First-and Second-Order Transitions

In first order transitions, such as melting, there is a discontinuity in the volume-temperature plot (see Fig. 2.22) or enthalpy-temperature plot at 

In rst order transitions, such as melting, there is a discontinuity in the volnme-temperature plot (see Fig. 2.19) or enthalpy-temperature plot at the transition temperatnre. In second-order transitions, only a change in slope occurs and there is thus a marked change in the rst derivative or temperamre coef cients, as illustrated in Fig. 2.20. The glass transition is not a rst-order ttansi-tion, as no discontinuities are observed at Tg when the speci c volume or enfropy of the polymer is measured as a function of temperature. However, the rst derivative of the property-temperature curve, i.e., the temperamre coef cient of the property (e.g., heat capacity and volnmeUic coef -cient of expansion), exhibits a marked change in the vicinity of Tg for this reason it is sometimes called a second-order transition. 

In first order transitions, siich as melting, there is a discontinuity in the volume-temperature plot (see Fig. 2.19) or enthalpy-temperature plot at the transition temperature. In second-order transitions, only a change in slope occurs and there is thus a marked change in the first derivative or temperature coefficients, as illustrated in Fig. 2.20. The glass transition is not a first-order transition, as no 

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It is easily shown that a first-order phase transition is obtained for cases were d < 0, whereas behaviour at the borderline between first- and second-order transitions, tricritical behaviour, is obtained for d = 0. In the latter case the transitional Gibbs energy is... 

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. 

CASSCF wave functions with their own sets of optimized orbitals, which where then not orthogonal to each other. The CAS State interaction CASSI, method made it possible to compute efficiently first and second order transition density matrices for any type of CASSCF wave functions [16, 17]. The method is used to compute transition dipole moments in spectroscopy and also in applications where it is advantageous to use localized orbitals, for example in studies of charge transfer reactions [18]. Today, the same approach is used to construct and solve a spin-orbit Hamiltonian in a basis of CASSCF wave functions [19]. 

The RASSI method can be used to compute first and second order transition densities and can thus also be used to set up an Hamiltonian in a basis of RASSCF wave function with separately optimized MOs. Such calculations have, for example, been found to be useful in studies of electron-transfer reactions where solutions in a localized basis are preferred [43], The approach has recently been extended to also include matrix elements of a spin-orbit Hamiltonian. A number of RASSCF wave functions are used as a basis set to construct the spin-orbit Hamiltonian, which is then diagonalized [19, 44],... 

The reasoning given above applies to most kinds of phase transition, for instance also for p—>a in the same system. An overview (not exhaustive) is given in Table 14.1 it also specifies the equilibrium temperature and the transition enthalpy. It should be added that it here concerns so-called first-order (phase) transitions. In section 16.1 the difference between first- and second-order transitions will be discussed. 

It is concluded in Ref. 217 that an analysis of the order of the herringbone transition entirely based on thermodynamic quantities might be very misleading, because it was shown that most of the data can be rationalized in terms of both first- and second-order transitions. Thus, an analysis along these lines would require systems which are orders of magnitude larger than those available in Refs. 56 and 217, but only this would allow to reliably estimate the latent heat and the order parameter jump at the transition. 

Both first- and second-order transitions are observed in polymers. Melting and allotropic transformations are accompanied by latent-heat effects and are known as first-order transitions. During second-order transitions, changes in properties occur without any latent-heat effects. Below the second-order-transition temperature (glass transition temperature) a rubberlike material acts like a true solid (see Chapter 1). Above this temperature the fixed molecular structure is broken down partially by a combination of thermal expansion and thermal agitation. The glass transition temperature of polystyrene is 100°C below 100°C polystyrene is hard and brittle, and above 100°C it is rubberhke and becomes easily deformed. 

Thermodynamic transitions can be characterized by corresponding changes in these parameters of state. First- and second-order transitions are both characterized by the thermodynamic equilibria on either side of the physical transition temperature. 

Fig. 6. First and second order transitions, (a) Behaviour of low molecular weight compounds, (b) Behaviour of glassy and semi-crystalline polymers.

Liquid crystals manifest a number of transitions between different phases uprm variation of temperature, pressure or a craitent of various compounds in a mixture. All the transitions are divided into two groups, namely, first and second order transitions both accompanied by interesting pre-transitional phenomena and usually described by the Landau (phenomenological) theory or molecular-statistical approach. In this chapter we are going to consider the most important phase transitions between isotropic, nematic, smectic A and C phases. The phase transitions in ferroelectric liquid crystals are discussed in Chapter 13. 

Thermal analysis procedures have been powerful tools for studying PTFE. McCrum s classic work with early dynamic methods [36] clearly showed the various first and second order transitions that were elucidated further by the work of Eby, Sinnott, and others. These transitions, of course, control much of the behavior of PTFE. 

Conformational studies on poly(decamethylene adipate) and poIy(tetra-methylene succinate) and the use of molecular orbital theory potential energy surfaces in determining polymer conformations for 2GT have also appeared. Crystallization, Crystallinity, and First and Second Order Transition Behaviour. The crystallization behaviour of 2GT, alone or in admixture with additives or modified by small amounts of co-monomers, continues to evoke considerable interest. Thus, X-iay fluorescence studies have been made on phenylmercurated... 

Crystallization, Crystallinity, and First and Second Order Transition Behaviour. Nylon 3 has been subjected to analysis by DTA and the crystallinity has been shown to increase on heating. Branching has been shown to affect the DTA trace observed and this is correlated with structural features. 

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