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Scripta Materialia 62 (2010) 455–457

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Modelling differential scanning calorimetry curves of precipitationin Al–Cu–Mg

E. Hersent, J.H. Driver* and D. Piot

Materials Centre and CNRS UMR 5146, Ecole des Mines de Saint Etienne, 158 Cours Fauriel, 42023 Saint Etienne, France

Received 19 November 2009; revised 4 December 2009; accepted 4 December 2009Available online 6 December 2009

The heat flux during a differential scanning calorimetry (DSC) experiment of a precipitation reaction is expressed analytically as afunction of the interfacial energy, the solid solution composition and the precipitate fraction. Using a physically based model ofnucleation, growth and dissolution, the respective contributions of these terms are compared for the case of S phase precipitationin AA2024 (Al–4% Cu–2% Mg). Good overall agreement is obtained for the experimental and simulated DSC curves if the inter-facial energy is tuned correctly.� 2009 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

Keywords: Differential scanning calorimetry (DSC); Aluminium alloys; Phase transformation kinetics; Simulation; Thermodynamics

Differential scanning calorimetry (DSC) is a pop-ular technique with which to study the thermodynamicsand kinetics of phase changes in materials. It is particu-larly useful for precipitation reactions in light alloys usedfor structural applications, where successive solid-statereactions of precipitation and dissolution can be analysedat increasing temperatures (e.g. [1]). The nucleation ordissolution of a phase in a DSC experiment is character-ized by a heat flow peak over the reaction temperaturerange, and the peak area gives a semi-quantitative evalu-ation of the phase quantities involved in the reaction. Thearea under the peak is proportional to the molar enthalpyof precipitation, and the displacement of the peak withheating rates can be linked to the activation energy.

Over the last 20 years, modelling simulations of pre-cipitation have made significant progress, in particularby using the numerical framework developed by Wagnerand Kampmann [2]. This method can be used to simu-late the evolution of the precipitate radius distributionduring any heat treatment and is now an established toolfor predicting the evolution of homogeneously nucleatedparticles in both simple and commercial alloys. Animportant application of this model is the simulationof precipitation in age-hardening aluminium alloys dur-ing welding [3].

Modelling DSC curves from the basic physical princi-ples of precipitation/dissolution has rarely been at-

1359-6462/$ - see front matter � 2009 Acta Materialia Inc. Published by Eldoi:10.1016/j.scriptamat.2009.12.009

* Corresponding author. E-mail: driver@emse.fr

tempted, perhaps because it requires a demandingaccuracy of both the thermodynamics and kinetics oftransformations. DSC curves for precipitate dissolutionhave been simulated, e.g. by Chen et al. [4], but to ourknowledge only Khan and Starink [5] have publishedsimulated DSC curves for precipitation and dissolutionin Al alloys. However, their analysis seems to have somedeficiencies: the units of their Figure 3 in W g�1 do notcorrespond to their equation for DH � dVs/dt inkJ mol�1, and more surprisingly the heating rate doesnot appear to influence the DSC curves.

In light of this, it seems that a proper model for theDSC precipitation experiment is needed, and that is thegoal of this article. We will first establish an analyticalexpression of the heat flux in a simple two-phase reaction(precipitation and dissolution). This will be applied to thecase of S (Al2CuMg) phase precipitation in AA2024 (Al–4% Cu–2% Mg) by simulating the heat flux due to S phaseprecipitation and dissolution in a DSC experiment.

We consider a system where the precipitates appearfrom a supersaturated solid solution. We suppose thatthe precipitates form a unique phase and that their inter-facial energy c is independent of the precipitate–matrixorientation relation. The total Gibbs energy is:

G ¼ gvss V ss þ

Xprecipitates

V p

!þ Dgv

p

Xprecipitates

V p þ y

�X

precipitates

Ap ð1Þ

sevier Ltd. All rights reserved.

Table 1. Summary of the input data used in the model.

XAl–CuðT Þ (J mol�1) �13760

XAl–MgðT Þ (J mol�1) 6821–3.077T

Dhmp (kJ mol�1) 75 kJ mol�1 [5]

c (J m�2) �0.10985 + 8.3074 � 10�4T

� 8.7143 � 10�7T2

q (kg m�3) 2.77 � 103

DAlCu (m2 s�1) 6:47 � 105e�

135 kJ mol�1

RT

XAl–iðT Þ is the sum of the Redlich–Kister coefficients of the binary Al–i

system.

456 E. Hersent et al. / Scripta Materialia 62 (2010) 455–457

where gvss is the Gibbs energy per volume of the solid solu-

tion, Dgvp is the difference of Gibbs energy per volume be-

tween the precipitates and the solid solution, Vss is thevolume of the solid solution, and Vp and Ap are the vol-ume and surface area of one precipitate. The total Gibbsenergy per unit volume of the system Gv

sys is:

Gvsys ¼ gv

ss þ Dgvp � fv þ c � A

V sysð2Þ

where fv is the volume fraction of the precipitates and Ais the total surface developed by the precipitates(P

precipitatesAp). The quantities which are referenced inthermodynamics database are quantities per mole, notper volume, and are obtained by dividing Eq. (2) bythe molar volume. For DSC one also divides by q, theweight per unit volume of the alloy, to give the totalGibbs energy per unit mass:

Gwsys ¼

1

qgm

ss

V mss

þDgm

p

V mp

� fv þ c � AV sys

!ð3Þ

In DSC experiments, the heat flow of a sample is notmeasured absolutely but relative to a reference which doesnot undergo a phase change in the temperature range. Foraluminium alloys, the chosen reference is usually high-purity aluminium. The heat flux per gram of a sample_q ¼ d

dt ðH � H REF Þ can be derived from Eq. (3) with thehelp of the standard relations H = G + TS andS ¼ �@G

@T

��P ; fixed composition

:

_q ¼ ddt

Gwsys � T

@Gwsys

@T

����P ; fixed composition

!� dH REF

dtð4Þ

H REF can be easily calculated from the molar Gibbs en-ergy of aluminium given in standard databases such asthat in Ref. [6]. The main alloying elements ofAA2024 are copper and magnesium, so the Gibbs molarenthalpy gm

ss can be approximated by:

gmss ¼ xAll

Alss þ xCul

Cuss þ xMgl

Mgss ð5Þ

It has been demonstrated for aluminium alloys thatthe regular solid solution model is sufficient to accu-rately calculate phase diagrams in the aluminium-richcorner [7]. The molar Gibbs energies of aluminium, cop-per and magnesium are thus given by:

lAlss ¼ lAl

PURE þ RT ln xAl

lCuss ¼ lCu

PURE þ RT ln xCu þ XAl–CuðT Þ � ðxAlÞ2

lMgss ¼ lMg

PURE þ RT ln xMg þ XAl–MgðT Þ � ðxAlÞ2ð6Þ

When the composition of each phase is fixed, thereaction rate is zero, and the volume fraction and pre-cipitate surface area are constant. As shown in theAppendix, the final expression of the heat flux _q can thenbe written:

_q ¼ ddt

hmss � hm

Al

V mþ

Dhmp

V mp

� fv þ c� @c@T

����P

� �� AV sys

!ð7Þ

This differs from the equation used by Khan andStarink [5], _q ¼ DH � d

dt ðfvÞ, by a term for the heat evo-lution due to compositional change in the solid solution(and which needs to be evaluated).

This expression was used to simulate the contributionof S phase to the heat flux during a DSC experiment onAA2024. Following Khan et al. [8], we assume, for sim-plicity, that the 2024 aluminium alloy is a pseudo-binarywith the concentrations of copper and magnesium in theAl-rich phase in constant ratio to each other during theentire process. To simulate S precipitation, the numericalframework for nucleation, growth and dissolution of par-ticles described by Myhr and Grong [9] has been appliedto obtain the theoretical particle size evolutions with timeand temperature. This is a physically based model of par-ticle nucleation and growth for different particle sizeclasses.

The model obviously requires a number of parame-ters to be known with good accuracy. The most criticalof these is the interfacial energy c. One solution adoptedby Kamp et al. [10] consists of calibrating the interfacialenergy with the evolution of the volume fraction mea-sured by small angle X-ray scattering during reversiontreatments. For the present alloy it is necessary to intro-duce a temperature dependence to achieve a nearly peakstrength independent of temperature during isothermaltreatments [8]. For the precipitation stage we use a rela-tion similar to that of Khan and Starink [5] as given inTable 1, for which c decreases with temperature (fromabout 0.08–0.05 J m�2 over the temperature of interest).

The shape of the S particles was taken to be a rod with aconstant aspect ratio [8]. Other parameters, such as thediffusion coefficient of Cu in Al, are given in the table.The average atomic volume of S phase (Al2CuMg) isnearly the same as that of the face-centred cubic Al matrix[11], so one can take V m

p � 4 � V m. Also, in a typical 2024aluminium alloy, the Al molar fraction is about 0.95, soxAl is approximated by unity in the following. The valuesof XAl–CuðT Þ and XAl–MgðT Þ are listed in Table 1.

Although the standard composition of the industrialalloy is about 4.2 wt.% Cu, approximately 0.8% Cu istied up in the intermetallic particles (stable to high tem-peratures) and therefore does not participate in the reac-tions: the active composition was taken as 3.4 wt.% Cu.

DSC curves were simulated assuming a quenched androom tempered alloy (i.e. containing GP zones of averageradius 0.6 nm and number density 2 � 1025 per m3). For astandard heating rate of 20 �C min�1 the simulated DSCcurve is given in Figure 1, together with the contributionsof each heat flux term (in dotted lines). This DSC curvefirst exhibits a small endothermic reaction at 150–250 �C, usually taken to be GP zone dissolution. Thereis then a large exothermic peak at 250–320 �C due to Sphase precipitation and finally a long S phase dissolutiontrough at 320–470 �C. It is also clear that the volume en-

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

100 150 200 250 300 350 400 450 500

Hea

t Flo

w (W

/g)

Temperature (°C)

experimentsimulation

Figure 2. DSC model prediction of the heat flow compared with anexperimental scan measured for a heating rate of 20 �C min�1.

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5H

eat f

low

(W/g

)

total heat flowvolume term

solid solution termsurface term

100 150 200 250 300 350 400 450 500Temperature (°C)

Figure 1. DSC model prediction of the different terms involved in theheat flow during heating AA2024 at 20 �C min�1.

E. Hersent et al. / Scripta Materialia 62 (2010) 455–457 457

thalpy term (DH) contributes about 90% of the heat flux,with most of the remaining 10% due the solid solutionterm. The contribution of the interface energy is verysmall, which is not surprising since only a very small frac-tion of the atoms is involved in the interface. Conse-quently, the volume enthalpy term can be taken as afirst-order approximation of the heat flux and the peakarea could then be used to determine the volume fractionof the precipitating S phase.

The simulated and experimental heat flow curves aregiven in Figure 2. The experimental DSC curve was ob-tained on a standard AA2024 alloy (T 351 condition)using a SETARAM model DSC 131; it is very similarto previous published DSC plots of this alloy [12]. Itcan be seen that the simulated DSC curve is very closeto the experimental one, with the peaks and troughs ofsimilar amplitude over the same temperature ranges.

This type of DSC simulation can in fact be consideredas a test or estimation of the interfacial energy. The reac-tion enthalpy for precipitate dissolution is given, to firstorder, by the area under the high-temperature dissolutioncurve. The other thermodynamic parameters are usuallyknown or can be estimated as done here (given that theinterfacial energies have little influence on the peak ampli-tudes). However, the kinetics, or the temperatures of theheat flow peaks in a DSC experiment, are then stronglydependent on the diffusion coefficients and the interfacialenergies. If the diffusion coefficients are known it shouldbe possible to deduce the interface energies by a fitting

procedure from the experimental DSC curves and thistype of model simulation. The strong influence of theinterfacial energy on the DSC simulation has beenchecked by changing its value by 10%: this produces achange in peak temperature of about 100 �C.

In conclusion, the expression for the heat flux duringa DSC experiment involving a two-phase reaction hasbeen refined to include the heat evolution due to compo-sitional change in the solid solution. It has been appliedto the formation and dissolution of S precipitates in a2024 aluminium alloy and shown to give good agree-ment with experimental DSC curves. Nevertheless, it isshown that, to first order, the expression of the heat fluxcan be approximated by the standard term V m

V mp

DHMAl

dfvdt . In a

wider context, it is suggested that the interfacial energiescan be deduced from DSC curves of precipitation reac-tions (assuming that the phases etc. are known) by com-parison with the temperature dependence of curvessimulated using physically based models.

Appendix

The goal of the appendix is to show that the enthalpyof

Dgmp

V mp

fv can be written asDhm

p

V mp

fv.

Dgmp

V mp

fv � T@

@T

Dgmp

V mp

fv

!�����P ; fixed composition

¼Dgm

p

V mp

fv � Tfv

V mp

@

@TDgm

p

� ����P ; fixed composition

because at this fixed composition, there is no exchangeof matter between phases, so the volume fraction is fixedand independent of temperature. The latter equation canbe simplified to:

fv

V mp

Dgmp � T

@

@TðDgm

p Þ� �

¼ fv

V mp

Dhmp

A similar reasoning can be applied to the term c A/Vsys.

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