Nanoparticle gel electrophoresis: Soft spheres in polyelectrolyte hydrogels under the Debye–Hückel approximation

Transient Gel Electrophoresis of a Spherical Colloidal Particle

The general theory is developed for the time-dependent transient electrophoresis of a weakly charged spherical colloidal particle with an electrical double layer of arbitrary thickness in an uncharged or charged polymer gel medium. The Laplace transform of the transient electrophoretic mobility of the particle with respect to time is derived by considering the long-range hydrodynamic interaction between the particle and the polymer gel medium on the basis of the Brinkman-Debye-Bueche model. According to the obtained Laplace transform of the particle's transient electrophoretic mobility, the transient gel electrophoretic mobility approaches the steady gel electrophoretic mobility as time approaches infinity. The present theory of the transient gel electrophoresis also covers the transient free-solution electrophoresis as its limiting case. It is shown that the relaxation time for the transient gel electrophoretic mobility to reach its steady value is shorter than that of the transient free-solution electrophoretic mobility and becomes shorter as the Brinkman screening length decreases. Some limiting or approximate expressions are derived for the Laplace transform of the transient gel electrophoretic mobility.

Gel Electrophoresis of a Hydrophobic Liquid Droplet with an Equipotential Slip Surface

A theoretical study has been carried out on the electrophoresis of charged dielectric liquid droplets with an equipotential and hydrodynamically slipping surface moving in a quenched polymeric charged hydrogel medium. The liquid inside the droplet is electrically neutral. The Brinkman-Debye-Bueche model is employed to study the gel electrophoresis of such a hydrophobic and equipotential liquid droplet considering the long-range hydrodynamic interaction between a migrating droplet and the gel skeleton. Within the weak field and Debye-Hückel electrostatic framework, we derive an original closed-form expression for electrophoretic mobility, which further recovers the existing mobility expressions derived under several limiting conditions. The derived expressions for electrophoretic mobility explicitly involve exponential integrals, which are not so convenient for practical applications. Thus, the exact forms of the electrophoretic mobility under various electrohydrodynamic conditions are further approximated to make them free from exponential integrals. The approximate forms are found to be in excellent agreement with the exact results with maximum relative errors of about 1.5%.

A simplified model for gel electrophoresis of a hydrophobic rigid colloid

Electrophoresis of a charged dielectric hydrophobic colloid embedded in a charged hydrogel medium is addressed. A slip velocity condition at the particle surface is considered. The characteristic of the gel electrophoresis is different compared with the free-solution electrophoresis due to the presence of immobile charges of the gel medium, which induces a strong background electroosmotic flow and modifies the Debye layer of the colloid. The gel electrophoresis of the dielectric hydrophobic charged colloid is made based on first-order perturbation analysis. A closed form solution involving simple exponential integrals for the mobility is derived, which reduces to several existing mobility expressions under limiting conditions such as for the gel electrophoresis of hydrophilic particles and a hydrophobic colloid in free-solution electrophoresis. We find that the mobility reversal is achieved by varying the Debye length or gel permeability. For the present first-order perturbation analysis, unlike free-solution electrophoresis, the particle dielectric permittivity is found to influence the mobility. One of the intriguing features of the present study is the derivation of the simplified mobility expression, which can be easily computed for a given set of parameter values.

Flocculation of Clay Suspensions by Anionic and Cationic Polyelectrolytes: A Systematic Analysis

The characteristics of clayey suspensions, majorly composed of quartz microparticles, in the presence of anionic and cationic polyelectrolytes were investigated using different techniques. A wide range of clay concentrations was used, i.e., from 0.07 to 1000 g/L for different experimental techniques, based on the fact that the clay concentration possible to analyze with selected experimental methods was significantly different. The optimum flocculant to clay ratio was defined as the ratio that gives the fastest initial floc growth by static light scattering or fastest initial settling velocity by settling column experiments. In case of anionic polyelectrolyte, it was observed that the optimum flocculant dose depends on the amount of cations present in the system. For suspensions made with demi-water, a lower optimum flocculant dose (<1 mg/g) than for suspensions prepared in tap water (2.28 mg/g) was observed. At these lower salinities, the supernatant remained turbid in all the experiments and was, therefore, not a good measure for optimal anionic based flocculation. The equilibrium floc size at a given shear rate was found to be independent on the shear history of the floc and only dependent on the current applied shear. This was confirmed by both light scattering and rheological analysis. In case of cationic polyelectrolyte, the optimum flocculant ratio (5–6 mg/g) corresponded to the ratio that gives the lowest electrophoretic mobility for each clay concentration and to the ratio that gives the fastest settling velocity for the highest clay concentrations (12–15 g/L), where static light scattering measurements were not possible. All investigation techniques, therefore, proved to be good indicators for predicting the optimum flocculant to clay ratio. For the lowest concentrations (1.75–8.7 g/L) studied by settling column measurements, the optimum flocculant ratio was observed to increase with decreasing clay concentration, for fixed mixing conditions. The optimum flocculant to clay ratio was not always corresponding to the clearest supernatant and the size of flocs at optimum dosage was dependent on the mixing efficiency. The equilibrium floc size at a given shear rate was found to be dependent on the shear history of the floc and the current applied shear. This was confirmed by both light scattering and rheological analysis.

Influence of slip velocity at the core of a diffuse soft particle and ion partition effects on mobility

Nonlinear effects on the electrophoresis of a soft particle, consisting of a rigid hydrophobic core coated with a diffuse polymer layer (PEL) suspended in an electrolyte medium, are studied. The impact of the ion partitioning effect arising due to the Born energy difference between the PEL and the electrolyte is approximated based on the equilibrium Boltzmann equation, with which the ion distribution and hence, the charge density is modified. The equations describing the electrokinetic transport comprising the Darcy-Brinkman extended Navier-Stokes equations which includes the ion partitioning effect coupled with the modified Nernst-Planck equations and Poisson equations for electric field are solved numerically. The present numerical model for the soft particle compares well with the existing theoretical solutions and experimental results in the limiting cases. A deviation from existing simplified models based on the Boltzmann distribution of ions occurs when the Debye layer polarization, relaxation and the electroosmosis induced by the PEL immobile charge become significant. The hydrophobicity of the inner core strongly influences the nonlinear electrokinetic effects by modifying the Debye layer, electroosmotic flow in the PEL and surface conduction. The results indicate that the ion partitioning can significantly increase the electrophoretic mobility of the soft particle by attenuating the shielding effect. When the Debye layer is in the order of the particle size the hydrophobicity of the core surface and the ion partitioning effect manifest the surface conduction, which implies that the Boltzmann distribution of ions is no longer valid. The core hydrophobicity and ion partitioning effect have influence on the condensation of the PEL immobile charge, which creates a significant impact on the mobility.

Gel electrophoresis of a soft particle

A review is given on the theory of the gel electrophoresis of a spherical soft particle, i.e., the electrophoresis of a spherical hard particle covered with an ion-penetrable surface layer of polyelectrolytes moving in a polymer gel medium, which may either be charged or uncharged. A detailed study is made on the fundamental electrokinetic equations describing the soft-particle gel electrophoresis due to the long-range hydrodynamic particle-gel interaction on the basis of the Brinkman-Debye-Bueche model. A general expression and several approximate analytic expressions are given for the electrophoretic mobility of a soft particle in a polymer gel medium.

Electrophoretic Mechanism of Au25(SR)18 Heating in Radiofrequency Fields

Gold nanoparticles in radiofrequency (RF) fields have been observed to heat. There is some debate over the mechanism of heating. Au₂₅(SR)₁₈ in RF is studied for the mechanistic insights obtainable from precise synthetic control over exact charge, size, and spin for this nanoparticle. An electrophoretic mechanism can adequately account for the observed heat. This study adds a new level of understanding to gold particle heating experiments, allowing for the first time a conclusive connection between theoretical and experimentally observed heating rates.

Electrokinetics of nanoparticle gel-electrophoresis

Gel-electrophoresis has been demonstrated in recent decades to successfully sort a great variety of nanoparticles according to their size, charge, surface chemistry, and corona architecture. However, quantitative theoretical interpetations have been limited by the number and complexity of factors that influence particle migration. Theoretical models have been fragmented and incomplete with respect to their counterparts for free-solution electrophoresis. This paper unifies electrokinetic models that address complex nanoparticle corona architectures, corona and gel charge regulation (e.g., by the local pH), multi-component electrolytes, and non-linear electrostatics and relaxation effects. By comprehensively addressing the electrokinetic aspects of the more general gel-electrophoresis problem, in which short-ranged steric interactions are significant, a stage is set to better focus on the physicochemical and steric factors. In this manner, it is envisioned that noparticle gel-electrophoresis may eventually be advanced from a nanoparticle-characterization tool to one that explicitly probes the short-ranged interactions of nanoparticles with soft networks, such as synthetic gels and biological tissues. In this paper, calculations are undertaken that identify a generalized Hückel limit for nanoparticles in low-conductivity gels, and a new Smoluchowski limit for polyelectrolyte-coated particles in high-conductivity gels that is independent of the gel permeability. Also of fundamental interest is a finite, albeit small, electrophoretic mobility for uncharged particles in charged gels. Electrophoretic mobilities and drag coefficients (with electroviscous effects) for nanoparticles bearing non-uniform coronas show that relaxation effects are typically weak for the small nanoparticles (radius ≈3-10 nm) to which gel-electrophoresis has customarily been applied, but are profound for the larger nanoparticles (radius ≳ 40 nm in low conductivity gels) to which passivated gel-electrophoresis experiments have recently been applied. To demonstrate its practical application, the model is applied to (pH charge regulating) carboxylated polystyrene nanospheres in low-density passivated agarose gels (weak steric effects). This furnishes a new theoretical interpretation of literature data for which a finite diffuse-layer-thickness, pH-charge regulation, high charge, and relaxation effects dominate over the steric influences.

Importance of core electrostatic properties on the electrophoresis of a soft particle

The impact of the volumetric charged density of the dielectric rigid core on the electrophoresis of a soft particle is analyzed numerically. The volume charge density of the inner core of a soft particle can arise for a dendrimer structure or bacteriophage MS2. We consider the electrokinetic model based on the conservation principles, thus no conditions for Debye length or applied electric field is imposed. The fluid flow equations are coupled with the ion transport equations and the equation for the electric field. The occurrence of the induced nonuniform surface charge density on the outer surface of the inner core leads to a situation different from the existing analysis of a soft particle electrophoresis. The impact of this induced surface charge density together with the double-layer polarization and relaxation due to ion convection and electromigration is analyzed. The dielectric permittivity and the charge density of the core have a significant impact on the particle electrophoresis when the Debye length is in the order of the particle size. We find that by varying the ionic concentration of the electrolyte, the particle can exhibit reversal in its electrophoretic velocity. The role of the polymer layer softness parameter is addressed in the present analysis.

Hydrogel charge regulation and electrolyte ion-concentration perturbations in nanoparticle gel electrophoresis

Gel electrophoresis of spherical nanoparticles (NPs) is studied using an electrokinetic model that couples the ion conservation equations to the Poisson and fluid momentum equations, thus including the so-called polarization and relaxation processes. This model is therefore the charged gel electrophoresis analogue of the well-known O’Brien and White solution of the standard electrokinetic model for free-solution electrophoresis. Results are provided for the small NPs (size around 10 nm) to which gel electrophoresis is relevant, because particles must be small enough to permeate the gel: these include the particle drag coefficient (or Brownian diffusivity), which is subject to hydrodynamic screening and electroviscous effects, and the electrophoretic mobility, which is subject to nonlinear electrostatic and charge polarization influences. Also addressed are the influences of charge-regulating gels and the accompanying particle-induced immobile charge-density perturbations. Ion-concentration perturbations attenuate the electrophoretic mobility and enhance the drag coefficient according to the particle charge and the mobility of the most abundant counterion. However, dynamic regulation of the hydrogel charge—termed the secondary immobile charge-density perturbation—has a negligible influence on the particle mobility, and may therefore be neglected for most practical purposes.

Corona charge regulation in nanoparticle electrophoresis

Nanoparticle (NP) size and charge play key roles in bioconjugation chemistry, imaging and drug delivery. Although the electrophoretic mobility and hydrodynamic size are routinely measured, interpreting these data can be extremely difficult. Here, the challenge is addressed via an electrokinetic model for spheres bearing a soft amphoteric corona, the charge of which is regulated by a multi-component electrolyte. The model is applied to NPs with a metallic core to which are grafted poly(ethylene glycol) chains with either weak acid or amphiprotic end groups. The results elucidate the separate roles of electrolyte pH and ionic strength on the electrophoretic mobility and diffusion coefficient. In this study, the forces were evaluated directly, rather than from the Stokeslet velocity disturbances. While the second-order convergence was demonstrated by both methods, the direct approach, which uses only the inner part of the global solution, furnished superior accuracy and robustness. This may benefit future attempts to model the dielectric and electroacoustic properties of these complex nanoparticulates.

Facile access to poly(DMAEMA-co-AA) hydrogels via infrared laser-ignited frontal polymerization and their polymerization in the horizontal direction

pH sensitive hydrogels are fabricated via infrared laser-ignited frontal polymerization (LIFP), and LIFP in the horizontal direction is employed towards dye adsorption.