Download The contribution of out-of-plane pore dimensions to the pore size

The contribution of out-of-plane pore dimensions to the pore
size distribution of paper and stochastic fibrous materials
W.W. Sampson∗
S.J. Urquhart
September 29, 2006
Abstract
Theory is presented for the distribution of the narrowest dimension of voids encountered in a path through a stochastic fibre network from one side of its plane to the other.
Using expressions from the literature we show that the mean out-of-plane pore dimension is always less than half the mean in-plane dimension such that out-of-plane pore
dimensions have a controlling influence on the measured pore size distribution.
An approximate model is derived which predicts that the overall mean pore dimension
of a network of fibres with given dimensions is inversely proportional to its areal density
and proportional to a simple function of porosity. Good agreement is demonstrated
between this model and experimental data. Regression analysis suggests that out-of-plane
pore dimensions account for more than 70 % of the measured pore size distribution.
Introduction
The pore size distribution of stochastic fibrous materials such as paper, fibrous filters and
nonwoven textiles is often modelled by considering the statistics of the polygons generated
by two-dimensional random networks of infinite lines or of finite length fibres; graphical
representations of such networks are shown in Figure 1. Mathematical analysis has provided
some statistics describing these structures; for example, the ³expected
´ number of sides per
π2
polygon is 4, the fraction of polygons that are triangles is 2 − 6 ≈ 0.355 [1] and the
fraction of polygons that are quadrilaterals is approximately 0.381[2]; the fractions of polygons
with more than four sides are known only approximately through simulation studies, see e.g.
[3, 4].
The distribution of pore sizes, as characterised by polygon area, or the radius of some
equivalent circle, is not known analytically, though some approximate models are available
and these suggest that pore area and pore radii have a gamma distribution [5–7]. Whereas
other workers have suggested that pore size in fibrous materials may exhibit a lognormal [4, 8]
or Weibull distribution [9], experimental data for the pore size of paper consistently shows
the coefficient of variation of pore size to be independent of the mean pore size [8, 10–12] and
this property characterises gamma distributed variables [13].
∗
Corresponding author: [email protected]
1
2
Figure 1: Graphical representations of two-dimensional fibre and line networks. Total length of lines in
each image is the same.
Much of the practical interest in such models arises from the use of stochastic fibrous
materials as barriers in filtration applications where manufacturers seek to optimise network
permeability and the efficiency of particle capture by altering the properties of fibres and
the porosity and weight per unit area, or ‘areal density’, of the network. Abdel-Ghani and
Davies [14] used simulation to investigate the capture of droplets by fibre mats placed in
series and demonstrated that for mats of a given porosity, the efficiency of particle capture
increased with decreasing fibre diameter. Using a thin network of fibres of infinite length and
finite width ω (m) to represent a single layer of a mat with open area and hence porosity ε,
Abdel-Ghani and Davies [14] give the mean polygon area as,
ā ≈
πε
2
2 ω
(1 − ε)
(1)
and observed good agreement with simulations. We note that the simulations of Clarenburg and Schiereck [15] reveal similar dependencies. Importantly, although Abdel-Ghani and
Davies [14] and Clarenburg and Schiereck [15] considered only networks of infinite lines, recent analytic models based on the statistics of networks of finite length fibres reveal the same
dependencies [16].
The structure of stochastic fibrous materials is rather orthotropic and this gives rise in turn
to dramatic differences in the in-plane and out-of-plane permeabilities of the network [17].
In comparing the results of a simulation with experimental data for the permeability of a
newsprint grade of paper, Termonia [18] suggested that the dimensions of voids perpendicular
to the plane of the network have a strong influence on permeability and diffusivity. These ‘pore
heights’ have been measured for different types of paper by Paavilainen [19] and Niskanen and
Rajatora [20] who found that their distribution was approximately exponential; they found
good agreement also between these experimental determinations of mean pore height h̄ and
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the expression,
ε
t̄
(2)
1−ε
where t̄ (m) is the mean vertical dimension, i.e. thickness, of the fibres.
Here we develop Equation (2) to derive approximate expressions for the distribution of
the smallest pore dimensions encountered in a path from one side of the network to the
other. We present also experimental data characterising the pore size in several families of
laboratory-formed paper samples and compare these with the theory.
h̄ =
Theory
Consider a path from one side of the plane of a fibre network to the other, such as that
followed by a fluid or particle passing through the network under some driving force acting
perpendicular to its plane. We assume that this path consists of alternate axial and lateral
movements, axial movements occurring in the direction of the driving force only. Further,
we assume that the transition between lateral and axial movements occurs only when the
path is obstructed by a fibre. A similar schema was considered by Foscolo et al. [22] in their
derivation of a tortuosity factor for isotropic random porous media, such as fluidised and
packed beds, this being given by the reciprocal of porosity.
Now, for each lateral movement, the void dimension that determines whether or not a
particle can pass is the pore height; similarly, for each axial movement this is determined
by some characteristic in-plane dimension of a pore, e.g. the radius of an inscribed circle, a
hydraulic radius or some equivalent diameter of a polygonal void.
If the probability density function for the characteristic in-plane dimension of voids
is f (din ) and the cumulative distribution function is F (din ) then the probability density function for the smallest voids selected at random from n independent and identical distributions
of voids is,
f (din , n) = n (1 − F (din ))n−1 f (din ) ,
(3)
with cumulative distribution function,
Z
F (din , n) =
din
0
f (din , n) .ddin .
(4)
Denoting the probability density of pore heights g(dh ) and the cumulative distribution function G(dh ), then the probability density function for the smallest pore heights selected at
random from n independent and identical distributions of pore heights is,
g(dh , n) = n (1 − G(dh ))n−1 g(dh ) ,
(5)
with cumulative distribution function,
Z
G(dh , n) =
0
dh
g(dh , n) .ddh .
(6)
Since any path from one side of the network to the other always begins and ends with
an out-of-plane movement, then a path that consists of n out-of-plane steps will consist
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of (n − 1) in-plane steps. We now unify the notation such that parameter d represents the
characteristic dimension of the smallest of (n − 1) pore heights and n in-plane pore diameters.
This parameter can be thought of as representing the size of the largest particle that could
pass from one side of the network to another and hence is that which we seek to measure, or
infer, using porometry techniques; it has probability density,
p(d) = (1 − G(d, n − 1)) f (d, n) + (1 − F (d, n)) g(d, n − 1) .
(7)
Equation (7) holds for any pair of distributions f (d) and g(d), and for stochastic fibrous
materials these can be described by the gamma and exponential distributions respectively,
allowing the probability density p(d) for given n to be stated explicitly in terms of the
parameters characterising the mean and variance of those distributions. Note that the terms
in parentheses in Equation (7) represent the fractions of the pores that contribute to p(d)
which come from the two source distributions and these will be influenced to a large extent
by the relative in-plane and out-of-plane dimensions of voids.
Now, Equation (1) can be used to characterise in dimensions of voids in an infinitesimally
thin slice in the plane of a network of finite thickness. Although fibres may bend so that along
their length they occupy space in adjacent planes through the thickness of the material, this
does not influence the applicability of Equation (1) since what is important is the probability
that a given point in space is occupied by a fibre or not, i.e. the fractional open area of
a planar section or the porosity of a section with finite thickness, these being numerically
equivalent. Note that any influence of fibre variables such as flexibility and length is accounted
for implicitly through their influence on porosity.
A convenient measure of the characteristic in-plane dimension of a polygon void is the
diameter of a circle with the same area. From Equation (1) we can estimate the mean
equivalent in-plane diameter of a void as,
√
2 ε
¯
ω̄ ,
(8)
din =
1−ε
where ω̄ (m) is the mean fibre width. Combining Equation (8) with Equation (2) for the
mean pore height yields, on manipulation,
√
h̄
ε t̄
=
.
(9)
¯
2 ω̄
din
Typically we expect t̄ ≤ ω̄ because fibres are either cylindrical or ribbon-like, the latter lying
in the plane such that t̄ < ω̄. Note that for cylindrical fibres of diameter ω, t̄ = ω̄ = π ω/4
√
such that h̄/d¯in = ε/2 and, since ε < 1, we may state that the mean pore height is always
less than half the mean in-plane dimension of pores. On this basis, we put forward the
hypothesis that the barrier properties of fibre networks and measurements of pore size are
influenced primarily by the distribution of pore heights. We now proceed to estimate the
overall mean pore diameter, d¯ on this basis.
If the pore height distribution is exponential with mean h̄, then the distribution of the
smallest pore heights selected at random from n independent and identical distributions of
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pore heights is exponential also with mean,
h̄
d¯ =
.
n
(10)
In a stochastic fibre network where the expected number of fibres covering a point is c̄, the
expected number of interfaces between fibres where pore heights can occur is (c̄ − 1). We
expect therefore,
n ∝ (c̄ − 1) ,
such that,
d¯ = K(ε)
h̄
,
c̄ − 1
(11)
where K(ε) is a porosity-dependent constant that accounts for the influence on n of tortuosity and contact between vertically adjacent fibres. Substitution of Equation (2) into
Equation (11) yields,
K(ε) ε
t̄ .
(12)
d¯ =
c̄ − 1 1 − ε
A network with mean areal density β̄ (g m−2 ) formed from from fibres of linear density δ
(kg m−1 ) and mean width ω̄ (m), has mean coverage given by
c̄ =
β̄ ω̄
δ
(13)
Typically, c̄ >> 1 so,
d¯ ≈
≈
K(ε) ε
t̄ ,
c̄ 1 − ε
K(ε) δ ε
t̄ .
β̄ ω̄ 1 − ε
(14)
(15)
If measures of pore size and the barrier properties of stochastic fibrous materials are influenced
primarily by the pore height distribution and Equation (15) holds, then we expect a plot of
mean pore diameter against the reciprocal of the mean areal density of the network to be
approximately linear with a gradient dependent upon the porosity of the network and the
morphology of the fibres. Similarly, any non-linearity in a plot of the mean pore diameter of
networks with constant areal density against ε/ (1 − ε) should be attributable to the relative
influences of tortuosity and fibre contact.
Experimental
Materials and sample preparation
Experimental paper samples were formed from natural cellulosic fibres obtained from wood
for industrial papermaking. The fibres had been obtained using either a Kraft chemical
pulping process or a Sulphite chemical pulping process. The fibres had been bleached and
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formed into dry sheets ready for dispersion into an aqueous suspension from which networks
can be formed. As received, such fibres make networks with high porosity and low strength.
Accordingly, in advance of making networks, fibre suspensions were subjected to mechanical
action in a Valley beater (ISO 5264-1:1979) to yield networks with a range of porosities.
Networks were formed by filtration of a fibre suspension over a standard woven wire fabric
using a British Standard Sheet Former; this equipment conforms to international standards
for forming paper in the laboratory (ISO 5269-1:2001).
Two families of networks were made with a range of areal densities; a Kraft spruce
pulp treated in the Valley beater was used to make networks with areal density between 50
and 70 g m−2 and a blend of Kraft spruce and pine fibres, prepared for papermaking by a commercial manufacturer was, used to make networks with areal density between 40 and 90 g m−2 .
In addition to the influence of areal density, Equation (15) shows the pore size to depend
upon the thickness of fibres and the porosity of the network. We chose to influence the
thickness of fibres by forming networks from fibres obtained from different wood species and
to vary porosity by processing these fibres in the Valley beater for different times. Accordingly
networks were made from three types of softwood and two types of hardwood fibres. One of
the softwoods was a spruce pulped by the sulphite process; all the other pulps were produced
by the Kraft process and were a spruce, a pine, a maple and a birch. In the analysis that
follows, the two spruce pulps are differentiated from each other by including the first letter
of the pulping process in parentheses after the fibre type.
After forming, the networks were pressed against blotters according to standard (ISO 52691:2001) and allowed to dry and condition in an atmosphere of 23 ± 1◦ C and 50 ± 2 % RH for
at least 24 hours.
The pore diameter distribution was measured using a capillary flow porometer, model
CFP 1500 AEX manufactured by PMI Inc. (Ithaca, NY, USA). The instrument automates
the saturated head gas drive technique described by Corte [23] and conforms to ASTM
standards [24]. Samples were saturated in a silicon oil of surface tension γ = 20.6 mN m−1
and the instrument used to record the flow rate of dry nitrogen, Q (m3 s−1 ) at a given
pressure, P (N m−2 ) as the silicon oil was displaced from successively smaller pores with
increasing pressure. The test was stopped when the flow rate increased in proportion to the
pressure indicating that all pores accessible to flow were open. For such data, Corte [23]
shows that the number of pores with diameter in an interval ∆d is given by,
µ
¶
dQ Q
−
P 3 ∆P ,
(16)
n(d) ∆d = k
dP
P
where k is a constant which can be expressed explicitly in terms of surface tension, contact
angle, network thickness and fluid viscosity. Pore size was determined using the Kelvin
equation assuming a contact angle of zero,
d=
4γ
.
P
(17)
Note that whereas the number of pores in a given class, as given by P
Equation (16), is dependent on parameter k, the fraction of pores in that class, i.e. n(d)/ n(d) is not. This is
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Spruce (K)
Pine/Spruce blend
Bliesner [10]
Gradient
187
311
271
Intercept
-2.2
-2.8
6.8
r2
0.702
0.706
0.953
Table 1: Regression data for influence of mean area density on mean pore diameter as plotted in Figure 2.
important since it allows us to state that any changes in pore size observed are not attributable to differences in their thickness.
The porometer measured flow through a circular area of diameter 24 mm, though a
larger area must be accomodated by the sample holder to minimise leaks, side-flows, etc. In
the holder, the samples were backed by a stainless steel wire mesh with 15 yarns per cm
such that it had a large open area with inter-yarn voids several orders of magnitude larger
than those within the sample. Given the inherent stochasticity of the networks, and to
ensure the best estimate of the areal density of the area being tested, samples were cut using
a circular punch of diameter 32 mm and weighed. The thickness of each test piece was
measured using an electronic micrometer (ISO 534:2005). Since fibre networks of this type
have irregular surfaces and are compressible the density obtained by dividing the areal density
by the thickness is termed an ‘apparent density’ (ISO 534:2005) and this was calculated for
the samples with a range of porosities. For each of the fibre types used to prepare samples
with a range of areal densities a plot of mean areal density against mean thickness was linear
with a non-zero intercept confirming that the samples could be considered to have constant
mean porosity independent of their areal density [25].
Results and Discussion
The mean pore diameter is plotted against the reciprocal areal density in Figure 2. In addition
to the data arising from the experiments described here, the data of Bliesner [10] for high
porosity networks of fibres obtained from an unspecified wood are included also. The solid
lines represent linear regressions on the data and the statistics characterising these are given
in Table 1; the data for the pine/spruce blend identified with a cross inside the data marker
and shown on the right hand side of the plot are for sheets with areal density around 40 g m−2 .
These samples were observed to exhibit pinholes, i.e. through-pores, such that flow would
be preferentially through these and the theory would not apply; accordingly these data were
excluded from the regression analysis that generated the line plotted.
On first inspection of Figure 2 it is clear that, to a first approximation, there is proportionality between the reciprocal of areal density and the mean pore diameter as predicted by
Equation (15). It is important to bear in mind that the mean pore diameter is calculated
as that of an equivalent circle assuming perfect wetting; such assumptions inevitably oversimplify the highly disordered structure that is being measured and as such, the non-zero
intercepts observed in Figure 2 might be expected. The coefficients of determination given in
Table 1 can be interpreted as the fraction of the variability in the data which is explained by
8
14
Spruce (K)
Pine/Spruce blend
Mean pore diameter,
d (µm)
12
Bliesner [10]
10
8
6
4
2
0
0
0.01
0.02
0.03
β (m2 g-1)
Reciprocal areal density, 1/
Figure 2: Mean pore diameter plotted against reciprocal areal density.
the model. So for these experiments we find that the model accounts for about 70 % of the
variability, i.e. around 70 % of the measured pore size distribution arises from pore height
distribution. The high coefficient of determination arising from regression on Bliesner’s data
probably arises from the small number of data considered, but is encouraging nonetheless.
The porosity of networks was estimated from their apparent density by assuming that
the fibres had the density of cellulose and that all void space within the network was between fibres; the treatment in the Valley beater typically causes wood pulp fibres to collapse
justifying this assumption. On this basis the porosity was calculated from,
ρ
,
(18)
ε=1−
ρcell
where ρ is the apparent density of the network and ρcell is assumed to be 1.5 g cm−3 .
The mean pore diameter is plotted against the function ε/(1 − ε) in Figures 3 and 4
for softwood and hardwood pulps respectively and in Figure 5 for the data of Corte and
Lloyd [8]. Due to the stochastic nature of the samples, there was inevitably some difference
in the densities and areal densities of the samples; accordingly, in Figures 3 and 4 the data are
shown with error bars that represent 95 % confidence intervals on the data as calculated from 6
repeats. For clarity one regression line only is shown in Figure 3, though regression outputs for
all data in Figures 3 to 5 are given in Table 2. The gradients observed for the different pulps
range from 1.9 to 4.7 supporting the dependence on pore size on fibre dimensions predicted
by Equation (15); further analysis of fibres with known dimensions would be required to
9
3.5
Pine
Spruce (S)
Mean pore diameter,
d (µm)
3.0
Spruce (K)
2.5
2.0
1.5
1.0
0.5
0
0
0.5
1.0
Porosity function,
1.5
2.0
ε/(1 - ε)
Figure 3: Mean pore diameter plotted against ε/(1 − ε) for softwood pulps.
Spruce (K)
Spruce (S)
Pine
All Softwood
Maple
Birch
Softwood [8]
Hardwood [8]
Gradient
4.7
4.3
1.9
3.9
3.0
4.7
13.8
22.9
Intercept
-4.8
-4.3
-1.6
-4.0
-2.6
-3.8
-6.7
-10.1
r2
0.995
0.938
0.828
0.876
0.978
0.787
0.922
0.974
Table 2: Regression data for influence of porosity on mean pore diameter as plotted in Figures 3 to 5.
10
3.5
Maple
Birch
Mean pore diameter,
d (µm)
3.0
2.5
2.0
1.5
1.0
0.5
0
0
0.5
1.0
Porosity function,
1.5
2.0
ε/(1 - ε)
Figure 4: Mean pore diameter plotted against ε/(1 − ε) for hardwood pulps.
11
9
Softwood
Mean pore diameter,
d (µm)
8
Hardwood
7
6
5
4
3
2
1
0
0
0.2
0.4
Porosity function,
0.6
0.8
1.0
ε/(1 - ε)
Figure 5: Mean pore diameter plotted against ε/(1 − ε) for data of Corte and Lloyd [8].
12
investigate this dependence further. Interestingly, the gradients observed for the two types
of Spruce fibre obtained by different pulping methods are similar.
Evidently, there is some non-linearity to the data shown in Figures 3 to 5 and this can
be attributed to the influence of porosity on parameter K(ε) in Equation (15). Nonetheless,
the coefficients of determination given in Table 2 suggest that for these samples around 80 %
or more of the observed variability can be attributed to the model and hence to contribution
of the pore height distribution to the measured distribution of pore sizes.
Conclusions
We have presented theory for the distribution of the narrowest dimension of voids encountered
in a path through a fibre network that consists of alternate lateral and axial movements
through voids with arbitrary distributions. Using expressions from the literature we have
shown that the ratio of the mean in-plane pore dimension to the mean out-of-plane dimension
is determined by the ratio of the mean in-plane and out-of-plane fibre dimensions such that the
mean out-of-plane pore dimension is always less than half the mean in-plane dimension. On
this basis, a model has been presented for a simplified system where the narrowest dimension
of voids encountered in a path through a fibre network is always that of an out-of-plane
pore. For such a system, the model predicts that the mean pore dimension is inversely
proportional to the areal density of the network, proportional to a simple function of porosity
and influenced also by fibre dimensions.
Several families of experimental networks have been prepared from fibres pulped from
different wood species with a range of areal densities and porosities. The mean pore diameter
of these networks has been measured using fluid porometry and analysis of the data arising
suggests that the simplified model accounts for more than 70 % of the variability observed.
We conclude that the pore size distribution of stochastic fibrous materials such as paper is
primarily influenced by the out-of-plane dimensions of voids.
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Determination of the pore size distribution in paper