{"content":[{"text":"method proceeds to ﬁnd a realization by evolving the microstructure such that the calculated","bbox":{"left":0.132353,"top":0.076103,"width":0.7762930000000001,"height":0.015109999999999998}},{"text":"correlation functions best match the target functions. This intriguing inverse problem is solved","bbox":{"left":0.132353,"top":0.099891,"width":0.776098,"height":0.015109999999999998}},{"text":"by minimizing an error based upon the distance between the target and calculated correlation","bbox":{"left":0.132353,"top":0.123108,"width":0.776489,"height":0.015529000000000015}},{"text":"functions. The two-phase medium can be a dispersion of particles in some matrix (liquid or","bbox":{"left":0.132353,"top":0.147315,"width":0.77644,"height":0.015110000000000012}},{"text":"solid) [6] or, more generally, a digitized image of a two-phase material.[7]","bbox":{"left":0.132353,"top":0.170952,"width":0.610587,"height":0.015110000000000012}},{"text":"An eﬀective reconstruction procedure enables one to generate accurate structures at will,","bbox":{"left":0.16098,"top":0.194739,"width":0.7476459999999999,"height":0.015110000000000012}},{"text":"and subsequent analysis can be performed on the image to obtain desired macroscopic prop-","bbox":{"left":0.132353,"top":0.218376,"width":0.776368,"height":0.015110000000000012}},{"text":"erties","bbox":{"left":0.132353,"top":0.242164,"width":0.044599,"height":0.015110000000000012}},{"text":"(e.g.,","bbox":{"left":0.185686,"top":0.242164,"width":0.04137199999999999,"height":0.015110000000000012}},{"text":"transport, electromagnetic and mechanical properties) of the media.","bbox":{"left":0.236274,"top":0.242164,"width":0.585491,"height":0.015110000000000012}},{"text":"This be-","bbox":{"left":0.836863,"top":0.242164,"width":0.07185799999999998,"height":0.015110000000000012}},{"text":"comes especially useful in generating three-dimensional structures from planar information","bbox":{"left":0.132353,"top":0.2658,"width":0.7764880000000001,"height":0.015110000000000012}},{"text":"when three-dimensional imaging techniques are not available:","bbox":{"left":0.132353,"top":0.289588,"width":0.5239199999999999,"height":0.015110000000000012}},{"text":"a “poor man’s” tomography","bbox":{"left":0.668626,"top":0.289588,"width":0.2402749999999999,"height":0.015110000000000012}},{"text":"experiment.[8, 3]","bbox":{"left":0.132353,"top":0.313224,"width":0.139803,"height":0.015110000000000012}},{"text":"Interestingly, the same procedure has been used to “construct” realizations of two-phase","bbox":{"left":0.16098,"top":0.337012,"width":0.747713,"height":0.015110000000000012}},{"text":"media from a hypothetical target correlation function.[7, 9, 10, 3] In this mode, the procedure is","bbox":{"left":0.132353,"top":0.360648,"width":0.776364,"height":0.015110000000000012}},{"text":"referred to as a construction algorithm. There are mnay diﬀerent types of statistical descriptors","bbox":{"left":0.132353,"top":0.384017,"width":0.776168,"height":0.015529000000000015}},{"text":"of two-phase media, [3] but the most basic one is the two-point correlation function, which","bbox":{"left":0.132353,"top":0.408072,"width":0.7760960000000001,"height":0.015110000000000012}},{"text":"gives the probability of ﬁnding two points in one of the phases (see deﬁnition below) and is","bbox":{"left":0.132353,"top":0.43186,"width":0.7763629999999999,"height":0.015110000000000012}},{"text":"obtainable from small-angle X-ray scattering.[11] The construction algorithm can be employed","bbox":{"left":0.132353,"top":0.455497,"width":0.776097,"height":0.015110000000000012}},{"text":"to determine if a prescribed two-point correlation function is in fact realizable.","bbox":{"left":0.132353,"top":0.479285,"width":0.654509,"height":0.015110000000000012}},{"text":"If such a two-","bbox":{"left":0.795489,"top":0.479285,"width":0.11323099999999997,"height":0.015110000000000012}},{"text":"point correlation function is realizable, then the procedure could be used to categorize classes","bbox":{"left":0.132353,"top":0.502921,"width":0.776168,"height":0.015109999999999957}},{"text":"of random microstructures, which would be a valuable accomplishment.","bbox":{"left":0.132353,"top":0.526709,"width":0.6039220000000001,"height":0.015109999999999957}},{"text":"However, not every","bbox":{"left":0.747059,"top":0.526709,"width":0.1614509999999999,"height":0.015109999999999957}},{"text":"hypothetical two-point correlation function corresponds to a realizable two-phase medium [3].","bbox":{"left":0.132353,"top":0.550345,"width":0.7764690000000001,"height":0.015109999999999957}},{"text":"Therefore,","bbox":{"left":0.132353,"top":0.574133,"width":0.08529399999999998,"height":0.015109999999999957}},{"text":"it is of great fundamental and practical importance to determine the necessary","bbox":{"left":0.226862,"top":0.574133,"width":0.681649,"height":0.015109999999999957}},{"text":"conditions","bbox":{"left":0.132353,"top":0.597769,"width":0.08499099999999998,"height":0.015109999999999957}},{"text":"that","bbox":{"left":0.226862,"top":0.597769,"width":0.03506999999999999,"height":0.015109999999999957}},{"text":"realizable","bbox":{"left":0.271372,"top":0.597769,"width":0.07849699999999998,"height":0.015109999999999957}},{"text":"two-point","bbox":{"left":0.359411,"top":0.597769,"width":0.07957999999999998,"height":0.015109999999999957}},{"text":"correlation","bbox":{"left":0.448626,"top":0.597769,"width":0.08923399999999998,"height":0.015109999999999957}},{"text":"functions must","bbox":{"left":0.547253,"top":0.597769,"width":0.1262470000000001,"height":0.015109999999999957}},{"text":"possess.[12, 3] We note","bbox":{"left":0.683136,"top":0.597769,"width":0.200264,"height":0.015109999999999957}},{"text":"in","bbox":{"left":0.892941,"top":0.597769,"width":0.015900999999999943,"height":0.015109999999999957}},{"text":"passing that this question is closely related to realizability issues of pair-correlation functions","bbox":{"left":0.132353,"top":0.621557,"width":0.7763630000000001,"height":0.015109999999999957}},{"text":"of many-particle systems.[13, 14, 15]","bbox":{"left":0.132353,"top":0.645193,"width":0.304116,"height":0.015109999999999957}},{"text":"One aim of this paper is to gather all of the known necessary conditions on the two-point","bbox":{"left":0.16098,"top":0.668981,"width":0.7478149999999999,"height":0.015109999999999957}},{"text":"correlation function of two-phase random media, also known as “random closed sets” in the","bbox":{"left":0.132353,"top":0.692617,"width":0.776537,"height":0.015109999999999957}},{"text":"ﬁeld of stochastic geometry.[16, 17] Some of these conditions are well-known in the physical","bbox":{"left":0.132353,"top":0.716405,"width":0.7760769999999999,"height":0.015109999999999957}},{"text":"sciences literature, but others are more arcane and are contained in obscure mathematical","bbox":{"left":0.132353,"top":0.740042,"width":0.776276,"height":0.015109999999999957}},{"text":"technical reports and/or proceedings. We also derive a new but simple positivity condition","bbox":{"left":0.132353,"top":0.763829,"width":0.776292,"height":0.015109999999999957}},{"text":"on the two-point correlation function.","bbox":{"left":0.132353,"top":0.787466,"width":0.321959,"height":0.015109999999999957}},{"text":"We consider some illustrative examples of proposed","bbox":{"left":0.468038,"top":0.787466,"width":0.440607,"height":0.015109999999999957}},{"text":"correlation functions and test whether they can correspond to realizable two-phase random","bbox":{"left":0.132353,"top":0.811254,"width":0.7765070000000001,"height":0.015109999999999957}},{"text":"media.","bbox":{"left":0.132353,"top":0.83489,"width":0.05509800000000001,"height":0.015109999999999957}},{"text":"2","bbox":{"left":0.515882,"top":0.926102,"width":0.009552000000000005,"height":0.015109999999999957}}],"images_bbox":[],"images_bbox_not_text_overlap":[]}