Entropy of a single chain

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. 

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. 

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... 

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. 

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... 

This assumption makes it possible to caleulate the change in entropy on deformation of a single chain for a specified macroseopie strain. A summation over all ehains gives the macroscopic change in entropy of the rubber block, and the subsequent application of Eq. (10.3.3) yields the desired force or stress corresponding to the imposed strain. Let us illustrate this process for some idealized situations. The more general case will be eonsidered later. 

At low concentrations, adsorption is a single-chain phenomenon. The adsorption takes place when the enthalpy gain by the monomer-surface contact with respect to the monomer-solvent contact surpasses the loss of the conformational entropy. In a good solvent the adsorption is not likely unless there is a specific interaction between monomers and the surface. At high concentrations, however, interactions between monomers dominate the free energy of the solution. The adsorption takes place when the enthalpy gain by the mono-... 

---