This article begins with a pertinent story from the writers’ tenure as a mediocre graduate student in Physics. A fellow student and I were working together on a homework set. We had spent the majority of the day on a particularly difficult problem that resulted in a lengthy equation expressed in terms of assorted variables. I turned to the back of the book to compare our result to that of the author’s and was astonished by the dissimilarity. I showed the ‘correct’ solution to my friend, a far better and more confident student than I. After looking at it he asked, “Did we apply the appropriate theorems?” I affirmed that we had. Next, he asked, “Did we make any mathematical errors?” I was confident that we had not. “Then,” he proclaimed, “our solution is correct we just are expressing it in different terms.”
I recall this incident when I encounter a debate over the ‘correctness’ of the results obtained for particle size measurements by two or more different analytical techniques. Provided that the instruments used are capable of producing highquality data, the pertinent questions, then, are, “was the sample properly prepared and properly presented to the instrument,” and “were the analytical parameters applied correctly”. If the answer to both is “yes,” then both analytical results probably are equally correct; they are just expressed in different terms.
Different Measurement Techniques and Different Results
If what has been stated thus far has not raised any questions, you probably don’t need to read the rest of this article. This article is intended to resolve questions users often have concerning comparisons of particle sizing results by different techniques. The techniques referenced are sieving, sedimentation, imaging (including microscopy and machine vision), electrozone sensing, and light scattering. The determination of particle size on the same sample by all of these techniques and others not mentioned will, in the majority of cases, yield different results for mean size, modal size, and quantity distribution by size.
Factors that Affect Particle Size Data
The following questions may seem simple, but take a close look at the answers. They often contain conditions and constraints that, if not abided by, will affect the accuracy of reported size data.
The Definition of a Particle
What defines or characterizes a “particle” and what limitations does the definition impose on particle sizing?
McGraw Hill’s Dictionary of Scientific and Technical Terms (third edition) defines a particle as “any relatively small subdivision of matter, ranging in diameter from a few angstroms to a few millimetres”. Particles that one wishes to measure for size may be composed of organic or inorganic molecules; they may be molecularly homogeneous or inhomogeneous; they may be in solid or liquid state; they may be isotropic or anisotropic; they may be of any shape; and may be suspended in various media. Molecular structure, homogeneity, state, isotropy, shape, and suspension medium associated with the particles under test all may cause different size measurement devices to respond differently to the same particle. When comparing the results from two different types of sizing instruments, one should know if any characteristic of the particle other than size, or any characteristic of the sample presentation could affect the reported size value. Error from these sources is associated with nonideal or even inappropriate application of the measuring instrument.
Definition of Particle Size
Is there a single, standard definition for “particle size” that can be applied to any particle?
Spherical Particles
There are many definitions, but none has been adopted as a comprehensive standard. To be able to apply a single rule to particle size determinations, and that rule enabling all techniques of sizing to agree is implausible for several reasons. If such a definition is based on geometry it must apply to both regular and irregular shapes and to the techniques used to obtain the measurement. The simplest case in respect to geometry is that of a sphere, and visual inspection (microscopy or image analysis) is the most straightforward measurement technique. When examining a sphere, its perimeter, projected crosssectional area, surface area, and volume can be described unambiguously by one linear dimension the diameter of the projected crosssection. Furthermore, the projected crosssectional diameter remains constant regardless of the angle of view; therefore a sphere is isotropic in a geometrical sense. No other regular or irregular shape projects the same crosssection at all angles of view, therefore neither surface area nor volume can be inferred from the crosssection of a non spherical particle.
Irregular Particles
The fact that an irregular particle can present a different crosssection depending on orientation is only one of the measurement problems. Another is that an irregularly shaped crosssection has different “diameters” depending on where the chord is drawn. To deal with these difficulties, definitions of ‘statistical geometric diameters’ were established. They are statistical because they have significance only when averaged over a large number of measurements. An example is Martin’s diameter, which is the length of the chord that divides the crosssectional shape into two equal areas. Another is Feret’s diameter, which is the distance between two parallel lines tangent to the projected crosssection.
Another approach was to extract a linear size dimension from the projected area of the particle; this is called the ‘area diameter’ and expresses particle size as the diameter of a circle that has the same projected area as the particle. Another was to determine the perimeter of the projected crosssection and assign to the particle the diameter of a circle having the same perimeter. If the volume of an irregular particle could be determined, the diameter of the particle was defined as the diameter of a sphere having the same volume  the volume diameter.
Different Sizes for the Same Particle
Obviously, several different ‘sizes’ could be obtained from a single, nonspherical particle due to various orientations or methods of measurement. Figure 1 is a series of illustrations of the particle; each illustration shows the particle rotated with a different face parallel to the page. Obviously, these are only a few of the possible orientations of the particle, and each orientation projects a different crosssection.
Figure 1. This example particle has seven plane facets, A through G. The particle is shown in seven views, one with each facet parallel to the page. These views represent only a few of the possible orientations and projected crosssections of the particle. In each, Feret's diameter, Martin's diameter, area diameter, and perimeter diameter differ. Only the volume diameter is constant.
Shortcomings of Statistical Models
As stated previously, the definitions of particle size are statistical geometric diameters and are applied to large numbers of particles where it is assumed that particles present themselves in random orientations to the measurement system. However, this was found not to be case. A good review of the shortcomings of using these definitions in actual practice is found in Chapter 3 of Mercer’s “Aerosol Technology in Hazard Evaluation”.
Without belaboring the subject, it should suffice for the present to state simply that the definition of ‘particle size’ depends on the technique of measurement. Information supporting this is given subsequently.
What do Particle Size Analyzers Actually Measure
So, what do particlesizing instruments actually measure?
Essentially all automated determinations of particle size are obtained indirectly from direct measurements of some parameter other than the complete geometry. These parameters are associated with a physical phenomenon in which the particle is involved. The direct measurement may be of some characteristic of the reaction of the particle to some action, or the measurement may be of the reaction of something that has interacted with the particle. An example of the former is the particle’s sedimentation velocity in a fluid, and of the latter is the pattern of light scattered by the particle. The parameter being directly measured is related to particle geometry by some law, theory or model describing the physical phenomenon.
Table 1 lists classes of particle characteristics related to ‘size’ and to a particular, measurable behavior that varies as a function of particle size. The table also lists other variables that affect the reported size.
Table 1. Characteristics associated with particles and related to size.




Geometrical

Area or perimeter of crosssection

Shape combined with orientation

Microscopy, image recognition, sieving

Displacement volume

Porosity, wettability

Electrozone sensing

Some linear dimension such as the diameter or statistical geometric diameter

Shape, orientation

Microscopy, image recognition

Hydrodynamic / Aerodynamic

Settling velocity
Drag (resistance to motion)

Shape and density combined with properties of the surrounding fluid
Reynolds number; molecular homogeneity

Elutriation and sedimentation

Optical

Light scattering characteristics

Refractive index, isotropy, shape + orientation, and surface detail of particle.
Refractive index of medium. Wavelength and polarity of incident light

Static light scattering (Mie and Fraunhofer diffraction)

A comprehensive list of particle sizing techniques is found at Duke Scientific Corporation's web site.
Equivalent Spherical Size
What is meant by the ‘size’ of a particle if something other than a geometrical attribute is being measured?
The behavior of a particle or the characteristics of some other entity after interacting with a particle are related to the particle’s geometry, but often in a very complex manner. A common way to describe the geometry of an irregular or regular particle is to compare the particle under test to a sphere of the same material and being of a size that causes it to exhibit the same behavior as the test particle. As an example, when sieving to determine size, the largest particles to pass through the screen are described in terms of the diameter of spheres that will just pass through the same apertures. This is the concept of ‘equivalent spherical size’ and is employed in almost all particlesizing techniques. An equivalent size may be related to any regular geometry, but the sphere is most often used due to its unique geometrical attributes as previously cited. Another reason for using the sphere as the reference shape in some cases is because the theory describing the behavior of a particle under certain conditions, or describing the results of an interaction with a particle has only been solved rigorously for spherical particles.
Consequently, when reference is made to particles of X mm equivalent spherical diameter, it means that the particles behave (pass through the same size aperture, settle at the same velocity, scatter light with the same intensity at the same angles, or displace the same volume of liquid) as do X mm spherical particles that otherwise have the same behaviorcontrolling properties as does the sample material.
How Important is the Equivalent Spherical Size
What has been said about particle sizing above was stated more concisely in the late 1940’s by Heywood, followed later by Beddow. Heywood characterized the results of particle sizing as being ‘somewhat dependent on the physical principles employed and the assumptions or conventions involved.’ He summarized current techniques as being ‘only able to measure and classify particles if the particles under test were imagined as spheres having some property equivalent to the test material.’ Essentially, things are still the same today, and we can add laser diffraction to the list of ‘physical principles employed.’ In the case of laser diffraction, the particles under test are imagined as spheres that scatter light in the same manner as do the particles of the test material.
Sieve Sizes and Mie Sizes
Since particle sizing techniques apply the common definition of a ‘spherical equivalency,’ can I expect to get the same measurement results for the same sample from each technique that applies this definition?
If the particles under test are of irregular shape, then the most probable outcome is that the results will differ. Why? Because two particles that, for example, settle with the same velocity (therefore, are the same Stokes size) can scatter light differently (therefore, have different Mie sizes). Two particles that pass through the same screen mesh have the same sieve size, but may have very different volumetric size as determined by electrozone sensing.
Assumption Made When Determining Particle Size
If all particles in the sample under test are spherical, then all particle size measuring techniques, in theory, should yield essentially the same results, provided the instrument is applied appropriately. But, being spherical in most cases is only one requirement of the theoretical model. Table 2 lists other assumptions about the particle system that also must be correct. If all of these requirements are satisfied and the value of all other parameters required by the model are known or controlled, then one can expect the results of measurements of a system of spherical particles by different measurement techniques to agree provided that the fundamental measurement data are of comparable quality (precision, accuracy, sensitivity, resolution, signaltonoise ratio, etc.)
Table 2. Size measurement techniques and the assumptions of the theoretical model.




Image recognition

Plane geometry

Particles are spherical, cubical or of other regular solid geometry

Known relationship between particle size and image size

Electrozone sensing

Ohm's law expressed in terms of electrolyte resistivity, crosssectional area of aperture, and volume of displaced electrolyte (particle volume)

Spherical particles that are much less conductive than the electrolyte

Size of aperture through which particles pass

Sedimentation

Stoke’s Law for the settling velocity of a spherical particle in a fluid medium

Spherical particle, laminar flow of fluid around settling particle, all particles in system of same density

Particle density, density and viscosity of medium at analysis temperature, gravity

Static light scattering

Mie theory of light scattering by a spherical particle (includes Fraunhofer theory)

Spherical particle, optically isotropic, no multiple scattering, monochromatic light, coherent light, plane wave

Refractive index of particle, refractive index of medium, wavelength of light, size and position of scattering pattern projected onto detector

The Effects of NonSpherical Particles
In Table 2, all theoretical models assume spherical particles. What are the effects if the sample particles are not spherical?
Measuring some attribute arising from a nonspherical particle, then reducing the data from those measurements using a spherical model will introduce error—that’s understandable. The magnitude of error depends on the technique and the data reduction method and, of course, the actual shape of the particle compared to the model. As has been mentioned, the mathematical complexities introduced by nonspherical geometry usually prevents models from being derived for other shapes, and these same complexities prevent predicting in exactly what manner the error will affect the reported values. So, depending on the sizing technique and the shape of the particles, the effect may be negligible or may be severe, and often cannot be predicted reliably.
Other Sources of Error
What are the effects on final results if the particles are spherical, but there is some other deviation from the assumptions or conditions of the theoretical model?
Assume that the sedimentation technique is to be used to determine particle size. In this case, the particles’ sedimentation velocities are measured and Stokes’ law is applied. In the application of Stokes’ law, it is assumed that the particle density, liquid density, and liquid viscosity are accurately known. If the value of any of these parameters contains error, say +/ 5%, then the error will affect the calculation of particle size in a predictable manner easily calculable by Stokes law.
Refractive Index and The Mie Theory
A more complex result of deviating from the assumptions of the model occurs in scattered light measurements reduced by Mie theory. A condition of using this theory is that the refractive index of the sample material be known. If the refractive index has an uncertainty of +/5%, the relationship of this error to the error exhibited in the reported size distribution is considerably more complex. It is unlikely that effects of this error can accurately be predicted, and may only be realized by reducing the experimental data multiple times using a number of values of refractive index covering the range of the uncertainty.
Methods to Minimise Errors
To varying extents, instrument design assures that conditions of the model are met. For example, the requirements for monochromatic and coherent light of a known wavelength for the Mie model in static light scattering instruments is satisfied by the design of the instrument. Determining the density of the sample material, or properly dispersing the sample are examples of experimental conditions that are under the control of the user. In other cases, there are preanalysis tests to determine if conditions are within the acceptable range of the model. Examples of this are calculating the Reynolds number of the settling particle to assure that it is in the laminar flow domain, or to monitor light extinction as sample concentration is varied to assure that there is neglibible multiple scattering when using the light scattering technique.
The Impact if Errors
In summary, the magnitude of error introduced by deviations from the assumptions of the model or by erroneous parameters depends on the magnitude of the error or degree of deviation, and the ‘sensitivity’ of the data reduction model to these variations. The effect on the reported data ranges from negligible to catastrophic.
Choosing the Right Particle Sizing Instrument for the Application
There is no single sizing technique that is superior in all applications. The instrument selected for an application must be suitable for the material to be measured and for the environment in which the instrument is to be used. It also must provide data to meet the specific needs of the application. This may mean fast, repeatable analyses, or it may mean highresolution and very accurate results. The determination of particle size distribution seldom is the ultimate objective; determining how particle size affects something else is usually the reason for the measurement. In this regard, the characteristic actually being measured and related to size may be more important than size, but size is the best way or the conventional way to express the characteristic. For example, sediment from an extinct river delta may be analyzed for size. But, what actually may be of interest is the deposition mechanism. In this case, size is a way of expressing sedimentation velocity and the sedimentation size (Stokes size) may be of more value than the size determined by light scattering (Mie size).
Literally, the bottom line is this: Different techniques are likely to produce different size results for the same particle, and all of them are likely to be correct. The best instrument for the application (the best size definition) may be the one that most closely relates particle size to the application of the particles.
This information has been sourced, reviewed and adapted from materials provided by Micromeritics Instrument Corporation.
For more information on this source, please visit Micromeritics Instrument Corporation.