Characterization of Particle-Size Distribution in Soils with a Fragmentation Model Marco Bittelli,* Gaylon S. Campbell and Markus Flury ABSTRACT to be best suited. The Shiozawa and Campbell model divides the particle distribution into two parts domi- Particle-size distributions (PSDs) of soils are often used to estimate nated by primary (sand and silt) and secondary (clay) other soil properties, such as soil moisture characteristics and hydraulic conductivities. Prediction of hydraulic properties from soil texture minerals, respectively. However, as pointed out by Bu- requires an accurate characterization of PSDs. The objective of this chan et al. (1993), the assumption of a lognormal distri- study was to test the validity of a mass-based fragmentation model bution in the clay fraction cannot be justified because to describe PSDs in soils. Wet sieving, pipette, and light-diffraction Shiozawa and Campbell (1991) had no data available techniques were used to obtain PSDs of 19 soils in the range of 0.05 in that range. to 2000 m. Light diffraction allows determination of smaller particle One of the latest developments in the study of PSDs sizes than the classical sedimentation methods, and provides a high in soils has focused on the use of fractal mathematics resolution of the PSD. The measured data were analyzed with a mass- to characterize particle sizes in soil (Turcotte, 1986; based model originating from fragmentation processes, which yields Tyler and Wheatcraft, 1992; Wu et al., 1993). However, a power-law relation between mass and size of soil particles. It was questions remain about the validity and applicability of found that a single power-law exponent could not characterize the PSD across the whole range of the measurements. Three main power- fractal concepts to PSDs. There has been some discus- law domains were identified. The boundaries between the three do- sion about the proper use and definition of the term mains were located at particle diameters of 0.51  0.15 and 85.3  “fractal” in the literature (Young et al., 1997; Pachepsky 25.3 m. The exponent of the power law describing the domain be- et al., 1997; Baveye and Boast, 1998). Different concepts tween 0.51 and 85.3 m was correlated with the clay and sand contents of fractals are used, and these concepts lead to different of the soil sample, indicating some relationship between power-law interpretations of fractal dimensions obtained. There- exponent and textural class. Two simple equations are derived to fore it is essential to clearly specify the type of fractal calculate the parameters of the fragmentation model of the domain model used. between 0.51 and 85.3 m from mass fractions of clay and silt. Particle- and aggregate-size distributions are often rendered as cumulative functions, either as number of particles larger than a certain diameter, or as mass P article-size distribution in soil is one of the more smaller than a certain diameter. These cumulative distri- fundamental soil physical properties. It is widely bution functions have been analyzed with power-law used for the estimation of soil hydraulic properties such relations and the exponents interpreted as fractal di- as the water-retention curve and saturated as well as mensions. Tyler and Wheatcraft (1989, 1992) analyzed unsaturated conductivities (Arya and Paris, 1981; particle-size data ranging from 0.5- to 5000-m radii, Campbell and Shiozawa, 1992). Generally, a conven- and observed that the fractal power law was not valid tional particle-size analysis involves the measurement across the entire extent of particle sizes. It is expected of the mass fractions of clay, silt, and sand. These frac- that there are lower and upper limits to the validity tions may be used to find the textural class using a of fractal relations (Turcotte, 1986). Wu et al. (1993) textural diagram, commonly in form of a textural trian- measured PSDs down to 0.02-m radius by using light- gle (e.g., Gee and Bauder, 1986). However, soil samples scattering techniques, and found a power-law relation that fall into a certain textural class may have consider- between number of particles and particle radius valid ably different PSDs. For example, the textural class of across a range of particle radii with a lower cutoff be- “clay” in the USDA classification scheme (Gee and tween 0.05 and 0.1 m and an upper cutoff between 10 Bauder, 1986) contains soil samples that vary in clay and 5000 m. Assuming that the exponent of a power- content between 40 and 100%. The size definitions of law relation is a fractal dimension, Wu et al. (1993) the three main particle fractions of clay, silt, and sand, found a dimension of D = 2.8  0.1 for the four soils used as diagnostic characteristics in most classification studied and suggested that this might be a universal schemes, are rather arbitrary, and they do not provide value of an underlying structure. Kozak et al. (1996) complete information on the soil PSD. analyzed PSDs of 2600 soil samples and found that for A more accurate description of texture is obtained 50% of the samples power-law scaling of particle num- by defining a PSD function. Commonly, PSDs are re- bers vs. size was not applicable across the whole range ported as cumulative distributions, and different func- of particle sizes between 2 and 1000 m. The authors tions have been proposed to fit experimental data. Bu- indicate that power-law scaling might be applicable for chan et al. (1993) fitted several of these models to a narrower range of particle sizes, although this was not experimental data and found the bimodal lognormal analyzed in their study. distribution proposed by Shiozawa and Campbell (1991) Most applications of fractal concepts to particle- and aggregate-size distributions are based on the fragmenta- M. Bittelli, G.S. Campbell, and M. Flury, Department of Crop and tion model of Matsushita (1985) and Turcotte (1986). Soil Sciences, Washington State University, Pullman, WA 99164. Re- ceived 26 Aug. 1998. *Corresponding author (bittelli@mail.wsu.edu). Abbreviations: PSD, particle-size distribution; RMSE, root mean square error. Published in Soil Sci. Soc. Am. J. 63:782–788 (1999). 782