Polymer-nanoparticle composites: from orientation phenomena to optoelectronic applications

Polymer-nanoparticle hybrids show synergistic effects, demonstrating both, the unique properties of nanosized structures and the good processability and functionalities of polymeric materials. This work shows the synthesis and application of block copolymers containing a soluble, functional block and a short anchor block, which efficiently binds to the surface of nanocrystals. We functionalized anisotropic, semiconducting nanoparticles, which can be dissolved in organic and polymeric matrices upon modification. The modified nanorods have the ability to form liquid crystalline phases, which behave similar to low molecular liquid crystals with a reversible clearing behaviour. These liquid crystalline phases could also be obtained in hole conducting matrices. For a macroscopic orientation of the nanorods, electric fields were applied and a switching (in analogy to known liquid crystals) to a homeotropic orientation was observed.rnBy introduction of dye molecules in the anchor block of a hole conducting block copolymer, all essential components of a solar cell can be combined in a single particle. Light absorption of the dye induces the injection of electrons into the particles, followed by a charging, that was monitored by a special AFM technique.rnLight emitting nanocrystals were functionalized analogously with a hole transporting polymer. The stability of the particles could be enhanced by the sterically stabilizing polymer corona and the particles showed improved properties in terms of processing. We applied these hybrid materials in light emitting devices, which showed better characteristics due to an improved hole injection and well dispersed emitting particles in the active device layer.rnThe work shows the broad spectrum of properties and applications based on the synergistic effects in hybrid and composite materials.

Supported Cu(II) polymer catalysts for aqueous phenol oxidation

Supported Cu(II) polymer catalysts were used for the catalytic oxidation of phenol at 30 degrees C and atmospheric pressure using air and H(2)O(2) as oxidants. Heterogenisation of homogeneous Cu(II) catalysts was achieved by adsorption of Cu(II) salts onto polymeric matrices (poly(4-vinylpyridine), Chitosan). The catalytic active sites were represented by Cu(II) ions and showed to conserve their oxidative activity in heterogeneous catalysis as well as in homogeneous systems. The catalytic deactivation was evaluated by quantifying released Cu(II) ions in solution during oxidation, from where Cu-PVP(25) showed the best leaching levels no more than 5 mg L(-1). Results also indicated that Cu-PVP(25) had a catalytic activity (56% of phenol conversion when initial Cu(II) catalytic content was 200 mg L(Reaction)(-1)) comparable to that of commercial catalysts (59% of phenol conversion). Finally, the balance between activity and copper leaching was better represented by Cu-PVP(25) due to the heterogeneous catalytic activity had 86% performance in the heterogeneous phase, and the rest on the homogeneous phase, while Cu-PVP(2) had 59% and CuO/gamma-Al(2)O(3) 68%.

Opportunities and challenges in the use of inorganic fullerene-like nanoparticles to produce advanced polymer nanocomposites

Polymer/inorganic nanoparticle nanocomposites have garnered considerable academic and industrial interest over recent decades in the development of advanced materials for a wide range of applications. In this respect, the dispersion of so-called inorganic fullerene-like (IF) nanoparticles, e.g., tungsten disulfide (IF-WS2) or molybdenum disulfide (IF-MoS2), into polymeric matrices is emerging as a new strategy. The surprising properties of these layered metal dichalcogenides such as high impact resistance and superior tribological behavior, attributed to their nanoscale size and hollow quasi-spherical shape, open up a wide variety of opportunities for applications of these inorganic compounds. The present work presents a detailed overview on research in the area of IF-based polymer nanocomposites, with special emphasis on the use of IF-WS2 nanoparticles as environmentally friendly reinforcing fillers. The incorporation of IF particles has been shown to be efficient for improving thermal, mechanical and tribological properties of various thermoplastic polymers, such as polypropylene, nylon-6, poly(phenylene sulfide), poly(ether ether ketone), where nanocomposites were fabricated by simple melt-processing routes without the need for modifiers or surfactants. This new family of nanocomposites exhibits similar or enhanced performance when compared with nanocomposites that incorporate carbon nanotubes, carbon nanofibers or nanoclays, but are substantially more cost-effective, efficient and environmentally satisfactory. Most recently, innovative approaches have been described that exploit synergistic effects to produce new materials with enhanced properties, including the combined use of micro- and nanoparticles such as IF-WS2/nucleating agent or IF-WS2/carbon fiber, as well as dual nanoparticle systems such as SWCNT/IF-WS2 where each nanoparticle has different characteristics. The structure–property relationships of these nanocomposites are discussed and potential applications proposed ranging from medicine to the aerospace, automotive and electronics industries.

Nanopartículas superparamagnéticas encapsuladas com polímeros para tratamento de câncer por hipertermia

O câncer é uma das maiores causas de mortalidade no Brasil e no mundo, com potencial de crescimento nas próximas décadas. Um tipo de tratamento promissor é a hipertermia magnética, procedimento no qual as células tumorais morrem pelo efeito do calor gerado por partículas magnéticas após a aplicação de campo magnético alternado em frequências adequadas. Tais partículas também são capazes de atuar como agentes de contraste para imageamento por ressonância magnética, um poderoso método de diagnóstico para identificação de células neoplásicas, formando a combinação conhecida como theranostics (terapia e diagnóstico). Neste trabalho foram sintetizadas nanopartículas de óxido de ferro por método de coprecipitação com posterior encapsulação por técnica de nano spray drying, visando sua aplicação no tratamento de câncer por hipertermia e como agente de contraste para imageamento por ressonância magnética. Para a encapsulação foram utilizadas matrizes poliméricas de Maltodextrina com Polissorbato 80, Pluronic F68, Eudragit® S100 e PCL com Pluronic F68, escolhidos com o intuito de formar partículas que dispersem bem em meio aquoso e que consigam atingir alvo tumoral após administração no corpo do paciente. Parâmetros de secagem pelo equipamento Nano Spray Dryer, como temperatura, solvente e concentração de reagentes, foram avaliados. As partículas formadas foram caracterizadas por Microscopia Eletrônica de Varredura, Difração de Raios-X, Análise Termogravimétrica, Espalhamento de Luz Dinâmico, Espectroscopia de Infravermelho, magnetismo quanto a magnetização de saturação e temperatura, citotoxicidade e potencial de aquecimento. Tais procedimentos indicaram que o método de coprecipitação produziu nanopartículas de magnetita de tamanho em torno 20 nm, superparamagnéticas a temperatura ambiente, sem potencial citotóxico. A técnica de nano spray drying foi eficiente para a formação de partículas com tamanho em torno de 1 μm, também superparamagnéticas, biocompatíveis e com propriedades magnéticas adequadas e para aplicações pretendidas. Destaca-se a amostra com Pluronic, OF-10/15-1P, que apresentou magnetização de saturação de 68,7 emu/g e interação específica com células tumorais.

Obtenção e caracterização de PLA reforçado com nanocelulose.

A celulose é o polímero natural renovável disponível em maior abundância atualmente. Por possuir estrutura semicristalina, é possível extrair seus domínios cristalinos através de procedimentos que ataquem sua fase amorfa, como a hidrólise ácida, obtendo-se assim partículas cristalinas chamadas nanopartículas de celulose (NCs). Estas nanopartículas têm atraído enorme interesse científico, uma vez que possuem propriedades mecânicas, como módulo de elasticidade e resistência à tração, semelhantes a várias cargas inorgânicas utilizadas na fabricação de compósitos. Além disso, possuem dimensões nanométricas, o que contribui para menor adição de carga à matriz polimérica, já que possuem maior área de superfície, quando comparadas às cargas micrométricas. Nanocompósitos formados pela adição destas cargas em matrizes poliméricas podem apresentar propriedades comerciais atraentes, como barreira a gases, melhores propriedades térmicas e baixa densidade, quando comparados aos compósitos tradicionais. Como se trata de uma carga com dimensões nanométricas, obtida de fontes renováveis, uma das principais áreas de interesse para aplicação deste reforço é em biopolímeros biodegradáveis. O poli(ácido lático) (PLA), é um exemplo de biopolímero com propriedades mecânicas, térmicas e de processamento superiores a de outros biopolímeros comerciais. No presente trabalho foram obtidas nanopartículas de celulose (NCs), por meio de hidrólise ácida, utilizando-se três métodos distintos, com o objetivo de estudar o método mais eficiente para a obtenção de NCs adequadas à aplicação em compósitos de PLA. Os Métodos I e II empregam extração das NCs por meio do H2SO4, diferenciando-se apenas pela neutralização, a qual envolve diálise ou neutralização com NaHCO3, respectivamente. No Método III a extração das NCs foi realizada com H3PO4. As NCs foram caracterizadas por diferentes técnicas, como difração de raios X (DRX), análise termogravimétrica (TG), espectroscopia vibracional de absorção no infravermelho (FTIR), microscopia eletrônica de transmissão (MET) e microscopia de força atômica (MFA). Os resultados de caracterização das NCs indicaram que, a partir de todos os métodos utilizados, há formação de nanocristais de celulose (NCCs), entretanto, apenas os NCCs obtidos pelos Métodos II e III apresentaram estabilidade térmica suficiente para serem empregados em compósitos preparados por adição da carga no polímero em estado fundido. A incorporação das NCs em matriz de PLA foi realizada em câmara de mistura, com posterior moldagem por prensagem a quente. Compósitos obtidos por adição de NCs obtidas pelo Método II foram caracterizados por calorimetria exploratória diferencial (DSC), análise termogravimétrica, microscopia óptica, análises reológicas e microscopia eletrônica de varredura (MEV). A adição de NCs, extraídas pelo Método II, em matriz de PLA afetou o processo de cristalização do polímero, o qual apresentou maior grau de cristalinidade. Além disso, a adição de 3% em massa de NCs no PLA foi suficiente para alterar seu comportamento reológico. Os resultados reológicos indicaram que a morfologia do compósito é, predominantemente, composta por uma dispersão homogênea e fina da carga na fase matriz. Micrografias obtidas por MEV corroboram os resultados reológicos, mostrando, predominantemente a presença de partículas de NC em escala nanométrica. Compósitos de PLA com NCs obtidas pelo Método III apresentaram aglomerados de partículas de NC em escala micro e milimétrica, ao longo da fase matriz, e não foram extensivamente caracterizados.

Synthesis and evaluation of novel ligands for quantum dot stabilisation and polymerisation studies

This thesis describes the design and synthesis of a variety of functionalised phosphine oxides and sulfides, based on the structure of trioctylphosphine oxide, synthesised for the purpose of surface modification of quantum dots. The ability of the ligands to modify the surface chemistry via displacement of the original hexadecylamine capping layer of quantum dots was evaluated. Finally the surface modified quantum dots were investigated for enhancement in their inherent properties and improved compatibility with the various applications for which they were initially designed. Upon the commencement of research involving quantum dots it became apparent that more information on their behaviour and interaction with the environment was required. The limits of the inherent stability of hexadecylamine capped quantum dots were investigated by exposure to a number of different environments. The effect upon the stability of the quantum dots was monitored by changes in the photoluminescence ability of their cores. Subtle differences between different batches of quantum dots were observed and the necessity to account for these in future applications noted. Lastly the displacement of the original hexadecylamine coating with the "designer" functionalised ligands was evaluated to produce a set of conditions that would result in the best possible surface modification. A general procedure was elucidated however it was discovered that each displacement still required slight adjustment by consideration of the other factors such as the difference in ligand structure and the individuality of the various batches of quantum dots. This thesis also describes a procedure for the addition of a protective layer to the surface of quantum dots by cross-linking the functionalised ligands bound to the surface via an acyclic diene metathesis polymerisation. A detailed description of the problems encountered in the analysis of these materials combined with the use of novel techniques such as diffusion ordered spectroscopy is provided as a means to overcome the limitations encountered. Finally a demonstration of the superior stability, upon exposure to a range of aggressive environments of these protected materials compared with those before cross-linking provided physical proof of the cross-linking process and the advantages of the cross-linking modification. Finally this thesis includes the presentation of initial work into the production of luminescent nanocrystal encoded resin beads for the specific use in solid phase combinatorial chemistry. Demonstration of the successful covalent incorporation of quantum dots into the polymeric matrices of non-functionalised and functionalised resin beads is described. Finally by preliminary work to address and overcome the possible limitations that may be encountered in the production and general employment of these materials in combinatorial techniques is given.

Grewia polysaccharide gum as a pharmaceutical excipient

Grewia gum is obtained from the inner stem bark of the edible plant Grewia mollis Juss (Fam. Tiliaceae) which grows widely in the middle belt region of Nigeria, and is also cultivated. The dried and pulverised inner stem bark is used as a thickening agent in some food delicacies in that region of the country. This ability of the material to increase solution viscosity has generated a lot of interest and is the catalysing momentum for this research. Such materials have been used as stabilizers or suspending agents in cosmetics, foods and liquid medications, and as mucoadhesives and controlled release polymeric matrices in solid dosage forms. The physicochemical characterization of candidate excipients forms an essential step towards establishing suitability for pharmaceutical application. For natural gums, this usually requires isolation of the gum from the storage site by extraction processes. Grewia polysaccharide gum was extracted and dried using techniques such as air-drying, freeze-drying or spray-drying. Component analysis of the gum showed that it contains five neutral sugars: glucose, galactose, rhamnose, arabinose and xylose. The gum contains traces of elements such as zinc, magnesium, calcium and phosphorus. At low substance weight, the gum hydrates in aqueous medium swelling and dispersing to give a highly viscous dispersion with pseudoplasmic flow behaviour. The method by which drying is achieved can have significant effect on some physicochemical properties of the gum. Consequently, the intrinsic viscosity and molecular weight, and parameters of powder flow were shown to differ with the method of drying. The gum has good thermal stability. In comparison with established excipients, grewia gum may be preferable to gum Arabic or sodium carboxymethylcellulose as a suspending agent in ibuprofen suspension formulations. The release retardant property of the gum was superior to guar and Metolose® in ibuprofen matrices. Similarly, carboxy methylcellulose, Methocel®, gum Arabic or Metolose® may not be preferable to grewia gum when controlled release of a soluble drug like cimetidine is indicated. The mucoadhesive performance of the gum compared favourably with excellent mucoadhesives such as hydroxypropyl methylcellulose, carboxymethylcellulose, guar and carbopol 971 P.

Desenvolvimento de bionanocompósitos: nano e microcristais de celulose com poli (álcool vinílico) e poli (ácido lático)

This work has the main objective to obtain nano and microcrystals of cellulose, extracted from the pineapple leaf fibres (PALF), as reinforcement for the manufacture of biocomposite films with polymeric matrices of Poly(vinyl alcohol) (PVA) and Poly(lactic acid) (PLA). The polymer matrices and the nano and microcrystals of cellulose were characterised by means of TGA, FTIR and DSC. The analysis was performed on the pineapple leaves to identify the macro and micronutrients. The fibers of the leaves of the pineapple were extracted in a desfibradeira mechanical. The PALF extracted were washed to remove washable impurities and subsequently treated with sodium hydroxide (NaOH) and sodium hypochlorite (NaClO) in the removal of impurities, such as fat, grease, pectates, pectin and lignin. The processed PALF fibers were hydrolysed in sulfuric acid (H2SO4) at a concentration of 13.5 %, to obtain nano and microcrystals of cellulose. In the manufacture of biocomposite films, concentrations of cellulose, 0 %, 1 %, 3 %, 6 %, 9% and 12% were used as reinforcement to the matrices of PVA and PLA. The PVA was dissolved in distilled water at 80 ± 5 oC and the PLA was dissolved in dichloromethane at room temperature. The manufacture of biocompósitos in the form of films was carried out by "casting". Tests were carried out to study the water absorption by the films and mechanical test of resistance to traction according to ASTM D638-10 with a velocity of 50 mm/min.. Chi-square statistical test was used to check for the existence of significant differences in the level of 0.05: the lengths of the PALF, lengths of the nano and microcrystals of cellulose and the procedures used for the filtration using filter syringe of 0.2 μm or filtration and centrifugation. The hydrophilicity of biocompósitos was analysed by measuring the contact angle and the thickness of biocompósitos were compared as well as the results of tests of traction. Statistical T test - Student was also applied with the significance level (0.05). In biodegradation, Sturm test of standard D5209 was used. Nano and microcrystals of cellulose with lengths ranging from 7.33 nm to 186.17 nm were found. The PVA films showed average thicknesses of 0.153 μm and PLA 0.210 μm. There is a strong linear correlation directly proportional between the traction of the films of PVA and the concentration of cellulose in the films (composite) (0,7336), while the thickness of the film was correlated in 0.1404. Nano and microcrystals of cellulose and thickness together, correlated to 0.8740. While the correlation between the cellulose content and tensile strength was weak and inversely proportional (- 0,0057) and thickness in -0.2602, totaling -0,2659 in PLA films. This can be attributed to the nano and microcrystals of cellulose not fully adsorbed to the PLA matrix. In the comparison of the results of the traction of the two polymer matrices, the nano and microcrystals have helped in reducing the traction of the films (composite) of PLA. There was still the degradation of the film of PVA, within a period of 20 days, which was not seen in the PLA film, on the other hand, the observations made in the literature, the average time to start the degradation is above 60 days. What can be said that the films are biodegradable composites, with hydrophilicity and the nano and microcrystals of cellulose, contribute positively in the improvement of the results of polymer matrices used.

Greffage d’un film mince de nano-TiO2 sur les fibres naturelles cellulosiques pour le renforcement de biocomposites polymériques

Abstract : Natural materials have received a full attention in many applications because they are degradable and derived directly from earth. In addition to these benefits, natural materials can be obtained from renewable resources such as plants (i.e. cellulosic fibers like flax, hemp, jute, and etc). Being cheap and light in weight, the cellulosic natural fiber is a good candidate for reinforcing bio-based polymer composites. However, the hydrophilic nature -resulted from the presence of hydroxyl groups in the structure of these fibers- restricts the application of these fibers in the polymeric matrices. This is because of weak interfacial adhesion, and difficulties in mixing due to poor wettability of the fibers within the matrices. Many attempts have been done to modify surface properties of natural fibers including physical, chemical, and physico-chemical treatments but on the one hand, these treatments are unable to cure the intrinsic defects of the surface of the fibers and on the other hand they cannot improve moisture, and alkali resistance of the fibers. However, the creation of a thin film on the fibers would achieve the mentioned objectives. This study aims firstly to functionalize the flax fibers by using selective oxidation of hydroxyl groups existed in cellulose structure to pave the way for better adhesion of subsequent amphiphilic TiO[subscript 2] thin films created by Sol-Gel technique. This method is capable of creating a very thin layer of metallic oxide on a substrate. In the next step, the effect of oxidation on the interfacial adhesion between the TiO[subscript 2] film and the fiber and thus on the physical and mechanical properties of the fiber was characterized. Eventually, the TiO[subscript 2] grafted fibers with and without oxidation were used to reinforce poly lactic acid (PLA). Tensile, impact, and short beam shear tests were performed to characterize the mechanical properties while Thermogravimetric analysis (TGA), Differential Scanning Calorimetry (DSC), Dynamic mechanical analysis (DMA), and moisture absorption were used to show the physical properties of the composites. Results showed a significant increase in physical and mechanical properties of flax fibers when the fibers were oxidized prior to TiO[subscript 2] grafting. Moreover, the TiO[subscript 2] grafted oxidized fiber caused significant changes when they were used as reinforcements in PLA. A higher interfacial strength and less amount of water absorption were obtained in comparison with the reference samples.

Evaluation of the genotoxicity of cellulose nanofibers

Background: Agricultural products and by products provide the primary materials for a variety of technological applications in diverse industrial sectors. Agro-industrial wastes, such as cotton and curaua fibers, are used to prepare nanofibers for use in thermoplastic films, where they are combined with polymeric matrices, and in biomedical applications such as tissue engineering, amongst other applications. The development of products containing nanofibers offers a promising alternative for the use of agricultural products, adding value to the chains of production. However, the emergence of new nanotechnological products demands that their risks to human health and the environment be evaluated. This has resulted in the creation of the new area of nanotoxicology, which addresses the toxicological aspects of these materials.Purpose and methods: Contributing to these developments, the present work involved a genotoxicological study of different nanofibers, employing chromosomal aberration and comet assays, as well as cytogenetic and molecular analyses, to obtain preliminary information concerning nanofiber safety. The methodology consisted of exposure of Allium cepa roots, and animal cell cultures (lymphocytes and fibroblasts), to different types of nanofibers. Negative controls, without nanofibers present in the medium, were used for comparison.Results: The nanofibers induced different responses according to the cell type used. In plant cells, the most genotoxic nanofibers were those derived from green, white, and brown cotton, and curaua, while genotoxicity in animal cells was observed using nanofibers from brown cotton and curaua. An important finding was that ruby cotton nanofibers did not cause any significant DNA breaks in the cell types employed.Conclusion: This work demonstrates the feasibility of determining the genotoxic potential of nanofibers derived from plant cellulose to obtain information vital both for the future usage of these materials in agribusiness and for an understanding of their environmental impacts.

A study on the Immobilizaton of Enzymes on Polymeric Supports

This work was focused to study the immobilization of enzymes on polymers. A large range of polymer matrices have been employed as supports for enzyme immobilization. Here polyaniline (PAN!) and poly(0~toluidine) (POT) were used as supports. PANI and POT provides an excellent support for enzyme immobilization by virtue of its facile synthesis, superior chemical and physical stabilities, and large retention capacity. We selected industrially important starch hydrolyzing enzymes a-amylase and glucoamylase for the study. In this work the selected enzymes were immobilized via adsorption and covalent bonding methods.To optimize the catalytic efficiency and stability of the resulting biocatalysts, the attempt was made to understand the immobilization effects on enzymatic properties. The effect of pH of the immobilization medium, time of immobilization on the immobilization efficiency was observed. The starch hydrolyzing activity of free 0:-amylase and glucoamylase were compared with immobilized forms. Immobilization on solid supports changes the microenvironment of the enzyme there by influences the pH and temperature relationship on the enzymatic activity. Hence these parameters also optimized. The reusability and storage stability of immobilized enzymes an important aspect from an application standpoint, especially in industrial applications. Taking in to consideration of this, the reusability and the long tenn storage stability of the immobilized enzyme investigated.

Barrier and Mechanical Properties of Clay-Reinforced Polymeric Nanocomposites

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

Multiscale morphology design of hybrid halide perovskites through a polymeric template

Hybrid halide perovskites have emerged as promising active constituents of next generation solution processable optoelectronic devices. During their assembling process, perovskite components undergo very complex dynamic equilibria starting in solution and progressing throughout film formation. Finding a methodology to control and affect these equilibria, responsible for the unique morphological diversity observed in perovskite films, constitutes a fundamental step towards a reproducible material processability. Here we propose the exploitation of polymer matrices as cooperative assembling components of novel perovskite CH3NH3PbI3 : polymer composites, in which the control of the chemical interactions in solution allows a predictable tuning of the final film morphology. We reveal that the nature of the interactions between perovskite precursors and polymer functional groups, probed by Nuclear Magnetic Resonance (NMR) spectroscopy and Dynamic Light Scattering (DLS) techniques, allows the control of aggregates in solution whose characteristics are strictly maintained in the solid film, and permits the formation of nanostructures that are inaccessible to conventional perovskite depositions. These results demonstrate how the fundamental chemistry of perovskite precursors in solution has a paramount influence on controlling and monitoring the final morphology of CH3NH3PbI3 (MAPbI3) thin films, foreseeing the possibility of designing perovskite : polymer composites targeting diverse optoelectronic applications.

Antimicrobial effect of polymeric biomaterials for bone infection treatment

Dissertação de mestrado em Bioquímica, apresentada à Faculdade de Ciências e Tecnologia da Universidade Nova de Lisboa, 2016.

Krylov subspace methods for approximating functions of symmetric positive definite matrices with applications to applied statistics and anomalous diffusion

Matrix function approximation is a current focus of worldwide interest and finds application in a variety of areas of applied mathematics and statistics. In this thesis we focus on the approximation of A^(-α/2)b, where A ∈ ℝ^(n×n) is a large, sparse symmetric positive definite matrix and b ∈ ℝ^n is a vector. In particular, we will focus on matrix function techniques for sampling from Gaussian Markov random fields in applied statistics and the solution of fractional-in-space partial differential equations. Gaussian Markov random fields (GMRFs) are multivariate normal random variables characterised by a sparse precision (inverse covariance) matrix. GMRFs are popular models in computational spatial statistics as the sparse structure can be exploited, typically through the use of the sparse Cholesky decomposition, to construct fast sampling methods. It is well known, however, that for sufficiently large problems, iterative methods for solving linear systems outperform direct methods. Fractional-in-space partial differential equations arise in models of processes undergoing anomalous diffusion. Unfortunately, as the fractional Laplacian is a non-local operator, numerical methods based on the direct discretisation of these equations typically requires the solution of dense linear systems, which is impractical for fine discretisations. In this thesis, novel applications of Krylov subspace approximations to matrix functions for both of these problems are investigated. Matrix functions arise when sampling from a GMRF by noting that the Cholesky decomposition A = LL^T is, essentially, a `square root' of the precision matrix A. Therefore, we can replace the usual sampling method, which forms x = L^(-T)z, with x = A^(-1/2)z, where z is a vector of independent and identically distributed standard normal random variables. Similarly, the matrix transfer technique can be used to build solutions to the fractional Poisson equation of the form ϕn = A^(-α/2)b, where A is the finite difference approximation to the Laplacian. Hence both applications require the approximation of f(A)b, where f(t) = t^(-α/2) and A is sparse. In this thesis we will compare the Lanczos approximation, the shift-and-invert Lanczos approximation, the extended Krylov subspace method, rational approximations and the restarted Lanczos approximation for approximating matrix functions of this form. A number of new and novel results are presented in this thesis. Firstly, we prove the convergence of the matrix transfer technique for the solution of the fractional Poisson equation and we give conditions by which the finite difference discretisation can be replaced by other methods for discretising the Laplacian. We then investigate a number of methods for approximating matrix functions of the form A^(-α/2)b and investigate stopping criteria for these methods. In particular, we derive a new method for restarting the Lanczos approximation to f(A)b. We then apply these techniques to the problem of sampling from a GMRF and construct a full suite of methods for sampling conditioned on linear constraints and approximating the likelihood. Finally, we consider the problem of sampling from a generalised Matern random field, which combines our techniques for solving fractional-in-space partial differential equations with our method for sampling from GMRFs.