Multiphase Thermodynamic System

Modem scaling theory is a quite powerful theoretical tool (appHcable to Hquid crystals, magnets, etc) that has been well estabUshed for several decades and has proven to be particularly useful for multiphase microemulsion systems (46). It describes not just iuterfacial tensions, but virtually any thermodynamic or physical property of a microemulsion system that is reasonably close to a critical poiat. For example, the compositions of a microemulsion and its conjugate phase are described by equations of the foUowiug form ... 

The mathematical basis of classic thermodynamics was developed by J. Willard Gibbs in his essay [1], On the Equilibrium of Heterogeneous Substances, which builds on the earlier work of Kelvin, Clausius, and Helmholtz, among others. In particular, he derived the phase mle, which describes the conditions of equilibrium for multiphase, multicomponent systems, which are so important to the geologist and to the materials scientist. In this chapter, we will present a derivation of the phase rule and apply the result to several examples. 

We have previously emphasized (Section 2.10) the importance of considering only intensive properties Rt (rather than size-dependent extensive properties Xt) as the proper state descriptors of a thermodynamic system. In the present discussion of heterogeneous systems, this issue reappears in terms of the size dependence (if any) of individual phases on the overall state description. As stated in the caveat regarding the definition (7.7c), the formal thermodynamic state of the heterogeneous system is wholly / dependent of the quantity or size of each phase (so long as at least some nonvanishing quantity of each phase is present), so that the formal state descriptors of the multiphase system again consist of intensive properties only. We wish to see why this is so. 

The systems to which thermodynamics have been applied have become more and more complex. The analysis and understanding of these systems requires a knowledge and understanding of the methods of applying thermodynamics to multiphase, multicomponent systems. This book is an attempt to fill the need for a monograph in this area. 

The analysis of a multiphase flow system is complex, in part because of the difficulties in assessing the dynamic responses of each phase and the interactions between the phases. In some special cases, the gas-solid mixture can be treated as a single pseudo-homogeneous phase in which general thermodynamic properties of a gas-solid mixture can be defined. This treatment provides an estimate for the bulk behavior of the gas-solid flow. The following treatment is based on the work of Rudinger (1980). 

The most stable compound of a multiphase binary system is often assumed to be the first to occur and grow at the A-B interface. The change, ArG , of the isobaric-isothermal potential (Gibbs free energy) in the reaction of formation of any compound from the elements under given conditions is usually considered to be a measure of its thermodynamic stability. The more negative the value of AfG°, the more stable the compound is. 

It appears relevant to note that many workers tend to overestimate the significance of thermodynamic predictions concerning the direction of the reaction-diffusion process. In fact, however, those only bear a likelihood character. Even if the free energy of formation of one compound from its constituents is -200 kJ mol-1, while that of the other is -20 kJ mol1, this does not necessarily mean, as often (tacitly or directly) assumed, that the former will occur first and the more so that its growth rate must be ten times greater than that of the latter. As exemplified with the growth rate of a compound layer in various diffusion couples of the same multiphase binary system, the opposite may well take place. 

Useful insights into the thermodynamics of a multiphase reacting system can be obtained by analyzing the situation when chemical and phase equilibrium (C PE) are achieved simultaneously. The Gibbs rule can be used to find the degrees of freedom F ... 

J. Gaydos and A. W. Neumann, "Line Tension in Multiphase Equilibrium Systems," in Applied Surface Thermodynamics (A. W. Neumann and J. K. Spelt, eds.). surfactant Science Series, Vol. 63. Marcel Dekker. New York. 1996. p 6 >-238. 

The method developed in this book is also used to provide input parameters for composite models which can be used to predict the thermoelastic and transport properties of multiphase materials. The prediction of the morphologies and properties of such materials is a very active area of research at the frontiers of materials modeling. The prediction of morphology will be discussed in Chapter 19, with emphasis on the rapidly improving advanced methods to predict thermodynamic equilibrium phase diagrams (such as self-consistent mean field theory) and to predict the dynamic pathway by which the morphology evolves (such as mesoscale simulation methods). Chapter 20 will focus on both analytical (closed-form) equations and numerical simulation methods to predict the thermoelastic properties, mechanical properties under large deformation, and transport properties of multiphase polymeric systems. 

There is also a correlation between the thermodynamic and transport properties (see Chapter 7). Thermodynamic properties of multiphase polymeric systems affect the flow, and vice-versa the flow affects the thermodynamics. As discussed in that chapter, the effects of stress can engender significant shift of the spinodal, AT 16°C. While at low stresses the effects can vary, i.e., the miscibility can either increase or decrease, at higher values an enhancement of miscibility is expected. The flow has also been used to establish whether the molten blends are miscible or not [Schlund and Utracki, 1987 Utracki and Schlund, 1987]. 

This new theory of the non-equilibrium thermodynamics of multiphase polymer systems offers a better explanation of the conductivity breakthrough in polymer blends than the percolation theory, and the mesoscopic metal concept explains conductivity on the molecular level better than the exciton model based on semiconductors. It can also be used to explain other complex phenomena, such as the improvement in the impact strength of polymers due to dispersion of rubber particles, the increase in the viscosity of filled systems, or the formation of gels in colloids or microemulsions. It is thus possible to draw valuable conclusions and make forecasts for the industrial application of such systems. 

An examination of all experimental results reveals that percolation theory displays distinct deficits when it comes to explaining the results observed. This shortcoming led to the development of a new theory that in addition to totally different processes responsible for the occurrence of conductivity, also postulates a fundamentally different view of the systems multiphase polymer systems are in a state of thermodynamic inequilibrium. 

In contrast to equilibrium thermodynamics, the thermodynamics of irreversible processes portray the application of thermodynamic methods as dynamic and therefore time-dependent procedures. The name Prigo-gine must be mentioned in relationship to this—he received for his work in this area the Nobel Prize in the year 1977. A new, very complex thermodynamics originated from his examination method for chemical reactions, and was developed by us, to come to a successful description of heterogenous multiphase polymer systems. This theory interprets crazing fracture energy dissipation and fracture mechanism in a totally new way on the basis of dissipative structures in polymer blends and their dynamics, For a list of abbreviations used in this section sec page 610,... 

The new non-equilibrium thermodynamic theory of heterogeneous polymer systems [37] is aimed at giving a basis for an integrated description for the dynamics of dispersion and blending processes, structure formation, phase transition and critical phenomena. Our new concept is derived from these more general non-equilibrium thermodynamics and has been worked out on the basis of experiments mainly with conductive systems, plus some orienting and critical examples with non-con-ductive systems [72d]. The principal ideas of the new general non-equilibrium thermodynamical theory of multiphase polymer systems can be outlined as follows. 

At the present stage, where we are just entering a non-equilibrium thermodynamical description of multiphase polymer systems, an ab initio theoretic derivation of equations (11.30) and (I l.3l)ff is still lacking. The a.m. thoughts and reformulations may at least lead to some important conclusions ... 

The concepts discussed in Section 14.4 describe the situation that will eventuate when a multiphase packaging system reaches thermodynamic equilibrium. However, the rate of mass transfer of permeant, sorbate, and migrant molecules in the polymer is not addressed by these equilibrium considerations. For example, if we consider a potential migration process, we know that eventually the migrant will be transferred to the food and it will finally reach equilibrium, but based on the equations presented in Section 14.4 we cannot predict how long the process will take. Similarly, these relationships will not allow us to estimate the shelf life of a product in a particular package system. For this, we need to look at diffusion. 

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. 

Reactive distillation occurs in multiphase fluid systems, with an important role of the interfacial transport phenomena. It is an inherently multicomponent process with much more complexity than similar binary processes. Multi-component thermodynamic and diffusional coupling in the phases and at the interface is accompanied by complex hydrodynamics and chemical reactions [4, 42, 43]. As a consequence, an adequate process description has to be based on specially developed mathematical models. However, sophisticated RD models are hardly applicable for plant design, model-based control and online process optimization. For such cases, a reasonable model reduction should be applied [44],... 

Leszek A. Utracki was bom and educated in Poland. After completing his postdoctorate at USC in Los Angeles with Robert Simha, he settled in Canada. He was a passionate researcher in the fields of thermodynamics, rheology, and processing of multicomponent/multiphase polymeric systems. Sadly, he passed away a few months after embarking on the work for the second edition of the Polymer Blends Handbook, during which he had made important contributions to the work. This edition of the handbook is dedicated to his memory. 

In this chapter we look at the basic relationships between COj in natural waters that are important in the environment and also investigate CO2 utihzing processes in the environment. As an example of natural systems we consider a boreal low mineral lake and discuss the effects of acidification of aquatic ecosystems in context of multiphase thermodynamics. 

This very brief review of phase equilibrium gives the reader the principles for dealing with two-phase systems. More background on the thermodynamic equilibrium of multiphase systems will sometimes be needed it is not covered in this book. The reader is advised to consult a multiphase thermodynamics book for that purpose. 

The equilibrium state of a system at constant temperature and pressure is characterized by a minimum in the Gibbs free energy of the system according to the second law of thermodynamics. For a multicomponent, multiphase (bulk) system, the minimum free energy corresponds to uniformity of the chemical potential (yu,) of each component throughout the system as demonstrated below. For a binary system, the molar free energy (G) and chemical potentials are related by... 

Statistical thermodynamics already provide an excellent framework to describe and model equilibrium properties of molecular systems. Molecular interactions, summarized for instance in terms of a potential of mean force, determine correlation functions and all thermodynamic properties. The (pair) correlation function represents the material structure which can be determined by scattering experiments via the scattering function. AU macroscopic properties of pure and mixed fluid systems can be derived by weU-estabhshed multiphase thermodynamics. In contrast, a similar framework for particulate building blocks only partly exists and needs to be developed much further. Besides equibbrium properties, nonequilibrium effects are particularly important in most particulate systems and need to be included in a comprehensive and complete picture. We will come back to these aspects in Section 4. 

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