Mean Particle Number Density Research Articles

Iron Oxide Nanocubes as a New Certified Reference Material for Nanoparticle Size Measurements.

The rational design and increasing industrial use of nanomaterials require a reliable characterization of their physicochemical key properties like size, size distribution, shape, and surface chemistry. This calls for nanoscale reference materials (nanoRMs) for the validation and standardization of commonly used characterization methods closely matching real-world nonspherical nano-objects. This encouraged us to develop a nonspherical nanoRM of very small size consisting of 8 nm iron oxide nanocubes (BAM-N012) to complement spherical gold, silica, and polymer nanoRMs. In the following, the development and production of this nanoRM are highlighted including the characterization by transmission electron microscopy (TEM) and small-angle X-ray scattering (SAXS) as complementary methods for size and shape parameters, homogeneity and stability studies, and calculation of a complete uncertainty budget of the size features. The determination of the nanocubes' edge length by TEM and SAXS allows a method comparison. In addition, SAXS measurements can also provide the mean particle number density and the mass concentration. The certified size parameters, area equivalent circular diameter and square edge length, determined by TEM with a relative expanded uncertainty below 9%, are metrologically traceable to a natural constant for length, the very precisely known (111) lattice spacing of silicon. Cubic BAM-N012 qualifies as a certified nanoRM for estimating the precision and trueness, validation, and quality assurance of particle size and shape measurements with electron microscopy and SAXS as well as other sizing methods suitable for nanomaterials. The production of this new iron oxide nanocube RM presents an important achievement for the nanomaterial community, nanomaterial manufacturers, and regulators.

Active particles with polar alignment in ring-shaped confinement.

We study steady-state properties of active, nonchiral and chiral Brownian particles with polar alignment and steric interactions confined within a ring-shaped confinement (annulus) in two dimensions. Exploring possible interplays between polar interparticle alignment, geometric confinement and the surface curvature, being incorporated here on minimal levels, we report a surface-population reversal effect, whereby active particles migrate from the outer concave boundary of the annulus to accumulate on its inner convex boundary. This contrasts the conventional picture, implying stronger accumulation of active particles on concave boundaries relative to the convex ones. The population reversal is caused by both particle alignment and surface curvature, disappearing when either of these factors is absent. We explore the ensuing consequences for the chirality-induced current and swim pressure of active particles and analyze possible roles of system parameters, such as the mean number density of particles and particle self-propulsion, chirality, and alignment strengths.

Compressibility effects in turbulent transport of the temperature field.

Compressibility effects in a turbulent transport of temperature field are investigated by applying the quasilinear approach for small Péclet numbers and the spectral τ approach for large Péclet numbers. The compressibility of a fluid flow reduces the turbulent diffusivity of the mean temperature field similarly to that for the particle number density and magnetic field. However, expressions for the turbulent diffusion coefficient for the mean temperature field in a compressible turbulence are different from those for the mean particle number density and the mean magnetic field. The combined effect of compressibility and inhomogeneity of turbulence causes an increase of the mean temperature in the regions with more intense velocity fluctuations due to a turbulent pumping. Formally, this effect is similar to a phenomenon of compressible turbophoresis found previously [J. Plasma Phys. 84, 735840502 (2018)JPLPBZ0022-377810.1017/S0022377818000983] for noninertial particles or gaseous admixtures. The gradient of the mean fluid pressure results in an additional turbulent pumping of the mean temperature field. The latter effect is similar to the turbulent barodiffusion of particles and gaseous admixtures. The compressibility of a fluid flow also causes a turbulent cooling of the surrounding fluid due to an additional sink term in the equation for the mean temperature field. There is no analog of this effect for particles.

Turbulent thermal diffusion in strongly stratified turbulence: Theory and experiments

Turbulent thermal diffusion is a combined effect of the temperature stratified turbulence and inertia of small particles. It causes the appearance of a non-diffusive turbulent flux of particles in the direction of the turbulent heat flux. This non-diffusive turbulent flux of particles is proportional to the product of the mean particle number density and the effective velocity of inertial particles. The theory of this effect has been previously developed only for small temperature gradients and small Stokes numbers (Phys. Rev. Lett. {\bf 76}, 224, 1996). In this study a generalized theory of turbulent thermal diffusion for arbitrary temperature gradients and Stokes numbers has been developed. The laboratory experiments in the oscillating grid turbulence and in the multi-fan produced turbulence have been performed to validate the theory of turbulent thermal diffusion in strongly stratified turbulent flows. It has been shown that the ratio of the effective velocity of inertial particles to the characteristic vertical turbulent velocity for large Reynolds numbers is less than 1. The effective velocity of inertial particles as well as the effective coefficient of turbulent thermal diffusion increase with Stokes numbers reaching the maximum at small Stokes numbers and decreases for larger Stokes numbers. The effective coefficient of turbulent thermal diffusion also decreases with the mean temperature gradient. It has been demonstrated that the developed theory is in a good agreement with the results of the laboratory experiments.

Sampling artifact in volume weighted velocity measurement. II. Detection in simulations and comparison with theoretical modeling

Measuring the volume weighted velocity power spectrum suffers from a severe systematic error, due to imperfect sampling of the velocity field from inhomogeneous distribution of dark matter particles/halos in simulations or galaxies with velocity measurement. This "sampling artifact" depends on both the mean particle number density $\bar{n}_P$ and the intrinsic large scale structure (LSS) fluctuation in the particle distribution. (1) We report robust detection of this sampling artifact in N-body simulations. It causes $\sim 12$% underestimation of the velocity power spectrum at $k=0.1$h/Mpc for samples with $\bar{n}_P=6\times10^{-3}$ (Mpc/h)$^{-3}$. This systematic underestimation increases with decreasing $\bar{n}_P$ and increasing $k$. Its dependence on the intrinsic LSS fluctuations is also robustly detected. (2) All these findings are expected by our theoretical modelling in paper I \cite{Zhang14}. In particular, the leading order theoretical approximation agrees quantitatively well with simulation result for $\bar{n}_P\gtrsim6\times 10^{-4}$(Mpc/h)$^{-3}$. Furthermore, we provide an ansatz to take high order terms into account. It improves the model accuracy to $\lesssim1$% at $k\lesssim0.1$h/Mpc over 3 orders of magnitude in $\bar{n}_P$ and over typical LSS clustering from $z=0$ to $z=2$. (3) The sampling artifact is determined by the deflection ${\bf D}$ field, which is straightforwardly available in both simulations and data of galaxy velocity. Hence the sampling artifact in the velocity power spectrum measurement can be self-calibrated within our framework. By applying such self-calibration in simulations, it becomes promising to determine the {\it real} large scale velocity bias of $10^{13}M_\odot$ halos with $\sim 1$% accuracy, and that of lower mass halos by better accuracy. ...[abridged]

Tangling clustering instability for small particles in temperature stratified turbulence

We study tangling clustering instability of inertial particles in a temperature stratified turbulence with small finite correlation time. It is shown that the tangling mechanism in the temperature stratified turbulence strongly increases the degree of compressibility of particle velocity field. This results in the strong decrease of the threshold for the excitation of the tangling clustering instability even for small particles. The tangling clustering instability in the temperature stratified turbulence is essentially different from the inertial clustering instability that occurs in non-stratified isotropic and homogeneous turbulence. While the inertial clustering instability is caused by the centrifugal effect of the turbulent eddies, the mechanism of the tangling clustering instability is related to the temperature fluctuations generated by the tangling of the mean temperature gradient by the velocity fluctuations. Temperature fluctuations produce pressure fluctuations and cause particle accumulations in regions with increased instantaneous pressure. It is shown that the growth rate of the tangling clustering instability is by \documentclass[12pt]{minimal}\begin{document}$\sqrt{\rm Re} \, (\ell _0 / L_T)^2 / (3 {\rm Ma})^4$\end{document} Re (ℓ0/LT)2/(3 Ma )4 times larger than that of the inertial clustering instability, where Re is the Reynolds number, Ma is the Mach number, ℓ0 is the integral turbulence scale, and LT is the characteristic scale of the mean temperature variations. It is found that depending on the parameters of the turbulence and the mean temperature gradient there is a preferential particle size at which the particle clustering due to the tangling clustering instability is more effective. The particle number density inside the cluster after the saturation of this instability can be by several orders of magnitude larger than the mean particle number density. It is also demonstrated that the evaporation of droplets drastically changes the tangling clustering instability, e.g., it increases the instability threshold in the droplet radius. The tangling clustering instability is of a great importance, e.g., in atmospheric turbulence with temperature inversions.

Mixing at the external boundary of a submerged turbulent jet

We study experimentally and theoretically mixing at the external boundary of a submerged turbulent jet. In the experimental study we use particle image velocimetry and an image processing technique based on the analysis of the intensity of the Mie scattering to determine the spatial distribution of tracer particles. An air jet is seeded with the incense smoke particles, which are characterized by a large Schmidt number and a small Stokes number. We determine the spatial distributions of the jet fluid characterized by a high concentration of the particles and of the ambient fluid characterized by a low concentration of the tracer particles. In the data analysis we use two approaches, whereby one approach is based on the measured phase function for the study of the mixed state of two fluids. The other approach is based on the analysis of the two-point second-order correlation function of the particle number density fluctuations generated by tangling of the gradient of the mean particle number density by the turbulent velocity field. This gradient is formed at the external boundary of a submerged turbulent jet. We demonstrate that probability density function of the phase function of a jet fluid penetrating into an external flow and the two-point second-order correlation function of the particle number density do not have universal scaling and cannot be described by a power-law function. The theoretical predictions made in this study are in qualitative agreement with the obtained experimental results.

Large Eddy Simulation of a dilute particle-laden turbulent flow over a backward-facing step

Dilute gas-particle turbulent flows over a backward-facing step are numerically simulated by Large Eddy Simulation (LES) for the continuous phase and Lagrangian particle trajectory method for the particle phase. Predicted results of mean velocities and fluctuating velocities of both phases agree well with the experimental data, and demonstrate that the main characteristics of the flow are accurately captured by the simulations. Characteristics of separation and reattachments as well as essential features of the coherent structure are obtained, in which the processes of vortex roll up, growth, pairing and breaking up are shown in details. Particle dispersions are then investigated through particles’ instantaneous distributions in coherent structure as well as the mean and fluctuating properties of particle number density (PND). The predicted mean PND agree well with experiment results. For small particles, the instantaneous distributions show much preferential concentration, while their mean PND shows more uniform distribution in downstream region. On the contrary, for large particles, their instantaneous distributions are much uniform (without clear preferential concentration) due to less effect of large eddy coherent, while their mean PND across the section is not uniform for more particles are distributed in the main flow region. The preferential concentration of particles by the large-scale eddies can lead to a high fluctuating PND.

Mean-field theory for a passive scalar advected by a turbulent velocity field with a random renewal time.

Mean-field theory for turbulent transport of a passive scalar (e.g., particles and gases) is discussed. Equations for the mean number density of particles advected by a random velocity field, with a finite correlation time, are derived. Mean-field equations for a passive scalar comprise spatial derivatives of high orders due to the nonlocal nature of passive scalar transport in a random velocity field with a finite correlation time. A turbulent velocity field with a random renewal time is considered. This model is more realistic than that with a constant renewal time used by Elperin et al. [Phys. Rev. E 61, 2617 (2000)], and employs two characteristic times: the correlation time of a random velocity field tau(c), and a mean renewal time tau. It is demonstrated that the turbulent diffusion coefficient is determined by the minimum of the times tau(c) and tau. The mean-field equation for a passive scalar was derived for different ratios of tau/tau(c). The important role of the statistics of the field of Lagrangian trajectories in turbulent transport of a passive scalar, in a random velocity field with a finite correlation time, is demonstrated. It is shown that in the case tau(c)<<tau<<tau(N) the form of the mean-field equation for a passive scalar is independent of the statistics of the velocity field, where tau(N) is the characteristic time of variations of a mean passive scalar field.

Passive scalar transport in a random flow with a finite renewal time: Mean-field equations

A mean-field equation for a passive scalar (e.g., for a mean number density of particles) in a random velocity field (incompressible and compressible) with a finite constant renewal time is derived. The finite renewal time of a random velocity field results in the appearance of high-order spatial derivatives in the mean-field equation for a passive scalar. We considered three models of a random velocity field: (i) a velocity field with a small renewal time; (ii) the Gaussian approximation for Lagrangian trajectories; and (iii) a small inhomogeneity of the velocity and mean passive scalar fields. For a small renewal time we recovered results obtained using the $\ensuremath{\delta}$-function-correlated in time random velocity field. The finite renewal time and compressibility of the velocity field can cause a depletion of turbulent diffusion and a modification of an effective drift velocity. For a compressible velocity field the form of the mean-field equation for a passive scalar depends on the details of the velocity field, i.e., the universality is lost. For an incompressible velocity field the universality exists in spite of the finite renewal time. Results by Saffman [J. Fluid Mech. 8, 273 (1960)] for the effect of molecular diffusivity in turbulent diffusion are generalized for the case of a compressible and anisotropic random velocity field. The obtained results may be of relevance in some atmospheric phenomena (e.g., atmospheric aerosols and smog formation).

Precipitation in small systems—II. Mean field equations more effective than population balance

Part I (Manjunath et al., 1994, Chem. Engng Sci. 49, 1451–1463) of this paper showed that the random particle numbers and size distributions in precipitation processes in very small drops obtained by stochastic simulation techniques deviate substantially from the predictions of conventional population balance. The foregoing problem is considered in this paper in terms of a mean field approximation obtained by applying a first-order closure to an unclosed set of mean field equations presented in Part I. The mean field approximation consists of two mutually coupled partial differential equations featuring (i) the probability distribution for residual supersaturation and (ii) the mean number density of particles for each size and supersaturation from which all average properties and fluctuations can be calculated. The mean field equations have been solved by finite difference methods for (i) crystallization and (ii) precipitation of a metal hydroxide both occurring in a single drop of specified initial supersaturation. The results for the average number of particles, average residual supersaturation, the average size distribution, and fluctuations about the average values have been compared with those obtained by stochastic simulation techniques and by population balance. This comparison shows that the mean field predictions are substantially superior to those of population balance as judged by the close proximity of results from the former to those from stochastic simulations. The agreement is excellent for broad initial supersaturations at short times but deteriorates progressively at larger times. For steep initial supersaturation distributions, predictions of the mean field theory are not satisfactory thus calling for higher-order approximations. The merit of the mean field approximation over stochastic simulation lies in its potential to reduce expensive computation times involved in simulation. More effective computational techniques could not only enhance this advantage of the mean field approximation but also make it possible to use higher-order approximations eliminating the constraints under which the stochastic dynamics of the process can be predicted accurately.

On the universal growth of two-dimensional aggregates of hydrophobed glass beads formed at the (aqueous solution of electrolyte)—air interfaces

Two-dimensional aggregation of surface-modified glass beads was carried out at water (aqueous solution of electrolyte)—air interfaces. The effects of particle hydrophobicity and electrolyte concentration on the structure of the forming aggregates and on the growth process were investigated. The growth was characterized by density functions which demonstrate the change in mean particle number density as a function of aggregate size. The aggregation yielded fractal or non-fractal structures, depending on aggregate size. The fractal character for small aggregates was found to be dependent on particle hydrophobicity as well as on electrolyte concentration. However, above a certain electrolyte concentration, where restructuring of the growing aggregates was hindered slightly, practically the same values of fractal dimensions were obtained for investigated systems having different hydrophobicities. This result was attributed to the universal aspect of two-dimensional growth.