Langmuir 2009, 25, 1961-1969
1961
Dynamic Dielectric Response of Concentrated Colloidal Dispersions: Comparison between Theory and Experiment† B. H. Bradshaw-Hajek,*,‡,§ S. J. Miklavcic,‡ and L. R. White‡ School of Mathematics and Statistics, UniVersity of South Australia, Mawson Lakes, SA 5082, Australia, and Department of Science and Technology, UniVersity of Linko¨ping, S-601 74, Norrko¨ping, Sweden ReceiVed September 4, 2008. ReVised Manuscript ReceiVed December 8, 2008 The cell-model electrokinetic theory of Ahualli et al. Langmuir 2006, 22, 7041; Ahualli et al. J. Colloid Interface Sci. 2007, 309, 342; and Bradshaw-Hajek et al. Langmuir 2008, 24, 4512 is applied to a dense suspension of charged spherical particles, to exhibit the system‘s dielectric response to an applied electric field as a function of solids volume fraction. The model’s predictions of effective permittivity and complex conductivity are favorably compared with published theoretical calculations and experimental measurements on dense colloidal systems. Physical factors governing the volume fraction dependence of the dielectric response are discussed.
1. Introduction In a recent series of papers,1-3 the authors proposed a selfconsistent cell-model theory of electrokinetics for concentrated suspensions of charged spherical particles. The cell-model theory, which is used to account approximately for electrical double layer interactions, is self-consistent in the sense that the model itself generates a natural set of boundary conditions through the application of physically motivated, volume-averaged constraints. To our knowledge, the cell-model provides the best means available of modeling electrical double layer interaction effects on the electrokinetics of concentrated charged particle systems. A more physically correct approach is the two-particle, perturbation method pursued by Ennis et al.4,5 Unfortunately, in practice, this approach leads only to first-order, volume fraction dependent expressions for the dynamic mobility and dielectric properties, as only two-particle interactions are incorporated explicitly. There has not yet been an announcement in the literature, that we are aware of, of a more accurate approach to determining the effect of particle-particle interactions on the mobility, conductivity, and dielectric response of suspensions under general conditions. An alternate approach for low particle permittivity and high κa systems has been advanced by O‘Brien et al.6 In our previous publications, attention focused on the influence of particle interactions on, first, the dynamic mobility of the particles1,2 and, second, on the conductivity of the suspension.3 The theoretical calculations of dynamic mobility were compared with experimental mobility measurements in ref 2, with quite reasonable agreement at both low (3%) and moderately high (25%) volume fractions of spherical silica particles in an aqueous electrolyte medium. It was thus demonstrated that the cell model is able to describe electrophoretic mobility measurements in † ‡ §
This work was supported by a grant from the Swedish Research Council. University of South Australia. University of Linko¨ping.
(1) Ahualli, S.; Delgado, A.; Miklavcic, S.; White, L. R. Langmuir 2006, 22, 7041. (2) Ahualli, S.; Delgado, A.; Miklavcic, S.; White, L. R. J. Colloid Interface Sci. 2007, 309, 342. (3) Bradshaw-Hajek, B. H.; Miklavcic, S. J.; White, L. R. Langmuir 2008, 24, 4512. (4) Ennis, J.; Shugai, A. A.; Carnie, S. L. J. Colloid Interface Sci. 2000, 223, 21. (5) Ennis, J.; Shugai, A. A.; Carnie, S. L. J. Colloid Interface Sci. 2000, 223, 37. (6) O’Brien, R. W.; Jones, A.; Rowlands, W. N. Colloids Surf., A 2003, 218, 89.
concentrated systems.2 Given that dielectric response measurements (conductivity and permittivity) appear to be a far more sensitive test, it is instructive to make a similar comparison with complex conductivity data. In this paper, we therefore supplement our electrokinetic cell-model analysis contained in refs 1-3 with information directed along three fronts. We compare our theoretical complex conductivity results with the published theoretical work of DeLacey and White7 which is correct to order φ but limited to lower frequencies. We also apply our model calculations of the complex conductivity3 to a number of dense colloidal systems that have been studied experimentally for which data on conductivity and permittivity is available in the literature.8,9 Third, we examine the physical reasons for the dependence of the dielectric response on volume fraction. Theories for the electrokinetics of dilute suspensions of spherical particles are well developed.10-16 The standard model for calculating the complex conductivity of dilute suspensions in an oscillating field was developed by DeLacey and White,7 based on the O’Brien and White10 model for a static electric field. This theory has been tested experimentally by a number of authors (for example, see refs 17 and 18) for low volume fraction suspensions. In dilute suspensions, the particle contribution to the suspension‘s complex conductivity can be difficult to measure19 because the particle contribution is approximately proportional to the concentration of particles. An electrokinetic theory valid for high volume fractions is therefore desirable. (7) DeLacey, E. H. B.; White, L. R. J. Chem. Soc. Faraday Trans. 2 1981, 77, 2007. (8) Midmore, B. R.; Hunter, R. J.; O’Brien, R. W. J. Colloid Interface Sci. 1987, 120, 210. (9) Minor, M.; van Leeuwen, H. P.; Lyklema, J. J. Colloid Interface Sci. 1998, 206, 397. (10) O‘Brien, R. W.; White, L. R. J. Chem. Soc. Faraday Trans. 2 1978, 74, 1607. (11) O’Brien, R. W. J. Colloid Interface Sci. 1986, 113, 81. (12) O’Brien, R. W. J. Fluid Mech. 1988, 190, 71. (13) O’Brien, R. W. J. Fluid Mech. 1990, 212, 81. (14) Mangelsdorf, C. S.; White, L. R. J. Chem. Soc. Faraday Trans. 1992, 88, 3567. (15) Rosen, L. A.; Baygents, J. C.; Saville, D. A. J. Chem. Phys. 1993, 98, 4183. (16) Ennis, J.; White, L. R. J. Colloid Interface Sci. 1996, 178, 446. (17) Rosen, L. A.; Saville, D. A. J. Colloid Interface Sci. 1990, 140, 82. (18) Carrique, F.; Zurita, L.; Delgado, A. V. J. Colloid Interface Sci. 1994, 166, 128. (19) Myers, D. F.; Saville, D. A. J. Colloid Interface Sci. 1989, 131, 448.
10.1021/la8028963 CCC: $40.75 2009 American Chemical Society Published on Web 01/20/2009
1962 Langmuir, Vol. 25, No. 4, 2009
Bradshaw-Hajek et al.
It is well-known that theoretical determinations of the frequency-dependent, complex permittivity do not always compare favorably with experiment, especially at the low frequency or static limit.9,17,20 There are two plausible explanations for this discrepancy. The first is the failure to accurately correct impedance measurements for electrode polarization.20,21 Second, since this discrepancy appears to be a general finding, independent of the particulars of the theoretical model employed, reasons that have been put forward refer to physical characteristics that have been excluded in the models, such as surface conduction or nonuniformity of the charge layer on the particle surface18,20,22 rather than a failure of the models per se. One of the conclusions we draw from the comparison herein is that if it can be accepted that the self-consistent cell-model of electrokinetics is the best means available to account for particle interactions, then further investigation of the above-mentioned second order effects are indeed warranted, at least in some experimental situations. A number of cell models have appeared in the literature (for example, see refs 23-25), differing only in the boundary conditions applied at the outer boundary. Historically, these boundary conditions have been invoked in an ad hoc manner. In our development of the cell model, we have taken the view that these boundary conditions must be derived from integral constraints over the suspension volume of the relevant physical quantities.1-3 This philosophy is also shared by the authors of a recent treatment of the cell model approach to colloidal electrokinetics.26
2. Fundamental Equations As described in detail in previous papers,1-3 the electrokinetic response of a charged particle suspension to a macroscopic oscillating electric field, E(t), is described by the following fundamental equations (see also refs 11-13):
∇2Ψ(r, t) ) Fs
1 F (r, t) εsε0 ch
(1)
∂u (r, t) ) - ∇ p(r, t) - Fch(r, t) ∇ Ψ(r, t) + ηs∇2u(r, t) ∂t (2) ∇ · u(r, t) ) 0 1 vj(r, t) ) u(r, t) - ∇ µj(r, t) λj ∂nj (r, t) + ∇ · [nj(r, t)vj(r, t)] ) 0 ∂t
µj(r, t) ) -zjeΦj(r, t) ) µj∞ + zjeΨ(r, t) + kBT
(4) (5) ln[nj(r, t)] (6)
N
∑ zjenj(r, t)
λj )
NAe2|zj| Λj∞
where NA is Avogadro’s number. The electrochemical potential functions Φj(r, t) describe the deviations of the local ion densities from their Poisson-Boltzmann expressions (see eq 6). For a more detailed explanation of these equations, see ref 1. The Debye parameter is given by N
κ2 )
e2z 2n∞
∑ εsε0jkBjT j)1
Under the influence of the volume averaged applied electric field, E, oscillating at angular frequency ω, the field variables deviate from their equilibrium values.11-13 The deviation is supposed sufficiently weak that only the first order perturbation correction in E is needed:
Ξ(r, t) ) Ξ(0)(r) + ξ(r)E · rˆe-iωt + O(E(t)2)
(8)
Here, Ξ represents any of the field variables, Φj, Ψ, Fch, µj, nj, and p, with Ξ(0) representing the respective field-independent equilibrium contributions. The first order correction, ξ(r), represents any of the functions, ψ(r), φj(r), and P(r), which correspond to perturbations to the electrostatic potential, electrochemical potential, and the hydrodynamic pressure, respectively. Symmetry ensures that the perturbed fluid velocity will have the form (in the reference frame of the central particle)27
(3)
with charge density
Fch(r, t) )
local charge density Fch, the electrochemical potentials µj, the drift velocities vj, (j ) 1, 2,..., N), the hydrodynamic flow field u, the pressure p, and the electrostatic potential, Ψ. In addition, e, εs, ε0, Fs, ηs, zj, λj, kB, and T are the electron charge, the fluid relative dielectric permittivity, vacuum permittivity, fluid mass density, fluid viscosity, valency and ionic drag coefficients of the jth ion type, Boltzmann constant, and temperature, respectively. The ion densities, n∞j , are those of the electrolyte reservoir in equilibrium with the field-free suspension. The drag coefficient, λj, is related to the ionic limiting conductance, Λ∞j , by
u(r) ) (ur, uθ, uφ) ) µ(ω)E 2h(r) 1 ∂(rh(r)) ˆ, 0 (9) E · rˆ, [E - E · rˆrˆ] · θ r r ∂r
(
where µ(ω) is the dynamic mobility and h(r) is a flow field function. Invoking the perturbation expansion eq 8 in the fundamental system (eqs 1-7) leads to a zeroth-order and a first-order system of equations. The zeroth-order contributions satisfy the field equations
{
(7)
j)1
Here, the electrolyte ion number densities are denoted nj, the (20) Jime´nez, M. L.; Arroyo, F. J.; van Turnhout, J.; Delgado, A. V. J. Colloid Interface Sci. 2002, 249, 327. (21) Hollingsworth, A. D.; Saville, D. A. J. Colloid Interface Sci. 2003, 257, 65. (22) Hill, R. J.; Saville, D. A.; Russel, W. B. J. Colloid Interface Sci. 2003, 263, 478. (23) Kozak, M. W.; Davis, E. J. J. Colloid Interface Sci. 1986, 112, 403. (24) Shilov, V. N.; Zharkikh, N. I.; Borkovskaya, Y. B. Kolloidn. Zh. 1981, 43, 434. (25) Dukhin, A. S.; Shilov, V. N.; Ohshima, H.; Goetz, P. J. Langmuir 1999, 15, 6692. (26) Zholkovskij, E. K.; Masliyah, J. H.; Shilov, V. N.; Bhattacharjee, S. AdV. Colloid Interface Sci. 2007, 134, 279.
)
nj(0)(r) ) nj∞ exp(-zjeΨ(0)(r)/kBT) N
F(0) ch (r) )
∑ zjenj(0)(r) j)1
p(0)(r) ) pamb +
N
∑
(10) (nj(0)(r) - nj∞)kBT
j)1
µj(0)(r) ) -zjeΦj(0)(r) ) µj∞ + kBT ln nj∞ and (27) Landau, L. D. Lifshitz, E. M. Fluid Mechanics; Pergamon Press: Oxford, 1966.
Response of Concentrated Colloidal Dispersions
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Langmuir, Vol. 25, No. 4, 2009 1963
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