The transition in random fiber networks from two-dimensional to asymptotically three-dimensional planar structure with increasing coverage c¯ (mean fiber length per unit area) is studied with a deposition model. Network geometry depends on the scale-free product of fiber length and c¯ at low c¯, and on another scale-free product of flexibility and the width-to-thickness ratio of fibers at high c¯. The structure becomes three-dimensional or decouples from the substrate faster when fibers are stiffer. Roughness of the free surface decreases with increasing fiber flexibility.Peer reviewe

Alava, Mikko J.

Niskanen, K. J.

arXiv

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This is an electronic reprint of the original article.
This reprint may differ from the original in pagination and typographic detail.
Author(s): Niskanen, K. J. & Alava, Mikko J.
Title: Planar Random Networks with Flexible Fibers
Year: 1994
Version: Final published version
Please cite the original version:
Niskanen, K. J. & Alava, Mikko J. 1994. Planar Random Networks with Flexible Fibers.
Physical Review Letters. Volume 73, Issue 25. 3475-3478. ISSN 0031-9007 (printed).
DOI: 10.1103/physrevlett.73.3475.
Rights: © 1994 American Physical Society (APS). This is the accepted version of the following article: Niskanen, K.
J. & Alava, Mikko J. 1994. Planar Random Networks with Flexible Fibers. Physical Review Letters. Volume
73, Issue 25. 3475-3478. ISSN 0031-9007 (printed). DOI: 10.1103/physrevlett.73.3475, which has been
published in final form at http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.73.3475.
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VOLUME 73, NUMBER 25 PH YS ICAL REVIEW LETTERS 19 DECEMBER 1994
Planar Random Networks with Flexible Fibers
K.J. Niskanen' and M. J. Alava
'KCL Paper Science Centre, P.O. Box 70, FIN-02151 Espoo, Finland*
and U S D .A. F.or. est Service, Forest Products Laboratory, One Gifford Pinchot Drive, Madison, Wisconsin 53705 23-98
'Helsinki University of Technology, Laboratory of Physics, FIN 0215-0 Espoo, Finland
(Received 2 May 1994; revised manuscript received 23 September 1994)
The transition in random fiber networks from two-dimensional to asymptotically three-dimensional
planar structure with increasing coverage c (mean fiber length per unit area) is studied with a deposition
model. Network geometry depends on the scale-free product of fiber length and c at low c, and on
another scale-free product of flexibility and the width-to-thickness ratio of fibers at high c. The structure
becomes three-dimensional or decouples from the substrate faster when fibers are stiffer. Roughness of
the free surface decreases with increasing fiber flexibility.
PACS numbers: 81.15.Lm, 64.60.—i, 81.30.Dz
The mechanical properties and structure of two-
dimensional random systems have recently been in-
vestigated intensively by computational and theoretical
methods [1,2], but experimental studies are still few in
number. However, planar fibrous materials, such as poly-
mer films, short-fiber composites, paper, and nonwovens,
provide promising possibilities for experimental work
[3], in addition to being technologically interesting. The
fibers in such materials are usually much longer than
the sheet or film is thick, and they are almost perfectly
aligned in the sheet plane [4]. The structure is therefore
planar, or "layered, " but no well-defined layers can be
separated because of the "undulations" of the fibers in
the thickness direction. If coverage (i.e., the mean fiber
length per unit area) c is low, the network is almost two
dimensional, and its properties must be governed by c.
At high c the three-dimensional "bulk" density replaces
c. However, e.g. , mechanical properties are in both
cases governed by the average number of connections
(interfiber bonds) per unit fiber length or "coordination
number" n [5]. Since the network is planar, we expect
that its mechanical properties are independent of c if
compared at a fixed n.
The statistical geometry of strictly two-dimensional
random fiber networks is well understood [6], but ana-
lytical models [7,8] for fiber networks with finite thick-
ness have so far not dealt accurately with the fact that the
bonding of fibers across intervening "layers" is controlled
by the random pore structure. In this Letter we employ a
simple computer simulation model to study the geometry
of random networks of flexible fibers that are deposited on
a two-dimensional flat substrate [growth in (2 + 1)D].We
concentrate on the coverage-driven transition from 2D to
3D network geometry with increasing coverage and show
that surface roughness [9] depends on fiber stiffness. The
simulation model allows the comparison of future experi-
ments with theoretical or numerical studies [10,11,12].
We start with a two-dimensional square lattice of linear
size L = 10, . . . , 1000. Periodic boundary conditions are
applied, except in the percolation study. The size L
affects properties only close to the percolation threshold
and, for surface roughness, at very high coverages. The
fibers are initially straight beams of unit width and
thickness, wf = tf = 1, and emphasis is given to high
fiber lengths lf » 1 [13]. Fibers are positioned and
aligned in the two principal directions at random [14] in
such a way that the local coverage c (number of fibers
covering the cell) is always an integer. The discretization
should not, of course, affect generic properties.
A "bending flexibility" Tf gives the largest allowed
vertical deflection of the fibers from one lattice cell to
the next (cf. Fig. 1). A high Tf gives a dense network, a
low Tf a sparse one. For example, the so-called wet fiber
flexibility (WFF) [15] of paper-making fibers is related
to Tf through Tf = [Cwf(WFF)]'l, where C depends on
experimental details. Measured values of the wet fiber
flexibility [16] and known dimensions of paper-making
fibers yield wfTf/tf ~ 1. In the simulations fibers are
placed down independently, one after another, as if they
were sedimenting from a dilute suspension. Each fiber
is kept straight and parallel to the substrate until the first
contact is made with the underlying network. Thereafter
the fiber is deformed so as to lie as low as possible-
not below the substrate —still obeying the deflection
constraint Tf. The finite width of real fibers implies that
they tend to follow a staircase pattern like that given by
the model (cf. Fig. 1). The cumulative buildup of the
network is a more serious departure from reality as it
does not distinguish between densification mechanisms
l~
I
FIG. 1. Two crossing fibers on the square lattice. The height
of the "steps" is = Tf = 1/3.
0031-9007/94/73(25)/3475(4)$06. 00 1994 The American Physical Society 3475
VOLUME 73, NUMBER 25 PH YS ICAL REVIEW LETTERS 19 DECEMBER 1994
such as hydrodynamic drag forces and the pressing of a
filtrated fiber mat. The construction results in different
top and bottom surfaces, but only the free top surface is
studied here.
At low mean coverage c the in-plane properties of a
planar network should be controlled by c/c, . , where 2,
is the percolation threshold [17,18]. We determined c,
numerically using the following rules: (1) the network
is connected when there exists a cluster of fibers that is
connected to at least one cell on two opposite boundaries
of the system; (2) two cells are connected by a fiber if
the fiber covers them both; and (3) two fibers covering
a cell are connected if the top surface of the lower fiber
is in contact with the bottom surface of the upper fiber.
In the case of zero fiber width, infinite flexibility (or
zero fiber thickness), and continuous spatial symmetry.
the percolation density is [19] c,. = 5.71//f. This should
also hold in our case despite the discrete orientations of
fibers due to the excluded area principle [20,21]. Indeed,
the computed threshold (cf. Fig. 2) is c,. = 5.7//& when
the fibers are long and flexible (/f ~ 30—40). The results
in Fig. 2 are independent of L at least for L ~ 800.
When the fibers are short, the nonzero width
lowers the percolation threshold at high T~,. we find c, =
4//f for /f = 7 and Tf ) 1, and c,. = 3.5//f for /f 3
and T~ ~ 5. A random walk argument suggests that in
the opposite limit of T~ 0 the radius r of a cluster
with N fibers is r —/f ~N and therefore the thickness of a
percolating cluster is -lf(L//f)2 )) L Thus the n. etwork
becomes three dimensional, and c,. no longer characterizes
its structure appropriately. Since the maximum vertical
deflection per unit length of a fiber is T~, crossover
between flexible and stiff fiber behavior should occur at
Tf ff //f our data 'suggest T& = 4ff /// with tf —1.
When coverage increases beyond the percolation
threshold, the number of bonds per fiber and, likewise,
the mechanical rigidity of the network increase [17,18].
We define the coordination number R as the average
bonded surface area of a fiber, divided by 2/fwf [5].
Figure 3(a) shows how n first increases with c and then
becomes asymptotically constant. Obviously the latter
12
10
lO
4
Q, l
FIG. 2. The percolation threshold c,, multiplied with the fiber
length lf against Tf for fiber lengths lf = 3 {{)),5 {x),7 ( ),
1 I {*},15 {A), and 21 (+); system size L = 100.
0,8-
0,6 I-
0,4-
*
I
F00
0,8
0,6
0,4 -,''
(b)
0
FIG. 3. Coordination number A against coverage { (a); and
against fiber flexibility T&- for ~ ~ (b). In (a) T, = 1 and
2 (filled and open symbols, respectively) and solid line gives
Eq. (1). In (b) solid line corresponds to Eq. (2) and dashed
line to F~ = T, . System size L = 1000. fiber length tt =—21.
happens when the substrate has no effect on n anymore.
The network has then become three dimensional, and
every new layer has the same structure as the preceding
one. However, in the case of T~ ~ this never happens,
and the coordination number is that of a two-dimensional
network at all coverages, given by [6]
)i =- l —[1 —exp( —( )]/(. ( I )
[cf. Fig. 3(a)]. The crossover from 2D to 3D can be
roughly located by observing that at low coverages there
are relatively few pores inside the network. Thus height
differences between neighboring cells are of the order
—J~/w/ since coverage is Poisson distributed [6]. Pores
start to develop inside the network when T, /l, —+2/w, .
Equation (1) then yields
A = I —(I —exp[ —(Tf n, /r() ])/{T/ vi//t() (2)
for the asymptotic 3D structure. The computed values
agree reasonably with Eq. (2) for large T, as Fig. 3(b)
shows (for w/ = t, = 1). For T& && I we can estimate
that on the average each fiber (with l& &) w f ) oil tile
surface of the network prevents new fibers from mak-
ing contact with others on /ftf/Tf sites, while bonding
with the fiber in question is possible in ~~ I~ cells. Hence
6 w f /f /(8'f /f + /fff /Tf ) (Tf 8'f /if) at high cover-
ages [dashed line in Fig. 3(b)j. The simulation results
in Fig. 3 were found to be independent of the fiber length
at least for It- ~ 5. The reproducible discontinuities at
Tf = 2 and 4 in Fig. 3(b) are artifacts of the discrete
model.
VOLUME 73, NUMBER 25 PH YS ICAL REVIEW LETTERS 19 DEcEMBER 1994
which follows as a generalization of Eq. (1).
The roughness o., of the free surface can be defined as
the standard deviation of the local surface height t, the
distance between the free surface and substrate. Through
trial and error we found that the simulation results can be
expressed as
Crt ctf [A(wf Tf /tf) + B(c)] (4)
0,8
0,6
Ic
0,4
0,2
In contrast to n, the three-dimensional bulk density
p decreases with increasing coverage (provided that
surfaces pores are excluded), since at low coverage
there are initially no pores inside the sheet. In reality
thickness measurements have finite spatial resolution, and
usually they underestimate the true density. A "coarse-
grained" or "apparent" density was calculated from the
model to mimic real measurements. The results were
similar to the n (c) curves in Fig. 3(a). At high coverages
(thicknesses) even an apparent density is independent of
surface roughness and equal to the "true" density. Since
the coordination number is tedious to measure directly,
the 3D bulk density is sometimes used to estimate it.
Our results (Fig. 4) indicate that if c is held constant
and Tf is varied, then n is indeed roughly proportional
to p. This also holds for apparent densities. However,
if Tf is held constant and coverage is increased, the true
3D density decreases and hence cannot be proportional
to n (cf. discussion above), but an apparent density may
still be.
As the local coverage is Poisson distributed [6], the
standard deviation of local coverage is o., = vie. This
holds very well in our simulations. The flat substrate
affects the network structure at small c, but as c grows
the structure of new layers becomes stable, independent
of c. Then, for example, the standard deviation of the
number of pores per cell (p) becomes o„~P(Tf)Jc.
The amplitude P(Tf) decreases with increasing Tf. Areas
of high coverage have few pores, and therefore P ( 1.
The average number of pores is directly related to the
coordination number through
p+1 [1 —exp( —c)]
C
10
0,1
0,01
0 1 2 3 4 5
FIG. 5. Function A(Tf), Eq. (1), for coverages c = 10 ( ),
100 (+), 10' (E), and 104 (0). Fiber length lf = 5, system size
L = 100. Inset shows function B(c) on log-log scale.
(of course, we have data only for tf = wf = 1). The
functions A and B are shown in Fig. 5 for system size
L = 100, Tf + 5, and c ~ 10 . They are independent of
fiber length as long as lf ~ 5. It was easy to decompose
o.
, into A and B since at large Tf, o., turned out to be a
function of c only (or A 0). The prefactor c in Eq. (4)
is motivated by the Poissonian deposition process. As
coverage increases, B becomes smaller and eventually o.,
grows as -o., = ~c. This corresponds to a decoupling of
the surface from the flat substrate. The decoupling is fast
with stiff fibers (A large), since stiff fibers cannot "feel"
the substrate from high above. For the same reason the
case of stiff fibers seems to correspond to a low surface
tension in ordinary growth models [9] as no long range
correlations occur in the surface height. Indeed, making
Tf small gives a rough surface. In fact, our model is
similar to the Void model [9,22]. Details of the growth
behavior at high coverage, including the effects of system
size, are beyond the scope of this study and will be
presented elsewhere.
Summarizing, we have presented a computer simulation
study of planar random fiber networks. The model pro-
vides a connection between the properties of real planar
fiber networks and ideal two-dimensional networks. The
results can be represented in closed form only in a few
cases, but the model is simple enough so that the numeri-
cal solution can be easily used, e.g. , in the analysis of ex-
perimental results.
This study was financially supported by the Technology
Development Centre of Finland and by the Academy of
Finland which is gratefully acknowledged.
0 0,2 0,4 0,6 0,8 1
FIG. 4. Coordination number n against density p when fiber
flexibility Tf varies. Coverage c = 10.5 (E), 31.5 ( ), and94.5 (0). Fiber length lf = 21, system size L = 1000.
*Present and permanent address.
[1] Statistical Models for the Fracture of Disordered Media,
edited by H. J. Heiiiiiann and S. Roux (North-Holland,
Amsterdam, 1990).
[2] M. Sahimi, Physica (Amsterdam) 1$6A, 160 (1992).
[3] J. Kertesz, Physica (Amsterdam) 191A, 208 (1992).
3477
VOLUME 73, NUMBER 25 PHYSICAL REVIE% LETTERS 19 DEceMev~ 1994
[4J B. Radvan, in The Fundamenta/ Properties of P'aper
Related to its Uses, edited by F. Bolam (British Paper
and Board Ind. Fed. , London, 1976},pp. 137—147.
[5] For paper the quantity corresponding to the coordina-
tion number is the relative bonded area (RBA) of the
fibers. See O. J. Kallmes, H. Corte. and G. Bernier. Tappi
J. 46, 502 (1963}.
[6] H. Corte and O. J. Kallmes, Tappi J. 43, 737 (1960).
[7] O. J. Kallmes, H. Corte, and G. Bernier, Tappi J. 44, 519
(1961).
[8] J. Gorres, C. S. Sinclair, and A. Tallentire, Paperi ja Puu
71, 54 (1989};J. Gorres and P. Luner, J. Pulp Paper Sci.
18, J127 (1992}.
[9] J. Krug and H. Spohn, in Solids Far,~rom Equilibria&n:
Growth, Morphologv, Defects, edited by C. Godreche
(Cambridge Univ. Press, Cambridge. 1991), pp. 479—
582.
[10] M. J. Alava and R. K. Ritala, J. Phys. Condens. Matter 2.
6093 (1990);M. J. Alava and R. K. Ritala, Phys. Scr. T33,
155 (1990).
[11] J.A. Astrom and K. J. Niskanen, Europhys. Lett. 21, 557
(1993).
[1'2] J. Astrom, S. Saarinen, K. Niskanen, and J. Kurkijarvi.
J. Appl. Phys. 75, 2383 (1994).
[13] Fiber thickness is arbitrary in our study. Fr&r paper-
making fiber» 1&q't& ~ /«'» f = '0 —~0.
[14J We use the R250 random number &gener rtor.
Kirkpatrick and E. Stol l. J. Comput. Phy». 40. :1:
(1981).
[15] R. Steadman and P. Luner. in Papennafi»r; Rarr' Maieri
air. edited by V. Punton (Mechanical Engineering Publ.
Ltd. . London, 1985), pp. 311-338.
[16] L. Paavilainen, Paperi ja Puu 46, 689 (1993).
(17] S. Kirkpatrick, Rev. Mod. Phy». 45, 574 (1973 r.
[18] L. M. Schwartz, S. Feng, M. F. Thorpe. ,md P. N. Scn.
Phys. Rev. 32, 4607 (1985).
[19J G. E. Pike and C. H. Seager, Phys. Rev. B 10.
(1974).
[ 0] I. Balberg, Phys. Rev. B 31, 4053 (1985 r.
[21] K. Tobochnik, M. A. Dubson. M. L. Wilson. and M. F.
Thorpe, Phys. Rev. A 40, 537() (1989).
[2'2] M. J. Void. J. Colloid Sci. 14. 168 (1959).
3478


English

Planar Random Networks with Flexible Fibers

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): Niskanen, K. J. & Alava, Mikko J.Title: Planar Random Networks with Flexible FibersYear: 1994Version: Final published versionPlease cite the original version:Niskanen, K. J. & Alava, Mikko J. 1994. Planar Random Networks with Flexible Fibers.Physical Review Letters. Volume 73, Issue 25. 3475-3478. ISSN 0031-9007 (printed).DOI: 10.1103/physrevlett.73.3475.Rights: © 1994 American Physical Society (APS). This is the accepted version of the following article: Niskanen, K.J. & Alava, Mikko J. 1994. Planar Random Networks with Flexible Fibers. Physical Review Letters. Volume73, Issue 25. 3475-3478. ISSN 0031-9007 (printed). DOI: 10.1103/physrevlett.73.3475, which has beenpublished in final form at http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.73.3475.All 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)VOLUME 73, NUMBER 25 PH YS ICAL REVIEW LETTERS 19 DECEMBER 1994Planar Random Networks with Flexible FibersK.J. Niskanen' and M. J. Alava'KCL Paper Science Centre, P.O. Box 70, FIN-02151 Espoo, Finland*and U S D .A. F.or. est Service, Forest Products Laboratory, One Gifford Pinchot Drive, Madison, Wisconsin 53705 23-98'Helsinki University of Technology, Laboratory of Physics, FIN 0215-0 Espoo, Finland(Received 2 May 1994; revised manuscript received 23 September 1994)The transition in random fiber networks from two-dimensional to asymptotically three-dimensionalplanar structure with increasing coverage c (mean fiber length per unit area) is studied with a depositionmodel. Network geometry depends on the scale-free product of fiber length and c at low c, and onanother scale-free product of flexibility and the width-to-thickness ratio of fibers at high c. The structurebecomes three-dimensional or decouples from the substrate faster when fibers are stiffer. Roughness ofthe free surface decreases with increasing fiber flexibility.PACS numbers: 81.15.Lm, 64.60.—i, 81.30.DzThe mechanical properties and structure of two-dimensional random systems have recently been in-vestigated intensively by computational and theoreticalmethods [1,2], but experimental studies are still few innumber. However, planar fibrous materials, such as poly-mer films, short-fiber composites, paper, and nonwovens,provide promising possibilities for experimental work[3], in addition to being technologically interesting. Thefibers in such materials are usually much longer thanthe sheet or film is thick, and they are almost perfectlyaligned in the sheet plane [4]. The structure is thereforeplanar, or "layered, " but no well-defined layers can beseparated because of the "undulations" of the fibers inthe thickness direction. If coverage (i.e., the mean fiberlength per unit area) c is low, the network is almost twodimensional, and its properties must be governed by c.At high c the three-dimensional "bulk" density replacesc. However, e.g. , mechanical properties are in bothcases governed by the average number of connections(interfiber bonds) per unit fiber length or "coordinationnumber" n [5]. Since the network is planar, we expectthat its mechanical properties are independent of c ifcompared at a fixed n.The statistical geometry of strictly two-dimensionalrandom fiber networks is well understood [6], but ana-lytical models [7,8] for fiber networks with finite thick-ness have so far not dealt accurately with the fact that thebonding of fibers across intervening "layers" is controlledby the random pore structure. In this Letter we employ asimple computer simulation model to study the geometryof random networks of flexible fibers that are deposited ona two-dimensional flat substrate [growth in (2 + 1)D].Weconcentrate on the coverage-driven transition from 2D to3D network geometry with increasing coverage and showthat surface roughness [9] depends on fiber stiffness. Thesimulation model allows the comparison of future experi-ments with theoretical or numerical studies [10,11,12].We start with a two-dimensional square lattice of linearsize L = 10, . . . , 1000. Periodic boundary conditions areapplied, except in the percolation study. The size Laffects properties only close to the percolation thresholdand, for surface roughness, at very high coverages. Thefibers are initially straight beams of unit width andthickness, wf = tf = 1, and emphasis is given to highfiber lengths lf » 1 [13]. Fibers are positioned andaligned in the two principal directions at random [14] insuch a way that the local coverage c (number of fiberscovering the cell) is always an integer. The discretizationshould not, of course, affect generic properties.A "bending flexibility" Tf gives the largest allowedvertical deflection of the fibers from one lattice cell tothe next (cf. Fig. 1). A high Tf gives a dense network, alow Tf a sparse one. For example, the so-called wet fiberflexibility (WFF) [15] of paper-making fibers is relatedto Tf through Tf = [Cwf(WFF)]'l, where C depends onexperimental details. Measured values of the wet fiberflexibility [16] and known dimensions of paper-makingfibers yield wfTf/tf ~ 1. In the simulations fibers areplaced down independently, one after another, as if theywere sedimenting from a dilute suspension. Each fiberis kept straight and parallel to the substrate until the firstcontact is made with the underlying network. Thereafterthe fiber is deformed so as to lie as low as possible-not below the substrate —still obeying the deflectionconstraint Tf. The finite width of real fibers implies thatthey tend to follow a staircase pattern like that given bythe model (cf. Fig. 1). The cumulative buildup of thenetwork is a more serious departure from reality as itdoes not distinguish between densification mechanismsl~IFIG. 1. Two crossing fibers on the square lattice. The heightof the "steps" is = Tf = 1/3.0031-9007/94/73(25)/3475(4)$06. 00 1994 The American Physical Society 3475VOLUME 73, NUMBER 25 PH YS ICAL REVIEW LETTERS 19 DECEMBER 1994such as hydrodynamic drag forces and the pressing of afiltrated fiber mat. The construction results in differenttop and bottom surfaces, but only the free top surface isstudied here.At low mean coverage c the in-plane properties of aplanar network should be controlled by c/c, . , where 2,is the percolation threshold [17,18]. We determined c,numerically using the following rules: (1) the networkis connected when there exists a cluster of fibers that isconnected to at least one cell on two opposite boundariesof the system; (2) two cells are connected by a fiber ifthe fiber covers them both; and (3) two fibers coveringa cell are connected if the top surface of the lower fiberis in contact with the bottom surface of the upper fiber.In the case of zero fiber width, infinite flexibility (orzero fiber thickness), and continuous spatial symmetry.the percolation density is [19] c,. = 5.71//f. This shouldalso hold in our case despite the discrete orientations offibers due to the excluded area principle [20,21]. Indeed,the computed threshold (cf. Fig. 2) is c,. = 5.7//& whenthe fibers are long and flexible (/f ~ 30—40). The resultsin Fig. 2 are independent of L at least for L ~ 800.When the fibers are short, the nonzero widthlowers the percolation threshold at high T~,. we find c, =4//f for /f = 7 and Tf ) 1, and c,. = 3.5//f for /f 3and T~ ~ 5. A random walk argument suggests that inthe opposite limit of T~ 0 the radius r of a clusterwith N fibers is r —/f ~N and therefore the thickness of apercolating cluster is -lf(L//f)2 )) L Thus the n. etworkbecomes three dimensional, and c,. no longer characterizesits structure appropriately. Since the maximum verticaldeflection per unit length of a fiber is T~, crossoverbetween flexible and stiff fiber behavior should occur atTf ff //f our data 'suggest T& = 4ff /// with tf —1.When coverage increases beyond the percolationthreshold, the number of bonds per fiber and, likewise,the mechanical rigidity of the network increase [17,18].We define the coordination number R as the averagebonded surface area of a fiber, divided by 2/fwf [5].Figure 3(a) shows how n first increases with c and thenbecomes asymptotically constant. Obviously the latter1210lO4Q, lFIG. 2. The percolation threshold c,, multiplied with the fiberlength lf against Tf for fiber lengths lf = 3 {{)),5 {x),7 ( ),1 I {*},15 {A), and 21 (+); system size L = 100.0,8-0,6 I-0,4-*IF000,80,60,4 -,''(b)0FIG. 3. Coordination number A against coverage { (a); andagainst fiber flexibility T&- for ~ ~ (b). In (a) T, = 1 and2 (filled and open symbols, respectively) and solid line givesEq. (1). In (b) solid line corresponds to Eq. (2) and dashedline to F~ = T, . System size L = 1000. fiber length tt =—21.happens when the substrate has no effect on n anymore.The network has then become three dimensional, andevery new layer has the same structure as the precedingone. However, in the case of T~ ~ this never happens,and the coordination number is that of a two-dimensionalnetwork at all coverages, given by [6])i =- l —[1 —exp( —( )]/(. ( I )[cf. Fig. 3(a)]. The crossover from 2D to 3D can beroughly located by observing that at low coverages thereare relatively few pores inside the network. Thus heightdifferences between neighboring cells are of the order—J~/w/ since coverage is Poisson distributed [6]. Poresstart to develop inside the network when T, /l, —+2/w, .Equation (1) then yieldsA = I —(I —exp[ —(Tf n, /r() ])/{T/ vi//t() (2)for the asymptotic 3D structure. The computed valuesagree reasonably with Eq. (2) for large T, as Fig. 3(b)shows (for w/ = t, = 1). For T& && I we can estimatethat on the average each fiber (with l& &) w f ) oil tilesurface of the network prevents new fibers from mak-ing contact with others on /ftf/Tf sites, while bondingwith the fiber in question is possible in ~~ I~ cells. Hence6 w f /f /(8'f /f + /fff /Tf ) (Tf 8'f /if) at high cover-ages [dashed line in Fig. 3(b)j. The simulation resultsin Fig. 3 were found to be independent of the fiber lengthat least for It- ~ 5. The reproducible discontinuities atTf = 2 and 4 in Fig. 3(b) are artifacts of the discretemodel.VOLUME 73, NUMBER 25 PH YS ICAL REVIEW LETTERS 19 DEcEMBER 1994which follows as a generalization of Eq. (1).The roughness o., of the free surface can be defined asthe standard deviation of the local surface height t, thedistance between the free surface and substrate. Throughtrial and error we found that the simulation results can beexpressed asCrt ctf [A(wf Tf /tf) + B(c)] (4)0,80,6Ic0,40,2In contrast to n, the three-dimensional bulk densityp decreases with increasing coverage (provided thatsurfaces pores are excluded), since at low coveragethere are initially no pores inside the sheet. In realitythickness measurements have finite spatial resolution, andusually they underestimate the true density. A "coarse-grained" or "apparent" density was calculated from themodel to mimic real measurements. The results weresimilar to the n (c) curves in Fig. 3(a). At high coverages(thicknesses) even an apparent density is independent ofsurface roughness and equal to the "true" density. Sincethe coordination number is tedious to measure directly,the 3D bulk density is sometimes used to estimate it.Our results (Fig. 4) indicate that if c is held constantand Tf is varied, then n is indeed roughly proportionalto p. This also holds for apparent densities. However,if Tf is held constant and coverage is increased, the true3D density decreases and hence cannot be proportionalto n (cf. discussion above), but an apparent density maystill be.As the local coverage is Poisson distributed [6], thestandard deviation of local coverage is o., = vie. Thisholds very well in our simulations. The flat substrateaffects the network structure at small c, but as c growsthe structure of new layers becomes stable, independentof c. Then, for example, the standard deviation of thenumber of pores per cell (p) becomes o„~P(Tf)Jc.The amplitude P(Tf) decreases with increasing Tf. Areasof high coverage have few pores, and therefore P ( 1.The average number of pores is directly related to thecoordination number throughp+1 [1 —exp( —c)]C100,10,010 1 2 3 4 5FIG. 5. Function A(Tf), Eq. (1), for coverages c = 10 ( ),100 (+), 10' (E), and 104 (0). Fiber length lf = 5, system sizeL = 100. Inset shows function B(c) on log-log scale.(of course, we have data only for tf = wf = 1). Thefunctions A and B are shown in Fig. 5 for system sizeL = 100, Tf + 5, and c ~ 10 . They are independent offiber length as long as lf ~ 5. It was easy to decomposeo., into A and B since at large Tf, o., turned out to be afunction of c only (or A 0). The prefactor c in Eq. (4)is motivated by the Poissonian deposition process. Ascoverage increases, B becomes smaller and eventually o.,grows as -o., = ~c. This corresponds to a decoupling ofthe surface from the flat substrate. The decoupling is fastwith stiff fibers (A large), since stiff fibers cannot "feel"the substrate from high above. For the same reason thecase of stiff fibers seems to correspond to a low surfacetension in ordinary growth models [9] as no long rangecorrelations occur in the surface height. Indeed, makingTf small gives a rough surface. In fact, our model issimilar to the Void model [9,22]. Details of the growthbehavior at high coverage, including the effects of systemsize, are beyond the scope of this study and will bepresented elsewhere.Summarizing, we have presented a computer simulationstudy of planar random fiber networks. The model pro-vides a connection between the properties of real planarfiber networks and ideal two-dimensional networks. Theresults can be represented in closed form only in a fewcases, but the model is simple enough so that the numeri-cal solution can be easily used, e.g. , in the analysis of ex-perimental results.This study was financially supported by the TechnologyDevelopment Centre of Finland and by the Academy ofFinland which is gratefully acknowledged.0 0,2 0,4 0,6 0,8 1FIG. 4. Coordination number n against density p when fiberflexibility Tf varies. Coverage c = 10.5 (E), 31.5 ( ), and94.5 (0). Fiber length lf = 21, system size L = 1000.*Present and permanent address.[1] Statistical Models for the Fracture of Disordered Media,edited by H. J. Heiiiiiann and S. Roux (North-Holland,Amsterdam, 1990).[2] M. Sahimi, Physica (Amsterdam) 1$6A, 160 (1992).[3] J. Kertesz, Physica (Amsterdam) 191A, 208 (1992).3477VOLUME 73, NUMBER 25 PHYSICAL REVIE% LETTERS 19 DEceMev~ 1994[4J B. Radvan, in The Fundamenta/ Properties of P'aperRelated to its Uses, edited by F. Bolam (British Paperand Board Ind. Fed. , London, 1976},pp. 137—147.[5] For paper the quantity corresponding to the coordina-tion number is the relative bonded area (RBA) of thefibers. See O. J. Kallmes, H. Corte. and G. Bernier. TappiJ. 46, 502 (1963}.[6] H. Corte and O. J. Kallmes, Tappi J. 43, 737 (1960).[7] O. J. Kallmes, H. Corte, and G. Bernier, Tappi J. 44, 519(1961).[8] J. Gorres, C. S. Sinclair, and A. Tallentire, Paperi ja Puu71, 54 (1989};J. Gorres and P. Luner, J. Pulp Paper Sci.18, J127 (1992}.[9] J. Krug and H. Spohn, in Solids Far,~rom Equilibria&n:Growth, Morphologv, Defects, edited by C. Godreche(Cambridge Univ. Press, Cambridge. 1991), pp. 479—582.[10] M. J. Alava and R. K. Ritala, J. Phys. Condens. Matter 2.6093 (1990);M. J. Alava and R. K. Ritala, Phys. Scr. T33,155 (1990).[11] J.A. Astrom and K. J. Niskanen, Europhys. Lett. 21, 557(1993).[1'2] J. Astrom, S. Saarinen, K. Niskanen, and J. Kurkijarvi.J. Appl. Phys. 75, 2383 (1994).[13] Fiber thickness is arbitrary in our study. Fr&r paper-making fiber» 1&q't& ~ /«'» f = '0 —~0.[14J We use the R250 random number &gener rtor.Kirkpatrick and E. Stol l. J. Comput. Phy». 40. :1:(1981).[15] R. Steadman and P. Luner. in Papennafi»r; Rarr' Maieriair. edited by V. Punton (Mechanical Engineering Publ.Ltd. . London, 1985), pp. 311-338.[16] L. Paavilainen, Paperi ja Puu 46, 689 (1993).(17] S. Kirkpatrick, Rev. Mod. Phy». 45, 574 (1973 r.[18] L. M. Schwartz, S. Feng, M. F. Thorpe. ,md P. N. Scn.Phys. Rev. 32, 4607 (1985).[19J G. E. Pike and C. H. Seager, Phys. Rev. B 10.(1974).[ 0] I. Balberg, Phys. Rev. B 31, 4053 (1985 r.[21] K. Tobochnik, M. A. Dubson. M. L. Wilson. and M. F.Thorpe, Phys. Rev. A 40, 537() (1989).[2'2] M. J. Void. J. Colloid Sci. 14. 168 (1959).3478

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Planar Random Networks with Flexible Fibers

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