Question

Consider a solution of charged, polymer molecules in a constant electric field \(\mathcal{E}\). The polymer is modeled as a chain of monomers of mass \(m\) connected by \(N\) massless rigid rods of fixed length \(\ell,\) which are freely jointed at the monomers so that adjacent rods may make any angle with respect to each other. The two monomers at the ends of the molecule have charge \(\pm q\), respectively, and the others are uncharged. The solvent molecules act as a heat bath for the polymer molecules, which can then be treated in the canonical ensemble. Assume that the solution is sufficiently dilute that different polymer molecules do not interact. The Hamiltonian for each molecule consists of the kinetic energy of the monomers plus the interaction with the field,

$$ H_{\mathrm{mol}}=\sum_{i=0}^{N} \frac{\mathbf{p}_{i}^{2}}{2 m}-q E_{0}\left(z_{N}-z_{0}\right) $$

where \(z_{0, N}\) are the coordinates of the charged end-monomers in the direction of the field (z). Calculate the mean and mean-square separations between the ends of a molecule. Find and explain the limiting behaviors at low and high temperatures.