Electrostatic interactions between diffuse soft multi-layered (bio)particles: beyond Debye-Hückel approximation and Deryagin formulation

We report a steady-state theory for the evaluation of electrostatic interactions between identical or dissimilar spherical soft multi-layered (bio)particles, e.g. microgels or microorganisms. These generally consist of a rigid core surrounded by concentric ion-permeable layers that may differ in thickness, soft material density, chemical composition and degree of dissociation for the ionogenic groups. The formalism allows the account of diffuse interphases where distributions of ionogenic groups from one layer to the other are position-dependent. The model is valid for any number of ion-permeable layers around the core of the interacting soft particles and covers all limiting situations in terms of nature of interacting particles, i.e. homo- and hetero-interactions between hard, soft or entirely porous colloids. The theory is based on a rigorous numerical solution of the non-linearized Poisson-Boltzmann equation including radial and angular distortions of the electric field distribution within and outside the interacting soft particles in approach. The Gibbs energy of electrostatic interaction is obtained from a general expression derived following the method by Verwey and Overbeek based on appropriate electric double layer charging mechanisms. Original analytical solutions are provided here for cases where interaction takes place between soft multi-layered particles whose size and charge density are in line with Deryagin treatment and Debye-Hückel approximation. These situations include interactions between hard and soft particles, hard plate and soft particle or soft plate and soft particle. The flexibility of the formalism is highlighted by the discussion of few situations which clearly illustrate that electrostatic interaction between multi-layered particles may be partly or predominantly governed by potential distribution within the most internal layers. A major consequence is that both amplitude and sign of Gibbs electrostatic interaction energy may dramatically change depending on the interplay between characteristic Debye length, thickness of ion-permeable layers and their respective protolytic features (e.g. location, magnitude and sign of charge density). This formalism extends a recent model by Ohshima which is strictly limited to interaction between soft mono-shell particles within Deryagin and Debye-Hückel approximations under conditions where ionizable sites are completely dissociated.     

Electrokinetics of soft polymeric interphases with layered distribution of anionic and cationic charges

Soft surface coatings attract increasing attention due to the versatile options they provide in numerous applications e.g. in the flourishing nanomedicine and nanobiotechnology areas. Optimisation of the performance of such ion- and solvent-permeable polyelectrolytic materials requires a detailed understanding of their electrostatic properties. This task is rendered difficult by the inherent non-uniform distribution of their structural charges. In this article, we review recent advances made in the measurement and theory of the electrokinetics (electrophoresis/streaming current) of soft surface coatings that carry spatially-separated cationic and anionic charges. Examples of such charge-stratified systems are polyelectrolyte-coated particles, polyelectrolyte multilayers, particles with zwitterionic interfacial functionality, microbial cells or hard–soft composite interfaces. It is shown here that the electrokinetic features of such colloidal systems are remarkably different from those of their counterparts with homogeneously distributed cationic and anionic charges. In particular, the interplay between electrostatic and hydrodynamic flow fields developed under electrokinetic conditions in the bulk and interfacial compartments of charge-stratified colloids/films are shown to induce a reversal of their electrokinetic response (electrophoretic mobility/streaming current) that depends on the concentration of monovalent electrolyte in solution. The prerequisites for occurrence of such spectacular behaviour are theoretically identified in terms of the Debye length, the spatial length scales defining charge layering, and the typical length for flow penetration within the colloids/films. Electrophoresis and streaming current results recently reported for poly(amidoamine) carboxylated nanodendrimers, natural rubber colloids and poly(ethyleneimine)-supported lipid bilayers are further discussed to illustrate the generic electrokinetic properties of soft interfaces defined by a given stratification of their anionic and cationic structural charges.     

Rates of ionic reactions with charged nanoparticles in aqueous media

A theory is developed to evaluate the electrostatic correction for the rate of reaction between a small ion and a charged ligand nanoparticle. The particle is assumed to generally consist of an impermeable core and a shell permeable to water and ions. A derivation is proposed for the ion diffusion flux that includes the impact of the equilibrium electrostatic field distribution within and around the shell of the particle. The contribution of the extra- and intraparticulate field is rationalized in terms of a conductive diffusion factor, f(el), that includes the details of the particle geometry (core size and shell thickness), the volume charge density in the shell, and the parameters defining the electrostatic state of the particle core surface. The numerical evaluation of f(el), based on the nonlinear Poisson-Boltzmann equation, is successfully complemented with semianalytical expressions valid under the Debye-Hückel condition in the limits of strong and weak electrostatic screening. The latter limit correctly includes the original result obtained by Debye in his 1942 seminal paper about the effect of electrostatics on the rate of collision between two ions. The significant acceleration and/or retardation possibly experienced by a metal ion diffusing across a soft reactive particle/solution interphase is highlighted by exploring the dependence of f(el) on electrolyte concentration, particle size, particle charge, and particle type (i.e., hard, core/shell, and entirely porous particles).     

Effects of dielectric gradients-mediated ions partitioning on the electrophoresis of composite soft particles: An analytical theory

In this work, we report original analytical expressions defining the electrophoretic mobility of composite soft particles comprising an inner core and a surrounding polymer shell with differentiated permeabilities to ions from aqueous background electrolyte and to fluid flow developed under applied DC field conditions. The existence of dielectric permittivity gradients operational at the core/shell and shell/solution interfaces is accounted for within the Debye–Hückel approximation and flat plate configuration valid in the thin double layer regime. The proposed electrophoretic mobility expressions, applicable to weakly to moderately charged particles with size well exceeding the Debye layer thickness, involve the relevant parameters describing the particle core/shell structure and the electrohydrodynamic features of the core and shell particle components. It is shown that the analytical expressions reported so far in literature for the mobility of hard (impermeable) or porous particles correspond to asymptotic limits of the more generic results detailed here. The impacts of dielectric-mediated effects of ions partitioning between bulk solution and particle body on the electrophoretic response are further discussed. The obtained expressions pave the way for a refined quantitative, analytical interpretation of electrophoretic mobility data collected on soft (nano)particles (e.g., functionalized dendrimers and multilayered polyelectrolytic particles) or biological cells (e.g., viruses) for which the classical hard core-soft shell representation is not appropriate.     

Electrophoresis of diffuse soft particles

A theory is presented for the electrophoresis of diffuse soft particles in a steady dc electric field. The particles investigated consist of an uncharged impenetrable core and a charged diffuse polyelectrolytic shell, which is to some extent permeable to ions and solvent molecules. The diffuse character of the shell is defined by a gradual distribution of the density of polymer segments in the interspatial region separating the core from the bulk electrolyte solution. The hydrodynamic impact of the polymer chains on the electrophoretic motion of the particle is accounted for by a distribution of Stokes resistance centers. The numerical treatment of the electrostatics includes the possibility of partial dissociation of the hydrodynamically immobile ionogenic groups distributed throughout the shell as well as specific interaction between those sites with ions from the background electrolyte other than charge-determining ions. Electrophoretic mobilities are computed on the basis of an original numerical scheme allowing rigorous evaluation of the governing transport and electrostatic equations derived following the strategy reported by Ohshima, albeit within the restricted context of a discontinuous chain distribution. Attention is particularly paid to the influence of the type of distribution adopted on the electrophoretic mobility of the particle as a function of its size, charge, degree of permeability, and solution composition. The results are systematically compared with those obtained with a discontinuous representation of the interface. The theory constitutes a basis for interpreting electrophoretic mobilities of heterogeneous systems such as environmental or biological colloids or swollen/deswollen microgel particles.     

Dynamic Electrophoretic Mobility of a Soft Particle

A theory of the dynamic electrophoretic mobility of a spherical soft particle (that is, a polyelectrolyte-coated spherical particle) in an oscillating electric field is presented. In the absence of the polyelectrolyte layer a spherical soft particle becomes a spherical hard particle, while in the absence of the particle core it tends to a spherical polyelectrolyte. The present theory thus covers two extreme cases, that is, dynamic electrophoresis of hard particles and that of spherical polyelectrolytes. Simple analytic mobility expressions are derived. It is shown how the dynamic electrophoretic mobility of a soft particle depends on the volume charge density distributed in the polyelectrolyte layer, on the frictional coefficient characterizing the frictional forces exerted by the polymer segments on the liquid flow in the polyelectrolyte layer, on the particle size, and on the frequency of the applied oscillating electric field. Copyright 2001 Academic Press.     

Calibration-free concentration determination of charged colloidal nanoparticles and determination of effective charges by capillary isotachophoresis

Although colloidal nanoparticles show an electrophoretic heterogeneity under the conditions of capillary electrophoresis, which can be either due to the particle-size distribution and/or the particle shape distribution and/or the zeta-potential distribution, they can form correct isotachophoretic zones with sharp-moving boundaries. Therefore, the technique of isotachophoresis permits to generate plugs in which the co-ions and counter ions of the original colloidal solution are removed and replaced by the buffering counter ions of the leading electrolyte. It is shown that analytical isotachophoresis can be used to measure directly, without calibration, the molar (particle) concentration of dispersed ionic colloids provided that the transference number and the mean effective charge number of the particles (within the isotachophoretic zone) can be determined with adequate accuracy. The method can also be used to measure directly the effective charge number of biomacromolecules or colloidal particles, if solutions with known molar (particle) concentration can be prepared. The validity of the approach was confirmed for a model solution containing a known molar concentration of bovine serum albumin.      

On the electrophoretic mobility of a cylindrical soft particle

A previous theory for the electrophoresis of a cylindrical soft particle (that is, a cylindrical hard particle covered with a layer of polyelectrolytes) [7], which makes use of the condition that the electrical force acting on the polymer segments is balanced with a frictional force exerted by the liquid flow, is modified by replacing this condition with an alternative and more appropriate boundary condition that pressure is continuous at the boundary between the surface layer and the surrounding electrolyte solution. The general mobility expression thus obtained is found to reproduce all of the approximate analytic mobility expressions derived previously. Received: 20 July 2000/Accepted: 21 August 2000     

Discrete charge effects on the Donnan potential and surface potential of a soft particle

The Donnan potential and surface potential of soft particles (i.e., polyelectrolyte-coated hard particles) in an electrolyte solution play an essential role in their electric behaviors. These potentials are usually derived via a continuum model in which fixed charges inside the surface layer are distributed with a continuous charge density. In this paper, for a plate-like soft particle consisting of a cubic lattice of fixed point charges, on the basis of the linearized Poisson–Boltzmann equation, we derive expressions for the electric potential distribution in the regions inside and outside the surface layer. This expression is given in terms of a sum of the screened Coulomb potentials produced by the point charges within the surface layer. We show that the deviation of the results of the discrete charge model from those of the continuous charge model becomes significant as the ratio of the lattice spacing to the Debye length becomes large.     

Influence of boundary on the effect of double-layer polarization and the electrophoretic behavior of soft biocolloids

The electrophoresis of a soft particle comprising a rigid core and a charged porous membrane layer in a narrow space is modeled. This simulates, for example, the capillary electrophoresis of biocolloids such as cells and microorganisms, and biosensor types of device. We show that, in addition to the boundary effect, the effects of double-layer polarization (DLP) and the electroosmotic retardation flow can be significant, yielding interesting electrophoretic behaviors. For example, if the friction coefficient of the membrane layer and/or the boundary is large, then the DLP effect can be offset by the electroosmotic retardation flow, making the particle mobility to decrease with increasing double layer thickness, which is qualitatively consistent with many experimental observations in the literature, but has not been explained clearly in previous analyses. In addition, depending upon the thickness of double layer, the friction of the membrane layer of a particle can either retard or accelerate its movement, an interesting result which has not been reported previously. This work is the first attempt to show solid evidence for the influence of a boundary on the effect of DLP and the electrophoretic behavior of soft particles. The model proposed is verified by the experimental data in the literature. The results of numerical simulation provide valuable information for the design of bio-analytical apparatus such as nanopore-based sensing applications and for the interpretation of relevant experimental data.     

Electrokinetic properties of noncharged lipid nanocapsules: influence of the dipolar distribution at the interface

Lipid nanocapsules (LNCs) containing poly(ethylene glycol) (PEG) were developed according to a phase inversion process without organic solvent. The distribution of PEG chains at the surface was determined due to electrokinetic properties, in order to correlate it with protein adsorption potentiality. In this aim, electrophoretic mobilities were measured as a function of ionic strength and pH, for particles differing by their size, dialysis effects, and the presence or not of lecithin in their shell. The study allowed the determination of the isoelectric point (pI) as well as the charge density (ZN) in relation with the dipolar distribution in the polyelectrolyte accessible layer (depth = 1/lambda), by using soft-particle electrophoresis analysis. These parameters pointed out that the PEG surface organization was dependent on the particle size. Moreover, this organization could be modified by dialyzing particles and/or by formulating them with or without lecithin. Lecithin was found to be present in the inner part of the polyelectrolyte layer and to play a role in the outer part disorganization. Dialyzing LNCs formulated with lecithin allowed to obtain stable and well-structured nanocapsules, ready to an in vivo use as drug delivery system.     

Adsorption of humic acids onto goethite: effects of molar mass, pH and ionic strength

In this paper, the LCD (ligand charge distribution) model is applied to describe the adsorption of (Tongbersven) humic acid (HA) to goethite. The model considers both electrostatic interactions and chemical binding between HA and goethite. The large size of HA particles limits their close access to the surface. Part of the adsorbed HA particles is located in the compact part at the goethite surface (Stern layers) and the rest in the less structured diffuse double layer (DDL). The model can describe the effects of pH, ionic strength, and loading on the adsorption. Compared to fulvic acid (FA), adsorption of HA is stronger and more pH- and ionic-strength-dependent. The larger number of reactive groups on each HA particle than on a FA particle results in the stronger HA adsorption observed. The stronger pH dependency in HA adsorption is related to the larger number of protons that are coadsorbed with HA due to the higher charge carried by a HA particle than by a FA particle. The positive ionic-strength dependency of HA adsorption can be explained by the conformational change of HA particles with ionic strength. At a higher ionic strength, the decrease of the particle size favors closer contact between the particles and the surface, leading to stronger competition with electrolyte ions for surface charge neutralization and therefore leading to more HA adsorption.     

Metal speciation dynamics in monodisperse soft colloidal ligand suspensions

A comprehensive theory is presented for the dynamics of metal speciation in monodisperse suspensions of soft spherical particles characterized by a hard core and an ion-permeable shell layer where ligands L are localized. The heterogeneity in the binding site distribution leads to complex formation/dissociation rate constants (denoted as k a (*) and k d (*), respectively) that may substantially differ from their homogeneous solution counterparts (k a and k d). The peculiarities of metal speciation dynamics in soft colloidal ligand dispersions result from the coupling between diffusive transport of free-metal ions M within and around the soft surface layer and the kinetics of ML complex formation/dissociation within the shell component of the particle. The relationship between k a,d (*) and k a,d is derived from the numerical evaluation of the spatial, time-dependent distributions of free and bound metal. For that purpose, the corresponding diffusion equations corrected by the appropriate chemical source term are solved in spherical geometry using a Kuwabara-cell-type representation where the intercellular distance is determined by the volume fraction of soft particles. The numerical study is supported by analytical approaches valid in the short time domain. For dilute dispersions of soft ligand particles, it is shown that the balance between free-metal diffusion within and outside of the shell and the kinetic conversion of M into ML within the particular soft surface layer rapidly establishes a quasi-steady-state regime. For sufficiently long time, chemical equilibrium between the free and bound metal is reached within the reactive particle layer, which corresponds to the true steady-state regime for the system investigated. The analysis reported covers the limiting cases of rigid particles where binding sites are located at the very surface of the particle core (e.g., functionalized latex colloids) and polymeric particles that are devoid of a hard core (e.g., polysaccharide macromolecules, gel particles). For both the transient and quasi-steady-state regimes, the dependence of k a,d (*) on the thickness of the soft surface layer, the radius of the hard core of the particle, and the kinetic rate constants k a,d for homogeneous ligand solutions is thoroughly discussed within the context of dynamic features for colloidal complex systems.     

Hydrolytic stability of nano-particle polyurethane dispersions: Implications to their long-term use

Nano-particle segmented polyurethane anionomer dispersions with ions either on the soft segment or on the hard segment were synthesised using 2,2-bis(hydroxymethyl) propionic acid and 5-sodiosulfo-1,3-benzenedicarboxylic acid as ionic centre. The resulting polyurethane dispersions were characterized for their particle size, reduced viscosity and hydrolytic stability in the presence of the aqueous phase during storage. At similar ionic contents, the polyurethanes that contain ionic groups on their soft segment had smaller particle sizes than those that contain ionic groups on the hard segment due to the effectiveness of the sulphonate ionic groups incorporated in the former. The reduced viscosity of the anionomers in dimethylformamide (DMF) showed typical polyelectrolyte effect that can be eliminated by the addition of LiBr. The hydrolysis study conducted over 2-years indicated that polyurethanes in which the ions were located on the hard segment had better hydrolytic stability in aqueous environment than those with ions located on the soft segment. We attributed this due to the fact that unsolvated hydrophobic polyester segments were packed in the interior of the particles while the strongly hydrated urethane segments with mutually repelling carboxylate ions were situated on the outside surface of the particles. The polyester groups prone to hydrolytic attack were thus protected against hydrolysis as effectively as in the dry solid form.     

Preparation and characterization of low dispersity anionic multiresponsive core-shell polymer nanoparticles

We prepared anionic multistimuli responsive core-shell polymer nanoparticles with very low size dispersity. By using either acrylic acid (AA) or methacrylic acid (MA) as a comonomer in the poly(N-isopropyl acrylamide) (PNIPAM) shell, we are able to change the distribution of negative charges in the nanoparticle shell. The particle size, volume phase transition temperature, and aggregation state can be modulated using temperature, pH, or ionic strength, providing a very versatile platform for applications in sensors, medical diagnostics, environmental remediation, etc. The nanoparticles have a glassy poly(methyl methacrylate) (PMMA) core of ca. 40 nm radius and a cross-linked PNIPAM anionic shell with either AA or MA comonomers. The particles, p(N-AA) and p(MA-N), respectively, have the same total charge but different charge distributions. While the p(MA-N) particles have the negative charges preferentially distributed toward the inner shell, in the case of the p(N-AA) particles the charge extends more to the particle outer shell. The volume phase transition temperature (T(VPT)) of the particles is affected by the charge distribution and can be fine-tuned by controlling the electrostatic repulsion on the particle shell (using pH and ionic strength). By suppressing the particle charge we can also induce temperature-driven particle aggregation.     

Tunable rheology of dense soft deformable colloids

In the last two decades, advances in synthetic, experimental and modeling/simulation methodologies have considerably enhanced our understanding of colloidal suspension rheology and put the field at the forefront of soft matter research. Recent accomplishments include the ability to tailor the flow of colloidal materials via controlled changes of particle microstructure and interactions. Whereas hard sphere suspensions have been the most widely studied colloidal system, there is no richer type of particles than soft colloids in this respect. Yet, despite the remarkable progress in the field, many outstanding challenges remain in our quest to link particle microstructure to macroscopic properties and eventually design appropriate soft composites. Addressing them will provide the route towards novel responsive systems with hierarchical structures and multiple functionalities. Here we discuss the key structural and rheological parameters which determine the tunable rheology of dense soft deformable colloids. We restrict our discussion to non-crystallizing suspensions of spherical particles without electrostatic or enthalpic interactions.     

Electrophoretic mobility of cylindrical soft particles

 A general theory for the electrophoresis of a cylindrical soft particle (i.e., a cylindrical hard colloidal particle coated with a layer of ion-penetrable polyelectrolytes) in an electrolyte solution in an applied transverse or tangential electric field is proposed. This theory unites two different electrophoresis theories for cylindrical hard particles and for cylindrical polyelectrolytes. That is, the general mobility expression obtained in this paper tends to the mobility expression for a cylindrical hard particle for the case where the polyelectrolyte layer is absent or the frictional coefficient in the poly-electrolyte layer becomes infinity, whereas it tends to that for a cylin-drical polyelectrolyte in the absence of the particle core. Simple approximate analytic mobility expressions are also presented. Received: 29 August 1996 Accepted: 7 November 1996     

Electrokinetic phenomena of soft particles

A review is given on the recent developments of theories of electrokinetics of soft particles (i.e., hard particles covered with an ion-penetrable surface layer of polyelectrolytes) with particular emphasis on the electrophoretic relaxation effects for highly charged soft particles and the effects of inhomogeneous distribution of polymer segments in the surface layer. The electrostatic interaction between soft particles is also discussed. Not only the interaction before contact of the surface layers of the interacting particles but also the interaction after their contact is considered.     

Polyion-induced aggregation of lipidic-coated solid polystyrene spheres: the many facets of complex formation in low-density colloidal suspensions

We have investigated the formation of a cluster phase in low-density colloidal systems formed by charged solid charged particles stuck together by an oppositely charged polyion. In analogy with what we have previously observed in the case of soft charged particles, also in this case the same basic phenomenology occurs, consisting of the presence of the two well-known characteristic phenomena of this class of colloids, that is, reentrant condensation and charge inversion. With the aim of comparing the cluster formation in both soft and solid charged particles, we have, in previous works, employed cationic liposomes (soft particles, lipidic vesicles built up by dioleoyltrimethylammonium propane [DOTAP] lipid) and, in the present work, polystyrene particles (solid particles) covered by the same lipidic bilayer as the one of the soft particles, so that the two classes of particles share electrostatic interactions of the same nature. These charged particle clusters, where the single aggregating particles maintain their integrity without undergoing a structural rearrangement, join to a class of different aggregated structures (lamellar or inverse hexagonal phases) observed as well in the polyion-induced aggregation of oppositely charged mesoscopic particles, in particular, lipidic vesicles. Our results show that the formation of relatively large, equilibrium clusters of particles which maintain their integrity, stuck together by a polyion which acts as an electrostatic glue, is one of the many facets of the complex phenomenology underlying the interactions of charged particles with oppositely charged objects.     

Numerical calculation of the electrophoretic mobility of concentrated suspensions of soft particles

The electrophoretic mobility of spherical soft particles in concentrated colloidal suspensions is numerically calculated. The particle is modeled as a hard core coated with an ion-penetrable membrane bearing a uniform distribution of fixed charges, while the high particle concentration is taken into account by means of a cell model. The network simulation method used makes it possible to solve the problem without any restrictions on the values of the parameters such as particle concentration, membrane thickness, fixed charge density in the membrane, viscous drag in the membrane, number and valence of ionic species, electrolyte concentration, etc. The theoretical model used is similar to the one presented by Ohshima [H. Ohshima, J. Colloid Interface Sci. 225 (2000) 233], except for the use of the Shilov-Zharkikh, rather than the Levine-Neale, boundary condition for the electric potential, and the inclusion in the force balance equation of an additional term corresponding to the force exerted by the liquid on the core of the moving particle [J.J. López-García, C. Grosse, J. Horno, J. Colloid Interface Sci. 265 (2003) 327]. The obtained results only coincide with existing analytical expressions for low particle concentrations, low particle charge, and when the electrolyte concentration is high, the membrane is thick, and its resistance to the fluid flow is high. This suggests that most interpretations of the electrophoretic mobility of soft particles in concentrated suspensions require numerical calculations.