What is the Effect of Sample Size on Particle Size Distribution Measurements?

Particle size distributions are the results generated by the AccuSizer. Rules of statistics govern the properties of the distributions. A vital question when employing a particle size analyzer is how much of a sample has to be measured to correctly define the resultant size distribution. This article addresses this issue, mathematically and experimentally.

Introduction

When particles in a sample are the same size, one particle can be measured to report the result. If the sample has a narrow distribution, such as 10-25 µm, then measurement of just a few particles can define the distribution. However, if the sample has a wide distribution, then it is necessary to measure thousands of particles to define the actual distribution.

The primary interest also needs to be determined. If a central value such as the meridian is of interest, then less particles can be measured than if the focus is on the edge of the distribution, e.g. the D90 or D95.

As with other aspects of particle size analysis, it is better to define what is required, and how the data is going to be used before starting the measurements. When determining the amount of sample to measure (how many particles to analyze when employing AccuSizer, a counting technique), some information on distribution width is needed before it is possible to outline the needed experimental parameters. This includes the number of particles, or the time period of the experiment.

The Standard Error Approach

The standard error (SE) of a sample with sample size n, is the standard deviation of the sample, divided by the square root of n as shown in equation 1.

 (1)

where

s = standard deviation
n = number of particles counted

So

 (2)

If the sample has a standard deviation of 2 and a 2% standard error is allowed, then the number of to-be-measured particles is

If the interest is in mean value, then a value of 5000 particles to measure is good to begin with.

ISO 13322-1 Approach

The issue of how many particles should be sampled for achieving satisfactory results is described in the standard ISO 13322-1, Particle Size Analysis—Image Analysis Methods—Part 1: Static Image Analysis Methods. The methodology described in the ISO standard is based on the work of Masuda and Innoya. A summary of this methodology is given below. The number of particles to be analyzed, n, is given by:

 (3)

Where:

δ is the relative error

K is numerically determined by the confidence limit, particle distribution, and other parameters.

Next

 (4)

Where:

 (5)

Where:

n* = number particles needed for measurements to realize a certain level of confidence

δ = relative error

σ = population standard deviation

c = β + α

β = basis number

α = constant

The relation between probability, P, and u is shown in Table 1:

Table 1. Relationship between P and u

P% u
50 0.67
75 1.15
80 1.28
90 1.64
95 1.96
97.5 2.24
99 2.58
99.5 2.81
99.8 3.09
99.9 3.29

The number of particles required, n*, analyzed using Equation (4) with admissible error of 5% as a function of the geometric standard deviation, sGSD, of the sample is presented in Table 2. Here, the probability, P, is taken as P = 0.95 (u = 1.96 in Table 1).

Table 2. Number of particles n* as a function of geometric standard deviation sGSD

σGSD n*(DMM) n*(Sauter) n*(DMV)
1.1 585 389 131
1.15 1460 934 294
1.2 2939 1808 528
1.25 5223 3103 843
1.3 8526 4920 1274
1.35 13059 7355 1750
1.4 19026 10504 2363
1.45 26617 14457 3096
1.5 36007 19295 3956
1.55 47358 25093 4952
1.6 60811 31919 6092

The first column in Table 2 denotes the width of the sample distribution expressed at the sGSD. As the maximum value of σGSD of 1.6 is not very wide, many samples can exceed this value, so the last row is considered for samples with an unknown width (σGSD). The second through fourth columns exhibit the number of to-be-analyzed particles, n*, based on which mean value is of interest. As the mass median diameter (DMM) is generally used, the second column is referenced for most samples.

When using the AccuSizer and converting to a volume distribution, about 60,000 particles are analyzed to achieve a high confidence level in the results.

Experimental Measurements

The discussed example of SE, and the methodology adopted in ISO 13322-1 means that between 5,000 and 60,000 particles need to be analyzed for realizing a high confidence level in the calculated mean value. To compare actual measurements to the theoretical limits, an experiment was performed. The sample chosen was hydrated alumina powder mixed in water and measured on an AccuSizer SIS system using a LE400 sensor, and dynamic range being 0.5-400µm. The analyzed sample volume was varied between 0.05-2 mL so as to realize variation in the analyzed number of particles. The D10, D50, and D90* as a function of number of particles analyzed, are shown in Table 3 and Figure 1.

Table 3. # particles sized vs. D10, D50 and D90

# Sized D10 (µm) D50 (µm) D90 (µm)
442 8.5 13.82 21.65
2579 8.3 13.82 20.99
5213 8.73 13.82 21.71
11364 9.47 15.39 25.63
13196 10 16.24 25.02
16748 17.41 24.07 44.11
37688 17.14 29.42 50.45

Figure 1. # particles sized vs. D10, D50 and D90

The data in Figure 1 allows us to draw many observations and conclusions including:

• The results at #Sized = 11,364 seem to be erroneous with all values under-reported.
• For achieving close to accurate results, at least 16,748 need to be analyzed.
• When analyzing too few particles, the error in the D10 or D50 is less than that in D90.
• If the experiment was stopped when 5,213 were analyzed, the results would have appeared to be accurate, but in reality they would have been wrong.

The high error in the D90 needs to be noted. A few large particles cause the biggest problems in many industries. In microelectronics industry, some large particles in a CMP slurry cause surface defects, reducing yields and profits. In CMP slurries, detection of the large particle count (LPC) needs sufficient statistics at the tail of the distribution – to the right of the D90 (Figure 2). More particles need to be analyzed than proposed, by theoretical or experimental results for accurate definition of the tail of the distribution.

Figure 2. 90% of the distribution lies below the D90, 50% below the D50, and 10% below the D10

Conclusions

It is risky to analyze just a few particles and think that the distribution is accurately defined. For reporting a mean value for very narrow distributions, it may be easy to analyze only a few particles. However, a large number of particles (tens of thousands) have to be analyzed for wider distributions to accurately define the true particle size distribution. The number should be higher when the D90 or tail of the distribution is of primary interest. The ideal analytical instrument for easy analysis of a large numbers of particles, for defining the distribution accurately and detecting tails of distributions is the Entegris AccuSizer.

This information has been sourced, reviewed and adapted from materials provided by Entegris

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