NON-EQUILIBRIUM PHENOMENA IN CONTINUOUS SYSTEMS

Chapter 5. Non-equilibrium Phenomena in Continuous Systems 2. Precipitation potential... 

In this chapter, we will consider simple non-equilibrium phenomena involving two fluxes and two forces. Such a situation arises in membrane transport phenomena (e.g. thermo-osmotic and electro-osmotic phenomena) involving two sub-systems separated by a membrane. In Chapters 5 and 6, transport phenomena in two flux-two force systems in continuous systems without a membrane would be discussed. 

Therefore, the simplest procedure to get the stochastic description of the reaction leads to the rather complicated set of equations containing phenomenological parameters / (equation (2.2.17)) with non-transparent physical meaning. Fluctuations are still considered as a result of the external perturbation. An advantage of this approach is a useful analogy of reaction kinetics and the physics of equilibrium critical phenomena. As is well known, because of their nonlinearity, equations (2.1.40) reveal non-equilibrium bifurcations [78, 113]. A description of diffusion-controlled reactions in terms of continuous Markov process - equation (2.2.15) - makes our problem very similar to the static and dynamic theory of critical phenomena [63, 87]. When approaching the bifurcation points, the systems with reactions become very sensitive to the environment fluctuations, which can even produce new nonequilibrium transitions [18, 67, 68, 90, 108]. The language developed in the physics of critical phenomena can be directly applied to the processes in spatially extended systems. 

From a physical point of view, the rhythmic phenomena are related to the fact that biological systems are maintained under far-from-equilibrium conditions through a continuous dissipation of energy [23]. However, non-equilibrium conditions can also give rise to more complicated behaviors. Chaotic dynamics, for instance, can arise either as a regular rhythmic process is destabilized and develops through a cascade of period-doubling bifurcations [24], by torus destruction in connection with the interaction of two or more rhythms, or via different types of intermittency... 

Studies carried out by Yoshida and coworkers have coupled this phenomena with oscillating chemical reactions (such as the Belousov-Zhabotinsky, BZ, reaction) to create conditions where pseudo non-equilibrium systems which maintain rhythmical oscillations can demonstrated, in both quiescent (4) and continuously stirred reactors (5). The ruthenium complex of the BZ reaction was introduced as a functional group into poly(N-isopropyl acrylamide), which is a temperature-sensitive polymer. The ruthenium group plays it s part in the BZ reaction, and the oxidation state of the catalyst changes the collapse temperature of the gel. The result is, at intermediate temperature, a gel whose shape oscillated (by a factor of 2 in volume) in a BZ reaction, providing an elegant demonstration of oscillation in a polymer gel. This system, however, is limited by the concentration of the catalyst which has to remain relatively small, and hence the volume change is small. 

In a series of impressive publications. Maxwell [95-98] provided most of the fundamental concepts constituting the statistical theory recognizing that the molecular motion has a random character. When the molecular motion is random, the absolute molecular velocity cannot be described deterministically in accordance with a physical law so a probabilistic (stochastic) model is required. Therefore, the conceptual ideas of kinetic theory rely on the assumption that the mean flow, transport and thermodynamic properties of a collection of gas molecules can be obtained from the knowledge of their masses, number density, and a probabilistic velocity distribution function. The gas is thus described in terms of the distribution function which contains information of the spatial distributions of molecules, as well as about the molecular velocity distribution, in the system under consideration. An important introductory result was the Maxwellian velocity distribution function heuristically derived for a gas at equilibrium. It is emphasized that a gas at thermodynamic equilibrium contains no macroscopic gradients, so that the fluid properties like velocity, temperature and density are uniform in space and time. When the gas is out of equilibrium non-uniform spatial distributions of the macroscopic quantities occur, thus additional phenomena arise as a result of the molecular motion. The random movement of molecules from one region to another tend to transport with them the macroscopic properties of the region from which they depart. Therefore, at their destination the molecules find themselves out of equilibrium with the properties of the region in which they arrive. At the continuous macroscopic level the net effect... 

Results of dynamic mechanical measurements are shown in Fig. 42 none of the gels show a real equilibrium plateau for G, so a real permanent network is not obtained. Moreover, the gels show thixotropic (two phase) behaviour, which follows from the observations that the gels are liable to a continuous stress relaxation upon applying a constant deformation and that the linearity of the dynamic moduli already disappears at strains of less than 1% the non-linear response, however, is almost instantaneous and, by turning back into the linear region, the initial values are quickly regained. These phenomena are typical for two-phase systems and especially for thixotropic ones. 

Ternary system consists of one volatile and two nonvolatile components, such phenomena as an azeotropy in liquid-gas equilibria and a formation of binary or ternary compounds are absent. Solid phases of volatile and each non-volatile components are completely immiscible and have the eutectic relations in equilibrium with fluid phases, whereas the solid phases of non-volatile components form a continuous solid solution. 

---