Crystal Growth Theory Research Articles

The Step Energy as a Habit Controlling Factor: Application to the Morphology Prediction of Aspartame, Venlafaxine, and a Yellow Isoxazolone Dye

In this paper, we report on the use of step energies in crystal morphology prediction as an improvement and computationally fast alternative to the Bravais, Friedel, Donnay, and Harker (BFDH) and attachment energy methods. One of the major improvements is a morphology prediction that is dependent on the driving force for crystallization. Step energies are calculated using STEPLIFT, an automated procedure to find the lowest step energies of infinitely long and straight steps. These steps are constructed by a combination of the connected nets of two nonparallel flat faces (F-faces), one representing the step terraces, the other the step edge. Step energies obtained in all relevant directions on a specific crystal face are used for a two-dimensional (2D) Wulff construction. This leads to the equilibrium shape of a 2D nucleus having the lowest energy, which can be used as a classification for the nucleation barrier, for example, in 2D birth and spread growth. Using the 2D nucleus energy, a link between the atomistic description of the steps and the macroscopic growth morphology is made using crystal growth theory to calculate growth rates as a function of the driving force for crystallization. This procedure is applied successfully to predict the crystal morphology of aspartame II-A, venlafaxine, and two polymorphs of a yellow isoxazolone dye.

Influence of deposition conditions on the microstructure of oxides thin films

Thin films of ZrO 2, HfO 2 and TiO 2 were deposited on kinds of substrates by electron beam evaporation (EB), ion assisted deposition (IAD) and dual ion beam sputtering (DIBS). Then some of them were annealed at different temperatures. X-ray diffraction (XRD) was applied to determine the crystalline phase and the grain size of these films, and the results revealed that their microstructures strongly depended on the deposition conditions such as substrate, deposition temperature, deposition method and annealing temperature. Theory of crystal growth and migratory diffusion were applied to explain the difference of crystalline structures between these thin films deposited and treated under various conditions.

Radial cordierite-bearing orbicular granite formed from the melting of pelitic hornfels in granitic magma, Tsukuba Mountains, Japan

Orbicular granite occurs at Minederayama in the Tsukuba Mountains, Ibaraki Prefecture, Japan. The orbs consist of a migmatitic core, originally pelitic hornfels, and a mantle that consists mainly of radial cordierite containing apatite inclusions and subordinate feldspar and micas. The orientation and size distribution of the orbs indicates that isolated orbs had floated within the host granitic magma. According to the geometrical selection theory of crystal growth, we propose that radial cordierite grew from the orb shells to the orb cores. Melting experiments demonstrate that the mineral assemblage of the orbicular granite reflects the incongruent melting of pelitic hornfels. Important factors in the formation of the orbicular granite are the unmixing between the incongruent melt of cordierite hornfels and host granite melt, and the infiltration of the incongruent melt into the host granite melt as the cordierite crystallized. The temperature of the host granite magma, as estimated from melting experiments and mineral assemblage, was above 730 °C at 0.3 GPa, which may represent superheated conditions for the host granitic magma. Cooling of the host granite magma placed the orbs under supercooled conditions as the radial cordierite grew from the shells of the orbs to the cores.

Fast Shape Evolution of TiN Microcrystals in Combustion Synthesis

Single-phase TiN crystalline powder consisting of microcrystals has been prepared by combustion synthesis. During the drastic combustion reaction with a high reaction temperature and rapid heating rate, TiN microcrystals undergo a fast shape evolution process and finally develop into various crystallization shapes, such as quasi-spherical grains, faceted cubes or pyramids, and dendrites. The formation mechanisms of different crystal shapes are discussed in light of basic crystal growth theories. It is proposed that, at the initial stage, quasi-spherical TiN crystals are formed first by diffusion-controlled continuous growth; by a faceting process, flat surfaces then appear in several low-index directions on the quasi-spherical crystals, and at last, well-faceted crystals are produced by two-dimensional nucleation and layered growth. In addition, interesting TiN dendrites are observed, probably arising from the fast cooling rate in combustion synthesis. The dendrites grow quickly in three orthogonal directions, with each growth unit exhibiting a 4-fold symmetry and sometimes a half-octahedral pyramid head.

Molecular dynamics simulations of the crystal–melt interfacial free energy and mobility in Mo and V

Molecular dynamics simulations, based on embedded-atom method potentials, have been used to compute thermodynamic and kinetic properties of crystal–melt interfaces in the bcc metals Mo and V. The interfacial free energy and its associated crystalline anisotropy have been obtained with the capillary fluctuation method and for both metals the anisotropy and the value of the Turnbull coefficient are found to be significantly lower than for the case of fcc materials. The interface mobility, or kinetic coefficient, which relates the isothermal crystallization rate to interface undercooling, was computed by non-equilibrium molecular dynamics simulations. Mobilities in the range 9-16 cm s−1K−1 are obtained. For Mo the mobility in the (110) crystallographic growth direction is larger than in the (100) and (111) directions, whereas for V the growth is found to be isotropic within numerical uncertainty. The kinetic-coefficient results are discussed within the framework of a density-functional-based theory of crystal growth.

TIM: a model protein for the study of crystal growth in space and gels

The understanding of the basic processes underlying the crystallisation of proteins requires the use of very well-characterised model systems, for which theory and models of crystal growth can be verified experimentally. Triose phosphate isomerase (Tm TIM) [1] can be produced in large amounts, and crystallises reproducibly. It has been used for experiments in microgravity and in gels, and it has been shown to produce crystals of marked improvement in these non-convective environments. It is thus an interesting model protein for studying the effect of convective mixing on protein crystallisation. We present here the phase diagrams for Tm TIM in function of temperature, protein, and precipitant concentration, and relate the results with the observed crystallisation behaviour. Solubility curves and the binodal curve in function of these tree parameters show that Tm TIM is a case of ‘retrograde solubility’. This phenomenon has been observed in many cases but only been described in detail in the case of human hemoblobin [2]. The phase diagrams form the basis for the understanding of the results from counterdiffusion experiments with Tm TIM, including improved crystal quality, and polymorfic variation in function of growth conditions.

On the determination of step energies. Theoretical considerations and application to an anisotropic Kossel model

Steps on surfaces are important in crystal growth theory, as the step free energy determines the two-dimensional nucleation rate, island growth, step flow and spiral growth. In this paper, it is illustrated that in general in lattice models the step energy of a single step cannot be determined directly by counting broken bonds. A new method is proposed that uses the geometry of a step together with the bonding topology, allowing for a straightforward determination of single-step energies for any case. The method is applied to an anisotropic Kossel model.

Deformation and stress from in-pore drying-induced crystallization of salt

The deformation and the fracture of porous solids from internal crystallization of salt is explored in the framework of the thermodynamics of unsaturated brittle poroelasticity. In the first place the usual theory of crystal growth in confined conditions is further developed in order to include both the deformation and the drying of the porous solid. The thermodynamics reveals the existence of a dilation coefficient associated with the crystallization process, and provides a solute–crystal equilibrium condition which involves the relative humidity, the supersaturation, and the salt characteristics. This thermodynamic condition and the mechanical equilibrium of the solution–crystal interface combine to give the current crystallization pore radius. Upscaling this information at the macroscopic scale, and taking into account the salt mass supplied by the invading solution, the approach leads to a quantitative analysis of the role of the pore size distribution on the crystal growth under repeated imbibition–drying cycles. The deformation and the fracture of the porous solid from drying-induced crystallization are then considered in the context of brittle poroelasticity. The current unsaturated macroscopic poroelastic properties are upscaled from the microscopic elastic properties of the solid matrix and from the current liquid, crystal and gas saturations. The adoption of a fracture criterion based on the elastic energy that the solid matrix can ultimately store finally leads to the determination of how long a stone can resist repeated cycles of drying-induced crystallization of salt.

Comparison of the Crystallization Kinetics of Canavalin and Lysozyme

Using dilatometry, we examined the crystallization kinetics of the jack bean storage protein, canavalin. We found that the kinetic rate law is first order in the supersaturation. At pH values above the pI = 5.2, where the protein is negatively charged, the apparent rate constant, k, increases as the pH approaches the pI, decreases with increasing temperature, and increases with increasing salt concentration. The rate constant is essentially independent of the initial protein concentration. Precisely the same patterns of behavior were observed by Kim et al. [Mol. Phys. 2003, 101, 2677−2686] in dilatometer experiments involving the crystallization of lysozyme even though lysozyme molecules are positively charged under crystal growth conditions. By combining dimensional analysis with a simple theory of protein crystal nucleation and growth, we show that k ≈ ω/cs, where ω is the rate coefficient that controls the rate of advance of the crystal facets, and cs is the solubility of the protein in the growth solu...

Grain Growth Control and Solid‐State Crystal Growth by Li2O/PbO Addition and Dislocation Introduction in the PMN–35PT System

Grain growth behavior and solid‐state single crystal growth (SSCG) in the Pb(Mg1/3Nb2/3)O3–35 mol% PbTiO3 (PMN–35PT) system have been investigated with varying Li2O/PbO ratios. The effect of dislocation density on crystal growth has also been studied. For SSCG, a BaTiO3 single‐crystal seed was embedded in a polycrystalline PMN–PT matrix. During annealing, a PMN–PT single crystal grew from the seed at the cost of the small matrix grains. Addition of Li2O dopant first enhanced and then reduced abnormal grain growth in the matrix. In the 2 mol% Li2O and 6 mol% PbO excess PMN–PT samples annealed at 1200°C, considerable single‐crystal growth occurred without formation of abnormally large grains in the matrix. Increasing the dislocation density in the BaTiO3 seed crystal resulted in enhanced growth of single crystals. These results were explained in terms of interface reaction‐controlled nucleation and growth, based on crystal growth theories.

Growth rate anisotropy and absorption studies on β-BaB2O4 single crystals grown by the top-seeded solution growth technique

Beta barium borate (β-BaB2O4; BBO) single crystals have been grown from Na2O flux by the TSSG technique and the observed growth rate anisotropy is reported. The symmetrical conoscopic interference pattern on the c-cut plate confirmed strain-free and optical homogeneity of the crystals. The observed growth habits of as-grown crystals are explained using crystal growth theories. The relative growth rate along different crystallographic directions of BBO can be described by R[100]=R[010]>R[001]. The absorption measurements show a nearly 9nm shift in fundamental absorption edges in X and Z cut samples. Band gap energies measured were 6.45 and 6.2eV along the X and Z directions, respectively. The absorption spectra near the fundamental absorption edges (AE) follow Urbach’s rule.

Phase field theory of crystal nucleation and polycrystalline growth: A review

We briefly review our recent modeling of crystal nucleation and polycrystalline growth using a phase field theory. First, we consider the applicability of phase field theory for describing crystal nucleation in a model hard sphere fluid. It is shown that the phase field theory accurately predicts the nucleation barrier height for this liquid when the model parameters are fixed by independent molecular dynamics calculations. We then address various aspects of polycrystalline solidification and associated crystal pattern formation at relatively long timescales. This late stage growth regime, which is not accessible by molecular dynamics, involves nucleation at the growth front to create new crystal grains in addition to the effects of primary nucleation. Finally, we consider the limit of extreme polycrystalline growth, where the disordering effect due to prolific grain formation leads to isotropic growth patterns at long times, i.e., spherulite formation. Our model of spherulite growth exhibits branching at fixed grain misorientations, induced by the inclusion of a metastable minimum in the orientational free energy. It is demonstrated that a broad variety of spherulitic patterns can be recovered by changing only a few model parameters.

Growth morphologies and mechanisms of non-equilibrium solidified MC carbide

Growth morphologies and mechanisms of the carbide of group IVB and VB elements (MC carbide), a typical faceted crystal, were studied with an estimated cooling rate from 102 to 105 K/s. Results showed that although the growth morphologies of the MC carbide vary remarkably with solidification cooling rate, the solid/liquid interface is always atomically smooth, and the growth mechanisms are always lateral growth. The growth mechanism transition from lateral to continuous growth mode, which was predicted by the classic crystal growth theory, was not observed for the TiC type MC carbide within the estimated cooling rate range of 102–105 K/s.

The probability of escaping interference

Consider two independent sequences of travelers arriving at opposite ends of a one-lane shared pathway. Each traveler attempts to traverse the entire pathway to the opposite end. An attempt fails if the traveler collides with an opposing traveler. In a collision, both opposing travelers are annihilated. We study the probability that a traveler manages to traverse the entire length of the one-lane shared pathway unobstructed. The dynamics of the travelers include the possibility of acting as bodyguards and "running interference" for a more recent arrival traveling in the same direction. This model was developed to address some questions in the theory of crystal growth. It may have possible applications in Particle Physics as well as to traffic at a one-lane bridge. This paper develops some properties of the model while focusing on the probability that a traveler crosses the entire pathway without interference.

Mechanisms of classical crystal growth theory explain quartz and silicate dissolution behavior

The central control of mineral weathering rates on biogeochemical systems has motivated studies of dissolution for more than 50 years. A complete physical picture that explains widely observed variations in dissolution behavior is lacking, and some data show apparent serious inconsistencies that cannot be explained by the largely empirical kinetic "laws." Here, we show that mineral dissolution can, in fact, be understood through the same mechanistic theory of nucleation developed for mineral growth. In principle, this theory should describe dissolution but has never been tested. By generalizing nucleation rate equations to include dissolution, we arrive at a model that predicts how quartz dissolution processes change with undersaturation from step retreat, to defect-driven and homogeneous etch pit formation. This finding reveals that the "salt effect," recognized almost 100 years ago, arises from a crossover in dominant nucleation mechanism to greatly increase step density. The theory also explains the dissolution kinetics of major weathering aluminosilicates, kaolinite and K-feldspar. In doing so, it provides a sensible origin of discrepancies reported for the dependence of kaolinite dissolution and growth rates on saturation state by invoking a temperature-activated transition in the nucleation process. Although dissolution by nucleation processes was previously unknown for oxides or silicates, our mechanism-based findings are consistent with recent observations of dissolution (i.e., demineralization) in biological minerals. Nucleation theory may be the missing link to unifying mineral growth and dissolution into a mechanistic and quantitative framework across the continuum of driving force.

Zinc Oxide Nanostructures: Morphology Derivation and Evolution

Zinc oxide nanostructures of various types, including nanobelts, nanoplatelets, nanowires, and nanorods, have been synthesized via well-developed routes by many research groups. However, so far, the underlying mechanism for the morphology derivation and evolution of the nanostructures has not been elucidated in depth. In this article, we report the systematic investigation of the morphology evolution characteristics of ZnO nanostructures from dense rods to dense nanoplatelets, nanoplatelet flowers, dense nanobelt flowers, and nanowire flowers in an evaporation-physical transport-condensation approach. Through the use of crystal growth theory, the determining factors for the formation of different nanostructural morphologies were found to be gas-phase supersaturation and the surface energy of the growing surface planes. Other experimental parameters such as the temperature at the source and the substrate, the temperature difference and the distance between the source and the substrate, the heating rate of the furnace, the gas flow rate, the ceramic tube diameter, and the starting material are all correlated with supersaturation and impose an effect on the morphology evolution. This finding may have an important impact on the qualitative understanding of the morphology evolution of nanostructures and the achieving of desired nanostructures controllably.

Theory of Conglomerate Crystallization in the Presence of Chiral Impurities

In the presence of an appropriate chiral impurity, the enantiomers of compounds that crystallize as conglomerates crystallize at different rates. During this process, the optical activity of the solution and the solid phase increases and then falls to zero. This transient optical activity happens on a time scale of about 15 min. In this paper, we present a general theoretical approach to describe this process. Our formulation is founded on basic theories of nucleation, crystal growth, and Langmuir adsorption. A computer code based on this theory is capable of describing both the time evolution of a single run of crystallization and the stochastic behavior observed in many crystallization runs; it also gives the crystal size distribution.

Quartz precipitation kinetics at 180°C in NaCl solutions—Implications for the usability of the principle of detailed balancing

The kinetics of heterogeneous quartz precipitation were determined at 180°C and a pH near 4 as a function of the degree of super saturation and dissolved NaCl concentrations. Stirred batch reactors were used and the changes in H 4SiO 4 concentrations over time were used to model the reaction rates. As the change in concentration in the batch experiment is not linear with time during all sampling intervals, the precipitation rates were retrieved by fitting the experimental observations to a forward model, i.e., a model that uses a prescribed rate law and compares the results with the experiments. Quartz precipitation occurs via an overall reaction, and therefore one can not presuppose the applicability of the predictions of Transition State Theory (TST). Therefore, different functions describing the dependence of the rate on deviation from equilibrium were examined in the forward model. The obtained precipitation rates were within error the same, regardless of the function that was used. By substituting the prediction of TST into the forward model, the change of quartz precipitation rate (at 180°C, pH 4 and ionic strength of 0.1M) with deviation from equilibrium was found to be r a t e = 1.9 ± 0.4 ⋅ 10 − 8 [ 1 − exp ⁡ ( Δ G r R T ) ] mol m − 2 s − 1 Within error, this observed rate agreed with predictions of quartz precipitation rates, which were calculated using the principle of detailed balancing, far from equilibrium quartz dissolution rates of Dove (1994) and equilibrium constant from Kharaka et al. (1988). Similar agreement between observations and predictions was obtained using the BCF (Burton Cabrera and Frank) crystal growth theory (Burton et al., 1951). In contrast to TST and BCF, a more general form of the rate dependence of deviation from equilibrium failed to predict the observed precipitation rates. Based on the observation that the principle of detailed balancing was verified for quartz dissolution and precipitation at 180°C and pH 4, we suggest that the principle of detailed balancing could be fruitfully applied to the huge database of quartz dissolution rates, together with appropriate thermodynamic data, for modeling quartz precipitation under natural conditions.

Observations on the geometries of etched fission and alpha-recoil tracks with reference to models of track revelation in minerals

The kinetic and atomistic theories of crystal growth and dissolution are used to interpret the shapes and orientations of fission-track, recoil-track and dislocation etch pits in tri-octahedral phlogopite and di-octahedral muscovite. An atomistic approach combined with symmetry considerations lead to the identification of the periodic bond chains that determine the etch pit morphologies and relative etch rates at a chemical level: O–Mg–O in phlogopite, O–Mg–O–Fe in biotite and O–Al–O in muscovite. Using first-order estimates of the bond strengths, it is possible to account for the relative track etch rates in these minerals. The reported, sometimes simultaneous, occurrence of triangular, polygonal and hexagonal etch pit contours in phlogopite, some of which violate the crystal symmetry, suggests that the cohesion of the phlogopite lattice is lost over a much larger radius than that of the track core around the trajectories of particles for which the energy loss exceeds a threshold value. This is interpreted as an indication of pronounced sublattice and anisotropic effects during track registration.

Imaging of defect-mediated surface morphology of GaN(0001¯) grown on sapphire by molecular beam epitaxy

In situ scanning tunneling microscopy and ex situ atomic force microscopy are used to study the surface morphology of GaN(0001¯) grown on sapphire substrates by plasma-assisted molecular beam epitaxy. Several types of defect-mediated surface morphologies are characterized, including surface terminations of screw and mixed dislocations, defect-induced grooves, and impurity-induced step bunching. In addition to the early theory of dislocation-enhanced crystal growth developed by Frank [F. C. Frank, Acta Crystallogr 4, 497 (1951)] step structures are also found to be critical to understand the formation of spiral growth hillocks around surface terminations of different types of threading dislocations. Growth experiments are also performed under different conditions, e.g., by varying III∕V flux ratio and growth temperature to further investigate the relationship between the final surface morphology and the early stages of growth, such as sapphire nitridation and the transition stage from the three-dimensional buffer layer growth under nitrogen-rich conditions to the two-dimensional active layer growth under slightly Ga-rich conditions.