porous media

Transp Porous Med (2014) 104:231251DOI 10.1007/s11242-014-0331-6

Spontaneous Inertial Imbibition in Porous Media Usinga Fractal Representation of Pore Wall Rugosity

Guodong Liu Meiyun Zhang Cathy Ridgway Patrick Gane

Received: 17 September 2013 / Accepted: 6 May 2014 / Published online: 24 May 2014 Springer Science+Business Media Dordrecht 2014

Abstract Considering the separable phenomena of imbibition in complex fine porous mediaas a function of timescale, it is noted that there are two discrete imbibition rate regimes whenexpressed in the LucasWashburn (LW) equation. Commonly, to account for this deviationfrom the single equivalent hydraulic capillary, experimentalists propose an effective contactangle change. In this work, we consider rather the general term of the Wilhelmy wetting forceregarding the wetting line length, and apply a proposed increase in the liquidsolid contact lineand wetting force provided by the introduction of surface meso/nanoscale structure to the porewall roughness. An experimental surface pore wall feature size regarding the rugosity areais determined by means of capillary condensation during nitrogen gas sorption in a groundcalcium carbonate tablet compact. On this nano size scale, a fractal structure of pore wall isproposed to characterize for the internal rugosity of the porous medium. Comparative modelsbased on the LucasWashburn and Bosanquet inertial absorption equations, respectively,

G. Liu (B) M. ZhangCollege of Light Industry and Energy, Shaanxi University of Science and Technology, and ShaanxiProvince Key Laboratory of Papermaking Technology and Specialty Paper, Xian 710021, Chinae-mail: [email protected]

M. Zhange-mail: [email protected]

G. Liu P. GaneDepartment of Forest Products Technology, School of Chemical Technology, Aalto University,P.O. Box 16300, 00076 Aalto, Finland

P. Ganee-mail: [email protected]; [email protected]

G. LiuTianjin Key Laboratory of Pulp & Paper, Tianjin University of Science & Technology,Tianjin 300457, China

C. Ridgway P. GaneOmya International AG, 4665 Oftringen, Switzerlande-mail: [email protected]

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for the short timescale imbibition are constructed by applying the extended wetting linelength and wetting force to the equivalent hydraulic capillary observed at the long timescaleimbibition. The results comparing the models adopting the fractal structure with experimentalimbibition rate suggest that the LW equation at the short timescale cannot match experiment,but that the inertial plug flow in the Bosanquet equation matches the experimental resultsvery well. If the fractal structure can be supported in nature, then this stresses the role of theinertial term in the initial stage of imbibition. Relaxation to a smooth-walled capillary thentakes place over the longer timescale as the surface rugosity wetting is overwhelmed by thepore condensation and film flow of the liquid ahead of the bulk wetting front, and thus to asmooth walled capillary undergoing permeation viscosity-controlled flow.

Keywords Imbibition Wetting line length Equivalent radius Meso and nanoscaleroughness Fractal structure

1 Introduction

The imbibition of a wetting liquid into a porous medium is an important frequently occurringphenomenon in both natural and industrial systems, as seen, for example, in soil science,petroleum recovery science, the papermaking and printing industries, and in the construc-tional materials industry. The earliest approach for describing imbibition dynamic was pro-posed by Bell and Cameron (1906) and Ostwald (1908), who found a dependency of absorp-tion volume on the square-root of time (t). The analytical equation and its solution werethen presented by Lucas (1918) and Washburn (1921) by means of balancing the Laplacemeniscus pressure relation with Poiseuilles equation of viscous resistive laminar flow, whichis well known as the LucasWashburn (LW) equation.

x2 =(

RehcLV cos 2

)t, (1)

where x is the distance travelled by the liquid with viscosity, , undergoing imbibition ina horizontal capillary of radius Rehc, representing the equivalent hydraulic (hydrodynamic)capillary of the sample, with a wetting force given as the liquid surface tension, LV, incontact with the solid surface having a meniscus boundary angle of .

Attempts to explain observed deviation from the LW standard imbibition model, based onthe single equivalent hydraulic capillary, has received increasing attention, with explorationof several alternative models (Schoelkopf et al. 2002). Frequently, the inertial effect is invokedas missing in the LW equation, a point recognised and stressed by Rideal (Rideal 1922). Onthe basis of Rideals research, Bosanquet (1923) in his equation of 1923 added the inertialimpulse drag effect associated with an accelerating fluid to complement the Poiseuille flow,expressed in the case where no external pressure is applied, as

ddt

( R2ehcx

dxdt

)+ 8x dx

dt= lLV cos (2)

Equation (2) introduces the wetting line length, l, (which for the simple cylindrical cap-illary = 2 Rehc) and the liquid density, , required to consider the inertial impulse effect.The expression in Bosanquet is more complicated than the LW equation, and in fact discon-tinuous at the instance of first absorption, such that solutions are approximated only to theshort timescale limit of the acceleration operator, limt0 d

2

dt2 . Later, researchers attempted

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Spontaneous Inertial Imbibition in Porous Media 233

to describe a real porous liquid-imbibing system by adding further refinements based on theBosanquet equation by a variety of methods, including substituting the discontinuity in theBosanquet rate equation at t = 0 with capillary entrance energy loss (Szekely et al. 1971),but the results eventually showed that they were either equivalent to that of Bosanquet orto be analytically unmanageable due to the number of unknown parameters when applyingthem to a realistic porous medium (Letelier and Leutheusser 1979; Levine et al. 1980). Inpractical application, the Bosanquet equation was, therefore, not widely accepted comparedwith the traditional LucasWashburn equation.

The lack of uptake of the Bosanquet model was also partly due to Bosanquets ownapplication of the concept to a single capillary, himself explaining that the inertial effect wastoo short lived to be of importance. As a result, many researchers thought that the impactof inertia on imbibition to be negligible because of its influence only over very short timeduring initial imbibition. Though this is true in general for the single capillary case, or whenstudying the imbibition process on a long timescale for some liquids, those who rejected theconcept overlooked the continuous acceleration and deceleration effects at the wetting frontin a complex geometry network structure. Thus, the cumulative effect of inertia should notbe ignored in an interconnected network structural material, and is required as an integralpart of the physics when analysing the process of dynamic forces acting during imbibitionat the wetting front, although the decay of imbibition rate from linear t to

t is intrinsic

to the eventual dominance of viscous drag within the saturated medium behind the wettingfront. Thus, comparing the LW and Bosanquet equations, it is readily identifiable that theBosanquet equation relaxes to the general form for the LW equation as t increases. Ganeet al. (2004), Schoelkopf et al. (2000) and Ridgway et al. (2001) highlighted this aspect, andestablished the expression of complexity of a porous network based on the coating materialstructure analysis of fine printing paper. The same workers then further expanded their modelapplication to other materials, like medicinal tablets, sand and building materials, providingreasonable correlation with the important first phase of liquid absorption and describing theselective pore filling phenomenon in terms of a preferred pathway of wetting.

In overlooking the important physics discussed above, researchers would often prefer toattempt some modification without invoking the complexity so as to enlarge the range ofapplicability of the LW equation in the early process of imbibition. Rabinovich et al. (2002)experimentally demonstrated the fluctuation of capillary forces with relation to nanoscalefeatures, which increased gradually with increasing roughness, and Stukan (2010) investi-gated the spontaneous imbibition of liquid in nanopores with different roughness resorting tomolecular dynamic simulation to match the LW equation. Dimitrov et al. (2007) providedthe molecular dynamics evidence for roughness (coarse-grained) models of nanotubes in ananopore construct with a further modified LW equation, adding the slip length, which effec-tively matches the Bosanquet plug flow solution (lim t 0). The work of Bico et al. (2002)and Qur (2008) in turn discussed the wetting of textured surfaces, providing a quantitativedescription of the changes of contact angle during the wetting, and indicated that the appar-ent contact angle for super-hydrophilic solids is greater than that of Wenzels contact anglecorrection. The primary reason for this can be understand as a smoothing effect of the liquidfilm covering the nano textured surface, which erases the solid roughness seen at the initialstage of wetting, in the bulk over time. This kind of film is formed by capillary condensation(Kohonen and Christenson 2000) in the mesoscale or nanoscale surface pores constitutingthe internal roughness. A parameter defining roughness of a pore surface should be seriouslyconsidered when initial wetting and imbibition are proceeding in a complex porous medium,as the roughness alters the effective contact angle, and further changes the Wilhelmy forcewetting line length. As the dynamic contact angle on the meso or nanoscale is not readily or

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reliably obtained from the various experimental methods, there is merit in seeking to makecomparison via wetting line length. Hammecker and Jeannette (1994) and Leventis (2000)modified the LW equation considering microscale structural parameters, but the predictedmodel still showed some deviations from the experimental results, which were thought tohave arisen from the profoundly complicated and irregular structure inside practical porousmedia. For this kind of complex situation, it is not easy to describe them using traditionalEuclidean geometry. However, the non-Euclidean sequential repetitive geometry offered byfractal theory (Mandelbrot 1983, pp. 32, 4949) is a powerful tool to describe stochasticstructure and has been applied to imbibition models of porous substrates (Cai et al. 2012; Yuand Li 2001), and many researches went on to point out that the interior structural parametersof porous media, such as roughness, specific surface area, effective capillary force (Li 2010)apparent contact angle (Onda et al. 1996) are all provided with fractal characteristics, in whichroughness was suspected more than other structural parameters to provide the dependencebeing sought to explain the faster than expected absorption on the short timescale.

In this paper, by considering the roughness, wetting line length and wetting force (inertialplug flow response) on the nanoscale pore or mesocale roughness of porous materials, weset out to generate a predictive model fitting for a real situation of experimental imbibitionof mineral oil into a medium-fine ground calcium carbonate tablet compact, and characterizefractal structure in the porous medium using fractal pore walls that are represented as a seriesof fractal half-capillaries. The resulting wetting line length increase is applied to the LW andBosanquet equations, respectively, for an exploratory range of fractal order and dimension.

2 Experimental Observation of Imbibition by Porous Materials

2.1 Spontaneous Imbibition Experiment

The sample material chosen for the imbibition experiment is ground calcium carbonate (GCC)(Hydrocarb 60 ME: Omya AG, Baslerstrasse, CH-4665 Oftringen, Switzerland), producedfrom Norwegian marble by wet grinding in the presence of polyacrylate dispersant, anddisplaying a broad particle size distribution having 60 w/w% of particles


Fig. 1 Gravimetric wetting and imbibition apparatus (Ridgway and Gane 2002)

liquid permeation and to minimise any interaction between the wax and absorbed liquid. Theprocedures for tablet making and absorption measurement are described in detail by Ridgwayand Gane (2002).

The liquid used for absorption is a mineral oil (PKWF 4/7 af new, Haltermann Products),which is typically found as an aromatic-free (af) main ingredient in offset printing ink.The surface tension, LV, of this liquid is 0.0274 kg s2(N m1), having a viscosity, , of0.0043 kg m1 s1 and density, , 804 kg m3. The contact angle of mineral oil and calciumcarbonate, , is assumed to be effectively zero following the data of Chibowski and Holysz(1997), who have shown that aliphatic alkanes completely wet a number of mineral surfacesincluding calcium carbonate. Moreover, the calcite is dispersed with a strongly hydrophilicpolyacrylate (Gane et al. 2004).

The rate of liquid uptake was measured using an automated microbalance, namely aPC-linked Mettler Toledo AX504 balance with a precision of 0.1 mg, capable of 10 measure-ments per second (Fig. 1). The chamber around the balance base plate enables a controlledatmosphere to be established, shielded from external air movement (Ridgway and Gane2002). The apparatus, gases and samples in this study were maintained at 23.0 1.5 C.Prior to the imbibition experiments, each sample was placed in a 5 dm3 chamber flushedthrough with dry nitrogen, and the sample left to equilibrate for 48 h. Due to low volatility,the chosen mineral oil did not require any evaporative correction.

2.2 The Separated Timescale Phenomena of Spontaneous Imbibition

To provide a universal absorption curve for the sample, the volume uptake, V (t), is normalisedto unit sample cross-section area, A. The imbibition volume per unit area for the samples isplotted as a function of time in Figs. 2 and 3.

Following normal practice, based on the assumption of the validity of the LW equation,the absorption data are plotted against

t , as shown in Figs. 4 and 5.

As is reported by Gane et al. (2004), the imbibition process is seen to be divided intotwo parts, a faster initial absorption and a later slower absorption. In neither of the case isthe sample near saturation. Therefore, these two regimes describe free capillary absorption.Each region can be approximated to a direct proportionality with

t , as shown in Fig. 4 for

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Fig. 2 The whole process ofimbibition

Fig. 3 The short timescale imbibition region plotted against linear time t

Fig. 4 The whole process of imbibition plotted as a function of

t

123


Fig. 5 The short timescale imbibition region plotted against

t

the slower and in Fig. 5 for the faster absorption, though, as is done increasingly frequently,the faster region can also be approximated reasonably well to linear t due to the short timesinvolved (Fig. 2).

The clear difference between the two phases of imbibition is the contrasting gradient ofabsorption rate, which can be considered as describing separable phenomena if each imbi-bition was to be expressed either with the well-known LW equation or with the Bosanquetequation, each having either a different equivalent capillary radius or changed wetting para-meters. At this point, we take up the hypothesis that this discretisation phenomenon is dueto the internal pore wall roughness of the sample acting to provide increased wetting linelength of the internal pore surface at short times and a subsequent pore condensation/slipplane effect for the longer time absorption until the point where the viscous drag takes overand defines Poiseuille flow through the equivalent smoothed capillary.

3 Proposing the Meso/nanosurface Wetting Hypothesis

The assumed internal roughness can be described as an equivalent increase in the wetting linelength, and thus an increase in the Wilhelmy capillary wetting force. At the very first initialphase of imbibition, the capillary force and inertial retardation force act together in oppositedirections to control the process of imbibition into the porous medium, and so the actingcapillary force must be per se very large. As the absorption progresses and the vapour in thesample becomes saturated ahead of the wetting front, the meso and nanoroughness inside thepores leads to capillary condensation such that the internal surface meso/nanopores is filledby the wetting liquid, which acts to reduce the surface roughness and leads to an equivalenttotally smooth fully wetting surface, thus reducing the effective wetting line length. Thesecond phase (long timescale) is then subsequently established as the equilibrium flow duringthe later stages of imbibition of the liquid as the smooth pore surface becomes established inbalance with the increased resistance of flow through the complex saturated structure behindthe wetting front. Considering the surface layer effect, the two phase imbibition process isfrequently understood from the point of view of contact angle which is increased from the firstphase to the second one. We challenge this interpretation, as the capillary condensation arisingfrom the saturated vapour ahead of the wetting front progressively erases the roughness on thesurface meso and nanoscale features, and so reduces the wetting line length and wetting force

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238 G. Liu et al.

Fig. 6 a Initial wetting including the increased wetting line (wetting force) associated with the surface mesoand nanoroughness, b pore condensation as the vapour saturates the surface ahead of the wetting front, actingto shorten the wetting line length to that of a smooth surface

to that of the perfectly smooth equivalent capillary. This can manifest an apparent change incontact angle on the micro and macroscale, but the underlying contact energetics does notchange, and so the nanoscale contact angle at the true liquidsurface interface is likely to befixed. We can capture the hypothesis above by considering the progressive loss of surfacerugosity by pore condensation. Figure 6 shows the case as the initial wetting front progressesinto the porous structure. At first, Fig. 6a, the wetting line length is long as the meniscusfront seeks to wet the pore surface roughness, thus creating a high wetting force. Eventually,the vapour saturates the pores ahead of the wetting front and liquid becomes condensed andadsorbed on the surface, leading to a fully wettable smooth surface, Fig. 6b.

That the two phase absorption phenomenon is usually seen when using strongly wettingliquids of low vapour pressure, such as alkanes and oils, or water in the case of a pre-driedmaterial, provides some support for this delayed pore condensation effect. We, therefore, seekto define this roughness and propose an imbibition model considering the experimentallyderived roughness to predict the imbibition of liquid in complex porous media.

3.1 A Wetting Model of Internal Nano/meso-featured Porous Material

We define an initial wetting line length l1 and resulting wetting force f1, where the poreroughness acts to increase the wetting line length above that of the smooth equivalent hydro-dynamic capillary, and l2 and f2 in the case where the pore condensation reduces the wettingline length and wetting force to that of the equivalent hydraulic smooth capillary. Thus, wehave the situation l1 l2 and f1 f2.

Generally, the imbibition of a liquid by a porous medium is simplified to correspond toan array of capillary tubes to account for different pore size. Since internal roughness coulderroneously be described as ink-bottle pores in porosimetric techniques, such as mercuryintrusion, we, therefore, rely here on the equivalent single hydraulic capillary model. Toincorporate the case as shown in Fig. 6a, we propose a theoretical capillary tube modelconsidering an internal meso and nanoscale roughness to simulate a complex system of aporous medium, Fig. 7. The equivalent capillary tube provided with surface rugosity executesthe two wetting regimes mentioned above, and the total process of imbibition naturallybecomes separated into the two discrete phases according to the model.

3.2 The Experimental Determination of the Pore Size over Condensation Area

The internal pore wall structural parameter of a porous sample over the meso and nanoscalecan be measured with a gas sorption analysis, in which nitrogen (or other gaseous vapour)is first adsorbed as a monolayer on the sample surface, and then condensed as liquid due tothe changes of relative pressure (p/p0). The condensation is detected as a hysteresis loopspanning the adsorption and desorption isotherm. The area of hysteresis within the loop

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Fig. 7 Modelling of capillarytube with inner surface roughnessover the meso and nanoscale

Fig. 8 Adsorptiondesorptionisotherm of 22 B GCC 6 SA,linear plot

defines the surface volume of gas condensed in the meso and nanoscale surface roughnessvoids. It is also possible to derive an equivalent pore size distribution from a Kelvin equationplot, which ascribes a pressure-driven energy change to the condensing gas according to thepore volume constriction. The method of Barrett, Joyner and Halenda (BJH) uses these datato calculate a pore size distribution from experimental isotherms using the Kelvin model ofpore filling. The adsorption or desorption branch of the isotherm in the chosen pressure rangeis generally used as initial data for BJH calculations.

The characteristics of nitrogen adsorption and desorption in the pore surface structure ofour experimental sample (22 B GCC 6 SA) are shown in Fig. 8, generated using a Tristar3020 sorption analyser (Micromeritics, Norcross, GA USA). The experimental samples werefirst dried in an oven at 105 C for at least 24 h. Then, air was expelled from the samples byexchange with nitrogen at 150 C over a period of 1 h. An amount of 0.9 g per sample wasthen taken and measured for nitrogen adsorption under reduced temperature. In terms of theKelvin equation and the hysteresis loop shown on the isotherm, the start point and end pointof the hysteresis loop were at the relative pressures (p/p0) 0.80 and 0.99, respectively. Thus,the pore size at these pressures, (p/p0) 0.80 and 0.99, could be calculated by the Kelvinequation (Gregg and Sing 1982),

log(

pp0

)= 2 VL

RT rm,(3)

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Table 1 The condensation areaand pore size of experimentalsamples

Sample Range of pore size over N2condensation area ()

22 A GCC 6 SA 3696622 B GCC 6 SA 53966Ref A GCC 36964Ref B GCC 27964

where (the surface tension of condensed nitrogen) = 8.88 mN m1, and VL (the volume ofa mol of condensate) = 34.68 cm3 mol1 at the instrumental temperature of T = 77.35 K. R,in this case, is the universal gas constant.

Thus, the area of the condensation hysteresis compares with a surface rugosity, i.e. surfacepore size covered internally on its walls by condensate, ranging in the case of sample 22 BGCC 6 SA from rm = 42 to 935 . Considering the thickness of the adsorption layer, thereal surface pore size, rp, is given by

rp = rm + tads, (4)where tads is the adsorbed film thickness. Thus, the practical pore size distribution is locatedto be in the range 53966 .

The condensation pore size ranges of all the samples are summarised in the Table 1.

4 Fractal Structure Modelling of Pore Wall Rugosity

It is very hard to characterize the nature or distribution of roughness scales formed by fineand irregular pores, but that these pores in many porous media follow a fractal relationshiphas been frequently proposed/reported in the last decade or so (Perrier et al. 2006; Datheand Thullner 2005; Perrier et al. 1999). As reported previously experimental studies bySchoelkopf et al. (2008), the particle size distribution of ground calcium carbonate approx-imates closely to a log-normal function. Therefore, the population occupancy as a functionof decreasing particle size itself follows a fractal relation. Ultimately, this particle size distri-bution is derived from the ground product which is formed from energetic breakdown of thecrystal fracture planes in the calcitic structure, and so we can also conclude that the fractureprobability mechanics follows a fractal rule in respect to the formation of the ground calciumcarbonate. Therefore, the particle surface rugosity, being defined by the crystal habit and itsinherent fracture planes manifest on the particle surface, can also reasonably be assumed tobe fractal. It is these particle surfaces that describe the internal surface structure of the pores,and so in turn this can be assumed to be approximated reasonably well by a fractal description.To express surface structural roughness as a fractal relationship in respect to a cylindricalpore, we can once again reduce the problem from that shown in Fig. 7 to the symmetry ofa cylinder, since the basic pore model follows this symmetry, and it adds nothing to makea more complex construct on the wall surface. Thus, a symmetry-maintaining constructioncan be developed using a fractal breakdown into a pore wall consisting of fractal-relatedhalf-pipes. The complexity of rugosity of the pore wall is thus increased at every fractal leveleither by the external growth or internal growth from the boundary of equivalent hydraulicradius, Rehc, as shown in Fig. 9.

In Fig. 9, we demonstrate the finding that there is an overlap problem when growingthe complexity of the rugosity outside (Fig. 9a) the original cylinder boundary but that the

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0r

1r

Model leads to overlap2r

0r

1r

2r

Model avoid overlap

(a) (b)

Fig. 9 The fractal pore wall model of equivalent capillary tube based on the external (a) and internal (b)growth

Fig. 10 The boundary condition of constant pore volume defines the relation between the structural radii forthe condition r0 r1

internal growth model (Fig. 9b) avoids this problem. We thus adopt the internal growth modelto match experimental imbibition and characterize fractal structure of porous media. To definethe total wetting force and wetting line length in a capillary whose wall is a constructed froma series of such half-pipes, it is necessary to establish the complete perimeter length. This isshown in Fig. 10

The perimeter length, l, for the first level fractal layer of half pipes following the sur-rounding perimeter of a complete pipe is thus defined by

l1 = (r0 r1)r1

r1 for r0 r1 (5)

for the specific condition of r0 r1, such that the linear diameters of the half pipes sumto approximate the curved circumference of the full pipe, under the boundary condition ofequal pore volume as the increased structure complexity is added. Therefore, applying theever fixed criterion of constant pore volume, and noting that the geometry of the cylindersatisfies this condition via the square of the radius, we obtain

R2ehc = 2 [

(r0 r1)22

+ (r0 r1)2r1

r212

]for r0 r1, (6)

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Fig. 11 Rugosity of the porewall described as a fractallyrelated curvature represented as afractal series of half pipes

where the term 2(r0 r1)/2r1 (=(r0 r1)/r1) enumerates how many half pipes constitutethe original perimeter of the complete pipe under the specific approximation that r0 r1.

To develop the full rugosity model, shown in Fig. 11, consisting of an extended series ofa structure within a structure, where the wall half pipes fulfil the fractal dimension criterionfor their radii, the added structure complexity is of half pipes added to the previous layer ofhalf pipes, such that the number of next layer j half pipes per previous layer j 1 half pipeis (r j1 r j )/2r j , the boundary condition of constant volume becomes

R2ehc(n) = 2

n

i=1

(ri1 ri )22

2 ri (ri1 ri )

ij=1

(r j1 r j )2 r j

+(

ni=1

(ri1 ri )2 ri

) r

2n

2

(7)

such that the perimeter length is given by

ln = n+1

2n1rn

ni=1

(ri1 ri )ri ,

(8)

where this wetting perimeter length of our imbibition model is made up of an array of fractalrelationship half radii, r . These structures following a fractal assembly rule are character-ized in that the length scale, N , of measurement used to define the structural boundaries orperimeter relate as a power law to the dimension of those boundaries L . If, for example, werepresent a cylindrical pore of radius R in respect to its cross-sectional perimeter, 2 R, andthen assume that on a diminishing measurement scale a fractal descriptor, Df , represents thepower law relation between this single pore and an array of fractally related half pores havingthe same total cross-sectional area, defined according to the expression

N (r) rDf namely: N (r) = C rDf (9)such that for two levels of measurement scales, Ni and Ni+1, two fractal levels of radius canbe identified, ri and ri+1, respectively, each following the fractal assembly parameter of thesystem, Df , i.e.

Ni = C rDfi , Ni+1 = C rDfi+1 (10)Expressing the terms in Eq. (10) as a ratio, we derive the term Xi according to

NiNi+1

=(

riri+1

)Df= Xi (11)

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Fig. 12 Schematic diagram of mathematical descriptor Eq. (13) including data of a 375 and b 8.0 106discrete levels

When considering n different length scales, the values Xi can be used to form the multi-plicative series

n1i=1

Xi =(

r1rn

)Df= N1

Nn(12)

such that

rn = r1 (

n1i=1

Xi

) 1Df= r1 Y 1/Dfn (13)

Based on this expression, it is possible to construct a mathematical descriptor space acrossall possible fractal dimensions Df , to reveal a three dimensional surface relating {r1, . . . , rn}set pairs via the computation of Yn over the predefined descriptor space 1 Df 2, whereDf = 1 describes a completely differentiable (smooth) perimeter and Df = 2 an array ofcylindrical pores that are so fine (equivalent to the ultimate roughness in perimeter) thatthere are effectively an infinite number of them to maintain the boundary condition of equalcross-sectional area to that of the starting single pore as shown in Fig. 12. As in Eqs. (9) and(10), we thus may express the fractal relation is respect to surface rugosity curvature, i.e. afractal set of {r1, . . . , rn} for a given fractal dimension Df .

In accordance with the wetting mechanisms shown schematically in Fig. 6, and respectivewetting line lengths, ln lehc, it is taken that the reduction of the imbibition rate from theinitial stage to the second stage relates to the decrease of wetting line length arising from thecondensation in the meso and nanopores, thus reducing the capillary force, fn fehc. Theequivalent radius Rehc of the second phase of experimental imbibition is given experimentallyfrom Eq. (14). The matching of the model with experimental imbibition at short times takesconsideration of the contribution to the wetting line length, ln , of the roughness related to themeso and nanoscale internal pore rugosity, derived from the value of the gas condensationover that area, and liquid dynamic viscous resistance using the equivalent hydrodynamiccapillary of LW calculated from the absorption experiment at long times, described underPoiseuille flow. In the process of fitting the experimental imbibition rate with fractal-relatedpore wall rugosity, all the structural parameters in the fractal series represent the structureper unit cross-sectional area of the sample and the full descriptive value of R2ehc is used. Thevalue for r0 can be refined for each complexity generated structure to match experimental

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244 G. Liu et al.

Fig. 13 Refinement to enlargethe starting radius r0 in order tomaintain bulk flow properties andpore volume

Table 2 The equivalent capillary radius of experimental samples

22 A GCC 6 SA 22 B GCC 6 SA Ref A GCC Ref B GCC

Rehc (m) 231.40 236.03 249.23 234.59

absorption rate in the initial stage via knowing the experimental Rehc in the second stage ofimbibition, as shown in Fig. 13.

The experimental Rehc for our porous samples is calculated from rearranging Eq. (1) bytaking into account the volume absorbed in the second long timescale stage 2

x2 =(

V (t)imbibed stage 2A R2ehc

)2=

(RehcLV cos

2

)t

R5ehc =(

V (t)imbibed stage 2A

)2/(

LV cos 2 t

) (14)

in accordance with the experimental imbibition data and LW equation, and listed in theTable 2.

5 Modelling Experimental Imbibition Using Fractal Pore Wall Based on LWEquation

For a given porous medium, as in the Experimental Sect. 2, it is possible to measure thevolume of a known liquid (viscosity and density ) imbibed by wicking as a function oftime. For a certain contact angle, the equilibrium imbibition is expressed using the LucasWashburn approximation, defining an equivalent hydraulic capillary of radius Rehc, whichassumes a static balance between the Laplace pressure and the Poiseuille viscous-controlledresistive flow. The volume imbibition Vimbibed(t) of liquid into the porous medium per unitarea A can thus be calculated based on the LW equation as

V (t)imbibed stage1/A = R2ehc

A

(RehcLVcos

2

)t (15)

In accordance with the fractal structure of the pore wall of the equivalent capillary (Sect. 4),the wetting of a half pipe is described by a Washburn wetting line similar to that of a fullcapillary of radius r modified according to the geometry (1/2), such that the wetting forcecan express as

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Table 3 The specific fractal and imbibition rate parameters used in Eq. (18) and shown in Fig. 14 comparedwith experiment

Sample Line no. in theplotted series(Fig. 14) providingbest fitting toexperiment

Fractaldimension (Df )

Fractal level,n (Eq. (12))

Slope of modelimbibition (Kcal)(ms0.5)

Slope of experimen-tal imbibition (Kexp)(ms0.5)

22 A GCC 6 SA 4 1.22 15 1.0118 103 1.01 10322 B GCC 6 SA 4 1.22 15 1.0632 103 1.05 103Ref A GCC 6 1.36 15 1.1245 103 1.11 103Ref B GCC 5 1.27 16 1.2640 103 1.27 103

fc = 12 (2rLV cos ) = rLV cos = lLV cos , (16)where l is the perimeter length of the cross-section of the half pipe. The total wetting forcein an equivalent capillary thus follows a relationship based on the structure of fractal-relatedpores forming a rugose capillary wall of the form

fn = lLV cos = n+1

2n1rn

ni=1

(ri1 ri )ri

LV cos , ri ri+1 (17)

and applying this to the LucasWashburn equation under the approximation that the wallstructure has no significant effect on the Poiseuille resistive flow,

Vimbibed(t)/A = R2ehc

A

(RehcLVcos

2

)t

A

2

n

i=1

(ri1 ri )22

2 ri (ri1 ri )

ij=1

(r j1 r j )2 r j

+(

ni=1

(ri1 ri )2 ri

) r

2n

2

)

n+12n1 rn

ni=1

ri1riri

LV cos 4

t

Df{ri ri+1} (18)

5.1 How Well Does LW Match Experimental Results?

In terms of the matching Eq. (18), the results of matching between modelling proposedwith imbibition experiments using the effective hydrodynamic capillary of LucasWashburndescribing the absorption behaviour calculated from the imbibition experiment at longtimescales are shown in Fig. 12. The detail information regarding plots is summarized inTable 3.

According to the area boundary condition, the volume of totally filled pores in our capillarytube with roughness and traditional smooth tube keeps constant. The fractal constructionwetting line is applied by an algorithm in Matlab (MathWorks, Cambridge Business Park,Cambridge CB4 0HH, U.K.) and the range of fractal parameters is sequentially tested to

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246 G. Liu et al.

Fig. 14 Imbibition rates calculated over the short timescale (stage 1) for the equivalent long timescalehydraulic pore using a fractally defined modification of the internal rugosity to increase the effective wet-ting line length

provide a selection of gradients ranging from across the numbered lines to represent theimbibition gradient. The resulting structures employ from 3,600 to 4,352 radii of half pipesin the range of fractal dimension Df = 12. Comparing the data matching in Table 3with the experimental absorption data, we see that it is possible to match the gradient of thecalculated line (Kcal) with the short-term slope of the experimental imbibition (Kexp). Thismeans that it is in principle possible to match experimental short timescale absorption usingthe half pipe fractal modification of the internal surface rugosity of the equivalent hydrauliccapillary defined for the steady-state long-term absorption dynamic according to LucasWashburn. The increased wetting line length, offered by the fractal model, thus accounts forthe increased absorption speed at short times whilst being able to define a single equivalentcapillary radius model by maintaining the pore volume criterion derived from the saturatedvolume. However, despite the apparent good fit, we have to recognise that correlation to thenitrogen adsorption/condensation fails in these matching fractal constructs. The pore sizes ofthe fractal structure are unrealistic, as it reaches to such a fine scale that it becomes a quantumlength (10(2030) m), i.e. less than the bond length between atoms. Since the theory ofwetting is by definition an energetic interaction between two continua, liquid molecules and

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Table 4 Limiting the fractal fineness to the realistic continuum region defined by the size scale observed innitrogen adsorption condensation

Sample Fractal dimension(Df ) limited to N2condensation size

n, limited to N2condensation size

Slope of modelimbibition line(Kcal) (ms0.5)

Slope of experimen-tal imbibition line(Kexp) (ms0.5)

22 A GCC 6 SA 1.40 4 6.5034 105 1.01 10322 B GCC 6 SA 1.38 4 6.8338 105 1.05 103Ref A GCC 1.39 4 7.9175 105 1.11 103Ref B GCC 1.38 4 7.3241 105 1.27 103

a solid surface, such length scales are totally out of range of the interaction theory and indeedof the pore size over the condensation area for nitrogen used to identify the scale of rugositydimension. We have, therefore, to limit the pore size to that area which is realistic to thephenomenon in question, as in Table 1. After refining the pore size of the fractal structurewithin this realistic criterion, it is no longer possible to match the experimental imbibitionrate, as seen in Table 4, with a deviation of 1.5 orders of magnitude.5.2 The Reason of Failed Matching

The results in Table 4 clearly demonstrate that, adopting a fractal series to describe an internalrugosity model of the porous structure, applied to the traditional equivalent hydraulic capil-lary, fails to provide a solution to the problem of the first absorption phase being significantlyfaster than that predicted by LW. In another work, we do show that a stochastic approachto the extended wetting line length, derived from a characteristic length expressed as theconstant volume equivalent pore with an extended convoluted perimeter distributed aboutthe mean of the equivalent cylindrical wall does provide a good match adopting the LW t regime (Liu et al. 2014).

The wetting line length from the fractal structure does not reach the complexity require-ments that would be needed to match experiment if the pore wall surface roughness size islimited to that identified by the nitrogen condensation hysteresis. For example, the experi-mental absorption rates Kexp of sample 22 B GCC 6 SA are 1.5103 and 3.38105 ms0.5at the first and second phase, respectively, which means the increase of wetting line lengthneeds to be about 1.43 m, i.e. equal to 964 times more than the smooth pipe (Rehc = 236.03m) circumference used to describe the equivalent capillary of the long timescale imbibi-tion. However, the fractal structure only gives an increase of wetting line length around 10times when the fractal-related pore size is confined to the region of capillary condensationwithin the porous medium. This kind of fractal construct, therefore, does not have enoughcomplexity of structure to generate the required increase in wetting line length.

To confirm that the principle of fractals fails per se, and not just the choice of half pipestructure, we also have checked other traditional famous fractal structures, for example,the snowflake and the Minkowski sausage (Mandelbrot 1983, pp. 32, 4949), as shown inFig. 15, and evaluated in Table 5. They are also not powerful enough in complexity to reachthe required idealised wetting line length to execute fitting. This confirms the findings ofother unpublished studies2 where, using the traditionally classical fractal structure, it was notpossible to fit with the experimental imbibition rate of porous media.

2 Mangin, P., Mandelbrot, B. and coworkers, personal communication

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Fig. 15 Traditional classicalfractal structures applied to theperimeter of the equivalentcapillary

Snowflake(n=1)

Minkowki sausage(n=1)

Table 5 The calculated increaseof wetting line length providedby a range of fractal structures

Fractal structure (lim-ited to N2 condensa-tion pore size region)

Increase of wetting linelength (m) {requiredfor fit, 1.43 106 m}

Initial scale(m)

Half pipes (used here) 1.367 104 0.1Snowflake 3.513 103 0.1Minkowski sausage 4.078 103 0.1

6 Matching Model of Experimental Imbibition Using Fractal Pore Wall Based onBosanquet Inertial Plug Flow

The main difference between the LW equation and the Bosanquet equation is that the roleof inertial force is ignored at the initial phase of imbibition by LW. The cumulative effectof inertia, as discussed in Sect. 1, also could be one of the possible causes for the failurefor matching between the fractal model and that of the experimental observation of first-stage imbibition. In the dynamic force system of the Bosanquet equation, the capillary forceand inertial force together work on the initial phase of imbibition, and the viscous drag willshow a dominance increasingly as imbibition proceeds with increased liquid amount and willdominate the flow resistance, eventually relaxing into the LW state, i.e. the LW equationis equivalent to the Bosanquet equation as the time of imbibition increases toward the secondphase over long timescale.

Thus, to test the match of the initial phase of imbibition using our model of the uniformequivalent capillary radius of the long timescale imbibition plus the half pipe fractal construct,adopting the size limiting boundary condition of limitation to the nitrogen condensationpore size, we apply the same geometry as for the previous fitting but now change the LW expression to that of the short timescale reduction of the Bosanquet equation offered byGane et al. (2004) and Schoelkopf et al. (2000). This approach, thus incorporates the effectiveinertial plug flow, which is equivalent to an instantaneous slip boundary condition within thefractal rugose structure. The matching term of short time solution of the Bosanquet equationis thus written as

Vimbibed(t)/A = R2ehc

A t

(2 LVcos

Rehc)

A

2

n

i=1

(ri1 ri )22

2 ri (ri1 ri )

ij=1

(r j1 r j )2 r j

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Table 6 Matching imbibition with experiment using the short timescale solution of the Bosanquet equationapplied to the fractal series of half pipes

Sample Fractaldimension(Df )

n, fractallevel

Slope of calculatedimbibition line (Kcal)(ms)

Slope of experimentalimbibition line (Kexp)(ms)

22 A GCC 6 SA 1.43 4 5.0390 104 5.0 10422 B GCC 6 SA 1.42 4 5.0205 104 5.0 104Ref A GCC 1.48 4 5.0363 104 5.0 104Ref B GCC 1.39 4 6.0461 104 6.0 104

+(

ni=1

(ri1 ri )2 ri

) r

2n

2

) t

n

2n1 n

i=1ri1ri

ri2 LVcos

rn

Df{ri ri+1} (19)

Following the matching principle of Sect. 5 and Eq. (19), the new matching results arelisted in Table 6, which suggest that they are almost matching perfectly when now consideringthe inertial wetting term in the initial imbibition phase. The continuity of this plug flow phaseis then related to the continuous rapid acceleration experienced in the meso and nanoscalechannels constituting the pore wall structure.

The internal wall surface pore size of sample 22 B GCC 6 SA, for example, at the givenfractal level is sitting in the range of 8996 . Although the finest pore size of 8 at thefourth fractal level, n = 4, is slightly out of the area of our proposed condensation limit,we can consider it to be of the same order of magnitude as the condensation limit of 54 .Similar arguments can be applied for the other three samples, which manifest an even closeragreement to the fine condensation limit. It is addressed, therefore, that a few finest poressmaller than 54 may exist of negligible volume, and assist in generating the cumulativeeffect of inertia. The deviation remains small.

7 Conclusions

In this work, we discussed wetting mechanisms, which are suspected to relate to the two phaseabsorption dynamic seen generally, but not exclusively, for low-vapour pressure liquids inreal porous systems. These are related to the surface roughness on the meso and/or nanoscalein porous media, and we proposed an imbibition model using a modified wetting line lengthand wetting force with the meso and nanoscale roughness exhibited in BET adsorption andcondensation hysteresis (nitrogen gas sorption analysis). Taking the thus defined pore wallroughness of a porous material into account, we went on to propose an equivalent capillarytube, having a radius defined by the long-term absorption phenomenon, and increased theinternal wetting line length, and related wetting force, to match the absorption rate of thereal porous system at short times using the LucasWashburn (LW) equation and Bosanquetequation, respectively.

The internal surface pore roughness leads to capillary condensation, as the liquid fills thebulk structure over time, ahead of the wetting front, and this acts to fill the fine pores to form a

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250 G. Liu et al.

liquid film. At the same time, the fine pore wall structure leads to high acceleration rates andallied inertial effects resulting in plug flow. The resultant film defines the wetting line lengthand wetting force as that of the smooth equivalent capillary defined in the latter timescale ofthe imbibition.

Due to the irregular pore size distribution in complex porous media, it is traditionally dif-ficult to define the representative liquid interactive structure. We chose, therefore, to considera material with a predominance of repeat structure behaviour arising from the assembly ofthe particle size distribution within a tablet compact consisting of ground calcium carbonatewith and without latex binder. A fractal structure characterizing pore wall rugosity was thusproposed using a series of fractal half-capillaries.

It was shown not to be possible to match the imbibition rate over the short timescaleregime with an equivalent capillary radius from the long timescale considering the changesof wetting line length related to the meso and/or nanoscale pore internal wall fractal rugosity,as the increase of wetting line length and wetting force is limited. The fractal structure failsto provide sufficient extra wetting force to match the practical physical situation during spon-taneous wetting assuming the viscosity-controlled LW dynamic. On the contrary, however,it was possible to provide a good match with experimental imbibition rate, within the sizelimited range defined by nitrogen condensation hysteresis, by adopting the short timescalesolution of the Bosanquet equation. This model proposes the plug flow regime associatedwith the rapid acceleration experienced in the pore wall surface meso and nanoscale features.The good fit with experiment supports the hypothesis that the long timescale dynamic isestablished within the equilibrium flow during the later stages of imbibition as the surfacecondensation of the liquid in the meso-roughness of the pore surfaces is completed. It alsosuggests that the inertial regime term and its cumulative effect play a key role in defining theimbibition of porous media which should not be ignored when studying the mechanism ofabsorption at the initial stage.

The fractal model described in this work provides an effective method to predict theimbibition rate of complex porous materials adopting a uniform equivalent capillary radiusand considering an inertial regime acting continuously in the fine scale pore wall surfaceroughness.

Acknowledgments The authors express their thanks to Dr. Philip Gerstner, formerly of Aalto University,for providing the sample tablet formed formulations, and to Dr. Carlo Bertinetto, Aalto University, for hissuggestions for the design of the matching algorithm of the fractal structure in MatLab. We also acknowledgethe Scientific Research Program Funded by the Foundation (No. 201309) of Tianjin Key Laboratory of Pulp &Paper (Tianjin University of Science & Technology), P. R. China, supported by Shaanxi Provincial EducationDepartment (Program No. 2014JK0636) and by the Research Projects of the Provincial Key Laboratory ofScience and Technology Department of Shaanxi Province (2011HBSZS014).

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Spontaneous Inertial Imbibition in Porous Media Using a Fractal Representation of Pore Wall RugosityAbstract1 Introduction2 Experimental Observation of Imbibition by Porous Materials2.1 Spontaneous Imbibition Experiment2.2 The Separated Timescale Phenomena of Spontaneous Imbibition

3 Proposing the Meso/nanosurface Wetting Hypothesis3.1 A Wetting Model of Internal Nano/meso-featured Porous Material3.2 The Experimental Determination of the Pore Size over Condensation Area

4 Fractal Structure Modelling of Pore Wall Rugosity5 Modelling Experimental Imbibition Using Fractal Pore Wall Based on L--W Equation5.1 How Well Does L--W Match Experimental Results?5.2 The Reason of Failed Matching

6 Matching Model of Experimental Imbibition Using Fractal Pore Wall Based on Bosanquet Inertial Plug Flow7 ConclusionsAcknowledgmentsReferences