Estimation of structural element sizes in sand and compacted blocks of ground calcium carbonate...

Transp Porous Med (2007) 66:403–419DOI 10.1007/s11242-006-0031-y

R E S E A R C H A RT I C L E

Estimation of structural element sizes in sandand compacted blocks of ground calcium carbonateusing a void network model

Giuliano M. Laudone · G. Peter Matthews ·Patrick A. C. Gane · Alexander G. Matthews ·Cathy J. Ridgway · Joachim Schoelkopf ·Stephen A. Huggett

Received: 30 November 2005 / Accepted: 30 March 2006© Springer Science+Business Media B.V. 2007

Abstract An algorithm has been developed for the calculation of the size of theeffective structural or skeletal elements which make up the solid phase of an unconsol-idated or consolidated porous block. It is based on a previously presented algorithm,but it has now been validated on unconsolidated samples and tested on consolidatedsamples. It also includes a virtual reality representation of the structures. First, anetwork model named Pore-Cor is made to reproduce the percolation behaviour ofthe experimental sample, by matching its simulated percolation characteristics to anexperimental mercury intrusion curve. The algorithm then grows skeletal elementsbetween the cubic pores and cylindrical throats of the void network model until theytouch up to four of the adjacent void features. The size distributions of the simulatedsolid elements are compared with each other and with experimentally determinedparticle size distributions, using a Mann–Whitney test. The algorithm was shown tosimulate skeletal elements with the correct trends in size distribution for two differentsand samples, provided the sand packed itself optimally under the applied mercurypressure. It was also applied to two samples of variously compressed calcium car-bonate powder, having fine and coarse particle size distributions respectively. Thesimulation demonstrated that on compressing the powder at the minimum force, theskeletal elements differed from the constituent particle sizes, as expected. The aver-age size of the skeletal elements increased as the compression force was increased on

G. M. Laudone · G. P. Matthews (B)Environmental and Fluid Modelling Group, University of Plymouth, 602 Davy Building, DrakeCircus Plymouth, PL4 8AA, UKe-mail: [email protected]

P. A. C. Gane · C. J. Ridgway · J. SchoelkopfOmya Development AG, CH-4665, Oftringen, Switzerland

S. A. HuggettSchool of Mathematics and Statistics, University of Plymouth, Plymouth, UK

A. G. MatthewsSt. John’s College, University of Cambridge, UK

404 Transp Porous Med (2007) 66:403–419

the calcium carbonate powders. The results suggest that the method could be usefulas a tool for probing the effect of compaction on aggregation or sintering, and forstudying other effects such as cementation in geological samples, where other moredirect techniques cannot be applied.

Keywords Void network model · Calcium carbonate · Sand · Compaction ·Tablet · Skeletal element

1 Introduction

When studying the structure of a porous solid, it is often of interest to know theeffective sizes of the elements which make up the solid matrix. In a geological study,for example, such information might assist in determining the degree of cementationof a reservoir sandstone. In the preparation of a sinter or powder compact, it mightallow the relative effects of pressure and temperature to be assessed, and hence theoptimum process parameters required to achieve a desired structure.

A standard method of estimating the sizes of constituent structural solid elementsis to study an electron micrograph such as that shown in Fig. 1.

It can be seen in the micrograph that some solid elements are almost discrete, andso their sizes can be readily estimated—in the present case diameters from around0.2 µm up to around 5 µm. However, other structural elements are agglomerated oron top of others, so that estimation of their sizes is very difficult and it requires thesample to be embedded in dye resin and polished (Thovert et al. 1993). Standard imageanalysis does not assist, because it requires the image colours to be converted into abinary distribution representing either the solid or the pore space, and, in so doing,

Fig. 1 Electron micrograph of a calcium carbonate tablet HC60 compressed under a pressure of57.6 MPa, 10 µm scale bar shown bottom left

Transp Porous Med (2007) 66:403–419 405

many visual depth clues are lost. The higher the degree of cementation, sintering orcompaction of the matrix, the greater the problem of estimation. Methods used forthe evaluation of particle size distributions include the grain reconstruction algorithm(Thovert et al. 2001) which grows particles until the right porosity is obtained, andthe Discrete Element Method (Sweeney and Martin 2003).

An alternative approach is to study the flow and pressure characteristics of intru-sion of the solid by a fluid, and infer from the relationship between applied pressureor suction the sizes of the particles that are obstructing or attracting the fluid. In anearly paper, Haynes discussed the behaviour of water in soil (Haines, 1927). However,as he noted, water wets with advance wetting fronts which jump forward from par-ticle to particle. Estimation of the sizes of the particles beneath the wetting fronts istherefore impossible. Non-wetting fluids, such as mercury, percolate more predictablyand therefore provide a more reliable estimate. Smaller particles contact other smallparticles with a greater degree of curvature at their point of contact, relative to largeparticles, so the connecting voids between them can be thought of as wider. Mayerand Stowe (1965) therefore assumed a structure such as that shown in Fig. 2, wheremercury intrudes a layer of smaller particles first, which are assumed to have wider,more easily penetrated voids between them. At higher pressures, the mercury canintrude the supposedly smaller voids between the larger particles. However, these

Fig. 2 A cross-section view through a solid as represented by the Mayer and Stowe method of esti-mating the particle size distributions. Mercury, shown grey, has intruded the first layer of particles(Mathews et al. 1999)


assumptions only apply to sorted arrangements of perfect spheres such as that shownin the figure. Even then they are counter-intuitive with respect to the size relationshipbetween pores-throats and particles in a real sample.

We present a more advanced approach, where firstly the sizes of the voids within asimulated three-dimensional network are adjusted so that the simulated andexperimental mercury intrusion curves match closely. The resulting void network isnot unique, so we generate several stochastic realisations which match the experimen-tal intrusion characteristics. Skeletal element sizes are then estimated by “growing”representative structural (or skeletal) elements in the solid phase between the sim-ulated voids. The algorithm is similar to one published previously (Mathews et al.1999). However it has now been given an interface to present its results in virtualreality.

The method is validated by applying it to unconsolidated sand, and checking thatthe diameters of the skeletal elements correspond with particle sizes measured bysieving. It is then applied to porous tablets formed by compressing finely groundcalcium carbonate of two different particle size distributions.

2 Theory

2.1 Generation of the pore network

The network model ‘Pore-Cor’1 has been previously used to model a range of materialssuch as soil, sandstone, catalysts and paper coating (Bodurtha et al. 2005; Johnson et al.2003; Ridgway and Gane 2002). It represents the void structure of porous mediumas a series of identical interconnected unit cells with periodic boundary conditions.Each unit cell comprises an array of 1000 nodes in a cubic-close-packed array equallyspaced at a distance S from each other in each Cartesian direction. Cubic pores arepositioned with their centres at each node, and are connected by cylindrical throats ineach Cartesian direction. The pores are of 100 discrete sizes distributed equally overa logarithmic scale. A typical unit cell is shown in Fig. 3.

A Boltzmann-annealed simplex (Johnson et al. 2003) is used to adjust four param-eters so that the mercury intrusion curve of the simulated structure closely matchesthat of the experimental sample. The four parameters are:

• connectivity, defined as the average number of connected throats per pore, up toa maximum of six (one connected to every face of the cubic pore);

• throat skew, defined as the percentage of throats of the smallest size in a distribu-tion of 100 sizes which is linear when plotted on a logarithmic size axis;

• pore skew, a scaling factor which bulks up the sizes of the pores to achieve theexperimental porosity;

• correlation level, which sets the level of local size-autocorrelation of the features,in the present case giving rise to vertical banding within each unit cell.

The simplex also takes into account three Boolean parameters, namely whether thenetwork can be drawn with no overlapping features, whether it can be adjusted to theexperimental porosity and whether the network is unfragmented.

1 Pore-Cor is a software name of the Environmental and Fluid Modelling Group, University ofPlymouth, PL4 8AA, UK


Fig. 3 Simulated void network and skeletal elements for HC60 compressed under a pressure of57.6 MPa

The adjustment of the fit to the experimental porosity is made by altering thespacing of the nodes, and hence altering the lengths of the throats joining the poreswhile keeping the pore sizes and throat diameters unchanged. To prevent overlap, thespacing between the pores, S, is not allowed to be less than 0.1% above the size dmax ofthe largest pore. This regular spacing is unrealistic, because in actual samples smallerparticles and voids tend to be packed closer together than large ones. By using thepore skew parameter, the size of the smaller pores can be increased up to the size ofthe largest pore dmax, to achieve the experimental porosity. This parameter providessome compensation for the unrealistic regular spacing. However, if too large, it cangenerate an anomalously large number of small particles.

2.2 Growth of the skeletal elements (Mathews et al. 1999)

In Fig. 4, we consider a cubic array of eight neighbouring cubic pores. Such an arrange-ment is shown in Fig. 4(a). The throats are omitted for clarity in Fig. 4 and are alsoignored in the calculation. Since the throat sizes are always smaller than, or at mostequal to, the size of the pores they connect, their omission from the calculation intro-


Fig. 4 Illustration of the algorithm used for the calculation of the particle size. (a) the 8 pores definingthe skeletal element space, the primary position Q and the 4 contact points, (b) the initial skeletalelement particle centred on the primary position grows and finds a first contact point, (c) the centreof the sphere moves away from the first contact point and finds a second contact point, (d) and (e)the growth of the sphere continues until the 4 contact points are found (the contact points are shownas vertices of a tetrahedron), or (f) until the sphere touches the outer cube formed by the centres ofthe 8 pores (Mathews et al. 1999)

duces a negligible error. The ‘primary position’ Q of the skeletal element is taken asbeing at the centre of a cube defined by the centres of the eight adjacent pores.

The algorithm first measures the distance between the primary position and theclosest corner of any of the eight cubic pores. It forms a sphere of this radius centredat the primary position. This sphere therefore has one contact point.

If q is a three-dimensional vector defining the primary position, Q, and c1 is thevector defining the position of the first contact point, C1, then the vector u

u = (1 + λ) q − λc1

defines the position, U, of a point lying in the line through Q and C1. By incrementingλ, it is possible to adjust u until the distance D1 between U and C1 is equal to thedistance D2 between U and one of the remaining seven inner-facing corners of thepores. This sets the second contact point, C2, with position vector c2. If the secondcontact point is the corner of the pore opposite to the first contact point the calculationis stopped, as the skeletal element cannot grow further without invading the volumeof the first two touching pores. If not co-linear, the three points, Q, C1 and C2, definea plane. It is possible to move the centre of the sphere away from the final position ofU in the plane defined by the three points, perpendicular to the line between C1 andC2. The position, V, of the centre of the sphere is defined by the position vector:

v = (1 + µ) u − µ

2(c1 + c2)

The parameter µ can be incremented until a third contact point C3 (with vector c3) isfound, equidistant from the sphere centre to C1 and C2. The diameter of the spherehas incremented to D3.


To find the fourth contact point, C4, with position vector, c4, the centre of thesphere is moved along the line

w = v + νn̂

where n̂ is the unit vector normal to the plane through C1, C2 and C3, given as:

n̂ = (c3 − c2) × (c2 − c1)

|c3 − c2‖c2 − c1| .

The parameter v is increased until the spherical skeletal element makes its fourthcontact point, which is equidistant from the centre of sphere to C1, C2 and C3. Itsdiameter is then D4.

The diameter of the growing sphere can never exceed the pore-row spacing S,Fig. 4(f). If all the diameters reached this limit, Du = S, the resulting representationof the skeletal elements would be a cubic close-packed array of mono-sized spheres.In a continuous solid structure the maximum percentage of the volume of solid phasewhich can be represented by the cubic close-packing of spheres is π/6, correspondingto 52.4 % of the solid phase.

The size of a skeletal element can also be crudely determined as a hard spherediameter Dh, which is the minimum diagonal distance between opposite poressurrounding the primary position:

Dh = √3

(

S − LI

2− LII

2

)

where LI and LII are the side lengths of the two diagonally opposed pores.In most cases, D4 > D3 > D2 ≈ Dh, but for some unusual pore configurations,

when the largest pores are all to one side of the primary position, D4 < Dh. Overall,the diameter was initially chosen as the maximum of Dh, D4, D3, and D2. Then theresulting value was compared with Du, and the minimum of these two values chosento prevent possible overlap with an adjacent skeletal element.

Intuitively, one would expect the size distribution of the skeletal elements todescend asymptotically to zero at minimum and maximum size. In fact, the simu-lated size distribution can show peaks at each end. At the larger end, no size can begreater than Du. So, as porosity �−> 0, D−> Du, as mentioned above, causing a peakat maximum size. Conversely, to match structures with large porosities, the pores arepacked together as closely as possible, leaving a minimum space between them of0.001 dmax. Therefore as �−> 1, D−> 0.001 dmax, causing a peak at minimum size.Also, high values of the pore skew parameter will tend to cause D−> 0.001 dmax , asmentioned previously, and so in the modelling the value of this parameter was keptas low as possible.

It is important to point out that the main aim of the network simulator is to modelthe void space of the porous samples, as explained in the previous section. The choicesof void network geometry and void network elements limit the skeletal elements tooccupy a cubic lattice, which cannot be physically expected for densely packed con-stituent particles. However, the trends of the sizes of the skeletal elements resultingfrom this algorithm give a useful approximation to the effective particle size distribu-tion within the porous material, especially when more direct methods and algorithmscannot be applied. In the case of paper coating (Laudone et al. 2005), for example,the samples are too thin (5 µm) to be sliced and studied by any of the image analysistechniques described in the Introduction section.


3 Experimental

3.1 Samples

Two different samples of Redhill sand,2 grades 30 and 65, were analysed by mercuryintrusion porosimetry. In order to investigate the effect of the high pressure mercuryintrusion on the packing of the samples, each sand sample was analysed ten timesunder the same experimental conditions. The particle sizes were measured by themanufacturer using 11 sieves, giving 12 size ranges or ‘bins’.

Also a series of calcium carbonate tablets were prepared using two different nat-ural CaCO3 powders from the same limestone source,3 Hydrocarb 60 (HC60) andHydrocarb 90 (HC90), having respectively 60 w/w% and 90 w/w% of particles withdiameter <2 µm as measured by sedimentation using a Sedigraph4 5100 instrument(Gane et al. 2000; Schoelkopf et al. 2000).

The tablets were prepared by applying a pressing force onto 60 g of the mineral pig-ment powders, on a circular cross sectional area of 17.35 cm2, using a hydraulic press atseven pressures in the range 57.6–259.4 MPa for HC60, and two pressures, 144.1 MPaand 201.7 MPa, for HC90. The resulting tablets had different porosities, as shown inTable 3. These tablets were chosen because they had already been extensively studiedwith respect to their porosity, mercury intrusion curves and their imbibition by wettingfluids other than water (Gane et al. 2000).

3.2 Mercury porosimetry

The sand and CaCO3 tablets were analysed by mercury intrusion porosimetry usingan Autopore5 III instrument.

The sand was assumed to be incompressible. For the CaCO3 tablets, the resultsof the mercury intrusion porosimetry measurements were corrected for sample com-pression by applying the equation of Gane et al. (1995), incorporated into the softwarePore-Comp6:

Vint = Vobs − δVblank +[

0.175(V1bulk) log10

(

1 + Papp

1820

)]

−V1bulk(1 − �1)

(

1 − exp

[

(P1 − Papp)

Mss

])

where Vint is the volume of intrusion into the sample, Vobs the intruded mercuryvolume reading, δVblank the change in the blank run volume reading, V1

bulk the samplebulk volume at atmospheric pressure, Papp the applied pressure, �1 the porosity atatmospheric pressure, P1 the atmospheric pressure and Mss the bulk modulus of thesolid skeletal material. The constants in this expression are relative to the pressureand the bulk modulus expressed in MPa.

2 Supplied by Sibelco (formerly Hepworth) Minerals and Chemicals Ltd, Redhill, Surrey, UK3 Limestone selected from Orgon, France by Omya AG, CH 4665 Oftringen, Switzerland4 Sedigraph is a product name of Micromeritics, USA5 Autopore is a product name of Micromeritics, USA6 Pore-Comp is a software name of the Environmental and Fluid Modelling group, University ofPlymouth, PL4 8AA, UK.


Fig. 5 Mercury intrusion porosimetry for Redhill sand samples

Table 1 Volume % porosity for Redhill sand samples

Sample Redhill 30 Redhill 30 Redhill 30 Redhill 65 Redhill 65 Redhill 65

Run 1 2 3 1 2 3Porosity (%) 34.9 40.7 41.9 39.6 44.4 45.6

Pressures Papp were then converted to the equivalent diameter of a cylindricalthroat which is just big enough to be percolated by mercury at that pressure, using theLaplace equation with a mercury interfacial tension of 0.485 N m−1 and an advancingcontact angle of 140◦.

4 Results

4.1 Sand

Figure 5 and Table 1 show the results of mercury intrusion porosimetry on the twotypes of sand. Only three of the ten runs on each are shown, namely the minimum(Run 1), intermediate (Run 2), and maximum intrusion (Run 3). It can be seen that thevolume of mercury intruded, and hence the effective porosity of the packed particles,differs substantially due to different particle packing in each run.

Figures 6 and 7 show the experimental particle sizes of the Redhill 30 and 65 sand,measured by sieving, plotted on logarithmic size axes. Also shown are the skeletal ele-ment size distributions generated from the first stochastic realisation of the simulatednetworks fitted to each of the six mercury intrusion curves shown in Fig. 5. Runs 2and 3 for each sample have a higher porosity, so must be more loosely packed. Tosimulate this higher porosity, the network model tends to bulk up the pores with thepore skew parameter and/or pack the pores as close together as possible. This givesrise to a peak at minimum size in Fig. 7, as explained above.


Fig. 6 Experimental and simulated size distributions for Redhill 30 sand. The size distributions cor-responding to different stochastic realisations of the structures based on run1 are also shown. Thedata labelled run2 and run3 correspond to the first stochastic realisation only

Fig. 7 Experimental and simulated size distributions for Redhill 65 sand. The peaks in the smallestsize range are due to artefacts, as discussed in the text. The simulated data correspond to the firststochastic realisation only, as explained in the text


Table 2 Sand: values of theMann-Whitney statistic P forpairwise comparison ofrandom sampling of the binnedsize distributions of theexperimental measurementsand simulated skeletalelements

Pairs of samples compared Redhill 30 Redhill 65compared

Exp and Run1 0.089 0.175Exp and Run2 0.000 0.000Exp and Run3 0.000 0.000Run1 and Run2 0.000 0.000Run1 and Run3 0.000 0.000Run2 and Run3 0.000 0.283

Table 3 Volume (%) porosity for HC60 and HC90 tablets

Sample: HC60 HC60 HC60 HC60 HC60 HC60 HC60 HC90 HC90

Compression/MPa: 57.6 86.5 115.3 144.1 172.9 230.6 259.4 144.1 201.7Porosity (%) 28.7 26.8 24.3 28.0 22.1 21.7 19.3 26.8 23.8

A statistical test is required to assess the similarity of the experimental and simu-lated results. As the distributions are skewed, because of truncation and/or artefactsat the top and bottom of each distribution, the Mann–Whitney non-parametric testhas been used. The two-tailed asymptotic significance P is derived from the ranks ofthe absolute values of the differences between the two variables, and treats top andbottom peaks of the size distribution more appropriately than a test based on variancealone. For the test, the simulated skeletal elements were grouped into the same sizebins as the experimental results.

It can be seen (Table 2) that for both Redhill 30 and Redhill 65, the experimentalparticle sizes are the same (P > 0.05) as the sizes of the skeletal elements for the cor-responding Run 1 — i.e. the run with the closest optimal particle packing. All othercomparisons are different (P < 0.05) except between the two poorly packed Redhill65 samples, Run 2 and Run 3.

Figure 6 also shows skeletal element distributions corresponding to different sto-chastic realisations of the structures based on run 1 mercury porosimetry intrusioncurve. All are identical (P > 0.05). This leads to the conclusion that the differentarrangements of pores and throats in the three-dimensional network do not affect theskeletal elements algorithm. This was verified with all the other samples presentedin this work, and, as a consequence, the redundant data corresponding to differentstochastic realisations of all the other samples presented in this work will be omitted.

4.2 Calcium carbonate

Figure 8 and Table 3 show the results of the mercury intrusion porosimetry on thecalcium carbonate tablets. Two anomalies can be observed. The porosity is expected todecrease for increasing tablet formation pressures, but, in Table 3, it can be observedthat the HC60 144.1 MPa tablet does not follow this trend. The Laplace diameter, cor-responding to the mercury intrusion pressure, is expected to decrease with increasingtablets formation pressures. In the case the HC60 86.5 MPa tablet the Laplace diam-eters, at which mercury intrusion is measured, are much smaller than those measuredfor tablets created applying higher formation pressures.


Fig. 8 Mercury intrusion porosimetry for HC60 and HC90 tablets

Table 4 Hydrocarb (HC)samples: non-zero values ofthe Mann-Whitney statistic P,as for Table 2. All othercombinations zero

Pairs of samples compared P

HC60 115.3 and 172.9 MPa 0.016HC90 144.1 and 230.6 MPa 0.008

The analysis on the sand samples showed how mercury porosimetry on unconsol-idated powder samples could give different results for the same sample, dependingon the degree of packing of the particles under the pressure applied by the intrudingmercury, and how this affects the results of the skeletal elements simulation. Theanalysis on unconsolidated calcium carbonate powders has therefore been omitted,and the attention has been focused on the effect of the tablets formation pressure onthe skeletal element size distribution.

Figure 9 represents the size distributions of the skeletal elements, modelled usingthe mercury intrusion curves for the Hydrocarb 60 tablets prepared under differentpressures.

All the size distributions were different from the experimental distribution (P < 0.05)

and they were all entirely different from each other (Table 4). The experimental par-ticle size distribution, measured by sedimentation, cannot be directly compared to thesimulated skeletal element distributions, as it measures the sizes of separate particles,while the algorithm simulates consolidated skeletal elements. As the tablets wereformed by pressure application only, without any binders, the particles were expectedto keep their original unconsolidated size, but, because of the applied pressure, largerclusters of consolidated skeletal elements could be observed. These clusters do con-tain pores separating the single constituent particles, but these pores are below therange of pore analysed by the network simulator. This explains the large differencebetween the experimental and simulated distributions.


Fig. 9 Particle size distributions for HC60 measured experimentally, and size distributions of skeletalelements derived from the modelling of tablets compacted under pressures in the range 57.6 to 259.4MPa. It is important to emphasise that the experimental particle size distribution cannot be directlycompared to the results of the simulations, as discussed in the text

Figure 10 shows the average size of the simulated skeletal elements, which increaseas the formation pressure of the tablets increases.

Thus the model implies that the lowest pressure (57.6 MPa) generates skeletal ele-ments of entirely different sizes to the constituent particles. It also shows an increasedaggregation of skeletal elements from 57.6 to 259.4 MPa. This effect can be confirmedby visual comparison of the simulated structures in Figs. 3 and 11.

A visual comparison between the sizes of the aggregated skeletal elements ofHC60 at 57.6 MPa shown in the electron micrograph, Fig. 1, and the skeletal elementsgenerated by the simulation algorithm, Fig. 9, confirms a realistic match.

Figure 12 shows the number of skeletal elements in the simulated unit cells of HC60tablets having a diameter in the range from 1 to 2 µm. This size range is free fromartefacts at maximum and minimum size, described earlier, so is a reliable indicatorof particle aggregation. A much clearer trend can be observed, with an increase innumber of relatively large skeletal elements in tablets created by applying a higherpressure. The only large deviation from such a trend is observed in the case of HC60144.1 MPa tablet, which is the also the one presenting an anomaly in the value ofmeasure porosity.

Figure 13 shows a comparison between the skeletal elements of HC90 under differ-ent pressing forces and the sizes of the separate particles. Once again, there is nosimilarity (P < 0.05) between the sizes of the separate particles and skeletal elements,and an increase in pressing force changes the distribution (P = 0) while increasingthe mean size by 36%, Fig. 10.


Fig. 10 Mean diameters of the skeletal elements for HC60 and HC90

Fig. 11 As for Fig. 2, but compressed under a pressure of 86.5 MPa


Fig. 12 Number of skeletal elements with a diameter in the range 1–2 µm in unit cell (dashed line)and porosity % (continuous line) for the HC60 tablets

Fig. 13 Comparison of the experimental particle sizes of HC90 and the skeletal elements in tabletsformed under different pressing forces


5 Conclusions

An algorithm for the calculation of the effective skeletal elements size distributionwithin a network model has been tested. It has been developed to include a virtualreality representation of the structures. The algorithm simulated skeletal elements ofthe correct size distribution for two different sand samples, but only for those samplesthat happened to pack optimally under the applied mercury pressure in the mercuryporosimeter.

For two different samples of compressed calcium carbonate, the simulation dem-onstrates that on pressing at the minimum pressure (57.6 or 144.1 MPa), the skeletalelements differ from the constituent particle sizes, as expected. It also shows a trendof increasing size as the pressing force is increased. For the coarser carbonate (HC60),this increase in average skeletal element size is strongly affected by the experimen-tal mercury intrusion porosimetry. The increase of skeletal elements size between aformation pressure of 57.6 and 86.5 MPa is larger than expected, and this is due toan anomalous trend in the mercury intrusion pressure for the HC60 86.5 MPa tablet.Also the reduction in average skeletal element size for the HC60 144.1 MPa tablet canbe explained by taking into account the anomalously high value of porosity measuredfor this tablet. For the finer carbonate (HC90), the mean size of the skeletal elementsincreases by 36% as the pressure is increased from 144.1 to 201.7 MPa, Fig. 10, whichis a statistically significant change, Table 4.

The number of skeletal elements in a unit cell with diameter in the range 1 µm to2 µm increases monotonically with pressure, Fig. 12, except for one sample with ananomalous porosity.

The derivation of representative structural or skeletal element sizes from mercuryintrusion is an indirect way of measuring them, and cannot compete with direct meth-ods where available. In this work we have demonstrated that the results are usefullyaccurate for unconsolidated sand samples. The consistent results for calcium carbon-ate skeletal elements suggest that, when no more direct technique is available, thisalgorithm can be used as useful indirect method for probing the effect of compactionon the structure of a consolidated particle matrix.

References

Bodurtha, P., Matthews, G.P., Kettle, J.P., Roy I.M.: Influence of anisotropy on the dynamic wettingand permeation of paper coatings, J. Colloid Interface Sci. 283, 171–189 (2005)

Gane, P.A.C., Kettle, J.P., Matthews, G.P., Ridgway, C.J.: Void space structure of compressible polymerspheres and consolidated calcium carbonate paper-coating formulations. Ind. Eng. Chem. Res.35, 1753–1764 (1995)

Gane, P.A.C., Schoelkopf, J., Spielmann, D.C., Matthews, G.P., Ridgway, C.J.: Observing fluid transportinto porous coating structures: some novel findings. Tappi J. 83, 77.

Haines, W.B.: Studies in the physical properties of soils. IV. A further contribution to the theory ofcapillary phenomena in soil. J. Agric. Sci. 17, 264–290 (1927)

Johnson, A., Roy, I.M., Matthews, G.P., Patel, D.: An improved simulation of void structure, waterretention and hydraulic conductivity in soil, using the pore-cor three-dimensional network. Eur.J. Soil Sci. 54, 477–489 (2003)

Laudone, G.M., Matthews, G.P., Gane, P.A.C., Ridgway, C.J., Schoelkopf, J.: Estimation of the effec-tive particle sizes within a paper coating layer using a void network model. Chem. Eng. Sci. 60,6795–6802 (2005)

Mathews, T.J., Matthews, G.P., Huggett, S.: Estimating particle size distributions from a network modelof porous media. Powder Technol. 104, 169–179 (1999)


Mayer, R.P., Stowe, R.A.: Mercury porosimetry—breakthrough pressure for penetration betweenpacked spheres. J. Colloid Sci. 20, 893–911 (1965)

Ridgway, C.J., Gane, P.A.C.: Dynamic absorption into simulated porous structures. Colloid. Surface.A Physicochem. Eng. Aspect. 206, 217–239 (2002)

Schoelkopf, J., Ridgway, C.J., Gane, P.A.C., Matthews, G.P., Spielmann, D.C.: Measurement and net-work modelling of liquid permeation into compacted mineral blocks. J. Colloid Interface Sci. 227,119–131 (2000)

Sweeney, S.M., Martin, C.L.: Pore size distributions, calculated from 3D images of DEM-simulatedpowder compacts. Acta Mater. 51, 3635–3649 (2003)

Thovert, J.F., Salles, J., Adler, P.M.: Computerised characterization of the geometry of real porousmedia: their discretization, analysis and interpretation, J. Microscopy 170, 65–79 (1993)

Thovert, J.F., Yousefian, F., Spanne, P., Jacquin, C.G., Adler, P.M.: Grain reconstruction of porousmedia: application to a low-porosity Fontainebleau sandstone. Phys. Rev. E 63(6)061307 (2001)

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