Electrophoresis of spheres with uniform zeta potential in a gel modeled as an effective medium

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.

Diffusiophoresis of a spherical particle in porous media

Recent experiments by Doan et al. (Nano Lett., 2021, 21, 7625-7630) demonstrated and measured colloid diffusiophoresis in porous media but existing theories cannot predict the observed colloid motion. Here, using regular perturbation method, we develop a mathematical model that can predict the diffusiophoretic motion of a charged colloidal particle driven by a binary monovalent electrolyte concentration gradient in a porous medium. The porous medium is modeled as a Brinkman medium with a constant Darcy permeability. The linearized Poisson-Boltzmann equation is employed to model the equilibrium electric potential distribution that is driven out-of-equilibrium under diffusiophoresis. We report three key findings. First, we demonstrate that colloid diffusiophoresis could be drastically hindered in a porous medium due to the additional hydrodynamic drag compared to diffusiophoresis in a free electrolyte solution. Second, we show that the variation of the diffusiophoretic motion with respect to a change in the electrolyte concentration in a porous medium could be qualitatively different from that in a free electrolyte solution. Third, our results match quantitatively with experimental measurements, highlighting the predictive power of the present model. The mathematical model developed here could be employed to design diffusiophoretic colloid transport in porous media, which are central to applications such as nanoparticle drug delivery and enhanced oil recovery.

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%.

Transient electrophoresis of a conducting spherical particle embedded in an electrolyte-saturated Brinkman medium

In this study, the time-dependent electrophoretic motion of a conducting spherical particle embedded in an arbitrary electrolyte solution saturated porous medium is investigated. The porous medium is uniformly charged and the embedded hard particle is charged with constant ζ -potential or constant surface charge density. The unsteady modified Brinkman equation with an electric force term, which governs the fluid velocity field, is used to model the porous medium and is solved by Laplace's transform technique. An analytical expression for the electrophoretic velocity of the spherical particle is obtained in Laplace transform domain as a function of the relevant parameters, and its inversion is obtained through numerical techniques. Also, in this study, the steady-state electrophoretic velocity is obtained analytically as linear functions of  ζ -potential (or surface density charge) and the fixed charge density. The steady-state electrophoretic velocity is displayed graphically for various relevant parameters and compered with the available data in the literature. Also, the numerical values of the transient electrophoretic velocity are plotted versus the nondimensional elapsed time and discussed for different values of the Debye length parameter, density ratio, permeability of the porous medium, and for high and nonconducting particles.

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.

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.

Study of release kinetics and degradation thermodynamics of ferric citrate liposomes

Ferric citrate liposome (FAC-Lip) with good sustained-released property was prepared by the rotary-evaporated film-ultrasonic method, and characterized by TEM, DLS, zeta potential and encapsulation efficiency (EE%). The effects of membrane material ratios (m_PC: m_chol = 8:1, 10:1 and 12:1) and drug lipid ratios (m_FAC: m_PC = 1:4, 1:6.5 and 1:8) on the release of FAC-Lip were examined. The in vitro release kinetic models and mechanisms of FAC-Lip in artificial gastric juice (SGF) and artificial intestinal juice (SIF) compared with free-FAC were determined. The thermal degradation in PBS was also determined. The results showed that FAC-Lip with membrane material ratio (10:1) and drug lipid ratio (1:6.5) had the optimal sustained-released property, unilamellar vesicles with uniform size (178 ± 2.12 nm), negative charge (-56 ± 3.51 mV) and high encapsulation efficiency (72.77 ± 0.42%). The in vitro release kinetic models of FAC-Lip were two-phase kinetics model and the release mechanisms were non-Fick diffusion both in SGF and SIF. The thermal degradation of FAC-Lip was an endothermic and spontaneous reaction. The results may be helpful in optimizing drug-liposome design, application in food and medicine industries, and furthermore, predicting and guiding medication in vivo.

Nasal delivery of nanoliposome-encapsulated ferric ammonium citrate can increase the iron content of rat brain

BACKGROUND

Iron deficiency in children can have significant neurological consequences, and iron supplementation is an effective treatment of choice. However, traditional routes of iron supplementation do not allow efficient iron delivery to the brain due to the presence of the blood-brain barrier. So an easily delivered iron formulation with high absorption efficiency potentially could find widespread application in iron deficient infants.

RESULTS

In this study, we have developed and characterized a nanovesicular formulation of ferric ammonium citrate (ferric ammonium citrate nanoliposomes, FAC-LIP) and have shown that it can increase brain iron levels in rats following nasal administration. FAC was incorporated into liposomes with high efficiency (97%) and the liposomes were small (40 nm) and stable. Following intranasal delivery in rats, FAC-LIP significantly increased the iron content in the olfactory bulb, cerebral cortex, striatum, cerebellum and hippocampus, and was more efficient at doing so than FAC alone. No signs of apoptosis or abnormal cell morphology were observed in the brain following FAC-LIP administration, and there were no significant changes in the levels of SOD and MDA, except in the cerebellum and hippocampus. No obvious morphological changes were observed in lung epithelial cells or tracheal mucosa after nasal delivery, suggesting that the formulation was not overtly toxic.

CONCLUSIONS

In this study, nanoscale FAC-LIP proved an effective system delivering iron to the brain, with high encapsulation efficiency and low toxicity in rats. Our studies provide the foundation for more detailed investigations into the applications of niosomal nasal delivery of liposomal formulations of iron as a simple and safe therapy for iron deficiency anemia. Graphical abstract The diagrammatic sketch of "Nasal delivery of nanoliposome-encapsulated ferric ammonium citrate can increase the iron content of rat brain". Nanoliposome-encapsulated ferric ammonium citrate (FAC-LIP) was successfully prepared and intranasal administration of FAC-LIP increased both the total iron contents and iron storage protein (FTL) expression in rat olfactory bulb, cerebral cortex, striatum and hippocampus, compared with those of FAC groups. Moreover, there was not overtly toxic affects to brain, lung epithelial cells and tracheal mucosa.

Determination of the Exact Particle Radius Distribution for Silica Nanoparticles via Capillary Electrophoresis and Modeling the Electrophoretic Mobility with a Modified Analytic Approximation

In this study, we use aqueous dispersions of amorphous silica nanoparticles of various sizes to investigate whether electropherograms recorded from capillary electrophoresis experiments can be converted directly into exact number-based particle radius distributions, provided that there is a relaxation effect-based size selectivity of the electrophoretic mobility and provided that the electrokinetic potential ζ of the particles can be regarded to be homogeneous over the surface of the particles, independent of the particle size. The results of this conversion procedure are compared with number-based particle radius distributions obtained from a large set of transmission electron microscopy (TEM) data. For this specific example, it is shown that the modified analytic approximation developed by Ohshima adequately describes the mobility-dependent relaxation effect and the electrophoretic mobility of the particle as a function of the reduced hydrodynamic radius and electrokinetic potential, which is a prerequisite for the presented procedure. Simultaneously, we confirmed that for the given Debye length/particle diameter ratio the electrokinetic surface charge density can be regarded to be size-invariant (including spherical geometry and planar limiting case). It is shown that the accuracy of the results of the developed method is comparable to that gained by a large set of TEM data, which is important when a precise description of the particle size distribution is needed to deduce conclusions regarding the underlying mechanism(s) of particle growth. The values obtained for the dispersion (width) of the distribution show only a small negative deviation, when compared with the TEM data (4-16%).

Electrophoretic and Electroosmotic Motion of a Charged Spherical Particle within a Cylindrical Pore Filled with Debye-Bueche-Brinkman Polymeric Solution

Electrophoretic and electroosmotic motion of a charged spherical particle within a cylindrical pore filled with a Debye-Bueche-Brinkman (DBB) polymeric solution is investigated theoretically, which is of high relevance in capillary electrophoresis as well as micro- and nanofluidic applications involving polymeric solutions in a micro- or nanopore. The DBB model describes the rheological response of a polymeric solution with a linear polymer dissolved in a homogeneous solvent. It is a well-known non-Newtonian model in liquid physics based on rigorous theoretical derivations. By Debye and Bueche, corresponding governing fundamental electrokinetic equations are solved numerically with a patched pseudo-spectral method based on Chebyshev polynomials. We found that the double-layer polarization effect reduces the particle mobility severely when the Debye parameter, κa, is around unity, especially in narrow pores. This is attributed to the extra confinement effect from the nearby wall, which tends to sweep the predominant counterions within the double layer to the wake of the moving particle, resulting in a motion-deterring induced electric field. The electrophoretic mobility in a polymer solution is smaller than that in an aqueous electrolyte solution in general as a result of the much stronger viscous drag effect in a polymer solution. Moreover, electroosmotic flow (EOF) as a result of a charged pore wall is found to exhibit a highly non-Newtonian behavior. Unlike the corresponding plug-like flow for a Newtonian solution, an axisymmetric flow with a large local maximum in the velocity profile in the region near the pore wall is observed. This radial-varying velocity profile offers a potential extra separation mechanism, which favors the elution of smaller particles in general. The results obtained here provide fundamental understandings and insights of the electrophoresis and electroosmosis phenomena in a cylindrical pore filled with polymeric solution.

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.

Quantum dots-hemin: Preparation and application in the absorption of heme iron

The absorption mechanism of heme iron remains unclear due to the limit of labeling techniques. Quantum dots (QDs) are powerful fluorescent probes resistant to photobleaching, however, there is no data about the application of QDs in heme iron absorption. Herein, we prepared hemin-coated CdSe/ZnS (QDs-hemin), and studied their absorption in vitro and in vivo. Results showed that QDs-hemin had uniform particle sizes, physiological stability and high joint efficiency. Moreover, QDs-hemin could be successfully absorbed gradually into the duodenum with the time using synchrotron radiation micro X-ray fluorescence and confocal laser scanning microscopy. Furthermore, QDs-hemin were observed to degrade in lysosomes, and their absorption was blocked by Heme Carrier Protein 1 (HCP1) antibody and HCP1 siRNA. All the results demonstrate that QDs can be a good tracer for heme iron and that HCP1 pathway is critical and predominant over the endocytosis pathway in the absorption mechanism.

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.

Electrophoresis of a charged colloidal particle in porous media: boundary effect of a solid plane

Electrokinetic treatments such as the electrophoretic technique have been applied successfully to various soil remediation and contaminant removal situations. To understand further the fundamental features involved, the electrophoretic motion of a charged particle in porous media is investigated theoretically in this study, focusing on the boundary effect of a nearby solid plane toward which the particle moves perpendicularly. The porous medium is modeled as a Brinkman fluid with a characteristic screening length (λ(-1)) that can be obtained directly from the experimental data. General electrokinetic equations are used to describe the system and are solved with a pseudospectral method based on Chebyshev polynomials. We found that the particle motion is deterred by the boundary effect in general. The closer the particle is to the boundary, the more severe this effect is. Up to a 90% reduction in particle mobility is observed in some situations. This indicates that a drastic overestimation (10-fold!) of the overall transport rate of particles may occur for large-scale in situ operations in porous media, such as soil remediation utilizing large planar electrodes, should a portable analytical formula valid for bulk systems only be used. Correction factors for various situations in porous media are presented as convenient charts with which to aid engineers and researchers in the field of environmental engineering, for instance, as a realistic estimation of the actual transport rate obtainable. In addition, the results of present study can be applied to biomedical engineering and drug delivery as well because polymer gels and skin barriers both have a porous essence.

Synthesis and characterization of a hydrogel with controllable electroosmosis: a potential brain tissue surrogate for electrokinetic transport

Electroosmosis is the bulk fluid flow initiated by application of an electric field to an electrolyte solution in contact with immobile objects with a nonzero ζ-potential such as the surface of a porous medium. Electroosmosis may be used to assist analytical separations. Several gel-based systems with varying electroosmotic mobilities have been made in this context. A method was recently developed to determine the ζ-potential of organotypic hippocampal slice cultures (OHSC) as a representative model for normal brain tissue. The ζ-potential of the tissue is significant. However, determining the role of the ζ-potential in solute transport in tissue in an electric field is difficult because the tissue's ζ-potential cannot be altered. We hypothesized that mass transport properties, namely the ζ-potential and tortuosity, could be modulated by controlling the composition of a set of hydrogels. Thus, poly(acrylamide-co-acrylic acid) gels were prepared with three compositions (by monomer weight percent): acrylamide/acrylic acid 100/0, 90/10, and 75/25. The ζ-potentials of these gels at pH 7.4 are distinctly different, and in fact vary approximately linearly with the weight percent of acrylic acid. We discovered that the 25% acrylic acid gel is a respectable model for brain tissue, as its ζ-potential is comparable to the OHSC. This series of gels permits the experimental determination of the importance of electrokinetic properties in a particular experiment or protocol. Additionally, tortuosities were measured electrokinetically and by evaluating diffusion coefficients. Hydrogels with well-defined ζ-potential and tortuosity may find utility in biomaterials and analytical separations, and as a surrogate model for OHSC and living biological tissues.

Review modeling the free solution and gel electrophoresis of biopolymers: The bead array-effective medium model

Free solution and gel electrophoresis is an extremely useful tool in the separation of biopolymers. The complex nature of biopolymers, coupled with the usefulness of electrophoretic methods, has stimulated the development of theoretical modeling over the last 30 years. In this work, these developments are first reviewed with emphasis on Boundary Element and bead methodologies that enable the investigator to design realistic models of biopolymers. In the present work, the bead methodology is generalized to include the presence of a gel through the Effective Medium model. The biopolymer is represented as a bead array. A peptide, for example, made up of N amino acids is modeled as 2N beads. Duplex DNA is modeled as a discrete wormlike chain consisting of touching beads. The technical details of the method are placed in three Appendices. To illustrate the accuracy and effectiveness of the approach, two applications are considered. Model studies on both the free solution mobility of 73 peptides ranging in size from 2 to 42 amino acids, and the mobility of short duplex DNA in dilute agarose gels are discussed.