Considering the number distribution weights each counted particle equally. As an example, consider the nine particles. Three particles are 1 μm, three are 2 μm, and three are 3 μm in size (diameter). Building a number distribution for these particles will generate a homogeneous distribution over the size, where each particle size accounts for one third of the total. If this same result were converted to a volume distribution, the result would appear differently where 75% of the total volume comes from the 3μm particles, and less than 3% comes from the 1μm particles. When presented as a volume distribution it becomes more obvious that the majority of the total particle mass or volume comes from the 3 μm particles.

Dear Zheng Li.

The problem of particle size measurement is a sophisticated and partially voluntaristic one. I can recommend you a "classic book" // T. Allen, Powder Sampling and Particle Size Determination, first ed., Elsevier Science, Boston, 2003 // or analogous to study this question in detail.

Best regards.

Alex J.

By analogy with the number-average and weight-average molecular masses of a polymer sample, the number-average particle size will tend to be more influenced by the population of smaller particles, and vice versa, the volume-average size will be more sensitive to presence of larger particles, as Jean-Serge has commented above.

In polymer science, both parameters (and even several additional ones) are used to characterize the sample weight distribution, a single parameter is not enough for complete description, but can be used depending on the certain task. I believe, situation is the same with the particles size distribution: if one is interested in a fraction of "extra-super-nano" particles, the number-average size should be used, and if the agglomerated or microparticles admixture is to be followed, the volume-average size is better option.

Also, the ratio of the volume-average and number-average sizes will characterize the width of the size distribution (that is, size dispersity), again, by analogy with the polymer sample.

Good question. What is best also depends on the measurement technique employed. It is good people make a point about what the size measurement basis is! As pointed out, TEM is number-based, while other techniques like small angle x-ray scattering (SAXS) are volume-based, and dynamic light scattering is volume-squared based. See the attached publication for more details.

Also,

Converting from one basis to another mathematically is hazardous, as noise will get amplified by a factor of 10^3 each conversion. See the standard from ASTM, E2490 on DLS, for a detailed discussion about that.

Article Challenges for physical characterization of silver nanoparti...

Many good replies above. I will try a very simple one without referring much to actually measuring size distributions. The number average is obtained if you sum up the sizes of each particle and then divide by the number of particles. The result may or may not be also the most frequent particle size present in your sample, depending on whether the distribution is Gaussian or not. It will not tell you in which size of particles most of your material in terms of mass or volume is present. For example, you could have in a sample one million of 1 nm particles and 10000 of 10 nm particles. Then, 99% of all particles are 1 nm, while 90% of the material in terms of its mass or volume is present in the form of 10 nm particles. So, a number average will be close to 1 nm, while a volume average will be close to 10 nm. This is why TEM gives you often completely different results than DLS. They look at different distributions, at least initially, before you start manipulating the results mathematically.

Thank you all for sharing the answers with me! A follow-up question would be both of the distribution can be acceptable for publication in terms of the requirement! For example, a nanoparticle suspension is not made of uniform nanoparticles, which contains 90% of 100 nm nanoparticles and 10% of 1000 nm nanoparticles! If such suspension is used to treat cancer cells and the effect of nanoparticles on cells is investigated! Then, we have better use number weighted size distribution to characterize the nanoparticles, instead of volume weighted distribution, because the 1000 nm particles would cause a huge bias on particle size of the suspension (smaller nanoparticles would definitely provide more effect on cancer cells due to the effecient cellular uptake via clathrin and/or caveolin endocytosis)!

Most people have a more intuitive liking for number based distributions. On the other hand, if you are a gold digger, you would be more interested in volume based ones, since they tell you where the gold is. Regarding the use of nanoparticles in cancer treatment, I am not sure your conclusion makes immediate sense to me. I guess it would, if a 100 nm particle is as effective and a 1000 nm particle.

apart fom that 1000 nm particles are not "nano"particles, the assumption that the smaller particles are better for uptake in cells is correct. Even more, nanoparticles are functioning mainly by their surface. The smaller the particles, the bigger the specific surface area (to be expressed in m²/g).

You know yourself from middle school how to calculate surface area of (e.g.) spheres with know radius. Then you know that the surface area A = f(r²)

:-)

Fortunately, for mono disperse distributions (very narrow, i.e. Dirac) there is virtually no difference between number weighted and volume weighted distributions.

However the difference can be huge between broadband distributions since one statistic is weighted by D^1 and there other one by D^3. Note that for for spheres it is easy to jump from one distribution to another one.

I agree with you it is really disappointing that a lot of papers do not provide this crucial information. But the things can be even worst! Number of users provide statistics obtained from the raw intensity based distribution... In this case the bias is related to D^6... in Rayleigh regime... I have noticed recently that some of the three important instruments providers make their demonstrations without any light scattering corrections (i.e. they provide only intensity distributions). This helps when their inversion software fails to reconstruct a "nice" distribution...

The basic problem is somewhat discussed in the book of R. Xu and one of our paper:

Xu R. Particlecharacterization:lightscatteringmethods. Dor-

drecht: KluwertAcademicPublisher; 2002.

F. Onofri, S. Barbosa, O. Toure, M. Woźniak, C. Grisolia

Journal of Quantitative Spectroscopy and Radiative Transfer 126:160-168 (2013)

https://www.researchgate.net/publication/235652037_Sizing_highly-ordered_buckyball-shaped_aggregates_of_colloidal_nanoparticles_by_light_extinction_spectroscopy

Article Sizing highly-ordered buckyball-shaped aggregates of colloid...

Considering the number distribution weights each counted particle equally. As an example, consider the nine particles. Three particles are 1 μm, three are 2 μm, and three are 3 μm in size (diameter). Building a number distribution for these particles will generate a homogeneous distribution over the size, where each particle size accounts for one third of the total. If this same result were converted to a volume distribution, the result would appear differently where 75% of the total volume comes from the 3μm particles, and less than 3% comes from the 1μm particles. When presented as a volume distribution it becomes more obvious that the majority of the total particle mass or volume comes from the 3 μm particles.

Physically, it is always better to weight the particle size statistics with what you expect to be the key property of your problem: number of particles (e.g.  for aggregation processes studies), surface (e.g. for chemical reactivity), volume (mass transfer) or a ratio volume/surface (drag and inertial effects,...). For spherical particles, all these distributions are simply related (see the two books below),  and they are equal for monodisperse samples.  Note that for nanoparticles, it is very common that people forget to specify whether it is a PSD in number, volume or intensity (i.e. light scattering, DLS,...). In other words, they miss a power of 3 (volume) or 6 (Intensity) to estimate the PSD (!)

XU, R., Particle characterization: Light scattering methods, Dordrecht, Kluwer Academic Publishers, Dordrecht, Nederland, 2002.

Onofri F., Barbosa S., "Chapter II: Optical particle characterization", in "Laser Metrology in Fluid Mechanics", Boutier, A (ed.), ISBN: 9781848213982, Wiley-ISTE, London (2012).

Any refereed article really should mention what type of particle size distribution is shown and how it was obtained... As indicated by the above posters, the weighting is different and makes a difference. In essence, number will often emphasize the smaller part of the distribution, however all types of transformation can be correct, and there is not one that is "better" than another. However different techniques measure different particle size distributions, and it is good to know which distribution was obtained from the 'raw data' and which are further transforms from that set.

https://www.materials-talks.com/blog/2017/01/23/intensity-volume-number-which-size-is-correct/

It depends on the application. In this case the follow up question from Zheng Li is regarding the potential to use engineered nanoparticles to treat cancer cells. Only nanoparticles (i.e., smaller than 0.1µm) can penetrate cell membranes and enter into organs to ostensibly detect and treat or deliver their payload that will destroy the cancer cells. The smaller the particle the better in this application.

How do we know?

If the damage caused by black carbon nanoparticles to our health is an indicator, then the smaller the better. Black carbon nanoparticles from combustion sources we now know deliver cytotoxic payloads of many chemicals (PAHs and organic materials) adsorped as they exit the chimneys, tailpipes, & smokestacks and mix in the troposphere and are inhaled and carried in the bloodstream (source: Barbara Panessa-Warren et al; Brookhaven National Laboratory, 2018). Also, confirmed is that it is the payloads that are toxic and not the black carbon itself.

Bernhard Wessling 's observation is correct regarding the smaller and more numerous (number weighted) the particles, then the surface area potential. Both the uptake and potential to deliver through contact the mechanism that will treat or destroy the cancer cells will be greater. The trick will be to enable the payload to be carried by sub 20 nm particles.

Note: "Sub-3 nm" particles are now being detected in great numbers both in ambient combustion source emissions and background air.

What would be your criteria for a parameter to be important? Published papers should give you an idea of how they have arrived at the size distribution they quote.