Numerical simulation of the Riesz fractional diffusion equation with a nonlinear source term

In this paper, A Riesz fractional diffusion equation with a nonlinear source term (RFDE-NST) is considered. This equation is commonly used to model the growth and spreading of biological species. According to the equivalent of the Riemann-Liouville(R-L) and Gr¨unwald-Letnikov(GL) fractional derivative definitions, an implicit difference approximation (IFDA) for the RFDE-NST is derived. We prove the IFDA is unconditionally stable and convergent. In order to evaluate the efficiency of the IFDA, a comparison with a fractional method of lines (FMOL) is used. Finally, two numerical examples are presented to show that the numerical results are in good agreement with our theoretical analysis.

Fundamental solution and discrete random walk model for time-space fractional diffusion equation

In this paper, we consider a time-space fractional diffusion equation of distributed order (TSFDEDO). The TSFDEDO is obtained from the standard advection-dispersion equation by replacing the first-order time derivative by the Caputo fractional derivative of order α∈(0,1], the first-order and second-order space derivatives by the Riesz fractional derivatives of orders β 1∈(0,1) and β 2∈(1,2], respectively. We derive the fundamental solution for the TSFDEDO with an initial condition (TSFDEDO-IC). The fundamental solution can be interpreted as a spatial probability density function evolving in time. We also investigate a discrete random walk model based on an explicit finite difference approximation for the TSFDEDO-IC.

Near infrared spectroscopy of stearic acid adsorbed on montmorillonite

The adsorption of stearic acid on both sodium montmorillonites and calcium montmorillonites has been studied by near infrared spectroscopy complimented with infrared spectroscopy. Upon adsorption of stearic acid on Ca-Mt additional near infrared bands are observed at 8236 cm-1 and is assigned to an interaction of stearic acid with the water of hydration. Upon adsorption of the stearic acid on Na-Mt, the NIR bands are now observed at 5671, 5778, 5848 and 5912 cm-1 and are assigned to the overtone and combination bands of the CH fundamentals. Additional bands at 4177, 4250, 4324, 4337, 4689 and 4809 cm-1 are attributed to CH combination bands resulting from the adsorption of the stearic acid. Stearic acid is used as a model molecule for adsorption studies. The application of near infrared spectroscopy to the study of this adsorption proved most useful.

An application of near-infrared and mid-infrared spectroscopy to the study of selected minerals

Near-infrared spectroscopy is a somewhat unutilised technique for the study of minerals. The technique has the ability to determine water content, hydroxyl groups and transition metals. In this paper we show the application of NIR spectroscopy to the study of selected minerals. The structure and spectral properties of two Cu-tellurite minerals graemite and teineite are compared with bismuth containing tellurite mineral smirnite by the application of NIR and IR spectroscopy. The position of Cu2+ bands and their splitting in the electronic spectra of tellurites are in conformity with octahedral geometry distortion. The spectral pattern of smirnite resembles graemite and the observed band at 10855 cm-1 with a weak shoulder at 7920 cm-1 is identified as due to Cu2+ ion. Any transition metal impurities may be identified by their bands in this spectral region. Three prominent bands observed in the region of 7200-6500 cm-1 are the overtones of water whilst the weak bands observed near 6200 cm-1in tellurites may be attributed to the hydrogen bonding between (TeO3)2- and H2O. The observation of a number of bands centred at around 7200 cm-1 confirms molecular water in tellurite minerals. A number of overlapping bands in the low wavenumbers 4500-4000 cm-1 is the result of combinational modes of (TeO3)2−ion. The appearance of the most intense peak at 5200 cm-1 with a pair of weak bands near 6000 cm-1 is a common feature in all the spectra and is related to the combinations of OH vibrations of water molecules, and bending vibrations ν2 (δ H2O). Bending vibrations δ H2O observed in the IR spectra shows a single band for smirnite at 1610 cm-1. The resolution of this band into number of components is evidenced for non-equivalent types of molecular water in graemite and teineite. (TeO3)2- stretching vibrations are characterized by three main absorptions at 1080, 780 and 695 cm-1.

Vibrational spectroscopy of intercalated kaolinites. Part I

The industrial application of kaolinite is closely related to its reactivity and surface properties. The reactivity of kaolinite can be tested by intercalation, i.e. via the insertion of low molecular weight organic compounds between the kaolinite layers resulting in the formation of a nano-layered organo-complex. Although intercalation of kaolinite is an old and ongoing research topic, there is a limited knowledge available on the reactivity of different kaolinites, the mechanism of complex formation as well as on the structure of the complexes formed. Grafting and incorporation of exfoliated kaolinite in polymer matrices and other potential applications can open new horizons in the study of kaolinite intercalation. This paper attempts to summarize (without completion) the most recent achievements in the study of kaolinite organo-complexes obtained with the most common intercalating compounds like urea, potassium acetate, dimethyl sulphoxide, formamide and hydrazine using vibrational spectroscopy combined with X-ray powder diffraction and thermal analysis.

PDMS spreading morphological patterns on substrates of different hydrophilicity in air vacuum and water

In paper has been to investigate the morphological patterns and kinetics of PDMS spreading on silicon wafer using combination of techniques like ellipsometry, atomic force microscope (AFM), scanning electron microscope (SEM) and optical microscopy. A macroscopic silicone oil drops as well as PDMS water based emulsions were studied after deposition on a flat surface of silicon wafer in air, water and vacuum. our own measurements using an imaging ellipsometer, which also clearly shows the presence of a precursor film. The diffusion constant of this film, measured with a 60 000 cS PDMS sample spreading on a hydrophilic silicon wafer, is Df = 1.4  10-11 m2/s. Regardless of their size, density and method of deposition, droplets on both types of wafer (hydrophilic and hydrophobic) flatten out over a period of many hours, up to 3 days. During this process neighbouring droplets may coalesce, but there is strong evidence that some of the PDMS from the droplets migrates into a thin, continuous film that covers the surface in between droplets. The thin film appears to be ubiquitous if there has been any deposition of PDMS. However, this statement needs further verification. One question is whether the film forms immediately after forced drying, or whether in some or all cases it only forms by spreading from isolated droplets as they slowly flatten out.

A comparison of different methods to determine diffusion length in dye-sensitized solar cells

A new steady state method for determination of the electron diffusion length in dye-sensitized solar cells (DSCs) is described and illustrated with data obtained using cells containing three different types of electrolyte. The method is based on using near-IR absorbance methods to establish pairs of illumination intensity for which the total number of trapped electrons is the same at open circuit (where all electrons are lost by interfacial electron transfer) as at short circuit (where the majority of electrons are collected at the contact). Electron diffusion length values obtained by this method are compared with values derived by intensity modulated methods and by impedance measurements under illumination. The results indicate that the values of electron diffusion length derived from the steady state measurements are consistently lower than the values obtained by the non steady-state methods. For all three electrolytes used in the study, the electron diffusion length was sufficiently high to guarantee electron collection efficiencies greater than 90%. Measurement of the trap distributions by near-IR absorption confirmed earlier observations of much higher electron trap densities for electrolytes containing Li+ ions. It is suggested that the electron trap distributions may not be intrinsic properties of the TiO2 nanoparticles, but may be associated with electron-ion interactions.

Characterisation of Ni carbonate-bearing minerals by UV-Vis-NIR spectroscopy

Four nickel carbonate-bearing minerals from Australia have been investigated to study the effect of Ni for Mg substitution. The spectra of nullaginite, zaratite, widgiemoolthalite and takovite show three main features in the range of 26,720–25,855 cm−1 (ν1-band), 15,230–14,740 cm−1 (ν2-band) and 9,200–9,145 cm−1 (ν3-band) which are characteristic of divalent nickel in six-fold coordination. The Crystal Field Stabilization Energy (CFSE) of Ni2+ in the four carbonates is calculated from the observed 3A2g(3F) → 3T2g(3F) transition. CFSE is dependent on mineralogy, crystallinity and chemical composition (Al/Mg-content). The splitting of the ν1- and ν3-bands and non-Gaussian shape of ν3-band in the minerals are the effects of Ni-site distortion from regular octahedral. The effect of structural cation substitutions (Mg2+, Ni2+, Fe2+ and trivalent cations, Al3+, Fe3+) in the carbonate minerals is noticed on band shifts. Thus, electronic bands in the UV–Vis–NIR spectra and the overtones and combination bands of OH and carbonate ion in NIR show shifts to higher wavenumbers, particularly for widgiemoolthalite and takovite.

Thermal analysis and hot stage Raman spectroscopy of the basic copper arsenate mineral : euchroite

The thermal analysis of euchroite shows two mass loss steps in the temperature range 100 to 105°C and 185 to 205°C. These mass loss steps are attributed to dehydration and dehydroxylation of the mineral. Hot stage Raman spectroscopy (HSRS) has been used to study the thermal stability of the mineral euchroite, a mineral involved in a complex set of equilibria between the copper hydroxy arsenates: euchroite Cu2(AsO4)(OH).3H2O → olivenite Cu2(AsO4)(OH) → strashimirite Cu8(AsO4)4(OH)4.5H2O → arhbarite Cu2Mg(AsO4)(OH)3. Hot stage Raman spectroscopy inolves the collection of Raman spectra as a function of the temperature. HSRS shows that the mineral euchroite decomposes between 125 and 175 °C with the loss of water. At 125 °C, Raman bands are observed at 858 cm-1 assigned to the ν1 AsO43- symmetric stretching vibration and 801, 822 and 871 cm-1 assigned to the ν3 AsO43- (A1) antisymmetric stretching vibration. A distinct band shift is observed upon heating to 275 °C. At 275 °C the four Raman bands are resolved at 762, 810, 837 and 862 cm-1. Further heating results in the diminution of the intensity in the Raman spectra and this is attributed to sublimation of the arsenate mineral. Hot stage Raman spectroscopy is most useful technique for studying the thermal stability of minerals especially when only very small amounts of mineral are available.

Thermogravimetric analysis and hot stage Raman spectroscopy of cubic indium hydroxide

The transition of cubic indium hydroxide to cubic indium oxide has been studied by thermogravimetric analysis complimented with hot stage Raman spectroscopy. Thermal analysis shows the transition of In(OH)3 to In2O3 occurs at 219°C. The structure and morphology of In(OH)3 synthesised using a soft chemical route at low temperatures was confirmed by X-ray diffraction and scanning electron microscopy. A topotactical relationship exists between the micro/nano-cubes of In(OH)3 and In2O3. The Raman spectrum of In(OH)3 is characterised by an intense sharp band at 309 cm-1 attributed to ν1 In-O symmetric stretching mode, bands at 1137 and 1155 cm-1 attributed to In-OH δ deformation modes, bands at 3083, 3215, 3123 and 3262 cm-1 assigned to the OH stretching vibrations. Upon thermal treatment of In(OH)3 new Raman bands are observed at 125, 295, 488 and 615 cm-1 attributed to In2O3. Changes in the structure of In(OH)3 with thermal treatment is readily followed by hot stage Raman spectroscopy.

Theoretical investigation of mechanisms of formation and interaction of nanoparticles

In this thesis an investigation into theoretical models for formation and interaction of nanoparticles is presented. The work presented includes a literature review of current models followed by a series of five chapters of original research. This thesis has been submitted in partial fulfilment of the requirements for the degree of doctor of philosophy by publication and therefore each of the five chapters consist of a peer-reviewed journal article. The thesis is then concluded with a discussion of what has been achieved during the PhD candidature, the potential applications for this research and ways in which the research could be extended in the future. In this thesis we explore stochastic models pertaining to the interaction and evolution mechanisms of nanoparticles. In particular, we explore in depth the stochastic evaporation of molecules due to thermal activation and its ultimate effect on nanoparticles sizes and concentrations. Secondly, we analyse the thermal vibrations of nanoparticles suspended in a fluid and subject to standing oscillating drag forces (as would occur in a standing sound wave) and finally on lattice surfaces in the presence of high heat gradients. We have described in this thesis a number of new models for the description of multicompartment networks joined by a multiple, stochastically evaporating, links. The primary motivation for this work is in the description of thermal fragmentation in which multiple molecules holding parts of a carbonaceous nanoparticle may evaporate. Ultimately, these models predict the rate at which the network or aggregate fragments into smaller networks/aggregates and with what aggregate size distribution. The models are highly analytic and describe the fragmentation of a link holding multiple bonds using Markov processes that best describe different physical situations and these processes have been analysed using a number of mathematical methods. The fragmentation of the network/aggregate is then predicted using combinatorial arguments. Whilst there is some scepticism in the scientific community pertaining to the proposed mechanism of thermal fragmentation,we have presented compelling evidence in this thesis supporting the currently proposed mechanism and shown that our models can accurately match experimental results. This was achieved using a realistic simulation of the fragmentation of the fractal carbonaceous aggregate structure using our models. Furthermore, in this thesis a method of manipulation using acoustic standing waves is investigated. In our investigation we analysed the effect of frequency and particle size on the ability for the particle to be manipulated by means of a standing acoustic wave. In our results, we report the existence of a critical frequency for a particular particle size. This frequency is inversely proportional to the Stokes time of the particle in the fluid. We also find that for large frequencies the subtle Brownian motion of even larger particles plays a significant role in the efficacy of the manipulation. This is due to the decreasing size of the boundary layer between acoustic nodes. Our model utilises a multiple time scale approach to calculating the long term effects of the standing acoustic field on the particles that are interacting with the sound. These effects are then combined with the effects of Brownian motion in order to obtain a complete mathematical description of the particle dynamics in such acoustic fields. Finally, in this thesis, we develop a numerical routine for the description of "thermal tweezers". Currently, the technique of thermal tweezers is predominantly theoretical however there has been a handful of successful experiments which demonstrate the effect it practise. Thermal tweezers is the name given to the way in which particles can be easily manipulated on a lattice surface by careful selection of a heat distribution over the surface. Typically, the theoretical simulations of the effect can be rather time consuming with supercomputer facilities processing data over days or even weeks. Our alternative numerical method for the simulation of particle distributions pertaining to the thermal tweezers effect use the Fokker-Planck equation to derive a quick numerical method for the calculation of the effective diffusion constant as a result of the lattice and the temperature. We then use this diffusion constant and solve the diffusion equation numerically using the finite volume method. This saves the algorithm from calculating many individual particle trajectories since it is describes the flow of the probability distribution of particles in a continuous manner. The alternative method that is outlined in this thesis can produce a larger quantity of accurate results on a household PC in a matter of hours which is much better than was previously achieveable.

Near infrared spectroscopy of benzoic acid adsorbed on montmorillonite

The adsorption of benzoic acid on both sodium and calcium montmorillonites has been studied by near infrared spectroscopy complimented with infrared spectroscopy. Upon adsorption of benzoic acid additional near infrared bands are observed at 8665 cm-1 and assigned to an interaction of benzoic acid with the water of hydration. Upon adsorption of the benzoic acid on Na-Mt, the NIR bands are now observed at 5877, 5951, 6028 and 6128 cm-1 and are assigned to the overtone and combination bands of the CH fundamentals. Additional bands at 4074, 4205, 4654 and 4678 cm-1 are attributed to CH combination bands resulting from the adsorption of the benzoic acid. Benzoic acid is used as a model molecule for adsorption studies. The application of near infrared spectroscopy to the study of adsorption has the potential for the removal of acids from polluted aqueous systems.