Comparison of a hydrogel model to the Poisson-Boltzmann cell model

Osmotic swelling behavior of surface-charged ionic microgels

In recent years, ionic microgels have garnered much attention due to their unique properties, especially their stimulus-sensitive swelling behavior. The tunable response of these soft, permeable, compressible, charged colloidal particles is increasingly attractive for applications in medicine and biotechnologies, such as controlled drug delivery, tissue engineering, and biosensing. The ability to model and predict variation of the osmotic pressure of a single microgel with respect to changes in particle properties and environmental conditions proves vital to such applications. In this work, we apply both nonlinear Poisson-Boltzmann theory and molecular dynamics simulation to ionic microgels (macroions) in the cell model to compute density profiles of microions (counterions, coions), single-microgel osmotic pressure, and equilibrium swelling ratios of spherical microgels whose fixed charge is confined to the macroion surface. The basis of our approach is an exact theorem that relates the electrostatic component of the osmotic pressure to the microion density profiles. Close agreement between theory and simulation serves as a consistency check to validate our approach. We predict that surface-charged microgels progressively deswell with increasing microgel concentration, starting well below close packing, and with increasing salt concentration, in qualitative agreement with experiments. Comparison with previous results for microgels with fixed charge uniformly distributed over their volume demonstrates that surface-charged microgels deswell more rapidly than volume-charged microgels. We conclude that swelling behavior of ionic microgels in solution is sensitive to the distribution of fixed charge within the polymer-network gel and strongly depends on bulk concentrations of both microgels and salt ions.

Assessing the Validity of NMR Relaxation Rates Obtained from Coarse-Grained Simulations of PEG-Water Mixtures

NMR relaxometry is a powerful and well-established experimental approach for characterizing dynamic processes in soft matter systems. All-atom (AA) resolved simulations are typically employed to gain further microscopic insights while reproducing the relaxation rates R₁. However, such approaches are limited to time and length scales that prevent to model systems such as long polymer chains or hydrogels. Coarse graining (CG) can overcome this barrier at the cost of losing atomistic details that impede the calculation of NMR relaxation rates. Here, we address this issue by performing a systematic characterization of dipolar relaxation rates R₁ on a PEG-H₂O mixture at two different levels of details: AA and CG. Remarkably, we show that NMR relaxation rates R₁ obtained at the CG level obey the same trends when compared to AA calculations but with a systematic offset. This offset is due to, on the one hand, the lack of an intramonomer component and, on the other hand, the inexact positioning of the spin carriers. We show that the offset can be corrected for quantitatively by reconstructing a posteriori the atomistic details for the CG trajectories.

Structure of swollen hollow polyelectrolyte nanogels with inhomogeneous cross-link distribution

HYPOTHESIS

Recently, it has become possible to synthesize hollow polyelectrolyte nano- and microgels. The shell permeability can be controlled by external stimuli, while the cavity can serve as a storage place for guest molecules. However, there is a lack of a detailed understanding at the molecular level regarding the role of the network topology, inhomogeneities of the distribution of cross-links, and the impact of the electrostatics on the structural response of hollow microgel to external stimuli. To bridge these gaps, molecular dynamics (MD) of computer simulations are used.

EXPERIMENTS

Here, we propose a fresh methodology to create realistic hollow microgel particles in silico. The technique involves a gradual change in the average local length of subchains depending on the distance to the center of mass of the microgel particles resulting in the microgels with a non-uniform distribution of cross-linking species. In particular, a series of microgels with (i) a highly cross-linked inner part of the shell and gradually decreased cross-linker concentration towards the periphery, (ii) microgels with loosely cross-linked inner and outer parts, as well as (iii) microgels with a more-or-less homogeneous structure, have been created and validated. Counterions and salt ions are taken into account explicitly, and electrostatic interactions are described by the Coulomb potential.

FINDINGS

Our studies reveal a strong dependence of the microgel swelling response on the network topology. Simple redistribution of cross-links plays a significant role in the structure of the microgels, including cavity size, microgel size, fuzziness, and extension of the inner and outer parts of the microgels. Our results indicate the possibilities of qualitative justification of the structure of the hollow microgels in the experiments by measuring the relative change in the size of the sacrificial core to the size of the cavity and by estimation of a power law function, [Formula: see text] , of the hydrodynamic radius of the hollow microgels as a function of added salt concentration.

Onset of criticality in hyper-auxetic polymer networks

Against common sense, auxetic materials expand or contract perpendicularly when stretched or compressed, respectively, by uniaxial strain, being characterized by a negative Poisson's ratio ν. The amount of deformation in response to the applied force can be at most equal to the imposed one, so that ν = - 1 is the lowest bound for the mechanical stability of solids, a condition here defined as "hyper-auxeticity". In this work, we numerically show that ultra-low-crosslinked polymer networks under tension display hyper-auxetic behavior at a finite crosslinker concentration. At this point, the nearby mechanical instability triggers the onset of a critical-like transition between two states of different densities. This phenomenon displays similar features as well as important differences with respect to gas-liquid phase separation. Since our model is able to faithfully describe real-world hydrogels, the present results can be readily tested in laboratory experiments, paving the way to explore this unconventional phase behavior.

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

We performed Metropolis Monte Carlo simulations to investigate the impact of varying acid and base dissociation constants on the pH-dependent ionization and conformation of weak polyampholyte microgels under salt-free conditions and under explicit consideration of the chemical ionization equilibria of the acidic and basic groups and their electrostatic interaction. Irrespective of their relative acid and base dissociation constant, all of the microgels undergo a pH-dependent charge reversal from positive to negative with a neutral charge at the isoelectric point. This charge reversal is accompanied by a U-shaped swelling transition of the microgels with a minimum of their size at the point of charge neutrality. The width of the U-shaped swelling transition, however, is found to depend on the chosen relative acid and base dissociation constants through which the extent of the favorable electrostatic intramolecular interaction of the ionized acidic and basic groups is altered. The pH-dependent swelling transition of the microgels is found to become broader, the stronger the intramolecular electrostatic interaction of the oppositely charged ionized species is. In addition, the intramolecular charge compensation of the acidic and basic groups of the microgels allows their counterions to abandon the microgel and the associated gain in translational entropy further amplifies the broadening of the pH-dependent swelling transition. The analysis of the radial ionization profiles of the acidic and basic groups of the differently composed microgels reveals a variety of radial ionization patterns with a dependence on the overall charge of the microgels.

Dynamical properties of different models of elastic polymer rings: Confirming the link between deformation and fragility

We report extensive numerical simulations of different models of 2D polymer rings with internal elasticity. We monitor the dynamical behavior of the rings as a function of the packing fraction to address the effects of particle deformation on the collective response of the system. In particular, we compare three different models: (i) a recently investigated model [N. Gnan and E. Zaccarelli, Nat. Phys. 15, 683 (2019)] where an inner Hertzian field providing the internal elasticity acts on the monomers of the ring, (ii) the same model where the effect of such a field on the center of mass is balanced by opposite forces, and (iii) a semi-flexible model where an angular potential between adjacent monomers induces strong particle deformations. By analyzing the dynamics of the three models, we find that in all cases, there exists a direct link between the system fragility and particle asphericity. Among the three, only the first model displays anomalous dynamics in the form of a super-diffusive behavior of the mean-squared displacement and of a compressed exponential relaxation of the density auto-correlation function. We show that this is due to the combination of internal elasticity and the out-of-equilibrium force self-generated by each ring, both of which are necessary ingredients to induce such a peculiar behavior often observed in experiments of colloidal gels. These findings reinforce the role of particle deformation, connected to internal elasticity, in driving the dynamical response of dense soft particles.

Solute diffusion in gels: Thirty years of simulations

In this review, we present a summary of computer simulation studies on solute diffusion in gels carried out in the last three decades. Special attention is paid to coarse-grained simulations in which the role of steric and electrostatic interactions on the particle diffusion can be evaluated. In addition, other important characteristics of particle diffusion in gels, such as the stiffness of the gel structure and hydrodynamic interactions, can be taken into account through coarse-grained simulations. Emphasis is placed on how simulation results help to test phenomenological models and to improve the interpretation interof experimental results. Finally, coarse-grained simulations have also been employed to study the diffusion controlled release of drugs from gels. We believe that scientific advances in this line will be useful to better understand the mechanisms that control the diffusive transport of molecules in a wide variety of biological systems.

Coarse-grained Monte Carlo simulations of nanogel-polyelectrolyte complexes: electrostatic effects

Coarse-grained Monte-Carlo simulations of nanogel-polyelectrolyte complexes have been carried out. The results presented here capture two phenomena reported in experiments with real complexes: (i) the reduction in size after absorbing just a few chains and (ii) the charge inversion detected through electrophoretic mobility data. Our simulations reveal that charge inversion occurs if the polyelectrolyte charge is large enough. In addition, the distribution of chains inside the nanogel strongly depends on whether charge inversion takes place. It should also be stressed that the chain topology has little influence on most of the properties studied here.

Ionic equilibria and swelling of soft permeable particles in electrolyte solutions

We discuss osmotic equilibria between soft permeable particles, of radius R and volume charge density ρ, and bulk electrolyte solutions of inverse Debye length κ. Existing models are based on a simplified assumption of weakly charged particles. Here we derive analytical approximations for the distribution of potentials, ions and pressure in a system, suitable even when ρ is quite large. Our theory is valid not only for "large" particles (κR≫ 1), where the central part is fully screened, but also for weakly screened "small" particles (κR≤ 1) with overlapping inner diffuse layers. Besides, we present novel coarse-grained simulations to validate the analysis and illustrate the variation of potential/ion profiles in response to changes in κR and ρ. Our simulations also allow us to argue that swelling of both "large" and "small" particles is uniform, although their inner non-uniform local pressure profiles are essentially and qualitatively different. These results are directly relevant for a variety of permeable charged objects, from polymer micro- and nanogels to more rigid porous colloids.

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

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

Crossover of the effective charge in ionic thermoresponsive hydrogel particles

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

Numerical insights on ionic microgels: structure and swelling behaviour

Recent progress has been made in the numerical modelling of neutral microgel particles with a realistic, disordered structure. In this work we extend this approach to the case of co-polymerised microgels where a thermoresponsive polymer is mixed with acidic groups. We compare the cases where counterions directly interact with microgel charges or are modelled implicitly through a Debye-Hückel description. We do so by performing extensive numerical simulations of single microgels across the volume phase transition (VPT) varying the temperature and the fraction of charged monomers. We find that the presence of charges considerably alters the microgel structure, quantified by the monomer density profiles and by the form factors of the microgels, particularly close to the VPT. We observe significant deviations between the implicit and explicit models, with the latter comparing more favourably to available experiments. In particular, we observe a shift of the VPT temperature to larger values as the amount of charged monomers increases. We also find that below the VPT the microgel-counterion complex is almost neutral, while it develops a net charge above the VPT. Interestingly, under these conditions the collapsed microgel still retains a large amount of counterions inside its structure. Since these interesting features cannot be captured by the implicit model, our results show that it is crucial to explicitly include the counterions in order to realistically model ionic thermoresponsive microgels.

Towards the realistic computer model of precipitation polymerization microgels

In this paper we propose a new method of coarse-grained computer simulations of the microgel formation in course of free radical precipitation polymerization. For the first time, we simulate the precipitation polymerization process from a dilute solution of initial components to a final microgel particle with coarse grained molecular dynamics, and compare it to the experimental data. We expect that our simulation studies of PNIPA-like microgels will be able to elucidate the subject of nucleation and growth kinetics and to describe in detail the network topology and structure. Performed computer simulations help to determine the characteristic phases of the growth process and show the necessity of prolongated synthesis for the formation of stable microgel particles. We demonstrate the important role of dangling ends in microgels, which occupy as much as 50% of its molecular mass and have previously unattended influence on the swelling behavior. The verification of the model is made by the comparison of collapse curves and structure factors between simulated and experimental systems, and high quality matching is achieved. This work could help to open new horizons in studies that require the knowledge of detailed and realistic structures of the microgel networks.

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

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

Numerical modelling of non-ionic microgels: an overview

Microgels are complex macromolecules. These colloid-sized polymer networks possess internal degrees of freedom and, depending on the polymer(s) they are made of, can acquire a responsiveness to variations of the environment (temperature, pH, salt concentration, etc.). Besides being valuable for many practical applications, microgels are also extremely important to tackle fundamental physics problems. As a result, these last years have seen a rapid development of protocols for the synthesis of microgels, and more and more research has been devoted to the investigation of their bulk properties. However, from a numerical standpoint the picture is more fragmented, as the inherently multi-scale nature of microgels, whose bulk behaviour crucially depends on the microscopic details, cannot be handled at a single level of coarse-graining. Here we present an overview of the methods and models that have been proposed to describe non-ionic microgels at different length-scales, from the atomistic to the single-particle level. We especially focus on monomer-resolved models, as these have the right level of details to capture the most important properties of microgels, responsiveness and softness. We suggest that these microscopic descriptions, if realistic enough, can be employed as starting points to develop the more coarse-grained representations required to investigate the behaviour of bulk suspensions.

Simulations of ionization equilibria in weak polyelectrolyte solutions and gels

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

Star polymers as unit cells for coarse-graining cross-linked networks

Reducing the complexity of cross-linked polymer networks by preserving their main macroscale properties is key to understanding them, and a crucial issue is to relate individual properties of the polymer constituents to those of the reduced network. Here we study polymer networks in a good solvent, by considering star polymers as their unit elements, and first quantify the interaction between their centers of masses. We then reduce the complexity of a network by replacing sets of its bridged star polymers by equivalent effective soft particles with dense cores. Our coarse graining allows us to approximate complex polymer networks by much simpler ones, keeping their relevant mechanical properties, as illustrated in computer experiments.

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

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

Condensation of Counterions Gives Rise to Contraction Transitions in a One-Dimensional Polyelectrolyte Gel

The equilibrium volume of a polyelectrolyte gel results from a balance between the tendency to swell caused by outbound polymer/counterion diffusion along with Coulomb interactions on the one hand; and, on the other, the elastic resilience of the cross-linked polymer network. Direct Coulomb forces contribute both to non-ideality of the equilibrated Donnan osmotic pressure, but also to stretching of the network. To isolate the effect of polyelectrolyte expansion, we have analyzed a "one-dimensional" version of a gel, a linear chain of charged beads connected by Hooke's law springs. As in the range of weak Coulomb strengths previously studied, the springs are significantly stretched by the repulsive interactions among the beads even when the Coulomb strength is strong enough to cause condensation of counterions. There is a quasi-abrupt transition from a stretched state to a partially collapsed state in a transition range between weak and strong Coulomb strengths. Fluctuations between stretched and contracted conformations occur within the transition range. As the solvent quality decreases past the transition range, a progressive collapse can result if the condensed counterions strengthen the spring constant.

Direct determination of forces between charged nanogels through coarse-grained simulations

In this work, electrostatic forces between charged nanogels are explored through coarse-grained simulations. These simulations allow us to explicitly consider the complex topology of these nanoparticles and provide reliable force values to examine highly charged nanogels of a few tens of nanometers. The results obtained here clearly reveal that the electrostatic interactions between these nanoparticles are not governed by the net charge of the nanogel, which includes not only the charge of the polymer network but also the charge of ions inside. Thus two theoretical procedures for predicting effective charges are also proposed and investigated. Both provide predictions of the same order and capture the behavior found for the effective charge obtained from simulations.

Internal structure and swelling behaviour of in silico microgel particles

Microgels are soft colloids that, by virtue of their polymeric nature, can react to external stimuli such as temperature or pH by changing their size. The resulting swelling/deswelling transition can be exploited in fundamental research as well as for many diverse practical applications, ranging from art restoration to medicine. Such an extraordinary versatility stems from the complex internal structure of the individual microgels, each of which is a crosslinked polymer network. Here we employ a recently-introduced computational method to generate realistic microgel configurations and look at their structural properties, both in real and Fourier space, for several temperatures across the volume phase transition as a function of the crosslinker concentration and of the confining radius employed during the 'in-silico' synthesis. We find that the chain-length distribution of the resulting networks can be analytically predicted by a simple theoretical argument. In addition, we find that our results are well-fitted to the fuzzy-sphere model, which correctly reproduces the density profile of the microgels under study.

Computer Simulations of Static and Dynamical Properties of Weak Polyelectrolyte Nanogels in Salty Solutions

We investigate the chemical equilibria of weak polyelectrolyte nanogels with reaction ensemble Monte Carlo simulations. With this method, the chemical identity of the nanogel monomers can change between neutral or charged following the acid-base equilibrium reaction HA ⇌ A- + H⁺. We investigate the effect of changing the chemical equilibria by modifying the dissociation constant K a . These simulations allow for the extraction of static properties like swelling equilibria and the way in which charge-both monomer and ionic-is distributed inside the nanogel. Our findings reveal that, depending on the value of K a , added salt can either increase or decrease the gel size. Using the calculated mean-charge configurations of the nanogel from the reaction ensemble simulation as a quenched input to coupled lattice-Boltzmann molecular dynamics simulations, we investigate dynamical nanogel properties such as the electrophoretic mobility μ and the diffusion coefficient D.

In Silico Synthesis of Microgel Particles

Microgels are colloidal-scale particles individually made of cross-linked polymer networks that can swell and deswell in response to external stimuli, such as changes to temperature or pH. Despite a large amount of experimental activities on microgels, a proper theoretical description based on individual particle properties is still missing due to the complexity of the particles. To go one step further, here we propose a novel methodology to assemble realistic microgel particles in silico. We exploit the self-assembly of a binary mixture composed of tetravalent (cross-linkers) and bivalent (monomer beads) patchy particles under spherical confinement in order to produce fully bonded networks. The resulting structure is then used to generate the initial microgel configuration, which is subsequently simulated with a bead-spring model complemented by a temperature-induced hydrophobic attraction. To validate our assembly protocol, we focus on a small microgel test case and show that we can reproduce the experimental swelling curve by appropriately tuning the confining sphere radius, something that would not be possible with less sophisticated assembly methodologies, e.g., in the case of networks generated from an underlying crystal structure. We further investigate the structure (in reciprocal and real space) and the swelling curves of microgels as a function of temperature, finding that our results are well described by the widely used fuzzy sphere model. This is a first step toward a realistic modeling of microgel particles, which will pave the way for a careful assessment of their elastic properties and effective interactions.

Solvent Induced Inversion of Core-Shell Microgels

The morphology of core-shell microgels under different swelling conditions and as a function of the core-shell thickness ratio is systematically characterized by mesoscale hydrodynamic simulations. With increasing hydrophobic interaction of the shell polymers, we observe drastic morphological changes from a core-shell structure to an inverted microgel, where the core is turned to the outside, or a microgel with a patchy surface of core polymers directly exposed to the environment. We establish a phase diagram of the various morphologies. Moreover, we characterize the polymer and microgel conformations. For sufficiently thick shells, the changes of the shell size upon increasing hydrophobic interactions are well described by the Flory-Rehner theory. Additionally, this theory provides a critical line in the phase diagram separating core-shell structures from the distinct two other phases. The appearing new phases provide a novel route to nano- and microscale functionalized materials.

Polymer Conformations in Ionic Microgels in the Presence of Salt: Theoretical and Mesoscale Simulation Results

We investigate the conformational properties of polymers in ionic microgels in the presence of salt ions by molecular dynamics simulations and analytical theory. A microgel particle consists of coarse-grained linear polymers, which are tetra-functionally crosslinked. Counterions and salt ions are taken into account explicitly, and charge-charge interactions are described by the Coulomb potential. By varying the charge interaction strength and salt concentration, we characterize the swelling of the polyelectrolytes and the charge distribution. In particular, we determine the amount of trapped mobile charges inside the microgel and the Debye screening length. Moreover, we analyze the polymer extension theoretically in terms of the tension blob model taking into account counterions and salt ions implicitly by the Debye–Hückel model. Our studies reveal a strong dependence of the amount of ions absorbed in the interior of the microgel on the electrostatic interaction strength, which is related to the degree of the gel swelling. This implies a dependence of the inverse Debye screening length κ on the ion concentration; we find a power-law increase of κ with the Coulomb interaction strength with the exponent 3/5 for a salt-free microgel and an exponent 1/2 for moderate salt concentrations. Additionally, the radial dependence of polymer conformations and ion distributions is addressed.

Competition between excluded-volume and electrostatic interactions for nanogel swelling: effects of the counterion valence and nanogel charge

The equilibrium distribution of monovalent and trivalent ions within a thermo-responsive charged nanogel is investigated using Monte Carlo simulations and Ornstein–Zernike equations.

Impact of volume transition on the net charge of poly-N-isopropyl acrylamide microgels

We explore the electrostatic properties of poly-N-isopropyl acrylamide microgels in dilute, quasi-de-ionized dispersions and show that the apparent net charge of these thermosensitive microgels is an increasing function of their size, the size being conveniently varied by temperature. Our experimental results obtained in a combination of light scattering, conductivity, and mobility experiments are consistent with those obtained in Poisson-Boltzmann cell model calculations, effectively indicating that upon shrinking the number of counterions entrapped within the microgels increases. Remarkably, this behavior shows that the electrostatic energy per particle remains constant upon swelling or deswelling the microgel, resulting in a square root dependence of the net charge on the particle radius.

Universal conformational properties of polymers in ionic nanogels

Polyelectrolyte gels are known to undergo significant conformational changes in response to external stimuli such as pH, temperature, or the dielectric constant. Specifically, an increase of the degree of ionization associated with an increasing number of counterions leads to swelling of the network. For a macroscopically large gel, which is electrostatically neutral in its interior, swelling is no longer governed by electrostatic interactions, but rather by the osmotic pressure of counterions. However, this electrostatic neutrality is typically violated for nanogels, because counterions are free to leave a gel particle. Although nanogel-swelling exhibits similar features as swelling of micro- and macrogels, another mechanism has to be relevant. Here, we use molecular dynamics simulations and scaling theory to unravel the structural properties of nanogels upon changing the electrostatic interactions. We demonstrate that the swelling of nanogels is governed by screened electrostatic interactions without a relevant contribution by the counterion osmotic pressure.

Temperature-sensitive nanogels in the presence of salt: explicit coarse-grained simulations

In this work, coarse-grained simulations of two charged thermo-shrinking nanogels (with degrees of ionization of 0.125 and 0.250) in the presence of 1:1 and 3:1 electrolytes have been explicitly performed through the bead-spring model of polyelectrolyte. In a first set of simulations, salt concentrations for 1:1 and 3:1 electrolytes ranged from 1 to 100 mM and from 0.167 to 16.7 mM, respectively, whereas temperature remained fixed at a value for which hydrophobic forces were negligible in our case (288 K). The sizes of swollen nanogels are smaller when trivalent cations are present, but they do not change significantly in the range of concentrations of 3:1 electrolyte studied here. It should be also stressed that trivalent cations neutralize the nanogel charge more efficiently. According to these results the electrostatic repulsion plays an important role. In a second set of simulations, the temperature varied from 288 to 333 K to study the effect of salt on the thermal response when hydrophobic forces are not negligible. For the nanogels with the lowest degree of ionization, the behavior of the radius with increasing the temperature can be described by a sigmoid function, which shifts towards lower temperatures in the presence of salt. This shift is more clearly observed for trivalent cations, even at low concentrations. For the nanogels with the highest degree of ionization, the effect of additional electrolyte is also noticeable. In this case, hydrophobic forces are not the only responsible for their shrinkage in the presence of trivalent cations. The surface electrostatic potential and the concentration of salt cations inside the nanogel have been computed from simulations and a modified Poisson-Boltzmann (PB) cell model. The thermosensitivity in size have certain influence on the sensitivity of these properties to temperature changes. The rich behavior of the surface electrostatic potential and the uptake of salt cations are successfully predicted by the modified PB cell model proposed (at least qualitatively). Particularly, the model is able to predict how the retention of salt cations depends on their charge and the ionic valence when nanogels shrink.

Mesoscale modelling of environmentally responsive hydrogels: emerging applications

We review recent advances in mesoscale computational modeling, focusing on dissipative particle dynamics, used to probe stimuli-sensitive behavior of hydrogels.