The official published version of the Article can be accessed from the link below - Copyright @ 2011 APSUsing atomic scale time-dependent density functional calculations we confirm that both diffusion-controlled and diffusionless crystallization modes exist in simple 2D systems. We provide theoretical evidence that a faceted to nonfaceted transition is coupled to these crystallization modes, and faceting is governed by the local supersaturation at the fluid-crystalline interface. We also show that competing modes of crystallization have a major influence on mesopattern formation. Irregularly branched and porous structures are emerging at the crossover of the crystallization modes. The proposed branching mechanism differs essentially from dendritic fingering driven by diffusive instability.This work has been supported by the EU FP7
Collaborative Project ENSEMBLE under Grant
Agreement NMP4-SL-2008-213669 and by the
Hungarian Academy of Sciences under Contract
No. OTKA-K-62588

Gyula I. Tóth

György Tegze

J. G. Ojalvo

László Gránásy

Physical Review Letters

English

Crossref

Faceting and Branching in 2D Crystal Growth

Tegze, G

Toth, GI

Granasy, L

Brunel University Research Archive

Faceting and Branching in 2D Crystal GrowthGyo¨rgy Tegze,1 Gyula I. To´th,1 and La´szlo´ Gra´na´sy1,21Research Institute for Solid State Physics and Optics, P.O. Box 49, H-1525 Budapest, Hungary2BCAST, Brunel University, Uxbridge, Middlesex, UB8 3PH, United Kingdom(Received 8 November 2010; published 11 May 2011)Using atomic scale time-dependent density functional calculations we confirm that both diffusion-controlled and diffusionless crystallization modes exist in simple 2D systems. We provide theoreticalevidence that a faceted to nonfaceted transition is coupled to these crystallization modes, and faceting isgoverned by the local supersaturation at the fluid-crystalline interface. We also show that competingmodes of crystallization have a major influence on mesopattern formation. Irregularly branched andporous structures are emerging at the crossover of the crystallization modes. The proposed branchingmechanism differs essentially from dendritic fingering driven by diffusive instability.DOI: 10.1103/PhysRevLett.106.195502 PACS numbers: 81.10.Aj, 89.75.KdAvast variety of natural crystals, including minerals andsnowflakes, display flat, smooth surfaces. Faceting is alsopresent in advanced, technologically important materials,such as ceramics [1], semiconductors [2], polymers [3],metallic nanostructures [4], colloid suspensions [5], andmonolayers of various surfactant molecules [6,7]. Theamazing complexity of faceted patterns has been generat-ing considerable interest for decades. Various crystalshapes; compact symmetric, dendritic, ramified fractaland needlelike morphologies have been investigated ex-perimentally [1–5]. While robust theoretical interpreta-tions of faceted equilibrium shapes exist, the dynamicalaspects of faceted pattern formation are less understood.The main difficulty in establishing a theoretical descriptionlies in the multiscale nature of faceted crystal growth, themesoscale behavior is influenced by the local arrangementof the particles. Russel et al. [8] have identified somemesoscale characteristics of crystal growth kinetics: (i) adiffusion-controlled (or ‘‘slow’’) growth mode that maylead to diffusive instability and dendritic branching inmonodisperse colloids [8,9], and (ii) a diffusionless (or‘‘fast’’) steady growth mode, in which crystalline orderingtakes place without significant density change. The exis-tence of these modes is verified experimentally by inves-tigating crystallization kinetics in hard-sphere suspensions[10,11]. Besides, in some organic glasses discontinuousenhancement of crystal growth is observed [12–14], whichis associated with diffusionless crystallization [15].Although various theories have been developed to explainthe existence of diffusion-controlled and diffusionlessrapid modes [14–16], none of them predicts a faceted-nonfaceted morphological transition often associatedwith different crystallization modes, when interpreting ex-periments [3]. Growth mode (i) and mode (ii) have alsobeen observed in simple mean-field models [17], althoughthese theories address anisotropic crystal growth in a phe-nomenological way. In contrast, molecular theories basedon the classical density functional technique (DFT) predictnot only the anisotropic bulk crystal properties, such as thecrystal structure and elastic constants, but also anisotropicsurface properties (e.g., surface tension [18]). A simpledynamical DFT [19], the phase-field crystal (PFC) model[20] is able to address crystallization kinetics up to themesoscale [21]. While the phase-field crystal model rep-resents the average of the microscopic states over a coarsegraining time, it does describe crystal defects. Grainboundaries and dislocations can be observed directly insimulations [20], while diffusing point defects (i.e., vacan-cies) are time averaged and are represented by a lowerfilling factor (lower integral density) at the lattice sites[22]. The Kirkendall effect that assumes vacancy diffusionhas also been investigated in the binary PFC model [23]. Ithas also been shown that the crystal growth anisotropy isinherent in the PFC model [24].In this Letter we investigate faceting during crystalgrowth using the PFC model, and interpret crystallinepattern formation in terms of two basic crystal growthmechanisms [type (i) and (ii)]. Although we simulatedsurface diffusion-coupled 2D crystallization (relevant di-rectly for 2D colloids, surfactant monolayers and islandformation on substrates, etc.), the conclusions we are goingto draw are more generic. None of the underlying processes(e.g., faceting, and diffusion-controlled growth) are spe-cific to these systems.The PFC model we use relies on the time-averagedparticle density , and applies on the diffusional time scale.The dimensionless free energy of the inhomogeneous sys-tem relative to a homogeneous reference state of densityrefL can be derived [25] from the Ramakrishnan-Yussoufftype DFT [26]:F ¼Zdrc2½þ ð1þr2Þ2c þ c44; (1)where c / ð refL Þ=refL is the scaled density differencerelative to refL . The model parameter  can be connected tosuch physical properties as the bulk moduli of the fluid andPRL 106, 195502 (2011) P HY S I CA L R EV I EW LE T T E R Sweek ending13 MAY 20110031-9007=11=106(19)=195502(4) 195502-1  2011 American Physical Societycrystalline phases at the reference density and temperature.In two dimensions, the homogeneous fluid and the hex-agonal crystal can coexist [20]. The transition layer be-tween these coexisting phases can be either thin or severalatomic layers thick. The interface thickness is tuned by themodel parameter . Crystallization and the diffusionaltransport are described by the time evolution of the particledensity (c ), and governed by a diffusive equation ofmotion:@c@¼ r2 Fcþ ; (2)where  is the dimensionless time, and  represents acolored Gaussian noise characterized by the correlatorhðr; Þðr0; 0Þi ¼ r2gðjr r0j; Þð 0Þ. Here controls the magnitude of the thermal fluctuations relativeto the free energy barriers and gðjr r0j=Þ is a highfrequency cutoff function [27] for wavelengths shorterthan the lattice spacing ().First, we investigate the dynamic properties of thecrystal-fluid interface as a function of the thermodynamicdriving force. For the sake of simplicity, we set the noisecorrelator to zero ( ¼ 0), and choose  ¼ 0:75 to en-sure faceted equilibrium shape. Crystal growth has beeninitiated using a crystal seed, while the initial particledensity (c 0) of the fluid phase has been used to tune thethermodynamic driving force of the phase transition.Some of the crystallization kinetics dependent interfaceproperties are summarized in Fig. 1. At low crystal growthrate, the interface is fairly thin and faceted [Fig. 1(a)],while at high growth rate it extends to several atomic layersand shows rounded corners [Fig. 1(b)]. Dynamical reduc-tion of the growth anisotropy has also been observed inmolecular monolayers and proved to be mainly of kineticorigin [28].The time evolution of the coarse-grained particle density( ~c ) profile shows distinct characteristics for faceted andnonfaceted interfaces. ( ~c has been obtained by two-dimensional finite impulse response filtering [29].) Atlow driving forces the time dependence of the interfacevelocity is parabolic (v / 1=2). Crystal growth is con-trolled by diffusive transport through an extending deple-tion layer [growth mode (i), see Fig. 1(c)]. At highsupersaturations crystal growth is limited by the rate ofparticle rearrangement at the interface from liquid state tosolid state [growth mode (ii)]: here the diffusion layershows a steady profile [Fig. 1(d)] and the interface velocityis constant. As opposed to growth mode (i), the densitydifference between the crystalline and liquid phases issubstantially reduced, which can be interpreted as ‘‘va-cancy trapping’’ into the crystal, analogously to the phe-nomenon of ‘‘solute trapping’’ in rapidly solidifyingalloys. The concentration of the quenched-in vacancies[30] can be characterized by the average crystal density.Following Aziz [31], we have defined an effective partitioncoefficient using the average solid and liquid densitiesat the interface as keff ¼ ~c S= ~c L and we have fittedkeff ¼ k1=ð1þv=vDÞ0 to our data. In the formula k0 ¼~c ES= ~cEL ¼ 0:649 is the equilibrium partition coefficient 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 0.01  0.1  1k effv(e)simulationAziz fit-0.6-0.55-0.5-0.45-0.4-0.35 0.2  1  5~ L x 1000(f) 0=-0.50320=-0.5033550=-0.5033750=-0.50350=-0.53 10 100 0.01  0.1  1d and d Dv(g)d  dDFIG. 1 (color online). Two modes of interface propagation:Close-up view of the density map at the interface for the (a) -diffusion-controlled ‘‘slow’’ (c 0 ¼ 0:53) (b) diffusionless‘‘fast’’ (c 0 ¼ 0:5032) growth modes. The coarse-grained den-sity profiles ( ~c ) are shown at different times for (c) the slow(c 0 ¼ 0:53) and (d) the fast (c 0 ¼ 0:5032) modes. Minimaof the curves correspond to the fluid density at the interface( ~c L), while the density steps left to the minima represent thedensity change across the fluid-crystalline interface. ~c S is thesolid density at the interface. (e)–(g) Merged data for c 0 ¼0:53; 0:5035; 0:503 375; 0:503 35; 0:5032 are plotted:(e) keff vs velocity. (f) ~c L vs time. (g) The interface thickness, d,and diffusion length, dD vs velocity.PRL 106, 195502 (2011) P HY S I CA L R EV I EW LE T T E R Sweek ending13 MAY 2011195502-2defined by the phase diagram of the system [20], v is theinterface velocity, while vD is the fitted characteristicinterface velocity for point-defect trapping. The effectivepartition coefficient vs interface velocity data are mergedinto master plots [Fig. 1(e)] for a sequence of simulations(c 0 ¼ 0:53; 0:5035; 0:503 375; 0:503 35 and0:5032). This master plot shows that the solid-liquiddensity difference at the interface decreases as the growthrate increases (keff increases towards 1 at high interfacevelocities). The density of the crystalline phase is dynami-cally adjusted to the fluid density, making steady growthfeasible. The Aziz formula is in convincing agreement withour data which implies that the density change correspondsto defect trapping. Similar behavior has been observed insimple field theoretic models [32].While at first glance mode switching in the dynamics ishidden, the faceted to nonfaceted transition associated withthis phenomenon is quite spectacular. A microscopic ex-planation of faceting can be given considering the localthermodynamic conditions at the interface. Close to equi-librium, in the faceted growth regime, the crystal growslayerwise; filling of a lattice site in an existing crystal layeris preferred to new layer formation via nucleation. At theinterface ~c L oscillates in time [Fig. 1(f)], where eachbreakdown from a local maximum corresponds to thenucleation of a crystal layer. Figure 1(f) also shows thatincreasing ~c L at the interface results in a vanishing barrierfor crystal layer nucleation, the interface becomes rough onthe atomic scale, and as a consequence, faceted to non-faceted transition occurs. Also note that for layer-by-layergrowth vacancy trapping is hardly feasible since the fillingof sites in the existing layers is preferred to new layerformation.Besides faceting, surface roughening [33] is manifestedin dynamical interface broadening [34] as the growth rateincreases [see Fig. 1(g)]. Figure 1(g) also shows that theinterface thickness crosses the diffusion length as the inter-face velocity increases. The mode switching occurs at thecrossover of these curves, at high driving forces the inter-face layer and the depleted zone overlap, due to trapping ofvacancies into the interface layer.Next, we examine mesoscale patterning near modeswitching, and compare the simulations to micrographsof 2D colloid crystals. The free growth of single crystalshas been investigated in the presence of thermal fluctua-tions ( ¼ 4 103). In the diffusion-controlled regimefaceted dendritic mesopatterns evolve [Fig. 2(c)]; branch-ing is driven by diffusive instability [35]. At higher super-saturations thermal fluctuation driven random switchingfrom fast to slow mode is observed along the interface.The coexisting slow and fast mode interfaces outlinefractal-like mesopatterns [Figs. 2(d) and 2(e)]. Contraryto diffusive instability driven branching, fractal-like armsare due to the difference in the propagation velocity forslow and fast mode interfaces. Pore formation via consecu-tive branching and merging of fractal arms has also beenobserved [see Fig. 2(e)], which is not common whenbranching is driven by diffusive instability. Such Swiss-cheese-like patterns are often observed in monolayers[5,36]. Simulations show a striking resemblance to mor-phologies observed in optical micrographs of colloid singlecrystals [5] [see Fig. 2(a) and 2(b)], and fractal-like growthin molecular monolayers [28].We note that the fast and slow modes can coexist evenin the zero noise limit ( ¼ 0). In this case, vacancytrapping and mode selection are governed by the localatomic arrangement at the solid-liquid interface that varieswith crystallographic orientation. Increasing the noiseFIG. 2 (color online). Single crystal growth patterns in thepresence of thermal fluctuations: (a),(b): dendritic and fractal-like 2D colloid crystals (micrographs by Skjeltorp [5]). (c),(d):Coarse-grained density maps for dendritic (c 0 ¼ 0:505) andfractal-like branched patterns (c 0 ¼ 0:504) in PFC simula-tions. (e) Fractal branching mechanism: Close-up view of mode-switching along the interface. Density waves in the liquidinherent to the fast mode, while depleted layer (dark) indicatesslow mode interface propagation.PRL 106, 195502 (2011) P HY S I CA L R EV I EW LE T T E R Sweek ending13 MAY 2011195502-3amplitude, the fast mode becomes increasingly preferred,while branching events become more frequent and theporosity of the growing crystal increases.Summarizing, using a microscopic theory, symbiosis offaceting with growth mode selection is predicted: if facetedgrowth exists at a temperature (), the slow mode, whichexhibits decreasing supersaturation necessarily leads tolayer-by-layer growth. Also layerwise growth largely pre-vents excess vacancy trapping into the crystal; thus, thedynamical adjustment of crystal density and the ‘‘fast’’mode interface propagation are not feasible. 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Faceting and branching in 2D crystal growth

Using atomic scale time-dependent density functional calculations we confirm that both diffusion-controlled

and diffusionless crystallization modes exist in simple 2D systems. We provide theoretical evidence that a faceted to nonfaceted transition is coupled to these crystallization modes, and faceting is governed by the local supersaturation at the fluid-crystalline interface. We also show that competing modes of crystallization have a major influence on mesopattern formation. Irregularly branched and porous structures are emerging at the crossover of the crystallization modes. The proposed branching mechanism differs essentially from dendritic fingering driven by diffusive instability

Tegze, György

Tóth, Gyula I.

Gránásy, László

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Faceting and branching in 2D crystal growth

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Brunel University Research Archive

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