Linear macromolecules, equilibrium

Fig. 3. Schematic diagram of the possible states of a liquid crystal forming material. Only the left side of the diagram corresponds to equilibrium. The states towards the right are increasingly metastable or unstable, but are often reached for flexible, linear macromolecules...

We turn to the non-equilibrium moments of co-ordinates and velocities of linear macromolecules. As a first step, we shall consider one-point second-order moments. The expressions for co-ordinates and velocities (4.7) with the same arguments can be used to make up proper combinations, and by averaging over the ensemble of realisation of random forces, we find the moments with accuracy to the first-order terms in velocity gradients. Then, by taking into account the properties of equilibrium moments and the antisymmetry of tensor cun, we find that... 

FIG. 2.12 Schematic drawing of the different macro-conformations possible in solid linear macromolecules, (a) Random, glassy (b) folded chain, lamellar (c) extended chain equilibrium (d) fringed micelle, mixture of (a) to (c) (after Wunderlich, 1970). 

As any ring-opening polymerization, the polymerization of lactams is characterized by competition between the intermolecular reaction resulting in a linear polyamide and the intramolecular reaction of cyclization. Thus, if thermodynamically feasible, as indicated by a negative value of the free-energy change of polymerization AGp, the conversion of lactam to the linear polyamide can be realized if an appropriate reaction path exists for a reasonable rate in a polymer-monomer equilibrium characterized by a preponderance of linear macromolecules. The corresponding equilibrium monomer concentration [M]g is related to the standard enthalpy and entropy ASp of polymerization and to... 

The extensive properties of the overall system that is not in equilibrium, such as volume or energy, are simply the sums of the (almost) equilibrium properties of the subsystems. This simple division of a sample into its subsystems is the type of treatment needed for the description of irreversible processes, as are discussed in Sect. 2.4. Furthermore, there is a natural limit to the subdivision of a system. It is reached when the subsystems are so small that the inhomogeneity caused by the molecular structure becomes of concern. Naturally, for such small subsystems any macroscopic description breaks down, and one must turn to a microscopic description as is used, for example, in the molecular dynamics simulations. For macromolecules, particularly of the flexible class, one frequently finds that a single macromolecule may be part of more than one subsystem. Partially crystalhzed, linear macromolecules often traverse several crystals and surrounding liquid regions, causing difficulties in the description of the macromolecular properties, as is discussed in Sect. 2.5 when nanophases are described. The phases become interdependent in this case, and care must be taken so that a thermodynamic description based on separate subsystems is still valid. 

The dilatometry at different pressures leads to a full p-V-T phase diagram. Linear macromolecules in the liquid state can reach equilibrium and have then been successfully described by a single p-V-T diagram. The semicrystalline and glassy... 

Figure 4.3. A schematic of the free enthalpy in the vicinity of the equilibrium melting temperature (left) and a plot of the linear crystallisation and melting rates of gaseous selenium (Sc2) (right). Selenium crystallizes or sublimes to and from selenium crystals made up of flexible, linear macromolecules. The process can be expressed as xSc2 (gaseous) Se2x...

CrystallinG MultiCOmponGnt Systems. A multicomponent system that crystallizes with a common crystal structure is said to form mixed crystals. Formation of mixed crystals is often possible for components that crystalbze in their pure state with the same crystal shapes (isomorphism). For linear macromolecules, a further condition must be fulfilled, the chain conformations of the components must match. For energy reasons, crystals usually can only be obtained with macromolecules close to their equilibrium conformations. In these conformations, close packing must be achieved with the second component, a rather rare event, although examples are known of limited solubility and cocrystallization with small molecule solvents (9). 

The bottom curve in Fig. 5.26 illustrates the ultimate result of thermal analysis of two crystalline linear macromolecules. Two isomers of 1,4-polybutadiene (PB) are analyzed. All data were extrapolated to 100% crystallinity and equilibrium. The entropy was then computed using Eq. (2) of Fig. 5.11, and adding the transition entropies at the equilibrium temperatures. Clearly, the trans isomer melts in two steps, while the cis isomer melts in a... 

Entropy of fusion data of pure, one-component systems of different molecular structures are collected in Tables 5.2-5.5. These will be discussed in connection with Fig. 1.9, progressing from spherical molecules to asymmetric molecules with conformational mobility. Naturally, as discussed in the previous section, the equilibrium data on linear macromolecules are often derived by extrapolation, to both the equilibrium melting temperature and crystallinity = 1.0). In Fig. 3.7 the equilibrium melting temperature is shown to be AHf/AS [Eq. (6)]. Heat of fusion measurements are thus able also to provide data on the entropy of fusion. This allows the development of quantitative lists of entropies of fusion. In this section an attempt is made to find the connection between the entropy of fusion and molecular structure. A typical example of experimental data on entropy changes with fusion is shown in Fig. 5.26. 

Melting and crystallization is discussed not only in this chapter, but also in several earlier chapters, since it is one of the prime subjects of thermal analysis of materials [see Sects. 3.4, 3.5, 4.6, 4.7.1, 4.7.2, and 5.S.2-5.5.4]. Melting can sometimes approach equilibrium conditions and is, in this case, describable by equilibrium thermodynamics (see Sect. 2.1.1). More often, however, it occurs under nonequilibrium conditions, as is described for thin lamellae in Fig. 4.28 (see also Figs. 4.29 and 4.30). In this section equilibrium and nonequilibrium melting will be analyzed with regard to the crystal size. Finally, equilibrium extension effects of linear macromolecules that were not analyzed in other sections are treated. 

The equilibrium melting temperature, 414.6 K, but it is in practice rarely reached for linear macromolecules. 

The principles of fractionation by precipitation and extraction are too well known to need description here. Earlier reviews (2—5) dealing with most of the experimental and theoretical aspects, we shall only consider some recent developments. The systems dealt with contain linear macromolecules built up of identical segments. Molecular structure differences other than chain length will be left out of consideration. The phase separations involved in fractionation are described as liquid-liquid separations, and the calculations related to the latter will be based on equations valid for phase equilibrium. A brief discussion of the comparatively new technique of fractional crystallization is given in Section 2. 

The ability of a cyclic monomer to polymerize according to the ring-opening mechanism is determined by two equally important factors-the conversion of monomer molecules into macromolecules (of linear or more complex topologies) must be allowed both thermodynamically and kinetically. In practical terms this means that (i) monomer-macromolecule equilibrium must be shifted to the right-hand (macromolecule) side and (ii) the corresponding polymerization mechanism should exist, that could enable conversion of the monomer molecules into the polymer repeating units, within the operable polymerization time (Equation 1.1). 

Casassa, E. R and Tagami, Y. An equilibrium theory for exclusion chromatography of branched and linear polymer chains, Macromolecules, 2, 14, 1969. 

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