Thermodynamics, equilibrium phase diagrams

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. 

FIGURE 9.26 The thermodynamic equilibrium phase diagram for a binary solid-liquid system. The eutectic temperature and species A mass fraction and a dendritic temperature and liquidus and solidus species A mass fractions are also shown. 

EQUILIBRIUM PHASE DIAGRAMS FROM AB INITIO THERMODYNAMICS... 

The identification of the superconducting phase YBagCug-O7 g provides an example in which knowledge of thermodynamics, i.e. the Gibbs phase rule and the theory of equilibrium phase diagrams coupled with X-ray diffraction techniques led to success. Further, the use of databases that can now be easily accessed and searched on-line provided leads to a preliminary structure determination. The procedures outlined here are among the basic approaches used in solid state chemistry research, but by no means are they the only ones. Clearly the results from other analytical techniques such as electron microscopy and diffraction, thermal... 

The science dealing with phase transitions is thermodynamics. Using thermodynamics, we may discuss which phase will eventually be formed when a material (of composition c and phase p) is maintained under the same conditions for an infinite time, and the phase reaches the minimum energy state (equilibrium state) under given thermodynamic conditions (temperature and pressure). Experimentally, a phase diagram (equilibrium phase diagram) is prepared, and we may use the data... 

Exact laws governing the sequence of occurrence of compound layers in a particular reaction couple have not so far been established. What is available is a few empirical rules predicting this sequence at a probability level of about 60 to 90 %. These are based either (i) on the structure of the equilibrium phase diagram of a binary system or (ii) on the thermodynamic properties of its compounds. 

The sequence of their occurrence is determined by the rates of chemical transformations at the interfaces. It cannot yet be theoretically predicted with full confidence for any particular reaction couple A-B. Having sufficient information on the equilibrium phase diagram, thermodynamics of chemical reactions, and the structure and physical-chemical properties of the compounds, it is possible to indicate those of them, which are most likely to occur and grow first at the A-B interface. 

Thermodynamic calculations have also been used to determine the equilibrium products (Mamyan and Vershinnikov, 1992 Shiryaev et ai, 1993), and to illustrate new possibilities for controlling the synthesis process, even for complex multicomponent systems. Correlating these calculations with the equilibrium phase diagram for each system provides a basis for predicting possible chemical interactions and even limits of combustion during CS of materials. Some examples are discussed in the following subsections. 

In this chapter, we focus on the fundamentals of the theory of solutions, which is needed for the understanding of the equilibrium phase diagrams. The thermodynamic analysis... 

Microemulsions are thermodynamically stable phases, which can be represented by clear areas in equilibrium phase diagrams. Nanoemulsions are really small emulsions, with the main characteristics of emulsions they are not thermodynamically stable and the way they are prepared has a great impact on their physical stability. The only difference with common emulsions is their very small droplet size, which ranges from 10 to 500 nm. Accordingly, nanoemulsions may look bluish, due to light diffusion (brown/yellow by transmission), just like microemulsions close to a critical point. 

Fig. 10 Thermodynamic and kinetic basis for solute depletion in the case of a binary alloy consisting of solvent A and solute B. (a) Binary equilibrium phase diagram with complete miscibility in the liquid state, partial miscibility in the solid state given by existence of a terminal solid solution. Cs is the composition along the solvus line. is the overall composition of the alloy, (b) Time-temperature-transformation diagram for precipitation of in an a matrix for the alloy shown in (a) with overall composition,...

In the liquid state (melt) antimony is fully soluble in lead. On cooling the melt, formation of an a-Pb solid—solution starts whose composition depends on temperature and Sb content in the melt. Figure 4.3 presents the equilibrium phase diagram of the Pb—Sb system [13]. It reflects the thermodynamic state of the system when the phases are in equilibrium. 

Whilst Lp phases have been accepted for years recently there has been debate about whether the state really does exist as a true thermodynamic equilibrium phase, based on very reasonable criticism of deficiencies in their location on properly determined phase diagrams [65]. However, in at least one case (the nonionic surfactant trioxyeth-ylene hexadecyl ether [66]) the Lp phase in water melts at a higher temperature than the crystalline surfactant. It forms on mixing water and the liquid surfactant just above the crystalline surfactant melting point, clear proof that it is the equilibrium state. 

Microporous framework solids are synthesised via solvent-mediated crystallisations from mixtures of reactive precursors. The reaction pathway is controlled by kinetic as well as thermodynamic considerations so that equilibrium phase diagrams, so relevant in the high-temperature preparation of ceramics, are not useful here. Rather, synthetic routes have been developed empirically via a major synthetic effort that continues today. The continuing industrial and academic interest in these materials provides a powerful incentive to understand the principles underlying their formation through the processes of gel formation and evolution, nucleation and crystal growth. 

The solidus portion of the pressure-dependent equilibrium phase diagram of the PrS2,o-PrSi,5 system constructed by methods (a) and (b) is shown in fig. 4. A series of new sulfur-deficient polysulfides, PrSi.9oo(2), PrSi.846(6)i PrSi 756(8), and PrSi.702(7), in addition to tlie strict stoichiometric PrS2 has been found, separated by miscibility gaps. The intermediate polysulfides are seen to have narrow fields of stability. Therefore, it is difficult to prepare a single-phase intermediate polysulfide, especially when the desired thermodynamic parameters are unknown. On the other hand, the information resulting from these diagrams on the nonstoichiometric polysulfide compositions opens... 

Once again, it should be mentioned that the reaction products and the reactants together comprise a quasi-binary system which can be read from the phase diagram for the case where local thermodynamic equilibrium is maintained at the phase boundaries. If local equilibrium at all the phase boundaries is not established, then it is possible that certain product phases do not form, even though they are present in the equilibrium phase diagram. The probability that local equilibrium at the phase boundaries will, in fact, occur becomes better and better as the product layer grows. Reactions as discussed here are most easily studied experimentally today by means of the electron probe (electron beam microanalysis). 

Liquid—Liquid Phase Separation, in contrast to solid-liquid phase separation, lowering temperature can induce liquid-liquid phase separation of a polymer solution with an upper critical solution temperature and when the crystallization temperature of the solvent is sufficiently lower than the phase separation temperature. In an equilibrium phase diagram of a polymer solution, the spin-odal curve divides the liquid-liquid phase separation region into two regions a thermodynamically metastable region (between the binodal and spinodal) and a thermodynamically unstable region (enclosed by the spinodal) (Fig. 11). Above the... 

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