General Mixing Rule Research Articles

Electrical properties of CaWO4–SiO2 composites

Composite materials (1 − f) CaWO4 – fSiO2 with volume fraction of SiO2f ≤ 1.0 were prepared by the solid state method. The obtained composites have been examined by SEM-EDS, the electrochemical impedance technique, conductivity measurements versus oxygen partial pressure and EMF measurements in order to evaluate ion transport numbers. It was shown that heterogeneous doping of the poor oxygen ion conductor CaWO4 with nano-SiO2 even at low concentration (f < 0.03) leads to an increase in ionic conductivity of the composites by more than an order of magnitude. The enhancement of conductivity in this system was explained by an additional contribution of the interface regions formed between the host and nanoparticles of dispersoid. A general mixing rule was applied for the calculation of electric conductivity of the (1 − f)CaWO4 – fSiO2 composite as a function of SiO2 concentration. Theoretical dependences were analyzed and compared with the experimental results.

Molecular-Based Bayesian Regression Model of Petroleum Fractions

Molecular reconstruction of petroleum fractions is for determining the detailed molecular compositions in the mixture from a few measurable bulk properties, e.g., density, Reid vapor pressure (RVP), molecular weight, and ASTM boiling point curves etc., which is a great challenge because the number of hydrocarbon compounds is much larger than that of the bulk properties. In this paper, a novel molecular reconstruction method is developed which includes two Bayesian regression models for bulk properties’ prediction and molecular reconstruction. By defining a characteristic function of bulk property and then establishing its general mixing rule with respect to compositions, the bulk property is predicted from a linear regression model with sigmoidal basis functions whose parameters can be estimated by maximizing a posterior distribution from a well-determined database containing bulk properties and molecular information on petroleum fraction samples. Furthermore, by developing a prior distribution of the molecular information with an assumption that the compounds in the hydrocarbon mixture have an independently and identically distributed (iid) gamma distribution and combining the likelihood function used in bulk properties’ prediction, the molecular information is thus reconstructed by maximizing a new posterior distribution. Case studies of naphtha fractions demonstrate the effectiveness of the proposed method.

Estimation of electrical properties of composite solid electrolytes of different morphologies

A generalized mixing rule was applied to calculation of electric properties (conductivity and dielectric permittivity) as a function of concentration and ac frequency for composites of different morphologies. Morphologies of three types were considered: (i) statistical mixture of ionic salt MX and oxide A with the same grains size of the components; (ii) composites in which A phase spreads along grain boundaries of MX and (iii) composites formed by filling the porous matrix of A with the ionic salt MX. Theoretical dependences were analyzed and compared to experimental results obtained earlier by authors or taken from the literature.

Influence of long-chain branching extent in polyethylenes on molecular weight and molecular weight distribution predicted via rheological analysis

This work highlights the potential of rheological analysis as an alternative powerful tool for estimation of molecular weight (MW) and molecular weight distribution (MWD) in terms of long-chain branching (LCB) extent in polyethylenes (PE). Different PE samples with different branching structure (mLLDPE, mVLDPE, LDPE, LLDPE, and LDPE/LLDPE blends) were prepared and their viscoelastic behavior was studied. The results showed that in metallocene samples a suitable agreement existed between the MWDs determined both experimentally and also by rheological data evaluation, while in the rest of samples this agreement was not so good. It seems that the presence of branching in LDPE and LLDPE makes this difference significant, while the absence of branching in metallocene samples caused suitable agreement between the results of predictions and measurements. In blend samples, this significant difference was attributed to existence of branching and also to noncompatibility of polymer components in some blend ratios. The results showed that normalized relaxation time spectrum is a good measure for qualitative comparison of MWDs of various samples. Applying Thimm Kernel function and relaxation modulus data in generalized mixing rule, the calculated MW and MWD of PE samples (without LCB) showed good agreement with the experimental results with high accuracy.

Microwave Backscatter and Extinction by Soft Ice Spheres and Complex Snow Aggregates

Abstract Coupled-dipole approximation (CDA) calculations of microwave extinction and radar backscatter are presented for nonhomogeneous (soft) ice spheres and for quasi-realistic aggregates of elementary ice crystal forms, including both simple needles and real dendrites. Frequencies considered include selections from the Dual-Frequency Precipitation Radar (DPR; 13.4 and 35.6 GHz) and the Global Precipitation Measurement (GPM) Microwave Imager (GMI; 18.7, 36.5, and 89.0 GHz), both slated for orbit on the GPM mission. The computational method is first validated against Mie theory using dipole structures representing solid ice spheres as well as stochastically generated “soft” ice spheres of variable ice–air ratio. Neither the traditional Bruggeman nor Maxwell Garnett dielectric mixing formula is found to correctly predict the full range of CDA results for soft spheres. However, an excellent fit is found using the generalized mixing rule of Sihvola with ν = 0.85. The suitability of the soft-sphere approximation for realistic aggregates is investigated, taking into account the spectral dependence of backscatter and/or extinction per unit mass at key DPR and GMI frequencies. Even when spheres of nonequal mass are considered, there is no single combination of fraction and mass that simultaneously captures all the relevant radiative properties. All four aggregate models do, however, exhibit a predictable power-law dependence of the mass extinction coefficient on the total particle mass. Dual-frequency mass extinction ratios are only very weakly dependent on particle masses; moreover, the ratio is found to be approximately proportional to frequency raised to the power 2.5. The dual-frequency backscatter ratio is found to be a predictable function of the aggregate mass for particles smaller than 3 mg. Above this size, the ratio is strongly sensitive to aggregate shape, a finding that raises concerns about the utility of dual-frequency backscatter ratio measurements whenever larger particles might be present in a volume of air. The validity of the Rayleigh–Gans approximation applied to radar backscatter from snow aggregates was also examined. Although the dual-frequency backscatter ratio was reasonably well reproduced, the absolute magnitude was not.

Mixing rules and effective potentials in gaseous mixtures

We generate exact mixing rules for dilute gaseous mixtures with nonconformal interactions by introducing an adequate effective potential. The general mixing rule leads to several approximations of the van der Waals type. The work uses nonconformal model pair potentials that accurately describe the pure fluid properties. The validity of the approximate mixing rules is assessed by comparison with exact results for binary mixtures over wide ranges of temperature and composition. Large differences in energy, size and potential profile are considered. The simplest mixing rules are straightforward generalizations of the van der Waals rules and one version, based on a square-well (SW) approximation, performs with high precision for all systems except for those with extreme differences in molecular size. The SW mixing rule is used to predict excess virial coefficients of mixtures involving Ar, Kr, H 2 and CF 4.

Composite solid electrolytes: recent advances and design strategies

The physical properties of composite solid electrolytes are briefly reviewed. The surface potential formation and the point defects equilibrium at free surfaces and interfaces are considered in frames of the unified Stern model. Special attention is drawn to true size effects due to the change of the bulk characteristics of ionic salts in the nanocomposites. The main thermodynamic reason of the nanocomposite formation relates to the adhesion energy, γ a. At sufficiently high γ a values, the ionic salt tends to spread along the oxide surface and to form a nanocomposite if the oxide is nanocrystalline or nanoporous. Analysis of the experimental data shows that non-autonomous interface phases, crystalline or amorphous, exist in some composites. The reason for the stabilization of non-equilibrium states is the complex influence of several factors, including the interfacial interaction between components, particle size effect, and elastic strains in the lattice of the ionic salt. The results of molecular dynamic simulations show that the main origins of the conductivity enhancement are the adsorption of ions to oxide surface with the formation of a space charge layer, the lattice deformation near interface, and an appearance of interdomain boundaries generated by misfit dislocations. The equations proposed earlier for the conductivity description are analyzed. Among them, the general mixing rule has a rather simple analytical form and provides appropriate description of the experimental data on both ordinary insulator–conductor composites and composite solid electrolytes in a broad concentration range. Main approaches for the improvement or creating new composite systems are analyzed.

Ionic conductivity of all-glass composites in the AgI–Ag 2O–P 2O 5 system

All-glass composites fabricated from mixtures of the superionic 33AgI·33Ag 2O·33P 2O 5 and 60AgI·20Ag 2O·20P 2O 5 glasses conduct Ag + ions. It is demonstrated that such composites can be easily formed by high-pressure methods. Their electric properties could be modelled by an equivalent electric circuit in which the ion conductivity of the bulk of a composite is represented by a single ‘effective resistance’. A related quantity—the effective ion conductivity, shows Arrhenius-like dependence. The effective activation energy and the effective preexponential factor both exhibit linear dependence on composition. The general mixing rule and the Uvarov model appeared to be unsuitable for the description of the effective conductivity of the all-glass composites as a function of composition and temperature. A new phenomenological model of effective conductivity is suggested and tested—proving its applicability.

Evaluation of Mixing Rules for Dielectric Constants of Composite Dielectrics by MC-FEM Calculation on 3D Cubic Lattice

Effective dielectric constants of diphase composite dielectrics are simulated by Monte Carlo-finite element method on three-dimensional lattice. Effective dielectric constants with coefficients of variation less than 5.5% are obtained for different ratios of dielectric constants of the two phases, ranging from 10 to 700. Various mixing rules and equations are fitted to these data and the accuracy and relevance of the fits are thoroughly examined. Modified logarithmic rule loses its physical basis when fitted to three-dimensional data. As the ratio of dielectric constants of the two phases increases, the parameters in general Lichterecker mixing rule, general Bruggeman's symmetric equation, general effective media equation and its modified form all increase or decrease monotonously. General effective media equation and its modified form give the best fits to the effective dielectric constants simulated. The simulation results for the dielectric constants of some composite systems are in good agreement with experimental data.

Response spectrum of incompatible acceleration, velocity and displacement histories by Praveen K. Malhotra, Earthquake Engng Struct. Dyn. 2001; 30(2):279–286

The Discussers would like to congratulate the Author for his contribution to the analysis of response spectra. As mentioned in the Original Paper, engineers often pay little attention to incompatibilities in the histories of ground acceleration, displacement, and velocity. As a result, computed response spectra may exhibit erroneous trends at short and long periods. Different techniques have been proposed to treat the problem, focusing primarily on the initial conditions of the oscillator [5,6,8,9,10,11]. The Author proposes an alternative technique: determining response spectra by combining the incompatible time histories using a 'mixing rule' (Equations (6)-(8) in the Original Paper, which allows for the proper asymptotic behaviour to be obtained at extreme periods. The scope of this discussion is three-fold: (i) to present a simplified version of the method, by applying the Author's idea only to initial conditions; (ii) to discuss a more general mixing rule; (iii) to propose alternative ways for estimating the effective frequency of ground motion. These items are presented to further validate the results provided by the Author and offer additional insight into the derivations presented in the Original Paper.

Estimation of composites conductivity using a general mixing rule

It is proposed to use a mixing rule for the estimation of conductivity, σ, in ‘conductor–insulator’ composites (M–A) and composite solid electrolytes (MX–A). The mixing rule in its classical form σ α =(1− f)· σ M α + f· σ A α (where f is the volume fraction of A; σ M,A are conductivity of the phases M and A; −1< α<1) is applicable only for some specific morphology of the composite. This equation may be generalized by introducing the dependence α( f)= α1·(1− f)+ α2· f in order to take variation of the morphology with f into account. For MX–A systems the general mixing rule has the form σ α( f) =(1− f− f S)· σ MX α( f) + f S· σ S α( f) + f· σ A α( f) , where f S is the volume fraction of high-conducting interface regions estimated in terms of the brick-wall model; σ S is the conductivity via the interface. The general mixing rules are rather simple, convenient, agree with effective medium models, qualitatively describe percolation effects and satisfactorily represent the experimental data in many composites.

Molecular weight distribution dependent kernels in generalized mixing rules

In this article a model is proposed for the kernel in the generalized mixing rule recently formulated by Anderssen and Mead [Anderssen, R. S. and D. W. Mead, J. Non-Newtonian Fluid Mech. 76, 299–306 (1998)]. In order to derive such a model, it is necessary to take into account the rheological significance of the kernel in terms of the relaxation behavior of the individual polymers involved. This naturally leads to examining how additional physical effects, which depend on the molecular weight distribution, can be included in the mixing rule. The advantage of this approach is that, without changing the generality of the Anderssen and Mead (1998) rule, the choice of the kernel can be used to enhance the physical and rheological significance of their mixing rule.

On the Rouse spectrum and the determination of the molecular weight distribution from rheological data

Maier et al. [Maier, D., et al., J. Rheol. 42, 1153–1173 (1998)] examined the reconstruction of binary molecular weight distributions from rheological data for a series of polystyrene mixtures using a generalized mixing rule recently introduced [Anderssen, R. S. and D. W. Mead, J. Non-Newtonian Fluid Mech. 76, 299–306 (1998)]. They found an unexpected high value for β(β=3.84±0.1), the mixing parameter, which is 1 for the reptation and 2 for the double reptation or entanglement models. This result can be understood when the relaxation time spectrum is decomposed into a Rouse and an entanglement part and only the latter is used for determination of the molecular weight distribution. Applying a procedure which separates the Rouse processes from the spectrum determined from measured dynamic shear moduli, β values are found which are in accordance with the double reptation theory and which give very good agreement between molecular weight distributions determined by size-exclusion chromatography and by rheological data evaluation.

Studies on polarizabilities and scattering behavior of small spherical particles

It is possible to relate the response of a sphere to an applied uniform static electric field with the scattering from a small spherical particle due to plane wave incidence. The limits up to which these relations between the polarizability and the extinction, scattering, and radar cross sections of a sphere are valid in the dynamic range are presented here. From the observations it can be concluded that radar cross section is a very good measure in predicting the polarizability. A related aspect studied here is the evaluation of the effective permittivity of a sparse mixture of spherical inclusions using a generalized Maxwell Garnett mixing rule. This is compared with extinction of a plane wave by a slab of n spherical inclusions sparsely located. The extinction by such a slab is calculated using the quasistatic approximation to Mie theory, and also using the full Mie theory, as the size of the inclusions is increased. The studies have been carried out for both lossless and lossy inclusions. The generalized mixing rule was found to be quite accurate in predicting the value of effective permittivity up to size parameters of 0.5 at least for small εr of the inclusion.

A generalized mixing rule for hard-sphere equations of state of Percus–Yevick type

An analogical approach has been used to extend to mixtures the radial distribution function of Percus–Yevick type of equations of state for hard spheres. The proposed approach follows the same formalism employed by Mansoori et al. to extend to mixtures the Carnahan–Starling equation of state. The generality of the proposed method permits to extend to mixtures equations of state with different singularities. In this work, it is applied to an equation of state which meets the correct low density and high density limits, previously proposed by Khoshkbarchi and Vera. The results show that both the radial distribution function and the resulting equation of state for a mixture of hard spheres, developed in this study, can accurately represent the computer simulation data while satisfying the correct limit at close-packing.

Prediction of Bulk Properties of Fischer−Tropsch Derived Liquids

Asymptotic behavior correlations (ABCs) based on the properties of pure n-paraffins are used to predict the bulk properties of liquid mixtures derived from the Fischer−Tropsch synthesis. Comparisons are made with literature data for congealing point, density, specific heat, viscosity, thermal conductivity, and surface tension. Mixture property predictions are based on appropriate average carbon numbers and on a general property mixing rule. Both ideal and nonideal solution behavior are exhibited. For the nonideal properties, viscosity and surface tension, mixing rules are proposed and parameters regressed to obtain reasonable agreement with mixture data. Property estimates based on the appropriate average carbon number are probably sufficient for most engineering applications. Methods are outlined for estimating different carbon-number averages based on congealing- and melting-point, density, and the Anderson−Schulz−Flory distribution when the carbon number distribution of the mixture is not available. The presence of other Fischer−Tropsch byproducts appears to have little effect on the accuracy of predictions made for high molecular weight mixtures.

Influence of Wool Content on Properties of Blended Fabrics

Examines wool/polyester blended fabrics with a view to finding the optimal blend in relation to suitable utility, surface and handling properties. Concludes that a simple general mixing rule is possible for use in producing the required optimum blends.