Impedance-Frequency Response of Closed Electrolytic Cells.

The electric AC response of electrolytic cells with DC bias is analyzed solving numerically the Poisson-Nernst-Planck equations and avoiding the commonly used infinite solution approximation. The results show the presence of an additional low-frequency dispersion process associated with the finite spacing of the electrodes. Moreover, we find that the condition of fixed ionic content inside the electrolytic cell has a strong bearing on both the steady-state and the frequency response. For example: the characteristic frequency of the high-frequency dispersion decreases when the DC potential increases and/or the electrode spacing decreases in the closed cell case, while it remains essentially insensitive on these changes for open cells. Finally, approximate analytic expressions for the dependences of the main parameters of both dispersion processes are also presented.

A program for the fitting of up to three Havriliak-Negami dispersions to dielectric data.

A new version of the DielParamFit program [J. Colloid Interface Sci. 419 (2014) 102-106] for the fitting of a superposition of dispersion terms to measured dielectric data is presented. It extends its applicability to a wide range of new systems by allowing the use of up to three Havriliak-Negami dispersions, each of which can be readily transformed into Debye, Cole-Cole, or Cole-Davidson terms. Moreover, it greatly enhances its usability by means of a graphic interface that displays the measured data together with the model spectra allowing to iteratively adjust the chosen terms and guess parameter values guided by a live view of the obtained results. The DielParamFit_2 program executable file for Microsoft Windows is available upon request from the author.

Numerical Solution of the Electrokinetic Equations for Multi-ionic Electrolytes Including Different Ionic Size Related Effects.

One of the main assumptions of the standard electrokinetic model is that ions behave as point-like entities. In a previous work (López-García, et al., 2015) we removed this assumption and analyzed the influence of finite ionic size on the dielectric and electrokinetic properties of colloidal suspensions using both the Bikerman and the Carnahan⁻Starling equations for the steric interactions. It was shown that these interactions improved upon the standard model predictions so that the surface potential, electrophoretic mobility, and the conductivity and permittivity increment values were increased. In the present study, we extend our preceding works to systems made of three or more ionic species with different ionic sizes. Under these conditions, the Bikerman and Carnahan⁻Starling expressions cease to be valid since they were deduced for single-size spheres. Fortunately, the Carnahan⁻Starling expression has been extended to mixtures of spheres of unequal size, namely the "Boublik⁻Mansoori⁻Carnahan⁻Starling⁻Leland" (BMCSL) equation of state, making it possible to analyze the most general case. It is shown that the BMCSL expression leads to results that differ qualitatively and quantitatively from the standard electrokinetic model.

Influence of steric interactions on the dielectric and electrokinetic properties in colloidal suspensions.

One of the main assumptions of the standard electrokinetic model is that ions behave as point like entities. In this work we remove this assumption and analyze the main consequences of finite ionic size on the dielectric and electrokinetic properties of colloidal suspensions. We represent the steric interactions by means of the Bikerman and the Carnahan-Starling equations and solve numerically the standard linearized electrokinetic equations in the stationary and the frequency domains, for surface charge density and electrolyte solution concentration values typically encountered in colloidal suspensions. In all cases the steric interactions improve upon the predictions of the standard model since the surface potential, the electrophoretic mobility, and the conductivity and permittivity increments increase. However, the corrections introduced by the Bikerman equation are generally small: less than 10% as compared to the standard model. On the contrary, the Carnahan-Starling equation leads to corrections to the surface potential versus surface charge and the electrophoretic mobility values that easily surpass 10% and can attain values as high as 50%. Corrections to the conductivity and permittivity increments are smaller but still non negligible.

Influence of the finite size and effective permittivity of ions on the equilibrium double layer around colloidal particles in aqueous electrolyte solution.

The equilibrium properties of the electrical double layer surrounding a charged spherical colloidal particle immersed in an aqueous electrolyte solution are examined taking into account the finite ion size. This study includes the representation of the steric interactions among ions using both the Bikerman and the Carnahan-Starling models, an account of all the effects related to the representation of hydrated ions as dielectric spheres (dependence of the electrolyte solution permittivity, on the local ion concentration, and appearance of the Born and the dielectrophoretic forces acting on the ions), and solution of the problem for both high and low surface charge values. We find that the Carnahan-Starling model together with effective ion permittivity related effects appears to be able to provide an interpretation to the electrokinetic potential vs. surface charge dependence in the case of colloidal particles suspended in aqueous electrolyte solutions. On the contrary, for electrode-electrolyte systems, both the Bikerman and the Carnahan-Starling models might be able to explain this dependence.

A program for the fitting of Debye, Cole-Cole, Cole-Davidson, and Havriliak-Negami dispersions to dielectric data.

The description and interpretation of dielectric spectroscopy data usually require the use of analytical functions, which include unknown parameters that must be determined iteratively by means of a fitting procedure. This is not a trivial task and much effort has been spent to find the best way to accomplish it. While the theoretical approach based on the Levenberg-Marquardt algorithm is well known, no freely available program specifically adapted to the dielectric spectroscopy problem exists to the best of our knowledge. Moreover, even the more general commercial packages usually fail on the following aspects: (1) allow to keep temporarily fixed some of the parameters, (2) allow to freely specify the uncertainty values for each data point, (3) check that parameter values fall within prescribed bounds during the fitting process, and (4) allow to fit either the real, or the imaginary, or simultaneously both parts of the complex permittivity. A program that satisfies all these requirements and allows fitting any superposition of the Debye, Cole-Cole, Cole-Davidson, and Havriliak-Negami dispersions plus a conductivity term to measured dielectric spectroscopy data is presented. It is available on request from the author.

Influence of the dielectrophoretic force in mixed electrical double layers.

The equilibrium properties of a charged plane immersed in an aqueous electrolyte solution are examined using a generalized Poisson-Boltzmann equation that takes into account the finite ion size by modeling the solution as a suspension of polarizable insulating spheres in water. This formalism is applied to a general solution composed of two or more counterion species with different valences, sizes, and effective permittivity values. It is shown that, due to the dependence of the dielectrophoretic force on the ion size and effective permittivity value, the concentration of the smaller counterion strongly increases while that of the larger one decreases in the immediate vicinity of the charged surface. As a result the surface potential value strongly increases as compared to the usual modified Poisson-Boltzmann theory that only includes steric interactions among ions. This effect is particularly important in the case of mixtures of univalent and divalent counterions, being significant even for relatively low surface charge values.

Extension of a classic low frequency dielectric dispersion theory of colloidal suspensions to include different counterion and co-ion valences, a broad frequency range, and the stagnant layer conductivity.

A unified extension of the classic Shilov-Dukhin theory of the low frequency dielectric dispersion of colloidal dispersions is presented. Purely analytical expressions for the AC dielectric and electrokinetic response over a broad frequency range including different counterion and co-ion valences and the presence of the stagnant layer conductivity are deduced. The obtained results are generally in good to very good agreement with available numerical data, showing that they should be useful for the interpretation of a broad range of experimental results without having to rely on numerical calculations.

Equilibrium properties of charged spherical colloidal particles suspended in aqueous electrolytes: finite ion size and effective ion permittivity effects.

The equilibrium properties of a charged spherical colloidal particle immersed in an aqueous electrolyte solution are examined using an extension of the Standard Electrokinetic Model that takes into account the finite ion size by modeling the aqueous electrolyte solution as a suspension of polarizable insulating spheres in water. We find that this model greatly amplifies the steric effects predicted by the usual modified Poisson-Boltzmann equation, which only imposes a restriction on the ability of ions to approach one another. This suggests that a solution of the presented model under nonequilibrium conditions could have important consequences in the interpretation of dielectric and electrokinetic data in colloidal suspensions.

Poisson-Boltzmann description of the electrical double layer including ion size effects.

The electrical double layer is examined using a generalized Poisson-Boltzmann equation that takes into account the finite ion size by modeling the aqueous electrolyte solution as a suspension of polarizable insulating spheres in water. We find that this model greatly amplifies the steric effects predicted by the usual modified Poisson-Boltzmann equation, which imposes only a restriction on the ability of ions to approach one another. This amplification should allow for an interpretation of the experimental results using reasonable effective ionic radii (close to their well-known hydrated values).

Extension of a classic thin double layer polarization theory of colloidal suspensions to include the stagnant layer conductivity.

A rigorous extension of the classic Dukhin-Shilov thin double layer polarization theory including the stagnant layer conductivity is presented. Precisely the same assumptions and approximations made in the original theory are maintained, and the same adsorption isotherms are used as in most of the existing numerical calculations. The obtained analytical results improve upon existing approximate extensions, mainly for low surface conductivities and high surface potentials and for high surface conductivities and low surface potentials. Moreover, they avoid the assumption that all the adsorbed ions in the stagnant layer must have a single sign. Finally, they present a very good agreement with numerical calculations specifically made using the same system parameters.

Extension of a classic theory of the low frequency dielectric dispersion of colloidal suspensions to the high frequency domain.

The classic Shilov-Dukhin theory of the low frequency dielectric dispersion of colloidal suspensions in binary electrolyte solutions [ Shilov , V. N. ; Dukhin , S. S. , Colloid J. 1970 , 32 , 245. ; Dukhin , S. S. ; Shilov , V. N. Dielectric Phenomena and the Double Layer in Disperse Systems and Polyelectrolytes ; Wiley : New York , 1974 ] was developed for the frequency range corresponding to the concentration polarization phenomenon: up to a few megahertz. While a few extensions to a broad frequency range including the Maxwell-Wagner-O'Konski dispersion exist, they all consist of modifications of the final results of the theory rather than modifications of its hypothesis, extending their validity to high frequencies. In this work we avoid this artificial process by providing a high frequency extension fully from within the theory.

Generalization of a classic theory of the low frequency dielectric dispersion of colloidal suspensions to electrolyte solutions with different ion valences.

The classic Shilov-Dukhin theory of the low frequency dielectric dispersion of colloidal suspensions in binary electrolyte solutions was developed for symmetric electrolytes: equal counterion and co-ion valences. A rigorous generalization of this theory to asymmetric electrolytes, such that the valence of counterions is double or half the valence of co-ions, is presented. This generalization is possible because analytical solutions of the intervening integrals exist for these two particular cases but do not exist in the general case of different counterion and co-ion valences, a result apparently overlooked or ignored in the past.

Generalization of a classic thin double layer polarization theory of colloidal suspensions to electrolyte solutions with different ion valences.

The classic Dukhin-Shilov theory for the thin double layer polarization of colloidal suspensions in binary electrolyte solutions was developed for symmetric electrolytes: equal counterion and co-ion valences. A rigorous generalization of this theory to asymmetric electrolytes, such that the valence of counterions is double or half the valence of co-ions, is presented. This generalization is possible because analytical solutions of the intervening integrals exist for these two particular cases but do not exist in the general case of different counterion and co-ion valences, a result apparently overlooked or ignored in the past.

Dependence of the dielectric properties of suspensions on the volume fraction of suspended particles.

It is shown that the fundamental expression for the complex permittivity epsilons* of a dilute suspension of monodispersed, spherical particles, epsilons*=epsilone*(1+3phid*), where epsilone* is the complex permittivity of the suspending medium and d* the dipolar coefficient, is strictly valid for any value of the volume fraction phi of particles in the suspension, provided that d* is interpreted as the ensemble average value of the dipolar coefficient of the particles and is defined in terms of the macroscopic electric field in the suspension.

Dependence of the broad frequency dielectric spectra of colloidal polystyrene particle suspensions on the difference between the counterion and co-ion diffusion coefficients.

Dielectric properties of four suspensions of spherical polystyrene particles were measured at 25 degrees C over a broad frequency range extending from 100 Hz to 10 MHz, using a HP 4192 A Impedance Analyzer. The instrument was coupled to a cell with parallel platinum black electrodes and variable spacing, and the quadrupole calibration method was used. The aqueous electrolyte solutions were prepared using equal concentrations of NaCl, KCl, NaAc, or KAc, so that the calculated Debye screening length and Zeta potential remained constant, while the conductivity changed. The polystyrene particles used (Interfacial Dynamics Corp., surfactant-free white sulfate latex) have a diameter of 1 micron and a surface charge density that is independent of the pH. The dielectric spectra were described using the Nettelblad-Niklasson expression combined with a Debye type high-frequency term and analyzed using the Shilov-Dukhin theory and numerical results. The theoretical prediction that the low-frequency dispersion parameters are determined by the co-ion diffusion coefficient was experimentally confirmed. This also allowed to justify an alternative definition of the characteristic time of the low-frequency dispersion. On the contrary, the prediction that the high-frequency dispersion parameters are determined by the diffusion coefficient of counterions could not be confirmed, possibly due to experimental problems. However, the zeta-potential values deduced from high-frequency data were compatible with values deduced from electrophoretic mobility measurements.

Numerical calculation of the dielectric and electrokinetic properties of vesicle suspensions.

The dielectric and electrokinetic properties of aqueous suspensions of vesicles (unilamellar liposomes) are numerically calculated in the 1 Hz to 1 GHz frequency range using a network simulation method. The model consists of a conducting internal medium surrounded by an insulating membrane with fixed surface charges on both sides. Without an applied field, the internal medium is in electric equilibrium with the external one, so that it also bears a net volume charge. Therefore, in the presence of an applied ac field, there is fluid flow both in the internal and in the external media. The obtained results are qualitatively different from those corresponding to suspensions of charged homogeneous particles, mainly due to the existence of an additional length scale (the membrane thickness) and the corresponding dispersion mechanism, charging of the membrane. Because of this dispersion, the shapes of the spectra change with the size of the particles (at constant zeta potential and particle radius to Debye length ratio) instead of merely shifting along the frequency axis. A comparison between the numerical results and those obtained using approximate analytical expressions shows deviations that are, in general, sufficiently large enough to show the necessity to use numerical results in order to interpret broad frequency range dielectric and electrokinetic measurements of vesicle suspensions.

Suspended particles surrounded by an inhomogeneously charged permeable membrane. Solution of the Poisson-Boltzmann equation by means of the network method.

The Poisson-Boltzmann equation is numerically solved for a suspended spherical particle surrounded by a permeable membrane that contains an inhomogeneous distribution of fixed charges. The calculations are carried out using the network simulation method, which makes it possible to solve the problem in the most general case, extending previous results (J.P. Hsu, Y.C. Kuo, J. Membrane Sci. 108 (1995) 107). Approximate analytical expressions for the counterion concentration and the electric potential in the membrane are also presented, together with criteria that determine their ranges of validity. The limiting case of a distribution of fixed charges in the membrane that reduces to a surface charge is also analyzed. It is shown that the solution for this case, considering a vanishingly small radius of the core, reduces to a superposition of solutions corresponding to a charged impermeable particle suspended in an electrolyte solution and to a cavity filled with a charged electrolyte solution.

Corrected results for the influence of size, zeta potential, and state of motion of dispersed particles on the conductivity of a colloidal suspension.

A correction of a recent work on the dependence of the DC conductivity of diluted colloidal suspensions on the size, zeta potential, and state of motion of dispersed particles (C. Grosse, S. Pedrosa, V.N. Shilov, J. Colloid Interface Sci. 251 (2002) 304) is presented. It is shown that the procedure used in that work to calculate the contribution of the particles to the conductivity of the suspension leads to a result that includes the variation of the conductivity of the dispersion medium. Revised analytical and numerical calculations are presented, which strongly reinforce the conclusions reached in the original work: The expression for the conductivity increment based on the value of the dipolar coefficient of the suspended particles (calculated taking into account their electrophoretic motion) appears to be valid over the whole range of particle sizes.

Field-induced disturbance of the double layer electro-neutrality and non-linear electrophoresis.

The influence of applied electric field on the field-induced variation of the electrolyte concentration (concentration polarization) disturbs the electro-neutrality of the system, represented by a dispersed particle and its electric double layer in electrolyte solution. The manifestation of this electro-neutrality disturbance in the non-linear electrophoresis was considered in the framework of a procedure of successive approximations in powers of the applied field strength. Analytic expressions describing the component of the electrophoretic velocity proportional to the cubic power of the applied field strength (cubic electrophoresis) were obtained for arbitrary values of the surface conductivity (of Dukhin number). The model restrictions are spherical non-conducting particle with homogeneous surface and thin double layer.