Equilibrium Macroscopic Description

The free energy of the drop on the substrate (in the small slope approximation), relative to the free energy of a vapor-covered substrate is 

For the case of small contact angles considered here, we can derive Young s law (Eq. (4.2)), without expiicitly solving for the drop profiie. To do this, one considers the constrained free energy, g of Eq. (4.4) as a function of two independent variables, h(x) and A,. Now, the minimization over h(x) requires fixing h = 0 a x = k so that these two variables are not completely independent. However, if we rescale the x coordinate and define u =xlk, the boundary conditions read  

This symmetry implies dh(u)/du = 0 at m = 0. With this scaiing, we can take the derivatives with respect to h and k independently. We define the spreading power, S by 

This is equivalent to Young s equation in the limit of small contact angles. 

We can also derive Young s equation by explicitly minimizing the grand potential of Eq. (4.4) with respect to the droplet shape. The resulting Euler-Lagrange equation is 

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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. 

Solutions are involved In every practical chemistry laboratory, in chemical analysis, biochemistry, and clinical chemistry, and in chemical synthesis. Filty years ago solution chemistry occupied a major fraction of physical chemistry textbooks and dealt mainly with classical thermodynamics, phase equilibria, and non-equilibrium phenomena, especially those related to electrochemistry. Much has happened in the intervening period, with tremendous advances in theor and the development of important new experimental techniques. Ucfuidi, Solutions, and Inttrfaces brings the reader through these developments from the classical macroscopic descriptions to the modern microscopic details. 

The macroscopic description of the adsorption on electrodes is characterised by the development of models based on classical thermodynamics and the electrostatic theory. Within the frames of these theories we can distinguish two approaches. The first approach, originated from Frumkin s work on the parallel condensers (PC) model,attempts to determine the dependence of upon the applied potential E based on the Gibbs adsorption equation. From the relationship = g( ), the surface tension y and the differential capacity C can be obtained as a function of E by simple mathematical transformations and they can be further compared with experimental data. The second approach denoted as STE (simple thermodynamic-electrostatic approach) has been developed in our laboratory, and it is based on the determination of analytical expressions for the chemical potentials of the constituents of the adsorbed layer. If these expressions are known, the equilibrium properties of the adsorbed layer are derived from the equilibrium equations among the chemical potentials. Note that the relationship = g( ), between and , is also needed for this approach to express the equilibrium properties in terms of either or E. Flere, this relationship is determined by means of the Gauss theorem of electrostatics. 

Physical kinetics applies to physical systems whose elements are far from equilibrium. It is the macroscopic description of the processes which occur in non-equilibrium systems. In physical kinetics, the changes in energy, momentum, and mass transfer in the various physical systems are investigated, as well as the influence of external fields on these systems. 

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. 

In the last two chapters of the book on Thermal Analysis of Polymeric Materials the link between microscopic and macroscopic descriptions of macromolecules will be discussed with a number of examples based on the thermal analysis techniques which are described in the prior chapters. Chapter 6 deals with single-component systems, Chap. 7 with multiple-component systems. It is shown in Sect. 6.2, as suggested throughout the book, that practically aU partially crystalline polymers represent nonequilibrium systems, and that thermodynamics can establish the equilibrium limits for the description. It was found, however, more recently, that equilibrium thermodynamics may be applied to local areas, often small enough to be called nanophases [1]. These local subsystems are arrested and cannot establish global equilibrium. 

Linear stability analysis describes the behavior of a system at near equilibrium. Hamiltonian dynamics show that classical mechanics is invariant to (—t) and (t). In a macroscopic description of dissipative systems, we use collective variables of temperature, pressure, concentration, and convection velocity to define an instantaneous state. The evolution equations of the collective variables are not invariant under time reversal for the reaction ... 

Second, the wide research devoted to fast reactions showed that sometimes the macroscopic description was quite inadequate and the microscopic level had to be resorted to. For instance, only the microscopic level permits the interpretation of the distinctly non-equilibrium processes in the interstellar space, in the Earth atmosphere, in combustion, chemical lasers, and even in engineering. 

Within the frameworks of this formalism to account for consistently nonlinear phenomenon complex nature is a success, such as memory effects and spatial correlations. In addition the earlier known solutions are not only reproduced, but their nontrivial generalization is given. Another important feature is connected with fractal structures self-similarity using. Unlike the traditional methods of system description on the basis of averaging different procedures, when microscopic level erasing occurs, in fractal conception medium self-affine structure and thus within the frameworks of this conception system micro and macroscopic description levels are united. Exactly such method is important for complex multicomponent systems, discovered far from thermodynamic equilibrium state [35], which are polymers [12], The authors of Refs. [31, 32] are attempted two indicated trends combination. 

With this link between the microscopic and macroscopic description of matter securely established, the next chapter of the book will concentrate on the description of the various theories needed for the understanding of thermal analysis, namely equilibrium and irreversible thermodynamics and kinetics. The Introduction will then be completed with a summary of the specific functions needed for the six basic branches of thermal analysis thermometry, differential thermal analysis, calorimetry, thermomechanical analysis, dilatometiy, and thermogravimetiy. 

The thermodynamic chain is more complex in the case of a ternary compound. To illustrate this, we consider the Ag2S-MS/M + interface (for example, Cty+, Cd +, or Pb " ISEs will operate according to this mechanism). A schematic view is given in Figure 10.1b. In this case, the SIC is a conductor for Ag+ ions. Equilibrium at the interface is ensured by the following exchange reactions, in a macroscopic description. 

Another more rigorous approach takes into account the equilibrium at the solid interface. In macroscopic description, we have... 

The macroscopic description of nonequilibrium states of fluid systems requires independent variables to specify the extent to which the system deviates from equilibrium and dependent variables to express the rates of processes. 

The time correlation function C(t) from (19) is a description of the transition statistics in the equilibrium system described in terms of the microscopic degrees of freedom. To make contact with a macroscopic description, appropriate for an experiment in which many molecules of type A and B are present in the sample, it is useful to consider the time evolution of the concentrations ca(i) = NA t)/V and cb (t) = IVb t)/V defined as the number of molecules per volume V of type A and B, respeclively. We imagine that the concentrations ca(1 ) and CB(t) can be determined experimentally in a time-resolved way. Since molecules can only convert into each other and are not created or destroyed, the total number of molecules N = +... 

At thermodynamic equilibrium fluctuations constitute a negligible correction to the macroscopic description of matter except near instability points such as phase transitions or critical points. Similarly, in nonequilibrium situations as well one expects that fluctuations become important only near points of nonequilibrium instability (see Ref. 17 for a detailed discussion). For nonlinear systems in which such instabilities and transitions can occur, the role of fluctuations is even more dramatic than at equilibrium, due to the existence in general of many distinct "phases" compatible with the external conditions imposed on the system. This fact has been known for several years now, since the first (numerical) demonstration on simple chemical models of multiple solutions to the governing macroscopic differential equations (44j 4 The variety of possible solutions... 

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... 

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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