A self-consistent mean-field model for polyelectrolyte gels

Adsorption of Mono- and Divalent Ions onto Dendritic Polyglycerol Sulfate (dPGS) as Studied Using Isothermal Titration Calorimetry

The effective charge of highly charged polyelectrolytes is significantly lowered by a condensation of counterions. This effect is more pronounced for divalent ions. Here we present a study of the counterion condensation to dendritic polyglycerol sulfate (dPGS) that consists of a hydrophilic dendritic scaffold onto which sulfate groups are appended. The interactions between the dPGS and divalent ions (Mg²⁺ and Ca²⁺) were analyzed using isothermal titration calorimetry (ITC) and showed no ion specificity upon binding, but clear competition between the monovalent and divalent ions. Our findings, in line with the latest theoretical studies, demonstrate that a large fraction of the monovalent ions is sequentially replaced with the divalent ions.

Research Progress of the Ion Activity Coefficient of Polyelectrolytes: A Review

Polyelectrolyte has wide applications in biomedicine, agriculture and soft robotics. However, it is among one of the least understood physical systems because of the complex interplay of electrostatics and polymer nature. In this review, a comprehensive description is presented on experimental and theoretical studies of the activity coefficient, one of the most important thermodynamic properties of polyelectrolyte. Experimental methods to measure the activity coefficient were introduced, including direct potentiometric measurement and indirect methods such as isopiestic measurement and solubility measurement. Next, progress on the various theoretical approaches was presented, ranging from analytical, empirical and simulation methods. Finally, challenges for future development are proposed on this field.

Thermodynamics of hydrogels for applications in atmospheric water harvesting, evaporation, and desalination

Most thermodynamic modeling of hydrogels is built on Flory's theories for the entropy of mixing and rubber elasticity, and Donnan's equilibrium conditions if polyelectrolyte polymer and mobile ions are involved. The entropy of mixing depends on the number of solvent and polymer molecules while the configurational entropy depends on the volume the polymer occupied. Flory's theory treated these two entropy terms in the Gibbs free energy on an equal basis: using the molecular numbers as the variable. I argue that the molecular number and volume are two independent thermodynamic variables and reformulate Flory's classical hydrogel thermodynamic model by minimizing the Helmholtz free energy of a combined system consisting of the hydrogel and its environment. This treatment enables us to unequivocally state that the osmotic pressure is the thermodynamic pressure of the solvent inside the hydrogel and to unambiguously write down the chemical potential of each species. The balance of the chemical potentials of the mobile species, including both the solvent and the mobile ions gives a set of equations that can be simultaneously used to solve for the equilibrium volume of the hydrogel, the osmotic pressure, and the Donnan potential, including their coupling. The model is used to study the thermodynamic properties of both pure and salty water in non-electrolyte and electrolyte hydrogels such as (1) the latent heat of evaporation, (2) the ability of hydrogels to retain water and to absorb water from the atmosphere, (3) the use of hydrogels for desalination via solar or forward osmosis, (4) the antifouling characteristics of hydrogels, and (5) melting point suppression and boiling point elevation, and solubility of salts in hydrogels. These properties are of interest in solar-driven interfacial water evaporation for desalination and wastewater treatment, atmospheric water harvesting, and forward osmosis. The reformulated thermodynamic framework will also be useful for understanding polymer electrolytes and ion transport in electrochemical and biological systems.

Tailoring intermolecular interactions in ion gels with rationally designed phosphonic acid polymers

Design strategies of phosphonic acid polymers established advanced ion gels with high ionic conductivity, mechanical strength, and self-healing ability via a configurable balance of ionic and hydrogen bonding interactions at the molecular level.

Polyelectrolyte Gels: Fundamentals, Fabrication and Applications

Polyelectrolyte gels are an important class of polymer gels and a versatile platform with charged polymer networks with ionisable groups. They have drawn significant recent attention as a class of smart material and have demonstrated potential for a variety of applications. This review begins with the fundamentals of polyelectrolyte gels, which encompass various classifications (i.e., origin, charge, shape) and crucial aspects (ionic conductivity and stimuli responsiveness). It further centralises recent developments of polyelectrolyte gels, emphasising their synthesis, structure-property relationships and responsive properties. Sequentially, this review demonstrates how polyelectrolyte gels' flourishing properties create attractiveness to a range of applications including tissue engineering, drug delivery, actuators and bioelectronics. Finally, the review outlines the indisputable appeal, further improvements and emerging trends in polyelectrolyte gels.

Universal description of steric hindrance in flexible polymer gels

In this work, the long-time diffusion of a solute in a chemically crosslinked and flexible hydrogel is computed from a bead-spring model of a polymeric network to assess the effect of steric obstruction. The relative diffusivities obtained for a wide variety of systems can be described by an exponential decay depending on a parameter that differs from that employed for rigid gels. The mathematical expression derived here can approximately predict the diffusivity in flexible gels if steric hindrance is the mechanism ruling diffusion.

Understanding the Interaction of Polyelectrolyte Architectures with Proteins and Biosystems

The counterions neutralizing the charges on polyelectrolytes such as DNA or heparin may dissociate in water and greatly influence the interaction of such polyelectrolytes with biomolecules, particularly proteins. In this Review we give an overview of studies on the interaction of proteins with polyelectrolytes and how this knowledge can be used for medical applications. Counterion release was identified as the main driving force for the binding of proteins to polyelectrolytes: Patches of positive charge become multivalent counterions of the polyelectrolyte and lead to the release of counterions from the polyelectrolyte and a concomitant increase in entropy. This is shown from investigations on the interaction of proteins with natural and synthetic polyelectrolytes. Special emphasis is paid to sulfated dendritic polyglycerols (dPGS). The Review demonstrates that we are moving to a better understanding of charge-charge interactions in systems of biological relevance. Research along these lines will aid and promote the design of synthetic polyelectrolytes for medical applications.

Computer modeling of polymer stars in variable solvent conditions: a comparison of MD simulations, self-consistent field (SCF) modeling and novel hybrid Monte Carlo SCF approach

Computer-aided modeling is a systematic approach to grasp the physics of macromolecules, but it remains essential to know when to trust the results and when not. For a polymer star, we consider three approaches: (i) Molecular Dynamics (MD) simulations and implementing a coarse-grained model, (ii) the self-consistent field approach based on a mean-field approximation and implementing the lattice model due to Scheutjens and Fleer (SF-SCF) and (iii) novel hybrid Monte Carlo self-consistent field (MC-SCF) method, which combines a coarse-grained model driven by a Monte Carlo method and a mean-field representation driven by SF-SCF. We compare the performance of these approaches under a wide range of solvent qualities. The MD approach is formally the most exact but suffers from reasonable convergence. The mean-field approach works similarly in all solvent qualities but is quantitatively least accurate. The MC-SCF hybrid allows us to combine the benefits of the simulation route and the effective performance of SCF. We consider the center-to-end distance R_ce, the radius of gyration R_g² of the star and the polymer density profiles φ(r) of polymer-segments in it. All three methods show a good qualitative agreement one to another. The MC-SCF method is in good agreement with the scaling predictions in the whole range of solvent quality values showing that it grasps the essential physics while remaining computationally in bounds.

Multivalent counterions accumulate in star-like polyelectrolytes and collapse the polymer in spite of increasing its ionization

We used computer simulations to explore the dissociative and conformational behaviour of branched weak polyelectrolytes with multivalent counterions. We compared simulated titration curves and chain sizes in the presence of added salt of various valencies, keeping the total charge of salt constant. We showed that multivalent counterions enhance ionization of the weak polyelectrolytes, in spite of collapsing of the chains. We provided evidence that such an effect is absent in systems with only monovalent counterions at the same ionic strength, and thus cannot be attributed to electrostatic screening. We attributed it to strong ion-ion correlations that we quantified by comparing potentials of mean force with the mean electrostatic potentials. Finally, we used the partition coefficient to quantify the ability of star-like polyelectrolytes to capture multivalent ions, that is important for water-treatment applications. Our work provides fundamental understanding of the mechanism of polyelectrolyte collapse and ionization response upon addition of multivalent ions.

Intramolecular micellization and nanopatterning in pH- and thermo-responsive molecular brushes

Conformational transitions and nanoscale self-organization triggered in double pH- and thermo-responsive molecular brushes by varying environmental conditions are studied by means of analytical mean-field theory and numerical Scheutjens-Fleer self-consistent field modelling. Such molecular brushes are composed of multiple thermo-responsive side chains end-grafted onto the main chain (backbone) and are capable of acquiring ionic charges via reversible (de)protonation of the monomer units. Competition of long-range Coulomb repulsion with short-range solvophobic interactions leads to complex patterns in the intramolecular self-organization of molecular brushes. In particular, we observed formation of pearl necklace-like structures with multiple dense nanodomains formed by weakly ionized collapsed side chains and stabilized by a fraction protruding into the solution and strongly ionized ones. Such structures are thermodynamically stable in a certain parameter range and can be termed as intramolecular micelles. The stimuli-induced intramolecular nanopatterning occurs via a sequence of quasi-first order phase transitions corresponding to splitting/fusion of collapsed domains accompanied by jumps in the average degree of ionization and macromolecular dimensions. A re-entrant sequence of transitions is observed when the salt concentration is used as a control parameter. These theoretical predictions provide guidelines for design of smart unimolecular devices, for example multicompartment nanocarriers of active substances or nanosensors.

Poly(sodium acrylate) hydrogels: synthesis of various network architectures, local molecular dynamics, salt partitioning, desalination and simulation

Various poly(sodium acrylate) hydrogels with different architectures, such as single networks, interpenetrating double networks and surface crosslinked hydrogels, are synthesized with a systematic change in their degree of crosslinking. The influence of these 3D structures on the absorbency of aqueous NaCl solutions is investigated. The local polymer mobility in water is probed in the form of transverse (T2) 1H-relaxation at a low field, which allowed confirming the structural aspects of the studied network topologies. Salt partitioning between the gel and the surrounding solution phase in NaCl solutions with an initial salt concentration of c0 = 0.017-0.60 mol L-1 (≙1-35 g L-1) is investigated. The data are compared with an idealized mean-field Donnan model, which fit the experimental findings only under the assumption of a drastically reduced effective charge density of feff ≈ 25 mol% independent of the hydrogel used. The unequal salt distribution allows desalination of salt water by applying an external pressure to a swollen hydrogel to recover its water which has a lower salinity. The specific energy needed to desalinate 1 m3 was estimated to be 6-18 kW h m-3. This value decreases with a lower degree of swelling independent of the network topology. Besides the experiments, simulations based on a Poisson-Boltzmann mean-field model and MD simulations are performed to determine the degree of swelling and salt partitioning as a function of c0 for different hydrogels. Both simulations describe qualitatively the experimental data, where deviations can be ascribed to model simplifications and the imperfect structure of the hydrogels synthesized via free radical polymerization.

Modeling Gel Swelling Equilibrium in the Mean Field: From Explicit to Poisson-Boltzmann Models

We develop a double mean-field theory for charged macrogels immersed in electrolyte solutions in the spirit of the cell model approach. We first demonstrate that the equilibrium sampling of a single explicit coarse-grained charged polymer in a cell yields accurate predictions of the swelling equilibrium if the geometry is suitably chosen and all pressure contributions have been incorporated accurately. We then replace the explicit flexible chain by a suitably modeled penetrable charged rod that allows us to compute all pressure terms within the Poisson-Boltzmann approximation. This model, albeit computationally cheap, yields excellent predictions of swelling equilibria under varying chain length, polymer charge fraction, and external reservoir salt concentrations when compared to coarse-grained molecular dynamics simulations of charged macrogels. We present an extension of the model to the experimentally relevant cases of pH-sensitive gels.

Simulations of ionization equilibria in weak polyelectrolyte solutions and gels

This article recapitulates the state of the art regarding simulations of ionization equilibria of weak polyelectrolyte solutions and gels. We start out by reviewing the essential thermodynamics of ionization and show how the weak polyelectrolyte ionization differs from the ionization of simple weak acids and bases. Next, we describe simulation methods for ionization reactions, focusing on two methods: the constant-pH ensemble and the reaction ensemble. After discussing the advantages and limitations of both methods, we review the existing simulation literature. We discuss coarse-grained simulations of weak polyelectrolytes with respect to ionization equilibria, conformational properties, and the effects of salt, both in good and poor solvent conditions. This is followed by a discussion of branched star-like weak polyelectrolytes and weak polyelectrolyte gels. At the end we touch upon the interactions of weak polyelectrolytes with other polymers, surfaces, nanoparticles and proteins. Although proteins are an important class of weak polyelectrolytes, we explicitly exclude simulations of protein ionization equilibria, unless they involve protein-polyelectrolyte interactions. Finally, we try to identify gaps and open problems in the existing simulation literature, and propose challenges for future development.

Influence of weak groups on polyelectrolyte mobilities

The ionization of dissociable groups in weak polyelectrolytes does not occur in a homogenous fashion. Monomer connectivity imposes constraints on the localization of the dissociated (charged) monomers that affect the local electric potential. As a result, the mean bare charge along a weak polyelectrolyte can vary depending on the proximity to topological features (e.g. presence of crosslinks or dangling ends). Using reaction-ensemble Monte-Carlo simulations we calculate the dissociation inhomogeneities for a few selected PE configurations, linear, rod-like, flexible four-arm star, and a star with stiff arms. An ensemble preaverage is used to obtain the annealed bare charge profile for these different polymer configurations. Using molecular dynamics simulations within a Lattice-Boltzman fluid, we investigate how the electrophoretic mobility is affected by the bare charge inhomogeneities arising from the annealed weak polyelectrolytes. Surprisingly, the mobility obtained for the situations corresponding to the predicted charge profile for annealed weak polyelectrolytes are not significantly different than the mobility obtained when all the monomers have an identical charge (under the constraint that the total polyelectrolyte bare charge is the same). This is also true for the stiff rod-like variants where conformational changes induced from the localization of the monomer charges are negligible. In salty solutions, we find that counterions are affected by the electric potential modulations induced by the topological features. Since the counterions crowd in regions where the electric potential caused by the dissociated monomers is highest, they wash-out the bare charge inhomogeneities and contribute to a more uniform effective backbone charge.

Local pH and Effective pK of a Polyelectrolyte Chain: Two Names for One Quantity?

In recent experiments, the "local pH" near polyelectrolyte chains was determined from the shift in the effective acidity constant of fluorescent pH indicators attached to the macromolecules. This indirect determination raises the question if the analyzed quantity was indeed the "local pH" and what this term actually means. In this study, we combined experiments and simulations to demonstrate that the shift in ionization constant is slightly lower than the difference between the pH and the "local pH". This offset is caused by correlations between fluctuations in chain conformation, small-ion distribution, and fluorophore ionization.

The Daoud and Cotton blob model and the interaction of star-shaped polymers

Since it was first proposed in 1982, the Daoud and Cotton (DC) model for star-shaped polymers was intensively used also for self-assembled copolymers and small colloids grafted with long polymers. We try to clarify the position of the DC model and focus on the star partition function which plays a central role in self-assembly and gives access to the star-star interaction. While the predicted star-star interaction agrees with scattering data by Likos et al. (Phys. Rev. Lett. 80, 4450 (1998)), an extensive simulation by Hsu et al. (Macromolecules, 37, 4658 (2004)) does not recover the prediction for the partition function. We try to reconcile this seemingly conflicting results. We discuss star-star interactions, star free energy in θ -solvents, mixing of A/B branches in copolymer stars, within or beyond the Daoud and Cotton blob model.

Effect of ionic strength and seawater cations on hagfish slime formation

The defensive slime of hagfish consists of a polyanionic mucin hydrogel that synergistically interacts with a fiber network forming a coherent and elastic hydrogel in high ionic strength seawater. In seawater, the slime deploys in less than a second entrapping large quantities of water by a well-timed thread skein unravelling and mucous gel swelling. This rapid and vast hydrogel formation is intriguing, as high ionic strength conditions generally counteract the swelling speed and ratio of polyelectrolyte hydrogels. In this work we investigate the effect of ionic strength and seawater cations on slime formation dynamics and functionality. In the absence of ionic strength skeins swell radially and unravel uncontrolled, probably causing tangling and creating a confined thread network that entraps limited water. At high ionic strength skeins unravel, but create a collapsed and dense fiber network. High ionic strength conditions therefore seem crucial for controlled skein unraveling, however not sufficient for water retention. Only the presence of naturally occurring Ca2+ or Mg2+-ions allowed for an expanded network and full water retention probably due to Ca2+-mediated vesicle rupture and cross-linking of the mucin. Our study demonstrates that hagfish slime deployment is a well-timed, ionic-strength, and divalent-cation dependent dynamic hydrogel formation process.

Monte Carlo simulations of weak polyelectrolyte microgels: pH-dependence of conformation and ionization

In this study, we investigated the effect of pH on single weak acidic polyelectrolyte microgels under salt-free conditions with (i) varying microgel concentration, (ii) varying content of acidic groups and (iii) different crosslinking densities using Monte Carlo simulations under explicit consideration of the protonation/deprotonation reaction. We assessed both global properties, such as the degree of ionization, the degree of swelling and the counterion distribution, and local properties such as the radial network ionization profile and the ionization along the polymer chains as a function of pH. We found a pronounced suppression of the pH-dependent ionization of the microgels, as compared to the ideal titration behavior and a shift of the titration curve to a higher pH originating in the proximity of acidic groups in the microgel. In contrast to macroscopic gels, counterions can leave the microgel, resulting in an effective charge of the network, which hinders the ionization. A decreasing microgel concentration leads to an increased effective charge of the microgel and a more pronounced shift of the titration curve. The number of acidic groups showed only a weak effect on the ionization behavior of the microgels. For two different microgels with different crosslinking densities, similar scaling of the gel size was observed. A distinct transition from an uncharged and unswollen to a highly charged and expanded polymer network was observed for all investigated microgels. The degree of swelling mainly depends on the degree of ionization. An inhomogeneous distribution of the degree of ionization along the radial profile of the microgel was found.

Maximizing the absorption of small cosolutes inside neutral hydrogels: steric exclusion versus hydrophobic adhesion

In this work the equilibrium absorption of nanometric cosolutes (which could represent drugs, reactants, small globular proteins and other kind of biomacromolecules) inside neutral hydrogels is studied. We specially focus on exploring, for different swelling states, the competition between the steric exclusion induced by the cross-linked polymer network constituting the hydrogel, and the solvent-induced short-range hydrophobic attraction between the polymer chains and the cosolute particle. For this purpose, the cosolute partition coefficient is calculated by means of coarse-grained grand canonical Monte Carlo simulations, and the results are compared to theoretical predictions based on the calculation of the excluded and binding volume around the polymer chains. For small hydrophobic attractions or large cosolute sizes, the steric repulsion dominates, and the partition coefficient decreases monotonically with the polymer volume fraction, ϕ_m. However, for large enough hydrophobic attraction strength, the interplay between hydrophobic adhesion and the steric exclusion leads to a maximum in the partition coefficient at certain intermediate polymer density. Good qualitative and quantitative agreement is achieved between simulation results and theoretical predictions in the limit of small ϕ_m, pointing out the importance of geometrical aspects of the cross-linked polymer network, even for hydrogels in the swollen state. In addition, the theory is able to predict analytically the onset of the maximum formation in terms of the details of the cosolute-monomer pair interaction, in good agreement with simulations too. Finally, the effect of the many-body attractions between the cosolute and multiple polymer chains is quantified. The results clearly show that these many-body attractions play a very relevant role determining the cosolute binding, enhancing its absorption in more than one order of magnitude.