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Geoderma 135 (2006) 118 – 132 www.elsevier.com/locate/geoderma 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 120 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 122 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. 126 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. 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