Advances in Colloid and Interface Science 201–202 (2013) 57–67

Contents lists available at ScienceDirect

Advances in Colloid and Interface Science

j ourna l homepage: www.e lsev ie r .com/ locate /c i s

Effects of non-equilibrium association–dissociation processesin the dynamic electrophoretic mobility and dielectric responseof realistic salt-free concentrated suspensions

Félix Carrique a,⁎, Emilio Ruiz-Reina b, Luis Lechuga c, Francisco J. Arroyo d, Ángel V. Delgado e

a Departamento de Física Aplicada I, Facultad de Ciencias, Universidad de Málaga, 29071 Málaga, Spainb Departamento de Física Aplicada II, Escuela Politécnica Superior, Universidad de Málaga, 29071 Málaga, Spainc Departamento de Matemática Aplicada, Escuela Politécnica Superior, Universidad de Málaga, 29071 Málaga, Spaind Departamento de Física, Facultad de Ciencias Experimentales, Universidad de Jaén, 23071 Jaén, Spaine Departamento de Física Aplicada, Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain

⁎ Corresponding author.E-mail address: carrique@uma.es (F. Carrique).

0001-8686/$ – see front matter © 2013 Elsevier B.V. All rihttp://dx.doi.org/10.1016/j.cis.2013.10.004

a b s t r a c t

a r t i c l e i n f o

Available online 14 October 2013

Keywords:Salt-free suspensionsCell modelsDynamic electrophoretic mobilityElectric permittivityAlpha and Maxwell–Wagner relaxation

Electrokinetic investigations in nanoparticle suspensions in aqueousmedia aremost often performed assuming thatthe liquidmedium is a strong electrolyte solutionwith specified concentration. The role of the ions produced by theprocess of charging the surfaces of the particles is often neglected or, at most, the concentrations of such ions areestimated in someway and added to the concentrations of the ions in the externally prepared solution. The situationhere considered is quite different: no electrolyte is dissolved in the medium, and ideally only the counterionsstemming from the particle charging are assumed to be in solution. This is the case of so-called salt-free systems.With the aim of making amodel for such kind of dispersions as close to real situations as possible, it was previouslyfound to consider the unavoidable presence of H+ and OH− coming fromwater dissociation, as well as the (almostunavoidable) ions stemming from the dissolution of atmospheric CO2. In this work, we extend such approach byconsidering that the chemical reactions involved in dissociation and recombination of the (weak) electrolytes insolution must not necessarily be in equilibrium conditions (equal rates of forward and backward reactions). Tothat aim, we calculate the frequency spectra of the electric permittivity and dynamic electrophoretic mobility ofsalt-free suspensions considering realistic non-equilibrium conditions, using literature values for the rate constantsof the reactions. Four species are linked by such reactions, namely H+ (from water, from the – assumed acidic –

groups on the particle surfaces, and from CO2 dissolution), OH− (from water), HCO3− and H2CO3 (again from

CO2). A cell model is used for the calculations, which are extended to arbitrary values of the surface charge, theparticle size, and particle volume fraction, in a wide range of the field frequency ω. Both approaches predict ahigh frequency relaxation of the counterion condensated layer and a Maxwell–Wagner–O'Konski electric doublelayer relaxation at intermediate frequencies. Also, in both cases an inertial decay of the electrophoretic mobility athigh ω takes place. The most significant difference between the present model and previous results based on theequilibrium hypothesis is by no means negligible: only in non-equilibrium conditions do we find a low-frequencyrelaxation (mostly noticed in permittivity data, while its significance is lower in dynamic mobility spectra). Thisnew relaxation presents all the characteristic features of the concentration polarization (or alpha) dispersion.These are: i) the average electric polarization of the system increases when the relaxation frequency is surpassed,contrary to the behavior after Maxwell–Wagner type relaxations; ii) the amplitude of the relaxation increaseswith surface charge, reaching a sort of saturation if the charge is too high; iii) the relaxation frequency increaseswith volume fraction while the relaxation amplitude decreases; iv) the characteristic frequency is reduced by theincrease in particle radius. All these facts confirm that the non-equilibrium approach seems to better describe thephysics of the system by giving rise to a concentration polarization kind of relaxation, only possible when ionscan accumulate on both sides of the particles as dictated by the field, and not as determined by equilibriumconditions in the dissociation–recombination reactions involved.

© 2013 Elsevier B.V. All rights reserved.

ghts reserved.

58 F. Carrique et al. / Advances in Colloid and Interface Science 201–202 (2013) 57–67

Contents

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 582. Fundamentals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 583. Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

3.1. General differences between equilibrium and non-equilibrium association–dissociation formalisms . . . . . . . . . . . . . . . . . 603.2. Reasons for the absence of an alpha-dispersion in realistic salt-free systems according to the EQ approach . . . . . . . . . . . . . . . . . 613.3. The role of particle charge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 633.4. Effects of particle volume fraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 633.5. Particle radius effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

4. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65Appendix 1. Derivation of the permittivity and dynamic mobility of a salt-free suspension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65Appendix 2. Supplementary data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

6767

1. Introduction

The considerable importance of concentrated disperse systems inmany industrial fields, ranging from pharmaceutical formulations tocosmetics, foods or paints has brought about a large number of scientificcontributions dealing with their electrokinetic characterization. This isso because it has been gradually proved that streaming potential orcurrent [1–3], sedimentation or centrifugation potentials [4,5], viscosityand electroviscous effects [6,7], or low-frequency dielectric dispersion[8–13] can all be applied to the characterization of such systems. Ofparticular interest in this respect, mainly because theoretical treatments[14–20] as well as commercial instruments, are available, are the so-called electroacoustic methods. There are essentially two approaches:one is based on determination of the ultrasound wave generatedwhen an alternating (ac) electric field is applied to a colloidal dispersion(which has received the name of Electrokinetic Sonic Amplitude, or ESA,see [15]); the second method, Colloid Vibration Potential (CVP) orCurrent (CVI) is reciprocal of the previous one, as it is based on theevaluation of the ac electric field produced when an ultrasonic pressurewave is applied to the system [19,20]. In both cases, the quantity ofinterest is the dynamic mobility, μ, which is the ac counterpart of theclassical electrophoretic mobility. This quantity must be obtained fromthe experimental electroacoustic data using suitable electrokineticmodels applicable to concentrated dispersed systems. Existingmethodsinclude approaches based on semi-empirical corrections [21,22] orexplicit consideration of particle–particle interactions [23] (in fact,valid for just moderately concentrated dispersions). However, cellmodels appear as the only fruitful focus [20,24–27].

A special case of concentrates, because of its singular phenomenology[28], is that of the so-called salt-free suspensions. They are defined asdisperse systems in which the only ions present in solution are thoseproduced by the charging of particle surfaces, typically by ionization orhydrolysis of chemical groups. Interest in this kind of systemswasmainlylaunchedby the fundamentalwork in obtaining ordered colloidal crystals,only possible if long-range electrostatic repulsions are not screened bysalts in solution [29,30]. In fact, the counterion concentrations are usuallylow in salt-free suspensions and electrical double layers (EDLs) aresubsequently thick (compared to the other length scale, that is, theparticle radius). For this reason, the study of these suspensions can beapplied to non-aqueous ones, much less understood than their aqueouscounterparts, but obviously characterized by negligible concentration ofcharged species, just as salt-free suspensions.

Another aspect to be considered is the fact that overlap of adjacentEDLs is very plausible for many usual values of the determinantparameters (volume fraction of solids, ϕ, particle radius, a, and surfacecharge density, σ), and these disperse systems will behave asconcentrated ones, even for very low ϕ values [29–33]. Another

remarkable feature that is magnified in salt-free suspensions is thecounterion condensation effect: a compact layer of counterions thatdevelops in a region close to the particle surface when the surface chargeis sufficiently high. This effect will play a very important role inthe electrokinetics of highly charged salt-free systems. In addition,there is a coupling between the ionic concentration in the liquid phaseand the particle volume fraction because of the release of ions fromthe particles' surface as they get charged.

After the original contributions of Imai andOosawa [34] andOosawa[35], recentworks byOhshimamust bementioned, inwhich this authorconsiders a wide variety of equilibrium and transport properties ofthese systems [36–40]. Extensions to arbitrary particle concentrationswere elaborated by Chiang et al. [41], and also by the present authors[42,43], who dealt with DC electrical conductivity, electrophoreticmobility or electroviscous effects, always using cell models [24,25]. Inthis contribution, we extend those studies to the investigation of thelow-frequency dielectric dispersion (LFDD) of concentrated salt-freesuspensions in the so-called “realistic” approach, that is, including inthe problem the presence in solution of H+ and OH− from waterdissociation, and also the ions and neutral species produced bydissolved atmospheric CO2, because of the aqueous nature of thesolutions typically used. The name given to this approach comes fromthe fact that these species, often ignored in the treatment of aqueoussalt-free systems, are extremely important if one wishes to elaborate arigorous model of the electrokinetics of this sort of systems for low tomoderate particle concentrations. In previous studies of both theequilibrium EDL of these systems [44] and different electrokineticphenomena taking place in them [45–48], an equilibrium scenario wasassumed for the chemical reactions of ionization of water and CO2

products. Recall that this means that the forward and backwardreactions proceed at the same rate.

In the present paper we remove that simplification, by consideringthe calculation of the frequency spectrum of the electric permittivityand dynamic electrophoretic mobility of realistic salt-free suspensions,including non-equilibrium association–dissociation processes for thechemical reactions involved.

2. Fundamentals

We will briefly review the most important theoretical aspects inorder to facilitate the reading and further analysis. According to theKuwabara's cell model [49], the presence of neighbor particles of agiven one can be accounted for by assuming that the particle (spherical,of radius a) is isolated in the center of a sphere of solution (the cell, ofradius b), and setting up proper boundary conditions at the cellboundary. These conditions contain the information about the effectsof particle–particle interactions on the electrokinetics of the whole

59F. Carrique et al. / Advances in Colloid and Interface Science 201–202 (2013) 57–67

system [25]. The size of the cell is determined by the volume fraction ofthe suspension, that is, ϕ=a3/b3. In addition to its radius, the particle ischaracterized by a surface charge density σ, and the solution, with massdensity ρs, viscosity ηs and relative permittivity εrs, contains addedcounterions (from surface group dissociation). These ions will beassumed to be H+, hence with valence z1=1 and diffusion coefficientD1 = 9.3 × 10−9 m2s−1. As mentioned, other species are also present,specifically, OH− (z2 = −1, D2 = 5.3 × 10−9 m2s−1) and HCO3

−

(z3 = −1, D3 = 1.2 × 10−9 m2s−1). In addition, the neutral speciesH2CO3 (the solution is saturated with CO2), with z4 = 0 and D4 =1.3 × 10−9 m2s−1 (estimated from Stokes law and using 0.18 nm asmolecular size [50]). The production of H+ from surface groupdissociation is assumed to be instantaneous, while the non-equilibriumassociation–dissociation processes for the chemical reactions in solutionare:

H2O ⇄K1

K−1

Hþ þ OH−

H2CO3 ⇄K2

K−2

Hþ þHCO−3

H2CO3 ⇄K3

K−3

CO2þH2O

ð1Þ

Here Ki andK−i (i=1, 2, 3) are the corresponding forward (units s−1)and backward (m3s−1) kinetic constants. As it has been previouslyreported [44], the dissociation of the bicarbonate ion HCO3

− into H+ andCO3

= can be neglected because of its minor quantitative importance.We assume that an alternating electric field E e −iωt of angular

frequency ω is applied to the suspension, and each particle will movewith electrophoretic velocity vee−iωt=μEe−iωtwhere μ is the dynamicelectrophoretic mobility.

Our target is the calculation of the complex electrical conductivityand the dynamic mobility of the suspension as a function of frequency,K∗(ω) and μ(ω), respectively. The former relates the average currentthrough the cell to the applied field:

jh i ¼ K� ωð ÞE ð2Þ

From this quantity the real ( ε′r ωð Þ ) and imaginary ( ε″r ωð Þ )components of the relative permittivity of the suspension follow readily[51]:

ε′r ωð Þ ¼ − Im K� ωð Þ½ �ωε0

ε″r ωð Þ ¼ Re K� ωð Þ½ �−K� ω ¼ 0ð Þωε0

ð3Þ

where ε0 is the permittivity of a vacuum. This calculation, as well as thatof μ(ω) requires knowledge on the cell boundary of the electricpotential, the ionic concentration and the fluid velocity. The referencesystem is fixed to the particle center, and spherical coordinates (r, θ,φ) will be used, with the polar axis (θ = 0) parallel to the field. Inprevious works [45,46] it was shown that:

μ ¼ 2h bð Þb

1

1þ ρp−ρs

ρs

� �ϕ

� �

K� ωð Þ ¼X3j¼1

z2j e2Dj

kBTdϕ j

drjr¼b

−2h bð Þb

zje

( )bj exp −

z jeΨ0 bð Þ

kBT

!þ iωεrsε0

dYdr

jr¼b

ð4Þ

where e is the elementary electric charge, kB is Boltzmann's constant, Tis the absolute temperature, ρp is the mass density of the solid, and bjrepresent the number concentration of the j-th ionic species at theposition of zero equilibrium electrical potential, chosen at r = b. Dueto the symmetry, the functions h(r), Y(r) and ϕj(r) contain informationabout the field-induced perturbations in the fluid velocity (v), theelectric potential (Ψ), and the electrochemical potential of species j

(μj), respectively. Such perturbations are assumed to be linear withthe field:

v j r; tð Þ ¼ v j rð Þe−iωt

v r; tð Þ ¼ v rð Þe−iωt

v rð Þ ¼ vr ; vθ; vφ� �

¼ −2rhE cosθ;

1rddr

rhð ÞE sinθ; 0� � ð5Þ

Ψ r; tð Þ ¼ Ψ0 rð Þ þ δΨ rð Þe−iωt

δΨ rð Þ ¼ −Y rð ÞE cosθð6Þ

μ j r; tð Þ ¼ μ0j þ δμ j rð Þe−iωt j ¼ 1;…;4ð Þ

δμ j rð Þ ¼ z jeδΨ þ kBTδnj

n0j

¼ −zj e ϕ j rð ÞE cosθ j ¼ 1;…;3ð ÞδμH2CO3

rð Þ ¼ −e ϕH2CO3rð ÞE cosθ j ¼ 4;H2CO3ð Þ

ð7Þ

Here E = |E|, nj and vj are the concentration in number and thevelocity of the j-th ionic species, respectively. The “0” superscriptindicates equilibrium quantities, and δX refers to field-inducedperturbations of the quantity X. Details of the algebraic manipulationrequired to obtain the system of differential equations and boundaryconditions to be solved are provided in Appendix 1. Here we wouldlike to just point out the fundamental difference between equilibriumreaction conditions and the non-equilibrium ones, basis of the presentapproach. The key point is the conservation equation:

∇ � nj r; tð Þv j r; tð Þh i

¼ σ j r; tð Þ−∂nj r; tð Þ∂t ; j ¼ 1;…;4ð Þ ð8Þ

This equation indicates that the balance for every individual speciesin the system includes a generation-recombination term, representedby the quantity σj: the concentration at any volume element changeswith time, not only because the species is driven in and out of thatvolume, but also because it can be produced or annihilated by chemicalreactions. The chemical equilibrium approach [47,48] is based on thefollowing continuity equation:

∇ �Xnj¼1

z jenj r; tð Þv j r; tð Þh i

¼ − ∂∂t ρel r; tð Þ½ � ð9Þ

which indicates that it is the total charge, instead of the number of eachindividual species, that is preserved. This was required by the fact thatchemical reactions and their associated ionic coupling precluded theconservation of the number of ions of each species. We note that inthat approach the equilibrium scheme for chemical reactions must bealso admitted as valid in the presence of the external AC field.

In our present approach, the functions σj(r,t) (j = 1,…,4) areexpressed as (see Eq. (1)):

σ1 r; tð Þ ¼ σHþ r; tð Þ ¼ K1nH2O r; tð Þ−K−1nHþ r; tð ÞnOH− r; tð Þh i

þþ K2nH2CO3

r; tð Þ−K−2nHCO−3r; tð ÞnHþ r; tð Þ

h iσ2 r; tð Þ ¼ σOH− r; tð Þ ¼ K1nH2O r; tð Þ−K−1nHþ r; tð ÞnOH− r; tð Þ

h iσ3 r; tð Þ ¼ σHCO−

3r; tð Þ ¼ K2nH2CO3

r; tð Þ−K−2nHCO−3r; tð ÞnHþ r; tð Þ

h iσ4 r; tð Þ ¼ σH2CO3

r; tð Þ ¼ − K2nH2CO3r; tð Þ−K−2nHCO−

3r; tð ÞnHþ r; tð Þ

h i−

− K3nH2CO3r; tð Þ−K−3nH2O r; tð ÞnCO2

r; tð Þh i

ð10Þ

where we have followed the procedure developed by Baygents andSaville [52] for weak electrolytes.

Fig. 1. Real (a) and imaginary (b) components of the relative permittivity, andmodulus ofthe dimensionless dynamic mobility (c) of suspensions of spheres in realistic salt-freesolutions for the indicated volume fractionsϕ, surface charge density σ and particle radiusa. Species in solution:H+ (from surface groups, water and carbonic acid dissociation), OH−

(from water dissociation), HCO3− and H2CO3 (from CO2 dissolution). The different

relaxation processes are labeled as: α (concentration polarization), MWO (Maxwell–Wagner–O'Konski relaxation), C (ion condensate relaxation), I (inertial).

60 F. Carrique et al. / Advances in Colloid and Interface Science 201–202 (2013) 57–67

3. Results and discussion

3.1. General differences between equilibrium and non-equilibriumassociation–dissociation formalisms

As a starting point, the physical phenomenology involved in thenon-equilibrium approach (NEQ hereafter) is best understood bycomparing some results to those obtained under the equilibrium (EQ)assumption. Thus, we will study the dielectric dispersion and dynamicmobility in our systems using both approaches: plots of the real andimaginary components of the relative permittivity (ε′r,ε″r) as a functionof the field frequency, f, and of the modulus and phase of the dynamicmobility (|μ(f)|,θμ(f)) will be discussed.

Fig. 1 shows the first example of comparison. The dielectricdispersion curves (Fig. 1a and b) shows two relaxations coincident inNEQ and EQ predictions, and a third one appearing only in the formerapproach. In previousworks [47,48], we demonstrated that the commonprocesses are of Maxwell–Wagner origin: the one at highest frequency(MHz region) only shows up for highly charged particles, and itsrelatively high frequency is an indication of short characteristic distances.All these features suggest that this relaxation is the manifestation of theexistence of a counterion condensate [36]. In addition, the middlefrequency relaxation (another decline in ε′r and maximum in ε″r ) isthe typical Maxwell–Wagner–O'Konski (MWO) process: when theMWO frequency is reached, the polarization of the counterionatmosphere cannot follow the field oscillation and cannot be built up.

The really different feature of the NEQ approach is its ability topredict a new low-frequency relaxation: this must be of “alpha” type,as expected from similarities with suspensions in the presence ofadded salt. Located in the kHz region, this relaxation comes from thedisappearance of concentration polarization in the EDL [53]: a veryclear clue to this can be found in the behavior of the dynamic mobilityspectra in Fig. 1c, where the modulus of the dimensionless mobility μ *(being μ ¼ 2εrsε0kBT

3ηseμ� ) is plotted as a function of frequency for both

approaches. Note that whereas the middle- and high-frequencyrelaxations are associated to successive increases in the mobilitymodulus |μ * |, the new low-frequency relaxation is associated to adecrease in |μ*|, a feature of alpha relaxation, as repeatedly demonstrated(see, e.g., [18]). In subsequent sections of this articlewewill try to explainthe reasons why the EQ approach is unable to predict the existence of analpha-relaxation process in realistic salt-free suspensions.

Before proceeding with our discussion, it may be of interest toconfirm that the approach here developed is correct, by comparing itsresults with analytical expressions valid in some limiting cases. Inparticular, considering that the overall ionic concentration in the EDLsis low in salt-free systems, we might approach a situation in whichsuch concentration is (with good approximation) constant. If the EDLis thick enough and the particle concentration sufficiently high, wecould reach the condition κ(b− a) N N 1, that is, the volume accessibleto ions is much smaller than the EDL volume (highly overlapped doublelayers). For this extreme situation there is an analytical solution to theproblem of dynamic mobility evaluation through colloid vibrationcurrent, CVI. This was carried out by Shilov et al. [54], in what theydenominated quasi-homogeneous model, in which ion concentrationsare constant (position independent) everywhere in the EDL.Considering the relationship existing between dynamic mobility andcolloid vibration current per unit pressure gradient ⟨∇P∞⟩ [54–56], it ispossible to compare the results of our computations of μ and theanalytical solutions provided in [54]:

CVI∇P∞h i ¼

μϕ ρp−ρs

� �ρs

ð11Þ

In Fig. 2 we plot the dependence of the vibration current on thefrequency of the sound wave applied to the suspension for data

presented in [54], corresponding to a suspension of spheres 1 μm inradius and 2 × 103 kg/m3, with a volume fraction of 0.2, and a chargedensity of 6.64 × 10−3 μC/cm2. The solution density is 103 kg/m3, itspermittivity 30 ε0, and its viscosity 1 mPa·s. Note that the agreementreached is very satisfactory for the frequency spectrum available tothe analytical model. At low frequencies, the complete theory includesMaxwell–Wagner relaxation (increase of CVI with frequency after theinitial plateau), which is not included in the analytical approximation,where only the hydrodynamic (inertial) relaxation is present.

Fig. 2. Modulus of the colloid vibration potential (CVI) per unit pressure gradient as afunction of the frequency of the sound wave. Conditions: 1 μm particle radius; surfacecharge density: 6.65 × 10−3 μC/cm2; particle density: 2 × 103 kg/m3; medium density:1 × 103 kg/m3; liquid viscosity: 1 mPa·s; relative permittivity of the medium: 30. Solidline: this work; dashed line, calculations in Ref. [54].

61F. Carrique et al. / Advances in Colloid and Interface Science 201–202 (2013) 57–67

At this point, and in order to reinforce arguments for theinterpretation of the low-frequency process as an alpha dispersion, weplot in Fig. 3 the real part of the average electric polarization Ptot(f)[13] (in dimensionless form P�

tot fð Þ ¼ 1ε0E

Ptot fð Þ ) as a function offrequency, for the cases considered in Fig. 1. The polarization isevaluated at an instant inwhich the field points to the right and reachesits maximum amplitude.

Note that both the EQ and NEQ polarizations start with a lowfrequency plateau, larger in value for the EQ approach. In this regionthe behavior of EQ and NEQ predictions is clearly different. If an alphaprocess is present, as it is for the NEQ formalism, its relaxation infrequency should increase the average polarization value, as observedin Fig. 3, due to the opposing effect it has on the MW polarization. Inother words, as the alpha process progressively disappears, the MWpolarization is less hindered, which rises the average polarization. Asfrequency is further increased, the first MW process starts to relax(the MW double layer relaxation). Whatever the formalism chosen,this MW relaxation corresponds to a decay of the average polarization.At higher frequencies we observe a second decay, which relates to thealready described MW relaxation of the ionic condensate [36].Summarizing, for the case analyzed in Fig. 3 an alpha relaxation process

Fig. 3.Real part of the dimensionless average electric polarization (or induced electricdipole moment per unit volume) as a function of frequency, plotted at an instantwhen the AC field points to the right and attains its maximum value. Systemsconsidered as in Fig. 1.

provokes an increase of themodulus of the average electric polarizationwhile MW relaxations imply two successive diminutions of suchquantity. Note also that both EQ and NEQ predictions of the averagepolarization values coincide in frequency once the alpha relaxation isover.

To conclude this first analysis of the NEQ alpha process, one has towonder the reason why the EQ model does not predict any alphaprocess. This will be addressed in the next section.

3.2. Reasons for the absence of an alpha-dispersion in realistic salt-freesystems according to the EQ approach

The question of the absence of the low-frequency dielectricdispersion in realistic salt-free suspensions when the EQ approach isused can be elucidated by considering that, apparently, the EQformalism does not permit the generation of a concentration gradientof neutral electrolyte at both sides of the particle outside the doublelayer. This is the key to the existence of alpha-relaxation, and noconcentration polarization phenomenon [48] seems to be generated.In our study the protons acting as counterions are forced to movefrom the left to the right part of our negative particle when the electricfield points to the right, thus leading to an increment of theirconcentration at the right-hand side of the particle. Simultaneously,carbonic anions acting as coions tend to move to that region from theouter region of the double layer, the bulk. The final result should be anincrement of both the proton and the carbonic anion concentrations,increasing the salt concentration in such region. The opposite will takeplace at the left-hand side of the particle: a reduction of theconcentration of protons and carbonic anions due to the migration ofthe former from the left to the right part of the particle and the anionsmoving further to the left forced by the field.

It must be taken into account that the carbonic acid is a weakelectrolyte. Although concentration polarization in these systemscannot be fully ruled out, Grosse and Shilov [57] showed previouslythat the alpha-relaxation is weak or even essentially absent in suchsolutions of weak electrolytes. In the past, these arguments supportedprevious EQ results, as alpha processes were not found by using thatformalism. Instead, the present NEQ study does confirm their existencefor realistic salt-free suspensions in aqueous solutions, which shouldbehave in principle as suspensions in weak electrolyte solutions.

When the electric field is imposed to the suspension, it induces amovement of ions that tends to generate a concentration gradient ofneutral electrolyte around the particle (usually called concentrationpolarization phenomenon), as it is well-known in the added salt case.The difference in our case is the effect of chemical reactions: they tendto hinder the concentration gradient so generated. But this hinderingeffect depends greatly on the kinetics of the chemical reactions: thereduction of the concentration polarization will be more effective thefaster the chemical reactions are, in comparison with the characteristictimes for ionic electromigration and diffusion fluxes. In the limitingcase of chemical equilibrium, the EQ case, forward and backwardreactions are instantaneous and the concentration polarization isnegligible. That is the reason why the EQ formalism is unable to predictany alpha relaxation. Instead, in the NEQ case, the velocity of chemicalreactions remains finite, and concentration polarization cannot bedisregarded, and therefore, we expect that an alpha relaxation processcould take place, as it is confirmed in the previous results in Figs. 1and 3. The importance of this alpha relaxation is greater the lower thekinetic rate constants of the chemical reactions.

A final study concerning the transition from EQ to NEQ formalismscan be useful. Such study might shed some light about the effects ofthe gradual incorporation of NEQ considerations on electrokinetic anddielectric suspension properties in comparison with EQ predictions.Such transition study has been carried out for a particular case as anexample, and the relative permittivity and dynamic electrophoreticmobility are plotted in Fig. 4, where we represent the ε′r and |μ * |

Fig. 4. Real part of the relative permittivity (a), andmodulus of the dimensionless dynamicmobility (b) as a function of frequency for the suspensions indicated. The dashed linesbetween theNEQandEQdependences correspond to increasing the forward andbackwardrate constants (while keeping their ratio constant and equal to the equilibrium value).

Fig. 5. Real (a) and imaginary (b) components of the relative permittivity of realistic salt-free suspensions of spherical particles 300 nm in radius and 1% volume fraction, for theindicated values of surface charge density. Red dashed lines: EQ; blue solid lines: NEQ.

62 F. Carrique et al. / Advances in Colloid and Interface Science 201–202 (2013) 57–67

dependences for both NEQ and EQ conditions, and we have alsoincluded (dashed lines in the Figure) the progression from one to theother by varying appropriately the forward and backward rateconstants in the NEQ association–dissociation equations. In all cases,the rate constant values, starting from their tabulated values, areincreased while maintaining a constant ratio between them,corresponding to the equilibrium ratio.

A remarkable aspect of Fig. 4 is that NEQ and EQ predictions onlydiffer in the low frequency region, where alpha processes take place.Both formalisms use the same linear perturbation procedure describedin Eqs. ((5)–(7)) in their calculations: perturbed quantities due to thepresence of the electric field are added to the corresponding onesobtained in its absence. At low frequencies, the ion responses perfectlyfollow the evolution of the varying electric field, but lag behind anddecrease their importance at increasing frequencies, when the systemundergoes relaxations which affect electrokinetic and dielectricproperties, as can be seen for example in Fig. 1a and b. When thealpha process has relaxed, the frequency behavior of the suspension ismainly related with the induced dipole moment created by themovement of the counterion atmosphere inside the EDL. When thefrequency increases, or the time between successive field inversionsshortens, all ionic transport processes inside the EDL becomeprogressively hindered, giving rise to the observed twoMWrelaxations.EQ and NEQ predictions show the same values in this frequency regionbecause the effect of the chemical reactions is negligible, as the EDL ismostly composed by counterions. The perturbations of coions in the

EDL are several orders of magnitude lower than that of counterions inboth formalisms, and the fact that they are in or out of chemicalequilibrium does not change appreciably the local counterionconcentration.

Summarizing, the difference between EQ and NEQ approaches takesplace in the low frequency region. This is because there is enough timefor all transport processes to develop and only the accepted relationsbetween ionic concentrations in both formalisms when chemicalreactions are present makes the difference. Unlike the EQ approach,where mass-action chemical equations connecting the correspondingion concentrations are obeyed under any circumstances (electric fieldapplied or not), the NEQ one allows that ion concentrations do notattain chemical equilibrium in the presence of the external electricfield. This is a consistent formation of a gradient of neutral electrolyte,which is the key point of the concentration polarization mechanismwhose relaxation as already pointed out is known as alpha process.Due to its simplicity, the EQ formalism has been used to study dynamicproperties in realistic salt-free suspensions as a first approximation, butit turns out that such suspensions require the more complex NEQpicture to deal with chemical reactions in solution.

In the following sections we will explore the behavior of the alphaprocess upon changing particle charge, particle radius, and volumefraction, in order to gain a more complete understanding of its role onelectrokinetic and dielectric properties of realistic salt-free concentratedsuspensions.

63F. Carrique et al. / Advances in Colloid and Interface Science 201–202 (2013) 57–67

3.3. The role of particle charge

In Fig. 5a and b we display a comparative EQ–NEQ study of the realand imaginary parts of the complex relative permittivity of a realisticsalt-free concentrated suspension of spherical particles of radius a =300nm, particle volume fraction ϕ=0.01, and different particle surfacecharge densities. Fig. 6a and b shows the modulus and phase of thedimensionless dynamic electrophoretic mobility for the same con-ditions. Because of the less intuitive behavior of the phase-frequencyrepresentation in Fig. 6b in comparison with that of the modulus(Fig. 6a), and given that all the information is already expressed in thelatter, we will focus our analysis only on the spectra of |μ * | becauseno information is lost.

Note that the extra relaxation predicted by the NEQ approach at thevery low-frequency end of the spectra is very much enhanced by theincrease in surface charge, and it appears to saturate for the highestcharge densities. Additionally, the characteristic frequency of thisrelaxation is essentially unaffected by the charge, again a verycharacteristic feature of alpha dispersion. The increase in σ above acertain value seems to favor the intensification of the counterioncondensation phenomenon, leaving the diffuse layer unchanged inspite of additional increases in surface charge. The outermost part ofthe EDL hardly changes once the condensation region is generated,and the alpha mechanism remains hence unmodified.

On the other hand, the alpha relaxation frequency in dilute systemsdepends on the inverse of the squared particle radius plus double layerthickness. The radius is constant and quite bigger than the double layerthickness contraction that commonly takes place when ionic strength

Fig. 6. Same as Fig. 5, but for the modulus and phase of the dimensionless dynamicmobility.

increases due to the enhancement of added counterions as particlecharge rises. Also the particle diffusion length is constant at a givenvolume fraction in the concentrated case. The predicted independencyof alpha relaxation frequency with particle charge displayed in Fig. 5 isthen expected.

With regard to the features of the NEQ dynamicmobilitymodulus ofFig. 6a in the low-frequency region, we reaffirm its alpha nature uponincreasing particle charge. Recall that the low frequency mobility istypically larger if a concentration polarization mechanism is acting,reducing the magnitude of the induced electric dipole moment, whichslows down the particle motion. As frequency increases and theconcentration polarization mechanism begins to relax, the net induceddipole moment is enhanced and the particle motion is subsequentlyslowed down. This is the reason for the decay observed in |μ * | beyondthe low frequency plateau, for frequencies around 10kHz. The mobilitywell is deeper the larger theσ, as expected for an alpha relaxation. If thefrequency is further increased, themagnitude of the net dipolemomentdiminishes, as double layer polarization is not fully developed. This isthe region of the first MW relaxation, the one associated to thepolarization of the double layer. The final result is that the mobilityincreases until the inertia of the particle and fluid progressively opposesany motion although a further increase of the mobility is even possiblewhen the MW of the ionic condensate relaxes at higher frequencies.Finally the inertia forces the particle to reduce its mobility to zero atvery high frequencies.

These qualitative interpretations are confirmed to a large extent byour analysis of the field-induced perturbations in ionic concentrationsaround a particle in a realistic salt-free suspension of the characteristicsdescribed in Fig. 1. The results are shown in the on-line SupplementaryInformationfile as snapshots of perturbations at the instant inwhich thefield points from left to right of the figures, and attains its maximumamplitude.

3.4. Effects of particle volume fraction

Let us now explore the effects of changing the volume concentrationof particles, ϕ, on both ε′r fð Þ (Fig. 7a) and the dimensionless mobilitymodulus |μ * (f)| (Fig. 7b). Let us point out that these data show thecharacteristics of an alpha process when the NEQ model is used. It iswell known that as the volume fraction increases, the magnitude ofthe dielectric increment of the alpha relaxation increases, reaches amaximum and subsequently decreases at large volume fractions [8,9].In addition, the alpha relaxation frequency is shifted to largerfrequencies upon increasing volume fraction. The latter two aspectsare observed in Fig. 6a. As volume fraction increases, additional chargeparticipates in the generation of a gradient of neutral electrolyte at theouter double layer and both sides of the particle. This mechanismgives rise to diffusive fluxes tending to minimize such concentrationgradient. The dielectric polarization associated with this processincreases until the volume fraction is large enough as to produceoverlapping between the concentration polarization regions of neigh-bor particles, partially compensating their effects, and diminishing thedielectric increment of the alpha process. Likewise, as volume fractionincreases, the radius of the cell shortens and so does the diffusionlength, a larger frequency being necessary for the relaxation of thealpha process. This frequency shift with volume fraction is clearlydisplayed in Fig. 7a reinforcing the alpha nature of the NEQ lowestfrequency process.

Regarding the behavior of the dynamic mobility, the presence of thealpha-decay in |μ * | in the 103–104 Hz frequency range is clearlyobservable at the lowest volume fraction explored (ϕb 5%, say). Whenϕ is further increased, the decrease associated to the alpha relaxationshifts to higher frequencies, as described. As a consequence, it mixeswith the MWO mobility elevation, which becomes predominant and isapparently the only feature observed, except of course for the finalinertial decay. Note as well that the elevation coming from the

Fig. 7. Real component of the relative permittivity (a) and modulus of the dimensionlessdynamic mobility (b) of realistic salt-free suspensions, plotted as a function of frequencyaccording to NEQpredictions. The particle radius is 300nm, and the surface charge densityis−5 μC/cm2. The curves are labeled with corresponding volume fractions.

Fig. 8. Frequency dependence of the real (a) and imaginary (b) components of the relativepermittivity, and modulus of the dimensionless dynamic mobility (c) of realistic salt-freesuspensions of spherical particles with the radii indicated (in nm). Surface charge densityσ=−5 μC/cm2, volume fraction ϕ=1%.

64 F. Carrique et al. / Advances in Colloid and Interface Science 201–202 (2013) 57–67

relaxation of the counterion condensate also mixes up with the typicalMWO process, since the increase of particle volume fraction bringsabout an increased concentration of counterions in solution. This inturn shifts ωMWO to larger values, since this characteristic frequency isroughly given by ωMWO ¼ 1−ϕð ÞKpþ 2þϕð ÞKs

1−ϕð Þεrpε0þ 2þϕð Þεrsε0 , where Kp and Ks are theconductivities of the particle and medium, respectively, and εrp therelative permittivity of the solids [47].

3.5. Particle radius effects

Let us finally analyze the dependences of the dynamic features of theEDL of spherical nanoparticles in realistic salt-free solutions with thesize of the particles. This parameter is essential in many aspects of thefrequency relaxation of both permittivity and AC mobility. We willfocus again on the NEQ approach exclusively, because it is in the alpharelaxation (absent in EQ predictions) that size effects are magnified.Our main results are shown in Fig. 8a and b for the real and imaginarycomponents of the permittivity and Fig. 8c for the dimensionlessdynamic mobility.

In the case of ε′r,ε″r the effect of size in the 1–10kHz region is veryremarkable. This is not surprising if a model is used that correctlyaccounts for EDL concentration polarization: indeed, the alpha processis very sensitive to the particle radius as it directly affects themagnitudeof the induced dipole moment by modifying the characteristic lengthsfor charge accumulation and depletion at both sides of the particles.

Also, it depends on particle charge, which increases at fixed surfacecharge density as particle radius rises. In spite of that, the totalcountercharge released by the particles to the solution decreases withparticle radius because for bigger particles there are fewer of them inthe unit volume of suspension (although they are more charged), atfixed volume fraction. The results in Fig. 8a and b point to the conclusionthat the observed increase in dielectric polarization is a consequence of

65F. Carrique et al. / Advances in Colloid and Interface Science 201–202 (2013) 57–67

the larger polarizability of bigger electrokinetic units. The latter effectseems to be more important than the expected decrease in dielectricpolarization associated with the reduction of the number of polarizablecomponents in the unit volume of suspension. On the other hand, ashift to lower frequencies is found for the relaxation peak appearingat the left side of the spectra in Fig. 8b as particle radius increases.This is a very well known result of the alpha relaxation process asits relaxation frequency depends basically on the inverse of thesquared radius.

Concerning the effect of particle size on themobilitymodulus, Fig. 8cshows clearly two significant features: one is the presence of “wells”in the mobility spectra, deeper the larger the radius. This is clearlyassociated to the relaxation of concentration polarization mechanism,which, when present, works against the braking effect of the induceddipole. The second effect, perhaps more obvious, refers to the oc-currence of inertia relaxation at lower frequencies when the radiusrises: by way of example, it shifts from ~5×108 to ~5×107Hz when ais increased from 100 to 500nm.

4. Conclusions

In this work a study of the dynamic electrophoretic mobility anddielectric response of realistic, concentrated salt-free suspensions ofspherical particles in the presence of an oscillating electric field (ac)has been carried out following two different approaches. The first one,that we call EQ, assumes an equilibrium scenario for chemical reactionsin solution, and it only predicts two Maxwell–Wagner relaxationprocesses associated with double layer and ionic condensate re-laxations. A common suspension in a low concentrated salt solutionis known to undergo three relaxation processes depending on theparticle charge. The additional relaxation process is associated withthe concentration polarization phenomenon by the generation of agradient of neutral electrolyte around the particles, and it is knownas alpha relaxation process. As the EQ model does not predict anyalpha process, reasons for such absence are addressed in this work.To elucidate this issue, a new and more rigorous scheme accountingfor non-equilibrium association–dissociation kinetics in solution(NEQ model) has been developed. This approach leads to theprediction of an alpha process in addition to the two MW's alreadypredicted by the EQ model. Reasons for this are given and analyzed,and different studies of the dependences of electrokinetic anddielectric properties on particle charge, radius and volume fractionhave been done in order to confirm the alpha nature of the newNEQ process. We believe that dealing with realistic salt-freeconcentrated suspensions under static or oscillating electric fieldsrequires NEQ models to account for chemical reactions in solution.The NEQ corrections have been shown to be very important inmany typical situations, and thus the use of low-frequency EQmethods has to be disregarded for a precise description of theelectrokinetics of realistic systems that include chemical reactionsin solution.

Acknowledgments

Financial support for this work by MICINN, Spain (projectFIS2010-18972, FIS2010-19493), and Junta de Andalucía, Spain(P08-FQM-3779, P08-FQM-3993), co-financed with FEDER (EuropeanFund for Regional Development) funds by the EU, is gratefullyacknowledged.

Appendix 1.Derivationof thepermittivity anddynamicmobility of asalt-free suspension

We will briefly review the main steps followed in the calculation ofthe complex relative permittivity εr∗(f) and dynamic mobility μ(f) inthe cell-model scenario. We assume a suspension of spheres of radius

a in an aqueous solution with relative permittivity εrs, mass density ρsand viscosity ηs. As mentioned in the text, realistic salt-free conditionsare admitted: this means that the solution contains:

• H+ ions (j=1) produced by dissociation of ionizable particle surfacegroups, water, and carbonic acid generated by dissolved atmosphericCO2

• OH− (j=2), produced exclusively by water dissociation• HCO3

−Z (j=3), from carbonic acid dissociation• H2CO3Z (j=4), from CO2 dissolution

An alternating (or AC) electric field of frequencyω, Ee−iωt is appliedto the suspension, and each particle is assumed to undergo oscillationsforced by the field, and represented by their AC electrophoretic velocityvee−iωt or, equivalently, by their dynamic mobility μ, given by vee−iωt=μEe−iωt. The reference system is fixed to the particle center, being thepolar axis (θ=0) parallel to the field. Considerations about the correctuse of such a reference system are provided in the SupplementaryInformation file (Section S1).

The fundamental equations connecting electrical potential Ψ(r,t),the number density of each species, nj(r,t), their drift velocity vj(r,t),the fluid velocity v(r,t), and the pressure P(r,t) are [10,17,22]:

∇2Ψ r; tð Þ ¼ −ρel r; tð Þεrsε0

ðA1Þ

ρel r; tð Þ ¼X3j¼1

z j e n j r; tð Þ ðA2Þ

ηs∇2v r; tð Þ−∇P r; tð Þ−ρel r; tð Þ∇Ψ r; tð Þ¼ ρs

∂∂t v r; tð Þ þ vp exp −iωtð Þh i ðA3Þ

∇ � v r; tð Þ ¼ 0 ðA4Þ

v j r; tð Þ ¼ v r; tð Þ− Dj

kBT∇μ j r; tð Þ j ¼ 1;…;4ð Þ ðA5Þ

μ j r; tð Þ ¼ μ∞j þ z jeΨ r; tð Þ þ kBT lnnj r; tð Þ j ¼ 1;…;4ð Þ ðA6Þ

∇ � nj r; tð Þv j r; tð Þh i

¼ σ j r; tð Þ−∂nj r; tð Þ∂t j ¼ 1;…;4ð Þ ðA7Þ

μj(r,t) is the electrochemical potential of j-th ionic species, with μj∞ itsstandard value. Eq. (A1) is Poisson's equation, where ρel(r,t) is theelectric charge density given by Eq. (A2). Eqs. (A3) and (A4) are theNavier–Stokes equations for an incompressible fluid flow at lowReynolds number in the presence of an electrical body force. Eq. (A5)means that the flows of ionic species and neutral molecules are due inpart to both the liquid flow and the gradient of the electrochemicalpotential defined in Eq. (A6). Recall that, as mentioned in the maintext, Eq. (A7) is substituted, in the case of the EQ approach, by acondition of total charge conservation, Eq. (9). In the present, NEQapproach, expressions are required for the ion creation and annihilationfunctions σj(r,t), (j=1,…,4):

σ1 r; tð Þ ¼ σHþ r; tð Þ ¼ K1nH2O r; tð Þ−K−1nHþ r; tð ÞnOH− r; tð Þh i

þþ K2nH2CO3

r; tð Þ−K−2nHCO−3r; tð ÞnHþ r; tð Þ

h iσ2 r; tð Þ ¼ σOH− r; tð Þ ¼ K1nH2O r; tð Þ−K−1nHþ r; tð ÞnOH− r; tð Þ

h iσ3 r; tð Þ ¼ σHCO−

3r; tð Þ ¼ K2nH2CO3

r; tð Þ−K−2nHCO−3r; tð ÞnHþ r; tð Þ

h iσ4 r; tð Þ ¼ σH2CO3

r; tð Þ ¼ − K2nH2CO3r; tð Þ−K−2nHCO−

3r; tð ÞnHþ r; tð Þ

h i−

− K3nH2CO3r; tð Þ−K−3nH2O r; tð ÞnCO2

r; tð Þh i

ðA8Þ

66 F. Carrique et al. / Advances in Colloid and Interface Science 201–202 (2013) 57–67

according to the procedure developed by Baygents and Saville for weakelectrolytes [52]. They represent the non-equilibrium association–dissociation processes for each particular species involved in chemicalreactions.

In realistic salt-free systems with spherical particles, the followingconsiderations are also required [44] (the quantities with a superscript“0” refer to equilibrium conditions in the absence of an applied electricfield):

1) Start from Poisson–Boltzmann equation for the equilibrium electricpotential:

1r2

ddr

r2dΨ0

dr

!¼ d2Ψ0

dr2þ 2

rdΨ0

dr¼ − ρ0

el

εrsε0

ρ0el rð Þ ¼

X3k¼1

zk e bk exp − zk e Ψ0 rð Þ

kBT

! ðA9Þ

where thequantities bk represent the concentration of the respectiveions at the position of zero equilibrium electrical potential. This ischosen at the outer surface of the cell r=b:

Ψ0 bð Þ ¼ 0 ðA10Þ

2) Electroneutrality of the cell:

4πa2σ þZ b

aρ0el rð Þ4πr2dr ¼ 0 ðA11Þ

3) The equilibrium dissociation constants are defined by:

Hþh i0OH−½ �0 ¼ K0

W

Hþ� 0 HCO−3½ �0

H2CO3½ �0 ¼ K02

ðA12Þ

where KW0 =[K1/K− 1][H2O]0, being expressed in (mol/L)2, and K2

0=[K2/K−2] in mol/L (see Eq. (1) and, e.g., ref. [44] for the values of theconstant K2

0 and the equilibrium concentration of dissolved carbonicacid [H2CO3]0).

4) In ref. [44] is shown the procedure to obtain the electric potentialprofile, using the boundary condition in Eq. (A10) together with

dΨ rð Þdr

jr¼b

¼ 0

dΨ rð Þdr

jr¼a

¼ − σεrsε0

ðA13Þ

The interested reader can find in ref. [44] a complete study of all thepossible cases and the methods of resolution of the resulting integro-differential Poisson–Boltzmann equations.

Substitution of the perturbation scheme described in Eqs. ((5)–(7))into the corresponding differential equations leads, after some algebra,to the following differential equations of the problem:

zHþeλHþ

n0Hþ LϕHþ þ i

ωλHþ

kBTϕHþ−Y½ �− e

kBTdΨ0

dr

!zHþ

dϕHþ

dr−2

λHþ

e rh

� �" #¼

¼ S KWe

kBTzHþ ϕHþ−Yð Þ þ zOH− ϕOH−−Yð Þ½ � þ

þ SC KCe

kBTn0H2CO3

zHCO−3

ϕHCO−3−Y

� �þ zHþ ϕHþ−Yð Þ−ϕH2CO3

h iðA14Þ

zOH−eλOH−

n0OH− LϕOH− þ i

ωλOH−

kBTϕOH−−Y½ �− e

kBTdΨ0

dr

!zOH−

dϕOH−

dr−2

λOH−

e rh

� �" #¼

¼ S KWe

kBTzHþ ϕHþ−Yð Þ þ zOH− ϕOH−−Yð Þ½ �

ðA15Þ

zHCO−3e

λHCO−3

n0HCO−

3LϕHCO−

3þ i

ωλHCO−3

kBTϕHCO−

3−Y

h i− e

kBTdΨ0

dr

!zHCO−

3

dϕHCO−3

dr−2

λHCO−3

e rh

!" #¼

¼ SC KCe

kBTn0H2CO3

zHCO−3

ϕHCO−3−Y

� �þ zHþ ϕHþ−Yð Þ−ϕH2CO3

h i ðA16Þ

LϕH2CO3þ i

ωλH2CO3

kBTϕH2CO3

¼ −SC KCλH2CO3

kBTzHCO−

3ϕHCO−

3−Y

� �þ zHþ ϕHþ−Yð Þ−ϕH2CO3

h iþ

þ λH2CO3

kBTχCH

KhϕH2CO3

ðA17Þ

LY rð Þ ¼ −X3j¼1

z2j e2n0

j rð Þεrsε0kBT

ϕ j rð Þ−Y rð Þh i

ðA18Þ

L Lh rð Þ þ γ2h rð Þh i

¼ − e2

kBTηsrdΨ0

dr

!X3j¼1

z2j n0j rð Þϕ j rð Þ ðA19Þ

where

γ ¼ffiffiffiffiffiffiffiffiffiffiffiiωρs

ηs

rðA20Þ

L ≡ d2

dr2þ 2

rddr

− 2r2

ðA21Þ

S ¼ K−1 m3s−1� �

; KW m−6� �

¼ K1

K−1n0H2O ⇒ K1 s−1

� �¼ S KW=n0

H2O ðA22Þ

SC ¼ K−2 m3s−1� �

; KC m−3� �

¼ K2

K−2⇒ K2 s−1

� �¼ SC KC ðA23Þ

Kh ¼ n0H2O m−3� �K−3 m3s−1

� �K3 s−1� � ; χCH s−1

� �¼ K−3 n

0H2O ðA24Þ

The appropriate boundary conditions are:

δΨp rð Þ ¼ δΨ rð Þ at r ¼ a ðA25Þ

εrs∇δΨ rð Þ � r̂−εrp∇δΨp rð Þ � r̂ ¼ 0 atr ¼ a ðA26Þ

v ¼ 0 atr ¼ a ðA27Þ

v j � r̂ ¼ 0 j ¼ 1;…;4ð Þ atr ¼ a ðA28Þ

ρmv′

� �D E¼ 1

Vcell

ZVcell

ρmv′

� �r′; t� �

dV ¼ 0 ðA29Þ

ω¼∇� v ¼ 0 atr ¼ b ðA30Þ

δnj rð Þ¼0 j ¼ 1;…;4ð Þ atr ¼ b ðA31Þ

δΨ rð Þ ¼ −E � r̂ at r ¼ b ðA32Þ

At the particle surface r=a, Eq. (A25) stands for the continuity of theelectric potential, Eq. (A26) is related to the discontinuity of the normalcomponent of the displacement vector, Eq. (A27) represents the non-slip condition for the fluid, and Eq. (A28) the impenetrability of theparticle for ions. Eq. (A29) expresses the condition of zero macroscopicmomentum per unit volume [23] where ρm is the local mass densityequal to ρs or ρp depending on whether it refers to the solution or tothe particle, respectively, and v′ the local velocity with respect to afixed Laboratory Reference System. At the outer surface of the cell,Eq. (A30) represents the null vorticity for the fluid velocity accordingto Kuwabara [49], and Eqs. (A31) and (A32) the Shilov–Zharkikh–

67F. Carrique et al. / Advances in Colloid and Interface Science 201–202 (2013) 57–67

Borkovskaya boundary conditions [22] for the perturbations of ionicconcentrations and electrical potential, respectively. In addition, theequation of motion for the unit cell imposes [17]

ρs

Z π

0

Z b

a

ddt

vr cosθ−vθ sinθþ μEð Þe−iωth i

2πr2 sinθ dr dθþ

þρp43πa3

ddt

μEe−iωth i

¼Z π

0σ rr cosθ−σ rθ sinθ½ �r¼b 2πb

2 sinθ dθðA33Þ

σrr and σrθ being the normal and tangential components of thehydrodynamic stress tensor [17].

By using the perturbation scheme of Eqs. ((5)–(7)), the above-mentioned boundary conditions can be expressed as:

h að Þ ¼ 0 ðA34Þ

dhdr

að Þ ¼ 0 ðA35Þ

d2hdr2

bð Þ þ 2bdhdr

bð Þ− 2b2

h bð Þ ¼ 0 ðA36Þ

d3hdr3

bð Þ þ 1bd2hdr2

bð Þ− 6b2

dhdr

bð Þ þ 6b3

h bð Þ− iωρs

ηs

h bð Þb

−μρp−ρs

� �ρs

ϕ−dhdr

bð Þ24

35

¼ ρ0el bð ÞY bð Þbηs

ðA37Þ

dϕ j

drað Þ ¼ 0 j ¼ 1;…;4ð Þ ðA38Þ

ϕ j bð Þ ¼ b j ¼ 1;…;3ð Þ; ϕ4 bð Þ ¼ 0 H2CO3ð Þ ðA39Þ

dYdr

að Þ− εrpεrs

Y að Þa

¼ 0 ðA40Þ

Y bð Þ ¼ b ðA41Þ

In addition, the calculation of the dynamic mobility (Eq. (4)) comesfrom the boundary condition (A29) for the velocity. In turn, the details ofthe calculation of the average current density through the suspension,required to obtainK∗(ω), are provided in Section S2 of the SupplementaryInformation file. The electrokinetic equations with the mentionedboundary conditions have been solved numerically by using ODE Solverroutines implemented in MATLAB©. More details can be found inprevious works from the authors [9,10,17].

Appendix 2. Supplementary data

Supplementary data to this article can be found online at http://dx.doi.org/10.1016/j.cis.2013.10.004.

References

[1] Chang MY, Robertson AA. Can J Chem Eng 1967;45:67.[2] O'Brien RW. J Colloid Interface Sci 1986;110:477.[3] Szymczyk A, Aoubrza B, Fievet P, Pagetti J. J Colloid Interface Sci 1999;216:285.[4] Ohshima H. J Colloid Interface Sci 1998;208:295.[5] Carrique F, Arroyo FJ, Delgado AV. Colloids Surf A 2001;195:157.[6] Ruiz-Reina E, Carrique F, Rubio-Hernández FJ, Gómez-Merino AI, García-Sánchez P. J

Phys Chem B 2003;107:9528.[7] Rubio-Hernández FJ, Carrique F, Ruiz-Reina E. Adv Colloid Interface Sci 2004;107:51.[8] Delgado AV, Arroyo FJ, González-Caballero F, Shilov VN, Borkovskaya YB. Colloids

Surf A 1998;140:139.[9] Carrique F, Arroyo FJ, Jiménez ML, Delgado AV. J Chem Phys 2003;118:1945.

[10] Carrique F, Ruiz-Reina E, Arroyo FJ, JiménezML, Delgado AV. Langmuir 2008;24:11544.[11] Midmore BR, Hunter RJ, O'Brien RW. J Colloid Interface Sci 1987;120:210.[12] Barchini R, Saville DA. J Colloid Interface Sci 1995;173:86.[13] Bradshaw-Hajek BH, Miklavcic S, White LR. Langmuir 2008;24:4512.[14] Marlow BJ, Fairhurst D, Pendse HP. Langmuir 1998;4:611.[15] O'Brien RW. J Fluid Mech 1990;212:81.[16] Ahualli S, Delgado AV, Miklavcic SJ, White LR. Langmuir 2006;22:7041.[17] Carrique F, Ruiz-Reina E, Arroyo FJ, JiménezML, Delgado AV. Langmuir 2008;24:2395.[18] Arroyo FJ, Carrique F, Ahualli S, Delgado AV. Phys Chem Chem Phys 2004;6:1446.[19] Dukhin AS, Shilov VN, Ohshima H, Goetz PJ. Langmuir 1999;15:3445.[20] Dukhin AS, Goetz PJ. Characterization of liquids. Nano- and microparticulates, and

porous bodies using ultrasound2nd. ed. Elsevier; 2010.[21] Delgado AV, González-Caballero, Hunter RJ, Koopal LK, Lyklema J. J Colloid Interface

Sci 2007;309:194.[22] OhshimaH. Theory of colloid and interfacial electric phenomena. Academic Press; 2006.[23] O'Brien RW, Jones A, Rowlands WN. Colloids Surf A 2003;218:89.[24] Shilov VN, Zharkikh NI, Borkovskaya YB. Colloid J 1981;43:434.[25] Zholkovskij EK, Masliyah JH, Shilov VN, Bhattacharjee S. Adv Colloid Interface Sci

2007;134–135:279.[26] Carrique F, Cuquejo J, Arroyo FJ, Jiménez ML, Delgado AV. Adv Colloid Interface Sci

2005;118:43.[27] Ahualli S, Jiménez ML, Carrique F, Delgado AV. Langmuir 2009;25:1986.[28] Sood AK. Solid State Phys 1991;45:1.[29] Medebach M, Palberg T. J Chem Phys 2003;119:3360.[30] Medebach M, Palberg T. Colloids Surf A 2003;222:175.[31] Wette P, Schöpe HJ, Palberg T. Colloids Surf A 2003;222:311.[32] Medebach M, Palberg T. J Phys Condens Matter 2004;16:5653.[33] Palberg T, Medebach M, Garbow N, Evers M, Fontecha AB, Reiber H, et al. J Phys

Condens Matter 2004;16:S4039.[34] Imai N, Oosawa F. Busseiron Kenkyu 1952;52:42.[35] Oosawa F. Polyelectrolytes. Dekker; 1971.[36] Ohshima H. J Colloid Interface Sci 2002;247:18.[37] Ohshima H. J Colloid Interface Sci 2002;248:499.[38] Ohshima H. J Colloid Interface Sci 2003;262:294.[39] Ohshima H. J Colloid Interface Sci 2003;265:422.[40] Ohshima H. Colloids Surf A 2003;222:207.[41] Chiang CP, Lee E, He YY, Hsu JP. J Phys Chem B 2006;110:1490.[42] Carrique F, Ruiz-Reina E, Arroyo FJ, Delgado AV. J Phys Chem B 2006;110:18313.[43] Ruiz-Reina E, Carrique F. J Phys Chem C 2007;111:141.[44] Ruiz-Reina E, Carrique F. J Phys Chem B 2008;112:11960.[45] Carrique F, Ruiz-Reina E. J Phys Chem B 2009;113:8613.[46] Carrique F, Ruiz-Reina E. J Colloid Interface Sci 2010;345:538.[47] Carrique F, Ruiz-Reina E, Arroyo FJ, Delgado AV. J Phys Chem B 2010;114:6134.[48] Arroyo FJ, Carrique F, Ruiz-Reina E, Delgado AV. Colloids Surf A 2011;376:14.[49] Kuwabara S. J Phys Soc Jpn 1959;14:527.[50] Mori T, Suma K, Sumiyoshi Y, Endo Y. J Chem Phys 2011;134:044319.[51] Delgado AV, González-Caballero F, Arroyo FJ, Carrique F, Dukhin SS, Razilov IA.

Colloids Surf A 1998;131:95.[52] Baygents JC, Saville DA. J Colloid Interface Sci 1991;146:9.[53] Dukhin SS, Shilov VN. Dielectric phenomena and the double layer in disperse

systems and polyelectrolytes. Wiley; 1974.[54] Shilov VN, Borkovskaja YB, Dukhin AS. J Colloid Interface Sci 2004;277:347.[55] O'Brien RW. J Fluid Mech 1988;190:71.[56] O'Brien RW, Canon DW, Rowlands WN. J Colloid Interface Sci 1995;173:406.[57] Grosse C, Shilov VN. J Colloid Interface Sci 2000;225:340.