ORCID Profile
0000-0003-4623-2811
Current Organisation
University of South Australia
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Publisher: Oxford University Press (OUP)
Date: 1988
Publisher: Elsevier BV
Date: 05-2001
Publisher: Springer Science and Business Media LLC
Date: 13-01-2013
Publisher: Springer Science and Business Media LLC
Date: 10-2018
Publisher: Elsevier BV
Date: 03-2018
Publisher: Springer Science and Business Media LLC
Date: 04-2011
Publisher: Cambridge University Press (CUP)
Date: 04-2013
DOI: 10.1017/S1446181113000217
Abstract: We review the work of the present authors to employ variational calculus to formulate continuous models for the connections between various carbon nanostructures. In formulating such a variational principle, there is some evidence that carbon nanotubes deform as in perfect elasticity, and rather like the elastica, and therefore we seek to minimize the elastic energy. The calculus of variations is utilized to minimize the curvature subject to a length constraint, to obtain an Euler–Lagrange equation, which determines the connection between two carbon nanostructures. Moreover, a numerical solution is proposed to determine the geometric parameters for the connected structures. Throughout this review, we assume that the defects on the nanostructures are axially symmetric and that the into-the-plane curvature is small in comparison to that in the two-dimensional plane, so that the problems can be considered in the two-dimensional plane. Since the curvature can be both positive and negative, depending on the gap between the two nanostructures, two distinct cases are examined, which are subsequently shown to smoothly connect to each other.
Publisher: SAGE Publications
Date: 11-2016
Abstract: In many nanotechnology areas, there is often a lack of well-formed conceptual ideas and sophisticated mathematical modeling in the analysis of fundamental issues involved in atomic and molecular interactions of nanostructures. Mathematical modeling can generate important insights into complex processes and reveal optimal parameters or situations that might be difficult or even impossible to discern through either extensive computation or experimentation. We review the use of applied mathematical modeling in order to determine the atomic and molecular interaction energies between nanoscale objects. In particular, we examine the use of the 6-12 Lennard-Jones potential and the continuous approximation, which assumes that discrete atomic interactions can be replaced by average surface or volume atomic densities distributed on or throughout a volume. The considerable benefit of using the Lennard-Jones potential and the continuous approximation is that the interaction energies can often be evaluated analytically, which means that extensive numerical landscapes can be determined virtually instantaneously. Formulae are presented for idealized molecular building blocks, and then, various applications of the formulae are considered, including gigahertz oscillators, hydrogen storage in metal-organic frameworks, water purification, and targeted drug delivery. The modeling approach reviewed here can be applied to a variety of interacting atomic structures and leads to analytical formulae suitable for numerical evaluation.
Publisher: IOP Publishing
Date: 05-2017
Publisher: AIP Publishing
Date: 15-01-2010
DOI: 10.1063/1.3289320
Abstract: In band structure calculations commonly used to derive the electronic properties of carbon nanotubes, it is generally assumed that all bond lengths are equal. However, hexagonal carbon lattices are often irregular and may contain as many as three distinct bond lengths. A regular (n,m) carbon nanotube will be metallic if p=(n−m)/3 for integer p. Here we analytically derive the generalized condition for metallic irregular carbon nanotubes. This condition is particularly relevant to small radius nanotubes and nanotubes experiencing small applied strains.
Publisher: Elsevier BV
Date: 2011
Publisher: Springer Science and Business Media LLC
Date: 17-05-2011
Publisher: Elsevier BV
Date: 2011
Publisher: ASME International
Date: 07-10-2008
DOI: 10.1115/1.2987874
Abstract: One approach to modeling fully developed shear flow of frictional granular materials is to use a yield condition and a flow rule, in an analogous way to that commonly employed in the fields of metal plasticity and soil mechanics. Typically, the yield condition of choice for granular materials is the Coulomb–Mohr criterion, as this constraint is relatively simple to apply but at the same time is also known to predict stresses that are in good agreement with experimental observations. On the other hand, there is no strong agreement within the engineering and applied mechanics community as to which flow rule is most appropriate, and this subject is still very much open to debate. This paper provides a review of the governing equations used to describe the flow of granular materials subject to the Coulomb–Mohr yield condition, concentrating on the coaxial and double-shearing flow rules in both plane strain and axially symmetric geometries. Emphasis is given to highly frictional materials, which are defined as those granular materials that possess angles of internal friction whose trigonometric sine is close in value to unity. Furthermore, a discussion is provided on the practical problems of determining the stress and velocity distributions in a gravity flow hopper, as well as the stress fields beneath a standing stockpile and within a stable rat-hole.
Publisher: ASME International
Date: 11-2012
DOI: 10.1115/1.4007521
Abstract: In this paper, we investigate methane encapsulation in five spherical fullerenes C60,C240,C540,C960, and C1500. We exploit the 6–12 Lennard-Jones potential function and the continuum approximation to model the surface binding energies between methane and spherical fullerenes of varying sizes. Our results show that for a methane molecule interacting inside a spherical fullerene, the binding energies are minimized at locations which become closer to the fullerene wall as the size of the fullerene increases. However, we find that the methane molecule would require an applied external force to overcome the repulsive energy barrier in order to be encapsulated into a C60 fullerene. The present modeling indicates that the optimal minimum energy for methane storage in any spherical fullerene occurs for a fullerene with radius ≃6.17 Å, with a corresponding potential energy of ≃0.22 eV which occurs for a fullerene bigger than a C60 but slightly smaller than a C240 as the ideal spherical fullerene for methane encapsulation. Overall, our results are in very good agreement with other theoretical studies and molecular dynamics simulations, and show that fullerenes might be good candidates for gas storage. However, the major advantage of the approach adopted here is the derivation of explicit analytical formulae from which numerical results for varying physical scenarios may be readily obtained.
Publisher: MDPI AG
Date: 07-03-2022
DOI: 10.3390/NANO12050887
Abstract: Simple and economical ferric ion detection is necessary in many industries. An europium-based metal organic framework has selective sensing properties for solutions containing ferric ions and shows promise as a key component in a new sensor. We study an idealised sensor that consists of metal organic framework (MOF) crystals placed on a polymer surface. A two-dimensional diffusion model is used to predict the movement of ferric ions through the solution and polymer, and the ferric ion association to a MOF crystal at the boundary between the different media. A simplified one-dimensional model identifies the choice of appropriate values for the dimensionless parameters required to optimise the time for a MOF crystal to reach steady state. The model predicts that a large non-dimensional diffusion coefficient and an effective association with a small effective flux will reduce the time to steady-state. The effective dissociation is the most significant parameter to aid the estimation of the ferric ion concentration. This paper provides some theoretical insight for material scientists to optimise the design of a new ferric ion sensor.
Publisher: Informa UK Limited
Date: 07-05-2010
Publisher: Elsevier BV
Date: 07-2011
Publisher: The Royal Society
Date: 08-07-1992
Abstract: In this paper we consider a number of axially symmetric flows of compressible granular materials obeying the Coulomb–Mohr yield condition and the associated flow rule. We pay particular attention to those plastic régimes and flows not included in the seminal work of Cox, Eason & Hopkins (1961). For certain plastic régimes, the velocity equations uncouple from the stress equations and the flow is said to be kinematically determined. We present a number of kinematically determined flows and the development given follows the known solutions applicable to the so-called ‘double-shearing’ model of granular materials which assumes incompressibility and for which the governing equations are almost the same. Similarly, for certain other plastic régimes the stresses may be completely determined without reference to the velocity equations and these are referred to as statically determined flows. In the latter sections of the paper we examine statically determined flows arising from the assumption that the shear stress in either cylindrical or spherical polar coordinates is zero. In the final section we present a numerical solution, which incorporates gravitational effects, for the flow of a granular material in a converging hopper. In addition, we examine the Butterfield & Harkness (1972) modification of the double-shearing model of granular materials which formally includes both the double-shearing theory and the Coulomb–Mohr flow rule theory as special cases. Moreover, for kinematically determined régimes, the velocity equations are the same apart from a different constant, while for statically determined régimes the governing velocity equations are slightly more complicated, involving another constant which is a different combination of the basic physical parameters. Thus some of the solutions presented here can be immediately extended to this alternative theory of granular material behaviour and therefore the prospect arises of devising experiments which might validate or otherwise one theory or the other.
Publisher: IEEE
Date: 02-2010
Publisher: Springer Science and Business Media LLC
Date: 25-10-2022
DOI: 10.1007/S00033-022-01862-1
Abstract: At various times in his life, Louis de Broglie firmly believed in the coexistence of both particle and an associated wave referring to “the theory of the double solution”, and an equation which he called “the guidance formula”. In an attempt to account for both particle and wave, the author has proposed a Lorentz invariant alternative to Newton’s second law which is developed in Hill (Zeitschrift fur angewandte Mathematik und Physik 69:133–145, 2018 Zeitschrift fur angewandte Mathematik und Physik 70:5–14, 2019 Zeitschrift fur angewandte Mathematik und Physik 70:131–153, 2019 Math Mech Solids 26:263–284, 2020 Math Mech Solids 25: 1763–1777, 2020 Zeitschrift fur angewandte Mathematik und Physik 72:1–14 Mathematics of particle–wave mechanical systems, Springer, Cham, 2022). Here, we summarise some of the major outcomes of this approach, including simple solutions of the proposed model exhibiting both sub-luminal and superluminal behaviour dependent upon the region of space-time, and two symmetrical sets of rate-differential relations for the two Lorentz invariants for a single spatial dimension x . One set follows the particle, while the other follows the wave, revealing a complete symmetry between the one-dimensional spatial physical force f and the force g in the “direction of time”. The existence of these symmetrical equations reinforces the parity and interchangeability of particles and waves.
Publisher: Elsevier BV
Date: 04-2007
Publisher: Elsevier BV
Date: 09-2012
Publisher: Springer Science and Business Media LLC
Date: 02-2012
Publisher: American Scientific Publishers
Date: 04-2011
Publisher: Springer Science and Business Media LLC
Date: 17-12-2018
Publisher: Elsevier BV
Date: 08-2008
Publisher: The Royal Society
Date: 08-02-2005
Abstract: We study a stochastic integral that arises during the implementation of the Milstein method for the numerical integration of systems of stochastic differential equations. The distribution of the integral can be written as the inverse Fourier transform of a characteristic function with essential singularities. This leads to a generalized integral that can be expressed as an infinite series involving the derivatives of Meixner polynomials. The generating function of the polynomials in combination with the Mittag–Leffler expansion theorem is used to obtain a novel series representation for the integral and the motivating problem in particular. This new form is rapidly convergent and, therefore, well suited to numerical work.
Publisher: SAGE Publications
Date: 29-11-2012
Abstract: The problem of finite elastic deformation of a long rectangular rubber block which is deformed in a perturbed cylindrical configuration is examined here, and is motivated from the problem of determining surface rippling that is observed in bent multi-walled carbon nanotubes. The problem of finite elastic bending of a tube is considerably more complicated than the geometrically simpler problem of finite elastic bending of a rectangular block. Accordingly, we examine here the simpler block problem which is assumed to be sufficiently long so that the out of plane end effects may be ignored. The general equations governing plane strain deformations of an isotropic incompressible perfectly elastic Mooney material, which models rubber-like materials, are used to determine small superimposed deformations upon the well-known controllable family for the deformation of rectangular blocks into a sector of a solid bounded by two circular arcs. Traction free boundary conditions are assumed in an average sense along the bounding circular arcs. Physically realistic rippling is found to occur and typical numerical values are used to illustrate the solution graphically.
Publisher: The Royal Society
Date: 04-2018
DOI: 10.1098/RSOS.171109
Abstract: We examine a static, spherically symmetric solution of the empty space field equations of general relativity with a non-orthogonal line element which gives rise to an opportunity that does not occur in the standard derivations of the Schwarzschild solution. In these derivations, convenient coordinate transformations and dynamical assumptions inevitably lead to the Schwarzschild solution. By relaxing these conditions, a new solution possibility arises and the resulting formalism embraces the Schwarzschild solution as a special case. The new solution avoids the coordinate singularity associated with the Schwarzschild solution and is achieved by obtaining a more suitable coordinate chart. The solution embodies two arbitrary constants, one of which can be identified as the Newtonian gravitational potential using the weak field limit. The additional arbitrary constant gives rise to a situation that allows for generalizations of the Eddington–Finkelstein transformation and the Kruskal–Szekeres coordinates.
Publisher: The Royal Society
Date: 08-01-2008
Abstract: Carbon nanotubes are nanostructures that promise much in the area of constructing nanoscale devices due to their enhanced mechanical, electrical and thermal properties. In this paper, we examine a gigahertz oscillator that comprises a carbon nanotube oscillating in a uniform concentric ring or bundle of carbon nanotubes. A number of existing results for nanotube oscillators are employed to analyse the design considerations of optimizing such a device, and significant new results are also derived. These include a new analytical expression for the interaction per unit length of two parallel carbon nanotubes involving the Appell hypergeometric functions. This expression is employed to precisely determine the relationship between the bundle radius and the radii of the nanotubes forming the bundle. Furthermore, several pragmatic approximations are also given, including the relationships between the bundle radius and the constituent nanotube radius and the oscillating tube radius and the bundle nanotube radius. We also present a simplified analysis of the force and energy for a nanotube oscillating in a nanotube bundle leading to an expression for the oscillating frequency and the maximum oscillating frequency, including constraints on configurations under which this maximum is possible.
Publisher: MDPI AG
Date: 04-06-2019
DOI: 10.3390/APP9112301
Abstract: After the discovery of circular formations of single walled carbon nanotubes called fullerene crop circles, their structure has become one of the most researched amongst carbon nanostructures due to their particular interesting physical properties. Several experiments and simulations have been conducted to understand these intriguing objects, including their formation and their hidden characteristics. It is scientifically conceivable that these crop circles, nowadays referred to as carbon nanotori, can be formed by experimentally bending carbon nanotubes into ring shaped structures or by connecting several sections of carbon nanotubes. Toroidal carbon nanotubes are likely to have many applications, especially in electricity and magnetism. In this review, geometry, construction, modelling and possible applications are discussed and the existing known analytical expressions, as obtained from the Lennard-Jones potential and the continuum approximation, for their interaction energies with other nanostructures are summarised.
Publisher: SAGE Publications
Date: 07-06-2020
Abstract: The existence of the so-called ‘dark’ issues of mechanics implies that our present accounting for mass and energy is incorrect in terms of applicability on a cosmological scale, and the question arises as to where the difficulty might lie. The phenomenon of quantum entanglement indicates that systems of particles exist that in idually display certain characteristics, while collectively the same characteristic is absent simply because it has cancelled out between in idual particles. It may therefore be necessary to develop theoretical frameworks in which long-held conservation beliefs do not necessarily always apply. The present paper summarises the formulation described in earlier papers (Hill, JM. On the formal origin of dark energy. Z Angew Math Phys 2018 69:133-145 Hill, JM. Some further comments on special relativity and dark energy. Z Angew Math Phys 2019 70: 5–14 Hill, JM. Special relativity, de Broglie waves, dark energy and quantum mechanics. Z Angew Math Phys 2019 70: 131–153.), which provides a framework that allows exceptions to the law that matter cannot be created or destroyed. In these papers, it is proposed that dark energy arises from conventional mechanical theory, neglecting the work done in the direction of time and consequently neglecting the de Broglie wave energy [Formula: see text]. These papers develop expressions for the de Broglie wave energy [Formula: see text] by making a distinction between particle energy [Formula: see text] and the total work done by the particle [Formula: see text], that which accumulates from both a spatial physical force [Formula: see text] and a force [Formula: see text] in the direction of time. In any experiment, either particles or de Broglie waves are reported, so that only one of [Formula: see text] or [Formula: see text] is physically measured, and particles appear for [Formula: see text] and de Broglie waves occur for [Formula: see text], but in either event both a measurable and an immeasurable energy exists. Conventional quantum mechanics operates under circumstances such that [Formula: see text] vanishes and [Formula: see text] becomes purely imaginary. If both [Formula: see text] and [Formula: see text] are generated as the gradient of a potential, the total particle energy is necessarily conserved in the conventional manner.
Publisher: Springer Science and Business Media LLC
Date: 2001
Publisher: Informa UK Limited
Date: 26-02-2019
Publisher: American Chemical Society (ACS)
Date: 29-06-2007
DOI: 10.1021/JP0721402
Publisher: American Physical Society (APS)
Date: 29-07-2008
Publisher: Springer Science and Business Media LLC
Date: 11-2013
Publisher: IOP Publishing
Date: 04-2019
Publisher: Elsevier BV
Date: 05-2008
Publisher: Institution of Engineering and Technology (IET)
Date: 2010
Publisher: IEEE
Date: 02-2010
Publisher: IOP Publishing
Date: 16-10-2007
Publisher: American Chemical Society (ACS)
Date: 27-06-2013
DOI: 10.1021/LA401446S
Abstract: 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.
Publisher: Elsevier BV
Date: 05-2009
Publisher: Institute of Electrical and Electronics Engineers (IEEE)
Date: 09-2011
Publisher: Springer Science and Business Media LLC
Date: 07-01-2012
Publisher: Elsevier BV
Date: 05-2011
Publisher: The Royal Society
Date: 05-2023
DOI: 10.1098/RSOS.230232
Abstract: Using the Lennard–Jones potential, we determine analytical expressions for van der Waals interaction energies between a point and a rectangular prism-shaped pore, writing them in terms of standard elementary functions. The parameter values for a new ferric ion sensor are used to compare these calculations with the cylindrical pore approximation for the interactions between an ion and a metal organic framework (MOF) pore. The results using the prismatic pore approximation predict the same qualitative outcomes as a cylindrical pore approximation. However, the prismatic approximation predicts lower magnitudes for both the interaction potential energy minimum and the force maximum, since the average distance from the centre-line to the surface of the prism is greater. We suggest that in some circumstances it is sufficient to use the simpler cylindrical approximation, provided that the cylinder radius is chosen so that the cross-sectional area is equal to the area of the prism pore opening. However, atoms at the nodes should remain approximated by semi-infinite lines. We also determine the interaction between a second ferric ion and a blocked MOF pore as expected, the second ferric ion experiences a force away from the pore, implying that approaching ferric ions can only occupy vacant MOF pores.
Publisher: American Scientific Publishers
Date: 11-2016
Publisher: Trans Tech Publications, Ltd.
Date: 09-2011
DOI: 10.4028/WWW.SCIENTIFIC.NET/MSF.700.85
Abstract: In this study, we investigate the internal mechanics for a two-state memory device,comprising a charged metallofullerene which is located inside a closed carbon nanotube.Assuming the Lennard-Jones interaction energy and the continuum approximation, the metallofullerenehas two symmetrically placed equal minimum energy positions. On one side theencapsulated metallofullerene represents the zero information state and by applying an externalelectrical field, the metallofullerene can be made to overcome the energy barrier of thenanotube, and pass from one end of the tube to the other, where the metallofullerene thenrepresents the one information state.
Publisher: IOP Publishing
Date: 19-06-2007
Publisher: Royal Society of Chemistry (RSC)
Date: 2015
DOI: 10.1039/C4RA13496F
Abstract: For nanoparticles penetrating biological tissue, modelling indicates that without external forces, carbon nanoparticles will remain trapped in lipid bilayers.
Publisher: Elsevier BV
Date: 04-2008
Publisher: Elsevier BV
Date: 11-1999
Publisher: Springer Science and Business Media LLC
Date: 06-01-2009
Publisher: IOP Publishing
Date: 10-03-2008
Publisher: American Scientific Publishers
Date: 11-2013
Publisher: Elsevier BV
Date: 02-2023
Publisher: IOP Publishing
Date: 07-2011
DOI: 10.1088/0957-4484/22/30/305403
Abstract: In this paper, we use applied mathematical modelling to investigate the storage of hydrogen molecules inside graphene-oxide frameworks, which comprise two parallel graphenes rigidly separated by perpendicular ligands. Hydrogen uptake is calculated for graphene-oxide frameworks using the continuous approximation and an equation of state for both the bulk and adsorption gas phases. We first validate our approach by obtaining results for two parallel graphene sheets. This result agrees well with an existing theoretical result, namely 1.85 wt% from our calculations, and 2 wt% arising from an ab initio and grand canonical Monte Carlo calculation. This provides confidence to the determination of the hydrogen uptake for the four graphene-oxide frameworks, GOF-120, GOF-66, GOF-28 and GOF-6, and we obtain 1.68, 2, 6.33 and 0 wt%, respectively. The high value obtained for GOF-28 may be partly explained by the fact that the benzenediboronic acid pillars between graphene sheets not only provide mechanical support and porous spaces for the molecular structure but also provide the higher binding energy to enhance the hydrogen storage inside graphene-oxide frameworks. For the other three structures, this binding energy is not as large in comparison to that of GOF-28 and this effect diminishes as the ligand density decreases. In the absence of conflicting data, the present work indicates GOF-28 as a likely contender for practical hydrogen storage.
Publisher: American Scientific Publishers
Date: 08-2011
Abstract: We investigate the internal mechanics for methane storage in a nanobottle, which is assumed to comprise a metallofullerene located inside a carbon nanobottle, which is constructed from a half-fullerene as the base, and two nanotubes which are joined by a nanocone. The interaction potential energy for the metallofullerene is obtained from the 6-12 Lennard-Jones potential and the continuum approximation, which assumes that a discrete atomic structure can be replaced by an average atomic surface density. This potential energy shows that the metallofullerene has two minimum energy positions, which are located close to the neck of the bottle and at the base of the nanobottle, and therefore it may be used as a bottle-stopper to open or to close the nanobottle. At the neck of the bottle, the encapsulated metallofullerene closes the nanobottle, and by applying an external electrical force, the metallofullerene can overcome the energy barrier of the nanotube, and pass from the neck of the nanobottle to the base so that the nanobottle is open. For methane storage, the metallofullerene serves the dual purposes of opening and closing the nanobottle, as well as an attractor for the methane gas. The analytical formulation gives rise to a rapid computational capacity, and enables the direct determination of the optimal dimensions necessary to ensure the correct working function of the nanobottle, and specific ranges for the critical parameters are formulated.
Publisher: Springer Science and Business Media LLC
Date: 20-12-2013
Publisher: AIP Publishing
Date: 22-02-2018
DOI: 10.1063/1.5019854
Abstract: We theoretically demonstrate that the junction between a phase-change material with a phase-invariant one can efficiently function as a conductive thermal diode. Analytical expressions for the heat flux and optimal rectification factor are derived and analyzed for junctions whose operations are driven by the thermal conductivity hysteresis of both VO2 and nitinol. It is shown that phase-change materials with higher thermal conductivity contrast, smaller thermal hysteresis, and faster phase transitions yield a conductive thermal diode with higher rectification of heat currents. Rectification factors of up to 19.7% and 18.8% are found for thermal diodes based on VO2 and nitinol, operating with a temperature difference between their terminals of 369.5 − 300 = 69.5 K and 388.2 − 273 = 115.2 K, respectively. These similar rectification factors could be enhanced by increasing the thermal conductivity variations of the diode terminals, and hence, the results obtained will be useful for guiding the development of phase-change materials capable of optimizing the rectification of conductive heat fluxes.
Publisher: Oxford University Press (OUP)
Date: 10-08-2011
DOI: 10.1093/QJMAM/HBR013
Publisher: Elsevier BV
Date: 11-2011
Publisher: Springer Science and Business Media LLC
Date: 22-06-2012
Publisher: Springer Science and Business Media LLC
Date: 07-2005
Publisher: Elsevier BV
Date: 02-2018
Publisher: Elsevier BV
Date: 05-2008
Publisher: IOP Publishing
Date: 23-03-2010
DOI: 10.1088/0957-4484/21/15/155305
Abstract: 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.
Publisher: Informa UK Limited
Date: 12-08-2008
Publisher: Springer Science and Business Media LLC
Date: 24-05-2018
Publisher: IEEE
Date: 02-2010
Publisher: IOP Publishing
Date: 11-09-2007
DOI: 10.1088/0953-8984/19/40/406209
Abstract: Continuum based models are presented here for certain boron nitride and carbon nanostructures. In particular, certain fullerene interactions, C(60)-C(60), B(36)N(36)-B(36)N(36) and C(60)-B(36)N(36), and fullerene-nanotube oscillator interactions, C(60)-boron nitride nanotube, C(60)-carbon nanotube, B(36)N(36)-boron nitride nanotube and B(36)N(36)-carbon nanotube, are studied using the Lennard-Jones potential and the continuum approach, which assumes a uniform distribution of atoms on the surface of each molecule. Issues regarding the encapsulation of a fullerene into a nanotube are also addressed, including acceptance and suction energies of the fullerenes, preferred position of the fullerenes inside the nanotube and the gigahertz frequency oscillation of the inner molecule inside the outer nanotube. Our primary purpose here is to extend a number of established results for carbon to the boron nitride nanostructures.
Publisher: American Scientific Publishers
Date: 03-2008
Publisher: Elsevier BV
Date: 08-2021
Publisher: Elsevier BV
Date: 05-2008
Publisher: Royal Society of Chemistry (RSC)
Date: 2010
DOI: 10.1039/B9NR00433E
Abstract: 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.
Publisher: Springer Science and Business Media LLC
Date: 18-06-2013
Publisher: The Royal Society
Date: 10-04-2008
Abstract: The classical Stefan problem for freezing (or melting) a sphere is usually treated by assuming that the sphere is initially at the fusion temperature, so that heat flows in one phase only. Even in this idealized case there is no (known) exact solution, and the only way to obtain meaningful results is through numerical or approximate means. In this study, the full two-phase problem is considered, and in particular, attention is given to the large Stefan number limit. By applying the method of matched asymptotic expansions, the temperature in both the phases is shown to depend algebraically on the inverse Stefan number on the first time scale, but at later times the two phases essentially decouple, with the inner core contributing only exponentially small terms to the location of the solid–melt interface. This analysis is complemented by applying a small-time perturbation scheme and by presenting numerical results calculated using an enthalpy method. The limits of zero Stefan number and slow diffusion in the inner core are also noted.
Publisher: Springer Science and Business Media LLC
Date: 2015
DOI: 10.1007/S11538-014-0056-2
Abstract: 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.
Publisher: Springer Science and Business Media LLC
Date: 27-11-2018
Publisher: AIP Publishing
Date: 18-04-2012
DOI: 10.1063/1.4704677
Abstract: We derive an analytic description of the spin susceptibility in finite length zigzag carbon nanotubes (CNT) with chirality (n, 0). The spin susceptibility is proportional to the Ruderman-Kittel-Kasuya-Yosida (RKKY) interactions which describes indirect carrier mediated exchange coupling between localized magnetic moments. We show that the strongest RKKY interactions are along the edges of the nanotube and in the thermodynamic limit at half filling with spin symmetry the shape of the susceptibility curve about the edge of the CNT can be determined solely by the lattice geometry represented by the parameter n and a parameter L which describes the nanotube length. We also show that the introduction of Zeeman splitting or doping may have no effect on the spin susceptibility, provided n is small. A detailed knowledge of magnetic interactions, such as RKKY interactions, in CNT is of vital importance to the development of nanotechnology applications.
Publisher: Elsevier BV
Date: 11-2009
Publisher: The Royal Society
Date: 08-11-2000
Publisher: Springer Science and Business Media LLC
Date: 30-03-2006
Publisher: Springer Science and Business Media LLC
Date: 06-10-2008
Publisher: IOP Publishing
Date: 18-03-2009
DOI: 10.1088/0953-8984/21/14/144214
Abstract: 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).
Publisher: IOP Publishing
Date: 21-05-2008
Publisher: Royal Society of Chemistry (RSC)
Date: 2015
DOI: 10.1039/C5RA08276E
Abstract: Variational calculus is employed to determine the folding behaviour of a single graphene sheet.
Publisher: Springer Science and Business Media LLC
Date: 05-04-2008
Publisher: Royal Society of Chemistry (RSC)
Date: 2016
DOI: 10.1039/C6RA02126C
Abstract: Mathematical modelling, comprising Lennard–Jones potential and calculus of variations, is utilized to obtain the energy equations arising from AFM probe and substrate, leading to deflection equations of AFM cantilever.
Publisher: American Physical Society (APS)
Date: 03-06-2009
Publisher: MDPI AG
Date: 21-12-2020
DOI: 10.3390/COMPUTATION8040107
Abstract: The Lennard–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–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.
Publisher: ASME International
Date: 02-2012
DOI: 10.1115/1.4006859
Abstract: In this study, we propose a new nanocomputer component. We investigate the mechanics of a multiwalled carbon nanotube, comprising two symmetrically placed inner tubes and a moveable tube of radius intermediate to the larger and the two smaller tubes. The larger tube has the two fixed smaller tubes located at its ends, and the moveable tube is assumed to be controlled by an applied voltage difference. The tube radii are purposely chosen so that electrons can jump from one tube to another and a current can flow from the larger tube to the moveable one and finally to one of the smaller tubes. The interaction energy for the system is obtained assuming the Lennard-Jones potential together with the continuum approximation. As expected, the system has two symmetrically placed equal minimum energy locations (i.e., the total interaction energies take on minimum values) and by adopting different electrical circuits, the design gives rise to the possibility of using the device either as a memory device or as logic gates. By applying a voltage input to produce an external electrical field and another voltage input to provide a charge on the moving tube, the moving tube provides an output signal which we assume is registered on a meter that is capable of measuring either voltage or charge. We present the basic design rules for such devices and we establish their feasibility for practical realization.
Publisher: Springer Science and Business Media LLC
Date: 19-09-2018
Publisher: Elsevier BV
Date: 09-2008
Publisher: Springer Science and Business Media LLC
Date: 25-02-2021
Publisher: AIP Publishing
Date: 10-04-2006
DOI: 10.1063/1.2185607
Abstract: The problem of electric field-induced force between spheres is fundamental to electrorheological fluids. Previously published experimental results indicate that the interaction force between two spheres under an external field is not adequately explained by the existing approximate and numerical methods. The specific models compared were dipole, dipole with local field corrections and a finite-element analysis. This letter employs an exact solution (via the equivalent multipole-moment method) to the electrostatic problem which accurately predicts the low-frequency experimental results at all measured interstices. The solution presented later is self-contained and addresses specifically the geometry of the previously mentioned experimental results. While more general solutions have been published previously, they are more complex than is required by this problem. The solution presented here is accurate for all sphere spacings, but in particular could apply to nano-spheres in close proximity.
Publisher: Oxford University Press (OUP)
Date: 1994
Publisher: American Chemical Society (ACS)
Date: 25-09-2008
DOI: 10.1021/JP803023Q
Publisher: Elsevier BV
Date: 02-2023
Publisher: American Chemical Society (ACS)
Date: 07-07-2009
DOI: 10.1021/JA9036302
Abstract: A new concept is described for methane and hydrogen storage materials involving the incorporation of magnesium-decorated fullerenes within metal-organic frameworks (MOFs). The system is modeled using a novel approach underpinned by surface potential energies developed from Lennard-Jones parameters. Impregnation of MOF pores with magnesium-decorated Mg(10)C(60) fullerenes, denoted as Mg-C(60)@MOF, places exposed metal sites with high heats of gas adsorption into intimate contact with large surface area MOF structures. Perhaps surprisingly, given the void space occupied by C(60), this impregnation delivers remarkable gas uptake, according to our modeling, which predicts exceptional performance for the Mg-C(60)@MOF family of materials. These predictions include a volumetric methane uptake of 265 v/v, the highest reported value for any material, which significantly exceeds the U.S. Department of Energy target of 180 v/v. We also predict a very high hydrogen adsorption enthalpy of 11 kJ mol(-1) with relatively little decrease as a function of H(2) filling. This value is close to the calculated optimum value of 15.1 kJ mol(-1) and is achieved concurrently with saturation hydrogen uptake in large amounts at pressures under 10 atm.
Publisher: Oxford University Press (OUP)
Date: 03-08-2017
DOI: 10.1093/QJMAM/HBX016
Abstract: Deoxyribonucleic acid (DNA) and carbon nanotubes (CNTs) constitute hybrid materials with the potential to provide new components with many applications in various technology areas, such as molecular electronics, field devices and medical applications. Using classical applied mathematical modelling, we investigate the suction force experienced by a double-stranded DNA (dsDNA) molecule which is assumed to be located on the axis near an open end of a semi-infinite single-walled CNT. We employ both the 6-12 Lennard-Jones potential and the continuum approximation, which assumes that a discrete atomic structure can be replaced by a surface with constant average atomic density. While most research in the area is dominated by molecular dynamics simulations, here we use elementary mechanical principles and classical applied mathematical modelling techniques to formulate explicit analytical criteria and ideal model behaviour. We observe that the suction behaviour depends on the radius of the CNT, and we predict that it is less likely for a dsDNA molecule to be accepted into the CNT when the value of the tube radius is ${ }12.9$ Å. The dsDNA molecule will be accepted into the CNT for radii lager than 13 Å, and we show that the optimal single-walled CNT necessary to fully enclose the DNA molecule has a radius of 13.56 Å, which approximately corresponds to the chiral vector numbers (20, 20). This means that the ideal single-walled CNT to be used to encapsulate a dsDNA is (20, 20) which has the required radius of 13.56 Å.
Publisher: Oxford University Press (OUP)
Date: 16-08-2007
Publisher: Springer Science and Business Media LLC
Date: 08-11-2014
DOI: 10.1007/S00249-013-0936-7
Abstract: 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.
Publisher: Springer Science and Business Media LLC
Date: 29-06-2010
Publisher: Springer Science and Business Media LLC
Date: 09-01-2009
DOI: 10.1007/S00249-008-0399-4
Abstract: In this paper, we model the mechanics of a collagen pair in the connective tissue extracellular matrix that exists in abundance throughout animals, including the human body. This connective tissue comprises repeated units of two main structures, namely collagens as well as axial, parallel and regular anionic glycosaminoglycan between collagens. The collagen fibril can be modeled by Hooke's law whereas anionic glycosaminoglycan behaves more like a rubber-band rod and as such can be better modeled by the worm-like chain model. While both computer simulations and continuum mechanics models have been investigated for the behavior of this connective tissue typically, authors either assume a simple form of the molecular potential energy or entirely ignore the microscopic structure of the connective tissue. Here, we apply basic physical methodologies and simple applied mathematical modeling techniques to describe the collagen pair quantitatively. We found that the growth of fibrils was intimately related to the maximum length of the anionic glycosaminoglycan and the relative displacement of two adjacent fibrils, which in return was closely related to the effectiveness of anionic glycosaminoglycan in transmitting forces between fibrils. These reveal the importance of the anionic glycosaminoglycan in maintaining the structural shape of the connective tissue extracellular matrix and eventually the shape modulus of human tissues. We also found that some macroscopic properties, like the maximum molecular energy and the breaking fraction of the collagen, were also related to the microscopic characteristics of the anionic glycosaminoglycan.
Publisher: Springer Science and Business Media LLC
Date: 21-12-2011
Publisher: Springer Science and Business Media LLC
Date: 16-07-2011
Publisher: Elsevier BV
Date: 06-2007
Publisher: Elsevier BV
Date: 08-2011
Publisher: American Scientific Publishers
Date: 04-2013
Publisher: Springer Science and Business Media LLC
Date: 03-2022
DOI: 10.1007/S00033-022-01697-W
Abstract: In this note, we derive an extension of the conventional Einstein variation of mass formula with a specific expression arising from a Lorentz invariant equation for the energy rate $$\\hbox {d}e/\\hbox {d}p$$ d e / d p where $$e = mc^2$$ e = m c 2 is the particle energy, $$p = mu$$ p = m u the particle momentum and u the velocity. This is the simplest one-parameter Lorentz-invariant extension of the Einstein mass–energy relation. Implicit in the new expression is space–time anisotropy such that the particle has different rest masses in the positive and negative x directions. While numerous experiments have been undertaken aimed at testing such hypothesis, and all indicate the veracity of the assumption that space is isotropic, nevertheless since it is generally believed that black-holes exist at the centres of galaxies, space must be intrinsically anisotropic in some sense. Finally, we note a very curious connection with both the conventional Einstein energy–mass expression $$e = e_0/ (1 - (u/c)^2)^{1/2}$$ e = e 0 / ( 1 - ( u / c ) 2 ) 1 / 2 and the new expression derived here with certain singular integral equations usually associated with aero-foil problems, fluid mechanics and punch problems in elasticity, and that this connection is not some vague intangible relationship, but involves an exact correspondence.
Publisher: Springer Science and Business Media LLC
Date: 09-05-2007
Publisher: Cambridge University Press (CUP)
Date: 07-2015
DOI: 10.1017/S1446181115000139
Abstract: We investigate the mechanics of a nano logic gate, comprising a metallofullerene which is located inside a square-shaped single-walled carbon nanotorus involving non-metallic, single-walled carbon nanotubes with perfect nanotoroidal corners. These are highly novel and speculative nanodevices whose construction, no doubt, involves many technical challenges. The energy for the system is obtained from the 6–12 Lennard-Jones potential with the continuous approximation. Our approach shows that there is not much difference between the energy when the metallofullerene is located in the tubes compared to when it is at the corners, and therefore the metallofullerene may be controlled by a small voltage. By applying two voltage inputs to produce external electric fields, one for the left–right motion and the other for the top–bottom motion, the metallofullerene can be moved to one of the four corners. Assuming that at the four corners there are charge detectors, the proposed device can be designed as a logic gate with different signals corresponding to particular gates.
Publisher: MDPI AG
Date: 22-05-2020
DOI: 10.3390/MATH8050841
Abstract: 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.
Publisher: MDPI AG
Date: 17-08-2018
DOI: 10.3390/NANO8080624
Abstract: 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.
Publisher: SAGE Publications
Date: 23-07-2022
DOI: 10.1177/10812865221107413
Abstract: The oscillon is a highly localized dynamical phenomenon occurring in a thin horizontal layer of granular material, which rests on a rigid plate oscillating in the vertical direction. The geometry is axially symmetric and physically resembles a splash of liquid due to a falling drop, except that it continually perpetuates itself and does not generate a spreading wave, as is the case for a liquid splash. The oscillon moves from “peak” to “crater” and “crater” to “peak” such that the time from “peak” to “peak” or “crater” to “crater” is twice the period of the oscillating plate. The physics of granular phenomena is not properly understood, and there is no continuum mechanical theory of granular materials which is widely accepted as accurately describing their behaviour. Here for a free-flowing (cohesion-less) granular material, under axially symmetric conditions, we present a partial continuum mechanical analysis assuming the Coulomb–Mohr yield function and non-dilatant double-shearing theory. We examine small perturbations superimposed upon a purely vertical vibration, and make the assumption that throughout the motion, the lower surface of the layer remains in contact with the rigid metal plate. We show how the temporal dependence, which decouples from the spatial structure, is governed by Mathieu’s equation for the physically relevant case of the rigid plate oscillating sinusoidally, and therefore stability is determined by certain key parameters. We explore a variety of possible forms for spatial dependence. The present axially symmetric analysis complements that presented by the authors for plane strain conditions, and we find, quite remarkably, that apart from constants, both flows are governed by similar fourth-order systems of ordinary differential equations. This means that for both plane strain and axially symmetry, analogous pattern forming conditions can operate.
Publisher: Springer Science and Business Media LLC
Date: 08-12-2007
Publisher: Oxford University Press (OUP)
Date: 1991
Publisher: Springer Science and Business Media LLC
Date: 28-06-2011
Publisher: The Royal Society
Date: 08-01-2005
Abstract: In this paper, we deal with the materials possessing angles of internal friction ϕ for which 1 − sin ϕ is close to zero, and we use the solution for sin ϕ = 1 as the leading term in a regular perturbation series, where the correction terms are of order 1 − sin ϕ . In this way we obtain approximate analytical solutions which can be used to describe the behaviour of real granular materials. The solution procedure is illustrated with reference to quasi–static flow through wedge–shaped and conical hoppers. For these two problems, the obtained perturbation solutions are shown to be graphically indistinguishable from the numerical solutions for high angles of internal friction, and for moderately high angles of internal friction the perturbation solutions still provide excellent approximations.
Publisher: Inderscience Publishers
Date: 2008
Publisher: Elsevier BV
Date: 10-2007
Publisher: Informa UK Limited
Date: 22-04-2022
Publisher: Springer Science and Business Media LLC
Date: 27-03-2014
Publisher: American Scientific Publishers
Date: 02-2009
Abstract: The melting of spherical nanoparticles is considered from the perspective of heat flow in a pure material and as a moving boundary (Stefan) problem. The dependence of the melting temperature on both the size of the particle and the interfacial tension is described by the Gibbs-Thomson effect, and the resulting two-phase model is solved numerically using a front-fixing method. Results show that interfacial tension increases the speed of the melting process, and furthermore, the temperature distribution within the solid core of the particle exhibits behaviour that is qualitatively different to that predicted by the classical models without interfacial tension.
Publisher: Springer Science and Business Media LLC
Date: 03-2018
Publisher: Elsevier BV
Date: 10-2010
Publisher: Oxford University Press (OUP)
Date: 08-2004
Publisher: Elsevier BV
Date: 12-2015
DOI: 10.1016/J.JTBI.2015.09.020
Abstract: 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.
Publisher: Cambridge University Press (CUP)
Date: 2018
DOI: 10.1017/S144618111700058X
Abstract: There are many fluid flow problems involving geometries for which a nonorthogonal curvilinear coordinate system may be the most suitable. To the authors’ knowledge, the Navier–Stokes equations for an incompressible fluid formulated in terms of an arbitrary nonorthogonal curvilinear coordinate system have not been given explicitly in the literature in the simplified form obtained herein. The specific novelty in the equations derived here is the use of the general Laplacian in arbitrary nonorthogonal curvilinear coordinates and the simplification arising from a Ricci identity for Christoffel symbols of the second kind for flat space. Evidently, however, the derived equations must be consistent with the various general forms given previously by others. The general equations derived here admit the well-known formulae for cylindrical and spherical polars, and for the purposes of illustration, the procedure is presented for spherical polar coordinates. Further, the procedure is illustrated for a nonorthogonal helical coordinate system. For a slow flow for which the inertial terms may be neglected, we give the harmonic equation for the pressure function, and the corresponding equation if the inertial effects are included. We also note the general stress boundary conditions for a free surface with surface tension. For completeness, the equations for a compressible flow are derived in an appendix.
Publisher: MDPI AG
Date: 15-01-2020
DOI: 10.3390/NANO10010152
Abstract: 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 (≈42 ° and ≈20 ° respectively) do not occur in the stack model.
Publisher: Springer Science and Business Media LLC
Date: 09-03-2011
Abstract: In this article, we investigate the storage of lithium ions between two parallel graphene sheets using the continuous approximation and the 6-12 Lennard-Jones potential. The continuous approximation assumes that the carbon atoms can be replaced by a uniform distribution across the surface of the graphene sheets so that the total interaction potential can be approximated by performing surface integrations. The number of ion layers determines the major storage characteristics of the battery, and our results show three distinct ionic configurations, namely single, double, and triple ion forming layers between graphenes. The number densities of lithium ions between the two graphenes are estimated from existing semi-empirical molecular orbital calculations, and the graphene sheets giving rise to the triple ion layers admit the largest storage capacity at all temperatures, followed by a marginal decrease of storage capacity for the case of double ion layers. These two configurations exceed the maximum theoretical storage capacity of graphite. Further, on taking into account the charge-discharge property, the double ion layers are the most preferable choice for enhanced lithium storage. Although the single ion layer provides the least charge storage, it turns out to be the most stable configuration at all temperatures. One application of the present study is for the design of future high energy density alkali batteries using graphene sheets as anodes for which an analytical formulation might greatly facilitate rapid computational results.
Publisher: MDPI AG
Date: 16-07-2018
DOI: 10.20944/PREPRINTS201807.0272.V1
Abstract: 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 which are, as far as possible, equal throughout the structure at the same distance from the conical apex. Predicted bond angles and bond lengths are shown to agree well with those obtained by relaxing the conventional rolled-up model using the LAMMPS software. 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 into account curvature effects close to the vertex, and from which we may determine an analytical formula which represents an improvement on that for the conventional rolled-up model.
Publisher: Springer Science and Business Media LLC
Date: 22-12-2007
Publisher: Institution of Engineering and Technology (IET)
Date: 02-2014
Publisher: SAGE Publications
Date: 15-09-2020
Abstract: The dark issues of cosmological mechanics imply that our accounting for mass and energy at this scale is incorrect. On the other hand, existing theory not only accounts for atomic physics, but does so to a very high degree of accuracy. de Broglie was first to propose a concrete physical picture of the co-existence of both particle and its associated wave. We have previously proposed a Lorentz invariant modification of Newton’s second law that applies when the particle energy itself is of comparable magnitude to the potential energy of the applied external field. A dual particle–wave formulation is developed that allows exceptions to the law that matter cannot be created or destroyed, and accommodates both particle and wave energies. Here we examine this formulation in a centrally or spherically symmetric gravitating environment. A logical analysis of the integrated rate-of-working equation gives rise to four distinct states of matter positive and negative energies with either non-zero rest mass or zero rest mass. This identification is meaningful only within the extended theory, and it has no meaning within conventional theory. We propose that dark matter and dark energy arise from a particular alignment of the spatial physical force f and the force g in the direction of time, such that the particle and wave energies coincide, and a consistent mathematical framework supports this proposal. Allowable potentials V( r, t) arise as the solutions of certain partial differential equations depending upon the assumed state of matter, and the mathematical framework corresponding to each state is briefly described and, where possible, some of the simpler and physically more interesting solutions are examined. The Newtonian gravitational potential arises from a variety of asymptotic limits, and can be identified as predominantly wave energy, which is entirely consistent with the post-Newtonian approximation.
Publisher: Springer Science and Business Media LLC
Date: 11-01-2022
Publisher: Springer Science and Business Media LLC
Date: 09-2014
Publisher: Springer Science and Business Media LLC
Date: 06-07-2007
Publisher: Informa UK Limited
Date: 11-2011
Publisher: Springer Science and Business Media LLC
Date: 31-07-2019
Publisher: Springer Science and Business Media LLC
Date: 08-2003
Publisher: Springer Science and Business Media LLC
Date: 08-2007
Publisher: American Physical Society (APS)
Date: 15-10-2007
Publisher: American Scientific Publishers
Date: 05-2016
Publisher: Royal Society of Chemistry (RSC)
Date: 2012
DOI: 10.1039/C2NR00042C
Abstract: 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.
Publisher: IOP Publishing
Date: 31-01-2008
DOI: 10.1088/0957-4484/19/7/075704
Abstract: 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 isible by 3.
Publisher: IOP Publishing
Date: 16-09-2010
Publisher: Elsevier BV
Date: 04-2011
Publisher: Elsevier BV
Date: 08-2009
Publisher: Australian Mathematical Publishing Association, Inc.
Date: 13-05-2023
DOI: 10.21914/ANZIAMJ.V63.16615
Abstract: 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.
Publisher: The Royal Society
Date: 03-10-2012
Abstract: We propose here two new transformations between inertial frames that apply for relative velocities greater than the speed of light, and that are complementary to the Lorentz transformation, giving rise to the Einstein special theory of relativity that applies to relative velocities less than the speed of light. The new transformations arise from the same mathematical framework as the Lorentz transformation, displaying singular behaviour when the relative velocity approaches the speed of light and generating the same addition law for velocities, but, most importantly, do not involve the need to introduce imaginary masses or complicated physics to provide well-defined expressions. Making use of the dependence on relative velocity of the Lorentz transformation, the paper provides an elementary derivation of the new transformations between inertial frames for relative velocities v in excess of the speed of light c , and further we suggest two possible criteria from which one might infer one set of transformations as physically more likely than the other. If the energy–momentum equations are to be invariant under the new transformations, then the mass and energy are given, respectively, by the formulae and where denotes the limiting momentum for infinite relative velocity. If, however, the requirement of invariance is removed, then we may propose new mass and energy equations, and an ex le having finite non-zero mass in the limit of infinite relative velocity is given. In this highly controversial topic, our particular purpose is not to enter into the merits of existing theories, but rather to present a succinct and carefully reasoned account of a new aspect of Einstein's theory of special relativity, which properly allows for faster than light motion.
Publisher: American Physical Society (APS)
Date: 24-05-2023
Publisher: Elsevier BV
Date: 08-2009
Publisher: Institution of Engineering and Technology (IET)
Date: 2010
Publisher: American Physical Society (APS)
Date: 06-10-2009
Publisher: Springer Science and Business Media LLC
Date: 16-10-2010
Publisher: Elsevier BV
Date: 2016
Publisher: IOP Publishing
Date: 13-01-2009
DOI: 10.1088/0953-8984/21/7/075301
Abstract: 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.
Publisher: Springer Science and Business Media LLC
Date: 27-07-2013
Publisher: Elsevier BV
Date: 02-2010
DOI: 10.1016/J.JTBI.2009.10.009
Abstract: Stiff polymers, such as single-stranded DNA, unstructured RNA and cellulose, are all basically extremely long rods with relatively short repeating monomers. The simplest model for describing such stiff polymers is called the freely jointed chain model, which treats a molecule as a chain of perfectly rigid subunits of orientationally independent statistical segments, joined together by perfectly flexible hinges. A more realistic model that incorporates the entropic elasticity of a molecule, called the worm-like chain model, has been proposed by assuming that each monomer resists the bending force. Some force-extension formulae for the worm-like chain model have been previously found in terms of interpolation and numerical solutions resulting from statistical mechanics. In this paper, however, we adopt a variational principle to seek the minimum energy configuration of a stretched molecule by incorporating all the possible orientations of each monomer under thermal equilibrium, i.e., constant temperature. We determine a force-extension formula for the worm-like chain model analytically. We find that our formula suggests new terms such as the free energy and the cut-off force of a molecule, which define a clear transition from the entropic regime to the enthalpic regime and the fracture of the molecule, respectively. In addition, we predict two possible phase changes for a stretched molecule, i.e., from a super-helix to a soliton and then from a soliton to a vertical twisted line. We show theoretically that a molecule must undergo at least one phase change before it is fully stretched into its total contour length. This new formula is used to fit recent experimental data and shows a good agreement with some current literature that uses a statistical approach. Finally, an instability analysis is adopted to investigate the sensitivity of the new formula subject to small changes in temperature.
Publisher: Springer Science and Business Media LLC
Date: 04-05-2011
DOI: 10.1007/S00894-011-1086-4
Abstract: 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.
Publisher: The Royal Society
Date: 11-10-2007
Abstract: The discovery of carbon nanotubes and C 60 fullerenes has created an enormous impact on possible new nanomechanical devices. Owing to their unique mechanical and electronic properties, such as low weight, high strength, flexibility and thermal stability, carbon nanotubes and C 60 fullerenes are of considerable interest to researchers from many scientific areas. One aspect that has attracted much attention is the creation of high-frequency nanoscale oscillators, or the so-called gigahertz oscillators, for applications such as ultrafast optical filters and nano-antennae. While there are difficulties for micromechanical oscillators, or resonators, to reach a frequency in the gigahertz range, it is possible for nanomechanical systems to achieve this. This study focuses on C 60 –single-walled carbon nanotube oscillators, which generate high frequencies owing to the oscillatory motion of the C 60 molecule inside the single-walled carbon nanotube. Using the Lennard-Jones potential, the interaction energy of an offset C 60 molecule inside a carbon nanotube is determined, so as to predict its position with reference to the cross-section of the carbon nanotube. By considering the interaction force between the C 60 fullerene and the carbon nanotube, this paper provides a simple mathematical model, involving two Dirac delta functions, which can be used to capture the essential mechanisms underlying such gigahertz oscillators. As a preliminary to the calculation, the oscillatory behaviour of an isolated atom is examined. The new element of this study is the use of elementary mechanics and applied mathematical modelling in a scientific context previously dominated by molecular dynamical simulation.
Publisher: Springer Netherlands
Date: 30-07-2012
Publisher: Elsevier BV
Date: 2016
Publisher: Trans Tech Publications, Ltd.
Date: 09-2011
DOI: 10.4028/WWW.SCIENTIFIC.NET/MSF.700.100
Abstract: Recently, the present authors proposed a model describing fullerene growth in carbonvapor for which the di usion equation is assumed outside the fullerene and a family of possibleStefan conditions describe the growing fullerene surface. Here we consider a related model, butwith the additional assumption that the fullerene surface atomic density is constant throughoutthe growth process, which is justi ed by experimental data. We show that this constraint onthe fullerene lattice structure limits the solution to one possible Stefan condition. The proposedmodel may be extended to any nanostructure with constant atomic surface density.
Publisher: The Royal Society
Date: 11-10-2007
Abstract: Owing to their unusual properties, carbon nanostructures such as nanotubes and fullerenes have caused many new nanomechanical devices to be proposed. One such application is that of nanoscale oscillators which operate in the gigahertz range, the so-called gigahertz oscillators. Such devices have potential applications as ultrafast optical filters and nano-antennae. While there are difficulties in producing micromechanical oscillators which operate in the gigahertz range, molecular dynamical simulations indicate that nanoscale devices constructed of multi-walled carbon nanotubes or single-walled carbon nanotubes and C 60 fullerenes could feasibly operate at these high frequencies. This paper investigates the suction force experienced by either an atom or a C 60 fullerene molecule located in the vicinity of an open end of a single-walled carbon nanotube. The atom is modelled as a point mass, the fullerene as an averaged atomic mass distributed over the surface of a sphere. In both cases, the carbon nanotube is modelled as an averaged atomic mass distributed over the surface of an open semi-infinite cylinder. In both cases, the particle being introduced is assumed to be located on the axis of the cylinder. Using the Lennard-Jones potential, the van der Waals interaction force between the atom or C 60 fullerene and the carbon nanotube can be obtained analytically. Furthermore, by integrating the force, an explicit analytic expression for the work done by van der Waals forces is determined and used to derive an acceptance condition, that is whether the particle will be completely sucked into the carbon nanotube by virtue of van der Waals interactions alone, and a suction energy which is imparted to the introduced particle in the form of an increased velocity. The results of the acceptance condition and the calculated suction energy are shown to be in good agreement with the published molecular dynamical simulations. In part II of the paper, a new model is proposed to describe the oscillatory motion adopted by atoms and fullerenes that are sucked into carbon nanotubes.
Publisher: Springer Science and Business Media LLC
Date: 16-03-2021
Publisher: IOP Publishing
Date: 14-01-2009
Publisher: Oxford University Press (OUP)
Date: 26-12-2008
Publisher: Trans Tech Publications, Ltd.
Date: 09-2012
DOI: 10.4028/WWW.SCIENTIFIC.NET/MSF.700.104
Abstract: We investigate the prospect of methane gas storage in carbon nanotubes, and in particular we determine the interaction energy between a methane molecule and (9, 5), (8, 8) and (10,10) carbon nanotubes. Employing the Lennard-Jones potential together with the continuous approximation, we determine analytically the interaction energy for a methane molecule inside a carbon nanotube. Our results indicate that larger tubes are highly favoured sites for methane storage although smaller tubes might be superior for methane adsorption at higher temperatures, especially in the range 400 − 500 K.
Publisher: Informa UK Limited
Date: 17-03-2010
Publisher: Springer Science and Business Media LLC
Date: 20-06-2018
Publisher: American Chemical Society (ACS)
Date: 29-11-2011
DOI: 10.1021/JP2069094
No related grants have been discovered for James Hill.