Crystallization second-order transition

The scheme in Figure 2 illustrates a possible alternative explanation for the observation that bond lengths in m-enol systems are intermediate between single and double bonds. If the molecules have statistically disordered enol systems, the hydrogen atoms of the hydrogen bond will be distributed over two positions in the crystal structure. Indeed this was the case for the C polymorph of naphtazarin above 110 K at this temperature there is a second-order transition to a state with an ordered enol hydrogen [2],... 

Among crystalline solids, typical second-order transitions are associated with abrupt intermolecular conformational, rotational, and vibrational changes and/or with abrupt changes in crystalline disorder and/or defects [7], These changes in crystalline solids are sometimes difficult to assign without the use of appropriate spectroscopic techniques such as solid-state NMR or a diffraction procedure such as single-crystal X-ray diffraction. 

Note 2 The divergence occurs at the point where the isotropic phase would be expected to undergo a second-order transition to the liquid-crystal phase, were it not for the intervention of a first-order transition to the liquid-crystal phase. 

Crystal phase transitions are a possible target with present day computational means, when the transition is a smooth one and does not involve melting of the mother phase and subsequent recrystallization into the daughter phase. For crystalline OL-norleucine, an MD simulation has provided a detailed picture of the mechanism of a solid-solid second-order transition between two polymorphic crystal forms, showing concerted molecular displacements involving entire bilayers [61]. 

When the free energies F of the two crystal structures are identical, the system is at a critical point. The identity of F does not imply identical fimctions (otherwise the two phases would be indistinguishable). Therefore, at the critical point first derivatives of F might differ and therefore enthalpy, volume, and entropy of the two phases would be different. These transformations are first-order phase transitions, according to Ehrenfest [105]. A discontinuous enthalpy imphes heat exchange at the transition temperature, which can easily be measured with DSC experiments. A discontinuous volume is evident under the microscope or, more precisely, with diffraction experiments on single crystals or powders. Some phase transitions are however characterized by continuous first derivatives of the free energy, whereas the second derivatives (specific heat, compressibility, or thermal expansivity, etc.) are discontinuous. These transformations are second-order transitions and are clearly softer. 

The vapour deposition method is widely used to obtain amorphous solids. In this technique, atoms, molecules or ions of the substance (in dilute vapour phase) are deposited on to a substrate maintained at a low temperature. Most vapour-deposited amorphous materials crystallize on heating, but some of them exhibit an intervening second-order transition (akin to the glass transition). Amorphous solid water and methanol show such transitions. The structural features of vapour-deposited amorphous solids are comparable to those of glasses of the same materials prepared by melt-quenching. 

Figure 4.23 Various degrees of complexity that can arise in JT systems. (After Gehring Gehring, 1975.) CF stands for crystal field. Single circle indicates first-order transition and a double circle indicates a second-order transition. A single circle with a broken circle indicates a transition which is first-order because of the existence of anharmonic forces.

Tg can be determined by studying the temperature dependence of a number of physical properties such as specific volume, refractive index, specific heat, etc. First-order transitions, such as the melting of crystals, give rise to an abrupt change or discontinuity in these properties. However, when a polymeric material undergoes a second-order transition, it is not the primary property (the volume), but its first derivative with respect to temperature, (the coefficient of expansion), which becomes discontinuous. This difference between a first and second-order transition is illustrated in Figure 10. 

At least two different glass transition temperatures have been reported for PVdF homopolymer. Owing to the large proportion of crystalline structure in this polymer and the rapid crystallization which occurs while heating quenched amorphous samples, it is difficult experimentally to obtain an unambiguous, well-defined second-order transition. Mandel-kem, Martin, and Quinn (16) reported a value below — 40°C based upon an extrapolation of the Tg data for vinylidene fluoride-chlorotri-fluoroethylene copolymers in accordance with the Fox equation (6),... 

Ehrenfest s concept of the discontinuities at the transition point was that the discontinuities were finite, similar to the discontinuities in the entropy and volume for first-order transitions. Only one second-order transition, that of superconductors in zero magnetic field, has been found which is of this type. The others, such as the transition between liquid helium-I and liquid helium-II, the Curie point, the order-disorder transition in some alloys, and transition in certain crystals due to rotational phenomena all have discontinuities that are large and may be infinite. Such discontinuities are particularly evident in the behavior of the heat capacity at constant pressure in the region of the transition temperature. The curve of the heat capacity as a function of the temperature has the general form of the Greek letter lambda and, hence, the points are called lambda points. Except for liquid helium, the effect of pressure on the transition temperature is very small. The behavior of systems at these second-order transitions is not completely known, and further thermodynamic treatment must be based on molecular and statistical concepts. These concepts are beyond the scope of this book, and no further discussion of second-order transitions is given. 

We all know that when a liquid transforms to a crystal, there is a change in order the crystal has greater order than the liquid. The symmetry also changes in such a transition the liquid has more symmetry than a crystal since the liquid remains invariant under all rotations and translations. Landau introduced the concept of an order parameter, , which is a measure of the order resulting from a phase transition. In a first-order transition (e.g., liquid-crystal), the change in is discontinuous, but in a second-order transition where the change of state is continuous, the change in is also continuous. Landau proposed that G in a second-order (or structured) phase transition is not only a function of P and T but also of and expanded G as... 

At temperatures below 300° K. deviations from linear p vs. T behavior may appear, particularly when the ratio of M to W03 is below 0.3. This has been observed on single crystals of copper-doped WOs (36), silver-doped WOs (5), and low-sodium Na WOs (20). It is postulated that a second-order transition occurs, which is related to the ferroelectric transition in W03. This transition which occurs in the neighborhood of 220° K. (it is spread over roughly 50°, probably because it occurs piecemeal as a domain-growth phenomenon), shows up in pure W03 in its resistivity behavior, its Hall coefficient, and its thermoelectric power (7). The carrier mobility drops significantly as the temperature is lowered through the transition, so it is probable that the rather steep rise in resistivity... 

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