Transition, first-order crystallization

McGrath, K. and Kidman, M., Spiral textures in lyotropic liquid crystals First order transition between normal hexagonal and lamellar gel phases, J. Phys. II (France), 3, 903-926 (1993). 

Figure 4.3b is a schematic representation of the behavior of S and V in the vicinity of T . Although both the crystal and liquid phases have the same value of G at T , this is not the case for S and V (or for the enthalpy H). Since these latter variables can be written as first derivatives of G and show discontinuities at the transition point, the fusion process is called a first-order transition. Vaporization and other familiar phase transitions are also first-order transitions. The behavior of V at Tg in Fig. 4.1 shows that the glass transition is not a first-order transition. One of the objectives of this chapter is to gain a better understanding of what else it might be. We shall return to this in Sec. 4.8. 

In Secs. 4.3 and 4.4 we discussed the thermodynamics of the crystal -> Uquid transition. This and other famiUar phase equilibria are examples of what are called first-order transitions. There are other less familiar but also well-known... 

Tra.nsitions and Relaxations. Only one first-order transition is observed, the melting poiat. Increasing the pressure raises mp. At low pressure, the rate of iacrease ia the melting poiat is ca 1.74°C/MPa (0.012°C/psi) at high pressures this rate decreases to ca 0.725°C/MPa (0.005°C/psi). Melting iacreases the volume by 8%. la the preseace of the HFP comonomer, crystal distortioa occurs with an iacrease ia iatramolecular distance that, ia turn, reduces the melting poiat (54). 

First of all the term stress-induced crystallization includes crystallization occuring at any extensions or deformations both large and small (in the latter case, ECC are not formed and an ordinary oriented sample is obtained). In contrast, orientational crystallization is a crystallization that occurs at melt extensions corresponding to fi > when chains are considerably extended prior to crystallization and the formation of an intermediate oriented phase is followed by crystallization from the preoriented state. Hence, orientational crystallization proceeds in two steps the first step is the transition of the isotropic melt into the nematic phase (first-order transition of the order-disorder type) and the second involves crystallization with the formation of ECC from the nematic phase (second- or higher-order transition not related to the change in the symmetry elements of the system). 

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. 

The crystal property perhaps most sensitive to structure is density and, conversely, the increase in density of a crystal with pressure is accompanied by significant structural changes. The latter can be of two types - either a smooth variation of the free parameters of the structure with pressure, or a first-order transition to a new structure type. 

Another important contribution by Landau is related to symmetry changes accompanying phase transitions. In second-order or structural transitions, the symmetry of the crystal changes discontinuously, causing the appearance (or disappearance) of certain symmetry elements, unlike first-order transitions, where there is no relation between the symmetries of the high- and low-temperature phases. If p(x, y, z) describes the probability distribution of atom positions in a crystal, then p would reflect the symmetry group of the crystal. This means that for T> T p must be consistent with... 

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.

Ramirez et al (1970) discussed a metal-insulator transition as the temperature rises, which is first order with no crystal distortion. The essence of the model is—in our terminology—that a lower Hubbard band (or localized states) lies just below a conduction band. Then, as electrons are excited into the conduction band, their coupling with the moments lowers the Neel temperature. Thus the disordering of the spins with consequent increase of entropy is accelerated. Ramirez et al showed that a first-order transition to a degenerate gas in the conduction band, together with disordering of the moments, is possible. The entropy comes from the random direction of the moments, and the random positions of such atoms as have lost an electron. The results of Menth et al (1969) on the conductivity of SmB6 are discussed in these terms. 

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. 

The crystal-mesophase transitions at Td are first order transitions. In the case of plastic crystals, they show usually the major heat of transition. In the condis crystals the magnitude of the transitions depends on the number of bonds which gain con-... 

As with liquid crystals (Sect. 5.1.2), only the equilibrium transition to the fully ordered crystal and the transition to the isotropic melt have been studied extensively16c 102). Both transitions are of the first order transition type, similar to that shown in Fig. 1. A series of transition data are collected in Table 7. Plastic crystals with motifs as small as... 

Figure 20 shows the phase diagram of polyethylene119). The existence range of the condis crystals increases with pressure and temperature. The enthalpy of the reasonably reversible, first order transition from the orthorhombic to the hexagonal condis phase of polyethylene is 3.71 kJ/mol at about 500 MPa pressure 121) which is about 80 % of the total heat of fusion. The entropy of disordering is 7.2 J/(K mol), which is more than the typical transition entropy of paraffins to their high temperature... 

Poly (diethyl siloxane) was suggested by Beatty et al. 1651 based on DSC, dielectric, NMR, and X-ray measurements to possess liquid crystalline type order between about 270 and 300 K. The macromolecule shows two large lower temperature first order transitions, one at about 200 K, the other at about 270 K166 ll,7). The transition of the possible mesophase to the isotropic liquid at 300 K is quite small and irre-producible, so that variable, partial crystallinity was proposed 165) [measured heat of transition about 150 J/mole1S8)], Very little can be said about this state which may even consist of residual crystals. It is of interest, however, to further analyze the high temperature crystal phase between 200 and 270 K. It is produced from the, most likely, fully ordered crystal with an estimated heat and entropy of transition of 5.62kJ/mol and 28J/(Kmol), respectively [calculated from calorimetric data 1S6)... 

The simplest definition of a first-order transition is one in which heat flows into or out of the material with no change in temperature. Examples are melting and boiling and their reversals, crystallization and condensation. 

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. 

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