Non-Hookean statistical mechanics of clamped graphene ribbons

Thermally fluctuating sheets and ribbons provide an intriguing forum in which to investigate strong violations of Hooke's Law: Large distance elastic parameters are in fact not constant but instead depend on the macroscopic dimensions. Inspired by recent experiments on free-standing graphene cantilevers, we combine the statistical mechanics of thin elastic plates and large-scale numerical simulations to investigate the thermal renormalization of the bending rigidity of graphene ribbons clamped at one end. For ribbons of dimensions $W\times L$ (with $L\ge W$), the macroscopic bending rigidity ${\kappa }_R$ determined from cantilever deformations is independent of the width when $W<{\ell }_{\mathrm{th}}$, where ${\ell }_{\mathrm{th}}$ is a thermal length scale, as expected. When $W>{\ell }_{\mathrm{th}}$, however, this thermally renormalized bending rigidity begins to systematically increase, in agreement with the scaling theory, although in our simulations we were not quite able to reach the system sizes necessary to determine the fully developed power law dependence on $W$. When the ribbon length $L>{\ell }_p$, where ${\ell }_p$ is the $W$-dependent thermally renormalized ribbon persistence length, we observe a scaling collapse and the beginnings of large scale random walk behavior.

Figure 1

(a) Our numerical simulations of a graphene strip of undeformed size $L\times W$ are performed with a coarse-grained model commonly used to study elastic membranes [42]. The strip is represented as an equilateral triangulation of a rectangle with bending and stretching energies defined along the edges and plaquettes of the triangulation. (b) For computational reasons, it is convenient to describe bending energy as a penalty of changing the dihedral angle between two triangles sharing an edge.

Figure 2

Scaling collapse for height fluctuations $〈{\left|h\left(\mathbf{q}\right)\right|}^2〉$ of a rectangular sheet of size $100\times 86.6a^2$ characterized by different values of bending rigidity $\tilde{\kappa }$ and spring constant $\epsilon$ that are defined in Eqs. (2) and (3). For large wave vectors $q\gg q_{\mathrm{th}}$ height fluctuations are well described with the harmonic approximation (dashed green line), where the renormalized bending rigidity ${\kappa }_R\left(q\right)$ can be approximated with the bare bending rigidity ${\kappa }_0$ [see Eq. (5)]. For small wave vectors $q\ll q_{\mathrm{th}}$ the renormalization of bending rigidity ${\kappa }_R\left(q\right)$ with the characteristic exponent $\eta$ becomes apparent. Solid black line corresponds to the perturbative renormalization group result from Ref. [16]. The dotted red line is adapted from the atomistic Monte Carlo simulations in Ref. [13], and the small bump at $q/q_{\mathrm{th}}\approx 5–10$ corresponds to wave vectors close to the edge of the first Brillouin zone.

Figure 3

Renormalization of ribbon elastic constants due to thermal fluctuations. (a) Snapshot of thermal fluctuations on a $W\times W$ patch of ribbon, the latter represented schematically, where color encodes the height of fluctuations with red (blue) describing positive (negative) height fluctuations. The effect of thermal fluctuations is to renormalize the bending rigidity ${\kappa }_R\left(W\right)$ and the Young's modulus $Y_R\left(W\right)$ on the scale of ribbon width $W$ according to Eqs. (7). (b) Coarse-grained ribbon is constructed with square blocks of size $W\times W$ with the renormalized elastic constants ${\kappa }_R\left(W\right)$ and $Y_R\left(W\right)$. (c) A coarse-grained representation of ribbon configurations tracks the orientation of an orthonormal triad $\left({\widehat{e}}_1\left(s\right),{\widehat{e}}_2\left(s\right),{\widehat{e}}_3\left(s\right)\right)$ relative to a laboratory frame $\left({\widehat{e}}_x,{\widehat{e}}_y,{\widehat{e}}_z\right)$ as a function of arc length $s$ along the ribbon [16].

Figure 4

Renormalized bending rigidity ${\kappa }_R\left(W\right)$ for ribbons of width $W$ extracted from simulation measurements of the persistence length ${\ell }_p$ [see Eq. (11)]. The scaling collapse for ribbons of various dimensions and bare bending rigidities ${\kappa }_0$ is consistent with Eq. (7a), where the red theoretical curve was obtained with renormalization group procedure described in Ref. [16]. The theoretical calculations were done with periodic boundary conditions across the ribbon width. We attempted to account for our different boundary conditions (clamped at one end and free at other) by shifting the theoretical curve along the horizontal direction. In these simulations, the value of ratio $W/{\ell }_{\mathrm{th}}$ was set by tuning the Young's modulus $Y_0$ of the ribbon.

Figure 5

Scaling collapse for ribbon height fluctuations $〈h^2\left(x\right)〉/{\ell }_p^2$, where $x$ represents the distance from the clamped end along the ribbon backbone. The red line indicates the analytically predicted curve for anisotropic ribbons in Eq. (13), which is in good agreement with results of numerical simulations with no adjustable parameters. The persistence length ${\ell }_p$ is measured from the simulations by computing the autocorrelation function of tangent vectors to the ribbon's midline as described in text.