Cell size distribution in a random tessellation of space governed by the Kolmogorov-Johnson-Mehl-Avrami model: Grain size distribution in crystallization
aa r X i v : . [ c ond - m a t . m t r l - s c i ] O c t Cell size distribution in a random tessellation of space governedby the Kolmogorov-Johnson-Mehl-Avrami model: grain sizedistribution in crystallization
Jordi Farjas ∗ and Pere Roura † GRMT, Department of Physics, University of Girona,Campus Montilivi, Edif. PII, E17071 Girona, Catalonia, Spain (Dated: October 22, 2018)
Abstract
The space subdivision in cells resulting from a process of random nucleation and growth is asubject of interest in many scientific fields. In this paper, we deduce the expected value and varianceof these distributions while assuming that the space subdivision process is in accordance with thepremises of the Kolmogorov-Johnson-Mehl-Avrami model. We have not imposed restrictions onthe time dependency of nucleation and growth rates. We have also developed an approximateanalytical cell-size probability density function. Finally, we have applied our approach to thedistributions resulting from solid phase crystallization under isochronal heating conditions.
PACS numbers: 81.30.-t, 81.10.Jt, 05.70.Fh, 02.50.EyKeywords: Cell size distribution; crystallization; grain size distribution; Kolmogorov-Johnson-Mehl-Avrami . INTRODUCTION In this paper we consider the subdivision of a D -dimensional Euclidean space into disjointregions created after a process of random nucleation and growth. Random subdivisions canbe obtained by several different methods, amongst which Poisson-Voronoi and Johnson-Mehl tessellations have been widely studied. The Poisson-Voronoi tessellation is obtainedby randomly picking several points, the seeds P i , by a Poisson process. Next, the spaceis subdivided in cells, C i , by the rule: C i contains all points in space closer to P i than toany other seed. This cellular structure is extensively applied in many diverse scientific fieldsincluding biology, computer science, materials science, astrophysics, medicine, agriculture, quantum field theory, and sociology. The space tessellation can be fully characterized by means of the probability densityfunction (PDF), f ( s ), which is the probability that a cell has a size between s and s + ds .The properties of the PDF of the Poisson-Voronoi tessellation have been extensively studiedboth theoretically and numerically. It is well known that the Poisson-Voronoi tessellation PDF is described by a gamma distribution f ( s ) = (cid:16) νE (cid:17) ν ν ) s ν − exp (cid:16) − νE s (cid:17) (1)where Γ is the gamma function, ν is a parameter that is dependent on the dimension D , i.e. ν = 2, 3.584 and 5.586 for D = 1, 2 and 3 respectively, and E is the expected cell size, E ≡ Z ∞ sf ( s ) ds (2)It is worth mentioning that Eq. (1) has been analytically derived for the one-dimensionalcase where ν =2 is an exact result. Conversely, for the two and three-dimensional cases,the validity of Eq. (1) is supported by analytical approximations and numerical fits.Our main interest is the characterization of grain morphology related to crystallization.In general, the crystallization of most materials takes place by means of a nucleation andgrowth mechanism: nucleation starts with the formation of small atom clusters of the newstable phase in the metastable phase. Subsequently, clusters with sizes greater than thecritical, or nuclei, start to grow by incorporating neighboring atoms of the metastable phase.During this growth, grains impinge upon each other. Finally, the structure of the new stablephase consists of disjoint regions or crystals separated by grain boundaries. The evolution2f crystallization and grain size distributions is entirely determined by the nucleation ratedensity I and the grain linear growth rate G . When nucleation takes place for a very shorttime, its rate may vanish before the onset of particle growth ( site saturated nucleation ). In this case, the crystal structure is equivalent to a Poisson-Voronoi tessellation providedthat nucleation is Poissonian through the whole space and growth is isotropic.Conversely, continuous nucleation takes place when nucleation and growth occur at thesame time. In general, there is an energy barrier for nucleation and growth to happen.Thus, I and G depend on temperature. For the particular case of isotropic and isothermaltransformations, where I and G are constant, the resulting crystal structure corresponds tothe well-known Johnson-Mehl tessellation. For this tessellation, Axe et al. have obtainedan analytical solution for the one-dimensional case while Mulheran has developed a simple(but not so accurate) relation for the two- and three-dimensional cases. Alternatively, Monte-Carlo simulations provide a powerful tool for the calculation of tessellations and PDFs undera wide variety of conditions. Under non-isothermal conditions, I and G depend on time by virtue of their tempera-ture dependence. Therefore, an infinite number of different tessellations/structures can beobtained by varying the thermal history. Unlike the Poisson-Voronoi and Johnson-Mehltessellations, the analytical results related to tessellations emerging from time dependentnucleation and growth rates are scarce. Indeed, as far as we know, the analytical models arelimited to time dependent nucleation rates. In particular, Jun et al. have derived ananalytical solution for the one-dimensional case. Particularly relevant to the present workare the results of Pineda et al. who have obtained an accurate analytical description forthe two- and three-dimensional cases.In the present work we will consider those transformations that fulfill the Kolmogorov-Johnson-Mehl-Avrami (KJMA) premises. No restrictions will be imposed on nucleationand growth rate time dependence. We will refer to these tessellations as KJMA tessel-lations . KJMA theory has been widely applied to describe systems undergoing first-orderphase transformations. For instance, DNA replication; crystallization of polymers, amor-phous materials and glasses; switching in ferroelectrics and ferromagnets; lattice-gas models; and film growth on solid substrates. In Section II we will describe the basicconcepts of KJMA theory and will focus our attention on those aspects that are useful tothe development of our work. Section III is devoted to the calculation of the expected value3nd variance of the distributions related to the KJMA tessellations. In Section IV we willderive a simpler approximate relation for the variance and will check its accuracy. As anapplication of the previous results, in Section V we will derive an approximate grain sizePDF which is the superposition of gamma distributions. Finally, at the end this sectionwe will verify that the grain radius PDF can be expressed as well as the superposition ofGaussian distributions.
II. THE KOLMOGOROV-JOHNSON-MEHL-AVRAMI THEORY
The KJMA theory describes in a very simple form the kinetics of transformationsgoverned by nucleation and growth that satisfy the following assumptions:i nucleation must be Poissonian through the entire space;ii the volume of an arbitrary grain is much smaller that the volume of the system;iii the crystal growth rate is isotropic.On the basis of these premises, Kolmogorov calculated the evolution of the transformedfraction, X ( t ), through the probability, p ( t ), that an arbitrary point O has not crystallized,i.e., the probability that no nuclei able to transform O will be formed during the time interval[0 , t ], X ( t ) = 1 − p ( t ) , (3a) p ( t ) = exp (cid:20) − g D Z t I ( τ ) r ( t, τ ) D dτ (cid:21) , (3b) r ( t, τ ) ≡ Z tτ G ( z ) dz, (3c)where g D is a geometrical factor related to the shape of the crystal – for a D -dimensionalsphere g D = π D/ / Γ( D/ r ( t, τ ) is the minimum distance between O and a nucleuscreated at τ , so that the nucleus would not transform O .Based on geometrical arguments, Avrami deduced the following relation: ∂ t v ( t, τ ) ∂ t v ex ( t, τ ) = 1 − X ( t )1 − X ( τ ) , (4)where ∂ t v ( t, τ ) and ∂ t v ex ( t, τ ) are respectively the actual and extended average volumetricgrowth rate at time t for grains nucleated at time τ . The word extended refers to the4olume a grain would attain if nuclei grew through each other and overlapped withoutmutual interference.The integration of Eq. (4) leads to dX ( t )1 − X ( t ) = dX ex ( t ) . (5)Finally, integration of Eq. (5) gives Avrami’s well-known formula X ( t ) = 1 − exp [ − X ex ( t )] . (6)The calculation of X ex ( t ) is straightforward and obtained by simply neglecting the impinge-ment between nuclei X ex ( t ) = g D Z t I ( τ ) r ( t, τ ) D dτ. (7)The combination of Eqs. (6) and (7) gives Eq. (3). As it is well known, Avrami and Kol-mogorov deduced the same relation using different approaches.Note that in Eq. (7) it is assumed that the nucleation rate is not affected by the shrinkingof the untransformed phase. In the calculation of X ex ( t ) the phantom nuclei are takeninto account. Avrami designated as phantom nuclei those nuclei that are formed in thetransformed fraction and therefore do not contribute to the formation of new grains. Indeed,the actual nucleation rate can be defined as I a ( t ) ≡ [1 − X ( t )] I ( t ) . (8)Concerning the limitations of the KJMA theory, it also holds in the case of anisotropicgrowth provided that the grains have a convex shape and are aligned in parallel. Moreover,the KJMA theory provides a good approximation when the anisotropy is moderate or forsoft impingement. However, KJMA theory fails when nucleation is non-random, whengrowth is anisotropic, when growth stops before crystallization is complete and whenthe incubation time is not negligible. III. STATISTICAL PROPERTIES OF THE KJMA CELL SIZE DISTRIBUTION
The cell size distribution is characterized by its PDF, f ( s ), the probability that a cell hasa size between s and s + ds . From its definition it is obvious that f ( s ) must be normalized Z ∞ f ( s ) ds = 1 . (9)5o analyze the properties of the cell size distribution, we will consider the contribution ofthe crystals formed at a time τ over a time interval dτ ( τ -crystals). We will call the cellsize distribution of the τ -crystals the τ -distribution. Accordingly, we define the PDF of the τ -crystals, f τ ( s ), as the probability that a τ -crystal has a volume between s and s + ds .From the definition of f τ ( s ), it is also apparent that f τ ( s ) must be normalized: Z ∞ f τ ( s ) ds = 1 . (10) f ( s ) is simply the addition of the contributions of the τ -crystals over the time interval inwhich their nucleation takes place f ( s ) = R ∞ I a ( τ ) f τ ( s ) dτ R ∞ I a ( τ ) dτ . (11)Note that the denominator in Eq. (11) ensures that f ( s ) is normalized if all f τ ( s ) arenormalized.In the following sections, we will present the expected grain size and the variance of the cellsize distribution and their relationship with the equivalent parameters of the τ -distributions. A. Expected grain size
It is well known that the expected grain size, E , is the inverse of the final grain density: E = (cid:18)Z ∞ I a ( τ ) dτ (cid:19) − . (12)Likewise, the expected value, E τ of a τ − distribution is simply the final average grain sizeof a τ -crystal normalized to the total volume: E τ = Z ∞ τ ∂ z v ( z, τ ) dz. (13)Introducing Eq. (4) into Eq. (13) leads to E τ = 11 − X ( τ ) Z ∞ τ [1 − X ( z )] ∂ z v ex ( z, τ ) dz, (14)where the extended average growth rate is given by ∂ z v ex ( z, τ ) = Dg D r ( z, τ ) D − G ( z ) . (15)Note than once the evolution of the transformed fraction, X ( t ), is known – i.e. thesolution of Eq. (3) – the calculation of E τ is straightforward.6esides, the final space fraction occupied by the τ -crystals, X τ , can be calculated fromthe integration over the entire space of the probability that a point P in the space belongs toa tau crystal nucleated at O . Since the system is homogeneous and isotropic, this probabilityonly depends on the distance b between O and P . Therefore, X τ = I ( τ ) Z P τ ( O, P ) dV P == Dg D I ( τ ) Z ∞ P τ ( b ) b D − db, (16)where P τ ( b ) is the probability that a point P , separated by a distance b from the nucleus O , belongs to the crystal nucleated at O . To calculate P τ ( b ), we will use the same approachthat Kolmogorov used for the deduction of Eq. (3). Since nucleation is Poissonian, P τ ( b ) isgiven by the probability that no nucleus is formed that could transform P before O does so. P would be transformed by O at the moment t b : b = r ( t b , τ ) = Z t b τ G ( z ) dz. (17)Thus, the nuclei formed at z that could transform P before O does so, are located in a D -sphere of radius r ( t b , z ) around P . Therefore, according to Eq. (3), P τ ( b ) is given by P τ ( b ) = exp (cid:20) − g D Z t b I ( z ) r ( t b , z ) D dz (cid:21) . (18)The previous integral spans the time interval [0 , t b ] since no nucleus formed after t b couldtransform P . Comparison of Eq. (7) with Eq. (18) gives P τ ( b ) = exp [ − X ex ( t b )] = 1 − X ( t b ) . (19)Finally, if we introduce the value of P τ ( b ) given by Eq. (18) into Eq. (16) and we changethe variable b by t b , we obtain X τ = Dg D I ( τ ) Z ∞ τ [1 − X ( t b )] r ( t b , τ ) D − G ( t b ) dt b . (20)Alternatively, the expected value, E τ is the ratio between the space fraction occupied bythe τ -crystals and the density of τ -crystals: E τ = X τ dτI a ( τ ) dτ . (21)7s expected, substitution of Eqs. (20) and (8) into Eq. (21) delivers Eq. (14). Moreover,the integration of X τ over the whole time interval where nucleation takes place gives thetotal transformed fraction, 1: Z ∞ I a ( τ ) E τ dτ = Z ∞ X τ dτ = 1 . (22)We will end this subsection verifying that the value of E evaluated from the τ -distributionscoincides with the value given at the beginning of this subsection [Eq. (12)]: E ≡ Z ∞ sf ( s ) ds = R ∞ I a ( τ ) (cid:0)R ∞ sf τ ( s ) ds (cid:1) dτ R ∞ I a ( τ ) dτ == R ∞ I a ( τ ) E τ dτ R ∞ I a ( τ ) dτ = 1 R ∞ I a ( τ ) dτ . (23) B. Variance of the grain size distribution
To determine the variance we will adapt the development of Gilbert for a Poisson-Voronoitessellation to our case. First, we define a new PDF, f ∗ ( s ), as the PDF of the crystals thatcontain a given arbitrary point O : i.e., if we pick an arbitrary point O , f ∗ ( s ) is the probabilitythat a crystal has a size between s and s + ds and contains the point O . Accordingly, f ∗ ( s )is proportional to f ( s ) and to s , because a large crystal has a proportionally greater chanceof containing the point O . Therefore, f ∗ ( s ) = s f ( s ) E . (24)The constant of proportionality, E − , has been deduced by imposing normalization: Z ∞ f ∗ ( s ) ds = 1 . (25)From the definition of f ∗ ( s ) it can be easily proved thatvar = Z ∞ ( s − E ) f ( s ) ds = E ∗ E − E (26)where E ∗ is the expected value of f ∗ ( s ).Once E ∗ is known, the calculation of the variance is simple. To obtain E ∗ we first analyzethe contribution of the τ -crystals. To do so, we define f ∗ τ ( s ) as the PDF of the τ -crystalsthat contain a given arbitrary point O : i.e., if we pick an arbitrary point O , f ∗ τ ( s ) is the8robability that a τ -crystal has a volume between s and s + ds and contains the point O .Accordingly, f ∗ τ ( s ) is proportional to f τ ( s ) and to s : f ∗ τ ( s ) ∝ f τ ( s ) s. (27)On the other hand, the integration of f ∗ τ ( s ) over all possible volumes is the probabilitythat an arbitrary point O belongs to a τ -crystal. This probability is the fraction of the spaceoccupied by the τ -crystals X τ : Z ∞ f ∗ τ ( s ) ds = X τ , (28)taking into account that, E τ = Z ∞ sf τ ( s ) ds, (29)and combining Eqs. (27), (28) and (21) we obtain f ∗ τ ( s ): f ∗ τ ( s ) = I a ( τ ) s f τ ( s ) . (30)Then the expected value of f ∗ τ , E ∗ τ , is E ∗ τ = R ∞ sf ∗ τ ( s ) ds R ∞ f ∗ τ ( s ) ds . (31)It can be easily verified that E ∗ is related with E ∗ τ through E ∗ = R ∞ sf ∗ ( s ) ds R ∞ f ∗ ( s ) ds = Z ∞ X τ E ∗ τ dτ. (32)Therefore the contribution of the τ -crystal to the expected value E ∗ is X τ E ∗ τ . In addition,this contribution is the integration over the entire space of the probability that a differentialvolume around a point P in the space belongs to the same τ -crystal as O . Since the systemis homogeneous and isotropic, this probability only depends on the distance b between O and P , X τ E ∗ τ = D g D Z ∞ P ∗ τ ( b ) b D − db, (33)where P ∗ τ ( b ) is the probability that two points, O and P , separated by a distance b belongto the same τ -crystal, P ∗ τ ( b ) = I ( τ ) Z P ∗ τ ( b, Q ) dV Q . (34)The integration domain covers the entire space, dV Q is the D -volume differential arounda point Q , I ( τ ) dV Q dτ is the probability that a nucleus is formed at Q at the time τ during9 (t ,z) O r(t ,z) P O PbV I Qr O r P q O FIG. 1: Schematic drawing of the calculation of P ∗ τ ( b, Q ). the time interval dτ and P ∗ τ ( b, Q ) is the probability that both O and P belong to the samecrystal nucleated at Q (see Fig. 1). For D = 2, dV Q = 2 r O dr O dθ O while for D = 3, dV Q = 2 πr O sin( θ O ) dr O dθ O in polar coordinates. P ∗ τ ( b, Q ) is given by the probability that no nucleus is formed that could transform O or P before Q does, then P ∗ τ ( b, Q ) = exp n − g D h Z t O I ( z ) r ( t O , z ) D dz ++ Z t P I ( z ) r ( t P , z ) D dz − Z t ′ I ( z ) V I g D dz io , (35a) r P = r O + b − r O b cos θ O , (35b) r x = r ( t x , τ ) = Z t x τ G ( z ) dz, for x = O, P (35c) r O + r P − b r ( t ′ , τ ) = Z t ′ τ G ( z ) dz, (35d)where r ( t O , z ) and r ( t P , z ) are the minimum distance between O , P and a nucleus created atthe time z , so that the nucleus would not transform O and P respectively (see Fig. 1). V I isthe volume intersection between two D -spheres of radius r ( t O , z ) and r ( t P , z ) centered at O and P , respectively (gray region in Fig. 1). The subtraction of the term V I is in accordancewith the fact that it has been accounted twice in the first and second integrals in Eq. (35a).For a particular set of values of the integration variables r O and θ O , r P is evaluated fromEq. (35b) while t O and t P are defined by Eq. (35c) and t ′ is defined by Eq. (35d). Notethat O and P are transformed by Q at the times t O and t P , respectively. Thus, any nucleus10ormed after t O and t P could not transform O or P , respectively, so the two first integralsin Eq. (35a) span the time interval [0 , t O ] and [0 , t P ], respectively. Additionally, it can beeasily verified that if z > t ′ , then the intersection between the D -spheres is null. Therefore,the last integral in Eq. (35a) spans the time interval [0 , t ′ ].Finally, Eq. (35) is simplified by substitution of Eq. (7) in the first and second integralsin Eq. (35a) P ∗ τ ( b, Q ) = exp [ − X ex ( t O ) − X ex ( t P )]exp h − R t ′ I ( z ) V I dz i = (cid:2) − X ( t O ) (cid:3)(cid:2) − X ( t P ) (cid:3) exp h − R t ′ I ( z ) V I dz i . (36)It can be easily proved that the variance of the τ -distributions, var τ , is given byvar τ = E ∗ τ E τ − ( E τ ) . (37)Finally, we will check if the variance of the distribution determined from the decomposi-tion of f ( s ) into τ -PDF, Eq. (11), gives the expected result, Eq. (26):var = R ∞ (cid:0)R ∞ s I a ( τ ) f τ ( s ) ds (cid:1) dτ R ∞ I a ( τ ) dτ − E == E (cid:20)Z ∞ (cid:18)Z ∞ sf ∗ τ ( s ) ds (cid:19) dτ (cid:21) − E == E (cid:20)Z ∞ E τ dτ (cid:21) − E = E ∗ E − E . (38)At this point, we would like to point out that the results obtained so far are exact andgeneral, i.e., we have not made any assumption concerning f ( s ) and f τ ( s ). IV. APPROXIMATE VARIANCE
According to our previous analysis, the exact calculation of the variance is reduced to thecalculation of the parameters E and E ∗ in Eq. (26). While the calculation of E is straight-forward, the evaluation E ∗ is more cumbersome. Indeed, when compared to Monte-Carloalgorithms, its numerical calculation is more complex without representing any significantreduction in computing time. That is because there are several integrals nested and, inparticular, the calculation of the intersection volume V I is complex. When r ( t O , z ) ≫ b or11 ≫ r ( t O , z ), V I tends towards being a D -sphere of radius r ( t O , z ) and 0, respectively. Onthe other hand, when r ( t O , z ) ≈ r ( t P , z ) the shape of V I roughly approaches a D -sphere.Since the width of V I (see Fig. 1) is r ( t O , z ) + r ( t P , z ) − b , we approximate V I by a D -sphereof diameter r ( t O , z ) + r ( t P , z ) − b : V I ≈ g D (cid:18) r ( t O , z ) + r ( t P , z ) − b (cid:19) D . (39)It is worth noting that the previous approximation also works for the limiting cases r ( t O , z ) ≫ b and b ≫ r ( t O , z ). Furthermore, for the one-dimensional case it can be easilyverified that Eq. (39) is exact (in Appendix A we derive P ∗ τ for D = 1). Finally, theapproximate solution (from here on approximation I) is obtained by substitution of Eqs. (7)and (39) into Eq. (36): P ∗ τ ( b, Q ) = exp [ − X ex ( t O ) − X ex ( t P )]exp [ − X ex ( t ′ )]= (cid:2) − X ( t O ) (cid:3)(cid:2) − X ( t P ) (cid:3)(cid:2) − X ( t ′ ) (cid:3) . (40)Therefore, the calculation of P ∗ τ ( b, Q ) is simple provided that the evolution of the trans-formed fraction, X ( t ), is known. Analytical exact solutions for X ( t ) are restricted to threeparticular situations under isothermal conditions: time-independent growth and nucleationrates, time-independent growth rate and nucleation rate proportional to a power of time, and site saturated nucleation. A quasi-exact solution of the KJMA model has recently beenobtained under continuous heating conditions. Moreover, there are numerical methodswhich allow a simple and fast calculation of X ( t ) for an arbitrary time dependence of thenucleation and growth rates. We have analyzed the distribution emerging from solid phase crystallization underisochronal heating conditions, i.e. heating at a constant rate, to check the accuracy of ap-proximation I , Eq. (40), in the case of time dependent nucleation and growth rates. To workwith realistic parameters we have taken those of amorphous silicon crystallization, inwhich the nucleation and growth rates are described by an Arrhenius temperature depen-dence I = I exp (cid:18) − E N K B T (cid:19) and G = G exp (cid:18) − E G K B T (cid:19) . (41)where T is the temperature in Kelvin and k B is the Boltzmann constant. In Table I wesummarize the corresponding parameters. When the temperature is raised at a constant12 ABLE I: Experimental nucleation and growth rates of amorphous silicon (Ref. 38,39,56).Nucleation Activation energy, E N I . × s − m − Growth Activation energy, E G G . × s − m rate β , the nucleation and growth rates become time dependent through Eq. (41). Underthose conditions, the kinetics is correctly described by the KJMA theory and there isgood agreement between experiment and theoretical predictions. For the calculation ofthe evolution of the transformed fraction, we have used the quasi-exact solution describedin Ref. 55. The numerical evaluation of the integrals has been performed by means of anextended midpoint algorithm. To confirm that the observed discrepancies are not relatedto numerical inaccuracies, we have performed several calculations with consecutive smallerintegration steps. Moreover, for the numerical integration over a semi-infinite interval, wehave imposed a minimum relative error of 10 − . To check the accuracy of the numericalcalculation, we have calculated the integral of X τ over the interval [0 , ∞ ) and have comparedthem to its predicted value, Eq. (22). Calculations that exhibit discrepancies larger than10 − were rejected.As is apparent from Fig. 1, the approximation of V I by a D -sphere of diameter equal toits width, Eq. (39), results in an underestimation of V I , which leads to an undervaluationof E ∗ τ and of var τ . The latter conclusion can be verified in Fig. 2, where the evolution of E ∗ τ and var τ with τ is shown. Although approximation I gives an accurate value of E ∗ τ , theapproximate value of var τ shows a significant deviation from the exact value. The reason isthat in the evaluation of var τ , Eq. (37), both terms in the difference have similar values. Thesame happens to the values of the variance and E ∗ ; the exact and approximate values of E ∗ are 3.93 and 3.69 respectively, while the exact and approximate variances are 3.56 and 3.22,respectively. (Space has been normalized to the space scaling factor ( G ( T P ) /I ( T P )) / ,where T P is the peak temperature.) Despite the significant discrepancy between the exactand approximate values of var τ , they have a nearly parallel evolution with τ . This resultis general and is related to the very similar dependency of the approximate and exact V I on the integration parameters. Therefore, the accuracy of approximation I can be analyzed13 Exact Approximation I v a r E* FIG. 2: E ∗ τ and var τ for three-dimensional growth and isochronal heating of 40 K/min. Time andspace have been normalized according to the time and space scaling factors, ( I ( T P ) G ( T P ) ) − / and ( G ( T P ) /I ( T P )) / , where T P is the peak temperature, i.e., the temperature at which thetransformation rate is maximum (see Ref. 35). The exact (black solid line) and the approximate(red dashed line) values are compared. Exact
Approximation I v a r EN/EG e x a c t v a r / app r o x i m a t e v a r FIG. 3: Exact (black triangles) and approximate (red circles) values of var and their ratio (solidblue squares) as a function of the ratio between nucleation and growth activation energies. through the relation between the exact and approximate values of var . To cover a widerange of distributions, we will recall the results given in Ref. 35. In this work it was shownthat the shape of the grain size PDF was practically insensitive to the heating rate, butit depends mainly on the ratio E N /E G , i.e., the relative evolution of the nucleation andgrowth rates with time. The limit E N /E G → E N /E G = 1 coincides with the isothermal case.14n Fig. 3 we have plotted the exact and approximate values of var as well as their ratio.(At this point it is worth recalling that, according to Eq. (41), a relation of one order ofmagnitude between the activation energies E N and E G would result in a huge difference inthe relative time evolution between the nucleation and growth rates.) First, we can easilyverify that var (and in general the variance) decreases with E N /E G . This means that thedistributions become broader as E N /E G increases. Indeed, when E G ≫ E N , during thefirst stages of the transformation, nucleation dominates over growth. Most of the nuclei areformed at the beginning and they grow at a slow rate. Thus, the average grain size and itsvariance diminishes when E N /E G diminishes. In contrast, when E N ≫ E G , during the firststages of crystallization, growth dominates and the nucleation rate increases progressively ascrystallization proceeds. Since the time left for growth is less for the nuclei that appear later,the density of small grains will be higher than that of larger grains. So the average grain sizeand the distribution variance increase with E N /E G . On the other hand, despite the largevariation of var , the total variation of the ratio between the exact and the approximatevar is very smooth – from 1.96 to 2.12. A similar behavior has been observed for the 2D-case where this rate evolves from 1.28 at E N /E G = 10 to 1.35 at E N /E G = 0. Hence, thedeviations of the approximate value of var from the exact value remain practically constant.This result is due to approximation I , which is based on a geometrical approach that is fairlyinsensitive to the relation between nucleation and growth rates.Since the ratio between the exact and the approximate var τ is nearly constant, we canobtain a significantly more accurate approximate value for var τ by simply multiplying itby the corresponding proportionality constant. This constant only depends on the growthdimensionality. We have chosen the values of 2.07 and 1.32 for the 3D and 2D cases,respectively. These values correspond to E N /E G = 1, i.e., they correspond to the isothermalcase with E G and E N constant in time. With this correction (from now on approximationII ), the relative error in the calculation of the variance diminishes to less than 2% andfor E N /E G ≥ E ∗ with respect to the ratio E N /E G . The exact and the approximate value obtained from approximation II of thevariance and E ∗ exhibit excellent agreement; the values overlap in such a way that they arenearly indistinguishable. Concerning the values obtained from approximation I , it is worthnoting that despite the significant error related to the calculation of var [Fig. 3] and of var τ ExactApproximation IApproximation II v a r i an c e / E EN/EG E * / E FIG. 4: Exact (black triangles), approximate (red circles) and corrected approximate (blue squares)values of the variance and E ∗ and as a function of the ratio between the nucleation and growthactivation energies. in general, the inaccuracies in the evaluation of the variance and E ∗ are significantly smaller.The reason is that both parameters depend exclusively on E τ and E [Eqs. (26) and (32)].From Fig. 2 it is clear that the error related to E τ is significantly smaller than the error inthe evaluation of var τ , while the calculation of E is exact.From now on, when will always use approximation II in the calculation of the approximatevalues of E ∗ and E ∗ τ . V. APPROXIMATE CELL SIZE PROBABILITY DENSITY FUNCTION
One application of the preceding analysis of the statistical properties of grain size dis-tribution is the derivation of a PDF. If we choose a set of f τ ( s ) such that their expectedvalue coincides with the result of Eq. (14) and their variance is equal to the value given byEq. (37), then the variance and expected values of the PDF obtained from Eq. (11) will beexact, i.e., the PDF obtained from Eq. (11) will have the same variance and expected valuesas the actual PDF. Indeed, Pineda et al., apply this approach in the case of tessellationsgenerated by random nucleation processes where the growth rate was assumed to be con-stant. The agreement between their approximate PDF and Monte-Carlo simulations wasremarkable. However, they did not notice that the expected value and the variance of theirapproximate PDF were exact. Concerning the particular choice of the f τ ( s ) functions, we16 Approximation III Approximation IV Monte-CarloD=3 f ( s ) • E s/E FIG. 5: Grain size distribution for three-dimensional growth and isochronal heating. Comparisonbetween the PDFs obtained from the exact calculation of E ∗ τ (black solid line), from the approxi-mate calculation of E ∗ τ (red dashed line) and from Monte-Carlo simulation (empty squares). will consider two cases: gamma and Gaussian distributions. A. Gamma distribution
Given that in a τ -distribution the nucleation events are simultaneous and the τ -nuclei arerandomly distributed, we can assume that f τ ( s ) is similar to the PDF resulting from a processof site saturated nucleation, i.e. that the PDF is that of a Poisson-Voronoi tessellation.As explained in the introduction, the gamma distribution is the exact PDF for the one-dimensional case while it provides a very accurate result for the two- and three-dimensionalcases: f τ ( s ) = (cid:18) ν τ E τ (cid:19) ν τ ν τ ) s ν τ − exp (cid:18) − ν τ sE τ (cid:19) , Since E τ is already the expected value, the problem comes down to the determination ofthe exponent ν τ . Indeed, ν τ can be calculated from the following property of the gammadistributions ν τ = ( E τ ) var τ . (42)To check the accuracy of the PDF we compare them to some Monte-Carlo simulations.The Monte-Carlo algorithm consists in dividing the space into a cubic lattice. Cells areassigned to nuclei randomly. The nucleation time of each nucleus is precisely calculated from17 Approximation III Approximation IV Monte-CarloD=2 f ( s ) • E s/E FIG. 6: Grain size distribution for two-dimensional growth and isochronal heating. Comparison be-tween the PDFs obtained from the exact calculation of E ∗ τ (black solid line), from the approximatecalculation of E ∗ τ (red dashed line) and from Monte-Carlo simulation (empty squares). the nucleation rate and it is recorded to evaluate the exact evolution of the grain growth.Finally, each cell is assigned to the nucleus that first reaches this cell. The evolution ofthe grain growth transformation is checked whenever the grain growth is equal to the sizeof a cell in order to avoid incorrect cell assignation due to shielding effects. In particularwe consider the crystallization of amorphous silicon under isochronal heating at 40 K/min(Table I). The results of the calculations for three- and two-dimensional growth are given inFigs. (5) and (6), respectively. We have calculated the PDF using the exact, Eq. (36), andapproximate, Eq. (40), values of E ∗ τ . We will refer to these PDFs as the approximate III and approximate IV PDFs, respectively. The validity of the selection of a gamma distributionfor the f τ ( s ) functions is confirmed by the good agreement between the calculated PDFand the Monte-Carlo simulations. The excellent agreement between the approximate III and the approximate IV PDFs is also noteworthy – the relative difference is less than 0.1%for s/E > .
05. Therefore, the approximate calculation of E ∗ τ is useful to obtain a simpleand accurate PDF for a KJMA tessellation. However, the complexity and the computingtime required for their evaluation is significantly different. For instance, the calculationof the approximate III PDF typically takes more than thirty times the time required forthe calculation of the approximate IV
PDF. The reason for this significant simplificationand reduction in computing time is that the approximate IV
PDF saves us from having to18
Approximation III Approximation IV Monte-Carlo f ( s ) • E s/E3D, site saturated nucleation FIG. 7: Grain size distribution for three-dimensional growth and site saturated nucleation. Com-parison between the PDFs obtained from the exact calculation of E ∗ τ (black solid line), from theapproximate calculation of E ∗ τ (red dashed line) and from Monte-Carlo simulation (empty squares). evaluate the inner and more complex integral of Eq. (36).To confirm this last conclusion, we have calculated the PDF for the case of site saturatednucleation . This case corresponds to E N /E G = 0 in Figs. (3) and (4) and is the case thatexhibits the greatest discrepancy between the exact and approximate values of the varianceand E ∗ . Therefore, it should give the worst agreement between approximate III and the approximate IV PDFs. As is apparent from Fig. (7) here again the agreement between thePDFs obtained from the exact calculation of E ∗ τ , from the approximate calculation of E ∗ τ andfrom Monte-Carlo simulation is excellent. Only small deviations of the approximate III fromthe approximate IV PDF are distinguishable for s ≈ E . Finally, for the one-dimensionalcase and for site saturated nucleation, both the approximate III and approximate IV PDFsturn into the exact PDF (see Appendix B).
B. Gaussian distribution
When analyzing the crystallization morphology, it is often better to use the grain radiusdistribution instead of the grain size distribution. The grain radius, r , of a grain of size s , is defined as the radius that will have a D -sphere of volume s . For instance, for three-19imensional growth r ≡ r s π . (43)Given the grain size PDF, the grain radius PDF, g ( r ), can be easily derived; for three-dimensional growth, g ( r ) = 4 πr f (cid:18) πr (cid:19) . (44)On the other hand, for site saturated nucleation and three-dimensional growth it hasbeen shown that g ( r ) is accurately described by a Gaussian distribution g ( r ) = 1 √ πσ exp (cid:18) − ( x − E g ) σ (cid:19) , (45)where E g is the expected value of the grain radius PDF and σ is the standard deviation σ = √ var g . (46)where var g is the variance of the grain radius distribution. The fitted parameters wereactually E g = 0 . /n / and σ = 0 . /n / , where n is the nuclei density.Conversely, g ( r ) can be determined by means of Eq. (44) where f ( s ) is a gamma distri-bution with ν = 5 . Under this approach, it can easily be proved that E g = (cid:18) π Eν (cid:19) / Γ( ν + 1 / ν )var g = (cid:18) π Eν (cid:19) / (cid:20) Γ( ν + 2 / ν ) − Γ( ν + 1 / Γ( ν ) (cid:21) , (47)where E = 1 /n is the expected value of the grain size distribution (see Appendix B).By substituting ν = 5 .
581 into the previous relations we obtain E g = 0 . /n / , var g =7 .
65 10 − /n / and σ = 0 . /n / , which are in good agreement with the parametersobtained from the Gaussian fit.Therefore, for a KJMA tessellation we can obtain g ( r ) as the superposition of g τ ( r )PDFs. In contrast with the previous subsection, we will now assume that g τ ( r ) are Gaussiandistributions: g ( r ) = R ∞ I a ( τ ) g τ ( r ) dτ R ∞ I a ( τ ) dτ . (48)To evaluate the g τ ( r ) PDFs we need to know their expected value, E g,τ , and their vari-ance, var g,τ . These parameters can be easily derived from the statistical parameters of the τ -distributions, E τ and var τ , by means of Eq. (47). In Fig. (8) we have plotted the grain20 Gamma distribution Gaussian distribution Monte-Carlo g (r) • E g r/E g FIG. 8: Grain radius distribution for three-dimensional growth and isochronal heating. Comparisonbetween the PDF obtained from Eq. (44) (black solid line), Eq. (48) (red dashed line) and Monte-Carlo simulation (empty squares). radius distribution obtained, from the grain size distribution [Eq. (44)] (where f ( s ) is thesuperposition of gamma distributions), as a direct superposition of g τ ( r ) Gaussian PDFs[Eq. (48)] and from Monte-Carlo simulations. This distribution corresponds to the crystal-lization of amorphous silicon under isochronal heating at 40 K/min (Table I). The goodagreement between the PDF calculated from Eqs. (44) and (48) confirms that the g τ ( r ) arecorrectly described by a Gaussian distribution. Finally, both approaches show good agree-ment with Monte-Carlo simulations, i.e., both approaches are useful for describing the grainradius PDF. VI. CONCLUSIONS
This paper deals with a subject of interest in many scientific areas, namely the cell-size distribution of space tessellations that emerge from first-order phase transformationsruled by nucleation and growth of the new stable phase. No restrictions are imposed onthe time dependency of the nucleation and growth rates, and the validity of our results islimited to transformations that obey the premises of the Kolmogorov-Johnson-Mehl-Avramimodel. We have derived some important statistical properties such as the expected valueand the variance. The approach used is an extension of the work of Gilbert. Additionally,21e have developed a significantly simpler relation for the calculation of the variance. Thediscrepancies between the exact and approximate variances are less than 2%.Like Pineda et al., we have derived an approximate grain size PDF as the superpositionof gamma distributions. We have proved that the expected value and variance derived fromthis approximate grain size PDF are exact. Moreover, we have checked its accuracy againstMonte-Carlo simulations for a system undergoing a crystallization under isochronal heatingconditions. The results show a remarkably good agreement between the approximate PDFand the Monte-Carlo simulations. Finally, we have shown that the grain radius PDF can beexpressed as the superposition of Gaussian distributions. APPENDIX A: ONE-DIMENSIONAL PDF
For the one dimensional case, the extended transformed fraction is X ex ( t ) = 2 Z t I ( τ ) (cid:20)Z tτ G ( z ) dz (cid:21) dτ, (A1)and the expected value E τ becomes E τ = 21 − X ( τ ) Z ∞ τ [1 − X ( z )] G ( z ) dz. (A2)With regard to the calculation of P ∗ τ ( b ), we have split the entire space into three regions.The first corresponds to Q located at the left side of O , in this case r P = r O + b and P ∗ τ ( b, Q ) = exp[ − X ex ( t P )] . (A3)The second region corresponds to Q located between O and P , then r P + r O = b and P ∗ τ ( b, Q ) = exp[ − X ex ( t O ) − X ex ( t P ) + X ex ( τ )] . (A4)And the third region corresponds to Q located at the right side of P , r O = r P + b and P ∗ τ ( b, Q ) = exp[ − X ex ( t O )] . (A5)Finally, combining Eqs. (A3), (A4) and (A5) with (34) we obtain P ∗ τ ( b ) = I ( τ ) n Z ∞ b [1 − X ( t O )] dr O ++ 11 − X ( τ ) Z b [1 − X ( t O )][1 − X ( t P )] dr O o , O = Z t O τ G ( z ) dz,b − r O = Z t P τ G ( z ) dz. (A6)Once P ∗ τ ( b ) is known, Eq. (33) combined with Eq. (37) delivers var τ . APPENDIX B: SITE SATURATED NUCLEATION
When nucleation is completed prior to crystal growth, the nucleation rate can be approx-imated to I ( t ) ≈ n δ ( t ) where n is the density of nuclei and δ is the Dirac delta function.In this case, the extended transformed fraction becomes X ex ( t ) = n g D (cid:20)Z t G ( z ) dz (cid:21) D . (B1)Then I a ( τ ) = n [1 − X ( τ )] δ ( τ ) (B2)and the PDF, Eq. (11), is reduced to f ( s ) = R ∞ n [1 − X ( τ )] δ ( τ ) f τ ( s ) dτ R ∞ n [1 − X ( τ )] δ ( τ ) dτ = f ( s ) . (B3)Therefore, in this case we only need to calculate E and var . Concerning E , Eq. (14), itcan be easily proved that it is simply E = 1 n Z ∞ exp[ − X ex ] dX ex = 1 /n . (B4)Indeed, according to Eq. (B3) E = E , and the expected value of E is 1 /n [Eq. (12)].Moreover, the fraction of space occupied by a τ -crystal, X τ , is reduced to δ ( τ ), as expected.With respect to var , its value is determined by E ∗ and E by means of Eq. (37). E ∗ is given by Eq. (33). We therefore need to evaluate P ∗ ( b ), the value of which depends onwhich relation for P ∗ ( b, Q ) we use: the exact one Eq. (36) or the approximate one Eq. (40).We will evaluate P ∗ ( b ) for the one-dimensional case because in this case the approximatesolution coincides with the exact one. Specifically, substitution of Eq. (B1) into Eq. (A6)leads to P ∗ ( b ) = n δ ( τ ) h Z ∞ b e − n r O dr O + Z b e − n b dr O i == n δ ( τ ) e − n b (1 /n + b ) . (B5)23hen, from Eqs. (33) and (B5) E ∗ = 32 1 n . (B6)And finally, from Eqs. (37) and (B6), var = (2 n ) − . (B7)If we choose a gamma distribution for the calculation of the cell-size PDF, from Eq. (42) weobtain ν = 2. Thus for site-saturated nucleation and one-dimensional growth we obtain agamma distribution with ν = 2 and E = 1 /n which agrees with the exact solution. ACKNOWLEDGMENTS
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