Surface Supersaturation Research Articles

Numerical simulation of the hydrodynamics and mass transfer in the large scale, rapid growth of KDP crystals—2: computation of the mass transfer

Numerical simulations of the hydrodynamics and mass transfer involved in the rapid growth of very large potassium di-hydrogen phosphate (KDP) crystals with linear dimensions up to 60 cm and masses greater than 270 kg have been performed. The computational geometry corresponds very closely to the 1000-l crystallizers currently in use at Lawrence Livermore National Laboratory (LLNL) for the growth of KDP crystals for the National Ignition Facility (NIF). In a related paper (J. Crystal Growth 222 (2000) 263), the temporal and spatial character of the three-dimensional, time-dependent, turbulent flow field was discussed. This paper presents the corresponding results concerning the mass transfer occurring at the crystal surface during the rapid growth process. Emphasis is placed on studying the surface concentration or supersaturation distribution, since it is critically involved in the processes of morphological instability and inclusion formation. The temporal and spatial evolution of the surface supersaturation is studied as a function of crystal size, growth rate, and rotation conditions.

Characterization of the effectiveness of silica anti-scalants

The presence of silica in brackish water greatly complicates the reverse osmosis desalting process. Threshold limits of silica scale precipitation are difficult to predict and there are no reliable techniques for evaluating the performance of silica anti-scalants (AS) used to extend water recovery limits. The aim of the present work is to illustrate a fundamentally oriented procedure for assessing the relative effectiveness of various anti-scalants. Rigorous analysis of inhibition data obtained in comparative tests of four commercial anti-scalants, dosed at the recommended dosage levels, revealed that there was no significant difference in their inhibitory effectiveness. The mitigating effect provided by all AS tested was increase of the threshold limit for the onset of scaling from a membrane surface supersaturation (s/s) level of 1.7–1.9 in the absence of an AS, to the level of 2.1–2.3, in the presence of the AS.

Transient behaviour of surface supersaturation caused by an abrupt change in bulk supersaturation

The transient behaviour of the surface supersaturation, caused by an abrupt (stepped) change in bulk supersaturation from one constant value σ 0 to another σ 1, is studied with the use of the surface diffusion theory of Burton, Cabrera and Frank (BCF). The presented analytical solution of the time-dependent BCF equation takes into account a phenomenological coefficient Λ s describing the kinetics of growth unit exchange between the step edge and the adsorbed layer. It is shown that the solution can be approximated by a simple expression, which enables an easy estimate of the time t s required to attain the steady-state value of the surface supersaturation. The time t s is independent of the difference between values σ 0 and σ 1 but it increases with increasing coefficient Λ s or increasing interstep distance. On the other hand, the average rate of change in the surface supersaturation during the transient state is determined neither by the coefficient Λ s nor by the interstep distance but it is proportional to the difference between values of σ 0 and σ 1 of bulk supersaturation.

Crystal growth in the presence of surface melting: supersaturation dependence of the growth of columnar ice crystals

We report measurements of the growth rates of the basal (0 0 0 1) and prism ( 1 0 1 ̄ 0 ) facets of ice as a function of surface supersaturation, in a regime where the growth dynamics are dominated by the presence of surface melting. Our measurements were made using freely falling columnar ice crystals, growing a solvent gas of air at a pressure of one atmosphere, with ambient supersaturation levels 1< σ ∞<11%. Condensation coefficients for the two facets were inferred by modeling the effects of both particle and heat diffusion, which also yielded surface supersaturation levels 0.75< σ surf<2.2%. Our results show a strong supersaturation dependence in the crystal growth, as for both facets the condensation coefficient exhibited a precipitous drop below σ surf≈1%. In addition, the condensation coefficient for the basal facet was found to be nearly a factor of 10 larger than for the prism facet. These results support a model in which the ice growth rates are governed by 2D nucleation at the interface between the crystalline solid and a quasi-liquid surface layer.

Spiral growth limited by surface kinetics a self-consistent model

A one-dimensional Stefan problem describing growth limited by surface processes is set and the time evolution of a spiral growing hillock calculated. The model is self-consistent in the sense that the critical surface supersaturation for the production of a new step, the slope and the growth rate of the hillock, strictly depend on a set of phenomenological parameters describing diffusion and matter exchange at the steps. The quasi-steady states calculated for several sets of values of the parameters, are illustrated and compared with classical models. Some criteria are proposed to correlate the features of the growth hillocks and the steps rates, to the kinetics.

The study of behaviour of surface supersaturation caused by variations in bulk supersaturation

Variations in the surface supersaturation σ s, caused by an abrupt (stepped) change in bulk supersaturation σ and subsequent exponential decrease or increase in σ, are studied with the use of the surface diffusion theory of Burton, Cabrera and Frank (BCF). The study is based on the analytical solution [M. Rak, Surf. Sci. 442 (1999) 149] of the BCF time-dependent equation. It is shown that the solution can be approximated by a simple expression convenient for the analysis. It is found that at the time t=0, the instantaneous rate of a change in the surface supersaturation σ s is determined only by the exchange of growth units between the crystal surface and the solution bulk, and is not affected by the surface diffusion. If an abrupt increase in bulk supersaturation σ at t=0 is followed by an exponential decrease in σ, then at t M>0 the function σ s of t has a maximum. In the opposite case, i.e. if an abrupt decrease in σ at t=0 is followed by an exponential increase in σ, then σ s has a minimum at t M>0. The time t M is estimated and an effect of variations in σ on value of t M is analysed.

Interface-supersaturation estimated from the shape of steps on the surface of InP MCE by LPE

Shape of the steps in the InP microchannel epitaxy (MCE) was investigated to determine the surface supersaturation, σ, during the MCE growth. σ near the stacking fault (SF) where 2-D nuclei are generated has been estimated. The ratio of the width-to-thickness (W/T ratio) of the MCE layer grown in the mode of 2-D nucleation at a SF is found to increase with decreasing σ. This dependence is similar to that of the growth from a spiral step but σ, where the W/T ratio starts to increase was found to move to a lower value. The mechanism by which the W/T ratio is determined is also discussed.

Solution-Induced Reconstructive Epitaxial Nucleation on Pseudoflat Surfaces of Fractal Gel-Grown Ammonium Chloride

The morphological manifestation of regular fractal patterns in ammonium chloride and the transition from ordinary to fractal morphology are studied at the crystallographic and molecular level by means of the Hartman-Perdok theory. The theoretically predicted growth forms as a function of growth conditions are in excellent agreement with experimental results for growth out of vapor ({211} and {110}) and out of polar solutions ({211}, {100}, and/or {111}), in the ordinary as well as fractal morphological regimes. The {211} surface is (2 1)-reconstructed due to internal or structural energy gain that persists in all growth conditions. The mechanism responsible for the onset of fractal growth in a gel containing water and agarose molecules is solution-induced reconstructive epitaxial nucleation (SIREN) between pseudoflat pyramidal {211} faces. SIREN is equivalent to the structural match between crystallites, anticipated in the macroscopic theory of nucleation and growth kinetics [Liu, X. Y.; Strom, C. S.; J. Chem. Phys.2000, 113. In press.]. It is associated with an increase in the effective surface supersaturation, brought about by the surface poisoning of the agarose inhibitors. It is characterized by a critical length parameter 〈l

Application of the principle of maximum entropy production to the analysis of the morphological stability of a growing crystal

The morphological stability of spherical and cylindrical crystals and an infinite plane growing from a supersaturated solution is studied using the principle of maximum entropy production in the Mullins and Sekerka approximation. In contrast to the first two geometries, the computational results for a plane agree completely with the results obtained on the basis of the classical linear perturbation theory. The concept of the binodal of a morphological transition is introduced in order to interpret the results for the sphere and cylinder. The boundaries of the metastable region are investigated. Morphological phase diagrams of stable-unstable growth are presented in terms of the variables surface energy and supersaturation as well as the variables crystal size and supersaturation. The physical nature of the appearance of metastability in this system is discussed.

“Bending” of steps on rapidly grown KH 2PO 4 crystals due to an inhomogeneous surface supersaturation field

The surfaces of large rapidly grown KH 2PO 4 (KDP) crystals can exhibit the formation of valleys in addition to the well-known growth hillocks and associated intervicinal boundaries between them. Understanding the mechanism of formation of these valleys is extremely important because they are observed to be precursors to the formation of solution inclusions on the crystal surface. Ex situ measurements using a precision coordinate measuring machine (CMM) have been made of the surface topography of rapidly grown KDP crystals. These measurements clearly indicate that these valleys are formed by the “bending” of macrosteps originating from a single growth hillock. Results from numerical simulations of the hydrodynamics and mass transfer for the conditions used in the rapid growth process show that the surface supersaturation field generated on these crystals is inhomogeneous due to the spatially and temporally varying boundary layer thickness on the rotating crystal surface. These numerical results are used together with a simple model for the propagation of steps in an inhomogeneous surface supersaturation field to simulate the resulting crystal surface features. The qualitative agreement between the surface measurements and the numerical simulations is shown to be quite good. The dependence of the valley formation on parameters such as crystal growth rate, rotation conditions, growth hillock location, and impurity level are studied.

An analysis of flow and mass transfer during the solution growth of potassium titanyl phosphate

The first fully three-dimensional solution flow and solute transport simulations are performed to model the potassium titanyl phosphate (KTP) growth system of Bordui et al. Steady flows and supersaturation fields for two crystal mounting geometries are computed using a stabilized finite element method implemented on a data-parallel supercomputer. Our results present a mechanistic picture of solute transport which is consistent with inclusion formation patterns obtained in experiments. The simulations also explain beneficial outcomes, in terms of better global mixing and more uniform surface supersaturation, observed for a crystal mounting geometry which strongly breaks cylindrical symmetry in the system.

Quantitative Measurement of Interstitial Flux and Surface Super-saturation during Oxidation of Silicon

AbstractDuring the oxidation of silicon, interstitials are generated at the oxidizing surface and diffuse into the silicon. Boron diffusion was used to map the local interstitial super-saturation, the ratio of interstitial concentration to the equilibrium concentration of interstitials I/I*, versus depth above buried Si0.795Ge0.2C0.005 layers during oxidation. The average interstitial supersaturation at the silicon surface, extrapolated from the depth profiles, is measured as, ∼24 and ∼11.5 for 750°C and 850°C respectively. Using the measured interstitial concentration at the surface, the silicon interstitial injection into the silicon is calculated for oxidation at 750°C and 850°C. Finally, it is found that the surface boundary condition remains fixed over an interstitial injection rate ranging over 4 orders of magnitude.

Investigation of silicon carbide physical vapor transport growth on the C-terminated face of 6H seeds

ABSTRACTSublimation growth of 6H-SiC has been studied with respect to surface morphology, growth temperature, supersaturation and growth rate. Growth was performed on the C-terminated face of 6H seeds mainly and for comparison also the Si-terminated face was used. Step bunching is observed dependent on different parameters and is strongly influenced by the seed orientation. The growth rate of 4H on the C-face is found to be higher than the rate of 6H grown on the Siface.

Transient behaviour of surface supersaturation caused by variation in bulk supersaturation

The time-dependent equation of Burton, Cabrera and Frank (BCF) is solved analytically for time-dependent bulk supersaturation tending exponentially to a constant value. As a result, the expression for the transient surface supersaturation is found. Special attention is focused on the particular case of the solution presented, corresponding to an abrupt change in bulk supersaturation from σ o to another constant value σ 1. It is shown that, in this particular case, the solution of the time-dependent BCF equation can be approximated by a simple expression which is accurate for large distances from the step edge. The study of the transient behaviour of surface supersaturation reveals that the time necessary to attain the steady-state value of the surface supersaturation does not depend on the difference between the values σ o and σ 1 of the bulk supersaturation. This time is proportional to the relaxation time for leaving the surface adsorption layer, and increases with increasing interstep distance.

Transient analysis of surface supersaturation after crystal face submersion using the analytical and transmission-line matrix (TLM) methods

The time-dependent equation of Burton, Cabrera and Frank (BCF) is analytically solved for transient-state growth conditions occurring immediately after submersion of crystal face in supersaturated solution. The time-dependent BCF model is also solved using the transmission-line matrix (TLM) numerical method. We find that the time necessary to attain the steady-state value of surface supersaturation does not depend on bulk supersaturation but increases with increasing interstep distance. Calculated results obtained by the two methods are shown to be in excellent agreement, indicating a promising potential of the TLM method for solving analytically intractable problems of interest.

Influence of the back-stress effect on the interstep distance and on the surface supersaturation

The back-stress effect is a reduction of local supersaturation in the centre of growth spiral caused by the presence of the rest of spiral step. The interstep distance approximations differ from one another in estimations of this effect. The influence of the interstep distance approximations on calculated supersaturation existing on the surface of growing crystal is analysed on the basis of the BCF surface diffusion theory. The analysis reveals that, although the back-stress effect does not essentially change the shape of the R( σ) curves, it affects strongly the surface supersaturation existing on the step terrace. It is shown that the back-stress effect should be taken into account even at low bulk supersaturations. A new modification of the Cabrera–Levine approximation for interstep distance is proposed. Replacing of bulk supersaturation by the theoretically found surface supersaturation makes the modified Cabrera–Levine approximation usable for both very low and higher bulk supersaturations.

Observation of two-dimensional nucleation on the {0 1 0} face of potassium hydrogen phthalate (KAP) crystals using ex situ atomic force microscopy

The surface topology of microcrystals of potassium hydrogen phthalate (KAP) grown at a relative supersaturation of approximately 26% has been investigated using ex situ atomic force microscopy (AFM). Two distinctly different plate-like habits were observed. The thicker of the two habits resulted from growth by a spiral mechanism on the {0 1 0} faces while a thinner, less common, plate-like morphology occurred due to growth by two-dimensional nucleation on this face. The nuclei, being both asymmetric and partially polygonised, are essentially identical in shape to that of the spirals observed on the thicker crystals. They range in size from about 1200 to 400 nm with the smallest of the nuclei being substantially larger than the estimated radius of the critical two-dimensional nucleus at this supersaturation. This is attributed to continued growth of the nuclei, during separation from the growth solution. The nuclei are distributed non uniformly across the surface of the crystals, being greatest in number at the edges and corners of the {0 1 0} face. This distribution reflects the expected variation in surface supersaturation across the crystal face.

Growth mode issues in epitaxy of complex oxide thin films

We discuss the hetero-epitaxy of complex oxides taking YBa2Cu3O7-δ as an example. Special emphasis is on films grown by pulsed laser deposition (PLD), but other growth techniques are also discussed. Lattice mismatch in three directions is responsible for the fact that a real single crystalline YBa2Cu3O7-δ thin film has not yet been obtained. Nevertheless, growth features typically observed on single crystals (e.g. growth spirals as well as repeated nucleation and growth) are often found on YBa2Cu3O7-δ thin films. We discuss how surface diffusion and supersaturation influence the surface morphology (as observed by atomic force and scanning tunnelling microcopies (AFM/STM)) of complex oxide thin films. In particular, we conclude that the kinetic energy of the impinging adatoms affects the surface diffusivity and therefore the growth mode. © 1998 Chapman & Hall

The kinetic growth model applied to micropipes in 6H-SiC

In this study we investigated the as-grown [0001] Si surface of modified Lely-grown 6H-SiC using atomic force microscopy. We found micropipes that lie in the center of growth spirals whose radii ranged between 25 and 6000 nm. The screw component of the Burgers vector of the micropipe, which is synonymous with the total step height of the growth spiral, ranged from 1 to 25 unit-cells (1.5–37.5 nm). We fitted Frank's theory of hollow core dislocations, as modified by Cabrera and Levine concerning kinetic effects, to these experimental results and obtained values for surface energy and supersaturation at the inner side of the micropipe.

Formation of micropipes in SiC under kinetic aspects

Abstract We measure the radii of micropipes at the {0001} surface of modified Lely grown 6H-SiC and the total step height of the accompanying growth spirals by using atomic force microscopy. The micropipes lie in the center of spirals; the total step height ranges between one and 19 unit-cells (1.5–28.5 nm). We fit Frank's theory of hollow core dislocations as modified with regard to kinetic effects by Cabrera and Levine to these experimental results and obtain values for surface energy and supersaturation near the emergence point of the micropipe.