On the distributions of mass, thickness and density in paper
C.T.J. Dodson
Y. Oba
W.W. Sampson
Department of Mathematics
Oji Paper Company Ltd.,
Department of Paper Science
UMIST, PO Box 88,
1-10-6 Shinonome, Koto,
UMIST, PO Box 88,
Manchester, M60 1QD, UK.
Tokyo, 135-8558, Japan.
Manchester, M60 1QD, UK.
[email protected]
[email protected]
[email protected]
June 6, 2000
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
r
12 S
tef f =
,
(1)
E
where tef f 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
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.
TMP
Chemical Softwood
Blend
Mean width, ω̄
µm
36.5
38.7
37.8
Mean Length, λ̄
mm
1.98
2.41
2.29
Coarseness, δ
g m−1 × 104
2.22
1.16
1.69
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
Variable
Furnish
Conditions
TMP
Chem. S/W.
Blend
Consistency
0.071
(%)
0.085
Settling
10
time
30
(s)
60
120
Grammage
40
(g m−2 )
60
Total Conditions
Number
3
2
4
2
48
Table 2: Sheet forming conditions.
TMP
Blend
Chem.
Gradient
0.969
1.007
0.992
Intercept
-32.6
-30.5
-19.7
r2
1.000
0.986
0.999
Table 3: Regression of thickness from laser tester on thickness from micrometer for Figure 1.
Results
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,
σ(x̃)
× 100 ,
(2)
x̄
where for a given property x, the coefficient of variation CV (x̃) is given as a percentage, σ(x̃) is the standard
deviation and x̄ is the mean value. Throughout we shall denote local grammage, β̃; local thickness, z̃ and
local density, ρ̃.
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
CV (x̃) =
3
Mean thickness from laser tester, z̄l , µm
175
TMP 40
Blend 40
Chem 40
150
TMP 60
Blend 60
Chem 60
125
100
75
50
75
100
125 micrometer,
150
Mean thickness
from
z̄m175
, 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 )
70
40
Thickness (µm)
120
0.40
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
Variance of thickness, σ 2 (z̃) (µm2 )
350
300
250
200
150
100
TMP 40
Blend 40
Chem 40
50
TMP 60
Blend 60
Chem 60
0
0
10
20
30
40
50
60
Variance of grammage, σ 2 (β̃) (g 2 m−4 )
Figure 3: Variance of local thickness plotted against variance of local grammage. The relationships are
dependent on pulp type but independent of grammage.
TMP
Blend
Chem.
Gradient
7.994
5.157
3.719
Intercept
48.878
52.046
21.825
r2
0.944
0.906
0.958
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 r2 values of 0.214 and 0.118 respectively. Nonetheless, the trend is interesting as
TMP
Blend
Chem.
40 g m−2
60 g m−2
Overall
Gradient
1.112
1.047
1.104
0.872
0.803
1.027
Intercept
1.394
2.113
1.452
4.191
3.431
2.029
r2
0.863
0.812
0.949
0.870
0.901
0.874
Table 5: Regression of coefficient of variation of thickness on coefficient of variation of grammage for Figure 4
5
Coefficient of variation of thickness, CV (z̃) (%)
20.0
TMP 40
Blend 40
Chem 40
17.5
TMP 60
Blend 60
Chem 60
15.0
12.5
10.0
7.5
5.0
5.0
7.5
10.0
12.5
15.0
Coefficient of variation of grammage, CV (β̃) (%)
Variance of density, σ 2 (ρ̃) (g 2 cm−6 × 104 )
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.
40
35
30
TMP 40
TMP 60
Blend 40
Blend 60
Chem 40
Chem 60
25
20
15
10
5
0
0
10
20
30
40
50
60
Variance of local grammage, σ 2 (β̃) (g 2 m−4 )
Figure 5: Variance of local density plotted against variance of local grammage.
6
Variance of density, σ 2 (ρ̃) (g 2 cm−6 × 104 )
40
35
30
TMP 40
TMP 60
Blend 40
Blend 60
Chem 40
Chem 60
25
20
15
10
5
0
0
50
100
150
200
250
300
350
Variance of local thickness, σ 2 (z̃) (µm2 )
Figure 6: Variance of local density plotted against variance of local thickness.
40 g m−2
60 g m−2
σ 2 (ρ̃) vs. σ 2 (β̃)
Gradient Intercept
r2
0.258
16.164
0.230
0.143
9.323
0.324
σ 2 (ρ̃) vs. σ 2 (z̃)
Gradient Intercept
r2
-0.002
21.426
0.002
0.002
13.023
0.000
Table 6: Regression of variance of density on variance of grammage for Figure 5 and on variance of thickness
for Figure 6
TMP
Blend
Chem.
CV (ρ̃) vs. CV (β̃)
Gradient Intercept
r2
0.623
3.845
0.571
0.512
4.370
0.428
0.284
4.981
0.467
CV (ρ̃) vs. CV (z̃)
Gradient Intercept
r2
0.584
2.796
0.718
0.565
2.414
0.706
0.297
4.247
0.601
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
Coefficient of variation of density, CV (ρ̃) (%)
15.0
TMP 40
TMP 60
TMP
12.5
Blend 40
Blend 60
Blend
Chem 40
Chem 60
Chem
10.0
7.5
5.0
5.0
7.5
10.0
12.5
15.0
Coefficient of variation of grammage, CV (β̃) (%)
Coefficient of variation of density, CV (ρ̃) (%)
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.
15.0
TMP 40
TMP 60
TMP
Blend 40
Blend 60
Blend
Chem 40
Chem 60
Chem
12.5
10.0
7.5
5.0
5.0
7.5
10.0
12.5
15.0
17.5
20.0
Coefficient of variation of thickness, CV (z̃) (%)
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
Mean density, CV (ρ̄) (g cm−3 )
0.6
0.5
0.4
TMP 40
Blend 40
Chem 40
TMP 60
Blend 60
Chem 60
0.3
5.0
7.5
10.0
12.5
15.0
Coefficient of variation of local grammage, CV (β̃) (%)
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
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