Abstract
Mass conservation in chemical species appears in a broad class of reaction-diffusion systems (RDs) and is known to bring about coarsening of the pattern in chemical concentration. Recent theoretical studies on RDs with mass conservation (MCRDs) reported that the interfacial curvature between two states contributes to the coarsening process, reminiscent of phase separation phenomena. However, since MCRDs do not presuppose a variational principle, it is largely unknown whether description of surface tension is operative or not. In this study, we numerically and theoretically explore the coarsening process of patterns in MCRDs in two and three dimensions. We identify the parameter regions where the homogeneous steady state becomes stable, unstable, and metastable. In the unstable region, pattern formation is triggered by usual Turing instability, whereas in the metastable region, nucleation-growth-type pattern formation is observed. In the later stage, spherical droplet patterns are observed in both regions, where they obey a relation similar to the Young-Laplace law and coarsen following the evaporation-condensation mechanism. These results demonstrate that in the presence of a conserved variable, a physical quantity similar to surface tension is relevant to MCRDs, which provides new insight into molecular self-assembly driven by chemical reactions.
Highlights
Reaction-diffusion systems (RDSs) are one of the most generic mathematical frameworks that give rise to spatiotemporal patterns and have been applied to various phenomena, ranging from physics, chemistry, biology, and geology to ecology [1,2,3,4]
We first consider the stability of the homogeneous steady solution (u0, v0). u0 and v0 are determined by the conditions f (u0, v0) = 0 and s0 = u0 + v0, where s0 ≡ sdr/V is the mean molecular concentration conserved through the dynamics [see Eq (3)], and V is the volume of the system
We found that the pattern dynamics in Mass-conserved reaction-diffusion systems (MCRDSs), starting from a uniform state toward an eventual single isolated domain, are classified into two types, similar to phase separation phenomena
Summary
Reaction-diffusion systems (RDSs) are one of the most generic mathematical frameworks that give rise to spatiotemporal patterns and have been applied to various phenomena, ranging from physics, chemistry, biology, and geology to ecology [1,2,3,4]. Two types of MCRDSs were mainly studied: in the first case, referred to as “Turing type” [11], multiple peaklike domains, illustrated in the left panel of Fig. 1(a), appear from a homogeneous initial state via usual (type-IIs) instability [5,6]; the second case is “wave-pinning type” [8], where the system is bistable with high and low chemical concentration states, and by exposing a finite (not infinitesimally small) perturbation to the homogeneous state, mesalike concentration profiles emerge as illustrated in the right panel of Fig. 1(a) Both systems exhibit coarsening, i.e., smaller peaks and mesas shrink and disappear while larger ones grow. By investigating the similarities and differences in coarsening process between MCRDSs and phase separation systems, we explore whether or not a concept of surface tension is relevant to MCRDSs
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