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MartensiteFractionDeterminationUsingCoolingCurveAnalysis

ARTICLEinSOLIDSTATEPHENOMENA·JUNE2011

DOI:10.4028/www.scientific.net/SSP.172-174.221

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PatricioFernandoMendez

UniversityofAlberta

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JohnGibbs

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ThomasKannengiesser

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Martensite fraction determination using cooling curve analysis

Diana Marcano1,a , Patricio Mendez2,b, John W. Gibbs3,c and Th. Kannengiesser1,d

1Federal Institute for Materials Research and Testing, Berlin-Germany2University of Alberta, Edmonton-Canada3Northwestern University, Chicago-USA

adianamarcano@bam.de, bpmendez@ualberta.ca, cjwgibbs@u.northwestern.edu,dthomas.kannengiesser@bam.de

Keywords: martensitic transformation, cooling curve analysis.

Abstract. This work presents a method of calculating the martensite fraction of an Fe-alloy,using cooling curve analysis (CCA). It is based on a differential heat balance equation whichtakes into account only convective exchange with the surroundings. By measuring a T(t) curveof an Fe-alloy and solving numerically the differential heat balance equation the martensitefraction can be calculated. It is found that calculated martensite fraction using this methodologyis comparable with results obtained using electron backscattering diffraction (EBDS).

Introduction

Thermal analysis techniques follows temperature changes in a sample under cooling andmonitoring in this way the phase transformations that occur. Normally, differential thermalanalysis (DTA) [1] is used for the thermal analysis of solidification. In DTA, heat exchangewith the environment (absorbed or evolved) due a phase transformation is calculated as thedifference of thermal events in the sample and a neutral reference. Recently, computer aidedcooling analysis has been used to determine thermo-physical properties of alloys like latentheat of transformation and solid-liquid fractions [2] and precipitations [3]. One advantage ofthis method is the acquisition of cooling curve data of the sample with the help of thermocou-ples and the afterwards processing of the data on the computer where different analysis canbe carried out. In this work, the Cooling curve analysis (CCA) method is applied to calculatethe martensite fraction of an Fe-alloy. This method also called the equation-based Newtonianmethod uses a heat balance equation together with a cooling curve data to calculate the marten-site fraction of an alloy upon cooling. The Newtonian method assumes Newtonian cooling ofthe sample, i.e. the thermal gradients across the sample are considered to be zero and heattransfer takes place by convention to the mold at a constant temperature Ta. In this case theheat exchange can be written as

Q = hA(T − Ta). (1)

Where A is the surface area of the sample and h is a heat transfer coefficient. The CCAmethod, was originally applied to calculate the solid fraction of a 45% wt Ag-Al alloy uponcooling [4], by comparing the derivative of the cooling curve in the transformation region tobaseline data that is interpolated from the single-phase regions. This baseline data is called azero curve. The zero curve, represent what the derivative of the cooling curve would have lookedlike in the solidification region if no thermal energy had been released by phase transformations.Differences between the derivative of the cooling curve and the zero curve are recorded as heatevolved due to phase transformations.

Analysis

Following the work from Gibbs and Mendez [3], the differential form of the heat balance equationEq.2 is used. It has the advantage to allow for a simple numerical solution of the equation.The heat balance equation can be written as a difference between heat input and heat losses,where external heat flow imposed to the system is taken into account through the r.h.s. ofEq.2. This form of the heat balance equation assumes a system with Newtonian cooling with aconstant heat transfer coefficient. This means the external heat flow is assumed to occur withthe surroundings only in a convective way. Conduction of heat is not taken into account inthe model and the effects of radiation do not deviate significantly the heat losses from beingproportional to (T-Ta). It was originally used to calculate liquid and solid phase fractions of asystem under solidification.

dQ

dt= m [fCpost + (1− f)Cpre]

dT

dt+ m∆H

df

dt. (2)

Where Cpost and Cpre are the heat capacities before and after the transformation, f is thetransformed fraction and ∆H is the latent heat of transformation. Regrouping constants, Eq.2can be rewritten as:

CLdf = gpre(1− f)dT + gpostfdT + (T − Ta)dt. (3)

gpre and gpost are quantities related to the specific heat capacities in the different phaseswhich are assumed to be temperature and time independent. CL is a constant related to thelatent heat of transformation. These quantities comprise the old variables as:

gpre =mCpre

hAgpost = mCpost

hACL =

m∆H

hA. (4)

Eq.3 is a differential equation which may be solved with different finite-differences algo-rithms. An explicit Euler forward integration scheme in order to calculate the transformingfraction upon cooling is used:

f it = f i−1

t +1

CL

[gpost

f i−1t

f ∗t

(T i − T i−1

)+ gpre

(1− f i−1

t

f ∗t

) (T i − T i−1

)]+

1

CL

(T i − Ta

) (ti − ti−1

). (5)

Where f t∗ represents an average value of f t over a range of temperatures below the transfor-

mation temperature range. Assuming initial values of CL and f t∗, f t can be calculated through

iteration, where in each iteration a more accurate value of f t∗ is found by requiring that at

known temperatures the conditions f t=0 and f t=1 are satisfied. As input for the calculation,a cooling curve T(t) and a known CL value are needed. Before the transformation occurs thefollowing conditions are valid: f t=0 and df =0. Therefore, it follows from Eq.3:

1

gpre

=d [ln(T i − Ta)]

dt. (6)

A similar equation can be written for gpost at the end of the transformation where f t=1. Inthis way, the inverses of gpre and gpost are fitted to the data as a function of time.A study of the thermodynamics of martensitic transitions should take into account some charac-teristic features of these transitions: surface energy contributions associated with the interfacesseparating different phases and elastic strain energy contributions related to the accommodationof the domains of the martensitic phase within the matrix. For a thermally-induced martensitic

transition without applied stress at a constant pressure, the heat exchange Q, can be writtenas:

Q = ∆H + Eel. (7)

Where Eel is the storage of elastic energy in the system. This means, for a martensitic trans-formation the heat exchanged Q, does not coincide with the enthalpy change ∆H (latent heatof transformation), due to the storage of elastic energy in the system. As pointed out by Ortin[8] only after a complete cycle, when the system has come back to the same thermodynamicstate from which it started, the different contributions ∆H and Eel can be evaluated from theexperimental heat Q. However, as observed in a previous work [9], the Eel (stored elastic energy)is very small when compared to ∆H. Neglecting this term, the martensite fraction f t, may stillbe estimated from the data. That is, the phase fraction calculation is based only on a thermalbalance with no contribution for strain energy.

Experimental Procedure

The selected specimen was a small Fe-alloy (0.04%C, 10% Cr and 10% Ni) cylinder. The samplewas heated with a heating rate of 5.8 ◦C/seg in a nitrogen atmosphere until 800 ◦C and allowedto rapidly cool in an oil bath. The temperature was recorded using a single type K thermocouple.Data was taken every 0.02 sec. The sample was heated and not tempered but instead afterreaching the highest temperature immediately removed from the oven and allowed to cool. It isto be noticed that the sample does not achieve the melting temperature and cooling starts inthe solid phase. In this work, only two phases were considered in the heat balance Eq. 2. As willbe shown below, it is observed in the electron backscattering diffraction results (EBSD) thatthe original material has mainly the phases: retained austenite, ferrite and martensite. Sinceonly the austenite phase transforms into martensite, only the volume fractions of martensiteand austenite were considered. To obtain the EBSD data, a LEO Gemini 1530 VP with EDX-Detector and EBSD-System was used. A region in the neighborhood of the thermocouple, 100µm2 big was choosen for analysis. A phase map was generated with 1 µm steps.

0

100

200

300

400

500

600

700

800

0 20 40 60 80 100 120−1

−0.5

0

T[º

C]

dT/d

t[ºC

/sec

]

Time[sec]

T(t)SMOOTHT(t)

dT/dt

Fig. 1: Cooling curve T(t) of alloy with first derivative. The smoothed curve T(t)SMOOTH ofT(t) was obtained using a filtering algorithm.

Results and discussion

In what follows, the cooling curve analysis and the martensite fraction determination is shownfor the Fe-alloy system described above. In order to obtain accurate results it is very importantto evaluate the quality of the data. Filtering of the data was required. In Fig. 1, the originalcooling curve T(t) is shown, together with its filtered version and its first derivative. The firstderivative was calculated after smoothing the CC using a simple filtering algorithm. From theCC, the quantity ln(T-Ta) is calculated and also its first derivative with respect to time. Sincethis variable is also a function of the temperature, an analytical fit of the data, in order to obtain1/gpre(T) and 1/gpost(T) is carried out. The functions are shown in Fig. 2 and the correspondingparameters are in Table 1.

Table 1: Functions 1/gpre and 1/gpost. A linear fitting was made in the regions before and afterthe transformation where g=a+b*T.

a b[◦C/sec] [1/sec]

gpre 0.00943128 -0.000396802gpost -0.0117472 -0.0000834918

Assuming that in the transformation region the heat capacity (Cp) and the latent heat oftransformation (∆H) are independent from the temperature, the martensite fraction f t wasdetermined according to Eq. 5. Results of this calculation are shown in Fig. 3. As can be ob-served the austenitic phase begins to transform at a temperature of approximately 90◦C, whichis comparable to experimental observed temperature of 93◦C [7]. A total martensite fraction of7% is determined within this methodology. In order to validate results, electron backscatteringdiffraction (EBSD) was carried out on the sample and a phase fraction was obtained. Theseanalysis (See Table 3) deliver a martensite fraction of 13% which is comparable with the resultobtained through the cooling curve analysis. In the microstructure, the zero solution regions,correspond to very low quality of the EBSD patterns, are identified as a martensite phase.These are areas where distortions of the crystal lattice are greater than surrounding areas [10].Lee et al. [5] by means of comparing both a measured cooling curve and the cooling curve ob-tained by means of a finite difference calculation obtained a temperature dependent martensiteenthalpy of transformation. This temperature dependent enthalpy (Eq. 8) was also used in thecalculation of the martensite fraction, results are shown in Fig. 3.

∆H[J/g] = 0.0007T 2 − 0.0014T − 90.97. (8)

A total martensite fraction of approximately 17% is determined within this methodology.The difference of about 5% between the calculated martensite fraction and the observed fractionwith EBDS can be attributed to different factors. As mentioned before, there is a contributionto the total enthalpy of transformation which is stored in the sample and which can not beevaluated from the experimental heat Q. The EBDS results are also dependent on the regionwhere the analysis was made. It is a matter of homogeneity inside the sample and how the heattreatment contributes to nucleation and growth of the martensite phase. It is to be observedthat the applied heat treatment does not modificate the martensite fraction inside the sample,as seen from the EBDS results. It has been observed in previous works [7] that the cooling ratedoes not affect the formation of martensite in this class of steel.

−0.08

−0.07

−0.06

−0.05

−0.04

−0.03

−0.02

−0.01

0

40 60 80 100 120 140

d[ln

(T−

Ta)

]/dt

T[ºC]

dlnT/dtgpre

gpost

Fig. 2: Time derivative of the log of the CC as a function of temperature. A linear fitting ofthe functions 1/gpost, 1/gpre ws carried out.

Table 2: Thermophysical properties used for the transformed fraction calculation. Literaturevalues taken from [5, 6].

Cp ∆H1 ∆H2 Cp ∆H3

[J/gK] [J/g] [J/g] [J/gK] [J/g]

0.45 -96.0 -38.09 0.44+6.58×10−4T-5.9×10−7T2 0.0007T2-0.0014T-90.97+7.78×10−9T3

f1trans, f2trans f1trans f2trans f3trans f3trans

Table 3: Electron Backscattering Diffraction (EBSD). Phases present in %.

Original Material Heat Treated Material

Iron bcc 84.2 84.8Iron fcc 3.4 1.5

Zero solutions 12.5 13.7

Summary

From the experimental T(t) curve and its further analysis, it is possible to calculate a marten-site fraction which is comparable to phase fraction determined using EBSD. The calculatedquantitites gpre, gpost relate to the heat capacities before and after the transformation. Theseare linear functions of the temperature. Using this methodology it was possible to calculatethe latent heat of formation up to elastic stored energy contributions and also to estimate thetemperature where the austenite phase begins to transform.

−1

1

3

5

7

9

11

13

15

17

19

40 50 60 70 80 90 100 110 120

f tran

s[%

]

T[ºC]

f1transf2transf3trans

Fig. 3: Martensite fraction as a function of temperature. The austenitic phase transforma-tion temperature is approximately 90◦C. f1trans (ft

∗=0.06996), f2trans (ft∗=0.17625) and f3trans

(ft∗=0.07553) were calculated with the values ∆H1, ∆H2 and ∆H3 respectively (Table 2).

References

[1] W.J. Boettinger and U.R. Kattner: Metall. Mater. Trans. A 33 (2002), p. 1779

[2] I. Haq, J. Shin and Z. Lee: Metals and Materials Int. Vol 10 No 1 (2004), p. 89

[3] J. Gibbs and P. Mendez: Scripta Materialia. Vol 58 (2008), p. 699

[4] J. Gibbs, M. Kaufman, R. Hackenberg and P. Mendez: Metall. Mater. Trans. A. Vol 41 (2010),p. 2215

[5] S. Lee and Y. Lee: Scripta Materialia. Vol 60 (2009), p. 1016

[6] S. Lit: Scripta Materialia. Vol 60 (2009), p. 1016

[7] Th. Kannengiesser, A. Kromm, M. Rethmeier, J. Gibmeier and Ch. Genzel: Adv. X Ray Anal.Vol 52 (2009), p. 755

[8] J. Ortin and A. Planes: Journal de Physique IV. Vol 1 (1991), p. 13

[9] X. He and L. Rong: Scripta Materialia. Vol 51 (2004), p. 7

[10] B. Jeong, R. Gauvin and S. Yue: Microsc. Microanal. Vol 8 (2002)