Mixing Law Research Articles

Linear and nonlinear viscoelastic properties of bidisperse linear polymers: Mixing law and tube pressure effect

In this manuscript, we extend the tube-based model that we developed for predicting the linear viscoelasticity of entangled polymers [van Ruymbeke et al., J. Non-Newtonian Fluid Mech. 128, 7–22 (2005)] to the prediction of the extensional rheology of monodisperse and bidisperse linear polymers and confront the results to experimental data. This model is based on the concepts of stretch-orientation separability [McLeish and Larson, J. Rheol. 42, 81–110 (1998)] and inter-chain pressure [Marrucci and Ianniruberto, Macromolecules 37, 3934–3942 (2004)]. In order to deal with polydisperse samples, a new mixing law is proposed. As it does not require knowledge of the full linear relaxation spectrum, the proposed model is a powerful predictive tool. Very good agreement is found between theoretical and experimental results. For bidisperse samples, the individual contribution of each component is determined, and it is shown that only few percent of long chains are enough to generate the strong strain hardening observed in the experimental data. Last, we discuss the value of the tube diameter relaxation time. For monodisperse samples, this parameter is found to scale with M2. However, for bidisperse samples, as it was already observed by Wagner et al. [J. Rheol. 52, 67–86 (2008)], the tube diameter relaxation time of the long component must be rescaled, which is contrary to the inter-chain pressure model and opens several new questions.

Acoustical Studies of Binary Liquid Mixtures of Cyclopentane with 1-Alkanol at Different Temperatures and Different Approaches for Ideal Mixing Laws

Speeds of sound have been measured in liquid mixtures of cyclopentane with 1-propanol, with 1-pentanol, and with 1-heptanol across the entire composition range at temperatures of (298.15, 308.15 and 318.15) K and atmospheric pressure. The experimental speed of sound data were used to estimate the isentropic compressibility κ S for all mixtures. The molar volumes were multiplied by the corresponding isentropic compressibilities to obtain estimates of the molar compressibilities K S,m. The corresponding $K_{S,\mathrm{m}}^{\mathrm{E}}$ values have also been calculated. Theoretical values of the speeds of sound were estimated using theories and empirical relations. Deviations of the speed of sound, u D, from the values calculated by different approaches for ideal mixing have been obtained for all mole fractions.

Binary blends of linear ethylene copolymers over a wide crystallinity range: Rheology, crystallization, melting and structure properties

This study deals with the physical properties of melt-compounded blends of three linear ethylene copolymers covering a large crystallinity range, namely 77% – 46% – 16% for the high density – linear low density – ultra low density copolymers, respectively. The melt behavior assessed from the zero-shear viscosity ( η o) reveals immiscibility of the three binary systems over the whole composition range. However, the change from positive to negative deviation of η o with respect to the log-additivity mixing law as a function of composition suggests a structural transition from partial miscibility at the interface of the phase-separated domains to incompatibility. Crystallization and melting behaviors of the blends corroborate the occurrence of phase separation in the three systems. For most blends, the temperature shift of the crystallization ( T c) and melting ( T m) peaks as compared to the ones of the pure copolymers yet indicates partial miscibility in the crystalline and/or in the amorphous regions. It is pointed out that miscibility in the amorphous phase resulting from partial miscibility in the melt may, on its own, entail T m depression of the crystals via surface free energy effect without necessarily implying cocrystallization and crystal thickness reduction. In several cases, the presence of intermediate endotherm and exotherm between the two main peaks of the melting and crystallization traces, respectively, discloses hybrid crystals assigned to a composition gradient at the interface of the phase-separated domains. A marked positive deviation of the upper T c from the linear mixing rule is observed for the three systems. A nucleating effect from the interface of the phase-separated domains is suggested to promote early crystallization in the upper T c phase. The SAXS data reveal electron density fluctuations at a much larger scale than that of the semi-crystalline structure demonstrating the occurrence of micro-phase separation in the melt prior to crystallization. Solubility of low T m chain species in the amorphous layers of the high T m phase is also evidenced. AFM and DMTA support micro-phase separation in the three systems and provide complementary information on the crystalline habits in the phase-separated domains of the blends.

The effective dielectric constant of plasmas — A mean field theory built from the electromagnetic ionic T-matrix

This work aims to obtain the effective dielectric constant tensor of a warm plasma in the spirit of the derivation of a mixing law. The medium is made of non point-like ions immersed in an electron gas with usual conditions relating the various lengths which define the problem. In this paper the ion dielectric constants are taken from their RPA responses as developed in a previous paper [1]. Furthermore the treatment of the screening effects is made through a mathematical redefinition of the initial problem as proposed in Ref. [1]. Here the complete calculation of the T-matrix describing the scattering of an electromagnetic wave on an isolated ion immersed in an “effective medium” is given. It is used for building , in the spirit of a mixing law, a self-consistent effective medium theory for the plasma dielectric tensor. We then extend the results obtained in Ref. [1] to higher orders in ion or dielectric inclusion densities. The techniques presented are generic and can be used in areas such as elasticity, thermoelasticity, and piezoelectricity.

Structural Relaxation Dynamics in Binary Glass-Forming Molecular Liquids with Ideal and Complex Mixing Behavior

The glass transition and structural relaxation dynamics of various binary glass-forming liquids are investigated with dielectric relaxation measurements across the entire composition range. Three categories of solutions with weak, intermediate, and strong mixing effects, namely methyl-m-toluate in methyl o-toluate, methyl m-toluate in di-n-butyl phthalate, and 1,2-propandiol in 2-hexylamine, are selected to address the mixing behaviors from near-ideal to nonideal cases. The glass transition temperatures, fragility indices, and stretching exponents of the solutions are determined and their composition dependence is the focus of this study. The experimental measurements show that mixing generally generates a negative deviation of fragility m relative to the composition average of the results of two neat components (ideal mixing law). This excess negative fragility proves to be a universal feature of binary systems, and the increase of the nonideal mixing degree results in a more pronounced negative deviation. In contrast, the composition dependence of the stretching exponents is more complex, and a transition from the negative to positive deviation is observed for substantial nonideal character. The study assists understanding the dynamics of multicomponent glass formers.

On the determination of thermal conductivity of sedimentary rocks and the significance for basin temperature history

ABSTRACT Thermal history is an important control on generation of gas and oil in source rocks. Therefore, a realistic treatment of thermal conductivity in rocks is essential for hydrocarbon generation modelling in sedimentary basins. We present two different techniques for thermal conductivity estimation: (1) upscaling, based on arithmetic, geometrical and harmonic mean values of averaged conductivity from anticipated mineral composition; (2) infometric regression of thermal conductivity and anisotropy from measurements in rocks of known porosity and mineral composition. The accuracy of the estimate depends on the technique and we expect that the most precise way to predict thermal conductivity of sedimentary rocks leads to considerable improvement in the accuracy of basin modelling and hydrocarbon prospect identification. Various thermal conductivity models based on porosity and lithology are used for modelling temperature history for sensitivity purposes. Models where matrix conductivity is determined from measured thermal conductivities based on the mineralogy give considerably lower strata temperatures than models where geometrically averaged matrix conductivity are determined from mineralogy. This is assumed to be caused by deficiency in the mixing laws not taking into account the texture effect of the sediments. The temperature histories estimated by the three mixing law models – arithmetic, geometrical and harmonic mean – are considerably different. We show that the mixing law models give a temperature history and predicted hydrocarbon maturation far off even if we have temperature measurement at the same location. For example the harmonic model predicts the onset of the maturation of the Jurassic formations 20 million years later than does the model based on measured conductivities.

Electromagnetic mixing laws: A supersymmetric approach

Abstract In this article we address the old problem of finding the effective dielectric constant of materials described either by a local random dielectric constant, or by a set of non-overlapping spherical inclusions randomly dispersed in a host. We use a unified theoretical framework, such that all the most important Electromagnetic Mixing Laws (EML) can be recovered as the first iterative step of a family of results, thus opening the way to future improvements through the refinements of the approximation schemes. When the material is described by a set of immersed inclusions characterized by their spatial correlation functions, we exhibit an EML which, being featured by a minimal approximation scheme, does not come from the multiple scattering paradigm. It is made of a pure Hori–Yonezawa formula, corrected by a power series of the inclusion density. The coefficients of the latter, which are given as sums of standard diagrams, are recast into electromagnetic quantities which calculation is amenable numerically thanks to codes available on the web. The methods used and developed in this work are generic and can be used in a large variety of areas ranging from mechanics to thermodynamics.

Poisson approximation of the mixed Poisson distribution with infinitely divisible mixing law

In this work, explicit upper bounds are provided for the Kolmogorov and total variation distances between the mixed Poisson distribution with infinitely divisible mixing law and the Poisson distribution. If μ and σ 2 are the mean and variance of the mixing distribution respectively, then the bounds provided here are asymptotically equal to σ 2 / ( 2 μ 2 π e ) and σ 2 / ( μ 2 π e ) for the Kolmogorov and the total variation distance respectively when μ → ∞ and σ 2 is fixed. Finally, as an application, the Poisson approximation of the negative Binomial distribution is considered.

Nonlinear elasticity of composite materials

We investigate the elastic properties of model composites, consisting in a dispersion of nonlinear (spherical or cylindrical) inhomogeneities into a linear solid matrix. Both phases are considered isotropic. Under the simplifying hypotheses of small deformation for the material body and of small volume fraction of the embedded phase, we develop a homogenization procedure based on the Eshelby theory, aimed at describing nonlinear features. We obtain the bulk and shear moduli and Landau coefficients of the overall material in terms of the elastic behavior of the constituents and of their volume fractions. The mixing laws for the nonlinear properties describe a complex scenario where possible strong amplifications of the nonlinearities may arise in some given conditions.

Homogenization of Maxwell equations and the Maxwell-Wagner dispersion

We study the behavior of an electromagnetic field in a porous medium, when the hard skeleton and porous fluid have different electrochemical properties. It is assumed that the pores are periodically distributed and the field is generated by an external source of current with alternating frequency. If the size of the periodical cell is small compared to the domain of measurements, oscillation of coefficients in the Maxwell equations is strong. This creates difficulties in calculating the resulting electric field. The essence of the homogenization method is in the substitution of the composite with a microstructure by an equivalent material with uniform properties. The coefficients oscillating fast with changing spatial variable are replaced by specific constant coefficients. A new law of mixing of components is deduced within this approach, and a numerical algorithm is suggested for calculation of the effective parameters, which is equally applicable for low and high frequencies. This mixing law was tested successfully on exact solutions for a composite with a complex structure for the fiber structure with chattered symmetry in the plane transversal to the fibers and using the known Bruggeman relation for spherical inclusions. The main objective of this work is to analyze the effect of the Maxwell‐Wagner dispersion and Archie’s statistical law. In this work, we ignore polarization induced by the double electric layer. Therefore, the results are applicable only to sandstones.

Dielectric permittivity simulation of random irregularly shaped particle composites and approximation using modified dielectric mixing laws

Finite difference quasielectrostatic modeling is used to predict the dielectric permittivity of composites consisting of irregular particles in a background matrix. Representations of particles having undulating surfaces described by sums of harmonic functions are created on the computer and subsequently packed into a three-dimensional cellular model space. Composite dielectric permittivities as a function of volumetric filling fraction and particle undulation amplitude were simulated using constituent permittivities similar to the low-field behavior of barium titanium oxide (particles) and polyvinylidene fluoride-trifluoroethylene-chlorofluoroethylene (terpolymer matrix). An increase in particle roughness (undulation amplitude) causes a more rapid increase in composite permittivity than that predicted by random spherical particle simulations. The dielectric behavior of irregular particle composites is also simulated over a wide range of ratios of particle permittivity to matrix permittivity, where both permittivities are purely real. An empirical mixing law, which is a modification of the Hanai equation with an exponent 1/μ instead of 1/3, is investigated and found to be in excellent agreement with the simulations. Additional empirical expressions that provide approximate values of μ in terms of the particle undulation amplitude and the ratio of constituent permittivities are developed. Together, the empirical expressions are potentially useful as a predictive mixing law for irregular particle systems.

Study of dielectric constants of binary composites at microwave frequency by mixture laws derived from three basic particle shapes

Powder mixture rules derived from the filler particles with the shapes of sphere, cylinder or rod, and lamella or disk with random distributions are studied in this paper. Three ceramic powders of fillers with dielectric constants of 10, 20, and 36, respectively, are adopted. The experimental dielectric constants of ceramic dispersions in the polyethylene matrix at microwave frequency are compared to those obtained by using different mixing laws. The mixing rules are also adopted to estimate the dielectric constants of pure ceramics from the measured dielectric constants of composites with various concentrations. The theory for the error of estimation is studied. In contrast to the traditional concept of obtaining the best curve fitting of mixture rule with the experimental data, this study conclude a very important concept on the powder mixture rule, that is, the most adequate mixture law for estimating the dielectric constants of pure ceramics requires both good curve fitting and potential of less error.

Improving Petrophysical Interpretation With Wide-Band Electromagnetic Measurements

Summary Because of their sensitivity to ionic content and surface texture, wide-band electromagnetic (WBEM) measurements of saturated rocks exhibit frequency dispersions of electrical conductivity and dielectric constant that are influenced by a variety of petrophysical properties. Factors as diverse as fluid saturation, porosity, pore morphology, thin wetting films, and electrically charged clays affect the WBEM response of rocks. Traditional dielectric mixing laws fail to quantitatively and practically integrate these factors to quantify petrophysical information from WBEM measurements. This paper advances a numerical proof of concept for useful petrophysical WBEM measurements. A comprehensive pore-scale numerical framework is introduced that incorporates explicit geometrical distributions of grains, fluids and clays constructed from core pictures, and that reproduces the WBEM saturated-rock response on the entire kHz-GHz frequency range. WBEM measurements are verified to be primarily sensitive (a) in the kHz range to clay amounts and wettability; (b) in the MHz range to pore morphology (i.e., connectivity and eccentricity), fluid distribution, salinity, and clay presence; and (c) in the GHz range to porosity, pore morphology and fluid saturation. Our simulations emphasize the need to measure dielectric dispersion in the entire frequency spectrum to capture the complexity of the different polarization effects. In particular, it is crucial to accurately quantify the phenomena occurring in the MHz range where pore connectivity effects are confounded with clay polarization and pore/grain shape effects usually considered in dielectric phenomena. These different sensitivities suggest a strong complementarity between WBEM and NMR measurements for improved assessments of pore-size distribution, hydraulic permeability, wettability, and fluid saturation.

The similarity solution for turbulent mixing of two-layer stratified fluid

Experiments are reviewed in which a two-layer salt-stratified tank of water was mixed by turbulence. The density profile began as a single step and evolved to a smooth mixed profile. The turbulence was generated by many excursions of a horizontally moving vertical rod with Richardson number Ri > 0.9 and Reynolds Number Re > 600. There was almost perfect collapse of all the profiles to one universal profile as a function of a similarity variable. We develop a theoretical model for a simple mixing law with a buoyancy flux that is a function of internal Richardson number Rii. A similarity equation is found. A flux law that increases with small Rii and decreases with large Rii is considered next. Since no analytical solution is known, the similarity concept is tested by numerically integrating the equations in space and time. With buoyancy flux monotonically increasing with internal Richardson number, the similarity approach is valid for a profile starting from a slightly smoothed step. However, a shock forms for a mixing law with higher initial Rii (so that buoyancy flux decreases with Richardson number) and the similarity approach is invalid for those initial conditions.

Finite difference time domain simulation of radar wave propagation through comet nuclei dielectric models

Abstract— The 90 MHz radar‐wave experiment, Comet Nucleus Sounding Experiment by Radiowave Transmission (CONSERT), on board the Rosetta mission (ESA, 2004) is expected to probe the nucleus of the comet 67P/Churyumov‐Gerasimenko (67P/C‐G) to reveal information on its physical properties, chemical composition, and internal structure. This investigation assesses the potential to recognize lithological structure in the comet nucleus with a radar experiment such as CONSERT. Radar simulations at 90 MHz were performed with a finite difference time domain (FDTD) method. The amplitude and losses of the transmitted and reflected electric field components of an incident radar pulse were evaluated as a function of time. Seven different dielectric models of sections of a hypothetical comet nucleus were used, representative of existing theories of comet nuclei. Values of dielectric constant assigned to these models are based on mixing laws for a porous mixture of ice and meteoritic dust, employing laboratory measured values of relative permittivity for mainly chondritic meteorites. Our results confirm that structural differences such as layers or inclusions are discernable from transmitted and reflected radar signals at 90 MHz, the central frequency of the CONSERT instrument. Such simulations help to constrain the ambiguities that might exist in future radar data associated with the nature of the comet nuclei, whether conglomerate or layered in nature.

High‐pressure viscosity and density of carbon dioxide + pentaerythritol ester mixtures: Measurements and modeling

AbstractThis article reports densities and viscosities up to 60 MPa and from 303.15 to 353.15 K on two binary mixtures of CO2 and lubricant pentaerythritol tetra‐2‐ethylhexanoate with 7.8% and 14.4% mass fraction of lubricant. Because CO2 and the lubricant are not in the same phase at atmospheric pressure and room temperature, a specially‐designed vibrating‐wire apparatus has been used. The experimental data, together with our previous data for six mixtures of CO2 and three polyolesters, have been used to test the predictive and correlation ability of several models: simple mixing laws, the self‐referencing method, and the hard‐sphere and free volume models. In general, it is not possible to predict the viscosity with deviations smaller than 10%. The hard‐sphere model and the self‐referencing model are the more adequate to represent the viscosity of these mixtures. Most of the models studied underestimate the viscosities over the investigated (T,p) range. © 2008 American Institute of Chemical Engineers AIChE J, 2008

Elastohydrodynamic study of vegetable oil–polyalphaolefin blends

AbstractTwo polyalphaolefins, of higher and lower viscosity than vegetable oils, were used to make binary blends of varying compositions with soy bean and canola oils. The pure oils and the blends were used in viscosity and film thickness investigations. The effects of composition and temperature on viscosity were found to agree well with the theoretical predictions of a simple mixing law. The film thicknesses of the various blends under elastohydrodynamic conditions were measured at 20 N load, and varying entrainment speeds and temperatures. From the data, pressure–viscosity coefficients, α, as a function composition and temperature were obtained. The resulting α values were compared with theoretical predictions. Experimental values of α as a function of composition showed a slight negative or no deviation from the values predicted by an ideal mixing model. On the other hand, experimental values of α displayed a mild decrease with increasing temperature, while the model predicted a sharp decrease with increasing temperature. Published in 2008 by John Wiley & Sons, Ltd.

Direct current electrical and microwave properties of polymer-multiwalled carbon nanotubes composites

We report the results of effective direct current (dc) resistivities and alternating current (ac) complex permittivity measurements carried out on two series of polymer∕multiwalled (MW) carbon nanotube (CNT) composite samples as function of the CNTs volume fraction and temperature. The CNTs have typical aspect ratio over 100:1 and are quasiuniformly dispersed in two types of polymer host [epoxy and polystyrene (PS)-cobutyl acrylate latex] according transmission electron microscopy characterization. A percolation threshold occurs in the composites with the PS latex matrix when the CNT volume fraction is ≈0.012. In contrast, the set of resistivity data for samples containing epoxy resin as host matrix is not representative of an intrinsically percolation transition. Atomic force microscopy, coupled to the measurement of the local electric resistances, permits us to study the agglomerate distribution in both types of composites. The differences in morphology between the two series of polymer∕MWCNTs are attributed to interface effects between the elongated filler nanoparticles and the polymer chains. Comparison of the measured effective dc and ac resistivities, at temperatures ranging from 30to300K, with Sheng’s model supports the fact that charge transport in the composites follows a thermal fluctuation induced tunneling mechanism, in which the tunneling of the electrons through the thermally induced fluctuating potential barrier formed by a thin insulating polymer layer separating MWCNTs aggregates. The spectral behavior of permittivity is consistent with a power-law form. Several mixing laws were tested to represent the filler fraction dependence of the effective permittivity in the microwave range of frequencies; however, none of them is able to quantitatively describe the sets of data. The basic deficiency of these formulas is that they make no explicit reference to the internal length scales in the composite samples.

Material Parameter Measurements for Microwave Antireflection Coating Development

The main steps for characterization and measurement of microwave absorbent materials in the 1–10 GHz range are introduced. The coaxial reflection-transmission type of material parameter measurement is analyzed in detail and the main measurement error is corrected. The microscopic material particle parameter measurement concept is also presented using different mixing rule laws to determine the material parameters of the single particles from the macroscopic parameters. Two-dimensional FDTD simulations have been used to model the behavior of mixed electric and magnetic type of material.

Petrophysical analysis of regional-scale thermal properties for improved simulations of geothermal installations and basin-scale heat and fluid flow

Development of geothermal energy and basin-scale simulations of fluid and heat flow both suffer from uncertain physical rock properties at depth. Therefore, building better prognostic models are required. We analysed hydraulic and thermal properties of the major rock types in the Molasse Basin in Southern Germany. On about 400 samples thermal conductivity, density, porosity, and sonic velocity were measured. Here, we propose a three-step procedure with increasing complexity for analysis of the data set: First, univariate descriptive statistics provides a general understanding of the data structure, possibly still with large uncertainty. Examples show that the remaining uncertainty can be as high as 0.8 W/(m K) or as low as 0.1 W/(m K). This depends on the possibility to subdivide the geologic units into data sets that are also petrophysically similar. Then, based on all measurements, cross-plot and quick-look methods are used to gain more insight into petrophysical relationships and to refine the analysis. Because these measures usually imply an exactly determined system they do not provide strict error bounds. The final, most complex step comprises a full inversion of select subsets of the data comprising both laboratory and borehole measurements. The example presented shows the possibility to refine the used mixing laws for Petrophysical properties and the estimation of mineral properties. These can be estimated to an accuracy of 0.3 W/(m K). The predictive errors for the measurements are 0.07 W/(m K), 70 m/s, and 8 kg/m^3 for thermal conductivity, sonic velocity, and bulk density, respectively.