The secret harmony of the random disorder, revealed in the paper sheet

Abstract

This study is about the properties of the sets of objects associated in a structure resulting from multiple-processes involving chance as are materials whose texture is unordered and random. Being a paper scientist the author refers to the sheet of paper which is a stochastic fibrous set whose porous texture can be considered as an archetype for many natural or artificial human structures. The paper properties are correlated with its texture by taking account the effect of chance occurring during its manufacturing process. The theoretical developments, the formalism and the application methods presented in this study have a general significance beyond the only paper material.A specific property of sets of objects randomly unorderly distributed in space is their interfaces orientation distribution. This distribution is usually obtained by the analysis of images sampled in the object sets. The density of orientation probability of the fibers or of the texture interfaces, weighted by their length or by their area, can be interpreted as the radius of curvature of an outline or of a warped surface which characterizes, from a global and statistical point of view, the texture geometry in two or three dimensions. This figure named by the author the "equivalent pore", is with its elliptical shape similar to the one of the mean pore defined by the mean directional chord between the interfaces in the texture. Different methods of "equivalent pore" establishing are analyzed : by conformal map of the fiber network or of the texture interfaces, by images stereometric analysis of texture tomographical cuts, by scattering and diffraction of a laser light beam impacting the fibrous texture or the material surface replica, by hard X-ray absorption and phase contrast at the European Synchrotron Radiation Facilities(ESRF), in Grenoble. The "equivalent pore" concept allows us to study random unordered sets behavior in strength fields while simplifying this analysis. Thus a phenomenon occurring in a plane set, in two dimensions, can be analyzed on its "equivalent pore" linear outline, and a phenomenon which occurs in volume in a three dimensional set can be analyzed on its "equivalent pore" warped surface. This concept has been applied for physical, mechanical, optical and ionic conduction properties of materials like papers, boards, felts, nonwoven textiles, polymer foams, metallic alloys with grain joints, geological grounds, and for the surface mapping of natural relief and of materials with different gloss, worn or roughness levels.The ellipse and the ellipsoïd, as well as multi-modes compositions of it, are the most appropriate figures to represent the "equivalent pore" of materials with a random unordered texture. The fact that a law, which defines the curvature of an elliptic deterministic geometrical configuration, is essential to represent interface orientation allocation of elements whose spatial distribution is probabilistic is a noteworthy fact that makes us wonder. This assertion is corroborated by fluid flow analysis through porous media. The global dissipated energy for fluid flow is distributed along the motion (translation and rotation) and fluid deformation components on the "equivalent pore" whose surface is conformal to the texture interfaces tangential space. The porous media being homogenous and the fluid particles indistinguishable each ones from the others, due to permanent stochastic exchanges from one fluid volume element into another, we conclude that their motion quantification is invariant on each point of the "equivalent pore" surface. This quantification can be represented by a group of elliptical cylinders.The "equivalent pore" and cylinders group surface intersections define the fluid particle probabilistic paths in the porous media. One singular geometrical configuration of the elliptic cylinders group union with the ellipsoidal "equivalent pore" fulfills the minimal energy dissipation requirements in the stress field applied to the fluid. The resulting paths are ellipsoïdo-cylindrical curves carpeting the ellipsoïd by a beam of isoperimétric lacets, in close or open loops depending on the possible pairing off the curves in their nodals and isoclinal tangency points. The flow laminarity and unrotationality are globally established in the porous media at the macroscopic scale, for Reynold's number small values, in agreement with the results found moreover. The so defined ellipsoïdo-cylindrical curve is a stationary curve at the average least action meaning, for a punctual mobile or a deformable volume element moving on the surface of the ellipsoid from a nodal spot to the other in an antipodal position. This curve establishes a gauge which characterizes, from a global point of view, the physical space embraced by the fluid during its flow, in dynamical balance conditions compatible with the stress field. The ellipsoïdo-cylindrical curve allows to carpet the ellipsoïd as a function of one cyclic angular variable, which is a new construction for the ellipsoïd. When the set of the objects is isotropic the curve is sphero-cylindrical whose plan mapping permit to decipher the tai-chi figure of the Taoïst Buddhism philosophy.This study of random unordered object sets, and more specially of fibrous porous media, allows to establish a structural link between their small scale texture stochastic disorder and the harmonious order that emerges from these sets at a larger scale. The largest chance possible, compatible with the stress field which put a strain on these sets, is the necessary and sufficient variable which allows to best globally adjust their behaviors according to the probabilistic and deterministic laws governing their evolution.The vocabulary used in this study is issued from the common language, most of the time adapted to the material class, it is transposable in other fields of interest. The text is completed by notes and a bibliography which refer to the works done following the presented concepts or in relation to this field of studies.