Problems of optimal transportation on the circle and their mechanical applications

We consider a mechanical problem concerning a 2D axisymmetric body moving forward on the plane and making slow turns of fixed magnitude about its axis of symmetry. The body moves through a medium of non-interacting particles at rest, and collisions of particles with the body's boundary are perfectly elastic (billiard-like). The body has a blunt nose: a line segment orthogonal to the symmetry axis. It is required to make small cavities with special shape on the nose so as to minimize its aerodynamic resistance. This problem of optimizing the shape of the cavities amounts to a special case of the optimal mass transfer problem on the circle with the transportation cost being the squared Euclidean distance. We find the exact solution for this problem when the amplitude of rotation is smaller than a fixed critical value, and give a numerical solution otherwise. As a by-product, we get explicit description of the solution for a class of optimal transfer problems on the circle.

Optimization of Li-ion solid electrolytes

The synthesis and optimization of two Li-ion solid electrolytes were studied in this work. Different combinations of precursors were used to prepare La0.5Li0.5TiO3 via mechanosynthesis. Despite the ability to form a perovskite phase by the mechanochemical reaction it was not possible to obtain a pure La0.5Li0.5TiO3 phase by this process. Of all the seven combinations of precursors and conditions tested, the one where La2O3, Li2CO3 and TiO2 were milled for 480min (LaOLiCO-480) showed the best results, with trace impurity phases still being observed. The main impurity phase was that of La2O3 after mechanosynthesis (22.84%) and Li2TiO3 after calcination (4.20%). Two different sol-gel methods were used to substitute boron on the Zr-site of Li1+xZr2-xBx(PO4)3 or the P-site of Li1+6xZr2(P1-xBxO4)3, with the doping being achieved on the Zr-site using a method adapted from Alamo et al (1989). The results show that the Zr-site is the preferential mechanism for B doping of LiZr2(PO4)3 and not the P-site. Rietveld refinement of the unit-cell parameters was performed and it was verified by consideration of Vegard’s law that it is possible to obtain phase purity up to x = 0.05. This corresponds with the phases present in the XRD data, that showed the additional presence of the low temperature (monoclinic) phase for the powder sintered at 1200ºC for 12h of compositions with x ≥ 0.075. The compositions inside the solid solution undergo the phase transition from triclinic (PDF#01-074-2562) to rhombohedral (PDF#01-070-6734) when heating from 25 to 100ºC, as reported in the literature for the base composition. Despite several efforts, it was not possible to obtain dense pellets and with physical integrity after sintering, requiring further work in order to obtain dense pellets for the electrochemical characterisation of Li Zr2(PO4)3 and Li1.05Zr1.95B0.05(PO4)3.