Swelling Behaviors of Hydrogels with Alternating Neutral/Highly Charged Sequences

Abstract

Polyelectrolyte gels comprising fixed ions exhibit swelling behaviors because of external solution conditions. Such behaviors are usually explained by using the Flory–Rehner model that considers the Donnan equilibrium. However, this model assumes a homogeneous distribution for fixed ions; therefore, its applicability to the case of heterogeneous distributions remains unclear. Here, we successfully designed a hydrogel with alternating neutral/highly charged sequences (i.e., tetrapoly­(acrylic acid)–poly­(ethylene glycol) gel). The Flory–Rehner model with the Donnan equilibrium was found to predict the swelling ratio only below pH 5.6. The distance between two neighboring fixed ions is larger than the Bjerrum length. The swelling behaviors above pH 5.6, where the fixed ions are more closely located than the Bjerrum length, can be explained by considering additionally the counterion condensation in the poly­(acrylic acid) units. These results indicate that the ionic interactions within the length scale of the network strand can determine the counterion condensation and that it is vital to estimate accurately the distance between neighboring fixed ions for predicting the swelling behaviors of polyelectrolyte gels.

Effect of Nonlinear Elasticity on the Swelling Behaviors of Highly Swollen Polyelectrolyte Gels

“…As indicated, the predictions clearly reproduce the swelling behaviors of the various Tetra-PAA-PEG gels under all external conditions examined, including the condition where the original FRGDM model fails. It is worth noting that the downward deviation at low-pH regions was attributed to the formation of the aggregation structure through hydrogen bonds between PAA and PEG, which was reported elsewhere as well [ 25 ]. This good correspondence indicates that the finite extensibility effect is essential for describing the elastic free energy of a highly swollen gel.…”

“…The network homogeneity in this gel was examined by spectroscopic and mechanical measurements, and it was found that the Tetra-PAA-PEG gel was a promising model for investigating the physicochemical properties of highly charged polyelectrolyte gels. In our previous study, we also investigated the swelling behaviors of the Tetra-PAA-PEG gel in various tuned external solutions, and validated the model describing the swelling behaviors of the Tetra-PAA-PEG gel through modification of the FRGD model with Manning’s counterion condensation model (i.e., the FRGDM model) [ 25 ].…”

Experimental Verification of the Balance between Elastic Pressure and Ionic Osmotic Pressure of Highly Swollen Charged Gels

“…When is 10 −4 M, the of the St-tetra-PEG gel with certain can be determined from the Donnan potential using Equation (14). Here, we additionally assume that γ is independent of , as in previous studies [ 16 , 21 , 32 ]. As should be inversely proportional to the swelling ratio Q , the of the St-tetra-PEG gel in the bath solution of any can be calculated by the following equation: …”

“…The other pressures of a charged gel, such as the pressure originating from electrostatic repulsion between the fixed ions, were also taken into account in the rigorous modeling of a charged gel, following some previous papers, which we assumed that such additional terms are excessively small to be ignorable relative to [ 16 , 17 ]. As the pressure of a charged gel should be balanced at its equilibrium swelling state, the equilibrium swelling state of charged gels can be expressed by Equation (1): is generally equal to its shear modulus, G [ 19 , 20 ], and can be determined from mechanical tests: follows the scaling relationship for the osmotic pressure of semidilute polymer solutions [ 21 ]: where is the polymer volume fraction, K is a parameter that depends on various factors such as the gel preparation conditions, and is the Flory exponent, which depends on the polymer–solvent affinity (0.5 for theta solvent and 0.588 for the athermal solvent). originates from the difference in the activity of the total mobile ions between the gel and the bath solution, , and is given by where k is the Boltzmann constant, T is the absolute temperature, and is the activity of the ion in the subsystem [ 16 ].…”

“…The obtained of the tetra-PEG gel corresponds well with its elastic properties, suggesting the high reliability of the determined [ 41 , 42 ]. The tetra-PEG-like system has been used to investigate the swelling mechanism of a moderately swollen charged gel [ 21 ]. Third, the molecular stent method was adopted for the independent control of the network structure and polyion density in the gel.…”