Schlogl’s model

In the chemical master equation, the steady-state probability distribution of the equUihrium steady state is a Poisson distribution. For Schlogl s model steady-state probability distributions become... 

Chemical reaction models of phase transition-like phenomena have extensively been studied. Schlogl s model (1972) of the second-order phase transition is... 

With the increased computational power of today s computers, more detailed simulations are possible. Thus, complex equations such as the Navier—Stokes equation can be solved in multiple dimensions, yielding accurate descriptions of such phenomena as heat and mass transfer and fluid and two-phase flow throughout the fuel cell. The type of models that do this analysis are based on a finite-element framework and are termed CFD models. CFD models are widely available through commercial packages, some of which include an electrochemistry module. As mentioned above, almost all of the CFD models are based on the Bernardi and Verbrugge model. That is to say that the incorporated electrochemical effects stem from their equations, such as their kinetic source terms in the catalyst layers and the use of Schlogl s equation for water transport in the membrane. 

The first model to describe the membrane in the above fashion was that of Bernardi and Verbrugge, "° which was based on earlier work by Verbrugge and Hill. " 214 model utilized a dilute solution approach that used the Nernst— Planck equation (eq 29) to describe the movement of protons, except that now v is not equal to zero. The reason is that, because there are two phases, the protons are in the water and the velocity of the water is give by Schlogl s equation ... 

One comment should be made regarding the form of the transport equations. In the literature, two-phase flow has often been modeled using Schlogl s equation [50, 51]. This equation is similar in form to Eq. (5.9), but it is empirical and ignores the Onsager cross coefficients. Equations (5.8) and (5.9) stem from concentrated-solution theory and take into account all the relevant interactions. Furthermore, the equations for the liquid-equilibrated transport mode are almost identical to those for the vapor-equilibrated transport mode making it easier to compare the two with a single set of properties (i.e., it is not necessary to introduce another parameter, the elec-trokinetic permeability). 

One of the early mechanistic models for a PEM fuel cell was the pioneering work of Bemardi and Verbrugge [45, 46]. They developed a one-dimensional, steady state, isothermal model which described water transport, reactant species transport, as well as ohmic and activation overpotentials. Their model assumed a fully hydrated membrane at all times, and thus calculated the water input and removal requirements to maintain full hydration of the membrane. The model was based on the Stefan Maxwell equations to describe gas phase diffusion in the electrode regions, the Nemst-Planck equation to describe dissolved species fluxes in the membrane and catalyst layers, the Butler Volmer equation to describe electrode rate kinetics and Schlogl s equation for liquid water transport. 

Abstract chemical models exhibiting nonlinear phenomena were proposed more than a decade ago. The Brusselator of PRIGOGINE and LEFEVER [54] has oscillatory (limit cycle) solutions, and the SCHLOGL [55] model exhibits bistability, but these models have only two variables and hence cannot have chaotic solutions. At least 3 variables are required for chaos in a continuous system, simply because phase space trajectories cannot cross for a deterministic system. As mentioned in the Introduction, the possibility of chemical chaos was suggested by RUELLE [1] in 1973. In 1976 ROSSLER [56], inspired by LORENZ s [57] study of chaos in a 3 variable model of convection, constructed an abstract 3 variable chemical reaction model that exhibited chaos. This model used as an autocatalytic step a Michaelis-Menten type kinetics, which is a nonlinear approximation discovered in enzymatic studies. Recently more realistic biochemical models [58,59] have also been found to exhibit low dimensional chaos. 

R. ScHLOGL and U. Schodel 146) have supplied another proof for the usability of the Nemst-Planck flux equations combined with the M.S.T. model, also for the case that an electric current flows through an ion-selective membrane. They determined the concentration profiles of the mobile ions for the case of a cation selective membrane on the basis of phenol sulfonic acid and NaCl solutions, under application of an electric current. 

Penner, S., Rupprechter, G., Sauer, H., Su, D.S., Tessadri, R., Podloucky, R., Schlogl, R., and Hayek, K. (2003) Pt/ceria thin film model catalysts after high-temperature reduction a (HR)TEM study, Vacuum 71(1-2), 71. 

We take the probability distribution to obey the master equation which has been used extensively. For the cubic Schlogl model ((2.7) with r = 3, s = 1) the master equation is [1,5]... 

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