Heterogeneous Materials - Diffusion, Laplace Spectrum, and Nuclear Magnetic Resonance

In this thesis diffusion in heterogeneous materials is studied using spectral analysisof the Laplace operator. Relations to the effective diffusion constant and therelaxation rate of the time-dependent diffusion coefficient for porous systems arederived from the Laplace operator’s spectrum. The Pad\ue9 approximation is thenexplained in terms of the Laplace operator spectrum. The calculations are madein a finite difference scheme with Neumann conditions defining the boundariesand validated by comparison with Brownian motion simulations. Furthermore,a new method to approximately solve the diffusion equation is presented. Themethod uses a mixture of free diffusion eigenfunctions and surface functions describingthe influence of the boundaries. The method is completely formulatedon the boundaries and the number of operations scale as O(s^2) for s numberof boundary points. The result is an approximation to the first low frequencyeigenfunctions and eigenvalues to the Laplace operator in bounded domains. Themethod is applied on diffusion NMR in the SGP-limit and also extended beyondthe SGP-limit to cover general gradient forms and pulse sequences. Finally, aniterative scheme to find exact eigenvalues and eigenfunctions to the Laplace operatorin bounded domains by surface integrals is presented with a theoreticalO(s log(s)) scaling