porosity and its measurement architectures within the

Characterization of Materials, edited by Elton N. Kaufmann.Copyright � 2012 John Wiley & Sons, Inc.

POROSITY AND ITS MEASUREMENT�

LAURA ESPINAL

Material Measurement Laboratory, National Institute ofStandards and Technology, Gaithersburg, MD, USA

INTRODUCTION

Porosity is one of the factors that influences the physicalinteractions and chemical reactivity of solids with gasesand liquids for many industrial applications. The exam-ples of industrially important porous materials includecatalysts, construction materials, ceramics, pharma-ceutical products, pigments, sorbents, membranes,electrodes, sensors, active components in batteries andfuel cells, and oil and gas bearing strata and rocks. Thevolume fraction of porosity (e) can be defined as thefraction of void space (Vn) relative to the apparent totalbulk volume (VT) of the sample (Equation 1) (Klobeset al., 2006). For a single phase material, the value ofVn can be obtained from the difference between thevolume of the solid (computed from its crystal latticedensity) and the apparent total bulk volume (VT) ofthe sample:

e ¼ Vn

VTð1Þ

Porosity in materials originates from different proces-singandsynthesis routes (Klobesetal., 2006;Rouquerolet al., 1994a). Synthesis of porous crystalline materials,such as zeolites and metal organic frameworks, leads tohighly regular intracrystalline pore networks in additionto inherent voids imparted by the presence of vacancies,grain boundaries, and interparticle spaces (Auerbachet al., 2003). As a simplifiedmeasure, pore size (orwidth)is referred to the smallest dimension within a given poreshape, that is, the width between two opposite walls fora slit-shaped pore and the diameter for a cylindricalpore (Rouquerol et al., 1999). Some zeolite crystal struc-tures with their characteristic pore sizes are shown inFigure 1. Porous materials can also be fabricatedthrough the bonding of finely divided nonporous parti-cles (e.g., sand) that form agglomerates at specific pres-sure and temperature conditions, which results ininterparticle void space (Rahaman, 2003). Figure 2shows a schematic of an agglomerate of polycrystallineparticles containing voids in between particles (Raha-man, 2003). Another route to porous solids constitutesdirect self-assembly of particles, examples of suchstruc-tures can be found in references (Deng et al., 2005;Warren et al., 2008).

Selective removal of certain elements in a multicom-ponentmaterial structure, via thermal decomposition orchemical etching, also leads to the formation of porous

architectures within the solid material (Mansooriet al., 2007; Rouquerol et al., 1994a). Figure 3 showsthe scanning electronmicroscopy (SEM) image of hollowsilica spheres prepared through chemical dissolution ofthe polystyrene nanoparticles used as template(Mansoori et al., 2007). The natural deposition of sedi-ments in geologically ancient seas also yields porousrocks and formations, which are composed of multipleminerals and contain intracrystalline and interparticlevoids (Dandekar, 2006). Porosity in rocks can also beinduced by natural geological processes following depo-sition, such asweathering or dissolution, dolomitization(a cation exchange reaction that increases porosity),and formation of fractures (Dandekar, 2006; Tiab andDonaldson, 2012). Figure 4 presents an example of thetypes of porosity found in sandstone formations (Tiaband Donaldson 2012).

Themost commoncharacteristics of porous solids arespecific surface area (SSA), average pore size, and pore-size distribution (Rouquerol et al., 1994a). Knowledge ofsuch material characteristics is critical for ensuringmaterial quality requirementsanddesigningnewporousmaterials for various industrial applications. SSA isdefined as the area of solid surface per unit mass ofmaterial. The surface area detectable by a particulartechnique depends on the size of the “yardstick” or probeused in the measurement (e.g., gas molecule of a givensize and light of a certain wavelength), experimentalconditions, and pore or surface model employed in thecalculations (Klobes et al., 2006; Rouquerol, 1994a).Therefore, the recorded values depend on both theinstrument and assumptions employed. Complexitiesalso exist for pore-sizemeasurements, particularly giventhe large variability and irregularity in pore shapes. Asshown in Figure 5, pores can have different shapes andaccessibility to gases or liquids: (a) closed pores isolatedfrom other pores, (b) ink-bottle-shaped pores that areclosed only at one end, (c) cylindrical pores that are openatbothends, (d) funnel-shapedopenpores, (e) cylindricalopen interconnected pores, and (f) cylindrical pores thatare open only at one end (Klobes et al., 2006; Rouquerol,1994a). The sequence in which pores sizes aremeasured(or detected) depends on the physical principle or detec-tionmethod,which ishighlyaffectedbyvariations inporeshapes and interconnectivity betweenpores. In the Inter-national Union of Pure and Applied Chemistry (IUPAC)thesystem forclassifyingporousmaterials, pore sizesaregrouped as follows: micropores (widths<2nm), meso-pores (2nm<widths<50nm), and macropores (widths>50nm) (Klobes et al., 2006; Rouquerol, 1994a, 1999).

TECHNIQUES

The large variety of porous structures and industrialapplications has led to the development anduse ofmanyexperimental techniques for determining the variouscharacteristics of porous solids (Rouquerol et al.,1994a). Gas sorption (Rouquerol et al., 1999; Lowellet al., 2006; Keller and Staudt, 2005; Asthana et al.,2006), liquid intrusion (Rouquerol et al., 1999; Lowell

*Disclaimer: Official contribution of the National Institute ofStandards and Technology; not subject to copyright in theUnited States.

et al., 2006; Ramachandran and Beaudoin, 2001;Sibilia, 1996, microscopy (Khayet and Matsuura, 2011;Reese, 2006), and x-ray and neutron scattering(Aligizaki, 2006; Feigin and Svergun, 1987; Silversteinet al., 2011;Wong, 1999) are themost commonmethodsfor porosity characterization, although other techniquessuchasthermoporometry (Rodriguez-Reinosoetal.,1991;Somasundaran, 2006; Brown and Gallagher, 2003),inverse size exclusion chromatography (Rouquerol,1994b; Unger et al., 2011; Striegel et al., 2009), positronannihilation spectroscopy (Petkov et al., 2003), nuclearmagnetic resonance (Wong,1999;Konsta-Gdoutos,2006;Stapf and Han, 2006), muon spin resonance (Lee et al.,1999), and ultrasonic attenuation methods (Thompsonand Chimenti, 1993; Kutz, 2002) can also provide anindependentassessmentofporosity.AsshowninFigure6,each method provides reliable information for differentpore-size ranges based on the specifics of the measuringphysical principle. The preference for using one methodover another depends on the expected range of pore sizes,

material properties, instrument availability, samplegeometry requirements, and final application. For exam-ple, gas separation sorbent materials (usually powders)are typically characterized using gas sorption methods(Keller and Staudt, 2005) while membranes via micros-copy (Li et al., 2008) and fluid flow (Khayet and Mat-suura, 2011). If resources are available, more than onemethodcanbeused for cross verification though therewillbe differences inherently associated with the use of differ-ent detection methods. For materials containing hierar-chical pore structures that span multiple length scalesfrom angstroms to millimeters, several instruments maybe needed to provide adequate characterization. In thisarticle, we provide a brief description of themost commonporosity measurement methods.

Gas Sorption

The physical adsorption of gases by porous materials atsubcritical temperatures, which involves weak intermo-lecular forces, has been widely used for micropore andmesopore analysis (Rouquerol et al., 1999; Lowellet al., 2006; Keller and Staudt, 2005). Prior to gas

Figure 1. Pore sizes of different zeolites (1nm¼10A�). The three-

letter acronyms shown represent the crystal structure of thezeolite displayed at the top of each column. Reprinted withpermission from Auerbach et al. (2003).

Figure 2. Schematic diagram of agglomerate consisting ofdense and polycrystalline particles. Reprinted with permissionfrom Rahaman (2003).

Figure 3. Scanning electron microscopy (SEM) image of hollowsilica spheres prepared by using polystyrene nanoparticles as atemplate.Polystyrenenanoparticlesare removedviadissolutionin toluene. Reprinted with kind permission from SpringerScience þ Business Media: (Mansoori et al., 2007).

Figure 4. Types of porosity found in geological sandstonereservoirs. Reprinted from Tiab and Donaldson (2012),Copyright 2012, with permission from Elsevier.

2 POROSITY AND ITS MEASUREMENT

adsorption experiments, the sample must be evacuatedor degassed to remove moisture, vapor, and any otherunwanted gas species from the pore surface. Sampleevacuation canbeachieved viaheatingunder vacuumorflushing with an inert gas. In order to avoid exceedingsample degradation temperatures during degassing, thethermal stability of the sample must be known. In thecase of samples sensitive to heating, water removal canbe accomplished in a dessicator (Lowell et al., 2006). Theevacuated sample of known mass is transferred to thesorption chamber, typically using a glove box to avoidthe exposure of the sample to air after degassing. Whileinside the sorption chamber, the sample is progressivelydosed with the known amounts of gas admitted into thesorption chamber in incremental pressure steps, allow-ing sufficient time to reach equilibrium at every step(Rouquerol et al., 1999). Once the final target adsorption

pressure step is achieved, usually saturation pressure(p0), the reverse process takes place (desorption) inwhich gas is progressively withdrawn until the initialstate is reached. The quantity of gas adsorbed (normal-ized by sample mass) at every step during the adsorp-tion and desorption process is recorded as a functionof the relative pressure (p/p0) of the adsorptive gas andthe resulting adsorption–desorption curve is known asa gas sorption isotherm. Some gas–adsorbent systemsexhibit hysteretic sorption behavior in which thedesorption branch of the isotherm follows a differentpath than the adsorption (Lowell et al., 2006). Basedon sorption data, pore models, and other assumptions,the surface area and pore-size distribution can beestimated (Rouquerol et al., 1994a). Although a priorisample evacuation is required, the great advantage ofthe gas sorption technique lies on its nondestructivenature (Lowell et al., 2006). An example of sorptionisotherms for calcined mesoporous silica is shown inFigure 7a (Mal et al., 2002).

For the measurement of the quantity of gas adsorbedat every pressure point, there are volumetric and gravi-metric techniques. Volumetric methods are generallyemployed for nitrogen at 77K and argon at 87K. Gravi-metric techniques are more common for studying theadsorption of vapors at temperatures near ambient con-ditions (Klobes et al., 2006). The pore-size distributionmeasurements are typically carried out using theadsorption branch of the isotherm although the desorp-tion branch can also be used (Lowell et al., 2006). Thepore-size range that can be measured with gas sorptionmeasurements depends on the size of the gas moleculeemployed as the probe and the capability of the equip-ment to achieve high relative pressures. In the case ofnitrogen, the smallest detectable pore size is about0.4nm and the largest is approximately 300nm(Klobes et al., 2006). However, for comparing pore-sizedistributions of materials containing micropores (poresizes<2nm), adsorptive molecules smaller than nitro-gen (e.g., argon) could provide more reliable data(Lowell et al., 2006). Anexample of pore-size distributionfor mesoporous silica is shown in Figure 7b (Mal et al.,2002).

The determination of pore-size distribution inmesoporous materials requires the use of methodsbased on the following Kelvin equation (Klobes et al.,2006; Lowell et al., 2006):

rk ¼ 2 �Vm � g � cos yR �T � ln p

p0

� � ð2Þ

where rk is the radius of curvature of the condensed gasinside the pore, g is the surface tension, Vm representsthe molar gas volume of an ideal gas, and y the contactangle.

For the calculation of the pore size, the value of rk iscorrected according to the assumed pore geometry (e.g.,cylindrical or slit-shapedpore) and thickness of the layerof gas adsorbed on the pore surface. For example, in

Figure 5. Schematic cross-section of a porous solid (Klobeset al., 2006).

Figure 6. Pore-size measurement range for different methods(Klobes et al., 2006).

POROSITY AND ITS MEASUREMENT 3

the case of a cylindrical pore, the radius (rp) is correctedby adding the thickness (rp¼ rk þ t), whereas in thecase of a slit-shaped pore, the slit-width (dp) is givenby dp¼ rk þ 2t. Several methods exist for estimating thethickness (Klobes et al., 2006).

Due to the characteristic molecular dimensions ofmicropores, which enhance interactions between gasmolecules and pore surface, separate models andmeth-ods of data analysis have been developed for microporeanalysis (Klobes et al., 2006; Rouquerol et al., 1999;Lowell et al., 2006) including Dubinin–Radushkevitch,Horvath–Kawazoe, t-plot method, and others. In addi-tion, employinga rangeof gasmolecules of different sizescan provide a more realistic picture of the pore micro-structure (Lowell et al., 2006).

TheBrunauer–Emmett–Teller (BET)methodhasbeenapplied for several decades to physisorption isothermdata to yield the values of the surface area. The first stepin calculating the BET surface area involves the calcu-lation of the monolayer capacity according the followingequation:

p

na p0�pð Þ ¼1

namC

þ C�1ð ÞnamC

� p

p0ð3Þ

where na is the amount adsorbed at the relative pressurep/p0, n

am represents the monolayer capacity, and C is a

constant (Klobes et al., 2006; Rouquerol et al., 1994a).The linear curve obtained from plotting p

na p0�pð Þ versuspp0, at relative pressures between 0.05 and 0.3, is used to

calculate the value of themonolayer capacity, nam . Based

on the nam obtained and the average molecular cross-

sectional area (am ) assumed for the adsorptivemolecule,the BET surface area can be estimated according to thefollowing equation:

SABET ¼ nam �L �am ð4Þ

where L is the Avogadro constant (Klobes et al., 2006).A common assumption for nitrogen is that the BET

monolayer is close-packed (Klobes et al., 2006), whichyields a value of am of 0.162nm2 (at 77K). The value ofthe surface area obtained excludes the contributionfrom those pores smaller than the gas molecules usedas probe.

Fully automated commercial instruments are com-monly used for gas sorption measurements except inlaboratories where custom units have been developed toenhance certain measurement features required tomeet specific research needs. Because porosity datadepend on both the experimental technique used andthe theoretical method employed for data analysis, cer-tified reference materials and standardized measure-ment procedures are used to calibrate and check theperformance of the sorption equipment. Certified refer-ence materials for the surface area and pore size areobtainable from the following internationally recognizedstandard organizations: Bundesanstalt f€ur Material-forschung und -pr€ufung (BAM, Germany), Institute forReference Materials and Measurements (IRRM, Euro-pean Community), National Institute of Standards andTechnology, (NIST,USA), andLGCStandards (LGC,UK).Standardized measurement procedures are availablefrom international organizations such as the Interna-tional Organization for Standardization (ISO, Geneva)and the American Society for Testing Materials (ASTMInternational).

Liquid Intrusion

Liquid intrusion by a nonwetting liquid is a very widelyaccepted method for analyzing meso and macroporescales (Rouquerol et al., 1994a), particularly between4nm and 60 mm. The most commonly used method isknownasmercury porosimetry. In caseswhere a sampleis known to react with mercury (e.g., zinc and gold),careful passivation of the sample or the use of an alter-native nonwetting fluid must be considered. The exper-imental procedure in mercury porosimetry is similar togas sorption. In a typical experiment, a sample ofknown mass is placed inside a sealed enclosure

7002.8 (nm)

3

2

1

00 2 4 6 8 10 12 14 16 18 20

Pore size (nm)

c

b

a

c

b

a

600

Por

e si

ze d

istr

ibut

ion

(cm

3 g–1

nm

–1)

V (c

m3 g

–1)

500

400

300

200

100

00.0 0.2 0.4

P/P0

0.6 0.8 1.0

(a) (b)

Figure 7. Nitrogen sorption isotherms (a) and pore-size distribution curves (b) of three differentcalcined mesoporous silica samples. Reprinted from (Mal et al., 2002), Copyright 2002, withpermission from Elsevier.


(penetrometer) and evacuated to remove unwanted spe-cies from the pore space (e.g., vapor, moisture, and air)via heating under vacuum or flushing with an inert gas.For heat-sensitive samples, prior drying in a dessicatormight be required (Lowell et al., 2006). After evacuation,the nonwetting fluid is progressively introduced into thechamber through incremental hydraulic pressure steps.The volume of mercury is recorded as a function ofapplied pressure (Klobes et al., 2006).

Based on the external pressure p (MPa) required topush themercury intoacylindrical poreof radius rp (mm),pore sizes can be obtained according to the followingWashburn equation (Klobes et al., 2006):

rp ¼ 2 � g � cos yp

ð5Þ

where g represents the surface tension of the liquid(N/m) and y the contact angle. For liquid mercury, thesurface tension at room temperature is approximately0.480N/mand the contact angle is typically assumed tobe 140� (Klobes et al., 2006). Amercury porosimeter canmeasure pore sizes from as low as 1.8nm to as high as200 mm (Lowell et al., 2006), depending on the specificpressure capabilities of the equipment and the contactangle assumed in the calculations. The lowest fillingpressure achievable by the instrument will determinethe largest pore size and the highest pressure will estab-lish the smallest pore size. Note that the radius rp repre-sents the size of an equivalent cylindrical pore thatrequires a given pressure to be filled. Therefore, thevalidity of pore-size measurements via mercury porosi-metry also depends on the pore geometry assumed.

A plot of the intruded (and extruded) volume of mer-cury (V) as a function of applied pressure (p) is known asa porosimetry curve. Based on the function V¼V(p), thetotal surface area can be estimated according to thefollowing equation (Klobes et al., 2006):

SAHg method ¼ 1

g cos y

ðVHg;max

VHg;0

p �dV ð6Þ

The values of surface area obtained from mercury poro-simetry exclude contributions from very small pores.Therefore, a complete agreement with the gas-sorption-basedsurfaceareameasurementsshouldnotbeexpected.As in the case of gas sorption equipment, commercialmercury porosimetry instruments are readily available.Also, certified referencematerials and standardmeasure-ment protocols for mercury porosimetry are accessiblefrom the international standard organizations mentionedearlier to enable equipment calibration and allow datacomparisons across laboratories worldwide.

Figure 8 displays an example of mercury porosimetrydata for a porous glass showing the volume of mercuryintruding into the pore space available as applied pres-sure increases from point A to B. Upon depressurization(from point B to C), the extrusion curve does not followthe same path, compared to intrusion, indicating somemercury is permanently retained in the pores (Klobes

et al., 2006). The permanent entrapment of mercuryupon extrusion makes this technique destructive.Another disadvantage of this method involves the usemercury,which shouldbehandledwith the samecare asother chemicals in the laboratory (Lowell et al., 2006).Local safety regulations should be checked for guide-lines on the proper use, handling, and disposal of mer-cury. Permissible exposure limits for mercury can befound in reference U.S. Department of Labor Occupa-tional Safety & Health Administration, OSHA.

Microscopy

Microscopy allows the reliable analysis of the pore geom-etry and sizes in the mesopore range or above via directobservations of thin cross-sections of solid materials.Depending on the specific pore-size resolution required,optical or electron microscopy can be used. Detailedequipment information can be found in Chapters ELEC-

TRON TECHNIQUES and OPTICAL IMAGING AND SPECTROSCOPY.Due to the wavelength of light available in opticalmicroscopy, its lateral resolution is in the order of200nm (Reese, 2006). Smaller features are typicallyanalyzed by electron and atomic force microscopy tech-niques. Sample preparation is an important consider-ation when analyzing samples via microscopy. For largesamples such as rocks and cement components, the useof thin sections is a very common practice. Althoughpowder samples are also suitable for analysis, the use ofthin sections allows for analysis of all sample compo-nents at the same level or depth. A variety of techniquescan be employed for improving contrast between matrixand pores (M€uller-Reichert, 2010; Hayat, 2000), forexample: surface replication, shadowing, or impregna-tion of the pore space with fluorescent resin. Noncon-ducting samples, such as polymeric membranes,require a thin metallic coating to avoid electrostaticcharging during electron microscopy analysis(Reese, 2006). The collection of a statistically large num-ber of samples and measurements is necessary for acomprehensive characterization of the pore geometry

Figure 8. Intrusion–extrusion curve (volume versus pressurecurve) of a porous glass (Klobes et al., 2006).


and size distribution. Numerous techniques and com-mercial software (Khayet and Matsuura, 2011; Bennettet al., 1991) are readily available for the automaticprocessing of 2D images of porous materials to yield theratio of the fractional areas of pore space to solidmatrix,pore-size distribution, distribution of pore lengths andwidths, and specific surface area. Figure 9 presents thepore-size distribution of two commercial distillationmembranes (Khayet and Matsuura, 2011), both basedon image analyses.

Image analysis via microscopy provides a direct mea-surement where no models or further assumptions arerequired. However, the requirement for having either aplane cross-sectional sample or application of a metalcoating makes microscopy-based techniques destruc-tive. In addition, the suitability of microscopy for aparticular sample depends on the effect of sample prep-aration requirements (e.g., thin sectioning, contrastenhancement method used, and metal coating proce-dure applied) on its pore structure. Another importantaspect relates to potential underestimation of the spe-cific surface area if the instrument employed cannotresolve the smallest pore features in the sample(Rouquerol et al., 1994a). Thin section samples arecommonly used in the petroleum industry to analyzethe pore structure of rocks and geological formations.However, additional measurements may be needed tocomplete the material’s evaluation. In the case of rocksamples, fluid flow measurements are needed for pre-dicting many of the important properties for the properassessment of porous media in the oil industry, forexample, permeability (Civan, 2000). Examples of car-bonate and sandstone thin sections are shown in Fig-ure 10 (Tsakiroglou et al., 2009).

Light, X-ray, and Neutron Scattering

Light, x-ray, and neutron scattering can be used forporosity characterization as solid material and voidspaces have different refractive indices (light sensitive)(Wong, 1999), electronic structure or density (x-ray sen-sitive) (Glatter and Kratky, 1982), and nuclear structureor neutron scattering length density (neutron sensitive)

(Elias, 2008). In brief, the pore-size distribution andsurface area can be deduced from measurements of theangular distribution of scattered intensities. The tech-niques are known as light scattering (staticand dynamic), small angle x-ray scattering (SAXS), andsmall angle neutron scattering (SANS). For detailedinformation on these methods, see article on DYNAMIC

LIGHT SCATTERING within the chapter on OPTICAL IMAGING

AND SPECTROSCOPY, and also see Chapters NEUTRON TECHNI-

QUES and X-RAY TECHNIQUES.In contrast to gas sorption techniques, scattering

techniques require no pretreatment of samples priorto data collection and thus can be analyzed in the pres-ence of a gas or air (Rouquerol et al., 1994a). However,material properties such as chemical composition andopacity must be considered for selecting the scatteringmethod to be used (Wong, 1999). For example, opaquematerials are not well suited for light scattering but arenot a problem for SAXS or SANS. In addition, the mate-rial used as sample containermust be transparent to theincident beam. Accordingly, the samples are placed inenclosures made of window materials suitable for everytechnique, for example, mica or thin polymeric film forSAXS (Aligizaki, 2006), and quartz for SANS (Rouquerolet al., 1994a). While dry powder samples can simply beplaced in between thin polymer films (Rouquerol et al.,1994a), liquid-containing samples may require the useof a special sample holder to keep the liquid in place. Thecharacterization of porous materials containing highZ-elements using SAXS requires sample thicknessesin the order of 0.25mm (Aligizaki, 2006) while thedesired path length for SANS is typically between 1 and5mm(Rouqueroletal.,1994a;Aligizaki,2006).Comparedto SANS, x-ray-based scattering provides higher sensitiv-ity to samples containing heavy elements. Neutron-basedscattering is ideal for materials containing light elementssuch as hydrogenous compounds. For absolute intensitymeasurements, the powder sample density and thicknessmust be recorded.

Both SAXS and SANS experiments require specialfacilities with a source of radiation, a monochromator,collimation system, sample holder, and a detector. Forequipment details see chapters on NEUTRON TECHNIQUES

and X-RAY TECHNIQUES. In a typical scattering experi-ment, the sample is bombarded with a monochromaticbeam of light, x-rays, or neutrons, of wavelength lo andintensity Io, which interacts with the sample compo-nents, including void spaces and solid matrix (Aligi-zaki, 2006). Such interaction produces a range ofscattering intensities, which aremeasured as a functionof the scattering angle (2ys) between the incident andscattered beams, as defined by d � l/2ys where d is thesize of the feature (e.g., pore diameter) and l the wave-length of radiation (e.g., light, x-rays, or neutrons)(Rouquerol et al., 1994a). A scattering curve can thenbe obtained by plotting the scattering intensity, I(Q) as afunction of the scattering angle 2ys or scattering vectorQ, (for a detailed definition of Q, see article SMALL

ANGLE NEUTRON SCATTERING). Pore structure informationis derived from analyses of small angle scattering datathrough the fitting of scattering data to models

35Fr

eque

ncy

(%)

Freq

uenc

y (%

)

(a) (b) 30

2520

1510

5

0

3025201510

dp (µm)dp (µm)

50 00.1

0.1 0.2

0.2 0.3

0.3 0.4

0.4 0.5

0.5 0.6

0.6 0.7

0.1 0.2

0.2 0.3

0.3 0.4

0.4 0.5

Figure 9. Pore-size distribution obtainedby software analysis offield emission scanning electronmicrographs of two commercialmembranes: (a) membrane A and (b) membrane B. The totalnumber of pores per image used in the calculations ranged from900 to 1300. Reprinted from Khayet and Matsuura (2011),Copyright 2011, with permission from Elsevier.


(Aligizaki, 2006) and the use of standard plots, numer-ical methods, and approximations (e.g., Guinier, Porod,and Zimm) (Glatter and Kratky, 1982) to extract char-acteristic slopes and intercepts. The interpretation ofscattering data requires prior knowledge of the expectedmorphological features in the sample in order to assumea suitable pore network geometry and reduce the num-ber of unknowns (Aligizaki, 2006). Methods for dataanalysis provide the size of various regionswith differentrelative densities via fitting to chosen models. While x-ray and neutron scattering are techniques sensitive tothe relative variation of the sample composition, thesemeasurements do not provide absolute densities. There-fore, results derived from these techniques should betreated with caution and be complemented with datafrom other methods such as microscopy and gas sorp-tion. Figure 11 presents data for two mesoporous silicasamples containing pores of different geometry and size,

with diameters of approximately 2nm (cylindrical) and13nm (spherical) (Smarsly et al., 2005).

Scattering techniques can be used to measure thesurface area between the hollow and solid structure aswell as the void-to-solid matrix volume fractions attrib-uted to the presence of intraparticle pores, particle sizes,and interparticle spaces. The type of radiation in eachtechnique determines the spatial resolution, whichranges from 5 to 50nm for SAXS, from 50 to 500nm forSANS, and from 200 to 20,000nm for light scattering.Scattering techniques cover mean pore sizes between1 and 1000nm (Rouquerol et al., 1994a). The value ofthe surface area obtained from scattering data excludescontributions outside the detectable pore-size range.Thus, complete agreement with gas sorption-basedsurface area measurements should not be expected.For highly crystalline materials, information on porewidths in the micropore range can be obtained from

Figure 10. Backscattered electronmicroscope images of thin section samples of: (a) carbonateand (b) sandstone. Black areas in the image represent pore spaces between grains. Reprintedfrom Tsakiroglou et al. (2009), Copyright 2009, with permission from Elsevier.

Figure 11. Transmission electron microscopy images(Left) and SAXS curves (Right) of two mesoporous silicasamples containing different pore geometries: cylindrical(Top) and spherical (Bottom). The Porod asymptote isindicated by the dashed line in both cases. Reprintedwith kind permission from Springer Science þ BusinessMedia: (Smarsly et al., 2005).


x-ray and neutron diffraction data (Ikeda et al., 2010;McCusker, 2004; Wessels et al., 1999; Wright, 2008;Cheetham et al., 1986; Newsam, 1991).

Scattering techniques have the great advantage ofbeing nondestructive and have particularly great sensi-tivity to the presence of closed pores (Rouquerolet al., 1994a; Wong, 1999), which are undetectable bygas sorption measurements as gas molecules cannotaccess closed pores. In cases where the material issuspected to have closed porosity, index-match lightscattering (Wong, 1999) or contrast-match SANS(Reese, 2006; Aligizaki, 2006) experiments can be per-formed in which the sample is analyzed before and afterimpregnation with a liquid of similar scattering proper-ties to the solid matrix. A material containing closedporosity will continue to show the presence of certainpores (closed pores) after impregnation.

SUMMARY

Due to the inherent difference in the physical principlesfor each method, perfect agreement between the valuesof total porosity, surface area, average pore size, anddistribution obtained for all techniques is not possible.Similarly, a perfect agreement between two techniqueswill not necessarily validate the values obtained. Withina given technique, the postsynthesis history of the sam-ple should be taken into account as many materialapplications require postsynthesis treatments (e.g.,annealing and mechanical forming), which can lead todifferent porosity. All techniques have advantages anddisadvantages, inparticularwith respect to the assump-tions made to derive the results. An approach to under-stand how these techniques differ from each other couldinvolve the evaluation of a referencemodel porousmate-rial using the techniques under consideration. Theselection criteria should extend beyond the expectedpore-size range and take into account the suitability ofsample preparation, material property, and samplegeometry requirements for every technique as well asthe intended material’s application.

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KEY REFERENCES

Lowell, et al., 2006. See above.Provides an overview of principles associated with the charac-

terization of solidswith regard to their surface area, pore size

and density based on gas adsorption and mercury

porosimetry.

Hayat, 2000. See above.Describes fundamental principles and techniques of electron

microscopy for materials with biological applications.

Rouquerol, et al., 1999. See above.Reviews theoretical and practical aspects of adsorption by pow-

ders and porous solids and presents some of the adsorptive

properties of common materials such as activated carbons,

oxides, clays and zeolites.Wong, P.-z. 1999. See above.Presents several techniques for characterizing porous media, in

particular scattering methods, from the standpoint of an

experimental physicist.


porosity and its measurement architectures within the

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