Isotropic polymers

When the polymer is unoriented, so that its properties are isotropic, the radial distribution function depends only on the magnitude and not on the direction of r. The intensity function is then also isotropic and depends only on the magnitude of q. Equation (4.14) can now be written (cf. Section B.5) as 

Necessity of a Wide q Range. Since in the sine transform (4.22) the integration is in principle to be performed from 0 to infinity in q, and since the interference 

Compton Scattering. It is difficult to eliminate the Compton-modified x-ray scattering reliably by experimental means, because the wavelength shift of the modified from the coherent scattering is rather small, especially at small scattering angles. The preferred practice is to include all the Compton-modified scattering in the measured intensity and then subsequently to subtract the Compton-modified intensity calculated theoretically from it. 

Normalization. The practice of subtracting the theoretical Compton-modified intensity works only if the measured intensity is available on an absolute scale. Similarly, before the intensity function I(q) can be converted into the interference function i(q) by means of Equation (4.13), I(q) must be on an absolute scale. In principle, the intensity can of course be measured from the beginning in absolute units by means of an instrument calibrated for absolute intensity, as discussed in Section 2.7. The more usual practice, however, is to measure the intensity first in relative units and then to scale it by the normalization constant determined according to the following criterion. The normalization condition is satisfied when the following is obeyed in the limit of large q 

In other words, the coherent scattering intensity per atom, I(q)/N, should be equal to the average independent scattering per atom, at large q where the effect of 

Over recent years a completely different approach has been adopted by Hu, Day, Stanford and Young [66-69], who have shown that, through the synthesis of specially designed copolymers, it is possible to prepare isotropic polymers for which the deformation can be followed using Raman spectroscopy. They have demonstrated that such materials can be used for the study of polymer surface and interface deformation [68,69] and their work in this area is reviewed below. 

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In packed beds of particles possessing small pores, dilute aqueous solutions of hydroly2ed polyacrylamide will sometimes exhibit dilatant behavior iastead of the usual shear thinning behavior seen ia simple shear or Couette flow. In elongational flow, such as flow through porous sandstone, flow resistance can iacrease with flow rate due to iacreases ia elongational viscosity and normal stress differences. The iacrease ia normal stress differences with shear rate is typical of isotropic polymer solutions. Normal stress differences of anisotropic polymers, such as xanthan ia water, are shear rate iadependent (25,26). 

In practical application to common isotropic polymer materials the IDF frequently exhibits very broad distributions of domain thicknesses. At the same time fits of the IDF curve to the well-known models for the arrangement of domains (cf. Sect. 8.7) are not satisfactory, indicating that the existing nanostructure is more complex. In this case one may either tit a more complex model85 on the expense of significance, or one may switch to the study of anisotropic materials and display their nanostructure in a multidimensional representation, the multidimensional CDF. Complex domain topology is more clearly displayed in the CDF than in the IDF. The CDF method is presented in Sect. 8.5.5. 

Pre-crystalline order in PET has been investigated by a number of different groups and in the present volume the issue is reviewed under different perspectives in two other contributions [25,26]. Nodular structures measuring ca. 7.5 nm and ca 15 nm apart in PET quenched from the melt close to Tg were first described by Geil [27,28]. Such structures are essentially amorphous, albeit characterized by some degree of orientational order and by a significantly higher density as compared to the fully isotropic polymer. They are indeed qualitatively compatible with the bundle model we propose [29]. Apparently... 

Orientation effects are strongly coupled to nonlinear behavior, discussed in Section V, and the stress-strain response discussed in Chapter 5, Orientation makes an initially isotropic polymer anisotropic so that five or nine modulus/compliance values arc required to describe the linear response instead of two, as discussed in Chapter 2. For an initially anisotropic polymer the various modulus/compliance components can be altered by the orientation. It may not be necessary to know all components for an... 

Indeed, it has been observed that the onset of yielding of isotropic polymers is approximately constant, 0.02< [<0.025, which implies that 0.04shear yield strain, the plastic shear deformation of the domain satisfies a plastic shear law. For temperatures below the glass transition temperature, the continuous chain model enables the calculation of the tensile curve of a polymer fibre up to about 10% strain [6]. Figure 7 shows the observed stress-strain curves of PpPTA fibres with different moduli compared to the calculated curves. 

When used in load-bearing applications, isotropic polymers may also fail because of low modulus. The moduli that must be considered in the design of functional polymers are Young s modulus E, shear modulus G, and bulk modulus K. Poisson s ratio P should also be considered. 

Here a is the elastic stress which arises from the change in the (dynamic) free energy in the macroscopic flow, while o(V) and a(S) are the viscous stresses produced by the polymer-solvent friction and the solvent-solvent friction, respectively. In concentrated isotropic polymer solutions, the elastic stress overwhelms the viscous stresses, so the latter are often neglected. However, it should be noticed that the viscous stresses may become significant in more dilute solutions as well as in nematic solutions where the elastic stress diminishes. 

In comparison to a conventional polymer and l.c. mentioned above, we will now discuss the PVT behavior of a l.c. side chain polymer, which has linked mesogenic moieties as side chains, and is very similar to the previous monomer. The experimental results are shown in Fig. 5. It is obvious, that the phase behavior of the l.c. polymer differs from that of a 1-l.c. and amorphous polymer. At high temperature we observe a transformation from the isotropic polymer melt into the l.c. phase, indicated by the jump in the V(T) curve. At low temperatures no crystallisation is observed but the bend in the curves signifies a glass transition. Obviously the phase behaviour is determined by the combination of l.c. and polymer properties. 

Fig. 4.44 Phase diagram for aqueous solutions of Pluronic P104 (PEOi7PP05,PFX)27) (Noolandi et al. 1996). Notation iso, isotropic (polymer poor) solution cubic, cubic phase hex[, hexagonal phase lam, lamellar phase, hex2, inverse hexagonal phase, cubicj, inverse cubic phase, iso2, isotropic (polymer rich) solution. The solid and dashed lines are calculated from the continuum and lattice versions of self-consistent field theory respectively.

The values of permeability coefficients for He, O2, N2, CO2, and CH4 in a variety of dense (isotropic) polymer membranes and the overall selectivities (ideal separation factors) of these membranes to the gas pairs He/N2,02/N2, and CO2/CH4 at 35°C have been tabulated in numerous reviews (Koros and Heliums, 1989 Koros, Fleming, and Jordan et al., 1988 Koros, Coleman, and Walker, 1992). Moreover, several useful predictive methods exist to allow estimation of gas permeation through polymers, based on their structural repeat units. The values of the permeability coefficients for a given gas in different polymers can vary by several orders of magnitude, depending on the nature of the gas. Thevalues oftheoverall selectivities vary by much less. Particularly noteworthy is the fact that the selectivity decreases with increasing permeability. This is the well-known inverse selectivity/permeability relationship of polymer membranes, which complicates the development of effective membranes for gas separations. 

The group contributions to the molar polarisation (for isotropic polymers) are given in Table 11.1. Application of Eqs. (11.7) and (11.8) permits the calculation of the dielectric... 

TABLE 11.1 Group contributions to molar dielectric polarisation (P) in isotropic polymers (cmVmol)... 

The moduli of polymers cover a wider range than those of other materials (from 105 N/m2 for rubber to 1010 N/m2 = 10 GPa for rigid polymers), which is one of the reasons why polymers are so versatile in application. Absolute stiffness and strength of polymers are much lower than those of metals, but on the basis of equal weight polymers compare favourably due to their much lower density. The specific moduli, defined as the moduli divided by the density, of isotropic polymers are of the order of one tenth of those of the stronger metals. The hyper-strong and hyper-stiff polymeric fibres such as fully extended polyethylene, stretched poly-(p-phenylene terephthalic amide) and carbon,... 

If an isotropic polymer is subjected to an imposed external stress it undergoes a structural rearrangement called orientation. In amorphous polymers this is simply a rearrangement of the randomly coiled chain molecules (molecular orientation). In crystalline polymers the phenomenon is more complex. Crystallites may be reoriented or even completely rearranged and oriented recrystallisation may be induced by the stresses applied. The rearrangements in the crystalline material may be read from the X-ray diffraction patterns. 

The theoretical treatment of the mechanical properties of fibres is, as a matter of fact, more complicated than that of isotropic polymers. Instead of two elastic parameters, e.g. the tensile modulus and the Poisson ratio, we now need five, because of the anisotropy. 

If an isotropic polymer is subjected to an imposed external stress at a suitable temperature (usually just above the glass-transition temperature) it undergoes a structural rearrangement called orientation. 

In the absence of a field the orientation of the non-linear species in an isotropic polymer will be random. An applied field will tend to orient the non-linear species, but this is opposed by their thermal motion. The macroscopic non-linearity of poled films is determined by the orientation of the nonlinear species, which can be calculated if the ground state dipole and the principal component of the first hyperpolarisability are assumed to be parallel, a reasonable approximation for axially elongated molecules. The probability distribution function of the molecular orientation can be written as ... 

The dynamic mechanical behaviour of ultra high modulus polypropylene is shown in Fig. 32. As in LPE, the modulus is temperature dependent, rising to a value of 25 GPa at —140 °C, which is rather more than half (he value of 42 GPa obtained from crystal measurements. Although the a and y relaxations of the isotropic polymer can be seen in the highly drawn material, the -relaxation is undetectable. On annealing, the modulus at high temperatures is markedly reduced, and a P-relaxa-... 

The intention of this brief survey has been to demonstrate that besides the "classical" aspects of isotropic polymer solutions and the amorphous or partially crystalline state of polymers, a broad variety of anisotropic structures exist, which can be induced by definable primary structures of the macromolecules. Rigid rod-like macromolecules give rise to nematic or smectic organization, while amphiphilic monomer units or amphiphilic and incompatible chain segments cause ordered micellar-like aggregation in solution or bulk. The outstanding features of these systems are determined by their super-molecular structure rather than by the chemistry of the macromolecules. The anisotropic phase structures or ordered incompatible microphases offer new properties and aspects for application. 

The molecular weight dependence of the critical concentration for the establishment of uniformly anisotropic solutions of PBG is shown in Table I for various solvents that we have examined. Volume fractions ij>) of polymer quoted in this compilation correspond to the B-point in the nomenclature of Robinson (28-29). The B-point differs from the A-point, a lower concentration where the anisotropic phase just begins to form and is in equilibrium with isotropic polymer solution. 

The raw data from a tensile test (Fig. 11-20) are obtained in terms of force and corresponding elongation for a test specimen of given dimensions. The area under such a force-elongation curve can be equated to the impact strength of an isotropic polymer specimen if the tensile test is performed at impact speeds. Show that this area is proportional to the work necessary to rupture the sample. 

Figure 14.3 (a) Isotropic polymer chain, (b) Chain extended and partially oriented by the action of local shear, (c) The conformation corresponding to (b) is maintained in glass (or crystal) due to the fact that new intermolecular forces have been established. Forces of an entropic nature cause the chain to adopt its original conformation when the sample is heated. 

In spite of the relative simplicity of the Tresca criterion, conditions for shear yielding in isotropic polymers are best summarized by the von Mises criterion (11),... 

For macroscopically isotropic polymers, the Tresca and von Mises yield criteria take very simple analytical forms when expressed in terms of the principal stresses cji, form surfaces in the principal stress space. The shear yield surface for the pressure-dependent von Mises criterion [Eqs (14.10) and (14.12)] is a tapering cylinder centered on the applied pressure increases. The shear yield surface of the pressure-dependent Tresca criterion [Eqs (14.8) and (14.12)] is a hexagonal pyramid. To determine which of the two criteria is the most appropriate for a particular polymer it is necessary to determine the yield behavior of the polymer under different states of stress. This is done by working in plane stress (ct3 = 0) and obtaining yield stresses for simple uniaxial tension and compression, pure shear (di = —CT2), and biaxial tension (cti, 0-2 > 0). Figure 14.9 shows the experimental results for glassy polystyrene (13), where the... 

No detectable signals can be obtained in the brittle fracture of isotropic polymers (even crystalline ones), but once plastic flow is induced (eg by small amounts of orientation above Tg before testing below Tg), radicals are readQy obtained during deformation. 

For isotropic polymer fluids, stress relaxation upon cessation of flow reflects the relaxation time scales during the previous flow. As the relaxation spectrum is determined by the microstructure such measurements can be used to probe the effect of shear on the structure. This turns out to be a rather insensitive technique in polymer fluids because of changes which already occur during the relaxation (161. 

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