Maxwell Mixture Research Articles

Nonlinear Transport in Inelastic Maxwell Mixtures Under Simple Shear Flow

The Boltzmann equation for inelastic Maxwell models is used to analyze nonlinear transport in a granular binary mixture in the steady simple shear flow. Two different transport processes are studied. First, the rheological properties (shear and normal stresses) are obtained by solving exactly the velocity moment equations. Second, the diffusion tensor of impurities immersed in a sheared inelastic Maxwell gas is explicitly determined from a perturbation solution through first order in the concentration gradient. The corresponding reference state of this expansion corresponds to the solution derived in the (pure) shear flow problem. All these transport coefficients are given in terms of the restitution coefficients and the parameters of the mixture (ratios of masses, concentration, and sizes). The results are compared with those obtained analytically for inelastic hard spheres in the first Sonine approximation and by means of Monte Carlo simulations. The comparison between the results obtained for both interaction models shows a good agreement over a wide range values of the parameter space.

On the extension of Maxwell's mixture formula to ellipsoidal particles

Abstract Fricke's generalization of the Maxwell mixture formula to the case of ellipsoidal particles is reexamined. It is shown that this generalization is based on two approximations which are not justified in the case of randomly oriented particles with high eccentricity. A new expression which avoids this objection is deduced. It tends to Fricke's result in the limit of low volume fractions, but reduces to the Maxwell formula for very high concentrations. When particularized to the case of needle shaped particles, it shows important deviations with respect to Fricke's formula, which are in qualitative agreement with the available experimental data.

On the decomposition of the Maxwell mixture formula into a sum of Debye type contributions

It is shown that, according to the Maxwell mixture formula, the generalised conductivity of a suspension of components whose conductivities are represented by a sum of Debye type contributions can also be expressed as a superposition of Debye terms. Their number is n+2 m+1, where n(m) is the number of Debye terms representing the particles (continuum) and the last addend appears when at least one of the components is conducting. The special case where n=m=1 is analysed in detail and the interpretation of data from ionic micelles in an electrolyte is discussed.

Study of Bound Water of Poly-Adenine Using High Frequency Dielectric Measurements

The high frequency dielectric constant of poly-adenine (poly-A) was measured between 1 MHz and 1 GHz. The purpose of these experiments was to investigate the state of water molecules that are bound to the charged groups of the poly-A molecule. Analysis of the data using the Maxwell's mixture equation revealed the dielectric constant of bound water higher than we expected. Using Onsager's internal field in Debye's equation, we calculated the dielectric constant of water in the vicinity of a charged ion. The result of this computation demonstrates that the dielectric constant of bound water is much smaller than the normal value only in the immediate proximity of charged ions (within 2 A). The dielectric constant increases rapidly to the normal value as the distance increases from 2 to 4 A. This observation indicates that charged sites of polyions have only short range interactions with the surrounding water molecules. However, this conclusion pertains only to rotary diffusion of bound water since dielectric measurement is unable to detect translational diffusion.

Dielectric studies on nonionic microemulsions

The dielectric properties of a series of O/W microemulsions were measured at frequencies between 1 MHz and 15.4 GHz. The microemulsions were composed of hexadecane, an emulsifier consisting of Tween 60 (a nonionic surfactant) and n-pentanol as cosurfactant, and various quantities of water or 0.1 N NaCl. The dielectric data include (a) the conductivity of salt-containing microemulsions at radiofrequencies, (b) the permittivity of the microemulsions at radiofrequencies (0.5 to 100 MHz), and the permittivity and conductivity of the microemulsions at microwave frequencies from 0.73 to 15.4 GHz. The dielectric permittivity values do not vary significantly with frequency below ca. 1 GHz. At higher frequencies dispersion phenomena appear because of the dielectric relaxation of the water in the system. The data are analyzed using the Hanai and Maxwell mixture theories, to determine the dielectric properties of the water and suspended phases. The results suggest that in suspensions containing 40–75% water, volume of water amounting to ca. 40–70% of the volume of the suspended oil/emulsifier is associated with the droplets, and has a dielectric relaxation frequency and ionic conductivity both reduced by a factor of 4–10 from that exhibited by the bulk water. For a mean droplet diameter of 74 Å, this is equivalent to a shell of ∼2 water monolayers, most of which is likely present as water of hydration of the ethylene oxide groups of the surfactant.

Microwave dielectric studies on proteins, tissues, and heterogeneous suspensions.

We summarize the results of several of our recent studies on the dielectric properties of protein solutions, tissues, and nonionic microemulsions at microwave frequencies extending to 18 GHz. The data in all cases are analyzed using the Maxwell mixture theory to determine the dielectric properties of the suspending water and the amount and dielectric properties of the water hydration associated with the suspended phase. The dielectric data from the protein solutions and tissues are broadly consistent with the results of previous studies at UHF frequencies; they indicate hydration values in the range of 0.4-0.6 g water/g protein. There is evidence of a dielectric relaxation process occurring at low-GHz frequencies that can be attributed in part to dielectric relaxation of the "bound" water in the system. The remaining solvent water appears to have dielectric properties close to, if not precisely the same as, those of pure water. The average relaxation frequency of the suspending water in the microemulsions is reduced from that of pure water, evidently reflecting an average of that of the water of hydration (approximately 5-6 GHz) and that of pure water. This reduced average relaxation frequency implies an increased average viscosity of the water and (by Walden's rule) accounts for the unexpectedly low ionic conductivity of the preparations.

Some observations on the dielectric properties of hemoglobin's suspending medium inside human erythrocytes

Dielectric permittivity and conductivity data for human erythrocytes exhibit two consecutive dispersions with frequency in the range from 0.1 to 250 MHz. The dispersion observed above 50 MHz yields a circle in the complex impedance plane with its center on the real axis, suggesting a Debye relaxation, for the erythrocyte interior medium, centered in the microwave frequency range as for bulk water. Moreover, the Maxwell mixture equation, indicates that this "free" water forms most of the suspending medium of hemoglobin macromolecules. These results extend to lower frequencies (earlier results obtained by Schwan) with biological tissues such as muscles, skin, or liver at microwaves frequencies.

Dielectric properties of brain tissue between 0.01 and 10 GHz

Dielectric permittivity and conductivity are reported for grey and white matter from dog brain tissue between 0.01 and 10 GHz. Between 0.01 and approximately 1 GHz, the permittivity decreases and conductivity increases as a power law of frequency. Above 1 GHz, the conductivity increases quadratically with frequency due to dipolar reorientation of free water molecules in tissue; the apparent rotational relaxation frequency at 37 degrees C is 21-24 GHz, slightly below the 25 GHz characteristic frequency of pure water at that temperature. The microwave data are analysed using the Maxwell mixture theory applicable for a suspension of nonconducting, low permittivity spheres in bulk water. From the increase in conductivity above 1 GHz, and the tissue permittivity at 2-4 GHz, the apparent volume fraction of water is approximately 0.70 and 0.55 for grey and white matter, respectively, about 10-15% lower than respective values from the literature.