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Theory and Fundamental Principles
CHAPTER
THEORY AND FUNDAMENTAL PRINCIPLES
1
The dielectric properties of materials are those electrical characteristics that determine the interaction of materials with electric fields. In radio-frequency (RF) and microwave heating of foods, agricultural products, and other dielectric materials, it is the interaction of the materials with the electric field component of the electromagnetic waves that produces the desired heating effects (Nelson and Trabelsi, 2014). Strictly speaking, radio frequencies range from about 10 kHz to about 100 GHz. These are the frequencies practicable for radio transmission; they span that portion of the electromagnetic spectrum between the audio frequencies and the infrared region (IEEE, 1990). Thus, the RF range includes those frequencies used for microwave heating. However, because RF dielectric heating applications were developed first in the frequency range of about 340 MHz, and microwave heating applications came later, there is a tendency— particularly in the food industry—to refer to the lower frequency applications as RF dielectric heating, and to refer to dielectric heating at microwave frequencies, about 1 GHz and higher, as microwave heating. Frequencies for RF communication are further designated as high frequency (HF, 330 MHz), very-high frequency (VHF, 30300 MHz), ultra-high frequency (UHF, 3003000 MHz), super-high frequency (SHF, 330 GHz), and extremely-high frequency (EHF, 30300 GHz).
1.1 DIELECTRIC PROPERTIES OF MATERIALS Dielectrics are a class of materials that are poor conductors of electricity, in contrast to materials such as metals that are generally good electrical conductors. Many materials, including foods, living organisms, and most agricultural products, conduct electric currents to some degree, but they are still classified as dielectrics. The electrical nature of these materials can be described by their dielectric properties, which influence the distribution of electromagnetic fields and currents in the region occupied by the materials, and which determine the behavior of the materials in electric fields. Thus, the dielectric properties determine how rapidly a material will warm up in RF or microwave dielectric heating applications. Their influence on electric fields also provides a means for sensing certain other properties of materials, which may be correlated with the dielectric
Dielectric Properties of Agricultural Materials and Their Applications. DOI: http://dx.doi.org/10.1016/B978-0-12-802305-1.00001-4 © 2015 Elsevier Inc. All rights reserved.
1
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CHAPTER 1 THEORY AND FUNDAMENTAL PRINCIPLES
properties, by nondestructive electrical measurements. Therefore, dielectric properties of agricultural products may be important for quality-sensing applications in the agricultural industry as well as in dielectric heating applications. A few simplified definitions of dielectric properties are useful in discussing their applications. A fundamental characteristic of all forms of electromagnetic energy is their propagation through free space at the velocity of light, c. The velocity of propagation v of electromagnetic energy in a material depends on the electromagnetic characteristics of that material and is given as: 1 v 5 pffiffiffiffiffiffi με
(1.1)
where μ is the magnetic permeability of the material and ε is the electric permittivity. For free space, this becomes: 1 c 5 pffiffiffiffiffiffiffiffiffiffi μo εo
(1.2)
where μo and εo are the permeability and permittivity of free space. Most food and agricultural products are nonmagnetic, so their magnetic permeability has the same value as μo . These materials, however, have different permittivities when compared to free space. The absolute permittivity εa can be represented as a complex quantity, εa 5 ε0a 2 jε00a
pffiffiffiffiffiffiffi where j 5 21. The complex permittivity relative to free space is then given as: εr 5
εa 5 ε0r 2 jε00r εo
(1.3)
(1.4)
where εo is the permittivity of free space (8:854 3 10212 farad=m); the real part ε0r is called the dielectric constant, and the imaginary part ε00r is called the dielectric loss factor. These latter two quantities are the dielectric properties of practical interest, and the subscript r will be dropped for simplification in the remainder of this book. The dielectric constant ε0 is associated with the ability of a material to store energy in the electric field in the material, and the loss factor εv is associated with the ability of the material to absorb or dissipate energy, that is, to convert electric energy into heat energy. The dielectric loss factor, for example, is an index of the tendency of the material to warm up in a microwave oven. The dielectric constant is also important because of its influence on the distribution of electric fields. For example, the electric capacitance of two parallel conducting plates separated by free space or air will be multiplied by the value of the dielectric constant of a material if the space between the plates is filled with that material. It should also be noted that ε 5 ε0 2 jεv 5 jεje2jδ where δ is the loss angle of the dielectric. Often, the loss tangent, tan δ 5 εv=ε0 , or dissipation factor, is ffi also used as a descriptive dielectric parameter, pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi and sometimes the power factor, tan δ= 1 1 tan2 δ, is used. The ac conductivity of the dielectric σ in S/m is σ 5 ωεo εv, where ω 5 2πf is the angular frequency, with frequency f in hertz (Hz). In this book, εv is interpreted to include the energy losses in the dielectric due to all operating dielectric relaxation mechanisms and ionic conduction.
1.2 VARIATION OF DIELECTRIC PROPERTIES
3
1.2 VARIATION OF DIELECTRIC PROPERTIES The dielectric properties of most materials vary with several influencing factors (Nelson, 1981, 1991; Nelson and Datta, 2001). In hygroscopic materials such as agricultural products, the amount of water in the material is generally a dominant factor. The dielectric properties also depend on the frequency of the applied alternating electric field, on the temperature of the material, and on the density, composition, and structure of the material. In granular or particulate materials, the bulk density of the airparticle mixture is another factor that influences the dielectric properties. Of course, the dielectric properties of materials are dependent on their chemical composition and especially on the presence of mobile ions and the permanent dipole moments associated with water and any other molecules making up the material of interest.
1.2.1 FREQUENCY DEPENDENCE With the exception of some extremely low-loss materials, that is, materials that absorb essentially no energy from RF and microwave fields, the dielectric properties of most materials vary considerably with the frequency of the applied electric fields. This frequency dependence has been discussed previously (Nelson and Datta, 2001; Nelson, 1973, 1991). An important phenomenon contributing to the frequency dependence of the dielectric properties is the polarization, arising from the orientation with the imposed electric field, of molecules which have permanent dipole moments. The mathematical formulation developed by Debye to describe this process for pure polar materials (Debye, 1929) can be expressed as: ε 5 εN 1
εs 2 εN 1 1 jωτ
(1.5)
where εN represents the dielectric constant at frequencies so high that molecular orientation does not have time to contribute to the polarization, εs represents the static dielectric constant, that is, the value at zero frequency (dc value), and τ is the relaxation time, the period associated with the time for the dipoles to revert to random orientation when the electric field is removed. Separation of Eq. (1.5) into its real and imaginary parts yields: ε0 5 εN 1 εv 5
εs 2 εN 1 1 ðωτÞ2
ðεs 2 εN Þωτ 1 1 ðωτÞ2
(1.6) (1.7)
The relationships defined by these equations are illustrated in Figure 1.1. Thus, at frequencies very low and very high with respect to the molecular relaxation process, the dielectric constant has constant values, εs and εN , respectively, and the losses are zero. At intermediate frequencies, the dielectric constant undergoes a dispersion, and dielectric losses occur with the peak loss at the relaxation frequency, ω 5 1=τ. The Debye equation can be represented graphically in the complex εv-versus-ε0 plane as a semicircle with locus of points ranging from (ε0 5 εs ; εv 5 0) at the low-frequency limit to (ε0 5 εN ; εv 5 0) at the high-frequency limit (Figure 1.2). Such a representation is known as a ColeCole diagram (Cole and Cole, 1941).
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CHAPTER 1 THEORY AND FUNDAMENTAL PRINCIPLES
εs ε⬘
ε⬙ ε∞ Log ω
FIGURE 1.1 Dielectric constant and loss factor for a material following the Debye relaxation (Nelson, 1973).
ε⬙
ε∞
ε⬘
εs
FIGURE 1.2 ColeCole diagram for a material following the Debye relaxation (Nelson, 1973).
Because few materials of interest here consist of pure polar materials with a single relaxation time, many other equations have been developed to better describe the frequency-dependent behavior of materials with more relaxation times or a distribution of relaxation times (Cole and Cole, 1941; Bottcher and Bordewijk, 1978; Davidson and Cole, 1951; Havriliak and Negami, 1967; Nigmatullin et al., 2006, 2008; Nigmatullin and Nelson, 2006). Water, in its liquid state, is a good example of a polar dielectric. The microwave dielectric properties of liquid water are listed in Table 1.1 for several frequencies at temperatures of 20 C and 50 C as selected from data in the literature (Hasted, 1973; Kaatze, 1989). Kaatze has shown that the dielectric spectra for pure water can be well represented by the Debye equation when using the relaxation parameters given in Table 1.2.
1.2 VARIATION OF DIELECTRIC PROPERTIES
5
Table 1.1 Microwave dielectric constant ε0 and dielectric loss factor εv of water at indicated temperatures (Hasted, 1973; Kaatze, 1989) 20 C
50 C
Frequency, GHz
ε0
εv
ε0
εv
0.6 1.7 3.0 4.6 7.7 9.1 12.5 17.4 26.8 36.4
80.3 79.2 77.4 74.0 67.4 63.0 53.6 42.0 26.5 17.6
2.75 7.9 13.0 18.8 28.2 31.5 35.5 37.1 33.9 28.8
69.9 69.7 68.4 68.5 67.2 65.5 61.5 56.3 44.2 34.3
1.25 3.6 5.8 9.4 14.5 16.5 21.4 27.2 32.0 32.6
Table1.2 Debye dielectric relaxation parameters for water (Kaatze, 1989)
Temperature, C 0 10 20 30 40 50 60
εs
εN
τ, ps
Relaxation Frequency, GHz
87.9 83.9 80.2 76.6 73.2 69.9 66.7
5.7 5.5 5.6 5.2 3.9 4.0 4.2
17.67 12.68 9.36 7.28 5.82 4.75 4.01
9.007 12.552 17.004 21.862 27.346 33.506 39.690
εs 5 static dielectric constant. εN 5 high-frequency dielectric constant. τ 5 relaxation time.
The relaxation frequency, ð2πτÞ21 , is provided in Table 1.2 along with the static and HF values of the dielectric constant, εs and εN , for water at temperatures between 0 C and 60 C. Thus, Eqs (1.6) and (1.7), together with relaxation parameters listed in Table 1.2, can be used to provide close estimates for the dielectric properties of water over a wide range of frequencies and temperatures. However, water in its pure liquid state appears in food products very rarely (Hasted, 1973). Most often it has dissolved constituents, is physically absorbed in material capillaries or cavities, or is chemically bound to other molecules of the material. Dielectric relaxations of absorbed water
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CHAPTER 1 THEORY AND FUNDAMENTAL PRINCIPLES
take place at lower frequencies than the relaxation of free water (Hasted, 1973), which occurs at about 19.5 GHz for water at 25 C. Depending upon the material structure, there may be various forms of bound water, differing in energy of binding and in dielectric properties. Moist material, in practice, is usually an inhomogeneous mixture, often containing more than one substance with unknown dielectric properties. Thus, it is difficult to understand and predict the dielectric behavior of such materials at different frequencies, temperatures, and hydration levels.
1.2.2 TEMPERATURE DEPENDENCE The dielectric properties of materials are also temperature dependent, and the nature of that dependence is a function of the dielectric relaxation processes operating under the particular conditions existing and the frequency being used. As temperature increases, the relaxation time decreases, and the loss-factor peak, and the accompanying dispersion noted for ε0 , illustrated in Figure 1.1, will shift to higher frequencies. Thus, in a region of dispersion, the dielectric constant will tend to increase with increasing temperature as a result of dielectric relaxation, whereas the loss factor may either increase or decrease, depending on whether the operating frequency is higher or lower than the relaxation frequency. However, for complex dielectrics such as agricultural materials, other mechanisms may mask or dominate the dielectric relaxation effects. The temperature dependence of εN is relatively small (Hasted, 1973), and while that of εs is larger, its influence is minor in a region of dispersion. Below and above the dispersion region, the dielectric constant tends to decrease with increasing temperature. Distribution functions can be useful in expressing the temperature dependence of dielectric properties (Bottcher and Bordewijk, 1978), but the frequency- and temperature-dependent behavior of the dielectric properties of most materials is complicated and can perhaps best be determined by measurement at the frequencies and under the other conditions of interest.
1.2.3 DENSITY DEPENDENCE Because the influence of a dielectric depends on the amount of mass interacting with the electromagnetic fields, the mass per unit volume, or density, will have an effect on the dielectric properties. This is especially notable with particulate dielectrics such as pulverized or granular materials. In understanding the nature of the density-dependence of the dielectric properties of particulate materials, relationships between the dielectric properties of solid materials and those of airparticle mixtures, such as granular or pulverized samples of such solids, are useful. In some instances, the dielectric properties of a solid may be needed when particulate samples are the only available form of the material. This was true for cereal grains, where kernels were too small for the dielectric sample holders used for measurements (You and Nelson, 1988; Nelson and You, 1989), and in the case of pure minerals that had to be pulverized for purification (Nelson et al., 1989). For some materials, fabrication of samples to exact dimensions required for the measurement of dielectric properties is difficult, and measurements on pulverized materials are more easily performed. In such instances, proven relationships for converting dielectric properties of particulate samples to those for the solid material are important. Several well-known dielectric mixture equations have been considered for this purpose (Nelson and You, 1990; Nelson, 1990, 1992).
1.2 VARIATION OF DIELECTRIC PROPERTIES
7
The notation used here applies to two-component mixtures, where ε represents the effective permittivity of the mixture, ε1 is the permittivity of the medium in which particles of permittivity ε2 are dispersed, and v1 and v2 are the volume fractions of the respective components, where v1 1 v2 5 1. Two of the mixture equations found particularly useful for cereal grains were the Complex Refractive Index mixture equation: ðεÞ1=2 5 v1 ðε1 Þ1=2 1 v2 ðε2 Þ1=2
(1.8)
and the Landau and Lifshitz, Looyenga equation: ðεÞ1=3 5 v1 ðε1 Þ1=3 1 v2 ðε2 Þ1=3
(1.9)
To use these equations to determine ε2 , one needs to know the dielectric properties (permittivity) of the pulverized sample at its bulk density (airparticle mixture density), ρ, and the specific gravity or density of the solid material, ρ2 . The fractional part of the total volume of the mixture occupied by the particles (volume fraction), v2 , is then given by ρ=ρ2 . Solving Eqs (1.8) and (1.9), respectively, for the complex permittivity of the solid material and substituting 1 2 j0 for ε1 (the permittivity of air), the permittivity of the solid materials can be calculated as: 1=2 2 ε 1v2 21 ε2 5 v2 1=3 3 ε 1v2 21 ε2 5 v2
(1.10)
(1.11)
It has been noted that Eqs (1.8) and (1.9) imply the linearity of ε1=2 and ε1=3 , respectively, with the bulk density of the mixture (Nelson, 1992). The Complex Refractive Index and the Landau and Lifshitz, Looyenga relationships thus provide a relatively reliable method for adjusting the dielectric properties of granular and powdered materials with characteristics like grain products from known values at one bulk density to corresponding values for a different bulk density. It follows from Eq. (1.8) that for an airparticle mixture, where ε1 5 1 2 j0, and because v1 5 1 2 v2 and v2 5 ρ=ρ2 , that: "
ðε2 Þ1=2 2 1 ρ2
ðεx Þ1=2 5
for a mixture of density ρx . Similarly,
"
ðε2 Þ1=2 2 1 ρ2
ðεy Þ1=2 5
!#
ρx 1 1
(1.12)
ρy 1 1
(1.13)
!#
for the same mixture of density ρy . Equating the slopes of these two lines (the terms in brackets in Eqs (1.12) and (1.13)) and solving for εx gives the following: "
εx 5
ρ ðεy1=2 21Þ x ρy
#2
11
(1.14)
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CHAPTER 1 THEORY AND FUNDAMENTAL PRINCIPLES
which provides an expression for the permittivity of the mixture at any given density ρx when the permittivity εy is known at density ρy . In an analogous way, it follows from Eq. (1.9) that: "
εx 5 ðε1=3 y 21Þ
#3 ρx 11 ρy
(1.15)
Either Eq. (1.14) or (1.15) should provide reliable conversions of permittivity from one mixture density to another, but the Landau and Lifshitz, Looyenga relationship (Eq. (1.9)) provided somewhat closer estimates within the range of measured densities in work with whole kernel wheat, ground wheat, and finely pulverized coal (Nelson, 1983); so Eq. (1.15) is preferred.
REFERENCES Bottcher, C.J.F., Bordewijk, P., 1978. Theory of Electric Polarization, Vol. II, Dielectrics in Time-Dependent Fields. Elsevier Scientific Publishing Company, Amsterdam, Oxford, New York, NY. Cole, K.S., Cole, R.H., 1941. Dispersion and absorption in dielectrics. I. Alternating current characteristics. J. Chem. Phys. 9, 341351. Davidson, D.W., Cole, R.H., 1951. Dielectric relaxation in glycerol, propylene glycol, and n-propanol. J. Chem. Phys. 19 (12), 14841490. Debye, P., 1929. Polar Molecules. The Chemical Catalog Co., New York, NY. Hasted, J.B., 1973. Aqueous Dielectrics. Chapman and Hall, London. Havriliak, S., Negami, S., 1967. A complex plane representation of dielectric and mechanical relaxation processes in some polymers. Polymer 8, 161210. IEEE, 1990. IEEE Standard Dictionary of Electrical and Electronics Terms. The Institute of Electrical and Electronics Engineers, Inc., New York, NY. Kaatze, U., 1989. Complex permittivity of water as a function of frequency and temperature. J. Chem. Eng. Data 34, 371374. Nelson, S.O., 1973. Electrical properties of agricultural products—a critical review. Trans. ASAE 16 (2), 384400. Nelson, S.O., 1981. Review of factors influencing the dielectric properties of cereal grains. Cereal Chem. 58 (6), 487492. Nelson, S.O., 1983. Density dependence of the dielectric properties of particulate materials. Trans. ASAE 26 (6), 18231825, 1829. Nelson, S.O., 1990. Use of dielectric mixture equations for estimating permittivities of solids from data on pulverized samples. In: Cody, G.D., Geballe, T.H., Sheng, P. (Eds.), Physical Phenomena in Granular Materials, vol. 195. Materials Research Society, Pittsburgh, PA, pp. 295300. Nelson, S.O., 1991. Dielectric properties of agricultural products—measurements and applications. IEEE Trans. Electr. Insul. 26 (5), 845869. Nelson, S.O., 1992. Estimation of permittivities of solids from measurements on pulverized or granular materials. In: Priou, A. (Ed.), Dielectric Properties of Heterogeneous Materials, vol. PIER 6. Elsevier, New York, Amsterdam, London, Tokyo, pp. 231271. Nelson, S.O., Datta, A.K., 2001. Dielectric properties of food materials and electric field interactions. In: Datta, A.K., Anantheswaran, R.C. (Eds.), Handbook of Microwave Technology for Food Applications. Marcel Dekker, Inc., New York, NY. Nelson, S.O., Trabelsi, S., 2014. Dielectric properties of agricultural products: fundamental principles, influencing factors, and measurement techniques. In: Awuah, G., Ramaswamy, H.S., Tang, J. (Eds.),
REFERENCES
9
Radio-Frequency Heating in Food Processing: Principles and Applications. CRC Press, Taylor & Francis Group, Boca Raton, FL. Nelson, S.O., You, T.-S., 1989. Microwave dielectric properties of corn and wheat kernels and soybeans. Trans. ASAE 32 (1), 242249. Nelson, S.O., You, T.-S., 1990. Relationships between microwave permittivities of solid and pulverised plastics. J. Phys. D Appl. Phys. 23, 346353. Nelson, S.O., Lindroth, D.P., Blake, R.L., 1989. Dielectric properties of selected minerals at 1 to 22 GHz. Geophysics 54 (10), 13441349. Nigmatullin, R.R., Nelson, S.O., 2006. Recognition of the “fractional” kinetics in complex systems: dielectric properties of fresh fruits and vegetables from 0.01 to 1.8 GHz. Signal Process. 86, 27442759. Nigmatullin, R.R., Arbuzov, A.A., Nelson, S.O., Trabelsi, S., 2006. Dielectric relaxation in complex systems: quality sensing and dielectric properties of honeydew melons from 10 MHz to 1.8 GHz. J. Instrum. 1, P10002. Nigmatullin, R.R., Osokin, S.I., Nelson, S.O., 2008. Application of fractional-moments statistics to data for two-phase dielectric mixtures. IEEE Trans. Dielectr. Electr. Insul. 15 (5), 13851392. You, T.-S., Nelson, S.O., 1988. Microwave dielectric properties of rice kernels. J. Microw. Power Electromagn. Energy 23 (3), 150159.
THEORY AND FUNDAMENTAL PRINCIPLES
1
The dielectric properties of materials are those electrical characteristics that determine the interaction of materials with electric fields. In radio-frequency (RF) and microwave heating of foods, agricultural products, and other dielectric materials, it is the interaction of the materials with the electric field component of the electromagnetic waves that produces the desired heating effects (Nelson and Trabelsi, 2014). Strictly speaking, radio frequencies range from about 10 kHz to about 100 GHz. These are the frequencies practicable for radio transmission; they span that portion of the electromagnetic spectrum between the audio frequencies and the infrared region (IEEE, 1990). Thus, the RF range includes those frequencies used for microwave heating. However, because RF dielectric heating applications were developed first in the frequency range of about 340 MHz, and microwave heating applications came later, there is a tendency— particularly in the food industry—to refer to the lower frequency applications as RF dielectric heating, and to refer to dielectric heating at microwave frequencies, about 1 GHz and higher, as microwave heating. Frequencies for RF communication are further designated as high frequency (HF, 330 MHz), very-high frequency (VHF, 30300 MHz), ultra-high frequency (UHF, 3003000 MHz), super-high frequency (SHF, 330 GHz), and extremely-high frequency (EHF, 30300 GHz).
1.1 DIELECTRIC PROPERTIES OF MATERIALS Dielectrics are a class of materials that are poor conductors of electricity, in contrast to materials such as metals that are generally good electrical conductors. Many materials, including foods, living organisms, and most agricultural products, conduct electric currents to some degree, but they are still classified as dielectrics. The electrical nature of these materials can be described by their dielectric properties, which influence the distribution of electromagnetic fields and currents in the region occupied by the materials, and which determine the behavior of the materials in electric fields. Thus, the dielectric properties determine how rapidly a material will warm up in RF or microwave dielectric heating applications. Their influence on electric fields also provides a means for sensing certain other properties of materials, which may be correlated with the dielectric
Dielectric Properties of Agricultural Materials and Their Applications. DOI: http://dx.doi.org/10.1016/B978-0-12-802305-1.00001-4 © 2015 Elsevier Inc. All rights reserved.
1
2
CHAPTER 1 THEORY AND FUNDAMENTAL PRINCIPLES
properties, by nondestructive electrical measurements. Therefore, dielectric properties of agricultural products may be important for quality-sensing applications in the agricultural industry as well as in dielectric heating applications. A few simplified definitions of dielectric properties are useful in discussing their applications. A fundamental characteristic of all forms of electromagnetic energy is their propagation through free space at the velocity of light, c. The velocity of propagation v of electromagnetic energy in a material depends on the electromagnetic characteristics of that material and is given as: 1 v 5 pffiffiffiffiffiffi με
(1.1)
where μ is the magnetic permeability of the material and ε is the electric permittivity. For free space, this becomes: 1 c 5 pffiffiffiffiffiffiffiffiffiffi μo εo
(1.2)
where μo and εo are the permeability and permittivity of free space. Most food and agricultural products are nonmagnetic, so their magnetic permeability has the same value as μo . These materials, however, have different permittivities when compared to free space. The absolute permittivity εa can be represented as a complex quantity, εa 5 ε0a 2 jε00a
pffiffiffiffiffiffiffi where j 5 21. The complex permittivity relative to free space is then given as: εr 5
εa 5 ε0r 2 jε00r εo
(1.3)
(1.4)
where εo is the permittivity of free space (8:854 3 10212 farad=m); the real part ε0r is called the dielectric constant, and the imaginary part ε00r is called the dielectric loss factor. These latter two quantities are the dielectric properties of practical interest, and the subscript r will be dropped for simplification in the remainder of this book. The dielectric constant ε0 is associated with the ability of a material to store energy in the electric field in the material, and the loss factor εv is associated with the ability of the material to absorb or dissipate energy, that is, to convert electric energy into heat energy. The dielectric loss factor, for example, is an index of the tendency of the material to warm up in a microwave oven. The dielectric constant is also important because of its influence on the distribution of electric fields. For example, the electric capacitance of two parallel conducting plates separated by free space or air will be multiplied by the value of the dielectric constant of a material if the space between the plates is filled with that material. It should also be noted that ε 5 ε0 2 jεv 5 jεje2jδ where δ is the loss angle of the dielectric. Often, the loss tangent, tan δ 5 εv=ε0 , or dissipation factor, is ffi also used as a descriptive dielectric parameter, pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi and sometimes the power factor, tan δ= 1 1 tan2 δ, is used. The ac conductivity of the dielectric σ in S/m is σ 5 ωεo εv, where ω 5 2πf is the angular frequency, with frequency f in hertz (Hz). In this book, εv is interpreted to include the energy losses in the dielectric due to all operating dielectric relaxation mechanisms and ionic conduction.
1.2 VARIATION OF DIELECTRIC PROPERTIES
3
1.2 VARIATION OF DIELECTRIC PROPERTIES The dielectric properties of most materials vary with several influencing factors (Nelson, 1981, 1991; Nelson and Datta, 2001). In hygroscopic materials such as agricultural products, the amount of water in the material is generally a dominant factor. The dielectric properties also depend on the frequency of the applied alternating electric field, on the temperature of the material, and on the density, composition, and structure of the material. In granular or particulate materials, the bulk density of the airparticle mixture is another factor that influences the dielectric properties. Of course, the dielectric properties of materials are dependent on their chemical composition and especially on the presence of mobile ions and the permanent dipole moments associated with water and any other molecules making up the material of interest.
1.2.1 FREQUENCY DEPENDENCE With the exception of some extremely low-loss materials, that is, materials that absorb essentially no energy from RF and microwave fields, the dielectric properties of most materials vary considerably with the frequency of the applied electric fields. This frequency dependence has been discussed previously (Nelson and Datta, 2001; Nelson, 1973, 1991). An important phenomenon contributing to the frequency dependence of the dielectric properties is the polarization, arising from the orientation with the imposed electric field, of molecules which have permanent dipole moments. The mathematical formulation developed by Debye to describe this process for pure polar materials (Debye, 1929) can be expressed as: ε 5 εN 1
εs 2 εN 1 1 jωτ
(1.5)
where εN represents the dielectric constant at frequencies so high that molecular orientation does not have time to contribute to the polarization, εs represents the static dielectric constant, that is, the value at zero frequency (dc value), and τ is the relaxation time, the period associated with the time for the dipoles to revert to random orientation when the electric field is removed. Separation of Eq. (1.5) into its real and imaginary parts yields: ε0 5 εN 1 εv 5
εs 2 εN 1 1 ðωτÞ2
ðεs 2 εN Þωτ 1 1 ðωτÞ2
(1.6) (1.7)
The relationships defined by these equations are illustrated in Figure 1.1. Thus, at frequencies very low and very high with respect to the molecular relaxation process, the dielectric constant has constant values, εs and εN , respectively, and the losses are zero. At intermediate frequencies, the dielectric constant undergoes a dispersion, and dielectric losses occur with the peak loss at the relaxation frequency, ω 5 1=τ. The Debye equation can be represented graphically in the complex εv-versus-ε0 plane as a semicircle with locus of points ranging from (ε0 5 εs ; εv 5 0) at the low-frequency limit to (ε0 5 εN ; εv 5 0) at the high-frequency limit (Figure 1.2). Such a representation is known as a ColeCole diagram (Cole and Cole, 1941).
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CHAPTER 1 THEORY AND FUNDAMENTAL PRINCIPLES
εs ε⬘
ε⬙ ε∞ Log ω
FIGURE 1.1 Dielectric constant and loss factor for a material following the Debye relaxation (Nelson, 1973).
ε⬙
ε∞
ε⬘
εs
FIGURE 1.2 ColeCole diagram for a material following the Debye relaxation (Nelson, 1973).
Because few materials of interest here consist of pure polar materials with a single relaxation time, many other equations have been developed to better describe the frequency-dependent behavior of materials with more relaxation times or a distribution of relaxation times (Cole and Cole, 1941; Bottcher and Bordewijk, 1978; Davidson and Cole, 1951; Havriliak and Negami, 1967; Nigmatullin et al., 2006, 2008; Nigmatullin and Nelson, 2006). Water, in its liquid state, is a good example of a polar dielectric. The microwave dielectric properties of liquid water are listed in Table 1.1 for several frequencies at temperatures of 20 C and 50 C as selected from data in the literature (Hasted, 1973; Kaatze, 1989). Kaatze has shown that the dielectric spectra for pure water can be well represented by the Debye equation when using the relaxation parameters given in Table 1.2.
1.2 VARIATION OF DIELECTRIC PROPERTIES
5
Table 1.1 Microwave dielectric constant ε0 and dielectric loss factor εv of water at indicated temperatures (Hasted, 1973; Kaatze, 1989) 20 C
50 C
Frequency, GHz
ε0
εv
ε0
εv
0.6 1.7 3.0 4.6 7.7 9.1 12.5 17.4 26.8 36.4
80.3 79.2 77.4 74.0 67.4 63.0 53.6 42.0 26.5 17.6
2.75 7.9 13.0 18.8 28.2 31.5 35.5 37.1 33.9 28.8
69.9 69.7 68.4 68.5 67.2 65.5 61.5 56.3 44.2 34.3
1.25 3.6 5.8 9.4 14.5 16.5 21.4 27.2 32.0 32.6
Table1.2 Debye dielectric relaxation parameters for water (Kaatze, 1989)
Temperature, C 0 10 20 30 40 50 60
εs
εN
τ, ps
Relaxation Frequency, GHz
87.9 83.9 80.2 76.6 73.2 69.9 66.7
5.7 5.5 5.6 5.2 3.9 4.0 4.2
17.67 12.68 9.36 7.28 5.82 4.75 4.01
9.007 12.552 17.004 21.862 27.346 33.506 39.690
εs 5 static dielectric constant. εN 5 high-frequency dielectric constant. τ 5 relaxation time.
The relaxation frequency, ð2πτÞ21 , is provided in Table 1.2 along with the static and HF values of the dielectric constant, εs and εN , for water at temperatures between 0 C and 60 C. Thus, Eqs (1.6) and (1.7), together with relaxation parameters listed in Table 1.2, can be used to provide close estimates for the dielectric properties of water over a wide range of frequencies and temperatures. However, water in its pure liquid state appears in food products very rarely (Hasted, 1973). Most often it has dissolved constituents, is physically absorbed in material capillaries or cavities, or is chemically bound to other molecules of the material. Dielectric relaxations of absorbed water
6
CHAPTER 1 THEORY AND FUNDAMENTAL PRINCIPLES
take place at lower frequencies than the relaxation of free water (Hasted, 1973), which occurs at about 19.5 GHz for water at 25 C. Depending upon the material structure, there may be various forms of bound water, differing in energy of binding and in dielectric properties. Moist material, in practice, is usually an inhomogeneous mixture, often containing more than one substance with unknown dielectric properties. Thus, it is difficult to understand and predict the dielectric behavior of such materials at different frequencies, temperatures, and hydration levels.
1.2.2 TEMPERATURE DEPENDENCE The dielectric properties of materials are also temperature dependent, and the nature of that dependence is a function of the dielectric relaxation processes operating under the particular conditions existing and the frequency being used. As temperature increases, the relaxation time decreases, and the loss-factor peak, and the accompanying dispersion noted for ε0 , illustrated in Figure 1.1, will shift to higher frequencies. Thus, in a region of dispersion, the dielectric constant will tend to increase with increasing temperature as a result of dielectric relaxation, whereas the loss factor may either increase or decrease, depending on whether the operating frequency is higher or lower than the relaxation frequency. However, for complex dielectrics such as agricultural materials, other mechanisms may mask or dominate the dielectric relaxation effects. The temperature dependence of εN is relatively small (Hasted, 1973), and while that of εs is larger, its influence is minor in a region of dispersion. Below and above the dispersion region, the dielectric constant tends to decrease with increasing temperature. Distribution functions can be useful in expressing the temperature dependence of dielectric properties (Bottcher and Bordewijk, 1978), but the frequency- and temperature-dependent behavior of the dielectric properties of most materials is complicated and can perhaps best be determined by measurement at the frequencies and under the other conditions of interest.
1.2.3 DENSITY DEPENDENCE Because the influence of a dielectric depends on the amount of mass interacting with the electromagnetic fields, the mass per unit volume, or density, will have an effect on the dielectric properties. This is especially notable with particulate dielectrics such as pulverized or granular materials. In understanding the nature of the density-dependence of the dielectric properties of particulate materials, relationships between the dielectric properties of solid materials and those of airparticle mixtures, such as granular or pulverized samples of such solids, are useful. In some instances, the dielectric properties of a solid may be needed when particulate samples are the only available form of the material. This was true for cereal grains, where kernels were too small for the dielectric sample holders used for measurements (You and Nelson, 1988; Nelson and You, 1989), and in the case of pure minerals that had to be pulverized for purification (Nelson et al., 1989). For some materials, fabrication of samples to exact dimensions required for the measurement of dielectric properties is difficult, and measurements on pulverized materials are more easily performed. In such instances, proven relationships for converting dielectric properties of particulate samples to those for the solid material are important. Several well-known dielectric mixture equations have been considered for this purpose (Nelson and You, 1990; Nelson, 1990, 1992).
1.2 VARIATION OF DIELECTRIC PROPERTIES
7
The notation used here applies to two-component mixtures, where ε represents the effective permittivity of the mixture, ε1 is the permittivity of the medium in which particles of permittivity ε2 are dispersed, and v1 and v2 are the volume fractions of the respective components, where v1 1 v2 5 1. Two of the mixture equations found particularly useful for cereal grains were the Complex Refractive Index mixture equation: ðεÞ1=2 5 v1 ðε1 Þ1=2 1 v2 ðε2 Þ1=2
(1.8)
and the Landau and Lifshitz, Looyenga equation: ðεÞ1=3 5 v1 ðε1 Þ1=3 1 v2 ðε2 Þ1=3
(1.9)
To use these equations to determine ε2 , one needs to know the dielectric properties (permittivity) of the pulverized sample at its bulk density (airparticle mixture density), ρ, and the specific gravity or density of the solid material, ρ2 . The fractional part of the total volume of the mixture occupied by the particles (volume fraction), v2 , is then given by ρ=ρ2 . Solving Eqs (1.8) and (1.9), respectively, for the complex permittivity of the solid material and substituting 1 2 j0 for ε1 (the permittivity of air), the permittivity of the solid materials can be calculated as: 1=2 2 ε 1v2 21 ε2 5 v2 1=3 3 ε 1v2 21 ε2 5 v2
(1.10)
(1.11)
It has been noted that Eqs (1.8) and (1.9) imply the linearity of ε1=2 and ε1=3 , respectively, with the bulk density of the mixture (Nelson, 1992). The Complex Refractive Index and the Landau and Lifshitz, Looyenga relationships thus provide a relatively reliable method for adjusting the dielectric properties of granular and powdered materials with characteristics like grain products from known values at one bulk density to corresponding values for a different bulk density. It follows from Eq. (1.8) that for an airparticle mixture, where ε1 5 1 2 j0, and because v1 5 1 2 v2 and v2 5 ρ=ρ2 , that: "
ðε2 Þ1=2 2 1 ρ2
ðεx Þ1=2 5
for a mixture of density ρx . Similarly,
"
ðε2 Þ1=2 2 1 ρ2
ðεy Þ1=2 5
!#
ρx 1 1
(1.12)
ρy 1 1
(1.13)
!#
for the same mixture of density ρy . Equating the slopes of these two lines (the terms in brackets in Eqs (1.12) and (1.13)) and solving for εx gives the following: "
εx 5
ρ ðεy1=2 21Þ x ρy
#2
11
(1.14)
8
CHAPTER 1 THEORY AND FUNDAMENTAL PRINCIPLES
which provides an expression for the permittivity of the mixture at any given density ρx when the permittivity εy is known at density ρy . In an analogous way, it follows from Eq. (1.9) that: "
εx 5 ðε1=3 y 21Þ
#3 ρx 11 ρy
(1.15)
Either Eq. (1.14) or (1.15) should provide reliable conversions of permittivity from one mixture density to another, but the Landau and Lifshitz, Looyenga relationship (Eq. (1.9)) provided somewhat closer estimates within the range of measured densities in work with whole kernel wheat, ground wheat, and finely pulverized coal (Nelson, 1983); so Eq. (1.15) is preferred.
REFERENCES Bottcher, C.J.F., Bordewijk, P., 1978. Theory of Electric Polarization, Vol. II, Dielectrics in Time-Dependent Fields. Elsevier Scientific Publishing Company, Amsterdam, Oxford, New York, NY. Cole, K.S., Cole, R.H., 1941. Dispersion and absorption in dielectrics. I. Alternating current characteristics. J. Chem. Phys. 9, 341351. Davidson, D.W., Cole, R.H., 1951. Dielectric relaxation in glycerol, propylene glycol, and n-propanol. J. Chem. Phys. 19 (12), 14841490. Debye, P., 1929. Polar Molecules. The Chemical Catalog Co., New York, NY. Hasted, J.B., 1973. Aqueous Dielectrics. Chapman and Hall, London. Havriliak, S., Negami, S., 1967. A complex plane representation of dielectric and mechanical relaxation processes in some polymers. Polymer 8, 161210. IEEE, 1990. IEEE Standard Dictionary of Electrical and Electronics Terms. The Institute of Electrical and Electronics Engineers, Inc., New York, NY. Kaatze, U., 1989. Complex permittivity of water as a function of frequency and temperature. J. Chem. Eng. Data 34, 371374. Nelson, S.O., 1973. Electrical properties of agricultural products—a critical review. Trans. ASAE 16 (2), 384400. Nelson, S.O., 1981. Review of factors influencing the dielectric properties of cereal grains. Cereal Chem. 58 (6), 487492. Nelson, S.O., 1983. Density dependence of the dielectric properties of particulate materials. Trans. ASAE 26 (6), 18231825, 1829. Nelson, S.O., 1990. Use of dielectric mixture equations for estimating permittivities of solids from data on pulverized samples. In: Cody, G.D., Geballe, T.H., Sheng, P. (Eds.), Physical Phenomena in Granular Materials, vol. 195. Materials Research Society, Pittsburgh, PA, pp. 295300. Nelson, S.O., 1991. Dielectric properties of agricultural products—measurements and applications. IEEE Trans. Electr. Insul. 26 (5), 845869. Nelson, S.O., 1992. Estimation of permittivities of solids from measurements on pulverized or granular materials. In: Priou, A. (Ed.), Dielectric Properties of Heterogeneous Materials, vol. PIER 6. Elsevier, New York, Amsterdam, London, Tokyo, pp. 231271. Nelson, S.O., Datta, A.K., 2001. Dielectric properties of food materials and electric field interactions. In: Datta, A.K., Anantheswaran, R.C. (Eds.), Handbook of Microwave Technology for Food Applications. Marcel Dekker, Inc., New York, NY. Nelson, S.O., Trabelsi, S., 2014. Dielectric properties of agricultural products: fundamental principles, influencing factors, and measurement techniques. In: Awuah, G., Ramaswamy, H.S., Tang, J. (Eds.),
REFERENCES
9
Radio-Frequency Heating in Food Processing: Principles and Applications. CRC Press, Taylor & Francis Group, Boca Raton, FL. Nelson, S.O., You, T.-S., 1989. Microwave dielectric properties of corn and wheat kernels and soybeans. Trans. ASAE 32 (1), 242249. Nelson, S.O., You, T.-S., 1990. Relationships between microwave permittivities of solid and pulverised plastics. J. Phys. D Appl. Phys. 23, 346353. Nelson, S.O., Lindroth, D.P., Blake, R.L., 1989. Dielectric properties of selected minerals at 1 to 22 GHz. Geophysics 54 (10), 13441349. Nigmatullin, R.R., Nelson, S.O., 2006. Recognition of the “fractional” kinetics in complex systems: dielectric properties of fresh fruits and vegetables from 0.01 to 1.8 GHz. Signal Process. 86, 27442759. Nigmatullin, R.R., Arbuzov, A.A., Nelson, S.O., Trabelsi, S., 2006. Dielectric relaxation in complex systems: quality sensing and dielectric properties of honeydew melons from 10 MHz to 1.8 GHz. J. Instrum. 1, P10002. Nigmatullin, R.R., Osokin, S.I., Nelson, S.O., 2008. Application of fractional-moments statistics to data for two-phase dielectric mixtures. IEEE Trans. Dielectr. Electr. Insul. 15 (5), 13851392. You, T.-S., Nelson, S.O., 1988. Microwave dielectric properties of rice kernels. J. Microw. Power Electromagn. Energy 23 (3), 150159.