Cole Equation Research Articles

Silicon MEMS probe using a simple adhesive bonding process for permittivity measurement

We developed a silicon MEMS probe for permittivity measurements using an adhesive bonding process. Only two photolithographic masks are required to fabricate the probe, which can be implemented through simple bonding processes using silicon substrates and a benzo cyclo butene (BCB) adhesive layer. Undoped silicon substrates with thicknesses of 300 µm are used as the dielectric layers of the proposed probe. BCB layers, which have good electrical properties at high frequencies as well as adhesive properties for the bonding process, play the role of bonding materials between the two silicon substrates. The length of the probe is 30 mm, and the aperture located at the tip of the probe is 1.1 mm × 0.62 mm. The permittivity of 0.5% saline was measured, and the results agreed with the values obtained through the Cole–Cole equation. To validate the feasibility of this probe for practical biological applications, we also performed in vivo measurements of the muscle, skin and blood of mice. Due to the simple fabrication process, the cost of the probe can be reduced in comparison with the previous micromachined probe (Kim et al 2005 J. Micromech. Microeng. 15 543–50) as well as the conventional laser machined probe. Low cost leads to disposability, which is an important factor for practical biomedical applications; and thus, coupled with the probe's capabilities of MMIC integration and CMOS compatibility, this probe has excellent potential in the field of microwave permittivity measurements.

Validation of three predictive equations for basal metabolic rate in adults

To cross-validate three predictive set of equations for basal metabolic rate (BMR) developed by Schofield (Schofield database), Henry (Oxford database) and Cole (Oxford database) using mean values for age, weight, height and BMR of published studies. Literature review of studies published from 1985 to March 2002. All studies selected used appropriate methods and followed conditions that met the criteria established for basal metabolism, were performed in healthy adults, and were not part of the Schofield or Oxford database. A total of 261 groups of men and women from 175 studies were selected and categorised in three age groups (18.5-29.9, 30.0-59.9, > or = 60 years old) and three body mass index (BMI) groups (normal weight, overweight and obese). Linear regression and concordance correlation analysis showed that the three sets of equations had the same association and agreement with measured BMR, across gender, age, and BMI groups. The agreement of all equations was moderate for men and poor for women. The lowest mean squared prediction errors (MSPRs) were given by Henry equations in men and Cole equations in women. Henry and Cole equations gave lower values than Schofield equations, except for men over 60 years of age. Henry equations were the most accurate in men. None of the three equations performed consistently better in women. These results support the use of Henry equations in men with a wide range of age and BMI. None of the proposed predictive equations seem to be appropriate to estimate BMR in women.

Impact of the displacement current on low-frequency electromagnetic fields computed using high-resolution anatomy models

We present a comparison of simulated low-frequency electromagnetic fields in the human body, calculated by means of the electro-quasistatic formulation. The geometrical data in these simulations were provided by an anatomically realistic, high-resolution human body model, while the dielectric properties of the various body tissues were modelled by the parametric Cole–Cole equation. The model was examined under two different excitation sources and various spatial resolutions in a frequency range from 10 Hz to 1 MHz. An analysis of the differences in the computed fields resulting from a neglect of the permittivity was carried out. On this basis, an estimation of the impact of the displacement current on the simulated low-frequency electromagnetic fields in the human body is obtained.

Reply to comment on ‘Gaint electrorheological activity of high surface area mesoporous cerium-doped TiO 2 templated by block copolymer’

We reply to recent comments about our experimental results by Choi et al. They reanalyzed our data of dielectric spectra using Cole–Cole equation and considered that this approach not only well fitted our dielectric data, but also deduced achievable polarizability that could be used to explain the ER enhancement in mesoporous ER system. Indeed, the Cole–Cole plot can provide further information that is difficult to be directly detected by dielectric spectra technique due to limitation of frequency range. We feel that the use of this approach may be reasonable to deduce some available information for further understanding ER effect.

Distribution of relaxation times in PMN single crystal

The dielectric permittivity of PMN single crystal was measured in the frequency range from 20 Hz to 3 GHz. Dielectric dispersion was observed in all investigated frequency range. From the frequency dependence of the real and imaginary parts of dielectric permittvity, the distribution of relaxation times was calculated. The low frequency limit of the distribution of the relaxation times diverge according to the Vogel -Fulcher law with the freezing temperature T 0 = 228 K, which is in good agreement with mean relaxation time temperature dependence obtained from Cole - Cole equation. Broadening of the relaxation time distribution function at low temperatures indicates the existence of two different components making contribution to the total dielectric spectrum of PMN.

Mössbauer spectroscopy and electrical transport properties of iron doped sodium lead borate glasses

Mössbauer Effect (ME) spectroscopy, density and molar volume as well as electrical transport measurements were employed to investigate the glass system: (70 − x) mol% B 2O 3· ( x) mol% Pb 3O 4·10 mol% Fe 2O 3·20 mol% Na 2O [with 0 ≤ x ≤ 35]. Both ME and density were measured at room temperature but conductivity measurements were carried out in the temperature range from 320 to 560 K at four fixed frequencies [0.12, 1, 10 and 100 kHz]. The ME results showed that, in the lead free sample only, about 80.8% of the total iron act mostly as glass network modifier (GNM), while the rest precipitated as α-Fe 2O 3. As lead oxide was just introduced into the glass, it assists to dissolve all iron ions through the glass network and iron ions start directly to occupy glass network former (GNF) positions. Both the density and the oxygen molar volume increased as lead oxide were increased. It was found also that the ac conductivity [ σ ( ω) ] is higher than the dc conductivity ( σ d c ) at low temperatures, whereas at high temperatures, σ ( ω) approaches σ d c at all frequencies. These results were explained in terms of the percolation path approximation (PPA) model, due to the macroscopic inhomogenity of the glass network. A good agreement has been obtained between the experimental results and the theoretical fitting of PPA at low frequencies, while at high frequencies the results showed slight deviation. The dielectric constant and the dielectric loss were found in good agreement with the theoretical fitting of Cole–Cole equation.

Bioimpedance dispersion width as a parameter to monitor living tissues

In the case of living tissues, the spectral width of the electrical bioimpedance dispersions (closely related with the α parameter in the Cole equation) evolves during the ischemic periods. This parameter is often ignored in favor of other bioimpedance parameters such as the central frequency or the resistivity at low frequencies. The object of this paper is to analyze the significance of this parameter through computer simulations (in the α and β dispersion regions) and to demonstrate its practical importance through experimental studies performed in rat kidneys during cold preservation. The simulations indicate that the dispersion width could be determined by the morphology of the extra-cellular spaces. The experimental studies show that it is a unique parameter able to detect certain conditions such as a warm ischemia period prior to cold preservation or the effect of a drug (Swinholide A) able to disrupt the cytoskeleton. The main conclusion is that, thanks to the α parameter in the Cole equation, the bioimpedance is not only useful to monitor the intra/extra-cellular volume imbalances or the inter-cellular junctions resistance but also to detect tissue structural alterations.

Dielectric relaxation study in electron-irradiated ferroelectric poly(vinylidene fluoride-trifluoroethylene) (80/20 mol%) copolymer

The relaxor ferroelectric (RFE) behavior in high-energy electron-irradiated poly(vinylidene fluoride-trifluoroethylene) [P(VDF-TrFE)] 80/20 mol % copolymer system is characterized over a broad frequency and temperature range. The dielectric properties remarkably vary with the irradiated dose in terms of the change from normal ferroelectric (FE) to RFE phase. During the RFE–paraelectric (PE) transition, the dielectric constants, as a function of temperature, can be described by the Vogel–Fulcher (V–F) relation. It has been found that the relationship between the real and imaginary part of dielectric constant in irradiated copolymer can be well fitted with modified Cole–Cole equation and Debye relaxation equation, exhibiting similar features as inorganic RFEs. © 2005 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 43: 2972–2980, 2005

Permittivity measurements up to 30 GHz using micromachined probe

We implemented a micromachined probe for the measurement of biological properties using MEMS technology, and experimentally showed the suitability of the micromachined probe in biological applications. The micromachined probe was fabricated on a silicon substrate, and to remove wave transmission through the silicon substrate, we etched the silicon substrate from beneath a lower ground and made the etched silicon surface conducting by using thermal evaporation of Cr/Au and a coating of conductive epoxy. The micromachined probe consists of a CPW and strip line between benzo cyclo butene (BCB) layers, which is known to be a material with high resistivity, low loss tangent, and low permittivity at high frequency. We measured the permittivity of a number of well-known liquids—0.5%, 0.9% and 1.3% saline, acetone, ethanol, and muscle and fat of pork—as biological samples using the micromachined probe after liquid calibration. The measured permittivity of 0.9% saline agreed well with the expected value of the Cole–Cole equation. In this paper, we first demonstrate that the micromachined probe can provide broadband measurement of measurable solid materials, such as biological samples, and also of well-known liquids at microwave frequencies. The size of the micromachined probe is 2000 µm (width) × 580 µm (thickness) × 30 000 µm (length), and the aperture size of the micromachined probe is only 650 µm × 70 µm. Therefore, we can extract the biological information from very small biological tissues and reduce radiation effects. Thus we show the feasibility of low-cost, small and portable permittivity measurement systems using a micromachined open-ended coaxial RF MEMS probe.

Study of different modes of dielectric relaxation in the ferro- and antiferroelectric phases of a newly-synthesized highly tilted chiral liquid crystal material 6H6B

Frequency- and temperature-dependent dielectric spectroscopy of a newly-synthesised AFLC compound in its planar orientation has been carried out in order to determine the different dielectric parameters and analyse the various modes of dielectric relaxation of the different phases. The compound is (S)-(+)-4′-(1-methylheptyloxycarbonyl)biphenyl-4-yl 4-(6-heptanoyloxyhex-1-oxy)benzoate (acronym 6H6B). It possesses paraelectric SmA* and ferroelectric SmC* phases, in addition to a wide-temperature-range antiferroelectric phase (∼23°C). The SmA* phase shows a soft mode of relaxation related to the amplitudons. Its relaxation frequencies are highly temperature-dependent and decrease with decrease in temperature. The dielectric strength is small (∼4–5), but increases with decrease in temperature, and follows the Curie law. The ferroelectric SmC* phase shows a Goldstone mode of relaxation related to the phasons. Its relaxation frequency is almost invariant (∼1–2 kHz) with temperature. The dielectric strength of the Goldstone mode is very large (∼130) and slowly increases with decrease in temperature; it shows a maximum just above the SmC*–SmCA* transition temperature. Two modes of dielectric relaxation have been clearly observed in the antiferroelectric phase. The lower-frequency mode observed in the kHz range has been assigned as the PL mode, whereas the higher-frequency mode observed at about 100 kHz has been assigned as the PH mode. Most of the observed relaxations follow a generalised Cole–Cole equation.

Electrical bioimpedance measurement during hypothermic rat kidney preservation for assessing ischemic injury

Non-heart-beating donors sustain an ischemic insult of unknown severity and duration, which can compromise the viability of the graft. This preliminary study aimed to assess whether electrical bioimpedance monitoring of cold preserved organs could be useful to identify kidneys that have suffered previous warm ischemia (WI). Two rat groups were studied during 24 h of preservation in University of Wisconsin solution (UW): a control cold ischemia group and another group subjected previously to 45 min of WI. Multi-frequency bioimpedance was monitored during preservation by means of a miniaturized silicon probe and the results were modeled according to the Cole equation. Tissular ATP content, lactate dehydrogenase in UW solution and histological injury were assessed. Renal function and cell injury, evaluated during 3 h of ex vivo reperfusion using the isolated perfused rat kidney model, demonstrated differences between groups. Bioimpedance results showed that the time constant and the high frequency resistivity parameters derived from the Cole equation were able to discriminate between groups at the beginning of the preservation (Δ τ ∼ 78%, Δ R ∞ ∼ 36%), but these differences tended to converge as preservation time advanced. Nevertheless, another of the Cole parameters, α, showed increasing significant differences until 24 h of preservation (Δ α ∼ 15%).

Development of a phantom compatible for MRI and hyperthermia using carrageenan gel—relationship between dielectric properties and NaCl concentration

A phantom has previously been developed containing carrageenan, agarose and gadolinium chloride (called CAG phantom) for MRI with 1.5 T. T1 and T2 relaxation times of this phantom are independently changeable by varying concentrations of relaxation-time modifiers to simulate relaxation times of the various types of human tissues. The CAG phantom has a T1 value of 202–1904 ms and a T2 value of 38–423 ms, when the GdCl3 concentration is varied from 0–140 µmol/kg and the agarose concentration is varied from 0–1.6%. A new phantom has now been developed (called CAGN phantom), made by adding an electric conductive agent, NaCl, to the CAG phantom for use in the areas of MRI and hyperthermia research. Dielectric properties of the CAGN phantom were measured and the results of experiments were expressed by the Cole–Cole equation in the frequency range of 5–130 MHz. The relationship between the conductivity of the CAGN phantom and the concentration of NaCl was expressed by a linear function in the frequency range of 1–130 MHz. The linear function involves a parameter of frequency and, when the frequency is 10 MHz, the conductivity of the CAGN phantom can be changed from 0.27–1.26 Sm−1 by increasing the NaCl concentration from 0–0.7%. The CAGN phantom developed can be employed in basic experiments for non-invasive temperature measurement using MRI and as a loading phantom for MRI with up to 3 T.

The skewed arc representation for the frequency dependence of complex permittivity

When a variable-frequency sinusoidal voltage is applied to a polar liquid dielectric, and the imaginary part of the complex permittivity plotted against the real part, Robert Cole and Kenneth Cole showed that, using the results of the analysis of Debye, a semicircle would result. This represents relaxational behaviour with a single relaxation time. Debye had ignored dipolar inertia, and also dipole–dipole interaction; he therefore expected his analysis to hold only for frequencies below the central frequency of the dispersion, and then only for dilute solutions of polar molecules in a non-polar solvent. In practice, even when these restrictions are not satisfied, in many cases Debye's analysis gives quite good predictions of experimental observations. However, for some liquids experiments do not yield a Cole–Cole semicircle but a skewed arc, i.e. a curve which tends to a semicircle at low frequencies, and to a straight line at high frequencies. Over 50 years ago such a curve was reported by Davidson and Cole for glycerol; it is represented by their empirical equation which is a simple modification of the equation given by Debye's original analysis. The Davidson–Cole equation can be analysed to yield a distribution of relaxation times. It is reasonable to interpret this to mean that some physical characteristic of the material must be such as to cause the distribution of relaxation times implicitly present in the Cole–Davidson equation. A satisfactory explanation of this causal relationship has yet to be put forward. In this paper such an explanation is suggested.

Relaxor Behavior of the 0.9BaTiO3-0.1La(Mg1/2Ti1/2)O3Solid Solution

Low-frequency dielectric response of air- and oxygen-sintered ceramics with the composition 0.9BaTiO3–0.1La(Mg1/2Ti1/2) O3 (0.9BT–0.1LMT) has been studied in the temperature range of 12–550 K. In comparison with pure BT, in 0.9BT–0.1LMT the dielectric permittivity maximum is shifted by almost 300 K toward lower temperatures. Both real and imaginary parts of dielectric permittivity of the solid solution, in the range 12–150 K, show a strong frequency-dependent behavior, which is typical of relaxors. On the basis of the model of exponential cluster size distribution and the Cole–Cole equation, the degree of interaction between the polar clusters was estimated. It was shown that the oxygen vacancies arising during sintering at high temperatures did not affect noticeably the relaxor properties of the material. The role of heterovalent La3+/Ba2+ and Mg2+/Ti4+ substitutions in the relaxor behavior formation is discussed.

Dielectric Spectroscopy of Aqueous Solutions of KCl and CsCl.

Results of a dielectric relaxation study of the aqueous solutions of KCl and CsCl at 25 °C are reported using measurements with a vector network analyzer (0.2 ≤ ν/GHz ≤ 20) and with waveguide interferometers (27 ≤ ν/GHz ≤ 89). Similar to previously reported data for NaCl, the spectra of both salts are well fitted by a Cole−Cole equation, although a double-Debye model is competitive at intermediate CsCl concentrations. The impact of the solute on the water structure, reflected in a decrease in the bulk-water relaxation time with rising electrolyte concentration, decreases in the sequence NaCl > KCl > CsCl and appears to be proportional to the surface-charge density of the cation. The interactions of K+ and Cs+ with their hydration shells are too weak to cause irrotational bonding of H2O molecules. It is argued that CsCl exhibits weak ion pairing, although the data are not sufficiently accurate to determine the extent of association and the nature of the ion pair formed.

Dielectric Spectroscopy of Aqueous Solutions of KCl and CsCl

Results of a dielectric relaxation study of the aqueous solutions of KCl and CsCl at 25 °C are reported using measurements with a vector network analyzer (0.2 ≤ ν/GHz ≤ 20) and with waveguide interferometers (27 ≤ ν/GHz ≤ 89). Similar to previously reported data for NaCl, the spectra of both salts are well fitted by a Cole−Cole equation, although a double-Debye model is competitive at intermediate CsCl concentrations. The impact of the solute on the water structure, reflected in a decrease in the bulk-water relaxation time with rising electrolyte concentration, decreases in the sequence NaCl > KCl > CsCl and appears to be proportional to the surface-charge density of the cation. The interactions of K+ and Cs+ with their hydration shells are too weak to cause irrotational bonding of H2O molecules. It is argued that CsCl exhibits weak ion pairing, although the data are not sufficiently accurate to determine the extent of association and the nature of the ion pair formed.

Dielectric relaxation spectroscopy of 2-propanol–water mixtures

The complex dielectric spectra of 2-propanol–water mixtures were determined at seven molar fractions of 2-propanol, X=0.03, 0.065, 0.14, 0.3, 0.5, 0.7, and 0.9 at 25 °C in the frequency range 0.1⩽ν/GHz⩽89 with the help of time domain reflectometry in 0.1⩽ν/GHz⩽25 and waveguide interferometry in 13⩽ν/GHz⩽89. In the alcohol-rich region of 0.3⩽X⩽1.0, a description of the ε*(ν) spectra requires the superposition of the three relaxation processes. The dominating low-frequency dispersion (j=1) follows a Cole–Cole equation. Additionally, two Debye equations (j=2 and 3) with the relaxation times of τ2∼10–20 ps and τ3∼1–2 ps are required to fit the high-frequency part of the spectrum. The three processes are assigned to the cooperative dynamics of the H-bond system (j=1), a rotation of singly H-bonded alcohol monomers at the ends of chainlike structure (j=2), possibly connected to the formation of bifurcate hydrogen bonds, and a flipping motion of free OH group (j=3). In the region of X<0.3, the intermediate alcohol monomer process becomes inseparable. Here, a two process model with a Cole–Cole equation for the main dispersion and a high-frequency Debye process for the fast switching mode gives the best fit. Based on the dielectric relaxation mechanism of the pure constituents proposed in the literatures [J. Barthel et al., Chem. Phys. Lett. 165, 369 (1990), and R. Buchner et al., Chem. Phys. Lett. 306, 57 (1999)], a composition-dependent relaxation behavior of the mixtures is discussed.

The Effect of Supercritical CO2 on the Macromolecules Parallel Conformation and Its Relation to the Electrical Conductivity and Dielectric Behavior of Epichlorohydrin Terpolymer

The effect of supercritical CO2 on the electrical conductivity of poly(epichlorohydrin–Ethylene oxide–Allyl glycidal ether) terpolymer is investigated using dielectric spectroscopy. Impedance measurements were carried out in the frequency range from 100–10 MHz and the temperature range of −35–70°C with intervals of 5°C. The experimental results of the dielectric constant and the dielectric loss were fitted with the Cole–Cole equation to obtain the maximum relaxation frequencies of the different relaxation processes. As a result of the CO2 treatment process, enhancement in the polymer chain mobility without noteworthy change in the glass transition temperature was determined. In addition, the level of the DC conductivity and the dielectric strength were increased. These effects were attributed to improvement in the chain dynamics, which arises from enhancement in the parallel conformation of macromolecules.

Structure, phase transitions and molecular motions in 4-aminopyridiniumperchlorate

The crystal structure of the 4-aminopyridinium perchlorate (4-apyH)ClO4 has beendetermined at 100 K by means of x-ray diffraction as monoclinic, with space groupP 21,with Z = 8.The crystal undergoes two structural phase transitions: one of first-order type,reversible, at 241/243 K (on cooling/heating respectively) and one ofweakly first-order type, irreversible, at 277 K (on heating). The crystaldynamics is discussed on the basis of the temperature dependence of the 1 Hnuclear magnetic resonance second moment(M2) and spin–latticerelaxation time T1.Both phase transitions are interpreted in terms of thechanges in the motional state of (4-apyH)+ cations and ClO4−anions. The dielectric dispersion studies disclose a relaxation process over thehigh-temperature phase (above 241 K) in the audio-frequency region. Thedielectric results are described by a Cole–Cole equation. The title crystal revealspyroelectric properties below 241 K. The ferroelastic domain structure of(4-apyH)ClO4 is observed over the whole temperature range studied.

Dielectric relaxor and ferroelectric relaxor: Bi-doped paraelectric SrTiO3

In this article, we report the evolution of the dielectric behavior from a dielectric relaxor to a ferroelectric relaxor with variation of Bi concentration in (Sr1−1.5xBix)TiO3 (0⩽x⩽0.2). In the doping range 0.0005⩽x⩽0.002, two dielectric modes A and B are induced. The temperature (Tm) where the permittivity maximum occurs for modes A and B is independent of Bi concentration and of dc electric fields. The complex permittivity of modes A and B follows the empirical Cole–Cole equation. The relaxation time for modes A and B follows the Arrhenius law. The dielectric possessing this type of dielectric behavior is named as a “dielectric relaxor.” At x⩾0.0033, an additional mode C appears, whose Tm increases with increasing Bi concentration. The complex permittivity for mode C does not follow the Cole–Cole equation. The relaxation time of mode C follows the Vogel–Fulcher law, indicating typical relaxor-ferroelectric behavior. In this work, we refer it to a “ferroelectric relaxor” mode. In the range of 0.0033⩽x⩽0.133, the coexistence of the dielectric-relaxor modes and the ferroelectric-relaxor mode is observed. In the samples doped with higher Bi concentration, modes A and B gradually merge into mode C, and only ferroelectric-relaxor behavior remains at x⩾0.133. This system provides a composition-controlled example of evolution from a “dielectric relaxor” to a “ferroelectric relaxor.” In addition, some controversial interpretations of the dielectric behavior of the Bi doped SrTiO3 solid solutions in the literature are discussed, and the polarization relaxation species of modes A and B are attributed to Bi ions.