Polymer RISM

We now apply the RISM  formalism to polymeric liquids . For simplicity we restrict the presentation to polyethylene , each CH₂ will be viewed as one ''atom''. It is clear that if boundary effects may be neglected, all monomers along the chain are equivalent, i.e. that $G_{\alpha \beta }(r)$ and all other functions do not depend on $\alpha$ or $\beta$. Remember that $\alpha$and $\beta$ each run from 0 to N, where (N+1) is the number of monomers per chain. The monomer-monomer radial distribution function then is

$$G(r)= \frac{1}{\rho _{m}}\sum_{\beta }G_{\alpha \beta }(r)\rho _{\beta }=G_{\alpha \beta }(r)$$ (2.35)

where $\rho _{m}=(N+1)\rho _{\beta }$ is the monomer density. Introducing Eq. (2.34) into Eq. (2.35) we get

 

$$\frac{1}{\rho _{m}}\{\omega (r)-\delta (r)\}+g(r)$$ G(r) = (2.36)

$$\omega (r) \sum_{\beta }\Omega _{\alpha \beta }(r)$$ = (2.37)

where $g(r)=g_{\alpha \beta }(r)$. Using $h(r)=h_{\alpha \beta }(r)$ and $c(r)=c_{\alpha \beta }(r)$ in Eq. (2.31) we derive

$$\hat{h}(k)=\hat{\omega}(k)\hat{c}(k)\hat{\omega}(k)+\rho _{m}\hat{\omega}(k) \hat{c}(k)\hat{h}(k)$$ (2.38)

$\hat{\omega}(k)$ may be calculated using the methods of Chapter 1. Some details are given in Appendix A.

We now apply the method to polyethylene, which will be modelled by a RIS model with characteristics given in section 2.6. The interaction between two monomers on different chains is modelled by a hard sphere interaction with diameter $\sigma = 3.9 \mbox{\AA}$. Moreover N=6416 and $\rho_m = 33.58 \mathrm{{nm^{-3}}}$. Calculations were done with PY closure.

In Fig. (2.3) is given the intermolecular distribution function. We clearly see what is called the ''correlation hole''  by de Gennes ; the intermolecular distribution gradually rises to its limit value 1, with only little structure. The scattering function is given in Fig. (2.4), and is in perfect agreement with X-ray experiments, except at very small values of k.

We give two more results, from MD simulations  this time. In Fig. (2.5 ) is given the intramolecular oxygen-oxygen correlation in PEO . The simulation was done with GROMOS united atom potentials. The box consisted of two chains of 800 monomers each; $\rho _{m}=14.45\mathrm{{nm^{-3}}}$ and T=400K. The corresponding intermolecular correlation function is given in Fig. (2.6).

  

Figure 2.3: Intermolecular distribution function.

  

Figure 2.4: Scattering function.

  

Figure 2.5: Intramolecular oxygen-oxygen correlation in polyethyleneoxide.

  

Figure 2.6: Intermolecular correlation function in polyethyleneoxide.