Soil Science Society of America Journal 63:782-788 (1999)
© 1999 Soil Science Society of America
DIVISION S-1-SOIL PHYSICS
Characterization of Particle-Size Distribution in Soils with a Fragmentation Model
Marco Bittellia,
Gaylon S. Campbella and
Markus Flurya
a Department of Crop and Soil Sciences, Washington State University, Pullman, WA 99164 USA
bittelli{at}mail.wsu.edu
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ABSTRACT
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Particle-size distributions (PSDs) of soils are often used to estimate other soil properties, such as soil moisture characteristics and hydraulic conductivities. Prediction of hydraulic properties from soil texture requires an accurate characterization of PSDs. The objective of this study was to test the validity of a mass-based fragmentation model to describe PSDs in soils. Wet sieving, pipette, and light-diffraction techniques were used to obtain PSDs of 19 soils in the range of 0.05 to 2000 µm. Light diffraction allows determination of smaller particle sizes than the classical sedimentation methods, and provides a high resolution of the PSD. The measured data were analyzed with a mass-based model originating from fragmentation processes, which yields a power-law relation between mass and size of soil particles. It was found that a single power-law exponent could not characterize the PSD across the whole range of the measurements. Three main power-law domains were identified. The boundaries between the three domains were located at particle diameters of 0.51 ± 0.15 and 85.3 ± 25.3 µm. The exponent of the power law describing the domain between 0.51 and 85.3 µm was correlated with the clay and sand contents of the soil sample, indicating some relationship between power-law exponent and textural class. Two simple equations are derived to calculate the parameters of the fragmentation model of the domain between 0.51 and 85.3 µm from mass fractions of clay and silt.
Abbreviations: PSD, particle-size distribution • RMSE, root mean square error
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INTRODUCTION
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PARTICLE-SIZE DISTRIBUTION in soil is one of the more fundamental soil physical properties. It is widely used for the estimation of soil hydraulic properties such as the water-retention curve and saturated as well as unsaturated conductivities (Arya and Paris, 1981; Campbell and Shiozawa, 1992). Generally, a conventional particle-size analysis involves the measurement of the mass fractions of clay, silt, and sand. These fractions may be used to find the textural class using a textural diagram, commonly in form of a textural triangle (e.g., Gee and Bauder, 1986). However, soil samples that fall into a certain textural class may have considerably different PSDs. For example, the textural class of "clay" in the USDA classification scheme (Gee and Bauder, 1986) contains soil samples that vary in clay content between 40 and 100%. The size definitions of the three main particle fractions of clay, silt, and sand, used as diagnostic characteristics in most classification schemes, are rather arbitrary, and they do not provide complete information on the soil PSD.
A more accurate description of texture is obtained by defining a PSD function. Commonly, PSDs are reported as cumulative distributions, and different functions have been proposed to fit experimental data. Buchan et al. (1993) fitted several of these models to experimental data and found the bimodal lognormal distribution proposed by Shiozawa and Campbell (1991) to be best suited. The Shiozawa and Campbell model divides the particle distribution into two parts dominated by primary (sand and silt) and secondary (clay) minerals, respectively. However, as pointed out by Buchan et al. (1993), the assumption of a lognormal distribution in the clay fraction cannot be justified because Shiozawa and Campbell (1991) had no data available in that range.
One of the latest developments in the study of PSDs in soils has focused on the use of fractal mathematics to characterize particle sizes in soil (Turcotte, 1986; Tyler and Wheatcraft, 1992; Wu et al., 1993). However, questions remain about the validity and applicability of fractal concepts to PSDs. There has been some discussion about the proper use and definition of the term "fractal" in the literature (Young et al., 1997; Pachepsky et al., 1997; Baveye and Boast, 1998). Different concepts of fractals are used, and these concepts lead to different interpretations of fractal dimensions obtained. Therefore it is essential to clearly specify the type of fractal model used.
Particle- and aggregate-size distributions are often rendered as cumulative functions, either as number of particles larger than a certain diameter, or as mass smaller than a certain diameter. These cumulative distribution functions have been analyzed with power-law relations and the exponents interpreted as fractal dimensions. Tyler and Wheatcraft (1989, 1992) analyzed particle-size data ranging from 0.5- to 5000-µm radii, and observed that the fractal power law was not valid across the entire extent of particle sizes. It is expected that there are lower and upper limits to the validity of fractal relations (Turcotte, 1986). Wu et al. (1993) measured PSDs down to 0.02-µm radius by using light-scattering techniques, and found a power-law relation between number of particles and particle radius valid across a range of particle radii with a lower cutoff between 0.05 and 0.1 µm and an upper cutoff between 10 and 5000 µm. Assuming that the exponent of a power-law relation is a fractal dimension, Wu et al. (1993) found a dimension of
for the four soils studied and suggested that this might be a universal value of an underlying structure. Kozak et al. (1996) analyzed PSDs of 2600 soil samples and found that for 50% of the samples power-law scaling of particle numbers vs. size was not applicable across the whole range of particle sizes between 2 and 1000 µm. The authors indicate that power-law scaling might be applicable for a narrower range of particle sizes, although this was not analyzed in their study.
Most applications of fractal concepts to particle- and aggregate-size distributions are based on the fragmentation model of Matsushita (1985) and Turcotte (1986). In this model, the fragmentation of an initially intact particle into smaller particles leads to a power-law relation between (i) number or (ii) mass of particles as a function of particle size. These two types of fragmentation relations are known as number-based and mass-based approaches (Turcotte, 1992). The power-law exponent of the number-based approach can be interpreted as fractal dimension (Matsushita, 1985; Turcotte, 1986). It is worth noting that the fragmentation model does not lead to a geometrical fractal with the fractal dimensions confined between Euclidian dimensions. The sorting of particles by size in the fragmentation model results in fractal dimensions ranging theoretically between the limits of 0 and 3 (Turcotte, 1986). Borkovec et al. (1993) experimentally determined fractal dimensions of fragmentation and surface areas of soil particles and found the two dimensions to be 2.8 ± 0.1 and 2.4 ± 0.1, respectively.
The objective of this study was to test the mass-based fragmentation approach proposed by Turcotte (1986) for characterizing PSDs, and to determine the range of particle diameters where power-law scaling is applicable. To test the general validity and the extent of power-law scaling it is of fundamental importance to have data that span several orders of magnitude. Traditional sedimentation and hydrometer techniques for the measurement of PSDs yield only limited data in the clay fraction smaller than 2 µm. Light-scattering methods overcome this problem and provide data between 0.05 to 1000 µm.
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Materials and methods
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Particle-Size Analysis
Nineteen soils were used in this study, five of them were from the USA and 14 from Switzerland. The soils were chosen such that they represent a wide variety of parent materials, weathering conditions, and textures. Characteristic properties of these soils are summarized in Table 1
. All soil samples were dried at 105°C, gently crushed, and passed through a 2-mm sieve. Each sample was tested for the presence of carbonates using cold 1 M HCl, and if carbonates were present, the sample was treated with 0.5 M sodium acetate at 75°C for at least 1 h. After acetate treatment, samples were washed with deionized water. The five soil samples from the USA were further pretreated by destroying organic matter using H2O2 (30%, w/w) at 65°C. The 14 Swiss soil samples were not pretreated for organic matter. The absence of pretreatment for organic matter could in some cases have affected the dispersion of particles for the Swiss soils, leading to incomplete segregation, and therefore to an underestimation of small particle fractions. Organic matter contents of the Swiss soils, determined with the Walkley-Black method (Nelson and Sommers, 1982), ranged from 0.4 to 2.2% by weight (Table 1).
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Table 1 Soil classification, geological parent material, percentage of sand, silt and clay by weight, and organic C content for the 19 soils used. Particle-size data were obtained by sieving and light-diffraction methods. Textural classes are according to the USDA classification
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After pretreatment, all samples were dried at 105°C for 24 h. Prior to particle-size analysis, all soil samples were dispersed in 1 g L-1 hexametaphosphate solution and shaken for 24 h to destroy aggregates. For the pipette analysis, samples were wet sieved with the hexametaphosphate solution at 1000-, 500-, 250-, 125-, and 53-µm mesh sizes. The material smaller than 53 µm was then analyzed by the pipette method (Gee and Bauder, 1986). To obtain four size classes between 2 and 50 µm, sedimentation techniques based on Stoke's law were used to obtain the following diameters: <2, <5, <10, and <20 µm. For the light-scattering technique, the soil samples were wet sieved down to a size of 250 µm for the Swiss soils and 500 µm for the U.S. soils. The particles passing the smallest sieve mesh were collected in a bucket, dried at 105°C, and subsequently analyzed by light diffraction. A 3-g aliquot of the dried material was introduced into an ultrasonic bath unit of 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.1
Suspension concentrations were adjusted to an obscuration of the primary beam of 
0.1 to 0.2%. The obscuration values were set to optimize between best signal/noise ratio and negligible multiple scattering effects. If the sample concentration is too low, the obscuration and the intensity of the scattered light are low, leading to noisy data. If the sample concentration is too high, then the light scattered from a particle may be scattered again by a second particle, causing errors in the final particle-size analysis. Prior to measurement, samples were dispersed by sonication in an ultrasonic bath for 25 min. A focal length of 300 mm was used with an ordinary Fourier Optics configuration, and a focal length of 45 mm was used for the inverse Fourier Optics configuration. The inverse configuration allows the accurate measurement of scattering at high angles in order to correctly measure the very fine particles (sizes down to 0.01 µm). Particle-size distribution was obtained by fitting full Mie scattering functions for spheres (Kerker, 1969).
Data Analysis
Soils are formed by weathering of geological parent material. The weathering results in a fragmentation of the initial solid rock or sediment. It has been recognized that the products of fragmentation in nature can often be described with fractal concepts. For different types of objects, a power-law relation between the number and size of objects has been proposed (Mandelbrot, 1982; Matsushita, 1985; Turcotte, 1986)
 | (1) |
where N(r > R) is the number of objects per unit volume having a radius r larger than R, C is a constant of proportionality, and D is the fractal dimension. For soil particles, Turcotte (1986) and Tyler and Wheatcraft (1992) pointed out that it is generally more convenient to express the number-based power law (Eq. [1]) as a mass-based form. The mass-based approach is compatible with data obtained from experimentation, where usually mass fractions rather than number fractions are measured. The mass-based form of Eq. [1] is expressed as (Turcotte, 1986; Tyler and Wheatcraft, 1992)
 | (2) |
where M(r < R) is the mass of soil particles with a radius smaller than R, MT is the total mass of particles with radius less than RL,upper, RL,upper is the upper size limit for fractal behavior, and v is a constant exponent. This power law can be related to the fractal number relation by taking incremental values as shown by Matsushita (1985) and Turcotte (1992). Taking the derivatives of Eq. [1] and [2] with respect to the radius R yields, respectively,
 | (3) |
and
 | (4) |
Assuming a constant density of soil particles, the volume of a particle with radius r is proportional to its mass m, hence r3 
m; therefore, for incremental particle numbers and masses we have
 | (5) |
Substituting Eq. [3] and [4] into [5] gives (Turcotte, 1992)
 | (6) |
from which it follows that
 | (7) |
Equation [7] relates the exponent v of the mass-based approach to the exponent D of the number-based approach. As shown below by our experimental data and discussed in the literature (Turcotte, 1992), the power-law relation given in Eq. [2] has also a lower limit of validity. The radius R of particles satisfying Eq. [2] is confined between RL,lower < R < RL,upper.
The mass-based fragmentation approach was used to analyze experimentally determined PSD data. The lower and upper limits RL,lower and RL,upper as well as the power-law exponent
were determined by the following procedure. A linear regression was used to fit Eq. [2] on a log-log plot to the experimental data. The entire range of experimental data was used first and the residuals were calculated. Subsequently, the upper- and lower-range data points were eliminated and new residuals and root mean square errors (RMSE) were calculated. In an iterative procedure, the RMSE error was minimized by eliminating data points at the upper and lower boundaries.
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Results and discussion
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Comparison between Pipette and Light-Diffraction Methods
Most of the textural data reported in the literature have been measured by sedimentation techniques, such as hydrometer or pipette. It is therefore illustrative to briefly compare experimental results obtained by pipette and light-scattering methods. The results obtained by the two techniques were in excellent agreement in our study. Figure 1
shows a qualitative comparison between pipette and light-scattering methods for four soils. Experimental differences in the cumulative fraction at a given particle size obtained by the two methods were in the order of 0.3 to 11.7%. Similar results were obtained by Wu et al. (1993), who found that sedimentation and light-scattering techniques were in good agreement for the majority of the soil samples used in their experiment.
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Fig. 1 Cumulative particle-size distributions for four soils obtained by two different experimental methods
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Characterization of Particle-Size Distribution
In Fig. 2
, cumulative mass fractions are plotted as a function of particle diameter on double logarithmic scale for four soils. The plots clearly show that a single power law cannot describe the data across the entire range of measured particle sizes. There is evidence that different power laws apply for three domains in all of the 19 soils. The solid lines in Fig. 2 are the curves of Eq. [2] fitted to the different domains of particle sizes on the log-log plots.
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Fig. 2 Log-log plots of particle-size distributions for four soil samples. Symbols denote experimental data, solid lines denote model fits
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Optimized parameters of the fragmentation model together with the median particle diameter for all the 19 soils are shown in Table 2
. The median diameter has been calculated from the measured PSDs by linearly interpolating the 50% quantile (e.g., Sokal and Rohlfs, 1995). The identified power-law domains separate the particle sizes in three classes, which we denote as clay, silt, and sand domains. The diameter boundaries between the clay and silt domains ranged from 0.33 to 0.99 µm, and between silt and sand domains from 45.3 to 126.7 µm. The average of the clay–silt domain boundary was 0.51 µm and of the silt–sand domain boundary was at 85.3 µm, with a coefficient of variation of 15 and 25%, respectively. The consistent occurrence of three power-law domains in all 19 soils and the close agreement of the domain scales indicate similarity between the different soils, particularly when considering the wide textural variability of the samples ranging from 0.3 to 46% clay. In a similar study on four soils, Wu et al. (1993) also found three domains where a power law was applicable, but the limits between the domains were located at 0.05 to 0.1 and 10 to 5000 µm. The consistency of the limits for the three domains needs therefore to be investigated across a greater number of soils.
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Table 2 Fragmentation fractal dimensions, median particle diameter, and cutoff boundaries, estimated from particle-size distribution data obtained by the light-diffraction method for the 19 soils
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We denote the three fractal dimensions determined in our study as Dclay, Dsilt, and Dsand. The fitted values of the fractal dimensions obey in all cases the relation: Dclay < Dsilt < Dsand. In the clay domain, the fractal dimension ranged from 0.118 to 1.21, in the silt domain from 1.728 and 2.792, and in the sand domain from 2.839 to 2.998 (Table 2). These data are consistent with the limits of the fragmentation approach given as 0 < D < 3 (Turcotte, 1986). The generally high value of the coefficient of determination R2 shows that the fragmentation models are good descriptions of the PSDs in the three domains. Some soils showed poor power-law agreement in the sand domain (e.g., Obfelden and Reckenholz). Experimental data in the sand as well as in the clay domain are limited by the experimental procedures, namely the maximum particle size as allowed by the 2-mm sieve mesh and the minimum particle size determined by the light-scattering technique.
The model used in the derivation of Eq. [2] is based on the fragmentation of an initially intact particle into smaller particles (Matsushita, 1985; Turcotte, 1986). An intact cubical particle of size h is fragmented into eight identical cubes of size h/2. Each of these smaller cubes is further divided in cubes with size h/4, and so forth. The fragmentation of a cube has a certain probability p, which is assumed to be constant for all orders of fragmentation. A cube can maximally disintegrate into eight smaller cubes
and minimally into one smaller cube
. As shown by Turcotte (1986), the fragmentation probability p is related to the fractal dimension D by
 | (8) |
where the range of possible fractal dimensions is 0 < D < 3. Figure 3
shows a plot of Eq. [8] along with values calculated from the analysis of our experimental data. A scale-independent fragmentation process would have a constant fragmentation probability. Evidently, fragmentation probabilities varied across almost the entire range of 1/8 < p < 1. It is interesting that the fractal dimensions for the three domains are typically Dclay < Dsilt < Dsand. It appears that for the 19 soils studied, the probability of fragmentation is scale dependent, and in particular it decreases with decreasing size of the particles. There is experimental evidence that fragmentation of soil and sediment aggregates is scale dependent (Perfect et al., 1993; Rasiah et al., 1993). Larger aggregates tend to fracture more easily than smaller aggregates (Perfect, 1997). Considering soil particles as products of a fragmentation process, our results are in qualitative agreement with observations from aggregate failure studies.
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Fig. 3 Fragmentation fractal dimensions D and probabilities p of fragmentation. The solid line represents Eq. [8], symbols are calculated with Eq. [8] from experimentally determined fractal dimensions
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Particle-size distribution measurements are strongly influenced by the experimental methods of dispersion of the soil particles. The dispersion itself can be regarded as a fragmentation process. Organic matter increases aggregate stability and hence leads to less fragmentation (Rasiah et al., 1993). Therefore the omission of organic matter removal in the Swiss soils probably leads to less dispersion of smaller particles. This explains the smaller clay fractions determined in the Swiss soils compared with the U.S. soils (Table 1). Based on the fragmentation model, we would also expect smaller fractal dimensions for the clay fraction of the Swiss soils compared with the U.S. soils; however, there is no evidence that this is the case (Table 2).
Following Tyler and Wheatcraft (1992), Dsilt vs. clay and sand fraction was plotted in Fig. 4
to demonstrate the relation between fractal dimension and soil texture. The fractal dimensions reported by Tyler and Wheatcraft (1992), plotted in this figure, were obtained by applying Eq. [2] to the entire range of the PSD data, which ranged from 1 to 50 µm, 0.5 µm to 5 mm, and 16 µm to 1 mm for different data sets. Fractal dimensions given by Tyler and Wheatcraft are therefore not directly comparable with our Dsilt, but nevertheless, Fig. 4 shows a trend between the D value and the clay and sand contents. The fractal dimension increases with clay content, and decreases with sand content. These results suggest that the power-law relation of Eq. [2] can be used to characterize PSD in soils, and may be an alternative to the conventionally used approaches, such as the lognormal distribution.
Calculation of Parameters of the Fragmentation Model from Mass Fractions of Clay and Silt
It is evident that no single power law can characterize the PSD of a soil across the entire scale usually measured in a particle-size analysis. For the majority of the samples, 46 to 86% (with an average of 71%) of the total mass is carried by particles with diameters between 0.51 and 85.3 µm, the silt domain of the distribution. On a log-log scale, the PSD of the silt domain is a straight line and is therefore characterized by two parameters, the intercept and the slope of the power-law distribution. If we know any two points on this line, we can calculate the model parameters of the silt domain. As Table 2 shows, the USDA boundaries between clay and silt (2 µm), and between silt and sand (50 µm) are within the silt domains for all 19 soils. Therefore we can use these standard particle-class fractions to calculate the two parameters of a power-law particle-size distribution in the silt domain. As the second parameter besides the fractal dimension D we choose the median particle diameter d50 of the PSD. The median particle diameter is chosen because it is often used in empirical relations to predict other soil properties, and as such is a useful parameter to know. The median particle diameter d50 and the fractal dimension of the silt domain Dsilt can be calculated from standard textural data as follows
 | (9) |
and
 | (10) |
where mclay and msilt are the mass fractions of clay and silt, log is the natural logarithm, and the median diameter d50 is given in micrometers. Note that the median d50 in Eq. [9] is a non-log-transformed parameter. Mass fractions of clay, silt, and sand are readily available for many soils, and from these data the median diameter and the fractal dimension of the silt domain can then be computed with Eq. [9] and [10]. Note that in the derivation of Eq. [9] and [10] we assumed the USDA textural definition; for other classification schemes, the equations have to be adapted accordingly.
We tested the validity of the two equations with our 19-soil data set. In Fig. 5
, experimental d50 and Dsilt values (Table 2) are plotted vs. values calculated based on standard particle-size fractions using Eq. [9] and [10]. There is a good agreement between experimental and calculated values, particularly for Dsilt. The RMSE in Fig. 5 shows that the d50 is not estimated as well as the Dsilt, and generally the model tends to underestimate the experimental value. The estimation of the fractal dimension is well performed by Eq. [10].
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Fig. 5 Experimental and calculated values of median diameter d50 and of fragmentation fractal dimension of the silt domain Dsilt for all 19 soils used in this study. Calculated values are from Eq. [9] and [10]. RMSE is the root mean square error
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Conclusions
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There is evidence that cumulative PSDs in soils follow a power-law distribution, consistent with a fractal fragmentation model. The mass-based approach suggested by Matsushita (1985) and Turcotte (1986) showed good agreement between the fractal model and our experimental data. Three main domains—a clay, silt, and sand domain—were identified where power-law scaling was applicable. The limits between the domains were relatively constant for different soil types, but do not coincide with the traditional boundaries between clay, silt, and sand. Fragmentation fractal dimensions of the three domains increased in the order: clay < silt < sand domain. A method is imposed to estimate the parameters of the fragmentation model of the PSD in the silt domain from standard textural data of clay, silt, and sand fractions.
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ACKNOWLEDGMENTS
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We thank Alan Busacca and Sandra Lilligren for assistance during the laboratory analyses. The manuscript benefitted from fruitful discussions with Claudio O. Stockle, Sally D. Logsdon, and Philippe Baveye.
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NOTES
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1 Reference to company name does not reflect endorsement of particular products by Washington State University. 
Received for publication August 26, 1998.
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1143 - 1150.
[Abstract]
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A. N. D. Posadas, D. Gimenez, M. Bittelli, C. M. P. Vaz, and M. Flury
Multifractal Characterization of Soil Particle-Size Distributions
Soil Sci. Soc. Am. J.,
September 1, 2001;
65(5):
1361 - 1367.
[Abstract]
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T. H. Skaggs, L. M. Arya, P. J. Shouse, and B. P. Mohanty
Estimating Particle-Size Distribution from Limited Soil Texture Data
Soil Sci. Soc. Am. J.,
July 1, 2001;
65(4):
1038 - 1044.
[Abstract]
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TOP
ABSTRACT
INTRODUCTION
