3.1 Physical Properties of Gas Hydrates
65
in contrast to 94 for I h at 273 K [1, 18]. Magnetic susceptibilities, µ, of clathrate
hydrates have not been reported.
The complex refractive index, n complex ≡ n + iκ, is also generally a complex
function that depends on the frequency (wavelength) of an electromagnetic wave.
The complex refractive index is related to the dielectric function through ε complex /ε 0
= n
2
complex , where ε 0 is the dielectric constant of vacuum, so ε
/ε 0 = n
2 – κ
2 and
ε
/ε 0 = 2nκ [44]. The refractive index of vacuum is defined to be n complex ≡ 1 with
no imaginary component. The imaginary part of a complex dielectric function or a
complex refractive index represents a loss of electromagnetic fields (attenuation or
absorption). The experimentally measurable absolute values of a dielectric function
and a refractive index are defined using the complex conjugate in each case: |ε complex |
2
≡ (ε
+ iε
)(ε
– iε
) for the dielectric function and |n complex |
2
≡ (n
+ iκ
)(n
– iκ
) for
the refractive index. For the refractive index, only the real part, n, over limited ranges
of wavelengths has been measured for clathrate hydrates. Using the wavelengths of λ
= 640, 756, and 844 nm, Bylov and Rasmussen reported n ≈1.346 for an sI-forming
methane hydrate and ≈1.350 for an sII-forming natural gas hydrate between –2.5
and 8.6 °C [45].
Electrical conductivity, σ, also generally depends on the AC frequency. Du Frane
et al. reported σ of sI-forming methane hydrate to be about ≈5 × 10
−5 Sm
−1 at 273 K
[46]. Small frequency dependence was observed between 100 kHz and 1 MHz [46].
The temperature dependence was regular, as a plot of log(σ ) versus (1/T ) yielded a
straight line of a negative slope [46].
One might wonder why we should worry about the electromagnetic properties
or the dielectric functions of clathrate hydrates which, at first glance, might not
appear terribly relevant. However, this is not so: like many things in life, the first
appearance of an entity can be deceiving. The van der Waals forces are ubiquitous and
exist in every material and substance, including clathrate hydrates. They dominate
many aspects of physical properties of the material and its interactions with its
surroundings. The origin of the van der Waals forces is electromagnetic forces that
arise between groups of electrons, as we will see in Chap. 4. The refractive index
of a material is a property determined by its electron density and its strength of
interactions with photons, which is none other than the dielectric functions. Simply
put, the higher the electron density and the stronger their interactions with photons,
the higher the refractive index. The adhesion, cohesion, and agglomeration properties
of clathrate hydrates are determined by the surface forces for which the van der Waals
forces are the major components. The thickness and the physical properties of quasiliquid layers, or their presence or absence on clathrate surfaces, are also determined
by the surface forces, as we will see in Chap. 5.
3.2 Thermodynamic Aspects of Gas Hydrates
The thermodynamic properties of clathrate hydrates that are central to their nucleation
are their surface and interfacial free energies. However, their descriptions require the
concept of disjoining pressure which we will not introduce until Sect. 4.1. As such,
65
in contrast to 94 for I h at 273 K [1, 18]. Magnetic susceptibilities, µ, of clathrate
hydrates have not been reported.
The complex refractive index, n complex ≡ n + iκ, is also generally a complex
function that depends on the frequency (wavelength) of an electromagnetic wave.
The complex refractive index is related to the dielectric function through ε complex /ε 0
= n
2
complex , where ε 0 is the dielectric constant of vacuum, so ε
/ε 0 = n
2 – κ
2 and
ε
/ε 0 = 2nκ [44]. The refractive index of vacuum is defined to be n complex ≡ 1 with
no imaginary component. The imaginary part of a complex dielectric function or a
complex refractive index represents a loss of electromagnetic fields (attenuation or
absorption). The experimentally measurable absolute values of a dielectric function
and a refractive index are defined using the complex conjugate in each case: |ε complex |
2
≡ (ε
+ iε
)(ε
– iε
) for the dielectric function and |n complex |
2
≡ (n
+ iκ
)(n
– iκ
) for
the refractive index. For the refractive index, only the real part, n, over limited ranges
of wavelengths has been measured for clathrate hydrates. Using the wavelengths of λ
= 640, 756, and 844 nm, Bylov and Rasmussen reported n ≈1.346 for an sI-forming
methane hydrate and ≈1.350 for an sII-forming natural gas hydrate between –2.5
and 8.6 °C [45].
Electrical conductivity, σ, also generally depends on the AC frequency. Du Frane
et al. reported σ of sI-forming methane hydrate to be about ≈5 × 10
−5 Sm
−1 at 273 K
[46]. Small frequency dependence was observed between 100 kHz and 1 MHz [46].
The temperature dependence was regular, as a plot of log(σ ) versus (1/T ) yielded a
straight line of a negative slope [46].
One might wonder why we should worry about the electromagnetic properties
or the dielectric functions of clathrate hydrates which, at first glance, might not
appear terribly relevant. However, this is not so: like many things in life, the first
appearance of an entity can be deceiving. The van der Waals forces are ubiquitous and
exist in every material and substance, including clathrate hydrates. They dominate
many aspects of physical properties of the material and its interactions with its
surroundings. The origin of the van der Waals forces is electromagnetic forces that
arise between groups of electrons, as we will see in Chap. 4. The refractive index
of a material is a property determined by its electron density and its strength of
interactions with photons, which is none other than the dielectric functions. Simply
put, the higher the electron density and the stronger their interactions with photons,
the higher the refractive index. The adhesion, cohesion, and agglomeration properties
of clathrate hydrates are determined by the surface forces for which the van der Waals
forces are the major components. The thickness and the physical properties of quasiliquid layers, or their presence or absence on clathrate surfaces, are also determined
by the surface forces, as we will see in Chap. 5.
3.2 Thermodynamic Aspects of Gas Hydrates
The thermodynamic properties of clathrate hydrates that are central to their nucleation
are their surface and interfacial free energies. However, their descriptions require the
concept of disjoining pressure which we will not introduce until Sect. 4.1. As such,