Dielectric relaxation spectroscopy of aqueous micellar electrolyte solutions: A novel application to infer Dukhin number and zeta potential of a micelle

The complex permittivities of aqueous SDS solutions, with and without the addition of sodium chloride (NaCl), are measured in the frequency range from 200 MHz to 14 GHz. The SDS concentrations are chosen such that the SDS molecules aggregate to micelles. In this frequency range, the measured spectra allow for the identification of two different relaxation processes. That is, the relaxation of the water molecules at frequencies above 1 GHz and the micellar relaxation at frequencies lower than 1 GHz. It is found that the addition of NaCl to the system mostly affects the micellar relaxation process. In detail, the time constant as well as the amplitude of the relaxation decrease by adding NaCl. These effects are attributed to the change in the solution conductivity that changes the properties of the micelle's electrical double layer. We also extract the Dukhin number of the micelles as a function of surfactant and electrolyte content from the measurements. The Dukhin number is a dimensionless group that describes the influence of the surface conductivity on a phenomena. A regression between Dukhin numbers and free sodium ions is found so that all data collapses on a single curve independent of the surfactant concentration. The surface conductivity is a manifestation of the electrical double layer and we use the Bikerman equation to infer the zeta potential of the micelles. Comparison to literature data shows very good agreement and proves that dielectric relaxation spectroscopy can be engaged to infer the zeta potential of micelles. Abbreviations: CMC critical micelle concentration, DRS dielectric relaxation spectroscopy, EDL electrical double layer     

Electrokinetic characterization of porous plugs from streaming potential coupled with electrical resistance measurements

The zeta potential of mixed nickel-iron oxide particles is evaluated by a new laboratory instrument. This latter allows the measurement of streaming potential together with the electrical resistance of porous plugs. The conductivity of electrolyte inside plug (pore conductivity) is deduced from electrical resistance measurements and is used together with streaming potential to evaluate the zeta potential by accounting for the surface conduction phenomenon. It is shown that neglecting the surface conduction phenomenon leads to a substantial underestimation of the zeta potential. The coupled measurements of streaming potential and plug electrical resistance yield zeta potential values that are in very good agreement with those obtained by electrophoresis. The densification of the porous plug with increasing pressure increments is put in evidence by the decrease in measured streaming potentials. Electrical resistance measurements make it possible to account for the increase in surface conductivity resulting from the more compacted structure of the plug. By doing so, the calculated zeta potential is found to be virtually independent of the pressure difference involved in streaming potential experiments, whereas the negligence of surface conduction phenomenon leads to a decrease in the apparent zeta potential with increasing pressure level.     

Zeta potential of ion-conductive membranes by streaming current measurements

Surface charge properties have a significant influence on membrane retention and fouling performance. As a key parameter describing the surface charge of membranes used in aqueous applications, zeta potential measurements on membranes of various types have attracted great attention. During the zeta potential characterization of a series of ion-conductive sulfonated poly(sulfone) membranes, it was found that the measured streaming current varied with the thickness of the sample, which is not predicted by the classical Smoluchowski equation. Moreover, for higher conductivity membranes with an increased concentration of sulfonate groups, the zeta potential tended toward zero. It was determined that the influence of membrane bulk conductance on the measured streaming current must be taken into account in order to correctly interpret the streaming current data for ion-conductive polymers and understand the relationship between membrane chemical composition and zeta potential. Extrapolating the measured streaming current to a membrane thickness of zero has proven to be a feasible method of eliminating the error associated with measuring the zeta potential on ion conductive polymer membranes. A linear resistance model is proposed to account for the observed streaming currents where the electrolyte channel is in parallel with the ion-conductive membranes.     

Estimation of zeta potential by electrokinetic analysis of ionic fluid flows through a divergent microchannel

The streaming potential is generated by the electrokinetic flow effect within the electrical double layer of a charged solid surface. Surface charge properties are commonly quantified in terms of the zeta potential obtained by computation with the Helmholtz-Smoluchowski (H-S) equation following experimental measurement of streaming potential. In order to estimate a rigorous zeta potential for cone-shaped microchannel, the correct H-S equation is derived by applying the Debye-Hückel approximation and the fluid velocity of diverging flow on the specified position. The present computation provides a correction ratio relative to the H-S equation for straight cylindrical channel and enables us to interpret the effects of the channel geometry and the electrostatic interaction. The correction ratio decreases with increasing of diverging angle, which implies that smaller zeta potential is generated for larger diverging angle. The increase of Debye length also reduces the correction ratio due to the overlapping of the Debye length inside of the channel. It is evident that as the diverging angle of the channel goes to nearly zero, the correction ratio converges to the previous results for straight cylindrical channel.     

Theoretical study of the electrokinetic and electrochemical behaviors of two-layer composite membranes

A theoretical study concerning the effect of structure (porosity, pore radius and layer thickness) and surface characteristics (zeta potential) of two-layer composite membranes on global streaming potential (SP_g), membrane potential (Em_g) and membrane conductivity (λ_g) is presented. To this end, each layer of the composite membrane (composed of a support layer and a filtering layer) was modeled as a bundle of identical capillary tubes with connections between pores of the two layers (the pores in the filtering layer being smaller than those of the support layer). The global parameters SP_g, Em_g and λ_g were calculated by using the theory of thermodynamics of irreversible processes and a space charge model. SP_g, Em_g and λ_g were expressed as a function of the individual parameters of each layer SP⁽ⁱ⁾, Em⁽ⁱ⁾ and λ⁽ⁱ⁾, respectively, the length fraction of the support layer, the porosity and pore radius ratios. It was shown that the electrokinetic (streaming potential and membrane conductivity) and electrochemical (membrane potential) behaviors of such composite membranes vary between that of single layers. For streaming potential, the results indicate that the contribution of the filtering layer to the global streaming potential is very little influenced by zeta potentials of both types of pores. It appears that the individual streaming potential of the filtering layer greatly dominates the global streaming potential. This is due to the fact that the streaming potential of the filtering layer is weighted by the pore radius ratio which is a predominant parameter in determining the global streaming potential. In contrast to the streaming potential, the contribution of the filtering layer to the global membrane potential (Em_g) or membrane conductivity (λ_g) depends more or less on the zeta potentials of both kinds of pores and the corresponding electrokinetic radii as well. As to the membrane potential, the contribution of the filtering layer to Em_g is all the more sensitive to the zeta potentials than the electrokinetic radii are small. The filtering layer greatly dominates the global membrane potential when its pores are narrow (with regard to the Debye length) and strongly charged. For the electrolyte conductivity inside pores, the smaller pores (inside the filtering layer) have an effect all the more dominant on the apparent membrane conductivity than their zeta potential is low and that of larger pores (inside the support layer) is high.     

Measurement of the zeta potential of planar solid surfaces by means of a rotating disk

A method for measuring the zeta potential of disks is described. Combining the hydrodynamic properties of a rotating disk, the solution of Laplace's equation for the potential, and the electrokinetic boundary condition, one obtains an equation that relates the zeta potential of the disk to the streaming potential in the disk's vicinity. Theory predicts a dependence of the streaming potential on the rotation rate raised to the 3/2 power. Theory also shows that placement of one reference electrode on the axis of rotation near the disk surface and the other far from the disk is favorable. Measurement of the streaming potential of silicon oxide and indium tin oxide in contact with a solution of potassium chloride demonstrated the expected 3/2 power dependence on rotation rate. The zeta potentials calculated from the combination of the theory and experimental data agreed with published values.     

Influence of roughness and capillary size on the zeta potential values obtained by streaming potential measurements

Streaming potential measurements are performed to determine the zeta potential of flat surfaces, particles, or fibers. Although the zeta potential is a well-defined property of solid surfaces in a liquid, there are indications that the absolute values of the zeta potential calculated using the Helmholtz-Smoluchowski equation are affected by surface roughness and—in case of particle or fiber assemblies—their packing density. The study at hand investigates these influences using flat polymer surfaces with different roughness and topography and assemblies of basalt spheres. It was found that increasing roughness of the flat surface and larger size or smaller number of particles in particle assemblies result in flatter slopes of the streaming potential versus pressure and thus lower apparent absolute values of the zeta potential. The interpretation of streaming potential measurements should therefore not focus on absolute zeta potential values but on trends in pH- and concentration-dependent measurements.     

Characterization of fractal particles using acoustics, electroacoustics, light scattering, image analysis, and conductivity

Fractals are aggregates of primary particles organized with a certain symmetry defined essentially by one parameter-a fractal dimension. We have developed a model for the interpretation of acoustic data with respect to particle structure in aggregated fractal particles. We apply this model to the characterization of various properties of a fumed silica, being but one example of a fractal structure. Importantly, our model assumes that there is no liquid flow within the aggregates (no advection). For fractal dimensions of less than 2.5, we find that the size and density of aggregates, computed from the measured acoustic attenuation spectra, are quite independent of the assumed fractal dimension. This aggregate size agrees well with light-scattering measurements. We applied this model to the interpretation of electroacoustic data as well. A combination of electroacoustic and conductivity measurements yields sufficient data for comparing the fractal model of the particle organization with a simple model of the separate primary particles. Conductivity measurements provide information on particle surface conductivity reflected in terms of the Dukhin number (Du). Supporting information for the zeta potential and Du can also be provided by electroacoustic measurements assuming thin double-layer theory. In comparing values of Du from these two measurements, we find that the model of separate solid particles provides much more consistent results than a fractal model with zero advection. To explain this, we first need to explain an apparent contradiction in the acoustic and electroacoustic data for porous particles. Although not important for interpreting acoustic data, advection within the aggregate does turn out to be essential for interpreting electrokinetic and electroacoustic phenomena in dispersions of porous particles.     

Polyelectrolyte adsorption layers studied by streaming potential and particle deposition

Adsorption of a cationic polyelectrolyte, polyallylamine hydrochloride (PAH), having a molecular weight of 70,000 on mica was characterized by the streaming potential method and by deposition of negative polystyrene latex particles. Formation of PAH layers was followed by determining the apparent zeta potential of surface zeta as function of bulk PAH concentration. The zeta potential was calculated from the streaming potential measured in the parallel-plate channel formed by two mica plates precovered by the polyelectrolyte. The experimental data were expressed as the dependence of the reduced zeta potential zeta/zeta0 on the PAH coverage Theta(PAH), calculated using the convective diffusion theory. It was found that for the ionic strength of 10(-2) M, the dependence of zeta/zeta0 on Theta(PAH) can be reflected by the theoretical model formulated previously for surfaces covered by colloid particles. The electrokinetic measurements were complemented by particle deposition experiments on PAH-covered mica surfaces. A direct correlation between the polymer coverage and the initial deposition rate of particles, as well as the jamming coverage, was found. For ThetaPAH > 0.3 the initial deposition rate attained the value predicted from the convective diffusion theory for homogeneous surfaces. The initial deposition rates for surfaces modified by PAH were compared with previous experimental and theoretical results obtained for heterogeneous surfaces formed by preadsorption of colloid particles. It was revealed that negative latex deposition occurred at surfaces exhibiting negative apparent zeta potential, which explained the anomalous deposition of particles observed in previous works. It was suggested that the combined electrokinetic and particle deposition methods can be used for detecting adsorbed polyelectrolytes at surfaces for coverage range of a percent. This enables one to measure bulk polyelectrolyte concentrations at the level of 0.05 ppm.     

Determining the Zeta Potential of Porous Membranes Using Electrolyte Conductivity inside Pores

The zeta potential is an important and reliable indicator of the surface charge of membranes, and knowledge of it is essential for the design and operation of membrane processes. The zeta potential cannot be measured directly, but must be deduced from experiments by means of a model. The possibility of determining the zeta potential of porous membranes from measurements of the electrolyte conductivity inside pores (lambda(pore)) is investigated in the case of a ceramic microfiltration membrane. To this end, experimental measurements of the electrical resistance in pores are performed with the membrane filled with KCl solutions of various pHs and concentrations. lambda(pore) is deduced from these experiments. The farther the pH is from the isoelectric point and/or the lower the salt concentration is, the higher the ratio of the electrolyte conductivity inside pores to the bulk conductivity is, due to a more important contribution of the surface conduction. Zeta potentials are calculated from lambda(pore) values by means of a space charge model and compared to those calculated from streaming potential measurements. It is found that the isoelectric points are very close and that zeta potential values for both methods are in quite good agreement. The differences observed in zeta potentials could be due to the fact that the space charge model does not consider the surface conductivity in the inner part of the double layer. Measurements of the electrolyte conductivity within the membrane pores are proved to be a well-adapted procedure for the determination of the zeta potential in situations where the contribution of the surface conduction is significant, i.e., for small and charged pores. Copyright 2001 Academic Press.     

Evaluating the surface charge of C18 stationary phases

The surface charge of four C18 stationary phases was investigated by measuring the flow induced streaming potential, a well known electrokinetic property of charged surfaces. Three of the stationary phases (Symmetry, Gemini, and Xterra-MS) had significantly positive streaming potentials at both pH 3 and 4.5. The fourth (Zorbax-SB) appeared to be essentially neutral at pH 3 and became negative at pH 4.5. Apparent zeta potentials ranged from approximately +16 to -4 mV. The retention behavior was also investigated using chloride as model anion and glycinamide (in its protonated form) as model cation. When the retention factor (k) of glycinamide was subtracted from k of chloride anion, the resulting delta k values showed very similar trends as apparent zeta potential values, suggesting that the simple chromatographic method could be used to estimate zeta potential values, or that the zeta potential values could be useful for ranking columns according to ion exchange or exclusion behavior. The anion exchange capacity of the Symmetry and Gemini columns was also estimated, using a published chromatographic procedure, and the results suggest about 2 microEq. capacity per gram of packing.     

Ionic strength-dependent pK shift in the helix-coil transition of grafted poly(L-glutamic acid) layers analyzed by electrokinetic and ellipsometric measurements

Surface-bound layers of poly(L-glutamic acid) prepared by a recently described "grafting-from" method were analyzed with respect to electrical charging and structural alterations upon variation of pH and concentration of the background electrolyte in aqueous solutions. The microslit electrokinetic setup (MES) was utilized for the combined determination of zeta potential and surface conductivity on the basis of streaming potential and streaming current measurements at polypeptide layers in contact with aqueous electrolyte solutions of varied composition. In situ ellipsometry was applied at similar samples immersed in identical aqueous solutions to investigate the influence of the solution pH on the structure of the polypeptide layers. Zeta potential and Dukhin number versus pH plots revealed the dissociation behavior of the surface-bound polypeptides indicating a significant shift of the pK of their acidic side chains correlating with the concentration of the background electrolyte potassium chloride and the related variation of the Debye screening length. Surface conductivity data pointed at a more expanded structure of the polypeptide layer in the fully dissociated state as an increased ion conductance in this part of the interface was determined. The occurrence of a strong increase of the thickness and a corresponding decrease of the refractive index for the coil state of the layer strongly supports the findings of the electrokinetic measurements. This fully reversible "switching" of the layer structure was attributed to helix-coil transitions within the grafted polypeptides induced by the dissociation of carboxylic acid functions of the polypeptide side chains. The shift of the "switching pH" of the surface-bound poly(L-glutamic acid) layers at varied concentrations of the background electrolyte was interpreted as a result of the pK shift of the carboxylic acid groups of the polypeptide side chains. The observed patterns prove that the electrostatic interactions causing this shift occur within but not between the grafted chains.     

Streaming potential generated by a long viscous drop in a capillary

The streaming potential generated by motion of a long drop of viscosity mu(d) = lambdamu in a uniform circular capillary filled with fluid of viscosity mu is investigated by means of a model previously used to study electrophoresis of a charged mercury drop in water. The capillary wall is at potential zeta c relative to the bulk fluid within it, and the surface of the drop is at potential zeta(d). Potentials are assumed to be sufficiently small so that the charge cloud is described by the linearized Poisson-Boltzmann equation, and the Debye length characterizing the thickness of the charge cloud is assumed to be thin compared with the gap h(0) between the drop and the capillary wall. Ions in the external fluid are not allowed to discharge at the surface of the drop, and the wall of the capillary has a nonzero surface conductivity sigma c. The drop is assumed to be sufficiently long so that end effects can be neglected. Recirculation of fluid within the drop gives rise to an enhanced streaming current when zeta(d) is nonzero, leading to an anomalously high streaming potential. This vanishes as the drop viscosity becomes large. If V is the velocity of the drop and gamma is the coefficient of interfacial tension between the two fluids, then the capillary number is Ca = mu V/gamma, and the gap varies as h(0)planck'sCa(2/3). When Ca is small, the gap h(0) is small and electrical conduction along the narrow gap is dominated by the surface conductivity sigma(c) of the capillary wall, which is constant. The electrical current convected by flowing fluid is proportional to Ca, as is the change in streaming potential caused by the presence of the drop. If sigma(c) = 0, then the electrical conductance of the gap depends on its width h(0) and on the bulk fluid conductivity sigma and becomes small as h(0) approximately equal to Ca(2/3) --> 0. The streaming potential required to cancel the O(Ca) convection current therefore varies as Ca(1/3). If sigma(c) = 0 and the drop is rigid (lambda --> infinity), then the change in streaming potential over and above that expected due to the change in pressure gradient is proportional to the difference in potentials zeta(c)-zeta(d).     

Electrokinetic fingerprinting of grafted polyelectrolyte layers--a theoretical approach

Electrokinetic fingerprinting (EF) was introduced by Marlow and Rowell [Marlow BJ, Rowel RL. Langmuir 1990;6:1088] for the comprehensive characterization of charged particle surfaces. Afterwards, EF was applied by many groups for the characterization of "hard" (i.e. non-swelling) surfaces. However, the advantages of EF could not yet utilized for the characterization of grafted polyelectrolyte layers (PL) since the theoretical background was not yet elaborated. A theory for the characterization of PL at complete dissociation of the functional groups was developed by Ohshima [Adv Colloid Interface Sci 1995;62:189] and later extended by Dukhin et al. [Dukhin S, Zimmermann R, Werner C. J Colloid Interface Sci 2005;286:761] for any degree of dissociation. Further progress in the characterization of soft surfaces may be achieved by combining EF and surface conductivity (SC) measurements. Both theory and experiment demonstrate that integrated measurements of SC and apparent zeta potential zeta(a) in broad ranges of pH and ionic strength provide information about Donnan potential Psi(D), surface charge, pK and surface potential Psi(0), while the interpretation is more uncertain, when only zeta(a) is measured. This advanced method of PL characterization is established for PL grafted on flat surfaces. When PL are formed on spherical particles, the SC may be measured by means of conductometry and/or dielectric spectroscopy. However, the current theories can only be applied within a rather narrow range of the practically relevant conditions. To overcome this limitation, an unified approach to the theory of electrophoresis for spherical particles with grafted PL was elaborated taking into account the existence of two different electrokinetic models for soft surfaces. While one model is focused on hydrodynamic permeability of soft surface and disregards surface current, another model considers the surface current and disregards electrokinetic water transport within the soft surface layer. Unification became possible through generalization of the capillary osmosis theory over soft surfaces.     

Theoretical analysis of electrokinetic flow and heat transfer in a microchannel under asymmetric boundary conditions

The main theme of the present work is to investigate the electrokinetic effects on liquid flow and heat transfer in a flat microchannel of two parallel plates under asymmetric boundary conditions including wall-sliding motion, unequal zeta potentials, and unequal heat fluxes on two walls. Based on the Debye-Huckel approximation, an electrical potential solution to the linearized Poisson-Boltzmann equation is obtained and employed in the analysis. The analytic solutions of the electrical potential, velocity distributions, streaming potential, friction coefficient, temperature distribution, and heat transfer rate are obtained, and thereby the effects of electrokinetic separation distance (K), zeta-potential level (zeta;(1)), ratio of two zeta potentials (r(zeta) identical with zeta;(2)/zeta;(1)), wall-sliding velocity (u(w)), and heat flux ratio (r(q) identical with q"(2)/q"(1)) are investigated. The present results reveal the effects of wall-sliding and zeta-potential ratio on the hydrodynamic nature of microchannel flow, and they are used to provide physical interpretations for the resultant electrokinetic effects and the underlying electro-hydrodynamic interaction mechanisms. In the final part the results of potential and velocity fields are applied in solving the energy equation. The temperature distributions and heat transfer characteristics under the asymmetrical kinematic, electric, and thermal boundary conditions considered presently are dealt with.     

Bulk conductivity of soft surface layers: experimental measurement and electrokinetic implications

Conductivity measurements were carried out on a family of polyacrylamide-co-sodium acrylate gels cross-linked with N,N'-methylenebisacrylamide in a homemade electrokinetic cell. The conductivity data allowed the equilibrium Donnan potential difference between the bulk gel and the bulk electrolyte solution to be estimated at various ionic strengths. The data were fit to a simple model assuming full dissociation of functional groups as well as to a more complete model (Dukhin, S. S.; Zimmerman, R.; Werner, C. J. Colloid Interface Sci. 2004, 274, 309) that accounts for the weak electrolyte nature of the acrylate groups fixed within the gel structure. The conductivity of the gel layers was observed to be significantly larger than the conductivity of the bulk electrolyte solutions at low ionic strengths. The increased conductivity reflects the enhanced concentration of ions within the gel structure due to Donnan equilibrium and the mobility of ions within the high water content gel layers. The electrokinetic implications of the bulk conductivity of gel-like soft surface layers are discussed in terms of the influence of the gel conductance on the resulting streaming potential.     

Conditions for similitude and the effect of finite Debye length in electroosmotic flows

Under certain conditions, the velocity field is similar to the electric field for electroosmotic flow (EOF) inside a channel. There was a disagreement between investigators on the necessity of the infinitesimal-Reynolds-number condition for the similarity when the Helmholtz-Smoluchowski relation is applied throughout the boundaries. What is puzzling is a recent numerical result that showed, contrary to the conventional belief, an evident Reynolds number dependence of the EOF. We show here that the notion that the infinitesimal-Reynolds-number condition is required originates from the misunderstanding that the EOF is the Stokes flow. We point out that the EOF becomes the potential flow when the Helmholtz-Smoluchowski relation is applied at the boundaries. We carry out a numerical simulation to investigate the effect of finiteness of the Debye length and the vorticity layer inherently existing at the channel wall. We show that the Reynolds number dependence of the previous numerical simulation resulted from the finiteness of the Debye length and subsequent convective transport of vorticity toward the bulk flow. We discuss in detail how the convection of vorticity occurs and what factors are involved in the transport process, after carrying out the simulation for different Reynolds numbers, Debye lengths, corner radii, and geometries.     

An improved method for determining zeta potential and pore conductivity of porous materials

An improved method based on streaming potential and streaming current was proposed to determine zeta potential and surface conductance of porous material simultaneously. In the electrokinetic generation mode, a resistor is connected to the generator and by measuring the voltage drop across resistors with different resistance, a true streaming current can be determined. The zeta potential and surface conductivity can be obtained simultaneously from their relation to streaming potential and streaming current. The electrode and ion concentration polarization effects during the measurement were also discussed. The resistance from channel ends to electrodes, which has typically been ignored in the literature, was shown to have a significant influence on the calculated zeta potential and surface conductance. Ignorance of this resistance would lead to underestimation of both zeta potential and surface conductance values.     

Transition from creeping via viscous-inertial to turbulent flow in fixed beds

This review is concerned with the analysis of flow regimes in porous media, in particular, in fixed beds of spherical particles used as reactors in engineering applications, or as separation units in liquid chromatography. A transition from creeping via viscous-inertial to turbulent flow is discussed based on macro-scale transport behaviour with respect to the pressure drop-flow rate dependence, in particular, the deviation from Darcy's law, as well as direct microscopic data which reflect concomitant changes in the pore-level hydrodynamics. In contrast to the flow behaviour in straight pipes, the transition from laminar to turbulent flow in fixed particulate beds is not sharp, but proceeds gradually through a viscous-inertial flow regime. The onset of this steady, nonlinear regime and increasing role of inertial forces is macroscopically manifested in the failure of Darcy's law to describe flow through fixed beds at higher Reynolds numbers. While the physical reasons for this failure still are not completely understood, it is not caused by turbulence which occurs at Reynolds numbers about two orders of magnitude above those for which a deviation from Darcy's law is observed. Microscopic analysis shows that this steady, nonlinear flow regime is characterized by the development of an inertial core in the pore-level profile, i.e., at increasing Reynolds number velocity profiles in individual pores become flatter towards the center of the pores, while the velocity gradient increases close to the solid-liquid interface. Further, regions with local backflow and stationary eddies are demonstrated for the laminar flow regime in fixed beds. The onset of local fluctuations (end of laminar regime) is observed at superficial Reynolds numbers on the order of 100. Complementary analysis of hydrodynamic dispersion suggests that this unsteady flow accelerates lateral equilibration between different velocities in fixed beds which, in turn, reduces spreading in the longitudial (macroscopic flow) direction.