Computer simulation of crystallization kinetics for the model with simultaneous nucleation of randomly-oriented ellipsoidal crystals

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Journal of Non-Crystalline Solids 171 (1994) 141-156

Computer simulation of crystallization kinetics for the model with simultaneous nucleation of randomly-oriented ellipsoidal crystals M . P . S h e p i l o v a,,, D . S . B a i k b a Research and Technological Institute o f Optical Materials, All-Russian Scientific Center 'S.1. Val,ilot, State Optical hzslitute ', 193171 St Petersburg, Russian Federation b Korea Electrotechnology Research Institute, Sungju-Dong 28-1, Changwon, Kyungnam 641-120, South Korea

Received 23 November 1993; revised manuscript received 17 January 1994

Abstract Methods of calculation of crystallization kinetics are discussed. The Kolmogorov model and some other models are reviewed. It is concluded that the problems of theoretical description of crystallization kinetics need further discussion. Computer simulation of crystallization kinetics for the non-Kolmogorov model with simultaneous nucleation of randomly-oriented ellipsoidal crystals has been carried out. From the results of this computer simulation, it follows that the Kolmogorov equation is practically exact for this model if the crystal anisotropy is not high (if the ratio, a, of the large principal diameter of ellipsoid to the small one is ~<2). At higher anisotropy (a/> 5), the results of computer simulation cannot be described by the Kolmogorov equation with sufficient accuracy; in particular, the kinetics do not correspond to the constant value n = 3 of the Avrami exponent, and the dimensionality of crystal growth cannot be established on the basis of Avrami's analysis. The results should be kept in mind in the discussion of experimental data on crystallization kinetics in the cases of growth of anisotropic crystals.

1. Introduction Experimental investigation of crystallization kinetics and comparison of the data with theoretical results calculated for various models of crystallization help one to understand the laws of nucleation and growth of crystals. However, in many cases (in particular for various processes of glass crystallization) the crystallization kinetics cannot be described within the framework of the existing theoretical models. Thus the problem of

* Corresponding author. Tel: +7-812 355 9046. Telefax: +7812 560 1022.

calculation of crystallization kinetics needs further discussion. For a description of crystallization kinetics, Kolmogorov [1] proposed a model (here this model will be called the K-model) and derived an equation for the non-crystallized volume fraction which is exact in this model. Unfortunately, the derivation of this equation has been given briefly without full proof, which led to numerous mistakes in application of the Kolmogorov method and the Kolmogorov equation. A strict and detailed analysis of the K-model and a discussion of related topics were given by Belenkii [2]. Because Refs. [1,2] are practically unknown to Englishspeaking readers, we present the description and

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some properties of the K-model in accordance with the analysis given by Belenkii [2]. The K-model is based on the following assumptions (premises). (A1) Crystallization is considered in a nonlimited medium. This means that the results of the theory can be applied to description of experiment if the mean size of crystals in a crystallized sample is small by comparison with the sizes of the sample. (A2) Nucleation of crystals begins at time t = 0 and occurs in a non-crystallized region; the probability of nucleation of one crystal in a non-crystallized volume, V, in the time interval, At, is

a(t)VAt +o(At)

(1)

and the probability of nucleation of two crystals is o(At), where o(At)/At ~ 0 at At -* 0, i.e., o(At) is an infinitesimal of higher order by comparison with At. The nucleation rate per unit volume, c~(t), is assumed to be independent of the coordinates. (A3) Before impingements, crystals have a geometrically similar and convex shape and the same spatial orientation; growth of crystals ceases at the points of mutual impingements, although it continues without change elsewhere. Hence, an isolated crystal is fully determined by its size which is measured from the point of nucleation along any fixed direction. (4) At any time, t, the growth rate in a fixed direction is the same for all the crystals, which continue to grow in this direction without impingements, and depends only on the time: v(t). Thus at time t an isolated crystal nucleated at time t' has a size R ( t ' , t) = fti'v(t") dt"

(2)

in this direction. Taking into account also assumption (A3), the volume of this crystal may be written as

where V0 is a shape factor. The spatial region occupied by such a crystal centered at a point X will be denoted as Z(X, t', t).

On the basis of these four assumptions, Kolmogorov [1] calculated the volume fraction of untransformed materials, q(t), as the probability of non-crystallization of a random point at time, t, and obtained the exact expression

q(t, =exp[- fota(t')V(t', t)d/'].

(4)

Two particular cases were noted by Kolmogorov [1]: (a) if the nucleation rate, a, and the growth rate, v, are independent of time, then

q(t) =exp[-¼Voau3t 4]

(a-model);

(5)

(b) if crystals are formed simultaneously at t = 0 with a mean number density, /3, and the growth rate, u, is constant, then

q(t) = exp[-Vo/3C3t 3]

(13-model).

(6)

The latter equation may be obtained by application of the Kolmogorov approach [1] to the case of simultaneous nucleation or by direct substitution of the nucleation rate a(t)=/360) in the general Eq. (4) [2] ( 6 0 ) is the Dirac 6-function). So some authors (see, for example, Refs. [3,4]) make a mistake when they declare that the case of simultaneous nucleation does not belong to the Kolmogorov model. Now let us give the main items of the Belenkii analysis [2] of the K-model. In the following discussion, it is suitable to use the term 'aggressor' for a crystal which is able to crystallize a given point, X, until a given time, t, in the absence of obstacles [2]. The region Z*(X, t', t), obtained by inverting the region Z(X, t', t) (see text below Eq. (3)) with respect to the point X, represents a region of possible nucleation of aggressor at time t'. Really, the distance from the point X to any point Y belonging to the region Z*(X, t', t) is smaller than or equal to the size of region Z(X, t', t) measured from the point X in the direction opposite to ~ ; therefore the region Z(Y, t', t), which may be obtained as a result of the translation )[~i7applied to the region Z(X, t', t) (in accordance with the assumptions (A3) and (A4)), contains the point X and, in the absence of obstacles, a crystal nucleated at point Y at time t' will crystallize the point

M.P. Shepiloc, D.S. Baik /Journal o f Non-Crystalline Solids 171 (1994) 141-156

X until time t. Similarly, a crystal nucleated at time t' outside the region Z * ( X , t', t) cannot be an aggressor. Note that the shape of the region Z * ( X , t', t) in the K-model is independent of the location of the point X because of assumption (A1). Two statements have been proved by Belenkii [2, pp. 19, 20] for the K-model which are a key to the derivation of the Kolmogorov Eq. (4): ($1) Nucleation of aggressor guarantees the crystallization of the point X until time t; in other words, a non-aggressor cannot screen the point X from an aggressor. ($2) In the absence of aggressors until time t' (t' < t), the region Z * ( X , t', t) of possible nucleation of aggressor at time t' is non-crystallized. The proof of statements ($1) and ($2) is based on a geometrical analysis of the laws of crystal growth which are determined by assumptions (A3) and (A4). In the Kolmogorov approach, the noncrystallized volume fraction, q(t), is calculated as the probability of non-crystallization of a random point X until time t. The statement ($1) means that the point X is not crystallized until time t if and only if aggressor is not nucleated in the time interval [0, t]. So

q(t) =P(B),

(7)

where P(B) is the probability that the aggressor is not nucleated until time t (event B). Let us divide the time interval [0, t] into i equal parts of duration A t ' = t / i . Then the event B can be r e p r e s e n t e d as the intersection of events BIB 2 ... B i where B m m e a n s that the aggressor is not nucleated in the mth part. Using conditional probabilities, we have

q( t ) = P( B) = P( B1B2... Bi) = P ( B1 ) P(B21BI )... P(BilB1B2... Bi_, ).

conditional probabilities in Eq. (9) may be expressed in terms of p~(t', t):

e(emI"l"2... Bm 1) = 1 - ~(t'm, t ) A t ' + o ( A t ' ) ,

bility of aggressor nucleation in the time interval [t', t ' + At'] provided that there were no aggressors until time t'. The quantity/z(t', t) must depend on the observation time, t, because the definition of aggressor includes this time. The

(9)

where t m = ( m - 1)At'. Substituting Eq. (9) in Eq. (8) and taking the logarithms of the left- and right-hand sides, we obtain i

In q(t) = Y'~ l n [ 1 - / x ( t ; , , , t ) A t ' + o ( A t ' ) ] In

1 i

tz( t m, t )At' + o( At')i m-I

= -f/dt'tx(t',

t).

(10)

Here we used the expansion of logarithm in vicinity of unity and passed to the limit A t ' ~ ~ in view of o(At') i = o ( A t ' ) t / A t ' ~ O. As a result, we have

q(t) = exp -

t)

(11)

The problem has been reduced to calculation of jx(t', t). By definition, ix(t', t) must be calculated on condition that there are no aggressors until time t'; thus, in accordance with the statement ($2), the region Z * ( X , t', t) must be taken as non-crystallized at time t' in this calculation. F r o m the above consideration of region Z * ( X , t', t), it follows that the volume of this region is equal to the volume V(t', t) (Eq. (3)) of the region Z ( X , t', t). Then the conditional probability of aggressor nucleation in time interval At' may be written on the basis of assumption (A2): /z(t', t ) A t ' + o ( A t ' )

(s) Let Ix(F, t)At' + o(At') be the conditional proba-

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= a ( t')V( t', t)At' + o( At'),

(12)

i.e., tx(t', t) = a ( t ' ) V ( t ' , t).

(13)

Substituting this equation into Eq. (l l), we obtain the Kolmogorov Eq. (4). In the derivation of Eq. (11), we did not follow

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Belenkii because, in our opinion, his derivation [2, pp. 18, 19] is incorrect. Belenkii ignored the dependence of ~(t', t) on time, t, i.e., he considered / 2 ( t ' ) - tz(t', t). He had obtained the differential equation d q / d t = - f z ( t ) q ( t ) with the initial condition q(0) = 1 and presented the solution in the form of Eq. (11). However, it is seen from Eqs. (13) and (3) (or from Eqs. (16) and (8) of Ref. [2]) that / 2 ( t ) = IZ(t, t ) = 0 and the solution is reduced to the certainly wrong one q ( t ) = 1. A mistake had been made in obtaining the differential equation. Really, Belenkii considered the absence of aggressors, relating to the time t, in the time interval [0, t] (even B) and the absence of aggressors, relating to the time t + dt, in the time interval [t, t + dt] (even C); then he represented the absence of aggressors, relating to the time t + dt, in the time interval [0, t + dt] as the intersection BC. This representation is false because the absence of aggressors, relating to the time t, in the time interval [0, t] does not mean the absence of aggressors, relating to the time t + dt, in the same interval [0, t]. In other words, a crystal which is nucleated at time t' ~ [0, t] and is not an aggressor with respect to time t can be an aggressor with respect to time t + d t. Thus this method is unacceptable. Therefore for derivation of Eq. (11) we used another method (Eqs. (8)(11)) which includes manipulations (10) proposed by Kolmogorov. It should be noted that, in derivation of Eq. (4), Kolmogorov mentioned (but did not prove) a statement equivalent to the statement (S1). However, he did not state explicitly that the absence of crystalline phase in the region Z * ( X , t', t), assumed in the calculation of q(t), is connected with the conditional character of the probabilities in the expression corresponding to the right-hand side of Eq. (8) and with the crystal growth laws which are accepted in the model and ensure the fulfilment of the statement ($2). For models where one or both of the assumptions (A3) and (A4) are violated, one or both of the statements (S1) and ($2) are not satisfied and a strict application of the Kolmogorov method, based on these statements, to calculation of q ( t ) becomes impossible [2]. Thus the results of some authors obtained by application of the Kol-

mogorov method for such models cannot be exact. Belenkii has generalized the K-model to the case of crystallization in a bounded space with an arbitrary region of nucleation [5]. In the generalized K-model, the assumptions (A3) and (A4) are satisfied (i.e., the type of crystal growth is the same as in the K-model and the statements (S1) and ($2) are satisfied), whereas the assumptions (A1) and (A2) are changed. It is assumed that crystallization occurs in a convex volume which can be unbounded in some or all directions. The region of nucleation is assumed to be a part of the volume of crystallization and can even have a different dimensionality. The nucleation rate defined in this region can depend on time and coordinates. For example, the case where crystals are nucleated on the surface of a convex sample is a particular case of this model. For the generalized K-model, Belenkii has obtained analytically the probability of non-crystallization of an arbitrary point as a function of time and position of this point. The fraction of non-crystallized volume can then be found by averaging over the volume. Note that some models considered in the literature before or even after the publication of Ref. [5] (see, for example, Ref. [6]) are particular cases of the generalized K-model. After the paper by Kolmogorov, the papers by Johnson and Mehl [7] and by Avrami [8-10] were published. Johnson and Mehl have applied an original statistical method of calculating q(t) for a model in which the nucleation rate and growth rate of spherical crystals are constant. This model is a particular case of the K-model (the a-model with spherical crystals) and the equation obtained by Johnson and Mehl is Eq. (5) written for a particular case of spherical crystals (i.e., with V0 = 4rr/3). Johnson and Mehl also assumed that their method "can be modified to apply to any assumed variation in nucleation rate and growth rate" and illustrated this modification for the model where the nucleation rate and growth rate vary linearly with time [7, p. 423]. However, the authors did not impose restrictions on the applicability of their method as Kolmogorov did. They only noted in discussion: " W e should expect some difficulty

M.P. Shepilov, D.S. Baik /Journal of Non-Crystalline Solids 171 (1994) 141-156 in a lack of sphericity in the growing grains" [7, p. 451]. Later Belenkii showed [2, p. 25] that the methods of Kolmogorov and Johnson and Mehl have the same region of strict applicability (which is the K-model) and lead to the same Eq. (4). For the a-model with spherical crystals, Johnson and Mehl calculated the crystal age distribution and used it for approximate representation of the crystal volume distribution [7, p. 425]. Let us note that the plot of this approximation [7, Fig. 7] is incorrect in a range of small volumes. This conclusion follows from Eq. (3) of Ref. [7]. The correct plot of the J o h n s o n - M e h l approximation, a review of the literature on crystal volume distribution and the results of computer simulation of crystal volume distributions for this and two other models are given in Refs. [11,12]. In particular, computer simulation has shown that the J o h n s o n - M e h l approximation cannot describe the crystal volume distribution even qualitatively for both small and large values of the volume. In discussion of the results of Johnson and Mehl, Avrami disagreed with their method [7, pp. 450-451]. He insisted that the contribution of phantom crystals (i.e., of imaginary crystals, 'nucleated' in already crystallized regions) must be eliminated from the extended volume (which is defined as the sum of the volumes of individual crystals treated as though every crystal continued growing, irrespective of the others, through other crystals). This elimination would have led to the appearance of a factor a ( t ' ) q ( t ' ) instead of a(t') in the right-hand side of Eq. (4). In their reply, Johnson and Mehl have shown [7, pp. 452-453] that consideration of phantom crystals is necessary for a correct treatment of impingements of crystals. Later Avrami accepted the J o h n s o n - M e h l method and used it in his papers [9,10]. By contrast with Johnson and Mehl, Avrami applied their method to the case of non-spherical (polyhedral) randomly oriented crystals and to cases where the growth of randomly oriented crystals in a three-dimensional medium is two-dimensional or one-dimensional (plate-like or needle-like crystals, respectively). However, these cases do

145

not belong to the K-model and therefore cannot be treated by the J o h n s o n - M e h l method (see Section 2). So Avrami did not propose a method, declared against the J o h n s o n - M e h l method, and later applied it outside the region of its applicability. Therefore, it is necessary to agree with Belenkii [2, pp. 11-12,24] that Avrami cannot be regarded as the original author of the theory of phase transformation kinetics. Let us also note two other details. First, Avrami's discussion of the crystal volume distribution in terms of the crystal age distribution [10] is based on the idea of Johnson and Mehl [7]. Second, it was shown [13] that the type of nucleation considered by Avrami (nucleation on active centers or preferred sites) [8,9] satisfies the assumption (A2) and can be treated by the Kolmogorov method. Following Avrami, some authors applied the J o h n s o n - M e h l method to models different from the K-model. As a result of such an application, Eq. (4) (or its particular case

q(t) = exp(-kt"),

(14)

the so-called Avrami equation with the Avrami exponent, n) is obtained which cannot be exact for these models. In these cases, it is desirable to use other approaches.

2. Some models that differ from the K-model

An interpretation of experiments on crystallization kinetics and other applications requires a study of models for which at least one of the statements (S1) and ($2) is not true because of violation of the assumptions (A3) or (and) (A4). Let us mention some of these models. A violation of statement ($2) is illustrated by the D-model t, where the radii of spherical crystals, R, grow by a diffusional law:

R ( t', t) = v;g( t - t ' ) ,

(15)

1 Note that the D-model with simultaneous nucleation of crystals is characterized by the same growth rate at any given time for all crystalsand can be considered as the K-model [14, p. 8]. Here, we speak about continuous nucleation.

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Fig. 1. The case where the phantom P grows out of the real crystal A for the model with diffusional growth (Eq. (15)) in which the statement ($2) is violated. The crystal nucleated at t = 0 at point A is shown at times t = t 1 (dashed line) and t = 2t I (solid line). The phantom "nucleated" at t = t I at point P inside the crystal is shown at time t = 2t a.

where t' and t are the times of nucleation and observation, respectively, and g is a constant. In this case, the growth rate decreases with an increase of the age of crystal, so the crystals nucleated at different times have different growth rates at a given time, and assumption (A4) is not satisfied. For the D-model, Belenkii showed [2, pp. 29-32] that statement ($1) is satisfied (i.e., q(t) can be calculated as the probability that an aggressor is not nucleated, and Eq. (11) is satisfied) and statement ($2) is violated (therefore one of the present authors has failed to calculate/x(t', t) and q(t) in Eq. (11) in a closed form [15]), and that the right-hand side of Eq. (4) gives the lower bound for q(t). An inaccuracy of Eq. (4) for the D-model may be clearly seen in the JohnsonMehl method. Indeed, the growth law (15) allows phantom crystals to outgrow real crystals (Fig. 1) and Eq. (4) includes contributions in crystallized volume such as non-physical outgrowth shown as the hatched area in Fig. 1 (in the K-model, phantom crystals always remain inside real crystals). So, for the D-model, the crystallized volume fraction considered in the Johnson-Mehl method is overestimated, and q(t) calculated by Eq. (4) is underestimated. In terms of the Kolmogorov method, the crystalline phase of non-aggressors can penetrate into the region Z*(X, t', t) [15] and the conditional probability of aggressor nucleation,/z(t', t), is lower than that calculated by Eq. (13); hence, Eq. (4) underestimates q(t). For the D-model in the absence of exact description,

one of the present authors obtained detailed analytical estimates of upper and lower bounds of q(t) [15]. In the case of constant nucleation rate, a, these estimates are in agreement with the results of computer simulation [16] and show that the Kolmogorov Eq. (4) (which is reduced to Eq. (14) with n = 5 / 2 and k = 87rag3~2~15 for this case) gives an insignificant error less than 0.008. Note that the crystallization kinetics in this model cannot be exactly described by Avrami's Eq. (14) with a constant value of n. Let us also note that the D-model (with simultaneous and continuous nucleation) and its generalization [17] may be used for a description of phase separation kinetics and kinetics of crystallization in non-stoichiometric systems [14,17]. In experiments, the shape of crystals is frequently non-spherical (and sometimes non-convex) and the orientation is random 2. In such cases, where one or both of the statements (S1) and ($2) are violated, Eq. (4) cannot be exact, and a special discussion of the transformation kinetics is needed. A violation of statement (S1) is illustrated by the following model with simultaneous nucleation. At any time before impingements, crystals are assumed to be the same (in shape and size) and randomly oriented; the shape is assumed to be non-spherical, convex and time-independent. A situation with violation of statement ($1) for a model of such a type is shown in Fig. 2(a) for time t. In the absence of the crystal S, the points in the hatched region 1 can be crystallized by the crystal A until time t. Therefore, the crystal A is an aggressor for these points. However, these points are not crystallized by time t because of the screening by the crystal S which is not an aggressor for them and will be called a screen. The question about the statement ($2) does not

2 As an exception to the rule, we can mention the case of locally the same orientation of ellipsoidal resistive zones in the theory of stability of superconducting magnet systems [18, pp. 51-53]. If the initial size of these zones is negligible (as assumed in eqs. (7.16) and (7.17) of Ref. [18]), their nucleation and growth are described within the scope of the K-model and the Kolmogorov equation is applicable to calculation of the volume fraction free of zones.

M.P. Shepilov, D.S. Baik /Journal of Non-Crystalline Solids 171 (1994) 141-156

-¢ a

b

,!A Fig. 2. The violations of the statements ($1) (a, b) and ($2) (c) for the models with simultaneous (a, b) and continuous (c) nucleation of randomly-oriented anisotropic (a, c) and nonconvex (b) crystals.

arise for this model since the nucleation is simultaneous. In Eq. (4), the possibility of screening the aggressor is not taken into account and the right-hand side of this equation underestimates the non-crystallized volume fraction. This conclusion may be also drawn from the Johnson-Mehl method where regions similar to those hatched in Fig. 2(a) are considered as crystallized in derivation of Eq. (4) whereas they are not crystallized. For this model, the exact dependence, q(t), may be calculated as the probability that, for a random point, X, a non-screened aggressor is not nucleated. However, analytical calculations in the three- or even two-dimensional case seem to be rather complicated [19] and have not been carried out until now. Therefore, computer simulation for a simple model of such a type (a two-dimensional model where simultaneously nucleated crystals grow as randomly-oriented ellipses) was undertaken [20]. The results have shown that screening effects become important at a late stage of crystallization if the anisotropy of the crystals is sufficiently high (namely, if the ratio of the

147

major and minor semi-axes of the ellipse is greater than 5) and then Eq. (4), which is reduced to Eq. (14) with n = 2, cannot give an acceptable description of the crystallization kinetics. A threedimensional analog of this model is considered in Section 3. Another example of violating the statement (S1) is given by the model with simultaneously nucleated non-convex crystals where screening effects take place even at the same orientation of crystals (Fig. 2(b)). An analytical treatment of the screening effects is possible for the following one-dimensional model [19]. It is assumed that 'crystals' are nucleated simultaneously at points of straight line x [ - ~ , oo] and grow symmetrically in both directions with rates independent of time; at points of impingements, the growth stops; different crystals have different growth rates; the probability that a crystal with growth rate between c and v + Av is nucleated in the interval Ax is a ( c ) A c A x + o(AvAx), and the probability of nucleation of two crystals is o(Ax). From the assumption about nucleation, it follows that the crystal centers are distributed at random with a mean number density oo

fl=f,

dv a ( c ) .

(16)

In this model, the assumption (A4) is not satisfied and that is why the statement (S1) is violated: a non-aggressor ('slow' crystal) can screen the considered point X from an aggressor ('rapid' crystal). The question about the statement ($2) does not arise because the nucleation is simultaneous. The fraction of non-crystallized length had been calculated [19] both with neglect of screening, q~(t) =

exp[-2tf=dva(v)v],

(17)

and in view of screening,

q(t)=

[

(1/fl)f ° dva(v) exp(-/3ct)

(18)

Eq. (17) is an analog of the Kolmogorov equation for this model and Eq. (18) gives an exact solution of the problem. Numerical examples [19] show that screening effects in this case may be

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appreciable if the growth rates of the crystals differ significantly. For instance, if the ratio of the growth rates of the rapid and slow crystals is equal to 5, the difference q ( t ) - ql(t) may achieve the value 0.09 at q l ( t ) -- 0.1. In the case of continuous nucleation of nonspherical randomly oriented crystals, not only the statement ($1) but also the statement ($2) are violated. A simple example of violating the statement ($2) at time, t, is shown in Fig. 2(c). At time t the points of the hatched region are not crystallized by the crystal Q nucleated at t " = 0, and therefore Q is a non-aggressor for these points. However, the point Q1, where the aggressor A may be nucleated at time t' = t / 2 , has been crystallized by the non-aggressor Q until time t'. Hence, ($2) is not satisfied. In other words, a phantom A can give a contribution to crystallization. To our knowledge, models of this type have not been correctly calculated in the literature. The above remarks concerning models with anisotropic non-oriented crystals are also related to Avrami's models with non-oriented plate-like and needle-like crystals. In addition, for the latter models, a similarity of crystals does not take place and this redoubles the difficulties of formulating and solving a problem. Bradley and Strenski [3] introduced a model for the description of crystallization kinetics in systems with two distinct stable phases. Crystals of one phase are assumed to grow faster than crystals of another phase. In this model, screening effects (retardation effects in the terminology of Bradley and Strenski) are present and the statement (S1) is not satisfied (the 'slow' crystal can screen a point from a 'rapid' aggressor). If nucleation is not simultaneous, the statement ($2) is also violated (points of possible nucleation of the rapid-phase aggressor are crystallized with some probability by the non-aggressors of the slow phase, nucleated earlier or, in other words, phantom crystals of the rapid phase can grow out of slow crystals). For an approximate description of the crystallization kinetics, Bradley and Strenski developed a mean-field theory which is a modification of the Johnson-Mehl method. Let us note that the same equations may be derived by the Kolmogorov method if screening effects

are ignored and the region of possible nucleation of the aggressor is assumed to be non-crystallized (this assumption means that the contribution of phantoms is included). Bradley and Strenski noted that their mean-field theory is not exact when the growth rates of phases are not equal and concluded that the inaccuracy is connected only with the retardation (screening) effects. Generally speaking, the latter conclusion is incorrect because the non-physical contribution of phantoms is also taken into account in the mean-field theory.

At first, let us consider the case of simultaneous nucleation where only the statement (S1) is violated. For this case, the one-dimensional variant of the model [3, section III.A] is a special case of the model calculated in Ref. [19] and the exact equation obtained by Bradley and Strenski is a particular case of Eq. (18). If the growth rates of phases differ significantly, the mean-field theory, which is reduced to Eq. (17) for this variant, leads to a significant error not only in the volume fraction of crystallized material (as noted in Ref. [19]) but also in the fraction of material in each of the two phases [3]. The reason is the full screening of rapid crystals by slow crystals. The screening in two- and three-dimensional cases must be insignificant for this model because rapid crystals can round and surround slow crystals and the shape of the crystallized region in view of retardation [3, Fig. 6] differs only slightly from the region constructed with neglect of screening (the latter region is simply the union of the volumes of crystals taken as though they grow independently). Really, the fraction of material in each of the two phases, obtained by computer simulation in two dimensions at time t = oo, is close to the results of the mean-field theory [3]. For the case of continuous nucleation where both statements (S1) and ($2) are violated, Bradley and Strenski succeeded in obtaining an exact description of the crystallization kinetics in one dimension. Possibly, this is the unique model with violation of statements ($1) and ($2) for which an exact solution of the problem can be found. The exact solution shows an anomalous power-law correction to the leading-order exponential decay of the untransformed fraction. The

149

M.P. Shepilov, D.S. Baik /Journal of Non-Crystalline Solids 171 (1994) 141-156

power-law exponent is a continuously varying function of the nucleation rates of phases. Nothing of this type appears in the mean-field theory which also gives a poor approximation for the volume fraction of crystallized material in each of the two phases at t = o0 (except for the case where the growth rates of the phases are approximately equal). In two dimensions, the volume fraction of crystallized material in each of the two phases at t -- ~ was calculated by computer simulation for the special case of the model with continuous nucleation and the results are close to those in the mean-field theory. In our opinion, in other special cases a difference may be somewhat greater because not only insignificant screening effects but also a phantom contribution depending on parameters of a problem determines this difference. As a result, the authors expect the mean-field theories to become increasingly good approximations for their models (with simultaneous and continuous nucleation) as the spatial dimension is increased. Andrienko, et al. [4] generalized the two-phase model of Bradley and Strenski to the case of arbitrary finite or infinite number of phases. For simultaneous nucleation in one dimension, the model shown in Ref. [4] is equivalent to the model [19] described above. The exact equation obtained in Ref. [4] for this case has a form (with notation used in Eq. (18)) oo

oo

nucleation of many phases in one dimension is calculated exactly also for time-dependent growth rates [21]. It should be pointed out that difficulties also arise in calculation of some properties of both the Kolmogorov and non-Kolmogorov models. For example, calculation of crystal volume distribution can be carried out only by computer simulation [11,12,22] (except for the simplest one-dimensional models [23]). A number of non-Kolmogorov models corresponding to processes of crystallization and phase separation need further discussion. In the following section, we consider the simplest three-dimensional model with anisotropic simultaneouslynucleated crystals.

3. Model

First, let us introduce a new term. If the growth of crystals is considered as mutually independent (as though they grow through each other), a crystal with a minimum time of crystallization of a given point will be called a concrete aggressor for this point. We will assume that crystals are nucleated simultaneously at time t = 0 at points which are randomly distributed with a mean number density, N, over a volume of medium. Before impingements, crystals are assumed to be ellipsoids, centered at these points (crystal centers), with principal semiaxes b(t), b(t), c(t), and

c(t)/b(t) × f,t+,,'tdx(x-vt-v't)

exp(-flx) (19)

and after simplification (integration with respect to x) is reduced to the above Eq. (18) derived earlier in Ref. [19]. In Ref. [4], computer simulation in two dimensions for the special cases of the models with simultaneous and continuous nucleation was also carried out and the conclusion about the closeness of the results of computer simulation and the mean-field theory was drawn. Let us note that the model with simultaneous

- y = constant.

(20)

That is, all crystals have the same fixed form and are ellipsoids of revolution (spheroids). If an origin of rectangular Cartesian coordinate system (x', y', z') is placed at a center of a crystal and the z'-axis is directed along the c-axis of the crystal, then the surface of the crystal before impingement is described by X'

2

yp

]2

Z'

2

The direction of the axis of revolution, c, may be specified by spherical coordinates (0, 4~) counted with respect to a fixed coordinate system (x, y, z).

M.P. Shepilov, D.S. Baik /Journal of Non-Crystalline Solids 171 (1994) 141-156

main. The question is about the character of growth of the real crystal C 2 after contact with the crystal C l at the point E. The answer depends on the microscopic mechanism of growth and may be given only on the basis of experimental data or simulation at the level of atoms or structural units. In our phenomenological model, this growth is assumed to be purely radial, so that all points of the hatched domain are crystallized by the crystal C 1 and their times of crystallization are greater than in the case of crystallization by the crystal C 2. Therefore, the non-crystallized volume fraction, qu(t), simulated below gives an upper bound for the non-crystallized volume fraction qr(t) which corresponds to some real situation. Another extreme case, where screening is ignored for points in the hatched region and is taken into account only for points in the region 1', gives a lower bound, qj(t), and will be considered in future. Apparently, in a real situation the crystal C 2 can round the point E and crystallize at least a part of the points of the hatched domain. In spite of the possibility of retardation effects caused by rounding (which are similar to those in the model of Bradley and Strenski [3]), these points will be crystallized earlier than in the present model, but later than in the future model, and the quantity qr(t) for a real system will be between the upper and lower bounds corresponding to these models. Any screening is ignored in calculating qK(t) by the Kolmogorov equation (see Eq. (26) below), so that qK(t) ~
c(t) = v l t ,

b(t)=v2t,

vJv 2=y.

151

this limitation is not essential. A more general case where the growth rate depends on time, t 1,

6(tl)=f(t,

),

[~(t,)=f(t,)/7,

(24)

can be reduced to the case of the growth law (23). Indeed, the shape and size of crystals at time t for the growth law (23) are the same as at the corresponding time t L (such that f(t,) = c~t) for the growth law (24); the laws of nucleation which determine positions and orientations of crystals are the same in both cases; hence, the non-crystallized volume fraction will be also the same. Therefore, if we have qi(t) for the growth law (23) (where qi(t) is qK(t), ql(t), qr(t) or qu(t)), the corresponding quantity, c~i(t), for the growth law (24) can be expressed as

qi( tl) = qi( f ( tl)/~,l).

(25)

Note that such a relation cannot take place in the case of continuous nucleation. A formal substitution of nucleation ( a ( t ) = NcS(t)) and growth (Eq. (23)) laws of our model in the Kolmogorov Eq. (4) leads to q K ( t ) = exp(--43"rrNv IU2 2t 3).

(26)

With neglect of screening, the same equation can be derived by direct application of the Kolmogorov method to our model in which the orientation of the region of possible nucleation of an aggressor depends on the orientation of the aggressor. This equation has the form of the Avrami Eq. (14) with Avrami exponent n = 3. Let us introduce the dimensionless variables p and z for distance and time which are more suitable for computer simulation:

p = rU 1/3,

"r = t[ Ut, lt,'22]1/3.

(27)

It can be shown that the mean number density of crystals in these variables is equal to unity, and the growth rates of spheroids are

,~2/3 (in the direction of the c-axis), 7-1/3 (perpendicular to the c-axis).

(28)

(23)

However, in the case of simultaneous nucleation,

In these variables, the Kolmogorov Eq. (26) has the universal form independent of parameters N,

M.P. Shepilov, D.S. Baik /Journal of Non-Crystalline Solids 171 (1994) 141-156

152

v l, and v 2 (or y): qI,:("r) : exp( - 4 v r 3 ) .

4. C o m p u t e r kinetics

simulation

(29)

of the crystallization

The general items of computer simulation of crystallization kinetics were described earlier [16]. A new development made in this work is connected with taking into account the screening effects in the three-dimensional case. Computer simulation was carried out in dimensionless variables (Eq. (27)) for a series of numerical experiments modeling the crystallization in a cubic sample of size L (for example, L = 8). It can be shown that the number of crystals in a sample is a random number with a Poisson distribution and the mean value L 3 (since the mean number density in dimensionless variables is equal to unity). The corresponding generator of pseudo-random numbers was used to find the number of crystals, Nj, in experiment j. The positions of the crystal centers, xi, yi, z~, i = 1, 2 , . . . , N~), were given by the random number generator with uniform distribution over the interval [0, L]. The right-hand side of Eq. (22) and the text below it show that the quantities cos 0 and ~b are random quantities with a uniform distribution over the intervals [0, 1] and [0, 2v], respectively, and the corresponding generators of pseudo-random numbers were used to determine cos 0 and ~b. In this way, the orientations (Oi, 49i) of the crystals were found. An algebraic relation was obtained for the time of crystallization of an arbitrary point by a crystal of arbitrary position and orientation. Computer simulation was carried out for oblate (3' < 1) and prolate (y > 1) spheroids. We use the following values of 3': y = 0.05, 0.1, 0.2, 0.5, 1, 2, 5, 10, 20.

(30)

Sometimes it is more suitable to speak about the anisotropy of crystals, a, which is defined as a =

(1/y y

(y 1)

.

(31)

For a fixed value of y, the non-crystallized volume fraction, q~(%), in numerical experiment j was calculated at given times, r m. Calculations of qj(%) were carried out by numerical integration over a volume of the cube which had the same center and orientation as the sample and size L 1 smaller than the size of the sample. In such a way, a surface layer with thickness d = ( L - L 1 ) / 2 was not taken into account in the calculation of qj(r m) and surface effects were partly excluded. The optimal value of d is determined by two opposite requirements (reducing surface effects and decreasing computing time) and increases with an increase in anisotropy, a. Since the calculations for all the values of y (Eq. (23)) were carried out simultaneously in the same program, in the choice of d we oriented ourselves to the cases with maximum anisotropy and used the value d = 3. The results of numerical experiments were averaged over a series. The statistical error of simulation of q(~-) was estimated for the confidence level 0.95. The maximum error was observed at q - 0 . 3 5 - 0 . 5 5 and was equal to =0.01 in the final results. By our estimates, the methodical error connected with surface effects and with the inaccuracy of numerical integration was approximately equal to the statistical error in the cases of high anisotropy (a = 20) and somewhat decreased with a decrease in anisotropy (by a factor of 0.5 at a -- 1). Two variants of the computing algorithm were realised. (1) In the first variant, the possibility of screening a concrete aggressor was considered in the calculation of the crystallization time of a point. At first, the times of crystallization of a given point by different crystals were calculated without taking into account the possibility of screening, i.e., on the assumption of independent growth of crystals through each other. Then we took the crystal corresponding to the minimum time ('the first concrete aggressor') and checked the fact of non-screening on the basis of consideration of pair impingements of this crystal with other ones (with possible 'screens'). If the screening did not take place, this minimum time was taken as the true time of crystallization for the

M.P. Shepilov, D.S. Baik / Journal of Non-Crystalline Sofids 171 (1994) 141-156

point. If the first concrete aggressor was screened, we excluded it from consideration for this point, and checked the fact of non-screening for the following crystal with minimum time of crystallization of point ('the second concrete aggressor'). This procedure was continued until the nonscreened aggressor was found, and the corresponding time was taken as the true time of crystallization. Results calculated for variant (1) are given in Figs. 4 - 9 as curves V1. (2) In the second variant, the effects of higher order were estimated. These effects are connected with the situation when the crystal, which may screen the aggressor, does not screen it because of interaction with a third crystal ('screening of screen' or 'screening of the second order'). These situations must be treated on the basis of an analysis of triple impingements (the concrete aggressor + the screen + third crystal). However, such a problem is difficult. Therefore, for estimation of these effects in variant (2), we

1

,

,

°0

•

.

6

l"

2

.

i

-z

i

-t

i

0

1 &,.tr z

.--~.c 3

. . . . . .

_'~

:2W

~

2

vz

i

i

i

i

%-

0

'

'"

0

0g

~6

-z

-~

0 &r

T

2.4

--~c 3 2

~

0

~ ' V 2 V! 0,'8

4.'6 ~

Z.'~

Fig. 5. Computer simulation results at 3' = 0.1 (the same notation as in Fig. 4 is used).

,

q

2.

153

i

restricted ourselves by consideration of double impingements (the aggressor + the screen, the screen + third crystal). The results calculated for variant (2) are given in Figs. 4 - 9 as curves V2. In the cases where the results obtained in variants (1) and (2) are close, the screening of second order is not essential and these results can be used for a quantitative description of the crystallization kinetics in our model. However, if the difference between the results of variants (1) and (2) is significant, these results can be used only for semi-quantitative or qualitative analysis of kinetics, since in this case the screening of the second order is essential and must be taken into account in a more accurate way than in variant (2).

5. Discussion of computer simulation results

i

Fig. 4. Results of computer simulation of the crystallization kinetics at 3, = 0.05 for the variants (1) (curves V1) and (2) (curves V2) and the results of using the Kolmogorov Eq. (29) (curves K) in three representations.

At first, it is necessary to note that for the wide interval of 3' 0.1 ~< 3' ~< 20,

(32)

M.P. ShepiloL,, D.S. Baik /Journal of Non-Crystalline Solids 171 (1994) 141-156

154

the results of variants (1) and (2) are rather close (Figs. 5-9). In particular, for 0.2 ~
(oral
or

3,>5

(fora>~5),

(34)

screening effects are appreciable (Figs. 4-9). Figs. 4-9 show that, for a given value of a (a >/5), the

a

1.o

0.5

b 2/~,/,=o.of"

~'~

-C"

o.~

b 2 [ ~ q ,'o.°~ "

ot

(33)

the Kolmogorov Eq. (29) gives a satisfactory description of the volume fraction non-crystallized. The difference between the simulated (variants (1) and (2)) and analytical (Eq. (29)) values of q(~-) is less than the statistical error (i.e., < 0.01). That is, we can say that the screening is negligible at small anisotropy (33). For y~<0.2

a

'

'

"E

K~

-g

-I

o 2_~,,

o

0.5

1.0

2.~

"'E

Fig. 6. C o m p u t e r s i m u l a t i o n results at y = 0.2 (the s a m e n o t a t i o n as in Fig. 4 is used).

-

i

-Z

---#. c

~.o

~

2.8 ~'~

1.0

' I{~':

vt,v,

i

-f

i

0 ~ f

2.6

,-..L,

i

0.5

i

i

t.0

"g

Fig. 7. C o m p u t e r s i m u l a t i o n results at y = 5 (the s a m e notation as in Fig. 4 is used).

screening and screening of the second order are more appreciable for oblate spheroids with y = 1/a than for prolate spheroids with y = a. For 3' = 0.2, 5 (a = 5), the { l n ( - l n q)-ln 7} plot of simulated results (Figs. 6(b), 7(b), curves V1, V2) deviates in the final stage of crystallization from the straight line with a slope 3 which presents the the Kolmogorov dependence (29). The slope of simulated { l n ( - l n q)-ln 7} curve decreases from 3 at the beginning of crystallization to = 2 ( f o r 3'=0.2) or =2.5 (for y = 5 ) at the late stage. For y = a = 10 (Fig. 8), the deviation of the simulated results from those obtained by the Kolmogorov equation is greater. The {ln(-ln q)In ~-} curve of the simulated results can be considered as consisting of two portions of straight line (with slopes 3 and - 2 . 2 ) connected by a transition region. For y = 0.1 (a = 10) (Fig. 5), the screening of second order is more appreciable than in the case of y = 10. However, in both variants (without and with screening of second order), the { l n ( - l n q)-

156

M.P. Shepiloc, D.S. Baik /Journal of Non-Crystalline Solids 171 (1994) 141-156

lated non-crystallized volume fraction, q(t), is an upper bound for the non-crystallized volume fraction in real systems (a model corresponding to the lower bound will be considered in future). Computer simulation of the crystallization kinetics for the examined model has shown the following. (1) If the anisotropy of the crystals is not high (if the ratio, a, of the large principal diameter of spheroid to the small one is ~<2), screening effects are negligible, and the Kolmogorov equation is practically exact. (2) For higher anisotropy (a >~5), the screening effects are appreciable, and the Kolmogorov equation is inapplicable. The {ln(-ln q)-ln t} plot in these cases is not a straight line and can be considered as consisting of two portions of a straight line (with different slopes) connected by curved line. In these cases, the Avrami analysis of the dimensionality of crystal growth, based on determination of the Avrami exponent, is inapplicable. (3) At high anisotropy (a >/20), screening of the second order becomes essential, and obtaining a quantitative description of crystallization kinetics in these cases seems to be difficult. These conclusions may be qualitatively related to the case of simultaneous nucleation of randomly oriented convex anisotropic crystals with a more complex form. The second of these conclusions has to be kept in mind during interpretation of the experimental data. Some authors connect the change in slope of the {In(-ln q)-ln t} plot in the course of crystallization with a change in the mechanism of nucleation and growth of the crystals. However, the results obtained have shown that the change of slope can take place within the framework of a fixed model of nucleation and growth. Similar conclusions may be expected in the more difficult case of continuous nucleation.

The authors thank the Korean Ministry of Science and Technology and Joongwon Electric Corporation for financial support of this work.

References [1] A.N. Kolmogorov, Izv. Akad. Nauk SSSR, Ser. Matem. 3 (1937) 355 (reprinted in: A.N. Kolmogorov, Teoriya Veroyatnostei i Matematicheskaya Statistika (Probability Theory and Mathematical Statistics) (Nauka, Moscow, 1986) p. 178 (in Russian)). [2] V.Z. Belenkii, Geometriko-Veroyatnostnie Modeli Kristallizatsii (Geometrical-Probability Models of Crystallization) (Nauka, Moscow, 1980) (in Russian). [3] R.M. Bradley and P.N. Strenski, Phys. Rev. B40 (1989) 8967. [4] Y.A. Andrienko, N.V. Brilliantov and P.L. Krapivsky, Phys. Rev. A45 (1992) 2263. [5] V.Z. Belenkii, Dokl. Akad. Nauk SSSR 278 (1984) 874. [6] M.C. Weinberg, J. Non-Cryst. Solids 142 (1992) 126. [7] W.A. Johnson and R.F. Mehl, Trans. AIME 135 (1939) 416. [8] M. Avrami, J. Chem. Phys. 7 (1939) 1103. [9] M. Avrami, J. Chem. Phys. 8 (1940) 212. [10] M. Avrami, J. Chem. Phys. 9 (1941) 177. [11] M.P. Shepilov and V.B. Bochkariov, Fiz. Khim. Stekla 15 (1989) 152. [12] M.P. Shepilov and V.B. Bochkariov, J. Non-Cryst. Solids 125 (1990) 161. [13] M.P. Shepilov, Fiz. Khim. Stekla 13 (1987) 489 (Sov. J. Glass Phys. Chem. 13 (1987) 252). [14] M.P. Shepilov, J. Non-Cryst. Solids 146 (1992) 1. [15] M.P. Shepilov, Fiz. Khim. Stekla 12 (1986) 110. [16] M.P. Shepilov and V.B. Bochkariov, Kristallografiya 32 (1987) 25 (Soy. Phys. Crystallogr. 32 (1987) 11). [17] M.P. Shepflov and V.B. Bochkariov, in: Proc. 16th Int. Congress on Glass, Madrid, 1992, Bol. Soc. Esp. Ceram. Vid. 31-C Vol. 2 (1992) 199. [18] V.R. Chechetkin and A.S. Sigov, Phys. Rep. 176 (1989) 1. [19] M.P. Shepilov, Kristallografiya 35 (1990) 298. [20] M.P. Shepilov and V.B. Bochkariov, Fiz. Khim. Stekla 17 (1991) 674. [21] P.L. Krapivsky, J. Chem. Phys. 97 (1992) 8817. [22] I. Orgzall and B. Lorenz, Scripta Metall. Mater. 26 (1992) 889. [23] J.L. Meijering, Philips Res. Rep. 8 (1953) 270. [24] U. Koster and U. Herold, in: Glassy Metals, ed. H.J. Guntherodt and H. Beck (Springer, Berlin, 1981) p. 225.