A snapshot of autonomously formed wave patterns in capillary tubes (inner diameter ) containing soft-silica gel with ferroin catalyst. This image was captured after immersing the tubes into the reactant solution.
Space-time representation of diffusion-driven wave patterns in a capillary tube: x axis, space; y axis, time; the time increases in the downwards direction. White and black indicate the oxidized (i.e., excited) and the reduced (i.e., unexcited) state of the ferroin catalyst, respectively. Each space-time frame consists of 256 horizontal data extracted along the center axis of the capillary at intervals (i.e., the span of each frame is 6 min) and each spatial datum is an average of two pixel rows. The tube length is .
Top: Digitally enhanced gray-scale density plots from tomography layers (tube cross section) 5, 15, 25, 35, 45, 65, and 95. Bottom: Core radius as a function of distance from the open capillary tube end and its fitted curve approximation (dashed line). The total tube length is .
Space-time plots from numerical 1D simulation exhibiting firing-type wave patterns (axes as in Fig. 2 ). Elevated levels of variable are shown as white and shades of gray. Equations (5)–(9) were integrated on a 1D grid using the explicit fourth-order Runge-Kutta method and utilizing three-point central-differences approximation for the 1D Laplacian operator (Ref. 36 ). Numerical parameters: grid length, 1024 points; grid spacing, ; time step (dimensionless), ; sampling interval, ; number of samples in each graph, 3333 (2083 time units). We considered a dimensionless space with immobilized catalyst and therefore set . Then, we fix the linear diffusion coefficients of and bromide to unity, , and use relative value (Table I in Ref. 37 ) for and bromomalonic acid. The initial conditions were uniformly throughout the medium and the values for other species set to zero except at the grid point on the left boundary . In order to simulate open tube end in contact with the BZR reactant solution, Dirichlet boundary conditions were applied to the left boundary with , , , and . The right boundary is zero flux (von Neumann conditions). The behavior remained unchanged when the time step was reduced to half.
Normalized concentration profiles from the simulation shown in Fig. 4 at (a) , (b) , (c) , and (d) . Preceding wave leaves a slowly decaying region of [W] in its wake which causes annihilation of the subsequent wave.
Article metrics loading...
Full text loading...