Rouse Frequency

Summary

Fit Rouse modes to a frequency dependent relaxation function

• Function

  Continuous Rouse model (valid for “large” \(N\)):

  \[\begin{split}G'(\omega) &= G_0 \dfrac 1 N \sum_{p=1}^N \dfrac{(\omega\tau_p)^2} {1 + (\omega\tau_p)^2}\\ G''(\omega) &= G_0 \dfrac 1 N \sum_{p=1}^N \dfrac{\omega\tau_p} {1 + (\omega\tau_p)^2}\\ \tau_p &= \dfrac{N^2 \tau_0 }{ 2 p^2}\end{split}\]

• Parameters

  • \(G_0 = ck_\mathrm B T\): “modulus”

  • \(\tau_0\): relaxation time of an elementary segment

  • \(M_0\): molar mass of an elementary segment

  where

  • \(c\): number of segments per unit volume

  • \(k_\mathrm B\): Boltzmann constant

  • \(T\): temperature

  • \(N=M_w/M_0\): number of segments par chain

  • \(M_w\): weight-average molecular mass