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Pneumatic conveying is the transportation of bulk solids in enclosed pipelines via a carrier gas, typically air. The local flow pattern in a pipeline is a function of the conditions, and slug flow can form under certain conditions. Slug flow is a naturally occurring, wave-like flow where the bulk material travels along the pipeline in distinct `slugs'. Establishing the environment for the formation of slugs within the conveying system is essential to maximise the overall system efficiency and minimise damage to the bulk material. MISG2021 considered a wide range of mathematical approaches to slug formation and travel. These two key problem areas have the most significant potential to impact the system design and efficiency. Critical interconnected facets of pneumatic conveying systems were investigated and an overview for future work was developed. Many of the avenues uncovered during the MISG2021 require more time for in-depth analysis. This analysis and framework will aid in optimising conveying system design and provide insight to construct more efficient pneumatic conveying systems. Refereed/Peer-reviewed

The Lennard&ndash, Jones potential and a continuum approach can be used to successfully model interactions between various regular shaped molecules and nanostructures. For single atomic species molecules, the interaction can be approximated by assuming a uniform distribution of atoms over surfaces or volumes, which gives rise to a constant atomic density either over or throughout the molecule. However, for heterogeneous molecules, which comprise more than one type of atoms, the situation is more complicated. Thus far, two extended modeling approaches have been considered for heterogeneous molecules, namely a multi-surface semi-continuous model and a fully continuous model with average smearing of atomic contribution. In this paper, we propose yet another modeling approach using a single continuous surface, but replacing the atomic density and attractive and repulsive constants in the Lennard&ndash, Jones potential with functions, which depend on the heterogeneity across the molecules, and the new model is applied to study the adsorption of coronene onto a graphene sheet. Comparison of results is made between the new model and two other existing approaches as well as molecular dynamics simulations performed using the LAMMPS molecular dynamics simulator. We find that the new approach is superior to the other continuum models and provides excellent agreement with molecular dynamics simulations.

The production of single dimensional carbon structures has recently been made easier using carbon nanotubes. We consider here encapsulated coronene molecules, which are flat and circular-shaped polycyclic aromatic hydrocarbons, inside carbon nanotubes. Depending on the radius of the nanotube, certain specific configurations of the coronene molecules can be achieved that give rise to the formation of stacked columns or aid in forming nanoribbons. Due to their symmetrical structure, a coronene molecule may be modelled by three inner circular rings of carbon atoms and one outer circular ring of hydrogen atoms, while the carbon nanotube is modelled as a circular tube. Using the continuous model and the Lennard-Jones potential, we are able to analytically formulate an expression for the potential energy for a coronene dimer and coronene inside a carbon nanotube. Subsequently, stacking of coronene molecules inside a nanotube is investigated. We find that the minimum energy tilt angle of coronenes in a stack differs from that of a single coronene within the same nanotube. More specifically, for both (18, 0) and (19, 0) zigzag carbon nanotube, we find that the minimum energy tilt angles of the single coronene case (&asymp, 42 ∘ and &asymp, 20 ∘ respectively) do not occur in the stack model.

Solar energy is an alternative source of energy that can be used to replace fossil fuels.Various types of solar cells have been developed to harvest this seemingly endless supply of energy,leading to the construction of solar cell devices, such as dye-sensitized solar cells. An importantfactor that affects energy conversion efficiency of dye-sensitized solar cells is the distribution ofdye molecules within the porous semiconductor (TiO2). In this paper, we formulate a continuummodel for the interaction between the dye molecule Tris(2,20-bipyridyl)ruthenium(II) (Ru(bpy)2+3 )and titanium dioxide (TiO2) semiconductor. We obtain the equilibrium position at the minimumenergy position between the dye molecules and between the dye and TiO2 nanoporous structure.Our main outcome is an analytical expression for the energy of the two molecules as a function oftheir sizes. We also show that the interaction energy obtained using the continuum model is in closeagreement with molecular dynamics simulations. Refereed/Peer-reviewed

We examine the van der Waals interactions between water molecules with both water channels, aquaporin-Z and glycerol channel GlpF. Here we model these problems using classical applied mathematics and obtain the potential energy for a water molecule interacting with the channels which we assume in both cases to have a flaired right cylindrical geometry. We propose a continuous model where all the atoms comprising the channels are assumed to be uniformly distributed within their volume. We model a water molecule as comprising two parts: firstly as a single point representing the location of the oxygen atom, and a spherical shell over which we assume a uniform distribution of the two hydrogen atoms. Our results indicate the spontaneous acceptance of water molecules into these channels. Refereed/Peer-reviewed

Aquaporin (AQP) is a family of membrane proteins that enable water and small individual molecules to permeate cell membranes. Examples of these protein channels are aquaglyceroporin and aquaporin-1 (AQP1). Here, we investigate the permeability of carbon dioxide $$(\hbox {CO}_2)$$ through both aquglyceroporin and AQP1 channels and explain their selectivity mechanisms. We provide a mathematical model which determines the molecular interaction potential between carbon dioxide molecule and an AQP channel. We evaluate this interaction using two approaches, namely discrete-continuum and completed discrete approaches. Both calculations agree well and our results indicate the acceptance of $$(\hbox {CO}_2)$$ molecule into these channels which is in good agreement with other recent studies.

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 Refereed/Peer-reviewed

In recent years, peptide nanotubes have been studied extensively due to their unique structure and their potential applications in many diverse fields such as biology, chemistry, material science and medicine. Here, we determine the Lennard-Jones interactions between peptide rings through simple applied mathematical modelling as a first step towards fully understanding the formation of the complex structure of peptide nanotubes. In this paper we study the interaction potential between a pair of cyclo[(-D-Ala-L-Ala)(4)-] and a pair of cyclo[(Gly-D-Ala)(4)] which are the simplest possible units comprising a peptide nanotube. We assume the continuum approximation that the atoms maybe smeared uniformly over the peptide ring providing an average atomic density and the Lennard Jones potential to obtain the interaction energy between two peptide rings. Our results show that the two peptide rings reach equilibrium when the perpendicular distance between the centre of the two rings is 4.45-5.38 angstrom, and the minimum energy is 14.86-23.65 kcal/mol occuring when the second ring tilts at an angle 0.2443-0.3316 radian from the first ring. These results are in agreement with several recent molecular dynamics studies. Refereed/Peer-reviewed

Carbon nanotubes have attracted considerable interest for their use as carriers of drugs across biological barriers. In this work, we investigate the encapsulation of L-Histidine amino acid inside a single-walled carbon nanotube, and determine the resultant interaction energy for various sizes of the nanotubes. In our model, the L-Histidine amino acid is accounted for in four parts; the inner ring group, the half outer group, the linear part and the cylinder group, all interacting with a single-walled carbon nanotube. We calculate the acceptance and suction energies which depend on the radius r of the carbon nanotube and the orientation angle Ø that the amino acid makes with the central axis of the nanotube. Our results indicate the acceptance of the L-Histidine amino acid into carbon nanotubes of r >3.7 Å, which is in good agreement with other recent studies. Refereed/Peer-reviewed

Benzene molecules confined in carbon nanotubes of varying radii are employed as semiconductors in electronic nanodevices, and their orientation determines the electrical properties of the system. In this paper, we investigate the interaction energy of all the possible configurations of a benzene molecule inside various carbon nanotubes and then we determine the equilibrium configuration. We adopt the continuous approach together with the semi-empirical Lennard-Jones potential function to model van der Waals interaction between a benzene molecule and a carbon nanotube. This approach results in an analytical expression, which accurately approximates the interaction energy and can be readily used to generate numerical data. We find that horizontal, tilted and perpendicular configurations on the axis of the carbon nanotube are all possible equilibrium configurations of the benzene molecule when the radius of the carbon nanotube is less than 5.580 . However, when the radius of the carbon nanotube is larger than 5.580 an offset horizontal orientation is the only possible equilibrium configuration of the benzene molecule. In the limiting case, the orientation of a benzene molecule on a graphene sheet can be derived simply by letting the radius of the carbon nanotube tend to infinity. Refereed/Peer-reviewed

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 C 60 -carbon nanotube bundle, C 60 -boron nitride nanotube bundle, B 36 N 36 -carbon nanotube bundle and B 36 N 36 -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.

At the nanoscale a number of very high frequency oscillating systems involving relative motion with respect to a carbon nanotube have been identified. In this paper, we study the two-body systems of an atom and a fullerene C60 orbiting around a single infinitely long carbon nanotube and a fullerene C60 orbiting around a fullerene C1500. The van der Waals interaction forces are modeled using the Lennard–Jones potential together with the continuum approach for which carbon atoms are assumed to be uniformly distributed over the surfaces of both the fullerenes and the carbon nanotube. Some analytical and perturbation solutions are obtained for the regime where the attractive term of the potential energy dominates. Certain circular orbiting radii of these nanoscale systems are estimated using a stability argument and the corresponding circular orbiting frequencies can then be calculated by investigating the minimum energy configuration of their effective potential energies. We find that the circular orbiting frequencies of the various proposed nano-systems are in the gigahertz range. Finally, the classification of their orbiting paths is determined numerically.

Nanostructures such as carbon nanotubes and fullerenes offer the means to create new mechanical devices operating at the nanoscale. Such devices include oscillators constructed from an inner carbon nanotube sliding inside another carbon nanotube. The resultant oscillatory frequency is found to be in the gigahertz range and they have applications in the computing industry for signalling devices, such as an ultra-fast optical filter. While most research in the area is dominated by molecular dynamics simulations, our approach here is to use elementary mechanical principles and classical applied mathematical modelling techniques to formulate explicit analytical criteria and ideal model behaviour. In this paper, we first investigate the suction force experienced by a single-walled carbon nanotube located near an open end of a semi-infinite single-walled carbon nanotube, using the Lennard-Jones potential and the continuum approximation. Second the equilibrium position of an offset inner tube with reference to the cross-section of the outer tube is determined Refereed/Peer-reviewed

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 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.These interactions are calculated based on the structure of the antiviral molecules. Our results show that the binding of fullerene C to the antiviral molecules increases the efficiency of the compound to prohibit the activity of HIV. Refereed/Peer-reviewed

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).

Aromatic hydrocarbon molecules encapsulated in carbon nanotubes have been proposed for applications as semiconductors. They can be formed by exploiting the van der Waals interaction as a simple method to incorporate molecules into carbon nanotubes. However, the existence of energy barriers near the open ends of carbon nanotubes may be an obstacle for molecules entering carbon nanotubes. In this paper, we investigate the encapsulation mechanism of a typical aromatic hydrocarbon, namely a benzene molecule, into a carbon nanotube in order to determine the dependence on radius of the tube. A continuous approach which assumes that the molecular interactions can be approximated using average atomic densities together with the semi-empirical Lennard-Jones potential function is adopted, and an analytical expression for the interaction energy is obtained which may be readily evaluated by algebraic computer packages. In particular, we determine the threshold radius of the carbon nanotube for which the benzene molecule will enter the carbon nanotube. The analytical approach adopted here provides a computationally rapid procedure for the determination of critical numerical values. Refereed/Peer-reviewed

n this paper, we investigate both pre- and post-buckling behaviors of multi-walled carbon nanotubes and multi-walled carbon nanopeapods by incorporating into the applied forces of a prescribed beam equation both van der Waals interactions between the adjacent walls of the nanotubes and the interactions between the fullerenes and the inner wall of the nanotube. Two beam theories are employed. First, we utilize Donnell's equilibrium equation to derive an axial stability condition for the multi-walled carbon nanotubes and multi-walled carbon nanopeapods. We then determine analytically the critical forces for single-walled and double-walled nanotubes and nanopeapods. Given the outer nanotube of a fixed radius, we observe that the critical force and strain derived from the axial buckling stability criterion decrease as a result of the molecular interactions between the adjacent layers of the nanotubes and the molecular interactions between the embedded fullerenes and the inner carbon nanotube, which is in agreement with existing literature. Next, we utilize an Euler-Bernoulli beam equation incorporating the curvature effect to obtain the post-buckled axial bending displacement for the multi-walled nanotubes and nanopeapods. We find that the interactions between molecules generate an inward force, which tends to resist any applied forces. While the inward force induced by the fullerenes to the inner wall of the nanotube vanishes as we increase the applied force, the inward force induced by the layers increases as the applied force increases. The main contribution of this paper is the incorporation of both van der Waals interactions and the curvature effect into prescribed beam theories to accurately measure the critical forces and the buckled displacements of multi-walled nanotubes and nanopeapods subject to a small external force. Our analysis is relevant to future nano devices, such as biological sensors and measuring devices for small forces arising from electrical charges or Casimir forces. Refereed/Peer-reviewed

Polycyclic aromatic hydrocarbons (PAHs) are emitted into the atmosphere as byproducts from combustion. Once emitted into the atmosphere, PAHs either can exist in the gas phase or they can be adsorbed onto particle surfaces, such as soot and charcoal. PAHs are one of the most widespread pollutants and are known carcinogens, mutagens and teratologies for humans. Unfortunately, they are easily inhaled when absorbed on the surfaces of airborne soot particles produced by the incomplete combustion of carbonaceous fuels. Studying the mechanism of adsorption of PAHs onto soot surfaces is therefore an important problem. In this work, we chose coronene (C24H12) to model the interaction between a typical PAH and agraphite surface. We comment that using conventional computational methods, such as full ab initio, isusually not feasible owing to the large molecules involved. Accordingly, we adopt an applied mathematical modelling approach and we therefore exploit the continuous atomistic approximation together withthe Lennard–Jones potential in order to investigate this problem. The major result of this study is an analytical expression for the interaction energy which we then use to describe the mechanism of adsorptionof a coronene molecule on a graphite surface. Refereed/Peer-reviewed

The interaction between aromatic rings is a fundamental problem in material science and biochemistry. These interactions are generally found to stabilise protein molecules and the double helical structure of DNA, and they also play an important role in the recognition processes in biological and non-biological systems. However, the complexity and variety in the structures and components of aromatic compounds are major obstacles to investigating their interactions. In this study, the simplest case of aromatic interactions, which is the interaction between two benzene rings, is modelled using a continuous approximation. Assuming a constant atomic surface density and modelling the structure of a benzene molecule as a combination of two rings, namely an inner carbon ring and an outer hydrogen ring, the van der Waals interaction between any two benzene rings can be obtained as the sum of four interactions. The major result obtained here is an analytical expression for the potential energy which can then be used to predict equilibrium configurations for two interacting benzene molecules. Moreover, we find that at sufficiently large distances between the two benzene molecules, the orientational angle phi at which the interaction energy is a minimum can be approximated by the arctan of the ratio of two separation distances in two mutually perpendicular directions. Refereed/Peer-reviewed

Refereed/Peer-reviewed