Barrier-free heterogeneous grain nucleation in polycrystalline materials: The austenite to ferrite phase transformation in steel

Acta Materialia 55 (2007) 4489–4498 www.elsevier.com/locate/actamat

Barrier-free heterogeneous grain nucleation in polycrystalline materials: The austenite to ferrite phase transformation in steel N.H. van Dijk a

a,*

, S.E. Offerman b, J. Sietsma b, S. van der Zwaag

c

Fundamental Aspects of Materials and Energy, Department of Radiation, Radionuclides and Reactors, Delft University of Technology, Mekelweg 15, 2629 JB Delft, The Netherlands b Department of Materials Science and Engineering, Delft University of Technology, Mekelweg 2, 2628 CD Delft, The Netherlands c Faculty of Aerospace Engineering, Delft University of Technology, Kluijverweg 1, 2629 HS Delft, The Netherlands Received 9 January 2007; received in revised form 4 April 2007; accepted 12 April 2007 Available online 1 June 2007

Abstract The process of grain formation during structural solid-state phase transformations has commonly been analysed in terms of the classical nucleation theory. In polycrystalline materials a new phase generally forms at grain boundaries in the parent phase microstructure. Under certain conditions the net interfacial energy to form a new phase can be relatively small due to the release of grain boundary energy from the parent phase. The nucleation process is then governed by cluster dynamics. We propose a simple model to predict the cross-over between different heterogeneous nucleation regimes and apply it to recent synchrotron X-ray diffraction experiments on ferrite nucleation in steel.  2007 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Classical nucleation theory; Heterogeneous nucleation; Solid-state phase transformation; Steel

1. Introduction Solid-state phase transformations often form an essential step in the processing of polycrystalline materials. A detailed understanding of these transformations is of both fundamental interest and technological importance for the formation of many polycrystalline materials. During a structural phase transformation two processes can be identified: nucleation and the subsequent growth of the newphase grains [1]. At the time of nucleation a small cluster of the new phase, which fluctuates in size, reaches a critical size beyond which it will no longer dissolve in the matrix of the parent phase. During growth the stable nuclei increase in size until the material is completely transformed. The process of nucleation is widely studied in a large variety of materials [2,3]. The classical nucleation theory [4] is commonly used to describe both homogeneous nucleation for systems without a preferred site for nucleation and hetero*

Corresponding author. Tel.: +31 15 2786775; fax: +31 15 2788303. E-mail address: [email protected] (N.H. van Dijk).

geneous nucleation for systems with preferred nucleation sites, but unfortunately often gives a poor quantitative prediction of the nucleation rate. Besides nucleation, phase condensation can involve spinodal decomposition, dispersed cluster growth and cluster aggregation (coalescence) [5]. The current understanding of nucleation is still limited due to the experimental difficulty of monitoring the formation of relatively small nuclei in the bulk of the material and the computational resources needed to evaluate the stability of the fluctuating clusters in the material. Recent advances in this field are laser confocal microscopy experiments [6] and numerical simulations [7] on the crystallization behaviour of colloids. Over the last decades the classical nucleation theory has been widely applied to describe the heterogeneous nucleation of ferrite grains during the austenite to ferrite transformation in steels [1]. During the structural phase transformation that occurs below the transformation temperature A3(1000–1200 K), the austenite phase (c-Fe) with a face-centred cubic (fcc) lattice structure and a high solubility of interstitial carbon transforms into the ferrite

1359-6454/$30.00  2007 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.actamat.2007.04.013

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phase (a-Fe) with a body-centred cubic (bcc) lattice structure and a low solubility of interstitial carbon. Over the years a variety of idealized model geometries have been proposed [8–12] to describe the initial stage of the ferrite nucleus and to characterize the relevant interfacial energies. However, surprisingly little experimental work has been done to test the validity of the proposed model geometries, or the basic assumptions of the classical nucleation theory itself. Rare examples of these studies are the ex situ investigations by Enomoto and Aaronson [9–12] and by Huang and Hillert [13]. Both studies revealed that austenite grain corners are the most favorable nucleation sites for the formation of ferrite grains. The difficulty with ex situ studies, however, is that the temperature quench, required to study the material at room temperature, can significantly alter the high temperature microstructure. Recent advances in instrumentation at synchrotron sources, like the European synchrotron radiation facility (ESRF), made it possible to use a high flux of hard X-rays (energies more than 50 keV) to monitor the transformation kinetics of individual grains within the bulk of the material with a high spatial and time resolution. The development of the three-dimensional X-ray diffraction (3DXRD) microscope at the ESRF has made it possible to study microstructures on the scale of individual (sub)grains in unprecedented detail [14–19]. In recent X-ray diffraction experiments on the transformation kinetics of the ferrite to austenite transformation in steels [17,20,21], we monitored the nucleation behavior of ferrite grains and showed that four different types of ferrite growth can be identified for individual ferrite grains within the austenite matrix. From the temperature dependence of the nucleation rate during continuous cooling we deduced a parameter W that relates the energy barrier for heterogeneous nucleation (DG*) to the driving force for nucleation (Dl). This parameter characterizes the net interfacial energy required to form a new grain. The experimentally observed value for parameter W was found to be two (or more) orders of magnitude smaller than the available model predictions [17,20]. In the present article, we reinvestigate heterogeneous nucleation for structural solid-state phase transformations. First, we will present a thermodynamic description of the nucleation process. Subsequently, a general classification of possible nucleation regimes is made, depending on the strength of the chemical driving force for the transformation and the net interfacial energy required to form the new phase within the matrix of the parent phase. Subsequently, the transformation kinetics for the different nucleation regimes is evaluated. The classification of these nucleation regimes is analysed in detail for medium-carbon steel, and compared with recent experimental synchrotron X-ray data on the kinetics of ferrite grain formation during isothermal transformations. We will show that the steadystate nucleation predicted by the classical nucleation theory only applies for a limited range of temperatures and ferrite fractions. For an undercooling of more than 5 K with respect to the transformation temperature, one of the main

assumptions in the classical nucleation theory required to reach a steady-state nucleation regime (that the energy barrier for nucleation DG* is larger than the thermal energy kBT) is found to break down. For an increasing undercooling, the energy barrier for nucleation becomes very weak and eventually vanishes. In this effectively barrier-free nucleation regime the cluster size distribution of the new phase and the resulting nucleation rate are intrinsically time-dependent and evolve during the transformation. 2. Thermodynamic description of nucleation When a system is cooled below the phase transformation temperature T0, the Gibbs free energy of the system can be lowered by the formation of a new phase. The formation of this new phase requires the creation of a new interface between the new phase and the parent phase, which costs interfacial energy. In the case of a solid-state phase transformation in a polycrystalline material, the new phase forms preferentially at the grain boundaries of the parent phase and, as a consequence, grain boundary energy of the parent phase is released during the formation of the new phase. Generally there is a net energy barrier to form the energetically favourable new phase. The transformation rate to reach thermodynamic equilibrium depends strongly on the size of the energy barrier to form the new phase with respect to the kinetic energy of the atoms. The change in Gibbs free energy between the new ferrite phase and the parent austenite phase, also known as the chemical driving force for the transformation, DGch = Dln is proportional to the number of atoms n in the cluster of the new phase and the difference in chemical potential Dl = lp  ln between the parent phase (lp) and the new phase (ln). The difference in chemical potential Dl generally increases for a growing undercooling DT = T0  T with respect to the transformation temperature T0. In polycrystalline materials the formation of a cluster of the new phase within the matrix of the parent phase leads to additional interface energy from the interfaces between the new phase and the parent phase, rnp, and a release of grain boundary energy from the boundaries between different grains of the parent phase, rpp. The net interfacial Gibbs P P free energy of the cluster amounts to DGs ¼ i Ainp rinp  j Ajpp rjpp , where Anp is the surface area of the newly formed interface between the new phase and the parent phase and App is the surface area of the consumed boundaries between different grains of the parent phase. The indices i and j are introduced to demonstrate that the interface and grain boundary energies generally vary over different parts of the grain surface, depending on the relative crystal orientation of the new and transforming (parent) grain with the neighbouring grains. Assuming that the shape of the cluster is independent of the size, the net interfacial Gibbs free energy scales with the total surface area of the cluster, and therefore as DGs  n2/3. The net Gibbs free energy needed to form the cluster interface of the new phase can therefore be

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expressed as DGs = DXn2/3, where DX is a proportionality constant that depends on the geometry of the cluster and the interfacial energies involved. The total change in Gibbs free energy of the cluster DG = DGch + DGs as a function of the size n is given by: DGðnÞ ¼ Dln þ DXn2=3 :

ð1Þ

This equation can be rewritten in terms of dimensionless parameters when the Gibbs free energy of the cluster is normalized by the kinetic energy kBT: cðnÞ ¼ an þ bn2=3 ;

ð2Þ

with c = DG/kBT, a = Dl/kBT and b = DX/kBT. In Fig. 1 the characteristic behaviour of c(n) as a function of n is shown. For a > 0 (T < T0) and b > 0 the relative Gibbs free energy c(n) shows a maximum c* = c(n*) at a critical cluster size n*: 8b3 27a3 an 4b3 c ¼ ¼ > 0: 2 27a2

n ¼

ð3Þ ð4Þ

In previous studies [17,20] we have shown that it is useful to introduce a parameter W, which expresses the relation between the chemical driving force per unit volume DGv = N0Dl and the activation energy for nucleation DG ¼ W=DG2v ¼ W=ðN 0 DlÞ2 , where N0 is the number density of atoms (1/N0 corresponds to the volume associated to a single atom). The parameter W combines all the relevant information on the formed interfaces and the consumed grain boundaries of the cluster and therefore, implicitly also the geometry of the cluster. From an evaluation of Eq. (4) we can see that W is directly related to DX:

W ¼ ðN 0 DlÞ2 DG ¼

4491

4 2 3 N DX ; 27 0

ð5Þ

where we have used DG* = c*kBT = 4DX3/27Dl2. 3. Classification of the nucleation behaviour A significant achievement of the widely applied classical nucleation theory [2–4] is that, based on Eq. (1), it gives an accurate description of the rate at which stable clusters of the new phase are formed during the phase transformation (nucleation rate). The main result of the classical nucleation theory, which will be discussed in the next paragraph, is that after an initial stage a constant (steady-state) nucleation rate is reached. This result is, however, only valid for c* > 1 and n* > 1. For c* < 1 the energy barrier for the formation of stable clusters is too weak to be effective, while for n* < 1 the concept of a critical cluster breaks down. In both cases, there is effectively no barrier for nucleation. As a consequence, the prediction of the classical nucleation theory that after an initial stage, a stable cluster size distribution is formed that leads to the steady-state nucleation no longer applies. It is important to note that for a solidstate phase transformation with a change in crystal structure the minimum critical cluster size n* for which the new phase can be distinguished from the parent phase is generally larger than 1. For instance, to distinguish a new bcc cluster from an fcc parent phase a minimum of four atoms are needed. In the following, we will for generality however, use n* = 1 as a critical limit.

6 2

n* > 1 γ* > 1

5

n* = 1

B

βn2/3

4

γ* = 1

β

A γ(n)

1

3

γ(n)

D 2 γ∗

1

C

n* < 1 γ* < 1

0 0

-αn

0

n*

1

2

3

4

α 1

10

100

1000

10000

n Fig. 1. Relative Gibbs free energy for a cluster of the new phase c(n) = DG(n)/kBT as a function of the cluster size n. The value of c(n) is a sum of the relative driving force an and the relative net interfacial energy bn2/3 (plotted for a = 0.002 and b = 0.03). A maximum value of c* is observed for the critical cluster size n*.

Fig. 2. Dependence of the critical cluster size n* and the relative energy barrier for nucleation c* = DG*/kBT as a function of parameters a = Dl/kBT and b = DX/kBT. Four different regions can be distinguished: (A) n* > 1 and c* > 1, (B) n* < 1 and c* > 1, (C) n* > 1 and c* < 1, (D) n* < 1 and c* < 1. Region A is described by the classical nucleation theory. In regions B, C and D there is effectively no energy barrier for nucleation. The lines correspond to n* = 1 (b = 3a/2) and c* = 1 (b = 3(a/2)2/3).

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In Fig. 2 the dependence of n* and c* on the parameters a (related to the driving force) and b (related to the interfacial energies) is evaluated. When we consider the two validity limits (c* > 1 and n* > 1) of the classical nucleation theory independently, four different regions can be distinguished. The lines c* = 1 and n* = 1 that separate these regions reflect a gradual cross-over between qualitatively different types of nucleation behaviour. Region A is described by the classical nucleation theory with an effective energy barrier for nucleation. The other three regions (B–D) correspond to effectively barrier-free nucleation. Region B applies to strongly undercooled systems with a high net interfacial energy DX [2]. Region D applies to spinodal decompositions and cluster aggregations, where the net interfacial energy is very weak or absent [5]. Region C qualitatively differs from regions B and D as the energy barrier for nucleation is weak, while the critical cluster size is still sizeable. In region C, the cluster growth is initially uphill (n* > 1), while in regions B and D it is downhill from the start (n* < 1). Region C occurs for only a limited range of parameters and to our knowledge has not yet been studied. As we will see later, this region may be relevant for heterogeneous nucleation in solid-state phase transformations where the net interfacial energy DX is relatively low. In these systems the energy needed to form an interface between the new and the old phase is nearly compensated for by the removal of the grain boundary between different grains of the parent phase for certain preferential nucleation sites (e.g., grain corners). For the barrier-free nucleation in regions B–D, no steady-state nucleation is found. The rate of formation of new stable clusters of the new phase is controlled by cluster dynamics and is intrinsically time-dependent. A description of the cluster dynamics for barrier-free nucleation is given after the predictions of the classical nucleation theory. 4. Classical nucleation theory According to the classical nucleation theory, one finds for c* > 1 and n* > 1 that after an initial non-stationary stage, a stable cluster size distribution is reached, leading to steady-state nucleation. In the steady-state nucleation regime the formation rate of stable clusters (n > n*) is expressed by [2–4]:  N_ ss ¼ N p x Zec

ð6Þ

where N_ ss is the steady-state nucleation rate for the formation of new grains per unit volume, Np is the density of potential nucleation sites, x* is the rate constant and Z is the so-called Zeldovich factor. For conditions in which nucleation is relatively slow compared with growth, the potential nucleation sites are mainly consumed by growth. In this case Np is proportional to the untransformed volume fraction. Alternatively, for relatively fast nucleation, growth has a negligible effect (site saturation), and therefore, Np is mainly controlled by the number of grains formed per unit volume N.

The rate constant x*, which describes the frequency with which atoms are added to (or removed from) the critical cluster (with n* atoms), depends strongly on the physical mechanism controlling the cluster dynamics. For solidstate phase transformations with a change in lattice structure, the atoms of the parent phase and those of the new phase are always in direct contact and only need to rearrange their position at the interface. In this case, the rate constant x for a cluster of n atoms can be written as [2]: xðnÞ ¼ nint m0 expðQ=k B T Þ ¼ n2=3 c0 m0 expðQ=k B T Þ;

ð7Þ

where nint is the number of atoms of the parent phase that can potentially cross the interface to form part of the cluster of the new phase, m0 is the attempt frequency of the atoms and Q is the energy barrier to cross the interface from the parent phase to the new phase. The number of atoms of the parent phase that can potentially cross the interface nint is given by the number of parent-phase atoms that are in direct contact with the new phase and is therefore given by nint = c0n2/3, where c0 is a constant that only depends on the geometry of the cluster of size n. For a spherical cluster it amounts to c0 = (36p)1/3. For crystalline materials the attempt frequency is given by m0 = (3/5)1/ 2 (kBhD/h), where hD is the Debye temperature of the parent phase and h is the Planck constant [22,23]. The energy barrier to cross the interface Q may, to a first approximation, be estimated by the energy barrier for self-diffusion of atoms in the parent phase. The rate constant for a critical cluster is given by x* = x(n*) and amounts to: x ¼ ðn Þ2=3 c0 m0 expðQ=k B T Þ  2 4b ¼ ð8Þ c0 m0 expðQ=k B T Þ: 9a2 The Zeldovich factor Z accounts for the fact that only cluster sizes with an energy within kBT from DG* = DG(n*) can effectively cross the energy barrier for nucleation. The reciprocal of the Zeldovich factor, D* = 1/Z, describes the effective width, in terms of n, of the energy barrier with a maximum DG* that has to be crossed by the clusters to enable their passage from sub-critical to super-critical. The nucleation rate of Eq. (6) now expresses that D* = 1/Z successive atom attachments with a frequency x* and a probability exp(c*) are required to form a nucleus. It is found [2] that for c* > 3 the function c(n) can be approximated accurately by a parabolic expansion around c*:c(n)  c* + (1/2) (n*  n)2(d2c /dn2)n*. The curvature at the critical cluster size n* corresponds to:  2  dc 2c 9a4 ¼  ¼  < 0: ð9Þ 2 dn2 n 8b3 3ðn Þ The resulting Zeldovich factor is now expressed as [2–4]: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffi  ffi 1 d2 c 1 c 9a4 Z¼  ¼  ¼ 2 2p dn n n 3p 16pb3 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi N 20 Dl4 : ð10Þ  12pk B T W

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The factor Z is smaller than 1 and depends weakly on temperature compared with the exponential temperature dependencies contributing to the nucleation rate. For 1 < c* < 3 the parabolic approximation is inaccurate and more general expressions should be used [2]. 5. Barrier-free nucleation For the barrier-free nucleation in regions B–D of Fig. 2, no steady-state nucleation is found. The rate of formation of stable clusters of the new phase is controlled by cluster dynamics and is intrinsically time-dependent. The cluster dynamics responsible for the formation of stable grains of the new phase in the matrix of the parent phase can be characterized by the following simplified process where only single atom attachment and detachments of the cluster are considered: k ðn;nþ1Þ

½CðnÞ ¢ ½Cðn þ 1Þ; k ðnþ1;nÞ

ð11Þ

where C(n) is the concentration of new-phase clusters of size n and C(n + 1) is the concentration of clusters of size n + 1. The clusters of size n have a transition rate k(n,n+1) to clusters of size n + 1 and the clusters of size n + 1 have a transition rate k(n+1,n) to clusters of size n. The transition rates k(n,n+1) and k(n+1,n) are directly related to the sizedependent relative Gibbs free energy of the cluster [24,25]: ecðnþ1Þ eW ¼x ; cðnÞ þe 1 þ eW ecðnÞ 1 ¼ x cðnÞ ¼x ; cðnþ1Þ 1 þ eW e þe

k ðn;nþ1Þ ¼ x k ðnþ1;nÞ

ecðnþ1Þ

ð12Þ ð13Þ

where W = c(n+1)  c(n) and x  n2/3 is the size-dependent rate constant for atom attachments and detachments introduced in Eq. (7). At the start of the transformation the potential nucleation sites are set to have a cluster size n = 1. The concentration of clusters therefore amounts to C(1) = Np and C(n) = 0 for n > 1 at t = 0. For each time step Dt = 1/x the cluster size distribution evolves according to the probabilities to attach or detach an atom to the cluster at a rate that is controlled by the transition rates k(n,n+1) and k(n+1,n). The system can then be described by a set of differential equations [24,25]:

In the cases that c* < 1 and/or n* < 1, no stable cluster size distribution will be reached as all clusters have sufficient kinetic energy to potentially form a stable grain. For systems with a weak or absent energy barrier for nucleation, a cluster becomes effectively stable when the energy gain with respect to the energy barrier for nucleation DG* becomes larger than kBT (DG < DG*  kBT). For a weak or absent energy barrier for nucleation (c* < 1), the condition c(n) = c*  1 shows a single solution for the cluster size n = n+. In the limit where c*  1, the cluster is stable beyond a size of n+  1/a. For nucleation regime A with c* > 1 and n* > 1, the condition c(n) = c*  1 has two solutions, n = n+ > n* and n = n < n*. The number of atom attachments required to form an overcritical nucleus corresponds to D* = n+  n = 1/Z for c* > 1 and to D* = n+ for c* < 1. Note that D* < n* for c* > 1 and D* > n* for c* < 1. An effective nucleation rate N_ eff can now be introduced that applies to both steady-state nucleation and barrier-free nucleation (independent of c* and n*) by considering the net number of clusters per unit of time that reaches the critical size n = n+ for which the thermal energy is insufficient to dissolve the cluster: þ þ þ þ N_ eff ¼ ½k ðn ;n þ1Þ Cðnþ Þ  k ðn þ1;n Þ Cðnþ þ 1Þ:

ð15Þ

The critical size n+ is defined by the condition c(n+) = c*  1 with n+ > n*. For nucleation regime A (c* > 1 and n* > 1) N_ eff approaches the steady-state nucleation rate after an initial time-dependent stage. In the limiting case where the net interfacial energy for the formation of a cluster of the new phase vanishes (DX = 0), the transition rates to attach and detach an atom to the cluster take the simple forms k ðn;nþ1Þ ¼ x=ð1 þ ea Þ and k ðnþ1;nÞ ¼ x=ð1 þ ea Þ, respectively. If we assume that the rate constant can be approximated by the dominant rate x+ = x(n+), the unbalance in the transition rates to attach or detach an atom to the cluster then amounts to k ðn;nþ1Þ  k ðnþ1;nÞ ¼ xþ tanhða=2Þ. The cluster size distribution is expected to show a maximum at the most probably  cluster size of k ðn;nþ1Þ  k ðnþ1;nÞ t and a maximum cluster size of x+t. As discussed before, the clusters can be considered stable for c(n) < 1, which is the case for n > n+ = 1/a. The maximum nucleation rate is expected at time tmax  nþ =½k ðn;nþ1Þ  k ðnþ1;nÞ , which roughly scales as tmax  a2  DT 2 for a small undercooling DT with respect to the

dCðnÞ ¼ k ðnþ1;nÞ Cðn þ 1Þ þ k ðn1;nÞ Cðn  1Þ dt  k ðn;n1Þ CðnÞ  k ðn;nþ1Þ CðnÞ for n > 1 dCð1Þ ¼ k ð2;1Þ Cð2Þ  k ð1;2Þ Cð1Þ dt

4493

ð14Þ

with the boundary conditions C(1) = Np and C(n) = 0 for n > 1 at t = 0. The kinetics of the cluster size distribution can in general only be evaluated numerically. In the case that c* > 1 and n* > 1, a stable cluster size distribution is reached after an initial time-dependent stage. This stable cluster size distribution then leads, according to the classical nucleation theory, to the steady-state nucleation rate of Eq. (6) [24,25].

transformation temperature T0. Future calculations will be needed to analyse the cluster dynamics in detail. In general, no steady-state nucleation is found for barrier-free nucleation. 6. The austenite to ferrite transformation in low-alloyed carbon steels The austenite (c) to ferrite (a) transformation is one of the most extensively studied examples of a structural solid-state phase transformation. As a consequence of the

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low solubility of carbon in the ferrite phase, the transformation temperature A3 depends strongly on the average interstitial carbon concentration in low-alloyed carbon steels. The formation of the ferrite phase within the polycrystalline austenite matrix is controlled by a heterogeneous nucleation and a subsequent growth of ferrite grains. The growth velocity of the ferrite grains is limited by the diffusion of interstitial carbon, which continuously piles up in the austenite phase as the transformation proceeds. Below the transformation temperature A3 a coexistence of both phases is found, characterized by a temperature-dependent equilibrium ferrite fraction faeq ðT Þ, until a second transformation temperature A1 is reached below which a carbide phase (Fe3C) is formed. The gain in chemical potential Dl below the transformation temperature A3 acts as the driving force for the heterogeneous nucleation of ferrite grains. This gain in chemical potential depends both on temperature and on the fraction transformed. The temperature dependence is to a good approximation described by [2,3]:   dDl DlðT Þ  ð16Þ ðA3  T Þ / ðA3  T Þ; dT where (A3  T) is the undercooling with respect to the transformation temperature A3 and (dDl/dT) is the temperature derivative of Dl, which is roughly constant. Once the new ferrite phase is formed, the parent austenite phase enriches in carbon due to the small solubility of carbon in ferrite xaC  0:022 wt.%. As a consequence, the transformed fraction of ferrite fa reduces the gain in chemical potential Dl. To a first approximation, the dependence of the gain in chemical potential Dl on the transformed fraction of ferrite fa can be approximated by:     fa fa Dlðfa Þ  Dlðfa ¼ 0Þ 1  eq / 1  eq ; ð17Þ fa fa where faeq is the equilibrium ferrite fraction at temperature T. In recent in situ synchrotron X-ray diffraction experiments [17,20] on the austenite-to-ferrite transformation, we analysed the ferrite nucleation kinetics during continuous cooling in three different medium-carbon steels. From the heterogeneous nucleation behaviour during continuous cooling, an experimental value of W = 5 · 108 J3/m6 was deduced for all three steels (the exact value for W depends on the used value for the activation energy Q as well as the difference in chemical potential Dl). As discussed before, the parameter W characterizes the net interfacial energy required to form a critical nucleus. The experimental value of W was found to be several orders of magnitude smaller than model predictions for all available models [17,20]. The value of W can easily be translated in the interfacial energy term introduced in Eq. (1), DX ¼ ½27W=4N 20 1=3 ¼ 9:73  1020  W1=3 ¼ 35:8  1023 J (Eq. (5)), where the atom density amounts to N0 = 8.56 · 1028 m3 for both the austenite and the ferrite phase.

The rate constant x of Eq. (7) in these low-alloyed carbon steels is controlled by the mobility of iron atoms, the number of iron atoms at the interface, and the attempt frequency. The activation energy to move an atom across the austenite–ferrite interface may, to a first approximation, be estimated by the experimental activation energy Q = 4.72 · 1019 J (284 kJ/mol) for self-diffusion of iron atoms in the austenite phase [26]. This is, however, expected to be an overestimation, since the activation energy to move an atom across the austenite–ferrite interface may be somewhat smaller (20–30%) than the value found within the austenite phase. The temperature-independent attempt frequency for atom movements at the austenite–ferrite interface can be estimated by the average frequency for atom displacements in the austenite phase m0 = (3/5)1/2 (kBhD/h) = 5.4 · 1012 s1, where hD = 335 K is the Debye temperature of austenite [27]. The average frequency for atom displacements in the ferrite phase (6.9 · 1012 s1) with a Debye temperature of hD = 430 K [28] is slightly higher, but of the same order of magnitude. For small cluster sizes the pile up of the interstitial carbon atoms at the interface of the austenite phase is negligible and therefore, the rate constant x is controlled by the mobility of iron atoms at the interface during nucleation. However, at some stage during the subsequent grain growth, the rate constant x will be limited by the diffusion of carbon due to the pile up at the interface. 7. Experimental heterogeneous nucleation of ferrite grains in low-alloyed carbon steel In the following, we will consider recent experimental data on heterogeneous nucleation of ferrite grains in C35 steel (0.364 wt.% C, 0.656 wt.% Mn, 0.305 wt.% Si, 0.226 wt.% Cu, 0.177 wt.% Cr, 0.092 wt.% Ni, 0.016 wt.% Mo, 0.017 wt.% Sn, 0.021 wt.% S, 0.014 wt.% P and balance Fe) as an example to evaluate the characteristics of a structural solid-state phase transformation in a polycrystalline metal. For this steel we recently reported in situ synchrotron X-ray diffraction experiments on the isothermal transformation from austenite to ferrite at several different transformation temperatures [21]. For C35 steel calculations with the thermodynamic database MTDATA indicate a transformation temperature for the austenite-to-ferrite transformation of A3 = 1033 K [21]. The associated temperature derivative for the change in chemical potential amounts to dDl/dT  0.43 · 1023 J K1. In combination with the experimental value of W = 5 · 108 J3 m6, deduced from continuous cooling experiments on the same steel [20], we find system parameters of a = Dl/kBT  0.0093 and b = DX/kBT  0.026 for the untransformed material at an undercooling of DT = A3  T = 30 K. This means that the system falls within region C of the phase diagram in Fig. 2. The critical nucleus size amounts to only n*  7 atoms and the relative energy barrier for nucleation is c* = DG*/kBT  0.030. The energy barrier for nucleation DG* = 4.15 · 1022 J is small

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compared with the kinetic energy of the atoms kBA3 = 1.43 · 1020 J, and therefore, the system shows effectively no energy barrier for heterogeneous nucleation in the initial stage of the isothermal transformation. When the isothermal transformation proceeds, the parameter a will decrease for increasing ferrite fractions according to Eq. (17), leading to a continuous increase in n* and c* until they both diverge when the equilibrium fraction of transformed ferrite is reached. A variation in the experimental value of W, caused by the experimental uncertainty in the activation energy Q for the movement of iron atoms across the austenite–ferrite interface, does not affect parameter a and has only a weak influence on parameter b (about 25% shift in the value of b for an uncertainty of a factor 2 in the used value of Q for the determination of W). For a sufficiently large undercooling nucleation regions C and D can therefore always be reached. In Fig. 3, examples of the relative Gibbs free energy c(n) as a function of the cluster size n are shown for the untransformed material at different isothermal transformation temperatures. In Table 1, the characteristic parameters for ferrite nucleation in C35 steel are shown for different isothermal transformation temperatures. The critical values for the relative energy barrier for nucleation c* and the critical nucleus size n* are calculated for fa = 0. The clusters with a size between n and n+ have an energy difference of less than kBT with respect to the critical energy for nucleation DG*. The value of n+ corresponds to the effective size beyond which the cluster is stable and will not dissolve any more. The value of n+ is significantly larger than n* for c* < 1, while it corresponds to n+  n* + 1/ (2Z) for c* > 1. For c* > 1, the value of 1/Z gives a good

1.2

C35

1.0

fα = 0

0.8

1028 K

γ(n)

0.6 0.4 1020 K

0.2 0.0 1003 K

983 K

-0.2

1

10

100

1000

10000

n Fig. 3. Relative Gibbs free energy c(n) = DG(n)/kBT for a cluster of the ferrite phase in C35 steel as a function of the cluster size n at several temperatures. Both the maximum relative Gibbs free energy c* and the critical cluster size n* significantly decrease for larger values of the undercooling A3  T.

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Table 1 Nucleation parameters for isothermal transformations of ferrite in C35 steel at different temperatures T T (K)

DT (K)

faeq

c*

n*

n

n+

1/Z

1/a

1028 1020 1003 983 965

5 13 30 50 68

0.06 0.13 0.29 0.41 0.49

1.08 0.16 0.030 0.011 0.006

1398 80 7 1 <1

7 – – – –

4703 733 200 100 68

4126 610 115 41 22

645 248 108 65 47

DT is the undercooling with respect to the transformation temperature A3 = 1033 K and faeq is the equilibrium ferrite fraction. The critical values for the relative energy barrier for nucleation c* and the critical nucleus size n* are calculated for fa = 0. The clusters with a size between n and n+ have an energy difference of less than kBT with the critical energy for nucleation. For c* > 1 the value of 1/Z gives a good estimate for the width of the nucleation barrier n+–n, while for c*  1the value of 1/a gives a better estimate.

estimate for the width of the nucleation barrier n+  n, while for c*  1 the value of 1/a gives a better estimate. For practical reasons, it is of interest to express the transitions between the different types of nucleation behaviour (indicated in the diagram of Fig. 2) in terms of the undercooling DT = A3  T and the fraction transformed fa =faeq , where faeq is the temperature-dependent equilibrium phase fraction of transformed ferrite. For W = 5 · 108 J3 m6 [17,20], the first phase line c* = 1 is expressed by: 1     1=2 fa dDl W DT 1  eq ¼ dT fa N 20 k B T  1  1=2 dDl W  ¼ DT c ¼1 ; ð18Þ dT N 20 k B A3 where DTc*=1 = (2.3 · 104 Km3/J3/2) · W1/2 = 5.1 K. The second phase line n* = 1 is expressed by: 1  1=3    fa dDl 2W DT 1  eq ¼ ¼ DT n ¼1 ; ð19Þ dT fa N 20 where DTn*=1 = (1.5 · 104 Km2/J) · W1/3 = 56 K. For a structure change from the fcc structure of austenite to the bcc structure of ferrite, one could consider that the new bcc structure can be identified from a minimum cluster size of n* = 4. In this case, DTn*=4 = (4)1/3DTn*=1 = 35 K. An uncertainty in the activation energy Q of a factor 2 causes a shift of 2 K in DTc*=1 and of 14 K in DTn*=1. In Fig. 4, the diagram of the nucleation regimes is shown for C35 steel as a function of two control parameters: the undercooling DT and the relative fraction transformed fa =faeq . For C35 steel the system can be in one of three different nucleation regimes (A, C and D), depending on the undercooling and the fraction transformed. For an undercooling DT smaller than DTc*=1 the system always shows an energy barrier for nucleation that is larger than the kinetic energy of the atoms (regime A). For an undercooling DT larger than DTc*=1 the nucleation initially shows either a weak barrier for DT < DTn*=1 (regime C) or no barrier for DT > DTn*=1 (regime D) and eventually shows a cross-over to c* > 1 (regime A) when the transformation

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1.0

a

A

γ* = 1

γ* > 1 n* > 1

0.8

C35

fα /fα

eq

C γ* < 1

0.6

n* > 1 n* = 1

b

0.4

D 0.2

0.0

γ* < 1 n* < 1

0

20

40

ΔT (K)

60

80

100

Fig. 4. Diagram of the nucleation regimes for C35 steel as a function the undercooling DT = A3  T and the fraction transformed fa =faeq . The system can be in one of three different nucleation regimes, depending on the undercooling and the fraction transformed. For c* > 1 and n* > 1 (regime A), classical nucleation is expected. For c* < 1 and n* > 1 (regime C) nucleation involves a weak energy barrier, while for c* < 1 and n* < 1 there is effectively no energy barrier for nucleation (regime D).

proceeds. The nucleation rate in regime D (no energy barrier for nucleation) is expected to be significantly higher than for regime C (weak energy barrier for nucleation), but due to the gradual nature of the cross-over between both regimes it will be difficult to distinguish the nucleation behaviour experimentally. In Fig. 5, the in situ synchrotron X-ray diffraction measurements on C35 steel [21] during isothermal transformations are shown at transformation temperatures with an undercooling of DT = 13, 30, 50 and 68 K. Prior to the isothermal transformation the sample was heated to the austenization temperature of 1233 K, held there for 10 min and subsequently rapidly cooled (at a rate of 6 K s1) to the isothermal transformation temperature in the austenite/ferrite phase region, where it was held until the transformation was finished. Fig. 5 shows the transformed ferrite fraction fa and the density of formed ferrite grains Na as a function of time t. During the limited time required to cool the sample from the equilibrium transformation temperature A3 to the isothermal transformation temperature some of the ferrite grains have nucleated and a significant ferrite fraction has formed. This strongly suggests that the nucleation of ferrite grains is initially barrier free for all temperatures. As the transformation proceeds, the nucleation rate slowly decreases, except for the lowest transformation temperature, where the equilibrium ferrite phase fraction is reached nearly instantaneously. We can now compare the experimental data for the isothermal phase transformation in C35 steel (Fig. 5) directly with the diagram of Fig. 4. In Fig. 6 the diagram for the nucleation regimes is shown as a function of temperature

Fig. 5. The measured ferrite fraction (a) and the ferrite nucleus density (b) in C35 steel as a function of the isothermal transformation time at four different annealing temperatures (data from [21]).

T and ferrite fraction fa. The equilibrium ferrite fraction as a function of temperature faeq ðT Þ can be deduced from the equilibrium phase diagram via the lever rule, and was estimated by the thermodynamic database MTDATA for the composition of our steel. The lines c* = 1 and n* = 1 that mark the different nucleation regimes are indicated by dashed lines. It is interesting to note that the temperature dependence of the lines for c* = 1 and n* = 1 closely follow that of the equilibrium ferrite phase fraction faeq ðT Þ with temperature shifts of DTc*=1 and DTn*=1, respectively. For comparison, the line n* = 4 is also indicated. The observed trajectories of the ferrite phase fraction during the isothermal transformations in Fig. 5 are indicated by the solid lines with solid circles at the start and end values. The initial part of the transformation, which is not registered in Fig. 5, is indicated by the dotted lines. In all cases, the barrier-free nucleation below n* = 1 (regime D) is too fast to be registered in the experiment. It is interesting to note that, for the three highest transformation temperatures, the equilibrium ferrite fraction is not reached within the time frame of 30–60 min. Towards the end of the experiments, the nucleation rate is significantly reduced but has not vanished. For all three temperatures, the system has just reached the cross-over line between the barrier-free nucleation regime (c* < 1) and the classical nucleation regime (c* > 1). For the lowest transformation temperature, the equilibrium ferrite fraction is reached nearly instantaneously. The high nucleation rate leads to a relatively high density of ferrite grains. In the last stage of the transformation, when the system has entered the classical nucleation regime (c* > 1), the nucleation of new ferrite grains becomes increasingly difficult. The grain growth continues, however, until the equilibrium ferrite fraction is reached. The relatively high density of nuclei

N.H. van Dijk et al. / Acta Materialia 55 (2007) 4489–4498

C35

eq

0.6

fα

γ* = 1

fα

0.4

n* = 4

0.2 n* = 1

0.0 940

960

980

1000

1020

1040

T (K) Fig. 6. Diagram of the nucleation regimes for C35 steel as a function of temperature T and ferrite fraction fa. The equilibrium ferrite fraction as a function of temperature faeq ðT Þ is indicated by a solid line. The cross-over lines for c* = 1, n* = 1 and n* = 4 that mark the different nucleation regimes are indicated by dashed lines. The observed trajectories of the ferrite phase fraction during the isothermal transformations (Fig. 5) are indicated by the solid lines with solid circles at the start and end values. The initial part of the transformation, which is not registered in Fig. 5, is indicated by the dotted lines.

observed for the transformation at T = 1003 K with respect to the transformation at T = 1020 K indicates some variation in the density of potential nucleation sites Np for successive transformations. Future cluster dynamics calculations are highly desirable to understand in more detail the heterogeneous nucleation process during structural solidstate phase transformations. 8. Conclusions We have investigated the nucleation behaviour for structural solid-state phase transformations in polycrystalline materials. As an example we used recent experimental in situ X-ray diffraction data on the austenite to ferrite transformation in a medium-carbon steel. The main conclusions are: 1. For structural solid-state phase transformations in polycrystalline materials the net interfacial energy to create a cluster of the new phase can be relatively small. This situation is especially relevant in cases when the new phase is formed in corners or on faces of the grains from the parent phase. In this case, the energy from the grain boundary between grains of the parent phase is released during the formation of the new phase. As a result, the energy barrier for nucleation DG* can become smaller than the thermal energy of the atoms kBT. The commonly applied steady-state nucleation, predicted by the classical nucleation theory, now no longer applies.

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2. We have developed a simple thermodynamic model to predict the characteristic properties of the crystallization behaviour from the size-dependent change in Gibbs free energy DG(n) for a cluster of the new phase in the matrix of the parent phase. Depending on the size of the energy barrier for nucleation relative to the kinetic energy of the atoms c* = DG*/kBT and the corresponding critical cluster size n*, four different types of nucleation behaviour can be identified. One of these types (c* > 1 and n* > 1) corresponds to the classical nucleation theory, which shows the well-known steady-state nucleation rate after an initial time-dependent nucleation stage. Two other types with n* < 1 (c* > 1 or c* < 1) effectively show no energy barrier for nucleation as the continuum model breaks down for n* < 1. These two types of nucleation behaviour have been observed for highly undercooled melts, spinodal decompositions and cluster aggregations. The last type (c* < 1 and n* > 1) shows a weak but finite energy barrier for nucleation. The last three types do not show the stable cluster size distribution nor the steadystate nucleation characteristic of the classical nucleation theory. For these types, the system is inherently unstable and therefore shows a time-dependent nucleation rate. 3. Recent synchrotron X-ray diffraction measurements allowed us to monitor the ferrite nucleation rate and the ferrite fraction simultaneously within the bulk of a medium-carbon steel. The deduced net interfacial energy to form a ferrite grain within the austenite matrix was found to be two orders of magnitude smaller than the best model predictions. The isothermal transformation kinetics of a C35 steel was compared with our predicted nucleation behaviour. We deduced that, depending on the transformation temperature and the fraction transformed, three types of nucleation can be observed: nucleation without an energy barrier for a large undercooling, nucleation with a weak energy barrier for a medium undercooling and classical nucleation with a steady-state nucleation rate for a small undercooling. We demonstrated that the nucleation regimes with a weak or absent energy barrier for nucleation describe the dominant transformation behaviour in medium-carbon steels. The observed barrier-free nucleation behaviour is expected to apply to a wide range of structural solid-state phase transformations in various polycrystalline materials. Acknowledgements The authors thank Dimo Kashchiev and Ignatz de Schepper for valuable discussions. S.E.O. acknowledges the financial support from the Dutch Foundation for Technical Sciences (STW) of the Netherlands Organization for Scientific Research (NWO). References [1] Christian JW. The theory of transformations in metals and alloys. Oxford: Pergamon; 1981.

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[2] Kashchiev D. Nucleation, basic theory with applications. Oxford: Butterworth; 2000. [3] Mutaftschiev B. The atomistic nature of crystal growth. Berlin: Springer-Verlag; 2001. [4] Kelton KF. In: Ehrenreich H, Turnbull D, editors. Solid State Physics, vol. 45. New York: Academic Press; 1991. p. 75. [5] Yang J, McCoy BJ, Madras G. J Chem Phys 2006;124:024713. [6] Gasser U, Weeks ER, Schofield A, Pusey PN, Weitz DA. Science 2001;292:258. [7] Auer S, Frenkel D. Nature 2001;409:1020. [8] Clemm PC, Fisher JC. Acta Metall 1955;3:70. [9] Enomoto M, Aaronson HI. Metall Trans A 1986;17A:1385. [10] Enomoto M, Aaronson HI. Metall Trans A 1986;17A:1399. [11] Lange III WF, Enomoto M, Aaronson HI. Metall Trans A 1988;19A:427. [12] Tanaka T, Aaronson HI, Enomoto M. Metall Trans A 1995;29A:547. [13] Huang W, Hillert M. Metall Mat Trans A 1996;27A:480. [14] Lauridsen EM, Jensen DJ, Poulsen HF, Lienert U. Scripta Mater 2000;43:561. [15] Poulsen HF, Nielsen SF, Lauridsen EM, Schmidt S, Suter RM, Lienert U, et al. J Appl Crystallogr 2001;34:751. [16] Margulies L, Winther G, Poulsen HF. Science 2001;292:2392. [17] Offerman SE, van Dijk NH, Sietsma J, Grigull S, Lauridsen EM, Margulies L, et al. Science 2002;298:1003.

[18] Schmidt S, Nielsen SF, Gundlach C, Margulies L, Huang X, Juul Jensen D. Science 2004;305:229. [19] Iqbal N, van Dijk NH, Offerman SE, Moret MP, Katgerman L, Kearley GJ. Acta Mater 2005;53:2875. [20] Offerman SE, van Dijk NH, Sietsma J, van der Zwaag S. In: Kawalla R, editor. Proceeding of the fifty-sisth Berg- und Hu¨ttenma¨nnischer Tag, Modellierung und Simulation von Prozessen der Stahlerzeugung und Stahlverarbeitung. Freiberg: TU Bergakademie Freiberg; 2005, 97. [21] Offerman SE, van der Zijden ACP, van Dijk NH, Sietsma J, Lauridsen EM, Margulies L, et al. In: Gundlach C, Haldrup K, Hansen N, Huang X, Juul Jensen D, Leffers T, Li ZJ, Nielsen SF, Pantleon W, Wert JA, Winther G, editors. Proceedings of the twentyfifth Risø international conference on materials science: evolution of deformation microstructures in 3D. Roskilde: Risø National Laboratory; 2004. p. 471. [22] Flynn CP. Point defects and diffusion. Oxford: Clarendon; 1972. [23] Schott V, Fa¨hnle M, Madden PA. J Phys: Condens Matter 2000;12:1171. [24] Kelton KF. Acta Mater 2000;48:1967. [25] Slezov VV, Schmelzer J. J Phys Chem Solids 1994;55:243. [26] Kucˇera J, Stra´nsky´ K. Mater Sci Eng 1982;52:1. [27] Matsumura O, Sakuma Y, Takechi Y. Scr Metall 1987;21:1301. [28] Jarestky J, Stassis C. Phys Rev B 1987;35:4500.