Nonequilibrium dynamic behavior

Although much work has been done to study equilibrium behavior in surfactant systems many nonequilibrium dynamic behavior are still far from well understood. When neat or concentrated surfactant is contacted with solvent complicated diffusion process occurs due to the presence of mesophase at the interface. Initially, at the interface, the formation, type and structure of the mesophase will influence the subsequent dynamics. In some cases the interface can become unstable during dissolution and rather striking instabilities form. To obtain a good understanding of such complicated nonlinear processes has relied on a systematic study of the equilibrium phase behavior in such systems. This has given us a firm basis on which to study the nonequilibrium behavior. 

In the above subsection it was demonstrated that the inclusion of electrostatic interactions into the pressure-area-temperature equation of state provides a better fit to the observed equilibrium behavior than the model with two-dimensional neutral gas. Considering this fact, we would like to devote our attention now to the character of the lipid film under the dynamical, nonequilibrium conditions. In the following we shall describe the dynamical behavior of the phospholipid(l,2-dipalmitoyl-3-sn-phosphatidylethanolamines DPPE) thin films in the course of the compression and expansion cycles at air/water interface. 

Even at steady state, efficiencies vary from component to component and with position in a column. Thus, if the column is not at steady state, then efficiencies also must vary with time as a result of changes to flow rates and composition inside the column. Thus, equilibrium-stage models with efficiencies should not be used to model the dynamic behavior of distillation and absorption columns. Nonequilibrium models for studying column dynamics are described hy, e.g., Kooijman and Taylor [AlChE 41, 1852 (1995)], Baur et al. [Chem. 

The computer modeling of the dynamical behavior of a solid or a liquid by following the motion of every atom and molecule is known as a molecular dynamics simulation. Such a simulation requires the specification of the positions and velocities of every molecule, which is not loiown a priori. In practice care is taken to start from an equilibrium distribution and slowly equilibrate to nonequilibrium forces, such as... 

The functional aspects, in terms of water dynamics and glass dynamics, and the appropriate kinetic description of the nonequilibrium thermomechanical behavior of food systems have been illustrated as a dynamics map, shown in Figure 8.13. 

The most studied and cited example of complex dynamic behavior in homogeneous catalysis is the Belousov - Zhabotinsky (BZ) reaction named after B. P. Belousov who discovered the reaction and A. M. Zhabotinsky who continued Belousov s early work. The reaction is theoretically important in that it shows that chemical reactions do not have to be dominated by equilibrium. These reactions are far from equilibrium and remain so for a length of time. In this sense, they provide an interesting chemical model of nonequilibrium biological phenomena, and the mathematical model of the BZ reactions themselves are of theoretical interest. 

These nuclear and electronic components, due to their different dynamic behavior, will give rise to different effects. In particular, the electronic motions can be considered as instantaneous and thus the part of the solvent response they originate is always equilibrated to any change, even if fast, in the charge distribution of the solute. On the contrary, solvent nuclear motions, by far slower, can be delayed with respect to fast changes, and thus they can give origin to solute-solvent systems not completely equilibrated in the time interval interested in the phenomenon under study. This condition of nonequilibrium will successively evolve towards a more stable and completely equilibrated state in a time interval which will... 

Here, <... > denotes an average over the equilibrium ensemble of initial conditions. C t) is the conditional probability to find the system in state B at time t provided it was in state A at time 0. According to the fluctuation-dissipation theorem [63], dynamics of equilibrium fluctuations are equivalent to the relaxation from a nonequilibrium state in which only state A is populated. At long timescales, these nonequilibrium dynamics are described by the phenomenology of macroscopic kinetics. Thus, the asymptotic behavior of C(t) is determined by rate constants and kg. At long times, and provided that a single dynamical bottleneck separating A from B causes simple two-state kinetics. 

The free-volume concept dates back to the Clausius [1880] equation of state. The need for postulating the presence of occupied and free space in a material has been imposed by the fluid behavior. Only recently has positron annihilation lifetime spectroscopy (PALS see Chapters 10 to 12) provided direct evidence of free-volume presence. Chapter 6 traces the evolution of equations of state up to derivation of the configurational hole-cell theory [Simha and Somcynsky, 1969 Somcynsky and Simha, 1971], in which the lattice hole fraction, h, a measure of the free-volume content, is given explicitly. Extracted from the pressure-volume-temperature PVT) data, the dependence, h = h T, P), has been used successfully for the interpretation of a plethora of physical phenomena under thermodynamic equilibria as well as in nonequilibrium dynamic systems. 

The rich phase behavior of amphiphilic systems and the complicated structure of many phases on mesoscopic length scales indicate interesting dynamical behavior of these systems. Three principle cases of the dynamical behavior of many-body systems can be distinguished in general the dynamics in the equilibrium state, the dynamics in a stationary state out of equilibrium, and the relaxation from a nonequilibrium state toward equilibrium. All three situations have been studied in some detail recently for amphiphilic systems. 

This section contains several models whose spatiotemporal behavior we analyze later. Nontrivial dynamical behavior requires nonequilibrium conditions. Such conditions can only be sustained in open systems. Experimental studies of nonequilibrium chemical reactions typically use so-called continuous-flow stirred tank reactors (CSTRs). As the name implies, a CSTR consists of a vessel into which fresh reactants are pumped at a constant rate and material is removed at the same rate to maintain a constant volume. The reactor is stirred to achieve a spatially homogeneous system. Most chemical models account for the flow in a simplified way, using the so-called pool chemical assumption. This idealization assumes that the concentrations of the reactants do not change. Strict time independence of the reactant concentrations cannot be achieved in practice, but the pool chemical assumption is a convenient modeling tool. It captures the essential fact that the system is open and maintained at a fixed distance from equilibrium. We will discuss one model that uses CSTR equations. All other models rely on the pool chemical assumption. We will denote pool chemicals using capital letters from the start of the alphabet. A, B, etc. Species whose concentration is allowed to vary are denoted by capital letters... 

Under these conditions, studying quantities such as Cp becomes problematic because their significance changes at an T interval near Tg, at which point the system falls out of equilibrium, and we are then faced with the problem of how to interpret a quantity such as Cp in a nonequilibrium state. How a measurement is performed affects the measured values. Thus, if we want to study well-defined equilibrium quantities in the liquid state, and still learn something about the GT, then we must study their dynamic behaviors. Note that in this situation these measurements are not in the linear-response regime (i.e., under nonequilibrium and nonlinear conditions). 

Molecular dynamics, in contrast to MC simulations, is a typical model in which hydrodynamic effects are incorporated in the behavior of polymer solutions and may be properly accounted for. In the so-called nonequilibrium molecular dynamics method [54], Newton s equations of a (classical) many-particle problem are iteratively solved whereby quantities of both macroscopic and microscopic interest are expressed in terms of the configurational quantities such as the space coordinates or velocities of all particles. In addition, shear flow may be imposed by the homogeneous shear flow algorithm of Evans [56]. 

As is well known, fluid dynamics is the study of motion and transport in liquids and gases. It is primarily concerned with macroscopic phenomena in nonequilibrium fluids and covers such behavior as diffusion in quiescent fluids, convection, laminar flows, and fully developed turbulence. 

Contrary to the accumulated knowledge on the static or quasi-static characteristics of thin lipid films at air/water interface, less attention has been paid to the dynamical or nonequilibrium behavior of the film. Studies on the dynamical characteristics of thin lipid films may be quite important, because the life phenomena are maintained under nonequilibrium conditions. According to the modern biochemistry [11,12], thin lipid membrane in living cells is not a rigid wall but a thermally fluctuating barrier with high fluidity. In the present section, we will show that thin lipid film exhibits the various interesting dynamical tc-A characteristics, such as the "overshoot hump", the "zero surface pressure", and the "flat plateau". 

Finally, it is very important to stress that the ESP is different from the solution reaction path (SRP) for the I2 reaction system [9], which is a much more faithful indicator of the reaction dynamics. The SRP is for example critical in understanding the vibrational relaxation behavior of the system[9],[41]. The ESP only finds its use, illustrated above, in helping decide which solvent coordinates should be considered as independent variables in the nonequilibrium calculation, and which solvent coordinates... 

Section 11 introduces two examples, one from physics and the other from biology, that are paradigms of nonequilibrium behavior. Section in covers most important aspects of fluctuation theorems, whereas Section IV presents applications of fluctuation theorems to physics and biology. Section V presents the discipline of path thermodynamics and briefly discusses large deviation functions. Section VI discusses the topic of glassy dynamics from the perspective of nonequilibrium fluctuations in small cooperatively rearranging regions. We conclude with a brief discussion of future perspectives. 

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