I examine the molecular dynamics of ice growth from water vapor, focusing on how the attachment kinetics can be augmented by edge-dependent surface diffusion. Although there are significant uncertainties in developing an accurate physical model of this process, it is possible to make some reasonable estimates of surface diffusion rates and admolecule density enhancements, derived from our basic understanding of ice-crystal growth processes. A quantitative model suggests that edge-dependent surface diffusion could substantially enhance terrace nucleation on narrow faceted surfaces, especially at the onset of surface premelting. This result supports our hypothesized mechanism for structure-dependent attachment kinetics, which readily explains the changes in snow crystal growth morphology with temperature depicted in the well-known Nakaya diagram. Many of the model features described here may be amenable to further quantitative investigation using existing computational models of the molecular structure and dynamics of the ice surface.
I examine a variety snow crystal growth experiments performed at temperatures near -2 C, as a function of supersaturation, background gas pressure, and crystal morphology. Although the different experimental data were obtained using quite diverse experimental techniques, the resulting measurements can all be reasonably understood using a single comprehensive physical model for the basal and prism attachment kinetics, together with particle diffusion of water vapor through the surrounding medium and other well-understood physical processes. As with the previous paper in this series, comparing a…
In this paper I examine snow crystal growth near -14 C in comparison with a comprehensive model that includes Structure-Dependent Attachment Kinetics (SDAK). Analyzing a series of ice-growth observations in air, I show that the data strongly support the model, which stipulates that basal growth is described by classical terrace nucleation on faceted surfaces in this temperature region. In contrast, prism growth exhibits a pronounced "SDAK dip" that substantially reduces the nucleation barrier on narrow prism facets (relative to that found on broad prism facets). I use these measurements to fur…
We present experimental data demonstrating the presence of structure-dependent attachment kinetics (SDAK) in ice crystal growth from water vapor near -5 C. Specifically, we find that the nucleation barrier on the basal edge of a thin-walled hollow columnar crystal is approximately ten times smaller than the corresponding nucleation barrier on a large basal facet. These observations support the hypothesis that SDAK effects play an important role in determining the growth morphologies of atmospheric ice crystals as a function of temperature.
In this paper I examine snow crystal growth near -4 C in comparison with a comprehensive model that includes Structure-Dependent Attachment Kinetics (SDAK). Together with the previous paper in this series that investigated growth near 14 C, I show that a substantial body of experimental data now supports the existence of pronounced 'SDAK dips' on basal surfaces near -4 C and on prism surfaces near -14 C. In both cases, the model suggests that edge-associated surface diffusion greatly reduces the nucleation barrier on narrow facet surfaces relative to that found on broad facets. The remarkable …
I examine a variety of snow crystal growth measurements taken at a temperature of -5 C, as a function of supersaturation, background gas pressure, and crystal morphology. Both plate-like and columnar prismatic forms are observed under different conditions at this temperature, along with a diverse collection of complex dendritic structures. The observations can all be reasonably understood using a single comprehensive physical model for the basal and prism attachment kinetics, together with particle diffusion of water vapor through the surrounding medium and other well-understood physical proce…
We present measurements of the diffusion-limited growth of ice crystals from water vapor at a temperature of -5 C, in air at a pressure of $p_{air}=1$ bar. Starting with thin, c-axis ice needle crystals, the subsequent growth morphologies ranged from solid prismatic columns to hollow columns to complex "fishbone" dendritic structures as the supersaturation was increased. We modeled the simpler morphologies using analytical techniques together with a cellular-automata method that yields faceted crystalline structures in diffusion-limited growth. We found that the molecular attachment coefficien…
I describe a semi-empirical molecular model of the surface attachment kinetics governing ice crystal growth from water vapor as a function of temperature, supersaturation, and crystal mesostructure. An important new hypothesis in this model is surface-energy-driven molecular diffusion enabled by a leaky Ehrlich-Schwoebel barrier. The proposed surface-diffusion behavior is sensitive to facet width and surface premelting, yielding structure-dependent attachment kinetics with a complex temperature dependence. By incorporating several reasonable assumptions regarding the surface premelting behavio…
We describe a novel crystal growth instability that enhances the development of thin edges, promoting the formation of plate-like or hollow columnar morphologies. This instability arises when diffusion-limited growth is coupled with structure-depdendent attachment kinetics, specifically when the nucleation barrier on a facet surface decreases substantially as the facet width approaches atomic dimensions. Experimental data are presented confirming the presence of this instability in the growth of ice from water vapor at -15 C. We believe this edge-enhancing effect plays an important role in det…
We present measurements of the diffusion-limited growth of ice crystals from water vapor at different supersaturation levels in air at a temperature of -15 C. Starting with thin, c-axis ice needle crystals, the subsequent growth morphologies ranged from blocky structures on the needle tips (at low supersaturation) to thin faceted plates on the needle tips (at high supersaturation). We successfully modeled the experimental data, reproducing both growth rates and growth morphologies, using a cellular-automata method that yields faceted crystalline structures in diffusion-limited growth. From thi…
I describe a new approach to the classification of snow crystal morphologies that focuses on the most common growth behaviors that appear in normal air under conditions of constant applied temperature and water-vapor supersaturation. The resulting morphological structures are generally robust with respect to small environmental changes and thus should be especially amenable to computational modeling. Because spontaneous structure formation depends on initial conditions, the choice of seed crystal can be an important consideration, and I have found that slender c-axis ice needles provide an exc…
We describe a comprehensive model for the formation and morphological development of atmospheric ice crystals growing from water vapor, also known as snow crystals. Our model derives in part from empirical measurements of the intrinsic ice growth rates as a function of temperature and supersaturation, along with additional observations and analyses of diffusion-driven growth instabilities. We find that temperature-dependent conformational changes associated with surface melting strongly affect layer nucleation dynamics, which in turn determines many snow-crystal characteristics. A key feature …
Graphene, a flat monolayer of carbon atoms packed tightly into a two dimensional hexagonal lattice, has unusual electronic properties which have many promising nanoelectronic applications. Recent Low Energy Electron Microscopy (LEEM) experiments show that the step edge velocity of epitaxially grown 2D graphene islands on Ru(0001) varies with the fifth power of the supersaturation of carbon adatoms. This suggests that graphene islands grow by the addition of clusters of five atoms rather than by the usual mechanism of single adatom attachment. We have carried out Kinetic Monte Carlo (KMC) simul…
Snow crystals growing from water vapor occasionally exhibit morphologies with three-fold (trigonal) symmetry, even though the ice crystal lattice has a molecular structure with six-fold symmetry. In extreme cases, thin platelike snow crystals can grow into faceted forms that resemble simple equilateral triangles. Although far less common than hexagonal forms, trigonal snow crystals have long been observed both in nature and in laboratory studies, and their origin has been an enduring scientific puzzle. In this paper I describe how platelike trigonal structures can be grown on the ends of slend…
This monograph reviews our current understanding of the physical dynamics of ice crystal growth, focusing on the spontaneous formation of complex structures from water vapor (called snow crystals) as a function of temperature, supersaturation, background gas pressure, and other extrinsic parameters. Snow crystal growth is a remarkably rich and rather poorly understood phenomenon, requiring a synthesis of concepts from materials science, crystal-growth theory, statistical mechanics, diffusion-limited solidification, finite-element modeling, and molecular surface processes. Building upon recent …
Facetted growth of snow crystals leads to a rich diversity of forms, and exhibits a 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 bran…
Burton-Cabrera-Frank (BCF) theory has proven to be a versatile framework to relate surface morphology and dynamics during crystal growth to the underlying mechanisms of adatom diffusion and attachment at steps. For an important class of crystal surfaces, including the basal planes of hexagonal close-packed and related systems, the steps in a sequence on a vicinal surface can exhibit properties that alternate from step to step. Here we develop BCF theory for such surfaces, relating observables such as alternating terrace widths as a function of growth conditions to the kinetic coefficients for …
The stacking sequence of hexagonal close-packed and related crystals typically results in steps on vicinal {0001} surfaces that have alternating A and B structures with different growth kinetics. However, because it is difficult to experimentally identify which step has the A or B structure, it has not been possible to determine which has faster adatom attachment kinetics. Here we show that in situ microbeam surface X-ray scattering can determine whether A or B steps have faster kinetics under specific growth conditions. We demonstrate this for organo-metallic vapor phase epitaxy of (0001) GaN…
We describe a numerical model of faceted crystal growth using a cellular automata method that incorporates admolecule diffusion on faceted surfaces in addition to bulk diffusion in the medium surrounding the crystal. The model was developed for investigating the diffusion-limited growth of ice crystals in air from water vapor, where the combination of bulk diffusion and strongly anisotropic molecular attachment kinetics yields complex faceted structures. We restricted the present model to cylindrically symmetric crystal growth with relatively simple growth morphologies, as this was sufficient …
We study a minimal stochastic model of step bunching during growth on a one-dimensional vicinal surface. The formation of bunches is controlled by the preferential attachment of atoms to descending steps (inverse Ehrlich-Schwoebel effect) and the ratio $d$ of the attachment rate to the terrace diffusion coefficient. For generic parameters ($d > 0$) the model exhibits a very slow crossover to a nontrivial asymptotic coarsening exponent $β\simeq 0.38$. In the limit of infinitely fast terrace diffusion ($d=0$) linear coarsening ($β$ = 1) is observed instead. The different coarsening behaviors are…
Step meandering instability in a Burton-Cabrera-Frank (BCF)-type model for the growth of an isolated, atomically high step on a crystal surface is analyzed. It is assumed that the growth is sustained by the molecular precursors deposition on a terrace and their decomposition into atomic constituents; both processes are explicitly modeled. A strongly nonlinear evolution PDE for the shape of the step is derived in the long-wave limit and without assuming smallness of the amplitude; this equation may be transformed into a convective Cahn-Hilliard-type PDE for the step slope. Meandering is studied…
We study the dynamics of island nucleation in the presence of adsorbates using kinetic Monte Carlo simulations of a two-species growth model. Adatoms (A-atoms) and impurities (B-atoms) are codeposited, diffuse and aggregate subject to attractive AA- and AB-interactions. Activated exchange of adatoms with impurities is identified as the key process to maintain decoration of island edges by impurities during growth. While the presence of impurities strongly increases the island density, a change in the scaling of island density with flux, predicted by a rate equation theory for attachment-limite…
I describe an adaptable apparatus for making precision measurements of the growth of faceted ice prisms from water vapor as a function of temperature, supersaturation, and background gas pressure. I also describe procedures for modeling growth data to disentangle a variety of physical effects and better understand systematic errors and measurement uncertainties. By enabling precise ice-growth measurements over a broad range of environmental conditions, this apparatus is well suited for investigating the molecular attachment kinetics at the ice/vapor interface, which is needed to understand and…
Oriented attachment (OA) has become a well-recognized mechanism for the growth of metal, ceramic, and biomineral crystals. While many computational and experimental studies of OA have shown that particles can attach with some misorientation then rotate to remove adjoining grain boundaries, the underlying atomistic pathways for this "Imperfect OA" process remain the subject of debate. In this study, molecular dynamics and in situ TEM were used to probe the crystallographic evolution of up to 30 gold and copper nanoparticles during aggregation. It was found that Imperfect OA occurs because (1) g…
We derive a set of algorithms for simulating the diffusion-limited growth of faceted crystals using local cellular automata. This technique has been shown to work well in reproducing realistic crystal morphologies, and the present work provides a more rigorous physical foundation that connects the numerical code to the physics of attachment kinetics and diffusion dynamics. We then apply these algorithms to examine a novel morphological transition in the growth of thin plate-like crystals.
We study a moving boundary model of non-conserved interface growth that implements the interplay between diffusive matter transport and aggregation kinetics at the interface. Conspicuous examples are found in thin film production by chemical vapor deposition and electrochemical deposition. The model also incorporates noise terms that account for fluctuations in the diffusive and in the attachment processes. A small slope approximation allows us to derive effective interface evolution equations (IEE) in which parameters are related to those of the full moving boundary problem. In particular, th…
Ice crystals growing in highly supersaturated air at temperatures near -5 C exhibit a distinctive, nonplanar dendritic morphology that has not been previously documented or explained. We examine this structure and identify its most prominent features in relation to the ice crystal lattice. Developing a full 3D numerical model that reproduces this robust morphology will be an interesting challenge in understanding diffusion-limited crystal growth in the presence of highly anisotropic surface attachment kinetics.
Images of the morphology of GaN (0001) surfaces often show half-unit-cell-height steps separating a sequence of terraces having alternating large and small widths. This can be explained by the $αβαβ$ stacking sequence of the wurtzite crystal structure, which results in steps with alternating $A$ and $B$ edge structures for the lowest energy step azimuths, i.e. steps normal to $[0 1 \bar{1} 0]$ type directions. Predicted differences in the adatom attachment kinetics at $A$ and $B$ steps would lead to alternating $α$ and $β$ terrace widths. However, because of the difficulty of experimentally id…
The electromigration-induced shape evolution of two-dimensional vacancy islands on a crystal surface is studied using a continuum approach. We consider the regime where mass transport is restricted to terrace diffusion in the interior of the island. In the limit of fast attachment/detachment kinetics a circle translating at constant velocity is a stationary solution of the problem. In contrast to earlier work [O. Pierre-Louis and T.L. Einstein, Phys. Rev. B 62, 13697 (2000)] we show that the circular solution remains linearly stable for arbitrarily large driving forces. The numerical solution …
As a two-dimensional material, graphene can be naturally obtained via epitaxial growth on a suitable substrate. Growth condition optimization usually requires an atomistic level understanding of the growth mechanism. In this article, we perform a mechanistic study about graphene growth on Ir(111) surface by combining first principles calculations and kinetic Monte Carlo (kMC) simulations. Small carbon clusters on the Ir surface are checked first. On terraces, arching chain configurations are favorable in energy and they are also of relatively high mobilities. At steps, some magic two-dimension…
We examine the shape of a an isolated, dislocation-free ice crystal when it is in equilibrium with the vapor phase in an isothermal closed environment, as a function of temperature. From our analysis we draw the following conclusions: 1) The equilibrium shape has not yet been definitively measured for ice crystals; 2) The surface energy anisotropy is likely cusp-like near the facet angles, and the size of the cusps can be estimated from crystal growth measurements; 3) The equilibrium shape is likely nearly spherical with only small faceted regions; 4) The time needed to reach equilibrium is li…
We propose a new type of morphological instability in the diffusion-limited growth of faceted crystals from the vapor phase that can explain the formation of thin ice plates at temperatures near -15 C. The instability appears when the attachment kinetics for facet growth depends strongly on the morphology of the facet surface. In particular, we propose that the condensation coefficient for growth of ice prism facets increases dramatically when the width of the facet approaches atomic dimensions. This model reconciles several conflicting measurements of ice crystal growth, and makes additional …
A one-dimensional continuum description of growth on vicinal surfaces in the presence of immobile impurities predicts that the impurities can induce step bunching when they suppress the diffusion of adatoms on the surface. In the present communication we verify this prediction by kinetic Monte-Carlo simulations of a two-dimensional solid-on-solid model. We identify the conditions where quasi one-dimensional step flow is stable against island formation or step meandering, and analyse in detail the statistics of the impurity concentration profile. The sign and strength of the impurity-induced st…
Features of the crystallization kinetics define directly the rate characteristics: the crystal nucleation rate, the crystal growth rate and the so-called kinetic rate factor known also as the attachment rate (of particles to the surface of a crystalline nucleus). We show that the kinetic rate factor as function of the reduced temperature follows a unified scaled power law. This scenario is confirmed by our simulation results for model atomistic systems (crystallizing volumetric liquids and liquid thin film) and by available experimental data for crystallizing polymers. We find that the exponen…
The quasistatic approximation is a useful but questionable simplification for analyzing step instabilities during the growth/evaporation of vicinal surfaces. Using this approximation, we characterized in Part I of this work the effect on stability of different mechanisms and their interplay: elastic step-step interactions, the Schwoebel barrier, and the chemical coupling of the diffusion fields on adjacent terraces. In this second part, we present a stability analysis of the general problem without recourse to the quasistatic approximation. This analysis reveals the existence of a supplementar…
We show that the decay of sinusoidal ripples on crystal surfaces, where mass transport is limited by the attachment and detachment of atoms at the step-edges, is remarkably different from the decay behavior that has been reported until now. Unlike the decreasing or at most constant rate of amplitude decay of sinusoidal profiles observed in earlier work, we find that the decay rate increases with decreasing amplitude in this kinetic regime. The rate of shape invariant amplitude relaxation is shown to be inversely proportional to both the square of the wavelength and the current amplitude. We ha…
Using recently proposed improvements of Reflection Electron Microsopy (REM) we study in perfectly controlled thermodynamics conditions spiral shapes and spirals on Si(111) surface. It is shown that the new method named low distortion reflection electron microscopy (LODREM) is a powerful instrument, resolving in much more details (compared with REM) growth or evaporation spirals at the crystal surface. More precisely, we examine the distance between two successive steps of a spiral at growth (or evaporation) with respect to the supersaturation (or undersaturation). It is found that this distanc…
Two-dimensional hexagonal boron nitride (h-BN) single crystals with various shapes have been synthesized by chemical vapor deposition over the past several years. Here we report the formation of three-leaf dart (3LD)-shaped single crystals of h-BN on Cu foil by atmospheric-pressure chemical vapor deposition. The leaves of the 3LD-shaped h-BN are as long as 18 μm and their edges are smooth armchair on one side and stepped armchair on the other. Careful analysis revealed that surface oxygen plays an important role in the formation of the 3LD shape. Oxygen suppressed h-BN nucleation by passivatin…
Al-induced layer exchange crystallization of amorphous Ge thin films has been demonstrated recently, and provides a suitable system to characterize, model and control Ge crystal growth on non-crystalline substrates. Direct observation of Ge transfer to the surface of Al through an interposed GeOx interfacial layer allows independent measurement of the density and average area of crystalline Ge islands formed on the surface. Based on these experimental observations, the Johnson-Mehl-Avrami-Kologoromov phase transformation theory is extended to model nanoscale nucleation and growth of Ge islands…
We study surface and bulk properties of porous films produced by a model in which particles incide perpendicularly to a substrate, interact with deposited neighbors in its trajectory, and aggregate laterally with probability of order $a$ at each position. The model generalizes ballistic-like models by allowing attachment to particles below the outer surface. For small values of $a$, a crossover from uncorrelated deposition (UD) to correlated growth is observed. Simulations are performed in 1+1 and 2+1 dimensions. Extrapolation of effective exponents and comparison of roughness distributions co…
We develop a mesoscopic dislocation dynamics model for vacancy-assisted dislocation climb by upscalings from a stochastic model on the atomistic scale. Our models incorporate microscopic mechanisms of (i) bulk diffusion of vacancies, (ii) vacancy exchange dynamics between bulk and dislocation core, (iii) vacancy pipe diffusion along the dislocation core, and (iv) vacancy attachment-detachment kinetics at jogs leading to the motion of jogs. Our mesoscopic model consists of the vacancy bulk diffusion equation and a dislocation climb velocity formula. The effects of pipe diffusion and the jog str…
In order to control the grain structure of multi-crystalline (mc) silicon during directional solidification, the development process of grain boundaries (GBs) with respect to the temperature gradient should be understood. A phase-field model incorporated with anisotropic interface energy and anisotropic attachment kinetic coefficient has produced the faceted shape of a growing silicon crystal, which is in agreement with experimental observation. The growth of coupled silicon grains under various growth velocities has been simulated to see the morphology of the solid-liquid front and the develo…
The Burton-Cabrera-Frank (BCF) model for the flow of line defects (steps) on crystal surfaces has offered useful insights into nanostructure evolution. This model has rested on phenomenological grounds. Our goal is to show via scaling arguments the emergence of the BCF theory for non-interacting steps from a stochastic atomistic scheme of a simplified kinetic solid-on-solid model in one spatial dimension. Our main assumptions are: adsorbed atoms (adatoms) form a dilute system, and elastic effects of the crystal lattice are absent. The step edge is treated as a front that propagates via probabi…
A sublimating vicinal crystal surface can undergo a step bunching instability when the attachment-detachment kinetics is asymmetric, in the sense of a normal Ehrlich-Schwoebel effect. Here we investigate this instability in a model that takes into account the subtle interplay between sublimation and step-step interactions, which breaks the volume-conserving character of the dynamics assumed in previous work. On the basis of a systematically derived continuum equation for the surface profile, we argue that the non-conservative terms pose a limitation on the size of emerging step bunches. This c…
Below the roughening transition, crystal surfaces exhibit nanoscale line defects, steps, that move by exchanging atoms with their environment. In homoepitaxy, we analytically show how the motion of a step train in vacuum under strong desorption can be approximately described by nonlinear laws that depend on local geometric features such as the curvature of each step, as well as suitably defined effective terrace widths. We assume that each step edge, a free boundary, can be represented by a smooth curve in a fixed reference plane for sufficiently long times. Besides surface diffusion and evapo…
In this work, we study the crystalline nuclei growth in glassy systems focusing primarily on the early stages of the process, at which the size of a growing nucleus is still comparable with the critical size. On the basis of molecular dynamics simulation results for two crystallizing glassy systems, we evaluate the growth laws of the crystalline nuclei and the parameters of the growth kinetics at the temperatures corresponding to deep supercoolings; herein, the statistical treatment of the simulation results is done within the mean-first-passage-time method. It is found for the considered syst…
The physics of ice crystal growth from the liquid phase, especially in the presence of salt, has received much less attention than the growth of snow crystals from the vapour phase. The growth of so-called frazil ice by solidification of a supercooled aqueous salt solution is consistent with crystal growth in the basal plane being limited by the diffusive removal of the latent heat of solidification from the solid--liquid interface, while being limited by attachment kinetics in the perpendicular direction. This leads to the formation of approximately disk-shaped crystals with a low aspect rati…
Numerous experimental data on the rapid solidification of binary systems exhibit the formation of metastable solid phases with the initial (nominal) chemical composition. This fact is explained by complete solute trapping leading to diffusionless (chemically partitionless) solidification at a finite growth velocity of crystals. Special attention is paid to developing a model of rapid solidification which describes a transition from chemically partitioned to diffusionless growth of crystals. Analytical treatments lead to the condition for complete solute trapping which directly follows from the…
