Kinetic phase transitions

V. P. Zhdanov, B. Kasemo. Kinetic phase transitions in simple reactions on solid surfaces. Surf Sci Rep 20 111-189, 1994. 

R. M. Zilf, E. Gulari, Y. Barshad. Kinetic phase transitions in an irreversible surface-reaction model. Phys Rev Lett 56 2553-2556, 1986. 

J. W. Evans. Kinetic phase transitions in catalytic reaction models. Langmuir 7 2514-2519, 1991. 

E. V. Albano. Monte Carlo simulation of a bimolecular reaction of the type A-t- (1/2) B2 —> AB. The influence of A-desorption on kinetic phase transitions. Appl Phys A 55 226-230, 1992. 

E. V. Albano. Finite-size effects in kinetic phase transitions of a model reaction on a fractal surface Scahng approach and Monte Carlo investigation. Phys Rev B 42 R10818-R10821, 1990. 

I. Jensen, H. C. Fogedby. Kinetic phase transitions in a surface-reaction model with diffusion Computer simulations and mean-field theory. Phys Rev A 2 1969-1975, 1990. 

A. Maltz, E. V. Albano. Kinetic phase transitions in dimer-dimer surface reaction models studied by means of mean-field and Monte Carlo methods. Surf Sci 277-A A-42S, 1992. 

D. A. Browne, P. Kleban. Equilibrium statistical mechanics for kinetic phase transitions. Phys Rev A 40 1615-1626, 1989. 

L. M. Martyushev, V. D. Seleznev, S. A. Skopinov. Reentrant kinetic phase transitions during dendritic growth of crystals in a two-dimensional medium with phase stratification. Tech Phys Lett 25 495, 1997. 

E. Temkin, Kinetic phase transition during a phase conversion in a binary alloy, Sov. Phys. Cryst., 15,1971, 773-80... 

The transition from a stable steady-state solution observed at large p to the oscillatory regime assumes the existence of the critical value of the parameter pc, which defines the point of the kinetic phase transition as p > pc, the fluctuations of the order parameter are suppressed and the standard chemical kinetics (the mean-field theory) could be safely used. However, if p < pc, these fluctuations are very large and begin to dominate the process. Strictly speaking, the region p pc at p > pc is also fluctuation-controlled one since here the fluctuations of the order parameter are abnormally high. 

What was said above is illustrated by Fig. 1.29 and Fig. 1.30 corresponding to the cases p > pc and p < pc respectively. To make the presented kinetic curves smooth, in these calculations the transformation rate A — B was taken to be finite. To make results physically more transparent, the effective reaction rate K (t) of the A —> B transformation is also drawn. The standard chemical kinetics would be valid, if the value of K (t) tends to some constant. However, as it is shown in Fig. 1.30, K(t) reveals its own and quite complicated time development namely its oscillations cause the fluctuations in particle densities. The problems of kinetic phase transitions are discussed in detail in the last Chapter of the book. 

Lastly, non-elementary several-stage reactions are considered in Chapters 8 and 9. We start with the Lotka and Lotka-Volterra reactions as simple model systems. An existence of the undamped density oscillations is established here. The complementary reactions treated in Chapter 9 are catalytic surface oxidation of CO and NH3 formation. These reactions also reveal undamped concentration oscillations and kinetic phase transitions. Their adequate treatment need a generalization of the fluctuation-controlled theory for the discrete (lattice) systems in order to take correctly into account the geometry of both lattice and absorbed molecules. As another illustration of the formalism developed by the authors, the kinetics of reactions upon disorded surfaces is considered. 

It should be reminded here that the most interesting feature of the original ZGB-model is the existence of kinetic phase transitions. Denoting the mole fraction of CO in the gas phase by Yco (and therefore Yqo2 = 1 - Yco), one finds a reactive interval 0.395 = y < Yco < Vi = 0.525 [2] in which both particle types are coexisting on the surface. For Yco < V and for Yco > 2/2 the surface is completely covered by O2 or CO, respectively. The phase transitions are found to be of the second order at y and of the first order at 2/2 Because of the irreversible character the model describes a poisoned state from which the system cannot escape and reaction comes to a stop. 

For S > Sc we obtain an infinite cluster for which in principle a reactive state exists. We use this fact to define the percolation threshold in a kinetic way for the particular reaction at hand as the transition point from the reactive (Rco2 > 0) to the non-reactive (Rco2 — 0) state. As we have shown above, this transition happens in such a way that the kinetic phase transition points of 2/1 and are approaching each other if S —> Sc [25]. At S = Sc the... 

First of all, we want to study the case in which all surface sites are activated (5 = 1). This means that a N2 molecule can adsorb at every pair of free surface sites. Figure 9.11 represents the behaviour of the surface coverages 0i of the various chemical species i as a function of the mole fraction of N2 in the gas phase, yn, for the case 5=1 and D = 1. The most prominent feature in this figure is the kinetic phase transition of the second order at yN = yl fa 0.21. For j/n < Vi the surface is nearly completely covered by... 

Figure 9.12 shows similar data of the surface coverages as in Fig. 9.11 but now we use a larger diffusion rate of D = 10. It can be seen that the value of yi is shifted to a larger value of yN (yi 0.23) which is closer to the stoichiometric ratio of 0.25. The kinetic phase transition sharpens also and it is now nearly of the first order. This can easily be understood from the... 

In Fig. 9.14 the coverages for D = 1 are shown. The kinetic phase transition is of the second order and the value of y is shifted to larger values of 2/n compared to the analogous case above. Over a whole interval of yN> h is... 

For the case 5=1 and D = 1 the results of the stochastic model are in good agreement with the CA model y = 0.262). This is understandable because the different definition of the reaction which leads to a difference in the blocking of activated sites cannot play significant role because all sites are activated. The diffusion rate of D = 10 leads nearly to the same reactivity as if we define the reaction between the nearest-neighbour particles. If the diffusion rate is considerably lowered (D = 0.1), the behaviour of the system changes completely because of the decrease of the reaction probability. This leads to the disappearance of the kinetic phase transition at y because different types of particles may reside on the surface as the nearest neighbours without reaction, a case which does not occur at all in the CA approach. 

Therefore the model avoids two main difficulties the large amount of computer time which is normally needed for simulations and the loss of structural information which occurs in simple theoretical models (mean-field models) which do not take into account the structural aspects of the adsorbate layer. Mean-field-kind models fail in the prediction of phase transitions of the second order because at these points the long-range correlations appear. They also fail in describing the system s behaviour in the neighbourhood of the point of first-order kinetic phase transition. 

For arbitrary parameters of the system, wi and ho differ dramatically from each other one of them is 1, and the other is close to zero. Within a narrow range of parameters, however, they have the same order of magnitude and one can refer to a kinetic phase transition between the two stable states it is analogous to the first-order phase transition in an equilibrium system with a potential (in the absence of quantum fluctuations) playing the role of the generalized free energy of the system [42,65,110]. This is the range of parameters that is of particular interest in the present chapter. 

One of the most important general features of fluctuations in a bistable system is the onset of a narrow zero-frequency spectral peak for parameter values lying in the range of the kinetic phase transition. This peak arises from... 

Thus, an increase in the value of the controlling parameter a can result either in the violation of the condition of aperiodicity in relaxation pro cesses at the preserved total stabihty of the system or, vice versa, in the vio lation of the condition of stationary state stability by which the system is transferred to the nonthermodynamic branch. It is important that the properties of the system state change jumpwise when passing through the bifurcation point, and thus these changes are called sometimes the kinetic phase transitions. 

In transferring from the precritical to the supracritical modes, the system symmetry changes spontaneously by analogy with thermodynamic phase transitions. This is why the transitions to form spatial dissipative structures in nonequihbrium systems are sometimes referred to as kinetic phase transitions. 

Despite its severe limitations, the model shows interesting behavior, including kinetic phase transitions of two types continuous (second order) and discontinuous (first order). These phenomena are observed in many catalytic surface reactions. For this reason, the ZGB model has been widely studied and serves as a starting point for many more realistic models. This forms the first reason why we discuss the ZGB model in this section. The second reason is that MC simulations and mean-field (MF) solutions for this model give different results. Cluster approximations to the MF solutions offer a better agreement between the two methods, and then only small discrepancies remain. The ZGB model is therefore a nice example to illustrate the differences between the two approaches. 

R. M. Ziff, E. Gulari, and Y. Barshad, Kinetic Phase Transitions in an Irreversible Surface-Reaction Model, Phys. Rev. Lett., 56 (1986) 2553. 

---