Hindawi Publishing CorporationJournal of Computational Methods in PhysicsVolume 2013 Article ID 174806 11 pageshttpdxdoiorg1011552013174806

Research ArticleQuantitative Modeling of Faceted Ice Crystal Growth fromWater Vapor Using Cellular Automata

Kenneth G Libbrecht

Department of Physics California Institute of Technology Pasadena CA 91125 USA

Correspondence should be addressed to Kenneth G Libbrecht kglcaltechedu

Received 27 March 2013 Accepted 9 September 2013

Academic Editor Xavier Ferrieres

Copyright copy 2013 Kenneth G Libbrecht This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

We describe a numerical model of faceted crystal growth using a cellular automata method The model was developed forinvestigating the diffusion-limited growth of ice crystals from water vapor when the surface boundary conditions are determinedprimarily by strongly anisotropicmolecular attachment kineticsWe restricted ourmodel to cylindrically symmetric crystal growthwith relatively simple growth morphologies as this was sufficient for making quantitative comparisons between models and icegrowth experiments Overall this numerical model appears to reproduce ice growth behavior with reasonable fidelity over a widerange of conditions More generally the model could easily be adapted for other material systems and the cellular automatatechnique appears well suited for investigating crystal growth dynamics when strongly anisotropic surface attachment kineticsyields faceted growth morphologies

1 Introduction

The formation of crystalline structures during solidificationyields a remarkable variety of morphological behaviorsresulting from the often subtle interplay of nonequilibriumphysical processes over a range of length scales In manycases seemingly small changes in surfacemolecular structureand dynamics at the nanoscale can produce large morpho-logical changes at all scales Some examples include freedendritic growth from the solidification of melts wheresmall anisotropies in the interfacial surface energy governthe overall characteristics of the growth morphologies [1 2]whisker growth from the vapor phase initiated by single screwdislocations and other effects [3] the formation of porousaligned structures from directional freezing of compositematerials [4] and a range of other pattern formation sys-tems [5 6] Since controlling crystalline structure formationduring solidification has application in many areas of mate-rials science much effort has been directed toward betterunderstanding the underlying physical processes and theirinteractions

We have been exploring the growth of ice crystals fromwater vapor in an inert background gas as a case study of

how complex faceted structures emerge in diffusion-limitedgrowth Although this is a relatively simple monomolecularphysical system ice crystals exhibit columnar and plate-like growth behaviors that depend strongly on temperatureand much of the phenomenology of their growth remainspoorly understood [7ndash9] Ice has also become something of astandard test system for investigating numerical methods offaceted crystal growth [10 11] A better understanding of icecrystal formation yields insights into the detailed molecularstructure and dynamics of the ice surface which in turncontributes to our understanding ofmany environmental andbiological processes involving ice [12ndash14]

In our investigation of how surface energy and attach-ment kinetics affect ice growth dynamics we needed a quan-titative numerical model that would allow us to ldquogrowrdquomodelcrystals for comparison with experimental measurements ofgrowth rates and morphologies Although proven numericalmethods for modeling diffusion-limited growth have beenavailable for years many of the existing methods are ill suitedfor modeling ice growth behavior For example phase-field[15 16] and front-tracking [17] methods have demonstratedthe ability to accurately model diffusion-limited growth forthe case of fast attachment kinetics and a weakly anisotropic

2 Journal of Computational Methods in Physics

surface energy which is typical of solidification from themeltThese systems typically yield unfaceted dendritic structureshowever in contrast to strongly faceted ice structures Earlymodels for the growth of faceted crystals [18 19] weregenerally too limited to allow quantitative comparisons withice growth data

Modeling diffusion-limited growth in systems withstrong surface anisotropies has proven difficult and onlyrecently have researchers demonstrated robust techniquescapable of generating structures that are both faceted anddendritic Gravner and Griffeath [10] described an especiallypromising cellular automata simulator that solves the dif-fusion equation by nearest neighbor relaxation including aset of parameterized nearest neighbor rules to define theboundary conditions at the crystal interface This methodyields a deterministic dendritic growth behavior in whichfaceting follows the symmetry of the predefined numericalgrid

Barrett et al [11] also developed a robust adaptive meshtechnique that generated faceted dendritic crystal growthpatterns In this work the authors found that a stronglyanisotropic surface energy was required to produce faceteddendritic growthwhile anisotropic attachment kinetics alonewere not sufficient to reproduce this behavior We havesuggested that the ice case is more likely described bythe opposite characteristics an essentially isotropic surfaceenergy together with strongly anisotropic attachment kinet-ics the latter dominating the growth behavior [20] In factthe relative roles played by these two physical effects is notyet known with certainty

The relative merits of different computational methodsfor modeling diffusion-limited growth in the presence ofstrong surface anisotropies are not presently well understoodand this is an area of current research Moreover our knowl-edge of the surface physics governing the growth of facetedmaterials is itself rather poor including the relative contri-butions of the anisotropies in surface energy and attachmentkinetics in different materials In our experience progress onboth these research fronts is linked better modelingmethodsallow better interpretation of growth experiments and thisin turn leads to a better understanding of the surface physicsinput into the models

Below we describe a cellular automata method for mod-eling diffusion-limited growth in the presence of stronglyanisotropic molecular attachment kinetics focusing on icegrowth from water vapor Our treatment of the surfaceboundary conditions derives from known surface physicsand thus replaces the somewhat unphysical parameterizationadopted in [10] We have found our model to be robust wellbehaved computationally straightforward and quite flexiblefor exploring ice growth behaviors The 2D cylindricallysymmetric model described below has been especially use-ful for investigating the simple growth morphologies oftenproduced in experiments allowing the rapid generation ofhundreds of model crystals with different input assumptionsThis model has allowed us to better examine the surfacephysical processes governing ice growth rates and it allowsstraightforward adaptation for use in other investigationsinvolving faceted diffusion-limited crystal growth

2 Modeling Ice Crystal Growth

A variety of physical processes are involved in the growthof ice crystals from water vapor in an inert background gasParticle diffusion plays a large role as water vapor moleculesmust diffuse through the gas to reach the growing crystalHeat diffusion is also present since the latent heat generatedby condensation must be removed from the crystal Surfaceenergy effects are present via the Gibbs-Thomson effect andsurface attachment kinetics govern the placement of watermolecules into the ice crystal lattice If the background gasis not inert surface chemical effects may also be significant

Our focus here will be on ice growth under near atmo-spheric conditions which allows a number of simplificationsin the problem Particle transport is described by the diffusionequation

120597119888

120597119905= 119863nabla

2119888 (1)

using the notation in [7] where 119888(119909) is the water moleculenumber density surrounding the crystal and119863 is the diffusionconstant The timescale for diffusion to adjust the vaporconcentration in the vicinity of a crystal is 120591diffusion asymp 119877

2119863

where 119877 is a characteristic crystal sizeThis is to be comparedwith the growth time 120591growth asymp 2119877V119899 where V119899 is the growthvelocity of the solidification front normal to the surface Theratio of these two timescales is the Peclet number 119901Peclet =119877V1198992119863 For typical growth rates of ice crystals we find

119901Peclet ≲ 10minus5 which means that diffusion adjusts the particle

density around the crystal much faster than the crystalshape changes In this case the diffusion equation reducesto Laplacersquos equation nabla2119888 = 0 which must be solved withthe appropriate boundary conditions Using this slow-growthlimit simplifies the problem considerably in comparison tomuch of the literature on diffusion-limited solidification

It is convenient to work with the supersaturation 120590(119909) =[119888(119909) minus 119888sat]119888sat where 119888sat is the equilibrium vapor densityabove a flat ice surface Then the diffusion equation becomes

120597120590

120597119905= 119863nabla

2120590 (2)

and the continuity equation at the interface gives

V119899=

119863

119888solid(119899 sdot nabla119888)

surf=119888sat119863

119888solid(119899 sdot nabla120590)

surf (3)

where 119888solid = 120588ice119898 is the number density for ice 120588iceis the ice density and 119898 is the mass of a water moleculeFor the outer boundary one typically assumes a constantsupersaturation 120590

infin

Including latent heating of the crystal wouldmean solvinga double diffusion problem involving both particle and heatdiffusion simultaneously complicating the problem consider-ably However since the thermal conductivity of ice is muchhigher than that of air heating raises the temperature of thecrystal roughly uniformly at least for simple morphologieswhich increases the effective 119888sat for the crystal Thus themain effect of heating can be approximated by a relatively

Journal of Computational Methods in Physics 3

small rescaling of 120590infin

[7] Furthermore many experimentsare done using ice crystals resting on a thermally conductingsubstrate and in this case the effects of latent heating areessentially negligible [7] For these reasons we have ignoredlatent heating in the present model with the understandingthat heating effects may be important in some experimentalcircumstances

The attachment kinetics are usually defined bywriting thegrowth velocity normal to a flat surface in terms of the Hertz-Knudsen formula [7]

V119899= 120572

119888sat119888solid

radic119896119879

2120587119898120590surf

= 120572Vkin120590surf

(4)

where 120590surf is the supersaturation at the surface and thisequation defines the ldquokinetic velocityrdquo Vkin The parameter120572 is known as the condensation coefficient and it embodiesthe surface physics that governs how water molecules areincorporated into the ice lattice collectively known as theattachment kinetics The attachment kinetics can be quitecomplex so in general 120572 will depend on 119879 and 120590surf andperhaps surface structure and geometry surface chemistryand so forth If molecules striking the surface are instantlyincorporated into it then 120572 = 1 otherwise we have 120572 le 1A faceted growth form usually indicates that the growth islimited by attachment kinetics so 120572 lt 1 on faceted surfacesFor a molecularly rough surface we expect 120572 asymp 1

One should note that the growth parameterization interms of 120572 assumes that the attachment kinetics can bedescribed as an intrinsically local process which may notalways be a valid assumption For the ideal case of an infinitedefect-free surface this parameterizationmust be valid sincein this limit it is little more than a mathematical redefinitionNonlocal effects however such as surface transport betweenfacets could make the parameterization invalid in somecircumstances

For example it has been shown that the assumption of awell-defined attachment coefficient is not valid for the growthof mercury whiskers [21 22] since a value of 120572 ≫ 1 wouldbe required at the whisker tip In this case the tip growth isenhanced by surface diffusion of molecules along the facetedsides of the whisker and onto the growing tip which is anintrinsically nonlocal process The Schwoebel-Ehrlich effectprovides a potential barrier that normally inhibits surfacediffusion around corners [23 24] the mercury case beingan exception Experimental evidence supports the hypothesisthat 120572 le 1 for ice crystal growth and that nonlocal growtheffects can be neglected [7] but this is not known withcertainty

For the remainder of our discussion we will assume that(1) ice growth is in the limit 119901Peclet ≪ 1 (2) latent heatingof the crystal is negligible (3) the surrounding environmentand the crystal are at a constant temperature and pressureso 119888sat and 119863 have a single value throughout the system and(4) the attachment coefficient 120572 is well defined at the surfaceWe retain that 120572 may depend on temperature the surfacesupersaturation and the orientation of the surface relative to

the crystal axesWe also retain that120572may depend on the localcrystal structure [25]

Our overarching goal was to develop a basic numericalmodeling tool that would allow us to compare different theo-retical pictures of surface growth processeswith experimentalmeasurements of ice growth rates over a range of condi-tions Our focus was therefore on producing quantitativecalculations of crystal growth rates and morphologies fromwell-defined physical inputs including known diffusion ratesthrough the surrounding gas known initial sizes and mor-phologies of test crystals and theoretical parameterizationsof surface attachment kinetics and surface energy effects Welimited our calculations to the growth of fairly simple mor-phologies as these are best suited for extracting informationabout surface growth processes from experimental data

3 The 1D Spherical Model

It is useful to begin by examining the simplest case ofthe growth of spherical crystals Starting with the diffusionequation in spherical coordinates

120597120590

120597119905= 119863nabla

2120590 = 119863(

2

119903

120597120590

120597119903+1205972120590

1205971199032) (5)

and assuming a one-dimensional array of pixels with uniformsize Δ119903 gives the propagation equation120590 (119903 120591 + Δ120591)

= (1 minus 2Δ120591) 120590 (119903)

+ Δ120591 [(1 +Δ119903

119903) 120590 (119903 + Δ119903) + (1 minus

Δ119903

119903) 120590 (119903 minus Δ119903)]

(6)where

120591 =119863

(Δ119903)2119905 (7)

We typically choose Δ120591 = 12 as larger values can lead tonumerical instabilities and this gives120590 (119903 120591 + Δ120591)

= Δ120591 [(1 +Δ119903

119903) 120590 (119903 + Δ119903) + (1 minus

Δ119903

119903) 120590 (119903 minus Δ119903)]

(8)

31 Boundary Conditions In this spherical model we areessentially assuming a single spherical ldquofacetrdquo for the inter-face with the surface growth rate given in (4)We let all pixelswith radial index 119894 le 119894ice be ice and we refer to the next pixelwith 119894 = 119894ice+1 as a ldquoboundaryrdquo pixel In one time step the icesurface grows an amount 120575119903 = VΔ119905 and this growth extractsa total mass

120575119898 = 41205871199032120588ice120572Vkin120590boundΔ119905 (9)

from the boundary pixel shell reducing the supersaturationin that shell by

120575120590 =119888ice120572Vkin120590bound

119888sat(1 minus

Δ119903

119903bound)Δ119903

119863Δ120591 (10)

4 Journal of Computational Methods in Physics

where 119903bound is the radius of the center of the boundary pixelThus we can write

120590bound (120591 + Δ120591) = 120590bound (120591) [1 minus (1 minusΔ119903

119903bound)120572Δ120585Δ120591]

(11)

for the mass drain of the boundary pixel where

Δ120585 =119888iceVkin119888sat

Δ119903

119863=Δ119903

1198830

1198830=119888sat119888ice

119863

Vkin= radic

2120587119898

119896119879119863

asymp 0145 120583m sdot (119863

119863air) radic

11989611987915

119896119879

(12)

11987915= minus15

∘C and 119863air asymp 2 times 10minus5m2sec is the diffusion

constant in air at a pressure of one barWe combine (6) and (11) to create a total propagation

algorithm for the boundary pixel

120590 (119903bound 120591 + Δ120591) = (1 minusΔ119903

119903bound)120590solidΔ120591

+ (1 minus 2Δ120591) 120590 (119903bound)

+ (1 +Δ119903

119903bound)120590 (119903bound + Δ119903) Δ120591

(13)

where

120590solid = 120590 (119903bound) (1 minus 120572Δ120585) (14)

To describe the ice growth we assume that a boundarypixel starts out with zero accumulated mass and it turnsinto an ice pixel when it has accumulated a mass Δ119898 =

120588ice41205871199032

boundΔ119903 After one time step it accumulates the massin (9) giving

120575119898

Δ119898= 120572120590 (119903) (1 minus

Δ119903

119903bound)119888sat119888iceΔ120591Δ120585 (15)

Thus we define an accumulated mass parameter 120582 for theboundary pixel where 120582 starts at zero and after each timestepbecomes

120582 (120591 + Δ120591) = 120582 (120591) + (1 minusΔ119903

119903bound)120572120590Δ120582 (16)

where

Δ120582 =119888sat119888iceΔ120591Δ120585 (17)

When 120582 becomes greater than one the boundary pixel turnsto ice

32 Adaptive Time Steps In practice (17) gives an exceedinglyslow growth rate because 119888sat119888ice is quite small so we speedup the code by using

Δ120582 = ΛΔ120591Δ120585 (18)

where Λ is an adjustable parameter Note that the number ofsteps needed to advance one pixel is (120572120590Δ120582)minus1 so the growthvelocity is

V119899=Δ119903

Δ119905120572120590ΛΔ120591Δ120585 (19)

We optimize the speed of the code by increasing Λ as muchas possible subject to criteria that limit the resulting errors inthe growth behavior Our first criterion is that it should takemore than119873

0time steps to change a boundary pixel from air

to ice where typically1198730asymp 10 and this gives

Λ lt1

120572120590Δ120591Δ1205851198730

(20)

We also want the Peclet number to be much less thanunity so the growth is slower than the time it takes for thesupersaturation field to stabilize as described above and thisgives

119901Peclet =119877V119899

2119863lt 119901Pecletmax

Λ lt2

120572120590Δ120585(Δ119903

119877) sdot 119901Pecletmax

(21)

where 119877 is the crystal radius We use 119901Pecletmax = 11198730 so

Λ = min [ 2

120572120590Δ120585(Δ119903

119877) sdot 119901Pecletmax

1

120572120590Δ120591Δ1205851198730

]

asymp 119860 speed1

Δ120585(120572120590surf)max(Δ119903

119877)

(22)

where 119860 speed = 21198730 asymp 02 is a constant speedup parameterThe quantities (120572120590surf)max and 119877 are computed as the crystalgrows

In practice we have found that this relation yields growthrates that are somewhat too fast when 119877 lt 119873

0Δ119903 owing to

finite pixel size effects We counter this by replacing 119877 in (22)with

1198771015840= radic1198772 + 1198772

0 (23)

where we set 1198770= 1198730Δ119903 This results in an improvement

in the accuracy of the growth for small crystals with only amodest increase in running time

The choice of Λ essentially means using an adaptive timestep where the physical time for each step equals

Δ119905 = Δ120582Δ120585Δ1199050 (24)

Δ1199050=1198830

Vkinasymp 070msec sdot (

Vkinminus15∘C

Vkin) (25)

In running this code we typically use Δ120591 = 12 and Δ120585 =1 which gives Δ119903 = 119883

0 We define the radial index from 119894 = 1

to 119873 and the radius of the center of each spherical shell is119903119894= (119894 minus 1)Δ119903

Journal of Computational Methods in Physics 5

33 Analytic Solutions If the outer boundary is at infinity thespherical growth case gives the analytic solution

120590 (119903) = 120590infinminus119877

119903

120572

120572 + 120572diff120590infin (26)

where 119877 is the radius of the sphere and 120572diff = 1198830119877 In thelimit 120572 ≪ 120572diff we have 120590(119903) asymp 120590

infin while 120572 ≫ 120572diff gives

120590(119877) asymp 0 The growth velocity in all cases is

V119899= 120572Vkin120590 (119877) = (

120572120572diff120572 + 120572diff

) Vkin120590infin (27)

With an outer boundary at 119903out and 120590(119903out) = 120590out we have

120590 (119903) = 120590out minus (1198771015840

119903minus1198771015840

119903out)120590out (28)

where

1198771015840= [

120574

119877minus

1

119903out]

minus1

120574 =120572 + 120572diff120572

(29)

and the growth velocity becomes

V119899= (

120572120572diff120572 + 120572diff

) Vkin120590out[1 minus119877

119903out

1

120574]

minus1

(30)

In computing V119899 we may have to solve the equation

120590 (119877) =120590out (120574 minus 1)

120574 minus 119877119903out(31)

when 120574 is itself a function of 120590(119877) We solve this by iterationusing

120590(119877)119894+1=1

2[120590119894+120590out (120574 (120590(119877)119894) minus 1)

120574 (120590(119877)119894) minus 119877119903out

] (32)

at each time step which quickly converges to 120590(119877)

34 Model Validation As an example that compares themodel growth of a spherewith the analytic result we useΔ120591 =12 Δ120585 = 1 119879 = minus15

∘C 119873 = 300 (giving 119903out = 435 120583m)120590out = 001 120572 = 1 and 119877initial = 651198830 = 094 120583m Resultsare shown in Figure 1 using different values of 119860 speed For themodel with 119860 speed = 1 the adaptive time steps were large so120590surf did not have time to relax fully to its analytic value as thecrystal grew with the outcome that the crystal grew too fastWith smaller 119860 speed values the supersaturation field relaxedmore fully and 119877(119905) was closer to the analytic result

35 Quantitative Comparisons with Experimental Data Inour experiments measuring ice growth rates the outerboundaries of the growth chamber are much larger thantypical crystal sizes so 119903out asymp infin In modeling such data werequire a certain modeling accuracy and the straightforward

0 50 100 150 200Time (s)

0

5

10

15

Radi

us (m

icro

ns)

Figure 1 A comparison of different models for the growth ofa spherical crystal (solid lines) together with the analytic result(dashed line) as described in the text The models used 119860 speed = 1

(top curve) 02 (middle) and 002 (lower)

route to achieving this is to run the code with some suitablysmall 119860 speed together with some suitably large 119903out Unfortu-nately the code converges rather slowly to analytic results asseen in the example in Figure 1 and we have found that thisstraightforward approach results in unnecessarily long runtimes

We have found an alternative operating strategy thatachieves good modeling accuracy with substantially shorterrun times In this strategy we select values of 119903out that are notespecially large and values of 119860 speed that are not especiallysmall and then adjust 120590out down to compensate This strategysacrificesmodel accuracy in exchange for increasedmodelingspeed and is thus a variant of the usual trade-off oneencounters in numerical modeling

Figure 2 shows an example of how this modeling strategycan be used The analytic curve uses the same parameters asin Figure 1 except with 119903out = infin The models also use thesame parameters as in Figure 1 again with 119903out = 435 120583mand we fixed 119860 speed = 02 The model with 120590out = 001 doesnot match the analytic curve which reflects systematic errorsin the numerical modeling The model growth is faster thanthe analytic result because 119903out is too small and 119860 speed is toolargeWe compensate for these systematic errors by adjusting120590out downward 30 percent to 0007 as seen in Figure 2

From this exercise we see that there are two ways toproduce accurate models for comparison with experimentaldata The first way is simply to use a large 119903out and asmall 119860 speed to reduce the modeling systematic errors to anacceptable level This method is the most straightforwardbut from a general perspective it is not the most efficientThe second approach is to use smaller 119903out and larger 119860 speedso the code runs quickly and then lowers 120590out slightly tocompensate This latter approach has proven quite useful inpractice especially when comparing experiments where 120590

infin

is itself not known with extremely high accuracy as is oftenthe case

6 Journal of Computational Methods in Physics

0 50 100 150 200Time (s)

5

10

Radi

us (m

icro

ns)

Figure 2 Another comparison of different models for the growthof a spherical crystal (solid lines) with the analytic result (dashedline) The analytic calculation used 120590

infin= 001 and 119903out = infin The

models used 119860 speed = 02 and 119903out = 435 120583m with 120590out = 001 (topcurve) 0007 (middle) and 0004 (lower) This shows that one canadjust 120590out in the model to approximately compensate for systematicmodeling errors

4 The 2D Cylindrically Symmetric Model

The 1D spherical model outlined above is useful for exam-ining the cellular automata method in detail but of courseit is of little use otherwise as analytic results can be com-puted for the general spherical case Adding one additionaldimension adds significant complexity and richness to thediffusion problem however analytic results are not pos-sible for calculating the growth of even rather simple 2Dmorphologies

We have found that a 2D cylindrically symmetric modelis especially useful for comparison with ice crystal growthexperiments In this case a simple hexagonal plate is approxi-mated by a circular disk while a hexagonal column becomesa cylinder The six prism facets are replaced by a singlecylindrical ldquofacetrdquo analogous to the spherical case aboveand we equate the attachment kinetics on this surface tothe attachment kinetics on a flat prism facet Other thanintroducing a small geometrical correction the cylindri-cal approximation appears to give reasonable quantitativeresults It can be applied only to simple morphologies suchas simple plates and columns hollow columns and cappedcolumns Fortunately experiments tend to focus on thesesimple morphologies as they are best suited for examiningsurface growth dynamics

The 2D diffusion equation in cylindrical coordinates is

120597120590

120597119905= 119863nabla

2120590 = 119863[

1

119903

120597120590

120597119903+1205972120590

1205971199032+1205972120590

1205971199112] (33)

for 120590(119903 119911 119905) giving the propagation equation

120590 (120591 + Δ120591) = (1 minus 4Δ120591) 120590 + Δ120591

times [(1 +Δ119903

2119903) 120590 (119903 + Δ119903) + (1 minus

Δ119903

2119903) 120590 (119903 minus Δ119903)

+ 120590 (119911 + Δ119911) + 120590 (119911 minus Δ119911) ]

(34)

where 120591 is given in (7)Including boundary conditions at the crystal surface

follows along the lines of the 1D calculation above AssumingΔ120591 = 14 and Δ119903 = Δ119911 gives the propagation equations for119903-type and 119911-type boundary pixels

120590 (119903bound 119911 120591 + Δ120591)

=1

4[(1 +

Δ119903

2119903) 120590 (119903 + Δ119903) + (1 minus

Δ119903

2119903) 120590solid

+120590 (119911 + Δ119911) + 120590 (119911 minus Δ119911) ]

120590 (119903 119911bound 120591 + Δ120591)

=1

4[(1 +

Δ119903

2119903) 120590 (119903 + Δ119903) + (1 minus

Δ119903

2119903) 120590 (119903 minus Δ119903)

+ 120590 (119911 + Δ119911) + 120590solid]

(35)

where

120590solid = 120590 (119903bound) (1 minus 120572Δ120585) (36)

Note that a corner boundary pixel with neighboring ice pixelsin both 119903 and 119911 will propagate using

120590 (119903bound 119911bound 120591 + Δ120591)

=1

4[(1 +

Δ119903

2119903) 120590 (119903 + Δ119903) + (1 minus

Δ119903

2119903) 120590solid

+ 120590 (119911 + Δ119911) + 120590solid]

(37)

which drains the supersaturation twice as fast as an ordinaryboundary pixel as expected

We then define an accumulated mass parameter 120582 where120582 starts at zero and after each timestep becomes

120582 (120591 + Δ120591) = 120582 (120591) +sum120572120590Δ120582 (38)

where the sum is over the number of neighboring ice pixelsand

Δ120582119911-neighbors =

119888sat119888iceΔ120591Δ120585

Δ120582119903-neighbors = (1 plusmn

Δ119903

2119903)119888sat119888iceΔ120591Δ120585

(39)

Journal of Computational Methods in Physics 7

41The 119903 = 0 and 119911 = 0 Special Cases In our model we definethe first row of pixels to have 119903 = 0 which leads to problemswith the (1 plusmn Δ1199032119903) factors Going back to the diffusionequation we use Gaussrsquos law to generate the propagationequation for 119903 = 0 pixels

120590 (0 120591 + Δ120591)

= (1 minus 6Δ120591) 120590 (0 120591)

+ Δ120591 [120590 (119911 + Δ119911) + 120590 (119911 minus Δ119911) + 4120590 (Δ119903)]

(40)

Unfortunately taking Δ120591 = 14 in this expression leads toon-axis propagation instabilities As a compromise betweenrunning speed and computational accuracy we use Δ120591 = 16for the on-axis propagation equation giving

120590 (0 120591 + Δ120591) =1

6[120590 (119911 + Δ119911) + 120590 (119911 minus Δ119911) + 4120590 (Δ119903)]

(41)

while we continue using Δ120591 = 14 for off-axis pixels Thiscauses the supersaturation field to relax slightly more slowlyon the 119903-axis but this fix apparently does not adversely affectthe modeling results

Similarly we define the first row of 119911 pixels to have 119911 = 0and for that row we use the propagation equation

120590 (120591 + Δ120591) =1

4[(1 +

Δ119903

2119903) 120590 (119903 + Δ119903)

+ (1 minusΔ119903

2119903) 120590 (119903 minus Δ119903) + 2120590 (119911 + Δ119911)]

(42)

42 Adaptive Time Steps Speeding up the code using adap-tive time steps again proceeds along the lines of the 1Dproblem We use

Δ120582 = ΛΔ120591Δ120585 (43)

with

Λ = 119860 speed1

Δ120585(120572120590surf)max(Δ119903

1198771015840max)

1198771015840

max = radic1198772

max + 1199112

max + 1198772

0

(44)

as above and the physical time step is that given in (25)

43 Neighbor Relations In all these expressions we mustchoose the attachment coefficient 120572 with some care asits value will depend on the number and orientation ofneighboring solid pixels We label a boundary pixel with(119873119903 119873119911) where119873

119903is the number of neighboring ice pixels in

the 119903 direction and 119873119911is the number of ice neighbors in the

119911 direction Both119873119903and119873

119911can take values 0 1 or 2 giving

nine cases for (119873119903 119873119911) The cases are

(0 0) the pixel is an air pixel

(0 1) one ice neighbor in the 119911 direction so 120572 =

120572basal the physical value appropriate for a basal facetsurface(1 0) one ice neighbor in the 119903direction so120572 = 120572prismfor a prism facet surface(1 1) a kink site where the growth will not benucleation limited since the corner provides a sourceof molecular steps We do not know a priori whatvalue to use for 120572 on this site but assume a constant120572 = 120572

11

(0 2) (1 2) (2 0) (2 1) and (2 2) these are allunusual cases where the growth will be fast so weassume that 120572 = 1

We index these possibilities with a single number bycomputing a boundary parameter 119861 = 2119873

2

119903+ 1198732

119911 We then

have 119861 = 0 for an air pixel 119861 = 1 for a basal facet 119861 = 2 for aprism facet 119861 = 3 for a (1 1) kink location and 119861 gt 3 for allother cases

If we consider the special case where 120572 is equal to someconstant value independent of the orientation of the surfacewith respect to the crystal lattice then the growth velocityshould equal V = 120572Vkin120590 for all surfaces For the basal or prismfacet surfaces in this constant-120572 case we take 120572basal = 120572prism =120572 while an analysis of the growth of a (11) surface shows thatwemust take 120572

11= 120572radic2 if the above algorithm is to produce

the correct growth velocity

44 Limitations on Grid Size Aswe pointed out in [26] thereare physical limits to how coarse the computing grid canbe made before instabilities appear or the growth deviatessubstantially from real growth Taking Δ120585 gt 1120572would cause120590solid to become negative which causes some concern in thatit may produce instabilities in the code With this limitationthe grid spacing could not be larger than Δ119903 = Δ119911 = 119883

0120572

For air at a pressure of one atmosphere and 120572 asymp 1 this givesthe pixel size119883

0asymp 0145 120583m

Physically we can gain some insights into these limita-tions from dendrite growth theory [7] We have 119883

0asymp 119877kin

(the latter from (28) in [7]) and a growing dendrite has a tipradius

119877tip asymp1198830

120572119904 (45)

where 119904 is the dimensionless solvability parameter which is oforder unity for ice crystal growth [7]The stability of the codethus limits the grid spacing to be no greater than the tip radiusof a growing dendritic structure From this we see that thecode can only function properly when the grid spacing is fineenough to allow the growth of physically realistic dendriticstructures the scale of which is given by solvability theory

In practice we typically assume that Δ119903 = Δ119911 = 1198830asymp

0145 120583m when comparing models with ice growth data Acoarser grid would reduce run times but at the risk of notreproducing physically relevantmorphological behaviorsWehave not yet explored in detail how different grid sizes affectthe modeling behavior

8 Journal of Computational Methods in Physics

45 Scaling Behavior If we run the code and produce somecomplex crystal shape the interpretation of our result stillcontains an ambiguity The crystal size is given in pixelswhere Δ119903 = Δ119911 = Δ120585119883

0is the pixel size The parameter

Δ120585 was fixed in the code but 1198830depends on the diffusion

constant 119863 which is not otherwise specified Similarly asingle time step in the code corresponds to a physical time

Δ119905 =1198832

0Δ1205852Δ120591

119863sim 119863 (46)

Thus we see that the growth behavior at different airpressures (different D) is determined once we know thegrowth at a single pressure (provided that 120590

infinis the same

at the different pressures) If the air pressure is half anatmosphere the growthmorphology (however complex) willbe the same as at one atmosphere except in the former casethe crystal will be double the size in double the time Thisscaling behavior nicely explains why ice crystal morphologyis generally simpler for smaller crystals andor for lower airpressures which has long been observed [7]

46 Analytic Solutions and Model Validation The case of aninfinitely long cylinder of radius 119877in has the analytic solution

120590 (119903) =120573 + log (119903119877in)

120573 + log (119877out119877in)120590119877out (47)

where the outer boundary has 120590(119877out) = 120590119877out and 120573 =

1198830119877in120572 giving the growth velocity

V =120572120572diffcyl

120572 + 120572diffcylVkin120590119877out

120572diffcyl =1

119861

1198830

119877in

(48)

with 119861 = log(119877out119877in)To compare this analytic result with numerical models of

a growing cylinder we created a version of our codewith peri-odic boundary conditions in 119911 thus modeling the growth ofan infinite cylinder Figure 3 shows results using 119879 = minus15∘C120590119877out = 001 119877init = 1 120583m 120572 = 1 119877out = 300 and 119883

0=

435 120583m Again we see that the numerical model converges tothe analytic result as 119860 speed goes to zero

5 Gibbs-Thomson Effect

The basic cellular automata code described above includes afairly realistic treatment of the attachment kinetics but doesnot include any effects of surface energy This is somewhatjustified for the case of faceted ice crystal growth as one canshow that attachment kinetic effects dominate the growthbehavior while surface energy effects are less important [7]Nevertheless surface energy effects are not always negligibleespecially at low supersaturations As we will demonstratebelow models with highly anisotropic attachment kineticsand no surface energy effects can exhibit the formation ofone-pixel-wide features that are not physically plausible To

5

0

10

Radi

us (m

icro

ns)

0 50 100 150 200Time (s)

Figure 3 The dashed line shows the analytic result for the growthof an infinite cylinder with 120572 = 1 119877out = 435 120583m and 120590

119877out = 001as described in the text Solid lines show numerical modeling resultsusing 119860 speed = 02 (top line) 002 (middle line) and 0002 (bottomline)

suppress these unphysicalmodelswemust include the surfaceenergy via the Gibbs-Thomson effect

For the ice case we are considering here the equilibriumvapor density above a curved surface can be written [27]

119888eq = 119888sat (1 + 120575120581) (49)

where 119888sat is the equilibrium (saturated) vapor density of a flatsurface 120581 is the surface curvature

120581 =1

1199031

+1

1199032

(50)

where 1199031 1199032are the principal radii of curvature of the surface

and

120575 =120574

119888solid119896119879asymp 1 nm (51)

where 120574 asymp 01 Jm2 is the surface energy of the icevaporinterface The anisotropy of 120574 is not well known but theavailable evidence suggests it is rather small [20] so for theremainder of our discussion we assume an isotropic surfaceenergy We believe this is a accurate assumption for ice but itis in stark contrast to the highly anisotropic ice surface energyassumed in [11]

From this we have the supersaturation over a curvedsurface

120590curved asymp 120590 minus120575

119877119888

(52)

to lowest order in 120575119877119888 where120590 is the normal supersaturation

relative to a flat interface and for later convenience we define119877119888= 120581minus1 Putting in some numbers a sharp ice needle might

have 119877119888asymp 1 120583m giving Δ120590 = 120590curved minus 120590 asymp 01 percent

which is small but not always negligible Moreover setting

Journal of Computational Methods in Physics 9

Nz = 7

Nr = 5

z

r

Figure 4 A graphical definition of the outer boundary pixels usedfor incorporating surface energy effects into our cellular automatamodel Here the blue pixels are ice while the clear pixels are airThe red pixels are boundary pixels having the largest 119903 and 119911 valuesdefined to be the ldquoouterrdquo boundary pixels

119877119888asymp 1198830asymp 015 120583m gives a rather large Δ120590 asymp 066 percent

indicating that one-pixel-wide features should not be presentat low supersaturations

Adding the Gibbs-Thomson effect requires an algorith-mic estimation of 119877

119888for all boundary pixels in our cellular

automata model We examined several possibilities but didnot find a suitable algorithm that could be used with arbitrarymorphologies in our cylindrically symmetrical model Sinceour primary goal was to model simple morphologies forcomparison with experimental data as described above wesettled for a simpler algorithm for such cases

Throughout our investigations we found that the overallgrowth behavior of faceted crystals with simplemorphologieswas determined primarily by the growth of the outermostfacet surfaces To apply the Gibbs-Thomson effect on thesesurfaces we defined the outer boundary pixels as shown inFigure 4 yielding the numbers 119873

119903and 119873

119911as the number of

outer-boundary pixelsUsing these outer boundary pixels as a proxy for esti-

mating surface curvature we defined 119877119888= Δ119903119873

1199032 for the

edges of plates 119877119888= Δ119903119873

1199112 for the edges of hollow columns

and 119877119888= Δ119903119873

1199114 for the tips of needles For all boundary

pixels that were not outer-boundary pixels we assumed that119877119888= infin Although these are crude estimations of surface

curvature we found them satisfactory for incorporating theGibbs-Thomson effect in our model in part because this is arelatively small effect compared to the more dominant role ofattachment kinetics

51 A Gibbs-Thomson Example The top image in Figure 5shows how our model can yield unphysical results if wedo not include the Gibbs-Thomson effect In this model weassumed the input parameters 119879 = minus15

∘C 120590infin

= 001

120572basal = exp(minus003120590surf) 120572prism = 1 an initial prism radius1198770= 2 120583m initial prism half-height119867

0= 5 120583m and a Gibbs-

Thomson parameter 120575 = 0 After growing this model for sixseconds the initial simple prism grew into a capped columnwhere the two capping plates each had a thickness of justone pixel as shown in the top image in Figure 5 Such smallstructural features are unphysical at this supersaturation andsimilar one-pixel-wide features often appeared in our modelswhen the supersaturation was low and the anisotropy inattachment kinetics was high

Figure 5 A test of the Gibbs-Thomson effect in our crystal growthmodel In each image the white pixels show ice while the brightnessof the surrounding region is proportional to the supersaturationModel crystals are reflection symmetric about the 119911 = 0 plane andthe images show 119911 ge 0 only Thin vertical lines show the initialcrystal size Using 120572prism = 1 and 120572basal ≪ 120572prism resulted in growthonly on the prism surfaces Full model parameters are given in thetext The top image displays the crystal after six seconds of growthwith a Gibbs-Thomson parameter 120575 = 0 This model exhibits a one-pixel-thick plate growing on the end of the column which is notphysically plausibleThemiddle and lower images show results using120575 = 03 and 10 nm respectively For these models the unphysicalplate growth does not appear

Including nonzero values of 120575 suppressed these smallfeatures as demonstrated in the lower two images in Figure 5From this example as well as additional tests with differentgrowth morphologies we found that our relatively simplealgorithm for including the Gibbs-Thomson effect yieldedreasonable results suppressing the unphysical growth of one-pixel-wide features Models with higher 120590

infinvalues or with

lower anisotropies in the attachment kinetics were generallyless affected by including the Gibbs-Thomson effect

6 An Example Comparison with Experiment

Figure 6 shows an example of how the cellular automatamethod can be used in laboratory investigations of growingice crystals In this experiment we first grew a thin ldquoelectricrdquoice needle as described in [28] and then transferred theneedle to a second growth chamber where we grew a thinplate-like crystal on the needle tip all in air at a pressureof one bar The hexagonal symmetry of the crystal andthe relatively simple growth morphology were well suitedfor analysis using our cylindrically symmetric model Byadjusting 120590

infinand the surface attachment coefficients for the

principal facets we were able to generate a model crystal

10 Journal of Computational Methods in Physics

50120583m

(a) (b)

Radi

us (120583

m)

60

50

40

30

20

10

0minus40 minus20

Plate radius

Needle radius

0 20 40Time (s)

60 80 100

(c)

Figure 6 An example showing our numerical modeling of a laboratory-grown ice crystal The composite image in (a) shows five successiveviews of a thin plate-like crystal growing on the end of a slender ice needle with a 50120583m scale bar The crystal is shown in a side view withillumination from behindThe composite image in (b) shows our numerical model of the same crystal Here the brightness around the crystalis proportional to supersaturation The data points in (c) shows measurements of the plate and needle radii (the latter at a distance of 50120583mfrom the base of the plate) as a function of timeThe lines in the graph are from the growth of the model crystal The inset image in (c) showsa more frontal view of a similar plate-on-needle crystal with a plate radius of 80120583m showing the thin sectored-plate morphology Note thatat earlier times (119905 lt 0 in the graph) the crystal was growing from a tapered needle to a column with nearly uniform radius along its lengthThe time axes were shifted so the plate began growing at 119905 asymp 0 for both the real and model crystals

that matched both the morphology and growth rates ofthe laboratory crystal In addition to the plate and needleradii note that the tapered neck of the needle just belowthe plate seen in Figure 6 was nicely reproduced by themodel crystal A discussion of the physical significance ofthe best-fit model parameters used for this model is left fora subsequent publication comparing the growth of manycrystals under different conditions This example is meantto demonstrate only that the behavior of faceted laboratory-grown ice crystals can be accuratelymodeledwith the cellularautomata method

7 Discussion

Numericalmodeling of structure formation during solidifica-tion has been well studied for decades especially cases whereweak surface anisotropies in diffusion-limited growth yieldrounded dendritic branching Modeling structures that areboth branched and faceted have beenmore difficult and onlyrecently researchers have demonstrated suitable numericalmethods [10 11] In the present work we have developeda numerical model aimed at investigating the formationof simple faceted morphologies during ice crystal growthOur model incorporates the cellular automata technique in[10] but uses a more physically realistic treatment of thesurface attachment kinetics and the Gibbs-Thomson effectUsing this model we are able to reproduce ice crystalgrowth behavior for simple growth morphologies over a

broad range of conditions We have been using this modelto aid in the interpretation of ice growth experiments inorder to extract information about the anisotropic surfaceattachment kinetics that governs ice crystal growth fromwater vapor Modifying the model for application to othermaterial systems is likely straightforward

At present we have limited our model to cylindricallysymmetric crystal growth and relatively simple growth mor-phologies for comparison with measurements A full three-dimensional extension of our model should be possibleas was demonstrated in [10] potentially allowing completemodeling of the many complex branched and faceted struc-tures that are commonly observed in ice crystal growth [7]Achieving this goal would likely require a substantially betterunderstanding of the surface attachment kinetics in ice thanpresently exists along with more advanced algorithms forincorporating all the relevant surface physics into a cellularautomata model

In general it appears that the cellular automata methodis well suited for modeling faceted crystal growth and itmay be the method of choice in material systems exhibitingstrongly anisotropic attachment kinetics Incorporatingmoresophisticated descriptions of surface growth processes willrequire substantially better algorithms than we have outlinedabove especially when extending the models to three dimen-sions Whether such models can reproduce the full rangeof complex structures appearing in faceted crystal growthremains to be seen

Journal of Computational Methods in Physics 11

References

[1] R Trivedi and W Kurz ldquoDendritic growthrdquo InternationalMaterials Reviews vol 39 no 2 pp 49ndash74 1994

[2] E A Brener ldquoThree-dimensional dendritic growthrdquo Journal ofCrystal Growth vol 166 no 1ndash4 pp 339ndash346 1996

[3] I Avramov ldquoKinetics of growth of nanowhiskers (nanowiresand nanotubes)rdquo Nanoscale Research Letters vol 2 no 5 pp235ndash239 2007

[4] H Zhang I Hussain M Brust M F Butler S P Rannard andA I Cooper ldquoAligned two- and three-dimensional structuresby directional freezing of polymers and nanoparticlesrdquo NatureMaterials vol 4 pp 787ndash793 2005

[5] M C Cross and P C Hohenberg ldquoPattern formation outsideof equilibriumrdquo Reviews of Modern Physics vol 65 no 3 pp851ndash1112 1993

[6] K Kassner Pattern Formation in Diffusion-Limited CrystalGrowth World Scientific Singapore 1996

[7] K G Libbrecht ldquoThe physics of snow crystalsrdquo Reports onProgress in Physics vol 68 no 4 pp 855ndash895 2005

[8] J Nelson ldquoGrowth mechanisms to explain the primary andsecondary habits of snow crystalsrdquo Philosophical Magazine Avol 81 no 10 pp 2337ndash2373 2001

[9] H R Pruppacher and J D Klett Microphysics of Clouds andPrecipitation Kluwer Academic Publishers Dordrecht TheNetherlands 1997

[10] J Gravner and D Griffeath ldquoModeling snow-crystal growtha three-dimensional mesoscopic approachrdquo Physical Review Evol 79 no 1 Article ID 011601 18 pages 2009

[11] J W Barrett H Garcke and R Nurnberg ldquoNumerical com-putations of faceted pattern formation in snow crystal growthrdquoPhysical Review E vol 86 no 1 Article ID 011604 14 pages2012

[12] S-Y Hong J Dudhia and S-H Chen ldquoA revised approach toice microphysical processes for the bulk parameterization ofclouds and precipitationrdquoMonthlyWeather Review vol 132 no1 pp 103ndash120 2004

[13] M Matsumoto S Saito and I Ohmine ldquoMolecular dynamicssimulation of the ice nucleation and growth process leading towater freezingrdquo Nature vol 416 no 6879 pp 409ndash413 2002

[14] J G Dash A W Rempel and J S Wettlaufer ldquoThe physicsof premelted ice and its geophysical consequencesrdquo Reviews ofModern Physics vol 78 no 3 pp 695ndash741 2006

[15] A Karma and W-J Rappel ldquoPhase-field method for com-putationally efficient modeling of solidification with arbitraryinterface kineticsrdquo Physical Review E vol 53 no 4 pp R3017ndashR3020 1996

[16] I Singer-Loginova andHM Singer ldquoThe phase field techniquefor modeling multiphase materialsrdquo Reports on Progress inPhysics vol 71 no 10 Article ID 106501 2008

[17] A Schmidt ldquoComputation of three dimensional dendrites withfinite elementsrdquo Journal of Computational Physics vol 125 no2 pp 293ndash312 1996

[18] E Yokoyama ldquoFormation of patterns during growth of snowcrystalsrdquo Journal of Crystal Growth vol 128 no 1ndash4 pp 251ndash257 1993

[19] S E Wood M B Baker and D Calhoun ldquoNew model forthe vapor growth of hexagonal ice crystals in the atmosphererdquoJournal of Geophysical ResearchD vol 106 no 5 pp 4845ndash48702001

[20] K G Libbrecht ldquoOn the equilibrium shape of an ice crystalrdquo2012 httparxivorgabs12051452

[21] G W Sears ldquoA growth mechanism for mercury whiskersrdquo ActaMetallurgica vol 3 no 4 pp 361ndash366 1955

[22] R L Parker R L Anderson and S C Hardy ldquoGrowth andevaporation kinetics and surface diffusion of K and Hg crystalwhiskersrdquo Applied Physics Letters vol 3 no 6 pp 93ndash95 1963

[23] S J Liu H Huang and C HWoo ldquoSchwoebel-Ehrlich barrierfrom two to three dimensionsrdquo Applied Physics Letters vol 80no 18 pp 3295ndash3297 2002

[24] Y Saito Statistical Physics of Crystal Growth World ScientificSingapore 1996

[25] K G Libbrecht ldquoExplaining the formation of thin ice-crystalplates with structure-dependent attachment kineticsrdquo Journal ofCrystal Growth vol 258 no 1-2 pp 168ndash175 2003

[26] K G Libbrecht ldquoPhysically derived rules for simulating facetedcrystal growth using cellular automatardquo 2008 httparxivorgabs08072616

[27] W Thomson ldquoOn the equilibrium vapor at a curved surface ofliquidrdquo Philosophical Magazine vol 42 no 282 pp 448ndash4521871

[28] K G Libbrecht T Crosby and M Swanson ldquoElectricallyenhanced free dendrite growth in polar and non-polar systemsrdquoJournal of Crystal Growth vol 240 no 1-2 pp 241ndash254 2002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Advances in Condensed Matter Physics

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Superconductivity

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Statistical MechanicsInternational Journal of

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Solid State PhysicsJournal of

 Computational  Methods in Physics

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Soft MatterJournal of

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PhotonicsJournal of

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Biophysics

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ThermodynamicsJournal of

2 Journal of Computational Methods in Physics

surface energy which is typical of solidification from themeltThese systems typically yield unfaceted dendritic structureshowever in contrast to strongly faceted ice structures Earlymodels for the growth of faceted crystals [18 19] weregenerally too limited to allow quantitative comparisons withice growth data

Modeling diffusion-limited growth in systems withstrong surface anisotropies has proven difficult and onlyrecently have researchers demonstrated robust techniquescapable of generating structures that are both faceted anddendritic Gravner and Griffeath [10] described an especiallypromising cellular automata simulator that solves the dif-fusion equation by nearest neighbor relaxation including aset of parameterized nearest neighbor rules to define theboundary conditions at the crystal interface This methodyields a deterministic dendritic growth behavior in whichfaceting follows the symmetry of the predefined numericalgrid

Barrett et al [11] also developed a robust adaptive meshtechnique that generated faceted dendritic crystal growthpatterns In this work the authors found that a stronglyanisotropic surface energy was required to produce faceteddendritic growthwhile anisotropic attachment kinetics alonewere not sufficient to reproduce this behavior We havesuggested that the ice case is more likely described bythe opposite characteristics an essentially isotropic surfaceenergy together with strongly anisotropic attachment kinet-ics the latter dominating the growth behavior [20] In factthe relative roles played by these two physical effects is notyet known with certainty

The relative merits of different computational methodsfor modeling diffusion-limited growth in the presence ofstrong surface anisotropies are not presently well understoodand this is an area of current research Moreover our knowl-edge of the surface physics governing the growth of facetedmaterials is itself rather poor including the relative contri-butions of the anisotropies in surface energy and attachmentkinetics in different materials In our experience progress onboth these research fronts is linked better modelingmethodsallow better interpretation of growth experiments and thisin turn leads to a better understanding of the surface physicsinput into the models

Below we describe a cellular automata method for mod-eling diffusion-limited growth in the presence of stronglyanisotropic molecular attachment kinetics focusing on icegrowth from water vapor Our treatment of the surfaceboundary conditions derives from known surface physicsand thus replaces the somewhat unphysical parameterizationadopted in [10] We have found our model to be robust wellbehaved computationally straightforward and quite flexiblefor exploring ice growth behaviors The 2D cylindricallysymmetric model described below has been especially use-ful for investigating the simple growth morphologies oftenproduced in experiments allowing the rapid generation ofhundreds of model crystals with different input assumptionsThis model has allowed us to better examine the surfacephysical processes governing ice growth rates and it allowsstraightforward adaptation for use in other investigationsinvolving faceted diffusion-limited crystal growth

2 Modeling Ice Crystal Growth

A variety of physical processes are involved in the growthof ice crystals from water vapor in an inert background gasParticle diffusion plays a large role as water vapor moleculesmust diffuse through the gas to reach the growing crystalHeat diffusion is also present since the latent heat generatedby condensation must be removed from the crystal Surfaceenergy effects are present via the Gibbs-Thomson effect andsurface attachment kinetics govern the placement of watermolecules into the ice crystal lattice If the background gasis not inert surface chemical effects may also be significant

Our focus here will be on ice growth under near atmo-spheric conditions which allows a number of simplificationsin the problem Particle transport is described by the diffusionequation

120597119888

120597119905= 119863nabla

2119888 (1)

using the notation in [7] where 119888(119909) is the water moleculenumber density surrounding the crystal and119863 is the diffusionconstant The timescale for diffusion to adjust the vaporconcentration in the vicinity of a crystal is 120591diffusion asymp 119877

2119863

where 119877 is a characteristic crystal sizeThis is to be comparedwith the growth time 120591growth asymp 2119877V119899 where V119899 is the growthvelocity of the solidification front normal to the surface Theratio of these two timescales is the Peclet number 119901Peclet =119877V1198992119863 For typical growth rates of ice crystals we find

119901Peclet ≲ 10minus5 which means that diffusion adjusts the particle

density around the crystal much faster than the crystalshape changes In this case the diffusion equation reducesto Laplacersquos equation nabla2119888 = 0 which must be solved withthe appropriate boundary conditions Using this slow-growthlimit simplifies the problem considerably in comparison tomuch of the literature on diffusion-limited solidification

It is convenient to work with the supersaturation 120590(119909) =[119888(119909) minus 119888sat]119888sat where 119888sat is the equilibrium vapor densityabove a flat ice surface Then the diffusion equation becomes

120597120590

120597119905= 119863nabla

2120590 (2)

and the continuity equation at the interface gives

V119899=

119863

119888solid(119899 sdot nabla119888)

surf=119888sat119863

119888solid(119899 sdot nabla120590)

surf (3)

where 119888solid = 120588ice119898 is the number density for ice 120588iceis the ice density and 119898 is the mass of a water moleculeFor the outer boundary one typically assumes a constantsupersaturation 120590

infin

Including latent heating of the crystal wouldmean solvinga double diffusion problem involving both particle and heatdiffusion simultaneously complicating the problem consider-ably However since the thermal conductivity of ice is muchhigher than that of air heating raises the temperature of thecrystal roughly uniformly at least for simple morphologieswhich increases the effective 119888sat for the crystal Thus themain effect of heating can be approximated by a relatively

Journal of Computational Methods in Physics 3

small rescaling of 120590infin

[7] Furthermore many experimentsare done using ice crystals resting on a thermally conductingsubstrate and in this case the effects of latent heating areessentially negligible [7] For these reasons we have ignoredlatent heating in the present model with the understandingthat heating effects may be important in some experimentalcircumstances

The attachment kinetics are usually defined bywriting thegrowth velocity normal to a flat surface in terms of the Hertz-Knudsen formula [7]

V119899= 120572

119888sat119888solid

radic119896119879

2120587119898120590surf

= 120572Vkin120590surf

(4)

where 120590surf is the supersaturation at the surface and thisequation defines the ldquokinetic velocityrdquo Vkin The parameter120572 is known as the condensation coefficient and it embodiesthe surface physics that governs how water molecules areincorporated into the ice lattice collectively known as theattachment kinetics The attachment kinetics can be quitecomplex so in general 120572 will depend on 119879 and 120590surf andperhaps surface structure and geometry surface chemistryand so forth If molecules striking the surface are instantlyincorporated into it then 120572 = 1 otherwise we have 120572 le 1A faceted growth form usually indicates that the growth islimited by attachment kinetics so 120572 lt 1 on faceted surfacesFor a molecularly rough surface we expect 120572 asymp 1

One should note that the growth parameterization interms of 120572 assumes that the attachment kinetics can bedescribed as an intrinsically local process which may notalways be a valid assumption For the ideal case of an infinitedefect-free surface this parameterizationmust be valid sincein this limit it is little more than a mathematical redefinitionNonlocal effects however such as surface transport betweenfacets could make the parameterization invalid in somecircumstances

For example it has been shown that the assumption of awell-defined attachment coefficient is not valid for the growthof mercury whiskers [21 22] since a value of 120572 ≫ 1 wouldbe required at the whisker tip In this case the tip growth isenhanced by surface diffusion of molecules along the facetedsides of the whisker and onto the growing tip which is anintrinsically nonlocal process The Schwoebel-Ehrlich effectprovides a potential barrier that normally inhibits surfacediffusion around corners [23 24] the mercury case beingan exception Experimental evidence supports the hypothesisthat 120572 le 1 for ice crystal growth and that nonlocal growtheffects can be neglected [7] but this is not known withcertainty

For the remainder of our discussion we will assume that(1) ice growth is in the limit 119901Peclet ≪ 1 (2) latent heatingof the crystal is negligible (3) the surrounding environmentand the crystal are at a constant temperature and pressureso 119888sat and 119863 have a single value throughout the system and(4) the attachment coefficient 120572 is well defined at the surfaceWe retain that 120572 may depend on temperature the surfacesupersaturation and the orientation of the surface relative to

the crystal axesWe also retain that120572may depend on the localcrystal structure [25]

Our overarching goal was to develop a basic numericalmodeling tool that would allow us to compare different theo-retical pictures of surface growth processeswith experimentalmeasurements of ice growth rates over a range of condi-tions Our focus was therefore on producing quantitativecalculations of crystal growth rates and morphologies fromwell-defined physical inputs including known diffusion ratesthrough the surrounding gas known initial sizes and mor-phologies of test crystals and theoretical parameterizationsof surface attachment kinetics and surface energy effects Welimited our calculations to the growth of fairly simple mor-phologies as these are best suited for extracting informationabout surface growth processes from experimental data

3 The 1D Spherical Model

It is useful to begin by examining the simplest case ofthe growth of spherical crystals Starting with the diffusionequation in spherical coordinates

120597120590

120597119905= 119863nabla

2120590 = 119863(

2

119903

120597120590

120597119903+1205972120590

1205971199032) (5)

and assuming a one-dimensional array of pixels with uniformsize Δ119903 gives the propagation equation120590 (119903 120591 + Δ120591)

= (1 minus 2Δ120591) 120590 (119903)

+ Δ120591 [(1 +Δ119903

119903) 120590 (119903 + Δ119903) + (1 minus

Δ119903

119903) 120590 (119903 minus Δ119903)]

(6)where

120591 =119863

(Δ119903)2119905 (7)

We typically choose Δ120591 = 12 as larger values can lead tonumerical instabilities and this gives120590 (119903 120591 + Δ120591)

= Δ120591 [(1 +Δ119903

119903) 120590 (119903 + Δ119903) + (1 minus

Δ119903

119903) 120590 (119903 minus Δ119903)]

(8)

31 Boundary Conditions In this spherical model we areessentially assuming a single spherical ldquofacetrdquo for the inter-face with the surface growth rate given in (4)We let all pixelswith radial index 119894 le 119894ice be ice and we refer to the next pixelwith 119894 = 119894ice+1 as a ldquoboundaryrdquo pixel In one time step the icesurface grows an amount 120575119903 = VΔ119905 and this growth extractsa total mass

120575119898 = 41205871199032120588ice120572Vkin120590boundΔ119905 (9)

from the boundary pixel shell reducing the supersaturationin that shell by

120575120590 =119888ice120572Vkin120590bound

119888sat(1 minus

Δ119903

119903bound)Δ119903

119863Δ120591 (10)

4 Journal of Computational Methods in Physics

where 119903bound is the radius of the center of the boundary pixelThus we can write

120590bound (120591 + Δ120591) = 120590bound (120591) [1 minus (1 minusΔ119903

119903bound)120572Δ120585Δ120591]

(11)

for the mass drain of the boundary pixel where

Δ120585 =119888iceVkin119888sat

Δ119903

119863=Δ119903

1198830

1198830=119888sat119888ice

119863

Vkin= radic

2120587119898

119896119879119863

asymp 0145 120583m sdot (119863

119863air) radic

11989611987915

119896119879

(12)

11987915= minus15

∘C and 119863air asymp 2 times 10minus5m2sec is the diffusion

constant in air at a pressure of one barWe combine (6) and (11) to create a total propagation

algorithm for the boundary pixel

120590 (119903bound 120591 + Δ120591) = (1 minusΔ119903

119903bound)120590solidΔ120591

+ (1 minus 2Δ120591) 120590 (119903bound)

+ (1 +Δ119903

119903bound)120590 (119903bound + Δ119903) Δ120591

(13)

where

120590solid = 120590 (119903bound) (1 minus 120572Δ120585) (14)

To describe the ice growth we assume that a boundarypixel starts out with zero accumulated mass and it turnsinto an ice pixel when it has accumulated a mass Δ119898 =

120588ice41205871199032

boundΔ119903 After one time step it accumulates the massin (9) giving

120575119898

Δ119898= 120572120590 (119903) (1 minus

Δ119903

119903bound)119888sat119888iceΔ120591Δ120585 (15)

Thus we define an accumulated mass parameter 120582 for theboundary pixel where 120582 starts at zero and after each timestepbecomes

120582 (120591 + Δ120591) = 120582 (120591) + (1 minusΔ119903

119903bound)120572120590Δ120582 (16)

where

Δ120582 =119888sat119888iceΔ120591Δ120585 (17)

When 120582 becomes greater than one the boundary pixel turnsto ice

32 Adaptive Time Steps In practice (17) gives an exceedinglyslow growth rate because 119888sat119888ice is quite small so we speedup the code by using

Δ120582 = ΛΔ120591Δ120585 (18)

where Λ is an adjustable parameter Note that the number ofsteps needed to advance one pixel is (120572120590Δ120582)minus1 so the growthvelocity is

V119899=Δ119903

Δ119905120572120590ΛΔ120591Δ120585 (19)

We optimize the speed of the code by increasing Λ as muchas possible subject to criteria that limit the resulting errors inthe growth behavior Our first criterion is that it should takemore than119873

0time steps to change a boundary pixel from air

to ice where typically1198730asymp 10 and this gives

Λ lt1

120572120590Δ120591Δ1205851198730

(20)

We also want the Peclet number to be much less thanunity so the growth is slower than the time it takes for thesupersaturation field to stabilize as described above and thisgives

119901Peclet =119877V119899

2119863lt 119901Pecletmax

Λ lt2

120572120590Δ120585(Δ119903

119877) sdot 119901Pecletmax

(21)

where 119877 is the crystal radius We use 119901Pecletmax = 11198730 so

Λ = min [ 2

120572120590Δ120585(Δ119903

119877) sdot 119901Pecletmax

1

120572120590Δ120591Δ1205851198730

]

asymp 119860 speed1

Δ120585(120572120590surf)max(Δ119903

119877)

(22)

where 119860 speed = 21198730 asymp 02 is a constant speedup parameterThe quantities (120572120590surf)max and 119877 are computed as the crystalgrows

In practice we have found that this relation yields growthrates that are somewhat too fast when 119877 lt 119873

0Δ119903 owing to

finite pixel size effects We counter this by replacing 119877 in (22)with

1198771015840= radic1198772 + 1198772

0 (23)

where we set 1198770= 1198730Δ119903 This results in an improvement

in the accuracy of the growth for small crystals with only amodest increase in running time

The choice of Λ essentially means using an adaptive timestep where the physical time for each step equals

Δ119905 = Δ120582Δ120585Δ1199050 (24)

Δ1199050=1198830

Vkinasymp 070msec sdot (

Vkinminus15∘C

Vkin) (25)

In running this code we typically use Δ120591 = 12 and Δ120585 =1 which gives Δ119903 = 119883

0 We define the radial index from 119894 = 1

to 119873 and the radius of the center of each spherical shell is119903119894= (119894 minus 1)Δ119903

Journal of Computational Methods in Physics 5

33 Analytic Solutions If the outer boundary is at infinity thespherical growth case gives the analytic solution

120590 (119903) = 120590infinminus119877

119903

120572

120572 + 120572diff120590infin (26)

where 119877 is the radius of the sphere and 120572diff = 1198830119877 In thelimit 120572 ≪ 120572diff we have 120590(119903) asymp 120590

infin while 120572 ≫ 120572diff gives

120590(119877) asymp 0 The growth velocity in all cases is

V119899= 120572Vkin120590 (119877) = (

120572120572diff120572 + 120572diff

) Vkin120590infin (27)

With an outer boundary at 119903out and 120590(119903out) = 120590out we have

120590 (119903) = 120590out minus (1198771015840

119903minus1198771015840

119903out)120590out (28)

where

1198771015840= [

120574

119877minus

1

119903out]

minus1

120574 =120572 + 120572diff120572

(29)

and the growth velocity becomes

V119899= (

120572120572diff120572 + 120572diff

) Vkin120590out[1 minus119877

119903out

1

120574]

minus1

(30)

In computing V119899 we may have to solve the equation

120590 (119877) =120590out (120574 minus 1)

120574 minus 119877119903out(31)

when 120574 is itself a function of 120590(119877) We solve this by iterationusing

120590(119877)119894+1=1

2[120590119894+120590out (120574 (120590(119877)119894) minus 1)

120574 (120590(119877)119894) minus 119877119903out

] (32)

at each time step which quickly converges to 120590(119877)

34 Model Validation As an example that compares themodel growth of a spherewith the analytic result we useΔ120591 =12 Δ120585 = 1 119879 = minus15

∘C 119873 = 300 (giving 119903out = 435 120583m)120590out = 001 120572 = 1 and 119877initial = 651198830 = 094 120583m Resultsare shown in Figure 1 using different values of 119860 speed For themodel with 119860 speed = 1 the adaptive time steps were large so120590surf did not have time to relax fully to its analytic value as thecrystal grew with the outcome that the crystal grew too fastWith smaller 119860 speed values the supersaturation field relaxedmore fully and 119877(119905) was closer to the analytic result

35 Quantitative Comparisons with Experimental Data Inour experiments measuring ice growth rates the outerboundaries of the growth chamber are much larger thantypical crystal sizes so 119903out asymp infin In modeling such data werequire a certain modeling accuracy and the straightforward

0 50 100 150 200Time (s)

0

5

10

15

Radi

us (m

icro

ns)

Figure 1 A comparison of different models for the growth ofa spherical crystal (solid lines) together with the analytic result(dashed line) as described in the text The models used 119860 speed = 1

(top curve) 02 (middle) and 002 (lower)

route to achieving this is to run the code with some suitablysmall 119860 speed together with some suitably large 119903out Unfortu-nately the code converges rather slowly to analytic results asseen in the example in Figure 1 and we have found that thisstraightforward approach results in unnecessarily long runtimes

We have found an alternative operating strategy thatachieves good modeling accuracy with substantially shorterrun times In this strategy we select values of 119903out that are notespecially large and values of 119860 speed that are not especiallysmall and then adjust 120590out down to compensate This strategysacrificesmodel accuracy in exchange for increasedmodelingspeed and is thus a variant of the usual trade-off oneencounters in numerical modeling

Figure 2 shows an example of how this modeling strategycan be used The analytic curve uses the same parameters asin Figure 1 except with 119903out = infin The models also use thesame parameters as in Figure 1 again with 119903out = 435 120583mand we fixed 119860 speed = 02 The model with 120590out = 001 doesnot match the analytic curve which reflects systematic errorsin the numerical modeling The model growth is faster thanthe analytic result because 119903out is too small and 119860 speed is toolargeWe compensate for these systematic errors by adjusting120590out downward 30 percent to 0007 as seen in Figure 2

From this exercise we see that there are two ways toproduce accurate models for comparison with experimentaldata The first way is simply to use a large 119903out and asmall 119860 speed to reduce the modeling systematic errors to anacceptable level This method is the most straightforwardbut from a general perspective it is not the most efficientThe second approach is to use smaller 119903out and larger 119860 speedso the code runs quickly and then lowers 120590out slightly tocompensate This latter approach has proven quite useful inpractice especially when comparing experiments where 120590

infin

is itself not known with extremely high accuracy as is oftenthe case

6 Journal of Computational Methods in Physics

0 50 100 150 200Time (s)

5

10

Radi

us (m

icro

ns)

Figure 2 Another comparison of different models for the growthof a spherical crystal (solid lines) with the analytic result (dashedline) The analytic calculation used 120590

infin= 001 and 119903out = infin The

models used 119860 speed = 02 and 119903out = 435 120583m with 120590out = 001 (topcurve) 0007 (middle) and 0004 (lower) This shows that one canadjust 120590out in the model to approximately compensate for systematicmodeling errors

4 The 2D Cylindrically Symmetric Model

The 1D spherical model outlined above is useful for exam-ining the cellular automata method in detail but of courseit is of little use otherwise as analytic results can be com-puted for the general spherical case Adding one additionaldimension adds significant complexity and richness to thediffusion problem however analytic results are not pos-sible for calculating the growth of even rather simple 2Dmorphologies

We have found that a 2D cylindrically symmetric modelis especially useful for comparison with ice crystal growthexperiments In this case a simple hexagonal plate is approxi-mated by a circular disk while a hexagonal column becomesa cylinder The six prism facets are replaced by a singlecylindrical ldquofacetrdquo analogous to the spherical case aboveand we equate the attachment kinetics on this surface tothe attachment kinetics on a flat prism facet Other thanintroducing a small geometrical correction the cylindri-cal approximation appears to give reasonable quantitativeresults It can be applied only to simple morphologies suchas simple plates and columns hollow columns and cappedcolumns Fortunately experiments tend to focus on thesesimple morphologies as they are best suited for examiningsurface growth dynamics

The 2D diffusion equation in cylindrical coordinates is

120597120590

120597119905= 119863nabla

2120590 = 119863[

1

119903

120597120590

120597119903+1205972120590

1205971199032+1205972120590

1205971199112] (33)

for 120590(119903 119911 119905) giving the propagation equation

120590 (120591 + Δ120591) = (1 minus 4Δ120591) 120590 + Δ120591

times [(1 +Δ119903

2119903) 120590 (119903 + Δ119903) + (1 minus

Δ119903

2119903) 120590 (119903 minus Δ119903)

+ 120590 (119911 + Δ119911) + 120590 (119911 minus Δ119911) ]

(34)

where 120591 is given in (7)Including boundary conditions at the crystal surface

follows along the lines of the 1D calculation above AssumingΔ120591 = 14 and Δ119903 = Δ119911 gives the propagation equations for119903-type and 119911-type boundary pixels

120590 (119903bound 119911 120591 + Δ120591)

=1

4[(1 +

Δ119903

2119903) 120590 (119903 + Δ119903) + (1 minus

Δ119903

2119903) 120590solid

+120590 (119911 + Δ119911) + 120590 (119911 minus Δ119911) ]

120590 (119903 119911bound 120591 + Δ120591)

=1

4[(1 +

Δ119903

2119903) 120590 (119903 + Δ119903) + (1 minus

Δ119903

2119903) 120590 (119903 minus Δ119903)

+ 120590 (119911 + Δ119911) + 120590solid]

(35)

where

120590solid = 120590 (119903bound) (1 minus 120572Δ120585) (36)

Note that a corner boundary pixel with neighboring ice pixelsin both 119903 and 119911 will propagate using

120590 (119903bound 119911bound 120591 + Δ120591)

=1

4[(1 +

Δ119903

2119903) 120590 (119903 + Δ119903) + (1 minus

Δ119903

2119903) 120590solid

+ 120590 (119911 + Δ119911) + 120590solid]

(37)

which drains the supersaturation twice as fast as an ordinaryboundary pixel as expected

We then define an accumulated mass parameter 120582 where120582 starts at zero and after each timestep becomes

120582 (120591 + Δ120591) = 120582 (120591) +sum120572120590Δ120582 (38)

where the sum is over the number of neighboring ice pixelsand

Δ120582119911-neighbors =

119888sat119888iceΔ120591Δ120585

Δ120582119903-neighbors = (1 plusmn

Δ119903

2119903)119888sat119888iceΔ120591Δ120585

(39)

Journal of Computational Methods in Physics 7

41The 119903 = 0 and 119911 = 0 Special Cases In our model we definethe first row of pixels to have 119903 = 0 which leads to problemswith the (1 plusmn Δ1199032119903) factors Going back to the diffusionequation we use Gaussrsquos law to generate the propagationequation for 119903 = 0 pixels

120590 (0 120591 + Δ120591)

= (1 minus 6Δ120591) 120590 (0 120591)

+ Δ120591 [120590 (119911 + Δ119911) + 120590 (119911 minus Δ119911) + 4120590 (Δ119903)]

(40)

Unfortunately taking Δ120591 = 14 in this expression leads toon-axis propagation instabilities As a compromise betweenrunning speed and computational accuracy we use Δ120591 = 16for the on-axis propagation equation giving

120590 (0 120591 + Δ120591) =1

6[120590 (119911 + Δ119911) + 120590 (119911 minus Δ119911) + 4120590 (Δ119903)]

(41)

while we continue using Δ120591 = 14 for off-axis pixels Thiscauses the supersaturation field to relax slightly more slowlyon the 119903-axis but this fix apparently does not adversely affectthe modeling results

Similarly we define the first row of 119911 pixels to have 119911 = 0and for that row we use the propagation equation

120590 (120591 + Δ120591) =1

4[(1 +

Δ119903

2119903) 120590 (119903 + Δ119903)

+ (1 minusΔ119903

2119903) 120590 (119903 minus Δ119903) + 2120590 (119911 + Δ119911)]

(42)

42 Adaptive Time Steps Speeding up the code using adap-tive time steps again proceeds along the lines of the 1Dproblem We use

Δ120582 = ΛΔ120591Δ120585 (43)

with

Λ = 119860 speed1

Δ120585(120572120590surf)max(Δ119903

1198771015840max)

1198771015840

max = radic1198772

max + 1199112

max + 1198772

0

(44)

as above and the physical time step is that given in (25)

43 Neighbor Relations In all these expressions we mustchoose the attachment coefficient 120572 with some care asits value will depend on the number and orientation ofneighboring solid pixels We label a boundary pixel with(119873119903 119873119911) where119873

119903is the number of neighboring ice pixels in

the 119903 direction and 119873119911is the number of ice neighbors in the

119911 direction Both119873119903and119873

119911can take values 0 1 or 2 giving

nine cases for (119873119903 119873119911) The cases are

(0 0) the pixel is an air pixel

(0 1) one ice neighbor in the 119911 direction so 120572 =

120572basal the physical value appropriate for a basal facetsurface(1 0) one ice neighbor in the 119903direction so120572 = 120572prismfor a prism facet surface(1 1) a kink site where the growth will not benucleation limited since the corner provides a sourceof molecular steps We do not know a priori whatvalue to use for 120572 on this site but assume a constant120572 = 120572

11

(0 2) (1 2) (2 0) (2 1) and (2 2) these are allunusual cases where the growth will be fast so weassume that 120572 = 1

We index these possibilities with a single number bycomputing a boundary parameter 119861 = 2119873

2

119903+ 1198732

119911 We then

have 119861 = 0 for an air pixel 119861 = 1 for a basal facet 119861 = 2 for aprism facet 119861 = 3 for a (1 1) kink location and 119861 gt 3 for allother cases

If we consider the special case where 120572 is equal to someconstant value independent of the orientation of the surfacewith respect to the crystal lattice then the growth velocityshould equal V = 120572Vkin120590 for all surfaces For the basal or prismfacet surfaces in this constant-120572 case we take 120572basal = 120572prism =120572 while an analysis of the growth of a (11) surface shows thatwemust take 120572

11= 120572radic2 if the above algorithm is to produce

the correct growth velocity

44 Limitations on Grid Size Aswe pointed out in [26] thereare physical limits to how coarse the computing grid canbe made before instabilities appear or the growth deviatessubstantially from real growth Taking Δ120585 gt 1120572would cause120590solid to become negative which causes some concern in thatit may produce instabilities in the code With this limitationthe grid spacing could not be larger than Δ119903 = Δ119911 = 119883

0120572

For air at a pressure of one atmosphere and 120572 asymp 1 this givesthe pixel size119883

0asymp 0145 120583m

Physically we can gain some insights into these limita-tions from dendrite growth theory [7] We have 119883

0asymp 119877kin

(the latter from (28) in [7]) and a growing dendrite has a tipradius

119877tip asymp1198830

120572119904 (45)

where 119904 is the dimensionless solvability parameter which is oforder unity for ice crystal growth [7]The stability of the codethus limits the grid spacing to be no greater than the tip radiusof a growing dendritic structure From this we see that thecode can only function properly when the grid spacing is fineenough to allow the growth of physically realistic dendriticstructures the scale of which is given by solvability theory

In practice we typically assume that Δ119903 = Δ119911 = 1198830asymp

0145 120583m when comparing models with ice growth data Acoarser grid would reduce run times but at the risk of notreproducing physically relevantmorphological behaviorsWehave not yet explored in detail how different grid sizes affectthe modeling behavior

8 Journal of Computational Methods in Physics

45 Scaling Behavior If we run the code and produce somecomplex crystal shape the interpretation of our result stillcontains an ambiguity The crystal size is given in pixelswhere Δ119903 = Δ119911 = Δ120585119883

0is the pixel size The parameter

Δ120585 was fixed in the code but 1198830depends on the diffusion

constant 119863 which is not otherwise specified Similarly asingle time step in the code corresponds to a physical time

Δ119905 =1198832

0Δ1205852Δ120591

119863sim 119863 (46)

Thus we see that the growth behavior at different airpressures (different D) is determined once we know thegrowth at a single pressure (provided that 120590

infinis the same

at the different pressures) If the air pressure is half anatmosphere the growthmorphology (however complex) willbe the same as at one atmosphere except in the former casethe crystal will be double the size in double the time Thisscaling behavior nicely explains why ice crystal morphologyis generally simpler for smaller crystals andor for lower airpressures which has long been observed [7]

46 Analytic Solutions and Model Validation The case of aninfinitely long cylinder of radius 119877in has the analytic solution

120590 (119903) =120573 + log (119903119877in)

120573 + log (119877out119877in)120590119877out (47)

where the outer boundary has 120590(119877out) = 120590119877out and 120573 =

1198830119877in120572 giving the growth velocity

V =120572120572diffcyl

120572 + 120572diffcylVkin120590119877out

120572diffcyl =1

119861

1198830

119877in

(48)

with 119861 = log(119877out119877in)To compare this analytic result with numerical models of

a growing cylinder we created a version of our codewith peri-odic boundary conditions in 119911 thus modeling the growth ofan infinite cylinder Figure 3 shows results using 119879 = minus15∘C120590119877out = 001 119877init = 1 120583m 120572 = 1 119877out = 300 and 119883

0=

435 120583m Again we see that the numerical model converges tothe analytic result as 119860 speed goes to zero

5 Gibbs-Thomson Effect

The basic cellular automata code described above includes afairly realistic treatment of the attachment kinetics but doesnot include any effects of surface energy This is somewhatjustified for the case of faceted ice crystal growth as one canshow that attachment kinetic effects dominate the growthbehavior while surface energy effects are less important [7]Nevertheless surface energy effects are not always negligibleespecially at low supersaturations As we will demonstratebelow models with highly anisotropic attachment kineticsand no surface energy effects can exhibit the formation ofone-pixel-wide features that are not physically plausible To

5

0

10

Radi

us (m

icro

ns)

0 50 100 150 200Time (s)

Figure 3 The dashed line shows the analytic result for the growthof an infinite cylinder with 120572 = 1 119877out = 435 120583m and 120590

119877out = 001as described in the text Solid lines show numerical modeling resultsusing 119860 speed = 02 (top line) 002 (middle line) and 0002 (bottomline)

suppress these unphysicalmodelswemust include the surfaceenergy via the Gibbs-Thomson effect

For the ice case we are considering here the equilibriumvapor density above a curved surface can be written [27]

119888eq = 119888sat (1 + 120575120581) (49)

where 119888sat is the equilibrium (saturated) vapor density of a flatsurface 120581 is the surface curvature

120581 =1

1199031

+1

1199032

(50)

where 1199031 1199032are the principal radii of curvature of the surface

and

120575 =120574

119888solid119896119879asymp 1 nm (51)

where 120574 asymp 01 Jm2 is the surface energy of the icevaporinterface The anisotropy of 120574 is not well known but theavailable evidence suggests it is rather small [20] so for theremainder of our discussion we assume an isotropic surfaceenergy We believe this is a accurate assumption for ice but itis in stark contrast to the highly anisotropic ice surface energyassumed in [11]

From this we have the supersaturation over a curvedsurface

120590curved asymp 120590 minus120575

119877119888

(52)

to lowest order in 120575119877119888 where120590 is the normal supersaturation

relative to a flat interface and for later convenience we define119877119888= 120581minus1 Putting in some numbers a sharp ice needle might

have 119877119888asymp 1 120583m giving Δ120590 = 120590curved minus 120590 asymp 01 percent

which is small but not always negligible Moreover setting

Journal of Computational Methods in Physics 9

Nz = 7

Nr = 5

z

r

Figure 4 A graphical definition of the outer boundary pixels usedfor incorporating surface energy effects into our cellular automatamodel Here the blue pixels are ice while the clear pixels are airThe red pixels are boundary pixels having the largest 119903 and 119911 valuesdefined to be the ldquoouterrdquo boundary pixels

119877119888asymp 1198830asymp 015 120583m gives a rather large Δ120590 asymp 066 percent

indicating that one-pixel-wide features should not be presentat low supersaturations

Adding the Gibbs-Thomson effect requires an algorith-mic estimation of 119877

119888for all boundary pixels in our cellular

automata model We examined several possibilities but didnot find a suitable algorithm that could be used with arbitrarymorphologies in our cylindrically symmetrical model Sinceour primary goal was to model simple morphologies forcomparison with experimental data as described above wesettled for a simpler algorithm for such cases

Throughout our investigations we found that the overallgrowth behavior of faceted crystals with simplemorphologieswas determined primarily by the growth of the outermostfacet surfaces To apply the Gibbs-Thomson effect on thesesurfaces we defined the outer boundary pixels as shown inFigure 4 yielding the numbers 119873

119903and 119873

119911as the number of

outer-boundary pixelsUsing these outer boundary pixels as a proxy for esti-

mating surface curvature we defined 119877119888= Δ119903119873

1199032 for the

edges of plates 119877119888= Δ119903119873

1199112 for the edges of hollow columns

and 119877119888= Δ119903119873

1199114 for the tips of needles For all boundary

pixels that were not outer-boundary pixels we assumed that119877119888= infin Although these are crude estimations of surface

curvature we found them satisfactory for incorporating theGibbs-Thomson effect in our model in part because this is arelatively small effect compared to the more dominant role ofattachment kinetics

51 A Gibbs-Thomson Example The top image in Figure 5shows how our model can yield unphysical results if wedo not include the Gibbs-Thomson effect In this model weassumed the input parameters 119879 = minus15

∘C 120590infin

= 001

120572basal = exp(minus003120590surf) 120572prism = 1 an initial prism radius1198770= 2 120583m initial prism half-height119867

0= 5 120583m and a Gibbs-

Thomson parameter 120575 = 0 After growing this model for sixseconds the initial simple prism grew into a capped columnwhere the two capping plates each had a thickness of justone pixel as shown in the top image in Figure 5 Such smallstructural features are unphysical at this supersaturation andsimilar one-pixel-wide features often appeared in our modelswhen the supersaturation was low and the anisotropy inattachment kinetics was high

Figure 5 A test of the Gibbs-Thomson effect in our crystal growthmodel In each image the white pixels show ice while the brightnessof the surrounding region is proportional to the supersaturationModel crystals are reflection symmetric about the 119911 = 0 plane andthe images show 119911 ge 0 only Thin vertical lines show the initialcrystal size Using 120572prism = 1 and 120572basal ≪ 120572prism resulted in growthonly on the prism surfaces Full model parameters are given in thetext The top image displays the crystal after six seconds of growthwith a Gibbs-Thomson parameter 120575 = 0 This model exhibits a one-pixel-thick plate growing on the end of the column which is notphysically plausibleThemiddle and lower images show results using120575 = 03 and 10 nm respectively For these models the unphysicalplate growth does not appear

Including nonzero values of 120575 suppressed these smallfeatures as demonstrated in the lower two images in Figure 5From this example as well as additional tests with differentgrowth morphologies we found that our relatively simplealgorithm for including the Gibbs-Thomson effect yieldedreasonable results suppressing the unphysical growth of one-pixel-wide features Models with higher 120590

infinvalues or with

lower anisotropies in the attachment kinetics were generallyless affected by including the Gibbs-Thomson effect

6 An Example Comparison with Experiment

Figure 6 shows an example of how the cellular automatamethod can be used in laboratory investigations of growingice crystals In this experiment we first grew a thin ldquoelectricrdquoice needle as described in [28] and then transferred theneedle to a second growth chamber where we grew a thinplate-like crystal on the needle tip all in air at a pressureof one bar The hexagonal symmetry of the crystal andthe relatively simple growth morphology were well suitedfor analysis using our cylindrically symmetric model Byadjusting 120590

infinand the surface attachment coefficients for the

principal facets we were able to generate a model crystal

10 Journal of Computational Methods in Physics

50120583m

(a) (b)

Radi

us (120583

m)

60

50

40

30

20

10

0minus40 minus20

Plate radius

Needle radius

0 20 40Time (s)

60 80 100

(c)

Figure 6 An example showing our numerical modeling of a laboratory-grown ice crystal The composite image in (a) shows five successiveviews of a thin plate-like crystal growing on the end of a slender ice needle with a 50120583m scale bar The crystal is shown in a side view withillumination from behindThe composite image in (b) shows our numerical model of the same crystal Here the brightness around the crystalis proportional to supersaturation The data points in (c) shows measurements of the plate and needle radii (the latter at a distance of 50120583mfrom the base of the plate) as a function of timeThe lines in the graph are from the growth of the model crystal The inset image in (c) showsa more frontal view of a similar plate-on-needle crystal with a plate radius of 80120583m showing the thin sectored-plate morphology Note thatat earlier times (119905 lt 0 in the graph) the crystal was growing from a tapered needle to a column with nearly uniform radius along its lengthThe time axes were shifted so the plate began growing at 119905 asymp 0 for both the real and model crystals

that matched both the morphology and growth rates ofthe laboratory crystal In addition to the plate and needleradii note that the tapered neck of the needle just belowthe plate seen in Figure 6 was nicely reproduced by themodel crystal A discussion of the physical significance ofthe best-fit model parameters used for this model is left fora subsequent publication comparing the growth of manycrystals under different conditions This example is meantto demonstrate only that the behavior of faceted laboratory-grown ice crystals can be accuratelymodeledwith the cellularautomata method

7 Discussion

Numericalmodeling of structure formation during solidifica-tion has been well studied for decades especially cases whereweak surface anisotropies in diffusion-limited growth yieldrounded dendritic branching Modeling structures that areboth branched and faceted have beenmore difficult and onlyrecently researchers have demonstrated suitable numericalmethods [10 11] In the present work we have developeda numerical model aimed at investigating the formationof simple faceted morphologies during ice crystal growthOur model incorporates the cellular automata technique in[10] but uses a more physically realistic treatment of thesurface attachment kinetics and the Gibbs-Thomson effectUsing this model we are able to reproduce ice crystalgrowth behavior for simple growth morphologies over a

broad range of conditions We have been using this modelto aid in the interpretation of ice growth experiments inorder to extract information about the anisotropic surfaceattachment kinetics that governs ice crystal growth fromwater vapor Modifying the model for application to othermaterial systems is likely straightforward

At present we have limited our model to cylindricallysymmetric crystal growth and relatively simple growth mor-phologies for comparison with measurements A full three-dimensional extension of our model should be possibleas was demonstrated in [10] potentially allowing completemodeling of the many complex branched and faceted struc-tures that are commonly observed in ice crystal growth [7]Achieving this goal would likely require a substantially betterunderstanding of the surface attachment kinetics in ice thanpresently exists along with more advanced algorithms forincorporating all the relevant surface physics into a cellularautomata model

In general it appears that the cellular automata methodis well suited for modeling faceted crystal growth and itmay be the method of choice in material systems exhibitingstrongly anisotropic attachment kinetics Incorporatingmoresophisticated descriptions of surface growth processes willrequire substantially better algorithms than we have outlinedabove especially when extending the models to three dimen-sions Whether such models can reproduce the full rangeof complex structures appearing in faceted crystal growthremains to be seen

Journal of Computational Methods in Physics 11

References

[1] R Trivedi and W Kurz ldquoDendritic growthrdquo InternationalMaterials Reviews vol 39 no 2 pp 49ndash74 1994

[2] E A Brener ldquoThree-dimensional dendritic growthrdquo Journal ofCrystal Growth vol 166 no 1ndash4 pp 339ndash346 1996

[3] I Avramov ldquoKinetics of growth of nanowhiskers (nanowiresand nanotubes)rdquo Nanoscale Research Letters vol 2 no 5 pp235ndash239 2007

[4] H Zhang I Hussain M Brust M F Butler S P Rannard andA I Cooper ldquoAligned two- and three-dimensional structuresby directional freezing of polymers and nanoparticlesrdquo NatureMaterials vol 4 pp 787ndash793 2005

[5] M C Cross and P C Hohenberg ldquoPattern formation outsideof equilibriumrdquo Reviews of Modern Physics vol 65 no 3 pp851ndash1112 1993

[6] K Kassner Pattern Formation in Diffusion-Limited CrystalGrowth World Scientific Singapore 1996

[7] K G Libbrecht ldquoThe physics of snow crystalsrdquo Reports onProgress in Physics vol 68 no 4 pp 855ndash895 2005

[8] J Nelson ldquoGrowth mechanisms to explain the primary andsecondary habits of snow crystalsrdquo Philosophical Magazine Avol 81 no 10 pp 2337ndash2373 2001

[9] H R Pruppacher and J D Klett Microphysics of Clouds andPrecipitation Kluwer Academic Publishers Dordrecht TheNetherlands 1997

[10] J Gravner and D Griffeath ldquoModeling snow-crystal growtha three-dimensional mesoscopic approachrdquo Physical Review Evol 79 no 1 Article ID 011601 18 pages 2009

[11] J W Barrett H Garcke and R Nurnberg ldquoNumerical com-putations of faceted pattern formation in snow crystal growthrdquoPhysical Review E vol 86 no 1 Article ID 011604 14 pages2012

[12] S-Y Hong J Dudhia and S-H Chen ldquoA revised approach toice microphysical processes for the bulk parameterization ofclouds and precipitationrdquoMonthlyWeather Review vol 132 no1 pp 103ndash120 2004

[13] M Matsumoto S Saito and I Ohmine ldquoMolecular dynamicssimulation of the ice nucleation and growth process leading towater freezingrdquo Nature vol 416 no 6879 pp 409ndash413 2002

[14] J G Dash A W Rempel and J S Wettlaufer ldquoThe physicsof premelted ice and its geophysical consequencesrdquo Reviews ofModern Physics vol 78 no 3 pp 695ndash741 2006

[15] A Karma and W-J Rappel ldquoPhase-field method for com-putationally efficient modeling of solidification with arbitraryinterface kineticsrdquo Physical Review E vol 53 no 4 pp R3017ndashR3020 1996

[16] I Singer-Loginova andHM Singer ldquoThe phase field techniquefor modeling multiphase materialsrdquo Reports on Progress inPhysics vol 71 no 10 Article ID 106501 2008

[17] A Schmidt ldquoComputation of three dimensional dendrites withfinite elementsrdquo Journal of Computational Physics vol 125 no2 pp 293ndash312 1996

[18] E Yokoyama ldquoFormation of patterns during growth of snowcrystalsrdquo Journal of Crystal Growth vol 128 no 1ndash4 pp 251ndash257 1993

[19] S E Wood M B Baker and D Calhoun ldquoNew model forthe vapor growth of hexagonal ice crystals in the atmosphererdquoJournal of Geophysical ResearchD vol 106 no 5 pp 4845ndash48702001

[20] K G Libbrecht ldquoOn the equilibrium shape of an ice crystalrdquo2012 httparxivorgabs12051452

[21] G W Sears ldquoA growth mechanism for mercury whiskersrdquo ActaMetallurgica vol 3 no 4 pp 361ndash366 1955

[22] R L Parker R L Anderson and S C Hardy ldquoGrowth andevaporation kinetics and surface diffusion of K and Hg crystalwhiskersrdquo Applied Physics Letters vol 3 no 6 pp 93ndash95 1963

[23] S J Liu H Huang and C HWoo ldquoSchwoebel-Ehrlich barrierfrom two to three dimensionsrdquo Applied Physics Letters vol 80no 18 pp 3295ndash3297 2002

[24] Y Saito Statistical Physics of Crystal Growth World ScientificSingapore 1996

[25] K G Libbrecht ldquoExplaining the formation of thin ice-crystalplates with structure-dependent attachment kineticsrdquo Journal ofCrystal Growth vol 258 no 1-2 pp 168ndash175 2003

[26] K G Libbrecht ldquoPhysically derived rules for simulating facetedcrystal growth using cellular automatardquo 2008 httparxivorgabs08072616

[27] W Thomson ldquoOn the equilibrium vapor at a curved surface ofliquidrdquo Philosophical Magazine vol 42 no 282 pp 448ndash4521871

[28] K G Libbrecht T Crosby and M Swanson ldquoElectricallyenhanced free dendrite growth in polar and non-polar systemsrdquoJournal of Crystal Growth vol 240 no 1-2 pp 241ndash254 2002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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ThermodynamicsJournal of

Journal of Computational Methods in Physics 3

small rescaling of 120590infin

[7] Furthermore many experimentsare done using ice crystals resting on a thermally conductingsubstrate and in this case the effects of latent heating areessentially negligible [7] For these reasons we have ignoredlatent heating in the present model with the understandingthat heating effects may be important in some experimentalcircumstances

The attachment kinetics are usually defined bywriting thegrowth velocity normal to a flat surface in terms of the Hertz-Knudsen formula [7]

V119899= 120572

119888sat119888solid

radic119896119879

2120587119898120590surf

= 120572Vkin120590surf

(4)

where 120590surf is the supersaturation at the surface and thisequation defines the ldquokinetic velocityrdquo Vkin The parameter120572 is known as the condensation coefficient and it embodiesthe surface physics that governs how water molecules areincorporated into the ice lattice collectively known as theattachment kinetics The attachment kinetics can be quitecomplex so in general 120572 will depend on 119879 and 120590surf andperhaps surface structure and geometry surface chemistryand so forth If molecules striking the surface are instantlyincorporated into it then 120572 = 1 otherwise we have 120572 le 1A faceted growth form usually indicates that the growth islimited by attachment kinetics so 120572 lt 1 on faceted surfacesFor a molecularly rough surface we expect 120572 asymp 1

One should note that the growth parameterization interms of 120572 assumes that the attachment kinetics can bedescribed as an intrinsically local process which may notalways be a valid assumption For the ideal case of an infinitedefect-free surface this parameterizationmust be valid sincein this limit it is little more than a mathematical redefinitionNonlocal effects however such as surface transport betweenfacets could make the parameterization invalid in somecircumstances

For example it has been shown that the assumption of awell-defined attachment coefficient is not valid for the growthof mercury whiskers [21 22] since a value of 120572 ≫ 1 wouldbe required at the whisker tip In this case the tip growth isenhanced by surface diffusion of molecules along the facetedsides of the whisker and onto the growing tip which is anintrinsically nonlocal process The Schwoebel-Ehrlich effectprovides a potential barrier that normally inhibits surfacediffusion around corners [23 24] the mercury case beingan exception Experimental evidence supports the hypothesisthat 120572 le 1 for ice crystal growth and that nonlocal growtheffects can be neglected [7] but this is not known withcertainty

For the remainder of our discussion we will assume that(1) ice growth is in the limit 119901Peclet ≪ 1 (2) latent heatingof the crystal is negligible (3) the surrounding environmentand the crystal are at a constant temperature and pressureso 119888sat and 119863 have a single value throughout the system and(4) the attachment coefficient 120572 is well defined at the surfaceWe retain that 120572 may depend on temperature the surfacesupersaturation and the orientation of the surface relative to

the crystal axesWe also retain that120572may depend on the localcrystal structure [25]

Our overarching goal was to develop a basic numericalmodeling tool that would allow us to compare different theo-retical pictures of surface growth processeswith experimentalmeasurements of ice growth rates over a range of condi-tions Our focus was therefore on producing quantitativecalculations of crystal growth rates and morphologies fromwell-defined physical inputs including known diffusion ratesthrough the surrounding gas known initial sizes and mor-phologies of test crystals and theoretical parameterizationsof surface attachment kinetics and surface energy effects Welimited our calculations to the growth of fairly simple mor-phologies as these are best suited for extracting informationabout surface growth processes from experimental data

3 The 1D Spherical Model

It is useful to begin by examining the simplest case ofthe growth of spherical crystals Starting with the diffusionequation in spherical coordinates

120597120590

120597119905= 119863nabla

2120590 = 119863(

2

119903

120597120590

120597119903+1205972120590

1205971199032) (5)

and assuming a one-dimensional array of pixels with uniformsize Δ119903 gives the propagation equation120590 (119903 120591 + Δ120591)

= (1 minus 2Δ120591) 120590 (119903)

+ Δ120591 [(1 +Δ119903

119903) 120590 (119903 + Δ119903) + (1 minus

Δ119903

119903) 120590 (119903 minus Δ119903)]

(6)where

120591 =119863

(Δ119903)2119905 (7)

We typically choose Δ120591 = 12 as larger values can lead tonumerical instabilities and this gives120590 (119903 120591 + Δ120591)

= Δ120591 [(1 +Δ119903

119903) 120590 (119903 + Δ119903) + (1 minus

Δ119903

119903) 120590 (119903 minus Δ119903)]

(8)

31 Boundary Conditions In this spherical model we areessentially assuming a single spherical ldquofacetrdquo for the inter-face with the surface growth rate given in (4)We let all pixelswith radial index 119894 le 119894ice be ice and we refer to the next pixelwith 119894 = 119894ice+1 as a ldquoboundaryrdquo pixel In one time step the icesurface grows an amount 120575119903 = VΔ119905 and this growth extractsa total mass

120575119898 = 41205871199032120588ice120572Vkin120590boundΔ119905 (9)

from the boundary pixel shell reducing the supersaturationin that shell by

120575120590 =119888ice120572Vkin120590bound

119888sat(1 minus

Δ119903

119903bound)Δ119903

119863Δ120591 (10)

4 Journal of Computational Methods in Physics

where 119903bound is the radius of the center of the boundary pixelThus we can write

120590bound (120591 + Δ120591) = 120590bound (120591) [1 minus (1 minusΔ119903

119903bound)120572Δ120585Δ120591]

(11)

for the mass drain of the boundary pixel where

Δ120585 =119888iceVkin119888sat

Δ119903

119863=Δ119903

1198830

1198830=119888sat119888ice

119863

Vkin= radic

2120587119898

119896119879119863

asymp 0145 120583m sdot (119863

119863air) radic

11989611987915

119896119879

(12)

11987915= minus15

∘C and 119863air asymp 2 times 10minus5m2sec is the diffusion

constant in air at a pressure of one barWe combine (6) and (11) to create a total propagation

algorithm for the boundary pixel

120590 (119903bound 120591 + Δ120591) = (1 minusΔ119903

119903bound)120590solidΔ120591

+ (1 minus 2Δ120591) 120590 (119903bound)

+ (1 +Δ119903

119903bound)120590 (119903bound + Δ119903) Δ120591

(13)

where

120590solid = 120590 (119903bound) (1 minus 120572Δ120585) (14)

To describe the ice growth we assume that a boundarypixel starts out with zero accumulated mass and it turnsinto an ice pixel when it has accumulated a mass Δ119898 =

120588ice41205871199032

boundΔ119903 After one time step it accumulates the massin (9) giving

120575119898

Δ119898= 120572120590 (119903) (1 minus

Δ119903

119903bound)119888sat119888iceΔ120591Δ120585 (15)

Thus we define an accumulated mass parameter 120582 for theboundary pixel where 120582 starts at zero and after each timestepbecomes

120582 (120591 + Δ120591) = 120582 (120591) + (1 minusΔ119903

119903bound)120572120590Δ120582 (16)

where

Δ120582 =119888sat119888iceΔ120591Δ120585 (17)

When 120582 becomes greater than one the boundary pixel turnsto ice

32 Adaptive Time Steps In practice (17) gives an exceedinglyslow growth rate because 119888sat119888ice is quite small so we speedup the code by using

Δ120582 = ΛΔ120591Δ120585 (18)

where Λ is an adjustable parameter Note that the number ofsteps needed to advance one pixel is (120572120590Δ120582)minus1 so the growthvelocity is

V119899=Δ119903

Δ119905120572120590ΛΔ120591Δ120585 (19)

We optimize the speed of the code by increasing Λ as muchas possible subject to criteria that limit the resulting errors inthe growth behavior Our first criterion is that it should takemore than119873

0time steps to change a boundary pixel from air

to ice where typically1198730asymp 10 and this gives

Λ lt1

120572120590Δ120591Δ1205851198730

(20)

We also want the Peclet number to be much less thanunity so the growth is slower than the time it takes for thesupersaturation field to stabilize as described above and thisgives

119901Peclet =119877V119899

2119863lt 119901Pecletmax

Λ lt2

120572120590Δ120585(Δ119903

119877) sdot 119901Pecletmax

(21)

where 119877 is the crystal radius We use 119901Pecletmax = 11198730 so

Λ = min [ 2

120572120590Δ120585(Δ119903

119877) sdot 119901Pecletmax

1

120572120590Δ120591Δ1205851198730

]

asymp 119860 speed1

Δ120585(120572120590surf)max(Δ119903

119877)

(22)

where 119860 speed = 21198730 asymp 02 is a constant speedup parameterThe quantities (120572120590surf)max and 119877 are computed as the crystalgrows

In practice we have found that this relation yields growthrates that are somewhat too fast when 119877 lt 119873

0Δ119903 owing to

finite pixel size effects We counter this by replacing 119877 in (22)with

1198771015840= radic1198772 + 1198772

0 (23)

where we set 1198770= 1198730Δ119903 This results in an improvement

in the accuracy of the growth for small crystals with only amodest increase in running time

The choice of Λ essentially means using an adaptive timestep where the physical time for each step equals

Δ119905 = Δ120582Δ120585Δ1199050 (24)

Δ1199050=1198830

Vkinasymp 070msec sdot (

Vkinminus15∘C

Vkin) (25)

In running this code we typically use Δ120591 = 12 and Δ120585 =1 which gives Δ119903 = 119883

0 We define the radial index from 119894 = 1

to 119873 and the radius of the center of each spherical shell is119903119894= (119894 minus 1)Δ119903

Journal of Computational Methods in Physics 5

33 Analytic Solutions If the outer boundary is at infinity thespherical growth case gives the analytic solution

120590 (119903) = 120590infinminus119877

119903

120572

120572 + 120572diff120590infin (26)

where 119877 is the radius of the sphere and 120572diff = 1198830119877 In thelimit 120572 ≪ 120572diff we have 120590(119903) asymp 120590

infin while 120572 ≫ 120572diff gives

120590(119877) asymp 0 The growth velocity in all cases is

V119899= 120572Vkin120590 (119877) = (

120572120572diff120572 + 120572diff

) Vkin120590infin (27)

With an outer boundary at 119903out and 120590(119903out) = 120590out we have

120590 (119903) = 120590out minus (1198771015840

119903minus1198771015840

119903out)120590out (28)

where

1198771015840= [

120574

119877minus

1

119903out]

minus1

120574 =120572 + 120572diff120572

(29)

and the growth velocity becomes

V119899= (

120572120572diff120572 + 120572diff

) Vkin120590out[1 minus119877

119903out

1

120574]

minus1

(30)

In computing V119899 we may have to solve the equation

120590 (119877) =120590out (120574 minus 1)

120574 minus 119877119903out(31)

when 120574 is itself a function of 120590(119877) We solve this by iterationusing

120590(119877)119894+1=1

2[120590119894+120590out (120574 (120590(119877)119894) minus 1)

120574 (120590(119877)119894) minus 119877119903out

] (32)

at each time step which quickly converges to 120590(119877)

34 Model Validation As an example that compares themodel growth of a spherewith the analytic result we useΔ120591 =12 Δ120585 = 1 119879 = minus15

∘C 119873 = 300 (giving 119903out = 435 120583m)120590out = 001 120572 = 1 and 119877initial = 651198830 = 094 120583m Resultsare shown in Figure 1 using different values of 119860 speed For themodel with 119860 speed = 1 the adaptive time steps were large so120590surf did not have time to relax fully to its analytic value as thecrystal grew with the outcome that the crystal grew too fastWith smaller 119860 speed values the supersaturation field relaxedmore fully and 119877(119905) was closer to the analytic result

35 Quantitative Comparisons with Experimental Data Inour experiments measuring ice growth rates the outerboundaries of the growth chamber are much larger thantypical crystal sizes so 119903out asymp infin In modeling such data werequire a certain modeling accuracy and the straightforward

0 50 100 150 200Time (s)

0

5

10

15

Radi

us (m

icro

ns)

Figure 1 A comparison of different models for the growth ofa spherical crystal (solid lines) together with the analytic result(dashed line) as described in the text The models used 119860 speed = 1

(top curve) 02 (middle) and 002 (lower)

route to achieving this is to run the code with some suitablysmall 119860 speed together with some suitably large 119903out Unfortu-nately the code converges rather slowly to analytic results asseen in the example in Figure 1 and we have found that thisstraightforward approach results in unnecessarily long runtimes

We have found an alternative operating strategy thatachieves good modeling accuracy with substantially shorterrun times In this strategy we select values of 119903out that are notespecially large and values of 119860 speed that are not especiallysmall and then adjust 120590out down to compensate This strategysacrificesmodel accuracy in exchange for increasedmodelingspeed and is thus a variant of the usual trade-off oneencounters in numerical modeling

Figure 2 shows an example of how this modeling strategycan be used The analytic curve uses the same parameters asin Figure 1 except with 119903out = infin The models also use thesame parameters as in Figure 1 again with 119903out = 435 120583mand we fixed 119860 speed = 02 The model with 120590out = 001 doesnot match the analytic curve which reflects systematic errorsin the numerical modeling The model growth is faster thanthe analytic result because 119903out is too small and 119860 speed is toolargeWe compensate for these systematic errors by adjusting120590out downward 30 percent to 0007 as seen in Figure 2

From this exercise we see that there are two ways toproduce accurate models for comparison with experimentaldata The first way is simply to use a large 119903out and asmall 119860 speed to reduce the modeling systematic errors to anacceptable level This method is the most straightforwardbut from a general perspective it is not the most efficientThe second approach is to use smaller 119903out and larger 119860 speedso the code runs quickly and then lowers 120590out slightly tocompensate This latter approach has proven quite useful inpractice especially when comparing experiments where 120590

infin

is itself not known with extremely high accuracy as is oftenthe case

6 Journal of Computational Methods in Physics

0 50 100 150 200Time (s)

5

10

Radi

us (m

icro

ns)

Figure 2 Another comparison of different models for the growthof a spherical crystal (solid lines) with the analytic result (dashedline) The analytic calculation used 120590

infin= 001 and 119903out = infin The

models used 119860 speed = 02 and 119903out = 435 120583m with 120590out = 001 (topcurve) 0007 (middle) and 0004 (lower) This shows that one canadjust 120590out in the model to approximately compensate for systematicmodeling errors

4 The 2D Cylindrically Symmetric Model

The 1D spherical model outlined above is useful for exam-ining the cellular automata method in detail but of courseit is of little use otherwise as analytic results can be com-puted for the general spherical case Adding one additionaldimension adds significant complexity and richness to thediffusion problem however analytic results are not pos-sible for calculating the growth of even rather simple 2Dmorphologies

We have found that a 2D cylindrically symmetric modelis especially useful for comparison with ice crystal growthexperiments In this case a simple hexagonal plate is approxi-mated by a circular disk while a hexagonal column becomesa cylinder The six prism facets are replaced by a singlecylindrical ldquofacetrdquo analogous to the spherical case aboveand we equate the attachment kinetics on this surface tothe attachment kinetics on a flat prism facet Other thanintroducing a small geometrical correction the cylindri-cal approximation appears to give reasonable quantitativeresults It can be applied only to simple morphologies suchas simple plates and columns hollow columns and cappedcolumns Fortunately experiments tend to focus on thesesimple morphologies as they are best suited for examiningsurface growth dynamics

The 2D diffusion equation in cylindrical coordinates is

120597120590

120597119905= 119863nabla

2120590 = 119863[

1

119903

120597120590

120597119903+1205972120590

1205971199032+1205972120590

1205971199112] (33)

for 120590(119903 119911 119905) giving the propagation equation

120590 (120591 + Δ120591) = (1 minus 4Δ120591) 120590 + Δ120591

times [(1 +Δ119903

2119903) 120590 (119903 + Δ119903) + (1 minus

Δ119903

2119903) 120590 (119903 minus Δ119903)

+ 120590 (119911 + Δ119911) + 120590 (119911 minus Δ119911) ]

(34)

where 120591 is given in (7)Including boundary conditions at the crystal surface

follows along the lines of the 1D calculation above AssumingΔ120591 = 14 and Δ119903 = Δ119911 gives the propagation equations for119903-type and 119911-type boundary pixels

120590 (119903bound 119911 120591 + Δ120591)

=1

4[(1 +

Δ119903

2119903) 120590 (119903 + Δ119903) + (1 minus

Δ119903

2119903) 120590solid

+120590 (119911 + Δ119911) + 120590 (119911 minus Δ119911) ]

120590 (119903 119911bound 120591 + Δ120591)

=1

4[(1 +

Δ119903

2119903) 120590 (119903 + Δ119903) + (1 minus

Δ119903

2119903) 120590 (119903 minus Δ119903)

+ 120590 (119911 + Δ119911) + 120590solid]

(35)

where

120590solid = 120590 (119903bound) (1 minus 120572Δ120585) (36)

Note that a corner boundary pixel with neighboring ice pixelsin both 119903 and 119911 will propagate using

120590 (119903bound 119911bound 120591 + Δ120591)

=1

4[(1 +

Δ119903

2119903) 120590 (119903 + Δ119903) + (1 minus

Δ119903

2119903) 120590solid

+ 120590 (119911 + Δ119911) + 120590solid]

(37)

which drains the supersaturation twice as fast as an ordinaryboundary pixel as expected

We then define an accumulated mass parameter 120582 where120582 starts at zero and after each timestep becomes

120582 (120591 + Δ120591) = 120582 (120591) +sum120572120590Δ120582 (38)

where the sum is over the number of neighboring ice pixelsand

Δ120582119911-neighbors =

119888sat119888iceΔ120591Δ120585

Δ120582119903-neighbors = (1 plusmn

Δ119903

2119903)119888sat119888iceΔ120591Δ120585

(39)

Journal of Computational Methods in Physics 7

41The 119903 = 0 and 119911 = 0 Special Cases In our model we definethe first row of pixels to have 119903 = 0 which leads to problemswith the (1 plusmn Δ1199032119903) factors Going back to the diffusionequation we use Gaussrsquos law to generate the propagationequation for 119903 = 0 pixels

120590 (0 120591 + Δ120591)

= (1 minus 6Δ120591) 120590 (0 120591)

+ Δ120591 [120590 (119911 + Δ119911) + 120590 (119911 minus Δ119911) + 4120590 (Δ119903)]

(40)

Unfortunately taking Δ120591 = 14 in this expression leads toon-axis propagation instabilities As a compromise betweenrunning speed and computational accuracy we use Δ120591 = 16for the on-axis propagation equation giving

120590 (0 120591 + Δ120591) =1

6[120590 (119911 + Δ119911) + 120590 (119911 minus Δ119911) + 4120590 (Δ119903)]

(41)

while we continue using Δ120591 = 14 for off-axis pixels Thiscauses the supersaturation field to relax slightly more slowlyon the 119903-axis but this fix apparently does not adversely affectthe modeling results

Similarly we define the first row of 119911 pixels to have 119911 = 0and for that row we use the propagation equation

120590 (120591 + Δ120591) =1

4[(1 +

Δ119903

2119903) 120590 (119903 + Δ119903)

+ (1 minusΔ119903

2119903) 120590 (119903 minus Δ119903) + 2120590 (119911 + Δ119911)]

(42)

42 Adaptive Time Steps Speeding up the code using adap-tive time steps again proceeds along the lines of the 1Dproblem We use

Δ120582 = ΛΔ120591Δ120585 (43)

with

Λ = 119860 speed1

Δ120585(120572120590surf)max(Δ119903

1198771015840max)

1198771015840

max = radic1198772

max + 1199112

max + 1198772

0

(44)

as above and the physical time step is that given in (25)

43 Neighbor Relations In all these expressions we mustchoose the attachment coefficient 120572 with some care asits value will depend on the number and orientation ofneighboring solid pixels We label a boundary pixel with(119873119903 119873119911) where119873

119903is the number of neighboring ice pixels in

the 119903 direction and 119873119911is the number of ice neighbors in the

119911 direction Both119873119903and119873

119911can take values 0 1 or 2 giving

nine cases for (119873119903 119873119911) The cases are

(0 0) the pixel is an air pixel

(0 1) one ice neighbor in the 119911 direction so 120572 =

120572basal the physical value appropriate for a basal facetsurface(1 0) one ice neighbor in the 119903direction so120572 = 120572prismfor a prism facet surface(1 1) a kink site where the growth will not benucleation limited since the corner provides a sourceof molecular steps We do not know a priori whatvalue to use for 120572 on this site but assume a constant120572 = 120572

11

(0 2) (1 2) (2 0) (2 1) and (2 2) these are allunusual cases where the growth will be fast so weassume that 120572 = 1

We index these possibilities with a single number bycomputing a boundary parameter 119861 = 2119873

2

119903+ 1198732

119911 We then

have 119861 = 0 for an air pixel 119861 = 1 for a basal facet 119861 = 2 for aprism facet 119861 = 3 for a (1 1) kink location and 119861 gt 3 for allother cases

If we consider the special case where 120572 is equal to someconstant value independent of the orientation of the surfacewith respect to the crystal lattice then the growth velocityshould equal V = 120572Vkin120590 for all surfaces For the basal or prismfacet surfaces in this constant-120572 case we take 120572basal = 120572prism =120572 while an analysis of the growth of a (11) surface shows thatwemust take 120572

11= 120572radic2 if the above algorithm is to produce

the correct growth velocity

44 Limitations on Grid Size Aswe pointed out in [26] thereare physical limits to how coarse the computing grid canbe made before instabilities appear or the growth deviatessubstantially from real growth Taking Δ120585 gt 1120572would cause120590solid to become negative which causes some concern in thatit may produce instabilities in the code With this limitationthe grid spacing could not be larger than Δ119903 = Δ119911 = 119883

0120572

For air at a pressure of one atmosphere and 120572 asymp 1 this givesthe pixel size119883

0asymp 0145 120583m

Physically we can gain some insights into these limita-tions from dendrite growth theory [7] We have 119883

0asymp 119877kin

(the latter from (28) in [7]) and a growing dendrite has a tipradius

119877tip asymp1198830

120572119904 (45)

where 119904 is the dimensionless solvability parameter which is oforder unity for ice crystal growth [7]The stability of the codethus limits the grid spacing to be no greater than the tip radiusof a growing dendritic structure From this we see that thecode can only function properly when the grid spacing is fineenough to allow the growth of physically realistic dendriticstructures the scale of which is given by solvability theory

In practice we typically assume that Δ119903 = Δ119911 = 1198830asymp

0145 120583m when comparing models with ice growth data Acoarser grid would reduce run times but at the risk of notreproducing physically relevantmorphological behaviorsWehave not yet explored in detail how different grid sizes affectthe modeling behavior

8 Journal of Computational Methods in Physics

45 Scaling Behavior If we run the code and produce somecomplex crystal shape the interpretation of our result stillcontains an ambiguity The crystal size is given in pixelswhere Δ119903 = Δ119911 = Δ120585119883

0is the pixel size The parameter

Δ120585 was fixed in the code but 1198830depends on the diffusion

constant 119863 which is not otherwise specified Similarly asingle time step in the code corresponds to a physical time

Δ119905 =1198832

0Δ1205852Δ120591

119863sim 119863 (46)

Thus we see that the growth behavior at different airpressures (different D) is determined once we know thegrowth at a single pressure (provided that 120590

infinis the same

at the different pressures) If the air pressure is half anatmosphere the growthmorphology (however complex) willbe the same as at one atmosphere except in the former casethe crystal will be double the size in double the time Thisscaling behavior nicely explains why ice crystal morphologyis generally simpler for smaller crystals andor for lower airpressures which has long been observed [7]

46 Analytic Solutions and Model Validation The case of aninfinitely long cylinder of radius 119877in has the analytic solution

120590 (119903) =120573 + log (119903119877in)

120573 + log (119877out119877in)120590119877out (47)

where the outer boundary has 120590(119877out) = 120590119877out and 120573 =

1198830119877in120572 giving the growth velocity

V =120572120572diffcyl

120572 + 120572diffcylVkin120590119877out

120572diffcyl =1

119861

1198830

119877in

(48)

with 119861 = log(119877out119877in)To compare this analytic result with numerical models of

a growing cylinder we created a version of our codewith peri-odic boundary conditions in 119911 thus modeling the growth ofan infinite cylinder Figure 3 shows results using 119879 = minus15∘C120590119877out = 001 119877init = 1 120583m 120572 = 1 119877out = 300 and 119883

0=

435 120583m Again we see that the numerical model converges tothe analytic result as 119860 speed goes to zero

5 Gibbs-Thomson Effect

The basic cellular automata code described above includes afairly realistic treatment of the attachment kinetics but doesnot include any effects of surface energy This is somewhatjustified for the case of faceted ice crystal growth as one canshow that attachment kinetic effects dominate the growthbehavior while surface energy effects are less important [7]Nevertheless surface energy effects are not always negligibleespecially at low supersaturations As we will demonstratebelow models with highly anisotropic attachment kineticsand no surface energy effects can exhibit the formation ofone-pixel-wide features that are not physically plausible To

5

0

10

Radi

us (m

icro

ns)

0 50 100 150 200Time (s)

Figure 3 The dashed line shows the analytic result for the growthof an infinite cylinder with 120572 = 1 119877out = 435 120583m and 120590

119877out = 001as described in the text Solid lines show numerical modeling resultsusing 119860 speed = 02 (top line) 002 (middle line) and 0002 (bottomline)

suppress these unphysicalmodelswemust include the surfaceenergy via the Gibbs-Thomson effect

For the ice case we are considering here the equilibriumvapor density above a curved surface can be written [27]

119888eq = 119888sat (1 + 120575120581) (49)

where 119888sat is the equilibrium (saturated) vapor density of a flatsurface 120581 is the surface curvature

120581 =1

1199031

+1

1199032

(50)

where 1199031 1199032are the principal radii of curvature of the surface

and

120575 =120574

119888solid119896119879asymp 1 nm (51)

where 120574 asymp 01 Jm2 is the surface energy of the icevaporinterface The anisotropy of 120574 is not well known but theavailable evidence suggests it is rather small [20] so for theremainder of our discussion we assume an isotropic surfaceenergy We believe this is a accurate assumption for ice but itis in stark contrast to the highly anisotropic ice surface energyassumed in [11]

From this we have the supersaturation over a curvedsurface

120590curved asymp 120590 minus120575

119877119888

(52)

to lowest order in 120575119877119888 where120590 is the normal supersaturation

relative to a flat interface and for later convenience we define119877119888= 120581minus1 Putting in some numbers a sharp ice needle might

have 119877119888asymp 1 120583m giving Δ120590 = 120590curved minus 120590 asymp 01 percent

which is small but not always negligible Moreover setting

Journal of Computational Methods in Physics 9

Nz = 7

Nr = 5

z

r

Figure 4 A graphical definition of the outer boundary pixels usedfor incorporating surface energy effects into our cellular automatamodel Here the blue pixels are ice while the clear pixels are airThe red pixels are boundary pixels having the largest 119903 and 119911 valuesdefined to be the ldquoouterrdquo boundary pixels

119877119888asymp 1198830asymp 015 120583m gives a rather large Δ120590 asymp 066 percent

indicating that one-pixel-wide features should not be presentat low supersaturations

Adding the Gibbs-Thomson effect requires an algorith-mic estimation of 119877

119888for all boundary pixels in our cellular

automata model We examined several possibilities but didnot find a suitable algorithm that could be used with arbitrarymorphologies in our cylindrically symmetrical model Sinceour primary goal was to model simple morphologies forcomparison with experimental data as described above wesettled for a simpler algorithm for such cases

Throughout our investigations we found that the overallgrowth behavior of faceted crystals with simplemorphologieswas determined primarily by the growth of the outermostfacet surfaces To apply the Gibbs-Thomson effect on thesesurfaces we defined the outer boundary pixels as shown inFigure 4 yielding the numbers 119873

119903and 119873

119911as the number of

outer-boundary pixelsUsing these outer boundary pixels as a proxy for esti-

mating surface curvature we defined 119877119888= Δ119903119873

1199032 for the

edges of plates 119877119888= Δ119903119873

1199112 for the edges of hollow columns

and 119877119888= Δ119903119873

1199114 for the tips of needles For all boundary

pixels that were not outer-boundary pixels we assumed that119877119888= infin Although these are crude estimations of surface

curvature we found them satisfactory for incorporating theGibbs-Thomson effect in our model in part because this is arelatively small effect compared to the more dominant role ofattachment kinetics

51 A Gibbs-Thomson Example The top image in Figure 5shows how our model can yield unphysical results if wedo not include the Gibbs-Thomson effect In this model weassumed the input parameters 119879 = minus15

∘C 120590infin

= 001

120572basal = exp(minus003120590surf) 120572prism = 1 an initial prism radius1198770= 2 120583m initial prism half-height119867

0= 5 120583m and a Gibbs-

Thomson parameter 120575 = 0 After growing this model for sixseconds the initial simple prism grew into a capped columnwhere the two capping plates each had a thickness of justone pixel as shown in the top image in Figure 5 Such smallstructural features are unphysical at this supersaturation andsimilar one-pixel-wide features often appeared in our modelswhen the supersaturation was low and the anisotropy inattachment kinetics was high

Figure 5 A test of the Gibbs-Thomson effect in our crystal growthmodel In each image the white pixels show ice while the brightnessof the surrounding region is proportional to the supersaturationModel crystals are reflection symmetric about the 119911 = 0 plane andthe images show 119911 ge 0 only Thin vertical lines show the initialcrystal size Using 120572prism = 1 and 120572basal ≪ 120572prism resulted in growthonly on the prism surfaces Full model parameters are given in thetext The top image displays the crystal after six seconds of growthwith a Gibbs-Thomson parameter 120575 = 0 This model exhibits a one-pixel-thick plate growing on the end of the column which is notphysically plausibleThemiddle and lower images show results using120575 = 03 and 10 nm respectively For these models the unphysicalplate growth does not appear

Including nonzero values of 120575 suppressed these smallfeatures as demonstrated in the lower two images in Figure 5From this example as well as additional tests with differentgrowth morphologies we found that our relatively simplealgorithm for including the Gibbs-Thomson effect yieldedreasonable results suppressing the unphysical growth of one-pixel-wide features Models with higher 120590

infinvalues or with

lower anisotropies in the attachment kinetics were generallyless affected by including the Gibbs-Thomson effect

6 An Example Comparison with Experiment

Figure 6 shows an example of how the cellular automatamethod can be used in laboratory investigations of growingice crystals In this experiment we first grew a thin ldquoelectricrdquoice needle as described in [28] and then transferred theneedle to a second growth chamber where we grew a thinplate-like crystal on the needle tip all in air at a pressureof one bar The hexagonal symmetry of the crystal andthe relatively simple growth morphology were well suitedfor analysis using our cylindrically symmetric model Byadjusting 120590

infinand the surface attachment coefficients for the

principal facets we were able to generate a model crystal

10 Journal of Computational Methods in Physics

50120583m

(a) (b)

Radi

us (120583

m)

60

50

40

30

20

10

0minus40 minus20

Plate radius

Needle radius

0 20 40Time (s)

60 80 100

(c)

Figure 6 An example showing our numerical modeling of a laboratory-grown ice crystal The composite image in (a) shows five successiveviews of a thin plate-like crystal growing on the end of a slender ice needle with a 50120583m scale bar The crystal is shown in a side view withillumination from behindThe composite image in (b) shows our numerical model of the same crystal Here the brightness around the crystalis proportional to supersaturation The data points in (c) shows measurements of the plate and needle radii (the latter at a distance of 50120583mfrom the base of the plate) as a function of timeThe lines in the graph are from the growth of the model crystal The inset image in (c) showsa more frontal view of a similar plate-on-needle crystal with a plate radius of 80120583m showing the thin sectored-plate morphology Note thatat earlier times (119905 lt 0 in the graph) the crystal was growing from a tapered needle to a column with nearly uniform radius along its lengthThe time axes were shifted so the plate began growing at 119905 asymp 0 for both the real and model crystals

that matched both the morphology and growth rates ofthe laboratory crystal In addition to the plate and needleradii note that the tapered neck of the needle just belowthe plate seen in Figure 6 was nicely reproduced by themodel crystal A discussion of the physical significance ofthe best-fit model parameters used for this model is left fora subsequent publication comparing the growth of manycrystals under different conditions This example is meantto demonstrate only that the behavior of faceted laboratory-grown ice crystals can be accuratelymodeledwith the cellularautomata method

7 Discussion

Numericalmodeling of structure formation during solidifica-tion has been well studied for decades especially cases whereweak surface anisotropies in diffusion-limited growth yieldrounded dendritic branching Modeling structures that areboth branched and faceted have beenmore difficult and onlyrecently researchers have demonstrated suitable numericalmethods [10 11] In the present work we have developeda numerical model aimed at investigating the formationof simple faceted morphologies during ice crystal growthOur model incorporates the cellular automata technique in[10] but uses a more physically realistic treatment of thesurface attachment kinetics and the Gibbs-Thomson effectUsing this model we are able to reproduce ice crystalgrowth behavior for simple growth morphologies over a

broad range of conditions We have been using this modelto aid in the interpretation of ice growth experiments inorder to extract information about the anisotropic surfaceattachment kinetics that governs ice crystal growth fromwater vapor Modifying the model for application to othermaterial systems is likely straightforward

At present we have limited our model to cylindricallysymmetric crystal growth and relatively simple growth mor-phologies for comparison with measurements A full three-dimensional extension of our model should be possibleas was demonstrated in [10] potentially allowing completemodeling of the many complex branched and faceted struc-tures that are commonly observed in ice crystal growth [7]Achieving this goal would likely require a substantially betterunderstanding of the surface attachment kinetics in ice thanpresently exists along with more advanced algorithms forincorporating all the relevant surface physics into a cellularautomata model

In general it appears that the cellular automata methodis well suited for modeling faceted crystal growth and itmay be the method of choice in material systems exhibitingstrongly anisotropic attachment kinetics Incorporatingmoresophisticated descriptions of surface growth processes willrequire substantially better algorithms than we have outlinedabove especially when extending the models to three dimen-sions Whether such models can reproduce the full rangeof complex structures appearing in faceted crystal growthremains to be seen

Journal of Computational Methods in Physics 11

References

[1] R Trivedi and W Kurz ldquoDendritic growthrdquo InternationalMaterials Reviews vol 39 no 2 pp 49ndash74 1994

[2] E A Brener ldquoThree-dimensional dendritic growthrdquo Journal ofCrystal Growth vol 166 no 1ndash4 pp 339ndash346 1996

[3] I Avramov ldquoKinetics of growth of nanowhiskers (nanowiresand nanotubes)rdquo Nanoscale Research Letters vol 2 no 5 pp235ndash239 2007

[4] H Zhang I Hussain M Brust M F Butler S P Rannard andA I Cooper ldquoAligned two- and three-dimensional structuresby directional freezing of polymers and nanoparticlesrdquo NatureMaterials vol 4 pp 787ndash793 2005

[5] M C Cross and P C Hohenberg ldquoPattern formation outsideof equilibriumrdquo Reviews of Modern Physics vol 65 no 3 pp851ndash1112 1993

[6] K Kassner Pattern Formation in Diffusion-Limited CrystalGrowth World Scientific Singapore 1996

[7] K G Libbrecht ldquoThe physics of snow crystalsrdquo Reports onProgress in Physics vol 68 no 4 pp 855ndash895 2005

[8] J Nelson ldquoGrowth mechanisms to explain the primary andsecondary habits of snow crystalsrdquo Philosophical Magazine Avol 81 no 10 pp 2337ndash2373 2001

[9] H R Pruppacher and J D Klett Microphysics of Clouds andPrecipitation Kluwer Academic Publishers Dordrecht TheNetherlands 1997

[10] J Gravner and D Griffeath ldquoModeling snow-crystal growtha three-dimensional mesoscopic approachrdquo Physical Review Evol 79 no 1 Article ID 011601 18 pages 2009

[11] J W Barrett H Garcke and R Nurnberg ldquoNumerical com-putations of faceted pattern formation in snow crystal growthrdquoPhysical Review E vol 86 no 1 Article ID 011604 14 pages2012

[12] S-Y Hong J Dudhia and S-H Chen ldquoA revised approach toice microphysical processes for the bulk parameterization ofclouds and precipitationrdquoMonthlyWeather Review vol 132 no1 pp 103ndash120 2004

[13] M Matsumoto S Saito and I Ohmine ldquoMolecular dynamicssimulation of the ice nucleation and growth process leading towater freezingrdquo Nature vol 416 no 6879 pp 409ndash413 2002

[14] J G Dash A W Rempel and J S Wettlaufer ldquoThe physicsof premelted ice and its geophysical consequencesrdquo Reviews ofModern Physics vol 78 no 3 pp 695ndash741 2006

[15] A Karma and W-J Rappel ldquoPhase-field method for com-putationally efficient modeling of solidification with arbitraryinterface kineticsrdquo Physical Review E vol 53 no 4 pp R3017ndashR3020 1996

[16] I Singer-Loginova andHM Singer ldquoThe phase field techniquefor modeling multiphase materialsrdquo Reports on Progress inPhysics vol 71 no 10 Article ID 106501 2008

[17] A Schmidt ldquoComputation of three dimensional dendrites withfinite elementsrdquo Journal of Computational Physics vol 125 no2 pp 293ndash312 1996

[18] E Yokoyama ldquoFormation of patterns during growth of snowcrystalsrdquo Journal of Crystal Growth vol 128 no 1ndash4 pp 251ndash257 1993

[19] S E Wood M B Baker and D Calhoun ldquoNew model forthe vapor growth of hexagonal ice crystals in the atmosphererdquoJournal of Geophysical ResearchD vol 106 no 5 pp 4845ndash48702001

[20] K G Libbrecht ldquoOn the equilibrium shape of an ice crystalrdquo2012 httparxivorgabs12051452

[21] G W Sears ldquoA growth mechanism for mercury whiskersrdquo ActaMetallurgica vol 3 no 4 pp 361ndash366 1955

[22] R L Parker R L Anderson and S C Hardy ldquoGrowth andevaporation kinetics and surface diffusion of K and Hg crystalwhiskersrdquo Applied Physics Letters vol 3 no 6 pp 93ndash95 1963

[23] S J Liu H Huang and C HWoo ldquoSchwoebel-Ehrlich barrierfrom two to three dimensionsrdquo Applied Physics Letters vol 80no 18 pp 3295ndash3297 2002

[24] Y Saito Statistical Physics of Crystal Growth World ScientificSingapore 1996

[25] K G Libbrecht ldquoExplaining the formation of thin ice-crystalplates with structure-dependent attachment kineticsrdquo Journal ofCrystal Growth vol 258 no 1-2 pp 168ndash175 2003

[26] K G Libbrecht ldquoPhysically derived rules for simulating facetedcrystal growth using cellular automatardquo 2008 httparxivorgabs08072616

[27] W Thomson ldquoOn the equilibrium vapor at a curved surface ofliquidrdquo Philosophical Magazine vol 42 no 282 pp 448ndash4521871

[28] K G Libbrecht T Crosby and M Swanson ldquoElectricallyenhanced free dendrite growth in polar and non-polar systemsrdquoJournal of Crystal Growth vol 240 no 1-2 pp 241ndash254 2002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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FluidsJournal of

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Advances in Condensed Matter Physics

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

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 Computational  Methods in Physics

Journal of

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Soft MatterJournal of

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Volume 2014

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PhotonicsJournal of

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Journal of

Biophysics

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ThermodynamicsJournal of

4 Journal of Computational Methods in Physics

where 119903bound is the radius of the center of the boundary pixelThus we can write

120590bound (120591 + Δ120591) = 120590bound (120591) [1 minus (1 minusΔ119903

119903bound)120572Δ120585Δ120591]

(11)

for the mass drain of the boundary pixel where

Δ120585 =119888iceVkin119888sat

Δ119903

119863=Δ119903

1198830

1198830=119888sat119888ice

119863

Vkin= radic

2120587119898

119896119879119863

asymp 0145 120583m sdot (119863

119863air) radic

11989611987915

119896119879

(12)

11987915= minus15

∘C and 119863air asymp 2 times 10minus5m2sec is the diffusion

constant in air at a pressure of one barWe combine (6) and (11) to create a total propagation

algorithm for the boundary pixel

120590 (119903bound 120591 + Δ120591) = (1 minusΔ119903

119903bound)120590solidΔ120591

+ (1 minus 2Δ120591) 120590 (119903bound)

+ (1 +Δ119903

119903bound)120590 (119903bound + Δ119903) Δ120591

(13)

where

120590solid = 120590 (119903bound) (1 minus 120572Δ120585) (14)

To describe the ice growth we assume that a boundarypixel starts out with zero accumulated mass and it turnsinto an ice pixel when it has accumulated a mass Δ119898 =

120588ice41205871199032

boundΔ119903 After one time step it accumulates the massin (9) giving

120575119898

Δ119898= 120572120590 (119903) (1 minus

Δ119903

119903bound)119888sat119888iceΔ120591Δ120585 (15)

Thus we define an accumulated mass parameter 120582 for theboundary pixel where 120582 starts at zero and after each timestepbecomes

120582 (120591 + Δ120591) = 120582 (120591) + (1 minusΔ119903

119903bound)120572120590Δ120582 (16)

where

Δ120582 =119888sat119888iceΔ120591Δ120585 (17)

When 120582 becomes greater than one the boundary pixel turnsto ice

32 Adaptive Time Steps In practice (17) gives an exceedinglyslow growth rate because 119888sat119888ice is quite small so we speedup the code by using

Δ120582 = ΛΔ120591Δ120585 (18)

where Λ is an adjustable parameter Note that the number ofsteps needed to advance one pixel is (120572120590Δ120582)minus1 so the growthvelocity is

V119899=Δ119903

Δ119905120572120590ΛΔ120591Δ120585 (19)

We optimize the speed of the code by increasing Λ as muchas possible subject to criteria that limit the resulting errors inthe growth behavior Our first criterion is that it should takemore than119873

0time steps to change a boundary pixel from air

to ice where typically1198730asymp 10 and this gives

Λ lt1

120572120590Δ120591Δ1205851198730

(20)

We also want the Peclet number to be much less thanunity so the growth is slower than the time it takes for thesupersaturation field to stabilize as described above and thisgives

119901Peclet =119877V119899

2119863lt 119901Pecletmax

Λ lt2

120572120590Δ120585(Δ119903

119877) sdot 119901Pecletmax

(21)

where 119877 is the crystal radius We use 119901Pecletmax = 11198730 so

Λ = min [ 2

120572120590Δ120585(Δ119903

119877) sdot 119901Pecletmax

1

120572120590Δ120591Δ1205851198730

]

asymp 119860 speed1

Δ120585(120572120590surf)max(Δ119903

119877)

(22)

where 119860 speed = 21198730 asymp 02 is a constant speedup parameterThe quantities (120572120590surf)max and 119877 are computed as the crystalgrows

In practice we have found that this relation yields growthrates that are somewhat too fast when 119877 lt 119873

0Δ119903 owing to

finite pixel size effects We counter this by replacing 119877 in (22)with

1198771015840= radic1198772 + 1198772

0 (23)

where we set 1198770= 1198730Δ119903 This results in an improvement

in the accuracy of the growth for small crystals with only amodest increase in running time

The choice of Λ essentially means using an adaptive timestep where the physical time for each step equals

Δ119905 = Δ120582Δ120585Δ1199050 (24)

Δ1199050=1198830

Vkinasymp 070msec sdot (

Vkinminus15∘C

Vkin) (25)

In running this code we typically use Δ120591 = 12 and Δ120585 =1 which gives Δ119903 = 119883

0 We define the radial index from 119894 = 1

to 119873 and the radius of the center of each spherical shell is119903119894= (119894 minus 1)Δ119903

Journal of Computational Methods in Physics 5

33 Analytic Solutions If the outer boundary is at infinity thespherical growth case gives the analytic solution

120590 (119903) = 120590infinminus119877

119903

120572

120572 + 120572diff120590infin (26)

where 119877 is the radius of the sphere and 120572diff = 1198830119877 In thelimit 120572 ≪ 120572diff we have 120590(119903) asymp 120590

infin while 120572 ≫ 120572diff gives

120590(119877) asymp 0 The growth velocity in all cases is

V119899= 120572Vkin120590 (119877) = (

120572120572diff120572 + 120572diff

) Vkin120590infin (27)

With an outer boundary at 119903out and 120590(119903out) = 120590out we have

120590 (119903) = 120590out minus (1198771015840

119903minus1198771015840

119903out)120590out (28)

where

1198771015840= [

120574

119877minus

1

119903out]

minus1

120574 =120572 + 120572diff120572

(29)

and the growth velocity becomes

V119899= (

120572120572diff120572 + 120572diff

) Vkin120590out[1 minus119877

119903out

1

120574]

minus1

(30)

In computing V119899 we may have to solve the equation

120590 (119877) =120590out (120574 minus 1)

120574 minus 119877119903out(31)

when 120574 is itself a function of 120590(119877) We solve this by iterationusing

120590(119877)119894+1=1

2[120590119894+120590out (120574 (120590(119877)119894) minus 1)

120574 (120590(119877)119894) minus 119877119903out

] (32)

at each time step which quickly converges to 120590(119877)

34 Model Validation As an example that compares themodel growth of a spherewith the analytic result we useΔ120591 =12 Δ120585 = 1 119879 = minus15

∘C 119873 = 300 (giving 119903out = 435 120583m)120590out = 001 120572 = 1 and 119877initial = 651198830 = 094 120583m Resultsare shown in Figure 1 using different values of 119860 speed For themodel with 119860 speed = 1 the adaptive time steps were large so120590surf did not have time to relax fully to its analytic value as thecrystal grew with the outcome that the crystal grew too fastWith smaller 119860 speed values the supersaturation field relaxedmore fully and 119877(119905) was closer to the analytic result

35 Quantitative Comparisons with Experimental Data Inour experiments measuring ice growth rates the outerboundaries of the growth chamber are much larger thantypical crystal sizes so 119903out asymp infin In modeling such data werequire a certain modeling accuracy and the straightforward

0 50 100 150 200Time (s)

0

5

10

15

Radi

us (m

icro

ns)

Figure 1 A comparison of different models for the growth ofa spherical crystal (solid lines) together with the analytic result(dashed line) as described in the text The models used 119860 speed = 1

(top curve) 02 (middle) and 002 (lower)

route to achieving this is to run the code with some suitablysmall 119860 speed together with some suitably large 119903out Unfortu-nately the code converges rather slowly to analytic results asseen in the example in Figure 1 and we have found that thisstraightforward approach results in unnecessarily long runtimes

We have found an alternative operating strategy thatachieves good modeling accuracy with substantially shorterrun times In this strategy we select values of 119903out that are notespecially large and values of 119860 speed that are not especiallysmall and then adjust 120590out down to compensate This strategysacrificesmodel accuracy in exchange for increasedmodelingspeed and is thus a variant of the usual trade-off oneencounters in numerical modeling

Figure 2 shows an example of how this modeling strategycan be used The analytic curve uses the same parameters asin Figure 1 except with 119903out = infin The models also use thesame parameters as in Figure 1 again with 119903out = 435 120583mand we fixed 119860 speed = 02 The model with 120590out = 001 doesnot match the analytic curve which reflects systematic errorsin the numerical modeling The model growth is faster thanthe analytic result because 119903out is too small and 119860 speed is toolargeWe compensate for these systematic errors by adjusting120590out downward 30 percent to 0007 as seen in Figure 2

From this exercise we see that there are two ways toproduce accurate models for comparison with experimentaldata The first way is simply to use a large 119903out and asmall 119860 speed to reduce the modeling systematic errors to anacceptable level This method is the most straightforwardbut from a general perspective it is not the most efficientThe second approach is to use smaller 119903out and larger 119860 speedso the code runs quickly and then lowers 120590out slightly tocompensate This latter approach has proven quite useful inpractice especially when comparing experiments where 120590

infin

is itself not known with extremely high accuracy as is oftenthe case

6 Journal of Computational Methods in Physics

0 50 100 150 200Time (s)

5

10

Radi

us (m

icro

ns)

Figure 2 Another comparison of different models for the growthof a spherical crystal (solid lines) with the analytic result (dashedline) The analytic calculation used 120590

infin= 001 and 119903out = infin The

models used 119860 speed = 02 and 119903out = 435 120583m with 120590out = 001 (topcurve) 0007 (middle) and 0004 (lower) This shows that one canadjust 120590out in the model to approximately compensate for systematicmodeling errors

4 The 2D Cylindrically Symmetric Model

The 1D spherical model outlined above is useful for exam-ining the cellular automata method in detail but of courseit is of little use otherwise as analytic results can be com-puted for the general spherical case Adding one additionaldimension adds significant complexity and richness to thediffusion problem however analytic results are not pos-sible for calculating the growth of even rather simple 2Dmorphologies

We have found that a 2D cylindrically symmetric modelis especially useful for comparison with ice crystal growthexperiments In this case a simple hexagonal plate is approxi-mated by a circular disk while a hexagonal column becomesa cylinder The six prism facets are replaced by a singlecylindrical ldquofacetrdquo analogous to the spherical case aboveand we equate the attachment kinetics on this surface tothe attachment kinetics on a flat prism facet Other thanintroducing a small geometrical correction the cylindri-cal approximation appears to give reasonable quantitativeresults It can be applied only to simple morphologies suchas simple plates and columns hollow columns and cappedcolumns Fortunately experiments tend to focus on thesesimple morphologies as they are best suited for examiningsurface growth dynamics

The 2D diffusion equation in cylindrical coordinates is

120597120590

120597119905= 119863nabla

2120590 = 119863[

1

119903

120597120590

120597119903+1205972120590

1205971199032+1205972120590

1205971199112] (33)

for 120590(119903 119911 119905) giving the propagation equation

120590 (120591 + Δ120591) = (1 minus 4Δ120591) 120590 + Δ120591

times [(1 +Δ119903

2119903) 120590 (119903 + Δ119903) + (1 minus

Δ119903

2119903) 120590 (119903 minus Δ119903)

+ 120590 (119911 + Δ119911) + 120590 (119911 minus Δ119911) ]

(34)

where 120591 is given in (7)Including boundary conditions at the crystal surface

follows along the lines of the 1D calculation above AssumingΔ120591 = 14 and Δ119903 = Δ119911 gives the propagation equations for119903-type and 119911-type boundary pixels

120590 (119903bound 119911 120591 + Δ120591)

=1

4[(1 +

Δ119903

2119903) 120590 (119903 + Δ119903) + (1 minus

Δ119903

2119903) 120590solid

+120590 (119911 + Δ119911) + 120590 (119911 minus Δ119911) ]

120590 (119903 119911bound 120591 + Δ120591)

=1

4[(1 +

Δ119903

2119903) 120590 (119903 + Δ119903) + (1 minus

Δ119903

2119903) 120590 (119903 minus Δ119903)

+ 120590 (119911 + Δ119911) + 120590solid]

(35)

where

120590solid = 120590 (119903bound) (1 minus 120572Δ120585) (36)

Note that a corner boundary pixel with neighboring ice pixelsin both 119903 and 119911 will propagate using

120590 (119903bound 119911bound 120591 + Δ120591)

=1

4[(1 +

Δ119903

2119903) 120590 (119903 + Δ119903) + (1 minus

Δ119903

2119903) 120590solid

+ 120590 (119911 + Δ119911) + 120590solid]

(37)

which drains the supersaturation twice as fast as an ordinaryboundary pixel as expected

We then define an accumulated mass parameter 120582 where120582 starts at zero and after each timestep becomes

120582 (120591 + Δ120591) = 120582 (120591) +sum120572120590Δ120582 (38)

where the sum is over the number of neighboring ice pixelsand

Δ120582119911-neighbors =

119888sat119888iceΔ120591Δ120585

Δ120582119903-neighbors = (1 plusmn

Δ119903

2119903)119888sat119888iceΔ120591Δ120585

(39)

Journal of Computational Methods in Physics 7

41The 119903 = 0 and 119911 = 0 Special Cases In our model we definethe first row of pixels to have 119903 = 0 which leads to problemswith the (1 plusmn Δ1199032119903) factors Going back to the diffusionequation we use Gaussrsquos law to generate the propagationequation for 119903 = 0 pixels

120590 (0 120591 + Δ120591)

= (1 minus 6Δ120591) 120590 (0 120591)

+ Δ120591 [120590 (119911 + Δ119911) + 120590 (119911 minus Δ119911) + 4120590 (Δ119903)]

(40)

Unfortunately taking Δ120591 = 14 in this expression leads toon-axis propagation instabilities As a compromise betweenrunning speed and computational accuracy we use Δ120591 = 16for the on-axis propagation equation giving

120590 (0 120591 + Δ120591) =1

6[120590 (119911 + Δ119911) + 120590 (119911 minus Δ119911) + 4120590 (Δ119903)]

(41)

while we continue using Δ120591 = 14 for off-axis pixels Thiscauses the supersaturation field to relax slightly more slowlyon the 119903-axis but this fix apparently does not adversely affectthe modeling results

Similarly we define the first row of 119911 pixels to have 119911 = 0and for that row we use the propagation equation

120590 (120591 + Δ120591) =1

4[(1 +

Δ119903

2119903) 120590 (119903 + Δ119903)

+ (1 minusΔ119903

2119903) 120590 (119903 minus Δ119903) + 2120590 (119911 + Δ119911)]

(42)

42 Adaptive Time Steps Speeding up the code using adap-tive time steps again proceeds along the lines of the 1Dproblem We use

Δ120582 = ΛΔ120591Δ120585 (43)

with

Λ = 119860 speed1

Δ120585(120572120590surf)max(Δ119903

1198771015840max)

1198771015840

max = radic1198772

max + 1199112

max + 1198772

0

(44)

as above and the physical time step is that given in (25)

43 Neighbor Relations In all these expressions we mustchoose the attachment coefficient 120572 with some care asits value will depend on the number and orientation ofneighboring solid pixels We label a boundary pixel with(119873119903 119873119911) where119873

119903is the number of neighboring ice pixels in

the 119903 direction and 119873119911is the number of ice neighbors in the

119911 direction Both119873119903and119873

119911can take values 0 1 or 2 giving

nine cases for (119873119903 119873119911) The cases are

(0 0) the pixel is an air pixel

(0 1) one ice neighbor in the 119911 direction so 120572 =

120572basal the physical value appropriate for a basal facetsurface(1 0) one ice neighbor in the 119903direction so120572 = 120572prismfor a prism facet surface(1 1) a kink site where the growth will not benucleation limited since the corner provides a sourceof molecular steps We do not know a priori whatvalue to use for 120572 on this site but assume a constant120572 = 120572

11

(0 2) (1 2) (2 0) (2 1) and (2 2) these are allunusual cases where the growth will be fast so weassume that 120572 = 1

We index these possibilities with a single number bycomputing a boundary parameter 119861 = 2119873

2

119903+ 1198732

119911 We then

have 119861 = 0 for an air pixel 119861 = 1 for a basal facet 119861 = 2 for aprism facet 119861 = 3 for a (1 1) kink location and 119861 gt 3 for allother cases

If we consider the special case where 120572 is equal to someconstant value independent of the orientation of the surfacewith respect to the crystal lattice then the growth velocityshould equal V = 120572Vkin120590 for all surfaces For the basal or prismfacet surfaces in this constant-120572 case we take 120572basal = 120572prism =120572 while an analysis of the growth of a (11) surface shows thatwemust take 120572

11= 120572radic2 if the above algorithm is to produce

the correct growth velocity

44 Limitations on Grid Size Aswe pointed out in [26] thereare physical limits to how coarse the computing grid canbe made before instabilities appear or the growth deviatessubstantially from real growth Taking Δ120585 gt 1120572would cause120590solid to become negative which causes some concern in thatit may produce instabilities in the code With this limitationthe grid spacing could not be larger than Δ119903 = Δ119911 = 119883

0120572

For air at a pressure of one atmosphere and 120572 asymp 1 this givesthe pixel size119883

0asymp 0145 120583m

Physically we can gain some insights into these limita-tions from dendrite growth theory [7] We have 119883

0asymp 119877kin

(the latter from (28) in [7]) and a growing dendrite has a tipradius

119877tip asymp1198830

120572119904 (45)

where 119904 is the dimensionless solvability parameter which is oforder unity for ice crystal growth [7]The stability of the codethus limits the grid spacing to be no greater than the tip radiusof a growing dendritic structure From this we see that thecode can only function properly when the grid spacing is fineenough to allow the growth of physically realistic dendriticstructures the scale of which is given by solvability theory

In practice we typically assume that Δ119903 = Δ119911 = 1198830asymp

0145 120583m when comparing models with ice growth data Acoarser grid would reduce run times but at the risk of notreproducing physically relevantmorphological behaviorsWehave not yet explored in detail how different grid sizes affectthe modeling behavior

8 Journal of Computational Methods in Physics

45 Scaling Behavior If we run the code and produce somecomplex crystal shape the interpretation of our result stillcontains an ambiguity The crystal size is given in pixelswhere Δ119903 = Δ119911 = Δ120585119883

0is the pixel size The parameter

Δ120585 was fixed in the code but 1198830depends on the diffusion

constant 119863 which is not otherwise specified Similarly asingle time step in the code corresponds to a physical time

Δ119905 =1198832

0Δ1205852Δ120591

119863sim 119863 (46)

Thus we see that the growth behavior at different airpressures (different D) is determined once we know thegrowth at a single pressure (provided that 120590

infinis the same

at the different pressures) If the air pressure is half anatmosphere the growthmorphology (however complex) willbe the same as at one atmosphere except in the former casethe crystal will be double the size in double the time Thisscaling behavior nicely explains why ice crystal morphologyis generally simpler for smaller crystals andor for lower airpressures which has long been observed [7]

46 Analytic Solutions and Model Validation The case of aninfinitely long cylinder of radius 119877in has the analytic solution

120590 (119903) =120573 + log (119903119877in)

120573 + log (119877out119877in)120590119877out (47)

where the outer boundary has 120590(119877out) = 120590119877out and 120573 =

1198830119877in120572 giving the growth velocity

V =120572120572diffcyl

120572 + 120572diffcylVkin120590119877out

120572diffcyl =1

119861

1198830

119877in

(48)

with 119861 = log(119877out119877in)To compare this analytic result with numerical models of

a growing cylinder we created a version of our codewith peri-odic boundary conditions in 119911 thus modeling the growth ofan infinite cylinder Figure 3 shows results using 119879 = minus15∘C120590119877out = 001 119877init = 1 120583m 120572 = 1 119877out = 300 and 119883

0=

435 120583m Again we see that the numerical model converges tothe analytic result as 119860 speed goes to zero

5 Gibbs-Thomson Effect

The basic cellular automata code described above includes afairly realistic treatment of the attachment kinetics but doesnot include any effects of surface energy This is somewhatjustified for the case of faceted ice crystal growth as one canshow that attachment kinetic effects dominate the growthbehavior while surface energy effects are less important [7]Nevertheless surface energy effects are not always negligibleespecially at low supersaturations As we will demonstratebelow models with highly anisotropic attachment kineticsand no surface energy effects can exhibit the formation ofone-pixel-wide features that are not physically plausible To

5

0

10

Radi

us (m

icro

ns)

0 50 100 150 200Time (s)

Figure 3 The dashed line shows the analytic result for the growthof an infinite cylinder with 120572 = 1 119877out = 435 120583m and 120590

119877out = 001as described in the text Solid lines show numerical modeling resultsusing 119860 speed = 02 (top line) 002 (middle line) and 0002 (bottomline)

suppress these unphysicalmodelswemust include the surfaceenergy via the Gibbs-Thomson effect

For the ice case we are considering here the equilibriumvapor density above a curved surface can be written [27]

119888eq = 119888sat (1 + 120575120581) (49)

where 119888sat is the equilibrium (saturated) vapor density of a flatsurface 120581 is the surface curvature

120581 =1

1199031

+1

1199032

(50)

where 1199031 1199032are the principal radii of curvature of the surface

and

120575 =120574

119888solid119896119879asymp 1 nm (51)

where 120574 asymp 01 Jm2 is the surface energy of the icevaporinterface The anisotropy of 120574 is not well known but theavailable evidence suggests it is rather small [20] so for theremainder of our discussion we assume an isotropic surfaceenergy We believe this is a accurate assumption for ice but itis in stark contrast to the highly anisotropic ice surface energyassumed in [11]

From this we have the supersaturation over a curvedsurface

120590curved asymp 120590 minus120575

119877119888

(52)

to lowest order in 120575119877119888 where120590 is the normal supersaturation

relative to a flat interface and for later convenience we define119877119888= 120581minus1 Putting in some numbers a sharp ice needle might

have 119877119888asymp 1 120583m giving Δ120590 = 120590curved minus 120590 asymp 01 percent

which is small but not always negligible Moreover setting

Journal of Computational Methods in Physics 9

Nz = 7

Nr = 5

z

r

Figure 4 A graphical definition of the outer boundary pixels usedfor incorporating surface energy effects into our cellular automatamodel Here the blue pixels are ice while the clear pixels are airThe red pixels are boundary pixels having the largest 119903 and 119911 valuesdefined to be the ldquoouterrdquo boundary pixels

119877119888asymp 1198830asymp 015 120583m gives a rather large Δ120590 asymp 066 percent

indicating that one-pixel-wide features should not be presentat low supersaturations

Adding the Gibbs-Thomson effect requires an algorith-mic estimation of 119877

119888for all boundary pixels in our cellular

automata model We examined several possibilities but didnot find a suitable algorithm that could be used with arbitrarymorphologies in our cylindrically symmetrical model Sinceour primary goal was to model simple morphologies forcomparison with experimental data as described above wesettled for a simpler algorithm for such cases

Throughout our investigations we found that the overallgrowth behavior of faceted crystals with simplemorphologieswas determined primarily by the growth of the outermostfacet surfaces To apply the Gibbs-Thomson effect on thesesurfaces we defined the outer boundary pixels as shown inFigure 4 yielding the numbers 119873

119903and 119873

119911as the number of

outer-boundary pixelsUsing these outer boundary pixels as a proxy for esti-

mating surface curvature we defined 119877119888= Δ119903119873

1199032 for the

edges of plates 119877119888= Δ119903119873

1199112 for the edges of hollow columns

and 119877119888= Δ119903119873

1199114 for the tips of needles For all boundary

pixels that were not outer-boundary pixels we assumed that119877119888= infin Although these are crude estimations of surface

curvature we found them satisfactory for incorporating theGibbs-Thomson effect in our model in part because this is arelatively small effect compared to the more dominant role ofattachment kinetics

51 A Gibbs-Thomson Example The top image in Figure 5shows how our model can yield unphysical results if wedo not include the Gibbs-Thomson effect In this model weassumed the input parameters 119879 = minus15

∘C 120590infin

= 001

120572basal = exp(minus003120590surf) 120572prism = 1 an initial prism radius1198770= 2 120583m initial prism half-height119867

0= 5 120583m and a Gibbs-

Thomson parameter 120575 = 0 After growing this model for sixseconds the initial simple prism grew into a capped columnwhere the two capping plates each had a thickness of justone pixel as shown in the top image in Figure 5 Such smallstructural features are unphysical at this supersaturation andsimilar one-pixel-wide features often appeared in our modelswhen the supersaturation was low and the anisotropy inattachment kinetics was high

Figure 5 A test of the Gibbs-Thomson effect in our crystal growthmodel In each image the white pixels show ice while the brightnessof the surrounding region is proportional to the supersaturationModel crystals are reflection symmetric about the 119911 = 0 plane andthe images show 119911 ge 0 only Thin vertical lines show the initialcrystal size Using 120572prism = 1 and 120572basal ≪ 120572prism resulted in growthonly on the prism surfaces Full model parameters are given in thetext The top image displays the crystal after six seconds of growthwith a Gibbs-Thomson parameter 120575 = 0 This model exhibits a one-pixel-thick plate growing on the end of the column which is notphysically plausibleThemiddle and lower images show results using120575 = 03 and 10 nm respectively For these models the unphysicalplate growth does not appear

Including nonzero values of 120575 suppressed these smallfeatures as demonstrated in the lower two images in Figure 5From this example as well as additional tests with differentgrowth morphologies we found that our relatively simplealgorithm for including the Gibbs-Thomson effect yieldedreasonable results suppressing the unphysical growth of one-pixel-wide features Models with higher 120590

infinvalues or with

lower anisotropies in the attachment kinetics were generallyless affected by including the Gibbs-Thomson effect

6 An Example Comparison with Experiment

Figure 6 shows an example of how the cellular automatamethod can be used in laboratory investigations of growingice crystals In this experiment we first grew a thin ldquoelectricrdquoice needle as described in [28] and then transferred theneedle to a second growth chamber where we grew a thinplate-like crystal on the needle tip all in air at a pressureof one bar The hexagonal symmetry of the crystal andthe relatively simple growth morphology were well suitedfor analysis using our cylindrically symmetric model Byadjusting 120590

infinand the surface attachment coefficients for the

principal facets we were able to generate a model crystal

10 Journal of Computational Methods in Physics

50120583m

(a) (b)

Radi

us (120583

m)

60

50

40

30

20

10

0minus40 minus20

Plate radius

Needle radius

0 20 40Time (s)

60 80 100

(c)

Figure 6 An example showing our numerical modeling of a laboratory-grown ice crystal The composite image in (a) shows five successiveviews of a thin plate-like crystal growing on the end of a slender ice needle with a 50120583m scale bar The crystal is shown in a side view withillumination from behindThe composite image in (b) shows our numerical model of the same crystal Here the brightness around the crystalis proportional to supersaturation The data points in (c) shows measurements of the plate and needle radii (the latter at a distance of 50120583mfrom the base of the plate) as a function of timeThe lines in the graph are from the growth of the model crystal The inset image in (c) showsa more frontal view of a similar plate-on-needle crystal with a plate radius of 80120583m showing the thin sectored-plate morphology Note thatat earlier times (119905 lt 0 in the graph) the crystal was growing from a tapered needle to a column with nearly uniform radius along its lengthThe time axes were shifted so the plate began growing at 119905 asymp 0 for both the real and model crystals

that matched both the morphology and growth rates ofthe laboratory crystal In addition to the plate and needleradii note that the tapered neck of the needle just belowthe plate seen in Figure 6 was nicely reproduced by themodel crystal A discussion of the physical significance ofthe best-fit model parameters used for this model is left fora subsequent publication comparing the growth of manycrystals under different conditions This example is meantto demonstrate only that the behavior of faceted laboratory-grown ice crystals can be accuratelymodeledwith the cellularautomata method

7 Discussion

Numericalmodeling of structure formation during solidifica-tion has been well studied for decades especially cases whereweak surface anisotropies in diffusion-limited growth yieldrounded dendritic branching Modeling structures that areboth branched and faceted have beenmore difficult and onlyrecently researchers have demonstrated suitable numericalmethods [10 11] In the present work we have developeda numerical model aimed at investigating the formationof simple faceted morphologies during ice crystal growthOur model incorporates the cellular automata technique in[10] but uses a more physically realistic treatment of thesurface attachment kinetics and the Gibbs-Thomson effectUsing this model we are able to reproduce ice crystalgrowth behavior for simple growth morphologies over a

broad range of conditions We have been using this modelto aid in the interpretation of ice growth experiments inorder to extract information about the anisotropic surfaceattachment kinetics that governs ice crystal growth fromwater vapor Modifying the model for application to othermaterial systems is likely straightforward

At present we have limited our model to cylindricallysymmetric crystal growth and relatively simple growth mor-phologies for comparison with measurements A full three-dimensional extension of our model should be possibleas was demonstrated in [10] potentially allowing completemodeling of the many complex branched and faceted struc-tures that are commonly observed in ice crystal growth [7]Achieving this goal would likely require a substantially betterunderstanding of the surface attachment kinetics in ice thanpresently exists along with more advanced algorithms forincorporating all the relevant surface physics into a cellularautomata model

In general it appears that the cellular automata methodis well suited for modeling faceted crystal growth and itmay be the method of choice in material systems exhibitingstrongly anisotropic attachment kinetics Incorporatingmoresophisticated descriptions of surface growth processes willrequire substantially better algorithms than we have outlinedabove especially when extending the models to three dimen-sions Whether such models can reproduce the full rangeof complex structures appearing in faceted crystal growthremains to be seen

Journal of Computational Methods in Physics 11

References

[1] R Trivedi and W Kurz ldquoDendritic growthrdquo InternationalMaterials Reviews vol 39 no 2 pp 49ndash74 1994

[2] E A Brener ldquoThree-dimensional dendritic growthrdquo Journal ofCrystal Growth vol 166 no 1ndash4 pp 339ndash346 1996

[3] I Avramov ldquoKinetics of growth of nanowhiskers (nanowiresand nanotubes)rdquo Nanoscale Research Letters vol 2 no 5 pp235ndash239 2007

[4] H Zhang I Hussain M Brust M F Butler S P Rannard andA I Cooper ldquoAligned two- and three-dimensional structuresby directional freezing of polymers and nanoparticlesrdquo NatureMaterials vol 4 pp 787ndash793 2005

[5] M C Cross and P C Hohenberg ldquoPattern formation outsideof equilibriumrdquo Reviews of Modern Physics vol 65 no 3 pp851ndash1112 1993

[6] K Kassner Pattern Formation in Diffusion-Limited CrystalGrowth World Scientific Singapore 1996

[7] K G Libbrecht ldquoThe physics of snow crystalsrdquo Reports onProgress in Physics vol 68 no 4 pp 855ndash895 2005

[8] J Nelson ldquoGrowth mechanisms to explain the primary andsecondary habits of snow crystalsrdquo Philosophical Magazine Avol 81 no 10 pp 2337ndash2373 2001

[9] H R Pruppacher and J D Klett Microphysics of Clouds andPrecipitation Kluwer Academic Publishers Dordrecht TheNetherlands 1997

[10] J Gravner and D Griffeath ldquoModeling snow-crystal growtha three-dimensional mesoscopic approachrdquo Physical Review Evol 79 no 1 Article ID 011601 18 pages 2009

[11] J W Barrett H Garcke and R Nurnberg ldquoNumerical com-putations of faceted pattern formation in snow crystal growthrdquoPhysical Review E vol 86 no 1 Article ID 011604 14 pages2012

[12] S-Y Hong J Dudhia and S-H Chen ldquoA revised approach toice microphysical processes for the bulk parameterization ofclouds and precipitationrdquoMonthlyWeather Review vol 132 no1 pp 103ndash120 2004

[13] M Matsumoto S Saito and I Ohmine ldquoMolecular dynamicssimulation of the ice nucleation and growth process leading towater freezingrdquo Nature vol 416 no 6879 pp 409ndash413 2002

[14] J G Dash A W Rempel and J S Wettlaufer ldquoThe physicsof premelted ice and its geophysical consequencesrdquo Reviews ofModern Physics vol 78 no 3 pp 695ndash741 2006

[15] A Karma and W-J Rappel ldquoPhase-field method for com-putationally efficient modeling of solidification with arbitraryinterface kineticsrdquo Physical Review E vol 53 no 4 pp R3017ndashR3020 1996

[16] I Singer-Loginova andHM Singer ldquoThe phase field techniquefor modeling multiphase materialsrdquo Reports on Progress inPhysics vol 71 no 10 Article ID 106501 2008

[17] A Schmidt ldquoComputation of three dimensional dendrites withfinite elementsrdquo Journal of Computational Physics vol 125 no2 pp 293ndash312 1996

[18] E Yokoyama ldquoFormation of patterns during growth of snowcrystalsrdquo Journal of Crystal Growth vol 128 no 1ndash4 pp 251ndash257 1993

[19] S E Wood M B Baker and D Calhoun ldquoNew model forthe vapor growth of hexagonal ice crystals in the atmosphererdquoJournal of Geophysical ResearchD vol 106 no 5 pp 4845ndash48702001

[20] K G Libbrecht ldquoOn the equilibrium shape of an ice crystalrdquo2012 httparxivorgabs12051452

[21] G W Sears ldquoA growth mechanism for mercury whiskersrdquo ActaMetallurgica vol 3 no 4 pp 361ndash366 1955

[22] R L Parker R L Anderson and S C Hardy ldquoGrowth andevaporation kinetics and surface diffusion of K and Hg crystalwhiskersrdquo Applied Physics Letters vol 3 no 6 pp 93ndash95 1963

[23] S J Liu H Huang and C HWoo ldquoSchwoebel-Ehrlich barrierfrom two to three dimensionsrdquo Applied Physics Letters vol 80no 18 pp 3295ndash3297 2002

[24] Y Saito Statistical Physics of Crystal Growth World ScientificSingapore 1996

[25] K G Libbrecht ldquoExplaining the formation of thin ice-crystalplates with structure-dependent attachment kineticsrdquo Journal ofCrystal Growth vol 258 no 1-2 pp 168ndash175 2003

[26] K G Libbrecht ldquoPhysically derived rules for simulating facetedcrystal growth using cellular automatardquo 2008 httparxivorgabs08072616

[27] W Thomson ldquoOn the equilibrium vapor at a curved surface ofliquidrdquo Philosophical Magazine vol 42 no 282 pp 448ndash4521871

[28] K G Libbrecht T Crosby and M Swanson ldquoElectricallyenhanced free dendrite growth in polar and non-polar systemsrdquoJournal of Crystal Growth vol 240 no 1-2 pp 241ndash254 2002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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ThermodynamicsJournal of

Journal of Computational Methods in Physics 5

33 Analytic Solutions If the outer boundary is at infinity thespherical growth case gives the analytic solution

120590 (119903) = 120590infinminus119877

119903

120572

120572 + 120572diff120590infin (26)

where 119877 is the radius of the sphere and 120572diff = 1198830119877 In thelimit 120572 ≪ 120572diff we have 120590(119903) asymp 120590

infin while 120572 ≫ 120572diff gives

120590(119877) asymp 0 The growth velocity in all cases is

V119899= 120572Vkin120590 (119877) = (

120572120572diff120572 + 120572diff

) Vkin120590infin (27)

With an outer boundary at 119903out and 120590(119903out) = 120590out we have

120590 (119903) = 120590out minus (1198771015840

119903minus1198771015840

119903out)120590out (28)

where

1198771015840= [

120574

119877minus

1

119903out]

minus1

120574 =120572 + 120572diff120572

(29)

and the growth velocity becomes

V119899= (

120572120572diff120572 + 120572diff

) Vkin120590out[1 minus119877

119903out

1

120574]

minus1

(30)

In computing V119899 we may have to solve the equation

120590 (119877) =120590out (120574 minus 1)

120574 minus 119877119903out(31)

when 120574 is itself a function of 120590(119877) We solve this by iterationusing

120590(119877)119894+1=1

2[120590119894+120590out (120574 (120590(119877)119894) minus 1)

120574 (120590(119877)119894) minus 119877119903out

] (32)

at each time step which quickly converges to 120590(119877)

34 Model Validation As an example that compares themodel growth of a spherewith the analytic result we useΔ120591 =12 Δ120585 = 1 119879 = minus15

∘C 119873 = 300 (giving 119903out = 435 120583m)120590out = 001 120572 = 1 and 119877initial = 651198830 = 094 120583m Resultsare shown in Figure 1 using different values of 119860 speed For themodel with 119860 speed = 1 the adaptive time steps were large so120590surf did not have time to relax fully to its analytic value as thecrystal grew with the outcome that the crystal grew too fastWith smaller 119860 speed values the supersaturation field relaxedmore fully and 119877(119905) was closer to the analytic result

35 Quantitative Comparisons with Experimental Data Inour experiments measuring ice growth rates the outerboundaries of the growth chamber are much larger thantypical crystal sizes so 119903out asymp infin In modeling such data werequire a certain modeling accuracy and the straightforward

0 50 100 150 200Time (s)

0

5

10

15

Radi

us (m

icro

ns)

Figure 1 A comparison of different models for the growth ofa spherical crystal (solid lines) together with the analytic result(dashed line) as described in the text The models used 119860 speed = 1

(top curve) 02 (middle) and 002 (lower)

route to achieving this is to run the code with some suitablysmall 119860 speed together with some suitably large 119903out Unfortu-nately the code converges rather slowly to analytic results asseen in the example in Figure 1 and we have found that thisstraightforward approach results in unnecessarily long runtimes

We have found an alternative operating strategy thatachieves good modeling accuracy with substantially shorterrun times In this strategy we select values of 119903out that are notespecially large and values of 119860 speed that are not especiallysmall and then adjust 120590out down to compensate This strategysacrificesmodel accuracy in exchange for increasedmodelingspeed and is thus a variant of the usual trade-off oneencounters in numerical modeling

Figure 2 shows an example of how this modeling strategycan be used The analytic curve uses the same parameters asin Figure 1 except with 119903out = infin The models also use thesame parameters as in Figure 1 again with 119903out = 435 120583mand we fixed 119860 speed = 02 The model with 120590out = 001 doesnot match the analytic curve which reflects systematic errorsin the numerical modeling The model growth is faster thanthe analytic result because 119903out is too small and 119860 speed is toolargeWe compensate for these systematic errors by adjusting120590out downward 30 percent to 0007 as seen in Figure 2

From this exercise we see that there are two ways toproduce accurate models for comparison with experimentaldata The first way is simply to use a large 119903out and asmall 119860 speed to reduce the modeling systematic errors to anacceptable level This method is the most straightforwardbut from a general perspective it is not the most efficientThe second approach is to use smaller 119903out and larger 119860 speedso the code runs quickly and then lowers 120590out slightly tocompensate This latter approach has proven quite useful inpractice especially when comparing experiments where 120590

infin

is itself not known with extremely high accuracy as is oftenthe case

6 Journal of Computational Methods in Physics

0 50 100 150 200Time (s)

5

10

Radi

us (m

icro

ns)

Figure 2 Another comparison of different models for the growthof a spherical crystal (solid lines) with the analytic result (dashedline) The analytic calculation used 120590

infin= 001 and 119903out = infin The

models used 119860 speed = 02 and 119903out = 435 120583m with 120590out = 001 (topcurve) 0007 (middle) and 0004 (lower) This shows that one canadjust 120590out in the model to approximately compensate for systematicmodeling errors

4 The 2D Cylindrically Symmetric Model

The 1D spherical model outlined above is useful for exam-ining the cellular automata method in detail but of courseit is of little use otherwise as analytic results can be com-puted for the general spherical case Adding one additionaldimension adds significant complexity and richness to thediffusion problem however analytic results are not pos-sible for calculating the growth of even rather simple 2Dmorphologies

We have found that a 2D cylindrically symmetric modelis especially useful for comparison with ice crystal growthexperiments In this case a simple hexagonal plate is approxi-mated by a circular disk while a hexagonal column becomesa cylinder The six prism facets are replaced by a singlecylindrical ldquofacetrdquo analogous to the spherical case aboveand we equate the attachment kinetics on this surface tothe attachment kinetics on a flat prism facet Other thanintroducing a small geometrical correction the cylindri-cal approximation appears to give reasonable quantitativeresults It can be applied only to simple morphologies suchas simple plates and columns hollow columns and cappedcolumns Fortunately experiments tend to focus on thesesimple morphologies as they are best suited for examiningsurface growth dynamics

The 2D diffusion equation in cylindrical coordinates is

120597120590

120597119905= 119863nabla

2120590 = 119863[

1

119903

120597120590

120597119903+1205972120590

1205971199032+1205972120590

1205971199112] (33)

for 120590(119903 119911 119905) giving the propagation equation

120590 (120591 + Δ120591) = (1 minus 4Δ120591) 120590 + Δ120591

times [(1 +Δ119903

2119903) 120590 (119903 + Δ119903) + (1 minus

Δ119903

2119903) 120590 (119903 minus Δ119903)

+ 120590 (119911 + Δ119911) + 120590 (119911 minus Δ119911) ]

(34)

where 120591 is given in (7)Including boundary conditions at the crystal surface

follows along the lines of the 1D calculation above AssumingΔ120591 = 14 and Δ119903 = Δ119911 gives the propagation equations for119903-type and 119911-type boundary pixels

120590 (119903bound 119911 120591 + Δ120591)

=1

4[(1 +

Δ119903

2119903) 120590 (119903 + Δ119903) + (1 minus

Δ119903

2119903) 120590solid

+120590 (119911 + Δ119911) + 120590 (119911 minus Δ119911) ]

120590 (119903 119911bound 120591 + Δ120591)

=1

4[(1 +

Δ119903

2119903) 120590 (119903 + Δ119903) + (1 minus

Δ119903

2119903) 120590 (119903 minus Δ119903)

+ 120590 (119911 + Δ119911) + 120590solid]

(35)

where

120590solid = 120590 (119903bound) (1 minus 120572Δ120585) (36)

Note that a corner boundary pixel with neighboring ice pixelsin both 119903 and 119911 will propagate using

120590 (119903bound 119911bound 120591 + Δ120591)

=1

4[(1 +

Δ119903

2119903) 120590 (119903 + Δ119903) + (1 minus

Δ119903

2119903) 120590solid

+ 120590 (119911 + Δ119911) + 120590solid]

(37)

which drains the supersaturation twice as fast as an ordinaryboundary pixel as expected

We then define an accumulated mass parameter 120582 where120582 starts at zero and after each timestep becomes

120582 (120591 + Δ120591) = 120582 (120591) +sum120572120590Δ120582 (38)

where the sum is over the number of neighboring ice pixelsand

Δ120582119911-neighbors =

119888sat119888iceΔ120591Δ120585

Δ120582119903-neighbors = (1 plusmn

Δ119903

2119903)119888sat119888iceΔ120591Δ120585

(39)

Journal of Computational Methods in Physics 7

41The 119903 = 0 and 119911 = 0 Special Cases In our model we definethe first row of pixels to have 119903 = 0 which leads to problemswith the (1 plusmn Δ1199032119903) factors Going back to the diffusionequation we use Gaussrsquos law to generate the propagationequation for 119903 = 0 pixels

120590 (0 120591 + Δ120591)

= (1 minus 6Δ120591) 120590 (0 120591)

+ Δ120591 [120590 (119911 + Δ119911) + 120590 (119911 minus Δ119911) + 4120590 (Δ119903)]

(40)

Unfortunately taking Δ120591 = 14 in this expression leads toon-axis propagation instabilities As a compromise betweenrunning speed and computational accuracy we use Δ120591 = 16for the on-axis propagation equation giving

120590 (0 120591 + Δ120591) =1

6[120590 (119911 + Δ119911) + 120590 (119911 minus Δ119911) + 4120590 (Δ119903)]

(41)

while we continue using Δ120591 = 14 for off-axis pixels Thiscauses the supersaturation field to relax slightly more slowlyon the 119903-axis but this fix apparently does not adversely affectthe modeling results

Similarly we define the first row of 119911 pixels to have 119911 = 0and for that row we use the propagation equation

120590 (120591 + Δ120591) =1

4[(1 +

Δ119903

2119903) 120590 (119903 + Δ119903)

+ (1 minusΔ119903

2119903) 120590 (119903 minus Δ119903) + 2120590 (119911 + Δ119911)]

(42)

42 Adaptive Time Steps Speeding up the code using adap-tive time steps again proceeds along the lines of the 1Dproblem We use

Δ120582 = ΛΔ120591Δ120585 (43)

with

Λ = 119860 speed1

Δ120585(120572120590surf)max(Δ119903

1198771015840max)

1198771015840

max = radic1198772

max + 1199112

max + 1198772

0

(44)

as above and the physical time step is that given in (25)

43 Neighbor Relations In all these expressions we mustchoose the attachment coefficient 120572 with some care asits value will depend on the number and orientation ofneighboring solid pixels We label a boundary pixel with(119873119903 119873119911) where119873

119903is the number of neighboring ice pixels in

the 119903 direction and 119873119911is the number of ice neighbors in the

119911 direction Both119873119903and119873

119911can take values 0 1 or 2 giving

nine cases for (119873119903 119873119911) The cases are

(0 0) the pixel is an air pixel

(0 1) one ice neighbor in the 119911 direction so 120572 =

120572basal the physical value appropriate for a basal facetsurface(1 0) one ice neighbor in the 119903direction so120572 = 120572prismfor a prism facet surface(1 1) a kink site where the growth will not benucleation limited since the corner provides a sourceof molecular steps We do not know a priori whatvalue to use for 120572 on this site but assume a constant120572 = 120572

11

(0 2) (1 2) (2 0) (2 1) and (2 2) these are allunusual cases where the growth will be fast so weassume that 120572 = 1

We index these possibilities with a single number bycomputing a boundary parameter 119861 = 2119873

2

119903+ 1198732

119911 We then

have 119861 = 0 for an air pixel 119861 = 1 for a basal facet 119861 = 2 for aprism facet 119861 = 3 for a (1 1) kink location and 119861 gt 3 for allother cases

If we consider the special case where 120572 is equal to someconstant value independent of the orientation of the surfacewith respect to the crystal lattice then the growth velocityshould equal V = 120572Vkin120590 for all surfaces For the basal or prismfacet surfaces in this constant-120572 case we take 120572basal = 120572prism =120572 while an analysis of the growth of a (11) surface shows thatwemust take 120572

11= 120572radic2 if the above algorithm is to produce

the correct growth velocity

44 Limitations on Grid Size Aswe pointed out in [26] thereare physical limits to how coarse the computing grid canbe made before instabilities appear or the growth deviatessubstantially from real growth Taking Δ120585 gt 1120572would cause120590solid to become negative which causes some concern in thatit may produce instabilities in the code With this limitationthe grid spacing could not be larger than Δ119903 = Δ119911 = 119883

0120572

For air at a pressure of one atmosphere and 120572 asymp 1 this givesthe pixel size119883

0asymp 0145 120583m

Physically we can gain some insights into these limita-tions from dendrite growth theory [7] We have 119883

0asymp 119877kin

(the latter from (28) in [7]) and a growing dendrite has a tipradius

119877tip asymp1198830

120572119904 (45)

where 119904 is the dimensionless solvability parameter which is oforder unity for ice crystal growth [7]The stability of the codethus limits the grid spacing to be no greater than the tip radiusof a growing dendritic structure From this we see that thecode can only function properly when the grid spacing is fineenough to allow the growth of physically realistic dendriticstructures the scale of which is given by solvability theory

In practice we typically assume that Δ119903 = Δ119911 = 1198830asymp

0145 120583m when comparing models with ice growth data Acoarser grid would reduce run times but at the risk of notreproducing physically relevantmorphological behaviorsWehave not yet explored in detail how different grid sizes affectthe modeling behavior

8 Journal of Computational Methods in Physics

45 Scaling Behavior If we run the code and produce somecomplex crystal shape the interpretation of our result stillcontains an ambiguity The crystal size is given in pixelswhere Δ119903 = Δ119911 = Δ120585119883

0is the pixel size The parameter

Δ120585 was fixed in the code but 1198830depends on the diffusion

constant 119863 which is not otherwise specified Similarly asingle time step in the code corresponds to a physical time

Δ119905 =1198832

0Δ1205852Δ120591

119863sim 119863 (46)

Thus we see that the growth behavior at different airpressures (different D) is determined once we know thegrowth at a single pressure (provided that 120590

infinis the same

at the different pressures) If the air pressure is half anatmosphere the growthmorphology (however complex) willbe the same as at one atmosphere except in the former casethe crystal will be double the size in double the time Thisscaling behavior nicely explains why ice crystal morphologyis generally simpler for smaller crystals andor for lower airpressures which has long been observed [7]

46 Analytic Solutions and Model Validation The case of aninfinitely long cylinder of radius 119877in has the analytic solution

120590 (119903) =120573 + log (119903119877in)

120573 + log (119877out119877in)120590119877out (47)

where the outer boundary has 120590(119877out) = 120590119877out and 120573 =

1198830119877in120572 giving the growth velocity

V =120572120572diffcyl

120572 + 120572diffcylVkin120590119877out

120572diffcyl =1

119861

1198830

119877in

(48)

with 119861 = log(119877out119877in)To compare this analytic result with numerical models of

a growing cylinder we created a version of our codewith peri-odic boundary conditions in 119911 thus modeling the growth ofan infinite cylinder Figure 3 shows results using 119879 = minus15∘C120590119877out = 001 119877init = 1 120583m 120572 = 1 119877out = 300 and 119883

0=

435 120583m Again we see that the numerical model converges tothe analytic result as 119860 speed goes to zero

5 Gibbs-Thomson Effect

The basic cellular automata code described above includes afairly realistic treatment of the attachment kinetics but doesnot include any effects of surface energy This is somewhatjustified for the case of faceted ice crystal growth as one canshow that attachment kinetic effects dominate the growthbehavior while surface energy effects are less important [7]Nevertheless surface energy effects are not always negligibleespecially at low supersaturations As we will demonstratebelow models with highly anisotropic attachment kineticsand no surface energy effects can exhibit the formation ofone-pixel-wide features that are not physically plausible To

5

0

10

Radi

us (m

icro

ns)

0 50 100 150 200Time (s)

Figure 3 The dashed line shows the analytic result for the growthof an infinite cylinder with 120572 = 1 119877out = 435 120583m and 120590

119877out = 001as described in the text Solid lines show numerical modeling resultsusing 119860 speed = 02 (top line) 002 (middle line) and 0002 (bottomline)

suppress these unphysicalmodelswemust include the surfaceenergy via the Gibbs-Thomson effect

For the ice case we are considering here the equilibriumvapor density above a curved surface can be written [27]

119888eq = 119888sat (1 + 120575120581) (49)

where 119888sat is the equilibrium (saturated) vapor density of a flatsurface 120581 is the surface curvature

120581 =1

1199031

+1

1199032

(50)

where 1199031 1199032are the principal radii of curvature of the surface

and

120575 =120574

119888solid119896119879asymp 1 nm (51)

where 120574 asymp 01 Jm2 is the surface energy of the icevaporinterface The anisotropy of 120574 is not well known but theavailable evidence suggests it is rather small [20] so for theremainder of our discussion we assume an isotropic surfaceenergy We believe this is a accurate assumption for ice but itis in stark contrast to the highly anisotropic ice surface energyassumed in [11]

From this we have the supersaturation over a curvedsurface

120590curved asymp 120590 minus120575

119877119888

(52)

to lowest order in 120575119877119888 where120590 is the normal supersaturation

relative to a flat interface and for later convenience we define119877119888= 120581minus1 Putting in some numbers a sharp ice needle might

have 119877119888asymp 1 120583m giving Δ120590 = 120590curved minus 120590 asymp 01 percent

which is small but not always negligible Moreover setting

Journal of Computational Methods in Physics 9

Nz = 7

Nr = 5

z

r

Figure 4 A graphical definition of the outer boundary pixels usedfor incorporating surface energy effects into our cellular automatamodel Here the blue pixels are ice while the clear pixels are airThe red pixels are boundary pixels having the largest 119903 and 119911 valuesdefined to be the ldquoouterrdquo boundary pixels

119877119888asymp 1198830asymp 015 120583m gives a rather large Δ120590 asymp 066 percent

indicating that one-pixel-wide features should not be presentat low supersaturations

Adding the Gibbs-Thomson effect requires an algorith-mic estimation of 119877

119888for all boundary pixels in our cellular

automata model We examined several possibilities but didnot find a suitable algorithm that could be used with arbitrarymorphologies in our cylindrically symmetrical model Sinceour primary goal was to model simple morphologies forcomparison with experimental data as described above wesettled for a simpler algorithm for such cases

Throughout our investigations we found that the overallgrowth behavior of faceted crystals with simplemorphologieswas determined primarily by the growth of the outermostfacet surfaces To apply the Gibbs-Thomson effect on thesesurfaces we defined the outer boundary pixels as shown inFigure 4 yielding the numbers 119873

119903and 119873

119911as the number of

outer-boundary pixelsUsing these outer boundary pixels as a proxy for esti-

mating surface curvature we defined 119877119888= Δ119903119873

1199032 for the

edges of plates 119877119888= Δ119903119873

1199112 for the edges of hollow columns

and 119877119888= Δ119903119873

1199114 for the tips of needles For all boundary

pixels that were not outer-boundary pixels we assumed that119877119888= infin Although these are crude estimations of surface

curvature we found them satisfactory for incorporating theGibbs-Thomson effect in our model in part because this is arelatively small effect compared to the more dominant role ofattachment kinetics

51 A Gibbs-Thomson Example The top image in Figure 5shows how our model can yield unphysical results if wedo not include the Gibbs-Thomson effect In this model weassumed the input parameters 119879 = minus15

∘C 120590infin

= 001

120572basal = exp(minus003120590surf) 120572prism = 1 an initial prism radius1198770= 2 120583m initial prism half-height119867

0= 5 120583m and a Gibbs-

Thomson parameter 120575 = 0 After growing this model for sixseconds the initial simple prism grew into a capped columnwhere the two capping plates each had a thickness of justone pixel as shown in the top image in Figure 5 Such smallstructural features are unphysical at this supersaturation andsimilar one-pixel-wide features often appeared in our modelswhen the supersaturation was low and the anisotropy inattachment kinetics was high

Figure 5 A test of the Gibbs-Thomson effect in our crystal growthmodel In each image the white pixels show ice while the brightnessof the surrounding region is proportional to the supersaturationModel crystals are reflection symmetric about the 119911 = 0 plane andthe images show 119911 ge 0 only Thin vertical lines show the initialcrystal size Using 120572prism = 1 and 120572basal ≪ 120572prism resulted in growthonly on the prism surfaces Full model parameters are given in thetext The top image displays the crystal after six seconds of growthwith a Gibbs-Thomson parameter 120575 = 0 This model exhibits a one-pixel-thick plate growing on the end of the column which is notphysically plausibleThemiddle and lower images show results using120575 = 03 and 10 nm respectively For these models the unphysicalplate growth does not appear

Including nonzero values of 120575 suppressed these smallfeatures as demonstrated in the lower two images in Figure 5From this example as well as additional tests with differentgrowth morphologies we found that our relatively simplealgorithm for including the Gibbs-Thomson effect yieldedreasonable results suppressing the unphysical growth of one-pixel-wide features Models with higher 120590

infinvalues or with

lower anisotropies in the attachment kinetics were generallyless affected by including the Gibbs-Thomson effect

6 An Example Comparison with Experiment

Figure 6 shows an example of how the cellular automatamethod can be used in laboratory investigations of growingice crystals In this experiment we first grew a thin ldquoelectricrdquoice needle as described in [28] and then transferred theneedle to a second growth chamber where we grew a thinplate-like crystal on the needle tip all in air at a pressureof one bar The hexagonal symmetry of the crystal andthe relatively simple growth morphology were well suitedfor analysis using our cylindrically symmetric model Byadjusting 120590

infinand the surface attachment coefficients for the

principal facets we were able to generate a model crystal

10 Journal of Computational Methods in Physics

50120583m

(a) (b)

Radi

us (120583

m)

60

50

40

30

20

10

0minus40 minus20

Plate radius

Needle radius

0 20 40Time (s)

60 80 100

(c)

Figure 6 An example showing our numerical modeling of a laboratory-grown ice crystal The composite image in (a) shows five successiveviews of a thin plate-like crystal growing on the end of a slender ice needle with a 50120583m scale bar The crystal is shown in a side view withillumination from behindThe composite image in (b) shows our numerical model of the same crystal Here the brightness around the crystalis proportional to supersaturation The data points in (c) shows measurements of the plate and needle radii (the latter at a distance of 50120583mfrom the base of the plate) as a function of timeThe lines in the graph are from the growth of the model crystal The inset image in (c) showsa more frontal view of a similar plate-on-needle crystal with a plate radius of 80120583m showing the thin sectored-plate morphology Note thatat earlier times (119905 lt 0 in the graph) the crystal was growing from a tapered needle to a column with nearly uniform radius along its lengthThe time axes were shifted so the plate began growing at 119905 asymp 0 for both the real and model crystals

that matched both the morphology and growth rates ofthe laboratory crystal In addition to the plate and needleradii note that the tapered neck of the needle just belowthe plate seen in Figure 6 was nicely reproduced by themodel crystal A discussion of the physical significance ofthe best-fit model parameters used for this model is left fora subsequent publication comparing the growth of manycrystals under different conditions This example is meantto demonstrate only that the behavior of faceted laboratory-grown ice crystals can be accuratelymodeledwith the cellularautomata method

7 Discussion

Numericalmodeling of structure formation during solidifica-tion has been well studied for decades especially cases whereweak surface anisotropies in diffusion-limited growth yieldrounded dendritic branching Modeling structures that areboth branched and faceted have beenmore difficult and onlyrecently researchers have demonstrated suitable numericalmethods [10 11] In the present work we have developeda numerical model aimed at investigating the formationof simple faceted morphologies during ice crystal growthOur model incorporates the cellular automata technique in[10] but uses a more physically realistic treatment of thesurface attachment kinetics and the Gibbs-Thomson effectUsing this model we are able to reproduce ice crystalgrowth behavior for simple growth morphologies over a

broad range of conditions We have been using this modelto aid in the interpretation of ice growth experiments inorder to extract information about the anisotropic surfaceattachment kinetics that governs ice crystal growth fromwater vapor Modifying the model for application to othermaterial systems is likely straightforward

At present we have limited our model to cylindricallysymmetric crystal growth and relatively simple growth mor-phologies for comparison with measurements A full three-dimensional extension of our model should be possibleas was demonstrated in [10] potentially allowing completemodeling of the many complex branched and faceted struc-tures that are commonly observed in ice crystal growth [7]Achieving this goal would likely require a substantially betterunderstanding of the surface attachment kinetics in ice thanpresently exists along with more advanced algorithms forincorporating all the relevant surface physics into a cellularautomata model

In general it appears that the cellular automata methodis well suited for modeling faceted crystal growth and itmay be the method of choice in material systems exhibitingstrongly anisotropic attachment kinetics Incorporatingmoresophisticated descriptions of surface growth processes willrequire substantially better algorithms than we have outlinedabove especially when extending the models to three dimen-sions Whether such models can reproduce the full rangeof complex structures appearing in faceted crystal growthremains to be seen

Journal of Computational Methods in Physics 11

References

[1] R Trivedi and W Kurz ldquoDendritic growthrdquo InternationalMaterials Reviews vol 39 no 2 pp 49ndash74 1994

[2] E A Brener ldquoThree-dimensional dendritic growthrdquo Journal ofCrystal Growth vol 166 no 1ndash4 pp 339ndash346 1996

[3] I Avramov ldquoKinetics of growth of nanowhiskers (nanowiresand nanotubes)rdquo Nanoscale Research Letters vol 2 no 5 pp235ndash239 2007

[4] H Zhang I Hussain M Brust M F Butler S P Rannard andA I Cooper ldquoAligned two- and three-dimensional structuresby directional freezing of polymers and nanoparticlesrdquo NatureMaterials vol 4 pp 787ndash793 2005

[5] M C Cross and P C Hohenberg ldquoPattern formation outsideof equilibriumrdquo Reviews of Modern Physics vol 65 no 3 pp851ndash1112 1993

[6] K Kassner Pattern Formation in Diffusion-Limited CrystalGrowth World Scientific Singapore 1996

[7] K G Libbrecht ldquoThe physics of snow crystalsrdquo Reports onProgress in Physics vol 68 no 4 pp 855ndash895 2005

[8] J Nelson ldquoGrowth mechanisms to explain the primary andsecondary habits of snow crystalsrdquo Philosophical Magazine Avol 81 no 10 pp 2337ndash2373 2001

[9] H R Pruppacher and J D Klett Microphysics of Clouds andPrecipitation Kluwer Academic Publishers Dordrecht TheNetherlands 1997

[10] J Gravner and D Griffeath ldquoModeling snow-crystal growtha three-dimensional mesoscopic approachrdquo Physical Review Evol 79 no 1 Article ID 011601 18 pages 2009

[11] J W Barrett H Garcke and R Nurnberg ldquoNumerical com-putations of faceted pattern formation in snow crystal growthrdquoPhysical Review E vol 86 no 1 Article ID 011604 14 pages2012

[12] S-Y Hong J Dudhia and S-H Chen ldquoA revised approach toice microphysical processes for the bulk parameterization ofclouds and precipitationrdquoMonthlyWeather Review vol 132 no1 pp 103ndash120 2004

[13] M Matsumoto S Saito and I Ohmine ldquoMolecular dynamicssimulation of the ice nucleation and growth process leading towater freezingrdquo Nature vol 416 no 6879 pp 409ndash413 2002

[14] J G Dash A W Rempel and J S Wettlaufer ldquoThe physicsof premelted ice and its geophysical consequencesrdquo Reviews ofModern Physics vol 78 no 3 pp 695ndash741 2006

[15] A Karma and W-J Rappel ldquoPhase-field method for com-putationally efficient modeling of solidification with arbitraryinterface kineticsrdquo Physical Review E vol 53 no 4 pp R3017ndashR3020 1996

[16] I Singer-Loginova andHM Singer ldquoThe phase field techniquefor modeling multiphase materialsrdquo Reports on Progress inPhysics vol 71 no 10 Article ID 106501 2008

[17] A Schmidt ldquoComputation of three dimensional dendrites withfinite elementsrdquo Journal of Computational Physics vol 125 no2 pp 293ndash312 1996

[18] E Yokoyama ldquoFormation of patterns during growth of snowcrystalsrdquo Journal of Crystal Growth vol 128 no 1ndash4 pp 251ndash257 1993

[19] S E Wood M B Baker and D Calhoun ldquoNew model forthe vapor growth of hexagonal ice crystals in the atmosphererdquoJournal of Geophysical ResearchD vol 106 no 5 pp 4845ndash48702001

[20] K G Libbrecht ldquoOn the equilibrium shape of an ice crystalrdquo2012 httparxivorgabs12051452

[21] G W Sears ldquoA growth mechanism for mercury whiskersrdquo ActaMetallurgica vol 3 no 4 pp 361ndash366 1955

[22] R L Parker R L Anderson and S C Hardy ldquoGrowth andevaporation kinetics and surface diffusion of K and Hg crystalwhiskersrdquo Applied Physics Letters vol 3 no 6 pp 93ndash95 1963

[23] S J Liu H Huang and C HWoo ldquoSchwoebel-Ehrlich barrierfrom two to three dimensionsrdquo Applied Physics Letters vol 80no 18 pp 3295ndash3297 2002

[24] Y Saito Statistical Physics of Crystal Growth World ScientificSingapore 1996

[25] K G Libbrecht ldquoExplaining the formation of thin ice-crystalplates with structure-dependent attachment kineticsrdquo Journal ofCrystal Growth vol 258 no 1-2 pp 168ndash175 2003

[26] K G Libbrecht ldquoPhysically derived rules for simulating facetedcrystal growth using cellular automatardquo 2008 httparxivorgabs08072616

[27] W Thomson ldquoOn the equilibrium vapor at a curved surface ofliquidrdquo Philosophical Magazine vol 42 no 282 pp 448ndash4521871

[28] K G Libbrecht T Crosby and M Swanson ldquoElectricallyenhanced free dendrite growth in polar and non-polar systemsrdquoJournal of Crystal Growth vol 240 no 1-2 pp 241ndash254 2002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

6 Journal of Computational Methods in Physics

0 50 100 150 200Time (s)

5

10

Radi

us (m

icro

ns)

Figure 2 Another comparison of different models for the growthof a spherical crystal (solid lines) with the analytic result (dashedline) The analytic calculation used 120590

infin= 001 and 119903out = infin The

models used 119860 speed = 02 and 119903out = 435 120583m with 120590out = 001 (topcurve) 0007 (middle) and 0004 (lower) This shows that one canadjust 120590out in the model to approximately compensate for systematicmodeling errors

4 The 2D Cylindrically Symmetric Model

The 1D spherical model outlined above is useful for exam-ining the cellular automata method in detail but of courseit is of little use otherwise as analytic results can be com-puted for the general spherical case Adding one additionaldimension adds significant complexity and richness to thediffusion problem however analytic results are not pos-sible for calculating the growth of even rather simple 2Dmorphologies

We have found that a 2D cylindrically symmetric modelis especially useful for comparison with ice crystal growthexperiments In this case a simple hexagonal plate is approxi-mated by a circular disk while a hexagonal column becomesa cylinder The six prism facets are replaced by a singlecylindrical ldquofacetrdquo analogous to the spherical case aboveand we equate the attachment kinetics on this surface tothe attachment kinetics on a flat prism facet Other thanintroducing a small geometrical correction the cylindri-cal approximation appears to give reasonable quantitativeresults It can be applied only to simple morphologies suchas simple plates and columns hollow columns and cappedcolumns Fortunately experiments tend to focus on thesesimple morphologies as they are best suited for examiningsurface growth dynamics

The 2D diffusion equation in cylindrical coordinates is

120597120590

120597119905= 119863nabla

2120590 = 119863[

1

119903

120597120590

120597119903+1205972120590

1205971199032+1205972120590

1205971199112] (33)

for 120590(119903 119911 119905) giving the propagation equation

120590 (120591 + Δ120591) = (1 minus 4Δ120591) 120590 + Δ120591

times [(1 +Δ119903

2119903) 120590 (119903 + Δ119903) + (1 minus

Δ119903

2119903) 120590 (119903 minus Δ119903)

+ 120590 (119911 + Δ119911) + 120590 (119911 minus Δ119911) ]

(34)

where 120591 is given in (7)Including boundary conditions at the crystal surface

follows along the lines of the 1D calculation above AssumingΔ120591 = 14 and Δ119903 = Δ119911 gives the propagation equations for119903-type and 119911-type boundary pixels

120590 (119903bound 119911 120591 + Δ120591)

=1

4[(1 +

Δ119903

2119903) 120590 (119903 + Δ119903) + (1 minus

Δ119903

2119903) 120590solid

+120590 (119911 + Δ119911) + 120590 (119911 minus Δ119911) ]

120590 (119903 119911bound 120591 + Δ120591)

=1

4[(1 +

Δ119903

2119903) 120590 (119903 + Δ119903) + (1 minus

Δ119903

2119903) 120590 (119903 minus Δ119903)

+ 120590 (119911 + Δ119911) + 120590solid]

(35)

where

120590solid = 120590 (119903bound) (1 minus 120572Δ120585) (36)

Note that a corner boundary pixel with neighboring ice pixelsin both 119903 and 119911 will propagate using

120590 (119903bound 119911bound 120591 + Δ120591)

=1

4[(1 +

Δ119903

2119903) 120590 (119903 + Δ119903) + (1 minus

Δ119903

2119903) 120590solid

+ 120590 (119911 + Δ119911) + 120590solid]

(37)

which drains the supersaturation twice as fast as an ordinaryboundary pixel as expected

We then define an accumulated mass parameter 120582 where120582 starts at zero and after each timestep becomes

120582 (120591 + Δ120591) = 120582 (120591) +sum120572120590Δ120582 (38)

where the sum is over the number of neighboring ice pixelsand

Δ120582119911-neighbors =

119888sat119888iceΔ120591Δ120585

Δ120582119903-neighbors = (1 plusmn

Δ119903

2119903)119888sat119888iceΔ120591Δ120585

(39)

Journal of Computational Methods in Physics 7

41The 119903 = 0 and 119911 = 0 Special Cases In our model we definethe first row of pixels to have 119903 = 0 which leads to problemswith the (1 plusmn Δ1199032119903) factors Going back to the diffusionequation we use Gaussrsquos law to generate the propagationequation for 119903 = 0 pixels

120590 (0 120591 + Δ120591)

= (1 minus 6Δ120591) 120590 (0 120591)

+ Δ120591 [120590 (119911 + Δ119911) + 120590 (119911 minus Δ119911) + 4120590 (Δ119903)]

(40)

Unfortunately taking Δ120591 = 14 in this expression leads toon-axis propagation instabilities As a compromise betweenrunning speed and computational accuracy we use Δ120591 = 16for the on-axis propagation equation giving

120590 (0 120591 + Δ120591) =1

6[120590 (119911 + Δ119911) + 120590 (119911 minus Δ119911) + 4120590 (Δ119903)]

(41)

while we continue using Δ120591 = 14 for off-axis pixels Thiscauses the supersaturation field to relax slightly more slowlyon the 119903-axis but this fix apparently does not adversely affectthe modeling results

Similarly we define the first row of 119911 pixels to have 119911 = 0and for that row we use the propagation equation

120590 (120591 + Δ120591) =1

4[(1 +

Δ119903

2119903) 120590 (119903 + Δ119903)

+ (1 minusΔ119903

2119903) 120590 (119903 minus Δ119903) + 2120590 (119911 + Δ119911)]

(42)

42 Adaptive Time Steps Speeding up the code using adap-tive time steps again proceeds along the lines of the 1Dproblem We use

Δ120582 = ΛΔ120591Δ120585 (43)

with

Λ = 119860 speed1

Δ120585(120572120590surf)max(Δ119903

1198771015840max)

1198771015840

max = radic1198772

max + 1199112

max + 1198772

0

(44)

as above and the physical time step is that given in (25)

43 Neighbor Relations In all these expressions we mustchoose the attachment coefficient 120572 with some care asits value will depend on the number and orientation ofneighboring solid pixels We label a boundary pixel with(119873119903 119873119911) where119873

119903is the number of neighboring ice pixels in

the 119903 direction and 119873119911is the number of ice neighbors in the

119911 direction Both119873119903and119873

119911can take values 0 1 or 2 giving

nine cases for (119873119903 119873119911) The cases are

(0 0) the pixel is an air pixel

(0 1) one ice neighbor in the 119911 direction so 120572 =

120572basal the physical value appropriate for a basal facetsurface(1 0) one ice neighbor in the 119903direction so120572 = 120572prismfor a prism facet surface(1 1) a kink site where the growth will not benucleation limited since the corner provides a sourceof molecular steps We do not know a priori whatvalue to use for 120572 on this site but assume a constant120572 = 120572

11

(0 2) (1 2) (2 0) (2 1) and (2 2) these are allunusual cases where the growth will be fast so weassume that 120572 = 1

We index these possibilities with a single number bycomputing a boundary parameter 119861 = 2119873

2

119903+ 1198732

119911 We then

have 119861 = 0 for an air pixel 119861 = 1 for a basal facet 119861 = 2 for aprism facet 119861 = 3 for a (1 1) kink location and 119861 gt 3 for allother cases

If we consider the special case where 120572 is equal to someconstant value independent of the orientation of the surfacewith respect to the crystal lattice then the growth velocityshould equal V = 120572Vkin120590 for all surfaces For the basal or prismfacet surfaces in this constant-120572 case we take 120572basal = 120572prism =120572 while an analysis of the growth of a (11) surface shows thatwemust take 120572

11= 120572radic2 if the above algorithm is to produce

the correct growth velocity

44 Limitations on Grid Size Aswe pointed out in [26] thereare physical limits to how coarse the computing grid canbe made before instabilities appear or the growth deviatessubstantially from real growth Taking Δ120585 gt 1120572would cause120590solid to become negative which causes some concern in thatit may produce instabilities in the code With this limitationthe grid spacing could not be larger than Δ119903 = Δ119911 = 119883

0120572

For air at a pressure of one atmosphere and 120572 asymp 1 this givesthe pixel size119883

0asymp 0145 120583m

Physically we can gain some insights into these limita-tions from dendrite growth theory [7] We have 119883

0asymp 119877kin

(the latter from (28) in [7]) and a growing dendrite has a tipradius

119877tip asymp1198830

120572119904 (45)

where 119904 is the dimensionless solvability parameter which is oforder unity for ice crystal growth [7]The stability of the codethus limits the grid spacing to be no greater than the tip radiusof a growing dendritic structure From this we see that thecode can only function properly when the grid spacing is fineenough to allow the growth of physically realistic dendriticstructures the scale of which is given by solvability theory

In practice we typically assume that Δ119903 = Δ119911 = 1198830asymp

0145 120583m when comparing models with ice growth data Acoarser grid would reduce run times but at the risk of notreproducing physically relevantmorphological behaviorsWehave not yet explored in detail how different grid sizes affectthe modeling behavior

8 Journal of Computational Methods in Physics

45 Scaling Behavior If we run the code and produce somecomplex crystal shape the interpretation of our result stillcontains an ambiguity The crystal size is given in pixelswhere Δ119903 = Δ119911 = Δ120585119883

0is the pixel size The parameter

Δ120585 was fixed in the code but 1198830depends on the diffusion

constant 119863 which is not otherwise specified Similarly asingle time step in the code corresponds to a physical time

Δ119905 =1198832

0Δ1205852Δ120591

119863sim 119863 (46)

Thus we see that the growth behavior at different airpressures (different D) is determined once we know thegrowth at a single pressure (provided that 120590

infinis the same

at the different pressures) If the air pressure is half anatmosphere the growthmorphology (however complex) willbe the same as at one atmosphere except in the former casethe crystal will be double the size in double the time Thisscaling behavior nicely explains why ice crystal morphologyis generally simpler for smaller crystals andor for lower airpressures which has long been observed [7]

46 Analytic Solutions and Model Validation The case of aninfinitely long cylinder of radius 119877in has the analytic solution

120590 (119903) =120573 + log (119903119877in)

120573 + log (119877out119877in)120590119877out (47)

where the outer boundary has 120590(119877out) = 120590119877out and 120573 =

1198830119877in120572 giving the growth velocity

V =120572120572diffcyl

120572 + 120572diffcylVkin120590119877out

120572diffcyl =1

119861

1198830

119877in

(48)

with 119861 = log(119877out119877in)To compare this analytic result with numerical models of

a growing cylinder we created a version of our codewith peri-odic boundary conditions in 119911 thus modeling the growth ofan infinite cylinder Figure 3 shows results using 119879 = minus15∘C120590119877out = 001 119877init = 1 120583m 120572 = 1 119877out = 300 and 119883

0=

435 120583m Again we see that the numerical model converges tothe analytic result as 119860 speed goes to zero

5 Gibbs-Thomson Effect

The basic cellular automata code described above includes afairly realistic treatment of the attachment kinetics but doesnot include any effects of surface energy This is somewhatjustified for the case of faceted ice crystal growth as one canshow that attachment kinetic effects dominate the growthbehavior while surface energy effects are less important [7]Nevertheless surface energy effects are not always negligibleespecially at low supersaturations As we will demonstratebelow models with highly anisotropic attachment kineticsand no surface energy effects can exhibit the formation ofone-pixel-wide features that are not physically plausible To

5

0

10

Radi

us (m

icro

ns)

0 50 100 150 200Time (s)

Figure 3 The dashed line shows the analytic result for the growthof an infinite cylinder with 120572 = 1 119877out = 435 120583m and 120590

119877out = 001as described in the text Solid lines show numerical modeling resultsusing 119860 speed = 02 (top line) 002 (middle line) and 0002 (bottomline)

suppress these unphysicalmodelswemust include the surfaceenergy via the Gibbs-Thomson effect

For the ice case we are considering here the equilibriumvapor density above a curved surface can be written [27]

119888eq = 119888sat (1 + 120575120581) (49)

where 119888sat is the equilibrium (saturated) vapor density of a flatsurface 120581 is the surface curvature

120581 =1

1199031

+1

1199032

(50)

where 1199031 1199032are the principal radii of curvature of the surface

and

120575 =120574

119888solid119896119879asymp 1 nm (51)

where 120574 asymp 01 Jm2 is the surface energy of the icevaporinterface The anisotropy of 120574 is not well known but theavailable evidence suggests it is rather small [20] so for theremainder of our discussion we assume an isotropic surfaceenergy We believe this is a accurate assumption for ice but itis in stark contrast to the highly anisotropic ice surface energyassumed in [11]

From this we have the supersaturation over a curvedsurface

120590curved asymp 120590 minus120575

119877119888

(52)

to lowest order in 120575119877119888 where120590 is the normal supersaturation

relative to a flat interface and for later convenience we define119877119888= 120581minus1 Putting in some numbers a sharp ice needle might

have 119877119888asymp 1 120583m giving Δ120590 = 120590curved minus 120590 asymp 01 percent

which is small but not always negligible Moreover setting

Journal of Computational Methods in Physics 9

Nz = 7

Nr = 5

z

r

Figure 4 A graphical definition of the outer boundary pixels usedfor incorporating surface energy effects into our cellular automatamodel Here the blue pixels are ice while the clear pixels are airThe red pixels are boundary pixels having the largest 119903 and 119911 valuesdefined to be the ldquoouterrdquo boundary pixels

119877119888asymp 1198830asymp 015 120583m gives a rather large Δ120590 asymp 066 percent

indicating that one-pixel-wide features should not be presentat low supersaturations

Adding the Gibbs-Thomson effect requires an algorith-mic estimation of 119877

119888for all boundary pixels in our cellular

automata model We examined several possibilities but didnot find a suitable algorithm that could be used with arbitrarymorphologies in our cylindrically symmetrical model Sinceour primary goal was to model simple morphologies forcomparison with experimental data as described above wesettled for a simpler algorithm for such cases

Throughout our investigations we found that the overallgrowth behavior of faceted crystals with simplemorphologieswas determined primarily by the growth of the outermostfacet surfaces To apply the Gibbs-Thomson effect on thesesurfaces we defined the outer boundary pixels as shown inFigure 4 yielding the numbers 119873

119903and 119873

119911as the number of

outer-boundary pixelsUsing these outer boundary pixels as a proxy for esti-

mating surface curvature we defined 119877119888= Δ119903119873

1199032 for the

edges of plates 119877119888= Δ119903119873

1199112 for the edges of hollow columns

and 119877119888= Δ119903119873

1199114 for the tips of needles For all boundary

pixels that were not outer-boundary pixels we assumed that119877119888= infin Although these are crude estimations of surface

curvature we found them satisfactory for incorporating theGibbs-Thomson effect in our model in part because this is arelatively small effect compared to the more dominant role ofattachment kinetics

51 A Gibbs-Thomson Example The top image in Figure 5shows how our model can yield unphysical results if wedo not include the Gibbs-Thomson effect In this model weassumed the input parameters 119879 = minus15

∘C 120590infin

= 001

120572basal = exp(minus003120590surf) 120572prism = 1 an initial prism radius1198770= 2 120583m initial prism half-height119867

0= 5 120583m and a Gibbs-

Thomson parameter 120575 = 0 After growing this model for sixseconds the initial simple prism grew into a capped columnwhere the two capping plates each had a thickness of justone pixel as shown in the top image in Figure 5 Such smallstructural features are unphysical at this supersaturation andsimilar one-pixel-wide features often appeared in our modelswhen the supersaturation was low and the anisotropy inattachment kinetics was high

Figure 5 A test of the Gibbs-Thomson effect in our crystal growthmodel In each image the white pixels show ice while the brightnessof the surrounding region is proportional to the supersaturationModel crystals are reflection symmetric about the 119911 = 0 plane andthe images show 119911 ge 0 only Thin vertical lines show the initialcrystal size Using 120572prism = 1 and 120572basal ≪ 120572prism resulted in growthonly on the prism surfaces Full model parameters are given in thetext The top image displays the crystal after six seconds of growthwith a Gibbs-Thomson parameter 120575 = 0 This model exhibits a one-pixel-thick plate growing on the end of the column which is notphysically plausibleThemiddle and lower images show results using120575 = 03 and 10 nm respectively For these models the unphysicalplate growth does not appear

Including nonzero values of 120575 suppressed these smallfeatures as demonstrated in the lower two images in Figure 5From this example as well as additional tests with differentgrowth morphologies we found that our relatively simplealgorithm for including the Gibbs-Thomson effect yieldedreasonable results suppressing the unphysical growth of one-pixel-wide features Models with higher 120590

infinvalues or with

lower anisotropies in the attachment kinetics were generallyless affected by including the Gibbs-Thomson effect

6 An Example Comparison with Experiment

Figure 6 shows an example of how the cellular automatamethod can be used in laboratory investigations of growingice crystals In this experiment we first grew a thin ldquoelectricrdquoice needle as described in [28] and then transferred theneedle to a second growth chamber where we grew a thinplate-like crystal on the needle tip all in air at a pressureof one bar The hexagonal symmetry of the crystal andthe relatively simple growth morphology were well suitedfor analysis using our cylindrically symmetric model Byadjusting 120590

infinand the surface attachment coefficients for the

principal facets we were able to generate a model crystal

10 Journal of Computational Methods in Physics

50120583m

(a) (b)

Radi

us (120583

m)

60

50

40

30

20

10

0minus40 minus20

Plate radius

Needle radius

0 20 40Time (s)

60 80 100

(c)

Figure 6 An example showing our numerical modeling of a laboratory-grown ice crystal The composite image in (a) shows five successiveviews of a thin plate-like crystal growing on the end of a slender ice needle with a 50120583m scale bar The crystal is shown in a side view withillumination from behindThe composite image in (b) shows our numerical model of the same crystal Here the brightness around the crystalis proportional to supersaturation The data points in (c) shows measurements of the plate and needle radii (the latter at a distance of 50120583mfrom the base of the plate) as a function of timeThe lines in the graph are from the growth of the model crystal The inset image in (c) showsa more frontal view of a similar plate-on-needle crystal with a plate radius of 80120583m showing the thin sectored-plate morphology Note thatat earlier times (119905 lt 0 in the graph) the crystal was growing from a tapered needle to a column with nearly uniform radius along its lengthThe time axes were shifted so the plate began growing at 119905 asymp 0 for both the real and model crystals

that matched both the morphology and growth rates ofthe laboratory crystal In addition to the plate and needleradii note that the tapered neck of the needle just belowthe plate seen in Figure 6 was nicely reproduced by themodel crystal A discussion of the physical significance ofthe best-fit model parameters used for this model is left fora subsequent publication comparing the growth of manycrystals under different conditions This example is meantto demonstrate only that the behavior of faceted laboratory-grown ice crystals can be accuratelymodeledwith the cellularautomata method

7 Discussion

Numericalmodeling of structure formation during solidifica-tion has been well studied for decades especially cases whereweak surface anisotropies in diffusion-limited growth yieldrounded dendritic branching Modeling structures that areboth branched and faceted have beenmore difficult and onlyrecently researchers have demonstrated suitable numericalmethods [10 11] In the present work we have developeda numerical model aimed at investigating the formationof simple faceted morphologies during ice crystal growthOur model incorporates the cellular automata technique in[10] but uses a more physically realistic treatment of thesurface attachment kinetics and the Gibbs-Thomson effectUsing this model we are able to reproduce ice crystalgrowth behavior for simple growth morphologies over a

broad range of conditions We have been using this modelto aid in the interpretation of ice growth experiments inorder to extract information about the anisotropic surfaceattachment kinetics that governs ice crystal growth fromwater vapor Modifying the model for application to othermaterial systems is likely straightforward

At present we have limited our model to cylindricallysymmetric crystal growth and relatively simple growth mor-phologies for comparison with measurements A full three-dimensional extension of our model should be possibleas was demonstrated in [10] potentially allowing completemodeling of the many complex branched and faceted struc-tures that are commonly observed in ice crystal growth [7]Achieving this goal would likely require a substantially betterunderstanding of the surface attachment kinetics in ice thanpresently exists along with more advanced algorithms forincorporating all the relevant surface physics into a cellularautomata model

In general it appears that the cellular automata methodis well suited for modeling faceted crystal growth and itmay be the method of choice in material systems exhibitingstrongly anisotropic attachment kinetics Incorporatingmoresophisticated descriptions of surface growth processes willrequire substantially better algorithms than we have outlinedabove especially when extending the models to three dimen-sions Whether such models can reproduce the full rangeof complex structures appearing in faceted crystal growthremains to be seen

Journal of Computational Methods in Physics 11

References

[1] R Trivedi and W Kurz ldquoDendritic growthrdquo InternationalMaterials Reviews vol 39 no 2 pp 49ndash74 1994

[2] E A Brener ldquoThree-dimensional dendritic growthrdquo Journal ofCrystal Growth vol 166 no 1ndash4 pp 339ndash346 1996

[3] I Avramov ldquoKinetics of growth of nanowhiskers (nanowiresand nanotubes)rdquo Nanoscale Research Letters vol 2 no 5 pp235ndash239 2007

[4] H Zhang I Hussain M Brust M F Butler S P Rannard andA I Cooper ldquoAligned two- and three-dimensional structuresby directional freezing of polymers and nanoparticlesrdquo NatureMaterials vol 4 pp 787ndash793 2005

[5] M C Cross and P C Hohenberg ldquoPattern formation outsideof equilibriumrdquo Reviews of Modern Physics vol 65 no 3 pp851ndash1112 1993

[6] K Kassner Pattern Formation in Diffusion-Limited CrystalGrowth World Scientific Singapore 1996

[7] K G Libbrecht ldquoThe physics of snow crystalsrdquo Reports onProgress in Physics vol 68 no 4 pp 855ndash895 2005

[8] J Nelson ldquoGrowth mechanisms to explain the primary andsecondary habits of snow crystalsrdquo Philosophical Magazine Avol 81 no 10 pp 2337ndash2373 2001

[9] H R Pruppacher and J D Klett Microphysics of Clouds andPrecipitation Kluwer Academic Publishers Dordrecht TheNetherlands 1997

[10] J Gravner and D Griffeath ldquoModeling snow-crystal growtha three-dimensional mesoscopic approachrdquo Physical Review Evol 79 no 1 Article ID 011601 18 pages 2009

[11] J W Barrett H Garcke and R Nurnberg ldquoNumerical com-putations of faceted pattern formation in snow crystal growthrdquoPhysical Review E vol 86 no 1 Article ID 011604 14 pages2012

[12] S-Y Hong J Dudhia and S-H Chen ldquoA revised approach toice microphysical processes for the bulk parameterization ofclouds and precipitationrdquoMonthlyWeather Review vol 132 no1 pp 103ndash120 2004

[13] M Matsumoto S Saito and I Ohmine ldquoMolecular dynamicssimulation of the ice nucleation and growth process leading towater freezingrdquo Nature vol 416 no 6879 pp 409ndash413 2002

[14] J G Dash A W Rempel and J S Wettlaufer ldquoThe physicsof premelted ice and its geophysical consequencesrdquo Reviews ofModern Physics vol 78 no 3 pp 695ndash741 2006

[15] A Karma and W-J Rappel ldquoPhase-field method for com-putationally efficient modeling of solidification with arbitraryinterface kineticsrdquo Physical Review E vol 53 no 4 pp R3017ndashR3020 1996

[16] I Singer-Loginova andHM Singer ldquoThe phase field techniquefor modeling multiphase materialsrdquo Reports on Progress inPhysics vol 71 no 10 Article ID 106501 2008

[17] A Schmidt ldquoComputation of three dimensional dendrites withfinite elementsrdquo Journal of Computational Physics vol 125 no2 pp 293ndash312 1996

[18] E Yokoyama ldquoFormation of patterns during growth of snowcrystalsrdquo Journal of Crystal Growth vol 128 no 1ndash4 pp 251ndash257 1993

[19] S E Wood M B Baker and D Calhoun ldquoNew model forthe vapor growth of hexagonal ice crystals in the atmosphererdquoJournal of Geophysical ResearchD vol 106 no 5 pp 4845ndash48702001

[20] K G Libbrecht ldquoOn the equilibrium shape of an ice crystalrdquo2012 httparxivorgabs12051452

[21] G W Sears ldquoA growth mechanism for mercury whiskersrdquo ActaMetallurgica vol 3 no 4 pp 361ndash366 1955

[22] R L Parker R L Anderson and S C Hardy ldquoGrowth andevaporation kinetics and surface diffusion of K and Hg crystalwhiskersrdquo Applied Physics Letters vol 3 no 6 pp 93ndash95 1963

[23] S J Liu H Huang and C HWoo ldquoSchwoebel-Ehrlich barrierfrom two to three dimensionsrdquo Applied Physics Letters vol 80no 18 pp 3295ndash3297 2002

[24] Y Saito Statistical Physics of Crystal Growth World ScientificSingapore 1996

[25] K G Libbrecht ldquoExplaining the formation of thin ice-crystalplates with structure-dependent attachment kineticsrdquo Journal ofCrystal Growth vol 258 no 1-2 pp 168ndash175 2003

[26] K G Libbrecht ldquoPhysically derived rules for simulating facetedcrystal growth using cellular automatardquo 2008 httparxivorgabs08072616

[27] W Thomson ldquoOn the equilibrium vapor at a curved surface ofliquidrdquo Philosophical Magazine vol 42 no 282 pp 448ndash4521871

[28] K G Libbrecht T Crosby and M Swanson ldquoElectricallyenhanced free dendrite growth in polar and non-polar systemsrdquoJournal of Crystal Growth vol 240 no 1-2 pp 241ndash254 2002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

Journal of Computational Methods in Physics 7

41The 119903 = 0 and 119911 = 0 Special Cases In our model we definethe first row of pixels to have 119903 = 0 which leads to problemswith the (1 plusmn Δ1199032119903) factors Going back to the diffusionequation we use Gaussrsquos law to generate the propagationequation for 119903 = 0 pixels

120590 (0 120591 + Δ120591)

= (1 minus 6Δ120591) 120590 (0 120591)

+ Δ120591 [120590 (119911 + Δ119911) + 120590 (119911 minus Δ119911) + 4120590 (Δ119903)]

(40)

Unfortunately taking Δ120591 = 14 in this expression leads toon-axis propagation instabilities As a compromise betweenrunning speed and computational accuracy we use Δ120591 = 16for the on-axis propagation equation giving

120590 (0 120591 + Δ120591) =1

6[120590 (119911 + Δ119911) + 120590 (119911 minus Δ119911) + 4120590 (Δ119903)]

(41)

while we continue using Δ120591 = 14 for off-axis pixels Thiscauses the supersaturation field to relax slightly more slowlyon the 119903-axis but this fix apparently does not adversely affectthe modeling results

Similarly we define the first row of 119911 pixels to have 119911 = 0and for that row we use the propagation equation

120590 (120591 + Δ120591) =1

4[(1 +

Δ119903

2119903) 120590 (119903 + Δ119903)

+ (1 minusΔ119903

2119903) 120590 (119903 minus Δ119903) + 2120590 (119911 + Δ119911)]

(42)

42 Adaptive Time Steps Speeding up the code using adap-tive time steps again proceeds along the lines of the 1Dproblem We use

Δ120582 = ΛΔ120591Δ120585 (43)

with

Λ = 119860 speed1

Δ120585(120572120590surf)max(Δ119903

1198771015840max)

1198771015840

max = radic1198772

max + 1199112

max + 1198772

0

(44)

as above and the physical time step is that given in (25)

43 Neighbor Relations In all these expressions we mustchoose the attachment coefficient 120572 with some care asits value will depend on the number and orientation ofneighboring solid pixels We label a boundary pixel with(119873119903 119873119911) where119873

119903is the number of neighboring ice pixels in

the 119903 direction and 119873119911is the number of ice neighbors in the

119911 direction Both119873119903and119873

119911can take values 0 1 or 2 giving

nine cases for (119873119903 119873119911) The cases are

(0 0) the pixel is an air pixel

(0 1) one ice neighbor in the 119911 direction so 120572 =

120572basal the physical value appropriate for a basal facetsurface(1 0) one ice neighbor in the 119903direction so120572 = 120572prismfor a prism facet surface(1 1) a kink site where the growth will not benucleation limited since the corner provides a sourceof molecular steps We do not know a priori whatvalue to use for 120572 on this site but assume a constant120572 = 120572

11

(0 2) (1 2) (2 0) (2 1) and (2 2) these are allunusual cases where the growth will be fast so weassume that 120572 = 1

We index these possibilities with a single number bycomputing a boundary parameter 119861 = 2119873

2

119903+ 1198732

119911 We then

have 119861 = 0 for an air pixel 119861 = 1 for a basal facet 119861 = 2 for aprism facet 119861 = 3 for a (1 1) kink location and 119861 gt 3 for allother cases

If we consider the special case where 120572 is equal to someconstant value independent of the orientation of the surfacewith respect to the crystal lattice then the growth velocityshould equal V = 120572Vkin120590 for all surfaces For the basal or prismfacet surfaces in this constant-120572 case we take 120572basal = 120572prism =120572 while an analysis of the growth of a (11) surface shows thatwemust take 120572

11= 120572radic2 if the above algorithm is to produce

the correct growth velocity

44 Limitations on Grid Size Aswe pointed out in [26] thereare physical limits to how coarse the computing grid canbe made before instabilities appear or the growth deviatessubstantially from real growth Taking Δ120585 gt 1120572would cause120590solid to become negative which causes some concern in thatit may produce instabilities in the code With this limitationthe grid spacing could not be larger than Δ119903 = Δ119911 = 119883

0120572

For air at a pressure of one atmosphere and 120572 asymp 1 this givesthe pixel size119883

0asymp 0145 120583m

Physically we can gain some insights into these limita-tions from dendrite growth theory [7] We have 119883

0asymp 119877kin

(the latter from (28) in [7]) and a growing dendrite has a tipradius

119877tip asymp1198830

120572119904 (45)

where 119904 is the dimensionless solvability parameter which is oforder unity for ice crystal growth [7]The stability of the codethus limits the grid spacing to be no greater than the tip radiusof a growing dendritic structure From this we see that thecode can only function properly when the grid spacing is fineenough to allow the growth of physically realistic dendriticstructures the scale of which is given by solvability theory

In practice we typically assume that Δ119903 = Δ119911 = 1198830asymp

0145 120583m when comparing models with ice growth data Acoarser grid would reduce run times but at the risk of notreproducing physically relevantmorphological behaviorsWehave not yet explored in detail how different grid sizes affectthe modeling behavior

8 Journal of Computational Methods in Physics

45 Scaling Behavior If we run the code and produce somecomplex crystal shape the interpretation of our result stillcontains an ambiguity The crystal size is given in pixelswhere Δ119903 = Δ119911 = Δ120585119883

0is the pixel size The parameter

Δ120585 was fixed in the code but 1198830depends on the diffusion

constant 119863 which is not otherwise specified Similarly asingle time step in the code corresponds to a physical time

Δ119905 =1198832

0Δ1205852Δ120591

119863sim 119863 (46)

Thus we see that the growth behavior at different airpressures (different D) is determined once we know thegrowth at a single pressure (provided that 120590

infinis the same

at the different pressures) If the air pressure is half anatmosphere the growthmorphology (however complex) willbe the same as at one atmosphere except in the former casethe crystal will be double the size in double the time Thisscaling behavior nicely explains why ice crystal morphologyis generally simpler for smaller crystals andor for lower airpressures which has long been observed [7]

46 Analytic Solutions and Model Validation The case of aninfinitely long cylinder of radius 119877in has the analytic solution

120590 (119903) =120573 + log (119903119877in)

120573 + log (119877out119877in)120590119877out (47)

where the outer boundary has 120590(119877out) = 120590119877out and 120573 =

1198830119877in120572 giving the growth velocity

V =120572120572diffcyl

120572 + 120572diffcylVkin120590119877out

120572diffcyl =1

119861

1198830

119877in

(48)

with 119861 = log(119877out119877in)To compare this analytic result with numerical models of

a growing cylinder we created a version of our codewith peri-odic boundary conditions in 119911 thus modeling the growth ofan infinite cylinder Figure 3 shows results using 119879 = minus15∘C120590119877out = 001 119877init = 1 120583m 120572 = 1 119877out = 300 and 119883

0=

435 120583m Again we see that the numerical model converges tothe analytic result as 119860 speed goes to zero

5 Gibbs-Thomson Effect

The basic cellular automata code described above includes afairly realistic treatment of the attachment kinetics but doesnot include any effects of surface energy This is somewhatjustified for the case of faceted ice crystal growth as one canshow that attachment kinetic effects dominate the growthbehavior while surface energy effects are less important [7]Nevertheless surface energy effects are not always negligibleespecially at low supersaturations As we will demonstratebelow models with highly anisotropic attachment kineticsand no surface energy effects can exhibit the formation ofone-pixel-wide features that are not physically plausible To

5

0

10

Radi

us (m

icro

ns)

0 50 100 150 200Time (s)

Figure 3 The dashed line shows the analytic result for the growthof an infinite cylinder with 120572 = 1 119877out = 435 120583m and 120590

119877out = 001as described in the text Solid lines show numerical modeling resultsusing 119860 speed = 02 (top line) 002 (middle line) and 0002 (bottomline)

suppress these unphysicalmodelswemust include the surfaceenergy via the Gibbs-Thomson effect

For the ice case we are considering here the equilibriumvapor density above a curved surface can be written [27]

119888eq = 119888sat (1 + 120575120581) (49)

where 119888sat is the equilibrium (saturated) vapor density of a flatsurface 120581 is the surface curvature

120581 =1

1199031

+1

1199032

(50)

where 1199031 1199032are the principal radii of curvature of the surface

and

120575 =120574

119888solid119896119879asymp 1 nm (51)

where 120574 asymp 01 Jm2 is the surface energy of the icevaporinterface The anisotropy of 120574 is not well known but theavailable evidence suggests it is rather small [20] so for theremainder of our discussion we assume an isotropic surfaceenergy We believe this is a accurate assumption for ice but itis in stark contrast to the highly anisotropic ice surface energyassumed in [11]

From this we have the supersaturation over a curvedsurface

120590curved asymp 120590 minus120575

119877119888

(52)

to lowest order in 120575119877119888 where120590 is the normal supersaturation

relative to a flat interface and for later convenience we define119877119888= 120581minus1 Putting in some numbers a sharp ice needle might

have 119877119888asymp 1 120583m giving Δ120590 = 120590curved minus 120590 asymp 01 percent

which is small but not always negligible Moreover setting

Journal of Computational Methods in Physics 9

Nz = 7

Nr = 5

z

r

Figure 4 A graphical definition of the outer boundary pixels usedfor incorporating surface energy effects into our cellular automatamodel Here the blue pixels are ice while the clear pixels are airThe red pixels are boundary pixels having the largest 119903 and 119911 valuesdefined to be the ldquoouterrdquo boundary pixels

119877119888asymp 1198830asymp 015 120583m gives a rather large Δ120590 asymp 066 percent

indicating that one-pixel-wide features should not be presentat low supersaturations

Adding the Gibbs-Thomson effect requires an algorith-mic estimation of 119877

119888for all boundary pixels in our cellular

automata model We examined several possibilities but didnot find a suitable algorithm that could be used with arbitrarymorphologies in our cylindrically symmetrical model Sinceour primary goal was to model simple morphologies forcomparison with experimental data as described above wesettled for a simpler algorithm for such cases

Throughout our investigations we found that the overallgrowth behavior of faceted crystals with simplemorphologieswas determined primarily by the growth of the outermostfacet surfaces To apply the Gibbs-Thomson effect on thesesurfaces we defined the outer boundary pixels as shown inFigure 4 yielding the numbers 119873

119903and 119873

119911as the number of

outer-boundary pixelsUsing these outer boundary pixels as a proxy for esti-

mating surface curvature we defined 119877119888= Δ119903119873

1199032 for the

edges of plates 119877119888= Δ119903119873

1199112 for the edges of hollow columns

and 119877119888= Δ119903119873

1199114 for the tips of needles For all boundary

pixels that were not outer-boundary pixels we assumed that119877119888= infin Although these are crude estimations of surface

curvature we found them satisfactory for incorporating theGibbs-Thomson effect in our model in part because this is arelatively small effect compared to the more dominant role ofattachment kinetics

51 A Gibbs-Thomson Example The top image in Figure 5shows how our model can yield unphysical results if wedo not include the Gibbs-Thomson effect In this model weassumed the input parameters 119879 = minus15

∘C 120590infin

= 001

120572basal = exp(minus003120590surf) 120572prism = 1 an initial prism radius1198770= 2 120583m initial prism half-height119867

0= 5 120583m and a Gibbs-

Thomson parameter 120575 = 0 After growing this model for sixseconds the initial simple prism grew into a capped columnwhere the two capping plates each had a thickness of justone pixel as shown in the top image in Figure 5 Such smallstructural features are unphysical at this supersaturation andsimilar one-pixel-wide features often appeared in our modelswhen the supersaturation was low and the anisotropy inattachment kinetics was high

Figure 5 A test of the Gibbs-Thomson effect in our crystal growthmodel In each image the white pixels show ice while the brightnessof the surrounding region is proportional to the supersaturationModel crystals are reflection symmetric about the 119911 = 0 plane andthe images show 119911 ge 0 only Thin vertical lines show the initialcrystal size Using 120572prism = 1 and 120572basal ≪ 120572prism resulted in growthonly on the prism surfaces Full model parameters are given in thetext The top image displays the crystal after six seconds of growthwith a Gibbs-Thomson parameter 120575 = 0 This model exhibits a one-pixel-thick plate growing on the end of the column which is notphysically plausibleThemiddle and lower images show results using120575 = 03 and 10 nm respectively For these models the unphysicalplate growth does not appear

Including nonzero values of 120575 suppressed these smallfeatures as demonstrated in the lower two images in Figure 5From this example as well as additional tests with differentgrowth morphologies we found that our relatively simplealgorithm for including the Gibbs-Thomson effect yieldedreasonable results suppressing the unphysical growth of one-pixel-wide features Models with higher 120590

infinvalues or with

lower anisotropies in the attachment kinetics were generallyless affected by including the Gibbs-Thomson effect

6 An Example Comparison with Experiment

Figure 6 shows an example of how the cellular automatamethod can be used in laboratory investigations of growingice crystals In this experiment we first grew a thin ldquoelectricrdquoice needle as described in [28] and then transferred theneedle to a second growth chamber where we grew a thinplate-like crystal on the needle tip all in air at a pressureof one bar The hexagonal symmetry of the crystal andthe relatively simple growth morphology were well suitedfor analysis using our cylindrically symmetric model Byadjusting 120590

infinand the surface attachment coefficients for the

principal facets we were able to generate a model crystal

10 Journal of Computational Methods in Physics

50120583m

(a) (b)

Radi

us (120583

m)

60

50

40

30

20

10

0minus40 minus20

Plate radius

Needle radius

0 20 40Time (s)

60 80 100

(c)

Figure 6 An example showing our numerical modeling of a laboratory-grown ice crystal The composite image in (a) shows five successiveviews of a thin plate-like crystal growing on the end of a slender ice needle with a 50120583m scale bar The crystal is shown in a side view withillumination from behindThe composite image in (b) shows our numerical model of the same crystal Here the brightness around the crystalis proportional to supersaturation The data points in (c) shows measurements of the plate and needle radii (the latter at a distance of 50120583mfrom the base of the plate) as a function of timeThe lines in the graph are from the growth of the model crystal The inset image in (c) showsa more frontal view of a similar plate-on-needle crystal with a plate radius of 80120583m showing the thin sectored-plate morphology Note thatat earlier times (119905 lt 0 in the graph) the crystal was growing from a tapered needle to a column with nearly uniform radius along its lengthThe time axes were shifted so the plate began growing at 119905 asymp 0 for both the real and model crystals

that matched both the morphology and growth rates ofthe laboratory crystal In addition to the plate and needleradii note that the tapered neck of the needle just belowthe plate seen in Figure 6 was nicely reproduced by themodel crystal A discussion of the physical significance ofthe best-fit model parameters used for this model is left fora subsequent publication comparing the growth of manycrystals under different conditions This example is meantto demonstrate only that the behavior of faceted laboratory-grown ice crystals can be accuratelymodeledwith the cellularautomata method

7 Discussion

Numericalmodeling of structure formation during solidifica-tion has been well studied for decades especially cases whereweak surface anisotropies in diffusion-limited growth yieldrounded dendritic branching Modeling structures that areboth branched and faceted have beenmore difficult and onlyrecently researchers have demonstrated suitable numericalmethods [10 11] In the present work we have developeda numerical model aimed at investigating the formationof simple faceted morphologies during ice crystal growthOur model incorporates the cellular automata technique in[10] but uses a more physically realistic treatment of thesurface attachment kinetics and the Gibbs-Thomson effectUsing this model we are able to reproduce ice crystalgrowth behavior for simple growth morphologies over a

broad range of conditions We have been using this modelto aid in the interpretation of ice growth experiments inorder to extract information about the anisotropic surfaceattachment kinetics that governs ice crystal growth fromwater vapor Modifying the model for application to othermaterial systems is likely straightforward

At present we have limited our model to cylindricallysymmetric crystal growth and relatively simple growth mor-phologies for comparison with measurements A full three-dimensional extension of our model should be possibleas was demonstrated in [10] potentially allowing completemodeling of the many complex branched and faceted struc-tures that are commonly observed in ice crystal growth [7]Achieving this goal would likely require a substantially betterunderstanding of the surface attachment kinetics in ice thanpresently exists along with more advanced algorithms forincorporating all the relevant surface physics into a cellularautomata model

In general it appears that the cellular automata methodis well suited for modeling faceted crystal growth and itmay be the method of choice in material systems exhibitingstrongly anisotropic attachment kinetics Incorporatingmoresophisticated descriptions of surface growth processes willrequire substantially better algorithms than we have outlinedabove especially when extending the models to three dimen-sions Whether such models can reproduce the full rangeof complex structures appearing in faceted crystal growthremains to be seen

Journal of Computational Methods in Physics 11

References

[1] R Trivedi and W Kurz ldquoDendritic growthrdquo InternationalMaterials Reviews vol 39 no 2 pp 49ndash74 1994

[2] E A Brener ldquoThree-dimensional dendritic growthrdquo Journal ofCrystal Growth vol 166 no 1ndash4 pp 339ndash346 1996

[3] I Avramov ldquoKinetics of growth of nanowhiskers (nanowiresand nanotubes)rdquo Nanoscale Research Letters vol 2 no 5 pp235ndash239 2007

[4] H Zhang I Hussain M Brust M F Butler S P Rannard andA I Cooper ldquoAligned two- and three-dimensional structuresby directional freezing of polymers and nanoparticlesrdquo NatureMaterials vol 4 pp 787ndash793 2005

[5] M C Cross and P C Hohenberg ldquoPattern formation outsideof equilibriumrdquo Reviews of Modern Physics vol 65 no 3 pp851ndash1112 1993

[6] K Kassner Pattern Formation in Diffusion-Limited CrystalGrowth World Scientific Singapore 1996

[7] K G Libbrecht ldquoThe physics of snow crystalsrdquo Reports onProgress in Physics vol 68 no 4 pp 855ndash895 2005

[8] J Nelson ldquoGrowth mechanisms to explain the primary andsecondary habits of snow crystalsrdquo Philosophical Magazine Avol 81 no 10 pp 2337ndash2373 2001

[9] H R Pruppacher and J D Klett Microphysics of Clouds andPrecipitation Kluwer Academic Publishers Dordrecht TheNetherlands 1997

[10] J Gravner and D Griffeath ldquoModeling snow-crystal growtha three-dimensional mesoscopic approachrdquo Physical Review Evol 79 no 1 Article ID 011601 18 pages 2009

[11] J W Barrett H Garcke and R Nurnberg ldquoNumerical com-putations of faceted pattern formation in snow crystal growthrdquoPhysical Review E vol 86 no 1 Article ID 011604 14 pages2012

[12] S-Y Hong J Dudhia and S-H Chen ldquoA revised approach toice microphysical processes for the bulk parameterization ofclouds and precipitationrdquoMonthlyWeather Review vol 132 no1 pp 103ndash120 2004

[13] M Matsumoto S Saito and I Ohmine ldquoMolecular dynamicssimulation of the ice nucleation and growth process leading towater freezingrdquo Nature vol 416 no 6879 pp 409ndash413 2002

[14] J G Dash A W Rempel and J S Wettlaufer ldquoThe physicsof premelted ice and its geophysical consequencesrdquo Reviews ofModern Physics vol 78 no 3 pp 695ndash741 2006

[15] A Karma and W-J Rappel ldquoPhase-field method for com-putationally efficient modeling of solidification with arbitraryinterface kineticsrdquo Physical Review E vol 53 no 4 pp R3017ndashR3020 1996

[16] I Singer-Loginova andHM Singer ldquoThe phase field techniquefor modeling multiphase materialsrdquo Reports on Progress inPhysics vol 71 no 10 Article ID 106501 2008

[17] A Schmidt ldquoComputation of three dimensional dendrites withfinite elementsrdquo Journal of Computational Physics vol 125 no2 pp 293ndash312 1996

[18] E Yokoyama ldquoFormation of patterns during growth of snowcrystalsrdquo Journal of Crystal Growth vol 128 no 1ndash4 pp 251ndash257 1993

[19] S E Wood M B Baker and D Calhoun ldquoNew model forthe vapor growth of hexagonal ice crystals in the atmosphererdquoJournal of Geophysical ResearchD vol 106 no 5 pp 4845ndash48702001

[20] K G Libbrecht ldquoOn the equilibrium shape of an ice crystalrdquo2012 httparxivorgabs12051452

[21] G W Sears ldquoA growth mechanism for mercury whiskersrdquo ActaMetallurgica vol 3 no 4 pp 361ndash366 1955

[22] R L Parker R L Anderson and S C Hardy ldquoGrowth andevaporation kinetics and surface diffusion of K and Hg crystalwhiskersrdquo Applied Physics Letters vol 3 no 6 pp 93ndash95 1963

[23] S J Liu H Huang and C HWoo ldquoSchwoebel-Ehrlich barrierfrom two to three dimensionsrdquo Applied Physics Letters vol 80no 18 pp 3295ndash3297 2002

[24] Y Saito Statistical Physics of Crystal Growth World ScientificSingapore 1996

[25] K G Libbrecht ldquoExplaining the formation of thin ice-crystalplates with structure-dependent attachment kineticsrdquo Journal ofCrystal Growth vol 258 no 1-2 pp 168ndash175 2003

[26] K G Libbrecht ldquoPhysically derived rules for simulating facetedcrystal growth using cellular automatardquo 2008 httparxivorgabs08072616

[27] W Thomson ldquoOn the equilibrium vapor at a curved surface ofliquidrdquo Philosophical Magazine vol 42 no 282 pp 448ndash4521871

[28] K G Libbrecht T Crosby and M Swanson ldquoElectricallyenhanced free dendrite growth in polar and non-polar systemsrdquoJournal of Crystal Growth vol 240 no 1-2 pp 241ndash254 2002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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FluidsJournal of

Atomic and Molecular Physics

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Advances in Condensed Matter Physics

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AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

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GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

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Physics Research International

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Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

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Soft MatterJournal of

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AerodynamicsJournal of

Volume 2014

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PhotonicsJournal of

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Journal of

Biophysics

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ThermodynamicsJournal of

8 Journal of Computational Methods in Physics

45 Scaling Behavior If we run the code and produce somecomplex crystal shape the interpretation of our result stillcontains an ambiguity The crystal size is given in pixelswhere Δ119903 = Δ119911 = Δ120585119883

0is the pixel size The parameter

Δ120585 was fixed in the code but 1198830depends on the diffusion

constant 119863 which is not otherwise specified Similarly asingle time step in the code corresponds to a physical time

Δ119905 =1198832

0Δ1205852Δ120591

119863sim 119863 (46)

Thus we see that the growth behavior at different airpressures (different D) is determined once we know thegrowth at a single pressure (provided that 120590

infinis the same

at the different pressures) If the air pressure is half anatmosphere the growthmorphology (however complex) willbe the same as at one atmosphere except in the former casethe crystal will be double the size in double the time Thisscaling behavior nicely explains why ice crystal morphologyis generally simpler for smaller crystals andor for lower airpressures which has long been observed [7]

46 Analytic Solutions and Model Validation The case of aninfinitely long cylinder of radius 119877in has the analytic solution

120590 (119903) =120573 + log (119903119877in)

120573 + log (119877out119877in)120590119877out (47)

where the outer boundary has 120590(119877out) = 120590119877out and 120573 =

1198830119877in120572 giving the growth velocity

V =120572120572diffcyl

120572 + 120572diffcylVkin120590119877out

120572diffcyl =1

119861

1198830

119877in

(48)

with 119861 = log(119877out119877in)To compare this analytic result with numerical models of

a growing cylinder we created a version of our codewith peri-odic boundary conditions in 119911 thus modeling the growth ofan infinite cylinder Figure 3 shows results using 119879 = minus15∘C120590119877out = 001 119877init = 1 120583m 120572 = 1 119877out = 300 and 119883

0=

435 120583m Again we see that the numerical model converges tothe analytic result as 119860 speed goes to zero

5 Gibbs-Thomson Effect

The basic cellular automata code described above includes afairly realistic treatment of the attachment kinetics but doesnot include any effects of surface energy This is somewhatjustified for the case of faceted ice crystal growth as one canshow that attachment kinetic effects dominate the growthbehavior while surface energy effects are less important [7]Nevertheless surface energy effects are not always negligibleespecially at low supersaturations As we will demonstratebelow models with highly anisotropic attachment kineticsand no surface energy effects can exhibit the formation ofone-pixel-wide features that are not physically plausible To

5

0

10

Radi

us (m

icro

ns)

0 50 100 150 200Time (s)

Figure 3 The dashed line shows the analytic result for the growthof an infinite cylinder with 120572 = 1 119877out = 435 120583m and 120590

119877out = 001as described in the text Solid lines show numerical modeling resultsusing 119860 speed = 02 (top line) 002 (middle line) and 0002 (bottomline)

suppress these unphysicalmodelswemust include the surfaceenergy via the Gibbs-Thomson effect

For the ice case we are considering here the equilibriumvapor density above a curved surface can be written [27]

119888eq = 119888sat (1 + 120575120581) (49)

where 119888sat is the equilibrium (saturated) vapor density of a flatsurface 120581 is the surface curvature

120581 =1

1199031

+1

1199032

(50)

where 1199031 1199032are the principal radii of curvature of the surface

and

120575 =120574

119888solid119896119879asymp 1 nm (51)

where 120574 asymp 01 Jm2 is the surface energy of the icevaporinterface The anisotropy of 120574 is not well known but theavailable evidence suggests it is rather small [20] so for theremainder of our discussion we assume an isotropic surfaceenergy We believe this is a accurate assumption for ice but itis in stark contrast to the highly anisotropic ice surface energyassumed in [11]

From this we have the supersaturation over a curvedsurface

120590curved asymp 120590 minus120575

119877119888

(52)

to lowest order in 120575119877119888 where120590 is the normal supersaturation

relative to a flat interface and for later convenience we define119877119888= 120581minus1 Putting in some numbers a sharp ice needle might

have 119877119888asymp 1 120583m giving Δ120590 = 120590curved minus 120590 asymp 01 percent

which is small but not always negligible Moreover setting

Journal of Computational Methods in Physics 9

Nz = 7

Nr = 5

z

r

Figure 4 A graphical definition of the outer boundary pixels usedfor incorporating surface energy effects into our cellular automatamodel Here the blue pixels are ice while the clear pixels are airThe red pixels are boundary pixels having the largest 119903 and 119911 valuesdefined to be the ldquoouterrdquo boundary pixels

119877119888asymp 1198830asymp 015 120583m gives a rather large Δ120590 asymp 066 percent

indicating that one-pixel-wide features should not be presentat low supersaturations

Adding the Gibbs-Thomson effect requires an algorith-mic estimation of 119877

119888for all boundary pixels in our cellular

automata model We examined several possibilities but didnot find a suitable algorithm that could be used with arbitrarymorphologies in our cylindrically symmetrical model Sinceour primary goal was to model simple morphologies forcomparison with experimental data as described above wesettled for a simpler algorithm for such cases

Throughout our investigations we found that the overallgrowth behavior of faceted crystals with simplemorphologieswas determined primarily by the growth of the outermostfacet surfaces To apply the Gibbs-Thomson effect on thesesurfaces we defined the outer boundary pixels as shown inFigure 4 yielding the numbers 119873

119903and 119873

119911as the number of

outer-boundary pixelsUsing these outer boundary pixels as a proxy for esti-

mating surface curvature we defined 119877119888= Δ119903119873

1199032 for the

edges of plates 119877119888= Δ119903119873

1199112 for the edges of hollow columns

and 119877119888= Δ119903119873

1199114 for the tips of needles For all boundary

pixels that were not outer-boundary pixels we assumed that119877119888= infin Although these are crude estimations of surface

curvature we found them satisfactory for incorporating theGibbs-Thomson effect in our model in part because this is arelatively small effect compared to the more dominant role ofattachment kinetics

51 A Gibbs-Thomson Example The top image in Figure 5shows how our model can yield unphysical results if wedo not include the Gibbs-Thomson effect In this model weassumed the input parameters 119879 = minus15

∘C 120590infin

= 001

120572basal = exp(minus003120590surf) 120572prism = 1 an initial prism radius1198770= 2 120583m initial prism half-height119867

0= 5 120583m and a Gibbs-

Thomson parameter 120575 = 0 After growing this model for sixseconds the initial simple prism grew into a capped columnwhere the two capping plates each had a thickness of justone pixel as shown in the top image in Figure 5 Such smallstructural features are unphysical at this supersaturation andsimilar one-pixel-wide features often appeared in our modelswhen the supersaturation was low and the anisotropy inattachment kinetics was high

Figure 5 A test of the Gibbs-Thomson effect in our crystal growthmodel In each image the white pixels show ice while the brightnessof the surrounding region is proportional to the supersaturationModel crystals are reflection symmetric about the 119911 = 0 plane andthe images show 119911 ge 0 only Thin vertical lines show the initialcrystal size Using 120572prism = 1 and 120572basal ≪ 120572prism resulted in growthonly on the prism surfaces Full model parameters are given in thetext The top image displays the crystal after six seconds of growthwith a Gibbs-Thomson parameter 120575 = 0 This model exhibits a one-pixel-thick plate growing on the end of the column which is notphysically plausibleThemiddle and lower images show results using120575 = 03 and 10 nm respectively For these models the unphysicalplate growth does not appear

Including nonzero values of 120575 suppressed these smallfeatures as demonstrated in the lower two images in Figure 5From this example as well as additional tests with differentgrowth morphologies we found that our relatively simplealgorithm for including the Gibbs-Thomson effect yieldedreasonable results suppressing the unphysical growth of one-pixel-wide features Models with higher 120590

infinvalues or with

lower anisotropies in the attachment kinetics were generallyless affected by including the Gibbs-Thomson effect

6 An Example Comparison with Experiment

Figure 6 shows an example of how the cellular automatamethod can be used in laboratory investigations of growingice crystals In this experiment we first grew a thin ldquoelectricrdquoice needle as described in [28] and then transferred theneedle to a second growth chamber where we grew a thinplate-like crystal on the needle tip all in air at a pressureof one bar The hexagonal symmetry of the crystal andthe relatively simple growth morphology were well suitedfor analysis using our cylindrically symmetric model Byadjusting 120590

infinand the surface attachment coefficients for the

principal facets we were able to generate a model crystal

10 Journal of Computational Methods in Physics

50120583m

(a) (b)

Radi

us (120583

m)

60

50

40

30

20

10

0minus40 minus20

Plate radius

Needle radius

0 20 40Time (s)

60 80 100

(c)

Figure 6 An example showing our numerical modeling of a laboratory-grown ice crystal The composite image in (a) shows five successiveviews of a thin plate-like crystal growing on the end of a slender ice needle with a 50120583m scale bar The crystal is shown in a side view withillumination from behindThe composite image in (b) shows our numerical model of the same crystal Here the brightness around the crystalis proportional to supersaturation The data points in (c) shows measurements of the plate and needle radii (the latter at a distance of 50120583mfrom the base of the plate) as a function of timeThe lines in the graph are from the growth of the model crystal The inset image in (c) showsa more frontal view of a similar plate-on-needle crystal with a plate radius of 80120583m showing the thin sectored-plate morphology Note thatat earlier times (119905 lt 0 in the graph) the crystal was growing from a tapered needle to a column with nearly uniform radius along its lengthThe time axes were shifted so the plate began growing at 119905 asymp 0 for both the real and model crystals

that matched both the morphology and growth rates ofthe laboratory crystal In addition to the plate and needleradii note that the tapered neck of the needle just belowthe plate seen in Figure 6 was nicely reproduced by themodel crystal A discussion of the physical significance ofthe best-fit model parameters used for this model is left fora subsequent publication comparing the growth of manycrystals under different conditions This example is meantto demonstrate only that the behavior of faceted laboratory-grown ice crystals can be accuratelymodeledwith the cellularautomata method

7 Discussion

Numericalmodeling of structure formation during solidifica-tion has been well studied for decades especially cases whereweak surface anisotropies in diffusion-limited growth yieldrounded dendritic branching Modeling structures that areboth branched and faceted have beenmore difficult and onlyrecently researchers have demonstrated suitable numericalmethods [10 11] In the present work we have developeda numerical model aimed at investigating the formationof simple faceted morphologies during ice crystal growthOur model incorporates the cellular automata technique in[10] but uses a more physically realistic treatment of thesurface attachment kinetics and the Gibbs-Thomson effectUsing this model we are able to reproduce ice crystalgrowth behavior for simple growth morphologies over a

broad range of conditions We have been using this modelto aid in the interpretation of ice growth experiments inorder to extract information about the anisotropic surfaceattachment kinetics that governs ice crystal growth fromwater vapor Modifying the model for application to othermaterial systems is likely straightforward

At present we have limited our model to cylindricallysymmetric crystal growth and relatively simple growth mor-phologies for comparison with measurements A full three-dimensional extension of our model should be possibleas was demonstrated in [10] potentially allowing completemodeling of the many complex branched and faceted struc-tures that are commonly observed in ice crystal growth [7]Achieving this goal would likely require a substantially betterunderstanding of the surface attachment kinetics in ice thanpresently exists along with more advanced algorithms forincorporating all the relevant surface physics into a cellularautomata model

In general it appears that the cellular automata methodis well suited for modeling faceted crystal growth and itmay be the method of choice in material systems exhibitingstrongly anisotropic attachment kinetics Incorporatingmoresophisticated descriptions of surface growth processes willrequire substantially better algorithms than we have outlinedabove especially when extending the models to three dimen-sions Whether such models can reproduce the full rangeof complex structures appearing in faceted crystal growthremains to be seen

Journal of Computational Methods in Physics 11

References

[1] R Trivedi and W Kurz ldquoDendritic growthrdquo InternationalMaterials Reviews vol 39 no 2 pp 49ndash74 1994

[2] E A Brener ldquoThree-dimensional dendritic growthrdquo Journal ofCrystal Growth vol 166 no 1ndash4 pp 339ndash346 1996

[3] I Avramov ldquoKinetics of growth of nanowhiskers (nanowiresand nanotubes)rdquo Nanoscale Research Letters vol 2 no 5 pp235ndash239 2007

[4] H Zhang I Hussain M Brust M F Butler S P Rannard andA I Cooper ldquoAligned two- and three-dimensional structuresby directional freezing of polymers and nanoparticlesrdquo NatureMaterials vol 4 pp 787ndash793 2005

[5] M C Cross and P C Hohenberg ldquoPattern formation outsideof equilibriumrdquo Reviews of Modern Physics vol 65 no 3 pp851ndash1112 1993

[6] K Kassner Pattern Formation in Diffusion-Limited CrystalGrowth World Scientific Singapore 1996

[7] K G Libbrecht ldquoThe physics of snow crystalsrdquo Reports onProgress in Physics vol 68 no 4 pp 855ndash895 2005

[8] J Nelson ldquoGrowth mechanisms to explain the primary andsecondary habits of snow crystalsrdquo Philosophical Magazine Avol 81 no 10 pp 2337ndash2373 2001

[9] H R Pruppacher and J D Klett Microphysics of Clouds andPrecipitation Kluwer Academic Publishers Dordrecht TheNetherlands 1997

[10] J Gravner and D Griffeath ldquoModeling snow-crystal growtha three-dimensional mesoscopic approachrdquo Physical Review Evol 79 no 1 Article ID 011601 18 pages 2009

[11] J W Barrett H Garcke and R Nurnberg ldquoNumerical com-putations of faceted pattern formation in snow crystal growthrdquoPhysical Review E vol 86 no 1 Article ID 011604 14 pages2012

[12] S-Y Hong J Dudhia and S-H Chen ldquoA revised approach toice microphysical processes for the bulk parameterization ofclouds and precipitationrdquoMonthlyWeather Review vol 132 no1 pp 103ndash120 2004

[13] M Matsumoto S Saito and I Ohmine ldquoMolecular dynamicssimulation of the ice nucleation and growth process leading towater freezingrdquo Nature vol 416 no 6879 pp 409ndash413 2002

[14] J G Dash A W Rempel and J S Wettlaufer ldquoThe physicsof premelted ice and its geophysical consequencesrdquo Reviews ofModern Physics vol 78 no 3 pp 695ndash741 2006

[15] A Karma and W-J Rappel ldquoPhase-field method for com-putationally efficient modeling of solidification with arbitraryinterface kineticsrdquo Physical Review E vol 53 no 4 pp R3017ndashR3020 1996

[16] I Singer-Loginova andHM Singer ldquoThe phase field techniquefor modeling multiphase materialsrdquo Reports on Progress inPhysics vol 71 no 10 Article ID 106501 2008

[17] A Schmidt ldquoComputation of three dimensional dendrites withfinite elementsrdquo Journal of Computational Physics vol 125 no2 pp 293ndash312 1996

[18] E Yokoyama ldquoFormation of patterns during growth of snowcrystalsrdquo Journal of Crystal Growth vol 128 no 1ndash4 pp 251ndash257 1993

[19] S E Wood M B Baker and D Calhoun ldquoNew model forthe vapor growth of hexagonal ice crystals in the atmosphererdquoJournal of Geophysical ResearchD vol 106 no 5 pp 4845ndash48702001

[20] K G Libbrecht ldquoOn the equilibrium shape of an ice crystalrdquo2012 httparxivorgabs12051452

[21] G W Sears ldquoA growth mechanism for mercury whiskersrdquo ActaMetallurgica vol 3 no 4 pp 361ndash366 1955

[22] R L Parker R L Anderson and S C Hardy ldquoGrowth andevaporation kinetics and surface diffusion of K and Hg crystalwhiskersrdquo Applied Physics Letters vol 3 no 6 pp 93ndash95 1963

[23] S J Liu H Huang and C HWoo ldquoSchwoebel-Ehrlich barrierfrom two to three dimensionsrdquo Applied Physics Letters vol 80no 18 pp 3295ndash3297 2002

[24] Y Saito Statistical Physics of Crystal Growth World ScientificSingapore 1996

[25] K G Libbrecht ldquoExplaining the formation of thin ice-crystalplates with structure-dependent attachment kineticsrdquo Journal ofCrystal Growth vol 258 no 1-2 pp 168ndash175 2003

[26] K G Libbrecht ldquoPhysically derived rules for simulating facetedcrystal growth using cellular automatardquo 2008 httparxivorgabs08072616

[27] W Thomson ldquoOn the equilibrium vapor at a curved surface ofliquidrdquo Philosophical Magazine vol 42 no 282 pp 448ndash4521871

[28] K G Libbrecht T Crosby and M Swanson ldquoElectricallyenhanced free dendrite growth in polar and non-polar systemsrdquoJournal of Crystal Growth vol 240 no 1-2 pp 241ndash254 2002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

Journal of Computational Methods in Physics 9

Nz = 7

Nr = 5

z

r

Figure 4 A graphical definition of the outer boundary pixels usedfor incorporating surface energy effects into our cellular automatamodel Here the blue pixels are ice while the clear pixels are airThe red pixels are boundary pixels having the largest 119903 and 119911 valuesdefined to be the ldquoouterrdquo boundary pixels

119877119888asymp 1198830asymp 015 120583m gives a rather large Δ120590 asymp 066 percent

indicating that one-pixel-wide features should not be presentat low supersaturations

Adding the Gibbs-Thomson effect requires an algorith-mic estimation of 119877

119888for all boundary pixels in our cellular

automata model We examined several possibilities but didnot find a suitable algorithm that could be used with arbitrarymorphologies in our cylindrically symmetrical model Sinceour primary goal was to model simple morphologies forcomparison with experimental data as described above wesettled for a simpler algorithm for such cases

Throughout our investigations we found that the overallgrowth behavior of faceted crystals with simplemorphologieswas determined primarily by the growth of the outermostfacet surfaces To apply the Gibbs-Thomson effect on thesesurfaces we defined the outer boundary pixels as shown inFigure 4 yielding the numbers 119873

119903and 119873

119911as the number of

outer-boundary pixelsUsing these outer boundary pixels as a proxy for esti-

mating surface curvature we defined 119877119888= Δ119903119873

1199032 for the

edges of plates 119877119888= Δ119903119873

1199112 for the edges of hollow columns

and 119877119888= Δ119903119873

1199114 for the tips of needles For all boundary

pixels that were not outer-boundary pixels we assumed that119877119888= infin Although these are crude estimations of surface

curvature we found them satisfactory for incorporating theGibbs-Thomson effect in our model in part because this is arelatively small effect compared to the more dominant role ofattachment kinetics

51 A Gibbs-Thomson Example The top image in Figure 5shows how our model can yield unphysical results if wedo not include the Gibbs-Thomson effect In this model weassumed the input parameters 119879 = minus15

∘C 120590infin

= 001

120572basal = exp(minus003120590surf) 120572prism = 1 an initial prism radius1198770= 2 120583m initial prism half-height119867

0= 5 120583m and a Gibbs-

Thomson parameter 120575 = 0 After growing this model for sixseconds the initial simple prism grew into a capped columnwhere the two capping plates each had a thickness of justone pixel as shown in the top image in Figure 5 Such smallstructural features are unphysical at this supersaturation andsimilar one-pixel-wide features often appeared in our modelswhen the supersaturation was low and the anisotropy inattachment kinetics was high

Figure 5 A test of the Gibbs-Thomson effect in our crystal growthmodel In each image the white pixels show ice while the brightnessof the surrounding region is proportional to the supersaturationModel crystals are reflection symmetric about the 119911 = 0 plane andthe images show 119911 ge 0 only Thin vertical lines show the initialcrystal size Using 120572prism = 1 and 120572basal ≪ 120572prism resulted in growthonly on the prism surfaces Full model parameters are given in thetext The top image displays the crystal after six seconds of growthwith a Gibbs-Thomson parameter 120575 = 0 This model exhibits a one-pixel-thick plate growing on the end of the column which is notphysically plausibleThemiddle and lower images show results using120575 = 03 and 10 nm respectively For these models the unphysicalplate growth does not appear

Including nonzero values of 120575 suppressed these smallfeatures as demonstrated in the lower two images in Figure 5From this example as well as additional tests with differentgrowth morphologies we found that our relatively simplealgorithm for including the Gibbs-Thomson effect yieldedreasonable results suppressing the unphysical growth of one-pixel-wide features Models with higher 120590

infinvalues or with

lower anisotropies in the attachment kinetics were generallyless affected by including the Gibbs-Thomson effect

6 An Example Comparison with Experiment

Figure 6 shows an example of how the cellular automatamethod can be used in laboratory investigations of growingice crystals In this experiment we first grew a thin ldquoelectricrdquoice needle as described in [28] and then transferred theneedle to a second growth chamber where we grew a thinplate-like crystal on the needle tip all in air at a pressureof one bar The hexagonal symmetry of the crystal andthe relatively simple growth morphology were well suitedfor analysis using our cylindrically symmetric model Byadjusting 120590

infinand the surface attachment coefficients for the

principal facets we were able to generate a model crystal

10 Journal of Computational Methods in Physics

50120583m

(a) (b)

Radi

us (120583

m)

60

50

40

30

20

10

0minus40 minus20

Plate radius

Needle radius

0 20 40Time (s)

60 80 100

(c)

Figure 6 An example showing our numerical modeling of a laboratory-grown ice crystal The composite image in (a) shows five successiveviews of a thin plate-like crystal growing on the end of a slender ice needle with a 50120583m scale bar The crystal is shown in a side view withillumination from behindThe composite image in (b) shows our numerical model of the same crystal Here the brightness around the crystalis proportional to supersaturation The data points in (c) shows measurements of the plate and needle radii (the latter at a distance of 50120583mfrom the base of the plate) as a function of timeThe lines in the graph are from the growth of the model crystal The inset image in (c) showsa more frontal view of a similar plate-on-needle crystal with a plate radius of 80120583m showing the thin sectored-plate morphology Note thatat earlier times (119905 lt 0 in the graph) the crystal was growing from a tapered needle to a column with nearly uniform radius along its lengthThe time axes were shifted so the plate began growing at 119905 asymp 0 for both the real and model crystals

that matched both the morphology and growth rates ofthe laboratory crystal In addition to the plate and needleradii note that the tapered neck of the needle just belowthe plate seen in Figure 6 was nicely reproduced by themodel crystal A discussion of the physical significance ofthe best-fit model parameters used for this model is left fora subsequent publication comparing the growth of manycrystals under different conditions This example is meantto demonstrate only that the behavior of faceted laboratory-grown ice crystals can be accuratelymodeledwith the cellularautomata method

7 Discussion

Numericalmodeling of structure formation during solidifica-tion has been well studied for decades especially cases whereweak surface anisotropies in diffusion-limited growth yieldrounded dendritic branching Modeling structures that areboth branched and faceted have beenmore difficult and onlyrecently researchers have demonstrated suitable numericalmethods [10 11] In the present work we have developeda numerical model aimed at investigating the formationof simple faceted morphologies during ice crystal growthOur model incorporates the cellular automata technique in[10] but uses a more physically realistic treatment of thesurface attachment kinetics and the Gibbs-Thomson effectUsing this model we are able to reproduce ice crystalgrowth behavior for simple growth morphologies over a

broad range of conditions We have been using this modelto aid in the interpretation of ice growth experiments inorder to extract information about the anisotropic surfaceattachment kinetics that governs ice crystal growth fromwater vapor Modifying the model for application to othermaterial systems is likely straightforward

At present we have limited our model to cylindricallysymmetric crystal growth and relatively simple growth mor-phologies for comparison with measurements A full three-dimensional extension of our model should be possibleas was demonstrated in [10] potentially allowing completemodeling of the many complex branched and faceted struc-tures that are commonly observed in ice crystal growth [7]Achieving this goal would likely require a substantially betterunderstanding of the surface attachment kinetics in ice thanpresently exists along with more advanced algorithms forincorporating all the relevant surface physics into a cellularautomata model

In general it appears that the cellular automata methodis well suited for modeling faceted crystal growth and itmay be the method of choice in material systems exhibitingstrongly anisotropic attachment kinetics Incorporatingmoresophisticated descriptions of surface growth processes willrequire substantially better algorithms than we have outlinedabove especially when extending the models to three dimen-sions Whether such models can reproduce the full rangeof complex structures appearing in faceted crystal growthremains to be seen

Journal of Computational Methods in Physics 11

References

[1] R Trivedi and W Kurz ldquoDendritic growthrdquo InternationalMaterials Reviews vol 39 no 2 pp 49ndash74 1994

[2] E A Brener ldquoThree-dimensional dendritic growthrdquo Journal ofCrystal Growth vol 166 no 1ndash4 pp 339ndash346 1996

[3] I Avramov ldquoKinetics of growth of nanowhiskers (nanowiresand nanotubes)rdquo Nanoscale Research Letters vol 2 no 5 pp235ndash239 2007

[4] H Zhang I Hussain M Brust M F Butler S P Rannard andA I Cooper ldquoAligned two- and three-dimensional structuresby directional freezing of polymers and nanoparticlesrdquo NatureMaterials vol 4 pp 787ndash793 2005

[5] M C Cross and P C Hohenberg ldquoPattern formation outsideof equilibriumrdquo Reviews of Modern Physics vol 65 no 3 pp851ndash1112 1993

[6] K Kassner Pattern Formation in Diffusion-Limited CrystalGrowth World Scientific Singapore 1996

[7] K G Libbrecht ldquoThe physics of snow crystalsrdquo Reports onProgress in Physics vol 68 no 4 pp 855ndash895 2005

[8] J Nelson ldquoGrowth mechanisms to explain the primary andsecondary habits of snow crystalsrdquo Philosophical Magazine Avol 81 no 10 pp 2337ndash2373 2001

[9] H R Pruppacher and J D Klett Microphysics of Clouds andPrecipitation Kluwer Academic Publishers Dordrecht TheNetherlands 1997

[10] J Gravner and D Griffeath ldquoModeling snow-crystal growtha three-dimensional mesoscopic approachrdquo Physical Review Evol 79 no 1 Article ID 011601 18 pages 2009

[11] J W Barrett H Garcke and R Nurnberg ldquoNumerical com-putations of faceted pattern formation in snow crystal growthrdquoPhysical Review E vol 86 no 1 Article ID 011604 14 pages2012

[12] S-Y Hong J Dudhia and S-H Chen ldquoA revised approach toice microphysical processes for the bulk parameterization ofclouds and precipitationrdquoMonthlyWeather Review vol 132 no1 pp 103ndash120 2004

[13] M Matsumoto S Saito and I Ohmine ldquoMolecular dynamicssimulation of the ice nucleation and growth process leading towater freezingrdquo Nature vol 416 no 6879 pp 409ndash413 2002

[14] J G Dash A W Rempel and J S Wettlaufer ldquoThe physicsof premelted ice and its geophysical consequencesrdquo Reviews ofModern Physics vol 78 no 3 pp 695ndash741 2006

[15] A Karma and W-J Rappel ldquoPhase-field method for com-putationally efficient modeling of solidification with arbitraryinterface kineticsrdquo Physical Review E vol 53 no 4 pp R3017ndashR3020 1996

[16] I Singer-Loginova andHM Singer ldquoThe phase field techniquefor modeling multiphase materialsrdquo Reports on Progress inPhysics vol 71 no 10 Article ID 106501 2008

[17] A Schmidt ldquoComputation of three dimensional dendrites withfinite elementsrdquo Journal of Computational Physics vol 125 no2 pp 293ndash312 1996

[18] E Yokoyama ldquoFormation of patterns during growth of snowcrystalsrdquo Journal of Crystal Growth vol 128 no 1ndash4 pp 251ndash257 1993

[19] S E Wood M B Baker and D Calhoun ldquoNew model forthe vapor growth of hexagonal ice crystals in the atmosphererdquoJournal of Geophysical ResearchD vol 106 no 5 pp 4845ndash48702001

[20] K G Libbrecht ldquoOn the equilibrium shape of an ice crystalrdquo2012 httparxivorgabs12051452

[21] G W Sears ldquoA growth mechanism for mercury whiskersrdquo ActaMetallurgica vol 3 no 4 pp 361ndash366 1955

[22] R L Parker R L Anderson and S C Hardy ldquoGrowth andevaporation kinetics and surface diffusion of K and Hg crystalwhiskersrdquo Applied Physics Letters vol 3 no 6 pp 93ndash95 1963

[23] S J Liu H Huang and C HWoo ldquoSchwoebel-Ehrlich barrierfrom two to three dimensionsrdquo Applied Physics Letters vol 80no 18 pp 3295ndash3297 2002

[24] Y Saito Statistical Physics of Crystal Growth World ScientificSingapore 1996

[25] K G Libbrecht ldquoExplaining the formation of thin ice-crystalplates with structure-dependent attachment kineticsrdquo Journal ofCrystal Growth vol 258 no 1-2 pp 168ndash175 2003

[26] K G Libbrecht ldquoPhysically derived rules for simulating facetedcrystal growth using cellular automatardquo 2008 httparxivorgabs08072616

[27] W Thomson ldquoOn the equilibrium vapor at a curved surface ofliquidrdquo Philosophical Magazine vol 42 no 282 pp 448ndash4521871

[28] K G Libbrecht T Crosby and M Swanson ldquoElectricallyenhanced free dendrite growth in polar and non-polar systemsrdquoJournal of Crystal Growth vol 240 no 1-2 pp 241ndash254 2002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

10 Journal of Computational Methods in Physics

50120583m

(a) (b)

Radi

us (120583

m)

60

50

40

30

20

10

0minus40 minus20

Plate radius

Needle radius

0 20 40Time (s)

60 80 100

(c)

Figure 6 An example showing our numerical modeling of a laboratory-grown ice crystal The composite image in (a) shows five successiveviews of a thin plate-like crystal growing on the end of a slender ice needle with a 50120583m scale bar The crystal is shown in a side view withillumination from behindThe composite image in (b) shows our numerical model of the same crystal Here the brightness around the crystalis proportional to supersaturation The data points in (c) shows measurements of the plate and needle radii (the latter at a distance of 50120583mfrom the base of the plate) as a function of timeThe lines in the graph are from the growth of the model crystal The inset image in (c) showsa more frontal view of a similar plate-on-needle crystal with a plate radius of 80120583m showing the thin sectored-plate morphology Note thatat earlier times (119905 lt 0 in the graph) the crystal was growing from a tapered needle to a column with nearly uniform radius along its lengthThe time axes were shifted so the plate began growing at 119905 asymp 0 for both the real and model crystals

that matched both the morphology and growth rates ofthe laboratory crystal In addition to the plate and needleradii note that the tapered neck of the needle just belowthe plate seen in Figure 6 was nicely reproduced by themodel crystal A discussion of the physical significance ofthe best-fit model parameters used for this model is left fora subsequent publication comparing the growth of manycrystals under different conditions This example is meantto demonstrate only that the behavior of faceted laboratory-grown ice crystals can be accuratelymodeledwith the cellularautomata method

7 Discussion

Numericalmodeling of structure formation during solidifica-tion has been well studied for decades especially cases whereweak surface anisotropies in diffusion-limited growth yieldrounded dendritic branching Modeling structures that areboth branched and faceted have beenmore difficult and onlyrecently researchers have demonstrated suitable numericalmethods [10 11] In the present work we have developeda numerical model aimed at investigating the formationof simple faceted morphologies during ice crystal growthOur model incorporates the cellular automata technique in[10] but uses a more physically realistic treatment of thesurface attachment kinetics and the Gibbs-Thomson effectUsing this model we are able to reproduce ice crystalgrowth behavior for simple growth morphologies over a

broad range of conditions We have been using this modelto aid in the interpretation of ice growth experiments inorder to extract information about the anisotropic surfaceattachment kinetics that governs ice crystal growth fromwater vapor Modifying the model for application to othermaterial systems is likely straightforward

At present we have limited our model to cylindricallysymmetric crystal growth and relatively simple growth mor-phologies for comparison with measurements A full three-dimensional extension of our model should be possibleas was demonstrated in [10] potentially allowing completemodeling of the many complex branched and faceted struc-tures that are commonly observed in ice crystal growth [7]Achieving this goal would likely require a substantially betterunderstanding of the surface attachment kinetics in ice thanpresently exists along with more advanced algorithms forincorporating all the relevant surface physics into a cellularautomata model

In general it appears that the cellular automata methodis well suited for modeling faceted crystal growth and itmay be the method of choice in material systems exhibitingstrongly anisotropic attachment kinetics Incorporatingmoresophisticated descriptions of surface growth processes willrequire substantially better algorithms than we have outlinedabove especially when extending the models to three dimen-sions Whether such models can reproduce the full rangeof complex structures appearing in faceted crystal growthremains to be seen

Journal of Computational Methods in Physics 11

References

[1] R Trivedi and W Kurz ldquoDendritic growthrdquo InternationalMaterials Reviews vol 39 no 2 pp 49ndash74 1994

[2] E A Brener ldquoThree-dimensional dendritic growthrdquo Journal ofCrystal Growth vol 166 no 1ndash4 pp 339ndash346 1996

[3] I Avramov ldquoKinetics of growth of nanowhiskers (nanowiresand nanotubes)rdquo Nanoscale Research Letters vol 2 no 5 pp235ndash239 2007

[4] H Zhang I Hussain M Brust M F Butler S P Rannard andA I Cooper ldquoAligned two- and three-dimensional structuresby directional freezing of polymers and nanoparticlesrdquo NatureMaterials vol 4 pp 787ndash793 2005

[5] M C Cross and P C Hohenberg ldquoPattern formation outsideof equilibriumrdquo Reviews of Modern Physics vol 65 no 3 pp851ndash1112 1993

[6] K Kassner Pattern Formation in Diffusion-Limited CrystalGrowth World Scientific Singapore 1996

[7] K G Libbrecht ldquoThe physics of snow crystalsrdquo Reports onProgress in Physics vol 68 no 4 pp 855ndash895 2005

[8] J Nelson ldquoGrowth mechanisms to explain the primary andsecondary habits of snow crystalsrdquo Philosophical Magazine Avol 81 no 10 pp 2337ndash2373 2001

[9] H R Pruppacher and J D Klett Microphysics of Clouds andPrecipitation Kluwer Academic Publishers Dordrecht TheNetherlands 1997

[10] J Gravner and D Griffeath ldquoModeling snow-crystal growtha three-dimensional mesoscopic approachrdquo Physical Review Evol 79 no 1 Article ID 011601 18 pages 2009

[11] J W Barrett H Garcke and R Nurnberg ldquoNumerical com-putations of faceted pattern formation in snow crystal growthrdquoPhysical Review E vol 86 no 1 Article ID 011604 14 pages2012

[12] S-Y Hong J Dudhia and S-H Chen ldquoA revised approach toice microphysical processes for the bulk parameterization ofclouds and precipitationrdquoMonthlyWeather Review vol 132 no1 pp 103ndash120 2004

[13] M Matsumoto S Saito and I Ohmine ldquoMolecular dynamicssimulation of the ice nucleation and growth process leading towater freezingrdquo Nature vol 416 no 6879 pp 409ndash413 2002

[14] J G Dash A W Rempel and J S Wettlaufer ldquoThe physicsof premelted ice and its geophysical consequencesrdquo Reviews ofModern Physics vol 78 no 3 pp 695ndash741 2006

[15] A Karma and W-J Rappel ldquoPhase-field method for com-putationally efficient modeling of solidification with arbitraryinterface kineticsrdquo Physical Review E vol 53 no 4 pp R3017ndashR3020 1996

[16] I Singer-Loginova andHM Singer ldquoThe phase field techniquefor modeling multiphase materialsrdquo Reports on Progress inPhysics vol 71 no 10 Article ID 106501 2008

[17] A Schmidt ldquoComputation of three dimensional dendrites withfinite elementsrdquo Journal of Computational Physics vol 125 no2 pp 293ndash312 1996

[18] E Yokoyama ldquoFormation of patterns during growth of snowcrystalsrdquo Journal of Crystal Growth vol 128 no 1ndash4 pp 251ndash257 1993

[19] S E Wood M B Baker and D Calhoun ldquoNew model forthe vapor growth of hexagonal ice crystals in the atmosphererdquoJournal of Geophysical ResearchD vol 106 no 5 pp 4845ndash48702001

[20] K G Libbrecht ldquoOn the equilibrium shape of an ice crystalrdquo2012 httparxivorgabs12051452

[21] G W Sears ldquoA growth mechanism for mercury whiskersrdquo ActaMetallurgica vol 3 no 4 pp 361ndash366 1955

[22] R L Parker R L Anderson and S C Hardy ldquoGrowth andevaporation kinetics and surface diffusion of K and Hg crystalwhiskersrdquo Applied Physics Letters vol 3 no 6 pp 93ndash95 1963

[23] S J Liu H Huang and C HWoo ldquoSchwoebel-Ehrlich barrierfrom two to three dimensionsrdquo Applied Physics Letters vol 80no 18 pp 3295ndash3297 2002

[24] Y Saito Statistical Physics of Crystal Growth World ScientificSingapore 1996

[25] K G Libbrecht ldquoExplaining the formation of thin ice-crystalplates with structure-dependent attachment kineticsrdquo Journal ofCrystal Growth vol 258 no 1-2 pp 168ndash175 2003

[26] K G Libbrecht ldquoPhysically derived rules for simulating facetedcrystal growth using cellular automatardquo 2008 httparxivorgabs08072616

[27] W Thomson ldquoOn the equilibrium vapor at a curved surface ofliquidrdquo Philosophical Magazine vol 42 no 282 pp 448ndash4521871

[28] K G Libbrecht T Crosby and M Swanson ldquoElectricallyenhanced free dendrite growth in polar and non-polar systemsrdquoJournal of Crystal Growth vol 240 no 1-2 pp 241ndash254 2002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

Journal of Computational Methods in Physics 11

References

[1] R Trivedi and W Kurz ldquoDendritic growthrdquo InternationalMaterials Reviews vol 39 no 2 pp 49ndash74 1994

[2] E A Brener ldquoThree-dimensional dendritic growthrdquo Journal ofCrystal Growth vol 166 no 1ndash4 pp 339ndash346 1996

[3] I Avramov ldquoKinetics of growth of nanowhiskers (nanowiresand nanotubes)rdquo Nanoscale Research Letters vol 2 no 5 pp235ndash239 2007

[4] H Zhang I Hussain M Brust M F Butler S P Rannard andA I Cooper ldquoAligned two- and three-dimensional structuresby directional freezing of polymers and nanoparticlesrdquo NatureMaterials vol 4 pp 787ndash793 2005

[5] M C Cross and P C Hohenberg ldquoPattern formation outsideof equilibriumrdquo Reviews of Modern Physics vol 65 no 3 pp851ndash1112 1993

[6] K Kassner Pattern Formation in Diffusion-Limited CrystalGrowth World Scientific Singapore 1996

[7] K G Libbrecht ldquoThe physics of snow crystalsrdquo Reports onProgress in Physics vol 68 no 4 pp 855ndash895 2005

[8] J Nelson ldquoGrowth mechanisms to explain the primary andsecondary habits of snow crystalsrdquo Philosophical Magazine Avol 81 no 10 pp 2337ndash2373 2001

[9] H R Pruppacher and J D Klett Microphysics of Clouds andPrecipitation Kluwer Academic Publishers Dordrecht TheNetherlands 1997

[10] J Gravner and D Griffeath ldquoModeling snow-crystal growtha three-dimensional mesoscopic approachrdquo Physical Review Evol 79 no 1 Article ID 011601 18 pages 2009

[11] J W Barrett H Garcke and R Nurnberg ldquoNumerical com-putations of faceted pattern formation in snow crystal growthrdquoPhysical Review E vol 86 no 1 Article ID 011604 14 pages2012

[12] S-Y Hong J Dudhia and S-H Chen ldquoA revised approach toice microphysical processes for the bulk parameterization ofclouds and precipitationrdquoMonthlyWeather Review vol 132 no1 pp 103ndash120 2004

[13] M Matsumoto S Saito and I Ohmine ldquoMolecular dynamicssimulation of the ice nucleation and growth process leading towater freezingrdquo Nature vol 416 no 6879 pp 409ndash413 2002

[14] J G Dash A W Rempel and J S Wettlaufer ldquoThe physicsof premelted ice and its geophysical consequencesrdquo Reviews ofModern Physics vol 78 no 3 pp 695ndash741 2006

[15] A Karma and W-J Rappel ldquoPhase-field method for com-putationally efficient modeling of solidification with arbitraryinterface kineticsrdquo Physical Review E vol 53 no 4 pp R3017ndashR3020 1996

[16] I Singer-Loginova andHM Singer ldquoThe phase field techniquefor modeling multiphase materialsrdquo Reports on Progress inPhysics vol 71 no 10 Article ID 106501 2008

[17] A Schmidt ldquoComputation of three dimensional dendrites withfinite elementsrdquo Journal of Computational Physics vol 125 no2 pp 293ndash312 1996

[18] E Yokoyama ldquoFormation of patterns during growth of snowcrystalsrdquo Journal of Crystal Growth vol 128 no 1ndash4 pp 251ndash257 1993

[19] S E Wood M B Baker and D Calhoun ldquoNew model forthe vapor growth of hexagonal ice crystals in the atmosphererdquoJournal of Geophysical ResearchD vol 106 no 5 pp 4845ndash48702001

[20] K G Libbrecht ldquoOn the equilibrium shape of an ice crystalrdquo2012 httparxivorgabs12051452

[21] G W Sears ldquoA growth mechanism for mercury whiskersrdquo ActaMetallurgica vol 3 no 4 pp 361ndash366 1955

[22] R L Parker R L Anderson and S C Hardy ldquoGrowth andevaporation kinetics and surface diffusion of K and Hg crystalwhiskersrdquo Applied Physics Letters vol 3 no 6 pp 93ndash95 1963

[23] S J Liu H Huang and C HWoo ldquoSchwoebel-Ehrlich barrierfrom two to three dimensionsrdquo Applied Physics Letters vol 80no 18 pp 3295ndash3297 2002

[24] Y Saito Statistical Physics of Crystal Growth World ScientificSingapore 1996

[25] K G Libbrecht ldquoExplaining the formation of thin ice-crystalplates with structure-dependent attachment kineticsrdquo Journal ofCrystal Growth vol 258 no 1-2 pp 168ndash175 2003

[26] K G Libbrecht ldquoPhysically derived rules for simulating facetedcrystal growth using cellular automatardquo 2008 httparxivorgabs08072616

[27] W Thomson ldquoOn the equilibrium vapor at a curved surface ofliquidrdquo Philosophical Magazine vol 42 no 282 pp 448ndash4521871

[28] K G Libbrecht T Crosby and M Swanson ldquoElectricallyenhanced free dendrite growth in polar and non-polar systemsrdquoJournal of Crystal Growth vol 240 no 1-2 pp 241ndash254 2002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of