Decay of metastable phases in a model for the catalytic oxidation of CO

We study by kinetic Monte Carlo simulations the dynamic behavior of a Ziff-Gulari-Barshad model with CO desorption for the reaction $\mathrm{C}\mathrm{O}+\mathrm{O}\to \mathrm{C}{\mathrm{O}}_2$ on a catalytic surface. Finite-size scaling analysis of the fluctuations and the fourth-order order-parameter cumulant show that below a critical CO desorption rate, the model exhibits a nonequilibrium first-order phase transition between low and high CO coverage phases. We calculate several points on the coexistence curve. We also measure the metastable lifetimes associated with the transition from the low CO coverage phase to the high CO coverage phase, and vice versa. Our results indicate that the transition process follows a mechanism very similar to the decay of metastable phases associated with equilibrium first-order phase transitions and can be described by the classic Kolmogorov-Johnson-Mehl-Avrami theory of phase transformation by nucleation and growth. In the present case, the desorption parameter plays the role of temperature, and the distance to the coexistence curve plays the role of an external field or supersaturation. We identify two distinct regimes, depending on whether the system is far from or close to the coexistence curve, in which the statistical properties and the system-size dependence of the lifetimes are different, corresponding to multidroplet or single-droplet decay, respectively. The crossover between the two regimes approaches the coexistence curve logarithmically with system size, analogous to the behavior of the crossover between multidroplet and single-droplet metastable decay near an equilibrium first-order phase transition.

Figure 1

Schematic of the algorithm. See discussion in the text.

Figure 2

Contour plot of the projection of ${10}^6$ MCSS of simulation onto the ternary phase diagram for $k=0.02$, $y=0.5332$, and $L=100$.

Figure 3

Order-parameter probability distribution, $P\left({\theta }_{\mathrm{C}\mathrm{O}}\right)$, shown vs $y$ for $k=0.03$ and $L=100$. The distribution for the value of $y$ closest to the coexistence value is shown with a bold line.

Figure 4

The order-parameter fluctuation measure $X_L$, shown vs $y$ for $k=0.02$ and four system sizes. The dotted lines are guides to the eye. The values of $X_L$ have an error of approximately 5%.

Figure 5

(a) Plot of $\ln \left(X_L^{\max }\right)$ vs $\ln \left(L\right)$ and (b) plot of $\ln (X_{bL}^{\max }∕X_L^{\max })∕\ln \left(b\right)$ with $L=20$ vs $1∕\ln \left(b\right)$, both for several values of $k$ and including all four system sizes. $X_L^{\max }$ is the maximum value of $X_L$, taken from figures similar to Fig. 4. The straight lines are the best linear fits to the data and give $X_L^{\max }\sim L^{d^{\prime }}$ with (a) $d^{\prime }=1.96\pm 0.02$, $1.91\pm 0.05$, and $1.58\pm 0.04$; and (b) $d^{\prime }=2.008\pm 0.005$, $2.006\pm 0.008$, and $1.52\pm 0.04$; for $k=0.02$, 0.03, and 0.04, respectively.

Figure 6

Dependence of the fourth-order reduced cumulant $u_L$ on $y$, for (a) $k=0.02$ and (b) $k=0.04$. The $P\left({\theta }_{\mathrm{C}\mathrm{O}}\right)$ histograms indicate that the minima of $u_L$ correspond to the transitions from unimodal to bimodal distribution. The maximum between them gives the coexistence point $y_2(k,L)$. The horizontal lines in the insets correspond to $u_L=2∕3$ (dashed) and $u_{\text{Ising}}^{*}\approx 0.61$ (dotted).

Figure 7

Critical CO adsorption rate $y_2(k,L)$, shown vs $1∕L^2$ for several values of the desorption rate $k$. The straight lines are weighted least-squares fits, yielding the estimated $y_2\left(k\right)$ for $L=\infty$.

Figure 8

Some points of the coexistence curve, analogous to the pressure vs temperature phase diagram for a fluid in equilibrium. The continuous line represents a quadratic fit to the data. The vertical dashed line indicates the estimate $k_c\approx 0.039$ from Ref. 9.

Figure 9

$⟨{\tau }_{\mathrm{p}}⟩$ as a function of (a) $y_{\mathrm{w}}$ with ${\theta }_{\mathrm{C}\mathrm{O}}^{*}=0.65$, and (b) ${\theta }_{\mathrm{C}\mathrm{O}}^{*}$ with $y_{\mathrm{w}}=0.475$; for $k=0.02$, $L=100$, and two different values of $y$.

Figure 10

$⟨{\tau }_{\mathrm{d}}⟩$ as a function of (a) $y_{\mathrm{w}}$ with ${\theta }_{\mathrm{C}\mathrm{O}}^{*}=0.45$, and (b) ${\theta }_{\mathrm{C}\mathrm{O}}^{*}$ with $y_{\mathrm{w}}=0.57$; for $k=0.02$, $L=100$, and two different values of $y$.

Figure 11

Snapshot configurations obtained at different (unevenly spaced) times, after a sudden change of $y$ from $y_{\mathrm{w}}=0.45$ to $y=0.538>y_2\left(k\right)$, i.e., ${\Delta }^{-1}\approx 200$, for $k=0.02$ and $L=100$. Here and in Figs. 12, 13, 14, the black dots represent lattice sites occupied by CO, while both adsorbed O atoms and empty lattice sites are white.

Figure 12

Snapshot configurations obtained at different (unevenly spaced) times, after a sudden change of $y$ from $y=0.65=y_{\mathrm{w}}$ to $y=0.5232<y_2\left(k\right)$, i.e., ${\Delta }^{-1}\approx 100$, for $k=0.02$ and $L=100$.

Figure 13

Snapshot configurations obtained at different (unevenly spaced) times, after a sudden change of $y$ from $y=0.45=y_{\mathrm{w}}$ to $y=0.5338>y_2\left(k\right)$, i.e., ${\Delta }^{-1}\approx 2000$, for $k=0.02$ and $L=100$.

Figure 14

Snapshot configurations obtained at different (unevenly spaced) times, after a sudden change of $y$ from $y=0.65=y_{\mathrm{w}}$ to $y=0.5315<y_2\left(k\right)$, i.e., ${\Delta }^{-1}\approx 500$, for $k=0.02$ and $L=100$.

Figure 15

Plot of distributions of ${\tau }_{\mathrm{p}}$ for $k=0.02$ and $L=100$, (a) ${\Delta }^{-1}\approx 200$ and (b) ${\Delta }^{-1}\approx 2000$. Note the very different timescales in (a) and (b).

Figure 16

Plot of distributions of ${\tau }_{\mathrm{d}}$ for $k=0.02$ and $L=100$, (a) ${\Delta }^{-1}\approx 100$ and (b) ${\Delta }^{-1}\approx 500$. Note the very different timescales in (a) and (b).

Figure 17

Log-linear plot of (a) $⟨{\tau }_{\mathrm{p}}⟩$ vs ${\Delta }^{-1}$ and (b) $⟨{\tau }_{\mathrm{d}}⟩$ vs ${\Delta }^{-1}$, shown for $k=0.01$ and different values of $L$. The solid inverted triangles indicate ${\Delta }_{1∕2}^{-1}$ for each value of $L$. Approximate data collapse is seen for ${\Delta }^{-1}<{\Delta }_{1∕2}^{-1}$.

Figure 18

Log-linear plot of (a) $L^2⟨{\tau }_{\mathrm{p}}⟩$ vs ${\Delta }^{-1}$ and (b) $L^2⟨{\tau }_{\mathrm{d}}⟩$ vs ${\Delta }^{-1}$, shown for $k=0.01$ and different values of $L$. The solid inverted triangles indicate ${\Delta }_{1∕2}^{-1}$ for each value of $L$. Approximate data collapse is seen for ${\Delta }^{-1}>{\Delta }_{1∕2}^{-1}$.

Figure 19

The relative standard deviation $r$ of (a) $⟨{\tau }_{\mathrm{p}}⟩$ and (b) $⟨{\tau }_{\mathrm{d}}⟩$, shown on a logarithmic scale vs ${\Delta }^{-1}$ for $k=0.01$. The behavior of $r$ crosses over from the approximate straight line expected from our postulate in the deterministic regime to $r\approx 1$ in the stochastic regime. The solid lines are weighted least-squares fits to the data in the linear region.

Figure 20

The estimates $1∕{\Delta }_{1∕2}$ for $1∕{\Delta }_{\mathrm{DSP}}$ for (a) poisoning and (b) decontamination as obtained from Fig. 19 and analogous data, shown vs $L$ on a logarithmic scale. The solid lines are weighted least-squares fits excluding the points corresponding to $L=40$.

Figure 21

(a) $⟨{\tau }_{\mathrm{p}}⟩∕L$ at $1∕{\Delta }_{1∕2}$ and (b) $⟨{\tau }_{\mathrm{d}}⟩∕L$ at $1∕{\Delta }_{1∕2}$, shown vs $L$ on a logarithmic scale. The solid straight lines are weighted least-squares fits. Excluding the points corresponding to $L=40$, the figures support the asymptotic behavior $⟨{\tau }_{\mathrm{p}}⟩\propto L\ln L$ at the DSP.

Figure 22

Decay times as functions of $y$ when the system evolves toward the low CO-coverage region (${\theta }_{\mathrm{C}\mathrm{O}}^{*}=0.45$, $y_{\mathrm{w}}=0.55$, left-pointing triangles) and when it evolves toward the high CO-coverage region (${\theta }_{\mathrm{C}\mathrm{O}}^{*}=0.65$, $y_{\mathrm{w}}=0.475$, right-pointing triangles) for $k=0.01$ and $k=0.04$, and $L=100$. As discussed in the text, the divergence is exponential in $1∕\mid y-y_2(k,L)\mid =1∕\Delta$.