Abstract
Restrictions to molecular motion by barriers (membranes) are ubiquitous in porous media, composite materials and biological tissues. A major challenge is to characterize the microstructure of a material or an organism non-destructively using a bulk transport measurement. Here we demonstrate how the long-range structural correlations introduced by permeable membranes give rise to distinct features of transport. We consider Brownian motion restricted by randomly placed and oriented membranes (dā1-dimensional planes in d dimensions) and focus on the disorder-averaged diffusion propagator using a scattering approach. The renormalization group solution reveals a scaling behaviour of the diffusion coefficient for large times, with a characteristically slow inverse square root time dependence for any d. Its origin lies in the strong structural fluctuations introduced by the spatially extended random restrictions, representing a new universality class of the structural disorder. Our results agree well with Monte Carlo simulations in two dimensions. They can be used to identify permeable barriers as restrictions to transport, and to quantify their permeability and surface area.
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Acknowledgements
We thank V. Kiselev and D. Sodickson for discussions. Research was supported by the Litwin Fund for Alzheimerās Research, and the National Institutes of Health Grants 1R01AG027852 and 1R01EB007656.
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D.S.N. carried out analytical calculations and wrote the manuscript. E.F. carried out numerical calculations. All authors discussed the results and implications and commented on the manuscript at all stages.
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Novikov, D., Fieremans, E., Jensen, J. et al. Random walks with barriers. Nature Phys 7, 508ā514 (2011). https://doi.org/10.1038/nphys1936
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DOI: https://doi.org/10.1038/nphys1936
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