Time domain dielectric spectroscopy: An advanced ...

Time domain dielectric spectroscopy: An advanced measuring system Yuri Feldmana) The Fredy and Nadine Herrmann Graduate School of Applied Science, Hebrew University of Jerusalem, Givat Ram, Jerusalem 91904, Israel

Alexander Andrianov Institute of Electronic Measurements ‘‘KVARZ,’’ 176, Prospect Gagarina, Nizhny Novgorod 603009, Russia, CIS

Evgeny Polygalov, Irina Ermolina, Grigory Romanychev, and Yuri Zuev Kazan Institute of Biology of the Russian Academy of Sciences, p/b 30, Kazan 420503, Russia, CIS

Bronislava Milgotin The Fredy and Nadine Herrmann Graduate School of Applied Science, Hebrew University of Jerusalem, Givat Ram, Jerusalem 91904, Israel

~Received 28 August 1995; accepted for publication 23 May 1996! The new time domain measurement system for dielectric measurements is described. The current model is comprised of an IBM PC-AT/486, ‘‘TDM-2,’’ a new time domain measurement system, a set of thermostabilized sample holders, and operation and analysis software. This system is designed for use in the measurement of dielectric parameters of liquid and solid materials over the frequency range 100 kHz–10 GHz. Software consists of programs of registration, accumulation and data collection, Fourier analysis, time domain treatment, analysis software: fast and reliable nonlinear curve fitting programs to determine spectroscopic parameters and correlation analysis in time domain. The system utilizes the difference method of measurement with the registration of primary signals with multiwindow nonuniform sampling. Such a system permits the overlap of a frequency range of five orders in a single measurement. © 1996 American Institute of Physics. @S0034-6748~96!00709-5#

I. INTRODUCTION

The recent appearance of time domain dielectric spectroscopy ~usually called TDS! precision measuring systems1–4 may help resolve the problems of accuracy and sensitivity associated with this method when investigating different kinds of substances. However, even in the case of two-channel bridge measuring systems it is necessary to make special calibration measurements and considerable calculations in order to overcome many of the disadvantages inherent in TDS such as shortcomings in the construction of coaxial lines, samplers, step generators, etc. These distortions and instabilities appear in the form of an altered shape of the incident pulse that cannot be compensated for either by the ordinary difference method5–8 or by apparatus correction of the pulse shape. In particular, these unwanted distortions have been found to be crucial in the case of the system with parallel time nonuniform sampling.9 Corrections of the incident pulse shape by apparatus adjustments in the case of nonuniform sampling of the signals do not lead to noticeable improvement because of such procedure limitations as time borders of the ‘‘time window.’’ As a rule, distortions increase outside time interval bounds. In addition, TDS systems inherently have various general hardware problems due to instability in the time axis including jitter, time drift, and variation of the sweep speed in the oscilloscope. These problems are all associated with the high-frequency limitations of the measuring system. On the other hand, the slow drifts of a!

Author to whom correspondence should be addressed; Electronic mail: [email protected]

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Rev. Sci. Instrum. 67 (9), September 1996

the base line and effects of truncation at a long finite time limit are primarily connected with low-frequency limitations.1–3,10 Recent developments in high-speed sampling techniques and digital processing, together with basic theory of TDS, have brought about a new generation of TDS measurement systems. We describe such an instrument based on time domain measuring set ~TDMS! specially designed for TDS hardware support. The system utilizes the difference method of measurement7,8 and the registration of primary signals with parallel time nonuniform sampling.9 II. BASIC PRINCIPLES OF THE METHOD

TDS is based on transmission line theory in the time domain which aids in the study of heterogeneities in the coaxial lines according to the change of the test signal shape.1–4 Until the line is homogeneous the shape of this pulse is not changed; but, in the case of heterogeneity in the line ~the dielectric put into it, for example! the signal is partly reflected from the air–dielectric interface and partly passes through it. The simplified block diagram of the setup common for most TDS methods ~except transmission techniques! is presented in Fig. 1. Differences include mainly the construction of the measuring cell and its position in the coaxial line. These lead to the different kinds of expressions for the values that are registered during the measurement and dielectric characteristics of the objects under study. For the configuration represented in Fig. 1, a rapidly increasing voltage step V 0 (t) arrives at the sampling head where the signal R(t) reflected from the dielectric sample is

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where R sc(t) and R ol(t) are the signals from the short circuit ˙ 9 (t) are the ˙ 8 (t) and K and open coaxial line respectively; K apparatus functions of incident and reflected signals, more˙ 9 (t). The difference between Eqs. ~5! and ~6! over K˙ 8 (t)ÞK gives one the following relation: R ol~ t ! 2R sh~ t ! 5D ~ t ! * r ol~ t ! 2D ~ t ! * r sh~ t ! , where D(t)5V (t) * K˙ 9 (t) is the setup function.

~7!

0

FIG. 1. Schematic drawing of the TDS system arrangement.

also registered. The R(t) signal goes to a sampler with the time delay corresponding to a double length of coaxial line from the sampler to the air–dielectric interface. Generally for the ideal system, the voltage applied to the sample is V ~ t ! 5V 0 ~ t ! 1R ~ t ! ,

~1!

where V 0 (t) and R(t) are the incident and the reflected signals, respectively ~Fig. 2!. The expression for the current flow through the sample is4,9,10 I~ t !5

1 @ V ~ t ! 2R ~ t !# , Z0 0

~2!

where Z 0 is the characteristic impedance of the empty line.

It is well known that in frequency domain, rsh~v!521 and rol~v!51, respectively, for the case of ideal coaxial line. In time domain it leads to the corresponding relations rsh(t)52 d (t) and rol(t)5 d (t). The convolution procedure with d function in Eq. ~7! leads to the following simple relation: R ol~ t ! 2R sh~ t ! 52D ~ t ! .

~8!

The correction of the registered signals can now be done both in frequency or time domain. In frequency domain one must apply the direct and inverse fast Fourier transform procedure of the signals or there will be differences before and after the correction, respectively. In time domain, one must resolve the deconvolution equation in order to eliminate the contribution of the setup function.

A. Setup function correction

In the real system, the test signal applied to the sample contains in itself the transfer function of the sampler and of the whole coaxial line. In fact, the incident signal can be registered in the measurement of the coaxial line terminated at a 50 V terminator. In this case one can write the real incident signal V 0 (t) in the sample cell as ˙ ~ t !, V 0 ~ t ! 5V 50 V ~ t ! 5Vˆ 50 V ~ t ! * K ~3! where Vˆ (t) is the incident pulse that can be registered in 50 V

the case of an ideal coaxial line terminated to a 50 V termi˙ (t) is the pulse transfer function of the circuit, called nator, K the apparatus function; the asterisk designates the procedure of deconvolution. In general, this function in itself contains all of the signal distortions present in the measuring system. For the reflected signal R(t) in the ideal case one can write R ~ t ! 5V 0 ~ t ! * r ~ t ! ,

B. Lumped capacitance approximation

The technique used in this work is called the lumped capacitance method.11–14 In this method the sample cell may be treated like a capacitor with capacitance e*~v!C 0 where e*~v! is the sample permittivity and C 0 is the capacitance of the empty capacitor, which can be calculated from its geometric parameters. The total current through the conducting dielectric is composed of the capacitor charging current I Q (t) and the low-frequency current between the capacitor electrodes I R (t). Since the active resistance at zero frequency of the cell containing sample is14 r5 lim t→`

˙ 8~ t ! 1 @ V ~ t ! *K ˙ 9 ~ t !# * r ~ t ! , R sc~ t ! 5V 0 ~ t ! * K 0 sc

~5!

˙ 8~ t ! 1 @ V ~ t ! *K ˙ 9 ~ t !# * r ~ t ! , R ol~ t ! 5V 0 ~ t ! * K 0 ol

~6!

~9!

the low-frequency current can be expressed as

~4!

where r(t) is the reflection coefficient of the line terminating at the sample holder. In the real measuring system one can write the expression for the signal reflected from the line terminated from the short circuit and the open coaxial line in the following way:

V~ t ! V 0 ~ t ! 1R ~ t ! 5Z 0 lim , I~ t ! t→` V 0 ~ t ! 2R ~ t !

I R~ t ! 5

V 0 ~ t ! 2R ~ t ! V ~ t ! V 0 ~ t ! 1R ~ t ! 5 lim . r Z0 t→` V 0 ~ t ! 1R ~ t !

~10!

Thus, relation ~2! can be written as I Q~ t ! 5

1 Z0

S

@ V 0 ~ t ! 2R ~ t !# 2 @ V 0 ~ t ! 1R ~ t !#

3 lim t→`

D

V 0 ~ t ! 2R ~ t ! . V 0 ~ t ! 1R ~ t !

~11!

Relations ~1! and ~11! present the basic equations, related to the experimental signals recorded. The electric charge of the capacitor filled with the sample Q(t) can be related to the dielectric response function F(t) and the applied voltage V(t) by the relation FIG. 2. Characteristic shape of the signal recorded during a TDS experiment. V 0 (t): incident pulse; R(t): reflected signal. Rev. Sci. Instrum., Vol. 67, No. 9, September 1996

S

Q ~ t ! 5C 0 e ` V ~ t ! 1

E

t

0

D

˙ ~ t2t 8 ! V ~ t 8 ! dt 8 , F Dielectric spectroscopy

~12! 3209


where C 0 is the empty cell capacitance. Taking into account that Q(t)5 * t0 I Q ( t )d t and performing a Fourier transform on Eq. ~12!, using the general equation12

e *~ v ! 5 e `1

E

`

0

F ~ t ! exp~ 2i v t ! dt,

L @ V 0 ~ t ! 2R ~ t !# 1 , i v C 0 Z 0 L @ V 0 ~ t ! 1R ~ t !#

~13!

where L is the operator of Laplace transform. Note that F ~ s ! 5L @ f ~ t !# 5

E

`

0

f ~ t ! e 2st dt,

s5x1i v ,

can be complex since x→0. Usually the various modifications of differential methods are used.4,7,8 The essence of the differential method is in recording standard sample signals with known dielectric characteristics and signals of the object being tested. This approach allows the compensation for undesirable reflections within the coaxial line and the increase in sensitivity while measuring weak signals ~for example, the measurement of dilute solutions of some samples with respect to a solvent8!. However, the ordinary difference method does not lead to the desired results because in addition to the two signals from the reference and unknown samples, a signal from the short-circuit line is necessary as well.7,8 Therefore, the distortions can only be partially compensated for.

C. Total difference method for lumped capacitance approximation

In order to obtain the relations for the total difference method it was much easier to perform further transformations in frequency domain. For lumped capacitance approximation relation ~13! can be written in the following form:12,13

e *~ v ! 5

1 v 0 ~ v ! 2r ~ v ! , i v Z 0 C 0 v 0 ~ v ! 1r ~ v !

~14!

where v 0~v! and r~v! are the Laplace transforms of the incident and reflection signals V 0 (t) and R(t), respectively, Z 0 is the characteristic impedance of the empty coaxial line, and C 0 is the capacitance of the empty sample cell terminated at the end of the coaxial line. Let us rewrite the above expression for the three different cases of measurements: (a) the empty cell, (r) the sample cell filled with reference sample, (x) the sample cell filled with unknown sample. After some simple transformations one obtains three equations in frequency domain:

e* n ~ v !@v 0 ~ v ! 1r n ~ v !# 5 ~ a /s !@v 0 ~ v ! 2r n ~ v !# ,

~15!

where a51/Z 0 C 0 , s5i v , n is equal to a, r, and x materials for each equation, respectively. The simple transformations of these three equations let us obtain the basic relationship for the total difference method in lumped capacitance approximation for the case when the reference sample is the nondespersive dielectric, er 5const and ea *~v![1, 3210

Rev. Sci. Instrum., Vol. 67, No. 9, September 1996

g * ~ v ! e r ~ 11 a /s ! 1 ~ a /s !~ e r 21 ! , g * ~ v !~ 11 a /s ! 2 ~ e r 21 !

~16!

where g *~ v ! 5

one arrives at the expression for the complex dielectric permittivity of the sample,

e *~ v ! 5

e* x ~ v !5

L @ R a ~ t ! 2R r ~ t !# , L @ R r ~ t ! 2R x ~ t !#

L is the operator of Laplace transform, and R a (t), R r (t), and R x (t) are the signals reflected from the coaxial line terminated at the sample holder with air, reference, and unknown dielectric, respectively. In general, this relation can be considered valid for the total difference method. The main feature of this relation is that Fourier transform of the setup function d( v )5L[D(t)] cancels out in both numerator and denominator of g *~v!, respectively. Thus, in the case of the total difference method it is not necessary to make the correction of registered signals. This correction is still valid if one is using the ordinary difference method.7,8 The ‘‘lumped capacitance’’ approximation has a strong limitation in the high-frequency range of TDS measurements, especially in the case of polar dielectrics being tested. If one takes into account the definite physical length of the sample and multiple reflections from the air–dielectric or dielectric–air interfaces relation ~9! must be written in the following form:1–4

e *~ v ! 5

L @ V 0 ~ t ! 2R ~ t !# c X cot X, i v ~ g d ! L @ V 0 ~ t ! 1R ~ t !#

~17!

where X5 v d A e * ( v )c, d is the effective length of the inner conductor, c is the velocity of light, and g is the capacitance per unit length of the cell to that of the matched coaxial cable. A similar relation to Eq. ~14! for the total reflection method was obtained previously.15 III. EXPERIMENTAL TOOLS A. Hardware

This TDMS has been specially designed for TDS hardware support. TDMS offers a comprehensive, high precision, and automatic measuring system. It has a small jitter factor ~,1.5 ps!, important for rising time; small flatness of incident pulse ~,0.5% for all amplitudes!; and a unique option for parallel time nonuniform sampling of the signal. It consists of a signal register, two-channel sampler and pulse generator. The generator produces 200 mV pulses with 10 ms duration and 28 ps rise time. Two sampler channels are characterized by an 18 GHz bandwidth and 1.5 mV noise ~rms!. Both channels are triggered by one common sampling generator which provides their time correspondence during operation. The form of the voltage pulse thus measured is digitized and averaged by the digitizing block of TDMS. In Fig. 3 one can see the block diagram of this unit. Time base is responsible for major metrology TDMS parameters. Two program-controlled devices provide the TDMS time scale: a digital delay unit ~DDU!, and an analog delay unit ~ADU!. A sequential DDU–ADU connection provides full time scale up to 10 240 ns with unlimited step and initial delay within this time. TDMS can form variable step discretization sweeps. It gives us the unique opportunity of special sweeps Dielectric spectroscopy


FIG. 3. Simplified block diagram of the TDMS system. FIG. 4. Circuit diagram of a TDS setup.

for evaluation of nonuniform sampling. Thus, in general, it is possible to measure complex dielectric permittivity over a frequency spectrum from the frequency equivalent of 10 ms to the frequency equivalent of 10 ps, taking into account that the high-frequency limit depends also on some other factors. The minimum discrete step is 2.5 ps. Due to refined design methods, time jitter of TDMS is less than 1.5 ps ~rms!. ADU nonlinearity and drift are compensated for by the addition of special devices to the basic circuitry. ADU drift measurement is performed by checking two reference points of 2 GHz sine. The 2 GHz generator is synchronized by quartz and a special calibration program. The first reference point must be chosen at the beginning of the ADU time band; the second point at the end. The difference between these values is proportional to the ADU drift. The sweep is given by means of a 16 bit digital-to-analog converter ~DAC! and ADU on a diode with the charge accumulation. A microprocessor program calculates this difference, inverts it and writes it to the TDAC buffer for drift compensation. The resulting error of a sweep within the range up to 40 ns does not exceed 5 ps. The pulse generator and sampler have their own independent drifts. Such drifts are eliminated by an autostabilization program called ‘‘auto center’’ through an autostabilization analog-to-digital converter ~ADC! control. For ADC control, two DACs are used: One controls the vertical position of the signal, the other the time delay position. The program checks the position of the reference signal, which is formed from the generator pulse in the microwave measuring circuit, and is received by any sampler channel. Two reference points ~one at the stable part of the reference signal, and the other one at the pulse edge! provide data for the program about vertical and time drifts. ADC control codes are calculated on the basis of this information analysis. Drift compensation and autostabilization programs run in ‘‘hidden’’ mode after each sweep cycle, allowing the reduction of overall signal drifts up to 1 ps and significantly increasing precision and performance of TDMS. The block diagram of described setup TDS-2 ~Dipole TDS, Ltd. Jerusalem! is presented in Fig. 4. B. Software

Measurement procedures, registration, storage, time referencing, and data analysis are carried out automatically. The Rev. Sci. Instrum., Vol. 67, No. 9, September 1996

process of operation is done in on-line mode and the results can be presented both in frequency and time domain. There are several features of the modern software that control the process of measurement and calibration. One can define the time windows of interest that may be overlapped by one measurement. During the calibration procedure the precise determination of front edge position is carried out and the setting of internal auto center on this positions applies to all following measurements. The precise determination and settings of horizontal and vertical positions of calibration signals are carried out also. All parameters are saved in a configuration file. This allows one to do a complete set of measurements with the same parameters without additional calibration. Some details of this software will be described in the following subsection. 1. The principle of building of the multiwindow time scale

The implemented time scale is the piecewise approximation of the logarithmic scale. It includes nw#16 sites with the uniform discretization step determined by the following formula:

d nw5 d 1 2 nw,

~18!

where d155 ps is the discretization step at the first site, and the number of points in each step except for the first one being equal to npw532. At the first site, the number of points npw152*npw. Doubling the number of points at the first site arises from the necessity to have the formal zero time position, which is impossible in the case of the strictly logarithmic structure of the scale. In addition, a certain number of points located in front of the zero time position are added. They serve exclusively for the visual estimation of the stability of the time position of a signal and are not used for the data processing. For the given version of the program this number is equal to 25. This is the number of the point where unprocessed signals join if the difference procedure is applied. The described structure of the time scale allows the overlapping of the time range from 5 ps to 10 ms during one measurement, which results in a limited number of registered readings. At will, the overlapped range can be shortened, Dielectric spectroscopy

3211


resulting in a decreasing number of registered points and reducing the amount of time required for data recording and processing. The major advantage of the multiwindow time scale is the ability to get more comprehensive information. The signals received by using such a scale contain information within a very wide time range and the researcher merely decides which portion of this information to use for data processing. Also, this scale provides for the filtration of registered signals close to the optimal one already at the stage of recording. 2. Calibration

For calibration, the short-circuit terminator must be used connected to the measuring line. Then, by using the appropriate choice, the calibration program bootstraps. The calibration procedure runs automatically if the base block works normally and the operator makes no error. Otherwise, the diagnostic part of the program determines a possible reason of failure and a message on how to operate in this case. The result of calibration is the formation of the time scale with the corresponding parameters and the recording of the detected parameters into the file MULTWIN.CFG. The calibration also determines the adjusted coefficients of the channel sensitivity, which later can be defined and used for normalization of the measured signals. 3. Measurements

The sample cell must be terminated at the measuring line, the appropriate menu choice must be made, and the measurement program must be run. With every access to the program the state of the base block is checked, and the adjusted coefficients of the channel sensitivity, base-line position, and the amplitude of the probing signal are defined. These parameters are all used for the signal scaling at the registration stage. If errors or failures in the base block operation are found and if they can be eliminated by the operator, a help message appears. 4. Data processing

The flow chart of the data processing software is presented in Fig. 5. It includes the options of signal corrections, correction of electrode polarization, and dc conductivity, and different fitting procedures both in time and frequency domain. For the case of nonuniform sampling the Samoloon algorithm12 of numerical Fourier transform is used, and can be presented in the following way: F~ v j !5

d1 12exp~ 2i v j d 1 !

N1

(

n52

$ f ~ n d 1 ! 2 f @~ n21 ! d 1 # %

N int

3exp~ 2i v j n d 1 ! 1 3

dk 12exp~ 2i v j d k !

( k52

S

exp 2i v j

k21

(

m51

Tm

D

Nk

(

n51

$ f ~ndk!

2 f @~ n21 ! d k # % exp~ 2i v j n d k ! , 3212


~19!

FIG. 5. Flow chart of data treatment software.

where N int is the number of uniform measurement intervals, dk is the sampling rate in uniform time interval ~k51,...,N int!, N k is the number of points in uniform interval k ~k51,...,N int!, v j is a frequency point, in which Fourier transform is calculated, and T m is the total current time interval. The fitting programs allow one to determine spectroscopic parameters and correlation analysis in time domain. C. Sample holders

A universal sample holder that can be used for both liquid and solid samples in the low- and high-frequency borders of the TDS method is unfortunately as yet unavailable. The choice of its configuration depends on the measurement method and data treatment procedure. In the framework of lumped capacitor approximation one can consider three general types of sample holders, as follows.4 ~a! A cylindrical capacitor filled with sample: Such a cell can also be regarded as a coaxial line segment with the sample having an effective g d ~g is the ratio of the vacuum capacitance per unit length of the cell to that of the matched coaxial cable! length characterized in this case by the corresponding spread parameters. This makes it possible to use practically identical cells for various TDS method modifications. For the total reflection method it is the most frequently used configuration.1–3,6,10,16,17 ~b! The second type is a plate capacitor terminated to the central electrode on the end of the coaxial line.4,12–14,18,19 ~c! The third type is an open-ended coaxial line sensor.4 In the case of lumped capacitance approximation ~a! and ~b! configurations have high-frequency limitations and for highly polar systems one must take into account the finite propagation velocity of the incident pulse or, in other words, the spread parameters of the cell.1 The choice of cell shape is determined to a great extent by the aggregate condition of Dielectric spectroscopy


FIG. 6. ~a! Sample cell for high-frequency measurements of liquids; ~b! sample cell for low-frequency measurements of liquids and solid disks.

the object studied. While cell ~a! is convenient to measure liquids, configuration ~b! is more suitable for the study of solid disks and films @for more details see Figs. 6~a! and 6~b!#. Both cell types can be used to measure powder samples. While studying anisotropic systems ~liquid crystals, for instance! the investigator may replace a coaxial line by a strip line or construct the cell with the configuration providing the measurements under various directions of the applied electric field.3,4 The ~c!-type cell is used only when it is impossible to put the sample into the ~a! or ~b! cell types.4,20 The fringing capacity of the coaxial line end is the working capacity for such a cell. The sample holders described above can be thermostabilized and in our case the temperature can be computer controlled over the temperature interval 260 to 1100 °C. IV. ERRORS

The error sources for various TDS modifications are practically identical21 and therefore can be classified as shown in Fig. 7. All of the sources are independent, so the total error is equal to the sum of all the components. We will classify two kinds of errors of different nature: hardware errors and computational errors. The level of hardware errors can be estimated by their influence on the measuring time signals. The evaluation of computational errors is based on the calculation of the permittivity spectrum of model signals in time domain produced by a hypothetical Debye-type dielectric with the known permittivity. A unique aspect of the technique proposed is differentiation of hardware errors relative to certain sources, while computed errors are considered

on the whole. The accuracy of the described system depends on the frequency range, dielectric permittivity and working capacitance of the sample cell22–24 ~see Table I!. The detailed description of the error analysis system of TDS measurement has been presented elsewhere.24,25 V. METHODICAL ASPECTS OF TDS MEASUREMENTS

Here we present some of the TDS measurement difficulties that can be found in the studying of different dielectric samples. It must be noted that the accuracy and resolution of the setup are strictly dependent on the polarity and dc conductivity of the sample. The most convenient and simple samples for TDS measurements are the pure polar nonconductive liquids that can be used as reference objects for TDS setup testing. Figure 8 presents the Cole–Cole plots of pure alcohols. The range of dielectric permittivities allows one to be in the framework of lumped capacitance up to the 8 GHz. The automatic fitting parameters are presented in Table II, and are in good agreement with those found in the

TABLE I. The accuracy of the system TDS-2. Frequency ~GHz! ,6 GHz

.6 GHz

Material ~dielectric permittivity!

e8 ~%!

e9 ~%!

e8 ~%!

e9 ~%!

3–15 15–30 30–80 .80

1–2 2–3 3–5 5

2–3 3–5 5–7 7–10

2–3 3–5 5–7 5–7

3–5 5–7 7–10 .10 FIG. 7. Classification of the error sources for various TDS modifications.


Dielectric spectroscopy

3213


FIG. 8. Cole–Cole plot for normal alcohols at 30 °C. ~1!j CH3OH, ~2! d C2H5 OH, ~3! m C3H7OH, ~4! l C4H9OH, ~5! 1C5H11OH, ~6! 3C6H13OH, ~7! *C8H17OH. 26,27

literature. The TDS-2 system shows reasonable accuracy for such a system when tested in both time and frequency domains. In contrast to simple samples, complex samples need a wide range of frequencies to be overlapped by one measurement. For example, biological systems, solutions, mixtures, microemulsions, etc., have several relaxation processes in different time scales. The multiwindow system is very useful for the study of such materials. Here we present some of the recent results on microemulsions obtained by the new TDS system described above. Microemulsions were prepared by dissolving 23.23% AOT, 24.79% H2O, and 51.94% n-decane ~% weight!. Temperature was controlled in a thermostabilized sample holder by a cooling thermostat HAAKE G with precision of 0.2 °C. The measurements were carried out in several time domain windows with a total time interval of T52.5 ms. All the measurements were carried out in the sample holder for high-frequency measurements @see Fig. 6~a!# with capacitance 0.5 pf ~pin length is 3 mm, pin diameter is 5 mm!. Even in the vicinity of the percolation threshold where dielectric permittivity reached high values the lumped capacitance approach gave reliable results up to 2 GHz. Continuation of the data treatment to the higher frequencies requires considering the multiple reflection in the sample holder and using the approach of distribute capacity.1–4 Figure 9 shows the three-dimensional picture of the spectrum of dielectric permittivity versus the temperature and frequency for ionic microemulsion during the percola-

FIG. 9. Three-dimensional representation of the dielectric permittivity of an ionic microemulsion vs temperature and frequency.

tion effect. The molecular mechanisms and the nature of this effect have been presented elsewhere.28 In the case of the low polar systems ~polymers, polar molecules in nonpolar solvent, etc.! the main difficulty of TDS setup resolution is the weak signal. There are two options for increasing the sensitivity of the spectrometer: One is to increase the working capacitor of sample cell and another is to increase the accumulation number. Thus, for the evaluation of the reliable dielectric spectrum for the samples of cross-linked natural rubber ~Fig. 10! one has to apply ;105 accumulation, which takes almost 40 min for one signal registration. Many dielectric materials are conductive. The TDS study of conductive samples is very complex because of electrode polarization and the need to correct for the effect of low frequency conductivity. Usually for a low conductive

TABLE II. Dielectric fitting parameters for normal alcohols at 30 °C.

Alcohol

es

e`

t ~ps!

Standard deviation

Methanol Ethanol Propanol Butanol Pentanol Hexanol Octanol

31.9 23.6 19.6 16.7 14.2 12.1 8.9

2.8 2.5 2.4 2.4 2.2 2.4 2.3

54 132 330 425 665 810 1190

0.4831021 0.4431021 0.2231021 0.1231021 0.6431022 0.2831022 0.1231022

3214


FIG. 10. Dielectric losses of natural rubber NR2 and NR5 ~Freiburg! crosslinked by sulfer 25 °C. ~Samples were prepared by Dr. H. Menge from Martin Luther University, Halle-Wittenberg, Germany.! Dielectric spectroscopy


dent and reflection signals V 0 (t) and R(t), respectively. A similar relationship can be written for the reference sample ~usually a buffer!: Z s ~ v ! 5Z 0

FIG. 11. The equivalent circuit for a conductive dielectric sample with electrode polarization impedence.

system the value of dc conductivity can be evaluated as described in Sec. II B. One of the greatest obstacles in TDS measurements of conductive systems is the parasite effect of electrode polarization. The accumulation of charge on the surfaces of the electrodes leads to the formation of electrode double layers. The magnitude of the capacitance caused by electrode polarization is so great that its correction is absolutely necessary to obtain accurate measurements with conductive samples, particularly aqueous biological systems. Many approaches29–31 have been tried in order to correct for the effect of electrode polarization; however, the nature of this phenomenon remains unclear, such that no simple technique for its correction is available. Here we briefly describe one way to correct this phenomenon. The ion layer near the electrodes can be considered in terms of a capacitor and resistor connected in series with the capacitor filled with sample ~Fig. 11!.32 This capacitor and resistor adds a complex impedance Z p to the system being tested. Thus, the following relation for the unknown impedance can be written in frequency domain: Z x ~ v ! 5Z 0

Rx v 0 ~ v ! 2r ~ v ! 5Z xp ~ v ! 1 , v 0 ~ v ! 1r ~ v ! 11i v e * x ~ v !C 0R x ~20!

where v 0~v! and r~v! are the Fourier transforms of the inci-

FIG. 12. Cole–Cole plot of the dielectric permittivity spectrum of a 7% erythrocyte suspension ~Ref. 32!: ~1! before taking into account the electrode polarization effect; ~2! after taking the electrode polarization effect into account. Rev. Sci. Instrum., Vol. 67, No. 9, September 1996

Rs v 0 ~ v ! 2r ~ v ! 5Z sp ~ v ! 1 . 11i v e s C 0 R s v 0 ~ v ! 1r ~ v !

~21!

The above relationship must meet the following requirements: R X 5R S 5R, Z sp 5Z xp 5Z p , and e s* ( v ) 5 e s 5 const. The combination of Eqs. ~20! and ~21! provides the basic formula for electrode polarization correction,

e* x ~ v !5

e s 2 @ A ~ v ! /i v C 0 R # 2A ~ v ! e s , A ~ v ! 1i v A e s C 0 R11

~22!

where A( v )5[Z x ( v )2Z s ( v )]/R, and R is the sample resistance as given by Eq. ~9!. It was found that the effect of electrode polarization is the same for both cell suspensions with small volume fractions ~less than 15%! and their supernatants ~see Fig. 12!. So, Z p must therefore be calculated from the additional measurement of the supernatant, which has the same conductivity as the cell suspension, but whose dielectric permittivity is known from high-frequency measurements.32

1

R. H. Cole, J. G. Berberian, S. Mashimo, G. Chryssik, A. Burns, and E. Tombari, J. Appl. Phys. 66, 793 ~1989!. 2 S. Mashimo, T. Umehara, T. Ota, S. Kuwabara, N. Shinyashiki, and S. Yagihara, J. Mol. Liq. 36, 135 ~1987!. 3 R. Nozaki and T. K. Bose, IEEE Trans Instrum. Meas. IM-39, 945 ~1990!. 4 Yu. D. Feldman, Yu. F. Zuev, E. A. Polygalov, and V. D. Fedotov, Colloid Polym. Sci. 270, 768 ~1992!. 5 R. H. Cole, S. Mashimo, and P. Winsor IV, J. Phys. C 84, 786 ~1980!. 6 S. Bone, Biochim. Biophys. Acta. 967, 401 ~1988!. 7 Yu. D. Feldman, Yu. F. Zuev, I. V. Ermolina, and V. A. Goncharov, Russ. J. Phys. Chem. 62, 269 ~1988!. 8 Yu. D. Feldman and V. D. Fedotov, Russ. J. Phys. Chem. 61, 1045 ~1987!. 9 I. V. Ermolina, E. A. Polygalov, G. D. Romanychev, Yu. F. Zuev, and Yu. D. Feldman, Rev. Sci. Instrum. 62, 2262 ~1991!. 10 D. Bertolini, M. Cassettari, G. Salvetti, E. Tombari, and S. Veronesi, Rev. Sci. Instrum. 61, 450 ~1990!. 11 Yu. D. Feldman, V. A. Goncharov, Yu. F. Zuev, and V. M. Valitov, Chem. Phys. Lett. 58, 304 ~1978!. 12 Yu. D. Feldman, V. A. Goncharov, Yu. F. Zuev, and V. M. Valitov, Chem. Phys. Lett. 65, 69 ~1979!. 13 V. A. Goncharov and Yu. D. Feldman, Chem. Phys. Lett. 71, 513 ~1980!. 14 M. J. C. Van Gemert, Adv. Mol. Relax. Processes 6, 123 ~1974!. 15 H. Nakamura, S. Mashimo, and A. Wada, Jpn. J. Appl. Phys. 21, 1022 ~1982!. 16 J. G. Berberian and R. H. Cole, Rev. Sci. Instrum. 63, 99 ~1992!. 17 J. G. Berberian, J. Mol. Liq. 56, 1 ~1993!. 18 N. E. Hager III, Rev. Sci. Instrum. 65, 887 ~1994!. 19 P. Dorenbos and H. W. den Hartog, J. Phys. E 21, 178 ~1988!. 20 C. Gabriel, E. H. Grant, and I. R. Young, J. Phys. E 19, 843 ~1986!. 21 A. W. Dawkins, R. J. Sheppard, and E. H. Grant, J. Phys. E 14, 1260 ~1981!. 22 M. A. Rzepecka and S. S. Stuchly, IEEE Trans Instrum. Meas. IM-24, 27 ~1975!. 23 K. Baba and T. Fujimura, Jpn. J. Appl. Phys. 25, 285 ~1986!. 24 G. Romanychev, I. Ermolina, E. Polygalov, Yu. Zuev, D. Upshinsky, and Yu. Feldman, Meas. Tech. R 35, 61 ~1992!. 25 G. Romanychev, I. Ermolina, E. Polygalov, Yu. Zuev, and Yu. Feldman ~unpublished!. 26 S. K. Garg and C. P. Smyth, J. Phys. Chem. 69, 1294 ~1965!. Dielectric spectroscopy

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Dielectric spectroscopy
