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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

This special Section of the ANZIAM Journal (Electronic Supplement) contains the refereed papers from the 2021 Mathematics in Industry Study Group (MISG-2021) held at the University of Newcastle from 27 to 30 January 2021. This report provides the equation-free outcomes.

This special Section of the ANZIAM Journal (Electronic Supplement) contains the refereed papers from the 2022 Mathematics in Industry Study Group (MISG2022) held at the University of Newcastle from 14--18 February 2022. This report provides the equation-free outcomes.

Aggregations abound in nature, from cell formations to locust swarms. One method of modelling these aggregations is the non-local aggregation equation with the addition of degenerate diffusion. In this article we develop a finite volume based numerical scheme for this style of equation and perform an error, computation time, and convergence analysis. In addition we investigate two methods for approximating the non-local component. References A. J. Bernoff and C. M. Topaz. Nonlocal aggregation models: A primer of swarm equilibria. SIAM Rev. 55.4 (2013), pp. 709–747. doi: 10.1137/130925669 R. Bürger, D. Inzunza, P. Mulet, and L. M. Villada. Implicit-explicit methods for a class of nonlinear nonlocal gradient flow equations modelling collective behaviour. Appl. Numer. Math. 144 (2019), pp. 234–252. doi: 10.1016/j.apnum.2019.04.018 J. A. Carrillo, A. Chertock, and Y. Huang. A finite-volume method for nonlinear nonlocal equations with a gradient flow structure. In: Commun. Comput. Phys. 17.1 (2015), pp. 233–258. doi: 10.4208/cicp.160214.010814a J. R. Dormand and P. J. Prince. A family of embedded Runge–Kutta formulae. J. Comput. Appl. Math. 6.1 (1980), pp. 19–26. doi: 10.1016/0771-050X(80)90013-3 J. von zur Gathen and J. Gerhard. Modern computer algebra. 3rd ed. Cambridge University Press, 2013. doi: 10.1017/CBO9781139856065 F. Georgiou, J. Buhl, J. E. F. Green, B. Lamichhane, and N. Thamwattana. Modelling locust foraging: How and why food affects group formation. PLOS Comput. Biol. 17.7 (2021), e1008353. doi: 10.1371/journal.pcbi.1008353 F. Georgiou, B. P. Lamichhane, and N. Thamwattana. An adaptive numerical scheme for a partial integro-differential equation. Proceedings of the 18th Biennial Computational Techniques and Applications Conference, CTAC-2018. Ed. by B. Lamichhane, T. Tran, and J. Bunder. Vol. 60. ANZIAM J. 2019, pp. C187–C200. doi: 10.21914/anziamj.v60i0.14066 F. Georgiou, N. Thamwattana, and B. P. Lamichhane. Modelling cell aggregation using a modified swarm model. Proceedings of the 23rd International Congress on Modelling and Simulation, MODSIM2019. Vol. 6. 2019, pp. 22–27. doi: 10.36334/modsim.2019.a1.georgiou J. E. F. Green, S. L. Waters, J. P. Whiteley, L. Edelstein-Keshet, K. M. Shakesheff, and H. M. Byrne. Non-local models for the formation of hepatocyte–stellate cell aggregates. J. Theor. Bio. 267.1 (2010), pp. 106–120. doi: 10.1016/j.jtbi.2010.08.013 R. J. LeVeque. Finite-volume methods for hyperbolic Pproblems. Cambridge Texts in Applied Mathematics. Cambridge University Press, 2002. doi: 10.1017/CBO9780511791253 C. F. Van Loan. Introduction to Scientific Computing: A Matrix Vector Approach Using MATLAB. 1997. url: https://www.pearson.com/us/higher-education/program/Van- Loan-Introduction-to-Scientific-Computing-A-Matrix-Vector- Approach-Using-MATLAB-2nd-Edition/PGM215520.html A. Mogilner and L. Edelstein-Keshet. A non-local model for a swarm. J. Math. Bio. 38.6 (1999), pp. 534–570. doi: 10.1007/s002850050158 C. M. Topaz, A. L. Bertozzi, and M. A. Lewis. A nonlocal continuum model for biological aggregation. Bull. Math. Biol. 68 (2006), p. 1601. doi: 10.1007/s11538-006-9088-6 C. M. Topaz, M. R. D’Orsogna, L. Edelstein-Keshet, and A. J. Bernoff. Locust dynamics: Behavioral phase change and swarming. PLOS Comput. Bio. 8.8 (2012), e1002642. doi: 10.1371/journal.pcbi.1002642

Dye-sensitized solar cells consistently provide a cost-effective avenue for sources of renewable energy, primarily due to their unique utilization of nanoporous semiconductors. Through mathematical modelling, we are able to uncover insights into electron transport to optimize the operating efficiency of the dye-sensitized solar cells. In particular, fractional diffusion equations create a link between electron density and porosity of the nanoporous semiconductors. We numerically solve a fractional diffusion model using a finite-difference method and a finite-element method to discretize space and an implicit finite-difference method to discretize time. Finally, we calculate the accuracy of each method by evaluating the numerical errors under grid refinement.

In this paper, ice melting under the impacts of water waves was studied numerically via smoothed particle hydrodynamics simulations. Effects due to the ice elasticity were also included. Accordingly, the melting of an ice plate, modeled as an elastic object and interacting with transitional water waves with wave height and wave steepness up to 0.32 m and 0.093, respectively, was simulated and analyzed. The simulations showed that water waves' effects on the ice melting are seen via overflow over the top surface and local fluid circulations in the submerged region due to water–ice interactions and wave motions. Those effects result in a melting amount of the ice plate up to 1.78 times higher than the ice in still water. The overflow contributes up to 25% of the total amount of the melted ice. In comparison, fluid convection in the submerged region also leads to an increase in about 43% in the ice-melting amount over the submerged region. The melting rate is seen highest at the early stage of the simulation period and then is constantly reducing. The melting rate of the ice is seen linearly varying with the initial water temperature.

Methane capture and storage are of particular importance for the development of new technology to reduce the effects of climate change and global warming. Carbon-based nanomaterials are among several porous nanomaterials proposed as potential candidates for methane storage. In this paper, we adopt a new continuum approach with functional Lennard-Jones parameters to provide interaction energies for methane inside carbon nanostructures, namely fullerenes, nanotube bundles, and nanocones. This study provides a significant improvement to previous continuum modeling approaches using the Lennard-Jones potential.

This paper investigates the potential use of carbon nanocones as selective filtration devices. Using a continuum approach and the Lennard-Jones potential, we determine the energy of truncated carbon nanocones interacting with ions (Na $$^{+}$$ and Cl $$^{-}$$ ) and water molecules. The Verlet algorithm is adopted to determine the dynamics of the ions and the water molecules as a result of the interaction with the nanocones. The acceptance energy is derived to determine the minimum and critical radii of the truncated nanocones that block the ions and allow only water molecules to pass through. Our results show that the channel with apex angle of $$19.2^{\circ}$$ and opening radius in the range 3.368–3.528 A gives highest suction energy.

Phagocytosis refers to a process in which one cell type fully encloses and consumes unwanted cells, debris or particulate matter. It has an important role in immune systems through the destruction of pathogens and the inhibiting of cancerous cells. In this paper, we combine cell–cell adhesion and predator–prey modelling to generate a new model for phagocytosis that can relate the interaction between cells in both space and time. Stability analysis for both homogeneous and non-homogeneous steady states is provided for one-dimensional model indicating the range of parameters that leads to phagocytosis. Finally, the paper presents numerical results for both one and two-dimensional models, which show excellent agreement with a real phenomenon of bacteria phagocytized by neutrophil cell.

One method of modelling cell-cell adhesion gives rise to a partial integro-differential equation. While non-adaptive techniques work in the numerical modelling of such an equation, there are also many opportunities for optimisation. The studied partial integro-differential equation has a tendency to produce aggregations leaving large regions where both the function value and derivative are equal to zero, leading to a higher resolution than needed and lower than desired resolution where the aggregations form. In order to overcome this we develop an adaptive scheme in both space and time using a modified form of Matlab's ode45 and finite volume methods to more efficiently simulate the studied partial integro-differential equation. We use our numerical scheme to simulate the problem presented by Armstrong et al. [J. Theor. Biol. 243 (2006), pp. 98--113] and compare results. References N. J. Armstrong, K. J. Painter, and J. A. Sherratt. A continuum approach to modelling cell-cell adhesion. J. Theor. Biol., 243:98113, 2006. doi:10.1016/j.jtbi.2006.05.030. R. J. LeVeque. Finite volume methods for hyperbolic problems. Cambridge Texts in Applied Mathematics. Cambridge University Press, 2002. doi:10.1017/CBO9780511791253. C. F. Van Loan. Introduction to scientific computing: A matrix vector approach using MATLAB. MATLAB Curriculum. Prentise Hall, 1997. J. A. Sheratt, S. A. Gourley, N. J. Armstrong, and K. J. Painter. Boundedness of solutions of a non-local reaction-diffusion model for adhesion in cell aggregation and cancer invasion. Eur. J. Appl. Math., 20(1):123144, 2009. doi:10.1017/S0956792508007742. M. S. Steinberg. On the mechanism of tissue reconstruction by dissociated cells, I. Population kinetics, differential adhesiveness and the absence of directed migration. P. Natl Acad. Sci. USA, 48(9):15771582, 1962a. doi:10.1073/pnas.48.9.1577. M. S. Steinberg. Mechanism of tissue reconstruction by dissociated cells, II: Time-course of events. Science, 137(3532):762763, 1962b. doi:10.1126/science.137.3532.762. M. S. Steinberg. On the mechanism of tissue reconstruction by dissociated cells, III. Free energy relations and the reorganisation of fused, heteronomic tissue fragments. P. Natl Acad. Sci. USA, 48(10):17691776, 1962c. doi:10.1073/pnas.48.10.1769. P. L. Townes and J. Holtfreter. Directed movements and selective adhesion of embryonic amphibian cells. J. Exp. Zool., 128(1):53120, 1955. doi:10.1002/jez.1401280105.

Dye-sensitized solar cells (DSSCs) are a novel approach to the renewable energy problem, allowing sunlight conversion while reducing production costs. Electricity generation is achieved through a series of chemical reactions designed to transport electrons as a means of creating a circuit. Current challenges facing their development is to reduce the impact of recombination reactions, whilst enhancing their efficiency. In this paper, we investigate the behaviour of DSSCs by solving a set of partial differential equations which are used to model the density of electrons in the conduction band of a DSSC's semiconductor and in the electrolyte couple. Previously, approaches to solve these equations are by assuming steady-state models and then making use of numerical methods. In this paper, we obtain full analytical solutions to these equations and based on values of parameters associated with DSSCs available in the literature, we derive results which can be used to determine the performance of DSSCs.

In this work, a newly developed Smoothed Particle Hydrodynamics (SPH) algorithm for nonlinear elasticity is combined with an incompressible SPH fluid solver to investigate the dynamics of a floating plate under impacts of regular water waves with a high steepness. Two scenarios of the plate's rigidity are investigated. The simulation results show that deformations of the stiffer plate mainly occur in a simple bending mode with small amplitudes, and the plate is almost submerged by a strong fluid flow over its surface. In the other scenario, the plate deforms more complexly with much higher deformation amplitudes but experiences a much weaker overwash. The more flexible plate is less resistant to wave motions and converts more wave energy into elastic deformations, and therefore, the overwash is less severe. A strong overflow exerts a pressure force onto the plate that alters the plate's dynamics and adds a viscous (damping) effect on the plate's elastic vibrations, especially in high-frequency modes. A rigorous examination of the numerical convergence and validation using the linear thin plate theory is also carried out. The new SPH algorithm for nonlinear elasticity shows its stability and reliability in evaluating finite and large elastic deformations. Therefore, it is promising for simulating elastic structures in fluid–structure interaction problems.

Locusts are short horned grasshoppers that exhibit two behaviour types depending on their local population density. These are: solitarious, where they will actively avoid other locusts, and gregarious where they will seek them out. It is in this gregarious state that locusts can form massive and destructive flying swarms or plagues. However, these swarms are usually preceded by the aggregation of juvenile wingless locust nymphs. In this paper we attempt to understand how the distribution of food resources affect the group formation process. We do this by introducing a multi-population partial differential equation model that includes non-local locust interactions, local locust and food interactions, and gregarisation. Our results suggest that, food acts to increase the maximum density of locust groups, lowers the percentage of the population that needs to be gregarious for group formation, and decreases both the required density of locusts and time for group formation around an optimal food width. Finally, by looking at foraging efficiency within the numerical experiments we find that there exists a foraging advantage to being gregarious., Author summary Locusts are short horned grass hoppers that live in two diametrically opposed behavioural states. In the first, solitarious, they will actively avoid other locusts, whereas the second, gregarious, they will actively seek them out. It is in this gregarious state that locusts form the recognisable and destructive flying adult swarms. However, prior to swarm formation juvenile flightless locusts will form marching hopper bands and make their way from food source to food source. Predicting where these hopper bands might form is key to controlling locust outbreaks. Research has shown that changes in food distributions can affect the transition from solitarious to gregarious. In this paper we construct a mathematical model of locust-locust and locust-food interactions to investigate how food distributions affect the aggregation of juvenile locusts, termed groups, an important precursor to hopper bands. Our findings suggest that there is an optimal food distribution for group formation and that being gregarious increases a locusts ability to forage when food becomes more patchy.

Dye-sensitized solar cells have continued to receive much attention since their introduction by O`Regan and Gr&auml, tzel in 1991. Modelling charge transfer during the sensitization process is one of several active research areas for the development of dye-sensitized solar cells in order to control and improve their performance and efficiency. Mathematical models for transport of electron density inside nanoporous semiconductors based on diffusion equations have been shown to give good agreement with results observed experimentally. However, the process of charge transfer in dye-sensitized solar cells is complicated and many issues are in need of further investigation, such as the effect of the porous structure of the semiconductor and the recombination of electrons at the interfaces between the semiconductor and electrolyte couple. This paper proposes a new model for electron transport inside the conduction band of a dye-sensitized solar cell comprising of TiO 2 as its nanoporous semiconductor. This model is based on fractional diffusion equations, taking into consideration the random walk network of TiO 2 . Finally, the paper presents numerical solutions of the fractional diffusion model to demonstrate the effect of the fractal geometry of TiO 2 on the fundamental performance parameters of dye-sensitized solar cells, such as the short-circuit current density, open-circuit voltage and efficiency.

© 2020 American Society of Civil Engineers. Column experiments that investigate the use of calcitic limestone as a potential material for permeable reactive barriers (PRBs), as well as its clogging behavior, are conducted under conditions that involve continuous acidic flow containing Al, Fe, and acidophilic bacteria. Results show that nonhomogenous biogeochemical clogging occurred toward the outlet, resulting in a 45% reduction of hydraulic conductivity at the inlet and 10% reduction at the outlet after the bicarbonate buffering period. A mathematical model developed to capture the reductions in longevity is presented. The model, which considers the effects of time-varying porosity, hydraulic conductivity, and head at a particular point on the horizontal flow path, is used for assessing the effect of coupled clogging in a calcitic porous medium.

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.

Dye-sensitized solar cells offer an alternative source for renewable energy by means of converting sunlight into electricity. While there are many studies concerning the development of DSSCs, comprehensive mathematical modelling of the devices is still lacking. Recent mathematical models are based on diffusion equations of electron density in the conduction band of the nano-porous semiconductor in dye-sensitized solar cells. Under linear diffusion and recombination, this paper provides analytical solutions to the diffusion equation. Further, Lie symmetry analysis is adopted in order to explore analytical solutions to physically relevant special cases of the nonlinear diffusion equations. While analytical solutions may not be possible, we provide numerical solutions, which are in good agreement with the results given in the literature.

Coarse aggregates increase inhomogeneity and alter rheological properties of self-compacting concrete. In this work, the effect of coarse aggregates on self-compacting concrete is studied by considering its pipe flow. Self-compacting concrete is modelled as suspension of coarse aggregates in cement mortar and a two-phase Smoothed Particle Hydrodynamics (SPH) model is employed. Cement mortar, a yield-stress fluid, is modelled by fluid SPH particles, while each coarse aggregate is represented by a group of solid SPH particles, which moves as a rigid body. Simulation results show that, at the same volume fractions, the flow rate is higher for larger coarse aggregates, that is up to 24% in the investigated cases. Both effective yield stress and effective plastic viscosity of the concretes increase with the coarse aggregate content. Effective yield stress is smaller for bigger coarse aggregates. The reduction in effective yield stress is a consequence of bigger averaged gap between larger coarse aggregates through which the cement mortar flows.

A mathematical model is proposed to describe the interaction between graphite, polyester and silica solid substrates whose surfaces are functionalized by hydroxyl group, carboxyl group and fluorine atoms. Assuming a relaxed form of these solid substrates, they are modelled as rectangular boxes, while the functional groups are modelled as boundary surfaces. Using the 6–12 Lennard–Jones potential together with a continuum approximation, the total van der Waals interaction energy is obtained as a function of the interfacial separation distance between two surfaces of the solid substrates. Here, the adhesion energy is the minimum energy calculated at the equilibrium separation distance. Our results reveal that among the substrates studied, graphite has the potential to be used as a base material in paints because of its strong adhesion when it is paired with either polyester or silica surfaces.

Continuum modeling using the Lennard-Jones potential has been shown to provide a good estimation for the interaction energy between regular-shaped homogeneous molecules comprising the same type of atoms. However, this method may not be accurate for heterogeneous molecules, which are made up of more than one chemical element. The traditional method to deal with this involves approximating the molecule via multiple surfaces in a piecemeal fashion. While this approach works well for small sized molecules, calculations become intensive for large sized molecules as a large number of sums from multiple surface interactions are involved. To address this issue, we propose a new model that approximates a heterogeneous molecule with a single surface or volume, where attractive and repulsive constants ( A and B) in the Lennard-Jones potential are replaced by functions A ( r ) and B ( r ), which depend on the parameterization of the surface r. We comment that this technique is suitable for regular-shaped nanostructures where their heterogeneity can be modeled by surface (or volume) parameterization. Validation of the new approach is carried out via two problems, namely, carbon nanotube–methane and carbon nanotube–coronene interactions. For coronene and methane, which are assumed to be radially symmetric, we propose A ( r ) and B ( r ) to be sigmoidal functions for which the interaction strength decreases from the inner region of the carbon atoms toward the outer region of the hydrogen atoms. Our results for both cases show that using the sigmoidal profiles for A ( r ) and B ( r ) gives rise to interaction energies that are in better agreement with those obtained from molecular dynamics studies compared to results using constant A and B. The new approach provides a significant improvement to the current continuum modeling using the Lennard-Jones potential.

Chemical vapor deposition is a popular technique for producing high-quality graphene sheets on a substrate. However, the cooling process causes the graphene sheet to experience a strain-induced, out-of-plane buckling. These wrinkles structures can have undesirable effects on the properties of the graphene sheet. We construct a pair of models to analyse the conformation structure of these wrinkles. An arch-shaped wrinkle is first modelled then expanded to incorporate self-adhesion between the wrinkle edges. Variational techniques are employed on both models to determine the optimal conformation for graphene supported on Cu and Ni substrates. We find these models predict a similar structure to experimental analysis of graphene wrinkles on these solid metal substrates.

Originated in 1991 by O‘Regan and Grätzel, dye-sensitized solar cells (DSSCs) provide alternative solutions for renewable energy problems. Earlier mathematical models for DSSCs are based on junction solar cells, which was first studied by Chapin et al in 1954. These equations were derived from Shockley’s work on modelling semiconductors in the late 1940s. However, it was pointed out by Cao et al and Gregg that diffusion model is more suitable for modelling DSSCs. Since the study by Södergren in 1994, the diffusion model has become prevalent in literature and the development of this model by including additional equations to incorporate electrolyte concentrations, time dependence for charge carrier densities and nonlinear diffusivity has shown to capture more complex processes of charge transport within DSSCs. In this paper, we review the development of the diffusion model for the charge carrier densities in a conduction band of DSSCs.

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 important factor that affects energy conversion efficiency of dye-sensitized solar cells is the distribution of dye molecules within the porous semiconductor (TiO 2 ). In this paper, we formulate a continuum model for the interaction between the dye molecule Tris(2,2 ′ -bipyridyl)ruthenium(II) (Ru(bpy) 3 2 + ) and titanium dioxide (TiO 2 ) semiconductor. We obtain the equilibrium position at the minimum energy position between the dye molecules and between the dye and TiO 2 nanoporous structure. Our main outcome is an analytical expression for the energy of the two molecules as a function of their sizes. We also show that the interaction energy obtained using the continuum model is in close agreement with molecular dynamics simulations.

The low bending rigidity of graphene facilitates the formation of folds into the structure. This curvature change affects the reactivity and electron transport of the sheet. One novel extension of this is the intercalation of small molecules into these folds. We construct a model incorporating a single-walled carbon nanotube into a sheet of folded graphene. Variational calculus techniques are employed to determine the minimum energy structure and the resulting curves are shown to agree well with molecular dynamics study. Graphical Abstract Using calculus of variations, the elastic bending energy and van der Waals energy are minimised giving rise to Euler-Lagrange equation for which analytical solutions are derived to determine the optimal curved sturctures of graphene wrapped around carbon nanotubes . Overall agreement between the analytical solutions (with different values of bending rigidities) and results from molecular dynamics simulations (grey) is shown here for (6,6), (8,8) and (10,10) armchair nanotubes, respectively.

Recently, organic nanostructures have attracted much attention, and amongst them peptide nanotubes are of interest in many fields of application including medicine and nanobiotechnology. Peptide nanotubes are formed by self-assembly of cyclic peptides with alternating L- and D-amino acids. Due to their biodegradability, flexible design and easy synthesis, many applications have been proposed such as artificial transmembrane ion channels, templates for nanoparticles, mimicking pore structures, nanoscale testing tubes, biosensors and carriers for targeted drug delivery. The mechanisms of an ion, a water molecule and an ion–water cluster entering into a peptide nanotube of structure cyclo[(-D-Ala-L-Ala-)$_{4}$] are explored here. In particular, the Lennard-Jones potential and a continuum approach are employed to determine three entering mechanisms: (i) through the tube open end, (ii) through a region between each cyclic peptide ring and (iii) around the edge of the tube open end. The results show that while entering the nanotube by method (i) is possible, an ion or a molecule requires initial energy to overcome an energetic barrier to be able to enter the nanotube through positions (ii) and (iii). Due to its simple structure, the D-, L-Ala cyclopeptide nanotube is chosen in this model; however, it can be easily extended to include more complicated nanotubes.

Cyclic peptide nanotubes occur during the self- assembly process of cyclic peptides. Due to the ease of synthesis and ability to control the properties of outer surface and inner diameter by manipulating the functional side chains and the number of amino acids, cyclic peptide nanotubes have attracted much interest from many research areas. A potential application of peptide nanotubes is their use as artificial transmembrane channels for transporting ions, biomolecules and waters into cells. Here, we use the Lennard-Jones potential and a continuum approach to study the interaction of a water molecule in a cyclo((-D- Ala-L-Ala)4-) peptide nanotube. Assuming that each unit of a nanotube comprises an inner and an outer tube and that a water molecule is made up of a sphere of two hydrogen atoms uniformly distributed over its surface and a single oxygen atom at the centre, we determine analytically the interaction energy of the water molecule and the peptide nanotube. Using this energy, we find that, independent of the number of peptide units, the water molecule will be accepted inside the nanotube. Once inside the nanotube, we show that a water molecule prefers to be off-axis, closer to the surface of the inner nanotube. Furthermore, our study of two water molecules inside the peptide nanotube supports the finding that water molecules form an array of a 1 � 2 � 1 � 2 file inside peptide nanotubes. The theoretical study presented here can facilitate thorough understanding of the behaviour of water molecules inside peptide nan- otubes for applications, such as artificial transmembrane channels.

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 $$_{60}$$ 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 $$_{60}$$ . These interactions are calculated based on the structure of the antiviral molecules. Our results show that the binding of fullerene C $$_{60}$$ to the antiviral molecules increases the efficiency of the compound to prohibit the activity of HIV.

In this paper, we construct a continuum model for graphene oxide based upon the Lerf–Klinowski structure to investigate the interaction forces between sheets of graphene oxide. We use the Lennard-Jones potential and coulombic potential to determine the total potential energy between sheets of graphene oxide. We analytically calculate the interaction forces within the system using sums of hypergeometric functions. Our model is then modified to investigate different levels of hydration and oxidation within the system. Our investigations are reconstructed using the LAMMPS molecular dynamics simulator and we find that the analytical solution quickly and effectively calculates results that match well against our simulation data and values taken from literature.

In this paper, we examine the nonlinear nanoelectromechanical effect on a doubly clamped suspended single-walled carbon nanotube which could be used for pressure sensor. Coulomb-blockade effects will be explored and investigated. We adopt the full expression of curvature term in the elastic energy and use a modified Euler’s method to determine the nanotube’s maximum displacement in all bending regimes. We find that while the approximate solution given by Sapmaz et al. [1] underestimates the maximum displacement of the buckled nanotube in the weak bending regime, the approximate solution fails to obtain the correct maximum displacement as given by our numerical solution. Accordingly, the effect of curvature must be properly addressed for this nanoelectromechanical system to be used as an accurate sensor.

In this chapter a number of simple interaction energies for linear objects are determined, such as for a single atom with a plane, and for spherical objects such as for a sphere interacting with a plane graphene sheet. Our principal concern is the evaluation of the Lennard-Jones potential function interaction energy between atoms, molecules and nanostructures. When calculating the interaction energy between molecules containing a number of atoms, the pairwise interactions may be summed to derive a total interaction energy that is given by a double summation over all the atoms in each molecule. Here we assume that all the atoms are smeared over ideal lines or surfaces representing the molecules in order to replace the explicit summations with line or surface integrals, and the task of calculating the interaction energy is then made considerably easier. It turns out that many of the molecules and nanostructures can be modelled by the basic geometric objects of points, straight lines, flat planes, spheres, and right circular cylinders, and therefore integrals over these surfaces are needed and the purpose of this chapter is to address these integrals in a systematic way to facilitate the evaluation of ideal Lennard-Jones interactions.

In this chapter mathematical preliminaries for various standard functions used throughout the book are presented, such as the gamma and beta functions, the hypergeometric function, and the associated Legendre functions. The essential modelling content of the book is the use of the continuum assumption for atomic surface densities and the replacement of double summations with double surface integrals involving atomic potential functions. For certain surfaces these integrals can often be evaluated in terms of well-known analytical functions, but generally these integrations are highly nontrivial. The mathematical perspective here is that many of the integrals can be identified from integral representations of the various hypergeometric functions including Appell’s hypergeometric function. These formulae are important since they often lead to evaluating the integral in terms of other special functions, such as Legendre polynomials, and through algebraic packages such as MAPLE and MATLAB the hypergeometric functions can be readily evaluated numerically. This chapter ensures that the reader is fully conversant with the major properties of all the special functions needed to undertake this process including the Dirac delta function, the Heaviside unit function, the gamma and beta functions, and the hypergeometric functions.

In general, particles tend to organise themselves into their most stable structures, which are the minimum energy configurations and this is also true at the atomic level. When two atoms are far apart an attractive force pulls them closer to each other, but if they are too close a repulsive force pushes them away. There exists a certain separation distance where the two atoms are most likely to be, namely where the interaction energy between them is at a minimum. This distance is termed the equilibrium distance. Ideally, atoms prefer to be apart by a distance equivalent to their equilibrium distance, as do molecules and other nanostructures. This chapter focuses on the determination of stable equilibrium configurations for nested carbon nanostructures. A nested structure refers to a structure that comprises both inner and outer components. We begin by examining the simple case where the inner structure is simply a single atom. Then we extend the analysis to more complicated structures, including fullerene@fullerene (carbon onion), fullerene@nanotube, nanotube@nanotube, and carbon onion@nanotube. We note that the convention X@Y denotes a nested structure for which molecule X is nested inside molecule Y. In the final parts of this chapter we look at the interaction of many body systems. For example, the case of a nested atom or molecule either inside a uniform concentric ring or a nanotube bundle.

Micromechanical oscillators, or resonators struggle to reach frequencies in the gigahertz range. However, nanoscale oscillators are able to reach these high frequencies and have therefore attracted much attention. The first such oscillator involved multiwalled nanotubes, in which the sliding of the inner-shell inside the outer-shell of a multiwalled carbon nanotube generated frequencies up to several gigahertz. Nano-oscillators can also be generated using other carbon nanomaterials; such as with a C60 fullerene and nanotube, double-walled nanotubes, and nanotube nanobundles. In this chapter, we examine the oscillatory behaviour of various nano-oscillators using the continuum approximation arising from the assumption that the discrete atoms can be smeared across each surface. This chapter utilises calculations for the suction force that are described in Chapter 5.