Chains equivalent random

Now, returning to Table 2, we find the value of c to be always larger than unity. Therefore, if we are concerned not only with the unperturbed dimensions but also with the thermodynamic interactions, the usual definition of the equivalent random chain, i. e. c = 1, can no longer be recommended. On the other hand, a rough equality may be found between (L yjn and cmin, except for the one case of the four-choice chain on the simple cubic lattice. The equality of c to (L2 0/w corresponds to the Kuhn condition, 153),... 

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. 

We shall first derive the average properties of an ideal polymer chain that is infinitely long, possesses negligibly small volume, and has freely jointed links. Next we shall examine the influence of fixed bond angles between adjacent links. The concept of the statistically equivalent random chain will then be introduced to rationalize the validity of using these model chains to represent the behavior of real polymer chains. Finally, the equation of state for a single polymer chain will be discussed. This equation is the starting point for equations (6-32) and (6-33)... 

It is known that for a poly-cz s-isoprene chain, each monomer unit is 0.46 nm long and r2 = 0.162 nm2. What is the statistically equivalent random chain for this macromolecule ... 

A fully extended chain, without distortion of bond angles or deformation of bonds, is represented in Figure 9(b). In this conformation, the value of the end-to-end distance is nZp, where Zp is the length of the bond vector projected on the chain axis. If the equivalent random chain is now required to have the same end-to-end distance in full extension and also the same mean-square displacement length as the real chain, then equations (33) and (34) result. The maximum extension ratio of unperturbed macromolecules may be defined as the ratio of the fully extended length nZp to the root-... 

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. 

In the derivation of eqn. (7) it was assumed that n (number of equivalent random links) is the same for all chains. For our samples (B2 system, Mw/Mn=1.45), this assumption is definitely not correct. Therefore, it is desirable to obtain birefringence results on networks prepared from monodisperse polymer (in that n is constant), before the validity of n itself is questioned. 

Thus each statistically equivalent random segment is equal to about three carbon-carbon bonds in the real polymethylene chain. 

In Edwards s theory (1966),12 the chains are represented by continuous curves. The interaction between chains is transformed so as to be replaced by an external random potential whose effect is mathematically equivalent. The chain behaviour in this external potential is studied in an approximate manner, and subsequently, the result is averaged over the set of potentials. 

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... 

Calculate the root-mean-square length of a polyethylene chain of M = 250000 g moP assuming that the equivalent random link corresponds to 18.5 C—C bonds. 

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. 

In the first two of these equations the subscript on the refractive index indicates that it refers to the mean refractive index and also distinguishes it from n in the third equation, which denotes the number of random links per chain. If the equivalent random link is taken as the structural unit. No is now the number of random links per unit volume and Q o and Aa are the mean value and the anisotropy of the polarisability of a random link. 

Let n and I be the num-ber and length of the equivalent random links and let the fully extended length of both chains be L. Then... 

The representation of a flexible polymer chain by an equivalent random flight model chain which is subsequently taken to be a continuous space curve leads directly to the introduction of a functional integral representation of polymer chains. 

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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