Analysis and Modeling of the Complex Dielectric Constant of Bound Water with Application in Soil Microwave Remote Sensing

: Complex dielectric constant (CDC) of bound water determines the accuracy of the complex dielectric constant of wet soil. According to electrical double-layer structure and dielectric properties, the bound water on clay particle surface is divided into strongly bound water and weakly bound water. Based on this classiﬁcation, models for the complex dielectric constants of bound water and soil are established taking into consideration factors such as temperature, moisture, texture, and microwave frequency. The results show that the fundamental reason why the complex dielectric constant of bound water is between that of ice and free water is the adsorption force which forms the electrical double-layer structure on the surface of clay particles. Low-concentration cationic solution could exist in free soil water and was found as the reason for the higher salinity and conductivity of free soil water, as compared to the measured soil solution. Results of soil CDC model are in good agreement with measured data across a wide range of microwave frequencies and soil temperature, moisture, and texture. The absolute root mean square error analysis also shows that the soil CDC model in this paper compared to the other models is more accurate.


Introduction
The physical process of soil moisture retrieval by microwave remote sensing involves two steps. The complex dielectric constant (CDC) of soil is first calculated using the radiation signal received by the sensor. Secondly, the moisture is obtained by the soil complex dielectric constant models which include the complex relationship between soil moisture and its CDC. Thus, the study of soil dielectric property via physical means is crucial for the soil moisture retrieval by microwave remote sensing [1][2][3].
The effective CDC of mixtures in nature is often expressed as a function of the CDC of inclusions and the background as well as the inclusion volume fraction [4]. Soil is a porous medium composed of air, dry soil, bound water, and free water [4][5][6]. Of the four, the CDC of air and dry soil are constant, and that of free water can be calculated by the Debye equation. The only unknown parameter is the CDC of bound water (ε b ). Thus, the ε b and the volume content of bound water (V b ) determine the accuracy of soil CDC. Bound water, also known as adsorbed water or hygroscopic water, is mainly found on the surface of clay particles, and forms as a result of the surface effect of these particles [7,8]. The dielectric property of soil water (mainly bound water) differs from that of the water extracted from the same soil. When soil minerals are exposed to water, the exchangeable cations on the surface of clay

The Electrical Double-Layer Model: Introduction and Application
Bound water is often treated as the electrical double-layer (EDL) of solution on the clay particle surface [5,7]. Hence, the existing Stern-Gouy electrical double-layer theory is used to describe the microscopic structure at the clay particle surface and calculate the microscopic physical quantities of soil. Although the theory of EDL in soil system has been discussed in many publications [17][18][19][20][21], quantitative calculations concerning DEL are still complicated. Here a derivation of the Stern-Gouy DEL and quantitative calculations for five soils with different textures are presented.
As shown in Figure 1, the EDL consists of surface charges and anti-ion solution. As the lattice of a clay particle is incomplete, the clay particle surface is negatively charged, and the anti-ion is positive. Based on the absorption force acting on the EDL, the anti-ion solution is further divided into the adsorption-layer and diffuse-layer. Some ions are immobilized to the clay particle surface by the van der Waals force, chemical bonds, and electrostatic attraction to form an adsorption-layer (also known as the Stern-layer). The remaining ions are dispersed in the solution phase by electrostatic force, forming the diffuse-layer (also known as Gouy-layer). The concentration of cations in the diffuse-layer decreases exponentially with the distance from the adsorption layer-diffuse layer interface. All equations (Equations (1)- (6))in this section are taken from Olphen [19,20] and Shang et al. [21]. The total amount of negative charges on the clay particle surface (σ) is the sum of charges in the adsorption layer (σ1) and diffuse layer (σ2): The amount of charges in the adsorption-layer can be approximated statistically using the number of cation adsorption sites and adsorption force, and is expressed as follows: [ ] where n1 is the number of adsorption sites per square centimeter on mineral particle surface.
Assuming that each site occupied by water molecules within a single layer is a potential cation adsorption site, the number of adsorption sites on the surface is then n1 ≈ 1015 sites/cm 2 . The following quantities are also defined: ν ≈ 2 is the average valence of cations in diffuse-layer of the five soil samples in Table 1; e = 1.602176634 × 10 −19 C is the charge on a single electron; NA = 6.02 × 10 23 is the Avogadro constant; Mw = 18.01528 g/mol is the molecular weight of the solvent; n0 is the concentration of soil solution; Φδ is the electric potential of the adsorption layer; Ψ = 0 is the specific The total amount of negative charges on the clay particle surface (σ) is the sum of charges in the adsorption layer (σ 1 ) and diffuse layer (σ 2 ): The amount of charges in the adsorption-layer can be approximated statistically using the number of cation adsorption sites and adsorption force, and is expressed as follows: where n 1 is the number of adsorption sites per square centimeter on mineral particle surface.
Assuming that each site occupied by water molecules within a single layer is a potential cation adsorption site, the number of adsorption sites on the surface is then n 1 ≈ 1015 sites/cm 2 . The following quantities are also defined: ν ≈ 2 is the average valence of cations in diffuse-layer of the five soil samples in Table 1; e = 1.602176634 × 10 −19 C is the charge on a single electron; N A = 6.02 × 10 23 is the Avogadro constant; M w = 18.01528 g/mol is the molecular weight of the solvent; n 0 is the Remote Sens. 2020, 12, 3544 4 of 20 concentration of soil solution; Φ δ is the electric potential of the adsorption layer; Ψ = 0 is the specific adsorption potential at clay particle surface; k = 1.38 × 10 −23 J/K is the Boltzmann constant; T is the thermodynamic temperature. The charge (σ 2 ) in the diffuse-layer can be expressed as function of the soil solution concentration (n 0 ) and the Stern potential (Φ δ ): where ε = 80 is the average dielectric constant of the diffuse-layer solution. When both the surface charge density (σ) and soil solution concentration (n 0 ) are known, the Stern potential (Φ δ ) could be obtained. All physical parameters pertaining EDL can be calculated from the Stern potential. The thickness (d) of the diffuse-layer is expressed as follows: The relationship between ion concentration in the diffuse-layer (n) and the distance (x) is: where x is the distance between the diffuse layer and the adsorption layer-diffuse layer interface, and the parameters β and K are defined as β = tanh( veΦ δ 4kT ), K = 8πn 0 v 2 e 2 εkT . The average ion concentration n in a sublayer of the diffuse-layer with thickness x 2 − x 1 is: The parameters of electrical double-layer and measurements of soil complex dielectric constants are obtained from [5,22]. The five different soil texture surveyed are representative of the sand and clay content in typical soil types, as per the classification of the U.S. Department of Agriculture, in which sand is any soil with particle diameter of d > 0.05 mm, silt is soil with particle diameter of 0.002 mm < d < 0.05 mm, and clay is soil with particle diameter of d < 0.002 mm.

Bound Water Model and Model Parameters
The complex dielectric constant (CDC) of bound water is closely related to the distance from the clay particle surface. The bound water film at the clay particle surface is tightly bound to these particles. The real part of the dielectric constant is close to that of ice [5]. As the thickness of bound water film increases, the CDC of the water layer farther away from the solid surface gradually approaches that of free water [5,23]. As such, the CDC of bound water is between that of ice and free water. According to the electrical double-layer structure, the bound water is divided into strongly bound water and weakly bound water. Strongly bound water is the very thin layer of adsorption solution at the surface of clay particle. Weakly bound water refers is the diffuse-layer solution outside the regime of strongly bound water but under the influence of the electric field generated by the surface charges. As the two types of bound water differ in their content and property, the methods of their CDC are also different.

Content of Strongly Bound Water
The adsorption-layer possesses certain particular properties. Although the type of exchangeable cations, concentration of soil solution, distribution of charges, and irregular shape of clay particle surface all affect the adsorption of cations, these factors have a weak influence on the thickness of the adsorption layer. Verwey and Overbeek [24] and Olphen [19,20] show that most of the adsorption-layer charges are found within 5 Å from the clay particle surface. Shang et al. [21], however, think that the thickness of the adsorption-layer varies between 5 and 6.5 Å. Tripathy et al. [25] suggest a reasonable thickness of adsorption-layer of 5 Å. Dobson et al. [5] use 3.6 Å as the thickness of the adsorption-layer to calculate the content of bound water. Based on the above, δ=5 Å is chosen as the adsorption-layer thickness in this paper. Treating the clay particle surface as planar [5], and taking in to consideration the linear relationship between specific surface area and the content of strongly bound water [26,27], the maximum volume of strongly bound water (V maxsb ) is: where A S is the specific surface area and ρ b is the soil bulk density. The exponent of 0.9 is included to correct for the content of strongly bound water with respect to the specific surface area.

Dielectric Constant of Strongly Bound Water
The complex dielectric constant of the adsorption-layer solution (ε sb ) is largely determined by the degree of binding between water molecules and clay particles and the nature of surface charges on clay particles [21]. Olphen [19,20] assigns a value of 6 to ε sb . Sridharan and Satyamurty [28] and Shang et al. [21] point out that ε sb could vary between 3 and 6 or 4 and 8, respectively. Wang and Schmugge [10] and Campbell [29] reach the conclusion that ε sb at the mineral surface is close to that of ice (ε sb = 3.5-3.8). Dobson et al. [5] assume that soil particles are covered by a thin layer of chemically bound water with a dielectric constant of ε sb = 3.2 + j0.1. As the diameter of bound water molecules is 2.5 [19,20] or 2.8 Å [12], the adsorption-layer thus contains roughly two layers of bound water film. The first layer has CDC close to that of ice, but the CDC of the second layer is difficult of assess. As shown in Figure 2, the relaxation times of the first two layers of bound water molecules on the clay particle surface are 5.0 × 10 −10 and 5.0 × 10 −11 s, respectively [12]. Assuming the CDC of the first layer of bound water equals that of ice (ε i = 3.2 + j0.1), and the CDC of the first two layers of water follow the Debye equation [30], the CDC of the second layer of water is found to be 16 + j0.5 using the relaxation time of the first two layers. This value is about five times the former. The CDC of strongly bound water adopts the following equation: where W is the soil moisture. Equation (8) necessarily implies that when soil moisture (W) is greater than the maximum content of strongly bound water (V maxsb ), the dielectric constant of strongly bound water adopts a constant value of 9 + j0.6. When soil moisture (W) is less than the maximum content of strongly bound water (V maxsb ), the dielectric constant of strongly bound water is determined by the ratio of soil moisture to the maximum content of strongly bound water.
where W is the soil moisture. Equation (8) necessarily implies that when soil moisture (W) is greater than the maximum content of strongly bound water (Vmaxsb), the dielectric constant of strongly bound water adopts a constant value of 9 + j0.6. When soil moisture (W) is less than the maximum content of strongly bound water (Vmaxsb), the dielectric constant of strongly bound water is determined by the ratio of soil moisture to the maximum content of strongly bound water.

Content of Weakly Bound Water
Taking into account the nonlinear relationship between the maximum content of bound water content (Vmaxb) and the specific surface area, the model for unfrozen water content is used here to calculate Vmaxb. At temperature below 0°C, unfrozen water is formed between ice and the soil particle surface due to adsorption and capillary action at the soil particle surface [31]. Based on its property, unfrozen water is categorized either as capillary water or bound water. As soil cools, capillary water would freeze before bound water, and the latter only begins to freeze when capillary water is completely frozen. The maximum content of bound water is readily computed using the model of unfrozen water content when the initial freezing point of bound water (T1) is known. Referencing the transition moisture [10], maximum content of bound water [6,13,14], and the initial freezing point of bound water [32][33][34], this paper defines the Vmaxb as approximately the unfrozen water content at −2 °C. Applying the unfrozen water model of Anderson and Tice (1973), the maximum volume of bound water is:

Content of Weakly Bound Water
Taking into account the nonlinear relationship between the maximum content of bound water content (V maxb ) and the specific surface area, the model for unfrozen water content is used here to calculate V maxb . At temperature below 0 • C, unfrozen water is formed between ice and the soil particle surface due to adsorption and capillary action at the soil particle surface [31]. Based on its property, unfrozen water is categorized either as capillary water or bound water. As soil cools, capillary water would freeze before bound water, and the latter only begins to freeze when capillary water is completely frozen. The maximum content of bound water is readily computed using the model of unfrozen water content when the initial freezing point of bound water (T 1 ) is known. Referencing the transition moisture [10], maximum content of bound water [6,13,14], and the initial freezing point of bound water [32][33][34], this paper defines the V maxb as approximately the unfrozen water content at −2 • C. Applying the unfrozen water model of Anderson and Tice (1973), the maximum volume of bound water is: where T 1 = 2 is the absolute value of the negative temperature, and the parameters C and D (C = 1.299A 0.552 ) are both related to the specific surface area of soil (A S ). The result of the original equation for unfrozen water [35] is given in mass percentage, and needs to be converted into volume fraction of water by multiplying it with soil bulk density (ρ b ) and the coefficient 0.01, which results in Equation (9). The content of weakly bound water (V wb ) is thus the difference between the maximum volume of bound water and that of strongly bound water:

Dielectric Constant of Weakly Bound Water
The complex dielectric properties of strongly bound water and weakly bound water in soil are different. Weakly bound water defined by this paper is the diffuse-layer solution. The dielectric property of unfrozen water has been experimentally proven to be similar to that of salt solutions [5,22]. Thus, the Debye equation for salt solutions is used here to calculate the dielectric constant of weakly bound water. As Figure 2 shows, the relaxation time varies slightly for water molecules in the 3rd to 10th layer on the clay particle surface (the first two layers of water molecules are the adsorption-layer solution, and the 3rd to 10th layer of bound water are the diffuse-layer solution). With the presence of such a moisture range in soil, the increment of bound water is the cause for the change in the soil dielectric constant in this moisture range [6,13,14]. For this reason, instead of the concentration of a salt solution, the average cation concentration in the 1st to 8th layer of water molecules in the diffuse-layer of five soil samples is used to calculate the CDC of weakly bound water (ε wb ): where S sw is the salinity, f is the frequency, and T is the soil temperature. The 1st to 8th layer of water molecules in the diffuse layer has a thickness of about 20 Å. The average cation concentration of the five soil samples is approximately 0.36 mol/L (Table 1), which is equivalent to a salinity of S sw ≈ 20.9 % of sodium chloride solution. Equation (11) shows the following: when V maxsb < W ≤ V maxb , the CDC of weakly bound water is equal to that of 20.9% sodium chloride solution. ε slat (S sw , f , T) is the Debye equation for salt solutions, and is expressed as [36]: where ε sw∞ is the dielectric constant at the high-frequency limit of salt solution, which is assumed to equal the value of 4.9. f is the wave frequency in Hz. ε sw0 is the static dielectric constant of pure water, and τ sw is the relaxation time of pure water in s. ε 0 is the dielectric constant for free space, which is 8.854 × 10 −12 F/m. σ i is the effective conductivity of salt solution in S/m. ε sw0 , τ sw , and σ i are all related to temperature and salt content: Remote Sens. 2020, 12, 3544 8 of 20

Dielectric Constant of Bound Water
The electrical double-layer (EDL) model describes the microscopic structure of clay particles. As shown in Table 1, despite the differences in soil texture, the input parameters (surface charge density (σ) and soil solution concentration (n 0 )) and output parameters (average diffuse-layer thickness (d) and parameter β) for the EDL model of the five soil samples are not significantly different. Soil texture mainly affects the specific surface area and not the parameters related to EDL. The average values of the EDL-related parameters for the five soil samples of Table 1 are therefore representative of a wide array of fixed-charge soils. Hence, this parametric model for the complex dielectric constant of bound water (ε bw ) established on the EDL structure is universal. ε bw is the product of the CDC of strongly bound water (ε sb ) and weakly bound water (ε wb ) with their respective volumes: The expression of ε wb changes with soil moisture. Equations (8)-(13) are known collectively as the model for the complex dielectric constant of bound water.

Soil Dielectric Mixing Model
Many natural substances are, strictly speaking, a mixture of different components. Their complex dielectric constants thus result from the combination of the CDC of each component [4]. The particle size, particle orientation, volume fraction, and dielectric constant of each component affect the dielectric constant of the whole mixture [4,[37][38][39][40]. Researchers have developed several dielectric models for CDC of natural mixtures [41][42][43][44][45][46]. For easier computation and application, the most concise of them, the four-component dielectric mixing model [5] is used in this paper, and its equation is: The subscripts soil, air, ss, f, and b refer to moist soil, air, dry soil, free water, and bound water, respectively. In the equation, V ss = ρ b /ρ s = 1 − P, ρ s is the specific density, and P is the porosity. [5]. ε b is calculated by Equation (13). As free water contains a certain amount of salt [10,15], 8% salt solution is used here as free soil water. α is the shape factor, reflecting the geometry of soil particles [47] and is closely related to the internal structure and depolarization factor of soil [48]. α adopts different values in different dielectric mixing models. In the semi-empirical soil dielectric mixing model [5], α = 0.65; in the generalized refractive mixing dielectric model, α = 0.5; in the Looyenga mixing model [46], α = 0.333. In the mixed model of this paper, α = 0.55 for the microwave frequency range of 1.4-4 GHz, and α=0.65 for the microwave frequency range of 4-18 GHz.
Similar to the CDC of bound water (ε b ), the soil complex dielectric constant (ε soil ) displays considerable piecewise characteristics. Equation (14) can be expressed as: Using Equation (15), the prediction of soil complex dielectric involves the following steps: 1. The moisture (W), temperature (T), specific surface area (A S ), specific density (ρ S ), and bulk density (ρ b ) of soil are measured. The volume of dry soil (V SS ), porosity (P), and air volume (V air ) are then calculated from these physical quantities.
2. The maximum content of strongly bound water (V maxsb ), maximum content of bound water (V maxb ), and content of weakly bound water (V wb ) are found from Equations. 7, 9, and 10. The content of free water (V f ) is calculated from soil moisture (W) and the maximum content of bound water (V maxb ).
3. The complex dielectric constants (ε sb and ε wb ) of strongly bound water and weakly bound water are calculated using Equations (8) and (11). The complex dielectric constants (ε b and ε f ) of bound water and free water are then calculated from Equations (12) and (13), respectively.
4. The volume fractions and complex dielectric constants of air, dry soil, bound water, and free water are substituted into the four-component soil dielectric mixing model (Equation (15)) to obtain the dielectric constant of soil.
Specific surface area (A S ) is an important input parameter of soil dielectric mixing model. Considering most of the measured soil data contain particle size distribution, a recommended algorithm in predicting specific surface area based on soil particle size distribution is as follows [49]: The soil complex dielectric mixing model developed here partitions soil water into strongly bound water, weakly bound water, and free water. This is a more precise classification than existing models [5,6,10]. In this study, the classification of bound water is based on the EDL structure at the clay particle surface. Four main assumptions are made in the calculation of the complex dielectric constant of bound water: (1) the dielectric property of the first layer of strongly bound water resembles that of ice; (2) the complex dielectric constant of the second layer of strongly bound water is about five times that of the first; (3) for V maxsb < V < V maxb , change in CDC of wet soil is caused by the increment of weakly bound water; (4) the CDC of weakly bound water is equivalent to that of salt solution.

Data and Results
The values of the soil complex dielectric constant used in this paper are from the measurements of five types of natural soil by [5,22]. These data consist of two parts: the first is the particle size, specific surface area, and surface charge density of these five types of soil (Table 1), and the second part is their measured soil complex dielectric constants (Figure 3) at 0-50% relative humidity and four individual microwave frequencies (1.4, 5, 10, and 18 GHz). These five types of natural soil represent a wide range of sand and clay content. The measured values of soil dielectric constant cover a broad array of soil moisture (0-50%), microwave frequencies (1.4-18 GHz), and details in soil microscopic physical quantities. Figure 3 shows the results of the soil complex dielectric mixing model (Equation (15)). The dielectric constant of bound water is calculated from Equation (13). The measured complex dielectric constants of the five soil samples agree well with the model results across the four individual microwave frequencies (1.4, 5, 10, and 18 GHz) and 0-50% relative humidity. Soil moisture is the main factor influencing soil complex dielectric constant. The latter shows significant change with soil moisture, and both its real part and imaginary part are strongly dependent on soil moisture. An inspection of the green curves (the real part of soil complex dielectric constant) of Figure 3a1-a5 reveals the maximum content of bound water (V maxb ). The slope of curve above this value behaves differently from that below this value. Above this value, the slope is almost invariant as moisture increases. Below this point, it increases gradually with moisture. This indicates the change in soil dielectric constant and the increase in of free water at W ≥ V maxb . Using the slope of the green curve in Figure 3a5, the soil dielectric constant can be partitioned into three segments: 0 < W < V maxsb (Stage I); V maxsb ≤ W < V maxb (Stage II); W ≥ V maxb (Stage III). At frequencies of 1.4 and 5 GHz (Figure 3a1-a5,b1-b5), the slope of soil complex dielectric constant (real part and imaginary part) of Stage III is significantly higher than that of Stage I or Stage II. At high moisture levels (Stage III), the increase in soil complex dielectric constant is due to free water increments, while at lower moisture levels (Stages I and II), this increase is due to bound water increments. The above result means the dielectric constant of free water is higher than that of bound.
At W < V maxb (Stage I), the curve is almost flat, which means at this stage, the dielectric constant of bound water is almost equal to that of dry soil. The slope of Stage II (V maxsb ≤ W < V maxb ) is greater than that of Stage I but less than that of Stage III. This means that strongly bound water possesses very different dielectric property compared to weakly bound water. At W < V maxb , the slope of the green curve increases gradually, indicating gradual rise in the dielectric constant of bound water with soil moisture. The dielectric constant of bound water at the proximity of clay particle surface resembles that of ice, while at farther distance, it approaches that of free water. Based on the above analysis and double layer structure, we define the bound water of Stage I as strongly bound water. At this stage, change in soil complex dielectric constant is caused by the increment of strongly bound water. The bound water of Stage II is a combination of strongly bound water with some weakly bound water, and change in the soil complex dielectric constant is caused by the increment of weakly bound water. The soil water of Stage III includes both bound water and free water, and change in the soil dielectric constant is caused by the increment of free water.
Remote Sens. 2020, 12, x FOR PEER REVIEW 10 of 21 Figure 3. Comparison between results of the soil mixing complex dielectric model (Equation (15)) and measured data for real part and imaginary part; the black vertical line represents the maximum bound water content (Vmaxb); the black vertical dotted line represents the maximum strongly bound water content (Vmaxsb). (a1)-(a5) represents five types of soil in Table 1 at 1.4 GHz, respectively; (b1-b5) represents five types of soil in Table 1 at 5 GHz, respectively; (c1-c5) represents five types of soil in Table 1 at 10 GHz, respectively; (d1-d5) represents five types of soil in Table 1 at 18GHz, respectively. Figure 3 shows the results of the soil complex dielectric mixing model (Equation (15)). The dielectric constant of bound water is calculated from Equation (13). The measured complex dielectric constants of the five soil samples agree well with the model results across the four individual microwave frequencies (1.4, 5, 10, and 18 GHz) and 0-50% relative humidity. Soil moisture is the main factor influencing soil complex dielectric constant. The latter shows significant change with soil moisture, and both its real part and imaginary part are strongly dependent on soil moisture. An inspection of the green curves (the real part of soil complex dielectric constant) of Figure 3a1-a5 reveals the maximum content of bound water (Vmaxb). The slope of curve above this value behaves differently from that below this value. Above this value, the slope is almost invariant as moisture increases. Below this point, it increases gradually with moisture. This indicates the change in soil  (15)) and measured data for real part and imaginary part; the black vertical line represents the maximum bound water content (V maxb ); the black vertical dotted line represents the maximum strongly bound water content (V maxsb ). (a1-a5) represents five types of soil in Table 1 at 1.4 GHz, respectively; (b1-b5) represents five types of soil in Table 1 at 5 GHz, respectively; (c1-c5) represents five types of soil in Table 1 at 10 GHz, respectively; (d1-d5) represents five types of soil in Table 1 at 18GHz, respectively. As shown in Figure 3, curves for the real part and imaginary part of the soil dielectric constant have almost the same intercept at W=0 across the various individual microwave frequencies. This means the dielectric constants of dry soil and air hardly change with microwave frequency. In Figure 3a5,b5,c5,d5, the soil complex dielectric constant displays substantial dispersion across the frequencies. Its real part decreases while the imaginary part increases as the frequency increases. Soil texture also affects the complex dielectric constant of soil. For soil samples 1, 3, and 5, the maximum content of strongly bound water and the maximum content of bound water (Table 2) both increase with soil specific surface area, indicating a higher level of bound water in soil with more clay content. The maximum content of strongly bound water has an approximately linear relation with soil specific surface area, while a nonlinear relationship exists between maximum bound water content and soil specific surface area. For each individual microwave frequency, the real part and the imaginary part of the dielectric constant of soil sample 5 (soil sample with the highest clay content) are smaller than other soil samples. This is mainly because of the higher level of bound water in a soil sample with more clay and the smaller real part of bound water dielectric constant than free water. For this reason, when soil moisture is the same, the soil sample with more clay would have a higher level of bound water and less free water, and a smaller soil dielectric constant (real part and imaginary part). Note: Maximum bound water content is from Equation (9); maximum strongly bound water content used Equation (7); complex dielectric constant of bound water is from Equation (13); physical properties of soils are shown in Table 1.
The salt content of free water needs to be considered when modeling the soil dielectric constant. Wang and Schmugge [10] and Liu et al. [15] both believe that the salt content of free soil water is higher than that of the salt concentration of real soil solution and it increases with the content of clay. This can be explained using the electrical double-layer model. In Figure 4, the cation concentration in the vicinity of the clay particle surface decreases exponentially with the distance to the surface. The transition moisture and maximum bound water content given by soil dielectric models [6,10,15] do not take into account the entire cationic solution and leave out the portions of low-concentration cationic solution dispersed in free water. As cationic solution only exists at the surface of clay particles, higher clay content would mean a greater proportion of cationic solution in free water, and higher salinity of the entire soil solution.
For the imaginary part of the soil complex dielectric constant at low frequency range, Liu et al. [15] suggest the introduction of an effective conductivity loss term related to soil water. The reason might be related to the unique properties of the imaginary part of the salt solution dielectric constant. In Figure 5, near 1.4 GHz frequency, the imaginary part of the salt solution dielectric constant is significantly higher than that of pure water, while in the 4-20 GHz range, little difference is seen between the imaginary parts of the two dielectric constants. The conclusion is that at the 1.4 GHz frequency, salt makes the prediction of the imaginary part more difficult for the dielectric constant of free water. As such, Liu et al. [15] adjust the conductivity term to negate the influence of cationic solution in free water.
Remote Sens. 2020, 12, x FOR PEER REVIEW 12 of 21 transition moisture and maximum bound water content given by soil dielectric models [6,10,15] do not take into account the entire cationic solution and leave out the portions of low-concentration cationic solution dispersed in free water. As cationic solution only exists at the surface of clay particles, higher clay content would mean a greater proportion of cationic solution in free water, and higher salinity of the entire soil solution.  Table 1.
For the imaginary part of the soil complex dielectric constant at low frequency range, Liu et al. [15] suggest the introduction of an effective conductivity loss term related to soil water. The reason might be related to the unique properties of the imaginary part of the salt solution dielectric constant. In Figure 5, near 1.4 GHz frequency, the imaginary part of the salt solution dielectric constant is significantly higher than that of pure water, while in the 4-20 GHz range, little difference is seen between the imaginary parts of the two dielectric constants. The conclusion is that at the 1.4 GHz frequency, salt makes the prediction of the imaginary part more difficult for the dielectric constant of free water. As such, Liu et al. [15] adjust the conductivity term to negate the influence of cationic solution in free water.    Table 1.
For the imaginary part of the soil complex dielectric constant at low frequency range, Liu et al. [15] suggest the introduction of an effective conductivity loss term related to soil water. The reason might be related to the unique properties of the imaginary part of the salt solution dielectric constant. In Figure 5, near 1.4 GHz frequency, the imaginary part of the salt solution dielectric constant is significantly higher than that of pure water, while in the 4-20 GHz range, little difference is seen between the imaginary parts of the two dielectric constants. The conclusion is that at the 1.4 GHz frequency, salt makes the prediction of the imaginary part more difficult for the dielectric constant of free water. As such, Liu et al. [15] adjust the conductivity term to negate the influence of cationic solution in free water.   [30] and [36], respectively. Figure 6 more explicitly shows changes in the real part and imaginary part of the complex dielectric constant of bound water (ε b ) with soil moisture in the three stages. In Stage I (0 < W < V maxsb ), the real part of the complex dielectric constant of strongly bound water shows linear change, and the imaginary part is almost 0. The slope of the curve becomes smaller with clay content increases. For all four frequencies, the real part of ε b of sandy loam (Sample 1) shows significantly larger slope than that of silty clay (Sample 5). In Stage II (V maxsb ≤ W < V maxb ), the change in ε b follows a curve, and the increase in ε b is produced by weakly bound water. The effect of soil texture is obvious, with smaller ε b occurring at higher amounts of clay content. Stage III (W ≥ V maxb ) has the maximum content of bound water, and the ε b is independent of soil moisture but is still affected by soil texture and temperature, and microwave frequency. The influence of microwave frequency on the ε b is largely manifested at Stages II and III, and is different on the real part and imaginary part. In soil samples of the same textural class, the imaginary part of the bound water dielectric constant decreases as frequency increases, while the imaginary part decreases first and then increases. all four frequencies, the real part of ɛb of sandy loam (Sample 1) shows significantly larger slope than that of silty clay (Sample 5). In Stage II (Vmaxsb ≤ W < Vmaxb), the change in ɛb follows a curve, and the increase in ɛb is produced by weakly bound water. The effect of soil texture is obvious, with smaller ɛb occurring at higher amounts of clay content. Stage III (W ≥ Vmaxb) has the maximum content of bound water, and the ɛb is independent of soil moisture but is still affected by soil texture and temperature, and microwave frequency. The influence of microwave frequency on the ɛb is largely manifested at Stages II and III, and is different on the real part and imaginary part. In soil samples of the same textural class, the imaginary part of the bound water dielectric constant decreases as frequency increases, while the imaginary part decreases first and then increases. Figure 6. The moisture dependence of the dielectric constant for bound water of five types of soil at frequencies of (a) 1.4 GHz, (b) 5 GHz, (c) 10 GHz, and (d) 18 GHz; the indicated moisture range for each soil extends between 0% and 20%; the first turning point of each curve is the maximum strongly bound water content (Vmaxsb), and the second turning point is the maximum bound water content (Vmaxb); the curves were drawn based on the results from the Equation (13); physical properties of soils are shown in Table 1.

Change of Bound Water Dielectric Constant with Soil Moisture
The complex dielectric constant of the bound water falls between those of ice and free water, primarily as a result of surface effect and the adsorption force it produces. The surface effect and adsorption force can be explained by the theory of electrical double-layer. The surface effect is the combined action generated by the molecules and negative charges on the surface of clay particles. Adsorption force is the comprehensive force generated by surface effect that makes strongly bound water and weakly bound water forming special properties. On the surface of clay particles, Van der Figure 6. The moisture dependence of the dielectric constant for bound water of five types of soil at frequencies of (a) 1.4 GHz, (b) 5 GHz, (c) 10 GHz, and (d) 18 GHz; the indicated moisture range for each soil extends between 0% and 20%; the first turning point of each curve is the maximum strongly bound water content (V maxsb ), and the second turning point is the maximum bound water content (V maxb ); the curves were drawn based on the results from the Equation (13); physical properties of soils are shown in Table 1.
The complex dielectric constant of the bound water falls between those of ice and free water, primarily as a result of surface effect and the adsorption force it produces. The surface effect and adsorption force can be explained by the theory of electrical double-layer. The surface effect is the combined action generated by the molecules and negative charges on the surface of clay particles. Adsorption force is the comprehensive force generated by surface effect that makes strongly bound water and weakly bound water forming special properties. On the surface of clay particles, Van der Waals force and valence force are formed by clay molecules, and electrostatic force is produced by negative charges. The adsorption force acting on the strongly bound water which is close to the surface of clay particle is the combined force of van der Waals force, valence force, and electrostatic force. The joint action of these forces stabilizes the dielectric property of strongly bound water. The adsorption force acting on the weakly bound water which is located in the outer layer of the strongly bound water is the electrostatic force, and this electrostatic force decreases exponentially with surface distance. The exponential decay of electrostatic force makes the complex dielectric constant of weakly bound water increase gradually and approach that of free water. Figure 7 shows the change of the complex dielectric constant of bound water (ε b ) with temperature at the three frequencies. Frequency dictates the change trend of ε b with temperature. At 1.4 GHz, for all three soil samples, the real part of ε b decreases as temperature increases, while the imaginary part increases with temperature. In contrast, at 10 GHz, the real part and the imaginary part of ε b change shows the opposite change with temperature increases. The real part increases with temperature, while the imaginary part decreases. At 18 GHz, the imaginary part of ε b first increases with temperature and then decreases. surface distance. The exponential decay of electrostatic force makes the complex dielectric constant of weakly bound water increase gradually and approach that of free water. Figure 7 shows the change of the complex dielectric constant of bound water (ɛb) with temperature at the three frequencies. Frequency dictates the change trend of ɛb with temperature. At 1.4 GHz, for all three soil samples, the real part of ɛb decreases as temperature increases, while the imaginary part increases with temperature. In contrast, at 10 GHz, the real part and the imaginary part of ɛb change shows the opposite change with temperature increases. The real part increases with temperature, while the imaginary part decreases. At 18 GHz, the imaginary part of ɛb first increases with temperature and then decreases.  Table 1.

Change of Bound Water Dielectric Constant with Soil Temperature
At a fixed frequency, soil texture has a slight impact on the change of the bound water dielectric constant with temperature, and significantly affects the value of the complex dielectric constant of bound water. In Figure 7, soil samples 1, 4, and 5 have 13.43%, 19.00%, and 47.38% of clay by mass, respectively. The real part and imaginary part of the bound water dielectric constant of sample 5 (with high clay content) are significantly lower than those of samples 1 and 4 (with lower clay content). As temperature changes, sample 5 also displays slightly smaller change in the real and imaginary part of its bound water dielectric constant than samples 1 and 4. The effect of temperature on soil dielectric constant occurs via a competitive mechanism [50]. Although this conclusion is based on the real part of soil dielectric constant in the Time Domain Reflection (TDR) frequency band (about 1.4 GHz), the same competitive mechanism is followed at other frequency bands. For example, at 10 GHz, at higher temperature, the thermal motion of molecules becomes more intense, bound water content decreases, and free water content increases. However, temperature elevation also leads to increase in the dielectric constants of bound water and free water. As a result, temperature rise only increases the real part of soil complex dielectric constant.

Comparison of Soil Dielectric Mixing Models
The CDC prediction with high average accuracy is very important for estimating soil moisture by microwave remote sensing, so several soil dielectric mixed models are compared in three frequency bands (Figure 8). In the range of 0-50% relative humidity, the predicted results of soil  Table 1.
At a fixed frequency, soil texture has a slight impact on the change of the bound water dielectric constant with temperature, and significantly affects the value of the complex dielectric constant of bound water. In Figure 7, soil samples 1, 4, and 5 have 13.43%, 19.00%, and 47.38% of clay by mass, respectively. The real part and imaginary part of the bound water dielectric constant of sample 5 (with high clay content) are significantly lower than those of samples 1 and 4 (with lower clay content). As temperature changes, sample 5 also displays slightly smaller change in the real and imaginary part of its bound water dielectric constant than samples 1 and 4. The effect of temperature on soil dielectric constant occurs via a competitive mechanism [50]. Although this conclusion is based on the real part of soil dielectric constant in the Time Domain Reflection (TDR) frequency band (about 1.4 GHz), the same competitive mechanism is followed at other frequency bands. For example, at 10 GHz, at higher temperature, the thermal motion of molecules becomes more intense, bound water content decreases, and free water content increases. However, temperature elevation also leads to increase in the dielectric constants of bound water and free water. As a result, temperature rise only increases the real part of soil complex dielectric constant.

Comparison of Soil Dielectric Mixing Models
The CDC prediction with high average accuracy is very important for estimating soil moisture by microwave remote sensing, so several soil dielectric mixed models are compared in three frequency bands (Figure 8). In the range of 0-50% relative humidity, the predicted results of soil dielectric mixing models have the same trend with the measured data, so the correlation coefficient between the results of each model and the measured data is very high. Considering that correlation coefficient has difficulty distinguishing the advantages and disadvantages of the model, we introduced the concept of absolute root mean square error (ARMSE) ( Table 3) to analyze the advantages and disadvantages of soil dielectric mixed model. ARMSE is the ratio of root mean square error from the modeling results and the measurements against the measurements. ARMSE = 1 n n t=1 (ε t,modl − ε t,measurement ) 2 1 n n t=1 ε t,measurement (17) where, n represents the measured number of soil moisture in the range of 0-50%. square error from the modeling results and the measurements against the measurements.  [5,10,13,15], respectively. In the model of Wang, we used 8% salt solution instead of free water.   Notes: Each ARMSE corresponds to a curve in Figure 8; average ARMSE is the average of all the models at a single frequency; Soil No. 1-5 represents five types of in Table 1.
According to the average ARMSEs of all models in three frequency bands, the average ARMSEs of the models in 1.4 GHz band are higher than 5 and 18 GHz, and the average ARMSEs of imaginary parts are obviously higher than that of real parts. This shows that the prediction difficulty of the 1.4 GHz band is higher than the other bands, and the prediction difficulty of imaginary part is higher than that of the real part. Considering that the dielectric properties of bound water are like salt solution, we analyzed the reasons of ARMSEs in different frequency bands according to the CDC of salt water and pure water ( Figure 5). The real parts of the CDC of pure water and salt water are about 80 at 1.4 GHz, and it gradually decreases to 35 with the increase of frequency to 18 GHz. The real part of CDC of bound water lies between ice and free water, so it is more difficult to predict bound water in 1.4 GHz band than in other bands. This may be the reason why the ARMSEs of the real parts of 1.4 GHz band are higher than that of 5 and 18GHz. At 1.4 GHz band, the imaginary part of the CDC of salt solution is obviously affected by the concentration. The difference of imaginary parts between pure water and salt solution can reach tens ( Figure 5). Therefore, the imaginary part of the CDC of bound water and content of bound water will significantly affect the imaginary part of soil dielectric constant, which is the reason why the average ARMSEs of the imaginary part in 1.4 GHz band are the largest and the prediction is the most difficult.
At 10 and 18 GHz bands, the average ARMSEs of the real parts of the models is around 0.1. . This phenomenon indicates that the effect of soil texture on the imaginary part of soil dielectric constant still needs to be improved. The influence of soil texture on prediction accuracy is obvious. At 18 GHz, the average ARMSE of the real part of sample 5 with the highest clay content is higher than that of the other four soils. The difference of the real part between bound water and free water at 18 GHz is the smallest, and the bound water content increases with the clay content, so this phenomenon shows that the bound water content calculated by each model is still not accurate enough. In summary, the soil complex dielectric constant model in this paper has good prediction results in all bands, especially in the imaginary part of 1.4 GHz which is used for soil moisture retrieval by microwave remote sensing [51]. As the imaginary part of the soil complex dielectric constant is a necessary parameter for calculating soil absorption coefficient, soil penetration depth, and air-soil interface reflectivity [6,30,51,52], the model in this paper can improve the accuracy of remote sensing inversion.
In order to get better results, some internal parameters of models do not conform to the actual situation. For example, in the Mironov2009 model, the real part of the dielectric constant of free water in 1.4 GHz band is 100, but it is actually 80. In the Liu model, the conductivity of free water in soil with low clay content is negative. These phenomena are inconsistent with the actual situation. Therefore, although the results of the mixed model of soil dielectric constant are good, it is necessary to deeply study each parameter. The ARMSEs of our model have a minimum value in almost every frequency range. These ARMSE values are stable and do not fluctuate greatly with soil texture and frequency. It shows that this model has strong adaptability.

Conclusions
Bound water is classified into strongly bound water and weakly bound water using the electrical double-layer (EDL) model for the clay particle surface, corresponding, respectively, to the adsorption-layer and the diffuse-layer solution. Applying this classification scheme, two models for the complex dielectric constant (CDC) of bound water and wet soil are established with microwave frequency and soil moisture, temperature, and texture as the independent variables. The models accurately describe the dielectric property of soil at the frequency range of 1.4-18 GHz and 0-50% moisture level.
The EDL structure is a representation of the microscopic structure of clay particles. The five soil samples studied have different textures but share similar physical properties of the EDL, such as surface charge density, soil solution concentration, average diffuse-layer thickness. Soil texture mainly affects the specific surface area of soil. Hence, it is reasonable to use averaged parameters of the five soil types in the construction of CDC for bound water.
The free water in the soil complex dielectric constant model has a higher content of salt than the actual soil solution. This is because free water contains a certain proportion of low-concentration cationic solution. As cationic solution is only found at the clay particle surface, higher clay content means a greater proportion of cationic solution in free water, thus higher salinity.
The complex dielectric constant of bound water falls between those of ice and free water, as a result of surface effect and adsorption force. The surface effect is the combined action generated by the molecules and negative charges on the surface of clay particles. Surface effect can produce the adsorption force. Clay molecules at the clay particle surface produce van der Waals and valence force, and the surface negative charges generate electrostatic force. The strongly bound water at close proximity to the clay surface corresponds to the adsorption-layer solution, and is predominantly under the action of the van der Waals force, valence force, and electrostatic force. Strongly bound water shows a stable dielectric property. Weakly bound water is the diffuse-layer solution and is mainly under the effect of electrostatic force. As electrostatic force decreases exponentially moving away from particle surface, the complex dielectric constant of weakly bound water increases gradually with the distance to the particle surface, eventually approaching that of free water. The nature and value of the adsorption force change with surface distance. When surface distance is less than thickness of adsorption layer, the adsorption force is essentially the resultant force of van der Waals force, chemical force and electrostatic force. When the surface distance is greater than thickness of adsorption layer and less than thickness of electrical double-layer which is the sum of the thickness of adsorption-layer and diffuse-layer, the essence of the adsorption force is electrostatic force.
Absolute root mean square error (ARMSE) is used to compare and analyze the results of existing soil CDC models. According to the average ARMSEs of all models in the three frequency band, the average ARMSEs of the models in 1.4GHz band are higher than 5 and 18 GHz, and the average ARMSEs in the imaginary part are obviously higher than that in the real part. The influence of soil texture on ARMSEs is obvious. At most frequency bands, the average ARMSE of the real part of sample 5 with the highest clay content is higher than that of the other four soils. The ARMSEs of our model have a minimum value in almost every frequency range and the ARMSE values do not fluctuate greatly with soil texture and frequency. It shows that model of this paper has strong adaptability.
The complex dielectric constant model of bound water established in this paper has limitations. This model is applicable only to fixed-charge soils possessing the similarity electrical double-layer structure. There are also shortcomings in the simulation of bound water, as the dielectric constant of real bound water changes as a smooth curve with soil moisture rather than as linear line segments. Therefore, more research should be performed on the dielectric property of bound water.