Published in Soil Sci. Soc. Am. J. 69:23-29 (2005).
© 2005 Soil Science Society of America
677 S. Segoe Rd., Madison, WI 53711 USA
Division S-1—Soil Physics
Extended Dual Composite Sphere Model for Determining Dielectric Permittivity of Andisols
Teruhito Miyamotoa,*,
Takeyuki Annakab and
Jiro Chikushic
a National Agricultural Research Center for Kyushu Okinawa Region, Nishigoshi, Kumamoto 861-1192, Japan
b Faculty of Agriculture, Yamagata Univ., Tsuruoka, Yamagata 997-8555, Japan
c Biotron Inst., Kyushu Univ., Hakozaki, Fukuoka 812-8581, Japan
* Corresponding author (teruhito{at}affrc.go.jp).
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ABSTRACT
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A dielectric mixing model was used for understanding the dependency of dielectric permittivity on water content and soil physical properties. We used a mixing model to interpret aggregate structural effects on the relationship between volumetric water content (
) and dielectric permittivity (
) of Andisols. Our objective was to determine the applicability of a mixing model to describe the dielectric permittivity of aggregated soil. A dual composite sphere model, proposed by Friedman in 1998, was extended by taking into account water distribution in aggregated soil and the processes of water filling in intra- and interaggregate pores. Hence, our extended model includes four-phase composite spheres and sigmoidal functions as a weight function in the model. Experimental data for two wet aggregates (0.1–0.25 and 1.0–2.0 mm in diam.) and two Andisols taken from Kumamoto and Miyazaki in Japan were used for the demonstration of model applicability. The addition of an additional layer in the composite sphere model improved the predictability of the model for the – relationship in a moisture range of less-than-critical water content. The –d/d curves estimated by the model were in better agreement with the experimental data than those of Friedman's model. In particular, applying the sigmoidal weight functions improved the estimation of –d/d curves in a moisture range higher than that of critical water content. Our adjusted model serves to improve the understanding of the relationship between the physical properties of aggregated soils and their dielectric permittivity.
Abbreviations: ASW, air-solid-water arrangement • AWSA, air-water-solid-water arrangement • EMA, effective medium approximation • SWA, solid-water-air arrangement • TDR, time domain reflectometry • WSWA, water-solid-water-air arrangement
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INTRODUCTION
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MACROSCOPIC DIELECTRIC properties of soil have been highlighted as a subject of soil physics since Topp et al. (1980) showed the relationship between volumetric water content () and dielectric permittivity (). The – relationships for various soils have been reported, and many attempts have also been made to model their – relationships by considering physical results. A mixing model has been used for understanding the dependency of the dielectric permittivity on water content and soil physical properties (Dirksen and Dasberg, 1993). Some mixing models based on physical theory have been proposed to describe the – relationship (De Loor, 1964, 1990; Birchak et al., 1974; Ansoult et al., 1985; Friedman, 1998; Tabbagh et al., 2000) and extended according to new experimental results. For example, the contribution of bound water (Dobson et al., 1985), the temperature effect (Or and Wraith, 1999), soil particle shape (Jones and Friedman, 2000), and particle-size distribution (Robinson and Friedman, 2001) were considered in the modeling.
From the modeling studies of dielectric permittivity of unsaturated soil, it was realized that distribution and geometry of the water phase played an important role on macroscopic dielectric permittivity of soil. For example, according to calculated results from mixing models that considered unsaturated soil as a three-dimensional network of capacitors, Friedman (1997) pointed out that the continuity of the water phase was an essential feature for modeling dielectric permittivity of unsaturated soil. Tabbagh et al. (2000) applied a numerical solution of Maxwell's equations based on a microscopic scale to reproduce the empirical function proposed by Topp et al. (1980) and also found that it was necessary to consider the continuity of the water phase. More recently, Jones and Or (2003) discussed the effects of bound water, particle shape, phase configuration, and porosity on permittivity by use of a three-phase mixing model. They found phase configuration had a substantial impact on the prediction of dielectric permittivity.
Soil structure is one of the major factors affecting various soil physical properties. Aggregate structure, composed of two pore systems, intra- and interaggregate pores, especially contributes to more complicated water distributions in unsaturated soils. Andisols are very unique soils in aggregate structure, with well-defined and stable intra- and interaggregate voids. Miyamoto et al. (2003) studied soil aggregate structure effects on dielectric permittivity of an Andisol. They found that the relationship between and d/d is partitioned into two moisture ranges, with the higher and lower moisture ranges corresponding to high and low values of d/d, respectively. In particular, the d/d values remained constant for the lower moisture range of less than the critical water content (c). It was speculated that the causes of this dielectric property for aggregated soil are the configuration of water in aggregates, the processes of water filling in intra- or interaggregate pores, and the low value of bound water adsorbed on soil surfaces. Modeling these primary mechanisms affecting the dielectric permittivity of aggregated soil is useful for better understanding their effects.
Various macroscopic dielectric models have been based on the assumption that multiple independent materials with different dielectric permittivites are background features. Actually, it is difficult to describe water distribution in aggregates and the processes of water filling in intra- or interaggregate pores. A feasible model may be a multiphase composite sphere model (Sihvola, 1997). Friedman (1998) has conceived that the solid, water, and air components form a mixture of composite spheres and that the mixture is composed of concentric shells with radically changing phases with the thickness of the water shell depending on the saturation degree and specific surface area of the medium. He showed that the dielectric properties of multiphase concentric shells differ according to the combination of the solid, water, and air components. Friedman's model might be applicable at describing the effect of water distribution in aggregated soil.
The objective of this study has been to determine the applicability of the mixing model for describing the dielectric permittivity of aggregated soils. Toward this end, we applied a composite sphere model to Andisols to provide some insight into the relationship between the physical properties of unsaturated soils and their dielectric behavior, and thus to indicate an interpretation of this complex problem.
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THEORETICAL CONSIDERATIONS
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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. Now, if a uniform electrical potential gradient, E, is applied at infinity, the problem to be solved becomes to determine of the matrix for which 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 with the boundary conditions provided as a continuous potential or a normal flux at the interfaces. This composite sphere model has been examined for two layers (Maxwell-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 water distribution in an unsaturated soil is commonly heterogeneous. Hence, it is difficult to describe the dielectric permittivity by a single composite sphere model. Friedman (1998) has found that a dual three-layer composite sphere model is suitable for describing the dielectric permittivity of natural unsaturated soils. This model contains the composite spheres of the solid-water-air arrangement (SWA) and the air-solid-water arrangement (ASW). Note that SWA, for example, means the sphere composed of solid, water, and air in order from the center toward the outer layer. For mixing the dual SWA-ASW system, was evaluated by the symmetric effective medium approximation (EMA) (Sen et al., 1981) with arbitrary linear weight functions depending on the saturation degree, S. The weight functions, fSWAw and fASWw were as
 | [1] |
where n is a porosity. From the aspect of the real configurations of three phases in soils, ASW and SWA should be more heavily weighted in high and low water content ranges, respectively. Applying the EMA equation to the dual SWA-ASW system results in the following solution for (Friedman, 1998).
 | [2] |
where
 | [3] |
ASW and SWA are dielectric permittivites using the three-phase composite sphere model with ASW and SWA arrangements, respectively.
Composite Sphere Model for Describing Aggregated Structure
Dielectric permittivity of an Andisol is affected by its aggregate structure (Miyamoto et al., 2003). In the – curve, there exists a critical water content at which d/d sharply changes. Moreover, the relationship between and d/d for aggregated soil was partitioned into two moisture ranges; that is, the d/d value was high and low in higher and lower moisture ranges, respectively. In particular, the d/d value was almost constant for the moisture range lower than the critical water content. This characteristic cannot be seen in the case of nonaggregated soils. Therefore, when applying a composite sphere model to aggregated soils, we should consider the water distribution in aggregates, the processes of water filling in intra- or interaggregate pores, and the dielectric permittivity of water held in intraaggregate pores. To describe these features of aggregated soils, we must adjust the composite sphere model by increasing the number of layers.
The pore system in aggregated soil is frequently classified into intra- and interaggregate pores. Because of this structure, the water retention curve of aggregate is often bimodal. First, we designed a five-layer composite model in which the third layer from the outside of the composite sphere was assumed as the solid phase, and thus the outside of the third layer represents the interaggregate pore and the inside represents the intraaggregate pore. Assuming that soil water is held mainly inside aggregates in a lower water content range and in interaggregate pores in a higher water content range, we were able to reduce the number of layers to four for describing the dielectric permittivity of aggregated soils (Fig. 2)
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Fig. 2. A schematic description of the four-phase composite sphere model for describing the dielectric permittivities of aggregated soils.
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Here we will consider a four-layer composite sphere model depicted in Fig. 1, where the boundary between subregions of permittivities k and k+1 is a sphere of radius Rk+1, (k = 1, 2, 3). Following Fiedman's (1998) derivation for three phases, we can derive a four-layer composite sphere model as follows:
 | [4] |
where 
i, (i = 1, 2, 3, 4), is the volumetric fraction of the ith composition, and Ri the radii of the sphere, and thus we can obtain
 | [5] |
Extension of Friedman's Model
We described the dielectric permittivity of aggregated soils by a dual composite sphere model following Friedman (1998). Minimum modification was applied to adjust Friedman's model to aggregated soils. Usually, the water phase is in contact with the solid phase in soil. Therefore, a combination of the air–water–solid–water (AWSA) arrangement and the water–solid–water–air (WSWA) arrangement for moisture ranges below and above the critical water content respectively must be suitable to describe the aggregate structural effects. Since the aggregated structure effects appear predominantly in the lower moisture range, the AWSA or WSWA arrangement was used instead of the SWA arrangement. That is, the ( < c) can be given by
 | [6] |
where
 | [7] |
and the ( > c) by
 | [8] |
where
 | [9] |
where AWSA, WSWA, ASW are the dielectric permittivities of AWSA, WSWA, and ASW arrangements, and fAWSAw, fWSWAw, and fASWw are weight functions for AWSA, WSWA, and ASW arrangements, respectively. In this model, the ASW arrangement describes the soil water adsorbed on the surface of aggregates and held around contact points between aggregates in the lower moisture range and entrapped air in the higher moisture range.
Weight Function
Calculating the dual composite sphere model, Friedman (1998) applied the arbitrary linear weight functions depending on the saturation degree. We also decided to use the linear weight functions as a first trial. However, we found that it is better to adopt more suitable weight functions reflecting water distribution in aggregates. In a moisture range lower-than-critical water content, soil water is held mainly in intraaggregate pores and around contact points between aggregates. The fraction of water content held in intraaggregate pores is much larger than that adsorbed on an aggregate surface and held around contact points between aggregates. Hence, the weight of AWSA is much larger than that of ASW at a low moisture content. On the basis of these considerations, two sigmoidal functions were adopted instead of the arbitrary linear weight functions. By assuming fAWSAw = fWSWAw, the one (SW1) was
 | [10] |
which Friedman (1998) used as a weight function for his model. The other one (SW2) was
 | [11] |
In summary, we tried to calculate for three cases of a four-layer composite model: with the arbitrary linear weight functions depending on the saturation degree (LW), and two different sigmoidal functions (SW1 and SW2).
Permittivity of Water close to Soil Surface
Bound water has been found to possess a lower dielectric permittivity than free water. The permittivity has been shown to be a function of distance from the solid surface. The reduced permittivity of bound water was estimated using the water film thickness approach taken by Friedman (1998). The dielectric permittivity of the water phase is represented using an exponential function starting from a minimum (min = 5.5) near the surfaces of solid phase and toward a maximum permittivity (max = 80.4, at 20°C) which describes the free water phase distant from the solid surfaces, resulting in
 | [12] |
where 
= 108 cm–1 and dw is the approximate thickness of the water film spread uniformly over the surface of the solid phase. The average thickness of the water film is calculated by dividing the volumetric water content by 
b and the soil surface area (SA):
 | [13] |
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MATERIALS AND METHODS
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Soil Samples
We used two Andisols (Hydric Pachic Melanudand) for experiments in this study. Soils were from experimental fields at the National Agricultural Research Center for Kyushu Okinawa Region in both Kumamoto and Miyazaki, Japan. The soils were passed through a 2-mm sieve and air dried. In addition to these soils, we prepared wet 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 was prepared into four aggregates as follows. A nest of five sieves were placed in a holder and suspended in a container of water. The mesh sizes of the sieves were 2.0, 1.0, 0.5, 0.25, and 0.1 mm. The sample clod was put on the top sieve of the nest, and then the nest was alternately moved up and down for a vertical distance of 38 mm at a rate of 30 cycles min–1 for 40 min. Thus, size fractions of the aggregates were 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 wet aggregates, of which aggregate sizes were 1.0 to 2.0 and 0.1 to 0.25 mm in diam., were used to determine the – relationship. Physical properties for soil samples used in this study are shown in Table 1.
Measurement of – Relationships
For measurement of the – relationships, we used four acrylic cylinders of 62.8 mm in diam. and 130 mm high. Each soil was packed into the cylinder until 110 mm high as uniformly as possible. When we packed the soils, we did not compress the soils directly but rapped at the wall of the cylinder to prevent breaking the aggregate structure of the samples. The measurement of was conducted using a TDR cable tester (Tektronix 1502B). For all measurements, we used the same three-rod TDR probe (3 mm in diam., 100 mm long, and 15 mm in space between center and outside rods). The TDR probe was inserted vertically into the soil columns. Waveform analysis was conducted using the WinTDR waveform analysis software (Or et al., 1997) that enabled automated TDR control, data acquisition, and waveform analysis. The measurement of was repeated three times for each sample. Water contents were measured by weighing the soil samples gravimetrically using an electronic balance. To prepare the samples with a higher water content, the soils were removed from the cylinders and received approximately 15 mL of distilled water. The soils were left undisturbed for at least 24 h to allow the moisture to become evenly redistributed. The wet soils were repacked into the same cylinders with the same bulk density and the measurements of and were conducted as mentioned above. These steps were repeated 10 to 12 times for each soil. The experiment was repeated twice following the same procedure. All these experiments were conducted in a laboratory in which room temperature was kept at 20°C.
Modeled Calculation of Four-Layer Composite Sphere Models
To evaluate their difference from the three-phase arrangement of the model, the four-layer models were calculated with the following conditions: (i) the porosity was 0.73 and the critical water content assumed to be the volumetric water content corresponding to the matric potential at 1555 kPa (c = 0.31), (ii) the dielectric permittivities for solid and air were assumed to be 5.5 and 1, respectively, and (iii) the permittivity for water was estimated using the water film thickness approach taken by Friedman (1998). In addition, a three-phase model with ASW and SWA arrangements employed by Friedman (1998) was also used for comparison.
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RESULTS AND DISCUSSION
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Calculation Results of Four-Layer Composite Sphere Model
Figure 3
shows the results calculated using the four-layer composite sphere model for two different ranges. In the moisture range from 0 to 0.31, the AWSA was estimated at a slightly higher than SWA. The calculated dielectric permittivity of the AWSA arrangement was significantly smaller than the measured one even at = 0.31 since the effect of the external air-phase layer encapsulating the solid layer was excessive on the estimation of . Contrary to these results, in the moisture range of 0.31 to saturation, the estimated dielectric permittivities greatly increased with . Moreover, although the curvature calculated by the WSWA arrangement was similar to that by the SWA arrangement, the calculated permittivities at saturation were significantly different from each other. Using WSWA and AWSA arrangements for high and low water content ranges, respectively, a moderate change of at critical water content could be described. Therefore, a combination of the four-layer composite sphere models make it possible to describe the aggregate structure effects on dielectric permittivity.
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Fig. 3. Calculated dielectric permittivities by using the two different four-layer composite sphere models (ASW, air-solid-water arrangement; AWSA, air-water-solid-water arrangement; SWA, solid-waterair arrangement; WSWA, water-solid-water-air arrangement). Dielectric permittivities calculated by SWA configuration are also shown.
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– Relationships for Aggregated Soils
Figure 4
shows the – relationships for four samples used in this study. The relationships were obtained in the water content range of 0.05 to 0.45 cm3 cm–3 for wet-aggregate soils and Andisols from Miyazaki, 0.09 to 0.54 cm3 cm–3 for Andisols from Kumamoto. Beyond these maximum water contents, uniform soil packing with a constant bulk density became difficult as the wetness increased.
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Fig. 4. Comparisons between model predictions and measured data of the relationship between volumetric water content and dielectric permittivity for four different soil samples. Three different weight functions, that is, two 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.
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There might be concern about breakage of aggregate structure during the experimental processes of sequential water additions and repacking. A small amount of aggregate breakage occurred during this experiment; however, for all soils the – relationships were on a moderately rising tendency to a critical value, but beyond that, the value steeply developed with . This characteristic was similar to the results reported by Miyamoto et al. (2003). Therefore, we judged that the effect of experimental processes on the – relationships was relatively small.
In the – relationships for all soils, there exists a critical water content at which d/d sharply changes. This characteristic was more obvious for wet-aggregate soils than that for the two Andisols. These results were explained by difference in pore size distribution. For wet-aggregate soils, the two clear peaks of pore size density made it possible to divide pore space into two distinguishable pore systems (Miyamoto et al., 2003). Hence, a clear change of water-configuration in pores, and thus , occurred in wet-aggregate soils. On the other hand, such an abrupt change of may not occur in two Andisols.
Comparison between Experimental Data and Calculations
For evaluating the proposed four-layer composite model with sigmoidal weight functions, we fitted the model to the experimental data for aggregated soils (Fig. 4). Two different types of sigmoidal function were used. In addition, the estimations based on both Friedman's model and the four-layer model with the linear weight function were obtained for comparison with the model with sigmoidal weight functions.
For wet-aggregate soils, the estimated – relationships of three models except the four-layer composite model with SW2 agreed well with the experimental data with a water content range of less-than-critical water content. However, beyond critical water content, the difference between estimation and experimental data became large. The Friedman's model tended to underestimate dielectric permittivity in a moisture range of less-than-critical water 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 with SW2 estimated the permittivity of the whole water content range better than the other three models, although this model tended to underestimate the permittivity around critical water content.
The estimated – relationships of all four models agreed well with the experimental data for Andisols from Kumamoto. In particular, the estimated – relationships of the four-layer models corresponded better with the measured data than Friedman's model in the range of lower-than-critical water content. These results were similar to those for wet-aggregate soils. For the range of higher-than-critical water content, the measured data were located between two – curves estimated by four-layer models with different sigmoidal weight functions. Compared with – relationships for wet-aggregate soils, the d/d values in the high water content range increased moderately.
For Andisols from Miyazaki, all models underestimated the – relationship. However, the curvature calculated by the four-layer model with SW1 fitted well, and the estimated values were slightly lower 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 we assumed s = 10 for the calculation of the mixing models, the estimated – relationship became closer to the measured data than that with the assumption of s = 5.5 (not shown). However, a detailed discussion of the nature of this effect is beyond the scope of this paper.
In contrast to the predicted – relationships, the calculated –d/d curves were quite different from each other among the four models (Fig. 5)
. The d/d values of the Friedman's model increased with monotonically in the moisture ranges between 0 and 0.5 m3 m–3. Contrarily, the calculated results for the four-layer models fitted satisfactorily to the experimental data in moisture ranges lower-than-critical water content. In particular, the d/d level was low in a low moisture range and increased rapidly nearing critical water content. However, the calculated dielectric permittivity for the four-layer model using linear weight function almost overlapped Friedman's model and monotonically increased with until near saturation in the higher moisture range. For the four-layer model employing SW2 sigmoidal weight functions, however, the d/d followed the observed data relatively well 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 – relationship in a moisture range of less-than-critical water content. In particular, the four-layer model could describe –d/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 between SWA and AWSA arrangements. Friedman (1998) employed the SWA arrangement as the more weighted of the two components in lower moisture content. The estimated permittivity based on this arrangement increased gradually and linearly in the low moisture range (Fig. 2 in Friedman, 1998). Contrary to this, the permittivity based on AWSA showed an increase in with in a very dry range and remained almost constant within the moisture range of less-than-critical water content (Fig. 3). An increase in the number of layers in the model becomes more suitable to describe the feature of aggregate structure effect on dielectric permittivity.
We introduced the sigmoidal weight functions for representing the situation that soil water is held mainly inside aggregate pores at a moisture range of lower-than-critical water content. Thus, the weight of AWSA must be much larger than that of ASW in the low moisture range. However, the sigmoidal weight functions improved the estimation of –d/d curves even in a moisture range higher than that of critical water content (Fig. 5). In this range, a continuous water phase could be expected in unsaturated aggregated soil. Since intraaggregate pores are almost saturated at critical water content, additional water would be easily connected with the water included in aggregate, and thus the air in the interaggregate pores would be entrapped by the water phase. Therefore, the use of the sigmoidal functions improve the prediction of –d/d curves because the weight of ASW remarkably increases near critical water content.
Applying the four-layer model with the sigmoidal function, we could well describe the – relationship for aggregated soils. In particular, the increase in the number of layers in the model resulted in a good fit of the estimated – relationship in a less-than-critical water content range. Moreover, we could estimate dielectric permittivity of the water phase by using the water film thickness approach taken by Friedman (1998). These results may suggest that the dielectric property of water held in aggregates is similar to that of water in mineral soils. That is, even though water is held inside an aggregate, the monomolecular layer of water is strongly bonded to the solid surface, and the second layer of water is held more loosely than the first layer. However, there is little experimental evidence concerning the dielectric permittivity of water inside an aggregate.
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CONCLUSION
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The four-layer composite sphere model using a sigmoidal weight functions can describe aggregate structure effects on dielectric permittivity. In modifying Friedman's model (Friedman, 1998), we considered the low value of bound water adsorbed on a soil surface, the water distribution in aggregated soils, and the processes of water filling in intra- and interaggregate pores. The addition of a layer in the composite sphere model improved the predictability of the model for the – relationship in the moisture range of less-than-critical water content. The –d/d curves estimated by the model were in better agreement with the experimental data than those of Friedman's model. In particular, applying the sigmoidal weight functions improved the estimation of –d/d curves in a moisture range higher-than-critical water content. Our improved model provides more insight into the relationship between the physical properties of aggregated soils and their dielectric permittivity.
Received for publication November 25, 2003.
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TOP
ABSTRACT
INTRODUCTION
