Numerical simulation of a new two-dimensional variable-order fractional percolation equation in non-homogeneous porous media

Percolation flow problems are discussed in many research fields, such as seepage hydraulics, groundwater hydraulics, groundwater dynamics and fluid dynamics in porous media. Many physical processes appear to exhibit fractional-order behavior that may vary with time, or space, or space and time. The theory of pseudodifferential operators and equations has been used to deal with this situation. In this paper we use a fractional Darcys law with variable order Riemann-Liouville fractional derivatives, this leads to a new variable-order fractional percolation equation. In this paper, a new two-dimensional variable-order fractional percolation equation is considered. A new implicit numerical method and an alternating direct method for the two-dimensional variable-order fractional model is proposed. Consistency, stability and convergence of the implicit finite difference method are established. Finally, some numerical examples are given. The numerical results demonstrate the effectiveness of the methods. This technique can be used to simulate a three-dimensional variable-order fractional percolation equation.

Cell migration and proliferation on homogeneous and non-homogeneous domains : modelling on the scale of individuals and populations

Cell migration is a behaviour critical to many key biological effects, including wound healing, cancerous cell invasion and morphogenesis, the development of an organism from an embryo. However, given that each of these situations is distinctly different and cells are extremely complicated biological objects, interest lies in more basic experiments which seek to remove conflating factors and present a less complex environment within which cell migration can be experimentally examined. These include in vitro studies like the scratch assay or circle migration assay, and ex vivo studies like the colonisation of the hindgut by neural crest cells. The reduced complexity of these experiments also makes them much more enticing as problems to mathematically model, like done here. The primary goal of the mathematical models used in this thesis is to shed light on which cellular behaviours work to generate the travelling waves of invasion observed in these experiments, and to explore how variations in these behaviours can potentially predict differences in this invasive pattern which are experimentally observed when cell types or chemical environment are changed. Relevant literature has already identified the difficulty of distinguishing between these behaviours when using traditional mathematical biology techniques operating on a macroscopic scale, and so here a sophisticated individual-cell-level model, an extension of the Cellular Potts Model (CPM), is been constructed and used to model a scratch assay experiment. This model includes a novel mechanism for dealing with cell proliferations that allowed for the differing properties of quiescent and proliferative cells to be implemented into their behaviour. This model is considered both for its predictive power and used to make comparisons with the travelling waves which result in more traditional macroscopic simulations. These comparisons demonstrate a surprising amount of agreement between the two modelling frameworks, and suggest further novel modifications to the CPM that would allow it to better model cell migration. Considerations of the model’s behaviour are used to argue that the dominant effect governing cell migration (random motility or signal-driven taxis) likely depends on the sort of invasion demonstrated by cells, as easily seen by microscopic photography. Additionally, a scratch assay simulated on a non-homogeneous domain consisting of a ’fast’ and ’slow’ region is also used to further differentiate between these different potential cell motility behaviours. A heterogeneous domain is a novel situation which has not been considered mathematically in this context, nor has it been constructed experimentally to the best of the candidate’s knowledge. Thus this problem serves as a thought experiment used to test the conclusions arising from the simulations on homogeneous domains, and to suggest what might be observed should this non-homogeneous assay situation be experimentally realised. Non-intuitive cell invasion patterns are predicted for diffusely-invading cells which respond to a cell-consumed signal or nutrient, contrasted with rather expected behaviour in the case of random-motility-driven invasion. The potential experimental observation of these behaviours is demonstrated by the individual-cell-level model used in this thesis, which does agree with the PDE model in predicting these unexpected invasion patterns. In the interest of examining such a case of a non-homogeneous domain experimentally, some brief suggestion is made as to how this could be achieved.

Mid- and near-infrared spectroscopic investigation of homogeneous cation distribution in MgxZnyAl(x+y)/2-layered double hydroxide (LDH)

In this report, a detailed FTIR fitting analysis was used to recognize Mg, Zn and Al homogeneous distribution in MgxZnyAl(x+y)/2-Layered double hydroxide (LDH) hydroxyl layer. In detail, OH-Mg2Al:OH-Mg3 ratios decreased from 95.2:4.8 (MIR) and 94.2:5.8 (NIR) to 58.9:41.1 (MIR) and 61.8:38.2 (NIR), when Mg:Al increased from 2.2:1.0 to 4.1:1.0 in MgAl-LDHs. These fitting results were similar with theoretical calculations of 94.3:5.7 and 59.0:41.0. In a further analysis of MgxZnyAl(x+y)/2-LDHs, OH bonded Zn2Mg, Zn2Al, MgZnAl, Mg2Al and Mg2Zn peaks were identified at 3420, 3430, 3445–3450, 3454 and 3545 cm-1, respectively. With the decrease of Mg:Zn from 3:1 to 1:3, metal-hydroxyl bands changed from OH-Mg2Al and MgZnAl (with a ratio of 49.4:50.6) to OH-MgZnAl and Zn2Al (with a ratio of 55.0:45.0). They were also similar with theoretical calculations of 47.6:52.4 and 54.6:45.4. As a result, these results show that there is an ordered cation distribution in MgxZnyAl(x+y)/2-LDH, and FTIR is feasible in recognizing this structure.

Homogeneous bent functions of degree n in 2n variables do not exist for n>3

We prove that homogeneous bent functions f:GF(2)^2n --> GF(2) of degree n do not exist for n>3. Consequently homogeneous bent functions must have degree 3.

The eight variable homogeneous degree three bent functions

We determine the affine equivalence classes of the eight variable degree three homogeneous bent functions using a new algorithm. Our algorithm applies to general bent functions and can systematically determine the automorphism groups. We provide a partial verification of the enumeration of eight variable degree three homogeneous bent functions obtained by Meng et al. We determine the affine equivalence classes of these functions.

Charge transport study of high mobility polymer thin-film transistors based on thiophene substituted diketopyrrolopyrrole copolymers

In this paper, we report on the device physics and charge transport characteristics of high-mobility dual-gated polymer thin-film transistors with active semiconductor layers consisting of thiophene flanked DPP with thienylene-vinylene-thienylene (PDPP-TVT) alternating copolymers. Room temperature mobilities in these devices are high and can exceed 2 cm2 V-1 s-1. Steady-state and non-quasi-static measurements have been performed to extract key transport parameters and velocity distributions of charge carriers in this copolymer. Charge transport in this polymer semiconductor can be explained using a Multiple-Trap-and-Release or Monroe-type model. We also compare the activation energy vs. field-effect mobility in a few important polymer semiconductors to gain a better understanding of transport of DPP systems and make appropriate comparisons.

Design and modification of three-component randomly incorporated copolymers for high performance organic photovoltaic applications

In this study we report the molecular design, synthesis, characterization, and photovoltaic properties of a series of diketopyrrolopyrrole (DPP) and dithienothiophene (DTT) based donor-acceptor random copolymers. The six random copolymers are obtained via Stille coupling polymerization using various concentration ratios of donor to acceptor in the conjugated backbone. Bis(trimethylstannyl)thiophene was used as the bridge block to link randomly with the two comonomers 5-(bromothien-2-yl)-2,5-dialkylpyrrolo[3,4-c]pyrrole-1, 4-dione and 2,6-dibromo-3,5-dipentadecyl-dithieno[3,2-b;2′,3′-d] thiophene. The optical properties of these copolymers clearly reveal a change in the absorption band through optimization of the donor-acceptor ratio in the backbone. Additionally, the solution processability of the copolymers is modified through the attachment of different bulky alkyl chains to the lactam N-atoms of the DPP moiety. Applications of the polymers as light-harvesting and electron-donating materials in solar cells, in conjunction with PCBM as acceptor, show power conversion efficiencies (PCEs) of up to 5.02%.

Synthesis of diketopyrrolopyrrole based copolymers via the direct arylation method for p-channel and ambipolar OFETs

In this paper, we have synthesized two novel diketopyrrolopyrrole (DPP) based donor-acceptor (D-A) copolymers poly{3,6-dithiophene-2-yl-2,5-di(2-octyl)- pyrrolo[3,4-c]pyrrole-1,4-dione-alt-1,5-bis(dodecyloxy)naphthalene} (PDPPT-NAP) and poly{3,6-dithiophene-2-yl-2,5-di(2-butyldecyl)-pyrrolo[3,4-c]pyrrole-1,4- dione-alt-2-dodecyl-2H-benzo[d][1,2,3]triazole} (PDPPT-BTRZ) via direct arylation organometallic coupling. Both copolymers contain a common electron withdrawing DPP building block which is combined with electron donating alkoxy naphthalene and electron withdrawing alkyl-triazole comonomers. The number average molecular weight (Mn) determined by gel permeation chromatography (GPC) for polymer PDPPT-NAP is around 23 400 g mol-1 whereas for polymer PDPPT-BTRZ it is 18 600 g mol-1. The solid state absorption spectra of these copolymers show a wide range of absorption from 400 nm to 1000 nm with optical band gaps calculated from absorption cut off values in the range of 1.45-1.30 eV. The HOMO values determined for PDPPT-NAP and PDPPT-BTRZ copolymers from photoelectron spectroscopy in air (PESA) data are 5.15 eV and 5.25 eV respectively. These polymers exhibit promising p-channel and ambipolar behaviour when used as an active layer in organic thin-film transistor (OTFT) devices. The highest hole mobility measured for polymer PDPPT-NAP is around 0.0046 cm2 V-1 s-1 whereas the best ambipolar performance was calculated for PDPPT-BTRZ with a hole and electron mobility of 0.01 cm2 V-1 s-1 and 0.006 cm2 V-1 s-1.

Synthesis, characterization and comparative OFET behaviour of indenofluorene-bithiophene and terthiophene alternating copolymers

The synthesis of alternating copolymers of tetraalkylindenofluorene with bithiophene and terthiophene using Suzuki polycondensation route is reported. We report on the optical and electrochemical properties of these copolymers. AFM analysis of the microscopic morphology of thin deposits showed that the copolymer with terthiophene units produced the more ordered films, with well-defined fibrillar structures, resulting from highly-regular dense packing due to strong π-π interchain interactions, in contrast to the amorphous bithiophene copolymer. Upon testing these materials in FETs the terthienyl copolymers displayed the higher charge mobilities among the studied compounds, with values of over 10-4 cm2 V-1 s-1 being obtained.

Furan containing diketopyrrolopyrrole copolymers : synthesis, characterization, organic field effect transistor performance and photovoltaic properties

In this work, we report design, synthesis and characterization of solution processable low band gap polymer semiconductors, poly{3,6-difuran-2-yl-2,5-di(2- octyldodecyl)-pyrrolo[3,4-c]pyrrole-1,4-dione-alt-phenylene} (PDPP-FPF), poly{3,6-difuran-2-yl-2,5-di(2-octyldodecyl)-pyrrolo[3,4-c]pyrrole-1, 4-dione-alt-naphthalene} (PDPP-FNF) and poly{3,6-difuran-2-yl-2,5-di(2- octyldodecyl)-pyrrolo[3,4-c]pyrrole-1,4-dione-alt-anthracene} (PDPP-FAF) using the furan-containing 3,6-di(furan-2-yl)pyrrolo[3,4-c]pyrrole-1,4(2H,5H)-dione (DBF) building block. As DBF acts as an acceptor moiety, a series of donor-acceptor (D-A) copolymers can be generated when it is attached alternatively with phenylene, naphthalene or anthracene donor comonomer blocks. Optical and electrochemical characterization of thin films of these polymers reveals band gaps in the range of 1.55-1.64 eV. These polymers exhibit excellent hole mobility when used as the active layer in organic thin-film transistor (OTFT) devices. Among the series, the highest hole mobility of 0.11 cm 2 V -1 s -1 is achieved in bottom gate and top-contact OTFT devices using PDPP-FNF. When these polymers are used as a donor and [70]PCBM as the acceptor in organic photovoltaic (OPV) devices, power conversion efficiencies (PCE) of 2.5 and 2.6% are obtained for PDPP-FAF and PDPP-FNF polymers, respectively. Such mobility values in OTFTs and performance in OPV make furan-containing DBF a very promising block for designing new polymer semiconductors for a wide range of organic electronic applications.

Isoindigo dye incorporated copolymers with naphthalene and anthracene: promising materials for stable organic field effect transistors

In this paper, we report the design and synthesis of isoindigo based low band gap polymer semiconductors, poly{N,N′-(2-octyldodecyl)-isoindigo-alt- naphthalene} (PISD-NAP) and poly{N,N′-(2-octyldodecyl)-isoindigo-alt- anthracene} (PISD-ANT). A series of donor-acceptor (D-A) copolymers can be prepared where donor and acceptor conjugated blocks can be attached alternately using organometallic coupling. In these polymers, an isoindigo dye acceptor moiety has been attached alternately with naphthalene and anthracene donor comonomer blocks by Suzuki coupling. PISD-NAP and PISD-ANT exhibit excellent solution processibility and good film-forming properties. Gel permeation chromatography exhibits a higher molecular mass with lower polydispersity. UV-vis-NIR absorption of these polymers exhibits a wide absorption band ranging from 300 nm to 800 nm, indicating the low band gap nature of the polymers. Optical band gaps calculated from the solid state absorption cutoff value for PISD-NAP and PISD-ANT are around 1.80 eV and 1.75 eV, respectively. Highest occupied molecular orbital (HOMO) values calculated respectively for PISD-NAP and PISD-ANT thin films on glass substrate by photoelectron spectroscopy in air (PESA) are 5.66 eV and 5.53 eV, indicative of the good stability of these materials in organic electronic device applications. These polymers exhibit p-channel charge transport characteristics when used as the active semiconductor in organic thin-film transistor (OTFT) devices in ambient conditions. The highest hole mobility of 0.013 cm2 V-1 s-1 is achieved in top contact and bottom-gate OTFT devices for PISD-ANT, whereas polymer PISD-NAP exhibited a hole mobility of 0.004 cm2 V -1 s-1. When these polymer semiconductors were used as a donor and PC71BM as an acceptor in OPV devices, the highest power conversion efficiency (PCE) of 1.13% is obtained for the PISD-ANT polymer.

4-hexylbithieno[3,2-b:2′3′-e]pyridine: An efficient electron-accepting unit in fluorene and indenofluorene copolymers for light-emitting devices

4-Hexylbithienopyridine has been prepared as a novel electron-accepting monomer for conjugated polymers. To test its electronic properties, alternating copolymers with fluorene and indenofluorene polymers have been prepared. The copolymers displayed reduction potentials about 0.5 V lower than for the corresponding fluorene and indenofluorene homopolymers, indicating much improved electron-accepting properties. Analysis of the microscopic morphology of thin films of the copolymers by AFM shows that they lack the extensive supramolecular order seen with the homopolymers, which is attributed to the bithienopyridine units disrupting the π-stacking. LEDs using these polymers as the emitting layer produce blue-green emission with low turn-on voltages with aluminum electrodes confirming their improved electron affinity. The indenofluorene copolymer displayed an irreversible red shift in emission at high voltages, which is attributed to oxidation of the indenofluorene units. This red shift occurred at higher potentials than for indenofluorene homopolymers in LEDs, suggesting that the heterocyclic moieties offer some protection against electrically promoted oxidation.

Supramolecular organization in fluorene/indenofluorene-oligothiophene alternating conjugated copolymers

A series of conjugated copolymers containing fluorene or indenofluorene units alternating with oligothiophene segments, with potential interest for use as the active layer in field-effect transistors, is investigated. Atomic force microscopy analysis of the morphology of thin deposits shows either the formation of fibrillar structures, which are the signature of long-range π stacking, or the presence of untextured aggregates, resulting from disordered assembly. These morphologies are interpreted in terms of the supramolecular organization of the conjugated chains. Molecular modeling simulations indicate that the commensurability between the lengths of the monomer units and the presence of alkyl side groups are the two key structural factors governing the chain organization into highly ordered assemblies. The most favorable structures are those combining fluorene (indenofluorene) units with unsubstituted bithiophene (terthiophene) segments.