Transformation thermodynamic continuous phase

The thermodynamics of phase equilibria is reviewed in Chapter 17 and the fundamental thermodynamic differences between conserved and nonconserved order parameters are reinforced with a geometrical construction. These order parameters are used in the kinetic analyses of continuous and discontinuous phase transformations. 

Classical thermodynamics deals with macroscopic thermal-mechanical properties and their relationships for massive assemblages of atoms or molecules (i.e., 10 fundamental particles) in terms of energy conversion and transformation. Studies of phase/chemical changes and equilibria involving nanoparticles are important areas where the classical thermodynamic approach is effective. Because quantum mechanical effects may be marked (e g., the energy of a nanoparticle may not be continuous) where there are only several hundred (or even only tens) of atoms in a nanoparticle, one may ask, Is classical thermodynamics still valid for nanoparticle systems ... 

We now briefly consider another important aspect of nonequilibrium thermodynamics, namely phase transformations and how they are modelled. Galenko and Jou198 develop a thermodynamic formalism for rapid phase transformations within a diffuse interface of a binary system in which the system is in a state of local nonequilibrium. The phase-field method, in which the phase- field variable O varies smoothly and continuously between one pure phase (in which O = +1) and another (in which -1), is used to derive... 

Hocculation as discussed in this section is a process by which suspended drops or particles aggregate in a continuous medium. Although free energy is reduced, the system remains thermodynamically unstable. In some suspensions, however, separation into two thermodynamically stable phases can occm. For example, both theory and experiment show that transformation from a disordered or fluid phase to an ordered or sohd phase occurs in uniform suspensions of noninteracting hard spheres when the volume fraction reaches a value of approximately 0.50. 

The thermodynamic differences between spinodal decomposition (a continuous phase transformation) and nucleation and growth (a discontinuous phase transformation) are illustrated in Figure 6.6. At a given temperature, the volume free... 

Having in mind Equation 13.3, such hysteresis cannot be explained by thermodynamics alone but has to involve a description of the kinetics of first-order phase transformation as well. In order to develop a theoretical description, we appHed the previous thermodynamic approach to the study of the kinetic decoding of back and forth transitions during temperature cycling of nanopowders and presented a numerical analysis within the framework of the standard kinetic equation approach [93-95]. As a continuation of our approximation in Section 13.4, we arbitrarily assumed that the new thermodynamically advantageous phase has strict stoichiometry Ci = 0.5. The composition in each initially supersaturated solid particle is equal to Co (so that the new phase nucleus has a composition different from that of the parent phase, Ci Co). 

A physical system in which phase transition(s) can occur is usually characterized by one or more long range order parameters (order parameter for short). For example, in nematic liquid crystals the order parameter is the quantity S = (P2(cos 0)) as defined in previous chapters " in ferromagnets the order parameter is the magnetization in a single domain and in liquid-gas systems the order parameter is the density difference between the liquid and gas phases. In each of the above cases the state of the system, at any fixed temperature, can be described by an equilibrium value of the order parameter and fluctuations about that value. A phase transition can be accompanied by either a continuous or a discontinuous change in the equilibrium value of the order parameter when the system transforms from one phase to the other. (For simplicity we will consider temperature as the only thermodynamic variable in this paper the pressure depedence of the various phenomena will be neglected). 

In this chapter we have shown that diffusive transformations can only take place if nuclei of the new phase can form to begin with. Nuclei form because random atomic vibrations are continually making tiny crystals of the new phase and if the temperature is low enough these tiny crystals are thermodynamically stable and will grow. In homogeneous nucleation the nuclei form as spheres within the bulk of the material. In... 

The various findings about fluoride and its interaction with the hydroxyapatite at the molecular level show that the relationship is complicated and multifaceted. The broad conclusion from the enormous volume of work that has led to our current understanding of the role of fluoride is that it is overwhelmingly beneficial. It promotes numerous desirable properties in tooth mineral, reducing solubility through action in both the saliva and in the mineral phase, it shifts the demineralisation/remineralisation equilibrium in favour of remineralisation, and through its actions in the solid state, ensures that the kinetically favoured OCP is transformed into the more thermodynamically stable hydroxyapatite. Research continues, and there is no doubt that there is still more to learn about the complexities of the interaction of fluoride with hydroxypatite under physiological conditions. 

In Chapter 3 we described the structure of interfaces and in the previous section we described their thermodynamic properties. In the following, we will discuss the kinetics of interfaces. However, kinetic effects due to interface energies (eg., Ostwald ripening) are treated in Chapter 12 on phase transformations, whereas Chapter 14 is devoted to the influence of elasticity on the kinetics. As such, we will concentrate here on the basic kinetics of interface reactions. Stationary, immobile phase boundaries in solids (e.g., A/B, A/AX, AX/AY, etc.) may be compared to two-phase heterogeneous systems of which one phase is a liquid. Their kinetics have been extensively studied in electrochemistry and we shall make use of the concepts developed in that subject. For electrodes in dynamic equilibrium, we know that charged atomic particles are continuously crossing the boundary in both directions. This transfer is thermally activated. At the stationary equilibrium boundary, the opposite fluxes of both electrons and ions are necessarily equal. Figure 10-7 shows this situation schematically for two different crystals bounded by the (b) interface. This was already presented in Section 4.5 and we continue that preliminary discussion now in more detail. 

In thermodynamic equilibrium a system may be composed of one or several physically distinct macroscopic homogeneous parts called phases, which are separated from one another by well-defined interfaces. These phases are determined by several parameters such as temperature, pressure, and electric and magnetic fields. By continuously varying the parameters it is possible to induce the transformation of the system from one phase to another. 

In non-fluoride-containing solutions, silicon is stable due to the presence of an oxide film and the electrode behavior can remain constant under a continuous cathodic polarization. The surface of a silicon electrode in fluoride-containing aqueous solution at the open circuit potential is also stable due to hydrogen adsorption. However, surface transformation can occur at cathodic potentials due to formation of hydrides. Thermodynamically, silicon hydride can be a stable phase at certain cathodic potentials as shown in Fig. 2.2. 

During calcination, the precursor undergoes a continuous crystallization and recrystallization process during which thermodynamically unstable or metastable phases are converted to more stable ones. X-ray diffraction patterns in Fig. 7 show phase transformation of hydrous titanium oxide from 100°C to 550°C.P l... 

Since zeolites are metastable crystallization products they are subject to Ostwald s rule which states that metastable phases are initially formed and gradually transform into the thermodynamically most stable product. The least stable zeolitic phase (that with the lowest framework density) is therefore formed first and consumed with further synthesis time at the expense of a more stable phase due to a continuous crystallization/redissolution equilibrium. 

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