Coupling to the Oscillating Belousov-Zhabotinsky Reaction

We now wish to test the above gel-reaction-diflfusion approach (Eqs 9.7-9.10) using the BZ reaction, which consists in the oxidation of malonic acid by bromate ions in acidic medium. The reaction proceeds only when catalyzed by a suitable metal ion. In the experiments, the chosen catalyst is the ruthenium tris(2,2 -bipyridine) that intervenes through its oxydo-reduction couple (Ru(bpy)3 /Ru(bpy)j ). In a batch reactor or a CSTR, this reaction exhibits well-documented periodic oscillations of concentrations in some region of the parameter space. 

Using responsive gels (e.g., N-isopropylacrylamide) in which this catalytic ion is covalently bound to the polymer network-so that the oscillatory properties are confined to the gel contents-and the unique capacity of the reaction to exhibit oscillatory regimes over extended periods of time even in batch conditions, Yoshida [8] has obtained time-periodic modulations of the volume of the gel in a constant environment, contrary to the usual on-off switching by externally controlled stimuli. 

An intuitive explanation of the oscillatory volume variations is given by Yoshida in Ref [8] it is based on hydrophobic effects that induce a difference of swelling rate between the oxidized and the reduced states of the catalyst. For a small piece of gel, the oscillations occur homogeneously [39]. But when the chemical wavelength is smaller than the system size, some inhomogeneous behavior in the form of waves may arise [8]. 

We restrict our presentation to the case of a spherical bead of gel submitted to the BZ reaction performed with feed and catalyst conditions analogous to the above-mentioned experiments (bonded catalyst). Each point of the sphere is defined by a triplet of variables (r, 9, ), and furthermore, we shall assume that spherical symmetry is maintained during the volume oscillations, that is, the deformations are only radial as in Ref. [27,40-42]. 

Starting from relations (9.8) and (9.9), we may derive the evolution equation for r(R, t). In the present geometry and conditions, this quantity represents the radial velocity for the motions of the particle labeled R, which is followed in its evolution 

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