ON THE GENERALISATION OF JMAK's THEORYPaulo R. Rios1,a, Elena Villa2,b

1Escola de Engenharia Industrial Metalurgica, Universidade Federal FluminenseAv. dos Trabalhadores, 420, Volta Redonda, RJ, 27255-125, BRASIL

2University of Milan, Dept. of Mathematics, via Saldini 50, 20133, Milano, Italyaprrios@id.uff.br, belena.villa@unimi.it

Keywords: microstructure, analytical methods, phase transformations, recrystallisation, cellular au-tomata, Stochastic Geometry

In their pioneer work Johnson-Mehl, Avrami and Kolmogorov (JMAK) developed well-known ana-lytical expressions to describe the transformation kinetics of nucleation and growth transformations.Their work and its subsequent development has been extensively applied by Vandermeer and cowork-ers to recrystallisation studies.With the help of methods from Stochastic Geometry it has been possibleto generalise JMAK's equations to situations well beyond their original assumptions. Although themo-tivation for this paper was recrystallisation the expressions derived here may be applied to nucleationand growth reactions in general.

IntroductionKolmogorov[1], Johnson-Mehl[2], Avrami[3], KJMA, formal kinetics methodology has been appliedto a diversity of transformation phenomena. Vandermeer and coworkers[4] substantially developedthe KJMA work building on the concept of microstructural path. Vandermeer and coworkers usedtheir Microstructural Path Method, MPM, extensively in many recrystallisation studies.

JMAK theory assumed that nuclei were uniform randomly distributed in space and that the velocityof the moving boundary was the same for every growing grain. Recently, Rios and Villa applied math-ematical methods from Stochastic Geometry and Geometric Measure Theory, that allow considerablegeneralisation of these assumptions. As a consequence, new analytical expressions were obtained. Al-though the motivation for this paper was recrystallisation the expressions derived here may be appliedto nucleation and growth reactions in general. For nuclei uniform randomly located in space JMAKshowed that the volume fraction transformed, VV (t), could be related to VE(t) by

VV (t) = 1− exp (−VE(t)) (1)

Two exact analytical expressions are normally associated with this early work. In the first of these thenucleation is supposed to be site-saturated, whichmeans that all possible nucleation sites are exhaustedat the very beginning of the reaction. One often describes this by saying that all nuclei are alreadypresent at t = 0. If the number of nuclei per unit of volume is NV , then

VV (t) = 1− exp(−4π

3NVG

3t3)

(2)

The other expression is for constant nucleation rate. For reasons of space we will concentrate on thesite-saturated case throughout this paper.

Mathematical background Set theory is the language of abstract mathematics. Therefore in order tobe able to use powerful mathematical results it is necessary to express intuitive microstructural con-cepts in terms of set theory. By doing this we can define and generalise well-known microstructuraldescriptors, such as “volume fraction". First, nucleation and growth is defined from what in Mathe-matics is usually called a birth-and-growth (stochastic) process. This is a dynamic germ-grain model[5], used to model situations in which nuclei (germs) are born in time and are located in space ran-domly, and each nucleus generates a grain evolving in time according with a given growth law. Since,

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in general, nucleation and growth are random in time and space, then the transformed region, Θt, atany time t > 0 is a random set [5] in R3. A mean volume density, VV (t, x), may be defined for thisrandom set, and it is proved that

VV (t, x) = P(x ∈ Θt) (3)

where P(x ∈ Θt) is the probability that a point x belongs to the transformed region in the instant t.The mean volume density can be integrated over a region A to yield the volume fraction transformedin that region, VV(t, A)

VV(t, A) =

∫AVV (t, x)dx

ν3(A)(4)

where ν3(A) is the volume of A or more precisely the 3-d Lebesgue measure of A. The volume frac-tion will be independent of A, i. e. VV (t) = VV(t, A), if the underlying point process is stationary,meaning invariant under translation. VV (t) is the usual “volume fraction" employed in JMAK theory.The obvious advantage of VV (t, x) over VV (t) is that VV (t, x) is able to describe inhomogeneous dis-tributions of the transformed region. Similar definitions apply to SV (t, x), the mean volume interfacialarea density of the transformed region as well as their corresponding extended quantities: VE(t, x) andSE(t, x), the mean extended volume density and the mean extended interfacial area density, respec-tively.

Spatial location of nuclei

The concept of point process formalises the idea of “nuclei location in space". Thus, the idea thatnuclei are uniform randomly located in space(Fig. 1 left) is equivalent to say that nuclei are locatedaccording to a homogeneous Poisson point process. Fig. 1 shows other examples of point processesthat can be used to describe nuclei locations other than uniform randomly. Fig. 1, middle, shows aninhomogeneous Poisson point process in which the nuclei density decreases from top to bottom. Fig.1 right shows nuclei located in clusters according to a Matern cluster process[5].

A particularly simple formula[6] may be obtained if the intensity, λ(x), is a harmonic function: itis twice continuously differentiable and it satisfies the Laplace's equation,∆f = 0. For site-saturationone has

VV (t, x) = 1− exp(−4πλ(x)G3t3

3

)(5)

Eq. (5) holds when the new phase is spherical, the nuclei are located according to a Poisson pointprocess with intensity λ(x) and grows with a constant velocity, G. Eq. (5) can be seen as a generali-sation of Eq. (2). It is possible[6] to generalise this expression for time-dependent velocity, G(t) andtime-dependent nucleation, λ(t, x), provided λ(t, x) remains harmonic with respect to x.

Fig. 1: Point processes: left: homogeneous Poisson ; middle: inhomogeneous Poisson; right: Materncluster process. The number of nuclei in Fig. 1 left and middle and right is the same: 100 nuclei.

138 Recrystallization and Grain Growth V

Another possibility is that nuclei may be located in space according to a Matern cluster process.The important parameters[7] in a Matern cluster process are: a)λp, the mean number of clusters perunit of area; b)nc, mean number of points per clusters and c)R, the cluster radius. Again, for sphericalgrains growing with a constant velocity, G, and site-saturated nucleation. The analytical expressioncannot be given for reasons of space but an example is depicted in the figure shown on Fig. 2. Fig. 2shows the effect of cluster radius for λp = 125 and nc = 10. The mean total number of nuclei per unitof volume is simply λpnc = 1250. There are two limiting situations. The first is when R → 0, all nc

points collapse to one single point and therefore the transformation approaches that of a homogeneousPoisson with intensity equal to λp = 125. The other limit is when the R → ∞. In this situation allclusters overlap and the transformation approaches that of a homogeneous Poisson with intensity equalto λpnc = 1250. Curves obtained for uniform randomly located nuclei with intensities(NV ) equal toλp = 125 and λpnc = 1250 are also shown in Fig. 2. It is interesting to remark that the classical “ln lnagainst ln" plot does not generally give straight lines for clusters. For details see [7].

Small specimens

General expressions for the mean volume density and the volume fraction were derived in [8] for bothsurface and bulk nucleation in a general Borel subset of R3. In practice, probably any specimen shapeof engineering interest is going to be a Borel set. An expression was also derived for the important caseof polyhedral shape in which surface nucleation may take place on the faces, edges and vertices of thepolyhedron as well as within the bulk. Moreover, explicit expressions were given for surface and bulknucleation for three specific shapes of engineering relevance: two parallel planes, an infinitely longcylinder and a sphere.

Fig. 3 shows the mean volume density for the transformation nucleating on the surface of a unitsphere, R = 1, with intensity λS = 10. Starting from the left hand side of the plot each curve corre-sponds to, from left to right(r is the distance between a spherical shell and the origin): r = 1 (long-dashline on the extreme left), r = 0.75(medium dash line), r = 0.5 (short dash line), r = 0.25 (dot line) andr = 0.0 (dash-dot line) on the extreme right. The solid black line corresponds to the volume fraction,VV(t). Although the volume fraction changes smoothly with time it is clear that the transformation ishighly heterogeneous. A point close to the sphere centre may take a while to transform as it needs “towait" for the transformation front to reach it.

0 50 100 150 2000.0

0.5

1.0

Poisson p

R=0.025 R=0.050 R=0.200 Poisson nc p

VV

time

Fig. 2: Mean volume density transformed as afunction of time for cluster radii ranging from0.025 to 0.200 units of length.

0 200 400 600 800 10000.0

0.2

0.4

0.6

0.8

1.0

time

mea

nvo

lum

ede

nsity

Fig. 3: Surface nucleation: mean volume den-sity, for the transformation nucleating on thesurface of a unit sphere.

Materials Science Forum Vol. 753 139

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Fig. 4: Mean volume density as a function oftime for transformations nucleated at randomparallel planes. The solid curve corresponds tothe theoretical solution and the full circles to thecomputer simulation.

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Fig. 5: The straight line without any symbolson it corresponds to the KJMA kinetics for aconstant, non-random velocity. The solid linescorrespond to the theoretical result. The sym-bols correspond toGodiksen et al.[13] simulateddata.

Planes and lines

Villa and Rios revisited a problem treated by Cahn in 1956[9]. They obtained a somewhat surprisingresult. Their mean volume density for site-saturation nucleation on random parallel planes was

VV (t) = 1− exp

−2SplanesV Gt

1∫0

{1− e−πλSG

2t2[1−z2]}dz

(6)

This is identical to that obtained by Cahn's methodology supposing that the planes are “randomly"distributed in space. It is not intuitive that such an expression should hold when the planes are parallelbut indeed this is precisely the case.

In fact, Villa and Rios new derivation has shown that Cahn's expression for transformations nucle-ated on random planes does not depend on the orientation of these planes but rather on the fact thanthe number of planes contained within a random sample of unit volume is a Poisson random variable.In other words it is valid if one has a Poisson plane process. As a result, Cahn's expression is valid forrandom parallel planes, a somewhat surprising result. Computer simulation illustrates this.

Figure 4 shows the influence of the number of nuclei per unit of area on the kinetics of the trans-formation nucleated on random parallel planes. The nucleation was site-saturated, i .e. all nucleationsites were exhausted at t = 0. The solid curve corresponds to the theoretical solution, Eq.(6), and thefull circles to the computer simulation.

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0.0

0.2

0.4

0.6

0.8

1.0

time HsL

vo

lum

efr

actio

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Fig. 6: Comparison between model(dashedline) and volume fraction(filled squares) ob-tained from experimental data of γ-fiber orien-tations.

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Fig. 7: Comparison between model(dotted line)and volume fraction(filled triangles) obtainedfrom experimental data of “other" orientations.

140 Recrystallization and Grain Growth V

Random velocity

In a recent review of recrystallisation experimental 3-d studies, Juul Jensen and Godiksen concludedthat there is compelling evidence to support that “every single grain has its own kinetics different fromthe other grains"[10]. Rios and Villa[11, 12] derived general expressions to deal with this situation.The velocity of each grain[11, 12] was assumed to be time-dependent and random of the type G(t) =G0g(t, α) with G0 and α non-negative independent random variables with probability distributionQ1(dξ) = q1(ξ)dξ and Q2(da) = q2(a)da, respectively.

When λ(x) is harmonic in the spatial region where the nucleation takes place, the mean volumedensity, VV (t, x), is given by

VV (t, x) = 1− exp{− 4

3πλ(x)E[G3

0]E[( t∫

0

g(τ, α)dτ)3]}

(7)

Where E[G30] is the expected value of G3

0. It is worthy of note that for a random G = G0, Eq. (7)becomes VV (t, x) = 1− exp

(−4

3πλ(x)E[G3

0]t3), which further generalises Eq. (2).

Simultaneous and sequential reactions

Two or more transformations may take place simultaneously but independently of each other. Oneexample of interest is the evolution of two or more texture components during recrystallisation. Moregenerally, the reactions might not start at the same time. For example, reaction 1 starts at t = 0whereasreaction 2 starts at t = τ > 0. These may be called sequential transformations, for reasons of space wewill not deal with these in this paper.Modelling simultaneous or sequential reactions normally involvestwo distinct but closely related objectives: 1)Predict the overall kinetics quantities of the combinedreactions from analytical models developed for each individual reaction; 2) Extract the kinetics for anindividual reaction from experimental measurements.

Recently[14], Rios and Villa derived exact analytical expressions to model simultaneous and se-quential reactions. Rios et al.[15] used these results to model the kinetics of recrystallisation of anIF steel studied by Magnussen et al.[16]. The main result is depicted in Figs. 6 and 7. As demon-strated elsewhere, the kinetics of the γ-fiber and of other texture components may be described as-suming site-saturated uniform randomly nucleation. Summarising, employing this new methodologyquantities suitable for theoretical analysis may be obtained from quantities measured experimentally.Conversely, experimentally measurable quantities may be predicted from theoretical ones.

Summary and Conclusions

The application of rigourous mathematical methods allowed generalising JMAK’s original expressionto situations well beyond their original assumptions. The generalised equations can deal with nucleilocated in space in many ways not only uniformly and randomly; transformations in small specimens,transformations nucleated on planes and lines; the situation in which the growth velocity is randomlydistributed throughout the growing regions, and when there are simultaneous and sequential transfor-mations.

Acknowledgments

P. R. Rios is grateful to the Conselho Nacional de Desenvolvimento Cientifico e Tecnologico, CNPq,and to the Fundacao deAmparo a Pesquisa do Estado do Rio de Janeiro, FAPERJ, for financial support.

Materials Science Forum Vol. 753 141

References

[1] N. A. Kolmogorov: Isvestiia Academii Nauk SSSR - Seriia Matematicheskaia Vol. (1937), p.333.

[2] W. A. Johnson, R. F. Mehl: Trans. AIME Vol. 135 (1939), p. 416.

[3] M. J. Avrami: J. Chem. Phys. Vol. 7 (1939), p. 1103.

[4] R. A. Vandermeer, R. A. Masumura and B. B. Rath: Acta Metall. Mater. Vol. 39 (1991) p. 383.

[5] D. Stoyan, W. S. Kendall and J. Mecke, Stochastic Geometry and its Application, 2nd edition,Wiley, Chichester, 1995.

[6] P. R. Rios and E. Villa: Acta Mater. Vol. 57 (2009), p. 1199.

[7] E. Villa and P. R. Rios: Acta Mater. Vol. 57 (2009), p. 3714.

[8] E. Villa and P. R. Rios: Acta Mater. Vol. 58 (2010), p. 2752.

[9] J. W. Cahn: Acta Metall. Vol. 4 (1956),p. 449.

[10] D. J. Jensen and R. Godiksen: Metall. Mater. Trans. A Vol. 39A (2008) p. 3065.

[11] P. R. Rios and E. Villa: Scr. Mater. Vol. 65 (2011) p. 938.

[12] E. Villa and P.R. Rios: Image Anal. Stereol. , in press.

[13] R.B. Godiksen, S. Schmidt and D. J. Jensen: Scr. Mater. Vol. 57 (2007) p. 345.

[14] P. R. Rios, E. Villa: Acta Mater. Vol. 58 (2011) p. 2752.

[15] P. R. Rios, E. Villa and S. C. Oliveira: Mater. Sci. Forum Vols. 706-709 (2012) p.149.

[16] H. Magnusson, D. J. Jensen, B. Hutchinson: Scr. Mater. Vol.44 (2001) p. 435.

142 Recrystallization and Grain Growth V

Recrystallization and Grain Growth V 10.4028/www.scientific.net/MSF.753 On the Generalisation of JMAK's Theory 10.4028/www.scientific.net/MSF.753.137

DOI References

[3] M.J. Avrami:J. Chem. Phys. Vol. 7(1939), p.1103.

http://dx.doi.org/10.1063/1.1750380