We present an analysis of areal mass density correlations in paper. Using β radiography, the local mass density of laboratory paper sheets has been measured. The real space density autocorrelation function calculated from the data reveals a nontrivial power law type of correlations with the decay exponent being roughly independent of the basis weight of the sheets. However, for low densities we find that correlations may extend at least an order of magnitude beyond the fiber length, whereas for heavier paper they quickly die out.Peer reviewe

Ala-Nissilä, Tapio

Alava, M. J.

Provatas, N.

Aaltodoc

This is an electronic reprint of the original article.
This reprint may differ from the original in pagination and typographic detail.
Author(s): Provatas, N. & Alava, M. J. & Ala-Nissilä, Tapio
Title: Density correlations in paper
Year: 1996
Version: Final published version
Please cite the original version:
Provatas, N. & Alava, M. J. & Ala-Nissilä, Tapio. 1996. Density correlations in paper.
Physical Review E. Volume 54, Issue 1. P. R36-R38. ISSN 1539-3755 (printed). DOI:
10.1103/physreve.54.r36.
Rights: © 1996 American Physical Society (APS). http://www.aps.org
All material supplied via Aaltodoc is protected by copyright and other intellectual property rights, and
duplication or sale of all or part of any of the repository collections is not permitted, except that material may
be duplicated by you for your research use or educational purposes in electronic or print form. You must
obtain permission for any other use. Electronic or print copies may not be offered, whether for sale or
otherwise to anyone who is not an authorised user.
Powered by TCPDF (www.tcpdf.org)
Density correlations in paper
N. Provatas,1 M. J. Alava,2 and T. Ala-Nissila1,3
1Research Institute for Theoretical Physics, University of Helsinki, P.O. Box 9, FIN–00014 University of Helsinki, Helsinki, Finland
2Laboratory of Physics, Helsinki University of Technology, Otakaari 1M, FIN–02150 Espoo, Finland
and Department of Physics and Astronomy, Michigan State University, East Lansing, Michigan 48824-1116
3Department of Physics, Brown University, Providence, Rhode Island
and Laboratory of Physics, Tampere University of Technology, Tampere, Finland
~Received 25 March 1996!
We present an analysis of areal mass density correlations in paper. Using b radiography, the local mass
density of laboratory paper sheets has been measured. The real space density autocorrelation function calcu-
lated from the data reveals a nontrivial power law type of correlations with the decay exponent being roughly
independent of the basis weight of the sheets. However, for low densities we find that correlations may extend
at least an order of magnitude beyond the fiber length, whereas for heavier paper they quickly die out.
@S1063-651X~96!51307-3#
PACS number~s!: 81.15.Lm, 61.43.Hv, 81.05.Rm, 82.70.Kj
The sedimentation of particles from a suspension poses a
formidable theoretical problem on a microscopic level @1#.
There are challenging theoretical problems associated with
the fundamentals of two-phase flow, and in particular with
hydrodynamic interactions arising between constituent par-
ticles @2–4#. These interactions have been shown to strongly
influence the structures formed by sedimentation @2,4#, as
compared to simple ballistic deposition models @5#.
A striking example of a material formed by such a pre-
cipitation process is ordinary paper @6#. Its existence begins
as a suspension of fibers in water, and it ends up forming a
disordered fiber network. The structure of paper is an inter-
esting problem with obvious commercial interest. The distri-
bution of the local mass density in a paper sheet reflects the
complicated physical processes during paper making and has
been a long-standing subject of study @6#. Traditionally, its
wavelength power spectrum has been used to characterize
the size of ‘‘flocs,’’ or formation in paper @6–9#.
In this paper, we address the fundamental question of how
random paper is by studying real space correlations in the
mass distribution of laboratory-made paper sheets. In par-
ticular, we investigate the existence and range of nontrivial
power law correlations. This is motivated by two reasons.
First, fibers in the suspension tend to floc together, giving
rise to structures often clearly visible even to the eye @10,11#
~see also Fig. 1!. Flocculation in such suspensions may take
place because of chemical or mechanical interactions be-
tween the fibers. A more complex element influencing floc-
culation is the hydrodynamics of the suspension. At present,
there is no microscopic understanding of this since even for
a static suspension of simple particles it is a formidable theo-
retical problem @4#. Second, power law correlations have im-
portant implications for recent experiments on front propa-
gation ~burning and imbibition!, where paper has been
assumed to be a uniformly random medium @12#. Our results
indeed show the existence correlations well approximated by
a power law type function that may persist up to about an
order of magnitude beyond the fiber length. We also find that
within the power law regime, the decay exponent is indepen-
dent of areal mass density.
Our analysis is based on hand-made laboratory paper
sheets from typical chemical pulp of industrial origin. The
pulp is first diluted in a suspension of about 0.12% by weight
of fibers. The paper making procedure is a standard one,
where the suspension is poured into a sheet mold. At this
stage, flocculation of fibers may take place, with the degree
of it depending on how long the suspension is allowed to
settle @13#. The liquid is then drained out by gravity through
the bottom of the mold, where a wire mesh collects the de-
posited fibers. The actual paper sheet thus forms on the
mesh. After formation, the sheets were wet pressed at 490
kPa and then dried in a drum. Three independent sets of
paper sheets, A , B , and C were produced, and for sets A and
C the pulp was fractionated @14#. Sheets of size of 16316
cm2 were prepared at several different final basis weights,
ranging from 6.6 to 120 g/m2.
A map of the local mass density was obtained using b
radiography @8,15#. Figure 1 shows an example of a radio-
graph from a sheet of basis weight of 80 g/m2. The radio-
FIG. 1. A 14317 cm2 sample radiograph of a paper sheet with
basis weight 80 g/m2. The gray scale shown on the side has ten
shades, starting from zero basis weight and increasing with steps of
14.5 g/cm2.
PHYSICAL REVIEW E JULY 1996VOLUME 54, NUMBER 1
541063-651X/96/54~1!/36~3!/$10.00 R36 © 1996 The American Physical Society
graphs typically have an imaging accuracy of about 0.2 mm
~comparable to the thickness of the sheet @15#!. Each set of
digitized data is roughly 90031600 pixels with a pixel size
of about 0.09 mm. As a check, the global mass distribution
was confirmed to be a Gaussian.
From the data, we calculated the two-point density corre-
lation function as
G~rW !5^@m~xW !2m¯ #@m~xW1rW !2m¯ #&, ~1!
where m(xW ) is the local mass density, m¯ its average and
^ & an average over each sheet. As m(xW ) can in this case be
assumed to be isotropic, G is symmetric @16#.
In Fig. 2 we show a log-log plot of Eq. ~1! for four dif-
ferent paper sheets whose basis weights increase from top to
bottom. For each basis weight studied, we tested three or
four independent samples and found consistent results. In all
cases studied, for r,l , G(r) follows to a good degree of
approximation the form
G~r !.r2a. ~2!
In Fig. 3 we plot a vs basis weight for set A , which spans
a large range of thicknesses. We find that a is essentially
independent of the basis weight, with an average value of
a50.3760.07. Data sets B and C show the same result,
with a slightly larger value of a . For comparison, we also
examined two commercial paper grades of newsprint type
finding similar results. The main difference was the anisot-
ropy of commercial paper. Due to the manufacturing process
there is a difference between the so called machine and cross
directions in the final paper that leads to different power law
exponents in the two directions @17#.
For r.l , we find that the range of G(r) depends on the
basis weight. In particular, the 7.2 g/m2 sheets exhibit power
law behavior for approximately 14 fiber lengths, whereas for
the 118.9 g/m2 sheets, no power law is seen beyond r'l .
Similar long range correlations are also observed for basis
weights up to about 26–30 g/m2 from set A . Analysis of the
curves using, e.g., running exponents from dln(G)/dln(r)
shows that for low density cases, the power law behavior is
rather consistent across and beyond the fiber length ~see Fig.
2!. For data sets B and C , all data obtained were of weight
greater than 40 g/m2 and thus showed no discernible power
law correlations beyond r'l . These results are qualitatively
consistent with other experiments in the sense that the struc-
ture of paper is known to become more homogeneous with
increasing basis weight @18#.
For completeness, we also analyzed the wavelength power
spectrum k2P(k) of our data, where P(k) is the Fourier
transform of G(r). This measure is commonly used in paper
physics to find the dominating wavelength component in the
density distribution @8#. While P(k) contains information
equivalent to G(r), the wavelength power spectrum for the
present case does not reveal the initial power law behavior
clearly. This is because of the additional k2 factor in its
definition, as well as cutoffs due to the finite size of the
sample and the finite range of the power law in real space.
Indeed, the wavelength power spectrum even for a two-
dimensional ~2D! uniformly random fiber network is just a
smooth function that peaks at an uninteresting value of
0.05 times the fiber length @6#.
The reason for nontrivial correlations must stem from the
underlying physical processes in paper making, such as floc-
culation of fibers in the suspension and sedimentation, as
explained in the Introduction. In particular, the persistence of
a constant power law with a'0.4 for all densities studied
here is quite interesting. This is because for a simple, uni-
formly random 2D fiber network, it can be shown that
a51 up to the fiber length, and no correlations exist beyond
this @19#. The nontrivial experimental value of a may be due
to enhanced flocculation in the suspension, or to enhanced
hydrodynamic activity in the formation phase. However, try-
ing to microscopically model the dynamics of fibers in a
suspension even for hand-made paper sheets is a prohibi-
tively difficult problem.
FIG. 2. A log-log plot of the correlation function G(r) for four
paper sheets of basis weights 7.2 g/m2, 70.4 g/m2, 86.3 g/m2, and
118.9 g/m2 ~from top to bottom!. The vertical line indicates the
average fiber length lA52.7 mm. Data are from set A , and the
curves have been shifted for clarity. For the lowest densities, the
power law extends up to about 14 times the fiber length.
FIG. 3. The power law exponent a vs basis weight of the paper
sheets for set A .
54 R37DENSITY CORRELATIONS IN PAPER
Recently, we have carried out an extensive study @19# of a
simple 2D fiber deposition model with the rule that if a de-
posited fiber overlaps with another fiber it is always ac-
cepted. However, if it lands on empty space it is accepted
only with a probability p , where 0,p,1. We find that this
model gives a constant nontrivial power law exponent
a,1 for all densities only if p is varied in such a way that
clustering is enhanced for high densities. However, the
model can at best only produce this power law up to about
the length of a typical cluster in the fiber network. A plau-
sible mechanism for obtaining long-range correlations at low
areal densities is the introduction of effective cluster-cluster
interactions. Such interactions could arise from hydrodynam-
ics of the suspension. In Ref. @20# it has been shown that
cluster-cluster interactions can indeed produce power law
type density correlations well beyond the range of the cluster
sizes. This is in agreement with the idea of enhanced floccu-
lation within the suspension.
In conclusion, we have investigated the two-point density
correlations in laboratory made paper sheets. Our results
demonstrate the existence of nontrivial power law correla-
tions. The power law exponent seems to be rather indepen-
dent of paper density, but the range of the correlations is not.
For basis weights greater than about 40 g/m2, power law
correlations only exist for distances of the order of the fiber
length, while for smaller basis weights they can persist for
distances more than ten times the fiber length.
The existence of nontrivial density correlations in paper
sheets is of particular importance for experiments in which
paper has been assumed to be uniformly random @12#. These
include kinetic roughening of burning fronts @21# and wet-
ting studies of directed percolation. Our results suggest that
paper is not as random as assumed. Power law correlations
extending to the centimeter range may induce undesirable
effects such as crossover behavior.
The paper-making process is a complicated problem. We
nevertheless hope to have demonstrated that the behavior of
the correlation functions gives important information about
paper structure and sedimentation in general. Many interest-
ing questions still remain, such as the true microscopic origin
of the power law correlations and the effect of the hydrody-
namics of the suspension on the paper structure.
We wish to thank the Paper Science Centre of the Finnish
Pulp and Paper Institute, and Lauri Salminen in particular for
technical assistance. This work has been supported by the
Academy of Finland through the MATRA program, and
M.J.A. in part by DOE Grant No. DE-FG02-090-ER45418.
@1# S. Kim and S. J. Karrila, Microhydrodynamics: Principles and
Selected Applications ~Butterworth-Heinemann, Boston,
1991!.
@2# I. Pagonabarraga and M. Rubi, Phys. Rev. Lett. 73, 114
~1994!.
@3# H. Hayakawa and K. Ichiki, Phys. Rev. E 51, R3815 ~1995!.
@4# S. Schwarzer, Phys. Rev. E 52, 6461 ~1995!.
@5# J. W. Evans, Rev. Mod. Phys. 65, 1281 ~1993!.
@6# M. Deng and C. T. J. Dodson, Paper: An Engineered Stochas-
tic Structure ~Tappi Press, Atlanta, 1994!.
@7# B. Norman and D. Wahren, Sven. Papperstidn. 75, 807 ~1972!.
@8# L. Haugland, B. Norman, and D. Wahren, Sven. Papperstidn.
77, 362 ~1974!.
@9# I. M. Kajanto, A. Komppa, and R. K. Ritala, Nordic Pulp
Paper Res. J. 3, 219 ~1989!.
@10# R. J. Kerekes and C. J. Schell, J. Pulp Paper Sci. 18, J32
~1992!.
@11# R. H. Zhao and R. J. Kerekes, Tappi J. 76, 183 ~1993!.
@12# A.-L. Baraba´si and H. E. Stanley, Fractal Concepts in Surface
Growth ~Cambridge University Press, Cambridge, 1995!.
@13# The sheets have been produced according to the Nordic stan-
dard SCAN C–26.
@14# Fractionation involves repeated distillation to remove short fi-
bers and fine particles. The typical fiber length distribution is
roughly Gaussian with an average of lA5lC52.7 mm
(lB52.6 mm!. The width and thickness distributions are usu-
ally Gaussian ~see @6#! with a typical mean width v530–35
mm and thickness of 6–10 mm.
@15# B. Norman and D. Wahren, Sven. Papperstidn. 77, 397 ~1974!.
@16# The orientational distribution of fibers in commercial paper is
usually asymmetric because of the flow fields at the wet end of
industrial paper machines.
@17# Note that when commercial paper has been coated or contains
filler materials, we do not expect our conclusions to hold.
@18# P. Wrist, in Formation and Structure of Paper, Transactions of
the 2nd Fundamental Research Symposium, Oxford, 1961, ed-
ited by F. Bolam ~Technical Section BPBMA, London, 1962!,
p. 839; W. W. Sampson, J. McAlpin, H. W. Kropholler, and C.
T. J. Dodson, 81st Annual Meeting ~Technical Section CPPA,
1995!, Part B.
@19# N. Provatas, M. Haataja, E. Seppa¨la¨, S. Majaniemi, M. Alava,
and T. Ala-Nissila ~unpublished!.
@20# N. Provatas, T. Ala-Nissila, and M. J. Alava, Phys. Rev. Lett.
75, 3556 ~1995!.
@21# J. Zhang, Y.-C. Zhang, P. Ahlstrom, and M. T. Levinsen,
Physica A 189, 383 ~1992!.
R38 54N. PROVATAS, M. J. ALAVA, AND T. ALA-NISSILA


English

Density correlations in paper

Aaltodoc Publication Archive

This is an electronic reprint of the original article.This reprint may differ from the original in pagination and typographic detail.Author(s): Provatas, N. & Alava, M. J. & Ala-Nissilä, TapioTitle: Density correlations in paperYear: 1996Version: Final published versionPlease cite the original version:Provatas, N. & Alava, M. J. & Ala-Nissilä, Tapio. 1996. Density correlations in paper.Physical Review E. Volume 54, Issue 1. P. R36-R38. ISSN 1539-3755 (printed). DOI:10.1103/physreve.54.r36.Rights: © 1996 American Physical Society (APS). http://www.aps.orgAll material supplied via Aaltodoc is protected by copyright and other intellectual property rights, andduplication or sale of all or part of any of the repository collections is not permitted, except that material maybe duplicated by you for your research use or educational purposes in electronic or print form. You mustobtain permission for any other use. Electronic or print copies may not be offered, whether for sale orotherwise to anyone who is not an authorised user.Powered by TCPDF (www.tcpdf.org)Density correlations in paperN. Provatas,1 M. J. Alava,2 and T. Ala-Nissila1,31Research Institute for Theoretical Physics, University of Helsinki, P.O. Box 9, FIN–00014 University of Helsinki, Helsinki, Finland2Laboratory of Physics, Helsinki University of Technology, Otakaari 1M, FIN–02150 Espoo, Finlandand Department of Physics and Astronomy, Michigan State University, East Lansing, Michigan 48824-11163Department of Physics, Brown University, Providence, Rhode Islandand Laboratory of Physics, Tampere University of Technology, Tampere, Finland~Received 25 March 1996!We present an analysis of areal mass density correlations in paper. Using b radiography, the local massdensity of laboratory paper sheets has been measured. The real space density autocorrelation function calcu-lated from the data reveals a nontrivial power law type of correlations with the decay exponent being roughlyindependent of the basis weight of the sheets. However, for low densities we find that correlations may extendat least an order of magnitude beyond the fiber length, whereas for heavier paper they quickly die out.@S1063-651X~96!51307-3#PACS number~s!: 81.15.Lm, 61.43.Hv, 81.05.Rm, 82.70.KjThe sedimentation of particles from a suspension poses aformidable theoretical problem on a microscopic level @1#.There are challenging theoretical problems associated withthe fundamentals of two-phase flow, and in particular withhydrodynamic interactions arising between constituent par-ticles @2–4#. These interactions have been shown to stronglyinfluence the structures formed by sedimentation @2,4#, ascompared to simple ballistic deposition models @5#.A striking example of a material formed by such a pre-cipitation process is ordinary paper @6#. Its existence beginsas a suspension of fibers in water, and it ends up forming adisordered fiber network. The structure of paper is an inter-esting problem with obvious commercial interest. The distri-bution of the local mass density in a paper sheet reflects thecomplicated physical processes during paper making and hasbeen a long-standing subject of study @6#. Traditionally, itswavelength power spectrum has been used to characterizethe size of ‘‘flocs,’’ or formation in paper @6–9#.In this paper, we address the fundamental question of howrandom paper is by studying real space correlations in themass distribution of laboratory-made paper sheets. In par-ticular, we investigate the existence and range of nontrivialpower law correlations. This is motivated by two reasons.First, fibers in the suspension tend to floc together, givingrise to structures often clearly visible even to the eye @10,11#~see also Fig. 1!. Flocculation in such suspensions may takeplace because of chemical or mechanical interactions be-tween the fibers. A more complex element influencing floc-culation is the hydrodynamics of the suspension. At present,there is no microscopic understanding of this since even fora static suspension of simple particles it is a formidable theo-retical problem @4#. Second, power law correlations have im-portant implications for recent experiments on front propa-gation ~burning and imbibition!, where paper has beenassumed to be a uniformly random medium @12#. Our resultsindeed show the existence correlations well approximated bya power law type function that may persist up to about anorder of magnitude beyond the fiber length. We also find thatwithin the power law regime, the decay exponent is indepen-dent of areal mass density.Our analysis is based on hand-made laboratory papersheets from typical chemical pulp of industrial origin. Thepulp is first diluted in a suspension of about 0.12% by weightof fibers. The paper making procedure is a standard one,where the suspension is poured into a sheet mold. At thisstage, flocculation of fibers may take place, with the degreeof it depending on how long the suspension is allowed tosettle @13#. The liquid is then drained out by gravity throughthe bottom of the mold, where a wire mesh collects the de-posited fibers. The actual paper sheet thus forms on themesh. After formation, the sheets were wet pressed at 490kPa and then dried in a drum. Three independent sets ofpaper sheets, A , B , and C were produced, and for sets A andC the pulp was fractionated @14#. Sheets of size of 16316cm2 were prepared at several different final basis weights,ranging from 6.6 to 120 g/m2.A map of the local mass density was obtained using bradiography @8,15#. Figure 1 shows an example of a radio-graph from a sheet of basis weight of 80 g/m2. The radio-FIG. 1. A 14317 cm2 sample radiograph of a paper sheet withbasis weight 80 g/m2. The gray scale shown on the side has tenshades, starting from zero basis weight and increasing with steps of14.5 g/cm2.PHYSICAL REVIEW E JULY 1996VOLUME 54, NUMBER 1541063-651X/96/54~1!/36~3!/$10.00 R36 © 1996 The American Physical Societygraphs typically have an imaging accuracy of about 0.2 mm~comparable to the thickness of the sheet @15#!. Each set ofdigitized data is roughly 90031600 pixels with a pixel sizeof about 0.09 mm. As a check, the global mass distributionwas confirmed to be a Gaussian.From the data, we calculated the two-point density corre-lation function asG~rW !5^@m~xW !2m¯ #@m~xW1rW !2m¯ #&, ~1!where m(xW ) is the local mass density, m¯ its average and^ & an average over each sheet. As m(xW ) can in this case beassumed to be isotropic, G is symmetric @16#.In Fig. 2 we show a log-log plot of Eq. ~1! for four dif-ferent paper sheets whose basis weights increase from top tobottom. For each basis weight studied, we tested three orfour independent samples and found consistent results. In allcases studied, for r,l , G(r) follows to a good degree ofapproximation the formG~r !.r2a. ~2!In Fig. 3 we plot a vs basis weight for set A , which spansa large range of thicknesses. We find that a is essentiallyindependent of the basis weight, with an average value ofa50.3760.07. Data sets B and C show the same result,with a slightly larger value of a . For comparison, we alsoexamined two commercial paper grades of newsprint typefinding similar results. The main difference was the anisot-ropy of commercial paper. Due to the manufacturing processthere is a difference between the so called machine and crossdirections in the final paper that leads to different power lawexponents in the two directions @17#.For r.l , we find that the range of G(r) depends on thebasis weight. In particular, the 7.2 g/m2 sheets exhibit powerlaw behavior for approximately 14 fiber lengths, whereas forthe 118.9 g/m2 sheets, no power law is seen beyond r'l .Similar long range correlations are also observed for basisweights up to about 26–30 g/m2 from set A . Analysis of thecurves using, e.g., running exponents from dln(G)/dln(r)shows that for low density cases, the power law behavior israther consistent across and beyond the fiber length ~see Fig.2!. For data sets B and C , all data obtained were of weightgreater than 40 g/m2 and thus showed no discernible powerlaw correlations beyond r'l . These results are qualitativelyconsistent with other experiments in the sense that the struc-ture of paper is known to become more homogeneous withincreasing basis weight @18#.For completeness, we also analyzed the wavelength powerspectrum k2P(k) of our data, where P(k) is the Fouriertransform of G(r). This measure is commonly used in paperphysics to find the dominating wavelength component in thedensity distribution @8#. While P(k) contains informationequivalent to G(r), the wavelength power spectrum for thepresent case does not reveal the initial power law behaviorclearly. This is because of the additional k2 factor in itsdefinition, as well as cutoffs due to the finite size of thesample and the finite range of the power law in real space.Indeed, the wavelength power spectrum even for a two-dimensional ~2D! uniformly random fiber network is just asmooth function that peaks at an uninteresting value of0.05 times the fiber length @6#.The reason for nontrivial correlations must stem from theunderlying physical processes in paper making, such as floc-culation of fibers in the suspension and sedimentation, asexplained in the Introduction. In particular, the persistence ofa constant power law with a'0.4 for all densities studiedhere is quite interesting. This is because for a simple, uni-formly random 2D fiber network, it can be shown thata51 up to the fiber length, and no correlations exist beyondthis @19#. The nontrivial experimental value of a may be dueto enhanced flocculation in the suspension, or to enhancedhydrodynamic activity in the formation phase. However, try-ing to microscopically model the dynamics of fibers in asuspension even for hand-made paper sheets is a prohibi-tively difficult problem.FIG. 2. A log-log plot of the correlation function G(r) for fourpaper sheets of basis weights 7.2 g/m2, 70.4 g/m2, 86.3 g/m2, and118.9 g/m2 ~from top to bottom!. The vertical line indicates theaverage fiber length lA52.7 mm. Data are from set A , and thecurves have been shifted for clarity. For the lowest densities, thepower law extends up to about 14 times the fiber length.FIG. 3. The power law exponent a vs basis weight of the papersheets for set A .54 R37DENSITY CORRELATIONS IN PAPERRecently, we have carried out an extensive study @19# of asimple 2D fiber deposition model with the rule that if a de-posited fiber overlaps with another fiber it is always ac-cepted. However, if it lands on empty space it is acceptedonly with a probability p , where 0,p,1. We find that thismodel gives a constant nontrivial power law exponenta,1 for all densities only if p is varied in such a way thatclustering is enhanced for high densities. However, themodel can at best only produce this power law up to aboutthe length of a typical cluster in the fiber network. A plau-sible mechanism for obtaining long-range correlations at lowareal densities is the introduction of effective cluster-clusterinteractions. Such interactions could arise from hydrodynam-ics of the suspension. In Ref. @20# it has been shown thatcluster-cluster interactions can indeed produce power lawtype density correlations well beyond the range of the clustersizes. This is in agreement with the idea of enhanced floccu-lation within the suspension.In conclusion, we have investigated the two-point densitycorrelations in laboratory made paper sheets. Our resultsdemonstrate the existence of nontrivial power law correla-tions. The power law exponent seems to be rather indepen-dent of paper density, but the range of the correlations is not.For basis weights greater than about 40 g/m2, power lawcorrelations only exist for distances of the order of the fiberlength, while for smaller basis weights they can persist fordistances more than ten times the fiber length.The existence of nontrivial density correlations in papersheets is of particular importance for experiments in whichpaper has been assumed to be uniformly random @12#. Theseinclude kinetic roughening of burning fronts @21# and wet-ting studies of directed percolation. Our results suggest thatpaper is not as random as assumed. Power law correlationsextending to the centimeter range may induce undesirableeffects such as crossover behavior.The paper-making process is a complicated problem. Wenevertheless hope to have demonstrated that the behavior ofthe correlation functions gives important information aboutpaper structure and sedimentation in general. Many interest-ing questions still remain, such as the true microscopic originof the power law correlations and the effect of the hydrody-namics of the suspension on the paper structure.We wish to thank the Paper Science Centre of the FinnishPulp and Paper Institute, and Lauri Salminen in particular fortechnical assistance. This work has been supported by theAcademy of Finland through the MATRA program, andM.J.A. in part by DOE Grant No. DE-FG02-090-ER45418.@1# S. Kim and S. J. Karrila, Microhydrodynamics: Principles andSelected Applications ~Butterworth-Heinemann, Boston,1991!.@2# I. Pagonabarraga and M. Rubi, Phys. Rev. Lett. 73, 114~1994!.@3# H. Hayakawa and K. Ichiki, Phys. Rev. E 51, R3815 ~1995!.@4# S. Schwarzer, Phys. Rev. E 52, 6461 ~1995!.@5# J. W. Evans, Rev. Mod. Phys. 65, 1281 ~1993!.@6# M. Deng and C. T. J. Dodson, Paper: An Engineered Stochas-tic Structure ~Tappi Press, Atlanta, 1994!.@7# B. Norman and D. Wahren, Sven. Papperstidn. 75, 807 ~1972!.@8# L. Haugland, B. Norman, and D. Wahren, Sven. Papperstidn.77, 362 ~1974!.@9# I. M. Kajanto, A. Komppa, and R. K. Ritala, Nordic PulpPaper Res. J. 3, 219 ~1989!.@10# R. J. Kerekes and C. J. Schell, J. Pulp Paper Sci. 18, J32~1992!.@11# R. H. Zhao and R. J. Kerekes, Tappi J. 76, 183 ~1993!.@12# A.-L. Baraba´si and H. E. Stanley, Fractal Concepts in SurfaceGrowth ~Cambridge University Press, Cambridge, 1995!.@13# The sheets have been produced according to the Nordic stan-dard SCAN C–26.@14# Fractionation involves repeated distillation to remove short fi-bers and fine particles. The typical fiber length distribution isroughly Gaussian with an average of lA5lC52.7 mm(lB52.6 mm!. The width and thickness distributions are usu-ally Gaussian ~see @6#! with a typical mean width v530–35mm and thickness of 6–10 mm.@15# B. Norman and D. Wahren, Sven. Papperstidn. 77, 397 ~1974!.@16# The orientational distribution of fibers in commercial paper isusually asymmetric because of the flow fields at the wet end ofindustrial paper machines.@17# Note that when commercial paper has been coated or containsfiller materials, we do not expect our conclusions to hold.@18# P. Wrist, in Formation and Structure of Paper, Transactions ofthe 2nd Fundamental Research Symposium, Oxford, 1961, ed-ited by F. Bolam ~Technical Section BPBMA, London, 1962!,p. 839; W. W. Sampson, J. McAlpin, H. W. Kropholler, and C.T. J. Dodson, 81st Annual Meeting ~Technical Section CPPA,1995!, Part B.@19# N. Provatas, M. Haataja, E. Seppa¨la¨, S. Majaniemi, M. Alava,and T. Ala-Nissila ~unpublished!.@20# N. Provatas, T. Ala-Nissila, and M. J. Alava, Phys. Rev. Lett.75, 3556 ~1995!.@21# J. Zhang, Y.-C. Zhang, P. Ahlstrom, and M. T. Levinsen,Physica A 189, 383 ~1992!.R38 54N. PROVATAS, M. J. ALAVA, AND T. ALA-NISSILA

N. Provatas

M. J. Alava

T. Ala-Nissila

Crossref

https://aaltodoc.aalto.fi/bitstream/handle/123456789/15808/A1_provatas_n._1996.pdf?sequence=1&isAllowed=y

Density correlations in paper

Abstract

Similar works

Full text

Available Versions

Aaltodoc

Aaltodoc Publication Archive

Crossref