Autocatalytic nature of the bainitic transformation in steels: a new hypothesis

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Autocatalytic nature of the bainitictransformation in steels: a newhypothesisS.M.C. van Bohemen aa Tata Steel Research Development & Technology , 1970 CAIJmuiden, The NetherlandsPublished online: 31 Aug 2012.

To cite this article: S.M.C. van Bohemen (2013) Autocatalytic nature of the bainitictransformation in steels: a new hypothesis, Philosophical Magazine, 93:4, 388-408, DOI:10.1080/14786435.2012.721570

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Autocatalytic nature of the bainitic transformation in

steels: a new hypothesis

S.M.C. van Bohemen*

Tata Steel Research Development & Technology, 1970 CA IJmuiden,The Netherlands

(Received 10 May 2012; final version received 13 August 2012)

To ensure improvements in predicting the kinetics of bainite formation, it isimportant to understand the autocatalytic nature of the transformation sothat this accelerating effect can be rigorously incorporated in kineticmodels. In the present paper, it is assumed that the broad faces of bainiticplates in particular provide new potential nucleation sites for autocatalyticnucleation. The dislocations in the austenite near a bainitic plate arethought to stimulate autocatalysis because carbon is assumed to pile up atthese regions and thereby other austenite–bainite interface regions maycontain less carbon which promotes nucleation. Based on these assump-tions, it is derived that the autocatalytic contribution is proportional to thevolume fraction of as-formed bainite, which is consistent with thedependence proposed by Entwisle [V. Raghavan and A.R. Entwisle,Special Report No. 93, The Iron and Steel Institute, London, 1965, p.30] onthe basis of empirical knowledge. In addition, it is assumed thatautocatalytic nucleation can also depend on the morphology of bainitedue to the associated difference in cementite precipitation. This newhypothesis for autocatalysis offers a viable explanation for the irregularvariation in kinetics associated with the transition from upper to lowerbainite measured for an alloy with eutectoid composition. Furthermore,comparison with experimental data of a Si-rich steel demonstrates that theisothermal kinetics of bainite formation can only be satisfactorily describedwhen the autocatalytic factor is inversely proportional to the thickness ofbainitic plates, which is consistent with the model proposed.

Keywords: bainitic steels; autocatalytic nucleation; phase transformationkinetics; interfaces; modelling

1. Introduction

Advanced high strength steels exhibiting an optimised balance between strength andformability have a complex microstructure containing various phase constituentssuch as ferrite, bainite, martensite and retained austenite. Understanding the kineticsof bainite formation is very important for the further development of such steels.Partly due to the complexity of its microstructure, the austenite to bainitetransformation is probably the least understood phase transformation in steels.

*Email: [email protected]

� 201 Taylor & Francis

http://dx.doi.org/10.1080/14786435.2012.721570Philosophical Magazine, 2013Vol. 93, No. 4, 388–408,

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Two alternative transformation mechanisms have been proposed to describe the

transformation kinetics: the diffusional [1] and the displacive model [2]. In the

present paper, the growth of bainite is assumed to be displacive, which means that

the overall transformation kinetics is primarily controlled by the nucleation rate.The formation of bainite in steels usually starts with nucleation at prior austenite

grain boundaries. In a bainitic microstructure, clusters of fine plates of ferritic bainite

can be identified which are called sheaves [2,3]. The growth of a sheaf requires the

nucleation of new plates [4]. Depending on the specific morphology, these bainitic

plates within a sheaf have also been denoted as bainitic sub-units, platelets or laths

[2–5]. Within such a sheaf, the bainitic plates are parallel and have an approximate

identical crystallographic orientation. After the start of the transformation due to

nucleation at prior austenite grain boundaries, the isothermal transformation

kinetics are often observed to increase. In the displacive school of thought, this

acceleration is assumed to be due to autocatalytic nucleation near the bainitic plates

that have just been formed [6,7]. In this manner, autocatalysis offers a viable

explanation for the development of the familiar sheaf structure of bainite. Various

kinetic models have been developed on the basis of nucleation and displacive growth

[2,6–9]. In the modelling approach described in [9], the number density of the

potential nucleation sites is assumed to be dependent on the temperature, which is

different compared to previous models and, in a subsequent paper, it was

demonstrated that the model can give satisfactory predictions of the bainite start

curves [10].Below the nose of the bainitic C curve in TTT diagrams, the rate of bainite

formation usually decreases regularly with decreasing temperature, i.e. the C curve

has a smooth appearance. However, for certain steels, it has been observed that the

change in kinetics is irregular [4,11,12], which is generally attributed to the transition

from upper to lower bainite. Based on such observations, it has been argued that

upper and lower bainite have separate C curves [13,14]; however, other investigations

[15,16] and published TTT diagrams [17–20] show that this phenomenon is certainly

not a general characteristic of steels. A thorough explanation for this irregular

behaviour has not been undertaken in the literature. The identification of a ‘‘lower

nose’’ in a TTT diagram means that the kinetics of lower bainite are relatively fast

compared to the extrapolated kinetics of upper bainite into the lower temperature

range [13,14]. Possibly related to this phenomenon, some steels exhibit a very strong

acceleration in the kinetics just above the Ms temperature [11,21–23]. At these low

temperature, the start of transformation shows a distinct ‘‘swing back’’ to shorter

times [11] and, therefore, Oka and Okamoto [23] called this reversal in kinetics

‘‘swing back’’ behaviour. Analysis of TTT diagrams reported in the literature [17–20]

indicates that this reversal in isothermal kinetics just above Ms is a quite atypical

phenomenon. In contrast, the acceleration of bainite kinetics below Ms seems a well

understood phenomenon and is thought to be associated with the accelerating effect

of the presence of martensite on the nucleation of bainite in austenite regions

adjacent to the as-formed martensite [11,24]. Consistent with these observations,

Steven and Haynes [25] applied the Greninger and Troiano [26] method to British

Standards (B.S.) En 30B steel samples and demonstrated that the presence of

martensite can also accelerate the bainite kinetics above the Ms temperature.

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In [10], the model for bainite formation proposed in [9] was thoroughly validatedusing published TTT diagrams, which gave a valuable insight into the activationenergy, in particular its dependence on composition. Most TTT diagrams showregular shaped C curves which could be accurately simulated using a temperature-independent value for the autocatalytic factor [10], which is consistent with theassumption made by other investigators [6,27]. However, the ongoing investigationin this area showed that the irregular C curves for certain alloys are a true fact, whichinitiated a re-assessment of the model and the factors controlling the transformationkinetics. In accordance with arguments made by Bhadeshia [28], it is believed that abetter treatment of autocatalytic nucleation is necessary to improve modelcalculations, in particular those for Si-rich and high-C steels.

In the present paper, a kinetic model is proposed which distinguishes betweeninitial nucleation events at �/� boundaries and those that follow autocatalyticallydue to the generation of �/� interface boundaries. Subsequently, autocatalysis isquantitatively described in terms of the thickness of bainitic plates and theeffectiveness of �/� interfaces. As a consequence, the contribution of autocatalysis tothe overall transformation kinetics is not only dependent on composition, but canvary with temperature as well. The modified model will be tested againstexperimental data of a plain carbon steel for which the trend in kinetics appearsto be irregular due to a transition from upper to lower bainite. Furthermore,comparison with experimental data from a Si-rich steel is made, which demonstratesthat the isothermal kinetics of bainite formation in that alloy can only besatisfactorily described when the degree of autocatalytic nucleation is assumed toincrease for finer bainitic plates.

2. Theory

2.1. Model

In previous work, it was proposed that the kinetics of the bainitic transformationas a function of time and temperature can be calculated using the differentialequation [9,10]:

df

dt¼ 1� fð Þ 1þ �fð Þ�

Z�

D��m Th � Tð Þexp �

Q�

RT

� �ð1Þ

describing the change in the bainite fraction f as a function of time t, which can alsobe written as (1�f )(1þ �f )� [9]. Other investigators [6,27] have reported similarexpressions as Equation (1), which is directly related to the nucleation rate [9].The autocatalytic factor � determines the relative contribution of autocatalysis to theoverall kinetics. In the derivation below in the present paper, it will be shown that �is not a material constant contrary to the often made assumption in many previousstudies [6,9,10,27]. Based on this new hypothesis for autocatalysis, which accountsfor a � dependent on the microstructural characteristics of bainite, it will bedemonstrated that the peculiar variation in kinetics sometimes observed for certainsteels can be explained.

Equation (1) shows that the temperature dependence of the kinetics is mainlygoverned by (Th�T) and the activation energy Q*. The austenite grain diameter

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is D�. For all calculations in this study, the constant � and the geometrical factor Z

are chosen as �¼ 1 nm and Z¼ 6 [9,10]; the attempt frequency � is assumed to be

1� 1013 s�1 [29]. The temperature Th is the highest temperature at which a displacive

transformation can occur [30] and, in this work, it is assumed that Th corresponds to

the start temperature of bainite. The factor (Th�T) accounts for the linear increase

in the number density of potential nucleation sites Ni as the temperature decreases.

Furthermore, Ni is inversely proportional to the austenite grain diameter D� [27] and,

according to the derivation in [9], Ni is assumed to have a weak composition

dependence, which is taken into account by the proportionality with �m, the kineticparameter describing the rate of martensite formation [31–33].

The factor (1þ �f ) in the nucleation rate equation was already introduced more

than 45 years ago for isothermal martensite nucleation [34]. Although various

measurements [29,35] have supported the proposed dependence, a detailed derivation

justifying the dependence on the volume of bainite has not been reported. In the

discussion of Entwisle’s paper [34], Christian [36] questioned whether this factor

properly accounts for autocatalysis, and he subsequently mentioned that ‘‘it is not

obvious that the autocatalytic effect ought to depend on the volume of the plate

formed; why should it not depend on the thickness.’’ Entwisle replied that it was not

obvious to him either; the volume dependence had simply been tried first, and

empirically this seemed to be so. Although modelling studies confirmed that the

majority of experimental data can be described by assuming that the autocatalytic

contribution depends linearly on the volume of bainite, the kinetics of some steels

exhibit a complex temperature dependence possibly due to varying autocatalytic

effects. Therefore, to accurately describe transformation kinetics for all alloys, it is

important to revisit this unresolved issue [28] and try to elucidate the origin of

autocatalysis.To determine which parameters control autocatalytic nucleation, it is important

to re-examine Equation (1) and distinguish the two contributions to the overall

nucleation rate: the term (1� f )� accounts for nucleation on pre-existing nucleation

sites, and autocatalytic nucleation is expressed by (1� f )�f�. It seems justified to

assume that the contribution of autocatalytic nuclei does not depend on the austenite

grain size and for that reason a new kinetic parameter � will be introduced, which is

independent of the austenite grain size D� and linked to �. On the other hand, it is

well understood that initial nucleation starts at austenite grain boundaries [27] and,

therefore, it seems appropriate to express the initial transformation rate in terms of

the area density of �/� boundaries, A��/V�, which leads to:

df

dt

� �i

¼ 1� fð ÞA��

V�� ð2Þ

A��/V� can be expressed in terms of the austenite grain size as Y/D�, with Y a

geometrical factor. Comparison with Equation (1) shows that the kinetic parameter

� can be written as:

� ¼ �Z�

Y�m Th � Tð Þexp �

Q�

RT

� �ð3Þ

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and is independent of the austenite grain size D�. It is noted that this kinetic

parameter � is linked to the rate parameter � [9,10] as �¼ �(D�/Y). The factor

(1� f ) in Equation (2) accounts for the decrease in the density of �/� grain

boundaries available for nucleation as the transformation progresses.After the initial nucleation at �/� boundaries, the transformation accelerates due

to autocatalytic nucleation near the bainitic plates that have just been formed. The

formation of bainitic plates has two effects important for autocatalysis: plastic

deformation of austenite [3], and the creation of �/� interface boundaries. In the

present study, the �/� phase boundaries are assumed to provide the new potential

nucleation sites. Accordingly, it is proposed that the transformation rate due to

autocatalysis can be written as:

df

dt

� �a

¼ 1� fð Þ�A��

V�� ð4Þ

with A��/V� the area density of �/� boundaries; the factor � expresses the

effectiveness of nucleation on �/� boundaries relative to the nucleation on �/�boundaries. This factor � is assumed to depend on the chemical composition, the

dislocation density and the morphology of bainite together with the associated

difference in cementite precipitation. These aspects will be discussed in more detail

later. The thickness w of a bainitic plate is determined by the chemical driving force

and the strength of the austenite [3]. For simplicity, the geometry of a plate is

assumed to be a disc having broad faces with an area S, which means that the volume

Vb¼S�w. It should be noted that assumptions regarding the exact geometry of

bainitic plates merely affects the dimensions of a bainite sheaf, not the underlying

mechanism of formation which is the principal subject of this study.The growth of a bainitic plate is stifled due to the shear and volume misfit [2]. The

austenite surrounding the plate is plastically deformed during growth and the

resulting dislocation debris eventually stops the transformation interface which

losses coherency [2,37]. Related to this, previous investigations have shown that the

bainite transformation is retarded in austenite that is strengthened due to

deformation [28]. This is usually called mechanical stabilisation. Observations of

dislocations in the austenite surrounding the bainitic plates in [4] support the

occurrence of mechanical stabilisation of austenite due to the shape change.In the present autocatalysis model, dislocations in the austenite near the tip of a

bainitic plate are considered to play an indirect role in stimulating autocatalysis.

It has been well established that carbon atoms have a preference to pile up near

dislocations [38]. This means that carbon is expected to partition mainly to austenite

near the tip of bainitic plates where most dislocations are expected and, therefore,

these austenite regions are chemically more stable compared to the broad faces.

In turn, a relative decrease in carbon content near the broad faces can stimulate the

autocatalytic nucleation in those austenite regions. These effects of dislocations are

accounted for by the factor �.Based on the stabilisation mechanism explained above, one can imagine that the

plastically deformed regions of austenite near a tip of a bainitic plate are relatively

stable. As a consequence, it seems justified to assume that only the two broad faces of

a bainitic plate with total area 2�S are effective in providing new sites for


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autocatalytic nucleation. When the approximation is made that only the broad facescontribute to autocatalytic nucleation, then the total area of �/� boundaries can beexpressed as:

A�� ¼X 2Vb

w¼

2V�

ð5Þ

in which is the average thickness of the bainitic plates, and V� is the volume ofbainite. By combining Equations (2), (4) and (5), it follows, using f¼V�/V�, that theoverall transformation rate can be written as:

df

dt

� �iþa

¼ 1� fð ÞA��

V�þ2�

f

� �� ð6Þ

which shows that autocatalysis is proportional to the volume fraction of bainite,consistent with the dependence proposed by Entwisle on the basis of empiricalknowledge [34]. It is also seen that the autocatalytic contribution is dependent on thebainitic plate thickness, which is in accordance with the suggestion made byChristian [36].

When �¼ �(D�/Y) is inserted in Equation (6), the expression becomesdf/dt¼ (1� f )(1þ �f )� [9,10] with the autocatalytic parameter � expressed by:

� ¼D�

Y

2�

� �ð7Þ

which is, contrary to the often made assumption in many previous studies [6,9,10,27],not a material constant but dependent on both the effectiveness factor � and theplate thickness . As a consequence of this relationship, the autocatalytic factor � canhave a significant temperature dependence since, for certain alloys, the thickness of

bainitic plates can decrease strongly with lowering temperature. Furthermore, thisderivation shows that the autocatalytic factor � is proportional to the austenite grainsize D�, a correlation that was already suggested in previous work on the basis ofmodel validation with experimental data [9]. Hence, the autocatalytic contribution,Equation (2), is insensitive to the austenite grain size. This proportionality between �and D� also means that for larger D� the autocatalytic contribution increases relative

to the contribution from initial nucleation on �/� boundaries.In summary, the derivation above shows that the factor (1þ �f ) can adequately

account for autocatalytic nucleation. However, the often made assumption of aconstant � is, in fact, only valid when the ratio �/ is virtually temperatureindependent. Equation (7) also indicates the dependence of autocatalysis on theparameter � describing the relative effectiveness of �/� boundaries. As will bediscussed in section 2.3 in more detail, it can be imagined that � has a temperaturedependence because the dislocation density as well as the morphology of bainite and

the associated characteristics of carbide precipitation vary with temperature.All these aspects determine the local carbon concentrations in the vicinity of �/�boundaries and thereby influence autocatalysis. The exact dependence of theautocatalysis factor on dislocation density or carbon concentration in the vicinity of�/� boundaries can not be formulated. Therefore, the combined influence ofdislocations and carbon concentrations are taken up in the effectiveness factor �,


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which is a parameter that can only be determined by comparison with experimental

data. It is also noted that dislocation densities in bainite can not be predicted, and theconcentration of carbon in the vicinity of �/� boundaries is even more difficult to

estimate.

2.2. Effect of plate thickness

In this subsection, it will be demonstrated how variations in the bainitic plate

thickness affect the autocatalytic parameter � (Equation (7)) and thereby thetransformation kinetics. Only a few researchers have investigated the variation in

bainite plate thickness with varying temperature [3,5,14,39,40]. On the basis of thedata reported by Singh and Bhadeshia [3], a neural network model was trained to

predict the plate thickness as a function of temperature, austenite strength anddriving force [41]. This program will be used to demonstrate the effect of the plate

thickness on the S shape of fraction–time curves or, equivalently, the appearance ofC curves in a TTT diagram. It was chosen to perform calculations for a medium

carbon steel with composition 0.5wt% C, 1wt% Mn, 0.2wt% Si, 0.2wt% Mo and0.8wt% Cr, denoted as 0.5C–Cr–Mo. These moderate additions Mn, Cr and Mo

ensure that the hardenability against ferrite formation is adequate. The variation ofthe bainitic plate thickness with temperature calculated using the neural network

program is plotted in Figure 1, which shows that changes almost 200% in thetemperature range from 300 to 500�C. It is noted that for other alloys the change in in a similar temperature interval can be much smaller [3], or larger [39]. Figure 1 alsoshows the temperature dependence of the autocatalytic parameter � calculated as a

function of the bainitic plate thickness using Equation (7). The factor � is assumedconstant and arbitrary chosen as 0.45.

In previous work, it was derived that by solving the differential equation

df/dt¼ (1� f )(1þ �f )�, which is equivalent to Equation (6), the bainite volumefraction can be calculated as a function of time according to:

f ¼1� exp �� 1þ �ð Þtð Þ

�exp �� 1þ �ð Þtð Þ þ 1ð8Þ

with � the rate constant, which can be expressed in terms of the effective activationenergy Qb as [10]:

� ¼ �Z�

D��mexp �

K1�

R

� �Th � Tð Þexp �

Qb

RT

� �ð9Þ

According to the procedure explained in [10], the kinetics can be predicted.The parameter K1 in Equation (9) is a material constant and � ¼ d(DGm)/dT , i.e. the

derivative of the maximum driving force DGm with respect to temperature.The empirical relationship for the composition dependence of Qb derived in [10]

yields Qb¼ 64.2 kJmol�1 for 0.5C–Cr–Mo. The material-independent relationshipK1�¼ (169.8 kJmol�1–Qb)/(705K) [10] leads to K1�¼ 150 Jmol�1K. Furthermore,

the bainite start temperature Th¼ 552�C [10] and �m¼ 0.016K�1 [42]. The austenitegrain size is chosen as D�¼ 20 mm.


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At a given temperature, the time t required to form a volume fraction bainite fcan be calculated using the inverse function of Equation (8), which has beendetermined as [10]:

t ¼� ln 1�f

1þ�f

� �� 1þ �ð Þ

ð10Þ

Since autocatalysis does not play a significant role at the beginning of transforma-tion, the distinct shape of the start curve is mainly determined by the temperaturedependence of �. However, with increasing bainite volume fraction, the influence ofautocatalytic nucleation becomes more prominent, which means that the variation ofboth � and � with temperature control the exact shape of the additional curves; forexample the halfway and finish curve.

To demonstrate the effect of �, Figure 2a shows the 2, 50 and 99 vol% curvescalculated for 0.5C–Cr–Mo (solid lines) using Equation (10) by accounting for theincrease in � with decreasing temperature as shown in Figure 1. For comparison, thedashed lines represent the kinetics calculated using a constant �¼ 20 for which the 2,50 and 99 vol% curves are seen to be parallel. The differences are highlighted withthe arrows showing the relative increase in autocatalytic effects with loweringtemperature. At 425�C, the temperature-dependent autocatalytic parameter � iscalculated as 28 (see Figure 1) and, consequently, the time required for 50 vol%transformation is only slightly shorter than given by the dashed line calculated forthe case of a constant � ¼ 20. This similarity in kinetics at 425�C is also seen inFigure 2b, which shows typical fraction–time curves calculated with Equation (8) forthe case where �¼ 20 (dashed lines) and, for comparison, curves demonstrating theeffect of � on the isothermal bainite kinetics (solid lines). Figure 2b shows largedifferences in the rate of transformation at 300�C between the solid line (�¼ 101) andthe dashed line (�¼ 20). The autocatalytic parameter does not only increase the

Figure 1. Temperature dependence of the autocatalytic parameter � calculated usingEquation (7). The variation in bainitic plate thickness in 0.5C–Cr–Mo has been calculatedwith a neural network program [41]; the factor � is assumed constant.


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overall transformation rate, � also determines the sigmoidal shape of the fraction–time curve. This is readily seen in Figure 2b by comparing the dashed sigmoidalcurve for 375�C (�¼ 20) with the solid curve for 300�C (�¼ 101). Both fractioncurves indicate that the transformation is finished after approximately 400 s;however, for �¼ 101, the S shape of the fraction–time curve is much morepronounced.

In summary, the gradual increase in � (Figure 1) has an almost negligibleeffect on the 2 vol% curve as seen in Figure 2a. However, due to the gradualincrease in �, the 50 and 99 vol% curves are changed considerably such that thedistance between the start curve and the additional curves decreases withdecreasing temperature, which is in qualitative agreement with published data forsome alloys, e.g. the TTT diagram of Fe–0.58C alloy investigated by Ohmoriet al. [43], and the medium-carbon alloy on page 184 in [19]. Although such typesof TTT diagram can be explained by assuming a gradual increase in �, other TTTdiagrams in the literature sometimes reveal that the C curves, in particular the 50and 99 vol% curves, have an irregular shape. This will be the subject of thefollowing section.

2.3. Upper and lower bainite

In the previous section, it was for simplicity assumed that the factor �, whichexpresses the effectiveness of �/� boundaries, is a constant. However, it seemsplausible to assume that the factor � can also be dependent on temperature; for

Figure 2. (a) The 2, 50 and 99 vol% curves calculated for 0.5C–Cr–Mo accounting for anincrease in � with decreasing temperature (see Figure 1) are represented by the solid lines.The 2, 50 and 99 vol% curves calculated using a constant �¼ 20 are also shown forcomparison (dashed lines). (b) Three fraction–time curves calculated for the case that�¼ 20 (dashed lines), and two curves demonstrating the effect of � on the isothermalbainite kinetics (solid lines).


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example, due to a change in the morphology of bainite. In the case of upper bainite,

the excess carbon will partition from bainitic plates to the surrounding austenite.

Initially, this will lead to an increased carbon concentration in the austenite near the

�/� interface boundaries and subsequently cementite or other carbides may be

formed. For lower bainite, the excess carbon will precipitate inside the bainitic ferrite

plates. The exact differences in �/� boundary conditions between upper and lower

bainite, and the associated differences in effectiveness for autocatalytic nucleation,

are difficult to specify a priori.In the case of conventional upper bainite, precipitates will form which are

thought to stimulate autocatalytic nucleation effects because precipitates are a sink

for carbon and lead to additional phase boundaries. On this point, the mechanism is

in agreement with models proposed for nucleation of acicular ferrite on inclusions

[44,45]. In such models, it is assumed that decarburisation of austenite close to the

inclusion–austenite interface occurs which may promote nucleation of acicular ferrite

[44]. For lower bainite, the thickness of bainitic plates is generally smaller and the

dislocation density is higher [3,4], both of which are considered to stimulate

autocatalytic nucleation. In the case that these interface conditions for upper and

lower bainite lead to similar values of �/, then the degree of autocatalysis is

effectively not dependent on temperature.Although the relative magnitude of autocatalytic nucleation for upper and

lower bainite is non-trivial, it is informative to perform calculations assuming that

the transition from upper to lower bainite is associated with a change in the

effectiveness factor �. Again, calculations will be performed for 0.5C–Cr–Mo

using the same parameters as discussed earlier, except for � which now is a

varying parameter. Figure 3a shows the variation in the autocatalytic parameter �with temperature calculated with Equation (7), due to both a gradual change in and a step-wise change in the factor � from 0.3 to 0.7, which is assumed to

account for a transition from upper to lower bainite at approximately 360�C. As

a consequence, it is seen that � increases strongly below approximately 360�C.

The solid lines in Figure 3b represent the 2, 50 and 99 vol% curves calculated

with Equation (10). Due to the sharp increase in � with decreasing temperature

below 360�C, the C curves do not exhibit the familiar regular shape, but reflect a

discontinuity in the kinetics around 360�C. Since autocatalysis has a minimal

effect on the start curve, the transition is only clearly visible in the 50 and 99

vol% curves. The curves calculated using a constant �¼ 20 are plotted for

comparison (dashed lines).In published literature, various TTT diagrams can be found for which the shapes

of C curves resemble the appearance of the C curves in Figure 3b; for example, the

TTT diagrams of the medium-carbon alloys on page 47, 124 and 185 in [19]. A more

detailed comparison with literature data will be given later in this paper when the

kinetic data measured for an eutectoid steel are discussed. It is important to point out

that the TTT diagrams of most steels [17–20] demonstrate that the halfway and finish

curves have a characteristic C shape and are virtually parallel to the start curve,

which is a strong indication that � is constant for such alloys. Moreover, the

calculations assuming a constant � performed for various B.S. En steels in [10]

showed very good agreement with experimental C curves [20].


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3. Experimental

Dilatometry experiments were conducted on a plain carbon steel with composition0.80wt% C, 0.61wt% Mn, 0.25wt% Si, 0.2wt% Ni and 0.2wt% Cr (Fe–0.80C).Samples of 10mm in length and 5mm in diameter were machined, and the dilatationwas measured using a Bahr 805A/D dilatometer. The Ac3 temperature of Fe–0.80C isapproximately 780�C. Samples were heated to a temperature of 950�C in 1min,austenitised for 1min at a pressure of less than 1� 10�5 mbar, and subsequentlycooled rapidly (� 60�C s�1) to the isothermal holding temperature. The bainitetransformation kinetics were measured isothermally at several temperatures in therange 260–425�C. The experimental results were reproducible, and also samples thatwere given a homogenisation treatment at 1050�C prior to the heat treatment showedidentical results to the as-received samples, which indicates that the distribution ofMn is homogeneous. In addition to the model validation against the kinetic dataof this plain carbon steel, model calculations will also be compared to literature dataof a Si-rich steel with composition 0.98wt% C, 1.89wt%Mn, 1.46wt% Si, 0.26wt%Mo, 1.26wt% Cr and 0.09wt% V [39,46], which will be designated as 0.98C–Si inthis work.

4. Results and discussion

4.1. Upper and lower bainite in Fe–0.80C

Figure 4 shows the experimental fraction curves measured for Fe–0.80C.The simulated transformation kinetics at each temperature have been calculatedaccording to Equation (8) using � as fit parameter (solid lines). The rate parameter� is calculated with Equation (9), analogous to the routine discussed earlier

Figure 3. (a) Temperature dependence of � calculated using Equation (7) as a function of agradual variation in and step-wise change in � from 0.3 to 0.7. (b) Calculated 2, 50 and 99vol% curves accounting for a sharp increase in � with decreasing temperature (solid lines) areirregular compared to the C curves calculated using a constant �¼ 20 (dashed lines).


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for 0.5C–Cr–Mo, using the values Th¼ 591�C, D�¼ 23 mm, �m¼ 0.014K�1, andQb¼ 83.2 kJmol�1. With the material-independent relationshipK1�¼ (169.8 kJ mol�1–Qb)/(705K) [10] it follows that K1�¼ 123 Jmol�1K.The differences with the values predicted using the empirical relationships [10],Th¼ 596�C and Qb¼ 81.4 kJmol�1, are within the estimated uncertainty rangesdiscussed in [10].

Since the transformation rate is most sensitive to autocatalytic nucleation athalfway transformation, it was decided to optimise the simulations in the rangef¼ 0.4–0.6. For each temperature, a good agreement with the data is observed,especially at 50% transformation, and the values of � derived from the best fits arelisted in Table 1. In the temperature range 260–340�C, the factor � is found todecrease strongly from 210 to 55. Above 340�C, a constant � of approximately 50 isobtained. The dashed line in Figure 4 represents the simulation at 260�C using�¼ 75, which is seen to strongly underestimate the kinetics. This comparisondemonstrates that it is essential to adjust the autocatalytic contribution withchanging temperature to ensure an satisfactory simulation of the kinetics.

The open circles in Figure 5 represent the experimental start curve (5 vol%) ofFe–0.80C; the error bars indicate the �2 vol% inaccuracy. The 5 and 50 vol% curvesindicated by the solid lines have been calculated with Equation (10) using the samevalues of � and � as used in the simulations shown in Figure 4. Some deviations areseen between the calculated 5% curve and the experimental start curve but the shapesof both curves are in qualitative agreement. Consistent with Figure 4, a very goodagreement with the 50 vol% transformation data (open triangles) is seen. In contrast,Figure 5 shows large discrepancies between the experimental data and the

Figure 4. Calculated kinetics of bainite formation compared to experimental data forFe–0.80C annealed at temperatures in the range 260–425�C. The best fits (solid lines) wereobtained using the values of � given in.


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simulations performed by assuming a constant � ¼ 75 in the whole temperature

range (dashed lines). Although a reasonable agreement is found for the beginning of

transformation, calculations assuming a constant � ¼ 75 lead to a rather poor

agreement with the 50 vol% data, especially below 300�C.To assess the temperature dependence of the � values derived from the best fits,

the variation of the bainitic plate thickness with temperature has been calculated

for Fe–0.80C using the neural network program described in [41]. It should be noted

that due to the high carbon concentration of this alloy the austenite yield strength

and driving force fall outside the validity range of the neural network. Thus, the

values of listed in Table 1 may be slightly imprecise; however, these predictions are

the best estimate possible, which is thought to be adequate for the present purpose.

Figure 6 shows that increases only 30% in the range 260–425�C, which is a

Figure 5. Solid lines represent the calculated 5 and 50 vol% curves accounting for increasingautocatalytic effects with lowering temperature. The experimental kinetics for Fe–0.80C at50% transformation (open triangles) are in good agreement with the simulations which areoptimised in this range by adjusting � (see Table 1). Calculations performed using a constant� ¼ 75 are shown for comparison (dashed lines).

Table 1. Model parameters for Fe–0.80C.

T (�C) � (mm) �

260 210 0.08 2.02275 170 0.09 1.84295 140 0.09 1.51310 105 0.09 1.13325 75 0.09 0.81340 55 0.10 0.66370 45 0.10 0.54410 45 0.11 0.59425 50 0.11 0.66


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relatively small change compared to the temperature dependence of calculated for0.5C–Cr–Mo discussed earlier. It is also seen in Figure 6 that the measured variationin of two Si-rich alloys is much larger [3,46]. This weak temperature dependence of calculated for Fe–0.80C means that the variation in � can be primarily attributed tochanges in the effectiveness factor �. The values of � calculated with Equation (7)are listed in Table 1, and are plotted on the right-hand y-axis of Figure 6. It is seenthat � increases strongly below approximately 340�C. It is thought that this rapidincrease in � is associated with the transition from upper to lower bainite, which isexpected to occur in this temperature range based on the transition temperature of350�C reported for similar plain carbon steels with 0.8–1.1wt% carbon [47,48].

The increase in � from approximately 0.6 in the temperature range of upperbainite towards values of about 2 below 300�C means that the �/� interfaceconditions for lower bainite are much more effective in stimulating autocatalysis. It isinteresting to mention that the results in Figure 5 are in very good agreement withthe TTT diagram of 0.90C–1.5Mn measured by Hawkins and Barford [12], whichshows a ‘‘lower nose’’ at approximately 330�C. Also, the TTT diagram of a 0.77%Calloy reported by Radcliffe and Rollason [11] shows a discontinuity in the C curvesat about 350�C. Although this atypical behaviour seems to be well established forthese plain carbon steels with 0.7–1.2wt% C, the transition from upper to lowerbainite is certainly not reflected as a discontinuity in the C curves for all steels.For example, Caballero et al. [16] investigated this phenomenon using a0.31C–1.2Mn alloy for which the TTT diagram shows a single C curve within thebainite temperature range for upper and lower bainite. Furthermore, the kinetics ofthe plain carbon steels with 0.4–0.6wt% C studied in [9] could be satisfactorilymodelled using a constant � for a wide temperature range.

In qualitative agreement with Figure 5, the TTT diagrams of various Si-rich steelsappear to exhibit a discontinuity in the C curves corresponding to the transition from

Figure 6. Temperature dependence of the plate thickness calculated for Fe–0.80C (solidcircles) using the methodology from [41], and the values of � (solid triangles) evaluated withEquation (7). Experimental values of for 0.46C–Si [3,14] and 0.98C–Si [46] are shown forcomparison (open symbols).


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upper to lower bainite [4,14]. For example, Chang [14] investigated the microstruc-

tures and isothermal bainite transformation kinetics of a 0.46C–Si steel. The

microstructure analysis in this study [14] indicated a transition from upper to lower

bainite at approximately 300�C and, correspondingly, an acceleration of the lower

bainite kinetics was found. It is seen in Figure 6 that the change in of 0.46C–Si steelis very small in the temperature range 260–350�C. This means that the acceleration in

the kinetics reported in Figure 11c of [14] is probably due to an increase in �associated with the transition from upper to lower bainite analogous to the

explanation given above for Fe–0.80C.

4.2. Incomplete transformation in 0.98C–Si

The model proposed in [10] can capture the incomplete transformation when a

simple procedure accounting for carbon enrichment in the remaining austenite is

applied to calculate the change in the temperature Th, which is a key parameter in the

model controlling the transformation kinetics. This extended model will be used to

describe the experimental data of the Si-rich steel 0.98C–Si [39]. The experimental

fraction data shown in Figure 7 have been determined by Garcia-Mateo et al. [46]

using X-ray diffraction after interrupted isothermal bainitic holding experiments in

the temperature range 200–325�C. The corresponding experimental TTT diagram of

0.98C–Si reported by Garcia-Mateo et al. [39] is shown in Figure 8. These published

data [39,46] are very useful for model validation, since for each temperature not only

the kinetics but also the bainitic plate thickness as well as the carbon content in the

retained austenite x�R has been measured (see Table 2).

Figure 7. Calculated kinetics of bainite formation compared to experimental data for0.98C–Si annealed at temperatures in the range 200–325�C.


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The bainite kinetics in 0.98C–Si have been simulated at each temperature bynumerical integration of df/dt¼ (1� f )(1þ �f )�, and calculating the change in � ineach time-step due to carbon enrichment. The rate parameter � at the start oftransformation has been calculated with Equation (9) using the values Th¼ 341�C,D�¼ 30 mm, �m¼ 0.011K�1, and Qb¼ 124.9 kJmol�1. As the transformationprogresses the carbon concentration in the remaining austenite x� will increasewhich leads to a higher Qb [10] and a lower Th [49]. The undercooling (Th�T) inparticular has a large effect on the transformation kinetics. This approach dictatesthat the transformation will cease when Th [49] equals the isothermal transformationtemperature.

The carbon concentration in the remaining austenite x�, which is an importantparameter determining the kinetics of bainite formation in Si-rich steels, is controlledby both the carbon content of the bainitic ferrite [50–52] and the degree of carbideprecipitation [40,53]. It has been observed that the carbon level in the bainitic ferrite

Figure 8. Calculated start and end curve for 0.98C–Si (solid lines) compared to theexperimental data [39]. The dashed line duplicated from [39] indicates the start curvecalculated with the model of Bhadeshia [2].

Table 2. Data from [39], values of � and x�b derived from the best fits (Figure 7), and otherevaluated parameters.

Data from [39] Model input Calculated at end of transformation

T (�C) (mm)x�bf

(wt%)x�R

(wt%)x�b

(wt%) � � te (days) fe x�e (wt%)

200 0.035 0.34 1.30 0.75 500 1.8 23 0.67 1.43250 0.055 0.28 1.65 0.55 280 1.5 3.2 0.59 1.58300 0.124 0.21 1.36 0.50 120 1.5 1.3 0.53 1.52325 0.185 0.21 1.10 0.45 80 1.5 1.0 0.26 1.16


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x�bf can be well above the concentration expected on the basis of para-equilibrium

with austenite [50]. This appears also to be significant for 0.98C–1.8Si [39] as shown

in Table 2. In addition, it seems justified to assume that some carbide precipitation

can occur in the remaining austenite despite the presence of Si [40,53]. The kinetics of

carbide precipitation during bainite formation in Si-rich steels is thought to depend

on the alloying content, the carbon content in the remaining austenite and the

transformation temperature [53]. As a consequence of such carbide precipitation the

carbon enrichment of the austenite will not be optimal. Furthermore, some Si-rich

steels annealed in the lower temperature range exhibited the existence of lower

bainitic structures containing interlath carbides [4,54], which also leads to less carbon

enrichment of the austenite.In the calculations the carbon content of bainite, x�b is adjusted to account for

various degrees of supersaturation and carbide precipitation, both of which depend

on the temperature at which bainite is formed. In the simulations, the Lever rule is

applied to calculate x� on the basis of x�b and the volume fraction bainite. Table 2

shows that values of x�b between 0.45 and 0.75 have been chosen to optimise the

agreement with the experimental data shown in Figure 7. The parameter x�b has a

strong effect on the final volume fraction bainite and the carbon concentration in the

austenite at the final stage of transformation. Also, the autocatalytic parameter � hasbeen adjusted for each temperature to improve the agreement with the experimental

data (see Table 2). Figure 7 shows that the simulations (solid lines) are in satisfactory

agreement with the experimental data. The differences between the best fits and the

experimental data are of the same order of magnitude as the scatter in the data

evaluated from the X-ray analysis. The differences observed between experimental

data and simulations might be related to the fact that the model assumes that carbon

is homogeneously distributed in the remaining austenite, the so-called mean field

approximation [7], whereas in the actual microstructure local variations in carbon

content may be present which can influence the overall transformation kinetics.For the range T¼ 200–325�C, the factor � is found to decrease from 500 to 80

(see Table 2). The dashed lines in Figure 7 represent the kinetics calculated assuming

a constant �¼ 120, the value of � found for the best fit at T¼ 300�C. Although the

start of the transformation can be reasonably well described at each temperature

using �¼ 120, the dashed lines in Figure 7 readily demonstrate that the overall

agreement becomes very poor for T¼ 250�C and T¼ 200�C when autocatalytic

effects are not assumed to increase. The parameter �, which has been calculated with

Equation (7) on the basis of � and , is approximately constant for all temperatures

as seen in Table 2; only for T¼ 200�C is a somewhat higher value of � found. This

means that the temperature dependence of � can be primarily attributed to the

variation in the plate thickness, which has been found to decrease from 185 to 35 nm

as the temperature is lowered from 325 to 200�C [39]. Although the fraction-time

curve for T¼ 325�C is not shown in [46], the start and end of the transformation at

this temperature can be determined from the TTT diagram reported in [39], which

has been used as input to perform the calculation for T¼ 325�C shown in Figure 7.

It is interesting to mention that also the kinetic data of other Si-rich steels exhibit

that the sigmoidal shape of the fraction curves becomes more pronounced with

decreasing temperature [55,56].


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Based on the same set of model parameters used for the model validation shownin Figure 7, the start curve (3 vol%) for 0.98C–Si has been calculated, which isrepresented by the solid line in Figure 8. Consistent with Figure 7, a good agreementwith the experimental start data from [39] (solid circles) is seen. Table 2 shows thecalculated time te corresponding to the bainite fraction fe at the end of the incompletetransformation. These calculated fractions and times agree well with the data givenin [39]. Only the time te calculated for the lowest temperature of 200�C isconsiderable longer than reported in [39], which is also seen in the TTT diagram ofFigure 8. Based on the overall agreement, it is argued that the bainite kinetics in0.98C–Si can be adequately simulated when the model accounts for both varyingautocatalytic effects depending on the plate thickness and carbon enrichment in theremaining austenite.

For each temperature, the carbon concentration in austenite at the end of theincomplete transformation x�e has been derived from the simulations. Table 2 showsthat these calculated values of x�e are in good agreement with the carbon contents inretained austenite x�R measured by Garcia-Mateo et al. [39], which supports thatreasonable values of x�b have been chosen to calculate the best fits. It also means thatthe carbon dependence of Th used in the simulations is properly accounting for theincomplete transformation phenomenon. Table 2 shows that the difference betweenx�b and the measured carbon content in bainitic ferrite x�bf [39] increases somewhatwith decreasing temperature, which seems plausible because the degree of carbideprecipitation can be expected to increase with decreasing temperature [4].

It is important to re-emphasise that in the calculations of the bainite kinetics inthe range 200–325�C only the concentration x�b and the autocatalytic factor � areadjusted. The time dependence of � at each temperature is controlled by x�, which inturn depends on f and x�b. It has been shown that the model can only adequatelydescribe the kinetics of the incomplete bainitic transformation in 0.98C–Si at alltemperatures when the autocatalytic factor � is assumed to increase with decreasingtemperature. The analysis of the results shows an approximate inverse dependencebetween � and , which strongly supports the validity of the autocatalysis mechanismproposed in this study.

4.3. Additional remarks

The model validation discussed above indicates that a temperature-dependent � isessential to properly simulate that the S shape of fraction–time curves becomes morepronounced with decreasing temperature. The temperature dependence of � can beattributed to the variation of the ratio �/. Although the bainitic plate thickness can be predicted with reasonable accuracy for a certain composition range, severalquestions exist with regard to the effectiveness of �/� interface boundaries describedby the model parameter �. For example, it is not known quantitatively how thechemical composition, carbide precipitation and dislocation density influence thenucleation conditions at �/� interface boundaries, i.e. the parameter �. This meansthat experimental input is required to determine the autocatalytic parameter � of acertain alloy. To enable accurate predictions for a wide variety of steels, further workis necessary, if at all possible, to elucidate the contributions of the aspects that


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control � or, at least, to result in empirical knowledge on the variation of � withcomposition and annealing.

Mechanical stabilisation is also an aspect of the bainitic transformation which isnot fully understood and can play a significant role in the overall transformationkinetics. The occurrence of mechanical stabilisation of the austenite is thought to beinduced by the transformation itself because the shape change needs to beaccommodated. The degree of mechanical stabilisation is expected to increase withincreasing fraction bainite, which offers a plausible explanation for the fact that theexperimental kinetics are often observed to be slower than predicted at the final stageof transformation. Furthermore, these effects due to mechanical stabilisation appearto be more prevalent at lower transformation temperatures.

Finally, it is worth mentioning that the description of nucleation in terms of areadensities might be a suitable start point for computational modelling of themicrostructure evolution in an austenitic matrix, e.g. by means of methods based oncellular automata or phase field modelling. Although the analytical model explainedin the present paper offers a simple way to calculate the volume fraction of bainite,such knowledge of phase fractions is not always sufficient. To correlate mechanicalproperties to microstructures, it is important to model the distribution of phaseconstituents. This can, in principle, be achieved with advanced techniques such asphase field modelling which can keep track of the grain boundaries. It is foreseenthat, by combining Equation (6) with such powerful simulation techniques, thetransformation kinetics can be calculated locally at the available grain boundaries inthe matrix.

5. Conclusions

A detailed derivation is proposed which shows that it is possible to calculate theacceleration of the bainite transformation kinetics due to autocatalysis by consid-ering changes in the �/� interface boundary area per unit volume as thetransformation progresses. In this model, the effectiveness of �/� interfaceboundaries (�) and the thickness of bainitic plates () determine the autocatalyticcontribution to the overall transformation kinetics: the autocatalytic factor �/�/.The derivation confirms Entwisle’s hypothesis [34] that the autocatalytic contribu-tion is proportional to the volume of bainite. Furthermore, the present modellingapproach yields that � is not a material constant but can vary with the isothermalholding temperature.

Comparison with experimental data of steel 0.98C–Si is made, which demon-strates that the isothermal kinetics of bainite formation can be satisfactorilydescribed when � is inversely proportional to . The modified model has also beentested against experimental data of a plain carbon steel Fe–0.80C for which theirregular temperature dependence of the kinetics can only be properly describedusing a varying �, which is probably a consequence of a change in � associated withthe transition from upper to lower bainite. To sum up, although the majority ofpublished experimental data can be simulated adequately with the assumption of atemperature-independent �, experimental data of 0.98C–Si and Fe–0.80C and someother alloys in the literature strongly indicate that � has a temperature dependence


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for certain steels. With the new autocatalysis model proposed in this study, all datacan be explained.

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Autocatalytic nature of the bainitic transformation in steels: a new hypothesis