We study the Dirichlet problem in a bounded plane domain for the heat equation with small parameter multiplying the derivative in t. The behaviour of solution at characteristic points of the boundary is of special interest. The behaviour is well understood if a characteristic line is tangent to the boundary with contact degree at least 2. We allow the boundary to not only have contact of degree less than 2 with a characteristic line but also a cuspidal singularity at a characteristic point. We construct an asymptotic solution of the problem near the characteristic point to describe how the boundary layer degenerates.

This thesis aims at a deeper understanding of Supercritical Fluid Chromatography (SFC). Although preparative SFC has started to replace Liquid Chromatography (LC) in the pharmaceutical industry - because of its advantages in speed and its less environmental impact - fundamental understanding is still lacking. Therefore there is no rigid framework to characterize adsorption or to understand the impact of changes in operational conditions.   In Paper I we demonstrated, after careful system verification, that most methods applied to determine adsorption isotherms in LC could not be applied directly in SFC. This was mainly due to operational differences and to the fact that the fluid is compressible which means that everything considered constant in LC varies in SFC.   In Paper II we showed that the most accurate methods for adsorption isotherm determination in LC, the so called plateau methods, do not work properly for SFC. Instead, methods based on overloaded profiles should be preferred.   In Paper III a Design of Experiments approach was successfully used to quantitatively describe the retention behavior of several solutes and the productivity of a two component separation system. This approach can be used to optimize SFC separations or to provide information about the separation system.   In Paper IV severe peak distortion effects, suspected to arise from injection solvent and mobile phase fluid mismatches, were carefully investigated using experiments and simulations. By this approach it was possible to examine the underlying reasons for the distortions, which is vital for method development.   Finally, in Paper V, the acquired knowledge from Paper I-IV was used to perform reliable scale-up in an industrial setting for the first time. This was done by carefully matching the conditions inside the analytical and preparative column with each other. The results could therefore provide the industry with key knowledge for further implementation of SFC. / This thesis aims at a deeper understanding of Supercritical Fluid Chromatography (SFC). Although preparative SFC has started to replace Liquid Chromatography (LC) in the pharmaceutical industry - because of its advantages in speed and its less environmental impact - fundamental understanding is still lacking. Therefore there is no rigid framework to characterize adsorption or to understand the impact of changes in operational conditions.   In Paper I-II it was demonstrated why most methods applied to determine adsorption isotherms in LC could not be applied directly for SFC. Methods based on extracting data from overloaded profiles should be preferred.   In Paper III a Design of Experiments approach was successfully used to quantitatively describe the behavior of several solutes in a separation system. This approach can be used to optimize SFC separations or to provide information about the separation system.   In Paper IV severe peak distortion effects often observed in SFC were carefully investigated and explained using experiments and simulations.   Finally, in Paper V, the prerequisites for performing reliable and predictable scale-up of SFC were investigated by small and large scale experiments. / <p>Paper 4 ("Evaluation of scale-up from analytical to preparative...") ingick som manuskript med samma titel i avhandlingen. Nu publicerad. </p>

The aim of this thesis is first and foremost to improve the fundamental knowledge of nonlinear and preparative separation theory by focusing on some of the remaining “white spots” on the theoretical chromatographic map. Secondly, the acquired knowledge is used to develop, validate and execute new methods for phase characterization in liquid chromatography. The methodology used in this thesis is a combination of experiments, fundamental nonlinear theory and systematic computer simulations. A fundamental knowledge of the molecular interactions between the compounds to be separated and the separation media requires the determination of adsorption isotherms over a broad concentration range to give a complete picture of all interactions in the separation system - weak as well as strong. In addition, such adsorption data is essential for optimization in preparative chromatography. For the first time, it has been experimentally shown that the injected molecules are not present in the detected peak when a small excess of molecules are injected into a chromatographic system equilibrated with a constant stream of identical molecules. Several experimental procedures for this method were developed such as (i) the optimal injection strategy and (ii) different labeling methods for visualizing the injected molecules. Remarkable phenomena in the single-component case, such as invisible peak deformation and deformed (invisible) frontal chromatograms, are reported, investigated, and explained. This phenomenon has asides from its future practical implementation, also a large didactic value. The accuracy of the ECP method is experimentally improved, and used to characterize the separation of protolytic compounds at different pH on modern commercially available silica and hybrid silica column packing materials. That investigation enables us to answer why basic compounds give a much more compact preparative peak profile at pH 11 than they yields at lower pH.

High Performance Liquid Chromatography (HPLC) is a widely used tech-nique both for detecting and purifying substances in academy and in the industry. In order to facilitate the use of, and knowledge in HPLC, character-ization of stationary phases is of utmost importance. Tailor made characteri-zation methods and workflows are steadily increasing the speed and accura-cy in which new separation systems and methods are developed. In the field fundamental separation science and of preparative chromatography there is always the need for faster and more accurate methods of adsorption isotherm determination. Some of that demand are met with the steadily increase of computational power, but the practical aspects on models and methods must also be further developed. These nonlinear characterization methods will not only give models capable of describing the adsorption isotherm but also actual values of local adsorption energies and monolayer saturation capacity of an individual interaction sites etc.The studies presented in this thesis use modern alkali stable stationary phas-es as a model phase, which will give an insight in hybrid materials and their separation mechanism. This thesis will include an update and expansion in using the Elution by Characteristic Points (ECP) method for determination of adsorption isotherms. The precision is even further increased due to the ability to use slope data as well as an increase in usability by assigning a set of guidance rules to be applied when determine adsorption isotherms having inflection points. This thesis will further provide the reader with information about stationary phase characterization and the power of using existing tech-niques; combine them with each other, and also what the expansion of meth-ods can revile in terms of precision and increased usability. A more holistic view of what benefits that comes with combining a non-linear characteriza-tion of a stationary phase with more common linear characterization meth-ods are presented.