Mean Particle Diameters. Part III: An Empirical Evaluation of Integration and Summation Methods for Estimating Mean Particle Diameters from Histogram Data

Mean particle diameters of histograms of size distributions were estimated by integration and by summation over the histogram intervals using several, partly model, size distributions. The Summation method, used in the Moment‐Ratio (M‐R) method, tends to be more accurate than the Integration method and is less sensitive to low values of the lower limit of size distributions. The mathematical equations for the Integration method appear to be difficult to apply in daily practice and their complexity may easily hide the physical background of a mean particle diameter. The Integration approach leads to peculiar saw‐tooth shaped histograms, which makes the calculation of (histograms of) density distributions of other particle quantities irreversible and the calculation of mean diameters ambiguous. The Summation method equations are straightforward and generally applicable. Owing to the use of midpoints of particle size classes, the method is both unambiguous and reversible. The order of a mean diameter appears to be a useful quantity for the visualization of properties of methods to estimate mean particle diameters. The Summation method is therefore proposed as the method of choice for standardization.     

Particle Mass Loading Effect on a Two-Phase Turbulent Downward Jet Flow

The particle mass loading effect on the flow structure of a two-phase turbulent jet flow was studied. A particle mass loading ratio ranging from 0 to 3.6 was used as the control parameter. The polystyrene solid particles used had nominal diameters of 210 and 780 μm. The flow Reynolds number, which was based on the pipe nozzle diameter and the fluid-phase centerline mean velocity, was 2 × 10⁴ in the current test. A two-color laser Doppler anemometer (LDA), combined with the amplitude discrimination method and the velocity filter method, was employed to measure the mean velocity distributions for the particle and fluid phases, and the turbulent intensities and Reynolds stresses of the flow. The two-phase jet flow field was measured from the initial pipe exit to 90 D downstream. Another one-component He? Ne laser LDA system was also applied to obtain the energy spectra and temporal correlations of the two-phase jet flow.     

Study of chemical shift in Ll and Lƞ X‐ray emission lines in different chemical forms of 48Cd and 50Sn compounds using WDXRF technique

Chemical shift in Ll and L? X‐ray emission lines of ₄₈Cd and ₅₀Sn elements in various chemical compounds was determined with high resolution wavelength dispersive X‐ray fluorescence (WDXRF) spectrometer. The positive and negative shifts were measured in ₄₈Cd compounds viz, CdS, CdB₄O₇, CdCl₂, Cd₃(PO₄)₂, CdCO₃, CdI₂ and CdO with reference to pure Cd foil and ₅₀Sn compounds viz, Sn(CrO₄)₂, SnO, SnO₂, SnCl₂, SnF₂, SnF₄ with reference to pure Sn foil. The measured energy shifts in Ll X‐ray emission lines range from ?0.47 to 1.82 eV and L? emission lines range from ?2.67 to 1 eV for both compounds. The effective charges (q, q ^/ , q ^// , and q ^/// ) were calculated from four models (Pauling method, Suchet method, Levine method and Batsonav method) and found to be linear dependence with chemical shift. The measured chemical shifts were correlated with effective charge, number of ligands and electronegativity of the central metal atom in the given compounds.     

Rank of quantized universal enveloping algebras and modular functions

We compute an intrinsic rank invariant for quasitriangular Hopf algebras in the case of general quantum groupsU _q (g). As a function ofq the rank has remarkable number theoretic properties connected with modular covariance and Galois theory. A number of examples are treated in detail, including rank (U _q (su(3))) and rank (U _q (e ₈)). We briefly indicate a physical interpretation as relating Chern-Simons theory with the theory of a quantum particle confined to an alcove ofg.     

Relativistic effects in one-dimensional hydrogen atom

The dependence of stationary levels of a Dirac particle in the regularized Coulomb potential V _??(z) = ?q/(|z| + ??) on the cutoff parameter ?? is studied. It is shown that, in 1 + 1 D, the energy spectrum of a Dirac particle in such a potential reveals some specific features which nonanalytically depend on the coupling constant q and are essentially relativistic in nature. These properties turn out to be most important for ?? ? 1, explicitly demonstrating the existence of a physically reasonable energy spectrum for an arbitrarily small ?? > 0 and, at the same time, the absence of regular limit ?? ?? 0 (hence, the absence of a well-defined spectral problem for the Dirac equation without regularization for arbitrary q in 1 + 1 D).     

Analysis of the Initial Slope of the Small‐Angle Scattering Correlation Function of a Particle

The small‐angle scattering correlation function of a particle γ(r) results from scattering experiments. This function possesses a well‐defined slope γ′(0) at the origin. This slope is defined by the particle volume V and the whole surface area S of the particle via γ′(0) = –S/(4V). In this paper it is demonstrated that this slope defines the mean chord length of the particle, . This theorem involves non‐convex particles, especially the case of particles with hollow parts. Consequently, for a large class of particle shapes the mean chord length is defined in terms of V and S. This extension of the Cauchy theorem is developed by closer analysis of the set covariance C(r), of the small‐angle scattering correlation function γ(r), and of the so‐called linear erosion P(r) near the origin r→0. The cases of a single hollow sphere, of two touching spheres, and of the single hollow cylinder are discussed.     

Two‐Phase Flow Characterization by Automated Digital Image Analysis. Part 2: Application of PDIA for Sizing Sprays

A series of experiments were conducted in order to assess the robustness and accuracy of a recently developed digital image analysis technique (PDIA). This paper investigates the application of the PDIA technique to the sizing of relatively small fuel droplets of diameters in the range 5 to 30 μm produced by a pressure‐swirl atomizer. The measurement performance of the PDIA system has been assessed in terms of individual object diameters and also number and volume probability density functions of diameter in comparison to phase Doppler anemometry (PDA) data obtained under identical conditions. PDIA measurements revealed good agreement with spray data obtained by PDA at a measurement location 36 diameters downstream from the nozzle orifice with differences in the arithmetic mean diameter, D₁₀ and volume mean diameter, D₃₀ of approximately 5 and 3% respectively. The PDIA technique was shown to detect the presence of very large, predominantly non‐spherical droplets whose diameters were in excess of 100 μm. These droplets, although few in number constitute a significant proportion of the total spray volume and would have otherwise been either erroneously measured or have passed through the probe volume undetected using PDA due to non‐sphericity. Smaller objects may also be measured correctly by both methods although sensitivity to signal‐to‐noise ratio, for both methods can generate spurious and contradictory errors.     

Particle size distribution and electrochemical properties of LiFePO4 prepared by a freeze-drying method

The electrochemical performance of carbon-coated nanocrystalline LiFePO₄ prepared by a freeze-drying method is examined. This method is based on the thermal decomposition of homogeneous phosphate-formate precursors. Structural and morphological characterization of LiFePO₄ is carried out by powder XRD, BET measurements, SEM and XPS analyses. The electrochemical behaviour is tested in model lithium cells using galvanostatic mode. By changing the solution concentration, the freeze-drying method allows preparing LiFePO₄ with mean particle sizes between 60 and 100 nm and different particle size distributions. The content of carbon appearing mainly on the particle surface depends on both the solution concentration and the annealing temperature. The effect of particle size distribution on the voltage profile of LiFePO₄ is also demonstrated. The specific capacity is mainly determined by the amount of carbon deposited on the particle surfaces.     

Multidimensional Particle Size Distributions and Their Application to Nonspherical Particle Systems in Two Dimensions

Particle science and technology evolve toward ever increasing complexity with respect to the multidimensional particle properties of size, shape, surface, internal structure, and composition. In this study, the theoretical background is elaborated for multidimensional particle size distributions (PSDs) by transferring the concepts known from 1D size distributions to anisotropic particles comprising at least two different length dimensions, e.g., nanorods and platelets. After introducing 2D PSDs, the calculation of differently weighted probability density functions including their interconversion is presented. This is necessary in order to compare data resulting from different measurement techniques which probe different physical properties and thus provide differently weighted PSDs. In addition, it is shown how 1D distributions with reduced content of information can be deduced from 2D PSDs. As a proof‐of‐concept and for illustration purposes, this approach is applied to a 2D Gaussian size distribution. Furthermore, a generalized scheme is suggested which outlines the conversion of number, surface, and volume weighted densities within the 2D space. The application of these methods to the more general n‐dimensional case is straightforward.     

Apollonian packings as physical fractals

The Apollonian packings (APs) of spheres are fractals that result from a space-filling procedure. We discuss the finite size effects for finite intervals s?∈?[s _min,?s _max] between the largest and the smallest sizes of the filling spheres. We derive a simple analytical generalization of the scale-free laws, which allows a quantitative study of such physical fractals. To test our result, a new efficient space-filling algorithm has been developed which generates random APs of spheres with a finite range of diameters: the correct asymptotic limit s _min/s _max?→?0 and the known APs' fractal dimensions are recovered and an excellent agreement with the generalized analytic laws is proved within the overall range of sizes.     

Electron correlation effects in N2 and CO studied by X-ray scattering and CISD calculations

Total X-ray scattering intensities σ_ee(q) for N₂ and CO have been measured as a function of momentum transfer using the energy dispersive method. Novel procedures to extract accurate σ_ee(q), which eliminate effects of polarization, inelastic scattering, anomalous dispersion, and molecular vibration, have been proposed. A simplified theoretical treatment based on configuration interaction singles and doubles (CISD) calculations has been suggested. This procedure makes it possible to apply combined theoretical and experimental X-ray scattering studies to larger molecules. The inclusion of f and g functions is crucial, and the σ_ee(q) calculated with the cc-pVQZ[5s4p3d2f1g] basis set almost reproduces the data based on more elaborate MR-CISD calculations within the experimentally most relevant region of up to q ? 3 au. In contrast to experimental electron scattering data, the X-ray scattering intensities agree well with the computed results.     

Ion‐ and positron‐acoustic solitons in magnetized dusty plasma with q‐non‐extensive electron and positron velocity distribution

In this study, the properties of ion‐ and positron‐acoustic solitons are investigated in a magnetized multi‐component plasma system consisting of warm fluid ions, warm fluid positrons, q‐non‐extensive distributed positrons, q‐non‐extensive distributed electrons, and immobile dust particles. To drive the Korteweg–de Vries (KdV) equation, the reductive perturbation method is used. The effects of the ratio of the density of positrons to ions, the temperature of the positrons, and ions to electrons, the non‐extensivity parameters q_e and q_p , and the angle of the propagation of the wave with the magnetic field on the potential of ion‐ and positron‐acoustic solitons are also studied. The present investigation is applicable to solitons in fusion plasmas in the edge of tokamak.     

Dimensional Reduction Over the Quantum Sphere and Non-Abelian q-Vortices

We extend equivariant dimensional reduction techniques to the case of quantum spaces which are the product of a K?hler manifold M with the quantum two-sphere. We work out the reduction of bundles which are equivariant under the natural action of the quantum group SU _q (2), and also of invariant gauge connections on these bundles. The reduction of Yang–Mills gauge theory on the product space leads to a q-deformation of the usual quiver gauge theories on M. We formulate generalized instanton equations on the quantum space and show that they correspond to q-deformations of the usual holomorphic quiver chain vortex equations on M. We study some topological stability conditions for the existence of solutions to these equations, and demonstrate that the corresponding vacuum moduli spaces are generally better behaved than their undeformed counterparts, but much more constrained by the q-deformation. We work out several explicit examples, including new examples of non-abelian vortices on Riemann surfaces, and q-deformations of instantons whose moduli spaces admit the standard hyper-K?hler quotient construction.     

New approach in theory of Clebsch-Gordan coefficients foru(n) andU q(u(n))

A new method for calculation of Clebsch-Gordan coefficients (CGCs) of the Lie algebrau(n) and its quantum analogU _q(u(n)) is developed. The method is based on the projection operator method in combination with the Wigner-Racah calculus for the subalgebrau(n−1) (U _q(u(n−1))). The key formulas of the method are couplings of the tensor and projection operators and also a tensor form of the projection operator ofu(n) andU _q(u(n)). It is shown that theU _q(u(n)) CGCs can be presented in terms of theU _q(u)(n−1)) q−9j-symbols. Presented at the 9th International Colloquium: “Quantum Groups and Integrable Systems”, Prague, 22–24 June 2000. Supported by Russian Foundation for Fundamental Research, grant 99-01-01163. Supported in part by the U.S. National Science Foundation under Grant PHY-9970769 and Cooperative Agreement EPS-9720652 that includes matching from the Louisiana Board of Regents Support Fund.     

Mean energies of hybridized valence states

A method of determining the relative signs of J _AB and J _BX in nuclear spin systems of the type AB₂X _q is described and applied to the AB₂X₃ proton spin system of 2-butyn-1-ol. Variation of the relative chemical shift v _A—v _B (through changes in temperature or in the composition of the solvent) causes certain pairs of ‘labile’ transitions in the X spectrum to coalesce on one side or other of the X spectrum. The information about the relative signs of the coupling constants is obtained by noting the two different critical values of v _A-v _B at which these coalescences occur. Since the method remains applicable even if the third coupling constant J _AX is vanishingly small, it is particularly useful in the determination of the signs of long-range spin coupling constants where the usual double resonance methods often break down. It is concluded that the long-range coupling J(H-C-C≡C-C-H) of 2-butyn-1-ol has a positive sign.     

Generalized inverse participation numbers in metallic-mean quasiperiodic systems

From the quantum mechanical point of view, the electronic characteristics of quasicrystals are determined by the nature of their eigenstates. A practicable way to obtain information about the properties of these wave functions is studying the scaling behavior of the generalized inverse participation numbers Z_q ~ N - ^{D_q (q - 1)}Z_q \sim N - ^{D_q (q - 1)} with the system size N. In particular, we investigate d-dimensional quasiperiodic models based on different metallic-mean quasiperiodic sequences. We obtain the eigenstates of the one-dimensional metallic-mean chains by numerical calculations for a tight-binding model. Higher dimensional solutions of the associated generalized labyrinth tiling are then constructed by a product approach from the one-dimensional solutions. Numerical results suggest that the relation D _q ^dd = dD _q ^1d holds for these models. Using the product structure of the labyrinth tiling we prove that this relation is always satisfied for the silver-mean model and that the scaling exponents approach this relation for large system sizes also for the other metallic-mean systems.     

A Simple,Scalable Process for the Production of Porous Polymer Microspheres by Ink‐Jetting Combined with Thermally Induced Phase Separation

Porous microspheres capable of delivering high payloads of biomolecules with suitable biodegradability and biocompatibility would be valuable in delivery systems to aid tissue regeneration. This study describes a facile, scalable technique to produce biodegradable porous microspheres by combining continuous ink‐jetting through a piezoelectric nozzle with thermally induced phase separation (TIPS). A selection of biomaterials is investigated to suit delivery in tissue engineering, the synthetic polyesters poly(lactic‐co‐glycolic acid) (PLGA), and poly caprolactone (PCL) and a natural polymer, gelatin. The parameters governing the microsphere production are determined experimentally and compared to theoretical predictions derived from the fluid mechanics and heat transfer during the ink‐jetting process. The microspheres produced have open interconnected pores with mean particle diameters of 80–200 μm and no significant skin region. The physical properties, such as the mean particle diameter, pore size, and surface area could be controlled by varying production parameters including the ink‐jetting pressure, nozzle height, and the size and oscillation frequency of the nozzle. The technique is demonstrated to successfully encapsulate a model hydrophobic molecule during microsphere production with uniform distribution. Porous PLGA microspheres are also used to achieve much higher adsorption capacities of a short peptide than non‐porous microspheres of the same material.     

q-Weyl coefficients forU q (3) andq-Racah coefficients forSU q (2)

We show that theq-Weyl coefficients of the quantum algebraSU _q (3) are equal to theq-Racah coefficients of the quantum algebraSU _q (2) (up to a simple phase factor). Using aq-analog of the resummation procedure we obtain also theq-analogues of all known general analytical expressions for the 6j-symbols (or the Racah coefficients) of the quantum algebraSU _q (2) starting from one such formula.Presented at the 4th Colloquium Quantum groups and integrable systems, Prague, 22–24 June 1995.The research described in this publication was supported in part by Grants No. MB1000 and No. NRC000 from International Science Foundation.