2022

The swelling of polyelectrolyte hydrogels has beenoften explained using simple models derived from the Flory-Rehnermodel. While these models qualitatively predict the experimentallyobserved trends, they also introduce strong approximations andneglect some important contributions.

Consequently, they some-times incorrectly ascribe the observed trends to contributions whichare of minor importance under the given conditions. In this work, weinvestigate the swelling properties of weak (pH-responsive)polyelectrolyte gels at various pH and salt concentrations, using ahierarchy of models, gradually introducing various approximations.For thefirst time, we introduce a three-dimensional particle-basedmodel which accounts for the topology of the hydrogel network, forelectrostatic interactions between gel segments and small ions, andfor acid-base equilibrium coupled to the Donnan partitioning of small ions.

This model is the most accurate one; therefore, we useit as a reference when assessing the effect of various approximations. As thefirst 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 nextstep, we use the mean-field approximation to replace particles by densityfields, combining the Poisson-Boltzmann equation withelastic stretching of the chain. Finally, we introduce an ideal gel model by neglecting the electrostatics while retaining all otherfeatures of the previous model.

Comparing predictions from all four models allows us to understand which contributions dominate athigh or low pH or salt concentrations. We observe that thefield-based models overestimate the ionization degree of the gel becausethey underestimate the electrostatic interactions.

Nevertheless, a cancellation of effects on the electrostatic interactions and Donnanpartitioning causes both particle-based andfield-based models to consistently predict the swelling of the gels as a function of pH andsalt concentration. Thus, we can conclude that any of the employed models can rationalize the known experimental trends in gelswelling, however, only the particle-based models fully account for the true effects causing these trends.

The full understanding ofdifferences between various models is important when interpreting experimental results in the framework of existing theories and forascribing the observed trends to particular contributions, such as the Donnan partitioning of ions, osmotic pressure, or electrostaticinteractions.