Phantom polymer

Equations 22.3-22.14 represent the simplest formulation of filled phantom polymer networks. Clearly, specific features of the fractal filler structures of carbon black, etc., are totally neglected. However, the model uses chain variables R(i) directly. It assumes the chains are Gaussian the cross-links and filler particles are placed in position randomly and instantaneously and are thereafter permanent. Additionally, constraints arising from entanglements and packing effects can be introduced using the mean field approach of harmonic tube constraints [15]. 

This means that the isopropyl group stabilises the secondary ion sufficiently (compared to but-l-ene) for hydride transfer reactions to be suppressed, at least at low temperature. At about -120 °C a high polymer of structure (IV) is formed. This is a true phantom polymer, since there exists no corresponding monomer. Evidently, at the very low temperature the propagation reaction which would lead to structure (III) becomes much slower than the isomerization reaction ... 

Vinyl cyclohexane and other similar monomers give analogous phantom polymers under similar conditions, and the reaction has been explored in great detail by Kennedy and his collaborators [9f, 74]. The occurrence of such relatively fast isomerizations makes it appear very likely that in these polymerizations the propagating species is a cation. 

The examples of self-similar functions considered above fall into an especially simple category all are functions of the generalized time or group parameter and functionals (either linear or nonlinear) of the initial condition. However, there are many self-similar physical properties and mathematical objects that depend on additional, unsealed variables. For example, the probability density for the distribution of end-to-end distances of a linear, ideal (phantom) polymer chain is given by the expression... 

Other monomers which were found to respond positively to anionic polymerization were the 2-furfurylidene ketones, which gave phantom polymers following a propagation mechanism involving the isomerization of the active species before each addition step [4d, 4e]. 

These structures occur in branched chains of low DP. Particularly interesting is unit 24 which predominates in polymers prepared at —20 °C with trifluoroacetic acid these products are in fact phantom polymers108 in which everything seems to have gone the wrong way. 

So far, we have not introduced a specific model of the polymer network chains. This problem can be rigorously solved for cross-linked polymer networks consisting of phantom chains [13], or even in the more general case of filled networks where the chains interact, additionally, with spherical hard filler particles [15]. 

The performance of such a probe can be demonstrated for polymer studies by heating a phantom sample up to 460 °C. The sample was made from a PTFE plug, two pieces of a PVC hose and a silicon rubber hose. The deformation and the... 

Fig. 2.1.7 Effect of high temperatures on a composite polymer phantom made from a polyethylene plug (white), a PVC hose (transparent) and a silicon rubber hose (yellow) shown at room temperature (left) and after...

The phantom network model contains a crucial deficiency, well known to its originators, but necessary for simplifying the mathematical analysis. The model takes no direct account of the impenetrability of polymer chains, not is the impossibility of two polymer segments occupying a common volume provided for in this model. Different views have been presented to remedy these deficiencies, no consensus has been reached on models which are both physically realistic and mathematically tractable. 

If network unfolding takes place so that distances between junctions connecting the ends of a polymer chain deform less than that of a phantom network, molecular dimensions change less than by any other of the models considered. This is easily seen from the data presented for a not equal to zero. 

O is the stress per unit unstrained area, G the shear modulus, A the deformation ratio, p the density of the dry network. iJ>2 volume fraction of polymer present in the network, V the volume at formation. A=1 for affine behaviour (expected) and 1-2/f for phantom behaviour(1,3). is the molar mass for the perfect network, essentially the molar mass of a chain of v bonds, the number which can form the smallest loop (5-7) see Figure 2. is equal to the... 

At the same time, the above mentioned chain-like structure leads to the fact that different parts of polymer molecules fluctuating in space cannot go through each other without chain rupture. For the system of non-phantom closed chains, this fact means that only those space conformations that can be transformed continuously into one another are available (see Fig. 1). The adequate mathematical language for description of those physical effects is elaborated in the mathematical discipline called topology. That is why we also call the effects connected with chain uncrossability the topological constraints. 

The real polymer chain may be usefully approximated for some purposes by an equivalent freely jointed chain. It is obviously possible to find a randomly jointed model which will have the same end-to-end distance as a real macromolecule with given molecular weight. In fact, there will be an infinite number of such equivalent chains. There is, however, only one equivalent random chain which will lii this requirement and the additional stipulation that the real and phantom chains also have the same contour length. 

Fig. 24. a Schematic view of the craze tip showing a molecule about to be drawn into two different fibrils, b Phantom fibril of undeformed polymers, diameter Dq, is ultimately drawn into a fibril of diameter D = The entangled chains in the isotropic network, represented... 

One can estimate the fraction of the original network chains that remain unbroken after fibrillation. Consider, for example, the cylinder of imdeformed polymer of diameter Do shown in Fig. 24b. After fibril formation this phantom fibril cylinder is drawn into a fibril of diameter D = The entangled chains in the... 

One can estimate the strand survival fraction q theoretically. The relevant parameters are the strand end-to-end distance d and the phmtom fibril diameter D, the diameter of starting polymer glass that is drawn into the final fibril, determined from smaU-angle electron (or X-ray) scattering. One can show that q is only a function of the ratio, Dg/d. The best method of calculation treats the strand as a Gaussian coil, with rms end-to-end distance d, and computes the probability that if one places one end at random in a cylindrical phantom fibril, the other end will be also inside For typical D s for polystyrene crazes (of the order of 14-20 nm at room temperature) the predicted values of q lie between 0.5 and 0.6, in satisfactory agreement with the experimental estimates (which include effects of the tie-fibrils not included in the theoretical method ° ). 

From the microscopic picture for the craze growth it seems clear that one important microscopic variable must be the mean number n of entangled strands within each fibril which survive the geometrically necessary strand loss associated with the interface formation. If the number of such strands is zero, the fibril will fail, since the polymer fluid which flows from the active zone into the fibril has no strain hardening capability and will not be able to support the relatively high tensile stresses necessary to propagate the interface. To obtain n one first estimates n, the total number of strands in the undeformed phantom fibril from which a craze fibril is drawn and which is given by ... 

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