Random-link model

As discussed further in the following section, it can be shown that the statistical distribution of end-to-end distances for any real chain reduces to the Gaussian form if the number of rotatable links is sufficiently large. By suitably choosing n and / for the freely jointed random-link model, both rms and the fully extended length can be made equal to the corresponding values for the real chain. These values define the equivalent freely jointed random chain. For example, if it is assumed that in a real polyethylene chain (i) the bonds are fixed at the tetrahedral angle and (ii) there is free... 

The following section is concerned with more realistic chains, but it is shown that the random-link model is still applicable for polymers in molten and certain other states. 

When polymer chains are cross-linked to form a non-flowing rubber, a molecular network is obtained. It is shown in section 3.3.4 that the freely jointed random-link model of polymer ehains is appKeable to rubbers provided that the equivalent random link is eorreetly ehosen. In considering the network the following simplifying assumptions will therefore be made, leading to the simplest form of the theory. 

To explain the difference between the experimental results and theory, Doherty et al. (4J have given an empirical and a theoretical hypothesis. The theoretical hypothesis concerns the question of the meaning to be attached to the concept of the "equivalent random link" in the statistical theory of the randomly-jointed chain. According to Doherty et al., the assumption that the optical properties of the chain are describable by a randomly jointed model, using the same value of n, as for the description of stress has no strictly logical foundation. 

This oversimplified random network model proved to be rather useful for understanding water fluxes and proton transport properties of PEMs in fuel cells. - - - It helped rationalize the percolation transition in proton conductivity upon water uptake as a continuous reorganization of the cluster network due to swelling and merging of individual clusters and the emergence of new necks linking them. ... 

It will be obvious that the treatment of the present section can also be applied to the subchain model (77). As is well-known, this model where every junction point of subchains is assumed to interact with the surroundings, seems to provide a more realistic description of the dynamic behaviour of chain molecules than the simple model used in the proceeding paragraphs, viz. the elastic dumb-bell model where only the end-points of the chain are assumed to interact with the surrounding. One of the important assumptions of the subchain model is that every subchain should contain enough random links for a statistical treatment. From this it becomes evident that the derivations given above for a single chain, can immediately be applied to any individual subchain. In particular, those tensor components which were characterized by an asterisk, will hold for the individual subchains as well. 

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)]. 

Fortunately, later calculations reviewed by Birshtein, Volkenstein, Gotlib and Ptitsyn (750) showed that the deviation of ZQ from Z is largest just for the free rotating chain, when compared with a great number of more realistic models. The conclusion was drawn by these authors that, for practical calculation purposes, Zg and Z0 can be equated. This means that, in principle, the optical anisotropy (oq — oq) of the random link, as used in previous sections, can be calculated with the aid of the experimentally accessible s (see below), when the bond polarizabilities are taken from the literature (757). In this way (oq — oq) is calculated as the anisotropy of a stretched piece of chain, containing s monomer units. This latter number is obtained by combining the first... 

From the point of view that the statistically coiled model chain is built up of rigid rods (random links), it seems that eq. (5.10) must be truncated, as eq. (5.11a). 

We now consider the random copolymer model in the presence of solvent— that is, for a copolymer volume fraction p0 < 1. We are not aware of previous work on this model in the literature, but will briefly discuss below the link to models of homopolymer/copolymer mixtures [57]. The excess free energy (86) then depends on two moment densities, rather than just one as in all previous examples. For simplicity, we restrict ourselves to the case of a neutral solvent that does not in itself induce phase separation this corresponds to X = 0, making the excess free energy... 

The literature data on the tortuosity factor r show a large spread, with values ranging from 1.5 to 11. Model predictions lead to values of 1/e s (8), of 2 (parallel-path pore model)(9), of 3 (parallel-cross-linked pore model)(IQ), or 4 as recently calculated by Beeckman and Froment (11) for a random pore model. Therefore, it was decided to determine r experimentally through the measurement of the effective diffusivity by means of a dynamic gas chromatographic technique using a column of 163.5 cm length,... 

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 ... 

We consider a random effects model to allow variability among studies, generalizing the standard fixed effects model. A logistic link can be specified as... 

One of the best-known of such schemes is the AFFINE deformation model for rubbers. The rubber is considered to be a network of flexible chains, and the macroscopic strain is imagined to be transmitted to the network such that lines joining the network junction points rotate and translate exactly as lines joining corresponding points marked on the bulk material. If we assume that the flexible chains consist of rotatable segments called random links , and that some statistical model can describe the configurational situation, it is then possible to obtain explicit expressions which relate the segmental orientation to the macroscopic deformation. 

Next we consider the statistical distribution of the end-to-end vector of the random flight model. Let (H, N) be the probability distribution function that the end-to-end vector of the chain consisting of N links is R. Given the conformational distribution for W( r ), [Pg.11]

The two major pore models that have been used extensively over the years for practical purposes are the parallel-pore model proposed by Wheeler in 1955 [5, 9] and the random-pore model proposed by Wakao and Smith in 1962 [34]. Among the more recent advanced models are the parallel cross-linked pore model [35] and pore-network models [36, 37]. 

Before concluding the discussion on the equivalent random link it may be mentioned that other model systems, such as the freely rotating tetrahedral chain model which was used to give eqn (3.6) may be corresponded to an rjc model. It can be shown that ... 

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