Peter Klobes, Klaus Meyer

and Ronald G. Munro

960-17

Porosity and Specific

Surface Area

Measurements for

Solid Materials

NIST recommended

practice guide

Special

Publication

960-17

National Institute of

Standards and Technology

Technology Administration

U.S. Department of Commerce
September 2006
i

Porosity and Specific

Surface Area

Measurements for

Solid Materials

Peter Klobes and Klaus Meyer

Bundesanstalt für Materialforschung und -prüfung

(BAM)

Ronald G. Munro

Materials Science and Engineering Laboratory,

National Institute of Standards and Technology

Special Publications 

September 2006

U.S. Department of Commerce

Carlos M. Gutierrez, Secretary

Technology Administration

Robert C. Cresanti , Under Secretary of 

Commerce for Technology

National Institute of Standards and Technology

Dr. William Jeffrey, Director

NIST Recommended Practice Guide

Special Publication 960-17
♦Porosity & Specific Surface Area Measurements

ii

Certain commercial entities, equipment, or materials may be identified in this

document in order to describe an experimental procedure or concept adequately.

Such identification is not intended to imply recommendation or endorsement

by the National Institute of Standards and Technology, nor is it intended to

imply that the entities, materials, or equipment are necessarily the best available for the purpose.

______________________________________

National Institute of Standards and Technology

Special Publication 960-17

Natl. Inst. Stand. Technol.

Spec. Publ. 960-17

89 pages (August 2006)

CODEN: NSPUE2

U.S. GOVERNMENT PRINTING OFFICE

WASHINGTON: 2006

For sale by the Superintendent of Documents

U.S. Government Printing Office

Internet: bookstore.gpo.gov Phone: (202) 512-1800 Fax: (202) 512-2250

Mail: Stop SSOP, Washington, DC 20402-0001
iii

Foreword

NIST SRM 1917 / CRM BAM-P127 (Mercury Porosimetry Standard), consisting of pellets of Al2O3 with a diameter of about 1 mm, was certified in August

2002 as a nanoporous reference material for mercury porosimetry. This reference material was the first jointly produced SRM/CRM of NIST and BAM.

To complement this international reference material, NIST and BAM extended

their collaboration to develop the present Guide addressing the broader metrological aspects of porosity and specific surface area measurements. The Guide

examines the practical issues that need to be considered when conducting a

measurement or analysis by one of the more commonly used porosimetry

techniques, especially the gas adsorption and mercury intrusion methods. The

Guide is written for persons who are not necessarily experts in the field, but

who do have some basic knowledge and familiarity of the issues involved.

Wherever appropriate, references to pertinent national and international

standards and other comprehensive sources of information are noted.

Acknowledgments

The authors gratefully acknowledge the comments, suggestions, and careful

reading of the manuscript provided by our colleagues Dr. Vincent Hackley and

Dr. Ajit Jillavenkatesa (NIST), Dr. Matthias Thommes (Quantachrome

Instruments, Boynton Beach, Florida, USA), and Dr. Jürgen Adolphs

(POROTEC GmbH, Hofheim, Germany).

Disclaimer

Certain commercial equipment, instruments, or materials may be identified in

this paper in order to specify an experimental procedure adequately. Such

identification is not intended to imply recommendation or endorsement by

either the National Institute of Standards and Technology or the Bundesanstalt

für Materialforschung und -prüfung, nor is it intended to imply that the

materials or equipment identified are necessarily the best available for the

purpose.

Foreword ♦
iv

♦Porosity & Specific Surface Area Measurements
v

Table of Contents ♦

Foreword . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .iii

Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .iii

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .vii

List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .ix

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1

1.1 Scope of the Present Guide . . . . . . . . . . . . . . . . . . . . . . . . . . .1

1.2 The Nature of Porous Solids . . . . . . . . . . . . . . . . . . . . . . . . . . .2

1.3 Important Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3

2. Porous and Highly Dispersed Solids . . . . . . . . . . . . . . . . . . . . . . . .4

2.1 General Definitions and Terminology . . . . . . . . . . . . . . . . . . . .4

2.1.1 Physical Adsorption (physisorption) . . . . . . . . . . . . . . . . . . . . .4

2.1.2 Sorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6

2.1.3 Desorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6

2.1.4 Multilayer Adsorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6

2.1.5 Monolayer Capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .7

2.2 Qualitative Description of a Porous Solid . . . . . . . . . . . . . . . . .7

2.2.1 Pore Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .7

2.2.2 Pore Classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .8

2.3 Quantitative Description of Porous Structures . . . . . . . . . . . . .9

2.3.1 Specific Surface Area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .10

2.3.2 Pore Size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .10

2.4 The Fractal Character of Porous Structures . . . . . . . . . . . . . .11

3. Pore Analysis Instruments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .14

4. General Procedures for Powder Sampling and Preparation . . . . .18

4.1 Sampling Alternatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .18

4.1.1 Cone and Quartering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .18

4.1.2 Table Sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .18

4.1.3 Spin Rifling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .18

4.2 Steps in a Sampling Procedure . . . . . . . . . . . . . . . . . . . . . . . .19

4.3 Special Preparation Steps . . . . . . . . . . . . . . . . . . . . . . . . . . . .20

4.3.1 Gas Adsorption Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .20
vi

♦Porosity & Specific Surface Area Measurements

4.3.2 Mercury Intrusion Method . . . . . . . . . . . . . . . . . . . . . . . . . . . .23

5. Specific Surface Area and Pore Analysis by Gas Adsorption . . . .23

5.1 Principle of Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . .24

5.2 Determination of Surface Area . . . . . . . . . . . . . . . . . . . . . . . .26

5.3 Mesoporosity and the Kelvin Equation . . . . . . . . . . . . . . . . . .27

5.4 Microporosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .29

5.4.1 Micropore Volume from the Langmuir Isotherm . . . . . . . . . . .31

5.4.2 Dubinin-Radushkevich Method . . . . . . . . . . . . . . . . . . . . . . . .33

5.4.3 Horvath-Kawazoe and Saito-Foley Methods . . . . . . . . . . . . . .34

5.4.4 Non Local Density Functional Theory (NLDFT) . . . . . . . . . . .38

6. Pore Analysis by Mercury Intrusion Techniques (Mercury

Porosimetry) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .40

6.1 Principle of Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . .40

6.2 Calculation of the Pore Size Distribution . . . . . . . . . . . . . . . . .41

6.3 Calculation of the Specific Surface Area . . . . . . . . . . . . . . . . .45

7. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .46

8. References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .47

Appendix 1: SRM/CRMs Worldwide . . . . . . . . . . . . . . . . . . . . . . . . .54

Appendix 2: Additional Illustrative Case Study Results . . . . . . . . . . .56

A2.1 CRM BAM-P107 (a microporous CRM) . . . . . . . . . . . . . . . . .56

A2.2 CRM BAM-PM-103 (a mesoporous CRM) . . . . . . . . . . . . . . .57

A2.3 CRM BAM-P123 (a macroporous CRM) . . . . . . . . . . . . . . . . .59

A2.4 CRM BAM-PM-122 (Hg porosimetry) . . . . . . . . . . . . . . . . . . .61

Appendix 3: Relevant Measurement Standards . . . . . . . . . . . . . . . .64

A3.1 AFNOR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .64

A3.2 ASTM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .65

A3.3 BSI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .66
vii

Table of Contents ♦

A3.4 CEN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .67

A3.5 DIN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .67

A3.6 ISO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .69

Appendix 4: Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .70

Appendix 5: Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .79

A5.1 Organizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .79

A5.2 Scientific Items . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .79

List of Figures

Figure 1.1:Pores in different aggregates (Gregg 1982): a) spherical

particles; b) plate like particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2

Figure 2.1: Schematic cross section of a porous solid (Rouquerol . . . .

1994) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .8

Figure 2.2: Determination of the fractal dimension from the monolayer

adsorption capacity for different test molecules (Noack) . . . . . . . . . .13

Figure 3.1: Measuring ranges of methods for pore size determination

(after Meyer 1997) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .14

Figure 3.2: Comparison of pore volume size distributions for Clear

Creek sandstone: curve (A) from mercury porosity; curve (B) from

photomicrography (sphere model) (Dullien 1975) . . . . . . . . . . . . . . . .17

Figure 4.1: Recombination table of the cross riffling scheme . . . . . . .20

Figure 4.2: Flowchart indicating steps in a typical sampling

procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .21

Figure 4.3a: Thermogravimetric control of degassing . . . . . . . . . . . .22

Figure 4.3b: Pressure control of degassing . . . . . . . . . . . . . . . . . . . .23
viii

Figure 5.1: Classification of isotherm types (Sing 1985) . . . . . . . . . .25

Figure 5.2: Linear isotherm plot of Ar on zeolite at 87.3 K . . . . . . . . .32

Figure 5.3: Logarithmic isotherm plot of Ar on zeolite at 87.3 K . . . . .32

Figure 5.4: Dubinin-Radushkevich DR-plot of an adsorption isotherm of

N2 at 77 K on activated carbon (Dubinin 1947) . . . . . . . . . . . . . . . . .34

Figure 6.1: Relative deviation in the calculated pore diameter dp as a

function of the mercury contact angle (reference value 140°) . . . . . .42

Figure 6.2: Schematic representation of the (a) advancing angle and

(b) static angle method for determination of the contact angle (Groen

2002) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .44

Figure 6.3: Intrusion/extrusion curve (volume vs. pressure curve) of a

porous glass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .45

Figure 6.4: Characteristic hysteresis loops for cylindrical holes and for

pores formed within aggregates (Webb 1997) . . . . . . . . . . . . . . . . . .46

Figure A2.1.1: The nitrogen (N2) Isotherm for CRM BAM-P107 . . . .57

Figure A2.2.1: Adsorption isotherm of nitrogen at 77 K on BAM-PM-103

(alumina type 60) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .58

Figure A2.2.2: Pore size distribution of BAM-PM-103 (alumina

type 60) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .58

Figure A2.2.3: Interlaboratory study: Specific pore volume (adsorption)

of CRM BAM-PM-103 (alumina type 60), the wide-dashed line corresponds to the mean of means (certified value) with standard deviation

(narrow-dashed lines); laboratory mean values with standard deviation,

individual values (single points), and standard deviation of means are

also given; laboratory number 5 is an outlier. . . . . . . . . . . . . . . . . . . .59

♦Porosity & Specific Surface Area Measurements
ix

Figure A2.3.1: Reference curve with simultaneous prediction bands at

significance level 0.95 for the material CRM BAM-P123 . . . . . . . . . .60

Figure A2.4.1: Certified mercury intrusion curve for CRM BAM-PM-122

with uncertainty intervals (logarithmic pressure axis) . . . . . . . . . . . . .62

Figure A.2.4.2: Certified mercury intrusion curve for CRM BAM-PM-122

with uncertainty intervals in the range between 8 MPa and 15 MPa

(linear pressure axis) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .63

Figure A2.4.3: Sharp pore size distribution for CRM BAM-PM-122 (not

certified) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .63

List of Tables

Table 3.1: Methods for the determination of total porosity (after Mikhail

and Robens 1983) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .15

Table 3.2: Methods for the determination of open porosity (Mikhail and

Robens 1983) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .16

Table 5.1: Recommended values for the calculation of liquid volumes

(ISO 15901-3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .31

Table 5.2: Adsorbent Parameters (according to DIN 66135-4) . . . . . .37

Table 5.3: Adsorptive Parameters (according to DIN 66135-4) . . . . .37

Table 5.4: Relation between the micropore diameter and the relative

pressure for nitrogen at 77.35 K in a carbon slit pore according to the

Horvath-Kawazoe approach (DIN 66135-4) using the materials constants in Tables. 5.2 and 5.3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .38

Table 5.5: Relation between the micropore diameter and the relative

pressure for argon at 87.27 K in a cylindrical zeolite pore according to

the Saito-Foley approach (DIN 66135-4) using the materials constants

in Tables. 5.2 and 5.3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .38

Table of Contents ♦
x

Table 6.1: Average contact angles of some materials, as measured with

the advancing angle and static angle method (Groen 2002). (n.d. = not

determined) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .43

Table A1.1: Producers / distributors of porous reference materials . .54

Table A1.2: Certified reference materials for porosity properties . . . .55

Table A2.1.1: Certified values for the microporous CRM BAM-P107 for

nitrogen adsorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .56

Table A2.2.1: Mesoporous CRM BAM-PM-103 certified values for the

gas adsorption method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .57

Table A2.3.1: Certified single values of BAM-P123 . . . . . . . . . . . . . .61

Table A2.4.1: Selected properties of CRM BAM-PM-122 for mercury

porosimetry (DIN 66133) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .62

Table A3.1: Selected national and international institutes that have

issued test methods for the characterization of porous solids . . . . . .64

♦Porosity & Specific Surface Area Measurements
1

1. Introduction

Properties such as specific surface area, pore size, and pore size distribution

are important material characteristics in many processing applications,

including those for catalysts, sintered materials, pharmaceutical products, and

chromatographic carriers. Improper pore and surface area analyses contribute

to poor product quality, high rejection rates, and economic losses.

Consequently, materials suppliers and manufacturers increasingly are using

measurements of porosity and specific surface area as an integral part of their

quality assurance programs (Gorman 2001). 

1.1 Scope of the Present Guide

The motivation for this guide stems from a joint program between the

Bundesanstalt für Materialforschung und -prüfung (BAM, Germany) and the

National Institute of Standards and Technology (NIST, United States) to develop and provide jointly certified reference materials (NIST SRM 1917 / CRM

BAM-P127). The present effort addresses issues and concerns pertaining to

the measurement of specific surface area and porosity for porous or highly dispersed (particulate) materials, such as powders and compacts, which are characterized by a large surface to volume ratio. 

In recent years, special interest has been focused on the ability to make meaningful comparisons among test results obtained by different individuals or

groups and at different times and locations (Malghan et al. 1995). The need

for consistency in these measured values has created a particular demand for

the calibration of the equipment used for testing and analysis. Concurrently,

the need for consistency also has created a need to understand and recognize

the limits of applicability of alternative measurement devices and methods

(see, for example, early round robin results, Munro et al. 1995). The use of

certified reference materials has evolved as an essential aspect of the approach

used to address both of these major concerns.

The present guide is by no means an exhaustive and comprehensive text on

pore analysis. Rather, this volume attempts to convey the practical issues that

need to be considered when conducting a measurement or analysis by one of

the more commonly used porosimetry techniques, especially the gas adsorption and mercury intrusion methods. The guide is written for persons who are

not necessarily experts in the field, but who do have some basic knowledge

and familiarity of the issues involved. Wherever appropriate, references to

pertinent national and international standards and other comprehensive sources

of information have been included. Much of the basis for the text presented

Introduction ♦
2

here stems from studies conducted at BAM, NIST, and other major facilities

and from the broad experience gained over many years of participation in

round robin tests and standards development efforts worldwide. 

1.2 The Nature of Porous Solids

Particles in finely divided powders contain many unsaturated bonds at the

surfaces of the particles. Due to this fact, some of these particles may bond

together weakly to form a loosely coherent secondary particle, generally called

an aggregate (Everett 1972; Everett 1986; Haber et al. 1991; Hackley et al.

2001). If the particles are joined more rigidly, as may occur under the influence of mechanical pressure and elevated temperature, these assemblages of

particles are called agglomerates. Some of the possible cavities that may form

in aggregates of spherical and plate-like particles are illustrated schematically

in Figure 1.1.

Porous and finely divided solids formed in this manner are distinguished by

large surface to volume ratios. These materials often exhibit enhanced

chemical reactivity, higher adsorption capacity, and lower sintering temperature. In many practical applications exploiting this large ratio, the size of the

accessible surface is as important as the chemical composition. For example,

certain natural minerals, such as zeolites, have well defined characteristic

cavities with pore sizes on the order of atomic dimensions. Their large specific surface areas allow them to possess not only a higher adsorption capacity,

but also an enhanced functionality with respect to the separation of gases.

♦Porosity & Specific Surface Area Measurements

Figure 1.1. Pores in different aggregates (Gregg 1982): a) spherical

particles; b) plate like particles.
3

These materials are especially important in areas such as the chemical

industry, materials engineering, ceramics production, and environmental

engineering. Examples of their applications include the development and

production of:

- catalysts - sintered materials

- adsorbents - building materials

- pigments - ceramics

- pharmaceutical products - filters

- chromatographic carriers

Significantly, every application appears to have an optimum specific pore size

distribution.

A further consideration for porous materials is the fractal nature of the

microstructure of the material. This issue is discussed in Chapter 2, Section 4.

In essence, measurements of surface features depend on the size of the

"gauge" that is used to probe the surface. Consequently, results derived from

different studies will be comparable only if the same gauge is used. For comparability, therefore, it is necessary to adhere strictly to a regulated procedure

which may include the standardization of instruments, detailed measurement

procedures (including such considerations as sample pretreatment, temperature

and duration of heating, and degassing), and the use of certified reference

materials for the calibration of the instruments.

1.3 Important Characteristics

The size and morphological structure of a grain can have a significant influence on the processing of raw materials and on the effectiveness of their use.

Such is the case, for example, for materials used as adsorbents, membranes,

and catalysts. Consequently, numerous measurement procedures have been

developed to characterize powders and porous solids with respect to these features (Sing et al. 1985; Rouquerol et al. 1994). Among the most commonly

determined characteristics are the particle size distribution, the specific surface

area (area per unit mass), the specific pore volume (pore volume per unit

mass), and the pore size distribution. Additionally, several types of density

measurements have been used as auxiliary information, especially for powders

(for example, tap density and envelope density).

Rather generally, these characteristics are derived quantities; i.e., they are

calculated from the measured results of other quantities. Often the relevant

measurements involve assumptions regarding both the quantities that are being

measured and the validity of the measurement procedures for the given

Introduction ♦
4

material. As such, the end results often are strongly dependent on the selected

measurement and analysis methods. This issue is discussed in Chapter 3. We

make the observation here to emphasize that the values derived from these

methods must be considered comparative or relative values. Indeed, quantities determined in this manner are known formally as procedural quantities

(Munro 2003). Such quantities do not have the theoretical basis to allow

their values to be calculated independently of the associated measurement

procedure. 

2. Porous and Highly Dispersed Solids

The use of porous materials has evolved over a long period of time and in a

wide variety of industries. In each application area, there have been efforts to

characterize and describe the materials in ways that are meaningful to that

area. As a result, different terminologies have evolved from different areas to

describe the same features. Likewise, different measurement procedures have

evolved to quantify what should be the same characteristic. A comprehensive

discussion of the appropriate nomenclature is given in earlier volumes of the

NIST Recommended Practice Guides (Hackley et al. 2001; see also

Jillavenkatesa et al. 2001). This issue also has been addressed by the

International Union of Pure and Applied Chemistry (IUPAC). In the interest

of clarity and consistency, certain selected terms important to porous and

dispersed materials are reviewed in this chapter, and a succinct glossary of

terms is given in Appendix 4.

2.1 General Definitions and Terminology

Several definitions and terms used for the characterization of porous or dispersed solids warrant discussion because of their special relevance or their

potential for confusion. The primary focus here is on sorption and intrusion

properties.

2.1.1 Physical Adsorption (physisorption) 

The physical adsorption of gases is an important part of several measurement

techniques used for determining specific properties of porous solids.

Adsorption, in general, occurs whenever a substance (the adsorptive, such as

an adsorbable gas) accumulates at an interface or in an interfacial region. If

the adsorption process involves only the weaker intermolecular forces that

result from van der Waals interactions or hydrogen bonding and if there is no

significant change in the electronic orbital states of the interacting species,

♦Porosity & Specific Surface Area Measurements
5

then the process is called physical adsorption. The term physical adsorption

is often contracted to physisorption. At constant temperature, the relation

between the amount of substance adsorbed and the equilibrium pressure of

the gas is known as the adsorption isotherm.

To form a clear picture of physical adsorption, it is convenient to regard the

interfacial layer as comprising two regions: the surface layer of the adsorbent

(often called the adsorbent surface) and the space adjacent to the surface (the

adsorption space) in which enrichment of the adsorptive can occur. The

adsorptive in the adsorption space is considered to be in a fluid phase. To

distinguish the adsorptive in the fluid phase from the adsorptive material

already adsorbed onto the surface, the material adsorbed onto the surface

is known as the adsorbate. 

It is sometimes useful to make a further distinction between external and internal regions of the adsorbent surface, especially when discussing porous adsorbents of high surface area. The envelope surrounding the discrete particles or

agglomerates of the adsorbent usually is regarded as the external surface.

However, it is difficult to define such surfaces precisely because the surfaces

rarely are smooth on an atomic scale. A suggested alternative convention is to

assume that the external surface includes all the prominences and any surface

cracks that are wider than they are deep. With that convention, the internal

surface of the adsorbent then consists of the cracks, pores, and cavities that are

deeper than they are wide and that are accessible to the adsorptive. The latter

restriction is a procedural condition that results in a demarcation between

internal and external surfaces that depends, in practice, on the method of

assessment and the nature of the pore size distribution. Pores and cracks

smaller in dimension than the absorptive may be, in effect, inaccessible. 

Porous and Highly Dispersed Solids ♦

Physisorption should not be confused with chemical

adsorption (chemisorption). Chemical adsorption is

an adsorption process in which significant chemical

bonds are formed at the interface or in the interfacial

region. These bonds change the electronic orbital

states of the interacting species and may affect

measurable properties. A significant heat of adsorption also may result from chemical adsorption.
6

2.1.2 Sorption

If molecules of the adsorptive penetrate the surface layer and enter into the

structure of the bulk solid, a different term, absorption, is used. However, in

some circumstances, it may be difficult, impossible, or even irrelevant to

distinguish between adsorption and absorption. It then becomes convenient to

use a more general term, sorption, that embraces both phenomena. In this

case, the generalized terms sorbent, sorbate, and sorptive also may be used.

2.1.3 Desorption

The counterpart of adsorption is called desorption. Desorption denotes the

process in which the amount of adsorbed material decreases. The terms

adsorption and desorption often are used adjectivally to indicate the measurement direction from which experimentally determined adsorption values have

been approached, e. g. the adsorption curve (or point) may be distinguished

from the desorption curve (or point). If the adsorption and desorption curves

do not coincide, then the process is said to exhibit adsorption hysteresis.

2.1.4 Multilayer Adsorption

When an adsorbate begins to form on a nascent surface, it is possible for each

molecule to adsorb directly onto the surface. More precisely, each molecule

occupies a position of closest approach to the adsorbent. In this state, the

adsorbate is said to form a monolayer. If adsorption persists in excess of

monolayer adsorption, we may visualize additional layers of adsorbate forming on top of the initial monolayer, each new layer being more remote from

the substrate than the preceding layer. Adsorption that progresses beyond

monolayer adsorption is called multilayer adsorption. For a given state of

multilayer adsorption, there may be more pore space available than is occupied by the adsorbate. If gas phase molecules from the absorptive enter and

occupy this residual pore space, a process called micropore filling is said to

♦Porosity & Specific Surface Area Measurements

The notion of external surface may also be used to

define a measure of surface roughness. If the nominal external envelope of the solid has a well defined

geometric shape, then the ratio of the area of the

external adsorbent surface to the area of the geometric shape may be taken as the roughness factor for

that surface.
7

occur. If, instead, a condensate separated from the gas phase by menisci forms

in the residual pore space, the special term capillary condensation is used. The

presence of capillary condensation often contributes to the

occurrence of adsorption hysteresis.

2.1.5 Monolayer Capacity

The amount of adsorbate needed to cover the surface of the adsorbent with a

monolayer of molecules is called the monolayer capacity and is denoted here

by the symbol nm. (We follow the convention here that quantities relating to a

monolayer of adsorbate are denoted by the subscript m.) The total surface

coverage then can be described with respect to a monolayer of adsorbate. For

both monolayer and multilayer adsorption, the surface coverage (θ) is defined

as the ratio of the amount of adsorbed substance to the monolayer capacity.

2.2 Qualitative Description of a Porous Solid

Any solid material that contains cavities, channels, or interstices may be

regarded as porous, but some care must be exercised in the choice of

terminology to avoid ambiguity. To aid in the classification of porous

materials, distinctions may be made with respect to the types of pores in the

material and with respect to the nature of the constitution of the material.

2.2.1 Pore Types

With reference to Figure 2.1, we can classify pores according to how accessible they are to an external fluid. In this context, one category (closed pores)

consists of pores that are inaccessible to an external fluid and totally isolated

from their neighbors, as in region (a). Closed pores influence macroscopic

properties such as bulk density, elasticity, mechanical strength, and thermal

conductivity, but they are inactive in processes such as fluid flow and adsorption of gases. On the other hand, pores that have a navigable channel of

communication with the external surface of the body (like b, c, d, e, and f)

are described as open pores. Open pores are further classified into "through

pores" and "blind pores." Through pores have an open channel that begins at

one location on the surface, extends into the particle, and re-emerges on the

surface at a different location (like the pore channels c-e-c and c-e-d). Blind

pores (also called dead-end or saccate pores) are open to the surface only at

one end (like b and f). A special distinction is made for blind pores like those

in region of g of Figure 2.1. While these small surface irregularities are technically blind pores, it is often more useful and convenient to consider them

separately as part of a distinct attribute, called surface roughness. A simple

Porous and Highly Dispersed Solids ♦
8

convention often used is to assume that surface irregularities are counted as

pores only if they are deeper than they are wide.

Pores may be classified further according to their shape. Common shape

terms include cylindrical (open c or blind f), ink-bottle (b), funnel (d), and slit

shapes. 

2.2.2 Pore Classes

Some materials are consolidated; i.e., distinguishable microscopic particles are

bound together in some manner to form a relatively rigid, macroscopic body

whose dimensions exceed those of the pores by many orders of magnitude.

The constituent particles themselves may be nonporous (e.g. sand), but the

macroscopic body will still have a porosity consisting of the surrounding

network of interparticle voids. In this case, the properties of the macroscopic

body can be expected to be dependent primarily on the size, shape, and

manner of packing of the constituent particles.

Other distinctions arise if the constituent particles themselves have a significant porosity. In such cases (e.g. spray-dried catalysts), it may then be necessary to distinguish between internal (or intraparticle) voids, and interparticle

voids. In general, internal pores will be smaller than the voids between

particles, both in size and in total volume; nevertheless, they often will provide

the dominant contribution to the surface area of the solid. Which void type

dominates depends on the volume/surface area ratio.

♦Porosity & Specific Surface Area Measurements

Figure 2.1. Schematic cross section of a porous solid (Rouquerol 1994).
9

There are three special classes of pores that are expressly distinguished (Meyer

1994). The first class involves pores that are an inherent feature of a

particular crystalline structure, such as occurs, for example, in zeolites and

some clay minerals. Being inherent to the crystal structure, these intracrystalline pores are generally of molecular dimensions and form highly

regular networks.

The second class of porous materials is distinguished by its sequential method

of processing. The material is formed in two nominal steps, first by loose

packing (i.e. aggregation) and then by consolidation (i.e. agglomeration) of the

small particles into a macroscopic solid. Some inorganic gels and ceramics

are formed in this manner. The final details of the resulting structure depend

on the original arrangement of the primary particles and on their sizes and

orientations.

The third distinguished class may be described as "subtractive" because certain

elements of an original structure are selectively removed to create pores.

Examples include porous metal oxides formed by the thermal decomposition

of hydroxides, carbonates, nitrates, and oxalates and porous glasses formed by

the chemical etching of multiphase solids. Many porous organic polymer

membranes are formed in this way. 

2.3 Quantitative Descriptions of Pore Structures

Porosity (more precisely, the volume fraction of porosity) may be defined

operationally as the fraction, ε, of the apparent specific volume of the sample

that is attributed to the pores: ε = Vp/V. The value of this fraction clearly

depends on the method used to determine the apparent specific volume, V,

which excludes interparticle voids (geometrical determination, fluid displacement), and on the method used to assess the specific pore volume Vp (adsorption and capillary condensation, fluid displacement, ultrasonics etc...). Some

methods only have access to open pores (e.g., those methods using a fluid),

whereas other methods may access both open and closed pores (e.g., methods

using adsorption or scattering of electromagnetic radiation). Moreover, for a

given method, the value determined experimentally depends on the size of the

molecular probe (fluid displacement, adsorption) or of the gauge (stereology).

Thus, a measured value of porosity is a reflection of both the physical state of

the material and the experimental method used for its determination. To

distinguish these three cases, it should be noted whether the specific pore

volume, Vp, is due to the open pores (leading to the "open porosity"), the

closed pores (leading to the "closed porosity"), or both types of pores together

(leading to the "total porosity").

Porous and Highly Dispersed Solids ♦
10

2.3.1 Specific Surface Area

The "specific surface area" (A) is defined as the area of solid surface per unit

mass of material. In practice, what is actually determined is the accessible (or

detectable) area of solid surface per unit mass of material. This distinction is

important because the value determined in a measurement is dependent on the

method, the experimental conditions employed, and the size of the probe

(e. g., the molecular size of the adsorbate or the wavelength of a radiation

probe). More significantly, the specific surface area is usually deduced from

measured quantities that must be interpreted using simplified models of the

measurement process. Consequently, the recorded value depends inherently

on the validity of the assumptions used in the model.

2.3.2 Pore Size

"Pore size," like specific surface area, it is not amenable to a precise definition. The problems already mentioned for the specific surface area are complicated further by the fact that the pore shape is usually highly irregular and

variable, leading to a variety of definitions of "the size." Moreover, pore

systems usually consist of interconnected networks. As a result, the detected

pore volume may depend on the sequence in which pores are encountered by

the probe substance. Consequently, quantitative descriptions of pore structure

usually are based on model systems.

To make the analysis tractable, systems with idealized pore shapes and sizes

often are used. In practice, the shapes of pores, whether known or assumed,

are preferably described in terms of a small number of shapes: cylinders

(which may be appropriate for activated oxides like alumina or magnesia),

prisms (some fibrous zeolites), cavities and windows (other zeolites), slits

(possible in clays and activated carbons), or spheres. (We note, however, that

when spherical shapes are considered, the pores are more likely to be the voids

left between solid spheres in contact with each other, as happens with gels

such as silica gel and zirconia gel during gelation). 

The limiting feature of a pore, for most applications, is the size of the smallest

dimension of the assumed pore shape. In the absence of greater precision, the

smallest dimension is referred to as the width of the pore, i.e., the width of a

slit-shaped pore and the diameter of a cylindrical pore. (To avoid a misleading

change in scale when comparing cylindrical and slit-shaped pores, the

convention is to use the diameter of a cylindrical pore, rather than its radius,

as its "pore width".) To facilitate discussions of pore sizes, the following

conventions are recommended by IUPAC (McNaught 1997):

♦Porosity & Specific Surface Area Measurements
11

• Micropores: widths < 2 nm

• Mesopores: 2 nm < widths < 50 nm

• Macropores: widths > 50 nm

2.4 The Fractal Character of Porous Structures

The dependence of microstructural characteristics on the characteristic length

scale of the measurement method hinders comparisons among results obtained

by different methods. With rough surfaces, for example, the measured specific surface area increases as the probe is changed from hydrocarbons to small

nitrogen molecules because there is an increasingly better covering of the

surface as the effective radius of the probe molecule decreases.

In the mid 1980s, the insight that irregular physical structures are described

only very inadequately by Euclidean geometry led to the introduction of

Mandelbrot's concept of fractal geometry (Mandelbrot 1975). The important

characteristic of the fractal model in porous materials is that it allows for a

characteristic quantity that is independent of length scales. 

The basic property of fractal objects is their so-called self-similarity (also

called dilation invariance or scale invariance). Regardless of the scale of

observation, one sees similar structures. Mathematically, scale invariance is a

special symmetry property. The mathematical description of self-similarity

leads to a non-integer, i.e. "fractal," dimension that is characteristic of the

system.

Applying that concept to specific surface area (A), for example, A can be

expressed as a limit derived from the use of probe molecules of different sizes.

A = lim nσD/2

n → ∞

σ → 0

where n = number of molecules in the monolayer, σ = area occupied by the

molecules, and the exponent D is the fractal dimension.

The fractal dimension for an ideally flat surface is 2, and for the three-dimensional body (volume structure), it is 3. As the roughness of a flat surface

increases, the portion of the surface extending into the 3rd dimension acquires

a greater significance. Hence, one expects the fractal dimension of a porous

material to be between 2 and 3. Mathematical fractals are defined constructively, therefore, self-similarity is found at all length scales. Physical fractal

objects possess only limited regions of scale invariance; the interval is bounded

Porous and Highly Dispersed Solids ♦
12

from below by the order of magnitude of atoms, and from above by the

respective macroscopic size of the object.

In recent years, numerous articles have been devoted to multilayer absorption

(Pfeifer, Wu et al. 1989) and to finding correlations between the fractal

dimension D and the characteristic properties of irregular or rugged solid

adsorbents (e.g. catalysts), and also of macromolecules and colloid aggregates.

Furthermore, there have been fractal descriptions of mass transport (diffusion,

Knudsen flow) through the micro- and mesopore system of porous solids as

well as of surface diffusion. Other work has centered on the fractal treatment

of chaotic reactions, percolation, and aggregation (Pfeifer, Obert et al. 1989).

Experimentally, the fractal dimension D can be determined from the slope of a

number of doubly logarithmic plots (log-log plots), of which we quote some

examples:

a) Monolayer adsorption of molecules of different size;

number (n) of molecules in the monolayer versus the area (σ) occupied

σ log n = (D/2)log σ + C

b) Determination of pores by N2 multilayer adsorption/desorption;

pore volume distribution (dV/dr) versus pore radius (r)

log (-dV/dr) = (2 - D) log r + C

c) Mercury porosimetry

First derivative of the pore volume with respect to the Hg intrusion

pressure (dV/dPHHg) versus PHg

log (dV/PHg) = (D - 4) log PHg + C

d) Plot of the BET specific surface area (SBET) of sieve fractions versus the 

mean particle diameter (d)

log SBET = (D - 3) log d + C

(The quantities C are additive constants and do not influence the slope).

A quasi-model-free thermodynamic method for calculating fractal dimensions

directly from experimental adsorption and mercury intrusion data has been

proposed by (Neimark 1990).

Fractal dimensions can also be determined by diffraction and scattering

experiments (e.g. small angle x-ray scattering). In addition, image processing

is applied in characterizing rocks and building materials.

♦Porosity & Specific Surface Area Measurements
13

The common feature of all these methods is that the intervals of self-similarity

are given by the linear sections of the log-log plots; sometimes, several linear

regions with different slopes may occur, indicating structural changes in the

transition from one interval to the next. Figure 2.2 shows the experimental

determination of the fractal dimension of a silica gel surface. The minimum

condition for a well-defined fractal dimension to exist is (Pfeifer 1984)

rmax/rmin > 2 1/D,

with rmax and rmin being, respectively, the upper and lower bounds of the

fractal region on the length scale. 

At the time of the pioneering articles on the application of fractal concepts to

problems of microstructure, it was hoped and assumed that the fractal

dimension would be a universal characteristic parameter for porous solids.

However, these great expectations have not been completely fulfilled, principally for two reasons. First, the accuracy of the experimental data used for

calculating the fractal dimension is often very limited. Therefore, in many

cases one cannot rely upon the fractal dimension as being a universal characteristic quantity for the texture of solids. Second, some scientists have

expressed doubts concerning the applicability of the fractal concept in today's

solid state research, at least in the broad way it was used in the past. Conner

and Bennett (Conner 1993), using the Menger sponge as a model fractal,

proposed that real catalysts (as used in industry) have pore systems without

Porous and Highly Dispersed Solids ♦

Figure 2.2. Determination of the fractal dimension frm the monolayer

adsorption capacity for different test molecules (Noack).
14

any fractal character, because the condition of self-similarity is not fulfilled

and probably cannot be realised for practical purposes. According to Conner

and Bennett, one can construct model systems without self-similarity and

therefore without fractal properties, the log-log plots of which yield exact

straight lines. A conclusion from this work would be that some former results

in the fractal dimension of pore systems should be examined critically.

Surfaces of real solids, on the other hand, can possess a fractal character, and

for those materials, fractal concepts can be very useful for understanding the

phenomena of surface roughness.

Even if the fractal concept is not relevant for all parameters of the void structure of solids (a matter of the description), we are still confronted with the

problems of scale and method dependence and the impossibility of obtaining

'absolute' results.

3. Pore Analysis Instruments 

Several methods have been developed for determining the surface area and the

pore size distribution in porous systems. The operations of these different

methods generally are based on different physical principles. It should be

expected, therefore, that they effectively represent probes of different sizes

and, hence, that the pore size ranges in which they are most reliable are necessarily different. Figure 3.1 compares the ranges of validity of a selection of

♦Porosity & Specific Surface Area Measurements

Figure 3.1. Measuring ranges of methods for pore size determination

(after Meyer 1997)
15

methods commonly used for pore characterization, and Tables 3.1 and 3.2

describe some of the advantages and disadvantages of these methods.

Table 3.1. Methods for the determination of total porosity (after Mikhail and

Robens 1983)

The practical significance of using methods with different detection limits is

illustrated quite graphically in Figure 3.2. The figure shows results from

Dullien and Dhawan who investigated the pore volume distribution of a sandstone sample. They used both mercury porosimetry and optical microscope

image analysis on a two-dimensional slice of the same sample. Apparently,

the narrow distribution resulting from mercury porosimetry was determined

mainly by the diameter of the pore opening, obviously quite narrow in this

sample. In contrast, the image analysis examined pores that had been cut open

when the specimen was sectioned for analysis. Apparently, the wide and flat

distribution function resulted from the size of the cut single pores.

Pore Analysis Instruments ♦

Procedure Range of pore

sizes Assumption Advantages Disadvantages

Calculation of

apparent

density

Small angle Xray scattering

Small angle

neutron

scattering

Quantitative

image analysis

Image analysis

system

Total

0.5 nm ≤ d ≤

100 nm

0.5 nm ≤ d ≤

100 nm

d ≥ 20 nm

d ≥ 20 nm

Spherical pores

Easy calculation

from

commonly known

property

Short time

method

Short time

method

Large pores can

be measured,

optical checking

is possible, also

microporosity can

be measured by

analysing TEM

photography

Size measurements of pores of

irregular shapes,

application on

ceramographic

sections,

figures,

microphotos,

view graphs

Special facilities

Special facilities,

limited to nuclear

centers

Limitation to

large pores, significant errors in

coarse grain

materials, area

information, for

spatial information large mathematical effort

Subjective measurement, area

information
16

Evidently, some care must be exercised when selecting a method. The

following observations have been offered by IUPAC (Rouquerol et al. 1994).

• The complexity of the porous texture of materials is such that even on theoretical grounds the concepts which can be used to describe the texture usually

entail the introduction of simplifying assumptions.

• No experimental method provides the absolute value of parameters such as

porosity, surface area, pore size, and surface roughness. Each gives a

characteristic value that depends on the principles involved and the nature of

the probe used (atom or molecule, radiation wavelength...). One cannot speak

of the surface area of an adsorbent but, instead, of its "BET-nitrogen surface

area", "equivalent BET-nitrogen surface area", "modified HJ-calorimetric surface area", "cumulative water thermoporometry surface area," etc.

• The selection of a method of characterization must start from the material

and from its intended use.

♦Porosity & Specific Surface Area Measurements

Procedure Range of pore

sizes Assumption Advantages Disadvantages

Xylene and water

impregnation

Liquid metal

impregnation

Air/He

penetration

N2 adsorption

Mercury intrusion

d > 100 nm

A few

nanometers

depending upon

gas applied

d < 50 nm

4 nm < d ≤

60 μm

Cylindrical pores

Simple method,

short time

True pore

network

Very short time

method, about 

1 min, good

reproducibility,

no influence on

sample

Correlation with

BET surface area

Much in use,

comparable data,

extended range

of pore sizes

No pore size

distribution,

specimens with

small diameter

No measurable

pore size distribution, carbon

burn-off

No pore size

distribution

Complicated

evaluation

program

Danger of

breaking pore

walls, large pores

are filled at

atmospheric

pressure

Table 3.2. Methods for the determination of open porosity (Mikhail and

Robens 1983)
17

• The method chosen must indeed assess a parameter related as directly as

possible to phenomena involved in the application of the porous material. In

this respect, it may often be advisable to select a method involving physical

phenomena similar or close to those involved during the practical application

(i.e. adsorption or capillary condensation method if the porous substance is to

be used as a desiccant, or a freezing point depression method if one is interested in the frost resistance of a construction material...) so that parameters

determined are appropriate.

• Rather than to "check the validity" of distinct methods, certified reference

porous materials are needed to establish how these methods differ and, of

course, to calibrate any individual equipment or technique.

• As a consequence, one must not look for a "perfect agreement" between

parameters provided by different methods. Such an agreement, when it occurs,

is not necessarily a proof of the validity of the derived quantities. Instead, one

must be aware of the specific, limited, and complementary significance of the

information delivered by each method of characterization of a porous solid.

Pore Analysis Instruments ♦

Figure 3.2. Comparison of pore volume size distributions for Clear Creek

sandstone: curve (A) form mercury porosity; curve (B) from

photomicrography (sphere model) (Dullien 1975).
18

4. General Procedures for Powder Sampling and Preparation

4.1 Sampling Alternatives

The goal of powder sampling is to collect a small amount of powder from the

bulk quantity such that this smaller fraction is representative of the characteristics of the entire bulk. Statistical sampling techniques (ISO 2859) have been

developed to help ensure that a representative sampling is obtained. Statistical

sampling requires the inspection of only a fraction of the bulk (thereby greatly

reducing the amount of time and effort spent in sampling) and provides the

advantage of calculating the level of certainty of detecting mistakes, upon

applying a particular level of inspection.

After the primary samples have been drawn from the bulk, they need to be further sub-divided to provide a specimen quantity appropriate to use in the laboratory. The most important techniques for dividing bulk samples into

laboratory samples are described in the following subsections (Allen 1981;

Jillavenkatesa 2001).

4.1.1 Cone and Quartering

This method requires that the powder be poured into a cone shaped heap on a

flat surface. The heap is flattened by a spatula and divided into 4 identical

volumes. One of these smaller volumes is taken and the procedure repeated to

produce 1/16 of the original material. This process is repeated until the sample size desired for the measurement is obtained.

4.1.2 Table Sampling

Table sampling involves powder flow along an inclined table with obstacles to

divert and control the powder flow. The sample is introduced through a feeder

at the top of an inclined plane from where the powder travels downward on

the table plane. Guides in the shape of triangular prisms are placed in rows

perpendicular to the direction of powder flow. The interaction of the powder

with these guides causes the stream of powder to be divided into different

fractions.

4.1.3 Spin Riffling (Van der Veen 1993)

A spinning (rotating) riffler is a rotary sample divider and causes division of

the sample powders into a series of bottles mounted on a circular holder. The

sample is introduced into the mass flow hopper while ensuring that the powder

does not form a heap. The material flows from the hopper to a vibratory

♦Porosity & Specific Surface Area Measurements
19

chute. The powder travels along the chute to the receivers (glass containers),

which are rotating in a circular motion at a constant speed. This causes the

containers to be filled uniformly. One can say that a spin riffler has a high

reliability in comparison with other techniques. Because of the low relative

standard deviation compared to other techniques, this cross riffling method is

often applied to the production of reference materials. 

4.2 Steps in a Sampling Procedure

In general, a sampling procedure involves the selection of techniques that

work best for the given powder system and must consider the nature of the

powder and whether granules or larger solid pieces will be used to determine

the specific surface area and the pore characteristics. The sampling procedure,

General Procedures ♦

Reference materials (RMs) of any type must be homogeneous with respect to the certified properties such that the

between-sample variation in the properties of the RM artifacts is as low as possible. The minimization of the

between-sample variation can be carried out by riffling

using a rotating sample divider (8 ports) applying the

cross-riffling scheme (see below) according to Van der

Veen (1993). Referring to Figure 4.1, the first step of this

procedure is a riffling step into 8 subsamples. After this

step, each of these subsamples is riffled again, yielding 64

small subsamples. Using a recombination of these 64

samples, 8 new larger samples are formed. Then, 8 additional rifflings are performed to yield a new set of 64 small

subsamples. In this manner, it may be expected that most

of the inhomogeneity that may have remained after the first

riffling (samples 1 trough 8) will have been removed. As

was shown by Van der Veen (1993), the advantage of this

riffling procedure is that even if the first 8 samples are not

very representative for the batch, the samples A to H will

be. To investigate the resulting homogeneity of the material, it is necessary to separate measurement uncertainty

from heterogeneity of the samples. If multiple measurements on the same specimen are possible (in the case of

nondestructive investigation methods), this may be

achieved by averaging the results. Averaging allows the

effect of measurement uncertainty to be minimized.
20

therefore, must take into consideration the powder characteristics, the size of

the bulk material, and the number of laboratory specimens to be prepared.

Homogeneity testing is a critical component of any sampling procedure

because it identifies inconsistencies between sub-samples and thus helps to

ensure that the sub-samples are representative of the bulk sample. Figure 4.2

shows a general flowchart of the steps in a typical sampling procedure.

4.3 Special Preparation Steps

Measurements using either the gas adsorption method or mercury intrusion

method require additional preparations to ensure proper results.

4.3.1 Gas Adsorption Method

Prior to the measurement of an adsorption isotherm, all the physisorbed

species should be removed from the surface of the adsorbent. This may be

achieved by outgassing, a procedure involving exposure of the surface to a

high vacuum at elevated temperatures, or by flushing the adsorbent with a

inert gas, typically with helium or nitrogen, which subsequently may be the

adsorbent.

♦Porosity & Specific Surface Area Measurements

Figure 4.1. Recombination table of the cross riffling scheme.
21

To obtain reproducible isotherms (ISO 9277), it is necessary to control the outgassing conditions such as temperature level, outgassing time, change in pressure over the adsorbent, and the residual pressure. Some samples require

careful attention when heat is applied due to the possibility of thermal damage.

In some cases, for example for zeolites, it is necessary to hold the temperature

under 100 °C for an extended period to allow all water to be removed. With

these microporous samples, reproducible isotherms are only obtained after one

or more adsorption - desorption cycles. The rate of desorption is strongly temperature dependent, and to minimize the outgassing time, the temperature

should be the maximum consistent with the avoidance of changes in the nature

of the adsorbent and with the achievement of reproducible isotherms.

Outgassing at a high temperature or under ultrahigh vacuum conditions, as

well as flushing with certain gases may lead to changes in the surface composition, e.g. through decomposition of hydroxides or carbonates, formation of

surface defects, or irreversible changes of the structure. The maximum

temperature at which the sample is not affected may be determined by

General Procedures ♦

Figure 4.2. Flowchart indicating steps in a typical sampling procedure.
22

thermogravimetric analysis (Figure 4.3a) or by trial experiments using

different degassing conditions of time and temperature.

The real sample mass can only be determined after degassing since the adsorbent contaminants can contribute significantly to the apparent initial mass of

the sample. Furthermore, the sample must be protected from exposure to the

atmosphere after degassing.

When vacuum conditions are used, the heated sample container must be

isolated from the pump and trap. When the vacuum line is closed, such as at

time ta in Figure 4.3b, the extent of degassing is indicated by the gas pressure.

If the pressure does not rise significantly over a period of 15 min to 30 min,

degassing is complete. An increase in pressure to a higher steady value

indicates that further degassing is needed. A rising pressure that does not

stabilize indicates a leak in the vacuum line. When degassing is complete, the

specimen is ready for measurement.

♦Porosity & Specific Surface Area Measurements

Figure 4.3a. Thermogravimetric control of degassing.
23

4.3.2 Mercury Intrusion Method

The additional preparation of the samples for investigation by the mercury

intrusion method is not as specific as in the case of gas adsorption. Samples

with high specific surface area may require pretreatment by heat and vacuum

to remove vapors, typically water vapor, which occupies small pores and prevents the intrusion of mercury. If the sample is heat sensitive, then drying in a

desiccator is required. Entrapment of small quantities of gas with the sample

in the penetrometer gives no obvious manifestations to the analytical results.

5. Specific Surface Area and Pore Analysis by Gas Adsorption

Gas adsorption measurements are widely used for the characterization of a

variety of porous solids. Of particular importance is the application of

physisorption (physical adsorption) for the determination of the surface area

and pore size distribution. Nitrogen (at 77 K) is the recommended adsorptive

Specific Surface Area & Pore Analysis ♦

Figure 4.3b. Pressure control of degassing.
24

for determining the surface area and mesopore size distribution, but it is necessary to employ a range of probe molecules to obtain a reliable assessment of

the micropore size distribution. An alternative technique to gas adsorption is

mercury porosimetry used for macropore size analysis. For operational

reasons, krypton adsorption (at 77 K) is usually adopted for the determination

of relatively low specific surface areas ( < 2 m2g-1), but this technique cannot

be employed for the study of porosity.

5.1 Principle of Measurement

Various procedures have been devised for determining the amount of gas

adsorbed. Volumetric methods are generally employed for measuring nitrogen

or krypton isotherms at cryogenic temperatures, 77 K, but gravimetric techniques are especially useful for studying the adsorption of vapors at, or near,

ambient temperature. The isotherm is usually constructed point-by-point by

the admission and withdrawal of known amounts of gas, with adequate time

allowed for equilibration at each point. Automated techniques have been

developed, which involve the slow continuous admission of the adsorptive and

thus provide a measure of the adsorption under quasi-equilibrium conditions.

Prior to the determination of the isotherm, it is usual to remove all physisorbed

material. The exact conditions required to attain a "clean" surface depend on

the nature of the system. For the determination of the surface area and mesopore size distribution by nitrogen adsorption, outgassing to a residual pressure

of about 13 mPa (10-4 torr) is considered to be acceptable. Determination of

the outgassed mass of the sample is one of the major sources of measurement

uncertainty and contributes to the discrepancy between different experiments.

Inorganic oxides are usually outgassed at temperatures from about 120 °C to

150 °C while microporous carbons and zeolites require higher temperatures

(about 300 °C).

Argon at its boiling point (87 K) is the adsorptive most commonly used in

connection with the micropore size analysis by means of the gas adsorption

method. For the evaluation of the measured isotherm, it is necessary to chose

among several evaluation procedures that differ according to the qualitative

nature of the isotherm. Six classes of isotherms, Figure 5.1, are usually

distinguished (Sing 1985).

Type I isotherms are observed for microporous solids having relatively small

external surfaces (e.g. activated carbons, molecular sieve zeolites, and certain

porous oxides). The limiting uptake is governed by the accessible micropore

volume rather than by the internal surface area.

♦Porosity & Specific Surface Area Measurements
25

Type II isotherm is the normal form obtained with a non-porous or

macroporous adsorbent. The type II isotherm represents unrestricted

monolayer-multilayer adsorption.

Type III isotherms are convex to the p/p0 axis over its entire range. This type

is not common, but there are a number of systems (e.g. nitrogen on

polyethylene) which exhibit isotherms with gradual curvature. In such cases,

the adsorbate-adsorbate interactions play an important role.

Type IV isotherms are distinguished by two characteristic features. This type

of isotherm exhibits a hysteresis loop, which is associated with capillary

condensation taking place in mesopores, and has a limiting uptake over a

range of high p/p0. The initial part of the type IV isotherm is attributed to a

monolayer-multilayer adsorption since it follows the same path as the

corresponding part of a type II isotherm obtained with the given adsorptive on

the same surface area of the adsorbent in a non-porous form. Type IV

isotherms are characteristic of many mesoporous industrial adsorbents.

Specific Surface Area & Pore Analysis ♦

Figure 5.1. Classification of isotherm types (Sing 1985)
26

Type V isotherms are uncommon. They are related to the Type III isotherm in

that the adsorbent-adsorbate interaction is weak, and they also exhibit

hysteresis as in Type IV.

Type VI isotherms exhibit stepwise multilayer adsorption on a uniform nonporous surface. The sharpness of the steps depends on the system and the

temperature. The step-height represents the monolayer capacity for each

adsorbed layer and, in the simplest case, remains nearly constant for two or

three adsorbed layers. Among the best examples of Type VI isotherms are

those obtained with argon or krypton on graphitized carbon blacks at liquid

nitrogen temperature.

Further subdivisions of the types I, II, III, and IV have been developed for

some applications (Rouquerol et al. 1999), but those finer subdivisions will

not be discussed here.

5.2 Determination of Surface Area

It is standard practice to apply the Brunauer-Emmett-Teller (BET) method to

derive the surface area from physisorption isotherm data. For this purpose it

is convenient to apply the BET equation in the form

where na is the amount adsorbed at the relative pressure p/p0, nm is the monolayer capacity, and C is a constant that is dependent on the isotherm shape.

According to the BET equation, a linear relation is given if p / na (p0 – p) is

plotted against p/p0 (i.e. the BET plot). In this manner, it is possible to obtain

nm, but the range of linearity of the BET plot is always restricted to a limited

part of the isotherm – usually not above p/p0 ≈ 0.3.

The second stage in the application of the BET method is the calculation of

the surface area, A(BET), often termed the "BET area" from na

m. This requires

a knowledge of the average area, am (i.e. molecular cross-sectional area),

occupied by each adsorbed molecule in the complete monolayer. Thus,

A(BET) = nm ⋅ L ⋅ am

where L is the Avogadro constant.

♦Porosity & Specific Surface Area Measurements
27

It is usually assumed that the BET nitrogen monolayer is close-packed, giving

am (N2) = 0.162 nm2 at 77 K. 

However, it must be kept in mind that a constant value of am (N2) is unlikely

and that caution needs to be exercised in dealing with surfaces that give rise to

either especially strong or weak adsorbent-adsorbate interactions which are

able to influence the packing.

A concern that must be recognized is that the theory underlying the BET

method is based on an oversimplified extension of the Langmuir mechanism to

multilayer adsorption. To obtain a reliable value of nm it is necessary that the

knee of the isotherm be fairly sharp (i.e. the BET constant C is not less than

about 100). A very low value of C (< 20) is associated with an appreciable

overlap of monolayer and multilayer adsorption; for that case, the application

of the BET analysis is of doubtful value.

In view of the complexity of physisorption, it has been found useful to apply

an empirical procedure for isotherm analysis. This approach makes use of

standard adsorption data obtained with non-porous reference materials and

attempts to interpret the differences in isotherm shape in terms of the various

mechanisms of physisorption.

5.3 Mesoporosity and the Kelvin Equation

Adsorption of gas into a porous solid takes place according to a number of

different mechanisms. For instance, in mesopores and macropores, multilayer

adsorption onto the pore walls occurs initially. At higher relative pressures,

capillary condensation takes place with the formation of a curved liquid-like

meniscus.

In the gas adsorption measurements, the quantity of gas adsorbed on a surface

is recorded as a function of the relative pressure of the adsorptive.

Measurements at a series of increasing relative pressures yield the adsorption

branch of the isotherm, while measurements at a series of decreasing relative

pressures determine the desorption branch of the isotherm. The pore size

distribution can be calculated either from the adsorption or the desorption

branches of the isotherm (or both), but results from the two branches will not

necessarily be in agreement. It is usually recommended that the analysis be

carried out using the adsorption branch of the isotherm.

The computation of the mesopore size distribution is generally carried out

using methods based on the Kelvin equation.

ASpecific Surface Area & Pore Analysis ♦
28

where:

rk = radius of curvature of the adsorptive condensed in the pore

po = saturation vapor pressure of nitrogen at the temperature of the liquid 

nitrogen

p = equilibrium pressure of the nitrogen adsorptive

σ = surface tension

Vm = molar gas volume of an ideal gas

θ = contact angle

For nitrogen at the boiling temperature, the Kelvin equation may be expressed

in the form:

Since condensation is considered to occur after an adsorbed layer has formed

on the pore walls, it is necessary to make allowance for the thickness of this

adsorbed film. In the case of cylindrical pores (Barrett 1951; ASTM 4641-94

e1), the allowance is made using:

rp = rk + ta

♦Porosity & Specific Surface Area Measurements

The minimum pore size that can be investigated is

limited by the size of the adsorptive molecule, about

0.4 nm in the case of nitrogen. The maximum pore

size is limited by the practical difficulty of determining

the amount of gas adsorbed at high relative pressure. In the case of nitrogen, the maximum size is

about 300 nm. Using nitrogen as the adsorptive,

comparative pore size distributions can be determined for micropores having widths less than 2 nm.

However other gases, such as argon, may provide

more reliable results.
29

where:

rp = cylindrical pore radius

ta = thickness of the adsorbed film

Various methods exist for the evaluation of ta as a function of relative pressure

and for calculating the pore size distribution. One of the most frequently used

equations is that of de Boer (de Boer 1965).

The required experimental data may be obtained in principle by volumetric,

gravimetric, or continuous flow methods. All of these methods can provide

data of comparable precision and reliability provided that the apparatus is

correctly calibrated and that care is taken to ensure that the system is at

equilibrium. However, for adsorption of nitrogen at liquid nitrogen

temperature, the static volumetric method is recommended because:

a) The sample is in direct contact with the wall of the sample holder and

therefore quickly reaches thermal equilibrium with the liquid nitrogen

cryostat; and

b) by monitoring the pressure in the system, it is straightforward to determine

when equilibrium between the adsorbate and adsorptive has been reached.

Finally, we note that the Kelvin equation cannot be used for pores less than

about 2 nm diameter. This limitation occurs because interactions with

adjacent pore walls become significant and the adsorbate can no longer be

considered a liquid having bulk thermodynamic properties.

5.4 Microporosity

According to the IUPAC Recommendations (Sing 1985), micropores are

defined as pores with internal width less than 2 nm. They are characterized by

the micropore volume and the micropore distribution. Different methods for

the characterization of micropores are available. Among the most important

methods are sorption methods. The field-proven methods of gas sorption are

of special interest with respect to the use of microporous material as specific

sorbents, molecular sieves, and carriers for catalysts. In this approach, of

Specific Surface Area & Pore Analysis ♦
30

course, the results of the adsorption measurements depend on the size of the

adsorptive gas molecules because some micropores may not be accessible to

larger molecules. Thus, exclusion effects may be observed.

In this section, methods for the evaluation of the volume of micropores and

the specific surface area of microporous material by low-temperature

adsorption gases are described. The methods use physisorbing gases which

can penetrate into the pores under investigation. The methods are not

applicable when chemisorption or absorption takes place.

The primary methods are described in the standards DIN 66135 and ISO

15901-3. These methods are based on the measurement of the adsorption of

gases at constant low temperature and the evaluation of the initial part of the

isotherm. The gases used are bound by physisorption at the solid surface, in

particular N2 at 77 K, Ar at 77 K or 87 K, or CO2 at 195 K. It should be

noted that these gas molecules have different sizes and the measurement

temperatures are different. Consequently, they are not expected to yield the

same values.

In micropores, the interaction potentials from opposite pore walls are overlapping and, hence, physisorption is stronger than in wide pores or at the external

surface. As a consequence, micropores are filled readily at very low relative

pressure. The filling of a significant portion of micropores is indicated by a

large and steep increase of the isotherm near its origin and subsequent bending

to a flat level. 

In general, adsorption in micropores at low relative and absolute pressure proceeds slowly. To determine a suitable pore size distribution in the micropore

range, many points with small increments of relative pressure should be

measured. For that purpose, the apparatus must be completely leak-free, and

wide connection tubes should allow for a rapid pressure adjustment.

The specific liquid volume Va of the adsorbate condensed in the pores at the

measuring temperature is calculated using

(5.1)

If the density ρg of the gas at pressure PApp and temperature TApp in the gas

dosing supply is unknown, the standard density ρg,STP (Table 5.1) should be

used, applying the general gas equation:

♦Porosity & Specific Surface Area Measurements
31

(5.2)

where

Va = specific volume of the adsorbate

Vg = specific volume of the gas

ρl = density of the liquefied adsorptive at measuring temperature

ρg,STP = gas density at STP

STP = standard temperature and pressure, specified as:

PSTP = 101325 Pa

TSTP = 273.15 K 

Table 5.1. Recommended values for the calculation of liquid volumes(ISO

15901-3)

5.4.1 Micropore Volume from the Langmuir Isotherm

Adsorption measurements of highly microporous materials result in a

Langmuir-type isotherm (type I of the IUPAC classification) if the external

surface and the volume of mesopores are negligible (Figure 5.2). The plateau

corresponds to the micropore quantity. The micropore volume can be calculated by use of Eq. (5.1) or (5.2). If macropores are also present, a steep increase

near p/po ≈ 1 may be observed. In general, the surface area of the macropores

can be neglected. For a better resolution of the steep initial part

of the curve, the isotherm of a microporous material, Va=f(p/po) or

Specific Surface Area & Pore Analysis ♦

Gas ρg,STP

mg/cm3

Measuring

temperature

K

ρl

g/cm3

ρg,STP / ρl

1

N2 1.2504 77 0.8086 1.546 ⋅ 10-3

Ar 1.7840 77 1.47 1.21 ⋅ 10-3

Ar 1.7840 87 1.40 1.27 ⋅ 10-3

CO2 1.9770 195 1.2778 1.547 ⋅ 10-3
32

ma=f(p/po), is favorably represented on a logarithmic scale of the relative

pressure (Figure 5.3).

♦Porosity & Specific Surface Area Measurements

Figure 5.2. Linear isotherm plot of Ar on zeolite at 87.3 K

Figure 5.3. Logarithmic isotherm plot of Ar on zeolite at 87.3 K
33

5.4.2 Dubinin-Radushkevich Method

The method originally developed to investigate the microporosity of activated

carbons (Dubinin 1955) can be used for any microporous material. Adsorption

isotherms of pure gases on microporous sorbents can be described by means of

Polanyi's potential theory. Each gas/sorbent system is

characterized by an adsorption potential E which is influenced in particular by

the chemical properties of the sorbent. The volume Va filled at a given relative

pressure p/po as a part of the total micropore volume Vmicro is a function of the

adsorption potential E:

Va=f(E) (5.3)

According to Dubinin, the adsorption potential equals the work required to

bring an adsorbed molecule into the gas phase. For T < Tcr, Polanyi's

potential yields

(5.4)

For the characteristic function (Dubinin 1969), the empirical relation can be

expressed as

(5.5)

The characteristic adsorption energy Eo characterizes the pore distribution,

while the affinity coefficient β unites all the curves for each gas on the same

adsorbent to one characteristic curve. The Dubinin isotherm then can be

written in logarithmic form to give a straight line:

(5.6)

(5.7)

Specific Surface Area & Pore Analysis ♦
34

To evaluate the data, the region of relative pressure 10-4 < p/p0 < 0.1 is

usually preferred. The data are registered in a diagram log10Va versus

(Figure 5.4). The slope of the regression line gives the parameter

D. The total micropore volume Vmicro can be calculated from the ordinate

intercept.

5.4.3 Horvath-Kawazoe (HK) and Saito-Foley (SF) Methods

Horvath and Kawazoe (HK) (Horvath 1983; Horvath 1998) described a

semi-empirical, analytical method for the calculation of effective pore size

distributions from nitrogen adsorption isotherms in microporous materials.

The original HK approach is based on the work of Everett and Powl (Everett

1976) and considers a fluid (nitrogen) confined to a slit-pore, such as can be

found in some carbon molecular sieves and active carbons. Everett and Powl

calculated the potential energy profiles for noble gas atoms adsorbed in a slit

between two graphitized carbon layer planes. The adsorbed fluid is considered as a bulk fluid influenced by a mean potential field that is characteristic

of the adsorbent-adsorbate interaction. The term mean field indicates that the

potential interactions between an adsorbate molecule and the adsorbent, which

♦Porosity & Specific Surface Area Measurements

Figure 5.4. Dubinin-Radushkevich DR-plot of an adsorption isotherm of N2

at 77 K on activated carbon (Dubinin 1947)
35

may exhibit a strong spatial dependence, are replaced by an average, uniform

potential field. Horvath and Kawazoe found by using thermodynamic

arguments, that this average potential can be related to the free energy change

of adsorption, yielding a relation between filling pressure and the effective

pore width dp = l – ds, where ds is the diameter of an adsorbent molecule, and

the separation between nuclei of the two layers is l.

(5.8)

with

(5.9)

The parameters d0, σ, As and Aa can be calculated using the following

equations:

(5.10)

(5.11)

(5.12)

(5.13)

where me = mass of electron

c = speed of light in vacuum

α = polarizability

χ = magnetic susceptibility

Specific Surface Area & Pore Analysis ♦
36

According to equation (5.8), the filling of micropores of a given size and

shape takes place at a characteristic relative pressure. This characteristic pressure is directly related to the adsorbent / adsorbate interaction energy. Saito

and Foley extended the HK method to calculate effective pore size distributions from argon adsorption isotherms at 87 K in zeolites (Saito 1991; Saito

1995). Like HK, Saito and Foley (SF) based their method on the Everett and

Powl potential equation (Everett 1976), but for a cylindrical pore geometry.

Following the logic of the HK derivation, Saito and Foley derived an equation

similar to the HK equation which related the micropore filling pressure to the

(effective) pore diameter dp = l - ds.

(5.14)

with

(5.15)

The parameters As and Aa can be calculated according to the equations (5.12)

and (5.13). 

Parameters αk and βk are constants defined as :

(5.16)

with α0 = β0 = 1. Ns is the number of atoms per unit area of pore wall. Na is

the number of atoms per unit area of a monolayer.

To perform the HK and SF calculations the values for the adsorbent

parameters αs, χs, ds, Ns as well as of the adsorptive parameters αa, χa, da and 

Na need to be known. The results are very sensitive with respect to the choice

of these constants. Examples for adsorptive and adsorbent model parameters

for application in connection with the Horvath-Kawazoe and Saito-Foley

method are given in Tables 5.2 and 5.3. The constants were taken from the

original papers of Horvath and Kawazoe (Horvath1983) and Saito and

Foley(Saito 1991 and 1995). Please note that other values for these

parameters can be found in the literature. Hence, to present meaningful 

♦Porosity & Specific Surface Area Measurements
37

Specific Surface Area & Pore Analysis ♦

Parameter Symbol Graphite

(a) and (b)

Zeolithe

(c) and (d) Unit

Polarizability αs 1.02 2.50 [10-24 cm3]

Magnetic

susceptibility χs 13.5 1.30 [10-29 cm3]

Surface

density

(atoms per

m2 of pore

wall)

Ns 3.84 1.31 [1019 m-2]

Diameter ds 0.34 0.28 [nm]

Table 5.2. Adsorbent Parameters (according to DIN 66135-4)

Parameter Symbol Graphite

(a) and (b)

Zeolithe

(c) and (d) Unit

Polarizability αa 1.46 1.63 [10-24 cm3]

Magnetic

susceptibility χa 2.00 3.25 [10-29 cm3]

Surface

density

(atoms per

m2 of

monolayer)

Na 6.70 8.52 [1018 m-2]

Diameter da 0.30 0.34 [nm]

Table 5.3. Adsorbent Parameters (according to DIN 66135-4)

(a) Horvath 1983, (b) Horvath 1998, (c) Saito 1991, (d) Saito 1995
38

results, it is crucial that the adsorbent and adsorptive parameters employed

in the HK and SF calculations always be stated in the analysis report.

Examples of results obtained by these methods and the constants from Tables

5.2 and 5.3 are given in Tables 5.4 and 5.5. 

Table 5.4. Relation between the micropore diameter and the relative pressure

for nitrogen at 77.35 K in a carbon slit pore according to the HorvathKawazoe approach (DIN 66135-4) using the materials constants in Tables. 5.2

and 5.3.

Table 5.5. Relation between the micropore diameter and the relative pressure

for argon at 87.27 K in a cylindrical zeolite pore according to the Saito-Foley

approach (DIN 66135-4) using the materials constants in Tables. 5.2 and 5.3.

5.4.4 Non-Local Density Functional Theory (NLDFT)

Two important methods have been developed to describe the sorption and

phase behavior of inhomogeneous fluids confined to porous materials:computer simulation and density functional theory (DFT). DFT is a computational

method that treats the intrinsic free energy of a system as a functional of the

particle distribution function (Evans 1992). These methods can be used to

calculate equilibrium density profiles of a fluid adsorbed on surfaces and in

pores from which the adsorption/desorption isotherms, heats of adsorption,

and other thermodynamic quantities can be derived. Pioneering studies on the

application of the DFT and molecular modeling by computer simulation to

study the adsorption and phase behavior of fluids in pores were performed by

Tarazona and Evans (Tarazona 1984; Tarazona 1985). Seaton et al. (Seaton

1989) were the first to apply the DFT to the calculation of the pore size

distribution (PSD) in both the meso-and micropore range. In that first attempt

to perform the PSD analysis based on statistical mechanical results, the so-

♦Porosity & Specific Surface Area Measurements

dp

[nm] 0.4 0.5 0.6 0.7 0.8 1.0 1.2 1.4 1.7 2.0

p/p0 1.8⋅10-7 1.2⋅10-6 1.7⋅10-4 9.6⋅10-4 3.2⋅10-3 1.4⋅10-2 3.5⋅10-2 6.3⋅10-2 1.1⋅10-1 1.6⋅10-1

dp

[nm] 0.4 0.5 0.6 0.7 0.8 1.0 1.2 1.4 1.7 2.0

p/p0 5.7⋅10-7 9.8⋅10-6 1.4⋅10-4 8.7⋅10-4 3.1⋅10-3 1.5⋅10-2 3.9⋅10-2 7.2⋅10-2 1.3⋅10-1 1.9⋅10-1
39

called local version of DFT was used. Although the local DFT provides qualitatively reasonable description of adsorption in the pores, it is quantitatively

inaccurate especially in the range of small pores (i.e., micropores). 

A significant improvement in accuracy was obtained with the Non-Local

Density Functional Theory (NLDFT) which was first used for the pore size

analysis of microporous carbons in 1993 by Lastoskie et al. (Lastoskie 1993).

Since then, the NLDFT has been applied frequently to the pore size analysis of

microporous and mesoporous materials (Olivier 1995; Ravikovitch 1997;

Ravikovitch 2001). 

The NLDFT method is commercially available for many adsorbent/adsorptive

systems. Unlike classical thermodynamic macroscopic models, the NLDFT

methods describe the behavior of fluids confined in the pores on a molecular

level which allows molecular properties of gases to be related to their adsorption properties in pores of different sizes. It follows that pore size

characterization methods based on the NLDFT approach are applicable to

the whole range of micropores and mesopores. 

Under experimental equilibrium conditions, the adsorbed fluid in a pore is in

equilibrium with a bulk gas phase. For such a system the grand canonical

ensemble provides the appropriate description of the thermodynamics. The

local density ρ(r) of the pore fluid is therefore determined by minimizing the

corresponding grand potential Ω[ρ(r)]. Once ρ(r) is known, other thermodynamic properties, such as the adsorption isotherm, heat of adsorption, free

energies, phase transitions, etc. can be calculated. The grand potential

function Ω[ρ(r)] is given by the following term:

Ω[ρ(r)] = F[ρ(r)] – ∫drρ(r)(μ – Vext (r))

where F[ρ(r)] is the free energy functional in the absence of any external field

and Vext(r) is the potential imposed by the walls, i.e., F[ρ(r)] depends of

course on all the interactions. 

To apply the theory practically to the calculation of the pore size distributions

from experimental adsorption isotherms, the theoretical model isotherms are

calculated using methods of statistical mechanics. In essence, those isotherms

are calculated by integrating the equilibrium density profiles, ρ(r), of the fluid

in the model pores. A set of isotherms calculated for a set of pore sizes in a

given range for a given adsorbate constitutes the model database. Such a set

of isotherms called a kernel can be regarded as a theoretical reference for a

given adsorption system and as such can be used to calculate pore size distriSpecific Surface Area & Pore Analysis ♦
40

butions from adsorption isotherms measured for the corresponding systems.

It is important to realize that the numerical values of a given kernel depend on

a number of factors such as the assumed geometrical pore model, values of the

gas-gas and gas-solid interaction parameters, and other model assumptions. It

is general practice to adjust interaction parameters (fluid-fluid and fluid-solid)

in such a way that the model would correctly reproduce fluid bulk properties

(e.g., bulk liquid-gas equilibrium densities and pressures, liquid-gas interfacial

tensions) as well as standard adsorption isotherms on well-defined non-porous

adsorbents. The calculation of pore size distribution is based on a solution of

the Adsorption Integral Equation (AIE), which correlates the kernel of theoretical adsorption/desorption isotherms with the experimental sorption isotherm:

where

N(p/p0) = experimental adsorption isotherm data

W = pore width

N(p/p0,W) = isotherm on a single pore of width W

f(W) = pore size distribution function

The AIE equation reflects the assumption that the total isotherm consists of a

number of individual "single pore" isotherms multiplied by their relative

distribution, f(W), over a range of pore sizes. The set of N(p/p0,W) isotherms

(kernel) for a given system (adsorptive/adsorbent) can be obtained by either

Density Functional Theory or by Monte Carlo computer simulation. The pore

size distribution is then derived by solving the AIE equation numerically. In

general, the solution of the AIE represents an ill-posed problem, which

requires some degree of regularization. The existing regularization algorithms

(Lawson 1995; Provencher 1982; Wahba 1977) allow meaningful and stable

solutions of this equation.

6. Pore Analysis by Mercury Intrusion Techniques (Mercury

Porosimetry)

Mercury porosimetry is a very widely accepted method for determining total

volume and pore size distribution in the meso- and macropore ranges.

6.1 Principle of Measurement

This method is based on the fact that a non-wetting liquid (i.e., the contact

♦Porosity & Specific Surface Area Measurements
41

angle, θ, between the liquid and the pore wall is such that π / 2 < θ≤π)

requires that a positive excess hydrostatic pressure ΔP be applied to enable it

to enter pores of radius r. ΔP varies inversely with r. The behavior of a nonwetting liquid in a porous sample was first described by E. W. Washburn

(Washburn 1921), and the governing basic equation bears his name. The

equation is directly applicable with mercury, the only known liquid really

suitable for porosity measurements.

Mercury does not wet most substances and will not penetrate pores by

capillary action. Liquid mercury has a high surface tension, i.e., molecular

forces in its surface film tend to contract its volume into a form with the least

surface area. Consequently, entry of mercury into pores requires applying

pressure in inverse proportion to the opening size. 

To measure the pore volume or the pore size distribution, a weighed sample is

enclosed in a stout bomb (penetrometer) and evacuated to remove air from the

pores. Mercury is than admitted to fill the cell and surround the sample and is

subjected to progressively higher pressures applied hydraulically. At each

pressure P, the corresponding volume V of mercury contained in the cell is

measured.

6.2 Calculation of the Pore Size Distribution

It is assumed that as the pressure is increased, mercury enters pores in

decreasing order of size. Thus, if ΔV is the volume intruded between P and

P+ΔP, it will equal the volume of pores with radii between r and r - Δr, with

(Washburn equation)

In this way, a volumetric distribution of pore sizes is obtained (Washburn

1921).

Washburn was the first to point out the fact that surface tension opposes the

Pore Analysis by Mercury Intrusion ♦

Caveat: The presumption that the pores are invaded

in decreasing order of size may be invalidated by

network effects, which imply that the sequential filling

of pores is dictated primarily by their mode of interconnection.
42

entrance into a small pore of any liquid having an angle of contact larger than

90°. Years later, Ritter and Drake (Ritter 1945) developed the method and

apparatus for measuring the penetration of mercury into pores.

This interpretation evidently depends on the validity of the geometrical and

other assumptions involved. The radius r is to be regarded not as a literal pore

dimension, but rather as the size of an equivalent cylindrical pore that would

be filled at the given pressure. 

Assuming that the pores are cylinders of different diameters is, of course, an

oversimplification, but it is the best representation available except in the case

of materials composed of stacked, thin sheets where slit-like openings

predominate.

A value of the factor σ cos θ must be assumed; customarily, σ is taken to be

the value for pure mercury, 484 mN m-1, despite the likelihood of contamination by hydraulic fluid. For many porous materials, the contact angle θ is

about 140°, but its exact value depends on the material to be tested and must

be known or measured prior to the analysis. Figure 6.1 shows the relative

deviation in the calculated pore diameter as a function of the mercury contact

angle. If the value of the contact angle is unknown, 140° is often assumed

♦Porosity & Specific Surface Area Measurements

Figure 6.1. Relative deviation in the calculated pore diameter dp as a

function of the mercury contact angle θ (reference value 140°).
43

(DIN 66131). 

For most metal oxides including, Al2O3 and SiO2, as well as for supported

catalysts, the contact angle (Groen 2002) is close to the presumed θ = 140°.

However, in zeolites the contact angle ranges from 127° to 140°, and

assuming 140° in those cases may lead to erroneous interpretation. 

The zeolite framework type seems to be the most important variable influencing the interaction between mercury and the zeolite. The most striking results

are found for cement-like materials and carbonaceous materials, which show a

relatively low (≤ 130°) and high (≥ 150°) contact angle, respectively.

Table 6.1 shows the measured mercury contact angles on some important

Pore Analysis by Mercury Intrusion ♦

Table 6.1. Average contact angles of some materials, as measured with the

advancing angle and static angle method (Groen 2002).

(n.d. = not determined)
44

classes of porous materials.

In Table 6.1, values for θ and its standard deviation (std.dev.) are based on

three replicate measurements.

Experimentally, the pressure can be varied between 10 kPa and 400 MPa (0.1

bar and 4 kbar), corresponding to cylindrical pore radii between 75 μm and

1.8 nm. (At high pressures, a blank correction must be applied to compensate

for the compressibility of mercury and for elastic distortion of the cell and

other component parts). Adequate control of temperature is essential, since the

pressure within a closed, liquid-filled cell will be strongly temperaturedependent. This is not usually provided for in commercial instruments, and

absolute calibration of pressure and volume transducers is not normally

available. At the high pressures employed, the sample may be deformed

elastically or even damaged irreversibly. On reducing the pressure, a

substantial volume of mercury may be retained in the pores. The method

therefore is destructive. Figure 6.3 shows a typical intrusion/extrusion curve

(volume vs. pressure curve) of a porous glass with a hysteresis loop that is

typical for cylindrical pores.

The accuracy of pore diameter measurements by mercury porosimetry when

the pores are right circular cylinders is exceptionally good, agreeing to at least

0.008 μm openings. However, true circular cross-section pores or pores of

any regular geometry rarely are encountered in real materials. To the contrary,

shapes of typical pores are slits, cracks, fissures, or highly irregular interstitial

♦Porosity & Specific Surface Area Measurements

Figure 6.2. Schematic representation of the (a) advancing angle and

(b) static angle method for determination of the contact angle (Groen 2002).
45

voids between particles. In order for the mathematics of pore characterization

to be manageable, it is convenient to treat the filling of these complex pores as

if they were right circular cylinders. This means that pore data from mercury

porosimetry should be understood and used as equivalent cylindrical dimensions, just as the particle size determined by sedimentation of nonspherical

particles is an equivalent spherical diameter. How pore shape alters basic

pressure-volume curves is illustrated in Figure 6.4.

6.3 Calculation of the Specific Surface Area

Assuming pores of cylindrical shape, a surface distribution may be derived

from the pore volume distribution. An estimate of the total surface area can be

made from the pressure/volume curve (Rootare 1967) without using a pore

model, but the material must be free of ink bottle pores:

From the function V = V(p) the integral may be calculated either graphically

or by means of a numerical method.

Pore Analysis by Mercury Intrusion ♦

Figure 6.3. Intrusion/extrusion curve (volume vs. pressure curve) of a porous

glass.
46

Because the surface of small pores is not measured, the results are not always

comparable with those derived from adsorption methods such as the BET

method. The recommended range (Rootare 1967) for the linear relation of

specific surface area measured by the BET and Hg porosimetry method is

approximately from 10 m2/g to 100 m2/g.

7. Conclusion

The present Guide addresses a full range of metrological issues pertaining to

methods commonly used to determine quantitative measures of porosity and

specific surface area in solid materials. These methods represent the best

standard methods currently applied worldwide to meet today's demands for

accuracy and reliability. In the natural course of evolving technology, some

methods cited in this Guide as being in various stages of development have

undoubtedly progressed to more advanced stages or have become approved

standard test methods. New methods, motivated by emerging technologies in

♦Porosity & Specific Surface Area Measurements

Figure 6.4. Characteristic hysteresis loops for cylindrical holes and for pores

formed within aggregates (Webb 1997).
47

nanomaterials and multilayer film structures, already are on the horizon or

under development. Many of the issues that will need to be addressed in those

new and innovative methods are fundamentally the same as those discussed in

the present Guide. Thus, the lessons learned here may also be of value and

service to the next generation of porosity and specific surface area

metrologies, and the Guide as a whole may serve as a substantial building

block for the evolution of those measurement sciences. 

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References ♦
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DIN 66132, Bestimmung der spezifschen Oberfläche von Feststoffen durch

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49

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♦Porosity & Specific Surface Area Measurements
51

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54

Appendix 1: SRM/CRMs Worldwide

♦Porosity & Specific Surface Area Measurements

Table A1.1. Producers / distributors of porous reference materials
55

Appendix 1 ♦

Table A1.2. Certified reference materials for porosity properties

*) Specific surface areas are measured using the N2 or Kr BET method at

77 K, except those marked by (*) with the permeation method and (**) using

the Wagner turbidimeter.
56

Appendix 2: Additional Illustrative Case Study Results

This appendix contains four additional case studies that may be of interest to

the reader.

Appendix 2.1 CRM BAM-P107 (a microporous CRM)

Appendix 2.2 CRM BAM-PM-103 (a mesoporous CRM)

Appendix 2.3 CRM BAM-P123 (a macroporous CRM)

Appendix 2.4 CRM BAM-PM-122 (Hg Porosimetry)

Appendix 2.1 CRM BAM-P107 (a microporous CRM)

Microporous zeolite (faujasite type) is a crystalline material with a well

defined and regular pore network. This material was certified by using DIN

66135-4 (Saito/Foley model for cylindrical pore geometry) for nitrogen

adsorption. Certified values were reported for the micropore width and the

micropore volume.

Table A2.1.1. Certified values for the microporous CRM BAM-P107 for

nitrogen adsorption

N = number of laboratories (outlier free)

x = average of averages (certified value)

s = standard deviation of laboratory averages

s /√ N = standard deviation of average of laboratory means,

standard uncertainty of the certified value

tn-1;0.975, s /√ N = half width of the confidence interval. The confidence

interval covers the true value (with certainty of 95 %).

The nitrogen isotherm for BAM-P107 is shown in Figure A2.1.1.

♦Porosity & Specific Surface Area Measurements

Property Unit N x s s /√ N tn-1;0.975, s /√ N

Specific

Micropore

Volume

cm3g-1 3 0.217 0.003 0.001 0.002

Median Pore

Width

cm 1 0.86 0.03 0.01 0.02
57

Appendix 2.2: CRM BAM-PM-103 (a mesoporous CRM)

This CRM is a porous and highly dispersed material for which interlaboratory

testing and statistical evaluation were used for the certification. The certified

values are given in Table A2.2.1. The manufacturer's designation for this

material was alumina type 60 (a mixture of transition aluminas). Figure

A2.2.1 shows the N2 isotherm (type IV according to the IUPAC classification), and Figure A2.2.2 shows the pore size distribution as calculated by

means of the BJH method. Figure A2.2.3 illustrates the statistical evaluation.

Table A2.2.1. Mesoporous CRM BAM-PM-103 certified values for the gas

adsorption method 

(*s: standard deviation of laboratory means)

Appendix 2 ♦

Figure A2.1.1. The nitrogen (N2) Isotherm for CRM BAM-P107

Material

Certified property (mean of laboratory means ± s)*

Specific Surface

Area

Method: BET

(DIN 66131)

Specific Pore

Volume

Method: BJH

(DIN 66134)

Hydraulic Pore

Radius (2V/ABET)

Method: BJH

(DIN 66134)

CRM BAM-PM-103

Alumina type 60

Adsorptive: nitrogen

at 77 K

156.0 ± 7.2 m2/g 0.250 ± 0.008 cm3/g 3.18 ± 0.08nm
58

♦Porosity & Specific Surface Area Measurements

Figure A2.2.1. Adsorption isotherm of nitrogen at 77 K on BAM-PM-103

(alumina type 60).

Figure A2.2.2. Pore size distribution of BAM-PM-103 (alumina type 60).
59

Appendix 2.3 CRM BAM-P123 (a macroporous CRM)

CRM BAM-P123 was the first macroporous material certified for the relatively low intrusion pressure region of mercury porosimetry. The material

(α-alumina) was certified by an interlaboratory comparison (26 laboratories

from 4 European countries) using mercury porosimetry according to DIN

66133. The pores of the material were verified by imaging analysis and

investigated by scanning electron microscopy.

Statistical data analysis was done for a designed experiment using 9 observed

pressure-volume curves of the filter tube pieces per laboratory. The

participating laboratories were qualified by means of a proficiency test using

standardized pressure-volume (p-V) curves. 

Appendix 2 ♦

Figure A2.2.3. Interlaboratory study: Specific pore volume (adsorption) of

CRM BAM-PM-103 (alumina type 60), the wide-dashed line corresponds to

the mean of means (certified value) with standard deviation (narrow-dashed

lines); laboratory mean values with standard deviation, individual values

(single points), and standard deviation of means are also given; laboratory

number 5 is an outlier. 
60

In contrast to powders, there is no possibility of homogenizing compact

samples. Therefore, the statistical analysis of the interlaboratory study

included main steps: 

• Evaluating the characteristics of the individual p-V curves by statistical

considerations and standardizing the curves using those characteristics;

• Analyzing the variance of the curve characteristics and the standardized

curves; and 

• Constructing an average reference intrusion curve (certified p-V curve)

with simultaneous confidence bands and prediction bands using the variance

analytical results for the curve characteristics and the standardized curves.

♦Porosity & Specific Surface Area Measurements

Figure A2.3.1. Reference curve with simultaneous prediction bands at

significance level 0.95 for the material CRM BAM-P123.
61

Vp, 1.4 MPa = Specific pore volume at 1.4 MPa

Appendix 2.4 CRM BAM-PM-122 (Hg Porosimetry)

CRMs for the high pressure range (from 0.1 MPa to 400 MPa) of the mercury

intrusion method need to have a sharp and narrow pore size distribution, a

graduation of the pore volumes and pore sizes of the CRMs, and a simple

handling of the materials.

For this CRM, a new certification philosophy was used. Instead of certifying

only single values of certain properties, such as the mean pore width or the

pore volume, the whole mercury intrusion curve (pore volume as a function of

intrusion pressure) was certified. This procedure resulted in a number of

advantages for the user, including the availability a large number of certified

data points in the typical working range of mercury porosimeters (0.1 MPa to

400 MPa), the possibility of assessing the pressure transducer performance

over a wide range of pressure, volume calibration of the porosimeter, and the

suitability of the CRM for 200 MPa instruments as well as for 400 MPa

instruments. The interlaboratory tests (DIN 66133) were carried out with 25

participating laboratories. To provide a strict comparison of the statistical treatment of the results of each laboratory, a fixed grid of data points, regularly

spaced on the logarithmic pressure scale, was used. Therefore, all of the

experimental mercury intrusion curves were transformed into this standard

grid of data points using a simple linear data interpolation. After calculating

the mean mercury intrusion curve for each laboratory, the mean (nonweighted) curve of laboratory mean curves and the standard deviation at each

data point leading to the uncertainty interval (Figures A2.4.1 and A2.4.2) were

calculated. Since the materials consisted of micro-beads, mercury porosimetry

measured some interparticle (intergranular) volume in addition to the real

intraparticle pores. Because of its more or less random character, the intergranAppendix 2 ♦

Quantity Certified

values

0.90

Confidence

Interval

0.95

Confidence

Interval

0.99

Confidence

Interval

Unit

Vp, 1.4 MPa 99.5 96.6 - 102.4 96.0 - 103.0 94.9 - 104.1 mm3g-1

p50 0.483 0.463 - 0.503 0.459 - 0.507 0.451 - 0.515 MPa

d50 3.05 2.92 - 3.18 2.89 - 3.21 2.85 - 3.25 μm

Table A2.3.1. Certified single values of BAM-P123.
62

ular volume could not be certified and had to be subtracted from the intrusion

curve at the beginning of the data evaluation. All calculations were carried out

on the basis of the Washburn equation (Washburn 1921) using values of 140°

for the angle of contact of mercury and 0.480 N/m for its surface tension.

The resulting certified mercury intrusion curve is shown in Figures A2.4.1 and

A2.4.2. The sharp pore size distribution of CRM BAM-PM-122 is seen in

Figure A2.4.3. 

♦Porosity & Specific Surface Area Measurements

Material

Specific Pore

Volume at

395 MPa

(mm3/g)

Median Pore

Width

(nm)

Specific

Surface Area

(not certified)

(m2/g)

CRM BAM-PM-122 porous glass

beads II

924.4 ± 17.2 139.0 ± 3.7 26.6

Figure A2.4.1. Certified mercury intrusion curve for CRM BAM-PM-122

with uncertainty intervals (logarithmic pressure axis).

Table A2.4.1. Selected properties of CRM BAM-PM-122 for mercury

porosimetry (DIN 66133).
63

Appendix 2 ♦

Figure A.2.4.2. Certified mercury intrusion curve for CRM BAM-PM-122

with uncertainty intervals in the range between 8 MPa and 15 MPa (linear

pressure axis).

Figure A2.4.3. Sharp pore size distribution for CRM BAM-PM-122

(not certified).
64

Appendix 3: Relevant Measurement Standards

Many national and international standards setting bodies have included norms

on the characterization of porous solids as part of their measurement standards

and procedures. Some of these norms are generic and only describe the

methods to be used to obtain certain typical parameters characterizing the

pore properties. Other norms are more specific and apply to specific classes

of materials, such as catalysts, ceramics, pigments, cements, and many more.

Table A3.1 lists several major institutions and their internet addresses. A listing of the most relevant standards from these organizations is given in the

following subsections:

Appendix 3.1 AFNOR Standards

Appendix 3.2 ASTM Standards

Appendix 3.3 British Standards

Appendix 3.4 CEN Standards

Appendix 3.5 DIN Standards

Appendix 3.6 ISO Standards

Table A3.1. Selected national and international institutes that have issued test

methods for the characterization of porous solids.

Appendix 3.1 AFNOR Standards

X-11-620 (1994) Détermination de la surface spécifique des solides par

adsorption de gaz á l'aide de la méthode BET

♦Porosity & Specific Surface Area Measurements

Abbreviation Institution Web address

AFNOR Association Française de

Normalisation

http://www.afnor.fr/

ASTM American Society for Testing

Materials

http://www.astm.org/

BSI British Standards

Institution

http://bsonline.techindex.co.uk/

CEN Comité Européenne de

Normalisation

http://www.cenorm.be/

DIN Deutsches Institut für

Normung

http://www2.din.de/

ISO International Organization

for Standardization

http://www.iso.ch
65

X-11-621 (1975) Détermination de l'aire massique (surface spécifique) des

poudres par adsorption de gaz - Méthode B.E.T: Mésure volumétrique par

adsorption d'azote à basse temperature

Appendix 3.2 ASTM Standards

D 1993-03 Standard Test Method for Precipitated Silicia Surface 

Area by Multipoint BET Nitrogen Adsorption

D2873-94(1999)e1 Standard Test Method for Interior Porosity of (PolyVinyl

Chloride) (PVC) Resins by Mercury Intrusion

Porosimetry

D4365-95(2001) Standard Test Method for Determining Micropore

Volume and Zeolite Area of a Catalyst

D4404-84(1998)e1 Standard Test Method for Determination of Pore Volume

and Pore Volume Distribution of Soil and Rock by

Mercury Intrusion Porosimetry

D4641-94(1999)e1 Standard Practice for Calculation of Pore Size

Distributions of Catalysts from Nitrogen Desorption

Isotherms

D3663-99 Standard Test Method for Surface Area of Catalysts and

Catalyst Carriers

D3766-86(2002) Standard Terminology Relating to Catalysts and

Catalysis

D3906-03 Standard Test Method for Determination of Relative

X-ray Diffraction Intensities of Faujasite-Type Zeoliteontaining Materials

D3942-03 Standard Test Method for Determination of the Unit

Cell Dimension of a Faujasite-Type Zeolite

D4222-98 Standard Test Method for Determination of Nitrogen

Adsorption and Desorption Isotherms of Catalysts by

Static Volumetric Measurements

Appendix 3 ♦
66

D4284-03 Standard Test Method for Determining Pore Volume

Distribution of Catalysts by Mercury Intrusion

Porosimetry

D4365-95(2001) Standard Test Method for Determining Micropore

Volume and Zeolite Area of a Catalyst

D4404-84(1998)e1 Standard Test Method for Determination of Pore Volume

and Pore Volume Distribution of Soil and Rock by

Mercury Intrusion Porosimetry

D4567-99 Standard Test Method for Single-Point Determination of

Specific Surface Area of Catalysts and Catalyst Carriers

Using Nitrogen Adsorption by Continuous Flow Method

D4641-94(1999)e1 Standard Practice for Calculation of Pore Size

Distributions of Catalysts from Nitrogen Desorption

Isotherms

D4780-95(2001) Standard Test Method for Determination of Low Surface

Area of Catalysts by Multipoint Krypton Adsorption

D5604-96(2001) Standard Test Methods for Precipitated Silica-Surface

Area by Single Point B.E.T. Nitrogen Adsorption

D5900-98(2002) Standard Specification for Physical and Chemical

Properties of Industry Reference Materials (IRM)

D6556-02a Standard Test Method for Carbon Black-Total and

External Surface Area by Nitrogen Adsorption

C1274-00 Standard Test Method for Advanced Ceramic Specific

Surface Area by Physical Adsorption

C493-98 Standard Test Method for Bulk Density and Porosity of

Granular Refractory Materials by Mercury Displacement

Appendix 3.3 British Standards

BS 7591 Porosity and pore size distribution of materials

BS 7591-1:1992 Method of evaluation by mercury porosimetry

♦Porosity & Specific Surface Area Measurements
67

BS 7591-2:1992 Method of evaluation by gas adsorption

BS 7591-4:1993 Method of evaluation by liquid expulsion

BS 4359 Determination of the specific surface of powders

BS 4359-1:1996 BET method of gas adsorption for solids (including

porous materials)

BS 4359-2:1982 Recommended air permeability methods

BS 4359-4:1995 Recommendations for methods of determination of metal

surface area using gas adsorption techniques

Appendix 3.4 CEN Standards

EN 623-2 (1993) Determination of density and porosity

EN 725-6 (1996) Determination of specific surface area

EN 725-7 (1995) Determination of absolute density

EN 725-8 (pr) Determination of tamped bulk density

EN 725-9 (pr) Determination of untamped bulk density

Appendix 3.5 DIN Standards

DIN 66131 (1993) Bestimmung der spezifischen Oberfläche von Feststoffen

durch Gasadsorption nach Brunauer, Emmett und Teller (BET) (Determination

of specific surface area of solids by gas adsorption using the method of

Brunauer, Emmett and Teller (BET))

DIN 66132 (1975) Bestimmung der spezifischen Oberfläche von Feststoffen

durch Stickstoffadsorption; Einpunkt-Differenzverfahren nach Haul und

Dümbgen (Determination of specific surface area of solids by adsorption of

nitrogen; single-point differential method according to Haul and Dümbgen)

DIN 66133 (1993) Bestimmung der Porenvolumenverteilung und der spezifischen Oberfläche von Feststoffen durch Quecksilberintrusion (Determination of

pore volume distribution and specific surface area of solids by mercury intrusion)

DIN 66134 (1997) Bestimmung der Porengrößenverteilung und der spezifischen Oberfläche mesoporöser Feststoffe durch Stickstoffsorption; Verfahren

nach Barrett, Joyner und Halender (BJH) (Determination of the pore size disAppendix 3 ♦
68

tribution and specific surface area of mesoporous solids by means of nitrogen

sorption - Method of Barrett, Joyner and Halender (BJH))

DIN 66135 (2001/2002) Mikroporenanalyse mittels Gasadsorption (Micropore

analysis by gas adsorption)

66135-1: Grundlagen und Messverfahren (Fundamentals and Testing

procedure)

66135-2: Auswerteverfahren - Isothermenvergleichsverfahren (Evaluation by

isotherms comparison)

66135-3: Bestimmung des Mikroporenvolumens nach Dubinin und

Radushkev (Determination of the micropore volume according to Dubinin and

Radushkevich)

66135-4: Bestimmung de Porenverteilung nach Horvath-Kavazoe und SaitoFoley (Determination of pore size distribution according to Horvath and

Kavazoe and Saito-Foley)

E DIN 66136 (2003-03) Bestimmung des Dispersionsgrades von Metallen

durch Chemisorption (Determination of the degree of dispersion of metals

using chemisorption)

66136-1: Grundlagen (Principles)

66136-2: Volumetrisches Verfahren (Volumetric method)

66136-3: Strömungsverfahren (Flow method)

DIN 66137 (2002) Bestimmung der Dichte fester Körper (Determination of

solid state density)

66137-1: Grundlagen (Principles)

66137-2: Gaspyknometrie (Gas pycnometry)

66137-3: Gasauftriebsverfahren (Gas bouancy method)

E DIN 66138 (2002) Messung der Sorption von Wasserdampf und anderen

Dämpfen an Feststoffen durch Wägung (Measurement of sorption of water

vapour and other vapours on solids by weighting)

66138-1: Messung des Feuchtegehalts (Measurement of the quantity of

moisture) 

66138-2: Messung der Wasserdampfsorptionsisotherme (Measurement of the

water vapour sorption isotherm)

DIN/ISO 9277(2003) Bestimmung der spezifischen Oberfläche von

Feststoffen durch Gasadsorption nach dem BET-Verfahren (Determination of

the specific surface area by gas adsorption using the BET-method)

♦Porosity & Specific Surface Area Measurements
69

Appendix 3.6 ISO Standards (International Organization for

Standardization, Geneva)

ISO/IEC Guide 2, Standardization and Related Activities - General Vocabulary

(1996).

ISO Guide 30, Terms and Definitions Used in Connection with Reference

Materials (2000).

ISO Guide 31, Contents of Certificates of Reference Materials (1996).

ISO Guide 32, Calibration of Chemical Analysis and Use of Certified

Reference Materials (1997).

ISO Guide 33, Uses of Certified Reference Materials (1989).

ISO Guide 34, Quality System Requirements for Reference Materials

Producers (1996).

ISO Guide 35, Certification of Reference Materials - General and Statistical

Principles (1989).

ISO 2859-0, Sampling Procedures for Inspection by Attributes -Part 0:

Introduction to the ISO 2859 Attribute Sampling System (1995).

ISO 8785, Geometrical Product Specification (GPS) – Surface imperfections

– Terms, definitions and parameters (1998).

ISO 9000, Qualitiy Management Systems-Fundamentals and Vocabulary

(2000).

ISO 9277, Determination of the Specific Surface Area of Solids by Gas

Adsorption Using the BET Method (1995).

ISO/WD 14488, Particle Size Analysis: Sample preparation-Sample splitting

(1999).

ISO/DIS 15901-1, Evaluation of Pore Size Distribution and Porosity of

Materials by Mercury Porosimetry and Gas Adsorption - Part 1: Mercury

Porosimetry (2004).

Appendix 3 ♦
70

ISO/DIS 15901-2, Evaluation of Pore Size Distribution and Porosity of

Materials by Mercury Porosimetry and Gas Adsorption - Part 2: Analysis of

Mesopores and Macropores by Gas Sorption (2004).

ISO/CD 15901-3, Evaluation of Pore Size Distribution and Porosity of

Materials by Mercury Porosimetry and Gas Adsorption - Part 3: Analysis of

Micropores by Gas Adsorption (2004).

Appendix 4: Glossary

For comprehensive collection of metrological terms defined by international

agreement see the Vocabulaire International des Terms Fondamentaux and

Généraux de Métrologie (BIPM, IEC, IFCC, ISO, IUPAC, IUPAP, and OIML,

1993), often referred to simply as the VIM. Terms related to the analytical

characterization of porous solids are given here for ease of reference. (See also

NIST Recommended Practice Guides SP 960-1 and SP 960-3 for additional

related terms).

acceptance number: in acceptance sampling, the maximum number of nonconforming items in a sample that allows the conclusion that the lot conforms

to the specification

acceptance sampling: sampling done to provide specimens for acceptance

testing

acceptance testing: testing performed to decide if a material meets

acceptance criteria

accepted reference value: a value that serves as an agreed upon reference for

comparison, and which is derived as (1) a theoretical or established value,

based on scientific principles, (2) an assigned or certified value, based on

experimental work of some national or international organization, or (3) a

consensus or certified value, based on collaborative experimental work under

the auspices of a scientific or engineering group

accuracy (of a test method): closeness of the agreement between the true

value (or an accepted standard value) of the property being tested and the

average of many observations made according to the test method

accuracy (of measurement): closeness of the agreement between the result of

a measurement and the value of the measurand

♦Porosity & Specific Surface Area Measurements
71

adsorbate: a molecular species that adheres to, or is adsorbed in, a surface

layer of a solid substance; a substance that is adsorbed at the interface, or into

the interfacial region, of a substrate material or adsorbent

adsorbed amount (n): the number of moles of an adsorbed gas (which may

depend on pressure and temperature)

adsorbent: the solid material or substrate on which adsorption occurs

adsorptive: the gas or vapor to be adsorbed

adsorption: the process by which a substance is accumulated at an interface

or in an interfacial region. (Should not be confused with absorption which

refers to accumulation inside a material)

agglomerate: rigid, consolidated assemblage of particles; an aggregate held

together by physical or electrostatic forces

agglomeration: formation of aggregates via physical (van der Waals,

hydrophobic) or electrostatic forces; the resulting structure is called an

agglomerate

aggregate: a cohesive mass consisting of particulate sub-units; a loose,

unconsolidated assemblage of particles

apparent density: the mass of a unit volume of powder, usually expressed as

grams per cubic centimeter, determined by a specific method; density of a

material, including closed and inaccessible pores, as determined by a stated

method

average: for a series of observations, the total divided by the number of

observations

batch: a definite quantity of some product or material produced under

conditions that are considered uniform (ASTM 456-96)

bias: in statistics, a constant or systematic error in test results (ASTM 456-96)

blind pore (dead end pore): a pore with a single connection with an external

surface

Appendix 4 ♦
72

bulk density: the mass per unit volume of a powder under nonstandard

conditions, for example, in a shipping container (not to be confused with

apparent density); includes closed and interparticle voids

bulk sample: one or more portions which (1) are taken from material that

does not consist of separately identifiable units and (2) can be identified after

sampling as separate or composed units (ASTM 456-96)

cage: intracystalline pore with windows allowing no passage of molecules bigger than a molecule of water (H2O)

calibrant: a reference material used for a calibration (ASTM E 135-04)

calibrate: to determine and record the relationship between a set of standard

units of measure and the output of an instrument or test procedure

(ASTM 456-96)

calibration: (1) the act or process of calibrating; the recorded relationship

resulting from calibrating, (2) the determination of a bias conversion factor of

an analytical process, under specified conditions, in order to obtain meaningful

results

cavity: an intracrystalline pore of finite extent and having at least one window

bigger than a water molecule

certificate of traceability: document stating the purpose, protocols, and

measurement pathways that support claims by a Traceable Reference Material

to have values traceable to specific standards or stated references; the

document references specific reports of analysis

certified reference material: reference material, accompanied by a certificate,

one or more of whose property values are certified by a procedure which

establishes its traceability to an accurate realization of the unit in which the

property values are expressed, and for which each certified value is

accompanied by an uncertainty as a stated level of confidence (VIM, 1993)

certified value: a value reported on an SRM/CRM certificate or report of

analysis for which all known or suspected sources of bias have been fully

investigated or included in an uncertainty statement or analysis.

(NIST SP 260-136)

♦Porosity & Specific Surface Area Measurements
73

channel: an intercrystalline pore allowing passage of molecules from one

distinguishable region to another.

closed pore: a pore not connected to the surface; a contiguous volume

enclosed by a solid substance

equilibrium adsorption pressure: the pressure of a substance in the gas state

in equilibrium with that substance in the adsorbate state

external surface area: area of surface of the particles, taking into account

their roughness, i.e. all cavities which are wider than deep, but not the surface

of the pore walls; the area of the smallest outermost surface forming a

boundary of a volume occupied by a solid substance

heterogeneous material: material in which sub-samples of the material will

have significantly different values of the characteristics under consideration,

compared to the mean values of the characteristics for the whole material

homogeneous material: material in which sub-samples of the material do not

have significantly different values of the characteristics under consideration

from the mean value of the characteristics for the whole material

hydraulic radius: mean value of the ratio, rh = Vp /A, for a porous solid,

where

rh = hydraulic radius,

Vp = pore volume determined at saturation, and

A = surface area (e.g. determined by the BET method)

increment: in sampling, a portion of material removed from a lot by a single

operation

information value: a value that will be of interest and use to the user of a

reference material, but for which insufficient information is available to assess

the uncertainty associated with the value. (NIST SP 260 -136)

ink bottle pore: a narrow necked open pore

interconnected pore: pore that communicates with other pores

Appendix 4 ♦
74

interlaboratory test comparison: organization, performance, and evaluation

of tests on the same or similar items or materials by two or more laboratories

in accordance with predetermined conditions(ISO Guide 2)

interlaboratory testing: the evaluation of a test method in more than one laboratory by analyzing data obtained from one or more materials that are as

homogeneous as practical (ASTM 456-96)

internal surface area: area of the pore walls, excluding the external surface

area

isotherm: in adsorption, the relation, at constant temperature, between the

amount adsorbed and the equilibrium pressure of the gas

laboratory sample: a portion of material taken to represent the lot sample, or

the original material, and used in the laboratory as a source of test specimens

laboratory sampling unit: a portion of material taken to represent one of the

lot sampling units or the original material and used in the laboratory as a

source of test specimens

macropore: pore of internal width greater than 50 nm

measurand: a quantity subject to measurement

measurement value: the numerical result of quantifying a particular property

or dimension

mesopore: pore of internal width between 2 nm and 50 nm

micropore: pore of internal width less than 2 nm

monolayer amount (nm): the number of moles of the adsorbate that forms a

monomolecular layer over the surface of the adsorbent

open pore: a cavity or channel with access to an external surface

physisorption (as distinct from chemisorption): a general phenomenon,

which occurs whenever an adsorbable gas (the adsorptive) is brought into

contact with the surface of a solid (the adsorbent)

pore size: generally, pore width; diameter or distance between opposite walls

♦Porosity & Specific Surface Area Measurements
75

of the pore; diameter of cylindrical pores; width of slit shaped pores; see also

micro-, meso-, and macropores

pore size distribution: represented by the derivatives or as a

function of rp; where Ap, Vp and rp are the wall area, volume and radius of the

pores; for cylindrical pores, rp is the pore radius; for other pore shapes, 2 rp

should be replaced by the width.

pore volume: volume of pores as determined by stated method (together, for

instance, with the nature of the probe molecule, the wavelength of the radiation used, the ultimate intrusion pressure, etc.)

porosity: ratio of the total pore volume Vp to the apparent volume V of the

particle or powder (excluding interparticle voids). In some cases one may

distinguish between open porosity (i.e. the volume of pores accessible to a

given probe molecule) and closed porosity. The methods used to measure pore

volume and apparent volume should be stated.

porous solid: solid with pores, cavities, or channels that are deeper than they

are wide

precision: the degree of agreement within a set of observations or test results

obtained as directed in a test method

precision (single operator, within-laboratory): the precision of a set of

statistically independent observations all obtained as directed in the method

and obtained over the shortest practical time interval in one laboratory by a

single operator using one apparatus and randomly draw specimens from one

sample of the material being tested (ASTM 456-96)

precision (multiple operators, within-laboratory): the precision of a set of

statistically independent test results all obtained in one laboratory using a

single sample of material and with each test result obtained by a different

operator, with each operator using one apparatus to obtain the same number of

observations by testing randomly drawn specimens over the shortest practical

time interval (ASTM 456-96)

precision (multiple operators, between-laboratory): the multi-laboratory,

single-sample, single-operator-apparatus-day (within-laboratory) precision of a

method; the precision of a set of statistically independent test results all of

Appendix 4 ♦
76

which are obtained by testing the same sample of material and each of which

is obtained in a different laboratory by one operator using one apparatus to

obtain the same number of observations by testing randomly drawn specimens

over the shortest practical time interval (ASTM 456-96)

reference material: a material or substance one or more of whose property

values are sufficiently homogeneous, stable, and well established to be used

for the calibration of an apparatus, the assessment of a measurement method,

or for assigning values to materials (VIM 1993)

reference standard: a reference material, generally of the highest metrological quality available, from which measurements are derived

reference value: a best estimate value of the true value provided on a

certificate, certificate analysis, or report of analysis where all known or

suspected sources of bias have not been fully investigated (NIST SP 260-136)

relative pressure: the ratio (p/p0) of the equilibrium adsorption pressure, p, to

the saturation vapor pressure, p0

right cylindrical pore: a cylindrical pore perpendicular to the surface

repeatability: a measure of precision and expressed as a standard deviation;

the precision of successive measurements of the same sample, or the closeness

of the agreement among the results of successive measurements of the same

measurand, carried out under the following conditions of measurement (called

repeatability conditions): the measuring instrument, the method of measurement, time, conditions of use, observer, and location.

reproducibility (of results of measurements): closeness of the agreement

between the results of measurements of the same measurand carried out under

changed conditions of measurement. A valid statement of reproducibility

requires specification of the conditions changed. The changed conditions may

include principle of measurement, method of measurement, observer,

measuring instrument, reference standard, location, conditions of use, and

time. Repeatability may be expressed quantitatively in terms of the dispersion

characteristics of the results. Results usually are understood to be corrected

results.

riffler: a device for dividing a stream of particles into representative samples

♦Porosity & Specific Surface Area Measurements
77

roughness factor: ratio of the external surface area to the area of the

geometrical envelope of the particles

sample: a portion of a material intended to be representative of the whole

(ASTM 135-98); a portion of a lot of materials which is taken for testing or

for record purposes; a group of specimens used, or observations made, which

provide information that can be used for making statistical inferences about

the population(s) from which the specimens were drawn

sample division: the process in sample preparation where the mass of the

sample is decreased by partition or extraction without change in particle size

(ISO/WD 14488-1999)

sampling plan: a procedure for obtaining a sample

sampling unit: an identifiable, discrete unit or subunit of material that could

be taken as part of a sample

saturation vapor pressure: the vapor pressure of the bulk liquified adsorbate

at the temperature of adsorption

specification: a precise statement of a set of requirements to be satisfied by

a material, product, system, or service that indicates the procedures for

determining whether each of the requirements is satisfied

specimen: a specific portion of a material or laboratory sample upon which a

test is performed or which is selected for that purpose

standard deviation, s: of a sample, a measure of the dispersion of variates

observed in a sample expressed as the positive square root of the sample variance

Standard Reference Material® (SRM®): a CRM issued by NIST that also

meets additional NIST-specified certification criteria and is issued with a

certificate or certificate of analysis that reports the results of its

characterizations and provides information regarding the appropriate use(s)

of material. (NIST SP 260-136) 

standardization: correlation of an instrument response to a standard of known

accuracy

Appendix 4 ♦
78

surface area: extent of total surface area, as determined by a given method

under stated conditions. It is essential to state the method used!

systematic sampling: the process of selecting units in a sample in accordance

with a specific order or location in time or space, or both

test method: a definitive procedure for the identification, measurement, and

evaluation of one or more qualities, characteristics, or properties of a material,

product, system, or service that produces a test result

test result: a value obtained by applying a given test method, expressed as a

single determination or a specified combination of a number of determinations

test sample: the sample prepared from the laboratory sample from which the

test portions are withdrawn

through pore: a pore which passes all the way through the sample

tortuosity: the path available for diffusion through a porous bed in relation to

the shortest distance across the bed

total porosity: ratio of volume of pores and voids to overall volume

traceability: the property of a result of a measurement or the value of a

standard whereby it can be related to stated references, usually national or

international standards, through an unbroken chain of comparisons all having

stated uncertainties (VIM, 1993); the ability to trace the history, application or

location of an item or activity and like items or activities by means of recorded

identification. (ANSI/ASQC A3-1987, A)

NIST Traceable Reference MaterialTM (NTRMTM): a commerciallyproduced reference material with a well-defined traceability linkage to existing

NIST standards for measurements. (NIST SP 260-136)

true density: the mass of a particle divided by its volume excluding pores and

interparticle voids

uncertainty: an indication of the variability associated with a measured value

that takes into account two major components of error: (1) bias and (2) the

random error attributed to the imprecision of the measurement process

(ASTM 456-96)

♦Porosity & Specific Surface Area Measurements
79

validation: proof, with reference materials or materials traceable to them, that

a method is acceptable for all elements in its scope (ASTM 135-98)

void: space between particles

Appendix 5 : Abbreviations

A5.1 Organizations

AFNOR: Association Française de Normalisation

ASTM: American Society for Testing Materials

BAM: Bundesanstalt für Materialforschung und -prüfung, Germany

BIPM: Bureau International des Poids et Mesures

BSI: British Standards Institution

CEN: Comité Européenne de Normalisation

DIN: Deutsches Institut für Normung

IEC: International Electrotechnical Commission

IFCC: International Federation of Clinical Chemistry

ISO: International Organization for Standardization

IUPAC: International Union of Pure and Applied Chemistry

IUPAP: International Union of Pure and Applied Physics

NIST: National Institute of Standards and Technology, USA

OIML: Organisation Internationale de Métrologie Légale

A5.2 Scientific items

BET method: Method of Brunauer, Emmett, and Teller

BJH method: Method of Barrett, Joyner, and Halenda

CRM: Certified Reference Material

DFT: Density Functional Theory

DR method: Method of Dubinin-Radushkevich

HK method: Method of Horvath-Kavazoe

RM: Reference Material

SF method: Method of Saito-Foley

SRM®: Standard Reference Material (NIST)

STP: Standard Temperature (273.15 K) and Pressure (101.325 kPa)

VIM: Vocabulaire International des Terms Fondamentaux and

Généraux de Métrologie (issued jointly by BIPM, IEC,

IFCC, ISO, IUPAC, IUPAP, and OIML)

Appendix 5 ♦