The entropy of a single chain

The expression just derived indicates the reason for the high extensibility of lightly cross-linked rubbers and serves to introduce the important concept of a mean square length, but yields no information on the probability of a chain having a particular end-to-end length. This latter problem was first analysed mathematically by Kuhn [4] and by Guth and Mark [5]. 

It is seen that the most probable end-to-end distance, irrespective of direction, is not zero, but it is a function of b, i.e. of the length I of the links and the number n 

Another important quantity is the root mean square chain length (r ) /  

The entropy of the freely jointed chain s is proportional to the logarithm of the number of configurations Q so that 

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Assuming that the chains within the network obey Gaussian statistics, the entropy of a single chain whose ends are at 0,0,0 and x0>y0,z0 (see Figure 13-49) can be written (Equation 13-43) ... 

We are using a lower case s to distinguish the entropy of a single chain from that of the network as a whole, which we will designate S.) If the chain is stretched to x,y,z then its entropy becomes (Equation 13-44) ... 

This section seeks to make a quantitative evaluation of the relation between the elastic force and elongation. The calculation requires determining the total entropy of the elastomer network as a function of strain. The procedure is divided into two stages first, the calculation of the entropy of a single chain, and second, the change in entropy of a network as a function of strain. 

The change in entropy of the ideal network caused by the deformation is the sum of the changes for all chains. It is shown in section 3.3.4 that the entropy of a single chain is given by... 

This equation is substituted into equation 2.21 to derive the entropy of a single chain (equation 2.22) and the force required to restore a chain to its equilibrium dimensions (equation 2.21). 

Quantitative evaluation of the stress-strain characteristics of the rubber network then involves calculation of the configurational entropy of the whole assembly of chains as a function of the state of strain. This calculation is considered in two stages calculation of the entropy of a single chain and calculation of the change in entropy of a network of chains as a function of strain. 

The entropy of a single chain can be calculated from the number of conformations which are possible for the given position of the jimction points. The entropy of the whole system is obtained by summing over the network. 

The Gaussian theory considers the number of possible conformations of a chain having a specified end-to-end distance. A more accurate non-Gaussian statistical treatment of the random chain is based on the distribution of sin j, i.e. of the angle between the direction of a random link and of the end-to-end vector. From the probability of finding n links in the range AGj, ri2 in A 2 and so on, the entropy of a single chain is derived [2b] as... 

Let us consider the conformation of a single chain in the special case of a disordered state with no vacancy. Fixing Ao=0, Ai = 1 in the theory developed above, the number Wh (n) of paths that visit all lattice points (cells) without overlap, referred to as Hamiltonian path, is found [21]. Within the theoretical framework (2.146) described in the preceding sections, the entropy of Hamiltonian paths is estimated by... 

We neglect the effect of chain contraction at high concentrations. Although the chain contraction decreases the confinement entropy, its effect is usually small (Problem 4.9). Therefore, we also use the confinement entropy of a single chain in the low concentration limit, k aSR Jdf, for the semidilute solution. 

VVThen two chemically different polymers are mixed, the usual result is a two-phase polyblend. This is true also when the compositional moities are part of the same polymer chain such as, for instance, in a block polymer. The criterion for the formation of a single phase is a negative free energy of mixing, but this condition is rarely realized because the small entropy of mixing is usually insufficient to overcome the positive enthalpy of mixing. The incompatibility of polymers in blends has important effects on their physical properties, which may be desirable or not, depending on the contemplated application. 

As the length of a polymer chain increases, x becomes very large. At extreme chain lengths, the total entropy of fusion is almost entirely due to the internal component. Note that since the value of 50 J/ K mol in Equation (19) becomes insignificant, it is possible to estimate the entropy associated with a single monomeric unit using Equation (20). 

Since C is a constant independent of r, the variation in entropy associated with the deformation for a single chain is... 

The loss of entropy of a single polymer chain attached to a colloidal particle, assumed to be a plate, on compression is found from the Boltzmann relationship to be... 

In the last section I have shown that shding of chains yields to an additional entropy which favors a finite fraction of amorphous tails. This idea can be generailzed to folded chain conformations as sketched in (Fig. 2.4). Here, I will consider a crystal made of a single chain. 

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