A semi-empirical approach to the modeling of the electrophoretic mobility in free solution: application to polystyrenesulfonates of various sulfonation rates

This work focuses on the understanding of the electrophoretic behavior of flexible chains of polystyrenesulfonates (PSSs) in free solution. It deals mainly with the variation of the electrophoretic mobility with (i) the polymerization degree (N) of fully sulfonated PSSs and (ii) the sulfonation rate of randomly sulfonated PSSs. In both cases, the electrophoretic mobility was modeled following a semi-empirical approach which involves parameters retaining a physical meaning. Fully sulfonated PSS oligomers, having a length smaller than or similar to the Debye length, exhibit a particular electrophoretic behavior, in-between that observed for multicharged small molecules and that for polyelectrolytes. The electrophoretic mobility of these oligomers increases strongly with N, which is attributed to a hydrodynamic coupling between monomers. Then the mobility is maximum for an N of about 10, for which the PSS oligomers are still in a rod-like conformation. Afterwards, as N increases and the PSSs are larger than the Debye length, the electrophoretic mobility decreases slowly until it reaches a constant value corresponding to the free-draining behavior. Next, the electrophoretic behavior of long PSS (N about 1,200) differing in their sulfonation rates was investigated. The effective charge rates were determined independently by conductimetric measurements and the mobilities were modeled as a function of the sulfonation rate. The PSS behavior observed was compared to the one previously reported for classical polyelectrolytes having hydrophilic backbones, such as copolymers of poly(acryamide-coacrylic acid). A specific behavior has been pointed out for these partially sulfonated PSSs, which is attributed to the hydrophobicity of their backbone. Finally, it is shown that separations of PSSs of different sulfonation rates can be obtained with electrolytes containing an anionic surfactant or methanol.     

Two independent measurements of Debye lengths in doped nonpolar liquids

Electric current measurements were performed between 2.5 cm x 7.5 cm parallel-plate electrodes separated by 1.2 mm of heptane doped with 0-15% w/w poly(isobutylene succinimide) (PIBS) having a molecular weight of about 1700. The rapid (microsecond) initial charging of the capacitor can be used to infer the dielectric constant of the solution. The much slower decay of current arising from the polarization of electrodes depends on the differential capacitance of the diffuse clouds of charge carriers accumulating next to each electrode and on the ohmic resistance of the fluid. Using the Gouy-Chapman model for the differential capacitance, Debye lengths of 80-600 nm were deduced that decrease with increasing concentration of PIBS. Values of the Debye lengths were confirmed by performing independent measurements of double-layer repulsion between a 6 microm polystyrene (PS) latex sphere and a PS-coated glass plate using total internal reflection microscopy in the same solutions. The charge carriers appear to be inverted PIBS micelles having apparent Stokes diameters of 20-40 nm. Dynamic light scattering reveals a broad distribution of sizes having an intensity-averaged diameter of 15 nm. This smaller size might arise (1) from overestimating the electrophoretic mobility of micelles by treating them as point charges or (2) because charged micelles are larger on average than uncharged micelles. When Faradaic reactions and zeta potentials on the electrodes can be neglected, such current versus time experiments yield values for the Debye length and ionic strength with less effort than force measurements. To obtain the concentration of charge carriers from measurements of conductivity, the mobility of the charge carriers must be known.     

Diffusiophoretic mobility of charged porous spheres in electrolyte gradients

The diffusiophoretic motion of a polyelectrolyte molecule or charged floc in an unbounded solution of a symmetrically charged electrolyte with a uniform prescribed concentration gradient is analytically studied. The model used for the particle is a porous sphere in which the density of the hydrodynamic frictional segments, and therefore also that of the fixed charges, is constant. The electrokinetic equations which govern the electrostatic potential profile, the ionic concentration distributions (or electrochemical potential energies), and the fluid velocity field inside and outside the porous particle are linearized by assuming that the system is only slightly distorted from equilibrium. Using a regular perturbation method, these linearized equations are solved for a charged porous sphere with the density of the fixed charges as the small perturbation parameter. An analytical expression for the diffusiophoretic mobility of the charged porous sphere in closed form is obtained from a balance between its electrostatic and hydrodynamic forces. This expression, which is correct to the second order of the fixed charge density of the particle, is valid for arbitrary values of kappaa and lambdaa, where kappa is the reciprocal of the Debye screening length, lambda is the reciprocal of the length characterizing the extent of flow penetration inside the particle, and a is the particle radius. Our result to the first order of the fixed charge density agrees with the corresponding solution for the electrophoretic mobility obtained in the literature. In general, the diffusiophoretic mobility of a porous particle becomes greater as the hindrance to the diffusive transport of the solute species inside the particle is more significant.     

Electrophoresis of a biocolloid covered with a cation-absorptive membrane

The electrophoretic behavior of a biocolloid covered with a charged membrane is theoretically analyzed in the present study. Here, the influences of nonuniformly distributed fixed groups, absorption of cations by fixed original functional groups, variation in dielectric constant in the electrophoretic system, and ionic sizes are considered. The results of numerical simulation suggest that a larger absolute value of the electrophoric mobility of biocolloids could be generated by larger membrane electricity. The absolute value of the electrophoric mobility for the nonlinear distribution of the fixed groups is larger than that for the linear distribution of the fixed groups. The absolute value of the electrophoric mobility increases with (1) the concentration of total fixed groups, (2) the cation-absorption equilibrium constant, (3) the nonuniform feature index for functional-groups distribution, (4) the dielectric constants of the inner uncharged membrane zone for only mobile cationic charge and for both mobile cationic and anionic charge, and (5) the effective size of the cations. An increase in the absolute value of the electrophoric mobility can also be resulted from a decrease in the following parameters: (1) the friction coefficient of the biocolloidal membrane phase, (2) the membrane thickness, (3) the dielectric constant of space for all charge and of outer uncharged membrane zone, (4) the effective sizes of anions and fixed groups, and (5) the number of cations and the fixed original functional groups involved in the formation of a cation-functional group complex.     

Transient electrophoresis in a suspension of charged particles with arbitrary electric double layers

The startup of electrophoretic motion in a suspension of spherical colloidal particles, which may be charged with constant zeta potential or constant surface charge density, due to the sudden application of an electric field is analytically examined. The unsteady modified Stokes equation governing the fluid velocity field is solved with unit cell models. Explicit formulas for the transient electrophoretic velocity of the particle in a cell in the Laplace transforms are obtained as functions of relevant parameters. The transient electrophoretic mobility is a monotonic decreasing function of the particle-to-fluid density ratio and in general a decreasing function of the particle volume fraction, but it increases and decreases with a raise in the ratio of the particle radius to the Debye length for the particles with constant zeta potential and constant surface charge density, respectively. On the other hand, the relaxation time in the growth of the electrophoretic mobility increases substantially with an increase in the particle-to-fluid density ratio and with a decrease in the particle volume fraction but is not a sensitive function of the ratio of the particle radius to the Debye length. For specified values of the particle volume fraction and particle-to-fluid density ratio in a suspension, the relaxation times in the growth of the particle mobility in transient electrophoresis and transient sedimentation are equivalent.     

Protonation of Lysozymes and Its Consequences for the Adsorption onto a Mica Surface

Several physicochemical properties of chicken egg white lysozyme (LSZ) in electrolyte solutions were determined. The hydrodynamic diameter of LSZ at an ionic strength of 0.15 M was found to be 4.0 nm. Using the determined parameters, the number of uncompensated (electrokinetic) charges, N(c), on the molecule surface was calculated from the electrophoretic mobility data. It was found that the N(c) = 2.8 at pH = 3.0 and an ionic strength of I = 0.15 M. At the lower ionic strength, I = 1 × 10(-3) M, this positive charge increased to N(c) = 5.6 at a pH = 3.0 The physicochemical characteristics were supplemented by the dynamic viscosity measurements. The intrinsic viscosity and the hydrodynamic diameter results were compared with theoretical predictions from Brenner's model. Using this approach, it was found that the effective molecule length of LSZ is equal to L(ef) = 5.6 nm. Additional information on the LSZ adsorbed films was obtained by the contact angle measurements. The notably large contact angles were measured on LSZ films formed under the conditions where both the LSZ and the mica were oppositely charged. The higher the positive zeta potential of LSZ, the greater the contact angle measured, which indicates that LSZ affinity for the adsorption on mica increases with its uncompensated charge. The adsorption dependence on the zeta potential of LSZ was explained, assuming a roughly uniform distribution of the net charge on the molecule surface. This assumption is supported by the results of depositing negatively charged, fluorescent latex particles onto the mica surface, which had been modified by LSZ adsorption. The highest latex coverage was formed on mica surfaces that had first been coated with LSZ solutions of lower pH, as a result of the increasing charge of LSZ monolayers in this condition.     

Determination of synthetic polypeptide conformations and molecular geometrical parameters by nonaqueous CE

The aim of this work was to study changes in homopolypeptide chain conformation as a function of the number of residues by the modeling of the electrophoretic mobility. For this purpose, the frictional coefficients of poly(N(epsilon)-trifluoroacetyl-L-lysine) with different number of residues (up to 11) were determined from the absolute ionic mobilities and modeled by the hydrodynamic frictional coefficient of an equivalent cylinder. This approach allowed determination of geometrical parameters of the polypeptide chain in a liquid phase (nonaqueous solution of the BGE). The fact that the BGE and analyte are dissolved in mixed (methanol-ACN) organic solvent implied to take into account different effects and corrections that are generally not considered in aqueous solvent: namely, the effect of ion-pairs between constituents of the BGE for the calculation of the ionic strength, the effect of ion-pairs between the solutes and the electrolyte counterions and the correction due to the dielectric friction (Hubbard-Onsager equations). In addition, the influence of the ionic strength on the electrophoretic mobility was corrected using the Pitts equation, and the effect of lateral charges due to a slight deprotonation of the -NH- group in the lateral chain was also considered. From this modeling, molecular geometrical parameters relative to the linear and helico?dal conformations were obtained with very good correlation coefficients. Interestingly, this work also points out that the use of ionic mobility modeling for extracting molecular geometrical parameters can also be applied to end-charged polypeptides with slightly charged lateral chains (3% of elementary charge per residue).     

Electrophoresis of hydrophilic/hydrophobic rigid colloid with effects of relaxation and ion size

This article deals with a semi‐analytical study on the electrophoresis of charged spherical rigid colloid by considering the effects of relaxation and ion size. The particle surface is taken to be either hydrophilic or hydrophobic in nature. In order to consider the ion size effect we have invoked the Carnahan and Starling model (J. Chem. Phys. 1969, 51, 635‐636). The mathematical model is based on Stokes equation for fluid flow, modified Boltzmann equation for spatial distribution of ionic species and Poisson equation for electric potential. We adopt a linear perturbation technique under a weak electric field assumption. An iterative numerical technique in employed to solve the coupled set of perturbed equations. We have validated the numerically obtained electrophoretic mobility with the corresponding analytical solution derived under low potential limit. Going beyond the widely employed Debye‐Hückel linearization, we have presented the results for a wide range of surface charge density, electrolyte concentration, and slip length to Debye length ratio. We have also identified several interesting features including occurrence of local maxima and minima in the mobility for critical choice of pertinent parameters.     

Electrophoresis and electric conduction in a salt-free suspension of charged particles

The electrophoresis and electric conduction of a suspension of charged spherical particles in a salt-free solution are analyzed by using a unit cell model. The linearized Poisson-Boltzmann equation (valid for the cases of relatively low surface charge density or high volume fraction of the particles) and Laplace equation are solved for the equilibrium electric potential profile and its perturbation caused by the imposed electric field, respectively, in the fluid containing the counterions only around the particle, and the ionic continuity equation and modified Stokes equations are solved for the electrochemical potential energy and fluid flow fields, respectively. Explicit analytical formulas for the electrophoretic mobility of the particles and effective electric conductivity of the suspension are obtained, and the particle interaction effects on these transport properties are significant and interesting. The scaled zeta potential, electrophoretic mobility, and effective electric conductivity increase monotonically with an increase in the scaled surface charge density of the particles and in general decrease with an increase in the particle volume fraction, keeping each other parameter unchanged. Under the Debye-Hückel approximation, the dependence of the electrophoretic mobility normalized with the surface charge density on the ratio of the particle radius to the Debye screening length and particle volume fraction in a salt-free suspension is same as that in a salt-containing suspension, but the variation of the effective electric conductivity with the particle volume fraction in a salt-free suspension is found to be quite different from that in a suspension containing added electrolyte.     

Dielectric friction as a mechanism for selectivity alteration in capillary electrophoresis using acetonitrile-water media

The mobilities of a series of aromatic carboxylates and sulfonates, ranging in charge from -1 to -4, were investigated as a function of acetonitrile concentration in the electrophoretic buffer. Absolute mobilities were determined by extrapolation of the effective mobilities to zero ionic strength according to the Pitts' equation. In general, anions of higher charge were more strongly influenced by ionic strength, with similarly charged anions experiencing ionic strength effects that were not significantly different at the 95% confidence level. Furthermore, the relative magnitudes of the Onsager slopes varied with acetonitrile content according to the z/(etaepsilon(1/2)) dependence in the electrophoretic effect of the Pitt's equation. Addition of acetonitrile to the electrophoretic media resulted in changes in the absolute mobilities of the anions. These acetonitrile-induced selectivity alterations were attributed to dielectric friction. As predicted by the Hubbard-Onsager model of dielectric friction, changes in sulfonate mobility were shown to correlate to changes in solvent viscosity (eta), dielectric constant (epsilon), and relaxation time (tau). The combined effects of ionic strength and dielectric friction caused analytes with higher charge-to-size ratios to be slowed to a greater extent upon addition of acetonitrile compared to those with lower charge-to-size. For example, at 75% acetonitrile and 20 mM ionic strength, a migration order reversal occurred between the triply and singly charged sulfonates.     

Electrophoretic mobility, zeta potential, and fixed charge density of bovine knee chondrocytes, methyl methacrylate-sulfopropyl methacrylate, polybutylcyanoacrylate, and solid lipid nanoparticles

The electrophoretic mobility and zeta potential of bovine knee chondrocytes (BKCs), methyl methacrylate-sulfopropyl methacrylate (MMA-SPM) nanoparticles (NPs), polybutylcyanoacrylate (PBCA) NPs, and solid lipid nanoparticles (SLNs) were investigated under the influences of Na+, K+, and Ca2+ with various ionic strengths. The fixed charge density in the surface layers of the four biocolloidal particles was estimated from the experimental mobility of capillary electrophoresis with a theory of soft charged colloids. The results revealed that, for a specific cationic species, the absolute values of the electrophoretic mobility, the zeta potential, and the fixed charge density decreased with an increase in ionic strength. For a constant ionic strength, the effect of ionic species on the reduction in the absolute values of the electrophoretic mobility, the zeta potential, and the fixed charge density followed the order Na+>K+>Ca2+ for the negatively charged BKCs, MMA-SPM NPs, and SLNs. The reverse order is true for the positively charged PBCA NPs.     

Electrokinetic effects of adsorbed neutral polymers

On the basis of a previously developed hydrodynamic model for adsorbed polymers the charge flow along a charged interface with adsorbed (uncharged) polymer is calculated. An effective electrokinetic layer thickness is defined and its dependence on the characteristics of the adsorbed polymer and the ionic strength is studied. It is found that tails are very important for the hydrodynamic effects considered because they effectively screen the solvent flow from inner parts of the absorbed layer. The electrokinetic layer thickness increases with decreasing ionic strength, and tends to a limit equal to the hydrodynamic thickness at very low ionic strength.     

Controlling the electrophoretic mobility of single-walled carbon nanotubes: a comparison of theory and experiment

The electrophoretic mobilities of single-walled carbon nanotubes (SWNTs) in agarose gels subjected to negatively charged covalent functionalization and noncovalent anionic surfactant adsorption are compared using a simplified hydrodynamic model. Net charges are calculated on the basis of estimated friction coefficients for cylindrical rodlike particles. The effects of functionalization with negatively charged 4-hydroxybenzene diazonium and anionic sodium cholate are quantified and compared with model predictions. The adsorption of Na+ counterions into the nonionic surfactant layer adsorbed on SWNTs (Triton-X-405) is shown to induce a positive charge and reverse the mobility under select conditions. This effect has not been identified or quantified for nanoparticle systems and may be important in the processing of these systems.     

Electrokinetic actuation of an uncharged polarizable dielectric droplet in charged hydrogel medium

Electrokinetic transport of an uncharged nonconducting microsized liquid droplet in a charged hydrogel medium is studied. Dielectric polarization of the liquid drop under the action of an externally imposed electric field induces a non-homogeneous charge density at the droplet surface. The interactions of the induced surface charge of the droplet with the immobile charges of the hydrogel medium generates an electric force to the droplet, which actuates the drop through the charged hydrogel medium. A numerical study based on the first principle of electrokinetics is adopted. Dependence of the droplet velocity on its dielectric permittivity, bulk ionic concentration, and immobile charge density of the gel is analyzed. The surface conduction is significant in presence of charged gel, which creates a concentration polarization. The impact of the counterion saturation in the Debye layer due to the dielectric decrement of the medium is addressed. The modified Nernst–Planck equation for ion transport and the Poisson equation for the electric field is considered to take into account the dielectric polarization. A quadrupolar vortex around the uncharged droplet is observed when the gel medium is considered to be uncharged, which is similar to the induced charge electroosmosis around an uncharged dielectric colloid in free-solution. We find that the induced charge electrokinetic mechanism creates a strong recirculation of liquid within the droplet and the translational velocity of the droplet strongly depends on its size for the dielectric droplet embedded in a charged gel medium.     

End effects for a rodlike polyelectrolyte molecule in salt solution

Electrostatic potentials around a single rodlike polyelectrolyte molecule are calculated by solving the nonlinear Poisson–Boltzmann equation numerically in the presence of externally added salt. The polyion is regarded as a cylinder with a finite length whose side surface is uniformly charged and end surfaces uncharged. The calculations show that the distance to which end effects extend is about half the Debye screening length and is almost independent of the surface charge density and concentration of added salt. For a long polyion whose length is much greater than the Debye length, the end effects can be neglected even for a polyelectrolyte with high surface charges, whereas they play an important role for a short polyion with a length of the same order as the Debye length. In addition, a strong charge condensation is found in the direction of the axis of the cylinder for a long polyion.     

End-labeled free-solution electrophoresis of DNA

DNA is a free-draining polymer. This subtle but "unfortunate" property of highly charged polyelectrolytes makes it impossible to separate nucleic acids by free-flow electrophoresis. This is why one must typically use a sieving matrix, such as a gel or an entangled polymer solution, in order to obtain some electrophoretic size separation. An alternative approach consists of breaking the charge to friction balance of free-draining DNA molecules. This can be achieved by labeling the DNA with a large, uncharged molecule (essentially a hydrodynamic parachute, which we also call a drag-tag) prior to electrophoresis; the resulting methodology is called end-labeled free-solution electrophoresis (ELFSE). In this article, we review the development of ELFSE over the last decade. In particular, we examine the theoretical concepts used to predict the ultimate performance of ELFSE for single-stranded (ssDNA) sequencing, the experimental results showing that ELFSE can indeed overcome the free-draining issue raised above, and the technological advances that are needed to speed the development of competitive ELFSE-based sequencing and separation technologies. Finally, we also review the reverse process, called free-solution conjugate electrophoresis (FSCE), wherein uncharged polymers of different sizes can be analyzed using a short DNA molecule as an electrophoretic engine.     

Electrophoretic mobilities of large organic ions in nonaqueous solvents: determination by capillary electrophoresis in propylene carbonate, N,N-dimethylformamide, N,N,-dimethylacetamide, acetonitrile and methanol

The mobilities of the monocharged permanent tertraphenylphosphonium cation and tetraphenylborate anion are determined by capillary zone electrophoresis in different organic solvents as a function of the ionic strength, I, of the background electrolyte. The nonaqueous solvents are propylene carbonate (PC), N,N-dimethylformamide (DMF), N,N,-dimethylacetamide (DMA), acetonitrile (MeCN) and methanol (MeOH). The ionic strength is between 5 and 50 mmol/L. The mobility as a function of I is in good agreement with the theory of Debye, Hückel and Onsager (DHO), extended by the ion size parameter as introduced by Falkenhagen and Pitts. The values of the limiting DHO slopes of the mobility vs. I curves (the slopes express the influence of the solvent on the reduction of the mobility with increase of I) decrease in the order MeCN > MeOH > DMF > DMA > PC. Absolute mobilities (obtained by extrapolation to I = 0) of a particular ion differ by a factor of about 7 between the solvents. However, constancy within 10% is observed for their Walden products (the absolute mobility multiplied with the solvent's macroviscosity). The role of dielectric friction on the mobility of the present monocharged, large analyte ions is discussed according to the theory of Hubbard and Onsager. Based on the radii of the ions, the static permittivity of the solvent and its permittivity at infinite frequency, and the relaxation time of polarization, an equal contribution of dielectric and hydrodynamic friction is predicted in MeOH as solvent. Experimental data are in contrast to this prediction, indicating the overestimation of dielectric friction, and the dominance of hydrodynamic friction on the migration of the analyte ions in all solvents under consideration.     

Effect of core charge density on the electrophoresis of a soft particle coated with polyelectrolyte layer

The electrophoresis of a charged soft particle with charged rigid core is considered under a weak imposed field condition. The rigid core of the soft particle is considered to have a finite dielectric permittivity and a fixed volume charge density. The electric potential distribution is determined by solving the Poisson-Boltzman equation out side the rigid core and a Poisson equation within the core along with continuity conditions on the core-shell interface. We have extended the analytic expression of Ohshima (Electrophoresis 27:526–533, 2006) for the electrophoretic mobility of a soft particle with a charged shell to include the effect of the volume charge density of the rigid core. Mobility based on the present expression matches exactly with the existing analytical solutions for a soft particle with an uncharged core. We have also made a comparison of our solution for mobility with an uncharged rigid core with the existing experimental results. The impact of the core charge density on the soft particle mobility is analyzed.     

Approximate analytic expressions for the electrophoretic mobility of spherical soft particles

Approximate analytic expressions are derived for the electrophoretic mobility of a weakly charged spherical soft particle consisting of the particle core covered with a surface layer of polymers in an electrolyte solution. The particle core and the surface polymer layer may be charged or uncharged. The obtained electrophoretic mobility expressions, which involve neither numerical integration nor exponential integrals, are found to be in excellent agreement with the exact numerical results. It is also found that the obtained mobility expressions reproduce all the previously derived limiting expressions and approximate analytic expressions for the electrophoretic mobility of a weakly charged spherical soft particle.     

Factors affecting selectivity of inorganic anions in capillary electrophoresis

Capillary zone electrophoretic separations of inorganic anions are largely governed by the intrinsic (infinite dilution) mobility of the anion. This in turn is a function of the hydrodynamic friction caused by the size of the ion and the dielectric friction caused by the charge density of the anion re-orienting the surrounding solvent. The influence of these factors on the mobility of anions is examined in both water and nonaqueous solvents. The influence of other experimental parameters, such as ionic strength, ion association, electroosmotic flow modifier concentration, and the addition of complexing agents such as polymeric cations, cyclodextrins, crown ethers and cryptands are also reviewed. From this discussion, some rules of thumb as to when different approaches will be most effective are drawn.