Thermal Analysis of Polymeric Materials

36 1 Atoms, Small, and Large Molecules

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This leaves as a final problem, the excluded volume. The excluded volume arises from the fact that a random flight can have unlimited revisits of any point in space, while a molecule modeled by the random flight would interfere with earlier segments at the same point. A certain volume of space is, thus, excluded for the remaining molecule, as indicated as entry d.

The summary of dimensions at the bottom of Fig. 1.33 demonstrates the effects of the various restrictions on the model compound. The random flight model gives for polyethylene already an answer within a factor of about 2.5. Comparison with the experiment is possible by analyzing the dimensions of a macromolecule in solution, as will be discussed in Chap. 7. One can visualize a solution by filling the vacuum of a random flight of the present discussion with the solvent molecules. The - temperature listed as condition for the experiment is the temperature at which the expansion of the molecule due to the excluded volume is compensated by compression due to rejection of the solvent out of the random coil. This compensation of an excluded volume is similar to the Boyle-temperature of a real gas as illustrated in

Fig. 1.34

Fig. 1.34. Not far from the -temperature, polymer solutions undergo phase separation. An equation describing the excluded volume effect gained from MonteCarlo simulation, is discussed in Sect. 1.3.6 and given in Fig. 1.39, below.

The discussion of the chain statistics permits one, thus, to have a more quantitative description of a flexible, linear macromolecule. The random coil of a sufficiently long molecule can be compared in mass-density and randomness to an ideal gas at atmospheric pressure. The elastic compression and expansion of gases are caused by changes in entropy. It will be shown below that corresponding behavior exists for the extension and contraction of random-coil macromolecules (entropy or rubber elasticity, see Sect. 5.6.5). Combining many random coils into a

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condensed phase (liquid), entanglements start to dominate structure-sensitive properties such as the viscosity (see Fig. 3.5). Macromolecules which are less flexible lose their typical plastic behavior. Their melting temperature becomes higher, the rubber elasticity is less pronounced, and ultimately a rigid macromolecule results (see Sect. 1.2.4).

Further discussion of the properties of macromolecules will recognize the cooperative behavior. The repeating units (or parts of them) have limited independent mobility and remain linked to the overall molecule. In the crystallization of a metal, for example, any suitably positioned atom on a crystal surface can be added to the growing new phase. For polymers, in contrast, only sequential crystallization is possible. It is thus not only that the molecules must disentangle themselves from the other random coils and straighten to a certain degree, the proper sequence (and, in addition, often the proper direction) must be established for the addition of a given repeating unit to a crystal. The chain statistics of a flexible molecule is thus at the root of the special properties of polymers. It should also be remembered that many biological and inorganic compounds are macromolecules and even many small molecules are flexible! Neglecting the special effects based on molecular shape and flexibility severely limits not only the understanding of the behavior of polymeric and flexible materials.

The second part of the discussion of chain statistics will establish, in addition, the link between structure and mobility. While molecular structure has a length scale based on atomic dimensions of less than one nanometer (10 9 m), it will be shown that the atomic time scale is in the picosecond range (10 12 s). Both of these units must be recognized to make a link between the macroscopic and microscopic material properties.

1.3.5 Steric Hindrance and Rotational Isomers

In this section, additional details on the conformations of macromolecules are given to further the understanding of flexible molecules. Details on computer simulation are presented to evaluate mobility. Increasingly stiff molecules are described to make the link to rigid macromolecules.

To begin the discussion, one can look at small molecules with two halves that can be rotated against each other about a sigma bond, a bond with cylindrical symmetry. The drawings of Fig. 1.35 are to scale, and, assuming the atomic sizes are fixed by the van der Waals radii, result in a solid atom model (see Fig. 4.23). The basic fact is that on overlapping, the electrons of different parts of the molecules show a strong repulsion at short distances. Strong steric hindrance to rotation for the dichloroethane, shown on the right, is obvious from the black area of overlap. Ethane, on the left, in contrast, shows no such steric hindrance. An analysis of the energy as a function of angle of rotation, U( ), shows that even in ethane the rotation around the C C-bond is not free, i.e., there is an intrinsic barrier to rotation. This barrier is particularly large for the sigma bonds of the small atoms of the second row of the periodic table (B, C, N, and O). The overall energy of rotation, thus, is written as:

U( ) = Usteric + Uintrinsic rotation.

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As was discussed with Fig. 1.35, there should only be minimal steric hindrance between the hydrogen atoms on rotation. The three conformations of the molecule at the staggered positions at the angles of 0, 120, and 240° are energetically most stable. These are called the rotational or conformational isomers, but are indistinguishable because of the identical hydrogens. Energetically least favorable are the eclipsed positions at 60, 180, and 300°. For molecules with larger side-groups, the

Fig. 1.37

steric hindrance must be added. The energetics of rotation about the center bond of n-butane is drawn in Fig. 1.37 as an example representing the shortest modelcompound for polyethylene. The potential-energy curve still shows three low-energy conformations (rotational isomers), but these are now of different energy and distinguishable. The trans conformation is lowest in energy. Compared to the thermal energy (3/2)RT, about 4 kJ mol 1 at room temperature, the barriers to rotation can occasionally be surmounted, but most of the time one expects the molecule to take one of the three staggered conformations.

The number of possible shapes a polyethylene molecule can now be calculated easily by assuming that each bond must be in one of these three rotational positions. Then the statistics of the molecule is based on the conformational entropy per rotatable bond of this type, k ln 3, where k is Boltzmann’s constant (see Sect. 2.2.4 and Fig. A.5.4). Per mole of bonds k is replaced by R, the gas constant. For a mol-

ecule with n rotatable bonds, Sconformation = n × 9.2 J K 1 mol 1. The calculation at the bottom of Fig. 1.37 reveals that the different conformations possible for a macro-

molecule with 20,000 flexible backbone bonds and three rotational isomers per bond are more than astronomical. This large amount of disorder (entropy) is at the root of the special behavior of polymers. It permits melting of the macromolecule without breaking into small parts, and is the reason for entropy elasticity (viscoelasticity and

40 1 Atoms, Small, and Large Molecules

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rubber elasticity), and also causes the special defects in polymer crystals, and the many thermodynamic nonequilibrium states (see Chaps. 5 7).

1.3.6. Monte Carlo Simulations

To understand the overall conformation of a large molecule, also called the macroconformation, one can use computers and simulate the molecule with a Monte Carlo program. Figure 1.38 illustrates the Monte Carlo technique [9]. First, one chooses a lattice for the generation of the macromolecule. Figure 1.39 illustrates a twodimensional square lattice with coordination number (CN) four. Then one places the first atom somewhere on the lattice. This leaves four choices for the placement of the second atom (wN = 4). Choosing one of these four possibilities leads to three as the number of choices for the third and most subsequent atoms, as shown in the figure.

Fig. 1.38

When in the course of generating the molecule one chooses an already occupied position, the simulation is discarded. This permits the representation of the excluded volume problem mentioned in Sect. 1.3.4. In the computation of the probability in Fig. 1.39 this termination of chains because of excluded volume is recognized by the factor FN. Results on 1123 surviving chains, generated on a tetrahedral lattice to simulate the conformations of a carbon backbone, are also given in Fig. 1.39 [10]. These data were also shown in Fig. 1.33 with the restriction and refinements of the random flight.

Many refinements and extensions of the Monte Carlo simulations are possible. More detailed information about the influence of the energetics of the trans and gauche conformations on the chain statistics can be obtained. In this case, one biases the probability of the choices of successive positions in favor of the trans conforma-

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Fig. 1.39

tion, using the known energy minima of the conformational isomers. It is also possible to simulate kinetics of changes of given conformation by considering the possible choices for rotation about the bonds of a molecule. Again, the energetics must be incorporated in the model. Other examples of Monte Carlo simulations include diffusion about obstacles, motion of the different molecules in mixtures, and during the crystallization process [11]. Limits are the growing computational needs for large molecules, difficulties in linking the computation steps with actual time scales, and the poor representation of a crystal by the chosen rigid lattices.

1.3.7 Molecular Mechanics Calculations

Molecular mechanics calculations are designed to find the energy of a given conformation of a molecule, U( ). Once the equation for the potential energy has been established, it can be minimized to identify the most stable macroconformation. For polyethylene, this conformation of minimum energy would obviously be the all- trans, extended-chain arrangement, i.e., it represents a zig-zag chain. In the calculation, all interactions important for the molecule must be included. Figure 1.40 is a typical example with parameters for the simulation by molecular mechanics of hydrocarbon molecules [13–15]. The equation in the figure considers steric hindrance, rotation, and valence angle distortion. Omitted is the less important change in bond length. Such simplifications are often included in the calculation to save computation time.

The term Usteric in Fig. 1.40 includes the van der Waals interactions between all atom pairs, including those involved in rotation and bending. The intrinsic hindrance

to rotation is given by Urot, as before for ethane. It is obvious that rotation about any one C C-bond must overcome the barriers to rotation which is similar to the ones for

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calculations are of importance for the prediction of crystal structures. The molecules in crystals frequently have the conformation of lowest energy. The results of Fig. 1.42 can be used to interpret the rotations of the phenylene groups in different crystal polymorphs (see Chap. 5). Although low-energy reaction-paths from one conformation to another can be calculated with molecular mechanics, actual dynamics often takes other paths.

Fig. 1.42

1.3.8 Molecular Dynamics Simulations

The ultimate simulation of a molecule involves full molecular dynamics (MD), as can be derived by step-wise integration of the Hamiltonian, H, in Fig. 1.43. The first term of H represents the kinetic energy, the others, the potential energies. The sums go over all atoms, i, of the system. Illustrated are p chains of 100 atoms each. The representation of the potential energy is similar to the just completed molecularmechanics treatment. The non-bonded (NB) van der Waals interaction is included only for the atoms not involved in the adjacent two, three, and four-body interactions (bond stretching, bending, and rotation, respectively), i.e., the short-range intramolecular NB interactions are included in V2 to V4. Furthermore, the Hamiltonian can be used to include static chains in the simulation. These contribute only to the NB interactions. If the full outermost ring of chains is immobilized, the simulation represents constant volume. Without static chains, it is close-to constant (zero) pressure. A set of parameters for paraffins and polyethylene are given in Fig. 1.44. These parameters are for the united-atom simplification that treats a CH2-group as a single unit of atomic mass 14.03 Da. This simplification saves computation time and does not affect many of the problems. The equations available for the MD simulation