Charged Porous Sphere Research Articles

Self-Assembly and In Situ Quaternization of Triblock Bottlebrush Block Copolymers via Organized Spontaneous Emulsification for Effective Loading of DNA.

Microspheres bearing large pores are useful in the capture and separation of biomolecules. However, pore size is typically poorly controlled, leading to disordered porous structures with limited performances. Herein, ordered porous spheres with a layer of cations on the internal surface of the nanopores are facilely fabricated in a single step for effective loading of DNA bearing negative charges. Triblock bottlebrush copolymers (BBCPs), (polynorbornene-g-polystyrene)-b-(polynorbornene-g-polyethylene oxide)-b-(polynorbornene-g-bromoethane) (PNPS-b-PNPEO-b-PNBr), are designed and synthesized for fabrication of the positively charged porous spheres through self-assembly and in situ quaternization during an organized spontaneous emulsification (OSE) process. Pore diameter as well as charge density increase with the increase of PNBr content, resulting in a significant increase of loading density from 4.79 to 22.5ngµg-1 within the spheres. This work provides a general strategy for efficient loading and encapsulation of DNA, which may be extended to a variety of different areas for different real applications.

Diffusiophoresis of a Charged Porous Particle in a Charged Cavity.

The quasi-steady diffusiophoresis of a charged porous sphere situated at the center of a charged spherical cavity filled with a liquid solution of a symmetric electrolyte is analyzed. The porous particle can represent a solvent-permeable and ion-penetrable polyelectrolyte molecule or floc of nanoparticles in which fixed charges and frictional segments are uniformly distributed, whereas the spherical cavity can denote a charged pore involved in microfluidic or drug-delivery systems. The linearized electrokinetic differential equations governing the ionic concentration, electric potential, and fluid velocity distributions in the system are solved by using a perturbation method with the fixed charge density of the particle and the ζ-potential of the cavity wall as the small perturbation parameters. An expression for the diffusiophoretic (electrophoretic and chemiphoretic) mobility of the confined particle with arbitrary values of a/ b, κ a, and λ a is obtained in closed form, where a and b are the radii of the particle and cavity, respectively; κ and λ are the reciprocals of the Debye screening length and the length characterizing the extent of flow penetration into the porous particle, respectively. The presence of the charged cavity wall significantly affects the diffusiophoretic motion of the particle in typical cases. The diffusio-osmotic (electro-osmotic and chemiosmotic) flow occurring at the cavity wall can substantially alter the particle velocity and even reverse the direction of diffusiophoresis. In general, the particle velocity decreases with an increase in a/ b, increases with an increase in κ a, and decreases with an increase in λ a, but exceptions exist.

Electrophoretic mobility of charged porous shells or microcapsules and electric conductivity of their dilute suspensions

Abstract An analysis is presented for the electrophoresis and electric conduction in a dilute suspension of charged spherical porous shells or permeable microcapsules with electric double layers of arbitrary thickness in an electrolyte solution. With the assumption that the system is slightly perturbed from equilibrium, the linearized electrokinetic equations governing the ionic electrochemical potential energy and fluid velocity distributions are solved as series expansions in the small fixed charge density of the porous shells. Explicit formulas for the electrophoretic mobility of the porous shells and effective electric conductivity of the suspension are derived. Both the electrophoretic mobility and the effective conductivity decrease monotonically with a decrease in the relative thickness of the porous shells, but these decreases are not conspicuous until the porous shells are quite thin. When the fluid permeability of the porous shells is smaller or the electric double layers are thicker, the effect of this relative thickness on the electrophoretic mobility and effective conductivity becomes more significant. In the limiting case of zero inner radius of the porous shells, our formulas reduce to the corresponding results obtained for the electrophoresis and electric conduction in a suspension of charged porous spheres.

Diffusiophoretic motion of an isolated charged porous sphere

Diffusiophoretic motion, the migration of a colloidal particle in response to an externally applied solute concentration gradient, is investigated theoretically in this study for an isolated charged porous sphere suspended in an unbounded medium of electrolyte solution. The porous sphere is treated as a Brinkman medium with a uniformly distributed fixed charge density. The resulted general electrokinetic equations adopting the full nonlinear Poisson equation are solved numerically with a pseudo-spectral method based on Chebyshev polynomials. In particular, the convection contribution of the ion flux is taken into account properly as well.Key parameters of electrokinetic interest are examined for their respective effect on the particle motion. The particle mobility is much smaller in general than the analytical prediction neglecting the convection-induced double layer polarization effect, which is by far the most important factor in determining the porous particle motion. A less charged particle may actually move faster than a highly charged one due to this effect. Visual demonstration of the polarization is provided. Formation of a separate axisymmetric vortex flow is be responsible for the observation that a particle may reverse its direction of motion across a threshold permeability. This implies that a porous polyelectrolyte (like a protein or a DNA) assuming a random coil conformation may tango back and forth as it makes gyrations in diffusiophoretic motion.

Diffusiophoresis in suspensions of charged porous particles.

An analysis of the diffusiophoretic motion in a suspension of charged porous spheres in an electrolytic solution with a macroscopic concentration gradient is presented. Each porous particle can be a solvent-permeable and ion-penetrable charged floc or polyelectrolyte molecule, in which the densities of the fixed charges and frictional segments are constant, surrounded by an arbitrary electric double layer. The multiparticle interaction effects are considered through the use of a unit cell model, which allows the overlap of adjacent double layers. The differential equations governing the electric potential, ionic concentration, and fluid velocity distributions inside and outside the porous particle in a unit cell are linearized by assuming that the system is only slightly deviated from equilibrium and then solved as power expansions in its dimensionless fixed-charge density. A closed-form expression for the diffusiophoretic velocity of the porous particle correct to the second order of the fixed charge density is obtained from a balance between the electrostatic and hydrodynamic forces acting on it. Detailed comparisons of the results for the multiparticle diffusiophoresis obtained from the cell model with various boundary conditions are made. The effect of particle interactions on the diffusiophoresis, which is a linear combination of electrophoresis and chemiphoresis, can be significant and complicated in typical situations. Although the electrophoretic mobility of the particles decreases with an increase in the particle volume fraction, their chemiphoretic mobility is not necessarily a monotonic function of it.

Nonlinear effects on electrokinetics of a highly charged porous sphere

The motivation of the present study is to provide a correct estimate of the electrophoretic mobility of a charged porous particle for wide-range electrokinetic parameters, such as particle charge density, permeability, and Debye length. Based on the Nernst–Planck equation, which takes into account the external electric field and fluid convection on ion transport, we have estimated the mobility of the particle by establishing a force balance. We have validated our results with the linear model due to Hermans and Fujita (K Nederl Akad Wet Proc Ser B 58:182–187, 1955) and the computed solution based on perturbation of the Poisson–Boltzmann model as obtained by Hsu and Lee (J Colloid Interface Sci 390:85–95, 2013). For the case of thin double layer, our computed results agree with the linear model even for large values of charge density of the particle. The linear model overpredicts our computed solution for mobility when the thick Debye layer is considered. However, a large discrepancy of the present model from the results based on the perturbation of the Boltzmann model is observed for all the cases considered. We have analyzed the double-layer polarization and counterion condensation through the distribution of counterions, net charge density, and the effective charge density of the particle.

Electrostatic interaction between two interpenetrating soft particles

A three-stage model of the electrostatic interaction between two interpenetrating charged spherical soft particles with no particle core (space-charged porous spheres) in an electrolyte solution is presented. That is, (i) interaction before contact of the two spheres, (ii) partial interpenetration, and (iii) full interpenetration, i.e., engulfing of one sphere by the other. This is an extension of the work of Dähnert and Rödenbeck (J. Colloid Interface Sci., 163 (1994) 229), who considered the interaction between two interpenetrating vesicle-like surface-charged particles, to the case of the interaction of space-charge porous spheres. Analytic expressions for the interaction energy and force between two interpenetrating weakly charged porous spheres as a function of particle separation are derived for the respective stages on the basis of the linearized Poisson–Boltzmann equations for the electric potential distribution.

Sedimentation of a Charged Porous Particle in a Charged Cavity

The sedimentation of a charged porous sphere at the center of a charged spherical cavity filled with an electrolyte solution is analyzed. The thickness of the electric double layers around the particle and cavity wall is arbitrary, and their relaxation effect is considered. Through the use of a set of linearized electrokinetic equations and a perturbation method, the ionic electrochemical potential energy, electric potential, and velocity fields in the fluid are solved with the fixed space charge density of the particle and surface charge density of the cavity as the small perturbation parameters, and an explicit formula for the sedimentation velocity is obtained. Due to the electroosmotic enhancement on the fluid recirculation in the cavity caused by the sedimentation-induced electric field, the presence of the surface charges on the cavity wall increases the sedimentation velocity of the porous particle. For the sedimentation of a porous particle in a cavity with their fixed charges of the same sign, the effect of electric interaction between the particle and cavity wall in general increases the sedimentation velocity. For the case of their fixed charges with opposite signs, the sedimentation velocity is increased/reduced if the magnitude of the fixed charge density of the cavity wall is relatively large/small. The effect of the surface charges at the cavity wall on the sedimentation of the porous particle increases with an increase in the permeability for fluid flow within the particle and with a decrease in the particle-to-cavity radius ratio (i.e., an increase in the surface area of the cavity wall relative to a given size of the particle, which enhances the fluid recirculation effect).

Electrophoresis of a Charged Soft Particle in a Charged Cavity with Arbitrary Double-Layer Thickness

An analysis for the quasi-steady electrophoretic motion of a soft particle composed of a charged spherical rigid core and an adsorbed porous layer positioned at the center of a charged spherical cavity filled with an arbitrary electrolyte solution is presented. Within the porous layer, frictional segments with fixed charges are assumed to distribute uniformly. Through the use of the linearized Poisson-Boltzmann equation and the Laplace equation, the equilibrium double-layer potential distribution and its perturbation caused by the applied electric field are separately determined. The modified Stokes and Brinkman equations governing the fluid flow fields outside and inside the porous layer, respectively, are solved subsequently. An explicit formula for the electrokinetic migration velocity of the soft particle in terms of the fixed charge densities on the rigid core surface, in the porous layer, and on the cavity wall is obtained from a balance between its electrostatic and hydrodynamic forces. This formula is valid for arbitrary values of κa, λa, r0/a, and a/b, where κ is the Debye screening parameter, λ is the reciprocal of the length characterizing the extent of flow penetration inside the porous layer, a is the radius of the soft particle, r0 is the radius of the rigid core of the particle, and b is the radius of the cavity. In the limiting cases of r0 = a and r0 = 0, the migration velocity for the charged soft sphere reduces to that for a charged impermeable sphere and that for a charged porous sphere, respectively, in the charged cavity. The effect of the surface charge at the cavity wall on the particle migration can be significant, and the particle may reverse the direction of its migration.

Electrophoresis of a single charged porous sphere in an infinite medium of electrolyte solution

Electrophoresis of a single charged porous sphere in an infinite medium of electrolyte solution is investigated theoretically. The porous sphere is treated as a Brinkman medium with uniformly distributed electric charges. General electrokinetic equations including the full nonlinear Poisson equation are employed as the governing equations, which are then solved with a pseudo-spectral method based on Chebyshev polynomials.Key parameters of electrokinetic interest are examined for their effects on particle motion. Motion-deterring nonlinear effects, the condensation effect and the double layer polarization effect, are separated from each other and examined in detail for their respective impact on the particle motion. The Debye–Hückel approximation is found to overestimate the particle mobility severely: Up to ten times for a porous sphere both highly charged and highly permeable in some situations, which is attributed to the polarization effect. Convenient charts of correction factors for this overestimation are provided to facilitate the usage by interested experimental researchers in the field of the study of polyelectrolytes, such as DNA and proteins, which are well modeled as charged porous spheres.

Sedimentation velocity and potential in a concentrated suspension of charged soft spheres

An analytical study of sedimentation in a suspension of charged spherical soft particles, each with the structure of a rigid core surrounded by a porous shell, in an electrolyte solution is presented. The porous shell of the particle is treated as a solvent-permeable and ion-penetrable surface layer of finite thickness, in which hydrodynamic frictional segments with fixed charges distribute uniformly. A unit cell model that allows the overlap of the electric double layers of adjacent particles is employed to account for the effect of particle interactions. When the system is slightly distorted from equilibrium, the electrokinetic equations governing the electric potential, ion concentration, and fluid velocity distributions are linearized and solved using a perturbation method with the fixed charge densities of the core surface and porous layer of each soft sphere as the small parameters. Explicit formulas for the sedimentation velocity and potential in the suspension are obtained to the second orders of these fixed charge densities with the relaxation effect of the double layers included. The sedimentation velocity and potential are not necessarily monotonic functions of the volume fraction of the particles; the particle concentration effects are significant, even in dilute suspensions. In the limiting cases, the analytical solutions of the sedimentation velocity and potential for charged soft spheres reduce to those for charged rigid spheres and charged porous spheres. It is shown that a suspension of charged but neutral soft spheres can produce sedimentation potential, and their sedimentation velocity differs from that of the uncharged soft spheres.

Diffusiophoresis of a Spherical Soft Particle in Electrolyte Gradients

An analytical study of the diffusiophoresis (consisting of electrophoresis and chemiphoresis) of a charged soft particle (or composite particle) composed of a spherical rigid core and a surrounding porous shell in an electrolyte solution prescribed with a uniform concentration gradient is presented. In the solvent-permeable and ion-penetrable porous surface layer of the particle, idealized frictional segments with fixed charges are assumed to distribute at a constant density. The electrokinetic equations that govern the electric potential profile, ionic concentration distributions, and fluid flow field inside and outside the porous layer of the particle are linearized by assuming that the system is only slightly distorted from equilibrium. Using a regular perturbation method, these linearized equations are solved with the fixed charge densities on the rigid core surface and in the porous shell as the small perturbation parameters. An analytical expression for the diffusiophoretic mobility of the soft 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 densities, is valid for arbitrary values of κa, λa, and r(0)/a, where κ is the reciprocal of the Debye screening length, λ is the reciprocal of the length characterizing the extent of flow penetration inside the porous layer, a is the radius of the soft sphere, and r(0) is the radius of the rigid core of the particle. It is shown that a soft particle bearing no net charge can undergo diffusiophoresis (electrophoresis and chemiphoresis), and the direction of its diffusiophoretic velocity is decided by the fixed charges in the porous surface layer of the particle. In the limiting cases of large and small values of r(0)/a, the analytical solution describing the diffusiophoretic mobility for a charged soft sphere reduces to that for a charged rigid sphere and for a charged porous sphere, respectively.

Electrophoretic motion of a charged porous sphere within micro- and nanochannels

Electrophoretic motion of a charged porous sphere within micro- and nanochannels is investigated theoretically. The Brinkman model and the full non-linear Poisson-Boltzmann equation are adopted to model the system, with the charged porous sphere resembling polyelectrolytes like proteins and DNA. General electrokinetic equations are employed and solved with a pseudo-spectral method. Key parameters of electrokinetic interest are examined for their respective effect as well as overall impact on the particle motion. We found, among other things, that the confinement effect of the channel can be so drastic that 75% reduction of particle mobility is observed in some situations for a poorly permeable particle. However, only 15% for the corresponding highly permeable particle due to the allowance of fluid penetration which alleviates the retarding shear stress significantly. In particular, an intriguing phenomenon is observed for the highly permeable particle: the narrower the channel is, the faster the particle moves! This was experimentally observed as well in the literature on DNA electrophoresis within nanostructures. The reason behind it is thoroughly explained here. Moreover, charged channels can exert electroosmosis flow so dominant that sometimes it may even reverse the direction of the particle motion. Comparison with experimental data available in the literature for some polyelectrolytes is excellent, indicating the reliability of this analysis. The results of this study provide fundamental knowledge necessary to interpret experimental data correctly in various microfluidic and nanofluidic operations involving bio-macromolecules, such as in biosensors and Lab-on-a-chip devices.

Electrophoresis of a charged porous sphere normal to an air–water interface

Electrophoretic behavior of a charged porous sphere normal to an air-water interface is investigated theoretically. The charged porous sphere is a very popular model of DNA and proteins, among other polyelectrolytes, based on the experimental observation of the DNA and protein conformations in a free solution. Two major motion-deterring effects are thoroughly investigated in particular: the boundary effect due to the presence of an air-water interface and the double layer polarization effect due to the convection-induced ion flux redistribution. The presence of the air-water interface is found to reduce the particle mobility in general, especially when the double layer is very thick or the particle is close to the interface. This boundary effect diminishes as the double layer gets very thin. However, an interesting phenomenon is observed that a particle closer to the interface may actually move faster than a farther one under some circumstances. The reason behind this phenomenon is attributed to the interaction between the boundary and polarization effects. This is demonstrated with evidence of corresponding contour plots. Convenient charts of correction factors are provided to facilitate the usage by interested experimental researchers.

Counterion condensation of a polyelectrolyte

Counterion condensation phenomenon of a highly charged polyelectrolyte, which is directly related to DNA condensation and RNA folding, has great importance and potential use in biological science, electrokinetic phenomenon, and medical science. To further understand this fascinating phenomenon, the distribution of the electric potential for a highly charged polyelectrolyte in an electrolyte solution is determined theoretically. The polyelectrolyte is modeled as a charged porous sphere and the full nonlinear Poisson–Boltzmann equation is used to describe the interaction between the counterions in the electrolyte solution and the backbone macroion of the polyelectrolyte itself. The solution is obtained with a pseudo-spectral method based on Chebyshev polynomials. The fraction of total charges condensed is analyzed in particular, with its dependence on the charged condition of the polyelectrolyte as well as the ionic strength of the solution investigated in detail. Comparison with limited experimental data available in the literature for DNA neutralization fraction is excellent in the asymptotic sense, suggesting the reliability of the analysis in this study, as well as the promising possibility of using the charged porous sphere to model a polyelectrolyte. Results presented here provide useful information in biological systems and can be utilized in practical applications such as DNA vaccines and gene delivery.

Electric Conductivity and Electrophoretic Mobility in Suspensions of Charged Porous Spheres

The electric conduction and electrophoresis of a suspension of charged porous spheres in an electrolyte solution with an arbitrary thickness of the electric double layers are analytically studied. The porous particle can be a solvent-permeable and ion-penetrable polyelectrolyte molecule or charged floc with uniformly distributed frictional segments and fixed charges. The effect of particle interactions is taken into account by employing a unit cell model, and the overlap of the electric double layers of adjacent particles is allowed. The electrokinetic equations, which govern the electrostatic potential profile, the ionic concentration (or electrochemical potential energy) distributions, and the fluid velocity field inside and outside the porous particle in a unit cell, are linearized by assuming that the system is only slightly distorted from equilibrium. Through the use of a regular perturbation method, these linearized equations are solved with the dimensionless density of the fixed charges as the small perturbation parameter. Analytical expressions for the electrophoretic mobility of each charged porous sphere and for the effective electric conductivity of the suspension correct to the first and second orders, respectively, of the fixed charge density are obtained in closed forms. The effect of particle interactions on the electrophoresis and electric conduction of the suspension can be significant in typical situations. Comparisons of the results of the cell model with different conditions at the outer boundary of the cell are made. The dependence of the electrophoretic mobility and the electric conductivity on the particle volume fraction and other properties of the particle−solution system is found to be quite complicated.

Electrophoresis in suspensions of charged porous spheres in salt-free media

Electrophoresis in either dilute or concentrated suspensions of charged porous spheres in salt-free media is investigated theoretically in this study. The Brinkman model and the Kuwabara’s unit cell model are adopted to simulate the porous structure and the suspensions, respectively. We found, among other things, that the polarization effect due to the convection flow within the porous sphere is a crucial factor in determining its electrophoretic behavior. An induced electric field opposite to the applied electric field is generated, which deters the particle motion significantly when the particle is highly permeable. Approximate analytical prediction for dilute suspensions neglecting convection flow can overestimate the mobility severely in this situation. The approximate analytical prediction is satisfactory when the permeability of particle is low, though. Counterion condensation happens at high fixed charge density which decreases the mobility drastically and the mobility approaches a constant value asymptotically. The mobility profile of the particles with increasing volume fraction can exhibit local minimum if the corresponding dimensionless parameter Q fix /( λa) 2 is high, where Q fix and λa are, respectively, the fixed charge density and the friction coefficient of the porous particles in dimensionless form. This is due to the overlapping of counterion clouds surrounding particles, which offsets the polarization effect, becomes significant as the suspension gets concentrated. No such phenomenon for low Q fix /( λa) 2, where the mobility profile decreases monotonously with increasing volume fraction. Comparison with experimental data available in the literature for polyelectrolyte suspensions is excellent, indicating the reliability of this analysis, as well as the success of using charged porous sphere to model a polyelectrolyte system.

Electrophoresis in concentrated dispersions of charged porous spheres

Electrophoresis in a concentrated suspension of charged porous spheres is investigated theoretically, taking into account of the polarization effect of double layer in particular. The double layer thickness and fixed charge density of the porous spheres are arbitrary and double layer overlapping is considered as well. A pseudo-spectral method based on Chebyshev polynomial is used to solve the resulted general electrokinetic equations. Local extrema are observed in the mobility profile as the double layer thickness varies, which is absent in previous theoretical studies neglecting polarization effect but agrees well with the experimental observations in the literature, indicating the importance of polarization effect in correctly interpreting the electrophoretic experimental results. A simple mutual transform of mobility predictions between Levine–Neale and Shilov–Zharkikh boundary conditions is established and confirmed by direct calculations, which facilitates the dialogue between theoretical and experimental researches. In general, the higher the fixed charge density, the more significant the polarization effect is.

Sedimentation velocity and potential in concentrated suspensions of charged porous spheres

The body-force-driven migration in a homogeneous suspension of polyelectrolyte molecules or charged flocs in an electrolyte solution is analyzed. 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 effects of particle interactions are taken into account by employing a unit cell model. The overlap of the electric double layers of adjacent particles is allowed and the relaxation effect in the double layer surrounding each particle is considered. The electrokinetic equations which govern the electrostatic potential profile, the ionic concentration (or electrochemical potential energy) distributions, and the fluid velocity field inside and outside the porous particle in a unit cell 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 symmetrically charged electrolyte with the density of the fixed charges as the small perturbation parameter. An analytical expression for the settling velocity of the charged porous sphere is obtained from a balance among its gravitational, electrostatic, and hydrodynamic forces. A closed-form formula for the sedimentation potential in a suspension of identical charged porous spheres is also derived by using the requirement of zero net electric current. The dependence of the sedimentation velocity and potential of the suspension on the particle volume fraction and other properties of the particle–solution system is found to be quite complicated.

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 κa and λa, where κ is the reciprocal of the Debye screening length, λ 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.