Pattern formation in a four-ring reaction-diffusion network with heterogeneity.

In networks of nonlinear oscillators, symmetries place hard constraints on the system that can be exploited to predict universal dynamical features and steady states, providing a rare generic organizing principle for far-from-equilibrium systems. However, the robustness of this class of theories to symmetry-disrupting imperfections is untested in free-running (i.e., non-computer-controlled) systems. Here, we develop a model experimental reaction-diffusion network of chemical oscillators to test applications of the theory of dynamical systems with symmeries in the context of self-organizing systems relevant to biology and soft robotics. The network is a ring of four microreactors containing the oscillatory Belousov-Zhabotinsky reaction coupled to nearest neighbors via diffusion. Assuming homogeneity across the oscillators, theory predicts four categories of stable spatiotemporal phase-locked periodic states and four categories of invariant manifolds that guide and structure transitions between phase-locked states. In our experiments, we observed that three of the four phase-locked states were displaced from their idealized positions and, in the ensemble of measurements, appeared as clusters of different shapes and sizes, and that one of the predicted states was absent. We also observed the predicted symmetry-derived synchronous clustered transients that occur when the dynamical trajectories coincide with invariant manifolds. Quantitative agreement between experiment and numerical simulations is found by accounting for the small amount of experimentally determined heterogeneity in intrinsic frequency. We further elucidate how different patterns of heterogeneity impact each attractor differently through a bifurcation analysis. We show that examining bifurcations along invariant manifolds provides a general framework for developing intuition about how chemical-specific dynamics interact with topology in the presence of heterogeneity that can be applied to other oscillators in other topologies.

Partition, Reaction, and Diffusion Coefficients of Bromine in Elastomeric Polydimethylsiloxane.

Experiments and models were used to determine the extent to which aqueous bromine permeated into, and reacted with, the elastomer polydimethylsiloxane (PDMS). Thin films of PDMS were immersed in bromine water, and the absorbance of bromine in the aqueous phase was measured as a function of time. Kinetics were studied as a function of mass and thickness of the immersed PDMS films. We attribute the decrease of bromine in solution to permeation into PDMS, followed by a combination of diffusion, reversible binding, and an irreversible reaction with PDMS. In order to decouple the irreversible reaction from the reversible processes, kinetics were also studied for bromine-passivated PDMS films. Fits of the models to a variety of experiments yielded the partition coefficient of bromine between the water and PDMS phases, the diffusion constant of bromine in PDMS, the irreversible reaction constant between bromine and PDMS, the molar concentration of the reactive sites within PDMS, and the on and off rates of reversible binding of bromine to PDMS. Developing a quantitative reaction-diffusion model accounting for the transport of bromine through PDMS is necessary for the design of microfluidic devices fabricated using PDMS, which are used in experimental studies of the nonlinear dynamics of reaction-diffusion networks containing Belousov-Zhabotinsky chemical oscillators.

Impact of PDMS-Based Microfluidics on Belousov-Zhabotinsky Chemical Oscillators.

Sub-nanoliter volumes of the Belousov-Zhabotinsky (BZ) reaction are sealed in microfluidic devices made from polydimethylsiloxane (PDMS). Bromine, which is a BZ reaction intermediate that participates in the inhibitory pathway of the reaction, is known to permeate into PDMS, and it has been suggested that PDMS and bromine can react ( J. Phys. Chem. A. 108, 2004, 1325-1332). We characterize the extent to which PDMS affects BZ oscillations by varying the volume of the PDMS surrounding the BZ reactors. We measure how the oscillation period varies with PDMS volume and compare with a theoretical reaction-diffusion model, concluding that bromine reacts with PDMS. We demonstrate that minimizing the amount of PDMS by making the samples as thin as possible maximizes the number of oscillations before the BZ reaction reaches equilibrium and ceases to oscillate. We also demonstrate that the deleterious effects of the PDMS-BZ interactions are somewhat mitigated by imposing constant chemical boundary conditions through using a light-sensitive catalyst, ruthenium, in combination with patterned illumination. Furthermore, we show that light can modulate the frequency and phase of the BZ oscillators contained in a PDMS matrix by 20-30%.