Theory of volume transition in polyelectrolyte gels with charge regularization

Effect of counterion size on polyelectrolyte conformations and thermodynamics

We present a theoretical model to study the effect of counterion size on the effective charge, size, and thermodynamic behavior of a single, isolated, and flexible polyelectrolyte (PE) chain. We analyze how altering counterion size modifies the energy and entropy contributions to the system, including the ion-pair free energy, excluded volume interactions, entropy of free and condensed ions, and dipolar attraction among monomer-counterion pairs, which result in competing effects challenging intuitive predictions. The PE self-energy is calculated using the Edwards-Muthukumar Hamiltonian, considering a Gaussian monomer distribution for the PE. The condensed ions are assumed to be confined within a cylindrical volume around the PE backbone. The dipolar and excluded volume interactions are described by the second and third virial coefficients. The assumption of freely rotating dipoles results in a first-order coil-globule transition of the PE chain. A more realistic, weaker dipolar attraction, parameterized in our theory, shifts it to a second-order continuous transition. We calculate the size scaling-exponent of the PE and find exponents according to the relative dominance of the electrostatic, excluded volume, or dipolar effects. We further identify the entropy- and energy-driven regimes of the effective charge and conformation of the PE, highlighting the interplay of free ion entropy and ion-pair energy with varying electrostatic strengths. The crossover strength, dependent on the counterion size, indicates that diminishing sizes favor counterion condensation at the expense of free ion entropy. The predictions of the model are consistent with trends in simulations and generalize the findings of the point-like counterion theories.

Counterion-controlled phase equilibria in a charge-regulated polymer solution

We study phase equilibria in a minimal model of charge-regulated polymer solutions. Our model consists of a single polymer species whose charge state arises from protonation-deprotonation processes in the presence of a dissolved acid, whose anions serve as screening counterions. We explicitly account for variability in the polymers' charge states. Homogeneous equilibria in this model system are characterised by the total concentration of polymers, the concentration of counter-ions and the charge distributions of polymers which can be computed with the help of analytical approximations. We use these analytical results to characterise how parameter values and solution acidity influence equilibrium charge distributions and identify for which regimes uni-modal and multi-modal charge distributions arise. We then study the interplay between charge regulation, solution acidity and phase separation. We find that charge regulation has a significant impact on polymer solubility and allows for non-linear responses to the solution acidity: Re-entrant phase behaviour is possible in response to increasing solution acidity. Moreover, we show that phase separation can yield to the coexistence of local environments characterised by different charge distributions.

Polyelectrolyte complexation of two oppositely charged symmetric polymers: A minimal theory

Interplay of Coulomb interaction energy, free ion entropy, and conformational elasticity is a fascinating aspect in polyelectrolytes (PEs). We develop a theory for complexation of two oppositely charged PEs, a process known to be the precursor to the formation of complex coacervates in PE solutions, to explore the underlying thermodynamics of complex formation, at low salts. The theory considers general degrees of solvent polarity and dielectricity within an implicit solvent model, incorporating a varying Coulomb strength. Explicit calculation of the free energy of complexation and its components indicates that the entropy of free counterions and salt ions and the Coulomb enthalpy of bound ion-pairs dictate the equilibrium of PE complexation. This helps decouple the self-consistent dependency of charge and size of the uncomplexed parts of the polyions, derive an analytical expression for charge, and evaluate the free energy components as functions of chain overlap. Complexation is observed to be driven by enthalpy gain at low Coulomb strengths, driven by entropy gain of released counterions but opposed by enthalpy loss due to reduction of ion-pairs at moderate Coulomb strengths, and progressively less favorable due to enthalpy loss at even higher Coulomb strengths. The total free energy of the system is found to decrease linearly with an overlap of chains. Thermodynamic predictions from our model are in good quantitative agreement with simulations in literature.

Multivalent binding proteins can drive collapse and reswelling of chromatin in confinement

Collapsed conformations of chromatin have been long suspected of being mediated by interactions with multivalent binding proteins, which can bring together distant sections of the chromatin fiber. In this study, we use Langevin dynamics simulation of a coarse grained chromatin polymer to show that the role of binding proteins can be more nuanced than previously suspected. In particular, for chromatin polymer in confinement, entropic forces can drive reswelling of collapsed chromatin with increasing binder concentrations, and this reswelling transition happens at physiologically relevant binder concentrations. Both the extent of collapse, and also of reswelling depends on the strength of confinement. We also study the kinetics of collapse and reswelling and show that both processes occur in similar timescales. We characterise this reswelling of chromatin in biologically relevant regimes and discuss the non-trivial role of multivalent binding proteins in mediating the spatial organisation of the genome.

Dipole-driven interlude of mesomorphism in polyelectrolyte solutions

Uniformly charged polyelectrolyte molecules disperse uniformly in aqueous electrolyte solutions, due to electrostatic repulsion between them. In stark contrast to this well-established result of homogeneous polyelectrolyte solutions, we report a phenomenon where an aqueous solution of positively charged poly(L-lysine) (PLL) exhibits precipitation of similarly charged macromolecules at low ionic strength and a homogeneous solution at very high ionic strength, with a stable mesomorphic state of spherical aggregates as an interlude between these two limits. The precipitation at lower ionic strengths that is orthogonal to the standard polyelectrolyte behavior and the emergence of the mesomorphic state are triggered by the presence of a monovalent small organic anion, acrylate, in the electrolyte solution. Using light scattering, we find that the hydrodynamic radius R_h of isolated PLL chains shrinks upon a decrease in electrolyte (NaBr) concentration, exhibiting the "anti-polyelectrolyte effect." In addition, R_h of the aggregates in the mesomorphic state depends on PLL concentration c_p according to the scaling law, [Formula: see text]. Furthermore, at higher PLL concentration, the mesomorphic aggregates disassemble by a self-poisoning mechanism. We conjecture that all these findings can be attributed to both intra- and interchain dipolar interactions arising from the transformation of polycationic PLL into a physical polyzwitterionic PLL at higher concentrations of acrylate. The reported phenomenon of PLL exhibiting dipole-directed assembly of mesomorphic states and the anti-polyelectrolyte effect are of vital importance toward understanding more complex situations such as coacervation and formation of biomolecular condensates.

On the nature of screening in Voorn–Overbeek type theories

By using a recently formulated Legendre transform approach to the thermodynamics of charged systems, we explore the general form of the screening length in the Voorn–Overbeek-type theories, which remains valid also in the cases where the entropy of the charged component(s) is not given by the ideal gas form as in the Debye–Hückel theory. The screening length consistent with the non-electrostatic terms in the free energy ansatz for the Flory–Huggins and Voorn–Overbeek type theories, derived from the local curvature properties of the Legendre transform, has distinctly different behavior than the often invoked standard Debye screening length, though it reduces to it in some special cases.

Extracting free energies of counterion binding to polyelectrolytes by molecular dynamics simulations

We use all-atom molecular dynamics simulations to extract ΔG^eff, the free energy of binding of potassium ions K⁺ to the partially charged polyelectrolyte poly(acrylic acid), or PAA, in dilute regimes. Upon increasing the charge fraction of PAA, the chains adopt more extended conformations, and simultaneously, potassium ions bind more strongly (i.e., with more negative ΔG^eff) to the highly charged chains to relieve electrostatic repulsions between charged monomers along the chains. We compare the simulation results with the predictions of a model that describes potassium binding to PAA chains as a reversible reaction whose binding free energy (ΔG^eff) is adjusted from its intrinsic value (ΔG) by electrostatic correlations, captured by a random phase approximation. The bare or intrinsic binding free energy ΔG, which is an input in the model, depends on the binding species and is obtained from the radial distribution function of K⁺ around the charged monomer of a singly charged, short PAA chain in dilute solutions. We find that the model yields semi-quantitative predictions for ΔG^eff and the degree of potassium binding to PAA chains, α, as a function of PAA charge fraction without using fitting parameters.

Theory of Charged Gels: Swelling, Elasticity, and Dynamics

The fundamental attributes of charged hydrogels containing predominantly water and controllable amounts of low molar mass electrolytes are of tremendous significance in biological context and applications in healthcare. However, a rigorous theoretical formulation of gel behavior continues to be a challenge due to the presence of multiple length and time scales in the system which operate simultaneously. Furthermore, chain connectivity, the electrostatic interaction, and the hydrodynamic interaction all lead to long-range interactions. In spite of these complications, considerable progress has been achieved over the past several decades in generating theories of variable complexity. The present review presents an analytically tractable theory by accounting for correlations emerging from topological, electrostatic, and hydrodynamic interactions. Closed-form formulas are derived for charged hydrogels to describe their swelling equilibrium, elastic moduli, and the relationship between microscopic properties such as gel diffusion and macroscopic properties such as elasticity. In addition, electrostatic coupling between charged moieties and their ion clouds, which significantly modifies the elastic diffusion coefficient of gels, and various scaling laws are presented. The theoretical formulas summarized here are useful to adequately capture the essentials of the physics of charged gels and to design new hydrogels with specified elastic and dynamical properties.

Spatial segregation of mixed-sized counterions in dendritic polyelectrolytes

Langevin dynamics simulations are utilized to study the structure of a dendritic polyelectrolyte embedded in two component mixtures comprised of conventional (small) and bulky counterions. We vary two parameters that trigger conformational properties of the dendrimer: the reduced Bjerrum length, [Formula: see text], which controls the strength of electrostatic interactions and the number fraction of the bulky counterions, [Formula: see text], which impacts on their steric repulsion. We find that the interplay between the electrostatic and the counterion excluded volume interactions affects the swelling behavior of the molecule. As compared to its neutral counterpart, for weak electrostatic couplings the charged dendrimer exists in swollen conformations whose size remains unaffected by [Formula: see text]. For intermediate couplings, the absorption of counterions into the pervaded volume of the dendrimer starts to influence its conformation. Here, the swelling factor exhibits a maximum which can be shifted by increasing [Formula: see text]. For strong electrostatic couplings the dendrimer deswells correspondingly to [Formula: see text]. In this regime a spatial separation of the counterions into core-shell microstructures is observed. The core of the dendrimer cage is preferentially occupied by the conventional ions, whereas its periphery contains the bulky counterions.

Ion-Induced Volume Transition in Gels and Its Role in Biology

Incremental changes in ionic composition, solvent quality, and temperature can lead to reversible and abrupt structural changes in many synthetic and biopolymer systems. In the biological milieu, this nonlinear response is believed to play an important functional role in various biological systems, including DNA condensation, cell secretion, water flow in xylem of plants, cell resting potential, and formation of membraneless organelles. While these systems are markedly different from one another, a physicochemical framework that treats them as polyelectrolytes, provides a means to interpret experimental results and make in silico predictions. This article summarizes experimental results made on ion-induced volume phase transition in a polyelectrolyte model gel (sodium polyacrylate) and observations on the above-mentioned biological systems indicating the existence of a steep response.

Overcharging of polyelectrolyte complexes: an entropic phenomenon

Overcharging in complex coacervation, in which a polyelectrolyte complex coacervate (PEC) initially containing equal moles of the cationic and anionic monomers absorbs a large excess of one type of polyelectrolyte species, is predicted using a recently developed thermodynamic model describing complexation through a combination of reversible ion binding on the chains and long-range electrostatic correlations. We show that overcharging is favored roughly equally by the translational entropy of released counterions and the binding entropy of polyelectrolytes in the polyelectrolyte complex, thus helping resolve competing explanations for overcharging in the literature. We find that the extent of overcharging is non-monotonic in the concentration of added salt and increases with both strength of ion-pairing between polyions and chain hydrophobicity. The predicted extent of overcharging of the PEC is directly compared with that of multilayers made of poly(diallyldimethylammonium), PDADMA, and poly(styrene-sulfonate), PSS, overcompensated by the polycation in two different salts: KBr and NaCl. Accounting for the specificity of salt ion interactions with the polyelectrolytes, we find good qualitative agreement between theory and experiment.

Experimental Evidence for Universal Behavior of Ion-Induced Volume Phase Transition in Sodium Polyacrylate Gels

Introduction of high valence counterions into polyelectrolyte gels results in a reversible volume phase transition. In the present work new experimental results are reported for the volume transition induced by calcium/sodium exchange in sodium polyacrylate gels. The effects of cross-link density, concentration of ionized groups on the network chains, composition of the equilibrium salt solution containing both mono- and divalent cations, and temperature on the swelling degree of these gels are systematically investigated. It is demonstrated that the normalized swelling data fall on a master curve, indicating that the ion-exchange-induced volume transition exhibits universal behavior in sodium polyacrylate gels. Model calculations made on the basis of the classical Flory-Rehner theory are in reasonable agreement with the measured dependencies.

Crossover of the effective charge in ionic thermoresponsive hydrogel particles

We use a generalized nonlinear Poisson-Boltzmann cell model that includes excluded-volume effects to investigate whether the effective charge (Z_{eff}) of charged thermosensitive hydrogel particles increases or decreases upon the hydrogel thermal collapse. We find the existence of a crossover charge, Z^{*}, that separates two regimes. For hydrogel bare charges below Z^{*} the system shows a behavior consistent with theories based on linear approximations, i.e., Z_{eff} increases in the collapsed state. However, for bare charges above Z^{*}, the system enters an anomalous regime, in which Z_{eff} decreases in the collapsed state. We show that diluted hydrogel suspensions at low ionic strength are more likely to follow the anomalous behavior. Our theory provides a full physical justification for the controversial theoretical and experimental results reported in this regard, and describes how the interplay between electrostatic, excluded-volume and entropic effects affects this crossover.

Understanding the Thermodynamics of the Binding of PAMAM Dendrimers to Graphene: A Combined Analytical and Simulation Study

We investigate the thermodynamics of the binding of a poly(amidoamine) dendrimer to an uncharged graphene sheet as a function of the pH level using umbrella sampling simulations and a mean-field theory for generations three and four. We find that the dendrimer strongly binds to the graphene sheet ( O (100) kcal/mol) from our potential of mean force (PMF) calculations. In specific, we find that the dendrimer binds the most at neutral pH (∼7) and the least at low pH (∼4). We explain this nonmonotonic nature of the dendrimer's adsorption by studying the interactions contributing to the PMF, i.e., the dendrimer-graphene, dendrimer-water, and dendrimer-ion interactions. We also corroborate our PMF calculations with molecular mechanics generalized Born surface area analysis and free energies obtained from a mean-field theory of Flory-Huggins-Debye-Hückel type [ Muthukumar , M. , J. Chem. Phys. 2010 , 132 , 084901 ], including electrostatic interactions. We find that the van der Waals interactions between the dendrimer and the graphene alone cannot capture the accurate trends in the binding free energies (BEs) as a function of pH. The solvent and the counterions present in the system are also found to have a major influence on these trends. We demonstrate that the dendrimer-graphene and dendrimer-water interactions become favorable, whereas the dendrimer-ion interaction becomes unfavorable, as the dendrimer binds to graphene. These opposing effects lead to the observed nonmonotonicity in the BE trends. Our theoretical model also reproduces these trends in the subinteractions contributing to the PMF. To the best of our knowledge, this is a novel attempt where an equivalence between theory and simulations is made in the context of the dendrimer's adsorption.

Effects of mono- and divalent cations on the structure and thermodynamic properties of polyelectrolyte gels

Measurements are reported on the effect of monovalent and divalent salts on the swelling behavior and supramolecular structure of sodium polyacrylate gels (NaPA) made by osmotic swelling pressure and small angle neutron scattering measurements. The swelling response of the gels in solutions of ten different monovalent salts is found to be practically identical indicating that the principal effect of monovalent ions is screening the electrostatic repulsion among the charged groups on the polyelectrolyte chains; i.e., chemical differences between the monovalent ions do not play a significant role. Introducing Ca2+ ions into the equilibrium NaCl solution results in a sharp volume transition of the gels. The threshold Ca2+ ion concentration at which the transition occurs increases with increasing NaCl concentration in the surrounding bath. It is demonstrated that the swelling behavior of NaPA gels exhibits universal properties in solutions containing both NaCl and CaCl2. Osmotic swelling pressure measurements reveal that both the second and third virial coefficients decrease with increasing CaCl2 concentration until the volume transition is reached. The macroscopic measurements are complemented by small angle neutron scattering that reveals the variation of the thermodynamic length scales as the volume transition is approached. The thermodynamic correlation length L increases with increasing CaCl2 concentration.

A simple model for electrical charge in globular macromolecules and linear polyelectrolytes in solution

We present a model for calculating the net and effective electrical charge of globular macromolecules and linear polyelectrolytes such as proteins and DNA, given the concentration of monovalent salt and pH in solution. The calculation is based on a numerical solution of the non-linear Poisson-Boltzmann equation using a finite element discretized continuum approach. The model simultaneously addresses the phenomena of charge regulation and renormalization, both of which underpin the electrostatics of biomolecules in solution. We show that while charge regulation addresses the true electrical charge of a molecule arising from the acid-base equilibria of its ionizable groups, charge renormalization finds relevance in the context of a molecule’s interaction with another charged entity. Writing this electrostatic interaction free energy in terms of a local electrical potential, we obtain an “interaction charge” for the molecule which we demonstrate agrees closely with the “effective charge” discussed in charge renormalization and counterion-condensation theories. The predictions of this model agree well with direct high-precision measurements of effective electrical charge of polyelectrolytes such as nucleic acids and disordered proteins in solution, without tunable parameters. Including the effective interior dielectric constant for compactly folded molecules as a tunable parameter, the model captures measurements of effective charge as well as published trends of pKa shifts in globular proteins. Our results suggest a straightforward general framework to model electrostatics in biomolecules in solution. In offering a platform that directly links theory and experiment, these calculations could foster a systematic understanding of the interrelationship between molecular 3D structure and conformation, electrical charge and electrostatic interactions in solution. The model could find particular relevance in situations where molecular crystal structures are not available or rapid, reliable predictions are desired.

Effect of counterion excluded volume on the conformational behavior of polyelectrolyte chains

Conformational behavior of a single strongly charged polyelectrolyte chain in a dilute solution is studied by molecular dynamics simulations. The novel feature of the model is variation of the excluded volume of counterions for investigating its effect on the chain conformation, especially in low-polar media. It has been confirmed that the chain with conventional counterions collapses into a dense globule with increasing electrostatic interactions. However, if the counterions are bulky enough, they prevent the chain collapse even in media with strong electrostatic interactions. They stay bound in the vicinity of the backbone of the chain that adopts a swollen conformation. In this conformation, the scaling relation for the polymer dimensions with the chain length is the same as for neutral macromolecules in a good solvent, however the polyelectrolyte chain complexed with bulky counterions has a larger gyration radius than its uncharged analogue due to the excluded volume of the counterions contributing to the chain rigidity. Study of the counterion mobility has shown that, similar to the conventional counterions, the bulky counterions do not form stable ion pairs with ions on the polymer chain even in media with strong electrostatic interactions, but rather freely move along the chain backbone. In solutions containing mixtures of counterions with a bimodal size distribution, the conformations of linear polyelectrolytes depend considerably on the fraction of bulky counterions. Furthermore, a kind of intramolecular microphase separation can take place within a polyelectrolyte globule with the formation of a core-shell particle: the smaller counterions concentrate within the globular core while the bulkier counterions form a shell on the globule surface. The stability of the core-shell globule depends on the relative size of the counterions as well as their fractions in the solution. Thus, fine tuning of the balance between the counterion excluded volume and the electrostatic interactions opens new ways for controlling the conformational behavior of polyelectrolytes.

Charge-regularized swelling kinetics of polyelectrolyte gels: Elasticity and diffusion

We apply a recently developed method [S. Sen and A. Kundagrami, J. Chem. Phys. 143, 224904 (2015)], using a phenomenological expression of osmotic stress, as a function of polymer and charge densities, hydrophobicity, and network elasticity for the swelling of spherical polyelectrolyte (PE) gels with fixed and variable charges in a salt-free solvent. This expression of stress is used in the equation of motion of swelling kinetics of spherical PE gels to numerically calculate the spatial profiles for the polymer and free ion densities at different time steps and the time evolution of the size of the gel. We compare the profiles of the same variables obtained from the classical linear theory of elasticity and quantitatively estimate the bulk modulus of the PE gel. Further, we obtain an analytical expression of the elastic modulus from the linearized expression of stress (in the small deformation limit). We find that the estimated bulk modulus of the PE gel decreases with the increase of its effective charge for a fixed degree of deformation during swelling. Finally, we match the gel-front locations with the experimental data, taken from the measurements of charged reversible addition-fragmentation chain transfer gels to show an increase in gel-size with charge and also match the same for PNIPAM (uncharged) and imidazolium-based (charged) minigels, which specifically confirms the decrease of the gel modulus value with the increase of the charge. The agreement between experimental and theoretical results confirms general diffusive behaviour for swelling of PE gels with a decreasing bulk modulus with increasing degree of ionization (charge). The new formalism captures large deformations as well with a significant variation of charge content of the gel. It is found that PE gels with large deformation but same initial size swell faster with a higher charge.

Two regions of microphase separation in ion-containing polymer solutions

The phenomenon of spinodal decomposition in weakly charged polyelectrolyte solutions is studied theoretically within the random phase approximation. A novel feature of the theoretical approach is that it accounts for the effects of ionic association, i.e. ion pair and multiplet formation between counterions and ions in polymer chains, as well as the dependence of local dielectric permittivity on the polymer volume fraction Φ. The main focus is on the spinodal instability of polyelectrolyte solutions towards microscopic phase separation. It has been shown that increasing the binding energy of ions decreases the classical microphase separation region (possible at low polymer concentrations) due to the effective neutralization of the chains. A qualitatively new type of microphase separation is found in the presence of a dielectric mismatch between polymer and solvent. This new branch of microphase separation is realized at high polymer concentrations where ion association processes are the most pronounced. Typical microstructures are shown to have a period of a few nanometers like in ionomers. The driving force for the microphase formation of a new type is more favourable ion association in polymer-rich domains where ionomer-type behavior takes place. Effective attraction due to ion association promotes microscopic as well as macroscopic phase separation, even under good solvent conditions for uncharged monomer units of polymer chains. Polyelectrolyte-type behavior at low Φ and ionomer-type behavior at high Φ result in the presence of two critical points on the phase diagrams of polyelectrolyte solutions as well as two separate regions of possible microscopic structuring. Our predictions on the new type of microphase separation are supported by experimental data on polymer solutions, membranes and gels.

Ionic Strength Responsive Sulfonated Polystyrene Opals

Stimuli-responsive photonic crystals (PCs) represent an intriguing class of smart materials very promising for sensing applications. Here, selective ionic strength responsive polymeric PCs are reported. They are easily fabricated by partial sulfonation of polystyrene opals, without using toxic or expensive monomers and etching steps. The color of the resulting hydrogel-like ordered structures can be continuously shifted over the entire visible range (405-760 nm) by changing the content of ions over an extremely wide range of concentration (from about 70 μM to 4 M). The optical response is completely independent from pH and temperature, and the initial color can be fully recovered by washing the sulfonated opals with pure water. These new smart photonic materials could find important applications as ionic strength sensors for environmental monitoring as well as for healthcare screening.

Ordinary-extraordinary transition in dynamics of solutions of charged macromolecules

The occurrence of the ubiquitous and intriguing "ordinary-extraordinary" behavior of dynamics in solutions of charged macromolecules is addressed theoretically by explicitly considering counterions around the macromolecules. The collective and coupled dynamics of macromolecules and their counterion clouds in salt-free conditions are shown to lead to the "ordinary" behavior (also called the "fast" mode) where diffusion coefficients are independent of molar mass and polymer concentration and are comparable to those of isolated metallic ions in aqueous media, in agreement with experimental facts observed repeatedly over the past four decades. The dipoles arising from adsorbed counterions on polymer backbones can form many pairwise physical cross-links, leading to microgel-like aggregates. Balancing the swelling from excluded volume effects and counterion pressure with elasticity of the microgel, we show that there is a threshold value of a combination of polymer concentration and electrolyte concentration for the occurrence of the "extraordinary" phase (also called the "slow" mode) and the predicted properties of diffusion coefficient for this phase are in qualitative agreement with well-known experimental data.

Salt contribution to the flexibility of single-stranded nucleic acid offinite length

Nucleic acids are negatively charged macromolecules and their structure properties are strongly coupled to metal ions in solutions. In this article, the salt effects on the flexibility of single-stranded (ss) nucleic acid chain ranging from 12 to 120 nucleotides are investigated systematically by the coarse-grained Monte Carlo simulations where the salt ions are considered explicitly and the ss chain is modeled with the virtual-bond structural model. Our calculations show that, the increase of ion concentration causes the structural collapse of ss chain and multivalent ions are much more efficient in causing such collapse, and both trivalent/small divalent ions can induce more compact state than a random relaxation state. We found that monovalent, divalent, and trivalent ions can all overcharge ss chain, and the dominating source for such overcharging changes from ion-exclusion-volume effect to ion Coulomb correlations. In addition, the predicted Na(+) and Mg(2+)-dependent persistence length l(p)'s of ss nucleic acid are in accordance with the available experimental data, and through systematic calculations, we obtained the empirical formulas for l(p) as a function of [Na(+)], [Mg(2+)] and chain length.

Swelling of pH-sensitive hydrogels

A model is derived for the elastic response of polyelectrolyte gels subjected to unconstrained and constrained swelling. A gel is treated as a three-phase medium consisting of a solid phase (polymer network), solvent (water), and solutes (mobile ions). Transport of solvent and solutes is modeled as their diffusion through the network accelerated by an electric field formed by ions and accompanied by chemical reactions (dissociation of functional groups attached to the chains). Constitutive equations (including the van't Hoff law for ionic pressure and the Henderson-Hasselbach equation for ionization of chains) are derived by means of the free energy imbalance inequality. Good agreement is demonstrated between equilibrium swelling diagrams on several pH-sensitive gels and results of simulation. It is revealed that swelling of polyelectrolyte gels is driven by electrostatic repulsion of bound charges, whereas the effect of ionic pressure is of secondary importance.

Sorption of metal ions by poly(ethylene glycol)/β-CD hydrogels leads to gel-embedded metal nanoparticles

Inorganic nanoparticles can be embedded within gels by selectively preloading them with suitable molecular precursors followed by reduction or another suitable reaction. Here, we exploit the selective sorption properties of cross-linked β-cyclodextrin/poly(ethylene glycol) hydrogels, in analogy with polyurethane foams, to preconcentrate metal salts (HAuCl4 and K2[Co(SCN)4]) and subsequently generate gel-embedded metal nanoparticles (10-50 nm). The nanoparticles are shown to be immobilized within the gel network as a consequence of their large dimensions in comparison to the gel network pore size. We suggest this is a useful approach for the generalized synthesis of hybrid soft-hard nanomaterials.

Counterion adsorption theory of dilute polyelectrolyte solutions: apparent molecular weight, second virial coefficient, and intermolecular structure factor

Polyelectrolyte chains are well known to be strongly correlated even in extremely dilute solutions in the absence of additional strong electrolytes. Such correlations result in severe difficulties in interpreting light scattering measurements in the determination of the molecular weight, radius of gyration, and the second virial coefficient of charged macromolecules at lower ionic strengths from added strong electrolytes. By accounting for charge-regularization of the polyelectrolyte by the counterions, we present a theory of the apparent molecular weight, second virial coefficient, and the intermolecular structure factor in dilute polyelectrolyte solutions in terms of concentrations of the polymer and the added strong electrolyte. The counterion adsorption of the polyelectrolyte chains to differing levels at different concentrations of the strong electrolyte can lead to even an order of magnitude discrepancy in the molecular weight inferred from light scattering measurements. Based on counterion-mediated charge regularization, the second virial coefficient of the polyelectrolyte and the interchain structure factor are derived self-consistently. The effect of the interchain correlations, dominating at lower salt concentrations, on the inference of the radius of gyration and on molecular weight is derived. Conditions for the onset of nonmonotonic scattering wave vector dependence of scattered intensity upon lowering the electrolyte concentration and interpretation of the apparent radius of gyration are derived in terms of the counterion adsorption mechanism.