# Dielectric Spectroscopy and Application of Mixing Models Describing Dielectric Dispersion in Clay Minerals and Clayey Soils

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## Abstract

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## 1. Introduction

_{V}), measured using time domain reflectometry (TDR):

^{3}cm

^{−3}).

- Free water, which exhibits bulk properties of a continuous background phase.
- Macroscopically confined water that exhibits bulk properties but is confined by the configuration of the solid, water and air phases, e.g., in aggregates or foams, as a discontinuous inclusion phase.
- Microscopically confined water where the dielectric response of the water is modified, for example, by:
- ○
- Extrinsic electric fields, including dipole forces, charged surfaces or hydration around ions.
- ○
- Extrinsic geometry, causing structural alteration due to being trapped, caged or structurally modified by proximity to a surface.

- Measure the dielectric dispersion of well-defined clay minerals and associated clayey soils, and to compare these with other unsaturated porous media where either macro- or microscopically confined water should be dominant.
- Use simple models to test the predictive capability of the geometrical modeling approach for water-saturated dispersive clayey soils.
- Compare the dispersive data for unsaturated clayey soils with the mixing model bounds we might expect for coarse granular materials.

## 2. Materials and Methods

#### 2.1. Clay Minerals

#### 2.2. Clayey Soils

^{−1}. The solution was extracted using a 2:1 dilution in deionized water at 25 °C and the EC of the extracts measured was: Cecil, 0.0366; Blount, 0.0385; and Okoboji, 0.0970 S m

^{−1}. Since the saturated gravimetric water content of clayey soils is generally assumed to be ~0.5 g g

^{−1}, the 2:1 dilution EC above was multiplied by 4 to obtain the water-saturated EC

_{w}used in fitting the dielectric mixing models. The bulk DC conductivity (σ

_{aDC}) of water-saturated clayey soils samples was also determined, as reported in the Supplementary Material (Figure S3).

#### 2.3. Vector Network Analyzer (VNA) Measurements

^{3}sample holder. Measurements spanned a 1 MHz–6 GHz bandwidth. The VNA was calibrated using the HP 85033C 3.5 mm Calibration Kit (Hewlett-Packard, Palo Alto, CA, USA) which includes the open, short and load (50 Ω) standards. Network analyzers can measure the reflection (one port) and transmission (two ports) characteristics of the medium of interest using a broad bandwidth signal [58]. For this study, reflection measurements were performed to obtain the dielectric properties of the clays and clayey soils. In order to obtain the complex permittivity from the reflection coefficient (S11) values provided by the VNA, the dielectric probe was calibrated using air, deionized water and a shorting block as standards, together with the calibration software supplied with the probe, following the steps reported in the probe manual [59].

## 3. Theory and Modeling

_{eff}) of a two-phase mixture of dielectric spheres in a uniform dielectric background is the MG [38] mixing equation [60], given as

_{e}is the permittivity of the background (water or air) and ε

_{i}is the permittivity of the granular inclusions. The model assumes that the inclusions and their respective electrical fields are non-interacting, which is an invalid assumption for densely packed granular materials. This interaction effect has been studied theoretically for periodic cubic lattices of spheres [61] and demonstrated experimentally for these cubic lattices [36].

^{i}) that describe the extent to which the inclusion polarization is reduced according to its shape and orientation with respect to the applied electrical field. The depolarization factors for a spheroid (ellipsoid with radii a, b, c, a ≠ b = c) with an aspect ratio of (a/b) can be approximated by the empirical function given by Jones and Friedman [7]:

^{a}

^{,b,c}= 1/3, 1/3, 1/3, for thin disks N

^{a}

^{,b,c}= 1, 0, 0 and for long needles N

^{a}

^{,b,c}= 0, 0.5, 0.5. Friedman and Robinson [66] found that a value of (a/b) equal to 0.466 was representative of quartz sand grains.

_{α}, in the model, which was defined as “the permittivity which an inclusion ‘feels’ in its surroundings in the mixture” [60]. It is expected to lie somewhere between the value of the bathing fluid, ε

_{e}(i.e., water or air) and ε

_{eff}. The model expression extended for complex values can be expressed as

^{i}= 1 − N

^{i}), giving rise to a different heuristic parameter for each principal axis and resulting in a single value of α = 2/3 for spheres. A value of α = 1 stands for the coherent potential approximation, assuming the inclusions “feel” around them a background of ε

_{eff}. Friedman and Robinson [66] found that a value of α = 0.2 accounted for the neighboring particle effects based on measurements in coarse, densely packed granular materials. Experimental evidence has further shown that this value does change depending on the packing of monosize spheres. Values of α ranged between 0.2 for the random packing of monosize spheres and 0.323 for the simple cubic periodic lattice [36].

^{−1}) is the electromagnetic angular frequency, ε

_{s}is the low-frequency limit of the dielectric value, ε

_{inf}is the high-frequency limit of the dielectric value, practically at a frequency high enough that the molecules cannot respond to the applied field (re-orient) and create polarization, and τ is the relaxation time (s), characterizing the time it takes the molecule dipoles to revert to their original random orientation after the external field is removed.

_{s}= 78.36, ε

_{inf}= 5.2 and τ = 8.27·10

^{−12}s [70]. As the frequency increases, the real part, ε′, varies from its static value, 78.36, towards that of high frequencies, in a sigmoidal pattern, passing through an inflection point at the relaxation frequency of 19.25 GHz (f

_{rel}= 1/2πτ). The imaginary part, ε″, is zero for the static and infinite frequency states, and has a bell shape (symmetric when ω is plotted on a log scale), with its maximal value, 36.58 ((ε

_{s}− ε

_{inf})/2), at the relaxation frequency.

_{e}in Equations (2)–(4) or (6) can be replaced by the Debye model for water. These models form the basis for testing geometrical effects on the frequency-domain dielectric response, onto which there is the potential to explore other phenomena. Based on soil solution measurements, the water phase of the studied water-saturated soils was considered as a conductive medium. To account for the conductivity effects on the effective dielectric response of soils, we added a term of conductivity to the imaginary part of the Debye model for the water phase, so that Equation (7) was extended to

_{wDC}is the EC

_{w}(ω ~ 0) of the solution and ε

_{0}is the permittivity of the vacuum (8.854 × 10

^{−12}F m

^{−1}).

## 4. Results and Discussion

#### 4.1. Clay Minerals and Clayey Soils Dispersion

^{8}Hz). Illite shows a similar response in Figure 2e,f, while montmorillonite exhibits the strongest dispersion (Figure 2g,h) for the real permittivity. While kaolinite and illite slightly level out at >10

^{8}Hz, montmorillonite still has distinct curvature in the real permittivity throughout the frequency range. Real and imaginary permittivities of water-saturated and nearly water-saturated (kaolinite, ~0.32–0.41 m

^{3}m

^{−3}; illite, ~0.55 m

^{3}m

^{−3}; montmorillonite, ~0.84–0.88 m

^{3}m

^{−3}) dispersive clays are substantially higher between 10

^{8}and 10

^{9}Hz, even exceeding the permittivity of pure water in some cases. Since this frequency range is common for many water content sensors, it is important to know the measurement frequency and sensor response in dispersive clayey soils in order to accurately relate dielectric response to water content.

_{V}= 0.56), can have a higher dielectric response at a lower water content (θ

_{V}= 0.53). This is consistent with the context in which interfacial polarization occurs, since in unsaturated soils, there are solid–water and air–water interfaces that contribute to the MW effect and thus the intensity of dispersion is not expected, in principle, to increase monotonically with water content.

#### 4.2. Modeling of Water-Saturated Clay Mineral Soils

^{3}m

^{−3}). We applied the Sihvola–Kong (SK) two-phase model (Equation (6)) with dielectric dispersion for the water phase (Equation (8)), where σ

_{wDC}= 0.2 S m

^{−1}, broadly in the middle of the three soils (Cecil = 0.14; Blount = 0.16; Okoboji = 0.40 S m

^{−1}), α = 0.2 and the particle shape was assumed to be spheres (Figure 4a) or oblate spheroids (Figure 4b). The electrically conductive water is considered as the background and the soil particles as non-conductive solid inclusions with a permittivity of 5. Similarly, MG (Equation (6) with α = 0) and PVS (Equation (6) with α

^{i}= 1 − N

^{i}) for spheroids and complex permittivities, and MWBH (Equation (4)) models are also presented in the frequency domain. The three dispersive soils show similar responses below 10

^{9}Hz. The Okoboji soil dielectric response is distinctly lower in the higher frequencies (>10

^{8}Hz) compared to the other two soils. The permittivity of talc exhibits a relatively dispersion-free response in comparison, a response that would also be expected in a sand, for example.

^{8}Hz. The particle aspect ratio, translated through N

^{i}in Equation (6), was adjusted to represent inclusion aspect ratios from spheres to oblate spheroids, although only that of a/b = 1/8 is presented in Figure 4b. The MG model assumes that the inclusions and their respective electrical fields are non-interacting, which is a valid assumption for diluted mixtures [36], constituting an upper bound for water-saturated soils. Nonetheless, this non-interaction is likely to explain the inability of the model to describe the MW dispersion. The DEMA in its different extensions, such as the MWBH formula, has proved very successful in the high-frequency limit (ω → ∞) [63], especially for media of broad particle size distribution [11,71,72]. However, its suitability in describing the interfacial polarization in water-saturated porous media below 10

^{8}Hz is limited. The DEMA is based on an infinitesimal sequential mixing of solid into the fluid background, which corresponds to an “infinitely wide” particle size distribution, i.e., a fractal granular medium. This means that the background phase connectivity is never interrupted, regardless of the porosity and the particle shape and orientation. In contrast, in fine-grained soils dominated by clay minerals, as the shape of the solid particles tends to be a thin disk, their ability to form barriers that interrupt the continuity of the conductive aqueous phase, where also charge accumulates, increases. This gives rise to portions of confined electrically conductive water, which are separated by assemblies of solid particles with a different conductivity and permittivity, not contributing to the in-phase conductivity of the medium, but increasing the dielectric polarization [37]. This is likely the cause for the MWBH model not being able to explain the dispersion shown by all soils. Evidence of the emergence of a percolation threshold as we approach disk-shaped particles supports this hypothesis [73], and can somehow help us understand how the geometry of the particles affects the manifestation of the MW effect.

_{w}and, for the case of SK, also α, from which a dispersion in the modeled real part below 10

^{8}Hz arises, increasing the low-frequency dielectric response. It is also noticeable that these models are very sensitive around this critical set of parameters; small changes in N

^{i}lead to large changes in the modeled real permittivity. From the computations, it has been observed that low-frequency dispersion is favored by the increase in ϕ, EC

_{w}and α and the approximation of the particles to the oblate shape. Conversely, the more oblate the particles are, the lower the permittivity is in the higher frequency range 10

^{8}–10

^{10}Hz.

^{9}Hz in the Blount soil. This was a little surprising as α of 0.2 described the effective static permittivity of coarse media with a lower porosity of 0.4 [66], so we were expecting α to be similar (or even smaller) for high-porosity soils (0.56 m

^{3}m

^{−3}), but a simple cubic packing of spheres shows this is not necessarily the case (ϕ = 0.476 m

^{3}m

^{−3}; α = 0.323). However, this finding supports that of Chen and Or [46], who found DEMA suitable for predicting the permittivity of medium-textured soils. It may suggest that the perturbation of the electrical field around an inclusion due to the particle size distribution is sufficient to manifest as an increase in α, even at the higher porosity. The shift between the modeled and experimental imaginary permittivity of the Blount sample can be linked to the possibility of an additional dispersion mechanism, i.e., different from conductivity and MW mechanisms.

^{8}–10

^{9}Hz range for all soils, sloping gently from left to right. One thought is that the effect is due to confined water (B in Figure 1, or the microscopic confinement due to caging as outlined in the Introduction), broadening the relaxation of the water phase, a hypothesis consistent with the findings of Calvet [75] (Figure S2) for the first layers of water in homo-ionic montmorillonites. Another dispersion mechanism candidate is the frequency-dependent surface conductivity (S in Figure 1). The relaxation contribution, disregarding the bulk conductivity contribution to the imaginary permittivity, is presented in the Supplementary Material, Figures S3 and S4.

#### 4.3. Comparison of Media with Confined Water

^{8}Hz) from the present study. Figure 6a shows the dielectric response of porous media without microscopically confined water that is electrostatically “bound” (hygroscopically adsorbed) (<0.01 g/g), whereas Figure 6b shows media that have substantial hygroscopic water (>0.04 g/g). Hygroscopic water contents of some media are reported in Table S2, Supplementary Material. In Figure 6a, the glass beads, sandy soils and Topp’s curve provide familiar references. The data of mica from Blonquist et al. [76] show the dielectric response due to the platy (oblate) particle shape of a layered sample that is aligned with or perpendicular to the applied electrical field. Pumice is presented from Blonquist et al. [12], which has a foam-like structure and high porosity. In addition, we have added kaolinite, kaolinitic Cecil soil and talc, but it is important to remember that these materials are repacked, so the geometry changes with each packing. Contrasting responses are observed, with talc having a very low dielectric response and kaolinite, with a similar mineral structure to talc, giving higher dielectric responses than Topp’s curve. We propose that the lower dielectric response of talc could be due to one of a number of mechanisms. The fact that it sits between the aligned and perpendicular mica at water contents beyond ~0.4 might indicate that it has something to do with the alignment of the tactoids with respect to the electrical field. Both kaolinite and talc are very platy materials and follow the same path as mica until a water content of ~0.2 for kaolinite and 0.4 for talc. One might expect the clays to pack perpendicular to the VNA fringing field initially, i.e., the principal axis of the ellipsoid parallel to the applied field, generating more of an open house of cards structure as they become wetter. Second, it is feasible that talc forms microaggregates and macroscopically confines the water as it is a very high porosity material. Thirdly, the clay mineral may microscopically cage the water, confining it, but both kaolinite and talc are relatively low surface area materials, so this is less likely. The fact that kaolinite produces a relatively high dielectric response must be due to the presence of the counter ions, giving rise to a substantial MW relaxation, as shown in Figure 2c. The change in the alignment of the particles is perhaps more consistent with the high-frequency response.

## 5. Conclusions

_{w}and contrast of conductivity and permittivity between phases. The MG and MWBH models were found inadequate to describe the strong dispersion associated with MW polarization in water-saturated clayey soils. In contrast, models that allow the inclusions to interact strongly, such as PVS (asymmetric effective medium approximation) and SK, were able to approximate the permittivity dispersion below 100 MHz. Fitting the SK model to the data resulted in a value of the heuristic parameter α, that accounts for particle interaction, of 0.425 with particle shape depolarization factors surprisingly consistent, with aspect ratios around 3:1, not as extreme as those reported for clay minerals in the literature, but being consistent with an averaged aspect ratio determined by the presence of also other more rounded particles in the soil. Our results suggest that MW relaxation processes are dominant in clayey soils, for example, the kaolinitic soil shows strong dispersion but has negligible microscopically confined water. Data presented suggest that phase composition (porosity and water content) and geometry (particle shape, orientation and size distribution) are the major factors determining the dielectric response and that confined or bound water is secondary.

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Contributors to dielectric loss in wet porous media covering a large frequency spectrum. Mechanisms include conduction (ionic conductivity), charged double layer, crystal water relaxation, ice relaxation, Maxwell–Wagner relaxation, surface conductivity, microscopically confined water relaxation; principle free water relaxation (Water 1), second free water relaxation (Water 2) (after Hasted [40], p. 238).

**Figure 2.**Real (left column) and imaginary (right column) permittivity measurements in (

**a**,

**b**) talc, (

**c**,

**d**) kaolinite, (

**e**,

**f**) illite and (

**g**,

**h**) montmorillonite clays at different water contents in a 10 MHz–6 GHz (10 MHz–3 GHz for talc) frequency bandwidth.

**Figure 3.**Real (left column) and imaginary (right column) permittivity measurements in (

**a**,

**b**) Cecil kaolinitic, (

**c**,

**d**) Blount illitic and (

**e**,

**f**) Okoboji montmorillonitic clayey soils at different water contents in a 10 MHz–6 GHz frequency bandwidth.

**Figure 4.**Real permittivity of Cecil, Blount and Okoboji water-saturated samples and unsaturated talc, and frequency-domain representation of Maxwell Garnett (MG), Polder van Santen (PVS), Maxwell–Wagner–Bruggeman–Hanai (MWBH) and Sihvola–Kong (SK) models for spherical (

**a**) and oblate spheroidal (

**b**) inclusions.

**Figure 5.**Modeling of the complex dielectric response of Cecil (

**a**), Blount (

**b**) and Okoboji (

**c**) soils using the SK model extended for complex permittivities (Equation (6)) and the extended Debye model (Equation (8)). Optimized parameters were: Cecil (α = 0.425, a/b = 0.355), Blount (α = 0.425, a/b = 0.3) and Okoboji (α = 0.425, a/b = 0.355).

**Figure 6.**Dielectric (K, ε’)—volumetric water content measurements of different porous media containing negligible (

**a**) and substantial (

**b**) hygroscopically adsorbed water. Glass Beads, Hyrum and Kidman are from Robinson et al. [77]; Pumice and Zeoponic data from Blonquist et al. [12]; Mica

_{⊤}(oblate principal axis perpendicular to the applied electrical field) and Mica

_{‖}(oblate principal axis parallel to the applied electrical field) data from Blonquist et al. [76]; JSC1 Martian from Robinson et al. [78]; and Talc, Kaolinite, Cecil, Okoboji and Montmorillonite are all VNA measurements at 10

^{8}Hz.

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**MDPI and ACS Style**

González-Teruel, J.D.; Jones, S.B.; Soto-Valles, F.; Torres-Sánchez, R.; Lebron, I.; Friedman, S.P.; Robinson, D.A. Dielectric Spectroscopy and Application of Mixing Models Describing Dielectric Dispersion in Clay Minerals and Clayey Soils. *Sensors* **2020**, *20*, 6678.
https://doi.org/10.3390/s20226678

**AMA Style**

González-Teruel JD, Jones SB, Soto-Valles F, Torres-Sánchez R, Lebron I, Friedman SP, Robinson DA. Dielectric Spectroscopy and Application of Mixing Models Describing Dielectric Dispersion in Clay Minerals and Clayey Soils. *Sensors*. 2020; 20(22):6678.
https://doi.org/10.3390/s20226678

**Chicago/Turabian Style**

González-Teruel, Juan D., Scott B. Jones, Fulgencio Soto-Valles, Roque Torres-Sánchez, Inmaculada Lebron, Shmulik P. Friedman, and David A. Robinson. 2020. "Dielectric Spectroscopy and Application of Mixing Models Describing Dielectric Dispersion in Clay Minerals and Clayey Soils" *Sensors* 20, no. 22: 6678.
https://doi.org/10.3390/s20226678