Ion Partition in Polyelectrolyte Gels and Nanogels.

Polyelectrolyte gels provide a load-bearing structural framework for many macroscopic biological tissues, along with the organelles within the cells composing tissues and the extracellular matrices linking the cells at a larger length scale than the cells. In addition, they also provide a medium for the selective transportation and sequestration of ions and molecules necessary for life. Motivated by these diverse problems, we focus on modeling ion partitioning in polyelectrolyte gels immersed in a solution with a single type of ionic valence, i.e., monovalent or divalent salts. Specifically, we investigate the distribution of ions inside the gel structure and compare it with the bulk, i.e., away from the gel structure. In this first exploratory study, we neglect solvation effects in our gel by modeling the gels without an explicit solvent description, with the understanding that such an approach may be inadequate for describing ion partitioning in real polyelectrolyte gels. We see that this type of model is nonetheless a natural reference point for considering gels with solvation. Based on our idealized polymer network model without explicit solvent, we find that the ion partition coefficients scale with the salt concentration, and the ion partition coefficient for divalent ions is higher than for monovalent ions over a wide range of Bjerrum length (lB) values. For gels having both monovalent and divalent salts, we find that divalent ions exhibit higher ion partition coefficients than monovalent salt for low divalent salt concentrations and low lB. However, we also find evidence that the neglect of an explicit solvent, and thus solvation, provides an inadequate description when compared to experimental observations. Thus, in future work, we must consider both ion and polymer solvation to obtain a more realistic description of ion partitioning in polyelectrolyte gels.

Coexistence of Crumpling and Flat Sheet Conformations in Two-Dimensional Polymer Networks: An Understanding of Aggrecan Self-Assembly.

We investigate the conformational properties of self-avoiding two-dimensional (2D) ideal polymer networks with tunable mesh sizes as a model of self-assembled structures formed by aggrecan. Polymer networks having few branching points and large enough mesh tend to crumple, resulting in a fractal dimension of d_{f}≈2.7. The flat sheet behavior (d_{f}=2) emerges in 2D polymer networks having more branching points at large length scales; however, it coexists with crumpling conformations at intermediate length scales, a feature found in scattering profiles of aggrecan solutions. Our findings bridge the long-standing gap between theories and simulations of polymer sheets.

Prestressed Composite Polymer Gels as a Model of the Extracellular-Matrix of Cartilage.

Articular cartilage is a composite hydrogel found in animal and human joints, which exhibits unique load-bearing properties that have been challenging to reproduce in synthetic materials and model in molecular dynamics (MD) simulations. We computationally investigate a composite hydrogel that mimics key functional properties of articular cartilage as a potential biomimetic model to investigate its unique load-bearing properties. Specifically, we find that the emergence of prestress in composite gels derives primarily from the stiffness of the polymer matrix and the asymmetry in the enthalpic interactions of the embedded particles and polymer matrix. Our MD simulations of the development of prestress agree qualitatively with osmotic pressure measurements observed in our model composite hydrogel material.

Molecular dynamics study of the swelling and osmotic properties of compact nanogel particles.

Owing to their great importance in materials science and other fields, we investigate the solution and osmotic properties of uncharged compact nanogel particles over a wide range of solvent quality and particle concentration by molecular dynamics (MD) simulations. We characterize the osmotic pressure by estimating the second and third virial coefficients, and by extension, we identify the θ-point where the second virial coefficient vanishes. Calculations of the structure factor indicate that these particles are similar to macrogels in that the particle-like scattering profile disappears at moderate concentrations. We also find that improving the solvent quality enhances the spatial segmental uniformity, while significant heterogeneous structure arises near the θ-point. Well below the θ-point where the second osmotic virial coefficient vanishes, these heterogeneous structures become less prevalent as the particles tend to collapse. We also investigate the degree of swelling and structure of compact nanogel particles with a variable excluded volume interaction and gel particle concentration. The osmotic modulus and the scaling exponents in good and θ-point conditions of these gels are characteristic of interacting randomly branched polymers, i.e., "lattice animals".

Influence of network defects on the conformational structure of nanogel particles: From "closed compact" to "open fractal" nanogel particles.

We propose an approach to generate a wide range of randomly branched polymeric structures to gain general insights into how polymer topology encodes a configurational structure in solution. Nanogel particles can take forms ranging from relatively symmetric sponge-like compact structures to relatively anisotropic open fractal structures observed in some nanogel clusters and in some self-associating polymers in solutions, such as aggrecan solutions under physiologically relevant conditions. We hypothesize that this broad "spectrum" of branched polymer structures derives from the degree of regularity of bonding in the network defining these structures. Accordingly, we systematically introduce bonding defects in an initially perfect network having a lattice structure in three and two topological dimensions corresponding to "sponge" and "sheet" structures, respectively. The introduction of bonding defects causes these "closed" and relatively compact nanogel particles to transform near a well-defined bond percolation threshold into "open" fractal objects with the inherent anisotropy of randomly branched polymers. Moreover, with increasing network decimation, the network structure of these polymers acquires other configurational properties similar to those of randomly branched polymers. In particular, the mass scaling of the radius of gyration and its eigenvalues, as well as hydrodynamic radius, intrinsic viscosity, and form factor for scattering, all undergo abrupt changes that accompany these topological transitions. Our findings support the idea that randomly branched polymers can be considered to be equivalent to perforated sheets from a "universality class" standpoint. We utilize our model to gain insight into scattering measurements made on aggrecan solutions.

Structure and conformational properties of ideal nanogel particles in athermal solutions.

We investigate the conformational properties of "ideal" nanogel particles having a lattice network topology by molecular dynamics simulations to quantify the influence of polymer topology on the solution properties of this type of branched molecular architecture. In particular, we calculate the mass scaling of the radius of gyration (Rg), the hydrodynamic radius, as well as the intrinsic viscosity with the variation of the degree of branching, the length of the chains between the branched points, and the average mesh size within these nanogel particles under good solvent conditions. We find competing trends between the molecular characteristics, where an increase in mesh size or degree of branching results in the emergence of particle-like characteristics, while an increase in the chain length enhances linear polymer-like characteristics. This crossover between these limiting behaviors is also apparent in our calculation of the form factor, P(q), for these structures. Specifically, a primary scattering peak emerges, characterizing the overall nanogel particle size. Moreover, a distinct power-law regime emerges in P(q) at length scales larger than the chain size but smaller than Rg of the nanogel particle, and the Rg mass scaling exponent progressively approaches zero as the mesh size increases, the same scaling as for an infinite network of Gaussian chains. The "fuzzy sphere" model does not capture this feature, and we propose an extension to this popular model. These structural features become more pronounced for values of molecular parameters that enhance the localization of the branching segments within the nanogel particle.

Comparative experimental and computational study of synthetic and natural bottlebrush polyelectrolyte solutions.

We systematically investigate model synthetic and natural bottlebrush polyelectrolyte solutions through an array of experimental techniques (osmometry and neutron and dynamic light scattering) along with molecular dynamics simulations to characterize and contrast their structures over a wide range of spatial and time scales. In particular, we perform measurements on solutions of aggrecan and the synthetic bottlebrush polymer, poly(sodium acrylate), and simulations of solutions of highly coarse-grained charged bottlebrush molecules having different degrees of side-branch density and inclusion of an explicit solvent and ion hydration effects. While both systems exhibit a general tendency toward supramolecular organization in solution, bottlebrush poly(sodium acrylate) solutions exhibit a distinctive "polyelectrolyte peak" in their structure factor, but no such peak is observed in aggrecan solutions. This qualitative difference in scattering properties, and thus polyelectrolyte solution organization, is attributed to a concerted effect of the bottlebrush polymer topology and the solvation of the polymer backbone and counterions. The coupling of the polyelectrolyte topological structure with the counterion distribution about the charged polymer molecules along with direct polymer segmental hydration makes their solution organization and properties "tunable," a phenomenon that has significant ramifications for biological function and disease as well as for numerous materials applications.

Disappearance of the polyelectrolyte peak in salt-free solutions.

We investigate the nature of the polyelectrolyte peak in salt-free solutions by molecular dynamics simulations using a minimal model of polyelectrolyte solutions that includes an explicit solvent and counterions and small angle scattering experiments. It is found that the polyelectrolyte peak progressively disappears as the strength of solvation for the charged species is increased and the scattering profiles start to resemble those of neutral polymer solutions. The disappearance of the polyelectrolyte peak coincides with the emergence of attractive interchain interactions over a wide range of length scales. These findings provide insights into the microscopic origin of the polyelectrolyte peak.

Design of nearly perfect hyperuniform polymeric materials.

Disordered hyperuniform materials are exotic amorphous systems that simultaneously exhibit anomalous suppression of long-range density fluctuations, comparable in amplitude to that of crystals and quasi-crystalline materials, while lacking the translational order characteristic of simple liquids. We establish a framework to quantitatively predict the emergence of hyperuniformity in polymeric materials by considering the distribution of localized polymer subregions, instead of considering the whole material. We demonstrate that this highly tunable approach results in arbitrarily small long-range density fluctuations in the liquid state. Our simulations also indicate that long-ranged density fluctuation of the whole polymeric material is remarkably insensitive to molecular topology (linear chain, unknotted ring, star, and bottlebrush) and depends on temperature in an apparently near universal fashion. Our findings open the way for the creation of nearly perfect hyperuniform polymeric materials.

Bottlebrush polymers in the melt and polyelectrolytes in solution share common structural features.

Uncharged bottlebrush polymer melts and highly charged polyelectrolytes in solution exhibit correlation peaks in scattering measurements and simulations. Given the striking superficial similarities of these scattering features, there may be a deeper structural interrelationship in these chemically different classes of materials. Correspondingly, we constructed a library of isotopically labeled bottlebrush molecules and measured the bottlebrush correlation peak position [Formula: see text] by neutron scattering and in simulations. We find that the correlation length scales with the backbone concentration, [Formula: see text], in striking accord with the scaling of ξ with polymer concentration cP in semidilute polyelectrolyte solutions [Formula: see text] The bottlebrush correlation peak broadens with decreasing grafting density, similar to increasing salt concentration in polyelectrolyte solutions. ξ also scales with sidechain length to a power in the range of 0.35-0.44, suggesting that the sidechains are relatively collapsed in comparison to the bristlelike configurations often imagined for bottlebrush polymers.

Influence of polymer topology on crystallization in thin films.

We investigate how varying molecular topology of polymers influences crystallization in thin polymer films. In particular, we simulate linear and star polymers of fixed mass having a progressively increasing number of arms (f ≤ 16) in a system where the linear polymer exhibits crystallization in a thin film geometry, but no apparent crystallization in the corresponding bulk material. The degree of crystallization of the polymer film at long times decreases progressively with increasing f, and no crystallization is observed beyond f = 8. Crystallization for smaller values of f develops as a sigmoidally shaped wavefront initiating from the supporting crystalline interface. We suggest that large shape fluctuations and the competition of length scales of star polymers with high f lead to inhibited crystallization.

Dynamic heterogeneity and collective motion in star polymer melts.

While glass formation of linear chain polymer melts has widely been explored, comparatively little is known about glass formation in star polymer melts. We study the segmental dynamics of star polymer melts via molecular dynamics simulations and examine the cooperative nature of segmental motion in star melts. In particular, we quantify how the molecular architecture of star polymers, i.e., the number of arms and the length of those arms, affects the glass transition temperature Tg, the non-Gaussian nature of molecular displacements, the collective string-like motion of monomers, and the role of chain connectivity in the cooperative motion. Although varying the number of arms f and the molecular mass Ma of the star arms can significantly influence the average star molecular shape, all our relaxation data can be quantitatively described in a unified way by the string model of glass formation, an activated transport model that derives from the Adam-Gibbs model, where the degree of cooperative motion is identified with the average length L of string-like particle exchange motions observed in our simulations. Previous work has shown the consistency of the string model with simulations of linear polymers at constant volume and constant pressure, as well as for thin supported polymer films and nanocomposites with variable polymer-surface interactions, where there are likewise large mobility gradients as in the star polymer melts studied in the present paper.

The Concentration Dependence of the Size and Symmetry of a Bottlebrush Polymer in a Good Solvent.

Bottlebrush polymers consist of a linear backbone with densely grafted side chains which impact the rigidity of the molecule. The persistence length of the bottlebrush backbone in solution is influenced by both the intrinsic structure of the polymer and by the local environment, such as the solvent quality and concentration. Increasing the concentration reduces the overall size of the molecule due to the reduction in backbone stiffness. In this study we map out the size of a bottlebrush polymer as a function of concentration for a single backbone length. Small-angle neutron scattering (SANS) measurements are conducted on a polynorbornene-based bottlebrush with polystyrene side chains in a good solvent. The data are fit using a model which provides both the long and short axis radius of gyration (R g,2 and R g,1, respectively), providing a measure for how the conformation changes as a function of concentration. At low concentrations a highly anisotropic structure is observed (R g,2/R g,1 ≈ 4), becoming more isotropic at higher concentrations (R g,2/R g,1 ≈ 1.5). The concentration scaling for both R g,2 and the overall R g are evaluated and compared with predictions in the literature. Coarse-grained molecular dynamics simulations were also conducted to probe the impact of concentration on bottlebrush conformation showing qualitative agreement with the experimental results.

Systematic investigation of synthetic polyelectrolyte bottlebrush solutions by neutron and dynamic light scattering, osmometry, and molecular dynamics simulation.

There is a great interest in the synthesis and characterization of polyelectrolytes that mimic naturally occurring bottlebrush polyelectrolytes to capitalize on the unique properties of this class of macromolecules. Charged bottlebrush polymers form the protective mucus layer in the lungs, stomach, and orifices of animals and provide osmotic stabilization and lubrication to joints. In the present work, we systematically investigate bottlebrush poly(sodium acrylates) through a combination of measurements of solution properties (osmometry, small-angle neutron scattering, and dynamic light scattering) and molecular dynamics simulations, where the bottlebrush properties are compared in each case to their linear polymer counterparts. These complementary experimental and computational methods probe vastly different length- and timescales, allowing for a comprehensive characterization of the supermolecular structure and dynamics of synthetic polyelectrolyte bottlebrush molecules in solution.

Influence of Branching on the Configurational and Dynamical Properties of Entangled Polymer Melts.

We probe the influence of branching on the configurational, packing, and density correlation function properties of polymer melts of linear and star polymers, with emphasis on molecular masses larger than the entanglement molecular mass of linear chains. In particular, we calculate the conformational properties of these polymers, such as the hydrodynamic radius R h , packing length p, pair correlation function g ( r ) , and polymer center of mass self-diffusion coefficient, D, with the use of coarse-grained molecular dynamics simulations. Our simulation results reproduce the phenomenology of simulated linear and branched polymers, and we attempt to understand our observations based on a combination of hydrodynamic and thermodynamic modeling. We introduce a model of "entanglement" phenomenon in high molecular mass polymers that assumes polymers can viewed in a coarse-grained sense as "soft" particles and, correspondingly, we model the emergence of heterogeneous dynamics in polymeric glass-forming liquids to occur in a fashion similar to glass-forming liquids in which the molecules have soft repulsive interactions. Based on this novel perspective of polymer melt dynamics, we propose a functional form for D that can describe our simulation results for both star and linear polymers, covering both the unentangled to entangled polymer melt regimes.

Superionic UO2: A model anharmonic crystalline material.

Crystalline materials at elevated temperatures and pressures can exhibit properties more reminiscent of simple liquids than ideal crystalline materials. Superionic crystalline materials having a liquidlike conductivity σ are particularly interesting for battery, fuel cell, and other energy applications, and we study UO2 as a prototypical superionic material since this material is widely studied given its commercial importance as reactor fuel. Using molecular dynamics, we first investigate basic thermodynamic and structural properties. We then quantify structural relaxation, dynamic heterogeneity, and average ion mobility. We find that the non-Arrhenius diffusion and structural relaxation time of this prototypical superionic material can be described in terms of a generalized activated transport model ("string model") in which the activation energy varies with the average string length. Our transport data can also be described equally well by an Adam-Gibbs model in which the excess entropy density of the crystalline material is estimated from specific heat and thermal expansion data, consistent with the average scale of stringlike collective motion scaling inversely with the excess entropy of the crystal. Strong differences in the temperature dependence of the interfacial mobility from nonionic materials are observed, and we suggest that this difference is due to the relatively high cohesive interaction of ionic materials. In summary, the study of superionic UO2 provides insight into the role of cooperative motion in enhancing ion mobility in ionic materials and offers design principles for the development of new superionic materials for use in diverse energy applications.

Polyelectrolyte association and solvation.

There has been significant interest in the tendency of highly charged particles having the same charge to form dynamic clusters in solution, but an accepted theoretical framework that can account for this ubiquitous phenomenon has been slow to develop. The theoretical difficulties are especially great for flexible polyelectrolytes due to the additional complex coupling between the polyelectrolyte chain configurations and the spatial distribution of the ionic species in solution. For highly charged polyelectrolytes, this leads to the formation of a diffuse "polarizable" cloud of counter-ions around these polymers, an effect having significant implications for the function of proteins and other natural occurring polyelectrolytes, as emphasized long ago by Kirkwood and co-workers. To investigate this phenomenon, we perform molecular dynamics simulations of a minimal model of polyelectrolyte solutions that includes an explicit solvent and counter-ions, where the relative affinity of the counter-ions and the polymer for the solvent is tunable through the variation of the relative strength of the dispersion interactions of the polymer and ions. In particular, we find that these dispersion interactions can greatly influence the nature of the association between the polyelectrolyte chains under salt-free conditions. We calculate static and dynamic correlation functions to quantify the equilibrium structure and dynamics of these complex liquids. Based on our coarse-grained model of polyelectrolyte solutions, we identify conditions in which three distinct types of polyelectrolyte association arise. We rationalize these types of polyelectrolyte association based on the impact of the selective solvent affinity on the charge distribution and polymer solvation in these solutions. Our findings demonstrate the essential role of the solvent in the description of the polyelectrolyte solutions, as well as providing a guideline for the development of a more predictive theory of the properties of the thermodynamic and transport properties of these complex fluids.

A comparative study of thermodynamic, conformational, and structural properties of bottlebrush with star and ring polymer melts.

Thermodynamic, conformational, and structural properties of bottlebrush polymer melts are investigated with molecular dynamics simulations and compared to linear, regular star, and unknotted ring polymer melts to gauge the influence of molecular topology on polymer melt properties. We focus on the variation of the backbone chain length, the grafting density along the backbone, and the length of the side chains at different temperatures above the melt glass transition temperature. Based on these comparisons, we find that the segmental density, isothermal compressibility, and isobaric thermal expansion of bottlebrush melts are quantitatively similar to unknotted ring polymer melts and star polymer melts having a moderate number ( f = 5 to 6) of arms. These similarities extend to the mass scaling of the chain radius of gyration. Our results together indicate that the configurational properties of bottlebrush polymers in their melt state are more similar to randomly branched polymers than linear polymer chains. We also find that the average shape of bottlebrush polymers having short backbone chains with respect to the side chain length is also rather similar to the unknotted ring and moderately branched star polymers in their melt state. As a general trend, the molecular shape of bottlebrush polymers becomes more spherically symmetric when the length of the side chains has a commensurate length as the backbone chain. Finally, we calculate the partial static structure factor of the backbone segments and we find the emergence of a peak at the length scales that characterizes the average distance between the backbone chains. This peak is absent when we calculate the full static structure factor. We characterize the scaling of this peak with parameters characterizing the bottlebrush molecular architecture to aid in the experimental characterization of these molecules by neutron scattering.

Influence of Ion Solvation on the Properties of Electrolyte Solutions.

It is widely appreciated that the addition of salts to water leads to significant changes in the thermodynamic and dynamic properties of these aqueous solutions that have great significance in biology and manufacturing applications. However, no theoretical framework currently exists that describes these property changes in an internally consistent fashion. In previous work, we developed a coarse-grained model of electrolyte solutions capable of reproducing basic trends on how salts influence the viscosity and water diffusion coefficient. The present work explores the predictions of this model for basic thermodynamic properties of electrolyte solutions, namely, the density, isothermal compressibility, and surface tension. On the basis of our model, we find that ion-specific effects on thermodynamics properties, and by extension the dynamics of electrolyte solutions, derive primarily from ion solvation.

Hidden Hyperuniformity in Soft Polymeric Materials.

We investigate the nature of long-range density fluctuations in melts of model "soft" polymers, specifically stars and bottlebrushes, over a wide temperature range by molecular dynamics simulation. The cores of the stars and the backbones of bottlebrush polymers are found to have a hyperuniform distribution; i.e., they exhibit anomalously small density fluctuations over a wide temperature range above the glass transition temperature. The hyperuniformity of these substituent polymer subregions is hidden since the fluid as a whole does not exhibit this property. These findings offer a strategy for the practical design of hyperuniform polymeric materials.