Front explosion in a periodically forced surface reaction

Resonantly forced oscillatory reaction-diffusion systems can exhibit fronts with complicated interfacial structure separating phase-locked homogeneous states. For values of the forcing amplitude below a critical value the front “explodes” and the width of the interfacial zone grows without bound. Such front explosion phenomena are investigated for a realistic model of catalytic CO oxidation on a Pt(110) surface in the 2:1 and 3:1 resonantly forced regimes. In the 2:1 regime, the fronts are stationary and the front explosion leads to a defect-mediated turbulent state. In the 3:1 resonantly forced system, the fronts propagate. The front velocity tends to zero as the front explosion point is reached and the final asymptotic state is a 2:1 resonantly locked labyrinthine pattern. The front dynamics described here should be observable in experiment since the model has been shown to capture essential features of the CO oxidation reaction.

Figure 1

Sketch of a phase front separating homogeneous domains of two resonantly locked phases. The interfacial zone is delimited by left and right profiles $h_L$ and $h_R$, respectively.

Figure 2

Snapshots of $u\left(\mathbf{r}\right)$ in the asymptotic state for $T_f=1.275\mathrm{s}$ without front explosion (left, $a=0.0138$) and with front explosion (right, $a=0.00864$). Black corresponds to small values of $u$ and white to large values of $u$.

Figure 3

Snapshot series of $u\left(\mathbf{r}\right)$ for $a=0.0130$ showing a typical front explosion. Left, $t=5T_f$; right, $t=20T_f$ with $T_f=1.275\mathrm{s}$. Color coding as in Fig. 2.

Figure 4

(Color online) Left: Number of defects $n_{\text{total}}$ as a function of time (same forcing as in the right panel of Fig. 2) for $\varphi (\mathbf{r},t)=\arctan \{[w(\mathbf{r},t)-w^0]∕[u(\mathbf{r},t)-u^0]\}$ with $(w^0,u^0)=(0.354,0.487)$ as the center of rotation. Right: Probability distribution of the number of defects with positive topological charge $n_{+}$ [(red) circles]. It is clearly different from a squared Poissonian [(black) triangles] [21] but shows very good agreement with a Gaussian [(blue) diamonds].

Figure 5

(Color online) Plot of the interface width ${\Delta }_0$ [(red) circles] and the interface growth rate $v_{\Delta }$ [(blue) stars] as a function of the forcing amplitude $a$ for $T_f=1.275\mathrm{s}$. The statistical errors are less than the size of the symbols. Three different regimes can be identified. See text for details.

Figure 6

(Color online) Dynamics of the fields $\mathbf{c}\left(\mathbf{r}\right)=\mathbf{(}u\left(\mathbf{r}\right),v\left(\mathbf{r}\right),w\left(\mathbf{r}\right)\mathbf{)}$ averaged over the interfacial zone $I\left(t\right)$, denoted as ${⟨\mathbf{c}⟩}_I$, for $T_f=1.275$ and different values of the forcing amplitude $a>a_2^{*}$ corresponding to different states of the system [$a=0.0173$ (2:1), $a=0.0155$ (4:1), $a=0.0147$ (8:1), $a=0.0138$ (chaos)]. The interface dynamics undergoes a period-doubling cascade into chaos while the dynamics of the homogeneous oscillations outside the interfacial zone (solid black curve) has only a single period-doubling bifurcation at $a_2^2\approx 0.0162$. The limit cycle of the homogeneous oscillations is basically the same for the subfigures denoted 4:1, 8:1, and chaos.

Figure 7

(Color online) 2:1 Arnold tongue for a single oscillator. Note the appearance of a period-doubling cascade within the tongue—well below the 1:1 tongue. For ${\omega }_f∕{\omega }_0=2$, only the first period-doubling bifurcation is present. Its location compares very well with the observed location for the spatially extended system.

Figure 8

Series of snapshots of the $u(\mathbf{r},t)$ field for $a=0.0354$ and $T_f=0.85\mathrm{s}$ showing the exploding front and the eventual formation of a labyrinthine pattern. From left to right and top to bottom: $t=40T_f$, $760T_f$, $4360T_f$, and $10600T_f$. Color coding as in Fig. 2. Note that due to the initial conditions and the no-flux boundary conditions certain symmetries persist in the pattern.

Figure 9

(Color online) 3:1 Arnold tongue for a resonantly forced single oscillator. The period-doubling bifurcation within the tongue is shown as a region shaded with parallel lines. Note that this region does not intersect the 3:1 resonance line (vertical line in the figure) in contrast to the case of 2:1 forcing. The transition from 3:1 to 2:1 locking (solid line with asterisks) is subcritical and the region of bistability is marked by the (orange) cross pattern.

Figure 10

(Color online) Plot of the mean interface width ${\Delta }_0$ (open circles), the mean front velocity $v_f$ (open diamonds), and the interface growth rate $v_{\Delta }$ (stars) as a function of the amplitude of the forcing $a$ for $T_f=0.85\mathrm{s}$. The statistical errors are less than the size of the symbols. Three different regimes can be identified. See text for details.

Figure 11

(Color online) Temporal evolution of the average width $⟨\Delta ⟩$ and position $⟨x_f⟩$ of the interfacial zone for $T_f=0.85$ and different values of the forcing amplitude $a>a_3^d$. The thin light (green) curves correspond to $a=0.03887$ and the thick black ones to $a=0.03865$. Two regimes are visible. At short times, a thick and slowly propagating front is present which transforms into a thin and fast propagating front at $t^c$. The transient time $t^c$ is shown as the dashed lines and increases with decreasing $a$.