Modeling Crystallization Dynamics when the Avrami Model Fails

Citation for this paper:

Gough, T. & Illner, R. (1999). Modeling Crystallization Dynamics when the Avrami Model Fails. VLSI Design, (9)4, 377-383. http://dx.doi.org/10.1155/1999/38517

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Modeling Crystallization Dynamics when the Avrami Model Fails Terry Gough & Reinhard Illner

1999

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Modeling Crystallization Dynamics

when the Avrami Model Fails

TERRY GOUGH andREINHARD ILLNERb’*

DepartmentofChemistry,

_Univbersity

_ofVictoria,P.O. Box3065, Victoria,B.C. V8W3V6, Canada,

e-mail:gought@uvvm.uvic.ca; DepartmentofMathematicsand Statistics, UniversityofVictoria, P.O.Box3045, Victoria,B.C. V8W3P4,Canada

(Received13August1997,"Infinalform1 December1998)

Recentexperimentsonthe formation of crystallineCO2fromanewlydiscovered binary

phase consisting ofCO2and_C2H2at90 Kfail to beadequatelysimulatedbyAvrami

equations. Thepurpose ofthis noteis to develop analternative to the Avramimodel whichcanmake accuratepredictionsfor these experiments. Thenewmodeluses empiri-cal approximations to the distribution densities of the volumes of three-dimensional

VoronoicellsdefinedbyPoisson-generated crystallization kernels(nuclei). Insideeach

Voronoicell,thegrowthofthecrystalisassumedtobelinear in diameter(i.e.,cubic in

volume)untilthe cellisfilledbythe CO2crystalsand the C2H2(thoughtofas awaste product).The cumulativegrowthcurve iscomputed by averagingthese individualgrowth

curves withrespect to thedistributiondensityofthevolumes of the Voronoi cells.

Agree-mentwiththe experimentsisexcellent.

Keywords: Crystallization,Avramimodel, binary phase

1. INTRODUCTION

Crystal growth is of obvious significance in semiconductor design. The classical approach to

model the formation of crystals is based on a

probabilistic argument and known as the Avrami

model [4]. We review the Avrami model in Sec-tion 2 but state here that the Avrami equations predicting conversion to crystal as a fraction of

total availablevolume are

( t)

e-gt

(1,1)

whereKandn are parameters.

The purpose ofthis note is to offer an

alterna-tive totheAvramimodelbasedonstatistical

prop-ertiesofVoronoidiagrams.Itisstraightforwardto see

(and well-known)

that the Voronoi diagrams

*Correspondingauthor,e-mail:rillner@math.uvic.caz

378 T. GOUGHANDR.ILLNER

associated with the crystallization kernels

(impu-rities referred to as nuclei) are important for the crystallization process: Crystals will form as

spherical globules and grow linearly with time in

diameter, hence cubic in volume, around the impurities, until these globules impinge into each

other; the latter occurs exactly at the interfaces

defining the Voronoi diagram, because these

interfaces are defined as being equidistant to the

two closest nuclei. See Figure 1.

One characteristic of Avrami equations is that they predict that regardlessofwhatnis,thelength of time required to convert the first half of the

reactant to product is shorter than the time

required for the second half of the conversion. Onecanargue that the slowing down occurswhen

expanding reaction fronts collide atthe boundary

separating adjacentVoronoicells. After such

colli-sions, the surface at which conversion occurs no

longer growsasthecube of time.

The experimentwhichinspiredusto look foran alternative totheAvramimodelwastheformation

and disintegrationof a metastablecrystalline

bina-ryphase

CO2.C2H2

undersuitable conditions: Mix-tures of these two gases are expanded through a

nozzle onto a zinc selenide window which is

mounted in the beam of an FTIR spectrometer andmaintained at 90K. The pulsedeposits afilm

approximately 200 molecules thick. The obtained

spectra depend markedly onthe conditions ofthe expansion.Ingeneral, absorptionscharacteristicof crystalline

CO2

and crystalline

_C2H2

are observed together with new features assigned to a mixed

phase. For 1:1 mole fractions the absorptions assigned to the purephases are absent, and it was

found that the intensityratioof thenewfeaturesis

independent of the mole fraction of

_CO2.

These

observations establishthe stoichiometry of thenew

phaseas

CO2.C2H

2

(for

moredetails,see

[3]).

Itwasthen observed from spectrarecorded over a

period of5 hours thatthe

CO2.C2H2

decomposes

into

CO2

and

C2H2.

Experimental data for

frac-tional conversion versus time, based on spectral intensities,aregiveninFigure2.

The

_CO2

formed hasasharpspectrumindicating crystallinity, whereas the

C2H2

has broad absorp-tions indicating a more amorphous state. It was

established [3] that the reaction is a solid state transformation. The plotin Figure 2 is sigmoidal, suggesting that an Avrami analysis might be ap-propriate.

However,

the decomposition does not

show the characteristic slowing down discussed earlier:Rewriting

(1.1)

as

(t)

e-:t"

taking logarithms

ln(1

99(t))

Kt and taking logarithms again yields

In(-

ln(1

(t)))

In

K

₊

n

In

t.

_(1.2)

So the left-handsideof

(1.2)

is a linear function in

Int. However,

plotting

ln(-ln(1-converted

frac-tion)) versusIn yields the curvegiveninFigure3,

which is obviouslynot linear.

The curve in Figure 3 indicates that nincreases

smoothly from at the beginning to 4 at its con-clusion.

A

least squaresfit to the overall data

pro-vided avalue ofn 3.3.

The basic idea put forward in this paper is to

produce approximations to the crystal growth

curves

(as

afractionof total

volume)

by averaging

the growth curves associated with the individual Voronoicells withrespectto thestatistical

proper-ties of a Voronoidiagram forwhichthe impurities

aregenerated byaPoissonprocess. Inthe absence

0.8

=

0.6 0 0.4 O o0.2 Decomposition of Carbon dioxide-Acetylene. 0 2 3 Time inHours. Avrami Plot -1.5 -1 -0.5 0 0.5 1.5 In(-In(1-conversion))

380 T. GOUGHAND R. ILLNER

ofother information, assuming that the impurities

are Poisson-distributed is natural.

Ourapproachcontains two inherent difficulties:

1. Thegrowthcurveofthepieceofcrystalinside a Voronoi cell depends on the geometry of that cell. The volumegrowthisassumed to becubic with time until aboundaryisreached,thenslow

down because there cannot be any growth

beyond that boundary, and stop completely once all the corners of the cell are filled. The

endphase of the growthwillthereforevaryfrom celltocell.

2. While the statisticalproperties ofVoronoi cells

are awidely investigated topic 1],theexact prob-ability distribution ofthe volumes ofVoronoi

cells associated with Poisson-distributedpoints

isnotknown.Intwo dimensions,thereare alot

of empiricalstudiesandmatches tothesestudies with variousad hoc approximations, like Gam-madistributions, Maxwell distributions or

log-normal distributions. Werefer to[1] for details.

An

empirical numerical study on the volume

distribution of three-dimensional Voronoi cells generated byaPoisson process isgivenin[2]. Wecannotdealwiththesedifficultiesrigorously;

weavoid thembymaking simplifying assumptions.

Asfor 1., we simplyassume that growthproceeds

as kt3untilthe cellis full, i.e., fora cell ofvolume

a > 0the growthcurveisgivenby

g(a,t)

{

kt3’

O

<_

<

(a/k)

1/3

a,

<_

Here, k denotes the (linear) growth rate of the radius oftheglobuleforminginsidethe cell.

Thisassumptionisclearly simplistic, butthe error is small for Voronoi cellswhich are regularin the

sense that they are well approximated by spheres

(we

refrain from trying to produce a rigorous approximation criterion).

A

mitigating factor in this context is the observation that the

crystal-lizationprocessinvolvestwo products, namely the pure

CO2

crystals,whichgrowfromthe nuclei, and

themoreamorphous C2H2,which will accumulate

attheboundariesandinthecornersoftheVoronoi

cells.

Asfor2.,wetried anumberofdensity distribu-tion funcdistribu-tionsforthe volume of Poisson-generated

Voronoicells.Itwillbe shown below that thereis a choice ofa probability density which reproduces theAvramimodel.

However,

using semi-empirical approximations as suggested in

[1]

and

[2]

pro-duced much better agreement with our

experi-mentaldata.

Theplan of this paper is as follows. We briefly

review the Avrami model in Section 2. InSection

3,weestablish ageneral formula forgrowthcurves

by averaging over the curves given in

(1.3)

with

respect to a. Several examples are discussed, and

one of them reproduces the Avrami model. For

other semi-empirical approximations to the Vor-onoi cell volume distribution, the general formula yields integrals which we evaluated by using the

MAPLE symbolic computation program.We

pre-sentthebest matchweobtained in thiswayto adata

setproducedin

[3]

in Section 4.

2. REVIEW OFTHE AVRAMI MODEL

A

direct derivation of the Avrami model from probabilistic considerationsworks as follows.

Assume

that N nuclei are equidistributed dis-tributed in a(macroscopic) Volume V> 0.Choose

anarbitrary butfixedreference pointPand letXbe thedistance to thenearestnucleus. Xis arandom variable, and

V

(2.1)

so thecumulative distribution function ofXis

FN(X)

P{X

x}

(1

7rx3

1

N

V

In the limit Nee, Voc such that

(N/V)

converges to

F(x)

e-

ax3.

If crystalline globules grow at speed v from each

nucleus, this calculationtranslates directlyintothe probability that crystallizationat time hasreached

the point P:

(t)- P{X < vt}-

1-e-

"3’3.

(2.2)

3. AVERAGED GROWTH CURVES

We now present an alternative approach which contains the Avrami equation

(2.2)

as a special

case. Suppose thatf(a), a

>_

O, is the density dis-tribution function for the volume of the Voronoi

cells (in

3D)

associated with Poisson distributed nuclei. We average the individual growth curves

given by

(1.3)

with respect to

_f

and compute a

macroscopic, observable growthcurve as

c?(t)

g(a,

t)

f

(a)

da kt

L

af

(a)

da

₊

kt3

fkt

_f

(a)

da

(3.1)

The upper limit o in the integrals on the right is

usedbecausewe canassumethat the largest

theoreti-cally possible Voronoi cellis ofmacroscopic scale relative to the typical one. The two-dimensional

analogueto

(3.1)

is k

(’)

f

af

(a)

daq-kl2

Sk

_f

(a)

da.

dO

Formula

(3.1)

is offered as an alternative to the

Avramiequation

(2.2).

Ofcourse,wehavetoknow what

_f

is in order to produce a usable equation. Unfortunately, the true

_f

is not known (but an

object ofstudy, see

[1]);

wediscuss a fewexamples

with ad hocchoicesfor

_f.

Thelastexampleisthen matched to experimental datainthe next section.

Example 3.1 To get a feeling for the type of

curves which

(3.1)

produces, we assume that

_f

is

the equidistribution on an interval [0, A], with A>O,i.e.,

f

(a)

X[0,A]

(a).

For this f, the integration in

(3.1)

can be done explicitly, and yields

p

{

k

237

k2 t6

for kt

<

A

for kt

>

A.

These curves already display the correct logistic-typegrowth,andtheyshowthe kind ofasymmetry aboutthe half-lifepoint (i.e., the point where half

ofthe substance has crystallized) which was seen

inthe experiment described in[3].

Example3.2 The equidistribution usedin

Exam-ple3.1 isclearlynot averyintelligent guess for the volumedistribution of Voronoi cells. Letusmake

a more systematic attempt, following the deriva-tion oftheAvrami model in Section 2.

Assume

thatNnuclei are Poisson distributed ina

(macroscopic) Volume V> 0.Chooseone ofthese

nucleiarbitrary butfixedandletXbe thedistance to its nearestneighbor. Xisarandom variable,and

so the cumulative distribution function ofXis

FN(X)- P{X

_<

_x}-

1-

(1

hence the density

fu(X)

is

fu(X)

F’

N(X)

4r

N-V

4

X3)

N-2 In the limit N-+

,

V-+ such that

(N/V)-+

A

> 0 (intensity of the Poisson process),

fN

382 T. GOUGHAND R.ILLNER

converges to

f

(x)

47r/kx2e

-’xx3

and

_FN

to

F

(x)

e-}’xx3

Weusethesetocompute the densityofthe random

variableS:=volumeof thespherewith radiusX/2; this is the largest sphere which will fit into the Voronoicell centered atthe chosen nucleus.Clearly,

P{S

<_

_s)-P

X

<_

s

e

-8"xs.

The associated densityis

g(s)-

8e

-.

(3.2)

We note forlater reference that

and hence the expected volume occupied by all suchspheresaroundallnuclei is

(N/SA)

(V/8).

So

7/8

ofthe available volume belongs to corners of

Voronoicellsin this sense.

Ifwe use g as obtained in

(3.2)

in the formula

(3.1),

we find kt

(,)

8i ,0

/

ae-8"Xada

_-+-

8/kt3

Skl3

e-8"Xada

8A

(3.3)

and modulo a normalizing factor this is the

Avramiequation

(2.2)

rediscovered.

Example 3.3 The lastexample which we present

uses the density

f(x)-

ctxZe

-/x2,

where a,

/3

are

parameters; they are linked because

_f

has to be a

probability density, so

_f

dependson onlyone free

parameter.Wearrive attheformula

k

o(t)

a

/

x3e-nX2dx

+

akt3

f

x2e-X2dx.

JO

(3.4)

The first integral is easy to compute; the second requires theerrorfunction tobe expressedinclosed form. Symbolicevaluationof

(t)

iseasily doneby

using a symbolic computation package like, e.g., MAPLE.

Weexperimentedwithmany otherdensities.The three given above were chosen for the following

reasons:3.1 isthe simplest example;3.2reproduces the Avrami equation; and 3.2 gave us the best

agreementwithexperiments,asweshowinthefinal section.

4. COMPARISON WITHTHE EXPERIMENT

The density from Example 3.3, with

/3--

1, gave the match to the data points which we present in

Figure4.

The agreement with the experimental data is

striking, suggesting that the assumptions under-lying the modeling processare realistic.

Infrared spectroscopy of the reactant and pro-duct phases shows that thereactantphase and the product carbondioxide are both highly crystalline

o:5

while the product acetylene phase is amorphous.

These observations are consistent withthe product consistingofcrystals ofcarbondioxideembeddedin aless rigid acetylenematrix. This meansthat each Voronoi cell willonly be half fullofrigidmaterial

andthis materialmay be freeto move.

Also, we point out that the density

f(x)-cx2e-x2 appearsto be agood approximation for the densitydistributionofthe volumes ofVoronoi

cellsassociated withPoisson-distributednuclei

(see

[2] for anempiricalstudy).

Wunderlich

[4]

hasreported experimental calori-metricdata for the crystallization ofacopolymer of ethylene terephthalateand ethylenesebacate. This

system resembles the present one in that

crystal-lizationdoesnotproceedto completion andsothe final product consists of crystalline regions

im-bedded inalessrigidamorphousmatrix. Thedata

were subjected to an Avrami analysis and n was

found to be 3.2. Furthermore, the data deviated

from the expectations of the model in the same sense as the present data. We conclude that the

present model would better describe the

crystal-lization of the copolymer, and presumably many

othersystems.

Acknowledgement

This research was supported by grants from the Natural Sciences and Engineering Research Coun-cil ofCanada. The authors are gratefulto Dennis Manke,who produced Figure with his

Voronoi-diagram producingsoftware.

References

[1] Okabe, A., Boots, B. and Sugihara, K. (1992). Spatial

Tesselations: ConceptsandApplications of Voronoi Dia-grams,J. Wiley.

[2] Quine,M. P.andWatson,D.F.(1984).Radial Generation of n-Dimensional Poisson Processes, J. Appl. Prob., 21,

548-557.

[3] Rowat, T. (1997). Stoichiometry and Stability of Binary Phase Crystals formed between Acetylene and Nitrous

Oxide/CarbonDioxide,Ph.D. Thesis,Universityof Victoria.

[4] Wunderlich, B. (1976). Macromolecular Physics, Vol. 2:

Crystal Nucleation, Growth, Annealing, p. 132 ft.

Aca-demicPress.

Authors’Biographies

Terry

Gough was born on October 12, 1939 in Portsmouth, England. He attended the

Ports-mouth Grammar School, and the University of

Leicester where he obtained his B.Sc. and Ph.D. The latter degree was obtained under the

super-vision of Professor M.C.R. Symons, ostensibly studyingion solvationandassociationusing

ultra-violet spectroscopy. In 1965he joined the

Depart-ment ofChemistry at the University ofWaterloo to establish anuclear magnetic resonancefacility.

In 1976 he began a collaboration with Giacinto

Scoles which lead to the formation ofthe Centre for Molecular Beams and Laser Chemistry at the

University ofWaterloo. In 1989 he moved to the University ofVictoria as Chairman ofChemistry where he is currently studying vibrational over-tones ofmolecular beams and the FTIR spectro-scopy oflarge molecular clusters.

Reinhard Illner was born on

January

2, 1950

inWabern,

Germany.

He studied Mathematics at

the Universities ofHeidelberg, Berkeley andBonn and obtained his Ph.D. under the supervision of

Professor J. Frehse in Bonn in 1976. He has held positions at the Universities of

Bonn,

Kaiserslau-tern, Duke University and the University of

Victoria, whereiscurrently full professor andchair of the Mathematicsand Statistics department. His main field of research is the mathematical theory

of Nonequilibrium Statistical Mechanics. He is

co-author of the monograph "The Mathematical