Estimation of the directional distribution of spatial fibre processes using stereology and confocal scanning laser microscopy

Fibrous structures like polymers, glass fibres, muscle fibres and capillaries are important components of materials and tissues. A spatial fibre process is the union of smoothly curved or linear one‐dimensional features of finite length, arranged in an unbounded three‐dimensional reference space according to some random mechanism. Design‐based stereology was combined with confocal scanning laser microscopy to study samples of fibre‐reinforced composites, which were considered as realizations of not necessarily isotropic fibre processes. The methods enable the unbiased estimation of the intensity and of the directional distribution of spatial fibre processes from arbitrarily directed pairs of registered parallel optical sections a known distance apart. The directions of fibres sampled by a frame of observation on the reference plane are estimated from the coordinates of the intersection points of the fibres with both optical planes using confocal scanning laser microscopy. The true directional distribution of the fibre process is estimated by weighting each measured direction by the reciprocal of its chance of being sampled, which can be inferred from the data. The concept of complete directional randomness for uniformly and independently distributed spatial directions is introduced. The cumulative distribution function of the angular distances between different directions and other exact relations are derived for complete randomness of vectorial and axial directions. A Monte Carlo method is constructed to test spatial fibre processes, whose fibres have negligibly small curvature, for complete directional randomness. Confocal scanning laser microscopy was used to study the angular distribution of glass fibres in a polymer composite which was subjected to increasing hydrostatic extrusion. The hypothesis of complete directional randomness had to be rejected for all samples with 1% probability of error. The directional distribution was of the bipolar type, with the principal axis directed parallel to the axis of extrusion. Progressive stretching of the material increased the degree of anisotropy of the glass fibres. Although presented for an application in polymer physics, the methods are general and may also be applied in biological investigations.