Novikov Dmitry S, Fieremans Els, Jensen Jens H, Helpern Joseph A
Center for Biomedical Imaging, Department of Radiology, New York University School of Medicine, New York, NY 10016, USA.
Nat Phys. 2011 Jun 1;7(6):508-514. doi: 10.1038/nphys1936.
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 nondestructively 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 (d - 1 dimensional planes in d dimensions) and focus on the disorder-averaged diffusion propagator using a scattering approach. The renormalization group solution reveals a scaling behavior 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 novel 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.
在多孔介质、复合材料和生物组织中，由屏障（膜）对分子运动的限制无处不在。一个主要挑战是使用体输运测量来无损地表征材料或生物体的微观结构。在此，我们展示了可渗透膜引入的长程结构相关性如何产生独特的输运特征。我们考虑受随机放置和定向的膜（d维空间中的d - 1维平面）限制的布朗运动，并使用散射方法关注无序平均扩散传播子。重整化群解揭示了长时间扩散系数的标度行为，对于任何d，其具有典型的缓慢平方根时间依赖性。其起源在于由空间扩展的随机限制引入的强结构涨落，代表了结构无序的一种新型普适类。我们的结果与二维蒙特卡罗模拟非常吻合。它们可用于识别作为输运限制的可渗透屏障，并量化其渗透率和表面积。
