Thermodynamic branch

In their subsequent works, the authors treated directly the nonlinear equations of evolution (e.g., the equations of chemical kinetics). Even though these equations cannot be solved explicitly, some powerful mathematical methods can be used to determine the nature of their solutions (rather than their analytical form). In these equations, one can generally identify a certain parameter k, which measures the strength of the external constraints that prevent the system from reaching thermodynamic equilibrium. The system then tends to a nonequilibrium stationary state. Near equilibrium, the latter state is unique and close to the former its characteristics, plotted against k, lie on a continuous curve (the thermodynamic branch). It may happen, however, that on increasing k, one reaches a critical bifurcation value k, beyond which the appearance of the... 

Figure 5.21 Bifurcation far from equilibrium, (a) Primary bifurcation is the distance from equilibrium, at which the thermodynamic branching of minimal entropy production becomes unstable. The bifurcation point or critical point corresponds to the concentration (b) Complete diagram of bifurcations. As the non-linear reaction moves away from equilibrium, the number of possible states increases enormously. (Adapted, with permission, from Coveney and Highfield, 1990).

In this chapter, we will try to answer the next obvious question can we find an explicit reaction rate equation for the general non-linear reaction mechanism, at least for its thermodynamic branch, which goes through the equilibrium. Applying the kinetic polynomial concept, we introduce the new explicit form of reaction rate equation in terms of hypergeometric series. 

The second motive of this chapter is concerned with evergreen topic of interplay of chemical kinetics and thermodynamics. We analyze the generalized form of the explicit reaction rate equation of the thermodynamic branch within the context of relationship between forward and reverse reaction rates (we term the corresponding problem as the Horiuti-Boreskov problem). We will compare our... 

Finally, we present the results of the case studies for Eley-Rideal and LH reaction mechanisms illustrating the practical aspects (i.e. convergence, relation to classic approximations) of application of this new form of reaction rate equation. One of surprising observations here is the fact that hypergeometric series provides the good fit to the exact solution not only in the vicinity of thermodynamic equilibrium but also far from equilibrium. Unlike classical approximations, the approximation with truncated series has non-local features. For instance, our examples show that approximation with the truncated hypergeometric series may supersede the conventional rate-limiting step equations. For thermodynamic branch, we may think of the domain of applicability of reaction rate series as the domain, in which the reaction rate is relatively small. 

Note that both Equations (56) and (60) result in the same series for the root of kinetic polynomial corresponding the "thermodynamic branch" (see Appendix 4 for the proof). 

Figure 6 Approximations of the thermodynamic branch steady-state multiplicity case (see Figure 1). Solid line is the first-term hypergeometric approximation. Circles correspond to the higher-order hypergeometric approximation (m = 3). Dashed line is the first-order approximation in the vicinity of thermodynamic equilibrium. Dash-dots correspond to the second-order approximation in the vicinity of thermodynamic equilibrium.

We are going to focus below on the series for the thermodynamic branch, i.e. the branch described by formula... 

Validity of the thermodynamic branch. To show that series (67) actually represents the thermodynamic branch we have to prove that all terms of this series include the cyclic characteristic C in positive degree. 

In the case of the quadratic equation, the convergence condition for the "thermodynamic branch" series is simply positive discriminant (Passare and Tsikh, 2004). For kinetic polynomial (48) this discriminant is always positive for feasible values of parameters (see Equation (49)). This explains the convergence pattern for this series, in which the addition of new terms extended the convergence domain. 

For certain types of the series the explicit inequalities involving "mirror reflections" of the discriminant were possible (Passare and Tsikh, 2004). The situation is clearer for series depending on fewer variables. For instance, applying Birkeland approach, we can reduce to two the number of parameters in the case of cubic equation. The "thermodynamic branch" corresponds to the Birkeland series (60) for p — 0 and q—1. The discriminant for cubic equation in Birkeland form is... 

We are going to compare our hypergeometric approximation of the thermodynamic branch to the classic rate-limiting step and linear equilibrium approximations (see Sections 3.1.1 and 3.1.2). 

Applying "kinetic polynomial" approach we found the analytical representation for the "thermodynamic branch" of the overall reaction rate of the complex reaction with no traditional assumptions on the rate limiting and "fast" equilibrium of steps. 

Proposition A4.2. Thermodynamic branch R — — [(Bo)/(Bi)] corresponds to the feasible solution in the vicinity of equilibrium. 

Por Keq > 0, condition (A4.6) could be satisfied only, if j = 0. There is a bijection between solution (A4.1) and condition (A4.6), and the case 7 = 0 corresponds to the only feasible solution (A4.2) (see Proposition A4.1). However, when p = 1, there is only one branch of solutions of kinetic polynomial vanishing at the equilibrium. As the thermodynamic branch satisfies the equilibrium condition (A4.0) and there are no other branches vanishing at the equilibrium (we proved in Appendix 3 that Bi O at the equilibrium (see also Lazman and Yablonskii, 1991), this branch should be feasible. By continuity, this property should be valid in some vicinity of equilibrium. 

Let now p> 1. In this case, we have p branches of kinetic polynomial zeros vanishing at the equilibrium. Which one corresponds to the thermodynamic branch ... 

APPENDIX 5. STURMFELS SERIES COINCIDES WITH THE BIRKELAND SERIES FOR THE CASE OF THERMODYNAMIC BRANCH ... 

Under those conditions P behaves as a Lagrangian in mechanics. Furthermore, as P is a nonnegative function for any positive value of the concentrations X,, by a theorem due to Lyapounov, the asymptotic stability of nonequilibrium steady states is ensured (theorem of minimum entropy production.1-23 These steady states are thus characterized by a minimum level of the dissipation in the linear domain of nonequilibrium thermodynamics the systems tend to states approaching equilibrium as much as their constraints permit. Although entropy may be lower than at equilibrium, the equilibrium type of order still prevails. The steady states belong to what has been called the thermodynamic branch, as it contains the equilibrium state as a particular case. 

Beyond the domain of validity of the minimum entropy production theorem (i.e., far from equilibrium), a new type of order may arise. The stability of the thermodynamic branch is no longer automatically ensured by the relations (8). Nevertheless it can be shown that even then, with fixed boundary conditions, nonequilibrium systems always obey to the inequality1... 

Fig. 2. Stability of the thermodynamic branch as a function of some parameter A that measures the chemical system s distance from equilibrium. In the linear range (i.e., for 0 sAssA ) the steady states belong to the thermodynamic branch (a) and are stable. Beyond this domain there may exist a threshold point Ac at which a new stable nonequilibrium branch of solutions (b) appears while the thermodynamic branch becomes unstable.

These values correspond to the homogeneous steady-state solution of (14), which belongs to the thermodynamic branch. We first determine the conditions under which this solution becomes unstable and subsequently the properties of the new regimes that branch off at the point of instability. 

When <0, the bifurcation diagram is as in Fig. 13. There exists a subcritical region in which three stable steady-state solutions may coexist simultaneously the thermodynamic branch and two inhomogeneous solutions. It must be pointed out that the latter are necessarily located at a finite distance from the thermodynamic branch. As a result, their evaluation cannot be performed by the methods described here. The existence of these solutions is, however, ensured by the fact that in the limit B->0, only the thermodynamic solution exists whereas for B Bc it can be shown that the amplitude of all steady-state solutions remains bounded. 

For p< 2/2 the only stationary solutions possible is the homogeneous state in which the concentrations are constant and equal to the value at the boundaries. This corresponds to the domain of stability of the thermodynamic branch. 

Fig. 23. Bifurcation diagram of KU2 in the case of Fig. 22. The lower branch of solutions corresponds to the thermodynamic branch. It tends to the asymptotic value K 12, which separates it from nonequilibrium types of solutions.

If we consider an experiment in which the system is set up at low flow rates, the system will settle to the lower branch (sometimes called the thermodynamic branch because it approaches the thermodynamic equilibrium state at the lowest flow rates). If the flow rate is now increased, the system moves along this branch, through the region of multistability, until it reaches the turning point. Beyond the fold, the system must jump suddenly to the... 

When P0 = i, the two roots of eqn (6.22) are exactly equal. The ignition and extinction points are coincident at ires = 64/27 multistability is lost. For larger inflow concentrations of B the stationary-state extent of reaction increases smoothly with the residence time and the distinction between the flow and thermodynamic branch is lost. 

In 1977. Professor Ilya Prigogine of the Free University of Brussels. Belgium, was awarded Ihe Nobel Prize in chemistry for his central role in the advances made in irreversible thermodynamics over the last ihrec decades. Prigogine and his associates investigated Ihe properties of systems far from equilibrium where a variety of phenomena exist that are not possible near or al equilibrium. These include chemical systems with multiple stationary states, chemical hysteresis, nucleation processes which give rise to transitions between multiple stationary states, oscillatory systems, the formation of stable and oscillatory macroscopic spatial structures, chemical waves, and Lhe critical behavior of fluctuations. As pointed out by I. Procaccia and J. Ross (Science. 198, 716—717, 1977). the central question concerns Ihe conditions of instability of the thermodynamic branch. The theory of stability of ordinary differential equations is well established. The problem that confronted Prigogine and his collaborators was to develop a thermodynamic theory of stability that spans the whole range of equilibrium and nonequilibrium phenomena. 

The steady-state solution that is an extension of the equilibrium state, called the thermodynamic branch, is stable until the parameter A reaches the critical value A,. For values larger than A<, there appear two new branches (61) and (62). Each of the new branches is stable, but the extrapolation of the thermodynamic branch (a ) is unstable. Using the mathematical methods of bifurcation theory, one can determine the point A, and also obtain the new solution, (i.e., the dissipative structures) in the vicinity of A, as a function of (A - A,.). One must emphasize that... 

Primary bifurcation, the first transition from the reference state on the thermodynamic branch, was defined and discussed in the paper by I. Prigogine. This phenomenon is nowadays well understood. Let us outline briefly its theoretical formulation for the reaction-diffusion equations1-2... 

Let Xs represent the uniform steady-state solution on the thermodynamic branch. Such a state always exists if v and D in equations (1) and (2) do not depend explicitly on space and time and if the system is subjected to symmetric boundary conditions. We introduce the deviation x from X.t,... 

Fig. 1. Bistability and hysteresis in the Edelstein mode 34 . In this figure xss ([X] ) is plotted as a function of a ([A]) for constant total enzyme concentration, [E], = 30, and [B] = 0.2. The upper and lower values (flow branch and thermodynamic branches, respectively) of jtss are stable. The intermediate branch (dashed line) is unstable. Suppose one starts with a solution of composition P, on the thermodynamic branch, and then proceeds to increase a. Stable solutions of x with increasing values result, until point N is reached. At this point an increase in a results in a transition to the flow branch. Decreasing a causes Xss to decrease along the flow branch, until point M is reached, whereupon an abrupt transition back to the thermodynamic branch occurs. Increases in a bring ss back to P, completing a hysteresis loop...

---