Solute diffusion in gels: Thirty years of simulations

Transport Properties of Self-Assembling G-Hydrogels: Evidence for a Tunable Fickian Diffusivity

The mixing of Guanosine (Gua) and Guanosine 5'-monophosphate (GMP) in water in selected compositions yields highly hydrated, transparent, and self-healing self-assembled supramolecular G-hydrogels, attractive for biomedical applications. This work investigates how hydrogel composition affects solute transport, including diffusion, binding, loading, and release properties, using a set of fluorescent probes with varying size and polarity. Although small/wide-angle X-ray scattering techniques showed that no structural changes are induced by probe addition, even when intercalation into G-quadruplexes is expected, the internal mesh structure of the hydrogel, modulated by the Gua:GMP ratio, directly impacts probe diffusivity and loading. Tighter networks (e.g., 1:1) slow diffusion and enhance retention compared to looser configurations (e.g., 1:4). Moreover, UV-visible titrations revealed markedly different binding affinities (Kb ≈ 5.7 × 104 M-1 for DAPI, 8.0 × 103 M-1 for ThT, and 1.4 × 102 M-1 for RhB), which are expected to result in lower diffusion coefficients and slower release, especially for DAPI and ThT. Indeed, diffusion coefficients, obtained via fluorescence recovery after photobleaching and time-resolved fluorescence spectroscopy, reach 90, 20, and 60 μm2/s for FITC-dextran, ThT, and RhB, respectively. Probe release kinetics, modeled via Weibull fitting, indicated sustained release with characteristic times (τ) between 9.6 and 23.2 h and β ≈ 1 in 1× PBS, consistent with predominantly Fickian diffusion. Remarkably, switching to 10× PBS significantly accelerated release (τ reduced by ≈ 40-50%), suggesting that ionic strength and/or pH changes critically affect not only probe-hydrogel interactions but also the internal gel architecture, altering porosity, mesh size, and network tortuosity, thus enhancing molecular mobility. Overall, the G-hydrogel system offers a structurally tunable and composition-dependent platform capable of finely regulating molecular transport and release profiles, making it highly suitable for controlled drug delivery and adaptive biomaterial applications.

Pattern Selection in Three-Precipitate Liesegang Systems

Liesegang patterns present a display of parallel stripes of precipitate that arise from the interdiffusion of coprecipitate ions in a gel medium. The bands observed in rocks are typical analogies of this phenomenon, and their composition is not restricted to the banded deposition of a single mineral. We here extend the study to a three-precipitate system, wherein Co2+, Ni2+, and Cd2+ cations are precipitated by the same anion (OH- from NH4OH) to form Co(OH)2, Ni(OH)2, and Cd(OH)2, respectively. The resulting pattern exhibits an alternation of compact mixed Co(OH)2 and Ni(OH)2 bands with granules of pure Cd(OH)2 between them. The obtained pattern confirms the generic type of (A + B)/C/(A + B)/C/(A + B)/C alternation conjectured analytically using the competitive particle growth (CPG) model and stability analysis.

The Influence of Local Constraints on Solvent Motion in Polymer Materials

The influence of obstacles in the form of polymer chains on the diffusion of a low-molecular-weight solvent was the subject of this research. Studies were performed by computer simulations. A Monte Carlo model-the Dynamic Lattice Liquid algorithm-based on the idea of cooperative movements was used. The tested materials were polymer networks with an ideal structure (with a uniform mesh size) and real, irregular networks (with a non-uniform mesh size) obtained numerically by copolymerization. The diffusion of the solvent was analyzed in systems with a polymer concentration that did not exceed 16%. The influence of the polymer concentration and macromolecular architecture structure on the mobility and character of the motion of the solvent was discussed. The influence of irregular network morphology on solvent dynamics appeared to be significantly stronger than that of regular networks and star-like polymers.

Interplay of Spatial and Topological Defects in Polymer Networks

Polymer networks are widely used in applications, and the formation of a network and its gel point can be predicted. However, the effects of spatial and topological heterogeneity on the resulting network structure and ultimately the mechanical properties, are less understood. To address this challenge, we generate in silico random networks of cross-linked polymer chains with controlled spatial and topological defects. While all fully reacted networks investigated in this study have the same number of end-functionalized polymer strands and cross-linkers, we vary the degree of spatial and topological heterogeneities systematically. We find that spatially heterogeneous cross-linker distributions result in a reduction in the network's primary loops with increased spatial heterogeneity, the opposite trend as observed in homogeneous networks. By performing molecular dynamics simulations, we investigated the mechanical properties of the networks. Even though spatially heterogeneous networks have more elastically active strands and cross-linkers, they break at lower extensions than the homogeneous networks and sustain slightly lower maximum stresses. Their shear moduli are higher, i.e., stiffer, than theoretically predicted, and higher than their homogeneous gel counterparts. Our results highlight that topological loop defects and spatial heterogeneities result in significantly different network structures and, ultimately, different mechanical properties.

Atomic Diffusion Rate Dominated Fabrication of High Entropy Phosphide with Heterogeneous Aggregation for Oxygen Evolution Reaction

The synthesis of high-entropy phosphide (HEP) remains a great challenge owing to the different migration rates of different metallic atoms. Herein, a new metal organic gel (MOG) precursor strategy is proposed for HEP synthesis by controlling the migration rate of different atoms in an organic gel. The MOG precursor with five kinds of metal and phosphor species homogeneously dispersing is formed through a facile solvothermal method, which is calcined at 900 °C to obtain carbon-supported HEP FeCoNiMnCdP (MPC-5). The difference in the atom radius and the influence of MOG on the migration rate result in heterogeneous aggregation of different atoms in the product, which increases the defects in the product to a certain extent. In addition, the presence of carbon and nitrogen in the gel simultaneously realizes carbon coating and nitrogen doping. Combining the above advantages, the MPC-5 shows excellent oxygen evolution reaction (OER) catalytic performance with an overpotential of 250 mV at 10 mA·cm-2, superior to many recently reported OER electrocatalysts. This work provides a new strategy to solve the differences in the migration rates of different metals to obtain pure phase high-entropy phosphides, which is conducive to the further development of high-entropy materials and their applications in the energy and catalysis fields.

Single-Particle Tracking in Poly(Ethylene Glycol) Diacrylate: Probe Size Effect on the Diffusion Behaviors of Nanoparticles in Unentangled Polymer Solutions

A thorough understanding of the relevant factors governing the transport of nanoparticles in poly(ethylene glycol) diacrylate (PEGDA) is crucial for many applications utilizing this polymer. Here, single-particle tracking (SPT) was used to systematically investigate the role of the probe size (3-200 nm) on the diffusion behaviors of individual fluorescent nanoparticles in semidilute and unentangled PEGDA solutions. The quantitative assessment of the SPT data via the recorded single-particle trajectories and diffusion coefficients (D) not only showed that the observed probe dynamics in PEGDA were temporally and spatially heterogeneous, but more importantly that the measured D were observed to be significantly reduced (vs in solvent) and strongly size-dependent. We explained these results based on a modified multiscale model for particle diffusion, built upon well-established hydrodynamics and obstruction theories. We furthermore showed that the presence of steric interactions and probe confinement effects in highly crowded, unentangled PEGDA microstructures can lead to deviations in the single-particle displacements from the expected Gaussian behavior, as revealed by the van Hove displacement distributions and the associated non-Gaussian parameters. This study has demonstrated the power of SPT methods in offering an advanced characterization of the transport characteristics in complex polymer structures, overcoming challenges posed by traditional characterization techniques.

Lecithin as an Effective Modifier of the Transport Properties of Variously Crosslinked Hydrogels

Transport properties are one of the most crucial assets of hydrogel samples, influencing their main application potential, i.e., as drug carriers. Depending on the type of drug or the application itself, it is very important to be able to control these transport properties in an appropriate manner. This study seeks to modify these properties by adding amphiphiles, specifically lecithin. Through its self-assembly, lecithin modifies the inner structure of the hydrogel, which affects its properties, especially the transport ones. In the proposed paper, these properties are studied mainly using various probes (organic dyes) to effectively simulate drugs in simple release diffusion experiments controlled by UV-Vis spectrophotometry. Scanning electron microscopy was used to help characterize the diffusion systems. The effects of lecithin and its concentrations, as well as the effects of variously charged model drugs, were discussed. Lecithin decreases the values of the diffusion coefficient independently of the dye used and the type of crosslinking. The ability to influence transport properties is better observed in xerogel samples. The results, complementing previously published conclusions, showed that lecithin can alter a hydrogel's structure and therefore its transport properties.

Feedback-controlled solute transport through chemo-responsive polymer membranes

Polymer membranes are typically assumed to be inert and nonresponsive to the flux and density of the permeating particles in transport processes. Here, we theoretically study the consequences of membrane responsiveness and feedback on the steady-state force-flux relations and membrane permeability using a nonlinear-feedback solution-diffusion model of transport through a slab-like membrane. Therein, the solute concentration inside the membrane depends on the bulk concentration, c₀, the driving force, f, and the polymer volume fraction, ϕ. In our model, the solute accumulation in the membrane causes a sigmoidal volume phase transition of the polymer, changing its permeability, which, in return, affects the membrane's solute uptake. This feedback leads to nonlinear force-flux relations, j(f), which we quantify in terms of the system's differential permeability, P_sys ^Δ∝dj/df. We find that the membrane feedback can increase or decrease the solute flux by orders of magnitude, triggered by a small change in the driving force and largely tunable by attractive vs repulsive solute-membrane interactions. Moreover, controlling the inputs, c₀ and f, can lead to the steady-state bistability of ϕ and hysteresis in the force-flux relations. This work advocates that the fine-tuning of the membrane's chemo-responsiveness will enhance the nonlinear transport control features, providing great potential for future (self-)regulating membrane devices.

Optimization of Media Change Intervals through Hydrogels Using Mathematical Models

Three-dimensional cell culture in engineered hydrogels is increasingly used in tissue engineering and regenerative medicine. The transfer of nutrients, gases, and waste materials through these hydrogels is of utmost importance for cell viability and response, yet the translation of diffusion coefficients into practical guidelines is not well established. Here, we combined mathematical modeling, fluorescent recovery after photobleaching, and hydrogel diffusion experiments on cell culture inserts to provide a multiscale practical approach for diffusion. We observed a dampening effect of the hydrogel that slowed the response to concentration changes and the creation of a diffusion gradient in the hydrogel by media refreshment. Our designed model combined with measurements provides a practical point of reference for diffusion coefficients in real-world culture conditions, enabling more informed choices on hydrogel culture conditions. This model can be improved in the future to simulate more complicated intrinsic hydrogel properties and study the effects of secondary interactions on the diffusion of analytes through the hydrogel.

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.

Tuning the permeability of regular polymeric networks by the cross-link ratio

The amount of cross-linking in the design of polymer materials is a key parameter for the modification of numerous physical properties, importantly, the permeability to molecular solutes. We consider networks with a diamond-like architecture and different cross-link ratios, concurring with a wide range of the polymer volume fraction. We particularly focus on the effect and the competition of two independent component-specific solute-polymer interactions, i.e., we distinguish between chain-monomers and cross-linkers, which individually act on the solutes and are altered to cover attractive and repulsive regimes. For this purpose, we employ coarse-grained, Langevin computer simulations to study how the cross-link ratio of polymer networks controls the solute partitioning, diffusion, and permeability. We observe different qualitative behaviors as a function of the cross-link ratio and interaction strengths. The permeability can be tuned ranging over two orders of magnitude relative to the reference bulk permeability. Finally, we provide scaling theories for the partitioning and diffusion that explicitly account for the component-specific interactions as well as the cross-link ratio and the polymer volume fraction. These are in overall good agreement with the simulation results and grant insight into the underlying physics, rationalizing how the cross-link ratio can be exploited to tune the solute permeability of polymeric networks.

Microgels at interfaces, from mickering emulsions to flat interfaces and back

In this review, we cover the topic of p(NIPAM) based microgels at interfaces, revisiting classical studies in light of the newest ones. In particular, we focus on their use as emulsifiers in the so-called mickering emulsions, i.e. Pickering emulsion stabilized by soft particles. Given the complexity of the experimental characterization and simulation of these soft particles at interfaces, the review is structured in progressive complexity levels, until we reach the highly interesting and promising responsiveness to stimuli of mickering emulsions. We start from the lowest level of complexity, the current understanding of the behavior of single microgels confined at a flat interface. Then, we discuss their collective behavior upon crowding, their responsiveness at interfaces, and their macroscopic properties as microgel films. Once we have the necessary characterization tools, we proceed to discuss the complex and convoluted picture of responsive mickering emulsions. The way is rough, with current controversial and contradicting studies, but it holds promising results as well. We state open questions worth of being tackled by the Soft Matter community, and we conclude that it is worth the trouble of continuing after the master theory of microgel interfacial activity, as it will pave the way to widely adopt responsive mickering emulsions as the worthy Pickering emulsion successors.