In life, you get out what you put in. This cannot apply more strongly than in the situation of particle size analysis, where poor results will be obtained if the material being measured is not sampled correctly.
Like most things in life you get out what you put in.This cannot apply more strongly than in the situation of particle sizing where "garbage in = garbage out" is an appropriate maxim.
If we consider a pack of cereal such as museli, it is quite obvious that the last two teaspoons taken from the packet are not the same as the first two. For systems like this, then it is clear even to the non-scientist that segregation has occurred ("the contents of the package may have settled during transit") and that the properties of the first and last samples from the packet are totally different. As scientists (hopefully!) we need to quantify these effects and decide when this phenomenon is important from a statistical point of view and when it is not. A more serious example of the "garbage in = garbage out" situation was that resulting from the confusion in UK Government laboratories between cow and sheep brains ("bovine or ovine?") resulting in the causes and transmittal mode of BSE being misinterpreted.This prompted the quotation within the editorial of the December 2001 issue of Chemistry in Britain that "Sample integrity and sound data analysis are essential cornerstones of valid analyses for decision making".
Thus before we even begin the statistical evaluation, we need to ask ourselves the reason or objective behind the analyses12 (because one analysis on its own would be statistically perfect):
We also need to ask ourselves whether we require a bulk powder measurement designed to investigate say, for example, flowability, filter blockage or dusting tendency or whether we are concerned with a primary size of the particle system as the latter may control dissolution rate, gas absorption and chemical reactivity.We need to be aware that the energies generated in any particle size "dispersion unit" or accessory are likely to be many times larger than those within a production environment. For example, a hydrocyclone operates only at a few ms-1 in contrast to the typical 70W of a low powered sonication bath.
A good illustration of the effect of segregation can be seen in any large mine spoil heap or, more conveniently, in small demonstrators available from Jenike and Johanson:
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A similar photograph of a heap of powder is present in the first figure of the first chapter of the standard particle size reference text1 but as no one bothers to read this chapter we will need to reinforce the science and conclusions therein.
The questions we need to consider are:
Some of the follow up questions in a production sense could concern batch uniformity-how do we get as many brasil nuts on the last spoon of museli as the first? Again, with a more serious example, how can we control the amount of active pharmaceutical a patient receives where our dosage form is made of active and excipient? It is obvious but if every particle in the system is identical then we would only need to remove one particle for it to be representative of the whole.
In reality this will never be the case.
The number of analyses needed for statistical validity is the easiest to answer and we will therefore tackle this first, although it is usually applied more often to chemical and physical measurements other than particle size determination.
The probable error or standard error (SE) in multiple measurements is given by:
SE = 2/3 [Σ(x - xm)2/n]0.5
where x is any sample measurement and xm the mean of the samples and n the number of samples.
Let us take an example of taking 5 samples in which the average particle size is (perfectly measured) 29.0, 27.5, 24.5, 22.0 and 20.5 microns.We may ask ourselves what is the number of measurements needed to give a precision of 1 μm on the mean? Excel handles this easily but in long-hand:
The mean is easily calculated as:
xm = Σx/5 = (29.0 + 27.5 + 24.5 + 22.0 + 20.5)/5= 24.7 μm
Σ(x - xm)2 = (29.0 - 24.7)2 + (27.5 - 24.7)2 + (24.5 - 24.7)2 +
(22.0 - 24.7)2 + (20.5 - 24.7)2 = 4.32 + 2.82 + (- 0.2)2 +
(- 2.7)2 + (- 4.2)2 = 18.49 + 7.84 + 0.04 + 7.29 + 17.64 = 51.3
The standard error is 2/3 X (51.3/5)0.5 = 2/3 X (10.26)0.5 = 2/3 X 3.20 = 2.1 μm
If a precision of ± 1μm is required then we can calculate the total number of samples to be taken as:
2.1/1.0 = (n/5)0.5 = 22.1
Thus 23 samples would be required.
This is not the usual situation that we are trying to deal with in particle size analysis. Indeed in the above situation we are so subject to the vagaries of the sampling method (are these independent samples?) and the dispersion method, that this answer is almost meaningless. Besides the British Standard Customer (BSC) will only take one measurement anyway…
The more important question that is usually requiring a statistical answer is the minimum amount of sample that would be needed to ensure that it could be representative of the whole or at least a rough guide to what we would expect to need. Clearly there will be a number of assumptions needed to answer this particular issue. It is an issue that has occupied chemical engineers since 1880 or so (e.g. Reed).Taggart's Handbook of Mineral Dressing (1948) gives a nonogram for minimum weight (based on type of ore-rich or lean-and particle size) based on the assumptions of Richard (1903) which has been the reference in the mining industry from thereon. Gy's work2, 3 (1953, 1982) is extensively quoted where the weight of sample required is proportional to da where the exponent is a variable between 2 and 3 but theoretically 3 (the number - volume/mass relationship).The constant of proportionality is dependent on the accuracy desired, the homogeneity and value of the ore.
Even later pragmatic work by Kraft (1978 and summarised in 4) is of little value as multiple determinations (he recommends 20 and talks of 1 kg shovels full!) are needed and this is not helpful to engineers seeking ballpark figures before beginning an analysis.
So we need to seek more practical solutions. Let us start from the basis that we require a standard error of 1% and that the mid-point of the highest size band is known (d99.5 is close enough unless the pedants insist on a more accurate geometrical midpoint assuming a logarithmic spacing of size classes).We will further assume that this highest size band contains 1% of the material by mass or volume (constant density assumed of a single component system).This is probably reasonable if the customer believes that 100 size classes equate to resolution! So we can calculate the number of particles required in this highest size band:
1/100 = 1/n0.5
leading to a need for 10000 particles in the highest size band.The number 10000 is interesting (although it does not provoke a large entry in the Book of Interesting Numbers5. In fact it provokes no entry at all in the aforementioned volume) as it is the same number of images (not particles) that NBS (National Bureau of Standards) stated that was needed for statistical validity in image analysis6:
"S(tandard error) is proportional to N-1/2 where N is the total number of particles measured…This consideration implies that image analysis may require the analysis of on the order of 10000 images to obtain a satisfactory limit of uncertainty" (p718, paragraph 1).
This minimum number of particles for 1% or better SE is then more useful as with knowledge or assumptions about the particle density, a spreadsheet is easily constructed and we are able to get a feel for the sort of sample size required:
| D (μm) | Diameter (cm) | Radius (cm) | Density (g/cm3) | Weight in top size fraction (g) | Total weight (g) (= Last column X100) |
|---|---|---|---|---|---|
| 1 | 0.0001 | 0.00005 | 2.5 | 1.31358E-08 | 1.31358E-06 |
| 10 | 0.001 | 0.0005 | 2.5 | 1.31358E-05 | 0.001313579 |
| 100 | 0.01 | 0.005 | 2.5 | 0.013135792 | 1.313579167 |
| 1000 | 0.1 | 0.05 | 2.5 | 13.13579167 | 1313.579167 |
| 10000 | 1 | 0.5 | 2.5 | 13135.79167 | 1313579.167 |
| 200* | 0.02 | 0.01 | 3.15 | 0.13240878 | 13.24 |
| 80 | 0.008 | 0.004 | 3.15 | 0.008474162 | 0.85 |
| This is the weight of 10000 particles | This is where 1% of the particles are in the top size band | ||||
| Assuming spheres |
*This represents a typical cement
More rigorous theoretical solutions are provided by Masuda and Gotoh7 and enhanced by Wedd8, but the figures crudely calculated above in the spreadsheet are in the same ballpark. We note that the old maxim that 75 or 100 μm provided the point at which sampling became the predominant error in particle size analysis is easily understood if a sample size of around 1g is assumed for many analytical techniques. See below:
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We can then decide on appropriate sampling regimes for the mass of material that we take as a (minimum) base sample calculated from the above spreadsheet. Note that it would be meaningless if we have material up to say 500 μm in the sample to try to measure 50 mg or so (1 particle?).At 1mm we need over 1 kg of material to have 10000 particles in the highest size band.To examine Allen's (frequently quoted) summary of sampling methods at this stage is extremely educational (and will lead on to answering the question "How do we take the sample as opposed to how should we take the sample?"):
| Reliability of selected sampling methods using a 60:40 sand mixture | |
|---|---|
| Sampling technique | Standard deviation |
| Cone and quartering | 6.81 |
| Scoop sampling | 5.14 |
| Table sampling | 2.09 |
| Chute slitting | 1.01 |
| Spinning riffling | 0.146 |
| Random variation | 0.075 |
Taken from Table 1.5 page 38,T.Allen Particle Size Measurement, 5th Edition Volume 1
Chapman and Hall 1997 ISBN 0 412 72950 4
The most common way that sample are taken within industry (scoop or spatula) is subject to an expected standard deviation of just over 5%.Thus if we are using this technique we could not expect to specify our material to better than 20% or so (3σ covering 99.7% of a Gaussian distribution is around 16% with this method). The choice is either to employ a spinning riffler for sample division or widen the permitted specification.Thus the recommendations in ISO133209 Section 6.4 which deal with repeatability allow for 3 consecutive aliquots a deviation of 3% for the D50 and 5% for the D10 and D90 in the tails of the distribution. This could only be achieved for samples that have either been correctly divided or were small enough that sufficient particles were in the highest size band.Thus this is a reasonable check on the homogeneity or otherwise of our sample rather than on the performance of the instrument. By the way, the doubling of the "permitted" deviations under 10 μm is more related to control of the dispersion of the material than to sampling.
Back to our important question on the best way to take a sample. Spinning rifflers have been around for at least 100 years. Here is an early version made of wood and buckets10:
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The modern version is with stainless steel trays or glass bottles:
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From companies such as Retsch or Microscal units can be obtained that will deal with 10 g - 100 kg of sample. These are conveniently and usually used in the laboratory in order to split a primary sample into a size suitable for measurement for, say, XRF or laser diffraction. One must realise at this stage that this method is the only one capable of producing less than 1% sample to sample variation. If one does not employ a rotary dividing technique and one wants less than 1% coefficient of variation sample-to-sample then this will certainly end in tears.The other alternative is to use scoop sampling and work with a larger acceptable specification.
Cone and quartering, although reasonably common, should never be employed11:
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Chute riffling is barely acceptable (just over 1% RSD):
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Riffle sample divider (Reproduced by permission from BS 5309, Part 4, 1976)
So what if we have a large mine dump or a glacial moraine ("so what?" you may say):
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What are the rules for sampling here? Allen provides two words of advice-don't and never. However, this is more flippant than practical (thus appeals to me!) and we need to outline the golden rules of sampling as expounded by Allen and others on many occasions:
The above photographs show the boulder-like particles (larger than a man together with the sub-micron rock flour in suspension at the Franz Josef Glacier in New Zealand). In all cases repeated samples (to reiterate, Kraft's recommendation is 20) should be taken to assess the homogeneity or otherwise and to get a feeling for the expected variations. Even this may have problems with friable or fragile material (e.g. needle-like pharmaceutical) where the simple act of sampling a barrel or container of powder with a spear is likely to cause damage to the particles:
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For mines and piles of materials then trenches can be dug and the literature (e.g. 1 and 11) contains details of these methodologies. However, actual measurements will indicate the huge spread of particle sizes in such situations and would be more practical than the flippant "Don't/never!" advice.
Slurry sampling is potentially much worse than sampling of dry powders in that segregation or settling is almost certain to be occurring. Extreme care and evaluation needs to be taken when sampling from a pipe or in a wet grinding situation as it is a virtual impossibility that the system is homogenous from a particle size point-of-view A Burt sampler is normally recommended for slurry or suspension sampling:
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It is reasonable now to examine some practical results of sampling that we have carefully undertaken at Malvern Instruments on purified teraphthalic acid (PTA) both as a dry powder and as a prepared slurry in water.The material is quite large and exhibits the potential for significant sampling errors.The recommendations of ISO13320 were followed in that, for riffled samples, the complete sampled sub-lot was used for the measurement. Repeatability was assessed for wet measurements (the marks on the plots indicating the full set of results) by taking 10 consecutive repeat measurements.
For successive scoop sampled aliquots we see standard deviations in line with those noted by Allen:
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We note too that the effect is only just seen at the D90 end of the distribution - there is virtually no effect on the smaller D10 and median, D50, points.There is a gradual fall in D90 as we remove the larger material (the nuts and raisins in the museli), which sits on top of the pile, but one can see why this type of (subtle) trend is rarely noted. In the case of a slurry sample we get an interesting step-effect:
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The initial samples were taken by pipette withdrawal from a well and continuously stirred (magnetic stirrer) slurry of the material contained in a beaker.This material in suspension shows little change.The step is at the point when no liquid remained and one started to sample, by spatula, the paste sitting at the bottom of the beaker. This shows segregation effects by the gradual rise in key parameters shown above.
In terms of samples extracted by spinning riffler and measured in their entirety we see no discernible trends in D90 over the 25 trays of the riffler:
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We note that in practical terms I cannot approach the wonderful (ideal?!) s.d. figures quoted by Allen but I can considerably improve on the values generated by simple scoop or slurry sampling and remove any systematic variation.
Last, but by no means least in practical terms let us examine some rules of thumb for product homogeneity.The author has witnessed a large pharmaceutical manufacturer being unable to control the dose of active ingredient with an inactive base material and this type of observation has been reported in the literature on numerous occasions13, 14.This was simply due to the large differences in sizes-micronized drug together with inactive up to 1000 μm or so.The general rule here is that we will reduce the propensity to "unmix" by mixing particle size distributions that do not differ by more than 0.3 - 1.4 (or 40% or so).The solution for the pharmaceutical customer above was obvious-micronize both the active and inactive.As an example if the drug is of mean size 50μm then to introduce a filler of 100 μm is inadvisable. Rather the filler should be within the range 15 - 70 μm. Small particles or a broad distribution aid packing and compaction strength and can help flowability issues. When small particles adhere to coarse ones then more homogenous mixtures can result but this could be in conflict of making smaller drugs for more rapid dissolution. As is usually the case a balance must be sought between particle size distribution, flowability and homogeneity.
A corollary to this relates to the recommendations within ISO13320 relating to verification of laser diffraction equipment. Here spherical materials of no more than 1 decade in diameter distribution are stated to be preferred.Although a 1 - 10 μm (or 10 - 100 μm) range in diameter does not appear significant it is equivalent to a 1000-fold difference in weight or volume so segregation is a real possibility unless the particles are particularly small where attractive forces bind them together.
Measurement by laser diffraction is characterised by a number of requirements that need to be met.
To avoid multiple scattering (a number/concentration/size constraint) then measurements are normally run at fairly low concentrations-typically 0.01 - 0.001 volume %.A pharmaceutical manufacturer may ask "What is the smallest amount of material that can be measured?" and may be wanting to minimise the amount of sample (expensive R & D material?) or exotic or dangerous or expensive solvent (at least for disposal). Hence the unwary customer and salesperson may be riding along the possibly dangerous track of specifying a small volume dispersion unit without remembering the requirements for adequate sample mass and also the need to keep material adequately in suspension. Failure to control either leads to lack of repeatability (consecutive measurements-the small number of large particles is not integrated a sufficient number of times or the larger particles are only occasionally "kicked" through the laser beam) and poor reproducibility (sub-sample to sub-sample variation based on the homogeneity or otherwise of the sample).
With a small volume unit we may only be using a small volume of dispersant and thus we could never load the amounts of material required for adequate sampling (at reasonable levels to ensure multiple scattering does not occur15) into the unit especially if the sample has any polydispersity. And if we sub-sample from our starting material there is an enormous danger that what we take (e.g. 50 mg) will certainly not be representative of the whole or if the sample has larger material present that the 10000 particles in the highest size band cannot be met.
Clearly it is the precision in the D90 or higher point of the frequency curve that becomes affected. Customers that try to specify a D99.5 or similar based on sieve type measurements are either deluding themselves or will have to select much wider tolerances on acceptable precision.The specification of a D100 (so if we don't find the single largest particle in the glacial moraine then the second biggest is the D100?) is so obviously ludicrous from a scientific point-of-view that such people need to be subjected to the Inquisition and burnt at the stake.The fact that some instrument manufacturers put such numbers on their analysis sheets shows that marketing and ignorance (normally mutually inclusive) rather than science and logic has played the main role in the decision making.
Thus from a sampling only point of view a small volume dispersion unit may only reasonably be selected if the particle size is small and/or narrow distribution. Of course a monodisperse sample would only require a single particle for statistical validity. Thus a larger sample dispersion unit even if it requires larger amounts of solvent may be the best statistical route to high precision with a fragile or friable material that is not capable of dry dispersion.There are constraints to on dry dispersion-the sample cannot be re-measured again or indeed repeat measurements taken on the same group of particles as is the case with wet-it's lost to the vacuum cleaner-which may be a problem if the sample is expensive and needs recovery. If we have plenty of sample and the material is capable of being suspended in air plus it will not exhibit attrition effects then dry would be a reasonable choice.The real answer to the pharmaceutical customer's question relates to how much the customer is prepared to lose and the desired degree of precision based on the material's particle size distribution. Indeed in certain cases if the entire universe of material (the whole lot!) is not taken for the analysis for a larger and/or polydisperse sample then the whole analysis may have dubious merit and it may be a case of garbage in - garbage out.
