КОЛЛОИДНЫЙ ЖУРНАЛ, 2007, том 69, № 6, с. 817-820
УДК 541.18
LOW-FREQUENCY DIELECTRIC RELAXATION OF LARGE COLLOIDAL
PARTICLES IN SUSPENSION © 2007 r. Zineb Mimouni, Hassan Chehouani
Laboratoire d'instrumentation, de métrologie et des procédés Faculté des Sciences et Techniques - Guéliz, Université Cadi Ayyad BP 549 Marrakech, Maroc Поступила в редакцию 09.01.2007 г.
The dielectric dispersion of an aqueous colloidal suspension is studied in the range of the low frequencies. The suspended particles are relatively large compared to those studied usually and consequently the required relaxation frequencies are low. The solvent is a mixture of water and heavy water without salt addition. Measurements are compared with the model of Havriliak-Negami. The suspension of the same particles (hydrated) re-suspended in a viscous solvent (glycerol) is also studied to show the variation in the parameters of Havriliak-Negami equation.
INTRODUCTION
The response of a colloidal suspension to a weak AC electric field is accompanied by the phenomena of polarization, conduction and diffusion. The physical mechanisms of these phenomena are related to the response time which cover a broad frequency range going from zero to several GHz, for the aqueous suspensions. The pioneer theoretical works on the double layer polarization theory were carried out by Dukhin and Shilov [1, 2]. The dielectric measurements constitute a very good tool for the study of the electrokinetic properties of a colloidal suspension, providing a macroscopic information on the suspension. However, the study of the dielectric behaviour of a colloidal suspension at low frequencies is a difficult problem due to the polarization of electrodes, which in some cases can mask the dielectric response of the colloidal system. This difficulty made the experimental studies at low frequencies to be rare. For example, we mention those of Grosse et al. [3] (from 10 kHz), Blum et al. [4] (from 5 Hz). The polarization of electrodes is due to the formation of a double electric layer at the electrode surfaces, which contribution is higher than the impedance of the suspension. The mechanism of this process is not clearly understood yet, however a certain number of experimental and theoretical treatments [5-7] were presented. In the literature, the aqueous suspensions of polystyrene particles (which diameter does not exceed several hundreds of nanometers) were studied in detail from the point of view of their dielectric relaxation. The present work describes and treats the experimental results obtained at low frequencies for suspensions of non-conducting spherical particles whose size plays important role as compared to that of the usually studied particles. The effect of the solvent change (from water to glycerol) is also studied.
THEORY
The equation suggested by Debye [8] for the variation of the complex permittivity £* with the frequency ro of an AC electric field is as follows
£* = +
es-e-
1 + i Ю1П'
(1)
where £s is the static dielectric permittivity, is the high frequency limit permittivity and TD is the relaxation time:
Td =
2 D'
(2)
Here Ds is the diffusion coefficient of the counter-ions and a is the radius of the particles. It was established that the Debye expression with a single relaxation time cannot describe the dielectric response of the majority of complex systems. Empirical relations were proposed to describe the dielectric spectra of these systems. The formula of Havriliak-Negami [9] is often used. The parameters a and P of this formula (which are the asymptotes of the power laws for the low and the high frequencies) are determined in the experiments. Experimental works showed that a and P depend on the temperature, the structure, the composition and other controlled physical parameters [10]. Models of dielectric dispersion proposed by Dukhin and Shilov [1, 2], DeLacey et al. [11], Fixman [12] and Grosse et al. [13] and other authors allowed to explain the experimental results for the dispersion of aqueous suspensions (with NaCl addition). Nettelblad et al. [14] mentioned that these models correspond to the Havriliak-Negami equation with a = 0.5 and P = 2. Another important parameter for a suspension is the Debye length which represents the thickness of the double electric layer:
-1 _ /e0ee Ds
к —
7 КОЛЛОИДНЫЙ ЖУРНАЛ том 69 < 6 2007
2
e
Fig. 1. Real part of relative permittivity versus frequency for dispersions with different volume fraction of particles: = 5% (1) and 2.5% (2). Curves (3) an (4) are calculated by the equation of Havriliak-Negami.
where 80 is the electric constant and 8e is the permittivity of the dispersion medium, Ge is the solvent conductivity.
EXPERIMENTAL Suspension
The colloidal particles dispersed at the origin in water, are polystyrene spheres (from Rhone-Poulenc, France) with diameter 2a of about 3 pm (checked by electronic microscopy). These spheres were obtained by the polymerization of styrene in the presence of initiator K2(SO4)2.
The sphere surface has a negative charge (ions SO4) resulting from the dissociation of the initiator. The K+ ions go in solution. The charge carried by each particle was determined by a conductimeter and is equal to N = 1.367 x 106 elementary charges, that corresponds to the charge surface density G0 = -31.7 mC m-2.
Measuring cell
In the design of the cell, we provide the variation of the distance d between the parallel electrodes (1 mm < < d < 10 mm). The cell was built on the same principle as that described by Schwan et al. [15]. Inside the cell, the temperature of the suspension is maintained constant using a system of temperature control. The electrodes have diameter 2R = 29.5 mm and were manufactured out of titanium covered by a thin layer of platinum to provide the ideal polarization. The total surface of the electrodes was increased by the deposition of platinum black, thus, the influence of the impedance of the electrodes, Ze, was reduced [16]. If the distance d is not too small, Ze is supposed to be independent of d. The upper electrode and the envelope of the cell were connected the negative terminal of the impedance analyzer (HP 4192A), while the bottom
electrode was connected to the positive terminal. All the other parts of the cell were connected to the ground. The position of the upper electrode relative to the bottom one was adjusted by a thumb screw. The diameter of the internal cylinder was 29.7 mm. The amplitude of the signal of measurement amounted to several mV.
Residual capacity
To calculate the residual capacity Cr, we measured the capacity of the cell filled with air for various distances d varying from 1 to 10 mm, the frequency used is 100 kHz. By writing the total capacity of the cell C
8 S*
in the form C = Cr + -°d— one obtains Cr = 9.44 pF and
S* = 7.278 10-4 m2, where S* is the measured effective surface of the electrode. The ratio S*/nR2 is equal to 1.065.
RESULTS AND DISCUSSION
At first, the dielectric permittivity of two aqueous suspensions was measured. The solvent was an equimolar mixture of water and heavy water. Figure 1 shows the dielectric permittivity of dispersions with volume fraction of particles = 5% and 2.5% as functions of frequency. The solid curves correspond to the real part of Havriliak-Negami equation calculated with a = 0.5, P = 2 and td = 56 x 10-5 s. The static real permittivity 8(0) (or 8s) is obtained from the formula [17]:
9 2 2
8(o) - 8e - "9-^v8e(Ka)2S2, (4)
where S is a parameter close to unity, 8e = 78, Ka = 49.4 (for the aqueous suspensions).
KOnnOHAHblH XyPHAH TOM 69 № 6 2007
LOW-FREQUENCY DIELECTRIC RELAXATION
819
Imaginary relative permittivity
Freguency, Hz
Fig. 2. Imaginary relative permittivity versus frequency dependences calculated by the equation of Havriliak-Negami for dispersions with different volume fraction of particles: 4>v = 5% (1) and 2.5% (2).
Imaginary relative permittivity
0 50 100 150 200 250 300 350 400 450 500
Real relative permittivity
Fig. 3. Cole-Cole diagrams for dispersions with different volume fraction of particles: = 5% (1) and 2.5% (2).
The imaginary part of is shown in Fig. 2, and Fig. 3 represents the Cole-Cole diagram.
The relaxation frequency fel (or 1/2tctd) decreases when the size of the particles increases. Thus, the relaxation frequency fall within the region where the parasitic polarization is enormous. The dielectric permittivity calculated by the Havriliak-Negami model with a = 0.5 and P = 2 coincides reasonably with the values measured for f > 1000 Hz. For the lower frequencies, the measured permittivity is higher than the calculated one but displays a similar behaviour.
The dielectric permittivity of particle dispersion in glycerol (glycerol 95%, hydrated particles 5%) was also measured. The original suspension was centrifuged to eliminate water. Figure 4 shows the measured dielectric permittivity and that calculated by the real part of the Havriliak-Negami equation with a = 0.78, P = 1.28 and TD = 56 x 10-5 s. The value of TD corresponds to that of aqueous suspension of the same particles. A larger value of TD in glycerol is expected since its viscosity is higher then than of water. The centrifugation of the original suspension does not completely eliminate water, thus the particles remain hydrated. The particle hydration gives a possible explanation to the fact that the diffusivity of the adsorbed ions is the same for the two solutions.
In the theoretical papers [1, 2, 11, 12] the low frequency dispersion is related to the diffusion of the ions in the dispersion medium. Thus, the change of a solvent implies a change in the ions diffusivity and as a result in the relaxation times. In the experiment presented, the change of the dispersion medium (water —► glycerol) leaves the relax-
ation time unchanged. This fact is close to the results of the paper [15] where the effects was described of low frequency relaxation (also dependent of the particle size)
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