CC BY
9Ponomarenko A. T., Ryvkina N. G., Travkin V. S.*, Tchmutin I. A., Gusev A. 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.
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 [12], 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
¿7 averaged in the REV mean velocity v Darcy velocity X generalised transport property
GREEK SYMBOLS
S constrictivity factor
complex dielectric constants of inhomogeneous media
£s static dielectric constant in Debye's equation s„. static dielectric constant of the j-components in Debye's equation
optical dielectric constant in Debye's equation sM optical dielectric constant of the j-components in
Debye's equation s' real part of complex dielectric constants s" imaginary part of complex dielectric constants r parameter in the equation (23) s, complex dielectric constants of the j-components porosity
A(^) local percolation probability ^ dynamic viscosity fj(<p) local porosity distribution p density of the fluid
conductivity of inhomogeneous media conductivity of the j-components t relaxation time in Debye's equation Tj relaxation time of the j-components in Debye's equation M electrostatic potential circular frequency
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 .
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
,
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/S , where S - is a «constrictivity factor», accounting for the effect of variation of pore area, l - is the actual pore path divided by the sample thickness.
б
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 p + p t ),
(1)
v=
dR dA
s
f d.
J кег _ f __u
з J f
2Pfü 2
АР L
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 hydraulic radius rhb. This assumption led them to formula
ф d
Г , =
hb
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
б(1-ф ) ,
where dp is the particle diameter. Meantime, a basis for a consistent hydraulic diameter for any systems is
_ 4Ф _ 4r
a (l-ф) 3(l-ф) P hb,
and Q is the flow of fluid through the cross-section area A). The advantage of this simple correlation is that it relies upon the permeability only constant or function k in any situation and media considered. The drawbacks of that kind of pressure loss equations were analysed elsewhere.
Mathematical simulation of physical processes in a highly non-homogeneous media, in general, calls for obtaining averaged characteristics of the medium and, consequently, the averaged equations. The averaging of processes in a randomly organized media can be performed in many different ways. If a physical model has several interdependent structurally organized levels of processes underway, it is expedient to employ one of the hierarchical methods of simulation (see, for example, [41]). The hierarchical principle of simulation consists of successively studying the processes at a number of structural levels.
At present time there are few theoretical approaches gaining more precise than just bulk description of transport phenomena in heterogeneous media. The most universal, comprehensive and mathematically accurate hierarchical theory is the so-called volume averaging theory (VAT) started independently in the works [42-45]. Until now the exclusive physical fields of development using the VAT were the thermal physics and fluid mechanics problems. The additional terms in the generalized VAT based transport governing equations [42,46-54], which are not seen in typical governing equation sets commonly used for heterogeneous media, determine the nature of the solutions, influensing factors and are the central issue of the closure problem. Treatment of the additional terms becomes crucial once the governing, averaged equations are written.
Methods for obtaining closure of the governing equations bulk processes in monodisperse regular and non-specified, random polydisperse media using conventional fluid mechanics coefficients and experimental correlations have been described in [50,51,54-59]
Among most of the well known and used porous media pressure loss dependencies which can be used instead of the Darcy law is one that was introduced in [60]. He suggested two types of effective bulk friction factors. One of them, the so-called kinetic energy friction factor fker is similar to the Fanning friction factor, f and is written as
u S .. ,
W V'
where av is the particle specific surface which is equal to the total particle surface divided by the volume of the particle and specific surface
= h (1 - Ф) .
This expression is strictly justified only when an equal or mean particle diameter is d =6/ a , which is exact for
L p I v '
spherical particles and often used as a substitute for granular media particles.
In models by Bird [61] and many other models used different characteristics length scales. It is not clear from evaluation of experimental data which is the most appropriate or generally valid. To address this, the simplified VAT (SVAT) momentum equation for flow of an incompressible fluid in a porous media [50,54,55,62] can be used as a consistent set of notions. The one-dimensional form of the momentum SVAT equation can be simplified for a regular morphology medium with constant porosity to the form
qp
dx
cf + Cdp~s^
T.p ir T. i p/2
Г т. i p fû2
where c. is the friction resistance coefficient and c, the
f dp
form drag, S is the cross flow form specific surface.
O' wp 1
The drag terms are combined into a single total drag
coefficient c, to model the flow resistance terms in the
d
general simplified momentum VAT equation and can be evaluated via the bulk pressure loss Fanning friction factor ff 5 cd (see more detail in [54]) which is conventionally thought from experiment.
Another problem of interest is that the whole matter of what is appeared to be named as the cross coefficient theoretical development for the momentum transport - electric field processes is up until now is based solely on the mathematical similarities in description of velocity (creeping) and electrical potential fields in a fluid phase located in an insulating porous solid phase. According to this approach, initially suggested in [63] the dependency for fluid static permeability in porous media can be written as
* .G!
dc 8A '
where A is the special parameter, which relates a pore volume to surface in porous me d ium and estimates through the known electrical field E (r ) , which is in turn found through an electrical potential E = -Vw. The
electrostatic potential field <p{r} is known to be described by the Laplace equation.
V w = 0,
inside of the fluid volume X with the boundary conditions
ô w ôn
ÔS
= 0, (w2 - w ) = !»
w
where w and w are the potentials on the both sides of medium applied that unit voltage drop is corresponds to the unit length in the primary direction
(w 2 ~wl)
L
= l.
The formation factor F is determined through the relationship (see, for example, [63, 64])
F =
X i (@(q)(2 ^
/
-1
a
f
i@ @ }
f °eff
(2)
persion on the dynamic and static permeability of parallel pore capillary morphologies and found out, that ''even if, in reality, wedgelike singularities are rounded off microscopically, large curvatures on the pore surface will produce strong surface concentration of the electric field''.
Also, they concluded that ''with regard to pore-size dispersion, it is clear that a theory based on a single-tube response function (as for the dynamic permeability k in [63,67] will be inadequate if the dispersion is wide enough''.
An exact flow resistance results obtained on the basis of VAT governing equations by [54] for the random pore diameter distribution for almost the same morphology as was used by [64] exposed the wide departure from the Darcy law based treatments with constant coefficients. That was shown even for the morphology where a single pore exists with diameter different from the all others. Meanwhile, using consistently the VAT based procedures [54] one can easily develop the needed variable, nonlinear permeability coefficient for Darcy dependency
where ar is the conductivity of fluids and &ell is the effective conductivity of porous medium filled with fluid. As it was shown in [65] the peculiarities of mathematical solution of transient Stokes equations to connect the fluid permeability kdc and formation factor F using the employment of solution decomposition for eigenfunctions and using decomposition of the constant unit vector e to seek the electrical field solution E. By presenting the fluid permeability kdc through the in series distributed solution in X the formula for permeability in porous media was established. As it is obvious from the algorithm and description of the found dependency for permeability kdc and two suggested parameters L (or A) and F it is clear that the porous medium flow problem was brought down with the help of the following used techniques:
1) Eigenfunction solutions which employ the Poisson
type equation infinite series solution;
and
2) Full scale Laplace equation solution for electric
field E in X .
It is worth to note here that the full problem solution of either one of these tasks only once does provide an actual complete solution for the entire closure problem for the VAT based porous medium fluid transport problem (see, for example, [54,66]). In the meantime, the major question of applicability of the electrical field measurements in porous media rests in the physical basics of the problem, but not in mathematical similarities of the equations.
One of the points, which couldn't be avoided - is how an absolute dielectric property could be justified for ordinary solid phase materials? Which is not true, meaning that the losses of electrical potential actually occur during an experiment. Another reality, as was shown in [65] is that, if in the medium which posses the real morphological features - as angles and steep curves of an interface surface - the problem of correspondence of kdc to parameters L (or LL) and F falls apart, and primarily due to the physics of phenomena. They studied the influence of pore roughness and pore-size dis-
X =
(
pd
S,
\
-1
u
2v
where cd derived for this particular morphology on the basis of the exact analytical (in laminar regime) or well established correlations for Fanning friction factor in tubes.
Wong and Pengra [68] in recent work made another attempt of justification for using electrokinetics cross-phenomena as the rigorous one and that was reported as the theoretically well proven fact. Authors consider the Onsager's reciprocal relations in the form
Je =- G .V®- Ln V P
j, = - l v® - l22 v p ,
where L =F0 is the no-pressure gradient conductivity of the saturated porous medium, and L22=k0/0 is the zero-electric-field permeability of the same medium. Deducing further the formula connecting the three transport coefficients Fr, streaming potential coefficient KS and electroosmosis coefficient KE ) authors note - «the significance of this analysis is that it shows that the pore radius and permeability can be rigorously determined by measuring the KS, KE and the conductivity Fr. This result is completely independent of any microscopic detail of the pore structure and does not rely on any empirical correlation». Meanwhile, this result is the complete mathematical consequence of the assigned linear dependencies in the Onsager's kinetic relations which are generally speaking only experimentally determined at low intensity regimes for conductivity and permeability coefficients and strictly not justified theoretically.
3. MODELS FOR DESCRIPTION OF DIELECTRIC PROPERTIES OF LIQUID-IMPREGNATED POROUS MEDIA
3.1. Composite approximation
Since liquid-impregnated porous media are two-component composite materials, the same mathematical
в
models, used for the common composites of two solid components can do computation of their dielectric properties. In all models the treatment is done for infinite 3D media and does not take into account any walls that can be present in real systems.
Effective-medium theories (EMT) are commonly used for calculation of the properties of composite media. Two calculation schemes have been used, depending on the morphology of the mixture. In the first scheme, being a symmetrical approach which is applicable to mixtures in which no clear assignment of a matrix component and a filler component can be made, both components of the mixture are embedded in an effective medium with properties of the mixture. It is assumed that the average dipolar field due to the inclusions vanishes. With this assumption the Bruggeman formula for the mixture dielectric constant ee is obtained:
a. s sо s si
-с )-- = C-
s + 2 s,
s + 2 s, .
(3)
s = s, + 3es
S2 S1 s, + 2s
(4)
r = r.
2r, + r2 - 2с (r, - r2 ) 2r, + r2 - с(r, - r2 )
(5)
s - s,
S ^ Si
= с
(б)
r = exp((l - c)ln(£, ) + c ln(s2 )). (8)
Which of the above equations better describes the properties of porous media, is a subject of discussion. For example, in work [22] in description of ee of glassy semi-crystalline polymer - polypropylene impregnated by aromatic fluid has been shown, that closest agreement is obtained with Lichteneker's equation [79] recommends asymmetric effective medium equation for description of ee of porous polymer, in participance epoxy, filled with air.
Apparently, the most effective equation should be selected individually for each porous medium.
3.2. Bergman-Milton theory
A different approach was used in [80-83]; it has subsequently been used in others works [31, 84, 85]. Bergman defines the variables s and f as:
f 2 l - Г / Г,
1
s = ■
The second effective medium scheme, an asymmetrical approach which is applicable to mixtures in which one component can be assigned as the matrix and the second component as being the filler, a two-phase element consisting of a filler particle covered with a layer of matrix material is embedded in an effective medium with properties of the mixture. Again it is assumed that the average dipolar field due to the inclusion vanishes. Asymmetrical EMT is given as
J Г 2
(9)
Bergman showed that the function f(s) is an analytic function where all the poles are on the real axis. The poles are simple poles and have positive residues. Accordingly, we can write:
f (s )=£■
В
where r and £2 are the complex dielectric constants of the components 1 and 2 and c is the volume fraction of the component 2. If one identifies component 2 with the liquid and component 1 with the pore matrix then where is the bulk porosity.
The asymmetrical approach with a different averaging of permittivity results in a so-called Maxwell-Garnett (MG) equation [69]
(10)
It can be shown that 0 < s < 1 for all s . This ren n
sult can be deduced from the fact that if the dielectric constant is real and positive for both the constituents, it also has to be real and positive for the composite. Normally, there is a pole at the origin (s^) =
0, having the residue:
7 = a / a, = 1/F , (11)
1.e., B1 is the reciprocal of the formation factor F. One can also deduce the following relations for the residues:
E В=s, X За
s (i - s)
(12)
Levy and Stroud [70] presented the version of this approach which allows the inclusions to be made of aniso-tropic components. Recently Sen and co-workers [71] have derived expressions based on a self-similar model of the pore space. Their equation for ee is identical to the so-called differential effective medium theory (DEMT). It reads
There are different modification of this expression for the case of non-spherical pores with identical orientation [71] and isotropically distributed orientations [23,72,73]. In Ref. [74] DEMT is extended to three component systems. The useful and convenient for experimental data reduction formula was suggested in the «generalized effective media» (GEM) approach by McLachlan (see, for example, [75-77].
Several papers [22,78] describe dielectric properties of porous media by the Looyenga equation:
r = (1 - c)s,"3 + csl13, (7)
as well as Lichteneker equation:
The poles and their residues do not depend on the dielectric properties of the constituent materials, but solely on the geometry of the composite. Having once determined the poles and the residues, is sufficient to determine ee for the same geometry but with different constituent materials (for example, when changing the pore fluid of a porous solid). However, this theory is so far only developed for two-phase systems, and, the interaction between the solid and the liquid makes real materials more complicated. Besides, Bergman-Milton theory now extends only to 2D systems, which limits its applicability.
3.3. Grain Consolidation Model
Another interesting model is the so called Grain Consolidation Model (GCM). This model was originally suggested by Roberts and Schwartz [86] with the main purpose of modelling sedimentary rocks. A porous material where the solid phase is made up of grains, is perhaps the natural composite to be simulated by this model. That does not prevent its use for other porous materials [21] describes how the GCM can be used to estimate the properties of impregnated paper.
The solid material is assumed to consist originally of spheres, either placed on a regular lattice or completely at random [86, 87]. These spheres are then allowed to
grow until they touch each other and then to continue growing - if the spheres are considered to be non-interpenetrable only at the surfaces which do not touch another sphere - until the desired volume fraction of solid material is attained (see Fig. 1) Here, the simplest method is to allow the grains to grow equally in all directions (except where the grains touch), but calculations have been made where the grains are allowed to grow differently in different directions [87] .
For equal grains on a lattice developed a useful calculation method [88], using Fourier transforms.
Electrical field in liquid-impregnated porous media he writes in the following form:
E = E,(r(r) + @(r)02(r) , (13)
where 0t (/-) equals one in material 1 and zero in material 2 (and vice versa for 0 ^ (/-) ). The value of Et in the part of space that is impregnated by material 2 is of no interest, which means that Et and E2 can be smooth. This considerably reduces the number of Fourier terms needed to obtain a good convergence. The functions are expressed in Fourier series:
@ (r) = 2
m
o, (r) = um
(14)
(15)
b xVVr C = 0
n / j / j an—m am
(16)
^ ! ^ 1 raUan~tnCain 0
(17)
Here, the index denotes the summation over the different materials, the index n denotes the summation in Fourier space of the expansions of the 0 functions and the m index denotes the summation over the expansion for Et and E2.
Shen et al. [88] solve Maxwell's equations for a finite number of Fourier components of the field, M, by using a larger, but finite, number of Fourier components of the geometry, N (they recommend that N — M > 2). This yields more equations than unknowns, and the whole system of equations is solved by the method of least squares.
The effective e in the j direction (if the composite is anisotropic) is obtained from:
(18)
where b is 7.KId times the vector [m„ m„> m_l> m,, m„ and m3 being the coordinates in reciprocal space, and d is the distance between neighbouring grains.
The formulas are analogous for E2(r) and 0, (/-) . Note that the C and T coefficients have two indices. The first index denotes the two different materials and the second index (which is a vector) denotes the position in Fourier space.
Maxwell's equations are then written as
In works [21,89] is shown how this model can be 9 used to estimate the dielectric properties of liquid impregnated natural sandstone and glass beads.
3.4. Local porosity theory
Recently a new approach to the calculation of effective transport quantities, local porosity theory (LPT), has been proposed [32, 90-92]. The LPT differs from previous theories mainly through its focus on a measurable geometric characterisation of arbitrary porous media. The geometric characterisation is not based on correlation functions but on local geometrical quantities such as local porosity <j> . Within the LPT an arbitrary porous medium is quantitatively characterised through the local porosity distribution fj(f) and the local percolation probability A(^) [32]. The local porosity distribution fj(p) is defined as the probability density to find a porosity between and inside a local measurement cell.
The local percolation probability A(^) is defined as the
a) R=1.0
Figure 1. Two stages of the uniform growth simple cubic Grain Consolidation Model
b) R=1.2
fraction of measurement cells with local porosity <j> which are percolating. A local cell is said to be percolating if it contains a connection between two opposite faces. Geometric characterisation through fj(f) and A(^)is different from more conventional correlation function approaches as was demonstrated recently [92].
According to LPT the effective dielectric constant £(&) is given by the following equation:
f Ai^i^^fM-y (s) M s) qs +
j / / \ . ^ / \
0' rc (а,ф) + 2r(а) , V ra (а, ф) - r (а )
■V
rB (а, ф ) + 2r (а )
(1 - У (ф))ß(ф)dS 2 о,
(l9)
where
10
sc {а, ф ) = s2 {а )
1-
_1zS_
"-[ri {а УГ2 {а )] 1 - "Гф
(20)
and
rB (®, S ) = Г (® )
i -
"-[ (®)/ r, (®)]-"(1 - S)
(22)
are the dielectric constant of a conducting C and nonconducting (blocking) B local geometry with porosity <j> at the circular frequency m. Morphological orientation of the LPT approach can be used in the closure of electrodynamics VAT governing equations in heterogeneous media [54].
3.5. Morphology simulating network theories
Among few specific morphology driven theories one needs to distinguish between the modeling processes on the network designed or assigned for the problem. There is a problem of how to connect porous medium morphology characteristics that can and are being measured and transport equation notions and coefficients. In this approach results are thought using substitution of porous medium properties and functions using artificial network simulation capabilities assuming that governing equations are stated for a homogenised bulk medium characteristics (see, for example, [93-95]). The concern is that the physical problem, which is simulated with the help of network modeling, must still be modeled on the basis of its initial physical and mathematical statements.
At the same time consistent research and development in porous medium morphology reveals that more and more realistic structures in the pore network image started being available (see, for example, [96]), with up to 5 degrees of randomness in morphology assignment; 1) pore surface roughness: 2) pore diameter; 3) pore length; 4) pore pathway between modes (for tortuosity); 5) pore cross-sectional shape. An important consideration, most often not dealt with, is determining which network morphology properties can be assigned externally, or from an effective properties point of view, and which can be justified. Some of the network characteristics should be calculated to match the problem features and not assigned.
Another network modeling approach when the network being considered as the lower scale (microscale) discrete system which in turn is the part of the more general upscale continuum simulation. Unfortunately, the latter approach is almost missed in the contemporary work and the results of the network scale modeling are being reduced and analysed as done for the continuum process thought. Numerical modeling and application of theoretical studies to this kind of approach is based on realities of multiscaled medium morphology. This is to be considered as significant tool for closure of the certain VAT equations using the network modeling approach. Considerable improvement of network models appeared recently when authors [97,98], for example) started to address the transport phenomena in networks, as written in each separate pore in the network using continuum equations of an adopted form. In this way, the field transport between the nodes in the network is no longer described by simple algebraic dependencies (as it usualy done), but with differential equations of mathematical physics (assuming that the physical scale considered is sufficient for continuum phenomenology). The next step in this direction would be when addressing the complex transport phenomena in nodes (sites) of network will be thought also through the employment of continuum models.
4. DIELECTRIC PROPERTIES OF LIQUID-IMPREGNATED POROUS MEDIA IN FREQUENCY DEPENDENT ELECTRICAL FIELDS
4.1. General remarks
Frequency dependence of dielectric properties of liquid-impregnated porous media is affected both by dielectric properties of components themselves and the interface.
Most homogeneous materials are dispersion free below 1 GHz, i.e., e' and a are independent of frequency. This is also the case for glass or rock and water (or water solutions) taken separately. The mixtures of glass (or rock) and water, however, shows strong dielectric dispersion below 1 GHz [24, 31]
The origin of this dispersion, called the MaxwellWagner effect [99, 100] is the difference in dielectric properties of the two or more components of the mixture. Charges build up at the boundary between the components. This charge build-up will tend to increase the polarisation and decrease the conductance of the material. Because the charge build-up is not instantaneous, the effective dielectric constant and conductivity changes with frequency. For dispersion-free components both e' and <J will have a constant high-frequency and a constant low-frequency level, corresponding to complete charge build-up (low frequency) and no charge build-up (high frequency). And so, inhomogeneous material with conducting or semiconducting inclusions in dielectric media behaves as dielectric with relaxation time . For composites with components, that do not have frequency dispersion it is possible to find parameters of Debye's equation £() , ^ and t for interface polarisation, from dielectric properties of components [101].
However in practice, components of heterogeneous systems can have their own relaxation transitions and this fact seriously complicates e' and a' spectra. Com-
posite material, containing conducting inclusions with their own relaxation, in dielectric matrix without dispersion is treated in [101]. It has been shown that processes of interfacial polarisation and characteristic relaxation are interdependent. The following expression describes dielectric constant of such systems:
r (® ) = r^ +
r. - r,
r — r
1 + iax, 1 + iax^
(23)
Parameters of Eq. (23) are presented in Table 1 for three particular cases r ^ t , r ~ t and t ! t (here r is relaxation time of characteristic transition of the filler, t = £0£ls I <r2s. corresponds to relaxation due to conductivity of the filler)
Analysis of the Table 1 shows that in the limiting case x ^ t the processes can be directly identified with the process of interfacial polarisation and the transition of the filler. Since r, < r2 , the first process will be observed at lower frequencies with respect to the second one. The expressions for r and £t. are the same as in the absence of the interfacial polarisation. In other limiting case ( t ! t) the first process is degenerate, while the second one should be identical to the interfacial polarisation. In similar systems at frequencies below interfacial relaxation process, no other relaxation processes are possible in principle.
The analysis of literature data shows, that in case porous media is sintered glass beads or rocks, and the liquid is water or water solution of NaCl, frequency of interfacial polarisation is in the interval 103 - 107 c/s (it depends on volume fraction and conductivity of the liquid) [23, 31], while frequency of orientation relaxation of water is 1010-3.1010 c/s [102]. For the system sintered glass beads - water solution of ethyl alcohol the frequency of interfacial polarisation is 105 -106 c/s [31], frequency of orientation polarisation - 109 - 1.7.1010 c/s, depending on the ratio of concentrations. For these systems the frequency of interfacial relaxation is much less the frequency of intrinsic relaxation process in conducting phase, or t «t. That is why in these systems exist two separate processes, with parameters, presented in Table 1. Hence, three separate cases will be considered,
a) dc and low frequency range,
b) interfacial polarisation range,
c) the range of orientation polarisation in the liquid
phase.
4.2. Dielectric properties at dc and low frequencies
For brine-saturated porous rocks [103] found an empirical relation that is widely known as Archie's law:
a = a2afl . (24)
In literature, Eq. (24) is often written using formation factor F. Originally, it was assumed that a = 1, but it was later found that this has to be modified. For impregnated porous samples made of sintered materials, for example sintered glass beads impregnated with salty water values of a different from unity have been obtained, as is shown in Table 2, together with values of m.
A number of theoretical papers predicts deviation from Archie' law at low porosities [104]. Experimental data, however, exhibit no deviation and experimental F () is a straight line in double logarithmic scales (see Figure 2).
At first formation factor was introduced for dc conduction. In further works [24,33,105] it was generalised on non-zero frequencies. In work by Nettelblad
and Niklasson [105] shown how Archie's law can be derived from equations of Maxwell-Garnett, symmetrical EMT and differential effective medium theory (DEMT), generalised on the case of arbitrary orientation of ellipsoidal particles and exponent m was found for several microgeometries. Figure 2 presents experimental and theoretical for sintered glass beads impregnated with salty water low-frequency formation factor as a function of porosity . Dielectric properties of solid particles, suspended in liquid were calculated in Refs. [73, 106] within the framework of DEMT, fraction (1-p) of the particles being nearly spherical with depolarisation factor L=1/3, and fraction p being disk-shaped of aspect ratio a/b ( a ^ b = c ) and the depolarisation factor L = 1 — S , S = n a 12b , for a field parallel to a-axis (see Figure 3).
It was found that even for small p dielectric constant at dc and low frequencies can be as high as 103-104 and equal to:
r '(o ) - f r. (i-r).
(25)
Conductivity in this case follows Archie's law with the following m:
m
=! (i-, )+jl
3V ' 28 2-q .
(26)
4.3. Dielectric properties in the range of interfacial polarisation
Interfacial polarisation in the system sintered glass beads impregnated with salty water is investigated in [31]. It was found that with increasing frequency real part of dielectric constant and formation factor decrease and functions s'(f') and F(f) have sigmoidal
Figure 2. The low-frequency formation factor as a function of porosity as measured by Holwech and Nost (squares) and predicted using DEMT (solid line) (Nettelblad and Niklasson, 1996).
Figure 4. Dielectric constant as a function of reduced fre- Figure 5. Formation factor as a function of reduced frequen-quency (/ / f2) with porosity as a parameter; f2 = a2 / s2 cy (/ / f2) with porosity as a parameter; f2 = a2 / s2
Figure 6. Experimental and theoretical curves for the dielectric dispersion. The solid circles correspond to the experimental real part of the effective dielectric constant, and the open circles correspond to the experimental inverse formation factor. The solid and dotted lines correspond to the dielectric dispersion calculated theoretically using LPT with different X (S ) (Haslund et al., 1994)
Figure 7. Formation factor as a function of frequency with Figure 8. Formation factor (a) and dielectric constant (b) water resistivity as a parameter. Specimen porosity 5.2% vol. as a function of reduced frequency (f / /,) with water resistivity as a parameter
a)
b)
Figure 9 The high-frequency values of the permittivity (a) and the formation factor (b) as functions of porosity, calculated according to DEMT equation (solid line). The experimental datafrom Nost et al. (1992) are shown as squares.
shape (see Figures 4 and 5). Similar results were found for samples of quarried calcite rock, saturated with salty water [24]. There was an attempt to describe frequency dependencies of r' and r" for liquid-impregnated porous media using DEMT [105]. A good correlation of experimental and theoretical results in the range of interfacial polarisation. Deviation is observed only at low frequencies for e(f) , presumably due to the fact that the model neglects percolation effects. The model of local porosity gives better results at low frequencies [33] (see Figure 6). At high frequencies the results practically coincide with DEMT.
Dependence of r' on porosity is complicated. Figure 4 shows that curves e'{f) for different porosities cross over. At low frequencies r™decreases with increasing , but at high frequencies the dependence is reversed. LPT can accommodate this crossover in qualitative agreement with experimental observation [32]. In Ref. [33] it was shown, that LPT successfully describes this effect, both qualitatively and quantitatively.
The formation factor decreases with increasing porosity (see Figure 5) in the whole frequency range, the difference between the low- and high frequency plateaus decreasing with increasing porosity. Formation factor becomes practically independent of frequency for porosities above 15 % [105].
In Ref. [31] Archie's law was shown to hold only at dc and in the low frequency plateau. In the range of interfacial polarisation F(^) should follow more complicated models.
Refs. [24,31] investigate the effect of conductivity and dielectric constant of liquid on the dielectric properties of liquid-impregnated porous media, using such systems as sintered glass beads impregnated with salty water [31] and calcite rock, saturated with salty water [24]. Increasing conductivity of liquid proportionally increases (see Figure 7), while increase of dielectric constant proportionally decreases the frequency of interfacial polarisation. Dependencies of F(f) and e'(f) on reduced frequency f I f2 (/, /s,) result practically in a master curve (see Figure 8).
4.4 Dielectric properties in the range of orientation polarisation
For the majority of homogeneous liquids the range of orientation polarisation is above 0.5 GHz, i.e. in the microwave range. Liquid-impregnated porous media are practically unexplored in this range. Several consideration can be found from the analysis, given in [101] (see paragraph 4.1). First, since for these systems t ! t (see Table 1), a relaxation process due to orientation polarisation in the liquid should exist, its frequency being shifted to higher side with respect to that of pure liquid. Second, the height of low frequency plateau on frequency dependence of r' and o"(or F) for the process of orientation polarisation coincides with high frequency plateau for interfacial polarisation.
Dependence of r' and F in the range of high frequency plateau for interfacial polarisation on porosity was calculated using DEMT [105]. A good correlation with experiment was found for sintered glass beads impregnated with salty water (see Figure 9).
Measurements and calculations of dielectric properties of calcite rock, saturated with salty water [24] show that in the range of high frequency plateau for interfacial polarisation r™ depends on prolateness of the solid phase particles, this dependence being less at lower frequencies.
Frequency dependencies of r' and F were theoretically analysed in the framework of LPT for different local porosities fj(q>) [32]. It was shown that in the region of high frequency plateau of interfacial polarisation r' and F are practically independent of local porosity distribution. Investigation of composites dielectric-conductor shows, that in this frequency region dielectric properties are practically independent of tortuosity of conductive channels [106]. Thus, dielectric properties of liquid impregnated porous media in the region of high frequency plateau of interfacial polarisation depend, mainly on the properties of components, porosity, prolateness and orientation of pores. This may the reason for the fact that practically all models give close results in this region and one should probably use the least laborious method of composite approximation in the region of orientation polarisation.
a)
b)
Figure 10. Dependence of the real and imaginary parts of complex permittivity on porosity of porous media from Ba3Co2Ti Fe O ,ferntefi°lled with a) water, b) heptane. Frequency 4.8 GHz.
a)
b)
Figure 11. Dependence of dielectric properties on porosity of porous media from Ba3CoJig 8Fe23 2O41ferrite, filled with ethyl alcohol (curve 1) and mixture 67% vol. water+37% vol. alcohol (curve 2), presented in coordinates porosity - r' / r", (a) and porosity - s" / r" (b), where r", and r" are the values of the upper Wiener boundary. Frequency 4.8 GHz.
In [19,107] investigated microwave dielectric properties of different liquid-impregnated porous ferrite media and the possibilities for using the methods of composite approximation to describe these properties. It was shown that the properties of dry or water- or heptane-impregnated porous ferrite media are adequately described by empirically derived mixing law:
VT = (1 -c)^ + , (27)
[24], related to the equations of composite approximation model (see Figure 10). It was also found that (see Figure 11) in porous ferrite media, impregnated with ethyl alcohol or water-alcohol mixtures the real and imaginary permittivities over the upper Wiener boundary [108]. At the same time, permittivity, calculated with
different equations of the composite approximation model lie within the Wiener boundaries [101]. This result means, that the composite approximation model is not always applicable in calculations of the dielectric properties in similar systems.
The results of studies porous media may be useful for developing new vacuum electric capacitor, cryogenic tanks, cryogenic pipelines, SVHI of space apparatus, space suits, fire protection suits, specisl heat-shielding equipment for the Ministry of Emergency Situations. This work rusults may be used to develop systems and ensure safety in the industries that present a potential danger for the environment. This work describes the mechanism of some interrelated mechanisms of the dominating processes observed in large cryogenic tanks with a residual hydrogen
16
atmosphere in HIC and in the presence of a small atmospheric leak [109]. Porous media can also be used as components of microwave devices of absorbing type [110].
5. CONCLUSION
Dielectric properties of liquid impregnated porous solids depends on composition of the system, properties of the components and structure of the porous media.
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, etc.
The review undertaken in present work exposed the current situation in understanding, studying, and practising of electrostatic and electrodynamics phenomena in dielectric porous solids and porous media filled with liquid. It is obvious, that the most advanced and accurate results can be obtained with the type of theory which describe physics of phenomena on each level of hierarchy mathematically correct for the bulk as well as for detail sensitive microscale features.
The analysis of literature data show that microwave dielectric properties of liquid impregnated porous media depend on dielectric properties of components, volume fraction and elongation of pores, being practically independent of local porosity distribution and tortuosity of pores.
Experimental data on the liquid impregnated porous media exhibit the relaxation, a characteristic of pure liquid, with frequency shifted toward higher frequencies band together with interfacial polarisation. The shift and other parameters of relaxation process are both depend on dielectric properties of liquid and porous media as well as the porosity itself.
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