Statistical mechanics of developable ribbons

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Entry: Chia Wei Hsu, AP 225, Fall 2010

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


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).


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 <math>l_p</math>.


Fig. 1. Schematics of a ribbon. The Frenet frame <math>({\mathbf t},{\mathbf n},{\mathbf b})</math> is sketched in red.

Define the center line of the ribbon (Fig 1), and consider the geometry of this center line in the Frenet frame <math>({\mathbf t},{\mathbf n},{\mathbf b})(s)</math> 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:

<math>{\mathbf t}'-\kappa {\mathbf n}, \quad {\mathbf n}'-\kappa {\mathbf t} + \tau{\mathbf b}, \quad {\mathbf b}'-\tau {\mathbf n}</math>

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

<math>F=\frac{1}{2} D w \int_0^L ds \frac{(\kappa ^2 + \tau ^2)^2}{\kappa ^2} </math>

Fig. 2. Typical confirmation of the modeled ribbon after <math>10^6</math> Monte Carlo steps. Left: <math>\beta^{-1}=0.1</math>; right:<math>\beta^{-1}=1</math>.

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

<math> \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}} </math>,

where <math>\beta=Dw/(ak_BT)</math>. This energy is used to carry out Monte Carlo simulations of a chain with <math>N</math> segments, with its conformation updated through pivot moves. For each chain the authors perform <math>10^6</math> Monte Carlo updates. Fig 2 shows the typical confirmation of such a ribbon.

Correlation Functions

Fig. 3. Tangent-tangent (left) and binormal-binormal (right) correlation functions from Monte Carlo simulations of a polymer chain of <math>N = 100</math> segments at <math>T = 0.1</math> (black), <math>0.2</math> (blue), <math>0.3</math> (red) and <math>0.4</math> (green). The graphs of <math><{\mathbf t}_n \cdot {\mathbf t}_0></math> have been translated along the vertical axis to avoid overlaps.

Fig 3 shows the tangent-tangent and binormal-binormal correlation for a chain of <math>N=100</math> segments at various temperatures. While <math><{\mathbf b}_n \cdot {\mathbf b}_0></math> always exhibits exponential decay, <math><{\mathbf t}_n \cdot {\mathbf t}_0></math> has oscillations superimposed on an exponential decay. They follow the expression

<math><{\mathbf t}_n \cdot {\mathbf t}_0> = e ^{-s/l_p} \cos ks </math>,

<math><{\mathbf b}_n \cdot {\mathbf b}_0> = e ^{-s/l_{\tau}} </math>,

where <math>l_p</math> is the persistence length, <math>k</math> is the wave number of oscillation, <math>l_{\tau}</math> is the torsional persistence length, and <math>s=na</math> is the arc-length coordinate. The authors show that in this model, the oscillatory term is always present, indicating that there is always a helical structure at finite temperature.

Persistence Length and Oscillation Wave Number

Fig. 4. Persistence length and torsional persistence length (left, in units of <math>a</math>) and Tangent-tangent correlation function wave number <math>k</math> (right, in units of <math>a^{-1}</math>) as a function of <math>\beta^{-1}</math>. Dots correspond to numerical data and lines to analytical predictions for the low temperature regime.

The authors then give an analytical estimate of the three quantities <math>l_p</math>, <math>l_{\tau}</math>, and <math>k</math>. Starting from the partition function, the authors derive that when <math>\beta \gg 1</math>,

<math>l_p = \left( \frac{75}{64}-\frac{25}{9\pi} \right) ^{-1} \frac{Dw}{k_BT} = 3.476 \frac{Dw}{k_BT}</math>,

<math>l_{\tau} = \frac{32}{5} \frac{Dw}{k_BT} = 6.4 \frac{Dw}{k_BT} </math>,

<math>k = \frac{5}{3}\sqrt{\frac{2}{\pi a}\frac{Dw}{k_BT}} </math>.

Fig. 4 compares these analytical expression to the numerical data. The persistence length <math>l_p</math> is a factor of <math>3.476</math> larger than that of a worm like chain.