Soil Science Society of America Journal 64:1581-1587 (2000)
© 2000 Soil Science Society of America
DIVISION S-1-SOIL PHYSICS
New Dielectric Mixture Equation for Porous Materials Based on Depolarization Factors
M.A. Hilhorsta,
C. Dirksenb,
F.W.H. Kampersc and
R.A. Feddesb
a IMAG-DLO, P.O. Box 43, NL-6700 AA, Wageningen, The Netherlands
b Dep. of Water Resources, Wageningen Agricultural Univ., Wageningen, The Netherlands
c DLO, P.O. Box 59, NL-6700 AB, Wageningen, The Netherlands
m.a.hilhorst{at}imag.wag-ur.nl
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ABSTRACT
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A change in the relative proportions of the constituents of a porous material like soil will cause a change in its electrical permittivity. The measured permittivity reflects the impact of the permittivities of the individual material constituents. Numerous dielectric mixture equations are published, but none of these equations are generally applicable. A new theoretical mixture equation is derived, using the principle of superposition of electric (E) fields. This mixture equation relates the measured permittivity to a weighted sum of the permittivities of the individual material constituents and includes depolarization factors to account for electric field refractions at the interfaces of the constituents. The depolarization factors are related to physical properties of the material. Most other mixture equations contain one or more empirical factors. The concept of the depolarization factor is comparable with that of the "shape factor" of particles as described by other authors. A special case of the new mixture equation, for which the depolarization factors equals one (no depolarizations), appeared equal to a mixture equation for fluids derived from using thermodynamics. The new mixture equation is compared with other mixture equations. Comparison of the new mixture equation with measured data for glass beads and fine sand was promising. Concluding, depolarization factors in the new mixture equation relate the microstructural and compositional material properties to the measured bulk permittivity of a material. Although not shown, depolarization factors can be calculated from physical material properties.
Abbreviations: E, electric [field] • FD, frequency domain • TDR, time domain reflectometry
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INTRODUCTION
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ONE PROMISING TECHNIQUE to characterize a porous material like soil is based on the interaction between its dielectric properties and its textural and compositional properties. Probably the most important drawback to using the dielectric properties of soil is the complexity of the dielectric theory, which involves a number of physical processes that are not well understood. Until now, no complete model is available that can describe the dielectric properties of soil.
Soil is a mixture of differently shaped granules, water films, air, and organic matter. For dielectric measurements a thoroughly mixed soil, with a volume large in comparison with the dimensions of its constituents, can be treated as a homogeneous body. The dielectric properties measured on a macroscopic scale (the bulk material) depend on the dielectric properties of the individual constituents at a microscopic scale (at the scale of a grain) and on different polarization phenomena. The relationship between macroscopic and microscopic soil properties can be described by a dielectric mixture equation. Many mixture equations have been published, but none are universally applicable. Mixture equations are often referred to as mixing formulas or mixing rules. The impact of microstructural and compositional soil properties on the distribution of the E field at a microscopic scale is complicated and not well understood. Some mixture equations are empirical and do not include the effect of E field refractions, which is essential for a good understanding of the relationship between the measured dielectric properties and those of the individual soil constituents. Most empirical equations are intended to find calibration curves, and thus, to fit measured water content data with measured dielectric data using an arbitrary function. A good example is the calibration function found by Topp et al. (1980), which was shown to be very useful. These equations are valuable for water content measurements but do not give insight into the relationship between macroscopic and microscopic soil properties. Other equations have a more theoretical basis but are only valid for static E fields or for frequencies >1 GHz. For an in-depth discussion of mixture equations the reader is referred to Cole and Cole (1941), Tinga et al. (1973), Wang and Schmugge (1980), Dobson et al. (1985), Priou (1992), and KobaConsider a dielectric between two metal plates. If an electric potential is applied to the plates, a static E field will result between the plates. A static E field is an E field that is constant as a function of time. As a result, the dielectric becomes polarized. The E field is represented by a field vector, E. For linear, isotropic, and homogeneous dielectrics, the polarization vector, P, is proportional to E and has the same direction. After a static E field is applied the material will find a new equilibrium. This causes a motion of bound charges to and from the plates. The material is polarized and energy is stored. The resulting current can be measured externally and is a measure of the ability to polarize the material. If an alternating potential, for example, a sine wave voltage with frequency f, is applied to the plates, the vectors E and P and the current are alternating as well. For a comprehensive treatment of electromagnetic wave theory and refraction of E fields, see Lorrain et al. (1988).
Polar molecules, like water, are fluctuating continuously due to thermal energy. This random process tends to neutralize the external E field. After removing a static E field, the stored energy dissipates within a certain time. This process is called dielectric relaxation. If an alternating E field is applied there will be a continuous competition between polarization and neutralization. A measure of this competition process is the dimensionless relative permittivity of a material, which is defined as the permittivity relative to that of free space
. This polarization process is frequency dependent and can be described by a complex representation of the relative permittivity. For short, in this paper we will refer to the complex relative permittivity as permittivity, denoted with 
, only. The permittivity is defined as
where
. The real part of the permittivity, ', is a measure of the polarizability of the material for a static E field. ' is usually referred to as dielectric constant. The imaginary part of the permittivity, '', represents the total energy absorption or dielectric loss.
Consider a soil containing only liquid water and solid grains. At the microscopic scale, the E field passes the interface between the liquid and the solid. The direction of Ewater and Esolid will change according to the refraction of the electric field lines crossing the liquid–solid interface. This is illustrated in Fig. 1
for water > solid where the field lines, going from water into the solid, bend to the normal. The resulting amplitude of Ewater will be smaller than that of Esolid, depending on the difference between water and solid, and on the shape of the grain. Surface roughness of the grains will result in scattered refractions leading to smaller amplitude of Ewater as well. No refraction takes place inside a homogeneous dielectric. Each point in a single homogeneous body in the soil, that is, each point in a soil grain or in a water filled pore, experiences the same field vector (i.e., Esolid in a grain and Ewater in the water). However, due to the refraction of E fields at the interface between two homogeneous bodies in the soil, each grain or water-filled pore will experience its own local field vector and hence its own polarization. At the macroscopic scale, only the mean field strength of the bulk soil, 
, and the resulting mean polarization, 
, can be observed, which are the results of all local field and polarization vectors.
In this paper, and denote the mean vectors measured with the soil between the two electrodes. They result from all local Es and Ps in the microscopic bodies. E and P denote the variables that would be measured if the soil, or a soil constituent, is replaced by a homogeneous dielectric of the same 1 that occupies the same volume and for which
and
. The variables E, , P, , and 1 will be used only at macroscopic scale. If E is not bold, we refer to an electric field that must be treated as a scalar.
The relationship between and all local Es in the mixture can be determined from the theories given by Böttcher (1952), De Loor (1956, 1990), Bordewijk (1973), Böttcher and Bordewijk (1978), Sihvola and Lindell (1988), and Sihvola (1996). From this, can be calculated. However, it is a rather difficult exercise to determine mathematically from the contributions of the local Es of soil constituents of arbitrary shapes.
A new dielectric mixture equation is derived below using the principle of superposition of E fields. This equation has a theoretical basis and contains parameters, which can, though not simply, be derived mathematically from the shapes and permittivities of the soil constituents. Although this study is focused on soil, the theory applies to porous materials in general.
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Theory
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Derivation of a New Mixture Equation
The relationship between P, E, and for a homogeneous material on a macroscopic scale is given by
 | (1) |
where is the permittivity of the material relative to the absolute permittivity of free space, 0. The proportionality factor between P and E is the electrical susceptibility ( - 1) of the dielectric.
As pointed out above, the macroscopic mean field strength of soil, , is the composite of the microscopic local field vectors of all soil constituents. Similarly for soil at macroscopic scale, the relationship between , , and is given by
 | (2) |
A soil consists of n constituents each denoted by the subscript i. Both n and i refer to the macroscopic scale. Note that we use the term constituents to refer to water, solids, and air, denoted by the subscripts "w", "s", and "a" respectively. The word particle refers to a single local part of a constituent that still can be treated as a homogeneous mass, for example, a soil particle or the water in a soil pore. Each dipole or molecule of a particle of the ith constituent with permittivity i experiences its own local Ei and Pi. On average, on a macroscopic scale, these particles experience the mean field strength i and the mean polarization i. Similar to Eq. [2], we may write for each constituent
 | (3) |
Using the principle of superposition of E fields and taking into account the assumptions and comments discussed by Böttcher (1952), Reynolds (1955), and Reynolds and Hough (1957), the total of all constituents appears to have the same direction and amplitude as the applied E field. These assumptions concern, for example, that boundary effects are negligible and the soil is isotropic. Most soils are isotropic, with the possible exception of plate-condensated clays. It is also assumed that for bulk soil at macroscopic scale, the number of particles of the ith constituent are large in number and randomly as well as uniformly distributed and have dimensions much smaller than at macroscopic scale. The latter assumption allows for a statistical treatment of the E fields. The voltage source for the E field between the plates and the meter to measure the resulting current through the soil cannot distinguish between the different soil constituents; hence, the calculated from this current is the result from the contribution of all the individual constituents. Thus, for the meter readings the soil may be replaced by a homogeneous dielectric of the same . Hence,
and Eq. [1] is equal to Eq. [2]. Since is not a vector, it can be seen from Eq. [1] and [2] that
as well. The relationship between macroscopic and microscopic polarization is illustrated in Fig. 2
.
The electric polarization is the dipole moment per unit volume at a given point. Therefore, i can be thought of as a polarization vector for the volume occupied by the ith constituent, situated at a point in the center of the soil under consideration. The ith constituent occupies a volume fraction vi of the soil with volume V, where vi is a dimensionless quantity. The soil contains n constituents, each with volume fraction vi. The summation of all volume fractions
. The contribution of this ith constituent to the macroscopic mean polarization of the soil is ivi. According to the principle of superposition, the macroscopic mean polarization of a soil containing n constituents can be written as
 | (4) |
With Eq. [3] substituted in Eq. [4] can be written as
 | (5) |
Replace the soil by a homogeneous dielectric of the same and apply the same E field. Hence
, and therefore
. With this, Eq. [1] may be equated with Eq. [5] resulting in,
 | (6) |
The coordinate system for the space between the plates can be chosen such that the y direction is perpendicular to the plates. Then, the x and z directions are parallel to the plates. As explained before, E is assumed to point in the same direction as i for each constituent, that is, in the y direction. The x and z components are zero. Now the field vectors may be treated as scalars (i.e., E and 
i, respectively). This allows the introduction of the depolarizion factor S. S of the ith constituent of the soil, Si, is defined by
 | (7) |
The effect of S is analogous to depolarization; therefore, S will be referred to as the dielectric depolarization factor. Si substituted in Eq. [6] will finally yield the new mixture equation
 | (8) |
for which Si remains to be determined either theoretically or experimentally. So far we developed the basic new mixture equation. Next we will illustrate the function of S and apply the new equation to soil.
Dielectric Depolarization Factors
The dielectric depolarization factor, Si, is a function of the difference between i of individual particles and of their surroundings. Furthermore, Si is a function of the shape and surface roughness of the particle. The relationship between Si, i, , and the shape of the particles can be illustrated by the following example (see also Fig. 3)
. Consider a homogeneous dielectric. Place a homogeneous sphere in the previously uniform electric field of the dielectric. Assume the permittivity of the dielectric is = 1 (e.g., air) and that of the sphere sphere = 5 (e.g., a glass bead or a soil grain). In this case the field lines crowd into the sphere and the field will be smaller inside than outside. If the radius of the sphere is much smaller than the surrounding medium the field vector inside the sphere, Esphere, is uniform and points in the same direction as the field vector of the surrounding medium E (Lorrain et al., 1988). Therefore, these field vectors may be treated as scalars (i.e., Esphere and E, respectively). In this example with only one sphere, the mean sphere is equal to Esphere. The function of the depolarization factor, Ssphere, is to relate Esphere to E of the surrounding medium. According to De Loor (1956)(p. 19) and Stratton (1941)(Sections 3.24 and 3.25) the relationship between Esphere and E is given by
 | (9) |
With Eq. [7], Eq. [9] can be rewritten as
 | (10) |
where A refers to the shape factor of the particle. According to De Loor (1956), for a sphere A = 1/3, for needles A < 1/3, and for discs A > 1/3. Thus, for this example with a sphere the depolarization factor would be Ssphere 
0.5. If the surrounding medium was not air but water, with = 80, then Ssphere 1.5. For = sphere, Ssphere = 1 as expected. Note that Eq. [9] can also be found using Laplace's equation as shown by Lorrain et al. (1988)(Chapter 12). So S inside the sphere is a function of the geometry and the differences between the permittivities in and outside the sphere.
The above example is an idealized special case and only meant to illustrate the function of Si and how it is related to the shape and the permittivities of the soil constituents. Soil is a complicated material and not homogenous. It is difficult to describe mathematically the shape of the water film in the pores, which cannot be approximated by spheroids. In addition, the shape of the water film will change if we add water to the soil matrix. It will not be easy to calculate the depolarization factor of the water component in the soil. In the example it was required that the radius of the sphere was much smaller than the surrounding medium, which will never be the case for soil. However, the importance of the concept of depolarization factors is that they mathematically can be related to physical material parameters.
New Mixture Equation Applied to Soil
This section is to demonstrate the application of the new mixture equation to soil. First we will make a simplification to Eq. [8]. Of course if the error due to this simplification becomes too large for a particular application, one is free to use Eq. [8] as it is.
In the above section we found for Si values between 1.5 and 0. There is no data available for S of bodies with irregular surfaces or for complex assemblages of bodies. Therefore, we assume for the following that Si has values between 1.5 and 0. The value of vi is most probably between 0.5 and 0. As a result the error on by neglecting
in Eq. [8] will be <1, which is small compared with > 25, a value for soils close to water saturation. The depolarization factors for dry soil will be close to one due to the small differences in permittivities of the constituents. The error made for dry soil with the latter approximations will be <<1. With this approximation Eq. [8] becomes
 | (11) |
As pointed out above, the depolarization factor, Si, depends on the changing shape of the water film as a function of water content, 
; that is, Si(). There is a depolarization factor for each ith soil constituent (for air, for solids, and water). Si is the average value of all the particles of the ith constituent. Note that Si will be different for each successive layer of water at the surface of the soil particles because of the binding of water in the electrical double layers. For simplicity of this treatment the amount of bound water is assumed negligible. Hence Eq. [11] can be written as
 | (12) |
With Eq. [12], of soil can be written as the sum of the permittivities of the fractions of its constituents by
 | (13) |
where 
is the porosity of the soil, w is the permittivity of water, s that of solids, and a that of air. Sw(), Ss(), and Sa() are the depolarization factors for water, solids, and air, respectively.
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Materials and methods
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Evaluation
The new mixture equation, Eq. [13], will be evaluated for glass beads and fine sand. The depolarization factors were estimated from the measured data for glass beads and applied to fine sand. We compared our own measurements and those found in the literature with the () relationships predicted according to Eq. [13]. These () relationships are often called calibration curves.
To obtain the () relationships, the procedure as described by Dirksen and Dasberg (1993) and Dirksen and Hilhorst (1994) was followed. The measured data were obtained with a frequency domain (FD) sensor and a time domain reflectometry (TDR) sensor. With the FD sensor the electrical capacitance of the soil between two electrodes was measured, and with the TDR sensor the transmission time of an electrical signal through the soil was measured. Both the capacitance and the transmission time are a function of . The FD sensor, described by Hilhorst and Dirksen (1994) and Hilhorst (1998), measured at a frequency of 20 MHz. The TDR measurements were carried out with a TDR cable tester (Model 1502B, Tektronix, Beaverton, OR) at an equivalent frequency of 150 MHz. The electrode configuration was the same for both sensors: three rods, 9.6 cm long, 0.2-cm diameter, and 1.0 cm spaced. This allowed as closely as possible a comparison of results obtained with the FD sensor and the TDR sensor. All measurements were made in the laboratory at room temperature. The FD sensor was used for both fine sand and glass beads, the TDR sensor only for fine sand. As shown by Dirksen and Hilhorst (1994), there is no difference in the () relationships for fine sand measured with the FD probe or with TDR. It was concluded that for fine sand is about constant between 20 and 150 MHz.
The dielectric properties at 20°C for glass beads, air, and water are s 5, a = 1, w = 80.3 (Kaatze, 1996), respectively. The average diameter of the glass beads is 0.2 mm and = 0.33. For the fine sand, = 0.438 and s = 3.5. According to Eq. [13], the value s = 3.5 for fine sand is reasonable, since both the TDR sensor and the FD sensor measured = 2.4 for = 0.002. The porosity was calculated from the sample volume and for saturation. All measured data are shown in Fig. 4
(denoted by f, g, and h). Curve f represents the FD data found for glass beads and curve g that for fine sand. Curve h is the TDR calibration curve for fine sand.
To calculate () according to Eq. [13], the depolarization factors Si() need to be known and will be estimated as follows.
Sa(). For dry glass beads, Sa is determined only by the relative small difference between a = 1 and s = 5, and thus Sa will be close to one. For wet glass beads, Sa is determined by the relative large difference between a = 1 and w = 80 of the air–water interface and, therefore, Sa will be <1. In addition, for 
the air in the pores disappears. The contribution of a to the actual will be <1 for dry glass beads and <<1 for water-saturated glass beads. Therefore, the error made by assuming Sa() 1 for an arbitrary water content can be regarded as acceptably small.
Ss(). For dry glass beads, = 0 and Sa = 1, and thus Eq. [13] reduces to
 | (14) |
For glass beads dried at ambient conditions, = 3.7 was measured. For the glass s 5. Hence Ss(0) 1. For water-saturated glass beads, there is no solid–air interface. And thus Ss() = 1.
Sw()
Sw applies to the molecules in the water film covering the particles. The shape and thickness of this water film changes with the water content, resulting in a varying depolarization factor, that is, Sw(). From measurements on water-saturated glass beads it was found that = 0.33 and = 28.5, from which Sw() 1 can be calculated by means of Eq. [13]. Sw() for < is difficult to determine due to the varying shape of the water films around the glass beads. We employ the form
 | (15) |
which gives a good fit with the () data measured for glass beads and fine sand (see Fig. 4). This empirical model includes the effect of . A thorough mathematical analysis is required to give Sw() a more solid basis.
With these estimates for Si substituted in Eq. [13], this mixture equation becomes
 | (16) |
Equation [16] is a special case of Eq. [13] for three-phase mixtures like soil and glass beads.
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Results and discussion
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Comparison with Topp's Calibration Curves
In the following we will compare Eq. [16] with measured data. It appears useful to compare calibration curves published in the literature with Topp's calibration curves (Topp et al., 1980) as a reference. In Topp's study two curves are given for glass beads:
ibration curve is a generally accepted calibration equation for TDR water content measurements which is accurate for sand in general, but represents an average for a number of sandy loam and clay loam soils:
Topp used w = 81.5 at 20°C as a reference instead of w = 80.3 (Kaatze, 1996) as used here. The calibration curves according to Eq. [16], the Topp curves, and our measurements are shown in Fig. 4.
Curve d for glass beads calculated from Eq. [16] agrees well with our FD measurements and with Topp's Curve a. Topp's Curve b is not in accordance with our measurements or calculation. The reason for this is not clear. Curve e for fine sand agrees well with the FD and the TDR measurements, while Topp's Curve c was slightly above the measured and predicted data.
Comparison with Other Mixture Equations
The new mixture Eq. [8] will now be compared with equivalent mixture equations found by other authors. Differences and similarities will be discussed.
For Si = 1, the new mixture equation (Eq. [8]) is identical to the one derived by Silberstein (1895) using thermodynamics for a mixture of fluids. He considered the mixture on one side of a semipermeable membrane stretched across the field in a parallel plate capacitor. On the other side of the membrane was a pure sample of one of the components in the mixture. He assumed the thermodynamic and electric energy to be independent. By considering the thermodynamic equilibrium of the system, for an infinitely small volume of pure substance passing through the membrane to dilute the mixture, he found the same equation as Eq. [8] if Si is set to one. This value is valid if we assume that each molecule of the different constituents in the fluid experiences the same electric field applied. This is the case if the fluid contains no microscopic bodies with surfaces at which E field refractions can take place. In Silberstein's model the polarization of each molecule is the direct result of the applied electric field. However, a mixture of fluids often contains clusters, permanent or temporally, of one of the constituents, which are equivalent to microscopic bodies. Each single molecule within a group of the same molecules experiences the same E field, which is refracted at the boundaries of this group. Thus Si = 1 is not generally applicable. This applies to fluids as well to other mixtures, like soil.
An important group of published mixture equations is based on Birchak's (Birchak et al., 1974) "exponential model":
 | (17) |
where the exponent 
is an empirical constant. This mixture equation is often used in soil science. For Si = 1 the new mixture equation (Eq. [8]) is equal to Eq. [17] for = 1, but Eq. [17] does not account for local refraction at microscopic scale like Eq. [8]. For = 0.5, Eq. [17] suggests similarity with the "refractive index" mixture model. This value of was proposed by Heimovaara (1993), White et al. (1994), and others. Whalley (1993) showed a possible physical basis for the refractive index model.
The first who derived the refractive index model for fluids was Philip (1897). He assumed the material nonmagnetic and found for a mixture of two fluids, denoted with the subscripts 1 and 2, the mixture equation
 | (18) |
Philip used Maxwell's (1873) relationship n2 = , where n is the refractive index of light. With v1 + v2 = 1, Eq. [18] can be written in terms of the refractive index
 | (19) |
which is equal to Eq. [17] for = 0.5. However, Philip (1897) showed that for fluids the use of the refractive index model can lead to erroneous results when calculating the permittivity of the constituents from the measured permittivity. The error increased with increasing . Compared with fluids, soil is a complicated material and not homogenous. If Philip was right, one may expect worse for soil. That his conclusions are right can be illustrated with measurements. For glass beads saturated with water, we measured = 28.5 (see also Fig. 4), while 20 can be calculated from the refractive index model with 0.33, w = 80, s 5, and = 0.5. Furthermore, the physical basis for = 0.5 is in contradiction with the reasonable fit with experimental data on various porous materials as reported in literature, for ranging from 0.8 to 0.33 (Landau and Lifshitz, 1960; Looyenga, 1965; Dobson et al., 1985; Jacobsen and Schjonning, 1995). It seems that adapts Birchak's model to the soil under consideration. Birchak's model lacks a parameter describing the effect of refractions of the E fields at microscopic scale and seems to correct for this.
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Conclusions
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The introduction of the depolarization factor for each constituent of a porous material in a new mixture equation for dielectrics is in our view the most accurate and effective way to relate the dielectric properties of a porous material measured on a macroscopic scale to those on a microscopic scale. In contrast to the empirical factor in Birchak's model (Birchak et al., 1974), depolarization factors are directly related to the E field refractions in the material, which in turn depend on its microstructural and compositional properties. It will not be easy to calculate the depolarization factor of the water component in the soil. They can be estimated empirically, though a mathematical solution is desired and possible. The importance of the concept of depolarization factors is that they are related to physical material parameters.
The new mixture equation was evaluated for fine sand and glass beads. The depolarization factors were derived from data measured for glass beads and next applied to soil. Predicted calibration curves of fine sand and glass beads were in reasonable agreement with the measured data and the data published by Topp et al. (1980). The new equation for soil allowed correction for porosity, while this is not possible with the calibration curve of Topp, Birchak's model, or the refractive index model.
The theory was applied to sand and glass beads, for which the amount of bound water is negligible. The layer of water molecules bound to the surface of a soil particle, in the electric double layer, is generally considered to be ice-like (Iwata et al., 1995). The permittivity will be lower than that of free water. This effect was not considered here; however, the new mixture equation can be expanded to include the effect of bound water.
The refractive index model does not account for local microscopic E field refractions. There is no solid theoretical foundation for = 0.5 in Birchak's model. The use of the refractive index mixture model can lead to erroneous results. Substitution of n2 for in analyzing dielectric mixtures is confusing since n and describe different physical phenomena. n describes the refraction and propagation of electromagnetic waves, while is the ratio of electric field strength to the electric displacement (Lorrain et al., 1988). Equating n2 with is only allowed under restricted conditions. Michels et al. (1961, p. 731) made a note that n2 = only applies to the case of nonconducting materials in the absence of permanent dipoles. In addition, for conducting materials n is a complex quantity that includes the absorption index. Therefore, one should avoid the usage of n for 
.
As discussed by many authors, plotting vs. results approximately in a linear relationship. However, this is not necessary. Computers and microprocessors are perfectly able to linearize water content calibration data.Kobayashi 1996; Michels 1961
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ACKNOWLEDGMENTS
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The funding of this research was supported, in part, by the European Commision, project WATERMAN, number FAIR1 PL95 0681.
Received for publication April 5, 1999.
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ABSTRACT
INTRODUCTION
, and their average electric field, 
