Complex wave patterns in an effective reaction–diffusion model for chemical reactions in microemulsions

Abstract

An effective medium theory is employed to derive a simple qualitative model of a pattern forming chemical reaction in a microemulsion. This spatially heterogeneous system is composed of water nanodroplets randomly distributed in oil. While some steps of the reaction are performed only inside the droplets, the transport through the extended medium occurs by diffusion of intermediate chemical reactants as well as by collisions of the droplets. We start to model the system with heterogeneous reaction-diffusion equations and then derive an equivalent effective spatially homogeneous reaction-diffusion model by using earlier results on homogenization in heterogeneous reaction-diffusion systems [S.Alonso, M.Bär, and R.Kapral, J. Chem. Phys. 134, 214102 (2009)]. We study the linear stability of the spatially homogeneous state in the resulting effective model and obtain a phase diagram of pattern formation, that is qualitatively similar to earlier experimental results for the Belousov-Zhabotinsky reaction in an aerosol OT (AOT)-water-in-oil microemulsion [V.K.Vanag and I.R.Epstein, Phys. Rev. Lett. 87, 228301 (2001)]. Moreover, we reproduce many patterns that have been observed in experiments with the Belousov-Zhabotinsky reaction in an AOT oil-in-water microemulsion by direct numerical simulations.

Spatio-temporal dynamics induced by competing instabilities in two asymmetrically coupled nonlinear evolution equations

“…Codimension-two bifurcations correspond to particular combinations of the parameter values where two types of instability appear simultaneously. In the proximity of such points, the associated dynamics have been extensively studied in the case of TuringHopf codimension-two bifurcation [7][8][9][10], and analyzed for the Turing-wave codimension-two bifurcation in a few instances [11,12]. In particular, in the latter case, two instabilities appear simultaneously with two character- istic spatial scales, which can be very different depending on the parameter values.…”

Wave Propagation in Excitable Media Through Randomly Distributed Heterogeneities: Simulations and Comparison to the Effective Medium Theory

“…However, for the case of small heterogeneities there are general methods to obtain effective reaction rates and diffusion coefficients for randomly heterogeneous reaction-diffusion systems with an arbitrary number of chemical species and linear [17] or nonlinear [18,19] kinetics. The effective medium theory can be also applied to excitable media [20] and it was able to reproduce the main characteristics of the patterns formed in the microemulsion system [21]. Other homogenization theories has been also previously employed for periodically inhomogeneous systems [22].…”

“…We apply a homogenization approach to obtain effective diffusion and reaction properties of a random heterogeneous system [17,18] and compare the analytical predictions with the results of numerical simulations in excitable systems [20] and with experiments [21]. Here, we employ a model of excitable media as in [20], however, we restrict the study to 1D and increase the characteristic size of the heterogeneities to evaluate the limits of the theory, originally developed for small heterogeneities [20].…”

Phase-locked scroll waves defy turbulence induced by negative filament tension

“…9 To date, much attention has been paid to spiral dynamics as they respond to various external fields such as dc and ac electric fields, [10][11][12][13][14][15] periodic forcing, [16][17][18][19][20] mechanical deformation, [21][22][23] and heterogeneity. [24][25][26][27][28][29] As reactiondiffusion (RD) systems exhibit mirror symmetry, CCW and CW spiral waves are physically identical and the response of spiral waves with opposite chirality to achiral fields is identical up to mirror symmetry. For example, the sense of drift perpendicular to a constant electric field will change for a spiral wave of opposite chirality.…”