Network representation of diffusion in multiscale porous media by D. Grebenkov, P. Levitz (Laboratory of Condensed Matter Physics, Ecole Polytechnique) Emails: denis.grebenkov@polytechnique.edu, pierre.levitz@polytechnique.edu Diffusive transport in porous media is ubiquitous in biology and material sciences. For instance, the geometrical structure of a cement paste essentially determines the related physico-chemical processes and the consequent hardening and mechanical strength of the concrete. The structure of sedimentary rocks influences the oil distribution and conditions the efficiency and cost of oil recovery. In both examples, porous media exhibit a complex hierarchical network of interconnected pores of sizes ranging from nanometers to millimeters. Modern X-ray microtomography and microscopy allow one to determine the morphology of various porous media with a spatial resolution ranging from ten nanometers to several microns. Although diffusive transport in the reconstructed structures can in principle be studied numerically, their complexity and multiscale character make complete numerical simulations too time-consuming for a routine characterization of porous materials. A coarser description of diffusive transport through a decomposition of the whole structure of the medium into a network of "elementary" pores is therefore required. For instance, each elementary pore could be represented by a site of the network, while a connection between two adjacent pores could be represented by a link between two sites. Diffusion inside each pore and between two adjacent pores is effectively modeled as jumps between the sites. The two fundamental problems are: 1) how to decompose a given medium into representative elementary pores (such a decomposition is far from being unique), and 2) how to assign transition rules on a network given that pore shapes and sizes are highly inhomogeneous. Once a reliable decomposition is established, analytical and numerical studies of random walks on networks can be undertaken. The rapidity of random walk algorithms should allow one for investigating diffusive transport in the whole sample at macroscopic time and length scales. The PhD thesis aims at developing these ideas through numerical simulations in model and realistic porous media encountered in the fields of catalysis, oil recovery and building materials. At first step, an off-lattice Monte Carlo algorithm will be implemented for simulating restricted diffusion in model shapes such as minimal surfaces, vycor pore glasses, etc. Bearing in mind oil-recovery applications, a special focus will be on models in which the geometry can be continuously varied from connected to disconnected configurations. The numerical simulations will allow us to determine the first-exit time statistics that answer the question: How long does it take for a diffusing particle started inside a pore to reach some parts of the pore boundary? The exit probability and the mean exit time will then be used to formulate the transition matrix and waiting times for random walks on the pore network. Once a reliable network representation of a porous medium is validated for model geometries, extensive numerical simulations will be started in order to investigate diffusive transport in realistic porous media. This study should allow us to give a rigorous formulation of classical notions such as "elementary pores" and "tortuosity". The numerical results can potentially be confronted with experimental measurement by pulsed-gradient spin-echo techniques also performed in the research group. The ultimate goal of the PhD thesis is to provide a reliable coarse description of diffusive transport in hierarchical porous media and to characterize it by few relevant geometry-based parameters. The candidate is expected to have a background in statistical physics and to be skilled in numerical methods and programming.