# Statistical mechanics of developable ribbons

Entry: Chia Wei Hsu, AP 225, Fall 2010

L. Giomi and L. Mahadevan, Phys. Rev. Lett. 104, 238104 (2010)

## Summary

The authors use analytical techniques and Monte Carlo simulations to study a model of long developable ribbons of finite width and very small thickness. They find that the tangent-tangent correlation functions always exhibit oscillatory decay, indicating an underlying helical structure. They also find the persistence to be over 3 times larger than that of a worm like chain (WLC).

## Background

The geometric separation of length scales often lead to constraints on the deformation of an object. For a plate, two dimensions are comparable and much larger than the third. For a rod, two dimensions are comparable and much smaller than the third. For a ribbon, all three dimensions are separated: its thickness is much smaller than is width, which is much smaller than its length. In such case the bending and torsional degrees of freedom are strongly coupled. Many nanostructures are ribbon like, such as DNA, certain secondary structures of proteins, and graphene ribbons. They cannot simply be modeled as a worm like chain (WLC), in which the polymer is assumed to have an inextensible backbone but with a flexibility governed by the persistence length $l_p$.

## Model

Fig. 1.

Define the center line of the ribbon (fig 1), and consider the geometry of this center line in the Frenet frame $({\mathbf t},{\mathbf n},{\mathbf b})(s)$ of the tangent, normal, and binormal vectors at each point along the center line parametrized by arc-length coordinate s. These three vectors evolves according to the Frenet-Serret equations:

${\mathbf t}'-\kappa {\mathbf n}, \quad {\mathbf n}'-\kappa {\mathbf t} + \tau{\mathbf b}, \quad {\mathbf b}'-\tau {\mathbf n}$

where $\kappa$ and $\tau$ are the curvature and torsion of the space curve, and the primes denotes derivative with respect to $s$. Consider a ribbon of length $L$ and width $w$. With the developability and the $w/L \ll 1$ assumptions, the elastic energy can be written as

$F=\frac{1}{2} D w \int_0^L ds \frac{(\kappa ^2 + \tau ^2)^2}{\kappa ^2}$

where $D$ is the bending rigidity. Now, discretize the chain into $N=L/a$ consecutive vertices $\{{\mathbf x}_i\}$ that are separated by distance $a$. With some algebra, the elastic energy can be written as

$\frac{F}{k_B T} = \beta \sum_{i=1}^N \frac{ \left( 1-{\mathbf t}_i \cdot {\mathbf t}_{i+1} + 1-{\mathbf b}_i \cdot {\mathbf b}_{i+1} \right)^2 }{1-{\mathbf t}_i \cdot {\mathbf t}_{i+1}}$,

where $\beta=Dw/(ak_BT)$. This energy is used to carry out Monte Carlo simulations of a chain with $N$ segments, with its conformation updated through pivot moves. For each chain the authors perform $10^6$ Monte Carlo updates.