Energy behaviour for DNA translocation through graphene nanopores.

Nanoparticles have considerable promise for many applications in electronics, energy storage, bioscience and biotechnologies. Here we use applied mathematical modelling to exploit the basic principles of mechanics and the 6-12 Lennard-Jones potential function together with the continuum approach, which assumes that a discrete atomic structure can be replaced by an average constant atomic surface density of atoms that is assumed to be smeared over each molecule. We identify a circular hole in a graphene sheet as a nanopore and we consider the molecular interaction energy for both single-strand and double-strand DNA molecules assumed to move through the circular hole in a graphene sheet to determine the radius b of the hole that gives the minimum energy. By minimizing the interaction energy, we observe that the single-strand DNA and double-strand DNA molecules penetrate through a graphene nanopore when the pore radii b> 7.8Å and b> 12.7Å, respectively. Our results can be adopted to offer new applications in the atomic surface processes and electronic sensing.

Modeling interactions between C60 antiviral compounds and HIV protease.

Fullerenes have generated a great deal of interest in recent years, due to their properties and potential applications in many fields, including medicine. In this paper, we study an antiviral fullerene compound which may be used to treat the human immunodeficiency virus (HIV). We formulate a mathematical model which can describe the interaction energy between the C[Formula: see text] antiviral compounds and the HIV. In particular, this paper predicts the energy and force arising from the interaction between HIV active region and the antiviral molecule which is attached to the external surface of a fullerene C[Formula: see text]. These interactions are calculated based on the structure of the antiviral molecules. Our results show that the binding of fullerene C[Formula: see text] to the antiviral molecules increases the efficiency of the compound to prohibit the activity of HIV.

Offset configurations for single- and double-strand DNA inside single-walled carbon nanotubes.

Nanotechnology is a rapidly expanding research area, and it is believed that the unique properties of molecules at the nano-scale will prove to be of substantial benefit to mankind, especially so in medicine and electronics. Here we use applied mathematical modelling exploiting the basic principles of mechanics and the 6-12 Lennard-Jones potential function together with the continuum approximation, which assumes that intermolecular interactions can be approximated by average atomic surface densities. We consider the equilibrium offset positions for both single-strand and double-strand DNA molecules inside a single-walled carbon nanotube, and we predict offset positions with reference to the cross-section of the carbon nanotube. For the double-strand DNA, the potential energy is determined for the general case for any helical phase angle ϕ, but we also consider a special case when ϕ = π, which leads to a substantial simplification in the analytical expression for the energy. As might be expected, our results confirm that the global minimum energy positions for a single-strand DNA molecule and a double-strand DNA molecule will lie off axis and they become closer to the tube wall as the radius of the tube increases.

High performance hydrogen storage from Be-BTB metal-organic framework at room temperature.

The metal-organic framework beryllium benzene tribenzoate (Be-BTB) has recently been reported to have one of the highest gravimetric hydrogen uptakes at room temperature. Storage at room temperature is one of the key requirements for the practical viability of hydrogen-powered vehicles. Be-BTB has an exceptional 298 K storage capacity of 2.3 wt % hydrogen. This result is surprising given that the low adsorption enthalpy of 5.5 kJ mol(-1). In this work, a combination of atomistic simulation and continuum modeling reveals that the beryllium rings contribute strongly to the hydrogen interaction with the framework. These simulations are extended with a thermodynamic energy optimization (TEO) model to compare the performance of Be-BTB to a compressed H2 tank and benchmark materials MOF-5 and MOF-177 in a MOF-based fuel cell. Our investigation shows that none of the MOF-filled tanks satisfy the United States Department of Energy (DOE) storage targets within the required operating temperatures and pressures. However, the Be-BTB tank delivers the most energy per volume and mass compared to the other material-based storage tanks. The pore size and the framework mass are shown to be contributing factors responsible for the superior room temperature hydrogen adsorption of Be-BTB.

Catalytic effect of graphene on the inversion of corannulene using a continuum approach with the Lennard-Jones potential.

The catalytic effect of graphene on the corannulene bowl-to-bowl inversion is confirmed in this paper using a pair-wise dispersion interaction model. In particular, a continuum approach together with the Lennard-Jones potential are adopted to determine the interaction energy between corannulene and graphene. These results are consistent with previous quantum chemical studies, which showed that a graphene sheet reduces the barrier height for the bowl-to-bowl inversion in corannulene. However, the results presented here demonstrate, for the first time, that the catalytic activity of graphene can be reproduced using pair-wise dispersion interactions alone. This demonstrates the major role that pair-wise dispersion interactions play in the catalytic activity of graphene.

Modelling peptide nanotubes for artificial ion channels.

We investigate the van der Waals interaction of D,L-Ala cyclopeptide nanotubes and various ions, ion-water clusters and C(60) fullerenes, using the Lennard-Jones potential and a continuum approach which assumes that the atoms are smeared over the peptide nanotube providing an average atomic density. Our results predict that Li(+), Na(+), Rb(+) and Cl(-) ions and ion-water clusters are accepted into peptide nanotubes of 8.5 Å internal diameter whereas the C(60) molecule is rejected. The model indicates that the C(60) molecule is accepted into peptide nanotubes of 13 Å internal diameter, suggesting that the interaction energy depends on the size of the molecule and the internal diameter of the peptide nanotube. This result may be useful for the design of peptide nanotubes for drug delivery applications. Further, we also find that the ions prefer a position inside the peptide ring where the energy is minimum. In contrast, Li(+)-water clusters prefer to be in the space between each peptide ring.

Dislodgement of carbon nanotube bundles under pressure driven flow.

Experimental and predicted flow rates through carbon nanotubes vary considerably but generally are reported to be well in excess of that predicted by the conventional Poiseuille flow, and therefore nanotubes embedded in a matrix might provide membranes with exceptional mass transport properties. In this paper, applied mathematical modelling is undertaken to estimate the three forces acting on a nanotube bundle, namely the molecular interaction force, the viscous force, and the static pressure force. In deducing estimates of these forces we introduce a modification of the notion of the effective dead area for a carbon nanotube membrane, and we calculate the total forces necessary to push one or more of the nanotubes out of the bundle, thus creating a channel through which further enhancement of flow may take place. However, careful analysis shows that the nett dislodgement force is entirely independent on the useable flow area, but rather depends only on the total cross-sectional area perpendicular to the flow. This rather surprising result is a consequence of the flow being steady and a balance of the viscous and pressure forces.

Variational model for a rippled graphene sheet.

The calculus of variations is utilised to study the behaviour of a rippled graphene sheet supported on a metal substrate. We propose a model that is underpinned by two key parameters, the bending rigidity of graphene γ, and the van der Waals interaction strength ξ. Three cases are considered, each of which addresses a specific configuration of a rippled graphene sheet located on a flat substrate. The transitional case assumes that both the graphene sheet length and substrate length are constrained. The substrate constrained case assumes only the substrate has a constrained length. Finally, the graphene constrained case assumes only the length of the graphene sheet is constrained. Numerical results are presented for each case, and the interpretation of these results demonstrates a continuous relationship between the total energy per unit length and the substrate length, that incorporates all three configurations. The present model is in excellent agreement with earlier results of molecular dynamics (MD) simulations in predicting the profiles of graphene ripples.

An idealized polyhedral model and geometric structure for silicon nanotubes.

In this paper, we introduce an idealized model of silicon nanotubes comprising an exact polyhedral geometric structure for single-walled silicon nanotubes. The silicon nanotubes considered here are assumed to be formed by sp(3) hybridization and thus the nanotube lattice is assumed to comprise only squares or skew rhombi. Beginning with the three postulates that all bond lengths are equal, all adjacent bond angles are equal, and all atoms are equidistant from a common axis of symmetry, we derive exact formulae for the geometric parameters such as radii, bond angles and unit cell length. We present asymptotic expansions for these quantities to the first two orders of magnitude. Because of the faceted nature of the polyhedral model we may determine a perceived inner radius for the nanotube, from which an expression for the wall thickness emerges. We also describe the geometric properties of some ultra-small silicon nanotubes. Finally, the values of the diameters for the polyhedral model are compared with results obtained from molecular dynamics simulations and some limited numerical calculations are undertaken to confirm the meta-stability of the proposed structures.

Oscillation of carbon molecules inside carbon nanotube bundles.

In this paper, we investigate the mechanics of a nanoscaled gigahertz oscillator comprising a carbon molecule oscillating within the centre of a uniform concentric ring or bundle of carbon nanotubes. Two kinds of oscillating molecules are considered, which are a carbon nanotube and a C(60) fullerene. Using the Lennard-Jones potential and the continuum approach, we obtain a relation between the bundle radius and the radii of the nanotubes forming the bundle, as well as the optimum bundle size which gives rise to the maximum oscillatory frequency for both the nanotube-bundle and the C(60)-bundle oscillators. While previous studies in this area have been undertaken through molecular dynamics simulations, this paper emphasizes the use of applied mathematical modelling techniques, which provides considerable insight into the underlying mechanisms of the nanoscaled oscillators. The paper presents a synopsis of the major results derived in detail by the present authors (Cox et al 2007 Proc. R. Soc. A 464 691-710 and Cox et al 2007 J. Phys. A: Math. Theor. 40 13197-208).

Carbon Nanocones with Curvature Effects Close to the Vertex.

The conventional rolled-up model for carbon nanocones assumes that the cone is constructed from a rolled-up graphene sheet joined seamlessly, which predicts five distinct vertex angles. This model completely ignores any effects due to the changing curvature, and all bond lengths and bond angles are assumed to be those for the planar graphene sheet. Clearly, curvature effects will become more important closest to the cone vertex, and especially so for the cones with the smaller apex angles. Here, we construct carbon nanocones which, in the assembled cone, are assumed to comprise bond lengths and bond angles that are, as far as possible, equal throughout the structure at the same distance from the conical apex. The predicted bond angles and bond lengths are shown to agree well with those obtained by relaxing the conventional rolled-up model using Lammps software (version: 11 September 2008). The major objective here is not simply to model physically realisable carbon nanocones for which numerical procedures are far superior, but rather, to produce an improved model that takes curvature effects close to the vertex into account, and from which we may determine an analytical formula which represents an improvement on the conventional rolled-up model.

Methane storage in molecular nanostructures.

We survey various molecular structures which have been proposed as possible nanocontainers for methane storage. These are molecular structures that have been investigated through either experiments, molecular dynamics simulations or mathematical modelling. Computational simulation and mathematical modelling play an important role in predicting and verifying experimental outcomes, but both have their limitations. Even though recent advances have greatly improved computations, due to the large number of atoms and force field calculations involved, computational simulations can still be time consuming as compared to an instantaneous mathematical modelling approach. On the other hand, underlying an ideal mathematical model, there are many assumptions and approximations, but such modelling often reveals the key physical parameters and optimal configurations. Here, we review methane adsorption for three conventional nanostructures, namely graphite, single and multi-walled carbon nanotubes, and nanotube bundles (including interstitial and groove sites), and we survey methane adsorption in other molecular structures including metal organic frameworks. We also include an examination of minimum binding energies, equilibrium distances, gravimetric and volumetric uptakes, volume available for adsorption, as well as the effects of temperature and pressure on the adsorption of methane onto these molecular structures.

Instability of C60 fullerene interacting with lipid bilayer.

Due to the large number of possible applications of nanoparticles in cosmetic and medical products, the possible hazards of nanoparticles in the human body are a major concern. A worst-case scenario is that nanoparticles might cause health issues such as skin damage or even induce cancer. As a first step to study the toxicity of nanoparticles, we investigate the energy behaviour of a C(60) fullerene interacting with a lipid bilayer. Using the 6-12 Lennard-Jones potential function and the continuous approximation, the equilibrium spacing between the two layers of a bilayer is predicted to be 3.36 Å. On assuming that there is a circular hole in the lipid bilayer, a relation for the molecular interaction energy is determined, involving the circular radius b of the hole and the perpendicular distance Z of the spherical fullerene from the hole. A graph of the minimum energy location Z ( min ) verses the hole radius b shows that a C(60) fullerene first penetrates through a lipid bilayer when b > 6.81 Å, and shows a simple circular relation [Formula: see text] for Z ( min ) positive and b ≤ 6.81 Å. For b > 6.81, the fullerene relocates from the surface of the bilayer to the interior, and as the hole radius increases further it moves to the centre of the bilayer and remains there for increasing hole radii. Accordingly, our modelling indicates that at least for the system with no external forces, the C(60) fullerene will not penetrate through the lipid bilayer but rather remains encased between the two layers at the mid-plane location.

The geometric structure of single-walled nanotubes.

In this paper, we survey a number of existing geometric structures which have been proposed by the authors as possible models for various nanotubes. Atoms assemble into molecules following the laws of quantum mechanics, and in general computational approaches to predicting the molecular structure can be arduous and involve considerable computing time. Fortunately, nature favours minimum energy structures which tend to be either very symmetric or very unsymmetric, and which therefore can be analyzed from a geometrical perspective. The conventional rolled-up model of nanotubes completely ignores any effects due to curvature and the present authors have proposed a number of exact geometric models. Here we review a number of these recent developments relating to the geometry of nanotubes, including both the traditional rolled-up models and some exact polyhedral constructions. We review a number of formulae for four materials, carbon, silicon, boron and boron nitride, and we also include results for the case when the bond lengths may take on distinct values.

Junctions between a boron nitride nanotube and a boron nitride sheet.

For future nanoelectromechanical signalling devices, it is vital to understand how to connect various nanostructures. Since boron nitride nanostructures are believed to be good electronic materials, in this paper we elucidate the classification of defect geometries for combining boron nitride structures. Specifically, we determine possible joining structures between a boron nitride nanotube and a flat sheet of hexagonal boron nitride. Firstly, we determine the appropriate defect configurations on which the tube can be connected, given that the energetically favourable rings for boron nitride structures are rings with an even number of sides. A new formula E = 6+2J relating the number of edges E and the number of joining positions J is established for each defect, and the number of possible distinct defects is related to the so-called necklace and bracelet problems of combinatorial theory. Two least squares approaches, which involve variation in bond length and variation in bond angle, are employed to determine the perpendicular connection of both zigzag and armchair boron nitride nanotubes with a boron nitride sheet. Here, three boron nitride tubes, which are (3, 3), (6, 0) and (9, 0) tubes, are joined with the sheet, and Euler's theorem is used to verify geometrically that the connected structures are sound, and their relationship with the bonded potential energy function approach is discussed. For zigzag tubes (n,0), it is proved that such connections investigated here are possible only for n divisible by 3.