Electrostatic hindrance to diffusion in flexible crosslinked gels: A coarse-grained simulation study.

In this work, we study how electrostatic forces slow down the diffusion of solute in flexible gels through coarse-grained simulations. The model used explicitly considers the movement of solute particles and polyelectrolyte chains. These movements are performed by following a Brownian dynamics algorithm. The effect of three electrostatic parameters characterizing the system (solute charge, polyelectrolyte chain charge, and ionic strength) is analyzed. Our results show that the behavior of both the diffusion coefficient and the anomalous diffusion exponent changes upon the reversal of the electric charge of one of the species. In addition, the diffusion coefficient in flexible gels differs significantly from that in rigid gels if the ionic strength is low enough. However, the effect of chain flexibility on the exponent of anomalous diffusion is significant even at high ionic strength (100 mM). Our simulations also prove that varying the polyelectrolyte chain charge does not have exactly the same effect as varying the solute particle charge.

Universal description of steric hindrance in flexible polymer gels.

In this work, the long-time diffusion of a solute in a chemically crosslinked and flexible hydrogel is computed from a bead-spring model of a polymeric network to assess the effect of steric obstruction. The relative diffusivities obtained for a wide variety of systems can be described by an exponential decay depending on a parameter that differs from that employed for rigid gels. The mathematical expression derived here can approximately predict the diffusivity in flexible gels if steric hindrance is the mechanism ruling diffusion.

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.

Maximizing the absorption of small cosolutes inside neutral hydrogels: steric exclusion versus hydrophobic adhesion.

In this work the equilibrium absorption of nanometric cosolutes (which could represent drugs, reactants, small globular proteins and other kind of biomacromolecules) inside neutral hydrogels is studied. We specially focus on exploring, for different swelling states, the competition between the steric exclusion induced by the cross-linked polymer network constituting the hydrogel, and the solvent-induced short-range hydrophobic attraction between the polymer chains and the cosolute particle. For this purpose, the cosolute partition coefficient is calculated by means of coarse-grained grand canonical Monte Carlo simulations, and the results are compared to theoretical predictions based on the calculation of the excluded and binding volume around the polymer chains. For small hydrophobic attractions or large cosolute sizes, the steric repulsion dominates, and the partition coefficient decreases monotonically with the polymer volume fraction, ϕm. However, for large enough hydrophobic attraction strength, the interplay between hydrophobic adhesion and the steric exclusion leads to a maximum in the partition coefficient at certain intermediate polymer density. Good qualitative and quantitative agreement is achieved between simulation results and theoretical predictions in the limit of small ϕm, pointing out the importance of geometrical aspects of the cross-linked polymer network, even for hydrogels in the swollen state. In addition, the theory is able to predict analytically the onset of the maximum formation in terms of the details of the cosolute-monomer pair interaction, in good agreement with simulations too. Finally, the effect of the many-body attractions between the cosolute and multiple polymer chains is quantified. The results clearly show that these many-body attractions play a very relevant role determining the cosolute binding, enhancing its absorption in more than one order of magnitude.

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

In this work the equilibrium distribution of ions around a thermo-responsive charged nanogel particle in an electrolyte aqueous suspension is explored using coarse-grained Monte Carlo computer simulations and the Ornstein-Zernike integral equation theory. We explicitly consider the ionic size in both methods and study the interplay between electrostatic and excluded-volume effects for swollen and shrunken nanogels, monovalent and trivalent counterions, and for two different nanogel charges. We find good quantitative agreement between the ionic density profiles obtained using both methods when the excluded repulsive force exerted by the cross-linked polymer network is taken into account. For the shrunken conformation, the electrostatic repulsion between the charged groups provokes a heterogeneous polymer density profile, leading to a nanogel structure with an internal low density hole surrounded by a dense corona. The results show that the excluded-volume repulsion strongly hinders the ion permeation for shrunken nanogels, where volume exclusion is able to significantly reduce the concentration of counterions in the more dense regions of the nanogel. In general, we demonstrate that the thermosensitive behaviour of nanogels, as well as their internal structure, is strongly influenced by the valence of the counterions and also by the charge of the particles. On the one hand, an increase of the counterion valence moves the swelling transition to lower temperatures, and induces a major structuring of the charged monomers into internal and external layers around the crown for shrunken nanogels. On the other hand, increasing the particle charge shifts the swelling curve to larger values of the effective radius of the nanogel.

The non-dominance of counterions in charge-asymmetric electrolytes: non-monotonic precedence of electrostatic screening and local inversion of the electric field by multivalent coions.

The asymptotic convergence of the thermodynamic and structural properties of unequally-sized charge-symmetric ions in strong electric fields was postulated more than thirty years ago by Valleau and Torrie as the dominance of counterions via the non-linear Poisson-Boltzmann theory [Valleau and Torrie, J. Chem. Phys., 1982, 76, 4623]. According to this mean field prescription, the properties of the electrical double layer near a highly charged electrode immersed in a size-asymmetric binary electrolyte converge to those of a size-symmetric electrolyte if the properties of counterions are the same in both instances. On the other hand, some of the present authors have shown that, in fact, counterions do not dominate the electrical properties of a spherical macroion in the presence of unequally-sized ions, symmetric in valence, if ion correlations and ionic excluded volume effects are taken into account consistently. These ingredients are neglected in the classical Poisson-Boltzmann picture. In the present work, we show the occurrence of the non-dominance of counterions in the opposite scenario, that is, when ions are equally-sized but asymmetric in valence. This is performed in the presence of highly charged colloidal surfaces of spherical and planar geometries for different ionic volume fractions. In addition to the phenomenon of non-dominance of counterions, our simulations and theoretical data also exhibit a non-monotonic order or precedence in the mean electrostatic potential, or electrostatic screening, at the Helmholtz plane of a charged colloid. This interesting behaviour is analyzed as a function of the coion's valence, the ionic volume fraction, and the charge and size of the colloidal particle. All these phenomena are explained in terms of the decay of the electric field near the colloidal surface, and by the appearance of a local inversion of both the electric field and the integrated surface charge density of the colloidal particle in the presence of monovalent counterions and multivalent coions.

Specific ion effects on the electrokinetic properties of iron oxide nanoparticles: experiments and simulations.

We report experimental and simulation studies on ion specificity in aqueous colloidal suspensions of positively charged, bare magnetite nanoparticles. Magnetite has the largest saturation magnetization among iron oxides and relatively low toxicity, which explain why it has been used in multiple biomedical applications. Bare magnetite is hydrophilic and the sign of the surface charge can be changed by adjusting the pH, its isoelectric point being in the vicinity of pH = 7. Electrophoretic mobility of our nanoparticles in the presence of increasing concentrations of different anions showed that anions regarded as kosmotropic are more efficient in decreasing, and even reversing, the mobility of the particles. If the anions were ordered according to the extent to which they reduced the particle mobility, a classical Hofmeister series was obtained with the exception of thiocyanate, whose position was altered. Monte Carlo simulations were used to predict the diffuse potential of magnetite in the presence of the same anions. The simulations took into account the ion volume, and the electrostatic and dispersion forces among the ions and between the ions and the solid surface. Even though no fitting parameters were introduced and all input data were estimated using Lifshitz theory of van der Waals forces or obtained from the literature, the predicted diffusion potentials of different anions followed the same order as the mobility curves. The results suggest that ionic polarizabilities and ion sizes are to a great extent responsible for the specific ion effects on the electrokinetic potential of iron oxide particles.

Using AFM to probe the complexation of DNA with anionic lipids mediated by Ca(2+): the role of surface pressure.

Complexation of DNA with lipids is currently being developed as an alternative to classical vectors based on viruses. Most of the research to date focuses on cationic lipids owing to their spontaneous complexation with DNA. Nonetheless, recent investigations have revealed that cationic lipids induce a large number of adverse effects on DNA delivery. Precisely, the lower cytotoxicity of anionic lipids accounts for their use as a promising alternative. However, the complexation of DNA with anionic lipids (mediated by cations) is still in early stages and is not yet well understood. In order to explore the molecular mechanisms underlying the complexation of anionic lipids and DNA we proposed a combined methodology based on the surface pressure-area isotherms, Gibbs elasticity and Atomic Force Microscopy (AFM). These techniques allow elucidation of the role of the surface pressure in the complexation and visualization of the interfacial aggregates for the first time. We demonstrate that the DNA complexes with negatively charged model monolayers (DPPC/DPPS 4 : 1) only in the presence of Ca(2+), but is expelled at very high surface pressures. Also, according to the Gibbs elasticity plot, the complexation of lipids and DNA implies a whole fluidisation of the monolayer and a completely different phase transition map in the presence of DNA and Ca(2+). AFM imaging allows identification for the first time of specific morphologies associated with different packing densities. At low surface coverage, a branched net like structure is observed whereas at high surface pressure fibers formed of interfacial aggregates appear. In summary, Ca(2+) mediates the interaction between DNA and negatively charged lipids and also the conformation of the ternary system depends on the surface pressure. Such observations are important new generic features of the interaction between DNA and anionic lipids.

Excluded volume effects on ionic partitioning in gels and microgels: a simulation study.

In this work the effect of volume exclusion on ionic partitioning in swollen and moderately collapsed gels has been studied through coarse-grained simulations. Our results show that finite size effects yield deviations from the classical theory of Donnan exclusion. At low or moderate reservoir electrolyte concentration these discrepancies become important if one of the ions has diameters of just a few nanometers. When the reservoir electrolyte concentration grows, volume exclusion can lead to a drastic failure of the ideal Donnan exclusion even for conventional hydrated monoatomic ions. In addition, an approximate analytical expression for the partition coefficient of ionic species including the volume exclusion associated with the polymer network and the neutralizing counterions has been proposed and tested. This theoretical approach also provides an expression for the Donnan potential difference that takes such effects into account. Good agreement between theory and simulations is found for slightly and moderately charged gels (both at low and high reservoir electrolyte concentrations). The theory also works acceptably for highly charged gels at high salt concentrations or for electrolytes with large counterions.

Polyelectrolyte adsorption onto like-charged surfaces mediated by trivalent counterions: a Monte Carlo simulation study.

Both experiments and theory have evidenced that multivalent cations can mediate the interaction between negatively charged polyelectrolytes and like-charged objects, such as anionic lipoplexes (DNA-cation-anionic liposome complexes). In this paper, we use Monte Carlo simulations to study the electrostatic interaction responsible for the trivalent-counterion-mediated adsorption of polyelectrolytes onto a like-charged planar surface. The evaluation of the Helmholtz free energy allows us to characterize both the magnitude and the range of the interaction as a function of the polyelectrolyte charge, surface charge density, [3:1] electrolyte concentration, and cation size. Both polyelectrolyte and surface charge favor the adsorption. It should be stressed, however, that the adsorption will be negligible if the surface charge density does not exceed a threshold value. The effect of the [3:1] electrolyte concentration has also been analyzed. In certain range of concentrations, the counterion-mediated attraction seems to be independent of this parameter, whereas very high concentrations of salt weaken the adsorption. If the trivalent cation diameter is doubled the adsorption moderates due to the excluded volume effects. The analysis of the integrated charge density and ionic distributions suggests that a delicate balance between charge inversion and screening effects governs the polyelectrolyte adsorption onto like-charged surfaces mediated by trivalent cations.

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.

Effect of calcium and magnesium on phosphatidylserine membranes: experiments and all-atomic simulations.

It is known that phosphatidylserine (PS(-)) lipids have a very similar affinity for Ca(2+) and Mg(2+) cations, as revealed by electrokinetic and stability experiments. However, despite this similar affinity, experimental evidence shows that the presence of Ca(2+) or Mg(2+) induces very different aggregation behavior for PS(-) liposomes as characterized by their fractal dimensions. Also, turbidity measurements confirm substantial differences in aggregation behavior depending on the presence of Ca(2+) or Mg(2+) cations. These puzzling results suggest that although these two cations have a similar affinity for PS(-) lipids, they induce substantial structural differences in lipid bilayers containing each of these cations. In other words, these cations have strong ion-specific effects on the structure of PS(-) membranes. This interpretation is supported by all-atomic molecular-dynamics simulations showing that Ca(2+) and Mg(2+) cations have different binding sites and induce different membrane hydration. We show that although both ions are incorporated deep into the hydrophilic region of the membrane, they have different positions and configurations at the membrane. Absorbed Ca(2+) cations present a peak at a distance ~2 nm from the center of the lipid bilayer, and their most probable binding configuration involves two oxygen atoms from each of the charged moieties of the PS molecule (phosphate and carboxyl groups). In contrast, the distribution of absorbed Mg(2+) cations has two different peaks, located a few angstroms before and after the Ca(2+) peak. The most probable configurations (corresponding to these two peaks) involve binding to two oxygen atoms from carboxyl groups (the most superficial binding peak) or two oxygen atoms from phosphate groups (the most internal peak). Moreover, simulations also show differences in the hydration structure of the membrane: we obtained a hydration of 7.5 and 9 water molecules per lipid in simulations with Ca(2+) and Mg(2+), respectively.

Computer simulations of thermo-sensitive microgels: quantitative comparison with experimental swelling data.

In this work, a quantitative comparison between experimental swelling data of thermo-sensitive microgels and computer simulation results obtained from a coarse-grained model of polyelectrolyte network and the primitive model of electrolyte is carried out. Polymer-polymer hydrophobic forces are considered in the model through a solvent-mediated interaction potential whose depth increases with temperature. The qualitative agreement between simulation and experiment is very good. In particular, our simulations predict a gradual shrinkage with temperature, which is actually observed for the microgels studied in this survey. In addition, the model can explain the swelling behavior for different contents of ionizable groups without requiring changes in the hydrophobic parameters. Our work also reveals that the abruptness of the shrinkage of charged gels is considerably conditioned by the number of monomeric units per chain. The swelling data are also analyzed with the Flory-Rhener theory, confirming some limitations of this classical formalism.

Effects of the internal structure of spheroidal divalent ions on the charge density profiles of the electric double layer.

In this work, the effects of the internal structure of charge for ions are analyzed by means of Monte Carlo simulations within a modified primitive model of electric double layer with spheroidal ions. The simulation results are compared to those obtained from a generalized Poisson-Boltzmann theory, where the separation of the charges within the spheroidal ions is considered. The spheroidal divalent ions have finite dimensions and two identical unitary charges separated by a distance of one diameter. Two structurally equivalent but oppositely charged ionic species are considered: coions and counterions. In the simulation, the number of particles is not fixed and the grand canonical ensemble is employed to reach the thermodynamic equilibrium. Meanwhile, the variational theory is applied to the analytical density functional. The fixed separation between charges within the spheroidal ions causes the orientational ordering of the spheroidal ions (with quadrupolar charge distributions) leading to very different charge distributions than those of the regular divalent ions from the primitive model of electrolyte. The internal structure of ions could be dramatically relevant for the modelling of large molecules, which are known to posses complex charge distributions.

Complexation of DNA with Thermoresponsive Charged Microgels: Role of Swelling State and Electrostatics.

Micro- and nanogels are being increasingly used to encapsulate bioactive compounds. Their soft structure allows large loading capacity while their stimuli responsiveness makes them extremely versatile. In this work, the complexation of DNA with thermoresponsive microgels is presented. To this end, PEGylated charged microgels based on poly-N-isopropylacrylamide have been synthesized, allowing one to explore the electrostatics of the complexation. Cationic microgels complexate spontaneously by electrostatic attraction to oppositely charged DNA as demonstrated by electrophoretic mobility of the complexes. Then, Langmuir monolayers reveal an increased interaction of DNA with swollen microgels (20 °C). Anionic microgels require the presence of multivalent cations (Ca2+) to promote the complexation, overcoming the electrostatic repulsion with negatively charged DNA. Then again, Langmuir monolayers evidence their complexation at the surface. However, the presence of Ca2+ seems to induce profound changes in the interaction and surface conformation of anionic microgels. These alterations are further explored by measuring adsorbed films with the pendant drop technique. Conformational changes induced by Ca2+ on the structure of the microgel can ultimately affect the complexation with DNA and should be considered in the design. The combination of microstructural and surface properties for microgels offers a new perspective into complexation of DNA with soft particles with biomedical applications.

Coarse-Grained Simulations of Release of Drugs Housed in Flexible Nanogels: New Insights into Kinetic Parameters.

The diffusion-controlled release of drugs housed in flexible nanogels has been simulated with the help of a coarse-grained model that explicitly considers polymer chains. In these in silico experiments, the effect of its flexibility is assessed by comparing it with data obtained for a rigid nanogel with the same volume fraction and topology. Our results show that the initial distribution of the drug can exert a great influence on the release kinetics. This work also reveals that certain surface phenomena driven by steric interactions can lead to apparently counterintuitive behaviors. Such phenomena are not usually included in many theoretical treatments used for the analysis of experimental release kinetics. Therefore, one should be very careful in drawing conclusions from these formalisms. In fact, our results suggest that the interpretation of drug release curves in terms of kinetic exponents (obtained from the Ritger-Peppas Equation) is a tricky question. However, such curves can provide a first estimate of the drug diffusion coefficient.

Synergic Thermo- and pH-Sensitive Hybrid Microgels Loaded with Fluorescent Dyes and Ultrasmall Gold Nanoparticles for Photoacoustic Imaging and Photothermal Therapy.

Smart microgels (µGels) made of polymeric particles doped with inorganic nanoparticles have emerged recently as promising multifunctional materials for nanomedicine applications. However, the synthesis of these hybrid materials is still a challenging task with the necessity to control several features, such as particle sizes and doping levels, in order to tailor their final properties in relation to the targeted application. We report herein an innovative modular strategy to achieve the rational design of well-defined and densely filled hybrid particles. It is based on the assembly of the different building blocks, i.e., µGels, dyes, and small gold nanoparticles (<4 nm), and the tuning of nanoparticle loading within the polymer matrix through successive incubation steps. The characterization of the final hybrid networks using UV-vis absorption, fluorescence, transmission electron microscopy, dynamic light scattering, and small-angle X-ray scattering revealed that they uniquely combine the properties of hydrogel particles, including high loading capacity and stimuli-responsive behavior, the photoluminescent properties of dyes (rhodamine 6G, methylene blue and cyanine 7.5), and the features of gold nanoparticle assembly. Interestingly, in response to pH and temperature stimuli, the smart hybrid µGels can shrink, leading to the aggregation of the gold nanoparticles trapped inside the polymer matrix. This stimuli-responsive behavior results in plasmon band broadening and red shift toward the near-infrared region (NIR), opening promising prospects in biomedical science. Particularly, the potential of these smart hybrid nanoplatforms for photoactivated hyperthermia, photoacoustic imaging, cellular internalization, intracellular imaging, and photothermal therapy was assessed, demonstrating well controlled multimodal opportunities for theranostics.

Influence of monovalent ion size on colloidal forces probed by Monte Carlo simulations.

The present work studies the role of ionic size in the interactions between the electrical double layers of colloids immersed into electrolyte solutions of monovalent ions. Such interactions are studied by means of Monte Carlo (MC) simulations and the classical Derjaguin-Landau-Verwey-Overbeek (DLVO) theory. Despite the omission of the steric effects and some other features of real electrolyte solutions, DLVO theory is known to work qualitatively well for 1:1 electrolyte solutions. However, this affirmation is based on previous tests where an ionic diameter around 0.4 nm was taken for all ionic species. In contrast, some experimental studies suggest that larger hydrated ions should be considered and even specified for each type of ion. In this work, the importance of ionic size is analyzed by applying the primitive model of electrolyte to the intermediate region between a pair of equally charged infinite planar surfaces. The double layer interactions were calculated from the ionic densities at the distance of closest approach to the charged surfaces, this method constitutes an alternative to the traditional calculations at the midplane. Our MC simulations predict the existence of negative net pressures for monovalent electrolytes in the case of zero charge density. In addition, MC simulations reveal some disagreements with theoretical predictions for ionic diameters larger than 0.4 nm. These discrepancies can become significant if surface charge density is large enough due to the restructuration of the double layer. The physical mechanisms for these deviations are also discussed.

Computer simulations of thermo-shrinking polyelectrolyte gels.

In this work, thermo-responsive polyelectrolyte gels have been simulated using polymer networks of diamond-like topology in the framework of the primitive model. Monte Carlo simulations were performed in the canonical ensemble and a wide collection of situations has been systematically analysed. Unlike previous studies, our model includes an effective solvent-mediated potential for the hydrophobic interaction between non-bonded polymer beads. This model predicts that the strength of the attractive hydrophobic forces increases with temperature, which plays a key role in the explanation of the thermo-shrinking behaviour of many real gels. Although this hydrophobic model is simple (and it could overestimate the interactions at high temperature), our simulation results qualitatively reproduce several features of the swelling behaviour of real gels and microgels reported by experimentalists. This agreement suggests that the effective solvent-mediated polymer-polymer interaction used here is a good candidate for hydrophobic interaction. In addition, our work shows that the functional form of the hydrophobic interaction has a profound influence on the swelling behaviour of polyelectrolyte gels. In particular, systems with weak hydrophobic forces exhibit discontinuous volume changes, whereas gels with strong hydrophobic forces do not show hallmarks of phase transitions, even for highly charged polyelectrolyte chains.

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.