Electrostatic correlations and the polyelectrolyte self energy

Ion Correlation-Driven Hysteretic Adhesion and Repulsion between Opposing Polyelectrolyte Brushes

Polyelectrolyte (PE) brushes are widely used in biomaterials and nanotechnology to regulate the surface properties and interactions. Here, we apply the electrostatic correlation augmented self-consistent field theory to investigate the interactions between opposing PE brushes in a mixture of 1:1 and 3:1 salt solutions. Our theory predicts a hysteretic feature of the normal stress induced by strong ion correlations. In the presence of trivalent ions, the force profile is discontinuous: repulsive in the compression branch and adhesive in the separation branch. The molecular origin of the hysteretic force is the coexistence of two collapsed modes: two separated condensed layers on each surface in the compression and a single bundled condensed layer in the separation. With the systematic inclusion of ion correlations, our theory captures well the hysteretic force, adhesive separation, "jump-in" and "jump-out" features, and the "specific ion effect", all in good agreement with the reported experimental results.

3D Printing of Aqueous Two-Phase Systems with Linear and Bottlebrush Polyelectrolytes

We formed core-shell-like polyelectrolyte complexes (PECs) from an anionic bottlebrush polymer with poly (acrylic acid) side chains with a cationic linear poly (allylamine hydrochloride). By varying the pH, the number of side chains of the polyanionic BB polymers (Nbb), the charge density of the polyelectrolytes, and the salt concentration, the phase separation behavior and salt resistance of the complexes could be tuned by the conformation of the BBs. By combining the linear/bottlebrush polyelectrolyte complexation with all-liquid 3D printing, flow-through tubular constructs were produced that showed selective transport across the PEC membrane comprising the walls of the tubules. These tubular constructs afford a new platform for flow-through delivery systems.

Statistical field theory of polarizable polymer chains with nonlocal dipolar interactions

The electromechanical response of polymeric soft matter to applied electric fields is of fundamental scientific interest as well as relevant to technologies for sensing and actuation. Several existing theoretical and numerical approaches for polarizable polymers subject to a combined applied electric field and stretch are based on discrete monomer models. In these models, accounting for the interactions between the induced dipoles on monomers is challenging due to the nonlocality of these interactions. On the other hand, the framework of statistical field theory provides a continuous description of polymer chains that potentially enables a tractable way to account for these interactions. However, prior formulations using this framework have been restricted to the case of weak anisotropy of the monomer polarizability. This paper formulates a general approach based in the framework of statistical field theory to account for the nonlocal nature of the dipolar interactions without any restrictions on the anisotropy or nonlinearity of the polarizability of the monomer. The approach is based on three key elements: (1) the statistical field theory framework, in which the discrete monomers are regularized to a continuous dipole distribution, (2) a replacement of the nonlocal dipole-dipole interactions by the local electrostatics partial differential equation with the continuous dipole distribution as the forcing, and (3) the use of a completely general relation between the polarization and the local electric field. Rather than treat the dipole-dipole interactions directly, the continuous description in the field theory enables the computationally tractable nonlocal-to-local transformation. Further, it enables the use of a realistic statistical-mechanical ensemble wherein the average far-field applied electric field is prescribed, rather than prescribing the applied field at every point in the polymer domain. The model is applied, using the finite element method, to study the electromechanical response of a polymer chain in the ensemble with fixed far-field applied electric field and fixed chain stretch. The nonlocal dipolar interactions are found to increase, over the case where dipole-dipole interactions are neglected, the magnitudes of the polarization and electric field by orders of magnitude as well as significantly change their spatial distributions. Next, the effect of the relative orientation between the applied field and the chain on the local electric field and polarization is studied. The model predicts that the elastic response of the polymer chain is linear, consistent with the Gaussian approximation, and largely unchanged by the orientation of the applied electric field, though the polarization and local electric field distributions are significantly impacted.

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.

Variational field theory of macroscopic forces in coulomb fluids

Based on the variational field theory framework, we extend our previous mean-field formalism [Y. A. Budkov and A. L. Kolesnikov, JStatMech 2022, 053205.2022], taking into account the electrostatic correlations of the ions. We employ a general covariant approach and derive a total stress tensor that considers the electrostatic correlations of ions. This is accomplished through an additional term that depends on the autocorrelation function of the local electric field fluctuations. Utilizing the derived total stress tensor and applying the mechanical equilibrium condition, we establish a general expression for the disjoining pressure of the Coulomb fluids, confined in a pore with a slit-like geometry. Using this equation, we derive an asymptotic expression for the disjoining pressure in a slit-like pore with non-electrified conductive walls. Present theory is the basis for future modeling of the mechanical stresses that occur in electrode pores with conductive charged walls, immersed in liquid phase electrolytes beyond the mean-field theory.

Electroresponse of weak polyelectrolyte brushes

End-tethered polyelectrolytes are widely used to modify substrate properties, particularly for lubrication or wetting. External stimuli, such as pH, salt concentration, or an electric field, can induce profound structural responses in weak polyelectrolyte brushes, which can be utilized to further tune substrate properties. We study the structure and electroresponsiveness of weak polyacid brushes using an inhomogeneous theory that incorporates both electrostatic and chain connectivity correlations at the Debye-Hückel level. Our calculation shows that a weak polyacid brush swells under the application of a negative applied potential, in agreement with recent experimental observation. We rationalize this behavior using a scaling argument that accounts for the effect of the surface charge. We also show that the swelling behavior has a direct influence on the differential capacitance, which can be modulated by the solvent quality, pH, and salt concentration.

A molecularly informed field-theoretic study of the complexation of polycation PDADMA with mixed micelles of sodium dodecyl sulfate and ethoxylated surfactants

The self-assembly and phase separation of mixtures of polyelectrolytes and surfactants are important to a range of applications, from formulating personal care products to drug encapsulation. In contrast to systems of oppositely charged polyelectrolytes, in polyelectrolyte-surfactant systems the surfactants micellize into structures that are highly responsive to solution conditions. In this work, we examine how the morphology of micelles and degree of polyelectrolyte adsorption dynamically change upon varying the mixing ratio of charged and neutral surfactants. Specifically, we consider a solution of the cationic polyelectrolyte polydiallyldimethylammonium, anionic surfactant sodium dodecyl sulfate, neutral ethoxylated surfactants (C[Formula: see text]EO[Formula: see text]), sodium chloride salt, and water. To capture the chemical specificity of these species, we leverage recent developments in constructing molecularly informed field theories via coarse-graining from all-atom simulations. Our results show how changing the surfactant mixing ratios and the identity of the nonionic surfactant modulates micelle size and surface charge, and as a result dictates the degree of polyelectrolyte adsorption. These results are in semi-quantitative agreement with experimental observations on the same system.

Numerical Techniques for Applications of Analytical Theories to Sequence-Dependent Phase Separations of Intrinsically Disordered Proteins

Biomolecular condensates, physically underpinned to a significant extent by liquid-liquid phase separation (LLPS), are now widely recognized by numerous experimental studies to be of fundamental biological, biomedical, and biophysical importance. In the face of experimental discoveries, analytical formulations emerged as a powerful yet tractable tool in recent theoretical investigations of the role of LLPS in the assembly and dissociation of these condensates. The pertinent LLPS often involves, though not exclusively, intrinsically disordered proteins engaging in multivalent interactions that are governed by their amino acid sequences. For researchers interested in applying these theoretical methods, here we provide a practical guide to a set of computational techniques devised for extracting sequence-dependent LLPS properties from analytical formulations. The numerical procedures covered include those for the determination of spinodal and binodal phase boundaries from a general free energy function with examples based on the random phase approximation in polymer theory, construction of tie lines for multiple-component LLPS, and field-theoretic simulation of multiple-chain heteropolymeric systems using complex Langevin dynamics. Since a more accurate physical picture often requires comparing analytical theory against explicit-chain model predictions, a commonly utilized methodology for coarse-grained molecular dynamics simulations of sequence-specific LLPS is also briefly outlined.

Association of two polyelectrolytes in salt solutions

The association of polyelectrolytes (PEs) in solution affects a wealth of structural and dynamic behaviors, and is also fundamentally important for an understanding of protein association and aggregation. Here, we theoretically study the association of two PE chains by addressing the stability and morphology of the non-spherical associates. Our theory predicts that an elongated pearl-necklace (PN) associate can be stable at high salt concentrations due to the screened electrostatic repulsion. This contradicts the implication of scaling theory. In addition, there is no one-to-one correspondence between the morphology of the associate and its constituting unimers, which is demonstrated by the existence of different association modes.

Wetting behavior of polyelectrolyte complex coacervates on solid surfaces

The wetting behavior of complex coacervates underpins their use in many emerging applications of surface science, particularly wet adhesives and coatings. Many factors dictate if a coacervate phase will condense on a solid surface, including solution conditions, the nature of the polymer-substrate interaction, and the underlying supernatant-coacervate bulk phase behavior. In this work, we use a simple inhomogeneous mean-field theory to study the wetting behavior of complex coacervates on solid surfaces both off-coexistence (wetting transitions) and on-coexistence (contact angles). We focus on the effects of salt concentration, the polycation/polyanion surface affinity, and the applied electrostatic potential on the wettability. We find that the coacervate generally wets the surface via a first order wetting transition with second order transitions possible above a surface critical point. Applying an electrostatic potential to a solid surface always improves the surface wettability when the polycation/polyanion-substrate interaction is symmetric. For asymmetric surface affinity, the wettability has a nonmonotonic dependence with the applied potential. We use simple scaling and thermodynamic arguments to explain our results.

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.

Structures of cationic and anionic polyelectrolytes in aqueous solutions: the sign effect

In this study, we use molecular dynamics simulation to explore the structures of anionic and cationic polyelectrolytes in aqueous solutions. We first confirm the significantly stronger solvation effects of single anions compared to cations in water at the fixed ion radii, due to the reversal orientations of asymmetric dipolar H₂O molecules around the ions. Based on this, we demonstrate that the solvation discrepancy of cations/anions and electrostatic correlations of ionic species can synergistically cause the nontrivial structural difference between single anionic and cationic polyelectrolytes. The cationic polyelectrolyte shows an extended structure whereas the anionic polyelectrolyte exhibits a collapsed structure, and their structural differences decline with increasing the counterion size. Furthermore, we corroborate that multiple cationic polyelectrolytes or multiple anionic polyelectrolytes can exhibit largely differential molecular architectures in aqueous solutions. In the solvation dominant regime, the polyelectrolyte solutions exhibit uniform structures; whereas, in the electrostatic correlation dominant regime, the polyelectrolyte solutions exhibit heterogeneous structures, in which the likely charged chains microscopically aggregate through counterion condensations. Increasing the intrinsic chain rigidity causes polyelectrolyte extension and hence moderately weakens the inter-chain clustering. Our work highlights the various, unique structures and molecular architectures of polyelectrolytes in solutions caused by the multi-body correlations between polyelectrolytes, counterions and asymmetric dipolar solvent molecules, which provides insights into the fundamental understanding of ion-containing polymers.

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.

Multiphase Coacervates Driven by Electrostatic Correlations

The liquid-liquid phase separation of a polyelectrolyte solution containing one type of negatively and two types of positively charged polymers with different charge densities is studied theoretically by random phase approximation (RPA). It is predicted that multicoacervate phases could coexist, driven purely by electrostatic correlations. The asymmetry of the linear charge density could induce an effective immiscibility between two positively charged polyelectrolytes, leading to the multiphase separation. Adding salt will induce the disappearance of the dilute phase, forming two coexisting complex phases, instead of fusion between coacervates. Raising temperature could either induce a two coexisting complex phase, or a dilute phase coexisting with a coacervate phase, depending on the bulk concentration. Our predictions are in good agreement with experiments and provide insights in the further designing of the multiphase coacervation system.

Nonelectrostatic Adsorption of Polyelectrolytes and Mediated Interactions between Solid Surfaces

Polymer-mediated interaction between two solid surfaces is directly connected to the properties of the adsorbed polymer layers. Nonelectrostatic interactions with a surface can significantly impact the adsorption of polyelectrolytes to charged surfaces. We use a classical density functional theory to study the effect of various polyelectrolyte solution properties on the adsorption and interaction between two like-charged surfaces. Our results show that nonelectrostatic interactions not only enhance polyelectrolyte adsorption but can also result in qualitatively different salt effects with respect to the adsorbed amount. In particular, we observe decreasing, increasing, and a previously unreported nonmonotonic behavior in the adsorbed amount of polymer with added salt under the conditions studied, although the nonmonotonic regime only occurs for a narrow range in the parameter space. With sufficient nonelectrostatic adsorption, the adsorbed polymer layers produce a long-range repulsive barrier that is strong enough to overcome dispersive interactions that cause surfaces to attract. Concurrently, a short-range bridging attraction is observed when the two polyelectrolyte layers span both the surfaces. Both the repulsive barrier and bridging attraction depend on the charge density of the polymer backbone and the bulk salt concentration but not on the chain length in the semidilute regime studied.

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.

Ion Pairing and the Structure of Gel Coacervates

A scaling model for the structure of coacervates is presented for mixtures of oppositely-charged polyelectrolytes of both symmetric and asymmetric charge-densities for different degrees of electrostatic strength and levels of added salt. At low electrostatic strengths, weak coacervates, with the energy of electrostatic interactions between charges less than the thermal energy, k B T, are liquid. At higher electrostatic strengths, strong coacervates are gels with crosslinks formed by ion pairs of opposite charges bound to each other with energy higher than k B T. Charge-symmetric coacervates are formed for mixtures of oppositely-charged polyelectrolytes with equal and opposite charge-densities. While charge-symmetric weak coacervates form a semidilute polymer solution with a correlation length equal to the electrostatic blob size, charge-symmetric strong coacervates form reversible gels with a correlation length on the order of the distance between bound ion pairs. Charge-asymmetric coacervates are formed from mixtures of oppositely-charged polyelectrolytes with different charge-densities. While charge-asymmetric weak coacervates form double solutions with two correlation lengths and qualitatively different chain conformations of polycations and polyanions, charge-asymmetric strong coacervates form bottlebrush and star-like gels. Unlike liquid coacervates, for which an increase in the concentration of added salt screens electrostatic interactions, causing structural rearrangement and eventually leads to their dissolution, the salt does not affect the structure of strong coacervates until ion pairs dissociate and the gel disperses.

Charge regulation of colloidal particles in aqueous solutions

We study the charge regulation of colloidal particles inside aqueous electrolyte solutions. To stabilize a colloidal suspension against precipitation, colloidal particles are synthesized with either acidic or basic groups on their surface. On contact with water, these surface groups undergo proton transfer reactions, resulting in colloidal surface charge. The charge is determined by the condition of local chemical equilibrium between hydronium ions inside the solution and at the colloidal surface. We use a model of Baxter sticky spheres to explicitly calculate the equilibrium dissociation constants and to construct a theory which is able to quantitatively predict the effective charge of colloidal particles with either acidic or basic surface groups. The predictions of the theory for the model are found to be in excellent agreement with the results of Monte Carlo simulations. This theory is further extended to treat colloidal particles with a mixture of both acidic and basic surface groups.

Statistical mechanical analysis of the electromechanical coupling in an electrically-responsive polymer chain

Polymeric materials that couple deformation and electrostatics have the potential for use in soft sensors and actuators with applications ranging from robotic, biomedical, energy, aerospace and automotive technologies. In contrast to the mechanics of polymers that has been studied using statistical mechanics approaches for decades, the coupled response under deformation and electrical field has largely been modeled only phenomenologically at the continuum scale. In this work, we examine the physics of the coupled deformation and electrical response of an electrically-responsive polymer chain using statistical mechanics. We begin with a simple anisotropic model for the electrostatic dipole response to electric field of a single monomer, and use a separation of energy scales between the electrostatic field energy and the induced dipole field energy to reduce the nonlocal and infinite-dimensional statistical averaging to a simpler local finite-dimensional averaging. In this simplified setting, we derive the equations of the most likely monomer orientation density using the maximum term approximation, and a chain free energy is derived using this approximation. These equations are investigated numerically and the results provide insight into the physics of electromechanically coupled elastomer chains. Closed-form approximations are also developed in the limit of small electrical energy with respect to thermal energy; in the limit of small mechanical tension force acting on the chain; and using asymptotic matching for general chain conditions.

Near-Critical Phase Behavior in Polyelectrolyte Solutions: Effect of Charge Fluctuations

The critical behaviors in polyelectrolyte (PE) solutions are studied by a renormalized Gaussian fluctuation theory. PEs are fully charged linear wormlike chains. The electrostatic interactions are considered in a continuum solvent while ignoring other interactions. The effects of temperature on the criticality are explored in a salt-free polyanion solution, and the effects of salt are explored in a symmetric mixed solution of polycation and polyanion. An unphysical phase coexistence (UPPC), in which a metastable dense phase coexists with an unstable dilute phase, always exists in the phase diagram. In the near-critical region, the UPPC could interfere with the real phase coexistence. The classical critical point is replaced by a "critical line", giving a flat top for the phase diagram. These behaviors are due to the effect of charge fluctuation in the near-critical region. Our results explain why the mean field approach overestimates the critical salt density by about 10% to experimental data in the study of coacervation between oppositely charged PEs.

Electrostatic contribution to colloidal solvation in terms of the self-energy-modified Boltzmann distribution

Electrostatic interactions make a large contribution to solvation free energy in ionic fluids such as electrolytes and colloidal dispersions. The electrostatic contribution to solvation free energy has been ascribed to the self-energy of a charged particle. Here we apply a variational field theory based on lower bound inequality to the inhomogeneous fluids of one-component charged hard-spheres, thereby verifying that the self-energy is given by the difference between the total correlation function and direct correlation function. Based on the knowledge of the liquid state theory, the self-energy specified in this study not only relates a direct correlation function to the Gaussian smearing of each charged sphere, but also provides the electrostatic contribution to solvation free energy that shows good agreement with simulation results. Furthermore, the Ornstein-Zernike equation leads to a set of generalized Debye-Hückel equations reflecting the Gaussian distributed charges.

A unified analytical theory of heteropolymers for sequence-specific phase behaviors of polyelectrolytes and polyampholytes

The physical chemistry of liquid-liquid phase separation (LLPS) of polymer solutions bears directly on the assembly of biologically functional dropletlike bodies from proteins and nucleic acids. These biomolecular condensates include certain extracellular materials and intracellular compartments that are characterized as "membraneless organelles." Analytical theories are a valuable, computationally efficient tool for addressing general principles. LLPS of neutral homopolymers is quite well described by theory, but it has been a challenge to develop general theories for the LLPS of heteropolymers involving charge-charge interactions. Here, we present a theory that combines a random-phase-approximation treatment of polymer density fluctuations and an account of intrachain conformational heterogeneity based on renormalized Kuhn lengths to provide predictions of LLPS properties as a function of pH, salt, and charge patterning along the chain sequence. Advancing beyond more limited analytical approaches, our LLPS theory is applicable to a wide variety of charged sequences ranging from highly charged polyelectrolytes to neutral or nearly neutral polyampholytes. This theory should be useful in high-throughput screening of protein and other sequences for their LLPS propensities and can serve as a basis for more comprehensive theories that incorporate nonelectrostatic interactions. Experimental ramifications of our theory are discussed.

On the origin of oscillatory interactions between surfaces mediated by polyelectrolyte solution

We use a numerical implementation of polymer classical density functional theory with an incompressibility condition to study the system consisting of nonadsorbing polyelectrolytes confined by two planar surfaces and quantify the effective interaction between the two planar surfaces as a function of the polyelectrolyte and salt concentrations. Our results indicate that for the uncharged surfaces (and weakly charged surfaces), the effective interaction primarily consists of a short-range attraction due to the depletion followed by repulsion due to the electric double layer overlapping and electrostatic correlations. For salt-free and low salt concentration systems, the magnitude of the repulsion barrier is determined by the overlap between the electric double layers, while at relatively high salt concentrations, the magnitude of the repulsion barrier is determined by the electrostatic correlations. Due to the competition between the electric double layer and the electrostatic correlations, the magnitude of the repulsion barrier varies nonmonotonically. In contrast, a mean-field Poisson-Boltzmann treatment of the electrostatics predicts a monotonically decreasing repulsion barrier with increasing salt concentration. At moderate salt concentrations, our theory predicts oscillatory interaction profiles. A comparison with the mean-field Poisson-Boltzmann treatment of electrostatics suggests that the oscillations are due primarily to electrostatic correlations.

Complex Coacervation in Asymmetric Solutions of Polycation and Polyanion

We study coacervation upon mixing two aqueous solutions of polyelectrolytes (PEs) with opposite charge, by considering asymmetric effects of PE composition and charge valency. The phase behavior, interfacial structure, and coacervate composition are investigated by a classical density-functional theory. We find two types of coacervation that are different in their density. Supernatant phase in low-density coacervation (LDCA) fully consists of small ions, while in high-density coacervation (HDCA) it contains a considerable amount of PE chains. Asymmetric PE composition could generate an electric double layer at the interface of coacervate. For HDCA, ion density changes monotonically, while for LDCA it shows a global minimum at the double layer, giving a low tension value. Charged species of high charge valency enhance the existence of double layer. Our results explained the coacervate structure of low interfacial tension, which is important for experiments and industrial applications.

Controlling Complex Coacervation via Random Polyelectrolyte Sequences

The utilization of chemical sequence control in polymeric materials is key to enabling material design on par with biomacromolecular systems. One important avenue for scalable sequence-controlled polymers leverages the random copolymerization of distinct monomers, with the statistical distribution of the monomeric sequence arising from reaction kinetics following a first-order Markov process. Here we utilize the framework of the random phase approximation (RPA) to develop a theory for the phase behavior of symmetric polyelectrolyte coacervates whose chemical sequences are dictated by simple statistical distributions. We find that a high charge "blockiness" within the random sequences favors the formation of denser and more salt-resistant coacervates while simultaneously increasing the width of the two-phase region. We trace these physical effects to the increased cooperativity of Coulomb interactions that results from increased charge blockiness in oppositely charged polyelectrolytes.

Small ion effects on self-coacervation phenomena in block polyampholytes

Self-coacervation is a phenomenon in which a solution of polyampholytes spontaneously phase separates into a dense liquid coacervate phase, rich in the polyampholyte, coexisting with a dilute supernatant phase. Such coacervation results in the formation of membraneless organelles in vivo and has further been applied industrially as synthetic encapsulants and coatings. It has been suggested that coacervation is primarily driven by the entropy gain from releasing counter-ions upon complexation. Using fully fluctuating field-theoretic simulations employing complex Langevin sampling and complementary molecular dynamics simulations, we have determined that the small ions contribute only weakly to the self-coacervation behavior of charge-symmetric block polyampholytes in solution. Salt partitioning between the supernatant and coacervate is also found to be negligible in the weak-binding regime at low electrostatic strengths. Asymmetries in charge distribution along the polyampholytes can cause net-charges that lead to "tadpole" configurations in dilute solution and the suppression of phase separation at low salt content. The field and particle-based simulation results are compared with analytical predictions from the random phase approximation (RPA) and postulated scaling relationships. The qualitative trends are mostly captured by the RPA, but the approximation fails at low concentration.

Influence of charge sequence on the adsorption of polyelectrolytes to oppositely-charged polyelectrolyte brushes

When a solution of polyanionic chains is placed in contact with a polycationic brush, the polyanions adsorb into the brush. We investigate the influence of the charge sequences of the free and bound species on the thermodynamics of polyelectrolyte adsorption. As model systems, we consider free and brush polyelectrolytes with either block or alternating charge sequences, and study the adsorption process using coarse-grained Langevin dynamics with implicit solvent, explicit counterions, and excess salt. Free energy, internal energy, and entropy of adsorption are computed using umbrella sampling methods. When the number of polyanions exceed the number of polycations, the brush becomes overcharged. Free chains adsorb most strongly when both free and tethered chains have a block charge sequence, and most weakly when both species have an alternating sequence. Adsorption is stronger when the free polyanion has a block sequence and the tethered polycation is alternating than in the reverse case of an alternating free polymer and a tethered block copolymer. Sequence-dependent effects are shown to be largely energetic, rather than entropic, in origin.

Complete Phase Diagram for Liquid-Liquid Phase Separation of Intrinsically Disordered Proteins

A number of intrinsically disordered proteins have been shown to self-assemble via liquid-liquid phase separation into protein-rich and dilute phases. The resulting coacervates can have important biological functions, and the ability to form these assemblies is dictated by the protein's primary amino acid sequence as well as by the solution conditions. We present a complete phase diagram for the simple coacervation of a polyampholyte intrinsically disordered protein using a field-theoretic simulation approach. We show that differences in the primary amino acid sequence and in the distribution of charged amino acids along the sequence lead to differences in the phase window for coacervation, with block-charged sequences having a larger coacervation window than sequences with a random patterning of charges. The model also captures how changing solution conditions modifies the phase diagram and can serve to guide experimental studies.

Narrow equilibrium window for complex coacervation of tau and RNA under cellular conditions

The mechanism that leads to liquid-liquid phase separation (LLPS) of the tau protein, whose pathological aggregation is implicated in neurodegenerative disorders, is not well understood. Establishing a phase diagram that delineates the boundaries of phase co-existence is key to understanding whether LLPS is an equilibrium or intermediate state. We demonstrate that tau and RNA reversibly form complex coacervates. While the equilibrium phase diagram can be fit to an analytical theory, a more advanced model is investigated through field theoretic simulations (FTS) that provided direct insight into the thermodynamic driving forces of tau LLPS. Together, experiment and simulation reveal that tau-RNA LLPS is stable within a narrow equilibrium window near physiological conditions over experimentally tunable parameters including temperature, salt and tau concentrations, and is entropy-driven. Guided by our phase diagram, we show that tau can be driven toward LLPS under live cell coculturing conditions with rationally chosen experimental parameters.

Tunable Coacervation of Well-Defined Homologous Polyanions and Polycations by Local Polarity

The ionic complexation of polyelectrolytes is an important mechanism underlying many important biological processes and technical applications. The main driving force for complexation is electrostatic, which is known to be affected by the local polarity near charge centers, but the impact of which on the complexation of polyelectrolytes remains poorly explored. We developed a homologous series of well-defined polyelectrolytes with identical backbone structures, controlled molecular weights, and tunable local polarity to modulate the solvation environment near charged groups. A multitude of systematic, accurate phase diagrams were obtained by spectroscopic measurements of polymer concentrations via fluorescent labeling of polycations. These phase diagrams unambiguously revealed that the liquidlike coacervation is more stable against salt addition at reduced local polarity over a wide range of molecular weights. These trends were quantitatively captured by a theory of complexation that incorporates the effects of dispersion interactions, charge connectivity, and reversible ion-binding, providing the microscopic design rules for tuning molecular parameters and local polarity.