Effective random link

Figure 8.6 shows a set of experimental data conforming to the prediction of equation (8.17). From the fit of the data to the equation the value of v, called the activation volume, can be obtained. In the equations above, the value of V is the activation volume per mole of sites and must be divided by Avogadro s number, Aa, to obtain the activation volume per site. The value found depends on the type of stress applied, i.e. whether it is shear, tensile or compressive, and is generally 2-10 times the volume of the effective random link deduced from data on dilute solutions. This suggests that yield involves the co-operative movement of a relatively large number of chain segments. 

The most interesting situation exists when the intrinsic optical anisotropy (%— sc2) of the random link is negative, i.e. when the polarizability (a2) perpendicular to the chain direction is the larger one. In this case the form effect, which always works in the same sense, counteracts the effect caused by the (negative) intrinsic anisotropy of the random links. An important feature of the form effect is that its contribution to... 

As has been pointed out (63), this is a rather artificial model and, moreover, its application is quite unnecessary. In fact, (a> can be calculated from the refractive index increment (dnjdc), as has extensively been done in the field of light scattering. This procedure is applicable also to the form birefringence effect of coil molecules, as the mean excess polarizability of a coil molecule as a whole is not influenced by the form effect. It is still built up additively of the mean excess polarizabilities of the random links. This reasoning is justified by the low density of links within a coil. In fact, if the coil is replaced by an equivalent ellipsoid consisting of an isotropic material of a refractive index not very much different from that of the solvent, its mean excess polarizability is equal to that of a sphere of equal volume [cf. also Bullough (145)]. 

Other difficulties are owing to the influence of the solvent. With stiff and bulky chains the so-called micro-form-effect becomes of importance, when the refractive index increment differs considerably from zero (7). In this case the random link approximately acts like a cylinder of length A and with a refractive index different from that of the solvent. Another effect occurs in good solvents which consist of anisotropic molecules. These molecules become oriented along the polymer chain, considerably contributing to its anisotropy [Frisman, Dadivanyan and Dyuzhev (752)]. In this way, the determination of the eigen anisotropy of weekly anisotropic polymer chains becomes rather doubtful. 

For the nonlinear step growth case above, eiTg, the crosslink density must be related to p. A relevant model, based on calculating the probabilities of finite chains being formed, has been published For the reaction of A -1- 2B2 (e.g., tetra-functional amine -b difunctional epoxy), A4 is considered to be an effective cross-linking site if three or more of its arms lead out to the infinite network. The probability of finding an effective crosslink is related to one minus the probability of a randomly chosen A leading to the start of a finite chain, which in turn is related to the extent of reaction. Application of this procedure to the system of Fig. 15 has been presented in detail The more complicated reaction of a tetrafunctional amine with a trifunctional epoxy was also considered. ... 

A number of workers have treated non-Gaussian networks theoretically in terms of this finite extensibility problem. The surprising conclusion is that the effect on simple statistical theory is not as severe as might be expected. Even for chains as short as 5 statistical random links at strains of up to 0.25, the equilibrium rubbery modulus is increased by no more than 20-30 percent (typical epoxy elastomers rupture at much lower strains). Indeed, hterature reports of highly crosslinked epoxy M, calculated from equilibrium rubbery moduh are consistently reasonable, apparently confirming this mild finite extmsibiUty effect. 

When the polymer is drawn, two effects contribute to the orientation of the random links ... 

The variation of birefringence with draw ratio for a set of uniaxially drawn samples of a certain polymer is found to be consistent with the simplest version of the affine rubber model when the draw ratio is less than 3.5. If the birefringence is 7.65 x 10 for draw ratio 3.0, calculate its value for a sample of draw ratio 1.5. If the birefringence for a very highly oriented sample is 0.045, what is the effective number of random links per chain ... 

The simplicity of (2.25) is to be contrasted with the complexity of the exact P(R n) for realistic models of flexible chains for all R (and for small n). When dealing with the complicated problems of nonideal polymer solutions, etc., it is therefore customary to replace the real polymer chain by the so-called Kuhn effective random flight chain. An effective chain is one with N (in general different from n) links of size A5 such that N lS.s = L and (R ) is as given by (2.29). This substitution of a real chain by its equivalent chain is often a necessity so that we may separate errors in principle from errors arising from a poor mathematical approximation to the exact P(R n) when dealing with problems which are not exactly soluble. This equivalent chain therefore provides us with reasonable approximations to the properties of real polymer chains, provided the physical properties of interest do not depend heavily upon those chain configurations with i > L or upon chain properties over small distances for which the real chain is stiff. 

In spite of this, the failure envelopes are normal. Thus Figure 1 shows the envelopes for several Solithane 113-300 compositions (lO), These envelopes can be fitted by the inverse Langevin approximation (ll) to the stress-strain curve, and from the curve fit both the number of effective chains per cm and the niimber of equivalent random links N can be determined (l2). The fit for two compositions is shown in Figure l8 and the results of such an analysis (13) are given in Table II. It can be seen that the chain concentration is almost constant but N increases, i.e, the chains effectively become stiffer as the concentration of prepolymer is increased. 05iis is -ttie only elastomer system we are aware of in which such a change can be effected at constant Ye ... 

Broad changes in the composition of the basic elastomer family do not influence the crosslink density or effective chain concentration but rather change the nature of the crosslink site so that the number of the equivalent random links per chain is changed. 

Figure 4 Effect of draw ratio A on the orientation factor / curve p = pseudoaffine deformation curve n = network deformation (the number of random links per network chain is N — 5.7)...

Radiation Effects. Polytetrafluoroethylene is attacked by radiation. In the absence of oxygen, stable secondary radicals are produced. An increase in stiffness in material irradiated in vacuum indicates cross-linking (84). Degradation is due to random scission of the chain the relative stabiUty of the radicals in vacuum protects the materials from rapid deterioration. Reactions take place in air or oxygen and accelerated scission and rapid degradation occur. 

Two types of well defined branched polymers are acessible anionically star-shaped polymers and comb-like polymers87 88). Such macromolecules are used to investigate the effect of branching on the properties, 4n solution as well as in the the bulk. Starshaped macromolecules contain a known number of identical chains which are linked at one end to a central nodule. The size of the latter should be small with respect to the overall molecular dimensions. Comb-like polymers comprise a linear backbone of given length fitted with a known number of randomly distributed branches of well defined size. They are similar to graft copolymers, except that backbone and branches are of identical chemical nature and do not exhibit repulsions. 

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]. 

Under these circumstances the probability that a given structural unit is cross-linked is not entirely independent of the status of other units in the same primary molecule. If an abnormally large fraction of some of the units of a given primary molecule are found to be cross-linked, the likelihood that it was formed toward the end of the polymerization process is enhanced hence the probability that one of its other units is cross-linked will be greater than the over-all p for the system. Calculations indicate that the magnitude of the non-randomness is not excessive below about 70 percent conversion. For most purposes its effect probably may be ignored without serious error, thus obviating a more elaborate theory which would take into account non-randomness of this nature. 

At present it is not clear whether extreme agitation, delirium, hyperthermia, and rhabdomyolysis are effects of cocaine that occur independently and at random among cocaine users, or whether these features are linked by common toxicologic and pathologic processes.20 Ruttenber and colleagues20 have examined excited delirium deaths in a population-based registry of all cocaine-related deaths in Dade County. This study has led to clear description of the cocaine delirium syndrome, its pattern of occurrence in cocaine users over time, and has identified a number of important risk factors for the syndrome. 

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