Schlogls Equation

The fluid dynamics and the net convection velocity developed under the influence of electrical potential and pressure gradient are described by Schlogl s equation of motion as (Bemardi and Verbrugge, 1991) 

Note that in order to estimate charge transport or electrical current in the electrode and electrolyte layers, it is necessary to solve for the electrical potential field in the electrode and electrolyte layers. 

The electrical potential can be written from the Nernst-Planck equation (Equation 7.7a) and Equation 7.9 

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Schlogl equation for water transport in polymer membrane ... 

While in ID model, Darcy and Schlogl equations are usually chosen in order to describe the momentum transfer in mnltiporous layer. As to the formation of ion and chemical components transferred in the membrane, Nemst-Planck and Stefan-Maxwell eqnation are simplified and applied. Butler-Volmer equation is responsible to the electrode reaction dynamics and current density. The pressure and voltage gradient in membrane is taken as constant. While modeling, reaction mechanisms and conservation equations vertical to flux can be established. In the meantime, parameters in other directions are assumed as invariable. 

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 ... 

Integration of the stationary electro-diffusion equations in one dimension. The integration of the stationary Nernst-Planck equations (4.1.1) with the LEN condition (4.1.3), in one dimension, for a medium with N constant for an arbitrary number of charged species of arbitrary valencies was first carried out by Schlogl [5]. A detailed account of Schlogl s procedure may be found in [6]. In this section we adopt a somewhat different, simpler integration procedure. 

Now consider the evolution in a situation as in fig. 34. There are three stationary macrostates 0fl, 0j>, c, of which 0a and 0C are locally stable and (j)b is unstable. Of course, even the pure macroscopist would not regard (j)b as a realizable state, on the ground that a system in (j>b would be caused to move into either 0a or 0C by the smallest perturbation. Systems having a macroscopic characteristic as in fig. 34 are called bistable . There are numerous examples the ones that occur most often in the literature are the laser (section 9 below), the tunnel diode 0, and the Schlogl reaction (X.3.6). The macroscopic rate equation for this reaction is... 

Exercise. One-step processes have been approximated by diffusion equations in VIII.5, but the danger has been pointed out in XI.5. Show for the Schlogl reaction (X.3.6) that it is incorrect to compute the relative stability (1.11) by using the potential function V in (XI.5.6) rather than U. 

The concentration of all ions in the two surface layers in the membrane are considered to be given. These are related to those of the outer solution by a set of Donnan relations analogous to equation (24). Schlogl calculated the fluxes, the profiles of the concentrations in the membrane and the membrane potential. 

R. Schlogl (144) obtained, through his general integration of the Nernst-Planck equations, also values for the diffusion potential. The approximations in the calculations are the same as those used for the fluxes (cf. 3.4). 

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. 

A further discussion of the Schlogl reaction provides an illustration of an important difference in the behavior of steady states in deterministic systems and in systems subject to stochastic fluctuations. In contrast to the deterministic Schlogl system analyzed earlier in terms of conventional chemical concentrations, the analysis of the stochastic system is carried out in terms of the probability that there are n x t) molecules in the system at time t in addition to nA = a and n-Q = b molecules of A and B that are held constant. Figure 6.2 illustrates the individual pathways and rates by which a system with n molecules could undergo a transition to a system containing either n -f 1 or n — 1 molecules. This leads to the master equation for each of the... 

However, in interdiffusion of ions of different mobilities, Fick s law fluxes would be unequal and disturb electroneutrality. Here, the first, minute deviation from local electroneutrality generates an electric potential gradient (diffusion potential) that produces electric transference of ions superimposed on diffusion. This is the mechanism by which the system manages to balance the fluxes so as to maintain electroneutrality (Schlogl and Helfferich, 1957 Helfferich, 1962a Helfferich and Hwang, 1988). The flux now obeys the Nernst-Planck equation (Nernst, 1888 1889 Planck, 1890)... 

This is the most widely used model for ion-exchange kinetics (Schlogl and Helfferich, 1957 Plesset et al., 1958). Combining the Nernst-Planck equation (Eq. [2]) with the constraints of electroneutrality and zero net charge transfer yields... 

Many researchers have derived improved equations concerning the membrane potential after Meyer-Sievers-Teorell e.g. Bonhoeffer,10 Schlogl-Helfferich,11 Nagasawa-Kobatake.12... 

The original version of the first Schlogl model contains a fourth step, the back reaction C —> B -I- U. We assume that the product C is immediately removed from the system. We consider two different ways of nondimensionalizing this rate equation, (i) Set t = k2 PnP ) t and p = 2/°/( iPa k- P, ). This is acceptable for all situations where k p > k p. Then we obtain the following nondimensionalized version of (1.66) ... 

Solve the kinetic equation (1.69) for the first Schlogl model. Confirm the results of the linear stability analysis for /r > 0 and /u. < 0. Determine the stability of... 

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). 

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... 

Nikonenko and Urtenov (54, 55] have presented an algorithm for the solution of the steady state transport equations in multicomponent systems, which is based on calculating first the electric field E and then using Eqs. (92 and 93). Their method is outlined here for the particular case when the LEN assumption is used. Unlike Schlogl s... 

In the case of nonhomovalent multiionic systems, the solution procedure is necessarily more complicated because there is no simple relation such as S2 = z Sq, a problem that was first encountered and solved in different ways by Schlogl [59] and Brady and Turner [60]. If the system contains N different ionic species, the N Nernst-Planck equations for the N flux densities J,- are replaced by... 

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. 

For further reading, particularly concerning solutions of the Nernst-Planck equation under rather general physical conditions in membrane systems, the very detailed monograph of R. SCHLOGL (1964) is recommended. 

The appearance of two stable steady states X, X3 allows the system to exist in two phases with different densities X and X3 of the species X. It may even happen that these two phases coexist in the same system separated by a phase boundary. The whole situation is very similar to the phenomenon of phase transitions in equilibrium systems such as gas-liquid or liquid-solid systems. According to this similarity, the phenomenon of different phases in a nonequilibrium system is called a nonequilibrium phase transition or a "dissipative structure". Clearly, the inclusion of coexistence between X and X3 and of phase boundaries into our theory requires the introduction of additional diffusion terms into the equation of motion (6.5) in order to account for spatial variations of X. The analogies between our autocatalytic system (for v = 2) and equilibrium phase transitions have been worked out by F. SCHLOGL (1972) on a phenomenological and by JANSSEN (1974) on a stochastic level. 

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]... 

This procedure is easy for a one-variable system because we know the solution of the stationary master equation to this approximation. For example, for the one-variable Schlogl model we have the elementary reaction steps... 

The numerical investigation of this equation shows that in the bistable regime the minimum in the diffusional dependence of the transition time between stationary states also occurs Cas in the case of the multivariate master equation ). Fig. 2 shows the results for the Schlogl model Cthe system size is 100 and 200D. 

Fig. 5. Prediction of equistability from Schlogl s analysis prediction of thermodynamic theory carried out for subdivisions of the inhomogeneous region into 100,200, or 400 boxes the large noise limit of for 200 boxes and the deterministic equation for 200 boxes. Reprinted from [29].

For one-dimensional systems the equal stability condition obtained from numerical solutions of the reaction diffusion equations agrees with Schlogl s... 

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