Dielectric Response of a Concentrated Colloidal ... - ACS Publications

11544

Langmuir 2008, 24, 11544-11555

Dielectric Response of a Concentrated Colloidal Suspension in a Salt-Free Medium Fe´lix Carrique,*,† Emilio Ruiz-Reina,‡ Francisco J. Arroyo,§ M. L. Jime´nez,| and ´ ngel V. Delgado| A Departamento de Fı´sica Aplicada I, Facultad de Ciencias, and Departamento de Fı´sica Aplicada II, Escuela UniVersitaria Polite´cnica, UniVersidad de Ma´laga, 29071 Ma´laga, Departamento de Fı´sica, Facultad de Ciencias Experimentales, UniVersidad de Jae´n, 23071 Jae´n, and Departamento de Fı´sica Aplicada, Facultad de Ciencias, UniVersidad de Granada, 18071 Granada, Spain ReceiVed July 11, 2008. ReVised Manuscript ReceiVed August 14, 2008 In this paper the complex dielectric constant of a concentrated colloidal suspension in a salt-free medium is theoretically evaluated using a cell model approximation. To our knowledge this is the first cell model in the literature addressing the dielectric response of a salt-free concentrated suspension. For this reason, we extensively study the influence of all the parameters relevant for such a dielectric response: the particle surface charge, radius, and volume fraction, the counterion properties, and the frequency of the applied electric field (subgigahertz range). Our results display the so-called counterion condensation effect for high particle charge, previously described in the literature for the electrophoretic mobility, and also the relaxation processes occurring in a wide frequency range and their consequences on the complex electric dipole moment induced on the particles by the oscillating electric field. As we already pointed out in a recent paper regarding the dynamic electrophoretic mobility of a colloidal particle in a salt-free concentrated suspension, the competition between these relaxation processes is decisive for the dielectric response throughout the frequency range of interest. Finally, we examine the dielectric response of highly charged particles in more depth, because some singular electrokinetic behaviors of salt-free suspensions have been reported for such cases that have not been predicted for salt-containing suspensions.

Introduction In everyday work in the colloidal industry, it is a well-known fact that concentrated colloidal suspensions are of much more practical interest than those considered as dilute. This special interest has motivated big efforts toward the understanding of the many phenomena displayed by such concentrated systems.1-13 In particular, their electrokinetic properties occupy a special place, regarding both the experimental characterization of the suspensions and the development of theoretical electrokinetic models to explain them. Recently, the case of concentrated suspensions in deionized (salt-free) media has been more deeply explored due to the special phenomenology associated with such systems.14 A salt-free suspension has no ions different from those stemming from the dispersed particles. These ions (the added counterions) may appear * To whom correspondence should be addressed. E-mail: [email protected]. † Departamento de Fı´sica Aplicada I, Universidad de Ma´laga. ‡ Departamento de Fı´sica Aplicada II, Universidad de Ma´laga. § Universidad de Jae´n. | Universidad de Granada.

(1) Levine, S.; Neale, G. H. J. Colloid Interface Sci. 1974, 47, 520. (2) Levine, S.; Neale, G. H.; Epstein, N. J. Colloid Interface Sci. 1976, 57, 424. (3) Ohshima, H. J. Colloid Interface Sci. 1998, 208, 295. (4) Ohshima, H. J. Colloid Interface Sci. 1997, 188, 481. (5) Carrique, F.; Arroyo, F. J.; Delgado, A. V. J. Colloid Interface Sci. 2001, 243, 351. (6) Ohshima, H. J. Colloid Interface Sci. 1999, 212, 443. (7) Ohshima, H. J. Colloid Interface Sci. 1997, 195, 137. (8) Marlow, B. J.; Fairhurst, D.; Pendse, H. P. Langmuir 1998, 4, 611. (9) Ohshima, H.; Dukhin, A. S. J. Colloid Interface Sci. 1999, 212, 449. (10) Dukhin, A. S.; Ohshima, H.; Shilov, V. N.; Goetz, P. J. Langmuir 1999, 15, 3445. (11) O’Brien, R. W. J. Fluid Mech. 1990, 212, 81. (12) Carrique, J.; Arroyo, F. J.; Jime´nez, M. L.; Delgado, A. V. J. Chem. Phys. 2003, 118, 1945. (13) Ruiz-Reina, E.; Carrique, F.; Rubio-Herna´ndez, F. J.; Go´mez-Merino, A. I.; Garcı´a-Sa´nchez, J. P. J. Phys. Chem. B 2003, 107, 9528. (14) Sood, A. K. Solid State Phys. 1991, 45, 1.

in the medium, for example, by dissociation of ionic groups on the surface of the particles.15,16 The resulting salt-free systems can form short- or long-ranged ordered phases with phase transitions at a relatively low volume fraction of particles. Due to the analogies with atomic or molecular crystals they are usually called colloidal crystals or glasses.14 In addition, some other specific effects have been described in the literature, including the counterion condensation effect and singular elastic behaviors associated with overlapping double layers and the crystalline order at low volume fractions, to mention a few.17-21 Among the most relevant previous studies on these systems are those of Ohshima dealing with surface charge density-surface potential relationships,22 stationary23,24 and dynamic25 electrophoresis, and the electrical conductivity or sedimentation field.26 Ohshima’s studies considered mainly the cases of high particle charge and low particle concentration. In a recent paper, Chiang et al.27 extended Ohshima’s model of the stationary salt-free electrophoretic mobility to an arbitrary particle concentration. Interest in these systems has also grown because of the particular behavior that they show as a consequence of the strong electrostatic interactions existing between particles.28 We also addressed the evaluation of the dc electrical conductivity and (15) Imai, N.; Oosawa, F. Busseiron Kenkyu 1952, 52, 42. (16) Oosawa, F. Polyelectrolytes; Dekker: New York, 1971. (17) Medebach, M.; Palberg, T. J. Chem. Phys. 2003, 119, 3360. (18) Medebach, M.; Palberg, T. Colloids Surf., A 2003, 222, 175. (19) Wette, P.; Scho¨pe, H. J.; Palberg, T. Colloids Surf., A 2003, 222, 311. (20) Medebach, M.; Palberg, T. J. Phys.: Condens. Matter 2004, 16, 5653. (21) Palberg, T.; Medebach, M.; Garbow, N.; Evers, M.; Fontecha, A. B.; Reiber, H.; Bartsch, E. J. Phys.: Condens. Matter 2004, 16, S4039. (22) Ohshima, H. J. Colloid Interface Sci. 2002, 247, 18. (23) Ohshima, H. J. Colloid Interface Sci. 2002, 248, 499. (24) Ohshima, H. J. Colloid Interface Sci. 2003, 262, 294. (25) Ohshima, H. J. Colloid Interface Sci. 2003, 265, 422. (26) Ohshima, H. Colloids Surf., A 2003, 222, 207. (27) Chiang, C. P.; Lee, E.; He, Y. Y.; Hsu, J. P. J. Phys. Chem. B 2006, 110, 1490.

10.1021/la802218j CCC: $40.75  2008 American Chemical Society Published on Web 09/23/2008

Dielectric Response of Concentrated Suspensions

Langmuir, Vol. 24, No. 20, 2008 11545

electrophoretic mobility in a salt-free concentrated suspension,29 analyzing the influence of different boundary conditions on such properties, with special emphasis on the Shilov-Zharkikh cell model.30-32 In the case of colloidal suspensions in salt solutions, the effect of the added counterions coming from the particle charging process is usually neglected because it is generally screened by the ions in the supporting solution for usual ionic concentrations. However, as was pointed out by Saville,33 even for the dilute case that effect should be taken into account. In the present work we study the dielectric response of a concentrated salt-free suspension in the presence of an applied oscillating electric field. In such systems, the added counterions are the only ionic species in the medium. We aim at investigating how they are distributed around the colloidal particles (when a field is applied) to properly predict the electrokinetic behavior of the system, in particular the above-mentioned dielectric response. In a very recent paper34 we studied the dynamic electrophoretic mobility of a spherical particle in a salt-free concentrated suspension. Many of the behaviors displayed along the frequency range (typically below 1 GHz) of the electric field were qualitatively related to the different relaxation mechanisms that such suspensions exhibit. In this new paper we explicitly analyze, using a cell model, such dielectric relaxations and try to understand the different charge polarization mechanisms underlying them.

Theory Electrokinetic Equations and Boundary Conditions. According to Kuwabara’s cell model,35 particle-particle interactions in concentrated suspensions can be taken into account by considering that each spherical particle is surrounded by a concentric spherical shell of solution of outer radius b, such that the particle/cell volume ratio in the unit cell is equal to the particle volume fraction φ throughout the whole suspension, i.e.

φ)(a/b)3

(1)

a being the radius of the spherical particle. We will assume that the so-called “slip plane”, the plane outside which the continuum equations of hydrodynamics are assumed to be applicable, coincides with the particle surface. Let us consider a charged spherical particle with mass density Fp and relative permittivity εrp bearing a surface charge density σ, immersed in an aqueous solution of relative permittivity εrm, mass density Fm, and viscosity ηm. The only ionic species are counterions of valence zc and drag coefficient λc, which, as mentioned above, reach the solution after the dissociation of ionic surface groups, leaving the countercharge linked to the surface of the particles. In the presence of an oscillating electric field Ee-iωtof angular frequency ω, the particle moves (relative to the laboratory) with a velocity ve, the electrophoretic velocity. The axes of a spherical coordinate system (r, θ, φ) are fixed at the center of the particle, with the polar axis (θ ) 0) parallel to the electric field. At every point r of the system, quantities such (28) Yamanaka, J.; Hayashi, Y.; Ise, N.; Yamagushi, T. Phys. ReV. E 1997, 55, 3028. (29) Carrique, F.; Ruiz-Reina, E.; Arroyo, F. J.; Delgado, A. V. J. Phys. Chem. B 2006, 110, 18313. (30) Shilov, V. N.; Zharkikh, N. I.; Borkovskaya, Y. B. Colloid J. 1981, 43, 434. (31) Borkovskaya, Y. B.; Zharkikh, N. I.; Dudkina, L. M. Colloid J. 1982, 44, 578. (32) Shilov, V. N.; Zharkikh, N. I.; Borkovskaya, Y. B. Colloid J. 1985, 47, 645. (33) Saville, D. A. J. Colloid Interface Sci. 1983, 91, 34. (34) Carrique, F.; Ruiz-Reina, E.; Arroyo, F. J.; Jime´nez, M. L.; Delgado, A. V. Langmuir 2008, 24, 3150. (35) Kuwabara, S. J. Phys. Soc. Jpn. 1959, 14, 527.

as the electrical potential Ψ(r,t), the number density of counterions nc(r,t), their drift velocity vc(r,t), the fluid velocity v(r,t), and the pressure P(r,t), must be evaluated to completely solve the problem. The fundamental equations connecting them are7,36-38

∇2Ψ(r, t) ) -

Fel(r, t) εrmε0

(2)

Fel(r, t) ) zce nc(r, t)

(3)

ηm ∇2v(r, t) - ∇ P(r, t) - Fel(r, t) ∇ Ψ(r, t) ) ∂ Fm [v(r, t) + ve exp(-iωt)] ∂t ∇ · v(r, t) ) 0 1 vc(r, t) ) v(r, t) ∇ µc(r, t) λc µc(r, t) ) µ∞c + zce Ψ(r, t) + kBT ln nc(r, t) ∇ · [nc(r, t) vc(r, t)] ) -

(4) (5) (6) (7)

∂ [n (r, t)] ∂t c

(8)

where e is the elementary electric charge, kB is Boltzmann’s constant, T is the absolute temperature, and µc(r) is the electrochemical potential of the counterion species, with µ∞c its standard value. The drag coefficient λc in eq 6 is related to the limiting ionic conductance Λ0c by36

λc )

NAe2|zc| Λ0c

kBT Dc

)

(9)

where NA is Avogadro’s number and Dc the counterion diffusion coefficient. These equations are the expressions of Poisson’s equation (2, 3), the Navier-Stokes equation (4, 5), the Nernst-Planck equation (6, 7), and ionic species conservation (8). Since we will be concerned with the linear response to the applied electric field, it will be assumed that the latter is weak enough for a first-order perturbation scheme to be valid (each quantity X is equal to its equilibrium value X0 plus a perturbation term with the same time dependence as the external field):

Ψ(r, t) ) Ψ0(r) + δΨ(r) e-iωt nc(r, t) ) n0c (r) + δnc(r) e-iωt µc(r, t) ) µ0c + δµc(r) e-iωt P(r, t) ) P0(r) + P(r) e-iωt

(10)

-iωt

v(r, t) ) v(r) e

vc(r, t) ) vc(r) e-iωt Fel(r, t) ) Fel0(r) + δFel(r) e-iωt Taking symmetry considerations into account permits us to introduce the quantities h(r), φc(r), and ψ(r), defined by7

2 1 d v(r) ) (Vr, Vθ, Vφ) ) - hE cos θ, (rh) E sin θ, 0 (11) r r dr δΨ(r) ) ψ(r) E cos θ (12)

(

)

(36) O’Brien, R. W.; White, L. R. J. Chem. Soc., Faraday Trans. 2 1978, 74, 1607. (37) Ohshima, H.; Healy, T. W.; White, L. R. J. Chem. Soc., Faraday Trans. 2 1983, 79, 1613. (38) DeLacey, E. H. B.; White, L. R. J. Chem. Soc., Faraday Trans. 2 1981, 77, 2007.

11546 Langmuir, Vol. 24, No. 20, 2008

δµc(r) ) zce δΨ + kBT

δnc n0c

Carrique et al.

) -zce φc(r) E cos θ (13)

Substituting eqs 10-13 into the basic differential equations, we obtain

L(Lh + γ2h) ) L φc(r) +

e dy b z 2e-zcy φc(r) ηmr dr c c

(

iωλc dy dφc 2λc h(r) z [φc(r) + ψ(r)] ) KBT dr c dr e r L ψ(r) )

(14)

)

zc2e2n0c (r) [φ (r) + ψ(r)] εrmε0kBT c

(15) (16)

where every perturbation term of order higher than 1 has been neglected. The quantities and operators shown in eqs 14-16 are

y) γ)




eΨ0 kBT




iωFm ) (i + 1) ηm

(17) ωFm 1 ) (i + 1) 2ηm δ

d2 2 d 2 + dr2 r dr r2

L≡

(18) (19)

The equilibrium volume charge density is given by

Fel0(r) ) zcen0c (r)

(20)

and the equilibrium counterion concentration

n0c (r) ) bc

(

zce Ψ0(r) exp kBT

)

(

∫ab bc exp -

)

zce Ψ0(r) 4πr2 dr ) kBT -4πa2σ zce

(22)

The equilibrium Poisson-Boltzmann equation will be given by

(

)

1 d 2 dΨ0 1 r )zce n0c (r) 2 dr dr ε ε r rm 0 with

(

(-4πa2σ) exp n0c (r) ) zce

∫a

b

(

0

zce Ψ (r) kBT

)

(23)

)

zceΨ0(r) exp 4πr2 dr kBT

εrm ∇ δΨ(r) · ˆr - εrp ∇ δΨp(r) · ˆr ) 0 v ) 0 at r ) a vc · ˆr ) 0 at r ) a Vr ) -ve · ˆr

r)b

at

ω) ∇ ×v)0

at

r)b

(26) at

r ) a (27) (28) (29) (30) (31)

where δΨp(r) is the perturbation in the electrical potential in the interior region of the solid particle, and analogously to eq 12, the inner electrical potential perturbation can be expressed by

δΨp(r) ) ψp(r) E cos θ

(32)

Equation 26 expresses the continuity of the electrical potential at the surface of the particle. Equation 27 expresses that the surface charge density is not perturbed by the field, so that the perturbation of the normal component of the displacement vector is continuous. Equation 28 means that the liquid cannot slip on the particle. Additionally, and due to the assumed insulating nature of the particle, the velocity of counterions in the normal direction to the particle surface is zero, as represented by eq 29, where rˆ is the normal vector outward of the surface. In the outer surface of the cell (r ) b), we will follow Kuwabara’s boundary conditions: in the radial direction, the velocity of the liquid will be minus the radial component of the electrophoretic velocity, as expressed by eq 30. Finally, eq 31 means that the liquid flow is free of vorticity on that surface. Concerning the ionic perturbation at the outer surface of the cell, two choices are frequently found in the literature (see the review paper by Carrique et al.,39 where the different boundary conditions are extensively discussed for the cell model approach):

δnc(b) ) 0

(33)

[∂δnc(r)/ ∂ r](b) ) 0

(34)

The first one (restricted here to just one ionic species) was first introduced by Shilov et al.30 and since then has been commonly used by many authors5,12,40-42 in electrokinetic cell models. The second one was used by Levine and Neale,1,2 Ding and Keh,43 Kozak and Davis,44,45 Ohshima,4-6 Keh and Hsu,46 etc. From symmetry considerations, the boundary conditions in eqs 26-34 transform into

|

εrp dψ ψ(a) ) 0 dr r)a εrma h(a) ) dφc dr

|

dh )0 dr r)a

|

r)a

)0

µb 2 L h(r)|r)b ) 0 h(b) )

(24)

and the boundary conditions are expressed by eq 22 and

dΨ0 (b) ) 0 dr

r)a

at

(21)

obeys the Boltzmann distribution. The prefactor bc corresponds to the concentration of counterions at a point where the electrical potential is set to zero and verifies the condition

∫ab n0c(r) 4πr2 dr )

δΨp(r) ) δΨ(r)

(25)

For the remaining fundamental equations the following boundary conditions apply:

(39) Carrique, F.; Cuquejo, J.; Arroyo, F. J.; Jime´nez, M. L.; Delgado, A. V. AdV. Colloid Interface Sci. 2005, 118, 43. (40) Lee, E.; Chu, J. W.; Hsu, J. J. Chem. Phys. 1999, 110, 11643. (41) Lee, E.; Yen, F. Y.; Hsu, J. P. J. Phys. Chem. B 2001, 105, 7239. (42) Carrique, F.; Arroyo, F. J.; Delgado, A. V. J. Colloid Interface Sci. 2002, 252, 126. (43) Ding, J. M.; Keh, H. J. J. Colloid Interface Sci. 2001, 236, 180. (44) Kozak, M. W.; Davies, E. J. J. Colloid Interface Sci. 1989, 127, 497. (45) Kozak, M. W.; Davies, E. J. J. Colloid Interface Sci. 1989, 129, 166. (46) Keh, H. J.; Hsu, W. T. Colloid Polym. Sci. 2002, 280, 922.

Dielectric Response of Concentrated Suspensions

φc(b) ) -ψ(b)

or

Langmuir, Vol. 24, No. 20, 2008 11547

[dφc(r)/dr](b) ) -[dψ(r)/dr](b) (35)

and the equation of motion for the unit cell leads to26,34

1 6 6 h′′′(b) + h′′(b) - 2 h
(b) + 3 h(b) b b b Fp - Fm iωFm h(b) φ - h
(b) ) -µ ηm b Fm

[

]

-

Fel0(b)

ψ(b) bηm

(36)

µ being the particle dynamic electrophoretic mobility defined by ve ) µE. An additional boundary condition for the perturbation quantity δΨ or ψ(r) at the outer surface of the cell is still required. Dukhin et al.47 proposed a Dirichlet-type electrical boundary condition according to the Shilov-Zharkikh cell model,30 showing the connection between the macroscopic, experimentally measured electric field 〈E〉 and local electrical properties

δΨ(r)|r)b ) -Eb cos θ

(37)

where E ) |〈E〉|, or in terms of the function ψ(r)

ψ(b) ) -b

(38)

This electrical boundary condition has been applied by many authors in different electrokinetic cell models.5,27,39,41,42,48 On the other hand, Levine and Neale introduced a different one in their study of electrophoresis in concentrated suspensions1

∇δΨ(r) · rˆ|r)b ) -E cos θ

(39)

where in this case E * |〈E〉|, with |〈E〉| ) -[ψ(b)/b]E. In terms of the function ψ(r), eq 39 transforms into

dψ (b) ) -1 dr

(40)

This boundary condition is implicit in Ohshima’s electrokinetic models for concentrated suspensions.4,6 As commonly found in the literature, the conditions for potential and ionic perturbations must be linked in two possible ways. For just one ionic species, the first set is

δnc(b) ) 0

and

δΨ(r)|r)b ) -Eb cos θ

(41)

and will be referred as the “bc1 boundary case” hereafter. The other one is

[∂δnc(r)/ ∂ r](b) ) 0 and ∇ δΨ(r) · ˆr|r)b ) -E cos θ

(42)

and will be designated as the “bc2 boundary case”. In terms of perturbed quantities they become

φc(b) ) -ψ(b)

and

ψ(b) ) -b (bc1 case)

(43)

[dφc(r)/dr](b) ) -[dψ(r)/dr](b) and [dψ(r)/dr](b) ) -1 (bc2 case) (44) The bc1 boundary case appears originally in the frame of the complete Shilov-Zharkikh cell model.30 The bc2 case joins the electrical Levine-Neale boundary condition with a Neumann condition proposed by Ding and Keh43 on the local perturbed counterion concentration at the outer surface of the cell. As mentioned, this condition was already implicit in preceding models by Levine and Neale1 and Ohshima.4,6 It has been reported (47) Dukhin, A. S.; Shilov, V. N.; Borkovskaya, Y. B. Langmuir 1999, 15, 3452. (48) Hsu, J. P.; Lee, E.; Yen, F. Y. J. Phys. Chem. B 2002, 106, 4789.

in a recent paper49 that bc1 predictions seem to be more reliable than bc2 predictions when compared to experimental results (see also refs 47 and 50). Hence, only the Shilov-Zharkikh set of boundary conditions will be explored in this paper according to the considerations given above. For dynamic electrophoresis in concentrated suspensions in salt solutions, the reader is encouraged to consider the recent review paper by Ahualli et al.51 on the consistent use of cell models and different boundary conditions. It is also worth mentioning the very recent study of electrokinetic phenomena in concentrated disperse systems by Zholkovskiy et al.,52 with special emphasis on building a general formulation and analysis of the spherical cell approach for different boundary conditions. One of its main achievements refers to the use of the Shilov-Zharkikh cell model for the electrokinetics of suspensions in salt solutions (our bc1 choice is of that kind but for just one ionic species), which the authors rigorously justify on wellfounded physical grounds. Nevertheless, a more in depth study on this topic is desirable, and it is likely that only experiments can help in supporting or refusing theoretical predictions based on the application of either set of boundary conditions to saltfree suspensions. In the following section we derive a general equation for the complex conductivity, allowing the dielectric response of a concentrated suspension in a salt-free solution to be obtained. Then we evaluate numerically the effect of the particle volume fraction, particle surface charge, particle radius, and counterion drag coefficient on the frequency dependence of the complex dielectric constant (or complex relative permittivity) of the suspension. Complex Conductivity and Dielectric Constant of the Suspension. The complex conductivity of a suspension K*(ω) can be defined as

〈i 〉 ) K*(ω) 〈E〉

(45)

in terms of the relation between the volume averages of the current density and electric field:

∫

1 i dV V V 1 ∇δΨ dV 〈E 〉 ) V V 〈i 〉 )

∫

(46) (47)

where V is the suspension volume and the current density is equal to the sum of the conduction and displacement terms

i(r, t) ) zce nc(r, t) vc(r, t) - iω D(r, t)

(48)

By following the same mathematical procedure as that used by the authors in ref 12 for the derivation of the dielectric response of a concentrated suspension in a salt solution, it is easy to obtain the complex conductivity of a concentrated suspension in a saltfree solution: (49) Cuquejo, J.; Jime´nez, M. L.; Delgado, A. V.; Arroyo, F. J.; Carrique, F. J. Phys. Chem. B 2006, 110, 6179. (50) O’Brien, R. W.; Rowlands, W. N.; Hunter, R. J. In Electroacoustics for Characterization of Particulates and Suspensions; Malghan, S. B., Ed.; NIST Special Publication 856; National Institute of Standards and Technology: Washington, DC, 1993. (51) Ahualli, S.; Delgado, A. V.; Miklavcic, S. J.; White, L. R. Langmuir 2006, 22, 7041. (52) Zholkovskiy, E. K.; Masliyah, J. H.; Shilov, V. N.; Bhattacharjee, S. AdV. Colloid Interface Sci. 2007, 134, 279–135.

11548 Langmuir, Vol. 24, No. 20, 2008

Carrique et al.

K*(ω) ) zce Ψ0(b) zc2e2 dφc 2h(b) + (b) zce bc exp λc dr b kBT

{[

] ( iωεrmε0

}

)

dψ b (b) dr ψ(b)

(49)

This equation is independent of the choice of boundary conditions at the outer surface of the cell. Choosing, as mentioned, the Shilov-Zharkikh conditions, eq 49 becomes

K*(ω) )

[

] (

)

zce Ψ0(b) zc2e2 dφc 2h(b) + (b) zce bc exp λc dr b kBT dψ iωεrmε0 (b) (50) dr

Recall that the knowledge of K* leads immediately to that of the complex relative permittivity εr*(ω) of the suspension

εr*(ω) ) εr′(ω) + iεr′′(ω)

(51)

according to the equation

K*(ω) ) K*(ω)0) - iωε0 εr*(ω) ) K*(ω)0) + ωε0 εr′′(ω) - iωε0 εr′(ω) (52) Finally, we can define a κa parameter for the salt-free case in analogy to the case of suspensions in salt solutions, with κ-1 meaning the double-layer thickness, but considering only the counterions neutralizing the particles, as follows:29

κa )




e2n¯czc2a2 ) εrmε0kBT




-

3σzceaφ εrmε0kBT(1 - φ)

(53)

where use has been made of the average counterion concentration defined as

nc ) -

σ4πa2 zce4π(b3 - a3) ⁄ 3

(54)

Results and Discussion As was indicated previously, the authors recently published a complete study dealing with the dynamic electrophoretic mobility of a spherical colloidal particle in a concentrated saltfree suspension.34 In that paper a general study was performed concerning the effects of particle charge, volume fraction of solids, particle radius, and counterion diffusion coefficient of the added counterions. Some remarkable features were found that could be related to the different relaxation processes exhibited by such suspensions: the so-called R′ relaxation and the Maxwell-Wagner-O’Konski (MWO) relaxation. The first one is connected to the relaxation of the diffusion currents generated by the counterion concentration gradient provoked by the field. The second relaxation (MWO) occurs when the field frequency is so high that such a gradient is greatly hindered. In the present paper we analyze in some depth the dielectric response of such systems as a function of the frequency in terms of the real and imaginary parts of the complex dielectric constant of the suspensions. We try to connect those relaxation processes to electrochemical properties of the suspension, giving physical support to our statements. In doing so, we analyze the perturbed counterion concentration at every point inside the cell and the diffusion current perturbation around a particle at different frequencies of the applied electric field. Dielectric Relaxation Mechanisms. The MWO relaxation mechanism is connected to the double-layer polarization in

response to the electric field. For low frequencies of the field, counterions (cations if the surface charge is negative) have time to migrate away from the particle on the right-hand side and to approach it on the left-hand side (between successive inversions of a field applied from left to right). At the steady-state situation, cations will accumulate on the left and will be depleted on the right, thus polarizing the double layer. Depending on the surface charge, it is possible that tangential transport of counterions in the diffuse layer compensates for this and provokes accumulation on the right and depletion on the left (see Figure 2 below). If the field frequency is high enough, the spherical symmetry of the ionic atmosphere is preserved, as ions cannot undergo significant displacements between field inversions. In such a situation the polarization disappears, and it is this phenomenon that is called MWO relaxation. As mentioned above, the R′ mechanism can be related to the relaxation of the diffusion currents generated by the abovementioned counterion concentration gradient provoked by the electric field. This is a slow process, as counterions must diffuse through distances on the order of the particle size, and as such, it shows all the features of the classical R-relaxation in saltcontaining suspensions.53,54 The difference is that it is not linked to neutral salt concentration gradients but rather to counterion concentration gradients. This is the reason we decided to call it R′, instead of R, the salt-free low-frequency process. However, and in spite of the resemblances, it is worth pointing out that for pure salt-free suspensions the absence of co-ions is absolute, and then, the relaxation mechanisms have to be revised and adapted to take account of this constraint. The strength and direction of the induced dipole will thus depend on the particle charge, volume fraction, counterion drag coefficient (or related counterion diffusion coefficient; see eq 9), and particle radius. Thus, it is mandatory to study the influence of all these parameters in the overall dielectric response of the salt-free suspension, and this is the aim of the following sections. In ref 34 we justified the use, for salt-free suspensions, of some classical expressions for the characteristic frequencies of the relaxation mechanisms that take place in common salt suspensions, once they were adapted and reinterpreted according to the singularities of the salt-free case. In particular, the characteristic relaxation frequencies of the R′ and MWO processes are given, respectively, by

ωR′ ≈

2Dc

ωMWO ) -

2

a

)

2kBT λca2

3σzceφ εrmε0aλc(1 - φ)

(55) (56)

The R′ relaxation frequency depends on the counterion diffusion coefficient (eq 9), but it is also controlled by some diffusion length, which in turn depends on the particle size and volume fraction. Therefore, eq 55 can be applied just for dilute or very low concentration suspensions, considering that it contains only the particle radius a, implicitly assuming that the diffusion distance of the counterions is volume fraction independent. However, it will help us in estimating the lowest frequency region that can be associated with the R′ process for a given particle radius. Note that, as the volume fraction is increased, the diffusion length is reduced, and hence, the R′ relaxation frequency should increase and separate from the approximate expression given by eq 55. (53) Dukhin, S. S.; Shilov, V. N. Dielectric Phenomena and the Double Layer in Disperse Systems and Polyelectrolytes; John Wiley and Sons: New York, 1974. (54) Lyklema, J. Fundamentals of Interface and Colloid Science Vol. II: SolidLiquid Interfaces; Academic Press: London, 1995.

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Figure 1. Schematic representation of the counterion (cation) concentration around a negatively charged particle in a salt-free concentrated suspension: (a) low-frequency and (b) high-frequency regimes.

Figure 2. Schemes of the low-frequency polarization of a negatively charged particle in a salt-free concentrated suspension in terms of the induced electric dipole moment (a, c) and perturbations in counterion concentration (b, d, e): (top) moderate particle charge, (center) very low particle charge, (bottom) intermediate particle charge to show the transition between the two polarization behaviors. Numerical computations for H+ counterions: a ) 100 nm, φ ) 0.1, and (b) σ ) -0.02 µC/cm2, (d) σ ) -0.0002 µC/cm2, and (e) σ ) -0.0033 µC/cm2.

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Figure 3. The arrows have directions and lengths indicating the perturbation of the diffusion current density of counterions at different frequencies for a negatively charged particle in a salt-free concentrated suspension at 25 °C (for the time shown the electric field is maximum and points to the right) (H+ counterions, σ ) -0.02 µC/cm2, a ) 100 nm, φ ) 0.1).

Figure 4. Perturbation of counterion concentration at different frequencies for a negatively charged particle in a salt-free concentrated suspension at 25 °C (for the time shown, the electric field is maximum and points to the right) (H+ counterions, σ ) -0.02 µC/cm2, a ) 100 nm, φ ) 0.1).

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Figure 5. Real and imaginary parts of the complex dielectric constant of a salt-free concentrated suspension at 25 °C as a function of frequency for different surface charge densities and two particle volume fractions (a ) 100 nm, H+ counterions): (a, b) φ ) 0.005, (c, d) φ ) 0.5.

On the other hand, since the MWO relaxation mechanism is connected with electromigration of ions, it will be a faster process than diffusion and will relax at larger frequencies. In Figure 1 we give a schematic representation of the counterion concentration around a negatively charged particle in the presence of low (Figure 1a) and high (Figure 1b) frequency ac fields. In Figure 1a we can observe electromigration counterion currents yielding the formation of a counterion concentration gradient around the particle and also the diffusion currents tending to diminish the latter. The frequency dependence of the net electric dipole (not represented in the figure) resulting from the competition of both contributions will be responsible for the observed low-frequency dielectric relaxation pattern. In Figure 1b only the electromigration currents are depicted, because at these frequencies the contribution of the diffusion component is supposed to be increasingly reduced. Two distinct behaviors corresponding to the cases of moderate to high and very small particle charge densities can be found in Figure 2. In these plots, opposite directions for the induced electric dipole moments at low and high charges are displayed at the given frequency, at the instant when the electric field, pointing to the right, reaches its maximum value. The schemes of the left panels are numerically confirmed in Figure 2b,d, where the perturbation in counterion concentration due to a low-frequency electric field is shown for a negatively charged particle. The transition between the two polarization patterns shown in (b) and (d) takes place gradually (Figure 2e): The accumulation on the right-hand side of the particle progressively decreases in magnitude and moves away from the surface; at the same time a depletion region appears close to the particle. The reverse happens on the left-hand side. In this transition situation the

competition between depletion and accumulation zones will likely lead to a negligible dipole moment. To support our statements about the relaxations of the R′ and MWO mechanisms, in Figures 3 and 4 we can observe, respectively, the frequency evolutions of the perturbation of counterion diffusion currents and counterion concentrations around a negative particle. In this and subsequent calculations we have chosen H+ as the counterion just because of its high diffusion coefficient, to magnify the behavior investigated. The frequent role of protons as potential-determining ions is hence not addressed in the present calculations. For simplicity only a few pictures corresponding to selected frequencies are shown. Note also that, around roughly 105 Hz, the diffusion contributions (from right to left) near the particle surface (north and south poles of the solid particle in Figure 3) start to considerably decrease as the frequency increases and tend to diminish their importance, as well as those diffusion currents pointing toward, or coming from, the far regions of the double layer, tending to reduce the counterion concentration gradient. This picture gives us a simple representation of the so-called R′ relaxation mechanism. Likewise, in Figure 4 we can see how the general polarization scheme of the perturbation in counterion concentration shown in the first low-frequency images no longer survives beyond more or less 1 MHz, yielding also an inversion of this increasingly small contribution to the total electric dipole moment. This can represent reasonably well the MWO relaxation mechanism. However, in many cases there is not a precise separation in frequency between both types of processes (R′ and MWO), especially at large volume fractions, where the particle diffusion length is small and no important counterion concentration gradient

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Figure 6. Real and imaginary parts of the complex dielectric constant of a salt-free concentrated suspension at 25 °C as a function of frequency for different particle volume fractions and three particle charge densities (a ) 100 nm, H+ counterions): (a, b) σ ) -0.1 µC/cm2, (c, d) σ ) -1.0 µC/cm2, (e, f) σ ) -10.0 µC/cm2.

can be developed owing to the overlap of neighboring double layers. Despite the complexity of the dielectric response of a salt-free suspension, in the following sections we will try to understand the role of the parameters affecting the global response, starting with the particle surface charge. Complex Dielectric Constant versus Particle Surface Charge at Fixed Particle Volume Fraction. In Figure 5 we plot the effect of the surface charge on the frequency dependence of the real and imaginary components of the complex dielectric constant for two well-separated volume fractions corresponding, respectively, to the dilute (Figure 5a,b) and concentrated (Figure 5c,d) particle concentration regimes. It is clear that, for dilute suspensions (Figure 5a,b), the complex dielectric constant shows both R′ and MWO relaxation processes associated with the successive hindering of diffusion currents and counterion polarization as the frequency increases. In addition,

the low-frequency value of the real part is higher the larger the particle charge (Figure 5a) and tends to a saturation value when such a charge is sufficiently high. This fact is another indication of the so-called “ionic condensation effect”, already described in the literature23 and first related to the observation of a saturation of the dc electrophoretic mobility as the particle charge increases. Note also that the low-frequency R′ process tends not only to saturate its magnitude as the particle charge increases but also to keep its relaxation frequency fixed (first peak in the plots of the imaginary part of the complex dielectric constant, Figure 5b). On the contrary, the MWO relaxation frequency increases with the particle charge due to the enhancement of electrical conductivity (second peaks in the same figure), in full agreement with the approximate predictions of eq 56. Our results also indicate that the low-frequency value of the real part of the dielectric constant decreases with the volume fraction for low

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Figure 7. Real (a) and imaginary (b) components of the complex dielectric constant of a salt-free concentrated suspension at 25 °C as a function of frequency for different particle radii (volume fraction φ ) 0.05, surface charge density σ ) -10.0 µC/cm2, H+ counterions).

Figure 8. Real and imaginary components of the complex dielectric constant of a salt-free concentrated suspension at 25 °C as a function of frequency for different counterions, two particle charge densities and volume fraction φ ) 0.1 (a ) 100 nm): (a, b) σ ) -1.0 µC/cm2, (c, d) σ ) -10.0 µC/cm2.

charged particles (compare parts a and c of Figure 5). Conversely, at high charge it is larger the higher the volume fraction. This fact can be explained considering the insulating nature of the material the particles are made of. At low volume fractions and low charges (Figure 5a), we are replacing material (the solution) of high dielectric constant (εrm ) 78.54 in our study) by an equivalent volume of material (the particles) of low dielectric constant (εrp ) 2), although the double-layer polarization partly compensates for this negative effect on the polarization. However, if the volume fraction is large and the particle charge has low or moderate values, the negative contribution to the electrical polarizability yields a reduction of the low-frequency dielectric

constant (Figure 5c). As the particle charge is further increased to a sufficiently high value, the electrical polarizability of the particle double-layer system increases as well, more than compensating the above-mentioned negative effect. As a result, an increase of the low-frequency dielectric constant is finally observed. However, the saturation scheme referred to above is not attained for the highest charge values studied: the addition of highly polarizable particles yields a remarkable increase of the low-frequency dielectric constant, in spite of the negative effect associated with the small value of the dielectric constant of the particles.

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On the other hand, the R′ relaxation frequency shifts to larger frequencies as the particle volume fraction increases. This is a result of the decrease in the counterion diffusion length when φ is raised (the diffusion fluxes have to describe shorter paths around the particles because of the presence of neighbors). Consequently, a larger frequency is necessary for the diffusion fluxes to relax. Also, both the magnitude and relaxation frequency of the MWO relaxation process increase with the particle charge because of the enhancement of conductivity. The final result is that both processes (R′ and MWO) tend to overlap for high charges and high volume fractions (Figure 5d). Complex Dielectric Constant versus Particle Volume Fraction at Fixed Particle Surface Charge Density. We now consider in Figure 6 the effect of varying the particle volume fraction from rather dilute suspensions to very concentrated ones for three different cases corresponding to low (Figure 6a,b), moderate (Figure 6c,d), and high (Figure 6e,f) surface charge densities. It is clear from these plots that the frequency dependence of εr′ is markedly dependent on both φ and σ. Thus, for the lowest particle charge tested (σ ) -0.1 µC/cm2, Figure 6a,b) the low-frequency value of εr′ decreases with the volume fraction, while this trend is reversed for highly charged particles (σ ) -10.0 µC/cm2, Figure 6e,f). The behavior is between these two extremes for moderate charge densities (σ ) -1.0 µC/cm2, Figure 6c,d). This is a manifestation of the mentioned balance between the negative effect on εr′ of the difference εrm - εrp and the positive contribution of the polarizability of the particle ionic atmosphere, more important the higher the σ. Another important feature in Figure 6 is the shift of both the R′ and MWO relaxations to larger frequencies as the volume fraction increases (see Figure 6b,d,f). This is mainly true for the R′ relaxation and obeys the considerable reduction of the diffusion length with increasing volume fraction. Besides, the magnitude of the R′ relaxation (difference between the low- and highfrequency values of εr′, related also to the height of the maxima in εr′′) first increases with the volume fraction and then attains a maximum at intermediate volume fractions (see Figure 6b) and gradually diminishes for the largest volume fractions analyzed. These facts can be connected with the increase in ionic strength of the liquid medium because of the shortening of the counterion confinement region as the volume fraction increases. Larger counterion concentration gradients can be developed in such situations, considering that the particle diffusion length is reduced. However, as the volume fraction is further increased, the double layers of neighbor particles start to overlap, thus reducing the magnitude of those counterion concentration gradients and therefore obscuring the R′ process. In addition, the relative importance of the R′ and MWO processes is strongly dependent on the particle charge: For low σ (Figure 6b), only the R′ relaxation is observed, as suggested by the fact that the behavior of the single peak observed is identical to that of the R-relaxation, well-described in salt-containing solutions.12 As the particle charge is raised, the subsequent increase in ionic strength progressively enhances the MWO process, practically the only one observed for σ ) -10.0 µC/cm2 (Figure 6e,f), which increases its magnitude with φ. Note, however, that for high φ the two relaxations are not clearly separated in frequency. Complex Dielectric Constant versus Particle Radius. Many interesting conclusions can be drawn from Figure 7 concerning the role of the particle size on the dielectric response of a saltfree suspension. According to eqs 55 and 56, a shift toward the high-frequency side of the spectra of both the R′ and MWO relaxation frequencies is expected for fixed both volume fraction and particle charge density as the particle radius is reduced. This

Carrique et al.

is indeed what is observed in the figure: Two well-separated peaks in εr′′(f) (R′ and MWO relaxations) are predicted for large particle size. As this decreases, the characteristic frequencies of the two processes increase, but this effect is more pronounced for the R′ relaxation (compare the dependences on particle radius in eqs 55 and 56). As a consequence, the two peaks eventually merge into a broader one. At the same time, reducing the size at constant σ and φ means increasing the conductivity of the medium (determined only by the charge dissociation on the particles). This yields a significant increase in the MWO relaxation amplitude, which explains the growing height of the εr′′(f) maxima when the radius is decreased. The convolution between the R′ and MWO peaks does not allow the evolution of the R′ amplitude with radius to be distinguished: it is likely that this amplitude will be roughly constant, as reducing the size should diminish the induced dipole strength, while the associated increase in ionic strength acts in the opposite direction. Role of Counterion Drag Coefficient. The calculations plotted in Figure 8 allow the study of the influence on the dielectric response of the counterion diffusion coefficient (or equivalently its drag coefficient; see eq 9) for moderate (Figure 8a,b) and high (Figure 8c,d) particle surface charges. Two immediate conclusions can be drawn from these plots concerning, first, the significant role played in the dielectric response by the faster counterion (H+) species as compared with the slow counterions (K+, Na+, Li+) and, second, the shift of both relaxation frequencies, R′ and MWO, to higher values the larger the counterion diffusion coefficient. These features are predicted by eqs 55 and 56, taking into account the different values of the diffusion coefficients for H+, K+, Na+, and Li+ counterions: Dc (10-9 m2/s) ) 9.34, 2.0, 1.36, and 1.05, respectively. It is also physically reasonable to explain the appearance of the R′ process (low-frequency shoulder in the dominating MWO peak, Figure 8b,d) as a consequence of the increasing importance of that relaxation if the particle charge is raised. It can also be deduced from these figures that faster ions will be less prone to accumulate, leading to smaller concentration gradients and electrical double-layer polarizabilities and, consequently, to a reduction of the dielectric response (compare H+ curves in Figure 8 with those for the other counterions).

Conclusions In this work, a cell model has been used to derive the complex dielectric constant of a concentrated colloidal suspension of spherical particles in a salt-free liquid medium. Special attention has been paid to the dependence of the dielectric response on the frequency of the applied electric field, particle surface charge density, volume fraction and radius, and counterion diffusion coefficient. The dielectric response of the suspension has been analyzed in terms of the competition of the different relaxation processes that take place as the frequency of the electric field increases, being related to (i) the relaxation of counterion diffusion fluxes (the slowest frequency process) provoked by counterion concentration gradients generated by the electric field at both sides of the particles (R′ relaxation process) and (ii) double-layer relaxation linked to the electromigration response (a faster frequency process) and its increasing lag in following the field oscillations (MWO process) as the frequency increases. The saltfree case has shown, as in electrophoresis, saturation of the lowfrequency response for very high particle charge, associated with the counterion condensation effect near the particle surface. Furthermore, the dependence of the different relaxation frequencies on such parameters as particle surface charge density, particle volume fraction, particle radius, and counterion diffusion

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coefficient is reasonably well captured by some approximate theoretical expressions. As the experimental data regarding the dielectric response of a salt-free concentrated suspension are quite insufficient, it would be worth designing experiments suited to confirming or refusing the principal predictions of this paper, and this will be the aim of a future work.

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Acknowledgment. Financial support for this work by MEC, Spain (Project FIS2007-62737) (cofinanced with FEDER funds by the EU), and Junta de Andalucı´a, Spain (Project FQM410), is gratefully acknowledged. LA802218J