Polyethylene elastic constants

Investigations of some of the elastic constants of uniaxially stretched polymers were made on nylon 66 and polyethylene terephthalate by... 

Table 4.3 Temperature dependences of the theoretically determined elastic moduli and some elastic constants of ideal single-crystal polyethylene, collected by Karasawa etal. (1991) from various sources...

There have been several theoretical determinations of elastic constants of ideally perfect crystalline polymers. Of these we consider here only that of Karasawa et al. (1991) for orthorhombic polyethylene. References to some other similar studies can also be foimd there. 

The perfect-crystal model of Karasawa et al. considers a basic force-field approach for the analysis of crystal properties. The model contains covalent-bonded interactions along the polymer chain as well as non-bonded van der Waals interactions between molecules and Coulombic interactions when relevant, all with appropriate temperature dependences. Table 4.3 lists some of the temperature-dependent elastic moduli and some elastic constants cy of ideal polyethylene, determined by Karasawa et al. (1991), which will be of interest to us in later chapters. These are shown also in Fig. 4.1. Of these Ec (= l/ssj) gives directly the main-chain Young s modulus of polyethylene. Also listed is the transverse shear elastic constant c e, which can be considered to be a good measure of... 

Fig. 4.1 Dependences of the Young s moduli i ( (parallel to the c-axis), E, (parallel to the h-axis), and (parallel to the a-axis), and the axial-transverse elastic constants C55 and the transverse shear modulus Ju (= cgg) of a perfect orthorhombic polyethylene crystal on temperature as calculated by Karasawa et al. (1991).

Our model repre.sentation of the oriented fiber is given in Fig. 3. The nodes in the figure represent the elementary repetition units of the polymer chains, i.e. methyl units for polyethylene. For very long chains, each node is made to correspond to more than one repetition unit (Termonia et al., 1985). The nodes are joined in the x- and z-directions by secondary bonds having an elastic constant Ki. These bonds account for the intermolecular vdW forces in polyethylene or hydrogen bonds in nylon. Only nearest-neighbor interactions are considered. In the y-direction, stronger forces with elastic constant K account for the primary bonds, i.e. C-C bonds in polyethylene. 

Strain dependence should also be considered when comparing acoustically measured elastic constants with statically measured values. As an example, for polyethylene at room temperature, the modulus is independent of strain up to a strain of about 10 (7). Beyond this point, the modulus decreases as the strain increases. Typically, acoustic measurements are made in the strain range 10 where the moduli are strain-independent, but static measin-ements... 

In a comprehensive study at room temperature, Hadley, Pinnock and Ward [20] determined the five independent elastic constants for oriented filaments of polyethylene terephthalate, nylon 6 6, low and high-density polyethylene and polypropylene. The orientation was determined in terms of draw ratio and optical birefringence. Subsequent studies indicated that it would have been appropriate to record not only the overall orientation, as derived from birefringence, but also the crystal orientation, obtainable from X-ray measurements. The results are summarized in Table 7.1 and Figures 7.9-7.13 (see Section 7.5 for discussion of the aggregate theory predictions). 

There have been no similar attempts to determine comprehensive sets of elastic constants for oriented fibres or monofilaments. Kawabata [21] devised an apparatus that used a linear differential transformer to measure diametral changes of 0.05 iim in single fibres of diameter 5 fxm subjected to transverse compression. Equation (7.2) above was then used to calculate the transverse modulus El = l/ ii. Results were obtained for poly(p-phenylene terephthalamide) (Kevlar) and high-modulus polyethylene (Tekmilon) fibres. Values of E were in the range 2.31 -2.59 GPa for Kevlar and a value of 1 -2 GPa was found for Tekmilon. 

More recently, Wilczynski, Ward and Hine [24] have proposed an inverse calculation method where the elastic constants of a fibre can be estimated from fibre resin composite and the elastic constants of the resin. The method was confirmed by measurements on polyethylene/epoxy and carbon fibre/epoxy resin composites. It has been applied [25] to the determination of the elastic constants of a new organic fibre, poly 2,6-dimidazo[4,5-6 4 5 -e]pyridinylene l,4(2,5-dihydroxy)pheny-lene (PIPD). This fibre is a lyotropic liquid crystalline fibre with a very high Young s modulus of 285 GPa and a much higher tensile strength (5.21 GPa) and compressive strength (5OOMPa) than other polyaramid fibres such as Kevlar. 

Table 7.4 Elastic constants of ultrahigh-modulus polyethylene...

In recent research, Al-Hussein et al. [25] prepared an oriented low-density polyethylene with a parallel lamellar stack morphology where the c axes of the crystalline lamellae were parallel to the lamellar plane normals. For this structure, explicit equations can be obtained for the elastic constants in terms of the crystalline volume fraction and the elastic constants of the crystalline lamellae (cji, C33, C44, etc.) and the amorphous layer (cji, cfj, c, etc.). 

Tadoroko et al. provide a new method for the calculation of the elasticity of an isolated helical chain and the distribution of strain energy to the internal co-ordinates under conditions that the rotational angle per monomer is constant. For the calculation of three-dimensional elastic constants, the space group symmetry of the unit cell may be used to reduce the memory size required for polymers such as poly(vinyl alcohol) which have unsymmetrical repeat units. The constraints imposed upon tie molecules in semi-crystalline polyethylene are considered to restrict them to three conformations, the deforma-... 

Although it Is the Young s modulus In the chain direction that Is of the greatest Importance for engineering applications. It must be remembered that a large number of elastic constants are needed to fully describe the elastic behaviour of a crystal [4]. Polydiacetylene single crystals usually possess monocllnic symmetry and therefore have 13 elastic constants. There have been several attempts to calculate the 9 elastic constants for orthorhombic polyethylene [42] but as yet there has been no similar calculation for a... 

Figure 6.10 The variation in phase angle with distance along a polyethylene monofilament for transmission of sound waves at 3000 Hz. (Reproduced from Chan, O.K., Chen, F.C., Choy, C.L, et al. (1978) The elastic constants of extruded polypropylene and polyethylene terephthalate ]. Phys. D, 11, 617. Copyright (1978) Institute of Physics.)...

Finally, it is of interest to compare the theoretical values for a uniaxially oriented sheet (calculated by averaging the stiffness values using the Voigt averaging scheme) with those obtained for a die-drawn rod and a sheet made by hot compaction of high modulus polyethylene fibres (Table 8.4). It can be seen that although, as expected, these materials have not reached full axial orientation so that the experimental values of C33 are much less than the theoretical value, the patterns of anisotropy are very similar, and some of the values for the other elastic constants are surprisingly close. 

At low temperatures, as discussed above (see Figure 8.17), a more conventional pattern of mechanical anisotropy is observed for low-density polyethylene. At the same time, the polar diagram of the modulus changes ([41], Figure 8.26) and S44 is no longer very much greater than the other elastic constants. These results are thus consistent with the aggregate model. 

The aggregate model predicts only that the elastic constants should lie between the Reuss and Voigt average values. In polyethylene terephthalate, it is clear that the experimental compliances lie approximately midway between the two bounds. For cold-drawn fibres, it has been shown that this median condition applies almost exactly [87]. 

All nine independent elastic constants have been determined for one-way drawn oriented polyethylene terephthalate sheet. The sheet was prepared by drawing isotropic sheet at constant width. It has been shown that there is then both a high degree of chain orientation in the draw direction and that the (100) crystal planes (which mainly reflect preferential orientation of the terephthalate residues in the chain) are preferentially oriented in the plane of the sheet. This type of orientation has been termed uniplanar axial. From the viewpoint of elastic anisotropy, the sheet possesses three orthogonal planes of symmetry and can be described as possessing orthorhombic symmetry. 

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. 

The high axial elastic modulus of polyethylene and polyamide 6 is due to the fact that these polymers have a preferred conformation that is fully extended, i.e. all-trans. The elastic deformation is caused by the deformation of bond angles and by bond stretching, both showing high elastic constants. Isotactic polypropylene and polyoxymethylene crystallize in helical conformations and therefore exhibit a maximum stiffness which is only 20% of the maximum stiffness of the all-trans polymers. The elastic deformation of a helical chain involves, in addition to the deformation of bond angles and bond stretching, deformation by torsion about the G bonds. The latter... 

Applications of the matrix method to the polyethylene crystal were described by Kitagawa (1968), Shiro and Miyazawa (1971). For orthorhombic polyethylene with the space group Pnam-Dl , the elastic constants Cn,C22, C33,C23,C3i and are for the Ag species, and C44, C55 and Qe are for the Big,B2g and B3, spedes, respectively. Accordingly, the treatment of elastic constants is significantly simplified by the use of internal symmetry-coordinate vector, internal symmetry-strain vector and external strain vector for each synunetry spedes. 

The elastic constants of the orthorhombic polyethylene crystal were calculated by Kitagawa (1968) as... 

---