Statistical Mechanics of Sadowsky Ribbons under applied Force and Torque

In our work, we study the statistical mechanics of the microscopic Sadowsky ribbon, a long developable ribbons of finite width and very small thickness. The Sadowsky ribbon is isometric to a flat strip at any temperature with a nonlinear coupling between the bending and torsional degrees of freedom. Our simulation was done using Monte Carlo algorithm and different equilibrium distribution of ribbon conformation was found for different values of applied force and torque. The shape of the ribbon characterized by its linking number, twist, and writhe, which are related via the Călugăreanu-White-Fuller theorem. At zero force and torque, the Sadowsky ribbon displays an underlying helical structure that disappears at large values of force and torque, indicating a phase transition from the ordered helical phase to the disordered entangled phase. The transition occurs at a finite Lifshitz point. The binormal-binormal correlation function exhibits a transition from exponential to oscillatory behavior at this critical point, and oppositely for the tangent-tangent correlation function. The plots of topological numbers as a function of force and torque are reminiscent to the role of magnetization in Ising models and can be viewed as order parameters of this ribbon system.