Ziff–Gulari–Barshad model

The Ziff–Gulari–Barshad (ZGB) model is a simple Monte Carlo method for catalytic reactions of oxidation of carbon monoxide to carbon dioxide on a surface using Monte-Carlo methods which captures correctly the essential dynamics: phase transitions between two poisoned states (either CO₂- or O-poisoned) and a steady-state in between. It is named after Robert M. Ziff, Erdogan Gulari, and Yoav Barshad, who published it in 1986.^[1]

Model definition

The model consists of three steps:

Adsorption of the reacting species CO and O₂
The actual reaction step on the surface: CO + O → CO₂
Desorption of the products.

The simplest implementation considers the catalyst as simple square two-dimensional lattice, but one can also consider other kinds of underlying lattices.^[2] When a gas-phase molecule touches an empty site, adsorption occurs immediately and the chemical reaction is also instantaneous. Furthermore, one assumes that the composition of the gas phase remains constant.

While these requirements would still allow a large number of models and corresponding behaviors, the two special assumptions of the ZGB model are: (i) CO molecules are adsorbed "standing" with the O touching the surface, and require thus only one free lattice site; (ii) O₂ molecules are adsorbed "flat" and require thus two adjacent free lattice sites for getting adsorbed.

Results and other work

When the ratio between O₂ and CO in the gas phase is increased, the model shows two phase transitions: A continuous one between a O-poisoned and a mixed state, and a discontinuous one between the mixed and a CO-poisoned state. The continuous transition belongs to the universality class of directed percolation.^[3] The model was modified several times.^[4] ^[5]

References

^ Ziff RM, Gulari E, Barshad Y (June 1986). "Kinetic phase transitions in an irreversible surface-reaction model". Phys Rev Lett. 56 (24): 2553–56. Bibcode:1986PhRvL..56.2553Z. doi:10.1103/PhysRevLett.56.2553. PMID 10033028.
^ Gao, Zhuo; Yang, Z. (March 1989). "Dynamic scaling behavior of the Ziff–Gulari–Barshad model on regular fractal lattices: The influence of lacunarity". Physical Review E. 59 (3): 2795–2800. Bibcode:1999PhRvE..59.2795G. doi:10.1103/PhysRevE.59.2795.
^ Grassberger, Peter (April 1995). "Are damage spreading transitions generically in the universality class of directed percolation?". Journal of Statistical Physics. 79 (1–2): 13–23. arXiv:cond-mat/9409068. Bibcode:1995JSP....79...13G. doi:10.1007/BF02179381. S2CID 18330976.
^ Beney, P; Droz, M; Frachebourg, L (21 July 1990). "On the critical behaviour of cellular automata models of nonequilibrium phase transitions". Journal of Physics A: Mathematical and General. 23 (14): 3353–3359. Bibcode:1990JPhA...23.3353B. doi:10.1088/0305-4470/23/14/031.
^ Albano, Ezequiel (July 1992). "Critical exponents for the irreversible surface reaction A+B→AB with B desorption on homogeneous and fractal media". Physical Review Letters. 69 (4): 656–659. Bibcode:1992PhRvL..69..656A. doi:10.1103/PhysRevLett.69.656. PMID 10046997.

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[1] Ziff RM, Gulari E, Barshad Y (June 1986). "Kinetic phase transitions in an irreversible surface-reaction model". Phys Rev Lett. 56 (24): 2553–56. Bibcode:1986PhRvL..56.2553Z. doi:10.1103/PhysRevLett.56.2553. PMID 10033028.

[2] Gao, Zhuo; Yang, Z. (March 1989). "Dynamic scaling behavior of the Ziff–Gulari–Barshad model on regular fractal lattices: The influence of lacunarity". Physical Review E. 59 (3): 2795–2800. Bibcode:1999PhRvE..59.2795G. doi:10.1103/PhysRevE.59.2795.

[3] Grassberger, Peter (April 1995). "Are damage spreading transitions generically in the universality class of directed percolation?". Journal of Statistical Physics. 79 (1–2): 13–23. arXiv:cond-mat/9409068. Bibcode:1995JSP....79...13G. doi:10.1007/BF02179381. S2CID 18330976.

[4] Beney, P; Droz, M; Frachebourg, L (21 July 1990). "On the critical behaviour of cellular automata models of nonequilibrium phase transitions". Journal of Physics A: Mathematical and General. 23 (14): 3353–3359. Bibcode:1990JPhA...23.3353B. doi:10.1088/0305-4470/23/14/031.

[5] Albano, Ezequiel (July 1992). "Critical exponents for the irreversible surface reaction A+B→AB with B desorption on homogeneous and fractal media". Physical Review Letters. 69 (4): 656–659. Bibcode:1992PhRvL..69..656A. doi:10.1103/PhysRevLett.69.656. PMID 10046997.

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