The stability of two-dimensional (2D) layers and membranes is subject of a long standing theoretical debate. According to the so called Mermin-Wagner theorem, long wavelength fluctuations destroy the long-range order for 2D crystals. Similarly, 2D membranes embedded in a 3D space have a tendency to be crumpled. These dangerous fluctuations can, however, be suppressed by anharmonic coupling between bending and stretching modes making that a two-dimensional membrane can exist but should present strong height fluctuations. The discovery of graphene, the first truly 2D crystal and the recent experimental observation of ripples in freely hanging graphene makes these issues especially important. Beside the academic interest, understanding the mechanisms of stability of graphene is crucial for understanding electronic transport in this material that is attracting so much interest for its unusual Dirac spectrum and electronic properties. Here we address the nature of these height fluctuations by means of straightforward atomistic Monte Carlo simulations based on a very accurate many-body interatomic potential for carbon. We find that ripples spontaneously appear due to thermal fluctuations with a size distribution peaked around 70 \AA which is compatible with experimental findings (50-100 \AA) but not with the current understanding of stability of flexible membranes. This unexpected result seems to be due to the multiplicity of chemical bonding in carbon.Comment: 14 pages, 6 figure

The thermal and mechanical stability of graphene is important for many potential applications in nanotechnology. We calculate the temperature dependence of the lattice parameter, elastic properties, and heat capacity by means of atomistic Monte Carlo simulations that allow us to go beyond the quasiharmonic approximation. We predict an unusual, nonmonotonic, behavior of the lattice parameter with a minimum at T approximately 900 K and of the shear modulus with a maximum at the same temperature. The Poisson ratio in graphene is found to be small approximately 0.1 in a broad temperature interval.

Structure and thermodynamics of crystalline membranes are characterized by the long wavelength behavior of the normal-normal correlation function G(q). We calculate G(q) by Monte Carlo and Molecular Dynamics simulations for a quasi-harmonic model potential and for a realistic potential for graphene. To access the long wavelength limit for finite-size systems (up to 40000 atoms) we introduce a Monte Carlo sampling based on collective atomic moves (wave moves). We find a power-law behaviour G(q) ∝ q −2+η with the same exponent η ≈ 0.85 for both potentials. This finding supports, from the microscopic side, the adequacy of the scaling theory of membranes in the continuum medium approach, even for an extremely rigid material like graphene. Collective phenomena involving infinitely many degrees of freedom are often characterized by scaling laws with power-law behavior of correlation functions. In three dimensional systems, this behavior occurs only at critical points [1,2,3]. In two dimensions (2D) the situation is different, and a whole temperature interval with "almost broken symmetry" and power-law decay of correlation functions frequently appears, the KosterlitzThouless (KT) transition in 2D superfluids and superconductors [4] being a prototype example. Existence of real long range order, where correlation functions remain non-zero in the limit of infinite distance, is forbidden in such cases by the Mermin-Wagner theorem [5] due to the divergence of the contribution of soft modes to relevant thermodynamic properties. The theory of flexible membranes [6] embedded in higher dimensions is an important part of the statistical mechanics of 2D systems. Here, we investigate the scaling behavior of crystalline flexible membranes by means of atomistic simulations, using graphene [7,8,9], the simplest known membrane, as an example.In the flat phase, the membrane in-plane and out-ofplane displacements are parametrized by a D-component 'stretching' phonon field u α (x), α = 1...D, and by a, where d is the space dimension and D is the membrane dimension. Softening of bending modes makes this situation very similar to the KT model. A minimal phenomenological model for membranes is just the elasticity theory described by the Hamiltonian [6, 10

We propose a bond order potential for carbon with built-in long-range interactions. The potential is defined as the sum of an angular and coordination dependent short-range part accounting for the strong covalent interactions and a radial long-range part describing the weak interactions responsible, e.g., for the interplanar binding in graphite. The short-range part is a Brenner type of potential, with several modifications introduced to get an improved description of elastic properties and conjugation. Contrary to previous long-range extensions of existing bond order potentials, we prevent the loss of accuracy by compensating for the additional long-range interactions by an appropriate parametrization of the short-range part. We also provide a short-range bond order potential. In Monte Carlo simulations our potential gives a good description of the diamond to graphite transformation. For thin ͑111͒ slabs graphitization proceeds perpendicular to the surface as found in ab initio simulations, whereas for thick layers we find that graphitization occurs layer by layer.

Plant cells are enclosed by a rigid cell wall that counteracts the internal osmotic pressure of the vacuole and limits the rate and direction of cell enlargement. When developmental or physiological cues induce cell extension, plant cells increase wall plasticity by a process called loosening. It was demonstrated previously that a class of proteins known as expansins are mediators of wall loosening. Here, we report a type of cell wall-loosening protein that does not share any homology with expansins but is a member of the lipid transfer proteins (LTPs). LTPs are known to bind a large range of lipid molecules to their hydrophobic cavity, and we show here that this cavity is essential for the cell wall-loosening activity of LTP. Furthermore, we show that LTP-enhanced wall extension can be described by a logarithmic time function. We hypothesize that LTP associates with hydrophobic wall compounds, causing nonhydrolytic disruption of the cell wall and subsequently facilitating wall extension.

We derive an analytical expression that describes the interaction energy between two graphene layers identically oriented as a function of the relative lateral and vertical positions, in excellent agreement with first principles calculations. Thanks to its formal simplicity, the proposed model allows for an immediate interpretation of the interactions, in particular of the potential corrugation. This last quantity plays a crucial role in determining the intrinsic resistance to interlayer sliding and its increase upon compression influences the frictional behavior under load. We show that, for these weakly adherent layers, the corrugation possesses the same nature and z dependence of Pauli repulsion. We investigate the microscopic origin of these phenomena by analyzing the electronic charge distribution: We observe a pressure-induced charge transfer from the interlayer region toward the near-layer regions, with a much more consistent depletion of charge occurring for the AA stacking than for the AB stacking of the two layers.

By atomistic modeling of moiré patterns of graphene on a substrate with a small lattice mismatch, we find qualitatively different strain distributions for small and large misorientation angles, corresponding to the commensurate-incommensurate transition recently observed in graphene on hexagonal BN. We find that the ratio of C-N and C-B interactions is the main parameter determining the different bond lengths in the center and edges of the moiré pattern. Agreement with experimental data is obtained only by assuming that the C-B interactions are at least twice weaker than the C-N interactions. The correspondence between the strain distribution in the nanoscale moiré pattern and the potential energy surface at the atomic scale found in our calculations, makes the moiré pattern a tool to study details of dispersive forces in van der Waals heterostructures. The study of these new hybrid materials is emerging as a strong research area.The superposition of periodic layered structures, with either slightly different lattice constants or different orientations, creates moiré patterns [3][4][5][6][7]. These patterns can yield a wealth of information about the lattice constant mismatch, strain and imperfections of the surface [8][9][10][11][12][13]. The moiré patterns imply a change of the interatomic distances that can affect properties that are important both for applications and for fundamental physics such as the quantum mechanics of electrons in quasi-periodic potentials [3][4][5][6].In recent years hexagonal boron nitride (h-BN) has become a standard substrate for graphene growth due to its flat surface without dangling bonds, the hexagonal lattice with a lattice constant only 1.8 % larger than that of graphene and the fact that h-BN is an insulator [14]. These properties have led to the realization of the first field effect transistor [15]. The difference in lattice constant leads to the appearances of moiré patterns, which can be observed experimentally [16][17][18].Usually, moiré structures are considered from a purely geometrical point of view for the superposition of two rigid lattices where the length L of the moiré patterns is found to depend on the angle θ and the lattice mismatch between the two layers aswhere p is the ratio between lattice constants and a the lattice constant of the substrate [19]. Strain due to the lattice mismatch and/or rotations have been considered in a continuum approach to study the modification of the electronic structures in tight binding calculations [20][21][22][23][24] and the pseudo-magnetic fields resulting from out-of-plane displacements [25,26]. Full atomic relaxation to minimal energy configurations is however necessary to make a detailed comparison to experimental structural information as obtained by scanning probe microscopy [7]. At the same time, we will show that this procedure allows to get quantitative information on the interplanar interactions. It is well known that dispersive forces are beyond the standard local density functional and generalized gradient correct...

We present an approach to the melting of graphene based on nucleation theory for a first order phase transition from the 2D solid to the 3D liquid via an intermediate quasi-2D liquid. The applicability of nucleation theory, supported by the results of systematic atomistic Monte Carlo simulations, provides an intrinsic definition of the melting temperature of graphene, Tm, and allows us to determine it. We find Tm ≃ 4510 K, about 250 K higher than that of graphite using the same interatomic interaction model. The found melting temperature is shown to be in good agreement with the asymptotic results of melting simulations for finite disks and ribbons of graphene. Our results strongly suggest that graphene is the most refractory of all known materials.