The fluctuation kinetics of formation of nanoparticles: Particle-size distribution

A stochastic simulation of the growth of particles on a uniform cubic lattice was performed by the Monte Carlo method. Changes in the width of the distribution (M _w /M _n ) as the size of particles increased were extremal in character. Distribution narrowing occurred much more slowly than in classic polymerization. An empirical equation relating the number of free vacancies of a growing particle and its mean size was obtained. The introduction of a stabilizer deactivating free vacancies of a growing particle caused the appearance of a critical phenomenon. At stabilizer concentrations higher than critical, large-sized particles could not form. At stabilizer concentrations close to critical, the particle-size distribution was bimodal. This resulted in an anomalously larger distribution width.     

Evolution of Voronoi/Delaunay Characterized Micro Structure with Transition from Loose to Dense Sphere Packing

Micro structures of equal sphere packing （ranging from loose to dense packing） generated numerically by discrete element method under different vibration conditions are characterized using Voronoi/Delaunay tessellation, which is applied on a wide range of packing densities. The analysis on micro properties such as the total perimeter, surface area, and the face number distribution of each Voronoi polyhedron, and the pore size distribution in each Voronoi/Delaunay subunit is systematically carried out. The results show that with the increasing density of sphere packing, the Voronoi//Delaunay pore size distribution is narrowed. That indicates large pores to be gradually substituted by small uniformed ones during densification. Meanwhile, the distributions of face number, total per/meter, and surface area of Voronoi polyhedra at high packing densities tend to be narrower and higher, which is in good agreement with those in random loose packing.     

Growth of gas phase nanoparticles with an accretion mechanism

Nano-droplet growth in a supersaturated vapor has been investigated in a gas aggregation source using laser-ionization time-of-flight mass spectrometry. During its propagation into an atomic vapor, a small particle grows by sticking atoms on its surface. This accretion process has been highlighted through the clustering of homogeneous particles M_n and heterogeneous M_n(M₂O) and M_n(MOH)₂ particles in a metallic vapor and a helium buffer gas (M = Na or K). A modelization is introduced so as to connect the measured cluster mass distributions to the pertinent physical parameters. The mass distribution width is particularly sensitive to the efficiency of the first steps in the growth sequence. We used this property to compare the ability of this vapor-condensed matter phase transition to occur around various homogeneous and heterogeneous nucleation seeds.     

Particle Sampling and Real Time Size Distribution Measurement in H2/O2/TEOS Diffusion Flame

Growth characteristics of silica particles have been studied experimentally using in situ particle sampling technique from H₂/O₂/Tetraethylorthosilicate (TEOS) diffusion flame with carefully devised sampling probe. The particle morphology and the size comparisons are made between the particles sampled by the local thermophoretic method from the inside of the flame and by the electrostatic collector sampling method after the dilution sampling probe. The Transmission Electron Microscope (TEM) image processed data of these two sampling techniques are compared with Scanning Mobility Particle Sizer (SMPS) measurement. TEM image analysis of two sampling methods showed a good agreement with SMPS measurement. The effects of flame conditions and TEOS flow rates on silica particle size distributions are also investigated using the new particle dilution sampling probe. It is found that the particle size distribution characteristics and morphology are mostly governed by the coagulation process and sintering process in the flame. As the flame temperature increases, the effect of coalescence or sintering becomes an important particle growth mechanism which reduces the coagulation process. However, if the flame temperature is not high enough to sinter the aggregated particles then the coagulation process is a dominant particle growth mechanism. In a certain flame condition a secondary particle formation is observed which results in a bimodal particle size distribution.     

Sizing Polydisperse Dispersions by Focused Beam Reflectance and Small Angle Static Light Scattering

In this work, two different methods for particle characterization, namely focused beam reflectance and small angle static light scattering, are quantitatively compared. The results are presented in the form of moment ratios of the particle size distribution, i.e., the number weighted diameter, D_1/0, and the volume weighted diameter, D_4/3, for a broad range of particle size distributions ranging from 20 to 400 μm. Various aqueous dispersions including narrow, broad, and bimodal particle size distributions of spherical shaped ceramic beads were used in the comparison. It was found that the moment ratios obtained by focused beam reflectance measurements and small angle static light scattering correlate well, in the case of spherical particles. Furthermore, it was found that the D_1/0 values obtained by focused beam reflectance measurements are more sensitive to the presence of a small fraction of fine particles in a bimodal distribution than those obtained by small angle static light scattering.     

Optimal Estimation of Qubit States with Continuous Time Measurements

We propose an adaptive, two step strategy, for the estimation of mixed qubit states. We show that the strategy is optimal in a local minimax sense for the trace norm distance as well as other locally quadratic figures of merit. Local minimax optimality means that given n identical qubits, there exists no estimator which can perform better than the proposed estimator on a neighborhood of size n ^−1/2 of an arbitrary state. In particular, it is asymptotically Bayesian optimal for a large class of prior distributions. We present a physical implementation of the optimal estimation strategy based on continuous time measurements in a field that couples with the qubits. The crucial ingredient of the result is the concept of local asymptotic normality (or LAN) for qubits. This means that, for large n, the statistical model described by n identically prepared qubits is locally equivalent to a model with only a classical Gaussian distribution and a Gaussian state of a quantum harmonic oscillator. The term ‘local’ refers to a shrinking neighborhood around a fixed state ρ ₀. An essential result is that the neighborhood radius can be chosen arbitrarily close to n ^−1/4. This allows us to use a two step procedure by which we first localize the state within a smaller neighborhood of radius n ^−1/2+ϵ, and then use LAN to perform optimal estimation.     

Cluster growth poised on the edge of break-up: Size distributions with small exponent power-laws

In many naturally occurring growth processes, cluster size distributions of power-law form n(s)∝s^−τ with small exponents 0<τ<1 are observed. We suggest here that such distributions emerge naturally from cluster growth, where size dependent aggregation is counterbalanced by size dependent break-up. The model used in the study is a simple reaction kinetic model including only monomer-cluster processes. It is shown that under such conditions power-law size distributions with small exponents are obtained. Therefore, the results suggest that the ubiquity of small exponent power-law distributions is related to the growth process, where aggregation driven cluster growth is poised on the edge of cluster break-up.     

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.     

Magnetic Properties of Dipolar Chains in Ferrofluids

We have investigated the dipole interaction energies per particle and the local dipole field distributions in a frozen-magnetization model of a ferrofluid chain in a saturating magnetic field. A lognormal distribution of particle diameters was assumed. The interaction energies were calculated for one-dimensional arrays of dipoles with moments parallel to the chain. We have computed the energies by various approximations related to the hard sphere particle diameter distribution. A similar approach was followed for the local field distributions. It was found that the energy per particle and mean local field were largely determined by the mean particle diameter, but the distribution of local fields was sensitive to both the mean diameter and the assumptions about spatial correlations between particles of different size. Detailed results are presented for water-soluble Fe₃O₄/PAA (polyacrylic acid).     

Structural properties of amorphous Fe2O3 nanoparticles

We have investigated the microstructure of amorphous Fe₂O₃ nanoparticles by using molecular dynamics (MD) simulations. Non-periodic boundary conditions with Born-Mayer type pair potentials were used to simulate a spherical model of different diameters of 2, 3, 4 and 5 nm. Structural properties of an amorphous model obtained at 350 K have been analyzed in detail through the partial radial distribution functions (PRPFs), coordination number distributions, bond-angle distributions and interatomic distances. Calculations showed that structural characteristics of the model are in qualitative agreement with the experimental data. The observation of a large amount of structural defects as the particle size is decreased suggested that surface structure strongly depends on the size of nanoparticles. In addition, surface structure of amorphous Fe₂O₃ nanoparticles have been studied and compared with that observed in the core and in the bulk counterpart. Radial density profiles and stoichiometry in morphous Fe₂O₃ nanoparticles were also found and discussed.     

Dynamically tessellating algorithm for analysis of pore size distribution in particle agglomerates

We describe a novel physical application of the OctTree data structure [P. Meagher, Comput. Graphics Image Process 19(2) (1982) 129–147] in a dynamically tessellating algorithm, in conjunction with an object-oriented, constructive solid geometry library (DOC), to efficiently determine pore size distributions in large multi-particle systems. We apply the DOC library to investigate the evolving dynamics of pore formation in multi-particle systems, such as a mixture of smooth hard cubes and spheres and a collection of frictional soft spheres. We demonstrate that the algorithm is able to provide insight into the effect of structural changes on the porosity network; for example, during the uniaxial compaction of soft spheres, we find the number density of pores increases while the mean volume of the pores decreases. This trend is responsible for a shift in the distribution of the pore volumes to favour smaller volumes. We anticipate that the DOC method will have wider applications in the area of granular materials for studying the changes in pore structure in both experimental and numerical systems as a complement to the analysis of particle packing.     

Interatomic potential effects on dynamical heterogeneities in liquid SiO2

Dynamical heterogeneities (DH) in low density liquid SiO₂ have been investigated by molecular dynamics (MD) method. Simulations were done in the basic cube under periodic boundary conditions containing 3000 particles with the pair interatomic potentials, which have a weak electrostatic interaction and a Morse type short range interaction (PMSI). We have evaluated the non-Gaussian parameter for the self part of the van Hove correlation function and we found a clear evidence of the existence of DH in low density liquid SiO₂. Moreover, the atomic displacement distribution (ADD) in a model has been obtained and it deviates from a Gaussian form. The results have been compared with those obtained in another liquid SiO₂ system with the Born-Mayer interatomic potentials (BMP) in order to observe the interatomic potential effects on the DH in the system and indeed, the effects are strong. Calculations showed that particles of extremely low or fast mobility have a tendency to form a cluster and mean cluster size of most mobile and immobile particles in PMSI models increases with decreasing temperature. In contrast, no systematic changes have been obtained for the most mobile and immobile particles in BMP models. Calculations show that there is no relation between local particle environment and particle mobility in the system.     

Magnetic properties of prussian blue modified Fe3O4 nanocubes

Size controlled cubic Fe₃O₄ nanoparticles in the size range 90–10 nm were synthesized by varying the ferric ion concentration using the oxidation method. A bimodal size distribution was found without ferric ion concentration and the monodispersity increased with higher concentration. The saturation magnetization decreased from 90 to 62 emu/g when the particle size is reduced to 10 nm. The Fe₃O₄ nanoparticles with average particle sizes 10 and 90 nm were surface modified with prussian blue. The attachment of prussian blue with Fe₃O₄ was found to depend on the concentration of HCl and the particle size. The saturation magnetization of prussian blue modified Fe₃O₄ varied from 10 to 80 emu/g depending on the particle size. The increased tendency for the attachment of prussian blue with smaller particle size was explained based on the surface charge. The prussian blue modified magnetite nanoparticles could be used as a radiotoxin remover in detoxification applications.     

Mean Particle Diameters. Part IV: Empirical Selection of the Proper Type of Mean Particle Diameter Describing a Product or Material Property

Mean particle diameters may be used to describe and to model physical, chemical or physiological properties of products or materials containing dispersed phases. An empirical method was developed to select the proper type of mean diameter from experimental data, if this mean diameter is not known a priori from theoretical reasoning. The present method uses mean diameters, D?_{p, q}, defined according to the Moment‐Ratio (M‐R) definition system. They are expressed as the 1/(p‐q)‐th power of the ratio of the p‐th and the q‐th raw moment of the number density distribution of the particle sizes. After calculation of the mean diameters, D?_{p, q}, the relationships between the product property and these mean diameters are investigated statistically. The selection method has been illustrated by four examples, three of which stem from a high shear granulation experiment in the field of detergent processing. The fourth example is concerned with a visual ranking of bubble size distributions of chocolate mousse samples. The data set of each example consists of a set of particle size distributions and the corresponding physical product properties that are influenced by the particle sizes. Hypotheses are formulated to explain the types of selected mean diameters. Application of the selection method gives mean diameters, D?_{p, q}, a clear physical look and identity, replacing their anonymity. Sharing worldwide results of applications of the newly developed selection method, will lead to a build‐up of knowledge of physical meanings and application areas for the types of the mean particle diameters. This will support decision‐making in product development. The examples used to develop the selection method clearly demonstrate the physical relevance of the previously developed nomenclature system for mean particle diameters, D?_{p, q}.     

Size distributions and synthesis of nanoparticles by photolytic dissociation of ferrocene

Iron-containing nanoparticles were made by laser-assisted (ArF excimer laser, λ=193 nm) photolytic dissociation of ferrocene (Fe(C₅H₅)₂ or FeCp₂) in argon and an oxygen/argon gas mixture. The particle-size distributions were obtained on-line by using differential mobility analysers (DMAs) and were found to be log-normal with a geometric standard deviation of 1.85. In argon, particle sizes between 3 and 100 nm were generated. The volumes of these particles were found to increase linearly with the increased repetition rate, fluence and beam size of the laser. These observations are explained on the basis of the residence-time approach model. Received: 23 November 1999 / Accepted: 19 September 2000 / Published online: 22 November 2000     

The metamorphosis in the emission angular profile from an inverted two-level atoms system

A rich variety of angular distributions in the cooperative emission from a sphere of inverted N two-level atoms are shown to result from the eigenstructure of the complex kernel of scalar photon theory exp(ik₀R)/(ik₀R). This angular distribution is sensitive both to the size of the sphere and to the instant of observation of the emission.     

Fluctuations of the Nodal Length of Random Spherical Harmonics

Using the multiplicities of the Laplace eigenspace on the sphere (the space of spherical harmonics) we endow the space with Gaussian probability measure. This induces a notion of random Gaussian spherical harmonics of degree n having Laplace eigenvalue E = n(n + 1). We study the length distribution of the nodal lines of random spherical harmonics.     

Simulation of bimodal size distributions for coarsening

We introduce two very simple models to decribe coarsening. Both models possesses two characteristics: long range diffusive interaction and fluctuations. Computer simulations show that they lead to bimodal particle size distributions. The appearance of bimodal distribution can be related to a discontinuous particle growth rate.     

Generation of CdS clusters using laser ablation: the role of wavelength and fluence

The formation of cationic clusters in the laser ablation of CdS targets has been investigated as a function of wavelength and fluence by mass spectrometric analysis of the plume. Ablation was carried out at the laser wavelengths of 1064, 532, 355, and 266 nm in order to scan the interaction regimes below and above the energy band gap of the material. In all cases, the mass spectra showed stoichiometric Cd _n S _n ⁺ and nonstoichiometric Cd _n S _n−1⁺, Cd _n S _n+1⁺, and Cd _n S _n+2⁺ clusters up to 4900 amu. Cluster size distributions were well represented by a log-normal function, although larger relative abundance for clusters with n=13, 16, 19, 34 was observed (magic numbers). The laser threshold fluence for cluster observation was strongly dependent on wavelength, ranging from around 16 mJ/cm² at 266 nm to more than 300 mJ/cm² at 532 and 1064 nm. According to the behavior of the detected species as a function of fluence, two distinct families were identified: the “light” family containing S₂⁺ and Cd⁺ and the “heavy” clusterized family grouping Cd₂⁺ and Cd _n S _m ⁺. In terms of fluence, it has been determined that the best ratio for clusterization is achieved close to the threshold of appearance of clusters at all wavelengths. At 1064, 532, and 355 nm, the production of “heavy” cations as a function of fluence showed a maximum, indicating the participation of competitive effects, whereas saturation is observed at 266 nm. In terms of relative production, the contribution of the “heavy” family to the total cation signal was significantly lower for 266 nm than for the longer wavelengths. Irradiation at 355 nm in the fluence region of 200 mJ/cm² has been identified as the optimum for the generation of large clusters in CdS.