Understanding the three-dimensional structure of fibrous materials using stereology

D. Lukas; J. Chaloupek    Technical University of Liberec, Czech Republic

Stereology is a unique mathematical discipline used to describe the structural parameters of fibrous materials found in textiles, geology, biology, fibrous composites, and in corn-grained solids, where fibre-like structures are created by the edges of grains in contact with each other. This chapter is compiled from lectures delivered to post-graduate students taking ‘Stereology of Textile Materials’ at the Technical University of Liberec (Lukas, 1999), and is relevant to students and researchers involved in interpreting flat images of fibrous materials in order to explain their behaviour, or to design new fibrous materials with enhanced properties. There are a number of excellent monographs on stereology, ranging from the basic to the expert. This chapter outlines an elementary technique for deriving most of the stereological formulae, avoiding those demanding either lengthy explanations or a specialised mathematical background. The chapter concentrates on the set of tools needed for a geometrical description of fibrous mass, and provides comprehensive references for further information on this relatively new field.

2.1 Introduction

Stereology was developed to solve various problems in understanding the internal structure of three-dimensional objects, such as fibrous materials, and especially textiles. The relevant geometrical features are mainly expressed in terms of volume, length, surface area, etc. (detailed in Section 2.1.1), and there are three main obstacles facing efforts to quantify these features. The first two difficulties are practical in nature and the third theoretical.

(i) The internal structure of an opaque object can only be examined in thin sections, comprising projections of its fibres. Sections of textile materials may be cut using sharp tools, or created virtually by applying the principles of tomography, confocal microscopy, etc.

(ii) The dimensions of an object under investigation are usually proportionately much greater than the characteristic dimensions of its internal structure; for instance, fibre diameters will be orders of magnitude smaller that the width and the length of the fabric they form. Hence, it is not practicable to study an entire object in detail.

(iii) Occasionally, investigators must determine an appropriate set of geometrical parameters to describe real structures and their properties. Specific parameters will be associated with either mechanical or adsorption properties of fibrous materials.

Various disciplines require information on the internal structures of objects, including biology, medicine, geology, material engineering and mathematics itself. The evolution of methods to quantify structural features laid the foundation for what is now known as stereology, and the concept has continued to evolve since it was proposed in 1961 by a small group of scientists at Feldberg in Germany, under the leadership of Hans Elias (Elias, 1963).

For the purposes of this chapter, the following definition of stereology is used:

The relationship between the geometrical quantities of an n-dimensional object and measurements of its sections and projections is quite logical and familiar. Figure 2.1 reminds us of the procedure for ascertaining the volume of a three-dimensional body. The volume of a three-dimensional body K, say V(K), may be expressed by a definite integral, laid out as:

(K)=∫0Ha(z)dz

where a(z) is the area of a planar cross-section of the body K and is perpendicular to the z-axis. H is the longitudinal length of the projection of the body on the z-axis. The left-hand side of the formula, i.e. the volume V(K), represents a parameter of the three-dimensional object. The right-hand side reveals another parameter of the body in question, a(z), which results from an analysis of its flat cross-section. The two-dimensional parameter a(z) symbolises the area of the flat section cut in the body K by a plane, normal to the z-axis, thus, expressing its cross-sectional area as a function of z. Thus the relationship between three- and two-dimensional parameters is established through integration.

The above relationship may also be demonstrated through Cavalieri’s principle. The conceptualisation was framed by Cavalieri, a student of Galileo in the 17th century (Naas and Schmidt, 1962; Russ and Dehoff, 2000), for two- and three-dimensional objects. For two dimensions, the principle states that the areas of two figures included between parallel lines are equal if the linear cross-sections parallel to and at the same distance from a given base line have equal lengths. For three dimensions, the principle states that the volumes of two solids included between parallel planes are equal if the planar cross-sections parallel to and at the same distance from a given plane have equal areas. This is illustrated in Fig. 2.2. Cavalieri’s principle thus provides further evidence of the relationship between the parameters of three-and two-dimensional objects and their sections.

Cauchy’s formula for surface area also supports the existence of the relationship between objects and their lower-dimension projections. According to this formula, the surface area S(K) of a three-dimensional convex body K is four times the mean area of its planar projection. This can easily be verified by considering a sphere of radius R, whose surface area S is 4πR2, and each of its planar projections has an area of πR2. These quantities are proportional to each other, being related by a factor of 4. A similar relationship for two-dimensional convex bodies will be established in Section 2.3.4. The definition of a convex body will be specified in Section 2.1.1.

However, these attempts to colligate the dimensional aspects of objects with their sections and projections are based only on geometry. Stereology involves statistical methodology in combination with geometry and gives us the ability to model geometrical relations where measurement is impractical or even impossible. To understand the effectiveness of this method, it is necessary to review an interesting experiment carried out in the 18th century.

In 1777, the French naturalist Buffon was attracted by the probability, P, that a randomly thrown needle, j, of length L(j) will hit a line among a given set of parallel lines in a plane with each of the neighbouring lines separated by a distance d, so as to conform to a precondition of d > L(j). The situation is depicted in Fig. 2.3. Buffon (1777) deduced P as 2L(j)/(π * d). The estimated value [P] of probability P, from a large number of throws, N, could be estimated through a relative frequency of hits. Precisely, the value of P equalled the limiting value of [P], while N tended to infinity, i.e. =limN→∞[ P ]. The relative frequency, [P], was defined as:

P ]=nN

where n is the number of positive trials, and N the total number of throws. From this relationship, an unbiased estimation of the distance between parallel lines, [d] can be obtained. The concept of ‘estimators’ will be detailed in sub-section 2.3.1.

d ]=2L(j)π[ P ]=2L(j)Nπn

The above relation [2.3] will be used in Section 3.3.2 to estimate the lengths of curves or fibrous materials in a plane.

Equation [2.3] can be verified by imagining a series of...