Compensating for Electrode Polarization in Dielectric Spectroscopy Studies of Colloidal Suspensions: Theoretical Assessment of Existing Methods

Abstract

Dielectric spectroscopy can be used to determine the dipole moment of colloidal particles from which important interfacial electrokinetic properties, for instance their zeta potential, can be deduced. Unfortunately, dielectric spectroscopy measurements are hampered by electrode polarization (EP). In this article, we review several procedures to compensate for this effect. First EP in electrolyte solutions is described: the complex conductivity is derived as function of frequency, for two cell geometries (planar and cylindrical) with blocking electrodes. The corresponding equivalent circuit for the electrolyte solution is given for each geometry. This equivalent circuit model is extended to suspensions. The complex conductivity of a suspension, in the presence of EP, is then calculated from the impedance. Different methods for compensating for EP are critically assessed, with the help of the theoretical findings. Their limit of validity is given in terms of characteristic frequencies. We can identify with one of these frequencies the frequency range within which data uncorrected for EP may be used to assess the dipole moment of colloidal particles. In order to extract this dipole moment from the measured data, two methods are reviewed: one is based on the use of existing models for the complex conductivity of suspensions, the other is the logarithmic derivative method. An extension to multiple relaxations of the logarithmic derivative method is proposed.

Keywords: colloidal suspension; complex conductivity and permittivity; electrode polarization.

Figure 1

Schematic representation of the distribution of particles (ions, colloidal particles) at a given time, when an AC electric field is applied to a solution (x = e) or suspension (x = s). Due to the blocking nature of the electrodes, ions and colloidal particles build-up close to the electrodes at low frequencies. The equivalent impedance of the cell ${\tilde{Z}}_{c,x}$ is then made up of two contributions: the wanted “true” impedance ${\tilde{Z}}_x$ of the bulk of the investigated solution or suspension and the contribution due to the polarization of the electrodes ${\tilde{Z}}_{EP}.$

Figure 2

(A) Conductivity K(S/m) and (B) relative permittivity (epsilon) of a solution of divalent salt as function of the applied electric field frequency, similar to MgCl2, for which D₁ = 1.4 × 10⁻⁹ m²/s and D₂ = 2.0 × 10⁻⁹ m² /s. The salt concentration is 0.5 mM. The electrodes are planar. The important characteristic frequencies associated with the system are ${\omega }_0=D_0{\kappa }^2$ and ω_P = 2κD₀/d where d = 10 × 10⁻³ m is the distance between the electrodes. The full solution found in this study, i.e., Equation (23) (red full line) is compared to the solution (blue dashed line) found in Chassagne et al. (2002), which was derived for the case ω ≪ ω₀. As expected, the two solutions overlap for ω ≪ ω₀. The green dotted line in the conductivity plot represents the theoretical conductivity, the value of which is given by Equation (29). The green dotted line in the epsilon plot represents the relative permittivity of water.

Figure 3

Equivalent circuit representation for a cell containing a solution or suspension, with blocking electrodes.

Figure 4

(A) Conductivity K(S/m) and (B) relative permittivity (epsilon) as function of frequency of a suspension of 100 nm colloidal spheres (ϕ = 1%, eζ/kT = 4) in a 1 mM electrolyte solution of monovalent salt solution for which $D_1=2\times 10^{-9}$ m²/s and $D_2=3\times 10^{-9}$ m²/s. Red curve: the suspension in the absence of electrode polarization, corresponding to ε_s and K_s from Equation (8). Dashed blue curve: the equivalent circuit model corresponds to the theoretical prediction provided that one takes R_b = 1/K_s, C_b = ε₀ε_s and C_EP = 0.

Figure 5

(A) Conductivity K(S/m) and (B): relative permittivity (epsilon) as function of frequency. Suspension with same properties as the one given in Figure 4. The suspending electrolyte solution (blue line) and suspension (red line) in the presence of electrode polarization. The magenta dashed line corresponds to the solution found in Figure 4 for no EP. The equivalent circuit of the suspension in the presence of electrode polarization was constructed by taking R_b = 1/K_s, C_b = ε₀ε_s and C_EP = ε₀ε_sκd/2 with d = 10 mm.

Figure 6

Enlargement of Figure 5. Above ω_b one finds that ε_c,s ≃ ε_s i.e., that the EP plays no role anymore. Note that for any frequency above ω_P the relation K_c,s ≃ K_s holds (see Figure 5). This implies that for frequencies above ω_b one has ${\tilde{K}}_{c,s}\simeq {\tilde{K}}_s$ (or equivalently ${\tilde{\epsilon }}_{c,s}\simeq {\tilde{\epsilon }}_s$). Similarly, above ω_b one has ${\tilde{K}}_{c,e}\simeq {\tilde{K}}_e$ which implies in particular for the present figure that ε_c,s ≃ ε_e.

Figure 7

(A) Conductivity K_c,s(S/m) and (B) relative permittivity (epsilon) ε_c,s of a suspension of 100 nm colloidal spheres (ϕ = 20%, eζ/kT = 4) in a 0.23 mM electrolyte solution of monovalent salt solution for which $D_1=2\times 10^{-9}$ m²/s and $D_2=1.98\times 10^{-9}$ m²/s. The spacing between electrodes is 10 mm. Red curve: the case where ε^* = ε_e; Dashed magenta curve: the case where ε^* = ε_s.

Figure 8

Relative permittivity (epsilon) of a suspension of 100 nm colloidal spheres (ϕ = 20%, eζ/kT = 4) in a 0.23 mM electrolyte solution of monovalent salt solution for which $D_1=2\times 10^{-9}$ m²/s and $D_2=1.98\times 10^{-9}$ m²/s. The spacing between electrodes is 10 mm. (A) The case where ε^* = ε_e. (B) The case where ε^* = ε_s Red curves: ε_sBlue curves: ε_s found by using the subtraction procedure

Figure 9

Schematic representation of a 4-electrode cell; the voltage difference at the outer electrodes, where EP takes place, is ${\tilde{V}}_o=\Delta {\tilde{V}}_{out}={\tilde{Z}}_o\tilde{I}$ whereas the voltage difference at the inner (probing) electrodes is ${\tilde{V}}_i=\Delta {\tilde{V}}_{in}$; as EP should be minimized at the inner electrodes, the voltage difference measured at the inner electrodes ${\tilde{V}}_i$ should be done with virtually zero inner current (${\tilde{I}}_i\simeq 0$), implying that the impedance ${\tilde{Z}}_i$ of the inner electrodes should be virtually infinite. One then obtains, from the measurement of ${\tilde{V}}_i$ and $\tilde{I}$ : ${\tilde{Z}}_s={\tilde{V}}_i/\tilde{I}.$

Figure 10

Comparison between ${\epsilon }_{D,s}^{\prime \prime }$, ${\epsilon }_{D,c,s}^{\prime \prime }$, $\left({\epsilon }_s^{\prime },{\epsilon }_s^{\prime \prime }\right)$ and $\left({\epsilon }_{c,s}^{\prime },{\epsilon }_{c,s}^{\prime \prime }\right)$ as function of frequency; suspension of 100 nm colloidal spheres (ϕ = 1%, eζ/kT = 4) in a 1 mM electrolyte solution of monovalent salt solution for which $D_1=2\times 10^{-9}$ m²/s and $D_2=1.98\times 10^{-9}$ m²/s. The spacing between the electrodes is 10 mm. Using the logarithmic derivative method (in blue) enables to better distinguish the relaxation processes associated with the colloidal particles.

Figure 11

Top: Logarithmic derivative ${\epsilon }_D^{\prime \prime }$ for a suspension consisting colloidal spheres (ϕ = 1%, eζ/kT = 4) in a 1 mM electrolyte solution of monovalent salt for which D₁ = 2 × 10⁻⁹ m²/s and D₂ = 3 × 10⁻⁹ m²/s. The electrode spacing is 10 mm. (left): 25 nm particles and (right): 250 nm particles. The fit with the dipolar coefficient was done between [ω_b = 2.5 × 10⁶-10¹⁰] rad/s using (ζ, a) as adjustable parameters. Bottom: Recalculated $\tilde{\beta }$ (blue) and original beta (red).

Figure 12

Top: Logarithmic derivative ${\epsilon }_D^{\prime \prime }$ for a suspension consisting of colloidal spheres(ϕ = 1%, eζ/kT = 4) in a 1 mM electrolyte solution of monovalent salt for which D₁ = 2 × 10⁻⁹ m²/s and D₂ = 3 × 10⁻⁹ m²/s. (left): 25 nm particles and (right): 250 nm particles. The electrode spacing is 10 mm, and the 2 HN fit was done between [2.5.10⁶–10¹⁰] rad/s. Bottom: recalculated $\tilde{\beta }$ [blue line, from Equation (74) and original beta (red)]: despite the inaccuracy in the 2 HN fit, the error in $\tilde{\beta }$ is small, see explanation in text.