Crystalline polymers energy calculations

Crystallization kinetics have been studied by differential thermal analysis (92,94,95). The heat of fusion of the crystalline phase is approximately 96 kj/kg (23 kcal/mol), and the activation energy for crystallization is 104 kj/mol (25 kcal/mol). The extent of crystallinity may be calculated from the density of amorphous polymer (d = 1.23), and the crystalline density (d = 1.35). Using this method, polymer prepared at —40° C melts at 73°C and is 38% crystalline. Polymer made at +40° C melts at 45°C and is about 12% crystalline. 

Calculations of conformational energy made by means of molecular mechanics fully confirm these conclusions. Such calculations were first introduced into the examination of synthetic crystalline polymers by Liquori and co-workers (175, 176) and were extensively used by Natta, Corradini, Allegra, Ganis, and co-workers (168, 177-179). The conformational energy map of isotactic poly-... 

Flory, Marie, and Abe (194) carried out a statistical mechanics analysis of vinyl polymers on the basis of a three rotational state model. Energy maps have been calculated both for m and r dyads as a function of the rotation angles around the bonds astride the methylene groups (CHR—CH2—CHR). These maps differ from those examined for crystalline polymers where rotations around... 

This work has shown that wide-angle X-ray scattering is a sensitive tool for evaluating the structure of non-crystalline polymers. The structure determination is only successfully approached by considering first the persistent conformation and then the packing which must to some extent be a consequence of such conformations. The method does not need to place any reliance on the semi-empirical conformational energy calculations. 

The extension of TDDFT and Tamm-Dancoff TDDFT to crystalline polymers is straightforward within the formalisms of Section 2.2.2. Figure 2-7 summarizes the results of TDDFT calculations of the photoconduction, photoemission, and optical absorption thresholds (energy gap, ionization energy, and excitation energy) of polyethylene as a function of basis set [50], The Slater-Vosko-Wilk-Nusair functional [116, 117] is used, but the following conclusion is unaltered... 

Priest (1973) and Straley (1973), in terms of the classical virial expansion, the Onsager theory (referred to in Section 2.1) and the curvature moduli theory, derived the elastic constants of rigid liquid crystalline polymers. The free energy varies according to the change of the excluded volume of the rods due to the deformation. The numerical calculation of elastic constants (Lee, 1987) are shown in Table 6.2. 

To obtain a reasonable prediction of miscibility, the values of the solubility parameters are to be measured to an accuracy better than 5j - 82 < 0.2 (J/mL). The measured values of 8j (where i = 1 or 2) far exceeds the magnitude of the critical difference of these parameters, 8j - 82 < 2 (J/mL). The calculated values of 8. are claimed to be precise within 0.8 (J/mL)i/2 [Coleman et al, 1990]. The solubility parameter approach is applicable to amorphous polymer systems. In order to adopt to highly crystalline polymers, the heat and entropy of fusion (Ah j and As p respectively) is to be dealt with in free energy of mixing equation [Van Krevelen and Hoftyzer, 1976] ... 

The most successful equation of state for semicrystalline polymers such as PE and PA stems from two unlikely sources (1) calculation of 5 = a of polymeric glasses at T< 80K [Simha et al., 1972] and (2) the Lennard-Jones and Devonshire (L-JD) cell model developed originally for gases and then liquids. Midha and Nanda [1977] (M-N) adopted the L-JD model for their quantum-mechanical version of crystalline polymers, taking into account harmonic and anharmonic contributions to the interaction energy. Simha and Jain (S-J) subsequently refined their model and incorporated the characteristic vibration frequency at T= 0 K from the low-Tglass theory [Simha and Jain, 1978 Jain and Simha, 1979a,b] ... 

The macroconformation of chainlike macro molecules in crystalline polymers is principally determined by two factors, which are inter- and intracatenary forces. Calculation of potential barriers of isolated molecules, that is, in a vacuum, is based exclusively on intracatenarily effective forces (see also Section 4.1.2). Microconformations calculated in this way correspond to an internal energy minimum. According to the equivalence principle, all structural units should adopt geometrically equivalent positions in relation to the crystallographic axes, whereby a monomeric unit, for example, may serve as a structural unit. Thus the regular sequence of microconformations should lead to a regular macroconformation. 

Some researchers use molecule computations to estimate the band gap from the HOMO-LUMO energy separation. This energy separation becomes smaller as the molecule grows larger. Thus, it is possible to perform quantum mechanical calculations on several molecules of increasing size and then extrapolate the energy gap to predict a band gap for the inhnite system. This can be useful for polymers, which are often not crystalline. One-dimensional band structures are... 

Mesoscale simulations model a material as a collection of units, called beads. Each bead might represent a substructure, molecule, monomer, micelle, micro-crystalline domain, solid particle, or an arbitrary region of a fluid. Multiple beads might be connected, typically by a harmonic potential, in order to model a polymer. A simulation is then conducted in which there is an interaction potential between beads and sometimes dynamical equations of motion. This is very hard to do with extremely large molecular dynamics calculations because they would have to be very accurate to correctly reflect the small free energy differences between microstates. There are algorithms for determining an appropriate bead size from molecular dynamics and Monte Carlo simulations. 

Polymers can be crystalline, but may not be easy to crystallize. Computational studies can be used to predict whether a polymer is likely to crystallize readily. One reason polymers fail to crystallize is that there may be many conformers with similar energies and thus little thermodynamic driving force toward an ordered conformation. Calculations of possible conformations of a short oligomer can be used to determine the difference in energy between the most stable conformer and other low-energy conformers. 

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