Spontaneous phase transitions

Consider a different kind of system, one consisting of the liquid and solid phases of a pure substance. At a given pressure, this kind of system can be in transfer equilibrium at only 

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Unlike melting and the solid-solid phase transitions discussed in the next section, these phase changes are not reversible processes they occur because the crystal stmcture of the nanocrystal is metastable. For example, titania made in the nanophase always adopts the anatase stmcture. At higher temperatures the material spontaneously transfonns to the mtile bulk stable phase [211, 212 and 213]. The role of grain size in these metastable-stable transitions is not well established the issue is complicated by the fact that the transition is accompanied by grain growth which clouds the inteiyDretation of size-dependent data [214, 215 and 216]. In situ TEM studies, however, indicate that the surface chemistry of the nanocrystals play a cmcial role in the transition temperatures [217, 218]. 

Oxides such as CaO, MgO, and Y2O2 are added to Zr02 to stabili2e the tetragonal phase at temperatures below the tetragonal to monoclinic phase-transition temperature. Without stabili2et, the phase transition occurs spontaneously at temperatures below 850—1000°C, and no fracture toughness enhancement occurs (25). 

This tells us immediately that, just as for Ising spins, we have a spontaneous magnetization and that there is an effective phase transition for T = 1 stored patterns will only be stable for temperatures T < 1. 

Langton s tentative answer to the question above is therefore We expect that information processing can emerge spontaneously and come to dominate the dynamics of a physical system in the vicinity of a critical phase transition. Langton speculates that the dynamics of phase transitions is fundamentally equivalent to the dynamics of information processing. 

Crystals with one of the ten polar point-group symmetries (Ci, C2, Cs, C2V, C4, C4V, C3, C3v, C(, Cgv) are called polar crystals. They display spontaneous polarization and form a family of ferroelectric materials. The main properties of ferroelectric materials include relatively high dielectric permittivity, ferroelectric-paraelectric phase transition that occurs at a certain temperature called the Curie temperature, piezoelectric effect, pyroelectric effect, nonlinear optic property - the ability to multiply frequencies, ferroelectric hysteresis loop, and electrostrictive, electro-optic and other properties [16, 388],... 

The appearance of spontaneous polarization in the case of LuTaO is related to volumetric irregularities and ordering of the Li+ - Ta5+ dipoles, as is in the case of the similar niobium-containing compound Li4Nb04F. It can be assumed that the main difference between the two compounds is that the irregularities and the Li+ - Ta5+ dipoles are thermally more stable compared to the niobium-containing system. This increased stability of the dipoles leads to the reversible phase transition at 660°C. 

The question arises as to how useful atomistic models may be in predicting the phase behaviour of real liquid crystal molecules. There is some evidence that atomistic models may be quite promising in this respect. For instance, in constant pressure simulations of CCH5 [25, 26] stable nematic and isotropic phases are seen at the right temperatures, even though the simulations of up to 700 ps are too short to observe spontaneous formation of the nematic phase from the isotropic liquid. However, at the present time one must conclude that atomistic models can only be expected to provide qualitative data about individual systems rather than quantitative predictions of phase transition temperatures. Such predictions must await simulations on larger systems, where the system size dependency has been eliminated, and where constant... 

Similar considerations apply to chemical or physicochemical equilibria such as encountered in phase transitions. A chilled salt solution may be stable (at or below saturation), metastable (supercooled to an extent not allowing nucle-ation), or unstable (cooled sufficiently to nucleate spontaneously). In the case of a solid, S, dispersed in a binary liquid, Li + L2, instability at the instant of formation gives way to a neutral or metastable condition wherein three types of contacts are established ... 

As the temperature is decreased, the chains become increasingly rigid zc then approaches 1 if we assume that there is only one fully ordered crystalline structure and Zconf for the liquid becomes smaller than 1. This means that, at this level of approximation, the disordered state becomes less favorable than the crystalline ground state. A first-order disorder-order phase transition is expected to occur under these conditions. Flory interpreted this phase transition as the spontaneous crystallization of bulk semiflexible polymers [12], However, since the intermolecular anisotropic repulsion essential in the Onsager model is not considered in the calculation, only the short-range intramolecular interaction is responsible for this phase transition. 

Figure 3 Landau free energy at different temperatures. Spontaneous symmetry breaking occurs fort < 0, giving rise to a second-order phase transition at t=0.

The infinite potential barrier, shown schematically in figure 10 corresponds to a superselection rule that operates below the critical temperature [133]. Above the critical temperature the quantum-mechanical superposition principle applies, but below that temperature the system behaves classically. The system bifurcates spontaneously at the critical point. The bifurcation, like second-order phase transformation is caused by some interaction that becomes dominant at that point. In the case of chemical reactions the interaction leads to the rearrangement of chemical bonds. The essential difference between chemical reaction and second-order phase transition is therefore epitomized by the formation of chemically different species rather than different states of aggregation, when the symmetry is spontaneously broken at a critical point. 

Figure 5.2 Graph of molar Gibbs function Gm as a function of temperature. Inset, at temperatures below r(meit) the phase transition from liquid to solid involves a negative change in Gibbs function, so it is spontaneous...

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