Extended Dual Composite Sphere Model for Determining Dielectric Permittivity of Andisols

Transcription

1 Published January, 2005 Extended Dual Composite Sphere Model for Determining Dielectric Permittivity of Andisols Teruhito Miyamoto,* Takeyuki Annaka, and Jiro Chikushi ABSTRACT ticle-size distribution (Robinson and Friedman, 2001) A dielectric mixing model as used for understanding the depen- ere considered in the modeling. dency of dielectric permittivity on ater content and soil physical From the modeling studies of dielectric permittivity properties. We used a mixing model to interpret aggregate structural of unsaturated soil, it as realized that distribution and effects on the relationship beteen volumetric ater content ( ) and geometry of the ater phase played an important role on dielectric permittivity (ε) of Andisols. Our objective as to determine macroscopic dielectric permittivity of soil. For example, the applicability of a mixing model to describe the dielectric permittiv- according to calculated results from mixing models that ity of aggregated soil. A dual composite sphere model, proposed by considered unsaturated soil as a three-dimensional net- Friedman in 1998, as extended by taking into account ater distri- ork of capacitors, Friedman (1997) pointed out that bution in aggregated soil and the processes of ater filling in intra- the continuity of the ater phase as an essential feaand interaggregate pores. Hence, our extended model includes four- ture for modeling dielectric permittivity of unsaturated phase composite spheres and sigmoidal functions as a eight function soil. Tabbagh et al. (2000) applied a numerical solution in the model. Experimental data for to et aggregates ( and of Maxell s equations based on a microscopic scale to mm in diam.) and to Andisols taken from Kumamoto and reproduce the empirical function proposed by Topp et Miyazaki in Japan ere used for the demonstration of model applicaal. (1980) and also found that it as necessary to conbility. The addition of an additional layer in the composite sphere model improved the predictability of the model for the ε relation- sider the continuity of the ater phase. More recently, ship in a moisture range of less-than-critical ater content. The dε/ Jones and Or (2003) discussed the effects of bound d curves estimated by the model ere in better agreement ith the ater, particle shape, phase configuration, and porosity experimental data than those of Friedman s model. In particular, ap- on permittivity by use of a three-phase mixing model. plying the sigmoidal eight functions improved the estimation of They found phase configuration had a substantial im- dε/d curves in a moisture range higher than that of critical ater pact on the prediction of dielectric permittivity. content. Our adjusted model serves to improve the understanding of Soil structure is one of the major factors affecting the relationship beteen the physical properties of aggregated soils various soil physical properties. Aggregate structure, and their dielectric permittivity. composed of to pore systems, intra- and interaggregate pores, especially contributes to more complicated ater M distributions in unsaturated soils. Andisols are very acroscopic dielectric properties of soil have unique soils in aggregate structure, ith ell-defined been highlighted as a subject of soil physics since and stable intra- and interaggregate voids. Miyamoto Topp et al. (1980) shoed the relationship beteen vol- et al. (2003) studied soil aggregate structure effects on umetric ater content ( ) and dielectric permittivity dielectric permittivity of an Andisol. They found that (ε). The ε relationships for various soils have been the relationship beteen and dε/d is partitioned into reported, and many attempts have also been made to to moisture ranges, ith the higher and loer moisture model their ε relationships by considering physical ranges corresponding to high and lo values of dε/d, results. A mixing model has been used for understand- respectively. In particular, the dε/d values remained ing the dependency of the dielectric permittivity on a- constant for the loer moisture range of less than the ter content and soil physical properties (Dirksen and critical ater content ( c ). It as speculated that the Dasberg, 1993). Some mixing models based on physical causes of this dielectric property for aggregated soil are theory have been proposed to describe the ε relationof the configuration of ater in aggregates, the processes ship (De Loor, 1964, 1990; Birchak et al., 1974; Ansoult ater filling in intra- or interaggregate pores, and et al., 1985; Friedman, 1998; Tabbagh et al., 2000) and the lo ε value of bound ater adsorbed on soil surfaces. extended according to ne experimental results. For Modeling these primary mechanisms affecting the di- example, the contribution of bound ater (Dobson et electric permittivity of aggregated soil is useful for better al., 1985), the temperature effect (Or and Wraith, 1999), understanding their effects. soil particle shape (Jones and Friedman, 2000), and parbased Various macroscopic dielectric models have been on the assumption that multiple independent ma- terials ith different dielectric permittivites are back- T. Miyamoto, National Agricultural Research Center for Kyushu Oki- ground features. Actually, it is difficult to describe ater naa Region, Nishigoshi, Kumamoto , Japan; T. Annaka, distribution in aggregates and the processes of ater Faculty of Agriculture, Yamagata Univ., Tsuruoka, Yamagata , Japan; J. Chikushi, Biotron Inst., Kyushu Univ., Hakozaki, Fukumay be a multiphase composite sphere model (Sihvola, filling in intra- or interaggregate pores. A feasible model oka , Japan. Received 25 Nov *Corresponding author (teruhito@affrc.go.jp). Abbreviations: ASW, air-solid-ater arrangement; AWSA, air-atersolid-ater Published in Soil Sci. Soc. Am. J. 69:23 29 (2005). arrangement; EMA, effective medium approximation; Soil Science Society of America SWA, solid-ater-air arrangement; TDR, time domain reflectometry; 677 S. Segoe Rd., Madison, WI USA WSWA, ater-solid-ater-air arrangement. 23

2 24 SOIL SCI. SOC. AM. J., VOL. 69, JANUARY FEBRUARY ). Friedman (1998) has conceived that the solid, (1998) has found that a dual three-layer composite sphere ater, and air components form a mixture of composite model is suitable for describing the dielectric permittivity of spheres and that the mixture is composed of concentric natural unsaturated soils. This model contains the composite shells ith radically changing phases ith the thickness spheres of the solid-ater-air arrangement (SWA) and the air-solid-ater arrangement (ASW). Note that SWA, for exof the ater shell depending on the saturation degree ample, means the sphere composed of solid, ater, and air in and specific surface area of the medium. He shoed order from the center toard the outer layer. For mixing the that the dielectric properties of multiphase concentric dual SWA-ASW system, ε as evaluated by the symmetric shells differ according to the combination of the solid, effective medium approximation (EMA) (Sen et al., 1981) ater, and air components. Friedman s model might be ith arbitrary linear eight functions depending on the saturation degree, S. The eight functions, f SWA and f ASW ere as applicable at describing the effect of ater distribution in aggregated soil. The objective of this study has been to determine f SWA 1 S, f ASW S, S /n [1] the applicability of the mixing model for describing the here n is a porosity. From the aspect of the real configura- dielectric permittivity of aggregated soils. Toard this tions of three phases in soils, ASW and SWA should be more end, e applied a composite sphere model to Andisols heavily eighted in high and lo ater content ranges, respec- tively. Applying the EMA equation to the dual SWA-ASW to provide some insight into the relationship beteen system results in the folloing solution for ε (Friedman, 1998). the physical properties of unsaturated soils and their dielectric behavior, and thus to indicate an interpretation of this complex problem. THEORETICAL CONSIDERATIONS Composite Sphere Model An unsaturated porous medium may be described as an array of spherical elements embedded in an infinite and macroscopically homogeneous matrix. Water content of the medium determines the dielectric permittivity (Friedman, 1998). The individual inclusions all have the same form, and can be represented by a single sphere in terms of electrical characteristics. The sphere is composed of multiple layers corresponding to the number of phases included in the medium. Therefore, it is sufficient to treat only one of them. No, if a uniform electrical potential gradient, E, is applied at infinity, the problem to be solved becomes to determine ε of the matrix for hich the composite sphere does not disturb this potential field (Fig. 1). Under quasistatic conditions, the electrical potential field can be determined by solving the Laplace equation ith the boundary conditions provided as a continuous potential or a normal flux at the interfaces. This composite sphere model has been examined for to layers (Maxell-Garnett, 1904; Sen et al., 1981), three layers (Tinga et al., 1973; Friedman, 1998), and multiple layers (Sihvola, 1989, 1997). Dual Composite Sphere Model (Friedman s Model) Soil ater distribution in an unsaturated soil is commonly heterogeneous. Hence, it is difficult to describe the dielectric permittivity by a single composite sphere model. Friedman here ε C 2 16 ε SWAε ASW 2 1/2 C 4 [2] C f SWA (ε ASW 2ε SWA ) f ASW (ε SWA 2ε ASW ) [3] ε ASW and ε SWA are dielectric permittivites using the three-phase composite sphere model ith ASW and SWA arrangements, respectively. Composite Sphere Model for Describing Aggregated Structure Dielectric permittivity of an Andisol is affected by its aggre- gate structure (Miyamoto et al., 2003). In the ε curve, there exists a critical ater content at hich dε/d sharply changes. Fig. 1. A schematic description of the four-phase composite sphere model. Fig. 2. A schematic description of the four-phase composite sphere model for describing the dielectric permittivities of aggregated soils.

3 MIYAMOTO ET AL.: DETERMINING DIELECTRIC PERMITTIVITY OF ANDISOLS 25 Moreover, the relationship beteen and dε/d for aggre- in the loer moisture range, the AWSA or WSWA arrangement gated soil as partitioned into to moisture ranges; that is, as used instead of the SWA arrangement. That is, the the dε/d value as high and lo in higher and loer moisture ε( c ) can be given by ranges, respectively. In particular, the dε/d value as almost constant for the moisture range loer than the critical ater ε C 2 16 ε AWSAε ASW 2 1/2 C content. This characteristic cannot be seen in the case of nonaggregated 4 [6] soils. Therefore, hen applying a composite sphere model to aggregated soils, e should consider the ater distribution here in aggregates, the processes of ater filling in intra- C f AWSA (ε AWSA 2ε ASW ) f ASW (ε ASW 2ε AWSA ) [7] or interaggregate pores, and the dielectric permittivity of ater held in intraaggregate pores. To describe these features of and the ε( c )by aggregated soils, e must adjust the composite sphere model ε C 2 16 ε WSWAε ASW 2 1/2 by increasing the number of layers. C The pore system in aggregated soil is frequently classified 4 [8] into intra- and interaggregate pores. Because of this structure, here the ater retention curve of aggregate is often bimodal. First, e designed a five-layer composite model in hich the third C f WSWA (ε WSWA 2ε ASW ) f ASW (ε ASW 2ε WSWA ) [9] layer from the outside of the composite sphere as assumed here ε AWSA, ε WSWA, ε ASW are the dielectric permittivities of as the solid phase, and thus the outside of the third layer AWSA, WSWA, and ASW arrangements, and f AWSA, f WSWA, represents the interaggregate pore and the inside represents and f ASW are eight functions for AWSA, WSWA, and ASW the intraaggregate pore. Assuming that soil ater is held arrangements, respectively. In this model, the ASW arrangemainly inside aggregates in a loer ater content range and ment describes the soil ater adsorbed on the surface of aggrein interaggregate pores in a higher ater content range, e gates and held around contact points beteen aggregates in ere able to reduce the number of layers to four for describing the loer moisture range and entrapped air in the higher the dielectric permittivity of aggregated soils (Fig. 2). moisture range. Here e ill consider a four-layer composite sphere model depicted in Fig. 1, here the boundary beteen subregions of permittivities ε k and ε k 1 is a sphere of radius R k 1,(k Weight Function 1, 2, 3). Folloing Fiedman s (1998) derivation for three Calculating the dual composite sphere model, Friedman phases, e can derive a four-layer composite sphere model (1998) applied the arbitrary linear eight functions depending as follos: on the saturation degree. We also decided to use the linear eight functions as a first trial. Hoever, e found that it is ε ε 1 3ε 1 {(ε 1 ε 2 )(2ε 2 ε 3 )(2ε 3 ε 4 )(φ 2 φ 3 φ 4 ) better to adopt more suitable eight functions reflecting ater 2(ε 1 ε 2 )(ε 2 ε 3 )(ε 3 ε 4 )[(φ 2 φ 3 φ 4 )φ 4 /(φ 3 φ 4 )] distribution in aggregates. In a moisture range loer-thancritical ater content, soil ater is held mainly in intraaggre- (ε 1 2ε 2 )(ε 2 ε 3 )(2ε 3 ε 4 )(φ 3 φ 4 ) (ε 1 2ε 2 ) gate pores and around contact points beteen aggregates. The (ε 2 2ε 3 )(ε 3 ε 4 )φ 4 }/{(2ε 1 ε 2 )(2ε 2 ε 3 )(2ε 3 ε 4 ) fraction of ater content held in intraaggregate pores is much larger than that adsorbed on an aggregate surface and held 2(2ε 1 ε 2 )(ε 2 ε 3 )(ε 3 ε 4 )[φ 4 /(φ 3 φ 4 )] 2(ε 1 ε 2 ) around contact points beteen aggregates. Hence, the eight (ε 2 ε 3 )(2ε 3 ε 4 )[(φ 3 φ 4 )/(φ 2 φ 3 φ 4 )] 2(ε 1 ε 2 ) of AWSA is much larger than that of ASW at a lo moisture content. On the basis of these considerations, to sigmoidal (ε 2 2ε 3 )(ε 3 ε 4 )[φ 4 /(φ 2 φ 3 φ 4 )] (ε 1 ε 2 )(2ε 2 ε 3 ) functions ere adopted instead of the arbitrary linear eight (2ε 3 ε 4 )(φ 2 φ 3 φ 4 ) 2(ε 1 ε 2 )(ε 2 ε 3 )(ε 3 ε 4 ) [(φ 2 φ 3 φ 4 )φ 4 /(φ 3 φ 4 )] (ε 1 2ε 2 )(ε 2 ε 3 )(2ε 3 ε 4 ) (φ 3 φ 4 ) (ε 1 2ε 2 )(ε 2 2ε 3 )(ε 3 ε 4 )φ 4 } [4] here φ i,(i 1, 2, 3, 4), is the volumetric fraction of the ith composition, and R i the radii of the sphere, and thus e can obtain φ 1 (R 3 1 R 3 2)/R 3 1, φ 2 (R 3 2 R 3 3)/R 3 1, φ 3 (R 3 3 R 3 4)/R 3 1, φ 4 R 3 4/R 3 1 [5] Extension of Friedman s Model We described the dielectric permittivity of aggregated soils by a dual composite sphere model folloing Friedman (1998). Minimum modification as applied to adjust Friedman s model to aggregated soils. Usually, the ater phase is in contact ith the solid phase in soil. Therefore, a combination of the air ater solid ater (AWSA) arrangement and the ater solid ater air (WSWA) arrangement for moisture ranges belo and above the critical ater content respectively must be suitable to describe the aggregate structural effects. Since the aggregated structure effects appear predominantly functions. By assuming f AWSA f AWSA f WSWA, the one (SW1) as f WSWA 1 3S 2 2S 3, f ASW 3S 2 2S 3 [10] hich Friedman (1998) used as a eight function for his model. The other one (SW2) as f AWSA f ASW f WSWA exp[ 10(S 0.5)] 1 1 exp[ 10(S 0.5)], [11] In summary, e tried to calculate for three cases of a four-layer composite model: ith the arbitrary linear eight functions depending on the saturation degree (LW), and to different sigmoidal functions (SW1 and SW2). Permittivity of Water close to Soil Surface Bound ater has been found to possess a loer dielectric permittivity than free ater. The permittivity has been shon to be a function of distance from the solid surface. The reduced permittivity of bound ater as estimated using the ater film thickness approach taken by Friedman (1998). The dielectric

4 26 SOIL SCI. SOC. AM. J., VOL. 69, JANUARY FEBRUARY 2005 permittivity of the ater phase is represented using an exponential function starting from a minimum (ε min 5.5) near the surfaces of solid phase and toard a maximum permittivity (ε max 80.4, at 20 C) hich describes the free ater phase distant from the solid surfaces, resulting in form analysis softare (Or et al., 1997) that enabled automated TDR control, data acquisition, and aveform analysis. The measurement of ε as repeated three times for each sample. Water contents ere measured by eighing the soil samples gravimetrically using an electronic balance. To prepare the samples ith a higher ater content, the soils ere removed from the cylinders and received approximately 15 ml of dis- tilled ater. The soils ere left undisturbed for at least 24 h to allo the moisture to become evenly redistributed. The et soils ere repacked into the same cylinders ith the same bulk density and the measurements of and ε ere conducted as mentioned above. These steps ere repeated 10 to 12 times for each soil. The experiment as repeated tice folloing the same procedure. All these experiments ere conducted in a laboratory in hich room temperature as kept at 20 C. ε ater d ε max d 1 ln ε max (ε max ε min )e d ε min [12] here 10 8 cm 1 and d is the approximate thickness of the ater film spread uniformly over the surface of the solid phase. The average thickness of the ater film is calculated by dividing the volumetric ater content by b and the soil surface area (S A ): d / b S A [13] MATERIALS AND METHODS Soil Samples We used to Andisols (Hydric Pachic Melanudand) for experiments in this study. Soils ere from experimental fields at the National Agricultural Research Center for Kyushu Okinaa Region in both Kumamoto and Miyazaki, Japan. The soils ere passed through a 2-mm sieve and air dried. In addition to these soils, e prepared et aggregates to determine the aggregate structure effects on dielectric permittivity measured clearly by time domain reflectometry (TDR). A clod of the soil sample from Kumamoto as prepared into four aggregates as follos. A nest of five sieves ere placed in a holder and suspended in a container of ater. The mesh sizes of the sieves ere 2.0, 1.0, 0.5, 0.25, and 0.1 mm. The sample clod as put on the top sieve of the nest, and then the nest as alternately moved up and don for a vertical distance of 38 mm at a rate of 30 cycles min 1 for 40 min. Thus, size fractions of the aggregates ere 1.0 to 2.0, 0.5 to 1.0, 0.25 to 0.5, and 0.1 to 0.25 mm in diameter. The et aggregates, of hich aggregate sizes ere 1.0 to 2.0 and 0.1 to 0.25 mm in diam., ere used to determine the ε relationship. Physical properties for soil samples used in this study are shon in Table 1. Measurement of ε Relationships For measurement of the ε relationships, e used four acrylic cylinders of 62.8 mm in diam. and 130 mm high. Each soil as packed into the cylinder until 110 mm high as uniformly as possible. When e packed the soils, e did not compress the soils directly but rapped at the all of the cylinder to prevent breaking the aggregate structure of the samples. The measurement of ε as conducted using a TDR cable tester (Tektronix 1502B). For all measurements, e used the same three-rod TDR probe (3 mm in diam., 100 mm long, and 15 mm in space beteen center and outside rods). The TDR probe as inserted vertically into the soil columns. Waveform analysis as conducted using the WinTDR ave- Table 1. Physical properties for soil samples used in this study. Particle Bulk Specific 1555-kPa Soil samples density density surface ater content Modeled Calculation of Four-Layer Composite Sphere Models To evaluate their difference from the three-phase arrangement of the model, the four-layer models ere calculated ith the folloing conditions: (i) the porosity as 0.73 and the critical ater content assumed to be the volumetric ater content corresponding to the matric potential at 1555 kpa ( c 0.31), (ii) the dielectric permittivities for solid and air ere assumed to be 5.5 and 1, respectively, and (iii) the per- mittivity for ater as estimated using the ater film thickness approach taken by Friedman (1998). In addition, a three- phase model ith ASW and SWA arrangements employed by Friedman (1998) as also used for comparison. RESULTS AND DISCUSSION Calculation Results of Four-Layer Composite Sphere Model Figure 3 shos the results calculated using the fourlayer composite sphere model for to different ranges. In the moisture range from 0 to 0.31, the AWSA as Mg m 3 m 2 g 1 m 3 m 3 Fig. 3. Calculated dielectric permittivities by using the to different Wet aggregate ( mm) four-layer composite sphere models (ASW, air-solid-ater arrange- Wet aggregate ment; AWSA, air-ater-solid-ater arrangement; SWA, solid-ater- ( mm) air arrangement; WSWA, ater-solid-ater-air arrangement). Di- Andisol (Kumamoto) electric permittivities calculated by SWA configuration are also Andisol (Miyazaki) shon.

5 MIYAMOTO ET AL.: DETERMINING DIELECTRIC PERMITTIVITY OF ANDISOLS 27 estimated at a slightly higher ε than SWA. The calcu- the ater content range of 0.05 to 0.45 cm 3 cm 3 for lated dielectric permittivity of the AWSA arrangement et-aggregate soils and Andisols from Miyazaki, 0.09 as significantly smaller than the measured one even to 0.54 cm 3 cm 3 for Andisols from Kumamoto. Beyond at 0.31 since the effect of the external air-phase these maximum ater contents, uniform soil packing layer encapsulating the solid layer as excessive on the ith a constant bulk density became difficult as the estimation of ε. Contrary to these results, in the moisture etness increased. range of 0.31 to saturation, the estimated dielectric per- There might be concern about breakage of aggregate mittivities greatly increased ith. Moreover, although structure during the experimental processes of sequenthe curvature calculated by the WSWA arrangement tial ater additions and repacking. A small amount of as similar to that by the SWA arrangement, the cal- aggregate breakage occurred during this experiment; culated permittivities at saturation ere significantly hoever, for all soils the ε relationships ere on a different from each other. Using WSWA and AWSA moderately rising tendency to a critical value, but bearrangements for high and lo ater content ranges, yond that, the ε value steeply developed ith. This respectively, a moderate change of ε at critical ater characteristic as similar to the results reported by content could be described. Therefore, a combination Miyamoto et al. (2003). Therefore, e judged that the of the four-layer composite sphere models make it possi- effect of experimental processes on the ε relationships ble to describe the aggregate structure effects on dielec- as relatively small. tric permittivity. In the ε relationships for all soils, there exists a ε Relationships for Aggregated Soils critical ater content at hich dε/d sharply changes. This characteristic as more obvious for et-aggregate Figure 4 shos the ε relationships for four samples soils than that for the to Andisols. These results ere used in this study. The relationships ere obtained in explained by difference in pore size distribution. For Fig. 4. Comparisons beteen model predictions and measured data of the relationship beteen volumetric ater content and dielectric permittivity for four different soil samples. Three different eight functions, that is, to different sigmoidal functions, SW1 and SW2, correspond to Eq. [10] and [11], respectively, and linear function (LW) is applied for the calculations of four-layer model.

6 28 SOIL SCI. SOC. AM. J., VOL. 69, JANUARY FEBRUARY 2005 better than the other three models, although this model tended to underestimate the permittivity around critical ater content. The estimated ε relationships of all four models agreed ell ith the experimental data for Andisols from Kumamoto. In particular, the estimated ε rela- tionships of the four-layer models corresponded better ith the measured data than Friedman s model in the range of loer-than-critical ater content. These results ere similar to those for et-aggregate soils. For the range of higher-than-critical ater content, the measured data ere located beteen to ε curves esti- mated by four-layer models ith different sigmoidal eight functions. Compared ith ε relationships for et-aggregate soils, the dε/d values in the high ater content range increased moderately. For Andisols from Miyazaki, all models underesti- mated the ε relationship. Hoever, the curvature cal- culated by the four-layer model ith SW1 fitted ell, and the estimated values ere slightly loer than the measured data. The permittivity of the solid phase for volcanic soils might be higher than that for mineral soils (Regalado et al., 2003). Once e assumed ε s 10 for the calculation of the mixing models, the estimated ε relationship became closer to the measured data than that ith the assumption of ε s 5.5 (not shon). Ho- ever, a detailed discussion of the nature of this effect is beyond the scope of this paper. In contrast to the predicted ε relationships, the cal- culated dε/d curves ere quite different from each other among the four models (Fig. 5). The dε/d values of the Friedman s model increased ith monotonically in the moisture ranges beteen 0 and 0.5 m 3 m 3. Contrarily, the calculated results for the four-layer models fitted satisfactorily to the experimental data in moisture ranges loer-than-critical ater content. In particular, the dε/d level as lo in a lo moisture range and increased rapidly nearing critical ater content. Hoever, the calculated dielectric permittivity for the fourlayer model using linear eight function almost overlapped Friedman s model and monotonically increased ith until near saturation in the higher moisture range. For the four-layer model employing SW2 sigmoidal eight functions, hoever, the dε/d folloed the observed data relatively ell even in the higher moisture range. The addition of a layer in the composite sphere model, that is, from three layers to four layers, improved the predictability of the model for the ε relationship in a moisture range of less-than-critical ater content. In particular, the four-layer model could describe dε/ d curves better than Friedman s model in the same moisture range. The reason for these results can be explained by the difference of calculated results beteen SWA and AWSA arrangements. Friedman (1998) employed the SWA arrangement as the more eighted of the to components in loer moisture content. The estimated permittivity based on this arrangement in- creased gradually and linearly in the lo moisture range (Fig. 2 in Friedman, 1998). Contrary to this, the per- mittivity based on AWSA shoed an increase in ε ith et-aggregate soils, the to clear peaks of pore size density made it possible to divide pore space into to distinguishable pore systems (Miyamoto et al., 2003). Hence, a clear change of ater-configuration in pores, and thus ε, occurred in et-aggregate soils. On the other hand, such an abrupt change of ε may not occur in to Andisols. Comparison beteen Experimental Data and Calculations For evaluating the proposed four-layer composite model ith sigmoidal eight functions, e fitted the model to the experimental data for aggregated soils (Fig. 4). To different types of sigmoidal function ere used. In addition, the estimations based on both Friedman s model and the four-layer model ith the linear eight function ere obtained for comparison ith the model ith sigmoidal eight functions. For et-aggregate soils, the estimated ε relationships of three models except the four-layer composite model ith SW2 agreed ell ith the experimental data ith a ater content range of less-than-critical ater content. Hoever, beyond critical ater content, the difference beteen estimation and experimental data became large. The Friedman s model tended to underestimate dielectric permittivity in a moisture range of lessthan-critical ater content. If aggregate-structural effects are taken into account, the predictability of the mixing model becomes improved. In contrast to these results, the four-layer composite model ith SW2 estimated the permittivity of the hole ater content range Fig. 5. Comparisons beteen model predictions and measured data of the relationship beteen volumetric ater content and dε/d for et-aggregate soil, hich aggregate size is 1.0 to 2.0 mm in diameter. Four model predictions are shon: four-layer model ith sigmoidal eight functions, Eq. [10] (solid line), four-layer model ith sigmoidal eight functions, Eq. [11] (dash dot line), fourlayer model ith linear eight functions (dash line), Friedman s model (dot line).

7 MIYAMOTO ET AL.: DETERMINING DIELECTRIC PERMITTIVITY OF ANDISOLS 29 in a very dry range and remained almost constant estimation of dε/d curves in a moisture range higherthan-critical ater content. Our improved model proithin the moisture range of less-than-critical ater content (Fig. 3). An increase in the number of layers in the vides more insight into the relationship beteen the model becomes more suitable to describe the feature physical properties of aggregated soils and their dielec- of aggregate structure effect on dielectric permittivity. tric permittivity. We introduced the sigmoidal eight functions for representing the situation that soil ater is held mainly REFERENCES inside aggregate pores at a moisture range of loer- Ansoult, M., L.W. De Backer, and M. Declercq Statistical than-critical ater content. Thus, the eight of AWSA relationship beteen apparent dielectric constant and ater con- must be much larger than that of ASW in the lo mois- tent in porous media. Soil Sci. Soc. Am. J. 49: Birchak, J.R., C.G. Gardner, J.E. Hipp, and J.M. Victor High ture range. Hoever, the sigmoidal eight functions im- dielectric constant microave sensing soil moisture. Proc. IEEE proved the estimation of dε/d curves even in a mois- 62: ture range higher than that of critical ater content Dirksen, C., and S. Dasberg Improved calibration of time do- (Fig. 5). In this range, a continuous ater phase could main reflectometry soil ater content measurements. Soil Sci. Soc. Am. J. 57: be expected in unsaturated aggregated soil. Since in- De Loor, G.P Dielectric properties of heterogeneous mixtures traaggregate pores are almost saturated at critical ater ith a polar constituent. Appl. Sci. Res. B11: content, additional ater ould be easily connected De Loor, G.P The dielectric properties of et soils. BCRS report ith the ater included in aggregate, and thus the air No :1 39. The Netherlands Remote Sensing Board, Delft. Dobson, M.C., F.T. Ulaby, M.T. Hallikainen, and M.A. El-rayes in the interaggregate pores ould be entrapped by the Microave dielectric behavior of et soil: II. Dielectric mixing ater phase. Therefore, the use of the sigmoidal func- models. IEEE Trans. Geosci. Remote Sens. GE-23: tions improve the prediction of dε/d curves because Friedman, S.P Statistical mixing model for the apparent dielectric constant of unsaturated porous media. Soil Sci. Soc. Am. J. the eight of ASW remarkably increases near critical 61: ater content. Friedman, S.P A saturation degree-dependent composite spheres Applying the four-layer model ith the sigmoidal model for describing the effective dielectric constant of unsaturated function, e could ell describe the ε relationship porous media. Water Resour. Res. 34: for aggregated soils. In particular, the increase in the Jones, S.B., and S.P. Friedman Particle shape effects on the effective permittivity of anisotropic or isotropic media consisting number of layers in the model resulted in a good fit of aligned or randomly oriented ellipsoidal particles. Water Resour. of the estimated ε relationship in a less-than-critical Res. 36: ater content range. Moreover, e could estimate di- Jones, S.B., and D. Or Modeled effects on permittivity measurements of ater content in high surface area porous media. Physica electric permittivity of the ater phase by using the B 338: ater film thickness approach taken by Friedman Maxell-Garnett, J.C Color in metal glasses and metal films. (1998). These results may suggest that the dielectric Trans. R. Soc. London. Ser. A. 203: property of ater held in aggregates is similar to that Miyamoto, T., T. Annaka, and J. Chikushi Soil aggregate structure effects on dielectric permittivity of an Andisol measured by of ater in mineral soils. That is, even though ater is held inside an aggregate, the monomolecular layer of time domain reflectometry. Vadose Zone J. 2: Or, D., B. Fisher, R.A. Hubscher, and J.M. Wraith WinTDR98 ater is strongly bonded to the solid surface, and the Users Guide. Utah State Univ., Logan. second layer of ater is held more loosely than the first Or, D., and J.M. Wraith Temperature effects on soil bulk dielectric permittivity measured by time domain reflectometry: A physilayer. Hoever, there is little experimental evidence concerning the dielectric permittivity of ater inside cal model. Water Resour. Res. 35: Regalado, C.M., R. Munoz Carpena, A.R. Socorro, and J.M. Hernánan aggregate. dez Moreno Time domain reflectometry models as a tool to understand the dielectric response of volcanic soils. Geoderma 117: CONCLUSION Robinson, D.A., and S.P. Friedman Effect of particle size distribution on the effective dielectric permittivity of saturated granular The four-layer composite sphere model using a sig- media. Water Resour. Res. 37: moidal eight functions can describe aggregate strucsedimentary rocks ith application to the dielectric constant of Sen, P.N., C. Scala, and M.H. Cohen A self-similar model for ture effects on dielectric permittivity. In modifying fused glass beads. Geophysics 46: Friedman s model (Friedman, 1998), e considered the Sihvola, A.H Self-consistency aspects of dielectric mixing theolo ε value of bound ater adsorbed on a soil surface, ries. IEEE Trans. Geosci. Remote Sens. 27: the ater distribution in aggregated soils, and the pro- Sihvola, A.H A revie of dielectric mixing models. Helsinki cesses of ater filling in intra- and interaggregate pores. Univ. of Technol. Electromagnetic Lab. Rep. 248:1 17. Tabbagh, A., C. Camerlynck, and P. Cosenza Numerical model- The addition of a layer in the composite sphere model ing for investigating the physical meaning of the relationship beimproved the predictability of the model for the ε teen relative dielectric permittivity and ater content of soils. relationship in the moisture range of less-than-critical Water Resour. Res. 36: ater content. The dε/d curves estimated by the Tinga, W.R., W.A.G. Voss, and D.F. Blossey Generalized ap- proach to multiphase dielectric theory. J. Appl. Phys. 44: model ere in better agreement ith the experimental Topp, G.C., J.L. Davis, and A.P. Annan Electromagnetic deterdata than those of Friedman s model. In particular, mination of soil ater content: Measurents in coaxial transmission applying the sigmoidal eight functions improved the lines. Water Resour. Res. 16: