Statistical Mechanics of Developable Ribbons

We investigate the statistical mechanics of long developable ribbons of finite width and very small thickness. The constraint of isometric deformations in these ribbonlike structures that follows from the geometric separation of scales introduces a coupling between bending and torsional degrees of freedom. Using analytical techniques and Monte Carlo simulations, we find that the tangent-tangent correlation functions always exhibit an oscillatory decay at any finite temperature implying the existence of an underlying helical structure even in the absence of a preferential zero-temperature twist. In addition, the persistence length is found to be over 3 times larger than that of a wormlike chain having the same bending rigidity. Our results are applicable to many ribbonlike objects in polymer physics and nanoscience that cannot be described by the classical wormlike chain model.

Figure 1

A twisted paper ribbon. In presence of twist the ribbon necessarily develops a nonzero curvature, as required by the Sadowsky functional. The Frenet frame ($\mathbf{t}$, $\mathbf{n}$, $\mathbf{b}$) and the material frame (${\mathbf{d}}_1$, ${\mathbf{d}}_2$, ${\mathbf{d}}_3$) are sketched in red and green, respectively. For a developable ribbon, the two frames coincide, so that the twist of the material frame and the torsion of the center line are identical.

Figure 2

The quantity $\Delta$ defined in the text as a function of $\beta$, calculated from the exact integral expression given on the last page. With exception for the two limits $\beta \to 0$, $\infty$, $\Delta$ is always negative and the associated eigenvalues ${\lambda }_{1,2}=\frac{1}{2}(b\pm \sqrt{\Delta })$ are complex and conjugate. In the inset the discrete Frenet frame.

Figure 3

Tangent-tangent (left) and binormal-binormal (right) correlation functions from Monte Carlo simulations of a polymer chain of $N=100$ segments at $T=0.1$ (black), 0.2 (blue), 0.3 (red) and 0.4 (green). The graphs of $⟨{\mathbf{t}}_n·{\mathbf{t}}_0⟩$ have been translated along the vertical axis to avoid overlaps.

Figure 4

The typical conformation of a ribbonlike polymer after ${10}^6$ Monte Carlo iterations. (Left) ${\beta }^{-1}=0.1$ and (right) ${\beta }^{-1}=1$. At low temperature the ribbon fluctuates around an underlying helical shape. The radius and the pitch of the helix are set by the persistence length ${\ell }_p$ and the wave number $k$ and are proportional to $\beta a$ and ${\beta }^{1/2}a$, respectively.

Figure 5

Persistence length and torsional persistence length (left) as a function of ${\beta }^{-1}$ (in units of $a$). (Right) Tangent-tangent correlation function wave number $k$ (in units of $a^{-1}$) as a function of ${\beta }^{-1}$. Dots correspond to numerical data and lines to analytical predictions for the low temperature regime. For biopolymers of Young modulus $E\approx 100\mathrm{MPa}$, $h\approx 1\mathrm{nm}$ and $w/a\approx 1$, the value of ${\beta }^{-1}$ in the plots spans the temperature range $T[0,{10}^3]\mathrm{K}$.