Interface propagation and nucleation phenomena for discontinuous poisoning transitions in surface-reaction models

J. W. Evans; T. R. Ray

doi:10.1103/PhysRevE.50.4302

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Interface propagation and nucleation phenomena for discontinuous poisoning transitions in surface-reaction models

J. W. Evans and T. R. Ray

Ames Laboratory and Department of Mathematics, Iowa State University, Ames, Iowa 50011

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Phys. Rev. E 50, 4302 – Published 1 December, 1994

DOI: https://doi.org/10.1103/PhysRevE.50.4302

Abstract

Here, we consider discontinuous nonequilibrium phase transitions to poisoned or ‘‘adsorbing’’ states in lattice-gas models of surface reactions. Specifically, we examine the monomer or CO-poisoning transition in the Ziff-Gulari-Barshad monomer-dimer reaction model for CO oxidation, modified to include adspecies diffusion. For CO pressures below the poisoning transition, we first characterize the propagation of and fluctuations at an interface between reactive and CO-poisoned states. Here, we utilize ideas from spatial contact models, reaction-diffusion theory, and kinetic roughening theory. Evolution is described by the Kardar-Parisi-Zhang equation, but with the nonlinearity and kinetic surface tension vanishing on approaching the transition. Next, again for CO pressures below the transition, we consider the evolution of a ‘‘nucleus’’ of the reactive state embedded in the CO-poisoned state, now exploiting concepts from epidemic theory. We elucidate the divergence and ‘‘sharpening’’ of the critical size of this nucleus, both approaching the transition and with increasing adspecies diffusion rates. The deviation from mean-field divergence approaching the transition is related to the vanishing of the kinetic surface tension. The sharpening is related to the reduced influence of fluctuations. Throughout this contribution, we focus on providing a unifying framework that can describe both the fluctuation-dominated behavior of the lattice-gas model for low adspecies diffusion rates, and the crossover to the deterministic mean-field behavior for high diffusion rates where the adlayer is well mixed or randomized.

References (41)

V. P. Zhadanov and B. Kasemo, Surf. Sci. Rep. 20, 111 (1994).
J. W. Evans and M. Sabella, in Trends in Statistical Physics (Research Trends, Trivandrum, India, 1994).
R. Dickman, Int. J. Mod. Phys. C 4, 271 (1993).
R. Imbihl, Prog. Surf. Sci. 44, 185 (1993); G. Ertl, Adv. Catal. 37, 213 (1990).
R. M. Ziff, E. Gulari and Y. Barshad, Phys. Rev. Lett. 56, 2553 (1986).
I. Jensen and H. Fogedby, Phys. Rev. A 42, 1969 (1990).
J. Mai, W. von Niessen and A. Blumen, J. Chem. Phys. 93, 3685 (1990); L. V. Lutsevich, V. I. Elokhin, A. V. Myshlyavtsev, A. G. Usov and G. S. Yablonskii, J. Catal. 132, 302 (1991).
J. W. Evans, J. Chem. Phys. 98, 2463 (1993).
J. D. Gunton and M. Droz, Introduction to the Theory of Metastable and Unstable States (Springer, Berlin, 1983); K. Kaski, K. Binder and J. D. Gunton, Phys. Rev. B 29, 3996 (1984).
D. Mollison, J. R. Stat. Soc. B 39, 283 (1977).
A. S. Mikhailov, Foundations of Synergetics I, Springer Series in Synergetics Vol. 51 (Springer, Berlin, 1990).
Dynamics of Fractal Surfaces, edited by F. Family and T. Vicsek (World Scientific, Singapore, 1991).
M. Kardar, G. Parisi and Y.-C. Zhang, Phys. Rev. Lett. 56, 342 (1986).
P. Grassberger and A. De La Torre, Ann. Phys. 122, 373 (1979).
I. Jensen, H. C. Fogedby and R. Dickman, Phys. Rev. A 41, 3411 (1990).
J. W. Evans and M. S. Miesch, Phys. Rev. Lett. 66, 833 (1991).
J. W. Evans, J. Chem. Phys. 97, 572 (1992).
B. J. Brosilow and R. M. Ziff, Phys. Rev. A 46, 4534 (1992); E. V. Albano, Appl. Phys. A 54, 2159 (1992); T. Tome and R. Dickman, Phys. Rev. E 47, 948 (1993); J. W. Evans, Langmuir 7, 2514 (1991).
M. Dumont, P. Dufour, B. Sente and R. Dagonnier, J. Catal. 122, 95 (1990).
M. Ehsasi, M. Matloch, O. Frank, J. H. Block, K. Christmann, F. S. Rys and W. Hirschwald, J. Chem. Phys. 91, 4949 (1989); H. P. Kaukonen and R. M. Nieminen, ibid. 91, 4380 (1989).
M. Tammaro, M. Sabella, and J. W. Evans (unpublished).
P. Meakin and D. J. Scalapino, J. Chem. Phys. 87, 731 (1987).
D. G. Valchos, L. D. Schmidt and R. Aris, Surf. Sci. 249, 248 (1991).
R. M. Ziff and B. J. Brosilow, Phys. Rev. A 46, 4630 (1992).
B. Chu, J. F. Schoenes and M. F. Fisher, Phys. Rev. 185, 219 (1969).
J. W. Evans and M. S. Miesch, Surf. Sci. 245, 401 (1991).
For h = inf, the mean-field rate equations yield $θ_{A}$ app $F_{+}$ [( $y_{A}$ - $y_{s +}$ )( exp lcurl $y_{A}$ t rcurl -1)], where $y_{s +}$ = case 2 over 3 and $F_{+}$ (x) =9x/(9x+2). However, the form $θ_{A}$ app $K_{+}$ [( $y_{A}$ - $y_{s +}$ )t] works better for small h.
R. Dickman, Phys. Rev. A 34, 4246 (1986).
J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields (Springer, New York, 1983).
P. Fischer and U. Titulaer, Surf. Sci. 221, 409 (1989).
M. Eden, in Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability, Berkeley, 1960, edited by J. Neyman (University of California Press, Berkeley, 1961).
J. Kertesz and D. E. Wolf, J. Phys. A 21, 747 (1988).
H. C. Kang and J. W. Evans, Surf. Sci. 271, 321 (1992).
Dynamical renormalization group analysis indicates that κ app $l^{alpha}$ depends on the length scale l of observation. See, e.g., C.-H. Lam and L. M. Sander, Phys. Rev. Lett. 71, 561 (1993).
J. Krug and P. Meakin, J. Phys. A 23, L987 (1990).
M. Tammaro and J. W. Evans (unpublished).
H. C. Kang and W. H. Weinberg, Phys. Rev. E 47, 1604 (1993).
Similar refinement of previous eta values (from Ref. [16]) is necessary for the monomer-dimer reaction with finite reaction rate k. For example, we find that η prime is no greater than 0.6 for k=1.
One might thus write κ app Δ ${}^{σ prime}$ ( $R^{star}$ $)^{α}$ $l_{c}^{2}$ / $τ_{c}$ , so then σ prime = γ - 1/2 φ (1- α ).
For h=0 and Δ >= 0.005, the relation $P_{g}$ app G (N/ $N^{star}$ -1) might have the form $P_{g}$ app (N/ $N^{star}$ $)^{chi}$ , for N << $N^{star}$ . Then χ = ν δ / phi for consistency with $N^{star}$ app Δ ${}^{- φ}$ and with $P_{g}$ app Δ ${}^{ν δ}$ (for fixed N). For h=0, we find that [16] $P_{g}$ app $N^{2.5}$ for N << $N^{star}$ and fixed DELTA, so φ = ( ν approx 0.9) ( δ approx 3.7)/ ( χ approx 2.5) approx 1.3, consistent with independent estimates for h=0 in Table I.
T. R. Ray, Ph.D. thesis, Iowa State University, 1993.

[c1] V. P. Zhadanov and B. Kasemo, Surf. Sci. Rep. 20, 111 (1994).

[c2] J. W. Evans and M. Sabella, in Trends in Statistical Physics (Research Trends, Trivandrum, India, 1994).

[c3] R. Dickman, Int. J. Mod. Phys. C 4, 271 (1993).

[c4] R. Imbihl, Prog. Surf. Sci. 44, 185 (1993); G. Ertl, Adv. Catal. 37, 213 (1990).

[c5] R. M. Ziff, E. Gulari and Y. Barshad, Phys. Rev. Lett. 56, 2553 (1986).

[c6] I. Jensen and H. Fogedby, Phys. Rev. A 42, 1969 (1990).

[c7] J. Mai, W. von Niessen and A. Blumen, J. Chem. Phys. 93, 3685 (1990); L. V. Lutsevich, V. I. Elokhin, A. V. Myshlyavtsev, A. G. Usov and G. S. Yablonskii, J. Catal. 132, 302 (1991).

[c8] J. W. Evans, J. Chem. Phys. 98, 2463 (1993).

[c9] J. D. Gunton and M. Droz, Introduction to the Theory of Metastable and Unstable States (Springer, Berlin, 1983); K. Kaski, K. Binder and J. D. Gunton, Phys. Rev. B 29, 3996 (1984).

[c10] D. Mollison, J. R. Stat. Soc. B 39, 283 (1977).

[c11] A. S. Mikhailov, Foundations of Synergetics I, Springer Series in Synergetics Vol. 51 (Springer, Berlin, 1990).

[c12] Dynamics of Fractal Surfaces, edited by F. Family and T. Vicsek (World Scientific, Singapore, 1991).

[c13] M. Kardar, G. Parisi and Y.-C. Zhang, Phys. Rev. Lett. 56, 342 (1986).

[c14] P. Grassberger and A. De La Torre, Ann. Phys. 122, 373 (1979).

[c15] I. Jensen, H. C. Fogedby and R. Dickman, Phys. Rev. A 41, 3411 (1990).

[c16] J. W. Evans and M. S. Miesch, Phys. Rev. Lett. 66, 833 (1991).

[c17] J. W. Evans, J. Chem. Phys. 97, 572 (1992).

[c18] B. J. Brosilow and R. M. Ziff, Phys. Rev. A 46, 4534 (1992); E. V. Albano, Appl. Phys. A 54, 2159 (1992); T. Tome and R. Dickman, Phys. Rev. E 47, 948 (1993); J. W. Evans, Langmuir 7, 2514 (1991).

[c19] M. Dumont, P. Dufour, B. Sente and R. Dagonnier, J. Catal. 122, 95 (1990).

[c20] M. Ehsasi, M. Matloch, O. Frank, J. H. Block, K. Christmann, F. S. Rys and W. Hirschwald, J. Chem. Phys. 91, 4949 (1989); H. P. Kaukonen and R. M. Nieminen, ibid. 91, 4380 (1989).

[c21] M. Tammaro, M. Sabella, and J. W. Evans (unpublished).

[c22] P. Meakin and D. J. Scalapino, J. Chem. Phys. 87, 731 (1987).

[c23] D. G. Valchos, L. D. Schmidt and R. Aris, Surf. Sci. 249, 248 (1991).

[c24] R. M. Ziff and B. J. Brosilow, Phys. Rev. A 46, 4630 (1992).

[c25] B. Chu, J. F. Schoenes and M. F. Fisher, Phys. Rev. 185, 219 (1969).

[c26] J. W. Evans and M. S. Miesch, Surf. Sci. 245, 401 (1991).

[c27] For h = inf, the mean-field rate equations yield $θ_{A}$ app $F_{+}$ [( $y_{A}$ - $y_{s +}$ )( exp lcurl $y_{A}$ t rcurl -1)], where $y_{s +}$ = case 2 over 3 and $F_{+}$ (x) =9x/(9x+2). However, the form $θ_{A}$ app $K_{+}$ [( $y_{A}$ - $y_{s +}$ )t] works better for small h.

[c28] R. Dickman, Phys. Rev. A 34, 4246 (1986).

[c29] J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields (Springer, New York, 1983).

[c30] P. Fischer and U. Titulaer, Surf. Sci. 221, 409 (1989).

[c31] M. Eden, in Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability, Berkeley, 1960, edited by J. Neyman (University of California Press, Berkeley, 1961).

[c32] J. Kertesz and D. E. Wolf, J. Phys. A 21, 747 (1988).

[c33] H. C. Kang and J. W. Evans, Surf. Sci. 271, 321 (1992).

[c34] Dynamical renormalization group analysis indicates that κ app $l^{alpha}$ depends on the length scale l of observation. See, e.g., C.-H. Lam and L. M. Sander, Phys. Rev. Lett. 71, 561 (1993).

[c35] J. Krug and P. Meakin, J. Phys. A 23, L987 (1990).

[c36] M. Tammaro and J. W. Evans (unpublished).

[c37] H. C. Kang and W. H. Weinberg, Phys. Rev. E 47, 1604 (1993).

[c38] Similar refinement of previous eta values (from Ref. [16]) is necessary for the monomer-dimer reaction with finite reaction rate k. For example, we find that η prime is no greater than 0.6 for k=1.

[c39] One might thus write κ app Δ ${}^{σ prime}$ ( $R^{star}$ $)^{α}$ $l_{c}^{2}$ / $τ_{c}$ , so then σ prime = γ - 1/2 φ (1- α ).

[c40] For h=0 and Δ >= 0.005, the relation $P_{g}$ app G (N/ $N^{star}$ -1) might have the form $P_{g}$ app (N/ $N^{star}$ $)^{chi}$ , for N << $N^{star}$ . Then χ = ν δ / phi for consistency with $N^{star}$ app Δ ${}^{- φ}$ and with $P_{g}$ app Δ ${}^{ν δ}$ (for fixed N). For h=0, we find that [16] $P_{g}$ app $N^{2.5}$ for N << $N^{star}$ and fixed DELTA, so φ = ( ν approx 0.9) ( δ approx 3.7)/ ( χ approx 2.5) approx 1.3, consistent with independent estimates for h=0 in Table I.

[c41] T. R. Ray, Ph.D. thesis, Iowa State University, 1993.

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Interface propagation and nucleation phenomena for discontinuous poisoning transitions in surface-reaction models

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