Cylindrical cell model for the electrostatic free energy of polyelectrolyte complexes

Associative phase separation (complex coacervation) in a mixture of oppositely charged polyelectrolytes can lead to different types of (inter-)polyelectrolyte complexes (soluble micelles, macroscopic precipitation). In a previous report [Langmuir 2004, 20, 2785-2791], we presented a model for the electrostatic free energy change when (weakly charged) polyelectrolyte forms a homogeneous complex phase. The influence of ionization of the polymer on the electrostatic free energy of the complex was incorporated but the influence of complex density neglected. In the present effort, cylindrical cells are assumed around each polyelectrolyte chain in the complex, and on the basis of the Poisson-Boltzmann equation, the electrostatic free energy is calculated as a function of the complex density. After combination with Flory-Huggins mixing free energy terms and minimization of the total free energy, the equilibrium complex density is obtained, for a given ratio of polycations to polyanions in the complex. The analysis is used in an example calculation ofpolyelectrolyte film formation by alternatingly applying a polycation and a polyanion solution. The calculation suggests that the often observed exponential growth of a polyelectrolyte film when the polymer is weakly charged has a thermodynamic origin: the polyelectrolyte complex shifts repeatedly between two equilibrium states of different densities and compositions. However, when the polyelectrolytes are strongly charged the difference in the compositions between the two equilibrium states is very small, and exponential growth by an absorption mechanism is no longer possible.     

The build-up of polyelectrolyte multilayers of microfibrillated cellulose and cationic polyelectrolytes

A new type of nanocellulosic material has been prepared by high-pressure homogenization of carboxymethylated cellulose fibers followed by ultrasonication and centrifugation. This material had a cylindrical cross-section as shown by transmission electron microscopy with a diameter of 5-15 nm and a length of up to 1 microm. Calculations, using the Poisson-Boltzmann equation, showed that the surface potential was between 200 and 250 mV, depending on the pH, the salt concentration, and the size of the fibrils. They also showed that the carboxyl groups on the surface of the nanofibrils are not fully dissociated until the pH has reached pH = approximately 10 in deionized water. Calculations of the interaction between the fibrils using the Derjaguin-Landau-Verwey-Overbeek theory and assuming a cylindrical geometry indicated that there is a large electrostatic repulsion between these fibrils, provided the carboxyl groups are dissociated. If the pH is too low and/or the salt concentration is too high, there will be a large attraction between the fibrils, leading to a rapid aggregation of the fibrils. It is also possible to form polyelectrolyte multilayers (PEMs) by combining different types of polyelectrolytes and microfibrillated cellulose (MFC). In this study, silicon oxide surfaces were first treated with cationic polyelectrolytes before the surfaces were exposed to MFC. The build-up of the layers was monitored with ellipsometry, and they show that it is possible to form very well-defined layers by combinations of MFC and different types of polyelectrolytes and different ionic strengths of the solutions during the adsorption of the polyelectrolyte. A polyelectrolyte with a three-dimensional structure leads to the build-up of thick layers of MFC, whereas the use of a highly charged linear polyelectrolyte leads to the formation of thinner layers of MFC. An increase in the salt concentration during the adsorption of the polyelectrolyte results in the formation of thicker layers of MFC, indicating that the structure of the adsorbed polyelectrolyte has a large influence on the formation of the MFC layer. The films of polyelectrolytes and MFC were so smooth and well-defined that they showed clearly different interference colors, depending on the film thickness. A comparison between the thickness of the films, as measured with ellipsometry, and the thickness estimated from their colors showed good agreement, assuming that the films consisted mainly of solid cellulose with a refractive index of 1.53. Carboxymethylated MFC is thus a new type of nanomaterial that can be combined with oppositely charged polyelectrolytes to form well-defined layers that may be used to form, for example, new types of sensor materials.     

Conformational effect on small angle neutron scattering behavior of interacting polyelectrolyte solutions: a perspective of integral equation theory

We present small angle neutron scattering (SANS) measurements of deuterium oxide (D(2)O) solutions of linear and star sodium poly(styrene sulfonate) (NaPSS) as a function of polyelectrolyte concentration. Emphasis is on understanding the dependence of their SANS coherent scattering cross section I(Q) on the molecular architecture of single polyelectrolyte. The key finding is that for a given concentration, star polyelectrolytes exhibit more pronounced characteristic peaks in I(Q), and the position of the first peak occurs at a smaller Q compared to their linear counterparts. Based on a model of integral equation theory, we first compare the SANS experimental I(Q) of salt-free polyelectrolyte solutions with that predicted theoretically. Having seen their satisfactory qualitative agreement, the dependence of counterion association behavior on polyelectrolyte geometry and concentration is further explored. Our predictions reveal that the ionic environment of polyelectrolyte exhibits a strong dependence on polyelectrolyte geometry at lower polyelectrolyte concentration. However, when both linear and star polyelectrolytes exceed their overlap concentrations, the spatial distribution of counterion is found to be essentially insensitive to polyelectrolyte geometry due to the steric effect.     

Interaction of cellulose-based cationic polyelectrolytes with mucin

Mucoadhesivity of water-soluble polymers is an important factor, when testing their suitability for controlled drug delivery systems. For this purpose, the interaction of new cationic cellulose polyelectrolytes with lyophilized mucin was investigated by means of turbidimetric titration, microscopy and measurement of zeta potential and particle size changes in the system. Results show that the cellulose derivatives interact with mucin. This interaction became stronger if cellulose macromolecules contained positively charged groups and an electrostatic interaction with the negatively charged mucin particles occurred. Under certain conditions flocculation of mucin particles by the cellulose polyelectrolyte was observed.     

Mean field theory of charged dendrimer molecules

Using self-consistent field theory (SCFT), we study the conformational properties of polyelectrolyte dendrimers. We compare results for three different models of charge distributions on the polyelectrolytes: (1) a smeared, quenched charge distribution characteristic of strong polyelectrolytes; (2) a smeared, annealed charge distribution characteristic of weak polyelectrolytes; and (3) an implicit counterion model with Debye-Huckel interactions between the charged groups. Our results indicate that an explicit treatment of counterions is crucial for the accurate characterization of the conformations of polyelectrolyte dendrimers. In comparing the quenched and annealed models of charge distributions, annealed dendrimers were observed to modulate their charges in response to the density of polymer monomers, counterions, and salt ions. Such phenomena is not accommodated within the quenched model of dendrimers and is shown to lead to significant differences between the predictions of quenched and annealed model of dendrimers. In this regard, our results indicate that the average dissociated charge α inside the dendrimer serves as a useful parameter to map the effects of different parametric conditions and models onto each other. We also present comparisons to the scaling results proposed to explain the behavior of polyelectrolyte dendrimers. Inspired by the trends indicated by our results, we develop a strong segregation theory model whose predictions are shown to be in very good agreement with the numerical SCFT calculations.     

Molecular dynamics simulations of multilayer polyelectrolyte films: effect of electrostatic and short-range interactions

The effect of the strength of electrostatic and short-range interactions on the multilayer assembly of oppositely charged polyelectrolytes at a charged substrate was studied by molecular dynamics simulations. The multilayer buildup was achieved through sequential adsorption of charged polymers in a layer-by-layer fashion from dilute polyelectrolyte solutions. The strong electrostatic attraction between oppositely charged polyelectrolytes at each deposition step is a driving force behind the multilayer growth. Our simulations have shown that a charge reversal after each deposition step is critical for steady multilayer growth and that there is a linear increase in polymer surface coverage after the first few deposition steps. Furthermore, there is substantial intermixing between chains adsorbed during different deposition steps. We show that the polymer surface coverage and multilayer structure are each strongly influenced by the strength of electrostatic and short-range interactions.     

Direct determination of the thermodynamics of polyelectrolyte complexation and implications thereof for electrostatic layer-by-layer assembly of multilayer films

Interpolyelectrolyte complex (IPEC) formation between poly(styrene sulfonate) (PSS) and poly(allylamine hydrochloride) (PAH) has been studied over a range of ionic strengths by isothermal titration calorimetry (ITC), turbidity titration, and electrostatic layer-by-layer assembly (ELBL). The results indicate that IPEC formation of PSS/PAH in aqueous solution is predominantly entropy-driven. The thermodynamic parameters suggest the formation of different types of complexes and aggregates due to salt-induced conformational changes in the polyelectrolyte conformation. Differences in polyelectrolyte behavior in the different salt-concentration regimes are described in terms of changes in the Debye screening length of the polyelectrolytes. The relationship of the results to the effect of salt concentration on the assembly of polyelectrolyte multilayer films (PEMs) is discussed.     

Structure of flexible and semiflexible polyelectrolyte chains in confined spaces of slit micro/nanochannels

Understanding the behavior of a polyelectrolyte in confined spaces has direct relevance in design and manipulation of microfluidic devices, as well as transport in living organisms. In this paper, a coarse-grained model of anionic semiflexible polyelectrolyte is applied, and its structure and dynamics are fully examined with Brownian dynamics (BD) simulations both in bulk solution and under confinement between two negatively charged parallel plates. The modeling is based on the nonlinear bead-spring discretization of a continuous chain with additional long-range electrostatic, Lennard-Jones, and hydrodynamic interactions between pairs of beads. The authors also consider the steric and electrostatic interactions between the bead and the confining wall. Relevant model parameters are determined from experimental rheology data on the anionic polysaccharide xanthan reported previously. For comparison, both flexible and semiflexible models are developed accompanying zero and finite intrinsic persistence lengths, respectively. The conformational changes of the polyelectrolyte chain induced by confinements and their dependence on the screening effect of the electrolyte solution are faithfully characterized with BD simulations. Depending on the intrinsic rigidity and the medium ionic strength, the polyelectrolyte can be classified as flexible, semiflexible, or rigid. Confined flexible and semiflexible chains exhibit a nonmonotonic variation in size, as measured by the radius of gyration and end-to-end distance, with changing slit width. For the semiflexible chain, this is coupled to the variations in long-range bond vector correlation. The rigid chain, realized at low ionic strength, does not have minima in size but exhibits a sigmoidal transition. The size of confined semiflexible and rigid polyelectrolytes can be well described by the wormlike chain model once the electrostatic effects are taken into account by the persistence length measured at long length scale.     

Molecular bottle brushes in a solution of semiflexible polyelectrolytes and block copolymers with an oppositely charged block: a molecular dynamics simulation

Using a coarse-grained model, we performed molecular dynamics simulations of the electrostatically driven self-assembly of strongly charged polyelectrolytes and diblock copolymers composed of oppositely charged and neutral blocks. Stoichiometric micelle-like complexes formed in a dilute solution represent cylindrical brushes whose conformation is determined by the linear charge density on the polyelectrolyte and by temperature. The core-shell morphology of the cylindrical brushes is proven. The core of these anisotropic micelles consists of an insoluble complex coacervate formed by the ionic chains and a shell made up of the neutral solvophilic blocks. As the concentration of macromolecules increases, the orientational ordering of ionic micelles takes place. The complexation can induce effective steric stiffening of the polyelectrolyte chains.     

Protein-polyelectrolyte cluster formation and redissolution: a Monte Carlo study

Aqueous solutions of proteins and oppositely charged polyelectrolytes were studied at different polyelectrolyte chain length, ionic strength, and protein-protein interaction potential as a function of the polyelectrolyte concentration. One of the protein models used represented lysozyme in aqueous environment. The model systems were solved by Monte Carlo simulations, and their properties were analyzed in terms of radial distribution functions, structure factors, and cluster composition probabilities. In the system with the strongest electrostatic protein-polyelectrolyte interaction the largest clusters were formed near or at equivalent amount of net protein charge and polyelectrolyte charge, whereas in excess of polyelectrolyte a redissolution appeared. Shorter polyelectrolyte chains and increased ionic strength lead to weaker cluster formation. An inclusion of nonelectrostatic protein-protein attraction promoted the protein-polyelectrolyte cluster formation.     

Titration of hydrophobic polyelectrolytes using Monte Carlo simulations

The conformation and titration curves of weak (or annealed) hydrophobic polyelectrolytes have been examined using Monte Carlo simulations with screened Coulomb potentials in the grand canonical ensemble. The influence of the ionic concentration pH and presence of hydrophobic interactions has been systematically investigated. A large number of conformations such as extended, pearl-necklace, cigar-shape, and collapsed structures resulting from the subtle balance of short-range hydrophobic attractive interactions and long-range electrostatic repulsive interactions between the monomers have been observed. Titration curves were calculated by adjusting the pH-pK(0) values (pK(0) represents the intrinsic dissociation constant of an isolated monomer) and then calculating the ionization degree alpha of the polyelectrolyte. Important transitions related to cascades of conformational changes were observed in the titration curves, mainly at low ionic concentration and with the presence of strong hydrophobic interactions. We demonstrated that the presence of hydrophobic interactions plays an important role in the acid-base properties of a polyelectrolyte in promoting the formation of compact conformations and hence decreasing the polyelectrolyte degree of ionization for a given pH-pK(0) value.     

Nanoparticle adsorption on a weak polyelectrolyte. Stiffness, pH, charge mobility, and ionic concentration effects investigated by Monte Carlo simulations

Monte Carlo simulations have been used to study two different models for a weak linear polyelectrolyte in the presence of nanoparticles: (i) a rodlike and (ii) a flexible polyelectrolytes. The use of simulated annealing has made it possible to simulate a polyelectrolyte chain in the presence of several nanoparticles by improving conformation sampling and avoiding multiple minima problems when dense conformations are produced. Nanoparticle distributions along the polymer backbone were analyzed versus the ionic concentration, polyelectrolyte stiffness, and nanoparticle surface charge. Titration curves were calculated and the influences of the ionic concentration, solution pH, and number of adsorbed nanoparticles on the acid/base polyelectrolyte properties have been systematically investigated. The subtle balance of attractive and repulsive interactions has been discussed, and some characteristic conformations are presented. The comparison of the two limit models provides a good representation of the stiffness influence on the complex formation. In some conditions, overcharging was obtained and presented with respect to both the polyelectrolyte and nanoparticle as the central element. Finally, the charge mobility influence along the polyelectrolyte backbone was investigated by considering annealed and quenched polyelectrolyte chains.     

Density functional theory for planar electric double layers: closing the gap between simple and polyelectrolytes

We report a nonlocal density functional theory (NLDFT) for polyelectrolyte solutions within the primitive model; i.e., the solvent is represented by a continuous dielectric medium, and the small ions and polyions by single and tangentially connected charged hard spheres, respectively. The excess Helmholtz energy functional is derived from a modified fundamental measure theory for hard-sphere repulsion, an extended first-order thermodynamic perturbation theory for chain connectivity, and a quadratic functional Taylor expansion for electrostatic correlations. With the direct and cavity correlation functions of the corresponding monomeric systems as inputs, the NLDFT predicts the segment-level microscopic structures and adsorption isotherms of polyelectrolytes at oppositely charged surfaces in good agreement with molecular simulations. In particular, it faithfully reproduces the layering structures of polyions, charge inversion, and overcharging that cannot be captured by alternative methods including the polyelectrolyte Poisson-Boltzmann equation and an earlier version of DFT. The NLDFT has also been used to investigate the influences of the small ion valence, polyion chain length, and size disparity between polyion segments and counterions on the microscopic structure, mean electrostatic potential, and overcharging in planar electric double layers containing polyelectrolytes.     

Ionic micelles in solutions of polyelectrolytes and block copolymers with oppositely charged blocks: Computer simulation

Complexation in solutions of strongly charged polyelectrolytes and diblock copolymers composed of oppositely charged and neutral blocks were studied via the molecular dynamics method. Stoichiometric micellar complexes formed in a dilute solution represent cylindrical brushes whose conformation is determined by the linear charge density on the polyelectrolyte and by temperature. As the concentration of macromolecules increases, the orientational ordering of anisotropic ionic micelles takes place. The complexation can induce the stiffening of the polyelectrolyte chain.     

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     

Synthesis and properties in solution of rodlike polyelectrolytes

Efficient synthetic strategies are described for the preparation of rodlike polyelectrolytes based on the intrinsically rigid poly(p-phenylene). Uncharged precursors were first prepared via the Suzuki coupling and then characterized by different methods of polymer analysis. Finally, they were transformed into polyelectrolytes using macromolecular substitution reactions. Depending on the substitution pattern, the obtained polyelectrolytes are either soluble or insoluble in water. Using water-soluble derivatives, the Poisson-Boltzmann cell model was tested by osmotic measurements and small-angle X-ray scattering. It is shown that the cell model provides a good first approximation of the distribution of the counterions around the macroion but still underestimates their correlation. Moreover, the PPP polyelectrolytes show a very pronounced polyelectrolyte effect. Since the rodlike PPPs are very rigid in shape, this observation proves that the polyelectrolyte effect is caused by long-range intermolecular electrostatic repulsion of the dissolved macroions rather than due to conformational changes.     

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)].     

Polyion complex nanomaterials from block polyelectrolyte micelles and linear polyelectrolytes of opposite charge: 1. Solution behavior

The present study investigates whether block polyelectrolyte micelles can form soluble complexes upon interaction with oppositely charged linear polyelectrolytes. The phase behavior and molecular characteristics of the complexes were examined by turbidimetry, phase analysis, dynamic light scattering, and sedimentation velocity techniques. At an excess of polyelectrolyte micelles, soluble complexes were formed either independently on the route of preparation or, for select linear polyelectrolytes, through routes that avoided macrophase separation. Such soluble complexes are in a thermodynamic equilibrium state for all polyion pairs. The hydrodynamic sizes and sedimentation coefficients did not depend on the chemical nature of the linear polyelectrolyte, but were determined by the charge ratios and the hydrodynamic properties of the initial micelles. At an excess of linear polyelectrolyte, complex solubility and molecular characteristics depended on the chemical nature of the linear polyelectrolyte. In this region, linear polyelectrolytes formed soluble complexes with micelles if soluble complexes could be formed with the corresponding linear analogues of the block polyelectrolyte.     

Electrostatic free energy of weakly charged macromolecules in solution and intermacromolecular complexes consisting of oppositely charged polymers

When oppositely charged polyelectrolytes are mixed in water, attraction between oppositely charged groups may lead to the formation of polyelectrolyte complexes (associative phase separation, complex coacervation, interpolymer complexes). Theory is presented to describe the electrostatic free energy change when ionizable (annealed) (macro-)molecules form a macroscopic polyelectrolyte complex. The electrostatic free energy includes an electric term as well as a chemical term that is related to the dissociation of the ionic groups in the polymer. An example calculation for complexation of polyacid with polybase uses a cylindrical diffuse double layer model for free polymer in solution and electroneutrality within the complex and calculates the free energy of the system when the polymer is in solution or in a polyelectrolyte complex. Combined with a term for the nonelectrostatic free energy change upon complexation, a theoretical stability diagram is constructed that relates pH, salt concentration, and mixing ratio, which is in qualitative agreement with an experimental diagram obtained by Bungenberg de Jong (1949) for complex coacervation of arabic gum and gelatin. The theory furthermore explains the increased tendency toward phase separation when the polymer becomes more strongly charged and suggests that complexation of polyacid or polybase with zwitterionic polymer (e.g., protein) of the same charge sign (at the "wrong side" of the iso-electric point) may be due (in part) to an induced charge reversal of the protein.