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.     

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)].     

Theory of inhomogeneous weakly charged polyelectrolytes

A theory of inhomogeneous multicomponent systems containing weakly charged polyelectrolytes is developed. The theory treats the polymer conformation and the electrostatics simultaneously using a functional integral representation of the partition function. A mean‐field approximation to the theory leads to two sets of coupled mean‐field equations: a Poisson‐Boltzmann type equation describing the electrostatic potential, and a set of self‐consistent field equations describing the equilibrium densities. Asymptotic forms of the theory at weak and strong segregation limits are derived. The theory can be used to study the interfacial properties, microphase structures, and adsorptions of a variety of weakly charged polyelectrolyte systems. As a simple example, the interface between the polymer‐rich and polymer‐poor phases of a polyelectrolyte solution is studied.     

Structure formation in films of weakly charged block polyelectrolyte solutions

A mean-field dynamic density functional theory is used to describe a phase diagram of concentrated solutions of weakly charged flexible block polyelectrolytes in a film. Electrostatics is taken into account by applying the local electroneutrality constraint (the Donnan membrane equilibrium approach). In the Donnan limit it is assumed that a salt added to the solution perfectly screens long-range electrostatic interactions. The phase diagram of a solution of a triblock polyelectrolyte in a film as a function of the solvent concentration and the charge of the polyelectrolyte (solvophilic) block is calculated for a given film thickness. The phase behavior of the block polyelectrolyte film arises from the interplay between surface-induced alignment and the electrostatically-driven structure formation. The observed mesoscopic structures (lamellar, perforated lamellar, cylindrical, micellar, and mixed phases) are oriented parallel to the surfaces for the considered case of morphologies unfrustrated by the film thickness. Structures with connections between parallel layers (bicontinuous, etc.) are not formed. As a result of surface-induced ordering, the region of ordered phases in a film is wider than in bulk and the phase boundary between ordered and disordered phases is more diffuse. As in the case of unconfined block polyelectrolyte solution, the solution in a film does not follow the lyotropic sequence of phases of such a block copolymer upon increase in the charge of the polyelectrolyte block. Upon changing the charge of the solvophilic copolymer block, transformations of copolymer morphology take place via change in curvature of polymeric domains. Due to confinement of a polyelectrolyte film, no swelling of solvophilic domains is observed.     

Mixed systems of hydrophobically modified polyelectrolytes: controlling rheology by charge and hydrophobe stoichiometry and interaction strength

Rheology and phase separation were investigated for aqueous mixtures of two oppositely charged hydrophobically modified polyelectrolytes. The typical phase separation, normally seen for oppositely charged polymer mixtures, is dramatically reduced by the presence of hydrophobic modification, and phase separation is only detected close to the point of charge neutralization. While the two polyelectrolytes separately can give high viscosities and a gel-like behavior, a pronounced maximum in viscosity and storage modulus with the mixing ratio of the polyelectrolytes is observed; the maximum is located between the points of charge and hydrophobe stoichiometry and reflects a combination of hydrophobic and electrostatic association. Lowering the charge density of the anionic polymer leads to a strengthened association at first, but at lower charge densities there is a weakened association due to the onset of phase separation. The strength of the electrostatic interaction was modified by adding salt. Increased ionic strength can lead to phase separation and to increased or decreased viscosity depending on the polyelectrolyte mixing ratio.     

Entropy and enthalpy of polyelectrolyte complexation: Langevin dynamics simulations

We report a systematic study by Langevin dynamics simulation on the energetics of complexation between two oppositely charged polyelectrolytes of same charge density in dilute solutions of a good solvent with counterions and salt ions explicitly included. The enthalpy of polyelectrolyte complexation is quantified by comparisons of the Coulomb energy before and after complexation. The entropy of polyelectrolyte complexation is determined directly from simulations and compared with that from a mean-field lattice model explicitly accounting for counterion adsorption. At weak Coulomb interaction strengths, e.g., in solvents of high dielectric constant or with weakly charged polyelectrolytes, complexation is driven by a negative enthalpy due to electrostatic attraction between two oppositely charged chains, with counterion release entropy playing only a subsidiary role. In the strong interaction regime, complexation is driven by a large counterion release entropy and opposed by a positive enthalpy change. The addition of salt reduces the enthalpy of polyelectrolyte complexation by screening electrostatic interaction at all Coulomb interaction strengths. The counterion release entropy also decreases in the presence of salt, but the reduction only becomes significant at higher Coulomb interaction strengths. More significantly, in the range of Coulomb interaction strengths appropriate for highly charged polymers in aqueous solutions, complexation enthalpy depends weakly on salt concentration and counterion release entropy exhibits a large variation as a function of salt concentration. Our study quantitatively establishes that polyelectrolyte complexation in highly charged Coulomb systems is of entropic origin.     

Human serum albumin self-assembly on weak polyelectrolyte multilayer films structurally modified by pH changes

Adsorption of proteins onto film surfaces built up layer by layer from oppositely charged polyelectrolytes is a complex phenomenon, governed by electrostatic forces, hydrogen bonds, and hydrophobic interactions. The amounts of the interacting charges, however, both in polyelectrolytes and in proteins adsorbed on such films are a function of the pH of the solution. In addition, the number and the accessibility of free charges in proteins depend on the secondary structure of the protein. The subtle interplay of all these factors determines the adsorption of the proteins onto the polyelectrolyte film surfaces. We investigated the effect of these parameters for polyelectrolyte films built up from weak "protein-like" polyelectrolytes (i.e., polypeptides), poly(L-lysine) (PLL), and poly(glutamic acid) (PGA) and for the adsorption of human serum albumin (HSA) onto these films in the pH range 3.0-10.5. It was found that the buildup of the polyelectrolyte films is not a simple function of the pure charges of the individual polyelectrolytes, as estimated from their respective pKa values. The adsorption of HSA onto (PLL/PGA)n films depended strongly on the polyelectrolyte terminating the film. For PLL-terminated polyelectrolyte films, at low pH, repulsion, as expected, is limiting the adsorption of HSA (having net positive charge below pH 4.6) since PLL is also positively charged here. At high pH values, an unexpected HSA uptake was found on the PGA-ending films, even when both PGA and HSA were negatively charged. It is suggested that the higher surface rugosity and the decrease of the alpha-helix content at basic pH values (making accessible certain charged groups of the protein for interactions with the polyelectrolyte film) could explain this behavior.     

Microphase separation of weakly charged block polyelectrolyte solutions: Donnan theory for dynamic polymer morphologies

A mean-field dynamic density functional theory for the phase behavior of concentrated weakly charged block polyelectrolyte solutions is developed, using the Donnan membrane equilibrium approach to account for electrostatic interactions. In this limit all long-range electrostatic interactions are canceled and the net charge density in any region on a coarse-grained scale is zero. The phase diagram of a model triblock polyelectrolyte in solution as a function of the charge of the solvophilic block and the solvent concentration is established. Different mesoscopic structures (lamellar, bicontinuous, hexagonal, micellar, and dispersed coexisting phases) are formed depending on the copolymer charge asymmetry. It is found that upon changing the charge of the solvophilic copolymer block the polyelectrolyte solution does not follow the lyotropic sequence of phases of this polymer. Upon increase in the charge of the solvophilic blocks, changes in copolymer morphology take place by means of change in curvature of polymeric domains.     

Phase diagram for the adsorption of weak polyelectrolytes at a soft charged surface

We have experimentally studied the adsorption of polyelectrolytes at oppositely charged surfaces. A weak flexible polyelectrolyte, poly(acrylic acid), was adsorbed from dilute solutions on a Langmuir film of a cationic amphiphile, dimethyldioctadecylammonium bromide. The polymer surface coverage, Gamma, at equilibrium was measured by two reflectivity techniques-ellipsometry and polarization modulated infrared reflection absorption spectroscopy (PM-IRRAS)-as a function of the surface charge density, sigma, and of the polymer ionization degree, alpha. Different adsorption regimes were evidenced. For weakly charged surfaces, sigma < sigma sat, Gamma increases with sigma and with 1/alpha, as expected for a neutralization of the surface by the adsorbed polymers. For highly charged surfaces, sigma > sigma sat, the adsorption of polyelectrolytes saturates. The mean orientation of the adsorbed chains also depends on the value of sigma: it is parallel to the surface for sigma < sigma (< sigma sat) and orthogonal to the surface for sigma > sigma. We have measured the values of sigma sat and sigma as a function of alpha and compared the results with existing theories.     

Molecular dynamics simulations of layer-by-layer assembly of polyelectrolytes at charged surfaces: effects of chain degree of polymerization and fraction of charged monomers

We performed molecular dynamics simulations of the electrostatic assembly of multilayers of flexible polyelectrolytes at a charged surface. The multilayer build-up was achieved through sequential adsorption of oppositely charged polymers in a layer-by-layer fashion from dilute polyelectrolyte solutions. The steady-state multilayer growth proceeds through a charge reversal of the adsorbed polymeric film which leads to a linear increase in the polymer surface coverage after completion of the first few deposition steps. Moreover, substantial intermixing between chains adsorbed during different deposition steps is observed. This intermixing is consistent with the observed requirement for several deposition steps to transpire for completion of a single layer. However, despite chain intermixing, there are almost perfect periodic oscillations of the density difference between monomers belonging to positively and negatively charged macromolecules in the adsorbed film. Weakly charged chains show higher polymer surface coverage than strongly charged ones.     

Simulations of surface forces in polyelectrolyte solutions

We have simulated interactions between charged surfaces in the presence of oppositely charged polyelectrolytes by coupling perturbations in the isotension ensemble to a free energy variance minimization scheme. For polymeric systems, this method completely outperforms configurationally biased versions of grand canonical simulations. Proper diffusive equilibrium between bulk and slit has been established for polyelectrolytes with up to 60 monomers per chain. A consequence of imposing diffusive equilibrium conditions, in contrast to previous more restricted models, is the possibility of surface charge inversion; ion-ion correlation and the cooperativity of monomer adsorption drive the formation of a polyion layer close to the surface, that overcompensates the nominal surface charge. This is observed even at modest surface charge densities, and leads to a build up of a long ranged electrostatic barrier. In addition, the onset of charge inversion requires very low bulk polymer densities. Due to screening effects, this leads to a higher and more long-ranged free energy barrier at low, compared to high, bulk densities. Oscillatory forces, reminiscent of those found in simple hard sphere systems, are resolved in the high concentration regime. As a consequence of a second surface charge inversion, the system "stratifies" to form a stable polyelectrolyte layer in the central part of the slit, stabilized by the adsorbed surface layers.     

Monte Carlo simulation and molecular theory of tethered polyelectrolytes

We investigate the structure of end-tethered polyelectrolytes using Monte Carlo simulations and molecular theory. In the Monte Carlo calculations we explicitly take into account counterions and polymer configurations and calculate electrostatic interaction using Ewald summation. Rosenbluth biasing, distance biasing, and the use of a lattice are all used to speed up Monte Carlo calculation, enabling the efficient simulation of the polyelectrolyte layer. The molecular theory explicitly incorporates the chain conformations and the possibility of counterion condensation. Using both Monte Carlo simulation and theory, we examine the effect of grafting density, surface charge density, charge strength, and polymer chain length on the distribution of the polyelectrolyte monomers and counterions. For all grafting densities examined, a sharp decrease in brush height is observed in the strongly charged regime using both Monte Carlo simulation and theory. The decrease in layer thickness is due to counterion condensation within the layer. The height of the polymer layer increases slightly upon charging the grafting surface. The molecular theory describes the structure of the polyelectrolyte layer well in all the different regimes that we have studied.     

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.     

A unified analytical treatment of the acid-dissociation equilibria of weakly acidic linear polyelectrolytes and the conjugate acids of weakly basic linear polyelectrolytes

 The influence of added sodium chloride concentration levels on the acid-dissociation equilibria of a weakly acidic linear polyelectrolyte and a conjugate acid of weakly basic linear polyelectrolyte has been investigated potentiometrically by use of polyacrylic acid (PAA) and poly(N-vinylimidazole) (PVIm) as examples of polyelectrolytes. Both equilibria are strongly influenced by the degree of dissociation of the polyacids as well as the concentration levels of sodium chloride due to an electrostatic effect originating from the negatively or positively charged polymer surfaces. These have been analyzed in a unified manner by taking accounts of two-phase properties of the charged linear polyions. Distribution of counterions and coions between a polyelectrolyte phase formed around the polymer skeleton and a bulk solution phase has been rationalized by a Donnan’s relation. Introduction of a volume term for the polyelectrolyte phase permits definition of averaged concentrations of mobile ions in the vicinity of the polyion molecules, which enables us to define hypothetical intrinsic acid-dissociation constants in the polyion domain. The intrinsic constants estimated by extrapolation of apparent acid-dissociation constants at zero-charge state are in good agreement with the acid-dissociation constants of the monomer analogs of the polymers, i.e., acetic acid for PAA and imidazole for PVIm, respectively. The difference between the apparent and intrinsic acid-dissociation constants for PVIm was much higher than that for PAA at defined degree of dissociation of the polyacids, even though the separations of the functionalities fixed on the linear polymers are approximately equal to each other. Received: 4 February 1997 Accepted: 26 May 1997     

Weak Polyelectrolytes as Nanoarchitectonic Design Tools for Functional Materials: A Review of Recent Achievements

The ionization degree, charge density, and conformation of weak polyelectrolytes can be adjusted through adjusting the pH and ionic strength stimuli. Such polymers thus offer a range of reversible interactions, including electrostatic complexation, H-bonding, and hydrophobic interactions, which position weak polyelectrolytes as key nano-units for the design of dynamic systems with precise structures, compositions, and responses to stimuli. The purpose of this review article is to discuss recent examples of nanoarchitectonic systems and applications that use weak polyelectrolytes as smart components. Surface platforms (electrodeposited films, brushes), multilayers (coatings and capsules), processed polyelectrolyte complexes (gels and membranes), and pharmaceutical vectors from both synthetic or natural-type weak polyelectrolytes are discussed. Finally, the increasing significance of block copolymers with weak polyion blocks is discussed with respect to the design of nanovectors by micellization and film/membrane nanopatterning via phase separation.     

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.     

Highly flexible polyelectrolyte nanotubes

A pressure-filter-template approach was employed to prepare polyelectrolyte nanotubes through layer-by-layer deposition in the alumina template. With the thicker wall, the ordered polymer nanotubes possess a high flexibility. The results demonstrate that the electrostatic interactions of polyelectrolytes play a key role in fabricating water-soluble charged polymer nanotubes. The structure of the polyelectrolyte nanotube was confirmed by SEM, TEM, and UV, respectively.     

Nematic ordering of rigid rod polyelectrolytes induced by electrostatic interactions: effect of discrete charge distribution along the chain

Similar to the Debye-Hu?ckel plasma, charged groups in solutions of rigid rod polyelectrolytes attract each other. We derive expression for the correlation free energy of electrostatic attraction of the rods within the random phase approximation. In this theory, we explicitly take into account positions of charged groups on the chains and examine both charge and polymer concentration fluctuations. The correlation free energies and the osmotic pressures are calculated for isotropic and completely ordered nematic phase. The results of the discrete model are compared with results of a continuous model. The discrete model gives rise to a stronger attraction between the charged groups both in the isotropic and nematic phases and to a stronger orienting action of the electrostatic forces.     

Poisson-Boltzmann theory of pH-sensitive (annealing) polyelectrolyte brush

We present a self-consistent field analytical theory of a polymer brush formed by weakly charged pH-sensitive (annealing) polyelectrolytes tethered to a solid-liquid interface and immersed in buffer solution of low molecular weight salt. We use the Poisson-Boltzmann framework, applied by us previously to polyelectrolyte (PE) brushes with quenched charge (Zhulina, E. B.; Borisov, O. V. J. Chem. Phys. 1997, 107, 5952). This approach allows for detailed analysis of the internal structure of annealing PE brush in terms of polymer density distribution, profiles of electrostatic potential and of local degree of chain ionization as a function of buffer ionic strength and pH without any assumptions on mobile ion distribution imposed in earlier scaling-type models. The presented analytical theory recovers all major asymptotic dependences for average brush properties predicted earlier. In particular, a nonmonotonic dependence of brush thickness on ionic strength and grafting density is confirmed and specified with accuracy of numerical coefficients including crossover regions. Moreover, the theory predicts qualitatively new effects, such as, e.g., disproportionation of tethered polyions into weakly charged concentrated proximal and strongly charged sparse distal brush domains at low salt and moderate grating densities. The presented results allow us to quantify responsive features of annealing PE brushes whose large-scale and local conformational properties can be manipulated by external stimuli.