modeling crystallization dynamics whenthe avrami model fails · vlsidesign 1999, vol. 9, no. 4, pp....

VLSI DESIGN1999, Vol. 9, No. 4, pp. 377-383Reprints available directly from the publisherPhotocopying permitted by license only

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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.


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


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


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


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.

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