Collective chemomechanical oscillations in active hydrogels

Abstract

We report on the collective response of an assembly of chemomechanical Belousov-Zhabotinsky (BZ) hydrogel beads. We first demonstrate that a single isolated spherical BZ hydrogel bead with a radius below a critical value does not oscillate, whereas an assembly of the same BZ hydrogel beads presents chemical oscillation. A BZ chemical model with an additional flux of chemicals out of the BZ hydrogel captures the experimentally observed transition from oxidized nonoscillating to oscillating BZ hydrogels and shows this transition is due to a flux of inhibitors out of the BZ hydrogel. The model also captures the role of neighboring BZ hydrogel beads in decreasing the critical size for an assembly of BZ hydrogel beads to oscillate. We finally leverage the quorum sensing behavior of the collective to trigger their chemomechanical oscillation and discuss how this collective effect can be used to enhance the oscillatory strain of these active BZ hydrogels. These findings could help guide the eventual fabrication of a swarm of autonomous, communicating, and motile hydrogels.

Keywords: chemomechanics; nonlinear dynamic; reaction–diffusion.

Fig. 1.

Experiments on the chemical oscillation of single BZ hydrogel beads. (A) Schematic of the experimental setup. (B) Classification of the gel in oscillating state (Osc.) and nonoscillating oxidized state (No Osc.) as a function of its radius for various BZ hydrogel beads compositions. The color code for the data points stands for the ruthenium catalyst concentration in the BZ hydrogel bead. The dark line represents the probability of being oxidized or oscillating as a function of the BZ hydrogel bead radius for all the composition of BZ hydrogel beads. We define the critical radius $R_{\mathrm{c},\exp }$ as the radius at which the probability for a BZ hydrogel bead to oscillate is $\frac{1}{2}$. We find $R_{\mathrm{c},\exp }$ = (225 $\pm 9)\mathrm{\mu }$m with the error bar corresponding to a $95\%$ CI. (C) Intensity as a function of time of a BZ hydrogel bead with a radius below $R_{\mathrm{c},\exp }$. The gel stays oxidized. (D) Intensity as a function of time of a BZ hydrogel bead with a diameter above $R_{\mathrm{c},\exp }$. For some BZ hydrogel beads, the oscillation stops before reaching the regime of steady state oscillation (SI Appendix, section 4).

Fig. 2.

Experiments on the collective chemical oscillation of BZ hydrogel beads. (A) Classification of the BZ hydrogel bead in oscillating state (Osc.) and nonoscillating oxidized state (No Osc) as a function of its radius for various BZ hydrogel bead compositions. In dark, probability of being oxidized or oscillating as a function of the BZ hydrogel bead radius for a single BZ hydrogel bead. (B) Image of collection of BZ hydrogel beads all smaller than $R_{\mathrm{c},\exp }$, the critical radius for a single BZ hydrogel bead to oscillate. The deep and light orange BZ hydrogel beads are made with [Ru²⁺] = 15 mM and [Ru²⁺] = 3 mM respectively. The scale bar is 250 $\mathrm{\mu }$m. (C) Ratio of period $\frac{T}{T_0}$ as a function of Ruthenium catalyst concentration in the steady state oscillation regime, i.e., when the period of oscillation is constant, for all BZ hydrogel bead sizes. $T_0$ is the mean oscillation period of the background BZ hydrogel beads with [Ru²⁺] = 3 mM.

Fig. 3.

Simulation results obtained with the theoretical reaction–diffusion model coupling the Vanag–Epstein model to additional loss terms of chemicals related diffusion or diffusion/reaction of chemicals. In all the simulations, [Ru²⁺] = 1 mM. (A) Phase diagram representing the chemical state of a BZ hydrogel bead as a function of the rates of loss of the activator, ${\alpha }_{{\mathrm{HBrO}}_2}$, and inhibitor ${\alpha }_{{\mathrm{Br}}^{-}}$. The green, red, and gray regions represent the states of the BZ hydrogel bead: oxidized, reduced, and oscillatory state, respectively. The dotted, dotted-dashed, and dashed lines correspond to a change in the radius of a BZ hydrogel bead. The slope corresponds to $\frac{L_{{\mathrm{HBrO}}_2}}{L_{{\mathrm{Br}}^{-}}}$ = 1, $\frac{L_{{\mathrm{HBrO}}_2}}{L_{{\mathrm{Br}}^{-}}}$ = 10, and $\frac{L_{{\mathrm{HBrO}}_2}}{L_{{\mathrm{Br}}^{-}}}$ = 100 respectively. (B) Simulation results of an oxidized BZ hydrogel bead (${\alpha }_{{\mathrm{Br}}^{-}}$ = 1; ${\alpha }_{{\mathrm{HBrO}}_2}$ = $5\times {10}^{-2}$), of a reduced BZ hydrogel bead (${\alpha }_{{\mathrm{Br}}^{-}}$ = 1; ${\alpha }_{{\mathrm{HBrO}}_2}$ = 5) and of an oscillating BZ hydrogel bead (${\alpha }_{{\mathrm{Br}}^{-}}$ = 1; ${\alpha }_{{\mathrm{HBrO}}_2}$ = $5\times {10}^{-1}$). (C) State of the oscillation of the BZ hydrogel bead as a function of its radius, for L_HBrO2 = 144 $\mathrm{\mu }$m, L_Br⁻ = 34.6 $\mathrm{\mu }$m, L_Br2 = 34.6 $\mathrm{\mu }$m for an isolated BZ hydrogel bead (black line), for a collection of BZ hydrogel beads (blue line). We observe a transition from a nonoscillating oxidized to an oscillating BZ hydrogel bead when its radius increases. The theoretical critical size is R_c,theo = 230 $\mathrm{\mu }$m for an isolated gel and R_c,collective = 38 $\mathrm{\mu }$m for a gel in the collective regime. (D) Ratio of period $\frac{T}{T_0}$ as a function of the Ruthenium catalyst concentration in the steady state oscillation regime for $\beta =0.002$ s⁻¹.

Fig. 4.

Mechanical oscillation of a BZ hydrogel bead with a radius $R$ = 150 $\mathrm{\mu }$m smaller than the critical size R_c,exp = $225\pm 9\mathrm{\mu }$m embedded in a layer of BZ hydrogel beads. (A) Picture of a chemomechanical BZ hydrogel bead (dark orange), with [Ru²⁺] = 6 mM and a bulk modulus of 10 kPa, surrounded by smaller BZ chemically active hydrogel beads. (B) Relative radius change of a BZ chemomechanical hydrogel bead as a function of time.