A novel numerical approximation for the space fractional advection-dispersion equation

In this paper, we consider a space fractional advection–dispersion equation. The equation is obtained from the standard advection–diffusion equation by replacing the first- and second-order space derivatives by the Riesz fractional derivatives of order β1 ∈ (0, 1) and β2 ∈ (1, 2], respectively. The fractional advection and dispersion terms are approximated by using two fractional centred difference schemes. A new weighted Riesz fractional finite-difference approximation scheme is proposed. When the weighting factor θ equals 12, a second-order accuracy scheme is obtained. The stability, consistency and convergence of the numerical approximation scheme are discussed. A numerical example is given to show that the numerical results are in good agreement with our theoretical analysis.

Numerical approximation for a variable-order nonlinear reaction–subdiffusion equation

Fractional reaction–subdiffusion equations are widely used in recent years to simulate physical phenomena. In this paper, we consider a variable-order nonlinear reaction–subdiffusion equation. A numerical approximation method is proposed to solve the equation. Its convergence and stability are analyzed by Fourier analysis. By means of the technique for improving temporal accuracy, we also propose an improved numerical approximation. Finally, the effectiveness of the theoretical results is demonstrated by numerical examples.

Deformation with coupled chemical diffusion

The deformation of rocks is commonly intimately associated with metamorphic reactions. This paper is a step towards understanding the behaviour of fully coupled, deforming, chemically reacting systems by considering a simple example of the problem comprising a single layer system with elastic-power law viscous constitutive behaviour where the deformation is controlled by the diffusion of a single chemical component that is produced during a metamorphic reaction. Analysis of the problem using the principles of non-equilibrium thermodynamics allows the energy dissipated by the chemical reaction-diffusion processes to be coupled with the energy dissipated during deformation of the layers. This leads to strain-rate softening behaviour and the resultant development of localised deformation which in turn nucleates buckles in the layer. All such diffusion processes, in leading to Herring-Nabarro, Coble or “pressure solution” behaviour, are capable of producing mechanical weakening through the development of a “chemical viscosity”, with the potential for instability in the deformation. For geologically realistic strain rates these chemical feed-back instabilities occur at the centimetre to micron scales, and so produce structures at these scales, as opposed to thermal feed-back instabilities that become important at the 100–1000 m scales.

Mathematical modelling of controlled drug release from polymer micro-spheres : incorporating the effects of swelling, diffusion and dissolution via moving boundary problems

Controlled drug delivery is a key topic in modern pharmacotherapy, where controlled drug delivery devices are required to prolong the period of release, maintain a constant release rate, or release the drug with a predetermined release profile. In the pharmaceutical industry, the development process of a controlled drug delivery device may be facilitated enormously by the mathematical modelling of drug release mechanisms, directly decreasing the number of necessary experiments. Such mathematical modelling is difficult because several mechanisms are involved during the drug release process. The main drug release mechanisms of a controlled release device are based on the device’s physiochemical properties, and include diffusion, swelling and erosion. In this thesis, four controlled drug delivery models are investigated. These four models selectively involve the solvent penetration into the polymeric device, the swelling of the polymer, the polymer erosion and the drug diffusion out of the device but all share two common key features. The first is that the solvent penetration into the polymer causes the transition of the polymer from a glassy state into a rubbery state. The interface between the two states of the polymer is modelled as a moving boundary and the speed of this interface is governed by a kinetic law. The second feature is that drug diffusion only happens in the rubbery region of the polymer, with a nonlinear diffusion coefficient which is dependent on the concentration of solvent. These models are analysed by using both formal asymptotics and numerical computation, where front-fixing methods and the method of lines with finite difference approximations are used to solve these models numerically. This numerical scheme is conservative, accurate and easily implemented to the moving boundary problems and is thoroughly explained in Section 3.2. From the small time asymptotic analysis in Sections 5.3.1, 6.3.1 and 7.2.1, these models exhibit the non-Fickian behaviour referred to as Case II diffusion, and an initial constant rate of drug release which is appealing to the pharmaceutical industry because this indicates zeroorder release. The numerical results of the models qualitatively confirms the experimental behaviour identified in the literature. The knowledge obtained from investigating these models can help to develop more complex multi-layered drug delivery devices in order to achieve sophisticated drug release profiles. A multi-layer matrix tablet, which consists of a number of polymer layers designed to provide sustainable and constant drug release or bimodal drug release, is also discussed in this research. The moving boundary problem describing the solvent penetration into the polymer also arises in melting and freezing problems which have been modelled as the classical onephase Stefan problem. The classical one-phase Stefan problem has unrealistic singularities existed in the problem at the complete melting time. Hence we investigate the effect of including the kinetic undercooling to the melting problem and this problem is called the one-phase Stefan problem with kinetic undercooling. Interestingly we discover the unrealistic singularities existed in the classical one-phase Stefan problem at the complete melting time are regularised and also find out the small time behaviour of the one-phase Stefan problem with kinetic undercooling is different to the classical one-phase Stefan problem from the small time asymptotic analysis in Section 3.3. In the case of melting very small particles, it is known that surface tension effects are important. The effect of including the surface tension to the melting problem for nanoparticles (no kinetic undercooling) has been investigated in the past, however the one-phase Stefan problem with surface tension exhibits finite-time blow-up. Therefore we investigate the effect of including both the surface tension and kinetic undercooling to the melting problem for nanoparticles and find out the the solution continues to exist until complete melting. The investigation of including kinetic undercooling and surface tension to the melting problems reveals more insight into the regularisations of unphysical singularities in the classical one-phase Stefan problem. This investigation gives a better understanding of melting a particle, and contributes to the current body of knowledge related to melting and freezing due to heat conduction.

Examining the influence of participant performance factors on contractor satisfaction : a structural equation model

Participant performance is critical to the success of projects. At the same time, enhancing the satisfaction of participants not only helps in problem solving but also improves their motivation and cooperation. However, previous research related to participant satisfaction is primarily concerned with clients and customers and relatively little attention has been paid to contractors. This paper investigates how the performance of project participants affects contractor project satisfaction in terms of the client's clarity of objectives (OC) and promptness of payments (PP), designer carefulness (DC), construction risk management (RM), the effectiveness their contribution (EW) and mutual respect and trust (RT). With 125 valid responses from contractors in Malaysia, a contractor satisfaction model is developed based on structural equation modelling. The results demonstrate the necessity for dividing abstract satisfaction into two dimensions, comprising economic-related satisfaction (ES) and production-related satisfaction (PS), with DC, OC, PP and RM having significant effects on ES, while DC, OC, EW and RM influence PS. In addition, the model tests the indirect effects of these performance variables on ES and PS. In particular, OC indirectly affects ES and PS through mediation of RM and DC respectively. The results also provide opportunities for improving contractor satisfaction and supplementing the contractor selection criteria for clients.

Multicomponent charge transport in electrolyte solutions

The work presented in this thesis investigates the mathematical modelling of charge transport in electrolyte solutions, within the nanoporous structures of electrochemical devices. We compare two approaches found in the literature, by developing onedimensional transport models based on the Nernst-Planck and Maxwell-Stefan equations. The development of the Nernst-Planck equations relies on the assumption that the solution is infinitely dilute. However, this is typically not the case for the electrolyte solutions found within electrochemical devices. Furthermore, ionic concentrations much higher than those of the bulk concentrations can be obtained near the electrode/electrolyte interfaces due to the development of an electric double layer. Hence, multicomponent interactions which are neglected by the Nernst-Planck equations may become important. The Maxwell-Stefan equations account for these multicomponent interactions, and thus they should provide a more accurate representation of transport in electrolyte solutions. To allow for the effects of the electric double layer in both the Nernst-Planck and Maxwell-Stefan equations, we do not assume local electroneutrality in the solution. Instead, we model the electrostatic potential as a continuously varying function, by way of Poisson’s equation. Importantly, we show that for a ternary electrolyte solution at high interfacial concentrations, the Maxwell-Stefan equations predict behaviour that is not recovered from the Nernst-Planck equations. The main difficulty in the application of the Maxwell-Stefan equations to charge transport in electrolyte solutions is knowledge of the transport parameters. In this work, we apply molecular dynamics simulations to obtain the required diffusivities, and thus we are able to incorporate microscopic behaviour into a continuum scale model. This is important due to the small size scales we are concerned with, as we are still able to retain the computational efficiency of continuum modelling. This approach provides an avenue by which the microscopic behaviour may ultimately be incorporated into a full device-scale model. The one-dimensional Maxwell-Stefan model is extended to two dimensions, representing an important first step for developing a fully-coupled interfacial charge transport model for electrochemical devices. It allows us to begin investigation into ambipolar diffusion effects, where the motion of the ions in the electrolyte is affected by the transport of electrons in the electrode. As we do not consider modelling in the solid phase in this work, this is simulated by applying a time-varying potential to one interface of our two-dimensional computational domain, thus allowing a flow field to develop in the electrolyte. Our model facilitates the observation of the transport of ions near the electrode/electrolyte interface. For the simulations considered in this work, we show that while there is some motion in the direction parallel to the interface, the interfacial coupling is not sufficient for the ions in solution to be "dragged" along the interface for long distances.

Fractional models for the migration of biological cells in complex spatial domains

Travelling wave phenomena are observed in many biological applications. Mathematical theory of standard reaction-diffusion problems shows that simple partial differential equations exhibit travelling wave solutions with constant wavespeed and such models are used to describe, for example, waves of chemical concentrations, electrical signals, cell migration, waves of epidemics and population dynamics. However, as in the study of cell motion in complex spatial geometries, experimental data are often not consistent with constant wavespeed. Non-local spatial models have successfully been used to model anomalous diffusion and spatial heterogeneity in different physical contexts. In this paper, we develop a fractional model based on the Fisher-Kolmogoroff equation and analyse it for its wavespeed properties, attempting to relate the numerical results obtained from our simulations to experimental data describing enteric neural crest-derived cells migrating along the intact gut of mouse embryos. The model proposed essentially combines fractional and standard diffusion in different regions of the spatial domain and qualitatively reproduces the behaviour of neural crest-derived cells observed in the caecum and the hindgut of mouse embryos during in vivo experiments.

First-principle studies of the formation and diffusion of hydrogen vacancies in magnesium hydride

Ab initio density functional theory (DFT) calculations are performed to study the formation and diffusion of hydrogen vacancies on MgH2(110) surface and in bulk. We find that the formation energies for a single H-vacancy increase slightly from the surface to deep layers. The energies for creating adjacent surface divancacies at two inplane sites and at an inplane and a bridge site are even smaller than that for the formation of a single H-vacancy, a fact that is attributed to the strong vacancy−vacancy interactions. The diffusion of an H-vacancy from an in-plane site to a bridge site on the surface has the smallest activation barrier calculated at 0.15 eV and should be fast at room temperature. The activation barriers computed for H-vacancy diffusion from the surface into sublayers are all less than 0.70 eV, which is much smaller than the activation energy for desorption of hydrogen on the MgH2(110) surface (1.78−2.80 eV/H2). This suggests that surface desorption is more likely than vacancy diffusion to be rate determining, such that finding effective catalyst on the MgH2 surface to facilitate desorption will be very important for improving overall dehydrogenation performance.

Atomic hydrogen diffusion in novel magnesium nanostructures : the impact of incorporated subsurface carbon atoms

Ab initio Density Functional Theory (DFT) calculations are performed to study the diffusion of atomic hydrogen on a Mg(0001) surface and their migration into the subsurface layers. A carbon atom located initially on a Mg(0001) surface can migrate into the sub-surface layer and occupy a fcc site, with charge transfer to the C atom from neighboring Mg atoms. The cluster of postively charged Mg atoms surrounding a sub-surface C is then shown to facilitate the dissociative chemisorption of molecular hydrogen on the Mg(0001) surface, and the surface migration and subsequent diffusion into the subsurface of atomic hydrogen. This helps rationalize the experimentally-observed improvement in absorption kinetics of H2 when graphite or single walled carbon nanotubes (SWCNT) are introduced into the Mg powder during ball milling.

Numerical solutions for thin film flow down the outside and inside of a vertical cylinder

We consider a model for thin film flow down the outside and inside of a vertical cylinder. Our focus is to study the effect that the curvature of the cylinder has on the gravity-driven instability of the advancing contact line and to simulate the resulting fingering patterns that form due to this instability. The governing partial differential equation is fourth order with a nonlinear degenerate diffusion term that represents the stabilising effect of surface tension. We present numerical solutions obtained by implementing an efficient alternating direction implicit scheme. When compared to the problem of flow down a vertical plane, we find that increasing substrate curvature tends to increase the fingering instability for flow down the outside of the cylinder, whereas flow down the inside of the cylinder substrate curvature has the opposite effect. Further, we demonstrate the existence of nontrivial travelling wave solutions which describe fingering patterns that propagate down the inside of a cylinder at constant speed without changing form. These solutions are perfectly analogous to those found previously for thin film flow down an inclined plane.

Computer modeling of diffusion in biological tissues

Molecular-level computer simulations of restricted water diffusion can be used to develop models for relating diffusion tensor imaging measurements of anisotropic tissue to microstructural tissue characteristics. The diffusion tensors resulting from these simulations can then be analyzed in terms of their relationship to the structural anisotropy of the model used. As the translational motion of water molecules is essentially random, their dynamics can be effectively simulated using computers. In addition to modeling water dynamics and water-tissue interactions, the simulation software of the present study was developed to automatically generate collagen fiber networks from user-defined parameters. This flexibility provides the opportunity for further investigations of the relationship between the diffusion tensor of water and morphologically different models representing different anisotropic tissues.

A study of the diffusion characteristics of normal, delipidized and relipidized articular cartilage using magnetic resonance imaging

This paper assesses the capacity to provide semipermeability of the synthetic layer of surface-active phospholipids created to replace the depleted surface amorphous layer of articular cartilage. The surfaces of articular cartilage specimens in normal, delipidized, and relipidized conditions following incubation in dipalmitoyl-phosphatidylcholine and palmitoyl-oleoyl-phosphatidylcholine components of the joint lipid mixture were characterized nanoscopically with the atomic force microscope and also imaged as deuterium oxide (D2O) diffused transiently through these surfaces in a magnetic resonance imaging enclosure. The MR images were then used to determine the apparent diffusion coefficients in a purpose-built MATLAB®-based algorithm. Our results revealed that all surfaces were permeable to D2O, but that there was a significant difference in the semipermeability of the surfaces under the different conditions, relative to the apparent diffusion coefficients. Based on the results and observations, it can be concluded that the synthetic lipid that is deposited to replace the depleted SAL of articular cartilage is capable of inducing some level of semipermeability.

Metrological performances of a diffusion charger particle counter for personal monitoring

Airborne particles have been shown to be associated with a wide range of adverse health effects, which has led to a recent increase in medical research to gain better insight into their health effects. However, accurate evaluation of the exposure-dose-response relationship is highly dependent on the ability to track actual exposure levels of people to airborne particles. This is quite a complex task, particularly in relation to submicrometer and ultrafine particles, which can vary quite significantly in terms of particle surface area and number concentrations. Therefore, suitable monitors that can be worn for measuring personal exposure to these particles are needed. This paper presents an evaluation of the metrological performance of six diffusion charger sensors, NanoTracer (Philips Aerasense) monitors, when measuring particle number and surface area concentrations, as well as particle number distribution mean when compared to reference instruments. Tests in the laboratory (by generating monodisperse and polydisperse aerosols) and in the field (using natural ambient particles) were designed to evaluate the response of these devices under both steady-state and dynamics conditions. Results showed that the NanoTracers performed well when measuring steady state aerosols, however they strongly underestimated actual concentrations during dynamic response testing. The field experiments also showed that, when the majority of the particles were smaller than 20 nm, which occurs during particle formation events in the atmosphere, the NanoTracer underestimated number concentration quite significantly. Even though the NanoTracer can be used for personal monitoring of exposure to ultrafine particles, it also has limitations which need to be considered in order to provide meaningful results.

Diffusion tensor of water in partially aligned fibre networks

Monte Carlo simulations were used to investigate the relationship between the morphological characteristics and the diffusion tensor (DT) of partially aligned networks of cylindrical fibres. The orientation distributions of the fibres in each network were approximately uniform within a cone of a given semi-angle (θ0). This semi-angle was used to control the degree of alignment of the fibres. The networks studied ranged from perfectly aligned (θ0 = 0) to completely disordered (θ0 = 90°). Our results are qualitatively consistent with previous numerical models in the overall behaviour of the DT. However, we report a non-linear relationship between the fractional anisotropy (FA) of the DT and collagen volume fraction, which is different to the findings from previous work. We discuss our results in the context of diffusion tensor imaging of articular cartilage. We also demonstrate how appropriate diffusion models have the potential to enable quantitative interpretation of the experimentally measured diffusion-tensor FA in terms of collagen fibre alignment distributions.

Investigations of the physical origins of etching LiNbO3 during Ti in-diffusion

We investigate the physical origins of etching observed during Ti diffusion. The relationship between observed etch depth and water vapor content in the annealing environment is quantified. The dynamics of the etching process are also identified. It is discovered that water vapor content is essential for etching and that there is a characteristic delay before etching is observed. From these observations we can conclude that the process is electrochemical in nature with ionic defects diffusing into the Ti strip from the lithium niobate and these defects catalyzing the dissociation of water into reactive ions.