Geometry-induced pulse instability in microdesigned catalysts: The effect of boundary curvature

We explore the effect of boundary curvature on the instability of reactive pulses in the catalytic oxidation of $\mathrm{CO}$ on microdesigned Pt catalysts. Using ring-shaped domains of various radii, we find that the pulses disappear (decollate from the inert boundary) at a turning point bifurcation, and we trace this boundary in both physical and geometrical parameter space. These computations corroborate experimental observations of pulse decollation.

Figure 1

(Color online) Bifurcation diagram of one-dimensional (1D) pulses with respect to the partial pressure of $\mathrm{CO}$. The variable $c$ is the propagating speed of the steady pulse (a representative one is shown in the inset). The turning point (saddle-node bifurcation) is indicated by a circle. Solid (dashed) lines denote stable (unstable) steady pulse solutions. $T=540\mathrm{K}$, $P_{{\mathrm{O}}_2}=1.33\times {10}^{-4}\mathrm{mbar}$. The inset shows the typical shape of $u$, $v$, and $w$ at the steady state marked by the arrow.

Figure 2

(Color online) Bifurcation diagram of pulses in a two–dimensional (2D) ring showing the influence of the $\mathrm{CO}$ partial pressure on the angular speed $\left(\omega \right)$ of the traveling pulse at stationarity. Point $H$, marked with a circle, denotes a Hopf bifurcation. Solid (dashed) lines denote stable (unstable) steady pulse solutions. The insets show profiles of O coverage (for the color version, the color is scaled from blue to red according to the coverage of oxygen from low to high). The leading eigenvalues and corresponding eigenvectors at points $H$ and $p$ are plotted in Fig. 3 below. The inner and outer dimensionless radii of the ring are 20 and 30, respectively. $T=540\mathrm{K}$, $P_{{\mathrm{O}}_2}=1.33\times {10}^{-4}\mathrm{mbar}$; $P_{\mathrm{CO}}$ is the bifurcation parameter.

Figure 3

(Color online) The corresponding eigenmodes of three leading eigenvalues at point $H$ and $p$ as in Fig. 2. (a)–(c) are eigenmodes for point $H$. $\lambda$= (a) $0.0005$, (b) and (c) $-0.0006\pm 0.2591i$. (d)–(f) are eigenmodes for point $p$. $\lambda$= (d) $0.0026$, (e) $0.1172$, and (f) $0.2324$.

Figure 4

(Color online) Bifurcation diagram varying the boundary curvatures while fixing ring width. $v_i$ is the local velocity of the propagating pulse at the inner boundary. $v_0$ is the propagating speed of a one-dimensional pulse. The increase of boundary curvature leads to a pulse instability that is associated with a Hopf bifurcation, followed again by a turning point. $T=540\mathrm{K}$, $P_{{\mathrm{O}}_2}=1.33\times {10}^{-4}\mathrm{mbar}$, $P_{\mathrm{CO}}=4.597\times {10}^{-5}\mathrm{mbar}$. The ring width $r_o-r_i$ is fixed at 8.

Figure 5

(Color online) Three leading eigenvalues and corresponding eigenmodes at the Hopf bifurcation point $H$ in Fig. 4. $\lambda$=(a) $-0.0012$; (b) and (c) $-0.0025\pm 0.2899i$.

Figure 6

(Color online) Bifurcation diagram of pulses in a 2D ring showing the influence of varying the inner ring radius $r_i$ on the traveling pulse stability. Point $H$, marked with a circle, denotes a Hopf bifurcation. Solid (dashed) lines denote stable (unstable) steady pulse solutions. For $r_i$ close to 30, steady pulse solutions are computed at discrete points. Insets show the profiles of O coverage. The dimensionless outer radius of the ring $r_o$ is fixed at 30. $T=540\mathrm{K}$, $P_{{\mathrm{O}}_2}=1.33\times {10}^{-4}\mathrm{mbar}$, $P_{\mathrm{CO}}=4.597\times {10}^{-5}\mathrm{mbar}$.

Figure 7

(Color online) Bifurcation diagram varying the inner ring radius $r_i$ while fixing $r_o$. The Hopf bifurcations are marked by circles. Solid (dashed) lines denote stable (unstable) steady pulse solutions. $v_i$ is the local velocity of the propagating pulse at the inner boundary. $v_0$ is the propagating speed of a one-dimensional pulse. The $v_i$ associated with the Hopf bifurcation point depends on $r_o$.

Figure 8

(Color online) Sequences of snapshots showing the decollation of propagating pulses from the inner boundary of a ring structure. (a) Numerical simulation. The time interval between each snapshot is $2\mathrm{s}$. Position-fixed arrows are used to show the change of the decollating pulse. The arrows are aligned to the boundary of the pulse in the first snapshot. The tip of the pulse at the inner boundary first slowly moves toward the left and shrinks at the same time; then it decollates from the boundary. (b) and (c) are experimental observations. The dark area is ${\mathrm{TiO}}_2$ and the gray area is $\mathrm{Pt}\left(110\right)$. The white propagating pulse is a $\mathrm{CO}$-rich one. Experimental conditions: $T=447\mathrm{K}$, $P_{{\mathrm{O}}_2}=4\times {10}^{-4}\mathrm{mbar}$; $P_{\mathrm{CO}}$= (b) $3.35\times {10}^{-5}$ and (c) $2.95\times {10}^{-5}\mathrm{mbar}$.