Introduction:
One of the least well understood parts of the analytical process is the collection and handling of heterogeneous samples. These include multiphase liquids and gases, but especially heterogeneous solid samples such as soils, sediments and rocks for minerals, nutrients and contaminants. There are essentially four separate but related steps. These are:
 Initial collection of the sample,
 Transportation to the place of analysis,
 Subsampling and preparation of the initial sample to an appropriate size for analysis and
 Actual analysis
The last of these is dealt with exhaustively by universities and instrument manufacturers. Both drive unceasingly toward speed, automation, precision, low cost, high sensitivity and smaller and smaller sample sizes. Considerably less attention is paid in designing and implementing the collection of the initial sample. This has been the purview of a relatively small and dedicated group of scientists, engineers and manufacturers. Even less thought and effort is put into taking a proper subsample of the initial sample for the purpose of the actual analysis (Steps 2 and 3). In this blog we will focus on Step 3 – the taking of a proper subsample for analysis.
General Principles:
Subsampling of a correctly taken sample from a much larger lot is best guided by the same principles that were used to take the initial sample. Taking of the initial sample is, after all, a “subsampling” of a much bigger lot – e.g. the box car, the truck load, four hours of weigh belt flow, etc. For heterogeneous solids, these principles have been thoroughly worked out by Dr. Pierre Gy, Dr. Francis Pitard and others (see Bibliography below). Their focus was originally on the mining industry, but over the past several decades these principles have been applied to agricultural, pharmaceutical and even environmental fields. In 2003, the USEPA published “Guidance for Obtaining Representative Laboratory Analytical Subsamples from Particulate Laboratory Samples,” EPA/600/R027. In 2012, ASTM updated D6323, “Laboratory Subsampling of Media Related to Waste Management Activities,” using the principles set forth by Gy and USEPA. These principles are now well established, not only for initial samples but also for doing subsampling prior to analysis.
Sampling and Preparation Equipment:
It is essential that we have the proper subsampling equipment onhand. That equipment must be able to collect the subsample without biasing the result. That includes the proper scoops, shovels, sieves, grinders and sample splitters. Unfortunately it is not possible to select the right equipment until the actual subsample size and condition (especially particle size) are known. This decision must be postponed until we work out an overall strategy and see what is required. Hence, our first approach must be a mental exercise where we ASSUME that we have the right equipment. Nevertheless, it is positively critical that we return to this topic after we have determined what it is that we are handling.
The SubSampling Strategy:
In general we will begin by looking at the sample that we receive from the field and the final subsample that will be needed in the actual analysis. In almost all cases, the sample received from the field is much too large to be used directly for analysis. We might receive several kg of marblesized rocks and the initial preparation step for analysis might call for 2.00 grams of sample ground to less than 1 mm in diameter. The question is now how do we go from the initial sample to the lab analysis sample without adding bias or undue variability? We will apply Gy’s principles for taking a “representative” sample from the initial sample, reducing the size of that aliquot and then taking a “representative” sample from that. We might find that we have to do this several times. Graphically we can look at this on a “nomagraph”:
At each step we will want to stay well below the variance goal of our measurement. Every time we make a new step we must “propagate” the error by adding the variances. It is also extremely important that we take subsamples “correctly” so that we only add to what Gy calls the “Fundamental Error” and we do not add to the problem with additional, uncontrolled errors or biases.
When correctly taking a subsample, the variance of the “Fundamental Error” (s^{2}_{FE}) may be described as:
Where:
s^{2}_{FE} = The variance of the Fundamental Error
M_{S} = The mass of the subsample in grams
M_{L} = The mass of the “lot” from which the subsample is taken
C = The sampling constant in g/cm it includes several factors
d^{3} = The diameter of the largest particles in cm
When M_{L} >> M_{S} the Equation 1 reduces to:
The sampling constant factor C is the product of four additional factors, c (compositional factor), f (shape factor), l (liberation factor) and q (granulometric factor). C is the product of these four factors:
The compositional factor, c, takes into account the concentration of the component of interest. The lower the concentration of interest the more difficult it is to get a low relative standard deviation. The compositional factor may be estimated by:
a_{L} = mass fraction of component
λ_{M} = density of the component bearing particles
λ_{g} = density of the background material
When a_{L }is small (<= 0.1), Equation 4 can be reduced to:
The liberation factor, l, is related to how completely the component of interest is separate from the components that are not of interest. The liberation factor is highest when the component of interest exists as pure particles of substance in an environment of “inert” particles. An example might be trying to determine the lead content of a berm at a shooting range. The lead exists is slivers and clumps of nearly pure lead in a big pile of sand. The liberation factor is based on the formula:
a_{L} = mass fraction of component in the lot
a_{max} = mass fraction of component in the largest fragment
There are other methods for estimating the liberation factor. Nevertheless, the liberation factor is rather difficult to measure. It can, however, be estimated using the following table for typical values:
l 
Type of Material 
1.0 
Analyte 100% “liberated.” This is recommended for use with environmental samples contaminated with pure or nearly pure components. 
0.8 
Very heterogeneous materials with easily distinguished particles with high concentration of the component of interest 
0.4 
Heterogeneous materials 
0.2 
Average materials 
0.1 
Homogeneous materials 
0.05 
Materials known to be very homogeneous 
Table 1
The shape factor, f, is a measure of the particles divergence from a perfect cube. Needles and flakes are the extremes of divergence. The factor is hard to measure but can be estimated by:
f 
Description 
>1.0 
Needlelike materials. Asbestos would be an example. 
1.0 
All the particles are cubes 
0.5 
All particles spheres. Most materials, including minerals and environmental samples, are more or less in this category 
0.2 
Soft, homogeneous solid particles, such as tar droplets or gold flakes 
0.1 
Flaky materials such as mica 
Table 2
The granulometric factor, g, accounts for the distribution of particle sizes. The more uniform the particle size distribution the higher the granulometric factor. This is difficult to measure but can be estimated from the table below:
g 
Description 
0.25 
Undifferentiated materials. This includes most soils and rocks that have been crushed to a maximum size but not differentiated by sieving. 
0.40 
Materials passing through a screen (i.e. some has been retained and rejected) 
0.50 
Materials retained on a screen 
0.60 – 0.70 
Materials retained between two screens 
0.75 
Naturally sized materials such as cereal grains and some sands 
1.00 
Uniformly sized materials such as ball bearings 
Table 3
For laboratory subsampling, Equations 2 and 3 turn out to be the most useful. Generally, a_{L }is around or below 0.1. Furthermore, we can usually use Tables 1 through 3 to estimate the values needed for calculating C. Our strategy would include evaluating our ability to take a correct subsample from the lot that was presented to the laboratory AND planning to take final subsamples for analysis. This is best illustrated by an example.
Example of Laboratory SubSampling Strategy:
Let us suppose that a correct sample of a heterogeneous stream of crushed rock has be taken during loading of a train car. Let us also suppose that our task is to measure the carbon content of the rock. Let us also suppose that the content (a_{L}) is roughly 0.1. And finally, let us target a relative standard deviation of the Fundamental Error at 5% or less. We will use Equations 3 and 4 for our calculations.
Our first task would be developing a strategy for taking a manageable subsample for transporting to the laboratory. It is likely that the correctly taken initial sample is many kilograms. Our method of carbon analysis might be a CHN analyzer that prefers a few milligrams of sample. Let us assume that our target is 0.004 gram for the final aliquot to go into the CHN analyzer. Let’s also assume that we have additional tests that are less stringent and could use up to 100 grams of sample for the analysis. Now our strategy might be twofold:
 Create a 100 gram subsamples for some analyses and
 Create a 0.004 g subsample from the above for CHN analysis.
We could do this by grinding the entire initial sample to a fairly course particle size for the 100 gram tests and then take 1  100 gram subsample and grind it to a much finer particle size for the CHN test.
Initially let’s start with our our target Fundamental Error is a relative standard deviation of 5%. Our first grinding for subsampling the initial sample could be calculated by:
SubSampling Example 


Grind a Sampling to 100 g 





sFE 
0.05 
Target Relative Standard Deviation 
sFE^2 
0.0025 
Target Fundamental Variance 



Ms 
100 
g 
aL 
0.1 
fraction 
λM 
2.6 
g/cm 
c 
26 
g/cm (Equation 5) 
l 
1 
(Table 1) 
f 
0.5 
(Table 2) 
g 
0.25 
(Table 3) 
C 
3.25 
g/cm (Equation 3) 



d^3 
0.076923 
cm 
d 
0.43 
cm 
d 
4.25 
mm 
We could easily meet our objectives for the first set of tests by a very coarse grind. Unfortunately, when we propagated the error for a second grind we would find that we could never do better than the 5% we accepted in this first step. Hence, we would want to make this first step much better than 5% relative standard deviation. A typical approach might be to grind the sample to pass through a Number 8 mesh (2.38 mm) for the first set of tests. If we did that, we would modify our sFE to 2% (see below):
SubSampling Example 


Grind a Sampling to 100 g 





Ms 
100 
g 
aL 
0.1 
fraction 
λM 
2.6 
g/cm 
c 
26 
g/cm (Equation 5) 
l 
1 
(Table 1) 
f 
0.5 
(Table 2) 
g 
0.25 
(Table 3) 
C 
3.25 
g/cm (Equation 3) 
d 
0.238 
cm 
d^3 
0.013481 
cm 



sFE^2 
0.0004 

sFE 
0.020932 

In order to have our overall propagated error less than 5% relative standard deviation, we need to shoot for an sFE^{2} of .0025 – 0004 ~ .0021 in the second grind. We would need to grind the 100 gram aliquot to 0.14 mm (100 mesh) and split it correctly down to 0.004 gram. This is shown below:
SubSampling Example 


Grind a Sampling to 0.004 g 








sFE^2 
0.0021 
Target Fundamental Variance 



Ms 
0.004 
g 
aL 
0.1 
fraction 
λM 
2.6 
g/cm 
c 
26 
g/cm (Equation 5) 
l 
1 
(Table 1) 
f 
0.5 
(Table 2) 
g 
0.25 
(Table 3) 
C 
3.25 
g/cm (Equation 3) 



d^3 
2.58E06 
cm 
d 
0.01 
cm 
d 
0.14 
mm 
Our strategy becomes:
 Grind the initial sample to pass a Number 8 mesh
 Split the ground material correctly into at least 2 – 100 gram aliquots
 Grind one of the 100 gram aliquots to pass Number 100 mesh
 Split the ground material correctly into several 0.004 gram aliquots
Back to Sampling Equipment:
Now that we have a strategy we can select equipment properly. The first two steps are probably pretty easy. There exists a lot of equipment to grind to 8 mesh and split into 100 gram aliquots (e.g. Riffle Splitters). The last two steps could be challenging based on the nature of the material (too hard, thermally unstable, sticky, stringy, etc.). The last step could be especially challenging. There exist microRiffle Splitters that can split samples easily to a few grams. Nevertheless, correctly selecting a 0.004 gram aliquot even from a 1 gram subsample can be difficult.
It is likely that we will want to select this final 0.004 gram aliquot by taking multiple scoops from a 1 gram pile. We should design our scoop with a flat bottom and square sides to avoid as much particle discrimination as possible. Here we should plan to do the best we can, but we should test our procedure for variance before declaring it “good.” A typical plan would be to run at least 10 replicates from a single pile to evaluate how we are doing. If after several trials we see that we can reduce the number to fewer than 10 without exceeding our variance goal we can do so later.
Summary:
This blog shows a typical approach for designing a laboratory subsampling plan. Many features have been left out of the discussion. There are a lot more details and guidance given by Gy, Pitard, USEPA and ASTM. Especially important is a thorough knowledge of the equipment and techniques available for correct physical sampling. The reader should consult these additional resources before declaring a sampling procedure good. We have dealt with Fundamental Error, but we have not dealt with the tremendous bias that can result from improper physical techniques. Improper technique can result in huge errors that are very difficult to detect and impossible to resolve.
Bibliography:
Gy, Pierre, Sampling for Analytical Purposes, John Wiley and Sons, New York, 1996.
Pitard, Francis, “Pierre Gy’s Theory of Sampling and C. O. Ingamells’ Poisson Process Approach,” Doctoral Thesis, Aalborg University, Campus Esbjerg, Denmark, June, 2009.
ASTM D632312, “Standard Guide for Laboratory Subsampling of Media Related to Waste Management Activities,” ASTM International, West Conshohockenn, PA, 2012.
Gerlach, Robert & Nocerino, John, “Guidance for Obtaining Representative Laboratory Analytical Subsamples from Particulate Laboratory Samples,” US Environmental Protection Agency, EPA/600/R03/027, November, 2003.
Comments