Electrophoresis of small particles and fluid globules in weak electrolytes

Influence of ion size effects on the electrokinetics of aqueous salt-free colloids in alternating electric fields

Electrokinetics is the science of the physical phenomena appearing at the solid-liquid interface of dispersed particles subjected to external fields. Techniques based on electrokinetic phenomena constitute an important set of tools for the electrical characterization of colloids because of their sensitivity to the properties of particle-solution interfaces. Their rigorous description may require inclusion of the effects of finite size of chemical species in the theoretical models, and, particularly in the case of salt-free (no external salt added) aqueous colloids, also consideration of water dissociation and possible carbon dioxide contamination in the aqueous solution. A new ac electrokinetic model is presented for concentrated salt-free spherical colloids for arbitrary characteristics of the particles and aqueous solution, including finite-size effects of chemical species by appropriate modifications of the chemical reaction equations to include such non-ideal aspects. The numerical solution of the electrokinetic equations in an alternating electric field has also been carried out by using a realistic non-equilibrium scenario accounting for association-dissociation processes in the chemical reactions. The results demonstrate the importance of including finite-size effects in the electrokinetic response of the colloid, mainly at high frequencies of the electric field, and for highly charged colloids. Findings of previous models for pointlike ions or for ideal salt-free colloids including finite ion size effects are recovered with the present model, for the appropriate limiting conditions.

Effects of Weak Electrolytes on Electric Double Layer Ion Distributions

Many common experimental systems have electric double layers containing weak electrolytes, including systems with buffers. The pH at the boundary of the diffuse layer is an important parameter for determining the physicochemical state of the system, including surface charge density. We show that the Boltzmann equilibrium relation can be used as an exact solution for weak electrolyte electric double layers. Using these results, we provide a closed-form relation for the maximum pH change in a buffered electric double layer, in terms of the boundary potential. Importantly, our results suggest that equilibrium electric double layer concepts developed for strong electrolytes can be expanded to include weak electrolytes.

Non-monotonic concentration dependence of the electro-phoretic mobility of charged spheres in realistic salt free suspensions

Using super-heterodyne Doppler velocimetry with multiple scattering correction, we extend the optically accessible range of concentrations in experiments on colloidal electro-kinetics. Here, we measured the electro-phoretic mobility and the DC conductivity of aqueous charged sphere suspensions covering about three orders of magnitude in particle concentrations and transmissions as low as 40%. The extended concentration range for the first time allows the demonstration of a non-monotonic concentration dependence of the mobility for a single particle species. Our observations reconcile previous experimental observations made on other species over restricted concentration ranges. We compare our results to the state-of-the-art theoretical calculations using a constant particle charge and the carefully determined experimental boundary conditions as input. In particular, we consider the so-called realistic salt free conditions, i.e., we respect the release of counterions by the particles, the solvent hydrolysis, and the formation of carbonic acid from dissolved neutral CO₂. We also compare our results to previous results obtained under similarly well-defined conditions. This allows identification of three distinct regions of differing density dependence. There is an ascent during the build-up of double layer overlap, which is not expected by theory, an extended plateau region in quantitative agreement with theoretical expectation based on a constant effective charge and a sudden decrease, which occurs way before the expected gradual decrease. Our observations suggest a relation of the non-monotonic behavior to a decrease in particle charge, and we tentatively discuss possibly underlying mechanisms.

Electrokinetic phenomena in a dilute suspension of spherical solid colloidal particles with a hydrodynamically slipping surface in an aqueous electrolyte solution

A review is given on the theory of the electrokinetics in a dilute suspension of spherical solid colloidal particles with a hydrodynamically slipping surface moving in an aqueous liquid medium containing electrolytes. For a solid particle with a slip surface, the Navier boundary condition is employed instead of the usual no-slip boundary condition on the particle surface. The effect of the hydrodynamic slip is characterized by the slipping length. The limiting case of zero slipping length corresponds to a hydrophilic surface. As the hydrophobicity of the particle surface increases, the slipping length increases. The limiting case of infinitely large slipping length corresponds to a completely hydrophobic surface. General formulas and approximate expressions of the electrophoretic mobility, the electrical conductivity, the sedimentation velocity and potential, and the diffusion constant are presented. The magnitudes of the electrophoretic mobility and the sedimentation potential, in particular, are found to increase with increasing slipping length. It is also shown that a spherical solid colloidal particle with a slip surface is hydrodynamically similar to a liquid drop.

Strong Deformation of the Thick Electric Double Layer around a Charged Particle during Sedimentation or Electrophoresis

The deformation of the electric double layer around a charged colloidal particle during sedimentation or electrophoresis in a binary, symmetric electrolyte is studied. The surface potential of the particle is assumed to be small compared to the thermal voltage scale. Additionally, the Debye length is assumed to be large compared to the particle size. These assumptions enable a linearization of the electrokinetic equations. The particle appears as a point charge in this thick-double-layer limit; the distribution of charge in the diffuse cloud surrounding it is determined by a balance of advection due to the particle motion, Brownian diffusion of ions, and electrostatic screening of the particle by the cloud. The ability of advection to deform the charge cloud from its equilibrium state is parametrized by a Péclet number, Pe. For weak advection (Pe ≪ 1), the cloud is only slightly deformed. In contrast, the cloud can be completely stripped from the particle at Pe ≫ 1; consequently, electrokinetic effects on the particle motion vanish in this regime. Therefore, in sedimentation the drag limits to Stokes' law for an uncharged particle as Pe → ∞. Likewise, the particle velocity for electrophoresis approaches Huckel's result. The strongly deformed cloud at large Pe is predicted to generate a concomitant increase in the sedimentation field in a dilute settling suspension.

Redistribution of mobile surface charges of an oil droplet in water in applied electric field

Most researches on oil droplets immersed in aqueous solutions assume that the surface charges of oil droplets are, similar to that of solid particles, immobile and distributed uniformly under external electric field. However, the surface charges at the liquid-liquid interface are mobile and will redistribute under external electric field. This paper studies the redistribution of surface charges on an oil droplet under the influence of the external electrical field. Analytical expressions of the local zeta potential on the surface of an oil droplet after the charge redistribution in a uniform electrical field were derived. The effects of the initial zeta potential, droplet radius and strength of applied electric field on the surface charge redistribution were studied. In analogy to the mobile surface charges, the redistribution of Al2O3-passivated aluminum nanoparticles on the oil droplet surface was observed under applied electrical field. Experimental results showed that these nanoparticles moved and accumulated towards one side of the oil droplet under electric field. The redistribution of the nanoparticles is in qualitative agreement with the redistribution model of the mobile surface charges developed in this work.

A mathematical model for electrical impedance spectroscopy of zwitterionic hydrogels

We report a mathematical model for ion transport and electrical impedance in zwitterionic hydrogels, which possess acidic and basic functional groups that carry a net charge at a pH not equal to the isoelectric point. Such hydrogels can act as an electro-mechanical interface between a relatively hard biosensor and soft tissue in the body. For this application, the electrical impedance of the hydrogel must be characterized to ensure that ion transport to the biosensor is not significantly hindered. The electrical impedance is the ratio of the applied voltage to the measured current. We consider a simple model system, wherein an oscillating voltage is applied across a hydrogel immersed in electrolyte and sandwiched between parallel, blocking electrodes. We employ the Poisson-Nernst-Planck (PNP) equations coupled with acid-base dissociation reactions for the charge on the hydrogel backbone to model the ionic transport across the hydrogel. The electrical impedance is calculated from the numerical solution to the PNP equations and subsequently analyzed via an equivalent circuit model to extract the hydrogel capacitance, resistance, and the capacitance of electrical double layers at the electrode-hydrogel interface. For example, we predict that an increase in pH from the isoelectric point, pH = 6.4 for a model PCBMA hydrogel, to pH = 8 reduces the resistance of the hydrogel by ∼40% and increases the double layer capacitance by ∼250% at an electrolyte concentration of 0.1 mM. The significant impact of charged hydrogel functional groups to the impedance is damped at higher electrolyte concentration.

Effects of non-equilibrium association-dissociation processes in the dynamic electrophoretic mobility and dielectric response of realistic salt-free concentrated suspensions

Electrokinetic investigations in nanoparticle suspensions in aqueous media are most often performed assuming that the liquid medium is a strong electrolyte solution with specified concentration. The role of the ions produced by the process of charging the surfaces of the particles is often neglected or, at most, the concentrations of such ions are estimated in some way and added to the concentrations of the ions in the externally prepared solution. The situation here considered is quite different: no electrolyte is dissolved in the medium, and ideally only the counterions stemming from the particle charging are assumed to be in solution. This is the case of so-called salt-free systems. With the aim of making a model for such kind of dispersions as close to real situations as possible, it was previously found to consider the unavoidable presence of H(+) and OH(-) coming from water dissociation, as well as the (almost unavoidable) ions stemming from the dissolution of atmospheric CO2. In this work, we extend such approach by considering that the chemical reactions involved in dissociation and recombination of the (weak) electrolytes in solution must not necessarily be in equilibrium conditions (equal rates of forward and backward reactions). To that aim, we calculate the frequency spectra of the electric permittivity and dynamic electrophoretic mobility of salt-free suspensions considering realistic non-equilibrium conditions, using literature values for the rate constants of the reactions. Four species are linked by such reactions, namely H(+) (from water, from the--assumed acidic--groups on the particle surfaces, and from CO2 dissolution), OH(-) (from water), HCO3(-) and H2CO3 (again from CO2). A cell model is used for the calculations, which are extended to arbitrary values of the surface charge, the particle size, and particle volume fraction, in a wide range of the field frequency ω. Both approaches predict a high frequency relaxation of the counterion condensated layer and a Maxwell-Wagner-O'Konski electric double layer relaxation at intermediate frequencies. Also, in both cases an inertial decay of the electrophoretic mobility at high ω takes place. The most significant difference between the present model and previous results based on the equilibrium hypothesis is by no means negligible: only in non-equilibrium conditions do we find a low-frequency relaxation (mostly noticed in permittivity data, while its significance is lower in dynamic mobility spectra). This new relaxation presents all the characteristic features of the concentration polarization (or alpha) dispersion. These are: i) the average electric polarization of the system increases when the relaxation frequency is surpassed, contrary to the behavior after Maxwell-Wagner type relaxations; ii) the amplitude of the relaxation increases with surface charge, reaching a sort of saturation if the charge is too high; iii) the relaxation frequency increases with volume fraction while the relaxation amplitude decreases; iv) the characteristic frequency is reduced by the increase in particle radius. All these facts confirm that the non-equilibrium approach seems to better describe the physics of the system by giving rise to a concentration polarization kind of relaxation, only possible when ions can accumulate on both sides of the particles as dictated by the field, and not as determined by equilibrium conditions in the dissociation-recombination reactions involved.

Flow patterns and deformation modes of coaxial liquid columns in transverse electric fields

Steady-state flow patterns and deformation modes of coaxial liquid columns in transverse electric fields are studied analytically. The governing creeping flow equations are solved for Newtonian and (mutually) immiscible fluids in the framework of leaky dielectric theory. A detailed analysis of the electric and flow fields is presented and it is shown that there will be four possible flow patterns in and around the columns, in terms of the direction of the external flow (top-to-sides/bottom-to-sides vs. sides-to-top/sides-to-bottom) and the number of vortices (single vortex vs. double vortices) in the shell, and that the senses of the net electric shear stresses at the inner and the outer interfaces and their relative importance are the key parameters in setting these patterns. Equilibrium shapes of the interfaces are also found and it is shown that there are four distinct modes of deformation, depending on the governing nondimensional parameters of the problem. The instability of the jet is also examined qualitatively using the observations pertaining the instability of single-phase drops and jets and the scaling arguments based on the present solution.

Electrohydrodynamics of a compound drop

The behavior of a compound drop, comprising two concentric fluid spheres, in a uniform electric field is studied analytically. The governing electrohydrodynamic equations are solved for Newtonian and immiscible fluids in the framework of leaky-dielectric theory and in the limit of small electric field strength and fluid inertia. A detailed analysis of the electric and flow fields is presented and it is shown that there will be four possible flow patterns in and around the globule, in terms of the direction of the external flow (pole-to-equator vs equator-to-pole) and the number of vortices (single-vortex vs double vortices) in the shell, and that the senses of the net electric shear stresses at the surfaces of the inner and the outer drops and their relative importance are the key parameters in setting these patterns. A circulation map is constructed, which is used to infer about the likelihood of the flow patterns and transition from one pattern to another for representative fluid systems. For small distortion from the spherical shape, the deformations of the inner and the outer drops are found using normal stress balances at the corresponding surfaces. It is shown that there will be four possible modes for the deformation of the compound drop, which are determined by the net normal electric and hydrodynamic stresses at the pertinent surfaces. The dynamic responses of the inner and the outer drops for representative fluid systems are studied using a deformation map, which characterizes the possibilities of the deformation modes and transition from one mode to another as a function of the fluid properties.

Nonlinear electrokinetic flow about a polarized conducting drop

In the thin-double-layer limit κa>>1, electrokinetic flows about free surfaces are driven by a combination of an electro-osmotic slip and effective shear-stress jump. An intriguing case is that of a highly conducting liquid drop of radius a, where the inability to balance the viscous shear by Maxwell stresses results in an O(κa) velocity amplification relative to the familiar electro-osmotic scale. To illuminate the inherent nonlinearity we consider uncharged drops, where the induced surface-charge distribution results in a fore-aft symmetric electrokinetic flow profile with no attendant drop translation. This problem is analyzed using a macroscale model, where the double layer is represented by effective boundary conditions. Because of the intense flow, ionic convection within the O(1/κ)-wide diffuse-charge layer is manifested by a moderate-zeta-potential surface-conduction effect. The drop deforms to a prolate shape in response to the combination of hydrodynamic forces and the effective electrocapillary reduction of the surface-tension coefficient, both mechanisms being asymptotically comparable. The flow field and the concomitant drop deformation are calculated using both a weak-field approximation and numerical simulations of the nonlinear macroscale model.

Electrokinetic Phenomena in concentrated disperse systems: general problem formulation and Spherical Cell Approach

Electrokinetic Phenomena in concentrated disperse and colloid systems have been studied employing Spherical Cell Approach for over three decades. The critical review of the advances in this area, which is conducted in the present paper, demonstrates a number of contradictions between the results reported by different authors. These contradictions are largely associated with imposition of boundary conditions at the outer boundary of the representative Spherical Cell. In order to establish a correct version of the Spherical Cell Approach, in the present paper, the theory of Electrokinetic Phenomena in concentrated suspensions is revisited by primarily focusing on the boundary conditions employed at the Spherical Cell outer boundary. To this end, a general mathematical problem is formulated for addressing the behavior of a planar layer of a macroscopically homogeneous disperse system under simultaneous influence of the pressure difference, gravitation and applied electric fields. On the basis of the general problem formulation, we present strict definitions of the kinetic coefficients which describe the system behavior. Making use of such definitions, some general relationships are rederived for the kinetic coefficients, namely, the Smoluchowski asymptotic expressions and the Onsager irreversible thermodynamic relationships. The general problem is reformulated for describing the electric, hydrodynamic and ion concentration fields inside the representative Spherical Cell. Using an original approach, a complete set of the boundary conditions is derived by employing the only assumption: the average over the disperse system volume is equal to the average over a representative Spherical Cell volume. A general method for predicting the kinetic coefficients is developed by employing the solution of the formulated problem. The developed method is combined with the method of small perturbation parameter using the normalized zeta potential. Final expressions for the kinetic coefficients are obtained while accounting for the terms proportional to zeta potential. The predictions are compared with results of other publications. On this basis, conclusions are made about the validity of different models proposed in the literature.

Primary electroviscous effect in a dilute suspension of charged mercury drops

The standard theory of the primary electroviscous effect in a dilute suspension of charged spherical rigid particles in an electrolyte solution (Watterson, I. G.; White, L. R. J. Chem. Soc., Faraday Trans. 2 1981, 77, 1115) is extended to cover the case of a dilute suspension of charged mercury drops of viscosity eta(d). A general expression for the effective viscosity or the electroviscous coefficient p of the suspension is derived. This expression tends to that for the case of rigid particles in the limit of eta(d) --> infinity. We also derive an approximate analytical viscosity expressions applicable to mercury drops carrying low zeta potentials at arbitrary kappaa (where kappa is the Debye-Hückel parameter and a is the drop radius) and to mercury drops as well as rigid spheres with arbitrary zeta potentials at large kappaa. It is shown that the large-kappaa expression of p for rigid particles predicts a maximum when plotted as a function of zeta potential. This result for rigid particles agrees with the exact numerical results of Watterson and White. It is also shown that in the limit of high zeta potential the effective viscosity of a suspension of mercury drops tends to that of uncharged rigid spheres given by Einstein's formula (Einstein, A. Ann. Phys. 1906, 19, 289), whereas in the opposite limit of low zeta potential the effective viscosity approaches that of a suspension of uncharged liquid drops derived by Taylor (Taylor, G. I. Proc. R. Soc. London, Ser. A 1932, 138, 41).

Numerical calculation of the electrophoretic mobility of colloidal particles in weak electrolyte solutions

A numerical calculation of the electrophoretic mobility of colloidal particles in weak electrolyte solution is presented. It is based on a previous work (C. Grosse, V.N. Shilov, J. Colloid Interface Sci. 211 (1999) 160-170), where the analytical theory of the thin double layer concentration polarization is generalized to the case of weak electrolytes, i.e., when the dissociation-recombination equilibrium and rate constants both have finite values. The analytical results are first completed by including terms corresponding to co-ions that were neglected in the original presentation. It is shown that these terms that have little bearing in the case of strong electrolytes, become quite important in the case when the electrolyte is weak. The problem is then solved using the network method, leading to numerical results for the electric potential and the concentrations of counterions, co-ions, and neutral ion pairs. Finally, the electrophoretic mobility is calculated both analytically and numerically. It is shown that the hypothesis of a weak electrolyte leads to changes of mobility with respect to the classical results that are even stronger than predicted analytically.

Conductivity, Permittivity, and Characteristic Time of Colloidal Suspensions in Weak Electrolyte Solutions

The analytical theory of the thin double-layer concentration polarization in dilute suspensions of colloidal particles, generalized by the authors to the case of weak electrolyte solutions [C. Grosse and V. N. Shilov, J. Colloid Interface Sci. 211, 160 (1999)], was used to determine the conductivity dispersion amplitude, the dielectric increment, and the characteristic time of the low-frequency dielectric dispersion (LFDD). It is shown that at constant ionic strength, the conductivity dispersion amplitude always diminishes for weak electrolytes. This is due to the increment in the zero-frequency dipolar coefficient, which occurs because the field-induced ion concentration change around the particle is lowered. On the contrary, while the dielectric increment and the characteristic time of the LFDD usually decrease, they can actually increase when the diffusion coefficient of co-ions is larger than that of counterions. The origin of this behavior is in the appearance of volume charge distributions outside the double layer, which do not vanish for weak electrolytes in the low-frequency limit. Copyright 2000 Academic Press.

Electrophoretic Mobility of Colloidal Particles in Weak Electrolyte Solutions

The analytical theory of the thin double layer concentration polarization in suspensions of colloidal particles is generalized to the case of weak electrolyte solutions, i.e., when the dissociation-recombination equilibrium and rate constants have both finite values. It is shown that under the action of a static applied field, regions near the particle appear where there is departure from the dissociation-recombination equilibrium. The resulting ion and ion-pair sources have a strong bearing on their flows, leading to a change of the electrolyte concentration gradients around the particle. This phenomenon also modifies the value of the particle electrophoretic mobility, which is dependent on the concentration polarization. At constant ionic strength, the theoretical maximum of the electrophoretic mobility versus zeta potential curve can substantially surpass in weak electrolyte solutions the corresponding value attained in strong electrolytes. Copyright 1999 Academic Press.

Measurements on the Diffusion Coefficient of Colloidal Particles by Taylor-Aris Dispersion

Taylor-Aris dispersion in narrow-bore capillaries is used to measure the diffusion coefficient of colloidal particles in aqueous suspension. The method is shown to yield accurate results for particles up to about 0.3 &mgr;m in diameter; the measurement time for larger particles is prohibitively long and impractical. For hydrophobic particles, interactions with the capillary walls can introduce error into the interpretation of the data. The measurements also suggest that buoyancy-driven particle motion can introduce error. Consequently, a method similar to capillary hydrodynamic fractionation was developed to establish when these factors were of negligible effect. The results constitute an order-and-a-half improvement in the sensitivity of the technique, which has been recently shown to work for nanometer-sized proteins. The data suggest that, when matched with the appropriate theory, dispersion in capillaries may be a useful probe of colloidal and gravitational interaction potentials. Copyright 1997 Academic Press. Copyright 1997Academic Press