pubs.acs.org/Langmuir © 2009 American Chemical Society
Role of Surface Conductivity in the Dynamic Mobility of Concentrated Suspensions Francisco J. Arroyo,† J. Cuquejo,‡ A. V. Delgado,§ M. L. Jimenez,§ and Felix Carrique*,‡ †
‡
Departamento de Fı´sica, Facultad de Ciencias Experimentales, Universidad de Ja en, 23071 Ja en, Spain, Departamento de Fı´sica Aplicada I, Facultad de Ciencias, Universidad de M alaga, 29071 M alaga, Spain, and § Departamento de Fı´sica Aplicada, Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain Received March 31, 2009. Revised Manuscript Received August 24, 2009
In this article, a cell model is used for the evaluation of the alternating current (ac) mobility (dynamic mobility) of spherical particles in suspensions of arbitrary volume fractions of solids. The main subject is the consideration of the role of the electrical conductivity (SLC or Kσi) of the stagnant layer (SL) on the mobility. It is assumed that the total surface conductivity (Kσ), resulting from both Kσi and the diffuse layer conductivity (Kσd), is constant in the cases considered and that it is the Kσi-Kσd balance that determines the SL effects. We first explore the effect of Kσi on the frequency dependence of the dynamic mobility. It is found that the mobility decreases on average, for any frequency, when Kσi increases. This is a consequence of stagnancy: ions in the SL, although contributing to the surface conductivity, do not drag liquid with them when they migrate and do not contribute to electro-osmotic flow or, equivalently, to electrophoresis. Three relaxations are observed in the mobility-frequency spectrum: inertial (the particle and liquid motions are hindered), Maxwell-Wagner-O’Konski (ions in the double layer cannot follow the field oscillations and can move only over a distance much smaller that the diffuse layer thickness), and the so-called alpha or concentration polarization process (the ions can rearrange around the particle, but they cannot form the electrolyte concentration field that appears at low frequency). Whereas the first two relaxations are little affected by Kσi, the alpha process undergoes significant changes. Thus, the mobility increases with frequency around the alpha relaxation region if Kσi is negligible, but it decreases with frequency in the same interval if Kσi is finite. With the aim of explaining this behavior, we calculate the capillary osmosis velocity field that is the fluid flow provoked by the concentration gradient around the particle. The calculations presented demonstrate that the velocity is reduced (for each frequency and position) when the SLC is raised. It is proposed that such a decrease adds to that due to the changes in the induced dipole moment of the particle, also favoring a decrease in the mobility. These tendencies are also present when the volume fraction of solids, φ, is modified, although higher φ values somewhat hide the effect of Kσi, as in fact observed with all features of electrokinetics associated with the phenomenon of concentration polarization.
1. Introduction Most fields where colloidal suspensions find industrial applications deal with moderate or high concentrations of solids. As in dilute suspensions, a characterization of the electrical state of the particle/solution interface is essential; however, the experimental techniques suited for that purpose have certain requirements associated with the fact that one must infer the properties of a single interface from measurements performed on a large collection. In a very high concentration range (close to the maximum packing fraction), methods such as electro-osmosis, streaming potential, and streaming current on plugs are the proper choices.1-4 In the intermediate range of the solids volume fraction (φ ≈ 10-20%), we can mention essentially two methods: one is electrical conductivity or permittivity measurements as a function of the frequency ω of an applied electric field; the other involves electroacoustic techniques providing information about the socalled dynamic or ac electrophoretic mobility, ue, of the particles.
In both cases, theories are available relating the experimentally measured quantity (electrical permittivity or conductivity, dynamic mobility) to the volume fraction, the particle size, a, the electric double layer thickness, κ-1, or the zeta potential, ζ. A description of the pertinent models can be found in refs 5-8. Most often, the classical or standard electrokinetic approach is used:1 the zeta potential is the only parameter characterizing the electrical state of the interface. It is the potential on an ideal surface (the electrokinetic or slip plane) that separates two regions: the Newtonian bulk solution and the stagnant layer, where the viscosity is so high that neither fluid nor ions can move relative to the surface. This means that, although an excess conductivity can be associated with the interface (the surface conductivity, Kσ), it will be attributed only to the diffuse region and it is called Kσd. An alternative approach, which has proven to be a significant advance over the classical model, is represented by the so-called finite-SLC (SLC: stagnant-layer conductivity, Kσi). It is assumed that, although the liquid is truly stagnant, ions can migrate by
*To whom correspondence should be addressed. E-mail:
[email protected]. (1) Lyklema, J. Fundamentals of Interface and Colloid Science; Academic Press: London, 1995; Vol. II. (2) Hunter, R. J. Foundations of Colloid Science; Oxford University Press: Oxford, U.K., 2001 (3) Delgado, A. V. Interfacial Electrokinetics and Electrophoresis; Surfactant Science Series; Marcel Dekker: New York, 2002; Vol. 106. (4) Werner, C.; Zimmermann, R.; Kratzm€uller, T. Colloids Surf., A 2001, 192, 205.
12040 DOI: 10.1021/la901132g
(5) Delgado, A. V.; Arroyo, F. J.; Gonzalez-Caballero, F.; Shilov, V. N.; Borkovskaya, Y. B. Colloids Surf., A 1998, 140, 139. (6) Carrique, F.; Arroyo, F. J.; Jimenez, M. L.; Delgado, A. V. J. Chem. Phys. 2003, 118, 1945. (7) O’Brien, R. W.; Jones, A.; Rowlands, W. N. Colloids Surf., A 2003, 218, 89. (8) Ahualli, S.; Delgado, A. V.; Miklavcic, S. J.; White, L. R. Langmuir 2006, 22, 7041.
Published on Web 09/18/2009
Langmuir 2009, 25(20), 12040–12047
Arroyo et al.
Article
diffusion and electromigration inside the stagnant layer. The total surface conductance, Kσ, will be the sum of its two components: K σ ¼ K σd þ K σi
ð1Þ
It is interesting that the electrokinetic role of these conductivities is associated with their values relative to that of the medium, Km, as represented by the Dukhin number, Du, and its diffuse and SL components, Dud and Dui: Du ¼
Kσ ¼ Dud þ Dui Km a
Dud, i ¼
K σd, i Km a
ð2Þ ð3Þ
The existence of a so-called anomalous surface conductance (that is, a surface conductivity not explained by charge transport in the diffuse layer) was first proposed by Sengupta et al.9,10 in the 1960s. According to Hunter,11 the first general model of conductivity beneath the electrokinetic plane (in the stagnant layer) was elaborated on by Dukhin and Derjaguin shortly thereafter.12-14 Because of the significance of these effects, there are abundant references in the literature regarding the theory and experimental determination of electrokinetic phenomena in the context of a finite SLC.15-22 Additionally, efforts have also been devoted to considering the effects of volume fraction, mainly using cell models:23-27 in these, hydrodynamic and electrical interactions between particles are indirectly considered by solving the problem for a single particle immersed in a concentric “cell” of liquid of radius b and selecting the proper boundary conditions on the cell surface. The size of the cell is such that the volume fraction of solids in it equals that of the overall suspension: φ ¼
3 a b
ð4Þ
A cell model combined with SLC was used in a previous paper28 for the evaluation of the electrical permittivity of suspensions. In the present work, we will extend that treatment and apply it to the calculation of the dynamic mobility ue. The article is (9) Sengupta, M.; Bose, A. K. J. Electroanal. Chem. 1968, 18, 21. (10) Sengupta, M.; Biswas, D. N. J. Colloid Interface Sci. 1969, 29, 536. (11) Hunter, R. J. Adv. Colloid Interface Sci. 2003, 100, 153. (12) Dukhin, S. S.; Derjaguin, B. V. In Surface and Colloid Science; Matijevic, E., Ed.; Wiley: New York, 1974; Vol. 7. (13) Derjaguin, B. V.; Dukhin, S. S.; Shilov, V. N. Adv. Colloid Interface Sci. 1980, 13, 141. (14) Dukhin, S. S.; Shilov, V. N. Adv. Colloid Interface Sci. 1980, 13, 153. (15) Kijlstra, J.; van Leeuwen, H. P.; Lyklema, J. J. Chem. Soc., Faraday Trans. 1992, 88, 3441. (16) Rosen, L. A.; Baygents, J. C.; Saville, D. A. J. Chem. Phys. 1993, 98, 4183. (17) Lyklema, J.; Minor, M. Colloids Surf., A 1998, 140, 33. (18) Carrique, F.; Arroyo, F. J.; Delgado, A. V. J. Colloid Interface Sci. 2001, 243, 351. (19) Dukhin, S. S.; Zimmermann, R.; Werner, C. Colloids Surf., A 2001, 195, 103. (20) Matsumura, H.; Verbich, S. V.; Dimitrova, M. N. Colloids. Surf., A 2001, 192, 331. (21) Moncho, A.; Martinez-Lopez, F.; Hidalgo-Alvarez, R. Colloids. Surf., A 2001, 192, 215. (22) Gupta, A. K.; Coelho, D.; Adler, P. M. J. Colloid Interface Sci. 2007, 316, 140. (23) Kuwabara, S. J. Phys. Soc. Jpn. 1959, 14, 527. (24) Levine, S.; Neale, G. H. J. Colloid Interface Sci. 1974, 47, 520. (25) Shilov, V. N.; Zharkikh, N. I.; Borkovskaya, Y. B. Colloid J. 1981, 43, 434. (26) Dukhin, A. S.; Ohshima, H.; Shilov, V. N.; Goetz, P. J. Langmuir 1999, 15, 3445. (27) Carrique, F.; Cuquejo, J.; Arroyo, F. J.; Jimenez, M. L.; Delgado, A. V. Adv. Colloid Interface Sci. 2005, 118, 43. (28) Carrique, F.; Arroyo, F. J.; Shilov, V. N.; Cuquejo, J.; Jimenez, M. L.; Delgado, A. V. J. Chem. Phys. 2007, 126, 104903.
Langmuir 2009, 25(20), 12040–12047
organized as follows: the basic aspects of the approach used will first be given. This will be followed by details of the calculation of ue and on how to take into account SLC in the model. We will finally discuss some significant results.
2. Fundamental Aspects A very detailed treatment was given in ref 28. It seems reasonable to give here only the main lines of the model. The starting point is writing the Navier-Stokes and Poisson-Boltzmann equations in order to describe the electrical potential, Ψ(r, t), and (incompressible) fluid velocity, v(r, t), distributions at position r and time t in a reference frame fixed at the (spherical) particle center: N P ezj nj ðr, tÞ r2 Ψðr, tÞ ¼ -
ηm r2 vðr, tÞ - rpðr, tÞ -
j ¼1
ð5Þ
εm ε0 N X
ezj nj ðr, tÞrΨðr, tÞ
j ¼1
¼ Fm
D ½vðr, tÞ þ UðtÞ Dt
r 3 vðr, tÞ ¼ 0
ð6Þ ð7Þ
In these equations, εmε0 is the electrical permittivity of the dispersion medium, nj(r, t) is the number concentration, ezj is the charge of each ionic species j (the solution contains N different ions), ηm is the viscosity of the dispersion medium, Fm is its density, p(r, t) is the pressure field, and U(t) is the velocity of the particle with respect to the laboratory system. In addition, the ionic velocities, vj(r, t), will have electromigration, diffusion, and convective contributions vj ðr, tÞ ¼ vðr, tÞ -
Dj rμj ðr, tÞ, kB T
j ¼ 1, ::::, N
ð8Þ
where the electrochemical potential of each species reads μj ðr, tÞ ¼ μ¥j þ zj eΨðr, tÞ þ kB T ln nj ðr, tÞ
ð9Þ
μ¥ j is the standard chemical potential of species j, Dj is its diffusion coefficient, kB is the Boltzmann constant, and T is the absolute temperature. If the applied field has an angular frequency ω (E exp(-iωt)), then all of the field-induced quantities will have the same frequency dependencies. To take phase differences into account, the quantities will be complex. Specifically, the perturbations of the electrochemical potentials, fluid velocity, and electrical potential can be written as28,29 δμj ðrÞ ¼ -zj eφj ðrÞE cos θ vðrÞ ¼ ðvr , vθ , vj Þ ¼
ð10Þ
2 1d ½rhðrÞE sin θ, 0 - hðrÞE cos θ, r r dr ð11Þ
δΨðrÞ ¼ -ΞðrÞE cos θ
ð12Þ
(29) Ohshima, H. Theory of Colloid and Interfacial Electric Phenomena; Academic Press: London, 2006.
DOI: 10.1021/la901132g
12041
Article
Arroyo et al.
where spherical polar coordinates (r, θ, j) are used, and φj(r), h(r), and Ξ(r) are the auxiliary functions depending only on the r coordinate because of the linearity assumed between the perturbations and the field and the spherical symmetry of the problem. Further details on how these substitutions lead to ordinary differential equations can be found in refs 6 and 27. At this point, one should consider how the existence of a finite SLC can be brought into consideration. As mentioned before, the development of a model of a dynamic stagnant layer or a dynamic Stern layer was originally due to Dukhin and Derjaguin.12-14 Extensions and additional contributions to the model and its numerical evaluations have also been proposed by Zukovski IV and Saville,30 Mangelsdorf and White,31-33 and Lopez-Garcı´ a et al.34,35 We will follow Mangelsdorf and White, who considered that, contrary to the assumptions of the standard electrokinetic model, normal motions of ions at the slip plane are possible: ions can penetrate or escape that layer, thus changing its charge. Likewise, they can move tangentially to the surface by either diffusion or electromigration. As in the classical approach, the thickness of the stagnant layer is taken to be the size of a (at least partially) dehydrated adsorbed ion. Such a thickness will be much smaller than any other length scale of the problem. From this, it will be considered that the solid radius and the position of the slip surface (relative to the particle center) are both equal to a. In addition, the stick condition on the solid prevents any liquid motion in the stagnant layer. The conductivity associated with ionic motion in that region can be estimated as28,31 K σi ¼
N X eFasgnðzj Þzj 2 103 cδj Dj zj eζ j ¼1 kB Texp kB T
ð13Þ
where c is the molar concentration of electrolyte and F is the Faraday constant and the basic parameter is zj e σ DSL Nj j exp aKj Dj kB T CSL " ð14Þ δj ¼ # ¥ N P ni zi e σ exp 1þ ζ - SL kB T C i ¼1 Ki where n¥ i is the bulk ionic concentration of type i ions and σ is the surface charge density. This allows us to establish a relationship between Kσi and - the diffusion coefficient of ions in the SL, DSL j - the number density, Nj, of sites available for adsorption of ions onto the SL - the dissociation constant, Kj, of a site Sj available for adsorption of the ionic species Xj Kj
Sj þ Xj T SXj
ð15Þ
- the specific (per surface area) capacitance of the stagnant layer, CSL. (30) (31) 2859. (32) 2441. (33) 2583. (34) (35) 384.
Zukoski, C. F., IV; Saville, D. A. J. Colloid Interface Sci. 1986, 114, 32. Mangelsdorf, C. S.; White, L. R. J. Chem. Soc., Faraday Trans. 1990, 86, Mangelsdorf, C. S.; White, L. R. J. Chem. Soc., Faraday Trans. 1998, 94,
3. Boundary Conditions and Mobility Evaluation Because of the importance of boundary conditions (in general, but particularly in cell models), we will give a short account of the physical assumptions and their analytical implications. We will do so both on the particle surface (r = a) and on the cell boundary (r = b). (i) Stick condition of the liquid, v = 0, at r = a: dh hðaÞ ¼ 0, ð16Þ ¼0 dr r ¼a (ii)
The equilibrium potential (Ψ0(r)) equals the zeta potential at r = a: ð17Þ Ψ0 ðaÞ ¼ ζ Alternatively, the surface charge density, σ, can be considered to be input: dΨ0 ðrÞ σ ¼ -ε0 εm ð18Þ dr r ¼a
(iii) There is an additional condition on the particle (or slip) surface regarding the ionic velocity. In the absence of SLC, vj ^r|r=a = 0, (^r is the unit vector of the radial coordinate of the reference system) or, equivalently, dφ/dr|r=a = 0. In the case of interest for this work, this condition must be substituted by an ion conservation equation:31 dφj 2δj φ ðaÞ ¼ 0, j ¼ 1, :::, N ð19Þ dr a j r ¼a
(iv) On the cell surface r = b, the conditions follow Kuwabara’s model:23 8 < vr ðbÞ ¼ -ue E cos θ ð20Þ ue b : hðbÞ ¼ 2 8 > > ðr vÞjr ¼b ¼ 0 < L½hðrÞjr ¼b ¼ 0 ð21Þ 2 > > :L d þ 2 d - 2 dr2 r dr r2 (v)
The condition of electroneutrality of the cell in equilibrium reads dΨ0 ¼0 ð22Þ dr r ¼b
(vi) The field-induced potential perturbation is, following the Shilov-Zharkikh-Borkovskaya25 original approach ( δΨðr ¼ bÞ ¼ -Eb cos θ ð23Þ ΞðbÞ ¼ b (vii) Similarly, according to Borkovskaya and Shilov36 and Lee et al.,37,38 the perturbations of the
Mangelsdorf, C. S.; White, L. R. J. Chem. Soc., Faraday Trans. 1998, 94, Lopez-Garcı´ a, J. J.; Grosse, C.; Horno, J. J. Phys. Chem. B 2007, 111, 8985. Lopez-Garcı´ a, J. J.; Grosse, C.; Horno, J. J. Colloid Interface Sci. 2009, 329,
12042 DOI: 10.1021/la901132g
(36) Borkovskaya, Y. B.; Shilov, V. N. Kolloid-Z. 1992, 54, 43. (37) Lee, E.; Chu, J. W.; Hsu, J. J. Colloid Interface Sci. 1999, 209, 240. (38) Lee, E.; Chu, J. W.; Hsu, J. J. Chem. Phys. 1999, 110, 11643.
Langmuir 2009, 25(20), 12040–12047
Arroyo et al.
Article
concentrations of the ionic species must be zero on the cell boundary: 8 > > >
φj ðbÞ ¼ ΞðbÞ > > : j ¼ 1, :::, N
ð24Þ
The physical justification for this choice of outer boundary conditions (eqs 23 and 24) has been recently discussed in detail by Zholkovskij et al.39 (viii) Finally, the consideration of the force balance on the cell leads to8,40 ! 2φðFp -Fm Þ hðbÞ d 2 1- ðLh þ γ hÞ γ b dr Fm
þ
2
r ¼b
F0el ðbÞ ¼0 ηm ð25Þ
where Fp is the density of the particle and sffiffiffiffiffiffiffiffiffiffiffi iωFm γ ¼ ηm
ð26Þ
In the following text, we will describe the most relevant aspects of our results concerning the role of the Kσd-Kσi balance on the ue(ω) dependence.
4. Results and Discussion a. Overall Effect of Kσi on the Dynamic Mobility. Figure 1 (in this and subsequent plots the frequency f = ω/2π will be used instead of ω) shows the frequency dependences of the modulus and phase of the dimensionless dynamic mobility, ue*, related to the mobility by
ue ¼
3ηm e ue 2εm ε0 kB T
ð27Þ
for given total surface conductance Kσ and different SL conductivities. Table 1 contains the choice of parameters for obtaining such data. Contrary to other conditions often selected by the present and previous authors (Kσd fixed, increasing Kσi, and hence increasing Kσ), we decided to keep Kσ constant. This was considered a more realistic approach: Kσ can be measured or estimated for a given experimental situation,41,42 and hence the job is to ascertain what fraction of that Kσ might be due to SLC. This is important because the contributions of Kσd and Kσi are not exchangeable. An increase in Kσ associated with an increase in Kσd provokes an increase in the mobility. On the contrary, the mobility will be reduced if the total surface conductivity increases as a consequence of a larger Kσi. The overall trend in Mod(ue*) can (39) Zholkovskij, E. K.; Masliyah, J. H.; Shilov, V. N.; Bhattacharjee, S. Adv. Colloid Interface Sci. 2007, 134-135, 279. (40) Ahualli, S.; Delgado, A. V.; Miklavcic, S. J.; White, L. R. J. Colloid Interface Sci. 2007, 309, 342. (41) Delgado, A. V.; Gonzalez-Caballero, F.; Hunter, R. J.; Koopal, L. K.; Lyklema, J. J. Colloid Interface Sci. 2007, 309, 194. (42) Jimenez, M. L.; Arroyo, F. J.; Carrique, F.; Delgado, A. V. J. Colloid Interface Sci. 2007, 316, 836. (43) Shilov, V. N.; Delgado, A. V.; Gonzalez-Caballero, F.; Horno, J.; Lopez-Garcı´ a, J. J.; Grosse, C. J. Colloid Interface Sci. 2000, 232, 141. (44) Ahualli, S.; Delgado, A. V.; Grosse, C. J. Colloid Interface Sci. 2006, 301, 660.
Langmuir 2009, 25(20), 12040–12047
Figure 1. Modulus (a) and phase (b) of the dimensionless dynamic mobility ue* of suspensions with total Kσ = 4.39 nS (Du = 3.18) and different values of stagnant-layer conductivity Kσi (increasing in the direction of the arrow), calculated using the parameters described in Table 1. Inset: Detail of the R-relaxation region.
Table 1. Parameters Used in the Calculations of the ue*( f ) Dependences in Figures 1 and 3a total Kσ (nS)/Du
ζ (mV)
eN+ (μC/cm2)
Kσd (nS)
Kσi (nS)
-161.5 0 4.39 0 -153.4 1 3.69 0.70 -122.7 5 1.86 2.53 -102.5 10 1.13 3.26 -84.0 20 0.68 3.71 0.732/0.523 -86.3 0 0.732 0 -78.4 1 0.580 0.152 -61.2 5 0.327 0.405 -50.0 10 0.209 0.523 -36.9 20 0.110 0.622 a Other parameters: radius a = 100 nm, [KCl] = 0.926 mM, eN- = 0, SL SL =130 μF/cm2, φ = DSL + /D+ =1, D- /D-= 0, pK+ = pK- =1, C 5%, Fp = 2200 kg/m3, and Fm = 997 kg/m3. 4.39/3.18
be at first glance explained by recalling the approximate expression for the DC mobility43,44 ue ðDCÞ ¼
2 εm ε0 ζð1 -CÞ 3 ηm
ð28Þ
where C is the induced dipole coefficient. If the field is alternating, the expression remains qualitatively valid provided that we DOI: 10.1021/la901132g
12043
Article
Arroyo et al.
Figure 2. (a and b) Dimensionless electro-osmotic and capillary osmosis velocity profiles, respectively, for the case Kσi = 0 (first line in
Table 1); the frequency of the electric field is 10 Hz . (c and d) The same for the maximum value used for Kσi (fifth line in Table 1). (e) Capillary osmosis velocity as a function of the dimensionless distance to the particle center, with total Kσ = 4.39 nS (Du = 3.18) and the parameters described in Table 1, at a constant frequency of 10 Hz. (f) Same as in plot e but with varying frequency of the electric field (constant radial distance = a þ k-1).
consider C to be a complex, frequency-dependent quantity and that we include the inertia of the particle and liquid if ω is high enough.45 Above a critical frequency ωi ≈ ηm/a2Fm, neither the particle nor the fluid can follow the rapid field oscillations and the mobility falls off rapidly. This is observed above ∼108 Hz in Figure 1. (45) O’Brien, R.W. J. Fluid Mech. 1990, 212, 81.
12044 DOI: 10.1021/la901132g
This decrease is preceded by a clear increase: it is the manifestation of the Maxwell-Wagner-O’Konski relaxation. For frequencies below that relaxation, ions in the double layer have time to accumulate on both sides of the particle before the field is inverted. This gives a positive contribution to the induced dipole coefficient43 and will provoke a mobility reduction; if the frequency goes above the characteristic frequency of the process, Langmuir 2009, 25(20), 12040–12047
Arroyo et al.
Article
ωMWO, then Mod(C) decreases and hence Mod(u*) e increases, as observed. An approximate expression for ωMWO is
ωMWO
Kσ þ ð2 þ φÞKm 2a ¼ ð1 -φÞεp ε0 þ ð2 þ φÞεm ε0 ð1 -φÞ
ð29Þ
with εp being the relative permittivity of the particles. Note how the value of ωMWO is controlled by both Kσ and Km. Because these are not changed in the different curves of Figure 1, it is reasonable to find the same ωMWO in all cases in this Figure. The mobility increase above ωMWO is observed whatever the Kσd/Kσi ratio, although the amplitude of the process is larger the smaller the value of Kσi. This can be explained by considering the opposite effects of R relaxation for low and high Kσi, to be described below. Indeed, a careful observation of the data in Figure 1 indicates an additional feature: for low or zero Kσi, both Mod(ue*) and phase(u*) e show a tiny increase before the MWO growth, whereas for high Kσi that rise becomes a decrease. The characteristic frequency of such a change in tendency is very close to that of another relaxation of the double layer: the so-called R relaxation, with a frequency that can be estimated as ωR ¼
2D L2
ð30Þ
where D is the counterion diffusion coefficient and L is some diffusion length, typically on the order of the particle radius if the volume fraction is not too high.46 Below this frequency, a gradient of electrolyte concentration (concentration polarization) is produced that opposes the MWO polarization; this leads to a decrease in Mod(C), and hence when ωR is surpassed, Mod(C) increases and the mobility is expected to decrease. We may wonder why such a decrease in Mod(u*) e is not observed at low Kσi whereas it is barely shown at high Kσi. An additional physical phenomenon can be mentioned to account for this: capillary osmosis.47 This is a liquid motion induced tangentially to the particle by the concentration gradient just mentioned: diffusion of ions (mainly counterions) in the diffuse double layer drags liquid in a direction opposite to that of the electro-osmotic motion directly induced by the field. In addition, the concentration gradient brings about a zeta potential gradient as well (the potential is smaller in absolute value at the high concentration side), which also makes the counterions migrate in the direction of the diffusion flow. Above ωR, the concentration polarization is absent and so is capillary osmosis, hence there should be an increase in the mobility. This may compensate for the decrease in mobility associated with the increase in C, past ωR, as mentioned above. If such a compensation occurs, then the mobility should increase when ω > ωR; this is observed for zero or low Kσi in Figure 1. On the contrary, if Kσi is large enough, then the role of capillary osmosis will be of little significance, and only the increase associated with C, and the corresponding decrease in mobility, is observed. To confirm these reasonings, we calculated the capillary osmosis velocity profile (vcap(r,θ)) using the following expression:47 2 3 2 zeΨ cosh 4kB T 7 4εm ε0 kB T 6 5rθ δ~ vcap ðr, θÞ ¼ ln4 n ze ηm cosh zeζ
ð31Þ
4kB T
(46) Dukhin, S. S.; Shilov, V. N. Dielectric Phenomena and the Double Layer in Disperse Systems and Polyelectrolites; Jerusalem Keter Publishing: Jerusalem, 1974. (47) Grosse, C.; Shilov, V. N. J. Phys. Chem. 1996, 100, 1771.
Langmuir 2009, 25(20), 12040–12047
Figure 3. Modulus (a) and phase (b) of the dimensionless dynamic mobility of suspensions with total Kσ = 0.732 nS (Du = 0.523) for the parameters shown in Table 1.
where δ~ n ¼ Kc
exp½ -ð1 þ iÞWðr=a - 1Þ 1 þ ð1 þ iÞWr=a Eea cos θ r2 =a2 1 þ W þ iW kB T ð32Þ
being sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ωa2 D þ D - ðz þ þ z - Þ W ¼ 2 D þz þ þ D -z -
ð33Þ
for a binary electrolyte. Kc is proportional to the variation of neutral salt concentration along the surface of the particle (see ref 47 for more details). The diffusion coefficient and valency of the counterion (co-ion) are Dþ and zþ (D- and z-), respectively, and z = zþ = -z -. The results have been plotted in Figure 2, in all cases at θ = π/2 for a negatively charged particle and for the maximum value of the electric field directed from left to right. Panels a and c in this Figure represent the electro-osmotic fluid velocity field veo in its dimensionless form:
veo, co ¼
ηm e veo, co εm ε0 kB TE
ð34Þ
Note that the (position-independent) electro-osmotic velocity far from the particle is equal (and opposite) to the particle’s DOI: 10.1021/la901132g
12045
Article
Figure 4. Modulus (a) and phase (b) of the dimensionless dynamic mobility of suspensions for different volume fractions. The parameters used correspond to the extreme cases in Figure 1: (-) Kσd = 4.39 nS nad Kσi = 0; (---) Kσd = 0.68 nS and Kσi = 3.71.
electrophoretic velocity with respect to the laboratory reference frame. The similarities of the velocity fields for zero and high Kσi are clear (compare Figure 2a,c), and they differ only in the velocity moduli at each position and are smaller when Kσd decreases. The calculation of the capillary osmosis velocity, vco, is plotted in Figure 2b,d for zero and finite Kσi, respectively. It can be observed that vco is opposed to the electro-osmotic velocity. In addition, although vco also decreases if Kσi increases, the fraction of veo represented by capillary osmosis is different for negligible (a, b) and high (c, d) Kσi. It is interesting to mention that the capillary osmosis is maximized at distances from the surface that are comparable to the particle radius (panel e), precisely the characteristic diffusion length of concentration polarization. Furthermore, it goes to zero for frequencies above (2-3) 105 Hz (panel f), thus confirming that it is associated with the concentration polarization process. A very clear feature of the data in Figure 1 (also in Figure 3, corresponding to lower Dukhin number) is the overall decrease of the mobility when Kσi is increased at constant Kσ. This can be understood if we examine the electrophoretic motion from the point of view of an observer sitting on the particle: the fluid moves relative to it because of electromigration and diffusive fluxes driven by the action of the field on the charged diffuse layer. The only contribution to electroosmotic flow is due to the diffuse double layer part (motions of ions in the SL do not drag fluid), which is increasingly lower as Kσi increases at constant Kσ and 12046 DOI: 10.1021/la901132g
Arroyo et al.
Figure 5. Same as Figure 4 but for different particle radii. The parameters used are described in Table 2 .
hence a reduction of fluid velocity will be observed; this explains the reduction observed for the mobility. b. Volume Fraction and Size Effects. Although unexpected at first glance, the dependences of ue on both the volume fraction of solids and the particle size are also affected by the Kσi/Kσ ratio. Figure 4 shows that, whatever the sources of Kσ, increasing φ leads to higher inertia frequencies. This is a manifestation of the fact that the presence of neighbor particles in the vicinity of a given one reduces the characteristic distance for inertia relaxation to values on the order of the particle-to-particle distance. The 6 decrease in ue associated with the R relaxation is clear when Kσi ¼ 0, as mentioned above, but it is apparent that increasing φ somewhat hides such a reduction, as a consequence of the fact that the concentration polarization loses its influence as the volume fraction is increased.6 Similar reasoning can be applied to the effect of particle size, shown in Figure 5. The choice of parameters is indicated in Table 2. The size significantly affects the relaxation frequencies and even the tendencies observed for both Mod(u*) e but such influences are clearly different for and phase(u*), e zero and finite Kσi. Specifically, the ranges of variation of both quantities are reduced if the stagnant layer is conductive, leaving a lower conductivity in the diffuse region. Also, the modulus of the mobility is very much reduced when Kσi 6¼ 0, as expected from the decrease that, according to Table 2, is required in the zeta potential if one wishes to keep Kσ constant, with finite Kσi. Nevertheless, such differences between negligible and finite Kσi are almost unobservable with the phase of u*. e Langmuir 2009, 25(20), 12040–12047
Arroyo et al.
Article
Table 2. Parameters Used in the Calculations of the Dimensionless Mobility in Figure 5a eNþ a (nm) ζ (mV) (μC/cm2)
Kσd (nS)
Kσi (nS)
Du
no SL conductivity
50 0 4.43 0 6.4 100 3.2 250 -161.9 1.28 500 0.63 finite SL conductivity 50 -84.0 20 0.68 3.75 6.4 100 -84.3 0.69 3.74 3.2 250 -84.4 0.69 3.74 1.28 500 -84.5 0.69 3.74 0.63 a Other parameters: volume fraction φ = 10%, [KCl] = 0.926 mM, SL SL = 130 μF/ eN- = 0, DSL þ /Dþ = 1, D- /D- = 0, pKþ = pK- = 1, C cm2, Fp = 2200 kg/m3, and Fm = 997 kg/m3.
5. Conclusions In this work, we have extended previous contributions to the evaluation of electrokinetic phenomena in the presence of
Langmuir 2009, 25(20), 12040–12047
nonzero stagnant-layer conductivity to the case of the dynamic electrophoretic mobility ue for arbitrarily concentrated suspensions. A cell model has been used for the calculation of ue as a function of the frequency of the applied electric field for different contributions of the diffuse (Kσd) and stagnant (Kσi) layers to the total surface conductivity Kσ. We have shown that increasing (Kσi) at constant Kσ changes the frequency trend of ue in the vicinity of the R relaxation; it is proposed that such changes are associated with modifications of the capillary osmosis velocity field. On the contrary, the Maxwell-Wagner-O’Konski and inertia relaxations are mostly affected by the volume fractions and are weakly dependent on the Kσi/ Kσ ratio. Acknowledgment. Financial support for this work by MICINN, Spain (project FIS2007-62737, cofinanced with FEDER funds by the EU)), and Junta de Andalucı´ a, Spain (projects P08FQM-3993 and P08-FQM-3779), is gratefully acknowledged.
DOI: 10.1021/la901132g
12047
