Nonlocal contour dynamics model for chemical front motion.

Pattern formation exhibited by a two-dimensional reaction-diffusion system in the fast inhibitor limit is considered from the point of view of interface motion. A dissipative nonlocal equation of motion for the boundary between high and low concentrations of the slow species is derived heuristically. Under these dynamics, a compact domain of high concentration may develop into a space-filling labyrinthine structure in which nearby fronts repel. Similar patterns have been observed recently by Lee, McCormick, Ouyang, and Swinney in a reacting chemical system. [email protected] [email protected] 1 In the study of chemical systems with both reactions and diffusion, one may discern two broad classes of spatial patterns: extended and compact. Extended patterns typically arise from supercritical symmetry-breaking bifurcations [1]. In the two dimensional case, which we consider here, they are often regular, periodic structures such as arrays of stripes, discs, or hexagons [2]. Compact patterns, or localized states, appear in systems with subcritical bifurcations via a nucleation process [3], and typically take the form of a single one of the repeating units found in extended systems. Both classes of patterns are often described in terms of a competition between two chemical species: an autocatalytic “activator” and its “inhibitor.” Localized states can exhibit a fingering instability [4]. One can imagine that these fingers may grow and branch until a complicated labyrinthine pattern fills the entire plane. Such structures may actually be metastable states; barriers to domain fission may prevent the pattern from evolving into the ground state (presumably a regular array of stripes) from an initial condition which is topologically different. Qualitatively similar kinds of pattern formation appear in other systems [5]. Complicated, labyrinthine pattern evolution in a chemical system has recently been observed by Lee, McCormick, Ouyang, and Swinney [6] in an iodate-ferrocyanide-sulfite reaction (see Fig. 1). The patterns are composed of regions with one of two different chemical compositions. This system appears to be bistable; if it is prepared with one of the two possible uniform compositions, it will persist in that state. Nontrivial pattern formation requires a nucleation site in an otherwise uniform background. These experiments also indicate that in the formation of patterns, nearby boundaries or fronts repel each other. We will use these results as a guide to the general features one would like to see in a model system [7]. It is natural to seek a representation of the dynamics of these and similar patterns in terms of the interface between domains of different composition; in order to capture the properties of repulsion and nonintersection, such a contour dynamics must be nonlocal, coupling segments of the interface that are distant in arclength yet close in space. The purpose of this letter is twofold: to give an intuitive construction of a fully nonlocal curve 2 dynamics from a set of reaction-diffusion equations and to show that it is useful for describing pattern formation. The evolution equation is written in terms of the intrinsic geometry of the curve, and is valid for pattern evolution far beyond the linear instability of localized states. In this simplified form, it is easier to identify the physics responsible for the destabilization of a compact pattern, the tendency of a pattern to grow or shrink, and the interaction between different portions of the interface. The curve dynamics is studied numerically and is shown to reproduce the qualitative features of the experimental patterns. The reaction-diffusion pair studied here is similar to models of spiral wave formation [8] and nerve impulse propagation [9]. With u the activator and v the inhibitor, we consider ut = D∇ u− F(u)− ρv , (1a) ǫvt = ∇ v − αv + βu . (1b) The nonlinear function F(u),with derivative F, embodies the autocatalytic nature of the activator. It is typically a polynomial in u with a double well structure whose minima, not necessarily of equal depth, we label u±. Patterns like that in Fig. 1 can then form in which regions of u ≃ u+ (e.g. white) are surrounded by a region of u ≃ u− (black). Inhibition of u is achieved for ρ > 0, while the couplings α and β reflect the self-limiting behavior of the inhibitor and its stimulation by the activator, respectively. The small number ǫ defines the fast-inhibitor limit. This limit is opposite to the limit assumed in phase-field models [10] and spiral wave dynamics [8]. The fast inhibitor assumption appears to be the simplest assumption that allows the elimination of one of the fields, giving rise to spatial nonlocality for the remaining field without introducing temporal nonlocality. A similar calculation can be done for the slow-inhibitor limit [11], but due to temporal nonlocality, the resulting equations are more complicated, obscuring the essential physics. We begin by discussing the interaction of fronts, illustrating the property of self-avoidance present in the reaction-diffusion pair (1), for certain parameter ranges. Figure 2 shows a simulation of a one-dimensional version of (1) with periodic boundary conditions. The 3 patterns are defined by the sharp interfaces of the u field. The parameters are such that the u ≃ u+ region expands into the u ≃ u− region. When the interfaces get too close, the exponential tails of the v field begin to overlap, causing a repulsion which stabilizes the final field configuration. In the derivation of the two-dimensional contour dynamics, we shall analytically see the source of this repulsion. To derive an interface evolution equation for two-dimensional patterns, we take the fastinhibitor limit, setting ǫ = 0 and thus slaving v to u, v(x, t) = β ∫ dxG(|x − x|)u(x) (2) where the Green’s function G(r) = K0(r/ξ)/(2π), and ξ = α . Substituting for v in equations (1a), we obtain the variational dynamics ut = −δF [u]/δu , with F [u]= ∫ dx { 1 2 D (∇u) + F (u) }