Uniaxially Oriented Polymers

The scattering intensity I(q), having cylindrical symmetry and therefore being independent of a, is also a function of only and . This means that as long as the z axis coincides with the symmetry axis, the x axis can be chosen arbitrarily. We therefore choose the x axis such that a is equal to zero. The dot product qr is then equal to R cos I + Z. Recognizing that the volume element dr is equal to R dR JO dZ, we can write equation (4.14) as 

Since g(R, Z) is independent of, the integration with respect to l can be performed immediately. Using the identity 

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Any extended part of a linear polymer molecule exhibits a strong anisotropy of many properties since its atoms and atomic groups are oriented and the macromolecule is actually a one-dimensional crystal . The parallel packing of these parts during the formation of a uniaxially oriented polymer substance imparts the anisotropie properties of a single molecule to the whole polymeric material. 

An approach simpler than the "general" method has been used by Bower and coworkers [52,53] and has proved to be useful to determine the order parameters of uniaxially oriented polymers. This approach, called "cylindrical" method, requires a smaller number of spectra that are easily measurable, but it assumes that the Raman tensor associated with the vibrational mode under study is cylindrical. In such case, at — a2 — a so that the depolarization ratio R so of an... 

MECHANICAL PROPERTIES OF UNIAXIALLY ORIENTED POLYMERS (FIBRES)... 

Figure 11.6. Schematic illustrations of brittle fracture, (a) Idealized limiting case of perfectly uniaxially oriented polymer chains (horizontal lines), with a fracture surface (thick vertical line) resulting from the scission of the chain backbone bonds crossing these chains and perpendicular to them. This limit is approached, but not reached, in fracture transverse to the direction of orientation of highly oriented fibers, (b) Isotropic amorphous polymer with a typical random coil type of chain structure. Much fewer bonds cross the fracture surface (thick vertical line), and therefore much fewer bonds have to break, than for the brittle fracture of a polymer whose chains are perfectly aligned and perpendicular to the fracture surface, (c) Illustration of a defect, such as a tiny dust particle (shown as a filled circle), incorporated into the specimen during fabrication, which can act as a stress concentrator facilitating brittle fracture.

As seen in Table 1, the decrease in permeability can be directly attributed to a dramatic reduction in the effective diffusion coefficient, while there is a much smaller effect on the apparent solubility. A similar dependence of the solubility and diffusion coefficients on the draw ratio has been observed in other uniaxially oriented polymers (35-37). Because the glass transition and density of the polystyrene samples were found independent of the draw ratio, they concluded that the reduction in diffusivity was due to anisotropic redistribution of the free volume during drawing. Using an expansion coefficient related to draw ratio, the polystyrene data were successfully correlated using the Cohen-Turnbull free volume theory. However, the situation was found to be more complex for PVC (i ) ... 

A uniaxially oriented polymer sample transmits 21% or 90% of infrared radiation corresponding to a particular absorption for radiation polarised parallel or perpendicular to the draw direction, respectively. If the transition dipole is parallel to the chain axis, calculate the value of P2(cos0)) for the chains. 

The emission and absorption axes usually either coincide or are separated by a small angle. If they coincide and both the exciting and the analysed fluorescent light are polarised parallel to the draw direction in a uniaxially oriented polymer, then a = = 9, where 6p is the angle between the absorption axis and the draw direction. The intensity observed will thus be proportional to (cos" Op. It is easy to show that, if the incident... 

The orientation-distribution function N 6) for a particular set of crystal planes in a uniaxially oriented polymer is found from WAXS measurements to be given approximately by N 6) = A(l + COS0). Calculate (P2(cos )) for this distribution. 

A particular sample of a uniaxially oriented polymer is composed of structural units each of which is transversely isotropic with respect to an axis Oz within the unit. The value of (cos 0) is 0.65, where 6 is the angle between the Oz axis of a typical unit and the draw direction. There are 4.2 x 10 structural units per m in the polymer and the polarisabilities of an individual unit for light polarised parallel and perpendicular to Oz are 2.6 x 10 and 1.9 x 10 F m, respectively. Calculate the polarisabilities of the sample for light polarised parallel and perpendicular to the draw direction and hence the birefringence of the sample. 

A highly uniaxially oriented polymer sample in the form of a thin film has mean optical polarisabilities of 2.3 x 10 and 1.7 x 10 F m per struetural unit for light polarised parallel and perpendicular to the draw direction, respectively it transmits 50% of infrared radiation at a particular wavelength 2 when the radiation is polarised parallel to the draw direction and transmits almost all... 

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... 

Fig. 6. Distribution of transition moments of a particular vibration in a uniaxially oriented polymer with respect to the draw direction (see t (t)...

Here at is the difference between the yield point in tension and in compression in the direction of orientation. Thus, according to equation 18, eight parameters are required to describe completely the yield stress of a uniaxially oriented polymer. 

The optical anisotropy, as characterized by the difference between the absorption of IR light polarized in the directions parallel and perpendicular to the reference axis (i.e., the direction of applied strain), is known as the IR linear dichroism of the system. For a uniaxially oriented polymer system [10, 28-30], the dichroic difference, A/4(v) = y4 (v) - Ax v), is proportional to the average orientation, i.e., the second moment of the orientation distribution function, of transition dipoles (or electric-dipole transition moments) associated with the molecular vibration occurring at frequency v. If the average orientation of the transition dipoles absorbing light at frequency is in the direction parallel to the applied strain, the dichroic difference AA takes a positive value on the other hand, the IR dichroism becomes negative if the transition dipoles are perpendicularly oriented. 

To further illustrate how extended ensembles can be designed to conduct MD simulations under various macroscopic constraints, we discuss here the NTLxPyycr ensemble. NTL yyOzz is an appropriate statistical ensemble for the simulation of uniaxial tension experiments on solid polymers [12] or relaxation experiments in uniaxially oriented polymer melts [13]. This ensemble is illustrated in Fig. 2. The quantities that are kept constant during a molecular simulation in this ensemble are the following ... 

Anisotropic friction was observed for all samples discussed in this paper. The anisotropy is directly correlated with the directionality of the polymer molecules at the surface of the specimens. The directionality was either confirmed experimentally by AFM (for uniaxially oriented polymers and the transcrystallized PEO) or reported in the literature in numerous diffraction studies for extended-chain polymer single crystals. The existence of regularly packed folds at the surface of lamellar polymer crystals is still a matter of discussions. 

DI Bower, ELV Lewis, IM Ward. Relationships among stress-induced Raman shifts for isotropic and uniaxially oriented polymers. Polymer 36 3473-3477, 1995. 

Figure 4.13 illustrates the use of polarised absorption spectroscopy to measure orientation for a uniaxially oriented polymer. For a single molecule, the absorbance is proportional to the square cosine of the angle between the vibration transition dipole moment and the IR electric field vector E). Thus if 0 = 90°, the absorbance is zero (note that the measurement is sensitive to the alignment of the vibrational dipole moment rather than the chain axis itself). The dichroic ratio R is calculated by ratioing the absorbance measured with E parallel toXs with that measured perpendicular toXs R = A A ). For an assembly... 

On this model, the birefringence Att of a uniaxially oriented polymer is given by... 

In the dynamic transmission mode, information about two-dimensional dynamic dichroic differences can be attained [46]. It is useful to study uniaxially oriented polymers such as strongly elongated fibres. However, the orientation of most polymer films should be considered biaxial, even with uniaxially drawn films, which still show slight biaxial orientation at the surface [52]. This fact suggests that the two-dimensional dynamic dichroic data obtained from transmission mode is not necessarily adequate for evaluating most (biaxially oriented) polymer films. 

For uniaxially oriented polymer samples, the chain direction (z-axis) of crystallites is parallel to the orientation direction. Suppose the momentum-transfer vector is in the xy plane but makes an an e

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Given limited applicability of earlier equations, Brown and co-workers (23, 24) have developed the following equation for uniaxially oriented polymers ... 

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