Spherical domains

At the beginning of this section we enumerated four ways in which actual polymer molecules deviate from the model for perfectly flexible chains. The three sources of deviation which we have discussed so far all lead to the prediction of larger coil dimensions than would be the case for perfect flexibility. The fourth source of discrepancy, solvent interaction, can have either an expansion or a contraction effect on the coil dimensions. To see how this comes about, we consider enclosing the spherical domain occupied by the polymer molecule by a hypothetical boundary as indicated by the broken line in Fig. 1.9. Only a portion of this domain is actually occupied by chain segments, and the remaining sites are occupied by solvent molecules which we have assumed to be totally indifferent as far as coil dimensions are concerned. The region enclosed by this hypothetical boundary may be viewed as a solution, an we next consider the tendency of solvent molecules to cross in or out of the domain of the polymer molecule. 

Figure 1.9 The spherical domain of a polymer molecule either expanding by imbibing solvent or contracting by excluding solvent.

We saw in Chap. 1 that the random coil is characterized by a spherical domain for which the radius of gyration is a convenient size measure. As a tentative approach to extending the excluded volume concept to random coils, therefore, we write for the volume of the coil domain (subscript d) = (4/3) n r, and combining this result with Eq. (8.90), we obtain... 

Morphology of the anionically synthesized triblock copolymers of polyfp-methyl-styrene) and PDMS and their derivatives obtained by the selective chlorination of the hard segments were investigated by TEM 146). Samples with low PDMS content (12%) showed spherical domains of PDMS in a poly(p-methylstyrene) matrix. Samples with nearly equimolar composition showed a continuous lamellar morphology. In both cases the domain structure was very fine, indicating sharp interfaces. Domain sizes were estimated to be of the order of 50-300 A. 

This is especially clear in the case of hard spheres each of which excludes a volume (47r/3)dg which is eight times its net volume. Thus, the portions of the spherical domains at distances between de/2 and de of the centers of two nearby molecules may overlap, with the result that the volume actually excluded by these two molecules is less than 2(47r/3)d. ... 

Figure 4.10 AX4, AX3E, and AX2E2 molecules (a) tangent sphere models or domain models with spherical domains B is a bonding pair and E is a lone pair and (b) conventional bond line structures.

Modeling the initial structure by spherical domains in a bcc-lattice1, the theoretical intensity along the ellipsoidal ridge as a function of the angle y1 between fiber axis and the direction of the radial beam is... 

Now again, a state of inhomogeneity in polymers, so especially interesting in films and interfaces, occur when discontinuities are built into the main valence chains and networks. Block polymers are the classic embodiments of this. Many periodic distances separating domains in such alternating or rhymthic copolymers have been reported. These indicate existence of phases in laminar domains and, in other cases, of spherical domains.(51) Cases are shown experimentally for styrene/isoprene copolymers and also for styrene/butadiene.(52,53,54)... 

A new set of theoretical equations based solely on the properties of networks and their interaction was derived by Yeo et al. [21]. Assuming spherical domains, a crosslink density of... 

Figure 13. Comparison between experimental scattering (O) and calculated curve (-) assuming spherical domains. (Reproduced with permission from Ref. 13. Copyright 1986 Butterworth Scientific Ltd.)...

Several models of polyelectrolyte thermodynamics have been proposed and have been recently reviewed [84], Historically, two general approaches [84,85] have been used to model polyelectrolyte thermodynamics, spherical and chain models. In the former approach the coiled polyions are treated as spherical domains with charge density distributed continuously within the sphere... 

Figure 13 shows the TEM micrograph of a star block containing 22% PpClSt. The morphology shows phase-separated domains of PpClSt and PIB. The PpClSt formed spherical domains with diffused interphase and are irregularly dispersed in the PIB matrix. The formation of irregular domains with diffused interphase is attributed to the presence of diblock and PpClSt contaminants. 

Segalman RA, Yokoyama H, Kramer EJ (2001) Graphoepitaxy of spherical domain block copolymer films. Adv Mater 13 1152... 

The morphology of a polyethylene blend (a homopolymer prepared from ethylene is a blend of species with different molar mass) after crystallisation is dependent on the blend morphology of the molten system before crystallisation and on the relative tendencies for the different molecular species to crystallise at different temperatures. The latter may lead to phase separation (segregation) of low molar mass species at a relatively fine scale within spherulites this is typical of linear polyethylene. Highly branched polyethylene may show segregation on a larger scale, so-called cellulation. Phase separation in the melt results in spherical domain structures on a large scale. 

Example 3.2 Consider a large number of uniformly charged solid particles initially kept in a spherical barrier of radius R with a symmetric density distribution. When the barrier is suddenly removed, the particles start to emerge from that spherical domain. The viscous drag in the gas is assumed to be negligible. Find the ratio of the force due to dipole to that due to electrostatic repulsion and show that for dilute suspensions, the dipole effect due to self-field is negligible. Also discuss the spreading of the solid particles in this simple symmetric system. 

It is a small step from van der Waals, electron-domain models of the C—H bonds of, e.g., biphenyl, cyclohexane, or methane (Figs. 1—3), to molecular models in which to a first, and useful, approximation each valence-shell electron-pair is represented by a spherical, van der Waals-like domain 7h (Non-spherical domains may be useful for describing, e.g., lone pairs about atoms with large atomic cores, -electrons, and the electron-pairs of multiple bonds vide infra.)... 

Sidgwick s discussion raises an important question What are the effective sizes and shapes of atoms in molecules From the viewpoint of the electride ion model of electronic structure, Sigdwick s circles for the fluoride ions in the first column of Fig. 15 are the wrong shape, if nearly the right overall size. In the electride-ion model a fluoride ion is composed of (approximately) spherical domains, but is not itself spherical, in the field of a cation, Fig. 16. Fig. 17 illustrates, correspondingly, the implied suggestion that, on the assumption that non-bonded interactions are not limiting, the covalency limits of an atom will be determined by the radius of the atom s core and by the effective radii, not of the overall van der Waals envelopes of the coordinated ions but, rather, by the radii of the individual, shared electron-pairs. 

Spherical-domain models of three-center bonds in localized-molecular-orbital models of a nonclassical carbonium ion, B4CI4, and TaeClfJ have been described 49,52) a drawing of a spherical-domain model of the methyl lithium tetramer, (LiCH, is shown in Fig. 31. Large, outer circles represent domains of electron-pairs of C—H bonds. Solid circles represent domains of Li+ ions. Shaded circles represent 4-center lithium-lithium-lithium-carbon bonds — i.e., electron-pair domains that touch, simultaneously, three lithium ions and the kernel of a carbon atom. The... 

Elastomers based on VDF and TFE-VDF-HFP consist of fine particles 16 to 30 nm in diameter in contrast to PTFE which has a rodlike microstructure in which the elementary fibrils are approximately 6 nm wide and the molecular chains are all extended.12 For example, the properties of a VDF-HFP elastomers such as their resilience and flexibility can be related to spherical domains with diameter approximately 25 nm that are interconnected.10 The diameter of these particles was found to be proportional to the molecular weight of the elastomer.12... 

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