Numerical and approximate studies on the theoretical dielectric relaxation of colloidal suspensions in the time domain

In this work we show numerical calculations on the dielectric behavior of colloidal suspensions in the time domain. The theory elaborated by DeLacey and White ((1981) J Chem Soc Faraday Trans 2 77:2007–2039) for dilute suspensions in the frequency domain, will be the basis for the present study. The different contributions, and their relative importance, to the transient current density generated in the suspensions after the application of a step electric field, are calculated from the dielectric response function associated to the DeLacey and White's model. In particular, we analyze the conduction and absorption current densities in the transient states upon changing the concentration of the supporting electrolyte in the suspension. With the aim of characterizing the response of the suspension for short times, an approximation to the distribution function of relaxation times that best fits the dielectric model, is calculated. Finally, an exhaustive analysis of the behavior of the dielectric response function is carried out, together with a comparison with other models in the time domain.     

Analysis of the dielectric permittivity of suspensions by means of the logarithmic derivative of its real part

Measurement of the dielectric permittivity of colloidal suspensions in the kilohertz frequency range (the so-called low-frequency dielectric dispersion) is a promising tool for the electrokinetic characterization of colloids. However, this technique is less used than would be desirable because of the difficulties associated with the measurements, the most important of which is the electrode polarization (EP). Recently (M. Wübbenhorst and J. Van Turnhout, Dielectrics Newsl. November (2000)) a method was proposed that appears capable of separating the unwanted electrode effects from the double-layer relaxation that we are interested in. The method, based on the logarithmic derivative of raw epsilon'(omega) data (epsilon'(omega) is the real part of the permittivity of the suspension for a frequency omega of the applied AC field), is first checked against the well-known theory of the AC permittivity of colloidal suspensions developed by DeLacey and White (E. H. B. DeLacey and L. R. White, J. Chem. Soc. Faraday Trans. 277, 2007 (1981)). We show that the derivative epsilon'(D)(omega)=-(pi/2)(partial differential epsilon'/partial differential ln omega) gives an excellent representation of the true imaginary part of the permittivity, epsilon'(omega). The technique is then applied to experimental data of the dielectric constant of polystyrene and ethylcellulose suspensions. We found that epsilon'(D) displays two separated behaviors when plotted against log omega in the frequency range 100 Hz-1 MHz: a monotonous decrease (associated with EP) followed by an absorption peak (associated with the double-layer relaxation, or alpha-relaxation). Interestingly, they are separated enough to make it possible to easily find the characteristic frequency of the alpha-relaxation. Fitting a relaxation function to epsilon'(D)(omega) after eliminating the part due to EP, we could calculate the real part epsilon'(omega) and compare it to the DeLacey and White (DW) theoretical predictions. A significantly better agreement between DW calculations and experimental epsilon'(omega) data is obtained when the logarithmic derivative method is used, as compared to the classical electrode-separation techniques.     

Dielectric response of a concentrated colloidal suspension in a salt-free medium

In this paper the complex dielectric constant of a concentrated colloidal suspension in a salt-free medium is theoretically evaluated using a cell model approximation. To our knowledge this is the first cell model in the literature addressing the dielectric response of a salt-free concentrated suspension. For this reason, we extensively study the influence of all the parameters relevant for such a dielectric response: the particle surface charge, radius, and volume fraction, the counterion properties, and the frequency of the applied electric field (subgigahertz range). Our results display the so-called counterion condensation effect for high particle charge, previously described in the literature for the electrophoretic mobility, and also the relaxation processes occurring in a wide frequency range and their consequences on the complex electric dipole moment induced on the particles by the oscillating electric field. As we already pointed out in a recent paper regarding the dynamic electrophoretic mobility of a colloidal particle in a salt-free concentrated suspension, the competition between these relaxation processes is decisive for the dielectric response throughout the frequency range of interest. Finally, we examine the dielectric response of highly charged particles in more depth, because some singular electrokinetic behaviors of salt-free suspensions have been reported for such cases that have not been predicted for salt-containing suspensions.     

Polarizability and complex conductivity of dilute suspensions of spherical colloidal particles with uncharged (neutral) polymer coatings

The polarizability of polymer-coated colloidal particles, as measured via dielectric relaxation spectroscopy, reflects on the degree to which convection, diffusion, and electromigration deform the equilibrium double layer. With a polymer coating, convection and electro-osmosis are resisted by hydrodynamic drag on the polymer segments. The electro-osmotic flow near the underlying bare surface is therefore diminished. Characteristics of the particles and the adsorbed polymer can, in principle, be inferred by measuring the frequency-dependent polarizability. In this work, "exact" numerical solutions of the electrokinetic equations are used to examine how adsorbed polymer changes the particle polarizability and, hence, the conductivity and dielectric constant increments of dilute suspensions. For neutral polymer coatings, the conductivity and dielectric constant increments are found to be very similar to those of the underlying bare particles, so the response depends mostly on the underlying bare particles. These observations suggest that dielectric spectroscopy is best used to determine the underlying surface charge, with characteristics of the coating inferred from the electrophoretic or dynamic mobility, together with the hydrodynamic radius obtained from sedimentation or dynamic light scattering. Addressed briefly are the effects of added counterions and nonspecific adsorption. The electrokinetic model explored in this work can be used to guide experiments (frequency and ionic strength, for example) to either minimize or maximize the sensitivity of the complex conductivity to the coating thickness or permeability.     

Dielectric relaxation behavior of colloidal suspensions of palladium nanoparticle chains dispersed in PVP/EG solution

Dielectric measurements were carried out on colloidal suspensions of palladium nanoparticle chains dispersed in poly(vinyl pyrrolidone)/ethylene glycol (PVP/EG) solution with different particle volume fractions, and dielectric relaxation with relaxation time distribution and small relaxation amplitude was observed in the frequency range from 10(5) to 10(7) Hz. By means of the method based on logarithmic derivative of the dielectric constant and a numerical Kramers-Kronig transform method, two dielectric relaxations were confirmed and dielectric parameters were determined from the dielectric spectra. The dielectric parameters showed a strong dependence on the volume fraction of palladium nanoparticle chain. Through analyzing limiting conductivity at low frequency, the authors found the conductance percolation phenomenon of the suspensions, and the threshold volume fraction is about 0.18. It was concluded from analyzing the dielectric parameters that the high frequency dielectric relaxation results from interfacial polarization and the low frequency dielectric relaxation is a consequence of counterion polarization. They also found that the dispersion state of the palladium nanoparticle chain in PVP/EG solution is dependent on the particle volume fraction, and this may shed some light on a better application of this kind of materials.     

Wide-frequency-range dielectric response of polystyrene latex dispersions

In this work we analyze the dielectric properties of dilute colloidal suspensions of nonconducting spherical particles with a thin electrical double layer from experimental data obtained by performing impedance spectroscopy experiments over a broad frequency range, from 20 Hz to 1 GHz. The electrode polarization correction was made by fitting a circuit model in the complex impedance plane (impedance spectrum) using a constant phase angle (CPA) element to fit the electrode polarization in series with the sample impedance. This simple procedure is found to be effective in eliminating the electrode contribution. The dielectric response shows two different dispersions, the alpha relaxation (counterion relaxation) that occurs at low kilohertz frequencies, and the delta relaxation (Maxwell-Wagner effect) found in the MHz range. These are reasonably well fitted over a broad frequency range by the theoretical expressions given by a simplified standard model (not including anomalous conduction) and a generalized model (including anomalous conduction) for the low-frequency dispersion, plus Maxwell-Wagner-O'Konski theory for the delta relaxation in the mid-frequency range. An analysis was also made of the need to include, for these latices, the effects of ion mobility in the Stern layer in order for the values of the zeta-potential obtained from electrophoretic and dielectric data to be compatible with each other.     

Sound velocity dispersion in room temperature ionic liquids studied using the transient grating method

Sound velocity is determined by the transient grating method in a range from 10(6) to 10(10) Hz in three room temperature ionic liquids, 1-butyl-3-methylimidazolium bis(trifluoromethanesulfonyl)imide, 1-butyl-3-methylimidazolium hexafluorophosphate, and N,N,N-trimethyl-N-propylammonium bis(trifluoromethanesulfonyl)imide. In all room temperature ionic liquids studied, the sound velocity increased with increasing frequency. The cause of this change is posited to be structural relaxation in the room temperature ionic liquids. Frequency dependence of the sound velocity is not reproduced by a simple Debye relaxation model. The sound velocity dispersion relation in 1-butyl-3-methylimidazolium hexafluorophosphate matches a Cole-Davidson function with parameters determined by a dielectric relaxation [C. Daguenet et al., J. Phys. Chem. B 110, 12682 (2006)], indicating that structural and reorientational relaxations are strongly coupled. Conversely, the sound velocity dispersions of the other two ionic liquids measured do not match those measured for dielectric relaxation, implying that structural relaxation is much faster than the reorientational relaxation. This difference is discussed in relation to the motilities of anions and cations.     

The dielectric response of a colloidal spheroid

In this article, we present a theory for the dielectric behavior of a colloidal spheroid, based on an improved version of a previously published analytical theory [C. Chassagne, D. Bedeaux, G.J.M. Koper, Physica A 317 (2003) 321–344]. The theory gives the dipolar coefficient of a dielectric spheroid in an electrolyte solution subjected to an oscillating electric field. In the special case of the sphere, this theory is shown to agree rather satisfactorily with the numerical solutions obtained by a code based on DeLacey and White's [E.H.B. DeLacey, L.R. White, J. Chem. Soc. Faraday Trans. 2 77 (1981) 2007] for all zeta potentials, frequencies and κa1 where κ is the inverse of the Debye length and a is the radius of the sphere. Using the form of the analytical solution for a sphere we were able to derive a formula for the dipolar coefficient of a spheroid for all zeta potentials, frequencies and κa1. The expression we find is simpler and has a more general validity than the analytical expression proposed by O'Brien and Ward [R.W. O'Brien, D.N. Ward, J. Colloid Interface Sci. 121 (1988) 402] which is valid for κa1 and zero frequency.     

The effect of the concentration of dispersed particles on the mechanisms of low-frequency dielectric dispersion (LFDD) in colloidal suspensions

There are two mechanisms which are currently used to explain the low-frequency (kHz range) dispersion of the dielectric permittivity of suspensions in electrolyte solutions (LFDD). The first, the surface diffusion mechanism (SDM), associates the LFDD with the diffusion of bound ions along the particle surface caused by the applied electric field. The second, the volume diffusion mechanism (VDM), follows from the generalization for alternating fields of the classical theory of the relaxation effect in electrophoresis and associates the LFDD with the diffusion of free ions in the diffuse double layer. It has been found that VDM is much more strongly dependent on particle concentration than SDM, opening new possibilities for the investigation of each of these two mechanisms separately. The reason is that when the concentration of particles in suspension increases, the characteristic length for the propagation of volume diffusion processes decreases together with the decrease of the free electrolyte volume, whereas the characteristic length for the surface diffusion remains constant. Correspondingly, when particle concentration is raised, the relaxation time of the VDM effect must decrease, whereas it must remain constant for the SDM mechanism. Thus, by varying the concentration of particles in suspension, one can separate the dispersion curves of SDM and VDM. A simple model is elaborated which can be useful to predict the volume fraction dependence of the parameters of LFDD; in particular, its amplitude and critical frequency. The results are compared with experimental data obtained with polymer latex dispersions of volume fractions ranging from 3 to 16%. It is found that the dielectric behaviour (the volume fraction dependence of both the amplitude and critical frequency of LFDD) of the dispersions is reasonably well explained with our model, thus demonstrating that VDM prevails in the systems studied. Experimental data previously found by other authors are also discussed in the light of the model presented.     

浓差极化的介电模型——复合膜/溶液体系的数值模拟

提出了具有电导率和介电常数线性分布的介质的介电模型, 并导出了其内部电的和结构性质的参数与宏观测量的电容和电导之间定量关系的理论表达式, 以模拟复合膜中的多孔层部分的介电弛豫行为. 大量的模拟计算描述并解释了多孔层介电谱随介电常数分布、厚度等性质而变化的规律. 进一步对具有层状构造的复合膜以及复合膜和溶液相组成的多层体系的弛豫行为进行了数值模拟, 比较了三个体系(多孔层、复合膜、复合膜/液相层状体系)的介电谱, 结果揭示了介电谱对各层性质的依赖关系. 所提出的电导率和介电常数线性分布的多孔层的介电模型, 也可用于具有其他电导率、介电常数分布规律的体系.     

Colloidal electroconvection in a thin horizontal cell. III. Interfacial and transient patterns on electrodes

Previously we have reported a family of convective patterns formed by charge-stabilized aqueous colloidal suspensions under constant (dc) vertical electric fields [Y. Han and D. G. Grier, J. Chem. Phys. 122, 164701 (2005); and ibid. 125, 144707 (2006)]. These patterns form in the bulk when electrokinetic forces act in the opposite direction to gravity. Here, we report on cellular patterns that silica colloidal spheres form on a horizontal electrode when electrokinetic forces act in the same direction as gravity. We suggest that these cellular patterns form as a result of bulk electroconvection mediated by charge injection into the supporting aqueous electrolyte. This charge-injection mechanism also accounts for some aspects of electroconvective pattern formation in our earlier reports. Cellular patterns reorganize themselves into distinct transient patterns after the driving voltage is turned off. These transients cast new light on the complex interplay between the motions of charged colloidal spheres and the ionic relaxation of water undergoing electrolysis.     

Theory of Frequency-Dependent Polarization of General Planar Electrodes with Zeta Potentials of Arbitrary Magnitude in Ionic Media

Electrode polarization effects have long aggravated the efforts of low frequency analysis, particularly those investigations carried out on biological material or in highly conductive media. Beginning from elementary equations of electrostatics and hydrodynamics, a comprehensive model is devised to account for the screening of a general planar electrode by an ionic double layer. The surface geometry of the planar electrode is left unspecified to include any type of micromachined array. Building on the previous work by DeLacey and White (1982, J. Chem. Soc. Faraday Trans. 2 78, 457) using a variational theorem, we extend their numerical results with compact analytic solutions, analogous to the Debye-Hückel potential for dc systems, but applicable now to dynamic ac experiments. The variational approach generates functions that are not restricted by perturbation expansions or numerical convergence, representing optimal approximations to the exact solutions. Copyright 2000 Academic Press.     

Rheological behavior of PMN gels for solid freeform fabrication

Lead magnesium niobate (PMN) is a relaxor ferroelectric material and have widespread applications in the manufacture of multilayer electronic devices such as ceramic capacitors, actuators and transducers. The dielectric constant of this electrostrictive material is much higher than the well known ferroelectric barium titanate. However, aqueous processing of PMN is not investigated yet especially for the novel wet shaping fabrication. In this study, concentrated aqueous colloidal PMN gels have been designed to use in the robocasting process. Concentrated PMN suspensions were stabilized by polyacrylic acid and then gelation induced by changing the pH or ionic strength of the suspension or by addition of a cationic polyelectrolyte to the system. Through this procedure it was essential to understand the solid–liquid transition under shear to explore the feasibility of forming without excessive use of polymers. Therefore, rheological response of the samples having a gel network was investigated. Results showed that gelation induced by cationic polyethylene imine or by multivalent salts were successful methods in preparation of PMN gels. However, gelation induced by changing the pH of the suspension was challenging due to ion dissolution from PMN surface.     

dc Electrokinetics for spherical particles in salt-free concentrated suspensions including ion size effects

We study the electrophoretic mobility of spherical particles and the electrical conductivity in salt-free concentrated suspensions including finite ion size effects. An ideal salt-free suspension is composed of just charged colloidal particles and the added counterions that counterbalance their surface charge. In a very recent paper [Roa et al., Phys. Chem. Chem. Phys., 2011, 13, 3960-3968] we presented a model for the equilibrium electric double layer for this kind of suspensions considering the size of the counterions, and now we extend this work to analyze the response of the suspension under a static external electric field. The numerical results show the high importance of such corrections for moderate to high particle charges, especially when a region of closest approach of the counterions to the particle surface is considered. The present work sets the basis for further theoretical models with finite ion size corrections, concerning particularly the ac electrokinetics and rheology of such systems.     

Temperature-Dependent Dielectric Relaxation of 2-Ethoxyethanol, Ethanol, and 1-Propanol in Dimethylformamide Solution Using the Time-Domain Technique

Complex reflection coefficients for 2-ethoxyethanol–dimethylformamide (DMF), ethanol–DMF, and 1-propanol–DMF mixtures at several temperatures from 20 to 50° and the frequency range 10 MHz to 10 GHz were determined by time-domain spectroscopy in reflection mode. Fourier transforms and least-squares fitting were used to obtain complex permittivity, static dielectric constant, and relaxation time. The excess dielectric parameters, Kirkwood correlation factors, and thermodynamic properties for the binary mixtures were also determined. The static dielectric constant for the mixtures was fitted well with the modified Bruggeman model.     

Deposition of cation-absorptive biocolloids onto a charged surface

Classical theoretical assumptions, which are implausible for describing biological behavior in real systems include uniform fixed charge distribution in colloidal outer membrane layer, uniform dielectric constant throughout the membrane phase, and point charge for ionic sizes. In the present study, absorption of cations by fixed functional groups in the membrane layer, variation in dielectric constant in a system, and effect of ionic sizes are considered to investigate the deposition of biocolloids each covered with an ion-penetrable membrane. The simulated results reveal that a larger numbers of cations involved in the formation of a cations-fixed groups complex, a smaller dielectric constant near a biological uncharged core, a larger dielectric constant of the membrane phase, a smaller cation-absorption equilibrium constant, a smaller concentration of total functional groups in the membrane layer, a thicker membrane, smaller cations, larger anions, and larger functional groups yield a faster rate of deposition.     

Two-Layer Model of Colloidal Gold Bioconjugates and Its Application to the Optimization of Nanosensors

An optical model of the conjugates of colloidal gold nanoparticles with biopolymers is analyzed in terms of two-layer spherical particle with the gold core and dielectric coating. The Mie theory was used to study the dependence of variations (caused by the adsorption of a biopolymer on the particle surface) in the extinction and light scattering (at 90°) spectra on the gold core diameter (d = 5–200 nm), the shell refractive index and thickness s (ratio s/d = 0–1). Some theoretical results by Templeton et al. (J. Phys. Chem. B, 2000, vol. 104, pp. 564–570) on the two-layer dipole model were corrected. It is shown that the dependence of spectral shifts of the extinction and scattering peaks on the conjugate structure is adequately described by the dipole approximation. In particular, we found the universal dependence of the normalized spectral shift of extinction maximum on the s/d ratio. Having in mind the optimization of conjugate–nanosensors, we studied the problem of what particle size is optimal for the transformation of biopolymer adsorption event into the variations in the spectral parameters of extinction and light scattering. Based on the calculations of extinction maximum values and positions, as well as on calculated differential extinction spectra, we concluded that a maximal conjugate efficiency corresponds to the core diameters of 40–80 nm. We also discussed the principles of conjugate–nanosensors optimization for the polymer shell structure.     

The response of a colloidal suspension to an alternating electric field

This paper is concerned with the calculation of the complex conductivity K* of a suspension, a quantity which may be determined experimentally from the measurement of the alternating current which flows between a pair of electrodes in the suspension due to an alternating voltage difference. A semi-analytic formula is derived for the complex conductivity of a dilute suspension of spherical particles with small dielectric constant which is reasonably accurate for ?-potentials of less than 50 mV. For such suspensions this formula represents a very economical alternative to the exact computer calculation of K* described by DeLacey and White (ref. 2). Although the formula for K* is derived for particles with fixed surface charge, it is shown that the formula can also be applied to a more general class of suspensions, in which the surface charge arises from the dissociation of a single type of surface group.     

Remarks on the determination of low-frequency measurements of the dielectric response of colloidal suspensions

Electrode impedance is a significant artifact in low frequency dielectric measurements involving conducting media. In their recent review article regarding the dielectric dispersion of aqueous colloidal systems, Grosse and Delgado [1] presented an electrode polarization model that provides a physical explanation of the effect of electrolyte concentration and mobility, electrode spacing, and frequency. Although the model properly predicts the undesired phenomenon, the low frequency scaling, often used to identify electrode polarization effects, is incorrect. The apparent dielectric constant actually follows an ω^? 2 frequency dependence for ω/κ²D ? 1, where κ^? 1 is the Debye length and D is an average ion diffusion coefficient. Strictly speaking, the predicted scaling with exponent ? 1.5 is applicable only for sufficiently high frequencies, where electrode polarization is insignificant. This letter is intended to help clarify matters: the asymptotic behavior of the polarization model is examined, and the approximate expressions representing the real part of the complex dielectric constant of a parallel plate cell containing electrolyte solutions or colloidal suspensions are discussed.