Ponomarenko A. T., Ryvkina N. G., Travkin V. S.*, Tchmutin I. A., GusevA. L.**, Shevchenko V. G.
Enikolopov Institute of Synthetic Polymeric Materials, Russian Academy of Sciences 117393, Moscow, Russia. * Mechanical and Aerospace Engineering Department, University of California, Los Angeles. ** Russian Federal Nuclear Center - All-Russian Research Institute of Experimental Physics.
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ABSTRACT
The paper presents a review of the structure and electrophysical properties of liquid-impregnated porous media. Principal parameters are considered, which reflect the internal structure of the systems and are used to describe porous media in the literature. The main physical processes, which govern the electrophysical properties of these systems are interface polarization and intrinsic dielectric relaxation in the liquid, with the frequency of the first process being practically always lower than that of the second one. The consequence of this ratio of frequencies of relaxation processes is the basic difference in behavior of liquid-impregnated porous media in three frequency regions, below, inside and above the interface polarization region. Principal electrophysical properties of porous media are reviewed and experimental data is classified in three frequency ranges. Different methods of calculating complex permittivity of porous media are compared: composite approximation, Bergman-Milton theory, Grain Consolidation Model, local porosity theory, called volume averaging theory, etc. The advantages and drawbacks of each model in different frequency ranges are outlined.
1. INTRODUCTION
At present, porous materials as the objects for research in physical chemistry, attract growing attention due to their importance in science and technology. Investigations of such systems are very important in cryogenics for development of insulation materials [1-2], in power engineering [3], for enhancing electrophysical [4] and thermal [5] properties, which can be most clearly seen in the works on high Tc superconductor ceramics. This is the reason for existence of extensive literature concerning different physical properties of porous media [6-12], in particular, their structure, heat and mass transfer [13-15].
Another aspect is the investigation of porous media with added functional ingredients, for example short conducting fibers, the basis for effective electromagnetic wave absorbers [16, 17]. This latter aspect is closely related to our research on diffraction structures with liquid media, where porous polymers are used to make casings of certain profile, which in its turn is filled with
polar liquids and liquid mixtures [18]. Finally, the third aspect is the investigation of the properties of porous and liquid media [19, 20] , combined in a single structure. This latter case is treated in this review.
Porous solid materials, in which the pores are filled with a liquid, are a class of materials with several applications. These materials are used in electrical equipment. A typical example is cellulose (paper or transformer board) impregnated with mineral oil which is used in transformers [21] We can also mention oil impregnated polypropylene used in high-energy density capacitors [22]. Sedimentary rocks may contain different amounts of salty water [23], as well as crude oil and natural gas. Geologists have tried to use electrical measurements in order to determine whether a particular rock formation contains amounts of hydrocarbon that could be exploited at a profit [24]. Building materials, such as cement are porous materials also. Even small amounts of pore water change their dielectric properties. Measurement of the dielectric response can be used for non-destructive testing of cement [25, 26]. Another type of liquid impregnated solid materials is the polymer composite materials
NOMENCLATURE
A cross-section area
a coefficient in the Archie's law
c volume fraction of the component 2
cd mean drag resistance coefficient in the REV
d distance between the neighbouring grains
dh capillary morphology characteristic hydraulic diameter
E electrical field
F formation factor
f frequency
ff Fanning friction factor
g acceleration of gravity
J electric volume current density
Jf fluid flow volume density
k permeability
KE electroosmosis coefficient
KS streaming potential coefficient
L depolarization factor
l the actual pore path divided by the sample
thickness
m power index in the Archie's law [mt, m2, m3] coordinates in reciprocal space
P hydrostatic pressure
Q flow of fluid
Sw specific surface of a porous medium
T tortuosity
t effective relaxation time of the inclusion material, due to conductivity of the filler
U averaged in the REV mean velocity v Darcy velocity X generalised transport property
GREEK SYMBOLS
S
e
M
P
a aj
T tj
M
constrictivity factor
complex dielectric constants of inhomogeneous media
static dielectric constant in Debye's equation static dielectric constant of the j-components in Debye's equation
optical dielectric constant in Debye's equation optical dielectric constant of the j-components in Debye's equation
real part of complex dielectric constants imaginary part of complex dielectric constants parameter in the equation (23) complex dielectric constants of the j-components porosity
local percolation probability dynamic viscosity local porosity distribution density of the fluid
conductivity of inhomogeneous media conductivity of the j-component relaxation time in Debye's equation relaxation time of the j-component in Debye's equation electrostatic potential circular frequency
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e
e
w
with disperse filler. Atmospheric water can be present in some amount in the pores between polymer phase and filler particles and affect conductivity and dielectric constant of the composite [27].
It is clear that the studies of the dielectric properties of solid-liquid systems are important in different fields of science. The purpose of this review is to illuminate the connection between dielectric properties, especially at nonzero frequencies, and characteristics of solid-liquid system, such as composition of the system, properties of the components and structure of the porous media.
2. MORPHOLOGICAL FEATURES AND MOMENTUM TRANSPORT IN LIQUID-IMPREGNATED POROUS MEDIA, AFFECTING THEIR DIELECTRIC PROPERTIES
Porous systems have a complicated and diverse structure [6,28,29]. A long-standing problem of considerable scientific and technological importance is to improve the understanding of geometric-dielectric correlation in liquid impregnated porous materials. The scientific problem is to find out which properties of the complicated random geometry of the pore space have a significant influence on the dielectric properties. Full description of a porous solid may require many parameters such as porosity, density, surface area, pore volume, pore size (mean diameter, pore size distribution), pore connectivity, pore shape, pore surface roughness, and others [30-31]
Modern models usually include the following parameters: porosity or local porosity distribution, permeability, tortuosity, etc.
For porous materials, one uses the concept of porosity to quantify the amount of pore space. The porosity of a sample is defined as the volume of the pores within the sample, divided by the total volume of the sample. One sometimes also uses the term «pore volume fraction» . It is assumed that the sample volume is big enough to yield no dependence of the porosity on the sample size. On a smaller, mesoscopic scale, one sometimes defines a local porosity [32-33], which is assumed to vary on a local scale in the macroscopically homogeneous sample. The porosity is in this paper denoted by 0 .
The pores in a porous solid are not smooth tubes running in straight lines throughout the material. Instead, the pores are tortuous; they bend and twist, making the actual pore path through the material much longer than the thickness of the specimen in question. This phenomenon is called tortuosity, and it impedes transport in the pores (such as electrical conduction or diffusion of gases).
Two properties of the pore - not running in a straight line and not having constant cross-section area - both have the same influence to the transport properties. Their effects are often [34-36] combined in the concept of tortuosity, which could be defined as:
T — X / X
1 ^ ideal ' ^ real '
where X is a transport property such as effective electrical conductivity, the index real denotes the specimen in question and the index ideal denotes the property of an ideal sample, having pores with constant cross-section running in straight lines through the specimen.
The following expression for tortuosity is presented in [37]: T — l2/8 , where 8 - is a «constrictivity factor», accounting for the effect of variation of pore area, l - is the actual pore path divided by the sample thickness.
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In some papers [38,39] , the dielectric properties of porous materials are related to their fluid permeability. This property is a measure of the ease by which a fluid, having a certain viscosity, is flowing through the medium when it is subjected to a pressure gradient. This quantity is dependent solely on the geometric structure of the pores. The permeability, k, is occurring as a constant in the so called Darcy's law [40]:
v =
V )(v P + pg ),
ß
(1)
where ^ is the dynamic viscosity, P is the hydrostatic pressure, p is the density of the fluid, g is the acceleration of gravity, v is the so called Darcy velocity (defined as
v =
dQ_
dA
Ф ,
/k
ker = / = = // =
2P/U 2
AP
U - the intrinsic averaged in the REV (representative elementary volume) velocity including turbulent regime. The problem is what to choose for the hydraulic diameter for a given porous media that properly represents its morphology. Bird et al. (1960) used the ratio of the volume available for flow to the cross section available for flow in their derivation of a hydrau
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