Free- and bound-water rotation in aqueous systems
Molecular rotation in polar liquids
Grain-boundary charging in porous materials
Ion conduction and double-layer electrode charging in liquid systems
Large molecule reorientation in polymer materials
We provide in-depth analysis and
interpretation as requested. We provide data in common dielectric
formats including complex permittivity as a function of frequency,
complex impedance diagrams, and Cole-Cole plots. We model data using
standard Debye, Cole-Cole, and Cole-Davidson models, to extract
molecular-level information on parameters such as relaxation time,
relaxation amplitude, and distribution of relaxation times. From
this we provide information on the state of processing of the material,
for such properties as viscosity, percent reaction, chemical state
of binding, etc.
MSI performs Broadband Dielectric Spectroscopy over the frequency
range 10 Hz to over 10 GHz. We use Low Frequency Impedance methods below 10 MHz and a combination
of TDR Dielectric and Microwave Cavity methods above 10 MHz. In aqueous materials we extend reliably
to the multi-GHz range to capture free-water behavior, using our TDR Smith-chart analysis.
Each of these areas is discussed below.
High-Frequency TDR Dielectric Spectroscopy
Multi-GHz TDR Smith-chart methods
TDR Dielectric Spectroscopy
Low-Frequency Dielectric Spectroscopy
Appendix - Smith chart basics
1. High-Frequency TDR Dielectric Spectroscopy
MSI uses Time Domain Reflectometry (TDR) Dielectric Spectroscopy for
high-frequency materials analysis, an innovative approach to high
frequency dielectric spectroscopy. The sensing electrodes are interrogated not with a continuous frequency
wave, but with a rapid voltage pulse containing a broad range of
frequencies at once [1-3]. The reflected pulse is converted to
complex permittivity by Laplace (Fourier) Transform, separating the
sensor response from connecting-line artifacts by propagation delay. An advantage is results can be interpreted in either
frequency or time domain, using calibration and frequency domain
analysis for high-quality scientific work, or direct analysis of the
reflected transient for robust field-grade monitoring.
TDR Dielectric Spectroscopy is related to conventional
Time-Domain-Reflectometry used in closed-circuit fault testing
[4-5]. However, TDR Spectroscopy focuses on a time and frequency
analysis from a lumped capacitance sensor while conventional TDR
focuses on spatial differences along a distributed transmission
TDR Background The expressions governing
TDR Dielectric Spectroscopy are described
in the literature [1,6]. An incident voltage pulse vi(t) propagating along
a transmission line of characteristic admittance Gc
encounters a terminating capacitive sensor of admittance Y producing
a reflected pulse vr(t). The terminating admittance Y is
related to the total current-to-voltage ratio Gc
(vi - vr)/(vi + vr),
where vi and vr are the Laplace transforms
of the incident and reflected pulses. The terminating admittance is
then related to sample
permittivity by Y=iωεCo so the permittivity is:
To establish a common time reference, the incident voltage is
substituted by the empty sensor reflection, by writing Equation 1
for both empty sensor and sample reflections and manipulating to
eliminate vi. The result is a reflection function
of similar form:
where vr,r and vr,x are the Laplace transforms for the
empty sensor and sample reflections. From this a differential expression can be written for
complex permittivity, or alternatively a bilinear form:
Where A, B, and C are complex parameters determined by calibration
with known reference standards. Additional methods such as nonuniform sampling and timing
control are described in the literature.
A typical TDR spectrum obtained in our laboratory for ethanol is
shown below. On the left is real permittivity and on the right is
the imaginary permittivity, both shown over a frequency range 100
MHz to near10 GHz. The data shows the expected dipolar relaxation
around 1 GHz, which continues to trail off in permittivity and loss
to around 10 GHz. A theoretical model based on Debye theory of
viscous rotation [7,8] is overlaid on both real and imaginary
components for comparison
The relaxation spectrum shifts with typical variations in material
parameters such as temperature, viscosity, molecular weight, mixture
concentration, etc. For example, the data below shows the
ethanol relaxation varying with temperature, with the loss peak
increasing to around 2 GHz at 55°C. Similar changes are seen
with other material variations such as addition of water or
substitution of different molecular-weight alcohols.
The data can also be presented in Cole-Cole or complex impedance
format, to further aid in the analysis . For example, the data
below shows a complex impedance arc in cement paste immediately
after mixing, at higher frequencies and shorter cure times than
allowed by low-frequency measurement. An arc in the complex
impedance plane demonstrates that material behaves as an electrolyte
resistance in parallel with an interelectrode capacitance, allowing
the bulk resistance to be quantified independent of electrochemical
effects at the electrodes.
2. Multi-GHz TDR Smith-chart methods
MSI extends measurement bandwidth to multi-GHz frequencies reliably
using inexpensive single-use sensors. The importance is capturing
the free-water relaxation occurring in aqueous systems, and
separating it from bound-water relaxation and other effects
occurring at lower frequencies. The ability to capture free water
response reliably opens a range of new applications, from industrial
process monitoring, to biotech, and other areas.
Despite its relative low cost and simplicity, TDR is an RF/microwave
measurement requiring RF/microwave levels of analysis. MSI recently
developed a TDR Smith Chart , in which the Laplace transform of
the sample reflection is divided by the Laplace transform of the
empty-sensor reflection and displayed in the complex plane, similar
to Vector Network Analyzer (VNA) methods.
The magnitude of this ratio is always one, for low-loss
materials, so the display traces a circle over the range of
frequencies, with the variation in phase appearing as a variation in
real and imaginary components. Deviations from this circle reveal
unwanted signal artifacts, isolating these
artifacts from normal signal response up to 10 GHz and above.
Transmission losses cancel, so movement across additional
Smith-chart circles of constant susceptance and conductance indicate
actual sensor response. The TDR Smith chart provides a quick
diagnostic, prior to time-consuming calibration, showing whether
transient data is artifact-free and following expected behavior, or
whether corrective steps must be taken.
Smith chart examples
Representative TDR Smith charts are shown below. Each shows the
ratio of sample-material Laplace transform to empty-sensor transform, or relative reflection coefficient
to 15 GHz. Each shows the signal tracing a semicircular arc around
the lower half of the complex plane, showing a reflection
coefficient with near constant magnitude
and increasing phase shift.
Frequency labels are shown at select points, starting at 2
GHz on the right and continuing to 15 GHz on the left.
An admittance Smith chart is used, with constant susceptance
and conductance circles originating from the left, treating the
sensor and sample material as a parallel circuit model. Both
susceptance and conductance are normalized to the characteristic
0.02 S/m line admittance in the usual manner, and labeled on the
An example for non-conducting liquids is shown on the left for
dichloromethane (ε' = 8.85), acetonitrile (ε' = 37.5),
and 0.7 M water/ethanol (10 < ε' < 45, measured with standard 3.5 mm
semi-rigid coax with a flat termination. The low-permittivity dichloromethane
traces a short arc around the lower right of the complex plane,
crossing circles of constant susceptance
(iωεCo/.02)slowly with frequency.
The high-permittivity acetonitrile traces a longer path around the lower half of
the complex plane, crossing circles of constant susceptance more rapidly
with frequency. The moderate-permittivity ethanol traces an intermediate path around the
complex plane, crossing circles of constant susceptance, but also
moving to the interior and crossing circles of constant conductance.
This results from the high ethanol loss, which causes the
signal to spiral inward with increasing frequency, crossing circles of
constant susceptance and constant conductance .
An example for conducting liquids is shown on the right for water
and salline solutions (ε’ ≈ 78). The
deionized water traces an arc around the perimeter of the complex
plane in the usual manner, while the saline shifts the arc to the
interior with increasing concentration, following circles of
Smith chart diagnostics
The TDR Smith chart reveals acquisition and analysis errors by
displaying results on a normalized unit circle in the complex plane, accentuating
small anomalies between real and imaginary components. Since the reflection coefficient is
a precursor to the reflection function used in bilinear
calibration, the TDR Smith chart is a valuable tool in detecting
errors upstream, before time-consuming calibration is performed.
One example is an
incorrect setting in baseline or integration cursors used in
the numerical Laplace integration. Below on the left is a TDR Smith
chart for acetonitrile and the 0.7 M water/acetone solution, in
which a 5 mv offset is introduced in the vertical baseline, about 1%
of the full 400 mv reflection. An
obvious artifact appears in the transformed display around 10 GHz,
with some distortion leading up to this frequency. Similar artifacts
occur for other types of acquisition and analysis errors, including
improper Laplace truncation, multiple reflections within input
lines, timing errors, and sensor damage. Each error propagates further into the calibration process,
appearing in the reflection function ρ(ω), the bilinear calibration
parameters A(ω) and B(ω), and the final calibrated permittivity
Another example is an internal reflection from sample boundaries. Below on the
right is a TDR Smith chart for a sensor with a 1 mm protruding pin
in a small beaker of water (ε’ = 78). When positioned near the
center of the beaker the red trace appears, when positioned near one
side the blue trace appears. An obvious difference is a small loop
appearing around 1 GHz, apparently representing a radiated signal
reflecting from sample boundaries. The reflection occurs because of
the high dielectric discontinuity between the water and surrounding
air, but is too small to be seen in the direct transient.
It is accentuated by the differential and bilinear methods
used in calibration, but is seen at an early stage in the TDR Smith
A third example is the approach to
which varies with pin length and sample permittivity.
Below on the left is a TDR Smith chart for 2 sensors in
water, one ground perfectly flat and the other with a 0.3 mm
protruding pin. A serrated shield surrounds the pin to prevent
radiation and sample boundary reflections. For the flat sensor the
signal traces a path around the lower half of the complex plane in
the usual manner, to around 10 GHz.
For the 0.4 mm pin the signal traces a more rapid path around
the complex plane, but begins to distort at around 7-8 GHz. The
distortion apparently represents the approach to pin resonance, and
occurs at this position on the Smith chart because the reflection
coefficient is defined as the ratio of the sample reflection to the
empty-sensor reflection, rather than the incident pulse. This is
done to eliminate timing differences between incident and reflected
pulses, but results in the TDR reflection coefficient being a relative
reflection coefficient between sample and empty-sensor reflections:
Which can be adjusted to an absolute reflection coefficient Гa by
multiplying by the term
which corrects the small difference between incident pulse and
empty-sensor reflection. The absolute reflection coefficient is shown on the right,
where the signal for the 0.4 mm pin crosses the negative real axis
into the inductive region at around 7-8 GHz. Obviously this
situation must be avoided, by adjusting the pin length and/or sample
3. Multi-GHz TDR Dielectric Spectroscopy
An example of multi-GHz TDR Dielectric Spectroscopy is the
monitoring of cement hydration  by following the free- and
bound-water relaxation spectrums at frequencies of 10 GHz and above.
An inexpensive capacitance sensor is made by terminating a
standard 3.5 mm semi-rigid coaxial line perfectly flat, and
immersing in fresh cement paste. The flat termination provides
approximately 20 femtofarads (ff) fringing capacitance, providing an
appropriate load admittance into a medium-permittivity material over
the range 100 MHz to 15 GHz.
A first step is the selection of calibration
liquids with similar permittivity/loss spectra. Any RF/microwave
measurement, be it VNA and TDR, relies on calibration with known
reference standards to remove artifacts originating in the
instrument and connecting lines. VNA requires 3 calibrations
generally open, short, and 50
ohms. TDR also requires 3 calibrations, where for materials
measurements we use the empty sensor and 2 known reference liquids,
whose permittivity and loss closely approximate the unknown
material. Since the
permittivity and loss of cement decrease during cure, as water is
consumed in reaction, we choose 2 reference liquids which
approximate the permittivity/loss at early cure and late cure,
assuming that the signal evolution during cure lies in between.
A good calibration for early cure is a mixture of saline and PMMA
microbeads. The saline provides the strong free-water relaxation
expected in fresh cement past, while the PMMA reduces the transition
amplitude from 78 to around 40. The saline also adds a strong ion
conductivity, similar to cement paste. A good calibration for late
cure is a low-permittivity solvent with a high relaxation frequency,
such as dichloromethane or THF. The high relaxation frequency
provides the slowly decreasing permittivity and rising loss expected
at late cure, typical of porous solids. A trace amount of zinc
nitrate adds a small conductivity at long transient times, keeping
the transient resolvable at long times and allowing calibration
parameters to be calculated over the entire range.
A second step is examination of the
TDR Smith charts at the two calibration limits, to correct any errors in sensor
response or acquisition and analysis. Smith charts for PMMA/saline
and THF/zinc nitrate are shown below, overlaid with cement cure data
at 0 and 72 hours, respectively. For the early calibration, the relative reflection
coefficient traces a rapid path to 15 GHz but does not cross
resonance; for the late calibration the reflection coefficient
traces a more gradual path due to the lower permittivity and
susceptance. In both cases the signal is smooth to 15 GHz, with no
sample boundary reflections, acquisition/analysis errors, or other
A third step is the calculation of corresponding reflection
functions, which essentially represent the uncalibrated permittivity. Each
reflection function ρ(ω) is calculated from its reflection
coefficient Γ(ω), according to:
Reflection functions for the saline/PMMA and the THF/zinc nitrate
are shown below, with the imaginary parts ρ’’ multiplied by εoω
to display the uncalibrated dielectric conductivity.
For saline/PMMA, the real permittivity on the left shows a
constant value to near 2-3 GHz and a roll-off around 10 GHz for the
free-water relaxation. The dielectric conductivity on the right
shows a flat region to near 1GHz due to ion conductivity and a rise
around 10 GHz for the free-water loss peak.
For the THF/zinc nitrate, the permittivity and conductivity
are much lower, and an additional feature appears around 1 GHz due
to the zinc-nitrate solute relaxation.
Each reflection function is overlaid with a model function to
generate bilinear calibration parameters. For the saline/PMMA, the relaxation
time is set to 8.2 ps for
water, with the relaxation amplitude adjusted to the lower
saline/PMMA volume ratio. For THF/zinc-nitrate, the relaxation time
is set to 5 ps for THF, to match the slowly falling permittivity and
rising loss over the range.A small solute relaxation is added to the THF/zinc nitrate
model at around 1 GHz. A
constant ion conductivity is added to both calibrations to match the
broad flat region below 100 MHz.
A final step is the generation of bilinear
parameters and the calibration of the cement reflection functions.
Bilinear parameters A and B are found by solving equation (3) for
both reflection functions and their corresponding model functions,
generating 4 simultaneous equations for real and imaginary
components. Details are described in the references . By then
applying parameters A and B to the reflection functions for curing
cement, the calibrated permittivity and conductivity at various cure
times is determined.
Results for hydrating cement paste are shown below, from about 100
MHz to 15 GHz. Separate
free-water relaxation and ion-conductivity regions are clearly seen
in the initial stages of cure. As cure proceeds the free water
permittivity and ion conductivity decrease, and a separate
bound-water region appears in the conductivity around 1 GHz,
representing water attaching to developing microstructure.
4. Low-Frequency Dielectric Spectroscopy
We also provide low-frequency dielectric and impedance spectroscopy
using standard HP4192 Impedance Analyzer methods. Samples are
placed in 4-wire capacitance cell and measured over a frequency
range 10 Hz to 10 MHz. Results can be modeled in the Cole-Cole
permittivity plane or complex impedance plane as appropriate.
An example of low-frequency dielectric spectroscopy is a cured epoxy
thermoset shown below. The epoxy shows polymer-chain relaxation in
the 1 kHz to 1 MHz range, with a broad roll-off in permittivity seen
on the left, and a similarly broad loss peak seen on the right.
Another example is a porous glass sample shown below. The sample
shows strong low-frequency dispersion due to surface currents along
pore edges accompanied by interfacial charging at grain boundaries
(Maxwell-Wagner effect). As the sample is heated to drive off
moisture the low-frequency dispersion disappears, and only reappears
as the sample is returned to ambient for a period of time.
TDR and low-frequency measurement can be combined over an extremely
wide frequency range . The data below shows the real
permittivity of curing cement over a frequency span of 9 decades,
from 10 Hz to 10 GHz. Two relaxations are seen in the figure below,
a low-frequency relaxation due to the mobility of free ions, and
high-frequency relaxation due to the mobility of bound water.
The high-frequency relaxation straddles both TDR and low-frequency
5. Appendix - Smith chart basics
The Smith chart is a convenient way of visualizing the reflection
coefficient on a transmission line, as well as the terminating
sensor impedance, all on one chart. The basic reflection
coefficient is plotted in the complex plane, with additional circles
of resistance and reactance indicating the changing sensor impedance. Smith
charts have been used for many years in VNA work, with numerous
references available in the literature .
For an admittance Smith
chart, used in parallel circuit models,
the starting point is the basic relation between incident and
reflected voltages and currents and the terminating sensor admittance Y(ω).
where the current difference in the numerator is replaced by the
voltage difference multiplied by the characteristic line admittance
(0.02 S/m). Both sides are then divided by the line admittance to
give a relation between the incident and reflected voltages in the
middle and the normalized terminating sensor admittance y(ω) on the left.
where the incident and reflected voltages in the
middle are then written in terms of the
reflection coefficient Γ on the right by dividing through by vi .
Substituting Γ = Γr + iΓi
and y = g + ib
in Equation 9
for the complex reflection coefficient and normalized terminating admittance,
separating real and imaginary parts, and manipulating some algebra, gives 2 equations:
Which are equations of circles
for Γi vs. Γr.
For the first set of circles the radius and offset are determined solely by
the conductance; for the second set the radius and offset are determined
solely by the susceptance. The 2 circle sets thus show
Γi varying with Γr
in a circular manner when either the conductance or susceptance
is held fixed*. Alternatively, plotting both circle sets for
differing values of conductance (red) and susceptance (blue) is equivalent to
adding 2 additional sets of gridlines, showing how the conductance and
susceptance vary as these circles are crossed.
*Γi vs. iΓr
may be varied at fixed conductance or susceptance
by either varying the frequency, as done here, or varying the transmission
line length, as done in antenna load-matching.
6. Technical References
Additional information on broadband TDR Dielectric Spectroscopy can be found at:
R. H. Cole, J. G. Berberian, S. Mashimo, G. Chrssikos, A. Burns,
and E. Tombari, "Time domain reflection methods for dielectric
measurements to 10 GHz", J. Appl. Phys 66, 793 (1989)
H. Fellner-Feldegg, J. Chem Phys. 73, 616 (1969).<.p>
N.E. Hager III, "Broadband Time-Domain-Reflectometry Dielectric
Spectroscopy using variable-time-scale Sampling", Rev. Sci.
Instrum. 65(4), April 1994, p 887. download pdf*
N.E. Hager III, R.C. Domszy, M.R. Tofighi, "Smith-chart
diagnostics for multi-GHz Time Domain Reflectometry Dielectric
Spectroscopy, Rev. Sci. Instrum. 83, 025108 (2012). download
Satoru Mashimo and Toshihiro Umehara, "Structures of water and
primary alcohol studied by microwave dielectric analysis", J.
Chem. Phys. 95 (9), 1 November 1991.
J. Barthel, K. Bachhuber, R. Buchner and H. Hetzenauer.
"Dielectric spectra of some common solvents in the Microwave
Region. Water and Lower Alcohols" Chem. Phys. Letters 165 (4) 19
January 1990 369.
N. E. Hager III and R. C. Domszy, "Monitoring of Cement
Hydration by Broadband TDR Dielectric Spectroscopy", J. Appl.
Phys. 96, 5117-5128 (2004). dowload pdf**
N.E. Hager III, R.C.
Domszy, M.R. Tofighi, “Multi-GHz Monitoring of Cement Hydration
Using Time-Domain-Reflectometry Dielectric Spectroscopy”, Fourth
International Symposium on Soil Water
Capacitance, Impedance and Time Domain Transmission (TDT),
Pointe Claire Quebec, July 2014.
William Hayt, John Buck,
“Engineering Electromagnetics” Eighth Edition, McGraw-Hill, NY,
A. K. Jonscher,
Dielectric Relaxation in Solids, Chelsea Dielectrics Press,
Arthur R. Von Hippel,
Dielectric Materials and Applications, Wiley, New York (1954).
*Copyright 2012 American Institute of Physics. This article may be
downloaded for personal use only. Any other use requires prior
permission of the author and the American Institute of Physics. The
following article appeared in Review of Scientific Instruments and
may be found at http://link.aip.org/link/?RSI/83/025108
**Copyright 2004 American Institute of Physics. This article may be
downloaded for personal use only. Any other use requires prior
permission of the author and the American Institute of Physics. The
following article appeared in Journal of Applied Physics and may be
found at http://link.aip.org/link/?jap/96/5117
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