Transverse instabilities in chemical Turing patterns of stripes B. Pen ˜ a, 1 C. Pe ´ rez-Garcı ´ a, 1, * A. Sanz-Anchelergues, 2 D. G. Mı ´ guez, 2 and A. P. Mun ˜ uzuri 2 1 Instituto de Fı ´sica, Universidad de Navarra, E-31080 Pamplona, Spain 2 Grupo de Fı ´sica No lineal, Universidad Santiago de Compostela, E-15782 Santiago, Spain Received 23 June 2003; published 18 November 2003 We present a theoretical and experimental study of the sideband instabilities in Turing patterns of stripes. We compare numerical computations of the Brusselator model with experiments in a chlorine dioxide–iodine– malonic acid CDIMA reaction in a thin gel layer reactor in contact with a continuously refreshed reservoir of reagents. Spontaneously evolving Turing structures in both systems typically exhibit many defects that break the symmetry of the pattern. Therefore, the study of sideband instabilities requires a method of forcing perfect, spatially periodic Turing patterns with the desired wave number. This is easily achieved in numerical simula- tions. In experiments, the photosensitivity of the CDIMA reaction permits control and modulation of Turing structures by periodic spatial illumination with a wave number outside the stability region. When a too big wave number is imposed on the pattern, the Eckhaus instability may arise, while for too small wave numbers an instability sets in forming zigzags. By means of the amplitude equation formalism we show that, close to the hexagon-stripe transitions, these sideband instabilities may be preceded by an amplitude instability that grows transient spots locally before reconnecting with stripes. This prediction is tested in both the reaction-diffusion model and the experiment. DOI: 10.1103/PhysRevE.68.056206 PACS numbers: 82.40.Ck, 82.40.Bj, 05.45.-a I. INTRODUCTION Half a century ago, Turing 1 developed a theory of mor- phogenesis which has had a profound impact on theoretical developments in pattern formation. Turing showed that sta- tionary concentration patterns may spontaneously develop in an open system containing two reacting substances provided one of them diffuses much faster than the other. Nowadays, the Turing mechanism is still considered a prototype for the formation of coherent patterns in nonequilibrium systems. Despite considerable efforts to verify Turing’s proposal ex- perimentally and to find stationary spatial patterns in a real chemical system, it took almost 40 years before the first ex- perimental evidence of convection-free Turing patterns was reported. The disparity in diffusion coefficients assumed in the Turing mechanism was hard to achieve because small molecules in aqueous solution have diffusion coefficients that do not differ substantially from each other. Castets et al., working with an open, continuously fed un- stirred reactor CFUR observed spatial pattern formation arising from a homogeneous steady state in the chlorite– iodide–malonic acid CIMA reaction 2. Since then, Turing patterns have been extensively studied in the CIMA reaction and in its variant, the chlorine dioxide–iodine–malonic acid CDIMA reaction 3. In these experiments, sufficient dif- ferences in the mobilities were achieved by using a macro- molecular indicator that partially immobilizes the ‘‘critical’’ species by reversible complexations. Depending on the con- trol parameters concentration of reactants and diffusion co- efficients, the dynamics of this reaction exhibits several kinds of steady spatially periodic patterns close to onset: hexagons, stripes, and ‘‘rhombs’’ 4. Usually the so-called black eyes arise as secondary modes far from threshold 4. Turing-like concentration patterns have also been ob- served during the irreversible polymerization of acrylamide in an oxygen atmosphere in the presence of methylene blue sulfide 5. But the main drawbacks of this system are that, once the polymerization is over, the pattern cannot be changed by further external perturbation, and that the driving instability mechanism is still under discussion 6–8. In a recent theoretical work 9, it was suggested that a certain class of electrochemical systems might exhibit Turing-type structures without suffering from the restriction on the dif- ferent rates of the transport processes. This therefore opens promising perspectives in the study of further Turing-like structures. The stability of patterns against spatial modulations is a key issue, because long-wave instabilities are pattern selec- tion mechanisms in systems with translation symmetry. In the case of a pattern of rolls, the Eckhaus or the zigzag instability may appear when the homogeneous translational invariance is spontaneously broken. Spatial modulations of patterns have been much studied in convective fluids 10,11, but, to our knowledge, they have scarcely been discussed in chemistry. On the theoretical side, the three instabilities of striped patterns cross roll, Eckhaus, and zigzag were well reproduced within the chemical Schnackenberg model 12. Experimentally, illumination and electric fields have been used to modify Turing-like patterns obtained during poly- merization in the acrylamide-methylene blue-sulfide–oxygen reaction, and the same system has been exposed to spatially periodic light perturbation 6. Recently, Mun ˜ uzuri et al. 13 have revealed the sensitivity of the CDIMA reaction to vis- ible light and proposed a simple model for its photosensitiv- ity. This study opened the possibility of controlling Turing patterns by illumination. Our main aim in the present work is to discuss the mecha- nism of modulational instabilities in chemical systems, by comparing numerical simulations of the Brusselator model *Electronic address: carlos@fisica.unav.es PHYSICAL REVIEW E 68, 056206 2003 1063-651X/2003/685/0562067/$20.00 ©2003 The American Physical Society 68 056206-1