BZ Waves

Into a small Erlenmeyer flask, introduce 7 mL of solution A, 3.5 mL of solution B, and 1 mL of solution C. Stopper the flask and allow to stir on a magnetic stirrer. The brown color is bromine, which forms from the oxidation of bromide by bromate. The bromine slowly disappears as it reacts with the malonic acid to form bromomalonic acid. When the solution has cleared, add 1.0 mL of ferroin (solution D) and stir. 

The reaction may oscillate between red and blue, but Ignore this. Use a pipette to transfer sufficient solution to form a thin (1 2-mm) layer in a clean Petri dish. Cover the dish and wail. To demonstrate the waves to the class using an overhead projector, it is essential to have a thin layer. Wetting the surface of a Petri dish can 

Safety and disposal information are the same as for the BZ oscillating reaction. 

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Figure 8. Velocity response function vfEj yielding the velocity of BZ waves, normalized to the field-free velocity v (0), as a function of the applied field E, normalized to the annihilation field E beyond which plane waves that would have propagated toward the negative electrode are destroyed. The smooth curve is that predicted by the theory outlined herein, and the dots are experimental values from...

This system is not in itself useful because the polymer is brominated and the reagents expensive but such a system has been used to generate periodic self-assembly of aggregates of acrylonitrile-derivatized gold nanocrystals (43). Kalyshyn et al. reported macroscopic patterns in crosslinked polyacrylamide with BZ waves (44). 

We note a significant difference between the liquid/liquid and the liquid/solid cases. For the liquid/solid case, convection in ascending fronts increases the front velocity but in the liquid/liquid case, convection slows the front. Convection increases the velocity of pH fronts and BZ waves. Why the difference between liquid/liquid frontal polymerization and other frontal systems In liquid/liquid systems the convection also mixes cold monomer into the reaction zone, which lowers the front temperature. The front velocity depends more strongly on the front temperature than on the effective transport coefficient of the autocatalyst. Convection cannot mix monomer into the reaction zone of a front with a solid product but only increases thermal transport so the velocity is increased. 

The most common method is to use a gel. Yamaguchi et al. (1991) compared a wide variety of gels for studying BZ waves. One of the simplest methods is to add ultrafine silica gel to the solution. This does not create a true crosslinked gel, but for wave studies in a Petri dish, especially for demonstrations (see Appendix 1 for recipes and Appendix 2 for lab experiments), the technique is ideal. For quantitative work, polyacrylamide gels are popular because they can be used to prepare the spatial analog of a CSTR. 

Maselko and Showalter, 1991). They were even able to study BZ waves on the surface of an individual bead (Maselko and Showalter, 1989) ... 

Schmidt, S. Ortoleva, P. 1981. Electric Field Effects on Propagating BZ Waves Predictions of an Oregonator and New Pulse Supporting Models, J. Chem. Phys. 74, 4488- 500. 

The reaction involving chlorite and iodide ions in the presence of malonic acid, the CIMA reaction, is another that supports oscillatory behaviour in a batch system (the chlorite-iodide reaction being a classic clock system the CIMA system also shows reaction-diffusion wave behaviour similar to the BZ reaction, see section A3.14.4). The initial reactants, chlorite and iodide are rapidly consumed, producing CIO2 and I2 which subsequently play the role of reactants . If the system is assembled from these species initially, we have the CDIMA reaction. The chemistry of this oscillator is driven by the following overall processes, with the empirical rate laws as given ... 

Figure A3.14.9. Reaction-diflfiision structures for an excitable BZ system showing (a) target and (b) spiral waves. (Courtesy of A F Taylor.)...

There exist many different CA models exhibiting BZ-like spatial waves. One of the simplest, and earliest, described in the next section, is a model proposed by Greenberg and Hastings in 1978 [green78], and based on an earlier excitable media model by Weiner and Rosenbluth [weiner46]. One of the earliest and simplest mathematical models of the BZ reaction, called the Orcgonator, is due to Field and Noyes [field74]. 

This reaction can oscillate in a well-mixed system. In a quiescent system, diffusion-limited spatial patterns can develop, but these violate the assumption of perfect mixing that is made in this chapter. A well-known chemical oscillator that also develops complex spatial patterns is the Belousov-Zhabotinsky or BZ reaction. Flame fronts and detonations are other batch reactions that violate the assumption of perfect mixing. Their analysis requires treatment of mass or thermal diffusion or the propagation of shock waves. Such reactions are briefly touched upon in Chapter 11 but, by and large, are beyond the scope of this book. 

The orbitals <]) j(k r) are Bloch functions labeled by a wave vector k in the first Brillouin zone (BZ), a band index p, and a subscript i indicating the spinor component. The combination of k and p. can be thought of as a label of an irreducible representation of the space group of the crystal. Thequantity n (k)is the occupation function which measures... 

Therefore, k+bm and k label the same representation and are said to be equivalent (=). By definition, no two interior points can be equivalent but every point on the surface of the BZ has at least one equivalent point. The k = 0 point at the center of the zone is denoted by T. All other internal high-symmetry points are denoted by capital Greek letters. Surface symmetry points are denoted by capital Roman letters. The elements of the point group which transform a particular k point into itself or into an equivalent point constitute the point group of the wave vector (or little co-group of k) P(k) C P, for that k point. 

Equation (3) shows that the space-group operator (R v) transforms a Bloch function with wave vector k BZ into one with wave vector R k, which either also lies in the BZ or is equivalent to ( ) a wave vector k in the first BZ. (The case Id = k is not excluded.) Therefore, as R runs over the whole R = P, the isogonal point group of G, it generates a basis ( 0kl for a representation of the space group G,... 

Equations (1) are the von Laue conditions, which apply to the reflection of a plane wave in a crystal. Because of eqs. (1), the momentum normal to the surface changes abruptly from hk to the negative of this value when k terminates on a face of the BZ (Bragg reflection). At a general point in the BZ the wave vector k + bm cannot be distinguished from the equivalent wave vector k, and consequently... 

According to Eq. (II.7), co = 0 for k = 0 in the center of BZ 1. With these values Eqs. (II. 1)—(II.3) lead to the relation UA = UB. This means that both sets of atoms vibrate with the same amplitude and in phase (because they have the same sign). A translation of the whole chain results which corresponds to an acoustical wave with X = °°. This is called a longitudinal acoustical branch (LA). 

Raman scattering can be considered as inelastic scattering of a photon by the phonons of a crystal. For a one-phonon process, energy conservation leads to the relations to, = to, + to in the Stokes case and to, = to, - to in the anti-Stokes case. The incident and the scattered photons, denoted by i and s, respectively, also have momentum. The momentum / of a photon is calculated from / = m c and A to = mc2 as I - hco/c — hk. Because the momentum is a vector, one must write I = hk. Now there arises the question whether a phonon also has momentum. With a mechanical wave in a crystal there is no motion of the center of gravity, therefore a phonon has no momentum in the usual sense. In spite of this a quasi-momentum can be ascribed to it. This question has been treated by Sussmann °). One may write down a relation for the k vectors which corresponds to conservation of momentum if all wave vectors of thejAonons are within BZ 1, namely fc, =ks + k for the Stokes case and ki = ks-k for the anti-Stokes case. 

Fig. 6. Time-dependent electronic population of the E2B2u state of Bz+ for a vertical transition. The wave-packet is located initially at the E state surface and seen to undergo an efficient radiationless transition at a time scale of 10-20 fs. The results of the full calculation (solid line) and of the propensity rule (dashed line) are compared.

Some of these chlorite oscillators exhibit particularly interesting or exotic phenomena. Batch oscillations in the absence of flow may be obtained in the systems numbered 3, 10 a and 13, while the chlorite-iodide-malonic acid reaction gives rise to spatial wave patterns as well. These latter, which are strikingly similar to those observed in the BZ reaction61 are shown in Fig. 12. Addition of iodide to the original chlorite-iodate-arsenite oscillator produces a system with an extremely complex phase diagram58, shown in Fig. 13, which even contains a region of tristability, three possible stable steady-states for the same values of the constraints. 

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