Response of thermalized ribbons to pulling and bending

Motivated by recent free-standing graphene experiments, we show how thermal fluctuations affect the mechanical properties of microscopically thin solid ribbons, which can be many thousand times wider than their atomic thickness. A renormalization group analysis of flexural phonons reveals that elongated ribbons behave like highly anisotropic polymers, where the two dimensional nature of ribbons is reflected in nontrivial power law scalings of the persistence length and effective bending and twisting rigidities with the ribbon width. With a coarse-grained transfer matrix approach, we then examine the nonlinear response of thermalized ribbons to pulling and bending forces over a wide spectrum of temperatures, forces, and ribbon lengths.

Figure 1

Renormalization of elastic constants (a) and the Poisson's ratio (b) for membranes under small isotropic tension. We chose parameters suitable for the graphene membrane at room temperature: $\kappa =1.1\text{eV}, Y=340\text{N}/\text{m}, \nu =\lambda /(2\mu +\lambda )=0.156, \sigma ={10}^{-7}\text{N}/\text{m}, {\ell }_{\mathrm{th}}\approx 2\text{nm}, {\ell }_{\sigma }/{\ell }_{\mathrm{th}}\approx 3\times {10}^5$.

Figure 2

Response of membranes to isotropic external tension $\sigma$. We chose parameters suitable for the graphene membrane of size $L=200\mu \text{m}$ at room temperature: $\kappa =1.1\text{eV}, Y=340\text{N}/\text{m}, \nu =\lambda /(2\mu +\lambda )=0.156, {\ell }_{\mathrm{th}}\approx 2\text{nm}$.

Figure 3

(a) Ribbon configuration with undeformed length $L>W$ can be described with orientations of material frame $\{{\mathbf{e}}_1,{\mathbf{e}}_2,{\mathbf{e}}_3\}$ attached to the ribbon relative to the fixed laboratory frame $\{\widehat{\mathbf{x}},\widehat{\mathbf{y}},\widehat{\mathbf{z}}\}$. (b-c) Initial ribbon orientations for studying the response $\langle z\rangle$ to external bending and pulling forces $F$.

Figure 4

Pulling and bending deflections $\langle z/L\rangle$ of ribbons with bending rigidities $A_2/A_1\to \infty$ and twisting rigidity $C/A_1=1$ in response to a fixed small external force $F{A}_1/{\left(k_{B}T\right)}^2=0.01$. The slope of $+2$ for bending when $L\ll {\ell }_p$ agrees with expectations for stiff cantilevers with, however, a bending rigidity greatly enhanced by a factor ${(W/{\ell }_{\mathrm{th}})}^{\eta }\gg 1$. The responses to pulling and bending forces agree when $L\gg {\ell }_p$.

Figure 5

Response of ribbons (neglecting quantum fluctuations) to a small bending force at various temperatures for fixed $W, L, F, \kappa$, and $Y$. Three regimes appear for the parameter choices, $F{L}^2/3W\kappa =0.01, Y{W}^2/\kappa ={10}^5, L/W={10}^2$.

Figure 6

Contributions of backbone stretching (blue line) and entropic elasticity (dashed black line) to $〈z/L〉$, describing the response to large ribbon pulling forces. We chose parameters $k_{B}T/\kappa =1/40$ (suitable for graphene at room temperature), and $W/{\ell }_{\mathrm{th}}={10}^4$ (200 $\mu \mathrm{m}$ width ribbon at room temperature).

Figure 7

One loop corrections to the renormalization of (a) $\kappa$ and (b) $Y$. Solid lines represent propagators for the out-of-plane displacements $f\left(\mathbf{q}\right)$ and dashed lines represent the momentum carried by the vertex $Y$.

Figure 8

One loop corrections to the renormalization of $2\mu +\lambda$. Here, the solid and wiggly lines represent propagators for the out-of-plane displacement $f\left(q\right)$ and for the in-plane displacement $u_x\left(q\right)$, respectively.

Figure 9

Definition of ribbon geometry with attached material frame $\{{\mathbf{e}}_1,{\mathbf{e}}_2,{\mathbf{e}}_3\}$ in the fixed laboratory frame $\{\widehat{\mathbf{x}},\widehat{\mathbf{y}},\widehat{\mathbf{z}}\}$.