Stress-controlled Poisson ratio of a crystalline membrane: Application to graphene

Abstract

We demonstrate that a key elastic parameter of a suspended crystalline membrane---the Poisson ratio (PR) $\nu$---is a non-trivial function of the applied stress $\sigma$ and of the system size $L$, i.e., $\nu ={\nu }_L(\sigma )$. We consider a generic 2D membrane embedded into space of dimensionality $2+d_c$. (The physical situation corresponds to $d_c=1$.) A particularly important application of our results is free-standing graphene. We find that at very low stress, where the membrane exhibits a linear response, the PR ${\nu }_L(0)$ decreases with increasing $L$ and saturates for $L\to \infty$ at a value which depends on the boundary conditions and is essentially different from the value $\nu =-1/3$ previously predicted by the membrane theory within a self-consisted scaling analysis. By increasing $\sigma$, one drives a membrane into a non-linear regime characterized by a universal value of PR that depends solely on $d_c.$ This universal non-linear PR acquires its minimum value ${\nu }_{\min }=-1$ in the limit $d_c\to \infty .$ With the further increase of $\sigma$, the PR changes sign and finally saturates at a positive non-universal value prescribed by the conventional elasticity theory. We also show that one should distinguish between the absolute and differential PR ($\nu$ and ${\nu }^{\mathrm{diff}}$, respectively). While coinciding in the limits of very low and very high stresses, they differ in general, $\nu \ne {\nu }^{\mathrm{diff}}$. In particular, in the non-linear universal regime, ${\nu }^{\mathrm{diff}}$ takes a universal value which, similarly to absolute PR, is a function solely of $d_c$ but is different from the universal value of $\nu$. In the limit $d_c\to \infty$, the universal value of ${\nu }^{\mathrm{diff}}$ tends to $-1/3$, at variance with the limiting value $-1$ of $\nu$. Finally, we briefly discuss generalization of these results to a disordered membrane.

21 pages, 9 figures

References in corpus (11)

• Electric Field Effect in Atomically Thin Carbon Films
• The electronic properties of graphene
• Finite temperature lattice properties of graphene beyond the quasiharmonic approximation
• Crumpling transition and flat phase of polymerized phantom membranes
• Graphene as a Prototype Crystalline Membrane
• Auxetic Nanomaterials: Recent Progress and Future Development
• Anomalous elasticity, fluctuations and disorder in elastic membranes
• Thermal fluctuations of free standing graphene
• All-Optical Blister Test of Suspended Graphene Using Micro-Raman Spectroscopy
• Quantum elasticity of graphene: Thermal expansion coefficient and specific heat
• Non-linear fluctuation effects in dynamics of freely suspended film

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• Three-loop order approach to flat polymerized membranes
• Disorder-induced rippled phases and multicriticality in free-standing graphene
• Spontaneous symmetry breaking and the flat phase of crystalline membranes
• Emergent continuous symmetry in anisotropic flexible two-dimensional materials
• Accessing negative Poisson`s ratio of graphene by machine learning interatomic potentials
• Disorder-driven transition to tubular phase in anisotropic two-dimensional materials
• Thermal fluctuations in crystalline membranes with long-range dipole interactions