Equivalent chain model

Muller investigated the relationship between the molecular orientation and the mechanical and optical anisotropies in fibrous as well as crosslinked rubberlike materials. The mechanical and optical properties of a crosslinked rubber network under deformation were theoretically analyzed with the so-called freely jointed equivalent chain model by Kuhn and Griin" and Treloar." Mechanical anisotropy in preoriented crystalline polymers was discussed by Raumann" " and by Ward et on the basis of the infinitesimal anisotropic elastic theory. Ward also expressed the... 

The different chain conformations observed in different polymorphic forms of a polymer are generally associated to nearly equivalent minima in the conformational energy maps, calculated for isolated chain models [2, 3],... 

Yamakawa and co-workers have formulated a discrete helical wormlike chain model that is mechanically equivalent to that described above for twisting and bending/79111 117) However, their approach to determining the dynamics is very different. They do not utilize the mean local cylindrical symmetry to factorize the terms in r(t) into products of correlation functions for twisting, bending, and internal motions, as in Eq. (4.24). Instead, they... 

Considering the large variation of / for the poly[2]catenand 51b, it is expected that little correlation will exist between the spatial orientation of neighboring monomer segments and that it will represent the closest synthetic equivalent of the freely jointed chain model [63]. In this model, a real polymer chain is replaced by an equivalent chain consisting of N rectilinear segments of length Z, the spatial orientations of which are mutually independent (Scheme 24) [63]. 

The theorem of renormalizability can be read in two ways. With the renormalized theory taken to be fixed, it implies the existence of a one-parameter class, parameterized by , of bare theories, all equivalent to the given renormalized theory and thus equivalent to each other. This aspect is related to universality a whole class of microscopic models yields the same scaling functions. In the next chapter we will use this aspect to get rid of the technical complications of the discrete chain model. We can however also interpret the theorem as establishing the existence of a one-parameter class of renormalized theories, all equivalent to a given bare theory. This class is parameterized by the length scale r or the scaling parameter... 

In this chapter we first show that the continuous chain model is renor-malizable by taking the naive continuous chain limit of the theorem of renor-malizability. We then argue that we can construct renormalization schemes for the continuous or the discrete chain models, equivalent in the sense that they yield the same renormalized theory (Sect. 12.1). In Sect. 12.2 we estab-... 

In Chap. 7 we have shown that the bare discrete chain or continuous chain models are naively equivalent only close to the 0-point. We thus might wonder whether the equivalence of the two models, shown above to one loop order, can hold generally. We thus have to show that starting from these different bare theories we nevertheless can construct identical renormalized theories. We consider the renormalized continuous chain limit (RCL), used in the theorem of renormalizability. 

Thus asymptotically the NCL and the RCL, if applied to the bare functions for d < 4, are identical. In other words, for fo < 1 the set of equivalent theories for i —> 0 reaches the 0-region, where the continuous chain model and the discrete chain model coincide. The same renormalized theory can therefore be constructed from both models. 

Concerning the first question we note that the result of any renormalization scheme based on the continuous chain model via a finite renormalization can be mapped on the renormalized theory derived from the discrete chain model, and vice versa. After renormalization the models are completely equivalent. 

We thus take the following attitude. For technical reasons we calculate the renormalized theory starting from the continuous chain model. By equivalence of the bare theories for fo < 1 we know that we can derive the same theory from the discrete chain model. Since the renormalized theory in the way we construct it should show no singularity at u, we can use it for u > u. This region however can be interpreted only in terms of the discrete chain model. 

This assumption is equivalent to considering the polymer molecule as a Gaussian chain. For a Gaussian chain the probability of the two ends colliding in three-dimensional space is proportional to its length to the power -3/2. For the Kuhn (or freely-jointed chain) model the same assumption maybe taken for sufficiently long chains [60]. For linear polymers in good solvents, no similar simple assumption can be adopted. To study cyclization one has to resort to more sophisticated mathematical treatments (see, e.g. [61]). 

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. 

In this structure the residues were arranged in three equivalent polypeptide chains, each wound in a left-handed helix with three residues per turn and a pitch of 9.5 A. While also a three-chain model, this structure differed from that of Pauling and Corey in that each chain coiled around its own axis, rather than around a common axis. The three chains were held together by hydrogen bonds as follows in each turn of each helix two of the three peptide nitrogens were hydrogen bonded to one of the carbonyl oxygens of each of the other two chains (see Fig. 9). This model had a... 

In the second type of semiflexible polymer molecule, rigid units are interspersed with flexible ones. Some examples of molecules of this kind are given in Chapter 11. The freely jointed chain model might be a suitable model for such semiflexible polymers. If a persistently flexible molecule and a freely jointed molecule are characterized by the same values of L and Xp (or, equivalently, of bx and Nk), then the gross statistical measures of the coil dimensions of the two such isolated chains will be the same, despite the differences in the type of flexibility. 

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