Stability in nonequilibrium systems

Since a definite function 82S leads to the stability condition, it operates as a Lyapunov function, and assures the stability of a stationary state. As the entropy production is the sum of the products of flows J and forces X, we have 

Since L(8X)2 is always positive, we always consider the stationary states described by the linear phenomenological equations as thermodynamically stable states. 

For systems not far from equilibrium, the total entropy production reaches a minimum value and also assures the stability of the stationary state. However, for systems far from equilibrium, there is no such general criterion. Far from equilibrium, we may have order in time and space, such as, appearance of rhythms, oscillations, and morphological structurization. 

They have shown that the first term on the right is negative definite even for cases for which the linear phenomenological equations do not hold. By introducing the linear phenomenological equations J, = LikXk with constant coefficients we get 

This relation shows that the contribution of the time change of forces to the entropy production is equal to that of the time change of flows. 

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In the preceding sections, various types of fluctuations and instabilities essential to corrosion were examined. As a result, it was shown that a corrosion system involves various kinds of problems of stability and instability. Unlike thermodynamic equilibrium systems, in nonequilibrium systems like corrosion systems, a drastic change in the reaction state should be defined as a bifurcation phenomenon. 

Perceived stability in ecological systems frequently takes the form of metastability achieved through structural and functional redundancy incorporated in space and time. Patterns that appear stable at one scale may be due to nonequilibrium and stochastic processes occurring at adjacent hierarchies of scale. 

Thermodynamics plays an important role in the stability analysis of transport and rate processes, and the nonequilibrium thermodynamics approach in particular may enhance and broaden this role. This chapter reviews stability analysis based on the conventional Gibbs approach and tbe nonequilibrium thermodynamics theory. It considers the stability of equilibrium, near-equilibrium, and far-from-equilibrium states with some case studies. The entropy production approach for nonequilibrium systems appears to be more general for stability analysis. One major implication of the nonequilibrium thermodynamics theory is the introduction of distance from global equilibrium as a constraint for determining the stability of nonequilibrium systems. When a system is far from global equilibrium, the possibility of new organized structures of matter arise beyond an instability point. 

The liquid junction potential (UP), as considered in relation to practical aspects of reference electrodes, is a rather bothering experimental problem. The additional and always unknown potential drop between the electrolytes of the electrode under smdy and of the reference electrode is harmful for the accuracy of potential measurements. In addition, the existence of this drop disturbs the equilibrium in the circuit (if any) and complicates stabilization of nonequilibrium systems. 

In contrast to a mixture of redox couples that rapidly reach thermodynamic equilibrium because of fast reaction kinetics, e.g., a mixture of Fe2+/Fe3+ and Ce3+/ Ce4+, due to the slow kinetics of the electroless reaction, the two (sometimes more) couples in a standard electroless solution are not in equilibrium. Nonequilibrium systems of the latter kind were known in the past as polyelectrode systems [18, 19]. Electroless solutions are by their nature thermodyamically prone to reaction between the metal ions and reductant, which is facilitated by a heterogeneous catalyst. In properly formulated electroless solutions, metal ions are complexed, a buffer maintains solution pH, and solution stabilizers, which are normally catalytic poisons, are often employed. The latter adsorb on extraneous catalytically active sites, whether particles in solution, or sites on mechanical components of the deposition system/ container, to inhibit deposition reactions. With proper maintenance, electroless solutions may operate for periods of months at elevated temperatures, and exhibit minimal extraneous metal deposition. 

The two main assumptions underlying the derivation of Eq. (5) are (1) thermodynamic equilibrium and (2) conditions of constant temperature and pressure. These assumptions, especially assumption number 1, however, are often violated in food systems. Most foods are nonequilibrium systems. The complex nature of food systems (i.e., multicomponent and multiphase) lends itself readily to conditions of nonequilibrium. Many food systems, such as baked products, are not in equilibrium because they experience various physical, chemical, and microbiological changes over time. Other food products, such as butter (a water-in-oil emulsion) and mayonnaise (an oil-in-water emulsion), are produced as nonequilibrium systems, stabilized by the use of emulsifying agents. Some food products violate the assumption of equilibrium because they exhibit hysteresis (the final c/w value is dependent on the path taken, e.g., desorption or adsorption) or delayed crystallization (i.e., lactose crystallization in ice cream and powdered milk). In the case of hysteresis, the final c/w value should be independent of the path taken and should only be dependent on temperature, pressure, and composition (i.e.,... 

We review in Section II the basic results of nonequilibrium thermodynamic stability theory and recall the thermodynamic and kinetic conditions necessary to the occurrence of cooperative coherent behaviors in chemical systems. We briefly indicate some known experimental systems that meet these requirements and in which dissipative structures... 

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. 

In the bottom-up approach, a large variety of ordered nano-, micro-and macrostructures may be obtained by changing the balance of all the attractive and repulsive forces between the structure-forming molecules or particles. This can be achieved by altering the environmental conditions (temperature, pH, ionic strength, presence of specific substances or ions) and the concentration of molecules/particles in the system (Min et al., 2008). As this takes place, the interrelated processes of formation and stabilization are both important considerations in the production of nanoparticles. In addition, as particles grow in size a number of intrinsic properties change, some qualitatively, others quantitatively some affect the equilibrium (thermodynamic) properties, and others affect the nonequilibrium (dynamic) properties such as relaxation times. 

Order arising through nucleation occurs both in equilibrium and nonequilibrium systems. In such a process the order that appears is not always the most stable one there are often competing processes that will lead to different structures, and the structure that appears is the one that nucleates first. For instance, in the analysis of the different possible structures in diffusion-reaction systems17-20 one can show, by analyzing the bifurcation equations, that there are several possible structures and some of them require a finite amplitude to become stable if this finite amplitude is realized through fluctuation, this structure will appear. In the formation of crystals (hydrates) the situation is similar the structure that is formed depends, according to the Ostwald rule, on the kinetics of nucleation and not on the relative stability. 

Christiansen et al. (54) applied the Naphtali-Sandholm method to natural gas mixtures. They replaced the equilibrium relationships and component vapor rates with the bubble-point equation and total liquid rate to get practically half the number of functions and variables [to iV(C + 2)]. By exclusively using the Soave-Redlich-Kwong equation of state, they were able to use analytical derivatives of revalues and enthalpies with respect to composition and temperature. To improve stability in the calculation, they limited the changes in the independent variables between trials to where each change did not exceed a preset maximum. There is a Naphtali-Sandholm method in the FraChem program of OLI Systems, Florham Park, New Jersey CHEMCAD of Coade Inc, of Houston, Texas PRO/II of Simulation Sciences of Fullerton, California and Distil-R of TECS Software, Houston, Texas. Variations of the Naphtali-Sandholm method are used in other methods such as the homotopy methods (Sec. 4,2.12) and the nonequilibrium methods (Sec. 4.2.13). 

The stability of transport and rate systems is studied either by nonequilibrium thermodynamics or by conventional rate theory. In the latter, the analysis is based on Poincare s variational equations and Lyapunov functions. We may investigate the stability of a steady state by analyzing the response of a reaction system to small disturbances around the stationary state variables. The disturbed quantities are replaced by linear combinations of their undisturbed stationary values. In nonequilibrium thermodynamics theory, the stability of stationary states is associated with Progogine s principle of minimum entropy production. Stable states are characterized by the lowest value of the entropy production in irreversible processes. The applicability of Prigogine s principle of minimum entropy production is restricted to stationary states close to global thermodynamic equilibrium. It is not applicable to the stability of continuous reaction systems involving stable and unstable steady states far from global equilibrium. The steady-state deviation of entropy production serves as a Lyapunov function. 

The general theory of thermodynamics of nonequilibrium processes also tells us the stability of nonequiUbrium stationary states with respect to spontaneous fluctuations of the internal thermodynamic parameters in the system. It turns out that this stability can also be investigated by analyzing the variations in the entropy production or energy dissipation rates on drawing the system away from its stationary state. 

Microstmctures are frequently present in a kinetically trapped nonequilibrium state, and their structures depend on the components and colloidal interactions based on their different chemical and physical properties, as well as on the procedure by which these components have been assembled. These structures are thermodynamically unstable and tend to reduce their free energy (surface area) with time. On the contrary, self-assembly nanostructures are thermodynamically stable, unless the molecules react with the environment or degrade. Most food systems are based on an interplay of kinetically stabilized and thermodynamic equilibrium structures. Some typical examples of structures at different length scales formd in food systems are shown in Figure 11.1. 

However, nonequilibrium processes occur in many systems containing emulsifiers at fluid interfaces (Figure 14.1). Thus, relaxation phenomena have great importance from a practical point of view in emulsifier films that stabilize food dispersions. The d)mamic phenomena and the development of intermolecular associations at the interface lead to alterations in surface properties that have measurable rheological consequences (Murray and Dickinson, 1996 Murray, 1998,2002 Bos and van Vliet, 2001) that is, surface... 

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