The asymmetric sandwich theorem

We discuss the asymmetric sandwich theorem, a generalization of the Hahn-Banach theorem. As applications, we derive various results on the existence of linear functionals that include bivariate, trivariate and quadrivariate generalizations of the Fenchel duality theorem. Most of the results are about affine functions defined on convex subsets of vector spaces, rather than linear functions defined on vector spaces. We consider both results that use a simple boundedness hypothesis (as in Rockafellar's version of the Fenchel duality theorem) and also results that use Baire's theorem (as in the Robinson-Attouch-Brezis version of the Fenchel duality theorem). This paper also contains some new results about metrizable topological vector spaces that are not necessarily locally convex.Comment: 17 page

The Purported Square Ice in Bilayer Graphene in a Nanoscale, Monolayer Object

The phase diagram of water is complex, and interfacial effects can stabilize unusual structures at the nanoscale. Here, we employ bond order accelerated molecular dynamics simulations to show that upon encapsulation within bilayer graphene, water can spontaneously adopt a two-dimensional (monomolecular) layer of “square ice” at ambient conditions, instead of an encapsulated water droplet. Free energy calculations show that this motif is thermodynamically stable up to diameters of approximately 15 nm due to enhanced hydrogen bonding and favorable binding to the graphene sheets. Entropic losses due to solidification and reduced graphene–graphene binding enthalpy are opposing thermodynamic forces that conspire to limit the maximum size, but modification of any of these thermodynamic factors should change the range of stability. Simulated core-level spectroscopy reveals unambiguous orientation dependent signatures of square ice that should be discernable in experiments

Ultra-strong Adhesion of Graphene Membranes

As mechanical structures enter the nanoscale regime, the influence of van der Waals forces increases. Graphene is attractive for nanomechanical systems because its Young's modulus and strength are both intrinsically high, but the mechanical behavior of graphene is also strongly influenced by the van der Waals force. For example, this force clamps graphene samples to substrates, and also holds together the individual graphene sheets in multilayer samples. Here we use a pressurized blister test to directly measure the adhesion energy of graphene sheets with a silicon oxide substrate. We find an adhesion energy of 0.45 \pm 0.02 J/m2 for monolayer graphene and 0.31 \pm 0.03 J/m2 for samples containing 2-5 graphene sheets. These values are larger than the adhesion energies measured in typical micromechanical structures and are comparable to solid/liquid adhesion energies. We attribute this to the extreme flexibility of graphene, which allows it to conform to the topography of even the smoothest substrates, thus making its interaction with the substrate more liquid-like than solid-like.Comment: to appear in Nature Nanotechnolog

Ultrathin Oxide Films by Atomic Layer Deposition on Graphene

In this paper, a method is presented to create and characterize mechanically robust, free standing, ultrathin, oxide films with controlled, nanometer-scale thickness using Atomic Layer Deposition (ALD) on graphene. Aluminum oxide films were deposited onto suspended graphene membranes using ALD. Subsequent etching of the graphene left pure aluminum oxide films only a few atoms in thickness. A pressurized blister test was used to determine that these ultrathin films have a Young's modulus of 154 \pm 13 GPa. This Young's modulus is comparable to much thicker alumina ALD films. This behavior indicates that these ultrathin two-dimensional films have excellent mechanical integrity. The films are also impermeable to standard gases suggesting they are pinhole-free. These continuous ultrathin films are expected to enable new applications in fields such as thin film coatings, membranes and flexible electronics.Comment: Nano Letters (just accepted

Selective Molecular Sieving through Porous Graphene

Membranes act as selective barriers and play an important role in processes such as cellular compartmentalization and industrial-scale chemical and gas purification. The ideal membrane should be as thin as possible to maximize flux, mechanically robust to prevent fracture, and have well-defined pore sizes to increase selectivity. Graphene is an excellent starting point for developing size selective membranes because of its atomic thickness, high mechanical strength, relative inertness, and impermeability to all standard gases. However, pores that can exclude larger molecules, but allow smaller molecules to pass through have to be introduced into the material. Here we show UV-induced oxidative etching can create pores in micrometre-sized graphene membranes and the resulting membranes used as molecular sieves. A pressurized blister test and mechanical resonance is used to measure the transport of a variety of gases (H2, CO2, Ar, N2, CH4, and SF6) through the pores. The experimentally measured leak rates, separation factors, and Raman spectrum agree well with models based on effusion through a small number of angstrom-sized pores.Comment: to appear in Nature Nanotechnolog

On the asymptotic reduction of a bifurcation equation for edge-buckling instabilities

Weakly clamped uniformly stretched thin elastic plates can experience edge buckling when subjected to a transverse pressure. This situation is revisited here for a circular plate, under the assumption of finite rotations and negligible bending stiffness in the pre-buckling range. The eigenproblem describing this instability is formulated in terms of two singularly perturbed fourth-order differential equations involving the non-dimensional bending stiffness ε>0. By using an extension of the asymptotic reduction technique proposed by Coman and Haughton (Acta Mech 55:179–200, 2006), these equations are formally reduced to a simple second-order ordinary differential equation in the limit ε→0+. It is further shown that the predictions of this reduced problem are in excellent agreement with the direct numerical simulations of the original bifurcation equations