Chemical Wave Packet Propagation, Reflection, and Spreading

scheme: Reactions 10-13 constitute the well-known Brusselator model.16 The additional steps 14 and 15 describe a reversible interconversion between the activator (X) and an unreactive form (Z). This interconversion may arise, for example, in the BZ-AOT system, from transfer of the activator from one phase (aqueous) to another (oil), where it is unreactive because of the absence of reaction partners.5 From reactions 10-15, we obtain three partial differential equations: where the variables u, V, and w are the dimensionless concentrations of the activator (X), inhibitor (Y), and “inactivator” (Z), whose corresponding diffusion coefficients are Du, DV, and Dw, respectively. The parameters c and d are rescaled17 from the rate constants kf and kb. The Brusselator kinetics16 are given by the functions f and g as The steady state of the model (16-18) is at (uss, Vss, wss) ) (a, b/a, ac/d). Linear stability analysis around this steady state yields the characteristic equation for the eigenvalues λ. Of the three eigenvalues, we may have either one real and one complex conjugate pair or three real. We are interested in the complex pair. The finite wave instability occurs when Re(λ) ) 0 at k ) kc * 0, and Re(λ) < 0 for all other k. This may happen when the diffusion coefficients are such that Du < DV , Dw, a case that arises in the BZ-AOT system, where nanometer-sized water droplets are dispersed in a continuous oil phase. The droplets carrying the activator, HBrO2, diffuse much more slowly (10-7 cm2/s level) than do small molecules such as the inactivator, BrO2, in the oil phase (10-5 cm2/s level).4 4. Results of Simulations A. Simulations and Calculations for an IP Wave Packet. Figure 2 illustrates an IP wave packet. The parameters were chosen so that the finite wave instability was close to the onset point (Re(λ) ) 0+, at k ) kc). The system was initialized to the homogeneous steady state. At t ) 0, we applied a narrow spike perturbation to the variable u. The Fourier spectrum of such a perturbation contains an abundance of wavenumbers. Because of the finite wavelength instability, waves within a narrowband around k ) kc can grow; outside of this narrowband, waves are suppressed. A local oscillation is induced around the center of the initial perturbation (x ) 0) but damps out because the Hopf mode is below onset (Re (λ)< 0, at k ) 0). A wave packet then forms in which the profile of the inner waves is sinusoidal and the amplitude is modulated by a Gaussian-like function (Figure 2). The wave packet slowly moves outward as new waves are formed, while the individual waves within the packet move much more rapidly toward the site of the original perturbation. We measured these two speeds in our simulations. At each moment, we fit the envelope with a Gaussian function. The movement of the maximum point gives the group velocity |Vg| ) 0.24 (Figure 2a-c). To find the phase velocity, we recorded snapshots and constructed space-time plots, which yielded Vp ) 10.53. It is also possible to calculate the wave speed from the dispersion relation. The results of our linear stability analysis are shown in Figure 3. The real part of the most positive eigenvalue is plotted in Figure 3a, where the finite wave instability is seen at kc ) 0.376, which is very close to onset. The corresponding imaginary part, which decreases monotonically, is shown in Figure 3b; its derivative appears in Figure 3c, where the group velocity is found to be Vg ) -0.24 at k ) kc. The negative sign for Vg implies that the component waves propagate inward, toward the “source”. The second derivative plotted in Figure 3d is related to the spreading of the packet envelope (eq 8). Figure 3e shows ω/k calculated from the data in Figure 3b, enabling us to read off the phase velocity as Vp ) 10.53. The derivative of the phase velocity (Figure 3f) determines the dispersion of the medium through which the waves propagate (eq 4). From the simulation in Figure 2, we can clearly see that the wave packet becomes broader with time. This effect results from dispersion, i.e., from the dependence of the phase velocity on A 98 k1 X + P (10) B + X 98 k2 Y + D (11) 2X + Y 98 k3 3X (12)