-
VLSI DESIGN1999, Vol. 9, No. 4, pp. 377-383Reprints available
directly from the publisherPhotocopying permitted by license
only
(C) 1999 OPA (Overseas Publishers Association) N.V.Published by
license under
the Gordon and Breach SciencePublishers imprint.
Printed in Malaysia.
Modeling Crystallization Dynamicswhen the Avrami Model Fails
TERRY GOUGH and REINHARD ILLNERb’*
Department of Chemistry, Univbersity of Victoria, P.O. Box 3065,
Victoria, B.C. V8W 3V6, Canada,e-mail: [email protected];
Department of Mathematics and Statistics, University of
Victoria,
P.O. Box 3045, Victoria, B.C. V8W 3P4, Canada
(Received 13 August 1997," In finalform 1 December 1998)
Recent experiments on the formation of crystalline CO2 from a
newly discovered binaryphase consisting of CO2 and C2H2 at 90 K
fail to be adequately simulated by Avramiequations. The purpose of
this note is to develop an alternative to the Avrami modelwhich can
make accurate predictions for these experiments. The new model uses
empiri-cal approximations to the distribution densities of the
volumes of three-dimensionalVoronoi cells defined by
Poisson-generated crystallization kernels (nuclei). Inside
eachVoronoi cell, the growth of the crystal is assumed to be linear
in diameter (i.e., cubic involume) until the cell is filled by the
CO2 crystals and the C2H2 (thought of as a wasteproduct). The
cumulative growth curve is computed by averaging these individual
growthcurves with respect to the distribution density of the
volumes of the Voronoi cells. Agree-ment with the experiments is
excellent.
Keywords: Crystallization, Avrami model, binary phase
1. INTRODUCTION
Crystal growth is of obvious significance insemiconductor
design. The classical approach tomodel the formation of crystals is
based on aprobabilistic argument and known as the Avramimodel [4].
We review the Avrami model in Sec-tion 2 but state here that the
Avrami equationspredicting conversion to crystal as a fraction
of
total available volume are
(t) e-gt (1,1)
where K and n are parameters.The purpose of this note is to
offer an alterna-
tive to the Avrami model based on statistical prop-erties of
Voronoi diagrams. It is straightforward tosee (and well-known) that
the Voronoi diagrams
*Corresponding author, e-mail: [email protected]
377
-
378 T. GOUGH AND R. ILLNER
associated with the crystallization kernels (impu-rities
referred to as nuclei) are important for thecrystallization
process: Crystals will form asspherical globules and grow linearly
with time indiameter, hence cubic in volume, around theimpurities,
until these globules impinge into eachother; the latter occurs
exactly at the interfacesdefining the Voronoi diagram, because
these
interfaces are defined as being equidistant to thetwo closest
nuclei. See Figure 1.One characteristic of Avrami equations is
that
they predict that regardless of what n is, the lengthof time
required to convert the first half of thereactant to product is
shorter than the timerequired for the second half of the
conversion.One can argue that the slowing down occurs when
FIGURE Voronoi cells.
-
MODELING CRYSTALLIZATION DYNAMICS 379
expanding reaction fronts collide at the boundaryseparating
adjacent Voronoi cells. After such colli-sions, the surface at
which conversion occurs nolonger grows as the cube of time.The
experiment which inspired us to look for an
alternative to the Avrami model was the formationand
disintegration of a metastable crystalline bina-ry phase CO2.C2H2
under suitable conditions: Mix-tures of these two gases are
expanded through anozzle onto a zinc selenide window which
ismounted in the beam of an FTIR spectrometerand maintained at 90
K. The pulse deposits a filmapproximately 200 molecules thick. The
obtainedspectra depend markedly on the conditions of theexpansion.
In general, absorptions characteristic ofcrystalline CO2 and
crystalline C2H2 are observedtogether with new features assigned to
a mixedphase. For 1:1 mole fractions the absorptionsassigned to the
pure phases are absent, and it wasfound that the intensity ratio of
the new features isindependent of the mole fraction of CO2.
Theseobservations establish the stoichiometry of the newphase as
CO2.C2H2 (for more details, see [3]).
It was then observed from spectra recorded over aperiod of 5
hours that the CO2.C2H2 decomposesinto CO2 and C2H2. Experimental
data for frac-tional conversion versus time, based on
spectralintensities, are given in Figure 2.The CO2 formed has a
sharp spectrum indicating
crystallinity, whereas the C2H2 has broad absorp-tions
indicating a more amorphous state. It was
established [3] that the reaction is a solid
statetransformation. The plot in Figure 2 is sigmoidal,suggesting
that an Avrami analysis might be ap-propriate. However, the
decomposition does notshow the characteristic slowing down
discussedearlier: 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-hand side of (1.2) is a linear function inInt.
However, plotting ln(-ln(1-converted frac-tion)) versus In yields
the curve given in Figure 3,which is obviously not linear.The curve
in Figure 3 indicates that n increases
smoothly from at the beginning to 4 at its con-clusion. A least
squares fit to the overall data pro-vided a value of n 3.3.The
basic idea put forward in this paper is to
produce approximations to the crystal growthcurves (as a
fraction of total volume) by averagingthe growth curves associated
with the individualVoronoi cells with respect to the statistical
proper-ties of a Voronoi diagram for which the impuritiesare
generated by a Poisson process. In the absence
0.8
= 0.600.4
O
o 0.2
Decomposition ofCarbon dioxide-Acetylene.
0 2 3Time in Hours.
Avrami Plot
-1.5 -1 -0.5 0 0.5 1.5In(-In(1-conversion))
FIGURE 2 The experiment. FIGURE 3 Testing the Avrami model.
-
380 T. GOUGH AND R. ILLNER
of other information, assuming that the impuritiesare
Poisson-distributed is natural.Our approach contains two inherent
difficulties:
1. The growth curve of the piece of crystal inside aVoronoi cell
depends on the geometry of thatcell. The volume growth is assumed
to be cubicwith time until a boundary is reached, then slowdown
because there cannot be any growthbeyond that boundary, and stop
completelyonce all the corners of the cell are filled. Theend phase
of the growth will therefore vary fromcell to cell.
2. While the statistical properties of Voronoi cellsare a widely
investigated topic 1], the exact prob-ability distribution of the
volumes of Voronoicells associated with Poisson-distributed
pointsis not known. In two dimensions, there are a lotof empirical
studies and matches to these studieswith various ad hoc
approximations, like Gam-ma distributions, Maxwell distributions or
log-normal distributions. We refer to [1] for details.An empirical
numerical study on the volumedistribution of three-dimensional
Voronoi cellsgenerated by a Poisson process is given in [2].
We cannot deal with these difficulties rigorously;we avoid them
by making simplifying assumptions.As for 1., we simply assume that
growth proceedsas kt3 until the cell is full, i.e., for a cell of
volumea > 0 the growth curve is given by
g(a,t) { kt3’ O 0 (intensity of the Poisson process), FN
-
MODELING CRYSTALLIZATION DYNAMICS 381
converges to
F(x) e-ax3.If crystalline globules grow at speed v from
eachnucleus, this calculation translates directly into
theprobability that crystallization at time has reachedthe point
P:
(t)- P{X < vt}- 1-e-"3’3. (2.2)
3. AVERAGED GROWTH CURVES
We now present an alternative approach whichcontains the Avrami
equation (2.2) as a specialcase. Suppose that f(a), a >_ O, is
the density dis-tribution function for the volume of the
Voronoicells (in 3D) associated with Poisson distributednuclei. We
average the individual growth curvesgiven by (1.3) with respect to
f and compute amacroscopic, observable growth curve as
c?(t) g(a, t) f(a) da
ktL af(a) da + kt3 fkt f(a) da(3.1)
The upper limit o in the integrals on the right isused because
we can assume that the largest theoreti-cally possible Voronoi cell
is of macroscopic scalerelative to the typical one. The
two-dimensionalanalogue to (3.1) is
k
(’) f af(a) da q- kl2 Sk f(a) da.dOFormula (3.1) is offered as
an alternative to theAvrami equation (2.2). Of course, we have to
knowwhat f is in order to produce a usable equation.Unfortunately,
the true f is not known (but anobject of study, see [1]); we
discuss a few exampleswith ad hoc choices for f. The last example
is thenmatched to experimental data in the next section.
Example 3.1 To get a feeling for the type ofcurves which (3.1)
produces, we assume that f isthe equidistribution on an interval
[0, A], withA>O,i.e.,
f(a) X[0,A] (a).
For this f, the integration in (3.1) can be doneexplicitly, and
yields
p { k 237 k2 t6 for kt < Afor kt > A.These curves already
display the correct logistic-type growth, and they show the kind of
asymmetryabout the half-life point (i.e., the point where halfof
the substance has crystallized) which was seenin the experiment
described in [3].
Example 3.2 The equidistribution used in Exam-ple 3.1 is clearly
not a very intelligent guess for thevolume distribution of Voronoi
cells. Let us makea more systematic attempt, following the
deriva-tion of the Avrami model in Section 2.
Assume that N nuclei are Poisson distributed in a(macroscopic)
Volume V > 0. Choose one of thesenuclei arbitrary but fixed and
let Xbe the distance toits nearest neighbor. X is a random
variable, and
so the cumulative distribution function of X is
FN(X)- P{X _< x}- 1- (1hence the density fu(X) is
fu(X) F’N-
N(X) 4r V4 X3)
N-2
In the limit N-+ , V-+ such that (N/V)-+A > 0 (intensity of
the Poisson process), fN
-
382 T. GOUGH AND R. ILLNER
converges to
f(x) 47r/kx2e-’xx3
and FN to
F (x) e-}’xx3
We use these to compute the density of the randomvariable S :=
volume of the sphere with radius X/2;this is the largest sphere
which will fit into theVoronoi cell centered at the chosen nucleus.
Clearly,
P{S
-
MODELING CRYSTALLIZATION DYNAMICS 383
while the product acetylene phase is amorphous.These
observations are consistent with the productconsisting ofcrystals
ofcarbon dioxide embedded ina less rigid acetylene matrix. This
means that eachVoronoi cell will only be half full of rigid
materialand this material may be free to move.
Also, we point out that the density f(x)-cx2 e-x2 appears to be
a good approximation forthe density distribution of the volumes of
Voronoicells associated with Poisson-distributed nuclei (see[2] for
an empirical study).Wunderlich [4] has reported experimental
calori-
metric data for the crystallization of a copolymer ofethylene
terephthalate and ethylene sebacate. Thissystem resembles the
present one in that crystal-lization does not proceed to completion
and so thefinal product consists of crystalline regions im-bedded
in a less rigid amorphous matrix. The datawere subjected to an
Avrami analysis and n wasfound to be 3.2. Furthermore, the data
deviatedfrom the expectations of the model in the samesense as the
present data. We conclude that thepresent model would better
describe the crystal-lization of the copolymer, and presumably
manyother systems.
Acknowledgement
This research was supported by grants from theNatural Sciences
and Engineering Research Coun-cil of Canada. The authors are
grateful to DennisManke, who produced Figure with his
Voronoi-diagram producing software.
References
[1] Okabe, A., Boots, B. and Sugihara, K. (1992).
SpatialTesselations: Concepts and Applications of Voronoi
Dia-grams, J. Wiley.
[2] Quine, M. P. and Watson, D. F. (1984). Radial Generationof
n-Dimensional Poisson Processes, J. Appl. Prob., 21,548-557.
[3] Rowat, T. (1997). Stoichiometry and Stability of BinaryPhase
Crystals formed between Acetylene and NitrousOxide/Carbon Dioxide,
Ph.D. Thesis, University ofVictoria.
[4] Wunderlich, B. (1976). Macromolecular Physics, Vol.
2:Crystal Nucleation, Growth, Annealing, p. 132 ft. Aca-demic
Press.
Authors’ Biographies
Terry Gough was born on October 12, 1939 inPortsmouth, England.
He attended the Ports-mouth Grammar School, and the University
ofLeicester where he obtained his B.Sc. and Ph.D.The latter degree
was obtained under the super-vision of Professor M.C.R. Symons,
ostensiblystudying ion solvation and association using ultra-violet
spectroscopy. In 1965 he joined the Depart-ment of Chemistry at the
University of Waterlooto establish a nuclear magnetic resonance
facility.In 1976 he began a collaboration with GiacintoScoles which
lead to the formation of the Centrefor Molecular Beams and Laser
Chemistry at theUniversity of Waterloo. In 1989 he moved to
theUniversity of Victoria as Chairman of Chemistrywhere he is
currently studying vibrational over-tones of molecular beams and
the FTIR spectro-scopy of large molecular clusters.
Reinhard Illner was born on January 2, 1950in Wabern, Germany.
He studied Mathematics atthe Universities of Heidelberg, Berkeley
and Bonnand obtained his Ph.D. under the supervision ofProfessor J.
Frehse in Bonn in 1976. He has heldpositions at the Universities of
Bonn, Kaiserslau-tern, Duke University and the University
ofVictoria, where is currently full professor and chairof the
Mathematics and Statistics department. Hismain field of research is
the mathematical theoryof Nonequilibrium Statistical Mechanics. He
isco-author of the monograph "The MathematicalTheory of Dilute
Gases", which appeared in 1994.
-
International Journal of
AerospaceEngineeringHindawi Publishing
Corporationhttp://www.hindawi.com Volume 2010
RoboticsJournal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume
2014
Hindawi Publishing Corporationhttp://www.hindawi.com Volume
2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume
2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttp://www.hindawi.com Volume
2014
Hindawi Publishing Corporation http://www.hindawi.com
Journal ofEngineeringVolume 2014
Submit your manuscripts athttp://www.hindawi.com
VLSI Design
Hindawi Publishing Corporationhttp://www.hindawi.com Volume
2014
Hindawi Publishing Corporationhttp://www.hindawi.com Volume
2014
Shock and Vibration
Hindawi Publishing Corporationhttp://www.hindawi.com Volume
2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttp://www.hindawi.com Volume
2014
Hindawi Publishing Corporationhttp://www.hindawi.com Volume
2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation http://www.hindawi.com
Volume 2014
The Scientific World JournalHindawi Publishing Corporation
http://www.hindawi.com Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume
2014
Modelling & Simulation in EngineeringHindawi Publishing
Corporation http://www.hindawi.com Volume 2014
Hindawi Publishing Corporationhttp://www.hindawi.com Volume
2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume
2014
Hindawi Publishing Corporationhttp://www.hindawi.com Volume
2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume
2014
DistributedSensor Networks
International Journal of