Modelling interactions between a PBB and fullerenes

Fullerenes have many uses including in medical and electronic nanodevices. High pressure liquid chromatography (HPLC) columns are generally used to extract a certain structure of fullerne from a mixture of them. In this paper, we investigate the interactions between various types of fullerenes and a station phase in HPLC known as pentabromobenzyl (PBB). The Lennard-Jones potential and a continuum approach are employed to determine the van der Waals energy of these interactions within the HPLC columns. The equilibrium configurations for any given distance between a fullerene and the centre of a PBB are obtained. Results of this study may assist the design of a chromatography column for fullerene separation.     

Modelling interaction between ammonia and nitric oxide molecules and aquaporins

Aquaporin is a family of small membrane-proteins that are capable of transporting nano-sized materials. In the present paper, we investigate the structure of these channels and provide information about the mechanism of individual molecules being encapsulated into aquaglyceroporin (GlpF) and aquaporin-1 (AQP1) channels by calculating the potential energy. In particular, we presents a mathematical model to determine the total potential energy for the interaction of the ammonia and nitric oxide molecules and different aquaporin channels which we assume to have a symmetrical cylindrical structure. We propose to describe these interactions in two steps. Firstly, we model the nitrogen atom as a discrete point and secondly, we model the three hydrogen atoms on the surface of a sphere of a certain radius. Then, we find the total potential energy by summing these interactions. Next, by considering the nitric oxide molecule as two discrete atoms uniformly distributed interacting with GlpF and AQP1 channels then gathering all pairs of interaction to determine the potential energy. Our results show that the ammonia and nitric oxide molecules can be encapsulated into both GlpF and AQP1 channels.     

Energy density functions for protein structures

In this paper, we adopt the calculus of variations to studythe structure of protein with an energy functional dependent on the curvature, torsion and theirderivatives with respect to the arc length of the protein backbone.Minimising this energy among smooth normal variations yieldstwo Euler–Lagrange equations, which can be reduced toa single equation. This equation is identically satisfied forthe special case when the free-energy density satisfies a certainlinear condition on the partial derivatives. In the case whenthe energy depends only on the curvature and torsion, it canbe shown that this condition is satisfied if the free-energydensity is a homogeneous function of degree one. Another simplespecial solution for this case is shown to coincide with anenergy density linear in curvature, which has been examinedin detail by previous authors. The Euler–Lagrange equationsare illustrated with reference to certain simple special casesof the energy density function, and a family of conical helicesis examined in some detail.     

Nanotube bundle oscillators: Carbon and boron nitride nanostructures

In this paper, we investigate the oscillation of a fullerene that is moving within the centre of a bundle of nanotubes. In particular, certain fullerene–nanotube bundle oscillators, namely C₆₀-carbon nanotube bundle, C₆₀-boron nitride nanotube bundle, B₃₆N₃₆-carbon nanotube bundle and B₃₆N₃₆-boron nitride nanotube bundle are studied using the Lennard–Jones potential and the continuum approach which assumes a uniform distribution of atoms on the surface of each molecule. We address issues regarding the maximal suction energies of the fullerenes which lead to the generation of the maximum oscillation frequency. Since bundles are also found to comprise double-walled nanotubes, this paper also examines the oscillation of a fullerene inside a double-walled nanotube bundle. Our results show that the frequencies obtained for the oscillation within double-walled nanotube bundles are slightly higher compared to those of single-walled nanotube bundle oscillators. Our primary purpose here is to extend a number of established results for carbon to the boron nitride nanostructures.     

Continuum modelling of spherical and spheroidal carbon onions

Carbon nanostructures are of considerable interest owing to their unique mechanical and electronic properties. Experimentally, a wide variety of different shapes are obtained, including both spherical and spheroidal carbon onions. A spheroid is an ellipsoid with two major axes equal and the term onion refers to a multi-layered composite structure. Assuming structures of either concentric spherical or ellipsoidal fullerenes comprising n layers, this paper examines the interaction energy between adjacent shells for both spherical and spheroidal carbon onions. The Lennard-Jones potential together with the continuum approximation is employed to determine the equilibrium spacing between two adjacent shells. We also determine analytical formulae for the potential energy which may be expressed either in terms of hypergeometric or Legendre functions. We find that the equilibrium spacing between shells decreases for shells further out from the inner core owing to the decreasing curvature of the outer shells of a concentric structure.     

Stress distributions in highly frictional granular heaps

The practice of storing granular materials in stock piles occurs throughout the world in many industrial situations. As a result, there is much interest in predicting the stress distribution within a stock pile. In 1981, it was suggested from experimental work that the peak force at the base does not occur directly beneath the vertex of the pile, but at some intermediate point resulting in a ring of maximum pressure. With this in mind, any analytical solution pertaining to this problem has the potential to provide useful insight into this phenomenon. Here, we propose to utilize some recently determined exact parametric solutions of the governing equations for the continuum mechanical theory of granular materials for two and three-dimensional stock piles. These solutions are valid provided sin = 1, where is the angle of internal friction, and we term such materials as highly frictional. We note that there exists materials possessing angles of internal friction around 60 to 65 degrees, resulting in values of sin equal to around 0.87 to 0.91. Further, the exact solutions presented here are potentially the leading terms in a perturbation solution for granular materials for which 1- sin is close to zero. The model assumes that the stock pile is composed of two regions, namely an inner rigid region and an outer yield region. The exact parametric solution is applied to the outer yield region, and the solution is extended continuously into the inner rigid region. The results presented here extend previous work of the authors to the case of highly frictional granular solids.     

Two approximate analytical solutions for the kinematically determined velocity equations for granular solids

For axially symmetric flows of dilatant granular materials, the velocity equations uncouple from the stress equations in certain plastic regimes, and assuming dilatant double shearing a closed set of three first-order partial differential equations are obtained. These supposedly simple equations are deceptive, because although they are simple in appearance, the determination of exact solutions is non-trivial. For one of the known families of solutions which has not been studied previously, the authors present the non-linear ordinary differential equation for the stress angle ψ and determine two small ψ approximations. Furthermore, the stream function and streamlines are obtained for ψ determined numerically and from the two small ψ approximations. For purposes of comparison, the streamlines for three further known exact solutions are also presented. In addition, we briefly examine the circumstances for which solutions of the velocity equations satisfy the principle of non-negative plastic work. For example, we are able to establish that in the case when the velocity equations are derived from a plastic potential, the solutions always satisfy the principle when the material has no cohesion.     

Modelling adsorption of a water molecule into various pore structures of silica gel

Silica gel is widely used in commercial applications as a water adsorbent due to its properties including hydrothermally stable, high water sorption capacity, low regeneration temperature, low cost and wide range of pore diameters. Since the water sorption capacity of silica gel strongly depends on the pore size and structure, which can be controlled during synthesis, this paper study the effect of pore shapes and dimensions of silica gel upon the adsorption of a water molecule aiming at maximising the water sorption capacity. In particular, we consider three types of pore structures, namely cylindrical, square prismatic and conical pores. On using the Lennard-Jones potential and a continuum approximation, we find that the minimum radii for a water molecule to be accepted into cylindrical, square prismatic and conical pores are 4.009, 3.7898 and 4.4575 Å, respectively. For cylindrical and square prismatic pores, the critical radii which maximise the adsorption energy are 4.5189 and 4.1903 Å, respectively. Knowledge of these critical pore sizes may be useful for the manufacturing process of silica gel that will maximise the water sorption capacity.     

Willmore energy for joining of carbon nanostructures

Numerous types of carbon nanostructure have been found experimentally, including nanotubes, fullerenes and nanocones. These structures have applications in various nanoscale devices and the joining of these structures may lead to further new configurations with more remarkable properties and applications. The join profile between different carbon nanostructures in a symmetric configuration may be modelled using the calculus of variations. In previous studies, carbon nanostructures were assumed to deform according to perfect elasticity, thus the elastic energy, depending only on the axial curvature, was used to determine the join profile consisting of a finite number of discrete bonds. However, one could argue that the relevant energy should also involve the rotational curvature, especially when its size is comparable to the axial curvature. In this paper, we use the Willmore energy, a natural generalisation of the elastic energy that depends on both the axial and rotational curvatures. Catenoids are absolute minimisers of this energy and pieces of these may be used to join various nanostructures. We focus on the cases of joining a fullerene to a nanotube and joining two fullerenes along a common axis. By comparing our results with the earlier work, we find that both energies give similar joining profiles. Further work on other configurations may reveal which energy provides a better model.