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A new multi-axial particle shape factor—application to particle sampling

Dosti S. Dihalu* and Bastiaan Geelhoed

Received 29th April 2011, Accepted 30th June 2011

DOI: 10.1039/c1an15364a

Because for a given sample size the sampling uncertainty increases with increasing particle mass, the

mass of a representative sample depends on the particle mass during chemical, physical and biological

analysis. Sampling theory can be used to formulate the quantitative relationship between the particle

mass and the corresponding mass or weight of a representative sample. But in practice, especially for

small particles, it is often easier to evaluate the particle size in dimension of length (e.g. mm) rather than

in dimension of mass (e.g. mg). In order to be able to apply sampling theory to predict the mass or

weight of a representative sample, a well-defined methodology that relates the mass of a particle to its

size is required. We here propose a new multi-axial shape factor which requires information of multiple

sizes of the particle of interest, whereas a uniaxial shape factor only needs one. In view of the

information loss that is implicit in the use of a one-dimensional shape factor like the Brunton shape

factor, the here-proposed new multi-axial shape is expected to perform better. Experimental data

confirm the better performance of the new shape factor. A multi-segment generalisation of the new

multi-axial shape factor is proposed.

Fig. 1 Illustration of the relationship between particle size and sampling

uncertainty for an equal sample size. The sample on the left contains 67%

dark particles and is taken from a population containing 68% dark

particles; the sample on the right contains 50% dark particles and is taken

Introduction

An eminent step for analyzing materials for chemical, physical

and biological analysis is the taking of representative samples. In

practice, the concentration of a property of interest in a sample

will inevitably deviate from the corresponding concentration in

the original population of particles. The sampling uncertainty is

a measure of the dispersion of the sample concentration around

the true value (i.e. the population concentration). Knowledge

about this sampling uncertainty will enable one to make reliable

decisions based on the samples.

It is well known that, for a given sample size, the sampling

uncertainty increases with increasing particle size.8,15,17–19 This

can be intuitively understood: when sampling a material con-

sisting of small particles of a small particle weight (denoted m)

a sample of a certain weight will contain more particles than

a sample of the same weight but taken from a population con-

sisting of larger particles of a higher weight (see Fig. 1).

Because in practice materials of widely varying particle size

are routinely sampled, from the above it is clear that the amount

of sample material collected should depend on the particle size:

for coarse grained material a large sample should be taken,

while for fine materials a smaller sample may already be suffi-

ciently representative. This is reflected in many harmonizing

standards like ISO 10381-8 (ref. 14) and CEN TR 15310-1

(ref. 4).

Faculty of Applied Sciences, Delft University of Technology, Mekelweg 15,2629 JB Delft, The Netherlands. E-mail: d.s.dihalu@tudelft.nl;b.geelhoed@tudelft.nl

This journal is ª The Royal Society of Chemistry 2011

The objective of sampling theory is to formulate the quanti-

tative relationship between sampling uncertainty, particle mass

and other relevant data. Brunton3 provided an early mathe-

matical study on the question of how large a sample minimally

must be as a function of the particle size. In order to accomplish

this, he proposed a shape factor (denoted fBrunton here) that can

from a population containing 67% dark particles. Therefore, the sample

on the left is more representative than the sample on the right. Generally,

samples containing more particles are more representative than samples

which contain fewer particles, although other factors are also of influence

on the representativeness.

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be used to predict the particle mass, if the sieve size (denoted

D here) of the particle is known and its density (r):

m ¼ rfBruntonD3 (1)

where m ¼ mass of the particle, r ¼ density, fBrunton ¼ Brunton

shape factor, D ¼ particle diameter (or particle size).

Brunton used for D the minimum sieve opening size through

which the particle can pass. Subsequently, the shape factor

fBrunton is defined as the ratio of the volume of a cubic particle of

sizeD and the volume of the particle. This definition was recently

also published.16

The shape factor fBrunton was later adopted by P. Gy in his

sampling theory, which is widely applied.18 Gy’s formula for the

variance of the Fundamental Sampling Error (FSE)13 is derived

under the assumption that the shape factor fBrunton is constant (of

course this constancy refers only to all particles of the population

from which the sample is taken), and is given by:

VarðFSEÞ ¼ fBruntonglcD3

M(2)

where Var(FSE)¼ the variance of the FSE (unitless quantity can

be expressed as a squared percentage), g ¼ size range factor

(unitless quantity), l ¼ the liberation factor (unitless quantity),

c¼ themineralogical factor (specified in the same units as density,

e.g. g cm�3 or kgm�3),M¼mass of the sample (specified inunits of

mass), D ¼ particle diameter (specified in units of length).

It is noted that in Gy’s theory of sampling (see e.g. ref. 12), Var

(FSE) is just one component of the variance of the total sampling

error (TSE). However, knowing the magnitude of the FSE

component is essential (in Gy’s theory) because the FSE error

component can never be eliminated, according to Gy’s theory. If

all other errors that are identified in Gy’s theory have been

eliminated (e.g. the Increment Delimitation Error, the Increment

Extraction Error, and the Increment Preparation Error, see for

example ref. 18), the FSE will then, according to Gy’s theory, be

the largest contribution to the sampling uncertainty. The fact

that this equation contains the fBrunton parameter demonstrates

the importance of predicting the particle mass using a well-

specified shape factor.

Nevertheless, it is clear that, in practice, particles will generally

differ in shape and that therefore the ability of a constant fBruntonto accurately predict the particle mass should not a priori be

assumed to be perfect. However, Gy’s formula (eqn (2)) is

derived from a more basic equation that has the particle mass as

parameter instead of the particle size (see also ref. 11). This can

still be seen from eqn (2): c is in units of density, whereas the

product of fcD3 scales with the particle mass. An improvement in

the way particle size is ‘‘translated’’ into particle mass will

therefore, when applied to the more basic equation in Gy’s

theory of sampling, lead to improvement of estimates for the

sampling variance. This will subsequently also lead to improve-

ments of estimates for the minimum sample mass which are

derived from Gy’s equation for the sampling variance (see e.g.

ref. 10).

The here-proposed new shape factor requires information of

a number of independent sizes of the particle (denoted here by Di

with i ¼ 1, ., N), whereas the Brunton shape factor only needs

one, which cube is used proportionally for mass determination.

3784 | Analyst, 2011, 136, 3783–3788

As a first step in the generalization of eqn (1) we propose:

m ¼ rfNYN

i¼1

D3=Ni (3)

where m ¼ mass of the particle (expressed in units of mass or

weight), r ¼ density (expressed in units of mass per volume),

N ¼ multi-axial dimensionality, which is used for calculating the

particle mass (an integer number to be selected such that

a sufficiently accurate prediction of particle mass is achieved),

fN ¼ here-proposed new multi-axial shape factor (a unitless

quantity), Di ¼ the ith independent particle size (or particle

diameter) of interest (expressed in units of length). The subscript i

ranges from 1 to N. P ¼ product of a sequence of factors.

It can be seen that eqn (3) is a direct generalization of eqn (1).

If the multi-axial dimensionality is set to one (N ¼ 1) eqn (3)

reduces to eqn (1) if we take f1 to be equal to fBrunton. It is noted

that each Di is raised to a power 3/N, assuring that eqn (3) is

dimensionally correct for all possible choices of N.

The next step in the generalization of the expression for the

particle mass is made by taking into account that a general model

for the shape of a particle is to regard a particle as being

a composition of its segments. Particles may thus be regarded to

consist of a number of segments of a certain mass. Each seg-

ment’s mass is given by the above equation (eqn (3)), but with

potentially different values ofN, r, fN and theDi’s. We also allow

for cavities by regarding them to be ‘‘particles’’ which add

negatively to the total particle mass.

The multi-segment generalization of eqn (3) that is proposed

here is:

m ¼XW

k¼1

skrk fNk

YNk

i¼1

D3=Nk

ik (4)

where Sk ¼ a summation over segments (k) of which the particle

is composed, fNk ¼ the multi-axial shape factor for segment k in

the particle. The subscript k ranges from 1 to W; the subscript N

is set to the multi-axial dimensionality of segment k, rk ¼ the

density of the material of which segment k is composed. If

segment k is a cavity, rk is the density of the embedding material

of the cavity k, sk¼ sign quantity that can either have the value of

�1 or 1. If sk¼�1 then segment k represents a cavity or a ‘‘hole’’;

otherwise segment k is filled with material of density rk,

W ¼ number of segments of the particle (should be selected

sufficiently high for an accurate prediction of particle mass),

Nk ¼ the multi-axial dimensionality of segment k, Dik ¼ the ith

particle size of interest of segment k. The subscript i ranges from

1 to Nk. The subscript k ranges from 1 to W.

Indeed, for Nk ¼ 1 and W ¼ 1, eqn (4) is equivalent to

Brunton’s equation (eqn (1)). For Nk ¼ 3 and W ¼ 1, eqn (4)

corresponds to the equation that can be deducted from Sneed

and Folk,20 Blott and Pye.2 Hence, forW ¼ 1, f3,1 is also denoted

by fS&F here.

m ¼ rfS&FDxDyDz (5)

whereDx,Dy, andDz are the three orthogonal axes of the particle

as identified by the method of Sneed and Folk.20 It is noted that

Sneed and Folk20 proposed to use a triangular diagram in which

each location specifies a specific combination ofDx,Dy andDz. It

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is noted that eqn (3) (and subsequently also the more general eqn

(4)) can only be used if fN (or fNk) is known. In the next section, it

will also be discussed how to measure fN.

The new particle shape factor can potentially be implemented

in sampling theories that have particle masses incorporated

directly, e.g., ref. 9 and 13, and the application of such theories in

international standards (e.g., ref. 4 and 14). In current standards

and sampling protocols, usually a default value of fBrunton ¼ 0.5

is taken. In the same way, the new multi-axial shape factor can

potentially also be used by assigning a suitable default value to it.

Hence, the multi-axial shape factor can in principle replace the

use of fBrunton in these sampling standards and protocols, which

underlines the importance of the new multi-axial shape factor.

Furthermore, the application of the novel multi-axial shape

factor might result in improved results for estimation method-

ologies of sampling theory, e.g. ref. 6, and particle descriptions in

general.1,5

Finally, it is noted that sometimes secondary sampling steps

are performed after comminution (crushing of particles). Due to

the breakage of particles during the comminution process, the

particle shape is expected to change and, consequently, the new

shape factor needs to be recalibrated after each comminution or

predicted using a (yet to be developed) model of how particle

breakage influences the particle shape.21

Experimental

Regarding the here-performed experiments, it is mentioned that

for all cases particles were modelled as a single segment (i.e. we

apply eqn (3), which is equivalent to eqn (4) for W ¼ 1). In

practice, the new shape factor can be calibrated by applying eqn

(3) in a rewritten form combined with measurements of density

(r), the particle sizes Di, and weighing of the particle mass:

fN ¼ m

rQN

i¼1

D3=Ni

(6)

The above equation shows the increased effort required when the

multi-dimensionality increases, because higher N requires the

determination of more Di’s. It is noted that an advantage of

selecting N ¼ 1 (i.e. using fBrunton) is that only one particle size

(the sieve size, which is expected to be equal to the median value

of the three orthogonal lengths Dx, Dy, Dz) needs to be deter-

mined experimentally per particle used in the calibration. For

N ¼ 1, this can generally be achieved by sieving. When N is

selected higher than one it is not certain if sieving can be applied

to find theDi’s. However, if there is a relation between the results

of a sieve analysis and the particle shape (see e.g., ref. 7), a single

sieving experiment may provide information that can be used to

infer in an indirect way values of the Di’s of the particles. This,

however, is still an undeveloped and untested idea. Manual

measurement of particle dimensions allows determinations of the

Di’s in a straightforward way. State-of-the-art automated

dynamic digital image analysis devices can in principle be applied

to automatically determine the Di’s but further discussion of this

is outside the scope of this article.

Generally, eqn (6) will not be applied to each particle indi-

vidually in a sample, because the effort required for doing so

would be prohibitive for large numbers of particles. The idea is to

This journal is ª The Royal Society of Chemistry 2011

rather use this equation for a sufficiently small number of

particles, thus providing an average value that can subsequently

be used in eqn (3) to predict the particle mass also for particles (of

the same type) for which fN was not directly measured using eqn

(6). The error that is possibly made during this process depends

on three factors:

(i) the number of particles that are used in the calibration (viz.,

is this number large enough so that the selected particles suffi-

ciently represent the population?),

(ii) the constancy of fN, and

(iii) whether or not N is selected appropriately: not too low so

that the particle shape can be described accurately and not too

high so that the inclusion of many irrelevant dimensions does not

distort the description of the particle shape. It is expected that for

irregularly shaped particles a higher N is required than for more

regular particles.

Generally, all three factors will be expected to play a role at the

same time. Hence, for example, in order to obtain more accurate

predictions of particle mass, a decision may have to be made

between increasing the effort spent on analyzing more particles

(i.e. reducing the influence of the error-generating factor (i)) and

increasing the effort spent per particle analyzed by increasing N

(thereby reducing the influence of the error-generating factor

(iii)).

Hence, the calibration of the new multi-axial shape factor fNfor N > 1 required slightly more effort than fBrunton. However,

this increased effort is justified by the increased accuracy of the

results obtained. It is also noted that in the current practice often

fBrunton is not measured, but assigned the default value of 0.5 (see

e.g. ref. 12, page 341) based on qualitative descriptions of the

particle shape.

In order to show the difference in accuracy of the mass esti-

mation with the use of a shape factor of a higher multi-axial

dimensionality in comparison to one-dimensionality, three

different dried materials were analyzed: (i) coffee beans, (ii)

lentils, and (iii) conchigliette rigate pasta (see Fig. 2). A total

number of 100 particles of each material type were measured

from three orthogonal axes Dx, Dy, and Dz according to the

following procedure:

(1) Find the longest length of the particle (Dx).

(2) Project particle along the axis found in step (1) on the plane

perpendicular to the axis found in step (1).

(3) Find the longest length of the projection (Dy).

(4) Projection along the axis found during step (3) on the plane

perpendicular to the axis found in step (3).

(5) Find the longest length of the projection obtained in step 4

(Dz).

The density was determined from the water displacement that

was observed after adding the particle of interest to a test tube

filled with water.

For each particle, the ‘‘true’’ particle mass was obtained by

accurate weighing. Next, eqn (6) was used to calculate fN of each

individual particle and the average of the individual fN’s was

calculated for each type of particle (coffee beans, lentils or pasta

particles). This resulted in three average values of fN (namely one

for coffee beans, one for lentils and one for pasta particles).

These average values of fN were subsequently used to predict

the individual particle masses using eqn (3). Table 1 shows the

Root Mean Squared Errors of these mass predictions (in which

Analyst, 2011, 136, 3783–3788 | 3785

Fig. 2 The irregularly shaped materials that were analyzed (from left to right): coffee beans, lentils, and conchigliette rigate pasta.

Table 1 Comparison of the Root Mean Squared Error (RMSE) forpredicting the particle masses of coffee beans, lentils and pasta particlesusing the shape

RMSE

With fBrunton With fS&F

Coffee beans 20% 9%Lentils 8% 4%Pasta particles 12% 5%

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the masses of particles of the same type are predicted using the

same value for fN).

From Table 1 it can be seen that the more accurate results are

obtained when the higher-dimensional shape factor is used,

because the RMSE’s of the predicted masses show smaller

residual errors when using the fS&F shape factor than when using

the fBrunton shape factor.

Sieving is a very efficient way of obtaining information about

the distribution of a single measurement (the median of Dx, Dy,

and Dz of a particle) for all particles in the lot and measuring Di

for all i between 1 and N for each particle of the lot may seem to

require an enormous amount of extra effort. Fortunately, this

effort can be reduced by classifying the particles in a finite

number of particle classes. In principle, if we can get a represen-

tative particle from each class, it suffices to measure only that

single particle for each class. However, from a practical stand-

point we recommend to subject more than one particle from each

class to the measurement process of the Di’s for a more reliable

estimate. Hence, the new equation can certainly get practical use

for estimating sampling error variance.

Simulations

In order to illustrate the potential improvement in particle mass

prediction for higher selected values of N, we perform a thought-

experiment in which we use fN for N ranging from one to five to

predict the particle mass for a mixture of spheroids. It is assumed

that particles are perfect spheroids and for numerical simplicity it

is furthermore assumed that the density of each particle is equal

to one. The lengths of the three orthogonal axes of each spheroid

particle are selected for each particle at random from the uniform

probability distribution on the interval [0.1; 2.0].

3786 | Analyst, 2011, 136, 3783–3788

It is noted that the sieve size D used when fBrunton is applied to

predict the particle mass was set to the median value of the three

orthogonal axes of each spheroid particle. For predicting the

particle mass using f2, the minimum and maximum values of the

three orthogonal axes were taken as D1 and D2.

The results of this thought-experiment can be found in Table 2

and show that whenever fS&F is used no error is made, whereas

the square root of the Mean Squared Error in the particle mass

prediction made by using fBrunton is found to be 284%. This

thought-experiment supports the proposition that, in practice,

fS&F can result in more accurate estimates of particle mass than

fBrunton.

The practical work using coffee beans, lentils and pasta

already showed that the mass of individual particles can be

estimated more accurately if three measurements of the size of

each particle are provided rather than a single measurement. It is

expected that, depending on the kind of particles, there exists an

optimum choice for the multi-axial dimensionality (N) which

results in the most accurate predictions of the particle mass

(lower MSE). This is confirmed by Table 2: the MSE becomes

lower as N increases from 1 to 3; reaches the optimum (zero

error) for N ¼ 3 case. The MSE increases for N > 3. This is as

expected: the shape of the perfect spheroids is fully determined by

Dx, Dy and Dz. Attempting to describe the shape of these perfect

spheroids with more than these three measures of dimension will

therefore inevitably lead to a deterioration of the accuracy of the

shape description. This is what the increasing MSE when going

fromN¼ 3 toN¼ 5 reflects. However, theN¼ 5 case still results

in a significantly lower MSE than the N ¼ 1 case.

We repeated the experiment performed in Table 2 for the

particle shapedepicted inFig. 3 (two cubes attached to eachother).

The sides of the cubes are the particle sizes (N ¼ 3; W ¼ 2). Ten

randomparticleswere createdby selecting eachDik randomly from

the interval [0.1; 2.0]. Theparticle sieve size (used for applicationof

fBrunton) was taken to be the second largest value in the set {D21 +

D22,D31,D12,D11,D32}. For application of fBrunton via eqn (1), the

RMSE ¼ 155.0%, whereas application of the combination

(f3,1¼ 1; f3,2¼ 1) via the here-proposedmore general eqn (4) results

in a perfect prediction of particle mass (RMSE ¼ 0%).

Conclusions

More accurate predictions of particle mass are in principle

possible when using the here-proposed new and more general

This journal is ª The Royal Society of Chemistry 2011

Table 2 Thought-experiment of predicting the particle mass of a set of 10 hypothetical particles using a constant shape factor for all 10 particles. Theparticles in the thought-experiment are perfect ellipsoids, Dx, Dy, Dz are randomly generated from a uniform probability distribution on [0.1; 2.0]. Forellipsoid particles fS&F¼p/6; the Brunton shape factor was obtained by averaging the fBrunton of each separate particle (calculated by eqn (6) withN¼ 1).The shape factor f5 was calculated by using D1 ¼ Dx, D2 ¼ Dy, D3 ¼ Dz, D4 ¼ (Dx + Dy + Dz)/3 and D5 ¼ (DxDyDz)

1/3

Particle no. Dx Dy Dz

Error2

N ¼1 (fBrunton)

N ¼2 (f2)

N ¼3 (fS&F)

N ¼5 (f5)

1 0.91 1.43 0.12 0.22 0.00 0.00 0.002 0.69 0.49 1.30 0.00 0.00 0.00 0.003 1.46 0.58 0.50 0.01 0.01 0.00 0.004 1.93 1.58 1.77 1.29 0.02 0.00 0.035 0.50 0.53 1.44 0.01 0.02 0.00 0.006 0.43 1.97 1.47 2.65 0.05 0.00 0.007 1.14 1.63 1.07 0.00 0.05 0.00 0.008 1.16 1.69 1.98 1.98 0.01 0.00 0.019 1.22 0.41 0.39 0.00 0.01 0.00 0.0010 1.60 1.31 1.45 0.33 0.01 0.00 0.01

ffiffiffiffiffiffiffiffiffiffiffiffiMSE

p¼ 284.1% 47.8% 0.0% 26.1%

Fig. 3 Example of a typical shape that is of particular interest for

application of the new shape factor. The particle consists of two

segments: segment 1 and segment 2. The Di,j’s that are to be used in eqn

(4) are indicated in the graph.

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multi-segment, multi-axial shape factor fNk instead of the

Brunton shape factor fBrunton. When the multi-axial dimension-

ality is set to three and for particles consisting of a single

segment, fNk becomes the Sneed and Folk shape factor fS&F.

In view of the information loss that is implicit in the use of

fBrunton, the fS&F is expected to perform better because there is

less loss of information. Experimental data confirm the better

performance of fS&F.

Selecting a higher multi-axial dimensionality (N) has the

potential of increasing the accuracy of the particle mass predic-

tion. However, selecting N higher also leads to an increased

required effort in determining Di for i ¼ 1, ., N. Hence, there

will be a trade-off between the required effort (which scales with

N) and the accuracy of the particle mass prediction. Also,

depending on the kind of particle, there may be an optimal choice

ofN which results in the most accurate predictions of the particle

mass. In view of this, future experiments are required to find out

which value of N is regarded to be optimal for specific cases.

More complex particle shapes can be accurately predicted using

the multi-segment generalization of the multi-axial shape factor

(fNk). A thought-experiment with randomly generated particles

consisting of two cubic segments confirms this.

This journal is ª The Royal Society of Chemistry 2011

Outlook

A future application of the new particle shape factor in sampling

theory could result in a new formula for the sampling variance

that contains the new shape factor. Based on the form of eqn (2),

an educated guess is:

VarðFSEÞ ¼fNglc

QN

i¼1

D3=Ni

M(7)

However, a more rigorous derivation is required, which may

result in a different equation.

Acknowledgements

The authors would like to thank Kevin Fung and Ole Thijs for

the effort they put in the size determinations of the particles used.

This work was performed as part of a project supported by the

Netherlands Technology Foundation STW, under STW grant

7457. The Netherlands Forensic Institute, Deltares, Nutreco,

Hosokawa Micron B.V., and Merck Sharp & Dohme are

members of the users’ committee of this project.

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This journal is ª The Royal Society of Chemistry 2011