Inter-fibre contacts in random fibrous materials: experimental verification of theoretical dependence on porosity and fibre width

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1 J Mater Si (2006) 41: DOI /s LETTER Inter-fibre ontats in random fibrous materials: experimental verifiation of theoretial dependene on porosity and fibre width W. J. Bathelor Æ J. He Æ W. W. Sampson Reeived: 30 January 2006 / Aepted: 24 August 2006 / Published online: 11 November 2006 Ó Springer Siene+Business Media, LLC 2006 The mehanial, optial and transport properties of industrially manufatured stohasti fibrous networks, suh as paper, non-woven fabris and fibrous filters, are strongly influened by the extent of ontat between fibres. The lassial referene strutures for modelling these materials are random fibre networks with fibre entres distributed aording to a two-dimensional point Poisson proess and with uniformly distributed fibre axes. The number of fibres overing a point in the plane of support of the network is a disrete random variable alled overage,, with Poisson probability, PðÞ ¼ e! for ¼ 0; 1; 2; 3... ð1þ The earliest work onsidering the statistial geometry of random fibre networks was derived for paper [1] and this has given rise to a signifiant body of literature, whih has been reently reviewed [2, 3]. The development of new heterogeneous fibrous materials for use in fuel ells and as saffolds for tissue engineering has provided new appliations for strutural models of stohasti fibrous materials, see e.g. [4, W. J. Bathelor J. He Australian Pulp and Paper Institute, Monash University, Melbourne, Vitoria 3800, Australia W. W. Sampson (&) North West Composites Centre, Shool of Materials, University of Manhester, Manhester, M60 1QD, UK w.sampson@manhester.a.uk Present Address: J. He Amor Researh and Tehnology Center, Melbourne, Vitoria, Australia 5], and we note that the theory is appliable to networks with arbitrary sale features [6]. Our interest here is the extent of ontat between fibres. When the mean overage,, is less than about 1, the number of inter-fibre rossings per fibre is a funtion of fibre length k (m), fibre width x (m) and mean overage [1]: n ¼ 2 p k for 1 ð2þ x The fration of the fibre surfae in ontat with other fibres, termed the frational ontat area (FCA), is a funtion of mean overage only [7]: U ¼ 1 1 e for 1 ð3þ Real networks typially have mean overage of 10 or more and hene a given pair of fibres may be obsured from ontat by the influene of nearby fibres. For suh networks, Kallmes et al. [8] derived expressions for the expeted number of ontats per fibre by onsidering the superposition of several two-dimensional networks and inluding fibre flexibility as a variable to weight the probability of ontat between fibres in nearby layers. The theory was diffiult to verify however sine the number of layers was not well-defined and meaningful measures of fibre flexibility are experimentally diffiult. Reently, Sampson [9] noted that for modelling purposes the mean overage of a layer ould be defined by the porosity of the network, e, suh that the number of layers in a network of a given mean overage was well defined and the influene of fibre flexibility was impliitly aounted for. The FCA of a network with infinite overage is [9],

2 8378 J Mater Si (2006) 41: U 1 ¼ 2 X 1 ¼1 1 P ðþþð1 eþ 2 X1 ¼1 1 P ðþ; ð4þ where the oeffiient 2 aounts for rossings above and below that fibre. The expeted number of rossings per fibre is therefore, where ¼ log ð1=eþ suh that U 1 is a funtion of porosity only. Note that the seond term on the right hand side of Eq. (4) differs from that given in [9] bya fator of 2; the influene on the predited FCA is typially small though Eq. (4) makes the theory preise. For a network of finite mean overage, the fibres in the outer surfaes an ontat others on one side only, so FCA is given by [10], U ¼ 1 1 U 1 ; ð5þ where for networks of mean mass per unit area, b (g m 2 ), formed from fibres of mass per unit length d (g m 1 ), we have ¼ bx=d. Sampson proeeded to show that if FCA was known, then for fibres of known length and width, the expeted number of ontats per fibre was given by n ¼ U U 1 ð6þ where F 1 is the FCA of a thin network alulated from Eq. (3) with mean overage,, determined by setting n ¼ 1 in Eq. (2). Equation (6) is not easily verified beause the extent of inter-fibre ontat is diffiult to measure in anything but the thinnest networks. Elias [11] ounted the number of ontats per fibre in low-weight networks of glass fibres and showed orrelations between the data obtained and the solid fration of the network, ð1 eþ. Experimental estimates of FCA obtained from image analysis of ross-setions of paper embedded in resin [12, 13] exhibit a dependene on porosity onsistent with Eq. (4) [9]. These tehniques are not readily appliable to the quantifiation of the number of ontats per fibre beause it is diffiult to obtain a setion along the full length of any fibre. This was reently addressed by He et al. [14] using onfoal sanning laser mirosopy (CLSM) and we extend their analysis here. It is easy to show that for a network of randomly oriented, retangular fibres of width x and infinite length, rossings have expeted area px 2 =2. For a network with FCA, F, formed from fibres with width, x (m) and length, k (m), the expeted total area of ontats on a given fibre is A ¼ 2 U kx; ð7þ n ¼ 2A px 2 ¼ 4 k p x U ð8þ Equation (8) provides an approximation to Eq. (6) and is more open to intuitive interpretation. We note the very reasonable dependenies suggested, i.e. the number of ontats per fibre is diretly proportional to the length of a fibre and to its FCA; the inverse proportionality to fibre width arises through the influene of fibre width on the overage of a network with a given number of fibre entres per unit area. We have analysed the data of He et al. [14] to obtain values for the number of rossings per fibre for fibres with a range of known widths and lengths formed into networks with known porosity. Full experimental details are provided in [14], so here we give only a summary. Sheets weighing 60 g m 2 were formed using a laboratory devie designed to produe paper samples with strutures similar to those obtained industrially [15]. All sheets were made from radiata pine fibres extrated in a laboratory pulping proess and proessed to yield fibres with different lengths and widths; some sheets were pressed to redue porosity independently of fibre dimensions. Prior to forming sheets, the fibres were stained with Aridine Orange so that they would fluorese under the CLSM. Samples were embedded in Struers Epofix resin and ross-setions were exposed by polishing the blok with abrasive papers. A 14 mm length of eah ross-setion, onsisting of 70 onseutive frames, was examined under a CLSM using a 60 oil-immersion lens. Fibres that were imaged along their major axis were identified and the number of fibres in ontat with eah of these was ounted. To aid identifiation of ontats, images were thresholded and binarised. A ontat was ounted between eah pair of fibre surfaes not separated by blak pixels and lassified as either a full or partial ontat. This is illustrated by example in Fig. 1 where fibres 2 and 3 make full ontats with the fibre of interest, fibre 1 makes a partial ontat, and fibre 4 makes no ontat. For eah of the samples, fibre ross-setional area and shape were also measured. This involved seleting a minimum of 200 individual fibres for imaging at the rosssetion surfae and 10 lm under the surfae. The angle at whih eah fibre ut the surfae was then alulated from the hange in position of the entre of mass of the fibre. Eah of the measured ross-setions was then orreted by this angle to obtain the true ross-setion

J Mater Si (2006) 41:8377 8381 8379 Fig. 1 Cross-setion image before (left) and after thresholding and binarisation (right).

3 J Mater Si (2006) 41: Fig. 1 Cross-setion image before (left) and after thresholding and binarisation (right). Fibre 2 and 3 in (B) make two full ontats, fibre 1 makes a partial ontat, and fibre 4 is not in ontat with the fibre of interest Fibre of interest 20 µm 20 µm and this orreted ross-setion was then fitted with a retangular bounding box, as shown in Fig. 2. The ratio of the fibre wall area to the bounding box area was alled the fill fator. Further details are given in [16]. The 95% onfidene interval of eah measured value was usually less than ±4%. To alulate the number of ontats using Eq. (8), we must alulate first the FCA using Eqs. (4) and (6) and so require a measure of the inter-fibre porosity. From inspetion of Fig. 1 we observe that a fration of the network voids exists within fibres. Aordingly the total porosity, as estimated from e ¼ 1 q network q solid ; ð9þ where q network and q solid are the densities of the network and solid phases, respetively, would overestimate the porosity. Reently, Bathelor and He [17] presented a linear orrelation between an experimental estimate of the FCA of paper and the sheet density weighted by the fill fator, f, given by the ratio of the ross setional area of the ell wall to the area of the bounding retangle with sides x and t, and illustrated in Fig. 2. This weighting aounts for the volume of the network rendered inaessible by the presene of a fibre so the aessible porosity is t ω Referene fibre Fig. 2 Determination of the fill fator, f, whih is given by the ratio of the ross setional area of the ell wall to that of the bounding retangle with sides x and t [16] e ¼ 1 q network f q solid : ð10þ We use this value of porosity to alulate F using Eqs. (4) and (5) and hene the expeted number of ontats per fibre via Eq. (8). Equations (6) and (8) are arrived at by dividing the expeted total area of ontat on a fibre by that of a full ontat. By definition, partial ontats exhibit a smaller area than full ontats so the theory will underestimate the total number of ontats observed in experiments. If the fration of a full ontat represented by a partial ontat is a non-skewed random variable then, on average, two partial ontats are equivalent to a full one. Aordingly, we determine the number of equivalent ontats from our experiments as the sum of the number of full ontats and half the number of partial ontats. The experimental data are summarised in Table 1 and the alulated and measured ontats are ompared in Fig. 3; these inlude also the data of Elias [11]. Sine the networks Elias studied were very thin, only the first term on the right hand side of Eq. (4) was used to alulate the FCA and hene the number of ontats; similarly, eah ontat was lassified as a full ontat. Clearly there is good agreement between theory and measurement, as indiated by the broken line with unit gradient. From Table 1, it an be seen also that the total number of ontats does not fall linearly with paper sheet density; we attribute this to the fat that inreasing density not only inreases the number of full and partial ontats but onverts some of the partial ontats to full ones also. The lowest density paper tested (192 kg m 3 ) still exhibited around 40 ontats per fibre, but almost all of these were partial. Any further redution in sheet density would rapidly redue the total number of ontats sine almost no full ontats would remain; aordingly,

4 8380 J Mater Si (2006) 41: Table 1 Fibre and network properties Fibre properties Network properties Number of ontats per fibre Width (lm) Length (mm) Coarseness ( 10 7 kg m 1 ) Fill fator Mean overage Density (kg m 3 ) Porosity a FCA Experiment Model Full Partial Effetive This study Elias [11] a For this study, porosity alulated using Eq. (10) assuming q solid = 1.5 kg m 3 ; for data of Elias, porosity alulated from solid fration given in [11] Experiment This study Elias [11] no ontats would be expeted when the sheet density is zero. Although we have onsidered experimental data for paper and glass fibre mats only, it is important to note that the theory is derived for general lasses of stohasti fibrous materials and therefore are appliable to networks formed from fibres with arbitrary dimensions. This is important beause it follows diretly from Eq. (8) that the expeted interval between rossings, i.e. the mean ligament length, is a funtion of aessible porosity through FCA and fibre width only. Sine ligaments in fibrous strutures represent the boundaries of voids and provide the available surfae of the struture, then the harateristi pore size and speifi surfae area also depend only on fibre width and network aessible porosity. Suh dependenies should aid targeted experimentation in the development of fibrous filters, non-woven omponents in fuel ells and nanofibrous saffolds for tissue engineering Model Fig. 3 Comparison of number of equivalent ontats per fibre measured experimentally with that alulated using the model. Error bars representing 95% onfidene intervals Referenes 1. Kallmes O, Corte H (1960) Tappi J 43(9):737; (1961) Errata 44(6): Dodson CTJ, Deng M (1994) Paper: an engineered stohasti struture. Tappi Press, Atlanta 3. Sampson WW (2001) The siene of papermaking, Transations XIIth Fundamental Researh Symposium, ed CF Baker, Pulp and Paper Fundamental Researh Soiety, Bury, p Berhan L, Yi YB, Sastry AM (2004) J Appl Phys 95: Yi YB, Berhan L, Sastry AM (2004) J Appl Phys 96: Eihhorn SJ, Sampson WW (2005) J Roy So Interfae 2(4): Kallmes O, Corte H, Bernier G (1963) Tappi J 46(8):493

5 J Mater Si (2006) 41: Kallmes O, Corte H, Bernier G (1961) Tappi J 44(7): Sampson WW (2004) J Mater Si 39(8): Sampson WW, Sirviö J (2005) J Pulp Paper Si 31(3): Elias TC (1967) Tappi J 50(3): Paavilainen L (1994) Pap ja Puu 76(3): Niskanen K, Rajatora H (2002) J Pulp Paper Si 28(7): He J, Bathelor WJ, Johnston RE (2004) Appita J 57(4): Xu L, Parker I (2000) Appita J 53(4): He J, Bathelor WJ, Markowski R, Johnston RE (2003) Appita J 56(5): Bathelor WJ, He J (2005) Tappi J 4(6):23

The Statistical Geometry of Fractional Contact Area in Random Fibre Networks

The Statistical Geometry of Fractional Contact Area in Random Fibre Networks The Statistial Geometry of Frational Contat Area in Random Fibre Networks W.W. SAMPSON Theory is presented for the frational ontat area of fibres in random networks. The theory onsiders first the statistial