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Paul A. WebbMicromeritics Instrument Corp.Norcross, GeorgiaJanuary 2001

An Introduction To The Physical Characterization of Materials by MercuryIntrusion Porosimetry with Emphasis On Reduction And Presentation ofExperimental Data

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CONTENTS

INTRODUCTIONSECTION I. THEORY AND METHOD OF MEASUREMENT

IntroductionFluid Dynamics and Capillary HydrostaticsCollecting Experimental DataMeasurement Transducers

SECTION II. OBTAINING INFORMATION ABOUT THE SAMPLE MATERIALPART A. INFORMATION OBTAINED USING VOLUME AND MASS MEASUREMENTS ONLY

Total Pore VolumeMaterial Volume and Density

Bulk and Envelope Volume and DensitySkeletal and True Volume and Density

Interstitial Void VolumePercent Porosity and Percent Porosity Filled

PART B. INFORMATION OBTAINED BY APPLICATION OF WASHBURN’S EQUATION

Pore Volume Distribution by Pore SizePore Area and Number of Pores

PART C. INFORMATION OBTAINED BY APPLICATION OF SPECIAL OR MULTIPLE MODELS

Particle Size DistributionPore Cavity to Pore Throat Size RatioThe Distribution of Pore Cavity Sizes Associated With a Pore Throat SizeMaterial Permeability and the Conductivity Formation FactorPore Fractal Dimensions

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INTRODUCTION

Mercury intrusion porosimetry is one of only a few analytical techniques that permits an analyst toacquire data over such a broad dynamic range using a single theoretical model. Mercuryporosimetry routinely is applied over a capillary diameter range from 0.003 µm to 360 µm— fiveorders of magnitude! This is equivalent to using the same tool to measure with accuracy and preci-sion the diameter of a grain of sand and the height of a 30-story building.

Not only is mercury porosimetry applicable over a wide range of pore sizes, but also the fundamen-tal data it produces (the volume of mercury intruded into the sample as a function of applied pres-sure) is indicative of various characteristics of the pore space and is used to reveal a variety ofphysical properties of the solid material itself.

The information that follows falls into three main categories: I) instrument theory and its applica-tion in data collection, II) information derived from reduced data, and III) presentation of theinformation. A glossary of terms also is included.

Understanding how a fluid behaves under specific conditions provides insight into exactly how amercury porosimeter probes the surface of a material and moves within the pore structure. Thisallows one to better understand what mercury intrusion and extrusion data mean in relation to thesample under test and allows one to understand the data outside of the bounds of the theoreticalmodel. It also allows one to make an educated comparison between similar data obtained usingother measurement techniques and theoretical models.

The information contained herein pertains for the most part to the general technique of mercuryporosimetry without regard to a specific instrument manufacturer or model. However,Micromeritics’ AutoPore series of porosimeters is used as a reference, particularly when examplesare required and details of data reduction are presented.

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SECTION I.THEORY AND METHOD OF

MEASUREMENT

IntroductionRoutine operation of an analytical instrument

does not require knowledge of the fundamentals ofinstrument theory. However, an in-depth understand-ing of the relationship between the probe and thesample allows one to interpret data outside of thestrict limitations of the theoretical model upon whichdata reduction is based. Although this may havelimited relevance for day-to-day quality or processcontrol applications, it is of extreme importance inresearch work and when developing analysis methodsfor control applications. For these reasons, thisdocument begins with information about how a non-wetting liquid (specifically, mercury) reacts inseeking equilibrium between internal and externalforces at the liquid-solid, liquid-vapor, and liquid-solid-vapor interfaces.Fluid Dynamics and Capillary HydrostaticsNote: Supporting information on fluid dynamics andhydrostatics can be found on the Internet at sites (1,2,3)cited in the Reference section of this document.

Consider a drop of liquid resting on a solidsurface as shown in Figure 1. The underside of theliquid is in contact with the solid surface. Theremainder of the surface of the liquid is in contactwith some other fluid above— typically, either itsown vapor or air. In this configuration, there areareas of liquid-solid, liquid-vapor, and solid-vaporinterfaces. There also exists a liquid-solid-vaporboundary described by a line.

Figure 1. Cross-section of a drop of non-wetting liquid restingon a solid surface. All interfaces are shown.

There is tension in each interface. The liquid-vapor interfacial tension is symbolized gl-v, the liquid-

solid gl-s, and the solid-vapor gs-v. The liquid-vaporand solid-vapor interfacial tensions also are referredto as surface tensions. Surface tension has dimen-sions of force per unit length and acts tangentially tothe interface.

The angle of contact of the liquid-vapor surface tothe solid-vapor surface at a point on the liquid-solid-vapor interface characterizes the interfacial tensionpresent between the solid, liquid, and vapor.Figure 2 shows five liquids of different surfacetensions resting on the same surface material. Differ-ent surface energies cause the liquids to assumedifferent contact angles relative to the solid surface.A liquid with low surface tension (low surfaceenergy) resting on a solid surface of higher surfacetension will spread out on the surface forming acontact angle less than 90o; this is referred to aswetting. If the surface energy of the liquid exceedsthat of the solid, the liquid will form a bead and theangle of contact will be between 90o and 180o; this isa non-wetting liquid relative to the surface.

Figure 2. Various liquids resting on a solid surface. The differentangles of contact are illustrated for wetting and non-wettingliquids.

Considering any point along the line that de-scribes the liquid-solid-vapor interface and indicatingall force vectors on that point results in a diagramsimilar to those of Figure 3. These illustrationsrepresent a time sequence (top to bottom) showingwhat happens when a liquid drop first is placed on ahorizontal surface until it achieves equilibrium. Onecan imagine the initial, somewhat spherical dropflattening and spreading over the surface prior tostabilizing. The contact angle begins at about 180o,and the liquid–vapor tension vector at the liquid-solid-vapor interface points at the angle of contact.As the contact angle decreases, the horizontal compo-nent of the liquid-vapor tension vector changes in

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magnitude and, if the contact angle decreases past90o, the horizontal component changes sign. Whenthe sum of the solid-vapor tension vector, liquid-solidtension vector, and horizontal component of theliquid-vapor tension vector equal zero, equilibriumoccurs and spreading ceases.

Figure 3. A droplet of liquid placed on a solid surface assumes acontact angle that balances the horizontal force components ofthe three tension vectors. For this example, è3 is the angle thatresults in equilibrium.

The surface of the liquid at the liquid-vapor interfaceassumes a curvature having two radii, r1 and r2, one inthe x-z plane, the other in the y-z plane, where thesolid surface is the x-y plane. This is another effectof surface tension. The surface molecules act like anelastic membrane pulling the surface into the smallestconfiguration, ideally a sphere where r1 = r2 = r.Surface tension contracts the surface and volume untilthe internal force Fi per unit area of surface As is inequilibrium with the external forces on the samesurface element. Since pressure, P, is force per unitarea (F/A), equilibrium can be expressed in terms ofinternal and external pressures. From the equationsof Young and Laplace for spherical surfaces, thedifference in pressure across the surface is

P” - P’ = γ (1/r1 + 1/r2) = 2γ /r (1)

where P” is the pressure on the concave side, P’ thepressure on the convex side, g the liquid-vaporsurface tension, and, since it is a spherical surface,r1 = r2.Interfacial tensions also cause liquids to exhibitcapillarity. If one end of a capillary tube is forced to

penetrate the vapor-liquid surface from the vaporside, a wetting liquid spontaneously enters thecapillary and rises to a level above the external liquid-vapor interface. A non-wetting liquid resists enteringthe capillary and that a level always below theexternal liquid-vapor level. In other words, a non-wetting liquid must be forced to enter a capillary.

Why does a non-wetting liquid resist entry into acapillary? Inside the capillary and along the linedescribing the vapor-liquid-solid boundary, theliquid-solid interface assumes an angle that results inequilibrium of forces. The contributing forces arethose of cohesion between the liquid molecules, andthe force of adhesion between the liquid moleculesand the walls of the capillary. The liquid-vaporinterface in the capillary (the meniscus) is concave fora wetting liquid and convex for a non-wetting liquid.In summary, there are three physical parametersneeded to describe the intrusion of a liquid into acapillary: a) the interfacial tension (surface tension) ofthe liquid-vapor interface, hereafter symbolizedsimply by g, b) the contact angle q, and c) the geom-etry of the line of contact at the solid-liquid-vaporboundary. For a circular line of contact, the geometryis described by pr2, where r is the radius of the circleor capillary.

Washburn (4) in 1921 derived an equationdescribing the equilibrium of the internal and externalforces on the liquid-solid-vapor system in terms ofthese three parameters. It states concisely that thepressure required to force a non-wetting liquid toenter a capillary of circular cross-section is inverselyproportional to the diameter of the capillary anddirectly proportional to the surface tension of theliquid and the angle of contact with the solid surface.This physical principal was incorporated into anintrusion-based, pore-measuring instrument by Ritter& Drake in 1945 (5). Mercury is used almost exclu-sively as the liquid of choice for intrusionporosimetry because it is non-wetting to most solidmaterials.

Washburn’s equation, upon which data reductionis based, assumes that the pore or capillary is cylin-drical and the opening is circular in cross-section. Ashas been stated, the net force tends to resist entry ofthe mercury into the pore and this force is appliedalong the line of contact of the mercury, solid, and(mercury) vapor. The line of contact has a length of2pr and the component of force pushing the mercuryout of the capillary acts in the direction cosq (seeFigure 4), where q is the liquid-solid contact angle.The magnitude of force tending to expel the mercury

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isFΕ = 2π rγ cosθ (2)

where γ is the surface tension.An external pressure on the mercury is required to

force its entry into the pore. The relationship be-tween force (F) and pressure (P) is P = F/area.Solving for force gives

FI = π r2P (3)

Where π r2 is the cross-sectional area of the poreopening.Balancing the intrusion and extrusion forces results inthe Washburn equation

-2πrγcosθ = πr2P (3)

or, in terms of diameter D,-πDγcosθ = (πD2P)/4 (4)

The relationship between applied pressure and theminimum size pore into which mercury will be forcedto enter is

D = -4γcosθ/P (5)

For a given liquid-solid system, the numerator isconstant, providing the simple relationship expressingthat the size of the pore into which mercury willintrude is inversely proportional to the appliedpressure. In other words, mercury under externalpressure P can resist entry into pores smaller than D,but cannot resist entry into pores of sizes larger thanD. So, for any pressure, it can be determined whichpore sizes have been invaded with mercury and whichsizes have not.

Collecting Experimental DataA typical mercury intrusion porosimetry test involvesplacing a sample into a container, evacuating thecontainer to remove contaminant gases and vapors(usually water) and, while still evacuated, allowingmercury to fill the container. This creates an environ-ment consisting of a solid, a non-wetting liquid(mercury), and mercury vapor. Next, pressure isincreased toward ambient while the volume ofmercury entering larger openings in the sample bulkis monitored. When pressure has returned to ambient,pores of diameters down to about 12 mm have beenfilled. The sample container is then placed in apressure vessel for the remainder of the test. Amaximum pressure of about 60,000 psia (414 MPa)is typical for commercial instruments and this pres-sure will force mercury into pores down to about0.003 micrometers in diameter. The volume ofmercury that intrudes into the sample due to anincrease in pressure from Pi to Pi+1 is equal to thevolume of the pores in the associated size range ri tori+1, sizes being determined by substituting pressurevalues into Washburn’s equation, Eq. 5.The measurement of the volume of mercury movinginto the sample may be accomplished in variousways. A common method that provides high sensitiv-ity is to attach a capillary tube to the sample cup andallow the capillary tube to be the reservoir for mer-cury during the experiment. Only a small volume ofmercury is required to produce a long ‘string’ ofmercury in a small capillary. When external pressurechanges, the variation in the length of the mercurycolumn in the capillary indicates the volume passinginto or out of the sample cup. For example, a capil-lary of 1 mm radius requires only 0.03 cm3 of mer-cury to produce a mercury column 1 mm in length.Therefore, volume resolution of 0.003 cm3 easilycould be obtained visually from a scale etched on thecapillary stem. However, electronic means of detect-ing the rise and fall of mercury within the capillaryare much more sensitive, providing even greatervolume sensitivity down to less than a microliter.The measurement of a series of applied pressures andthe cumulative volumes of mercury intruded at eachpressure comprises the raw data set. A plot of thesedata is called the intrusion curve. When pressure isreduced, mercury leaves the pores, or extrudes. Thisprocess also is monitored and plotted and is theextrusion curve. According to the shape of the poresand other physical phenomena, the extrusion curveusually does not follow the same plotted path as the

Figure 4. Capillary action of a wetting and non-wetting liquidrelative to the walls of a capillary. The g indicates the directionof the interfacial tension (force) vector.

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intrusion curve. Therefore, the intrusion curve andextrusion curve contain different information aboutthe pore network.When to collect the data point is an important consid-eration when measuring intrusion and extrusioncharacteristics. Since the intrusion process involvesmoving a mass of mercury into a confined porespace, the process is not instantaneous as exemplifiedby the Hagen-Poiseuille law

Q = V/t = (πr4/8η)(∆P/l) (6)

where Q = flow of the liquid, V the volume of liquid, ttime, r the capillary radius, η the liquid viscosity and∆P/l the pressure drop per unit length of capillary.

However, long and tortuous pore channels resultin smaller Q values, therefore requiring more time tofill the same volume as would be the case for poresystems having higher Q values. To obtain highlyresolved and highly accurate data, the intrusionprocess must be allowed to equilibrate before chang-ing pressure and probing the next smaller-sized poreclass. Expressed another way, high-resolution datacollection, particularly in the small pore size range,requires a pressure step, that is, pressure is raised tothe next pressure, then held until flow ceases. Scan-ning mode, in which pressure is continually changed,is best employed for very large pores or for screeningpurposes.

Measurement TransducersFrom the above discussion, it is clear that a

mercury porosimeter measures only applied pressureand the volume of mercury intruded into or extrudedfrom the sample bulk. Pressure measurements areobtained by pressure transducers that produce anelectrical signal (current or voltage) that is propor-tional to the amplitude of the pressure applied to thesensor. This analog electrical signal is converted intodigital code for processing by the monitoring com-puter.

The transducer that detects mercury volume isintegrated into the sample holder assembly as previ-ously exemplified and shown in Figure 5.

Figure 5. Cross-section of a penetrometer in which pressure hasforced some mercury into the pores of the sample and about50% of the stem capacity has been used.

The sample cup has a capillary stem attached andthis capillary serves both as the mercury reservoirduring analysis and as an element of the mercuryvolume transducer. Prior to the beginning of eachanalysis, the sample cup and capillary are filled withmercury. After filling, the main source of mercury isremoved leaving only the mercury in the sample cupand capillary stem, the combination being referred toas the penetrometer. Pressure is applied to the mer-cury in the capillary either by a gas (air) or a liquid(oil). The pressure is transmitted from the far end ofthe capillary to the mercury surrounding the sample inthe sample cup.

The capillary stem is constructed of glass (anelectrical insulator), is filled with mercury (an electri-cal conductor), and the outer surface of the capillarystem is plated with metal (an electrical conductor).The combination of two concentric electrical conduc-tors separated by an insulator produces a co-axialcapacitor. The value of the capacitance is a functionof the areas of the conductors, the dielectric constantof the insulator, and other physical parameters. In thecase of this particular capacitor, the only variable isthe area of the interior conductor as mercury leavesthe capillary and enters the sample voids and pores,or as it moves back into the capillary when pressure is

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reduced. This is mechanically analogous to amercury thermometer in which case mercury movesin and out of a calibrated capillary from a large bulbat one end. A small volume of mercury entering orleaving a small capillary causes the length (and area)of the mercury column to change significantly, thusproviding volume-measuring sensitivity and resolu-tion. In the case of the thermometer, the change involume is proportional to the change in temperatureby the coefficient of volumetric expansion of mer-cury.

The capacitance value of the stem is monitored bya capacitance detector that, similar to the pressuretransducer electronics, produces an electrical signalthat is proportional to capacitance. Capacitancemeasurements are transformed into volume measure-ments by knowledge of the diameter of the precisioncapillary and the equation governing coaxial capaci-tors.

SECTION II.

OBTAINING INFORMATION ABOUT THESAMPLE MATERIAL

PART A. Information Obtained Using Volumeand Mass Measurements Only

Part A discusses characteristics of the sample thatcan be deduced directly from the intrusion volumescombined with physical properties of the sample andsystem. Detailed Pressure data are not required, nor isWashburn’s equation.

Total Pore VolumeTotal pore volume is the most direct determina-

tion of a physical property by mercury intrusion,involving only the volume of mercury entering thesample bulk and not requiring Washburn’s equationor a pore model. At the lowest filling pressure,intrusion is considered nil and no pore volume ofinterest has been filled. Pressure is increased tomaximum; at this pressure mercury has been forcedinto all voids of the sample accessible to the mercuryat maximum pressure. The volume of mercuryrequired to fill all accessible pores is considered thetotal pore volume. Dividing this value by the mass ofthe sample gives total specific pore volume in units ofvolume per unit mass.

Material Volume and DensityThe concepts of volume and density seem simple atfirst consideration. However, they become complexwhen an attempt is made to rigorously define eachterm. The Glossary section of this document con-

tains various definitions. For a more completediscussion of density and volume, refer toMicromeritics’ publication, ”Volume and Density forParticle Technologists.”(6)

Bulk and Envelope Volume and Density: Bulkvolume (as applied to a collection of pieces) is thesum of the volumes of the solids in each piece, thevoids within the pieces, and the voids among thepieces. Envelope volume (as applied to a singlepiece) is the volume of a particle or monolith aswould be obtained by tightly shrinking a film aroundit. Therefore, it is the sum of the volumes of the solidcomponents, the open and closed pores within eachpiece, and the voids between the surface features ofthe material and the close-fitting imaginary film thatsurrounds the piece. Bulk and skeletal densitiesfollow from dividing the respective material mass byvolume.

Mercury porosimetry tests in general incorporatethe majority of steps in Archimedes’ displacementmethod for volume determination. Including theremaining steps requires additional weights otherwisenot necessary for pore characterization. Examples ofdensity determinations are given below. Powderedand solid (monolithic) forms of sample materials areconsidered separately because of a slight but impor-tant difference between volumes determined for asolid object compared to that of a finely dividedpowder.

Assume, then, that two samples are to be ana-lyzed, one a solid piece of irregularly shaped mate-rial, and the other a quantity of powder, both ofknown mass. Prior to the tests, empty sample con-tainers are weighed, filled with mercury, and thenweighed again. From these measurements and thedensity of mercury, the exact volume of each con-tainer is calculated. After the samples have beenloaded into the containers and the containers refilledwith mercury, the mercury surrounds the samples.Being a non-wetting liquid with only atmosphericpressure applied, it does not enter small indentions,cracks, and crevices on the surface nor into poreswithin the structure of the material. In the case offine powders, mercury does not invade the interpar-ticle voids.

The weight of the surrounding mercury in eachcase is calculated from values obtained by reweighingthe filled sample container and subtracting the weightof the empty sample container and sample. Mercuryvolume follows from density and weight. The differ-

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ence in volumes of mercury in the sample containerbefore and after introducing the sample is equal to thebulk volume (in the case of the solid piece), orenvelope volume (in the case of the powder).Whether the term ‘bulk’ or ‘envelope’ applies de-pends on the sample form (powder, single piece,granules, etc.) as well as the applications-specificdefinitions of these terms.

Skeletal and True Volume and Density: A poremay have access to the surface (open pores), or maybe isolated from the surface (closed or blind pores).Skeletal volume, as applied to discrete pieces of solidmaterial, is the sum of the volumes of the solidmaterial in the pieces and the volume of closed poreswithin the pieces. True volume is the volume only ofthe solid material, excluding the volume of openpores and closed pores. Skeletal and true densitiesfollow from dividing the respective material mass byvolume.

If a sample contains both open and closed pores,at maximum applied pressure, only open pores in thesample are filled with mercury. The volume ofmercury intruded into the pores subtracted from thebulk or envelope volume of the sample gives itsskeletal volume. The volume measured is the truevolume if the sample contains no blind pores and allpore space is filled. Finely grinding materials withclosed pores (when appropriate) may allow truevolume and density also to be determined. If so, thenthe volume measurements obtained prior to and aftergrinding provide a means for obtaining the totalvolume of closed pores by subtracting true volumefrom skeletal volume.

If the sample contains pores smaller than theminimum pore size into which mercury can intrude atmaximum instrument pressure, then skeletal and truevolumes cannot be obtained accurately. This mayrepresent a small, perhaps insignificant, volumepercent when using an instrument capable of generat-ing 60 kpsia (414 MPa). For these samples, volumesdetermined by gas pycnometry are smaller than thoseobtained by mercury porosimetry because gases suchas helium and nitrogen can penetrate into microporeswhere mercury cannot. The difference in skeletaldensity obtained by mercury porosimetry and by gaspycnometry serves as a good approximation of porevolume in the range from essentially the size of thegas molecule to the lower size represented by thehighest pressure obtained by mercury porosimetry.

When measuring volume (density) by mercury

porosimetry, it should be recognized that the valueobtained is pressure-dependent up to the pressure atwhich all particle voids and pores are filled. Thepressure required for total pore filling may be onlyseveral thousand psi, but may require the full pressurerange of the instrument. At higher pressures, one mustbe aware of material compressibility, which willreduce the reported skeletal volume. More informa-tion about compressibility is presented in a subse-quent section.

Interstitial Void VolumeInterstitial void volume, sometimes called inter-

particle void, is the space between packed particles.Such voids were taken into consideration in thedefinition of envelope volume, above. These voidstypically are larger than voids in the individualparticles and therefore fill at lower pressure. Beinglarger, they also hold more mercury than particlepores. This means that the rate of intrusion ofmercury with increasing pressure is greater whenfilling the interstitial void than when filling poreswithin the sample material. The completion ofinterparticle void volume filling is indicated by anabrupt change in filling rate observed on the intrusioncurve. The total volume of the interparticle voids isthe volume of mercury intruded at the inflectionpoint.

Percent Porosity and Percent Porosity FilledKnowing the bulk or envelope volume (VB or VE)

of a sample and total porosity (VPt) allows percentporosity to be calculated by the simple relationship

P% = (VPt/V*) x 100%. (7)

The type of volume used in the equation for V*

determines what volumes are considered ‘porosity.’Volumes associated with interstitial voids and withporosity can be differentiated as discussed previously.

In some applications, it is desired to know whatpercent of total porosity has been filled (or remainsunfilled, or has emptied) at a certain pressure or poresize boundary. This information also is readilyavailable from mercury porosimetry data since, oncetotal intrusion volume (total porosity) is known, thecumulative intrusion scale can be represented in unitsof percent porosity.

PART B. INFORMATION OBTAINED BY APPLICATION

OF WASHBURN’S EQUATION

The characteristics of a sample discussed in PartA, with the exception of percent porosity filled at a

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pore size boundary, are not dependent on any geo-metrical pore model. When the sizes of pores are tobe determined and associated with volume determina-tions, Washburn’s equation and a pore model arerequired. Characteristics that require models arediscussed in Part B.

Pore Volume Distribution by Pore SizeRather than pumping the system immediately tomaximum pressure as in the example of obtainingtotal pore volume, pore size distribution analysesachieve maximum pressure by a series of smallpressure steps or by controlled-rate scanning. In stepmode, pressure and volume are measured afterintrusion (or extrusion) equilibration is achieved. Thecumulative intrusion volume of mercury at eachmeasured pressure is determined by subtracting thevolume of mercury remaining in the stem from theoriginal volume.

Applying Washburn’s equation to each measuredpressure provides the pore size associated with eachpressure, that is, the pore size class interval bound-aries. In the great majority of cases and in the form ofWashburn’s equation derived and presented as Eq. 5,the pore is considered to be a right circular cylinder.

The volume of mercury intruded as the result of eachpressure step is the difference between the respectivecumulative intrusion volumes. This value and theassociated pressure (pore size) values yield a table ofpore size intervals and incremental volumes associ-ated with each interval. This is sufficient data for abar graph of pore volume versus size class. For acontinuous curve, only a single size value is requiredto represent the size class. This value may be theupper or lower size boundary or some representativesize between the two boundaries (the average size, forexample).

Pore Area and Number of PoresSince the Washburn model is based on cylindricalcapillaries, the pore shape is assumed to be cylindri-cal with a circular opening. Therefore, the equivalentcylindrical pore size is obtained from the data. Inprinciple, any pore opening geometry (cavity cross-section) is applicable as long as the equation is knownfor equilibrium between the external pressure appliedover the spanned area of the opening and for theresistive force produced by interfacial tension aroundthe perimeter of the opening (the solid, liquid, vaporinterface).

A cylindrical pore model is used almost exclu-

sively in practice. From this model, pore wall area iseasily determined from incremental pore volume VIii

by the equation

AWi = Di/4VIi (8)

where Di is the representative diameter for the size class(the average class diameter, for example). A certain in-cremental pore volume and diameter implies a porelength L since the relation between the length, diam-eter, and volume of a cylinder is

L = 4V/πD2 (9)

If a representative diameter is combined with arepresentative pore length (thickness of the sample,for example), then the number of pores in the sizerange can be calculated because each pore wouldhave a specific volume, VPi, and the total volume ofall pores in the class is known. The number of poresof a specific size, then, is

N =VIi/VPi (10)

Rootare and Prenzlow (7) take a modelessapproach to obtaining surface area information frommercury porosimetry curves. Their calculations beginwith an expression of the reversible work required toimmerse in mercury a unit area of a solid surface.That expression is simply the difference in the surfacetension of the solid-vacuum interface (γ s-v) and theinterfacial tension of the solid-mercury interface (γ s-l),which reduces to γ L cos θ, where γL is the surface freeenergy of liquid mercury in vacuo and θ is the contactangle between the mercury and solid surface. Differ-entiation over area (a) gives

dW = γL cos θ da (11)

In a porosimeter, the work is supplied by pressure Pforcing a volume V of mercury into a pore, thereforework is PdV. Substitution into Eq. 11 and subsequentintegration yields

a = -∫ PdV ( γL cos θ )−1 (12)

PART C. INFORMATION OBTAINED BY APPLICATION OF

SPECIAL OR MULTIPLE MODELS

Mercury porosimetry has been employed for decadesto characterize sample materials in regard to thephysical parameters described in Parts A and B,above. More recently, theoretical models of mercuryintrusion and extrusion mechanisms have beenintroduced allowing additional information about thesample material to be extracted. These methods ofdata reduction are the subject of this part.

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Particle Size DistributionIn 1965, Mayer and Stowe (8) published a paper

expanding the work of Frevel and Kressley (9) on themercury breakthrough pressure required to penetrate abed of packed spheres and the subsequent filling ofthe interstitial void. This work relates particle size tobreakthrough pressure. Later work by Pospech andSchneider (10) led to a method for determining thesize distribution of particles from the intrusion data inthe range of interstitial filling.

This method is based on models of penetration offluids into the void spaces of a collection of uniformsolid spheres packed in a regular manner. The forcesresisting penetration of mercury between particlesoriginates from interfacial tensions just as withpenetration of mercury into capillaries. However, thegeometric model is considerably different andtherefore is not described by the Washburn equationof equilibrium. The simplest geometry exists whenthe particles are monosized spheres and the shapes ofthe void necks and void cavities of such a system areexemplified in Figure 6. The figure shows the rangeof angular relationships between spheres centered atW, X, Y, and Z in a plane section.

Figure 6. Ordered packing of spheres with cross-sectional view of the shapes of mercury filling theresulting voids. Also shown is the 3-dimensionalshape of the void in a rhombohedral packing.

Regardless of the actual particle shape, theparticle size distribution derived from this method isthe size distribution of spheres that, when applied tothe mathematical model, most closely reproduces theexperimental penetration data. The size unit, then, is‘equivalent spherical size.’ How closely the resultscompare to that obtained by other methods of particlesizing depends largely on how closely the samplematerial conforms to the model of closely packedspheres.

Pore Cavity to Pore Throat Size RatioFor many materials, porosity is composed of a

network of interconnected voids of various sizes.Small pores at the surface may connect to large poreswithin the material; the openings are referred to aspore throats and the spaces within the material as porecavities. There are numerous methods for extractingpore shape information from mercury injection data.These often are based on specific pore models andmay require some knowledge of the pore structure inorder to select the appropriate method. A few of thesemethods are discussed here.

Most pore shape evaluation methods are based onthe hypothesis that hysteresis in mercury porosimetryis attributable to pore shape. Indeed, if each porewere a simple, uniform cylinder and the intrusion andextrusion contact angles are known and applied, onewould expect there to be no hysteresis since theintrusion and extrusion processes both are controlledby the same mechanism and known parameters.However, hysteresis loops of various shapes often areobserved and many do not close when pressure isreduced from some elevated value to ambient. Thisis attributed to large cavities being interconnected bysmaller pore throats. This applies even if the shape ofthe throats and cavities are cylindrical. Although largepores fill at low pressures, a large cavity connected tothe surface by a small throat cannot fill until thepressure is sufficient to fill the smaller connectingthroat. Upon decompression, the small throat emp-ties at the same pressure at which it filled, but thelarge cavity behind the throat remains filled becausethe internal forces in this volume of relatively largeradius is insufficient to overcome the external forcesat the current pressure (see Section I). The pore willempty at the lower pressure associated with its radius,or it may not empty at all if the path to the surface iscomposed of pores of inappropriate sizes.

So, intrusion of mercury into a cavity is con-trolled by the size of the pore throat radius while theradius of the cavity and its connectivity controlsextrusion of mercury from the cavity. A way tocharacterize the relationship between pore throat andcavity existing in a material is by the ratio of thesesizes and is calculated in the following manner.

Assume that at some pressure Px relating to poresize Rx that Vx cm3 of mercury is contained by poresin the sample. Intrusion continues to pressure Pz atwhich all pores are filled (Vz). Then, pressure isreduced and extrusion begins. If there is hysteresis

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(as is typical), pressure must fall to a value lower thanPx, say Pw, before sufficient mercury extrudes toreduce the volume of mercury within the sampleagain to Vx. The pore size that relates to Pw is Rw,and this is the cavity size that is paired with throatsize Rx,; the pore throat to pore cavity size ratio, then,is Rx/Rw.

Intrusion volume is expressed in terms of fractionof total porosity filled in order to normalize data forsample-to-sample comparisons. The fraction ofporosity filled is the ratio of the cumulative intrusionvolume at any point to the total intrusion volume.The pore cavity to pore throat ratio method does notrelate the cavity size to the size of the throat throughwhich it was filled; rather it relates the fraction ofporosity filled to the pore sizes that control this stateduring intrusion and extrusion.

The Distribution of Pore Cavity Sizes AssociatedWith a Pore Throat Size

Reverberi and Ferraiolo (11) devised a means bywhich to determine the distribution of cavity sizesconnected to a pore throat of a specific size. Theyconstructed a pore model that relates hysteresis solelyto the existence of inkwell- shaped pores. Aninkwell-shaped pore has a cylindrical pore throat(radius R1) that connects to a larger cylindrical porecavity (radius R2).

The test is performed first by allowing intrusionto occur up to maximum pressure Pz, thereby fillingall throats and cavities from size Rz (at minimumpressure) to Ra (at maximum pressure) with Vz unitsof mercury. The intrusion curve is a record ofcumulative intrusion at each point in the pressureramp. Pressure is then lowered a few percent to somesub-maximum pressure Py that corresponds to capil-lary size Ry. This empties only pores and porecavities with diameters between Rz and Ry. Thedifference in the cumulative intrusion observed at Py

on the intrusion branch and Py on the extrusion branchis the volume of cavities with sizes greater than Ry.Pressure is increased again to Pz. The intrusion curvewill not take the same path as the original intrusioncurve if mercury was trapped since these poresalready are filled. Pressure is reduced from Pz to apressure Px where Px<Py<Pz. The total intrudedvolume remaining in the pore space is recorded. Porethroats and cavities in the size range Rx to Rz empty.The pressure is raised again to maximum. Theintrusion path retraces neither path Pa to Pz nor path Py

to Pz because a different set of pores is being filled.

The pressure increased again to maximum,reduced to Pw, raised to maximum, reduced to Pv, andso on until the range of interest is sampled. The datathus obtained are reduced to yield a three-dimensionaldata set providing the distribution of pore cavityvolumes versus cavity sizes behind each of a series ofpore throat sizes.

Material Permeability and the ConductivityFormation Factor

Permeability is the inherent ability of a porousmedium to transmit a fluid and is a property of thematerial. It relates to porosity in the sense that it is the“proportionality constant” linking fluid flow rate toapplied pressure across a porous medium. Therehave been many attempts over the years to relatepermeability to some relevant microstructurally-defined length scale. (12) The work of Katz andThompson (13, 14) provides an important contribu-tion to mass transport studies in facilitating theprediction of fluid permeability of materials frommercury injection data.

Katz and Thompson introduced two expressionsfor calculating absolute permeability (k) using datafrom a single mercury injection capillary pressurecurve. The equations were derived from percolationtheory. The first provided rigorous results by incorpo-rating the measured conductivity formation factor(conductance ratio) σ/σo as a parameter. Here σ is therock conductivity at characteristic length Lc, and σo

the conductance of brine in the pore space. Mercuryporosimetry data are used to determine the character-istic length Lc of the pore space. The equation forpermeability is

k = 1/226(Lc)2 σ/σo (13)

The second equation provides for an estimation ofσ/σo to be obtained from the same mercury intrusiondata as were used to determine k. This equation, inaddition to Lc

, requires the value of the length (poresize) at which hydraulic conductance is maximum(Lmax), and the fraction of total porosity φ filled at Lmax

(symbolized by S(Lmax)). The equation is

k = (1/89)(Lmax)2(Lmax/Lc)φS(Lmax) (14)

Although appearing similar, the Katz-Thompsonexpression for conductivity formation factor isfundamentally different from the classical Archie’slaw. For Equations 13 and 14, k is reported in unitsof millidarcys and has fundamental units of area.

As pressure is increased, mercury is forced to

13

invade smaller and smaller pore openings in thepermeable material. Ultimately, a critical pressure isreached at which the mercury spans the sample. Thisconduction path is composed of pores of diameterequal to and larger than the diameter calculated fromthe Washburn equation for the critical pressure. Thisdiameter, Lc in equation 13, is a unique transportlength scale and dominates the magnitude of thepermeability.

To obtain this characteristic length Lc from themercury intrusion data set, pressure is determined atthe point of inflection in the rapidly rising range ofthe cumulative intrusion curve. This inflection pointwas determined experimentally by Katz and Thomp-son to correspond closely to the pressure at whichmercury first spans the sample and the point at whichpercolation begins. This pressure point is defined asthe threshold pressure (Pt). The value of Lc is the porediameter (or length scale) calculated from theWashburn equation for Pt. Having obtained Lc issufficient if σ/σo has been measured independentlyand is known.

To use the second form of the equation, whichestimates σ/σo, the cumulative intrusion volume Vt atthreshold pressure (Pt) is determined. Then, thequantity Vt is subtracted from each intrusion volumevalue at each pressure in the data set from thresholdpressure to maximum pressure. Data points prior tothe threshold pressure are excluded. The net volume(Vc - Vt in cm3) times the diameter-cubed(micrometer3) for the corresponding pressure iscalculated as a function of pore diameter(micrometers). This is the hydraulic conductancefunction, or permeability path. The pore diametercorresponding to the maximum y-value is Lmax, thecumulative volume of mercury intruded at thisdiameter is VLmax. The fraction S(Lmax) is calculated asthe ratio of VLmax/Vt and is the fractional volume ofconnected pore space composed of pore width of sizeLmax and larger.

For both data reduction methods (using enteredvalues for σ/σo or calculating an estimated σ/σo), theequations depend on a judicious choice of the point atwhich inflection occurs and the resulting values ofthreshold pressure (Pt) and threshold volume (Vt).The computer program performs the first approxima-tion and the initial value for permeability is reportedon this basis. However, the intrusion curve should beinspected and the choice of the appropriate inflectionpoint confirmed. There are cases in which there are

multiple inflection points and a judgment isneeded to determine which most likelyrepresents the onset of percolation, or if themethod is even applicable to the sample.Therefore, permeability calculations usuallyrequire a second pass through the datareduction routine using adjusted param-eters. It should be noted also that a changein permeability value would affect the re-ported value for tortuosity.

In the literature, one frequently finds referencesto the K-T method, both in regard to determiningpermeability by mercury injection, and to theirmethod of identifying the percolation region, theoptimum path for permeability, and the thresholdpressure from mercury intrusion data. These param-eters are in other methods discussed in this document.

Pore Fractal Dimensions

Many natural and man-made materials havecomplex pore structure that changes only in size, butnot in geometrical shape, over a range of porevolumes. This type of geometry has the quality of‘self-similarity’ and can be described in terms of itsfractal dimensions.

Angulo, Alvarado, and Gonzalez (15) published amethod by which mercury porosimetry data can bereduced to extract information about the fractalcharacteristics of the pore space of the samplematerial. For their method to be applicable, theremust be one or more linear regions on a log-log plotof intrusion volume versus pressure. The plotdescribes the volume scaling of the pore space andlinearity implies that pore volume has fractal dimen-sions. The equation describing a linear region isdetermined and the value of the fractal dimension ofthe material in the linear range is extracted from theinverse log of the equation. Two regions of linearityare found in the data from some materials; this isattributed to there being two different processesrelated to pore space. The lower pressure linearregion occurs during ‘backbone formation’ whenmercury is finding a conductivity path through thepore structure. The adjacent linear region at higherpressure occurs during ‘percolation’ where flowthrough the medium is optimized.

For fractal analysis to be of value, the implica-tions of fractal geometry must be appreciated. Foranyone unfamiliar with fractal geometry, a simple,instructive exercise can be performed. First, draw a

14

square of dimension L, where L is any dimension.The square will be the conserved geometry, so everysub-element also will be a square, only smaller.Divide the original square into four squares of equalsize (L/2 x L/2). Then, remove any one of the smallsquares. Divide each of the remaining three squaresinto four smaller squares (L/4 x L/4). From each ofthe three new groups, remove one square. Continuethis process n times, where n is a positive integer.The result is a set of very small squares forminglarger squares, all but the smallest having voids ofvarious sizes. The dimension of the smallest squarefrom which all other squares are composed, is calledthe length scale. The dimension of the smallestsquare is calculated using the formula L/2n, where n isthe number of times the original square was divided.The numerator is the original length of a side of thesquare, and the number 2 in the denominator is thenumber of equal parts into which the original lengthwas divided. The number N of remaining squares atthe end of the exercise equals 3n, where the number 3is derived from the number of remaining squares aftereach divide-and-discard step.

In general, a self-similar set of this type is ex-pressed by the equation

N(δ) = δ-D (15)

where N(δ) is the number of elements of dimensionδ, δ is the smallest dimension after n divisions, and Dis the fractal dimension.

Taking the logarithm of this equation provides asolution for D, which is

D = log[N(δ)]/log(1/δ) (16)

For the example given, N(δ) = 3n and δ = 1/2n.Substituting these values into equation 10 gives

D = log(3n)/log(2n)

= nlog(3)/nlog(2)

= log(3)/log(2) = 1.58 (17)

Note that the fractal dimension does not dependupon the number of iterations, n. Ultimately, itdepends upon the fraction by which the length scale isdivided and the number of remaining elements aftereach iteration. The total number of elements and thelength scale, however, are functions of n.

Following the example above, the total area of theremaining object is equal to the number of minorelements times the area of each, or

A = N(δ) δ2 (18)

Since N(δ) = δ-D,

A = δ(2-D) (19)

where D in this case is the area fractional dimension.

An expression for volumetric fractal dimension isderived as above, yielding

V = δ(3-D) (20)

If a sample has pore space that is fractal, then thevolume of mercury intruded with increasing pressurealso will scale as a fractal. Using a pressure scalerather than a length scale, and subtracting the thresh-old pressure Pt from the capillary pressure P, Equa-tion 14 is expressed as

V = (P-Pt)(3-D) (21)

Taking the logarithm of this equation gives

log(V) = (3-D)log(P-Pt) (22)

where (3-D) is the slope of the log(V) vs. log(P-Pt)plot. Pt is the pressure identified by Katz and Th-ompson (see the section on permeability) as thepressure at which mercury first percolates and spansthe porous medium.

The equations developed above are based onregular geometry, as are most models, the cylindricalpore model of the Washburn equation, being anexample. The application of this data reductionmethod indicates, by the occurrence of a linear rangeon the log(V) vs. log(P-Pt) plot, that pore volumefilling is indicative of pores of fractal dimension.Reduction of data in that area provides the fractaldimension.

Fractal dimensions often are compared with othereffects of porosity to test for correlation. For ex-ample, materials with fractal geometry often havequite different fluid transport characteristics than domaterials having random geometry. Therefore, fractaldimensions can be important physical parameterswhen studying reservoir rocks and catalysts and othermaterials through which fluid must flow. The factthat the porosity has or does not have fractal geom-etry also may be indicative of the physical processesthat formed the material.

In practice, determinations of the fractal dimen-sions of the backbone formation and percolationregions are accomplished after the log(V) vs. log(P-Pt) plot has been examined and the boundaries of the

15

linear regions identified. Another variable that mayrequire adjustment after the analysis is Pt, the thresh-old pressure. Since these determinations only can bemade after seeing certain reduced data, parameteradjustments and a second pass through the datareduction routine is required, this time only producingthe reports on which changes are expected.

Pore Tortuosity and Tortuosity Factor

The terms tortuosity and tortuosity factor expresstwo different characteristics of a material. Tortuosityis the ratio of actual distance traveled between twopoints to the minimum distance between the sametwo points. Tortuosity factor is commonly used in thearea of heterogeneous catalysis and is the ratio oftortuosity to constriction.

The tortuosity factor characterizes the efficiencyof diffusion of fluids through a porous media.Diffusion-controlled processes are of particularimportance in catalysts where the solid supportusually contains a pore network with pores rangingfrom micro- through to macropores. The manner inwhich pores interconnect can have a profound effecton the accessibility of reactants to the active sites andon the removal of products.

Also, characterization of the pore structure,including tortuosity, of porous filtration membranessuch as microfiltration and ultrafiltration membranesis an important determination. Modeling the transportof contaminants through soil or other porous matricesalso depends on knowledge of the materials tortuos-ity.

In 1998, Jörgen Hager (16) in his Ph. D. thesis atLund University (Sweden) derived an expression formaterial permeability based on a capillary bundlemodel in which pores are homogeneously distributedin random directions. Using the Hagen-Poiseuillecorrelation for fluid flow in cylindrical geometries,making substitutions with measurable parameters, andcombining with Darcy’s law, he derived an expres-sion for material permeability in terms of total porevolume, material density, pore volume distribution bypore size, and material tortuosity. All parameters buttortuosity are obtainable from mercury porosimetrytests.

As presented previously, Katz and Thompson alsoderived an expression for material permeability basedon measurements obtainable from mercuryporosimetry. Their expression does not depend onknowledge of material tortuosity. Therefore, combin-

ing the Hager and Katz-Thompson expressionsprovides a means for determining tortuosity fromdata collected by mercury porosimetry.

Obviously, if the Katz-Thompson value forpermeability is used for tortuosity, permeability mustfirst be determined appropriately. This means that theK-T point of inflection must have been accuratelyidentified. Furthermore, for minimum error, thevalue of σ/σ

o should be known and entered as a

parameter in the K-T data reduction routine. How-ever, the K-T method can be used to estimate σ/σ

o, but

estimation diminishes the reliability of the calculatedtortuosity value.

The dependency of tortuosity determinations onpermeability determinations means also that anyrecalculation that affects the reported permeabilityalso affects the reported value for tortuosity. There-fore, if permeability is recalculated for any reason (abetter estimate of the inflection point, for example),then the system should be allowed also to recalculateand report tortuosity using the updated value ofpermeability.

Material CompressibilityIt has long been noted that for many materials the

intrusion curve at near maximum pressure takes asudden upward swing. In some cases, the apparentuptake of mercury by the sample actually is caused bymercury filling the void in the sample cup producedby the collapse or compression of the sample mate-rial. If the extrusion curve follows the intrusion curvein this region, the material is demonstrating restitutionor elasticity and returning to its original shape orvolume. If the extrusion curve fails to retrace theintrusion curve, the material, to some extent, ispermanently deformed. A pore filled with mercuryapplies pressure to the pore walls essentially with thesame pressure as applied by the bulk mercury sur-rounding the sample. Therefore, structural collapse isnot likely caused by collapse of the open porestructure, but more likely is due to voids that areinaccessible to the mercury. However, there is noway mercury porosimetry can determine whether theupward swing in the intrusion curve was caused bymaterial compression, void collapse, filling of openpores, or a more common combination of these.

Compressibility data are more reliable whenworking with non-porous materials and, in thisapplication, any apparent uptake of mercury in thehigh or low pressure regions may be attributed tomaterial compression. However, before data reduc-

16

tion can be performed, there must be available a“blank run” file consisting (at least ideally) of a runmade with the same penetrometer that is to be used inthe compressibility test and on the same instrumentports as will be used in the compressibility run. Thepressure range of the blank run must, at a minimum,fully encompass the planned range to be used in thecompressibility measurement. This greatly improvesthe accuracy of the data by eliminating any deviationsin the baseline caused by components in the system.

Quantifying compressibility consists of firstidentifying the range of data in which compressibilityoccurs, or in which the compressibility function is tobe determined. The default range is the entireintrusion curve and this may need to be changed byhaving pre-knowledge of the desired range, or byinspection of the intrusion curve after the analysis.By the latter method, a second pass through the datareduction routine is required if the calculation range ischanged from the pre-analysis values.

The compression function is expressed as aquadratic equation derived from the change involume with pressure. Assume that at each experi-mental pressure, Pn, the corresponding blank cor-rected intrusion, V(Pn), can be computed using thesecond order polynomial expression

V(Pn) = Vo + B*Pn + C*Pn2 . (23)

where Vo is the exact volume of the sample materialcomputed as the ratio of the sample weight andthe sample density supplied by the user or,alternatively, supplied as the pre-measured samplevolume by the user; B is the linear pressurecoefficient of volumetric compressibility; and C isthe quadratic pressure coefficient of volumetriccompressibility.

The instrument will determine the best fit of thisquadratic equation to the experimental data byadjusting the B and C appropriately. A plot of theexperimental data overlaid with the predicted data canbe inspected for goodness of fit. If the two plotsclosely agree, then the derived function expresses thecompression response of the material to pressure inthe range of calculation.

It is expected that the knowledgeable user willneed to fully take into account various limitations inthe art and technique. A blank correction run willprovide a reference that will compensate for most ofthese variables. Perhaps the least controllable vari-

able is temperature increases which create a “ther-mometer effect” in the penetrometer system and arecaused by compression of the hydraulic fluid aspressure is built. The amount of temperature increasedepends upon the exact rate and time pattern withwhich the pressure is increased but can be as high as50 C if maximally rapid pressurization of a smallpenetrometer is done.

Relation of Mercury Porosimetry to OtherPorosimetry Techniques

Gas sorption and mercury porosimetry arecomplementary techniques. Physical adsorptiontechniques can extend the lower size measurementdown to about 0.00035 mm diameter, thus probingthe intraparticle microstructure. Mercury porosimetryis paired with the gas sorption technique to obtainporosity information in the large size range (greaterthan about 0.3 mm diameter up to about 360 mm),which is not attainable by gas sorption. When usingtwo different techniques (two very different models),one should not expect necessarily to obtain the sameresults in the overlapping or common range of bothinstruments. However, comparable results have beenreported (17) for some materials.

One potential difficulty in comparing porositydata sets obtained in the smaller size range by mer-cury porosimetry with data gathered in the samerange by gas sorption is that sample compression is apossibility with mercury porosimetry. If samplecompression occurs, an apparent uptake of mercury issuperimposed on the intrusion curve leading to anerroneous indication of pore volume, and one notreproduced by the gas adsorption analysis.

As was mentioned in the sections on volume anddensity, mercury porosimetry data can be comparedwith gas pycnometry data to reveal additional poros-ity not detected by the mercury intrusion methodalone. Indeed, comparing data obtained by differenttechniques for the same material characteristics canreveal information not attainable by either methodalone. The requirement, of course, is that the mea-surement theories of both techniques be well under-stood.

17

GLOSSARY

Angle of contact: see Contact Angle

Apparent quantity: find under name of specific quan-tity, i.e. Volume, Apparent

Archie’s Law: An empirical equation relating the elec-trical conductivity of a porous material and expressedas

ρR = ρ

FAφ-m

where ρR is the electrical property of the rock, ρ

F the

electrical property of the fluid, φ the porosity of themedium, and A and m depend on the geometry of thepores.

Area, Incremental Specific: The total area of pores withthe size class defined by the increment boundaries. Thisvalue is calculated using incremental volume rather thancumulative volume. Therefore, the incremental porearea between the boundaries i and j is calculated by

Aij = 4(Vj-Vi)/Dm,

where Dm is the mean diameter (see definition).

Area, Cumulative Pore: The summation of pore areaover a range of pore sizes. Pore area is calculated fromthe geometry of a right circular cylinder beginning withCumulative Volume (as measured) and equals πD2h/4,where D is the diameter of the pore and h is its depth.Since the area of a right circular cylinder equals π Dh,the relation between the volume and area is A= 4V/D.

Backbone: That part of the spanning cluster of con-nected pores and voids that takes the most direct paththrough the medium.

Backbone Formation: Development of the main paththrough which the fluid will percolate.

Bulk quantity: find under name of specific quantity, i.e.Volume, Bulk

Capillarity: The action that causes the elevation or de-pression of a liquid surface in contact with a solid. It iscaused by the relative attraction of the liquid moleculesto each other and to the molecules of the solid. Sameas capillary action.

Capillary Pressure: The pressure differential across themeniscus; the driving force for capillarypenetration.

Characteristic Length: The pore diameter calculated

from the pressure at which percolation through the po-rous material first occurs.

Compact: Verb form- To increase the bulk density of agranular material by the compression.

Noun form- A tablet or briquette resulting fromcompressing a granular or powdered material.

Conductivity Formation Factor: The ratio of the elec-trical conductance of a rock permeated with brine tothe electrical conductance of brine. The reduction inconductivity is caused by the presence of the insulatingsolid phase and therefore is related the porosity. It alsois affected by pore tortuosity and interconnectivity.

Connectivity: The degree to which pores, fractures, andvoids are joined to form continuous paths through amedium. Directly related to Percolation.

Contact angle: The angle between the liquid and thesolid surface at the liquid-solid-vapor interface and tan-gent to the curve of the droplet. The contact angle of aliquid on a smooth, homogeneous surface; depends onthe surface energy of solid and liquid. The higher thesurface energy of the solid substrate, the better itswettability and the smaller the contact angle. Relatedto the Surface Tension by the Young’s Law.

Cumulative quantity: Total quantity accumulated (sum-mation) over a range of operation (compare with Incre-mental quantity). For specific definitions, look underthe name of the specific quantity, i.e. Volume, Cumula-tive.

Darcy: A unit of permeability. Equal to the flow of 1ml of fluid of 1 centipoise viscosity in 1 second under apressure gradient of 1 atmosphere across a 1cm2 and lcm long section of porous material. Has units of area(cm2). 1 darcy = 1x10 -12 m2 or 1x10 -8 cm2 . 1millidarcy (md) = 1x10 -15 m 2 or 1x10 -11 cm2 .

Darcy’s Law: A law describing the rate of fluid flowthrough a porous medium having specific physical prop-erties. Mathematically,

Q = k∆P/ηl

Where Q is the rate of flow (ml/sec), ∆P the pressuregradient, η the fluid viscosity, and l the length (thick-ness of sample) The material permeability (hydraulicconductivity) k is

k = r2/8

where r is the radius of the pore. Darcy’s Law is validonly for steady-state, laminar fluid flow. See the Hagen-Poiseuille equation.

18

Density, Apparent (or Particle Density) is the massdivided by the volume including both closed pores andopen pores. This is in contrast to the mass per unitvolume of the individual particles, which is a highervalue.

Density, Bulk (or Packing Density) are terms used inpowder technology. Bulk density or packing density isthe mass of particles composing the bed divided by thebulk volume of the bed.

Density, Packing see Density, Bulk

Density The mass of a substance per unit volume (D =m/V). Volume may be defined in different ways ac-cording to how it is measured. Therefore, the defini-tion of density is determined by the definition of vol-ume. Definitions can vary somewhat from industry toindustry.

Density, Bulk: The mass of the bulk quantity dividedby the bulk volume; see Volume, Bulk.

Density, Envelope: The mass of the specimen dividedby the envelope volume; see Volume, Envelope.

Density, Particle (or Envelope Density) is the mass ofthe particle divided by the volume of the particle in-cluding closed pores but excluding open pores.

Density, True (or Substance Density) is the mass di-vided by the solid volume or true skeletal volume. It isusually determined after the substance has been reducedto a particle size so small that it accommodates no in-ternal voids.

Density, Theoretical is similar to true density excepttheoretical density includes the requirement that the solidmaterial has an ideal regular arrangement at the atomiclevel.

Density, Skeletal is usually the mass divided by the skel-etal volume remaining after the volume of all open poreslarger than 0.005 micrometers have been subtracted.

Diameter / Radius: In mercury porosimetry, this is thedimension of the circular (or other) pore model. Forevery collected pressure point, there is a correspondingdiameter obtained from the Washburn equation. Ra-dius, when used, is obtained by dividing the calculateddiameter by two.

Diameter, Mean: The average diameter within a classor data range. Over a given range of diameters (pres-sures) from Di to Dj, the mean diameter is calculated asDi-Dj/2.

Differential quantity: In general, the magnitude of the

change in a quantity, or difference. Generally, the dif-ference between two quantities, but in mercuryporosimetry, the differential quantity always will be poresize (diameter or radius). It is calculated by subtractingthe size of the pores corresponding to one pressureboundary of the increment from that of the other pres-sure boundary. For specific definitions, look under thename of the specific quantity, i.e. Volume, Differential.

Diffusion: As used in this document, the transport ofmass by the spontaneous movement of particles (ions)through the liquid filling the pores.

Distribution (number, area, and volume): In mercuryporosimetry, pore size calculated from pressure is theindependent variable and this represents the distribu-tion range. That which is distributed is the dependentvariable, either the number, area, or volume of pores ateach value throughout the range. For example, the porevolume distribution by pore sizes (diameter or radius).

Equilibration: The state at which mercury has ceasedto flow into the pore space at the current pressure.

Fluid: A substance having no resistance to deforma-tion when subjected to a shearing force.

Formation Factor: See Conductivity Formation Fac-tor:

Fractal: A shape that is composed of smaller replicasof the same shape, that is, having self similarity. Asimple example is a square area composed of smallsquares that are themselves composed of smallersquares, ad infinitum. Characterized by a power lawdistribution.

Geometric quantity: A dimension of an object of regu-lar geometry such as a sphere or cylinder. For specificgeometries (shapes), look under the name of the shape,i.e. “Volume, Geometric”

Greenware: Ceramic ware that has not been fired. Alsoapplied to powdered metal compacts prior to sintering.

Hagen-Poiseuille Equation: An expression describ-ing the laminar flow of fluid through a single cylindri-cal tube (capillary or pore) under the influence of a pres-sure gradient and as a function of fluid viscosity anddensity, capillary length and diameter, and flow veloc-ity. Often expressed in terms of flow quantity, Q as

Q = V/t = (πr4/8η)(∆P/l)

Where V/t is the volume rate of flow, r the radius of thecapillary (pore), η the liquid viscosity, and ∆P/l the pres-sure differential over a length 1 of capillary. See Darcy’s

19

Law.

Hydraulic Conductivity: Permeability in relation to thefluid; unlike permeability, hydraulic conductivity takesinto account the particular fluid that is present in themedium. See Permeability

Hydraulic Radius: The ratio of volume to surface areaof a porous material. Also, the ratio of the cross-sec-tional area of flow to the perimeter of the channel.

Hysteresis: In mercury porosimetry, hysteresis is usedto describe the failure of the extrusion curve to retracethe intrusion curve, that is, at the same pressure on thetwo curves, the quantity of mercury contained in thepore system differs. The extrusion curve with no hys-teresis will exactly retrace the intrusion curve, and withhysteresis, will always have volume values greater thanthe intrusion curve at the same pressure. An error con-dition exists if the extrusion curve dips below the intru-sion curve. The two curves seldom close to form a hys-teresis loop as is required for gas adsorption -desorp-tion curves.

Incremental quantity: A quantity summed between thetwo boundaries that define the increment (compare withCululative quantity). For specific definitions, lookunder the name of the specific quantity, i.e. Volume,Incremental.

Interface: As used in this document, the boundary be-tween any two phases, gas (vapor), liquid, or solid.These include, vapor-liquid, vapor-solid, liquid-liquid,liquid-solid, and solid-solid.

Interfacial Energy: The free energy of the surfaces atthe interface of two phases resulting from differencesin the tendencies of each phase to attract its own mol-ecules. Also known as surface energy. See also, Sur-face Tension and Surface Energy.

Intergranular porosity: Void space between particles.See Interstitial Space.

Internal Energy (Forces): That portion of the total en-ergy of a substance that is due to the kinetic and poten-tial energy of the individual molecules; for example,that possessed by a compressed fluid.

Interstitial Space (Voids): The void space formed be-tween two or more particles packed together.

Katz-Thompson Method: A method by which to deter-mine the permeability of a porous medium using dataobtained by mercury porosimetry.

Lithogenesis: The formation of rocks.

Log differential quantity: The difference betweenthe logarithms of two quantities. In mercury porosimetrydiscussions, log differential quantities refer to quanti-ties calculated using the log of the differential pore size(diameter or radius). For example, log(Di) - log(Dj).

Lognormal distribution: A distribution in which thelog values of sizes are distributed in a normal distribu-tion.

Macropore: Pores with diameters exceeding 0.05 µm.

Meniscus: The free surface of a liquid-vapor boundarynear the walls of the containing vessel (a pore or capil-lary) and which assumes a curvature due to surface ten-sion.

Mesopore: Pores with diameters between 0.002 µmand 0.05 µm.

Micropore: Pores with diameters equal to or less than0.002 µm.

Millidarcy: 10-3 Darcy; see Darcy.

Model (Theoretical): A model, in the sense used in thisdocument, is a mathematical or physical system thatobeys specific conditions and whose behavior is usedto help understand an analogous physical system. Inmercury porosimetry, one theoretical model is that ofsystem cylindrical pores.

Optimum path for Permeability: The pore size at whichthe product of intrusion volume and the cube of thepore size is maximum. The peak of the hydraulic con-ductance function in the Katz-Thompson method.

Pascal: Unit of pressure. To convert Pascals to otherunits, use the table below.

To convert Pascals to… multiply by…

atmosphere 9.869 x 10-6

bar 1 x 10-5

dynes/cm2 10

kg/cm2 1.020 x 10-5

psi or lb/in2 1.4508 x 10-4

torr or mm Hg 7.5028 x 10-3

Percolate (Percolation): The movement of a fluidthrough a porous medium.

Penetrometer: A combination of a sample holder (cup)and the analytical mercury reservoir (stem) used inmercury porosimetry. Also called a dilatometer.

20

Permeability: The capacity of a material to transmitfluid. The conductance of fluid flow that a porous orfractured medium exhibits. Under special conditions,also referred to as Hydraulic Conductivity.

Pore Cavity: Any void laying beneath the surface andconnected to the surface by a smaller void or pore.

Pore, Closed: Strictly, a pore that has no conduit to thesurface. In regard to mercury porosimetry, a pore thathas no conduit to the surface of sufficient size for mer-cury to invade at maximum pressure.

Pore, Ink Well: A pore system composed of a smallcylindrical pore that opens into a larger cylindrical pore.

Pore, Open: A pore that is on the surface or has a con-duit to the surface of the particle or specimen.

Pore Throat (neck): The opening at the surface (some-times called simply ‘pore’). Also, a pore through whicha larger pore (cavity) is accessed.

Porosity: A dimensionless unit symbolized by φ andequal to the ratio of the void volume to the total vol-ume (V

V /V

T ) of the porous medium, or the fraction of

the total volume of a porous medium occupied by voidspace. It also may be considered the storage capacityof a medium such as reservoir rock.

Power Law: A distribution of the form y = c xn.

Pressure: In general, the force pre unit area, F/A. Inmercury porosimetry, this value (measurement) is con-sidered raw data, but actually is reduced to some extentby head pressure correction and transducer offset cor-rection. Nevertheless, it is the most fundamental pres-sure data used in further data reduction. For every col-lected pressure point, Pi, there is a corresponding col-lected volume data point Vi. In certain data reductionroutines, pressure data may be derived by interpolationbetween two measured pressure data points.

Pressure, Threshold: The pressure at which fluid firstpercolates through a porous medium.

PSIA: Units of absolute pressure. Pounds per SquareInch Absolute. To convert PSIA to other units, use thetable below.

To convert PSIA to…. multiply by..

atmosphere 0.06805

bar 0.06893

dynes/cm2 6.8927 x 104

kg/cm2 7.0309 x 10-2

mtorr or micron Hg 5.171 x 104

Pa or N/m2 6.8927 x 103

torr or mm Hg 51.71

Radius / Diameter: In mercury porosimetry, this is thedimension of the circular (or other) pore model. Forevery collected pressure point, there is a correspondingdiameter obtained from the Washburn equation. Ra-dius, when used, is obtained by dividing the calculateddiameter by two.

Saturation Zone (Saturated Zone): The region of porespace completely filled with fluid.

Self-similarity: Having the same properties at differentsize scales.

Sinter: In powder metallurgy, to form a coherent bondedmass by heat and pressure without melting.

Surface Energy: The energy per unit area of surface.Compare with definitions of Surface Tension and In-terfacial Energy.

Surface Tension: The internal force acting on the sur-face that tends to contract the surface into a configura-tion of minimum surface area. It is due to an unbalancein molecular forces at the interface of two materials.The difference in molecular forces between a liquid andsolid determines the contact angle. Also, the work dWnecessary to increase the surface area by dA. Alsoknown as interfacial tension or interfacial force. Seealso, Surface Energy and Interfacial Energy.

Tortuosity: The ratio of the length of the path describedby the pore space to the length of the shortest path acrossa porous mass; the minimum value is 1.

Tortuosity Factor: Tortuosity factor is commonly usedin the area of heterogeneous catalysis and is the ratio oftortuosity to constriction, where constriction is a func-tion of the ratio of cross-sectional areas of the conduit.

Transport Properties: The properties of a material thatare associated with the transport of mass through thematerial.

Unsaturated Zone: The region of the pore space that ispartly filled with fluid.

Viscosity: A measure of a fluid’s ability to resist defor-mation. High viscosity fluids flow more slowly thanlow viscosity fluids.

Volume: The size of a specific region in three-dimen-

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sional space.

Volume, Bulk: Applied to finely divided samples andgranulated solid materials. Includes all pores, open andclosed, and includes interstitial space between particles.In regard to a single solid specimen, often referred to asenvelope volume; see envelope volume.

Volume, Cumulative: In mercury porosimetry, the totalvolume accumulated or accounted for over the range ofpressures or pore sizes. Considered raw data but mayhave been blank-corrected. For every cumulative vol-ume data point (Vi) collected, there is a correspondingpressure point (Pi) collected.

Volume, Differential Specific: The incremental specificvolume divided by the difference between the pore sizeat one boundary of the increment and the pore size atthe other boundary. Incremental volume is used ratherthan cumulative because the calculation involves anincremental step in pore size and volume intruded andnot the cumulative volume intruded at a specific poresize.

Volume, Envelope: The volume enclosed within a ‘formfitting’ surface that covers or envelops a single solidspecimen of material. It includes the volume of smallsurface irregularities, open pores, and closed pores.Compare with Volume, Bulk.

Volume, Geometric is a theoretically ideal volume, de-rived from calculations with linear dimensions of regu-lar shapes such as the radii (R) and heights (h) of cylin-ders (V=πR2h), the radii of spheres (V = 4/3 π R3), andthe lengths (l), heights (h), and widths (w) of rectangu-lar cubes (V = lwh). It does not include surface irregu-larities that exist on “real” objects, nor pores or voids.

Volume, Incremental: For a given range of pressureand cumulative volume data pairs, (Pi,Vi) to (Pj,Vj),incremental volume is Vi-Vj (corresponding to a pres-sure difference of Pi - Pj, which converts to a size rangefrom Di to Dj). In certain data reduction routines, in-cremental volume data may be calculated using valuesderived by interpolation between two measured volumedata points.

Volume, Skeletal (or True Volume): The volume of thesolid material only. Skeletal volume may be determinedby mercury porosimetry or helium pycnometry and mustbe performed on either non-porous materials or mate-rials with no closed pores. The skeletal volume as de-termined by mercury porosimetry and heliumpycnometry may differ because mercury cannot intrudeinto small micropores, therefore including these voids

in the reported skeletal volume.

Volume, Specific: The volume (intrusion or extrusion,cumulative or incremental) divided by the sampleweight. This produced units of volume per unit weightand normalizes the data for convenient comparison tosimilar analyses using different quantities of samplematerial.

Volume, True: see Volume, Skeletal

Washburn Equation: A mathematical expression ofdynamic equilibrium between external forces tendingto force a liquid into a capillary of size R and the inter-nal forces repelling entry into the capillary. Assuminga circular cross-section for the capillary opening spannedby a non-wetting liquid such a mercury, the equationcan be resolved in terms of the external pressure P andthe diameter of the capillary D, the smallest size intowhich the liquid can be forced to enter at the prevailingpressure.

Young’s law: An expression of the relationship of thesolid-gas, solid-liquid, and liquid-gas interfacial ten-sions and the contact angle. Expressed mathematicallyasγ

s,g = γ

s,l + γ

l,g cos θ

where γs,g

is the solid-gas interfacial tension, γs,l the solid-

liquid interfacial tension, γ

l,g the liquid gas interfacial

tension, and θ the contact angle

Young-Laplace Equation: An expression of thepressure differential (capillary pressure) across theliquid-gas interface (the meniscus). At equilibrium,∆p = γ (1/R

1 + 1/R

2)

where R1 and R

2 are the radii of curvature of the inter-

face and γ is the surface tension of the liquid-gas inter-face. Within pores, the two radii are assumed to beequal to the pore size, D/2. Therefore, capillary pres-sure across the interface in a pore of size D is given by

∆p = γ (2/D + 2/D) = 4γ /D.

NOTE: The expression for capillary pressure is derivedunder the assumption that the fluids are in static equi-librium, i.e. there is no flow into or from the pore.

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REFERENCES

1. Dirk T. Mayer and Claudia Dittmer, Printing Inksand Related Subjects, University of Printing and Me-dia, Stuttgart, Germany, World Wide Web: www.hdm-stuttgart.de/projekte/printing-inks

2. John M. Cimbala, Fluid Mechanics ElectronicLearning Supplement, Penn State University, Depart-ment of Mechanical and Nuclear Engineering, WorldWide Web: www.spike.me.psu.edu/~cimbala/Learning/Fluid

3 Richard Zartman, Texas Tech University, Depart-ment of Plant and Soil Science, World Wide Web:www.pssc.ttu.edu/pss43365335/class%20days/day_14.htm

4. E.W. Washburn, Proc. Nat. Acad. Sci., 7, 115(1921)

5. H.L. Ritter and L.C. Drake, Pore-Size Distribu-tion in Porous Materials, Industrial and EngineeringChemistry, 17, 782 (1945)

6. Paul A. Webb, Volume and Density for ParticleTechnologists., Micromeritics Instrument Corporation(2000)

7. H.M. Rootare and C.F. Prenzlow, Surface Areasfrom Mercury Porosimeter Measurements, Journal ofPhysical Chemistry, 71, 2733 (1967)

8. R.P. Mayer and R.A. Stowe, J. Colloid InterfaceSci. 20, 893 (1965)

9. L. K. Frevel and L. J. Kressley, Modifications inMercury Porosimetry, Analytical Chemistry, Vol. 35,(1963)

10. R. Pospech and P. Schneider, “Powder ParticleSizes from Mercury Porosimetry”, Powder Technol. 59,163 (1989)

11. A. Reverberi, G. Ferraiolo, and A. Peloso, Deter-mination by Experiment of the Distribution Functionof the Cylindrical Macropores and Ink bottles in Po-rous Systems, Annali di Chimica, 56, 12, 1552 - 1561(1966)

12. E.J. Garboczi, Permeability Length Scales and

the Katz-Thompson Equation, National Institute of Stan-dards and Technology, World Wide Web, http://ciks.cbt.nist.gov/garbocz/paper10/node2.html

13. A.J. Katz and A.H. Thompson, “A QuantitativePrediction of Permeability in Porous rock”, Phys. Rev.B, (1986)

14. A.J. Katz and A.H. Thompson, “Prediction ofRock Electrical Conductivity From Mercury InjectionMeasurements”, Journal of Geophysical Research, Vol.92, No. B1, Pages 599-607, January 10, 1987

15. R.F. Angulo, V. Alvarado, and H. Gonzalez,“Fractal Dimensions from Mercury Intrusion CapillaryTests”, II LAPEC, Caracas, March 1992

16. Jörgen Hager, Steam Drying of Porous Media,Ph.D. Thesis, Department of Chemical Engineering,Lund University, Sweden (1998)

17. Sari Westermarck, Use of Mercury Porosimetryand Nitrogen Adsorption in characterization of the PoreStructure of Mannitol and Microcrystalline CellulosePowders, Granules and Tablets, Academic Dissertation,Department of Pharmacy, University of Helsinki (2000)

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