Elastic constants oriented polymers

The high viscosity of the PMMA damps out orientational contributions so that the yqeo that is measured is thought to be =60-90% electronic. This has been ascertained by measuring the electric field induced second harmonic generation (EFISH) below the Tg of the polymer. From this can be obtained the microscopic elastic constant, which can in turn be used to estimate the magnitude of the two orientational contributions to yQEO- Details are provided else where (13,16). 

Assume that the degree of the ordering of liquid crystalline polymers is high and the orientational distribution function is simply Gaussian, Odijk (1986) developed the analytical formulae for elastic constants... 

In analog to the approach used by Odijk when dealing with elastic constants, Lee (1988) took the orientation distribution function approximately as Gaussian. When the system is highly ordered, the asymptotic expression can be deduced for viscosities of liquid crystalline polymers, e.g., the Miesowicz viscosities (in the unit of fj) are expressed by... 

Equations (A.7) also show that, in general, the prediction of a property that depends on a tensor of rank I will require knowledge of orientation averages of order /. The elastic constants of a material are fourth-rank tensor properties thus the prediction of their values for a drawn polymer involves the use of both second- and fourth-order averages, in the simplest case P ico O)) and P (x>s6)), and thus provides a more severe test of the models for the development of orientation. The elastic constants are considered in section 11.4. 

The prediction of the elastic constants of an oriented polymer provides a further, more stringent, test for the deformation models because these properties depend in a more complex way on the details of the distribution... 

The velocity of sound propagation in a material is simply related to the stiffness constants and density. It is a very convenient method in practice and gives simple and unambiguous results in the case of oriented glassy polymers, as in the work of Treloar on polymethylmethacrylate. From a single experiment, two of the five elastic constants may be determined. The principle of the method is shown in Fig. 1. 

Two distinct types of macroscopic theoretical model for the low strain mechanical behaviour of oriented solid polymers will be considered in this chapter. First, models which predict the changes in elastic constants with the development of orientation these will be referred to as orienting element models. Secondly, models which seek to explain the mechanical behaviour of both isotropic and oriented polymers in terms of a two phase material with separate components representing crystalline and amorphous fractions these we shall call composite structure models. 

The calculation of the elastic constants of the partially oriented polymer can be made in two ways. One can assume either uniform strain throughout the aggregate, which involves a summation of stiffness constants, or uniform stress, which implies a summation of compliance constants. In the former case the tractions across the boundaries of the unit do not satisfy the stress equilibrium conditions in the latter case there is discontinuity... 

The equations which predict the elastic constants of a partially oriented polymer involve orientation functions to define the orientation of the aggregate units. For example, the average extensional compliance S33 for a transversely isotropic aggregate of transversely isotropic structural units is given by S 33 = Su sin 0+S33Cos" 0-l-(2Si3-l-544) sin 0cos 9. 

In this equation Sn, S33, S13 and S44 are the elastic compliance constants for the completely oriented polymer the quantities imFd, cos" 0 and sin 0cos 0 define averages of trigonometrical functions involving the angle 0 between the symmetry axis of the unit and the draw direction, which is the symmetry axis of the aggregate. 

In a series of related publications, Hennig has reported the measurements of elastic constants for oriented polymers which are either amorphous or of low crystallinity. In his earliest work." Hennig showed that in polyvinyl chloride and polymethylmethacrylate the relationship 3/Eo = S33 + 2S11, where is the modulus of the isotropic polymer, holds to a good approximation. Results for the anisotropy of the linear compressibility y in polyvinyl chloride, polymethylmethacrylate, polystyrene and polycarbonate were also reported. In this experiment Hennig measured the linear compressibility parallel to the draw direction 7ii, and that in the plane perpendicular to the draw direction Vi. For uniaxially oriented polymers yn = 2Si3 + S33 = S i -I-Si2-I-S 3. It was... 

Interpretation of mechanical measurements in terms of molecular structure was until fairly recently confined essentially to identification of the temperatures of the major viscoelastic relaxations through extensional or torsional dynamic mechanical studies. Now, however, investigations of the elastic constants and their temperature dependence—allied with dynamic mechanical, creep and both wide and small angle X-ray diffraction— are yielding fairly detailed pictures of the interrelation of the crystalline and less well ordered regions of some oriented solid polymers. 

There appears to be no useful formalism capable of giving a rigorous and succinct description of the complicated behaviour observed in oriented polymers and therefore the formalism of classical elastic theory is retained for describing fime dependent finite strain deformation accepting the lack of rigour and investigating its utility. The individual compliance constants Siy are allowed to become functions of time and of stress and/or strain in any one experiment. 

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. 

The preconditions for the use of polymer liquid crystals in display applications are that they exhibit bulk optical properties dependent on the molecular orientation in the mesophase and that this orientation may be altered on application of an external field. In this chapter we shall be concerned with electric or optical fields only. The particular optical property, i.e. (a) the birefringence, (b) the dichroism or (c) the scattering power, defines the display construction in terms of the use of polarized (a and b) or non-polarized (b and c) light, whereas the ability to switch from one orientation to another depends on the anisotropic electric permittivity and the orientational elastic constants. The dynamics of the induced orientation will depend, additionally, on the viscosity constants of the material. 

The simplest model for dilute polymer solutions is to idealize the polymer molecule as an elastic dumbbell consisting of two beads connected by a Hookean spring immersed in a viscous fluid (Fig. 2.1). The spring has an elastic constant Hq. Each bead is associated with a frictional factor C and a negligible mass. If the instantaneous locations of the two beads in space are riand r2, respectively, then the end-to-end vector, R = ri — ri, describes the overall orientation and the internal conformation of the polymer molecule. The polymer-contributed stress tensor can be related to the second-order moment of R. There are two expressions namely the Kramers expression and the Giesekus expression, respectively (Bird et al. 1987b) ... 

All other terms are zero. In a liquid, there is only one elastic constant K. Note that most polymeric solids are isotropic, either because they are amorphous or because they are polycrystalline, with a random orientation of the crystallites (see Amorphous Polymers Semicrystalune Polymers). 

In practical applications of oriented polymers we are concerned with films or fibres, which reduces the number of independent elastic constants to nine or six, respectively. It is also convenient at this stage to limit the discussion to the compliance constants because these can be related directly to the readily measured engineering elastic constants, such as Young s moduli, shear moduli and Poisson s ratios. 

The technique of Brillouin spectroscopy (Section 5.3.3 above) has been applied to determine the elastic constants of oriented polymer fibres. Early studies of this nature were undertaken by Kruger and co-workers [26, 27] on oriented polycarbo-... 

In recent years there has been increased interest in the theoretical calculation of the elastic constants for ideal and fully oriented polymers based on knowledge of their crystal structures. This increased interest arises from the major developments in computational methods and from the success achieved in producing very highly oriented polymers with reasonably high stiffness. 

Can the elastic constants of the isotropic polymer be deduced from measurements on the most highly oriented sample ... 

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