In the present dissertation the formation of microstructure, the kinetics of densification and the formation of surface layers developing during liquid phase sintering of silicon carbide are studied. The focus is on the additive systems Al2O3 plus Y2O3 and AlN plus Y2O3. Phase and especially liquid phase formation in both of the systems SiC, Al2O3 , Y2O3 and AlN, Al2O3 , Y2O3 are investigated in detail examining 12 espectively 17 different compositions per system. Melting temperatures have been determined by TG/DTA, in both systems for the first time. Phase composition of samples was analysed by the combination of XRD, SEM and EDX. In the system SiC, Al2O3 , Y2O3 the formation of the phases expected from the quasibinary Al2O3 , Y2O3 could be observed thus silicon carbide has to be in equilibrium with the oxide additives. The low solubility of SiC in the oxide melt, which was suggested by Hoffmann and Nader, could be confirmed. In the system AlN, Al2O3 , Y2O3 the formation of phases as stated by Medraj was confirmed, except for the dimension of the stability region of the γ- spinel and YAG which is wider in the present work. For the first time diffusion coefficients of the species Y3+ and Al3+ in the oxide melt formed by Al2O3 and Y2O3 at temperatures above 1825 ◦ C were determined. The values are in the order of 2 · 10−6 cm2 /s which results in a diffusion length of 14.1 μm for a diffusion time of one second. This allows the fast equilibration of Y and Al deficiencies. Kinetics of densification was modeled by kinetic field, master curve and thermokinetic method, based on detailed experimental investigation of the shrinkage during liquid phase sintering of SiC. It could be proved that the first 30 − 40 % of densification are controlled by solid phase reactions which accelerate particle rearrangement without presence of a liquid phase. During the remaining 60 − 70 % of densification a liquid is present, resulting in the predominance of mechanisms of liquid phase sintering. The models deliver activation energies in the range from 608 KJ/mol to 1668 kJ/mol and allow, within the scope of validity of each method the prediction of densification during liquid phase sintering of silicon carbide. When sintering silicon carbide with Al2O3 plus Y2O3 the formation of several surface layers, depending on atmosphere, maximum temperature, dwelling time and amount and composition of additives was observed. In nitrogen atmosphere with low partial pressures a surface layer consisting of AlN is forming whilst at high partial pressures SiAlON- polytypes occur. After sintering in Argon or Ar-CO- atmosphere three main types of surface layers are present. One consists of alumina, one contains only YAG and one shows highly porous, additive depleted regions. An explanation for the formation of the several surface layers could be given by the combination of the determined diffusion coefficients with the results achieved in the thermodynamics part. The results achieved in this work can be a contribution to the knowledge based design of the production process of liquid phase sintering of silicon carbide.

In this thesis, we consider two models from physics, which are characterized by the interplay of thermodynamical and fluid mechanical phenomena: demixing (spinodal decomposition) and Rayleigh--Bénard convection. In both models, we investigate the dependencies of certain intrinsic quantities on the system parameters. The first model describes a thermodynamically driven demixing process of a binary viscous fluid. During the evolution, the two components of the mixture separate into two domains of the different equilibrium volume fractions. One observes a clear tendency: Larger domains grow at the expense of smaller ones, and thus, the average domain sizes increases --- a phenomenon called coarsening. It turns out that two mechanisms are relevant for the coarsening process. At an early stage of the evolution, material transport is essentially mediated by diffusion; at a later stage, when the typical domain size exceeds a certain value, due to the viscosity of the mixture, a fluid flow sets in and becomes the relevant transport mechanism. In both regimes, the growth rates of the typical domain size obey certain power laws. In this thesis, we rigorously establish one-sided bounds on these growth rates via a priori estimates. The second model, Rayleigh--Bénard convection, describes the behavior of a fluid between two rigid horizontal plates that is heated from below and cooled from above. There are two competing heat transfer mechanisms in the system: On the one hand, thermodynamics favors a state in which temperature variations are locally minimized. Thus, in our model, the thermodynamical equilibrium state is realized by a temperature with a linearly decreasing profile, corresponding to pure conduction. On the other hand, due to differences in the densities of hot and cold fluid parcels, buoyancy forces act on the fluid. This results in an upward motion of hot parcels and a downward motion of cold parcels. We study the dependence of the average upward heat flux, measured in the so-called Nusselt number, on the temperature forcing encoded by the container height. It turns out that the efficiency of the heat transport is independent of the height of the container, and thus, the Nusselt number is a constant function of height. Using a priori estimates, we prove an upper bound on the Nusselt number that displays this dependency --- up to logarithmic errors. Further investigations on the flow pattern in Rayleigh--Bénard convection show a clear separation of length scales: Along the horizontal top and bottom plates one observes thin boundary layers in which heat is essentially conducted, whereas the large bulk is characterized by a convective heat flow. We give first rigorous results in favor of linear temperature profiles in the boundary layers, which indicate that heat is indeed essentially conducted close to the boundaries.

Im Rahmen der vorliegenden Arbeit wurde ein Spritzgießwerkzeug mit einem neuartigen Mehrschichtverbundheizsystem zur dynamischen Temperierung entwickelt und umgesetzt. Dabei wurde das ausgewählte Heiz‐ und Kühlsystem unter theoretischen und praktischen Gesichtspunkten betrachtet und für den variothermen Fertigungsprozess verifiziert. Aus den ersten durchgeführten praktischen Versuchen zeigte sich, dass dieses Heizsystem zur dynamischen Temperierung von Formwerkzeugen geeignet ist. Anschließend wurden mit dem realisierten Spritzgießwerkzeug Versuchskörper mit spezieller Oberflächenstrukturierung und variierenden Werkzeugwandtemperaturen angefertigt und untersucht. Ziel war es, über diese Strukturierung eine Beeinflussung der Glasfaserverteilung im Formteilrandbereich zu erreichen und die tribologischen Eigenschaften bei Kunststoff‐Kunststoff‐Gleitpaarungen hinsichtlich Reibung und Verschleiß zu verbessern. Mit einer kleinen Auswahl an Strukturen und entsprechenden thermoplastischen Polymermaterialien wurden praktische Versuche zur tribologischen Prüfung durchgeführt. / In the present work an injection mould was developed and implemented with a novel multilayer composite heating system for dynamic temperature control. Here the selected heating and cooling system was considered from a theoretical and practical point of view and verified for the variothermal manufacturing process. The first practical tests showed that this heating system is suitable for the dynamic temperature control of tools. Subsequently, with this injection mould, test specimens with a special surface structure and varying mould wall temperatures were produced and examined. The aim was to achieve through this structuring an impact on the distribution of glass fibres in the edge region of mouldings and improve the tribological properties of plastic‐plastic‐pairings in terms of friction and wear. With a small selection of structures and corresponding thermoplastic polymeric materials practical experiments for tribological testing were performed.

Im Niedertemperaturplasma einer Stickstoff-Hochfrequenzgasentladung werden mit Hilfe der plasmadiagnostischen Methoden LIF und OES räumliche Konzentrationsprofile der N2-Triplett-Zustände A, B und C in verschiedenen Schwingungsniveaus studiert. Dabei wird die Druckabhängigkeit im Bereich 30 bis 100 Pa und die Abhängigkeit von der Brennspannung bis 250 V untersucht. Die deutlich unterschiedlichen räumlichen Verteilungen der verschiedenen Spezies werden mit Hilfe einer Modellierung beschrieben, die sowohl Elektronenstoßanregung in der Plasmarandschicht als auch Diffusion, Stoßabregung, Kaskadenprozesse sowie Deaktivierung durch Wandstöße berücksichtigt. Dabei zeigt sich, daß vor allem Wandstöße für die räumliche Verteilung der langlebigen Spezies A(v=0,1) von zentraler Bedeutung sind. Die kurzlebigen C und B finden sich hingegen hauptsächlich in der Plasmarandschicht. Durch die Kombination der LIF mit Rayleighstreulichtexperimenten werden absolute Teilchendichten der drei Triplett-Zustände bestimmt.

In the present dissertation the formation of microstructure, the kinetics of densification and the formation of surface layers developing during liquid phase sintering of silicon carbide are studied. The focus is on the additive systems Al2O3 plus Y2O3 and AlN plus Y2O3. Phase and especially liquid phase formation in both of the systems SiC, Al2O3 , Y2O3 and AlN, Al2O3 , Y2O3 are investigated in detail examining 12 espectively 17 different compositions per system. Melting temperatures have been determined by TG/DTA, in both systems for the first time. Phase composition of samples was analysed by the combination of XRD, SEM and EDX. In the system SiC, Al2O3 , Y2O3 the formation of the phases expected from the quasibinary Al2O3 , Y2O3 could be observed thus silicon carbide has to be in equilibrium with the oxide additives. The low solubility of SiC in the oxide melt, which was suggested by Hoffmann and Nader, could be confirmed. In the system AlN, Al2O3 , Y2O3 the formation of phases as stated by Medraj was confirmed, except for the dimension of the stability region of the γ- spinel and YAG which is wider in the present work. For the first time diffusion coefficients of the species Y3+ and Al3+ in the oxide melt formed by Al2O3 and Y2O3 at temperatures above 1825 ◦ C were determined. The values are in the order of 2 · 10−6 cm2 /s which results in a diffusion length of 14.1 μm for a diffusion time of one second. This allows the fast equilibration of Y and Al deficiencies. Kinetics of densification was modeled by kinetic field, master curve and thermokinetic method, based on detailed experimental investigation of the shrinkage during liquid phase sintering of SiC. It could be proved that the first 30 − 40 % of densification are controlled by solid phase reactions which accelerate particle rearrangement without presence of a liquid phase. During the remaining 60 − 70 % of densification a liquid is present, resulting in the predominance of mechanisms of liquid phase sintering. The models deliver activation energies in the range from 608 KJ/mol to 1668 kJ/mol and allow, within the scope of validity of each method the prediction of densification during liquid phase sintering of silicon carbide. When sintering silicon carbide with Al2O3 plus Y2O3 the formation of several surface layers, depending on atmosphere, maximum temperature, dwelling time and amount and composition of additives was observed. In nitrogen atmosphere with low partial pressures a surface layer consisting of AlN is forming whilst at high partial pressures SiAlON- polytypes occur. After sintering in Argon or Ar-CO- atmosphere three main types of surface layers are present. One consists of alumina, one contains only YAG and one shows highly porous, additive depleted regions. An explanation for the formation of the several surface layers could be given by the combination of the determined diffusion coefficients with the results achieved in the thermodynamics part. The results achieved in this work can be a contribution to the knowledge based design of the production process of liquid phase sintering of silicon carbide.

In this thesis, we consider two models from physics, which are characterized by the interplay of thermodynamical and fluid mechanical phenomena: demixing (spinodal decomposition) and Rayleigh--Bénard convection. In both models, we investigate the dependencies of certain intrinsic quantities on the system parameters. The first model describes a thermodynamically driven demixing process of a binary viscous fluid. During the evolution, the two components of the mixture separate into two domains of the different equilibrium volume fractions. One observes a clear tendency: Larger domains grow at the expense of smaller ones, and thus, the average domain sizes increases --- a phenomenon called coarsening. It turns out that two mechanisms are relevant for the coarsening process. At an early stage of the evolution, material transport is essentially mediated by diffusion; at a later stage, when the typical domain size exceeds a certain value, due to the viscosity of the mixture, a fluid flow sets in and becomes the relevant transport mechanism. In both regimes, the growth rates of the typical domain size obey certain power laws. In this thesis, we rigorously establish one-sided bounds on these growth rates via a priori estimates. The second model, Rayleigh--Bénard convection, describes the behavior of a fluid between two rigid horizontal plates that is heated from below and cooled from above. There are two competing heat transfer mechanisms in the system: On the one hand, thermodynamics favors a state in which temperature variations are locally minimized. Thus, in our model, the thermodynamical equilibrium state is realized by a temperature with a linearly decreasing profile, corresponding to pure conduction. On the other hand, due to differences in the densities of hot and cold fluid parcels, buoyancy forces act on the fluid. This results in an upward motion of hot parcels and a downward motion of cold parcels. We study the dependence of the average upward heat flux, measured in the so-called Nusselt number, on the temperature forcing encoded by the container height. It turns out that the efficiency of the heat transport is independent of the height of the container, and thus, the Nusselt number is a constant function of height. Using a priori estimates, we prove an upper bound on the Nusselt number that displays this dependency --- up to logarithmic errors. Further investigations on the flow pattern in Rayleigh--Bénard convection show a clear separation of length scales: Along the horizontal top and bottom plates one observes thin boundary layers in which heat is essentially conducted, whereas the large bulk is characterized by a convective heat flow. We give first rigorous results in favor of linear temperature profiles in the boundary layers, which indicate that heat is indeed essentially conducted close to the boundaries.:1 Introduction 2 Coarsening rates in binary viscous fluids 2.1 Background from physics 2.2 Background from mathematics 2.3 The model 2.4 The gradient flow structure 2.5 Heuristics 2.6 Numerical simulations 2.7 Main results 2.8 Preliminaries 2.9 Proof of upper bounds on coarsening rates 2.10 Appendix: Well-posedness and regularity of solutions 3 Scaling of the Nusselt number 3.1 Background from physics 3.2 The model and the Nusselt number 3.3 Heuristics 3.4 Main results 3.5 Scaling law in the linear regime 3.6 Preliminaries and review 3.7 Upper bound using the background field method 3.8 Upper bound using the maximum principle 3.9 Appendix: Some elementary estimates 4 The laminar boundary layer 4.1 Background, model, and motivation 4.2 Main results 4.3 Preparation: Bounds on the velocity field 4.4 On the energy distribution 4.5 Bounds on the second order derivatives of the temperature field 4.6 Bounds on the third order derivatives of the temperature field

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Im Rahmen der vorliegenden Arbeit wurde ein Spritzgießwerkzeug mit einem neuartigen Mehrschichtverbundheizsystem zur dynamischen Temperierung entwickelt und umgesetzt. Dabei wurde das ausgewählte Heiz‐ und Kühlsystem unter theoretischen und praktischen Gesichtspunkten betrachtet und für den variothermen Fertigungsprozess verifiziert. Aus den ersten durchgeführten praktischen Versuchen zeigte sich, dass dieses Heizsystem zur dynamischen Temperierung von Formwerkzeugen geeignet ist. Anschließend wurden mit dem realisierten Spritzgießwerkzeug Versuchskörper mit spezieller Oberflächenstrukturierung und variierenden Werkzeugwandtemperaturen angefertigt und untersucht. Ziel war es, über diese Strukturierung eine Beeinflussung der Glasfaserverteilung im Formteilrandbereich zu erreichen und die tribologischen Eigenschaften bei Kunststoff‐Kunststoff‐Gleitpaarungen hinsichtlich Reibung und Verschleiß zu verbessern. Mit einer kleinen Auswahl an Strukturen und entsprechenden thermoplastischen Polymermaterialien wurden praktische Versuche zur tribologischen Prüfung durchgeführt. / In the present work an injection mould was developed and implemented with a novel multilayer composite heating system for dynamic temperature control. Here the selected heating and cooling system was considered from a theoretical and practical point of view and verified for the variothermal manufacturing process. The first practical tests showed that this heating system is suitable for the dynamic temperature control of tools. Subsequently, with this injection mould, test specimens with a special surface structure and varying mould wall temperatures were produced and examined. The aim was to achieve through this structuring an impact on the distribution of glass fibres in the edge region of mouldings and improve the tribological properties of plastic‐plastic‐pairings in terms of friction and wear. With a small selection of structures and corresponding thermoplastic polymeric materials practical experiments for tribological testing were performed.