Papers by Keyword: Anomalous Diffusion

Abstract: The second order equation (also known as Fick’s equation) is derived from a classical well-known theory, but it is not enough to model all applications of interest. Recently, fractional equations and higher order equations began to receive more attention, demanding increased research efforts. They are used to simulate the diffusion process in many important applications in sciences, such as chemistry, heat and mass transfer, biology and ecology. In this work, the sensitivity analysis is performed for a recently developed anomalous diffusion model in order to evaluate the possibility of estimating a set of parameters that are part of the fourth order equation model, including the parameters representing the variation of the fraction of particles that are allowed to diffuse using a sigmoid function. Finally, after the sensitivity analysis the Inverse Problem approach is used to estimate viable parameters that are necessary for simulation in the cases considered. The differential equation was approximated using the Finite Difference Method, and that solution was implemented in the RStudio platform. The Sensitivity Matrix was calculated and the Inverse Problem was solved using the same RStudio platform, and the Simulated Annealing Method.

Abstract: The classical concept of diffusion characterized by Fick’s law is well suited for describing a wide class of practical problems of interest. Nevertheless, it has been observed that it is not enough to properly represent other relevant applications of practical interest. When in a system of particles their spreading is slower or faster than predicted by the classical diffusion model, such a phenomenon is referred to as anomalous diffusion. Time fractional, space fractional and even space-time fractional equations are widely used to model phenomena such as solute transport in porous media, financial modelling and cancer tumor behavior. Considering the effects of partial and temporary retention in dispersion processes a new analytical formulation was derived to simulate anomalous diffusion. The new approach leads to a fourth-order partial differential equation (PDE) and assumes the existence of two concomitant fluxes. This work investigates the behavior of the bi-flux approach in one dimensional (1D) medium evaluating the mean square displacement for different cases in order to classify the diffusion process in normal, sub-diffusive or super-diffusive.

Abstract: Nanoscale systems show a wide variety of physical properties that cannot be observed in the bulk. Using atom probe tomography, it is possible to study nanostructured materials with almost atomic resolution in all three dimensions. In this article, we will present a short review of the latest atom-probe measurements carried out at University of Münster with particular focus on diffusion and segregation measurements in triple junctions and interface analysis.

Abstract: The sorption and transport of water in two porous building materials, clay brick and autoclaved aerated concrete, was studied in detail. The evolution of the distribution of liquid in the porous medium was analysed in terms of the Boltzmann transform method and anomalous diffusion equation proposed by Küntz and Lavallée [1]. The apparent moisture diffusion coefficients of water were determined from the total water profiles using a modified Boltzmann-Matano analysis, and a good agreement with literature values was found. The application of anomalous diffusion model to building materials indicates that the previous 1/ 2 t relation is not entirely accurate to estimate the volume of absorbed water. This result has particular relevance for evaluating the durability of building structures.

Abstract: Fisher’s model for grain boundary diffusion considers the lattice and the grain boundary on the same basis by presuming the validity of Fick’s second law for both cases, despite the significant structural differences between them. Recent studies [1-3] have, however, shown that grain boundary diffusion is profoundly different from lattice diffusion. We propose an alternative mathematical formulation that incorporates these structural differences and consequently models grain boundary diffusion phenomena more accurately than Fisher’s model. This is achieved by considering possible deviations from the classical random walk for solute atoms diffusing through grain boundaries. This formalism can also be applied to surface diffusion and triple junction diffusion.

Abstract: Pulse loading of diffusion couples leads to the formation of the broad metastable solid solutions. Under higher temperatures, combined with high deformation rates, intermetallics also can form. Possible mechanisms of this phenomenon are discussed. Formation of nanostructure under uniaxial compression/decompression (observed in MD simulations) seems to be one of the possibilities.