Aggregate size distributions in migration driven growth models

The kinetics of aggregate growth through reversible migrations between any two aggregates is studied. We propose a simple model with the symmetrical migration rate kernel at which the monomers migrate from the aggregates of size k to those of size j. The results show that for the case, the aggregate size distribution approaches a conventional scaling form; moreover, the typical aggregate size grows as in the case and as in the case. We also investigate another simple model with the asymmetrical rate kernel ( ), which exhibits some scaling properties quite different from the symmetrical one. The aggregate size distribution satisfies the conventional scaling form only in the case of and , and the typical aggregate size grows as .Received: 14 October 2003, Published online: 23 December 2003PACS: 82.20.-w Chemical kinetics and dynamics - 68.43.Jk Diffusion of adsorbates, kinetics of coarsening and aggregation - 89.75.Da Systems obeying scaling laws     

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.     

City size distributions for India and China

This paper studies the size distributions of urban agglomerations for India and China. We have estimated the scaling exponent for Zipf’s law with the Indian census data for the years of 1981-2001 and the Chinese census data for 1990 and 2000. Along with the biased linear fit estimate, the maximum likelihood estimate for the Pareto and Tsallis q-exponential distribution has been computed. For India, the scaling exponent is in the range of [1.88, 2.06] and for China, it is in the interval [1.82, 2.29]. The goodness-of-fit tests of the estimated distributions are performed using the Kolmogorov-Smirnov statistic.     

Dew nucleation and growth

Dew is the condensation of water vapor into liquid droplets on a substrate. It is characterized by an initial heterogeneous nucleation on a substrate and a further growth of droplets. The presence of a substrate that geometrically constrains the growth is the origin of the peculiarities and richness of the phenomenon. A key point is the drop interaction through drop fusion or coalescence, which leads to scaling in the growth and gives universality to the process. As a matter of fact, growth dynamics are only dependent on substrate and drop dimensionality. Coalescence events lead to temporal and spatio-temporal fluctuations in the substrate coverage, drop configuration, etc., which give rise to a very peculiar dynamics. When the substrate is a liquid or a liquid crystal, the drop pattern can exhibit special spatial order, such as crystalline, hexatic phases and fractal contours. Condensation on a solid substrate near its melting point can make the drop jump.The applications of monitoring dew formation are manifold. Examples can be found in medicine (sterilization process), agriculture (green houses) and hydrology (production of drinkable water). To cite this article: D. Beysens, C. R. Physique 7 (2006).     

An NMR protocol for determining ice crystal size distributions during freezing and pore size distributions during freeze-drying

Nuclear magnetic resonance water proton relaxometry is widely used to investigate pore size distributions and pore connectivity in brine-saturated porous rocks and construction materials. In this paper we show that, by replacing water with acetone, a similar method can be used to probe the porous structure of freeze-dried starch gels and therefore the ice crystal size distribution in frozen starch gels. The method relies on the observation that the starch surface acts as a powerful relaxation sink for acetone proton transverse magnetization so that Brownstein-Tarr theory can be used to extract the pore size distribution from the relaxation data. In addition the relaxation time distribution is found to depend on the spectrometer frequency and the Carr-Purcell-Meiboom-Gill pulse spacing, consistent with the existence of large susceptibility-induced field gradients within the pores. The potential of this approach for noninvasively measuring ice crystal size distributions during freezing and pore size distributions during freeze-drying in other food systems is discussed.     

Rate-equation approach to island capture zones and size distributions in epitaxial growth

Understanding and predicting the effects of correlations between island size and the rate of monomer capture has been shown to be the central problem in predicting the island-size distribution in submonolayer growth. Here we summarize a method which involves a self-consistent coupling of evolution equations for the capture-zone distributions with rate equations for the island-size distribution. The method has been successfully applied to irreversible submonolayer growth in both one and two dimensions to predict the size-dependent capture numbers and island-size distributions.     

Droplet size and velocity distributions for spray modelling

Methods for constructing droplet size distributions and droplet velocity profiles are examined as a basis for the Eulerian spray model proposed in Beck and Watkins (2002,2003) [5], [6]. Within the spray model, both distributions must be calculated at every control volume at every time-step where the spray is present and valid distributions must be guaranteed. Results show that the Maximum Entropy formalism combined with the Gamma distribution satisfy these conditions for the droplet size distributions. Approximating the droplet velocity profile is shown to be considerably more difficult due to the fact that it does not have compact support. An exponential model with a constrained exponent offers plausible profiles.     

Cluster size distributions in equilibrium

Equilibrium size distributions for finite system containingM monomers in a volumeV undergoing coagulation (polymerization)/fragmentation, are determined for coagulation rateK _ij and fragmentation rateF _ij , corresponding to the processA _i +A _j A _i+j (hereA _k denotes a cluster containingk monomeric units). It is shown that microscopic detailed balance impliesF _ij /K _ij =a _i a _j /a _i+j , where anda _k are arbitrary. The mean and most probable size distributionc _k = ^–1 a _k e ^–k coincide in the thermodynamic limitM, V, =M/V fixed. If , 2<<3, then the theory predicts a (solgel) phase transition at a well defined value ofq = ^–1, with critical exponents = and = –2. The gel fraction exponent assumes its classical value unity, probably due to the neglect of spatial fluctuations. Finally it is indicated how the theory can in principle be extended to account for isomerism and cyclization.     

Multidimensionally resolved pore size distributions

A novel method of determining median pore size and pore size distributions as a function of spatial position inside a porous sample is described. Pore sizes have been measured with 1-, 2- and 3-dimensional spatial resolution, using NMR cryoporometry in conjunction with magnetic resonance imaging techniques. The method is suitable for pore diameters in the range of 30 Å to over 2000 Å pore diameter, and is based on the technique of freezing a liquid in the pores and measuring the melting temperature by nuclear magnetic resonance. Since the melting point is depressed for crystals of small size, the melting point depression gives a measurement of pore size.     

Crystal growth of small-molecule organic semiconductors with nucleation additive

In this study, we employ a nucleation additive 4-octylbenzoic acid (OBA) with an optimized solvent evaporation method to regulate crystal orientation and grain width of small-molecule organic semiconductors. When 6,13-bis(triisopropylsilylethynyl) pentacene (TIPS pentacene) was utilized as a benchmark material to mix with the additive, a self-assembled OBA interfacial layer was formed and promoted uniform deposition of nucleation seeds. As a result, the TIPS pentacene/OBA blend crystalline film exhibited crystal alignment in long range order, attributing to a 11-fold reduction of the crystal misorientation angle and a 4-fold increase of the grain width. We further discussed the important correlation between the effective hole mobility, grain boundaries, grain width and length, and nucleation sites. Organic thin film transistors were fabricated to test charge transport, yielding a hole mobility of up to 0.17 cm²/V. This work provides a new pathway to modulate the nucleation and crystallization events of organic semiconductors, and can potentially be applied to optimize the thin film morphology and electrical performance of organic semiconducting materials in general.     

Grain size of MnBi films

MnBi films were prepared by annealing Bi-Mn composite films. The grain size of the MnBi films was strongly influenced by the temperatureT _s of the film substrate during evaporation of Bi and Mn. For 55-nm-thick films, the numberN of MnBi grains per mm² decreased from 4×10⁶ per mm² forT _s =+90°C to 1 per mm² forT _s =−40°C. The diameters of the largest grains observed were about 5 mm. For interpretation of this result, the formation process of MnBi films was observed magneto-optically and electronmicroscopically. It was found thatN was determined by the structure of Bi in the Bi-Mn composite films and that this strongly depended onT _s .     

Evolution of cluster size distribution during nucleation with rapidly changing dynamic gas processes

A non-steady-state solution of the Zel’dovich equation for the cluster size distribution function has been obtained. The solution is applicable in the entire range of cluster sizes. The relaxation time of the quasi-steady-state distribution and the partial time lags of cluster formation were derived and analyzed. The dependence of these times on cluster size was investigated. Expressions for the quasi-steady-state flux and the induction times have been derived. The accuracy of these calculations is confirmed by comparison with numerical calculations by other authors.     

Size selected growth of nanodots: effects of growth kinetics and energetics on the formation of stationary size distributions

The size selection of nanodots during the growth is studied by using a reaction kinetic model, where reaction rates depend on the dot size. The characteristic feature of the reaction rates is the energetics, where the free energy of dots has a minimum at the certain dot size. The model equations are solved by using a particle coalescence simulation method. We find phenomenologically three distinct stages of growth. First, during the initial deposition stage, distributions with high density of small dots occur. Second, there is an intermediate and short-lived stationary state, which is controlled by kinetics of growth. Third, a long-lived stationary state is obtained, with nearly Gaussian size distributions, mostly determined by the energetics of the growth but also significantly affected by the kinetics. In the final stage, size selection and narrowing of the distributions occur. It is also shown that in the final stage of growth the Fokker-Planck type continuum model describes well the evolution of the distributions and the size selection.     

Simulation of random packing of spherical particles with different size distributions

A numerical model for a loose packing process of spherical particles is presented. The simulation model starts with randomly choosing a sphere according to a pregenerated continuous particle-size distribution, and then dropping the sphere into a dimension-specified box, and obtaining its final position by using dropping and rolling rules which are derived from a similar physical process of spheres dropping in the gravitational field to minimize its gravity potential. Effects of three different particle-size distributions on the packing structure were investigated. Analysis on the physical background of the powder-based manufacturing process is additionally applied to produce optimal packing parameters of bimodal and Gaussian distributions to improve the quality of the fabricated parts. The results showed that higher packing density can be obtained using bimodal size distribution with a particle-size ratio from 1.5 to 2.0 and the mixture composition around n ₂:n ₁=6:4. For particle size with a Gaussian distribution, the particle radii should be limited in a narrow range around 0.67 to 1.5.     

Log-normal size distributions of ultrafine metal particles

We have produced particles with median diameters well below 5 nm of Al, Fe, Co and Sn, using inert gas evaporation. The logarithm of the particle volume has a Gaussian distribution which can be explained by a theoretical model for coalescence of liquid particles. The standard deviations of these log-normal distributions, as inferred from electron microscopy, always lie close to a common value regardless of the kind of metal or method of evaporation.     

Grain growth in ice

Summary Grain growth is studied in polycrystalline ice, consisting of elongated grains, of (200÷300) μm mean width and (2÷3) mm mean length . The samples are annealed at different temperatures, between 0°C and −10°C. It is found that is not affected by annealing, while increases with the annealing time. Below the melting point, (t) tends to a limit value . This behaviour is related to the pinning action of air bubbles, which would be similar to that found for solid inclusions in metals. By assuming where is the mean bubble diameter andf is the volume fraction of air dissolved in water, reasonable values are found for . The activation energy of the phenomenon is evaluated on the basis of the present and of Jellinek and Gouda's results. It is foundQ=0.6 eV, which value approximately coincides with that for bulk self-diffusion as it occurs for metals, several degrees below the melting point. This coincidence suggests that, for ice, grain growth would be controlled by bulk impurity diffusion up to the very melting point. Instituto de Matemática, Astronomía y Física, Universidad Nacional de Córdoba and CONICET.