Clausius-Mossotti Research Articles

Generalized Clausius–Mossotti Formula for Random Composite with Circular Fibers

An important area of materials science is the study of effective dielectric, thermal and electrical properties of two phase composite materials with very different properties of the constituents. The case of small concentration is well studied and analytical formulas such as Clausius–Mossotti (Maxwell–Garnett) are successfully used by physicists and engineers. We investigate analytically the case of an arbitrary number of unidirectional circular fibers in the periodicity cell when the concentration of the fibers is not small, i.e., we account for interactions of all orders (pair, triplet, etc.). We next consider transversely-random unidirectional composite of the parallel fibers and obtain a closed form representation for the effective conductivity (as a power series in the concentration v). We express the coefficients in this expansion in terms of integrals of the elliptic Eisenstein functions. These integrals are evaluated and the explicit dependence of the parameter d, which characterizes random position of the fibers centers, is obtained. Thus we have extended the Clausius–Mossotti formula for the non dilute mixtures by adding the higher order terms in concentration and qualitatively evaluated the effect of randomness in the fibers locations. In particular, we have proven that the periodic array provides extremum for the effective conductivity in our class of random arrays (“shaking” geometries). Our approach is based on complex analysis techniques and functional equations, which are solved by the successive approximations method.

Local field corrections for light absorption by fullerenes

With the aim of analysing measurements of absorption of electromagnetic radiation by fullerene molecules in the gas phase, in liquid solution, and in solid films, we consider the classical problem of correction for enhancement of the local field due to polarisation of the medium. We argue that for crystalline films the Clausius-Mossotti relation between the molecular polarisability and the dielectric function of the medium should be reasonably accurate, despite the fact that the polarised fullerene molecules are far from being point dipoles. For fullerene molecules in solution, the field derived by Onsager from a cavity model should be a good approximation, and we discuss and correct errors in earlier developments of this model. Owing to the large polarisability of fullerenes, the local field corrections are very important, leading to large changes in the relative strength of absorption lines and also to small shifts in the positions of these lines. Application of the corrections leads to improved consistency between measurements in the gas phase, in solution, and in the solid.

Dual capacitor technique for measurement of through-plane modulus of thin polymer films

Polymer thin films are widely used as coatings and interlevel dielectrics in microelectronic applications. In thin-film structures, stresses are generated due to interaction with adjacent layers and film shrinkage due to solvent evaporation or curing. This causes polymer chain orientation resulting in anisotropic (direction dependent) film properties. The dual capacitor technique has been developed to measure in situ, the through-plane (z) stress-strain behavior of thin polymer films. A parallel plate capacitor device and an interdigitated electrode structure were used as sensors to detect changes in dielectric permittivity and thickness of thin polymer films under compression. The analytical and finite element models used to interpret the capacitance measurements have been presented. The Clausius–Mossotti equation was used to determine the volume change in the film from the permittivity measurements. Results have been reported for 10–14 μm thick, Cyclotene 4026-46 benzocyclobutene films and 10–12 μm thick films of polyimide PI-2611. The Cyclotene 4026-46 films were found to be mechanically isotropic, whereas the PI-2611 films were highly anisotropic. © 2000 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys 38: 1634–1644, 2000

Dispersion formula and continuous absorption of water vapor

A full Lorentz line shape of the radiation absorption is used in order to account for the self-broadening continuum absorption of the water vapor. The Clausius–Mossotti formula of dispersion has been changed to take into consideration the influence of the medium on the imaginary part of the susceptibility. Absorption along path lengths is described by a hypothetical nonlinear equation. The method developed allows us to retrieve the supper-and sub-Lorentzian behavior of the continuum absorption of water in the region 700–1200 cm −1, the nonlinear density dependence of the water self-broadening continuum and then to model the strong temperature dependence of line absorption cross sections. An experiment by Burch and Alt (1984) is taken for a detailed consideration and a good coincidence of the data calculated and measured is observed. The extrapolation of the data from 700 to 1200 cm −1 is applied to the frequency and temperature dependence of the absorption coefficient in the transparency windows 2400–3000 cm −1, in a microwave region, and for a high pressure equal to 25 atm. The empirical temperature and pressure dependence of the interaction has been obtained. The selective continuum simulated is quite different from the conventional representation, especially, for the saturated spectra.

Dipole polarization and charge transfer effects in the lattice dynamics and dielectric properties of ionic crystals

We extend our formalism of an ab initio description of electronic screening in solids with a strong ionic component of bonding by introducing electronic density polarization effects of dipole type in addition to charge transfer polarization processes. The latter have been found earlier to be very important for the lattice dynamics and the electron-phonon interaction in the high-temperature superconductors. The dipole deformations are calculated with the Sternheimer method and the influence on the polarizability induced by the crystalline environment and by self-interaction corrections is studied. We further derive detailed expressions for the various coupling coefficients appearing in the dynamical matrix in our formalism and apply this theory in a first step in view of the ultimate goal, namely, the high-temperature superconductors, to the calculation of complete dispersion curves for the alkali halide NaCl and the alkaline-earth oxide MgO taking into account for the first time dipole polarization and charge transfer effects together. The agreement of our calculated results with the experiments is good. As a main effect we find a renormalization of the LO modes and a reduction of the LO-TO splitting by both types of polarization processes. The dipole polarization, however, dominates the electronic screening by far. Finally, we investigate the macroscopic dielectric constant and the transverse effective charges and discuss the results in context with the Clausius-Mossotti relation.

Investigation of a microwave differential cavity resonator device for the measurement of humidity in gases

A resonant cavity based microwave differential device has been developed as a sensor for the measurement of small quantities of water vapor in gases. The presence of a contaminant is assessed by induced variations in the relative permittivity due to a shift in the resonant frequency of the measuring resonator. The measured output signal is related to the difference in the reflection coefficients of the measuring resonator and the reference one. A simple modeling approach of the system shows the proportionality between the difference of the reflection coefficients of the cavities and the variation in the dielectric constant. The evaluation of the minimum detectable change in the permittivity is possible using the Clausius–Mossotti equation for a binary gas mixture at a given concentration and pressure. The detection threshold is then determined by taking into account the signal-to-noise ratio of the differential setup. The estimation of the limit of detection for some practical moisture contaminated gases yields values in the low ppm level at T=293 K and atmospheric pressure. Experimental data suggest a detection threshold of 6 ppm for moisture.

Electrical properties of viscose-kraft fibre mixtures

ABSTRACT We used mixture handsheets of pine kraft and viscose fibres to study how density and moisture content affect the electrical properties of paper. When the viscose content increased from 0% to 70%, paper density decreased from 650 kg/m 3 to 280 kg/m 3 . The effect of density on the dielectric constant of paper was consistent with the Clausius-Mossotti relation provided that the relative humidity was low or the measurement frequency high. Under these conditions, the water absorbed by paper had little effect on the dielectric constant or dielectric loss factor. When the moisture content of paper exceeded a certain limit, the dielectric constant began to increase rapidly. Presumably this happened because some of the adsorbed water resembles bulk water in its electrical properties and therefore has a much higher dielectric constant than dry cellulose. The threshold moisture content increased with increasing measurement frequency. The viscose fibres absorbed 2-3%-units more water than the kraft fibres at any given relative humidity. In spite of this, increasing the viscose content decreased the dielectric constant of paper. This indicates that the water molecules are more closely bonded and have less freedom to rotate in viscose fibres than in kraft fibres. The dielectric loss term was practically zero at low moisture contents, approximately below the same threshold that governs the dielectric constant. The dc resistivity decreased as an exponential function of increasing moisture content, but viscose fibres gave a lower conductivity than kraft fibres at a fixed moisture content. Hence, the water absorbed by viscose fibres influenced the electrical properties of paper in all respects less than the water absorbed by kraft fibres.

Calculation of the pressure dependence of anion polarizabilities in crystals

A program has been developed in Maple V that allows for the calculation of anion polarizabilities of ionic crystals (based on the Clausius–Mossotti treatment) and its pressure dependence. It has been applied to alkali halides for which the results may be compared to those obtained from experimental data. It opens the possibility of its further use for the evaluation of the pressure dependence of the dielectric constant from the knowledge of experimental data at p=0; we here apply this to alkaline-earth fluorides obtaining very good results.

Electrostriction of Near-CriticalSF6in Microgravity

We used interferometry to measure the electric-field-induced (i.e., electrostrictive) increase of the density of sulfur hexafluoride near its critical point. The results at the temperatures T(sub c) + 5 mK, T(sub c) + 10 mK, and T(sub c) + 30 mK with T(sub c) = 319 K agree with a calculation based on the Clausius-Mossotti relation and the restricted cubic model equation of state. These measurements were performed in microgravity so that the electrostrictive density changes (less than or equal to 3.5% in this work) would not be complicated by stratification of the fluid's density induced by the Earth's gravity.

Electrostriction of a near-critical fluid in microgravity

We used interferometry to measure the electric-field-induced (i.e., electrostrictive) increase of the density of sulfur hexafluoride (SF6) near its critical point. The results at three temperatures (T(c)+5.0 mK, T(c)+10.0 mK, T(c)+30.0 mK with T(c)=319 K) agree with a calculation based on the Clausius-Mossotti relation and the restricted cubic model equation of state. To measure electrostriction, an inhomogeneous electric field (< or =26 kV/cm) was applied to the SF6 sample by charging a fine wire that passed through it. These measurements were performed in microgravity so that the small electrostrictive density changes (< or =3.5% in this paper) would not be complicated by stratification of the fluid's density induced by the Earth's gravity. The predicted shifts of the critical temperature and density resulting from the electric field were too small to detect.

Electronic polarizability of a Sm3+ ion estimated from refractive indexes and molar volumes of molten SmCl3

The refractive indexes and the molar volumes of molten SmCl3 were measured by goniometry and dilatometry, respectively. The electronic polarizability of a Sm3+ ion was estimated from these data to be 1.14×10−30m3 according to the Clausius–Mossotti equation. The correlation between the electronic polarizability of an ion and the ionic radius was also discussed.

Electrical and optical properties of dense composites

Effective cluster models have been developed that treat disordered suspensions of monodisperse metal spheres as mixtures of isolated spheres and compact clusters of spheres using the Clausius–Mossotti equation [Phys. Rev. B 42, 9319 (1990); J. Appl. Phys. 71, 3926 (1992)] and the Bruggeman equation [J. Appl. Phys. 78, 6165 (1995)]. These effective cluster models are adapted to suspensions of dielectric particles with arbitrary complex permittivity. The models are compared with the coated sphere model of Sheng [Phys. Rev. Lett. 45, 60 (1980); J. Opt. Soc. Am. B 15, 1022 (1998)]. Model calculations are compared with the measurements of the optical transmission spectra and low frequency electrical conductivity of Au–SiO2 films by Cohen, Cody, Coutts, and Abeles [Phys. Rev. B 8, 3689 (1973)], and with the low frequency permittivity measurements on suspensions of Ag spheres in KCl of Grannan, Garland, and Tanner [Phys. Rev. Lett. 46, 375 (1981)]. The models accurately predict the percolation thresholds seen in the electrical measurements and are in good agreement with all of the experiments over the entire range of volume loading.

Density functional theory applied to the calculation of dielectric constant of low-k materials

Abstract The interest of low-k dielectric materials to reduce capacitance in multilevel metal interconnects of integrated circuits is well known in the semiconductor industry. The use of these materials (especially hydrogen silsesquioxane (HSQ) and methyl silsesquioxane (MSQ): intermetal dielectric applications in the back end of line fabrication) leads to a reduction of the dielectric constant from k≅4 in a traditional intermetal dielectric material of silicon dioxide to a value of ≅2.5–3. The physical difference between HSQ or MSQ and a-SiO2 is the presence of Si–H bonds (for HSQ) or Si–CH3 bonds (for MSQ) and the density of the material. A theoretical calculation of bond polarizability (Si–H or Si–CH3) associated to experimental values of electric dipole densities can lead, using the Clausius-Mossotti relationship, to the calculation of the dielectric constant. After validation of the calculation methods both on simulation and experimental values, it is shown that for a constant density, the difference between the materials could be due to the bond polarizability and furthermore that this difference accounts, in part, for the value of dielectric constant. Consequently, even if densification remains the main parameter explaining low-k values, the polarizability of building units of these materials is not negligible.

Refractive indexes and electronic polarizabilities of molten HoCl 3–NaCl and HoCl 3–KCl mixtures

The refractive indexes of molten HoCl 3–NaCl and HoCl 3–KCl mixtures were measured by goniometry using visible light at eight wavelengths and expressed as functions of both temperature and wavelength into empirical formulas by a least-squares method. The electronic polarizabilities of ions in the mixture melts were obtained from the data of refractive indexes and molar volumes, and the polarizability of a Ho 3+ ion was evaluated to be about 0.98×10 −30 m 3 according to the Clausius–Mossotti equation. The isotherms of these properties show little deviation from their additivity line.

Structure and pressure variation of the anion polarizability and dependent properties in solid magnesium oxide

Ab initio electronic structure computations yield the polarizability αA(R) of the O2- ion in the 4-coordinated B3, B1 rocksalt and 8-coordinated B2 phases of solid MgO over a wide range of closest cation-anion separations R. In all three phases and in the point lattices for the B1 and B2 structures, the anion size as measured by (r2)tot, the mean-square distance of all the anion electrons, controls αA(R) in exactly the same way since these values all lie on the same smooth curve when plotted against 4((r2)tot)2/90. The Birch-Murnaghan equation of state yields the pressure for which any R is the equilibrium distance. For pressures occurring in the mantle of the earth, αA(R) is substantially reduced from its value under ambient conditions. For each of the three phases, the, variation of the O2- polarizability over the entire range of distances of any possible practical importance can be represented by a single exponential increase with increasing R. The R dependence of αA(R) modifies that of the interionic dispersive attractions and high frequency dielectric constant governed by the Clausius—Mossotti relation. The analytical expression for the R dependence of αA(R) is used to derive the derivative with respect to distance of both these properties, and hence the pressure dependnece of the dielectric constant and refractive index at zero pressure. The numerical values presented for these latter derivatives are more than than ten times smaller than would be predicted if αA(R) were taken to be constant.

Optical absorption of island films of noble metals: wave length of the plasma absorption band

Experimental measurements of the positions of the optical plasma resonance band for thin island films of noble metals agree closely with positions calculated from a simple equation with no disposable parameters. The equation includes the volume fraction of particles, the refractive index of the substrate, and the real part of the dielectric constant of the metal. This equation is derived from the Lorentz–Lorenz or Clausius–Mossotti conditions. The particles in the films are right cylinders, and have polarizabilities consistent with the equation. There is no need to assume fitting parameters involving particle shapes and sizes and their distributions.

Evanescent light scattering: The validity of the dipole approximation

In near-field optics the very concept of dipole is often used to represent either an elementary source or a scattering center. The most simple and widely used example is that of a small spherical particle whose polarizability is assumed to conform to the Clausius-Mossotti relation. While in conventional, far-field optics this approximation is known to be valid provided that the object is much smaller than the wavelength, its extension to near-field optics requires some precautions. Indeed, in the case of the scattering, by a spherical object, of an evanescent field generated, for instance, by total internal reflection or by a surface polariton, the strong-field gradient may increase the contribution to the polarizability of multipoles higher than the dipole. Such high-order multipoles are seldom considered in near-field optics because they complicate considerably any scattering calculation. In this paper we derive, for a spherical particle, the contributions of multipole orders up to the hexadecapole. This serves to illustrate the relative importance of each order. Moreover, within the framework of the coupled dipole method, we study, self-consistently, the problem of the scattering of an evanescent field by the sphere. We show that, with an initial field decreasing exponentially, the dipole approximation can be misleading. @S0163-1829~98!01428-3#

Dielectric Properties of an Immiscible Polymer Blend with Electrorheological Effect

Dielectric measurement was carried out on an immiscible blend of a side-chain type polysiloxane liquid crystalline polymer (LCP) and a poly-dimethylsiloxane (PDMS), which has a large electrorheological (ER) effect, in the frequency range from 100 Hz to 10 MHz and over the temperature range from 25°C to 105°C. Two relaxation processes were observed; one is due to the mesogenic motion, i.e., the δ-process of LCP and the other to the Maxwell-Wagner effect originating in the conductivity of LCP dispersed in PDMS. The results were analyzed in terms of the Clausius-Mossotti (CM) approximation and the local field exerted on droplets of LCP was discussed.

Measurement of the C 60 molecular polarizability of the excited singlet state using Z-scan

We report on a method of determining the C 60 molecular polarizability α e of the excited singlet state under 23 picosecond excitation at 532 nm using the Z-scan technique. Based on the Clausius–Mossotti equation, the sign and the magnitude of the real part and the imaginary part of α e are respectively measured to be 1.2×10 −22 and 7.2×10 −23 esu. We also obtain the nonlinear refractive cross section σ r of 4.2×10 −17 cm 2.

Phase border and density determinations in the critical region of (carbon dioxide+ethane) determined from dielectric permittivity measurements

We have determined the (p,T, ρ,x) phase boundaries in three mixtures of {(1−x)CO2+xC2H6}, withx=(0.25166, 0.49245, and 0.73978), in the vicinity of the critical locus from measurements of the relative dielectric permittivity εr. We combined accurate measurements of εrwith the Clausius–Mossotti relationship to determine the density. Measurements of εrwere made with two apparatuses: a reentrant cavity resonating at 375 MHz, and a coaxial cylindrical capacitor and bridge operating at a frequency of 1 kHz. The phase border was determined with uncertainties of approximately 0.5 per cent in density and 0.06 K in temperature, the latter corresponding to a fractional uncertainty of 1.2·10−3in pressure. The uncertainties in temperature and pressure resulted from the size of the temperature steps that were used; smaller values could be attained. The uncertainty in density resulted from uncertainty in the parameters relating the density to εr.