Fig. 1: Non-equilibrium coupling of stimuli across time.

a Traditionally, a responsive polyacrylic acid (PAA) hydrogel contracts and swells directly in-phase with the presence or absence of an acid stimulus (yellow). Here, hydrogel contraction tilts an array of embedded microplates (gray). b In contrast to this rapid reversibility, divalent cations (Cu²⁺, blue) contract the PAA gel by forming a kinetically stable complex with two carboxylate (COO⁻) groups, remaining in the gel after removal of Cu²⁺ from the environment. A subsequent acid stimulus then competes for COO⁻ groups and triggers dissociation of the Cu²⁺ on a timescale determined by its delivery rate (τ_H+). The ensuing dynamics of diffusion, complexation, and mechanical deformations in the presence of the entering and exiting stimuli can lead to scenarios depicted in c-d: c Competition between transient water influx, induced by released Cu²⁺, and the mechanical relaxation time of the gel (τ_⊥) creates traveling osmotic swelling waves reporting the speed of an oncoming acid front when τ_H+ < τ_⊥; d Competition between the diffusion and transient recomplexation of released Cu²⁺ (top, curved blue arrow) and its re-release by oncoming acid creates rate-sensitive traveling color waves when the acid progression rate is smaller than the Cu²⁺ diffusion rate (bottom right, narrow blue band).