Determining the Gaussian Modulus and Edge Properties of 2D Materials: From Graphene to Lipid Bilayers

The dominant deformation behavior of two-dimensional materials (bending) is primarily governed by just two parameters: bending rigidity and the Gaussian modulus. These properties also set the energy scale for various important physical and biological processes such as pore formation, cell fission and generally, any event accompanied by a topological change. Unlike the bending rigidity, the Gaussian modulus is, however, notoriously difficult to evaluate via either experiments or atomistic simulations. In this Letter, recognizing that the Gaussian modulus and edge tension play a nontrivial role in the fluctuations of a 2D material edge, we derive closed-form expressions for edge fluctuations. Combined with atomistic simulations, we use the developed approach to extract the Gaussian modulus and edge tension at finite temperatures for both graphene and various types of lipid bilayers. Our results possibly provide the first reliable estimate of this elusive property at finite temperatures and appear to suggest that earlier estimates must be revised. In particular, we show that, if previously estimated properties are employed, the graphene-free edge will exhibit unstable behavior at room temperature. Remarkably, in the case of graphene, we show that the Gaussian modulus and edge tension even change sign at finite temperatures.

Figure 1

Gaussian modulus and edge properties play a central role in physical processes that involve topological changes or deformation of an open edge, e.g., pore formation, structural deformation of a finite nanoribbon, cell fission and fusion.

Figure 2

We extract the Gaussian modulus and edge properties from MD simulations for lipid membrane DPPC by fitting our analytical results. Our fit is found to be more sensitive to edge tension compared to the edge moduli, and hence, the green dashed line yields a better fit.

Figure 3

Fluctuations of the free edge of graphene monolayer with size of $L=8.4\mathrm{nm}$.

Figure 4

This figure shows the predictions of edge fluctuations of graphene if the currently accepted values of Gaussian modulus [25] and edge properties [46] in the literature are used. As is evident, our atomistic simulations cannot be captured by those properties.