Structure-property relations for equilibrium swelling of cationic gels

Highlights

• Governing equations are reported for equilibrium swelling of cationic gels.

• The ability of the model to predict observations is confirmed by simulation.

• Relations are established between material constants and composition of a gel.

Abstract

Governing equations are developed for equilibrium swelling of a cationic polyelectrolyte gel in aqueous solutions with various pH and molar fractions of a monovalent salt. Material constants are determined by fitting observations on several homo- and copolymer gels in two types of tests: when swelling is conducted in water baths with varied pH and a fixed molar fraction of salt, and when experiments are performed in water baths with a fixed pH and varied concentration of salt. Phenomenological relations are suggested for the effect of total volume fraction of monomers in a pre-gel solution, molar fraction of ionic monomers, and molar fraction of cross-linker on adjustable parameters in the model.

Introduction

Hydrogels are three-dimensional networks of polymer chains connected by covalent bonds, physical cross-links and hydrogen bonds. When a dry gel is immersed into a water bath, it swells retaining structural integrity and ability to withstand large deformations. Stimuli-responsive gels form an important class of hydrogels whose equilibrium degree of swelling and kinetics of water uptake are strongly affected by temperature, pH, ionic strength, electric field, and light [1], [2]. These materials have recently became a focus of attention as they demonstrate potential for a wide range of “smart” applications including biomedical devices, drug delivery carriers, scaffolds for tissue engineering, filters and membranes for selective diffusion, sensors for on-line process monitoring and soft actuators [3], [4], [5], [6], [7], [8].

In polyelectrolyte gels, functional groups (attached to main or side chains of the polymer network) are ionized when pH of water is altered. Depending on the charge of bound groups, anionic, cationic, and ampholyte gels are distinguished. This study deals with water uptake by cationic gels whose functional groups are ionized due to proton exchange reaction with mobile hydronium ions.

We focus on the analysis of equilibrium swelling of cationic gels prepared by cross-linking polymerization of monomers in an aqueous solution. Composition of an homopolymer gel is characterized by two parameters: (i) volume fraction of monomers ${\varphi }_{\text{m}}$ in a pre-gel solution and (ii) molar fraction of cross-linker ${\rho }_{\text{c}}$ (the ratio of molar mass of a cross-linker to the molar mass of monomers). Composition of a copolymer gel is described by three parameters: (i) volume fraction of monomers ${\varphi }_{\text{m}}$ in a pre-gel solution, (ii) molar fraction of ionic monomers ${\rho }_{\text{i}}$ (the ratio of molar mass of ionic monomers to the total molar mass of monomers), and (iii) molar fraction of a cross-linker ${\rho }_{\text{c}}$. The effect of these quantities on degree of swelling of cationic gels under unconstrained swelling was analyzed experimentally in [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], to mention a few. To examine how degree of swelling is affected by parameters ${\varphi }_{\text{m}}$, ${\rho }_{\text{i}}$, and ${\rho }_{\text{c}}$, two types of water uptake tests are conventionally employed. In the first type of experiments, degree of swelling Q is measured as a function of pH of water, whereas in the other type of tests, this quantity is determined as a function of molar fraction $\theta$ of a monovalent salt in an aqueous solution with a fixed pH. In this study, a constitutive model is reported that allows observations in one type of experiments to be predicted by using experimental data in the other type of tests, and its adjustable parameters are determined by fitting experimental swelling diagrams.

Modeling and simulation of the mechanical response of pH-sensitive gels under three-dimensional deformations with finite strains accompanied by swelling has attracted substantial attention in the past decade when it has been recognized that their equilibrium water uptake is strongly affected by geometrical constraints [21], [22]. Coupled equations for the elastic behavior of polyelectrolyte gels and transport of solvent and solutes were recently developed in [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33].

In our previous study [33], a model has been derived for anionic gels that allows an equilibrium swelling diagram in water uptake tests with varying pH to be re-calculated into that in experiments with varying $\theta$ and vice versa. The aim of this work is (i) to report an analog of this model for cationic gels, (ii) to validate the governing equations by comparison of results of simulation with observations when both experimental dependencies $Q(\text{pH})$ and $Q(\theta )$ are provided, and (iii) to develop phenomenological relations that describe the effect of ${\varphi }_{\text{m}}$, ${\rho }_{\text{i}}$, and ${\rho }_{\text{c}}$ on adjustable parameters in the model.

The novelty of the present study consists in the following: (i) it is demonstrated that the effect of molar fraction of salt on degrees of ionization of anionic and cationic gels is described by different equations, and (ii) structure-property relations are established for a cationic gel that allow its equilibrium degree of swelling Q to be predicted for a given composition (total molar fraction of monomers in a pre-gel solution, concentration of ionized functional groups, and concentration of a cross-linker). The latter is of essential importance for design of pH-sensitive gels with required swelling abilities.

The exposition is organized as follows. Governing equations for equilibrium unconstrained swelling of a cationic gel are reported in Section 2. A detailed derivation of these relations is presented in Supporting Information. Validation of the model is performed in Section 3. In Section 4, material constants are determined by fitting experimental water uptake diagrams on several homopolymer and copolymer gels, and phenomenological relations are suggested to describe how adjustable parameters are influenced by ${\varphi }_{\text{m}}$, ${\rho }_{\text{i}}$, and ${\rho }_{\text{c}}$. Concluding remarks are formulated in Section 5.