On the distributions of mass, thickness and density in paper

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1 On the distributions of mass, thickness and density in paper C.T.J. Dodson Department of Mathematics UMIST, PO Box 88, Manchester, M60 1QD, UK. Y. Oba Oji Paper Company Ltd., Shinonome, Koto, Tokyo, , Japan. June 6, 2000 W.W. Sampson Department of Paper Science UMIST, PO Box 88, Manchester, M60 1QD, UK. Abstract 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 12 S t eff = E, (1) where t eff 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 1

2 spherical platens to scan a square area of side 77 mm. In conjunction with β-radiographic measurements of local grammage for the same areas, Schultz-Ekland et al. produced thickness and grammage maps for calendered and uncalendered samples and determined the coefficients of variation of thickness, grammage and density at the 150 µm scale; histograms for each of these properties for a CTMP sheet were essentially Normal in shape. Whilst the correlation between local grammage and local thickness was good, the measurement of thickness used a direct contact technique, and absolute values showed some sensitivity to platen diameter and measuring force. More recently, Izumi and Yoshida [9] have developed a non-contact method to measure a thickness distribution map using two-sided laser triangulation techniques. This technique has been applied in this study and is discussed further below. Since many paper properties are dependent on the mean density, we have the expectation that the distribution of local densities will affect the local values of those properties. Such distributions are important as a distribution of densities infers a distribution of, e.g. local compliances and local degrees of bonding. We note the recent results of Wu et al. [10] who showed a good relationship between bonding and tensile properties in small zones, and those of Popil [11] who showed relationships between mean density and dynamic z-directional compressibility. Dodson and Sampson have recently presented analytic results for the variance of porosity, and hence density, in three dimensional random fibre networks [12]. In that article, expressions were presented also which, assuming that the relationship between local grammage and local thickness is described by the bivariate normal distribution, allowed the variance of local porosity in two and three dimensional networks to be expressed in terms of the coefficients of variation of local grammage and thickness and their covariance. The model has been tested and shown to be valid for laboratory formed papers with varying degrees of formation [13, 14]. Here we present an experimental investigation into the relationships between local grammage, local thickness and local density. Standard statistical methods are used to identify the interdependence of their distributions. Experimental Handsheets were formed in a British Standard Sheet Former from a TMP, a Chemical Softwood pulp and a 50:50 blend of the two fibres. Fibre length and coarseness were measured for each pulp using a Kajaani FS-200 fibre length analyser, fibre width was measured using a light microscope with a calibrated eyepiece graticule; data are summarised in Table 1. Mean width, ω Mean Length, λ Coarseness, δ µm mm g m TMP Chemical Softwood Blend Table 1: Properties of fibres used to prepare sheets Standard handsheets were formed from each furnish; flocculated sheets were formed for each furnish by increasing the time between stirring and forming and by increasing the consistency in the forming chamber to five times the standard. In all, 48 sets of handsheets were formed; conditions are summarised in Table 2. For each sample, the local averages of thickness and grammage of 1 mm zones were measured within a 50 mm 50 mm area and the samples marked to allow zone by zone comparison. Thickness was measured using the two sided laser triangulation device described by Izumi and Yoshida [9]; measurements were made at 0.5 mm intervals and the average of 2 2 adjacent readings was taken to give the local average thickness of a 1 mm square. Sheets were marked to allow measurement at the same locations of the local grammage of 1 mm diameter circular zones using an Ambertec β-formation tester. Mean sheet thickness was measured also using a standard paper thickness micrometer. 2

3 Results Variable Conditions Number Furnish TMP Chem. S/W. 3 Blend Consistency (%) Settling 10 time 30 (s) Grammage 40 (g m 2 ) 60 2 Total Conditions 48 Table 2: Sheet forming conditions. Gradient Intercept r 2 TMP Blend Chem Table 3: Regression of thickness from laser tester on thickness from micrometer for Figure 1. The mean thickness measured using the laser triangulation tester is plotted against that measured using the micrometer in Figure 1. The relationships are linear with approximately unit gradient and the thickness measured by the laser tester is systematically lower than that obtained using the micrometer; regression data is given in Table 3. The sheets formed from the TMP and blended furnish have greater thickness than the sheets formed from the chemical softwood pulp. The systematic difference in thickness recorded by the two instruments for the TMP and the blend is similar to the fibre widths measured for the two pulps, whereas that for the chemical softwood is approximately half a fibre width. This is consistent with the micrometer platen measuring the thickest parts of the measurement zone whereas the laser tester measures both these and lower regions. The smaller difference in measured thickness observed for the chemical softwood may be attributed to the lower coarseness of these fibres when compared to the TMP; low coarseness fibres being more likely to collapse. The coarser TMP fibres seem to dominate the surface structure of the blended furnish, resulting in the observed difference between the two testers. Knowledge of the local averages of grammage and thickness for each 1 mm zone in a given sample allowed direct determination of the local average density of each zone. Examples of density maps for grammage, thickness and density are given in Figure 2 for a 45 g m 2 sheet formed from the chemical softwood pulp. The data allowed calculation of the mean and variance and hence the coefficient of variation at the 1 mm scale for each property according to, CV ( x) = σ( x) 100, (2) x where for a given property x, the coefficient of variation CV ( x) is given as a percentage, σ( x) is the standard deviation and x is the mean value. Throughout we shall denote local grammage, β; local thickness, z and local density, ρ. The variance of local thickness is plotted against that of local grammage in Figure 3, and coefficients of variation for the same properties are shown in Figure 4; regression data is given in Tables 4 and 5 respectively. The variance of local thickness shows an affine relationship with that of grammage and is dependent on pulp type but not on mean grammage. The contribution of the two pulps to the relationship for the blended furnish is approximately equal, the gradient for the blend falling between that of the individual pulps. The sensitivity to pulp type is largely removed when considering the coefficients of variation of local grammage and thickness, though a dependence on mean grammage is introduced. This is likely to be due to the reduced 3

4 Mean thickness from laser tester, zl, µm Mean thickness 100 from 125micrometer, 150 z m 175, µm 200 Figure 1: Mean thickness from laser thickness tester plotted against those measured using standard thickness tester. The micrometer preferentially measures thicker regions and therefore gives a higher estimate of thickness than the laser tester. 40 Areal Density (g m 2 ) Thickness (µm) Density (g cm 3 ) 0.85 Figure 2: In-plane distributions of grammage, thickness and density. Example shown is for a network formed from Chemical softwood fibres with a mean grammage of 45 g m 2. Each image represents the same 50 mm 50 mm region. 4

5 350 Variance of thickness, σ 2 ( z) (µm 2 ) Variance of grammage, σ 2 ( β) (g 2 m 4 ) Figure 3: Variance of local thickness plotted against variance of local grammage. dependent on pulp type but independent of grammage. The relationships are Gradient Intercept r 2 TMP Blend Chem Table 4: Regression of variance of thickness on variance of grammage for Figure 3 influence of surface structure to the measurement of thickness with increasing grammage. The coefficient of variation of local thickness was between 6 % and 48 % greater than that of local grammage for a given sample; the larger differences typically being observed for sheets with worse formation. Plots of the variance of density against those of grammage and thickness are given in Figures 5 and 6 respectively; regression data is given in Table 6. It is clear that the variance of density is independent of the variance of thickness and weakly dependent on the variance of grammage. The coefficient of variation of local density is plotted against that of local grammage in Figure 7 and that of local thickness in Figure 8; associated regression data is given in Table 7. The correlations are not as strong as those between the coefficients of variation of local thickness and local grammage, though show a greater sensitivity to furnish. The coefficient of variation of local density is more strongly correlated to that of thickness than that of grammage. It is interesting to note that the gradients for the TMP and blended furnish are similar and significantly greater than for the chemical pulp. The result suggests that the TMP has a greater influence on the bulk structure of the blended furnish than the chemical pulp. The influence of formation on the mean density of sheets is shown in Figure 9. The lines represent linear regressions on the data. There is no correlation for the chemical pulp, and the correlations are weak for the TMP blended furnish with r 2 values of and respectively. Nonetheless, the trend is interesting as Gradient Intercept r 2 TMP Blend Chem g m g m Overall Table 5: Regression of coefficient of variation of thickness on coefficient of variation of grammage for Figure 4 5

6 Coefficient of variation of thickness, CV ( z) (%) Coefficient of variation of grammage, CV ( β) (%) Figure 4: Coefficient of variation of local thickness plotted against coefficient of variation of local grammage. The broken lines represent the linear regressions for the two grammage classes. In all cases, the coefficient of variation of local thickness is greater than that of local grammage. Variance of density, σ 2 ( ρ) (g 2 cm ) Variance of local grammage, σ 2 ( β) (g 2 m 4 ) Figure 5: Variance of local density plotted against variance of local grammage. 6

7 Variance of density, σ 2 ( ρ) (g 2 cm ) Variance of local thickness, σ 2 ( z) (µm 2 ) Figure 6: Variance of local density plotted against variance of local thickness. σ 2 ( ρ) vs. σ 2 ( β) σ 2 ( ρ) vs. σ 2 ( z) Gradient Intercept r 2 Gradient Intercept r 2 40 g m g m Table 6: Regression of variance of density on variance of grammage for Figure 5 and on variance of thickness for Figure 6 CV ( ρ) vs. CV ( β) CV ( ρ) vs. CV ( z) Gradient Intercept r 2 Gradient Intercept r 2 TMP Blend Chem Table 7: Regression on coefficient of variation of density on coefficient of variation of grammage for Figure 7 and on coefficient of variation of thickness for Figure 8 7

8 Coefficient of variation of density, CV ( ρ) (%) TMP 40 Blend 40 Chem 40 TMP 60 Blend 60 Chem 60 TMP Blend Chem Coefficient of variation of grammage, CV ( β) (%) Figure 7: Coefficient of variation of local density plotted against coefficient of variation of local grammage. The gradients for the TMP and blended furnishes are greater than those for the chemical pulp, suggesting that TMP dominates the bulk structure in the blended furnish. Coefficient of variation of density, CV ( ρ) (%) TMP 40 Blend 40 Chem 40 TMP 60 Blend 60 Chem 60 TMP Blend Chem Coefficient of variation of thickness, CV ( z) (%) Figure 8: Coefficient of variation of local density plotted against coefficient of variation of local thickness. As in Figure 7, the gradients of the regressions suggest that TMP dominates the bulk structure in the blended furnish. 8

9 0.6 Mean density, CV ( ρ) (g cm 3 ) Coefficient of variation of local grammage, CV ( β) (%) Figure 9: Mean density plotted against coefficient of variation of local grammage. Mean density decreases with worsening formation for the TMP and blended furnish. it suggests that for fibres with high coarseness and a low potential to collapse, an improvement in formation will lead to an increase in sheet density, i.e. at a given grammage the sheet will be thinner. Conclusions The pulps studied show an affine relationship between the variance of local thickness and that of local grammage; these relationships are dependent on pulp type but are independent of sheet grammage. The coefficients of variation of thickness and grammage also exhibit affine relationships, though these are independent of pulp type and somewhat dependent on mean grammage; the lower grammage sheets exhibiting a higher coefficient of variation of local thickness at a given coefficient of variation of local grammage than the higher grammage sheets. The result confirms the intuitive expectation that the variability in surface structure has a greater influence on thickness variability at low grammages. The variance of local density is correlated to that of local grammage, and greater density variation is observed in the low grammage sheets. Interestingly, there is no correlation between the variance of local density and that of local thickness. The coefficient of variation of local density however exhibits a good correlation with both the coefficient of variation of local grammage and that of local thickness. In both cases the relationships are insensitive to sheet grammage but sensitive to pulp type. The data suggests that the bulk structure in the blended furnish is dominated by the higher coarseness TMP fibres. The mean density of the TMP and blended furnishes increases with improved formation. Acknowledgments The authors wish to acknowledge the support of Oji Paper Company Ltd., Japan for access to equipment and financial support of the PhD studies of Yasuhiro Oba. 9

10 References [1] B. Norman. The Formation of Paper Sheets. Ch. 6 in Paper Structure and Properties (eds. J.A. Bristow and P. Kolseth). Marcel Dekker, New York, [2] M. Deng and C.T.J. Dodson. Paper: An Engineered Stochastic Structure. Tappi Press, Atlanta [3] K. Niskanen. Paper Physics. Fapet Oy, [4] Thickness (caliper) of paper, paperboard and combined board. Tappi Test Method T411 om-89. Tappi Press, Atlanta, [5] T. Yamauchi. Measurement of paper thickness and density. Appita J. 40(5): , [6] W.A. Wink and G.A. Baum. A rubber platen caliper gauge A new concept in measuring paper thickness. Tappi J. 66(9): , [7] V.C. Setterholm. Factors that affect the stiffness of paper. In Fundamental properties of Paper Related to its Uses, Trans. Vth Fund. Res. Symp. (ed. F. Bolam), pp BPBIF, London, [8] O. Schultz-Eklund, C. Fellers and P-A. Johansson, Method for the local determination of the thickness and density of paper. In proc. Tappi International Paper Physics Conference, Kona, 1991: pp Tappi Press, Atlanta, [9] H. Izumi and Y. Yoshida. Development and application of a system for determining paper nonuniformity. In proc. 66th Pulp and Paper Research Conference, Japan, [10] A. Wu, M.T. Kortschot, Y. Chen and R. Kwong. Variability in the local structure and properties of paper. In proc. Tappi International Paper Physics Conference, San Diego, 1999: pp Tappi Press, Atlanta, [11] R.E. Popil. Observations of dynamic paper compressibility. In proc. Progress in Paper Physics: A Seminar, Vancouver, 1998: ppf1. [12] C.T.J. Dodson and W.W. Sampson. Spatial statistics of stochastic fibre networks. J. Statist. Phys. 96(1/2): , [13] Y. Oba. Z-directional structural development and density variation in paper. PhD thesis, Department of Paper Science, UMIST, UK, [14] C.T.J. Dodson, Y. Oba and W.W. Sampson. Bivariate normal thickness density structure in real near-planar stochastic fibre networks. Preprint, Available via WWW at: 10