Diffusion-limited Crystal Growth Research Articles

Orientation and morphology of solid-state dewetting holes

We report on the orientation and morphology of solid-state dewetting holes obtained from a kinetic Monte Carlo model with nearest neighbor and next nearest neighbor interactions on a cubic lattice. The morphologies found in simulations share strong similarities with those of diffusion-limited crystal growth: We find compact shapes, isotropic and anisotropic seaweed shapes, and dendritic shapes. Some of these shapes have been observed in solid-state dewetting experiments with various semi-conductors or metals on insulating substrates. The sides of the fingers can exhibit two types of zigzag instabilities with ${90}^{\ensuremath{\circ}}$ or ${45}^{\ensuremath{\circ}}$. Due to fourfold symmetry, dewetting hole shapes are found to exhibit two possible orientations with fingers along the (110) or (100) directions. We find that the rotation transition from one orientation to the other can be observed in our model by varying anisotropy, temperature, or dewetting strength. The two orientations correspond, respectively, to experimental observations of dewetting for silicon and germanium on insulator and to the dewetting of metal films with different reducing gas flow rate.

The Importance of Salt-Enhanced Electrostatic Repulsion in Colloidal Crystal Engineering with DNA.

Realizing functional colloidal single crystals requires precise control over nanoparticles in three dimensions across multiple size regimes. In this regard, colloidal crystallization with programmable atom equivalents (PAEs) composed of DNA-modified nanoparticles allows one to program in a sequence-specific manner crystal symmetry, lattice parameter, and, in certain cases, crystal habit. Here, we explore how salt and the electrostatic properties of DNA regulate the attachment kinetics between PAEs. Counterintuitively, simulations and theory show that at high salt concentrations (1 M NaCl), the energy barrier for crystal growth increases by over an order of magnitude compared to low concentration (0.3 M), resulting in a transition from interface-limited to diffusion-limited crystal growth at larger crystal sizes. Remarkably, at elevated salt concentrations, well-formed rhombic dodecahedron-shaped microcrystals up to 21 μm in size grow, whereas at low salt concentration, the crystal size typically does not exceed 2 μm. Simulations show an increased barrier to hybridization between complementary PAEs at elevated salt concentrations. Therefore, although one might intuitively conclude that higher salt concentration would lead to less electrostatic repulsion and faster PAE-to-PAE hybridization kinetics, the opposite is the case, especially at larger inter-PAE distances. These observations provide important insight into how solution ionic strength can be used to control the attachment kinetics of nanoparticles coated with charged polymeric materials in general and DNA in particular.

The "speed limit" for macromolecular crystal growth.

A simple “diffusion‐to‐capture” model is used to estimate the upper limit to the growth rate of macromolecular crystals under conditions when the rate limiting process is the mass transfer of sample from solution to the crystal. Under diffusion‐limited crystal growth conditions, this model predicts that the cross‐sectional area of a crystal will increase linearly with time; this prediction is validated by monitoring the growth rate of lysozyme crystals. A consequence of this analysis is that when crystal growth is diffusion‐limited, micron‐sized crystals can be produced in ~1 s, which would be compatible with the turnover time of many enzymes. Consequently, the ability to record diffraction patterns from sub‐micron sized crystals by X‐ray Free Electron Lasers and micro‐electron diffraction technologies opens the possibility of trapping intermediate enzyme states by crystallization.

Reassessment of Crystal Growth Theory of Graphite in Cast Iron

The problem of graphite crystallization and growth in cast iron has recently received increased attention. As most of the published literature describe analysis of room temperature graphite, there is a legitimate concern that the crystallization of graphite is concealed by recrystallization and growth in solid state occurring after solidification. To avoid confusion in the interpretation of room temperature graphite morphology, the authors used Field Emission Gun SEM on deep-etched interrupted solidification (quenched) specimens to reveal the morphology of graphite at the very beginning of solidification, when the graphite is in contact with the liquid. Information from related phenomena, such as crystallization of hexagonal structure snowflakes and metamorphic graphite, as well as of diamond cubic structure silicon crystals in aluminum alloys is incorporated in the analysis. Research discussing graphite produced through gas-solid and solid-solid transformations is also examined. Because the faceted growth of graphite is the result of diffusion-limited crystal growth in the presence of anisotropic surface energy and anisotropic attachment kinetics, a variety of solidification morphologies are found. The basic building blocks of the graphite aggregates are hexagonal faceted graphite platelets generated through the growth of graphene layers. As solidification advances, the platelets thicken through layer growth, and then aggregate through mechanisms that may include foliated/tiled-roof crystals and dendrites, curved-circumferential, cone-helix, helical, and columnar or conical sectors growth.

On Precipitation of Sparingly Soluble Fluoride Salts

The classical approaches for nucleation rate and diffusion-limited crystal growth are checked for their validity for sparingly soluble salts, i.e., calcium fluoride and strontium fluoride. Particle size distributions are calculated by population balance modeling as a function of the initial supersaturation and free lattice ion ratio. Theoretical results are compared with experimentally determined particle sizes which are measured by dynamic light scattering technique. Before measurement, suspensions were stabilized by dilution with deionized water or surfactant solution. The dilution of suspensions was done after solid liquid equilibrium had been reached. Consequently, particle formation mechanisms, such as nucleation and crystal growth which depend on supersaturation, are assumed to be unaffected. If supersaturation is calculated correctly by considering activities and ion complex formation, we found that experimental results are predictable by modeling with classical approaches. Hereby, interfacial ener...

Controlled Recrystallization of Tubular Vinpocetine Crystals with Increased Aqueous Dissolution Rate and In Vivo Bioavailability

Vinpocetine was a BCS II drug, whose clinical applications had suffered from low oral bioavailability because of its inefficient dissolution in the GI tract. As the dissolution rate depended on the surface area of the drug crystals, we herein explored shape-controlled recrystallization via antisolvent process as an excipient-free strategy to improve the bioavailability of VIN. By adjusting the water/ethanol ratio, initial VIN concentration, and temperatures, morphologies of the crystalline products could be finely tuned from three-dimensional cubes and tubes, to two-dimensional frizzled plates, and finally to zero-dimensional microparticle clusters. Morphology analysis and in situ FBRM surveillance of the growing process suggested that a diffusion-limited crystal growth mechanism was responsible for the shape variation of VIN products. Finally, we tested the in vitro dissolution efficiency as well as the in vivo bioavailability of recrystallized VIN crystals. Results manifested that the tubular crystal sh...

Reexamination of crystal growth theory of graphite in iron-carbon alloys

Most analysis of graphite morphology in cast iron-carbon alloys is performed on samples cooled to room temperature. This raises the concern that the crystallization of graphite is obscured by subsequent recrystallization and growth in solid state. To bring clarity to this issue, the authors used Field Emission Gun Scanning Electron Microscopy on deep-etched interrupted solidification (quenched) specimens to reveal the morphology of graphite growing in contact with the liquid at the very beginning of solidification.To understand the complexity of graphite crystallization in iron alloys, the analysis included evidence from the crystallization of materials with analogous hexagonal structure, such as of snowflakes and metamorphic graphite, and from the crystallization of diamond cubic structure silicon crystals in aluminum-based alloys. Information from research discussing graphite produced through gas-solid (chemical vapor deposition) and solid-solid (graphite in steel) transformations was also exploited.The large variety of graphite solidification morphologies described in this and earlier papers derives from the complexities of its faceted growth during crystallization, a diffusion-limited crystal growth process, in the presence of anisotropic surface energy and anisotropic attachment kinetics. It was confirmed that the basic building blocks of the graphite aggregates are hexagonal faceted graphite platelets generated through the growth of graphene layers. As solidification advances, the platelets thicken through layer growth through two-dimensional or screw dislocation nucleation. Depending on bulk composition, local supersaturation and undercooling, the platelets aggregate through a variety of mechanisms including tiled-roof and foliated crystals and dendrites, curved-circumferential, cone-helix, helical (macro-spiral), and polyhedral pyramidal (or conical) sectors growth. The final graphite shape of graphite spheroids is affected by the crystallography of the nucleus, as it affects the initial growth of the graphite platelets.

Lattice Boltzmann simulation of water isotope fractionation during ice crystal growth in clouds

We describe a lattice Boltzmann (LB) method for simulating water isotope fractionation during diffusion-limited ice crystal growth by vapor deposition from water-oversaturated air. These conditions apply to the growth of snow crystals in clouds where the vapor composition is controlled by the presence of both ice crystals and water droplets. Modeling of water condensation with the LB method has the advantage of allowing concentration fields to evolve based on local conditions so that the controls on grain shapes of the condensed phase can be studied simultaneously with the controls on isotopic composition and growth rate. Water isotope fractionation during snow crystal growth involves kinetic effects due to diffusion of water vapor in air, which requires careful consideration of the boundary conditions at the ice-vapor interface. The boundary condition is relatively simple for water isotopes because the molecular exchange rate for water at the interface is large compared to the crystal growth rate. Our results for the bulk crystal isotopic composition are consistent with simpler models using analytical solutions for radial geometry. However, the model results are sufficiently different for oxygen isotopes that they could affect the interpretation of D-excess values of snow and ice. The extent of vapor oversaturation plays a major role in determining the water isotope fractionation as well as the degree of dendritic growth. Departures from isotopic equilibrium increase at colder temperatures as diffusivity decreases. Dendritic crystals are isotopically heterogeneous. Isotopic variations within individual snow crystals could yield information on the microphysics of ice condensation as well as on the accommodation or sticking coefficient of water associated with vapor deposition. Our results are ultimately a first step in implementing LB models for kinetically controlled condensation or precipitation reactions, but needs to be extended also to cases where the molecular exchange rate is comparable to the crystal growth rate. This approach could also be applicable to aerosol chemical evolution.

Robust Nonsticky Superhydrophobicity by the Tapering of Aligned ZnO Nanorods

The robust nonsticky superhydrophobicity of aligned nanoneedle films is reported. A facile, efficient, cheap, and available method based on the diffusion-limited crystal growth principle is proposed for controlling the tapering of ZnO nanorods, the profiles of which can be tuned effectively by synergetic control over reaction time and temperature in an extremely strong alkaline reaction system. The synthesized nanoneedle, nanopencil, and nanorod arrays are chosen for studying the effects of nanoscale topography on anti-droplet-sticking ability. After silanization, all of them show excellent quasi-static anti-droplet-stickiness, and water adhesion along the normal and lateral directions can be greatly reduced by the tapering of nanorods and eliminated by sharp nanoneedles. However, their antisticking stability is distinct under the droplet impact: the nanoneedle sample is still nonsticky but the nanorod sample loses its antisticking ability. Only ensuring the liquid/air interface is in the suspended nonwetting state is insufficient to obtain robust nonsticky surfaces, which also require extremely low solid-liquid van der Waals attraction.

46. Modelling the cryopreservation process: The effect of the cell size distribution

Cryopreservation modelling starts from the pioneering work by Mazur in 1963. Later in the nineties continuous improvements led to a complete physico-chemical theory describing IIF, where the water transport model is coupled to the classical nucleation theory of ice nucleation and diffusion limited crystal growth. Following the developed theoretical approach, the PIIF has been defined to allow the comparison of model results with typical experimental data obtained through cryo-microscopy. Since this, researchers have been operating in a cul-de-sac , being unable or even uninterested to overcome the predictive limitations that this theoretical approach shows. Above all, the PIIF adopted so far is strictly valid only for the case of a single ice nucleus per cell and growing extremely fast. Moreover, since in this theoretical approach all the cells in a population are assumed of the same average size, resorting to the “sporadic nucleation” of identical cells (i.e. stochastic criterion) is mandatory to explain the experimental evidence that cells subjected to the same freezing cycle show different IIF temperatures. On the other hand, it is well known that even a purified population of cells does not comprise cells of identical size. Needless to say that, larger cells are characterised by a smaller surface-to-volume ratio than smaller cells. As such, larger cells are expected to lose less water via osmosis. Therefore, at any cooling rate, the largest cells of a population are expected to reach the super-cooled conditions earlier than the smallest. Following these considerations, in this work a size distributed cell population has been taken into account through a Population Balance approach, and the “sporadic nucleation” has been dropped in favour of a fully deterministic criterion: differences in cell volumes are now invoked to explain the experimental evidence that cells subjected to the same freezing cycle show different IIF temperatures. Accordingly, a new general definition of PIIF has been provided, able to take into account not only the ice nucleation but also the ice crystal growth of an unlimited number of ice nuclei per cell. Experimental validation of this novel modelling approach has been provided by comparisons with suitable experimental data taken from the literature regarding rat hepatocytes. The new definition of PIIF appears to be far more consistent with the experimental measurements than the previous theoretical interpretations. In addition, the novel modelling approach has provided an original and more sound explanation for well-known experimental evidences. In particular, while the stochastic model for a population of identical cells needs to invoke two different ice nucleation mechanisms (namely SCN and VCN, surface and volume catalysed nucleation mechanisms) in order to describe the experimental evidence of the so called “break point for IIF incidence”, the novel deterministic model is able to describe such experimental behaviour by considering only one heterogeneous nucleation mechanism of IIF taking place in a population of differently sized cells. The Fondazione Banco di Sardegna, Italy, and The Sardinia Regional Government are gratefully acknowledged for the financial support.

Numerical computations of faceted pattern formation in snow crystal growth

Faceted growth of snow crystals leads to a rich diversity of forms with remarkable sixfold symmetry. Snow crystal structures result from diffusion-limited crystal growth in the presence of anisotropic surface energy and anisotropic attachment kinetics. It is by now well understood that the morphological stability of ice crystals strongly depends on supersaturation, crystal size, and temperature. Until very recently it was very difficult to perform numerical simulations of this highly anisotropic crystal growth. In particular, obtaining facet growth in combination with dendritic branching is a challenging task. We present numerical simulations of snow crystal growth in two and three spacial dimensions using a computational method recently introduced by the present authors. We present both qualitative and quantitative computations. In particular, a linear relationship between tip velocity and supersaturation is observed. In our computations, surface energy effects, although small, have a pronounced effect on crystal growth. We compute solid plates, solid prisms, hollow columns, needles, dendrites, capped columns, and scrolls on plates. Although all these forms appear in nature, it is a significant challenge to reproduce them with the help of numerical simulations for a continuum model.

Preparation and Growth Mechanism of ZnO Nanotubes Array

ZnO nanotubes array has been fabricated by a facile hydrothermal treatment of ZnO seeded silicon substrate in a sealed system. The as prepared ZnO nanotubes are single crystalline and have a [0001] preferential growth direction. Shape and size of the ZnO product is found to be sensitive to reaction conditions. Diffusion-limited crystal growth is proposed to explain the formation mechanism of ZnO nanotubes.

Multiscale simulations of micro‐structure selection in binary alloy solidification

AbstractWe demonstrate that the competition of dendritic crystals in a solidifying sample gives rise to two qualitatively different micro‐structure solutions depending on the density ρ of crystals in the melt. Here we show for the first time, that there is a non‐steady transition from one to the other. The precise ρ‐dependence of the transition point is determined by the Biot number. Our investigation is based on a scaling analysis for the tip velocity of the dendritic crystals, which we assume to be aligned in an array and to be coupled via the transport of heat. We develop our analytical solutions based upon the asymptotic Kruskal‐Segur reduction to a differential equation in the complex plane. It is supported by numerical simulations of a multiscale model of alloy growth. On the one hand this solution can be used to improve the accuracy of applied solidification simulations. On the other hand it yields additional insight in the universality of diffusion limited crystal growth in the presence of competing micro‐structures. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Diffusion-limited crystal growth in metallic glasses under continuous heating

Mathematical analysis for diffusion-limited growth of a spherical crystal in an infinite matrix under linear heating conditions has been performed, and the exact and approximated solutions of the problem have been obtained. The approximated expression derived for evolution of the growing crystal radius at constant heating rate q formally coincides with the well-known parabolic law derived for isothermal conditions if the real time is replaced by T2/(qQ), with Q being the activation energy of diffusion. Despite a relatively large difference between the exact and approximated values of the crystal size, the accuracy of the volume fraction crystallized is within 1%. A good agreement of the calculated results with the relevant experimental data for the Fe–B metallic glasses implies the validity of the simple analytical relations derived for description of the diffusion-limited crystal growth in metallic glasses at constant rate heating.

A Theoretical Model of Intracellular Devitrification

Devitrification of the intracellular solution can cause significant damage during warming of cells cryopreserved by freezing or vitrification. Whereas previous theoretical investigations of devitrification have not considered the effect of cell dehydration on intracellular ice formation, a new model which couples membrane-limited water transport equations, classical nucleation theory, and diffusion-limited crystal growth theory is presented. The model was used to explore the role of cell dehydration in devitrification of human keratinocytes frozen in the presence of glycerol. Numerical simulations demonstrated that water transport during cooling affects subsequent intracellular ice formation during warming, correctly predicting observations that critical warming rate increases with increasing cooling rate. However, for cells with a membrane transport activation energy less than approximately 50 kJ/mol, devitrification was also affected by cell dehydration during warming, leading to a reversal of the relationship between cooling rate and critical warming rate. Thus, for low warming rates (less than 10°C/min for keratinocytes), the size and total volume fraction of intracellular ice crystals forming during warming decreased with decreasing warming rate, and the critical warming rate decreased with increasing cooling rate. The effects of water transport on the kinetics of intracellular nucleation and crystal growth were elucidated by comparison of simulations of cell warming with simulations of devitrification in H2O-NaCl-glycerol droplets of constant size and composition. These studies showed that the rate of intracellular nucleation was less sensitive to cell dehydration than was the crystal growth rate. The theoretical methods presented may be of use for the design and optimization of freeze-thaw protocols.

Diffusion-limited crystal growth in silicate systems: similarity with high-pressure liquid-phase sintering

In natural partially molten silicate systems as well as in material systems, textural adjustments occur to reduce the total free energy of the system, including those contained in surfaces and interfaces. To assess the effect of water content on textural equilibration and crystal growth in solid–liquid silicate systems we conducted wetting-angle and diffusion experiments. These experiments provide information about the extent of the interfacial energy reduction associated with dissolution along grain edges. The wetting angle exhibits a constant median value of ca. 50°, reflecting overall reduction of interfacial energy of the two-phase silicate system during melt penetration. The nature and extent of the breakdown reaction, the proportion of the new phases, and the crystallographic development are related to the initial water content. The solid was initially impermeable, so the liquid–solid equilibration involves dissolution producing crystallite rearrangements on a local scale, with immediate and nearby reprecipitation. Because the process of dissolution/precipitation requires transport of the atomic components, liquid penetration, and the rate-limited growth of the new phases are found to be consistent with capillary-driven diffusion.

Carbothermal reduction and nitridation of silica: nuclei planar growth controlled by silicon monoxide diffusion on the reducer surface

The carbothermal reduction and nitridation of silica as a source of low-cost silicon nitride powders prepared from cheap raw materials, lamp black and fumed silica, was investigated in the temperature range of 1200–1450°C. Carbon and silica with a specific surface area of 27 and 176 m 2 g −1, respectively, were wet mixed to form gels. The degree of mixing of carbon and silica, as given by microscopic observation, and the Si 3N 4 reaction yield were optimised in the study of a set of mixed gels that were dispersed using water and ethanol as dispersing liquids. The Si 3N 4 reaction yield at 1450°C was further increased to a value of about 96% of the conversion ratio of silica into Si 3N 4 and silica loss as SiO(g) was controlled to a value of below 1% by using lamp black with a specific surface area of 390 m 2 g −1. Quantitative analysis of the reaction kinetics of the carbothermal reduction and nitridation of silica reveals that it is of exponential type, as given by the Johnson–Mehl–Avrami law for processes controlled by fast nucleation followed by diffusion-limited crystal growth. By selecting carbon sources of high specific surface area, the Si 3N 4 nuclei density number is increased, yielding nearly spherical particles of Si 3N 4 powders with sub-micron sizes.

Crystallization of Supercooled Zr41Ti14Cu12Ni10B23 Melts During Continuous Heating and Cooling

AbstractThe crystallization behavior of the bulk glass forming Zr41Ti14Cu12Ni10Be23 liquid was studied under different heating and cooling rates. Investigations were performed in high purity graphite crucibles since heterogeneous surface nucleation at the container walls does not effect the crystallization of the bulk sample. A rate of about 1 K/s is sufficient to circumvent crystallization of the melt while cooling from the equilibrium melt. In contrast, upon heating a rate of more than 150 K/s is necessary to avoid crystallization of Zr41Ti14Cu12Ni10Be23 samples. The difference between the critical heating and cooling rate is discussed within classical nucleation theory and diffusion limited crystal growth. The calculated difference of the critical heating and cooling rate can be explained by the fact that nuclei formed during cooling and heating are expose to different growth rates.

A crystal growth rate of equation for ancient dolomites; evidence for millimeter-scale flux-limited growth

ABSTRACT A linear correlation exists between the width of cathodoluminescent zones (CL) and the size of individual crystals in the Seroe Domi Fm (Miocene, Netherlands Antilles), Burlington-Keokuk Fm (Mississippian), Fort Payne Fm (Mississippian), and Saluda Fm (dovician). If individual zones, correlated by CL character and stratigraphy, are isochronous, then the width of a zone is directly proportional to the rate of crystal growth, Linear correlation coefficients for between-zone thickness and crystal size in individual thin sections range from 0.98 to 0.41, indicating a size dependence governing individual dolomite crystal growth rates. Mononuclear, polynuclear, spiral dislocation, and diffusion-limited crystal growth models operating under homogeneous or inhomogeneous kinetic conditions were considered as possible crystal growth mechanisms. Of the models considered, only mononuclear growth may result in a correlation between zone width and crystal radius in chemically homogeneous systems. However, this relationship is not linear. Any of the growth laws considered may produce a linear size-dependent growth rate when there are temporally constant differences between crystals in the kinetic factors governing growth. Differences in the kinetic factors governing growth could arise from stable variations in the chemistry of pore fluids related to permeability variations in the rock. The radius-rate data give a crystal growth rate equation of the form dr/dr = kr. This equation is consistent with a model wherein the rate of crystal growth is determined by stable variations in the chemistry of pore fluids. This implies that the kinetics of dolomite crystal growth is flux limited and therefore no slower than many diagenetic reactions. The data also support the hypothesis that thin-section-scale variations in dolomite crystal size can result from variations in crystal growth rate.

Batch crystallization under continuous cooling: analytical solution for diffusion limited crystal growth

The kinetic equations of nucleation and crystal growth in a multicomponent system undergoing continuous cooling are solved for asymptotically slow cooling rates meaning that the maximum supercooling ΔTm is much smaller than the exponent A in the nucleation rate I exp(−AT−2). The growth rate is controlled by chemical diffusion so that it depends not only on the supercooling but also on the crystal size. The periods of metastable cooling, nucleation and relaxation are related as 1:p−1:p−1/3 where p = AΔT−2m 30–80 is a parameter characterizing the maximum supercooling Tm. It is a weak (logarithmical) function of all the parameters including the cooling rate Ṫ. The crystal sizes are determined by the density of nuclei produced during the nucleation period and depend on the cooling rate approximately as Ṫ−0.5. The theoretical predictions are found in a good agreement with experimental data for AlCu, AlSi, SnPb, and CaMgSi2O6CaAl2Si2O8NaAlSi3O8.