Dynamics of a Long Polymer Backbone

For the study of the dynamics of a polymer molecule in dilute solution a continuous wire model is proposed and investigated. This model is a refinement of the wormlike chain model introduced by Kratky and Porod such that the configurational energy depends not only on the curvature $\kappa$ but also on the torsion $\tau$ , both of which may be considered as functions of the arc lengths measured from one end. The physical basis of considering such energy dependence is given from a general differential–geometrical viewpoint, and also by the explicit calculation of the elastic energy of an ideally thin wire, which is found to be $${\displaystyle {\int }_0^l}\mathrm{ds}\frac{1}{8}{\mathrm{\pi R}}^4{\mathrm{[E(\kappa \hspace{0.17em}-\hspace{0.17em}\kappa }}_0{)}^2{\text{\hspace{0.17em}+\hspace{0.17em}2μ(τ\hspace{0.17em}−\hspace{0.17em}τ}}_0{)}^2]$$ , where $E$ and $\mu$ are Young's modulus and the modulus of rigidity of the wire material; $R$ is the radius of the circular cross section of the wire; ${\kappa }_0\mathrm{(s)}$ and ${\tau }_0\mathrm{(s)}$ are the curvature and torsion of the space curve characterizing the wire of minimum energy. The dynamics of the model is formulated with the aid of Hamilton's principle of least action. In particular the wave propagation along the axis of a helical coil is investigated in detail. It is shown that the measurement of the propagation speed of a longitudinal wave along a polymer chain can lead to the quantitative estimate of the bending and twisting characteristics of the chain.