Langevin dynamics simulation of counterion distribution around isolated flexible polyelectrolyte chains

Dilute polyelectrolyte solutions: recent progress and open questions

Polyelectrolytes are a class of polymers possessing ionic groups on their repeating units. Since counterions can dissociate from the polymer backbone, polyelectrolyte chains are strongly influenced by electrostatic interactions. As a result, the physical properties of polyelectrolyte solutions are significantly different from those of electrically neutral polymers. The aim of this article is to highlight key results and some outstanding questions in the polyelectrolyte research from recent literature. We focus on the influence of electrostatics on conformational and hydrodynamic properties of polyelectrolyte chains. A compilation of experimental results from the literature reveals significant disparities with theoretical predictions. We also discuss a new class of polyelectrolytes called poly(ionic liquid)s that exhibit unique physical properties in comparison to ordinary polyelectrolytes. We conclude this review by listing some key research challenges in order to fully understand the conformation and dynamics of polyelectrolytes in solutions.

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.

Mechanical deformation affects the counterion condensation in highly-swollen polyelectrolyte hydrogels

Polyelectrolyte gels can generate electric potentials under mechanical deformation. While the underlying mechanism of such a response is often attributed to changes in counterion-condensation levels or alterations in the ionic conditions in the pervaded volume of the hydrogel, the exact molecular origins are largely unknown. By using all-atom molecular dynamics simulations of a polyacrylic acid hydrogel in explicit water as a model system, we simulate the uniaxial compression and uniaxial stretching of weakly to highly swollen (i.e., between 60-90% solvent content) hydrogel networks and calculate the microscopic condensation levels of counterions around the hydrogel chains. The counterion condensation under deformation is highly non-monotonic. Ionic condensation around the constituting chains of the deformed hydrogel tends to increase as the chains are stretched. This increase reaches a maximum and decreases as the chains are strongly stretched. The condensation around the collapsed chains of the hydrogel is weakly affected by the deformation. As a result, both compressing and stretching the model hydrogel lead to an overall increase in the counterion condensation. The effect vanishes for weakly swollen hydrogels, for which most ions are already condensed. The simulations with single, stretched polyelectrolyte chains show a qualitatively similar response, suggesting the effect of chain elongation on the ionic distribution throughout the hydrogel. Notably, this deformation-induced counterion condensation phenomenon does not occur in a polyelectrolyte solution at its critical concentration, indicating the role of hydrogel topology constraining the chain ends. Our results indicate that counterion condensation in a deforming polyelectrolyte hydrogel can be highly heterogeneous and exhibit a rich behaviour of electrostatic responses.

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.

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.

Explicit characterization of counterion dynamics around a flexible polyelectrolyte

The article presents a comprehensive study of counterion dynamics around a generic linear polyelectrolyte chain with the help of coarse-grained computer simulations. The ion-chain coupling is discussed in the form of binding time, mean-square displacement (MSD) relative to the chain, local ion transport coefficient, and spatiotemporal correlations in the effective charge. We have shown that a counterion exhibits subdiffusive behavior 〈δR^{2}〉∼t^{δ}, δ≈0.9 w.r.t. chain's center of mass. The MSD of ions perpendicularly outward from the chain segment exhibits a smaller subdiffusive exponent compared to the one along the chain backbone. The effective diffusivity of ion is the lowest in chain's close proximity, extending up to the length-scale of radius of gyration R_{g}. Beyond R_{g} at larger distances, they attain diffusivity of free ion with a smooth cross-over from the adsorbed regime to the free ion regime. We have shown that the effective diffusivity drastically decreases for the multivalent ions, while the crossover length scale remains the same. Conversely, with increasing salt concentration the coupling-length scale reduces, while the diffusivity remains unaltered. The effective diffusivity of adsorbed-ion reveals an exponential reduction with electrostatic interaction strength. We further corroborate this from the binding time of ions on the chain, which also grows exponentially with the coupling strength of the ion-polymer duo. Moreover, the binding time of ions exhibits a weak dependence with salt concentration for the monovalent salt, while for multivalent salts the binding time decreases dramatically with concentration. Our work also elucidates fluctuations in the effective charge per site, where it exhibits strong negative correlations at short length-scales.

The Role of Sulphonic and Phosphoric Pendant Groups on the Diffusion of Monovalent Ions in Polyelectrolyte Membranes: A Molecular Dynamics Study

Lithium-ion consumption has risen significantly in recent years due to its use in portable devices. Alternative sources of lithium, which include the recovery from brine using the sustainable and eco-friendly electrodialysis technology, has been explored. This technology, however, requires effective cation-exchange membranes that allow the selective permeation of lithium ions. In this study, we have investigated, via molecular dynamics simulations, the role of the two common charged groups, the sulfonic and the phosphoric groups, in promoting the adsorption of monovalent ions from brine comprising Li+, Na+, Mg2+, and Ca2+ ions. The analysis of the mean square displacement of the ions revealed that Li+ and Na+ ions exhibit superior diffusion behaviors within the polyelectrolyte system. The O-atoms of the charged groups bind strongly with the divalent ions (Mg2+ and Ca2+), which raises their diffusion energy barrier and consequently lowers their rate of permeation. In contrast, the monovalent ions exhibit weaker interactions, with Na+ being slightly above Li+, enabling the permeation of Li+ ions. The present study demonstrates the role of both charged groups in cation-exchange membranes in promoting the diffusion of Li+ and Na+ ions, and could serve as a guide for the design of effective membranes for the recovery of these ions from brine.

Recent progress in the science of complex coacervation

Complex coacervation is an associative, liquid-liquid phase separation that can occur in solutions of oppositely-charged macromolecular species, such as proteins, polymers, and colloids. This process results in a coacervate phase, which is a dense mix of the oppositely-charged components, and a supernatant phase, which is primarily devoid of these same species. First observed almost a century ago, coacervates have since found relevance in a wide range of applications; they are used in personal care and food products, cutting edge biotechnology, and as a motif for materials design and self-assembly. There has recently been a renaissance in our understanding of this important class of material phenomena, bringing the science of coacervation to the forefront of polymer and colloid science, biophysics, and industrial materials design. In this review, we describe the emergence of a number of these new research directions, specifically in the context of polymer-polymer complex coacervates, which are inspired by a number of key physical and chemical insights and driven by a diverse range of experimental, theoretical, and computational approaches.

Force driven transition of a globular polyelectrolyte

We have systematically studied behavior of a flexible polyelectrolyte (PE) chain with explicit counterions, subjected to a constant force at the terminal ends. Our simulations reveal that in the hydrophobic regime, a PE globule abruptly opens to a coil state beyond a critical force F_c. At the transition point, the polymer shape shows large scale fluctuations that are quantified in terms of end-to-end distance R_e. These fluctuations suggest that the system coexists in globule and coil states at the transition, which is also confirmed from the bimodal distribution of R_e. Moreover, the critical force associated with the globule coil transition exhibits a nonmonotonic behavior, where surprisingly, F_c decreases with Bjerrum length l_B in the limit of small l_B, followed by an increase in the larger l_B limit. Furthermore, this behavior is also validated from a theory adopted for the PE. From the free energy analysis, we have demonstrated that predominantly, the competition between the intrachain repulsive energy, counterion's translational entropy, and adsorption energy leads to the novel feature of nonmonotonic behavior of force.

Polyelectrolyte Conformation Controlled by a Trivalent-Rich Ion Jacket

The configuration of charged polymers is heavily dependent on interactions with surrounding salt ions, typically manifesting as a sensitivity to the bulk ionic strength. Here, we use single-molecule mechanical measurements to show that a charged polysaccharide, hyaluronic acid, shows a surprising regime of insensitivity to ionic strength in the presence of trivalent ions. Using simulations and theory, we propose that this is caused by the formation of a "jacket" of ions, tightly associated with the polymer, whose charge (and thus effect on configuration) is robust against changes in solution composition.

Designing Electrostatic Interactions via Polyelectrolyte Monomer Sequence

Charged polymers are ubiquitous in biological systems because electrostatic interactions can drive complicated structure formation and respond to environmental parameters such as ionic strength and pH. In these systems, function emerges from sophisticated molecular design; for example, intrinsically disordered proteins leverage specific sequences of monomeric charges to control the formation and function of intracellular compartments known as membraneless organelles. The role of a charged monomer sequence in dictating the strength of electrostatic interactions remains poorly understood despite extensive evidence that sequence is a powerful tool biology uses to tune soft materials. In this article, we use a combination of theory, experiment, and simulation to establish the physical principles governing sequence-driven control of electrostatic interactions. We predict how arbitrary sequences of charge give rise to drastic changes in electrostatic interactions and correspondingly phase behavior. We generalize a transfer matrix formalism that describes a phase separation phenomenon known as "complex coacervation" and provide a theoretical framework to predict the phase behavior of charge sequences. This work thus provides insights into both how charge sequence is used in biology and how it could be used to engineer properties of synthetic polymer systems.

Tuning the collapse transition of weakly charged polymers by ion-specific screening and adsorption

The experimentally observed swelling and collapse response of weakly charged polymers to the addition of specific salts displays quite convoluted behavior that is not easy to categorize. Here we use a minimalistic implicit-solvent/explicit-salt simulation model with a focus on ion-specific interactions between ions and a single weakly charged polyelectrolyte to qualitatively explain the observed effects. In particular, we demonstrate ion-specific screening and bridging effects cause collapse at low salt concentrations whereas the same strong ion-specific direct interactions drive re-entrant swelling at high concentrations. Consistently with experiments, a distinct salt concentration at which the salting-out power of anions inverts from the reverse to direct Hofmeister series is observed. At this so called isospheric point, the ion-specific effects vanish. Furthermore, with additional simplifying assumptions, an ion-specific mean-field model is developed for the collapse transition which quantitatively agrees with the simulations. Our work demonstrates the sensitivity of the structural behavior of charged polymers to the addition of specific salt beyond simple screening and shall be useful for further guidance of experiments.

Interfacial properties of polymeric complex coacervates from simulation and theory

Polymeric complex coacervation occurs when two oppositely charged polyelectrolytes undergo an associative phase separation in aqueous salt solution, resulting in a polymer-dense coacervate phase and a polymer-dilute supernatant phase. This phase separation process represents a powerful way to tune polymer solutions using electrostatic attraction and is sensitive to environmental conditions such as salt concentration and valency. One area of particular research interest is using this to create nanoscale polymer assemblies, via (for example) block copolymers with coacervate-forming blocks. The key to understanding coacervate-driven assembly is the formation of the interface between the coacervate and supernatant phases and its corresponding thermodynamics. In this work, we use recent advances in coacervate simulation and theory to probe the nature of the coacervate-supernatant interface. First, we show that self-consistent field theory informed by either Monte-Carlo simulations or transfer matrix theories is capable of reproducing interfacial features present in large-scale molecular dynamics simulations. The quantitative agreement between all three methods gives us a way to efficiently explore interfacial thermodynamics. We show how salt affects the interface, and we find qualitative agreement with literature measurements of interfacial tension. We also explore the influence of neutral polymers, which we predict to drastically influence the phase behavior of coacervates. These neutral polymers can significantly alter the interfacial tension in coacervates; this has a profound effect on the design and understanding of coacervate-driven self-assembly, where the equilibrium structure is tied to interfacial properties.

Individual circular polyelectrolytes under shear flow

Individual circular polyelectrolytes in simple shear flow are studied by means of mesoscale hydrodynamic simulations, revealing the complex coupling effects of shear rate, electrostatic interaction, and circular architecture on their conformational and dynamical properties. Shear flow deforms the polyelectrolyte and strips condensed counterions from its backbone. A decrease in condensed counterions alters electrostatic interactions among charged particles, affecting shear-induced polymer deformation and orientation. Circular architecture determines the features of deformation and orientation. At weak electrostatic interaction strengths, the polyelectrolyte changes its shape from an oblate ring at small shear rates to a prolate ring at large shear rates, whereas strong electrostatic interaction strengths are associated with a transition from a prolate coil to a prolate ring. Circular polyelectrolytes exhibit tumbling and tank-treading motions in the range of large shear rates. Further study reveals a similarity between the roles of intramolecular electrostatic repulsion and chain rigidity in shear-induced dynamics.

Effect of grafting density on conformation of poly(acrylic acid) in solution by dielectric spectroscopy

The effect of grafting density of poly(ethylene oxide) and dodecyl groups on the conformation of poly(acrylic acid) in solution was clarified by dielectric relaxation spectroscopy over a frequency range from 40 Hz to 110 MHz. Two distinct dielectric relaxations were found after the elimination of electrode polarization, and valuable information about the conformations and interfacial electrokinetic properties of molecules was obtained by analyzing the dielectric spectra based on a refined double-layer polarization model. The critical aggregation concentration was determined by the concentration dependency of dielectric parameters. The results based on zeta potential suggested that the ionization performance and behavior of counterion condensation were strongly influenced by the grafting density of poly(ethylene oxide) and dodecyl groups. The concentration dependency of correlation length, ratio of zeta potential, ratio of linear density of counterions, and the Debye length showed that the chain length and degree of intermolecular aggregation were also strongly influenced by the grafting density. We revealed the role of grafting density with respect to electrostatic interaction in determining the chain conformation of polyelectrolytes in solution.

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.

A “turn-off” red-emitting fluorophore for nanomolar detection of heparin

A simple fluorophore bearing a diethylaminocoumarin donor and a pyridinium acceptor was synthesized and utilized for the ultra-sensitive detection of heparin.

50th Anniversary Perspective: A Perspective on Polyelectrolyte Solutions

From the beginning of life with the information-containing polymers until the present era of a plethora of water-based materials in health care industry and biotechnology, polyelectrolytes are ubiquitous with a broad range of structural and functional properties. The main attribute of polyelectrolyte solutions is that all molecules are strongly correlated both topologically and electrostatically in their neutralizing background of charged ions in highly polarizable solvent. These strong correlations and the necessary use of numerous variables in experiments on polyelectrolytes have presented immense challenges toward fundamental understanding of the various behaviors of charged polymeric systems. This Perspective presents the author's subjective summary of several conceptual advances and the remaining persistent challenges in the contexts of charge and size of polymers, structures in homogeneous solutions, thermodynamic instability and phase transitions, structural evolution with oppositely charged polymers, dynamics in polyelectrolyte solutions, kinetics of phase separation, mobility of charged macromolecules between compartments, and implications to biological systems.

Transfer matrix theory of polymer complex coacervation

Oppositely charged polyelectrolytes can undergo a macroscopic, associative phase separation in solution, via a process known as complex coacervation. Significant recent effort has gone into providing a clear, physical picture of coacervation; most work has focused on improving the field theory picture that emerged from the classical Voorn-Overbeek theory. These methods have persistent issues, however, resolving the molecular features that have been shown to play a major role in coacervate thermodynamics. In this paper, we outline a theoretical approach to coacervation based on a transfer matrix formalism that is an alternative to traditional field-based approaches. We develop theoretical arguments informed by experimental observation and simulation, which serve to establish an analytical expression for polymeric complex coacervation that is consistent with the molecular features of coacervate phases. The analytical expression provided by this theory is in a form that can be incorporated into more complicated theoretical or simulation formalisms, and thus provides a starting point for understanding coacervate-driven self-assembly or biophysics.

Facilitated dissociation of transcription factors from single DNA binding sites

The binding of transcription factors (TFs) to DNA controls most aspects of cellular function, making the understanding of their binding kinetics imperative. The standard description of bimolecular interactions posits that TF off rates are independent of TF concentration in solution. However, recent observations have revealed that proteins in solution can accelerate the dissociation of DNA-bound proteins. To study the molecular basis of facilitated dissociation (FD), we have used single-molecule imaging to measure dissociation kinetics of Fis, a key Escherichia coli TF and major bacterial nucleoid protein, from single dsDNA binding sites. We observe a strong FD effect characterized by an exchange rate [Formula: see text], establishing that FD of Fis occurs at the single-binding site level, and we find that the off rate saturates at large Fis concentrations in solution. Although spontaneous (i.e., competitor-free) dissociation shows a strong salt dependence, we find that FD depends only weakly on salt. These results are quantitatively explained by a model in which partially dissociated bound proteins are susceptible to invasion by competitor proteins in solution. We also report FD of NHP6A, a yeast TF with structure that differs significantly from Fis. We further perform molecular dynamics simulations, which indicate that FD can occur for molecules that interact far more weakly than those that we have studied. Taken together, our results indicate that FD is a general mechanism assisting in the local removal of TFs from their binding sites and does not necessarily require cooperativity, clustering, or binding site overlap.

Conformation transitions of a single polyelectrolyte chain in a poor solvent: a replica-exchange lattice Monte-Carlo study

The thermodynamic behaviors of a strongly charged polyelectrolyte chain in a poor solvent are studied using replica-exchange Monte-Carlo simulations on a lattice model, focusing on the effects of finite chain length and the solvent quality on the chain conformation and conformation transitions. The neutralizing counterions and solvent molecules are considered explicitly. The thermodynamic quantities that vary continuously with temperature over a wide range are computed using the multiple histogram reweighting method. Our results suggest that the strength of the short-range hydrophobic interaction, the chain length, and the temperature of the system, characterized by ε, N, and T, respectively, are important parameters that control the conformations of a charged chain. When ε is moderate, the competition between the electrostatic energy and the short-range hydrophobic interaction leads to rich conformations and conformation transitions for a longer chain with a fixed length. Our results have unambiguously demonstrated the stability of the n-pearl-necklace structures, where n has a maximum value and decreases with decreasing temperature. The maximum n value increases with increasing chain length. Our results have also demonstrated the first-order nature of the conformation transitions between the m-pearl and the (m-1)-pearl necklaces. With the increase of ε, the transition temperature increases and the first-order feature becomes more pronounced. It is deduced that at the thermodynamic limit of infinitely long chain length, the conformational transitions between the m-pearl and the (m-1)-pearl necklaces may remain first order when ε > 0 and m = 2 or 3. Pearl-necklace conformations cannot be observed when either ε is too large or N is too small. To observe a pearl-necklace conformation, the T value needs to be carefully chosen for simulations performed at only a single temperature.

Potential of mean force and transient states in polyelectrolyte pair complexation

The pair association between two polyelectrolytes (PEs) of the same size but opposite charge is systematically studied in terms of the potential of mean force (PMF) along their center-of-mass reaction coordinate via coarse-grained, implicit-solvent, explicit-salt computer simulations. The focus is set on the onset and the intermediate transient stages of complexation. At conditions above the counterion-condensation threshold, the PE association process exhibits a distinct sliding-rod-like behavior where the polymer chains approach each other by first stretching out at a critical distance close to their contour length, then "shaking hand" and sliding along each other in a parallel fashion, before eventually folding into a neutral complex. The essential part of the PMF for highly charged PEs can be very well described by a simple theory based on sliding charged "Debye-Hückel" rods with renormalized charges in addition to an explicit entropy contribution owing to the release of condensed counterions. Interestingly, at the onset of complex formation, the mean force between the PE chains is found to be discontinuous, reflecting a bimodal structural behavior that arises from the coexistence of interconnected-rod and isolated-coil states. These two microstates of the PE complex are balanced by subtle counterion release effects and separated by a free-energy barrier due to unfavorable stretching entropy.

Simulation of self-assembly of polyzwitterions into vesicles

Using the Langevin dynamics method and a coarse-grained model, we have studied the formation of vesicles by hydrophobic polymers consisting of periodically placed zwitterion side groups in dilute salt-free aqueous solutions. The zwitterions, being permanent charge dipoles, provide long-range electrostatic correlations which are interfered by the conformational entropy of the polymer. Our simulations are geared towards gaining conceptual understanding in these correlated dipolar systems, where theoretical calculations are at present formidable. A competition between hydrophobic interactions and dipole-dipole interactions leads to a series of self-assembled structures. As the spacing d between the successive zwitterion side groups decreases, single chains undergo globule → disk → worm-like structures. We have calculated the Flory-Huggins χ parameter for these systems in terms of d and monitored the radius of gyration, hydrodynamic radius, spatial correlations among hydrophobic and dipole monomers, and dipole-dipole orientational correlation functions. During the subsequent stages of self-assembly, these structures lead to larger globules and vesicles as d is decreased up to a threshold value, below which no large scale morphology forms. The vesicles form via a polynucleation mechanism whereby disk-like structures form first, followed by their subsequent merger.

Development of the modern theory of polymeric complex coacervation

Oppositely charged polymers can undergo the process of complex coacervation, which refers to a liquid-liquid phase separation driven by electrostatic attraction. These materials have demonstrated considerable promise as the basis for complex, self-assembled materials. In this review, we provide a broad overview of the theoretical tools used to understand the physical properties of polymeric coacervates. In particular, we discuss historic theories (Voorn-Overbeek, Random Phase Approximation), and then describe recent developments in the field (Field Theoretic, Counterion Release, Molecular Simulation, and Polymer Reference Interaction Site Model methods). We provide context for these methods, and map out the patchwork of theoretical models that are used to describe a diverse array of coacervate systems. We use this review of the literature to clarify a number of important theoretical challenges remaining in our physical understanding of complex coacervation.

Electrostatic Correlations in Polyelectrolyte Solutions

The major attribute of polyelectrolyte solutions is that all chains are strongly correlated both electrostatically and topologically. Even in very dilute solutions such that the chains are not interpenetrating, the chains are still strongly correlated. These correlations are manifest in the measured scattering intensity when such solutions are subjected to light, X-ray, and neutron radiation. The behavior of scattering intensity from polyelectrolyte solutions is qualitatively different from that of solutions of uncharged polymers. Using the technique introduced by Sir Sam Edwards, and extending the earlier work by the author on the thermodynamics of polyelectrolyte solutions, extrapolation formulas are derived for the scattering intensity from polyelectrolyte solutions. The emergence of the polyelectrolyte peak and its concentration dependence are derived. The derived theory shows that there are five regimes. Published experimental data from many laboratories are also collected into a master figure and a comparison between the present theory and experiments is presented.

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.

Modeling competitive substitution in a polyelectrolyte complex

We have simulated the invasion of a polyelectrolyte complex made of a polycation chain and a polyanion chain, by another longer polyanion chain, using the coarse-grained united atom model for the chains and the Langevin dynamics methodology. Our simulations reveal many intricate details of the substitution reaction in terms of conformational changes of the chains and competition between the invading chain and the chain being displaced for the common complementary chain. We show that the invading chain is required to be sufficiently longer than the chain being displaced for effecting the substitution. Yet, having the invading chain to be longer than a certain threshold value does not reduce the substitution time much further. While most of the simulations were carried out in salt-free conditions, we show that presence of salt facilitates the substitution reaction and reduces the substitution time. Analysis of our data shows that the dominant driving force for the substitution process involving polyelectrolytes lies in the release of counterions during the substitution.

Spatial Rearrangement and Mobility Heterogeneity of an Anionic Lipid Monolayer Induced by the Anchoring of Cationic Semiflexible Polymer Chains

We use Monte Carlo simulations to investigate the interactions between cationic semiflexible polymer chains and a model fluid lipid monolayer composed of charge-neutral phosphatidyl-choline (PC), tetravalent anionic phosphatidylinositol 4,5-bisphosphate (PIP₂), and univalent anionic phosphatidylserine (PS) lipids. In particular, we explore how chain rigidity and polymer concentration influence the spatial rearrangement and mobility heterogeneity of the monolayer under the conditions where the cationic polymers anchor on the monolayer. We find that the anchored cationic polymers only sequester the tetravalent PIP₂ lipids at low polymer concentrations, where the interaction strength between the polymers and the monolayer exhibits a non-monotonic dependence on the degree of chain rigidity. Specifically, maximal anchoring occurs at low polymer concentrations, when the polymer chains have an intermediate degree of rigidity, for which the PIP₂ clustering becomes most enhanced and the mobility of the polymer/PIP₂ complexes becomes most reduced. On the other hand, at sufficiently high polymer concentrations, the anchoring strength decreases monotonically as the chains stiffen-a result that arises from the pronounced competitions among polymer chains. In this case, the flexible polymers can confine all PIP₂ lipids and further sequester the univalent PS lipids, whereas the stiffer polymers tend to partially dissociate from the monolayer and only sequester smaller PIP₂ clusters with greater mobilities. We further illustrate that the mobility gradient of the single PIP₂ lipids in the sequestered clusters is sensitively modulated by the cooperative effects between anchored segments of the polymers with different rigidities. Our work thus demonstrates that the rigidity and concentration of anchored polymers are both important parameters for tuning the regulation of anionic lipids.

Counter-ion distribution around flexible polyelectrolytes having different molecular architecture

We explore the monovalent counter-ion distribution around flexible highly-charged polyelectrolytes with different molecular architectures (linear chains, stars, and unknotted and trefoil rings) using molecular dynamics simulations that include an explicit solvent that interacts with the polyelectrolyte. In particular, we find that the molecular topology influences the fraction of counter-ions transiently associating with the polyelectrolyte on a scale of the order of the chain segments, forming a "condensed" counter-ion interfacial layer. As with the hydrogen bonding of water to proteins and other polymers, the persistence time of these interfacial "bound" counter-ions is relatively short, O(1 ps), and we characterize the fluctuations in the number of the counter-ions populating the interfacial layer. We also find that the counter-ions are distributed in a non-uniform fashion on the polyelectrolyte backbone, forming dynamical clusters whose form and average size is sensitive to molecular architecture. In addition, we find that the residual bound counter-ions, not located in either the interfacial layer or the bulk solution, form a diffuse ionic cloud around the polyelectrolyte due to the uncompensated polyelectrolyte charge along the backbone. Generally charge valence strongly influences the extent of the diffuse counter-ion cloud, but in the case of monovalent counter-ions, we find that the size of the diffuse counter-ion cloud nearly coincides with the polyelectrolyte radius of gyration, independent of molecular topology.

Morphology-enhanced conductivity in dry ionic liquids

The size polarity and tail stiffness of amphiphilic ionic liquid molecules can be tailored to obtain 3D continuous ionic channels possessing isotropic conductivities.

Electrophoretic mobilities of counterions and a polymer in cylindrical pores

We have simulated the transport properties of a uniformly charged flexible polymer chain and its counterions confined inside cylindrical nanopores under an external electric field. The hydrodynamic interaction is treated by describing the solvent molecules explicitly with the multiparticle collision dynamics method. The chain consisting of charged monomers and the counterions interact electrostatically with themselves and with the external electric field. We find rich behavior of the counterions around the polymer under confinement in the presence of the external electric field. The mobility of the counterions is heterogeneous depending on their location relative to the polymer. The adsorption isotherm of the counterions on the polymer depends nonlinearly on the electric field. As a result, the effective charge of the polymer exhibits a sigmoidal dependence on the electric field. This in turn leads to a nascent nonlinearity in the chain stretching and electrophoretic mobility of the polymer in terms of their dependence on the electric field. The product of the electric field and the effective polymer charge is found to be the key variable to unify our simulation data for various polymer lengths. Chain extension and the electrophoretic mobility show sigmoidal dependence on the electric field, with crossovers from the linear response regime to the nonlinear regime and then to the saturation regime. The mobility of adsorbed counterions is nonmonotonic with the electric field. For weaker and moderate fields, the adsorbed counterions move with the polymer and at higher fields they move opposite to the polymer's direction. We find that the effective charge and the mobility of the polymer decrease with a decrease in the pore radius.