Interaction force between two charged plates immersed in a solution of charged particles. Coupling between double layer and depletion forces

When two parallel plates are immersed in a solution of small charged particles, the center of the particles is excluded from a region of thickness D/2 near the plate, where D is their diameter. The approach which Langmuir developed for the double layer repulsion in the presence of an electrolyte with ions of negligible size is extended to the case in which one of the "ions" is a charged particle of finite, relatively small size. A general expression for the force generated between the two charged plates immersed in an electrolyte solution containing relatively small charged particles is derived. In this expression, only the electrical potential at the middle distance between the plates is required to calculate the force. A Poisson-Boltzmann equation which accounts for the volume exclusion of the charged particles in the vicinity of the surface is solved to obtain the electrical potential at the middle between the two plates. Starting from this expression, some results obtained previously for the depletion force acting between two plates or two spheres are rederived. For charged plates immersed in a solution of an electrolyte and charged small particles, the effects of the particle charge, particle charge sign, particle size, and volume fraction of the particles on the force acting between the two plates are examined.     

Polyelectrolyte brushes

The simple scaling theory of weakly-charged polyelectrolyte brush (the layer of polyelectrolyte chains grafted at one end onto an impermeable surface) immersed into a good solvent has been developed.The asymptotic scaling dependences of the free layer thickness on charge density and solvent strength are obtained. The behavior of polyelectrolyte brush subjected to normal and tangential external forces is considered. New “polyelectrolyte effect” is predicted: shear of a free polyelectrolyte brush leads to a decrease in brush thickness in contrast to the case of a free neutral brush. Such behavior is equivalent to that of a neutral brush subjected to external normal stretching force. This force in the case of polyelectrolyte brush is created by the osmotic pressure of mobile counterions neutralizing grafted chain charges.     

Diblock polyampholytes grafted onto spherical particles: Monte Carlo simulation and lattice mean-field theory

Spherical brushes composed of diblock polyampholytes (diblock copolymers with oppositely charged blocks) grafted onto solid spherical particles in aqueous solution are investigated by using the primitive model solved with Monte Carlo simulations and by lattice mean-field theory. Polyampholyte chains of two compositions are considered: a copolymer with a long and a short block, A100B10, and a copolymer with two blocks of equal length, A50B50. The B block is end-grafted onto the surface, and its charge is varied, whereas the charge of the A block is fixed. Single-chain properties, radial and lateral spatial distributions of different types, and structure factors are analyzed. The brush structure strongly depends on the charge of the B block. In the limit of an uncharged B block, the chains are stretched and form an extended polyelectrolyte brush. In the other limit with the charges of the blocks compensating each other, the chains are collapsed and form a polyelectrolyte complex surrounding the particles. At intermediate charge conditions, a polyelectrolyte brush and a polyelectrolyte complex coexist and constitute two substructures of the spherical brush. The differences of the brush structures formed by the A100B10 and A50B50 polyampholytes are also analyzed. Finally, a comparison of the predictions of the two theoretical approaches is made.     

Three-dimensional discrete charge effects on the electrostatic interaction between two soft particles

An expression for the electrostatic interaction energy between two parallel plate-like soft particles (i.e., hard particles covered with an ion-penetrable surface layer of polyelectrolytes) in an electrolyte solution is derived by using the linearized Poisson-Boltzmann equation. This expression is based on a discrete charge model in which the surface layer consists of a cubic lattice of fixed point charges. We show that the deviation of the results of the discrete charge model from those of the conventional smeared charge model becomes significant as the ratio of the lattice spacing to the Debye length becomes large. As this ratio decreases, on the other hand, the discrete charge model approaches a smeared charge model, leading to the Donnan-potential regulated interaction model.     

Diffusion of cationic polyelectrolytes into cellulosic fibers

The penetration of cationic polyelectrolytes into anionic cellulosic fibers was evaluated with fluorescent imaging techniques in order to clarify the mechanism and time scales for the diffusion process. The bulk charge of the cellulosic fibers indirectly creates a driving force for diffusion into the porous fiber wall, which is entropic in nature due to a release of counterions as the polyelectrolyte adsorbs. The individual bulk charges in the fiber cell wall also interact with the diffusing polyelectrolyte, such that the polyelectrolyte diffuses to the first available charge and consequently adsorbs and remains fixed. Thus, subsequent polyelectrolyte chains must first diffuse through the adsorbed polyelectrolyte layer before adsorbing to the next available bulk charges. This behavior differs from earlier suggested diffusion mechanisms, by which polyelectrolytes were assumed to first adsorb to the outermost surface and then reptate into the pore structure. The time scales for polyelectrolyte diffusion were highly dependent on the flexibility of the chain, which was estimated from calculations of the persistence length. The persistence length ultimately depended on the charge density and electrolyte concentration. The charge density of the polyelectrolyte had a greater influence on the time scales for diffusion. High charge density polyelectrolytes were observed to diffuse on a time scale of months, whereas the diffusion of low charge density polyelectrolytes was measured on the order of hours. An influence of the chain length, that is, steric interactions due the persistence length of the polyelectrolyte and to the tortuosity of the porous structure of the fiber wall, could only be noted for low charge density polyelectrolytes. Increasing the electrolyte concentration increased the chain flexibility by screening the electrostatic contribution to the persistence length, in turn inducing a faster diffusion process. However, a significant change in the diffusion behavior was observed at high electrolyte concentrations, at which the interaction between the polyelectrolyte charges and the fiber charges was almost completely screened.     

Strong and weak adsorptions of polyelectrolyte chains onto oppositely charged spheres

We investigate the complexation of long thin polyelectrolyte (PE) chains with oppositely charged spheres. In the limit of strong adsorption, when strongly charged PE chains adapt a definite wrapped conformation on the sphere surface, we analytically solve the linear Poisson-Boltzmann equation and calculate the electrostatic potential and the energy of the complex. We discuss some biological applications of the obtained results. For weak adsorption, when a flexible weakly charged PE chain is localized next to the sphere in solution, we solve the Edwards equation for PE conformations in the Hulthen potential, which is used as an approximation for the screened Debye-Huckel potential of the sphere. We predict the critical conditions for PE adsorption. We find that the critical sphere charge density exhibits a distinctively different dependence on the Debye screening length than for PE adsorption onto a flat surface. We compare our findings with experimental measurements on complexation of various PEs with oppositely charged colloidal particles. We also present some numerical results of the coupled Poisson-Boltzmann and self-consistent field equation for PE adsorption in an assembly of oppositely charged spheres.     

Interaction between colloids with grafted diblock polyampholytes

The interaction between composite colloidal particles composed of a spherical core and grafted AB-diblock polyampholytes (diblock copolymers with oppositely charged blocks) are investigated by using a coarse-grained model solved with Monte Carlo simulations. The B block is end-grafted onto the core of the colloid and its linear charge density is varied, whereas the linear charge density of the A block is fixed. The brush structure of a single colloid, the mean force between two colloids, and the structure of solutions of such colloids have been determined for different linear charge densities of the B blocks and block lengths. Many features of the present system are controlled by the charge of the B blocks. In the limit of uncharged B blocks, (i) the grafted chains are stretched and form an extended polyelectrolyte brush, (ii) a strong repulsive force is operating between two colloids, (iii) and the solution is thermodynamic stable and displays strong spatial correlation among the colloids. In the limit where the charges of the two types of blocks exactly compensate each other, (i) the chains are collapsed and form a polyelectrolyte complex surrounding the cores, (ii) an attractive force appears between two colloids, and (iii) strong colloid clustering appears in the solution. These features become more pronounced as the length of the polymer blocks is increased, and a phase instability occurs at sufficiently long chains. A comparison with properties for other related colloidal particles is also provided.     

Surface properties of bottle-brush polyelectrolytes on mica: effects of side chain and charge densities

Surface properties of a series of cationic bottle-brush polyelectrolytes with 45-unit-long poly(ethylene oxide) side chains were investigated by phase modulated ellipsometry and surface force measurements. The evaluation of the adsorbed mass of polymer on mica by means of ellipsometry is complex due to the transparency of mica and its birefringence and low dielectric constant. We therefore employed a new method to overcome these difficulties. The charge and the poly(ethylene oxide) side chain density of the bottle-brush polymers were varied from zero charge density and one side chain per segment to one charge per segment and no side chains, thus spanning the realm from a neutral bottle-brush polymer, via a partly charged brush polyelectrolyte, to a linear fully charged polyelectrolyte. The adsorption properties depend crucially on the polymer architecture. A minimum charge density of the polymer is required to facilitate adsorption to the oppositely charged surface. The maximum adsorbed amount and the maximum side chain density at the surface are obtained for the polymer with 50% charged segments and the remaining 50% of the segments carrying poly(ethylene oxide) side chains. It is found that brushlike layers are formed when 25-50% of the segments carry poly(ethylene oxide) side chains. In this paper, we argue that the repulsion between the side chains results in an adsorbed layer that is non-homogeneous on the molecular level. As a result, not all side chains will contribute equally to the steric repulsion but some will be stretched along the surface rather than perpendicular to it. By comparison with linear polyelectrolytes, it will be shown that the presence of the side chains counteracts adsorption. This is due to the entropic penalty of confining the side chains to the surface region.     

Electrokinetic flow in a capillary with a charge-regulating surface polymer layer

An analytical study of the steady electrokinetic flow in a long uniform capillary tube or slit is presented. The inside wall of the capillary is covered by a layer of adsorbed or covalently bound charge-regulating polymer in equilibrium with the ambient electrolyte solution. In this solvent-permeable and ion-penetrable surface polyelectrolyte layer, ionogenic functional groups and frictional segments are assumed to distribute at uniform densities. The electrical potential and space charge density distributions in the cross section of the capillary are obtained by solving the linearized Poisson-Boltzmann equation. The fluid velocity profile due to the application of an electric field and a pressure gradient through the capillary is obtained from the analytical solution of a modified Navier-Stokes/Brinkman equation. Explicit formulas for the electroosmotic velocity, the average fluid velocity and electric current density on the cross section, and the streaming potential in the capillary are also derived. The results demonstrate that the direction of the electroosmotic flow and the magnitudes of the fluid velocity and electric current density are dominated by the fixed charge density inside the surface polymer layer, which is determined by the regulation characteristics such as the dissociation equilibrium constants of the ionogenic functional groups in the surface layer and the concentration of the potential-determining ions in the bulk solution.     

Interactions of polyelectrolyte brushes with oppositely charged surfactants

Using Brownian dynamics simulations, we study the effect of the charge ratio, the surfactant length, and the grafting density on the conformational behavior of the complex formed by the polyelectrolyte brush with oppositely charged surfactants. In our simulations, the polyelectrolyte chains and surfactants are represented by a coarse-grained bead-spring model, and the solvent is treated implicitly. It is found that varying the charge ratio induces different morphologies of surfactant aggregates adsorbed onto the brush. At high charge ratios, the density profiles of surfactant monomers indicate that surfactant aggregates exhibit a layer-by-layer arrangement. The surfactant length has a strong effect on the adsorption behavior of surfactants. The lengthening of surfactant leads to a collapsed brush configuration, but a reswelling of the brush with further increasing the surfactant length is observed. The collapse of the brush is attributed to the enhancement of surfactants binding to polyelectrolyte chains. The reswelling is due to an increase in the volume of adsorbed surfactant aggregates. At the largest grafting density investigated, enhanced excluded volume interactions limit the adsorption of surfactant within the polyelectrolyte brush. We also find that end monomers in polyelectrolyte chains exhibit a bimodal distribution in cases of large surfactant lengths and high charge ratios.     

Diblock polyampholytes grafted onto spherical particles: effect of stiffness, charge density, and grafting density

The structure of spherical brushes formed by symmetric diblock polyampholytes end-grafted onto small spherical particles in aqueous solution is examined within the framework of the so-called primitive model using Monte Carlo simulations. The properties of the two blocks are identical except for the sign of their charges. Three different chain flexibilities corresponding to flexible, semiflexible, and stiff blocks are considered at various polyampholyte linear charge densities and grafting densities. The link between the two blocks is flexible at all conditions, and the grafted segments are laterally mobile. Radial and lateral spatial distribution functions of different types and single-chain properties are analyzed. The brush structure strongly depends on the chain flexibility. With flexible chains, a disordered polyelectrolyte complex is formed at the surface of the particle, the complex becoming more compact at increasing linear charge density. With stiff blocks, the inner blocks are radially oriented. At low linear charged density, the outer blocks are orientationally disordered, whereas at increasing electrostatic interaction the two blocks of a polyampholyte are parallel and close to each other, leading to an ordered structure referred to as a polyampholyte star. As the grafting density is increased, the brush thickness responds differently for flexible and nonflexible chains, depending on a different balance between electrostatic interactions and excluded volume effects.     

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.     

Structure and collapse of a surface-grown strong polyelectrolyte brush on sapphire

We have used neutron reflectometry to investigate the behavior of a strong polyelectrolyte brush on a sapphire substrate, grown by atom-transfer radical polymerization (ATRP) from a silane-anchored initiator layer. The initiator layer was deposited from vapor, following treatment of the substrate with an Ar/H(2)O plasma to improve surface reactivity. The deposition process was characterized using X-ray reflectometry, indicating the formation of a complete, cross-linked layer. The brush was grown from the monomer [2-(methacryloyloxy)ethyl]trimethylammonium chloride (METAC), which carries a strong positive charge. The neutron reflectivity profile of the swollen brush in pure water (D(2)O) showed that it adopted a two-region structure, consisting of a dense surface region ～100 ? thick, in combination with a diffuse brush region extending to around 1000 ? from the surface. The existence of the diffuse brush region may be attributed to electrostatic repulsion from the positively charged surface region, while the surface region itself most probably forms due to polyelectrolyte adsorption to the hydrophobic initiator layer. The importance of electrostatic interactions in maintaining the brush region is confirmed by measurements at high (1 M) added 1:1 electrolyte, which show a substantial transfer of polymer from the brush to the surface region, together with a strong reduction in brush height. On addition of 10(-4) M oppositely charged surfactant (sodium dodecyl sulfate), the brush undergoes a dramatic collapse, forming a single dense layer about 200 ? in thickness, which may be attributed to the neutralization of the monomers by adsorbed dodecyl sulfate ions in combination with hydrophobic interactions between these dodecyl chains. Subsequent increases in surfactant concentration result in slow increases in brush height, which may be caused by stiffening of the polyelectrolyte chains due to further dodecyl sulfate adsorption.     

Molecular dynamics simulations of polyelectrolyte adsorption

We have performed molecular dynamics simulations of polyelectrolyte adsorption at oppositely charged surfaces from dilute polyelectrolyte solutions. In our simulations, polyelectrolytes were modeled by chains of charged Lennard-Jones particles with explicit counterions. We have studied the effects of the surface charge density, surface charge distribution, solvent quality for the polymer backbone, strength of the short-range interactions between polymers and substrates on the polymer surface coverage, and the thickness of the adsorbed layer. The polymer surface coverage monotonically increases with increasing surface charge density for almost all studied systems except for the system of hydrophilic polyelectrolytes adsorbing at hydrophilic surfaces. In this case the polymer surface coverage saturates at high surface charge densities. This is due to additional monomer-monomer repulsion between adsorbed polymer chains, which becomes important in dense polymeric layers. These interactions also preclude surface overcharging by hydrophilic polyelectrolytes at high surface charge densities. The thickness of the adsorbed layer shows monotonic dependence on the surface charge density for the systems of hydrophobic polyelectrolytes for both hydrophobic and hydrophilic surfaces. Thickness is a decreasing function of the surface charge density in the case of hydrophilic surfaces while it increases with the surface charge density for hydrophobic substrates. Qualitatively different behavior is observed for the thickness of the adsorbed layer of hydrophilic polyelectrolytes at hydrophilic surfaces. In this case, thickness first decreases with increasing surface charge density, then it begins to increase.     

Counterion distributions and effective interactions of spherical polyelectrolyte brushes

We consider the electrosteric repulsion of colloidal particles whose surface carries a dense layer of long polyelectrolyte chains (spherical polyelectrolyte brushes). The theory of electrosteric repulsion of star polyelectrolytes developed recently is augmented to include particles with a finite core radius. It is shown that most of the counterions are confined within the brush layer. The strong osmotic pressure thus created within the brush layer dominates the repulsive interaction between two such particles. Because of this the pair interaction potential between spherical polyelectrolyte brushes can be given in terms of an analytic expression. The theoretical predictions are compared with available experimental data and semi-quantitative agreement between the two is found.     

The bridging force between two plates by polyelectrolyte chains

The interaction between particles in a colloidal system can be significantly affected by their bridging by polyelectrolyte chains. In this paper, the bridging is investigated by using a self-consistent field approach which takes into account the van der Waals interactions between the segments of the polyelectrolyte molecules and the plates, as well as the electrostatic and volume exclusion interactions. A positive contribution to the force between two plates is generated by the van der Waals interactions between the segments and the plates. This positive (repulsive) contribution plays an important role in the force when the distances between the plates are small. With increasing van der Waals interaction strength between segments and plates, the force between the plates becomes more repulsive at small distances and more attractive at large distances. When the surfaces of the plates have a constant surface electrical potential and a charge sign opposite to that of the polyelectrolyte chains, the force between the two plates becomes less attractive as the bulk polyelectrolyte concentration increases. This behavior is due to a higher bulk counterion concentration dissociated from the polyelectrolyte molecules. At short distances, the force between plates is more repulsive for stiffer chains. A comparison between theoretical and experimental results regarding the contraction of the interlayer separation between the platelets of vermiculite clays against the concentration of poly(vinyl methyl ether) was made.     

On the mechanism of uptake of globular proteins by polyelectrolyte brushes: a two-gradient self-consistent field analysis

We present model calculations for the interaction of a protein-like inhomogeneously charged nanoscale object with a layer of densely grafted polyelectrolytes ("polyelectrolyte brush"). The motivation of this work is the recent experimental observation that proteins that carry an overall negative charge are absorbed into negatively charged polyelectrolyte brushes. Two-gradient self-consistent field (2G-SCF) calculations have been performed to unravel the physical mechanism of the uptake of protein thus effected. Our results prove that an overall neutral, protein-like object can electrostatically be attracted and therefore spontaneously driven into a polyelectrolyte brush when the object has two faces (patches, domains), one with a permanent positive charge and the other with a permanent negative charge. Using a 2G-SCF analysis, we evaluate the free energy of insertion, such that the electric dipole of the inclusion is oriented parallel to the brush surface. An electroneutral protein-like object is attracted into the brush because the polyelectrolyte brush interacts asymmetrically with the charged patches of opposite sign. At high ionic strength and low charge density on the patches, the attraction cannot compete with the repulsive excluded-volume interaction. However, for low ionic strengths and sufficiently high charge density on the patches, a gain on the order of k(B)T per charge becomes possible. Hence, the asymmetry of interaction for patches of different charges may result in a total attractive force between the protein and the brush. All results obtained herein are in excellent agreement with recent experimental data.     

Poisson-Boltzmann theory of the charge-induced adsorption of semi-flexible polyelectrolytes

A model is suggested for the structure of an adsorbed layer of a highly charged semi-flexible polyelectrolyte on a weakly charged surface of opposite charge sign. The adsorbed phase is thin, owing to the effective reversal of the charge sign of the surface upon adsorption, and ordered, owing to the high surface density of polyelectrolyte strands caused by the generally strong binding between polyelectrolyte and surface. The Poisson-Boltzmann equation for the electrostatic interaction between the array of adsorbed polyelectrolytes and the charged surface is solved for a cylindrical geometry, both numerically, using a finite element method, and analytically within the weak curvature limit under the assumption of excess monovalent salt. For small separations, repulsive surface polarization and counterion osmotic pressure effects dominate over the electrostatic attraction and the resulting electrostatic interaction curve shows a minimum at nonzero separations on the Angstrom scale. The equilibrium density of the adsorbed phase is obtained by minimizing the total free energy under the condition of equality of chemical potential and osmotic pressure of the polyelectrolyte in solution and in the adsorbed phase. For a wide range of ionic conditions and charge densities of the charged surface, the interstrand separation as predicted by the Poisson-Boltzmann model and the analytical theory closely agree. For low to moderate charge densities of the adsorbing surface, the interstrand spacing decreases as a function of the charge density of the charged surface. Above about 0.1 M excess monovalent salt, it is only weakly dependent on the ionic strength. At high charge densities of the adsorbing surface, the interstrand spacing increases with increasing ionic strength, in line with the experiments by Fang and Yang [J. Phys. Chem. B 101, 441 (1997)].     

Density-functional theory of spherical electric double layers and zeta potentials of colloidal particles in restricted-primitive-model electrolyte solutions

A density-functional theory is proposed to describe the density profiles of small ions around an isolated colloidal particle in the framework of the restricted primitive model where the small ions have uniform size and the solvent is represented by a dielectric continuum. The excess Helmholtz energy functional is derived from a modified fundamental measure theory for the hard-sphere repulsion and a quadratic functional Taylor expansion for the electrostatic interactions. The theoretical predictions are in good agreement with the results from Monte Carlo simulations and from previous investigations using integral-equation theory for the ionic density profiles and the zeta potentials of spherical particles at a variety of solution conditions. Like the integral-equation approaches, the density-functional theory is able to capture the oscillatory density profiles of small ions and the charge inversion (overcharging) phenomena for particles with elevated charge density. In particular, our density-functional theory predicts the formation of a second counterion layer near the surface of highly charged spherical particle. Conversely, the nonlinear Poisson-Boltzmann theory and its variations are unable to represent the oscillatory behavior of small ion distributions and charge inversion. Finally, our density-functional theory predicts charge inversion even in a 1:1 electrolyte solution as long as the salt concentration is sufficiently high.     

Analytical expressions for the electrical potential near planar, cylindrical, and spherical surfaces for symmetric electrolytes

The conjecture of Tuinier (J. Colloid Interface Sci. 258 (2003) 45) for the electrical potentials near a cylindrical surface and near a spherical surface under the conditions of symmetric electrolyte and large scaled radius are derived by solving the corresponding Poisson-Boltzmann equation. The surface charge density-surface potential relations for these surfaces are also derived under the conditions of constant surface potential. We show that the level of surface charge density for planar, cylindrical, and spherical surfaces follows the order spherical surface > cylindrical surface > planar surface.