Article pubs.acs.org/Langmuir

Electric Permittivity and Dynamic Mobility of Dilute Suspensions of Platelike Gibbsite Particles M. Alejandro González, Á ngel V. Delgado, Raúl A. Rica, María L. Jiménez, and Silvia Ahualli* Department of Applied Physics, School of Science, University of Granada, 18071 Granada, Spain S Supporting Information *

ABSTRACT: In this work we discuss the electrokinetic evaluation of model platelike particles. By model particles we mean homogeneous and controlled size and shape. The electrokinetic analysis in such complex geometries cannot be limited to a single data point as in usual electrophoresis in constant (dc) fields. The information can be made much richer if alternating (ac) fields with a sufficiently wide range of frequencies are used. In this case, two techniques can be applied: one is the determination of the frequency spectrum of the electric permittivity or dielectric constant (low-frequency dielectric dispersion), and the other is the electroacoustics of suspensions and the determination of the frequency dispersion of the electrophoretic mobility (dynamic or ac mobility). In this work, these techniques are used with planar gibbsite (γ-Al(OH)3) particles, modeled as oblate spheroids with a small aspect ratio. As in other laminar minerals, a particular charge distribution, differing between edges and faces, gives rise to very peculiar electrokinetic behavior. It is found that pH 7 approximately separates two distinct field responses: below that pH the dielectric dispersion and dynamic mobility data are consistent with the existence of individual, highly charged platelets, with charge mainly originating on edge surfaces. At pH 4, a low-frequency relaxation is observed, which must originated from larger particles. It is suggested that these are individual ones bridged by negatively charged fiberlike structures, coming from the partial decomposition of gibbsite particles. On the other side of the measured pH spectrum, the overall charge of the particles is low, and this probably produces aggregates with a relatively large average size, with relaxation frequencies on the low side. This is confirmed by dynamic mobility data, showing that a coherent picture of the nanostructure can be reached by combining the two techniques.

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been solved in some cases:4 rigid spheres,5 soft spheres,6 planar interfaces,7 cylinders and spheroids,8,9 typically in dilute suspensions, whereby any particle−particle interactions are considered to be negligible. The analysis of concentrated suspensions of spheres has received attention as well, and this is justified by the fact that such suspensions are probably more technologically interesting.10−12 On the contrary, the electrokinetics of nonspherical particles has been scarcely explored, considering the experimental and theoretical difficulties involved. Shape effects are very often neglected in the investigation of disperse systems, as nonspherical model particles (with controlled geometry, polydispersity, and composition) are not readily accessible, unlike the wide variety of spheres even commercially available. In addition, the theoretical description of electrokinetic phenomena in suspensions of particles with geometry other than spherical (spheroidal, cylindrical, planar), although developed under certain limiting situations in the past, has received attention and an exact description.13−21 For that reason, the scarce existing experimental data have typically been described in terms of

INTRODUCTION An almost universally used set of techniques for the characterization of nanoparticles in suspension is described under the general name of electrokinetic methods.1−3 They are all based on the evaluation of the response of the dispersed system to externally applied fields mostly tending to induce a relative motion between the suspension medium (typically an aqueous solution and less often a nonaqueous liquid) and the nanoparticles (in most cases, solid, but also with different degrees of softness). Although the experimental techniques to be used are more or less accessible in most laboratories, the interpretation of electrokinetic phenomena requires a suitable theoretical treatment, which at the end should provide information about such quantities as the surface (in reality the electrokinetic or zeta) potential (ζ) or charge density, the surface conductivity, the properties of the stagnant (conductive or not) layer on the particles, and eventually even some effective size. A fundamental aspect of the electrokinetics of colloidal particles deals with the frequency of the electric dipole induced by the externally applied electric field, oscillating with the frequency of the field. This response provides a huge amount of information about the solid/liquid interface and even about the microstructure of the system. The problem is old, and it has © 2015 American Chemical Society

Received: April 6, 2015 Revised: July 2, 2015 Published: July 2, 2015 7934

DOI: 10.1021/acs.langmuir.5b01136 Langmuir 2015, 31, 7934−7942

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We are referring to the field-induced concentration gradient of neutral electrolyte on both sides of the particle along the direction of the field. This polarization induces large diffusion electric currents around the particles, lagging behind the electric field, and these displacement currents are macroscopically observed as an increased electric permittivity.7,25 The main finding in the analysis of the dielectric dispersion of nanoparticles suspensions is that the low-frequency permittivity undergoes a significant relaxation in the few-kHz frequency range, also present in the induced dipole moment (Figure S2): this is the alpha relaxation, with frequency ωα, well described by Dukhin and Shilov.7,26 When the frequency is below the indicated range, the phenomenon of concentration polarization can take place, as there is time for ions to move a distance on the order of a particle radius between two field oscillations. At still higher frequencies, the Maxwell−Wagner or Maxwell−Wagner-O’Konski relaxation is observable. Below its corresponding characteristic frequency, ωMWO, counterions can arrange themselves to polarize the double layer at distances on the order of an EDL thickness due to the different conductivities of the EDL itself and the medium. More information can be found in section 2 of the SI. One of the mathematical complications of the solution in the case of geometries other than spherical is precisely the description of the polarization of the EDL of the particles in the presence of an alternating electric field. This has been solved by Shilov et al.,18,27 by Fixman,8 and by Mishchuk28 for several geometries (see section 3 of the SI). Unfortunately, experimental tests of these descriptions are very scarce.20,29−32 In addition, in a series of theoretical works by Loewenberg and O’Brien14,15 an approximate theory was proposed for the evaluation of the dynamic mobility of dilute suspensions of charged colloidal spheroids and cylinders, valid for axial ratios (r = a/b; here a is the semiaxis parallel to the axis of revolution and b is the semiaxis in the perpendicular direction, see Figure 1) in a wide range, including prolate (r > 1) and oblate (r < 1) spheroids. Furthermore, it is assumed that the EDL is thin, i.e., its thickness (the Debye length κ−1) is much smaller than the smallest particle dimension. A detailed account of this model is described in section 3 of the SI. An additional complicating factor regarding these particles is that they are typically inhomogeneously charged. The case of platelike clay particles is most typical:30,33 a clear distinction can be established between the face surfaces (pH-independent and negative) and edge surfaces of the plates (typical oxide surfaces with a point of zero charge around pH 7). The interactions between similar or dissimilar surfaces can produce complex structures whose electrokinetic behavior is even more unpredictable. In spite of the complexities mentioned, we aim in this paper at contributing information on the DD and the dynamic mobility of nonspherical particles, with approximately planar shape, which will be modeled as strongly anisometric oblate spheroids with gibbsite (γ-Al(OH)3) composition. We will consider these particles because they can be prepared in a controlled way in the form of planar units, quite stable and with a well-characterized chemistry. We will focus on how the pH and ionic strength affect their electrokinetic response and to what extent such a response can give clues regarding their intrinsic surface charge heterogeneity and their subsequent structural organization in dilute suspensions.

qualitative models or under the assumption that an equivalentsphere formulation suffices, something strictly valid only in the so-called Smoluchowski limit (low surface conductivity and electric double layer, EDL, thickness much smaller than the particle radius7). Notwithstanding these limitations, correct characterizations are mandatory for many applications, in which an exact knowledge of the electrical state of the particle surface, and from this the electrostatic interaction between particles, is a necessary requirement. From the collection of electrokinetic techniques available, electrophoresis is probably the most widely used.1 The existence of quite a number of instruments commercially built and of sufficiently detailed theoretical treatment can suffice to explain it. However, when it comes to the effect of shape or size, its sensitivity is limited. As an example, in Figure S1 of the accompanying Supporting Information (SI hereafter), the theory of O’Brien and Ward13 has been applied to the calculation of the deviation between the electrophoretic motilities of spheroids and spheres of the same radii, for different axial ratios and zeta potentials. Note that the differences are hardly larger than 10%, except for extreme conditions (low axial ratios and high zeta potentials). This result indicates that just electrophoretic mobility measurements are not informative enough about geometrical effects on the electrokinetics of particles. Fortunately, the sensitivity to the geometry of the particles (in addition to their electrical surface properties) is increased to a large extent when the applied fields are harmonically alternating in a suitable frequency range. The electrical double layer polarization processes are extremely dependent on the frequency of the field. There are three electrokinetic techniques in which the frequency variation of the external action is in their very essence. Two are typically recognized as electroacoustic ones: they are characterized by the fact that either an alternating field induces an ultrasound pressure wave or, conversely, a pressure wave induces an alternating field. In the first case, we call the phenomenon ESA (electrokinetic sonic amplitude); in the second one, the phenomenon is designated CVP or CVI (colloid vibration potential or current).22−24 As a result, it is possible to evaluate the frequency spectrum of the so-called dynamic electrophoretic mobility, ue, a complex quantity that can be considered to be the alternating current (ac) counterpart of the direct current (dc) or standard electrophoretic mobility, as discussed above. Interestingly, the electroacoustic response is a collective one, and the measurements can be carried out without the need to dilute the sample and hence alter its state. In addition, the existing experimental techniques provide significant information on the particle size distribution, making use of the high-frequency relaxation of the mobility or the attenuation of an acoustic wave through the suspension. The mobility spectrum is determined by the properties of the particle itself (such as size, shape, chemical composition, and surface charge) and by the polarization state of the EDL. The third technique, although strictly not an electrokinetic one, is equally informative: it is based on the evaluation of the dielectric dispersion (DD) or frequency dependence of the dielectric constant of the suspensions. In this case the frequency dependence of the EDL polarization is directly probed by measuring the complex permittivity of the disperse system (actually, by measuring its complex impedance). The main contribution to the dielectric relaxation processes in suspensions comes from the concentration polarization phenomenon. 7935

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Although the process of obtaining the real and imaginary components of ε*(ω) appears straightforward, in reality it is not, as when samples are conducting, the phenomenon of electrode polarization (EP) can be very perturbing at low frequencies. This is the accumulation of ions around the electrodes when an electric field is applied. The ions screen the applied field, and hence, the potential drop in the sample is greatly reduced in an uncontrolled way. The polarization process is well known and has been studied for decades.35−37 In spite of this, it has not received a definite solution yet, although it may provoke the complete masking of the sample contribution to the real part of the electric permittivity. Fortunately, there is still a broad frequency interval in which the electric permittivity can be corrected from EP by a number of techniques.38−40 In the majority of cases the correction is a time-consuming and tedious task. In this sense, a new method was proposed in this field by Jiménez et al.41 and afterward applied in different systems.42−44 It is based on the similarity between the imaginary part of the electric permittivity and the logarithmic derivative (LD) of the real part, defined as

εD″(ω) = −

(1)

This makes LD the most suitable quantity to characterize the electric permittivity spectra of the suspension since (i) contrary to ε″(ω), εD″ (ω) is free of the conductive contribution and (ii) if electrodes are detached enough, the EP contribution can be better separated from the actual spectra of the suspension than in the case of the real part.41 Once the LD data is separated from the unwanted EP perturbation, the experimental ε′(ω) spectrum can be recovered. Some examples of the data treatment and the application of the LD method to two cases where the DD behaviors are clearly different are shown in section 4 of the SI, Figures S6 and S7. In addition, the data can be fitted to the LD of a known dielectric dispersion function (Cole−Cole45 in our case, eq [S.30]). From this, one can extract the relevant parameters of the dispersion, e.g., the maximum of the relaxation peak in the LD data and the low-frequency dielectric increment. Finally, dynamic electrophoretic mobility ue(ω) values were obtained, in the frequency range of 1−18 MHz, using a commercial electroacoustic spectrometer, the Acoustosizer II, manufactured by Colloidal Dynamics (USA). This device is based on the analysis of the ESA signal or the ultrasound wave generated when an alternating electric field is applied to a suspension containing particles with a density different from that of the dispersion medium.22,23 Details are provided in the accompanying Supporting Information.

Figure 1. High-resolution transmission electron microscope picture of the gibbsite particles used. Bar length: 200 nm.

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2 ∂ε′(ω) π ∂ ln ω

EXPERIMENTAL SECTION

Materials. Gibbsite particles were obtained using the procedure described by Wijnhoven.34 Briefly, the method involves the aging of an aqueous solution of HCl (0.09 M), aluminum sec-butoxide (0.08 M) from FlukaChemika, Germany, and aluminum isopropoxide from Across Organics, Belgium (0.08 M), during 7 days at room temperature and, furthermore, 72 h at 85 °C. Cleaning is carried out by dialysis against deionized water (Milli-Q Academic, Millipore, France). All other chemicals were purchased from Sigma-Aldrich Spain, and they were used as received. As shown in Figure 1, obtained by HRTEM microscopy (Philips CM20, The Netherlands), the particles have a clear platelike, hexagonal geometry, with thin edges. From pictures like this, it was found that the distribution of dimensions yields a volume average diameter (±S.D.) of 2b = 250 ± 50 nm and an average thickness of 2a = 6.2 ± 0.8 nm, that is, an aspect ratio of r = a/b = 0.025. For our experiments, we prepared aqueous suspensions of the synthesized gibbsite particles at 2% v/v concentration of solids in solutions of different KCl concentrations and pH values. Methods. The classical, or dc, electrophoretic mobility determinations were carried out in a Malvern NanoZS device (Malvern Instruments, U.K.), based on the PALS (phase analysis light scattering) technique. The dielectric dispersion of the gibbsite suspensions was determined by measuring the complex impedance of a volume of suspension contained in a glass cell limited by circular, parallel electrodes made of platinized platinum 2 cm in diameter. Micrometer screws are used for electrode distance setting, and the cell body is surrounded by a thermostatization chamber consisting of a concentric cylinder through which water is pumped by a ThermoHaake DC10 thermostat (Germany). The sample temperature was 25.0 ± 0.1 °C in all experiments. Details on the cell construction can be found in ref 35. The complex impedance Z*(ω) measurements were performed in a HP4284A (Hewlett-Packard, USA) LCZ meter for frequencies ranging between 1 kHz and 1 MHz. The complex conductivity K*(ω) of the suspensions was obtained from the impedance after calibrating the cell using conductivity standards (eq [S.1]) and taking into account the relationship between complex conductivity and permittivity (as described in the SI, eqs [S.3, S.4]).

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RESULTS AND DISCUSSION Average Surface Potential: dc Mobility. Figure 2 gives a clue to the electrical properties of gibbsite particles. The

Figure 2. Electrophoretic mobility of gibbsite particles as a function of pH for the indicated ionic strengths in solution. 7936

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Figure 3. Logarithmic derivative of the real part of the permittivity of gibbsite suspensions as a function of frequency for the pH values indicated. Ionic strengths: 0.5 mM (a) and 1 mM (b).

electrophoretic mobility is plotted vs pH for two ionic strengths. The most noticeable feature, in addition to the positive overall charge of the particles up to pH 9, suggesting an isoelectric point (or pH of zero zeta potential) pHiep close to 10, is the nonmonotonous mobility−pH trend. In fact, ue changes very little up to pH 5 and displays a faster decrease beyond that pH. With the understanding that a single electrophoretic mobility curve cannot be that informative, we can try to explain the behavior of ue in Figure 2 on the basis of existing models of the surface charge generation in gibbsite. With the aim of modeling their rheological data on gibbsite suspensions, Wierenga et al.46 proposed a single-site model based on the crystal structure of the mineral as a layering of Al−OH (with each Al ion coordinated with six hydroxyls). The simple charge-generation reactions are

Al−OH1/2 − + H+ XoooooooooY Al−OH 21/2 + log K = 10

Al 2−O− + H+ XooooooooooY Al 2−OH 0 log K = 12.3

Al 2−OH + H XoooooooooooY Al 2−OH 2+ 0

+

log K =−1.5

(3)

With such a distinction and the proper choice of the equilibrium constants, the authors predict that the edge surfaces generate positive charge through a wide pH range, with a point of zero charge at pH 10, and face surfaces are uncharged up to pH 9 and become negative beyond that point. The pH 9 range is controlled by the planar surfaces. This is in clear contrast to Wierenga et al.’s data, as no isoelectric point (point of zero charge, actually) of edges can be expected below pH 10. In the case of the rather complete experimental study performed by Adekola et al.,50 apart from a clear influence of sample origin and treatment on the charge−pH dependence, the authors found that, although according to the MUSIC model the faces are not active toward protons in the mentioned pH range, they could adsorb species and build surface charge in a different way. In particular, the adsorption of Al3+ or Al13 (abbreviated denomination for the complex solute: Al13O4(OH)4(H2O)127+) can provide the faces with positive charge. However, such side effects will be strongly sample-dependent, and it is likely that models based on pure charge-generation reactions based on the presence of active sites on edges or plates will not be applicable to all gibbsite samples. Nevertheless, as mentioned above, we can consider a more complete set of electrokinetic data in order to explain our results. This will be dealt with in the next section. Dielectric Dispersion of Gibbsite Suspensions. The extreme sensitivity of DD to pH variations is clearly seen in Figure 3, where the logarithmic derivative of the relative permittivity is plotted as a function of frequency for pH ranging between 4 and 9 and two KCl concentrations (0.5 mM in a, 1 mM in b). The mere observation of this raw data indicates that pH 7 separates two clearly distinct behaviors in the dielectric dispersion: below that pH the high-frequency relaxation (roughly centered at 2 × 105 rad/s) is clearly observed, while for pH 7 and above, this relaxation is mixed up with a lowerfrequency process. The effect is more noticeable at 1 mM ionic strength, when particle−particle interactions are more efficiently screened. This is more clearly seen in Figure 4 and in Table 1, where the fitting parameters of Cole−Cole

Al−OH 2+ + OH− ⇄ Al−OH + H 2O ⇄ Al−O− + H3O+ (2)

In addition, the same authors propose that the crystal structure of gibbsite must determine that the acidity of the edges is different from that of the faces so that the isoelectric points of both kinds of surfaces are likely to be different. In fact, TEM micrographs obtained with mixed gibbsite-gold nanoparticle suspensions confirmed that the gibbsite particles have two distinct values of pHiep, namely, pHe ∼7 (ascribed to edges) and pHf ∼10 (for faces), in a similar fashion to clay particles.47 These arguments are justified by the fact that the negative gold particles adsorb preferentially on the edges of gibssite at pH 4 and on the faces of crystals at pH 7. Our results in Figure 2 are coherent with these experimental observations as long as pHe is assumed to be around 6 to 7. These results must in any case be reinterpreted, first by using more elaborate charging models and then by considering the DD data. The first task can be accomplished by making use of the MUSIC (multisite complexation) model, as described by Hiemstra et al.48,49 and Adekola et al.50 The former authors distinguished between singly coordinated Al−OH and Al−OH2 groups and doubly coordinated Al2−OH and Al2−OH2 groups. The latter are present in the crystal interior and on the planar surfaces, whereas singly and doubly coordinated hydroxyls coexist on the edge surfaces. The corresponding chargegeneration reactions are 7937

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Langmuir distributions (eq[S.30]) for the experimental DD data are displayed.

It must be kept in mind that the maxima displayed by the imaginary component of the permittivity (or by ε″D) are very sensitive to the average size and geometry of the particles (eq[S.8], SI). Specifically, large particles will show larger alpha relaxation and at lower frequency than smaller ones. Results in Figures 3 and 4 and Table 1 show first of all that, in accordance with the rheological findings of Wierenga et al.,46 there is an intermediate pH value (around 6 to 7 in our case) for which the electrokinetics of gibbsite changes clearly. Furthermore, at pH 4 the low- and high-frequency relaxations are well separated, which suggests the coexistence of large structures (maxima around 2 × 104 rad/s) and individual particles (2 × 105 rad/s). According to Wierenga et al.,46 at pH 4 partial dissolution of gibbsite occurs, releasing fibril-shaped aluminates. These negatively charged complexes may behave as bridges between positive faces, leading to the appearance of large structures. On the other hand, at pH 5, only a wide relaxation is observed: the highly charged particles repel each other, leading to relaxations controlled by individual gibbsite particles and hence characterized by high ωα values. The wide relaxation peak comes from the proximity in frequency of the relaxations of the parallel and perpendicular orientations of the particles18 (Figure S5). At still higher pH values, the overall values of the permittivity are greatly reduced, an indication of the reduction of the particle charge because of the increased neutralization of the faces. It is worth mentioning that at pH 7 and above the low-frequency peak is the predominant one, in agreement with our interpretation based on progressively more negative edge charge interacting attractively with still positive faces: the typical house-of-cards structure can be expected in such a situation (large average size).This is well known and often reported in classical studies of clay nanoparticle suspensions, where the distinct charge signs on edges and faces favor electrostatic attraction between both types of surfaces. On the other hand, the high-frequency relaxation decreases in amplitude due to the decrease in the overall charge. This description is qualitatively coherent, but a more quantitative analysis can be performed if the Grosse et al.’s18 model (section 3 of SI) is used to fit the data. This is illustrated in Figure 5. The data at pH 7 are in accordance with a description where the high-frequency peak is calculated by means of the model for spheroids while the low-frequency peak can be described as a result of the relaxation of a spherical

Figure 4. Data in Figure 3 for two pH values and the two ionic strengths.

Table 1. Best-Fit Parameters of the Relaxation Spectraa pH

Δε′l (0)

ωα,l

4 5 6 7 8 9

130

16 000

110 130 68

29 000 29 000 31 000

4 5 6 7 8 9

160

15 000

80 240 250 180

17 000 19 000 18 000 18 000

1 − αl

Δε′h(0)

KCl 0.5 mM 0.94 200 160 150 0.82 78 0.72 34 0.85 51 KCl 1.0 mM 0.94 280 230 0.93 180 0.83 80 0.72 40 0.75 20

ωα,h

1 − αh

154 000 154 000 140 000 150 000 150 000 150 000

0.73 0.75 0.73 0.8 0.9 0.8

140 000 140 000 140 000 140 000 140 000 140 000

0.73 0.72 0.7 0.8 0.9 0.9

a

Subscripts l and h refer to the relaxations at low and high frequencies, ′ (0): amplitude of the corresponding relaxation. ωα,l,h respectively. Δεl,h (rad/s): reciprocal of the characteristic time of the Cole−Cole distribution (eq [S.30]). 1 − αl,h: broadness of the distribution. In all cases the uncertainty is given by the last significant digit.

Figure 5. Real part of the relative permittivity (Δε′, half-filled symbols) and its logarithmic derivative (εD″ , full symbols) plotted as a function of frequency for gibbsite suspensions. The solid lines are the best fit to Grosse et al.’s18 model (pH 5 and high-frequency relaxation of pH 7) and to the spherical particle model5 (pH 7, low-frequency relaxation). 7938

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not differ by more than a factor of 2 with respect to TEM pictures. Such a difference could be associated with the extremely laminar shape of the particles, for which the condition of a thin double layer on the edge surfaces (κa ≫ 1) is not fulfilled. The model can also be used for the calculation of the surface charge quantities, namely, zeta potential and surface conductivity, This gives us the chance to establish a comparison with the DC mobility data. Figure 7 shows the results: as observed, the overall decrease in both the potential and the conductivity with pH is coherent with a description based on the tendency of faces to get neutralized as the pH is increased, and this is confirmed with very similar data on the surface conductivities found with the two completely independent sets of observations. The differences observed at pH 4 to 5 come from the simplified approach used for this comparison. We are trying to explain the two types of results with a single parameter (in fact, two related parameters: zeta potential and surface conductivity). But the important point of this investigation is that dielectric dispersion is sensitive to features of the system which go unnoticed for electrophoretic mobility. In Figure 2 we do not observe important differences between the electrophoretic mobilities for the pH range of 4 to 5. Being “single-point” measurements, one can mistakenly conclude that the zeta potential of the individual particles has not changed. In reality, it might undergo small changes, which are hidden by the averaging performed in electrophoresis measurements. However, the structure of the permittivity spectra is much more informative. At pH 4, it accounts for an alpha relaxation process related to a comparatively large structure (the low-frequency relaxation), while the alpha relaxation of single particles is still observable. The latter occurs at the same frequency as it does at higher pH, and hence it can be ascribed to the same process, that is, the alpha relaxation of a single particle. For this reason, it can be used for calculating the zeta potential and size of single particles. In this way we obtain that the zeta potential of the single particle increases when the pH decreases. This result cannot be reached with the single measurement obtained with electrophoresis, which does not distinguish between large aggregates and small particles, and it is rather an average of all of the particles and aggregates in the suspension. One can say that the formation of structures involving several particles might perhaps hinder the elevation of the electrophoretic mobility, even if the zeta potential increases.

aggregate. In the case of pH 5, if the formation of such aggregates is less likely, then we can expect that it is mostly individual particles that contribute; in this case, the Grosse’s model suffices. Figure 6 gives the necessary information on the

Figure 6. Large semiaxis of gibbsite particles obtained by fitting the DD data in Figure 3 to the oblate spheroid particle model. In the cases of two relaxations, only the one at higher frequency (assumed to be due to single-particle processes) is considered. The small axis is always given the fixed value 2a = 6.2 nm.

best-fit dimensions. In the cases of two relaxations, only the high-frequency one, obtained with the spheroids model, is presented; the small axis is always 2a = 6.2 nm. Note from Figure 1 that some polydispersity is present in our particles. The effect of such polydispersity on dielectric determinations was studied by Carrique et al.,51 who reported that (for spheres, although the results appeared quite general, since they were based on very general arguments) the dielectric behavior of a polydisperse system does not differ from that of a monodisperse suspension with a single size close to the volume average of the distribution. Hence, our study can be analyzed in terms of a single relaxation time (for each particle dimension), with the understanding that such time is controlled by the (volume) average of the particle dimensions in the system. This is how the fitted values of the size parameters must be interpreted. Note that the fitted dimensions show better agreement with the values obtained by electron microscopy in the case of 1 mM KCl solutions. This is so because under such conditions we are closer to the thin EDL approximation required by the model. Even in the worst situation (0.5 mM KCl) the fitted sizes do

Figure 7. Surface conductivity (left) and zeta potential (right) obtained from the DD (red squares) and the DC mobility (black circles) results using the electrokinetic model for oblate spheroids. Open symbols, KCl 0.5 mM; closed symbols, KCl 1 mM. 7939

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Figure 8. Effect of pH on the real and imaginary components of the dynamic mobility of gibbsite particles. Ionic strength 1 mM KCl.

Figure 9. Effect of ionic strength on the real and imaginary components of the dynamic mobility of gibbsite particles; pH 5.

Dynamic Mobility of Gibbsite Suspensions. As mentioned in the Supporting Information, the electroacoustic experiments probe a frequency interval far beyond that accessible to DD but still very useful, as it contains information on the surface conductivity of the particles (through the Maxwell−Wagner−O’Konski, or MWO relaxation; see section 2 of the SI file) and on the in situ particle size distribution. Figure 8 shows the main features of the effect of pH on the real and imaginary components of the mobility at 0.5 mM KCl. Figure 9 includes the effects of ionic strength at pH 5. The data presented confirm the DC mobility and, mainly, DD results in a number of features. First, clearly distinct behaviors are found at acidic and basic pH values. Thus, the MWO relaxation, associated with the disappearance of surface conductivity effects (the mobility increases clearly at frequencies around 30 Mrad/s), is mainly noticed at acidic pH values, and it is very weak at pH 8 and, particularly, at pH 9, proving the low charge of the particles under such conditions. This is in clear agreement with the estimations of surface conductivity found from both DC mobility and DD data (Figure 7). Second, the inertial decrease of the mobility is best observed in the imaginary component of the mobility, as the high-frequency elevation of Im(ue) corresponds to the beginning of the inertial relaxation. This elevation is less intense and occurs at higher frequencies at acidic pH, an indication of the stability of the suspensions and of the presence of rather small (likely individual gibbsite platelets) dispersed units under such conditions. The fact that pH 7 (absent due to the difficulties in measuring this sample by the presence of aggregates) constitutes the imaginary boundary

between both kinds of behaviors and separates regions of strong and weak electrokinetic response is also clear. Using the theory of O’Brien and Loewenberg14 for the dynamic mobility of spheroidal particles, it is possible to fit the size and the zeta potential (Table 2) from the experimental Table 2. Fitted Values of the Zeta Potential (ζ) and Large Semiaxis (b) from the Experimental Values of Dynamic Mobility in Figure 8 pH

ζ (mV)

b (nm)

4 5 6 8 9

84 101 79 22 18

165 177 163 307 500

data. As before, this is realistic particularly for 1 mM solutions, when the assumption of a thin double layer is fulfilled. At acidic pH the values obtained for the large semiaxis agree very well with those observed in the TEM pictures and obtained in DD data (Figure 6). Also, the presence of structures at high pH (manifested in the large particle sizes in Table 2) is in accordance with DD, where the low-frequency relaxation peak is also associated with the formation of particle aggregates in suspensions of inhomogeneously charged nanoparticles. The dynamic mobility data reported in this article are coherent with the “house-of-cards” kind of structure. In particular, the detection of a low-frequency relaxation attributable to structure formation was also found for a different system of nonspherical particles (elongated goethite) in ref 32. 7940

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It is also worth mentioning that when the ionic strength is above 0.5 mM (Figure 9), the peak corresponding to MWO relaxation weakens, as expected from a reduced effect of surface conductivity (smaller Du number, as inferred from Figure 7). Furthermore, the relaxation moves to higher frequencies as the ionic strength is increased, indicating an increased importance of the solution conductivity Km, in accordance with eq [S.12].

CONCLUSIONS In spite of the complexity of the system investigated (nonspherical particles with nonhomogeneous surface charge), we have shown that the use of a combination of electrokinetic techniques (dc and ac mobilities and dielectric dispersion) can yield a worth of in situ information about the microstructure of the suspension under different conditions of pH and ionic strength. This has been made possible not only by means of experimental observations but thanks to the use of theoretical models suitable for nonspherical geometries. Although they are based on simplifying assumptions (the most important is the supposition of thin double layers), the values of the zeta potential and even the average size of the particles are internally coherent. It has been found that pH 7 separates two distinct behaviors: while under acidic conditions the particles remain rather stable, almost as individual platelets, the formation of structures at basic pH is also evident, as suggested by the increased size and reduced surface potential estimated from dynamic mobility and DD. These two techniques prove to be extremely informative about the structure and dynamics of dispersed systems, even in situations far from the ideal case of dilute suspensions of large spheres. ASSOCIATED CONTENT

S Supporting Information *

Fundamentals of dielectric dispersion and dynamic mobility and a short account of theoretical predictions and experimental approaches. The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/ acs.langmuir.5b01136.

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REFERENCES

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Article

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Present Address

(R.A.R.) ICFO-Institut de Ciències Fotòniques, 08860 Castelldefels, Barcelona, Spain. Author Contributions

The manuscript was written through the contributions of all authors. All authors have given approval to the final version of the manuscript. All authors contributed equally. Funding

Funding provided by Junta de Andalucia,́ Spain, and MINECO, Spain. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS Financial support of this investigation by Junta de Andalucia,́ Spain (grant no. PE2012-FQM0694) and MINECO, Spain (project no. FIS2013-47666-C3-1-R) is gratefully acknowledged. 7941

DOI: 10.1021/acs.langmuir.5b01136 Langmuir 2015, 31, 7934−7942

Article

Langmuir

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DOI: 10.1021/acs.langmuir.5b01136 Langmuir 2015, 31, 7934−7942