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Austenite Grain Size Effects on Isothermal Allotriomorphic Ferrite Formation in 0.37C–1.45Mn–0.11V Microalloyed Steel. Carlos Capdevila1, Francisca G.
Materials Transactions, Vol. 44, No. 6 (2003) pp. 1087 to 1095 #2003 The Japan Institute of Metals

Austenite Grain Size Effects on Isothermal Allotriomorphic Ferrite Formation in 0.37C–1.45Mn–0.11V Microalloyed Steel Carlos Capdevila1 , Francisca G. Caballero1 and Carlos Garca de Andre´s1 1

Department of Physical Metallurgy, Centro Nacional de Investigaciones Metalu´rgicas (CENIM), Consejo Superior de Investigaciones Cientficas (CSIC), Avenida Gregorio del Amo 8, E-28040 Madrid, Spain

Allotriomorphic ferrite is the morphology of ferrite formed at relatively small undercooling below the Ae3 temperature. Because it is the first transformation in austenite decomposition during cooling, allotriomorphic ferrite affects indirectly the subsequent austenite phase transformations, and then its study is of vital importance. The present paper is concerned to the theoretical and experimental study of the isothermal decomposition of austenite in allotriomorphic ferrite in a modern medium carbon vanadium–titanium microalloyed forging steel. This paper deals with the isothermal austenite-to-allotriomorphic ferrite as a whole, considering the specific role of different features such as prior austenite grain size (PAGS) and isothermal temperature, in nucleation and growth processes independently. (Received January 14, 2003; Accepted April 24, 2003) Keywords: medium carbon steel, microalloyed steel, allotriomorphic ferrite, isothermal growth kinetics

1.

Introduction

The crude steel production by the major steel-producing countries is of the order of 850 millions tonnes per annum, the vast majority of which is subsequently fabricated into plates, bars, rods, strips, sheets and coils, using large scale deformation processes, in which the cooling rate from austenite phase field is relatively slow. Such products are destined for use in the construction, power generation, ship building and automotive industries (amongst others), where structural materials of high strength, good toughness, low cost and weldability are required. Usually, the microstructure of these steels is a mixture of allotriomorphic ferrite and pearlite. The mechanical properties are derived from these microstructural constituents. A great number of attempts, from purely empirical to semiempirical models, have been made to predict the kinetics of ferrite transformation.1,2) However, models recently developed are becoming less empirical since they rely on thermodynamic and phase transformation theories.3) Unemoto et al.4) developed a methodology to simulate the allotriomorphic ferrite transformation under isothermal conditions. Reed and Bhadeshia5) reported a thermodynamic model couple with a simplified kinetic theory. This model can reproduce the C-curve behavior typical of those parts of the time-temperature-transformation diagrams that are due to allotriomorphic ferrite in low-carbon multicomponent steel. In principle, most of these models are able to predict the kinetics of allotriomorphic ferrite for low-carbon low-alloy steels. But, the level of agreement between predicted and calculated allotriomorphic ferrite kinetics is less satisfactory for medium carbon microalloyed steels. On the other hand, the role of PAGS on the isothermal allotriomorphic ferrite transformation is worth studying. A reduction in the austenite grain size should lead to an increase in the rate of transformation because of the greater number density of grain boundary nucleation sites.6) However, it is not clear if PAGS would affect the growth kinetics. This work represents an attempt to clarify and model the role PAGS on the isothermal decomposition of austenite in

allotriomorphic ferrite, so the part of the time-temperaturetransformation (TTT) diagram that is due to this reaction product can be estimated as a function of temperature, alloy chemistry and PAGS. Emphasis is placed on performing this for multicomponent steels, because it is these that are relevant both practically and commercially. 2.

Materials and Experimental Procedures

The chemical composition of the steels studied is presented in Table 1. The isothermal decomposition of austenite has been analysed by means of a high-resolution dilatometer DT1000 Adamel-Lhomargy described elsewhere.7) The dimensional variations of a cylindrical specimen (12 mm in length, 2 mm in diameter) are transmitted via an amorphous silica pushrod. These variations are measured by a linear variable differential transformer (LVDT) sensor in a gas-tight enclosure enabling to test under vacuum or in an inert atmosphere. The dilatometer is equipped with a low thermal inertia radiation furnace for heating. The heat radiated by two tungsten filament lamps is focussed on the specimen by means of a bi-elliptical reflector. The temperature is measured with a 0.1 mm diameter Chromel-Alumel (Type K) thermocouple welded to the specimen. Cooling is carried out by blowing a jet of helium gas directly onto the specimen surface. The helium flow-rate during cooling is controlled by a proportional servovalve. These devices ensure an excellent efficiency in controlling the temperature and holding time of isothermal treatments and also ensure fast cooling in quenching processes. Dilatometric samples were austenitised in vacuum (1 Pa) at a constant rate of 5 C/s. Austenitization conditions and their corresponding prior austenite grain size (PAGS) are listed in Table 2. After austenitisation, specimens were isothermally

Table 1 Chemical composition (mass%) of the steel. C

Si

Mn

Cr

Al

Ti

V

Cu

Mo

0.37

0.56

1.45

0.04

0.024

0.015

0.11

0.14

0.025

C. Capdevila, F. G. Caballero and C. G. de Andre´s

1088 Table 2

Austenitisation conditions. Prior austenite grain size (PAGS) and grain boundary surface area per unit volume (SV ) results.

Austenitisation

Austenitisation

Prior austenite grain

Austenite grain surface per

temperature, T / C

time, t /s

size, PAGS/mm

unit volume, SV /mm1

1250

60

76  12

0:039  0:007

1000

60

11  5

0:25  0:03

transformed at temperatures of 700, 640 and 600 C at different holding times and subsequently quenched to room temperature. Specimens were grounded and polished using standardised metallographic techniques. A 2% nital etching solution was used to reveal the ferrite microstructure by optical microscopy. Previous works with this steel8,9) indicated the presence of TiN and VC in the as-received microstructrure. MTDATA10) calculations also indicates that at the austenitisation temperature range studied, VC has already dissolved. Moreover, the variation in volume fraction of TiN between the two austenitisation temperatures tested is very small according to calculations, which allows us to assume no significant changes in volume fraction of these particles for the different studied transformation temperatures. The PAGS was estimated by means of the thermal etching method.11) This method consists in revealing the austenite grain boundaries in a pre-polished sample by the formation of grooves in the intersections of austenite grain boundaries with the polished surface, when the steel is exposed to a high temperature in an inert atmosphere. These grooves decorate the austenite grain boundary and make it visible at room temperature in the optical microscope. Once the austenite grain boundaries are revealed, binary images of the microstructures are processed using and image analyser.12) Measurements are made of at least 200 grains randomly selected from the population of grains revealed on crosssections of the steel. The PAGS results are analysed in terms of mean values of the equivalent circle diameter PAGS ¼ ð4A=Þ1=2 ,13) being A the equivalent circle area. Table 2 shows the average PAGS results for two austenitisation conditions. Figure 1 shows the microstructures from which the values listed in Table 2 were obtained. The grain boundary surface area per unit volume SV was measured on the same micrographs used to determine the PAGS, by using the stereological relationship SV ¼ 2PL , where PL is the density of the intersection points of austenite grain boundaries with a circular test grid.14) Table 2 also shows the SV results for the tested steel. The volume fraction of allotriomorphic ferrite VV was statistically estimated by a systematic manual point counting procedure15) for several isothermal times. A grid superimposed on the microstructure provides, after a suitable number of placements, an unbiased statistical estimation of the VV . 3.

Results and Discussion

3.1 Transformation kinetics The evolution of the transformation kinetics during the isothermal decomposition of austenite in allotriomorphic ferrite could be analysed by means of the following approach which closely follows Cahn’s analysis.5–16) The allotrio-

Fig. 1 Austenite grain structure revelaed by means of thermal etching after austenitisation at (a) 1250 C (PAGS ¼ 76 mm), and (b) 1000 C (PAGS ¼ 11 mm).

morphs are considered as cylinders having their faces parallel to the austenite grain boundary plane. The cylinders are assumed to grow towards the centre of the grain with their length, L, varying parabolically with time (t) through the equation17) L ¼ ðt  Þ1=2

ð1Þ

where  is the incubation time for allotriomorphic ferrite formation, i.e. the minimum time at which some allotriomorphs are formed, and  is the parabolic growth rate constant which will be subject of further study in forwards sections of this paper. Therefore, the extended volume fraction is described by18) VVex ¼ 2R20 NV L ¼ 2R20 NV ðt  Þ1=2

ð2Þ

where NV is the number of allotriomorphs per unit volume of austenite grain. On the other hand, it is necessary to take into account the geometrical impingement to avoid the overestimation of the

Austenite Grain Size Effects on Isothermal Allotriomorphic Ferrite Formation

i

dVV ¼ ð1  VV Þ dVVex

20

(a)

15

Vv (vol%)

volume fraction of allotriomorphic ferrite, since an allotriomorph can not invade the volume occupied by another one previously. Ferrite nucleation is not random, but it is produced predominantly on the austenite grain boundaries. Thus, the spatial distribution of ferrite nuclei is clustered and the classical Johnson-Mehl equation19) is not applicable. The following phenomenological equation has been suggested for a non random impingement by Hillert:20) ð3Þ

where i > 1 for non random geometrical impingement. An exponent i ¼ 2 has been used in several cases where non random geometrical impingement has been produced.20–25) Therefore, the use of a value of i ¼ 2 has been accepted in the present study. Thus, it is obtained: 2

dVV ¼ ð1  VV Þ dVVex

10

5

76 microns 11 microns 0

ð4Þ

0

6

12

1/2

24

30

36

/s

30

(b)

25

ð6Þ 20

Vv (vol%)

The integration of the above equation gives, VV ¼ ¼ 2R20 NV ðt  Þ1=2 1  VV

ð5Þ

18

(t-τ)

Therefore, deriving in eq. (2) and subsituting in eq. (4), the following expression is obtained: dVV ¼ ð1  VV Þ2 R20 NV ðt  Þ1=2 dt

1089

where  is the transformation ratio. It might be obtained from the combination of eqs. (5) and (6) an expression which describe the evolution with time of the volume fraction of allotriomorphic ferrite, i.e. the transformation rate. In this sense, this rate is expressed below, dVV VV ð1  VV Þ ð7Þ ¼ 2ðt  Þ dt

3.2 Classical nucleation theory Nucleation does not occur with equal probability in all regions of the assembly. In reality, nucleation will be favored on inhomogeneities within the material, such as defects, grain boundaries and impurities. These are high energy sites due to the surface are created between them and the matrix, and nucleation of more coherent particles in their place can lead to a reduction in free energy. In the case of ferrite nucleating from austenite, prior austenite grain boundaries are favoured

10

76 microns

5

11 microns 0

0

6

12

18

(t-τ)

1/2

24

30

36

/s

25

(c)

20

Vv (vol%)

Figure 2 shows the evolution of the allotriomorphic ferrite volume fraction with isothermal holding time at three different temperatures for both selected PAGS. The volume fraction of allotriomorphic ferrite was determined by means of quantitative metallographic techniques from micrographs as shown in Fig. 3. Likewise, Fig. 4 shows the evolution of the transformation rate with time at three isothermal temperatures and for the two PAGS. It is conclude from this figure that the PAGS strongly affects the transformation kinetics. In other words, the PAGS not only affects the evolution of the volume fraction of allotriomorphic ferrite, but also affects the rate at which the transformation of austenite in allotriomorphic ferrite proceeds. Since the isothermal decomposition of austenite in allotriomorphic ferrite is a nucleation and growth processes, in the following sections it will be analysis the influence of PAGS on nucleation processes and in growth processes, separately.

15

15

10

5

76 microns 11 microns

0

0

6

12

18

24

30

36

(t-τ) /s 1/2

Fig. 2

Evolution of VV with time at (a) 700 C, (b) 640 C, and (c) 600 C.

1090

C. Capdevila, F. G. Caballero and C. G. de Andre´s 0.1

(a)

76 microns 11 microns

dVv/dt / s

-1

0.08

0.06

0.04

0.02

0

0

0.1

0.2

1/(t-τ) / s

0.3

-1

0.15

(b)

76 microns 11 microns

dVv/dt / s

-1

0.1

0.05

0

0

0.05

0.1

0.15

0.2

1/(t-τ) / s -1 0.15

(c)

76 microns 11 microns

Fig. 3 Microstructures obtained at 640 C for (a) 30 s, (b) 60 s, and (c) 1000 s. (AF: allotriomorphic ferrite; IF: idiomorphic ferrite; P: pearlite; M: martensite).

sites. If the effects of strain energy are unimportant or neglected, (Lee et al.26) demonstrated that strain energy would only influence the critical nucleus shape if it was a large fraction of the volume free energy change for nucleation), then the driving force for nucleation at a particular site will depend on the reduction in free energy due to removal of the high energy grain boundary surface. Destruction of this surface provides the energy for formation of the new nucleus surface and volume. There are three types of grain boundary sites to consider: grain faces, edges and corners. For each type of site, the most important factors are the density of sites and the probability of nucleation at that site (in other words, how energetically favorable it would be to form a nucleus).

dVv/dt / s

-1

0.1

0.05

0

0

0.05

0.1

1/(t-τ) / s

0.15

0.2

-1

Fig. 4 Evolution of dVV =dt with (t  ) for two PAGS at (a) 700 C, (b) 640 C, and (c) 600 C.

The first theoretical treatment of this was due to Cahn,16) who utilized the geometrical calculations of Clemm and Fisher.27) Cahn justified a nucleation rate per unit grain boundary area

Austenite Grain Size Effects on Isothermal Allotriomorphic Ferrite Formation

ðK j G þ QÞ na j exp  2 h kB T

)

1.0E+08

ð8Þ

where kB is the Boltzmann’s constant; h is the Planck’s constant, Q is the activation energy for atoms crossing the austenite/ferrite nucleus interface; G is the critical free energy change to nucleation; na is the number of atoms per unit area; j represents the number of sites of a particular type available for nucleation, and K2j is a ‘shape factor’ for the sites of different dimensionality j. A value of j ¼ 3 for homogeneous nucleation, 2 for faces, 1 for edges and 0 for corners. The factor j is approximately equal to ð=d Þ3j , where  is the effective grain boundary thickness (reasonable value of 2:5  1010 m28)) and d is the austenite grain size. As discussed by Cahn16) and Christian29) the shape factor is a function of the ratio of the free energy required to form a grain boundary nucleus to that needed for homogeneous nucleation. Sites of lower dimensionality lead to smaller ratios, so that corner sites are more favourable than edges, which are more favourable than faces, i.e. K22 > K21 > K20 is expected. However, the relative nucleation rates do not necessarily increases in the same order as the activation energies for nucleation decreases, since the density of sites also decreases as the mode of nucleation changes from faces, to edges, and finally to grain boundary corner. The number of atoms per unit area on the three types of sites can be expressed as nfa ¼ na ð=d Þ, nea ¼ na ð=d Þ2 , and nca ¼ na ð=d Þ3 where na is the number of nucleation sites per unit grain boundary area and the supscripts f , e and c indicates faces, edges and corners, respectivley.16) This will obviously always give a greater number of face sites than edges and then corners, as would be expected from consideration of an ideal equiaxed grain shape and also reflects the decrease in number of possible sites per unit volume as austenite grain size increases. This means that, with decreasing temperature, corners, edges, and finally faces make the greatest contribution to the nucleation rate. Lange et al.30) proposed a model to calculate the classical nucleation rate of allotriomorphic ferrite based on a traditional disk-shaped ‘pillbox’ nucleus. In that case, the incubation time for allotriomorphic ferrite is estimated as: ¼

12kB Ta4  DC xv2 G2v

ð9Þ

where DC is the diffusivity of carbon in austenite; v is the volume of an atom of iron in ferrite; a is the average of the lattice parameters of both phases, ferrite and austenite; x is the average carbon content in mole fraction; Gv is the volume free energy change associated to the formation of the nucleus; and  is the interfacial energy per unit area of the nucleus. Therefore, it is clear from eq. (9) that the incubation time is independent of the number of nucleation sites in which embryos are able to be formed, which subsequently lead to form a allotriomorphic ferrite nucleus, i.e. it is independent of PAGS. It is possible to make approximations to eq. (8). For example, at temperatures well below the Ae3 , nucleation at faces is likely to dominate; eq. (8) then approximates to

-2 -1

j¼0

(

/m s

¼

j¼2 X kB T

HET

IBHET

Nucleation Rate IB

in a steady state of the form:

1.0E+06

1091

76 microns

Ae 3

11 microns

1.0E+04 1.0E+02 1.0E+00 1.0E-02 1.0E-04

500

600

700

800

o

Temperature T / C Fig. 5

Evolution of IBHET with temperature for the two selected PAGS.

IBHET ¼

  kB T f ðK 2 G þ QÞ na exp  2 h kB T

ð10Þ

The value nfa ¼ na ð=d Þ where na is the number of nucleation sites per unit grain boundary area, and might be expressed as K=2 where K is a constant (reasonable value of 1:0  109 28)) and  is the effective grain boundary thickness. A value for Q of 240 kJ mol1 , equal to the activation energy for self-diffusion of iron, was determined.5) The 3 activation energy for nucleation, G ¼  =G2V , where 2 31)  ¼ 0:022 J m ) and GV is the maximum chemical free energy change per unit volume available for nucleation under paraequilibrium.5) Finally, K22 is a constant taking into account the shape of the critical nucleus, and a value of 2:6  103 has been considered.5) Figure 5 shows the evolution of IBHET with the temperature for the two selected PAGS. The purpose of this figure is to point out how much the PAGS affects nucleation of ferrite. In this sense, as the austenite grain diameter increase from 11 up to 76 mm, the nucleation rate is reduced in seven times. So far, it has been established that the incubation time for the formation of the first allotriomorphic ferrite nucleus it is independent of the total number of nucleation sites and hence of the PAGS. However, the rate at which the allotriomorphic ferrite nucleus are formed in those nucleation sites available is strongly influenced by the PAGS, which lead to site saturation becomes faster as PAGS is decreased. This result is consistent with the ones presented at Figs. 2 and 4, in which it is clear that the faster transformation rate, the small PAGS is. 3.3 Allotriomorphic ferrite growth kinetics The austenite grain surface is almost entirely covered by allotriomorphic ferrite when site saturation has occured, i.e. all the nucleation sites are occupied. In such a case, hard impingement is produced and the allotriomorphic ferrite grains hinder each other’s growth along directions parallel to the respective austenite grain boundaries, so only perpendicular growth is allowed. The allotriomorphic ferrite growth

C. Capdevila, F. G. Caballero and C. G. de Andre´s

perpendicular to the austenite grain boundaries directions can be described by eq. (1) and the transformation ratio () by eq. (6). Assuming that ferrite nucleates primarily on grain boundaries, Grong32) reported that NV is given as: NV ¼ na SV

76 microns 11 microns

ð11Þ

where na is the number of nucleation sites per unit grain boundary area and SV is the grain boundary surface area per unit volume. Thus, SV indicates the nucleation sites available at the austenite grain boundary. Since hard impingement is produced, SV is given by: SV

¼

NV R20

3

(a)

ξ /SV / µm

1092

2

1

0

R20

where is the lateral surface of an allotriomorph considered as a cylinder of radius R0 . Combining eqs. (6) and (12), the following result is obtained: ð13Þ

SV

The values of  and are experimentally determined for the studied steel at PAGS of 76 and 11 mm, respectively. Therefore, from the slope of the plot (=2SV ) versus ðt  Þ1=2 a value of the parabolic growth rate constant is obtained (Fig. 6). From this figure it is concluded that the value of the parabolic growth rate constant it is independent of PAGS, and it is strongly influenced by the isothermal temperature. Moreover, the experimentally obtained values of parabolic growth rate constant for allotriomorphic ferrite has been used to test the calculated ones from the thermodynamic and kinetic theory. The value of  can be obtained by numerical solution from the equation following:33) !   1=2   2  C  C   DC ffi  exp ð14Þ ¼ 2 erfc pffiffiffiffiffiffi  4DC  C   C  2 DC DC

is the diffusivity of carbon in austenite, C is the where overall carbon content, C is the austenite solute content at the interface, and C  is the ferrite solute content at the interface. According to Bhadeshia,34) the consideration of paraequilibrium is a good approach for the kinetics of this transformation. In that case, partitioning of substitutional solute atoms does not have time to occur and the adjoining phases have identical X=Fe atom ratios, where X represents the substitutional solute elements. Then, the substitutional lattice is configurationally frozen, but interstitial solutes such as carbon are able to partition and attain equilibration of chemical potential in both phases. Hence, the values of C and C in eq. (14) refer to carbon concentrations and they were calculated according to the procedure reported by Shiflet et al.35) C and C  values as well as  values are listed in Table 3. Figure 7 shows the evolution of  with isothermal temperature. Superimposed to the theoretical curve, the experimentally obtained values of  are presented. It could be concluded from this figure that there is an excellent agreement between experimental and calculated results, and therefore, it might be concluded that the above mentioned method to obtain the parabolic growth rate constant is very reliable in the case of site saturation.

4

8

12

(t- τ )

-1/2

16

20

24

/s

3

(b)

76 microns 11microns

ξ /SV / µm

 Þ

1=2

0

2

1 y = 0.2986x 2 R = 0.826 0 0

4

8

12

(t- τ )

-1/2

16

20

24

/s

3

(c)

76 microns 11 microns

ξ /SV / µm



2SV ðt

y = 0.1098x 2 R = 0.9372

ð12Þ

2

1 y = 0.2131x 2 R = 0.8872 0 0

4

8

12

(t- τ )

-1/2

16

20

24

/s

Fig. 6 Parabolic growth rate constant graphically obtained at (a) 700 C, (b) 640 C, and (c) 600 C.

3.4

Model for isothermal allotriomorphic ferrite formation So far, it has been analysed the role of PAGS on nucleation

Austenite Grain Size Effects on Isothermal Allotriomorphic Ferrite Formation

1093

Table 3 Calculated values of C  , C  and . C  /mass%

C  /mass%

/m s1=2

700

0.014

0.500

2:88  107

640 600

0.016 0.016

0.903 1.195

4:85  107 4:76  107

Parabolic Constant α / m s

-1/2

4.00E-07

1 0.8 f( α ,I,t)

T/ C

0.6 0.4 11 microns 76 microns

0.2

3.50E-07

0

3.00E-07

0

0.25

2.50E-07

0.5

0.75

1

t/t total

2.00E-07

Fig. 8 Fraction of the total transformation time needed for site saturation for both PAGS studied.

1.50E-07 1.00E-07

started. By contrast, a value of f ð; I; tÞ ¼ 1 means that saturation of nucleation sites has been reached. The expression for f ð; I; tÞ is37)

5.00E-08 0.00E+00

500

600

700

800

o

Temperature T / C

Z1 f ð; I; tÞ ¼

Comparison between calculated and experimental values of .

and growth kinetics of allotriomorphic ferrite. The aim of this section is to present a model for the evolution of allotriomorphic ferrite volume fraction with time at a certain temperature which incorporate the facts presented in previous sections. Since allotriomorphs are assumed as cylinders nucleated on the grain boundary, the formed volume fraction of allotriomorphic ferrite (VV ) is determined as follows. Let us consider a series of arbitrary planes parallel to the boundary and spaced a distance y apart. If the half thickness of the particle (Z) exceeds the distance y then the area of intersection with the plane can be obtained. Thus, VV is determined evaluating the total of such areas of intersection for all the particles growing from the boundary. Moreover, no allowance has been made for the overlap of particles emanating from different regions of the boundary. The following equation is proposed to describe the evolution of VV with time (t > ) during the isothermal decomposition of austenite36,37)    2SV ðt  Þ1=2 f ð; I; tÞ ð15Þ VV ¼ Ve 1  exp  where Ve is the maximum volume fraction of allotriomorphic ferrite, and is the supersaturation in carbon which can be estimated from the phase diagram as follows C  C ð16Þ ¼  C  C The process of nucleation is taking into account in eq. (15) by means of the function f ð; I; tÞ, which takes values between 0 and 1. A value of f ð; I; tÞ ¼ 0 means that nucleation has not

ð17Þ

0 1=2

Fig. 7

1  expf4:5I2 t2 ½1  4 gd

where ¼ y=ðt Þ corresponding to the ratio between the distance to the arbitrary plane and the half thickness of the allotriomorph at time t. The nucleation rate (I) has been calculated according to eq. (10). The value of the expression f ð; I; tÞ tends to unity when the saturation of the nucleation sites in the austenite grain boundary is reached. Figure 8 illustrate the evolution of the nucleation function (the expression f ð; I; tÞ in eq. (17)) with the homologous time (a ratio between time and the total time need to reach the equilibrium volume fraction of allotriomorphic ferrite) at a temperature of 600 C. This figure shows that the time required to saturate the total number of sites available for allotriomorphic ferrite nucleation (f ð; I; tÞ ¼ 1) is almost the 75% of the transformation time for a PAGS of 11 mm. By contrast, for a PAGS of 76 mm this time is almost negligible. Therefore, for a coarse austenite grain size, the time necessary to saturate the austenite boundaries is so small that the formation of the majority of ferrite essentially involves the thickening of layers of grain boundary ferrite. Hence, it is a reasonable approach to consider this transformation mainly in terms of growth for modeling of isothermal austenite decomposition in allotriomorphic ferrite. Finally, the results presented in this work are all included in the calculation of the time-temperature-transfomration (TTT) diagrams for allotriomorphic ferrite for two very different PAGS, which are shown in Fig. 9. It is clear from the figure that the aparition of the first allotriomorphs, i.e. the incubation processes, are not related with the value of PAGS. By contrast, Ve is reached faster as PAGS decreases.

C. Capdevila, F. G. Caballero and C. G. de Andre´s

1094

Incubation 50%Ve 100%Ve 750

(a)

o

Temperature T / C

Ae3 700

650

600

550 1.E+00 1.E+01 1.E+02 1.E+03 1.E+04 1.E+05

Time t / s Incubation 50%Ve 100%Ve 750

(b)

700

o

Temperature T / C

Ae3

650

and theoretical analysis. It has been concluded that PAGS does not affect the parabolic growth rate constant, and hence does not influence on growth kinetics. Likewise, it has been found a good agreement between the parabolic growth rate constant values experimentally obtained, and the theoretical values derived from the thermodynamic and kinetic theory. (4) Due to the excellent agreement between experimental and calculated results for the parabolic growth rate constant, the consideration of paraequilibrium is a reasonable assumption for modeling the isothermal decomposition of austenite in allotriomorphic ferrite. (5) Since for a coarse PAGS the time required to complete the transformation is much longer than that needed for the saturation of nucleation sites in the austenite grain boundary, it is a sensible approach to consider the formation of allotriomorphic ferrite during the isothermal decomposition of austenite essentially in terms of growth. Acknowledgements The authors acknowledge financial support from Spanish Ministerio de Ciencia y Tecnologa (PETRI 1995-0436-OP). Carlos Capdevila would like to express his gratitude to the Consejo Superior de Investigaciones Cientifcas for financial support as a Post-Doctoral contract (I3P PC-2001-1). Francisca G. Caballero would like to thank the Consejera de Educacio´n. D.G. de Investigacio´n de la Comunidad Auto´noma de Madrid (CAM) for the financial support in the form of a Postdoctoral Research Grant. REFERENCES

600

550 1.E+00 1.E+01 1.E+02 1.E+03 1.E+04 1.E+05

Time t / s Fig. 9 Transformation temperature region for allotriomorphic ferrite (a) PAGS ¼ 11 mm, and (b) PAGS ¼ 76 mm.

1) 2) 3) 4) 5) 6) 7) 8)

4.

Conclusions

(1) The influence of PAGS on the allotriomorphic ferrite formation kinetics has been analysed in this paper. It has been experimentally demonstrated that PAGS strongly affects the isothermal allotriomorphic ferrite formation kinetics. The finer PAGS, the faster transformation rate is. (2) Bearing in mind that the isothermal decomposition of austenite in allotriomorphic ferrite is a nucleation and growth process, the nucleation and growth kinetics has been separately analysed. It has been concluded that PAGS do not affect the incubation time for allotriomorphic ferrite, but strongly affects the nucleation rate. (3) The parabolic growth rate constant for allotriomorphic ferrite transformation has been measured through a combination of quantitative metallogaphic techniques

9) 10) 11) 12)

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