Dynamics of crystal size distributions with size-dependent rates

The growth and dissolution dynamics of nonequilibrium crystal size distributions (CSDs) can be determined by solving the governing population balance equations (PBEs) representing reversible addition or dissociation. New PBEs are considered that intrinsically incorporate growth dispersion and yield complete CSDs. We present two approaches to solving the PBEs, a moment method and a numerical scheme. The results of the numerical scheme agree with the moment technique, which can be solved exactly when powers on mass-dependent growth and dissolution rate coefficients are either zero or one. The numerical scheme is more general and can be applied when the powers of the rate coefficients are non-integers or greater than unity. The influence of the size dependent rates on the time variation of the CSDs indicates that as equilibrium is approached, the CSDs become narrow when the exponent on the growth rate is less than the exponent on the dissolution rate. If the exponent on the growth rate is greater than the exponent on the dissolution rate, then the polydispersity continues to broaden. The computation method applies for crystals large enough that interfacial stability issues, such as ripening, can be neglected. (C) 2002 Elsevier Science B.V. All rights reserved.

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.

Vaporization and collision modeling of liquid fuel sprays in a co-axial fuel and air pre-mixer

Droplet collision occurs frequently in regions where the droplet number density is high. Even for Lean Premixed and Pre-vaporized (LPP) liquid sprays, the collision effects can be very high on the droplet size distributions, which will in turn affect the droplet vaporization process. Hence, in conjunction with vaporization modeling, collision modeling for such spray systems is also essential. The standard O'Rourke's collision model, usually implemented in CFD codes, tends to generate unphysical numerical artifact when simulations are performed on Cartesian grid and the results are not grid independent. Thus, a new collision modeling approach based on no-time-counter method (NTC) proposed by Schmidt and Rutland is implemented to replace O'Rourke's collision algorithm to solve a spray injection problem in a cylindrical coflow premixer. The so called ``four-leaf clover'' numerical artifacts are eliminated by the new collision algorithm and results from a diesel spray show very good grid independence. Next, the dispersion and vaporization processes for liquid fuel sprays are simulated in a coflow premixer. Two liquid fuels under investigation are jet-A and Rapeseed Methyl Esters (RME). Results show very good grid independence in terms of SMD distribution, droplet number distribution and fuel vapor mass flow rate. A baseline test is first established with a spray cone angle of 90 degrees and injection velocity of 3 m/s and jet-A achieves much better vaporization performance than RME due to its higher vapor pressure. To improve the vaporization performance for both fuels, a series of simulations have been done at several different combinations of spray cone angle and injection velocity. At relatively low spray cone angle and injection velocity, the collision effect on the average droplet size and the vaporization performance are very high due to relatively high coalescence rate induced by droplet collisions. Thus, at higher spray cone angle and injection velocity, the results expectedly show improvement in fuel vaporization performance since smaller droplet has a higher vaporization rate. The vaporization performance and the level of homogeneity of fuel-air mixture can be significantly improved when the dispersion level is high, which can be achieved by increasing the spray cone angle and injection velocity. (C) 2012 Elsevier Ltd. All rights reserved.

On the Two-Step Mechanism for Synthesis of Transition-Metal Nanoparticles

The two-step particle synthesis mechanism, also known as the Finke-Watzky (1997) mechanism, has emerged as a significant development in the field of nanoparticle synthesis. It explains a characteristic feature of the synthesis of transition metal nanoparticles, an induction period in precursor concentration followed by its rapid sigmoidal decrease. The classical LaMer theory (1950) of particle formation fails to capture this behavior. The two-step mechanism considers slow continuous nucleation and autocatalytic growth of particles directly from precursor as its two kinetic steps. In the present work, we test the two-step mechanism rigorously using population balance models. We find that it explains precursor consumption very well, but fails to explain particle synthesis. The effect of continued nucleation on particle synthesis is not suppressed sufficiently by the rapid autocatalytic growth of particles. The nucleation continues to increase breadth of size distributions to unexpectedly large values as compared to those observed experimentally. A number of variations of the original mechanism with additional reaction steps are investigated next. The simulations show that continued nucleation from the beginning of the synthesis leads to formation of highly polydisperse particles in all of the tested cases. A short nucleation window, realized with delayed onset of nucleation and its suppression soon after in one of the variations, appears as one way to explain all of the known experimental observations. The present investigations clearly establish the need to revisit the two-step particle synthesis mechanism.

Coalescence of drops in stirred dispersion. A white noise model for coalescence

Coalescence between two droplets in a turbulent liquid-liquid dispersion is generally viewed as a consequence of forces exerted on the drop-pair squeezing out the intervening continuous phase to a critical thickness. A new synthesis is proposed herein which models the film drainage as a stochastic process driven by a suitably idealized random process for the fluctuating force. While the true test of the model lies in detailed parameter estimations with measurement of drop-size distributions in coalescing dispersions, experimental measurements on average coalescence frequencies lend preliminary support to the model.

Optical and Physical Properties of Atmospheric Aerosols over the Bay of Bengal during ICARB

Simultaneous and collocated measurements of total and hemispherical backscattering coefficients (σ and β, respectively) at three wavelengths, mass size distributions, and columnar spectral aerosol optical depth (AOD) were made onboard an extensive cruise experiment covering, for the first time, the entire Bay of Bengal (BoB) and northern Indian Ocean. The results are synthesized to understand the optical properties of aerosols in the marine atmospheric boundary layer and their dependence on the size distribution. The observations revealed distinct spatial and spectral variations of all the aerosol parameters over the BoB and the presence of strong latitudinal gradients. The size distributions varied spatially, with the majority of accumulation modes decreasing from north to south. The scattering coefficient decreased from very high values (resembling those reported for continental/urban locations) in the northern BoB to very low values seen over near-pristine environments in the southeastern BoB. The average mass scattering efficiency of BoB aerosols was found to be 2.66 ± 0.1 m2 g−1 at 550 nm. The spectral dependence of columnar AOD deviated significantly from that of the scattering coefficients in the northern BoB, implying vertical heterogeneity in the aerosol type in that region. However, a more homogeneous scenario was observed in the southern BoB. Simultaneous lidar and in situ measurements onboard an aircraft over the ocean revealed the presence of elevated aerosol layers of enhanced extinction at altitudes of 1 to 3 km with an offshore extent of a few hundred kilometers. Back-trajectory analyses showed these layers to be associated with advection from west Asia and western India. The large spatial variations and vertical heterogeneity in aerosol properties, revealed by the present study, need to be included in the regional radiative forcing over the Bay of Bengal.

Analysis of microstructure and texture evolution in pure magnesium during symmetric and asymmetric rolling

Asymmetric rolling of commercially pure magnesium was carried out at three different temperatures: room temperature, 200 degrees C and 350 degrees C. Systematic analysis of microstructures, grain size distributions, texture and misorientation distributions were performed using electron backscattered diffraction in a field emission gun scanning electron microscope. The results were compared with conventional (symmetric) rolling carried out under the same conditions of temperature and strain rate. Simulations of deformation texture evolution were performed using the viscoplastic self-consistent polycrystal plasticity model. The main trends of texture evolution are faithfully reproduced by the simulations for the tests at room temperature. The deviations that appear for the textures obtained at high temperature can be explained by the occurrence of dynamic recrystallization. Finally, the mechanisms of texture evolution in magnesium during asymmetric and symmetric rolling are explained with the help of ideal orientations, grain velocity fields and divergence maps displayed in orientation space.

High Rate Capability of a Dual-Porosity LiFePO4/C Composite

Mesoporous intercalation compounds consisting of two differentdistributions of pores represent a potentially attractive material for high-rate cathodes. A mesoporous LiFePO4/C composite with two sizes of pores is prepared for the first time via a solution-based polymer templating technique. The precursor of the LiFePO4/C composite is heated at different temperatures in the range from 600 to 800 degrees C to study the effect of crystallinity, porosity, and morphology on the electrochemical performance. The composite is found to attain reduction in the surface area, carbon content, and porosity upon increasing temperature. Nonetheless, the composite prepared at 700 degrees C with pore-size distributions of around 4 and 50 nm exhibits a high rate capability and stable capacity retention upon cycling.

The effect of cooling rate on the structure and properties of closed-cell aluminium foams

The application of different cooling rates as a strategy to enhance the structure of aluminium foams is studied. The potential to influence the level of morphological defects and cell size non-uniformities is investigated. AlSi6Cu4 alloy was foamed through the powder compact route and then solidified, applying three different cooling rates. Foam development was monitored in situ by means of X-ray radioscopy while foaming inside a closed mould. The macro-structure of the foams was analysed in terms of cell size distribution as determined by X-ray tomography. Compression tests were conducted to assess the mechanical performance of the foams and measured properties were correlated with structural features of the foams. Moreover, possible changes in the ductile brittle nature of deformation with cooling rate were analysed by studying the initial stages of deformation. We observed improvements in the cell size distributions, reduction in microporosity and grain size at higher cooling rates, which in turn led to a notable enhancement in compressive strength. (C) 2010 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

A new model for the breakage frequency of drops in turbulent stirred dispersions

A model of drop breakage in turbulent stirred dispersions based on interaction of a drop with eddies of a length scale smaller than the drop diameter has been developed. It predicts that, unlike the equal breakage assumed by earlier models, a large drop reduces in size due to stripping of smaller segments off it through unequal breakage. It is only when the drop nears the value of the maximum stable drop diameter that it breaks into equal parts. This new model of drop breakage, coupled with the pattern of interaction of drops with eddies of different sizes existing in the vessel, has been used to evaluate not only the breakage frequency, but also the size distribution of the daughter droplets(which was hitherto assumed). The model has been incorporated in the population balance equation and the resulting cumulative size distributions compared with those availble in the literature.

An investigation of well-characterized small gold clusters by photoelectron spectroscopy, tunneling spectroscopy, and cognate techniques

After microscopic characterization of the size distributions of gold clusters, deposited on carbon substrates by vacuum evaporation or by soft landing, Au(4f') binding energy of the clusters has been measured as a function of the mean cluster size. Similar measurements have been carried out on Au clusters prepared from sols by chemical means and high-nuclearity cluster compounds. In general, small clusters with a mean diameter of $2 nm show significantly larger binding energies than the bulk metal value, due to the onset of nonmetallicity. Nonmetallicity manifests itself in terms of a tunneling conductance gap only in clusters of diameter ;5 1 nm containing 40 atoms or fewer.

On the spurious increase in particle volume fraction during coarsening simulations

The problem of spurious increase in volume fraction of second-phase particles during computer simulations of coarsening is examined. The origin of this problem is traced to the use of too long a time step (used for numerical integration of growth rates with respect to time) which leads to small particles with large negative growth rates shrinking to negative radii at the end of the time step. Such a shrinkage to negative sizes has the effect of pumping solute into the system. It is therefore suggested that the length of the time step be chosen in accordance with the size of the smallest particle present in the system. It is shown that spurious increase in particle Volume has a significant effect on the particle size distributions in the scaling regime (making them broader and more skewed in the Lifshitz-Slyozov-Wagner model). Its effect on coarsening kinetics, however, is found to be small.

Growth of Nanometric Gold Particles in Solution Phase

The time evolution of colloidal gold particles in the nanometric regime has been investigated by employing electron microscopy and electronic absorption spectroscopy. The particle size distributions are essentially Gaussian and show the same time dependence for both the mean and the standard deviation, enabling us to obtain a time-independent universal curve for the particle size. Temperature dependent studies show the growth to be an activated process with a barrier of about 18 kJ mol(-1). We present a phenomenological equation for the evolution of particle size and suggest that the growth process is stochastic.

Phase diagram of a two-species lattice model with a linear instability

We discuss the properties of a one-dimensional lattice model of a driven system with two species of particles in which the mobility of one species depends on the density of the other. This model was introduced by Lahiri and Ramaswamy (Phys. Rev. Lett., 79, 1150 (1997)) in the context of sedimenting colloidal crystals, and its continuum version was shown to exhibit an instability arising from linear gradient couplings. In this paper we review recent progress in understanding the full phase diagram of the model. There are three phases. In the first, the steady state can be determined exactly along a representative locus using the condition of detailed balance. The system shows phase separation of an exceptionally robust sort, termed strong phase separation, which survives at all temperatures. The second phase arises in the threshold case where the first species evolves independently of the second, but the fluctuations of the first influence the evolution of the second, as in the passive scalar problem. The second species then shows phase separation of a delicate sort, in which long-range order coexists with fluctuations which do not damp down in the large-size limit. This fluctuation-dominated phase ordering is associated with power law decays in cluster size distributions and a breakdown of the Porod law. The third phase is one with a uniform overall density, and along a representative locus the steady state is shown to have product measure form. Density fluctuations are transported by two kinematic waves, each involving both species and coupled at the nonlinear level. Their dissipation properties are governed by the symmetries of these couplings, which depend on the overall densities. In the most interesting case,, the dissipation of the two modes is characterized by different critical exponents, despite the nonlinear coupling.

Continued self-similar breakup of drops in viscous continuous phase in agitated vessels

Drop breakup inviscous liquids in agitated vessels occurs in elongational flow around impeller blade edges. The drop size distributions measured over extended periods for impellers of different sizes show that breakup process continues up to 15-20 h, before a steady state is reached. The size distributions evolve in a self-similar way till the steady state is reached. The scaled size distributions vary with impeller size and impeller speed, in contrast with the near universal scaling known for drop breakup in turbulent flows. The steady state size of the largest drop follows inverse scaling with impeller tip velocity. The breadth of the scaled size distributions also shows a monotonic relationship with impeller tip velocity only. (C) 2011 Elsevier Ltd. All rights reserved.