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Geoderma 135 (2006) 118 – 132
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Laser diffraction, transmission electron microscopy and image
analysis to evaluate a bimodal Gaussian model for particle size
distribution in soils
Linda Pieri, Marco Bittelli ⁎, Paola Rossi Pisa
Department of AgroEnvironmental Science and Technology, University of Bologna, Italy
Received 19 April 2005; received in revised form 20 October 2005; accepted 22 November 2005
Available online 23 January 2006
Abstract
The Shiozawa and Campbell Gaussian bimodal model describes the particle size distribution (PSD) in soils as a weighted sum
of two fractions: the primary minerals (sand and silt) and the secondary minerals (clay) fraction, each described by a Gaussian
function.
This model was developed and tested using traditional sedimentation techniques analysis for PSD such as sieving and
hydrometer. Because of the lack of particle size distribution data in the clay range, Shiozawa and Campbell set the mean and the
standard deviation in the clay fraction as a constant. Today, the availability of laser diffraction (LD) techniques makes it possible to
overcome this limit and test the model by using a soil dataset that includes the clay fraction distribution.
This paper describes the results of the test of the Shiozawa and Campbell Gaussian bimodal model on eight samples, six of them
from different locations in Washington State (USA) and two from a hillside area of Northern Italy. PSD analysis was performed
with sedimentation techniques, small-angle laser diffraction apparatus and transmission electron microscopy, the latter allowing
measurement of very fine particles (sizes down to 0.05 μm).
To test the effect of the PSD technique on the particle-size measurement and therefore on the model reliability, a comparison
between sedimentation techniques and LD was performed. Moreover a validation of the LD method in the clay range was
performed by comparison of LD to Transmission Electron Microscopy and Image Analysis methodologies.
The results from the bimodal model showed that the model provides a good characterization of PSD for five of the eight
samples analyzed only, revealing that more complex distributions are required for a loam, a silt loam and for a clay soil, where
multimodal modes were found.
The comparison between sedimentation technique and LD showed that the volume percentage of the clay-size fraction obtained
by laser diffraction was lower than the mass percentage of the clay-size fraction measured by pipette. The silt fraction displayed the
opposite trend. Transmission Electron Microscopy and Image Analysis of the clay fraction showed that Laser Diffraction provides
an overestimation of the mean diameter in the clay fraction, when particles are assumed to be represented as spheres.
© 2005 Elsevier B.V. All rights reserved.
Keywords: Particle size distribution; Bimodal Gaussian model; Laser diffraction; Transmission electron microscopy
⁎ Corresponding author. Department of AgroEnvironmental Science
and Technology, University of Bologna. Viale Fanin, 44 - 40127
Bologna, Italy.
E-mail address: [email protected] (M. Bittelli).
0016-7061/$ - see front matter © 2005 Elsevier B.V. All rights reserved.
doi:10.1016/j.geoderma.2005.11.009
1. Introduction
Particle Size Distribution (PSD) is a soil property that
provides fundamental information about the size and the
L. Pieri et al. / Geoderma 135 (2006) 118–132
distribution of the soil mass fraction. It is commonly
used for soil classification (Gee and Bauder, 1986) as
well as for the estimation of other soil properties, such as
the water retention curve and the soil thermal conductivity (Campbell and Shiozawa, 1992; Campbell et al.,
1994; van Genuchten et al., 1999; Wösten et al., 2001).
A particle size analysis usually involves the measurement of the fractions of clay, silt and sand. Once
the values of the three fractions are known, a textural
triangle can be used for soil classification. While the
textural triangle and the size fractions have been extensively used for soil classification, neither the former nor
the latter provide adequate PSD characterization (Shiozawa and Campbell, 1991; Bittelli et al., 1999).
A better characterization of soil texture can be
obtained by describing the PSD by means of mathematical models. Many alternative models have been proposed to characterize PSD.
Among them, Shiozawa and Campbell (1991) presented a bimodal lognormal Gaussian distribution to
characterize the PSD of various soil samples. Since
the traditional sedimentation methods for PSD analysis
did not provide detailed data in the clay range, Shiozawa and Campbell (1991) set the mean and the standard deviation as constants in the clay fraction. Buchan
et al. (1993) pointed out the lack of measurement of
PSD in the clay fraction, noting that the assumption of
arbitrary and constant mean and standard deviation is a
limitation for the proposed model.
In the last few decades, there have been considerable
research efforts to develop alternative techniques that
would provide more detailed particle size characterization in the clay range as well (diameter b 2 μm). Because
of its mineralogical properties and high specific surface,
the clay fraction is usually the most important fraction
affecting solute adsorption and exchange (Hillel, 1998).
It is therefore very important to correctly describe PSD
in this size range. Laser diffraction (LD) techniques
available today are powerful methods for particle size
measurement and can be successfully used for broadparticle size distribution analysis (Martin and Montero,
2002). Consequently the lack of PSD data in the clay
range can now be overcome by using these techniques,
where PSD can be measured down to 0.05 μm with as
many as 25 size classes below 2 μm (Wu et al., 1993).
Because of the availability of this technique, it is
now possible to test the Gaussian model in the clay
range as well, and test if this model is applicable to
soil PSD data.
The purpose of this paper is: (a) to test the Shiozawa
and Campbell (1991) model by using a soil dataset
encompassing a wide range of textural classes and
119
providing a broad particle size analysis, and (b) to
verify the applicability of a bimodal Gaussian model
when more detailed information on the clay fractions
are available.
1.1. The effect of the measuring technique on PSD
analysis
While LD has been progressively more utilized for
PSD analysis, there are still debates regarding the validity
and applicability of this method, especially when compared to the common sedimentation-based techniques.
The traditional techniques used to measure PSD in
soils are based on sedimentation analysis, where the
particle size is determined by measuring its settling
time into a liquid (Gee and Bauder, 1986). The two
most common sedimentation methods are the pipette
and the hydrometer, which provide comparable results
if similar pre-treatment protocols are followed (Walter
et al., 1978). Usually the pipette also requires a measurement of the sand fraction by wet and dry sieving.
The sedimentation methods have several disadvantages: (a) small ranges and limited number of
size classes when compared to other techniques such
as LD, (b) a lack of reliable data at smaller sizes
(b2 μm) due to Brownian motion effects on sedimentation times (Loveland and Whalley, 2001), (c)
long analysis time and, (d) assumptions about particle density because of the mass-based nature of the
analysis (Clifton et al., 1999).
On the other hand, most of the PSD databases have
been implemented using data from sedimentation-based
measurements, therefore most of the soil classification
and characterization have been based on these techniques. However, because of the experimental limitations,
many alternative methods have been developed and
tested (Allen, 1997). Among them, LD is a promising
method because it overcomes many of the disadvantages
of the sedimentation techniques. LD has the following
advantages: (a) it provides a wide range of size classes
including many data points b2 μm, (b) it is fast (usually,
one sample analysis after pre-treatment takes between 5
to 15 min), and (c) it is independent of the particle
density because it provides a volume-based distribution.
Comparisons between sedimentation methods and
LD have been performed by several authors, however
there is still disagreement between results. Konert and
Vandenberghe (1997) found that LD ‘underestimated’
the clay fraction when compared to the sieve and pipette
method. These authors found a coefficient of determination R2 = 0.91 for 158 soils, by applying the relationship y = 0.361x − 0.232 in the clay fraction, where x is
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L. Pieri et al. / Geoderma 135 (2006) 118–132
the clay fraction determined by sieving and pipette, and
y is the clay fraction determined by LD.
Eshel et al. (2004) also found that for 40 of the 42
soils analyzed, LD yielded a smaller clay fraction than
the pipette method, however the relationship for their
soil samples in the clay fraction was y = 0.345x + 2.69,
with R2 = 0.72.
On the other hand Wu et al. (1993) and Muggler et
al. (1997) reported good agreement between LD and
sedimentation techniques. Dur et al. (2004), using LD
and transmission electron microscopy (TEM), found
good agreement between LD and TEM when measuring
the distribution of number, volume and surface area of
particles in the clay fraction. However these authors
found that the representation of particles in the clay
range as flat discs provided a better representation of
PSD than the common spherical representation.
As pointed out by Eshel et al. (2004), the lack of
agreement between different studies could be due to
several sources of error, inherent to the two different
techniques. LD is independent on particle density, while
sedimentation methods are dependent on particle density, therefore uncertainty about particle density can
affect PSD. For instance, particle density can vary in
the same soil sample from 1.6 to 2.9 Mg m− 3 (Clifton et
al., 1999), therefore the assumption behind the sedimentation methods, of a single particle density, can be a
source of error. On the other hand, LD methods are
susceptible to the variation of the refractive index,
which can vary from sample to sample based, for example, on the soil colour. In their study Muggler et al.
(1997) used a refractive index (real part) of nr = 1.56,
while Eshel et al. (2004) used a value of nr = 1.5, yielding differences in their PSD analysis.
However, the most important factor affecting the PSD
analysis is probably the irregularity of particle shapes
(Konert and Vandenberghe, 1997; Dur et al., 2004). For
simplicity most of the particle size analysis represent a
particle as an equivalent sphere, such to represent it by
using a single dimension (an equivalent diameter) for
every size class (Allen, 1997). This assumption might
have different effects on PSD analysis depending on the
size fraction and on the chosen technique.
Usually particles in the sand and silt fractions are
more likely to resemble a sphere, while in the clay
fraction particles are often flat disc-shaped depending
on the type of clay minerals characterizing the clay-size
fraction. In sedimentation, a non-spherical particle tends
to settle with the maximum cross-sectional dimension
perpendicular to the direction of motion (Krumbein,
1942). This assumption results in decreasing the equivalent diameter (longer settling times) with ‘overestima-
tion’ of the clay fraction. In LD techniques, a nonspherical particle reflects a cross-sectional area which
is larger than a theoretical sphere of the same volume
(Jonasz, 1987). This effect results in increasing the
equivalent diameter, with ‘underestimation’ of the clay
fraction, because a particle is assigned to a larger size
section of the distribution. Considering these inherent
technique limitations, the results obtained by Konert
and Vandenberghe (1997) and Eshel et al. (2004) (yielding a smaller clay fraction for LD when compared to the
pipette method) seem to agree.
However the first question one may ask might be: is
the sedimentation technique that ‘overestimates’ the
clay fraction or is the laser diffraction technique that
‘underestimates’ it, or both? Indeed one could argue that
there is not a ‘universal’ or ‘true’ method for PSD and
that both methods reflect a different property of the
material depending on the characteristics of the measuring methodology.
Allen (1997) suggested that analysis of microscope
images of soil particles could be regarded as an absolute
method of particle-size analysis because, with this
method, individual particles are directly observed and
measured. Since the test and the parameterization of any
PSD model, therefore also of the bimodal Gaussian
model, are based on the reliability of the measurement,
in this work we: (a) compared the PSD analysis
obtained by sieving and pipette to LD techniques in
the range of 0.05 to 1000 μm, and (b) compared the
PSD analysis in the clay range as measured by LD and
Transmission Electron Microscopy and Image Analysis.
2. Theory
The Gauss function used to describe the PSD is:
"
# "
#
1
ðx−lÞ2
f ðxÞ ¼
exp −
ð1Þ
2r2
rð2 pÞ1=2
where μ is the mean of x and σ is the standard deviation.
For a lognormal distribution the particle diameter x is
replaced by its logarithm. Usually, the PSD is represented by a cumulative curve which, for a Gaussian
model, is obtained by integration of Eq. (1) with respect
to x:
1
ðx−lÞ
FðxÞ ¼
1 þ erf pffiffiffi
for ðx N lÞ
ð2Þ
2
r 2
FðxÞ ¼
1
ðx−lÞ
1−erf pffiffiffi
2
r 2
for ð xV lÞ
ð3Þ
L. Pieri et al. / Geoderma 135 (2006) 118–132
where erf( ) is the error function. For the computation of
the error function we used a numerical approximation
from Abramowitz and Stegun (1970):
erf ðxÞ ¼ 1−ð0:3480242 T −0:0958 T
þ 0:7478556T 3 Þexpð−x2 Þ
2
ð4Þ
where T = 1 / (1 + 0.47047x).
The approximation of the error function, as
described in Eq. (4), was tested by computing values
of T, which ranged from 0.34 to 0.99 for F1(x) and from
0.266 to 0.99 for F2(x). These values provided a satisfying approximation of the error function (Abramowitz
and Stegun, 1970).
In a bimodal distribution the sample is divided into
two fractions, the primary fraction (sand and silt) and
the secondary fraction (clay), each being described by a
Gaussian function.
The distribution is then a weighted sum of the two
fractions:
FðxÞ ¼ eF1 ðxÞ þ ð1−eÞF2 ðxÞ
ð5Þ
where F1(x) represents the cumulative Gaussian function for the secondary minerals (clay), F2(x) represents
the Gaussian function for the primary minerals (sand
and silt), and ε is the limit between the secondary
minerals (clay fraction) and the primary minerals
(sand and silt).
The mean and the standard deviation of the bimodal
distribution are also weighted sums of the two fractions,
called bimodal parameters. The model was fit to experimental PSD data by using a non-linear least square
fitting procedure (Marquardt, 1963), where μ1, μ2, σ1,
and σ2 were fitting parameters, while ε was set equal to
the clay fraction at 2 μm. The subscript 1 and 2 for the
mean and the standard deviation are the Gaussian parameters for the clay (F1) and the silt and sand fraction
(F2) distributions respectively.
Since for many soils, only the clay, silt and sand
fractions are available, Shiozawa and Campbell (1991)
calculated the log mean and the log standard deviation
following the approach proposed by Shirazi et al.
(1988) (referred as SBH, Shirazy–Boersma–Hart), to
obtain the mean and the standard deviation from three
fractions only:
ln l ¼ my ln dy þ mt ln dt þ md ln dd
ð6Þ
ln r2 ¼ my ðln dy Þ2 þ mt ðln dt Þ2
þ md ðln dd Þ2 −ðln lÞ2
ð7Þ
where dy, dt and dd are the geometric mean for clay, silt
and sand (the subscripts y, t and d refer to the last letters
121
of the words clay, silt and sand), which were calculated
from set limits:
dy ¼ ð0:01 2Þ1=2 ¼ 0:141; dt ¼ ð2 50Þ1=2
¼ 10 and dd ¼ ð50 2000Þ1=2 ¼ 316:2:
The lower and the upper limits for the clay fraction
were set at 0.01 and 2, while 2 and 50 are the lower and
upper limits for the silt fraction and 50 and 2000 are the
ones for sand, based on the USDA classification.
The rationale behind the use of geometric instead of
arithmetic parameters is that in a natural soil sample
there is a wide range of particle sizes making the geometric scale more suitable than the arithmetic one. The
application of geometric statistical parameters for soil
sample is strictly connected with the distribution of the
logarithm of particle size as described in the original
paper of Shirazi and Boersma (1984). These authors
also proposed a reformulation of the textural triangle
based on the additional mean and standard deviation
information.
Eq. (5) is used when a detailed PSD distribution is
available, and it has been used in this study to test the
bimodal model, while Eqs. (6) and (7) are used when
the three size fractions only are available.
3. Materials and methods
3.1. Samples pre-treatment
Eight soil samples were used in this study, six of
them from different locations in the State of Washington (USA) and two from the Emilia–Romagna region
in northern Italy. The soils were selected to embody a
variety of different textures. The samples used to measure the whole PSD (clay, silt and sand fractions
together) were dried at 105 °C, as described in classical
pre-treatment protocols (Gee and Bauder, 1986). The
samples used to measure the clay-size PSD only were
instead dried at room temperature. This was done to
avoid clay structural changes due to clay structuralwater removal. The samples were gently crushed, and
dry sieved at 2 and 1 mm mesh-size. To eliminate the
carbonate cements, each sample was washed with 50
ml of 0.5 m of NaOAc and was left overnight on a hot
plate at 75 °C. Four centrifugations and decantations
were needed to eliminate all the NaOAc in excess. The
soil samples were pre-treated by addition of 5 ml of
H2O2 (30%, w/w) at 65 °C, to remove the organic
matter. We checked that the samples did not release
additional carbon dioxide, after additional treatment
with H2O2, to assure complete removal of organic
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L. Pieri et al. / Geoderma 135 (2006) 118–132
material. Prior to particle size analysis, the samples
were dispersed to remove aggregates by adding a
sodium hexametaphosphate solution and shaking the
samples for 24 h.
3.2. Particle size distribution by laser diffraction
Particle size distribution analysis was performed
with a small-angle light scattering apparatus (Malvern
Master Sizer MS20, Malvern, England), equipped with
a low-power (2 mW) Helium-Neon laser with a wavelength of 633 nm as the light source. The apparatus has
active beam length of 2.4 mm, and it operates in the
range 0.05 to 1000 μm. The sample obscuration was
adjusted to an optimal value of 45%.
Usually, the real part of the soil minerals refractive
index (nr) ranges between 1.5 to 1.71, while for some
soil minerals (Hematite), this value can be as high as 3.2
(Press, 2002). However, for most soil minerals a value
of refractive index nr = 1.53 and of adsorption coefficient ni = 0.1 are considered suitable (Jonasz, 1991). The
reference refractive index for standard deionized water
was 1.33. A 3-g aliquot of the sample was introduced
into the ultrasonic bath.
Particle size distribution was obtained by fitting full
Mie scattering functions for spheres (Kerker, 1969). We
selected the Mie theory approach, instead of the Fraunhofer one, because it provides a better estimate of particle size in the clay fraction (deBoer et al., 1987).
3.3. Wet sieving and pipette
For the pipette analysis, samples were wet sieved at
1000-, 500-, 250-, 125- and 53-μm mesh sizes. The soil
material smaller than 53-μm was then analyzed by the
pipette method (Gee and Bauder, 1986). Separation
between size classes was also performed for image
analysis. Wet sieving was used to separate the sand
fraction, while pipette was used for separation of the
clay fraction as described in the international standard
ISO 11277. To obtain a better clay separation, pipetting
was repeated 8 times on each sample.
3.4. Transmission electron microscopy and image
analysis
After separation and measurement, the clay fractions
were diluted to obtain diluted suspension, necessary for
particle sizing and counting. Clay suspended samples
(20 μl) were deposited on coated Cu grids and dried at
room temperature. The TEM microscope was a Jeol
12EX, equipped with a CCD camera. The diameters of
the soil particles were determined by image analysis.
We collected images at magnifications of 4000 and
15 000.
Image analysis was performed with the AnalySis
Software and the Scion Image Analysis Software. The
Scion Image Analysis software counts and measures
objects in binary or threshold images. It does perform
a scanning across the image until it finds the boundary
of an object, it outlines the object area and it measures
it. The measurement is performed by counting the number of pixels belonging to a certain threshold grey or
colour level value. The spatial calibration is performed
by using a calibration tool, calibrated to a known distance printed on the TEM images. The software allows
one to choose among different measuring and analyzing
options including: particle counting, area measurement,
particle coordinate, particle perimeter, particle equivalent diameter and so forth.
Microscope images are projected areas whose
dimensions depend on the particle orientation on the
slide. Particles in stable orientation tend to present the
maximum area to the observer, therefore for clay discshaped particles under a TEM, the unknown dimension
is usually the thickness. As suggested by other authors
(Srodon and Elsass, 1994), we assumed that the discshaped particles had thickness of one-tenth of its diameter. The diameter of the particles was computed from
the TEM and image analysis measured area by using the
projected area equivalent diameter.
As pointed out by Allen (1997) the projected area
equivalent diameter (A = (π / 4)d2) is the one that provides a better estimation of particle size in microscopy.
Since the area (A) of a particle is measured, and solving
for the particle diameter d = (4A / π)1 / 2, it is then possible to compute the particle volume V = A × d / 10. Multiplying the volume of one particle (Vi) for the total
number (ni) of particles of that size class divided by
the total number of particle (N), a volume based distribution of that size class (Vi) is obtained, given by:
Φ = Vi × (ni / N), for a given size class di. To evaluate the
effect of the spherical assumption on the PSD measurement we computed a volume-based PSD from the TEM
and image analysis for flat disc-shaped particles, and we
compared the results to the spherically based LD
analysis.
3.5. Particle size distribution in the clay fraction by
laser diffraction
To measure the PSD of the separated clay fraction,
we used the same small-angle light scattering apparatus (Malvern Master Sizer MS20, Malvern, England)
L. Pieri et al. / Geoderma 135 (2006) 118–132
123
Fig. 1. Density and cumulative distribution fractions for eight soils with different textural composition, obtained by sieving-pipette (Sedimentation)
and laser diffraction (LD) techniques. The sedimentation data are presented only in cumulative form.
124
L. Pieri et al. / Geoderma 135 (2006) 118–132
Fig. 2. Comparison of the clay (a), silt (b) and sand (c) fractions obtained by the pipette and the laser diffraction methods for 8 soils. The trendline
represents a linear fitting between pipette and laser diffraction data.
L. Pieri et al. / Geoderma 135 (2006) 118–132
which was used for the PSD analysis of the whole
sample. The Malvern Master Sizer allows the use of a
differential scattering apparatus which provides a
more precise measurement of smaller particles (diameter b2 μm). Clay fractions were diluted in sodium
hexametaphosphate and kept in suspension by a recirculation pump.
4. Results and discussion
4.1. Particle size distribution as measured by pipette
and LD
Fig. 1 shows the density and cumulative distribution
fractions for the eight soil samples, measured by wetsieving and pipette (black solid dots) and laser diffraction (lines). For all the soil samples measured, the LD
method yielded a smaller clay fraction than the pipette,
as shown in Fig. 2a, where the measured points are
below the 1:1 line.
On the other hand, the LD method yielded a larger
silt fraction, with measured points above the 1:1 line
(Fig. 2b). For the sand fraction, the trend showed a good
agreement between LD and pipette, except for two
samples, where the sand fractions measured by pipette
and LD differed considerably (Fig. 2c).
These trends are consistent with the ones presented
in previous works, however our relations (Fig. 2a) differed from the one of Eshel et al. (2004) where the
relationships for the clay fraction was y = 0.345x
+ 2.69, where y is the volume percentage obtained by
the LD and x is the mass percentage obtained by the
pipette; as well from the one of Konert and Vandenberghe (1997). In both cases, our results showed less
pronounced differences between the LD and the pipette,
when compared to other works.
125
Table 1 shows the fractions of sand, silt and clay
as measured by laser diffraction, predicted by the
equations in Fig. 2 and the relative errors. The
maximum error obtained by using these equations
is 7.5% for the sand fraction, 10.5% for the silt
fraction and 2.9% for the clay fraction. The differences in the relationships between LD and pipette
fractions found by different authors may indicate that
it is difficult to establish a unique relationship
between the LD and the pipette. Beuselinck et al.
(1987) pointed out that there is not a unique relationship between the PSD's obtained by LD and
pipette, and that the different mineralogy and particles' shapes strongly affect the differences between
the two methods.
The sources of error between the two methodologies can be due to causes inherent to the methods
and therefore varying for different soil samples. First,
the pipette method is strongly dependent on the
particle density and its variations within the sample,
while the LD is independent on particle density. As
previously discussed, the assumption of a single particle density for the pipette method is a source of
error. Second, the variety of physical and chemical
properties of soil particles have a different effect on
the two measurement systems, for instance the colour
of the sample affects the refractive index in the LD
technique, which is difficult to address. Third, the
particles' shape changes size measurement and data
interpretation.
Overall our observations are in agreement with other
studies. However, because of the variability of the relative differences between the two methods, it may be
difficult to obtain unique relationships between sedimentation-based and LD methods for PSD data, which
hold for all soil samples analyzed.
Table 1
Fraction of clay, silt and sand measured by laser diffraction calculated using the equations in Fig. 2, and the relative error of the calculated values
Measured
L-Soil
Walla Walla
Palouse
Royal
Salkum
Red Bluff
Ozzano 1a
Ozzano 8a
Predicted
Error
Clay
Silt
Sand
Clay
Silt
Sand
Clay
Silt
Sand
3.295
13.020
17.980
6.096
28.140
45.320
20.480
17.643
11.082
77.860
68.040
61.440
59.370
36.596
42.310
59.342
85.660
9.120
13.980
32.464
12.490
18.111
37.210
23.015
3.228
12.689
17.497
5.978
27.345
43.997
19.920
17.170
12.131
86.034
75.166
67.862
65.571
40.338
46.691
65.540
92.129
9.802
15.029
34.911
13.426
19.472
40.016
24.747
0.946
2.540
2.687
1.936
2.826
2.920
2.734
2.680
9.471
10.498
10.473
10.452
10.445
10.306
10.355
10.445
7.552
7.473
7.504
7.537
7.497
7.517
7.540
7.527
The relative error was calculated by computing the difference between measured and predicted values as percentage of the measured value.
Particle sizes measured by pipette and light diffraction methods.
Classes are according to USDA methods.
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L. Pieri et al. / Geoderma 135 (2006) 118–132
4.2. Particle size distribution in the clay fraction as
measured by LD, TEM and image analysis
Fig. 3 shows the PSD of the clay fraction (b2 μm) by
laser diffraction. The clay fractions occur as distinct
unimodal domains. The main modes have a mean of
0.39 μm for the L-Soil, 0.53 μm for the Walla Walla and
0.33 μm for the Red Bluff sample. These data were used
to compare the clay distribution obtained by laser diffraction to the one obtained by TEM and image analysis. The pipette method does not provide a distribution
within the clay range because the smallest data point for
particle size is usually the one at 2 μm, therefore preventing inclusion of a particle clay-size distribution
analysis using sedimentation methods.
Transmission electron microscopy and image analysis were utilized to measure particle size distribution of
three soil samples. Fig. 4 shows TEM images for the LSoil, Walla Walla and Red Bluff samples at two magnifications of 4000 and 15 000. Previous mineralogical
analysis of these samples showed that the main clay
minerals in the clay fraction are kaolinite, illite and
smectite (Bittelli et al., 2004).
Image analysis of the TEM images produces a number distribution of particles, as shown in Fig. 5a, from
which we derived a disc-shaped volume based distribution. Fig. 5b shows the comparison between the TEM
and the LD obtained distribution. When particle sizes
are represented as flat disc-shaped particles, TEM pro-
duces a smaller mean diameter than the LD method
(which represents the particles as volume fractions of
spheres), confirming that volume based distribution
obtained with LD shifts the particle size toward larger
fractions.
The differences we found were of 0.17 μm for LSoil, 0.26 μm for Walla Walla and 0.10 μm for Red
Bluff.
We could not measure a particle size distribution in
the clay range by using the pipette method and therefore we cannot say which of the two methodologies
(LD or pipette) is closer to the results obtained by
TEM and image analysis. The TEM analysis produced
more narrow distributions when compared to the LD
method.
4.3. Application of the bimodal Gaussian model
Fig. 1 shows the experimental distribution functions
of the eight soil samples, while Fig. 6 depicts the
experimental and the fitted cumulative distribution
function.
The probability distribution function in Fig. 1
shows that three of the eight soil samples (Salkum,
Red Bluff, and Ozzano 1a) displayed a multimodal
distribution with three modes, while the other five
soils displayed a bimodal distribution. The Red Bluff
sample is the one with a more distinct tri-modal
distribution. Model parameters are given in Table 2.
Fig. 3. Particle size distribution of the clay fraction (b2 μm) by laser diffraction for three soil samples.
L. Pieri et al. / Geoderma 135 (2006) 118–132
127
Fig. 4. TEM images of the L-Soil, Walla Walla and Red Bluff soil clay fraction, at magnification of 15 000 (a) and 4000 (b).
The parameters listed in the table under Bimodal
model are the ones obtained by fitting Eq. (5) to the
experimental data.
The means and standard deviations listed as SBH
(which stands for Shirazy–Boersma–Hart) are obtained
by using Eqs. (6) and (7). An overall bimodal weighted
mean representing both modes has been computed by
using μw = εμ1 + (1 − ε)μ2 (where the subscript w for the
mean stands for weighted). The means and standard
deviations listed as SBH-modified are obtained by
using Eqs. (8) and (9), which are described in the
following paragraphs.
The parameter ε in Table 2, which was obtained by
the experimental clay fraction, was kept constant during
fitting, therefore the model was reduced to a 4 parameters one (the two means and the two standard deviations). The bimodal logarithmic means of the particle
diameter (μ1) for the secondary minerals (clay fraction)
range from 0.001 to 0.325 (which is equivalent to a nonlogarithmic range between 1.001 to 1.384 μm). The
128
L. Pieri et al. / Geoderma 135 (2006) 118–132
Fig. 5. Particle size distribution by number of the clay fraction for the three soils (a). Comparison between LD and TEM and image analysis (IA) of
particle size distribution by volume (b). The boxes indicate the mean diameters for the two methods.
logarithmic means of the particle diameter (μ2) for the
primary minerals (sand and silt fractions) range between
2.708 for the Palouse soils and 5.669 for the L-Soil soil
(which is equivalent to a non-logarithmic range between
14.99 to 289.74 μm). As expected, L-Soil, being the
sample with the larger sand fraction, displays the larger
value of mean diameter in the primary fraction.
The value of logarithmic means of the particle diameter (μ1) for the clay fraction indicates that the choice
of a geometric mean equal to 0.141 μm (as proposed by
Shiozawa and Campbell (1991)) tends to underestimate
the mean for the clay fraction, and consequently also the
overall bimodal weighted mean. From our experimental
data the mean of the clay fraction is better represented
by a mean value between the two limits 0.01 and 2
which equal to 1.005 μm corresponding to a logarithmic
mean of 0.0049 μm. Based on these results, Eqs. (6) and
(7) are re-written as
ln lm ¼ ½my ð0:0049Þ þ mt ð2:3Þ þ md ð5:76Þ
ð8Þ
ln rm ¼ ½my ð0:0049Þ2 þ mt ð2:3Þ2
þ md ð5:76Þ2 −ln lm 1=2
ð9Þ
which we labelled SBH-modified in Table 2 (the subscript m stands for modified).
The SBH-modified model gave good estimation of
the mean when compared to the weighted bimodal
mean (μw = εμ1 + (1 − ε)μ2). To test the SBH-modified
model, we calculated the relative error of the logarithmic mean computed with the original SBH model, and
the weighted bimodal mean ((μw − ln μ) / μw); and the
relative error of the logarithmic mean computed with
the SBH-modified model, and the weighted bimodal
mean ((μw − ln μm) / μw).
The SBH-modified displayed a smaller relative error
except for the Ozzano 8a, showing that the modified
version of the SBH model provides an improved estimates of the bimodal parameters. Consequently, Eqs.
(8) and (9) can be used to derive soil PSD parameters,
when the sand, silt and clay fractions only are available. As an example, Table 3 gives representative silt,
sand and clay fractions for the 12 textural classes and
shows mean particle diameter and standard deviation
obtained by using Eqs. (8) and (9). The multimodal
(three modes) behaviour of some of the samples analyzed is likely due to the different pedological origins
of the soil particles, making the bimodal Gaussian
approach not applicable to the whole range of soil
textures.
Other authors found similar results, showing that the
bimodal approach is applicable only to certain textural
L. Pieri et al. / Geoderma 135 (2006) 118–132
129
Fig. 6. Cumulative particle size distribution of eight soils with different textural composition. Points are experimental values, solid lines are
prediction using the bimodal Gaussian model. SSE is the sum of square errors between the experimental data and the fitted model.
130
L. Pieri et al. / Geoderma 135 (2006) 118–132
Table 2
Measured properties of eight soils: fractions of sand (md), silt (mt) and clay (my), modelled bimodal distribution parameters and SBH model
parameters
Soil
L-Soil
Walla Walla
Palouse
Royal
Salkum
Red Bluff
Ozzano 1a
Ozzano 8a
Sand
Silt
Clay
0.857
0.111
0.033
0.130
0.779
0.091
0.140
0.680
0.180
0.325
0.614
0.061
0.119
0.597
0.284
0.181
0.366
0.453
0.371
0.424
0.205
0.231
0.593
0.176
Bimodal model
ε
μ1
σ1
μ2
σ2
SSE
0.033
0.249
0.582
5.699
1.055
0.065
0.091
0.001
0.323
2.835
0.832
0.070
0.180
0.001
0.376
2.708
1.129
0.080
0.061
0.058
0.361
3.562
0.699
0.020
0.284
0.001
0.549
2.718
1.123
0.040
0.453
0.001
0.579
2.832
1.905
0.090
0.205
0.110
0.585
2.865
1.090
0.030
0.176
0.325
0.402
2.446
0.778
0.040
Model ln means
SBH
SBH-modified
μw = εμ1 + (1 − ε)μ2
5.125
5.180
5.535
2.362
2.540
2.578
2.018
2.370
2.221
3.163
3.280
3.351
1.502
2.059
1.957
0.996
1.884
1.549
1.429
2.156
2.301
1.926
2.377
2.072
Model ln standard deviations
SBH
5.189
SBH-modified
1.442
2.541
1.407
2.371
1.617
3.283
1.800
2.060
1.692
1.886
2.094
2.158
2.095
2.379
1.821
Relative errors
(μw − ln μm) / μw
(μw − ln μ) / μw
0.014
0.084
0.067
0.092
0.020
0.056
0.052
0.233
0.218
0.357
0.062
0.379
0.148
0.071
0.063
0.074
Natural logarithms of the means and standard deviation are listed.
classes. For example, Hwang et al. (2002) found that
only about half of the USDA texture triangle can be
adequately described by a bimodal Gaussian model, and
that the suitability of different PSD models appears to
be influenced by soil textural classes and/or by the soil
clay content. Hwang et al. (2002) pointed out that all of
silty clay, silty clay loam, and silt loam among the
USDA texture triangle could be properly modelled
Table 3
Representative natural logarithms of the geometric mean and standard
deviation for the twelve textural classes calculated using Eqs. (8) and
(9)
Texture
md
mt
my
μ
σ
Sand
Loamy sand
Sandy loam
Loam
Silt loam
Silt
Sandy clay
Clay loam
Silty clay loam
Sandy clay
Silty clay
Clay
0.92
0.81
0.65
0.42
0.08
0.06
0.60
0.32
0.08
0.53
0.10
0.20
0.05
0.12
0.25
0.40
0.79
0.87
0.13
0.34
0.58
0.07
0.45
0.20
0.03
0.07
0.1
0.18
0.13
0.07
0.27
0.34
0.34
0.40
0.45
0.60
5.40
4.93
4.31
3.33
2.27
2.34
3.75
2.62
1.79
3.21
1.61
1.61
1.21
1.75
2.05
2.20
1.28
1.03
2.54
2.34
1.57
2.75
1.75
2.25
Fractions of sand (md), silt (mt) and clay (my) were chosen within a
textural class.
with lognormal PSDs; but more complex distributions
were required for sandy clay loam, sandy clay, and
much of the clay region.
5. Conclusions
We analyzed 8 soil samples with very different particle size distributions. We first compared the PSD
obtained by LD and pipette methods. Differences were
found between the two methods, and we propose conversion equations that allow to convert data from one
type of measurement to the other. It appears that the LD
methods tend to ‘underestimate’ the clay fraction when
compared to the traditional sedimentation techniques,
and therefore conversion equations need to be applied
when the two methods are compared.
However, we believe that the LD method is a valid
method for PSD analysis, even though it provides data
that are not fully comparable with the classical sedimentation methods. In particular, many of the LD's features
are very advantageous and therefore we foresee that this
method may become a widely spread technique for PSD
analysis. LD technique provides a wide range of size
classes including many data points in the clay-size
range, it is independent of the particle density because
it provides a volume-based distribution, and it is fast.
L. Pieri et al. / Geoderma 135 (2006) 118–132
We compared the PSD analysis obtained by laser
diffraction and images obtained by using transmission
electron microscopy and image analysis techniques in
the clay fraction. Differences were found when particles
were represented as flat disc-shaped, in comparison to
the laser diffraction analyses which represent the particles as spheres. This assumption results in underestimation of the clay fraction by laser diffraction because
particles are assigned to a larger size fraction, when
compared to transmission electron microscopy and
image analysis techniques. The representation of soil
particles as volume equivalent discs might improve the
laser diffraction method for PSD analysis.
The bimodal lognormal distribution function proposed by Shiozawa and Campbell provided a good
PSD characterization for five of the eight soils analyzed.
Three of the eight samples displayed a multimodal
distribution, requiring other mathematical characterization of the PSD curve. However, when the soils are
bimodal, the present model provides a good characterization and it allows to derive PSD parameters from
three size fractions (sand, silt and clay) only. Based on
our experimental results we propose two improved
equations to obtain the logarithmic mean and standard
deviation, which provide better PSD characterization
when compared to the original model.
Acknowledgements
We thank Alan Busacca and Markus Flury for use of
the laser diffraction analyzer at the Department of Crop
and Soil Sciences, Washington State University, Pullman, WA. We are thankful to Youjun Deng for help
with the experiments in Transmission Electron Microscopy. We are also grateful to two anonymous reviewers
for suggestions and critiques in a previous version of
this paper.
References
Abramowitz, M., Stegun, I.A., 1970. Handbook of Mathematical
Functions. Dover Publications, New York.
Allen, T., 1997. Particle Size Analysis by Image Analysis, 5th ed.
Particle Size Measurement, vol. 1. Chapman and Hall, London,
pp. 269–306.
Beuselinck, L.G., Govers, G., Poesen, J., Degraer, G., 1987. Grain-size
analysis laser diffractometry: comparison with the sieve-pipette
method. Catena 32, 193–208.
Bittelli, M., Campbell, G.S., Flury, M., 1999. Characterization of
particle-size distributions in soils with a fragmentation model. Soil
Sci. Soc. Am. J. 63, 782–788.
Bittelli, M., Flury, M., Roth, K., 2004. Use of dielectric spectroscopy
to estimate ice content in frozen porous media. Water Resour. Res.
40, doi:10.1029/2003WR002343 (W04212).
131
Buchan, G.D., Grewal, K.S., Robson, A.B., 1993. Improved models of
particle-size distribution: an illustration of model comparison
techniques. Soil Sci. Soc. Am. J. 57, 901–908.
Campbell, G.S., Shiozawa, S., 1992. Prediction of hydraulic properties
of soils using particle-size distribution and bulk density data. In:
van Genuchten, M.T., Leij, F.J., Lund, L.J. (Eds.), Indirect
Methods for Estimating the Hydraulic Properties of Unsaturated
Soils. University of California, Riverside, pp. 317–328.
Campbell, G.S., Jungbauer, J.D., Bidlake, W.R., Hungerford, R.D.,
1994. Predicting the effect of temperature on soil thermal
conductivity. Soil Sci. 158, 307–313.
Clifton, J., Mcdonald, P., Plater, A., Oldfield, F., 1999. An
investigation into the efficiency of particle size separation
using Stokes' measurement. Earth Surf. Process. Landf. 24,
725–730.
deBoer, G.B., de Weerd, C., Thoenes, D., Goossens, H.W., 1987. Laser
diffraction spectrometry: Fraunhofer versus Mie scattering. Part.
Charact. 4, 14–19.
Dur, J.C., Elsass, F., Chaplain, V., Tessier, D., 2004. The relationship
between particle-size distribution by laser granulometry and image
analysis by transmission electron microscopy in a soil clay
fraction. Eur. J. Soil Sci. 55, 265–270.
Eshel, G., Levy, G.J., Mingelgrin, U., Singer, M.J., 2004. Critical
evaluation of the use of laser diffraction for particle-size
distribution analysis. Soil Sci. Soc. Am. J. 68, 736–743.
Gee, G.W., Bauder, J.W., 1986. Particle-size analysis, In: Klute, A.
(Ed.), Methods of Soil Analysis. Part 1. Physical and Mineralogical Methods, 2nd edn. American Society of Agronomy,
Madison, Wisconsin, pp. 383–411.
Hillel, D., 1998. Environmental Soil Physics. Academic Press, San
Diego.
Hwang, S.I., Lee, K.P., Lee, D.S., Powers, S.E., 2002. Models for
estimating soil particle-size distributions. Soil Sci. Soc. Am. J. 66,
1143–1150.
Jonasz, M., 1987. Nonsphericity of suspended marine particles
and its influence on light scattering. Limnol. Oceanogr. 32,
1059–1065.
Jonasz, M., 1991. Size, shape, composition and structure of
microparticle from light scattering. In: Syvitske, J.P.M. (Ed.),
Principles, Methods, and Application of Particle Size Analysis.
Cambridge University Press, Cambridge, UK, pp. 143–162.
Kerker, M., 1969. The Scattering of Light and Other Electromagnetic
Radiation. Academic Press, New York.
Konert, M., Vandenberghe, J., 1997. Comparison of laser grain size
analysis with pipette and sieve analysis: a solution for the
underestimation of the clay fraction. Sedimentology 44, 523–535.
Krumbein, W.C., 1942. Settling velocity of flume behavior of
nonspherical particles. Trans. Am. Geophys. Union 41, 621–633.
Loveland, P.J., Whalley, W.R., 2001. Particle size analysis. In: Smith,
K.A., Mullins, C.E. (Eds.), Soil and Environmental Analysis,
Physical Methods. Marcel Dekker Inc., New York, pp. 281–314.
Marquardt, D.W., 1963. An algorithm for least-squares estimation of
non-linear parameters. J. Soc. Ind. Appl. Math. 11, 431–441.
Martin, M.A., Montero, E., 2002. Laser diffraction and multifractal
analysis for the characterization of dry soil volume-size distributions. Soil Tillage Res. 64, 113–123.
Muggler, C.C., Pape, T., Buurman, P., 1997. Laser grain-size
determination in soils genetic studies. 2. Clay content, clay
formation and aggregation in some Brazilian soils. Soil Sci. 162,
219–228.
Press, C., 2002. Handbook of Chemistry and Physics. CRC Press,
Boca Raton, Florida.
132
L. Pieri et al. / Geoderma 135 (2006) 118–132
Shiozawa, S., Campbell, G.S., 1991. On the calculation of mean
particle diameter and standard deviation from sand, silt, and clay
fractions. Soil Sci. 152, 427–431.
Shirazi, M.A., Boersma, L., 1984. A unifying quantitative analysis of
soil texture. Soil Sci. Soc. Am. J. 48, 142–147.
Shirazi, M.A., Boersma, L., Hart, J.W., 1988. A unifying quantitative
analysis of soil texture: improvement of precision and extension of
scale. Soil Sci. Soc. Am. J. 52, 181–190.
Srodon, J., Elsass, F., 1994. Effect of the shape of fundamental
particles on XRD characteristic of illitic material. Eur. J. Mineral.
6, 1–10.
van Genuchten, M.T., Leij, F.J., Wu, L., 1999. Characterization and
Measurement of the Hydraulic Properties of Unsaturated Porous
Media. University of California, Riverside, CA.
Walter, N.F., Hallberg, G.R., Fenton, T.S., 1978. Particle size analysis
by the Iowa State University, soil survey lab. In: Hallberg, G.R.
(Ed.), Standard Procedures for Evaluation of Quaternary Materials
in Iowa. Iowa Geological Survey, Iowa City, pp. 61–74.
Wösten, J.H.M., Pachepsky, Y.A., Rawls, W.J., 2001. Pedotransfer
functions: bridging the gap between available basic soil data and
missing soil hydraulic characteristics. J. Hydrol. (Amst.) 251,
123–150.
Wu, Q., Borkovec, M., Sticher, H., 1993. On particle-size distributions
in soils. Soil Sci. Soc. Am. J. 57, 883–890.