Electrophoresis and Dielectric Dispersion of Spherical Polyelectrolyte

Article pubs.acs.org/Langmuir

Electrophoresis and Dielectric Dispersion of Spherical Polyelectrolyte Brushes Silvia Ahualli,† Matthias Ballauff,‡ Francisco J. Arroyo,§ Á ngel V. Delgado,*,† and María L. Jiménez† †

Department of Applied Physics, School of Sciences, University of Granada, 18071 Granada, Spain F-12 Soft Matter Functional Materials, Helmholtz-Zentrum Berlin, 14109 Berlin, Germany § Department of Physics, School of Experimental Sciences, University of Jaén, 23071 Jaén, Spain ‡

S Supporting Information *

ABSTRACT: Spherical polyelectrolyte brushes (SPBs) consist of a rigid core on which polyelectrolyte chains are grafted in such a way that in certain conditions (low ionic strength and high charge of the chains) the polymer chains extend radially toward the liquid medium. Because of the hairy-like structure of the polymer brushes, the typical soft-particle approach used for explaining the behavior of polyelectrolyte-coated particles must be modified, using the assumptions that the density of charged segments in the polymer chains decreases with the squared distance to the rigid core surface and that the same happens to the friction between the brushes and the surrounding fluid. Interest in clarifying the electrokinetics of these systems is not just academic. It has recently been found experimentally (Jiménez et al., Sof t Matter 2011, 7, 3758−3762) that the response of concentrated suspensions of spherical polyelectrolyte brushes in the presence of alternating electric fields shows a number of unexpected features. Both dielectric and dynamic electrophoretic mobility spectra (respectively, dependences of the electric permittivity and the AC electrophoretic mobility on the frequency of the applied field) showed very special aspects, with giant values of the mobility and an unusually strong dielectric relaxation in the kHz region. In the present paper we give a full account of the electrodynamics of such systems, based on a cell model for describing the hydrodynamic and electrical interactions between the particles. It is found that the lowfrequency dynamic mobility of SPBs is much higher than that of rigid particles of comparable size and charge, making any interpretation based on zeta potential estimations of very limited applicability. The very characteristic feature of SPBs in concentrated suspensions, namely, the enhanced alpha relaxation, can be explained by considering an adequate description of the field-induced perturbations in the counterion and co-ion concentrations, well developed both outside and inside the soft layer in the case of brush-coated particles. It can be also pointed out that the dynamic electrophoretic mobility of SPBs increases with the volume fraction of particles, as a consequence of the large thickness of the brush. Predictions are also shown for the effects of friction coefficient and charge of the polyelectrolyte layer. The results compare well with experimental spectra of the dynamic mobility and electric permittivity of moderately concentrated suspensions of SPBs consisting of a 50 nm polystyrene core with grafted poly(styrene sulfonate) chains some 140 nm in length.

I. INTRODUCTION We have recently witnessed an increase in the use of polyelectrolyte layers for coating rigid particles for a number of applications,1−6 for instance, for providing stability to dispersed particles in high ionic strength solutions, where electric double layers (EDLs) are strongly screened, and electrostatic repulsions are largely diminished. In other cases, such layers constitute a biodegradable or biocompatible coating, making the resulting structures suitable for drug transport and controlled release. In spite of such an apparently simple description, these structures present a rich phenomenology and complicated physics. In fact, many models rely on simplifications (such as planar interface, low Donnan potentials, thick homogeneous layers), and numerical methods appear necessary for a full understanding of the electrokinetics of these systems, as noted recently by Barbati and Kirby.7 If the polyelectrolyte © 2012 American Chemical Society

chains are long and sufficiently charged, they coat the core, forming a distribution similar to a radial brush emerging from the surface, a structure known as spherical polyelectrolyte brush (SPB), as shown in Figure 1. This conformation is possible if electrostatic repulsions keep the chains away from each other, and hence it will be most likely present in conditions of low ionic strength. Screening of such repulsions if the electrolyte concentration is high may lead to chain folding and a tendency toward rigid-like behavior.6−11 The charge distributed along the chains as a consequence of the dissociation of their ionizable groups is compensated for by counterions in solution. In the case of highly charged Received: June 19, 2012 Revised: October 25, 2012 Published: October 30, 2012 16372

dx.doi.org/10.1021/la302483e | Langmuir 2012, 28, 16372−16381

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which the density of charged segments goes smoothly to zero as one approaches the outer limit of the layer.17 This can be considered as a limiting case of our calculations, in which the curvature is arbitrary, so that the density will change with the distance to the surface, the polymer and core charges are arbitrary, and, more important, the volume fraction can be moderate, leading to interactions between brushes from neighbor particles. In the work mentioned,17 Ohshima finds a clear elevation of the electrophoretic mobility of polyelectrolyte-coated particles, a characteristic of the electrokinetics of soft particles which is magnified in the case of determination in the presence of alternating fields, as will be shown below. In this paper, we give full details of our model for the dynamics of SPBs under the action of an alternating electric field, considering in addition the possibility of highly concentrated suspensions, in which compression of the brushes of neighbor particles must be contemplated. The predictions of the model will be described, with emphasis on the features of the frequency dependences of both the electric permittivity and the dynamic mobility for different charge densities and friction coefficient of the brush, and a wide range of volume fractions of particles. We will first describe (section II) the basic equations of the model and provide indications on the methods used to solve them. In section III we will summarize the results, in terms of the effects on the dynamic mobility and the electric permittivity of the particle volume fraction, drag coefficient, and polymer charge inside the soft layer. Comparisons with experimental results will also be provided.

Figure 1. Schematic representation of the structure of a spherical polyelectrolyte brush. Rc is the core radius, LB is average length of the polymer chains, b is cell radius defined in eq 1, and d = b − Rc is the distance between the core surface and the cell boundary.

counterions6 their binding to the chains can be so intense that such ions are immobile, making the liquid layer inside the brushes essentially stagnant and leading to an electrodynamic behavior similar to that of a rigid particle with radius Rc + LB, where Rc is the radius of the particle’s core and LB is the thickness of the brushes (Figure 1). If counterions are monovalent, the weaker interactions with the chain charged groups will make them prone to diffusive migration, and hence, a sort of EDL will be created in the brush volume. The liquid in the latter can be set into motion by tangential electric fields, much the same as in the electro-osmosis phenomenon in the vicinity of rigid particles.12 This effect on the electro-osmotic displacement of liquid will equally manifest in its reciprocal phenomenon, electrophoresis, and, in general, in any process where the EDL polarization is present, as can be said of any electrokinetic phenomenon. This was considered in our previous contribution,13 where the AC electrophoretic mobility and the dielectric dispersion of suspensions of SPBs were experimentally investigated. Roughly, their electrokinetic response was found to be highly amplified by the presence of the charged brush, and in addition, the effect of the volume fraction of particles, ϕ, on the response to AC fields was equally unexpected. For these reasons, consideration of the electrokinetics of such systems cannot be based on a simple elaboration of that of soft particles, i.e., particles consisting of a rigid core and a homogeneous deformable shell.14,15 The understanding of the experimental results led us to modify the theory of soft particles considering a specific distribution of the volume charge density of the polymers and of the friction of the liquid against them (a very general treatment was provided by Duval and Ohshima16): an r−2 dependence is assumed for both quantities. It is proposed that, because the charge is not homogeneously distributed in the SPB, EDL polarization inside the brush volume can occur, and in addition, the large charge density of the polymer brush leads to anomalously large dielectric relaxation amplitudes, characteristic of SPB suspensions. It must be noted that the same type of polyelectrolyte configuration adsorbed on a planar interface would lead to a constant density of polymer segments up to the layer limit: this can be modeled as a step function, which Ohshima calls “hard” in distinction to a soft step function, in

II. EQUATIONS OF THE MODEL Cell models have been used in the last years to describe the electrokinetics of concentrated systems of colloidal particles.18−20 Although it is a mean field approach, it is also an effective way to solve such a complicated problem.21−25 The basis of the cell method is the consideration of a representative spherical cell for the system, in which a single core particle (also spherical) is surrounded by the same electrolyte solution as in the bulk of the real system (Figure 1). The relation between the radius of the core, Rc, and the radius of the cell, b, is chosen in such a way that the volume fraction of solids in the cell is the same as in the overall suspension ϕ=

Rc3 b3

(1)

The key point of this model is that the hydrodynamic and electrical interactions between neighbor particles are simulated by a proper choice of boundary conditions for electric potential, fluid velocity and ionic concentration on the cell surface (r = b). More details can be found in ref 14. Poisson’s equation for the electric potential, Ψ, takes different forms for the region inside the polyelectrolyte layer and outside it, so as to take into account the volume charge density of the layer, ρpol ⎧ N ez n ρ ⎪−∑ j j − pol R c < r < R c + L B εmε0 ⎪ ⎪ j = 1 εmε0 ∇2 Ψ = ⎨ N ez n ⎪ j j −∑ Rc + LB < r < b ⎪ ⎪ ε ε j=1 m 0 ⎩

(2)

where εm is the relative electric permittivity of the solution, ε0 is the permittivity of the vacuum, e is the electron charge, and zj 16373


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and nj are the valence and concentration of ion j (j = 1, ..., N), respectively. Because of the particular distribution of polyelectrolyte chains (Figure 1), we consider here that the volume charge density associated to them decays with the inverse of the square of the radial distance, in such a way that the integral of the charge density in the volume occupied by the brushes is the total charge of the polymer, Qpol. The mathematical expression for the density of brushes we propose here is ρpol =

Q pol 1 4πL B r 2

vj = u −

(3)

(9)

where the form of the spherical components of the velocity is justified by the axial symmetry (no vφ component) and by the linearity with the field and by the continuity equation (the condition ∇·v = 0 is automatically fulfilled by that choice; details can be found in ref 27). Boundary conditions coming from physical constraints have to be established on the surface of the rigid core, at the brush/ solution interface, and on the cell boundary.18,21,24,28,29 a. On the Rigid Particle Surface. The electric potential must satisfy the Gauss law

(4)

−εmε0

dΨ(0) dr

= r = R c+

Qc 4πR c 2

(10)

where Ψ0(r) is the equilibrium potential distribution and Qc is the core charge. The nonslip condition (v(R+c ) = 0) for the fluid on the surface leads to

(5)

h(R c+) = 0 dh dr

=0 r = R c+

(11)

In addition, the particle is also impenetrable to ions vj(r) ·r|̂ r = R c+ = 0

(12)

so that dϕj

= −∇·(nj vj)

(8)

⎛ 2h(r ) ⎞ 1 d(rh(r )) v = (vr , vθ , vφ) = ⎜ − Ecos θ , E sin θ , 0⎟ ⎝ ⎠ r r dr

where λ0 = (3πσg)1/2, and σg is the grafting density; this expression assumes that the friction sites are spherical and homogeneously distributed along the chains.26 In addition, the fact that the density of charged segments can be very high close to the core surface suggests that the diffusion coefficient of counterions should also be affected by a similar r dependence:7 if D0 is the value of any diffusion coefficient far from the interface, the dependence D(r) = D0(r2/(Rc + LB)2) is proposed when the packing density is very high (as it happens with our experimental data, see below). Finally, we consider the continuity equations for each ionic species

∂t

(7)

where the time dependence will be identical to that of the field, and ξ(r) equals ψ(r), ϕj(r), and p(r), when Ξ(r) refers respectively to the electric potential Ψ(r), the chemical potential function Φj(r), and the pressure. Similarly, the velocity will have the following dependence:23

In eq 4, δm is the mass density of the liquid medium, ηm is the viscosity of the latter, and P is the pressure field. The velocity V of the particle with respect to the laboratory frame must be introduced to account for the fact that v is the fluid velocity with respect to the particle. Since the drag force increases with the number of friction sites,25 it is logical to assume that the drag on the liquid will also follow an r‑2 decay, as the polyelectrolyte charge density does (eq 3). In addition, it is usual to characterize the drag force by the friction parameter λ = (γ/ηm)1/2 (in turn, the inverse of the Brinkman screening length). Hence, we can write that

∂nj

∇μj

δ Ξ(r) = ξ(r )E·r ̂

⎧ ⎪−∇P + η ∇2 (v + V) R c < r < R c + L B m ⎪ N ⎪ − ∑ ez n ∇Ψ − γ v j j ⎪ j=1 ∂(v + V) ⎪ δm =⎨ ∂t ⎪ 2 ⎪−∇P + ηm∇ (v + V) R c + L B < r < b N ⎪ ⎪ − ∑ ezjnj∇Ψ ⎪ j=1 ⎩

R λ = λ0 c r

NAe 2|zj|

where Λ∞ is the limiting equivalent conductance of the j corresponding ionic species, NA the Avogadro number and μj = −zjeΦj = μ∞ j + ezjΨ + kT ln nj the electrochemical potential. From the set of differential coupled eqs 2, 4, and 6, we can obtain the fluid velocity distribution and the profiles of ion concentration, electric potential profile, and pressure at any time in the cell. The numerical solutions are obtained by considering only the linear terms with respect to the applied field. In addition, taking into account the spherical symmetry of the problem, the perturbations induced by the alternating (AC) electric field, E exp(−iωt), on the quantities of interest are of dipolar type, and we can write the general form of such perturbed quantities as follows:

The incompressible fluid velocity v(r,t), governed by Navier−Stokes equation, must also be written for the two regions. This is because in the polymer brush an additional drag force, −γv, is exerted by the polyelectrolyte chains on the interstitial fluid

∇·v = 0

Λ∞ j

dr

(6)

where convective, diffusive, and electromigration contributions to the velocity vj of the jth ionic species are taken into account

= 0, r = R c+

j = 1, ..., N (13)

Finally, the continuity of both the potential and the normal electric displacement implies 16374


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− r = R c+

εc ψ (R c ) = 0 εmR c

boundary. Hence instead of using the typical boundary condition of zero perturbation of ionic concentrations on the surface r = b, we impose that the derivatives of these quantities are zero.24

(14)

where εc is the relative permittivity of the core. b. At the Brush/Solution Interface (r = Rc + LB). In this case, we have to consider the continuity of the potential, the normal displacement vector, the fluid velocity, the ion concentration and the radial component of the ion velocity. In terms of the auxiliary functions these can be written −

∂δnj(r) ∂r

= r = R c+ LB

−

dψ dr

r = R c + L B+

h(R c + L B−) = h(R c + L B+) dh dr

= r = R c + L B−

dh dr

ue =

r = R c + L B+

dr

= r = R c + L B−

r = R c + L B+

(15)

zj 2e 2

(16)

r = R c + L B+

r = R c + L B−

(17)

c. Conditions on the Outer Boundary of the Cell. For the equilibrium problem, the electroneutrality of the cell can be written, using the Gauss law =0 r=b

(21)

(22)

(18)

The set of conditions regarding the average values of the perturbations of potential and pressure can be written, respectively ⟨−∇δ Ψ(r)⟩ = E ⇒ ψ (b) = −b

(19)

⟨∇δP(r)⟩ = 0 ⇒ p(b) = 0

(20)

dψ dr

N

dϕj dr

− r=b

2h(b) ∑ zjen(0) j (b) j=1

b

r=b

(23)

(24)

III. RESULTS AND DISCUSSION a. General Features of the Model. The general behaviors of the spectra of the dynamic mobility and the permittivity of SPB suspensions are far from trivial. Because of that, very valuable information can be obtained on the particle characteristics: core charge and size as well as charge, density, and size of the polymer brush. In fact, the AC electrokinetics of hard (rigid), soft (composed by a rigid core plus a homogeneous polymer shell), and brush particles are significantly different for otherwise comparable conditions. Figure 2 and the S1 constitute proofs of this. Figure 2 displays the real (a) and imaginary (b) components of the dynamic mobility of suspensions of the three kinds of particles, as a function of frequency, for frictionless polyelectrolyte layers. In these and subsequent representations, the general parameters used are detailed in Table 1. It can be seen that the dynamic mobility trends are similar in all cases and that the mobility spectra of SPBs are qualitatively similar to those of either rigid21,30,31 or soft14,28 particles. We also point out that the increase in the concentration ϕ of either soft particles or SPBs is assumed to provoke a compression of the polymer layer, and this is implemented by specifying that the thickness of the brush is set equal to (d − 2 nm = b − Rc − 2 nm): increasing the volume fraction (eq 1) means decreasing b (hence decreasing d ≡ b − Rc) and consequently reducing the average polymer chain length. Note that the low-frequency mobility of the rigid (or hard) particles is much lower in absolute value than that of either the

From the continuity of pressure we obtain the following equality:

⎡ ⎛ iωδm ⎞⎤ d = ⎢ ⎜⎜Lh + h⎟⎟⎥ ηm ⎠⎥⎦ ⎢⎣ dr ⎝

r=b

where δc is the core mass density, K* is the complex conductivity, λj = [(|z|e2NA)/Λj∞], and ε′ and ε″ are respectively the real and imaginary components of the complex relative permittivity of the suspension (ε* = ε′ + iε″).

Lh|r = R c + LB− = Lh|r = R c + LB+

⎡ ⎛ ⎤ ⎛ dh iωδm ⎞ h⎞ ⎢ d ⎜⎜Lh + + ⎟⎥ h⎟⎟ − λ 2⎜ ⎝ dr ηm ⎠ r ⎠⎥⎦ ⎢⎣ dr ⎝

r=b

dψ dr

K * = K *(ω = 0) − iωε0ε* = K *(ω = 0) + ωε0ε″ − iωε0ε′

d2 2 d 2 + − 2 2 r dr dr r

λj

n(0) j (b)

+ iωεmε0

∇ × v|r = R c + LB− = ∇ × v|r = R c + LB+

dΨ(0) dr

∑ j=1

Also, the components of the hydrodynamic stress tensor (σ̃H = −pI ̃ + η(∇v + (∇v)T) are continuous at the interface, leading to

L≡

=−

2h(b) 1 b 1 + ϕ δc − δm δ N

K* =

dϕj dr

dr

m

ϕj(R c + L B−) = ϕj(R c + L B+) dϕj

r=b

dϕj

From this set of differential equations and boundary conditions, it is possible to calculate the dynamic or electrophoretic mobility ue of the particles and the relative electric permittivity ε* of the suspension when it is subjected to the AC field of frequency ω. The details are in ref 14. The macroscopic complex quantities of interest, namely the dynamic mobility and the electrical permittivity, can be written as

+

ψ (R c + L B ) = ψ (R c + L B ) dψ dr

=0⇒

where the brackets denotate the average of the quantity in the cell volume. As the brush length is typically larger than the core radius and the suspensions considered are concentrated, the limits of the polyelectrolyte layer are typically close to the cell 16375


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a homogeneously (soft) or inhomogeneously (brush) coated particle. These are compared to the potential profile around an uncoated rigid core with a radius equal to Rc + LB. The charge of the hard particle is chosen such that the mobility is the same in the three cases (ue = −3 × 10−8 m2/(V s)), and the drag coefficient of the soft and brush particles is supposed to be large enough to make them hydrodynamically equivalent to the rigid ones. The experimental mobility is obtained assuming Qpol = −5 × 10−16 C, but we need to double this charge to reach the same mobility with the hard particle model. This is a consequence of the largely different potential profiles in the three cases, illustrated in Figure S1: as an example, the zeta potential of the rigid particle (∼−80 mV) is far from realistic values of the potentials involved in the actual core/shell particles. Hence, the concept of zeta potential (even if denominated “effective” or “equivalent”) is largely meaningless when dealing with the latter particles). Figure 2 exhibits the inertial relaxation and, in some cases, the Maxwell−Wagner−O’Konski (MWO) process. The latter (increase in the absolute value of the real component, minimum in the imaginary part) occurs at comparatively low frequencies (of the order of a few MHz): it is the manifestation of the decrease in the dipole moment of the particle when the MWO relaxation frequency is surpassed; the ions cannot follow the field oscillations along distances long enough for the interface to get polarized.19,32 It is worth to mention that the MWO relaxation is apparently absent in the case of soft particles: the presence of a thick uniformly charged polyelectrolyte layer in this case does not allow a double layer polarization. Another process is observed at higher frequencies: this is called inertial relaxation, and it corresponds to the tendency of the mobility to zero at frequencies around f in = 2 × 108 Hz. It is due to the fact that the particle and fluid inertia do not allow them to respond to the field oscillations. The decrease of the mobility with volume fraction is also a known fact, and it comes from the braking effect of neighbor particles interacting both hydrodynamically and electrically with any given one.32 As shown in Figure 3, the dielectric spectra of the three kinds of particles considered are very different. Both rigid particles and SPBs exhibit enhanced permittivities at low frequencies and two relaxations in the 104−107 Hz frequency range. Previous results33 indicate that the high frequency relaxation is the MWO process, whereas the low-frequency (and much more noticeable) one is the α or concentration polarization relaxation. Such relaxations are indicative of systems with high charge and broadly developed diffusion processes. On the other hand, larger relaxation amplitudes are predicted for both α and MWO relaxations in SPBs than in rigid particles suspensions. This can be understood on a more quantitative basis if the profiles of the real part of the ion concentration perturbations, Re[δni], are plotted in some representative cases, as in Figure S2. Here, a comparison can be established between the effect of the field on the concentration polarization of the EDLs of hard, soft, and SPB particles. This figure demonstrates that a gradient of neutral electrolyte concentration is established around the three kinds of particles, with the high concentration sides on the right poles of the clouds. However, the characteristic dimension and the magnitude of the gradient is significantly different in the three cases. Because in the case of the soft layer the equilibrium space charge distribution is homogeneous, the difference between the transport number of co-ions and

Figure 2. Real (a) and imaginary (b) parts of the dynamic mobility of SPBs as a function of frequency for different particle volume fractions. Core radius: 50 nm; polymer layer thickness: (d − 2 nm) (d is defined in Figure 1); ionic strength: 5 × 10−4 mol/L KCl; dimensionless friction parameter λ0Rc = 0; total charge of the polymer layer: Qpol = −5 × 10−16 C; total charge on the rigid particle: Qc = 0. Solid lines: hard particles; open symbols: soft particles; full symbols: SPBs. Single arrow: MWO relaxation; double arrow: inertial relaxation.

Table 1. Values of the Quantities Used in the Calculations of Dynamic Mobility and Permittivity of the SPB Suspensions variable

value

T εm εc δm δc ηm z1 z2 Λ∞ 1 Λ∞ 2 Qc

298 K 78.5 2 997 kg/m3 103 kg/m3 0.8904 mPa·s 1 −1 73.5 × 10−4 S·m2/eq 76.3 × 10−4 S·m2/eq 0

soft or brush-coated ones, the differences amounting to almost a factor of 10 at low volume fractions. This is a manifestation of the existence of distributed charge throughout the polyelectrolyte layer volume. Let us consider the plots in Figure S1, where the electric potential is plotted as a function of the dimensionless radial distance κr (where κ is the inverse of the Debye length) around 16376


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Figure 4. Imaginary part of the relative permittivity of the suspensions of soft particles (open symbols) and SPBs (full symbols) considered in Figure S3 (bottom).

limit of the layer to the cell boundary decreases up to a critical volume fraction ϕcr = 0.06 above which the layer starts to be compressed. This shrinking is modeled by assuming that the layer thickness is (d − 2 nm). This assumption comes from the fact that the repulsion between neighbor brushes becomes comparable to that of the chains of the same particle when interparticle distances are smaller than the EDL thickness (around 13 nm for 0.5 mM salt concentration), and forces the chains to shrink. To a first approximation, it can be considered that the brush structure is not perturbed for cell sizes larger than Rc + LB + 2 nm, but that it is compressed for larger particle concentrations. At the onset of compression and beyond, LB = d − 2 nm. Our calculations (Figure 5) show the unexpected result (not found in rigid particles, Figure 2) that the dynamic electrophoretic mobility of SPBs increases with volume fraction when some critical value of this quantity (smaller when the charge increases) is surpassed. This fact was confirmed by experimental results as Figure 7 shows. Here, we present values of the dynamic mobility obtained with the ESA (Electrokinetic Sonic Amplitude) technique, using an Acoustosizer II (Colloidal Dynamics, U.S.A.). The samples were suspensions of SPBs consisting of a polystyrene rigid core 50 nm in radius and poly(styrene sulfonate) chains extending 169 nm from the core surface in water or 143 nm in dilute electrolyte solution, as deduced from dynamic light scattering data reported in refs 9 and 13. At the volume fraction values shown in Figure 7, the chains occupy the available volume between the core and the cell surface, since, as abovementioned, they shrink when the SPBs concentration increases above the estimated critical mass fraction (CMF) of 2.8%. In addition, data in Figure 5 show that both the α and MWO relaxations are clearly visible at low volume fractions and become almost unobservable above the critical concentration. At low ϕ, we observe two successive steps in the mobility at frequencies around 50 kHz and 10 MHz due to and MWO processes, before the inertial decay. Both processes are apparently absent for the largest volume fractions for different reasons. It is well-known22 that the amplitude of the former relaxation decreases with volume fraction beyond a certain value, depending on the surface charge. In addition, the MWO dispersion is shifted to higher frequencies and merges with the inertial decay, giving rise to an almost constant mobility in a

Figure 3. Real (a) and imaginary (b) parts of the permittivity increment (eq 24) as a function of frequency for different particle volume fractions. Particle radius: 50 nm; polymer layer thickness: (d − 2 nm); ionic strength: 5 × 10−4 mol/L KCl; dimensionless friction parameter: λ0Rc = 0; total charge of the polymer layer: Qpol = −5 × 10−16 C. Solid lines: hard particles; open symbols: soft particles; lines + symbols: SPBs (lines are a guide to the eye). The arrows indicate the α-relaxation processes.

counterions along the radial coordinate is smaller than in the case of the SPBs, and as a consequence both the concentration of perturbed ions and the region where diffusion is maximum (limit of the space charge region) are smaller. The situation of the ionic atmosphere of SPBs is radically different: since the perturbation is not forced to go to zero at the cell boundary (only its gradient, eq 21), the cloud of neutral salt can be well developed outside the soft layer. Also, since charge is not homogeneously distributed inside the brush, the cloud of neutral electrolyte can be formed even inside the polyelectrolyte brush. This leads to a considerable concentration polarization and, from it, to a large increase in the lowfrequency permittivity, as in fact has been found experimentally.13 The model predictions shown in Figure 4 confirm this fact very clearly. The effects of increasing polyelectrolyte thickness on the dielectric spectra of soft particles and SPBs are opposite: contrary to soft model predictions, the amplitude of the relaxation increases when LB rises in the case of SPBs. b. Effect of Volume Fraction. With the aim of making theoretical predictions as close as possible to the situation suggested by experiments on similar systems,34,35 we will consider a 100 nm thick polyelectrolyte layer and assume that as the volume fraction increases the empty distance from the 16377


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Figure 5. Real (a) and imaginary (b) parts of the dynamic mobility of SPBs as a function of frequency for different particle volume fractions. Core radius: 50 nm; polymer layer thickness: 75 nm up to 6% volume fraction, and (d − 2 nm) beyond this value; ionic strength: 5 × 10−4 mol/L KCl; dimensionless friction parameter: λ0Rc = 0; total charge of the polymer layer: Qpol = −1 × 10−15 C.

Figure 6. Real (top) and imaginary (bottom) components of the relative permittivities of the suspensions described under Figure 5.

in the cases considered in Figure 5. Note that the electric permittivity exhibits a larger relaxation due to the α process than that of MWO, whereas inertia is not observable in dielectric dispersion. It is worth noting that the permittivity can be almost four times larger than that of the supporting aqueous solution, in agreement with our experimental results as we show in Figure 7 (details in ref 13). c. Effect of the Friction Coefficient and Polymer Charge. The values of the dimensionless friction coefficient (λ0Rc) are, together with those of the charges of both the core and the polymer (to be considered below), determinant of the physical properties of the nanostructure. Recall that the larger λ0Rc the stronger the friction of the liquid inside the polyelectrolyte layer, either soft or brush. With this, it is expected that, as shown in Figure S3, the dynamic mobility is strongly decreased by the increase in λ0Rc, no matter if the space charge is homogeneously distributed (soft particles) or decreases radially (SPBs). However, since in the SPB model the drag is not homogeneous inside the brush layer, the fluid motion is as a whole less affected by changes in the layer friction, and hence the mobility decrease is not as deep as in the soft particle model. This is the reason why we observe that homogeneous soft layer models applied to brush improperly account for the drag force. This is another example of the importance of considering a detailed structure of the polymer layer in order to correctly characterize the brush. The differences between both kinds of particles regarding the characteristic relaxation frequencies of the MWO and inertial dispersions can be better appreciated if the imaginary parts are

wide frequency range. The fact that the MWO relaxation frequency moves to the high-frequency side of the spectrum is a manifestation of the EDL overlap: as the particle concentration increases, the effective MWO polarization distance is shortened and its frequency rises in turn. The most relevant result in the above study of volume fraction effects is the increase of the mobility with particle concentration (Figure 5), a result never found either theoretically or experimentally for hard spheres or soft particles with a thin soft layer. As already noted in ref 13 and as it can be seen in Figure 7, where such an increase was reported for the first time, this is the result of the balance between two counteracting effects. On one hand, the increase in particle concentration leads to stronger electric and hydrodynamic interactions that tend to reduce the mobility; on the other hand, such an increase in the volume fraction (reduction in the cell radius b) forces a larger portion of liquid to move in the polyelectrolyte region (the region outside the brush is thinner), where convection is favored by counterions. In addition, when shrinkage of the brush (for volume fractions above 6%) is established, the counterion concentration inside the layer will increase, producing a concomitant rise in the electro-osmotic fluxes. The effect of the friction coefficient of the layer on these fluxes will be discussed in the following section. The α relaxation can be better explored through the dielectric spectra. Figure 6 show the results of our calculations 16378


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parameter. It can be suggested that this is a manifestation of the fact that as the parameter increases the polymer layer becomes stiffer (from the point of view of liquid penetration), tending to become comparable to the core in this respect. This brings about an apparent increase in the volume fraction of “solids” (core plus polymer layer) and hence a tendency toward higher inertial frequencies (as in purely rigid particles21). This also justifies the appearance of two relaxation frequencies in the inertial region (as shown in the inset in Figure S4): one could be related to the core and the other to the increasingly rigid brush layer. The effect of the friction parameter on the permittivity is almost negligible and will not be shown. On the contrary, one can expect that this quantity will be strongly affected by the charge in the polymer layer. Like in other electrokinetic phenomena, the electric charge present at the interface will be determinant of the observed behavior. Note in Figure S5 that the SPBs display the two expected trends in their mobility: the MWO increase is more significant if the polymer charge is raised, and the lowfrequency mobility behaves the same way. It is important to observe the extremely high values that the mobility can attain at the maximum prior to the inertial relaxation. The dependence of the permittivity on the polymer charge is, as mentioned, very remarkable (Figure S5). The low-frequency values of this quantity can be extremely high (as also found experimentally, see Figure 7), and in cases of large polymer charge, even the MWO relaxation, hardly observable in the case of rigid particles, is very noticeable. As observed, the α relaxation does not change its position with the amount of charge, whereas the MWO peak tends to higher frequencies, because of the overall increase in the interfacial conductivity. d. Comparison with Experimental Data. At this point, it is worth comparing our theoretical predictions with experimental results we found for concentrated systems of the polyelectrolyte brushes above-mentioned (50 nm rigid polystyrene core, densely coated with long poly(styrene sulfonate) chains). Figure 7a shows the dynamic mobility results, and Figure 7b depicts the frequency spectra of the electric permittivity, obtained from the complex impedance of the sample using a four terminal impedance analyzer HP4284A (Hewlett-Packard, U.S.A.). Our model fits the data reasonably well, considering the complexity of the system and the limited number of adjustable parameters used: particle charge: −5 × 10−16 C; charge of the polyelectrolyte layer plus condensed counterions: −8.5 × 10−14 C, dimensionless friction parameter: λ0Rc = 110; polyelectrolyte layer thickness (in nm): 156, 136, 94, 85, 75, and 55, for volume fractions (%) 2, 3, 4, 5, 6, and 7, respectively. These parameters are not modified for the fittings of the two phenomena considered. It must be pointed out that two independent sets of electrokinetic data have rarely been interpreted using the same physical parameters for both. Of course, as much as the model is complete, the agreement is better. It is not only a matter of data fitting: the physical predictions of our model agree with the experimental findings: the mobility reaches high lowfrequency values (not expected for rigid systems) and increases with volume fraction, the alpha relaxation is broad and giant, and its frequency increases with volume fraction as well. Not only do we observe larger relaxations but also broader ones for coated particles than those typically found in suspensions of hard particles. These experimental results cannot be explained

Figure 7. (a) Symbols: real part of the experimental values of the dynamic mobility of SPBs (polystyrene sulfonate chains grafted on polystyrene core) as a function of frequency for the volume fractions indicated, in 0.5 mM NaCl. (b) Symbols: experimental values of the imaginary component of the relative electric permittivity as a function of the frequency for suspensions of SPBs with the indicated values of volume fraction in 0.5 mM NaCl. The lines in panels a and b correspond to the best-fit to our theoretical predictions, obtained using the following parameters: Particle charge: −5 × 10−16 C; charge of the polyelectrolyte layer plus condensed counterions: −8.5 × 10−14 C, dimensionless friction parameter: λ0Rc = 110; polymer layer thicknesses: 156, 136, 94, 85, 75, and 55 nm for volume fractions ϕ = 2, 3, 4, 5, 6, and 7%, respectively. The inset in panel b) allows the comparison between measured (full symbols, E) and fitted (open symbols, T) relaxation frequencies, fα, as a function of ϕ.

considered, as shown in Figure S4. Note that in this figure we use a different set of parameters in order to exhibit the main features. When the friction is increased, the maximum corresponding to the MWO relaxation is shifted to lower frequencies, whereas the minimum of the inertial decline has the opposite tendency. The former fact can be explained considering that the convective contribution to the surface conductivity of the particle is reduced when λ0Rc increases. Considering that33 2σm ωMWO = (Du + 1) εp + 2εm (25) where Du = κσ/Rcσm is the Dukhin number that relates the surface conductivity, κσ, and the medium conductivity, σm, it is clear that this frequency must be reduced by the increase in λ0Rc. The inertial reduction of the mobility occurs, on the contrary, at higher frequencies the higher the friction 16379


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IV. CONCLUSIONS We have modeled the electrokinetics of concentrated suspensions of particles consisting of rigid cores surrounded by spherical polyelectrolyte brushes, that is, highly charged polymer chains densely grafted onto the core. We have shown that a correct account of the brush structure is necessary in order to properly describe their behavior. The inhomogeneity of the charge distribution at the polymer layer is the key to predict huge dielectric increments at low frequency, and in turn, it is negligible if a homogeneous distribution of the brush is imposed. On the other hand, both the brush thickness and the drag coefficient inside it greatly affect the dynamic mobility and the position of the inertial decay. We have also shown that the combination of electric permittivity and dynamic mobility spectra is a useful tool to determine the main in situ characteristics of the SPB structures. Our results suggest that it is in fact possible to obtain values of such parameters as the polymer charge, the drag coefficient (in turn, the Brinkman length), the spatial distribution of chains, and the size of both core and brush layer. This is especially interesting in the case of having short Brinkman lengths, since in this case the brush thickness cannot be determined by other experimental techniques such as dynamic light scattering. A comparison with experimental data confirms most of the model results.

using the soft model for the coated particles, since in such a case the spectra would be almost flat, with negligible frequency dispersion (Figure 3). This is in agreement with dynamic mobility calculations when a nonvanishing drag coefficient is considered (Figure S3), where it was found that no polarization is possible in the case of these particles, and hence no relaxation is observed. In more detail, we can focus our attention at the very high values of the amplitude of the relative permittivity relaxation (Figure 7b). For instance, for a concentration of 4% the SPB suspensions show a relaxation amplitude above 1900, which is more than ten times higher than that characteristic of a suspension of rigid particles with the same size than the core of the SPBs, namely, 158 (ref.13). These experimental values are rather well fitted with our model, which predict amplitudes of 1740 and 116, respectively. In this and the other cases, a discrepancy is observed in the low-frequency behavior of the permittivity, which may be an indication of the fact that in that frequency range the parasitic effects of the electrode polarization can be very noticeable. This is rather common in the interpretation based on physical models (not just the fitting with relaxation functions) of the dielectric behavior of suspensions of any kind. None of these phenomena is so significant in dynamic mobility determinations below the inertia relaxation, because the very low frequency range is not swept with this technique. The best-fit values of the polymer layer thickness found for the different volume fractions, are consistent with hydrodynamic radius determinations (around 150 nm, as mentioned above) in the most dilute suspensions, with increasingly lower values upon increasing ϕ. This means that chains shrink already when the volume fraction is above 2%. The free counterion charge density that we obtain corresponds to a Manning parameter ξM = 3,36 very close the theoretical one ξM = [(|zpolz|lB)/bpol] = 2.5 where zpol and z are respectively the valences of the charged group in the chain and of the counterion, lb is the Bjerrum length, and bpol is the separation between two neighbor fixed charged sites in the polyelectrolyte chain. Also, the Brinkman screening length on the particle surface that we have obtained from the fitting parameter, λ−1, is 0.5 nm, again similar to the value obtained theoretically of 1 nm. Beyond the ability of the model to properly fit the experimental data, the information that can be extracted from our electrokinetic analysis, namely, the charge of the polymer and the in situ thickness of the chain, is very valuable, and not accessible by other experimental techniques when the systems are as concentrated as in our case. That is the key point of this work: it is very usual to use (DC) electrophoretic mobility determinations in the evaluation of the electrical state of the interface, and eventually in quantifying the electrokinetic charge density. However, this is clearly insufficient in the present case, as theories elaborated for hard particles,37 linking the mobility with some surface (electrokinetic) potential cannot be used in SPBs suspensions, as shown in this work. Even more elaborated treatments, suited for soft particles, are of limited applicability. It can be concluded that a model along the lines described here is actually necessary for finding values of the particle and polymer charge, the layer friction, and its thickness, all for different volume fractions, from a set of experimental values of, for instance, dynamic electrophoresis, dielectric dispersion, or, even better, both.

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ASSOCIATED CONTENT

S Supporting Information *

Figures S1−S7 are provided. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*Telephone: +34958243209. Fax: +34958243214. E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS Financial support by Project P08-FQM-3993 (Junta de Andaluciá and FEDER funds) and MICINN, Spain (Project FIS2010-19493) is greatly appreciated. One of us (M.L.J.) also acknowledges support from Ministerio de Economiá y Competitividad, Spain, for her Ramón and Cajal Contract.

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REFERENCES

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Electrophoresis and Dielectric Dispersion of Spherical Polyelectrolyte

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