Equilibrium Microscopic Description

We first consider the case where the substrate is uniformly covered by a layer of fluid , but that this layer is not necessarily at the equilibrium fluid density. In general, the layer will not be homogeneous in density as one approaches the vapor — i.e., if the normal to the substrate is in the direction, the layer density will vary with . Our goal is first to find the relationship between the thickness of the layer and the interactions of the fluid molecules with the substrate. We consider the free energy per unit area, /o, of the fluid on the surface to be given by an virial expansion of the form 

The total interfacial free energy per unit area, consists of the sum of /o and the free energy per unit area that comes from the liquid-vapor interface. In equilibrium, one minimizes the total free energy subject to the conservation constraint — i.e., one works at fixed chemical potential. As explained in the discussion of the gas-fiquid interface in Chapter 2, the appropriate bulk free energy to minimize to find the interfacial profile is the grand potential per unit area, gs, which is written  

We consider the total grand potential per unit area, gt, given by 

The Euler-Lagrange equation arising from the minimization of fg with respect to the density profile, n(z), is 

The calculation of the profile is similar to that of the bulk gas-liquid interface, discussed in Chapter 2. We consider the case where ris n. From Eq. (4.47) with the form of W given by Eq. (4.45) one has 

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Linear response theory is an example of a microscopic approach to the foundations of non-equilibrium thennodynamics. It requires knowledge of tire Hamiltonian for the underlying microscopic description. In principle, it produces explicit fomuilae for the relaxation parameters that make up the Onsager coefficients. In reality, these expressions are extremely difficult to evaluate and approximation methods are necessary. Nevertheless, they provide a deeper insight into the physics. 

At r > Tr, the relaxation of a non-equilibrium surface morphology by surface diffusion can be described by Eq. 1 the thermodynamic driving force for smoothing smoothing is the surface stiffness E and the kinetics of the smoothing is determined by the concentration and mobility of the surface point defects that provide the mass transport, e.g. adatoms. At r < Tr, on the other hand, me must consider a more microscopic description of the dynamics that is based on the thermodynamics of the interactions between steps, and the kinetics of step motion [17]. 

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. 

This model of the liquid will be characterized by some macroscopic quantities, to be selected among those considered by classical equilibrium thermodynamics to define a system, such as the temperature T and the density p. This macroscopic characterization should be accompanied by a microscopic description of the collisions. As we are interested in chemical reactions, one is sorely tempted to discard the enormous number of non-reactive collisions. This temptation is strenghtened by the fact that reactive collisions often regard molecules constituting a minor component of the solution, at low-molar ratio, i.e. the solute. The perspective of such a drastic reduction of the complexity of the model is tempered by another naive consideration, namely that reactive collisions may interest several molecular partners, so that for a nominal two body reaction A + B —> products, it may be possible that other molecules, in particular solvent molecules, could play an active role in the reaction. 

Let us first ask the question how can a species exist in clusters of molecules at equilibrium with its own monomers We necessarily must consider the fluctuation of any system about its equilibrium. Under normally liquid conditions, the density fluctuations are small in extent (localized) and persist for short times. A microscopic description of the phenomenon may be that of temporary formation of weak bonds, which are then quickly dissipated. If, however, these fluctuations lead to clusters that have a nontransient existence, then... 

Considerable effort has been made during the last two decades to develop a "microscopic" description of gas diffusion in polymers, which is more detailed than the simplified continuum viewpoint of Fick s laws. It has been known for a long time that the mechanism of diffusion is very different in "rubbery" and "glassy" polymers, i.e., at temperatures above and below the glass-transition temperature, Tg, of the polymers, respectively. This is due to the fact that glassy polymers are not in a true state of thermodynamic equilibrium, cf. refs. (1,3,5,7-11). Some of the models and theories that have been proposed to describe gas diffusion in rubbery and glassy polymers are discussed below. The models selected for presentation in this review reflect only the authors present interests. 

The basic, macroscopic theories of matter are equilibrium thermodynamics, irreversible thermodynamics, and kinetics. Of these, kinetics provides an easy link to the microscopic description via its molecular models. The thermodynamic theories are also connected to a microscopic interpretation through statistical thermodynamics or direct molecular dynamics simulation. Statistical thermodynamics is also outlined in this section when discussing heat capacities, and molecular dynamics simulations are introduced in Sect 1.3.8 and applied to thermal analysis in Sect. 2.1.6. The basics, discussed in this chapter are designed to form the foundation for the later chapters. After the introductory Sect. 2.1, equilibrium thermodynamics is discussed in Sect. 2.2, followed in Sect. 2.3 by a detailed treatment of the most fundamental thermodynamic function, the heat capacity. Section 2.4 contains an introduction into irreversible thermodynamics, and Sect. 2.5 closes this chapter with an initial description of the different phases. The kinetics is closely link to the synthesis of macromolecules, crystal nucleation and growth, as well as melting. These topics are described in the separate Chap. 3. 

The extensive properties of the overall system that is not in equilibrium, such as volume or energy, are simply the sums of the (almost) equilibrium properties of the subsystems. This simple division of a sample into its subsystems is the type of treatment needed for the description of irreversible processes, as are discussed in Sect. 2.4. Furthermore, there is a natural limit to the subdivision of a system. It is reached when the subsystems are so small that the inhomogeneity caused by the molecular structure becomes of concern. Naturally, for such small subsystems any macroscopic description breaks down, and one must turn to a microscopic description as is used, for example, in the molecular dynamics simulations. For macromolecules, particularly of the flexible class, one frequently finds that a single macromolecule may be part of more than one subsystem. Partially crystalhzed, linear macromolecules often traverse several crystals and surrounding liquid regions, causing difficulties in the description of the macromolecular properties, as is discussed in Sect. 2.5 when nanophases are described. The phases become interdependent in this case, and care must be taken so that a thermodynamic description based on separate subsystems is still valid. 

Summary of Chapter 4. At the end of this discussion of thermal analysis tools it may be worthwhile to attempt a brief summary. The basic theory of thermal analysis is well represented by macroscopic equilibrium and nonequiUbrium thermodynamics, and the connection to the microscopic description is given by statistical thermodynamics and kinetics. All of these theories are highly developed, but they have not been applied to their fullest in the description of materials. The reason for this failure to... 

The next important advance in the theory, and the one that provided the foundation for all later work in this field, was made by Boltzmann, who in 1872 derived an equation for the time rate of change of the distribution function for a dilute gas that is not in equilibrium—the Boltzmann transport equation. (See Boltzmann and also Klein. " ) Boltzmann s equation gives a microscopic description of nonequilibrium processes in the dilute gas, and of the approach of the gas to an equilibrium state. Using the Boltzmann equation. Chapman and Enskog derived the Navier-Stokes equations and obtained expressions for the transport coefficients for a dilute gas of particles that interact with pairwise, short-range forces. Even now, more than 100 years after the derivation of the Boltzmann equation, the kinetic theory of dilute gases is largely a study of special solutions of that equation for various initial and boundary conditions and various compositions of the gas.t... 

The macroscopic theories of matter consist of equilibrium thermodynamics, irreversible thermodynamics, and kinetics. Of these, kinetics provides an easy link to the microscopic description via its molecular models. The thermodynamic theories are also connected to a microscopic interpretation through statistical thermodynamics. 

Temperature, pressure, and volume are the major variables of state for the description of the transitions. Next is entropy, discussed before in Figs. 2.3 and 2.4. The magnitude of entropy can also be estimated from the microscopic description of the states. One expects the entropy of the gas to be much larger than the entropies of the condensed phases because of the high degree of disorder in the dilute, gaseous state. The crystal, on the other hand, because of its order, should have the smallest entropy in fact, the third law of thermodynamics (Sect. 2.1.1) sets the entropy of an ideal, equilibrium crystal equal to zero at the absolute zero of temperature. 

In this chapter, we have examined a number of simple analjdical models which are extremely useful for the study of equilibrium states. However, these models involve parameters with unknown physical meaning. In addition, they are often not sufficiently accurate. For that reason, it is worth considering the modeling of solutions by a microscopic description. 

The potential V R) introduces electron-electron (e-e) interactions, and Taylor expansion of t(R) or V(R) about equilibrium generates electron-phonon (e-ph) coupling. Conjugated polymers abundantly illustrate [12,13] e-ph and e-e contributions whose joint analysis is difficult mathematically. But a joint analysis will undoubtedly emerge, and this review is a step in that direction. We seek sufficiently powerful e-ph descriptions for detailed fits of vibrational spectra and sufficiently accurate correlated states to understand excitations, including a host of recent nonlinear optical (NLO) spectra. Both e-ph and e-e interactions appear naturally in models, and both lead to characteristic susceptibilities. A related issue for vibrational spectra is the precise identification of IT-electronic contributions. We will emphasize the advantages of models for microscopic descriptions of conjugated polymers. To develop these themes. 

In most circumstances the spatiotemporal dynamics of reacting systems constrained to lie far from equilibrium can be described adequately by reaction-diffusion equations. These equations are valid provided the phenomena of interest occur on distance and time scales that are sufficiently long compared to molecular scales. Naturally, the complete microscopic description of the reacting medium, whether near to or far from equilibrium, must be based on the full molecular dynamics of the system, as embodied in Newton s or Schrodinger s equations of motion. 

The first group starts from a microscopic description of the liquid by the molecular variables and usually estimates the equilibrium average of only a few variables on the basis of a proper statistical model. This procedure allows the evaluation of some dynamic parameters of the liquid susceptibiUty, as introduced previously in the microscopic model. Recently this approach deeply benefits from computer simulations that enable the extraction of a valid molecular picture of the liquid dynanaics from the OKE signal [58,59]. 

A statistical ensemble can be viewed as a description of how an experiment is repeated. In order to describe a macroscopic system in equilibrium, its thennodynamic state needs to be specified first. From this, one can infer the macroscopic constraints on the system, i.e. which macroscopic (thennodynamic) quantities are held fixed. One can also deduce, from this, what are the corresponding microscopic variables which will be constants of motion. A macroscopic system held in a specific thennodynamic equilibrium state is typically consistent with a very large number (classically infinite) of microstates. Each of the repeated experimental measurements on such a system, under ideal... 

Progress in the theoretical description of reaction rates in solution of course correlates strongly with that in other theoretical disciplines, in particular those which have profited most from the enonnous advances in computing power such as quantum chemistry and equilibrium as well as non-equilibrium statistical mechanics of liquid solutions where Monte Carlo and molecular dynamics simulations in many cases have taken on the traditional role of experunents, as they allow the detailed investigation of the influence of intra- and intemiolecular potential parameters on the microscopic dynamics not accessible to measurements in the laboratory. No attempt, however, will be made here to address these areas in more than a cursory way, and the interested reader is referred to the corresponding chapters of the encyclopedia. 

The reason for this enliancement is intuitively obvious once the two reactants have met, they temporarily are trapped in a connnon solvent shell and fomi a short-lived so-called encounter complex. During the lifetime of the encounter complex they can undergo multiple collisions, which give them a much bigger chance to react before they separate again, than in the gas phase. So this effect is due to the microscopic solvent structure in the vicinity of the reactant pair. Its description in the framework of equilibrium statistical mechanics requires the specification of an appropriate interaction potential. 

In a microscopic equilibrium description the pressure-dependent local solvent shell structure enters tlirough... 

A key problem in the equilibrium statistical-physical description of condensed matter concerns the computation of macroscopic properties O acro like, for example, internal energy, pressure, or magnetization in terms of an ensemble average (O) of a suitably defined microscopic representation 0 r ) (see Sec. IVA 1 and VAl for relevant examples). To perform the ensemble average one has to realize that configurations = i, 5... 

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