ESTIMATION OF THE DIRECTIONAL DISTRIBUTION OF SPATIAL FIBER PROCESSESUSING STEREOLOGY AND CONFOCAL SCANNING LASER MICROSCOPY

Fibrous structures like polymers, glass fibres, muscle fibres and capi llaries are important components of materials and tissues. A spatial f ibre 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 ster eology was combined with confocal scanning laser microscopy to study s amples of fibre-reinforced composites, which were considered as realiz ations of not necessarily isotropic fibre processes. The methods enabl e the unbiased estimation of the intensity and of the directional dist ribution of spatial fibre processes from arbitrarily directed pairs of registered parallel optical sections a known distance apart. The dire ctions of fibres sampled by a frame of observation on the reference pl ane are estimated from the coordinates of the intersection points of t he fibres with both optical planes using confocal scanning laser micro scopy. The true directional distribution of the fibre process is estim ated by weighting each measured direction by the reciprocal of its cha nce of being sampled, which can be inferred from the data. The concept of complete directional randomness for uniformly and independently di stributed spatial directions is introduced. The cumulative distributio n function of the angular distances between different directions and o ther exact relations are derived for complete randomness of vectorial and axial directions. A Monte Carlo method is constructed to test spat ial fibre processes, whose fibres have negligibly small curvature, for complete directional randomness. Confocal scanning laser microscopy w as used to study the angular distribution of glass fibres in a polymer composite which was subjected to increasing hydrostatic extrusion. Th e hypothesis of complete directional randomness had to be rejected for all samples with 1% probability of error. The directional distributio n was of the bipolar type, with the principal axis directed parallel t o the axis of extrusion. Progressive stretching of the material increa sed the degree of anisotropy of the glass fibres. Although presented f or an application in polymer physics, the methods are general and may also be applied in biological investigations.