D[3,2] vs. D[4,3]: Which parameter should you choose to report particle size?

Particle size distribution (PSD) is a critical parameter for materials scientists across industries to understand and control. Failure to do so can have wide-ranging consequences – from dissolution failures during pharma scale-up to defects in final cathode coatings.

But conducting particle size analysis is only the beginning. The metrics you prioritize and actions you take based on their outcomes change depending on the materials you’re handling.

In this blog, we’ll break down the difference between two important ways of calculating mean particle diameter: D[3,2] and D[4,3] mean diameter values. We’ll cover what they are, when to use them, and which analysis techniques will work best for your materials.

Why particle diameter and particle size distribution matter to your materials

PSD impacts many essential material properties, including:

• Flowability
• Reactivity
• Dissolution rate
• Bioavailability
• Sintering behavior

For example, in additive manufacturing, controlling the particle size distribution of metal powders ensures maximum packing density, helping reduce defects like porosity and balling.

Figure 1. Particle size distribution versus packing density

PSD is similarly important when manufacturing cement materials. For average cement materials, between 60% and 70% of the material should be comprised of fine particles between 3µm and 30µm. This produces a higher surface area, which helps to ensure good compressive strength and curing qualities in the final cement.

The right balance here is essential. Too many particles larger than 50µm could cause incomplete hydration, reducing mechanical strength; too many particles smaller than 2µm could cause the cement to set too rapidly, generating excess heat and causing cracking.

Both these defects spell disaster for cement plants, as they could lead to costly downtime or rejected batch shipments.

Understanding D[3,2] and D[4,3] metrics in particle size analysis can help you spot defects like this before they become costly problems. Here’s how to use them in your PSD calculations.

How D[3,2] and D[4,3] work in particle size distribution calculations

D[3,2] and D[4,3] are two different expressions of the average particle size of a powder or dispersion. The “D” stands for particle diameter – or equivalent spherical diameter for irregular particles – while the two numbers refer to the mathematical method used to calculate the average.

The general formula for calculating the mean diameter D[m,n] from a volume-based particle size distribution such as a laser diffraction measurement is:

Here, d_i is the midpoint of each size class in the distribution and V_i is the volume of the sample in that size class.

The important thing about D[3,2] and D[4,3] mean values is that they calculate the average particle diameter based on different priorities:

• D[3,2] gives you the average particle size according to surface area
• D[4,3] gives you the average particle size according to volume/mass

Here’s what that means for your calculations.

How is D[3,2] calculated?

D[3,2], also known as the Sauter mean diameter (SMD), is the average particle size of your sample, weighted by surface area. It calculates the average particle size by representing the particle diameter of a hypothetical sphere that has the same volume-to-surface-area ratio as the total particle population.

The equation you use to calculate D[3,2] is:

Because of its emphasis on calculating surface area, D[3,2] is especially sensitive to fine particles. Common uses include predicting the reactivity of cement clinker or forecasting dissolution behaviors in pharmaceutical drugs.

How is D[4,3] calculated?

D[4,3] is also known as the De Brouckere mean diameter. It is a volume-weighted mean that reflects the particle diameters where the majority of the sample’s mass is concentrated.

The equation you use to calculate D[4,3] is:

Because of its focus on volume/mass, D[4,3] is sensitive to large particles and agglomerates. It is therefore commonly used to monitor milling efficiency in mining and cement plants, as well as bulk consistency and agglomeration in manufacturing.

D[3,2] vs. D[4,3]: Summary table

Below is a quick reference for the difference between D[3,2] and D[4,3] mean particle diameter values.

	D[3,2]	D[4,3]
Also known as	Sauter mean diameter (SMD)	De Brouckere mean diameter
Weighted by	Surface area	Volume/mass
Equation
Sensitive to	Fine particles	Large particles & agglomerates
Reveals	Reactivity	Where most of the mass is
Common uses	Predicting dissolution behavior in pharmaceuticals Controlling reactivity, e.g. in cement clinker	Milling control Bulk consistency Agglomeration detection

How to use D[3,2] and D[4,3] in your decision-making

The particle diameter mean value you prioritize in daily monitoring and decision-making depends heavily on the materials you’re working with. Here are the main scenarios in which you would use D[3,2], D[4,3], or both values together.

When to use D[3,2]

You should prioritize D[3,2] mean particle diameter values when you care most about:

• Reactivity
• Dissolution
• Surface-driven behavior

This is common for particle size in pharmaceuticals – for instance, when formulating an inhaled drug, developers need the surface area to be within a specific range for optimal absorption into the lung.

When to use D[4,3]

You should focus on D[4,3] mean particle diameter values when your priority is to assess:

• Bulk density and packing behavior
• Flowability and performance in processes such as milling, tableting, mixing, and more
• Detecting large particles and agglomerates

This is common for battery materials engineers, for instance, during incoming quality control on cathode powders. Particle size distribution is critical for batteries because agglomerates in cathode powders can cause defects in the final electrode coatings.

When to use both D[3,2] and D[4,3]

Often, the best insights come from looking at D[3,2] and D[4,3] values together to understand the general shape of your particle size distribution before analyzing the full distribution curve.

This is especially useful for rapid assessments during routine QC and process control:

• D[3,2] and D[4,3] values close together suggest a narrow particle size distribution and, therefore, a consistent particle population.
• When D[4,3] is much larger than D[3,2], this suggests a broad distribution, which could mean the presence of large agglomerates, a large population of fine particles, or both. The full distribution curve can give you further details.

If your materials are behaving unexpectedly during R&D or QC, looking at the D[3,2] and D[4,3] mean particle diameter values together can give you an initial indication of what’s going on.

3 techniques for measuring particle size distribution – and how they affect your D[3,2] and D[4,3] calculations

There are many techniques for analyzing particle size distribution that will deliver the information necessary to arrive at D[3,2] and D[4,3] values. When choosing the right particle sizing method for your application, you’ll likely consider three main techniques: laser diffraction, dynamic light scattering, and sedimentation.

1. Laser diffraction

The first technique for measuring average particle diameter is laser diffraction. Laser diffraction measures particle size distributions by shining a laser beam through a dispersed particulate sample and measuring the angular variation in the intensity of the light scattered. Large particles scatter light at smaller angles relative to the beam, while small particles scatter light at larger angles.

The benefits of using laser diffraction instruments like the Mastersizer range to calculate PSD include:

• Large measurement range, from 0.01 µm to 3500 µm
• Rapid measurement times, typically less than 10 seconds
• Robust instrumentation, suitable for industrial environments
• Repeatable measurement, usually better than 0.5% variation for standard samples

Laser diffraction is natively volume-weighted but can measure both D[3,2] and D[4,3] mean values reliably and in compliance with ISO 13320 and relevant pharmacopeial standards.

2. Dynamic light scattering (DLS)

Another common technique for assessing average particle diameter is dynamic light scattering, or DLS.

DLS instruments like the Zetasizer range measure the size distribution of particles in suspension by shining a laser through a liquid sample and analyzing the intensity of the fluctuations in scattered light caused by Brownian motion.

However, because DLS is intensity-weighted by default, it is highly sensitive to noise in the conversion from an intensity-weighted mean to a volume- or surface area-weighted mean. It is especially sensitive to large particles and agglomerates, as larger particles scatter significantly more light.

It is therefore less commonly used for calculating D[3,2] and D[4,3] averages, although it is still a common technique for D[3,2] calculation in nanomedicine and drug delivery, where the particles are typically in the 10-500 nm range. This is below the lower limit of most laser diffraction instruments – though not the Mastersizer 3000+.

3. Sedimentation

Finally, a traditional method for particle size distribution analysis is sedimentation, for instance, with an instrument like the Micromeritics SediGraph.

Sedimentation analysis determines the particle diameter range of a sample by measuring how fast particles sink through a liquid under gravity. This is governed by Stokes’ Law, whereby larger particles settle faster.

It is a volume-based method by nature and is typically employed where other techniques have limitations for measuring very dense materials – such as titanium dioxide in pigment and coatings manufacturing – or where mass-based results are specifically required.

It is well-established across industries and regions and suitable for handling very large samples. However, it is a slower analysis method, making it less common in routine industrial QC.

3 techniques for D[3,2] and D[4,3] calculation: Summary table

Choosing the right technique for average particle diameter analysis depends on several factors, including which metric you’re prioritizing and what materials you’re working with. Use this table to identify the best candidate.

	Laser diffraction	Dynamic light scattering	Sedimentation
Instrument	Mastersizer	Zetasizer	SediGraph
Measurement range	0.1 µm – 3,500 µm	0.3 nm to 10 µm	0.1 µm to 300 µm
Weighting	Volume	Light scattering intensity	Mass/volume
D[3,2] reliability	High	Lower: Converting intensity to volume introduces significant margin for error for fine particles	High
D[4,3] reliability	High	Moderate: Large particles dominate intensity signal so the volume conversion is more stable, but can still be unreliable for broad distributions	High
Best for	– Wide range of particle sizes – Fast analysis using wet or dry dispersion – Applications sensitive to coarse particles	– Small particles – Colloidal suspensions	– Legacy technique for high concentration wet dispersion – Applications sensitive to fine particles
Typical applications	– Pharmaceuticals – Cement – Batteries	– Nano-pharmaceuticals – Biotechnology	– Minerals – Cement – Pigments and coatings

For reliable D[3,2] and D[4,3] particle diameter values, we have the technology you need

Accurate D(3,2) and D(4,3) mean particle diameter calculations deliver essential insights into materials under development or on your process line.

With reliable instruments like the Mastersizer range, Zetasizer, and SediGraph, R&D, process, and QC engineers can make data-driven decisions that protect processes and profit.

To learn more about our powerful combination of solutions for particle size analysis and which one best suits your workflow, contact our team.

Frequently asked questions (FAQs)

Below are the answers to some commonly asked questions about D[3,2] and D[4,3] mean particle diameter measurement.

What is D[3,2] particle size?

D[3,2], also known as the Sauter mean diameter (SMD), describes average particle size as the diameter of a hypothetical sphere that has the same volume-to-surface-area ratio as the total particle population.

The equation you use to calculate D[3,2] from a volume-based distribution is:

What does D[4,3] mean?

D[4,3], also known as the De Brouckere mean diameter, is a volume-weighted mean that reflects the particle diameters where most of a sample’s mass is concentrated.

The equation you use to calculate D[4,3] from a volume-based distribution is:

What is D in particle size?

“D” in particle size calculations typically stands for “particle diameter”, often defined as equivalent spherical diameter for irregular-shaped particles. In particle diameter mean values like D[3,2] and D[4,3], the numbers in brackets refer to the exponents in the formula used to calculate the average: D[4,3] uses d⁴ and d³, while D[3,2] uses d³ and d².

How to measure particle diameter?

There are many methods for measuring particle diameter. One common method is laser diffraction on instruments like the Mastersizer, which measures particle size distribution by passing a laser beam through a dispersed sample and measuring the angular variation in the intensity of the scattered light.

Other techniques include dynamic light scattering for nanoscale particles and sedimentation for dense materials.