Measurements of Complex Permittivity of Geological
Materials Mixtures At Rf Frequencies
Antonio Sarri, Matteo Batisti
Matteo Bientinesi
Measurement Laboratory
IDS Ingegneria dei Sistemi SpA
Pisa, Italy
[email protected]
CPTM Consorzio Polo Tecnologico Magona
Cecina (LI), Italy
[email protected]
Abstract— This paper presents a measurement technique
specifically developed to address the problem of measuring the
complex permittivity of mixtures of geological materials at RF
frequencies (10MHz – 1GHz). The main challenge of
measuring this kind of mixtures is that each component could
have a different state of aggregation (solid, liquid) and
conglomeration (e.g. homogeneous, porous, granular). The
paper describes the data acquisition as well as the processing
technique, reporting some examples of results that show the
applicability and effectiveness of the proposed approach to
different kind of geological materials mixtures.
Keywords – permittivity;
materials; mixtures; RF
I.
measurements;
geological
INTRODUCTION
Dielectric properties of materials (i.e. permittivity and
permeability) at RF frequencies are of interest for a wide
number of applications, from the aerospace and defense to
the chemical industry.
Several techniques have been proposed in literature and
different commercial and custom-made instrumentation and
fixtures have been developed to address the problem [1] [2]
[3].
The main parameters to be considered to select the most
suitable instrumental set up and data processing technique
are:
•
Frequency range of interest;
•
State of aggregation of the material (i.e. solid, liquid,
semi-solid, granular etc.);
•
Non-destructive nature of testing;
•
Possibility to manufacture samples with tight
dimensional tolerances.
Among the different fixtures, the most used are:
•
Coaxial probe;
•
Transmission line;
•
Free space;
•
Resonant cavity;
•
Parallel plate.
Furthermore, several instruments are commercially
available, such as:
•
Network analyzers;
•
Impedance analyzers;
•
LCR meters.
These devices allow the measurement of the dielectric
properties of materials (by applying an inversion algorithm),
from few MHz up to tens of GHz.
Nevertheless, each instrument and technique has its own
specific features and limitations; furthermore each one
requires specific assumption regarding the sample under test
(e.g. semi-infinite thickness, non-magnetic, isotropic and
homogeneous, flat surface of sample, etc.), in order to assure
the nominal measurement accuracy and repeatability.
II.
MATERIALS OF INTEREST AND MEASUREMENTS
CHALLENGES
The materials of interest are homogenous granular
materials (e.g. sand, silicon carbide, aluminum oxide, etc.) or
mixtures of sand, clay, water and bitumen. It is worth noting
that, giving that grain size is much less than the wavelength
at which measurements are performed, granular materials
can be considered as a mixture of a solid homogeneous
material and air.
The nature of these materials represents a measurement
challenge, for two main reasons:
1. Due to the state of aggregation, it is often not
possible to manufacture samples of desired shape with tight
dimensional tolerances, as usually done for solid materials
(to be fitted in coaxial cells or similar fixtures); at the same
time samples cannot be considered liquid (thus allowing the
use of open coaxial probes);
2. We are interested in the complex permittivity of
each component of a mixture as well as in deriving the
complex permittivity of mixtures of a given type but with
different composition (i.e. percentage in weight or even type
of components), that possibly are not available for testing.
simpler equivalent circuit model of this set up is represented
by the parallel of a capacitance C p and conductance G [2].
Regarding the first challenge, several techniques have
been proposed in literature, e.g. coaxial cells or open coaxial
probes modified to improve the contact between the probe
end and the sample surface [5] [6].
The instrument is capable to measure directly the
complex admittance of the resulting circuit (as a function of
frequency) and directly compute the complex dielectric
permittivity ε r according to the following equations:
Our approach is to manufacture custom sample holders
with PTFE chassis and very tiny glass covers (see Fig. 1): the
material of interest is placed in between glass covers so that
a multilayered material was obtained and measured.
⎛ Cp
G ⎞
⎟
Y = G + jωC p = jωC0 ⎜⎜
−j
C
ω
C0 ⎟⎠
⎝ 0
εr =
Cp
C0
−j
tC p
G
t
=
+j
ωC0 Aε 0
ωR p Aε 0
(1)
(2)
where t is the distance between electrodes and A is
their surface.
It is worth nothing that the thickness of MUT (equal to
t ) has to be known (by direct measurement).
The technique is very accurate in case of solid samples
that fit perfectly in between the electrodes (i.e. with no air
gaps); on the other hand, any lack of contact result in very
poor accuracy and eventually in the impossibility to have any
reliable measurement.
Figure 1.
Custom sample holder with PTFE chassis and very tiny glass
covers
To address the second challenge, we focus our attention
on the properties of each single component of a mixture and
on the relation between the complex permittivity of each
component and that of the mixture. This kind of relationship
is known in literature as “mixing models” [7] [8] [9].
III.
MEASUREMENTS TECHNIQUE OVERVIEW
The technique and results presented in this paper cover
the frequency range 10MHz–1GHz. Furthermore, the
complex permittivity has been measured over a range of
temperatures, from 20°C to 125°C (i.e. over the boiling
temperature of water to evaluate the effect of possible water
evaporation), using a climatic chamber.
The adopted set up was based on a commercial
Impedance/Material analyzer with a capacitive fixture
capable to work at high temperature.
Experimental physicochemical characterization of materials
is also necessary to get input data to be used in the post
processing.
A. Permittivity measurements
In the standard case of an Impedance/Material Analyzer,
the Material Under Test (MUT) is hold between the
electrodes of a parallel plate capacitive fixture, thus
representing the dielectric of the resulting capacitor; the
Furthermore, the accuracy strongly depends on the
accurate knowledge of the sample thickness.
B. Physicochemical Characterization
In order to process the results of the dielectric properties
measurements on granular materials and mixtures, it is
essential to know with the maximum grade of precision the
volumetric composition of the samples. Thus a methodology
for the characterization of the volumetric composition was
set up for both types of materials of interest, as described in
the following.
1) Volumetric composition of granular materials: the
granular materials considered have no internal porosity and
can be seen as a mixture of the material composing the
grains and of air sorrounding the grains. The void grade of
each granular material was estimated by:
• measuring the tapped bulk density (ρb) of the
granular material (according to the ASTM D7481-09
[11], method B);
•
measuring the absolute density (ρabs) of the grains
(according to the ASTM D854-10 method [12]);
•
calculating the void grade φ with the following
formula:
ϕ = 1−
ρb
ρ abs
(3)
2) Volumetric composition of mixtures: the absolute
density of the different components (inorganic matrix,
granular solids, bitumen, acqueous solution) of each mixture
was first measured with methods analogues to those
described in the previous paragraph.
For reconstructed mixture samples, the massive
composition is obtained directly during the preparation of the
sample by weighting each component before mixing. For
natural occurring mixture (e.g. soil samples), the massive
composition was determined by Thermogravimetric Analysis
(TGA) with the following procedure:
•
•
•
the water mass fraction is taken to be equal to the
percentage weight loss of a representative sample
after a treatment at 180°C for 1 h in nitrogen flux;
the bitumen mass fraction is taken to be equal to the
percentage weight loss of a representative sample
after a high temperature thermal cycle (rising from
35°C to 800°C in N2 flux with a rate of 10°C/min,
then permanence at 800°C for 10 min in N2 and for 5
min in air) minus the water content determined in the
previous step;
Figure 2.
Equivalent circuit representing respectively the actual MUT
and the glass layers.
The purpose of the so called de-embedding technique is
to derive the unknown properties of the MUT itself by the
knowledge of the properties of the glass and the thickness of
the MUT and of the glass layers.
Glass properties can be directly measured with the
standard technique, being each glass layer an homogeneous
solid sample.
The corresponding equations are:
Z MUT = Z LAYERS − 2Z GLASS =
1
2
−
YLAYERS YGLASS
(6)
the organic matrix content is the rest.
The instrument used for TGA analysis was the Perkin
Elmer Pyris 1 TGA.
Once the density (ρi) and the mass fraction (xi) of each
component i is known, the volumetric fraction ( x vol
j ) of each
component can be calculated by the following equation:
ρj
n
∑
xi
(4)
i =1 ρ i
IV.
(7)
YGLASS = GGLASS + jωC p GLASS
(8)
where Z is the impedance, Y is corresponding inverse (i.e.
the admittance) and the subscript indicate the sample (layers
is the multilayer).
xj
x vol
j =
YLAYERS = G LAYERS + jωC p LAYERS
PROCESSING TECHNIQUES
To address the problem of measuring the complex
permittivity of mixtures and components, as well as inferring
the permittivity of a hypothetical mixture from the measured
permittivity of its components, we developed specific
techniques for the processing of the measured data.
A. De-embedding
As anticipated, we manufactured custom sample holders
with PTFE chassis and very tiny glass covers in which the
material of interest was placed in between the glass covers.
Using this fixture, the direct measurement obtained by
the Impedance/Material Analyzer is the admittance of a
multilayered sample composed of the actual MUT put in
between the two glass layers.
The resulting multilayered structure can be modeled as
the series of the equivalent circuit representing respectively
the actual MUT and the glass layers (see fig. 2).
Once YMUT is known, the complex permittivity is
calculated with (1) and (2). It is worth nothing that, in order
to achieve the desired accuracy, the code developed to apply
(1) and (2) uses values of t and A slightly different from
the geometrical ones that are derived by calibration with
standard material (e.g. PTFE or Rexolite).
To verify the validity of this approach, we use solid
samples and compare the estimation of their permittivity
(obtained with the de-embedding procedure by putting the
solid sample in the sample holder) with the direct
measurement of the solid sample. Some relevant results are
reported in section V.
By applying this processing to several materials, we
found out some practical applicability conditions (mainly
consequence of computational issues):
•
the de-embedding processing is necessary when the
impedance of the MUT is of the same order of
magnitude of that of glass;
•
on the contrary, the de-embedding processing is not
strictly necessary when the impedance of the MUT
is much less of that of glass, being the latter
negligible so that the measured impedance of the
multilayer material is almost equal to the actual
impedance of the MUT (within measurement
tolerances).
It is worth nothing that the above practical guideline can
be considered valid separately for the real and imaginary part
of impedance.
B. Inverse and Direct Mixing
To address the different aspects of complex permittivity
of mixtures, we considered and assessed several mixing
models found in literature [7][8][9], i.e.:
•
Linear model;
•
Time-Propagation (TP) model;
•
Refractive Mixing Dielectric Model (RMDM)
•
Generalized Refractive Mixing Dielectric Model
(GRMDM);
•
GRMDM with volume fraction (same as GRMDM
but including the volume fraction of mixture in the
equations).
by using measurements of single component or some data
from literature [8].
V.
RESULTS
The following figures show some examples of results (in
terms of real and imaginary part of relative permittivity, i.e.
ε’ and ε”). These results are obtained by applying the postprocessing techniques described in the previous section:
• De-embedding (see Fig. 6 and Fig. 7)
• Inverse Mixing (see Fig. 5)
• Direct Mixing on a mixture (see Fig. 3 and Fig.
4).
The first case of interest is the measure of granular
materials performed by preparing a two solid component
mixture (i.e. granular MUT and wax). In this case the most
suitable model appears to be the TP model (see [9] for
equations).
By inverting the equations of the TP model (inverse
mixing), it is possible to compute the dielectric permittivity
of the MUT from the direct measurement of the dielectric
permittivity of a solid sample composed by wax only and the
dielectric permittivity of a solid sample composed by the
mixture of MUT and wax (with known volume fractions).
Figure 3. Permittivity (real part ε’) of mixture 1 (bitumen, sand, salt
water) at 20°C, comparison between direct masurement and calculation
through direct mixing.
To verify the validity of this approach, we compare the
dielectric permittivity of the same MUT obtained from
mixtures with different volumetric percentage of wax; the
method is considered to be accurate if the different
estimations of dielectric permittivity agree each other (within
the measurement accuracy of the standard measurement).
Some results are reported in section V.
The other case of interest concerns natural occurring
mixtures, where typically one or more components of the
mixture is liquid (e.g. groundwater). In particular, we focus
on the case of a mixture composed by sand, groundwater and
bitumen, for which the most suitable model appears to be the
GRMDM (see [7] for equations).
To verify the validity of this approach, we apply the
GRMDM model (direct mixing) to a mixture that can be
found in nature in solid form, first by measuring a sample of
the natural mixture directly and then by measuring the single
components of the mixture and using this data as the input of
the mixing model. The method is considered to be accurate if
the direct measurement and the permittivity resulting from
mixing equations agree within the measurement accuracy of
the standard measurement. Some results are reported in
section V. Once validated, the model can be used to derive
the complex permittivity of mixtures not available for testing
Figure 4. Permittivity (imaginary part ε") of mixture 1 (bitumen, sand,
salt water) at 20°C, comparison between direct masurement and calculation
through direct mixing
VI.
CONCLUSIONS
We address the problem of measuring the complex
permittivity of mixtures of geological materials at RF
frequencies (10MHz – 1GHz). In order to measure materials
having different states of aggregation we used custom
sample holders (made with PTFE and glass) and a deembedding processing, while to deal with different kind of
mixtures we evaluate and implement different mixing
models found in literature. The experimental results fully
demonstrated the applicability and reliability of the presented
approaches to practical cases of interest. The measured data
have been also used as input for the simulation of the process
of radiofrequency/microwave heating for thermal heavy oil
recovery based on a novel tight-shell conceptual design [4].
Figure 5. Permittivity (real part ε’) of Silicon Carbide at 20°C calculated
through inverse mixing from two different mixtures (wax 20% and 30% in
volume).
ACKNOWLEDGMENT
The authors would like to thank ENI S.p.A. for support
and for providing some of the materials under test. The
authors would also like to thank Mauro Bandinelli,
Alessandro Cerutti, Luigi Petarca and Francesca Signorini
for help and valuable suggestions.
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Figure 6. Permittivity (real part ε’) of Silicon Carbide from 20°C to
125°C; comparison of the values calculated through de-embedding of
sample holder and through inverse mixing from mixture with wax.
Figure 7. Permittivity (imaginary part ε") of Silicon Carbide from 20°C to
125°C; comparison of the values calculated through de-embedding of
sample holder and through inverse mixing from mixture with wax.
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