Influence of anisotropy on the dynamic wetting and ... - Semantic Scholar
Journal of Colloid and Interface Science 283 (2005) 171–189 www.elsevier.com/locate/jcis
Influence of anisotropy on the dynamic wetting and permeation of paper coatings Paul A. Bodurtha a , G. Peter Matthews a,∗ , John P. Kettle b , Ian M. Roy a a Environmental and Fluid Modelling Group, University of Plymouth, Plymouth PL4 8AA, UK b SCA Graphic Research, SE-850 03 Sundsvall, Sweden
Received 16 June 2003; accepted 14 August 2004 Available online 2 December 2004
Abstract A void network model, named Pore-Cor, has been used to study the permeation of an ink solvent into paper coating formulations coated onto a synthetic substrate. The network model generated anisotropic void networks of rectangular cross-sectional pores connected by elliptical cross-sectional throats. These structures had porosities and mercury intrusion properties which closely matched those of the experimental samples. The permeation of hexadecane, used as an analogue for the experimental test oil, was then simulated through these void structures. The simulations were compared to measurements of the permeation of mineral oil into four types of paper coating formulation. The simulations showed that the inertia of the fluid as it enters void features causes a considerable change in wetting over a few milliseconds, a timescale relevant to printing in a modern press. They also showed that in the more anisotropic samples, fast advance wetting occurred through narrow void features. It was found that the match between experimental and simulated wetting could be improved by correcting the simulation for the number of surface throats. The simulations showed a more realistic experimental trend, and much greater preferential flow, than the traditional Lucas–Washburn and effective hydraulic radius approaches. 2004 Elsevier Inc. All rights reserved. Keywords: Anisotropy; Paper coatings; Absorption; Porous media; Modelling; Wetting; Permeation; Mercury porosimetry; Pore-Cor; Shale
1. Introduction If a porous solid is made up from a homogeneous packing of unsorted spherical particles bound together, the resulting void structure will normally be structurally isotropic—i.e., it will have approximately the same structural characteristics in any direction, and hence approximately the same permeation characteristics [1]. However, a sample made up from non-isometric particles such as clay platelets, packed in an aligned manner, would be structurally anisotropic [2]. Consequently, the rate of wetting would vary with direction. Such anisotropic permeation is relevant to the printing of paper coated with aligned anisotropic clay particles, and to other processes such as the subterranean migration of fluids in shale [3]. * Corresponding author. Fax: +44(0)1752-233021.
E-mail address: [email protected] (G.P. Matthews). 0021-9797/$ – see front matter 2004 Elsevier Inc. All rights reserved. doi:10.1016/j.jcis.2004.08.072
When designing paper coatings with optimum printing properties, manufacturers have a wide choice of mineral pigments to choose from, ranging from isotropic precipitated calcium carbonates, through ‘blocky’ clays such as Georgia kaolin, to ‘platey’ clays such as Speswhite with a high aspect factor. Each type of pigment can also be prepared with a wide range of particle sizes, and each size distribution can have a different range and skew. The problem when assessing the ink absorption properties of different pigments is that if one compares particles with different aspect factors, their size distribution changes also, and it is difficult to separate one effect from the other. So in this work a void network model has been used to highlight the effects of anisotropy and aspect factor which are otherwise hidden within experimental trends. The network model, named Pore-Cor (pore-level properties correlator), can accommodate changes in void sizes and connectivity between samples, such that additional changes caused by anisotropy can
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be revealed and studied. It has been used previously for modelling the behaviour of fluids in a wide range of materials such as porous calcium carbonate blocks [4,5] and soils [6]. A new version of the network model has been used, which generates an anisotropic three-dimensional void network of pores with a rectangular cross-section connected by throats with an elliptical cross-section. The equation for wetting was an inertially modified version of the Lucas–Washburn equation developed by Bosanquet [7]. Previous researchers have shown the dependence of wetting on the thickness of the porous layer [8] and the size of the applied droplet of fluid [9]. Others have used an effective hydraulic radius approximation [10] or the equation of state approach [11]. However, the wetting of such simple systems is totally different from that in more complex networks [12]. Nevertheless, a series of approximations had to be made to make the computation time manageable, in practice less than 30 h on a 1 GHz computer. Wetting and consequent permeation start at the contact point or surface between the fluid reservoir or droplet and the porous solid. There is an almost instantaneous wetting jump when a wetting fluid contacts a wettable porous sample [4]. Next, the fluid has to accelerate from rest into the various void features. Many researchers have improved Bosanquet’s description of initial, inertial, and continuity influences [8], and some have included a kinetic energy term [13] or used full Navier–Stokes equations [14,15]. It has been shown previously that the time-step for the wetting calculation in our network model must be no more than 1 ns [4], and consequently the calculation time would have been unmanageable if these more sophisticated effects had been included. Improvements to overcome a singularity at the tube entrance at time zero have been suggested by Szekely et al. [16], but are insignificant under the conditions of this work. The way the fluid subsequently permeates through the structure is then controlled by a further series of factors, all of which are approximated in this work. These are the sizes and shapes of the voids [17], their connection into the network [18–20], wetting jumps [21,22], the surface energy between the fluid and the solid phase [23], and the density, viscosity and applied pressure of the fluid [24–26]. Modern printing presses feed paper through a printing nip, a few centimetres long, at speeds of up to 15 m s−1 . So the fluid contact time in the nip is of the order of a few milliseconds. At longer times, of the order of seconds to tens of seconds, there may be problems due to back-transfer of ink onto other rollers or ink smearing, but these effects are beyond the scope of this publication.
adhesive to bind the pigment particles together and to the substrate [27]. Typically, the weight fraction of pigment is 80–95% [28]. Assuming a pigment density of 2.6 g cm−3 and a binder density of 1.0 g cm−3 , the volume fraction of the pigment is about 70%. Four types of commercially available pigments were studied: Speswhite and Amazon 90 SD (both Kaolin clays), and OpacarbA40 (OpA40) and Albaglos (both precipitated calcium carbonates or PCCs). Speswhite is a primary coarsegrained kaolin mineral supplied from Imerys, UK, while Amazon 90 SD is a secondary fine-grained kaolin supplied from Caulim da Amazônia S.A., Brazil. Both have a platey or foliated mineral structure. By contrast, the PCC group contains chemically synthesised calcium carbonate minerals belonging to either a calcitic or an aragonitic crystalline polymorph. OpA40 is aragonitic in structure, with an acicular (or needle-like) crystal habit. Albaglos contains a calcitic structure, with a rhombohedral crystal habit. The aragonite and Albaglos pigments were supplied by Specialty Minerals Nordic Oy, Finland. In the coating preparation, sodium polyacrylate (Dispex N 40 from Allied Colloids Ltd., UK) was used as a dispersant for the pigments. A carboxylated styrene-butadiene latex (DPL 935 from Dow Rheinmünster GmbH, Germany), with a diameter of 0.15 µm and a glass transition temperature T g of 8 ◦ C, was used as a binder, and carboxymethyl cellulose (CMC), with an average molecular weight of 45000 (FF5 from Noviant, Finland), was used as water retention aid. The kaolin pigments were received in a dry form and were dispersed in water under conditions suggested by the manufacturers. The samples for the permeation experiments were coated onto an impermeable plastic backing known as Synteape (Arjo Wiggins Ltd., UK). This has a slightly rough, hooked surface on which the coating can adhere, producing the flexible but stable surface necessary for the permeation experiments. The pigment suspensions contained 1 pph CMC and 10 pph latex. All suspensions were prepared in a conventional manner to 60% solids content by weight and a pH value of 8.5. They were drawn down onto the Synteape using a wire-wound rod in a bench coater (K-coater, RK PrintCoat Instruments Ltd., Royston, UK), to a coat weight of around 80 g m−2 in each case. Synteape is permeable from the side, so for porosimetry the suspensions were benchcoated onto the aluminium backing foils to a coat weight of about 18 g m−2 . All coating and drying was carried out at room temperature. Each coated sample strip had dimensions of 4 × 25 cm.
2. Materials and methods
2.2. Techniques for characterising particle and void size distributions
2.1. Samples In its simplest form a paper coating consists of mineral pigments, binders and air voids. The binders act as an
Particle diameters and the widths of their size distributions were measured by X-ray monitoring of sedimentation rate (Micromeritics Sedigraph 5000), and by light scattering
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(Malvern Mastersizer 1000) [29]. No light scattering measurements were available for Albaglos. The aspect ratios of the pigment particles were estimated from image analysis of atomic force micrographs [29]. Aspect ratio is defined for plate-like structures as the ratio of the maximum dimension of the particle and its thickness; the more platey a particle, the higher its aspect ratio. For acicular (rod-like) structures, aspect ratio is defined as the length of the particle divided by the radial diameter. A scanning electron microscope (Cambridge 360) was used to obtain two types of view of the void structure of the samples, namely that manifest at the surface of the sample and that visible in a cross-section, at magnifications around 40,000×. For the surface views, the samples were gold plated, whereas for the cross-sections, the samples were embedded in epoxy resin and viewed in back-scatter mode [30]. The micrographs were used to gain semi-quantitative estimates of appropriate aspect factors, and to estimate the number density of surface throats. For mercury porosimetry, the samples were first removed from the aluminium foil backing, thus forming thin flakes approximately 1 cm × 1 cm × 8 µm. Mercury intrusion curves for collections of similar fragments were then measured using a Micromeritics Autopore III mercury porosimeter up to a maximum mercury pressure of 414 MPa (60,000 psia), with an equilibration time at each pressure of 60 s. The mercury intrusion measurements were corrected for the compression of mercury, expansion of the glass sample chamber or ‘penetrometer’, and compressibility of the solid phase of the sample, by use of the following equation from Gane et al. [31]: 1 P log10 1 + Vint = Vobs − δVblank + 0.175 Vbulk 1820 1 (P − P ) 1 − Vbulk (1) (1 − Φ 1 ) 1 − exp . Mss Vint is the volume of intrusion into the sample, Vobs the intruded mercury volume reading, δVblank the change in the 1 the sample bulk volume at blank run volume reading, Vbulk atmospheric pressure, P the applied pressure, Φ 1 the porosity at atmospheric pressure, P 1 the atmospheric pressure and Mss the bulk modulus of the solid sample. 2.3. Fluids The oil used was a non-setting test oil designed for use as the solvent or ‘vehicle’ for offset printing inks. It was supplied by Haltermann Products, a division of Ascot Speciality Chemicals, and contained a mixture of paraffinic, naphthenic and aromatic hydrocarbons (mainly C16 –C18 chains). Its density ρ was measured gravimetrically to be approximately 1025.3 kg m−3 , and its viscosity η was measured by a control stress rheometer (AR1000, TA Instruments) to be about 7.4 mPa s at 20 ◦ C, suggesting mainly paraffinic components.
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The exact formulation of the ink was commercially confidential, so modelling was carried out using hexadecane, a paraffinic oil with similar physical properties. It had a surface tension γ of 0.0246 N m−1 , a density ρ of 773.31 kg m−3 , and a viscosity η of 3.3 mPa s at 20 ◦ C [32]. The contact angle θ was 40◦ , estimated from comparing contact angles of similar mineral oils [33]. The properties of both the test oil and hexadecane were tested on a freshly ground calcite crystal surface, using wet grinding by 1200 grit paper sprayed with tap water to prevent oxidation. Both the test oil and hexadecane were fully wetting, in that no fluid contact angle was visible. On aged surfaces, which have adsorbed atmospheric aerosols, smoke or vapours including water, the contact angle of both fluids increases to around 40◦ as published [33]. This behaviour on aged surfaces was taken to be that which was most representative of actual paper coatings. 2.4. Method for measuring permeation The method for measuring liquid permeation properties involved the novel use of an ink surface interaction tester (ISIT, SeGan Ltd., UK). The procedure involved a sample roller in contact with a gravure roller. The gravure roller (IGT Testing Systems, Amsterdam) comprised an array of small pits in a polished surface, with sizes and spacings such that complete transfer of the fluid onto a substrate of the same surface area would produce a film of thickness 1.5 µm. Examination of the roller using a confocal laser scanning microscope showed that the radius r of each pit was around 50 µm, with a depth s of approximately 10 µm. The footprint of the gravure roller in contact with the mounted coating strip was measured by the use of carbon paper, and found to be approximately 1 mm in the direction of rotation. The length of the footprint was then used to calculate the time of contact between the gravure roller and the coating strip, for each speed setting. The use of a gravure roller had the benefit that flooding of the sample was avoided, and that there was minimal uncertainty in the mode of transfer of the fluid onto the sample. Since the fluid was recessed in the gravure pattern, the fluid itself had a delivery pressure of zero when in contact with the sample. The gravure roller was loaded with oil, and any excess removed so that a known volume of fluid was available for transfer. The sample strip was mounted onto the sample roller. The sample roller and the gravure roller were pressed together under a known contact pressure so that the beginning of the sample strip was in contact with the gravure roller. Then the rollers were turned, still in non-sliding contact with each other, through one revolution at an exactly controlled speed. Oil transferred from the gravure roller to the sample strip. The sample strip was then demounted from the sample roller. A jig and scalpel was then used to cut out an exactly reproducible rectangular area (20 × 40 mm) of the saturated sample, while avoiding compressing the sample and keeping
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away from edges or partially saturated areas at the beginning and the end of the sample strip. The cut rectangle was then weighed, and the amount of fluid uptake calculated by comparison with an equal area of dry sample. The above procedure was followed for permeation measurements at speed settings between 0.1 and 2.0 m s−1 , with a new coating strip being used for each test run. The fluid was delivered from the gravure roller to the sample in the order of 0.01 s (t∞ ), as the results below show. So the fluid velocity v = s/t∞ was of the order of 1 mm/s. On the basis of the dimensions of the gravure roller pits and the fluid properties described above, the Reynolds number (2ρrv/η) was of the order of 0.01, i.e., five orders of magnitude lower than the turbulence threshold of around 2000. The compressing action of the roller on the coating structure was accounted for by a series of experiments using Amazon coating as a test sample. The amount of fluid transferred to the coating was measured at a range of contact pressures from 100 to 600 N at a slower contact speed of 0.5 m s−1 to emphasise any pressure-induced effects. A higher transfer of oil was observed at 400 N and above, and could be attributed to a ‘sponge’ effect, with enhanced uptake being caused by the compression and subsequent relaxation of the coating structure. Although such an effect may well occur during a commercial printing process, it is also dependent on the sample elasticity, which brings in another experimental variable. Therefore all the other experiments reported in this work were carried out at a contact pressure of 100 N to avoid this effect.
3. Void network modelling The network model generates a void structure comprising an infinite array of repeating, identical unit cells. Each unit cell is made up from 1000 cubic pores, in a 10 × 10 × 10 array, equally spaced from its neighbouring pores in the Cartesian coordinates x, y and z by the ‘pore row spacing’ q. Hence, each unit cell is a cube of side length 10q. Cylindrical throats connect each pore and there can be up to six throats per pore—i.e., one on each face. The number of throats connected to a particular pore is termed the ‘pore coordination number,’ and the arithmetic mean of this quantity over the whole unit cell is the connectivity. Individual pore coordination numbers may range from 0 to 6, and a typical value of the connectivity for matching the percolation properties of a real sample is 3.5. The percolation characteristics and the porosity of the overall void structure are matched to the experimental mercury intrusion curve of a chosen sample by adjusting the skews of the throat and pore size distributions, and the connectivity. The void network model is also capable of simulating size-correlated networks such as layered or laminated structures [34]. However, electron microscopy suggested that the structures had no short-range size correlation,
and it was not possible to fit any correlated structures generated by our model to the mercury intrusion curves. In order to simulate anisotropic structures, five modifications were made to the model: (i) the geometry of the unit cell was made anisotropic, (ii) the porosity calculation was modified, (iii) the percolation equations were modified, (iv) the equations for wetting were modified, and (v) the graphical routines were altered so that the geometry of the anisotropic structures could be viewed in both static and virtual reality modes. Modifications (i)–(iv) are now described. With regard to (v), many examples of static snapshots of the structures are presented in this paper. They are available in color on http://www.sciencedirect.com and can be viewed in virtual reality at http://www.pore-cor.com/virtual_reality. htm. 3.1. Generation of an anisotropic unit cell The unit cell was made anisotropic by rescaling the z-axis relative to the x- and y-axes. When the z-axis was rescaled to make features smaller, the pores became tetragonal, and the throats in the x and y directions gained ellipsoidal crosssections with the major axis in the x or y direction. The effect is illustrated in Fig. 1, in which the z-axis is shown vertically. Alternatively, the z-axis was scaled so that all features had larger dimensions with respect to this axis with the dimensions remaining constant on the other two axes (Fig. 2). Such a transformation results in rod-like pores, and the throats in the x and y direction become ellipsoidal in cross-section, but in this case with the major axis in the z direction. The scaling factor was termed the ‘aspect factor’ α, and the rescaled pore row spacings qx , qy and qz in the x, y and z directions were q, q and αq, respectively. The aspect factors α of the structures shown in Figs. 1 and 2 are 0.6 and 1.5, respectively. 3.2. Porosity calculations for anisotropic void structures The porosity Φ of a porous medium is defined as the percentage of void space within the total bulk medium. For anisotropic void networks, the porosity equation for a unit cell generated by the network model was modified so that it referred to the volume of throats with an elliptic crosssection and pores with a rectangular cross-section: 3 1000 n=1 Ln,x Ln,y Ln,z + i=1 πr1,n,i r2,n,i hn,i , (2) Φ= 1000qx qy qz where Ln,x , Ln,y and Ln,z refer to the width of the nth pore in the x, y and z directions, r1,n,i and r2,n,i are the two radii of the nth ellipsoidal throat in direction I , and hn,i is length of the nth throat in the ith Cartesian direction. 3.3. Percolation The Laplace equation gives the diameter d of a cylindrical throat in an incompressible solid exposed to mercury, which is intruded when the pressure applied to the mercury
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Fig. 1. Simulation of the absorption of oil (hexadecane) into Speswhite at 0.1 s and at an aspect factor 0.6. Degree of wetting as shown on wetting scale bar.
Fig. 2. Simulation of the absorption of oil into OpA40 at 0.1 s and at an aspect factor 1.5. Same wetting scale bar as in Fig. 1.
is greater than or equal to P : d=
4γ cos ϑ . P
(3)
γ is the interfacial tension between mercury and air (assumed 480 dyn cm−1 ), and θ is the contact angle between the edge of the advancing convex mercury meniscus and the solid surface (assumed 140◦ ). The use of this equation has many well-known shortcomings discussed by
Van Brakel [35], such as uncertainties in the values of γ and θ . For anisotropic structures, the Laplace equation was used to calculate an ‘effective’ intruded throat diameter (deff ) for every applied pressure of mercury. The effective throat size is related to the two diameters (d1 and d2 ) of an elliptical throat by the relation deff =
2 4γ cos ϑ 2(d2 α) = , = P 1/d1 + 1/d2 1+α
(4)
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Fig. 3. Experimental and simulated mercury intrusion curves for the clay and the PCC coating samples.
where α=
d1 . d2
For isotropic structures, with α = 1, Eq. (4) reduces to Eq. (3) and deff = d1 = d2 , as expected. To simulate percolation, mercury was initially applied at the xy plane at maximum z of the unit cell, at zero applied pressure. The applied pressure was then increased. At each pressure, all features which satisfied Eq. (4) and which were connected to the surface or to another filled feature were instantaneously filled with mercury [36]. This corresponded to percolation down from the top surface of the unit cells shown, for example, in Figs. 1 and 2. Only one hundred different sizes were used for the throats, so the simulated intrusion curves showed some step-wise behaviour (Fig. 3). Periodic boundary conditions made the simulation equivalent to applying mercury to a sheet of infinite width in the x direction and breadth in the y direction, with a thickness equal to the size of one unit cell in the z direction. The porosimetry of the coating was studied on collections of thin flakes, as described above, in which the thickness was around three orders of magnitude smaller than the other dimensions. Therefore it was reasonable to assume that percolation through the edges of the samples was negligible compared to percolation through their faces, and that there was a realistic match between the boundary conditions of the experimental and simulated percolation. 3.4. Simple wetting
void features can be represented by a single cylinder of radius R, known as the effective hydraulic radius (EHR). Then distance travelled, x, by a liquid front in time, t, is Rγ t cos θ x= (5) . 2η 3.5. Wetting of anisotropic structures Bosanquet [7] improved some of the many approximations implicit in Eq. (5) by considering the inertial and viscous forces which act as a fluid enters a capillary tube of radius r from an infinite reservoir (supersource). Balancing these forces with the capillary force, he showed that dx d dx = Pe πr 2 + 2πrγ cos θ, (6) πr 2 ρx + 8πηx dt dt dt where Pe is the external pressure applied at the entrance of the capillary tube. (On setting ρ = 0, Pe = 0, r = R, and integrating, this equation reduces to Eq. (5), as expected.) For elliptical throats with major radius r1 = d1 /2 and minor radius r2 = d2 /2, it follows readily from geometrical considerations of the forces acting that Eq. (6) becomes dx d dx πr1 r2 ρx + 8πηx dt dt dt = Pe πr1 r2 + 4r1 E(e2 )γ cos θ.
(7)
E(e) is the elliptical integral: π/2 E(e) = (1 − e sin2 θ )1/2 dθ.
(8)
0
The simplest model of wetting is that of Lucas and Washburn, who equated the capillary wetting force to the force required for laminar flow as described by the Poiseuille equation. An additional approximation is to assume that all the
The eccentricity, e, is defined as r2 e = 1 − 22 . r1
(9)
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Abramowitz and Stegun [37] have derived a polynomial approximation for E(e): 1 E(e) = (1 + a1 w + a2 w2 ) + (b1 w + b2 w2 ) loge w + error, (10) where w = 1 − e, |error| < 4 × 10−4 , a1 = 0.4630151, a2 = 0.1077812, b1 = 0.2452727, and b2 = 0.0412496. It follows from Eqs. (9) and (10) that
2 4 r r2 2 E(e2 ) = 1 + a1 + a2 r1 r1
4 2 r2 2 r2 r1 + b1 (11) + b2 . loge r1 r1 r2 By integration of Eq. (7) and letting a=
8η ρr1 r2
and b =
Pe πr2 + 4E(e2 )γ cos θ , πρr2
it can be shown that
2b 1 2 2 −at x2 − x1 = t − (1 − e ) , a a
(12)
where x1 is the initial position and x2 is the position after time t. The modified Bosanquet equation was used to calculate the wetting flux in each throat in the void network after every time-step, using a predictor–corrector method [4]. Each throat filled at a constant rate according to its size and shape, according to the Eqs. (7), (11) and (12). Once full, this constant fill rate was assumed to continue, and the excess liquid filled the pore attached to the end of the throat. The pore could be filled by the excess fluid from more than one throat, which could start to flow into it at different times. Once a pore was full, it started to fill the throats connected to it that were not already full and which were not already filling from other pores. These throats filled at their own flow rates, again calculated from Eqs. (7), (11) and (12). Thus although the wetting of the ellipsoidal cylindrical throats was calculated explicitly, the pores acted simply as volumes, and could have been of any shape. They were represented tetragonally, as being the most convenient shape for joining up to six throats aligned on the three Cartesian axes.
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If at any stage the outflow of a pore exceeded the inflow then a mass conservation restriction was applied which removed this imbalance and restricted the fluid flow further into the network. The permeation, Π(t), was quantified as the volumetric amount of fluid absorbed per unit area, Va (t): Π(t) =
Va (t) . qx qy
(13)
4. Experimental results 4.1. Characterisation of the particles and voids in the coatings The characteristics of the particles within the four samples, as measured by sedimentation and light scattering, are shown in columns 3–6 of Table 1. The aspect ratios, estimated from atomic force microscopy, are shown in column 7. Fig. 3 shows the mercury intrusion curves after correction for sample compression, Eq. (1), and conversion of the applied pressure to an effective throat entry diameter, Eq. (4). It can be seen that the two clays had markedly different intrusion curves. The PCC samples were more similar to each other, and had higher porosities than the clay coatings. The particles in the Speswhite coating have the highest aspect ratio, and the coating process favours a horizontal orientation during packing, resulting in slit-like pores (Figs. 4a and 4b). The Amazon particles have a lower aspect ratio, resulting in voids which are less slit-like than for Speswhite. The particles in OpA40 coating have elongated prismatic shape, they pack together to give pores which may also have a rod-like structure (Figs. 5a and 5b). The Albaglos particles, due to their calcitic morphology and rhombohedral shape, create almost spherical particles which, in turn, form an isotropic porous network structure. These and other electron micrographs of the surfaces were examined, covering areas of surface of around 40 µm2 . Each micrograph was split into four equal areas, and a count made in each area of the number of throats at the surface, which appeared to lead into the bulk of the material (Fig. 4a).
Table 1 Characteristics of particles and coatings Sample information
Sedimentation
Sample Sample type name
Width of particle size Median particle Width of particle Aspect ratio Median particle diameter, d50 (µm) distribution (µm) (particles) diameter, d50 (µm) distribution (µm)
PCC PCC Clay Clay
0.80 0.39 0.23 0.60
Albaglos OpA40 Amazon Speswhite
Light scattering
0.22 0.37 0.38 0.85
a Value estimated visually. See text for definitions of parameters.
– 0.49 0.48 3.28
Atomic force Mercury porosimetry microscopy
– 0.36 0.44 0.57
1a 2 8 16
Porosity (%) Median throat diameter, d50 (µm) 37.6 38.3 30.0 25.8
0.198 0.192 0.064 0.119
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(a)
(b) Fig. 4. Speswhite coating: scanning electron micrographs of (a) surface scan showing method of counting surface throats and (b) resin-embedded cross-sectional scan. Scale bars 2 and 5 µm, respectively.
This was done visually rather than by image analysis, because image analysis algorithms find it difficult to identify throats rather than surface pores in a surface electron micrograph. The results are shown in Table 2, together with the % standard errors of the mean, which arose chiefly from the heterogeneity of the surface structures rather than the subjectivity of the visual count. It can be seen that the number density of surface throats (or surface capillaries) is lower for Speswhite than for the Amazon coating, while a smaller number density is observed for Albaglos than for OpA40 coating.
4.2. Experimental permeation results The absorption curves of the ISIT permeation experiments (Fig. 6a) were extrapolated to time zero. The intercept at time zero was assumed to be due to almost instantaneous surface film transfer [4], which was then subtracted to give the oil uptake additional to this transfer. Also, taking the gravure roller footprint as 1 mm as described previously, allowed an absolute contact time to be estimated (Fig. 6b). It can be seen that there is a faster permeation rate into Ama-
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(a)
(b) Fig. 5. OpA40 coating: scanning electron micrographs of (a) surface scan and (b) resin-embedded cross-sectional scan. Scale bars 2 and 5 µm, respectively.
Table 2 Estimates of number density of surface throats and the correction factor, f , used for each sample Number density of surface throats (µm−2 )
Sample information Sample type
Sample name
Experimental
Experimental % standard error of mean
Model
Surface throat number density correction factor, f
PCC PCC Clay Clay
Albaglos OpA40 Amazon Speswhite
1.4 3.02 4.09 0.66
11 4 4 15
14.84 13.54 10.66 7.89
0.094 0.22 0.38 0.084
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(a)
(b) Fig. 6. Amount of oil transferred from gravure roller to coating samples at 100 N contact pressure. (a) Raw results and (b) the results are corrected for film transfer and with respect to absolute contact time.
zon and OpA40 at between 1 and 2 ms than for the other samples.
5. Simulation results 5.1. Matching of the model to the experimental characteristics Fig. 3 shows the fits of the simulated to the experimental percolation characteristics, which are close enough to generate meaningful differences in structure between the different types of sample. The asymptotes at low throat entry diameter are the simulated and experimental porosities, which
agree to within 0.1%. Following estimation of aspect factors, as described below, three characteristic parameters of the network model, namely connectivity, throat skew and pore skew, were adjusted to achieve the best fit between the simulated and experimental intrusion curves (Fig. 3 and Table 3). Although experimental techniques are available to measure the sizes and aspect ratios of particles in a porous solid, it is impossible in practice to measure the aspect factors of the voids. We therefore allocated values of aspect factor (rather than aspect ratio) to the experimental samples, on the basis of visual inspection of the micrographs. The three criteria used were: (i) whether the particles were rodlike (implying voids with aspect factor > 1), isotropic (aspect factor = 1) or plate-like (aspect factor < 1); (ii) that
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Table 3 Characteristic parameters of the network model used to generate the simulated structures for each sample Sample type
Sample name
Minimum throat entry diameter (µm)
Maximum throat entry diameter (µm)
Connectivity
Throat skew
Pore skew
Porosity (%)
Aspect factor (voids)
PCC PCC Clay Clay
Albaglos OpA40 Amazon Speswhite
0.0036 0.0036 0.0036 0.0036
2.39 2.38 2.31 2.32
5 3.4 4.6 4.8
1.3 0.78 1.84 1.63
2 1.6 3.1 1.8
37.6 38.3 30 25.8
1 1.5 0.8 0.6
Fig. 7. Simulated absorption of oil per unit area (µm3 /µm2 ) for clay coating samples. Speswhite and Amazon simulated at their estimated (α = 0.6 and 0.8, respectively) and isotropic (α = 1) aspect factors, with Speswhite also being simulated at an aspect factor of 0.01.
aspect factors for voids should be less than the aspect ratios of the particles, because the particles are not completely aligned; and (iii) that the aspect factors should give porerow spacings which were realistic relative to the dimensions of the experimental sample. The resulting values of the aspect factor are listed in Table 3. The sensitivity of the choice of value was tested by running simulations with a range of other values as well, including an aspect factor of 1 for each anisotropic sample. Despite the close match between the experimental and simulated mercury intrusion curves and porosities, and realistically assigned aspect factors, there was no correspondingly close match between the measured and simulated number densities of surface throats; Table 2 shows that the simulated number densities were significantly higher. Improving the match by decreasing the aspect factor to make the particles more platey proved unrealistic in another sense. For Speswhite, for example, an aspect factor of 0.1 provided a match, but created a simulated structure with a pore row spacing around order of magnitude larger than implied by the electron micrographs. So a simple correction factor, f , was used, where f = experimental number density /modelled number density.
(14)
This correction is valid while the wetting front is contained in the surface throats. However, it becomes increasingly invalid as the front progresses throughout the structure, as the body of the modelled structure would then have a porosity which was a factor of f greater than the experimental sample. Therefore the results below are compared to experiment both with and without the correction factor. 5.2. Volume averaged extent of permeation A number of trials were initially carried out without considering the correction factor f . The wetting algorithm was first tested on Speswhite clay, and it generated a set of absorption curves for a range of aspect factors from 0.01 to 1.0 (Fig. 7). It can be seen that for Speswhite there was generally a decrease in the amount of oil absorbed per unit area with a decrease in aspect factor, i.e., as the simulated voids became less rod-like and more slit-like. This effect was especially noticeable for the lowest aspect factors at the shortest times. However, several of the curves cross over each other at various times. It can also be seen that the curves show a series of steps and points of inflexion, corresponding to the filling of successive layers. The points of inflexion for the absorption simulations with lower aspect factors occur earlier than those with the higher aspect factors.
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Fig. 8. The amount of oil absorbed for the PCC samples, OpA40 and Albaglos, at their estimated (α = 1.5 and 1.0, respectively) and isotropic (α = 1) aspect factors.
Fig. 9. Modelled absorption rates of the four coatings, before applying the correction factor, compared to the experimental absorption rates at a time t of approximately 0.002 s.
The wetting algorithm was then used to calculate the amount of oil absorption for each coating structure, using the estimated aspect factors. ‘Control’ simulations were also performed for comparison. These had the experimental percolation and porosity of a particular experimental sample, but were isotropic (α = 1.0). The use of aspect factors of less than 1 for the clays caused the simulated absorption of oil to decrease relative to isotropic structures. It can be seen in Fig. 7 that Speswhite absorbs a higher amount of oil than Amazon. However, as time proceeds, the unit cell becomes saturated with fluid and the flow is impeded, reducing the absorption rate. As a result, the curve for Speswhite is shown to converge on to Amazon clay. Fig. 8 shows the amount of oil absorbed into the precipitated calcium carbonate coatings, using both estimated and isotropic aspect factors (which are both the same for Alba-
glos). The simulated absorption for Albaglos is shown to be lower than that for OpA40 control, up to a time of 0.007 s. However, comparing simulations with the appropriate estimated aspect factor, the simulation for Albaglos is shown to be lower than OpA40 for a much longer period time of 0.02 s. A comparison of the vertical scales of Figs. 7 and 8 shows that overall the simulated absorption for the clays is less than half of that predicted for the PCCs. To investigate whether the trends in the simulations match those of the experiments, we chose a time at which to compare the results. Inspection of Fig. 6b suggests that 0.002 s is a time which reveals the trends in the experimental results before they plateau. The resulting comparison is shown in Fig. 9. It can be seen that there is some correlation between the simulated and experimental trends for Speswhite, Albaglos and OpA40, but not for Amazon.
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Fig. 10. The modelled absorption rates of the four coatings, after applying the correction factor, compared to the experimental absorption rates at a time t of approximately 0.002 s.
Fig. 11. Detail of Fig. 1, showing permeation into Speswhite at 0.1 s. Same wetting scale bar as in Fig. 1.
The correction factor f for each coating structure was then applied to the simulated oil absorption curves, and the results are shown in Fig. 10. This gives a significant improvement in the agreement, as Amazon is now shown to have a modelled absorption rate higher than for the experimentally low permeable coating samples. 5.3. Fine structure and effect of inertia All the wetting results described so far, both experimental and simulated, have been volume or mass-averaged mea-
sures of the extent of permeation and the position of the wetting front. However, the steps in Figs. 7 and 8 show that there is also a fine structure within the simulated wetting which bears closer examination. A detail of Fig. 1, shown in Fig. 11, reveals advance wetting into small features. If the inertia of the fluid is ignored by using the Lucas– Washburn equation in each feature rather than the integrated Bosanquet equation (12), the wetting at 0.1 s changes from that shown in Fig. 11 to Fig. 12. It can be seen that there is a substantial difference in the extent of the filling if fluid inertia is ignored, and that the advance wetting in the small
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Fig. 12. As in Fig. 14, but ignoring the effects of inertia. Same wetting scale bar as in Fig. 1.
Fig. 13. Comparison of EHR and present model (left-hand axis), with experimental wetting (right-hand axis).
features is less noticeable. Fig. 13 shows that the Lucas– Washburn and EHR approach predicts wetting fronts two orders of magnitude faster than the present model, and also the wrong trends for the samples. Fig. 14 shows that there is a lack of advance wetting in Amazon. Fig. 15 shows a detail of Fig. 2 and reveals some advance wetting in OpA40, which is not evident in Albaglos (Fig. 16). Speswhite and OpA40, the samples which show advance wetting, both have a much broader void size distribution than Amazon and Albaglos (Table 1). This rela-
tionship is in agreement with the findings of Schoelkopf et al. [38] for porous calcium carbonate blocks, and of Ridgway and co-workers [4] for void networks with broad size distributions. 5.4. The effect of anisotropy and inertia on the wetting profile of a structure The effect of anisotropy on the wetting profile, when the inertial term is included or disregarded in the wetting simu-
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Fig. 14. The permeation of Amazon sample at 0.1 s.
Fig. 15. Detailed image of Fig. 2, showing permeation into OpA40 at 0.1 s. Same wetting scale bar as in Fig. 1.
lation, was examined by modelling Speswhite and OpA40. These two samples were chosen as they exhibited two different types of void shapes: slit-like and rod-like pores, respectively. Wetting simulations using both Bosanquet and Lucas–Washburn equations, which include and disregard inertia, were tested on the two structures at their isotropic and respective estimated aspect factor. The results for Speswhite and OpA40 are shown in Figs. 17 and 18, respectively.
The effect of anisotropy on Speswhite, when the inertial term is included in the simulation, caused a reduction in absorption of 23% (Fig. 17), while only a 3% reduction was observed when the inertial term is disregarded. Meanwhile, the effect of anisotropy on OpA40, when the inertial term is included in the simulation, caused an increase in absorption of about 21% (Fig. 18), while only a 4% reduction was observed when the inertial term is disregarded. In summary,
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was taken to be the layer where the number of empty voids outweighs the number of voids containing fluid. The wetting profile in Fig. 19 shows the vertical extent of the advanced wetting to be relatively close to the vertical extent of the wetting front, having a distance A for the advanced wetting of only approximately 6 µm. However, when the estimated aspect factor is applied to the structure, Fig. 20 shows a much longer advanced wetting distance A of approximately 11 µm. Hence, the effects of anisotropy caused further advanced wetting and preferential flow, not only in the large features but also in the narrow features of the structure, irrespective of inertia.
6. Summary
Fig. 16. The permeation of Albaglos sample at 0.1 s. Same wetting scale bar as in Fig. 1.
anisotropy (from isotropic to slit-like or rod-like pores) only showed a significant change in permeation when inertial effects are introduced in the wetting simulation. To examine the effect of anisotropy without inertia, the advance wetting was best visualised by rendering the isotropic and anisotropic structures as wire frames, Figs. 19 and 20, respectively. The voids containing any amount of fluid are outlined in blue colour, while voids containing no fluid are outlined in red. The transition line of where the wetting front ends and where the advanced wetting begins
We set out with the ambitious task of fitting a network model of void structure to a wide range of the experimental properties of four paper coating samples. The modelled void structures matched the mercury intrusion (percolation) properties and porosities of the experimental samples, in itself an achievement rarely seen in the literature. When additional properties were included, namely the spacing of the void features in the body of the sample, the number density of the surface throats, and the anisotropy of the voids, a compromise had to be struck, and a correction factor applied. The improvement caused by the correction shows that a more sophisticated method of packing the void features in the model is desirable. The model’s overestimate of the number of surface throats is probably due to the fact that a slight overlap of platey particles at the surface in an experimental sample occludes many of the surface throats, whereas in the model, no such overlap can take place.
Fig. 17. Simulated permeation curves for Speswhite at its estimated (α = 0.6) and isotropic (α = 1.0) aspect factor, and the results using both Bosanquet and Lucas–Washburn equations, including and disregarding inertia.
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Fig. 18. Simulated permeation curves for OpA40 at its estimated (α = 1.5) and isotropic (α = 1.0) aspect factor, and the results using both Bosanquet and Lucas–Washburn equations, including and disregarding inertia.
Fig. 19. Simulation of the absorption of oil into OpA40 at 0.1 s, applying an isotropic aspect factor (α = 1.0) and disregarding inertia (Lucas–Washburn equation). The structure is rendered into a wire frame in which the voids containing any amount of fluid are outlined with a blue colour, while voids containing no fluid are outlined with a red colour.
Despite these difficulties, models of the void structures were produced which provided a major improvement over the EHR approach, and which were of sufficient realism to provide insights into the wetting process in the samples. It was shown that the inertia of the fluid as it enters void features causes a considerable change in the extent of permeation over a few milliseconds, a timescale relevant to the time scale of printing in a modern press. Com-
parison of the void structures with anisotropic structures (all of which match the percolation and porosity criteria) provided insights into the precise nature of wetting into a range of experimental coating samples. For example, in the more anisotropic sample of each type, namely Speswhite and OpA40, fast advanced wetting occurred through narrow void features. This phenomenon has also been observed by other workers, as cited above. It would not be visible by any ex-
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Fig. 20. Simulation of the absorption of oil into OpA40 at 0.1 s, applying an estimated aspect factor (α = 1.5) and disregarding inertia (Lucas–Washburn equation). The structure is rendered into a wire frame in which the voids containing any amount of fluid are outlined with a blue colour, while voids containing no fluid are outlined with a red colour.
perimental technique which measured the volume averaged position of the wetting front. It can be categorised as preferential flow, although preferential flow is usually an experimentally observed quantity and therefore advance wetting into low-volume features tends to be overlooked. It is difficult to express the detail of the wetting in static snapshots, so we have made the two structures mentioned above available as supplementary on-line material, with viewing instructions at http://www.pore-cor.com.virtual_reality.htm. There is also more information to be considered about the simulation, such as a sensitivity analysis of the fitting parameters, and the tracking of all phenomena with time through the structures. However, such studies are outside the scope of this work, which is concerned with the overall comparison of the Darcy wetting front compared to that inferred from experiment. Generic findings from this work are currently being used in the design of new high performance paper coatings.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17]
Acknowledgments This project was funded by the University of Plymouth Science Faculty from its block research grant, with additional support from SCA, Sundsvall, Sweden. We are grateful for academic and experimental support from Sven Lohmander (formerly of STFi, Stockholm, Sweden), Edward Seyler (SeGan Ltd., Cornwall, UK), Patrick Gane and Joachim Schoelkopf (Omya, Oftringen, Switzerland), and Eirian Jones, (University of Plymouth, UK). The virtual reality programming is by Alexander Matthews.
[18] [19] [20] [21] [22] [23] [24] [25] [26] [27]
S.L. Bryant, P.R. King, D.W. Mellor, Trans. Por. Med. 11 (1993) 53. V.S. Patwardhan, C. Tien, AIChE J. 32 (1986) 321. D.J. Sherwood, Proc. R. Soc. Math. Phys. Sci. 445 (1994) 679. J. Schoelkopf, C.J. Ridgway, P.A.C. Gane, G.P. Matthews, D.C. Spielmann, J. Colloid Interface Sci. 227 (2000) 119. C.J. Ridgway, J. Schoelkopf, G.P. Matthews, P.A.C. Gane, P.W. James, J. Colloid Interface Sci. 239 (2001) 417. A. Johnson, I.M. Roy, G.P. Matthews, D. Patel, European J. Soil Sci. 54 (2003) 477. C.H. Bosanquet, Philos. Mag. Ser. 6 (45) (1923) 525. A. Marmur, Adv. Colloid Interface Sci. 39 (1992) 13. A. Marmur, J. Colloid Interface Sci. 122 (1988) 209. E. Chibowski, L. Holysz, J. Adhes. Sci. Technol. 11 (1997) 1289. L.-H. Lee, J. Adhes. Sci. Technol. 7 (1993) 583. T.X. Lu, D.R. Nielsen, J.W. Biggar, Water Resour. Res. 31 (1995) 11. N. Ichikawa, Y. Satoda, J. Colloid Interface Sci. 162 (1994) 350. A. Borhan, K.K. Rungta, J. Colloid Interface Sci. 155 (1993) 438. A.I. Moshinskii, Colloid J. 59 (1997) 62. J. Szekely, A.W. Neumann, Y.K. Chuang, J. Colloid Interface Sci. 35 (1971) 273. S.X. Ma, G. Mason, N.R. Morrow, Colloids Surf. A Physicochem. Eng. Aspects 117 (1996) 273. P.M. Adler, C.G. Jacquin, J.A. Quiblier, Int. J. Multiphase Flow 16 (1990) 691. G.R. Jerauld, S.J. Salter, Trans. Por. Med. 5 (1990) 103. R.L. Portsmouth, L.F. Gladden, Trans. Inst. Chem. Eng. 70 (1992) 63. W.B. Haines, J. Agric. Sci. 17 (1927) 264. L. Paterson, S. Painter, M.A. Knackstedt, W.V. Pinczewski, Physica A 233 (1996) 619. V.M. Starov, V.V. Kalinin, V.I. Ivanov, Colloids Surf. A Physicochem. Eng. Aspects 91 (1994) 149. C. Coulliette, C. Pozrikidis, J. Fluid Mech. 358 (1998) 1. I. Ginzbourg, P.M. Adler, Trans. Por. Med. 20 (1995) 37. J. Schoelkopf, P.A.C. Gane, C.J. Ridgway, G.P. Matthews, Colloids Surf. A Physicochem. Eng. Aspects 206 (2002) 445. J.E. O’Shea, Pigments for Paper, Tappi Press, Atlanta, 1997.
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[28] P. Lepoutre, Prog. Org. Coat. 17 (1989) 89. [29] S. Lohmander, Nord. Pulp Paper Res. J. 15 (2000) 221. [30] J.W. McCoy, in: Tappi Proceedings—1998 Coating/Papermakers Conference, 1998, p. 121. [31] P.A.C. Gane, J.P. Kettle, G.P. Matthews, C.J. Ridgway, Ind. Eng. Chem. Res. 35 (1996) 1753. [32] CRC Handbook of Chemistry and Physics, CRC Press, Boca Raton, FL, 1979. [33] Y. Ishigami, Y. Gama, Y. Sano, S. Lang, F. Wagner, Biotechnol. Lett. 16 (1994) 593.
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[34] G.P. Matthews, A.K. Moss, C.J. Ridgway, Powder Technol. 83 (1995) 61. [35] J. van Brakel, S. Modry, M. Svata, Powder Technol. 29 (1981) 1. [36] G.P. Matthews, A.K. Moss, M.C. Spearing, F. Voland, Powder Technol. 76 (1993) 95. [37] M. Abramowitz, I.E. Stegun, Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables, Gannon Distributing Co., 1972. [38] C.J. Ridgway, P.A.C. Gane, J. Schoelkopf, J. Colloid Interface Sci. 252 (2002) 373.
Influence of anisotropy on the dynamic wetting and permeation of paper coatings Paul A. Bodurtha a , G. Peter Matthews a,∗ , John P. Kettle b , Ian M. Roy a a Environmental and Fluid Modelling Group, University of Plymouth, Plymouth PL4 8AA, UK b SCA Graphic Research, SE-850 03 Sundsvall, Sweden
Received 16 June 2003; accepted 14 August 2004 Available online 2 December 2004
Abstract A void network model, named Pore-Cor, has been used to study the permeation of an ink solvent into paper coating formulations coated onto a synthetic substrate. The network model generated anisotropic void networks of rectangular cross-sectional pores connected by elliptical cross-sectional throats. These structures had porosities and mercury intrusion properties which closely matched those of the experimental samples. The permeation of hexadecane, used as an analogue for the experimental test oil, was then simulated through these void structures. The simulations were compared to measurements of the permeation of mineral oil into four types of paper coating formulation. The simulations showed that the inertia of the fluid as it enters void features causes a considerable change in wetting over a few milliseconds, a timescale relevant to printing in a modern press. They also showed that in the more anisotropic samples, fast advance wetting occurred through narrow void features. It was found that the match between experimental and simulated wetting could be improved by correcting the simulation for the number of surface throats. The simulations showed a more realistic experimental trend, and much greater preferential flow, than the traditional Lucas–Washburn and effective hydraulic radius approaches. 2004 Elsevier Inc. All rights reserved. Keywords: Anisotropy; Paper coatings; Absorption; Porous media; Modelling; Wetting; Permeation; Mercury porosimetry; Pore-Cor; Shale
1. Introduction If a porous solid is made up from a homogeneous packing of unsorted spherical particles bound together, the resulting void structure will normally be structurally isotropic—i.e., it will have approximately the same structural characteristics in any direction, and hence approximately the same permeation characteristics [1]. However, a sample made up from non-isometric particles such as clay platelets, packed in an aligned manner, would be structurally anisotropic [2]. Consequently, the rate of wetting would vary with direction. Such anisotropic permeation is relevant to the printing of paper coated with aligned anisotropic clay particles, and to other processes such as the subterranean migration of fluids in shale [3]. * Corresponding author. Fax: +44(0)1752-233021.
E-mail address: [email protected] (G.P. Matthews). 0021-9797/$ – see front matter 2004 Elsevier Inc. All rights reserved. doi:10.1016/j.jcis.2004.08.072
When designing paper coatings with optimum printing properties, manufacturers have a wide choice of mineral pigments to choose from, ranging from isotropic precipitated calcium carbonates, through ‘blocky’ clays such as Georgia kaolin, to ‘platey’ clays such as Speswhite with a high aspect factor. Each type of pigment can also be prepared with a wide range of particle sizes, and each size distribution can have a different range and skew. The problem when assessing the ink absorption properties of different pigments is that if one compares particles with different aspect factors, their size distribution changes also, and it is difficult to separate one effect from the other. So in this work a void network model has been used to highlight the effects of anisotropy and aspect factor which are otherwise hidden within experimental trends. The network model, named Pore-Cor (pore-level properties correlator), can accommodate changes in void sizes and connectivity between samples, such that additional changes caused by anisotropy can
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be revealed and studied. It has been used previously for modelling the behaviour of fluids in a wide range of materials such as porous calcium carbonate blocks [4,5] and soils [6]. A new version of the network model has been used, which generates an anisotropic three-dimensional void network of pores with a rectangular cross-section connected by throats with an elliptical cross-section. The equation for wetting was an inertially modified version of the Lucas–Washburn equation developed by Bosanquet [7]. Previous researchers have shown the dependence of wetting on the thickness of the porous layer [8] and the size of the applied droplet of fluid [9]. Others have used an effective hydraulic radius approximation [10] or the equation of state approach [11]. However, the wetting of such simple systems is totally different from that in more complex networks [12]. Nevertheless, a series of approximations had to be made to make the computation time manageable, in practice less than 30 h on a 1 GHz computer. Wetting and consequent permeation start at the contact point or surface between the fluid reservoir or droplet and the porous solid. There is an almost instantaneous wetting jump when a wetting fluid contacts a wettable porous sample [4]. Next, the fluid has to accelerate from rest into the various void features. Many researchers have improved Bosanquet’s description of initial, inertial, and continuity influences [8], and some have included a kinetic energy term [13] or used full Navier–Stokes equations [14,15]. It has been shown previously that the time-step for the wetting calculation in our network model must be no more than 1 ns [4], and consequently the calculation time would have been unmanageable if these more sophisticated effects had been included. Improvements to overcome a singularity at the tube entrance at time zero have been suggested by Szekely et al. [16], but are insignificant under the conditions of this work. The way the fluid subsequently permeates through the structure is then controlled by a further series of factors, all of which are approximated in this work. These are the sizes and shapes of the voids [17], their connection into the network [18–20], wetting jumps [21,22], the surface energy between the fluid and the solid phase [23], and the density, viscosity and applied pressure of the fluid [24–26]. Modern printing presses feed paper through a printing nip, a few centimetres long, at speeds of up to 15 m s−1 . So the fluid contact time in the nip is of the order of a few milliseconds. At longer times, of the order of seconds to tens of seconds, there may be problems due to back-transfer of ink onto other rollers or ink smearing, but these effects are beyond the scope of this publication.
adhesive to bind the pigment particles together and to the substrate [27]. Typically, the weight fraction of pigment is 80–95% [28]. Assuming a pigment density of 2.6 g cm−3 and a binder density of 1.0 g cm−3 , the volume fraction of the pigment is about 70%. Four types of commercially available pigments were studied: Speswhite and Amazon 90 SD (both Kaolin clays), and OpacarbA40 (OpA40) and Albaglos (both precipitated calcium carbonates or PCCs). Speswhite is a primary coarsegrained kaolin mineral supplied from Imerys, UK, while Amazon 90 SD is a secondary fine-grained kaolin supplied from Caulim da Amazônia S.A., Brazil. Both have a platey or foliated mineral structure. By contrast, the PCC group contains chemically synthesised calcium carbonate minerals belonging to either a calcitic or an aragonitic crystalline polymorph. OpA40 is aragonitic in structure, with an acicular (or needle-like) crystal habit. Albaglos contains a calcitic structure, with a rhombohedral crystal habit. The aragonite and Albaglos pigments were supplied by Specialty Minerals Nordic Oy, Finland. In the coating preparation, sodium polyacrylate (Dispex N 40 from Allied Colloids Ltd., UK) was used as a dispersant for the pigments. A carboxylated styrene-butadiene latex (DPL 935 from Dow Rheinmünster GmbH, Germany), with a diameter of 0.15 µm and a glass transition temperature T g of 8 ◦ C, was used as a binder, and carboxymethyl cellulose (CMC), with an average molecular weight of 45000 (FF5 from Noviant, Finland), was used as water retention aid. The kaolin pigments were received in a dry form and were dispersed in water under conditions suggested by the manufacturers. The samples for the permeation experiments were coated onto an impermeable plastic backing known as Synteape (Arjo Wiggins Ltd., UK). This has a slightly rough, hooked surface on which the coating can adhere, producing the flexible but stable surface necessary for the permeation experiments. The pigment suspensions contained 1 pph CMC and 10 pph latex. All suspensions were prepared in a conventional manner to 60% solids content by weight and a pH value of 8.5. They were drawn down onto the Synteape using a wire-wound rod in a bench coater (K-coater, RK PrintCoat Instruments Ltd., Royston, UK), to a coat weight of around 80 g m−2 in each case. Synteape is permeable from the side, so for porosimetry the suspensions were benchcoated onto the aluminium backing foils to a coat weight of about 18 g m−2 . All coating and drying was carried out at room temperature. Each coated sample strip had dimensions of 4 × 25 cm.
2. Materials and methods
2.2. Techniques for characterising particle and void size distributions
2.1. Samples In its simplest form a paper coating consists of mineral pigments, binders and air voids. The binders act as an
Particle diameters and the widths of their size distributions were measured by X-ray monitoring of sedimentation rate (Micromeritics Sedigraph 5000), and by light scattering
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(Malvern Mastersizer 1000) [29]. No light scattering measurements were available for Albaglos. The aspect ratios of the pigment particles were estimated from image analysis of atomic force micrographs [29]. Aspect ratio is defined for plate-like structures as the ratio of the maximum dimension of the particle and its thickness; the more platey a particle, the higher its aspect ratio. For acicular (rod-like) structures, aspect ratio is defined as the length of the particle divided by the radial diameter. A scanning electron microscope (Cambridge 360) was used to obtain two types of view of the void structure of the samples, namely that manifest at the surface of the sample and that visible in a cross-section, at magnifications around 40,000×. For the surface views, the samples were gold plated, whereas for the cross-sections, the samples were embedded in epoxy resin and viewed in back-scatter mode [30]. The micrographs were used to gain semi-quantitative estimates of appropriate aspect factors, and to estimate the number density of surface throats. For mercury porosimetry, the samples were first removed from the aluminium foil backing, thus forming thin flakes approximately 1 cm × 1 cm × 8 µm. Mercury intrusion curves for collections of similar fragments were then measured using a Micromeritics Autopore III mercury porosimeter up to a maximum mercury pressure of 414 MPa (60,000 psia), with an equilibration time at each pressure of 60 s. The mercury intrusion measurements were corrected for the compression of mercury, expansion of the glass sample chamber or ‘penetrometer’, and compressibility of the solid phase of the sample, by use of the following equation from Gane et al. [31]: 1 P log10 1 + Vint = Vobs − δVblank + 0.175 Vbulk 1820 1 (P − P ) 1 − Vbulk (1) (1 − Φ 1 ) 1 − exp . Mss Vint is the volume of intrusion into the sample, Vobs the intruded mercury volume reading, δVblank the change in the 1 the sample bulk volume at blank run volume reading, Vbulk atmospheric pressure, P the applied pressure, Φ 1 the porosity at atmospheric pressure, P 1 the atmospheric pressure and Mss the bulk modulus of the solid sample. 2.3. Fluids The oil used was a non-setting test oil designed for use as the solvent or ‘vehicle’ for offset printing inks. It was supplied by Haltermann Products, a division of Ascot Speciality Chemicals, and contained a mixture of paraffinic, naphthenic and aromatic hydrocarbons (mainly C16 –C18 chains). Its density ρ was measured gravimetrically to be approximately 1025.3 kg m−3 , and its viscosity η was measured by a control stress rheometer (AR1000, TA Instruments) to be about 7.4 mPa s at 20 ◦ C, suggesting mainly paraffinic components.
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The exact formulation of the ink was commercially confidential, so modelling was carried out using hexadecane, a paraffinic oil with similar physical properties. It had a surface tension γ of 0.0246 N m−1 , a density ρ of 773.31 kg m−3 , and a viscosity η of 3.3 mPa s at 20 ◦ C [32]. The contact angle θ was 40◦ , estimated from comparing contact angles of similar mineral oils [33]. The properties of both the test oil and hexadecane were tested on a freshly ground calcite crystal surface, using wet grinding by 1200 grit paper sprayed with tap water to prevent oxidation. Both the test oil and hexadecane were fully wetting, in that no fluid contact angle was visible. On aged surfaces, which have adsorbed atmospheric aerosols, smoke or vapours including water, the contact angle of both fluids increases to around 40◦ as published [33]. This behaviour on aged surfaces was taken to be that which was most representative of actual paper coatings. 2.4. Method for measuring permeation The method for measuring liquid permeation properties involved the novel use of an ink surface interaction tester (ISIT, SeGan Ltd., UK). The procedure involved a sample roller in contact with a gravure roller. The gravure roller (IGT Testing Systems, Amsterdam) comprised an array of small pits in a polished surface, with sizes and spacings such that complete transfer of the fluid onto a substrate of the same surface area would produce a film of thickness 1.5 µm. Examination of the roller using a confocal laser scanning microscope showed that the radius r of each pit was around 50 µm, with a depth s of approximately 10 µm. The footprint of the gravure roller in contact with the mounted coating strip was measured by the use of carbon paper, and found to be approximately 1 mm in the direction of rotation. The length of the footprint was then used to calculate the time of contact between the gravure roller and the coating strip, for each speed setting. The use of a gravure roller had the benefit that flooding of the sample was avoided, and that there was minimal uncertainty in the mode of transfer of the fluid onto the sample. Since the fluid was recessed in the gravure pattern, the fluid itself had a delivery pressure of zero when in contact with the sample. The gravure roller was loaded with oil, and any excess removed so that a known volume of fluid was available for transfer. The sample strip was mounted onto the sample roller. The sample roller and the gravure roller were pressed together under a known contact pressure so that the beginning of the sample strip was in contact with the gravure roller. Then the rollers were turned, still in non-sliding contact with each other, through one revolution at an exactly controlled speed. Oil transferred from the gravure roller to the sample strip. The sample strip was then demounted from the sample roller. A jig and scalpel was then used to cut out an exactly reproducible rectangular area (20 × 40 mm) of the saturated sample, while avoiding compressing the sample and keeping
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away from edges or partially saturated areas at the beginning and the end of the sample strip. The cut rectangle was then weighed, and the amount of fluid uptake calculated by comparison with an equal area of dry sample. The above procedure was followed for permeation measurements at speed settings between 0.1 and 2.0 m s−1 , with a new coating strip being used for each test run. The fluid was delivered from the gravure roller to the sample in the order of 0.01 s (t∞ ), as the results below show. So the fluid velocity v = s/t∞ was of the order of 1 mm/s. On the basis of the dimensions of the gravure roller pits and the fluid properties described above, the Reynolds number (2ρrv/η) was of the order of 0.01, i.e., five orders of magnitude lower than the turbulence threshold of around 2000. The compressing action of the roller on the coating structure was accounted for by a series of experiments using Amazon coating as a test sample. The amount of fluid transferred to the coating was measured at a range of contact pressures from 100 to 600 N at a slower contact speed of 0.5 m s−1 to emphasise any pressure-induced effects. A higher transfer of oil was observed at 400 N and above, and could be attributed to a ‘sponge’ effect, with enhanced uptake being caused by the compression and subsequent relaxation of the coating structure. Although such an effect may well occur during a commercial printing process, it is also dependent on the sample elasticity, which brings in another experimental variable. Therefore all the other experiments reported in this work were carried out at a contact pressure of 100 N to avoid this effect.
3. Void network modelling The network model generates a void structure comprising an infinite array of repeating, identical unit cells. Each unit cell is made up from 1000 cubic pores, in a 10 × 10 × 10 array, equally spaced from its neighbouring pores in the Cartesian coordinates x, y and z by the ‘pore row spacing’ q. Hence, each unit cell is a cube of side length 10q. Cylindrical throats connect each pore and there can be up to six throats per pore—i.e., one on each face. The number of throats connected to a particular pore is termed the ‘pore coordination number,’ and the arithmetic mean of this quantity over the whole unit cell is the connectivity. Individual pore coordination numbers may range from 0 to 6, and a typical value of the connectivity for matching the percolation properties of a real sample is 3.5. The percolation characteristics and the porosity of the overall void structure are matched to the experimental mercury intrusion curve of a chosen sample by adjusting the skews of the throat and pore size distributions, and the connectivity. The void network model is also capable of simulating size-correlated networks such as layered or laminated structures [34]. However, electron microscopy suggested that the structures had no short-range size correlation,
and it was not possible to fit any correlated structures generated by our model to the mercury intrusion curves. In order to simulate anisotropic structures, five modifications were made to the model: (i) the geometry of the unit cell was made anisotropic, (ii) the porosity calculation was modified, (iii) the percolation equations were modified, (iv) the equations for wetting were modified, and (v) the graphical routines were altered so that the geometry of the anisotropic structures could be viewed in both static and virtual reality modes. Modifications (i)–(iv) are now described. With regard to (v), many examples of static snapshots of the structures are presented in this paper. They are available in color on http://www.sciencedirect.com and can be viewed in virtual reality at http://www.pore-cor.com/virtual_reality. htm. 3.1. Generation of an anisotropic unit cell The unit cell was made anisotropic by rescaling the z-axis relative to the x- and y-axes. When the z-axis was rescaled to make features smaller, the pores became tetragonal, and the throats in the x and y directions gained ellipsoidal crosssections with the major axis in the x or y direction. The effect is illustrated in Fig. 1, in which the z-axis is shown vertically. Alternatively, the z-axis was scaled so that all features had larger dimensions with respect to this axis with the dimensions remaining constant on the other two axes (Fig. 2). Such a transformation results in rod-like pores, and the throats in the x and y direction become ellipsoidal in cross-section, but in this case with the major axis in the z direction. The scaling factor was termed the ‘aspect factor’ α, and the rescaled pore row spacings qx , qy and qz in the x, y and z directions were q, q and αq, respectively. The aspect factors α of the structures shown in Figs. 1 and 2 are 0.6 and 1.5, respectively. 3.2. Porosity calculations for anisotropic void structures The porosity Φ of a porous medium is defined as the percentage of void space within the total bulk medium. For anisotropic void networks, the porosity equation for a unit cell generated by the network model was modified so that it referred to the volume of throats with an elliptic crosssection and pores with a rectangular cross-section: 3 1000 n=1 Ln,x Ln,y Ln,z + i=1 πr1,n,i r2,n,i hn,i , (2) Φ= 1000qx qy qz where Ln,x , Ln,y and Ln,z refer to the width of the nth pore in the x, y and z directions, r1,n,i and r2,n,i are the two radii of the nth ellipsoidal throat in direction I , and hn,i is length of the nth throat in the ith Cartesian direction. 3.3. Percolation The Laplace equation gives the diameter d of a cylindrical throat in an incompressible solid exposed to mercury, which is intruded when the pressure applied to the mercury
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Fig. 1. Simulation of the absorption of oil (hexadecane) into Speswhite at 0.1 s and at an aspect factor 0.6. Degree of wetting as shown on wetting scale bar.
Fig. 2. Simulation of the absorption of oil into OpA40 at 0.1 s and at an aspect factor 1.5. Same wetting scale bar as in Fig. 1.
is greater than or equal to P : d=
4γ cos ϑ . P
(3)
γ is the interfacial tension between mercury and air (assumed 480 dyn cm−1 ), and θ is the contact angle between the edge of the advancing convex mercury meniscus and the solid surface (assumed 140◦ ). The use of this equation has many well-known shortcomings discussed by
Van Brakel [35], such as uncertainties in the values of γ and θ . For anisotropic structures, the Laplace equation was used to calculate an ‘effective’ intruded throat diameter (deff ) for every applied pressure of mercury. The effective throat size is related to the two diameters (d1 and d2 ) of an elliptical throat by the relation deff =
2 4γ cos ϑ 2(d2 α) = , = P 1/d1 + 1/d2 1+α
(4)
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Fig. 3. Experimental and simulated mercury intrusion curves for the clay and the PCC coating samples.
where α=
d1 . d2
For isotropic structures, with α = 1, Eq. (4) reduces to Eq. (3) and deff = d1 = d2 , as expected. To simulate percolation, mercury was initially applied at the xy plane at maximum z of the unit cell, at zero applied pressure. The applied pressure was then increased. At each pressure, all features which satisfied Eq. (4) and which were connected to the surface or to another filled feature were instantaneously filled with mercury [36]. This corresponded to percolation down from the top surface of the unit cells shown, for example, in Figs. 1 and 2. Only one hundred different sizes were used for the throats, so the simulated intrusion curves showed some step-wise behaviour (Fig. 3). Periodic boundary conditions made the simulation equivalent to applying mercury to a sheet of infinite width in the x direction and breadth in the y direction, with a thickness equal to the size of one unit cell in the z direction. The porosimetry of the coating was studied on collections of thin flakes, as described above, in which the thickness was around three orders of magnitude smaller than the other dimensions. Therefore it was reasonable to assume that percolation through the edges of the samples was negligible compared to percolation through their faces, and that there was a realistic match between the boundary conditions of the experimental and simulated percolation. 3.4. Simple wetting
void features can be represented by a single cylinder of radius R, known as the effective hydraulic radius (EHR). Then distance travelled, x, by a liquid front in time, t, is Rγ t cos θ x= (5) . 2η 3.5. Wetting of anisotropic structures Bosanquet [7] improved some of the many approximations implicit in Eq. (5) by considering the inertial and viscous forces which act as a fluid enters a capillary tube of radius r from an infinite reservoir (supersource). Balancing these forces with the capillary force, he showed that dx d dx = Pe πr 2 + 2πrγ cos θ, (6) πr 2 ρx + 8πηx dt dt dt where Pe is the external pressure applied at the entrance of the capillary tube. (On setting ρ = 0, Pe = 0, r = R, and integrating, this equation reduces to Eq. (5), as expected.) For elliptical throats with major radius r1 = d1 /2 and minor radius r2 = d2 /2, it follows readily from geometrical considerations of the forces acting that Eq. (6) becomes dx d dx πr1 r2 ρx + 8πηx dt dt dt = Pe πr1 r2 + 4r1 E(e2 )γ cos θ.
(7)
E(e) is the elliptical integral: π/2 E(e) = (1 − e sin2 θ )1/2 dθ.
(8)
0
The simplest model of wetting is that of Lucas and Washburn, who equated the capillary wetting force to the force required for laminar flow as described by the Poiseuille equation. An additional approximation is to assume that all the
The eccentricity, e, is defined as r2 e = 1 − 22 . r1
(9)
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Abramowitz and Stegun [37] have derived a polynomial approximation for E(e): 1 E(e) = (1 + a1 w + a2 w2 ) + (b1 w + b2 w2 ) loge w + error, (10) where w = 1 − e, |error| < 4 × 10−4 , a1 = 0.4630151, a2 = 0.1077812, b1 = 0.2452727, and b2 = 0.0412496. It follows from Eqs. (9) and (10) that
2 4 r r2 2 E(e2 ) = 1 + a1 + a2 r1 r1
4 2 r2 2 r2 r1 + b1 (11) + b2 . loge r1 r1 r2 By integration of Eq. (7) and letting a=
8η ρr1 r2
and b =
Pe πr2 + 4E(e2 )γ cos θ , πρr2
it can be shown that
2b 1 2 2 −at x2 − x1 = t − (1 − e ) , a a
(12)
where x1 is the initial position and x2 is the position after time t. The modified Bosanquet equation was used to calculate the wetting flux in each throat in the void network after every time-step, using a predictor–corrector method [4]. Each throat filled at a constant rate according to its size and shape, according to the Eqs. (7), (11) and (12). Once full, this constant fill rate was assumed to continue, and the excess liquid filled the pore attached to the end of the throat. The pore could be filled by the excess fluid from more than one throat, which could start to flow into it at different times. Once a pore was full, it started to fill the throats connected to it that were not already full and which were not already filling from other pores. These throats filled at their own flow rates, again calculated from Eqs. (7), (11) and (12). Thus although the wetting of the ellipsoidal cylindrical throats was calculated explicitly, the pores acted simply as volumes, and could have been of any shape. They were represented tetragonally, as being the most convenient shape for joining up to six throats aligned on the three Cartesian axes.
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If at any stage the outflow of a pore exceeded the inflow then a mass conservation restriction was applied which removed this imbalance and restricted the fluid flow further into the network. The permeation, Π(t), was quantified as the volumetric amount of fluid absorbed per unit area, Va (t): Π(t) =
Va (t) . qx qy
(13)
4. Experimental results 4.1. Characterisation of the particles and voids in the coatings The characteristics of the particles within the four samples, as measured by sedimentation and light scattering, are shown in columns 3–6 of Table 1. The aspect ratios, estimated from atomic force microscopy, are shown in column 7. Fig. 3 shows the mercury intrusion curves after correction for sample compression, Eq. (1), and conversion of the applied pressure to an effective throat entry diameter, Eq. (4). It can be seen that the two clays had markedly different intrusion curves. The PCC samples were more similar to each other, and had higher porosities than the clay coatings. The particles in the Speswhite coating have the highest aspect ratio, and the coating process favours a horizontal orientation during packing, resulting in slit-like pores (Figs. 4a and 4b). The Amazon particles have a lower aspect ratio, resulting in voids which are less slit-like than for Speswhite. The particles in OpA40 coating have elongated prismatic shape, they pack together to give pores which may also have a rod-like structure (Figs. 5a and 5b). The Albaglos particles, due to their calcitic morphology and rhombohedral shape, create almost spherical particles which, in turn, form an isotropic porous network structure. These and other electron micrographs of the surfaces were examined, covering areas of surface of around 40 µm2 . Each micrograph was split into four equal areas, and a count made in each area of the number of throats at the surface, which appeared to lead into the bulk of the material (Fig. 4a).
Table 1 Characteristics of particles and coatings Sample information
Sedimentation
Sample Sample type name
Width of particle size Median particle Width of particle Aspect ratio Median particle diameter, d50 (µm) distribution (µm) (particles) diameter, d50 (µm) distribution (µm)
PCC PCC Clay Clay
0.80 0.39 0.23 0.60
Albaglos OpA40 Amazon Speswhite
Light scattering
0.22 0.37 0.38 0.85
a Value estimated visually. See text for definitions of parameters.
– 0.49 0.48 3.28
Atomic force Mercury porosimetry microscopy
– 0.36 0.44 0.57
1a 2 8 16
Porosity (%) Median throat diameter, d50 (µm) 37.6 38.3 30.0 25.8
0.198 0.192 0.064 0.119
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(a)
(b) Fig. 4. Speswhite coating: scanning electron micrographs of (a) surface scan showing method of counting surface throats and (b) resin-embedded cross-sectional scan. Scale bars 2 and 5 µm, respectively.
This was done visually rather than by image analysis, because image analysis algorithms find it difficult to identify throats rather than surface pores in a surface electron micrograph. The results are shown in Table 2, together with the % standard errors of the mean, which arose chiefly from the heterogeneity of the surface structures rather than the subjectivity of the visual count. It can be seen that the number density of surface throats (or surface capillaries) is lower for Speswhite than for the Amazon coating, while a smaller number density is observed for Albaglos than for OpA40 coating.
4.2. Experimental permeation results The absorption curves of the ISIT permeation experiments (Fig. 6a) were extrapolated to time zero. The intercept at time zero was assumed to be due to almost instantaneous surface film transfer [4], which was then subtracted to give the oil uptake additional to this transfer. Also, taking the gravure roller footprint as 1 mm as described previously, allowed an absolute contact time to be estimated (Fig. 6b). It can be seen that there is a faster permeation rate into Ama-
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(a)
(b) Fig. 5. OpA40 coating: scanning electron micrographs of (a) surface scan and (b) resin-embedded cross-sectional scan. Scale bars 2 and 5 µm, respectively.
Table 2 Estimates of number density of surface throats and the correction factor, f , used for each sample Number density of surface throats (µm−2 )
Sample information Sample type
Sample name
Experimental
Experimental % standard error of mean
Model
Surface throat number density correction factor, f
PCC PCC Clay Clay
Albaglos OpA40 Amazon Speswhite
1.4 3.02 4.09 0.66
11 4 4 15
14.84 13.54 10.66 7.89
0.094 0.22 0.38 0.084
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(a)
(b) Fig. 6. Amount of oil transferred from gravure roller to coating samples at 100 N contact pressure. (a) Raw results and (b) the results are corrected for film transfer and with respect to absolute contact time.
zon and OpA40 at between 1 and 2 ms than for the other samples.
5. Simulation results 5.1. Matching of the model to the experimental characteristics Fig. 3 shows the fits of the simulated to the experimental percolation characteristics, which are close enough to generate meaningful differences in structure between the different types of sample. The asymptotes at low throat entry diameter are the simulated and experimental porosities, which
agree to within 0.1%. Following estimation of aspect factors, as described below, three characteristic parameters of the network model, namely connectivity, throat skew and pore skew, were adjusted to achieve the best fit between the simulated and experimental intrusion curves (Fig. 3 and Table 3). Although experimental techniques are available to measure the sizes and aspect ratios of particles in a porous solid, it is impossible in practice to measure the aspect factors of the voids. We therefore allocated values of aspect factor (rather than aspect ratio) to the experimental samples, on the basis of visual inspection of the micrographs. The three criteria used were: (i) whether the particles were rodlike (implying voids with aspect factor > 1), isotropic (aspect factor = 1) or plate-like (aspect factor < 1); (ii) that
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Table 3 Characteristic parameters of the network model used to generate the simulated structures for each sample Sample type
Sample name
Minimum throat entry diameter (µm)
Maximum throat entry diameter (µm)
Connectivity
Throat skew
Pore skew
Porosity (%)
Aspect factor (voids)
PCC PCC Clay Clay
Albaglos OpA40 Amazon Speswhite
0.0036 0.0036 0.0036 0.0036
2.39 2.38 2.31 2.32
5 3.4 4.6 4.8
1.3 0.78 1.84 1.63
2 1.6 3.1 1.8
37.6 38.3 30 25.8
1 1.5 0.8 0.6
Fig. 7. Simulated absorption of oil per unit area (µm3 /µm2 ) for clay coating samples. Speswhite and Amazon simulated at their estimated (α = 0.6 and 0.8, respectively) and isotropic (α = 1) aspect factors, with Speswhite also being simulated at an aspect factor of 0.01.
aspect factors for voids should be less than the aspect ratios of the particles, because the particles are not completely aligned; and (iii) that the aspect factors should give porerow spacings which were realistic relative to the dimensions of the experimental sample. The resulting values of the aspect factor are listed in Table 3. The sensitivity of the choice of value was tested by running simulations with a range of other values as well, including an aspect factor of 1 for each anisotropic sample. Despite the close match between the experimental and simulated mercury intrusion curves and porosities, and realistically assigned aspect factors, there was no correspondingly close match between the measured and simulated number densities of surface throats; Table 2 shows that the simulated number densities were significantly higher. Improving the match by decreasing the aspect factor to make the particles more platey proved unrealistic in another sense. For Speswhite, for example, an aspect factor of 0.1 provided a match, but created a simulated structure with a pore row spacing around order of magnitude larger than implied by the electron micrographs. So a simple correction factor, f , was used, where f = experimental number density /modelled number density.
(14)
This correction is valid while the wetting front is contained in the surface throats. However, it becomes increasingly invalid as the front progresses throughout the structure, as the body of the modelled structure would then have a porosity which was a factor of f greater than the experimental sample. Therefore the results below are compared to experiment both with and without the correction factor. 5.2. Volume averaged extent of permeation A number of trials were initially carried out without considering the correction factor f . The wetting algorithm was first tested on Speswhite clay, and it generated a set of absorption curves for a range of aspect factors from 0.01 to 1.0 (Fig. 7). It can be seen that for Speswhite there was generally a decrease in the amount of oil absorbed per unit area with a decrease in aspect factor, i.e., as the simulated voids became less rod-like and more slit-like. This effect was especially noticeable for the lowest aspect factors at the shortest times. However, several of the curves cross over each other at various times. It can also be seen that the curves show a series of steps and points of inflexion, corresponding to the filling of successive layers. The points of inflexion for the absorption simulations with lower aspect factors occur earlier than those with the higher aspect factors.
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Fig. 8. The amount of oil absorbed for the PCC samples, OpA40 and Albaglos, at their estimated (α = 1.5 and 1.0, respectively) and isotropic (α = 1) aspect factors.
Fig. 9. Modelled absorption rates of the four coatings, before applying the correction factor, compared to the experimental absorption rates at a time t of approximately 0.002 s.
The wetting algorithm was then used to calculate the amount of oil absorption for each coating structure, using the estimated aspect factors. ‘Control’ simulations were also performed for comparison. These had the experimental percolation and porosity of a particular experimental sample, but were isotropic (α = 1.0). The use of aspect factors of less than 1 for the clays caused the simulated absorption of oil to decrease relative to isotropic structures. It can be seen in Fig. 7 that Speswhite absorbs a higher amount of oil than Amazon. However, as time proceeds, the unit cell becomes saturated with fluid and the flow is impeded, reducing the absorption rate. As a result, the curve for Speswhite is shown to converge on to Amazon clay. Fig. 8 shows the amount of oil absorbed into the precipitated calcium carbonate coatings, using both estimated and isotropic aspect factors (which are both the same for Alba-
glos). The simulated absorption for Albaglos is shown to be lower than that for OpA40 control, up to a time of 0.007 s. However, comparing simulations with the appropriate estimated aspect factor, the simulation for Albaglos is shown to be lower than OpA40 for a much longer period time of 0.02 s. A comparison of the vertical scales of Figs. 7 and 8 shows that overall the simulated absorption for the clays is less than half of that predicted for the PCCs. To investigate whether the trends in the simulations match those of the experiments, we chose a time at which to compare the results. Inspection of Fig. 6b suggests that 0.002 s is a time which reveals the trends in the experimental results before they plateau. The resulting comparison is shown in Fig. 9. It can be seen that there is some correlation between the simulated and experimental trends for Speswhite, Albaglos and OpA40, but not for Amazon.
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Fig. 10. The modelled absorption rates of the four coatings, after applying the correction factor, compared to the experimental absorption rates at a time t of approximately 0.002 s.
Fig. 11. Detail of Fig. 1, showing permeation into Speswhite at 0.1 s. Same wetting scale bar as in Fig. 1.
The correction factor f for each coating structure was then applied to the simulated oil absorption curves, and the results are shown in Fig. 10. This gives a significant improvement in the agreement, as Amazon is now shown to have a modelled absorption rate higher than for the experimentally low permeable coating samples. 5.3. Fine structure and effect of inertia All the wetting results described so far, both experimental and simulated, have been volume or mass-averaged mea-
sures of the extent of permeation and the position of the wetting front. However, the steps in Figs. 7 and 8 show that there is also a fine structure within the simulated wetting which bears closer examination. A detail of Fig. 1, shown in Fig. 11, reveals advance wetting into small features. If the inertia of the fluid is ignored by using the Lucas– Washburn equation in each feature rather than the integrated Bosanquet equation (12), the wetting at 0.1 s changes from that shown in Fig. 11 to Fig. 12. It can be seen that there is a substantial difference in the extent of the filling if fluid inertia is ignored, and that the advance wetting in the small
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Fig. 12. As in Fig. 14, but ignoring the effects of inertia. Same wetting scale bar as in Fig. 1.
Fig. 13. Comparison of EHR and present model (left-hand axis), with experimental wetting (right-hand axis).
features is less noticeable. Fig. 13 shows that the Lucas– Washburn and EHR approach predicts wetting fronts two orders of magnitude faster than the present model, and also the wrong trends for the samples. Fig. 14 shows that there is a lack of advance wetting in Amazon. Fig. 15 shows a detail of Fig. 2 and reveals some advance wetting in OpA40, which is not evident in Albaglos (Fig. 16). Speswhite and OpA40, the samples which show advance wetting, both have a much broader void size distribution than Amazon and Albaglos (Table 1). This rela-
tionship is in agreement with the findings of Schoelkopf et al. [38] for porous calcium carbonate blocks, and of Ridgway and co-workers [4] for void networks with broad size distributions. 5.4. The effect of anisotropy and inertia on the wetting profile of a structure The effect of anisotropy on the wetting profile, when the inertial term is included or disregarded in the wetting simu-
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Fig. 14. The permeation of Amazon sample at 0.1 s.
Fig. 15. Detailed image of Fig. 2, showing permeation into OpA40 at 0.1 s. Same wetting scale bar as in Fig. 1.
lation, was examined by modelling Speswhite and OpA40. These two samples were chosen as they exhibited two different types of void shapes: slit-like and rod-like pores, respectively. Wetting simulations using both Bosanquet and Lucas–Washburn equations, which include and disregard inertia, were tested on the two structures at their isotropic and respective estimated aspect factor. The results for Speswhite and OpA40 are shown in Figs. 17 and 18, respectively.
The effect of anisotropy on Speswhite, when the inertial term is included in the simulation, caused a reduction in absorption of 23% (Fig. 17), while only a 3% reduction was observed when the inertial term is disregarded. Meanwhile, the effect of anisotropy on OpA40, when the inertial term is included in the simulation, caused an increase in absorption of about 21% (Fig. 18), while only a 4% reduction was observed when the inertial term is disregarded. In summary,
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was taken to be the layer where the number of empty voids outweighs the number of voids containing fluid. The wetting profile in Fig. 19 shows the vertical extent of the advanced wetting to be relatively close to the vertical extent of the wetting front, having a distance A for the advanced wetting of only approximately 6 µm. However, when the estimated aspect factor is applied to the structure, Fig. 20 shows a much longer advanced wetting distance A of approximately 11 µm. Hence, the effects of anisotropy caused further advanced wetting and preferential flow, not only in the large features but also in the narrow features of the structure, irrespective of inertia.
6. Summary
Fig. 16. The permeation of Albaglos sample at 0.1 s. Same wetting scale bar as in Fig. 1.
anisotropy (from isotropic to slit-like or rod-like pores) only showed a significant change in permeation when inertial effects are introduced in the wetting simulation. To examine the effect of anisotropy without inertia, the advance wetting was best visualised by rendering the isotropic and anisotropic structures as wire frames, Figs. 19 and 20, respectively. The voids containing any amount of fluid are outlined in blue colour, while voids containing no fluid are outlined in red. The transition line of where the wetting front ends and where the advanced wetting begins
We set out with the ambitious task of fitting a network model of void structure to a wide range of the experimental properties of four paper coating samples. The modelled void structures matched the mercury intrusion (percolation) properties and porosities of the experimental samples, in itself an achievement rarely seen in the literature. When additional properties were included, namely the spacing of the void features in the body of the sample, the number density of the surface throats, and the anisotropy of the voids, a compromise had to be struck, and a correction factor applied. The improvement caused by the correction shows that a more sophisticated method of packing the void features in the model is desirable. The model’s overestimate of the number of surface throats is probably due to the fact that a slight overlap of platey particles at the surface in an experimental sample occludes many of the surface throats, whereas in the model, no such overlap can take place.
Fig. 17. Simulated permeation curves for Speswhite at its estimated (α = 0.6) and isotropic (α = 1.0) aspect factor, and the results using both Bosanquet and Lucas–Washburn equations, including and disregarding inertia.
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Fig. 18. Simulated permeation curves for OpA40 at its estimated (α = 1.5) and isotropic (α = 1.0) aspect factor, and the results using both Bosanquet and Lucas–Washburn equations, including and disregarding inertia.
Fig. 19. Simulation of the absorption of oil into OpA40 at 0.1 s, applying an isotropic aspect factor (α = 1.0) and disregarding inertia (Lucas–Washburn equation). The structure is rendered into a wire frame in which the voids containing any amount of fluid are outlined with a blue colour, while voids containing no fluid are outlined with a red colour.
Despite these difficulties, models of the void structures were produced which provided a major improvement over the EHR approach, and which were of sufficient realism to provide insights into the wetting process in the samples. It was shown that the inertia of the fluid as it enters void features causes a considerable change in the extent of permeation over a few milliseconds, a timescale relevant to the time scale of printing in a modern press. Com-
parison of the void structures with anisotropic structures (all of which match the percolation and porosity criteria) provided insights into the precise nature of wetting into a range of experimental coating samples. For example, in the more anisotropic sample of each type, namely Speswhite and OpA40, fast advanced wetting occurred through narrow void features. This phenomenon has also been observed by other workers, as cited above. It would not be visible by any ex-
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Fig. 20. Simulation of the absorption of oil into OpA40 at 0.1 s, applying an estimated aspect factor (α = 1.5) and disregarding inertia (Lucas–Washburn equation). The structure is rendered into a wire frame in which the voids containing any amount of fluid are outlined with a blue colour, while voids containing no fluid are outlined with a red colour.
perimental technique which measured the volume averaged position of the wetting front. It can be categorised as preferential flow, although preferential flow is usually an experimentally observed quantity and therefore advance wetting into low-volume features tends to be overlooked. It is difficult to express the detail of the wetting in static snapshots, so we have made the two structures mentioned above available as supplementary on-line material, with viewing instructions at http://www.pore-cor.com.virtual_reality.htm. There is also more information to be considered about the simulation, such as a sensitivity analysis of the fitting parameters, and the tracking of all phenomena with time through the structures. However, such studies are outside the scope of this work, which is concerned with the overall comparison of the Darcy wetting front compared to that inferred from experiment. Generic findings from this work are currently being used in the design of new high performance paper coatings.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17]
Acknowledgments This project was funded by the University of Plymouth Science Faculty from its block research grant, with additional support from SCA, Sundsvall, Sweden. We are grateful for academic and experimental support from Sven Lohmander (formerly of STFi, Stockholm, Sweden), Edward Seyler (SeGan Ltd., Cornwall, UK), Patrick Gane and Joachim Schoelkopf (Omya, Oftringen, Switzerland), and Eirian Jones, (University of Plymouth, UK). The virtual reality programming is by Alexander Matthews.
[18] [19] [20] [21] [22] [23] [24] [25] [26] [27]
S.L. Bryant, P.R. King, D.W. Mellor, Trans. Por. Med. 11 (1993) 53. V.S. Patwardhan, C. Tien, AIChE J. 32 (1986) 321. D.J. Sherwood, Proc. R. Soc. Math. Phys. Sci. 445 (1994) 679. J. Schoelkopf, C.J. Ridgway, P.A.C. Gane, G.P. Matthews, D.C. Spielmann, J. Colloid Interface Sci. 227 (2000) 119. C.J. Ridgway, J. Schoelkopf, G.P. Matthews, P.A.C. Gane, P.W. James, J. Colloid Interface Sci. 239 (2001) 417. A. Johnson, I.M. Roy, G.P. Matthews, D. Patel, European J. Soil Sci. 54 (2003) 477. C.H. Bosanquet, Philos. Mag. Ser. 6 (45) (1923) 525. A. Marmur, Adv. Colloid Interface Sci. 39 (1992) 13. A. Marmur, J. Colloid Interface Sci. 122 (1988) 209. E. Chibowski, L. Holysz, J. Adhes. Sci. Technol. 11 (1997) 1289. L.-H. Lee, J. Adhes. Sci. Technol. 7 (1993) 583. T.X. Lu, D.R. Nielsen, J.W. Biggar, Water Resour. Res. 31 (1995) 11. N. Ichikawa, Y. Satoda, J. Colloid Interface Sci. 162 (1994) 350. A. Borhan, K.K. Rungta, J. Colloid Interface Sci. 155 (1993) 438. A.I. Moshinskii, Colloid J. 59 (1997) 62. J. Szekely, A.W. Neumann, Y.K. Chuang, J. Colloid Interface Sci. 35 (1971) 273. S.X. Ma, G. Mason, N.R. Morrow, Colloids Surf. A Physicochem. Eng. Aspects 117 (1996) 273. P.M. Adler, C.G. Jacquin, J.A. Quiblier, Int. J. Multiphase Flow 16 (1990) 691. G.R. Jerauld, S.J. Salter, Trans. Por. Med. 5 (1990) 103. R.L. Portsmouth, L.F. Gladden, Trans. Inst. Chem. Eng. 70 (1992) 63. W.B. Haines, J. Agric. Sci. 17 (1927) 264. L. Paterson, S. Painter, M.A. Knackstedt, W.V. Pinczewski, Physica A 233 (1996) 619. V.M. Starov, V.V. Kalinin, V.I. Ivanov, Colloids Surf. A Physicochem. Eng. Aspects 91 (1994) 149. C. Coulliette, C. Pozrikidis, J. Fluid Mech. 358 (1998) 1. I. Ginzbourg, P.M. Adler, Trans. Por. Med. 20 (1995) 37. J. Schoelkopf, P.A.C. Gane, C.J. Ridgway, G.P. Matthews, Colloids Surf. A Physicochem. Eng. Aspects 206 (2002) 445. J.E. O’Shea, Pigments for Paper, Tappi Press, Atlanta, 1997.
P.A. Bodurtha et al. / Journal of Colloid and Interface Science 283 (2005) 171–189
[28] P. Lepoutre, Prog. Org. Coat. 17 (1989) 89. [29] S. Lohmander, Nord. Pulp Paper Res. J. 15 (2000) 221. [30] J.W. McCoy, in: Tappi Proceedings—1998 Coating/Papermakers Conference, 1998, p. 121. [31] P.A.C. Gane, J.P. Kettle, G.P. Matthews, C.J. Ridgway, Ind. Eng. Chem. Res. 35 (1996) 1753. [32] CRC Handbook of Chemistry and Physics, CRC Press, Boca Raton, FL, 1979. [33] Y. Ishigami, Y. Gama, Y. Sano, S. Lang, F. Wagner, Biotechnol. Lett. 16 (1994) 593.
189
[34] G.P. Matthews, A.K. Moss, C.J. Ridgway, Powder Technol. 83 (1995) 61. [35] J. van Brakel, S. Modry, M. Svata, Powder Technol. 29 (1981) 1. [36] G.P. Matthews, A.K. Moss, M.C. Spearing, F. Voland, Powder Technol. 76 (1993) 95. [37] M. Abramowitz, I.E. Stegun, Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables, Gannon Distributing Co., 1972. [38] C.J. Ridgway, P.A.C. Gane, J. Schoelkopf, J. Colloid Interface Sci. 252 (2002) 373.
