Elastic constants, of polymers

Fig. 7. Relative change of the lattice parameter in TS-6 along the chain direction as a function of conversion X. Indices m and p refer to monomer and polymer, respectively. The full curve is derived from a continuous dilatometric measurement the dashed curve is calculated according to Ref. by minimizing the elastic energy assuming a ratio Ep/E = 5.0 for the elastic constants of polymer and monomer, respectively. Open circles are X-ray data from Ref. crosses represent data from optical absorption studies...

Next we note that there are two physieally different sources of temperature and pressure dependence of the elastic constants of polymers. One, in common with that exhibited by all inorganic crystals, arises from anharmonic effects in the interatomic or intermolecular interactions. The second is due to the temperature-assisted reversible shear and volumetric relaxations under stress that are particularly prominent in glassy polymers or in the amorphous components of semi-crystalline polymers. The latter are characterized by dynamic relaxation spectra incorporating specific features for different polymers that play a central role in their linear viscoelastic response, which we discuss in more detail in Chapter 5. 

The different experimental methods for sound wave propagation and for measuring the mechanical response or elastic constants of polymers are summarized below with an attempt to give an idea of the different time scales involved. 

In addition to the adiabatic or isothermal difference, acoustically determined elastic constants of polymers differ from static values because polymer moduli are frequency-dependent. The deformation produced by a given stress depends on how long the stress is applied. During the short period of a sound wave, not as much strain occurs as in a typical static measurement, and the acoustic modulus is higher than the static modulus. This effect is small for the bulk modulus (on the order of 20%), but can be significant for the shear and Young s modulus (a factor of 10 or more) (5,6). 

Marsh, D. 2001. Elastic constants of polymer-grafted lipid membranes. Biophys. J. 81 2154. 

Brillouin spectroscopy enables the elastic constants of polymers to be determined at frequencies of several gigahertz, i.e. three orders of magnitude higher than those pertaining to ultrasonic measurements, which are known as hypersonic frequencies. 

Tashiro, K., Kobayashi, M. and Tadokoro, H. (1978) Calculation of three-dimensional elastic constants of polymer crystals. 2. Application to orthorhombic polyethylene and poly(vinyl alcohol). Macromolecules, 11, 914. 

Table 8 presents a survey of the basic elastic constants of a series of polymer fibres and the relation with the various kinds of interchain bonds. As shown by this table, the interchain forces not only determine the elastic shear modulus gy but also the creep rate of the fibre. 

The ratio of elastic constants Ku, calculated for the S-effect according to the equation (4) appeared to be (Kn (polymer XIV)/Kn (polymer XIII)) x 1 100 and (Ku (polymer XVI)/Kn (polymer XV)) x 1 36. Yet, as we have just indicated, taking into account molecular masses of the LC polymers and reducing k, values for various polymers to equal values of DP one may come to substantially different values for ratios of constants presented. It is necessary to note that up to date no quantitative data on the determination of elastic constants of LC polymers has been published (excluding the preliminary results on Leslie viscosity coefficients for LC comb-like polymer127)). Thus, one of the important tasks today is the investigation of elastic and visco-elastic properties of LC polymers and their quantitative description. 

The sets of equations are solved by the assumption of periodic waves and, by expansion in powers of the wave number, a relation is found for the limiting case of long waves so that the elements of the dynamical matrix elastic constants of the continuum. It is also possible to derive the Raman frequencies from the lattice dynamics analysis but this does not seem to have been done for polymer crystals, though they have been derived for example, for NaCl and for diamond. 

There have been many efforts for combining the atomistic and continuum levels, as mentioned in Sect. 1. Recently, Santos et al. [11] proposed an atomistic-continuum model. In this model, the three-dimensional system is composed of a matrix, described as a continuum and an inclusion, embedded in the continuum, where the inclusion is described by an atomistic model. The model is validated for homogeneous materials (an fee argon crystal and an amorphous polymer). Yang et al. [96] have applied the atomistic-continuum model to the plastic deformation of Bisphenol-A polycarbonate where an inclusion deforms plastically in an elastic medium under uniaxial extension and pure shear. Here the atomistic-continuum model is validated for a heterogeneous material and elastic constant of semi crystalline poly( trimethylene terephthalate) (PTT) is predicted. 

For liquid crystalline polymers, the elastic constants are determined not only by the chemical composition but also by the degree of polymerization, i.e., the length of the molecular chain. One main aim of this section is to address the effects of molecular chain length on the elastic constants of liquid crystalline polymers. Figure 6.1 shows the three typical deformations of nematic liquid crystalline polymers. The length and flexibility of liquid crystalline polymers make the elastic constants of liquid crystalline polymers quite different from those of monomer liquid crystals. 

The above qualitative description illustrates that the molecular length has an important effect on the elastic constants of liquid crystalline polymers. 

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. 

Several experiments were carried out to investigate the elastic constants of nematic polymers. They were essentially in agreement with the theory. But the available data are insufficient for checking theoretical predictions. Systematic and careful experiments are required to investigate the relationship of elastic constants to the flexibility and molecular length. We will introduce some measurement techniques and experimental data for elastic constants. 

The elastic constants of liquid crystalline polymers can be measured in terms of the Frederiks transitions under the presence of a magnetic or electric field. Raleigh light scattering is also a method for measuring the elastic constants. Those techniques successfully applied to small molecular mass liquid crystals may not be applicable to liquid crystalline polymers. This is why very few experimental data of elastic constants are available for liquid crystalline polymers. 

Three kinds of Frederiks transitions can be used to measure the three elastic constants of nematic polymers. The magnetic threshold fields Hci are functions of Ku(i = 1,2,3) respectively,... 

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