Thermodynamics of Uniform Systems

In this chapter we discuss uniform systems, the properties of which change only in time. Similarly as in [1-8] our main aim here is to demonstrate the method of rational thermodynamics and application of its principles in simple material models. In other words, the main aim is pedagogical—to begin with simple issues, demonstrations, and examples. Nevertheless, even this chapter contains practical results which can be applied on many simple real systems. Among others the principal results of classical equilibrium thermodynamics will be obtained and this will be shown also for reacting mixtures and heterogeneous (multiphase) uniform systems. 

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These ideas of asymptotic functional self-similarity and of a critical scale find their expression in the postulatory formulations of the thermodynamics of uniform systems due to Tisza [11] and Callen [12]. Thus, the internal energy of a single-component system is written in the form... 

The importance of FIPI is twofold. It can be used to promote phase inversion without changing the thermodynamics of the system to obtain a higher entropy state, or it is possible to delay phase inversion while reducing the system entropy. The characteristics of the microstructure formed (such as emulsion droplet size) are dependent on the type of microstructure and deformation (shear, extension, or combined), as well as the deformation rate. To maximize the fluid micro-structure/flow field interactions, the flow fleld must be uniform, which requires the application of the flow field over a small processing volume, which can be achieved by using MFCS mixers or CDDMs. 

An essential basis for the study of boiling heat transfer is the thermodynamics of multiphase systems. Here, it is normal practice to consider systems at thermodynamic equilibrium, in which the temperature of the system is uniform. Of course, as we will see, departures from such thermodynamic equilibrium are important in many instances. In what follows, the thermodynamic equilibrium of a single-component material is first considered. In many applications of boiling (particularly in the process and petroleum industries), multicomponent mixtures (for example, mixtures of hydrocarbons or refrigerants) are important, and the subject of multicomponent equilibrium is dealt with in the final part of this section. 

For a PVnr system of uniform T and P containing N species and 7T phases at thermodynamic equiUbrium, the intensive state of the system is fully deterrnined by the values of T, P, and the (N — 1) independent mole fractions for each of the equiUbrium phases. The total number of these variables is then 2 + 7t N — 1). The independent equations defining or constraining the equiUbrium state are of three types equations 218 or 219 of phase-equiUbrium, N 7t — 1) in number equation 245 of chemical reaction equiUbrium, r in number and equations of special constraint, s in number. The total number of these equations is A(7t — 1) + r -H 5. The number of equations of reaction equiUbrium r is the number of independent chemical reactions, and may be deterrnined by a systematic procedure (6). Special constraints arise when conditions are imposed, such as forming the system from particular species, which allow one or more additional equations to be written connecting the phase-rule variables (6). 

Although corrosion is due to the thermodynamic instability of a metal in a specific environment, and although in many metal/environment systems attack will tend to be uniform, there are a variety of factors associated with the metal, the environment and the geometry of the system that may result in the attack being localised. 

However, further analysis of the behavior of the system in LC cells cast doubt on this interpretation. First, while intuitively attractive, the idea that relaxation of the polarization by formation of a helielectric structure of the type shown in Figure 8.20 would lower the free energy of the system is not correct. Also, in a thermodynamic helical LC phase the pitch is extremely uniform. The stripes in a cholesteric fingerprint texture are, for example, uniform in spacing, while the stripes in the B2 texture seem quite nonuniform in comparison. Finally, the helical SmAPF hypothesis predicts that the helical stripe texture should have a smaller birefringence than the uniform texture. Examination of the optics of the system show that in fact the stripe texture has the higher birefringence. 

The procedures are similar to those used for the ground state energy. A general static external potential is treated in perturbation theory and the expansions are rearranged and resummed. The unperturbed system is of uniform density and fully extended and the thermodynamic limit is taken at the outset. Particular care is required to treat the chemical potential correctly. The result for D = 3 and arbitrary two-body interaction is [28,29]... 

Polymer-polymer systems exhibit phase behavior similar to other mixtures, such that an initially uniform system separates into two or more phases as a result of small change in thermodynamic variable. Two mechanisms can be envisioned to explain this phenomenon nucleation and growth (NG), and spinodal decomposition (SD). 

Thus the response of a spatially uniform system in thermodynamic equilibrium is always characterized by translationally invariant and temporaly stationary after-effect functions. This article is restricted to a discussion of systems which prior to an application of an external perturbation are uniform and in equilibrium. The condition expressed by Eq. (7) must be satisfied. Caution must be exercised in applying linear response theory to problems in double resonance spectroscopy where non-equilibrium initial states are prepared. Having dispensed with this caveat, we adopt Eq. (7) in the remainder of this review article. 

The treatment of systems in a magnetic field follows very closely the development of the thermodynamics of systems in an electrostatic field. We again consider two systems. One is a long solenoid with uniform windings, in empty space. The second is the same solenoid in which the total volume within the solenoid is filled with isotropic matter. Edge effects are neglected. Ferromagnetic effects and hysteresis are omitted. 

Many polymer blends or block polymer melts separate microscopically into complex meso-scale structures. It is a challenge to predict the multiscale structure of polymer systems including phase diagram, morphology evolution of micro-phase separation, density and composition profiles, and molecular conformations in the interfacial region between different phases. The formation mechanism of micro-phase structures for polymer blends or block copolymers essentially roots in a delicate balance between entropic and enthalpic contributions to the Helmholtz energy. Therefore, it is the key to establish a molecular thermodynamic model of the Helmholtz energy considered for those complex meso-scale structures. In this paper, we introduced a theoretical method based on a lattice model developed in this laboratory to study the multi-scale structure of polymer systems. First, a molecular thermodynamic model for uniform polymer system is presented. This model can... 

To establish the molecular thermodynamic model for uniform systems based on concepts from statistical mechanics, an effective method by combining statistical mechanics and molecular simulation has been recommended (Hu and Liu, 2006). Here, the role of molecular simulation is not limited to be a standard to test the reliability of models. More directly, a few simulation results are used to determine the analytical form and the corresponding coefficients of the models. It retains the rigor of statistical mechanics, while mathematical difficulties are avoided by using simulation results. The method is characterized by two steps (1) based on a statistical-mechanical derivation, an analytical expression is obtained first. The expression may contain unknown functions or coefficients because of mathematical difficulty or sometimes because of the introduced simplifications. (2) The form of the unknown functions or unknown coefficients is then determined by simulation results. For the adsorption of polymers at interfaces, simulation was used to test the validity of the weighting function of the WDA in DFT. For the meso-structure of a diblock copolymer melt confined in curved surfaces, we found from MC simulation that some more complex structures exist. From the information provided by simulation, these complex structures were approximated as a combination of simple structures. Then, the Helmholtz energy of these complex structures can be calculated by summing those of the different simple structures. 

This chapter deals with the thermodynamics of one-phase systems, and it is understood that the phase is homogeneous and at uniform temperature. The basic structure of thermodynamics provides the tools for the treatment of more complicated systems in later chapters. This book starts with the fundamentals of thermodynamics, but the reader really needs some prior experience with thermodynamics at the level of undergraduate thermodynamics (Silbey and Alberty, 2001). Legendre transforms play an important role in this chapter, and the best single reference on Legendre transforms is Callen (1960, 1985). Other useful references for basic thermodynamics are Tisza (1966), Beattie and Oppenheim (1979), Bailyn (1994), and Greiner, Neise, and Stocker (1995). 

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