Kinetic mechanisms microscopic level

The plan of this chapter is the following. Section II gives a summary of the phenomenology of irreversible processes and set up the stage for the results of nonequilibrium statistical mechanics to follow. In Section III, it is explained that time asymmetry is compatible with microreversibility. In Section IV, the concept of Pollicott-Ruelle resonance is presented and shown to break the time-reversal symmetry in the statistical description of the time evolution of nonequilibrium relaxation toward the state of thermodynamic equilibrium. This concept is applied in Section V to the construction of the hydrodynamic modes of diffusion at the microscopic level of description in the phase space of Newton s equations. This framework allows us to derive ab initio entropy production as shown in Section VI. In Section VII, the concept of Pollicott-Ruelle resonance is also used to obtain the different transport coefficients, as well as the rates of various kinetic processes in the framework of the escape-rate theory. The time asymmetry in the dynamical randomness of nonequilibrium systems and the fluctuation theorem for the currents are presented in Section VIII. Conclusions and perspectives in biology are discussed in Section IX. 

The mechanism for cross-linking of thermosetting resins is very complex because of the relative interaction between the chemical kinetics and the changing of the physical properties [49], and it is still not perfectly understood. The literature is ubiquitous with respect to studies of cure kinetic models for these resins. Two distinct approaches are used phenomenological (macroscopic level) [2,5,50-72] and mechanistic (microscopic level) [3,73-85]. The former is related to an overall reaction (only one reaction representing the whole process), the latter to a kinetic mechanism for each elementary reaction occurring during the process. 

The only way to validate kinetic models is to measure experimentally the degree of cure as a function of time and temperature. It can be done on both macroscopic and microscopic levels by monitoring chemical, physical (refractive index [135], density [136], and viscosity [137]), electrical (electrical resistivity [138,139]), mechanical, and thermal property changes with time [140,141]. The most-used techniques to monitor cure are presented in the next two subsections. 

Before we go on and discuss these objectives in more detail, it might be appropriate to consider the relation between molecular reaction dynamics and the science of physical chemistry. Normally physical chemistry is divided into four major branches, as sketched in the figure below (each of these areas are based on fundamental axioms). At the macroscopic level, we have the old disciplines thermodynamics and kinetics . At the microscopic level we have quantum mechanics , and the connection between the two levels is provided by statistical mechanics . Molecular reaction dynamics encompasses (as sketched by the oval) the central branches of physical chemistry with the exception of thermodynamics. 

Before starting to discuss (116), we make an observation. The fast time evolution (116) is also observed in driven systems that cannot be described on the level Ath- For example, let us consider the Rayleigh-Benard system (i.e., a horizontal layer of a fluid heated from below). It is well established experimentally that this externally driven system does not reach thermodynamic equilibrium states but its behavior is well described on the level of fluid mechanics (by Boussinesq equations). This means that if we describe it on a more microscopic level, say the level of kinetic theory, then we shall observe the approach to the level of fluid mechanics. Consequently, the comments that we shall make below about (116) apply also to driven systems and to other types of systems that are prevented from reaching thermodynamical equilibrium states (as, e.g., glasses where internal constraints prevent the approach to Ath)-... 

In a non-equilibrium gas system there are gradients in one or more of the macroscopic properties. In a mono-atomic gas the gradients of density, fluid velocity, and temperature induce molecular transport of mass, momentum, and kinetic energy through the gas. The mathematical theory of transport processes enables the quantification of these macroscopic fluxes in terms of the distribution function on the microscopic level. It appears that the mechanism of transport of each of these molecular properties is derived by the same mathematical procedure, hence they are collectively represented by the generalized property (/ ... 

We have found a free energy for the oxidation at 1000 K of the order of -75 KJmol and an activation free energy of about 30 KJmol . A plausible mechanism for this chemical reaction was given, that may explain at microscopic level the phenomenological first order kinetics with respect to NO3 found experimentally. 

Bruemmer et al. (55) studied Ni, Zn, and Cd sorption on goethite, a porous iron oxide known to have defects within the structure in which metals can be incorporated to satisfy charge imbalances. At pH 6, as reaction time increased from 2 hours to 42 days (at 293K), sorbed Ni increased from 12 to 70% of Ni removed from solution, and total increases in Zn and Cd sorption over this period increased 33 and 21%, respectively. The kinetics of Cd, Zn, and Ni were described well with a solution to Pick s second law (a linear relation with the square root of time). Bruemmer et al. (55) proposed that the uptake of the metal follows three-steps (i) adsorption of metals on external surfaces (ii) solid-state diffusion of metals from external to internal sites and (iii) metal binding and fixation at positions inside the goethite particle. They suggest that the second step is the rate-limiting step. However, they did not conduct microscopic level experiments to confirm the proposed mechanism. In view of more recent studies, it is likely that the formation of metal-nucleation products could have caused the slow metal sorption reactions observed by Bruemmer et al. (55). 

An understanding of both the microscopic structure and dynamics of matter is essentia] for a full description of its macroscopic properties. In this regard, recent studies of the internal dynamics of paraffin- and polyethylene-like crystals by simulation with supercomputers have established a link between microscopic motion and macroscopic effect [1-9]. It was shown that conformational, rotational, and diffusional motion can start considerably below the melting or disordering transition temperatures. These types of motion underlie much of the observed macroscopic properties of polymers, and thus it is essential to develop an understanding of how such motion occurs on the microscopic level (mechanisms) and the rates of changes it introduces (kinetics). 

Another, less widely appreciated idealization in chemical kinetics is that phenomena take place instantaneously—that a change in [A] at time t generates a change in [fi] time t and not at some later time t + z. On a microscopic level, it is clear that this state of affairs cannot hold. At the very least, a molecular event taking place at point x and time t can affect a molecule at point x only after a time of the order of x — x f jlD, where D is the relevant diffusion constant. The consequences of this observation at the macroscopic level are not obvious, but, as we shall see in the examples below, it may sometimes be useful to introduce delays explicitly in modeling complex reaction networks, particularly if the mechanism is not known in detail. 

A major success of statistical mechanics is the ability to predict the thermodynamic properties of gases and simple solids from quantum mechanical energy levels. Monatomic gases have translational freedom, which we have treated by using the particle-in-a-box model. Diatomic gases also have vibrational freedom, which we have treated by using the harmonic oscillator model, and rotational freedom, for which we used the rigid-rotor model. The atoms in simple solids can be treated by the Einstein model. More complex systems can require more sophisticated treatments of coupled vibrations or internal rotations or electronic excitations. But these simple models provide a microscopic interpretation of temperature and heat capacity in Chapter 12, and they predict chemical reaction equilibria in Chapter 13, and kinetics in Chapter 19. 

Mechanisms. Mechanism is a technical term, referring to a detailed, microscopic description of a chemical transformation. Although it falls far short of a complete dynamical description of a reaction at the atomic level, a mechanism has been the most information available. In particular, a mechanism for a reaction is sufficient to predict the macroscopic rate law of the reaction. This deductive process is vaUd only in one direction, ie, an unlimited number of mechanisms are consistent with any measured rate law. A successful kinetic study, therefore, postulates a mechanism, derives the rate law, and demonstrates that the rate law is sufficient to explain experimental data over some range of conditions. New data may be discovered later that prove inconsistent with the assumed rate law and require that a new mechanism be postulated. Mechanisms state, in particular, what molecules actually react in an elementary step and what products these produce. An overall chemical equation may involve a variety of intermediates, and the mechanism specifies those intermediates. For the overall equation... 

Ray Kapral came to Toronto from the United States in 1969. His research interests center on theories of rate processes both in systems close to equilibrium, where the goal is the development of a microscopic theory of condensed phase reaction rates,89 and in systems far from chemical equilibrium, where descriptions of the complex spatial and temporal reactive dynamics that these systems exhibit have been developed.90 He and his collaborators have carried out research on the dynamics of phase transitions and critical phenomena, the dynamics of colloidal suspensions, the kinetic theory of chemical reactions in liquids, nonequilibrium statistical mechanics of liquids and mode coupling theory, mechanisms for the onset of chaos in nonlinear dynamical systems, the stochastic theory of chemical rate processes, studies of pattern formation in chemically reacting systems, and the development of molecular dynamics simulation methods for activated chemical rate processes. His recent research activities center on the theory of quantum and classical rate processes in the condensed phase91 and in clusters, and studies of chemical waves and patterns in reacting systems at both the macroscopic and mesoscopic levels. 

Derivations of equation (4) involve a microscopic viewpoint. The reasoning, in its simplest form, is that the reaction rate is proportional to the collision rate between appropriate molecules, and the collision rate is proportional to the product of the concentrations. Implicit in this picture is the idea that equation (4) will be valid only if equation (1) represents a process that actually occurs at the molecular level. Equation (1) must be an elementary reaction step, with v[ molecules of each molecular species i interacting in the microscopic process equation (4) will not be meaningful if equation (1) is the overall methane-oxidation reaction CH -1- 2O2 CO2 -1- 2H2O, for example. Thus, there are two basic problems in chemical kinetics the first is to determine the reaction mechanism, that is, to find the elementary steps by which the given reaction proceeds, and the second is to determine the specific rate constant k for each of these steps. These two problems are discussed in Sections B,2 and B.3, respectively. 

Our approach to the study of the departure from equilibrium in chemical reactions and of the "microscopic theory of chemical kinetics is a discrete quantum-mechanical analog of the Kramers-Brownian-motion model. It is most specifically applicable to a study of the energy-level distribution function and of the rate of activation in unimolecular (dissociation Reactions. Our model is an extension of one which we used in a discussion of the relaxation of vibrational nonequilibrium distributions.14 18 20... 

The possibility of measuring the small fluctuations in / a k occurring at nerve membrane level led to the realisation that such phenomena could provide insight into the microscopic mechanisms of ionic permeability changes. This technique had been successfully employed to examine membrane potential fluctuations and membrane current fluctuations due to the chemically mediated open-closed kinetics of ionic channels [63-68]. 

---