Two universality classes of the Ziff-Gulari-Barshad model with CO desorption via time-dependent Monte Carlo simulations

We study the behavior of the phase transitions of the Ziff-Gullari-Barshad (ZGB) model when CO molecules are adsorbed on a catalytic surface with a rate $y$ and desorbed from the surface with a rate $k$. We employ large-scale nonequilibrium Monte Carlo simulations along with an optimization technique based on the coefficient of determination, in order to obtain an overview of the phase transitions of the model in the whole spectrum of $y$ and $k$ ($0\le y\le 1$ and $0\le k\le 1$) with precision $\mathrm{\Delta }y=\mathrm{\Delta }k=0.001$. Successive refinements reveal a region of points belonging to the directed percolation universality class, whereas the exponents $\theta$ and $\beta /{\nu }_{\parallel }$ obtained agree with those of this universality class. On the other hand, the effects of allowing the CO desorption from the lattice on the discontinuous phase transition point of the original ZGB model suggest the emergence of an Ising-like point previously predicted by Tomé and Dickman [Phys. Rev. E 47, 948 (1993)]. We show that such a point appears after a sequence of two lines of pseudocritical points which leads to a unique peak of the coefficient of determination curve in $y_c=0.554$ and $k_c=0.064$. In this point, the exponent $\theta$ agrees with the value found for the Ising model.

Figure 1

(a) Coefficient of determination $r$ as function of $y$ and $k$ for the ZGB model with desorption of CO molecules. (b) A highlight of the coefficient of determination $r$ as function of $y$ and $k$ for the ZGB model with desorption of CO molecules. We can observe an extension that ends after the discontinuous phase transition point ($y\approx 0.53$) of the model without desorption ($k=0$). This extension comes from continuous phase transition of the original ZGB model ($y\approx 0.39$ and $k=0$).

Figure 2

Refinement process of the coefficient of determination for (a) $r\ge 0.98$, (b) $r\ge 0.99$, (c) $r\ge 0.999$, and (d) $r\ge 0.9995$.

Figure 3

(a) A small part of region 2 ($0.44\lesssim y\lesssim 0.56$ and $k\lesssim 0.09$) corresponding to the two curves that, after the refinement, lead to a single point. The vertical dashed lines correspond to different values of $y$: (i) $y=0.530$, (ii) $y=0.536$, (iii) $y=0.542$, and (iv) $y=0.552,$ where the coefficient of determination $r$ is calculated as function of $k$. (b) $r\times k$ for these values of $y$. We can observe that the double peaks observed in plots (i), (ii), and (iii) culminate in an single peak shown in plot (iv). The inset in plot (iv) is a zoom in order to verify the existence of a unique peak.

Figure 4

(a) The time evolution of $\rho \left(t\right)$ in $log\times log$ scale for one particular case $y=0.387$ and $k=0.1$, when all sites are initially filled with CO molecules. We can observe that after a transient, the system decays according to the power law given by Eq. (7) (the inset plot highlights the robustness of this power law). (b) The initial growing of $\rho \left(t\right)$ when the initial lattice is fully occupied by O atoms except for a vacant site located at the center of the lattice. We show the power laws for $k=0.1,0.2,...,0.9$ and $y=0.388$.

Figure 5

Time evolution of $\rho \left(t\right)$ for $y=0.388$ when we start with all sites vacant (${\rho }_0=1$) for $k=0.1,0.2,...,0.9$.

Figure 6

Time evolution of density of vacant sites, $\rho \left(t\right)$ vs $t$, for $y=0.554$ and $k=0.064$ when we start the simulation with all sites occupied by CO molecules.