960-

Particle Size 

Characterization

Ajit Jillavenkatesa

Stanley J. Dapkunas

Lin-Sien H. Lum

NIST recommended

Pu
Particle Size

Characterization

Ajit Jillavenkatesa

Stanley J. Dapkunas

Lin-Sien H. Lum

Materials Science and 

Engineering Laboratory

January 2001

U.S. Department of Commerce

Donald L. Evans, Secretary

Technology Administration

Karen H. Brown, Acting Under Secretary of 

Commerce for Technology

National Institute of Standards and Technology

Karen H. Brown, Acting Director

UNITED STATES OF AMERICA

DEPARTMENT OF COMMERCE

Special Publication 960-1
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 960-1

Natl. Inst. Stand. Technol.

Spec. Publ. 960-1

164 pages (January 2001)

CODEN: NSPUE2

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iii

Preface ◆

PREFACE

Determination of particle size distribution of powders is a critical step in almost

all ceramic processing techniques. The consequences of improper size analyses

are reflected in poor product quality, high rejection rates and economic losses.

Yet, particle size analysis techniques are often applied inappropriately, primarily

due to a lack of understanding of the underlying principles of size analysis, or

due to confusion arising from claims and counter-claims of the analytical ability

of size determination techniques and instruments.

This guide has been written to address some of these issues and concerns in

this regard. The guide is by no means an exhaustive and comprehensive text on

particle size analysis, but attempts to convey the practical issues that need to

be considered when attempting analysis by some of the more commonly used

techniques in the ceramics manufacturing community. The document is written

to guide persons who are not experts in the field, but have some fundamental

knowledge and familiarity of the issues involved. References to pertinent

international standards and other comprehensive sources of information have

been included. Data and information from studies conducted at the National

Institute of Standards and Technology, and experience gained over years of

participation in international round robin tests and standards development, have

been used in developing the information presented in this text.

The authors would like to thank and acknowledge the considerable help and

contributions from Steve Freiman, Said Jahanmir, James Kelly, Patrick Pei

and Dennis Minor and of the Ceramics Division at NIST, for providing critical

reviews and suggestions. Thanks are also due to Ed Anderson, Tim Bullard

and Roger Weber (Reynolds Metals Co.), Mohsen Khalili (DuPont Central

Research and Development), Robert Condrate (Alfred University) and

Robert Gettings (Standard Reference Materials Program, NIST) for their

role as reviewers of the document. Leslie Smith, Director of the Materials

Science and Engineering Laboratory at NIST, is thanked for his support in

the production of this guide.

It is our hope that this guide will be added to and revised over the years to

come. Please direct your comments and suggestions for future additions and

about this text to:

Stephen Freiman Said Jahanmir

Chief, Ceramics Division Group Leader, Ceramics Division

National Institute of National Institute of

Standards & Technology Standards & Technology

Gaithersburg, MD 20899 Gaithersburg, MD 20899

USA USA

.........................
v

Table of Contents ◆

Preface ................................................................................................. iii

1. Introduction to Particle Size Characterization ..................................1

2. Powder Sampling .............................................................................7

3. Size Characterization by Sieving Techniques ............................... 27

4. Size Characterization by Gravitational

Sedimentation Techniques .............................................................49

5. Size Characterization by Microscopy-Based Techniques .............69

6. Size Characterization by Laser Light

Diffraction Techniques ...................................................................93

7. Reporting Size Data .....................................................................125

Glossary. Terms Related to Particle Size Analysis

and Characterization ..........................................................................139

Annex 1. Some Formulae Pertaining to Particle

Size Representation ...........................................................................161

Annex 2. NIST RM/SRMs Related to Particle Size

Characterization .................................................................................165
1

1. INTRODUCTION TO PARTICLE SIZE

CHARACTERIZATION

Knowledge of particle sizes and the size distribution of a powder system is a

prerequisite for most production and processing operations. Particle size and

size distribution have a significant effect on the mechanical strength, density,

electrical and thermal properties of the finished object. Significant production

losses can be incurred due to high rejection rates if size and size distribution

of powders being used in a process are not adequately controlled. In most

instances powder suppliers provide size and size distribution information,

but that information needs to be checked for quality control purposes. Also,

the validity of the supplied size information needs to be evaluated if the

powder batch has been divided into smaller batches based upon the user’s

requirements. In certain processing operations, powder size and size

distribution may have to be monitored at different stages of a processing

operation for suitable process control.

Powder size and size distribution can be determined using numerous

commercially available instruments, or using instruments designed for

operations in very specific environments. Some of these instruments can be

used in an on-line mode of operation (i.e., integrated into the manufacturing

process) or can be used off-line. The number of instruments that can be

used in both modes, with minimal adjustments is steadily increasing. Some

manufacturers now provide accessories that enable off-line instruments to be

adapted to on-line operations. Most instruments operate in a batch mode of

analysis. There are few instruments that can be used in a continuous mode.

Instruments can be used for analysis of dry powders and powders dispersed in

suspension. Whenever possible, it is desirable to analyze powders dispersed in

a state similar to that in which they will be used. Instruments are available that

enable analysis over a fairly narrow size distribution or over a very broad size

range. The choices available for particle size determination instruments can

be quite confusing. Table 1.1 illustrates some of the existing and emerging

techniques for size analysis, the physical principle on which the technique is

based, and the general sample concentration and size range over which these

techniques are applicable.

Factors to be considered before selecting an instrument for size and size

distribution analysis include1, but are not limited to:

1. the amount of sample available;

2. the desired number of points on the size distribution (for data interpretation

and/or resolution);

3. number and frequency of analyses required (laboratory, fast-response on

on-line methods);

Introduction to Particle Size Characterization ◆
2

◆ Introduction to Particle Size Characterization

Table 1.1. able 1.1.

Particle Size Analysis Instruments Based on Different Physical Principles

Instrumental Technique Physical Principle Volume Fraction Range (%) Size Range(μm) 

Acoustic Attenuation Spectroscopy (N) [Ulltrasonic Attenuation Spectroscopy] 

ultrasonics > 1 0.05 to 10 

Centrifugal Sedimentation-Optical (E) sedimentation ID 0.01 to 30 Centrifugal Sedimentation-X-Ray (E) sedimentation ID 0.01 to 100 Electrical Resistance Zone Sensing (E) [Particle Counting, Coulter Counter] 

particle counting < 0.1 0.4 to 1200 

Electroacoustic Spectroscopy (N) [Electrokinetic Sonic Amplitude] 

ultrasonics > 1 0.1 to 10 

Gas Absorption Surface Area Analysis (E) [BET Absorption] 

surface area analysis NA NA 

Laser Light Diffraction (E) [Static Light Scattering, Mie Scattering, Elastic Light Scattering] 

electromagnetic wave interaction 

and scattering 

0.01 to 5 0.04 to 1000 

Light Microscopy (E) particle counting NA > 1.0 Quasi-Elastic Light Scattering (E) [Dynamic Light Scattering, Photon Correlation Spectroscopy, Optical Beating Spectroscopy] 

 <0.005 0.003 to 6 

Scanning Electron Microscopy (E) particle counting NA >0.1 X-Ray Gravitational Sedimentation (E) sedimentation ID 0.5 to 100 Colloid Vibration Current (N) [Single Frequency] 

ultrasonics > 1 <10 

Electrokinetic Sonic Amplitude (N) [single frequency] 

ultrasonics > 1 <10 

Microelectrophoresis (E) [Laser Light Scattering, Quasi-Elastic Light Scattering] 

electromagnetic wave interaction 

and scattering 

<0.001 0.1 to 1 

Sieving (E) sieving NA 5 to 100,000 E: Established Technique N: Emerging Technique ID: Instrument Dependent Information compiled by V. A. Hackley, Ceramics Division, NIST
3

4. extent of human intervention and degree of operator skill required;

5. cost of instrument and other accessories, such as those needed for sample

preparation, evaluation of data, etc.;

6. degree of automation required.

It is important to understand that as different instruments are based on different

physical principles, there are bound to be differences in the size results obtained

from these instruments. Furthermore, even when using instruments based on

the same physical principle, manufacturers’ use of proprietary algorithms,

components and variations in adaptations of the same basic physical principle

can give rise to significant variations in the measured size. Thus, comparison of

size and size distribution results from different instruments should be conducted

with extreme caution. Suitable protocols can be developed for comparison of

size results from different instruments. The development of such protocols

would require extensive analysis of standard, well characterized powders

similar to that of interest. The analysis would have to be conducted over a

broad size range on all the instruments to be used, and under conditions similar

to those under which the powder of interest would be analyzed. Statistical

considerations for deviations in particle shape would have to be factored into

the development of such protocols.

Given these considerations, numerous users prefer to consider particle size

and size distribution results as relative measurements, where results from one

run can be compared with those from another, obtained on the same instrument

under similar conditions. The use of particle size and size distribution data as

relative measurements is very significant when monitoring size distribution as

a process control indicator. In such instances the magnitude and/or direction

of change is indicative of changes in the manufacturing process, which

can significantly impact on the quality of the product. Often, users define

acceptable limits for these fluctuations and monitor size and/or size distribution

data to ensure that it lies within these limits. In these applications the precision

of the measurement is of far greater interest than the accuracy. Absolute

measurements can be conducted with some degree of reliability when using

techniques such as microscopy-based analysis. In these cases, the particles

being counted for size analysis can be visually examined. However, in such a

case the reliability of the absolute measurement can be affected by the number

of particles that are counted, the representative nature of the particles included

in the analysis, the shape of the particles, the state of dispersion and the

specimen preparation technique followed.

The use of primary and secondary standards for calibration can significantly

improve the reliability of analysis. Standards for calibration can also be

developed from products used by the analyst on a regular basis. In these

instances it is critical to compare the material used as the standard against
4

a primary standard to ensure tractability. The development of protocols for

sample preparation, analysis and data interpretation particular to the material

system and/or the instrument is a good practice. Reliable and reproducible

size analysis can be conducted by thoroughly understanding the instrument

manufacturer’s instructions for instrument operation. However, the experience

gained by the operator through experimentation with different material systems

and analysis under different operating conditions is invaluable in understanding

the limitations and capabilities of the instrument. A schematic flowchart of the

various stages involved in size analysis is illustrated in Figure 1.1.

Effective communication between powder manufacturers/suppliers and

customers is a very critical factor in the development of robust size

measurement procedures and protocols. This communication includes

information about the raw material used for powder manufacture, procedures

used for powder production, instruments and instrumental parameters used

for analysis, information about sample preparation and powder characteristics.

Establishing mutually acceptable protocols and procedures engenders a level

of trust that can drastically reduce the amount of time and resources that needs

to be devoted for analysis of each powder lot. Incorporating appropriate audits

also enable tracking performance and conformity to the agreed upon protocols.

Such procedures are now also required for ISO 9000 certification.

This guide is designed to provide a general user, not very familiar with issues

related to particle size characterization, guidelines on various aspects of particle

size and size distribution analysis. While by no means an exhaustive resource,

this guide concentrates on some techniques commonly used in the ceramic

manufacturing industry. The techniques examined in this guide include sieving,

gravitational sedimentation, laser light diffraction, and microscopy-based

methods. The capabilities and limitations of these techniques are examined,

together with discussions of the general principles on which these techniques

are based. Procedures and precautions generally followed for sample

preparation and analysis are presented, and sources of error bias and other

variations are identified. A chapter on sampling of materials discusses some

fundamental issues concerning representative sampling applicable to techniques

described herein. The reader should note that this guide is applicable to size

and size distribution analysis of particles rather than crystallites. This note of

caution is mentioned as often particle size and crystallite sizes can be very

similar. This is encountered when Bayer derived calcined alumina powders are

fully ground to their ultimate crystallite size2. The same powders when tested in

a partially ground or unground state indicate significant difference between the

particle size and the crystallite size.

All data reported in this guide have uncertainties that are estimated as ± one

standard deviation, unless otherwise specified.

Introduction to Particle Size Characterization ◆
5

Figure 1.1. Figure 1.1.

Flowchart of General Procedures Followed for Particle Size and

Size Distribution Analysis

◆ Introduction to Particle Size Characterization

Sampling from bulk 

Sub-sample from bulk samples 

for specimen 

Disperse powder 

(for both dry and wet) 

Specimen preparation 

Instrument calibration 

and set-up

Conduct size analysis (including checks 

for reproducibility) 

Post processing of data 

(if necessary) 

Data interpretation 

Reporting of results 
6

References

1. L. Svarovsky, Characterization of Powders, in Principles of Powder

Technology, M. Rhodes, ed., John Wiley and Sons, Chichester,

UK (1990) p. 35.

2. E. Anderson, Personal Communications (2000).

Introduction to Particle Size Characterization ◆
7

2. POWDER SAMPLING

1. Introduction

2. Techniques of Bulk Sampling

2.1 Large containers sampling

2.2 Bag sampling

2.3 Heap sampling

3. Sample Sub-division

3.1 Cone and quartering

3.2 Scoop sampling

3.3 Table sampling

3.4 Chute splitting

3.5 Spin riffling

4. Practical Application

5. Relevant Standards

1. Introduction

Reliable powder sampling constitutes the first step of most powder

characterization and processing procedures. The goal of powder sampling

is to collect a small amount of powder from the bulk quantity, such that this

smaller fraction best represents the physical and chemical characteristics of

the entire bulk. Thus, the samples necessarily need to be representative of the

bulk. The consequences of incorrect and/or non-representative sampling can

be significant and result in poor characterization, systems/process failure, large

fraction of defective parts, high rejection rates, and customer dissatisfaction.

It is critical to spend the required amount of time, effort and resources in

developing sampling procedures that are robust and reliable. It is just as

important to recognize that a sampling procedure that works well for one

material system may perform inadequately for another material system. The

sampling procedure should be developed or adapted considering the following

issues:

(a) quantity of powder from which samples are being obtained,

(b) amount of sample required,

(c) powder characteristics, including but not limited to flow characteristics of

the powder, shape of the powder particles, size of the powder particles,

tendency to segregate, surface chemistry that may cause the powder to

be hygroscopic, etc.,

Powder Sampling ◆
8

(d) mechanical strength of the powders, i.e., are the particles friable and thus,

likely to fracture during transport or during sampling,

(e) mode by which powders are transported,

(f) possibility of powder contamination, and acceptable limits of contamination,

(g) duration of time needed to conduct the sampling procedure (this factor

requires careful consideration, as sampling procedures can be cumbersome

and time-consuming; the desire to save time and money should be balanced

with the need for a high level of certainty about the representative nature of

the samples), and

(h) skill and knowledge of operators conducting the sampling process.

The following basic guidelines can be followed to aid obtaining representative

samples. These are colloquially known as the “golden rules of sampling”1.

1. A powder should be sampled when in motion.

2. The whole stream of powder should be taken for many short increments

of time in preference to part of the stream being taken for the whole of

the time.

It is also important to note that the entire quantity of the sample powder

should pass through the device used for sampling. Sampling procedures can be

supplemented by the use of established and proven mathematical techniques.

Such techniques of “statistical sampling” require the inspection of only a

fraction of the bulk and thus greatly reduce the amount of time and effort

spent in sampling. They also provide the advantage of calculating the level of

certainty of detecting mistakes, upon applying a particular level of inspection15.

A typical example of the need for sampling is illustrated where a process

engineer may be asked to measure the particle size distribution (PSD) of a

powder before and after processing operations. A starting batch of 500 kg

(bulk) of powder may be contained in several bags, but the test specimen

required for analysis is only 2 g. The issue to be addressed would be what

sampling processes can be developed or followed to derive from the 500 kg lot

test samples of 2 g, such that these samples are representative of the larger

lot? Considering the issues mentioned above, it may be decided that powder

samples would have to be withdrawn from different regions of different bags

(seemingly contradictory to the first golden rule, the application of which would

be impractical here), blended to homogenize the powders, and then reduced

into smaller sub-samples. Using a randomization technique, some of these

sub-samples would be selected, blended again and checked for homogeneity.

Using an iterative process of random sub-sample selection and blending, the

quantity of the sub-sample lot would finally be reduced to the desired amount.

◆ Powder Sampling
9

This process, though time consuming, would best ensure the development of

a representative sub-sample.

As powder batches can vary from a few grams to several tons, there

are various devices and techniques that have been developed to aid in

representative sampling of powders. The design of these incorporate the

“golden rules” to the greatest possible extent. Together with statistical sampling

procedures, these can provide a high degree of reliability and homogeneity in

the samples produced. Some of these techniques are fairly independent of the

operator, while others are strongly influenced by human factors. It is thus

necessary to develop and implement practices and controls in these procedures

that document and track any error or variation that may be introduced by

human factors.

2. Techniques of Bulk Sampling

Issues of bulk sampling arise when dealing with powder quantities larger

than about 25 kg. Powders in bulk quantities can be shipped in containers of

various sizes and shapes, including bags, drums, carboys, hopper cars, etc.

Due to the large quantity of powder involved, generating a small amount that is

representative of the bulk can be extremely challenging. Often, the techniques

used for sampling combine experience, gained over the years dealing with

material systems, with simple sampling devices and mathematical procedures.

This section describes some of the issues associated with bulk sampling1-3.

2.1 Large Container Sampling

Powder sampling from large containers (ranging from a hundred kilograms to

hundreds of kilograms) is typically encountered when manufacturers receive

powders from raw material suppliers, or powder manufacturers ship out

powders in bulk quantities to customers. Typical examples would include

manufacturers of paints and pigments, certain segments of ceramic component

and whitewares manufacturing industries, etc. Due to the large quantities of

powder involved, sampling at this stage is mostly to answer questions of a

“go – no go” nature, i.e., does the powder batch meet the customer’s minimal

size requirements such that the lot is acceptable or not?

The ideal location for sampling would be from the moving stream of powder

when filling the container. However, as this may not be practical or feasible for

the customer, a mutually agreed upon procedure developed by the customer

and the supplier may have to be used by the supplier. This requires significant

trust between the participants and a proven track record for the supplier.

Regular audits need to be conducted to ensure that the sampling procedures

meet the agreed upon requirements. In a similar manner, it is just as practical to

Introduction ◆
10

sample the powder from a moving stream upon unloading from the container.

This would require that the entire container be discharged, and the powder

stored.

Samples can also be obtained through a static sampling procedure. A static

sampling procedure refers to sampling from powder loads at rest. This requires

the use of a sample thief. A sample thief is a long rod with a sampling chamber

at the end of the rod. The sample chamber can be opened and closed with

controls at the operator’s end of the rod. The sample thief is placed at several

specifically selected areas. With the chamber closed, the thief is inserted into

the selected area. The thief is opened and filled with the powder. The chamber

is closed and then removed from the area. This procedure is repeated until all

the selected sites are sampled. This sampling method is based upon subjective

decisions as to the best areas for removal of the sample. Sampling should be

done at several sites throughout the container. The choice of sites to be

sampled is dictated by understanding of factors such as possible powder flow

and segregation during transport. Sampling of fine powder less than about

2 μm by a sample thief may not be suitable as these powders do not flow

well into the thief and the use of these often leads to cross contamination.

For powders having a large particle size distribution, fine powders will tend to

segregate into the interstitial spaces between large particles, causing the large

particles to rise to the top of the container. Heterogeneous packing of powders

may occur due to irregular motion during transport. Once all the samples have

been obtained, the selected samples may be combined into one bulk sample.

This bulk sample can then be divided using a spinning riffler or other subsample dividing processes to obtain the necessary smaller laboratory sample.

Allen1 provides some guidelines for sampling from wagons and containers

encountered in bulk shipments.

2.2 Bag Sampling

When sampling powders that are shipped in bags, it is best to obtain a

representative sample while the powder is in motion. This can be done while

the powder is being poured out of the bag. If obtaining samples when the

powder is in motion is not possible, a static process as described previously can

be used. Using a sampling probe, samples are taken from several specific spots

within each bag. Powder samples should be obtained from the bottom, center

and top of the bags to account for any sample segregation. Powders should

also be sampled from the front and the rear of the bags, depending on how the

bags were stacked during transport. Particular attention should be given to the

corners of the bags as these tend to trap powder particles in a disproportionate

manner. Variations may occur between the bags. To reduce this variability,

powder samples should be taken from different bags with the help of a random

◆ Powder Sampling
11

number selection process. This process involves assigning a number to each

sampling spot, and sampling from some or all of these spots in a random order.

The random order to be followed can be created by any simple random number

generator. The advantage of following such a process is to minimize any

operator bias during the sampling process. After all the samples have been

selected, the powders are blended to produce a combined sample. This leads

to a powder sample that is a more reliable representation of the entire batch

of powders in all the bags. This sampling procedure is based on the assumption

that the powders in all the bags will be blended together prior to their use in

subsequent processing.

2.3 Heap Sampling

Ensuring representative sampling from a heap is one of the hardest tasks in

practical sampling. Segregation of the powders is very prevalent when forming

the heap. Segregation occurs due to differences in flow characteristics of

powders and their differences in particle motion due to size. The fine particles

tend to stay close to the center of the heap while the coarse particles segregate

to the peripheral regions of the heap. This segregation is a problem from a

sampling point of view. To reduce the extent of segregation, the material

should by thoroughly mixed with a scoop by turning the heap over several

times. With the last turn, the heap should be formed into a conical pile by

placing each scoop of material on top of the previous one. Bias in sampling

can be minimized by sampling while pouring the powder to form the heap.

Thus, samples may be obtained when the material is being poured out from

a container to form the heap.

3. Sample Sub-Division

After samples have been obtained from the bulk, they still need to be further

sub-divided in order to reduce the sample size to a manageable quantity. In the

following sub-sections, some of the common sampling devices used for dividing

bulk samples into laboratory samples are discussed1,4. The cone and quartering,

and scoop sampling methods are relatively simple methods of dividing the

material using very simple tools. These are very widely used techniques;

however, both methods are significantly influenced by operator error, and

should be used only if application of other techniques of sample subdivision is

not practical. Both methods require that the material be formed into a heap

before sampling. Segregation of the material may contribute to errors in

obtaining a representative sample. Table sampler, chute splitter, and spinning

riffler are more efficient sampling devices. Kaye4 describes studies performed

to determine the efficiencies of scoop sampling, cone and quartering, the chute

riffler and the spinning riffler methods. The spinning riffler was the most

efficient of the methods studied.

Techniques of Bulk Sampling ◆
12

3.1 Cone and Quartering

The cone and quartering 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 four 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 specimen size desired for conducting the measurement is

obtained. This sampling technique is best for powders that have poor flow

characteristics and thus, minimal segregation. This technique is strongly

operator-dependent. Errors can occur due to differences in the manner the

heap is formed and sub-divided. An improperly formed heap will lead to the

propagation of the error introduced initially, with every further round of sample

sub-division.

3.2 Scoop Sampling

The scoop sampling technique is a widely used, simple technique for sample

division. An operator, using a scoop, extracts the laboratory sample from some

portion of the bulk sample. This technique is only appropriate for materials that

are homogenous and exhibit poor flow characteristics. One of the drawbacks

of this technique is that all the bulk material does not go through the sampling

process. The other drawback is the dependency on the operator to decide

where to scoop the material and what quantity of the sample to extract. If

the sample material is taken from the surface, the sampling is more likely to

be atypical due to segregation of the material. To eliminate the segregation,

the material should be shaken in a container before pouring into a heap and

the scooping the sample. Allen and Kaye1,4 describe this shaking method.

3.3 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. A series of holes is drilled into the table surface, to allow the

powder to fall through. Guides in the shape of triangular prisms are placed in

rows perpendicular to the direction of powder flow. These prisms serve to

divide and guide the powder moving along the table. The interaction of the

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

different fractions. Some of the powder quantity falls through the holes. The

remaining powder passes on to the next series of prisms and holes where more

of the powder is removed. The procedure is continued until the remaining

powder reaches the bottom of the plane. The powder that is collected at the

end of the table is the sample. A table sampling device is shown in Figure 2.1.

◆ Powder Sampling
13

In this table sampler, there are four prisms in the front, two in the center

and one at the end. Errors can result from variations in the initial feed of

the material at the top of the table. The initial feed also needs to be mixed

thoroughly such that there is no segregation of the material that is present in

the feed. After powder separation, it is critical to blend.

Figure 2.1.

A Schematic Diagram of a Table Sampler1

(Reproduced with permission from Kluwer Academic Publishers)

3.4 Chute Splitting

In the chute splitter device, the sample powder is introduced in a V-shaped

trough with two powder collecting trays placed on either side of the trough.

Upon introduction, the sample powder flows through a series of chutes feeding

into the two collection trays. This process is repeated with powder from one

of the trays until the desired amount of sample is obtained. An example of a

chute splitter is shown in Figure 2.2. This splitter contains twelve chutes that

feed into two trays. Each time the material goes through the splitter, the initial

sample is reduced in half. In order to obtain a good representative sample,

precautions should be in place to ensure sample loading into the trough without

causing powder segregation. Operator bias can affect the results through

unequal splitting of the sample by the method of feeding the material.

Sample Sub-Division ◆
14

Figure 2.2. Figure 2.2.

Chute Splitter (Figure from presentation by M. Kahili,

Sampling Particulate Materials)

3.5 Spin Riffling

A spinning riffler (see Figure 2.3) causes division of the sample powders into

a series of containers 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 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. The operator can control the velocity of the circular motion and

also the amplitude of the vibratory motion of the feeder. Thus, different

powders with varying flow characteristics can be sub-divided using the same

device. The vibratory motion of the feeder causes the powders to flow into

the receivers at a uniform controlled rate. The spinning riffler is ideal for

sub-division of free flowing powder. The quantity of powders that can be

obtained as samples depends on the volume of the glass containers that can

be accommodated on the riffler. Spinning rifflers are available in different sizes,

enabling sub-division of powders from a few milligrams to grams. A drawback

of this technique is that the number of sample containers that can be filled at

one time is often small. The maximum number of containers that can be filled

in one cycle is dictated by the number that can be accommodated on the

circular holder.

3.6 Comparison of Different Sampling Techniques

Allen and Khan5 conducted a study to determine the reliability of sampling

devices. The material tested was a mixture having a volume fraction of 60 %

coarse sand and 40 % fine sand. The coarse and fine binary mixtures were

◆ Powder Sampling
15

Mass feed

hopper

Vibratory

feeder

Spinning riffler

Figure 2.3. Figure 2.3.

A Schematic Diagram of a Spinning Riffler1

(Reproduced with permission from Kluwer Academic Publishers)

prepared by sieving and blending. The coarse grade was composed of 420 μm

to 500 μm size sand and the fine grade was composed of 120 μm to 250 μm

size sand. It was determined that the spinning riffler was the most reproducible

technique for sample sub-division, while the cone and quartering method was

the least. Reproducibility of the technique is based on the standard deviation in

size distribution observed between different sub-samples generated from the

same primary sample. The ability of the spinning riffler to divide the mixture

uniformly was far superior to the other methods. Different sampling methods

with associated relative standard deviations are shown in Table 2.1. It is

obvious from the table that sub-samples produced by spin riffling had the least

standard deviation in size distribution between different sub-samples of the

same powder lot. Allen1 discussed other studies that also show spin riffling to

be the most reliable technique. Table 2.2 shows a summary of the advantages

and disadvantages for each of the sampling devices.

Method Relative Standard Deviation (%) 

Cone & Quartering 6.81 

Scoop Sampling 5.14 

Table Sampling 2.09 

Chute Riffling 1.01 

Spin Riffling 0.125 

Table 2.1. able 2.1.

Relative Reliability of Sampling Devices

Sample Sub-Division ◆
16

Figure 2.4. Figure 2.4.

Flowchart Indicating Steps in a Typical Sampling Procedure

◆ Powder Sampling

Obtain samples by established sampling plan

Blend the collected sample powder 

Divide blended powders into appropriately 

sized sub-samples

Conduct homogeneity tests on sub-samples 

If homogeneity tests are acceptable, use samples for representative analysis. 

If unacceptable, re-examine sampling plan and repeat sampling procedure 

Identify volume of bulk material to be sampled 

Determine powder characteristics 

(e.g., flowing, non-flowing, homogeneous, heterogeneous, etc.) 

Determine quantity of sample powder needed 

Identify sampling technique/tool to be used 

Develop sampling 

plan using tools of 

random sampling 

and statistical 

sampling. Sampling 

plan should 

identify: 

a. quantity of 

powder per 

sample 

b. number of 

samples 

c. acceptable 

variation 

among the 

samples. 
17

Table 2.2. able 2.2.

Comparison of Sampling Devices

3.7 General Procedure

A general sampling procedure involves identification of certain critical

parameters such as the size of the bulk sample, the size of each laboratory

sample, the number of laboratory samples, powder characteristics, available

tools, and development and selection of procedures that work best for the given

powder system. Homogeneity testing is a critical component of any sampling

procedure as it identifies inconsistencies between sub-samples and thus helps

to ensure that the sub-samples are representative of the bulk sample. It is

necessary to establish practical limits for acceptable results in homogeneity

testing. The use of random sampling and statistical sampling tools can greatly

aid in the sampling process, by reducing the number of sub-samples needed,

while enhancing the reliability of the homogeneity tests. Figure 2.4 shows

the steps that are involved in the design of a typical sampling process for

dry powders.

4. Practical Application

Various techniques for sampling have been described previously. A practical

example of a sampling procedure used in the production of a NIST Standard

Reference Material (SRM) is discussed here. This example will provide

an illustration of the sample division process and the homogeneity study

associated with the SRM certification. The homogeneity test measurements

were conducted to detect any significant differences in the properties of the

selected samples. Any significant differences are an indication that the various

sub-samples are not similar to each other and cannot be considered to be

representative of the bulk material. Each of the samples tested was randomly

selected. Randomly selected samples are important as they minimize operator

Sampling Device Advantages Disadvantages 

Cone & Quartering Good for powders with poor 

flow characteristics 

Operator-dependent 

Scoop Reliable for homogeneous 

and non-flowing powder 

Particle segregation 

Table Ability to separate large 

quantity of material 

Dependent on the initial 

feed 

Chute Ability to reduce powder 

samples in half after one 

pass 

Operator bias 

Spinning riffler Reliable for free flowing 

powder 

Inability to do large 

quantity of powder 

efficiently 

Sample Sub-Division ◆
18

bias in selecting samples for testing. The number of randomly selected samples

depends on the total number of samples in the lot and the type of measurement

test conducted.

The study presented here involves the development of SRM 1982: Thermal

Spray Zirconia Powder-Particle Size Distribution13,14. The size range of the

powder system used in SRM 1982 is from 10 μm to 150 μm. Certification was

conducted by scanning electron microscopy, laser light scattering, and sieving

methods.

In 1992, at a NIST-sponsored Thermal Spray Workshop, powder

characterization and quality control were identified as critical areas for

producing cost-effective thermal spray coatings. Thus, the production of a

Particle Size Distribution (PSD) Standard Reference Material was initiated

and completed in 1996 to provide a reliable and accessible PSD reference

material for a thermal-sprayed yittria stabilized zirconia (YSZ) powder.

Different processes can be used for the production of YSZ powder,

including sol-gel precipitation and sintering; spray drying and sintering or

plasma densification; fusion and crushing or agglomeration and sintering.

Powder samples by each of these manufacturing processes were obtained

for evaluation as a suitable candidate for SRM production and certification.

The desired properties of the YSZ powder for SRM production were identified

to be (a) well defined PSD, (b) particles of spherical shape, (c) absence of

extensive agglomeration, and (d) high mechanical strength and low friability.

After a thorough evaluation of YSZ powders for the properties listed above,

a candidate material was chosen. It was a powder produced by a spray drying

and sintering process.

The particle shape and degree of agglomeration of the candidate powder

determined by SEM imaging at different magnifications is shown in Figure 2.5.

The low friability of the powder was demonstrated by comparing the PSD of

the powder before and after ultrasonication with a high wattage sonic horn.

Figure 2.6 shows the size distribution curve of the powder determined by laser

light scattering before and after ultrasonication. The very minimal change in

the shape and position of the distribution curve is indicative of the absence

of significant fracture (and thus, size reduction) during ultrasonication. The

powders were thus considered to possess an adequate mechanical strength

such that the particles would not fracture during normal handling and

transportation.

A 9.08 kg batch of powder was first split into 16 sub-samples using a spinning

riffler. These sub-samples were designated A to P. Each was further riffled to

produce 16 smaller bottles, designated A-1 to A-16. The contents of five of the

smaller bottles (A-1 to A-16) were combined and riffled for the third time into

◆ Powder Sampling
19

20 μm 10 μm

250 μm 100 μm

Figure 2.5. Figure 2.5.

SEM Micrographs of YSZ Powder at Different Magnifications to

Illustrate Powder and Particle Morphology13

Figure 2.6. Figure 2.6.

Changes in PSD of YSZ Powder Upon Ultrasonication13

Sonicated

Not Sonicate

Sonication at 45 W for 2 min

100

80

CUMULATIVE VOLUME, %

PARTICLE SIZE, μm

0.1 1.0 10.0 100.0

60

40

20

0

Practical Application ◆

Not Sonicated

Sonicated

Sonication at 45 W for 2 min
20

16 containers. Each container was weighed to ensure evenness of splitting. It

was determined that each sample bottle should have 11 + 0.5 g. The containers

were numbered consecutively from #1. This packaging scheme is summarized

in Table 2.3. A randomized set of 100 bottles from the last stage was selected

for sample homogeneity testing and PSD certification analyses.

Table 2.3. able 2.3.

Packaging Scheme for SRM 1982

The PSD of 15 stage 4 samples from the randomized set were determined in

triplicate and in a random sequence using a laser light scattering instrument.

The same specified specimen preparation procedure was used for each

specimen. The applied instrument and specimen related parameters were

(a) sample size for each analysis of 0.2 g, and (b) relative refractive index

of 1.8 with respect to water (refractive index of water was 1.33 with zero

imaginary component of refractive index). The particle size data were

calculated on a volume basis assuming a normal distribution.

Although the laser light instrument produces a continuous plot of weight

percentage finer than a given diameter, five cumulative percentiles (10 %,

25 %, 50 %, 75 %, and 90 %) were selected as a representative data set. The

homogeneity data were statistically analyzed and are shown in Figure 2.7.

After certification analyses, SRM 1982 was accepted and made available to

the public.

◆ Powder Sampling

Stage 1 9.08 kg of Zirconia Powder (starting powder)

Stage 2 16 sub-samples of powder (label A to P)

Stage 3 Each stage 2 sub-samples ( A) split into 16 smaller containers

(label A-1 to A-16)

Stage 4 Five smaller containers (A-1 to A-16) were recombined and

split into 16 bottles
21

120

PARTICLE SIZE, μm

BOTTLE NUMBER

0 100 200 300

d90

d75

d50

d25

d10

400 500 600 700

100

80

60

40

20

Figure 2.7. Figure 2.7.

Measure of Variability Between Bottles by a Laser Light Instrument 13

5. Relevant Standards

Numerous American Society for Testing and Materials (ASTM) standards

have been developed for sampling processes. Many of the standards discussed

here are not written specifically for ceramic powders. However, they can be

used for guidance in sampling bulk shipments or dividing bulk materials into

smaller samples.

ASTM C322 Standard Practice for Sampling Ceramic Whiteware Clays6 is

designed for sampling of bulk and bagged shipments of ceramic whiteware

clays. This standard provides a detailed sampling procedure for bulk shipments

(e.g. obtaining samples of 4.5 kg from containers that can hold upto 30 tons

of bulk material). The standard, however, does not describe any procedure for

further sample sub-division, and such a procedure is needed as 4.5 kg is too

large a quantity for most laboratory analyses.

ASTM D1900 Standard Practice for Carbon Black—Sampling Bulk

Shipments7 shows examples of how to obtain samples from hopper cars with

a sample thief. The standard provides general procedures for sampling in

three possible scenarios: (a) sampling from the top ports of a hopper car,

(b) sampling a hopper car during loading, and (c) sampling from the hopper

car while unloading.

Techniques for reduction of large samples to sizes more appropriate for testing

are covered in ASTM C702 Standard Practice for Reducing Samples of

Practical Application ◆
22

Aggregate to Testing Size8. The standard describes three methods, namely,

(a) using a mechanical splitter, (b) quartering, and (c) miniature stockpile

sampling (for damp fine aggregate only). The apparatus for each technique is

described along with the procedures to be followed. Guidelines for choosing

which method depend on the nature of the sample. Aggregate material can be

either fine, coarse or mixtures of coarse and fine aggregates. The mechanical

splitter is similar to a chute riffler. The procedure describes how the aggregate

samples can be introduced into the chute. A procedure for further reduction

of the sample for sizes suitable for testing is described. The quartering method

has two procedures where either one or a combination of both may be used.

The first technique is a simple cone and quartering method. The second method

uses a canvas blanket to roll the material for better mixing and then follows the

cone and quartering procedure to divide the sample. A procedure for dividing

the sample is described if the surface beneath the canvas blanket is uneven.

The latter method is used specifically for damp fine aggregates.

ASTM D75 Standard Practice for Sampling Aggregates9 is designed for

sampling of coarse and fine aggregates from different sources. This standard

provides procedures for sampling materials from a flowing aggregate stream

(bins or belt charge). Procedures are also presented for sampling materials

from conveyor belts, stockpiles, transportation units, and roadways. The

minimum mass of field sample that is required for analysis depends on the

maximum nominal size of the aggregates. This standard includes a table

showing this relationship.

ASTM B215-96 Standard Practices for Sampling Finished Lots of Metal

Powders10 is written for selecting representative samples of metal powders.

Two different practices are described in this standard. The first practice is for

powders that are in the process of being packaged from blenders or storage

tanks using an apparatus similar to a chute splitter. The second practice is

for powders already packaged in containers using a keystone sampler that is

similar to a sample thief. This standard provides operating procedures for both

samplers. The number of containers to be sampled is dependent on the total

number of containers in the lot. A table showing this relationship is included in

the standard.

A very comprehensive standard for sampling is the British Standard BS 3406:

Part 1, Determination of Particle Size Distribution: Part 1. Guide to powder

sampling11. The standard includes descriptions of different sampling devices

for the sub-division of the laboratory sample; it describes each sampling

device and presents a simple operating procedure. The standard has a

section on the suggested sampling devices for different types of powder:

free-flowing powders, non-free flowing powders and non-flowing powders.

Flow characteristics of powders are determined by their flow through a

◆ Powder Sampling
23

60° funnel with a 10 mm diameter orifice. Non-free flowing powders do not

flow readily through the funnel unless assisted, or may flow slowly but leave

a residue sticking to the sides of the funnel. A non-flowing powder does not

flow from the funnel. A section on the methods for preparation of very

small test portions is included. This section describes how to sample from

suspensions and pastes. A comparison of the dry powder sampling and wet

powder sampling techniques is included in this standard.

A draft standard, ISO/WD 14888: Sample Splitting of Powders for Particle

Size Characterization12 is currently being prepared by the International

Organization for Standardization (ISO). This draft describes methods to obtain

test samples from bulk material for measurement of size and size distribution.

It contains statistical background on sampling and calculation of sampling

sequence variances. General procedures for sampling and recommended

sampling sequence practices are included. Detailed statistical sampling

information pertaining to the spinning riffler and cone and quartering methods

is given. A section on suspension sampling is also included.

Relevant Standards ◆
24

References

1. T. Allen, Particle Size Measurement, 3rd ed., Chapman and Hall, London,

(1981).

2. J.T. Jones and M.F. Berard, “Powder characteristics: Part one, sampling,”

Ceramic Industry, 66, October 1996.

3. A. Liebermann, Fine particle characterization methods in liquid suspension,

in particle characterization in technology: Volume I. Applications and

Microanalysis, J.K. Beddow, ed., CRC Press, Boca Raton (1984) p. 188.

4. B.H. Kaye, An Investigation into the Relative Efficiency of Different

Sampling Procedure, Powder Metallurgy, 9, 213 (1962).

5. T. Allen and A.A. Khan, Critical evaluation of powder sampling

procedures, Chem. Engr., 238, CE108 (1970).

6. ASTM C322-82 Standard Practice for Sampling Ceramic Whiteware

Clays, American Society for Testing and Materials, West Conshohocken,

PA (1982).

7. ASTM D1900-94 Standard Practice for Carbon Black — Sampling Bulk

Shipments, American Society for Testing and Materials, West

Conshohocken, PA (1982).

8. ASTM C 702-98 Standard Practice for Reducing Samples of Aggregate

to Testing Size, American Society for Testing and Materials, West

Conshohocken, PA (1998).

9. ASTM D75-97 Standard Practice for Sampling Aggregates, American

Society for Testing and Materials, West Conshohocken, PA (1997).

10. ASTM B215-96 Standard Practices for Sampling Finished Lots of

Metal Powders, American Society for Testing and Materials, West

Conshohocken, PA, USA 1996.

11. BS 3406: Part 1: 1986 British Standard Methods for Determination of

particle size distribution Part 1. Guide to Powder Sampling, British

Standards Institute, London (1986).

12. ISO/WD: 14888 Sample Splitting of Powders for Particle Size

Characterisation International Organization for Standardization, Geneva.

13. P. Pei, J. Kelly, S. Malghan, and S. Dapkunas, Analysis of Zirconia Powder

for Thermal Spray: Reference Material for Particle Size Distribution

Measurement, in Thermal Spray: Practical Solutions for Engineering

Problems, C.C.Berndt ed., ASM International, Materials Park, OH (1996).

◆ Powder Sampling
25

14. Certificate of Analysis, Standard Reference Material®1982, Zirconia

Thermal Spray Powder – Particle Size Distribution, Standard Reference

Materials Program, National Institute of Standards and Technology,

Gaithersburg, MD (1996).

15. ISO 2859-Statistical Sampling. http://www.iso-9000.co.uk/9000qfa9.html,

International Organization for Standardization, Geneva (2000).

References ◆
27

3. SIZE CHARACTERIZATION BY

SIEVING TECHNIQUES

1. Introduction

2. General Procedures

3. Reporting of Size Data

4. Sources of Error and Variation

5. Practical Concerns

6. Calibration

7. Relevant Standards

1. Introduction

Sieving is one of the oldest techniques of powder classification (i.e.,

separation based on size or some other physical characteristic) that is still in

use today. It is among the most widely used and least expensive techniques

for determination of particle size distribution over a broad size range. The

preponderance of applications using this technique can be attributed to the

relative simplicity of the technique, low capital investment, high reliability, and

low level of technical expertise required in its application, characteristics not

often associated with other techniques of size analysis. Sieving can be used

over a very broad size range, from just over 100 mm to about 20 μm1. A test

sieve is a measuring device designed to retain particles larger than a designated

size, while allowing smaller particles to pass through the device2.

Size determination by sieving is applicable to free flowing dry powders,

and some carefully prepared slurries. The method, in the most general sense,

consists of shaking (agitating) the sample through a stacked series of sieves

with decreasing mesh size. The mesh with the largest aperture is at the

top, and that with the smallest aperture is at the bottom of the stack. Size

distribution is reported as the mass of the material retained on a mesh of given

size, but may also be reported as the cumulative mass retained on all sieves

above a mesh size or as the cumulative mass fraction above a given mesh size.

Sieves are typically classified as coarse, medium, or fine, based upon the size

of the aperture openings3. Coarse sieves have aperture openings in the size

range of 100 mm to 4 mm, medium sieves range from 4 mm to 200 μm, and

fine sieves extend below 200 μm. Numerous standards define the size of

aperture openings in sieves. Three ASTM standards, E 11-951, E 161-964 and

E 323-805, define specifications for various types of sieves. E 11-95 addresses

requirements and specifications for sieves made by weaving of wire cloth to

form sieve openings. E 161 defines specifications for electroformed sieves,

while E 323 defines the specifications for perforated plate sieves.

Size Characterization by Sieving Techniques ◆
28

Perforated plate sieves are primarily used for coarse powders and have

aperture sizes ranging from 125 mm to 1 mm. The size of the aperture

openings in consecutive sieves progress by about 4√2 larger than the previous

sieve. The apertures (or perforations) are either circular or square in shape.

The circular apertures are patterned in a staggered manner, while square

apertures can be arranged in either a staggered manner or rectilinear manner

where the perforations line up in both planar dimensions. Perforated sieves

are typically formed by punching rigid metal sheets that have not been coated

with any material. Woven wire sieves are formed by weaving wire of material,

such as stainless steel, brass or bronze to form square shaped openings. The

resulting “cloth” is then clamped into a hollow cylinder. The aperture openings

in woven wire sieves follow the same ratio of about 4√2 to 1. Table 3.1 lists

the American Standard Sieve size descriptors and the nominal corresponding

opening sizes which allow the passage of a particle of that size or smaller.

Identification of sieves by their sieve number refers to the number of wires

per linear inch forming that sieve. The requirement of uncoated wires to be

used for weaving the sieve cloth holds true here. The use of uncoated material

forming the sieve is to avoid any possible contamination of the sample by wear

debris abraded from the sieve material. Electroformed sieves are manufactured

by a technique involving the deposition of a photosensitive layer on to a metal

surface. The photosensitive layer is then exposed to the image of the sieve

pattern, and the photosensitive material and underlying substrate are selectively

etched, so that only the area of the metal corresponding to the apertures is

dissolved away. As aperture sizes get smaller than 1 mm, the electroformed

sieves often exhibit improved precision (with respect to aperture size,

separation of aperture centers, and shape of apertures) than woven wire or

perforated sieves. This fact is magnified as the aperture sizes get into the tens

of micrometers size range. However, it is important to note that as the ability

to reproduce aperture openings on different sieves with the same nominal

aperture size is limited, reproducing the results of analysis on different sieves

with the same aperture size may not be easy. It has been observed that the

coefficient of variance may vary between 10 % for sieves with small aperture

widths and 5 % for sieves with larger aperture widths6.

Some other pertinent international standards relating to specification of sieves

are the ISO R-565-1972(E): Woven wire and perforated plate test sieves, and

the BS 410: Woven wire test sieves.

◆ Size Characterization by Sieving Techniques
29

Sieving Sieving Techniques: echniques:

General Information and Capabilities

Size Range: Size Range: From about 125 mm to 20 μm (see text for size categories).

Specimen Type: Dry, free-flowing powders. Some instruments are

designed for analysis of slurries.

Operating Mode: Operating Mode: Mostly off-line in batch mode. Some automated

systems have been designed for on-line process and quality control

monitoring.

Ease of operation: Ease of operation: Relatively simple. Usually minimal sample preparation

required.

Sources of Error and Variations: ariations: Can arise during the sieving process

(blinding of sieves), due to inadequate and/or improper maintenance of

sieves, during/due to material transfer to and from sieves.

Powder characteristics may cause particles to agglomerate or fracture

during analysis.

Significant bias may be introduced during analysis of acicular particles.

See Ishikawa diagram (Figure 3.3).

Strengths: Strengths: Applicable over a broad size range.

Based on simple principles, and thus does not require highly skilled

operators.

Low capital investment.

Minimal sample preparation.

Limitations: Limitations: Requires long analysis times. Analysis times get longer as

sieves with finer aperture openings are used.

Extent of automation and computerization is relatively limited.

Mechanical motion during sieving affects repeatability and

reproducibility of results.

When particles have high aspect ratios (e.g., needle shaped particles),

PSDs are susceptible to large uncertainties due to measurement of

smallest diameter by this method.

Introduction ◆
30

ASTM Sieve Designation 

Aperture 

Opening 

Designation Aperture 

Opening 

Designation Aperture 

Opening 

Designation 

125 mm 5 in. 9.5 mm 3/8 in. 425 μm No. 40 

106 mm 4.24 in. 8.0 mm 5/16 in. 355 μm No. 45 

100 mm 4 in. 6.7 mm 0.265 in. 300 μm No. 50 

90 mm 31⁄2 in. 6.3 mm 1⁄4 in. 250 μm No. 60 

75 mm 3 in. 5.6 mm No. 3 1⁄2 212 μm No. 70 

63 mm 21⁄2 in. 4.75 mm No. 4 180 μm No. 80 

53 mm 2.12 in. 4.00 mm No. 5 150 μm No. 100 

50 mm 2 in. 3.35 mm No. 6 125 μm No. 120 

45 mm 13⁄4 in. 2.80 mm No. 7 106 μm No. 140 

37.5 mm 11⁄2 in. 2.36 mm No. 8 90 μm No. 170 

31.5 mm 11⁄4 in. 2.00 mm No. 10 75 μm No. 200 

26.5 mm 1.06 in. 1.7 mm No. 12 63 μm No. 230 

25.0 mm 1.00 in. 1.4 mm No. 14 53 μm No. 270 

22.4 mm 7/8 in. 1.18 mm No. 16 45 μm No. 325 

19.0 mm 3⁄4 in. 1.00 mm No. 18 38 μm No. 400 

16.0 mm 5/8 in. 850 μm No. 20 32 μm No. 450 

13.2 mm 0.53 in. 710 μm No. 25 25 μm No. 500 

12.5 mm 1⁄2 in. 600 μm No. 30 20 μm No. 635 

11.2 mm 7/16 in. 500 μm No. 35 

(Adapted in part from ASTM E11-95) 

Table 3.1. able 3.1.

ASTM Sieve Designation and Corresponding Nominal Aperture Openings

2. General Procedures

Sieving instruments have evolved with various modifications, but continue to be

based upon the same general principle. The goal during modification has been

to make the sieving process more efficient and minimize errors and variations

arising during sieving. Examples of these modifications include automation of

the sieving process to eliminate human error arising due to operator variability,

fatigue, etc., application of force by either mechanical or pneumatic means to

enable passage of particles through the sieves while minimizing blinding of the

sieves, creation of fluidized beds by application of an upward air draft (air-jet

sieving) to enable better particle separation on the sieve and consequently,

improved passage through the sieve openings.

Sieving processes can be generally categorized as dry sieving or wet sieving

processes. Typically wet sieving is conducted for analysis of particles finer

than about 200 mesh (75 μm), where powder surfaces develop enough of a

surface charge and show an enhanced tendency to agglomerate6. Wet sieve

analysis of particles even about 10 μm is conducted very routinely. Dry sieving

analyses are to an extent still conducted by hand. However, analysis of large

quantities of material, or analysis on a frequent basis, requires some degree of

automation. Most modern sieving instruments have automated the gyratory/

◆ Size Characterization by Sieving Techniques
31

oscillatory motion of the sieve stacks. The vibratory motion produced by

gyrations or oscillation of the sieve stack can be further enhanced by a tapping

motion applied to the top of the nest of sieves. The Tyler Ro-tap incorporates

the combination of the forces generated by the two actions. Air-jet sieving

is another technique utilizing a combination of negative air pressure and

pneumatic updrafts to create a fluidized bed of particles above the sieve.

Instruments based on this technique combine an upward air draft through a

sieve with a simultaneous negative pressure below the sieve, to enable the

fine particles to pass through the sieve, while preventing blinding of the sieve

by coarser particles. An added advantage of this technique is the prevention

of segregation of particles on the sieve that can hamper the smooth flow of

undersize particles through the sieve. This technique has been promoted for

brittle and friable particles that would otherwise fracture under the action

of strong forces. An instrument that combines the pneumatic action with

the mechanical vibratory motion is the Allen-Bradley (ATM) sonic sifter.

The pneumatic column passing through the sieves raises the particles over

the screen, while the mechanical motion shakes and separates the particles,

allowing free passage of the fines through the sieve. Such a motion also helps

reduce the amount of sieving time. The pneumatic pulse is applied as an

oscillatory pulse in the vertical direction. The combination of these two forces

also helps prevent blinding of sieves. Some instruments are designed to be

vibrated using an electromagnetic coupling system.

Wet sieving techniques follow the same general principle of application of

suitable force to enable the flow of fluid and dispersed powders through the

aperture openings. Some of the errors and sources of variation associated with

this action are discussed later in this chapter. Important facts to be considered

while using this technique are controlling the state of dispersion of particles in

the fluid to prevent agglomeration, control of the viscosity and surface tension

of the fluid to allow smooth flow through the apertures and regular care and

maintenance of the screens to prevent wear, corrosion or other damage to the

sieves. Wet sieving techniques are often aided by the use of ultrasonic agitation

or light vacuum to aid the flow of the fluid through the screens. Extreme

caution should be exercised while applying these forces, and the sieves should

be examined for damage immediately after the use of such techniques. Most

automatic sieving machines are designed for operation in a wet sieving mode.

Automatic machines are most commonly used in industrial settings, and the

preference for wet sieving is due to the ability to rapidly transport the powder

and weigh the amount of powder retained on each screen.

Some recommendations made by Allen3 for preparing powders for sieving

and for the analysis technique itself, include:

● Reducing the charge load (i.e., the amount of sample added to the top

sieve) to reduce the time required for analysis. However, the load should be

General Procedures ◆
32

maintained at a level that ensures adequate accuracy during weighing of

the different fractions, so as not to be affected by the amount of powder

loss during sieving.

● Removal of the fine fraction by pre-sieving using the screen with the

smallest aperture opening. This helps reduce excessive powder loss

during sieving, and also the amount of time for sieving.

● Incorporating motion that minimizes the blocking of sieve openings by

entrapped particles (blinding). This can be achieved by a jolting action to

dislodge particles wedged in the apertures, or by vertical air flow upwards

through the sieves.

● Materials that may agglomerate should be pretreated to prevent

agglomeration prior to sieving.

● Friable and brittle powders should preferably be sieved in an instrument

without jarring forces as may be encountered in sieves incorporating

oscillating and/or jolting motion. Air jet sieving or sieving by a sonic sifter

provide a relatively gentle sieving action.

A general procedure to conduct dry sieve analysis of a powder system would

include the following steps:

1. Classify the powder by sieving through single-screens to limit particle size

distribution over a series of four to six consecutive mesh sizes. Such an

action might result in numerous fractions of finite size distribution, where

each fraction will have to be analyzed separately.

2. Select for each fraction of interest, sieves covering that size range in order

to have a nest of five or six sieves.

3. Examine each sieve for any damage, wear, deformities, etc., to screen.

Check to ensure that screen openings are not clogged and that screens fit

snugly into sieve housing. Any space between the screen and the cylinder

will result in powder loss.

4. Stack sieves such that the screen with coarsest aperture opening is at the

top, and the screen with finest aperture opening is at the bottom. Stack

the collection pan at the bottom of the nest.

5. Load charge on to the top screen.

6. Sieve according to established protocol, i.e., sieve for a given duration of

time, and/or sieve until the quantity passing through the sieve is less than

some fraction of the original mass in a given period of time (e.g., sieving

until the quantity passing through sieve is less than 0.5 % of the total mass

(ASTM D45228). While developing a sieving protocol for a material system

◆ Size Characterization by Sieving Techniques
33

that would be analyzed often, it is useful to sieve a given quantity of sample

for varying amounts of time, e.g., 10 min to 30 min at 5 min intervals.

Sieving time is selected based on the time period after which no significant

difference in mass change is observed. This technique of fixing the

sieving time has been followed at NIST for the development of Standard

Reference Materials (SRMs) for particle size distribution. Experience

through studies conducted at NIST has indicated that sieving for a fixed

period of time is more convenient than sieving until less than a certain

fraction of the total mass passes through the sieve in a given period of

time. The time and effort to disassemble and assemble a complete set of

sieves to ensure that no powder is lost while weighing can be considerable

and is not very efficient when attempting to analyze numerous samples.

7. Separate carefully each sieve and transfer the powder retained on each

screen on to a weighing paper. If necessary, lightly brush the underside of

the screen to release any particles adhering to the screen. This should be

done with extreme caution so as not to damage the wire screens.

8. Weigh each fraction on mass balance with adequate sensitivity. In case

total mass after sieving reflects powder loss greater than 0.5 % of the

initial charge mass, the test should not be considered. Sieves should be

examined for damage or improper fit.

9. Plot results either as a histogram representing percentage mass retained on

each screen as a function of aperture opening size, and/or as cumulative

mass percentage finer or coarser as a function of aperture opening size.

10. Clean sieves, examine for damage and store in a safe environment.

A quick check to ensure that no powder has been lost during the sieving

process, is to compare the mass of the powder retained after sieving with the

mass of the original charge. This total mass of powder retained after sieving

should be the same or very close to the amount of charge with which the sieve

analysis was started. Any significant differences in the two quantities is an

indication of loss of powder. Total mass loss over 1 % can introduce significant

error in the size distribution results and affect the reproducibility of the results.

The above-mentioned procedure is a general procedure that may have to be

modified to suit the material system being examined, and to accommodate any

instrument specific requirements. Manufacturers of sieving instruments provide

detailed instructions specific to the operation of that instrument. Wet sieving

would require the development of a similar procedure. Additional steps that

would have to be considered would be the amount of sample and dispersant to

be added and the level of dilution that would be required to provide a suitable

fluid medium. Most manufacturers specify the level of dilution. If not specified,

the level of dilution should be such that no significant increase occurs in the

General Procedures ◆
34

Figure 3.1. Figure 3.1.

Results of Inter-Laboratory Round-Robin Test on

Particle Size Determination of Silicon Nitride by Sieve Analysis

viscosity of the fluid. The application of agitation, ultrasonic energy, vacuum or

control of flow by pumps is specific to the instrument and is specified by the

manufacturer.

It is critical to examine sieving procedures to ensure their applicability to

different material systems. Factors that are often ignored or deemed

insignificant, but that have a marked impact on the results include monitoring

and control of humidity, frequency at which sieves are vibrated, force of jolt

or impact, etc. It is also important to consider whether a set of procedures

that are applicable to one material system can be applied to another material

system comprised of particles with a size distribution in the same range. Based

on the powder characteristics, there can be significant differences in surface

properties leading to formation of aggregates or charging effects that may

greatly influence the observed results. A sieving procedure developed for

an inter-laboratory round-robin test is given in Table 3.2. Participants in this

study were supplied with the candidate material and a procedure for analysis.

Figure 3.1 shows the reported results of size distribution of silicon nitride, a

candidate material in this study. Results from various laboratories reporting

the cumulative mass percentage as a function of the aperture opening size

is plotted. With the exception of one laboratory (lab 14), the results from all

laboratories were in close agreement. The mean d50 for the results from all of

◆ Size Characterization by Sieving Techniques

100

90

Lab 1

Lab 3

Lab 13

Lab 14

Lab 15

Lab 20

Lab 32

0 50 100 150

Aperture Opening (μm)

Cumulative Mass Fraction (%)

200 250 300

80

70

60

50

40

30

20

10

0
35

the laboratories was 66.41 μm with an overall standard deviation of 4.91 μm.

Considering that different laboratories used sieving instruments from different

manufacturers, the results are in generally good agreement. The variation in

the results can be attributed to differences in operational characteristics of

instruments, variations due to operators, any possible variations arising from

sample handling (samples in different temperature and humidity conditions),

differences in accuracy and precision of weighing instruments, differences in

conditions of screens, etc.

Table 3.2. able 3.2.

Sieving Procedure Used in Inter-Laboratory Round-Robin Test

General Procedures ◆

1. Weigh each clean sieve.

2. Stack the clean sieves with a collection pan at the bottom and the sieve

with the largest opening at the top.

3. Weigh the sample.

4. Place the weighed sample on the top sieve.

5. Cover the sieve and set in the sieving apparatus.

6. Sieve for 30 min,

O R

Sieve until the quantity passing through each sieve becomes less than

0.2 % of the mass of the charge in 1 min or less than 10 mg in several

minutes.

7. Collect the retained sample on the sieve.

8. Calculate the sample mass on the sieve.

9. Compare the total mass with the starting mass of the powder. The mass

loss of the sieving should be within the range of ± 2 %. If the loss is out

of this range, the test should be re-examined.

10. Calculate the size distribution.

3. Reporting of Size Data

Results from sieve analysis can be represented as a cumulative graph of

mass fraction coarser than or finer than a particular size, or as a histogram

of the mass fraction on each screen plotted against the screen size and thus

an indication of particle size. Either of these techniques of representation can

be used to visualize the particle size distribution. Size data for most powders

analyzed by sieving is best reported on a mass basis. This minimizes any

errors that may be introduced during transformation from one physical basis to

another. However, for material systems comprising of spherical particles that

do not change in density with a change in size, the mass basis can readily be

converted to a volume basis and represented accordingly. The same conversion
36

Figure 3.2. Figure 3.2.

Sieving Data From Sub-Sample Analyzed During Certification of

NIST SRM 1018b

may not be observed. Table 3.3 represents one set of data generated during the

development of NIST SRM 1018b. The quantity measured after sieve analysis

is the mass of glass beads retained on the sieve. This can also be expressed

as the fraction of the cumulative mass of glass beads. With these values, it is

possible to generate a plot representing the percentage of cumulative mass

that is finer or coarser than a particular size. The same data are represented

in a graphical format in Figure 3.2, which indicates a histogram representing

the mass percentage retained on each sieve and the cumulative mass percent

coarser than a particular size.

0

5

10

15

20

25

pan 250 300 355 425 500 600 710

0

10

20

30

40

50

60

70

80

90

100

Nominal aperture size (μm)

Cumulative mass fraction coarser

Mass fraction on sieve

may lead to error if applied to non-spherical particles, due to errors in

calculating the correct volume of the particle. An important factor to keep

in mind while using histogram based representation is the width of size

classes being used in the histogram. The width of the size classes should be

the separation of the nominal aperture openings between two consecutive

sieves. In case of a very broad class width, there may be significant loss of

resolution in the size distribution and finer details, including small changes that

◆ Size Characterization by Sieving Techniques Cumulative Mass Fraction Coarser

Nominal Aperture Size (μm)
37

US Sieve No Nominal Size 

opening (μm) 

Mass on sieve 

(g) 

Mass fraction 

on sieve (%) 

Cumulative 

mass fraction 

coarser (%) 

Cumulative 

mass fraction 

finer (%) 

25 710 5.28 6.16 6.16 93.84 

30 600 11.29 13.17 19.33 80.67 

35 500 14.23 16.60 35.93 64.07 

40 425 10.7 12.49 48.42 51.58 

45 355 8.97 10.47 58.89 41.11 

50 300 10.20 11.9 70.79 29.21 

60 250 20.11 23.47 94.26 5.74 

PAN 4.92 5.74 100 0 

4. Sources of Error and Variation

As in any technique of size analysis, errors and variation can be introduced

during the numerous stages involved in the analysis technique. Propagation of

errors introduced during the early stages can lead to significant variations in

the observed results from the true values. It is very important to recognize the

possible sources of error and variation, and the stages at which these can be

introduced. Early detection can help prevent and control some of these sources

of variation. Operating procedures should be designed keeping in perspective

the sources of error, reasons for propagation and means to prevent, control

and/or correct for the same. Potential sources of error and variation that can

be encountered during sieve analysis are indicated in the Ishikawa diagram2

in Figure 3.3. An Ishikawa diagram (also known as fish-bone diagram) is a

total quality management (TQM) tool that is used to plot all components and

sub-processes that constitute a process (e.g., a manufacturing process, or

an analysis procedure). By identifying all of the components and steps that

are involved in the process, it becomes easier to recognize the various sources

of variation associated with each of the steps and thus identify some of the

sources of error also. While this diagram considers sources of error and

variation arising during the analysis, the probability of introducing variation

arises during the sampling process itself. Some commonly encountered

sources of error are discussed briefly in the following paragraphs.

1. Sampling and specimen-related sources of error

a. Errors due to non-representative sampling of the bulk material to create test

specimens. This source of error can be very significant when sampling from

large quantities of bulk (e.g., sampling from box-car loads or other large

More information about representation of size data and some of the issues

associated with transformation of data from one basis to another is covered in

a later chapter on reporting of size data.

Table 3.3. able 3.3.

Size Data Obtained by Sieve Analysis of Glass Beads During

Development of a NIST Size Standard (Total Mass Sampled = 85.70 g)

Reporting of Size Data ◆
38

containers typically used in shipping where significant segregation of

powders occurs during transport and storage).

b. Errors arising from analysis of inadequately prepared powders (e.g.,

powders containing agglomerates which may not breakup during the

sieve analysis, or powders that are very brittle and may fracture into finer

fractions during sieve analysis). During wet sieve analysis, it is necessary

to ensure that the powders are suitably dispersed in the liquid medium, and

do not form agglomerates.

c. During wet sieving, it is critical to control the viscosity and surface tension

of the liquid, to ensure adequate and proper flow of the dispersant with the

sample through the sieves. Liquids with very high viscosity will not flow

through finer apertures. Similarly, inadequate control of the surface tension

can cause temporary blinding of the apertures, preventing smooth flow

through the sieves.

d. Analysis of powders that develop surface electrical charges during sieving

with metallic sieves needs to be avoided. The generation of charges often

prevents the passage of particles through aperture openings, where they

would otherwise pass through in the absence of these electrical charges.

2. Instrument- and analysis-related sources of error

Potential errors and variations arising during this stage are mapped in the

Ishikawa diagram.

a. The selection of the appropriate type of screen and material with which the

screen is made is extremely important. Screens used should be properly

calibrated and examined to ensure that the screens are in good operational

condition. Errors can be introduced due to blinding of screens, corrosion

or damage to screens that may have changed the shape and size of the

aperture openings. Other factors that can effect the aperture shape and

opening size are changes in tension of the wires and wear of the wires.

These may occur either as a natural course over a period of time, or may

be due to the use of stiff brushes or rough handling of screens during

cleaning. While selecting screens to be used in the stack, it is necessary to

ensure that the sieves cover the appropriate size range and that they consist

of aperture openings with appropriate size separation (i.e., the series of the

sieves used should be selected with care).

b. Along with care and attention to the sieves, proper maintenance and

operational procedures for the sieve shakers are just as important. Regular

maintenance procedures should be developed to ensure that all mechanical

moving components are well lubricated, timers are working, any belts or

chains are appropriately tensioned. For instruments with air jet or sonic

◆ Size Characterization by Sieving Techniques
39

sifting capabilities, these components should be checked and maintained as

per manufacturer’s recommendations.

c. Loss of sample during analysis, or during removal for weighing should

be avoided at all costs. Loss of sample during analysis can occur due to

entrapment of powder between the screen and the frame, or due to spaces

between the sieves if stacked incorrectly. While weighing the samples,

the balance used should be selected on the basis of desired precision and

accuracy. This in turn depends upon the amount of sample charge. The

desired resolution of the final result also dictates the type of balance to be

used for analysis.

d. Reproducibility of results may be poor when using sieves of different

shapes or different sieving techniques due to the nature of vibrations

generated in the shakers and transmitted through the screens to the powder

in question. Comparison of results from different instruments should be

done with extreme caution ensuring that sieving conditions have been

reproduced to the greatest extent possible.

5. Practical Concerns

Although sieving is a robust technique and a generally rugged measurement

method, sieves do require careful handling during use and regular maintenance.

Sieves should be kept clean and dry and stored in containers or cabinets that

limit contamination, corrosion and impact by unintended objects. Cleaning can

be done by gentle brushing to dislodge adherent particles, but particular care

must be taken with mesh sizes greater than 80 (sieves with aperture openings

smaller than 150 μm) to prevent damage to delicate woven wires. Low power

ultrasonic cleaning in a mild detergent and water solution is often preferred.

Agitation of sieves is a source of variation in measurements of size distribution.

Clearly, the energy imparted to the sieve or stack of sieves is the most general

description of this procedural step. Duration of agitation is an often measured

agitation parameter, whereas amplitude and periodicity are not. Automated

mechanical agitation devices, such as sieve shakers, provide more consistent

agitation than manual methods. In all cases, duration of agitation should be

accurately controlled and recorded.

Measuring a relatively small amount of material which is representative of a

larger lot requires care in obtaining samples. Dry powders with a range of sizes

can settle during shipment with finer material at the bottom of the container,

and other handling can also segregate powders by size. The tendency for

powders to separate by size makes sampling a critical step in measuring a

useful size distribution. Guidelines recommended by Allen to counteract

segregation in a moving stream include sampling a powder when it is in motion

Sources of Error and Variation ◆
40

◆ Size Characterization by Sieving Techniques

Figure 3.3. Figure 3.3.

Ishikawa Diagram of Potential Variation Sources in Sieve Analysis2

Potential variationsources inequipment used insieve analysis forparticle sizing

Weighing systems Sieve shakers

Precision

Capacity Data acquisition

ComputerInterface Manual

Environment

Mounting

Gage capability Type

Electronic Mechanical

Frequency Type

 Air jet Gyratory Gyratory with

tapping

 Sonic sifter

Electromagnetic

Maintenance

Installation/mounting

Timer

 Electronic

Manual

rpm

Tapping method

Cleaning devices

Brushes

Type

 Brass Synthetic Natural

bristle

Parameters

 Force of application

Condition of brush

Compressed air

Acid

Sieving aidsduring testing

 Metal Plastic Glass beads

Leather

Ultrasonic cleaning

Screens

Size

Diameter

Calibration

Media

◆ ◆

◆ ◆ ◆ ◆

◆ ◆

◆ ◆

◆

◆◆

◆◆◆◆

◆

◆◆◆◆

◆

◆◆◆◆◆

◆

◆◆◆◆

◆◆

OpticalReference

materials

Data

reduction

Frame

Dimensio

nal

Stack setting

Stack selection Stack height

Construction

Media

Condition

Blinding

 Cleanliness

DamageTension

 Wear

Type

ElectroformedPerforated

plate

 Woven Synthetic

Metal

Frame

Shape

 Round Other

Material

 Brass

Stainless

Other

◆ Dimensional
41

and sampling an entire stream of powder for many short increments of time,

rather than sampling a fraction of the stream for all of the time.

The amount of time for analysis can become a major cause for concern if

the operational procedure is improperly designed. Considering the fact that as

aperture openings become smaller, the amount of time for analysis gets larger,

it is necessary to consider the powder size range, the desired size resolution

and sizes of sieves in the stack before designing a sieving experiment.

6. Calibration

The measurement of particle size is ultimately based on primary standards of

length. The use of light or electron microscopy-based techniques that can be

calibrated to standards certified for length serve as the primary particle size

measurement methods to which indirect methods, such as sieving, light

scattering or other methods are often calibrated.

Sieve calibration is most practically conducted by sieving a material with a

known size distribution and determining the similarity of the test results to

that of the test material. Standard Reference Materials have been developed

for sieve calibration by the National Institute of Standards and Technology.

These materials are certified by electron and optical microscopy methods for

dimension and are intended for sieve calibration. Table 3.4 lists the appropriate

NIST SRMs. A series of publications known as the SP 260 series, sponsored

by the NIST SRM7 program, is available to assist users in the application of

SRMs. A more general, but applicable text discussing the general concepts

of precision and accuracy as applied to SRMs, and touching upon their impact

on quality assurance and measurement processes is available8. Another

standard, ASTM D1214-89 (Reapproved 1994)9 can be followed for analysis

of glass spheres by sieving techniques. This standard test method is designed

specifically for sieve analysis of glass spheres used in reflective road and

pavement markings and other industrial uses. A limitation of this standard is

its use of non-SI units for reporting of values (i.e., inch-pound units).

SRM Material Particle size diameter, nominal (μm) 

1003b Glass beads 10 to 60 (600 to 325 mesh)

1004b Glass beads 40 to 150 (270 to 120 mesh)

1017b Glass beads 100 to 400 (140 to 45 mesh)

1018b Glass beads 220 to 750 (60 to 25 mesh)

1019b Glass beads 750 to 2450 (20 to 10 mesh)

Table 3.4. able 3.4.

NIST Standard Reference Materials Designated for Calibration of Sieves

Practical Concerns ◆
42

Other proposed techniques for calibration of sieves have included techniques

of optical and photometric examination of sieves to record the distribution of

aperture sizes in that particular screen. A primary concern with this technique

has been the lack of instruments that can examine and record the distribution

of aperture openings in a rapid, automated manner10. Other techniques of

calibration involving the recording of sieve analysis results of a material

system with log-normal distribution on a log-probability paper to ensure a

straight line fit, and the technique of counting and weighing of undersize

fraction passing through a sieve, as suggested by Andreasen and Anderson,

have been discussed by Allen3. BCR quartz size standards are available from

the National Physical Laboratory (NPL), London, for calibration of sieves

based on the technique of Andreasen and Anderson.

7. Relevant Standards

There are numerous normative standards pertaining to specifications for sieves,

and describing standard techniques for sieving, reflecting the experience gained

with this technique over the years that it has been used. Just as interesting is

the availability of standards pertaining to sieve analysis of a wide range of

materials including ceramic powders, materials for powder metallurgy, soils and

fertilizers, industrial chemicals, minerals and other raw materials, spices, green

coffee, pigments, porcelains, enamels, clays, concrete, fibers, etc. Standards

are typically available defining specifications for sieves that would be used for

size analysis or defining standard techniques to be followed for analysis of a

particular system. Some standards are also available specifying terminology

for use in context of sieving. ISO 2395:199011 Test Sieves and Test SievingVocabulary is a standard ISO guide defining terminology used in context of

sieving. The ASTM E1638-9412 is an equivalent Standard Terminology Module

Relating to Sieve Analysis, compiled and adopted by the ASTM.

Standards pertaining to specifications for sieves and standard techniques

to be followed for sieve analysis of materials are available from every major

international standards organization. ASTM E1919-9713 has a comprehensive

listing of the pertinent standards available from ASTM, ISO, BSI, DIN

(German standards), AFNOR (French standards) and JSA/JIS (Japanese

standards).

There are three ASTM standards defining specifications for different types

of sieves:

● ASTM E11-951 Standard Specification for Wire Cloth and Sieves for

Testing Purposes.

● ASTM E323-80 (Reapproved 1996)5 Standard Specification for

Perforated-Plate Sieves for Testing Purposes.

◆ Size Characterization by Sieving Techniques
43

● ASTM E161-964 Standard Specification for Precision Electroformed

Sieves.

These standards describe the requirements for sieve construction including

details on aperture openings, means to measure the same, wire diameter,

frame diameter and techniques for maintenance and cleaning of sieving. ISO

standards similar to the above mentioned ASTM standards are also available:

● ISO 3310-1:199014 Test Sieves – Technical requirements and testing –

Part 1: Test sieves of metal wire cloth.

● ISO 3310-2:199015 Test Sieves – Technical requirements and testing –

Part 2: Test sieves of perforated metal plate.

● ISO 3310-3:199016 Test Sieves – Technical requirements and testing –

Part 3: Test sieves of electroformed sheets.

BS 410 (1986) Specification for test sieves was long considered the

pertinent British standard and has been withdrawn. However, BS 410-

2:200017 (ISO 3310-2:1999) Test sieves; Technical requirements and testing:

Test sieves of perforated metal plate, is still available from the British

Standards Institute.

There are standards available that define standard test methods for

characterization of particular class of materials by a particular set of sieves.

Some of these standards applicable to ceramics and related material systems

adopted by the ASTM are:

● ASTM C136-96a18 Standard Test Method for Sieve Analysis of Fine

and Coarse Aggregates, applicable for the determination of particle size

distribution of fine and coarse aggregates by sieve analysis,

● ASTM C285-88 (Reapproved 1994)19 Standard Test Method for Sieve

Analysis of Wet-Milled and Dry-Milled Porcelain Enamel, for determination

of the fineness of frit in wet- or dry-milled procelain enamels and other

ceramic coatings on metals by use of the No. 200 or No. 325 mesh,

● ASTM C325-81 (Reapproved 1997)20 Standard Test Method for Sieve

Analysis of Ceramic Whiteware Clays, for use in testing shipments of clay

and for plant control tests,

● ASTM C925-79 (Reapproved 1995)21 Standard Test Method for Precision

Electroformed Wet Sieve Analysis of Nonplastic Ceramic Powders, for

particle size distribution determination of pulverized alumina and quartz for

particle sizes from 45 μm to 5 μm by wet sieving,

● ASTM D1921-9622 Standard Test Method for Particle Size (Sieve

Analysis) of Plastic Materials, applicable to particle size determination of

Relevant Standards ◆
44

plastic materials in powdered, granular or pelleted forms by dry sieving,

with the lower limit of measurement at about 38 μm.

Some pertinent ISO Standards are:

● ISO 2591-1:198823 Test Sieving – Part 1: Methods using test sieves of

woven wire cloth and perforated metal plate, describing standard techniques

of sieving applicable to a broad range of materials,

● ISO 4497:198324 Metallic Powders – Determination of particle size by dry

sieving,

● ISO 8130-1:199225 Coating Powders – Part 1: Determination of particle

size distribution by sieving,

● ISO 6274:198226 Concrete–Sieve analysis of aggregates.

Another comprehensive source of information on testing techniques and

applications of sieve analysis is the ASTM Manual on Testing Sieving

Methods27. This manual replaces the ASTM STP 447 B edition that has

been in circulation and widely used for quite a few years.

◆ Size Characterization by Sieving Techniques
45

References

1. ASTM E11-95 Standard Specification for Wire Cloth and Sieves for

Testing Purposes, American Society for Testing and Materials, West

Conshohocken, PA (1995).

2. S. Nushart, Sieve Analysis: More Ways to Help Your Numbers Come Out

Right, Powder and Bulk Eng., 37, February 1999.

3. T. Allen, Particle Size Measurement, 4th ed., Chapman and Hall, London,

(1993) 192-213.

4. ASTM E161-96 Standard Specification for Precision Electroformed

Sieves, American Society for Testing and Materials, West Conshohocken,

PA (1996).

5. ASTM E323-80 (1996) Standard Specification for Perforated-Plate Sieves

for Testing Purposes, American Society for Testing and Materials, West

Conshohocken, PA (1996).

6. R. Davies, Particle size measurement: Experimental Techniques in

Handbook of Powder Science and Technology, M. E. Fayed and L. Otten,

eds., Van Nostrand Reinhold Company Inc., New York (1984) p. 31.

7. Standard Reference Materials Program, Office of Measurement

Services, Technology Services, National Institute of Standards and

Technology, Gaithersburg, MD 20899-0001; Phone (301)-975-6776, Fax

(301)-948-3730; email srminfo@nist.gov; URL http://ts.nist.gov/srm

8. J. K. Taylor, Handbook for SRM Users, (NIST Special Publication 260-

100), N.M. Trahey, ed., National Institute of Standards and Technology,

Gaithersburg, MD (1993).

9. ASTM D1214-89 (Reapproved 1994) Standard Test Method for Sieve

Analysis of Glass Spheres, American Society for Testing and Materials,

West Conshohocken, PA (1994).

10. K. Leschonski, Proc. Particle Size Analysis Conf., Chem. Soc. Analyt.

Div., London (1977).

11. ISO 2395:1990 Test Sieves and Test Sieving-Vocabulary, International

Organization for Standardization, Geneva (1990).

12. ASTM E1638-94 Standard Terminology Relating to Sieve Analysis,

American Society for Testing and Materials, West Conshohocken, PA

(1994).

References ◆
46

13. ASTM E1919-97 Standard Guide for Worldwide Published Standards

Relating to Particle and Spray Characterization, American Society for

Testing and Materials, West Conshohocken, PA (1997).

14. ISO 3310-1:1990 Test Sieves – Technical Requirements and Testing –

Part 1: Test sieves of metal wire cloth, International Organization for

Standardization, Geneva (1990).

15. ISO 3310-2:1990 Test Sieves – Technical Requirements and Testing –

Part 2: Test sieves of perforated metal plate, International Organization

for Standardization, Geneva (1990).

16. ISO 3310-3:1990 Test Sieves – Technical Requirements and Testing –

Part 3: Test sieves of electroformed sheets, International Organization for

Standardization, Geneva (1990).

17. BS 410-2:2000 (ISO 3310-2:1999) Test Sieves; Technical Requirements

and Testing: Test Sieves of Perforated Metal Plate, British Standards

Institute, London (2000).

18. ASTM C136-96a Standard Test Method for Sieve Analysis of Fine and

Coarse Aggregates, American Society for Testing and Materials, West

Conshohocken, PA (1996).

19. ASTM C285-88 (Reapproved 1994) Standard Test Method for Sieve

Analysis of Wet-Milled and Dry-Milled Porcelain Enamel, American

Society for Testing and Materials, West Conshohocken, PA (1994).

20. ASTM C325-81 (Reapproved 1997) Standard Test Method for Sieve

Analysis of Ceramic Whiteware Clays, American Society for Testing and

Materials, West Conshohocken, PA (1997).

21. ASTM C925-79 (Reapproved 1995) Standard Test Method for Precision

Electroformed Wet Sieve Analysis of Nonplastic Ceramic Powders,

American Society for Testing and Materials, West Conshohocken, PA

(1995).

22. ASTM D1921-96 Standard Test Method for Particle Size (Sieve Analysis)

of Plastic Materials, American Society for Testing and Materials, West

Conshohocken, PA (1996).

23. ISO 2591-1:1988 Test Sieving – Part 1: Methods Using Test Sieves of

Woven Wire Cloth and Perforated Metal Plate, International Organization

for Standardization, Geneva (1988).

24. ISO 4497:1983 Metallic Powders – Determination of Particle Size by Dry

Sieving, International Organization for Standardization, Geneva (1983).

◆ Size Characterization by Sieving Techniques
47

25. ISO 8130-1:1992 Coating Powders – Part 1: Determination of Particle

Size Distribution by Sieving, International Organization for Standardization,

Geneva (1992).

26. ISO 6274:1982 Concrete-Sieve Analysis of Aggregates, International

Organization for Standardization, Geneva (1982).

27. ASTM Manual 32, Manual on Testing Sieving Methods: Guidelines for

Establishing Sieve Analysis Procedures, 4th ed., American Society for

Testing and Materials, West Conshohocken, PA (1998).

28. ASTM D452-91 Standard Test Method for Sieve Analysis of Surfacing

for Asphalt Roofing Products, American Society for Testing and Materials,

West Conshohocken, PA (1997).

References ◆
49

4. SIZE CHARACTERIZATION BY

GRAVITATIONAL SEDIMENTATION TECHNIQUES

1. Introduction

2. General Procedures

2.1 Powder sampling

2.2 Dispersion

2.3 Test for dispersion

2.4 Instrument set-up and analysis

2.5 Data representation

3. Sources of Error, Variations and Other Concerns

4. Relevant Standards

5. Practical Application: Dispersion Procedure for Silicon Nitride in Aqueous

Suspension

1. Introduction

Gravitational sedimentation-based techniques are amongst the most

commonly used techniques for particle size and size distribution determination.

Instruments based on this type of technique are routinely used in industrial

and research applications. Simply put, instruments based on the principle of

gravitational sedimentation measure the velocity with which particles in a

fluid settle due to the gravitational forces acting on the particle, against the

buoyancy of the fluid and other drag forces acting against the settling of

particles. This relationship is expressed by Stokes’ law, which is valid only

for particles settling under conditions of terminal velocity and non-turbulent

flow. The terminal velocity can be expressed by the Stokes’ equation as:

L

d P L g v

η

ρ ρ

18

( ) 2 − =

where, ν is settling velocity of the particles,

d is the particle diameter,

ρP and ρL are the particle and fluid density respectively,

g is the acceleration due to gravity,

ηL is the viscosity of the fluid.

Thus, time required for the particles to settle in a column of height H can be

expressed as:

d g

H

t

P L

L

( )

18

2 ρ ρ

η

− =

Gravitational Sedimentation Techniques ◆
50

In most instances, the particles upon introduction into the fluid reach

terminal velocity very rapidly and in a very short travel distance.

Conventional instruments use either a collimated x-ray beam (x-ray

gravitational sedimentation) or light beam (photo-gravitational sedimentation)

as a probe to determine the concentration of particles at a given depth in

the fluid column. In both cases, the attenuation in the intensity of the beam

due to either absorption (x-ray) or scattering (light) of the probe signal is used

to measure particle concentration at that height. The intensity of the beam

after attenuation is compared with the intensity of the beam in the absence of

any particles, measured as a baseline signal. Thus, knowing the particle density

and by determining the concentration of the particles at various depths, the

settling velocities of the particles can be determined and the particle diameter

corresponding to that velocity calculated. As settling times increase with

decreasing particle sizes, the settling times can be greatly reduced by the

application of an external centrifugal force. The settling times can then be

expressed as15:

2 2

0

( )

18 ln

ρ ρ ω

η

P L

t

d

r

r

t −

















=

where, ω is the angular velocity of the centrifuge,

r0 and rt are the radial positions of the particles before and after

centrifuging.

The velocity of the particle depends on its position in the suspension as well as

its size. The particle velocities increase as the particles move away from the

axis of rotation. Allen and others1-4 have described the different centrifugal

sedimentation techniques. Standard test methods of size analyses incorporating

this principle are also widely practiced.5-6

Some commercial instruments incorporate the centrifugal action with

gravitational sedimentation using either a photo or an x-ray probe. These

instruments provide the advantage of faster data acquisition times over those

without centrifuging capabilities. This is particularly valuable when analyzing

powder systems with particles finer than about 1 μm. The use of centrifugal

action also enables reliably extending the lower size limit of the instrument to

about 50 nm. Claims of lower size limits of 10 nm are influenced by factors

such as particle density, solids loading, x-ray mass absorption coefficient and

the use of any optical constants.

Gravitational sedimentation techniques are applicable for a wide variety of

powder systems. A basic requirement is that all the particles comprising the

powder system be of the same density. Under such a condition, the rate at

◆ Gravitational Sedimentation Techniques
51

which different particles settle in the fluid column is influenced only by the

particle size. For increased sensitivity, it is necessary that there be an adequate

density difference between the density of the particles and the density of the

suspension. In some cases, this may require the use of non-aqueous fluids, and

thus necessitate knowledge of the fluid’s optical properties and specific gravity.

Consideration of the fluid surface tension and wetting behavior is critical while

choosing the non-aqueous fluid. Furthermore, changing the suspension used

in the system would also require calculating the Reynold’s number to ensure

that all particles settling in the fluid would do so under laminar flow conditions.

Jones7 describes the limitations of the Stokes’ Law. Instruments with an x-ray

probe exhibit improved sensitivity for materials with high x-ray absorption

coefficient. As a general rule, elements with atomic numbers less than 13

do not have high x-ray absorption coefficients, though these increase as the

atomic number approaches 20. For instruments equipped with a photo probe

(photo-sedimentation-based instruments), knowledge of the materials’

refractive index and extinction coefficient values is critical. For powders

sizes less than about 2 μm, the use of the imaginary component of the

refractive index may be necessary. Incorrect values of real and/or imaginary

components of the refractive indices can lead to significant error in the

measured size values. Instruments not equipped with centrifugal capabilities

are strongly affected by Brownian motion when analyzing particles finer than

about 0.5 μm. Depending upon the density and the size of this fine fraction,

the random thermal motion can keep the particles from settling, leading to

a skewed distribution indicative of a greater fine fraction than is actually

present. Keeping the particles well dispersed is just as critical as particleparticle interaction can lead to hindered settling or give rise to significant

drag forces which, in turn, affect the settling rate of particles. Centrifugal

forces minimize the effects of Brownian motion and drag forces otherwise

encountered. Allen8 describes the limits of the gravitational sedimentation

methods relating to concentration limits, upper and lower size limits.

Instruments based on these techniques report the measured size data of

powders in terms of the equivalent spherical diameter. This measure represents

the diameter of a sphere that would settle at the same terminal velocity as the

particle in question, under the same experimental conditions. Though this can

be very representative of the particle system being analyzed, if the particles

are approximately equi-axed, this measure becomes rather non-representative

as the aspect ratio of the particles increases.

Introduction ◆
52

Gravitational Sedimentation Instruments: Gravitational Sedimentation Instruments:

General Information, Capabilities and Limitations

Size Range: Typically about 0.1 μm to 300 μm (lower limit of 0.001 μm

and upper limit up to 1000 μm are possible with certain instruments

and/or under certain conditions).

Specimen Type: Powders dispersed in suspension. Suspensions can be

aqueous or non-aqueous, depending on the desired viscosity and

density contrast.

Operating Mode: Off-line or on-line, but only batch mode.

Ease of Operation: Relatively simple, requires a very basic

understanding of fundamental principles.

Sources of Error and Variation: Can arise during sample preparation,

instrument set-up and operation. Some extent of operator dependence,

with possible influence of operator techniques on accuracy.

Strengths: Simple specimen preparation.

Rapid analysis, except below certain size limits.

Robust technique, well suited for industrial environments.

Relatively inexpensive instrumentation.

Does not require highly skilled operators.

Can be used over a broad size range with minimal changes.

Limitations: Cannot be used for analysis of powder batch with particles

of different densities.

Analysis times increase with increasing fraction of fine particles (less

than about 1 μm, for gravitational sedimentation with centrifugal action).

Sensitivity and accuracy of instruments with x-ray probes is strongly

influenced by materials mass absorption coefficient. Sensitivity and

accuracy of instruments with photo probes is influenced by the value

of optical constants used.

Accuracy of analysis is strongly influenced by physical factors such as

solids loading, particle-particle interaction, and in certain size ranges,

convection currents and Brownian motion.

Reduced sensitivity, accuracy and resolution for materials with density

close to that of the dispersion medium, and under conditions of laminar

flow of particles.

◆ Gravitational Sedimentation Techniques
53

Obtain baseline X-ray intensity signal

Load sample in cell to concentration level 

recommended by instrument manufacturer 

Obtain data under parameters chosen for 

experiment 

Apply any correction factors to data, if necessary 

Convert data to form desired for interpretation of results 

Obtain suitable laboratory samples from bulk powder 

Disperse sample in suspension 

(ensure absence of agglomeration, adequate density contrast) 

Keep suspension agitated/stirred to prevent settling or agglomeration 

Flush sample cell and plumbing in instrument 

Sedimentation techniques for particle size measurements can be classified as

either incremental or cumulative measurements. In the incremental method,

the solids concentration of the suspension is determined at a given height

and time. The rate at which the solids (powder) settle out of the suspension

is determined by the cumulative method. Both measurement methods can

introduce the powder as a line-start or homogeneous technique. The powder

can be introduced as a thin layer on top of a column of liquid (line-start

technique) or the powder is uniformly dispersed in liquid (homogeneous

technique). Allen and others 1-4 described details of the different sedimentation

methods and techniques.

Introduction ◆

Figure 4.1. Figure 4.1.

Flowchart Indicating Steps in Size Analysis by X-Ray Gravitational Sedimentation
54

2. General Procedures

The general procedures required for size analysis by x-ray gravitational

sedimentation are illustrated in the flowchart indicated in Figure 4.1.

These steps include obtaining a representative sample weighing between

a few milligrams and a few grams (dictated by powder density and sample

concentration), dispersion and stabilization in a suitable medium, instrument

set-up, obtaining data, and post-processing of this data/information to include

any necessary corrections and conversion of data to a format suitable for

data analysis.

2.1 Powder Sampling

It is critical to ensure that the sample used for testing is representative of the

entire batch, and the sampling procedures followed ensure the preservation of

that aspect. Specimen preparation is another critical area and a major source

of error during analysis. Errors and variations may result due to changes in the

nature of the powder during specimen preparation. The nature of the sample

may change primarily due to either agglomeration of the powder, or fracture

of the particles during attempts to disperse the powder in a suitable dispersion

medium. A more detailed discussion on issues of sampling can be referred to

in Chapter 2 on Powder Sampling.

2.2 Dispersion

A well-dispersed suspension is a key factor in the reliable and reproducible

measurement of particle size and size distribution. Procedures have to be

developed for dispersing materials depending upon the physical and chemical

characteristics of the powder and the medium in which the powder is to be

dispersed. Thus, different material systems will require different dispersion

procedures. The selection of the dispersion medium requires consideration of

physical properties such as specific gravity of the fluid, viscosity of the fluid

and any possible changes in viscosity during fluid flow (e.g., thixotropic or

dilatant characteristics). The density of the powder being analyzed should be

known, and the dispersion fluid should be so selected that there is adequate

density difference between the fluid and the particles, so that the particles will

settle in the fluid. The CRC Handbook of Chemistry and Physics, and other

similar physical data handbooks are commonly used sources of this information.

Usually, the most reliable sources of such information are the manufacturers

and suppliers of these materials. The influence of dispersion agents that may

be added to the dispersion medium should also be carefully considered.

One of the properties related to the surface chemistry and the electrochemical

behavior of the particles is the iso-electric point (i.e.p.). The i.e.p. is the pH

◆ Gravitational Sedimentation Techniques
55

value at which the zeta potential of a system is zero. The zeta potential is

the potential at the hydrodynamic slippage plane created in the electrical

double-layer surrounding a particle with a charged surface. Expressed

simply, at a pH value equal to the i.e.p. of the system the positive and

negative charges surrounding the particle surface are balanced (zero net

charge). At a suspension pH equal to i.e.p., the particles in suspension will

not stay dispersed and will spontaneously agglomerate. The i.e.p. value also

provides a range of pH values in which suspensions can be dispersed and

it determines whether a material is acidic or basic. As a general rule, the

suspension pH should be at least 3 pH units from either side of the i.e.p.

in order to aid dispersion. For example, silicon nitride has an i.e.p. of

approximately 6 to 7 and thus, suspensions should be stable at around pH 9

to pH 10.

The selection of a dispersing agent is also influenced by the surface chemistry

of the material to be dispersed. An acidic polyelectrolyte will work best with

a basic solid, while a basic (cationic) polymer is the preferred dispersing agent

for an acidic solid. A basic solid is a material that has an iso-electric point

greater than pH 7. Similarly, an acidic solid has an i.e.p. below pH 7. For

example, alumina with an i.e.p of 9 is a basic solid. The preferred dispersing

agent is an acidic polyelectrolyte such as polymethacrylic acid. Low molecular

weight polymers (MW 5000 to 10000) are the preferred dispersing agents

because of their minimal effect on the solution viscosity.

A study was conducted at NIST to determine the influence of ultrasonication

on the state of dispersion of ceramic powders during particle size

measurement. The method involves following the change in the particle size

distribution as a function of the ultrasonication time used to disperse the

powder in dilute suspensions. The ultrasonication energy was kept constant.

If agglomerates are broken up due to the energy applied during ultrasonication,

then the particle size distribution should shift toward the finer size as the

ultrasonication time is increased. The results of this study showed that

the measured median diameter of the powder system decreased as the

ultrasonication time was increased, within limits. Longer ultrasonication times

helps to eliminate the errors associated with dispersion procedures. However,

ultrasonication also causes an increase in the temperature of the suspension,

and possible evaporation. Figure 4.2 shows that the median diameter of an

alumina and silicon nitride powder system decreased as the ultrasonication

time was increased, with the ultrasonication energy input kept constant. This

trend was observed till a limit of 1 min after which no significant reduction was

observed. Increasing the ultrasonication energy input to the suspension also

results in a finer median particle size from the deagglomeration of the particles 9.

General Procedures ◆
56

Figure 4.2. Figure 4.2.

Median Diameter as a Function of Time of

Ultrasonication for Alumina and Silicon Nitride Powder

Time of Ultrasonication, s 

0 20 40 60 80 100 120 140

Diameter d50, μ

m

0.0

0.2

0.4

0.6

0.8

1.0

Alumina

Silicon Nitride

A case study describing studies conducted at NIST involving dispersion of

silicon nitride in water is included in Section 5 of this chapter. This example is

specific to silicon nitride-water system, and usually dispersion procedures and

suspension preparation techniques should be developed specifically for the

material system being analyzed.

2.3 Test for Dispersion

After dispersing a sample for testing, it is just as important to ensure that the

powders remain well dispersed in suspension prior to analysis. A method that

allows a qualitative determination of the state of dispersion involves examining

suitable specimens using a microscope. This technique involves placing a glass

coupon coated with a thin layer (ideally, monolayer) of a cationic, anionic or

non-ionic flocculant depending on the material studied in the suspension of

interest. The coupon is kept in the suspension for a period of time, removed

and rinsed in distilled water to wash away the suspension fluid. Ideally, those

particles in suspension that come in contact with the flocculant surface adhere

to the coupon, while the rest of the particles are washed off upon rinsing.

Agglomerates that may be present in the suspension will also exhibit a similar

behavior and some agglomerates may adhere to the coupon. The slide is then

examined under an optical microscope. This technique enables a qualitative

judgement only, and works for particles in the 0.5 μm to 50 μm size range.

◆ Gravitational Sedimentation Techniques
57

Particles finer than this size range are too small to be resolved using an optical

microscope, while particles larger than this range will spontaneously settle due

to the particle mass.

2.4 Instrument Set-up and Analysis

Before attempting any size analysis, it is absolutely essential to ensure that

the tubing comprising the plumbing system of the instrument is clear of any

powders that may contaminate the sample of interest. Any kinks in the tubing

should be cleared, and the tubing around peristaltic pumps checked for wear

and tear. Similarly, the sample cell window should be checked to ensure that

no powder is adhering on to the window, and that the window is free from

scratches. Cell windows get scratched by the abrasive action of powders

during fluid flow in the instrument. Scratches on cell windows act as sites

for powder adhesion due to the roughened surface, and also cause light to

be scattered in instruments that use photo probes.

Instruments should be calibrated as recommended by the instrument

manufacturers. The frequency with which these instruments are calibrated

depend upon various factors, including how frequently the instrument is

used, the nature of application (e.g., quality control, process monitoring or an

occasional size analysis requirement). Instruments used for critical applications

should be calibrated at least once daily. Some instrument manufacturers supply

calibration standards, while others recommend third-party standard materials.

In all cases, the standards chosen should be either primary standards or

secondary standards traceable to primary standards. Examples of primary

standards include SRMs produced by NIST. A discussion of the SRMs is in

Section 4 of this chapter. Examples of secondary standards include two garnet

material systems with mass median diameters of 3.77 μm and 11 μm produced

by Micromeritics Instrument Corp., the manufacturer of the Sedigraph 5100,

an x-ray gravitational sedimentation system. One garnet material (3.77 μm)

is traceable to NIST SRM 1978.

Prior to actual analysis, the instrument should be setup as per the

manufacturer’s recommended procedures. Set-up procedures may include

establishing powder and suspension medium information, such as powder

density, specific gravity and viscosity of suspension, determining the nature

of analysis, resolution of analysis, etc. A procedure common to most

instruments irrespective of the manufacturer is the establishment of baseline

data. A baseline signal for the intensity of the probe transmitted through the

suspension fluid in the absence of any sample is first obtained. This signal

intensity at different heights of the sample window is compared with the

signal intensity that is obtained with the sample powder present in the

suspension to determine the settling rates of the powders during analysis.

General Procedures ◆
58

Samples should be analyzed as soon as possible after preparation. Specimens

that have been prepared and allowed to stand may undergo changes in the

state of dispersion, and in physical and chemical properties of the powder being

analyzed. A global raw material producer aims to analyze samples in less than

20 s after preparation. Powders while in suspension often leach or dissolve on

the surface and may indicate non-representative behavior or characteristics

upon analysis.

Data obtained from sample analysis may need to be processed further.

Processing operations typically include transforming differential distributions

to cumulative distributions or vice versa. Some instruments might require the

application of certain correction factors. Post-processing of data might also

include transformation of data from a mass basis to a volume basis or other

bases of interest. Most modern instruments come equipped with software

that enables such post-processing operations. While using pre-programmed

software or transformations, it is a good practice to attempt to understand the

nature and design of the algorithms being used. These algorithms are often

based on assumptions that may not necessarily be applicable to the material

system under study and may lead to significant errors in the observed results.

2.5 Data Representation

The particle size data from the gravitational sedimentation can be presented

either in tabular or graphical form. In both forms, the data can be displayed

as a plot of the cumulative mass percent finer (or cumulative mass percent

coarser) than a particular fraction as a function of the equivalent spherical

diameter or as the mass fraction contained in given size intervals expressed

as a plot of the mass fraction versus the corresponding size interval. It is good

practice to include both cumulative and fractional size distribution data, as such

representations allow easy interpretation of whether the powder is of single

or multi modal distributions, and also enable judging characteristics such as

degree of grinding. Size and size distribution data are always expressed as

a function of the equivalent spherical diameter, as the particles settling in the

sedimentation column are considered to have a mass equal to that of a sphere

settling with the same sedimentation velocity. Thus, for spherical particles, the

equivalent spherical diameter is close to that of the particles in question, but,

for irregular particles, there will be a significant difference between the two

values. As the aspect ratio of the particle increases, the magnitude of this

difference also increases correspondingly. Tabulated data are useful in the

determination of statistical information about the particle size distribution of

the sample and in determining representative size/size distribution measures

such as the median diameter (d50), the d10 and d90 values. The particle size

distribution data are expressed on a mass basis. A detailed discussion on

◆ Gravitational Sedimentation Techniques
59

representation of size data is covered in Chapter 7 of this guide. Figure 4.3

shows a graphical representation of particle size data of a silicon nitride

powder system plotted as a measure of the cumulative mass percent finer

than a given fraction as a function of the equivalent spherical diameter.

Equivalent Spherical Diameter (μm)

100.0 10.0 1.0 0.1

Cumulative Mass Finer, (%)

0

10

20

30

40

50

60

70

80

90

100

Figure 4.3. Figure 4.3.

Example of Graphical Representation of

Particle Size Distribution of a Silicon Nitride Powder System

3. Sources of Error, Variations and Other Concerns

Errors and variations in results and analysis may arise at various stages of

sample preparation and analysis. These errors and variations can be attributed

to various causes including those related to specimen and sample preparation,

instruments and equipment, and those due to human factors and intervention.

Errors due to human factors are mostly observed during specimen preparation

and handling and during instrument setup. These errors can be minimized with

the development and adoption of robust operational procedures and techniques.

As with any case of classification of errors, those encountered here too can

be classified as random or systematic errors. Some commonly encountered

sources of error and variation are:

Errors and variation associated with specimen/sample preparation

● Non-representative sampling of powders for specimen preparation is a

common source of error. It is important to ensure that samples are obtained

from powder lots representative of the entire bulk. This may require that

the powder lot be split into smaller batches using some reliable statistical

sampling technique.

General Procedures ◆
60

● Errors often arise during specimen preparation due to inadequate dispersion

of powders. Formation of agglomerates during specimen preparation should

be avoided, and similarly fracture of particles due to excessive energy input

during dispersion should be prevented. Techniques used for dispersing the

particles should be reliably tested to ensure that they do not lead to a

change in the nature of the powders.

● Contamination of the specimen powder during sample preparation should be

carefully avoided. A commonly observed contaminant is wear debris from

the ultrasonic probe (typically Ti) over a period of time. Cavitation near the

probe tip is a common factor contributing to the rapid degradation of the

probe tip.

● Samples should be tested as soon as possible after preparation to minimize

the effects of changes in sample properties due to various chemical and

surface phenomenon including dissolution, leaching, re-precipitation, etc.

● Errors and variations may also arise due to the concentration effects of the

suspension. Particle-to-particle interactions observed at high concentrations

give rise to a wide variety of physical effects. Depending upon the surface

chemistry of the powder, high solids concentrations can lead to hindered

settling of powders, causing a reduction in the settling velocity. This

decrease in the particle velocity could shift the particle size distribution

lower and produce an effect that indicate a greater concentration of fines

than is actually present. Conversely, particle-to-particle interactions at a

high concentration may also lead to agglomeration of powders, causing an

apparent increase in the coarse fraction. Also, at high concentrations, some

fine particles may be entrapped in the wake created by the settling of larger

particles, skewing the settling velocity of the finer particles.

Errors and variation associated with instrumentation/equipment Errors and variation associated with instrumentation/equipment

● Improper instrument set-ups due to errors in calibration, lack of calibration,

and incorrect baseline signals are factors that contribute to systematic errors

in the observed results. Similarly scratches, abrasions or other forms of

damage to the sample cell can lead to significant errors in the observed size

results. These errors can readily be minimized, if not totally avoided, by

following the instrument manufacturer’s recommended procedures for

instrument set-up and maintenance.

● Analysis of particles finer than about 0.5 μm can be strongly influenced by

Brownian motion. For particles of the same size, the influence of Brownian

motion is much greater on particles with lower density, rather than those

with higher density. The actual size limit at which random thermal motion

becomes a significant factor is strongly influenced by particle density, the

temperature at which analysis is conducted and fluid specific gravity.

◆ Gravitational Sedimentation Techniques
61

Particles with high density are influenced by Brownian motion at a smaller

size limit than particles with low density. Brownian motion interferes with

the settling of particles, and increases the settling time of the particles. For

very fine particles, Brownian motion may also keep the fines suspended for

an indefinite time, grossly skewing the representation of the finer fraction

actually present.

● Defective or damaged tubing, forming the plumbing and flow system

of an instrument can lead to significant errors in analysis. Two commonly

encountered errors that can be directly traced to defective/damaged

tubing are air bubbles trapped in the fluid flow system and powder

particles trapped or sticking to the walls of the tubing. The latter causes

contamination of the sample and may lead to errors in size analysis due to

differences in particle density or may cause agglomeration of some of the

existing particles. The existence of air bubbles can lead to turbulence in

fluid flow and directly interfere with the settling of particles. It is necessary

to regularly examine the tubing to ensure no damage or kinks that can trap

powder or pinholes that may lead to aspiration of air into the flow system.

● The values of physical properties of the materials being studied should be

correctly known as these values are used for calculation of the size results.

These include the value of density of powder, specific gravity and viscosity

of the suspension medium, temperature of the sample cell, and in some

cases, refractive indices of the material and dispersion medium.

The need for correction factors should also be considered. The use of

correction factors is most frequently encountered with instruments based on

principles of photo-sedimentation. The correction factors would account for

scattering of light due to physical phenomenon such as multiple scattering

of light. In these techniques, analysis of powders finer than about 2 μm will

require a precise knowledge of the real and imaginary component of the

refractive index. Determination of the imaginary component of the refractive

index for a particular material can be time-consuming and prone to errors.

A detailed discussion on issues associated with the role of the imaginary

component of the material refractive index and the determination of the same

is present in Chapter 6 on Size Determination by Laser Light Diffraction.

The gravitational sedimentation method should not be used as the sole

technique to obtain absolute size data, unless the powder contains spherical

particles only. The method is very useful for observing the trend of the size

distribution when changes are made to the processing parameters. Excessively

long sedimentation time, especially in the sub-micron range, can be a source

of concern.

Sources of Error, Variations and Other Concerns ◆
62

4. Relevant Standards

Numerous normative standards and standard materials have been developed

for the measurement of particle size and size distribution using gravitational

sedimentation techniques.

An ASTM standard test method applicable to ceramic materials is ASTM

C958, Test Method for Particle Size Distribution of Alumina or Quartz by

X-ray Monitoring of Gravity Sedimentation10. This standard is for the

determination of particle size distribution of alumina or quartz powders with

average particle diameter from 2.5 μm to 10 μm. The dispersion procedure

recommends that the powder be suspended in sodium hexametaphosphate

and be dispersed in an ultrasonic bath. Our experience with using an ultrasonic

bath is that the amount of energy output generated and transferred by the

ultrasonic bath is insufficient to break some agglomerates.

ASTM B761, Standard Test Method for Particle Size Distribution of

Refractory Metals and Their Compounds by X-ray Monitoring of Gravity

Sedimentation11, has been developed for determination of particle size

distributions of refractory metal powders. This standard covers sample

preparation and gives a detailed procedure for analysis. Included in this

standard is a sample dispersion procedure for refractory metal powders.

Other international agencies for standards have also developed standard test

methods for gravitational sedimentation techniques. One of these is the BS

3406, British Standard Method for Determination of Particle Size Distribution

– Part 2: Recommendation for Gravitational Liquid Sedimentation Methods

for Powders and Suspensions12. This is a very comprehensive standard that

includes methods for both incremental and cumulative techniques of analysis.

The incremental techniques section includes a description of the pipette,

photo-sedimentation and x-ray sedimentation methods. Techniques using the

sedimentation balances are covered under cumulative methods. The standard

includes a brief description of the underlying principles of each method and

calculations needed to compute the particle size. It also covers the preparation

of a suspension and the dispersion procedure. A table of suitable suspending

liquids and dispersing agents for a range of material has been included.

An ISO working draft designated ISO/WD 13317-1, Determination of

Particle Size Distribution by Gravitational Liquid Sedimentation Methods –

Part 1: General Principles and Guidelines13 is being compiled for gravitational

sedimentation. This draft is applicable to gravitational sedimentation techniques

in general, and provides general principles for this method, including an

explanation of incremental and cumulative methods using line-start and

homogeneous techniques. It discusses the size limitations of these techniques

and includes a discussion on how particle size is calculated. A general

◆ Gravitational Sedimentation Techniques
63

discussion on sample dispersion is included. Some general criteria to consider in

the selection of the suspension fluid are also discussed. The draft also provides

methods to examine the state of the dispersion. A section on the contents of

the analysis report is included in this draft.

Another working draft, ISO/WD 13317-3 Determination of Particle Size

Distribution by Gravitational Liquid Sedimentation Methods – Part 3: The

X-ray Gravitational Technique14, is being developed specifically for instruments

based on the x-ray sedimentation method. This draft includes discussions

on the principle of gravitational x-ray method along with a description of the

instrument, preparation for analysis and the instrument procedure to be

followed.

Similar standard methods developed for sedimentation-based techniques are the

German Standard DIN 66 111, Particle Size Analysis: Sedimentation Analysis:

Principles; DIN 66 115, Test Method for Particle Size Analysis, Sedimentation

Analysis in the Gravitational Field. Standard test methods developed by the

French standards institute, AFNOR, include NF X 11-681 Test Method for

Particle Size Analysis – Particle Size Analysis by Gravity Sedimentation in

a Liquid Medium; and NF X 11-683 Test Method for Particle Size Analysis –

Particle Size Analysis of a Powder by Variable Height Gravity Sedimentation

in a Liquid-Method Using X-ray Adsorption Measurement. There are also

standards developed by the Japanese Standards Institute, JIS Z8820, General

Rules for the Determination of Particle Size Distribution by Sedimentation

in Liquid.

A list of possible SRMs for calibration of instruments based on gravitational

sedimentation is listed in Table 4.1.

SRM Material Material Particle Size Diameter (μm) Particle Size Diameter (μm)

659 Silicon Nitride 0.2 to 10

1978 Zirconia 0.33 to 2.19

SRM 659 is a silicon nitride powder that has equiaxed primary particles with

a minimal amount of large agglomerates. Similarly, SRM 1978 is a zirconium

oxide powder comprised of granular, irregular-shaped primary particles

consisting of minimal amounts of large agglomerates. These SRMs are

specifically certified as standards for particle size determination by gravity

sedimentation methods, and can be used to calibrate and evaluate instruments

based on these techniques.

Relevant Standards ◆

Table 4.1. able 4.1.

NIST Standard Reference Materials for Gravity Sedimentation
64

5. Practical Application: Dispersion Procedure for

Silicon Nitrade in Aqueous Suspension

An example of a dispersion procedure developed for an international

inter-laboratory round-robin test is illustrated in Table 4.2. There are many

factors to consider when developing a specific procedure. Some of these

include concentration of the suspension, the pH of the suspension, the power

and duration of the ultrasonication. The volume fraction of the suspension

should be dilute (no more than 2 %). Often the optimum volume fraction is

determined by trial and error. This optimal value is such that the reduction in

intensity of the x-ray beam when it travels through the sample lies within a

value recommended by the instrument manufacturer. The pH of the suspension

will be different depending on the material being analyzed. Knowledge of the

i.e.p. of the specific powder is necessary in developing a stable suspension.

Usually, adjusting the pH of the suspension alone will not disperse the material.

Energy needs to be applied to break up any agglomerate. The amount of

ultrasonication energy may vary depending on the material. The required

energy should have the capacity to break soft agglomerates. In this example,

a total ultrasonication time of 3 min was used. To prevent overheating of the

sample, ultrasonication was done in 3 cycles, with each cycle comprising of

power applied for 1 min and shut-off for another minute. The output power

of the ultrasonic probe depends on the diameter of the probe used. The

temperature of the suspension needs to be monitored. Overheating of the

sample can also be minimized by placing the suspension in an ice bath.

Each material may need a specific dispersion procedure because the

surface chemistry of each material is different. It is preferable to keep

the suspension circulating to prevent any settling.

◆ Gravitational Sedimentation Techniques
65

Table 4.2. able 4.2.

Example of a Procedure for Silicon Nitride used for Sample Dispersion

in an International Inter-Laboratory Round-Robin Test

Practical Application ◆

Some instruments require an optimum concentration to enable the instrument

to operate efficiently. The user needs to experiment with the mass fraction of

the suspension to produce the optimum concentration for the instrument.

Slurry Preparation Procedure for a 4 % Mass Fraction of Solids

● Weigh 4.0 g of silicon nitride powder.

● Dispense 96 g of de-ionized water into a 150 cm3 beaker.

● Measure the pH and adjust the liquid to a pH of 9, by adding ammonium

hydroxide of 1 M concentration.

● Stir for 2 min.

● Measure the pH of the solution and readjust if needed.

● Add silicon nitride powder slowly to the pH-adjusted liquid.

● Stir the slurry while adding the powder so that it does not settle.

● Measure the pH of the slurry and readjust if necessary.

● Stir for 10 min to allow for equilibration.

● Measure the pH of the slurry after stirring and readjust the pH if needed.

Ultrasonication Procedure

The following ultrasonication procedure is recommended for the 100 g slurry

in a 150 cm3 beaker for a 19 mm diameter probe.

● Prepare the ultrasonic probe as per the manufacturer recommendation

(including tuning and polishing the tip).

● Rinse the probe well with distilled water.

● Place the probe 20 mm above the bottom of the beaker.

● Adjust the output power to 30 W (use the calibration curve provided by

the manufacturer). Note that these power values are approximate since

each instrument is not calibrated.

● Apply power for the appropriate period of time.

● Rinse the probe with distilled water after ultrasonication and make sure

no powder residue remains on the probe.

● Measure the temperature and the pH of the slurry after ultrasonication.

● Allow the slurry to return to room temperature by stirring. Do not allow

the temperature of the slurry to exceed 35 oC during ultrasonication.
66

References

1. T. Allen, Particle Size Measurement, 3rd ed., Chapman and Hall, London

(1981).

2. C. Bernhardt, Particle Size Analysis: Classification and Sedimentation

Methods, 1st Eng. lang. ed., Chapman and Hall, London (1994).

3. T. Allen, Sedimentation Techniques of Particle Size Measurement in

Particle Size Analysis, 1985 Proceedings of the Fifth Particle Size Analysis

Conference, P.J. Lloyd, ed., John Wiley and Sons, Chichester (1987) p. 25.

4. L. Bunville, Commercial Instrumentation for Particle Size Analysis in

Modern Methods of Particle Size Analysis, H.G. Barth, ed., John Wiley

and Sons, New York (1984) p. 1.

5. BS 3406: Part 6 (1985) British Standards Methods for Determination

of Particle Size Distribution, Recommendations for Centrifugal Liquid

Sedimentation Methods for Powders and Suspensions, British Standards

Institute, London (1985).

6. ISO/WD 13318-1 Determination of Particle Size Distribution by

Centrifugal Liquid Sedimentation Methods – Part 1: General Principles and

Guidelines, International Organization for Standardization, Geneva (1997).

7. J.T. Jones, When is Stokes’ Law Not a Law? Ceramic Industry, August

1997.

8. T. Allen, The Limits of Gravitational Sedimentation, Powder Technology,

18, 131 (1977).

9. S.G. Malghan, Statistical Analysis of Parameters Affecting the

Measurement of Particle-Size Distribution of Silicon Nitride Powders by

Sedigraph, Powder Technology, 73, 275 (1992).

10. ASTM C958 Test Method for Particle Size Distribution of Alumina or

Quartz by X-ray Monitoring of Gravity Sedimentation, American Society

for Testing and Materials, West Conshohocken, PA (1997).

11. ASTM B761 Standard Test Method for Particle Size Distribution of

Refractory Metals and Their Compounds by X-ray Monitoring of Gravity

Sedimentation, American Society for Testing and Materials, West

Conshohocken, PA (1998).

12. BS 3406: Part 2: 1984 British Standard Methods for Determination of

Particle Size Distribution Part 2. Recommendations for Gravitational

Liquid Sedimentation Methods for Powders and Suspensions, British

Standards Institute, London (1984).

◆ Gravitational Sedimentation Techniques
67

13. ISO/WD 13317-1 Determination of Particle Size Distribution by

Gravitational Liquid Sedimentation Methods – Part 1: General Principles

and Guidelines, International Organization for Standardization, Geneva

(1996).

14. ISO/WD 13317-3 Determination of Particle Size Distribution by

Gravitational Liquid Sedimentation Methods – Part 3: The X-ray

Gravitational Technique, International Organization for Standardization,

Geneva (1996).

15. J.S. Reed, Introduction to the Principles of Ceramic Processing, 1st ed.,

John Wiley and Sons, New York (1988).

References ◆
69

5. SIZE CHARACTERIZATION BY

MICROSCOPY-BASED TECHNIQUES

1. Introduction

2. General Procedures

2.1 Number of particles to count

2.2 Powder dispersion for specimen preparation

2.3 Specimen preparation

2.4 Techniques of size analysis

3. Sources of Error

4. Practical Concerns

5. Relevant Standards

1. Introduction

Microscopy-based techniques for particle size characterization provide

a powerful tool for characterization of particle size, size distribution and

morphology. They involve direct observation of particles and the consequent

determination of size based on a defined measure of diameter (see Chapter 7,

Reporting Size Data, Table 7.3). Typically, the calculated sizes are expressed

as the diameter of a sphere that has the same projected area as the projected

image of the particle. The calculated sizes or size distributions can then be

converted to, or expressed in, different measures (area, mass or volume

distributions) with suitable precautions. When compared to other techniques

of particle size analysis, a significant advantage of microscopy-based

techniques lies in the ability to determine the particle shape, in addition to

making a direct measurement of size. Numerous commercial instruments

and software packages are available for particle size and size distribution

analysis. Some commercial instruments are sold as complete packages

including the microscope, accessories, dedicated computer systems and

software algorithms for analysis. Some packages are available as software

add-ons to existing microscopes. Packages with a wide range of functions and

levels of complexity are available. While some packages are limited to size

determination of well-dispersed particles, other packages have the ability to

discern individual particle sizes and shapes from clusters and agglomerates –

even the ability to distinguish and determine particles of varying chemical

compositions and phases. The analytical capabilities of numerous software

packages can be enhanced by the use of additional mathematical algorithms

or software systems that can be tailored to meet the needs of individual users.

Microscopy-Based Techniques ◆
70

◆ Microscopy-Based Techniques

Microscopy-based techniques can be applied to study a wide range of

materials with a broad distribution of powder sizes, ranging from the nanometer

to the millimeter scale. Instruments used for microscopy-based techniques

have conventionally included optical light microscopes, scanning electron

microscopes (SEM) and transmission electron microscopes (TEM). The choice

of the instrument to be used is determined by the size range of the powders

being studied, magnification, and resolution that is desired. However, the cost

of analysis is also observed to generally increase as the size of the powders

decreases due to requirements of higher magnification, improved resolution,

greater reliability and reproducibility. The cost of size analysis also depends

upon the material system being studied, as that dictates the techniques of

specimen preparation and image analysis that need to be followed.

Optical microscopes tend to be more affordable, and comparatively easier

to operate and maintain than electron microscopes, but are more limited in

magnification and resolution. This limits most reproducible analysis by optical

microscopy to approximately 1 μm particles1 with a theoretical resolution

limit of about 0.2 μm (dependent upon wavelength of illumination source).

However, in practice this limit is not achieved and BS340619 recommends

sizing by optical microscopy for particles that are a minimum of 10 times

larger than the resolution limit of the objective lens present in the microscope.

Scanning electron microscopes enable examination of details at higher

magnification and resolution, and are thus suitable for particles in the size range

of about 0.1 μm to 1000 μm (the lower limit depending upon the quality of the

instrument being used). Transmission electron microscopes enable analysis in

the size range of 0.01 μm to 10 μm1, retaining suitable resolution to enable

distinguishing surface characteristics. Well-maintained SEMs can regularly be

used at resolutions of about 10 nm, while the limit on the resolving power of a

TEM is a minimum of 5 nm. In all these cases the magnification and resolving

power of the instrument used will be dependent upon the nature of the sample,

the instrument operational parameters and the state of technology incorporated

in the instrument. The theoretical limit of resolving two discrete points

separated by a distance d, can be expressed as:

NA

d 0.61λ =

where, λ is the wavelength of the illumination source (expressed in μm) and

NA is the numerical aperture. Figure 5.1 illustrates a guideline indicating

the relationship between the size of the particle to be determined and the

instrument suitable for analysis in that size range.

The skill level required to operate a TEM and to prepare specimens for the

instrument is significantly higher than that associated with the use of a SEM.
71

When comparing these techniques with each other, sample preparation for

optical microscopy-based analysis is relatively easy and can be achieved in

a short period of time. Sample preparation for SEM analysis requires greater

effort and time to ensure suitable mounting of the sample. Samples for TEM

techniques are prepared as replicas or thin films, depending on the size of

the powders being analyzed. These techniques require the greatest amount

of effort and time. However, SEM and TEM techniques offer significant

advantages in terms of the greatly enhanced depth of focus that enables

viewing of particles with varying sizes in the same field of focus (particles of

significantly different sizes can be observed simultaneously without the need to

refocus on each particle). Optical microscopes conversely have a very limited

depth of focus and thus, require constant correction in the focus field when

studying particles of varying height profiles lying on the sample stub. As a

general rule, sample preparation for optical microscopy tends to be relatively

simple and can be accomplished rapidly, while sample preparation for TEM is

far more involved and requires suitable training and a considerable amount

of time.

Figure 5.1. Figure 5.1.

Relationship Between Particle Size and Instrument Suitable for Analysis;

based on information in Ref. [2]

Introduction ◆

Sea sand

Oil soot

Ragweed/Pollen

Potato starch

Corn starch

Red blood cells

Bacteria

TiO2 pigment

Carbon black

Tobacco smoke

Viruses

0.01

10–3

10–6

10–9

10–12

10–15

10–18

10–21

0.1

Log Particle Mass (g)

1.0

Log Particle Diameter (μm)

Transmission Electron Microscopy

Mono Objective Optical

Microscopy

Scanning Electron

Microscopy

Stereobinocular Microscopy

10.0 100
72

◆ Microscopy-Based Techniques

2. General Procedures

The general procedures followed during particle size analysis by microscopybased techniques are illustrated in the flowchart in Figure 5.2. While size

analysis by microscopy-based techniques enables direct observation of the

powders constituting the material system under study, they can lead to

erroneous results if adequate precautions are not taken during sampling and

sample preparation. It is absolutely essential that the sample being analyzed is

representative of the bulk of the powder. While this is true for any size analysis

technique, it has an even greater implication, when sample size gets smaller.

Thus, any bias towards either fine particles or coarse particles in the observed

field will tend to shift the calculated size and size distribution in that direction.

These shifts are not a significant problem, when studying particle systems of

relatively uniform sizes. However, this is not true for most real systems, where

particles tend to have a broad size distribution. An illustration of this issue can

be observed in Figure 5.3. This figure shows the SEM image of a bi-modal

mixture of dispersed commercially available silica spheres. This system is a

volume fraction mixture of 10 % 1.0 μm (nominal size) and 10 % 0.5 μm

(nominal size) silica particles. The number of 0.5 μm (nominal size) particles

contained in this volume will be far greater than the number of 1.0 μm (nominal

size) particles. In a uniform, well-dispersed system, this ratio will be preserved.

The same ratio needs to be maintained if pipetting out a sample to create the

microscopy specimen. Often, particle settling or agglomeration may cause

particles of a particular size fraction to settle at the bottom, or to be present in

a disproportionate amount in the specimen, leading to a bias in the counting and

thus a skewed result. The solution to this problem lies in ensuring that (a) the

sample is well dispersed, (b) the specimen being studied is representative of the

sample, and (c) counting a large number of particles to ensure that statistical

errors due to inadequate representation are controlled.
73

Figure 5.2. Figure 5.2.

Flowchart Indicating Steps to be Followed in

Particle Size Analysis by Microscopy-based Techniques

General Procedures ◆

Sample powder to obtain suitable sub-sample

Disperse sample in appropriate fluid 

Prepare specimen 

(slide, stub or grid) 

Check instrument calibration and 

conduct other pertinent checks 

Observe and capture images 

of regions of interest 

Conduct image analysis of particles 

Process image analysis data 

Calculate and report powder size 

and size distribution data 
74

◆ Microscopy-Based Techniques

Figure 5.3. Figure 5.3.

SEM Image of 20 % volume fraction SiO2 Dispersion Comprised of a Bimodal

Mixture of 10 % 1.0 μm (nominal) and 10 % 0.5 μm (nominal) Powders

2.1 Number of Particles to Count Number of Particles to Count

The statistical errors typically encountered when using microscopy-based

techniques are due to counting a disproportionately larger fraction of coarse or

fine particles giving rise to a bias towards that fraction, than is actually present

in the whole powder. This bias is also enhanced in the calculated standard

deviation of the powder size distribution around a central measure. The number

of particles to be counted to ensure statistical accuracy depends on whether

the distribution is characterized on a number basis or weight basis. Allen3 states

that number-based analyses are simpler than weight-based analyses and that

formulae can be used to calculate the number of particles to be sized in order

to achieve a given accuracy for both distributions.

A more comprehensive mathematical procedure to determine the number of

particles to be counted, in order to minimize the error associated with size

1 0.5 5 μm
75

determination, has been proposed by Masuda and Iinoya4. They have studied

the scatter of experimental data due to a particle size distribution. The theory

they have put forward has been verified by computational simulation and may

be included in the ISO standard on particle size analysis by image analysis

methods (currently in draft stage)14.

From this theory, in order to calculate the number of particles to be counted

so as to keep the error within a desired range, the following steps can be

followed.

1. Determine the desired range of error, and thus, determine δ.

For an error of ± 5 %, δ = 0.05; for an error of ± 10 %, δ = 0.1

2. Note the variance of the powder distribution (σ2).

3. Determine the value of the non-zero exponential constant defining the

powder distribution, α.

The value of this exponent can be found by considering the equation

describing the shape of the powder distribution to be y = K Dp

α.

Thus, for a log-normal distribution α = 2.

4. Determine the basis number for the distribution, β.

For a powder distribution expressed on a number basis β = 0, and for a

mass basis β = 3.

5. Calculate the constant c, such that, c = β + α/2

6. Calculate the value of ω, where ω = u2α2σ2(2c2σ2 + 1),

where,

2 1 2 2 + ≡ασ σ

δ

c

n

u

7. The number of particles to be counted (n*) can then be determined by

substitution into the following equation:

log n* = -2 log δ + log ω

(Also see Appendix for further details.)

As an example,

Consider a powder having a log-normal size distribution of variance (σ2)

0.2209 (such that α = 2). If 90 % of the data is desired to be included in the

relative error of ± 5 %, then for such a calculation, δ = 0.05 and P = 0.95.

From equations 1 and 2 (see Appendix), P = 0.90 will generate a corresponding

value of 1.64 for u. As a number count basis corresponds to a β value of 0,

thus, in this case c = 1.

General Procedures ◆
76

◆ Microscopy-Based Techniques

Substituting these values in equation 5, ω = 1.27 u2,

Thus, log n*= -2 log 0.05 + log 4.879

n* = 1951.53

or, approximately 2000 particles need to be analyzed to get the error within the

desired range.

If it is desired that 99 % of the data should be included in the relative error of

± 5 %, then P = 0.99. In such a case, using the above described procedure:

log n*= -2 log 0.05 + log 8.454

n* = 3381.45

or, approximately 3400 particles need to be analyzed to get the error within

the desired range.

The above theory provides some guidelines for estimating the number of

particles that need to be analyzed in order to have results with some degree of

confidence. However, the assumptions that have been considered in developing

this theory do not always hold good for all “real” powder distributions and the

user should have a good estimate of the values of terms such as α and σ2 of

the powder distribution.

The development and use of procedures involving large number of particle

counts are followed in the development of NIST SRMs. SRM 1004b5, a

particle size distribution standard comprising of glass beads with a size

distribution of 40 μm to 150 μm, required the counting and size determination

of about 20,000 individual particles to ensure representative sampling. These

20,000 particles were classified into seven size fractions corresponding to

seven standard sieve sizes and the results developed into a cumulative

distribution curve. Figure 5.4 shows a representative image used for the

development of a WC-Co Standard Reference Material (NIST SRM 1985,

to be certified for particle size distribution, 20 μm to 50 μm), where a similar

procedure involved the counting of a large number of particles, due to the

broad size range of the particles present.

Counting such a large number was not necessary in the case of a dispersion

comprised of uniformly sized 1.0 μm SiO2 powders. A typical image obtained

from the SEM analysis of this powder system is shown in Figure 5.5(a).

Table 5.1 lists the results from image analysis of this powder system. After

counting over 3000 particles, the mean and median diameters were computed

for 250, 500, 750, 1000, 1500 and 2000 particles selected randomly. It is

observed that the calculated sizes and distribution statistics are similar when

fewer particles are counted and, also, when large numbers of particles are
77

20 μm

Figure 5.4. Figure 5.4.

Representative Image of Particle Size Standard NIST SRM 1985,

Comprising Powders with a Broad Size Distribution

counted. These precautions, incorporated to ensure representative and

unbiased sampling, hold true for all microscopy-based techniques. Due to the

large number of particles being observed, size analysis would be greatly aided

by the use of automated or semi-automated computer systems and algorithms.

The use of such systems is gaining increasing acceptance with the falling costs

of computer hardware and software.

Table 5.1. able 5.1.

Calculated Diameters of Powder System Comprised of

Uniform SiO2 Particles (Nominal Size of 1.0 μm)

Random Sampling Summary

Number of Particles Sampled 

 250 500 750 1000 1500 2000 

Mean Dia. 

(μm) 

1.04 1.04 1.04 1.04 1.04 1.04 

Median Dia. 

(μm) 

1.03 1.04 1.03 1.04 1.04 1.04 

Std. Dev. 0.03 0.04 0.04 0.04 0.04 0.04 

General Procedures ◆
78

◆ Microscopy-Based Techniques

Figure 5.5. Figure 5.5.

(a) Raw Image of Nominally 1.0 μm SiO2 Particles of

Uniform Size Distribution (Table 5.1), (b) Same Image After

Processing by NIH Image6 for Particle Size Analysis and Identification

(a) 10 μm

(b) 10 μm
79

2.2 Powder Dispersion for Specimen Preparation

In order to ensure that the analysis represents the true picture of the powder

size and size distribution, it is necessary that the specimens are prepared from

sample powders that are in a physical state similar to that in which they would

ultimately be used. Thus, when analyzing powders to be used in slip casting,

specimens should ideally be made from the dispersed powders constituting the

slip. Similarly, specimens should be ideally made from dry powders when the

powders are to be used in such a state (e.g., if powders are to be compacted

to form a green body). Irrespective of the physical state of application, the

techniques used for specimen preparation should be reproducible and should

ensure that the specimens are representative of the sample.

Specimen preparation for any microscopic technique (optical, SEM or TEM)

requires that the particles be well dispersed. This is often achieved by

dispersing the powders in solution and then preparing the specimens, but

often the nature of the sample being studied will require that the sample

should not or cannot be dispersed in suspension. Irrespective of whether the

powders are sampled in a dry state or when in suspension, the primary goal of

specimen preparation should be to produce specimens that are representative

of the powder being sampled. The number of particles touching each other or

interacting with each other should be minimized, and should be proportional to

that which exists in the powder.

It is necessary to ensure that no contaminants are introduced during specimen

preparation and storage and that the processes followed during specimen

preparation do not lead to fragmentation or segregation of the powders, both

of which would lead to erroneous results. The use of ultrasonication devices

such as ultrasonic baths and ultrasonic probes are recommended, but with

caution, to ensure dispersion of powders. The ultrasonication energy and

time should be decided upon with great care to ensure that particle dispersion

alone is achieved. Power settings that are too high can cause fragmentation

of particles or heating of the fluid constituting the suspension. Also,

ultrasonication at too low power may not cause adequate dispersion of the

powders. Figure 5.6 indicates the influence of varying ultrasonication duration

on the size distribution of a 2 % (by volume) silica suspension. At 0.25 min,

the size distribution is extremely large (indicative of possible presence of

agglomerates) and after 3 min of ultrasonication, the size distribution does

not show any significant changes. The ultrasonic probe transfers 30 W of

energy into the suspension. The use of ultrasonic horns and probes to disperse

suspensions can lead to the introduction of impurities from the probe/horn

during the process of dispersion. This is more of a problem when using a

probe that has been in service for a while and starts to exhibit signs of pitting

and corrosion due to interaction with abrasive particles and other harsh

General Procedures ◆
80

◆ Microscopy-Based Techniques

environments. The generation of heat during ultrasonication also needs to

be controlled by either cycling the operation of the source or by placing the

suspension in a controlled temperature bath. Excessive heating can cause

irreversible changes in the sample.

The use of ball mills to break up agglomerates present in a powder suspension

is a feasible alternative to the use of ultrasonication devices. The obvious

advantage to this technique is its applicability to large samples, where the use

of ultrasonic horns and probes is impractical due to limitations of sample size

and quantity. The limitations of this technique lie in the initial experiments

to determine appropriate operational parameters. The speed of the mill

and the milling time need to be determined to ensure that no attrition or

agglomeration of the sample powder is occurring. Excessive milling would

lead to fragmentation and increase the concentration of fines. Possible

introduction of contaminants from the milling media and the mill itself is

another shortcoming of this technique.

Often agglomeration of powders during specimen preparation occurs when

droplets of the medium containing the dispersed powder are allowed to dry

on the sample stage. A slow drying process causes the particles to aggregate

due to capillary forces pulling the particles together and may lead to errors in

discerning naturally occurring agglomerates from those formed during sample

preparation. Furthermore, it is good practice to ensure that specimens are

labeled with a suitable shelf-life period, after which the dispersion medium

holding the particles or the particles themselves may change, and the specimen

may not be representative.

Working with powders in a dry state requires greater precautions in order to

prevent aggregation and settling of powders in preferred orientations during

specimen preparation. Uniform powders that are relatively coarse can be

spread out on a glass slide and then picked up on to a tape with an adhesive

surface. The powder particles can be spread out by either tapping the slide

gently or by spreading them out using a brushing action with light strokes.

In some instances, the powders may be spread out directly on to the adhesive

surface of the tape itself. These techniques are reliable only if the particles

that adhere to the tape are representative of the bulk. Uniform spread of the

particles will not easily be attained and may lead to errors if the particles are

too fine, or have a marked aspect ratio that leads to settling along preferred

orientations. Some of these techniques have been used at NIST in the

development of SRMs made up of glass spheres (e.g., SRM 1004b, 1017b,

1018b and 1019b).

Specimens can also be prepared by dispersing a small quantity of the sample

powder (typically a few milligrams) in a non-polar liquid, such as isopropanol,

ethanol or acetone. A drop of the suspension is then placed on the surface to
81

Ultrasonication time (min)

01234567

Measured Size (μm)

0

1

2

3

4

5

6

7

8

9

10

Mean Dia.

Median Dia. (d50)

d10

d90

Figure 5.6. Figure 5.6.

Influence of Ultrasonication Time Upon Particle Size Distribution

of a 2 % Volume Fraction SiO2 Suspension. Ultrasonic Probe

Transmitting Approximately 30 W Used

be studied (such as a glass slide, SEM sample stub or mounting tape). Despite

the relative ease and simplicity of this technique, there are numerous possible

sources of error associated with this. It should be ensured that the small

quantity of powder added to the organic dispersant is representative of the

bulk quantity of the powder. Dispersing the powders in the fluid may require

the use of an ultrasonic bath, and excessive ultrasonication can cause

undesirable fragmentation of the powder. The fluid to be used should be

selected by judging the nature of the interaction between the powders and

the fluid, and the drying rates of the fluid.

In certain instances, specimens can be prepared by filtering the powders

through a suitably sized filter. In this technique, the powders are dispersed

typically in distilled water and then vacuum filtered, such that the particles are

embedded on to the filter. The filter is then suitably mounted on the observation

stage and the powders observed accordingly. The important issue to be kept

in mind when following this technique of specimen preparation is to ensure

that dispersing the powder in water does not change the characteristics of the

powder being studied, and thus, does not cause agglomeration of the powder.

In order to decide the filter size to be used, it is essential that the operator has

some prior knowledge of the powder size distribution, so that there is neither a

General Procedures ◆

Ultrasonication Time (min)

Measured Size (μm)
82

◆ Microscopy-Based Techniques

loss of the fine fraction nor too slow a filtration rate. This technique requires

that the powder concentration being dispersed is dilute enough, such that

particle deposition on the filter will not lead to too many particles overlapping

each other. Particles with a significant aspect ratio will tend to settle along the

long axis of the particle and may cause a significant bias in the observed size

distribution. If possible, these particles should be excluded from the count.

2.3 Specimen Preparation

Preparation of specimens for optical microscopy is relatively easy, as, in

some instances, the particles may still be dispersed in the fluid medium, when

viewed under a microscope. In such a case, the suspension is placed in a

groove or depression in the slide and covered with a cover slip through which

the specimen is viewed. Such a technique cannot be followed when attempting

to prepare specimens for examination by electron microscopy due to the high

vacuum conditions maintained in the microscope. Numerous techniques of

varying levels of complexity are available for preparation of samples for optical

microscopy. Allen describes some of the techniques and their application7.

Some specimen preparation techniques for examination by TEM use a

conducting metallic grid (primarily, copper) on to which the powders are

introduced by different techniques8,9. In order to hold the powder on the grid,

the powder is deposited on a formvar film which is then placed on the grid.

Specimens for examination by electron microscopy require that the powders

be coated with a conducting film, which prevents electron charge build up at

any of the non-conducting surfaces of the specimen. TEM specimens are

typically coated with carbon films, while SEM specimens surfaces are coated

with Au-Pd, Au or carbon films. The deposited film thickness may vary from

about 5 nm to about 20 nm, depending on the material system being studied and

the technique used. Other techniques of specimen preparation for examination

by electron microscopy-based techniques, are also available10.

2.4 Techniques of Size echniques of Size Analysis

Manual, semi-automatic or automatic techniques for particle size analysis are

all influenced by the skill and procedures used by the operators11. Manual

techniques of analysis are based on the use of a marking device such as a

cross-hair being moved along the particle to obtain a linear dimensional

measure of the particle. Other manual techniques involve the use of a graticule

with inscribed circles of varying diameters that are used to evaluate the size of

the particle, by selecting a circle of suitable diameter that can be circumscribed

or inscribed with the particle of interest. Variations to this basic principle have

been described by Allen12.
83

Semi-automatic techniques of analysis require human intervention but, offer

to some degree, the ease of automated analysis. This reduces the tedium and

some of the random errors associated with manual counting and analysis. Some

of the techniques require the projection of the image of the particles on to a

suitable medium12, where they are compared with spots of known sizes, while

others are based on the image shear technique. In the image shear technique

two images of the same particle are created by the splitting of the light beam in

the microscope. Size analysis is achieved by adjusting the amount of shear to

that of the particle diameter, such that the two images only touch each other,

without any overlap or marked separation between the two projected images13.

The shear distances are set for certain values and numerous particles are then

examined, enabling particle classification into various sizes.

Other semi-automatic techniques of size analysis include variations to a basic

principle of recording the image, converting the image to a gray-scale image,

thereby introducing suitable density contrast between the particles of interest

and the background. The image is then digitized and the area of the particles

then calculated. Figure 5.5 indicates the application of a similar principle to a

powder system of silica particles. Figure 5.5(a) shows the raw image obtained

using SEM, while 5(b) shows the processed image with the particles identified.

In processing the raw image, the software is designed to fit an ellipse to the

identified feature, subject to constraints (size, contrast, interior hole-filling)

defined by the user. Table 5.2 shows the information that would be obtained

and then derived for an image processed in a similar manner. The first column

identifies the feature on the processed image, and the second and third columns

the dimensions of the major and minor axes of the feature calculated by the

software and expressed in terms of pixels. The fourth column represents the

area of the feature, also expressed in terms of square pixels. Knowing the

calibration factor for the image and the magnification of the image, the area

calculated by the software algorithm is expressed as the diameter in microns in

the sixth column. From the basic information calculated by the software, other

factors of interest such as the aspect ratio can also be calculated. Analysis

based on this software has been used in the certification on NIST SRM 1982,

a spray-dried zirconia particle size distribution standard (Figure 5.7).

Particles can be counted in semi-automated techniques by a process

sometimes called the “Guard Ring Technique.” This process involves defining

a measurement frame inside the image frame and establishing criteria for

particles falling within or intersecting the edges of the measurement frame14.

Typically, measurement frames are square or rectangular in shape. In some

metallurgical applications, circular frames have been used. When using square

or rectangular measurement frames, two adjacent edges of the frame can

be defined as reject edges, i.e., any particles intersecting these edges get

excluded. Any particles within the frame and/or intersecting the other

General Procedures ◆
84

◆ Microscopy-Based Techniques

two adjacent edges are counted for the analysis. Counting can also be

accomplished by defining a point in the particle, or along the perimeter of

the particle as a feature count point. The same feature count point on every

particle in the image field is examined, and particles with that point in the

measurement field are included, while those particles that have the feature

count point outside the measurement field are excluded. The use of such

techniques helps ensure consistency in selection and counting of particles,

and are most effective when developed on simple criteria.

Automatic image analysis techniques use a combination of image capture and

viewing techniques, followed by image interpretation using algorithms based

mostly on Fourier and fractal shape analysis. The strengths of this technique

lie in the ability to manipulate the image to enhance certain features, improve

contrast and thereby reduce some of the errors that arise due to poor image

quality or particle discernment. Similarly, the use of computers and powerful

algorithms reduces the amount of time required for analysis, and thereby

enable rapid size determination of a large number of particles. The drawbacks

of this technique lie in the high cost of equipment and software needed for this

analysis. A major drawback to automatic image analysis techniques is the

Figure 5.7. Figure 5.7.

Comparison of Particle Size Distribution for NIST SRM 1982

reduction of operator intervention and decision making. This can often lead

to analysis of unwanted features, or analysis using parameters that are

inappropriate for that particular system. However, most modern systems

do allow some degree of human intervention and control.

100

90

80

70

60

50

40

30

20

10

0

20 40

Diameter (μm)

SRM 1982 Particle Size Distribution

Cumulative Mass Fraction (%)

SEM

Sieve

Laser Diffraction

60 80 100 120
85

Table 5.2. able 5.2.

Table of Data Generated from Semi-Automated Image Analysis System

Maj.Axis Min. Axis Area(pix2

) Dia. (pix) Dia. (μm) Aspec. Rat

1 22.29 21.88 383 22.08 1.09 1.02

2 22.23 21.42 374 21.82 1.08 1.04

3 21.96 21.75 375 21.85 1.08 1.01

4 44.6 21.64 758 31.06 1.53 2.06

5 23.42 22.56 415 22.98 1.13 1.04

6 21.59 20.88 354 21.23 1.05 1.03

7 23.6 22.55 418 23.07 1.14 1.05

8 22.99 22.43 405 22.70 1.12 1.02

9 45.29 22.18 789 31.69 1.56 2.04

10 22.76 22.33 399 22.53 1.11 1.02

3. Sources of Error

Various sources of error, both systematic and random, have been mentioned

and discussed in the above sections. A thorough understanding of these is

essential in order to be able to identify the occurrence of these and develop

procedures to minimize their impact on the final results. It is also necessary

to understand that while these errors cannot be eliminated, implementation of

standard procedures and protocols can help characterize the errors and the

magnitude of their influence on the measured results.

Some of the common sources of errors are summarized below:

Errors associated with specimen/sample preparation

● Non-representative sampling of powders

● Inadequate or improper dispersion of powders during specimen preparation

● Agglomeration or fracture of primary particles during specimen preparation

● Contamination of specimen/introduction of artifacts during specimen

preparation/storage

● Specimen degradation during storage.

Errors associated with instrumentation/equipment and image analysis

● Improper calibration of microscope magnification

● Microscope operation under less than ideal parameters (e.g., accelerating

potential or condenser lens current set too high or too low, errors due to

stigmatism)

● Charging of specimen surface while observing under an electron beam

General Procedures ◆
86

◆ Microscopy-Based Techniques

● Operating microscope in regions of “empty magnification” (i.e., no gain in

resolution beyond a certain magnification limit)

● Improper calibration of image analysis hardware/software

● Errors in establishing gray scale threshold in image analysis techniques

● Counting of inadequate number of particles.

Errors due to human and other factors

● Operator fatigue

● Operator skill level and experience (in use of equipment, techniques

associated with analysis)

● Lack of understanding of equipment and software/hardware capabilities

and limitations

● Errors due to inadequate or incorrect statistical analysis.

The above list identifies some commonly encountered sources of errors.

Development of procedures that address the above issues can help alleviate

the impact of some of these sources of errors. Successful procedures are

those that are simple and lead to reproducible results.

4. Practical Concerns

Some of the practical concerns pertaining to sample and specimen preparation

have been discussed in the previous sections. Errors in analysis can be reduced

by ensuring that the powders are well dispersed, and do not agglomerate

when introduced on to a specimen stage. If agglomeration is observed, such

agglomerates should not be included in the analysis. Other important concerns

arise when analyzing irregularly shaped particles or particles with a significant

aspect ratio. Irregularly-shaped particles may come to rest on the sample stub

in a preferred orientation and may thus bias the analysis. Care should be taken

to ensure that such orientation of particles is minimized and, if unavoidable,

procedures should be designed to factor in such a bias in the results. An

extreme example of such preferred orientation may be observed in analyzing

fine fibers or fiber-shaped particles. If the system being studied consists

primarily of such particles, then the measure used to express the particle size

should reflect some information about the particle shape and aspect ratio.

When using projected area values of the particles in the image to generate a

size distribution curve, anisotropic particles at rest in a preferred orientation

can introduce a significant error if that size distribution is expressed on a mass

or volume basis.

Other concerns during size analysis by microscopy-based techniques arise

due to systematic errors or artifacts associated with the instruments used for

observing the particles. The microscopes being used should be adjustable such
87

that the effects of aberrations, flare, stigmatism, can be eliminated using

standard practices (see next section, Relevant Standards) or manufacturerrecommended procedures. Most importantly, the magnification scale of the

microscopes should be calibrated. Magnification scales can be calibrated by

using size standards pertinent to that range, or magnification standards such

as SRM 484g that can be used to calibrate the magnification scale of an SEM

in the range of 1000x to 20000x. SRM 2090 (currently in preparation) will

be a new magnification standard made of a silicon chip with well-defined line

separations. SRM 2069b is a SEM performance standard of graphitized rayon

fibers. Optical microscopes can be calibrated using standards such as SRMs

2059, 475 and 476, all of which comprise of opaque lines and clear spaces

in a substrate. These standards have varying line widths and pitch values.

When calibrating the magnification power, the procedure used should be a

standardized technique that takes into account all possible sources of variation,

and calibration should be done not just for the magnification power that will be

used in the analysis, but for a range of magnification powers.

5. Relevant Standards

Standards covering various aspects of size analysis by microscopy-based

techniques have been developed or are under development. A draft standard

addressing general issues of size analysis by imaging techniques is ISO/CD

13322, Particle Size Analysis – Image Analysis Methods14. This standard

defines some terms and symbols that are commonly used and encountered

in image analysis. The standard also describes some techniques for sample

preparation of both dry powders and powders in suspension. Issues and

requirements for recording the image of the specimen and further image

analysis are discussed.

ASTM E20 Practice for Particle Size Analysis of Particulate Substances in

the Range of 0.5 μm to 75 μm by Optical Microscopy is another general

standard, which was discontinued in 1994. Other pertinent ASTM standards

include the ASTM E175-82(1995)15, a standard defining terminology for

microscopy-related applications. The definitions in this standard cover a

broader range of terms (including instrumentation, errors and units) than the

ISO/CD 13322 in its present draft form. ASTM E766-9816 is a standard

practice for calibrating the magnification of a SEM and is particularly useful

in ensuring that the instrument is truly observing specimens at the indicated

magnification values. As there are numerous factors (SEM accelerating

potential, condenser lens current, working distance, etc.) that can lead to

discrepancies between the stated magnification and the true value, the use

of this standard may help minimize some of these errors.

Other standards pertaining to size analysis by optical microscopy include

the French standard, NF X11-661: Test Method for Particle Size Analysis –

Practical Concerns ◆
88

◆ Microscopy-Based Techniques

Determination of Particle Size of Powders – Optical Microscope17. NF X11-

696 Test Methods for Particle Size Analysis Through Image Analysis18 can be

applied for general image analysis techniques, and the British Standard, BS

3406: Methods for Determination of Particle Size Distribution, Part 4: Guide

to Microscopy and Image Analysis Methods19 is a very comprehensive and

descriptive guide designed for size determination by microscopy-based

techniques. This guide describes some standard symbols and terms pertaining

to microscopy. Some of the terms and symbols described here are also found

with similar notations in the draft ISO/CD 13322. The standard has a section

on sample preparation describing some important factors to keep in mind while

preparing the samples, to ensure reproducibility and also minimize errors and

bias during analysis. It describes the resolution limits of the three microscope

systems based on the basic mathematical formulae for resolution. A section on

calibration of the microscope system describes means to set up the instrument

for analysis. It then describes means to conduct measurements using manual

methods with graticules and strip counts, and with image analyzers. Means to

calculate the size distribution and assess the accuracy and precision of the

calculated values are also described in detail. The guide also has annexes

describing the equipment used in these techniques, together with some

background information on the techniques, and highlights some issues to be

addressed when deciding on the type of image analysis system to be used.
89

Appendix

From the calculations of Masuda and Iinoya4,14, if P represents the probability

that the experimental data may be in the range of relative error -δ to +δ, then:

( | |) 2

1

u

P = Φ − −

where, Φ(-|u|) represents a normal probability distribution function of the form:

2 1 2 2 + ≡ασ σ

δ

c

n

u

' 2

1 ( ) 2

'2

z e dz

z z

∫

−∞

−

Φ ≡ π

and,

where, n is the size of a random sample,

α is the non-zero exponential constant that defines the powder distribution,

i.e., (y = K Dp

α ),

σ2 is the variance of the distribution, and

c is a constant of the form c = β+α/2, where β is 0 when computing the

population-mean particle diameter on count basis and 3 when computing on

mass basis.

Thus, defining n* as the number of particles that needed to be counted, the

following equations can be used to determine n*:

log n*= -2 log δ + log ω (4)

where, ω ≡ u2α2σ2 (2c2σ2 +1) (5)

and, δ is the desired relative error (e.g., if the desired relative error is 5 %,

δ is 0.05).

(1)

(2)

(3)

Appendix ◆
90

◆ Microscopy-Based Techniques

References

1. R. Davies, Particle Size Measurement: Experimental Techniques in

Handbook of Powder Science and Technology, M.E. Fayed and L. Otten,

eds., Van Nostrand Reinhold Company, New York (1984) p. 31.

2. The Particle Atlas, 2nd ed., Vol. IV, The Particle Analyst’s Handbook,

W.C. McCrone and J.G. Delly, eds., Ann Arbor Science Publishers Inc.,

Ann Arbor (1973) p. 804.

3. T. Allen, Particle Size Measurement, 4th ed. Chapman and Hall, London

(1993) p. 243.

4. H. Masuda and K. Iinoya, Theoretical study of the scatter of experimental

data due to particle size distribution, J. Chem. Engg. Japan, 4 (1), 60

(1971).

5. Certificate, Standard Reference Material® 1004b, Glass Beads–Particle

Size Distribution, Standard Reference Materials Program, National

Institute of Standards and Technology, Gaithersburg, MD.

6. NIH Image, developed by the National Institutes of Health, Bethesda, MD

(http://rsb.info.nih.gov/nih-image/).

7. T. Allen, Particle Size Measurement, 4th ed. Chapman and Hall, London,

(1993) p. 219.

8. Ibid., 224-227.

9. D.G. Drummond, (ed.) The Practice of Electron Microscopy, Royal

Microscopical Society, London (1950).

10. T. Allen, Particle Size Measurement, 4th ed. Chapman and Hall, London

(1993) p. 228.

11. I.F. Nathan, M.I. Barnett and T.D. Turner, Operator Error in Optical

Microscopy, Powder Tech., 5, (2), 105-110 (1972).

12. T. Allen, Particle Size Measurement, 4th ed. Chapman and Hall, London,

(1993) p. 228.

13. R. Davies, Particle Size Measurement: Experimental Techniques in

Handbook of Powder Science and Technology, M.E. Fayed and L. Otten,

eds., Van Nostrand Reinhold Company, New York (1984) p. 35.

14. ISO/CD 13322 Particle Size Analysis – Image Analysis Methods,

International Organization for Standardization, Geneva.
91

15. ASTM E175-82(1995) Standard Terminology of Microscopy, American

Society for Testing and Materials, West Conshohocken, PA (1995).

16. ASTM E766-98 Standard Practice for Calibrating the Magnification

of a Scanning Electron Microscope, American Society for Testing and

Materials, West Conshohocken, PA (1998).

17. NF X11-661 Test Method for Particle Size Analysis – Determination of

Particle Size of Powders – Optical Microscope, Normalization Francaise

(AFNOR), Paris.

18. NF X11-696 Test Methods for Particle Size Analysis Through Image

Analysis, Normalization Francaise (AFNOR), Paris.

19. BS 3406: Methods for Determination of Particle Size Distribution, Part 4:

Guide to Microscopy and Image Analysis Methods, British Standard

Institute, London (1993).

References ◆
93

6. SIZE CHARACTERIZATION BY LASER LIGHT

DIFFRACTION TECHNIQUES

1. Introduction

2. General Principles

2.1 Instrumentation

2.2 Mie scattering

2.3 Fraunhofer approximation

2.4 Rayleigh and Rayleigh–Gans scattering

2.5 Role of refractive index

3. General Procedures

3.1 Sampling

3.2 Dispersion and homogenization

3.3 Optical alignment and other instrument checks

3.4 Background signal collection and determination of obscuration levels

3.5 Sample analysis

3.6 Data output and post-processing

4. Sources of Error, Variations and Other Concerns

5. Relevant Standards

6. Appendix

1. Mie scattering theory

2. Fraunhofer approximation

3. Rayleigh scattering

4. Rayleigh–Gans scattering

5. Inversion procedures

1. Introduction

Size determination techniques using a light beam as a probe are broadly

classified into imaging and non-imaging types. Some aspects of the imagingbased techniques are discussed in Chapter 5 on Size Characterization by

Microscopy-based Techniques. Techniques constituting the non-imaging methods

of size determination can further be classified into numerous groups based

upon the physical principle describing the interaction between the light beam

and the particle and the information that can be deduced from that interaction.

Laser diffraction instruments fall within this category of non-imaging type of

instruments. Manufacturers produce instruments that are based upon the same

general principles of laser diffraction but the use of different components,

Laser Light Diffraction Techniques ◆
configurations, and algorithms gives rise to a broad selection of instruments

with analysis capabilities covering a wide range of sizes and size distributions.

The underlying assumption in the design of laser diffraction instruments is

that the scattered light pattern formed at the detector is a summation of

the scattering pattern produced by each particle that is being sampled.

Deconvolution of the resultant pattern can generate information about the

scattering pattern produced by each particle and, upon inversion, information

about the size of that particle. In order for this to be true, multiple scattering

of light should not be permitted. Multiple scattering refers to the phenomenon

where light scattered from one particle interacts with another particle and

gets scattered again. This places an inherent limitation on the concentration

of particles that can be analyzed. The concentration limit is determined by the

design of the instrument and varies depending on the nature of the incident

light source, optical geometry and light detection system. Laser diffraction

instruments can be used for the analysis of dry powders, powders dispersed

in aqueous or non-aqueous dispersants, and also finely divided aerosols. These

instruments are mostly used for analysis of small sample quantities (typically, a

few milligrams to a gram), though some are designed for, or have accessories

that enable sampling from a moving stream and are thus, suited for size

characterization and process control of an industrial production line.

Instruments are designed with capabilities for applying various optical models

for deconvolution of the scattered pattern and size determination. Those using

the Fraunhofer model for size analysis have typically a lower size limit of about

2 μm and an upper limit of about 8000 μm. However, it should be kept in mind

that the Fraunhofer model is not valid below a size range of about 2 μm to

4 μm. In such instances it is better to use instruments that have capabilities for

the Mie optical model. Instruments using this model typically enable analysis

to about 0.1 μm, and reliable analysis in even finer size ranges is possible

depending upon the configuration of the optical elements and the detector array

used in the instrument. The difference between these two models is discussed

later in this chapter and refers to the mathematical solutions used to solve the

scattering problems.

◆ Laser Light Diffraction Techniques

94

The interaction of a particle and light incident upon it gives rise to four different

but inherently related scattering phenomena, namely, diffraction, refraction,

reflection and absorption of the incident beam1. The magnitude of each

phenomenon will vary depending upon the nature and size of the particle and

the beam. Size analysis by interpretation of the scattered light patterns formed

due to diffraction of the incident light is of primary interest in this chapter.

Diffraction of light occurs at the surface of the particle and can be thought of

as the bending of light waves by the surface of the particle. Diffraction arises

due to slight differences in the path length of the light waves created upon

interaction with the particle surface. These differences in the path length cause
Laser Diffraction Instruments: Laser Diffraction Instruments:

General Information, Capabilities and Limitations

Size Range: Typically about 0.04 μm to 8000 μm (limits vary for different

instruments).

Specimen Type: Powders dispersed in suspension. Some instruments are

designed for dry powder specimens.

Operating mode: Usually off-line, but some instruments have accessories

to enable, or are designed for, on-line use.

Ease of operation: Relatively simple. Some understanding of basic

principles and algorithm design is required.

Sources of Error and Variations: Can arise during sampling, specimen

preparation, instrument set-up and operation, and user dependent.

Significant variations in calculation algorithms.

Strengths: Rapid analysis.

Relatively simple specimen preparation.

Some instruments can be used for both dry powders and powders in

suspension.

Some instruments can be used on-line and off-line.

Relatively inexpensive.

Do not require highly skilled operators.

Limitations: Instrument performance and operation is highly dependent on

instrument design (e.g., laser sources of different wavelengths, differing

number and positions of detectors).

Given the above reason, comparison of results from different

instruments may be misleading.

May require a precise knowledge of optical properties of specimen.

Cannot distinguish between dispersed particles and agglomerates.

Significant error and bias may be introduced in the particle size

distribution results, if powder particle shapes deviate from spherical

configurations.

Instrument design precludes analysis of concentrated suspensions.

Introduction ◆

95
96

◆ Laser Light Diffraction Techniques

constructive and destructive interference between the sinusoidal light waves

leading to characteristic diffraction patterns. The diffracted waves are then

scattered in different directions. The direction of scatter depends on the size

and shape of the particle. Large, spherical particles scatter mostly in the

forward direction. As the particle size gets smaller, the scattering occurs over

a broader range of angles. This description is a very simplistic perspective.

In practice, scattering is significantly more complex and is influenced by the

nature of polarization of the incident light, optical properties of the particle

and surface roughness of the particle. Numerous texts deal in detail with the

underlying mathematics and physics of scattering and diffraction2-4. Laser

light scattering instruments are often referred to as Fraunhofer diffraction

instruments, because instruments based on early designs processed the

diffracted light scattered in the forward direction by the particles using the

Fraunhofer approximation to deduce particle size from the scattered signal.

The development of instruments based on laser diffraction has been closely

related to developments in technology that have enabled the miniaturization of

laser light sources and photo-detector arrays and systems. Early particle size

analysis instruments based on this principle used simple configurations of

optical elements, such that the light scattered only in the forward direction, over

a relatively narrow angle from the optic axis, was focussed on the detector

system. Detector systems used in these instruments comprised of individual

detector elements rather than multi-element arrays and thus, needed to be

placed at discrete distances from each other. This also limited the number of

detectors that could be placed to receive the scattered signal and so early

instruments would process signals from a very narrow segment of the

scattering cross-section and were thus limited for size analysis over a range of

about 300 μm to 2 μm5. Size determination of particles finer and coarser than

these limits has been possible due to developments of both suitable hardware

and signal processing algorithms. Multi-element detectors and arrays have

enabled positioning of more detector elements closer to each other and over a

wider scattering angle, contributing to the improved sensitivity and resolution in

more recent instruments. The development of better optical configurations and

techniques makes possible the analysis of signals scattered over a wider range

of angles off the optic axis, and thus the analysis of finer particles. Processing

of the signals scattered from the fine particles by algorithms based on the

Mie theory provide a more realistic estimation of the size of the fine particles.

Instrument resolution and sensitivity for size calculation of the fine particles

have been improved by other techniques including the use of 90o scattering

configurations (additional detector elements at a 90o angle to the incident

beam), and observing the changes in polarization of light after scattering.

Instruments for particle size determination based on closely related principles

of light scattering are also available and may be confused with light diffraction
97

instruments. These instruments include those based on single particle light

interaction methods6, where the interaction of a single particle with the light

beam causes a reduction in the intensity of the light beam due to absorption

and scattering of light by the particle. The reduction in transmitted light

intensity (extinction) is translated into an electrical signal the strength of

which correlates to the particle size. By the very nature of this technique,

these instruments can be used only on extremely dilute suspensions. Other

instruments are based upon the principle of quasi-elastic light scattering

(QELS)7, a generic term encompassing a wide range of specific methods

including photon correlation spectroscopy, dynamic light scattering, and

heterodyne spectroscopy. These instruments are designed to make use of the

Brownian motion of particles in suspension. Consequently, the measurement

of variations in the intensity and frequency of the scattered light (incident light

is monochromatic in nature) can be correlated to the Brownian motion and

other dynamics of the particles that cause the scattering8. Because of the

dependence on Brownian motion and relaxation time associated with the

random motion, these techniques are ideally suited for particles smaller than

1 μm and require very dilute suspensions. Thus, only small quantities of

sample powder (order of a few milligrams) are needed.

2. General Principles

Light diffraction instruments are based on three basic assumptions:

(a) The particles scattering the light are spherical in nature,

(b) There is little to no interaction between the light scattered from different

particles (i.e., no multiple scattering phenomena), and

(c) The scattering pattern at the detectors is the sum of the individual

scattering patterns generated by each particle interacting with the incident

beam in the sample volume.

Deviations from these assumptions will introduce some degree of error due to

the inability of the mathematical algorithms for the deconvolution and inversion

procedures to account for the deviations. The assumption of spherical particle

shape is particularly important as most algorithms in commercial instruments

use the mathematical solution for Mie, Fraunhofer and Rayleigh scattering

from spherical particles. At present even though mathematical solutions for

cylindrical systems are available, these solutions have not been incorporated

into commercially available programs. However, those interested can attempt

to develop such algorithms based on published mathematical solutions3.

Similarly, the assumption of the lack of multiple scattering is critical due to the

nature of the mathematical models which are used to deconvolute the resultant

diffraction patterns. Most algorithms and mathematical models do not take into

Introduction ◆
98

◆ Laser Light Diffraction Techniques

consideration the effects of multiple scattering on the resultant diffraction

patterns and thus are unable to account for such interactions. The need

to prevent multiple scattering also places an inherent limitation on the

concentration of particles present in the sample volume.

2.1 Instrumentation

Light diffraction instruments comprise of a light source, typically a low power

(approximately 10 mW Helium-Neon, in the region of 632 nm wavelength)

laser source, optical elements to process the incident beam, a sample cell

within which the sample is introduced. Some sample cells have built-in

ultrasonicators or agitators to keep the specimen powders dispersed and to

prevent agglomeration. Sample cells also possess pumping systems to keep the

specimen circulating. Light diffraction instruments lack the ability to distinguish

between well-dispersed powders and agglomerates, and thus, prevention of

agglomeration is a key factor in ensuring reliability and reproducibility. Light

scattered from the sample is then focussed on to a detection system, that can

be a multi-element array or numerous detectors placed at discrete locations.

The detectors convert the scattered light intensity incident upon them into

electrical signals that are then processed to obtain information about the

particle size and size distribution. Figure 6.1 is a schematic diagram of the

different components. Conversion of a scattering pattern into size distribution

information requires the use of optical models and inversion procedures

(detailed discussion in the Appendix at the end of this chapter).

Figure 6.1. Figure 6.1.

Schematic Diagram of Components in a Typical Laser Diffraction Instrument

High Angle Detector

Elements

Low Angle

Detector

Elements

OUTPUT

DEVICE Computer System for Signal Processing and 

Analysis

Optics

Sample

Cell

Laser Source
99

The need for optical models arose in order to explain the scattering of light

from spherical particles. Gustav Mie published a theoretical analysis for

scattering of light from spherical particles in 19089. Lorenz is credited with

having derived a solution for the same problem independently in 1890. Solutions

to scattering of light by spherical particles of varying sizes can be thought to

fit into the following categories:

(i) When the particle diameter (d) >> wavelength of incident light (l), the

Fraunhofer model may be used, and this represents one limiting case of

the Mie theory.

(ii) When d << λ, the solutions for scattering are best represented by Rayleigh

scattering models.

(iii) When d < λ, the Rayleigh–Gans optical model best represents the solutions

to the scattering of light by the particles.

While the Fraunhofer model is independent of the material properties, the

Rayleigh and Rayleigh–Gans models, which can be derived from the Mie

scattering theory, are very much dependent on the optical properties, and thus

require the use of both the real and imaginary components of the material’s

refractive index. The value of the real component of the refractive index can

be found with relative ease, but determining the value of the imaginary

component can be far more difficult.

The above distinctions represent solutions to general scattering problems of

light by spheres under varying boundary conditions. Numerous texts3, 4, 10

detail the mathematical operations involved to obtain these solutions. The

following descriptions together with the details presented in the Appendix at

the end of this chapter attempt to collect this information and present it in a

manner suitable for a brief description of the basic principles and includes

references from various texts3, 4, 10-13.

2.2 Mie Scattering

The Mie scattering theory is a comprehensive mathematical solution to

scattering of incident light by spherical particles. The theory can be extended

to consider scattering from particles with different shapes and aspect ratios.

This theory indicates the necessity for a precise knowledge of the real and

imaginary components of the refractive index of the material being analyzed,

to determine particle size and size distribution. Most modern instruments

based on laser light diffraction for particle size determination utilize this theory

(with some proprietary variation) for modeling the diffraction patterns that

General Principles ◆
100

◆ Laser Light Diffraction Techniques

are formed. The Mie theory is applicable when the particle size is equal to,

or smaller than, the wavelength of the incident light. (See Appendix at the

end of the chapter for more information and theoretical treatment of the

Mie scattering theory).

2.3 Fraunhofer Approximation

The Fraunhofer approximation (also referred to as the Fraunhofer theory)

is applicable when the diameter of the particle scattering the incident light is

larger than the wavelength of the radiation. The Fraunhofer theory can be

derived from the Mie scattering theory, or can be derived independently

considering simple diffraction of light from two points giving rise to a phase

difference between the two diffracted beams. The resultant diffraction pattern

is marked by a series of maxima and minima. The spatial separation between

these can be used to calculate the size of the particle. In the case of diffraction

from numerous particles, the resultant diffraction pattern is the summation of

the intensities of each pattern corresponding to the particle giving rise to that

pattern. By its very nature, this model does not need any information about

the refractive index of the particle and so is extremely useful for analysis

of powders coarser than about 1μm to 2 μm. (See Appendix at the end of

the chapter for derivation of the Fraunhofer approximation, from the

Mie theory.)

2.4 Rayleigh and Rayleigh–Gans Scattering

These two scattering models are based on the Rayleigh scattering model that is

applicable when the size of the particle is much smaller than the wavelength of

the incident light. The Rayleigh–Gans theory, however, is applicable when the

particle diameter is not significantly smaller than the wavelength of the incident

light. Like the Mie theory, both these models require the use of both the real

and imaginary component of the material’s refractive index. (See appendix at

the end of chapter for a brief discussion on the derivation of the Rayleigh

and Rayleigh–Gans theory.)

2.5 Role of Refractive Index

An important point to be made here is the need for a precise knowledge of

the optical constants of the material in certain instances. As is observed from

equations 10-13, 17 and 19 presented in the Appendix, the solutions by the Mie,

Rayleigh and Rayleigh–Gans theories all require the definition of the complex

refractive index (m = n -in ́, where n is the real component of the refractive
101

index and n ́ the imaginary component of the same). However, equation 15,

which represents the Fraunhofer solution, does not include the term m, and

indicates no dependence and, hence, there is no need for a knowledge of

the optical property. It is for this reason that optical models based on the

Fraunhofer solution do not require the user to specify the refractive index of

the material being studied, but those based on the Mie models do require the

user to specify the values of both the real and imaginary components. Thus,

the Fraunhofer model is preferred when analyzing powder systems containing

mixtures of different materials, or when analyzing particles with inherent

heterogeneity in density distribution. However, applicability of this model

with the appropriate size criterion also has to be met.

Though there are various sources for obtaining the values of the refractive

indices of materials14-17, caution should be exercised while using them. The

wavelength at which these values have been determined and reported may

not be the same wavelength at which the instrument is operating and may thus

introduce systematic errors. For materials that occur in different stable phases,

the value used should correspond to that particular phase. For instance, it has

been reported that the real component of the refractive index of α-Al2O3,

occurring as natural corundum is 1.77, while that γ-Al2O3 is 1.70. In most

instances, it is relatively easy to find the value of the real component of the

refractive index. It is however, far more difficult to obtain the value of the

imaginary component of the refractive index of the material. Some tables

exist for determination of these values, but often trial and error procedures

of size determination using a microscopy-based technique and an instrument

using laser diffraction have to be utilized. Such a procedure could introduce

significant errors in the value determined and may be biased by the algorithm

incorporated in the laser diffraction instrument. Furthermore, the imaginary

component is affected by factors such as the surface roughness of the particle,

density heterogenities in the particle. To enable repeatability and reproducibility

of experiments, it is a good practice to report the wavelength of the incident

radiation used when reporting refractive index values or size values based

upon the use of the refractive index information of a material.

Figure 6.2 shows the influence of changing the value of the imaginary

component of the refractive index (based on Mie models) of a powder system

comprised of nominally 1 μm spherical SiO2 particles, at a fixed value of the

real component of refractive index of 1.44 and 1.43 (data generated during

studies at NIST). The plots represent the change in the value of the d10, d50

and d90 size fractions, and thus represent a change in the measured size

distribution, as measured by a laser diffraction instrument.

General Principles ◆
102

◆ Laser Light Diffraction Techniques

Figure 6.2. Figure 6.2.

Influence of Imaginary Component of Refractive Index on Size Distribution

Measured by a Laser Light Diffraction Instrument (Using Mie Models)

The trends in this plot indicate the importance of using the precise value of the

imaginary component of the refractive index. Considering the use of a real

component of 1.44, upon changing the imaginary component from 0.01 to 1.0,

the width of the distribution between d90 and d10 changes from about 0.88 μm

to approximately 0.4 μm, almost a 50 % decrease. The median value of the

powder distribution (d50) increases from about 0.84 μm to almost 1.0 μm. Put

together, this shows that the calculated width of the size distribution is much

narrower upon using an imaginary component of 1.0, than when using a value

of 0.01. Based upon analysis by SEM and optical microscopy, we know that

the size distribution of this powder system is very narrow and close to that

calculated at an imaginary component of 1.

Huffman has addressed the application of optical constants calculated for bulk

materials to small particles. The use of these values is an important issue as

the optical properties of the small particles may be significantly different from

those of the bulk materials, even without entering the regime where breakdown

of optical constants occur. These variations are most pronounced in the

Imaginary component of RI

0.01 0.10 1.00

Measured size (μm)

0.2

0.6

1.0

1.4

d90

d50

d 1.43 10

1.44

1.43

1.44

1.43

1.44

Imaginary Component of RI

Measured Size (μm)
103

ultraviolet region for metals, and the infrared region for insulators. These

deviations are observed to occur in the proximity of frequencies close to the

surface mode resonance. The conditions under which optical constants for

the bulk can be used for small particles have been identified by Huffman as:

(a) the constants are measured with a high degree of accuracy, (b) the

particles of interest are of the same homogeneous state as the bulk material,

i.e., both particle and bulk are crystalline, or both are amorphous, and (c) the

particles are well dispersed such that there is no particle-particle contact

or agglomeration. The same holds true also when the particle size is large

enough to negate any quantum size effects and the damping of free electron

oscillations is not determined by the particle size.

3. General Procedures

Procedures to be followed for particle size determination by laser light

diffraction instruments are to a great extent specific to the instrument being

used and are usually very well defined by the manufacturer. The procedures

are designed to work best with the algorithms used in the instrument and are

optimized for the instrument set-up. As a result, these procedures vary from

one instrument to another. This also requires a fair amount of work by the user

to find the set of parameters (e.g., real and imaginary components of refractive

index, solids loading, signal collection time) that appears to work best for that

instrument. However, irrespective of the instrument being used there are

certain general procedures that can be adopted in order to minimize various

systematic errors. Some of these general procedures are discussed in this

section. Figure 6.3 shows the general steps involved in the determination of the

particle size and size distribution of a powder system by laser light scattering.

3.1 Sampling

Powder sampling is conducted in a manner which ensures that the specimen

being examined is representative of the entire batch. Various techniques for

sampling are discussed in Chapter 2. While most laser diffraction instruments

are set up for batch samples, there are numerous instruments that are set-up

for on-line sampling or have accessories that can sample from an on-line

stream. An example of an on-line sampling system would be the use of laser

diffraction instruments in a cement production plant where tight control of the

powder size during grinding operations is desired. In such instances, wherever

practical, the guideline of sampling the entire stream for some time, rather

than the some of the stream all the time, should be practiced. In case of

batch samples, it should be ensured that the specimen is representative of

the powder system under study.

General Principles ◆
104

◆ Laser Light Diffraction Techniques

Powder sampling 

(section 3.1)

Dispersion and 

homogenization 

Specimen stabilization 

(section 3.2)

Optical path alignment and other instrument checks 

(section 3.3)

Background signal collection 

(section 3.4)

Determination of obscuration/solids loading 

(section 3.4)

Sample analysis (signal collection) 

(section 3.5)

Sample analysis 

(signal deconvolution, inversion and data computation)

(section 3.5)

Size and size distribution data output 

(section 3.6)

Post-processing of data (if any) 

(section 3.6)

Figure 6.3. Figure 6.3.

Flowchart Indicating Steps Involved in Particle Size Determination

by Laser Light Diffraction Technique
105

3.2 Dispersion and Homogenization

The state of the powder being sampled also plays a role in deciding the

sampling procedure to be used. Flow of dry powders may be significantly

different from that of powders dispersed in a liquid. Powders in both cases

may segregate in the case of a broad size distribution causing some size

fractions to be present in a non-representative manner. Agglomeration of

these powders may be observed depending upon the surface characteristics

of these powders and the stability of the dispersions. Sampling procedures

that are followed should take into account these factors. Development of

suitable dispersion procedures during or immediately prior to the analysis

helps ensure that powders are well dispersed and agglomeration is minimized.

Most instruments are designed for analyzing samples dispersed in fluids such

as water or isopropanol or other low hazard fluids. Instruments have special

cells designed for flammable liquids or explosive vapors. Accessories may

be available from the instrument manufacturer for analysis of powders in

a dry state, obviating the need for dispersion in a fluid. While using such an

accessory it is still essential to ensure that the powders are not agglomerated

and particle-particle bridging is minimized. Procedures recommended by the

manufacturer should be followed in such instances. Additional procedures to

prevent agglomeration or segregation may be designed and implemented by

the user.

Ensuring a good degree of dispersion prior to sample analysis is an important

step to ensure reliable and reproducible size analysis. Due to the inability of

the instrument to distinguish between agglomerates and primary particles,

this process can help reduce some bias and other systematic errors in the

calculated powder size distribution. Sample dispersion can be achieved by

numerous methods including ultrasonication, milling, pH stabilization, addition

of dispersion agents, etc.

Regardless of the technique followed, it is critical to ensure that the process

of dispersion does not lead to creation of stable air bubbles, fracture of

brittle particles or agglomeration. The occurrence of any of these will lead

to skewed particle size distribution results. The use of ultrasonic probes to

disperse powders is well suited for this technique due to the small quantities

of sample powders needed for analysis. While ultrasonication does lead to

dispersion, the specimens may still need to be stabilized in such a state by

either controlling the pH of the suspension or adding dispersion agents. After

stabilization, it is preferable to keep the suspension agitated to prevent any

settling or agglomeration. Instrument manufacturers are making available

ultrasonic kits that can be attached to the specimen chamber to ultrasonicate

the specimens prior to analysis.

Important factors that are to be considered for determining ultrasonication

parameters are the ultrasonication duration, energy, diameter of the probe/horn

General Procedures ◆
106

◆ Laser Light Diffraction Techniques

and coupling between horn and sample. Time and power output to be used

for ultrasonication are dictated by the nature of the material being dispersed.

Ultrasonication duration can extend from a few seconds to a few minutes.

Ultrasonication should be carried out until no further reduction in particle size

distribution is observed. In some instances, this might call for ultrasonication

durations of a few minutes. For systems that tend to generate heat upon

ultrasonication for such lengths of time, the energy can be applied in fractional

duty cycles (e.g., 50 % on and 50 % off during the duty cycle), or the beaker

containing the sample being ultrasonicated can be placed in an ice bath.

Close attention should be paid to the nature of the coupling observed between

the ultrasonic probe and the sample. The nature of the coupling defines the

efficiency of power transfer from the probe to the material system. The depth

of the probe into the sample influences the coupling. It has been observed that

glass beakers provide better reflection of ultrasonic energy and thus lead to

better coupling than polystyrene beakers. Determination of ultrasonication

parameters is mostly by a trial-and-error process, designed to ensure

minimization of the particle size distribution, without causing fracture of

the primary particles.

Figures 6.4 and 6.5 (studies at NIST) indicate the need to create and maintain

stable dispersions for particle size analysis by laser diffraction. Figure 6.4

shows the influence of the duration of ultrasonication on a suspension of

nominally 1 μm spherical SiO2 powder system. The specimens were

ultrasonicated for 15 s, 1 min, 3 min and 6 min and then stabilized at a pH

value of 8. The graph represents the observed size distribution (d10, d50 or

median size and d90) and a measure of the mean size as a function of

ultrasonication duration. A significant difference is observed upon increasing

the ultrasonication duration from 15 s to 1 min. Ultrasonication for 3 min

causes a small reduction in the mean particle size and size distribution.

Ultrasonication for 6 min does not produce any significant improvement in

the results. However, ultrasonication for 6 min causes significant heating

of the suspension. Thus, excessive ultrasonication may cause changes in the

physical and chemical state of the powder system being analyzed. From this

information, ultrasonication duration of 3 min was chosen to be the desired

value for this material system.

Figure 6.5 shows the change in the measured size of a nominally 1 μm

spherical SiO2 powder system that has been ultrasonicated for 3 min. at 30 W

and adjusted to different pH levels prior to size analysis by laser diffraction.

As this powder system has an i.e.p near 2.5, it is expected that the particles

will agglomerate at pH values close to the i.e.p and be well dispersed at pH

values further away from the i.e.p. Thus, at low pH values of the dispersion,

the measured mean and median sizes are significantly larger than the expected

values. At higher pH values this size is closer to the expected value.
107

Ultrasonication time (minutes)

01234567

Measured Size (μm)

0

1

2

3

4

5

6

7

8

9

10

Mean Dia.

Median Dia. (d50)

d10

d90

Figure 6.4. Figure 6.4.

 Influence of Ultrasonication Duration Upon Particle Size Distribution,

Measured by a Laser Diffraction Instrument

Furthermore, the associated uncertainty in the measured value is significantly

higher at lower pH values. Each point represents the average value of 3 sets

of measurements.

After dispersing the powders, it is essential to ensure that the dispersions are

stabilized in such a state. Stabilization is usually achieved by addition of suitable

additives to the dispersion. Controlling pH by addition of a suitable acid or

base to keep the pH of the dispersion well away from the i.e.p of the system

constitutes a technique of electrostatic stabilization. Addition of suitable

deflocculants to physically prevent the coagulation of particles is known as

steric stabilization. Some polyelectrolytes often used as deflocculants include

sodium polyacrylate, ammonium polyacrylatae, sodium silicate, sodium

carbonate, tetrasodium pyrophosphate, sodium polysulfonate and ammonium

citrate. For milling techniques of size reduction and dispersion, it is desirable

to use polyelectrolytes with short polymeric chains. This prevents damage to

the polymeric chains during the milling operations, which would reduce the

efficacy of the deflocculant.

3.3 Optical Alignment and Other Instrument Checks

Other general procedures to be followed prior to analysis include ensuring

optical alignment and establishing a background or threshold signal level.

It is absolutely critical to perform these procedures as any deviations will

lead to significant errors in the calculated results. Optical alignment includes

General Procedures ◆

Ultrasonication Time (minutes)

Measured Size (μm)
108

◆ Laser Light Diffraction Techniques

corrections in vertical, horizontal or angular positions of the laser source, the

optical elements including lenses and the sample cell and the detector arrays

so that the incident beam is aligned with the optic axis of the instrument. Loss

of alignment occurs primarily due to ambient vibrations transmitted through the

structures such as laboratory benches, tables, etc., and those introduced during

the sample pumping process. Alignment procedures are instrument-specific and

clearly defined by the instrument manufacturer. In some cases, the alignment

procedure may require manual intervention by the operator, or may be

conducted electronically with minimal operator intervention.

3.4 Background Signal Collection and Determination of Background Signal Collection and Determination of

Obscuration Levels

A background signal or threshold should be obtained immediately after optical

alignment. By obtaining a background signal the current levels at each detector

element in the absence of diffracted light is established. When analyzing a

specimen, the current signals at the detector due to diffraction are compared

with those determined during the background scan. Furthermore, the intensity

of the laser beam in the absence of a specimen is determined during the

background scan. By comparing the signal intensity of the laser beam without

any sample present to the signal intensity with a specimen loaded in the sample

cell, the obscuration level (extent of attenuation of incident beam intensity due

to presence of particles) can be calculated.

Obscuration values are required to calculate the sample concentration present

in the specimen. Most instruments are designed to operate within a particular

obscuration value range. Means to determine whether the sample loading is

within this range are specified by the instrument manufacturer. This includes

either a number range or a bar graph specific to that instrument. Higher

particle concentrations that may cause high levels of obscuration can lead

to multiple scattering, while low solids loading corresponding to low levels of

obscuration may cause inadequate signal strength at the detector elements.

The impact of changes in the obscuration level on the determined particle

size distribution is a function of the particle system being studied, the optical

model used, and the algorithm used for deconvolution and inversion.

Figure 6.6 (studies at NIST) indicates the effect of varying obscuration

levels on the calculated particle size distribution of a nominally 1 μm SiO2

powder system. In the case of the instrument used in this study, the

manufacturer recommended range indicates obscuration levels of 7 % to be

low and 13 % to be high. An obscuration level of 10 % is in the middle of the

range specified for the instrument. As is observed there are small differences

in the calculated size distribution at these varying obscuration levels. Though

the observed differences are small for the quoted example, these difference

can be significant for other material systems.
109

pH value

123456789

Measured Dia. (μm) 0

1

2

3

4

5

6

7

8

Mean Dia.

Median Dia. (d50)

Figure 6.5. Figure 6.5.

Graph Representing Variation in Measured Size of Nominally 1μm SiO2

Powder System After Ultrasonication and Stabilization at Varying pH Levels

Figure 6.6. Figure 6.6.

Effect of Varying Obscuration Levels on Calculated Particle Size

Distribution of Nominally 1.0 μm SiO2 Powder System

Laser obscuration (%)

6 7 8 9 10 11 12 13 14

Measured Dia. (μm)

0.7

0.8

0.9

1.0

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

Median Dia. (d50)

d10

d90

Mean Dia.

Median Dia. (d50)

d10

d90

Mean Dia.

General Procedures ◆

pH Value

Laser Obscuration (%)

Measured Dia. (μm) Measured Dia. (μm)
110

◆ Laser Light Diffraction Techniques

3.5 Sample Analysis

In numerous cases, the results of the particle size analysis are only as good

as the optical model chosen to interpret and convert the diffraction pattern into

particle size distributions. Instruments are set up to enable the user to select

from a predetermined set of optical models. These optical models are usually

a combination of values representing the real and imaginary components of the

refractive index of the material being analyzed. In some instruments, the user

may be asked to make some selections about the expected shape and width

of the size distributions. Some instruments allow the operator to create optical

models based on their choice of real and imaginary components of the

refractive indices. The importance of knowing the correct values and the

associated difficulties have been discussed previously. Instruments that are

designed for size analysis of powders coarser than about 2 μm may not have

these requirements, as the algorithms in these instruments are based on a

Fraunhofer diffraction model and thus totally independent of material

properties. Figure 6.7 (studies at NIST) shows how changes in optical models

can create significant changes in the calculated particle size distribution. The

graphs show the powder size distribution of a hydroxyapatite powder system

represented as a differential volume distribution. The five curves represent

size distribution results calculated using different optical models including the

Fraunhofer model, labeled as such. The remaining four graphs are based on

Mie models, and in this case the real component of the refractive index of

the powders was reliably determined to be 1.63. However, there was some

uncertainty about the correct value for the imaginary component. As is

observed, significant differences are observed in both the shape and magnitude

of size distribution depending upon the refractive index value chosen. For

imaginary values of 0 and 0.01, the lower range of the distribution is calculated

to be in the vicinity of 1 μm. For the Fraunhofer model and the Mie models

with imaginary components of 0.1 and 1.0, the lower range is found to be

smaller than 0.1 μm. The choice of the “correct model” in this case has to

be made with the aid of other techniques, such as microscopic determination,

that would enable a direct examination of the powder system.

3.6 Data Output and Post-Processing

Particle size distribution results are typically expressed as a function of the

equivalent spherical diameter. This is based on the assumption of spherical

particles giving rise to the diffraction process. Size distributions are typically

expressed on a volume basis. This basis though not always accurate does

simplify matters to a great extent. A very precise representation would require

the distribution to be expressed on the scattering cross-sectional area basis.

If the particles are spherical or are assumed to be so, then the volume
111

Figure 6.7. Figure 6.7.

Influence of Optical Model on Calculated Particle Size and Size Distribution

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

0.1 1 10 100

Equivalent spherical diameter(um)

Differential Vol. (%)

Fraunhofer

1.63/1.0

1.63/0.1

1.63/0.01

1.63/0.0

representation can be easily converted to a cross-sectional area, and vice

versa. Size distributions can also be expressed in terms of a number or

surface area basis, but it should be remembered that such expressions are

derived from the volume basis. Thus, any deviations from the spherical nature

of the powders will introduce significant error and/or bias in the particle

size distribution representation. For these very reasons, when reporting size

distributions in a quantitative manner (e.g., using the mean value and the d10,

d50 and d90 values), these numbers should be calculated from the volume basis

distribution graph.

It is a good practice to represent the results of the particle size distribution

as both a cumulative finer than and fraction distribution graph as a function

of the observed particle size. This enables the user to observe the nature

of the distribution (unimodal, bimodal, etc.) the fraction of the distribution

in a particular size range of interest, and be able to determine the size

corresponding to a particular fraction (d10, d50 or d90, etc.). Availability of this

information enables conversion from one format or distribution (e.g., volume

based distribution to mass based or number based distribution) to another with

relative ease.

General Procedures ◆

Equivalent Spherical Diameter (μm)

Differential Vol. (%)
112

◆ Laser Light Diffraction Techniques

4. Sources of Error, Variations and Other Concerns

Errors and variations in the observed results can be introduced in various

stages of analysis, right from sampling to data interpretation. Some of these

have been discussed in previous sections. Most instrument manufacturers list

sources of error particular to the instrument and suggest means and techniques

to avoid them. The following discussion lists briefly in a general manner the

various stages at which errors and other variations can be introduced in the

analysis process.

1. Sampling and specimen preparation related

a. Errors introduced due to use of non-representative samples, i.e.,

errors due to incorrect sampling procedures. This holds true for both

dry powders and powders dispersed in suspensions. Proper sampling

procedures can ensure that these errors can be minimized, if not

eliminated completely.

b. Analysis of powders finer or coarser than the detection limits of the

instrument being used. Even when analyzing powders with dimensions

close to the upper and lower detection limits, it is a good practice to

verify the validity of these limits using suitable primary or secondary

standards.

c. Errors introduced upon analysis of non-spherical powders. Due

to the assumptions of particle sphericity, any deviations from this shape

will cause bias and errors to be introduced in the particle size and size

distribution results. Unless analyzed by appropriate algorithms designed

as part of the instrument software, or designed for use on the obtained

scattering patterns, it may be expected that the magnitude or error will

be magnified as the deviation from spherical shapes is increased.

d. Errors associated with optical properties of the material. In most

instances it is extremely helpful to know the optical properties and some

physical properties such as density of the material being tested. Most

instruments require the user to provide this information for calculation

of the particle size and size distribution and any errors in these values

are reflected in the calculated results.

e. Reliable procedures for creating and maintaining stable

dispersions of the powders should be developed. The particles should

remain dispersed even in the sample cell, as these instruments lack

the ability to distinguish between primary particles and agglomerates.

These procedures should not cause fragmentation of friable particles

or lead to formation of stable bubbles that can interfere with the
113

measurement and bias the calculated result. Closely related to this is the

issue of sample stability, where certain powders may change size over

a period of time due to dissolution or precipitation mechanisms. In such

cases, samples should be prepared immediately prior to analysis and

discarded after analysis.

2. Instrument and procedure related

a. Errors introduced due to non-aligned or misaligned optics. The need

for ensuring proper alignment has been discussed in detail in the previous

sections. Instrument manufacturers specify procedures and checks to

ensure proper alignment.

b. Errors due to lack of background signals or errors in procedures

for obtaining background signals. In this instance, the detector

elements use an incorrect background signal to ratio the diffracted signal.

Thus, if the dispersion medium has been changed then an incorrect signal

will be used leading to errors in the calculated size and size distribution.

c. Light leakage due to stray or extraneous light in the instrument will

cause additional signals at the detector elements that will be analyzed

as diffracted signals from the particles and be included in the size

distribution results.

d. The use of incorrect optical models will have a significant effect on the

calculated size distributions. These effects may be manifest not only in

the range of the size distribution, but also in the shape of the distribution.

The importance of using the correct optical model has been discussed in

detail in the preceding sections. Determining whether the optical model

used is appropriate can be quite complex. Some manufacturers are

working on software routines that will enable determination of a

theoretical scattering pattern from the calculated size distribution and

based on the chosen optical model. A comparison of the calculated

scattering pattern with the observed scattering pattern can provide a

check of the model used. In numerous instances, microscopy-based

techniques may have to be used to compare the calculated size

distributions. In these cases however, appropriate precautions need

to be observed (see Chapter 5).

e. Software related bias may cause significant errors in the calculated

results. Most errors would arise due to the design of the deconvolution

and inversion algorithms that may be based on assumptions not reflected

or applicable to the particle system under study. An example of such an

error would be the use of a model-dependent inversion procedure (i.e.,

inversion procedure that assumes a particular shape for the powder

distribution) on a multi-modal powder system.

Sources of Error, Variations and Other Concerns ◆
114

◆ Laser Light Diffraction Techniques

f. Errors due to non-linear detector responses arise when sample

loading is either too high or too low. This issue has been discussed in

terms of the obscuration of the incident beam and the need to ensure

obscuration in the range specified by the manufacturer. Put differently,

obscuration levels outside the range specified by the manufacturer

correspond to the non-linear response range of the detector elements,

and the current signals generated by the diffracted beam are not

proportional to the diffracted signal incident on the detector.

Errors and precautions pertaining to representation and interpretation of size

data are discussed in greater detail in Chapter 7 on Reporting Size Data.

5. Relevant Standards

ISO 13320-1 Particle Size Analysis – Laser Diffraction Methods –

Part 1: General Principles14, is an international standard covering general

principles and procedures pertaining to particle size analysis using laser

diffraction techniques. This standard is designed to be in two parts, with

Part 2 pertaining to the validation of inversion procedures used for processing

the diffracted light signals into particle size and size distribution data. Part 1

defines some terms and symbols used in conjunction with this technique of size

analysis. Elements constituting the instrument and their interaction with other

components in the system are described in general terms. Descriptions about

operational procedures for size analyses include selection criteria for specimen

dispersion fluids, some techniques for sample preparation and dispersion and

precautions to be observed while following these procedures. Other discussions

include procedures and precautions to be observed for the actual measurement,

ensuring reproducibility and accuracy, sources of random and systematic errors

and means to identify the occurrence of the same, and factors that could affect

the resolution of the instrument. It is recommended that reporting of results

be along the guidelines set forth in ISO 9276-122 (described in the chapter on

Reporting Size Data). Attached annexes have concise and useful descriptions

of the underlying theory of laser diffraction and the need for different

diffraction models, and a useful listing of the refractive indices of different

liquids and solids. This listing is a compilation of values from various sources in

the literature. The imaginary component of the refractive index of only certain

materials is reported in this compilation. However, for most applications, the

user will need to determine the appropriate value of the imaginary component

of the refractive index. The determination of such a value is not a trivial task

and is compounded by the fact the surface roughness of the particles and any

heterogeneity in density distribution of the particle will influence the value of

the imaginary component.
115

ASTM B822-97 Standard Test Method for Particle Size Distribution of Metal

Powders and Related Compounds by Light Scattering23 is an ASTM standard

test method for size determination of particulate metals and compounds by

laser diffraction. The test method is applicable over a size range of 0.1 μm to

1000 μm for both aqueous and non-aqueous dispersions. An important issue

stressed in this standard is that even though different instruments are based

upon the same basic principle, significant differences are present in the

arrangement of the optical components (including the collection lenses and the

number and position of detector elements), and the algorithms used to interpret

the scattering pattern and convert it to particle size data. These differences

may lead to different results from different instruments, and thus, comparison

of results from different instruments should be done with caution. Sources of

interference that can lead to random and/or systematic errors are identified

and information about sampling, calibration and general procedures for

measurement are defined.

ASTM E1458-92 Standard Test Method for Calibration Verification of Laser

Diffraction Particle Sizing Instruments Using Photomask Reticles24 is a very

useful standard outlining the procedures to be followed to ensure that the laser

diffraction instrument is operating within the manufacturer specified tolerance

limits. This procedure is based upon the use of a photomask made up of a

two-dimensional array of thin, opaque circular discs on a transparent substrate,

simulating the cross-section of spherical particles of varying diameters that

would diffract the incident light beam. A photomask is used as it is difficult

to generate, in a reproducible manner, an aerosol containing a controlled

population and distribution of droplets or particles that can be used for

calibrating the instrument. This test method can be used to compile calibration

data that can be used to check the instrument operation. This standard

describes the photomask reticle to be used and the procedures to be followed

for the calibration process. The precision and bias that can be expected from

this technique of calibration are also described. The occurrence of artifacts

and other sources of error that can influence the calibration procedure are

explained and discussed.

NIST has developed various size standards, available as SRMs, for calibration

and performance evaluation of laser diffraction instruments. SRM 1690, 1691,

1692, 1960, 1961 and 1963 comprise of polystyrene latex spheres dispersed in

water and can be selected based on the desired size distribution. In addition to

these, SRM 1003b, 1004a, 1017b. 1018b and 1019b comprise of glass beads of

varying particle size distributions ranges and sizes. SRM 1982 comprising of

zirconia was developed using various laser diffraction instruments in an interlaboratory round-robin study. Currently various standards are being developed

for certification as SRMs. The development of these has been based on

measurements using laser diffraction instruments and the standards are made

Relevant Standards ◆
116

◆ Laser Light Diffraction Techniques

of WC-Co powders and a zeolite-based system. In addition to the NIST

Standard Reference Materials, numerous secondary standards are available

from instrument manufacturers and vendors of scientific supplies. These

secondary standards are all calibrated against primary standards developed

by international standards accreditation agencies.
117

Appendix

The following sections present a slightly more detailed discussion of the

underlying theory of the various scattering processes. The references quoted in

these discussions delve into significantly greater detail than what is presented

here.

1. Mie Scattering

Derivations of the Mie solutions arise from Maxwell’s equations for

electromagnetic radiation. Representing the electric and magnetic induction

vectors as E and H respectively, and, μ being the permeability and ε the

permittivity, Maxwell’s equations can be represented as:

∇ • E = 0 (1)

∇ • H = 0 (2)

∇ x E = i ω μ H (3)

∇ x H = -i ω ε E (4)

Assuming, k2 ≡ ω2 ε μ, then equations 3 and 4 can be rewritten as:

∇2E + k2E = 0 (5)

∇2H + k2H = 0 (6)

To solve the above set of equations, separation of variables leads to a

general solution of the form12:

[ cos( ) cos( ) ( ) cos( ) cos( ) ( )

0

A m P z kr B m P z kr n

m

l

m

l

l

l

m l

n

m

l

m ψ ∑ ∑ l φ θ φ θ

∞

= −=

= +

where, Pl

m (x) are the associated Legendre polynomials and zn

(x) are the

spherical Bessel functions, and ψ is a scalar function.

Expressing the boundary conditions of a sphere (representing the particle) and

the incident plane wave in vector spherical harmonics, the complex plane wave

(Ec ) can then be expressed in the form:

( )

0

o

mn

o

mn

e

mn

e

mn

l

m nl

o

mn

o

mn

e

mn

e Ec = ∑∑ BmnM + B M +A N + A N =−

∞

=

(8)

(7)

The above expression can further be simplified under the condition that the

incident plane wave is polarized such that m = 1 is the only contributing factor,

and so equation 8 now is of the form:

∑ [ ] ∞

=

− +

+ = 0

(1)

1

0(1)

1 ( 1)

2 1

n

e

n n

n

c M iN

n n

n E i (9)

Appendix ◆
118

◆ Laser Light Diffraction Techniques

The vector spherical harmonics expressed in equation 9 can be used to arrive

at two scalar components, each of which has a spherical Bessel function and

this information can be used to calculate the initial electromagnetic field outside

the sphere.

Applying the boundary conditions that the transverse components of the total

electric and magnetic induction fields at the particle surface are zero enable

calculating two components for the internal and the scattered fields, which

correlate to the scattered wave outside the sphere and the wave produced

inside the sphere itself.

Representing the constants for these solutions as an

 and bn and πn(x) and τn(x)

as supplementary functions, the amplitude functions are expressed as:

∑ [ ] ∞

=

+

+

+ = 1

1 cos( ) cos( ) ( 1)

2 1 ( ) n

n n n n a b

n n

n S θ π θ τ θ

and,

The extinction and scattering efficiencies are resolved to have the form:

∑ [ ] ∞

=

+

+

+ = 1

2 cos( ) cos( ) ( 1)

2 1 ( ) n

n n n n b a

n n

n S θ π θ τ θ

that arise from the linear relationship existing between the scattered and

incident amplitudes as per Maxwell’s equations10:

∑

∞

=

= + +

1

2 (2 1)( ) 2

n

ext n n R n a b

x

Q

∑

∞

=

= + +

1

2 2

2 (2 1)( ) 2

n

sca n n n a b

x

Q

and,

S1 and S2 are elements of the amplitude scattering matrix of the form:

















4 1

2 3

S S

S S















































 =













 − −

−

( )

0

( )

0

4 1

2 ( ) / 2 3

1

0

( )

( )

0 2

r

l

i d s

r

l

E

E

S S

S S

e

id

E

E π λ

λ

π

E0

(l) and E0(r), E(l) and E(r) are components of the electric vector for the incident

and scattered wave, respectively. The scattered wave is separated from the

incident wave by a phase difference of 2πi(d-s)/λ . In case of scattering for

(10)

(11)

(12)

(13)

(14)
119

homogeneous, spherical scatterers, the elements S3 and S4 vanish, and thus

only solutions for S1 and S2.

Equations 10 to 13 represent the analytical solution as per Mie theory. Thus,

for scattering of unpolarized light by a spherical particle, the scattered intensity

can be represented by the equation14:

(15)

where, I0 is the intensity of the incident light, k represents the wavenumber

2π/λ, and a is the distance from the scatterer to the detector. S1(θ) and S2(θ)

represent the complex functions describing the change in amplitude of the

perpendicular and parallel polarized light as a function of the angle θ, measured

in the forward direction.

2. Fraunhofer Approximation

When the particle diameter is larger than the wavelength, scattering occurs

mostly in the forward direction(i.e., θ is small) and so equation 14 simplifies to:

[ ] [ ][ ]2

2

2

2 2 1

0 ( ) ( ) 2 (θ ) S θ S θ

k a

I I = +

2

4 1

2 2

0

sin

( sin )

2 ( ) 



 



 = θ

θ θ

x

J x

x

k a

I I

2

2 4 1

2

2

1 sin

( sin ) as, (S ) (S ) 



 



 = = θ

θ

x

J x

x

The term x represents a dimensionless size parameter of the form x = πD/λ,

(D being the particle diameter) and J1 is a Bessel function of the order unity.

Equation 16 represents the Fraunhofer approximation of the Mie theory.

3. Rayleigh Scattering

When the particle diameter is much smaller than the wavelength of the incident

radiation, the solution to the scattering problem is given by Rayleigh’s solution4.

The scattered intensity in this case is the sum of the scattered components

S1(θ) and S2(θ), and thus can be represented as:

2

2 4

2

2

4

2

2

2 cos

2

2 1 ( )

a a

I θ

λ

π

α λ

π θ α 



 



  +  



 =

(16)

(17)

(18)

Appendix ◆
120

◆ Laser Light Diffraction Techniques

In the above equation, α represents particle polarizability. Furthermore,

for homogeneous spheres of volume V and refractive index relative to the

surrounding medium of m, as m–1 → 0 (i.e., anamolous diffraction model),

equation 18 can be simplified as:

π

α

4

3

( 2)

( 1)

2

2 V

m

m

+

− = (19)

4

2( 1) 4 ( 1) 2 Vm

V

α = m − ≅ − (20)

The value of α from equation 19 can then be substituted into equation 18 to

calculate the scattered intensities.

4. Rayleigh–Gans Scattering

In case of Rayleigh–Gans scattering, which may be thought of as a special

condition of the Rayleigh scattering, the particle diameter is not significantly

smaller than the wavelength of the incident radiation. Under such conditions,

the scattered intensity is still a sum of the intensities for the horizontal and

vertically polarized components. Thus, equation 18 now has the form:

( ) θ

π

θ

2

2 2 3

4 2 2

0 (sin cos ) 1 cos

3

8

( 1) + 



 



 − 



 



 − = u u u

r u

k V m I I (21)

In the above equation, u = 2πD/λ(sin(θ/2)) and k = 2π/λ. Equation 21 reduces

to equation 18 (the Rayleigh solution) when the term (3/u3)(sin u –u cos u)

tends to 1.

5. Inversion Procedures

The calculation of the particle size distribution of the powder system being

examined, requires the application of an inversion procedure to the diffraction

pattern formed at the detector system in the instrument. The calculated

size distribution is very dependent on the nature of the algorithm used for

processing the scattering intensity data. The algorithms can be designed

to make no assumption about the functional form and shape of the size

distribution, in which case the algorithms are said to be model independent.

Often, the algorithms may be designed with certain assumptions of the

shape. Some of the commonly assumed shape distributions including normal,

lognormal or the Rosin–Rammler distribution. This is particularly true of earlier

instruments which had limited computational power and memory. Assumptions

of certain shapes for the distribution reduced the time and resources required
121

































+

































































=

































n n nm m n

m

m

n q

q

q

a a a

a a a

a a a

L

L

L

ε

ε

ε

...

...

...

...

...

... ... ... ...

... ... ... ...

...

...

...

...

2

1

2

1

1 2

21 22 2

11 12 1

2

1

for deconvolution of the intensity patterns and calculating the size distribution.

With decreasing costs of computer memory and increasing computational

power, more systems are being designed to be model independent.

Boxman18 et al. describe various inversion procedures. In a model-independent,

direct inversion technique, the recorded scattering pattern can be expressed as

a set of linear equations of the form:

The above equation can be vectorially represented as:

L = A q + L = A q + ε (23)

where, L represents the integrated light intensity measured as a function of the

scattering angle,

A is the scattering matrix comprised of the calculated scattering

coefficients aij ,

q is the solution vector that contains fraction of particles in each size class,

ε is the associated random measurement error.

n is the number of detector elements, and

m is the number of particle size classes.

The solution for the vector q, calculated by the least squares methods is:

q = ( AT A)-1 AT L (24)

Boxman et al. claim that the above process may not be the most efficient, as

there maybe a significant error arising from the measurement errors contained

in L and the systematic errors contained in A. A suggested method to

overcome these drawbacks is to add an additional matrix that would suppress

the oscillatory behavior of the solution vector and lead to early convergence.

The use of such an additional matrix follows the Phillips–Twomey Inversion,

and in such a case, equation 24 can be represented as:

q = ( AT A + γ H)-1 AT L (25)

In equation 25, H represents the smoothing matrix, and, γ a smoothing scalar

that represents the amount of smoothing included in the final solution. This

(22)

Appendix ◆
122

◆ Laser Light Diffraction Techniques

procedure gives the user greater control on deciding the amount of smoothing

to be included in the algorithm for deconvolution. The use of non-negativity

constraints and incorporation of intensity fluctuation information have been

shown to increase the stability of the linear equations and thus arrive at a

more reliable solution. Other techniques to improve sensitivity of instruments

to particles with particular characteristics have been presented19. These

techniques use a combination of statistical means and deconvolution

procedures.

Detailed reviews about various aspects of light diffraction including the general

theory, applications and related studies are available in the works of Krathovil20

and Kerker21.
123

References

1. H. Hildebrand, Refractive Index Considerations in Light Scattering Particle

Size Measurements in Advances in Process Control Measurements for

the Ceramic Industry, A. Jillavenkatesa and G. Onoda, eds., American

Ceramic Society, Westerville, OH (1999) p. 379.

2. M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of

Propogation Interference and Diffraction of Light, 7th ed. Cambridge

University Press, Cambridge (1999).

3. L.P. Bayvel and A.R. Jones, Electromagnetic Scattering and Its

Applications, Applied Science Publishers, Essex (1981).

4. H.C. Van de Hulst, Light Scattering by Small Particles, John Wiley &

Sons, Inc., New York (1962).

5. E.C. Muly and H.N. Frock, Industrial Particle Size Measurement Using

Light Scattering, Opt. Eng., 19 (6), 861 (1980).

6. BS 3406-7(1998) British Standard Methods for Determination of Particle

Size Distribution; Part 7: Recommendation for Single Particle Light

Interaction Methods, British Standards Institute, London (1998).

7. B.E. Dahneke, Measurement of Suspended Particles by Quasi-Elastic

Light Scattering, John Wiley and Sons, NY (1983).

8. R. Pecora, Doppler shifts in light scattering from pure liquids and polymer

solutions, J. Chem. Phys., 40, 1604 (1964).

9. G. Mie, Beträge zur Optik trüber Medien, speziell kolloidaler

Metallösungen, Ann. Physik., 25, 377 (1908).

10. R.M. Goody and Y.L. Yung, Atmospheric Radiations: Theoretical Basis,

Oxford University Press, New York (1989).

11. T. Allen, Particle Size Measurement, 4th ed., Chapman and Hall, London

(1993).

12. E.W. Weisstein, Eric Weisstein’s Treasure Trove of Physics, http://

www.treasure-troves.com/physics/

13. C.F. Bohren and D.R. Huffman, Absorption and Scattering of Light by

Small Particles, John Wiley & Sons, New York (1983).

14. ISO/FDIS 13320-1: 1999, Particle Size Analysis-Laser Diffraction

Methods – Part 1: General Principles, International Organization for

Standardization, Geneva (1999).

References ◆
124

◆ Laser Light Diffraction Techniques

15. Relative Refractive Index Tables, Horiba Instruments, Irvine, CA.

16. R.C. Weast, ed. CRC Handbook of Chemistry and Physics, CRC Press,

Inc., Boca Raton, FL (1984).

17. E.D. Palik, Handbook of Optical Constants of Solids, Academic Press,

New York (1985).

18. A. Boxman, H. Merkus, P.J.T. Verheijen and B. Scarlett, Deconvolution

of Light-Scattering Patterns by Observing Intensity Fluctuations, Appl.

Optics, 30 (33), 4818 (1991).

19. Z. Ma, H.G. Merkus, J.G.A.E. de Smet, P.J. Verheijen and B. Scarlett,

Improving the Sensitivity of Forward Light Scattering Technique to Large

Particles, Part. Syst. Charact., 16, 77 (1999).

20. J.P. Krathovil, Light Scattering, Anal. Chem., 36 (5), 458R (1964).

21. M. Kerker, Scattering of Light and Other Electromagnetic Radiation,

Academic Press, New York (1969).

22. ISO 9276-1: 1998, Representation of Results of Particle Size Analysis –

Part 1: Graphical Representation, International Organization for

Standardization, Geneva (1998).

23. ASTM B822-97 Standard Test Method for Particle Size Distribution of

Metal Powders and Related Compounds by Light Scattering, American

Society for Testing and Materials, West Conshohocken, PA (1997).

24. ASTM E1458-92 Standard Test Method for Calibration Verification of

Laser Diffraction Particle Sizing Instruments Using Photomask Reticles,

American Society for Testing and Materials, West Conshohocken, PA

(1992).
125

Reporting Size Data ◆

7. REPORTING SIZE DATA

1. Introduction

2. Standards for the Reporting of Particle Size Data

2.1 ASTM E1617-97

2.2 ISO 9276-1:1998

3. Particle Diameter

4. Representation of Size Data

5. Summary

1. Introduction

Reporting particle size and particle size distribution information is as critical

a component as designing and conducting the experiments that generate

this information. The main consideration while reporting size and size

distribution data is to ensure that all pertinent information is presented,

such that the experiments conducted to generate this information can

be reproduced when necessary. This feature is of particular interest for

comparing and communicating the particle size and size distribution data

between different laboratories or between suppliers and customers. Thus,

effective communication of size and size distribution data requires that

pertinent information be provided in the most efficient manner.

At present, there are over 400 commercially available particle size analysis

instruments. These instruments are based on a wide range of physical

principles, and thus the information generated from these instruments tends to

be of varied nature. Some instruments report size results as diameters, while

some as surface area or projected area. The nature of the physical principle

used may bias the results depending on the technique and parameters (mass,

number, surface area, etc.) used for data representation. For example, results

from instruments based on the principle of gravitational sedimentation tend to

be more accurate when reporting particle size distribution on a mass basis than

surface area. Similarly, results from instruments using the electrozone sensing

principle tend to be more representative when represented on a number basis

than mass basis. Particle size and size distribution data can be presented in

varied forms (i.e., by a single number representing a particular feature of

the distribution, in a tabular manner and/or graphically). Each representation

has its own distinct advantages and shortcomings, and the choice of the

representation to be used should be made judiciously. Various texts1,2 discuss

these strengths and shortcomings, and also, the means to convert from one

method of representation to another. Some of the more commonly used

representations will be discussed briefly in this text.
126

◆ Reporting Size Data

2. Standards for the Reporting of Particle Size Data

As of November 1999, there are two implemented standards that are pertinent

to reporting particle size data. The ASTM E1617-973, Standard Practice for

Reporting Particle Size Characterization Data, was originally published as

E1617-94 and refers to means for reporting particle size measurement data.

The ISO standard, ISO 9276-1:1998, on Representation of Results of Particle

Size Analysis – Part 1: Graphical Representation4 is the second standard that is

recommended for the reporting of particle size data. Other pertinent ISO draft

standards in various stages of implementation are:

● ISO/DIS 9276-2 Representation of Results of Particle Size Analysis –

Part 2: Calculation of average particle sizes/diameters and moments from

particle size distributions;

● ISO/CD 9276-3 Representation of Results of Particle Size Analysis –

Part 3: Calculation of means and moments of particle size distributions;

● ISO/DIS 9276-4 Representation of Results of Particle Size Analysis –

Part 4: Characterization of a classification process;

● ISO/WD 9276-5 Representation of Results of Particle Size Analysis –

Part 5: Validation of calculations relating to particle size analyses using

the logarithmic normal probability distribution.

2.1 ASTM E1617-97

The ASTM E1617-97 standard is designed to promote the use of a standard

reporting technique to help reduce inconsistencies that arise when attempting to

compare particle size measurement data obtained from different measurement

techniques and instruments. This standard requires that all pertinent information

relating to the material being analyzed, the instrument being used for analysis,

the methodology used for generating data, and the statistical approaches for

evaluating the data be categorized into three levels. The nature of the report

then dictates what levels need to be reported. The exchanging of analysis

results between two laboratories for the purpose of comparison may thus

require reporting all pertinent details, while the reporting of size data from a

raw materials supplier to a customer might not require the same level of detail,

depending upon the application.

Level I reporting gives fundamental information about the material being

analyzed. This is of a general nature and includes details such as the material

type, sample source and sample lot number pertaining to material identification.

With respect to size analysis, this level of reporting requires information about

the basic measurement principle of the instrument used for analysis (physical

principle such as sedimentation, laser light scattering or optical analysis).
127

Reporting of Particle Size Data ◆

Information about size data needs to be defined by a “minimum reporting of

size parameters,” where the parameters being reported are a set of mutually

agreed upon parameters between the supplier of the information and the party

receiving the information. This level of information also requires that any bias

in data arising due to the sensing technique used by the instrument be indicated

by a suitable choice of diameter (see Section 3). By requiring the reporting

of the physical bases for the reported parameters (e.g., number, volume or

surface area distribution) this standard can help minimize errors that may arise

during data interpretation. A final requirement at this level of reporting is the

range of size measurement.

Level II reporting conveys in greater detail the information presented in

Level I reporting. In addition to the information required for Level I, Level II

requires that the instrument used for particle size analysis be explicitly

identified, together with the appropriate software version, where applicable.

In terms of instrumentation, this level of reporting needs a description of the

experimental parameters under which the instrument was used. The basis by

which calculations are performed by the instrument (where available from the

manufacturer) needs to be reported together with any additional calculations

conducted on the particle size data. The report also requires quantification

of some basic sample statistics, including the standard deviation, number of

degrees of freedom and confidence interval, and the calculations by which this

statistical information is generated. The technique used for sample preparation

also needs to be described in detail for this level of size reporting.

Level III reporting requires the greatest detail about both instrument

parameters and sample parameters. Level III encompasses all of the

information required in Level I and Level II reports together with information

and description of calibration and operations conducted on the instrument,

where these operations are either non-standard operations or are not

specifically asked for. The operating procedures followed while conducting

the size analysis need to be mentioned; if any of these procedures are

designated standardized techniques, then appropriate standard designations

need to be mentioned. This level of reporting also requires, if available, a

statement of precision and bias of the particle size measurement data.

Thus, this standard method of size data reporting requires the users to spell

out all pertinent information that may be required for comparison of particle

size data from one instrument with that from another. This technique of data

reporting also has significant advantages in record-keeping and tracking, when

size and distribution data need to be referred to at a later date. Depending upon

the application of the powder size and distribution data, the level of detail at

which these data are represented can be defined by the user.
128

◆ Reporting Size Data

An important procedure not spelled out in this standard is the mention of

the sampling technique used to create the sample for analysis. As sampling

procedures vary depending upon the size of the initial batch, the nature of

the material, the mode by which the batch is transported, and the nature

of the technique and instrument being used for analysis, it is imperative that

the technique and any instruments used for sampling be defined.

2.2. ISO 9276-1: 1998

This standard concentrates on issues pertaining to the graphical

representation of particle size analysis data, particularly as represented in

the form of histograms, density distributions and cumulative distributions.

The distributions in questions apply not only to powders, but also to droplets

and/or gas bubbles, over any size range. The standard further recommends

a standard nomenclature to be followed to generate density and cumulative

distributions from the measured raw data. The reference to density in this case

is not to the physical property of a material, but to the fractional distribution.

An important issue highlighted in this standard is the common use of the

symbols “x” and “d” to represent the particle size. While using the term “x,”

the standard also states that the term “d” can be used for the same purpose.

The standard defines a nomenclature to distinguish between density

distributions and cumulative distributions on different bases, by the use of the

term q(bases)(x) to define the density distribution and the term Q(bases)(x) to define

the cumulative distribution. The various bases used in this standard are number,

length, surface or projected area and volume/mass. Recognizing that there

is no single definition of particle size, this standard defines particle size as

“the diameter of a sphere having the same physical properties; this is known

as equivalent spherical diameter.” The standard also defines the measures

as either cumulative measures or density measures, either of which can be

derived from the other by suitable integration of differentiation operations over

the appropriate size range. As size distributions are often spread out over a

very wide size range, the use of linear abscissa for representing the size

range may not be practical. This standard defines equations to be used for

representing cumulative and density distributions on a logarithmic abscissa.

3. Particle Diameter

The term “particle diameter” as a quantitative measure of the particle size

should be used with caution. As is obvious, the term “diameter” would be well

representative of the size of spherical particles, but “real” powders tend to be

more irregular in shape with a distribution of sizes and thus “diameter” may be

interpreted differently by numerous operators. Allen2, Svarovsky5, and other
129

authors have termed the various expressions for diameter as equivalent sphere

diameters, equivalent circle diameters and statistical diameters depending upon

the technique by which this number is calculated. Just as important is the

notation for diameter, where traditional notation followed in the United States of

America has used the character “d” followed by a subscript to define the manner

in which the diameter has been computed. ISO standards are now referring to

diameter using the character “x” with the use of the subscript still intact.

Equivalent sphere diameters refer to the diameters of regular, homogeneous

spheres that would have the same property as the particle being measured.

Thus, the use of the term dv (or xv) would refer to volume diameter and can

be defined as the diameter of a regular, homogeneous sphere that has the same

volume as the particle being measured. An important issue put forth by Allen

is that determination of a particle size should be conducted by a technique such

that the obtained results represents a property of the powder that is critical

for the powder application, e.g., powders used in chemical reactions should be

characterized by their surface area. Equivalent spherical diameters represent

three-dimensional properties of the real particle, and thus are most reliable

when used for practical applications that are influenced by volume-based

properties (density, surface area, etc.) (see Table 7.1).

Table 7.1. able 7.1.

Definitions of Volume-Based Diameters2,5

Drag diameter (dd): Diameter of a sphere that has the same

resistance to motion in the same fluid (having same viscosity), moving

at the same velocity as the particle

Free-falling diameter (df

): Diameter of a sphere having the same

density and the same free-falling speed as the particle in the same fluid

Surface diameter (ds

): Diameter of a sphere having the same

surface area as the particle

Surface volume diameter (dsv): Diameter of a sphere having the

same surface to volume ratio as the particle

Stokes’ diameter (dSt): Diameter of a free falling sphere (i.e.,

particle having attained the Stokes velocity) in the laminar flow region

Sieve diameter (dA): Diameter of a sphere equivalent to the size of

the minimum square aperture through which the particle will pass

Volume diameter (dv): Diameter of a sphere having the same

volume as the particle.

Particle Diameter ◆
130

◆ Reporting Size Data

Equivalent circle diameters are used (see Table 7.2) when size determination

is conducted by evaluating the projected profile of a particle. Thus, these

diameters are used mostly for optical or electron microscopy-based techniques,

where the profile of a three-dimensional particle is observed on a twodimensional plane. While these expressions are very useful for observing

and reporting the sizes of spherical particles, the results of irregular particles

can be erroneous if adequate precautions are not taken. The more irregular a

particle, the greater the possibility of surface details being obscured. Also, the

orientation of the irregular particle at rest and the resultant projected image

can significantly influence the calculated diameter. A relatively easy solution is

to sample a large number of random particles in order to minimize variations.

The number of particles that need to be sampled, however, needs to be

evaluated on a case-by-case basis, and can vary over a wide range (from as

few as a few hundred particles to many thousands of particles). Increasing

availability of commercial software for image analysis, to some extent, can

significantly ease and automate the process of size analysis. However, a

thorough evaluation of the strengths and limitations of any software for image

analysis must be conducted preferably using certified size standards. Other

limitations of this group of definitions include the inability of the definition to

describe the shape of the particle, especially when particles have a significant

aspect ratio, and the inability to often discern two distinct particles as such due

to the nature of the projected image.

Table 7.2. able 7.2.

Size Definitions Based on Equivalent Circle Diameter2,5

Projected area diameter (da): Diameter of a circle having the same

area as the projected area of the particle at rest in a stable position

Projected area diameter (dp): Diameter of a circle having the same

area as the projected area of the particle at rest in a random orientation

Perimeter diameter (dc): Diameter of a circle having the same

perimeter as the projected area of the particle, irrespective of the

particle orientation.

The definitions in the third group, the statistical diameters, are probably the

most useful to define the shape of a particle (and thus, the most useful for

particles with a marked aspect ratio), as these involve the calculation of a

linear dimension measured with respect to a defined direction. The use of

these expressions is common while using microscopy-based techniques of

analysis, but also tend to be tedious (see Table 7.3). Automating the analysis

tasks, using commercial software and algorithms, can help reduce the time

and effort involved in conducting these analyses.
131

Table 7.3. able 7.3.

Size Definitions Based on Statistical Diameters2,5

Feret’s diameter (dF): Mean value of the distance between pairs

of parallel tangents to the projected outline of the particle

Martin’s diameter (dM): The mean chord length of the projected

outline of the particle

Unrolled diameter (dR): The mean chord length through the center

of gravity of the particle

Maximum chord diameter (dCH): The maximum length of a line

limited by the particle contour

Shear diameter (dSH): Particle width obtained with an image

shearing device.

Particle sizes are often expressed using terms such as mean diameter, median

diameter and modal diameter. Though these expressions are determined from

the distribution size range of the particle diameters, it is appropriate to define

these terms in this section.

Mean Diameter: Mean diameter of a particle system can be defined for

different particle characteristics of the system under study. The arithmetic

mean is calculated by dividing the sum of the diameters of all the individual

particles in the distribution, by the total number of particles in the distribution.

The geometric mean is calculated by calculating the nth root of the products

of the diameters of the n particles constituting the system under study.

Thus, the mean diameter of a powder can be its length mean diameter, surface

mean diameter, volume mean diameter or moment mean diameter. Heywood6

has defined mathematical expression for various mean diameters, expressed

according to the measured characteristics, namely, number, length, surface

volume and weight.

Median Diameter: The median diameter is the size at which half of all

the particles in the distribution are coarser than the expressed diameter, and

the remaining are finer. Thus, the median diameter is the 50 % size of a

cumulative distribution curve of the particle sizes.

Modal Diameter: The most frequently occurring size value of the

particles under study represents the modal diameter, and thus, in a graphical

representation, is the size that is represented by the peak of the differential

size distribution curve.

Particle Diameter ◆
132

◆ Reporting Size Data

The definition used to express the particle size should be chosen with care as

often different powder distributions can give the same mean value (arithmetic

average), but have significantly different shapes of distribution and size range

over which the distribution is spread. Often, the use of more than one definition

(e.g., stating both the mean and modal diameters) may help alleviate that

confusion and may also give some information about the skewness of the

distribution.

Figure 7.1 represents the particle size distribution of a powder obtained by

laser light scattering. The plot shows the differential volume as a function of

the particle size and indicates the occurrence of the different size measures

(mode, mean and median diameters) for the same distribution.

Particle size (μm)

0.1 1.0 10.0 100.0

Differential Volume (%)

0

1

2

3

4

5

6

7

8

Modal Dia.

Median Dia.

Mean Dia.

Figure 7.1. Figure 7.1.

Occurrence of Different Particle Size Measures of a Powder System

4. Representation of Size Data

Particle size and size distribution data can be represented in either a tabular or

graphical form. The choice of representation used can be dependent on what

information is being expressed and the applications to which this information

will be used. Very often, both tabular and graphical representations are used to

Particle Size (μm)
133

convey this information, allowing the end-user to transform the data to enable

conversions or comparisons with data from other sources. However, care

should be taken when attempting transformation from one form of distribution

to another form, due to errors associated with the assumption of a constant

shape factor throughout the distribution, which is not often the case with real

materials. Both forms of representation are closely related as information

displayed in one form is derived from the other.

At a very basic level, tabular representations typically convey a series of size

ranges and the corresponding quantity of powder observed in that size range.

Thus, the size range can be the size difference between two adjacent sieves,

and the corresponding quantity would be the mass of powder on each of the

sieve. In case of analysis by laser light scattering, the size range may be the

differences in sizes corresponding to the positions of consecutive detector

elements, and the associated sample quantity would be the volume of sample

giving rise to the signal incident on that element. Allen2 and Reed7 give suitable

examples of the organization of data in typical tabular representations.

Table 7.4. able 7.4.

Size Data Obtained by Sieve Analysis of Glass Beads During

Development of a NIST Size Standard (total mass sampled = 85.70 g)

Table 7.4 shows a typical representation of size and size distribution data that

would be obtained from sieve analysis. This information provides enough raw

data to conduct some basic statistical analysis and obtain information about the

size distribution, such as mean, median and standard deviation. This information

however, reveals very little about the shape of the distribution which can be

obtained from plotting these results.

Graphical representation of size and size distribution results can be obtained

either as differential distribution or cumulative distribution of the size parameter,

as a function of the size. When plotting distribution data, the plots are often

drawn as histograms, with the frequency of occurrence as the ordinate and

the size or size range over which this frequency is observed as the abscissa.

The greatest drawback of this representation arises from the finite width of the

US Sieve No. Nominal Size 

Opening (μm) 

Mass on Sieve 

(g) 

Mass Fraction 

on Sieve (%) 

Cumulative 

Mass Fraction 

Coarser (%) 

Cumulative 

Mass Fraction 

Finer (%) 

25 710 5.28 6.16 6.16 93.84 

30 600 11.29 13.17 19.33 80.67 

35 500 14.23 16.60 35.93 64.07 

40 425 10.7 12.49 48.42 51.58 

45 355 8.97 10.47 58.89 41.11 

50 300 10.20 11.9 70.79 29.21 

60 250 20.11 23.47 94.26 5.74 

PAN 4.92 5.74 100 

Representation of Size Data ◆
134

◆ Reporting Size Data

abscissa. If the size range chosen is too wide, then slight variations in the

size distribution will not be observed and the histogram will not have adequate

resolution. More common representations involve plotting the size distribution

data as differential distribution curves and cumulative distribution curves.

Figure 7.2. Figure 7.2.

Cumulative and Differential Distribution Plots

of Nominal 1.0 μm SiO2 Spheres

Eq. Diff. Cum.

Spherical Volume fraction

Dia.(μm) (%) finer

(%)

0.375 0 0

0.412 0 0

0.452 0 0

0.496 0 0

0.545 0 0

0.598 0 0

0.656 0 0

0.721 1.05 1.05

0.791 11.6 12.65

0.868 24.9 37.55

0.953 24.8 62.35

1.047 12.1 74.45

1.149 1.87 76.32

1.261 2.75 79.07

1.384 11.7 90.77

1.52 8.45 99.22

1.668 0.78 100

1.832 0 100

2.011 0 100

2.207 0 100

2.423 0 100

2.66 0 100

2.92 0 100

Particle Dia. (μm)

0.1 1.0 10.0 100.0

Differential vol(%)

0

5

10

15

20

cpf

0

20

40

60

80

100

120

Differential Volume (%)

Cum. Fraction Finer (%)

d90

d50

d10

(median dia.)

Table 7.5.

Size Distribution Data

from Coulter LS 230TM

Particle Size Analyzer

Table 7.5 shows the size distribution data obtained from a Coulter LS230TM

laser light scattering particle size analyzer. The data in the left column

correspond to the calculated equivalent spherical diameters. The data in the

center column are the differential volumes corresponding to each of the size

diameter and the data in the right column are calculated as the cumulative

volume obtained from sequential summation of the differential volume. These

data are plotted as differential and cumulative volume distribution curves in

Figure 7.2. As is observed, the differential distribution plot gives a very good

representation of the relative percentage frequency distribution (by volume)

and thus can be used to estimate what are the relative amounts of particles of
135

different size ranges. The cumulative distribution, on the other hand, makes

it easy to determine the median size (d50) and the percentage distribution

contained between any two sizes2. It also enables the determination of sizes

corresponding to the d10 and d90 values with relative ease.

The d10, d50 and d90 values are size values corresponding to the cumulative

distribution at 10 %, 50 % and 90 %. Thus, the d10 value represents a size

value below which 10 % of the cumulative distribution is present, and similarly

d90 represents a size below which 90 % of the cumulative distribution occurs.

Thus, from these definitions, the d50 value corresponds to the median diameter.

Often, d10, d50 and d90 values are quoted as a measure of the size distribution

of the powder. However, this does not reflect the true shape of the distribution

curve and can be misleading. In an attempt to overcome this drawback some

have included the d25 and d75 values.

Particle Dia. (μm)

0.1 1.0 10.0 100.0

Differential Vol. (%)

0

5

10

15

20

25

30

cpf

0

20

40

60

80

100

120

Differential Volume(%)

Cum. Fraction Finer(%)

d10 = 0.849 μm

d25 = 0.907 μm

d50 = 0.992 μm

d75 = 1.125 μm

d90 = 1.504 μm

Figure 7.3. Figure 7.3.

Differential and Cumulative Distribution Plots of

Powder of Bimodal Size Distribution

A cumulative distribution plot is of limited value when the powder under study

may have variations in the relative distributions at different sizes, giving rise to

a multi-modal fractional distribution curve, rather than a uni-modal as shown in

Figure 7.2. In such an instance, the cumulative distribution curve would show

an inflection, but the relative amounts cannot be easily determined. Similarly,

Representation of Size Data ◆
136

◆ Reporting Size Data

the use of d10, d25, d50, d75 or d90 values would indicate very little, other

than some idea of the range of powder sizes. In such instances the use of

differential distribution curves would be ideal to estimate the relative amounts

of powder of varying sizes. An example of this occurrence is shown in Figure

7.3, which shows cumulative and differential distributions for a SiO2 powder

system comprising of spherical particles. The d10, d25, d50, d70 and d90 values

are also listed for reference. The differential distribution shows two distinct

size distributions giving rise to a bimodal distribution. The ability to obtain

such information can have critical consequences in various manufacturing

and quality control processes.

The choice of the technique used to represent size data is also determined

by how that data will be used. A fitting example would be representation of

size data of clay powders for certain whitewares manufacturing applications.

In such instances, discrete numbers such as the d10, d50 and d90 values would

not convey much more than the size distribution of the powders. However,

the slope of the cumulative distribution graph would enable determination of

the packing of the powders and also some idea about the viscosity and flow

behavior of the powders, all of which would directly impact the forming

process and the properties of the final product.

5. Summary

The reporting of particle size and particle size distribution data can be done

in various forms and techniques. The first factor to be considered is the

property that is being used to characterize the particle size and size distribution.

One should be aware of the errors and bias associated with measurement

instruments and techniques when attempting to characterize a particular

property. Similarly, conversion from distributions representing a particular

property to that representing another property should be conducted with

extreme care and, if possible, avoided all together. In attempting such

transformations, the aspect ratio of the particles should be considered and

their influence on the transformation factored in. The technique used for

reporting size data should preferably address the shape and extent of

distribution and some representative characteristic such as median size

or moment of distribution.

While reporting of the measured data satisfies one aspect of size reporting, a

comprehensive report would necessarily require reporting of various aspects of

sampling, sample preparation, instrumental parameters used and any additional

statistical analysis. While often cumbersome and superfluous, this additional

information would enable testing for reproducibility by other testing facilities

or personnel and would also provide a basis for the development of good

laboratory practices for an organization.
137

References

1. C. Bernhardt, Particle Size Analysis: Classification and Sedimentation

Methods, 1st Eng. Lang. ed., Chapman and Hall, London (1994).

2. T. Allen, Particle Size Measurement, 4th ed., Chapman and Hall, London

(1990).

3. ASTM E1617-97 Standard Practice for Reporting Particle Size

Characterization Data, American Society for Testing and Materials,

West Conshohocken, PA (1997).

4. ISO 9276-1: 1998, Representation of Results of Particle Size Analysis –

Part 1: Graphical representation, International Organization for

Standardization, Geneva (1998).

5. L. Svarovsky, Characterization of Powders, in Principles of Powder

Technology, M.J. Rhodes, ed., John Wiley &Sons, Chichester (1990) p. 35.

6. H. Heywood, Pharmaceutical Aspects of Fine Particles and Their

Evaluation. II. Evaluation of Powders, Pharm. J., 191 (5211), 291

(1963).

7. J.S. Reed, Introduction to the Principles of Ceramic Processing, John Wiley

& Sons, New York (1988).

References ◆
139

Glossary ◆

GLOSSARY

TERMS RELATED TO PARTICLE SIZE ANALYSIS

AND CHARACTERIZATION*

accepted reference value4: 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

accuracy2: of a test method, the degree

of agreement between the true value of

the property being tested (or an accepted

standard value) and the average of many

observations made according to the

test method, preferably by many

observers

accuracy of measurement7: closeness

of the agreement between the result of a

measurement and the value of the

measurand

aerated bulk density6: the minimum

bulk density of a motionless powder

agglomerate1: several particles held

together by weak cohesive forces

agglomeration: formation of aggregates

via physical (van der Waals, hydrophobic)

or electrostatic forces; the resulting

structure is called an agglomerate

aggregate1: a cohesive mass consisting

of particulate subunits

* EDITORIAL COMMENTS

1. The definition of terms in this glossary does not reflect in any way the National Institute

of Standards and Technology’s (NIST) preference or endorsement of these expressions.

2. Most of the definitions expressed here have been adapted as recommendations from various

national and international standards agencies.

3. The use of multiple definitions for the same term reflects the varied usage of these terms

for different applications and contexts.

ASTM: American Society for Testing and

Materials

aberration10: any error that results in

image degradation. such errors may be

chromatic, spherical, astigmatic, comatic,

distortion, or curvature of field; and can

result from design or execution, or both

absolute powder density6: the mass

of powder per unit of absolute powder

volume

absolute powder volume6: the volume

occupied by a powder excluding all pores

and voids

absolute volume6: the volume of the solid

matter after exclusion of all the spaces

(pores and voids)

acceptable quality level acceptable quality level2 (AQL or p1):

in acceptance sampling, the maximum

fraction of nonconforming items at which

the process average can be considered

satisfactory; the process average at which

the risk of rejection is called the producer’s

risk

acceptance number2: in acceptance

sampling, the maximum number of

nonconforming items in a sample that

allows the conclusion that the lot conforms

to the specification

acceptance sampling2: sampling done to

provide specimens for acceptance testing;

testing performed to decide if a material

meets acceptance criteria
140

◆ Glossary

aggregation1: a general term defined as

any process by which particles collect

to form a cohesive mass or cluster; the

resulting structure is called an aggregate

air jet sieving6: a device in which a

portion of powder in a sieve is fluidized

by air passing upwards through it from a

rotating slit. At the same time a negative

pressure is applied to the bottom of the

sieve which removes the fine particles to

a collecting device

Analysis of Analysis of Variance (ANOVA) 2: a

procedure for dividing the total variation of

a set of data into two or more parts, one of

which estimates the error due to selecting

and testing specimens and the other part(s)

possible sources of additional variation

angle of repose6: the angle to the

horizontal assumed by the surface of a

poured heap of powder at rest

angular aperture10: the angle between the

most divergent rays that can pass through a

lens to form the image of an object

aperture size16: the dimension defining an

opening in a sieving medium

apparent density17: the mass of a unit

volume of powder, usually expressed as

grams per cubic centimeter, determined by

a specified method

apparent particle density6: the mass of

a particle divided by its apparent volume

apparent particle volume6: the total

volume of the particle excluding open pores,

but including closed pores

apparent powder density6: the mass of a

powder divided by its apparent volume

apparent powder volume6: the total

volume of solid matter, open pores and

closed pores and interstices

assignable cause2: a factor which

contributes to variation and is feasible to

detect and identify

attribute data2: observed values or

determinations which indicate the presence

or absence of specific characteristics

auto-delineation8: the process which

attempts to predict the true edge of unsharp

images of particles by reference to the local

gray level values; this is used to improve

the accuracy of measurements and to make

them less sensitive to incorrect gray level

thresholding

average2: for a series of observations, the

total divided by the number of observations

BSI: British Standards Institute

balanced incomplete block design (BIB) 4:

an incomplete block design in which each

block contains the same number k of

different versions from the t versions of

a single principal factor arranged so that

every pair of versions occurs together in

the same number, λ, of blocks from the

b blocks

batch4: a definite quantity of some product

or material produced under conditions that

are considered uniform

bias2: in statistics, a constant or systematic

error in test results

From ASTM E 131-983: Bias is determined

by the equation:

Bias = ‘ = (1/n) Σi=1n ei

where, n = number of observations for

which the accuracy is determined,

ei= the difference between a measured

value of a property and its accepted

reference value, and

‘ = the mean value of the ei

binary image8: a digitized image consisting

of an array of pixels, each of which has a

value of either 0 or 1; these values are

normally represented by dark and bright

regions on the display screen or by the use

of two distinct colors
141

Glossary ◆

binomial distribution2: the frequency

distribution which has the probability

function:

P(r) = n!/ [r!(n-r)!] pr

 qn-r

where: P(r) = probability of obtaining

exactly r “successes” in n independent

trials,

p = probability, constant from trial to

trial, of obtaining a “success” in a single

trial, and,

q = 1 – p

blending17: the thorough intermingling of

powders of the same nominal composition

(not to be confused with mixing)

blinding6: the building up of small

particles on a screening surface reducing the

aperture size or closing them completely

block2: in experimenting, a group of units

that is relatively homogeneous within itself,

but may differ from other similar groups

Boolean operation6: a logical procedure

where two binary images are compared

pixel by pixel to produce a third binary

image

bridge width6: the distance between the

nearest edges of two adjacent holes in a

perforated plate

bridging6: the formation of arches of

particles keyed or jammed across the

direction of flow

Brownian motion6: the random movement

of small particles in a disperse phase caused

by the bombardment by molecules of the

surrounding medium (thermally induced

motion – ed. note)

bulk density17: the mass per unit volume

of a powder under nonstandard conditions,

for example, in a shipping container (not to

be confused with apparent density)

bulk sample2: 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

calibrant19: a reference material used for a

calibration

calibrate2: to determine and record the

relationship between a set of standard units

of measure and the output of an instrument

or test procedure

calibration : (1) the act or process of

calibrating; the recorded relationship

resulting from calibrating2; (2) the

determination of a bias conversion factor

of an analytical process, under specified

conditions, in order to obtain meaningful

results6; (3)(in context of microscopy):

the set of operations which establish,

under specified conditions, the relationship

between magnification values indicated

by the SEM and the corresponding

magnification values determined by

examination of a reference material11

centrifugal sedimentation6: (1) the

classification of particles by their rate of

fall in a fluid accelerated by centrifugal

force; (2) the grading by mass of suspended

particles accelerated by centrifugal force

certification scheme5: certification

system as related to specified products,

processes or services to which the same

particular standards and rules, and the same

procedure apply (ISO Guide 2)

certified reference material certified reference material11: 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 uncertainity

as a stated level of confidence (ISO Guide

30:1992)

characteristic4: a property of items

in a sample or population which, when

measured, counted or otherwise observed,

helps to distinguish between the items

chromatic aberration10: a defect in a

lens or lens system as a result of which the

lens possesses different focal lengths for

radiation of different wavelengths
142

◆ Glossary

classification17: separation of a powder

into fractions according to particle size

cluster sampling4: when the primary

sampling unit comprises of a bundle of

elementary units or a group of subunits,

the term cluster sampling may be applied

coagulation1: formation of aggregates

induced by the addition of electrolyte to

a suspension; the resulting structure is

termed the coagulum or coagulate, while the

electrolyte additive is termed the coagulant

coagulum, coagulate1: in a suspension,

aggregates formed by addition of

eletrolyte

coarse particle1: state of subdivision

implying that the particles have average

dimensions greater than roughly 100 μm

Coefficient of ficient of Variation, CV: (1) a

measure of the dispersion of observed

values equal to the standard deviation for

the values divided by the average of the

values; may be expressed as a percentage

of the average2; (2) (relative measure

[%] for precision) standard deviation

divided by mean value of population

and multiplied by 100 (for normal

distributions of data the median is equal

to the mean)14

coincidence6: detection of two or more

particles as a single particle

colloidal (colloid) 1: state of subdivision

implying that the particles have at least in

one direction a dimension roughly between

1 nm and 1 μm. Colloids are significantly

affected by Brownian (thermal) motion

when suspended in a liquid

coma10: a lens aberration occurring in

that part of the image field that is some

distance from the principal axis of

the system; it results from different

magnification in the various lens zones.

Extra-axial object points appear as short

comet-like images with the brighter

small head toward the center of the field

(positive coma) or away from the center

(negative coma)

comminution18: the act or process of

reduction in particle size, usually but

not necessarily by grinding or milling

completely randomized design: a design

in which the treatments are assigned at

random to the full set of experimental units

completely randomized factorial completely randomized factorial

design4: a factorial experiment (including

all replications) run in a completely

randomized design

complex refractive index14 (Np): refractive

index of a particle, consisting of a real and an

imaginary (absorption) part, Np = np – ikp

Component of Component of Variance2: a part of a total

variance identified with a specified source

of variability

composite design4: a design developed

specifically for fitting second order

response surfaces to study curvature,

constructed by adding further selected

treatments to those obtained from a 2n

factorial (or its fraction)

cone and quartering6: a method of

subdividing a dry powder sample (see

chapter on Sampling for description)

confidence interval confidence interval2: an interval estimate

of a population parameter computed so that

the statement “the population parameter

lies in this interval” will be true, on average,

in a stated proportion of the times such

statements are made

confidence level confidence level19: the probability that

the true value lies within a stated range

(the confidence interval)

confidence limits2: the two statistics that

define the ends of a confidence interval

confounded factorial design4: a factorial

experiment in which only a fraction of

the treatment combinations are run in

each block and where the selection of

the treatment combinations assigned to

each block is arranged so that one or more

prescribed effects is (are) confounded with

the block effect(s), while the other effects

remain free from confounding
143

Glossary ◆

confounding4: combining indistinguishably

the main effect of a factor or a differential

effect between factors (interactions) with

the effect of other factor(s), block factor(s)

or interaction(s)

consensus5: general agreement,

characterized by the absence of substantial

issues by any important part of the

concerned interests and by a process that

involves seeking to take into account the

views of all parties concerned and to

reconcile any conflicting arguments,

(ISO Guide 2)

control chart factor4 : a factor, usually

varying with sample size, to convert

specified statistics or parameters into

a central line value or control limit

appropriate to the control chart

control limits2: predetermined

ranges based on the variability of past

observations between which the instrument

data for a test must fall to be considered

valid

control control4 (evaluation): an evaluation

to check, test, or verify; (authority):

the act of guiding, directing, or managing;

(stability): a state of process in which

the variability is attributable to a constant

system of chance causes

correction7: value added algebraically to

the uncorrected result of a measurement

to compensate for systematic error

correction factor7: numerical factor

by which the uncorrected result of a

measurement is multiplied to compensate

for systematic error

critical difference2: the observed

difference between two test results, which

should be considered significant at the

specified probability level

cumulative oversize distribution plot6:

a plot obtained by recording the amount of

the oversize particles (y axis) against the

particle size (x axis) for several different

size levels

cumulative undersize distribution plot6:

a plot obtained by recording the amount of

the undersize particles (y axis) against the

particle size (x axis) for several different

size levels

deconvolution14: mathematical procedure

whereby the size distribution of a particle

ensemble is inferred from measurements of

their scattering pattern

depletion stabilization1: mechanism

in which particle aggregation is inhibited

by the presence of free (non-adsorbed)

polymer due to the creation of high-energy

depletion zones (depleted of polymer

compared to the bulk solution) between

closely interacting particle surfaces

depth of field10: the depth or thickness of

the object space that is simultaneously in

acceptable focus

depth of focus10: the depth or thickness of

the image space that is simultaneously in

acceptable focus

design of experiments4: the arrangement

in which an experimental program is to

be conducted, and the selection of the

levels (versions) of one or more factors or

factor combinations to be included in the

experiment; synonyms include experiment

design and experimental design

detection8: the process by which the

digitized gray level image is converted into

a binary image representing the particles to

be measured

determination value2: the numerical

quantity calculated by means of the test

method equation from the measurement

values obtained as directed in a test method

deviation4: the difference between a

measurement or quasi-measurement and

its stated value or intended level

diffraction14: spreading of light around the

contour of particle beyond the limits of its

geometrical shadow with a small deviation

from rectilinear propagation
144

◆ Glossary

diffusion diameter6: the diameter of a

particle calculated from photon correlation

spectroscopy, using the Stokes-Einstein

equation

digitization8: the process of converting

the continuously varying voltage output of

a video signal into a discrete array of pixels

whose values correspond to the intensities

of the parts of the original image that they

represent

discrete sample2: one or more units taken

from a material that consists of separately

identifiable units

discrete variate2: a variate that is a

measurement based on a scale that has a

limited or finite number of steps; such as a

count, a rating scale, or a ratio of successes

to total observations.

dispersing agent1: a substance which

when present in small amounts facilitates

the dispersion of aggregates to form a stable

suspension; for example, polyelectrolytes

are often used as dispersing agents in

ceramic processing.

dispersion1: in general, a two-phase

system in which particles of any kind (solid,

liquid, gas) are dispersed in a continuous

phase of a different composition or state;

more specifically in the field of ceramics,

dispersion is used to describe a suspension

of solid particles in a liquid medium

drift19: in instrumental methods of

quantitative analysis, a gradual change in

instrument response from start to

completion of a set of determinations

dry milling18: the process of reducing

the particle size of a substance by milling

without the use of a liquid medium

dry screening18: the process of separating

small sizes of granular or powdered solids

from coarser particles by passing them

through a screen of desired mesh size while

in the dry state

dry sieving6: the sieving of powders

without the aid of a liquid

duplicate2: in experimenting or testing, one

of two or more runs with the same specified

experimental or test conditions but with

each experimental or test conditions not

being established independently of all

previous runs

edge6: the boundary between regions of

interest and background

edge finding9: one of the many edge

detection methods used to detect transition

between objects and background

effective aperture10: the diameter of the

entrance pupil; it is the apparent diameter

of the limiting aperture measured from

the front

effective particle density6: the mass of

a particle divided by its volume including

open pores and closed pores

effective solid density6: the density of

powder particles as determined by a given

fluid displacement method

effective solid volume6: the mass of the

particles divided by the effective solid

density

electrostatic stabilization1: mechanism

in which particle aggregation is inhibited

by the presence of a mutually repulsive

electrostatic potential that surrounds each

particle

electrosteric stabilization1: mechanism in

which particle aggregation is inhibited by

the combined effects of electrostatic and

steric interactions; usually associated with

the adsorption of polyeletrolytes onto the

particle surface

elongation shape factor6: the ratio of the

length of a rectangle with two sides parallel

to the longest dimension of a particle to its

width

envelope volume6: the external volume

of a particle, powder or monolith such as

would be obtained by tightly shrinking a

film to contain it
145

Glossary ◆

equivalent circular diameter9: the

diameter of a circle having the same area

as the projected image of the particle

(also known as Haywood Diameter)

equivalent diameter6: the diameter of a

sphere which behaves like the observed

particle relative to or deduced from a

chosen property

error (of measurement) 7: result of a

measurement minus the values of the

measurand

error of the first kind2, α: in a statistical

test, the rejection of a statistical hypothesis

when it is true

error of the second kind2, β: in a

statistical test, the rejection of a statistical

hypothesis when it is false

evolutionary operation (EVOP) 4:

a sequential form of experimentation

conducted in production facilities during

regular production

experimental error2: variability

attributable only to a test method itself

extinction14: attenuation of a light beam

traversing a medium through absorption

and scattering

factor2: in experimenting, a condition or

circumstance that is being investigated to

determine if it has an effect upon the result

of testing the property of interest

factorial experiment (general) 4:

in general, an experiment in which all

possible treatments formed from two or

more factors, each being studied at two or

more levels (versions) are examined so that

interactions (differential effects) as well as

main effects can be estimated

fine particle1: state of subdivision

implying that the particles have average

dimensions lying roughly between 1 μm

and 100 μm

fineness18: a measurement number

designation of the particle size of a material,

usually reported as passing a screen of

a particular standard size

fines18: the portions of a powder composed

of particles smaller than a specified size

floc22: loose assembly of particles in a liquid

flocculation1: formation of aggregates in a

suspension mediated by polymeric species,

either attached to the particles or free in the

suspending medium; the resulting structure

is called a floc, while the polymer additive

is termed a flocculant; polymer bridging is a

flocculation process

flowability6: the ease with which a powder

flows; usually expressed as an index

focal plane10: a plane through the focal

point perpendicular to the principal axis

of a lens or mirror

fouling6: the building up of particles onto

surfaces because of the particles’ stronger

attraction to the surfaces than to the fluid

in which they are suspended

Fourier transform (FT)3: the

mathematical process used to convert an

amplitude-time spectrum to an amplitudefrequency spectrum, or vice versa

fraction17: the portion of a powder sample

which lies between two stated particle

sizes; synonymous with cut

fraction factorial design4: a factorial

experiment in which only an adequately

chosen fraction of the treatments required

for the complete factorial experiment is

selected to be run

Fraunhofer diffraction15: the optical

theory that describes low-angle scattering

of light by particles that are large compared

to the wavelength of the incident light

free falling velocity22: velocity of a

particle through a still liquid at which the

effective accelerating force of the particle is

balanced by the drag exerted by the liquid

on the particle

frequency distribution; differential frequency distribution; differential

distribution; density distribution6:

a plot exhibiting a series of observations

of values as a function of the frequency of

occurrence
146

◆ Glossary

frequency distribution2: of a population,

a function that, for a specific type of

distribution, gives for each value of a

random discrete variate, or each group of

values of a random continuous variate, the

corresponding probability of occurrence

frequency distribution2: of a sample,

a table giving for each value of a discrete

variate, or for each group of values of a

continuous variate, the corresponding

number of observations

F-test2: a test of statistical significance

based on the use of George W. Snedecor’s

F-distribution and used to compare two

sample variances or a sample variance and

a hypothetical value

grading6: separation of a powder into

particle size fractions

granule1: state of subdivision generally

referring to dry particulates with

dimensions lying in the 50 μm to 200 μm

range; typically granules are aggregates of

finer particulates, produced, for instance,

by spray-drying for ease of handling during

subsequent consolidation processes

gravitational sedimentation6: the grading

of mass of suspended particles by mass

accelerated by gravity

gravity feed18: the movement of materials

from one container to another container or

location by force of gravity

green density6: the apparent density of a

green compact

gray image9: an image in which multiple

values, not just the “black” and the “white”

of a binary image, are permitted for each pixel

halo error6: of light microscopy and image

analysis; the oversizing of objects due to halo

halo6: of light microscopy and image

analysis; a dark or light false perimeter of

a particle image

hard-aggregate1: an aggregate which

cannot be easily redispersed by the

application of moderate mechanical

agitation (shaking, stirring or

ultrasonication) and/or chemical treatment;

typically, hard-aggregates consist of

particulate subunits which have been

chemically bonded or fused

heterogeneous material heterogeneous material20: material in

which a spot sample will have significantly

different value of the characteristics under

consideration from the mean value of the

characteristic for the whole material

heterodisperse1: describes a dispersed

system in which multiple size distribution

modes occur; e.g., bimodal, trimodal, etc.

homogeneous material homogeneous material20: material in

which a spot sample will not have a

significantly different value of the

characteristic under consideration from

the mean value of the characteristics for

the whole material

ISO: International Organization for

Standardization

image analysis9: a processing and data

reduction operation which will yield a

numerical or logical result from an image

image processing/image enhancement6:

operations on an image which produces a

modified image

immersed density12: the mass of the

powder per unit volume of suspension

medium that it displaces

immersed particle density6: the mass

of a particle per unit volume of suspension

medium displaced

impaction sampling6: the process of

selectively removing particles from a

gas-borne stream by obstructing the flow

of gas so that particles of higher momentum

collect on a solid surface

impactor6: a device to remove particles

selectively from a gas-borne stream by

impaction onto a solid collection surface

impinger6: a device to remove particles

selectively from a gas-borne stream into a

liquid medium
147

Glossary ◆

incomplete block design4: a design in

which the experiment space is subdivided

into blocks in which there are insufficient

experimental units available to run a

complete set of treatments or replicate

of the experiment

increment19: in sampling, a portion of

material removed from a lot by a single

operation

inspection5: process of measuring,

examining, testing, gaging, or using other

procedures to ascertain the quality or state,

detect errors or defects, or otherwise

appraise materials, products, services,

systems or environments to a preestablished standard

instrument line shape (ILS) function3:

the Fourier transform of the function by

which an interferogram is weighted

interaction2: the condition that exists

among factors when a test result obtained

at one level of a factor is dependent on the

level of one or more additional factors

intercorrelation coefficient, (rXX) 3 :

a measure of the linear association between

the values of the same type of variable

expressed as a correlation coefficient (r)

interference2: in testing, an effect due to the

presence of a constituent or characteristic

that influences the measurement of another

constituent or characteristic

interlaboratory test comparison5:

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 testing2: the evaluating of

a test method in more than one laboratory

by analyzing data obtained from one or

more materials that are as homogeneous as

practical

intermediate precisions4: the closeness

of agreement between test results obtained

under specified intermediate precision

conditions

intermediate precision conditions4:

conditions under which test results are

obtained with the same test method using

test units or test specimens taken at random

from a single quantity of material that is as

nearly homogeneous as possible, and with

changing conditions such as operator,

measuring equipment, location within

the laboratory, and time

inversion6: the reversal of a binary image

by the conversion of all the 1’s to 0’s and

0’s to 1’s

inversion procedure6: a back calculation

method used to obtain a result (e.g., of

particle size distribution from a light

scattering energy pattern)

isokinetic sample6: sample taken at the

same velocity as that in the process under

test

konisampler6: a thermal precipitator fitted

to a microscope for observing particle while

being sampled

laboratory sample2: 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 unit2: 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

least difference of practical importance, least difference of practical importance,

δ2: the smallest difference based on

engineering judgement deemed to be of

practical importance when considering

whether a significant difference exists

between two statistics or between a

statistic and a hypothetical value

levigation6: the classification of particles

according to their movement through a

separating fluid

line start technique12: a technique in

which a thin layer of homogeneous

suspension is floated on the surface of

the sedimentation liquid
148

◆ Glossary

lot2: in acceptance sampling, that part

of a consignment shipping consisting of

materials from one production lot

lot sample2: one or more shipping units

taken at random to represent a lot and used

as a source of laboratory sampling units or

test specimens, or both

lower control limit4 (LCL): control limit

for points below the central line

lower critical Stokes diameter22: Stokes

diameter at which the root mean square

displacement due to thermal diffusion

(Brownian Motion) equals 5 % of the

sedimentation distance

lower tolerance limit4 (LTL): tolerance

limit that defines the lower conformance

boundary for an individual unit of a

manufacturing service operation

margin6: the region between the outside

edges of the outside rows of holes and the

edges of a perforated plate

mass distribution6: the distribution of

mass of particles as a function of their

size

matched test sieve16: a test sieve that

reproduces the results of a master test

sieve within defined limits for a designated

material

mean square2: in analysis of variance, a

contraction of the expression “mean of the

squared deviation from the appropriate

average(s)” where the divisor of each sum

of squares is the appropriate degrees of

freedom

measurement value2: the numerical result

of quantifying a particular property or

dimension

median2: for a series of observations,

after arranging them in order of magnitude,

the value that falls in the middle when

the number of observations is odd or

the arithmetic mean of the two middle

observations when the number of

observations is even

mesh16: in a piece of woven wire cloth, the

nominal number of apertures or fraction

thereof, per inch

method of attributes4: measurement of

quality by the method of attributes consists

of noting the presence (or absence) of some

characteristic or attribute in each of the

units in the group under consideration, and

counting how many units do (or do not)

possess the quality attribute, or how many

such events occur in the unit, group, or area

method of least squares4: a technique of

estimation of parameter which minimizes

Σe2, where, e is the difference between the

observed values and the predicted value

derived from the assumed model

method of variables4: measurement of

quality by the method of variables consists

of measuring and recording the numerical

magnitude of a quality characteristic for

each of the units in the group under

consideration

Mie scattering15 : the complex

electromagnetic theory that describes the

scattering of light by spherical particles

with diameters that are close to the

wavelength of the incident light. The real

and imaginary indices of light refraction

of the particles are needed

minimum particle size1: operationally

defined as the smallest achievable average

particle size (mean or median) measured

in a well-dispersed suspension

minus sieve17: the portion of a powder

sample which passes through a standard

sieve of specified number

mixing17: the thorough intermingling of

powders of two or more materials

mode2: the value of the variate for which

the relative frequency in a series of

observations reaches a local maximum

model matrix14: matrix containing light

scattering vectors from unit volumes

of different size classes, scaled to the

detector’s geometry, as derived from

model computation
149

Glossary ◆

monodisperse1: describes a dispersed

system in which all the particles are of the

same or nearly the same size, forming a

narrow (unimodal) distribution about a

median value; a particle distribution may

be considered monodisperse if at least 90 %

of the distribution lies within 5 % of the

median size

moving range2: the difference without

regard to the sign between two successive

observations

nest of sieves6: a set of test sieves

assembled together

nominal mesh size16: the designated

dimension of the opening of a sieve

nonconformity2: an occurrence of failing

to satisfy the requirements of the applicable

specification; a condition that results in a

nonconforming item

nonparametric2: a term referring to a

statistical technique that does not assume

the nature of the underlying frequency

distribution is known

normal distribution2: the distribution that

has the probability function:

observation2: the process of determining

the presence or absence of attributes or

making measurements of a variable; a result

of the process of determining the presence

or absence of an attribute or making a

measurement of a variable

open area16: the ratio of the total area of

the apertures to the total functional area

of the screening media, expressed as a

percentage

optical axis10: the line formed by the

coinciding principal axes of a series of

optical elements comprising an optical

system; it is the line passing through the

centers of curvature of the optical surfaces

optical model optical model14: theoretical model used for

computing the model matrix for optically

homogeneous spheres, with, if necessary,

a specified complex refractive index,

e.g. Fraunhofer diffraction, anomalous

diffraction, Mie scattering

oversize13: portion of material composed

of particles that are larger than some

specific size

oversize powder17: particles coarser than

the maximum permitted by a given particle

size specification

parameter2: in statistics, a variable that

describes a characteristic of a population

or mathematical model

partially balanced incomplete block

design4 (PBIB): an incomplete block

design in which each block contains the

same number k, of different versions from

the t versions of the principal factor

particle22: a discrete element of material

regardless of its size

particle size17: the controlling lineal

dimension of an individual particle as

determined by analysis with sieves or

other suitable means

particle size distribution6: a description

of the size and frequency of particles in a

population

f x dx e dx

x

2

2

2

( )

2

1 ( ) σ

μ

σ π

− −

=

where, x = random variate

μ = mean of the distribution, and

σ= standard deviation of the

distribution

number distribution6: the distribution by

number of particles as a function of their

size

numerical aperture8: product of the

refractive index of the object space and the

sine of the semi-aperture of the cone of rays

entering the entrance pupil of the objective

lens from the object point

obscuration (optical concentration) 14:

percentage or fraction of light that is

attenuated due to extinction (scattering

and/or absorption) by the particles
150

◆ Glossary

particulate1: of or composed of distinct

particles

perforated plate6: a sieving medium

consisting of a plate with uniformly sized

holes in symmetrical arrangement

phase modulation3: modulation

produced by rapid oscillation of one

mirror of a scanning interferometer through

an amplitude which is smaller than the

shortest wavelength in the spectrum to

produce an interferogram which is, to a

good approximation, the first derivative

of the conventional interferogram

pitch11: the separation of two similar

structures, measured as the center to center

or edge to edge distance

pixel9: a “pixel,” or “picture element,”

corresponds to an individual sample in a

digital image which has been formed by

uniform sampling in both the horizontal

and vertical directions

plain weave6: the weave in which every

warp wire crosses alternately above and

below every weft wire and vice versa

plate thickness6: the thickness of the

plate sieving medium after perforation

Poisson distribution2: the distribution

which has as its probability function:

P(r) = e-μmr

 / r!

where, P(r) = probability of obtaining

exactly r occurrences of an event in one

unit, such as a unit of time or area,

μ = both mean and variance of the

distribution, and

e = base of natural logarithms

polydisperse1: describes a dispersed

system in which many particle sizes occur;

more specifically, a system in which less

than 90 % of the size distribution lies

within 5 % of the median size

population4: the totality of items or units

of material under consideration

powder1: dry collection of particulates

with a macroscopic consistency

powder volume6: the apparent volume

of a powder as measured under specific

conditions

practice2: a definitive procedure for

performing one or more specific operations

or functions that does not produce a test

result

precision2: the degree of agreement within

a set of observations or test results obtained

as directed in a test method

precision2: under conditions of between

laboratory precision, the multi-laboratory,

single-sample, single-operator-apparatusday (within-laboratory) precision of a

method; the precision of a set of

statistically independent test results all of

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

precision2: under conditions of singleoperator precision, the single-operatorlaboratory-sample-apparatus-day precision

of a method; 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

precision2: under conditions of withinlaboratory precision with multiple

operators, the multi-laboratory, singlelaboratory-sample, single-apparatus-day

(within-laboratory) precision of a method;

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
151

Glossary ◆

pressed density6: the apparent density of

a compact

primary particles1: smallest identifiable

subdivision in a particulate system; primary

particles may also be subunits of aggregates

principal component analysis3: a

mathematical procedure for resolving sets

of data into orthogonal components whose

linear combinations approximate the original

data to any desired degree of accuracy

probability function2: of a continuous

variate, the mathematical expression whose

definite integral gives the probability that a

variate will take value within the two limits

of integration

probability function2: of a discrete

variate, the mathematical expression which

gives the probability that a variate will take

a particular value

probability level probability level2: a general term that

reflects the stated proportion of times an

event is likely to occur

probability sample4: a sample of which

the sampling units have been selected by a

chance such that, at each step of selection,

a specified probability of selection can be

attached to each sampling unit available for

selection

process average2: for the items produced,

the true and unknown level of (1) the

fraction of nonconforming items, or

(2) a characteristic of the items as

determined by a specific test method

production lot2: that part of one

manufacturer’s production made from the

same nominal raw material under essentially

the same conditions and designed to meet

the same specifications

proficiency testing5: determination of

laboratory testing performance by means

of interlaboratory test comparisons

(ISO Guide 2)

quality5: totality of characteristics of an

entity that bear on its ability to satisfy

stated or implied needs

random cause2: one of the many factors

which contribute to variation, but which

are not feasible to detect and identify since

they are random in origin and usually small

in effect

random error7: result of a measurement

minus the mean that would result from

an infinite number of measurements of

the same measurand carried out under

repeatability conditions

random error of result4: a component of

the error which, in the course of a number

of test results for the same characteristic,

varies in an unpredictable way

random sampling2: the process of

selecting units for a sample of size n in

such a manner that all combinations of n

units under consideration have an equal or

ascertainable chance of being selected as

the sample

randomization4: the procedure used to allot

treatments at random to the experimental units

so as to provide a high degree of independence

in the contributions of experimental error to

estimates of treatment effects

randomized block experiment2: a kind

of experiment which compares the average

of k different treatments that appear in

random order in each of b blocks

randomized block factorial design4:

a factorial experiment run in a randomized

block design in which each block includes

a complete set of factorial combinations

receiver pan6: a pan which fits snugly

under a sieve to collect fine particles

reference material reference material11: 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 (see ISO Guide 30:1992)

reference standard11: a reference material,

generally of the highest metrological quality

available, from which measurements are

derived
152

◆ Glossary

refractive index3: the phase velocity of

radiant powder in a vacuum divided by the

phase velocity of the same radiant power

in a specified medium. When one medium

is a vacuum, n is the ratio of the sine of the

angle of incidence to the sine of the angle

of refraction

rejection number: in acceptance

sampling, the minimum number of

nonconforming items in a sample that

requires the conclusion that the lot does

not conform to the specification

relative refractive index (m) 14: complex

refractive index of a particle, relative to

that of the medium, m = Np/nm

repeatability6: a measure of precision

and expressed as a standard deviation; the

precision of successive measurements of

the same sample carried out subject to all

of the following conditions:

(a) the same measuring instrument;

(b) the same method of measurement;

(c) the same conditions of use;

(d) the same observer;

(e) the same location;

(f) repetition over a short period of time

repeatability (of results of

measurements)7: closeness of the

agreement between the results of successive

measurements of the same measurand

carried out under the same conditions of

measurement

NOTES

1. These conditions are called

repeatability conditions.

2. Repeatability conditions include:

X the same measurement procedure;

X the same observer;

X the same measuring instrument,

used under the same conditions

X the same location;

X repetition over a short period of

time.

3. Repeatability may be expressed

quantitatively in terms of the

dispersion characteristics of the

results.

replicate2: in experimenting or testing, one

of two or more runs with the same specified

experimental or test conditions and with

each experimental or test conditions being

established independently or all previous

runs; to repeat a run so as to produce a

replicate

reproducibility6: a measure of precision

and expressed as a standard deviation; the

precision of measurements of the same

sample where individual measurements are

carried out under changing conditions, such

as the following:

(a) the measuring instrument;

(b) the method of measurement;

(c) time;

(d) the conditions of use;

(e) the observer;

(f) the location

reproducibility (of results of

measurements) 7: closeness of the

agreement between the results of

measurements of the same measurand

carried out under changed conditions of

measurement.

NOTES

1. A valid statement of reproducibility

requires specification of the

conditions changed.

2. The changed conditions may include:

X principle of measurement;

X method of measurement;

X observer;

X measuring instrument;

X reference standard;

X location;

X conditions of use;

X time.
153

Glossary ◆

3. Repeatability may be expressed

quantitatively in terms of the

dispersion characteristics of the

results.

4. Results are here usually understood

to be corrected results.

residual error4: the difference between

the observed result and the predicted value

(estimated treatment response); observed

result minus predicted value

resolution6: a quantitative expression

of the ability of a measuring device to

distinguish meaningfully between closely

adjacent values of the quantity indicated

response variable4: the variable that

shows the observed results of an

experimental treatment

Reynolds number6: the dimensionless

number which defines the flow pattern of

a fluid surrounding a particle

riffler6: a device for dividing a stream

of particles into representative samples

robustness4: insensitivity of a statistical

test to departures from underlying

assumptions

ruggedness4: insensitivity of a test

method to departures from specified test

or environmental conditions

run2: in experimenting or testing, a single

performance or determination using one of

a combination of experimental or test

conditions

sample19: a portion of a material intended

to be representative of the whole

sample2: (1) a portion of a lot of

materials which is taken for testing or

for record purposes; (2) 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, final sample, final21: sample obtained or

prepared under the sampling plan for

possible subdivison into identical portions

for testing, reference or storage

sample, gross19: a single large sample

obtained by placing in a single container

two or more increments taken from a lot

sample, laboratory19: a subsample

selected from the properly prepared gross

sample of a lot for submission to one or

more laboratories for chemical analysis

sample, prepared19: a sample on which

comminution, division, blending or other

procedures have been performed to make

the sample ready for analysis

sample, spot20: a sample of specified

quantity taken from a specified place in the

bulk material, gross sample or laboratory

sample, or at a specified place and time in

a stream and representative of its own local

environment

sample, test19: a subsample selected from

the properly prepared laboratory sample

which has a suitable sample weight or

volume for one or more determinations

by chemical analysis

sample, umpire19: a laboratory sample

submitted to a laboratory of recognized

capability chosen to resolve a difference

sample division21: the process in sample

preparation where the mass of the sample is

decreased by partition or extraction without

change in particle size

sampling dimension21: a dimensionless

number characterizing the distribution of

sampling increments in space relative to

the powder; the volume of each sampling

increment is not taken into consideration

sampling fraction4, f: the ratio f of the

number of sampling units selected for the

sample to the number of sampling units

available

sampling increment21: a quantity of

material taken as sample from a larger

quantity
154

◆ Glossary

sampling plan2: a procedure for obtaining

a sample

sampling plan result2: the number

obtained for use in judging the acceptability

of a lot when applying a sampling plan

sampling sequence21: the sequence of

sampling partitions and extractions and

necessary sample preparation steps that

result in the reduction of the primary

sample to representative test portion

sampling unit2: an identifiable, discrete

unit or subunit of material that could be

taken as part of a sample

sampling with replacement4: a procedure

used with some probability sampling plans

in which a selected unit is replaced after any

step in selection so that this sampling unit

is available for selection again at the next

step of selection, or at any other succeeding

step of the sample selection procedure

scattering14: general term describing the

change in propagation of light at the

interface of two media

scattering pattern14: angular or spatial

pattern of light intensities originating from

scattering, or the related energy values

taking into account the sensitivity and the

geometry of the detector elements

screen6: a sieve or sieving medium used

for separating particles in a manufacturing

process

screen mesh18: the average number of

openings per linear inch of material

screening design4: a balanced design,

requiring relatively minimal amounts of

experimentation, to evaluate the lower

order effects of a relatively large number

of factors in terms of contributions to

variability or in terms of estimates of

parameters for a model

sensitivity criterion2: a quantitative

measure of the relative merit of two test

methods, that combines the estimated

variance of each method with the ability

of the method to measure differences in the

property of interest

sensitivity ratio2, SR: in comparing two

test methods, the ratio of the sensitivities of

the test methods with the large sensitivity

in the numerator

shading6: the combined effect of uneven

illumination, optical path and camera

characteristics, which causes a variation

in the image background light intensity

distribution of an image

sieve16: a plate, sheet, or woven wire

cloth, or other device, with regularly

spaced apertures of uniform size, mounted

in a suitable frame or holder, for use in

separating material according to size.

The terms sieve and screen can be used

interchangeably

sieve analysis: (1) the act of obtaining

a particle size distribution using sieves16;

(2) the particle size distribution of a

particulate or granular solid or sample

thereof, when determined by passage

through and retention on a graded set

of sieves18

sieve classification17: the separation of

powder into particle size ranges by the use

of a series of graded sieves

sieve fraction17: the portion of a powder

sample which passes through a standard

sieve of specified number and is retained

by some finer sieve of specified number

sieve mesh number6: the number of

apertures occurring in the surface of a sieve

per linear inch

sieve size of a particle6: the smallest sieve

aperture through which a particle will pass

if presented in the most favorable attitude

sieving6: the operation of determining the

size distribution of particles in a powder

by using a series of sieves

sieving diameter6: the diameter of the

aperture of the mesh of the sieve which

just allows the passage of the particle

sieving medium6: a network of perforated

sheets used for separating particles

according to their size
155

Glossary ◆

sifter6: a type of screening machine having

a rotary motion substantially in the plane of

the screening surface, used for the screening

of powders

significance level significance level2, α: the stated upper

limit for the probability of a decision being

made that a hypothesis about the value of a

parameter is false when in fact it is true

single sampling2: in acceptance sampling,

a sampling plan for which the decision to

accept or reject a lot is based on a single

sample

size fraction6: the portion composed of

particles between two given size limits,

expressed in terms of mass, volume surface

area or numerical frequency

smoothing, relaxation6: the processing

of a digitized image whereby each pixel is

considered in turn and its value modified by

reference to its neighbors

specification2: 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

specimen18: a specific portion of a material

or laboratory sample upon which a test is

performed or which is selected for that

purpose

spherical aberration10: a lens defect

in which the image forming rays passing

through the outer zones of the lens focus

at a distance from the principal plane,

different from that of the rays passing

through the center of the lens

sphericity shape factor6: the ratio of

the surface area of the particle to that

of a sphere of the same volume

spinning riffler6: a rotary sample divider

(description in Chapter 2 on Powder

Sampling)

stable suspension1: a suspension that

has sufficient kinetic stability to prevent

the occurrence of significant aggregation

as measured over a relevant time frame;

stability may be ascertained by suitable

experimental means, such as particle size,

turbidity, or sedimentation measurements

stage graticule; reticle6: a reference scale

used to calibrate an eyepiece graticule

standard deviation2, 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 error of calibration3 (SEC):

a measure of the calibration accuracy

determined by the following equation:

1/ 2

1

2

1

1

















− − = ∑=

n

i i e n p

SEC

where:

n = the number of observations in the

calibration data set,

p = the number of independent

variables in the calibration, and

ei

 = the difference between a measured

value of a property and its accepted value

standard error of performance3 (SEP):

a measure of accuracy determined by the

following equation:

1/ 2

1

2 ( ) 1

1 



 



 − − = ∑=

− n

i i e e n SEP

where:

n = the number of observations for

which the accuracy is determined,

ei

 = the difference between a measured

value of a property and its accepted value,

and

‘ = is the mean of all ei

Standard Reference Material Standard Reference Material® (SRM®):

Certified reference material (CRM) issued

by NIST. A SRM is a well-characterized

material produced in quantity to improve

measurement science. It is certified for

specific chemical or physical properties,

and is issued by NIST with a certificate that

reports the results of the characterization

and indicates the intended use of the

material.
156

◆ Glossary

standardization5: correlation of an

instrument response to a standard of known

accuracy

state of statistical control state of statistical control2: a condition in

which a process, including a measurement

process, is subject only to random variation

statistic2: a quantity that is calculated from

observation on a sample and that estimates

a parameter of a population

steric stabilization1: mechanism in which

particle aggregation is inhibited by the

presence of an adsorbed polymer layer that

is firmly anchored to the particle surface so

as not to desorb during particle collisions in

general, a steric stabilizer has one portion of

its structure that exhibits low solubility in

the dispersion medium and/or high affinity

for the particle surface, and the other

portion is soluble in the medium

stereology6: the quantitative study of

three dimensional structures from two

dimensional sections or projections of them

Stokes diameter13: equivalent diameter

of the spherical particle that has the same

density and terminal settling velocity as

the real particle in the same liquid under

creeping (laminar) flow conditions

Stokes’ law6: the equation which

determines the free-falling velocity attained

by a particle under viscous flow conditions

and allows calculation of particle size

Stokes-Einstein equation6: the equation

which relates the pace of Brownian motion

to the diameter of the particle in motion

subsieve fraction17: particle all of which

will pass through a 44 μm (no. 325)

standard sieve

sum of squares2: in analysis of variance,

a contraction of the expression “sum of the

squared deviations from the appropriate

average(s)” where the average(s) of interest

may be the average(s) of a specific subset(s)

of data or of the entire set of data

surfactant6: a substance which reduces

surface tension

suspension1: a liquid in which solid

particles are dispersed

systematic error7: mean that would result

from an infinite number of measurements

of the same measurand carried out under

repeatability conditions minus the value of

the measurand

systematic error of result4: a component

of the error, which in the course of a

number of test results for the same

characteristic, remains constant or varies

in a predictable way

systematic sampling2: the process of

selecting units in a sample in accordance

with a specific order or location in time

or space, or both

tap density17: the mass per unit volume

of a powder that has been placed in a

specified container and has been settled

under specified conditions by mechanical

action

terminal settling velocity13: velocity of

a particle through a still liquid at which

the force due to gravity on the particle is

balanced by the drag exerted by the liquid

on the particle

test determination4: (1) the process

of deriving from one or more test

observations (observed values) the presence

or absence of an attribute or the value of a

characteristic or dimension of a single test

specimen, or (2) the attribute (presence or

absence) or value derived from the process

test method2: 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 portion20: the portion withdrawn

from the test sample and entirely used in

the test or observation
157

Glossary ◆

test result2: a value obtained by applying

a given test method, expressed as a single

determination or a specified combination

of a number of determinations

test sample20: the sample prepared from

the laboratory sample from which the test

portions are withdrawn

test sieve: (1) a sieve that meets all of

the minimum requirements of ASTM

Committee E-1116; (2) a frame supporting

a sieving medium intended for particle size

analysis, and conforming to a standard

specification6

test sieving6: the use of test sieves in a

prescribed manner to perform separation

of particles into size classes

threshold6: a gray level value which

separates regions of interest from

background in a gray level image

throughput3: the vector product of the

area and solid angle of a beam at its focus

and the square of the refractive index of the

medium in which the beam is focussed

tolerance limits (specification limits)4:

limits that define the conformance

boundaries for an individual unit of

manufacturing or service operation

tolerance specification4: the total

allowable variation around a level or state

(upper limit minus lower limit), or the

maximum acceptable excursion of a

characteristic

tolerances2: in mathematics, prescribed

limits of variation for specified properties

of a particular material based on observed

values obtained by specified test methods

and on samples that are representative of

the material

traceability5: the property of a result

of a measurement whereby it can be

related to appropriate standards, generally

international or national standards, through

an unbroken chain of comparisons (ISO

Guide 2); 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)

treatment4: a combination of the levels

(versions) of each of the factors assigned

to an experimental unit, synonym

treatment combination

treatment combination2: in

experimenting, one set of experimental

conditions

true density; true particle density6:

the mass of particle divided by its volume,

excluding open pores and closed pores

trueness4: the closeness of agreement

between the population mean of the

measurements or test results and the

accepted reference value

t-test2: a test of statistical significance based

on the use of Student’s t-distribution and

used to compare two sample averages or a

sample average and a hypothetical value

Type A operating characteristic curve2:

an operating characteristic curve which

describes the operation of a sampling plan,

where the size of the lot being sampled is

taken into consideration

Type B operating characteristic curve2:

an operating characteristic curve which

describes the operation of a sampling plan,

where items are drawn at random from a

theoretically infinite process

ultrasonic probe6: a point source of

ultrasonic vibration

ultrasonication1: application of highenergy, high-frequency sonic energy to a

suspension in order to disperse aggregates

uncertainity4: 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
158

◆ Glossary

underflow6: that portion of the feed

material which has passed through a

screening surface

undersize13: portion of material

composed of particles that are smaller

than some specific size

undersize control screen6: a screen

used for the removal of undesirable fines

from a material

universe (population)4: the totality of

the set of items, units, or measurements,

etc., real or conceptual, that is under

consideration

unstable suspension1: a suspension

that lacks kinetic stability as measured

over a relevant time frame; a highly unstable

suspension is one that is subject to rapid

aggregation

upper control limit4 (UCL): control

limit for points above the central line

upper tolerance limit4 (UTL) (upper

specification limit): a tolerance limit

applicable to the upper conformance

boundary of an individual unit of a

manufacturing or service operation

validation19: proof, with reference

materials or materials traceable to them,

that a method is acceptable for all elements

in its scope

variable2: a quantity to which any of the

values in a given set may be assigned

variables data2: measurements which

vary and may take any of a specified set

of numerical values

variance2 (s2): of a sample, a measure

of the dispersion of variates observed

in a sample expressed as a function of

the squared deviations from the sample

average

variance2 (σ2): of a population, a measure

of the dispersion of members of the

population expressed as a function of the

sum of the squared deviations from the

population mean

variate2: a measured value that includes

a random error of measurement; a variable

with which a probability distribution is

associated

verification5: checking or testing to

assure conformance with the specification

vibrating screen6: a screen oscillated by

mechanical, electrical or ultrasonic means

volume distribution6: the distribution by

volume of particles as a function of their size

warp6: all wires running lengthwise of

the cloth as woven

weave6: the pattern formed when warp

and weft wires cross each other

wedge wire screen6 (profile wire deck):

a screening surface comprising wires

of wedge shaped cross-section spaced

from each other at a fixed dimension.

The underflow thus passes through an

aperture of increasing cross-section

weft; shoot6: all wires running crosswise

of the cloth as woven

well-dispersed1: a state that describes

a stable suspension in which the

minimum particle size has been achieved;

operationally defined by the observance

of a constant (minimum) particle size

distribution during the dispersion process

wet milling18: the grinding of materials

with sufficient liquid to form a slurry

wet sieving6: the sieving of particles

with the aid of a liquid

wire diameter6: sieving, the diameter of

the wire in the woven cloth

wire screen6: a screen produced by a wire

weaving process (or by pressure welding)

of two layers of parallel wires set at right

angles to form apertures of nominally

uniform size

woven wire cloth6: a sieving medium of

wires which cross each other to form the

apertures
159

Glossary ◆

References

1. V.A. Hackley, Nomenclature for Ceramic Particle Dispersion Systems,

National Institute of Standards and Technology, Gaithersburg, MD (1999).

2. ASTM D123-96a Standard Terminology relating to Textiles, American

Society for Testing and Materials, West Conshocken, PA (1996).

3. ASTM E131-98 Standard Terminology Relating to Molecular

Spectroscopy, American Society for Testing and Materials, West

Conshocken, PA (1998).

4. ASTM 456-96 Standard Terminology Relating to Quality and Statistics,

American Society for Testing and Materials, West Conshocken, PA

(1996).

5. ASTM E1887-97 Standard Terminology Relating to Conformity

Assessment, American Society for Testing and Materials, West

Conshocken, PA (1997).

6. BS 2955: 1993 (98) Glossary of Terms Relating to Particle Technology,

British Standards Institute, London (1998).

7. NIST Technical Note 1297, Guidelines for Evaluating and Expressing

the Uncertainty of NIST Measurement Results, National Institute of

Standards and Technology, Gaithersburg, MD (1994).

8. BS3406: Part 4; British Standard Methods for Determination of Particle

Size Distribution, Optical Microscope Method, British Standards Institute,

London (1993).

9. ISO/CD 13322 Particle Size Analysis – Image Analysis Methods,

International Organization for Standardization, Geneva (1999).

10. ASTM E175-82 (Reapproved 1995) Standard Terminology of

Microscopy, American Society for Testing and Materials, West

Conshocken, PA (1995).

11. ASTM E766-98 Standard Practice for Calibrating the Magnification of

a Scanning Electron Microscope, American Society for Testing and

Materials, West Conshocken, PA (1998).

12. BS 3406: Part 6; British Standard Methods for Determination of Particle

Size Distribution, Recommendations for Centrifugal Liquid Sedimentation

Methods for Powders and Suspensions, British Standards Institute, London

(1985).

References ◆
160

◆ Glossary

13. ISO/DIS 13317-1 Determination of Particle Size Distribution by

Gravitational Liquid Sedimentation Methods – Part 1: General principles

and guidelines, International Organization for Standardization, Geneva

(2000).

14. ISO/FDIS 13320-1: 1999 Particle Size Analysis – Laser Diffraction

Methods – Part 1: General principles, International Organization for

Standardization, Geneva (1999).

15. ASTM B822-97 Standard Test Method for Particle Size Distribution of

Metal Powders and Related Compounds by Light Scattering, American

Society for Testing and Materials, West Conshocken, PA (1997).

16. ASTM E1638-94 Standard Terminology Relating to Sieve Analysis,

American Society for Testing and Materials, West Conshocken, PA

(1994).

17. ASTM B243-99 Standard Terminology of Powder Metallurgy, American

Society for Testing and Materials, West Conshocken, PA (1999).

18. ASTM C1145-98 Standard Terminology of Advanced Ceramics,

American Society for Testing and Materials, West Conshocken, PA

(1998).

19. ASTM E135-98 Standard Terminology Relating to Analytical Chemistry

for Metals, Ores, and Related Materials, American Society for Testing and

Materials, West Conshocken, PA (1998).

20. BS 3406: Part 1; British Standard Methods for Determination of Particle

Size Distribution, Guide to Powder Sampling, British Standards Institute,

London (1985).

21. ISO/WD 14488 Particle Size Analysis: Sample Preparation-Sample

Splitting, International Organization for Standardization, Geneva (1999).

22. ISO/DIS 13317-1 Determination of particle size Distributions by

Gravitational Liquid Sedimentation Methods, Part1: General principles and

guidelines, International Organization for Standardization, Geneva (1999).
161

Annex 1 ◆

ANNEX 1

SOME FORMULAE PERTAINING TO PARTICLE SIZE

REPRESENTATION

1. Mean Particle Size

For a powder system comprising of spherical particles, with a fractional

distribution function fN(a), the total number of particles in the system (N) are

thus defined by1:

∫ =

max

min

( )

x

x

N f N x dx

The length of all the particles in the system can be summed and represented

as:

∫ =

max

min

( )

x

x

N L xf x dx

where, x is the length of each particle.

Thus, the mean length, xL = L/N

N

xf x dx

x

x

x

N

L

∫ = −

max

min

( )

Similarly, a mean size can be defined in terms of the surface area and volume,

and so mean surface area, xA can be calculated as:

N

x f x dx

x

x

x

N

A

∫ = −

max

min

2 ( )

and the mean volume, xV as:

3

max

min

3 ( )

N

x f x dx

x

x

x

N

V

∫ = −
162

◆ Annex1

2. Size Distribution Functions

Size distributions of powders can be represented by various mathematical

functions, including1:

(i) Log-Normal Distribution: Typically observed for granulated powders

produced by spray-drying, milling of fine grained minerals and calcined

aggregates:

where, f(ln x) is the frequency function in a number, volume or mass basis,

xg is the geometric mean size, and,

σg is the geometric standard deviation indicating the range of sizes.

(ii) Rosin-Rammler Distribution: Typically observed for some milled, coarsegrained materials:

where, FM(x) is the cumulative mass fraction finer than size x,

xRR is the size parameter for which FM(x) =0.37, and,

n-1 is an index of the dispersion of size.

(iii) Gaudin-Schuhmann (GS) Distribution: Typically observed for coarse

industrial minerals produced by processes of industrial crushing and

grinding:

where, xmax is the apparent maximum particle size obtained by

extrapolation, and n is the constant indicating the range of sizes present in

the powder.

( )2

2

2 ln

ln ln

exp 2 ln

1 (ln )

g

i g

g

x x

f x

π σ σ





 



 −

= −

n

RR

M x

x F x 













 ( ) =1− exp−

n

M x

x F x 













 =

max

( )
163

Annex 1 ◆

3. Relationship between Fractional and

Cumulative Distributions

Fractional distributions can be calculated from the obtained cumulative

ln xNL = ln xgN + 0.5 ln2 σg

ln xNS = ln xgN + 1.0 ln2 σg

ln xNV = ln xgN + 1.5 ln2 σg

ln xNM = ln xgN + 2.0 ln2 σg

distribution, or vice versa, from the relationship:

where, f(x) is the function describing the fractional distribution, and F(x) is

the cumulative fraction finer than x.

In this case, the distributions can be on a number or mass basis.

Conversion between Different Basis of Representation

For a powder system following a log-normal distribution of particle size, the

following equations can be used for conversion from one distribution to

another2:

dx

dF x f x ( ) ( ) =

where,

xgN is the geometric mean (median) of the number distribution,

σg is the geometric standard deviation,

xNL is the length mean diameter of the number distribution of the particles,

xNS is the surface area mean diameter of the number distribution of the

particles,

xNV is the volume mean diameter of the number distribution of the

particles, and

xNM is the moment mean diameter of the number distribution of the

particles.
164

◆ Annex1

References

1. J. S. Reed, Introduction to the Principles of Ceramic Processing, John Wiley

and Sons, New York (1988).

2. T. Allen, Particle Size Measurement, 4th ed., Chapman & Hall, London

(1993).
ANNEX 2

NIST RM/SRMs RELATED TO PARTICLE SIZE

CHARACTERIZATION

* Reference Material (RM) # RM/SRM under development 

Standard SRM No. Certified Value Attribute

Silicon Nitride

Powder

659 0.3 μm to 3.0 μm Particle Size

Distribution

Silicon Nitride

Powder

1899

1900

10.67 m2/g

2.79 m2

/g

Surface Area

Zirconia

Powder

1978 0.3 μm to 2.2 μm Particle Size

Distribution

Zirconia Thermal

Spray Powder

1982 20 μm to 90 μm Particle Size

Distribution

Alumina and Silicaalumina Powder

8571*

8572*

to be replaced

158 m2

/g

291 m2

/g

Surface Area

Tungsten Carbide

Cobalt Powder

1984/1985# 10 μm to 60 μm Particle Size

Distribution

Zeolite Powder 2850#, 2851#,

2853#

0.3 μm to 3.0 μm Particle Size

Distribution

Silica Sand 8010* 40 μm to 400 μm Particle Size

Distribution

Glass Beads 1021#, 1003, 1004#,

1017, 1018, 1019

1 μm to 2450 μm Particle Size

Distribution

Etched Si chip 2090 0.2 μm to 3000 μm SEM Magnification

Graphitized Rayon

Fibers

2069 b 2 mm to 4mm SEM Performance

Sales and information about SRMs can be obtained from:

Standard Reference Materials Program

National Institute of Standards and Technology

100 Bureau Drive

Room 204, Building 202

Gaithersburg, MD 20899-2322

USA

Tel: (301) 975-6776

Fax: (301) 948-3730

email: srminfo@nist.gov

http://ts.nist.gov/srm

165
166

◆ Annex 2
January 2001