Kinetics of solid-state transformation subjected to anisotropic effect: Model and application

Available online at www.sciencedirect.com

Acta Materialia 59 (2011) 3276–3286 www.elsevier.com/locate/actamat

Kinetics of solid-state transformation subjected to anisotropic effect: Model and application S.J. Song, F. Liu ⇑, Y.H. Jiang, H.F. Wang State Key Laboratory of Solidification Processing, Northwestern Polytechnical University, Xi’an, Shaanxi 710072, People’s Republic of China Received 9 January 2011; accepted 1 February 2011 Available online 2 March 2011

Abstract In the present work, a growing particle subjected to anisotropic effect, if not influenced by other particles, is assumed to be an isotropically growing particle with constant volume. Accordingly, how to describe the anisotropic growth just becomes how to solve the blocking effect arising from the anisotropic growth. Following the statistical description of Johnson–Mehl–Avrami–Kolmogorov kinetics, the blocking effect was investigated further. Consequently, a series of analytical models for solid-state transformation, where a particle undergoes 1-scale blocking, k-scale blocking and infinite-scale blocking, were developed. On this basis, it was analytically proved for the first time that the classical phenomenological equation accounting for the anisotropic effect (f ¼ 1 ½1 þ ðn  1Þxe 1=n1 ) corresponds to an extreme case where a particle encounters infinite-scale blocking. From the model analysis, the anisotropic effect on the transformation depends on two factors: the non-blocking factor c and the blocking scale k. From the model calculations, the Avrami exponent, subjected to the anisotropic effect, changes as a function of the transformed fraction, whereas the effective activation energy is not affected by the anisotropic effect. The present models were adopted to describe isothermal crystallization of amorphous Fe33Zr67 ribbons; good agreement with the published results was achieved. Ó 2011 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Extended fraction; Anisotropic growth; Blocking effects; Probability

1. Introduction In studies of phase transformations involving nucleation and growth, the classical Johnson–Mehl–Avrami–Kolmogorov (JMAK) equation [1–5] often plays an important role. This equation provides an expression for the fraction of material transformed as a function of time f(t) in terms of the nucleation and growth rates. It is known that the JMAK formula is exact, provided the conditions imposed in the original derivations are not violated, such as: isothermal transformation, either pure site saturation at time t = 0 or pure continuous nucleation; high driving force (large undercooling or superheating); and randomly dispersed nuclei which grow isotropically [6–11]. In these cases, the kinetic parameters, the Avrami exponent n, the effective ⇑ Corresponding author. Tel.: +86 29 88460374; fax: +86 29 88491000.

E-mail address: [email protected] (F. Liu).

activation energy Q and the pre-exponential factor K0 all hold constant with respect to time and temperature. Recently, a modular model for transformation kinetics [8–16] was proposed which includes, but is not restricted to, the classical JMAK description [11,12]. The model recognizes three mechanisms, i.e. nucleation, growth and impingement of growing new-phase particles, and is applicable to both isothermal and non-isothermal transformations. By choosing suitable nucleation and growth mechanisms, in particular a mixture of nucleation modes (e.g. the mixture of pre-existing nuclei and continuous nucleation), the model even leads to analytical formulations for the degree of transformation which exhibit the framework of the JMAK equation but with time-dependent kinetic parameters n(t), Q(t), K0(t) (isothermal transformation) or temperature-dependent kinetic parameters n(T), Q(T), K0(T) (isochronal transformation) [13–15]. This implies that a transformation can still be considered

1359-6454/$36.00 Ó 2011 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.actamat.2011.02.001

S.J. Song et al. / Acta Materialia 59 (2011) 3276–3286

“iso-kinetic” in the sense that the prevailing transformation mechanism does not change throughout the transformation process, in spite of the change in n and Q with transformation. In this modular model, the effects due to anisotropic growth and non-random nuclei distribution have also been considered as two contrary modes for impingement [11,12], which merely modify the relation between the transformed fraction f and the extended fraction xe, applying the phenomenological factors for impingement. However, the calculation of xe still follows the essential JMAK-like restrictions, e.g. randomly dispersed nuclei and isotropic growth. Under practical conditions, unfortunately, the JMAKlike restrictions are often violated. For example, the transient nucleation [17,18], the spatially correlated nucleation [19,20], the necessarily associated size-dependent growth [21,22] or the non-parallel anisotropic growth with blocking up to all relevant orders [23,24] all lead to deviations from the JMAK-like kinetics. On this basis, some extensions based on JMAK-like theory have been made, especially in the cases subjected to the anisotropic effect. Calculation of transformation kinetics involving anisotropic particles is a much more challenging problem than that for isotropic particles. Up to now, two approaches have been proposed to deal with the anisotropic effect. One approach is the phenomenological extension of JMAK-like formulation by adding (one or more) new variables which provide freedom to improve the agreement where anisotropic growth occurs [11,12,25–27]; this approach changes only the relation between f and xe, but does not change xe itself (e.g. the modular model mentioned above, or see Section 2.2). The other approach is, according to the physical essence of anisotropic effect, devoted to deriving an analytical description with physically realistic variables (e.g. the growth rate anisotropy gr and the orientation /) [28–34]; this approach does reduce xe. Furthermore, computer modeling and simulations have also been used to analyze the anisotropic effect. Shepilov and co-workers performed computer simulations to investigate the growth of randomly distributed and oriented ellipsoidal particles in two-dimensional (2D) [35] and three-dimensional (3D) [23] spaces, where the mutual blocking of growing particles in the first and the second order was studied. In the first-order treatment, the possibility that a third particle hinders the blocker in blocking the aggressor (second-order blocking) was not accounted for. Subsequently, with the Monte Carlo method, Pusztai and Gra´na´sy [24] and Kooi [36,37] also studied the mutual blocking of anisotropically growing particles up to all relevant orders, and Kooi proposed an analytical model to describe the blocking effect. On the basis of Kooi’s model, the deviations from JMAK-like kinetics due to the anisotropic effect were investigated further by Liu and Yang [38]. Actually, a proper analytical treatment in the spirit of JMAK-like theory has not yet been developed to describe such transformation. In the present work, a statistical analysis for the blocking effect arising from the anisotropic

3277

growth is performed. Analytical models for solid-state transformation, where the particle undergoes 1-scale blocking, k-scale blocking and infinite-scale blocking, are developed. In Section 2, a theoretical background essential for the current models is summarized. In Section 3, a philosophical description of the statistical analysis and a derivation of the current models are presented. In Section 4, first, new expressions for the Avrami exponent nnew and effective activation energy Qnew subjected to the anisotropic effect are obtained and discussed; then the present model calculation, which illustrates the contributions of non-blocking factor c and blocking scale k (see Sections 3 and 4) to the anisotropic effect, is discussed; and finally, the model fit to crystallization of amorphous Fe33Zr67 ribbons at 663 K is performed. Several brief concluding remarks are summarized in Section 5. 2. Theoretical background 2.1. JMAK kinetics In the JMAK description, nucleation and the growth are modeled as two statistical processes. The original derivation of the JMAK equation rests on calculating the probability that a randomly chosen point in space (e.g. the origin point O) will have remained untransformed by a given time t. The probability that a particle nucleated at time s will grow to the origin point O at time t is expressed as [5] I ¼ N_ ðsÞdsY ðs; tÞ

ð1Þ

where N_ is the steady-state nucleation rate per unit volume, N_ ðsÞds is the probability for a particle nucleated in the time interval [s, s + ds] per unit volume, and Y(s, t) is the volume of a particle at time t when it was nucleated at time s. Accordingly, q(t), the probability of the random point O untransformed at time t, can be obtained, from Eq. (1), as [5,23]  Z t  qðtÞ ¼ exp  N_ ðsÞY ðs; tÞds ð2Þ 0

And thus, the JMAK equation describing the temporal evolution of transformed fraction follows f ðtÞ ¼ 1  exp½xe ðtÞ

ð3Þ

where the extended fraction xe obeys Z t N_ ðsÞY ðs; tÞds xe ¼

ð4Þ

0

As such, the probability that a particle nucleated at time s will transform the origin point O at time t corresponds to the differential form of xe (i.e. dxe). For the sake of generalization, dxe, defined herein as a comprehensive probability factor, i.e. the increment of xe incorporating all the prevalent modes of nucleation and growth, will be used to derive the current models (see Sections 2.3 and 3).

3278

S.J. Song et al. / Acta Materialia 59 (2011) 3276–3286

2.2. Modular transformation kinetics Modular transformation kinetics expands the JMAK theory with time- or temperature-dependent kinetics parameters. A detailed description is shown in Refs. [11– 16]. Here, the main results for isothermal transformation needed for this paper are summarized. 2.2.1. Nucleation modes The term “site saturation” is used here for the case of initial nucleation site saturation where the number of supercritical particles of the new phase formed does not change during the transformation: all nuclei, of number N per unit volume, are present at t = 0 already. This implies for the nucleation rate [8,9,11,13] N_ ¼ N  dðt  0Þ

ð5Þ 

with dðt  0Þ denoting Dirac functions, and N the number of nuclei per unit volume. The “continuous nucleation” rate per unit volume is at large undercooling only determined by the rate of jumping of atoms through the interface between the nucleus of critical size and the parent phase, which can be given by an Arrhenius term:   Q N_ ðT Þ ¼ N 0 exp  N ð6Þ RT

perature- and time-independent activation energy for growth, m the growth mode parameter (m = 1 for interface-controlled growth; m = 2 for volume diffusion-controlled growth), and d the dimensionality of the growth (d = 1–3) [7,9–11]. For interface-controlled growth, m0 is a temperatureindependent interface velocity constant, and QG represents the energy barrier at the interface. For volume diffusioncontrolled growth, m0 equals the pre-exponential factor for diffusion D0, and QG represents the activation energy for diffusion QD [11,12]. 2.2.3. Extended fraction For a wide range of nucleation and growth mechanisms, by extensive calculations from Eq. (4), the following general analytical expression for the extended fraction xe can be obtained as [12]    nðtÞQðtÞ nðtÞ xe ðtÞ ¼ K 0 ðtÞ tnðtÞ exp  ð11Þ RT In general, for isothermal transformation, the kinetic parameters n, Q and K0 are functions of time t and depend on model parameters such as N and N0, QN and QG. Explicit expressions for n, Q and K0, in terms of general nucleation and growth modes, are given in Refs. [12,13]. 2.2.4. Impingement and transformed fraction If the nuclei of the new phase distribute randomly in the parent phase and the subsequent growth is isotropic, a relation between the real transformed fraction and the extended transformed fraction can be given as [1–6]

where N0 is a temperature-independent nucleation rate, and QN is the temperature-independent activation energy for nucleation. If the so-called nucleation index a (P1) is introduced, the nucleation rate can be written   Q N_ ðT Þ ¼ aN 0 ta1 exp  N ð7Þ RT

df ¼ ð1  f Þ dxe

“Mixed nucleation” represents a combination of site saturation and continuous nucleation modes [12,13]. Thus,   QN  _ N ðT ; tÞ ¼ N dðt  0Þ þ N 0 exp  ð8Þ RT

Integrating Eq. (12) gives Eq. (3) above. Considering the blocking effect of particles arising from the anisotropic growth, one phenomenological approach accounting for this impingement has been proposed [36,39] by modification of Eq. (12) as

which after introduction of the nucleation index a becomes   Q N_ ðT ; tÞ ¼ N  dðt  0Þ þ aN 0 ta1 exp  N ð9Þ RT

df ¼ ð1  f Þn dxe

where N and N0 represent the relative contributions of the two modes of nucleation. 2.2.2. Growth modes For isotropic growth, diffusion-controlled and interfacecontrolled growth modes can be given as a compact form. At time t, the volume Y of a particle nucleated at time s is given by Z t d=m 0 Y ¼g mdt ð10Þ s

where g is a particle-geometry factor and m(t) = m0exp (QG/RT(t)) (at large undercooling), with QG as the tem-

ð12Þ

ð13aÞ

where n P 1. Impingement due to Eq. (13a)1 is more severe, i.e. the difference between f and xe is greater, than that due to Eq. (12), and increases with n [12]. For n > 1, integrating Eq. (13a) yields 1

f ¼ 1  ½1 þ ðn  1Þxe n1

ð13bÞ

As shown in Section 1, impingement due to Eq. (13a) implies that the anisotropic effect changes only the relation between f and xe, but does not change xe itself. A comparison between the phenomenological approach (i.e. Eq. (13)) 1 Although n is an adjustable parameter, Eq. (13a) or (13b) is applied universally to account for some real process, such as solid-state precipitation [39,40], spatially correlated nucleation [19,20], phantom overgrowth [41–43].

S.J. Song et al. / Acta Materialia 59 (2011) 3276–3286

and the computer simulations was performed [27,36], which showed that the two approaches can be consistent with each other at the early stage of the transformation, but deviated substantially at the late stage. This implies that the phenomenological approach could not provide a physically realistic description for the anisotropic growth. The present models will show that Eq. (13) corresponds to the extreme case where the anisotropic effect occurs. 2.3. Transformation kinetics for randomly oriented anisotropic particles Following the JMAK-style approach (see Section 2.1 above), the blocking or shielding effect may be described, simply, as follows [28]. If a particle nucleates at s at a distance r from a randomly chosen point (i.e. denoted as the origin) and grows towards the origin with speed v, then if all the particles exhibit isotropic growth, the origin will be transformed at t (t > s) provided (t  s)v > r. However, if the particle growth is anisotropic, the origin will probably not be transformed at t even if (t  s)v > r is satisfied. This stems from the fact that a second growing particle (blocker), which reaches the origin at a time greater than t, may block the path of the first particle (aggressor), as shown in Fig. 1. Generally, the blocking or shielding effect arising from the anisotropic growth of particles adds difficulties to modeling the transformation kinetics for randomly oriented anisotropic particles. Ignoring the blocking or shielding effect, as is standard for JMAK-type derivations (i.e. the anisotropic growth of particles proceeds in an infinite space, and the growth of one particle is not interfered with by the others), Weinberg and Birnie III [29] proposed a transformation model for randomly distributed and randomly oriented particles in two and three dimensions, considering both the extreme cases of pre-existing nuclei and continuous nucleation. This model generalizes the JMAK expression to arbitrarily shaped particles and provides a rigorous verification that the orientation distribution of particles will not modify the kinetics of the transformation, at least while the model assumptions are not violated. Accordingly, the transformed fraction accounting for the anisotropic growth depends only on the particle volume, but is independent

Fig. 1. Schematic diagram depicting a statistical treatment of the blocking effect arising from the anisotropically growing particles (here, randomly oriented elliptical particles). The broken circle indicates that an anisotropically growing particle (aggressor A) is equivalently considered as an isotropically growing particle with invariable volume. For a detailed description of the statistical treatment, see Section 3.1.

3279

of the particle orientation. The probability that a randomly oriented anisotropic particle will transform the origin can still be expressed in the form of Eq. (1), i.e. dxe, only considering the volume Y of the anisotropic particle. However, added difficulties appear once the blocking effect is included. It warrants the introduction of a function of non-blocking probability PNB [28,30–33]. Thus, the probability that a particle (aggressor) is initially not interfered with by other particles and subsequently transforms the origin can be expressed as the product of the two-condition probability, PNB and dxe [30] I ¼ P NB dxe

ð14Þ

Analogous to the probability of the origin untransformed, the derivation of PNB rests on calculating the probability that a particle (blocker) will interfere with the aggressor. Since PNB is a complicated function considering the nucleation time, positions and orientations of all possible blockers, as well as whether or not the blocking will occur [28,30–33], it is interesting to see whether this complicated effect can be reasonably approximated by a relatively simple factor, in order to represent more easily such transformation kinetics. Here, as an attempt, a time-dependent function S(t) is introduced to replace PNB, thus, Eq. (14) becomes I ¼ SðtÞdxe ðtÞ

ð15Þ

where S(t) represents all the orientation-, time- and position-averaged values of the non-blocking probability factors. At t = 0, S(t) = 1. As we know a priori, the blocking effect due to anisotropic growth causes a relative retardation of the transformation, so S(t) is a decaying function of t. In the following section, the study aims to derive the expression for S(t). 3. Statistical analysis and model derivation 3.1. Philosophical description As shown in Section 2.3, for randomly oriented anisotropic particles neglecting the blocking effect, the JMAK equation (i.e. Eq. (3) involving Eq. (4)) still holds, and the transformed fraction depends only on the particle volume, but is independent of the particle orientation. Therefore, an anisotropically growing particle, which is not interfered with by other particles, can be equivalently considered as an isotropically growing particle with invariable volume. Thus, the growth rate equation v = v0exp(QG/ RT) (see below Eq. (10)) for the isotropically growing particle corresponds just to the averaged rate of anisotropic growth at all orientations. This equal-volume treatment avoids dealing with the growth rate anisotropy and the random orientations. Accordingly, how to study the anisotropic growth just becomes how to solve the problem of the blocking effect. Following the above philosophy, as illustrated schematically in Fig. 1, a statistical treatment of the blocking effect

3280

S.J. Song et al. / Acta Materialia 59 (2011) 3276–3286

is proposed and is described as follows. A particle (aggressor) nucleating at t = s and growing anisotropically towards the origin by a rate dY can be considered as an isotropically growing particle with invariable volume. As the transformation proceeds, the aggressor is progressively interfered with by the first blocker (t = t1), the second blocker (t = t2), and up to the Nth blocker until it arrives at the origin at time t = t (s < t1 < t2    < t). Accordingly, the aggressor grows by a rate of cdY after t = t1, of c2dY after t = t2 and, finally, of cNdY after t = tN; see Fig. 1. Physically, dY and c are defined as the averaged volume increment and the non-blocking factor (i.e. the unblocked part of the averaged volume increment) for a single particle, respectively. Actually, one time blocking decreases the volume increment by 1  c times.2 Please note that the above statistical elucidation for blocking effect implies that all the blockers will be effective in interfering with the aggressor; the event that a blocker may be interfered with by other particles (blocker-blockers) is ignored.3 Denote the probability function for a particle to encounter only N-scale blocking as pN(t). In particular, p0(t) indicates the probability of the particle being unblocked. The overall probability that a particle has been blocked before reaching the origin is the sum of probabilities that it has been blocked only once, twice and so on (i.e. p1(t) + p2(t) +    + pN(t), if the N-scale blocking occurs without any higher-scale blocking). On this basis, the two events, a particle being blocked and unblocked, are complementary events and, accordingly, the probabilities of the two events satisfy p1 ðtÞ þ p2 ðtÞ þ    þ pN ðtÞ ¼ 1  p0 ðtÞ

ð16Þ

And then, the probability that the aggressor, undergoing the N-scale blocking without any higher-scale blocking, will transform the origin at time t, i.e. the increment of new extended fraction xen, can be expressed as dxen ¼ p0 ðtÞdxe þ p1 ðtÞcdxe þ p2 ðtÞc2 dxe þ    þ pN ðtÞcN dxe ð17Þ It should be mentioned that Eq. (17) represents the statistical contributions of different degrees of anisotropic effect to the increment of the extended fraction. This

2 In order to make the statistical treatment tractable, c is assumed constant (0 < c < 1), i.e. the effect of each blocking on the averaged volume increment always holds constant. Of course, c only represents the averaged effect between two successive blockings, since growth terminates at all points of contact, but continues unabated elsewhere when impingement occurs. Under practical conditions, the effect of each scale blocking on average volume increment is certainly not always the same. 3 In Refs. [23,24,36,37], the influence of blocker-blockers (i.e. the mutual blocking of anisotropically growing particles up to all relevant orders) is considered, using Monte Carlo simulations. It appears that such events will contribute only to a higher-order correction. And thus, in the current statistical analysis, it appears to be a reasonable hypothesis to consider the effect as negligible; the first-order blocking (as treated in Refs. [28–34]) is merely considered.

strongly implies that the anisotropic effect on the transformation depends on not only c, but also n; the coexistence of multiple blocking prevails. Comparing Eqs. (17) and (15), one can obtain a timedependent, averaged, non-blocking probability function S(t): SðtÞ ¼

N X

ci pi ðtÞ

ð18Þ

i¼0

Then, according to Eq. (2), the probability that the origin O will have remained untransformed after a given time t is given by  Z t  0 0 Sðt Þdxe ðt Þ ð19Þ qðtÞ ¼ exp  0

with xe as the JMAK-like extended fraction neglecting the blocking effect (i.e. equivalent isotropic growth), as a function of t. Finally, the transformed fraction f is just equal to 1  q(t). Note that the current model still follows some restrictions of JMAK-like equation, e.g. the nuclei are randomly dispersed in an infinite space. In the following sections, the probability functions pN(t) and S(t) under different conditions are derived. First, the extreme case where an aggressor undergoes infinite-scale blocking is studied, where a rigorous expression for the transformed fraction is derived, to demonstrate clearly that the phenomenological equation (i.e. Eq. (13a) or (13b)) corresponds to the extreme case with n = 2  c. Then the kinetic equations for finite-scale blocking (e.g. the cases of 1-scale blocking and k-scale blocking under practical conditions) are obtained. 3.2. Infinite-scale blocking 3.2.1. Derivation for probability function pN(t) As for particles subjected to infinite-scale blocking, the blocking scale N approaches infinity. As stated in Section 2.3, the probability that a particle (aggressor) will transform the origin at time t can be given as dxen ¼ Sðxe ðt0 ÞÞdxe ðt0 Þ

ð20Þ

Here it is assumed that the probability that any blocker (without resorting to other blockers) will reach to the edge of the aggressor at t is still expressed as Eq. (20). Thus, the probability for a particle unblocked until time t, p0(t), can be expressed as  Z xe  p0 ðtÞ ¼ exp  Sðx0e Þdx0e ð21Þ 0

Following the basic statistical treatment for scale dimensionality, the probability for any aggressor nucleated in the time interval [s, s + ds], to grow from s to t1 without blocking and to be blocked in the interval [t1, t1 + dt1], and to grow to the origin at t without further blocking, as shown in Fig. 1, is given by

S.J. Song et al. / Acta Materialia 59 (2011) 3276–3286

 Z p1 ðt1 Þdt1 ¼ exp 

t1

 Sðt0 Þdxe ðt0 Þ Sðt1 Þdxe ðt1 Þ

0

 Z t  0 0  exp  Sðt Þdxe ðt Þ

ð22Þ

t1

Integrating Eq. (22), the probability of a particle being blocked only once until the origin transformed at t, i.e. p1(t), can be written as  Z t1  Z t exp  Sðt0 Þdxe ðt0 Þ Sðt1 Þdxe ðt1 Þ p1 ðtÞ ¼ 0 t

0

 Z  Sðt0 Þdxe ðt0 Þ  exp 

ð23Þ

t1

Simplification of Eq. (23) gives Z xe   Z xe  0 0 0 0 p1 ðtÞ ¼ Sðxe Þdxe exp  Sðxe Þdxe 0

3281

The summation term at the right-hand side of Eq. (29) is just the Taylor’s series expansion, such as the form 1 X 1 ½ktN ¼ expðktÞ ð30Þ N ! N ¼0 Accordingly, Eq. (29) can be rewritten as  Z t  Sðxe ðtÞÞ ¼ exp  ð1  cÞSðxe ðsÞÞdxe ðsÞ

As an ordinary differential equation (ODE) about xen and xe, in combination with dxen = S(xe(t))dxe, Eq. (31) can be further rewritten as dxen ¼ exp½ð1  cÞxen  dxe

ð32Þ

Solving Eq. (32) leads to ð24Þ

0

Analogously, the probability of a particle being blocked only twice until the origin transformed at t can be expressed as   Z t2  Z t Z t2 0 0 0 0 p2 ðtÞ ¼ Sðt Þdxe ðt Þ exp  Sðt Þdxe ðt Þ 0 0 0  Z t  Sðt2 Þdxe ðt2 Þ exp  Sðt0 Þdxe ðt0 Þ ð25Þ t2

xen ¼

1 ln½1 þ ð1  cÞxe  1c

ð33Þ

Applying dxen = S(xe(t))dxe once again, it can be obtained from Eq. (33) as 1 Sðxe Þ ¼ ð34Þ 1 þ ð1  cÞxe Furthermore, the transformed fraction f accounting for the infinite-scale blocking arising from the anisotropic growth can be expressed as 1

Simplification of Eq. (25) gives Z xe 2  Z xe  1 0 0 0 0 Sðxe Þdxe exp  Sðxe Þdxe p2 ðtÞ ¼ 2 0 0

f ¼ 1  expðxen Þ ¼ 1  ½1 þ ð1  cÞxe 1c ð26Þ

Consequently, the probability function for each particle undergoing the Nth blocking until the origin transformed at t, i.e. pN(t), can be obtained as Z xe N  Z xe  1 pN ðtÞ ¼ Sðx0e Þdx0e exp  Sðx0e Þdx0e ð27Þ N! 0 0 Clearly, the probability for a particle subjected to the Nth blocking obeys the Poisson distribution. Furthermore, the sum of pN(t) satisfies 1 X pN ðtÞ ¼ 1 ð28Þ N ¼0

3.2.2. Function of non-blocking probability and transformed fraction Inserting Eq. (27) into Eq. (18) with N approaching infinity yields 1 X cN pN ðtÞ Sðxe ðtÞÞ ¼ N ¼0

 Z t N 1 X 1 c ¼ Sðxe ðsÞÞdxe ðsÞ N! 0 N ¼0  Z t   exp  Sðxe ðsÞÞdxe ðsÞ 0

ð31Þ

0

ð29Þ

ð35Þ

Note that Eq. (35) is exactly the same as the phenomenological Eq. (13b) with n = 2  c (0 < c < 1). This strongly shows that the phenomenological treatment corresponds just to the extreme case where the particle encounters infinite-scale blocking. Under practical conditions, however, it is impossible for a particle to encounter infinite-scale blocking; generally, only finite-scale blocking can appear. Thus, the transformed fraction predicated by Eq. (35) is overestimated by the infinite-scale blocking. 3.3. One-scale blocking If a particle encounters 1-scale blocking without any higher-scale blocking, two events, i.e. the probability of the particle unblocked p0(t) and the probability of the particle once blocked p1(t), occur and follow according to Eq. (16): p0 ðtÞ þ p1 ðtÞ ¼ 1

ð36Þ

So, the probability that a particle (aggressor) will transform the origin at time t, i.e. dxen, can be, according to Eq. (17), given as dxen ¼ Sðxe Þdxe ¼ p0 ðtÞdxe þ p1 ðtÞcdxe

ð37Þ

where the p0(t) can still be given as Eq. (21). Thus,  Z xe  0 0 p1 ðtÞ ¼ 1  p0 ðtÞ ¼ 1  exp  Sðxe Þdxe

ð38Þ

0

3282

S.J. Song et al. / Acta Materialia 59 (2011) 3276–3286

From Eq. (37), the function of non-blocking probability S(xe) can be obtained as  Z xe  0 0 Sðxe Þ ¼ exp  Sðxe Þdxe 0   Z xe  0 0 Sðxe Þdxe þ c 1  exp  ð39Þ 0

Eq. (39) is still a differential Requation for xen and xe, with x S(xe) = dxen/dxe and xen ¼ 0 e Sðx0e Þdx0e , whose analytical solution follows c  1 þ expðcxe Þ c Meanwhile, S(xe) can be obtained as

xen ¼ ln

Sðxe Þ ¼

c expðcxe Þ c  1 þ expðcxe Þ

ð40Þ

ð41Þ

(see Eq. (34)). The transformed fraction f accounting for the 1-scale blocking arising from the anisotropic growth can be expressed as c f ¼ 1  expðxen Þ ¼ 1  ð42Þ c  1 þ expðcxe Þ 3.4. k-scale blocking Analogously, if a particle encounters k-scale blocking without any higher-scale blocking, k + 1 events, i.e. the probability of the particle unblocked p0(t) and the probabilities of the particle blocked only once p1(t), twice p2(t), . . .and k-scale pk(t), occur and follow from Eq. (16): k X

pi ðtÞ ¼ 1

ð43Þ

i¼0

The probability of a particle encountering the previous (k  1) scales blocking satisfies Eq. (27): pi ðtÞ ¼

1 i!

Z 0

xe

i  Z Sðx0e Þdx0e exp 

xe

 Sðx0e Þdx0e ;

i ¼ 0; 1; . . . ; k  1

0

ð44Þ

But the kth blocking satisfies the probability as follows: Z xe i  Z xe  k1 X 1 0 0 0 0 Sðxe Þdxe exp  Sðxe Þdxe pk ðtÞ ¼ 1  i! 0 0 i¼0 ð45Þ Applying dxen ¼ Sðxe Þdxe , S(xe) can be, according to Eqs. (18), (44), and (45), rewritten as the following differential equation: k1 X dxen 1 i ½xen  expðxen Þðci  ck Þ ¼ ck þ i! dxe i¼0

ð46Þ

Since Eq. (46) is a non-linear ODE, the exact analytical solution to this equation is not feasible. Numerical calculations have shown that the transformed fraction accounting for the k-scale blocking arising from the anisotropic growth can still be obtained through f = 1  exp(xen(t)); see Appendix A.

4. Discussion and application As shown in Section 3, the derivation of the present model accounting for the blocking effect arising from anisotropic growth only rests on the relation between the new extended fraction xen and the JMAK-like xe, but does not depend on the prevailing modes of nucleation and growth, as well as the transformation path, i.e. isothermally or isochronally conducted process. Therefore, the present models, which describe the kinetics of transformations subjected to the anisotropic effect, are expected, in conjunction with the modular transformation kinetics, to provide physically realistic explanations for real experiments. Owing to space limitation, however, abundant applications of the present models, especially for isochronal transformation, as well as a comparative study between the present model regarding the one-dimensional process of anisotropic growth and the exact model proposed by Weinberg and Birnie III will be performed elsewhere. The present work focuses mainly on the blocking effect arising from anisotropic growth. In the following sections, first, the time-dependent Avrami exponent (for isothermal transformation) as well as the invariant effective activation energy are shown; then, the contribution of non-blocking factor c and blocking scale k to the anisotropic effect is discussed; finally, a published experiment, i.e. isothermal crystallization of amorphous Fe33Zr67 ribbon at 663 K, is considered as a representative example to show application of the present model. 4.1. Avrami exponent and effective activation energy Generally, for isothermal transformation, the Avrami exponent is often evaluated by plotting ln(ln(1f)) vs. ln t (from one single transformation), whereas the effective activation energy is evaluated by plotting ln t vs. 1/T (from multi-transformations performed at different temperatures) [11,12]. Applying f = 1  exp(xen(xe)), in combination with the statistical treatment of blocking effect, a new expression for the Avrami exponent nnew is according to the present model deduced as nnew ¼

d lnð lnð1  f ÞÞ d ln xen d ln xe xe ¼ ¼ Sðxe Þn d ln t d ln xe d ln t xen

ð47Þ

where, n actually corresponds to the analytical expression obtained from the modular model for transformation, assuming combinations of different nucleation and growth modes (Eq. (11)) (see Refs. [44–46]). For transformations incorporating the anisotropic growth, it is known a priori that nnew is declined with the fraction transformed, since the transformation is retarded by the blocking effects. Analogously, a new expression of effective activation energy for transformation incorporating the anisotropic growth can be obtained as follows Qnew ¼

d ln t d lnð lnð1  f ÞÞ=dð1=T Þ d ln xe =dð1=T Þ R¼ R¼Q R¼ d lnð lnð1  f ÞÞ=d ln t d ln xe =d ln t dðT1 Þ ð48Þ

S.J. Song et al. / Acta Materialia 59 (2011) 3276–3286

It should be noted that, in Eq. (48), Q is, according to the modular model (Eq. (11)) [11,12,44–46], expressed as (d/mQG + (n  d/m)QN)/n. This strongly indicates that the effective activation energy is not affected by the blocking effect; see Section 4.3. From now on, nnew and Qnew can be defined as the generalized n and Q; in the following sections, nnew tends to n only if the blocking effect is alleviated, and Qnew holds equal to Q covering throughout the whole transformation, independent of the blocking effect.

3283

Assuming continuous nucleation and 3D interface-controlled growth with model parameters N0 = 1  1028 m3 s1, v0 = 1  1010 m s1, QN = 200 kJ mol1, and QG = 300 kJ mol1, for isothermal transformation at T = 800 K, the evolution of f with t and the evolution of nnew with f are obtained, subjected to the same-scale blocking but different c (e.g. only 1-scale blocking occurring, but

c = 0.2, 0.4, 0.6, 0.8; see Fig. 2), and subjected to differentscale blocking but the same c (e.g. c = 0.4 and k-scale blocking with k = 0–3; see Fig. 3). Note that the application of continuous nucleation is aimed solely at elucidating briefly the contributions of non-blocking factor c and blocking scale k to the anisotropic growth. Evaluation of the current model with regard to mixed nucleation is shown in Section 4.3. Figs. 2 and 3 show clearly that the transformation accounting for the blocking effect arising from anisotropic growth depends on not only c, but also k. From Eq. (47), xe/xen and S(t) act as the two key factors to evaluate the Avrami exponent nnew, from which the effect of anisotropic growth can thus be reflected. Applying the same model parameters as those used for Figs. 2 and 3, the evolution of xe/xen and S(t) with f, as well as the evolution of nnew and n with f, subjected to c = 0.4 and 1-scale blocking, is calculated and shown in Fig. 4. Clearly, xe/xen increases monotonously with f, but S(t) declines with f, which implies that three basic regimes of transformation could be identified. At the initial stage of transformation, xe/xen and S(t) both approach unity; the equivalent

Fig. 2. Evolution of f with t and nnew with f for transformation subjected to the same-scale blocking but different c (here, only 1-scale blocking is occurring and c = 0.2, 0.4, 0.6, 0.8), assuming continuous nucleation and 3D interface-controlled growth. Values of the model parameters are given in Section 4.1.

Fig. 3. Evolution of f with t and nnew with f for transformation subjected to different-scale blocking, but the same c (here, c = 0.4 and k-scale blocking with k = 0–3), assuming continuous nucleation and 3D interfacecontrolled growth. Values of the model parameters are given in Section 4.1.

4.2. Anisotropic effect: non-blocking factor c and blocking scale k

3284

S.J. Song et al. / Acta Materialia 59 (2011) 3276–3286

isotropic particles each grow independently, without the blocking effect. At the middle stage of transformation, a severe blocking effect occurs; nnew does have a minimum value at some time. At the final stage of transformation, however, the blocking effect is alleviated, so that nnew tends to the value where the blocking has not occurred, i.e. n; see Fig. 4 and Section 4.3. 4.3. Model fit to isothermal crystallization of amorphous Fe33Zr67 ribbon at 663 K Isothermal crystallization of melt-spun amorphous Fe33Zr67 ribbon was described by Sun et al. [47], where the transformed fraction as a function of t, as well as the evolution of nnew and Qnew with f (deduced using the Avrami plot (see Fig. 8 in Ref. [47] or Fig. 1 in Ref. [48]) and Eq. (48) (see Fig. 9 in Ref. [47])), was presented. Experimental details regarding specimen preparation and differential scanning calorimetry analysis are given in Ref. [47]. In the present work, the transformed fraction as a function of t, as well as the evolution of nnew and Qnew with f, at T = 663 K, is adopted for model analysis, i.e. fits of the present model to the obtained results; see Fig. 5a–c. Values for the fitting parameters are given in Table 1. The model parameters N0, N, v0, QG, QN and a were determined by fitting the present model, i.e. Eqs. (46) and (47) in combination with the modular model (Section 2.2), to the f–t and nnew–f curves as obtained for the holding temperature, simultaneously. The fitting starts with an assumed set of initial values of N0, N, v0, QG, QN and a, as well as adopting d/m = 2 (with reference to Ref. [47]). Substitution of values for these model parameters into Eqs. (9)–(11), further into Eqs. (46) and (47), allows the values of f and nnew to be calculated. The least squares difference between the calculated f and nnew values and those obtained experimentally was minimized using a downhill

Fig. 5. Evolution of (a) f with t, (b) nnew with f and (c) Qnew with f, for the crystallization of amorphous Fe33Zr67 alloy at T = 663 K: symbols, experimentally deduced results from Ref. [47]; line, fitted or calculated results from the present models (e.g. Eq. (46)) with 2-scale blocking. Values of the model parameters and error are given in Table 1.

Fig. 4. Evolution of xe/xen and S(t) with f, as well as evolution of n and nnew with f, for transformation subjected to 1-scale blocking and c = 0.4, assuming continuous nucleation and 3D interface-controlled growth. Values of the model parameters are given in Section 4.1.

simplex fitting procedure by altering the values for the model parameters. The goodness of the fits (indicated as “error” in Table 1) was calculated as the sum of the absolute differences between the measured and the fitted f and nnew values, divided by the sum of the absolute values of the fitted f and nnew values, of both curves fitted simultaneously.

S.J. Song et al. / Acta Materialia 59 (2011) 3276–3286

3285

Table 1 Kinetic parameters as determined by fitting the present models to the f–t and nnew–f curves obtained from the isothermal crystallization of amorphous Fe33Zr67 ribbons at T = 663 K. k-scale blocking 1 2 3 4 5

N (m3)

N0 (m3 s1) 20

6.70  10 6.64  1020 6.83  1020 6.91  1020 6.43  1020

25

3.23  10 3.24  1025 3.20  1025 3.21  1025 3.10  1025

QN (kJ mol1) 5

2.58  10 2.53  105 2.53  105 2.53  105 2.50  105

QG (kJ mol1) 5

2.96  10 2.98  105 2.97  105 2.97  105 2.96  105

v0 (m s1)

a

c

Error (%)

9.11  1010 8.98  1010 9.19  1010 9.24  1010 8.71  1010

3.062 3.063 3.057 3.056 2.953

0.0826 0.2943 0.46 0.56 0.648

4.33 4.03 5.20 6.35 7.07

the value (n; see Figs. 5b and 6) where the blocking effect is severely alleviated at the end of transformation. Using the fitted parameters given in Table 1, Qnew is calculated in terms of Q = (d/mQG + (n  d/m)QN)/n, and subsequently compared with the value experimentally deduced; see Fig. 5c. Clearly, both evolutions of the effective activation energy with f show that, for f = 0.1–0.9, Qnew holds almost equivalent to Q. This strongly supports the conclusion below Eq. (48), i.e. the effective activation energy is not affected by the blocking effect arising from anisotropic growth. 5. Conclusion Fig. 6. Evolution of xe/xen and S(t) with f, as well as evolution of n and nnew with f, to analyze the best model fit to the crystallization of amorphous Fe33Zr67 alloy at T = 663 K, applying the fitted parameters corresponding to k = 2 in Table 1.

From Table 1 and Fig. 5a, the combination of mixed nucleation (incorporating nucleation index a  3)4 and 2D interface-controlled anisotropic growth (with 2-scale blocking, k = 2 and c  0.3) fit the isothermal crystallization of amorphous Fe33Zr67 ribbons at T = 663 K best. Applying the fitted model parameters corresponding to k = 2 (Table 1), the evolution of xe/xen and S(t) with f analogous to Fig. 4, as well as the evolution of n and nnew with f, are calculated and illustrated in Fig. 6, where n is according to the modular model incorporating the nucleation index (a = 3) calculated. As is highlighted in Section 4.2, a continuous increase in xe/xen is accompanied by a continuous decrease in S(t) with f. This can be basically reflected from Figs. 5b and 6, where, at the initial stage of transformation, nnew is increased owing to the prevailing isotropic growth and, at the middle stage, nnew is decreased owing to the strengthened blocking effect (i.e. arising from the anisotropic growth), but after the occurrence of the minimum nnew value, it almost returns to

4 For transformation assuming mixed nucleation and 2D interfacecontrolled growth, the Avrami exponent should be within (2, 3). Regarding Fig. 5b, the abnormal nnew (>3) in this case should be ascribed to increased nucleation rate, which can be interpreted using the so-called nucleation index a (Table 1). A detailed description for the nucleation index is available in Refs. [12,16]. This is not our focus in the present work.

A statistical process accounting for the blocking effect arising from the anisotropically growing particles was performed. An anisotropically growing particle, if not influenced by other particles, is considered as an isotropically growing particle with invariable volume. Accordingly, how to study the anisotropic growth just becomes how to solve the problem of blocking effect. On this basis, a series of analytical models for solid-state transformation, where a particle undergoes 1-scale blocking, k-scale blocking and infinite-scale blocking, were obtained. It is concluded that the anisotropic effect on the transformation depends on two factors: the non-blocking factor c and the blocking scale k. The phenomenological equation accounting for the anisotropic effect (i.e. Eq. (13)) corresponds to a limiting case where a particle encounters infinite-scale blocking. The Avrami exponent, subjected to the anisotropic effect, changes as a function of the transformed fraction, whereas, the effective activation energy is not affected by the anisotropic effect. The present models were successfully applied to the isothermal crystallization of amorphous Fe33Zr67 ribbon at 663 K and a good fit was achieved. Acknowledgements The authors are grateful for financial support of the Free Research Fund of State Key Laboratory of Solidification Processing (09-QZ-2008 and 24-TZ-2009), the 111 project (B08040), the Natural Science Foundation of China (Grant No. 51071127, 50901059), the Huo Yingdong Yong Teacher Fund (111502), the Fundamental Research Fund of Northwestern Polytechnical University (JC200801) and National Basic Research Program of China (973 Program)

3286

S.J. Song et al. / Acta Materialia 59 (2011) 3276–3286

2011CB610403. S.J.S. is also grateful to the Doctorate Foundation of Northwestern Polytechnical University (No. CX201008).

[3] [4] [5] [6]

Appendix A Here, the numerical solution to the ODE (Eq. (46)) is described. In order to make the numerical calculation more accurate and tractable, the classical fourth-order Runge– Kutta (RK) method in combination with the built-in MATLABÒ [49] solver ODE45 is adopted. First, Eq. (46), a differential equation about xen and independent variable xe, is rewritten as the following equation about independent variable t " # k1 X dxen dxen dxe 1 dxe i k i k ½xen  expðxen Þðc  c Þ ¼ ¼ c þ i! dt dxe dt dt i¼0 ðA1Þ The rate of extended fraction of equivalent isotropic particles dxe/dt, for isothermal transformation, can be obtained from the modular model as stated in Section 2.2, as follows:   d=m dxe d Q d ¼ gN  m0 exp  G tm1 ; m dt RT

for site saturation nucleation ðA2Þ

   d=m dxe QN QG d ¼ gN 0 exp  tm ; m0 exp  dt RT RT for continuous nucleation

ðA3Þ

For mixed nucleation representing a combination of site saturation and continuous nucleation, the rate dxe/dt can be expressed as the linear summation of Eqs. (A2) and (A3). And then, applying the classical fourth-order RK method with the initial-value condition xen = 0 when t = 0, the value of xen can be obtained. Furthermore, since the fitting of the present model to experiments is based on MATLAB, the built-in function ODE45, a modified RK method which can choose the step size at each step in an attempt to achieve the desired accuracy [50], is used as the numerical solver of Eq. (A1) in the fitting program. References [1] Johnson WA, Mehl RF. Trans Am Inst Min (Metall) Eng 1939;135:1. [2] Avrami M. J Chem Phys 1939;7:1109.

[7] [8] [9] [10] [11]

[12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45] [46] [47] [48] [49] [50]

Avrami M. J Chem Phys 1940;8:212. Avrami M. J Chem Phys 1941;9:177. Kolmogorov AN. Izv Akad Nauk SSSR Ser Fiz 1937;3:355. Christian JW. The theory of transformation in metals and alloys, part 1 equilibrium and general kinetics theory. 2nd ed. Oxford: Pergamon Press; 2002. Mittemeijer EJ. J Mater Sci 1992;27:3977. Mittemeijer EJ, Sommer F. Z Metall 2002;93:5. Kempen ATW, Sommer F, Mittemeijer EJ. J Mater Sci 2002;37:1321. Kempen ATW, Sommer F, Mittemeijer EJ. Acta Mater 2002;50:1319. Mittemeijer EJ. Fundamentals of Materials Science: The Microstructure–Property Relationship Using Metals as Model Systems, Chap. 9: Phase Transformations. Heidelberg: Springer Verlag; 2010. Liu F, Sommer F, Bos C, Mittemeijer EJ. Int Mater Rev 2007;52:193. Liu F, Sommer F, Mittemeijer EJ. J Mater Sci 2004;39:1621. Liu F, Sommer F, Mittemeijer EJ. Acta Mater 2004;52:3207. Liu F, Sommer F, Mittemeijer EJ. J Mater Res 2004;19:2586. Liu F, Sommer F, Mittemeijer EJ. J Mater Sci 2007;42:573. Kashchiev D. Surf Sci 1969;14:209. Kelton KF, Greer AL, Thompson CV. J Chem Phys 1983;79:6261. Tomellini M, Fanfoni M. Phys Rev B 2008;78:014206. Tomellini M. J Mater Sci 2010;45:733. Weinberg MC, Kapral R. J Chem Phys 1989;91:7146. Bradley RM, Strenski PN. Phys Rev B 1989;40:8967. Shepilov MP, Baik DS. J Non-Cryst Solids 1994;171:141. Pusztai T, Granasy L. Phys Rev B 1998;57:14110. Lement BS, Cohen M. Acta Metall 1956;4:469. Hillert M. Acta Metall 1959;7:653. Starink MJ. J Mater Sci 2001;36:4433. Birnie III DP, Weinberg MC. J Chem Phys 1995;103:3742. Weinberg MC, Birnie III DP. J Non-Cryst Solids 1995;189:161. Weinberg MC, Birnie III DP. J Chem Phys 1996;105:5138. Birnie III DP, Weinberg MC. Physica A 1996;230:484. Birnie III DP, Weinberg MC. Physica A 1996;223:337. Weinberg MC, Birnie III DP. J Non-Cryst Solids 1996;202:290. Weinberg MC, Birnie III DP. J Non-Cryst Solids 1997;219:89. Shepilov MP, Bochkariov VB. Fiz Khim Stekla 1991;17:674. Kooi BJ. Phys Rev B 2004;70(22):224108. Kooi BJ. Phys Rev B 2006;73(5):054103. Liu F, Yang GC. Acta Mater 2007;55:1629. Starink MJ. Int Mater Rev 2004;49:191. Starink MJ, Zahra AM. Acta Mater 1998;46:3381. Tagami T, Tanaka SI. Acta Mater 1997;45:3341. Tagami T, Tanaka SI. Acta Mater 1998;46:1055. Fanfoni M, Tomellini M, Volpe M. Phys Rev B 2002;65(17):172301. Liu F, Song SJ, Xu JF, Wang J. Acta Mater 2008;56:6003. Liu F, Song SJ, Sommer F, Mittemeijer EJ. Acta Mater 2009;57:6176; Liu F, Song SJ, Sommer F, Mittemeijer EJ. Acta Mater 2010;58:2291. Liu F, Nitsche H, Sommer F, Mittemeijer EJ. Acta Mater 2010;58:6542–53. Sun NX, Zhang K, Zhang XH, Liu XD, Lu K. Nano Mater 1996;7:637. Sun NX, Liu XD, Lu K. Scripta Mater 1996;34:1201. Matlab. . Dormand JR, Prince PJ. J Comput Appl Math 1980;6:19.