Controlled pattern formation in the CDIMA reaction with a moving boundary of illumination Mads Kærn, a Razvan Satnoianu, b Alberto P. Mun ˜uzuri c and Michael Menzinger* d a Center for BioDynamics, Department of Biomedical Engineering, Boston University, Boston, Massachusetts 02215, USA b Department of Mathematics, City University London, London, UK EC1V 0HB and Centre for Mathematical Biology, Mathematical Institute, University of Oxford, Oxford, UK OX1 3LB c Group of Nonlinear Physics, Faculty of Physics, University of Santiago de Compostela, Santiago de Compostela, 15706, Spain d Department of Chemistry, University of Toronto, Toronto, ON M5S 3H6, Canada Received 15th October 2001, Accepted 9th January 2002 First published as an Advance Article on the web A reaction–diffusion (RD) system that grows axially as one of its boundaries moves is equivalent to a boundary- forced open flow in which all species have identical flow coefficients. Depending on the flow or growth rate, f, and on the intrinsic spreading velocity, c 0 , of the RD structure, such systems are either absolutely (f < c 0 ) or convectively (f > c 0 ) unstable. We previously showed how periodic boundary forcing of an axially growing domain could be used to control the formation of space-periodic structures in biological morphogenesis. This paper proposes, as a chemical equivalent of an axially growing embryo, the design of a continuously fed unstirred flow reactor (CFUR), characterized by a photo-chemically controlled moving boundary. Using the Turing-unstable CDIMA system as an example, we illustrate by simulations the kinds of wave structures that are expected to arise in the absolutely and convectively unstable regimes when boundary forcing is either constant or time-periodic. 1. Introduction It has been known for some time 1–4 that the spatio-temporal dynamics of so-called convectively unstable open flow systems is determined by the temporal dynamics at its inflow bound- ary. Recent theoretical studies 5–9 have shown how a constant boundary forcing causes space-periodic structures to form spontaneously in plug-flow reactors containing an oscillating chemical medium. The differential transport that is usually associated with the breaking of spatial symmetry 10,11 is not required in this case and all species may have identical flow and diffusion coefficients. This phenomenon was verified experimentally 12 and it was shown that the space-periodic structure is actually a phase wave generated as the flow distri- butes the temporal oscillation in space. 12 Also investigated was the case of periodic boundary forcing. 13–16 In that case, volume elements entering the flow at different times have differ- ent phases. The results are upstream or downstream travelling phase waves of constant 13,14 or oscillatory 15 velocity. We recently extended our analysis of periodically forced open flows of oscillatory media to open flows of Turing- unstable and bistable media 17 where the interacting activator and inhibitor have different diffusion coefficients but equal flow coefficients f. In general, the flow converts a periodic bound- ary forcing with frequency o (period T ¼ 2p/o) into a spatial mode with wave-number k ¼ o/f, (wavelength l ¼ fT). This mode may be amplified and maintained when the imposed wave number lies within an appropriate range supported in the absence of a flow and when the flow system is linearly or nonlinearly convective unstable. 17 Convectively unstable con- ditions arise when the flow velocity is greater than the intrinsic velocity c 0 with which a perturbation spreads in the absence of a flow. In the absolutely unstable regime, the mode imposed by the boundary forcing competes with the intrinsic mode as well as with modes that are excited either by noise within the system or boundary noise. The latter, rather than the boundary for- cing, may determine its spatio-temporal dynamics and produce a so-called noise-sustained structure. 1,2 The local control of global patterning by the boundary for- cing of convectively unstable open flow systems has key impli- cations, for instance in biological morphogenesis. This is because any system in which a boundary moves relative to a medium, for instance the cells in an axially growing tissue, from a mathematical point of view, is equivalent to an open flow where all species have identical flow coefficients. 13,14,17 This equivalence is easy to verify. Consider a boundary that moves through a stationary medium with velocity f. Now change to the reference frame where the boundary remains sta- tionary. In this reference frame the medium moves away from the boundary, i.e. downstream, at a constant velocity +f,asit would in an open flow system where all the flow coefficients are equal. Whether it is the medium or the boundary that actually moves is irrelevant, as long as there is a relative motion between them. This relative motion between the medium and its boundary is what resolves a time-periodic boundary forcing into a space-periodic mode. This paper investigates the control of pattern formation by a moving boundary of illumination in the light-sensitive 20,21 chlorine dioxide iodide malonic acid (CDIMA) reaction 18,19 in a continuously fed unstirred reactor (CFUR). 22,23 We sug- gest how an experimental investigation of patterns that are controlled at a moving boundary may be carried out. The dynamics of the CDIMA reaction is known to be controllable by illumination with visible light. According to the current mechanism, 20,21 iodine atoms produced by photo-dissociation of molecular iodine initiate reduction of chlorine dioxide and DOI: 10.1039/b109387h Phys. Chem. Chem. Phys., 2002, 0, 000–000 1 This journal is # The Owner Societies 2002 PCCP