Study of the absorption of organic compounds on biopolymers

This work has been conducted in order to determine the solubility and diffusion coefficients of different aromatic substances in two different grades of polylactic acid (PLA), Amorphous (PDLLA) and Crystalline (PLLA); in particular the focus is on the following terpenes: Linalool, α-Pinene, β-Citronellol and L-Linalool. Moreover, further analyses have been carried out with the aim to verify if the use of neat crystalline PLA, (PLLA), a chiral substrate, may lead to an enantioenrichment of absorbed species in order to use it as membrane in enantioselective processes. The other possible applications of PLA, which has aroused interest in carry out the above-mentioned work, concerns its use in food packaging. Therefore, it is interesting and also very important, to evaluate the barrier properties of PLA, focusing in particular on the transport and absorption of terpenes, by the packaging and, hence, by the PLA. PLA films/slabs of one-millimeter thickness and with square shape, were prepared through the Injection Molding process. On the resulting PLA films heat pretreatment processes of normalizing were then performed to enhance the properties of the material. In order to evaluate solubility and diffusion coefficient of the different penetrating species, the absorption kinetics of various terpenes, in the two different types of PLA, were determined by gravimetric methods. Subsequently, the absorbed liquid was extracted with methanol (MeOH), non- solvent for PLA, and the extract analyzed by the use of High Performance Liquid Chromatography (HPLC), in order to evaluate its possible enantiomeric excess. Moreover, PLA films used were subjected to differential scanning calorimetry (DSC) which allowed to measure the glass transition temperature (Tg) and to determine the degree of crystallinity of the polymer (Xc).

The Feynman-Kac formula and applications to PDEs

The thesis presents a probabilistic approach to the theory of semigroups of operators, with particular attention to the Markov and Feller semigroups. The first goal of this work is the proof of the fundamental Feynman-Kac formula, which gives the solution of certain parabolic Cauchy problems, in terms of the expected value of the initial condition computed at the associated stochastic diffusion processes. The second target is the characterization of the principal eigenvalue of the generator of a semigroup with Markov transition probability function and of second order elliptic operators with real coefficients not necessarily self-adjoint. The thesis is divided into three chapters. In the first chapter we study the Brownian motion and some of its main properties, the stochastic processes, the stochastic integral and the Itô formula in order to finally arrive, in the last section, at the proof of the Feynman-Kac formula. The second chapter is devoted to the probabilistic approach to the semigroups theory and it is here that we introduce Markov and Feller semigroups. Special emphasis is given to the Feller semigroup associated with the Brownian motion. The third and last chapter is divided into two sections. In the first one we present the abstract characterization of the principal eigenvalue of the infinitesimal generator of a semigroup of operators acting on continuous functions over a compact metric space. In the second section this approach is used to study the principal eigenvalue of elliptic partial differential operators with real coefficients. At the end, in the appendix, we gather some of the technical results used in the thesis in more details. Appendix A is devoted to the Sion minimax theorem, while in appendix B we prove the Chernoff product formula for not necessarily self-adjoint operators.