# Moisture and temperature effects on the microwave dielectric behavior of soils

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Ann Arbor, MI 48106 f MOISTURE AND TEMPERATURE EFFECTS ON THE MICROWAVE DIELECTRIC BEHAVIOR OF SOILS John Oliver Curtis Dartmouth College Hanover, New Hampshire June 1992 MOISTURE AND TEMPERATURE EFFECTS ON THE MICROWAVE DIELECTRIC BEHAVIOR OF SOILS A Thesis Submitted to the Faculty in partial fulfillment of the requirements for the degree of Doctor of Philosophy by John Oliver Curtis DARTMOUTH COLLEGE Hanover, New Hampshire June, 1992 Examining Committee: /^^j(efiairman) Dean of Graduate Studies ABSTRACT A comprehensive experimental and analytical study of the complex dielectric response of moist soils over the 100 MHz-18 GHz frequency range was undertaken. The experimental setup included a square cross-section coaxial sample holder and a vector network analyzer system for signal generation and detection as well as an external bath for sample temperature control. Soils chosen for study included a poorly-graded Ottawa sand, a clean well-graded tan sand, a clean poorly-graded silt, and a nearly-pure nonswelling clay mineral, kaolinite. Complex dielectric constant data for these soils were collected at ten different temperatures ranging from -10® C to +40° C as well as at numerous volumetric moisture contents varying from nearly saturated to nearly dry. Two model interpretations were applied to the data. One was an equivalent circuit that contained separate soil and water elements. This circuit simulated both low frequency and high frequency loss mechanisms, and the elements were arranged to represent both series and parallel electrical response of the soil constituents. The second model assumed a fractal pore structure for small pore spaces and hypothesised that the critical volumetric moisture content at which the soil dielectric response went through a transition from series to parallel behavior was equivalent to the field capacity of the soil. Testing of this hypothesis against the soils used in this study led to the conclusion that the small-pore fractal dimension of soils tends toward 2.0 for coarse substances like the clean sands and toward 3.0 for very fine substance like the kaolinite, with normal soils falling in the 2.5 to 2.7 range. i f ACKNOWLEDGEMENTS I am deeply indebted to my advisor, Prof. John Walsh, for his willingness to take me on as a student and for remaining committed to this study. In spite of the incredible amount of time and energy consumed by teaching and the quest for research support for his students, he has always been able to keep a better focus on what I was doing than L I also wish to acknowledge Dr. Lewis E. Link, Technical Director at the US Army Cold Regions Research and Engineering Laboratory in Hanover, for introducing me to Prof. Walsh and for his continued encouragement and support. The springboard for anything accomplished within this study was the recent work done by Dr. Jeffrey Campbell during his doctoral and post-doctoral studies at Dartmouth. Jeff also provided very useful and refreshing critical comments on the models presented in this dissertation. Only one who has undertaken a serious graduate study program as both a spouse and parent can understand how much the family suffers from neglect. My wife, Leigh, and children, David, Lauren, and Jensen, will never recover time lost, but they have my commitment for the future. I dedicate this dissertation to the memory of my father, John, a man with a good heart who did not live to experience Jensen. He would have enjoyed the graduation ceremony. ii CONTENTS ABSTRACT i ACKNOWLEDGEMENTS ii LIST OF TABLES vu LIST OF FIGURES viu CHAPTER 1: INTRODUCTION 1 Reasons for Studying the Electrical Properties of Soils 1 Soil moisture measurements 1 Subterranean investigations 3 Remote sensing of the environment 4 Others 5 The Complex Dielectric Constant 6 Problem Statement 9 Scope of Research Effort 10 CHAPTER 2: THE ELECTRICAL PROPERTIES OF WATER 11 A Summary of Data on Pure Water 11 Microwave Frequency Loss Mechanisms 13 Breaking of hydrogen bonds 13 The Debye relaxation model 16 The Cole-Cole relaxation model 23 iii CHAPTER 3: THE ELECTRICAL PROPERTIES OF MOIST SOILS Data on Variations in Frequency, Moisture and Temperature 26 28 The structure of clay minerals 32 Dispersive behavior in moist soils 33 Existing Data on the Effects of Moisture Content 43 The Effects of Sample Temperature on Dielectric Properties 56 Bound vs Free Water . 61 Hydrogen bonding 62 van der Waal's forces 62 Hydration of exchangeable cations 62 Osmosis 63 Radio Frequency Loss Mechanisms 64 Free Water Relaxation 65 Bound Water Relaxation 66 Maxwell-Wagner Effect 66 Surface Conductivity 67 Charged Double Layers 67 Ionic Conductivity 68 Activation Energy Data 68 CHAPTER 4: MODELS FOR SOIL ELECTRICAL BEHAVIOR 72 Mixing Models 75 Equivalent Circuits 80 Homogeneous Materials 80 Mixtures 89 Percolation Transition or Long-Range Connectivity 92 Fractal Models of Electrical Behavior 94 Modeling of Electrodes 94 Fractal Pore-Filling Model 96 iv CHAPTER 5: A COAXIAL APPARATUS FOR DIELECTRIC MEASUREMENTS 99 The Measurement System 99 The Governing Equation 102 Experimental Procedure 108 Calibration 108 Sample preparation and Measurements 110 Sanity Checks Ill Empty Holder 112 Water 114 Ethylene Glycol 117 CHAPTER 6: EXPERIMENTAL RESULTS 120 Summary of Data Collected 120 Dispersion in Soils as a Function of Moisture Content 122 Single Frequency Observations 131 Temperature Effects 131 Moisture Effects 138 CHAPTER 7: DATA ANALYSES 143 Equivalent Circuit Representation 143 Fractal Geometry Model and Critical Water Content 157 A Fractal Model of Pore-Size Distribution 158 The Fractal Model and Pressure Plate Data 159 The Fractal Model Related to Particle-Size Distribution Data 163 Fractal Model Applied to This Study 173 CHAPTER 8: CONCLUSIONS 184 APPENDIX A; SKIN DEPTH CALCULATIONS 187 v APPENDIX B: REFLECTION AND REFRACTION AT PLANE INTERFACES . 193 The Fresnel Coefficients 197 Reflection From Lossless Media 198 Reflection From Lossy Media 200 APPENDIX C: FRACTAL MODELS OF SOIL STRUCTURE 202 Fractals 202 Soil Structure 204 APPENDIX D: SOIL PROPERTIES 206 REFERENCES 211 vi LIST OF TABLES No. Page 1. Electrical property measurements 29 2. Electrical property models 73 3. Fractal cutoff moisture content vs measured field capacity moisture 172 4. Estimating CC for the soils in this study 175 vii LIST OF FIGURES No. Page 1. The complex dielectric constant for liquid water as a function of temperature (analytical models after Ray, 1972) 12 2. Point charge model of the free water molecule (after Mitchell, 1974) 14 3. Graphical representation of Debye equations inspace 21 4. Equivalent circuit for Debye model 23 5. Representative Cole-Cole plots 25 6. Equivalent circuit for the Cole-Cole model 25 7. Radar backscatter coefficient vs. moisture content (Ulaby, et al, 1974) 27 8. Dielectric constant of clays vs frequency (Smith, 1971) 34 9. Dielectric constant of soils vs frequency (Lundien, 1971) 38 10. The dielectric constant of Goodrich clay (Hoekstra and Delaney, 1974) 40 11. Dielectric constant of Suffield silty clay (Hoekstra and Delaney, 1974) 41 12. Dielectric constant of loam vs frequency (Hallikainen, et al, 1985) 42 13. Dielectric constant of Manchester silt (Campbell, 1988) 44 14. Apparent dielectric constant of a silt loam vs gravimetric moisture (Lundien, 1966) 45 15. Apparent dielectric constant of a clay vs gravimetric moisture (Lundien,1966) 46 16. Dielectric constant of a sand vs volumetric moisture (Lundien, 1971) 48 17. Dielectric constant of a silt vs volumetric moisture (Lundien, 1971) 49 18. Dielectric constant of a clay vs volumetric moisture content(Lundien, 1971) . . 50 19. Dielectric constant of soils vs volumetric moisture at 10°C (Hoekstra and Delaney, 1974) 51 20. Dielectric constant of soils vs volumetric moisture (Hallikainen, et al, 1985) . . 52 21. Dielectric constant for soils vs volumetric moisture (Campbell, 1988) 53 viii 22. Dielectric constant of soils vs temperature at three volumetric moisture contents (Hoekstra and Delaney, 1974) 57 23. Dielectric constant of soils vs temperature at several volumetric moisture contents and a frequency of 0.5 GHz (Delaney and Arcone, 1982) 58 24. Dielectric constant for two soils as a function of frequency and temperature (Hallikainen, et al, 1985) 25. Microscopic capacitive elements in clayey soils 59 63 26. Dielectric loss mechanisms for heterogeneous moist materials (after Hasted (1973)) 65 27. Dielectric loss and conductance in Na-montmorillonite as a function of temperature (Hoekstra and Doyle, 1971) 70 28. Dielectric losses at 1 MHz (after Campbell, 1988) 71 29. Several mixing formulas plotted against real data (Wang and Smugge, 1980) . . 79 30. Simple equivalent circuits 82 31. The Debye equivalent circuit and its response 84 32. The Cole-Cole equivalent circuit and its response 85 33. An equivalent circuit for impure water (von Hippel, 1954) 87 34. Water model including DC conductivity and optical permittivity 88 35. Equivalent circuit for clay film studies (Bidadi, et al 1988) 89 36. An equivalent circuit model for saturated media (Sachs and Spiegler 1964; Smith 1971) 90 37. Campbell's (1988) percolation model grid for moist soil 93 38. The "finite modified Sierpinski electrode" (Sapoval, et al, 1988) 95 39. Scaled critical water content vs pore size (Campbell, 1988) 98 40. Experimental Measurement System 99 41. One of the brass, coaxial sample holders 101 42. Sample geometry and voltage and current notation 103 43. Calibration and measurement planes 109 44. Empty sample holder measurements 113 ix 45. Dispersion curves for water 115 46. Dispersion measurements of ethylene glycol 118 47. Data collected at 20° C 121 48. Dispersion in a poorly-graded sand 124 49. Dispersion in a well-graded sand 125 50. Dispersion in silt (first set) 126 51. Dispersion in silt (second set) 128 52. Dispersion in kaolinite (continued) 129 53. Temperature effects for poorly-graded sand (8 GHz) 132 54. Temperature effects for well-graded sand (8 GHz) 133 55. Temperature effects for silt 134 56. Temperature effects for kaolinite 136 57. Moisture effects for non-frozen soils at 100 MHz, 20° C 139 58. Moisture effects for non-frozen soils at 800 MHz, 20° C 140 59. Moisture effects for non-frozen soils at 2 GHz, 20° C 141 60. Moisture effects for non-frozen soils at 8 GHz, 20° C 142 61. The three-path equivalent circuit used for this study 145 62. Equivalent circuit model-data comparisons for tan sand 148 63. Equivalent circuit model-data comparisons for silt 151 64. Equivalent circuit model-data comparisons for clay 154 65. Fractal model applied to real soil desorption data (after Arya and Paris, 1981) . 161 66. Comparison of measured and predicted pore-size distributions; 70% silty clay, 30% sandy loam (Arya and Paris, 1981) 167 67. Comparison of measured and predicted pore-size distributions; loam 40-50 cm depth (Arya and Paris, 1981) 168 68. Comparison of measured and predicted pore-size distributions; 40% silty clay, 60% sandy loam (Arya and Paris, 1981) 169 69. Comparison of measured and predicted pore-size distributions; loam 20-30 cm depth (Arya and Paris, 1981) X 170 70. Comparison of measured and predicted pore-size distributions; 20% silty clay, 80% sandy loam (Arya and Paris, 1981) 171 71. Predicted pore distribution curves for Ottawa sand 176 72. Predicted pore distribution curves for tan sand 177 73. Predicted pore distribution curves for tan silt 178 74. Predicted pore distribution curves for kaolinite 179 75. Inferred fractal dimension for tan sand 181 76. Inferred fractal dimension for tan silt 182 77. Inferred fractal dimension for kaolinite 183 Al. Skin depth as a function of wavelength, permittivity, and loss tangent 189 A2. Skin depth nomograph (Albrecht, 1966) 191 A3. Attenuation in moist soils (Hoekstra and Delaney, 1974) 192 Bl. Wave vectors at a plane interface 194 B2. Reflection amplitudes and phase shifts for non-magnetic lossless materials . . . 199 B3. Reflection amplitudes and phase shifts for non-magnetic lossy materials 201 CI. A fractal snowflake of dimension 1.5 (Mandelbrot, 1983) 203 C2. Fractal islands and lakes of dimension 1.6131 (Mandelbrot, 1983) 203 C3. The Menger sponge, fractal dimension 2.7268 (Mandelbrot, 1983) 205 C4. A fractal representation of soil fabric (Moore and Krepfl, 1991) 205 Dl. Gradation curve for Ottawa sand 207 D2. Gradation curve for tan sand 208 D3. Gradation curve for tan silt 209 D4. Gradation curve for kaolinite 210 xi MOISTURE AND TEMPERATURE EFFECTS ON THE MICROWAVE DIELECTRIC BEHAVIOR OF SOILS CHAPTER 1: INTRODUCTION New and improved methods of remote sensing have increased our understanding of our planet's origins, its resources, and those processes that contribute to its dynamic (on a large time scale) nature. Scientists and engineers from many disciplines are constantly exploring new ways to quantify the Earth's properties for their particular applications. Among these methods are measurements of electromagnetic energy in many different wavelength regimes, both passive and active. The development of small powerful sources and ultra-sensitive receivers along with data processing capabilities has fostered renewed interest in improved measurements of the Earth within the microwave region of the spectrum where wavelengths in air range from a few millimeters to several meters. Natural terrain surfaces consist of bare soils, rocks, vegetation, and water. It is with the hope of making a meaningful contribution to the understanding of microwave interactions with natural terrain that this dissertation proposal is offered. In particular, this research focuses on the measurement and modeling of the electrical properties of well-characterized soils. Reasons for Studying the Electrical Properties of Soils Soil moisture measurements The dominant factor that controls the electrical behavior of soils is the presence of water (Topp, Davis, and Annan, 1980). Obviously one would hope to take advantage of this experimental fact to develop a means of accurately and quickly measuring the moisture in soils without having to collect numerous samples in the field, weigh them, dry them for extended periods of time, and weigh them again to obtain either a 1 gravimetric (weight of water/weight of dry soil) moisture content or a more useful volumetric (volume of water/volume of soil sample) moisture content. Numerous attempts have been made to develop a useful method for measuring soil moisture content. All have met with varying degrees of success and none has proven accurate under all conditions. For example, a technique that essentially amounts to burying radar transmit and receive antennas in the soil and relating the measurements of attenuated received signals to moisture content (Birchak, et al, 1974) is destructive to the soil fabric (the way in which soil particles are arranged), is very much controlled by the fabric and the size distribution of particles, and precludes the use of the same instrument in multiple locations (Nature is not homogeneous) as well as the ability to easily repair defective equipment. Another approach for making field measurements of soil moisture taken by some researchers (that also has numerous application in the biomedical field) is that of measuring the change in fringe capacitance of an open-ended coaxial probe (Thomas, 1966; Brunfeldt, 1987; Gabriel, et al, 1986). When pressed against a soil whose properties are unknown, the resulting change in capacitance produced by the impedance mismatch is related to electrical properties through calibration relationships. Problems arising from these measurements include the need to have proper contact between the probe tip and the soil surface, the fact that the volume of material associated with the fringe capacitance is quite small (on the order of a cubic centimeter or less), and that calibration conditions simply cannot account for all of the dielectric loss mechanisms that exist in natural soils. The losses in moist soils can be highly frequency dependent over a range of several frequency decades on the electomagnetic spectrum. A recent variation on the open-ended probe measurement scheme involves the use of a waveguide section instead of a coaxial device (Parchomchuk, et al, 1990). The concern over small sample volumes can be overcome with a redesign of the open-ended coaxial probe that replaces the solid outer conductor with several pointed tines (Campbell, 1988) that allows for the probe to be pushed into the surface of soft soils. The volume of soil enclosed by such probes can easily be tens of cubic centimeters. The tined coaxial probes that have been built to-date operate in a frequency 2 range that is very much subject to the material-dependent loss mechanisms alluded to above. Research continues that is directed to better understanding these various mechanisms and to fabricate a probe that operates in a frequency range that is not subject to such material-dependent anomalies. Subterranean investigations There are many types of electrical measurements in soils (and rocks) that are used to understand what lies beneath the Earth's surface (Telford, et al, 1984). One can collect most of these methods under three headings; namely, those employing natural electrical sources, those that measure soil resistivity, and those that utilize propagating electromagnetic waves. Resistivity measurements. Resistivity data in soils are collected by injecting known currents (usually of a frequency less than 60 Hz; AC required to minimize effects of charge buildup on the probe) into the ground and measuring potential differences across pairs of nearby electrodes. Assuming homogeneous media and uniform resistivity, it is possible to calculate from the potential differences an apparent resistivity of the earth material. These measurements will be affected by the presence of water in the soil or rock, by the presence of mineral compounds that could go into ionic solution with available water, and by the physical structure of the subsurface terrain itself. This being the case, resistivity measurements are useful as a measure of subsurface water volume, the locations of mineral deposits, and subsurface structure. Electromagnetic wave propagation. Another method of making electrical subsurface measurements involves the transmission of electromagnetic waves into the soil and detection of energy that results from waves reflected from subsurface anomalies. Having the ability to track wave propagation in time, either by pulsing the source or sweeping over a known frequency band in some controlled manner, means that electromagnetic wave propagation methods of subterranean investigations are particularly useful for locating the depth of electrical anomalies such as the water table in sandy soil (Olhoeft, 1983; Stewart, 1982; Wright, et al, 1984), buried pipes or wires, or cavities 3 such as tunnels or caves (Ballard, 1983). Other applications include the delineation of stratified media (Lundien, 1972) and determination of the thickness of ice and frost layers (Jakkula, et al, 1980; O'Neill and Arcone, 1991). There are some practical bounds on the utility of radio frequency systems to conduct subterranean investigations due to the phenomenon of "skin depth", a measure of the attenuation of the electromagnetic energy as it travels through the medium. For relatively low frequency sources (on the order of 200 MHz) it can be shown that low loss soils such as dry sands can possess a skin depth on the order of 10-15 meters, while high loss soils such as wet silts and clays may have skin depths on the order of only a few centimeters. Modem radio frequency receivers are extremely sensitive devices, often having a dynamic range of 50 to 100 decibels. The signal at skin depth represents about an 8.7 dB loss in power or a two-way loss at the receiver for reflected signals of about 17.4 dB. It is certainly not inconceivable that radio frequency receivers should be capable of successfully detecting reflected signals from subterranean anomalies at depths of two to three skin depths or more. Remote sensing of the environment Virtually all remote sensing of our environment from airborne or spacebome platforms involves the measurement of electromagnetic radiation from the Earth's surface and/or atmosphere. Passive surveillance involves measurements of emitted radiation and that reflected from natural sources such as the sun, the atmosphere, and surrounding terrain. Active remote sensing measurement systems include a source for illuminating the target of interest. Whether passive or active, whether visual, thermal infrared, microwave, or millimeter wave sensors are utilized, remote sensing is the collection and interpretation of electromagnetic radiation, and as such, demands an understanding of the dielectric properties of those materials being observed. Environmental remote sensing applications form a list that grows yearly as electronic components are improved and data collection and processing hardware and software become faster, more reliable, and less costly. For example, satellites can 4 provide world-wide surveys of land-use patterns to monitor the threat of urbanization, waste disposal, and erosion of the land (Colwell 1983). Similar systems (including those mounted in aircraft) can monitor the health of vegetation to keep abreast of such things as loss of forest and the potential for food shortages. Sea traffic in the far northern and southern shipping lanes can be made safer through the use of airborne and spacebome sensors to detect ice hazards. One of the more obvious applications for microwave remote sensing devices is that of conducting surface moisture surveys to help predict ground water availability and the potential for flooding. Attempts have been made to relate soil moisture to both laboratory reflectance data (Lundien 1966) and to airborne sensor backscatter measurements (Ulaby 1974; John 1992). Careful airborne sensor measurements might provide a first approximation to the complex dielectric constant of the soil near the surface. Because of the difference in dielectric behavior of liquid water and various forms of ice, it may be possible to use airborne sensors to detect freezing and thawing (Wegmuller, 1990) in remote locations that could be used to predict spring runoff conditions and all that that encompasses for agricultural applications, the effects on the fishing industry, and the anticipation of flooding in built-up areas. Military analysts are concerned about soil moisture conditions because of its impact on trafficability, the ability of vehicles to move effectively over natural terrain. Another remote sensing application is the mapping of exposed soils and rocks in remote areas of the world from high-flying aircraft or satellites which might prove useful for geomorphological studies (Swanson 1988) or even mineral exploration. A less obvious, but recent, application of microwave remote sensing in soils dealt with archeological surveys in desert areas (Berlin, et al, 1986; McCauley et al 1986) Others While the above paragraphs emphasize some of the most obvious and useful applications of a better understanding of soil electrical properties, others have been noted 5 in the literature. For example some researchers have attempted to relate electrical property measurements to the physical properties of soils (Campbell and Ulrichs, 1969; Hayre 1970; Arulanandan and Smith 1973; Madden 1974). And, of course, nothing has been said about the military's need to better understand the microwave response of soils that form the backgrounds to military targets; i.e., when and why does clutter become a source of target-like signatures. Outside of the topic of soils, studies of the microwave response of foodstuffs has direct application to quality control concerns in the food industry (Nelson 1973; Nelson 1983). Another new application of this technology that is closely related to the discussion on soil moisture is that of detecting liquid ground contaminants, either near the surface or at arbitrary depths using a specially-fabricated probe. If, as will be argued later, polarizable liquids can be characterized by a unique frequency of peak losses due to the dielectric relaxation phenomenon, then a probe could be designed to measure losses over a frequency span that is broad enough to detect a peak loss frequency and, coupled with the results of a thorough experimental program, to identify the particular contaminant. The Complex Dielectric Constant Background information for this study would not be complete without a definition of terms that will be used. Consider, for the moment, Ampere's law written for linear, isotropic materials and for current density divided into a component due to the motion of free charges such as electrons and ions and one due to other factors such a time variation of polarization and magnetization (Gaussian units are used throughout the text. For other units see the excellent Appendix on Units and Dimensions in Jackson (1975)). (1) 6 where H = the magnetic field Jf = the current density due to free charges D = the electric displacement. If one also assumes that the magnetic contribution to free charge forces is small relative to that of the electric field, then one can rewrite (1) using Ohm' law. Jf = oE (2) where a is the conductivity of the medium. Furthermore, having assumed a linear isotropic material, one can write the electric displacement in terms of the electric field as D = €E (3) where € is the electric permittivity of the material. (It is understood that if € is frequency dependent. Equation (3) is not rigorously correct in the time domain, but rather that the electric displacement and electric field are related through Fourier transforms of the frequency-dependent permittivity (Jackson, 1975, p. 307). Adding this notational complexity would only serve to cloud the qualitative development intended within this section.) Then Ampere's law can be written as VxH = + -1-^ c (4) cot Thus, the flow of current in a medium is both proportional to the applied electric field and to the time rate of change of that field. By analogy with simple electrical circuits, permittivity is a measure of the electrical capacity or capacitance of the media, its ability to store charge. Capacitor plates in a vacuum collect and give up charge, thus, there is a permittivity of free space, GQ (unity in Gaussian units). Capacitor plates filled with a polarizable material exhibit a higher capacitance due to material polarization such that the material permittivity can be written as 7 (5) € = €o(l + 471%^) where %g is the electric susceptibility for linear materials and relates the material polarization to electric field. The value of material permittivity normalized to the permittivity of free space is what is called the dielectric constant (or relative permittivity). € = -^ = 1 + 47T:Xe (6) From this point on in the text, 6 will refer to the relative permittivity. As will be shown by the data, the dielectric constant of real materials is frequency dependent (a characteristic referred to as "dispersion"). At relatively low frequencies, polarizable particles, which are effectively electric dipoles, are able to rotate or align themselves with the changing electric field with little loss in energy. However, as will be discussed later, there are several other mechanisms at work that do result in electrical energy losses, particularly at frequencies less than 100 MHz. As the frequency goes up, the dipoles cannot keep up completely with the electric field, resulting in a phase shift in current that peaks in the 10-20 GHz frequency range and is quite temperature dependent. Finally, as the frequency goes even higher, the dipoles do not even respond, resulting in a reduction of real capacitance to its optical value which is due to electronic polarizability (Feynman, 1964). This frequency-dependent loss mechanism can be modeled by a complex dielectric constant € = where, , reflects the phase lag in the particle motion. Finally, then. Ampere's law could be written as Vx5 = + \ c dE C/ d t (8) If one further assumes that the material is subjected to a sinusoidal electric field 8 (9) then (06 Vx# = 15° ^ c C (10) c The first two terms on the right hand side of Equation 10 represent the conductive nature of the material, the flow of energy that is associated with losses. The third term is a measure of its energy flow due to its polarizability. All of the terms within the square brackets are collectively referred to as the admittance of the material, being the ratio of current to voltage. The first two terms represent the conductance of the material, and the last term represents its susceptance. It is clear, then, that (coe'V 4tt:) can be thought of as something like a dielectric conductivity. It appears that electrical property measurements reported in the literature seldom, if at all, distinguish between the two physically different loss mechanisms. In fact, the terms, , and , 4ito/(*) , are often used interchangeably. This observation, coupled with the fact that the engineering community uses a notational and sign convention for dealing with dielectric properties that is different than that used by physicists and chemists, makes reading of the literature often quite burdensome. Problem Statement The overall objective of this research is to measure and model the dielectric response of moist soils in ways that will test the current understanding of loss mechanisms over a broad range of moisture levels, frequencies, and material temperatures and perhaps suggest new ways of thinking about losses as a function of these variables. Of particular interest is the issue of long-range electrical connectivity of water in the soil-water-air mixture that has been proposed as the mechanism for an abrupt 9 change in dielectric response of a given soil between low and high volumetric moisture contents (Campbell, 1988). What is first required to help achieve this objective is an apparatus that will allow for the measurement of attenuation in moist soils over a broad spectrum of frequencies while maintaining a required sample temperature. Numerous samples of the same soil will be tested at different volumetric moisture contents to make certain that any transition in electric response due to increasing moisture content can be observed. Two approaches to modeling soil electrical behavior will be exercised and compared to experimental results. One of these models will be an extension of a previous fractal approach to explaining how long-range connectivity can occur at saturation levels well below 100 percent (Campbell, 1988). The other will be an equivalent circuit representation of moist soil behavior for which the circuit parameters will be related to physical properties of the soil/water/air mixture. Scope of Research Effort Because so much of the electrical response of moist soils is due to the presence of water, Chapter 2 of this dissertation describes what is known about the behavior of pure water. Chapter 3 summarizes what past investigations have discovered about the electrical properties of moist soils and will include several examples of others' experimental data. Chapters 4 and 5 deal with a description of existing models and a description of the experimental setup used in this study. Chapter 6 is a summary of data collected, and Chapter 7 discusses model interpretations. The main body of the text is concluded in Chapter 8 with a summary of conclusions drawn from the analysis of those data. Several appendices are included that contain calculations of electrical skin depth, a discussion of plane interface reflection and refraction phenomena, a brief description of fractal geometry concepts, and a description of the properties of soils used in these studies. The report closes with a listing of supporting journal articles and textbooks. 10 CHAPTER 2: THE ELECTRICAL PROPERTIES OF WATER Certainly, over the range of the electromagnetic spectrum from D.C. to millimeter waves, the single most important factor in determining the electrical behavior of soils is the presence of water. Whether as a solvent that provides a medium and path for ionic conduction or as particles that bond to clay changing the electrochemical characteristics of the soil, or as a source of dipoles that lead to high frequency residual dielectric conduction, water has a major impact on how the moist soil interacts with electromagnetic fields. For these reasons, a study of the electrical properties of moist soils should include a study of the electrical properties of water. A Summary of Data on Pure Water The available literature on electrical measurements of water show a consensus on several points. First is that the so-called static (frequency < 100 MHz) dielectric behavior of pure, deionized water is very well defined over a broad range of temperatures (Hasted, 1973). Second is that there is a loss mechanism (dipole relaxation) at higher frequencies (greater than 1 GHz) that can be modeled in a rather simple fashion and for which data is highly repeatable (Kaatze 1986). Third is that fresh water exhibits low-frequency conductivity losses. Finally, sea water and saline solutions have low frequency losses that are a couple of orders of magnitude higher than fresh water. There seems to be little, if any, information on the electrical behavior of pure, deionized water in the frequency range between 100 MHz and 1 GHz. Figure lis a graphical representation of the complex dielectric constant for liquid fresh water as a function of temperature. These curves were plotted from empirical relationships developed by Ray (1972) for which excellent correlation between the empirical fits and actual data was clearly demonstrated. A description of the models for losses at higher frequencies follows. 11 100 10Q • 80 • \\ 70 € / 6 a 7 ID 9 11 LOG(F) " Hz TEMP 0 20 40 Figure 1. The complex dielectric constant for liquid water as a function of temperature (analytical models after Ray, 1972). 12 Microwave Frequency Loss Mechanisms Breaking of hydrogen bonds The precise physical structure of liquid water is unknown. We accept a physical model of free water molecules (H2O) as consisting of two hydrogen atoms bonded to an oxygen atom in a V-shape with an angle of 104.5 degrees formed by lines through the hydrogen protons and meeting at the center of the oxygen nucleus. Viewing such a molecule along the normal to the plane through the bond angle using the point charge model of the water molecule shown in Figure 2, one can compute that the free water molecule has a dipole moment with a value of 1.83x10"" electrostatic units. The molecule has two positively charged "comers" at the hydrogen protons and two negatively charged "comers" at the electron pairs which are not shared. Hydrogen bonds are formed when one of these positive comers bonds to the negative comer of a neighboring molecule with the resultant sharing of a hydrogen proton. Each free molecule, therefore, has the capability of having four neighboring molecules bonded to it. In fact, this model might predict that all of the H^O molecules might be bonded together. Of course this can't be so because of the liquid nature of water. This is not even true for ice (Hasted 1973). Further experimental evidence is needed to help postulate a reasonable pure liquid water physical stmcture. Given that the individual water molecules are dipoles (which is supported by the fact that pure water does exhibit a relatively high static permittivity of around 80), then one could reasonably assume that the dominant high-frequency loss mechanism shown in Figure 1 is, in fact, associated with the dipoles not being able to keep up with the rapidly changing electric field. Switching off a static electric field applied to a sample of water would produce a finite, albeit small, delay in the material returning to an unpolarized state (relaxation). Such behavior could be modeled by P a (11) where 13 I P = the polarization of the material t = time T = a characteristic of the material called its relaxation time. e = electronic charge +e +e -2e Oxygen . Nucleus -2e Distance from oxygen nucleus to I H'-H" « 0.97 . C-C* - 0.374. a-a* • 0.111. Distance z Figure 2. Point charge model of the free water molecule (after Mitchell, 1974). 14 The rate of change of polarization is then dt a —P T (12) Carrying this argument a bit further, assume that the relaxation process behaves as a temperature dependent chemical reaction for which rate of reaction = ic[S] (13) where k = the rate constant, or the probability of the reaction taking place. [S] = the concentration of reacting material Now, Arrhenius postulated that the rate constant goes like (Grimshaw, 1971) where A = the activation energy, or the energy needed to overcome some equilibrium state R = the universal gas constant T = the absolute temperature. Drawing parallels between the hypothesized time rate of change of material polarization and Arrhenius' rate of chemical reaction by thinking of polarization as concentration and the inverse of the relaxation time as the rate constant, one could write that (15) •u or (16) 15 Then if measurements of l/t (radial frequency at peak loss) as a function of temperature plotted semilogarithmically as a straight line, one could estimate the activation energy required for that process to take place from the slope of the curve. Ray's empirical fit for liquid water includes a relationship between the relaxation wavelength and temperature that looks just like equation 15. A simple calculation results in an estimate for activation energy of water to be about 5.4 kcal/mole which is about the energy required to break a hydrogen bond (Cotton and Wilkinson, 1972). In other words, hydrogen bond breaking becomes a likely candidate for describing the relaxation process in liquid water, and therefore, an indication of water structure. The Debye relaxation model The experimental data on the electrical behavior of water clearly demonstrate a frequency-dependent response or an anomalous dispersion within the 1-100 GHz frequency range. The preceding section pointed out that the structure of liquid water is some collection of permanent dipoles; i.e., particles subject to realignment in the presence of an oscillating electromagnetic field. An obvious analogy found in classical mechanics for an oscillating particle whose behavior changes with frequency is the harmonic oscillator. What follows is the development of a simple model for the dipole relaxation loss seen in liquid water based on the harmonic oscillator. Using the notation and following (with some extension) the treatment of optical dispersion in materials presented by Reitz, Milford, and Christy (1980), consider the dipole to behave like a one-dimensional, damped, forced oscillator where the positive charge, e, moves and the negative charge remains fixed. 16 (17) where X = a measure of the displacement G = a viscosity coefficient C = a spring resistance coefficient e = the value of charge at each end of the dipole Eta = the local electric field. If the driving force was zero and the particle was given some initial displacement, Xo, then x{t) = XQG ct ® = XQG t 1 (18) which is consistent with the previous description of relaxation behavior and activation energies. Now assume that both the driving force (the local electric field) and the resulting displacement vary sinusoidally with amplitudes Ag and Ax, respectively, and that the inertia forces are negligable. One can then write the equation of motion as + 471 vx) (i*) or (20) For the single dipole, the dipole moment is eA^. Therefore the polarization of the material, P, which is the number of dipole moments per unit volume and is proportional 17 to the amplitude of the driving force through the susceptability, % , goes like P = NeA^ = (21) where N = the number of dipoles/unit volume. Then one has Ne^ (22) which shows that the electric susceptibility of the medium is both complex and frequency dependent. But we want to model this behavior in terms of the dielectric constant. One must then pose the relationship between dielectric constant and susceptibility as € = €„ + 471% (23) where €„ replaces the unity term in the usual definition for linear dielectrics and represents the high frequency limit on the real part of the dielectric constant. Combining the last two expressions, one can now write Ne^ (24) 1 - icOT Referring to the static (CO = 0) dielectric constant as Eg , one then has that 18 f or finally, Recalling that € = the above expression can be solved for e' = and , gp ~ €« + (27) 1 + (tot)' g// = (€o - €.) WT (28) 1 + (wty These are the often-referenced Debye equations (Debye, 1929) for modeling the dielectric behavior of materials made up of polar molecules. Plotted on semilogarithmic scales, these equations approximate the anomalous behavior shown for liquid water on Figure 1. At (OT = 1, €''Ms maximized, which says that the maximum loss due to the dielectric relaxation mechanism occurs at a frequency equal to the inverse of the relaxation time. Eliminating (*) x from these equation allows one to write the expression J _ (ep + e.) [e'f = Thus, the Debye equations, when drawn in ^0 - €. (29) space result in a circle centered at 19 (€o + €00) 2 ' e" = 0 and having a radius of {e^ - €„)/2 . Such a plot, shown in Figure 3, is known as Cole-Cole diagram. Three different relaxation frequencies representing the three temperatures shown in the figure were chosen based on Ray's empirical model. One should further note that at O) =1/t (the inverse of the relaxation time) = e + (ep - e.) 2 + e. 2 and g// = ^0 " i.e., the peak relaxation loss is represented by the point at the top of the Cole-Cole diagram. 20 IDD A TBiP D — — — 20 — 40 / / \ I 0 I 1 I I I 10 I I I I I 20 » I I I I 3D I I I >( 40 I I 1 I I SO I I 1 I I 60 I 1 I I I 1 I I 1 70 Figure 3. Graphical representation of Debye equations in 21 I I 90 I I I I I I 1 I 1 90 space. I I 100 What has been demonstrated in the preceding paragraphs is that a simple mechanical model can do a very good job in representing the high-frequency dispersive dielectric behavior of liquid water. Of coarse a simple electrical analog can also be develops. In terms of simple circuit elements, the equivalent circuit that precisely models the Debye equations is shown in Figure 4 (Cole and Cole 1941) where € = e'+ie^' is taken to be the equivalent capacitance of the circuit. Simple models like this will prove to be very helpful in analyzing the behavior of the complex dielectric constant in moist soils. O a Figure 4. Equivalent circuit for Debye model. 22 The Cole-Cole relaxation model Cole and Cole (1941) observed that a good bit of experimental data on both polar liquids and polar solids were not fit by the semicircular €^/ space predictions of Debye's equations. Rather, it seemed that over large frequency intervals, the data seemed to be best fit by circular arcs; i.e., pieces of circles in 6% space that were centered below the €^'=0 line in the manner shown in Figure 5. These curves on Figure 5 were derived from Ray's empirical model for which artificially large alpha values were selected. The vertical lines on the figure reflect the contribution of the conductivity term added to the Cole-Cole model by Ray and were not part of the original development. They do, however, give an indication of what real material responses may look like. Using a series of geometrical and analytical arguments, the authors showed that linear materials that behaved in the manner shown in Figure 5 can be described by the relationship e - = —5°—— oo) 1 - (iWT)^"" Expressions for e'' and i-a = found by substituting = cosl^\ - 2 / \ 2 are those slightly modified by Ray (1972) to produce the curves in Figure 1. 23 100 ' 90 80 a = 70 -16.8129 + 0.25 (T + 273) 60 50 TBiP ---- 0 — — — 20 —— 40 40 30 20 N \ \ 10 > \ a%/2 = 0.196 lad = ' ' ' "^1-1 'J M J'Jg 10 20 I 30 I I 40 I I I I I 50 I I I I I I I I I I 40" C I I I GO Figure 5. Representative Cole-Cole plots. 24 I 1I 'I 'I 'I 'I 1I I I I I 8D 9D Furthermore, it is clear from the form of equation 30 that a simple equivalent circuit can still be drawn for materials that follow the behavior described above. Figure 6 shows that the resistance in the Debye equivalent circuit now becomes a complex impedance. O T(i(OT)"" ^0 o Figure 6. Equivalent circuit for the Cole-Cole model. 25 CHAPTER 3: THE ELECTRICAL PROPERTIES OF MOIST SOILS A quick review of available dielectric data on dry soils and wet soils leads to the conclusion that the dominant factor that controls the electrical behavior of soils is the presence of water. Over a broad frequency range, the real part of the complex relative dielectric constant (referred to in this text as permittivity) of dry soil minerals changes very little, covering a range of values from about 2 to about 6 (Nelson, Lindroth, and Blake 1989; Ulaby, et al 1990). However, when considered from the perspective of the idealized electromagnetic wave reflection phenomenon for lossless materials as described in Appendix B, this can translate into reflection coefficients that vary from -0.17 to 0.42. In terms of power (which is proportional to radar backscatter coefficients) these small variations in dry soil properties result in reflectances that span values from 0.029 to 0.176, or an increase of about 8 dB. When water is added to the soil fabric, further substantial changes in reflectance at the soil/air interface can take place. Take the permittivity of water to be about 80 (at low frequencies). Then for normal incidence electromagnetic waves and lossless media, one can calculate a reflection coefficient of about -0.8 and a reflectance of about 0.64. Now, take the permittivity of a very moist soil to be about 20. Then under ideal conditions, its reflectance at normal incidence could be as much as 0.4. In other words, a very moist soil could theoretically reflect more than thirteen times the power (or more than 11 dB) than can the driest soils. While airborne scatterometers may easily see this much variation in measurements over large areas of so-called homogeneous terrain due to surface roughness effects and interference phenomena, these differences in electrical properties of soils could be quite significant for close-up measurements with ground-based electromagnetic devices. As an example of the effect of soil moisture on real-world situations, consider the data reported by Ulaby, Cihlar, and Moore (1974) which was collected with a groundbased radar scatterometer located in an unplanted, plowed field of clay loam soil. As shown in Figure 7, the power returned to the scatterometer while looking nearly straight down at the soil surface (0 degree incidence) increased by 20 dB as moisture contents increased from 4.3% to 36.3% (average values in the first five centimeters). 20 dB means a 100-fold increase in power. The arguments above would indicate an anticipated bound of about 11 or 12 dB increase. It is possible that surface roughness affects could account for the larger field measurement numbers. 20 16 Frequency 4.7 GHz Polarization VV Incidence Angle 0° 10° - 30° CQ 12 TD 8 S i 4 8 o t o f -4 -8 -12 -16 4.3 15.8 24,0 30.2 Percent Moisture Content by Weight, 36.3 40.0 Figure 7. Radar backscatter coefficient vs. moisture content (Ulaby, et al, 1974). 27 Data on Variations in Frequency. Moisture and Temperature At this point it would be prudent to assess what is already known about the electrical behavior of moist soils. What are the conditions under which data have already been collected? What kinds of soils have been studied? Is there general agreement on results of previous measurements? Have the measurements led to any insight as to the loss mechanisms involved? While the governing equations for the propagation of electromagnetic waves in linear, homogeneous, isotropic media have been accepted since the late 1800's, only in the last 30 years or so has any serious attention been given to the complex dielectric response of heterogeneous mixtures such as soils. Nevertheless, all measurements of soil electrical properties are still interpreted in a macroscopic sense as if the soil is truly homogeneous and isotropic. The following table identifies some of the most relevant work on measuring the electrical response of materials that ultimately apply to this study on the response of soils. No such list could ever hope to be complete, but it does serve to provide an historical perspective and a proper point of departure for the work to be conducted in this and future studies. Paragraphs that follow will highlight several of these contributions. 28 Table 1 (continued) Electrical Property Measurements Authorfs") Year Mat'lfst Freqfs) Lane, Saxton 1952 water, alcohols 9.35,24, 48 GHz Grant, et al 1957 water .58,1.74,3,3.65 coaxial Line 9.3,23.8 GHz Cole-Cole model fits Lundien 1966 sand, silt clay .3, 9.4 34.5 GHz depth-of-penetration studies de Loor 1968 moist organic substances 1.2,3,3.8,6 8.6,9.4,11.5 16 GHz Carroll, et al 1969 desert soils 0-3 Hz resistivity array .0001-100 MHz capacitance cell Campbell, Ulrichs 1969 lunar soil 100 Hz-1 MHz capacitance cell Saint-Amant, Strangway 1970 dry rocks, 50 Hz-2 MHz powdered rocks capacitance cell Maxwell-Wagner losses observed in dry minerals Lundien 1971 sand, silt clay free-space transmission established correlations between moisture content and permittivity 1.07-1.5 GHz Techniquefs') waveguide reflectance Motivation/Interpretation alcohols and water are polarizable bound water vs free water studies; develop test materials with known complex electrical properties Table 1 (continued) elevated temperature behavior Table 1 (continued) Electrical Property Measurements Authorfs) Year Mat'lCs) Freqfs) TechniqueCst Motivation/Interpretation Hoekstra, Doyle 1971 Na-mont 100 Hz 9.8 GHz Smith 1971 kaolinite 2-60 MHz illite montmorillonite Nelson 1972 fruits 8-12 GHz Strangway, et al 1972 lunar soil 100 Hz-1 MHz capacitance cell impact of residual moisture Olhoeft, et al 1974 lunar soil 100 Hz-1 MHz capacitance cell behavior at elevated temperatures Olhoeft 1977 illite-rich permafrost 10 Hz-1 MHz multiple loss mechanisms in unfrozen water Hall, Rose 1977 kaolinite .0002-10 MHz capacitance cell attributed peak losses to Debye relaxation mechanism Topp, et al 1980 soils, glass beads 1 MHz-1 GHz time-domain reflection moisture is the dominant factor Delaney, Arcone 1982 silt, sand .1-5 GHz time-domain reflection looking for attenuation properties slotted waveguide low frequency free charge losses; high frequency mechanisms: proton mobility, dipole rotations, H-bond ruptures capacitance cell Maxwell-Wagner loss mechanism slotted waveguide moisture content relationships capacitance cell Table 1 (concluded) Electrical Property Measurements Authorfsl Year Mat'lCs) Freqfs) Technique's) Waite, et al 1984 silt 1.5, 6 GHz Hallikainen, et al 1985 silts, sandy loam 1-2, 4-6 GHz waveguide trans, select freq. free-space trans. from 4-18 GHz soil texture is a factor; liquid water exists at sub-zero temperatures Pissis 1985 cellulose DC thermal depolarization attempts to distinguish free from bound water response El-Rayes, Ulaby 1987 vegetation .2-20 GHz open-ended coaxial rapid data collection (fringe capacitance) Bidadi, et al 1988 Na, Li-mont 30-100 Hz capacitance cell low-water content Maxwell-Wagner behavior Campbell 1988 sand, silt, clay .001-1.5 GHz coaxial probe, resonant cavity low-freq ionic cond. losses; fractal long-range connectivity Nelson, et al 1989 mineral powders 1,2.45,5.5, 11.7,22 GHz slotted waveguide observed some dispersion Ulaby, et al 1990 dry rocks .5-18 GHz open-ended coaxial non-dispersive permittivity; (&inge capacitance) loss factor decreases bistatic reflectance Motivation/Interpretation reflectance increases w/ moisture The structure of clav minerals Although this section may seem out of place, it is important to set the stage for discussions of soil electrical behavior by briefly addressing the structure of the soil elements that probably are most influential in determining their electrical response; namely, the clay minerals. Clay minerals are layered silicates whose fundamental building blocks are tetrahedral sheets in which the tetrahedra are linked at their comers and octahedral sheets in which the octahedra are linked along their edges (Moore and Reynolds, 1989). Each tetrahedron is formed from four oxygen ions normally surrounding Si"*"^ cations (which can also be replaced by Al^^ or Fe'^ cations). Each octahedron is formed from six oxygen ions (or hydroxyl ions) surrounding a cation which is normally either AP^, Mg^"^, Fe^+, or Fe^+. Most clay minerals fit into structural classifications referred to as 1:1 or 2:1. A 1:1 structure is comprised of a tetrahedral sheet joined to an octahedral sheet. The mechanism for this bonding is that the apical oxygens of the tetrahedra replace two out of every three anions in the octahedral sheet. The 2:1 structure is generated by a second tetrahedral sheet bonding to the opposite side of the octahedral sheet. Slight differences in sheet dimensions (or anion spacing) due to various combinations of cations can cause distortions of the layered structure, sometimes so severe as to result in tubular geometries. Clay mineral layers formed by the tetrahedral and octahedral sheets are sometimes electrically neutral but most often are somewhat negatively charged due to the substitution of lesser valence cations for Si"*"^ in the tetrahedral sheet and/or Al^+ in the octahedral sheet. Furthermore, because of the finite lateral dimensions of clay mineral crystals, there are unsatisfied bonds at the edges of the layers that also result in layer charge imbalance. The following simplified descriptions of the clay minerals that will be referred to in this study can now be given. Kaolinite is a 1:1 structure mineral which, if in a very pure form, will have little or no layer charge due to cation exchange but will attract ions or polar molecules such as water to its edges. Illite, montmorillonite, and hectorite are 2:1 minerals. There seem to be many opinions as to what illite really is in terms of a structural formula, but it is generally accepted that it is a mica material whose layers have a half-unit-cell charge imbalance of about 0.8 and whose interlayer spaces are occupied by cations (usually potassium) whose spatial distribution nearly balances the layer charges. Montmorillonite and hectorite have smaller layer charge imbalances and the interlayer cations can attract water which results in a swelling of the layered structure and the creation of hydrogen bonds between the water molecules and the anions of the tetrahedral sheet surfaces (Moore and Reynolds, 1989). These swelling clays have a greater affinity for water which should cause much different electrical responses than for the non-swelling kaolinite. Dispersive behavior in moist soils Smith (1971) conducted a series of tests on the electrical behavior of saturated clays up to a maximum frequency of about 60 MHz using a capacitive bridge measurement apparatus. He looked at three distinct clay types: a montmorillonite, an illite, and a kaolinite. A representative set of Smith's measurement results is given in Figure 8 in terms of the real part of the dielectric constant and conductivity vs. frequency. Of particular interest is the observation that clays with higher basic structural water content have larger complex dielectric constants than those with less water. No conclusions should be drawn from consolidation differences as density should clearly affect the dielectric properties of the basic mineral themselves by providing varying degrees of particle contact and, hence, varying electrical path lengths through the matrix. Lundien (1971) reported permittivity and conductivity measurements over a range of frequencies between 10 MHz and 1.5 GHz. At the lowest frequencies he used a capacitive bridge setup. Data at about 300 MHz was collected from radar reflectance measurements. The higher frequency data were collected using what was referred to as a microwave interferometer but was, in fact, a free-space transmission measurement apparatus with the sample faces tilted to the radar path. IN 4 Hontnniilloflite; #/e = 457 Alllite; «/c°il 160 O KMlinile: w/e = 60 140 £• o d 100 A 2 3 S 7 10 U 20 30 SO 70 FREQUENCY (HHz) UOSOO OjOOlSO OJOOOSO FREQUENCY (Mb) (a) For different clay minerals Figure 8. Dielectric constant of clays vs frequency (Smith, 1971) (continued). 34 O Paste. #'c - 159* 0 Oj = 5.0 n Ki 'em^ , w,e = 42* <r^ -15.0 K( cn^ , w 'c:32t I ft s * \, V 1 N" \\ 1 \1 \ "s > 1N < V 10 15 20 1 3 0 4 0 5 0 70 100 FREQUENCY (MHz) —r 1 1 1—r . • c E isn 4 • "i«us Kt M?. w.e = M% O"i * SiO Kg . a'c:42% o oj «15.0 Kg/oif. w. c = 32% OMIS &00M 0.0012 0.0010 0.0000 10 15 20 » 40 50 70 FREQUENCY (MHz) (b) For illite with different moisture contents Figure 8. Dielectric constant of clays vs frequency (Smith, 1971) (continued). 35 i U U X 3 0 4 0 S 0 70 100 FREQUENCY (KHz) mi ANa* lllito; w/c = t2 fl.OOU o K* lllita; w/t'Sl O u* '• = (0 4 yA I 1 OJNU // /* o OJOU LOOM ojom ir' J IS ' / y 1 / • y a 30 70 FREQUENCY (HHz) (c) For illite with different cations Figure 8. Dielectric constant of clays vs frequency (Smith, 1971) (concluded). 36 Lundien's results are shown in Figure 9 for three types of soil: a very poorly graded sand, a well-graded silt, and a high clay content locally-available soil (50% of particles by weight < .0075 mm). Conclusions are difficult to draw because the data include a broad range of moisture levels. The Long Lake clay is described by Lundien as being composed mostly of montmorillonite particles. A very nice set of moist soil electrical measurements over a broad range of frequencies were conducted by Hoekstra and Delaney (1974) using a 7mm coaxial line for frequencies up to about 3 GHz and a series of slotted waveguide devices from about 5 GHz to about 20 GHz. Copies of their reported data as a function of frequency are shown in Figures 10 and 11 for two different clays. Figure 10 (a) shows data for one gravimetric moisture content and two temperatures, while Figure 10 (b) shows data for on temperature and two moisture conditions. Qualitatively they observed a Debye-like dispersion in the moist soils but at a lower frequency than for bulk liquid water. They also observed that increasing temperatures reduced the frequency of maximum dielectric loss (which is inconsistent with the response of water) as did increasing water content. They further attempted to fit a modified Debye relaxation equation to the data for one of the clays (Figure 11) with partial success. They concluded that there was no difference between the relaxation of water in sandy soil and that for clays which contradicts the thesis that chemical bonding of the water dipoles to clay particles should cause a shift of the relaxation frequency. 37 ^0 3.088 0/CM^ w A( VOLUMETR C WATER COhiTENT RANG MOISTURE CONTENT RA MCE: 4.0 TO o \ 0 . -4—- 5 1—0 a. YUMA SAND 3 VOLUMETR IC WATER CO NTENT RANC E: 0.048 TO 0.077 G/CM*® MOISTURE CONTENT RA NGE: 4.7 T() 5.8 •/. ) 1 I 1 0 \ % 8 1 8— 0 b. OPENWOOD STREET SILT 0 VOLUMETR C WATER CO NTENT RANG E: 0.103 TO D. 120 G/CM' MOISTURE CONTENT R ANGE: 7.9 TC 10.5% 1 O c L \ / % 0 • — OO — - 0.6 0.6 1 1 -1-8- ^ 0 1.0 FREQUENCY, GHz c. LONG LAKE CLAY Figure 9. Dielectric constant of soils vs frequency (Lundien, 1971) (Continued). 38 O O \ 1 1 O C \ oo 0 •- 0.02 0 0 / VOLUMETR IC WATER CC NTENT RANG E: 0.054 TO 0088 G/CM* MOISTURE CONTENT R>INGE: 4 .0 TO 5.5 V. ) 0 g. YUMA SAND 0.06 I D__0 O D ) CCD —_ — ) O \ \ 0.02 "-0 o2 \ 8 O X 0 04 0 0 0 VOLUMETR C WATER CO NTENT RANC,E: 0.048 TO 0.077 G/CM^ MOISTURE CONTENT RA NGE: 4.7 TO 5.8 b. OPENWOOD STREET SILT 0.20 0^^^ — CD \ § OO VOLUMETR IC WATER CC INTENT RANC E: 0.103 TO 0.120 C/CM^ MOISTURE CONTENT R/» NGE: 7.9 TO lO.S •/. D n 3—--8 8 8 Oy 0 3 0.6 0.6 1.0 1.2 FREQUENCY; GHz c. LONG LAKE CLAY Figure 9. Dielectric constant of soils vs frequency (Lundien, 1971) (Concluded). 39 14.0 12.0 iOO 8.0 K » ">P 6.0 Oo o +24 -10' 10' (a) 10% gravimetric moisture p.io K 0,05 0.10 •. J A L 0.05 —. I I !• I i Frequency. Hz (b) 24°C Figure 10. The dielectric constant of Goodrich clay (Hoekstra and Delaney, 1974) 40 12.0 10,0 6,0 2,0 . Frequenay, Hz (a) 10% moisture, 24°C 14.0 12.0 • 10,0 K * 8,0 6,0 • 2.0 K: 10" Frequency, Hz (b) 10% moisture, -10°C Figure 11. Dielectric constant of Suffield silty clay (Hoekstra and Delaney, 1974). 41 Another extensive set of experimental data was collected by Hallikainen, et al (1985) using waveguide transmission techniques in the 1-2 and 4-6 GHz bands and a free space transmission technique at eight selected frequencies between 4 and 18 GHz. They conducted measurements on soil types ranging from sand loam (51% sand, 13% clay) to a silty clay (5% sand, 47% clay). Figure 12 contains their published results for one soil (42% sand, 49.5% silt, 8.5% clay) collected at an ambient temperature of about 23°C, ' 0.374 cm' cm' Field 2: Loam Temperature: 23°C Free-Space Method: 3 - 1 8 GHz Waveguide Method: 1.4 GHZ * Field 2; loam Temperature: 23®C Free-Space Method: 3-18 GHz m. • 0.368 cm' cm' Waveguide Method: 1.4 GHz m." 0.368 cm' cifl? fflv* 0.374 cm' cnR, 6 6 8 10 12 Frequency ((GHzl. 8 10 12 Frequency ((GHzl Figure 12. Dielectric constant of loam vs frequency (Hallikainen, et al, 1985). 42 Another set of frequency-dependent data is that collected by Campbell (1988) using a tined coaxial probe and a device for precisely controlling sample moisture content and sample temperature. Frequency was limited to 50 MHz for these measurements. The objective of these studies was to examine the low frequency dielectric loss mechanisms in moist soils. Figure 13 is a summary of Campbell's measurements for one type of silty soil that shows the low frequency dispersion of this soil as a function of volumetric water content. Obviously, one is not able to ascertain the changes in Debye type relaxation because of the frequency limitations of these data. Of particular note was the conclusion drawn by Campbell that the predominant loss mechanism in this frequency range was that of ionic conductivity and not the Maxwell-Wagner effect hypothesized by Smith (1971). These mechanisms will be discussed in a later section. Existing Data on the Effects of Moisture Content Because of the obvious applications of remote sensing technology to the determination of soil moisture levels, a good deal of data has been collected and reported in the literature. An early example is that of Lundien (1966) in which he used four different radar scatterometers and large specially prepared soil samples to measure the apparent dielectric constant. "Apparent" in this context means that the dielectric constant was computed from the square of the Fresnel coefficients (Appendix B) assuming no imaginary components (no losses). Lundien's results for two soil types are shown in Figures 14 and 15 in which he suggests an exponential fit might be in order. The data are limited in quantity because of the difficulty of sample preparation and, as such, do not reveal what later researchers believe to be a bilinear type of response. 43 30 ! • MAN 31.10 RE • MAN 25.3 RE a MAN 20.01 RE « MAN 14.58 RE • MAN 9.43 RE • MANORE A MAN 4.83 RE 20 - BEtotonagaBBmomaBainnEEtoaBBQma I 0 1 5 10 H *«#» ******** * * * *^***^<*****!.>*** til i k cpc 10' • • • A • *>• °CL MAN 30.1OIM MAN 20.01 IM MAN 9.43 IM MAN 4.83 IM 0 IM MAN OIM 8 10^ 0 1 5 I loO I i^nnD I O™' 10" —: 10 1 1 1 20 30 FREQUENCY (MHz) 40 50 Figure 13. Dielectric constant of Manchester silt (Campbell, 1988). 44 50 A 1 0.3 GHz 0/ - 1 /°/ 40 5 9 GHz 30 - 20 / /o / —/ ^ / / / / / % ' Z s 8 20 30 40 so 60 • A 10 20 a. P —BAND 30 40 50 MOISTURE CONTENT, % io MOISTURE CONTENT, % b. C-BAND w I(t : / / ^.30 9.4 GHz I 20 a. /6 < y / OJ / / / 34. 5 GHz " z O / / ^ O / / 9 / 10 20 30 40 50 60 10 20 30 40 MOISTURE CONTENT, % MOISTURE CONTENT^ % c. X"BAND d. Ka-BAND 50 Figure 14. Apparent dielectric constant of a silt loam vs gravimetric moisture (Lundien, 1966). 45 60 50 X o 0 40 5.9 GHz - 30 0.3 GHz _ y 0 0 o-p iU— > HZ § 10 20 30 40 50 60 MOISTURE CONTENT, % 0 S 1o a. P-BAND 9 4 GHz - 0 10 20 30 40 50 60 MOISTURE CONTENT, % • b. C-BAND k QC 3/ # 0C GHz - ^ 30 Q. < O^ 0— n / o 0 X 0 c % 20 30 10 40 20 30 40 MOISTURE CONTENTj % MOISTURE CONTENT, % d. Ka-BAND c. X-BAND Figure 15. Apparent dielectric constant of a clay vs gravimetric moisture (Lundien,1966). 46 50 60 Lundien's later work (1971) using free space transmission measurement techniques is summarized in Figures 16, 17, and 18. The moisture content reported here is volumetric (volume of water/volume of sample) as opposed to his earlier gravimetric (weight of water in sample/dry weight of soil in sample; = irigP^j-yl Pw^ter ) data. Polarization labels on these figures refer to the orientation of the antennas with regard to the horizon. The data show no polarization effects, even for the clay which under compaction can assume a somewhat regular platelet structure or fabric. The quantity of data collected is much more substantial than the previous results reported above, and in some cases, shows a bilinear response. One can argue from these data that soil fabric does not play a major role in determining the complex dielectric response of moist soils. Hoekstra and Delaney (1974) also found little difference in the dielectric response of their sand, silt, and clay samples as shown in the data on Figure 19. From an engineering perspective, their conclusion is valid. However, if one wishes to examine the physical mechanisms behind the electrical losses in moist soils, one would have to have more data on each soil type than is shown in Figure 19. Hallikainen, et al (1985) reported results of measurements on five different soil types. Figure 20 shows curves of dielectric constant vs volumetric water content drawn from polynomial fits to the available data. One could easily argue that these results indicate some differences in electrical behavior due to soil texture at relatively low frequencies (< 5 GHz) but hardly any differences at higher frequencies. Their assumption of a polynomial fit (second order in volumetric moisture content) precludes any discussion of bilinear behavior. 47 LEGEND COMPACTIVE effort: 0 5.74 N/CM2 0 H.83 N/CM2 9 10.47 N/CM2 EFFECT OF WATER CONTENT ON CONDUCTIVITY YUMA SAND FREQUENCY = 1.412 GHz ,o<« 0.1 O.Z 0.3 0.4 0.4 0.8 0.7 0.3 0.4 O.i O .e 0.7 N5 o 0.1 0.2 Figure 16. Dielectric constant of a sand vs volumetric moisture (Lundien, 1971). 48 LEGEND EFFECT OF WATER CONTENT ON CONDUCTIVITY COMPACTIVE EFFORT: 0 5.74 N/CM2 0 11.63 N/CM2 9 16.47 N/CM2 OPENWOOD STREET SILT FREQUENCY = 1.412 GHz o o f y* 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.5 0.6 0.7 i i o > O • 0.1 0.2 0.3 0.4 Figure 17. Dielectric constant of a silt vs volumetric moisture (Lundien, 1971). 49 LEGEND EFFECT OF WATER CONTENT ON CONDUCTIVITY COMPACTIVE EFFORT; O i . l A N/C m 2 0 11.63 N/CM2 V 16.47 N/CM2 LONG LAKE CLAY FREQUENCY = 1.412 GHZ o o A o^ < N5 • o "< < > ] A VCP^ O o > r o Figure 18. Dielectric constant of a clay vs volumetric moisture content(Lundien, 1971). 50 28 • A • a 24 O •i Monchesier Fmp Sand Monclester Fine Sand Suffjel<^ SHty Cloy Goodricn Cloy Foirbonfcs Silt O.IO Sutfield Silty Cloy Goodrich Cloy O Foirbonks Silt a p.20 Woler Content, O.IO 0.30 M. 5 X JO* Hz. - A • o 12 O 14 0.20 Woler gHjO/cm' .6. Manchester Fine Sond Suffield Silty Cloy Goodrict) Cloy Fairbonks Silt 14 12 6 • o o 0.30 Content, q HjO/cm* 4 X 10* Hz. Monctiester Fine Sand Suffield Silty Cloy Goodrich Cloy Foirbonks Silt 10 10- /C' K' O.IO 0.20 O.IO 0.30 Water Content, g HjO/em' c. 0.20 Woter Content, g HjO/cm' 10 X 10* Hz. d. 26 V IC Hz. Figure 19. Dielectric constant of soils vs volumetric moisture at 10°C (Hoekstra and Delaney, 1974). 51 • 0.30 40 silt CUy 1 - Field 1 - Sandy Loam ? - Field 2 - Loam 3 - Field 3 - Silt Loam ,<( - fieia4 - Silt loam 5 - field 5 - Silly Clay S4ndy low lOM Silt LoM Silt lo*# Silt/ Clay 47.6 47.4 30 Frequency: 5 GHz T'2yc Frequency: 1.4 GHz T-zyc Z5 o 20 10 0.2 0.3 0.4 Volumetric Moisture m, 0.2 0.3 0.4 Volumetric Moisture m, (a) T ss 30 ' T:—' • (b) r -I : r 1- Field 1 - Sandy Loom 2-Field 2-Loam 3-Field 3-SIIt Loom 4-Field O-Sllt Loom 5-Field 5-SUty Cloy 1 - Field 1-Sandy Loom 2-Field 2-Loom 5-Field 3-silt Loan 4-Field 4-Sllt Loom 5-Field 5-SlItv Cloy Frequency: J86Hz T-23*C 25 - Frequency: 106Hz I-23-C % 20 ais 10 0.0 0.2 0.3 0.1 Volunetrlc Moisture n, 0.2 0.3 0.1 Volumetric Moisture n, 0.6 (d) (C) Figure 20. Dielectric constant of soils vs volumetric moisture (Hallikainen, et al, 1985). 52 Another source of experimental data on moisture content variation is that reported by Campbell (1988) and shown in Figure 21 for three different soils. Clearly, at low frequencies (<50 MHz) dielectric energy loss goes up with moisture content and is very dependent upon soil type at the upper end of the frequency range of these experiments. al 1 MHZREE 2MHZREE SMHZREE 10MHZREE 20 MHZREE SO MHZ REE 11818 tfi® Q 10 -I r 10 20 30 % WATER BY VOLUME QQ Q 1! '• • QQ 1- —1 1 Qq BQ, OBq 1 r- 10 20 30 % WATER BY VOLUME 1 MHZ LI 2MHZLT 5MHZLT 10MHZLT 20MHZLT 50MHZLT 40 (a) Hart sand Figure 21. Dielectric constant for soils vs volumetric moisture (Campbell, 1988) (continued). 53 eflbr • • B • • a 1 MHZ REE 2MHZ REE 5 MHZREE 10MHZREE 20 MHZREE 50 MHZREE a • B • B a IMHZLT 2MHZLT SMHZLT 10MHZLT 20MHZLT 50MHZLT mi T r 10 20 % WATER BY VOLUME % 2.0- I BP Q •D ° *•••***** IUrn "n" "oB O BO^bb"""" 1.0- I a • .j^gBisisissHBSSflsssHgiflse^^• •• 0.0 f 1 1 1 1 1 10 20 % WATER BY VOLUME 1— 30 (b) Manchester silt Figure 21. Dielectric constant for soils vs volumetric moisture (Campbell, 1988) (continued). 54 1MHZREE 2MHZREE 5MHZREE 10MHZREE 20 MHZ REE 50 MHZREE U 60 §SgSS-S -I 1 1 r 20 30 40 % WATER BY VOLUME 12 o qo Q 10- a 1I 8 - • 1 MHZ LT 2MHZLT a 5MHZLT « 10MHZLT • 20MHZLT a 50 MHZLT * 6 - * qD 4- Qq a B DO® •"'* • "•/••-a- H SSSSa 5 SoSa Boo -I 10 1 1 r 20 30 40 % WATER BY VOLUME (d) Fort Edwards clay Figure 21. Dielectric constant for soils vs volumetric moisture (Campbell, 1988) (concluded). 55 The Effects of Sample Temperature on Dielectric Properties While several of the preceding figures contain information on the effects of sample temperature on dielectric properties, special attention should be given to the work done by Hoekstra and Delaney (1974). While they were not able to conclude anything on the temperature dependence of the characteristic frequency of Debye-type relaxation, they did report on some very interesting dielectric behavior near the freezing point of water. Figure 22 shows how frozen soils, in general, have a lower dielectric response than do unfrozen soils, and that increasing moisture content reveals some very anomalous behavior near 0°C. Some earlier results reported by Hoekstra and Doyle (1971) support the same contention. Another very informative study was carried out by Delaney and Arcone (1982) that involved the cross-plotting of extensive time domain reflectometry data to produce the curves shown in Figure 23 for a silt and a sand. Hallikainen, et al (1985) also reported measurements made at four different sample temperatures ranging from +23°C to -24°C. Their results, shown in Figure 24, reveal the same relative insensitivity to temperature at sub-freezing conditions as were reported by others. 56 -20 •b*" 1 10 —w 1 o-o-o-o—v ——o—. I '' ' 0 10 Temperoiure, ®C "o ; o——o 0.05 1 20 30 (a) Goodrich clay Temperoture, ®C (b) Fairbanks silt Figure 22. Dielectric constant of soils vs temperature at three volumetric moisture contents (Hoekstra and Delaney, 1974). 57 32 0.55 fl- HgO/cm* 0% 24 0.29 K' 4 T T .0.21 O.IT 136 -25 20 -20 25 (a) Fairbanks silt 24 0.27 0.17 0.17 -25 •20 -15 -10 20 25 (b) North Slope sand Figure 23. Dielectric constant of soils vs temperature at several volumetric moisture contents and a frequency of 0.5 GHz (Delaney and Arcone, 1982). 58 20 18 Field 1 51.5%Sand. 35.1%Sllt. 13.4%Clay Volumetric Wetness 0.24 cmVcm^ Bulk Density: 1.54 gcm^ 16 14 S 12 23* C <s 10 o -18®C nor -18°C 8 12 Frequency i (GHz) -24* C 16 (a) Silty sand Figure 24. Dielectric constant for two soils as a function of frequency and temperature (Hallikainen, et al, 1985) (continued). 59 Field 5 5%Sand, 47.6%$ilt. 47.4%Clay Volumetric Wetness: 0.36 cm^/cm^ Bulk Density: 1.42 g cm^ 5 12 c o <u -II*C 23° C -18°C -24° C Frequency ((GHz) (b) Silty clay Figure 24. Dielectric constant for two soils as a function of frequency and temperature (Hallikainen, et al, 1985) (concluded). 60 Bound vs Free Water In spite of the mixed bag of conclusions regarding the dependence of soil dielectric response on soil texture many researchers contend that the dependence is there (Wang and Smugge, 1980). If moist soils with different texture do, in fact, possess different dielectric properties for the same levels of moisture content, it must be because of different soilwater bonding mechanisms. Clay particles offer a relatively large amount of electrostatically-charged surface area (van Olphen 1963; Dobson, et al 1985) that could support numerous bonding mechanisms. The unique structure of clayey soils also present many opportunities to alter the dielectric response of the soil as a whole by the creation of microscopic capacitive elements. Even if the surface charge density of sand particles is not significantly different than that of clay particles (van Olphen 1963), the rounded shape of the sand particles and their smaller specific surface area (mVg) must lead to different mechanical as well as electrical behavior. It may even be possible to relate the physical behavior of moist soil to its dielectric response. For example, soil scientists have long associated varying levels of soil tension, the vacuum needed to draw water from a sample, to varying degrees of soil water bonding. Wang and Smugge (1980) took the concept of soil tension a bit further by correlating a particular level of tension, known as the wilting point, to the transition moisture content in moist soils. The transition moisture content is the break point in the bilinear dielectric response referred to in an earlier section. There is no universal agreement on a single mechanism for soil-water bonding. Rather, several hypotheses are offered, each of which can be both supported and refuted by experimental evidence (Mitchell 1974). 61 Hydrogen bonding The most easily supportable picture of soil-water bonding is that of hydrogen bonding, which, as discussed previously, is the sharing of a proton by two electronegative atoms. Mitchell (1974) points out that soil particle surfaces are usually composed of either a layer of oxygens or a layer of hydroxyls. He argues that the shared proton is supplied by the water molecule if the surface layer is oxygens, while the hydroxyl supplies the proton in the other situation. In either case, activation energy measurements do provide some supporting evidence of this mechanism. Newman (1987) further explains that unsatisfied bonds at the edges of clay particles present further bonding opportunities to the water molecules depending on the pH of the fluids in the moist soil. van der Waal's fnrr.p.f: A second proposed mechanism for soil-water bonding is that of electrostatic attraction combined with the dipole nature of the water molecule to cause several layers of molecules to migrate to the negatively-charged surfaces of the soil particles. In clayey soils with platelet surfaces nearly parallel it is possible to create microscopic dielectricfilled capacitive elements through the van der Waal attraction mechanism if one also has cations available to complete the dipole path as shown in Figure 25. Hydration of exchangeable cations A third soil-water bonding mechanism is a parasitic one in which cations are attracted to the negatively-charged particle surfaces. Newman (1987) views this as the primary water adsorption mechanism in swelling clays. This mechanism assumes that the cations take their water of hydration along with them for the ride. Naturally one must have cations available in solution for this model to have any validity, and the concentration of cations determines the amount of water that can be bonded. Dobson, et al, (1985) were motivated by such a mechanism to model the contribution of bound water in a four-component dielectric mixing model. In reality, they used a double-layer model for ion distribution near clay particle surfaces to modify bulk conductivity for free water losses and simply assumed values of the complex dielectric constant for bound water. CATION \ / 'G)O "G) (G*'G^©'G>G) G^G^OG^G) Figure 25. Microscopic capacitive elements in clayey soils Osmosis The fourth mechanism (and that least supported by data) is that due to osmotic forces. Again assuming that there are a large number of cations in solution and that their attraction to the negatively charged particle surfaces is stronger than the attraction of the dipole water molecules to the surfaces, this model predicts that there will be a higher concentration of cations closer to the particle surfaces. The existence of another concentration of cations on a nearby surface then results in a volume of lower particle concentration between the surfaces and the resulting migration of water molecules to that 63 volume. Whatever the mechanism is, water is attracted to soil particles, particularly clay particles, as proven by the extraordinary vacuums required to drain soil samples. In the immediate vicinity of the soil particle surface, the concentration of water molecules should be higher than in the bulk, or free, water occupying the void spaces (Martin 1960). Thus, one has to believe, even if the bonding mechanism was that of hydrogen bonds, that bound water should exhibit a different dielectric response than free water in the soil matrix. Radio Frequency Loss Mechanisms Clearly, water in contact with soil particles must behave differently (electrically) than free water. This shows up very dramatically at radio frequencies through a variety of loss mechanisms that don't exist for the dry minerals or for pure, liquid water. Not only does the water bind to the soil particle surfaces, but also salts are dissolved to produce a source of conducting ions (Mitchell, 1974), and, as already mentioned, microscopic capacitive elements can be created simply from the platelet-like structure of clay particles. These and other phenomena lead to a number of loss mechanisms in moist soils within the radio frequency range that are discussed in the following paragraphs. Hasted (1973) presented a particularly useful schematic, shown in Figure 26, for the relative magnitude and frequency range of application of each of what are believed to be the major contributors to dielectric loss in heterogeneous moist materials. The absolute magnitudes of these effects should not be regarded as applying to any particular material. A very brief description of each mechanism follows. More detailed expositions may be found in other sources (Campbell 1988; Hasted 1973; Mitchell 1974). 64 ionic conductivity charged double layers crystal water relaxation ice relaxation Maxwell-Wagner effect surface conductivity bound water relaxation free water relaxation log€ 2 0 1 2 3 4 5 6 . 7 8 9 K ) II 12 13 log V( Hz) ^ Figure 26. Dielectric loss mechanisms for heterogeneous moist materials (after Hasted (1973)). Free Water Relaxation The inability of free water dipolar molecules to keep up with an applied alternating field has already been discussed in Chapter 2 of this study. Dipole relaxation in free water is the dominant loss mechanism at microwave frequencies. 65 Bound Water Relaxation Experimental evidence exists (Muir, 1954; Hoekstra and Delaney, 1974) that indicates a lowering of the critical dipole relaxation frequency in water that is bound to solid particle surfaces. One can think of this physically in terms of the simple mechanical analog described in Chapter 2 and imagine that, for all other factors kept constant, an increase in the viscosity coefficient will result in an increase in the relaxation time or a lowering of the relaxation frequency. Hall and Rose (1978) proposed bound water relaxation as the dominant loss mechanism in carefully prepared kaolinite clays in the 10-100 kHz range of the spectrum, but their conclusions seem to contradict the observations of others and the implications of Figure 26. Maxwell-Wagner Effect Another loss mechanism often referred to in the literature as the dominant mechanism at radio frequencies is one termed the Maxwell-Wagner effect (Campbell 1988; Bidadi, et al 1988). It originates in the experimental observation that there is dispersion in a suspension of conducting particles in non-conducting dielectric media much like that observed in a parallel-plate capacitor filled with two different dielectrics. The Maxwell-Wagner effect is seen as an accumulation of charge at the interface of dissimilar materials during the flow of current through heterogeneous material due to a discontinuity in dielectric constant values. This charge buildup is time dependent and results in dispersion as the frequency of the alternating current increases to a point where the buildup and relaxation cannot keep up. When modeled as series capacitors, one having zero conductivity and the other a non-zero conductivity leading to a dielectric loss through equation 10 (assuming the polarization loss is zero), the Maxwell-Wagner effect can be described by Debye-like relationships (Hasted 1973; Campbell 1988). For example, for equal thickness capacitors, the relaxation time expression is found to go like 66 X = ^ (31) 4no^ Thus the relaxation frequency is seen to get smaller as conductivity gets smaller. One application of the Maxwell-Wagner model (Bidadi, et al 1988) had water as the conductive medium and the clay platelets as the non-conductive medium. The high capacitance of the closely separated platelets was used to argue for the observed enhancement of clay-water permittivity above the level for pure water. Surface Conductivity Another radio frequency loss mechanism in moist heterogeneous materials is that of surface conductivity. One possible scenario for this loss is a rapid intermolecular proton exchange within the bound water layer on soil particles (Hoekstra and Doyle 1971; Fripiat, et al 1965; von Hippel 1988c). One application of this model (Schwan 1962) led to the calculation of a Debye-like relaxation time T = —fsL4K03 where 6^ and (32) refer to the permittivity and conductivity of the aqueous solution in which nonconducting particles were immersed. Again, as with the Maxwell-Wagner effect, reducing the conductivity of the aqueous solution reduces the relaxation frequency. Charged Double Layers If one accepts that cations can be attracted to negatively-charged soil particle surfaces, then one can possibly think of the cations surrounding the charged particle like an electron cloud surrounding a nucleus. The charged double layer loss mechanism then has a parallel in the concept of atomic polarizability. As the electric field oscillates, so 67 too does the ion cloud about the particle. As frequency goes up, the ability of the cloud to distort goes down with the resulting lowering of the dielectric loss. Ionic Conductivity The most dominant loss mechanism in moist heterogeneous materials at low frequencies is that due to ionic conductivity, the 4itG/w term in equation (10). Of course, this conductivity is brought about by dissolving salts in the soil and providing a path through the pore spaces for the transport of electrical energy to take place. As a result, low frequency ionic conductivity in moist soils would have to be related in some way to the degree of saturation, or how the pore spaces fill with water. Activation Energy Data In the earlier description of the structure of water, arguments were made relating the frequency of peak loss in a Debye-type dielectric relaxation process to the activation energy associated with that process. The slope of a semi-logarithmic plot of that relaxation frequency vs the inverse of the absolute temperature is proportional to the activation energy. Hoekstra and Doyle (1971) argued that linear plots of the log of the dielectric loss term vs the inverse of absolute temperature can also be obtained when the variation in dielectric loss with temperature is caused by changes in the thermal energy of a charge carrier. In fact the dynamical theory of sorption (Hasted 1973) relates the time of residence of a water molecule on a surface to the heat of adsorption by the same kind of exponential relationship as used for the earlier activation energy development. Certainly, if loss is inversely proportional to this residence time by virtue of the molecules being free to reorient in the alternating electric field, then Hoekstra's and Doyle's arguments are valid. Thus, if the dielectric loss at any given frequency increases with sample temperature, the semilogarithmic plot mentioned above can lead to an estimate of the energy required to bring about such a change, which, in turn, may help identify the loss mechansims. 68 In their 1971 paper Hoekstra and Doyle did, in fact, measure a linear relationship between the natural logarithm of loss terms and the inverse of absolute temperature for Na-montmorillonite samples. Their results are reproduced in Figure 27 (a) for a frequency of 9,8 GHz and in Figure 27 (b) for a frequency of 0.1 MHz. The activation energy associated with the low-frequency measurements was about 12 kcal/mole, while that associated with the high-frequency measurements was about 6 kcal/mole as long as the temperature exceeded -52°C. The high-frequency results are consistent with the free water dielectric relaxation mechanism discussed earlier and attributed to the breaking of hydrogen bonds. However, the low-frequency measurements indicate another mechanism for energy dissipation, most likely free charge carriers coupled with Maxwell-Wagner effects. Campbell (1988) used the same approach as did Hoekstra and Doyle to estimate the activation energy in soils above freezing temperatures. Data he measured at 1 MHz are reproduced in Figure 28. Campbell calculated an activation energy of about 7 kcal/mole and argued that the dominant loss mechanism at this frequency was the Maxwell-Wagner effect. 69 10 5 6 4 3x10" (a) 9.8 GHz, 0.69 g/g water content E I 1 J 4.5x10 3.6 4.0 4.8 5.2x10"' Vt (b) 0.1 MHz, 0.65 g/g water content Figure 27. Dielectric loss and conductance in Na-montmorillonite as a function of temperature (Hoekstra and Doyle, 1971) 70 y = 9.281e+4 ' 10'^(-831.089x) R = 0.99 0.0026 0.0028 I 1 1 r 0.0030 0.0032 0.0034 0.00313 1/KELT Figure 28. Dielectric losses at 1 MHz (after Campbell, 1988). 71 0.0038 CHAPTER 4: MODELS FOR SOIL ELECTRICAL BEHAVIOR Moist soils are a heterogeneous mixture of mineral particles, water, air, and possibly any number of organic substances. The mineral particles cover a broad range of sizes, varying from as small as tens of microns to several centimeters. Water is not pure and, as discussed in the previous section, in combination with the structural features of the mineral particles, can show anomalous dielectric behavior at radio frequencies unlike that of pure water. Nevertheless, the only practical approach to predicting soil response to either physical or electrical driving forces is that a real soil must be modeled as a continuum with homogeneous properties. As with the review of related electrical property measurements in the previous chapter, a review of electrical property models for soils and water would be remiss without a summary table of significant contributions. The table follows on the next two pages, and many of the entries in the table will be expounded on in succeeding sections. 72 Table 2 (continued) Electrical Property Models AuthorCs") Year Mat'lfs) Debye 1929 liquids Cole, Cole 1941 liquids, solids FreqCs^ — Description classical anomalous dispersion model with single peak loss frequency (or relaxation time) modification to Debye's model to account for a distribution of relaxation times Malmberg, Maryott 1956 water < 1000 Hz polynomial function relating permittivity to temp. Reynolds, Hough 1957 het. mixtures — mixture model incorporating ratio of field strength within particles to the macroscopic average Looyenga 1965 het. mixtures — symmetrical weighted mixture formula de Loor 1968 het. mixtures — begins to focus on water Stogryn 1971 saline water — Debye model with empirical relationships for static permittivity and relaxation time Ray 1972 ice, water Cole-Cole model plus DC conductivity and temperature dependence Arulanandun, Smith 1973 clays 2-60 MHz equivalent circuit applications Olhoeft, Strangway lunar soils > 100 MHz permittivity is a nonlinear function of density 1975 Table 2 (Concluded) Electrical Property Models Authorfs") Year Mat'lfs) Freqfs) Description Klein, Swift 1977 sea water 1.4, 2.65 GHz similar to Stogryn Wang, Smugge 1980 clays, sands 1.4, 5 GHz Troitsldi, Stepanov 1980 soils .6, 1, 10 MHz dependency of permittivity on specific surface and moisture at fixed frequencies Shutko, Reutov 1982 soils 1-3 GHz Katz, Thompson 1985 sandstones Manabe, et al 1987 water < 100 GHz single Debye relaxation model with temperature dependance Ulaby, El-Rayes 1987 vegetation .2-20 GHz mixture model distinguishing free and bound water Sapoval, et al 1988 theoretical an attempt to relate the observed response of porous electrodes to a fractal description of its surface Dissado, Hill 1989 theoretical frequency-dependent fractal dimensions of electrodes mixture model that accounts for moisture content, a transition moisture content, and soil porosity surveyed existing models to obtain best fits under selected conditions electrical conductivity modeled with fractal geometry concepts Mixing Models The most common approach to modeling the dielectric response of moist soils that attempts to draw upon a combination of the physical structure of the soil and experimental observations of the electrical response of laboratory samples is that of mixture theory in which the effective dielectric constant for the material takes the functional form (33) where = frequency-dependent dielectric constant of the mixture €^(f) = frequency-dependent dielectric constant of the ith component of the mixture In fact, the approach that is often taken is that the mixture dielectric lies somewhere between that of a parallel plate capacitor that is filled with a mix of fibers extending from one plate to the other and that of a capacitor filled with sheets of material whose interfaces are parallel to the plates (Hasted 1973). The former is called parallel mixing, and the latter is called series mixing. For example, if one took the soil to consist of air, water, and some homogeneous soil mineral, the two mixing models would take the form PARALLEL MIXING (34) 75 SERIES MIXING A = 5 + ^ + 5 (35, where the subscripts are self-explanatory and the W; represent weighting factors. The weighting factors are most commonly taken to be the volume fraction occupied by that substance. One reference to an early generalization of the mixture model for three components was given by Ansoult, deBacker, and DeClercq (1984) as In this case, k = +1, represents parallel mixing, and, k = -1, represents series mixing. From an even more physical basis many researchers have developed dielectric constant mixture models that consider the effects of local electric fields on the induced internal fields of specially shaped homogeneous dielectric particles. Reynolds and Hough (1957) argued that a general formulation for a three component system could be written as either Gm = + €262^2 + (37) or (e»,-«l)8lfl + (€.-€2)624 + (€„-€3)83f3 = 0 where the 6^ are the volume fractions of material "i", and the (38) are the local field ratios; that is, the ratio of the field within the dielectric component to the average field in the sample. From Stratton (1941) an homogeneous electric field induces a field inside a spheroid in such a way that 76 3 COS^ttj. fi = E J=1 (39) 1 where the a^. are angles between the field and the axes of the spheroid, the Aj are depolarization factors related to the shape of the spheroid, and €* is the dielectric constant of the homogeneous material into which the spheroid is inserted. DeLoor (1968) published the same result but less general in the sense that he assumed a random orientation of spheroids for which cos^a^-=l/3 . One of the more well-known mixture formulae is that associated with Boettcher which, for a two-component mixture, can be written as ^2 _ g (^1 ^2) ' («i+2€„) (40) which comes from Equations (37) and (39) by setting Aj = 1/3 (spheres) and €* = . In this formula, is the dielectric constant of the particles and Eg is the dielectric constant for the host medium. Another popular formula comes from Equations (38) and (42) and €*=€3 , Aj = 1/3 (spheres). This yields Rayleigh's formula. ^2 _ § (^1 ^2) i^+262 (€1+262) (41) Bruggeman took the differential form of Rayleigh's formula and integrated it between the limits of €% and to arrive at the formula - (€i-€2)(1-6I) -\^2j 77 (42) Looyenga (1965) converted Boettcher's formula into a differential equation on volume fraction 6 ^ and integrated between the limits of 0 and 1 to give the symmetrical form (similar to Equation (36)) Birchak, et al (1974) utilized a symmetrical form much like Equation (43) except that the powers were taken to be 1/2 instead of 1/3. One positive feature of the mixture models is that they all come from the same origin; namely, consideration of the average electric field iand electric displacement in a mixture of otherwise homogeneous materials in which the variables are physically measurable quantities of volume fractions and the dielectric constants of the components. Thus there is a good bit of physics involved in developing these formulae. The approximations involved in obtaining different formulae include the particle shapes and orientation distribution and what value gets assigned to the host medium dielectric constant when computing the field ratios. All of these mixture formulae, as well as many others (Reynolds and Hough 1957) have been applied to particular data sets with satisfactory results. It would, however be impossible to say that one formula applies to all situations and materials. The converse is also true; i.e., one cannot expect all of these formulae to fit a limited set of data. Wang and Smugge (1980) exercised several two-component mixture models (probably resulting in slight overestimates at low moisture contents due to ignoring air voids) to calculate the real part of the complex dielectric constant as a function of moisture content and to compare the span of model predictions to real data collected at 1.4 GHz. Figure 29 shows the results of those calculations. Curve 3 is the bounding simple parallel mixing formula (Equation 34); curve 1 is Rayleigh's formula 78 (Equation 41); curve 2 is Boettcher's formula (Equation 40); and curve 4, which gives the best fit is Birchak's model (Equation 41). Looyenga's results (Equation 43) have also been added as'L' symbols. O YUMA SAND lUNOIEN. 1971 + VERNON ClAY LOAM • MILLER ClAY NEWTON. 1977 80 0.2 0.3 0.4 VOLUMETRIC WATER CONTENT (em'/cm'l 0.5 0.6 Figure 29. Several mixing formulas plotted against real data (Wang and Smugge, 1980). 79 Campbell (1988) has demonstrated that the dielectric response of real materials is bounded by the series and parallel models. It is not surprising that Looyenga's model for spherical particles is a good fit to much of the data (Hasted 1973) as it clearly lies between these limits. Dobson, et al (1985) have carried mixture models to the fourcomponent level by introducing different parameters for soil particles, air, free water, and water tightly bound to the soil particles. Equivalent Circuits It is often advantageous to think of mixtures in terms of their electrical analogs, particularly when one wants to consider different electrical paths through the medium and some semiempirical way to account for the weighting of those paths. The following paragraphs take a careful look at equivalent circuits for both individual components of mixtures and the mixtures themselves. They range from quite simple models to quite complex circuits. However, with today's readily available computational power, exercising these models is very straightforward. Homogeneous Materials The complex dielectric constant and conductivity of a linear material relate an applied electric field to current flow within the material through Ampere's law (Equation 10). Current flow in the material can be modeled by a two terminal network of lumped circuit elements. Furthermore, because real materials do exhibit both energy storage (capacitive) and energy loss (free charge movement and dielectric relaxation) mechanisms it is natural to imagine that simple equivalent circuits will have to include both capacitive and resistive elements. The following paragraphs describe models of the shunt response of two-terminal equivalent circuits which represent the flow of current (both conduction and displacement) between the inner and outer conducting surfaces and through the dielectric mixture being measured. The series response, or current flow along the inner and outer conducting surfaces, which would be represented by series inductive and resistive elements, is not modeled. As examples of simple equivalent circuits, consider the two circuits shown in Figure 31, one having constant value elements in series and the other with elements in parallel. Drawing upon the very useful concept of equivalent capacitances for each of the = i/(x>R ; elements in these and the following circuits( = -l/co^L ), the frequency-dependent dielectric response of the circuits in Figure 31 can be shown to be SIMPLE SERIES MODEL e/ C € // (Oi?C2 (45) SIMPLE PARALLEL MODEL (46) 81 (a) Series W (b) Parallel Figure 30. Simple equivalent circuits 82 The physical interpretation of the parameters contained in these equations is that the capacitor accounts for the dielectric polarization of the material and the resistor is representative of ionic conduction or other losses that decrease with increasing frequency. Both parameters are adjustable to provide the best fit the amplitude and phase response of the material as it is measured over a broad range of frequencies. However, neither of these circuits adequately model the high frequency behavior of water ith regard to the real part of the dielectric constant. The parallel model, when placed in series with another parallel model, has been used to model the Maxwell-Wagner losses in saturated soils (Smith 1971; Bidadi, et al 1988). As reported in an earlier section, Cole and Cole (1941) used equivalent circuits to describe both the ideal Debye relaxation model and their own interpretation of real data. To help make this section of the study self-contained, their circuits are reproduced in Figures 31 and 32. 83 Q LOG[F) - Hz Figure 31. The Debye equivalent circuit and its response. 84 o €o - €« T(iCi)T)"^ ^0 "" 100 -1D0 O 90 90 T = 20® C 70 70 a =0.125 60 GO 40 40 30 20 \ % / / 2^ 20 % 10 ; I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I B 9 10 11 LOG[F) - Hz Figure 32. The Cole-Cole equivalent circuit and its response. 85 The corresponding governing equations for the Debye and Cole-Cole models are DEBYE MODEL ./ _ (47) Co Co[i+(wac2)2] g// _ Co[l+((Oi?C2)2] COLE-COLE MODEL 1 + ^2 ""sin- an 2 (48) air ^ ((01.^^2(1-.) 1 + 2((OTo)^""sin 2 e" = ^2 ((OTo)^""COS an' 2 1 + 2(cox o)'""sin-^ where 1/Tq is the radial frequency at the peak loss. It is even possible to approximate the dielectric response of impure water through the use of equivalent circuits, von Hippel (1954) showed the following circuit (Figure 33) as an approximation to the frequency response of water exhibiting low frequency ionic conductivity. The frequency response of the dielectric constant in von Hippel's water model is governed by the following set of equations. 86 VON HIPPEL'S WATER MODEL (with ionic conductivity) e/= ^ Co[i + ^// ^ [1 + c'l CoO)i?Ji + ((oi?2q^] 85 75 tan 8 100 /i/if 53000 ohms 0.085 ohm 0.1 0.01 RC Circuit 0.001 Frequency in cycles per second Figure 33. An equivalent circuit for impure water (von Hippel, 1954). In fact, there is no reason not to include another parallel capacitor to von Hippel's water model to account for the high frequency permittivity, ending up with a four element model shown in Figure 34 for which the governing equations are 87 e' = ii + (54) C„[l + (wKzC,):] €" = [1 + C^oS^[l + -r100 10Q • 90 90 Cj/Co — 4 Cj/Co = 76 80 Ri = 5.888x10® s/cm 70 70 Rj = 1.25x10"" s/cm GO 60 50 40 40 A 30 30 \\ % \ I r\ - 20 10 20 -r -r "T" nr 11 12 13 14 "T IS LOG(F) « Hz Figure 34. Water model including DC conductivity and optical permittivity 88 .// Mixtures It certainly seems reasonable to use combinations of homogeneous material equivalent circuits to model the frequency dependent dielectric response of mixtures such as moist soils. In fact, such models would be particularly useful for studying the effects of water content in nonsaturated soils if there could be a sensible weighting of the equivalent circuit elements to account for varying moisture contents. Campbell (1988) demonstrated quite clearly that if one considers the behavior of the real part of the complex dielectric constant as a function of water content for soils, it falls between the limits for a three component system (soil, water, and air) created by assuming, on the one hand, that the three components can be represented by equivalent capacitances in series and, on the other hand, by equivalent capacitances in parallel. Bidadi, et al (1988) extended this concept a bit further to successfully model the complex dielectric response of clay films as a Maxwell-Wagner mechanism using the equivalent circuit shown in Figure 35. The upper frequency limit of their study was only about 100 MHz. R R 1 2 AAA AAA •o o c c Figure 35. Equivalent circuit for clay film studies (Bidadi, et al 1988) 89 Their reasoning for choosing such an arrangement of elements is the following. They wanted to model the physical situation of easy conducting paths through the clay interstices that could be interrupted by thin insulating barriers (clay platelets, for example), this being their interpretation of the Maxwell-Wagner effect. Small values of R, and Cj allow for easy current flow, while large values of R; and C2 account for the barriers. A similar model was proposed by Eicke, et al (1986) in which the barriers (actually, the solvent) were taken as lossless (R; = 0). Backing up further in time, Sachs and Spiegler (1964), and later, Smith (1971), applied even more complicated equivalent circuits to the study of the complex dielectric response of saturated materials. Theirs was a very physically satisfying approach to the problem in which they assumed a combination of equivalent circuits in series representing a discontinuous electrical path through the soil-water system (Bidadi's reasoning for interrupted current flow) and of equivalent circuit elements in parallel representing continuous paths for effective current flow. The resulting combined circuit is shown in Figure 36 and was applied to data that did not exceed 100 MHz. Solution 1-d w Figure 36. An equivalent circuit model for saturated media (Sachs and Spiegler 1964; Smith 1971). 90 The geometrical parameters a, b, c, and d were empirically established by Sachs and Spiegler for their work by obtaining a best fit to low frequency conductivity data. One has to believe, however, that these parameters must be related to physical quantities. For example, it seems likely that d could be associated with porosity. For saturated media, the higher the porosity, the less chance for series-like interruptions in effective current flow. Likewise, b should be related to dry density. The greater the sample dry density, the more opportunity there is for conductive paths through the soil matrix. And finally, c should somehow relate to free water content in the soil sample. The greater the amount of free water, the greater would be the opportunity for current to flow through, unimpeded by the solid matrix. The governing equations for the circuit representation in Figure 36 are THE THREE-PATH EQUIVALENT CIRCUIT ./ _ + N2D2) ^ 3cf(l Co(DI2 + g// ^ ^ Co (51) Co ad{l-d)(N2Dj_ - N^Dz) ^ b ^ c CQ{D^^ + Cq(^Rs Co(oi? where N, = N, = D, = w2 - 1 + ®»rC„) + dcj D, - 0>[<1R„ + (l-C^i?J It certainly seems reasonable to include another set of equivalent circuits to those of Figure 36 to account for air in non-saturated soils. Perhaps even more simply, allow a+b-hc < 1 and, 1-d, to be replaced by x-d, where x < 1. 91 Furthermore, it is undoubtedly possible to arrange a limited number of electrical circuit elements in such a way as to approximate almost any dielectric response in a single homogeneous material. What is really needed is an equivalent circuit for which the elements not only allow a good reproduction of spectral response over a broad range of frequencies, but also have sensible physical meaning or whose values can be measured in the laboratory. Percolation Transition or Long-Ranpe Connectivity As pointed out in Chapter 3, experimental data on moisture content variations in soils shows that the real part of the dielectric constant at low frequencies increases more rapidly with increasing moisture at higher levels of moisture content than it does at lower moisture levels. Some of the data can almost be viewed as bilinear, where below some critical value of moisture content, the rate of increase in dielectric constant is nearly constant. Above that critical value, the rate is again nearly constant but somewhat higher. Wang and Smugge (1980) argued that this transition point was a function of soil texture; i.e., a function of the amount of sand and clay. They also related the wilting point of the soil (the volumetric water content under a pressure of 15 bars at standard temperature) to its texture, and , in so doing, were able to show a strong correlation between the transition point and wilting point. In fact, Wang and Smugge went on to develop these observed soil texture effects into empirical relationships for the electrical properties of soils. Campbell (1988) viewed this phenomenon by electrical analogy as a transition between series mixing (Equation 35) at lower water contents and parallel mixing (Equation 34) at moisture levels above the critical value. He borrowed from the percolation theory the concept of a percolation transition in moisture (Zallen, 1983) at which level one or more continuous, low resistance electrical paths through the porous media suddenly appear. He referred to this transition as the appearance of "long-range connectivity". 92 Campbell attempted to create a computerized model of moist soil passing through the transition point using the grid shown in Figure 37. The grid was initially set up with a random distribution of soil and air squares to provide a given porosity. The air squares were then randomly replaced with water squares as water content was allowed to increase. A dielectric constant was then calculated for each column of soil, air , and water squares extending from one capacitor plate to the other by the series model (Equation 35) and, finally, an effective dielectric constant for that moisture content was calculated as the average value for all of the columns. + + + + + + + + + + + + + + + + + + mm SEPARATION I WATER LENGTH I Figure 39. Campbell's (1988) percolation model grid for moist soil. 93 While this model simulated the increasing rate of change in real dielectric constant with increasing water content, it did not, in fact, simulate long-range connectivity because it was only a one-dimensional model. Unless the dry soil grid contained several columns of only air spaces, there was no way to simulate long-range connectivity. Campbell also pointed out that another shortcoming of this model was its inability to scale properly. Doubling capacitor dimensions gave a different result. Fractal Models of Electrical Behavior Modeling of Electrodes In recent years fractals (see Appendix C for definitions) have been utilized to study the electrical behavior of fluid-saturated porous electrodes. The fundamental thought behind the use of fractal concepts is that the number of elements used to construct a fractal geometry scales as a power law of the element length or relative size (Mandelbrot, 1983). It seems to have become fashionable to also talk about a fractal geometry as one for which "physical quantities scale as a power law of the length scale....on which they are measured" (Brouers and Ramsamugh, 1988). For example, if there is a region of the electromagnetic spectrum where conductivity of a sample of material is observed to vary as a power law in frequency (hence, wavelength), then that sample is thought to possess a fractal geometry. In fact, Brouers and Ramsamugh (1988) combined low frequency permittivity and conductivity data for brine-saturated alumina ceramics with some statistical models to calculate fractal dimensions of 2.65 ± 0.10 for five different ceramic samples, independent of sample porosity. As part of a study on the frequency response of porous battery electrodes, Sapoval, Chazalviel, and Peyriere (1988) also observed that the impedance of rough or porous electrodes often goes like a power law in frequency (constant phase angle behavior) and is, in fact, related to the fractal dimension of the electrode surface in some cases. The model they used to associate fractal dimension to electrode structure was that called a "finite modified Sierpinski electrode" as shown in Figure 38. The Sierpinski electrode is something like a one-dimensional Menger sponge that is discussed in Appendix C. Sapoval, et al, obtained an expression for the admittance of a single pore within the electrode and used the fact that the total pore admittance is the sum of the admittances of all the pores to analytically study the effect of fractal dimensions on the electrode response. Figure 38. The "finite modified Sierpinski electrode" (Sapoval, et al, 1988). Dissado and Hill (1989) used the same model to simulate analytical response functions and to attempt to prove that there is one form of self-similarity at high frequencies (within individual fractal pores) and another at low frequencies (sum of all pores). 95 Fractal Pore-Filling Model Campbell (1988) observed that, up to the time of his publication, fractal concepts had been applied to the bound water and saturated regimes for porous media, but not to the broad regime of moisture contents below saturation. He was searching for a way to model the dielectric response of moist soils at moisture" contents less than saturation and decided that it might be possible to do so using fractal techniques. His assumptions were: (a) Water within pore spaces accumulates in uniformly-sized fractal clusters. (b) The pore spaces are all of equal characteristic size, 1 , which is equal to the average particle size of the soil. (c) The dielectric constant of the soil-water-air mixture increases with increasing moisture content in a series mixing fashion until the size of the water clusters reaches the size limitation on pores at which time long-range water connectivity in the soil mixture is established; i.e., the percolation threshold is reached. (d) Once long-range connectivity is established, additional moisture enters in a non-fractal manner and the dielectric constant rises in a parallel mixing fashion. (e) The soil is neither a clean gravel (which precludes water cluster spanning of the pore spaces) nor a pure clay (dominated by bound water effects). The key assumptions made by Campbell that allowed a rather simple expression between critical water content (the moisture content at which long-range connectivity is established) and soil porosity to be developed were assumptions (a) and (b). From (a), Campbell wrote that the volume of water contained in a cluster of size, r, and fractal dimension, D, was 96 {cluster water volume) a r^ (5%) Thus the volumetric water content also goes as r®. Mandelbrot, himself, found no argument against the mass density relationship that stated that mass within a fractal cluster of size, R, went as (Mandelbrot, 1983) (cluster mass) a and, if one accepts that individual water molecules all have the same size and mass, then Equation 52 is very believable. At such time as the clusters just span the pores of size, 1 , to produce long-range connectivity, the volumetric water content can be written as ^critical « -I" Campbell took the soil sample porosity, (|) , to be defined in three-space as (J) a i3 (55) He finally concluded from the ratio of these last two quantities that ^ a 4) (56) and that a log-log plot of 6^/<j) vs i should be a straight line with slope equal to D3. If there is no difference in how the fractal water clusters grow in different soils, then D is the same for any type of soil that fits within the bounds presented by assumption (e). In fact, Campbell plotted critical water content data for five soils at a frequency of 50 MHz (shown in Figure 39) and found a fractal dimension of 2.4 that is similar to others' calculations of about 2.5 for aggregate clustering. He made no attempt, nor is there sufficient data, to draw any conclusions about the effects of soil texture or signal frequency on critical water content. 97 SCALED CRITICAL WATER CONTENT VS. PORE SIZE 10' y = 0.9818 * x'^-0.6051 R = 0.92 N. i Ig 10' I Q • N i 10^ •| q-3q+0 •J Q-2e+0 ijQ-le+0 10Oe+O PORE SIZE (mm) Figure 39, Scaled critical water content vs pore size (Campbell, 1988). Campbell's fractal pore filling model is very simplistic but, in his own words, "..the similar dielectric behavior of soils with huge ranges in particle size and shape of particles argues against a complicated model which considers many different soil parameters.." (Campbell, 1988). He went on to admit that the weakest point in his model was, in fact, assumption (b); i.e., that the characteristic pore size equals the median particle size of the soil. This point is examined closely in a future section. 98 CHAPTER 5: A COAXIAL APPARATUS FOR DIELECTRIC MEASUREMENTS The Measurement System A schematic of the measurement system used to conduct these experiments is shown in Figure 40. The electronic equipment that was available is not that which would normally be used for making material electrical property measurements, but it was successfully adapted for this use. HP B5iac • HP 9000 COMPUTER NETWORK ANALYZER HP 8340B HP 8511A SYNTHESIZED SWEEPER FREQUENCY CONVERTER REFKace RB'LECTED DUAL DIRECTIONAL COUPLER ' TmNSMinH) TRANSMISSION CELL CARON 20B5 CIRCULATING BATH Figure 40. Experimental Measurement System 99 'SOURCE The Hewlett-Packard 85 IOC Vector Network Analyzer (Hewlett-Packard, 1991) is a laboratory device designed to measure the magnitude and phase characteristics of electronic networks and/or components. It consists of a detector unit that measures signals in up to four different channels at an intermediate frequency of 100 kHz and a data processing unit that provides a display of the desired quantities. The HP 85 IOC directs the signal source (in this case, an HP 8340B Synthesized Sweeper) to generate a sinusoidal signal at a specific frequency and directs a test set ( an HP 8511A Frequency Converter) to act as the receiver for up to four channels of information which undergoes a first frequency conversion to about 20 MHz through the process of harmonic mixing. The combined dual-directional coupler and frequency converter serve as a substitute for a special test set, called the S-Parameter Test Set, which is usually used for such measurements (Hewlett-Packard, 1985). The S-Parameter Test Set contains power splitters, coupling devices, and means of establishing equal circuit electrical lengths where appropriate that are transparent to the user. In truth, the S-Parameter Test Set is a bit of overkill, as it provides information on measurements made in the reverse direction. As long as one believes that the material response is not dependent on which direction current flows through the sample, then the reverse measurements should be redundant. Figure 41 is a photograph of one of the two nearly-identical sample holders that were fabricated for these measurements. It was made from 4.5 mm brass stock and provides a square cross-section coaxial test volume that is 7.5 mm square and 40 mm long. A piece of copper bus wire was soldered into place as the center conductor, connecting the two 50 ohm SMA-type connectors at each end of the sample holder. The value of sample volume for each holder was estimated, by length measurements, to be 2.23 cm\ 100 Figure 41. One of the brass, coaxial sample holders. The choices for laboratory techniques for making material electrical property measurements include resonant cavity, waveguide, and coaxial with some new work being done on hybrids of the latter two (Taherian, et al, 1991). Resonant cavity applications are limited to small volumes of material which could present a problem with trying to measure the homogeneous response of substances like sands which are highly non-homogeneous at small scales. Waveguide measurements are very sensitive to how the sections are mated (and how repeatable are those connections). While they involve adequate volumes of material, sample preparation is not likely to produce a uniform density for granular materials, as the substance must be poured into one end and tamped to fill the volume. Dielectric seals would be required to provide a plane interface and must be properly accounted for. The literature also indicates that any phase shifts due to non-ideal calibrations are non-linear functions of frequency and must be accounted for during processing of the data. Waveguards also possess a cutoff frequency, below which 101 plane waves cannot propagate. As is true for waveguides, the usual coaxial devices (headless air lines) suffer from the problems of uniform sample preparation, and, in fact, are best suited to measuring solid cylinders of material that must be precisely machined to provide a snug fit within the measurement cavity. However, coaxial lines allow plane waves to propagate at all frequencies. Therefore, the sample holder used in these experiments provides a practical tradeoff among sample volume, repeatability of connections, and the ability to prepare uniform samples. One area of concern with this sample holder deals with the development of higherorder transverse modes. Theory says that a coaxial device should support only the transverse electromagnetic (TEM) mode as long as the wavelength in the material is greater than twice the cavity width (Stratton, 1941). Thus, if the cavity is empty, the highest frequency allowed for these measurements would be (•^max) • = ^ = -^ = 3x10^° cm/sec ^ 20 GHz (57) A nX ^/Txl.5 cm/cycle while for pure water 3 X 1 0 ^ °. = 2 . 2 4 GHz (58) y/WOxl.5 Plane wave conditions for measurements of moist soils should, therefore, be allowed to some intermediate frequency. The Governing Equation Consider the geometry of the coaxial measurement device and the voltages in all regions as shown schematically in Figure 42. A relatively simple governing equation that relates the sample properties and dimension to quantities that can be measured by the network analyzer system can be developed by proper consideration of the boundary conditions. 102 OUTER CONDUCTOR 2" «*CABLE J 2^ V, , I, Vj . Ig SAMPLE , Zg CENTER CONDUCTOR ^3 , I3 CABLE ^ Figure 42. Sample geometry and voltage and current notation. Using the customary notation for spatial variation of transverse voltages within the cables and sample and considering a snapshot in time (without loss of generality, let time be zero), one can write: ^2 = 0 < V3 = X < d (58) X > d where the k's are the complex propagation constants and contain all of the material electrical property information (see Appendix A). Similar expressions for the center conductor current can be written by taking advantage of the definition of the characteristic impedance of the coaxial line as being the ratio of transverse voltage to the conduction current along the center conductor (Stratton, 1941). 103 X < 0 I2 0 < X < d J = X > d where the Z's are the characteristic impedances within each section of the line. The boundary conditions that can be applied to the geometry shown in Figure 51 are that the tangential field components at the interfaces are continuous and, from the steady state continuity relationship, that the currents are also continuous across the interfaces. In equation form these boundary conditions reduce to: = V2 ® x=0 = V'3 @ x=d Vinp + (60) (61) ; + ^'2 = I2 @ x=0 J2 = J3 ® x=d = y+2 + y-2 : 104 (62) Not being able to measure or V~2 but certainly being able to obtain a reading on the other voltages, the two voltage amplitudes inside the material can be eliminated by adding together Equations 60 and 62 and by taking their difference: V\ = ^ + -^(Vinp - V-i) (64) - y-i) (65) ^0 Substituting these results into Equation 61, one has + V\ + Vinp + Normalizing to - y-J (66) - -i{V^ and collecting terms, l + -^)cos(i:^d) - i V" 1-- :sin(jCgd) (67) ^inp; A similar relationship can be written from Equation 62. y1-. ^inpj !cos(kgd) - 1 l+-|^]sin(ic^d)= ^inpj ^0 ^inp 105 <"> It is further convenient to eliminate the characteristic impedances by solving each of the last two equations for Z/Zg and setting the two expressions equal: (69) Simplifying, inp/ (70) Now, following an appropriate calibration of the measurement apparatus, the network analyzer effectively measures V'lfVinp and • They are designated, using standard 'S', or scattering parameter notation, as and S21, respectively. The definition of S parameters requires that there be no current traveling into the sample from the right side of Figure 42. This implies that all circuitry and connectors on the transmission side of the sample are perfectly matched so that no energy is reflected back to the sample. Although this is an assumption for this derivation, the fact that proper equipment was not available for these measurements and that a less-than desirable calibration technique was used (described in the next section) makes this idealized configuration quite unlikely. 106 Using the relationships between the propagation constant and material properties (Appendix A) and assuming non-magnetic materials, one can then write down the final form of the governing equation: COS V I I = 1 + F21)' - Fii)' = (71) This form of the governing equation was reported by Campbell (1990). An earlier derivation, although arriving at a different solution technique, has also been recently discovered (Hewlett-Packard, 1985). The process for calculating values of the complex dielectric constant can now be outlined: a) At a given frequency, measure the complex values of Si, and Sji. b) Compute cos"^(IV^) = . c) Square both sides of this expression and solve for € . The procedure is not as straightforward as it seems because of a problem with principal values of the cos"' function. One method for solving for the cos ' function yields (Zucker, 1972): cos"^(IV^) = -i ln(re^®) = 0 -i ln(r) (72) where re^® = + ^{W^f - 1 A great deal of time was spent developing a calculation procedure for Equation 72 that would yield the correct values of and . However, it was later discovered that the calculation routine for the inverse complex cosine function that resides in the HP BASIC compiler within the HP9000 computer that serves as the overall system controller gave exactly the same results. The only difficulty is that, depending on where W, lies in 107 complex space, the cos'^(WJ value computed using the available routines returns real and imaginary components that are both positive, both negative, or having opposite signs. A set of rules had to be developed to select the sensible results; namely, those that would yield positive values of e' and . The real part of (0 dsfe/c is a measure of the number of wavelengths (expressed in radians) of signal that can be contained within the sample holder at the given radial frequency. The value of 0 that is returned by the above solution algorithm is a principal value, being bounded by ±7r . However, any multiple of 2TZ can be added to 0 and give the same result. It turns out that there is a sensible way of determining how many multiples of 2TZ to add. One can either estimate the properties of the material to establish the correct value of 0 , or one can use the value of "group delay" which is calculated by the network analyzer to establish the correct value of 0 . Group delay is defined as the signal transit time through a test device (Hewlett-Packard, 1991). What the designers of the network analyzer assumed is that the phase for the wave in the material is a term like " (O t ". Therefore, group delay, or the derivative of phase with respect to radial frequency is a number with dimensions of time and interpreted as the time required to transit the sample. An estimate of the initial value of the number of cycles (multiples of 2 Tt ) to add to 0 was taken as the truncated product of group delay and frequency. Experimental Procedure Calibration Because of the arrangement and type of measurement apparatus used for these studies, the most appropriate calibration was found to be the One-Path, 2-Port calibration combined with the adapter-switching technique (Hewlett-Packard, 1991). During this procedure reflection and transmission cables were first connected with a thru adapter from an HP85052B, 3.5mm Calibration Kit and forward transmission and forward match measurements were conducted. Next, an equivalent adapter (male-female) from the same 108 Kit was connected to the input (reflection) cable and three reflection standards were measured. These standards were an offset short, an offset open, and a broadband load, all from the same Calibration Kit. The net result of the calibration procedure is that the S,i measurement effectively takes place at the bead interface in the SMA connector on the reflection side of the sample holder, and the S21 measurement effectively takes place at the bead interface in the connector on the transmission side of the holder. Due to the physical separation of the calibration planes and the interface between the sample and the connector on the end of the holder, an additional phase shift needed to be added to the S,, parameter and subtracted from the S21 parameter to make the measured quantities consistent with what was defined in the earlier derivation of the govering equation. Referring to the sketch in Figure 43, the reflected signal must travel an additional distance of 2a beyond the calibration plane to be recorded in the frequency converter. The transmitted signal travels an additional distance of a+b. For these measurements, a=b and the required phase shift was found to be about 80 picoseconds from measurements of the empty holder. REFLECTION TRANSMISSION CALIBRATION PLANE MEASUREMENT PLANES \ X y CALIBRATION PLANE CABLE CABLE SAMPLE HOLDER Figure 43. Calibration and measurement planes. 109 Sample preparation and Measurements Soil samples were prepared from one of four distinctly different soils. These consisted of a poorly-graded sand, a well-graded sand, a poorly-graded clean silt, and almost pure kaolinite, a non-swelling clay mineral. The soils and their associated properties were obtained from the Geotechnical Laboratory at the US Army Engineer Waterways Experiment Station. Each soil type batch was made as homogeneous as possible by drying, pulverizing and mixing thoroughly. Samples were then taken to determine the usual engineering properties of grain size distribution and plasticity indices. The results of those measurements are contained in Appendix D, along with the results of specific surface measurements on each sample that were conducted by a commercial laboratory. The normal measurement procedure was the following: (a) The dry, empty sample holder was carefully weighed. (b) A nearly-saturated sample of soil was prepared using distilled, deionized water and, following a reasonable amount of time for the silt and clay to reach some sort of equilibrium, was placed in the sample holder. The holder was tapped against a flat, hard surface to uniformly settle the sample. (c) The sample and holder were carefully weighed, then sealed and connected to the measurement system. (d) The temperature of the bath was set at -10°C. For each new temperature setting, the sample was given ten minutes to reach equilibrium once the bath had reached the desired setting. The ten minute wait was found experimentally to be more than adequate for sample equilibrium to be achieved. 110 (e) The frequency range for the dual directional coupler being used was stepped through and calculations of 6^ and were made. (f) The temperature of the sample was changed and measurements repeated in the following sequence: -10, -5, -2, 0, 2, 5, 10, 20, 30, and 40°C. (g) The holder was unsealed and its cover loosened. The sample and holder were periodically weighed to determine approximate values of moisture content. When the sample was thought to be at about the desired volumetric moisture content, the holder was resealed and reconnected, the sample temperature was taken back to 10°C and the whole measurement sequence repeated. (h) After all measurements were conducted, the sample was carefully removed, dried and weighed to determine its original dry density and to facilitate calculations of volumetric moisture. As explained earlier, measurements had to be conducted over two different frequency ranges because of the unavailability of a single dual directional coupler that could cover the entire range desired. Sanity Checks In order to develop confidence in the experimental procedure materials were measured whose responses are known experimentally or can be modelled very accurately. These measurements also served to confirm the amount of phase shift required to collapse the calibration and measurement planes as discussed in last section. Data were collected for empty sample holders, and samples of distilled water and ethylene glycol (ethanediol). Each is discussed in a following section. Ill Empty Holder The first question to answer is 'How well does this experimental procedure measure the properties of air?'. Ideally, the real part of the complex dielectric constant for dry air should be unity, and the imaginary part should be zero. Figure 44 shows the results of measuring empty sample holders. The vertical dashed line at 2 GHz separates the low-frequency apparatus measurements from the high-frequency measurements alluded to earlier in this chapter. Considering all of the experimental variables that could have an impact on any measurement such as measurement system drift, connector and connection variability, outer conductor discontinuities created by the sample holder cover, etc., the agreement of empty holder (air) measurements with theoretical predictions is quite remarkable. The anomalous behavior centered about 3.75 GHz and 7.5 GHz is associated with the mathematical solution technique. At 3.75 GHz the empty sample holder should be spanned by about a half of a wavelength, at 7.5 GHz it should hold about a full wavelength, and so on. The number of wavelengths contained within the sample holder for a material with no loss is proportional to {(XID\F^/ C , or the cosine function argument in Equation 71. A half wavelength in the sample holder will result in the complex vector representing the right-hand-side of Equation 71 approaching the negative real axis in complex space. A full wavelength corresponds to the vector having made a complete revolution in complex space and approaching the positive real axis. But these two conditions are branch cuts for the cos"'(z) function and are, therefore, inherently unstable. Much of the data that follow in this chapter show the same anomalous behavior whenever integer multiples of half-wavelengths exist within the sample holders. The severity of these peaks decreases with increasing frequency because at low frequencies the computation of € from Equation 71 involves the division of the right-hand-side of the equation by very small numbers. Any inaccuracies are amplified at low frequencies (< 1 GHz). This fact also had a very definite negative impact on low-frequency measurement calibrations. Instrumentation drift was often enough to ruin a calibration at 112 the lowest frequencies in just a few hours. Better measurement equipment; namely, an Sparameter test set and a high-quality set of cables and connectors, would help alleviate this low-frequency measurement problem. 2.0o 1 1 I 1 1 10 ru 0 TEMP I •2.0 1.8 -1.8 1.6; D I 1.4; 1.4 1.2- 1.2 1 e' 1.0 ryv^ 1 * 0.0- 0.8 0.6- 0.6 0.4 0.4 0.2 •0.2 0.01 0.01 • • 1 1 1 1 1 11 0.10 1 1 1 1 1 1 1 11 1 1.00 1 1 1 1 1 11 10.00 FREQUENCY ~ GHz Figure 44. Empty sample holder measurements. 113 1 1 1 1 1 1 111 •0.0 100.00 Water Measured values of the complex dielectric constant for samples of distilled, deionized water are shown in Figure 45. For comparison, Ray's (1972) empirical fits previously shown in Figure 1 are duplicated here. Several observations are in order. First of all, the comparison of measured data to the accepted standard is quite good (less then 4% difference), especially for the imaginary term. The sample holder volume was evidently not completely filled with water due to the fact that the holder cover must be pressed down upon the water meniscus, causing water to be squeezed from the holder in an unpredictable way. It is also possible that some evaporation of the sample took place during the 2-3 hour measurement period. Both of these phenomena could account for the lower measured values of e' . The hypothesis that the sample holder was never completely full was supported by weight measurements that indicate the holders were never more than 95% full during these tests. Another observation is that because of the good correlation with idealized data, the impact of higher order modes must be relatively small as one would expect (Stratton, 1941). As pointed out at Equation 58, higher-order transverse modes should appear above 2.3 GHz for pure water, but these data that extend to 4-5 GHz continue to behave well. (Water data are cut off at 4-5 GHz because the noise floor of the network analyzer measurement system was reached.). The same anomalies observed at half-wavelength intervals in the empty holders also occur in the water with very little impact being observed beyond one and a half wavelengths. The low frequency tail on may possibly be associated with the ionic conductivity loss mechanism as a result of either trace quantities of ions that escape the deionization process or as a result of ions that originate from unclean and/or oxydized outer or inner conductor surfaces. The magnitude of low-frequency losses are approximately those associated with fresh water conductivity as shown in Figure 1, but the actual loss mechanism has to be viewed as purely speculative. 114 TEMP 5 40 10 50 20 60 30 90- •90 80 40 20- 10- I 0.01 I I I II I 0.10 "T 1.00 10.00 FREQUENCY ~ GHz (a) these measurements Figure 45. Dispersion curves for water (continued). 115 100.00 TEMP 10 40 30 60 •90 90- 60 \\ /\ / / / //, tttt 0.01 0.10 T 1 I I I I I T T 1.00 1 I I I I I 10.00 FREQUENCY ~ GHz (b) Ray's (1972) fit Figure 45. Dispersion curves for water (concluded). T—r 100.00 Ethylene Glycol Data were collected on several different alcohols: ethylene glycol (ethanediol), methanol, and ethanol. However, due to surface tension problems and rapid evaporation rates, only the results for ethylene glycol (HOCH2CH2OH, molecular weight 62.07, specific gravity 1.1088 at 20° C relative to water at 4° C, < .05% water) are reproduced here (Figure 46). Measured values of both e/ and are consistently lower than Jordan's (1978) results for frequencies greater than 1 GHz. This is not a surprising observation in that the experimental procedure does suffer from the shortcoming that it is difficult to completely fill the sample holder volume with liquids because of surface tension problems, and it is difficult to maintain the volume of liquid during the 2-3 hour measurement period due to evaporation. Although ethanediol was least affected by these problems, sample weights for the high frequency measurements indicate that the holder was 90-95% full during the experiments. At low frequencies, measurements revealed another loss mechanism, most likely ionic conductivity. In summary, results of measurements of air, water, and ethylene glycol are quite good, resulting in a high degree of confidence in the experimental procedure. Differences between these measurements and more precise measurements reported in the literature can be explained by physical and/or chemical factors. One must conclude, however, that improvements would have to be made to the experimental procedure before it could be used to accurately measure the electrical properties of a variety of fluids. In particular it would be necessary to find a way to completely fill the sample holder and to maintain that fluid level throughout the measurements. The only current concerns with measurements of heterogeneous mixtures of soil and water are cleanliness of the sample holder and uneven drying of the sample throughout its volume. 117 10 20 TEMP 0 — 10 30 — AO 40- 1% rrrr 0.01 prq- 10.00 0.10 I 100.00 FREQUENCY " GHZ (a) these measurements Figure 46. Dispersion measurements of ethylene glycol (continued). 118 TEMP ————• JO ——— 20 30 '' 40 40 • 35 30 25 20 15 10 \ \ \\\ 5 0 I 11111| TTvr I I I 1 iif 40 35 30 25 .// SO 15 10 // y -r-r-rrn| 0.01 0.10 i i i i rrii|—-i—i"r-i-rriij 1.00 10.00 100.00 FREQUENCY " GHz (b) Jordan's data (1978) Figure 46. Dispersion measurements of ethylene glycol (concluded). 119 CHAPTER 6: EXPERIMENTAL RESULTS Summary of Data Collected As was mentioned in the previous chapter, attention in this study was focused on four different soils, two sand, a silt, and a pure non-swelling clay mineral. Nearly saturated samples were prepared, measured, then incrementally dried and remeasured until the samples could be dried no further by conventional methods. For each moisture content, measurements were normally made at ten different temperatures, which took between three and four hours to complete. There was no control over sample density. Figure 47 serves as an indication of both data quantity and quality. On this figure are symbols that represent data collected for the four primary soil materials at one temperature, 20° C. For most of these data points, data were collected at nine other sample temperatures. Although they are not shown on this figure, data were also collected for the empty holder, for water, alcohols, and a swelling clay mineral called hectorite. Results for hectorite are not reported because physical property information is not currently available for that material. The total number of data sweeps (a sweep of frequencies) collected during these experiments easily exceeds five hundred. The reference above to data quality is in terms of the repeatability of sample dry density as a result of the crude sample preparation technique described in the last chapter. The strong clustering of the data points for each material type indicates how closely sample dry density matched for all samples. Only one string of data is significantly shifted from the others, and that is the data for kaolinite. The four data points that lie to the right of the others represent a sample with a dry density approximately 20% greater than the other clay mineral samples. The starting point for this sample was a batch of kaolinite that had been allowed to dry somewhat before the sample was inserted into the sample holder. 120 D 'o 0 to 00 0 • ^00 0 a 0 0 * , n n « ° ^ 0 / 0 0 8 I 0 0 ^ 0 :n n D 0 0 -4—I—I—I—I—I—I—I—I—I—I—I—1—I—I—I—I—I—I—I—I—r4—I—I—I—I—I—I—I—I—I—I—r2 3 4 SAMPLE MASS ^ ^ Ottsws Sind nnn Sand 000 (g) Sill Figure 47. Data collected at 20° C, 121 000 Kaollnlte Dispersion in Soils as a Function of Moisture Content If the electrical behavior of moist soils is controlled by the amount of and purity of water and how it fills the interstices, then one should be able to observe highfrequency dielectric relaxation and low frequency losses due to conductivity or MaxwellWagner effects or bound water, and these phenomena should be more pronounced as moisture content increases. One should also observe different responses in different soils at the same moisture content because of such factors as specific surface and the ways in which water molecules are attracted to the solid surfaces. Figures 48 through 52 summarize dispersion measurements for the four soil types. Each figure contains data for sample temperatures of -10°C and -f-20°C. These two temperatures were chosen simply to compare soils with frozen water and soils with liquid water. Remember that the vertical dashed line at 2 GHz represents the break point between measurements made with the low-frequency apparatus and those made with the high-frequency setup. Two figures contain silt data, the second one representing results for a slightly different experimental procedure. In the latter case, both frequency range sweeps were conducted before the sample was incrementally dried. It is very satisfying the results for the two setups overlap almost exactly. Conclusions that may be drawn from these data include the following: (a) Frozen soils do not exhibit high-frequency dielectric relaxation, but do reveal some low-frequency losses. (b) Non-frozen soils do demonstrate high-frequency relaxation which does increase with increasing moisture content. These results are quite consistent with those published by Hallikainen, et al (1985). (c) Low-frequency losses in non-frozen soils are also proportional to moisture content. Values are consistent with those of Campbell (1988). Minimum losses in non-frozen soils occur in the 1-2 GHz range which agrees with Hallikainen's 122 (1985) observations. (d) Comparison of sand and silt high-frequency losses at comparable moisture contents demonstrates that there is less free water in the silt because of its larger specific surface (84 mVg versus 10 mVg). (e) If the relative magnitudes and frequency ranges of different loss mechanisms published by Hasted (1973) and reproduced in Figure 29 of this report are correct, then bound water relaxation is a good candidate for some of the low frequency losses seen in these data. One cannot, however, discount ionic conductivity or Maxwell-Wagner losses, or both, as contributing loss mechanisms. (c) The electrical response of frozen soils appears to be relatively insensitive to moisture content within the frequency range measured in this report, but does show some dependence on soil type at low frequencies. From the results of their excellent measurements on real soils ranging from sandy loams to heavy clays, Topp, Davis, and Annan (1980) reported, that in the frequency range of 20 MHz to 1 GHz, the apparent permittivity "was strongly dependent on (moisture content) and only weakly on soil type, density, temperature, and frequency". While the measurements reported in this study have not dealt with density effects, and they do support the contention that temperature effects are minimal for unfrozen soils, they clearly demonstrate that soil type and frequency are important variables. 123 OTTAWA SAND . Temp=20 /|0 MOIS 34.1 31.2 19.3 35 30 25 20 15 10 I 0.01 I I ^— 0.10 I I 111 ll— -1—1—I I I I "1— mnj 100.00 10.00 1.00 20- 18 16 14 12 .// 10 8 6 4 2 0 1—I I I I ii| -I—I—I I 1111 0.01 1.00 0.10 1—I—I I I 1111 10.00 FREQUENCY " GHz Figure 48. Dispersion in a poorly-graded sand. 124 100.00 SAND , Temp=20 40HOIS 14.3 27.2 11.2 24.6 7.3 20.0 35 17.7 29.2 — 30 25- 20 15 10- TTTI 0.01 0.10 1 1—I I I I ll| I I III r- 100.00 10.00 1.00 20- 18- 16- 14- 12 .// 10 8 6 4 2 A 0- -1—I 1 ).01 0.10 1.00 1.00 I I 1 1 1 1 1 1 — I 10.00 FREQUENCY " GHZ Figure 49. Dispersion in a well-graded sand. 125 I I 1 1 T 100.00 SILT Temp=-10 16.8 40.6 I I I I I I I I I I 1 1 1 10.00 100.00 10,00 100.00 .// 0.01 0.10 1.00 FREQUENCY " GHz Figure 50. Dispersion in silt (first set) (continued). 126 23.3 • — • — 45.7 29.8 46.2 SILT . Temp=20 MOIS I " 0.01 I I I 1' I I 1 1 1 " 0.10 T " ' I I ' l l l l l J I I l l l ' I I I J 1.00 10.00 I l ' l l l l l l | ' 100.00 I 0.01 0.10 10.00 1.00 100.00 FREQUENCY ~ GHz Figure 50. Dispersion in silt (first set) (concluded). 127 SILT , Temp=20 HOIS FREQUENCY * GHz Figure 51. Dispersion in silt (second set). 128 KAOLINITE Temp=-10 40 35 30 0.0 ; 49.0 • 4.4 • 55.0 — • • •34.7 • 62.B •— • 41.6 64.3 • 45.3 • 65.6 25 J € 20 MOIS 20.50 56.10 31.80 63.90 ' 11.80 - 20.10 • 40.60 - 45.20 15 10 I I l l | 0.01 I TTTJ I I 1 1l l -1—I I I 1111 0.10 1.00 10.00 100.00 0.10 1.00 10.00 100.00 20 H 10- 1614- 12 .// 10 8 6 4 2 00. FREQUENCY " GHz Figure 52. Dispersion in kaolinite (continued). 129 KAOLINITE Temp=20 0.0 • ^9.0 4.A • 55.0 '34.7 •62.0 ——- 41.6 ' — - 64.3 45.3 65.6 — 1.90 — 20.50 - 56.10 —— 3.84 —— 31.00 63.90 — 11.00 40.60 40 * MOTS :\-4. 35 \ 30 \ 25 20 15 10 I I— I— I— KII I 111n 0.01 0.10 I I I I M i l I 1111 10.00 1.00 100.00 so le \ 16 \ 14 ,\ \\ 12 .// 10 // // /. 8 .//. / 6 //. / 4 2 0 I 0.01 I I 1 1 1 1 0.10 111'j i i 1 i i 111|™r • "i"t i i 1111 1.00 10.00 FREQUENCY ~ GHz Figure 52, Dispersion in kaolinite (concluded). 130 100.00 — 20.10 • • 45.20 Single Frequency Observations Temperature Effects Data collected at -10, -5, -2, 0, 2, 5, 10, 20, 30, and 40°C is quite adequate for visualizing the temperature-dependent behavior of the complex dielectric constant at selected frequencies. These results are shown on Figures 53 through 56 and lend themselves to several observations: (a) Liquid water appears at temperatures less than 0°C, probably somewhere between -2°C and -5°C. This is consistent with results reported by Delaney and Arcone (1982). (b) The real part of the dielectric constant for non-frozen soils is approximately independent of temperature for all soil types. (c) The imaginary part of the dielectric constant for non-frozen soils is not independent of temperature, showing a rise with temperature at low frequencies and a drop with temperature at high frequencies. Because the low frequency data show a rise in the loss term with increasing temperature, an estimate of the activation energy involved in the loss mechanism can be made following the method of Hoekstra and Doyle (1971) and Campbell (1988) which was discussed earlier in Chapter 3. Using their formula (73) Kelvin and the temperature effects data at 100 MHz yields an average (over moisture contents) activation energy of 3.1 Kcal/mole for silt and 3.3 Kcal/mole for kaolinite. These 131 numbers are within the range of values attributed to hydrogen bond breakage. 40 35 30 25 15 I I I I I I -10 0 I I I I I I I I I I I I I I I I I I I I I I I I I I I I' 10 20 30 40 TEMPERATURE'DEG C Figure 53. Temperature effects for poorly-graded sand (8 GHz). 132 40 • 35MC 13.4 31.3 30 25 g! 20 15- 10- I I I I I I I I I I I I I I I I I I I I I II 1 I I I I I I I I•'I I I I • I ' I -10 0 10 20 30 40 40- 35- 30- 25- e" 20 15- 10- 1111111111»111111111111111111111111111111111111 -10 0 10 20 30 40 TEMPERATURE " DEG C Figure 54. Temperature effects for well-graded sand (8 GHz). 133 40 H MC 14.0 35.4 . 26.6 42.3 31.1 47.8 32.1 35 30 35 30 25 .// 20 15 10- 04 i"i r -10 0 r'l' I• I 'I I"I" m ~ n 10 20 • I'I'' 30 40 TEMPERATURE " DE6 C (a) 100 MHz Figure 55. Temperature effects for silt (continued). 134 40 • MC 35 7.2 .35.9 9.3 36,0 IB.a 40.6 23.3 • — •"•45.7 29.8 — —46.2 30 25 e 20 15 10 0-•j-t-n -10 I I I I I I < I 1 I I I I I I I I I 20 10 30 40 40 35 30 25 e." 20 15 10 0- I I I I I I I I I I I I I I -10 10 20 30 40 TEMPERATURE " DEG C (b) 8 GHz Figure 55. Temperature effects for silt (concluded). 135 25 20 15 10 5 0 -n-r 10 I I I I I I I I I 20 10 30 nr 40 40 MC 35 ' 0.0 49.0 4.4 55.0 ——— 34.7 • • . . 62.B —— — 45.3 65.6 64.3 30 25 20 15 r 10 5 0 10 0 10 20 30 40 TEMPERATURE " OEG C (a) 100 MHz igure 56. Temperature effects for kaolinite (continued). 136 MC 40 1.90 20.50 56.10 3.84 31.80 63.90 11.80 40.60 20.10 • • • • 45.20 35 30 25 20 15 J 10 I II I I I I I I I I I I II II II I I I 1 I I -10 I II III I 20 10 I II I I I I I I I I I I 30 40 40 35 30 25 20 15 10 0-10 I I I I II IN 10 20 30 I I I II 40 TEMPERATURE ~ DEG C (b) 8 GHz Figure 56. Temperature effects for kaolinite (concluded). 137 Moisture Effects Figures 57-60 summarize data at selected frequencies as a function of volumetric moisture content for each of the materials tested and at +20° C. What can said immediately from an examination of the data on moist soils presented in this format is: (a) The real part of the dielectric constant for all non-frozen soils is best described by some non-linear relationship with volumetric moisture. Bilinear fits such as those sketched in on each figure would break at critical values of moisture content in the range of 30-35% for the silt and clay minerals and in the 10-15% range for the sands. Wang and Schmugge (1980) reported critical moisture contents in the 20-30% range for the real soils that they studied. (b) The capacitive nature of non-frozen soils as reflected in the real part of the dielectric constant varies with soil type. 138 Silt 000 * A * Kaolinite VOLUMETRIC MOISTURE CONTENT " X Figure 57. Moisture effects for non-frozen soils at 100 MHz, 20° C. 139 * * * Kaolinite 000 10 " oU^: °°° °n * * ** * Y « ' « « I i • I I I' l 'i' i ' i I I I I I ' l ' i 1 1 " : I I I 1 1 I I I I 1 1 1 I I I I 1 1 I I ' l r i 1 1 I 1 1 I I ' l ' i i T i I I I I I I' 0 5 10 15 20 25 30 35 40 45 50 55 60 65 VOLUMETRIC MOISTURE CONTENT " % Figure 58. Moisture effects for non-frozen soils at 800 MHz, 20° C. 140 40 35 30 25 20 15 10 5 0 40 35 n n n Ottawa Sand 000 Sand o oo *** Kaolinlte 30 Silt 25 20 15 10 5 Q, g * 0 I • 1 1 1 1 r i 1 1 I 1 1 n 1 1 I • I r i 1 1 1 1 1 1 1 1 1 1 1 • 1 1I I n I • I n I I 1 1 1 1 1 1 1 1 1 1 1 n 1 1 1 1 1 5 10 15 20 25 30 35 40 45 50 55 60 65 VOLUMETRIC MOISTURE CONTENT " % Moisture effects for non-frozen soils at 2 GHz, 20° C. 141 40- 35 30 25 20 15- 10 11111111111111 H111111111111111111111111111111111 n 111111111111111 0 5 10 15 20 25 30 35 40 15 50 55 60 65 40- 35 n n D Ottawa Sand ^ Sand Q oo *** 30 Silt Kaollnlte 25 20 15 10 Oiff no >» A I I I I I I I I I I I I I IT |"l I I I'jTI'l 'I'l II II I I I I I 'I IIII I llll|llll|llll|' 0 5 10 15 20 25 30 35 40 45 50 55 60 65 VOLUMETRIC MOISTURE CONTENT " % Figure 60. Moisture effects for non-frozen soils at 8 GHz, 20° C. 142 CHAPTER 7: DATA ANALYSES Contained within the following paragraphs are two approaches to interpreting the data shown in the previous chapter. One is an adaptation of the classical technique of representing the electrical response of an heterogeneous mixture by an equivalent circuit that incorporates the concept of series and parallel behavior discussed earlier as well as a model for water behavior that accounts for dielectric relaxation. The new features of this model are the broad frequency band of simulations through a modification of the equivalent circuit for water and the attempt to fix the values of as many circuit elements as possible while giving physical interpretations to the remaining parameters. The other analysis technique involves the application of fractal geometry to model the structure of the porous media and the association of a change in the fractal dimension with the onset of long-range connectivity. While the model has been used in basic research on the properties of coal, this is its first application to soils. A new hypothesis that the soil moisture content (or cumulative pore volume) that bounds the fractal behavior is equivalent to the critical moisture that defines a transition from series to parallel electrical response is tested and yields fractal dimensions for the soils tested within this study that are supported by other references. Equivalent Circuit Representation If one accepts that the electrical response of soil-water mixtures is bounded on the lower end by a "series-like" behavior and on the upper end by a "parallel-like" behavior (Campbell, 1988), then the most sensible simulation should include both elements. With the requirement for both series and parallel response, one very sound approach to modeling the response of moist soil by the method of equivalent circuits is the three-path model first suggested by Sachs and Spiegler (1964) and adapted by Smith (1971). However, the specific representation used by Smith and shown on Figure 36 is not adequate to simulate both low-frequency loss mechanisms and losses at higher frequencies due to dielectric relaxation. A variation of the three-path model was adopted for this 143 study using the following reasoning. Data from the last chapter and from other sources show clearly that the response of moist soils is controlled by the presence of water in the samples. Most of the data show a high-frequency anomalous loss highly correlated with moisture content and a conductivity-like low frequency loss mechanism. In fact the data do not say whether the low frequency losses are due to a conductivity effect in free water or a Maxwell-Wagner mechanism caused by the presence of the soil particles as suggested by many. Lower frequency measurements coupled with a much more thorough understanding or control of the soil chemistry would help answer this last concern. Nevertheless, the best equivalent circuit for the liquid paths has to be that suggested in Figure 38 that includes a highfrequency permittivity, a DC conductivity, and relaxation losses in the 10r20 GHz range of frequency. As for the contribution by the soil, it is felt that a good representation for its electrical response would be one that accounts for a constant wave speed at all frequencies while including a low-frequency loss term. Therefore, the parallel circuit shown in Figure 30 (b) was chosen as the most appropriate representation for the soil component of the mixture. One implication of this selection is that one cannot model anomalous losses within the soil that result from the interaction of the soil and water particles. However, in the range of frequencies covered by these experiments it is probably not possible to detect the difference in losses due to one mechanism or the other. Combining these two sets of elements, the final three-path equivalent circuit used to simulate the response of the soils tested in this study is that shown in Figure 61, where the parameters a, b, c, and d have the same meaning as discussed previously. A thought process for selecting the model parameters is discussed later, and the results of a number of simulations will be shown, but first, a brief presentation of the governing equation for this circuit is given. As was done in previous sections, the macroscopic complex dielectric constant of the moist soil material is modeled as the equivalent capacitance of the circuit shown in Figure 61. Using the w subscript for water and the s subscript for soil, the macroscopic 144 dielectric properties can be written € = + jb€g + (71) where e g (74) e sol I SOI I water c, vz water c. J *1 1-d J Figure 61. The three-path equivalent circuit used for this study 145 In terms of the simple circuit capacitances and resistances, €3 = e/g + + i 1 (75) COi? and (76) 1 + 1 + [^R„2^w2)^ A reasonable set of circuit parameters begins with the same values chosen for water that are shown on Figure 34. They allow for a dielectric relaxation peak at about 16.75 GHz and for the finite values of conductivity losses at frequencies less than 1 MHz. Although the data measured in this study did not go as low as 1 MHz, it was discovered by trial and error that the low frequency conductivity term helped improve the model's fit to data through its interaction with the soil element response. As for the soil parameters, a static permittivity, C„ was taken to be about 2. That only leaves the soil resistivity, R,, to be considered. It so happens that I allowed soil resistivity to be a variable; however, it is rather easy to estimate its range in the following way. The model has associated the low frequency losses with the soil element. They, in turn, go like l/(cojRg) . A typical value of the loss term is 10 at 2 GHz for wet silt. This leads to an estimate of the resistivity as about SxlO^'^s/cm. To obtain values of the weighting parameters, a, b, c, and d, as well as the value of R„ a simple iterative code was developed that minimized the difference between circuit predictions and real data. Smith (1971) used a similar approach to calculate the parameters associated with his three-path model and hinted that it was a rather sophisticated calculation process. I did not find it necessary to do anything more complicated than to increment each parameter in both directions and to calculated the difference between all data points and predicted values for each parameter change taken 146 one at a time. That parameter increment that caused the greatest reduction in difference was applied at each step. Smith (1971) also chose to allow all of the parameters in his three-path model to vary except for the permittivity of water. I chose to hold most of the electrical parameters fixed because they have physical meaning. The permittivities selected for both the water element and the soil element reflect observed values that are associated with wave velocity in each media. The value of resistivity in the water element that results in the anomalous behavior is necessary to locate the peak losses due to dielectric relaxation. In the absence of data at higher frequencies, the frequency of peak loss was taken to be constant for all moisture contents. The soil resistivity was allowed to be variable to help optimize the fits to low frequency loss data. Figures 62-64 contain a comparison of model fits to real data for tan sand, silt, and kaolinite samples at three different volumetric moisture conditions. Besides demonstrating the ability to simulate the real response of the soil samples quite well, it is gratifying to note sensible trends in the weighting parameters. Obviously, as moisture levels rise, the contribution to sample response from the parallel (or long-range connectivity) water element should increase as reflected in an increase in the parameter, c. Because water dominates the electrical response, the soil-related parameter, b, should drop as moisture increases. Similarly, if the series elements can account for MaxwellWagner like behavior in the soil-water-air mixture, then it is also sensible for the losses contributed by the soil particles in the presence of an increasing amount of water to increase as well. This would be reflected in an increase in the parameter, d, which, in general, is observed. 147 40 35 30 25 20 15 IB Doamooo an cipcpBi>BoaoD° ouoawwaRQ 5 Q I .1 I _J L_ 1 10 100 FREQUENCY (GHz) 40 as C w l= 2 4 Cw2"= 76 Rs 5.07110793561E-9 Cs 30 25 =• = Rwl = Ruj2= 20 15 10 I- PMMllrfHT mg gea*WWtjliL n .1 5.888888E-8 1.25E-13 B =• .951482080462 C = .0393254985868 D = .0970299 , (0 100 (a) 7 percent moisture Figure 62. Equivalent circuit model-data comparisons for tan sand (continued). 148 4B 35 30 25 20 IS IB owimuo aauJijjuuixtfiPu uoqi 5 a _J .1 40 1 FREQUENCY (GHz) 10 im as = 2 C w l= 4 Cw2= 76 Cs 30 25 Rs 28 Rwl = Rw2: B = IS IB au nn -J = 2.1550865132lE-S 5.888888E-8 1.25E-13 .914355577452 C ' .0835669675074 D = .09801 RwrtiUP"< IQQ IQ (b) 14 percent moisture Figure 62. Equivalent circuit model-data comparisons for tan sand (continued). 149 .] 4B 1 100 10 FREQUENCY (GHz) as " 2 Cwi «= 4 Cw2= 76 Cs 30 h •= 25 Rs 2B Rwl" Rw2= 5.888888E-8 1.25E-13 B = .795455493409 C = .198338903211 D = .099 15 IB 1.0217900642GE- 5 B (c) 27 percent moisture Figure 62. Equivalent circuit model-data comparisons for tan sand (concluded). 150 4B 35 30 25 • 28 - 15 • IB - "(tfOawjUQ Dat,cc(Sa%% 5 • B - J .1 40 I I I ] 10 FREQUENCY (GHz) 100 35 . Cs = 2 Cwl= 4 Cw2= 76 25 • Rs 4.01007541109E-10 28 RwlRw2= 5.888888E-8 1.25E-13 B = .869456101922 C = .0194597770088 D = .0351609206558 30 • • IS • IB • = 5 • B - .] ] im iQ (a) 10 percent moisture Figure 63. Equivalent circuit model-data comparisons for silt (continued). 151 4B - 35 30 - Z5 - 20 - ]5 - IB 5 • 0 - .] 1 100 10 FREQUENCY (GHz) 4B - as • Cs 30 • 25 • 20 - 15 • IB 5 \ = 2 C w l= 4 Cw2= 76 Rs 2,14242679558E-10 = R w l= 5.888888E-8 Rui2 = 1.25E-13 B « .77923619408 C = .121208323022 D = .0501166414118 \ '•"*^0 0 ri B _J __L IQ IQQ (b) 25 percent moisture Figure 63. Equivalent circuit model-data comparisons for silt (continued). 152 40 r- 35 A 38 25 20 J5 IB S _J B .1 1 100 10 FREQUENCY (GHz) 40 35 = 2 Cu)l = 4 Cs 30 25 J! € Cw2= 76 Rs 9.10474624347E-11 = Ru)l = 20 Rw2= B = 15 18 5.888888E-8 1.25E-13 .620209064457 C = .293510390969 D = ,09801 _J .1 Km IQ (c) 39 percent moisture Figure 63. Equivalent circuit model-data comparisons for silt (concluded). 153 40 35 38 25 e' 20 15 10 aaoewoDMmeu u'daortuciHBgO I l_ .] J 10 FREQUENCY (GHz) I 4EI _1 100 as ' 2 Cwl= 4 Cw2= 76 Rs 3.10090155884E-9 Cs 30 25 = Rwl= 2B Rw2= B = 15 IB 5.888888E-8 1.25E-13 .988829689491 C = .00676569027829 D = .0730840153457 5 0 .] a •ndgtunomuJo 1 10 IQQ (a) 3.5 percent moisture Figure 64. Equivalent circuit model-data comparisons for clay (continued). 154 4B • 35 • 3B 25 • 2B •flu mujuiygJ5 • 10 - 5 D J .1 I I } 10 . I 100 FREQUENCY (GHz) 4B as Cs = 2 C w 1• = 4 Cw2= 76 25 Rs 7.2538305G.4GE-10 2B • Rwl= Rw2= 5.888888E-8 i.25E-13 B = .712059299585 C = .157215804939 D •= .0738222377229 3D 15 IB = S B .1 1 (b) 42 percent moisture Figure 64. Equivalent circuit model-data comparisons for clay (continued). 155 40 35 SB - Z5 - 20 - 15 - IB - 5 • _J B - .1 40 1 FREQUENCY (GHz) 100 10 35 - = 2 C w l= 4 Cw2= 76 Rs 5.19282102695E- Cs 30 • 25 • 20 • IS • la - = Ruj1"= 5.888888E-8 Rw2= I.25E-13 B = .52898037522 C = .451648706606 D = .120798813955 5 • 0 IQ 100 (c) 64 percent moisture Figure 64. Equivalent circuit model-data comparisons for clay (concluded). 156 While the arrangement of elements in the three-path model is physically reasonable and the simulations are very accurate, a number of questions concerning its application remain unanswered. For example, when does one hold circuit element parameter values constant and, if not, how should those parameters be allowed to vary? It might seem preferable, at first thought, to hold all circuit element values fixed and allow only the weighting constants to vary. In fact, this approach was tried, but with only limited success. If only the weighting coefficients could vary, then they would have to account for all of the dramatic changes that take place chemically within the mixture, and they don't appear to be able to do so. In other words, by allowing only weighting coefficients to change implies that the properties of the basic constituents do not change as the mixture changes. Even if the selected parameters for dry soil particles and unbound liquid water are quite accurate, weighting parameters, by themselves, cannot be expected to account for the loss mechanisms due to bound water, or due to MaxwellWagner type effects, or enhanced conductivity due to salts going into solution, or enhanced capacitive response set up by the parallel plate-like structure of the clay particles being filled by water in the interlayer spaces or simply between particles in nonswelling clays. If, then, some of the circuit parameters are allowed to vary, which ones should? Why let the resistivity of the soil element vary and not the water? A number of simulations were tried with R, fixed and R^i free. These results were unsatisfactory for the mixtures that showed increasing simulation values of values with decreasing frequency, with the becoming constant with decreasing frequency. In summary, the three-path equivalent circuit is a promising tool for exploring the complex dielectric behavior of heterogeneous mixtures. Simulations done in this study very accurately reproduce measured responses. The model accounts for both series and parallel electrical responses and possesses some parameters that can be fixed to values measured in other experiments. There are still, however, unresolved issues such as what physical or chemical properties can be associated with the parameters that do vary that should provide a rich opportunity for further research. 157 Fractal Geometry Model and Critical Water Content Campbell (1988) suggested that water may fill the pore spaces of a soil sample in a fractal manner. Pursuit of this concept, particularly as it relates to the distribution of pore sizes in a soil sample, has led to the discovery of related research in pore size distributions (Sridharan, et al 1971; Arya and Paris, 1981) and in the modeling of porous media as fractal geometries (Friesen and Mikula, 1987) that lends itself to a relatively simple fractal interpretation of pore size distribution data. The difference between the following study and Campbell's model is that rather than assuming porefilling water clusters have a fractal geometry, this new approach models the soil structure, itself, using fractal concepts and makes certain assumptions to relate volumetric soil moisture content to the fractal model behavior. A Fractal Model of Pore-Size Distribution The porous soil-water-air structure is taken to be described by a sponge-like structure similar to that shown in Figure C3. Much of what follows has been adopted from Friesen and Mikula (1987), where additional references are given to some earlier work. Their application was toward a better way of quantifying the porous structure of coal. Consider, first, a unit solid cube of soil minerals. Divide the cube equally into m^ subcubes of size d=l/m. Next remove the subcubes necessary to give the basic structure generator shape. (In the case of the Menger sponge shown in Figure C3, m=3, and the number of subcubes removed is 7.) That leaves a number of remaining cubes = N^. But by the definition of fractals given in equation C2, = (d)-^ (77) where D is the fractal "dimension" of the porous media. Now repeat the entire process on each subcube. After this second step, the new subcubes have a dimension of d^ and the number of remaining subcubes is N^^. Continuing to the kth subdivision, we have a particle size of d"^ and the number of remaining subcubes is In other words, the volume of remaining solid material at step k is Vk = (78) Noting that at each step, the particle size is identical to the smallest pore size, and renaming the particle size (d)"' as 1^, and the pore volume is just Vpor, = 1 - {hf" What I now have is a fractal model for a porous medium that relates pore volume (which has to be related to moisture content) and pore size. If I had such a pore-size distribution relationship, then I would have a graphical technique for determining the fractal dimension by plotting the slope of the distribution vs pore size on a log-log plot: / log V = constant - (2-D)log(i) The Fractal Model and Pressure Plate Data Unfortunately, simple pore-size distribution data do not exist. What is available is a limited amount of porosimetry and/or pressure plate data which can be interpreted as pore distribution data through the capillary relationship (Bear and Verruijt, 1987) p = --^COS(0) 159 (82) where p = equilibrium pressure 6 = surface tension of non-wetting fluid 6 = contact angle between fluid meniscus and pore surface D = capillary diameter Substituting the capillary relationship into the fractal model for pore distribution, Vpoxa = 1 - (-f)"" = 1 where K and K' respresent a lumping of constants. Then ^^pore _ -Kfm-2)p^-'^ dp (84) from which / log j = constant - (D-4)log(p) Therefore, the slope of a log-log plot of the derivative of a porosimetry or pressure plate curve is proportional to D-4. As a test of this relationship, the actual pressure plate data from Arya and Paris (1981) for five different soil samples were digitized, their slopes estimated by difference methods, and the results plotted on Figure 65. In general these curves reveal a bilinear response with the break point occurring at approximately the field capacity of 0.333 bars. At pressures higher than field capacity but less than 15 bars, which soil scientists refer to as the wilting point (Wang and Schmugge, 1980), a straight line fit to the data on Figure 72 would result in a fractal dimension for the soil structure of approximately 2.67. The break in the slope of these data roughly corresponds to volumetric moisture contents of 25-40 percent. 160 Arya and Paris Desorption Data Q• -1 -a • I Field Capacity O O O dV.m dP Wilting Point -5 -6 -7 Capillary Gravitational Forces j Forces -8 - "T I I I I I I I I j 'I•I—I—I—I—I—I—I—I—I—I—I—I—I n I 'm I I r T I I I I I I I I I I I I I I I 4 log(P) * * * 70% s-c, 30% sO O O loom, 20-30 cm d I p 5 (cm •Q n l O B n i ' 40-50 cm d 20% s-c, 80% s- tr ir ir ^^^ 40% s-c, 60% B- I Figure 65. Fractal model applied to real soil desorption data (after Arya and Paris, 1981) 161 An interesting question is whether or not one might be able to predict these observed moisture contents from ancillary data. For example, if the fractal behavior of the soil is, in fact, associated with water being held by capillary forces, then the field capacity can be identified with a certain pore diameter through the capillary equation. Taking conditions for water and clean glass (contact angle equal zero degrees and surface tension equal to 72 dynes/cm at 25°C) as an approximation for the elements in moist soil, one can calculate that the range of pressures from infinity to field capacity cover a range of pore diameters up to about 8 microns. Furthermore, if one assumes as did Campbell (1988) that the smallest pores in soil fill first with water, then one could always estimate the moisture content below which the soil structure can be modeled as fractal in the following way. First of all, we have by definition (86) where Wv = volumetiic moisture content V„ = volume of water in a sample Vt = total volume of the sample Vv = volume of voids in a sample n = sample porosity S = degree of saturation of sample But the sample porosity can be rewritten in terms of sample dry density and particle density, and the degree of saturation is (under the pore-filling assumptions) just the integral of the pore distribution curve from zero volume to the cumulative volume associated with the particular pore diameter. The cutoff value of volumetric moisture content for fractal soil sti*ucture response can then be written simply as 162 .5, (87) where Sg refers to the degree of saturation (or the decimal value of cummulative pore volume) at the 8 micron mark on a pore-size distribution curve. In summary, then, if one assumes that soil fills with water beginning with the smallest pores, and if one accepts the small-pore fractal model, then the moisture content below which the soil possesses a fractal structure can be estimated from the sample dry density and cumulative pore volume given by the 8 micron pore diameter mark on the pore-size distribution curve. The Fractal Model Related to Particle-Size Distribution Data In the absence of porosimetry and pressure plate data (which is the case for this research), it would be advantageous to estimate pore volume distibutions from other measurements that are more readily available. At least one simple model for pore distributions does exist; namely, the physicoemperical model offered by Arya and Paris (1981). Their research centered on the need for a model to predict soil moisture characteristics from simple data like particle-size distributions and bulk (dry) density, and their key assumptions were: (a) The particle-size distribution curve can be broken into segments covering a small range of diameters and that the solid fraction within each segment, or particle-size range, has a bulk density equal to that of the natural-structure sample. (b) Each solid fraction consists of uniform-size spheres having the mean radius of the fraction. 163 (c) The pores in each segment are uniform-size cylindrical capillary tubes whose radii are related to the mean particle radius for that segment. If there are n, spherical particles of size R, in the ith particle-size segment of the distribution curve, then Kn where Vp^ = solid volume of the ith fraction per unit sample mass IVj = solid mass/unit sample mass in the ith fraction which comes from the particlesize distribution curve Pp = particle density Arya and Paris then let the total pore volume for the ith fraction be represented by a single pore of radius r, and length h; that threads the volume occupied by the particle-size fraction. Thus V„ = TCZ/j]; = W, (89) P/ where = void volume of the ith particle-size fraction per unit sample mass e = void ratio = volume voids/volume solids Not knowing exactly how this single pore threads the mass of spherical particles, Arya and Paris assumed that the total pore length would be something greater than n;2R; and introduced an empirical factor, a , defined by 164 Aj = (90) Then dividing Equation 89 by Equation 88 and substituting Equation 90, one arrives at their final relationship between pore radius and particle radius in the ith particle-size fraction 1/2 = (91) Ri where % comes from Equation 88 and the manner in which the particle-size distribution curve gets subdivided. By assuming that the pores fill with water beginning with the smallest size and accumulating the total pore volume as one progresses with the calculation, a volumetric water content can be calculated = E ^VjPb j=l At the ith particle-size fraction, the capillary suction comes from Equation 82. Thus one can track a soil tension vs volumetric water content. Arya and Paris did this for several soils and adjusted the empirical parameter, a , for each to give the best fit to the experimental data. For five different soils, the best fit values of a ranged from 1.35 to 1.39. In summary, given some knowledge of the a parameter that first appears in Equation 90, we have a model for converting a particle-size distribution curve into a pore-size distribution curve and an opportunity to apply the fractal geometry model and compute a limiting volumetric moisture content, below which the soil structure can be taken as a fractal geometry. As a test of this model, consider its reapplication to the data that Arya and Paris provided in their paper. Figures 66 to 70 contain a reproduction of their particle-size distribution curves for five of the soils and soil mixtures that they studied along with a 165 pore size distribution deduced from their soil tension data using the standard capillary relationship. The contact angle was taken to be zero degrees and the surface tension for pure water at 25°C was used to convert pressure data to pore diameters. Maximum volumetric moisture (100% saturation) was taken to be defined by the dry density of the sample as before. The solid line on each figure is the particle-size distribution curve, and the dashed line is the pressure-plate derived pore-size distribution. The open circles on the curves represent the results of calculating a pore-size distribution curve using the model above. 166 100 90 80 H « 60 § CO a w 40 PM 100 1 0 1 2 -3 -4 -5 log (Diameter) ~ millimeters Figure 66. Comparison of measured and predicted pore-size distributions; 70% silty clay, 30% sandy loam (Arya and Paris, 1981). 167 100 CO 20- O, 0.87 OO CD 1 0 100 1 2 -3 4 -5 log (Diameter) ~ millimeters Figure 67. Comparison of measured and predicted pore-size distributions; loam 40-50 cm depth (Arya and Paris, 1981). 168 100 I H o/ o. 3 50 C/} CO O/' i o /• 100 1 0 -1 2 -3 -4 -5 log (Diameter) " millimeters Figure 68. Comparison of measured and predicted pore-size distributions; 40% silty clay, 60% sandy loam (Arya and Paris, 1981). 169 100 c 90 o o 80 o I H 70 ^ 60 o / / / / o o ^ ^ / H 7^ T-.° 40 1 1 / 50 \ Sg = 0 . 5 8 \ o'. * 1 1 30 /A\ / 20 ^ 6 1 X. 10 1 1 oP • 11 11 1 11 1 1 1 1 1 11 11 111 -1 -2 -3 -A log (Diameter) ~ millimeters Figure 69. Comparison of measured and predicted pore-size distributions; loam 20-30 cm depth (Arya and Paris, 1981). 170 -5 100 i B I Sg = 0 . 5 6 1 Pi 100 -1 -2 -3 -4 log (Diameter) ~ millimeters Figure 70. Comparison of measured and predicted pore-size distributions; 20% silty clay, 80% sandy loam (Arya and Paris, 1981). 171 Other than the large pore size discrepancy for the sample shown in Figure 70, the two techniques for calculating pore-size distributions (one from pressure-plate data and one from pore-size distribution data) produce quite similar results. As a brief aside, it is interesting to note that Campbell's earlier assumption of median pore size equalling median particle size does not hold for these data. One might even say that based on this limited model exercise, the median pore size is approximately an order of magnitude smaller than the median particle size. Now, how well does the fractal model for small pore structure apply to the Arya and Paris data? From their data I can obtain the moisture content for each sample at field capacity, and from the previous figures, I can read Sg, the degree of saturation at a pore diameter of 8 microns. Equation 87 can then be used to compute the moisture content at the limits of fractal soil structure (particle density is taken to be 2.65 g/cc). These results are summarized in the following table and show a remarkable correlation. Table 3 Fractal Cutoff Moisture Content vs Measured Field Capacity Moisture (data from Arya and Paris, 1981) Sample Dry Density Peg Sat-8 mic Mvlww {Mvlndd cpaci^ B 1.400 0.81 0.38 0.39 C 1.416 0.87 0.41 0.42 D 1.480 0.67 0.30 0.33 E 1.456 0.58 0.26 0.28 F 1.517 0.56 0.24 0.25 172 Fractal Model Applied to This Study Is the fractal structure model compatible with the data collected in this study? Most importantly, can I correlate the electrical response of the test soils with the fractal model? First of all, let me state the hypothesis: (a) Soil water fills the smallest pores first in a dry sample. (b) Soil structure can be modeled by a fractal geometry in the range of pore sizes in which water is held by capillary forces. (c) Within the small-pore fractal range, the fractal dimension can be calculated from knowledge of either the pore volume-pressure relationship (Equation 85) or the pore volume-pore size relationship (Equation 81). (d) The fractal model fails at pressures less than field capacity (1/3 bar), which, using the capillary equation and assumed values of surface tension and contact angle for water, is equivalent to a pore diameter of about 8 microns (in other words, the fractal model applies for pore diameters less than 8 microns). (e) The upper bound on moisture content associated with the small-pore fractal model is the field capacity, which can be estimated from the product of sample porosity (which is calculated from dry density and particle density) and the degree of saturation at the 8 micron mark on the pore volume-pore size curve. (f) The critical moisture content at which there is a transition from series to parallel-like electrical behavior is identical to the field capacity. 173 Items (a) through (e) were examined in the previous sections. It is item (f) that contains the physics of the relationship between soil moisture levels and the electrical response of the soil. What it says is that long-range connectivity becomes achievable when the pores governed by the capillary equation become filled with water. At moisture contents higher than this value, free water exists throughout the sample; i.e., the sample is no longer just "moist," it is now "wet". The ideal set of data required to test this hypothesis would include enough electrical response measurements as a function of moisture content to determine the critical moisture content and field capacity measurements taken from pressure plate data at the same sample densities. Unfortunately, these data do not exist. The best that I can do is to use the electrical property measurements in this study and estimate the field capacity of my soil samples from the particle-size distribution curves using the Arya-Paris model to generate a pore-size distribution. This requires values of the fitting parameter, oc . Although Arya and Paris (1981) were able to use values of the model parameter that fell within the range of 1.35 to 1.39 for the five soils discussed in the previous section, I would argue that these values may not be suitable for all soil samples. Their soils were well-graded and all contained some sand as well as some clay. The soils used in this study are, relative to the Arya-Paris soils, poorly graded. Recalling that the a parameter was a way of accounting for the fact that the length of a pore associated with an assumed spherical particle is something more than twice the particle radius, it seems likely that it could vary considerably for soils that are poorly graded and for which the basic particle shapes are distinctly different. For example, I see no reason for not believing that for the flat plate-like structure of kaolinite, that the pore length contribution from each particle should be about the particle diameter. In other words, a for kaolinite should be on the order of 1.0. At the other extreme, Ottawa sand in a close-pack geometry should possess pore channels that hug the surface of the almost-spherical particles which means each particle contributes a path length much greater than its diameter. This would imply the largest values of a . 174 Since I have no knowledge of the a parameter for my soils, I will take the reverse approach to testing the above hypothesis. I will assume that the critical moisture content is the field capacity, which, in turn, is represented by the degree of saturation shown on the pore volume vs pore size curve at the 8 micron diameter mark. Following the generation of several pore-size distribution curves from my particle-size distributions, I can estimate a for each soil by the value that yields a pore-size distribution that matches the critical moisture and the 8 micron intersection. With this value of a I can generate the most likely pore-size distribution curves for each of my soils and graphically determine the soil fractal dimension for each soil type using the relationship in Equation 81. The final test of the hypothesis will be to compare those values for firactal dimensions with data for similar soils that are published in the open literature. First of all, Figures 71 to 74 show the results of applying Arya and Paris' model to each soil used in this study with a allowed to vary from 1.0 to 2.0 in increments of 0.1, Secondly, values of critical moisture were assigned to the four soils based on the moisture-related data shown on Figures 57-60. These values, along with soil densities are used to calculate the required degrees of saturation shown in Table 4. Armed with this information, one can return to Figures 71 to 74 and estimate the value of a associated with each soil that are also shown in the table. Table 4 Estimating tt Soil (Mv)criUcal Pdry for the Soils in This Study n P part a J] a 00 Ottawa sand 0.0 1.7 2.67 0.36 0.0 tan sand 0.10 1.7 2.66 0.36 0.28 1.8 tan silt 0.35 1.4 2.71 0.48 0.73 1.3 kaolinite 0.35 0.9 2.61 0.66 0.53 0.9 * assumed to be like that of the tan sand 175 100 90 80 § 70 60 50 40 30 1 /mmp• ^iiiiiiiiil iiiijiiiii iiiiii i iiiij# illlnllili I a = 1.0 • 1 * « I i « a = 2.0 liM/M... 20 10 > 1'*1'' 11 i * w \ \\ i#7' f I I I I 11 TT-rr -2 -3 IIIII IIIIIII -4 log (Diameter) ~ millimeters Figure 71. Predicted pore distribution curves for Ottawa sand. 176 -5 PERCENT SMALLER BY WEIGHT PERCENT SMALLER BY VOLUME 100 1.0 90 V' g H <± 30 2.0 100 1 0 1 2 3 4 log (Diameter) ~ millimeters Figure 73. Predicted pore distribution curves for tan silt. 178 5 100 \ « = 10 / 90 ,.V,/ \ •\ \ ' • \ ... A 80 / /• i• / . f , i / / i ' y' ' • r \ \ ''l I' i 'L 'l 70 < . / 60 % 50 : / / : . & / 40 i Y £ '/ • / •'V / / 30 / A/ / : / / 20 10 , 1 « 4 / / A/ /:// / / l /y' • • , < « I . I « « \ 1 1 1 1 • « / * • 1 ^1 1 0 -1 -2 -3 -4 log (Diameter) ~ millimeters Figure 74. Predicted pore distribution curves for kaolinite. 179 -5 Finally, I can now take the slopes of the calculated pore-size distribution curve for each soil (using the estimated a value) and infer from those results a fractal dimension for the soil structure in the manner described in a previous section. Figures 75 to 77 show the results of this exercise and include the estimate of the soil fractal dimensions. Ottawa sand is not reported because it appears to have no critical moisture electrical response. Although very little data exist in the literature on fractal dimensions of minerals and/or soils, the extremes are supported by at least a couple of references. For example, Pfeifer (1984) calculated a fractal dimension for kaolinite of 2.92 using a dye adsorption and photometric measurement technique. On the other hand, he and other colleagues (Farin, Avnir, and Pfeifer, 1984) report from another source a dimension of about 2.15 for quartz sands using a nitrogen adsorption technique. In summary, the above exercise does not prove, conclusively, that the critical moisture content of a soil that identifies a transition from series to parallel electrical response is equivalent to its field capacity, but it does demonstrate that assuming this hypothesis is true results in a fractal structure model for the soil that is supported by a limited amount of data reported in the literature. 180 8 a = 1.8 = SLOPE == 2-D = 0, D = 2.0 / ... i o o °o 0 ' o 9,,.o. O o' oo OcA) o < io o 1 1 1 • 1 1 1 1 1 1 1 1 1 1 1 i 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 i 1 1 1 1 1 1 1 1 1 1 1 1 II 1 rr ri-i-i -6 -5 _4 _3 _2 log (Diameter) " millimeters Figure 75. Inferred fractal dimension for tan sand. 181 —J 8 a = 1.3 SLOPE = 2-D = -.5, D = 2.5 Q& I I I IIII I I I I I II I I I I I I II I II I I I II I I I I I I I I I I I I j I I I III I I -6 -5 -4 -3 -2 log (Diameter) ~ millimeters Figure 76. Inferred fractal dimension for tan silt. 182 -1 8 a = 0.9 SLOP z = 2-D = -.9, D = 2.9 4 3 o o o o Oo o 9o o o o 11111111111111 1111111 1111111111111111111 111111111111111111 -6 -5 -A -3 -2 log (Diameter) " millimeters Figure 77. Inferred fractal dimension for kaolinite. 183 —1 CHAPTERS: CONCLUSIONS The response of moist soils (or any other moist heterogeneous mixture) to active microwave sensors is controlled by surface geometry and the dielectric properties of the medium. A review of the literature reveals that what data do exist on dielectric properties suffer from a number of shortcomings. In most cases, the soils are simply not properly characterized in terms of their physical parameters such as the distribution of particle sizes, the dry density under which tests were conducted, or the temperature of the soil, all of which have some impact on the dielectric properties. The volumetric moisture content of the soil, which is the usually the most important factor controlling dielectric response, is often not specified or is not computable. And, finally, the data are usually collected at a limited number of frequency values and over a very limited range of frequencies. Attempts to model the dielectric response of these soils are also limited in several respects. Many models are strictly empirical and, therefore, apply only to that data set with its few variations. Equivalent circuit models have been successfully applied to data over small frequency range, but the model parameters seldom have any physical meaning. This study began with an attempt to supplement existing data on the complex dielectric properties of moist soils. A coaxial measurement apparatus for which measurements are controlled by a vector network analyzer was fabricated to allow property measurements in nominally moist soils to frequencies above 10 GHz which allowed one to see the beginning of the dipole loss mechanism. Sample temperatures were controlled by an external bath, and sample moisture contents were varied by incremental drying of the samples. Actual test moisture values were determined after each series of measurements were completed by weighing the completely dry sample. Distilled, deionized water was used to saturate dry soil samples. Soil samples were chosen to reflect a broad span of responses. They varied from a poorly-graded clean sand to a very pure non-swelling clay mineral. The physical properties of the soil samples were characterized by the development of grain-size distribution curves and measurements of specific gravity and specific surface. With 184 hindsight either mercury porosimetry measurements or pressure plate measurements should have been conducted to obtain a relationship between pore volume and pore size. As expected, the soil dielectric response was found to be a strong function of volumetric moisture content for non-frozen soils. Low frequency losses due to either ionic conductivity or Maxwell-Wagner effects, or both, were quite apparent, as were high frequency losses due to the free-water dipole relaxation loss mechanism. Minimum losses occur in the 1-2 GHz range. Sample temperature did not have a significant effect on the dielectric response of non-frozen soils over a range of temperatures from 0° C to 40° C at any frequency. Data collected on frozen soils at -10°C and -5° C also did not show any significant difference. The pronounced transition from frozen to nonfrozen soils took place within the -5° C to 0° C range, but the spacing of data within this range was not enough to completely define that transition. Data collected in the study supported the claim of previous researchers that there is a definite transition in normal soils from a series-like capacitive response to a parallel like response with increasing volumetric moisture content. For the silt and clay samples, that transition occurred in the 30-35 % moisture content range, v/hile for the well-graded sand, it occurred in the 10-15% range. For equal moisture contents, then, the sands will always have a higher permittivity than silts and non-swelling clays. Two approaches were taken to modeling the dielectric response of these soils. One was an equivalent circuit that allowed for both series-like and parallel-like responses of soil and water elements. An iterative solution technique was used to arrive at the weighting parameters for each element that varied in a sensible way with moisture content. The permittivities assigned to the circuit elements were held fixed at acceptable values and excellent fits to the data over the entire range of frequencies were achieved for all of the soil types. The second model was one that assumed the small pore geometry of a porous medium like soils could be described by fractal methods. The result was an expression for pore volume as a function of pore size or capillary pressure that included a fractal dimension of the soil structure. When applied to a set of pressure plate data taken from the open literature, this model showed that the fractal geometry assumption held for the small-pore portion of the data and collapsed at the field capacity of the soils (1/3 bar pressure, or 8 micron pore diameter). The model goes on to hypothesize that the critical moisture content of the soils at which its electrical response makes a transition from series to parallel behavior is, in fact, the moisture content at field capacity. Lacking a measured pore volume distribution for these soils in terms of either pressures or pore size, a pore volume-pore size distribution was estimated from the measured particle size distribution curves and used to evaluate bounds on the small-pore fractal dimension of the idealized soils used in this study. The results indicate that while real soils are likely to have fractal dimensions on the order of 2.5 to 2.7, the value for coarse sands tends toward 2,0 and the value for fine clay minerals tend toward 3.0. 186 APPENDIX A: SKIN DEPTH CALCULATIONS Obviously, skin depth effects preclude the utility of practical electromagnetic devices to penetrate to distances comparable to acoustic devices, but nevertheless, low frequency devices have some utility (Berlin, et al, 1986; Farr, et al, 1986; Schaber, et al, 1986). As a means of quantifying the penetration of electromagnetic waves into earth media in terms of the complex dielectric constant of the media, consider the following arguments. Let the one-dimensional propagation of an electromagnetic wave into some medium be described by an amplitude function like g i(kx-w t) (Al) where i = the symbol designating the imaginary quantity \/-l k = the complex wave number (0 = the radial frequency. Furthermore, let (A2) where p is called the phase constant and governs how a lossless medium propagates a wave, and a is called the attenuation factor and controls how the amplitude of the wave decreases with distance traveled through the medium. Then the amplitude of the wave goes like (A3) i.e., a traveling wave combined with an attenuation factor. 187 If skin depth is defined as that distance over which the wave amplitude decreases by a factor of 1/e, then the skin depth, 8 , is 5=— a (A4) But the wave number can also be written (A5) V c where v = the phase velocity in the medium, c = the speed of light N = the complex index of refraction, Thus 2 _ ^ C (OH // (A6) or the skin depth is inversely proportional to the imaginary part of the complex index of refraction. However, this study focuses on the complex dielectric constant, € , as the measure of soil electrical characteristics. e=e^+ie^^=N^ (A7) Following several substitutions and simplifications, one can show that _1 2 6= 271 188 (A8) where is the free-space wavelength. A visualization of this equation for the range of values of permittivity ( ) and loss tangent ( €^V ) normally found in soils is provided in Figure Al. This figure does not represent any new science, as the same information has been presented in other forms (von Hippel, 1954). It is, however, a more condensed way of visualizing the skin depth that has not been observed elsewhere in the literature. Figure Al is valid for any interpretation of loss tangent; i.e., whether one is speaking of a loss mechanism due to actual charge migration or one due to dielectric relaxation. 100. 10. 1. .01 .1 1. 10. SKIN DEPTH / WAVELENGTH Figure Al. Skin depth as a function of wavelength, permittivity, and loss tangent. 189 Another useful skin depth relationship found in the literature is the nomograph generated by Albrecht (1966) and reproduced on Figure A2. Albrecht first developed an empirical relationship that associated ground conductivity to gravimetric moisture content and ground temperature at low frequencies. He then related skin depth to wavelength and conductivity by the usual equation for highly conductive materials (assumes that the conductivity phenomenon dominates the loss mechanism in soils at low frequencies): 6 (A9) Ttjxca where the units used are MKS, and the skin depth is in meters and the ground conductivity is in units of mho/meter. The magnetic permeability was taken to be 4 %X10 Henrys/meter. The way to use this nomograph is to select a moisture content and move on a horizontal line to the appropriate temperature curve. Then move on a vertical line to the appropriate frequency curve. From that a point a horizontal line to the right hand scale reveals the desired value of skin depth. There is no way to relate Albrecht's results to those of Figure Al, because it begins with an unknown empirical relationship among moisture, temperature, and conductivity. Hoekstra and Delaney (1974) also published some useful data on plane wave attenuation in moist soils at higher frequencies. These data are shown in Figure A3. They concluded that attenuation was relatively independent of soil fabric and that either passive or active microwave sensors could obtain a measureable response from only about the first five centimeters of ground below the surface. 190 >-4 2 10"^ 2 ^ lOT" 2 CONDUCTIVITY * 10" (mho/m) Figure A2. Skin depth nomograph (Albrecht, 1966). 191 Freq 4.0 «10". 5.0 «10' .30 .10. WATER CONTENT (g/cc) Figure A3. Attenuation in moist soils (Hoekstra and Delaney, 1974). 192 APPENDIX B; REFLECTION AND REFRACTION AT PLANE INTERFACES Remote sensing of the environment usually involves the interaction of electromagnetic energy waves with solid media at some air-media interface. For example, an active radar transmits waves toward a surface (usually at some oblique angle) and measures what is reflected back toward itself (backscatter) or toward some other receiver (bistatic reflection). Passive devices receive energy emitted by the surface (and the atmosphere within the antenna's beamwidth) due to internal sources as well as what is reflected from natural sources such as the sun, the atmosphere, and surrounding terrain. An idealization of the wave interaction phenomenon is shown in Figure B1 where an incident plane wave, defined by its wave vector, kj , gives rise to a reflected wave, iCjj , and a refracted (or transmitted) wave, k.^ . (Nearly every textbook on electromagnetic theory has a development on reflected and refracted plane waves. What follows was taken primarily from Jackson (1975) with a slight change in notation. Units are cgs.) The wave vectors are related to the dielectric properties of both media by l^ll = I-^rI = -fc = — = 141 = ic, = ^ ^ where <*) = radial frequency N = complex index of refraction € = complex dielectric constant = magnetic permeability (hereafter taken to be unity) c = speed of light in a vacuum (3x10' m/sec) 193 (»!) 2^-2 (B2) The plane of incidence is taken to be that plane formed by the k j and i5 vectors; i.e., the plane of the paper. MATERIAL 2 MATERIAL 1 Figure Bl. Wave vectors at a plane interface. Under these ideal conditions of plane waves at a smooth interface, one can write the following expressions for the three sets of electromagnetic energy fields. 194 INCIDENT 1 J.Ci REFRACTED (TRANSMITTED) £„ = E, 'r •"To Br = = -yW REFLECTED 4 = 4= = ^4x4 195 By insisting that the phase factors ( k'X terms) are all equal at z=0, one is led to Snell's law and the fact that 62=8^ . sinGj _ icj. _ sinSy k P2E2 _ \ (B9) N What one desires to have are expressions for the amplitudes of the reflected and refracted waves, for it is the wave amplitude (or voltage amplitude) that is measured by a receiver unit. Amplitude relationships can be found by applying the electromagnetic boundary conditions; namely, that the normal components of electric displacement, D , and magnetic induction, B , are continuous across the interface as are the tangential components of the electric field, E , and the magnetic field, H . These conditions may be written as: (BIO) (BID = 0 (B12) xi5 196 0 (B13) The Fresnel Coefficients At this point two further idealizations are made before finally writing down the expressions for reflected and refracted wave amplitudes. One is to assume that the incident electric field vector is perpendicular to the plane of incidence (into the paper with respect to Figure Bl), and the other is to assume it is parallel to the plane of incidence (lying within the paper with respect to Figure Bl). In radar parlance, the terms are "horizontal polarization" and "vertical polarization", respectively. Due to redundancies generated by Snell's law, one only needs the third and fourth boundary conditions expressed in equations B6 to arrive at the final expressions for amplitude ratios, which are known as the Fresnel reflection coefficients. HORIZONTAL POLARIZATION E. '0 _ 2COS0J E, (B14) cosGj. + —. E .^0 _ E, COS0J - —. 1^2 \ COS0J + — . 197 - sin^0, - sin^e. HlGi (B15) - Sin20, VERTICAL POLARIZATION '0 2COS0J _ (B16) E, COS0J + \^2^1 COS0J ^0 _ E, \ 1 - lifisin^Sj. V-2^2 \^2^1 _ f i l l [^1^2 COS0J + ^ j4fi (B17) sin^0j Reflection From Lossless Media The previous equations apply to lossy as well as lossless media. For lossless media, the dielectric constant and index of refraction are real quantities. Refracted waves propagate without attenuation, and both refracted and reflected waves retain the polarization of the incident wave. Of practical value to the remote sensing arena is the variation with incidence angle of the reflection coefficient in a lossless media. Because received power in a sensor goes as the square of the amplitude of the signal and is proportional to the radar cross sections of the object being illuminated, scientists and engineers have a simple model between electrical properties of the media and its response to electromagnetic energy. What is deduced from the sensor measurements and a knowledge of the surface slopes is an apparent dielectric constant or apparent permittivity of the medium. Figure B2 is a representation of how the apparent permittivity of a medium affects the reflected amplitude ratio as a function of incidence angle. The angle at which the reflected wave is totally eliminated in the vertical polarization case is referred to as the Brewster angle ( tan0g = = y(H2^2)/(M'i^i) )• For angles less than the 198 Brewster angle, the reflected wave suffers a phase reversal. The values of dielectric constant were chosen to span the range of values from dry soil to pure water. HORIZ POL VERT POL e =80+10 e =10+10 e 0 =2+10 10 20 30 40 50 E0 78 8B 90 INCIDENCE ANGLE, DEGREES a. Normalized reflected voltage amplitudes. Hpol ; Vpoi X" for angles > + 180. + 160. Brewster 0 + 140. + 120. 1 «—I II + 100. +80. II ! +G0. +40. 1 i II Vj : vu +20. +0. -20. -40. -60. -80. -100. -120. -140. -160. I -180. 10 20 30 40 50 G0 70 60 • I 90 INCIDENCE ANGLE, DEGREES b. Phase shifts. Figure B2. Reflection amplitudes and phase shifts for non-magnetic lossless materials. 199 Reflection From Lossy Media Appendix A gave some indication of what happens to electromagnetic waves incident on a lossy media ( > 0 . The refracted, or transmitted, wave is attenuated. Furthermore, for oblique angles of incidence, surfaces of constant amplitude are no longer parallel to the surfaces of constant phase, resulting in what is referred to as an inhomogeneous wave (Stratton 1941). In fact, the index of refraction for lossy media becomes a function of the angle of incidence of the incoming wave. As lossiness goes up the refracted wave vectors get closer and closer to normal to the interface. Similarly, it is possible to rationalize the second expressions in equations B7 and B8 to obtain ratios of reflected amplitudes to incident amplitudes. One representation of these relationships may be found in Stratton (1941). The effects of adding a loss term to the dielectric constant are to enhance reflectance and to reduce the destructive interference mechanism that causes the total loss of reflected energy at the Brewster angle. A visual comparison of the effects of loss factors in materials is shown in Figure B3. It is not inconceivable that if one had good reflectance data at a single frequency over a large range of incidence angles (enough to include the minimum on the vertical polarization curve) and at the wavelength of the sensor could assume smooth interface conditions, then one could iteratively obtain a first approximation to the complex dielectric constant for the material near the surface. 200 HORIZ POL VERT POL e =10+110 5 § M_i Q. e =10+10 U b a. _L I ' _L I e 10 20 30 40 50 B0 IVi.,1 70 80 96 INCIDENCE ANGLE, DEGREES a. Normalized reflected voltage amplitudes. HORIZ POL + 180. + 160. + 140. VERT POL e =10+10 + 120. + 100. +80. +60. +40. £ =10+110 +20. +0. -20. -40. e =10+10 -60. -80, -100. -120. -140. 6 =10+110 -160. -180. 0 10 20 30 40 50 BQ 70 80 90 INCIDENCE ANGLE, DEGREES b. Phase shifts. Figure B3. Reflection amplitudes and phase shifts for non-magnetic lossy materials. 201 APPENDIX C: FRACTAL MODELS OF SOIL STRUCTURE Fractals In recent years there has been some very interesting work done on the conductive behavior of porous media using the concept of self-similar geometries, or fractals, to model the behavior of certain physical quantities of that media. Mandelbrot, the guru of fractals, defines self-similar shapes as those in which a certain part of the shape can be broken up into N smaller parts, each looking like the original, but reduced in size by a fractional factor labelled, r (Mandelbrot, 1983). The fractal dimension, D, of this selfsimilar shape can be written as D = (Gl) or N = 1 (C2) Mandelbrot insisted that the exponent, D, be thought of as a dimension because it arose from a method of measuring the perimeter of an object having an irregular boundary. Fractals are often used to generate complex, or textured, curves or surfaces from very simple initiator geometries. For example consider Figure CI in which the initiator (or the initial segment of the geometry which will be fractalized) is a solid square (precisely stated, each side of the square is an initiator), and the generator (or the desired geometry) is the broken line segment as shown. Moving clockwise from the initiator, the first two applications of the generator are shown at full scale, while the next two are enlarged to show the edge detail. Figure C2 shows another construction initiated with a solid square in which the generator results in the formation of both "lakes" and "islands". 202 Figure CI. A fractal snowflake of dimension 1.5 (Mandelbrot, 1983). #- • rn rn I UU . niU' UV W ftft N=18 r=l/6 D~1.6131 OR BK . "ow s [R s " .»« Bh> BK # BR , BA # * Ub MM S #1 t] BM E #- #' * * *• « Figure C2. Fractal islands and lakes of dimension 1.6131 (Mandelbrot, 1983). 203 Soil Structure A third example of a fractal construction that is useful for studying porous media, and may be particularly useful for studying soil structure, is that shown in Figure C3 in which the initiator is a cube with square holes centered on each face and joining in the center. The generator is the same object but reduced by a factor of 1/3. Therefore, in the first stage, the initiator is divided into 20 smaller cubes, resulting in a fractal dimension of 2.7268. Mandelbrot referred to this construction as a Menger sponge. Friesen and Mikula (1987) used the Menger sponge to model the fractal dimension of the surface area of coal in such a way that the slope of the isotherm (fractional volume change vs pressure for porosimetry measurements) plotted on a log-log scale is linearly related to the fractal dimension. At least one recent study attempted to measure the fractal dimensions of sandstone solid-void interfaces (Katz and Thompson, 1985). Measured values of interface fractal dimensions on five different samples using electron microscope techniques varied from 2.57 to 2.87. Work has been done very recently on modeling the microstructure of soil using fractals (Krepfl, Moore, and Lee, 1989; Moore and Krepfl, 1991). A two dimensional representation of a three-dimensional model using hexagonal-shaped flakes as the generator is shown in Figure C4, Such models may be useful as a representation of pure clays. This particular fractal structure is strongly reminiscent of microphotographs of the pure kaolinite mineral structure. 204 Figure C3. The Menger sponge, fractal dimension 2.7268 (Mandelbrot, 1983). generator Figure C4. A fractal representation of soil fabric (Moore and Krepfl, 1991). 205 APPENDIX D: SOIL PROPERTIES Particle size distribution curves for each of the four soils used in these experiments were developed by technicans in the Geotechnical Laboratory at the US Army Engineer Waterways Experiment Station, Vicksburg, MS, and are shown on the following four figures. Also included on each figure is the specific surface for that soil which was measured by laboratory personnel at Soil Analytical Services, Inc., in College Station, TX, using an ethylene glycol adsorption technique. 206 U,S. SWIDWD SIEVE OPENING « INCHES I J 1 1 1 i 1 1 100 US. SBNOARD SIQE NUieEHS 1 1 y 1 1 f HYDROMETER 1 1 1 1 10 90 I i D N) O -0 fi 80 20 70 30 0' 40 I ^ 50 50 E 1 i i u. 60 t 40 ca ^30 70 20 80 0 10 1 c« 90 1 0 100 500 COBBLES u. 50 10 avRSE FL GRAVEL 1 GS nuE 2.67 5 0.1 I 0.5 GRAIN SIZE IN MUJMEIERS CONGE 1 NATW^ SAND ItCIMM 1 0.05 0.01 100 0.001 0.005 SILT or CLAY FINE OROS! PROJECT CLASSFICAHON SAND (SP). WHfTE GRADATION CURVE specific area = 10 m^/g I LABORATORY USAE WES - STF/GL BORING NO. OEPTH/ELEV SAND (WHITE) SAMPLE NO. DATE 19 AUG 91 U.S. SIWOARO SIEVE NUVEERS US. srmm soe opening n inocs 6 100 43 " "' ' 2 1 15 1 43 ~1 ~3 3 4 6 810 HYDROMETER IB 20 30 40 50 70 100 140 200 T—I—nTTT II I I If 90 80 70 i s \ 0' 1 I 50 Q s 00 i 3 ^ 40 a! 30 s 20 CD S 10 0 1 CL 100 500 COARSE COBBLES PL LL 10 50 PI GRAVEL S 0.1 1 05 GRAIN SIZE IN MlilMETERS 1 fNE CQWSE 1 ^ 2.66 NATW.X SAND KOIUN 1 0.05 0.01 0.005 0.001 SILT or CLAY FINE ORG.% PROJECT OASSFCATION SAND (SP), BROWN m 7/ specific area = 10 m^/g SORING NO. DEPTH/ELEV GRADATION CURVE LABORATORY USAE WES - STF/GL SAND CTAN) SAMPLE NO. DATE 19 AUG 91 U.S. swrow® SIEVE OPENINS N WCHES 100 1 3 - 1 3 1 1 1 r 1 1 1 1 1 1 U& 5ONDAR0 SIEVE NUkCERS HYDROMETER n 1 T- 1 T 1 90 i 80 1 70 I I 60 % t ^ 50 \ i c > \ g 40 \ S 30 \ 20 \ 10 100 500 COBBLES LL PL 27 10 50 5 22 1 GS 0.05 SAND GRAVEL tXWS 0.1 I 0.5 CRAM SIZE N MILUMETERS FWE 2.71 CQWSE 1 NWW.X IfSIUM 1 0.01 0.001 0.005 SILT or CWY FINE CROS PROJECT O/SSFICAHON CLAYEY SILT (ML). BROWN; TRACE OF SAND specific area = 84 m^/g GRADATION CURVE LABORATORY USAE WES - STF/GL SORING NO. DEPTH/ELEV SILT (TAN) SAMPLE NO. DATE 19 AUG 91 U5. SWNOARO SEVE MAEERS VS. snwcw© 9E\€ OPENING NINOCS 1 100 1 1 3 1 1 1 r mDROMETER 3 1 1 1 1 —r t —T -I" 1 K -v N 90 \ \ 80 It A \ \ 70 5 60 \: \ 'i \ ° 50 L. g 40 L 20 10 ' 100 500 COBBLES LL PL 47 10 50 CQ«SE 30 17 GRAVEL 1 TOE GS 2.61 5 0.1 1 0J5 GRAIN SIZE N MIUMEIERS CCWBE 1 NATW.S SAND MEDIUM 1 0.05 0.01 0.001 0.005 SLT or CLAY FINE CRG.J! 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