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Modeling Stochastic Fibrous Materials with Mathematica: Intrinsic Correlation in Planar Poisson Line Processes
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2006 Wolfram Technology Conference
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Champaign, IL
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There is considerable interest in the materials science community in the structure of stochastic fibrous materials and the development of models of such materials to inform our understanding of the dependence of their mechanical and transport properties on their structure. A significant body of theory has arisen from the application of statistical geometry and probability theory to the study of materials such as nonwoven textiles, fibrous filters, and paper, see for example [1,2]. More recently, theory describing stochastic fibrous networks is seeing application by scientists and engineers studying micro- and nano-fibrous electrospun networks as biomaterials for tissue culture and by those developing technical nonwoven textiles for application as proton exchange membranes in high-efficiency fuel cells. In a planar Poisson line process, such as that illustrated above, the distances between adjacent pairs of intersections have an exponential distribution, though in commercially realized networks physical interaction between fibers yield intercrossing distances distributed according to a gamma distribution [a]. Whereas previous models for the pore size distribution in stochastic fibrous networks have assumed that the lengths of the adjacent intercrossing distances that provide polygon sides are independent [4,5], preliminary inspection reveals that in regions of high density there are many short intercrossing distances, and that in regions of low density there are fewer but longer inter-crossing distances. Of course, this seems intuitively reasonable, and it carries with it an important implication: there is positive correlation between nearby polygonal side lengths and this tends to yield more regular polygons, simply from the random variations in the local density that arise from the underlying Poisson point process for fiber centers.
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materials science, stochastic fibrous materials, statistical geometry, stochastic fibrous networks, nano-fibrous electrospun networks, proton exchange membranes, planar Poisson line process
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| TechConf2006_slideshowSampsonR2.zip (4.9 MB) - ZIP archive [for Mathematica 5.2] |
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