A guide to equivalent spherical diameter for particle sizing
Understanding equivalent spherical diameter: Definition, importance, and uses in particle size measurement
Learn about the sphere equivalent theory, different measurement types, and the instruments you need for successful particle sizing
The physical properties of particulate materials can influence a wide range of material behaviors, including reaction and dissolution rates, how easily ingredients flow and mix, or compressibility and abrasivity. But measuring the dimensions of particles isn’t always simple. Particles are three-dimensional objects, and unless they are perfect spheres (e.g. emulsions or bubbles), they cannot be fully described by a single dimension such as a radius or diameter. There are, therefore many ways to express particle size, such as mesh size and sieve fractions.
One method that simplifies the measurement process and simplifies comparisons of particle size across various materials – without having to use complicated three-dimensional modelling – is equivalent spherical diameter.
In this blog, we’ll explain how equivalent spherical diameter works, including how sphere equivalent differs from other particle size metrics and when to choose imaging analysis instead
What is equivalent spherical diameter in particle size analysis?
Equivalent spherical diameter describes particle size by converting a specific measured property of a particle into the diameter of a hypothetical sphere that would give the same measurement. Common properties used to calculate sphere equivalents include:
- Volume
- Surface area
- Settling velocity
- Light scattering intensity or diffraction pattern
The defining property varies with the measurement technique used; the variety of available particle sizing techniques means that different measurements can produce different results for the same sample.
What types of equivalent spherical diameter measurements are there?
Here are some of the most common ways to measure the sphere equivalent diameter, and the relevant instruments.
| Spherical diameter | Technique | Instrument |
|---|---|---|
| Volume-equivalent diameter | Laser diffraction | Mastersizer 3000+ |
| Surface area diameter | BET surface area | Micromeritics TriStar II Plus |
| Hydrodynamic diameter (diffusion speed) | Dynamic light scattering | Zetasizer Advance |
| Stokes diameter (density and settling velocity) | Sedimentation | Micromeritics Sedigraph |
Which equivalent spherical diameter measurement should I use?
When selecting which equivalent spherical diameter to use, the decision comes down to which physical behavior matters most for your application. This also influences how you summarise a size distribution as a single mean value – for instance, choosing between D[3,2] and D[4,3] values.
| Spherical diameter | Most important for | Key applications |
|---|---|---|
| Volume-equivalent diameter | Detecting oversized materialFollowing dispersion or milling processesUnderstanding particle packing | PharmaceuticalsCementMinerals and miningPigments and coatingsBattery materials |
| Surface area diameter | Dissolution rateReactivityCatalysisAdsorptionCoating | Drug deliveryCatalysts |
| Hydrodynamic diameter (diffusion speed) | Measuring nanoparticles where sedimentation and sieving aren’t practicalMonitoring aggregation during formulation or storage | NanoparticlesDrug deliveryColloidal stability assessment |
| Stokes diameter (density and settling velocity) | When particles separate under gravity or centrifugal force FiltrationSedimentationClassification | Minerals processingPigmentsCement |
How does the sphere equivalent differ from other particle size metrics?
By reporting one dimension for a three-dimensional particle, equivalent spherical diameter is affected by both particle size and particle shape.
Image analysis allows multiple parameters to be reported for each particle, describing the size and shape of particles in more detail, for example:
- Feret diameter (or maximal ferret diameter) is the furthest distance between any two points on a particle. For a spherical particle this would be the same as the spherical equivalent diameter, but as particles become irregular these values will start to differ.
- Particle length and width are particularly useful for understanding the behavior of needle shaped particles. Aspect ratio is a dimensionless parameter derived from length and width, a cube has an aspect ratio of 1 whereas a needle like particle has an aspect ratio approaching zero.
- Most imaging techniques will also report a circular or spherical equivalent diameter which is useful for comparison to techniques like laser diffraction.
The case for and against using equivalent spherical diameters
The spherical equivalent concept works very well for looking at bulk changes in particle size and for comparing materials of similar shape. However, it may not always be the best solution for highly irregularly shaped particles, such as needles or plates. For highly irregular particles, effects such as broadening of the particle size distribution can occur.
When particle shape is important, imaging solutions like Morphologi 4 may be more appropriate. However, many modern analytical workflows use both, with equivalent spherical diameter for rapid quantitative sizing and imaging for morphological validation or troubleshooting anomalous behavior.
For example, the Jan De Nul Group, a global market leader in dredging and marine construction, used both Mastersizer and Morphologi instruments for efficient sand particle characterization.
The bottom line
Selecting the best measurement type for your application requires an understanding of the different spherical equivalent diameter techniques. Start by defining your goal, from detecting agglomerates to monitoring dissolution rates. We’ll help you take care of the rest.
Spherical diameter: Frequently asked questions (FAQs)
Understanding the spherical equivalent diameter doesn’t have to be complicated. Here are the answers to some frequently asked questions about spherical diameter.
What is an equivalent diameter?
Equivalent Spherical Diameter, also known as equivalent sphere theory, is the diameter of a sphere that would produce the same results as an irregularly shaped particle using a specific analytical technique. The equivalent spherical diameter depends on the technique as each measures a different physical property of the particle.
What is Stokes diameter?
Stokes diameter (Dst) is the diameter of a sphere that would settle through a viscous fluid at the same terminal velocity as the particle being measured. This is measured via sedimentation and is best suited to roughly equidimensional particles in the 1–100 µm range. Because it is based on hydrodynamic behavior rather than geometry, irregular particles typically report a smaller Stokes’ diameter than their true dimensions, since non-spherical shapes experience greater drag than an equivalent sphere.
Why am I getting different equivalent spherical diameters for the same sample?
Each technique measures a different physical property of the particle and relates that to particle size, so each technique will report a slightly different result. The Malvern Panalytical portfolio offers solutions for all common variants of equivalent spherical diameter:
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