2 2 ðRn2 Þ Dr Dθ Dθ
1 1 Pe1 R2 Pe2
ðr n0 Þðrcos θÞ,
ð26Þ
Dδφ ¼ 0, Dr
where a symbol with an asterisk represents a dimensionless quantity. δφ*=δφ/(kBT/e), gj*=gj/(kBT/e), φ*=φe*þδφ*, and n*j = nj/n10. Pej is the corresponding Peclect number of ion j, representing the effect of convection. ψ*=ψ/[ε(kBT/e)2a/μ] is
at r ¼
-1
b a
ð27Þ
at r ¼ 1
ð28Þ
We assume a constant perturbed ionic concentration gradient across the outer virtual surface of the unit cell. Thus, we have nj ¼ nje þrn0 3 zB,
at r ¼ b
ð29Þ
where n1e = n10 exp(-φe*) and n2e = (n10/R) exp(Rφe*) are equilibrium concentration of cations and anions, respectively. In the dimensionless form, eq 29 can be rewritten as follows:
ðδφþg1 Þ ¼ -ðr n0 Þðr cos θÞ , f 1 ðδφþg2 Þ ¼ ðr n0 Þðr cos θÞ R
at r ¼
b a
ð30Þ
The particle surface is impermeable to ions. Therefore
Dg1 Dg2 ¼ ¼ 0, Dr Dr
atr ¼ 1
ð31Þ
For convenience, we assume that the fluid is flowing toward a stationary particle with a scaled velocity U*. The bulk liquid can not penetrate the particle surface, and Kuwabara39 unit cell model is adopted to describe the concentrated dispersion. On the basis of these assumptions, the following conditions are assumed: Dψ ψ ¼ 0, ¼ 0, Dr
E 2 ψ ¼ 0, ð24Þ
1 1 þ Pe1 RPe2
(r*n0*)=rn0/(n10/a) refers to the dimensionless applied concentration gradient. Furthermore, since the particle under consideration is dielectric, we have
1 ψ ¼ r 2 U sin2 θ, 2
4
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ðKaÞ2 fexp½ -ðφe þδφþg1 Þ r δφ ¼ ð1þRÞ 2
δφ ¼ -
ð21Þ
(B) Perturbed System. Substituting eq 6 into eq 1, we obtain the governing equation of induced electric potential from eq 1 and eq 7 in the dimensionless form:
-exp½Rðφe þδφþg2 Þgþ
at r ¼ b
Substituting eq 5 into eq 26, we have:
1þB Dφe B φ ¼1 A Dr 1þB e
ð25Þ
We assume that the net ionic flux across the virtual surface of a cell is zero, that is
Combining eq 16 and eq 19 gives:
Dφe ¼ 0, Dr
D2 sin θ D 1 D ¼ 2 þ 2 r Dθ sin θ Dθ Dr
2
at r ¼ 1
ð32Þ
at r ¼ b=a
ð33Þ
at r ¼ b=a
ð34Þ
13
The approach of Prieve and Roman is adopted where it was assumed that the concentration of solute is only slightly nonuniform over the length scale a, that is, a|rn0| , n10. In this case, the present problem can be decomposed into two subproblems, both are of linear nature. In the first subproblem a particle moves with a constant velocity in the absence of the applied concentration gradient, and in the second subproblem it is fixed in the space Langmuir 2010, 26(1), 47–55
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when the concentration gradient is applied. If we let F1 and F2 be respectively the forces acting on the surface of a particles in these two subproblems, then F1 = f0 1(r*n0*) and F2 = f20 U*, where f10 and f20 are proportional constants. Consequently, the scaled diffusiophoretic mobility U*m can be expressed as
Um ¼
0
U f1 ¼ - 0 r n0 f2
ð35Þ
Given the values of r*n0* and U*, F1 and F2 are calculated first through solving the entire set of electrokinetic equations; f10 and * is f20 are then determined directly by their definitions, and Um calculated by eq 35. A pseudospectral method based on Chebyshev polynomial14-18 is adopted for the solution of the governing equations, subject to the associated boundary conditions. Details of the pseudospectral method employed in analyzing electrokinetic phenomena can be found elsewhere, such as Lee et al.40 It proves to be a very powerful and suitable method for the fields of interest.
Results and Discussion We first compare our computation results with experimental data available in literature41 for the limiting case at least. In convenience, the scaled diffusiophoretic mobility U*m was normalized by a reference factor U0 = (2/3), which is the same as provided by Ebel et al.41 Note that the experimental data provided by Ebel et al. were obtained for very dilute suspensions of colloids with various constant potential surfaces. As a result, we have to reduce our general model to the limiting case of the very dilute and constant surface potential situation. This is done by setting A very large (=1000 in actual calculations) and adjusting B to fit the various constant potentials corresponding to each data point provided by Ebel et al. in eq 20, which is a simple linear combination of both constant potential and constant charge cases. Moreover, a/b is set to 1/5 which yields a volume fraction j (=(a/b)3) about 0.008. Note that eq 20 is valid for low surface potential situation only, if both the values of A and B are given by experimental data directly, whereas the data provided were obtained in the range of φ*e around -2 to -5, which is generally considered high. However, the resultant constant potentials by varying the value of B are indeed exactly the same as reported in the experimental study. As shown in Figure 2, the agreement is excellent for this limiting case when the surface potential is kept as constant and the particle concentration is very dilute. This indicates the correctness of the algorithm used here, which is based on the pseudospectral method, as well as the reliability of the cell model adopted here for the modeling of concentrated suspensions. As we would like to compare further with general charge-regulation situation, with both A and B given by experimental data directly, no pertinent experimental can be found in literature, to the best of our knowledge. Effect of Functional Groups on Particle Surface. The most important factor in determining the eventual diffusiophoretic mobility of a charged-regulated colloid suspension is A (=ae2Ns/εkBT), which represents the number of the functional groups on particle surface in the dimensionless form. For a 0.1 μm sized particle, for example, if the temperature T is 25 C and A= 1, then NS is of the order 1015 site/m2 in the typical system considered. Figures 3a and 4a depict the scaled diffusiophoretic * calculated in this study as a function of κa at various mobility Um values of A. For the convenience of illustration purpose, KCl (41) Ebel, J. P.; Anderson, J. L.; Prieve, D. C. Langmuir 1988, 4, 396–406.
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Figure 2. Comparison with the experimental data at constant surface potential and dilute suspension where the solid line is the numerical results in this study, and the open symbols are the experiment data from Ebel et al.41
(Pe1 = Pe2 = 0.26) and NaCl (Pe1 = 0.39, and Pe2 = 0.26) are chosen as the representative electrolyte solutions. The scaled particle surface potential φ*e as a function of κa is also presented in Figures 3b and 4b. The negative value of φ*e indicates a negatively charged condition on particle surface. Figures 3a and * increases with the 4a reveal that for a fixed value of κa, Um increase of A. This mobility behavior is also observed in the corresponding studies of electrophoresis.35 Briefly speaking, a large A implies that the number of the functional groups on particle surface, Ns, is large, which gives a higher surface charge density and higher surface potential on particle surface as indicated in Figures 3b and 4b, hence the resultant electric force upon the particle increases accordingly. The higher the electric force, the higher the particle velocity trends to be. This explains why a pronounced increase of scaled mobility at medium κa is observed in Figures 3a and 4a with the increase of A. Effect of Ionic Diffusion Coefficients. Comparing the results of Figure 3a with those of Figure 4a further, we observe that the particle always moves faster in NaCl solution than in KCl solution. Similar to the limiting case of constant potential,14,15 the reason behind this is that diffusiophoresis of the charged particles in KCl solution comes solely from chemiphoresis,42 which is independent of any electric property of the charged particles themselves, since the diffusion velocities are identical for cations and anions (Pe1 = Pe2 = 0.26). The term “chemiphoresis” implies that the motion of charged particles is due to the presence of the solute gradient which leads to an osmotic pressure gradient around the charged particles, which gives rise to a diffusiophoretic particle motion in the same direction of the bulk concentration gradient of the electrolytes. Note that the concentration gradient is pointed toward the higher concentration region by definition, hence the actual direction of ions diffusion would be in the opposite direction of the gradient. In NaCl solution, however, an extra induced electric field arises as a result of different diffusion mobilities of sodium ions and chloride ions (Pe1 = 0.39 and Pe2 = 0.26). Since the diffusivity of Cl- is larger than that of Naþ, more negatively charged chloride ions will cluster in the lower NaCl concentration region hence generate an induced electric field in the opposite direction as that of bulk concentration (42) Dukhin, S. S. Adv. Colloid Interface Sci. 1993, 44, 1–134.
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Figure 3. (a) Variation of scaled mobility Um * as a function of κa at various A for the case B=10, a/b=1/2, and Pe1=Pe2=0.26. Dashed line: mobility in the study of constant surface potential when φe*=-3. (b) Variation of scaled surface potential φe* as a function of κa at various A for the case B = 10, a/b = 1/2, and Pe1 = Pe2 = 0.26. Dashed line: scaled surface potential in the tudy of constant surface potential when φe* = -3.
gradient of electrolytes. Depending on the charged condition on the particle surface, this induced electric force will reinforce or compete against the bulk concentration gradient. This phenomenon can be explained by the governing equations and the corresponding boundary conditions in two separate ways: the contribution from the ionic concentration gradient in eq 30, and the two Peclet numbers in eq 27 are not identical (Pe1=0.39 and Pe2 =0.26). Thus, the eventual speed and direction of particle motion is determined by the balance of these two forces. Besides, according to eq 27, the direction of the induced electric force is determined by the specific charged conditions of a particle and the relative magnitudes of Pe1 and Pe2. In the present study, we assume that hydrogen ions dissociate from the particle surface, as shown in eq 9, hence the particles in the suspension are always negatively charged. For a negatively charged particle, the effect of this induced electric field will give rise to an enhancement of particle motion as it pushes the particle in the same direction as 52 DOI: 10.1021/la902113s
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Figure 4. (a) Variation of scaled mobility Um * as a function of κa at various A for the case B = 10, a/b = 1/2, Pe1 = 0.39, and Pe2 = 0.26. Dashed line: mobility in the study of constant surface potential when φe*=-3. (b) Variation of scaled surface potential φe* as a function of κa at various A for the case B = 10, a/b=1/2, Pe1 =0.39, and Pe2 = 0.26. Dashed line: scaled surface potential in the study of constant surface potential when φe* = -3.
that of the bulk concentration gradient of electrolytes as well. Hence the diffusiophoretic mobility in NaCl solution is always larger than that in KCl solution. Effect of Double Layer Thickness. Figures 3a and 4a also reveal that the profile of U*m has a local maximum for each value of A as κa varies. This can be elaborated as follows. A small κa implies a thick double layer, and the degree of overlapping between adjacent double layers is serious, which leads to a small U*. m As κa increases, higher ion concentration nj0 is resulted in according to eq 8. In other words, the amount of electrolytes increases accordingly in the vicinity of the colloid surface, hence the effect of osmotic pressure due to bulk concentration gradient exerted upon it increases simultaneously. Moreover, as κa increases, the effect of double layer overlapping from neighboring particles decreases, and the particle mobility increases as a result. For the purpose of comparison, corresponding limiting case of constant surface potential under the same conditions are presented in Figures 3a and 4a as well. For simplicity, we suppose Langmuir 2010, 26(1), 47–55
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that φ*e on particle surface is -3.0 (roughly -76.8 mV) to represent the limiting case of constant surface potential, as it is roughly the average value of φe* (-2 to -5) in the experimental study by Ebel et al.41 As clearly seen from these figures, the mobility behavior of charged-regulated particles is very similar to that of constant surface potential in low to medium ranges of κa. However, we notice that in Figures 3a and 4a that if we adopt the constant surface potential assumption to represent the particle surface electric situation, the diffusiophoretic mobility of colloid suspension may be underestimated at low κa in comparison with charge-regulated surface condition in this specific case. This is because lower κa implies lower concentration of ions in the bulk phase, which induces more hydrogen ions to dissociate from particle surface, hence results in a higher surface charge density or higher surface potential on particle surface than that predicted in the case of constant surface potential, as shown in Figures 3b and 4b. However, since one may run into constant surface potential values other than the one discussed here, it would be overgeneralization to draw any conclusion here in terms of absolute values. In particular, as shown in Figure 3a or 4a, we notice the appearance of local maxima over some ranges of κa. Similar behaviors can also be observed in the study of electrophoresis.30,31 The explanation there is also applicable here. With the increase of κa, more counterions will accumulate near the particle surface, and the dissociation of functional groups on particle surface will be repressed due to this accumulation of counterions. Hsu et al.31 pointed out that this common ion effect results in a low |φe*|, as suggested by Figures 3b and 4b. From Figures 3b and 4b, we notice that |φ*| e always decreases with the increase of κa. If the number of functional groups represented by A is small, the value of |φe*| even reduces to zero at large κa. Therefore, the electric driving force acting on particle surface diminishes with the increase of κa, leading to a decreasing particle velocity instead. Moreover, we also observe that the diffusiophoretic mobility under the assumption of constant surface potential may be overestimated at large κa in comparison with the present charge-regulating situation, as shown in Figures 3a and 4a. In the present analysis, the true particle surface potential varies ionic concentration in the bulk phase, that is, double layer thickness κa around the charged particles. Figures 3b and 4b indicate that the particle surface potential |φe*| decreases with the increase of κa due to the common ion effect, the shift of equilibrium to the left in the dissociation reaction. However, if we adopt constant surface potential as particle surface property, this common ion effect disappears since the particle surface potential always remains a constant value regardless of values of κa. As shown in Figures 3b and 4b, |φ*| e is fixed at -3.0 at arbitrary κa in the situation of constant surface potential. We thus conclude that the assumption of constant surface potential on particle surface yields a mobility profile quite different from that from a charge-regulated surface, with the former may increase monotonously with κa, at least for the specific |φ*| e value examined, whereas the latter exhibits local maxima in general. Indeed, the actual particle surface electric situation strongly influences the diffusiophoretic behavior of colloidal particles. In addition, the intersection of profiles of diffusiophoretic mobilities with constant surface potential and that with chargeregulated surface in Figures 3a and 4a represents that identical results are possible at some specific κa. This explanation is also applicable in the corresponding analysis of Figures 3b and 4b, where the intersection of |φe*| of two cases appears at exactly the same κa. Effect of Bulk Concentration of [Hþ]. The variation of * and scaled surface potential scaled diffusiophoretic mobility Um Langmuir 2010, 26(1), 47–55
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Figure 5. (a) Variation of scaled mobility U*m as a function of B at various κa for the case A = 500, a/b = 1/2, and Pe1 = Pe2 = 0.26. (b) Variation of scaled surface potential φe* as a function of B at various κa for the case A = 500, a/b = 1/2, and Pe1 = Pe2 = 0.26.
φe* as a function of B at various κa is illustrated in Figures 5 and 6, respectively. B denotes the ratio of bulk concentration of [Hþ] to the dissociation constant Ka. Figure 5b or 6b reveals that for a fixed κa, |φe*| decreases with the increase of B. This can be explained by resorting to the definition of B, which implies that the larger its value, the higher the [Hþ]0, and the higher the surface concentration of [Hþ] as suggested by eq 11. The dissociation of functional groups on particle surface is again the dominating factor behind it as previously discussed in the analysis of the effect of varying A. To keep the equilibrium of the dissociation reaction, more [Hþ] leads to a lower [A-] and a lower surface charge density and hence a lower |φe*|, as indicated by Figure 5b or 6b. Moreover, if we examine closely the variation of |φ*| e at each κa, we observe that the higher the value of B, the slower the variation of the rate of the decrease of |φe*|. The reason is that few H þ can dissociate from the particle surface when the concentration of Hþ is too high. Hence most of the functional groups on particle surface maintain electrically neutral, which indicates a slow variation of particle surface potential. Taking a closer look at eq 18, we also find that the right-hand side in eq 18 tends to zero with increasing B. The variation of |φ*| e becomes a weak function of B as B gets large. Note that the equilibrium potential DOI: 10.1021/la902113s
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Figure 7. Variation of scaled mobility Um * as a function of κa at various j for the case A = 100, B = 10, and Pe1 = Pe2 = 0.26.
Figure 6. (a) Variation of scaled mobility Um * as a function of B at various κa for the case A = 500, a/b = 1/2, Pe1 = 0.39, and Pe2 = 0.26. (b) Variation of scaled surface potential φ* e as a function of B at various κa for the case A = 500, a/b = 1/2, Pe1 = 0.39, and Pe2 = 0.26.
φ*e at particle surface plays the role of particle surface potential. Therefore, the dependence of the diffusiophoretic velocity on the surface potential can be obtained in Figures 5a and 6a. The higher the surface potential is, the larger the electric driving force exerted on the particle surface, and this results in a faster velocity. In particular, we notice that the behavior of the diffusiophoretic mobility U*m corresponding to κa = 8.0 is different from the rest of the profiles in Figure 5a and 6a. Instead of assuming the highest values as in Figure 5b and 6b for surface potential, the mobility now is smaller than that of κa = 3.0 throughout the range of B investigated. This behavior is a key feature of double layer polarization, which agrees very well to the corresponding case of constant surface potential,14,15 where the polarization of the double layer strongly influences the direction of particle movement. A microscopic electric field arising from the polarization effect normally tends to retard the diffusiophoretic velocity. According to our analysis in the electrophoresis of chargedregulated particles,31 polarization effect must be considered when the particle surface potential and the double layer thickness are finite. Judging from the figures shown here, we conclude that the 54 DOI: 10.1021/la902113s
Figure 8. Variation of scaled mobility U*m as a function of κa at
various j for the case A = 100, B = 10, Pe1 = 0.39, and Pe2 = 0.26.
polarization effect still plays an important role in the diffusiophoresis of charged-regulated particle dispersion. Effect of Particle Volume Fraction. Finally, we examine the influence of the particle concentration, measured by the volume fraction j = (a/b)3 of the colloidal particle in the electrolyte solution. The results are shown in Figures 7 and 8. Note that in spite of the mathematical correctness, it is physically unrealistic to have volume fraction larger than the maximum packing factor with spheres, which is 0.74. Moreover, sphere volume fractions of 0.5 or larger correspond to glassy and crystallized colloid dispersions, not free-flowing liquids for actual experiments. In fact, the cell model when used for volume fraction larger than, e.g., 0.5, in many theoretical modelings yields predictions that increasingly deviate from experimental results. The different phase-transitions as particle concentrations increases before the maximum attainable packing, and the corresponding particle-particle interactions are not well accomplished by the cell model. In any case, it is better than nothing and it is usually found making predictions Langmuir 2010, 26(1), 47–55
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beyond 0.4 or 0.5 volume fractions, but never beyond the maximum packing factor with spheres. With the above discussion in mind, we present the results for volume fraction from 0.125 up to 0.5. Double layer overlapping is allowed in the current analysis. It is found that the diffusiophoretic mobility decreases with increasing volume fraction. This is mainly due to the hindrance effect of neighboring particles: the higher the concentration of colloidal particles, the more significant the overlapping of the neighboring double layers, leading to a greater hydrodynamic resistance for fluid flow in diffusiophoresis. However, if κ f ¥, U* becomes independent of j. This is because as κa is large, the electric double layer surrounding the particles is thin, hence the influence of nearby particles on its motion is negligible electrostatic-wise.
(ii)
(iii)
Conclusion The diffusiophoretic behavior of a concentrated dispersion of charged-regulated particles with arbitrary double layer thickness is investigated theoretically here. A general mixed-type condition is resulted on the particle surface when the surface potential is low. And the conventional models, which assume either constant surface potential or constant surface charge, can be recovered easily as the special cases of the present analysis. When the surface potential is high, the current charge-regulation model remains as the general expression of what really happens physically on the particle surface nonetheless, with a much more complicated nonlinear surface condition though. In particular, the behavior of biological cells, which carry dissociable functional groups, and particles which are capable of exchanging ions with the surrounding medium can be simulated by the present model. The effect of particle volume fraction and double layer polarization is also examined. In summary, we conclude the following: (i) In general, the larger the number of dissociated functional groups on particle surface is, the higher
Langmuir 2010, 26(1), 47–55
(iv)
(v)
(vi)
the particle surface potential and hence the larger the magnitude of the particle mobility. The electric property on particle surface depends on both the concentration of dissociated hydrogen ions and the concentration of electrolyte in the solution. The electric potential on particle surface turns to be the dominant factor in the determination of the eventual particle diffusiophoretic mobility. The assumption of constant surface potential on particle surface yields a mobility profile quite different from that from a charge-regulated surface, with the former may increase monotonously with κa, at least for the specific |φe*| value examined, whereas the latter exhibits local maxima in general. Indeed, the actual particle surface electric situation strongly influences the diffusiophoretic behavior of colloidal particles. When the concentration of counterions goes high inside the double layer, the dissociation of ionizable groups on particle surface is repressed by this common ion effect. The diffusiophoretic mobility exhibits a local maximum as a result, and then decreases rapidly with increasing κa. The magnitude of diffusiophoretic mobility declines with the volume fraction of particles in general due to the hindrance effect and electrostatic interaction between neighboring particles, as consistent with previous investigations for limiting constant potential surface. The effect of double layer polarization is significant in general and can not be neglected when the double layer thickness is finite.
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