**The short answer to the titular question is this: radius or diameter. But what type of radius or diameter, and what to do with non-spherical particles? This article presents some ideas for individuals who have never undertaken particle sizing.**

## Length vs. Mass

If you are a particle technologist, then the only answer is the length. However, at a recent national biochemistry meeting, a group of protein chemists regularly referred to the molecular weight of a globular protein, a relative molar mass, as the protein’s “size”. Overhearing this, other groups were surprised as it is usual to refer to the size of a globular protein in nanometers. Indeed, apart from these protein chemists, the majority of all chemists refer to particle size in terms of a length in nanometers or microns.

## To Be or Not To Be a Sphere

Of all the three-dimensional particles, the sphere is by far the most significant in terms of particle sizing. This is not because most particles are spheres, although many come close to a spherical form (spherical micelles, liposomes, oil-in-water, and water-in-oil emulsions, un-aggregated latex, monoclonal antibodies, etc.). Moreover, many more particles are nearly spherical, particularly if measurements are averaged over rotationally diffusing particles.^{1}

Over the timescales of many types of measurements, a rotationally averaged size is being quantified, and therefore a sphere typically represents an acceptable approximation. Additionally, if highly irregular, long particles fragment as a result of abrasion, they fragment into shorter particles, becoming more globular rather than less.

For example, water and wind exposure forms smooth, globular rocks out of the uneven shards of volcanic debris. Likewise, jagged and/or irregular particles round off as they are stirred or mixed during their approach to final product status.

In accordance with the first law of thermodynamics, a liquid body encountering no external forces will constitute a sphere as a way of minimizing its surface area for a given volume of material. In this way, ignoring external forces, liquid droplets form spheres. This accounts for why even cooling planets formed into first order, spherical objects.

However, there is a more significant reason for the significance of spheres. This is that numerous second-order differential equations describing the physics of the automated methods used in the measurement of particle size are exactly soluble for spheres. Although this is equivalent to fitting nature into what is conveniently achievable, remarkably, it is consistently successful, particularly for quality control purposes.

## A Quick Tour of Spherical Geometry

The volume is either 4πr^{3}/3 or πd^{3}/6 wherein twice the radius r equals the diameter d. An introductory mathematics textbook will contain these simple formulae and represents the first step for anyone wishing to learn about particle sizing. For completion, the surface area of a sphere is either 4πr^{2} or πd^{2}, concepts that the Greeks theorized around 2,500 years ago. The simple factor of two that relates radius to diameter is occasionally the source of a 100% error.

Guesswork is sometimes required if a result (mean size, for instance) or specification for size does not identify it as radius or diameter, or if a graph is not labeled. The asserted capability to measure up to 5 micron in radius is identical to the asserted capability of measuring 10 micron in diameter. This error, which occurs with reasonable regularity, must always be identified, particularly as it is sometimes used to deceive, especially in advertising brochures.

## The Equivalent Spherical Diameter, ESD

There are two categories: geometric equivalent and technique equivalent.

### Geometric ESDs

Begin by considering a static, two-dimensional image of a particle. As it is two-dimensional, what appears to be a circle may actually correspond to a thin disc-like particle and not a sphere, unless shadows disclose a more space-filling structure. When the image was captured, assuming it was not a 3-D holographic image, it can be supposed that the particles settle in order to become stable on a flat surface.

In this way, if they did not land on their faces, most discs would fall over if they landed on an edge. This can be tested by throwing coins into the air and seeing how they land.

There are numerous different geometrically defined ESDs that can be assigned to an irregular particle. A useful method of determining the ESD of an arbitrarily shaped particle is to draw circles around the actual image until it is just completely enclosed. Subsequently, one must either assign the diameter of the enclosing circle to that of the particle (d_{e}), or locate a circle which embodies an area equal to that of the measured particle area.

This is easily undertaken by counting pixels and is assisted by computer programs that permit ever-improving precision with smaller and smaller pixels. Considering the area of the drawn circle, one must then assign its diameter to that of the particle (πd^{2}/4). This ESD must be labeled d_{A}. Alternatively, it is possible to trace the image’s perimeter and assign that to the diameter of the circle with the identical perimeter (πd_{P}).

These all represent geometric equivalent diameters. There are many more choices based on chords (Martin’s diameter, d_{M}) and parallel tangents (Feret’s diameter, d_{F}). It is important to bear in mind that the same particle can embody multiple types of geometric ESDs and, if correctly labeled, these should not be equal as particle shape irregularity increases. Moreover, the ratio of two such geometric ESDs for the same particle expresses information regarding shape and space-filling.

As particle shape increases, description using a single variable becomes more complex. Consequently, interpreting the “size” established by image analysis is more complicated than an automated machine based on an ESD determined by the technique. What is meant by this type of ESD?

### Technique ESDs

Imagine a stack of sieve plates. The mass of all of the particles that remain on a specific plate (after satisfactory shaking), for which hole size^{2} is d_{S}, are said to constitute the cumulative increment by mass larger than diameter d_{S}. Picture a particle moving radially outward in a centrifuge or falling under gravity. Its velocity is quantified and subsequently set equal to that of a sphere that would have moved in an identical manner. The consequent diameter is known as the Stokes diameter, d_{St}, due to the fact that the motion is described by Stokes Law.

Think of a tumbling, rotating particle for which a diffraction pattern is registered on a detector. The pattern is subsequently set equal to that of a sphere that would yield the nearest diffraction pattern. This is known as the laser or laser diffraction particle size and must be labeled d_{LD}.

Finally, the use of dynamic light scattering should be considered, as a way of establishing the translational diffusion coefficient of a submicron particle. Subsequently, the so-called hydrodynamic diameter or radius, corresponding to the measured diffusion coefficient, should be calculated.

There must be two subscripts herein: H for hydrodynamic and DLS for the technique (d_{H,DLS}). Practically speaking, although double subscripts such as this are seldom seen*, *if they were, what was actually being measured would be clearer.

In each of these instances, as well as numerous others, the size of the sphere is assigned in such a way that it would yield an identical result to the particle itself. These represent the so-called technique ESDs (or ESRs). However, in practice, either d or r is observed, and this can generate confusion during result comparison.

Among spheres, if techniques were equally precise, then a subscript would be unnecessary. However, among irregular shapes, utilizing subscripts enables one to understand that the “sizes” should not be equal. Moreover, as with geometric ESDs, ratios of technique ESDs provide information on the shape.

In contrast to image analysis, there is only one definition for a given method (ignoring specialized flow orientation methods). This represents the downside of image analysis: There are no broadly applicable, simple guidelines to consult when selecting the technique to characterize the particle size.

## The Promise and Heart Break of Image Analysis

For particularly long rods, the aspect ratio, AR, is characterized as the length divided by the diameter, L/d, with the reciprocal sometimes referred to as the AR. For less regular particles, the longest dimension divided by the shortest dimension, or its reciprocal, yields the aspect ratio. Given the L and d for each particle in a distribution, the aspect ratio can be computed. A particle’s performance could be correlated with L, d or even with AR.

However, picture a more highly irregular particle, smooth or jagged. There are a number of possible statistical descriptors of size and shape. The majority of modern software provides many choices that can be utilized, which is precisely the issue. Which size parameter or subset of size parameters will correlate with particle performance?

In certain disciplines, these answers are understood, while in many others they are not. Image analysis generates substantial amounts of data, but it does not necessarily generate helpful information. There are other problems posed by image analysis and these are considered in further detail in an additional section entitled “A Guide to Choosing a Particle Sizer”.

## Three Types of Radii

The first type of radius is the commonly known hard-sphere, geometric radius, R_{s}. This radius is most easily measured utilizing image analysis. The second type, as previously mentioned, is the hydrodynamic radius, R_{h}, which can be obtained using dynamic light scattering (DLS). This radius is obtained via a comparison of a sphere with the translational diffusion coefficient actually under measurement. For example, a solid, hard-core particle might embody a surface that is coated with long-chain polymers or surfactants that project far out into the liquid, often referred to as “hairy” particles.

In this instance, their radii are considerably larger than those of their cores. Thirdly, there is the gyration radius, R_{g}, which is acquired using static light scattering (SLS). It is notable that the R_{g} acquired using SLS quantifications is independent of shape assumptions, whereas all R_{h} values assume a sphere.

Ratios of R_{g}/R_{h} suggest shape: 0.77 a sphere; 1.54 a random-coil polymer.

## Summary

After initially establishing that particle size represents a length and not the mass of a protein, it is necessary to establish if the results are provided for a single statistical variable, or represent multiple variables involved, using image analysis. If it is a single statistical variable, does it represent a true diameter, or an ESD determined either geometrically (image analysis) or by comparison with what a sphere would yield using an automated technique (sieving, laser diffraction, centrifugation, zone counters, etc.)?

Finally, is it a radius or a diameter (a true one or ESD/ESR) that is being examined? Answers to these questions will enable a more effective comparison of numerical results, which is the subject of the next application note in this series.

## References and Further Reading

- The rotational diffusion coefficient, DR, for a sphere of radius 1 micron in water at 25 °C is 0.18 s-1 and it varies inversely with the cube of radius. Thus, a 100 nm radius particle is diffusing (rotating) 180 times per second. If the measurement time is a second or longer, the results are rotationally averaged.
- Sieve sizes are a complete topic in themselves. Often, they are not circular holes. Abraded holes as well as particles broken by abrasion may be problems. Sifting long enough to ensure all smaller particles made it through is an issue. Finally, for highly irregular shapes, if the particle can be oriented by sifting, then it is the smaller dimension that is determined Think of a distribution of long rods of varying lengths and varying, but much smaller diameters. Although unlikely, you would be determining the size distribution of the rod diameters and learn nothing about the distribution of rod lengths if you could sift them such that they all stood on end when passing through the sieves’ holes.

This information has been sourced, reviewed and adapted from materials provided by TESTA Analytical Solutions.

For more information on this source, please visit TESTA Analytical Solutions.