Applications of computational thermodynamics

J. Âgren et al.: Applications of computational thermodynamics

J. Âgren\. H. Hayes^, L. Hôglund\. R. Kattner^, B. Legendre^, R. Schmid-Fetzer^

^ Royal Institute of Technology Materials Science and Engineering, Stockholm, Sweden; ^University of Manchester an UMIST Materials Science Centre, U.K.; ^NIST, Gaithersburg, MD, USA; ''Laboratoire de Chimie Physique Minérale Châtenay-Malabry, France; ^ Institut fïir Métallurgie, TU Clausthal, Germany

Applications of computational thermodynamics The major tools used in applying computational thermodynamics to varions problems in materials science are briefly presented and several practical examples are given as illustrations. Solutions to industrial problems, pertaining to the processing of and microstructure development in several différent materials, are shown with answers given in graphical form. Solutions to kinetic problems linked with diffusion are also treated. The last section is devoted to the problem of interfacing between thermodynamic computations and applications oriented software. Keywords: Computational thermodynamics; Phase diagram calculation; Phase equilibrium

1 Introduction Two groups at the previous workshop [OOKat] focused on the "use of thermodynamic software in process modelling and new applications of thermodynamic calculations". A broad range of examples was given to illustrate the wide spectrum of applicability of phase diagrams and thermodynamic calculations to industrial and practical problems in materials science. The purpose of the présent article is to présent detailed examples to show how the link between calculation and practical application is actually achieved. It is hoped that the examples given w i l l help and encourage other users of the Calphad method to interact with non-specialist users and vice versa. The aim is to help the Calphad community address more easily the types of frequently asked questions relating to practical problems such as those that arise in industry. Solving such problems involves several distinct steps. First we wiU identify the problem and présent the information required to solve it. Then the problem must be put in a form that can be solved by calculation and the calculations carried out. Finally, the calculated results must be explained to the questioner in a form that can be understood by the non-specialist. Simply handmg out phase diagrams is not a sufficient way of explaining results of the calculations. In the first section we focus on problems that can be solved by using only thermodynamic data. In the second section we présent more sophisticated answers that also involve using of micro-kinetic data. 2 Tools and examples involving thermodynamic data only 2.1 Tools 2.1.1 Single point equilibrium calculations Some practical problems can be solved by means of calculations that involve thermodynamics alone whereas 128

many other problems can only be solved by using the modynamics in combination with kinetic considération The basic tool used to obtain phase equilibrium informa tion from thermodynamic data is the single equilibriu calculation. This kind of calculation is carried out b setting the degrees of freedom to zéro. In the most com mon case this means fïxing the total composition of th System, its température and pressure in order to min mise the Gibbs energy of the system. Single poin equilibrium calculations are the cote of any kind of ca culations that involve stepping in one dimension or map ping in two or more dimensions. Single point calcula tions further provide the derivatives that are needed fo the évaluation of effects resulting from changes in th ambient conditions. 2.1.2 Stepping equilibrium calculations

In one-dimensional stepping calculations one of the var ables that was fixed for the single point equiUbrium calcul tion is stepped. A t each step a single point equilibrium ca culation is carried out. The most common application the stepping formaUsm is to step along the température axi The graphical représentation of the results is not a pha diagram but a property diagram displaying phase fraction phase compositions, enthalpies etc. as a function of the st variable. A weU-known example for this kind of calculation the lever-rule calculation of equiUbrium solidificatio The soHdifïcation path from this calculation correspond to the case where complète diffusion occurs in the liqu and soUd phases. In addition, stepping equiUbrium calc lations can be used for many other applications by usin another variable for stepping, such as the concentratic or activity (partial pressure) of a component or the overa pressure. ; 2.1.3 Scheil solidification calculations

The Scheil soUdifïcation calculation is another type of ca culation, which uses the single point equiUbrium calcu tion. The Scheil path of soUdification is based on the a sumptioil' that diffusion occurs only in the liquid phase ai is absent in the soUd phases. This path produces the woi case of microsegregation with the lowest final freezing ter perature. Together with the lever-rule (equiUbrium) soUdi cation thèse two solidification paths form the two limitii cases of solidification. The Scheil soUdification path is usuaUy approximated I stepping through the température interval of Lnterest and £ suming local equiUbrium to exist at each température ste At each température step the fraction of soUd phas © Cari Hanser Verlag, Miinchen

Z. Metallkd. 93 (2002)

J. Âgren et al.: Applications of computational thermodj'namics

formed is removed from the calculation and the composition of the remaining Hquid phase is used as the Uquid composition for the next température step. Simultaneously the fractions of the solid phases that have been removed are summed to give the total fraction of solid formed during soUdification. Since no diffusion is aUowed in the soUd phases, stepping is usuaUy stopped after a eutectic equiUbrium has been encountered or a predetermined fraction of soUd phase has formed. However, calculation of the enthalpy may be continued if the results are needed for other calculations such as process simulations. The calculation of the enthalpy vs. température during Scheil soUdification requires that the variations in the compositions within the accmnulated soUd phases be considered. Ideally the calculated compositions and incréments of soUd phase fraction are stored for each température step and the enthalpy is then calculated for the current température from the summation of the enthalpies of ail soUdified "layers" and the remaining Uquid phase. This strategy is appUed in code developed by Schalin [98Sch] who also noted that this procédure is very time consuming. Boettinger et al. [98Boe] avoided this problem by averaging the composition for each individual soUd at each température step. The enthalpy is then calculated using the average composition and the total fraction soUd of each phase. A comparison of the results obtained from thèse two stratégies for a 0.83 Ni-0.15 A l - 0 . 0 2 Ta alloy revealed only insignificant différences [97Sch]. The graphical représentation of the results of a Scheil soUdification calculation is similar to the case of stepping equiUbrium calculations, i. e., a property diagram. Although the two types of calculations given above are Umiting cases for the simulation of solidification behaviour, expérience has shown that they are very useful approximations. In case of low-alloy or ferritic steels, where diffusion occurs easily, the lever raie soUdification is a good approximation whereas for alloys where soUd diffusion is insigiUficant, a Scheil calculation is a suitable approximation [95Boe, 97Sau]. However, for more realistic treatment of soUdification, diffusion should be considered [95Boe, 97Meu]. 2.1.4 Phase diagrams Phase equilibrium calculations done by mapping two (or more) state variables resuit in phase diagrams. Reasonable sélections for the axes of a phase diagram have been discussed by Hillert [98Hil]. In multicomponent Systems the phase diagram (section) is often used by experts for checking the internai consistency of results or to estimate the sensitivity of stepping calculations to variations in other state variables. There is, however, a wide variety of direct appUcations of phase diagrams [OOKat], especiaUy with binary Systems or temary isothermal sections. 2.2 Examples 2.2.1 How does the constitution of a given aUoy vary with temperamre? Question: The variation of the constitution of an alloy as a function of temperamre is a cential question in materials science because the properties and performance of a materiZ. MetalUcd. 93 (2002) 2

al dépend on it. For example, one might wish to know the séquence of the phases, their relative amounts and compositions, that would form during the equilibrium cooling of an alloy of fixed composition from a chosen temperamre above Uquidus down to room tem.perature. Also microstractural changes at elevated températures form the basis for heat treatment but may also cause dégradation of the material. Answer: As a first approximation one may consider the equilibrium state at each temperamre. One might wish to know for example how the relative amounts of the phases présent and their compositions dépend on the applied température during high température solution annealing of a two-phase soUd aUoy. The foUowing example on duplex stainless steels illustrâtes this type of application. Background: Commercial grades of stainless steels can be broadly classified into four différent types, namely martensitic, fenitic (body-centred cubic, BCC), austenitic (face-centred cubic, FCC), and duplex (austenite + ferrite) [92Lle]. AU stainless steels contain at least 12 % Cr (aU percentages are in mass %) for overaU corrosion résistance, plus various amounts of N i and C and other aUoying additions, e.g., Mo and N in the case of the duplex grades. Différent microstractures and thus différent physical and mechaïUcal properties are produced by adjusting the amounts of Cr, Ni, C, etc. présent and the heat treatment appUed. Ferritic stainless steels typically contain 10 -18 % Cr with 1 % N i whereas the austenitic grades have Cr and N i contents from 16-23 % Cr with 6-19 % Ni. The martensitic grades typically contain 1 2 - 1 4 % Cr plus 1 % N i and 0.1-0.3 % C. The origins of thèse différent classes of stainless steels can be seen from Fig. 1 which shows the 70 wt.% iron isopleth for the F e - C r - N i system, computed using the MTDATA System [94Dav]. (Trade names are used in this paper for completeness only and their use does not constitote an endorsement by NIST). For temperamres in the range 727-1477 °C three différent phase fields can be seen. For Cr contents up to about 20 % a broad single-phase FCC austeiUtic (y) région is évident; with Cr contents from approximately 20-26 % there is a two-phase austenite plus ferrite (a) région and at Cr contents above 26 % a single-phase ferritic field exists. This example is concemed with the (y + a) two-phase région, the source of the duplex stainless steels. Thèse materials exhibit a very high résistance to stress corrosion cracking combined with high strength and toughness: properties that resuit directly from two-phase (y + a), i. e., the 'duplex' microstracture. This duplex microstracture is produced by solution anneaUng the steèl in the (y + a) phase field at températures in the région of 777 °C and is then retained during cooUng to room température by rapid water quenching. Fig. 1 shows why this quenching is necessary, since at températures below about 727 °C a séries of two- and three-phase fields exist, which, in addition to austenite and ferrite, mvolve the a phase and the,,Cr-rich ferrite phase known as a'. Both of thèse phases cause embrittlement and must be avoided. It should be noted that although any steel whose total composition lies in the (y -i- a) phase field would be two-phase on rapid quenching to room température, no information about the relative amounts of the two phases présent or their compositions can be obtained directly from Fig. 1 since the tielines that join the equiUbritmi austenite and ferrite compositions do not Ue in the plane of the isopleth [90Hay]. This information can, however, be obtained for any total composi129

J. Âgren et al.: Applications of computational thermodynamics

1600 Liquid 1400-

a

1200I-

Y

1000a+Y 800-

. 600

1

Va-f-yX a+a \a+c!+y 5

10

15 20 Ni (wt.%)

25

30

Fig. 1. 70 mass% Fe isopleth for the F e - C r - N i system from 70 %Fe-30 %Cr to 70 %Fe-30 %Ni.

tion and temperamre from a single point equilibrium calculation. One-dimensional stepping along the temperamre axis allows diagrams such as phase fraction versus temperatitre and phase composition versus température to be plotted. The two examples of computed phase fraction vs. température diagrams in Figs. 2 and 3 illustrate how two F e C r - N i stainless steels with différent compositions would behave during equilibrium cooling from the liquid phase. From Fig. 2, computed for 6 % N i and 24 % Cr, it is seen that primary ferrite begins to form at 1488 °C, the liquidus température, the solid is then completely ferritic down to 1197 °C where ferrite begins to transform to austenite as the steel enters the two-phase (y + a) phase field. The percentage of austenite increases as the température decreases reaching a maximum of approximately 49% at about 844 °C. The a phase then becomes stable at about 777 °C as the System «nters Xheia + a+y) phase field. The Fe-rich a phase reappears at approximately 657 °C and a décom-

400

600

800

1000

1200

1400

1600

T(°C) Fig. 2. Eqmhbrium mass fractions (NP) of the various phases in Fe24Cr-6Ni from 400 to 1600 °C.

130

poses eutectoidally at 447 °C to give y + a plus the Cr-rich ferrite phase, a'. Calculations can also be made for possible low température metastable equilibria, see [90Hay]. Fig. 3 shows the phase balance during equilibrium cooling of a 9 % N i and 21 % Cr Fe - Cr - N i stainless steel. Compared with Fig. 2, increasing the Cr content and decreasing the N i content causes only a small increase in the liquidus température to 1505 °C and soUdification to 100 % ferrite is completed at 1449 °C. However, the stabiUsing effect on austeiUte of the higher N i and lower Ci contents compared to the previous example can be seen from the large increase, from 1197 to 1405 °C, in the température at which austenite first appears on cooling. This is the point where the steel composition enters the two-phase (y -I- a) phase field to generate the duplex microstrucmre, The higher N i content also increases the maximum austenite content to over 90 % at 907 °C. The a phase begins te form at 822 °C as the system enters a narrow three phase (y + a + a) région below which there is a short température range over which cr is in equilibrium with y. a ii replaced by the Cr-rich ferritic phase, a' at the temary eutectoid temperatmre 447 °C, where the system enters the {a + y + a') région. Figs. 2 and 3 enable the appropriate solution annealing température for each steel to be selected to give any chosen yla contents in the final water quenched duplex stainless steel.

Ui practice, in addition to Fe, Cr and Ni, several extra al loying éléments such as Mo, N , W, and Cu, are présent i i commercial duplex stainless steels to enhance properties Small amounts of Mn, Si and C ( < 0.03 %) are also inevi tably présent giving a total of ten or more éléments. Me is added to increase the résistance to pitting corrosion and being a ferrite stabiUser, has to be balanced by the ad dition of an austenite former to maintain the phase bal ance. Nitrogen is the most commonly used austenite stabi User and has the added advantage of increasing thi weldabiUty of the material. Each of the aUoying élément présent wiU affect the overall phase balance and aU hav to be included in the calculation [94Lon]. As an illustra tion, Fig. 4 shows the computed phase fractions vs. tem perature diagram for a F e - C r - N i - M o - N duplex stain

600

800

1000 1200 T(°C)

1400

1600

Fig. 3. Equilibrimn massfractionsof the various phases in Fe-21CrNi from 600 to 2000 °C.

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J. Âgren et al.: Applications of computational thermodynamics

Fig. 4. Equilibrium mass fractions of the various phases in a F e 25Cr-6.5Ni-3.5-Mo-0.25N duplex stainless steel from 600 to 1500 °C. HCP = hexagonal close-packed

Fig. 6. Equilibrium composition in massfractionof ferrite versus temperature for same steel as in Fig. 4 from 1100 to 1500 °C.

less Steel containing in mass %: 25 Cr, 6.5 Ni, 3.5 Mo, 0.25 N , balance Fe. Introducing M o and N can cause additional phases, such as the Laves phase, the chi-phase and carbonitrides based on Cr2N, to become stable in addition to sigma. Over the température range from 1227 down to 927 °C Fig. 4 shows that the phase balance changes from 70 % ferrite plus 30 % austenite to 60 % austenite plus 40 % ferrite. The 50% point occurs at 1047 °C. The a phase is stable from 907 to 427 °C and the Laves phase and the a' phase below 447 °C. Thus, to produce a duplex stainless steel containing 50 % each of austenite and ferrite for this overall composition would require solution annealing at 1047 °C followed by rapid water quenching to room température to avoid the formation of the embrittling intermetaUics {a. Laves-phase, a!), which would otherwise occur in slowly cooled materials.

tion on the individual phase compositions is also of importance in designing duplex stainless steels with high pitting corrosion résistance. This phase composition information is also obtainable using the stepping equilibrium calculation method. Figs. 5 and 6 show the ferrite and austenite compositions vary over the temperamre range of interest for the 2 5 C r - 6 . 5 N i - 3 . 5 M o - 0 . 2 5 N steel. It is seen that, as expected, Cr and Mo partition preferentially in the ferrite phase whereas N i and N partition preferentiaUy in austenite. The pitting résistance équivalent (PRE) ntmiber defined as [94Cha]:

In addition to knowing how the austenite and ferrite fractions vary with aimealing température used, informa-

PRE = 1 %Cr + 3.3 %Mo + 16 % N Typical values for the PRE number are 32 to 33 for 22Cr5Ni, 38 to 39 for 25Cr-7Ni and more than 40 for the super duplex stainless steels. The computed phase compositions from the stepping equilibrium calculations enable the expected PRE values for each phase to be obtained in addition to the global value calculated from the steel composition. The PRE values of the austenite and ferrite phases of the 2 5 C r - 6 . 5 N i - 3 . 5 M o - 0 . 2 5 N steel are shown in Fig. 7. 2.2.2 How sensitive is the limiting austenitising température of a low-aUoy steel for a variation in alloy content?

1500

Fig. 5. Equihbrium composition in mass fraction of austenite versus température for same steel as in Fig. 4 from 1100 to 1500 °C.

Z. Metallkd. 93 (2002) 2

Question: A low-carbon low-alloy steel is austenitised by heating to températures where one obtains a single-phase austenitic structure. The question is primarily related to the multicomponent multidknensional phase diagram. Answer: For a spécifie alloy we may directly calculate this température by setting pressure P = 100 kPa, fixing the content of the various aUoy éléments and require that the fraction of ferrite is 0 which is the condition of the austenite phase boundary of ferrite formation. We may subsequently repeat the calculation for perturbations in alloy content and evaluate the coefficients kj =

dT dcj 131

J. Âgren et al.: Applications of computational thermodynamics

It should be emphasised that P is the total pressure. Suppc the gas could be considered as a mixtture of CO, H 2 and î The overall composition of the gas is then fixed by givi the mole fraction of two of the three components. Howev usuaUy the éléments are defined as components in equil rium calculations and thus, the gas is a 4-component syst( with components C, O, H and N . It may be convenient regard rather C, O 2 , H 2 and N 2 as components. Such redéfinition of components is easily made in some comm cial software. The mole fractions of the gas compone; are then transformed accordingly = a 4 : = 0.5x^co):

< ^

0.54-/(1.54^+4-+4-)

1500

Fig. 7. Pitting résistance équivalent (PRE) of both austenite and ferrite as a function of température for the same steel as in Fig. 4.

and represent the variation of the austenitising température (7^) with a linear expression

As an example we shall consider the steel F e - l C r 0.5Mn-0.5Si-0.20C. Using Thermo-Calc [85Sun] we obtain directly r ( ° C ) - 832 - 10.20A% Cr - 32.31A% M n -i- 38.21 A% Si -281.77A%C AU percentages are in mass%. A variation in C content thus has a much stronger effect than a variation in the content of any of the other éléments. This expression may be compared with the Umiting case obtained for the "binary" F e 0.20C steel where the concentration of the substitutional éléments approach zéro: r (°C) = 836 - 9.9A% Cr - 29.72A% M n + 45.97A% Si -291.94A%C 2.2.3 How is a steel affected by a certain gas atmosphère? Question: This question often occurs in coimection with carburising during case hardening or when analysing the effect of fossil-fuel waste gases on steel. The most important parameters are the carbon activity and the oxygen partial pressure of the gas. Too high carbon activity will lead to carburisation and carbide -formation whereas too low carbon activity w i l l cause decarburisation. Too high oxygen partial pressure wiU cause oxidation. Ui the thermodynamic calculation we would like to fix the température and the composition of the gas as well as the iiUtial composition of the steel. However, in practice the situation usuaUy is such that the composition of the reactive gas is not affected by the reaction with the aUoy because there is usuaUy a continuons flow of gas by which fresh gas is continuously added. Answer: The conditions of the calculation are set up in the foUowing way: pressure P and température Tare fixed. 132

•4T) 4r

=

^llO--^^co

where the "primed" mole fractions are based on the comj nents C, O 2 , H 2 and N 2 and are fixed as conditions. We w; the gas phase to take part in the equiUbrium but not in mass balance and thus we apply as an extra condition tl it shaU be présent but witii a total amount of 0 mol. The éléments C, O, N and H wiU react with the steel a their amount in the steel, as carbides, oxides, etc. or sim] as a soUd solution in Fe, wiU come out as a resuit from 1 equiUbrium calculation. The other éléments of the steel v thus remain constant and we may give their amounts, mass or number of moles, as conditions. The procédure wiU now be illustrated with the low-al steel F e - l C r - 0 . 5 M n - 0 . 5 S i - 0 . 2 0 C and a gas phase w 4 O C O - 2 O N 2 - 4 O H 2 . The température is 930°C (1203 and the total pressure is 100 kPa. For the gas phase we tl obtain

1.54o

-4T -4^ =

1.2 and

= 0.4/1.2 = 0.3333 eas'

^2 ;as/

= 0 . 5 * 0 . 4 / 1 . 2 = 0.1667

4. = 4T' =

0 . 4 / 1 . 2 = 0.3333 0 . 2 / 1 . 2 = 0.1667

As the four mole fractions sum up to unity we can oiUy three of them as conditions. We consider 100 g of the initial steel. The amount of Cr, M n and Si should remain in the material and we tl give as conditions: mpe = 100 - 1 - 0.5 - 0.5 - 0.2 = 97.8 mci = l,mMn = 0.5, msi = 0.5

The amount of C, O, N and H should not be fixed but v come out as a resuit of the equiUbrium calculation. 1 graphite phase w i l l be suspended because it does not fo that easily. In this report the calculation was performed w Thermo-Calc [85Sun] using the SGTE (Scientifîc Grc Thermodata Europe) substance and solution databa [87Ans]. The resulting output from Thermo-Calc is sho in Table 1. The calculation involves a large number of g eous species and only the most important ones and th content are Usted. As can be seen, the carbon activity re tive to graphite is high (1.67) and graphite would form

Z. Metallkd. 93 (2002

J. Âgren et al.: Applications of computational thermodynamics

Table 1. Output from the Thermo-Calc software.

Conditions: T = 1203, P = 101325, X(GAS, H2) = 3.3333E-1, X(GAS, C l ) = 3.3333E-1, X ( G A S , 02) = 1.66667E-1, B ( C R l ) = 1, B(MN1) = 5 E - 1 , B(SIl) = 5 E - 1 , B(FE1) = 97.8 FDŒD PHASES GAS = 0 D E G R E E S OF F R E E D O M 0 Température 1203.00, Pressure 1.013250E + 05 Number of moles of components 2.32750E + 00, Mass 1.06168E -i- 02 Total Gibbs energy -1.16819E + 05, Enthalpy 7.60470E + 04, Volume 1.41051E-05 Component

Moles

W-Fraction

Activity

Potential

Ref

H2

O.OOOOE + 00 5.3011E-01 4.2440E-05 O.OOOOE + 00 1.7512E + 00 1.9232E-02 9.1012E-03 1.7803E-02

O.OOOOE + 00 5.9967E-02 1.1198E-05 O.OOOOE + 00 9.2118E-01 9.4191E-03 4.7095E-03 4.7095E-03

3.9270E-01 1.6699E-I-00 2.0162E-01 6.5946E-21 2.6539E-03 1.2602E-05 2.1027E-06 2.6207E-07

-9.3494E + 03 5.1291E + 03 -1.6018E + 04 -4.6479E + 05 -5.9331E-F04 -1.1284E-F05 -1.3075E + 05 -1.5158E + 05

GAS C_S GAS GAS SER SER SER SER

Cl

N2 02 FEl CRI MNl SU GAS#1

Stams F I X E D Number of moles O.OOOOE + 00,

02 N2

3.63357E-01 3.18112E-01

Cl H2

2.72751E-01 4.57795E-02

Driving force Mass O.OOOOE + 00, FEl CRI

O.OOOOE + 00

Mass fractions:

1.37272E-10 1.68059E-12

MNl SIl

O.OOOOOE + 00 O.OOOOOE + 00

Constimtion: ClOl H2 N2 C1H4 H201 C102 CIHINI HCN H3N1 C2H4 C1H1N1_HNC C2H2 H C1H201 C2H6 C1H3 cmiNioi C4H8

3.97420E-01 3.92695E-01 2.01615E-01 4.07108E-03 2.38663E-03 1.74002E-03 4.90461E-05 2.12294E-05 5.63052E-07 1.75663E-07 1.30293E-07 1.30044E-07 7.94683E-08 3.32201E-08 .2.69248E-08 1.60298E-08 7.67573E-09

CEMENTITE#1

Status E N T E R E D

Driving force

Number of moles 2.0701E + 00, Mass 9.2844E + 01 FEl Cl

9.17581E-01 6.69414E-02

CRI MNl

1.05991E-02 4.87238E-03

N2 H2

Status E N T E R E D

FCC Al#l

9.46286E-01 3.75257E-02

Z . Metallkd. 93 (2002) 2

Cl MNl

1.13728E-02 3.57476E-03

Mass fractions:

6.4I656E-06 0,09000E + 00

02 SU

Driving force

Number of moles 2.5738E-01, Mass 1.3324E + 01 FEl SU

O.OOOOE + 00

CRI N2

O.OOOOOE + 00 O.OOOOOE + 00 O.OOOOE + 00

Mass fractions:

1.19660E-03 4.45174E-05

H2 02

O.OOOOOE + 00 O.OOOOOE + 00

133

J. Âgren et al.: Applications of computational thermodynamics

10° Cementite 0]

to

SI 1 0 - ^ QO C

BCC

o t3

Oxide

101-3 500

600

700

800

900

1000

T(°C) Fig. 8. Carbon and nitrogen activities as function of température for reaction between the alloy steel Fe-lCr-0.5Mn-0.5Si-0.20C and a flowing gas phase containing 40CO-20N2-40H2 at 100 kPa total pressure.

was not suspended. The oxygen partial pressure is very low, 6.6x 10~^^ Pa and no oxide forms at this température. On the other hand, there is a massive cementite formation due to the high carbon activity. The calculation can be repeated for a séries of températures and the resuit may be plotted. Fig. 8 shows the carbon and nitrogen activities as function of température. Fig. 9 shows the same information as Fig. 8 but with a différent scaUng. The curve at the bottom shows the oxygen partial pressure as a function of température. Fig. 10 shows the mass fraction of the various phases as function of température. As can be seen there is a massive cementite formation at ail températures whereas the oxide starts to form below 920 °C. 2.2.4 What is the heat of melting of quatemary Mg alloys? Question: What is the heat of melting of the quatemary magnésium alloy M g - 2 0 A l - l M n - 5 C e (mass%)? Back103 Carbon 10°-j Nitrogen

10"^10'^-

£ 10

I

10-12 H

500

600

700

800

900

10-2* H 10,-27. 500

600

700

800

900

1000

T(°C) Fig. 9. Same as Fig. 8 but with différent scahng. The curve at the bottom shows the oxygen partial pressure as a function of température.

134

1100

T(°C) Fig. 10. Mass fraction of the various phases in the steel in Fig. function of température.

ground: We need to know this because problems arise die casting of M g alloys. The die does not heat up suË ciently even after a number of shots. Thèse problems dor arise with A l - a l l o y s , and we know the basic problem is th the heat of melting of pure Mg is only 61 % of that of pu A l , with respect to volume. This information for the allo; is cracial, so we need accurate figures. Ansvrer: Sorry, there is no simple answer like "8888 mol". This is because an acmal "latent heat" can reasonab only be defined for a nonvariant reaction (pure elemei congment melting alloy/compound or eutectic soUdific tion). I f the liquidus température, J L , and the solidus ter perature, Ts, are différent there is always a contributic from the heat capacity over the température interval (TL Ts). This effect wiU be detailed below for the cases of eqi librium and Scheil solidification. It will be shown that acc rate numerical data should be taken directly from the tôt enthalpy curve of the alloy.

For simpficity we start with the equilibrium enthal curves of four binary M g - A l alloys in Fig. 11. Abo 650 °C the enthalpy at a fixed température decreases w alloy content, reflecting the négative enthalpy of mixing the liquid phase. In the sohd state the positive enthalpy mixing in the (Mg) solid solution causes the mcrease of firom curve 1 to curve 2. With further increasing alloy co tent, curves 3 and 4, H decreases again, due to the presem of y phase with its négative heat of formation. Generally, we wiU define a "heat of melting", A//L_S, the enthalpy différence between liquidus and solidus po of the entire alloy, exactly AHi^s ="H{Ti., / L = 1) - H{Ts, / L - 0)

Oxygen

10-2^

1000

(

where H is the intégral molar enthalpy of the aUoy and/L the molar fraction of liquid. Only for pure M g and Mg 33.3Al (eutectic aUoy, ail compositions in mass%) A^L-s also the "latent heat of melting", A/fiatent- The n merical values in Table 2 clearly show how A H L - S i creases from 8.5 to 10.8 kJ/mol for M g - 1 0 . 9 A l due to Û broad L -i- (Mg) primary soUdification interval of 135 ° This even overcompensates the reverse contribution firo

Z. MetalUcd. 93 (2002)

J. Âgren et al.: Applications of computational thermodynamics

30-

1 2 3 4

Pure Mg Mg-AI10.9 Mg-AI23.8 Mg-AI33.3

o E

400

500

600

700

T e m p é r a t u r e in ° C Fig. 11. Equilibrium enthalpy of four binary Mg-Al alloys. Inset shows the partial M g - A l phase diagram up to the (Mg) + y eutectic.

the enthalpies of mixing, detailed above for curves 1 and 2. The alloy Mg-23.8A1 shows additional secondary (eutectic) soUdification, however, since this narrows down the soUdification interval to 77 °C, the value of AHL_S decreases again. And it decreases further to a minimum of 6.6 kJ/mol for the eutectic aUoy. Clearly the width of the soUdification interval adds a massive contribution of heat capacity to the heat of melting. This effect cannot be separated uniquely from latent heat if the heat capacities of Uquid and soUd are différent. This is demonstrated in Fig. 12 for the quatemary M g - 2 0 A 1 l M n - 5 C e aUoy. The solidification occurs in 12 steps (a-1) over an interval of 319 °C with primary AlgMus (step a) and so on, as detailed in Table 3. A t any température point m the soUdification interval we could define a hypothetical latent heat as the différence between the extrapolated H curves of Uquid and solid, shown dashed in Fig. 12. Numerical values of Aiïiatent-hyp would vary between 6.6 kJ/mol at TL = 755 °C, 7.8 kJ/mol at Ts = 436 °C, and 7.2 kJ/mol at r = 600 °C in the middle of the interval. So AHiatent-hyp is not unique because of différences and because T cannot be selected uniquely. A n additional compUcation arises because the Cp values of the supercooled Uquid and super-

heated solid are often not physically supported but just mathematical extrapolations. Furthermore, for the H curve of superheated soUd we have to make an additional arbitrary décision: take the frozen-in multiphase constitution at Ts or Select the equilibrium solid phase(s) during soUdification? This is why a simple number for the latent heat cannot be given. Also the value of A H L _ S from Table 2 would be misleadingly large (18.0 kJ/mol) since it contains the heat capacity contributions from the soUdification interval. For any appUcations to heat transfer calculations, like in die casting, the total enthalpy curve H(T) of the aUoy as given in Fig. 12 should be used. I f the soUdification does not foUow the global equiUbrium (sometimes denoted as lever mie) but the ScheU conditions, we have to note some additional intricacies. For simpUcity, let us assume that the Scheil soUdification occurs in 100 separate small steps, that is, after complète soUdification we have 100 soUd layers of différent composition. Within each step the homogeneous residual Uquid is in thermochemical equiUbrium only with the most recently solidified layer. SoUd state diffusion between the layers is blocked. A t say, / L = 40 % we have 60 soUd layers which are aU at the same température Tas the residual liquid, since complète thermal equiUbrium is another reasonable assumption in Scheil model. However, each of the solidified layers has its individual contiibution to the total enthalpy content. The total enthalpy can be calculated by one of the stratégies that were akeady discussed in Section 2.1.3. As pointed out above, there is no unique way to define a "latent heat" over a temperamre interval and that also appUes to the Scheil soUdification. However, a simple approximation can be used which is em^ployed in version TC-M of the Thermo-Calc software. We wiU again step down in T, although stepping down in / L or in the extracted heat might also be done. After each step we may eut off the already soUdified part and disregard it. A "latent heat évolution of soUdification", DELH, in the next step may be calculated at {T-ôT) as DELH = DELH +f^x

30-

{H-H(L))

1

1

(2)

1

1

25-

20Table 2. The "heat of melting", Aiï^L~5. of Mg and four alloys. It may contain contributions from heat capacity. L = liquid, S = solid. Alloy (mass %)

TK-C)

•''6.6 kj

"

f

,. - » '

15-

/

,.'-''7.2 kJ

Ts (°C) (kJ/mol)

Mg

650

650

8.5

Mg-10.9 Al

593

458

10.8

Mg-23.8A1

513

436

8.6

Mg-33.3AI

436

436

6.6

Mg-20Al-lMn-5Ce

755

436

18.0

Z. MetalUcd. 93 (2002) 2

1

10-

5400

1

1 500

1 600

1

1 700

800

T(°C) Fig. 12. EquiUbrium enthalpy of the quatemary magnésium alloy Mg-20Al-lMn-5Ce (mass %).

135

J. Âgren et al.: Applications of computational thermodynamics

Table 3. Equilibrium solidification of the quatemary M g - 2 0 A l - l M n - 5 C aUoy. Phases

Step

H(rsurt)-H(rcnd) Od/mol)

Tstart (°C)

(°C)

L + AlgMnj

755

667

B

L + AlgMns + AlzCe

667

634

1.5

C

L + AlgMns + AljCe + AljCe

634

629

0.2

D

L + AlgMnj + AljCe

629

620

0.4

E

L + AlgMns + AliiCes + AlsCe

620

615

0.2

F

L + AlgMns + AliiCes

615

550

2.6

G

L + AlgMns + AliiCes + (Mg)

550

543

1.0

A

Tend

3.2

H

L + AlgMns + AlnMn4 + AluCea + (Mg)

543

543

0.4

I

L + AlnMn4 + AlnCes + (Mg)

543

455

6.3

J

L + AliiMoi + Al4Mn + AluCcs + (Mg)

455

455

0.2

K

L + AUMn + AliiCes + (Mg)

455

436

0.9

L

L + Al4Mn + AlnCe3 + (Mg) + Mg^Aljz

436

436

1.1 18.0

Total

DELH is zéro at the start of the calculation (T = Ti) and is incremented recursively by this formula. The total amount of Uquid remaining at any step,/L, is calculated recursively as the product of ail Uquid fractions, starting w i t h / L = 1 . H is the enthalpy of the System (new soUd layer + residual Uquid) for 1 mole of material. H(L) is the enthalpy of the Uquid for 1 mole of material, taken after the same equiUbrium calculation as done for H at (T-ôT). The corresponding cur\'e for our quatemary aUoy M g 2 0 A l - l M n - 5 C e is shown in Fig. 13. Thirteen différent steps (phase fields) occur under Scheil conditions, the first seven are identical to the equiUbrium soUdification, detaUed as a to g in Table 3. SoUdification terminâtes at 316 °C, weU below the value of 436 °C for the same alloy in equilibrium. The cumulated value of D E L H in Fig. 13 is 6.4kJ/mol, which is at least in the same range as

AHiatent-hyp Fig- 12. The problems of defining a "laten heat" over a température range are not resolved, though They are just accumulated by the numerical intégration starting at a "fresh" Uquidus point at each step. This corre sponds to the arbitrary sélection of the hypothetical isother mal soUdification at TL (with 6.6 kJ/mol in Fig. 12) and re peating that for any foUowing layer. One should be aware of two other inaccuracies in Eq. (2 FoUowing the idea of an "isothermal" soUdification a (T-ÔT), the amount should refer to that of the previou residual Uquid at T, f^°. \n fact, / L ° = ôfs +A, where ôfs i the molar fraction of the new soUd layer at (T-ôT). S Eq. (1) should be rewritten as

DELH = DELH -f/L° X

The second point is that H(LIQUID) refers to the Uqui composition at (T-ST) and not to the Uquid composition a T, which is the tme initial composition for this soUdificatio step, Strictly, one has to take an amount ôfs out of the initi Uquid, supercool it to (T-ôT) and calculate its enthalpy, sa ôfs X //(supercooled Uquid). The enthalpy of the just solid fied layer at the same température is fif x H - fi^xH{L] The corrected incrémental formula is D E L H = D E L H +fif -ôfs

Température in °C Fig. 13. The so-called "latent heat évolution" during Scheil soUdification of aUoy Mg-20Al-lMn-5Ce according to Eq. (2), see discussion in text.

136

(3

(H-HÇL))

xH-fi^x

X //(supercooled Uquid)

H(h)

(4

This, however, involves the additional calculation c //(supercooled L ) . The error compared to Eq. (3 ôfs X (//(L) - //(supercooled L)), should become small fc smaU composition shifts. fil simimary, for both equiUbrium and Scheil soUdific; tion, a true "latent heat" should only be defined for nonva iant soUdification. I f a température interval (Ti^-Ts) oc curs, the enthalpy différence between Uquidus and soUdv point could be defined as "heat of melting", but this is mi leadingly large because of the heat capacity contributioi For subséquent use in soUdification and heat transfer mo( elUng the total enthalpy function HÇT) of the aUoy shouj be used. — • Z. Metallkd. 93 (2002)

J. Âgren et al.: Applications of computational thermodynamics

2.2.5 How does solidification affect solder alloys? Question: What is the effect of Pb contamination fiom pretinned leads/pads on the soUdification behaviour of Pb-free solders and how is it affected by différent cooUng rates? A n s w e r : The Pb contamination is a resuit from pre-tinning leads/pads with a Sn-Pb aUoy and then soldering with Pb-fiee solder, such as a S n - B i alloy which is used as an example in the présent discussion. It has been estimated that the Pb contamination can be up to 6 mass % Pb [OOMoo]. Three S n - B i aUoys with such contamination were smdied: 0.898Sn-0.042Bi-0.06Pb, 0.856Sn-0.084Bi-0.06Pb and 0.454Sn-0.486Bi-0.06Pb, the compositions are given as mass fractions. The lowest température at which the Uquid phase can occur is 138 °C in the S n - B i system and 99 °C in the S n - B i - P b system, which are the binary and temary eutectic temperattires, respectively. A temary eutectic température of 99 °C is undesirably low and, therefore, the formation of temary eutectic during soldering should be avoided. In Sections 2.1.2 and 2.1.3 it was shown that lever-mle equiUbrium and Scheil soUdification are the two limiting cases that can be utiUsed to describe soUdification behaviour. Calculations were carried out for the three S n - B i Pb compositions. The thermodynamic description used for the présent calculations was developed by Yoon and Lee [98Yoo]. The Uquidus projection together with the various soUdification paths is shown in Fig. 14. Although Fig. 14 provides an overview of the composition changes of the Uquid phase and the phase fields encountered for the différent solidification paths, it provides no information, unless specially marked, for other quantities, such as the fraction soUd formed or the température. It should be noted that the lever-mle and ScheU paths for the 0.454Sn-0.486Bi0.06Pb aUoy are essentially identical. This behaviour results from the fact that for Bi-rich alloys the soUd phases do not reveal significant composition changes, i. e., the behaviour dtiring soUdification is very simUar to that of the formation of stoichiometric phases. Another classiçal manner for the graphical représentation of phase diagrams is the isopleth. The isopleth for a constant mass fraction of 0.06Pb is shown in Fig. 15 The températures and the encotmtered phase fields for the lever-mle soUdification can be easUy obtained from this diagram. However, since the tie-lines are usuaUy not in the plane of the isopleth, no information on the phase fractions or phase compositions can be obtained. This kind of diagram also does not provide information for the Scheil soUdification. The most useful information about température and phase fractions can be obtained from Fig. 16. The slope changes in the curves also indicate the beginning and/or end of the formation of a phase. It can be seen from Figs. 14 and 15 that using a Bi-rich alloy for soldering of Pb-contanUnated lead/pads results, independent of cooUng history, in the formation of the low melting temary eutectic. Bi-rich aUoys should, therefore, be avoided, especiaUy i f the product could be exposed to elevated appUcation températures. The other two alloys only contain eutectic after Scheil soUdification. The amount of eutectic in thèse two alloys, as can be seen in Fig. I6a, is fairly smaU. It should be possible to avoid the formation of eutectic by slower cooUng of the solder joint, thus allowmg a soUdification path that is doser to the lever-mle equiUbZ. MetalUid. 93 (2002) 2

0.3

Sn

0.4

0.5

0.6

1.0 Bi

0.7

M a s s Fraction Bi

Fig. 14. Calculated liquidus projection of the Sn-Bi-Pb system. The dashed lines show the lever-rule ( ) and the Scheil ( ) soUdification paths for the alloys 0.898Sn-0.042Bi-0.06Pb (a), 0.856Sn0.084Bi-0.06Pb (b) and 0.454Sn-0.486Bi-0.06Pb (c). The symbols (*, +) indicate the mass fraction of sohd phase formed in incréments of 0.2. Except for the lever-rule paths of alloys a and b where soUdification end on the Unes of 2-fold saturation {^), soUdification ends at the temary eutectic (•).

rium path. In the case of the Sn-rich aUoy, the mass fraction of eutectic predicted by the calculation is very small (0.013) and it may not form at ail during soldering or can be dissolved by diffusion after production. As pointed out in Section 2.2.4. a value for the latent heat cannot be clearly defined for multicomponent aUoys. However, enthalpy vs. température curves can be easUy obtained from thermodynamic calculations. Calculated enthalpy vs. temperamre curves of the three aUoys for lever-mle and 260

I

240 H

I

a b

Ç

220 L 1 O

\

\)

o

\ \ \

Jr4-(Sn) +(Pb)

\

y /

\

l\ — \-1 \ 1 -r^(Sn) + ^ / '

80

0

1/ 0.1

1

/

L + (BI)

\

L + ( S n ) + (Bi) (Sn) + (Bi)->

(Sn) + E|+ (Bi) 0.2

0.3

0.4

1

0.5

0.6

0.7

0.8

0.9

Mass Fraction Bi Fig. 15. Calculated isopleth for a constant mass fraction of 0.06 Pb. The dashed fines represent the alloys 0.898 Sn-0.042Bi-0.06Pb (a), 0.856Sn-0.084Bi-0.06Pb (b) and 0.454Sn-0.486Bi-0.06Pb (c).

137

J. Âgren et al.: Applications of computational thermodynamics

Scheil solidification are shown in Fig. 16b. Although the total enthalpy change during the lever-rule and Scheil soUdification is of the same order of magnimde, more significant enthalpy différences between the two solidification mechanisms occur over certain temperamre régimes. This kind of information could, for example, be utiUsed in computational process simulations. 2.2.6 What are the best conditions for the préparation of a n - V I compound? Question: In some cases it is necessary to take in account the influence that changes in pressure have on the phase diagram and to deduce the conséquences of this on the préparation conditions of compounds. Answer: The individual I I - V I compounds exhibit interesting physical properties as semiconductors. Furthermore when combined together as solid solutions they give physical properties which change with the alloy composition. For example, alloys of CdTe-ZnTe, CdTe-ZnSe exhibit properties which reflect the relative amount of the component compounds. To prépare such alloys requires a knowledge of the multicomponent phase diagram. Of course the first step is the optimisation of the constiment binary aUoys. In ail thèse diagrams a compound with equi-atomic composition exists, and thèse compounds have high melting tem-

peramres, compared to the éléments. For Cd-Te, the mel ing température of cadmium and telluritmi are respectivel 321 and 459.5 °C and their température of vaporisation fo a pressure of 100 kPa are: 767 and 988 °C. It is easy to imagine that vapour may interfère with th liquidus curve. In this case it is necessary to optimise th phase diagram for the condensed phases, and then the ga phase is added using thermodynamic data for cadmium v pour and for tellurium vapour [1998Feu]. This enables th phase diagram to be calculated for différent pressure (0.1-10 MPa). For a pressure of 100 kPa we may observ on Fig. 17 four invariant reactions: L