The pH-dependent Swelling of Weak Polyelectrolyte Hydrogels Modeled at Different Levels of Resolution

Abstract

The swelling of polyelectrolyte hydrogels has been often explained using simple models derived from the Flory-Rehner model. While these models qualitatively predict the experimentally observed trends, they also introduce strong approximations and neglect some important contributions. Consequently, they sometimes incorrectly ascribe the observed trends to contributions which are of minor importance under the given conditions. In this work, we investigate the swelling properties of weak (pH-responsive) polyelectrolyte gels at various pH and salt concentrations, using a hierarchy of models, gradually introducing various approximations. For the first time, we introduce a three-dimensional particle-based model which accounts for the topology of the hydrogel network, for electrostatic interactions between gel segments and small ions and for acid-base equilibrium coupled to the Donnan partitioning of small ions. This model is the most accurate one, therefore, we use it as a reference when assessing the effect of various approximations. As the first approximation, we introduce the affine deformation, which allows us to replace the network of many chains by a single chain, while retaining the particle-based representation. In the next step, we use the mean-field approximation to replace particles by density fields, combining the Poisson-Boltzmann equation with elastic stretching of the chain. Finally, we introduce an ideal gel model by neglecting the electrostatics while retaining all other features of the previous model. Comparing predictions from all four models allows us to understand which contributions dominate at high or low pH or salt concentrations. We observe that the field-based models overestimate the ionization degree of the gel because they underestimate the electrostatic interactions. Nevertheless, a cancellation of effects on the electrostatic interactions and Donnan partitioning causes that both particle-based and field-based models consistently predict the swelling of the gels as a function of pH and salt concentration. Thus, we can conclude that any of the employed models can rationalize the known experimental trends in gel swelling, however, only the particle-based models fully account for the true effects causing these trends. The full understanding of differences between various models is important when interpreting experimental results in the framework of existing theories and for ascribing the observed trends to particular contributions, such as the Donnan partitioning of ions, osmotic pressure or electrostatic interactions.

The pH-Dependent Swelling of Weak Polyelectrolyte Hydrogels Modeled at Different Levels of Resolution

“…Figure (left panel) shows that, as the pH in the reservoir is increased, the intersection with P = 0 shifts to higher volumes, indicating an increased swelling of the gel as its ionization increases. Figure (right panel) shows that different models yield slightly different shapes of P ( L ) curves; however, they follow the same general trend as a function of pH and added salt (see ref for other values of pH and c 0 ). In the following, we show how the ionization of the gel at free swelling equilibrium is related to the pH value in the reservoir, and how these two quantities determine the pH-dependent swelling of the gel.…”

“…For brevity, we provide below just a simplified description of our models, focusing mainly on differences between them. The full technical details, including the values of all parameters, are provided in ref .…”

“…The system is coupled to a reservoir by inserting and deleting ion pairs, formally represented as chemical reactions where ⌀ denotes an empty set. The equilibrium constants of these reactions are determined by the reservoir pH and salt concentration, as detailed in refs and . All particle-based simulations are performed using the simulation software ESPResSo …”

“…As in the case of the particle-based single-chain CGM, we assume an affine transformation of the network and set the cylinder height to the value of L z = R e + b that leads to the cylinder volume The radius of the cylinder is then given by For simplicity, we assume that chain monomers are homogeneously distributed inside an inner cylinder of radius R in , so that the concentration of monomers at a distance from the cylinder axis is where θ is the Heaviside step function. We fix the inner cylinder radius R in by matching the radii of gyration in the x – y plane of the cylindrical distribution and of a stretched chain in the direction perpendicular to the stretching force (see ref for details).…”

“…The Langevin function rather than the commonly employed Gaussian stretching approximation is used because of the high extension that the chains can reach at high ionization degrees . A detailed discussion of the different contributions can be found in ref . The pressure in the reservoir is calculated from the concentrations in the reservoir using the ideal gas law …”