Mode Description of Chain Statistics

In a real chain segment-segment correlations extend beyond nearest neighbour distances. The standard model to treat the local statistics of a chain, which includes the local stiffness, would be the rotational isomeric state (RIS) [211] formalism. For a mode description as required for an evaluation of the chain motion it is more appropriate to consider the so-called all-rotational state (ARS) model [212], which describes the chain statistics in terms of orthogonal Rouse modes. It can be shown that both approaches are formally equivalent and only differ in the choice of the orthonormal basis for the representation of statistical weights. In the ARS approach the characteristic ratio of the RIS-model becomes mode dependent. 

As Allegra et al. [213, 214] have shown, for polyolefines it can be well approximated by  

The higher the mode number, the smaller becomes the square of the Fourier components. Using periodic boundary conditions and considering that a segment vector is given by the difference of adjacent position vectors the statistical average (R(qf) in Fourier space becomes  

With this equation in place we now may evaluate the statistical average for any distance k= h-j along the chain, where h and j denote the position of chain segments. Fourier transforming r n ) (Eq. 3.3) we obtain a probability distribution in Fourier space  

Like in the Rouse model from the probability distribution Prob the free energy is obtained by taking the logarithm and finally the force exerted on a segment h (x-component) follows by taking the derivative of the free energy  

---