On the distributions of mass, thickness and density in paper

Data is presented from experimental measurements of the local grammage and thickness made on laboratory formed sheets with a range of structures. We observe that the distributions of local grammage and local thickness are strongly correlated. The correlation between the variances of thickness and grammage are dependent on pulp type but independent of mean sheet grammage, whereas the correlation between their coefficients of variation is independent of pulp type and dependent on mean sheet grammage. A correlation is shown also between the variance of local density and that of grammage and this is sensitive to mean grammage. The variance of local density is however independent of the variance of local thickness. The coefficient of variation of local density is correlated to the coefficients of variation of local grammage and thickness; the correlations being sensitive to pulp type and apparently dominated by the properties of high coarseness fibres in a blended furnish. We note also that the mean density of sheets containing high coarseness fibres increases with improved formation. Introduction It is well established that the performance of paper in end-use applications is dependent to some extent on the structure, or formation, of the sheet [1]. The distribution of mass density has been the subject of extensive experimental and theoretical research [2]. It is well established also that many paper properties are strongly correlated, for a given pulp type, to the mean apparent density of the sheet [3]. By definition, the mean apparent density of paper is given by the ratio of the mean grammage to the mean thickness. As a consequence of the surface nonuniformity and compressibility of the sheet, the mean thickness of paper is a notoriously difficult property to measure. The standard technique uses a micrometer to measure the separation of parallel hard circular platens applied around a sheet or stack of sheets at a standard load and standard rate of loading [4]. A study of the measurement techniques for sheet thickness was presented by Yamauchi [5] who compared the standard technique with mercury buoyancy and pyncnometric techniques and with a modified platen technique using soft rubber platens, as developed by Wink and Baum [6]. The highest values were obtained using the standard technique, where the thickest regions of the sheet only are measured; values obtained using the soft platens, which conform to some extent to the sheet surface, were lower; the mercury techniques, where mercury covers the sheet surface well but does not penetrate into the bulk, gave markedly lower values. An alternative measure of thickness, based on the mechanical properties of the sheet was proposed by Setterholm [7], such that teff = √ 12S E , (1) where teff is the effective thickness (m), S is the bending stiffness (N m) and E is the tensile stiffness (N m−1). Whilst such techniques allow determination of a mean thickness under a given set of conditions, they provide no information on its distribution. A technique for measuring the thickness of small zones was presented by Schultz-Ekland et al. [8]. The technique used a pair of small diameter mutually opposing