Gentle Dragging of Reaction Waves

Using a recently realized “addressable catalyst surface” [Science 294, 134 (2001)] we study the interaction of chemical reaction waves with prescribed spatiotemporal fields. In particular, we study how a traveling chemical pulse is “dragged” by a localized, moving temperature heterogeneity as a function of its intensity and speed. The acceleration and eventual “detachment” of the wave from the heterogeneity is also explored through simulation and stability analysis.

Figure 1

Experimental (a) and computed (b) dependence of the leading angle of the dragged pulse on the dragging speed (normalized with the natural pulse speed). Experimental conditions: $T=456\mathrm{K}$, $P_{\mathrm{C}\mathrm{O}}=3.9\times {10}^{-5}\mathrm{m}\mathrm{b}\mathrm{a}\mathrm{r}$, $P_{{\mathrm{O}}_2}=3.0\times {10}^{-4}\mathrm{m}\mathrm{b}\mathrm{a}\mathrm{r}$. The dragging direction was about $4°$ off the fast diffusion direction. The angle is marked by white lines in (a) in the $0.7\times 0.4{\mathrm{m}\mathrm{m}}^2$ insets. Computational parameters: $T=540\mathrm{K}$, $P_{\mathrm{C}\mathrm{O}}=4.52\times {10}^{-5}\mathrm{m}\mathrm{b}\mathrm{a}\mathrm{r}$, $P_{{\mathrm{O}}_2}=1.33\times {10}^{-4}\mathrm{m}\mathrm{b}\mathrm{a}\mathrm{r}$, and $D_{\mathrm{C}\mathrm{O}}=1\mu {\mathrm{m}}^2/\mathrm{s}$; $180\times 60\mu {\mathrm{m}}^2$ domain; boundary conditions: periodic in $x$ and no flux in $y$. $A$ is used in (b) to visually discriminate the stable ($A=0$) and the unstable ($A$ proportional to the spot-pulse tip visual distance) branch.

Figure 2

(a) Computed average CO coverage vs dragging speed for different heterogeneity “strengths” ($\Delta T$ at the center of the laser spot). Insets show selected oxygen coverage and dragging temperature 1D profiles. Spectra corresponding to points 1, 2 (low $\Delta T$) and 6 (high $\Delta T$) are shown in (b). $T=540\mathrm{K}$, $P_{\mathrm{C}\mathrm{O}}=4.52\times {10}^{-5}\mathrm{m}\mathrm{b}\mathrm{a}\mathrm{r}$, $P_{{\mathrm{O}}_2}=1.33\times {10}^{-4}\mathrm{m}\mathrm{b}\mathrm{a}\mathrm{r}$, and $D_{\mathrm{C}\mathrm{O}}=1\mu {\mathrm{m}}^2/\mathrm{s}$; 1D, $80\mu \mathrm{m}$ long, periodic domain.

Figure 3

Computed types of pulse detachment [phase portraits, (a) and (b), and snapshots, (1)–(6)] at different dragging speeds. In (a) and (b) we plot the real versus the imaginary part of the first fast Fourier transform component of the computed CO coverage along the line marked in (1). Representative $u(x,y)$ snapshots at the onset of the detachment are marked in (a) and (b) and shown in (1)–(6). The remaining conditions are as in Fig. 1b.

Figure 4

(a) Experimental (size: $1\times 1.2{\mathrm{m}\mathrm{m}}^2$, $T=493\mathrm{K}$, $P_{\mathrm{C}\mathrm{O}}=7.0\times {10}^{-5}\mathrm{m}\mathrm{b}\mathrm{a}\mathrm{r}$, $P_{{\mathrm{O}}_2}=3.0\times {10}^{-4}\mathrm{m}\mathrm{b}\mathrm{a}\mathrm{r}$), and (b) computational snapshots, comoving with the laser spot, of pulse initiation/detachment, (c) $x\mathrm{\text{−}}t$ plot of transient pulse detachment beyond point 4 in Fig. 1a, and (d) a computational analogue. (e) “Snake” dragging (size: $1\times 1{\mathrm{m}\mathrm{m}}^2$, $T=501\mathrm{K}$, $P_{\mathrm{C}\mathrm{O}}=7.3\times {10}^{-5}\mathrm{m}\mathrm{b}\mathrm{a}\mathrm{r}$, $P_{{\mathrm{O}}_2}=3.0\times {10}^{-4}\mathrm{m}\mathrm{b}\mathrm{a}\mathrm{r}$).