Electroacoustic and Dielectric Dispersion of. Concentrated Colloidal Suspensions. S. Ahualli, M. L. Jiménez, A. V. Delgado. Department of Applied Physics.
IEEE Transactions on Dielectrics and Electrical Insulation
Vol. 13, No. 3; June 2006
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Electroacoustic and Dielectric Dispersion of Concentrated Colloidal Suspensions S. Ahualli, M. L. Jiménez, A. V. Delgado Department of Applied Physics University of Granada, Faculty of Science 18071 Granada, Spain
F. J. Arroyo Department of Physics University of Jaén, Faculty of Experimental Sciences 23071 Jaén, Spain and F. Carrique Department of Applied Physics II University of Málaga, Faculty of Science 29071 Málaga, Spain
ABSTRACT The determination of the permittivity of colloidal dispersions has been demonstrated to be a very useful technique to characterize the electrical state of the solid/liquid interface. This is so because of its sensitivity to such features as particle size and shape, state of aggregation, surface charge, and so on. The dielectric dispersion data are particularly suited to the evaluation of the surface characteristics when the suspension contains a high solids concentration, as in such conditions the usual electrokinetic techniques linked to optical methods are not applicable. The same advantage is shared by electroacoustic techniques, where it is a collective response of the system that matters. In this case, the experimental quantity is the dynamic (or ac) electrophoretic mobility. In this work, both methods will be used for the investigation of the electrical surface characteristics of concentrated alumina suspensions with volume fractions of solids ranging between 1 and 20%. The data will be analyzed in the frame of the so-called cell model, often used to analyze the hydrodynamic and electrical interactions between particles in concentrated suspensions. We will show that the model is capable of properly describing both their dielectric dispersion and dynamic mobility, and that the surface (zeta) electric potential calculated from electroacoustic data can be used as an input parameter to reproduce the permittivity. This will demonstrate the internal coherence and accuracy of the cell model and give clues to improve our understanding of the electrokinetic behavior of concentrated slurries.
Index Terms — Acoustoelectric effects, concentrated colloidal dispersions, electrophoresis, permittivity measurement.
1 INTRODUCTION THE dielectric spectroscopy of colloidal dispersions is a powerful tool for the precise characterization of the electrical properties of interfaces. Its determination can be carried out by performing measurements of the impedance of a conductivity cell immersed in the suspension. One difficulty involved in these experiments is that the frequency interval where dielectric relaxation takes place (kHz) can include the region where electrode polarization interferes with the results. Manuscript received on 27 June 2005, in final form 15 March 2006.
Since impedance measurements inform about a collective response of the system, they would be an excellent tool for the analysis of electrokinetics of concentrated suspensions. Both the frequency dependence of the dielectric constant and the amplitude of its relaxations are very sensitive to such properties of particles as the surface charge, size and shape. On the other hand, electroacoustic techniques are recently gaining interest from the colloid science community as powerful tools to obtain information on the electrical state of many interfaces, particularly in a number of situations where traditional electrokinetic methods are of limited use. One of such situations is that of concentrated suspensions and slurries,
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S. Hawalli et al.: Electroacoustic and Dielectric Dispersion of Concentrated Colloidal Suspensions
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§a· φ=¨ ¸ (1) ©b¹ In this paper, our aim is to report experimental dynamic mobility and electric permittivity data on suspensions of alumina of different volume fractions, checking these results against the predictions of cell models. In a previous work [9], we have carried out a similar study on the dynamic mobility as a function of electrolyte concentration.
centrifugation at 8500×g in a Kontron T-124 centrifuge, and redispersion in the KCl solution. The volume fraction of solids, φ, in the suspensions prepared was determined from the dry weight of known volumes using ȡp = 2.4 g/cm3 as the density of alumina. A
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“first-principles” approach, considering a particle interacting with its nearest neighbors [1]. Another evaluation, first carried out by Dukhin et al. [2,3], is based on a so-called “cell model”, that has also been successfully applied to the analysis of a variety of electrokinetic phenomena in concentrated suspensions, including electrophoresis, sedimentation potential or dielectric dispersion [4-8]. In this sort of calculation, the hydrodynamic and electrical interactions between particles are accounted for in an averaged form by solving the electrokinetic equations for a single spherical particle of radius a, enclosed by a spherical shell of solution of radius b, concentric with the particle. The radius of the sphere is chosen in such a way that the volume fraction of solids is the same in the cell and in the suspension:
2È10-4 mol/l KCl solution. The process was carried out by
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proportional to the ESA signal, and relate it to the zeta potential, the particle shape and dimensions, the ionic composition of the medium, and, very important, the volume fraction of solids, φ. It can be said that dynamic mobility is the ac (alternating current) counterpart of classical dc (direct current) electrophoresis. Two independent approaches have been followed to solve specifically the problem of the volume fraction dependence of ud* in the case of concentrated dispersions. O’Brien et al. use a
polydisperse. The average hydrodynamic radius of Alumina A was 165 ± 30 nm and for Alumina B it was 288 ± 50 nm, as measured by photon correlation spectroscopy in a Malvern 4700c PCS device (Malvern Instruments, U. K.). The original suspensions were repeatedly washed with
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in which we can get no information on the behavior of individual particles by performing, for instance, microelectrophoresis experiments. There are two such techniques. One involves the generation of a pressure wave when an ac electric field is applied to the suspension: we speak of the ESA effect, used in this work. The second method, reciprocal of ESA, is based on the determination of the electric potential induced by the passage of a sound wave through the system. It is called colloid vibration potential (CVP) or colloid vibration current (CVI), depending on the quantity measured. Current electroacoustic technology allows to obtain the dynamic mobility, ud* , a complex quantity that is in fact
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Figure 1. Comparison between the modulus of the dynamic mobility of alumina suspensions in 0.2 mM KCl (symbols) and the corresponding theoretical predictions obtained from cell model (solid lines), for the zeta potentials indicated. The dc mobility measured in dilute suspensions is shown as a short dotted line labelled “ue”. A and B correspond to alumina samples A and B, respectively.
2 EXPERIMENTAL 2.1 MATERIALS Two different samples of hydrated alumina were used in this investigation, both kindly provided by Dispersion Technology Inc. (USA). Sample A was given to us in the form of a concentrated suspension in an unknown dispersion medium, while sample B was a dry powder. SEM micrographs showed that in both cases the particles were crystalline and moderately
2.2 METHODS The classical or dc electrophoretic mobility of the particles, ue, was measured in dilute suspensions (φ ∼ 10-4), using a Malvern Zetasizer 2000 (Malvern Instruments, U.K.). These measurements provided an alternative or complementary way to evaluate the electrokinetic or zeta potential of the particles.
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The dynamic or ac mobility of the particles, ud* , was measured A 5
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for a frequency range between 1 and 18 MHz in a Acoustosizer II (Colloidal Dynamics, USA). For the determination of the dielectic properties of the samples we used a HP4284A LCZ meter operating between 100 Hz and 1 MHz, connected to a conductivity cell with variable electrode distance. Because of the high solids concentrations typically used, we performed the measurements keeping the systems flowing through the conductivity (and, eventually, also through the electroacoustic) cell, by pumping them from a stirred and thermostatted beaker. The temperature was kept constant at 25.0±0.1°C using a Haake F3K water circulator. For the frequencies of interest in the determinations of permittivity in colloidal suspensions, an important source of perturbation is the polarization of the electrode/solution interface. We used some procedures to minimize this unwanted effect: i) measure at different separations between the electrodes; ii) use calibration solutions and perform relative measurements (quadrupole method, see Ref. [10]); iii) work with the logarithmic derivative of the real permittivity, so as to separate the frequency dependences of the electrode and sample contributions to the measured permittivity [11]. Specifically, the results presented in this work were obtained using methods (ii) for getting a general picture of the permittivity-frequency trends, and (iii) to obtain the relaxation frequency and the low-frequency permittivity.
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3 RESULTS 3.1 DYNAMIC MOBILITY The modulus and phase of the complex dynamic mobility of A and B alumina particles with varying volume fractions (φ ∼ 2.5 – 30 %) as obtained from the ESA instrument are shown in Figures 1 and 2. Note that, as Figure 1 shows, increasing the volume fraction provokes a reduction in the mobility. This was to be expected, as it is predicted by all theoretical models [1-8, 12] and justified because of the hydrodynamic and electrostatic interactions of a given particle with its neighbors. In addition, increasing the frequency f can bring about two distinct behaviors in the mobility: for frequencies on the order of 2-3 MHz, u d*
goes through a maximum, that occurs at higher
frequencies the more concentrated the suspension. If we increase the frequency even further, the modulus of the mobility monotonously decreases. On the other hand, phase angle data in Figure 2 show consistently negative values of that angle, i.e., the velocity lags behind the applied ac field. This must be due to two phenomena that, for certain frequency ranges, affect the electrophoretic motion. One is the particle and fluid inertia, that become increasingly important for frequencies above
f I ≈ η / ρ p a 2 (§ 5 MHz for our particles), where Ș is the
Figure 2. Comparison between the phase angle of the dynamic mobility (symbols) and the corresponding theoretical predictions obtained from cell model (solid lines) for the same suspensions as in Figure 1. A and B correspond to alumina samples A and B, respectively.
viscosity of the liquid medium. Because of this inertia, the phase lag of the mobility will become increasingly negative with frequency for f > fI . This inertial effect also explains the decrease observed in u d* beyond the maxima in Figure 1. The other phenomenon that contributes to the value of the mobility is the induction of an electrical dipole at the solid/liquid interface, that depends on the frequency and the zeta potential. Roughly speaking [12-14], the dynamic mobility can be expressed as: ε * ud* = m ζ (1 − C * )Ginertia (ω) (2) η where C* is the induced dipole coefficient, and Ginertia (Ȧ) is the *
complex function that accounts for the particle and fluid inertia above mentioned, for an applied field of frequency Ȧ. The induced dipole moment has a somewhat complex frequency dependence, that can be summarized as shown in Figure 3, where we have plotted 1 − C * : at relatively low frequencies (in the kHz region), a decrease is observed, associated to the
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so-called Į-relaxation. This curve has been obtained for a single spherical particle, using the DeLacey and White’s theory [15].
a lower curvature indicates that the electrocaoustic measurements are very sensitive to the larger polydispersity of sample B. The cell model, elaborated for monodisperse particles, can hardly reproduce such flat dependences.
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At this frequency, the diffusion processes in the ionic atmosphere surrounding the particles (the electrical double layer) cannot follow the field changes and the induced dipole moment increases. At still higher frequencies, the MaxwellWagner (MW) relaxation gives rise to a decrease in the dipole moment, hence the increase in 1 − C * and in the modulus of the mobility, observed in Figure 1. This relaxation may also provoke a phase lead, observable in some cases in Figure 2. Let us now consider how the cell model can describe the effects of volume fraction on the dynamic mobility. In Figures 1 and 2, we have superimposed on the experimental points the best fit relationships between the modulus (or phase) of the mobility and the frequency of the field, obtained from the model, using the zeta potential (ȗ) of the interface as the only parameter. The best-fit ȗ values are included in the Figures. It must be mentioned that, according to Shilov et al [16], the zeta potential is a suitable parameter describing the electrical state of the interface as long as the double layer thickness and/or the particle concentration are such that the interparticle distance is larger than the double layer thickness. If this condition is not fulfilled, it is the surface charge and not the potential that remains physically sound. Throughout this paper, we assume that we are far from the strong-overlap situation. Note that in general the cell model is capable of properly fitting the data, although sometimes it fails to reach the highfrequency side because of convergence limitations of our computer program. In the case of sample A, the model can reproduce the maxima in the modulus of the mobility at frequencies close to the experimental ones (Figure 1A). However, the fitting is not as good for sample B. The fact that
ud* -frequency (and, mostly, phase-frequency) curves display
An equally promising and useful procedure is the comparison between permittivity and dynamic mobility data obtained on exactly the same suspensions. Thus, Figure 4 shows the real part of the relative permittivity, ε’(ω)/ε0, of A and B Alumina suspensions as a function of the frequency of the applied electric field for the volume fractions indicated. Note the rough fall corresponding to the low frequency or α-relaxation in the range 104-105 Hz. The Maxwell-Wagner (MW) relaxation cannot be observed because its amplitude is much smaller than the low-frequency one. Figure 2 shows that MW relaxation is clearly noticeable in dynamic mobility data, so that both techniques probe the frequency dependence of the electrokinetic quantities in a complementary fashion. The effect of volume fraction on the dielectric relaxation is appreciated in Figure 4: both the very low frequency (ωĺ0) electric permittivity, ε’(0), and the alpha relaxation frequency, ȦĮ increase with the volume fraction φ. 3.3 COMPARISON BETWEEN BOTH TECHNIQUES We will now check if these experimental data are well reproduced by the theoretical predictions of the cell model, already verified for the ac mobility results. This requires a previous knowledge of the zeta potential of the particles, a task often performed by additionally using dc electrophoresis measurements in dilute suspensions of the same particles and the same ionic medium. The present work explores the interesting possibility of using the same concentrated suspensions in both electrophoresis and dielectric dispersion experiments: the former will be used to estimate the zeta potential, which in turn will be used to calculate the permittivity using the same cell model. This offers the possibility of testing both the theory itself and the joint use of the two electrokinetic techniques. Figures 5 and 6 show, respectively, the values of the low-frequency relative permittivity ε’(0)/ε0, and of the Į-relaxation frequency as a function of the volume fraction of suspensions. Using the zeta potentials obtained from dynamic mobility, the cell model allows to explain the overall features of the dielectric relaxation in the suspensions, and to predict the variation of the relaxation amplitude and the increase of the relaxation frequency with volume fraction. However, similar to previous findings regarding the comparison of permittivity and dc electrophoresis, it is clear that the electrokinetic model tends to underestimate ε’(0) and to overestimate ωα , the differences amounting to about one order of magnitude. These differences have been explained considering that the classical or standard electrokinetic model has fundamental limitations:
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the hypothesis that the stagnant layer (extending between the solid surface and the shear plane) is non-conducting has been revised and so-called dynamic-Stern-layer models have been elaborated that demonstrate a better quantitative agreement between different techniques [6-8]. The behavior of ε’(0)/ε0 (Figure 5) is similar to the predictions of our previous work [17]: although the permittivity of the particles is much smaller than that of the medium, only for φ larger than 0.14 is the permittivity of the suspension lower than that of the dispersion medium; for lower concentrations of particles, the double layer polarization more than compensates this, and very high values of the relative permittivity are found at low frequencies. In the case of sample B (Figure 5B), the theoretical predictions are very similar to the experimental data; however, the maximum reached by ε’(0) occurs at different volume fractions. This diffrence may be due to the fact that the zeta potential obtained from ud* has
some degree of uncertainty because of the polydispersity of the sample, that we mentioned above. It must be taken into account that the maximum in the permittivity-volume fraction dependence is very sensitive to the value of ȗ [17]. Let us finally mention that both the calculations and the eexperimental data demonstrate that the relaxation frequency increases with the volume fraction of particles (Figure 6). In order to justify this, we must recall that the high values of ε’(0) are due to the diffusive currents of counterions in the electical double layer; the shorter the characteristic diffusion length, the higher the relaxation frequency. It is reasonable to admit that the length will decrease (and the frequency will increase) by the presence of closer neighbor particles as we increase the volume fraction, as in fact observed.
CONCLUSIONS We can conclude that using the zeta potential obtained from electroacoustics is a reasonable approximation to the true electrical potential at the solid/liquid interface, better, in any case, than that deduced from dc electrophoresis in dilute
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Figure 6. Experimental (full squares) and theoretical (open symbols) data of the Į-relaxation frequency of alumina suspensions. The latter were calculated using the zeta potential obtained from electroacoustic measurements (circles) and from electrophoresis (triangles). A and B as in Figure 4. Note the different scales of the left and right ordinate axes.
suspensions. The cell model used appears to properly describe the main features of the volume fractions dependence of electrokinetic quantities. The fact that it underestimates the permittivity and overestimates the relaxation frequency is a point open for further investigation, perhaps in relation with boundary conditions for ions and liquid in the vicinity of the solid surface.
ACKNOWLEDGMENT Financial support from the Spanish Ministry of Science and Technology and Feder Funds (EU) (FIS-2005-06860-C0201/02) is acknowledged. S.A. would like to thank the Spanish Ministry of Education for a grant covering this research.
[15]
[16]
[17]
S. Kuwabara, “The forces experienced by randomly distributed parallel circular cylinders or spheres in a viscous flow at small Reynolds numbers”, J. Phys. Soc. Jpn., Vol. 14, pp. 527-535, 1959. H. Ohshima, “Dynamic electrophoretic mobility of spherical colloidal particles in concentrated suspensions: approximation of nonoverlapping double layers”, Colloids Surf. A, Vol. 159, pp. 293-297, 1999. F. Carrique, F. J. Arroyo and A. V. Delgado, “Sedimentation velocity and potential in a concentrated colloidal suspension. Effect of a dynamic Stern layer”, Colloids Surf. A, Vol. 195, pp. 157-169, 2001. F. Carrique, F. J. Arroyo and A. V. Delgado, “Electrokinetics of concentrated suspensions of spherical colloidal particles: effect of a dynamic Stern layer on electrophoresis and DC conductivity”, J. Colloid Interface Sci., Vol. 243, pp. 351-361, 2001. F. Carrique, F. J. Arroyo and A. V. Delgado, “Electrokinetics of concentrated suspensions of spherical colloidal particles with surface conductance, arbitrary zeta potential, and double layer thickness in static electric fields”, J. Colloid Interface Sci., Vol. 252, pp. 126-137, 2002. A. V. Delgado, S. Ahualli, F. J. Arroyo and F. Carrique, “Dynamic electrophoretic mobility of concentrated suspensions. Comparison between experimental data and theoretical predictions”, Colloids Surf. A, Vol. 267, pp. 95-102, 2005. C. Grosse and M. Tirado, “Measurement of the dielectric properties of polystyrene particles in electrolyte solution”, Mater. Res. Soc. Symp. Proc., Vol. 430, pp. 287-298, 1996. M.L. Jiménez, F.J. Arroyo, J. van Turnhout, A.V. Delgado, “Analysis of the dielectric permittivity of suspensions by means of the logarithmic derivative of its real part”, J. Colloid Interface Sci., Vol. 249, 327-335, 2002. F. J. Arroyo, F. Carrique, S. Ahualli and A. V. Delgado, “Dynamic mobility of concentrated suspensions. Comparison between different calculations”, Phys. Chem. Chem. Phys., Vol. 6, pp. 14461452, 2004. R. W. O’Brien, “Electro-acoustic effects in a dilute suspension of spherical particles”, J. Fluid Mech., Vol. 190, pp. 71-86, 1988. A. S. Dukhin, V. N. Shilov, H. Ohshima and P. J. Goetz in A. V. Delgado (Ed.), Interfacial Electrokinetics and Electrophoresis, Surfactant Science Series, Marcel Dekker, Inc., New York, 2002, Chapter 17. E. H. B. DeLacey and L. R. White, “Dielectric response and conductivity of dilute suspensions of colloidal particles”, J. Chem. Soc., Faraday Trans. II, Vol. 77, pp. 2007-2039, 1981. V. N. Shilov, Y. B. Borkovskaja and A. S. Dukhin, “Electroacoustic theory for concentrated colloids with overlapped DLs at arbitrary ța. 1 Application to nanocolloids and nonaqueous colloids”, J. Colloid Interface Sci., Vol. 277, pp.347-358, 2004. F. Carrique, F. J. Arroyo, M. L. Jiménez and A. V. Delgado, “Dielectric response of concentrated colloidal suspensions”, J. Chem. Phys., Vol. 118, pp. 1945-1956, 2003.
S. Ahualli was born in Tucumán, Argentina in 1971. She received the B.S. (Physics) degree from the University of Tucumán, Argentina in 2001, and the PHOTO M.S.C. (Electroacustic phenomena) degree from the 24mm X University of Granada, Granada, Spain in 2004. 30mm Currently, she is pursuing a Ph.D. degree in the Department of Applied Physics of the Unversity of Granada. Her areas of interest include electrokinetic and interface phenomena, particularly, electroacoustic and dielectric properties of concentrated suspensions.
REFERENCES [1]
[2]
[3]
R.W. O’Brien, A. Jones and W. N. Rowlands, “A new formula for the dynamic mobility in a concentrated colloid”, Colloids Surf. A, Vol. 218, pp. 89-101, 2003. A.S. Dukhin, H. Ohshima, V.N. Shilov and P.J. Goetz, “Dynamic electrophoretic mobility in concentrated disperse systems. Cell model”, Langmuir, Vol. 15, pp. 3445-3451, 1999. A.S. Dukhin, V. N. Shilov, H. Oshima and P. J. Goetz, “Electroacoustic phenomena in concentrated dispersions. New theory and CVI experiments”, Langmuir, Vol. 15, pp. 6692-6698, 1999.
M.L. Jiménez was born in Seville, Spain in 1975. She received the B.S. (Physics) and the M.S.C. (Permittivity of suspensions) degrees from the University of Granada, Spain in 1998 and 2000, respectively She defended her Ph.D. dissertation in 2003, dealing with the permittivity of suspensions of nonspherical particles. Her areas of interest include electrokinetic phenomena, dielectric properties of suspensions, and Kerr effects. She is presently doing her post-doc research in the University of Milano, Italy.
IEEE Transactions on Dielectrics and Electrical Insulation
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A. V. Delgado was born in Badajoz, Spain in 1956. He received the B.S. (Physics) and Ph.D. (Electrokinetics) degrees from the University of Granada, Spain in 1978 and 1984, respectively. He worked as an associate professor between 1986 and 1988 and since 1998 he is Professor at the Department of Applied Physics of the same University. His research is devoted to the electrical properties of disperse systems. A former member of the Editorial Board of the Journal on Colloid and Interface Science, he has been the Editor of a multiauthor book on Electrokinectic Phenomena, and coauthor of a number of publications in the field.
F.J. Arroyo was born in Granada, Spain, in 1969. He received the B.S. (Physics) and Ph.D. (Electrokinetics) degrees from the University of Granada, Spain in 1994 and 1998, respectively. He worked as assistant professor between 2000 and 2004 in the Universities of Jaén and Granada, Spain. Since 2004 he is an Associate Professor at the Department of Physics of the University of Jaén. His research is devoted to the dielectric properties of dispersed systems.
663 F. Carrique was born in Almería, Spain, in 1963. He received the B.S. (Physics) and Ph.D. (Electrokinetics) degrees from the University of Granada, Spain in 1988 and 1993, respectively. He worked as assistant professor between 1989 and 1996 and since 1996 he is an Associate Professor at the Department of Applied Physics of the University of Málaga. His research is devoted to the electrical properties of disperse systems. He has coauthored a number of papers on the calculation of the dielectric properties of dispersed systems.
