Some results for polyethylene

Polyethylene  was modelled with a RIS model : $\vartheta =112^{\circ }$, $\varphi _{+}=120^{\circ }$, $\varphi _{t}=0^{\circ }$, $\varphi _{-}=-120^{\circ }$, $\epsilon _{1}(\varphi _{+})=\epsilon _{1}(\varphi _{-})=500\mathrm{cal/mol}$ and $l=1.54\mbox{\AA}$.

Using methods similar to those of section 4, the second and fourth moment of the end-to-end vector  were calculated. For chains of length up to 17monomers these quantities were calculated exactly by generating all configurations and averaging R² respectively R⁴. In Fig. (1.3) is plotted $\langle R^{4}\rangle /\langle R^{2}\rangle ^{2}$ as a function of the number of monomers in the chain. Notice that for large Nthe graph approaches a limit value of 5/3:

$$\lim_{N\rightarrow \infty } \frac{\langle R^{4}\rangle _{N}}{(\langle R^{2}\rangle _{N})^{2}}=\frac{5}{3} .$$ (1.45)

In Fig. (1.4) the radius of gyration is plotted as a function of N. For chains consisting of 1000 monomers or more the radius of gyration can be written like

 

R_g=bN^1/2. (1.46)

Notice that this behaviour sets in at the same value of N at which $\langle R^{4}\rangle /\langle R^{2}\rangle ^{2}$ attains its plateau value 5/3. Many more results may be found in the book of Flory .

  

Figure 1.3:

  

Figure 1.4: