M A T E R I A L S C H A R A C T E R I Z A T I O N 6 0 ( 2 0 0 9 ) 1 4 – 2 0

Evaluation of permeable pore sizes of macroporous materialsusing a modified gas permeation method

Hyunjung Kima,1, Yoseop Hanb, Jaikoo Parkb,⁎aDepartment of Chemical and Environmental Engineering, University of California, Riverside, Riverside, CA 92521, USAbDepartment of Geoenvironmental System Engineering, Hanyang University, #17, Heangdang-dong, Seongdong-gu, Seoul 133-791,Republic of Korea

A R T I C L E D A T A

⁎ Corresponding author. Tel.: +82 2 2220 0416;E-mail addresses: kshjkim@gmail.com (H. K

1 Tel.: +1 951 529 3210; fax: +1 951 827 5695.

1044-5803/$ – see front matter © 2008 Publisdoi:10.1016/j.matchar.2008.05.002

A B S T R A C T

Article history:Received 23 December 2007Received in revised form25 April 2008Accepted 9 May 2008

The sizes of permeable pores in porous materials, assuming three-dimensional reticulation,were evaluated using a modified gas permeation method for two important purposes: todevelop a technique for characterizing pores in macroporous materials, and to selectivelyestimate the dimensions of permeable pores influencing fluid flow. The range of pore sizesmeasuredusing thismethodwasbetween30and80μm,withameanvalueof about 50μm.Thevalidity of using amodified gas permeationmethodwas justified by the fact that the fluid flowalways followed the Hagen–Poiseuille's law. Moreover, the distribution of pore sizesdetermined by this method was smaller than that determined by the image analysis method.

© 2008 Published by Elsevier Inc.

Keywords:Porous materialsPermeable poresPore size distributionModified gas permeation method

1. Introduction

Porous materials have been widely used in applications, suchas molten metal filtration, hot gas filtration, and catalystsupport, where fluid transport is required [1–5]. In order tomeet the stringent requirements of these applications, it isessential for control purposes that the pore structure of porousmaterials be characterizable. Pore characteristics, such as poresize, porosity and pore morphology, are among the mostfundamental properties of porousmaterials, and are intimatedby secondary properties, such as the specific surface area,water absorption, and permeability. Pores are classified intotwo forms, either open or closed. Open poresmay be classifiedas permeable pores, i.e., they open on both sides of the porousmaterials, and non-permeable. Permeable pores contribute tofluid flow through porous materials. Therefore, the character-ization of permeable pores in porousmaterials is important toapplications such as filter technology.

fax: +82 2 2281 7769.

im), jkpark@hanyang.ac.

hed by Elsevier Inc.

So far, diverse methods such as image analysis [6,7], mercuryintrusion [8,9], the gas permeation method (GPM) [1], and othershave been used to determine pore size distributions in porousmaterials. In the case of image analysis, data analysis requires ahigh level of technical skill to achieve satisfactory quantitativeresults. In addition, pores that can be deformed by high entrancepressures have an effect on the size distributionwhenmeasuredusing the mercury intrusion method. Furthermore, the two me-thods have an inherent limitation, in that non-permeable poresaremisidentified as permeable poreswhenmeasuredwith thesetechniques. In contrast, GPM relies on a straightforward experi-mental procedure in which only permeable pores are detected.

Increments of entrance pressure in the conventional gaspermeationmethod (CGPM) causenon-uniformity of fluid flowin permeable pores [9]. In the case of a modified gaspermeation method (MGPM) to be discussed in the followingsections, it is possible to control the flow regime bymeasuringflow rates under a constant low entrance pressure. For the

kr (J.K. Park).

Fig. 1 –A schematic diagram for calculating the pore size of the porous support through the gas permeation method.

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above-mentioned reasons, theMGPMcanbe applied to analyzethe size distribution of permeable pores in macroporousmaterials. This study focuses on evaluating pore size distribu-tions in porousmaterials using theMGPM, and on verifying thevalidity of the method.

2. Background

2.1. Modified Gas Permeation Method

Fig. 1 shows a schematic diagram of the pore size calculationused by the gas permeation method. In CGPM, the pores of aspecimen are initially filled with a wetting fluid and a non-wetting fluid is used as a flowing medium. Generally, water isused as a wetting fluid. The sizes of the pores are calculatedusing the following equation [10]:

DPi ¼4gcosh

dið1Þ

whereΔPi is thedifferential pressure needed to expelwater frompores of diameter di, θ is the contact angle between the supportand the water, and γ is the surface tension of the water. In thisstudy, 73 mN/m and 10° [11] are the values of γ and θ,respectively. The size of all pores was measured by changingthe arbitrary differential pressure (ΔPcur) from the minimumdifferential pressure (ΔPmin), related to the maximum porediameter (dmax), to the maximum differential pressure (ΔPmax),related to the minimum pore diameter (dmin) (Fig. 1).

Fig. 2(a) shows the relationship between the measureddifferential pressure (ΔP′) and permeated flow rate (Q) in CGPM[10]. The solid and broken lines represent two different types offlow curves, referring to specimens with inner pores that areeither saturatedwith water or not saturated, respectively. Fromthe solid line depicted in Fig. 2(a), we know that point A forms aboundary between non-linear and linear flow rates. Here, thestarting and ending points of the non-linear curve mean thatwater is expelled from the largest or smallest pores, respec-

tively. During this step, the gas permeability gradually increasesand the flow curve passes point A, and shows linear behavior inagreement with Darcy's law.

The relationship between the measured differential pres-sure and permeated flow rate, generated for specimens withlarge pores or with a high fluid velocity in inner channels, isillustrated in Fig. 2(b) [9]. As themeasured differential pressureincreases, the permeability of the specimen decreases and aflow transition occurs corresponding to a change from Darcyto non-Darcy flow at point B. Therefore, CGPM is not accurateenough to evaluate the pore sizes of porous supports contain-ing large pores, because the flow is non-Darcian in this region.

The graph showing the ratio between the permeated flowrate and the measured differential pressure according to thevariation of entrance pressure is plotted in Fig. 2(c), for the caseof a porous support inwhichnopores are saturatedwithwater.For the unsaturated porous support, the permeability gradu-ally decreased with increasing entrance pressure. Specifically,this means that increased entrance pressure causes the flowregime to depart from Darcy's flow predictions. To solve theproblemsmentioned above, the permeated flow rate has to bemeasured under a constant low entrance pressure. Thisenables the fluid velocity in the pore channels to decrease,and the fluid flow to obey Darcy's relation [9]. The methodstated above is a modification of the gas permeation method.

In the case of the MGPM, the relationship between themeasured differential pressure and permeated flow rate isdepicted in Fig. 2(d). Here, ΔP′ signifies the differential pressuremeasured under constant low pressure after pores of the sizecorresponding to the arbitrary applied differential pressure areopen. In particular, MGPM can differentiate the applieddifferential pressure from the measured differential pressureand maintain a constant ratio of the measured differentialpressure versus the permeated flow rate (Fig. 2(d) α).

2.2. Flow Equation

In a porous support, the flow equation for the fluid passingthrough cylindrical channels of arbitrary diameters depends

Fig. 2 –The relationship between permeated flow rate and measured differential pressure (a and b) for the conventional and(d) themodified gas permeationmethod. (c) The ratio of the measured differential pressure and permeated flow rate versus theentrance pressure in the porous support. Applied conditions are as follows: (a–c) various and (d) constant entrance pressure.

Fig. 3 –The schematic equipment diagram for the evaluationof pore size in the porous support.

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on the circumstances of the fluid flow. Before determining asuitable flow equation, therefore, the characteristics of thefluid flowmust be determined. In this study, the flow equationwas determined on the basis of the Reynolds number (Re) andthe motion of gas molecules.

The fluid flow through the inner channels of the poroussupport is classified by three types of regimes in terms of Re[12,13]; 1) Laminar (Reb2100), 2) Transition (2000bReb4000), 3)Turbulent (ReN4000). Re is calculated using Eq. (2).

Rei ¼davem

� uave ð2Þ

where, Rei is Re of the fluid passing i th and (i+1) th channels,dave is the average pore diameter of i th and (i+1) th pores, ν isthe kinematic viscosity of fluid (=μf /ρ), and uave is the averagevelocity of the fluid flowing in the channels corresponding tothe diameter of dave. μf and ρ represent the absolute viscosityand density of the fluid, respectively.

In case of a compressible fluid in the laminar regime, thepermeated flow rate along a capillary tube is dependant on themean free path (λ) [14,15] of a gas molecule. λ is calculatedusing Eq. (3):

k ¼ 1p

ffiffiffi2

p � ktPd2m

ð3Þ

where k, T, P, and dm denote Boltzmann constant, absolutetemperature, entrance pressure, and the diameter of a gasmolecule, respectively. If λ is much less than the pore size, thedominant flow can be described by the Hagen–Poiseuilleequation [16]. If λ is much greater than the pore size, Knudsen

diffusion [16] becomes the dominant mechanism. When λ isapproximately equal to the pore size, the flow falls into atransition region between Knudsen diffusion and the Hagen–Poiseuille relation [17].

3. Experimental Procedure

A specimen 60 mm in diameter and 6 mm in thickness wastested. It was prepared using the foaming and gelcasting

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method previously described [2,3,5,18,19]. The starting mate-rial was cordierite (Koyritsu Ceramic Materials Co. Ltd., Japan),and sodium lauryl sulfate (CH3(CH2)11OSO3Na, Samchun PureChemical Co., Korea) was used as the foaming agent.

Prior to making the measurements, the dry specimen wasboiled for 10 min in water to remove any internal gases [20].Fig. 3 shows a schematic diagram of the experimentalapparatus used to evaluate the pore size of the porous support.As depicted in Fig. 3, the test was conducted with air flowingthrough the specimen. The permeated flow rate was mea-sured under a constant entrance pressure of 4000 Pa, whichwas lower than the entrance pressure applied to displace thewater saturated in the largest pores.

Fig. 5 –Two models of the continuous pore structure;(a) 1-directional flow model and (b) 2-dimensional modelof 3-directional flow.

4. Results and Discussion

4.1. Determination of a Flow Equation

The Hagen–Poiseuille equation is applicable provided thatlaminar flow through cylindrical channels predominates andthat the λ of the gas molecule is much smaller than the poresize [13]. Scanning micrographs, pictured in Fig. 4(a) and (b),show that the pores were almost circular. The Re calculation

Fig. 4 –Scanning electron microscope (SEM) images of themicrostructures of the porous support; (a)×30 and (b) ×100.

results indicate that the fluid passing through the innerchannels of the porous support always obeyed the laminarregion in the experimental conditions employed in this study(Reb2100). In addition, the value of λ calculated using Eq. (3)was 1.68 μm for the gas molecules, which is smaller than thesmallest pore size. From the resultsmentioned above, the flowequation for the porous supportmodelwas found to follow theHagen–Poiseuille's law [9]:

qi ¼pd4i

128Ags� DPi

Dxð4Þ

where di is the pore diameter, ΔPi/Δx is the pressure drop of theporous support, μg is the absolute viscosity of the gas, and τ isthe tortuosity of pores. By assuming that all pores are straighttubes, we could set the tortuosity to unity [9].

4.2. Correlation Between the Measured Differential Pressureand the Permeated Flow Rate

As illustrated in Fig. 4(b), the porous support used in this studyhad a 3-dimensional network structure of interconnectedchannels. Therefore, it is essential to simplify this complexpore geometry, as shown Fig. 5(a) and (b). If the inner channelsare considered to be confined to one-dimension and thepressure applied to the specimen is uniform, the variation ofthe permeated flow ratewith time can be depicted as shown inFig. 6(a). In Fig. 6(a), Qi is the flow rate measured at the ith

Fig. 6 –The variation of the permeated flow rate versus time:(a) ideal and (b) actual.

Fig. 8 –The relationship between the permeated flow rate andthe measured differential pressure at representative times.

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applied differential pressure, and ti1, defined as the equili-brium time, is the time elapsed between the application of theith differential pressure and the point at which the permeatedflow rate becomes constant. In reality, the pore modelassumed in this study is 2-dimensional (Fig. 5(b)). If all theconditions are ideal as shown in Fig. 5(a), variations of thepermeated flow rate according to time should resemble thoseshown in Fig. 6(a). However, the actual result was closer to anon-linear curve, as shown in Fig. 6(b). Therefore, it wasnecessary to determine the representative flow rate at anarbitrary time. In Fig. 6(b), ti1 andQi1 denote this representativetime and the flow rate at ti1, respectively. Similarly, ti2 andQtotal represent the time and flow rate measured immediatelyafter the water filling all pores is expelled. The left and rightsides of ti1 refer to the water that is expelled from larger andsmaller pores than those related to the differential pressureapplied at that time. In particular, the right side results fromthe structural features of the pores in the porous support.

From the results regarding the variation in the permeatedflow rate versus time, it was observed that the value of thepermeated flow rate at each applied differential pressure wasnot constant (data not shown). Specifically, increased applieddifferential pressure resulted in a gradual increase in the flowrate. When the minimum applied differential pressure wasapplied, the permeated flow rate was constant up to certaintime,whichwas considered to be a steady state.Meanwhile, themaximum applied differential pressure was determined at thepoint where the slope was close to infinity as shown in Fig. 6(a).The representative flow rate of the fluid passing through eachchannel was obtained at intervals between the minimum andmaximum applied differential pressure values. If the applieddifferential pressure is incremented, the water filling thechannels should be simultaneously expelled at each represen-tative time. This is based on two assumptions: first, thepermeated flow rate at each representative time follows themodel shown in Fig. 6(a). Second, the tortuosity of each channelis unity. On the basis of the above principle, the fluid velocity

Fig. 7 –The relationship between the measured differentialpressure and the permeated flow rate according time(applied differential pressure: 3500 Pa).

through channels was found to be proportional to the applieddifferential pressure, and the rate of change of permeated flowrate was constant as shown by Eq. (5):

AQrv

Atrv¼ const: ð5Þ

where trv and Qrv are the representative time and permeatedflow rate, for a fluid flowing through each channel.

Fig. 7 represents the relationship between the measureddifferential pressure and permeated flow rate with respect totime under a constant applied differential pressure of 3500 Pa,where A is the initial, and B the final flow rate measuredimmediately after all pores are open. During the variation of theflow rates between point A and B, thewater filling each channelis stepwise expelled. More specifically, the measured differen-tial pressure and the permeated flow rate at representative time

Fig. 9 –Flowchart for the calculation of pore size distribution.

Fig. 10 –Cumulative and fractional pore size distributions ofporous support.

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for each pore are determined by line C. The result derived fromFig. 7 can be represented by Eq. (6).

AQrv

A DPrvð Þ ¼ const: ð6Þ

where ΔPrv and Qrv denote the differential pressure and thepermeated flow rate measured at representative times. As aresult, themeasureddifferential pressurewasdeterminedusingEqs. (5) and (6). The relationship between the measureddifferential pressure and permeated flow rate at representativetimes produced the results illustrated in Fig. 8. These results areconsistent with the model depicted in Fig. 2(d), indicatingMGPM,unlikeCGPM,enables the fluid to flowunderDarcy's law.

4.3. Evaluation of Pore Size Distribution

The pore size distribution of the porous support can be obtainedfrom both the relationship between the measured differentialpressure and the permeated flow rate, and by the flow ratethrough one channel as calculated by the Hagen–Poiseuille'sequation [10]. The procedure for evaluating the pore sizedistribution is illustrated in Fig. 9. Where, ΔP, Q, q, and N arethemeasureddifferential pressure, thepermeated flow rate, thecalculated flow rate passing through one channel, and thenumberofopenpores. IfQi−1 andQiare flowrates correspondingto the measured differential pressures of ΔPi−1 and ΔPi, the netpermeated flow rate (ΔQi) can be defined as the flow through thepores with a size distribution between diameter di−1 and di, andis equal to (Qi−Qi−1). Therefore, the number of pores (Ni) in thisrange can be calculated from (ΔQi/qi).

Because of the geometric complexity of the pore structure,the absolute number of pores is overestimated. The mainreasons are as follows: first, the flow rate calculated from theHagen–Poiseuille equation assumes that a channel in theporous support is straight, and its diameter is equal to thesmallest part of a channel [21]. Second, the permeated flowrate measured by experiment reflects the real pore geometry.Accordingly, the concept of volume fraction (fv) should be

introduced to eliminate the overestimation, and (fv,j) for poresof diameter dj can be calculated using Eq. (7).

fv;j ¼Qj=qj� �

d2jPn

i¼1Qi=qið Þd2i

ð7Þ

Fig. 10 shows the volume fraction of pores calculated usingEq. (7). The sizes of the pores ranged from 30 to 80 μm and theaverage pore size was 50 μm.

Through a comparison between MGPM and the imageanalysis method, it was found that the pores evaluated byMGPM were representative of the smallest part of eachchannel [7]. Average pore size determined by the imageanalysis method ranged from 100 to 200 μm [7]. From thetwo results mentioned above, we confirmed that the poresizes evaluated by MGPM represent the smallest diameters ofthe arbitrary channels within the porous support.

5. Conclusions

In this study, a modified method, MGPM, was tested toevaluate the pore sizes of a porous support prepared by thefoaming method. The following conclusions were reached:

1. MGPM can be applied to evaluate pore sizes in a poroussupport. Thiswasverified by a thoroughexaminationofReofthe fluid flowing inner channels of the support and the λ ofgas molecules.

2. The inner channels of the porous supports have a 3-dimensional network structure, which means that the flowthrough them is not constant with respect to time. At eachinstant, however, the permeated flow through each channelvaried linearly. The measured differential pressures at eachrepresentative time decreased linearly as the permeationflow rate increased.

3. The size of the pores evaluated by MGPM in a poroussupport ranged from 30 to 80 μm,with the average pore sizemeasured to be approximately 50 μm, which correspondsto the smallest diameter of each pore channel.

Acknowledgements

This work was supported by the Korea Science and Engineer-ing Foundation Grant (No. 2005-215-D00166).

R E F E R E N C E S

[1] Roy SK. Characterization of porosity in porcelain-bondedporous alumina ceramics. J Am Ceram Soc 1969;52:543–8.

[2] Park JK, Lee JS, Lee SI. Preparation of porous cordierite usinggelcasting method and its feasibility as a filter. J Porous Mater2002;9:203–10.

[3] Kim H, Lee S, Han Y, Park J. Preparation of dip-coated TiO2

photocatalyst onceramic foampellets. JMater Sci 2006;41:6150–3.[4] Furuta S, Katsuki H, Komarneni S. Modification of porous

silica with activated carbon and its application for fixation ofyeasts. J Porous Mater 2001;8:43–8.

20 M A T E R I A L S C H A R A C T E R I Z A T I O N 6 0 ( 2 0 0 9 ) 1 4 – 2 0

[5] Park JK, Park JH, Park JW, Kim HS, Jeong YI. Preparation andcharacterization of porous cordierite pellets and use as adiesel particulate filter. Sep Purif Technol 2007;55:321–6.

[6] PrattWK.Digital imageprocessing.NewYork:Wiley-IntersciencePublication; 1991. p. 261–662.

[7] Park JK, Lee SH. Observation and segmentation of gray imagesof surface cells in open cellular ceramic foams. J Ceram SocJpn 2001;109:580–6.

[8] Sakai K. Determination of pore size and pore size distribution.2. Dialysis membrane. J Membrane Sci 1990;96:91–130.

[9] Lee Y, Jeong J, Youn IJ, Lee WH. Modified liquid displacementmethod for determination of pore size distribution in porousmembrane. J Membrane Sci 1997;130:149–56.

[10] Dallaire S, Angers R. Porosity and permeability of sinteredMgO. J Can Ceram Soc 1982;51:29–34.

[11] Muster TH, Prestidge CA, Hayes RA.Water adsorption kineticsand contact angles of silica particles. Colloids Surf, APhysicochem Eng Asp 2001;176:253–6.

[12] Skjetne E, Auriault JL. New insight on steady, non-linear flowin porous media. Eur J Mech B, Fluids 1999;18:131–45.

[13] Roberson JA, Crowe CT. Engineering fluid mechanics.New York: John Wiley & Sons, Inc.; 1997.

[14] Laidler KJ, Meiser JH. Physical chemistry. NewYork: HoughtonMifflin Co., Inc.; 1999.

[15] Creutz E. The permeability minimum and the viscosity ofgases at low pressure. Nucl Sci Eng 1974;53:107–9.

[16] Hernandez A, Calvo JI, Pradanos P, Tejerina F. Pore sizedistributions in microporous membranes. A critical analysisof the bubble point extended method. J Membrane Sci1996;112:1–12.

[17] Schofield RW, Fane AG, Fell CJD. Gas and vapour transportthrough microporous membranes. 1. Knudsen–Poiseiulletransition. J Membrane Sci 1990;53:159–71.

[18] Lee JS, Park JK. Preparation of porous ceramic pellet by pseudodouble-emulsion method from 4-phase foamed slurry.J Mater Sci Lett, 2001;20:205–7.

[19] Lee JS, Park JK. Processing of porous ceramic spheres bypseudo double-emulsion method. Ceram Int 2003;29:271–8.

[20] ChaoWJ, Chou KS. Studies on the control of porous propertiesin the fabrication of porous supports. Key Eng Mater1996;115:93–108.

[21] McGuire KS, Lawson KW, Lloyd DR. Pore size distributiondetermination from liquid permeation through microporousmembranes. J Membrane Sci 1995;99:127–37.