The statistical mechanics of twist-storing polymers

This thesis investigates the statistical mechanics of a torsionally constrained polymer. The polymer is modeled as a fluctuating rod with bend stiffness $Ak\sb{\rm B}T$ and twist stiffness $Ck\sb{\rm B}T.$ In such a model, thermal bend fluctuations couple geometrically to an applied torque through the relation Lk = Tw + Wr. A general treatment of this coupling is complicated, and so we restrict ourselves to explorations of two limiting cases. In the first case, we consider polymers which are nearly extended under tension. We develop a theory based on a phantom chain model to describe the geometric coupling and compare its predictions to recent single $\lambda$-DNA molecule experiments. Quantitative agreement between theory and experiment is obtained using C = 109 nm. The analysis we develop further predicts a thermal reduction of the effective twist rigidity induced by bend fluctuations. In the second case, we consider the zero tension limit in which the polymer retracts into an entropic coil. Here self-avoidance needs to be included explicitly in our model. We therefore apply de Gennes\u27 mapping of self-avoiding walks onto the $N \to 0$ limit of the O(N) spin model and include the geometric coupling through a chemical potential for W scRITHE. We find that this modification does not destabilize the Wilson-Fisher fixed point which characterizes normal self-avoiding walks. From this result, we conclude that the polymer\u27s long-wavelength statistics are unmodified by the presence of a small chemical potential for W scRITHE