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.