These may help:

http://www.horiba.com/scientific/products/particle-characterization/education/sz-100/particle-size-by-dynamic-light-scattering-resources/what-is-z-average/

http://macro.lsu.edu/HowTo/MALVERN/FAQ_HTML/what_is_the_z_average/what_is_the_z_average_.htm

You're not comparing apples with apples. Indeed, a closer analogy (but still not perfect) is that you're comparing apples with apple seeds...The z-average relates to the intensity distribution (proportional to volume squared or r6). Thus the z-average will always be much larger than the number average if we have any polydispersity in the system. The z-average will be equal to the volume and number means only when all the particles are identical. Take a look at:

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

Number distributions are calculated by knowing information on every particle in the distribution - an impossible task as can be shown by simple calculation. For example, how many 100 nm particles would be present in 1 g of any material if the material was subdivided thus? What if they were 10 nm? Number-based techniques such as NTA look at individual particles in the distribution and can provide number distributions thus. Note that, if you ate cornflakes for breakfast this morning, I doubt you calculated the number of flakes you would eat. However, you may have weighed the cereal (if on a diet) or put the flakes into a bowl of constant volume.

Note that number distributions from DLS are 'deprecated' in the international standards (ISO22412:2017 & ASTM E2490-09(15)).

The Zetasizer measures with an ensemble technique so there is no need to know the number of particles in colloidal suspension and the Zetasizer does not count particles.

Hi,

Explanation in the Manual; e.g. here (page 11-5):

https://www.chem.uci.edu/~dmitryf/manuals/Malvern%20Zetasizer%20ZS%20DLS%20user%20manual.pdf

Thanks Alan for your answer and my purpose of asking this question is that whether particle size distribution (PSD) by number obtained through Malvern software is relevant or not.

Same aspect as you told that Z-average obtained by Malvern software is PSD by intensity which is proportional to r^6 while average particle size determined by number is proportional to r which may be seen in word document and Excel sheet attached below.

By calculation in the same sheet, average particle size (APS) by intensity is 882 nm and average particle size is 137 nm. so it means that by this calculation, APS by intensity should be six times the APS by number.

APS by intensity for this sample = 6 * APS by number

APS by intensity for this sample = 6 * 137 = 822 nm.

so, by theoretical calculation, APS by intensity for this sample is 822 nm which is almost same as z-average given by Malvern software.

so even as number distributions from DLS are 'deprecated' in the international standards (ISO22412:2017 & ASTM E2490-09(15)) but APS by number is almost same as compared to APS by intensity and only its six times less.

I don't know about particle size distribution. whether the PSD by intensity will be different as compared to PSD by intensity.

but whether some one quotes the PSD by number.. whether theoretically it will be wrong.

please specify..

The international standards, that I mentioned above, tell you that the most robust DLS parameters are the z-average and the polydispersity index (PDI). Anything else is based on various assumptions which may or may not be reasonable. The raw information (other than photon count) is the correlogram and its (modeled/assumed) exponential decay. As an exponential decay is a 2-parameter model then there's not a lot of information that can be derived from such a plot. This means that DLS is a low resolution technique (but powerful all the same). The deconvolution of the correlogram for a distribution is an ill-conditioned or ill-posed mathematical problem with very small change in data leading to the possibilities of very different results/outcomes. For wider distributions the assumptions in conversion can turn around and bite you. Provencher (who wrote Contin) referred to this deconvolution as a 'hopeless problem'.

Go back to the corn flake analogy in my first posting. Why do you need to count the numbers of cornflakes you eat? Why do you need a number distribution?

If you can justify a number distribution then you'll need a (slower) instrument or technique to count particles.

I don't understand what you mean by 'relevant' in your comment 'whether particle size distribution (PSD) by number obtained through Malvern software is relevant or not'. Please explain.

Your paragraph:

'By calculation in the same sheet, average particle size (APS) by intensity is 882 nm and average particle size is 137 nm. so it means that by this calculation, APS by intensity should be six times the APS by number.

APS by intensity for this sample = 6 * APS by number

APS by intensity for this sample = 6 * 137 = 822 nm'

...is completely incorrect from a scientific perspective (indeed you use 6r and not r6 even if your math and derivation was correct). The z-average arises from the cumulants analysis and not by (the type of) calculation as you show. You need to understand the differences between measured and calculated and derived values and the assumptions therein.

Last, but no means least, your Word attachment ('Malvern Explanation' - when it is nothing of the sort!) shows a cropped report with the comment 'Refer to quality report'. What does the Quality report say? There are a lot of explanations in here regarding the quality of the data. You also have an RI for the particle of 1.10 which is not possible. As you may know the optical properties are required for an intensity to volume or number (deprecated) conversion. Your descriptor appears to refer to Ag (silver) nanoparticles. The RI for silver at 632.8 nm is: RI = 0.135 - 3.99.i [Reference: P Winsemius, F F van Kampen, H P Lengkeek, C G van Went, J. Phys. F 6, 1583 (1976)]

The answer to your posed question ('Is there any difference betwwen Z-average (d.nm) calculated by instrument itself and calculation of average by number particle size distribution?') is 'Yes'.

Alan F Rawle makes many good points. One is about the robustness of using cumulants. It's an example of why using the more complicated algorithms can mislead you (true for other techniques, too). The main reason is that users of the instruments and the people making decisions based on the results do not understand the operating principle and the assumptions that the instrument's software has to make. These assumptions have been decided by the manufacturer!

If you don't know what assumptions are being made, how can you interpret the data with a high level of confidence?

At the most primitive level, DLS characterizes the timescale of random fluctuation of light scattered from diffusing particles. The process that evolved over decades is autocorrelation. There aren't many assumptions required to create the correlation function. The cumulants method is a simple fit to the correlation function that requires no assumptions. It's merely a way to represent the correlation function by one or two parameters (z-average and PDI) for human convenience. Of course, you have to assume that these parameters accurately describe the sample, but it isn't an assumption related to the calculation itself. The information contained in the correlation function fundamentally relates to the diffusion coefficient, D, not the size. i.e., the instrument measures diffusion. Size is calculated from D. This is where the first set of assumptions come in - viscosity, temperature etc. It is also assumed that the particles are spherical. Can you be sure that yours are?

The more advanced ways to estimate the size distribution (NNLS, CONTIN etc) make lots of assumptions that you will almost certainly get wrong or a realistically are impossible to satisfy.

For cumulants and the others, the graphical representation of the distribution is a misleading feature of every instrument. There's a belief that you are seeing the actual distribution. In fact, all you are are seeing is a distribution plotted based on the estimated z-average and PDI (for cumulants). Visually useful - yes. Misleading - yes, unless you know where it comes from.

So, should you avoid the less robust calculations? No. But you must know why they are less robust. Do you need them? In an industrial environment, I'd say probably not. Trying to control formulation manufacture, process changes, batch failures etc. doesn't usually require it. In fact, you could get by with just reporting diffusion coefficient. It's as valid a "critical quality attribute" as the size (if not better). Indeed, you can directly compare correlation functions without calculating anything. You get tens or hundreds of data points instead of two.

Know your instrument, know your science and know the needs of your organization!

(FYI, I give webinars on this very subject for zeta potential measurement).