new perspectives in mercury porosimetry

8/9/2019 New Perspectives in Mercury Porosimetry

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Advances in Colloid and Interface Science .76 77 1998 341 372

New perspectives in mercury porosimetry

Carlos A. Leon y Leon Quantachrome Corporation, 1900 Corporate Dr., Boynton Beach, FL 33426, USA

Abstract

Seventy-six years ago, Washburn pioneered the concept that the structure of porous solidscould be characterized by forcing a non-wetting liquid to penetrate their pores. At that timeWashburn postulated that the minimum pressure P required to force a non-wetting liquidlike mercury to penetrate pores of size R is given by P K R, where K is a constant.Nowadays that very same concept constitutes the backbone of mercury porosimetry, atechnique applied routinely to the characterization of all kinds of solids. Despite itsperceived fundamental and practical limitations, mercury porosimetry will continue to be

regarded as a standard measure of macro- and mesopore size distributions for years to . .come. This is so because this time-tested technique is 1 conceptually much simpler, 2 .experimentally much faster, and 3 unique in its ability to evaluate a much wider range of

pore sizes, than any alternative method practised currently e.g. gas sorption, calorimetry,.thermoporometry, etc. . Clearly it would be desirable to derive as much structural informa-

tion as possible from simple mercury porosimetry experiments. Surprisingly, relatively fewattempts have been made in the open literature to extract much information beyond poresize distributions from mercury porosimetry data. This contribution emphasizes the need todevelop concerted efforts towards expanding the interpretation of mercury porosimetry databy examining the virtues and flaws of various reported attempts to generate particle sizedistributions, inter- and intraparticle porosities, pore tortuosities, permeabilities, throat poreratios, fractal dimensions and compressibilities from mercury intrusion and or extrusioncurves. 1998 Published by Elsevier Science B.V. All rights reserved.

Keywords: Mercury porosimetry; Characterization; Pores; Particles

Tel: 1 561 7314999; fax: 1 561 7329888; e-mail: [email protected]

0001-8686 98 $19.00 1998 Published by Elsevier Science B.V. All rights reserved. . P I I S 0 0 0 1 - 8 6 8 6 9 8 0 0 0 5 2 - 9


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( )C.A. Leon y Leon Ad . Colloid Interface Sci. 76 77 1998 341 372 342

Contents

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3422. Mercury porosimetry as a characterization tool . . . . . . . . . . . . . . . . . . . . . . . . .343

2.1. Theoretical foundation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3432.1.1. Hysteresis phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3442.1.2. Studies of ideal pore systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .345

2.2. Experimental approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3472.2.1. Scanning versus stepwise data acquisition . . . . . . . . . . . . . . . . . . . . . .3482.2.2. Contact angle measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3502.2.3. Mercury purity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3512.2.4. Blank corrections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .352

2.3. Range of applicability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3542.3.1. Sample types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3542.3.2. Pressure and pore size limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .354

2.4. Interpretation of mercury porosimetry data . . . . . . . . . . . . . . . . . . . . . . . .3542.4.1. Particle size distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .354

.2.4.1.1. Mayer Stowe MS theory . . . . . . . . . . . . . . . . . . . . . . . . . .355 .2.4.1.2. Smith Stermer SS theory . . . . . . . . . . . . . . . . . . . . . . . . . .3562.4.2. Inter- and intraparticle porosity . . . . . . . . . . . . . . . . . . . . . . . . . . . .3582.4.3. Pore tortuosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3612.4.4. Permeability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3632.4.5. Throat pore ratios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3642.4.6. Fractal dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3672.4.7. Sample compressibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .369

3. Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .369 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .370References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .370

1. Introduction

Mercury porosimetry is a well established technique for the characterization of porous materials 1 3 . This fact is most clearly manifested in documents such as

.the International Union of Pure and Applied Chemistrys IUPACs Recommenda -tions for the Characterization of Porous Solids , in which it is stated that mercuryporosimetry is widely accepted as a standard measure of total pore volume and

pore size distribution in the macro- and mesopore ranges 4 . Moreover, practicalapplications of the technique have led to the development of internationally

. recognized standard procedures ASTM, DIN revolving around its use 5 8 . Yet

the full potential of mercury porosimetry as a characterization tool is far frombeing uncovered.

The use of mercury porosimetry is much too often limited to the evaluation of pore volumes or pore size distributions only. Over the years, numerous literaturereports have described theories and procedures designed to obtain additionalinformation from mercury porosimetry curves. However, judging from the relativelyfew pertinent publications and research reports presented at international meet-


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ings 9 , these approaches have not received the attention they might deserve.Clearly, it would be greatly beneficial to expand the capabilities of a technique thathas already proven to be fast, simple, reliable, broad in scope and, hence, ideallysuited for industrial and research applications alike.

The purpose of this contribution is to call attention to some approaches that inour experience can or have already turned mercury porosimetry into a much morepowerful characterization tool. Their examination is preceded by an overview of the theoretical foundations and experimental approaches that define mercury

porosimetry as a technique including a discussion of how certain practical con-siderations can impact its applicability to the characterization of solid materials in

.general and is aided by selected illustrative examples.

2. Mercury porosimetry as a characterization tool

Mercury porosimetry is by far the most popular method employed for the characterization of relatively large pores, in particular macropores 10 12 . Com-

.pared to alternative pore size characterization methods e.g. gas sorption 1 3mercury porosimetry is based on a simpler principle the Washburn equation Sec.

. 2.1 , is much faster producing full pore size distributions in minutes compared to.hours in gas sorption tests and, perhaps more importantly, is unique in that it

.covers a very wide range of pore sizes, including large pores 0.5 m that aredifficult or impossible to probe reliably by other techniques. Yet much like othertechniques, mercury porosimetry analyses yield cumulative raw data from whichdifferential pore size distributions are numerically derived.

2.1. Theoretical foundation

The principles governing conventional mercury porosimetry calculations have been described in a number of publications 1 3,10 15 . Almost two centuries ago,

Laplace 16 reasoned that the work required to expand a nonspherical fluidsurface of principal radii of curvature R and R is equal to the work done to the1 2concave side of the surface, and managed to derive the following equilibriumexpression:

. . P 1 R 1 R 11 2

where is the fluid surface tension and P is the pressure across the interface. Later on Young 17 derived an expression describing the mechanical equilibrium

of liquid drops on surfaces 1,15 which Washburn 18 linked to Laplaces equationin order to postulate that, for cylindrical pores of equivalent radius R wetted witha fluid of contact angle ,

. . P 2 cos R 2

The latter equation predicts the behaviour of liquids confined in capillaries. Accordingly, for wetting liquids 90 , so P P P 0 and the liquidsliquid gas


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rise in the capillaries to develop a column whose head pressure compensates forthat pressure difference. Conversely, non-wetting liquids like mercury for which

.90 180 tend to recede in capillaries and must be forced hydraulically tofill them.

The Washburn equation clearly provides a simple and convenient relationshipbetween applied pressure and pore size. Pore size distributions are thereforegenerated by monitoring the amount of non-wetting mercury intruded into pores as

a function of increasing applied pressure 1 . The choice of cylindrical poregeometry was one of mathematical convenience in order to avoid the complexitiesof having to deal with mean radii and contact angles in pores of irregular crosssections. For that reason mercury porosimetry has traditionally treated solids asporous materials containing a bundle of capillaries of various sizes. Not very mucherror is introduced by the presence in solids of interconnected channels, providedthat all pores are equally accessible to the exterior mercury reservoir; that is,access to pores of a given size D must always be possible through pores of size

D. If the pores have only smaller entrances, they can only be filled upon .reaching a pressure higher than that required by their actual inner dimensions.Once the pressure required to fill the largest entrance was reached, the filling of the entire pore would ensue. In such case, the calculated pore size distribution

. would be biased towards smaller than actual inner pore sizes. The converse wouldbe true upon mercury extrusion, and could presumably lead to entrapment and

hysteresis in so-called ink-bottle pores 19,20 . The total content of ink-bottlepores, when estimated from the end point of the depressurization curve, has beenreported to vary widely from a negligible fraction of total pore volume in a

silica alumina gel to 80% in an activated carbon 15 .

2.1.1. Hysteresis phenomenaMany explanations have been offered to account for the experimental observa-

tion that mercury extrusion curves do not overlap intrusion curves 1,3,13,21 28 .Currently, three explanations appear to be favored by different groups in the

. .literature: i the ink-bottle pore assumption 20 , as described in Section 2.1; iinetwork effects, i.e. an extension of the ink-bottle concept which is substantiated

.by complex computer simulations 13,23 26 ; and iii a pore potential theory whereby mercury is not subjected to pore wall interactions during its initialintrusion but is partly held in pores upon extrusion as a function of wall interac-

tions 1,27,28 . According to the latter mechanism, the work done to expand the .mercury surface area upon initial intrusion consists of two parts: i the irreversible

work of intrusion, associated with mercury entrapment in potential wells or . .ink-bottle pores; and ii the reversible work of intrusion, which is greater than iii

the reversible work of extrusion. The latter is quite feasible thermodynamically 1,15,27,28 and is ascribed to the extrusion of mercury being facilitated by a

decrease in the tensile extrusion contact angle relative to the compressiveEintrusion contact angle . In this context, good quantitative agreement has beenI

reported 1,27,28 upon comparing the results of pore potential predictions withcontact angle changes. Moreover, it is an experimental fact that hysteresis gaps can


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Fig. 1.

.in many cases be completely eliminated within ca. 1% by modifying the extrusioncontact angle until the extrusion curve and a second intrusion curve overlap each

.other 1,25,27,28 . This fact is illustrated for a typical material tested a silica inFigs. 1 and 2. Fig. 1a presents standard mercury intrusion extrusion volumes as a

.function of pore sizes calculated using a constant contact angle of 140 .I EFig. 1b illustrates the excellent overlap obtained after shifting the extrusion angle . to 106.5 . Fig. 2a,b confirms the excellent overlap in pore size distributionsEcalculated before and after shifting the extrusion contact angle, respectively.

Exceptions to this general observation have been attributed to materials having multiple or variable contact angles 1,3 . Interestingly, if contact angle shiftsdominated hysteresis phenomena in mercury porosimetry, attempting to infer pore

.shapes from hysteresis curves as done through network models would be futile.

2.1.2. Studies of ideal pore systemsOne way to substantiate the validity of contact angle hysteresis in mercury


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.Fig. 1. Mercury intrusion extrusion curves as a function of pore size for a typical material; a assuming . 140 ; b after shifting to 106.5 ; Symbols: circles, intrusion; triangles, extrusion. ForI E E

clarity, only one out of every 100 data points collected is shown.

porosimetry is through the analysis of ideal pore systems. For instance, a material with uniformly sized, straight cylindrical pores would not be subjected to ink-bottleor network effects, and should therefore show no hysteresis upon mercury intru-sion extrusion experiments. On the other hand, hysteresis loops in such ideal poresystems would be consistent with an unavoidable shift in intrusion extrusion

contact angles. Over a decade ago, Lowell 29 performed pertinent mercury

porosimetry experiments on Nuclepore brand polycarbonate membrane filters 30 .These filters, made by bombarding polycarbonate films with laser beams, exhibitrather uniform straight cylindrical pores perpendicular to the surface, as confirmed

by SEM pictures 31 . Mercury porosimetry experiments confirmed that a clear intrusion extrusion hysteresis gap occurred in these ideal materials 29 . In spite of

a possible influence of sample compression on the results 32 , the study of idealpore systems could prove to be invaluable to confirm the validity of different


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Fig. 2.

models of hysteresis phenomena. Unfortunately, MCM41s and related materials .appear to be mechanically too weak to withstand exposure to high pressures 33 .

2.2. Experimental approach

Even though Washburn postulated his equation in 1921, it was not until the late1940s that the usefulness of mercury porosimetry experiments began to be realized.

At around that time Ritter and Drake 34,35 popularized the use of the techniquethrough the introduction of a porosimeter capable of reaching 10 000 psi. Note: 1

3 .psi 6.895 10 MPa. Nowadays commercial porosimeters can reach 60 000 psiand perform intrusion extrusion experiments automatically. The basic componentsand the operation of modern mercury porosimeters have been described in detail

elsewhere 1,3 . Among the different experimental approaches available, perhapsthe most controversial issues are the choice of scanning versus stepwiseintrusion extrusion data acquisition and the choice or determination of contactangles. These and other experimental aspects are briefly reviewed in turn next.


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.Fig. 2. Pore size distributions calculated from the curves shown in Fig. 1; a assuming 140 ;I E .b after shifting to 106.5 . Symbols: circles, intrusion; triangles, extrusion. For clarity, only one outEof every 100 data points collected is shown.

2.2.1. Scanning ersus stepwise data acquisitionEarly porosimeters operated exclusively in stepwise fashion, i.e. raising or

lowering the applied pressure in steps. This approach had several drawbacks, . .including: a an inability to hold the pressure constant at any given step; b the

.need for long analysis times; c the poor resolution obtained as a result of collecting relatively few data points in order to construct pore size distribution

.curves; and d the possibility of skipping important information in pressure rangescorresponding to the filling of narrow or multimodal pore sizes. All these draw-backs were eliminated by the introduction of the scanning technique pioneered by

Quantachrome 1,3 . In addition, Quantachrome porosimeters were designed tobalance mercury compression under pressure and its expansion due to compressive

heating of the hydraulic oil 1 . This advantage leads to blank runs whose intru-sion extrusion volumes are within 0.5% of full scale throughout the entire experi-


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mental pressure ranges, thereby making blank run corrections unnecessary see. Section 2.2.4 1,3 . Surprisingly, the application of the stepwise approach, including

the need for blank corrections, seems to have found a niche in the literature. Supporters of the stepwise approach often invoke Darcys law 1 to justify qualita-

tively that the degree of mercury intrusion should be a function of time at anypressure. If so, stepwise curves could be claimed to approach a more equilibratedcondition than scanning curves. However, the quantitative application of Darcys

law in combination with the well-known Poiseuilles law 1,3 leads, after allowingfor a gradual increase in mercury viscosity with pressure up to 18% between 1 and

.60 000 psi 36 , to an expression of the form

10 5 . 2 2. .t 2.21 10 7.2 10 P L D 3

. where t in seconds is the time it takes for mercury to flow into cylindrical pores .of length L and diameter D both in microns as a function of applied pressure

.difference P in psi . The predictions of the above equation for pore lengths of .100 m probably an upper limit for most industrial powders and D valuescovering the range probed by mercury porosimetry are illustrated in Fig. 3. Alsoshown in Fig. 3 are the intrusion pressure differences P required to fill pores of diameter D according to the Washburn equation. As an illustration, note thatpores with D 3 m would take about 0.01 s to fill at 10 psi, if they could befilled at 10 psi; however, the Washburn equation indicates that pores with D 3

m could only be filled at 100 psi, and by the time that pressure difference isreached those pores would be filled in only 0.001 s or so. Fig. 3 shows clearly thatthe times required to fill 100 m-long pores with mercury vary between about 10 5s for the largest intrudable pores to 1 s for the smallest intrudable pores. Thesequantitative predictions confirm that the filling of pores with mercury occursalmost instantaneously and is not at all limited by viscous effects within theexperimental range of typical mercury porosimeters. The same conclusion wasreached on experimental grounds by others who, e.g. confirmed the insensitivity of

mercury intrusion and extrusion scanning curves to the applied scanning rates 25 .It is therefore not surprising to find a fair agreement between porosimetry spectraobtained via scanning and stepwise approaches on Quantachrome porosimeters, as

.shown in Fig. 4. Yet the calculated pore size distributions Fig. 5 differ consider-ably for curves collected at approximately equal analysis times. This is so becausethe stepwise approach can only generate tens of data points within the same timeframe and pressure range in which the scanning approach collects thousands.

In spite of all the above arguments, if one still chooses to apply the stepwise

method to generate mercury porosimetry data, it is advisable to minimize the pressure drop during each pause 5 . This is desirable in order to prevent the curves

from falling too deeply within the hysteresis loop and thus undergo irreversiblechanges. Time-related phenomena upon mercury intrusion extrusion have been

ascribed to a slow and relatively minor mercury vapor transfer mechanism 1,3 , butin some instruments their magnitude is considerable, and it could be related toadditional mercury intrusion extrusion induced by heat dissipation effects.


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Fig. 3. Time required to fill 100 m-long cylindrical pores of diameter D with mercury at 293.15 K .predicted by Darcy-Poiseuille laws .

Nonetheless, many commercial porosimeters allow for an automatic repressuriza-tion to the target pressure when the pressure changes. Particularly when samples with relatively narrow pore size distributions are analyzed, the extent of depressur-

ization and repressurization can affect the precision of the results 5 . It is alsoadvisable, therefore, to confirm that stepwise data are truly insensitive to thechoice of equilibration parameters by comparing the results of tests performedusing different equilibration parameters. A necessary condition to ensure equilib-rium is the insensitivity of the results to the choice of experimental variables.

2.2.2. Contact angle measurementThe exact value of the contact angle between mercury and solid surfaces

depends on many factors, including the solid surface chemistry, cleanliness, androughness, as well as the purity of the mercury and whether the mercury isadvancing or retreating on the solid surface. Ritter and Drake originally adopted

140 as an average contact angle representative of a wide variety of materials .34,35 . However, in many cases this is a crude approximation see Table 1 that

can lead to errors in pore size estimates of the order of up to 50% 3 . For utmostaccuracy in mercury porosimetry calculations, contact angles should be measuredfor each material tested. However, assigning the same contact angle value to

compare the data of chemically similar materials is qualitatively useful 1 . There are a wide variety of methods available to measure contact angles 1,3 .


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Fig. 4. Comparison of scanning and stepwise curves obtained on a Quantachrome Poremaster analysis.times of ca. 15 min in both cases . Symbols: triangles, scanning data; circles, stepwise data; open

symbols, intrusion; filled symbols, extrusion.

.Preferably scanning curves should be processed using dynamic e.g. advancing 1contact angles, whereas stepwise curves may be more compatible with static or

. Sessile-drop contact angles 3,15 . In either case, every effort should be made to .assure the purity of the mercury used see Section 2.2.3 .

2.2.3. Mercury purityHigh purity mercury should be employed because its purity affects both contact

.angles Section 2.2.2 and surface tension values required for data interpretation. Itis advisable to employ acid washed, dried and distilled preferably doubly- or

.triply-distilled mercury, even though its cost can be relatively high. Recyclingmercury directly after decanting the contents of spent sample cells is not advisable.Similarly, spent hydraulic oil should not be recycled because it carries particles that


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Fig. 5. Comparison of pore size distributions calculated from the mercury intrusion and extrusioncurves shown in Fig. 4. Symbols: triangles, scanning data; circles, stepwise data; open symbols, intrusion;filled symbols, extrusion.

contaminate the mercury and change its dielectric and flow properties. Constantlevels of impurities in the mercury employed should not be assumed to lead to

constant shifts in intrusion or extrusion data. However, small levels of impuritiesmay introduce uncertainties as small as those expected of pure surface tension values as a function of pore size 42,43 .

2.2.4. Blank correctionsSubtracting an empty cell run from an actual sample analysis is often invoked as

a means to correct primarily for the apparent volume intruded due to compression


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Table 1Examples of contact angles reported in the literature

.Material Contact angle Reference

Alkali borosilicate glass 153 37 Aluminum oxide 127,142 37, this study Calcite 146 38 Carbon 155, 162 37,40 Cement 125 40 Clay minerals 139 147 37 Coal 142 38 Glass 135 140 38 Mica 126 38 Nickel 130 41 Oxide-type surfaces 140 38 Paraffin wax 149 39 Pyrite 146 38 Quartz 132 147 37 Steel 154 39 Titanium dioxide 141, 160 37 Zinc oxide 141 37

of mercury, but also of the sample itself, of the sample cell, and of the rest of the instrument components 3,44 . Properly designed instruments are built so as to

minimize the latter two effects 1 . Accordingly, the compressibility of a sample can .be accurately determined Section 2.4.7 provided that the mercury compression

does not affect the volume in the dilatometer stem significantly. 1 . .

The variation in mercury compressibility in psi with pressure P in psiHg at ambient temperature can be derived from experimental data 45 47 as: Hg2.7735 10 7 6.5331 10 13 P . This implies that a typical sample cell filled withca. 6 cc of mercury could experience a drop in mercury level within the dilatometer

.stem 0.2 cm i.d. of at most 0.5% of the total stem height at pressures up to 60000psi. This is the order of magnitude typically observed at 60 000 psi during blankruns performed on Quantachrome porosimeters.

In any event, utmost accuracy would be achieved by subtracting from the sampleanalysis results those of an analogous test on a nonporous sample of similar bulk

. volume and compressibility if realistically possible , in order to correct the resultsfor compressibilities and temperature changes. This is so because the compressibil-

.ities of the various components in the system will augment even if negligibly themeasured intrusion values while the pressure-induced heating and consequentexpansion of the system reduces the measured volumes. For a given porosimeterdesign, either one of these effects could dominate. Hence, the results of blank testscan be used to assess a given porosimeter in terms of the relative importance of an

. apparent intrusion compressibility dominant or an apparent extrusion heating.dominant on the sample analyses.


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2.3. Range of applicability

Mercury porosimetry is unique in its ability to complement the pore size ranges other pore characterization approaches 1 3 . Two aspects that often raise ques-

tions are the types of samples that can be analyzed and the pore size ranges thatare accessible to mercury. Both aspects are discussed in this section.

2.3.1. Sample typesIn principle, mercury porosimetry is applicable to all kinds of solid materials. In

practice, materials whose structure could compress or even collapse at highpressures require corrections for compressibility or analyses at relatively lowpressures. On the other hand, certain metals show a tendency to amalgamate

readily with mercury 44 . One could thus investigate mercury amalgamationmechanisms with, e.g. gold or silver, by monitoring apparent intrusion rates. Lessnoble metals show less of a tendency to amalgamate with mercury because they are

protected by a thin surface oxide layer that can delay amalgamation rates suffi-ciently to allow regular intrusion measurements to be made.

2.3.2. Pressure and pore size limitsMercury porosimeters measure intrusion and extrusion pressures and volumes,

and calculate pore sizes from measured pressures. Hence, the pore size rangescovered by standard porosimeters are limited by the pressures they can achieve.Modern porosimeters allow the commencement of intrusion tests at pressures as

low as ca. 0.5 psi 3 . This initial pressure is required to force mercury to fill thesample cell. Regardless of the filling angle, due to their own height all samples are

.invariably subjected to a mercury head pressure of the order of 0.1 psi that could induce some intrusion prior to commencing an experiment 1 . To reduce the latter

possibility, intrusion experiments are often commenced at pressures slightly higherthan 0.5 psi but seldom higher than 1 psi. On the other hand, the upper pressurelimit of commercial porosimeters is set by safety design margins to 60 000 psi.

The estimation of pore size limits from experimental pressure limits, using theWashburn equation, requires using an assumed or measured contact angle ; see

.Section 2.2.2 . Table 2 lists typical pore size ranges calculated from differentassumed contact angles. From the Washburn equation, it follows that instrumentsthat work in the same pressure range cannot cover different pore size ranges. Thisobservation again highlights the importance of paying close attention to the valueof the contact angle used when converting applied pressures to calculated poresizes.

2.4. Interpretation of mercury porosimetry data

2.4.1. Particle size distributionThe notion that mercury intrusion curves yield information about pores between

particles has led several researchers to postulate that the same intrusion curvesalso contain structural information about the dimensions of the particles them-


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Table 2Effect of assumed contact angle on calculated pore size range covered by mercury porosimeters

aContact K Calculated upper Calculated lowerWashburn . .angle m.psi pore size limit, pore size limit,

. .m pressure 1 psi m pressure 60 000 psi

140 213.4 213.4 0.00356130 179.1 179.1 0.00298

a Pore size K pressure.Washburn

selves. Two theories in particular have found general acceptance in the literature: the simple mercury breakthrough theory of Mayer and Stowe 48 and the integral

patchwise approach of Smith and Stermer 49 . Both are described in turn next.

( ) 2.4.1.1. Mayer Stowe MS theory. The manner in which mercury penetrates a bed of uniform spherical particles was examined in detail by Mayer and Stowe 48 , who

postulated that the breakthrough pressure P required to force mercury tobpenetrate the void spaces between packed spheres of diameter D is given by

K . P 4b D

where K is the so-called MS proportionality constant and is the surface tensionof mercury. The dimensionless MS constant K was shown to be a complex functionof the mercury contact angle with, and the packing arrangement of, the particles.

Using intraparticle porosity as a measure of packing arrangement, it can bea .shown that for randomly packed spheres 37.5% and a typical mercuryacontact angle of 140 , K 10.73. In general, K was found to increase with

and to decrease with 50 . On the other hand, independent work confirmed thata . the average particle coordination number N can be estimated from 51c

. N 5c Hg1

He

. . where and are the bulk particle and helium true densities of theHg Hematerial.

The validity of the MS theory has been confirmed experimentally by quantitativecomparison with particle size distributions derived from independent techniquessuch as X-ray sedimentation and electron microscopy. The best agreement amongdifferent techniques was found for solids with narrow monomodal particle sizedistributions and for materials with relatively few interparticle voids. Particle shapeis generally thought to play a minor role in the results, although dimensionless


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intrusion or extrusion particle shape factors f have occasionally been introducedin the MS expression as follows:

fK .

P 6b D

( ) 2.4.1.2. Smith Stermer SS theory. Upon applying Mayer and Stowes approach,Smith and Stermer observed that even for narrow particle size distributionsmercury intrusion curves generally do not exhibit unique breakthrough pressures

52 . These deviations were attributed to the fact that particles of a given size donot pack in separate regions of a powder bed, as was implicitly assumed by others

48 . Smith and Stermer generalized Mayer and Stowes approach by postulating 49 that the total volume of mercury V intruded into a bed of particles of i

different sizes at any given pressure P is the sum of volumes intruded betweeni

particles of each size D according to

. . .V K P , D F D d D 7Hi i . where K P , D is a kernel which describes mercury intrusion between particlesi

.of fixed diameter D and F D is the particle size distribution function. Usingexperimental V and P values and a generalized kernel function, Smith andi iStermer adopted a numerical approach in order to solve the previous equation for

. F D . Their numerical approach called for the division of the expected particle size .range into discrete intervals within which the distribution function F D for its j

.average pore size D could be evaluated: j

. . .V K P , D F D D 8i i j j j

If the summation is applied in the interval 1 j N , and the equation is solved .for 1 i M , evaluating F D involves solving simultaneously a system of M by

N equations subject to a non-negativity constraint such as Lawson and Hansons . Non-Negative Least Squares NNLS approach 53 , i.e. making the function

2 . . . E V P V P d D 9H experiment theoryconverge within a minimum accuracy E. To minimize the oscillatory behavior of

the calculated distribution function, a smoothing term can be added 54 as follows:

2 2 . . . . E V P V P d D F D d D 10H Hexperiment theory where 10 and is the so-called regularization parameter. To speed up theconvergence of the iterative solution of the above summations, a relaxation


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parameter, , can also be incorporated by correcting successive iteration values asfollows:

. . . . . F D F D F D F D 11 k 1 k k 1 k

. where F D is the iteration value before applying the relaxation approach. k 1Typical values fall between 0 and 2, with 1 implying that the relaxationapproach is not applied.

The particle size distributions predicted using the MS and the SS methods havealready been applied successfully to a wide variety of materials. Fig. 6 shows acomparison of MS and SS model predictions for a particle size reference material .PSRM 4000 supplied by Quantachrome. The agreement among particle size

.distributions estimated from X-ray sedimentation Microscan II , laser diffraction .CILAS 1064 , the MS and the SS methods for this reference material is remark-ably good.

An example of how these methods can already be used to address problems of industrial relevance is that of the characterization of carbon blacks. Carbon blacksare made up of very fine and compact particles formed usually through thehigh-temperature pyrolysis of gaseous hydrocarbons. These nanometer-sized parti-cles tend to agglomerate and thus generate a complex structure that makes theirparticle size analysis very difficult. Currently the most widely used approach tocharacterize particle sizes of carbon blacks is by electron microscopy. However, thetechnique is laborious, time consuming, expensive, and it requires technicians withspecial skills. Moreover, the results are uncertain because of representative sam-pling errors and because they depend on the number of particles counted and onthe assumption that 2-dimensional pictures can characterize 3-dimensional mor-phology. Mercury porosimetry is free of all these undesirable limitations, and canprovide satisfactory results in a few minutes per sample. The validity of the

technique was tested using a series of reference carbon blacks available from.Titan Products routinely used by ASTM Committee D-24 to monitor the precision

of their standard test methods. Fig. 7 shows the MS particle size distributionsobtained for these carbon blacks, and Table 3 compares their estimated modalparticle diameters with those expected from their typical size ranges determined byelectron microscopy tests. The results are again quite encouraging, considering the

speed and simplicity of the mercury porosimetry technique both of which are.essential in industrial carbon black manufacturing facilities . Moreover, the mer-

cury porosimetry technique provides additional information such as pore sizedistributions and, perhaps more importantly, mercury intrusion surface areas, given

the critical relevance of surface areas especially external or nonmicroporous.surface areas to the reinforcing ability of different commercial carbon black

grades. An obvious advantage of the MS method in the previous examples is its higher

resolution compared to that of the highly computationally-demanding SS method .see Fig. 6 . It is noteworthy that the MS and the SS methods do not necessarily yield identical results. Work is in progress to establish the sensitivity of each


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.Fig. 6. Comparison of particle size distribution predicted for a reference material PSRM 4000 usingthe MS and the SS methods. Continuous curve, MS method; filled circles, SS method.

method to parameters such as sample type, particle size range considered, multi-modal distributions, surface chemistry, etc.

2.4.2. Inter- and intraparticle porosity .Porosity is defined as the percentage of void space in a solid. Total porosity is

. .often evaluated from mercury density and helium density values asHg Hefollows:

Hg . 100 1 12 He

This definition accounts for all pores smaller than those filled by mercury ca. 14.5


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Fig. 7. Particle size distributions estimated for series SRB-5 reference carbon blacks using the MSmethod. Symbols: crosses, SRBA5; filled circles, SRBB5; x-marks, SRBC5; open triangles, SRBD5; opensquares, SRBE5; dots, SRBF5.

.m in diameter at standard pressure and at the same time accessible to helium . .i.e. open pores greater than ca. 3 A in diameter , whose total pore volume V isp

1 1 .V 13p

Hg He

. In powders, both intrapore spaces between particles and interpore voids within.particles contribute to the total porosity. The demarcation between interior pores

and voids between particles is often unclear. In fact, some materials possessintraparticle and interpores of the same magnitude, in which case making adistinction between them is quite complicated. But more often than not distinction


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Table 3Comparison of mean particle sizes of series SRB-5 reference carbon blacks expected from electron

. .microscopy as cited in ASTM D 1765 and estimated using mercury porosimetry MS method

Reference Typical average particle Modal particle . .carbon black diameter nm diameter nm

. . .grade cited in ASTM D1765 MS method

.SRB A5 N135 11 19 19 .SRB B5 N330 26 30 28 .SRB C5 N220 20 25 20 .SRB D5 N762 61 100 51 .SRB E5 N660 49 60 59 .SRB F5 N683 49 60 65

is clear because the sizes of interpores greatly exceed those of intrapores. Hence,bimodal distributions are frequently observed, in which case the boundary could bedefined as the cumulative intruded volume curves inflection point. On the otherhand, it is worth noting that the volume of interpores is related to the packingcharacteristics of the sample bed. Therefore, loose beds will yield different inter-pore volumes than tapped beds, with the latter yielding more reproducible resultsthan the former ones.

To distinguish among the various types of porosities present in a powderedsample, the following equations can be used:

V a . .Interparticle Porosity % 100 14V bV V a b

. .Intraparticle Porosity % 100 15V V c bV a . .Mercury Intrusion Porosity % 100 16V c

where V is the mercury volume intruded at any given pressure, V is the mercurya b volume intruded at a user-defined Intrapore Filling Pressure Limit usually at the

inflection point of the plateau between low and high pressure steps in mercury.intrusion curves , and V is the mercury volume intruded at the maximum experi-c

mental pressure attained.Fig. 8 illustrates an instance in which both intra- and interpore spaces appear to

be present on a sample, judging from the shape of the mercury intrusion curve .Fig. 8a . This kind of curve invariably leads to a seemingly bimodal pore size

.distribution Fig. 8b . In the past, people experienced in mercury porosimetry would recognize the lower pressure intrusion as being related to intrapore volumefilling. However, they had no direct way of confirming if that was truly the case orif they were actually dealing with a bimodal distribution of pore sizes. The sampleshown in Fig. 8 is in fact a reference material currently being used for particle size


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Fig. 8.

distribution round-robing tests by ASTM Committee D-32 on Catalysts. Its particlesize distribution, calculated from mercury intrusion data up to its first inflection

.point ca. 400 psi , agrees fairly well with the distribution expected from laser .diffraction CILAS 1064 measurements see Table 4. This indicated that one is

indeed dealing with a monomodal distribution of pore sizes in a material whosepore and particle size distributions can be easily, quickly and reliably estimatedfrom a single mercury intrusion experiment.

2.4.3. Pore tortuosityWhen modeling the diffusion of fluids in porous solids it is often reported that . the effective or measured diffusivity D differs from the theoretical or bulkeff

.fluid diffusivity D by a factor related to the structure of the solid as follows:b

D b c . D 17eff


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Fig. 8. Determination of particle size distribution, pore size distribution and porosities of a reference . catalyst via a single mercury intrusion experiment. a Cumulative intrusion curve only one of each 100

. . experimental data points shown ; b Differential intrusion curve only one of each 10 experimental data.points shown .

where is the pore volume fraction and is the tortuosity factor. This effectivectortuosity factor lumps all deviations from straight diffusion paths into a singledimensionless parameter which usually falls between about 1 and 7, with a value of 2 being associated with nonintersecting cylindrical pores. Being an average value,the effective tortuosity is thought by some to be relatively insensitive to changes in

pore structure characteristics 39 . Nonetheless, using Ficks first law to describe

fluid diffusion through cylindrical paths, Carniglia derived the following expression 55 :

. 2.23 1.13V 18c co Hg

where V is the total specific pore volume which can be approximated by theco.mercury volume intruded at the maximum experimental pressure attained, V andc


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Table 4 .Comparison of particle size distribution values obtained by laser diffraction CILAS 1064 and mercury

.intrusion MS method for a reference catalyst

Weight % Laser diffraction data Mercury intrusion data . . . .finer than CILAS 1064 m MS method m

diameter listed

10% 34.0 34.8 .50% median 60.6 58.5

90% 98.4 124.2

is the bulk or particle density of the solid. Carniglia pointed out that thisHgrelationship was generally applicable within 0.05 V 0.95, so that if cylin-co Hgdrical pores truly prevailed in a material, its tortuosity should not deviate greatlyfrom 2. However, experimental values significantly larger than 2.23 led

Carniglia 55 to expand the previous equation into a more generalized form:1 E . . . 2.23 1.13V 0.92 y 19c co Hg

where

4 V i . y 20 s di

and

S total surface area;

V change in pore volume within a pore size interval i;i d average diameter within a pore size interval i; andi . E pore shape exponent 1 for cylinders .

Using S as the BET surface area, Carniglia reported fair agreement betweentortuosity factors computed with the above model and calculated from experimen-tal diffusion measurements for a wide variety of metal oxides and catalysts, with

few E values falling below 1.0 55 .

2.4.4. PermeabilityThe permeability K of a solid is defined as the ability of the solid to let fluids

travel across it 13 . Various experimental methods are available to evaluate the

permeability of porous solids, but they often produce different results and semiem- .pirical correlations at best among them. This is so because different approaches

are affected to different degrees by factors such as head pressure, molecule poreopening size ratio, and by adsorption. The use of mercury intrusion for permeabil-ity determinations can be regarded as qualitatively useful when comparing, e.g.applications involving gas or liquid permeabilities. A match among the variouspermeability estimates is in principle possible if a slip correction is allowed as is


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often done to match the results of liquid and gas permeability estimates 39 . A fundamental limitation of the mercury-permeability method is that it needs toassume a model for the pore structure. This model usually adopted is a medium

consisting of a bundle of parallel capillaries 39 , which is widely regarded as anadequate frame of reference. Permeability, like tortuosity, is affected by bedpacking efficiency; the most consistent results are again obtained on fully tappedsamples or, alternatively, by compressing a powdered sample into a tablet by

applying a compression force known to minimize interparticle porosity effects 39 .Numerous theoretical attempts have been made to relate the permeability K of asolid to intrinsic and more readily measurable properties, such as porosity and pore

diameters. One such approach 56 models the flow of fluids across straightcylindrical channels in a bed of powder by combining Darcys and Poiseuilles laws

1 to obtain

d2p . K 2132

. where is the powder bed porosity and d is the average mean volume diameterpof the pores. Correction for nonuniform pore characteristics leads to an expressionof the general form

d2p . K 2216

where can be taken to represent the effective tortuosity of the pores. If the poresare assumed to be straight cylindrical capillaries then 2.

As an illustration of the usefulness of quantifying mercury permeabilities,consider the pore size distributions of two different kinds of filter paper shown inFig. 9. One filter paper has more and larger pores than the other, and could thusbe expected to be more permeable. But by how much? If pore sizes or pore volumes did not correlate directly with filter performance, one would perhaps wonder if tortuosity effects were involved. Yet the tortuosities of these samples,

calculated from the same mercury intrusion experiments, were similar see Table.5 . Table 5 also shows the calculated mercury permeabilities for these samples,

which differ by a factor of ca. 20. These calculated permeabilities are much lower 2.than those calculated for, e.g. commercial paper towels 10 000 nm or labora-

2.tory kimwipes 1000 nm , as expected. Being more akin to filter applications, itremains to be confirmed whether simple quantitative permeability estimates such

as those made here correlate better with the performance of filters or of othermaterials. Nonetheless, it seems clear that even relative permeability estimates canprovide valuable information about the structure of porous samples in general.

2.4.5. Throat pore ratiosSeveral computer simulations and models in the literature have treated porous

.solids as a network of empty chambers pores interconnected by a network of


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Fig. 9. Pore size distribution of various kinds of paper. Symbols: cross, filter 1; filled circle, filter 2.

. smaller channels throats 13,57,58 . Indeed, network models are often applied tothe characterization of solids such as porous rocks and agglomerated particles

13,58 . One inference from network models is the relationship between mercury

Table 5Comparison of mercury permeabilities and tortuosities of two industrial filter papers

Filter paper Modal pore Pore Mercury2 . .sample diameter m tortuosity permeability nm

1 42 2.12 10.512 2 2.19 0.41


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intrusion and extrusion curves and the shape of pores and throats 57 . From apurely geometric point of view, the shapes of mercury intrusion curves could in

.principle be related to pore constrictions throats , while those of mercury extru-sion curves could be related to the shapes of cavities beyond the constrictions .pores . In such a case, the throat pore size ratio R would vary as a function of TP,i

.the void fraction filled with mercury as follows:i

D P f cos I E E I . R 23TP,i / / / / D P f cos E I I Ei i where D, P , f and are the throat or pore size, the applied pressure, the poreshape factor and the mercury contact angle, with the subscripts I and E referring tointrusion and extrusion curves, respectively, and

V a . 24iV c

. with V and V defined previously see Section 2.4.2 . If the shape factors area c .assumed to be the same for throats and pores as is often done implicitly 57 , and

the intrusion and extrusion contact angles are assumed to be constant, R isTPsimply given by the ratio of extrusion to intrusion pressures, P P , at any givenE I

. The variations in P P ratios with have been correlated with pore shapei E I i characteristics of a variety of agglomerated microparticles 57 . On the other hand,

if the intrusion and extrusion contact angles are allowed to vary 1 , the relativecontributions of geometric and surface chemical effects on the shapes of mercury

.porosimetry curves can be assessed upon examination of the resulting D DI E i versus curves.

iFor reference purposes, a characteristic throat pore ratio, R , can be definedTPCas

Dmax,I . R 25TPC Dmax,E

where D and D are the modal throat and pore sizes, respectively.max,I max,E An example of the kind of information that can be extracted from variations in

Pore Throat ratios is shown in Fig. 10. As pointed out by Conner et al. 57 , if mercury intrusion extrusion hysteresis occurred solely due to differences in fixed

.intrusion and extrusion contact angles Section 2.1.1 , plots like Fig. 10 could be .expected assuming f and f to be constant to yield horizontal lines throughoutI E

. Fig. 10 shows that, although many samples pharmaceutical excipients andi.papers being the cases in point exhibit fairly horizontal lines, there are exceptions

to such behaviour, and that these exceptions are not necessarily related to the type of material tested. It was suggested for agglomerates 57 that the ratios are related

to packing characteristics and to the shapes of their corresponding void spaces. Accordingly, as plotted in Fig. 10 D D , in order to work with ratiosi I E


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Fig. 10. Variation in throat pore ratios of selected materials. Symbols: x-marks, pharmaceuticalexcipient 1; open triangles, pharmaceutical excipient 2; open squares, paper 1; crosses, paper 2.

.between 0 and 1 , low ratios would be indicative of voids resembling those betweenrods, and high ratios would resemble more those expected of voids between plates.Similarly, slopes in these plots would suggest changes in packing arrangementsand or pore shapes.

2.4.6. Fractal dimensionThe fractal dimension D of a solid is a parameter that characterizes the degree

of roughness of its surface 59 . Perfectly flat surfaces expose areas that can becalculated as a function of a characteristic dimension, e.g. 4 R2 for a nonporoussphere of radius R. Any surface roughness or porosity would increase the spheressurface area up to an extreme point in which the sphere would be so porous that its


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entire volume could be occupied by pore walls. At this hypothetical point, thesurface area would be proportional to the volume of the sphere, i.e. 4 3 R3. Real

Dsolids expose areas which, following fractal arguments 59 , are proportional to R , with their fractal dimension D ranging between 2 for flat surfaces and about 3 forextremely rough surfaces.

For pore wall surfaces in general it was shown 60,61 that the pore sizedistribution function, dV d r , could be expressed as

dV 2 D . . k r 261d r

where k is a proportionality constant, r is the pore radius, and D is the fractal1dimension. It follows from the Washburn equation that fractal dimensions can bederived from mercury porosimetry data according to

dV D 4. . k P 272d P

where k is another proportionality constant and P is the applied pressure. Taking2logarithms on both sides of the expression yields

dV . . . .log log k D 4 log P 282 /d P

. .Hence, D values can be derived from the slope of log d V d P versus log P plots. .Smooth surfaces D 2 would present slopes close to 2, whereas rougher

.surfaces for which D approaches 3 would present slopes approaching 1.

Inspection of typical mercury porosimetry curves reveals that constant slopes in theappropriate range are often encountered at pressure regions which coincide withthe filling or emptying of pores of distinct sizes. Hence, each step of a cumulativeintrusion or extrusion volume versus pressure plot can yield a unique D value thatcharacterizes the particular range and type of pores being filled or emptied at givenpressure ranges. Once pores of a distinct size become filled or emptied, themercury volume ceases to change significantly and d V d P decreases, therebycomplicating fractal calculations in the transition regions between pore ranges of materials with multimodal pore size distributions.

More recently, Neimark proposed a thermodynamic method for the evaluation .of surface fractal dimensions D from mercury porosimetry or gas sorption datafs

61 . The approach essentially involves expressing the interface area of the adsor- .bed or the intruding fluid S as a fractal function of the radius of curvature of the

.meniscus of the pores affected at equilibrium , i.e.

2 D f s. .S K 29

where K is a proportionality constant. In gas sorption experiments, is given by the Kelvin equation and S is calculated from the Kiselev equation 1,2 . The


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successful application of the thermodynamic approach in the capillary condensa-tion region of gas sorption isotherms led to its incorporation into commercial

. software under the name Neimark-Kiselev NK Method 62 . It is anticipated that the analogous approach, when using the Rootare Prenzlow equation 63 to

calculate the interface area S, will result in its inclusion in commercial mercury porosimetry software as the NRP method 64 .

2.4.7. Sample compressibilityMercury intrusion involves subjecting samples to hydrostatic pressures which are

applied equally in all directions. This means that upon intrusion the walls of allpores penetrated by mercury at any given pressure are uniformly affected bysimilar stresses. Hence, a collapse of the pore walls as they are filled with mercuryis in general unlikely. On the other hand, solid samples could in principle compressand thus generate additional volume for mercury intrusion to take place. Theinfluence of sample compressibility on mercury porosimetry data can be assessed

by means of a solid compressibility factor , defined as the fractional change insolid volume V per unit of pressure,s

1 dV dV s . 30HeV d P d P s

11Most solids exhibit very low compressibilities typically of the order of 102 . m N which vary fairly linearly with pressure 46,65 . This implies that a typical

sample could compress by about 0.5% of its original volume when subjected to anapplied pressure of 60 000 psi. In agreement with these observations, mercuryporosimetry curves often exhibit small yet finite d V d P slopes at pressures higherthan those required to fill all accessible pores in many solids. Hence, the compress-ibility factor can be estimated from the slope of the linear portions of high-pres-sure mercury intrusion or extrusion curves. Similarly, a bulk modulus of elasticity . .1 can be derived as a correlator of the elastic reversible deformation or

.plastic irreversible deformation behavior of solids in general. As illustrated in Table 6, ongoing work on oxygen-plasma treated activated

. .carbons Garca et al., in preparation . has uncovered a not unexpected relation-ship between crystalline XRD parameters such as the average crystallite height Lcand mercury compressibility. Interestingly, Table 6 also shows that there appears tobe an inverse correlation between compressibility and the adjusted extrusion

.contact angle itself a function of surface chemistry 1 that yields overlappingmercury intrusion and extrusion curves. The implications of the latter correlation

.are to be discussed elsewhere Garca et al., in preparation .

3. Concluding remarks

The classical interpretation of mercury porosimetry data has been largely re-stricted to the generation of pore volumes and pore size distributions. This has


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Table 6Characterization of oxygen-plasma treated activated carbon via mercury porosimetry

. . . .Burnoff SA BET SA Hg Lc XRD V Hg Compressibility Extrusion contactpore2 2 9 2 . .% m g m g nm cc g 10 m N angle

0.00 1050 138.3 1.22 1.95 1.31 101.53.12 1022 139.2 1.18 1.49 1.38 102.97.40 1021 138.1 1.23 1.71 1.31 100.39.40 1010 129.6 1.21 1.77 1.23 99.1

19.40 955 125.0 1.40 1.83 1.19 98.9

proven to be an unnecessary handicap imposed on a technique that is ideally suitedto handle the generation of high quality, high throughput characterization data.The usefulness of applying alternative literature models to extract informationfrom mercury porosimetry curves has been demonstrated. Among the most promis-

ing models tested thus far, those generating particle size distributions, permeabili-ties and compressibilities have already led to useful correlations with performanceand other properties of the porous materials investigated. Undoubtedly the incor-poration of such literature models into commercial software will provide answers tomany questions that could not be resolved through the application of the Wash-burn equation alone. It is expected that the development of reliable standardreference materials and the analysis of solids of ideal structures will contributetremendously towards the long overdue maturation of a technique as versatile andpowerful as mercury porosimetry has already proven to be.

Acknowledgements

The author is privileged to have enjoyed very insightful discussions with Profes- .sor S. Lowell founder of Quantachrome Corporation on the nature of mercury

porosimetry and other solid characterization techniques. Thanks are also due to DrM.A. Thomas for useful comments and to Dr J.M.D. Tascon and his research

.group at CSIC-Oviedo Spain for supplying the oxygen-plasma treated carbonsamples.

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new perspectives in mercury porosimetry

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