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DIVISION S-9—SOIL MINERALOGY

Calculation of the Dielectric Properties of Temperate and Tropical Soil Minerals from Ion Polarizabilities using the Clausius–Mosotti Equation

Dep. of Plants, Soils and Biometerology, Utah State Univ., Logan, UT 84322

^* Corresponding author (darearthscience{at}yahoo.com)

	ABSTRACT

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Soil dielectric properties are commonly used to estimate soil water content, however, the relative permittivity of common soil minerals is often unknown or estimated to be 5, similar to quartz (4.6). In this study, the Clausius–Mosotti model is used to estimate mineral relative permittivity from atomic polarizability for common soil minerals. Atomic polarizabilities are used to calculate the average mineral polarizability using the oxide additivity rule. Results of these calculations are compared with measurements found in the literature for rock and mineral samples. Predictions correspond well with measurements on single crystals or obtained using the immersion method to determine relative permittivity (better than 10%). The application of this simple calculation technique allows estimates of relative permittivity to be obtained for minerals for which it is difficult to measure relative permittivity directly, such as soil clay minerals.

	INTRODUCTION

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ELECTROMAGNETIC measurement techniques have become popular for obtaining estimates of soil water content. Time domain reflectometry is commonly used for sample scale measurements in the laboratory and field (Topp et al., 2002; Robinson et al., 2003) and techniques such as ground penetrating radar are being used to obtain two-dimensional estimates of profile soil water content in field applications (Binley et al., 2001; Huisman et al., 2001). Such techniques measure the propagation velocity of electromagnetic waves, which are closely related to the relative permittivity, referred to here after simply as the permittivity, of the medium through which they travel:

[1]

where c is the velocity of light (3 x 10⁸ ms^–1), _o is the permittivity of vacuum (8.854 pF m^–1), _r is the relative permittivity, µ_o is the magnetic permeability of vacuum (1.257 x 10^–6 H), and µ_r is the relative magnetic permeability.

A strong relationship between the permittivity measured by these methods and water content of soils arises due to the difference between the permittivity of the soil constituents and water. Air has a permittivity of 1, water is 78.5 at atmospheric pressure and 25°C, and soil minerals commonly fall in the category of 4.5 to 10. To obtain the best determination of water content, grain scale models are being developed and tested that predict the permittivity of porous media and soils based on the soil constituents and their geometrical arrangement (Sen et al., 1981; Friedman, 1998; Jones and Friedman, 2000; Robinson and Friedman, 2002a). An important input for these models is the value of the permittivity of the mineral phase. An average value is difficult to measure directly and thus the mineral permittivity is often assigned a value of 5 based on reported values of rock permittivity (Friedman, 1998). Hence the mineral permittivity essentially remains a fitting parameter, which is undesirable for testing models.

A recent application of electromagnetic sensors has been to search for and locate buried objects such as unexploded ordnance (UXO). In this application, techniques such as ground penetrating radar exploit the contrast between the dielectric properties of a target and the dielectric properties of the background soil, usually under wet conditions (Das et al., 2001). In temperate soils dominated by quartz, standard calibrations between permittivity and water content may suffice to indicate dielectric contrast. However, in tropical soils where metal oxides may be dominant these relationships may need adjustment. To place bounds on the possible adjustment that might be required an initial step is to estimate permittivity values of oxide minerals often found in tropical soils; in this paper we provide such estimates.

	MATERIALS AND METHODS

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Methods of Measuring Mineral Dielectric Permittivity
A variety of methods exist for measuring the permittivity of minerals. These are in increasing order of accuracy, repacked granular sample (Nelson, 1992), dielectric immersion (Schmidt, 1902; Andeen et al., 1970; Robinson and Friedman, 2003), the three-terminal guarded electrode method (Lowndess and Martin 1969), and the two-terminal method (Subramanian et al., 1989). The repacked granular sample method is fast. However, it relies on the use of a dielectric mixing model to estimate the permittivity of a granular sample packed in air, by extrapolating the permittivity values back to zero porosity. This method is undesirable as the accuracy can vary depending on the choice of model (Robinson and Friedman, 2003). The immersion technique has been found to have accuracies of about 10% and better for coarse-grained materials according to Robinson and Friedman (2003). Measurement becomes difficult in clays due to adsorption of the immersion dielectric on the clay surfaces and it's reduced permittivity, air entrapment in samples can also create difficulties (Robinson, 2004). Smectites present a particular problem as the immersion dielectric fluid can be adsorbed onto the internal surface of the mineral and so have its permittivity reduced. The assumption of a bulk permittivity for the liquid then becomes invalid. Accuracies of the three terminal and two terminal methods have been found to be 0.5 and 0.01%, respectively (Bussey et al., 1964; Andeen et al., 1970). These techniques however, are only suited to single crystal samples that are cut and prepared for direct measurement. This technique cannot be used to measure the permittivity of clay minerals for example.

Minerals in Soils
Soils tend to pose two distinct problems for the determination of mineral permittivity. The mineralogy is often fine grained and is usually a mixture of different minerals. As previously discussed clay minerals can present particular problems for dielectric measurement methods. In the case of the second problem, results indicate (Robinson and Friedman, 2002b) that the effective permittivity of the mineral phase of a soil can be predicted using the arithmetic average of the mineral components, and their permittivities, so long as the mineral permittivity contrast is less than about 4.

Tectosilicates
The silica group exists as seven distinct polymorphs in nature (Drees et al., 1989). Quartz is the most dominant form in soils, especially in temperate zone soils. Disordered cristobalite is also found in some soils. Measurements of the permittivity of quartz appear to range in the geology literature between 4.19 and 5.0 (Keller, 1989). Due to its importance in soils a number of calculations are performed from data for silica minerals. Calculations are also performed for orthoclase feldspar. Feldspar is a primary mineral but often found in many soils inherited from parent materials (Huang, 1989).

Phyllosilicates
Phyllosilicates found in soils are in the mineral groups of kaolin (kaolinite, dickite, and nacrite), mica (muscovite, biotite, phlogopite, and illite), and the smectites (montmorillonite, beidellite, and nontronite) (Allen and Hajek, 1989). Other phyllosilicates found occasionally in soils include, talc and pyrophyllite. Literature permittivity values for the clay minerals vary greatly with reported values between 5 and 10 (Olhoeft, 1981). The variation in these values is most likely due to poor measurement method and the effects of hygroscopic water contained in samples; this raises observed values of permittivity. Calculations are performed on a range of phyllosilicates to give a feel for expected permittivity values.

Oxides and Hydroxides
Oxides and hydroxide minerals are an important component of many tropical soils (Allen and Hajek, 1989). The aluminum oxide gibbsite and the iron oxide goethite are considered the most common oxide minerals found in soils. Lepidocrocite tends to be associated with gley soils where as hematite tends to be found in well-drained soils in warmer climates. Other oxides including, magnetite, ilmenite, and anatase can also be found in soils but tend to result from weathering of parent material that contains those minerals.

	THEORETICAL CONSIDERATIONS

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Polarization
Polarization mechanisms in solids can be divided into three major categories: electronic, ionic, and orientational; the frequency range in which these mechanisms primarily operate is presented in Fig 1. In the radio and microwave frequency bandwidths all these mechanisms can operate. Orientational polarization is assumed to be negligible in dehydrated soil minerals as it arises due to the presence of permanent dipoles. It is a mechanism associated with water and other polar liquids. It is thus the combination of electronic and ionic polarizabilities that is of most interest to determine the mineral polarizability. The link between the mineral dielectric polarizability, _D, and the measured real part of the relative permittivity, ^'_r, is given by the Clausius–Mosotti equation:

[2]

View larger version (20K): [in this window] [in a new window]   Fig. 1. A schematic diagram of the major polarization mechanisms in the frequency spectrum. The contribution to mineral permittivity comes mostly from electronic and ionic polarizability. Relaxation of orientational polarization usually occurs in the microwave frequency region or lower (1010 Hz) and ionic relaxation usually takes place in the infrared (1013 Hz).

The Additivity Rule
The additivity rule allows the summation of molecular (Heydweiller, 1920) or ion (Shannon, 1993) polarizabilities to give the substance dielectric polarizability. Conversely, complex solids can be broken down to reveal molecular polarizabilities:

[3]

where M and M' represent cations and X represents the complimentary anion in the substance, for instance _D(Mg₂SiO₄) = 2_D(MgO) + _D(SiO₂). Chemical formulae can be further subdivided into individual ion polarizabilities, :

[4]

So that _D(Mg₂SiO₄) = 2(Mg²⁺) + (Si⁴⁺) + 4(O^2–). Most recently Shannon (1993) used measurements of the permittivity of high quality crystals of known orientation to develop a data set of ion polarizabilities. This data set was then used to predict the permittivity of minerals of known composition and structure. The approach proved accurate for simple oxides allowing permittivity calculation to within 2 to 3% of measured values. This level of accuracy is better than can be measured using techniques such as immersion or repacking of granular samples. Shannon (1993) used deviation between measured and modeled results to gain insight into polarization mechanisms in minerals that weren't described by the Clausius–Mosotti model.

	RESULTS AND DISCUSSION

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The values of ion polarizabilities presented in Table 1 were used in the Clausius–Mosotti equation to predict mineral permittivity of a range of soil minerals according to:

[5]

View this table: [in this window] [in a new window]   Table 2. Tectosilicates, values used for the molar volume, Vm, and mineral polarizability, D.

View this table: [in this window] [in a new window]   Table 3. Average permittivity values of minerals found in the literature.

The value calculated for kaolinite of 5.0 is in close agreement with the immersion measurement value of 5.1. The predicted value for Talc of 5.4 is also very close to the immersion-measured value of 5.3. The modeled results are exceptionally encouraging for the phyllosilicates. The general trend would indicate that the modeled permittivity values of these minerals fall into a permittivity range between approximately 3 and 6, with the more common soil minerals falling into a range between 5 and 6. This corresponds well with the measurements of Robinson (2004) who also found that values tended to fall in the 5 to 6 range. Clearly, the mineral composition is important and more refined tests might be conducted based on known mineral composition.

Oxides and Hydroxides
Metal oxides and hydroxides of Fe and Al are found predominantly in highly weathered tropical soils. As a result there has been much less focus on the dielectric properties of these soils by scientists working predominantly in the more temperate zones. However, the results (Table 5) would indicate that these minerals are likely to have permittivity values two to five times higher than quartz. Literature values for these minerals are again scarce, however some can be compared. In the case of the Fe minerals permittivity estimates between 18.9, (Nelson et al., 1989) and 25 (Olhoeft, 1981) have been presented for Hematite, in close agreement with our value of 23.9. Nelson et al. (1989) also presented a value of 13.6 for a goethite and Olhoeft (1981) presented a value of 11.7, both of which are close to our calculated value of 12.7. A value of 20 for magnetite has been presented by Jantzen (1963, as cited by Young and Frederikse, 1973). The measured value for magnetite is higher but this may have something to do with the magnetic properties of the magnetite affecting the measurement.

Despite the assumptions about chemical composition and mineral density the predicted values tended to be within 20% of the best available measured values for these minerals in the literature. In general it appeared that the calculated values formed a lower bound for the expected mineral permittivity. The method of calculating mineral permittivity values from ion polarizabilities appears to work well for the soil minerals presented. This modeling approach is a useful tool for estimating the permittivity of minerals that can be particularly difficult to measure such as the Smectites. Improved testing of the reliability of this method could be achieved using better-defined mineral samples for measurements.

	CONCLUSIONS

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Permittivity values are calculated from ion polarizabilities for a selection of minerals commonly found in soils. The calculated values are in excellent agreement with values measured for single crystals or using the immersion method. The calculated values tend to be within 10% of measured values. This is good considering the exact mineral compositions for many of the measured minerals are not known exactly. The agreement between permittivity values estimated using repacked samples and those calculated can differ by as much as 100%. Poorer agreement is ascribed to the poor accuracy of the permittivity estimates, obtained using this repacking method, the calculations are considered to be more accurate. This method of calculating permittivity provides a useful tool for estimating the permittivity of minerals that due to size, shape, and layer charge can me hard to measure.

	ACKNOWLEDGMENTS

 
The author acknowledges funding provided in part by a USDA NRI grant 2002-35107-12507.

Received for publication October 20, 2003.

	REFERENCES

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