Modeling Polymer Gels Exhibiting Self-Oscillations Due to the Belousov−Zhabotinsky Reaction

Polymer gels exhibit a unique capability of undergoing significant volume changes in response to chemical, mechanical, and electrical stimulation.1-3 For example, gels can undergo rhythmic swelling and deswelling in response to variations in the pH of the surrounding solution caused by an oscillatory chemical reaction.4,5 In recent experiments, the pH oscillations were generated by the Landolt reaction6 or by an enzymatic reaction.7 Periodic volume changes were also observed in a pHresponsive swollen gel, which exhibited bistability with diffusion of a reactive substrate into the gel.8 In all of these studies, the cause of the complex behaviorsthe chemical reactionswas external to the gels. Furthermore, the chemical reactions were carried out under nonequilibrium conditions provided, for example, by precisely controlled continuous-flow stirred tank reactors.6,8 The only known responsive polymer gels that exhibit autonomous volume oscillations under stationary conditions are those involved in the Belousov-Zhabotinsky (BZ) reaction.9,10 In these systems, a ruthenium catalyst, which participates in the BZ reaction, is covalently bonded to the polymer chain, and the rest of the reagents are dissolved in the solvent.11,12 The BZ reaction in the swollen gel generates periodic redox changes of the metal ion catalyst, and the gel undergoes an expansion and contraction at the oxidized and reduced states of the catalyst, respectively.11 The cause of oscillations is purely internal because all of the necessary ingredients are present within the swollen gel; there is no chemical reaction outisde the gel. Self-oscillating gels driven by the BZ reaction have been fabricated not only as submicron-sized beads13 but also in larger, centimeter-sized cylinders and rectangular slabs, in which traveling waves of swelling were observed.12,14 These selfoscillating polymer gels could potentially be utilized in fabricating miniature peristaltic pumps, autonomously functioning membranes for pulsating drug release, and distributed actuator systems.4,12,15 Despite their potential utility and intrinsically fascinating behavior, there have been only a few theoretical studies of polymer gels exhibiting self-oscillating reactions.16,17 The studies are restricted to considering spatially uniform swelling and deswelling behavior. In the case of the bistable chemical reaction,18 the model for the nonuniform radial swelling of the 3D spherical gel does not adequately capture the gel elasticity.19 It is noteworthy that there are currently no models in the literature to describe the BZ reactions in polymer gels. Herein, we present a model for a polymer gel undergoing the BZ reaction. Using this model, we first pinpoint regions in parameter space where the gels exhibit self-oscillations and then, after incorporating the spatially nonuniform gel dynamics, study onedimensional traveling waves in the reacting gel. The polymer gel, which contains the anchored catalyst, occupies a volume fraction φ. Correspondingly, 1 φ is the volume fraction of solvent, which contains the remaining dissolved BZ reagents. To mathematically describe the BZ reaction, we adapted the well-known “Oregonator” model,10 implementing the version introduced by Tyson and Fife.20 According to this model, the reaction kinetics can be described in terms of two dimensionless variables, u and V, where u is the concentration of reagent in solution and V gives the concentration of the oxidized catalyst. We modified the original Oregonator model to explicitly take into account how the evolution of these species depends on φ (see below) and thereby describe the effect of gel swelling and deswelling on the reaction kinetics. The final form of the equations depends on the size of the gel. If the gel particle is sufficiently small that the solvent transport can be considered as essentially instantaneous, then the dimensionless equations that describe the dynamics of a reacting gel undergoing the BZ reaction have the following form: