calphad

Computer Coupling of Phase Diagrams and Thermochemistry 31 (2007) 53–74www.elsevier.com/locate/calphad

Applications of computational thermodynamics — the extension from phaseequilibrium to phase transformations and other properties

Andre Costa e Silvaa,∗, John Agrenb, Maria Teresa Clavaguera-Morac, D. Djurovicd, TomasGomez-Aceboe, Byeong-Joo Leef, Zi-Kui Liug, Peter Miodownikh, Hans Juergen Seiferti

a EEIMVR-UFF, Volta Redonda and IBQN, Rio de Janeiro, Brazilb Department of Materials Science and Engineering, KTH, Stockholm, Sweden

c Departament de Fısica, Universitat Autonoma de Barcelona, 08193-Bellaterra, Spaind Max-Planck-Institut fur Metallforschung, Stuttgart, Germany

e CEIT and TECNUN, University of Navarra, Spainf Department of Materials Science and Engineering, Pohang University of Science and Technology, Pohang 790-784, Republic of Korea

g Department of Materials Science and Engineering, The Pennsylvania State University, 209 Steidle Building, University Park, PA 16803, USAh ThermoTech and Sente Ltd, Surrey Technology Centre, 40 Occam Road, Guildford, GU2 7YG, UK

i University of Florida, Department of Materials Science and Engineering, PO Box 116400, Gainesville, FL 32611-6400, USA

Received 14 February 2006; accepted 17 February 2006Available online 23 March 2006

Abstract

Complex equilibria and phase transformations involving diffusion can now be calculated quickly and efficiently. Detailed examples aregiven for cases which involve varying degrees of non-equilibrium and therefore time-dependence. Despite very good agreement between suchcalculations and experimental results, many potential end-users are still not convinced that such techniques could be usefully applied to their ownspecific problems. Friendly graphic interface versions of calculating software are now generally available, so the authors conclude that the mostlikely source of the reluctance to use such tools lies in the formulation of relevant questions and the interpretation of the results. Although thepotential impact of such tools was foreseen many years ago [M. Hillert, Calculation of phase equilibria, in: Conference on Phase Transformations,1968], few changes in the relevant teaching curricula have taken into account the availability and power of such techniques.

This paper has therefore been designed not only as a collection of interesting problems, but also highlights the critical steps needed toachieve a solution. Each example includes a presentation of the “real” problem, any simplifications that are needed for its solution, the adoptedthermodynamic formulation, and a critical evaluation of the results. The availability of such examples should facilitate changes in subject matterthat will both make it easier for the next generation of students to use these tools, and at the same time reduce the time and effort currently neededto solve such problems by less efficient methods.

The first set of detailed examples includes the deoxidation of steel by aluminum; heat balance calculations associated with ladle additions tosteel; the determination of conditions that avoid undesirable inclusions; the role of methane in sintering atmospheres; interface control during thephysical vapour deposition of cemented carbide; oxidation of γ -TiAl materials; and simulation of the thermolysis of metallorganic precursors forSi–C–N ceramics and interface reaction of yttrium silicates with SiC-coated C/C–SiC composites for heat shield applications.

A second set of examples, more dependent on competitive nucleation and growth, includes segregation and carburization in multicomponentsteels and features a series of sophisticated simulatons using DICTRA software.

Interfacial and strain energies become increasingly important in defining phase nucleation and morphology in such problems, but relativelylittle information is available compared to free energy and diffusion databases. The final section therefore demonstrates how computationalthermodynamics, semi-empirical atomistic approaches and first-principles calculations are being used to aid filling this gap in our knowledge.c© 2006 Elsevier Ltd. All rights reserved.

∗ Corresponding author.E-mail addresses: [email protected] (A. Costa e Silva), [email protected] (J. Agren), [email protected] (M.T. Clavaguera-Mora),

[email protected] (D. Djurovic), [email protected] (T. Gomez-Acebo), [email protected] (B.-J. Lee), [email protected] (Z.-K. Liu),[email protected] (P. Miodownik), [email protected] (H.J. Seifert).

0364-5916/$ - see front matter c© 2006 Elsevier Ltd. All rights reserved.doi:10.1016/j.calphad.2006.02.006

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http://dx.doi.org/10.1016/j.calphad.2006.02.006

54 A. Costa e Silva et al. / Computer Coupling of Phase Diagrams and Thermochemistry 31 (2007) 53–74

1. Introduction

The last few years have seen a dramatic advance incomputer technology, yielding a tremendous increase inthe calculation power available to the PC user. Over thesame period, CALPHAD (Computer Calculation of PhaseDiagrams) techniques have evolved from just performingcomplex equilibrium calculations relevant to materials scienceto simulating phase transformations involving diffusion. Bothtasks can now be performed quickly and efficiently and producevery good results. This has a large effect on the ability to solvecomplex problems in several key areas: materials processing,the prediction of material behavior under industrial conditions,and supporting the design of new materials.

Since 1997, the series of Ringberg Workshops onComputational Thermodynamics have produced two reportson the status and evolution of “Applications of ComputationalThermodynamics” [2,3]. The first report focused on reviewingapplications and highlighting the development needs in eacharea of application, whereas the second report, while alsoreviewing applications, attempted to formulate the review inthe form of questions and answers in an attempt to improvethe interface between the final user and the providers ofthermodynamic and allied property information.

This reflects the observation that properly translating theconditions of a “practical” problem into a well-formulatedthermodynamic or diffusion problem (within the limitations ofthe CALPHAD technique) is not a trivial question. One sourceof difficulties in communication between users and providershas traditionally been ascribed to the user–software interface,but it is the opinion of the authors that more fundamentalquestions are probably just as relevant as, or more importantthan, problems related to the user–software interface. Thishas been confirmed in recent years, as much more user-friendly graphic interface versions of calculating software hasbecome available. This has made it more evident that the realdifficulties experienced by end-users lie in formulating thecorrect questions, as well as interpreting the results of thecalculations.

It is apparent that part of the problem is related to difficultiesin understanding the thermodynamic concepts involved in theproblems of interest. If that is true, the best route to achievingmore efficient utilization of the current generation of noveltechniques lies in paying more attention to the education andtraining of the potential users in the relevant areas.

Computational thermodynamics has already been success-fully introduced as an educational tool in several programs andschools [4,5]. In addition, user courses are offered by virtu-ally all the commercial software suppliers. However the avail-ability of these advanced software tools has not yet producedmuch change in the way the thermodynamics, physical chem-istry and diffusion courses are taught. Although the impact ofsuch tools on the relevant curricula was foreseen many yearsago [1], change in these curricula has been extremely slow.

The present account is aimed at providing examples fromthe background experience of the participants of this group thatmay help to rectify this situation. The critical steps in solving

each problem are highlighted: (a) the definition of the “real”problem; (b) when simplifications are needed, and the rationalebehind the selection of the relevant part of the problem; and (c)the thermodynamic formulation of the problem. The results arethen presented and critically evaluated.

It is hoped that this form of presenting problems may helpin several different aspects; firstly, to help identify the basicknowledge necessary for the efficient use of these techniques;secondly, to accelerate changes in curricula that will make iteasier for the next generation of students to use these tools;and thirdly to appreciably reduce the time and effort presentlyused on teaching how such problems are solved by less efficientmethods.

2. Selected application examples

One of the important aspects when applying computationalthermodynamics to the solution of a real problem concernsthe effect of time. As thermodynamics is concerned withequilibrium, one must decide how the information onequilibrium will be used. If one expects the system to comesufficiently close to equilibrium in the time-frame relevantto the problem, or if the interest is in evaluating what canhappen in a given system, then time-independent calculationsare sufficient. If, however, the time available for the processesto occur is not long enough for equilibrium to prevail, either atime-dependent solution is used or the equilibrium informationhas to be considered with additional care. For this reason, theexamples are grouped, taking into consideration the effect oftime on their solution.

2.1. Time-independent problems

2.1.1. Aluminum deoxidation of steel [6]Practical and experimental results indicate that the

deoxidation of liquid steel using aluminum is a fast process,where equilibrium is reached in a very short time. Thus,time-independent equilibrium calculations are sufficient in thiscase. Without taking into consideration aluminum losses, theamount of aluminum required for steel deoxidation is definedby the initial oxygen content of the steel to be deoxidized andthe acceptable oxygen content at the end of the deoxidationprocess. The classical solution to this problem is a two-step calculation: first, the aluminum dissolved in the steelat the end of the deoxidation process is calculated, usingthe oxide solubility product at the deoxidation temperature(here, knowledge of the oxide to be formed is necessary);then the aluminum consumed in forming oxide is calculatedbased on the oxide stoichiometry, as indicated in Fig. 1. Thestrategy of breaking down the aluminum addition into twoparts is only necessary because of the information used for thecalculation.

Unfortunately, this strategy does not help the understandingof the equilibrium issue. This is a consequence of the classicalapproach to the problem. A different approach can be proposedif a tool that calculates equilibrium is available. In this case,the main concepts involved are: (a) the system composed of the

A. Costa e Silva et al. / Computer Coupling of Phase Diagrams and Thermochemistry 31 (2007) 53–74 55

Table 1Calculation procedure to define the aluminum addition required for steel deoxidation [6]

Conditions to calculate equilibrium Calculation result

Constituents of the system Fe–Al–O, C = 3 (a) Phases present: 998.7 kg of liquid steel containing O = 6 ppm%Al = 0.02% 1.3 kg of alumina (Al2O3)

Conditions to be fixed (F = 5 = C + 2): Total size of the system = 106 g (1 ton of steel) (b) Total amount of aluminum in the system: 875 g%O total = 600 ppm%O in solution in steel = 6 ppmT = 1873 K (1600 ◦C)P = 105 Pa (1 atm)

Fig. 1. Schematic presentation of the deoxidation process on the iron-richcorner of the 1873 K isotherm of the Fe–O–Al system.

steel in the ladle plus the aluminum to be added is assumed tobe a closed system, (b) equilibrium will be reached, (c) the totaloxygen content of the system is given by the oxygen contentin the steel before deoxidation, and (d) the equilibrium amountof oxygen in the liquid-steel phase should be the acceptableoxygen content at the end of the deoxidation process.

Furthermore, one must consider that the system of equationsthat must be solved to calculate any equilibrium is onlymathematically determined when the number of equations isequal to the number of variables [7]. Thermodynamically, thisis expressed in the form of the familiar Gibbs Phase Rule:

P + F = C + 2 F = C − P + 2

where F is the number of degrees of freedom, C is the numberof system components, and P is the maximum number of stablephases. If no stable phase is prescribed in a calculation, onehas to fix C + 2 conditions to be able to solve the equilibriumproblem [8].

This is sufficient to calculate the total amount of aluminumin the system, as well as the aluminum in solution in the steeland the oxide formed and its quantity, as presented in Table 1.

Except for eventual losses associated with the processof adding the aluminum and reoxidation from the slag, thecalculated values are in accordance with the results found inindustry [9].

2.1.2. Heat balance — the effect of additions to the steelladle [6]

When alloying additions are made to liquid steel in theladle, not only composition changes occur, but there is alsoan important thermal effect. These changes are due not onlyto “sensible heat” needed to bring the alloy to the steeltemperature but also to enthalpy variation associated with thedissolution of the alloying elements in steel. The traditionalsolution requires the knowledge or estimation of the heatcapacities of the alloy to be added and of the liquid steel, aswell as information on the enthalpies of mixing. Furthermore,if the elements present in the alloy form compounds (as inthe case of Fe–Si, for instance), the enthalpy change relatedto the dissociation of the compounds must be considered. Tosimplify the calculations, a dilute solution is assumed and athermal effect (in ◦C/kg/t) is derived for any given ferro-alloy composition (e.g. [10]). Such an approximate value isfrequently adopted for all similar alloys due to calculationdifficulties.

Solving this problem involves two important observations:(a) both the initial and the final states of the system (before andafter alloying) are equilibrium states, and (b) if the heat lossesoccurring during the mixing process are negligible, the processis adiabatic, i.e. heat is conserved. As the process occurs atconstant pressure, this is expressed as constant enthalpy, i.e. thetotal enthalpy is the same before and after mixing. Thus, if oneconsiders, for instance, an addition of 70 wt.%Si ferro-siliconalloy to raise the silicon content of the steel to 0.3%, one needsto know at what temperature the final steel will have the sameenthalpy as the initial steel plus the room-temperature ferro-alloy. Table 2 shows the steps in this calculation.

2.1.3. Equilibrium calculation for inclusionsAlumina and high-melting aluminate non-metallic inclu-

sions can be detrimental to the fatigue resistance of steels.Higher-plasticity inclusions are desirable in these cases. A typ-ical example is steel for engine valve spring applications. Inthis case, it is necessary to adjust the chemical composition ofthe steel in order to guarantee that the desired inclusions willbe formed. As the precipitation of oxide inclusions in steel oc-curs close to equilibrium, this is a simple equilibrium problem.However, it is almost impossible to solve this with the classicaltools available. Although the classical dilute solution formalismused in steel-making can handle the behavior of the solutes inthe steel well, there is no simple way to describe the complex


Table 2Step-by-step calculation of the heat effect of silicon addition to liquid steel [6]

Conditions for calculating equilibrium constituents of the system: Fe–Si, C = 2 Calculation result

(a) First step: steel without silicon. Conditions to be fixed (F = 4 = C + 2): (a) Phases present: 1000 kg of liquid steel with enthalpy= 1.34 × 109 J

Total size of the system = 106 g (1 ton of steel)%Si in solution in steel = 0T = 1873 K (1600 ◦C)P = 105 Pa (1 atm)

(b) Second step: Fe–Si alloy to add 3 kg of Si. Conditions to be fixed (F = 4): (b) Phases present: Solid Si (DIA) and FeSi enthalpy= −1.46 × 106 J

Total size of the system = 4286 g, i.e., the weight of ferro-alloy, calculated as 3000 gof Si/70 wt.% Si in the ferro-alloy.T = 298 K (25 ◦C)P = 105 Pa (1 atm)

(c) Third step: Si containing steel. Conditions to be fixed (F = 4): (c) Phases present: 1.0043 × 106 g of liquid steelcontaining %Si = 0, 3T = 1870.31 K (1597.16 ◦C)

Total size of the system = 106 + 4286 g%Si in steel = 0.3Enthalpy = 1.34 × 109 J + (−1.46 × 106 J)P = 105 Pa (1 atm)

Fig. 2. Calculated equilibrium between steel (0.8% C, 0.6% Mn e 0.3% Si) andoxide in the MnO–SiO2–Al2O3 system at 1823 K. The solid line indicates steelAl content [11].

behavior of the oxides, even in the simplest acceptable systemin this case: the MnO–SiO2–Al2O3 and the CaO–SiO2–Al2O3ternaries. This calculation can be performed using computa-tional thermodynamics as indicated in Fig. 2 [11]. The calcu-lation indicates that the aluminum content in the steel is criticalfor the achievement of the desired equilibrium, i.e. for theavoidance of high melting point, undesirable inclusions (indi-cated by silica and mullite in Fig. 2). By adjusting the ladleslag composition and controlling aluminum inputs in the refin-ing steps, the process can be adjusted to maintain the aluminumcontent within these narrow limits.

2.1.4. Role of methane in sintering atmospheresIt is generally assumed that species in the gas phase are in

thermodynamic equilibrium; accordingly, their potentials and

the gas speciation can be deduced with a simple equilibriumcalculation using, as an input, the initial gas composition,total pressure, and temperature. However, sometimes kineticsplay an important role on these reactions, and the equilibriumcalculations have to be taken into account with care.

Some examples of this fact are the CH4 decomposition usedin sintering atmospheres, or in CVD deposition.

During sintering of high-performance and close dimensionalsteel components, nitrogen/hydrogen atmospheres are widelyused. Sometimes, small amounts of hydrocarbons are addedto these atmospheres. Among other functions, one of theroles of the sintering atmospheres during sintering of ferrouscomponents is the reduction of metallic oxides and the controlof the chemical composition, particularly the carbon content.However, small variations in atmosphere composition andfurnace temperature can lead to large differences in thecharacteristics of the components.

In N2–H2 atmospheres, the oxygen potential can be relatedto the pH2O/pH2 ratio, by means of the chemical equilibrium

H2 + 1/2O2 = H2O.

In industrial practice, it is common to express the watercontent of an atmosphere as its dew point, instead of the waterpartial pressure; in principle, the dew point can be calculated asthe temperature for a hypothetical equilibrium between the gasphase and ice, at constant total pressure. However, it is morecommon to use empirical correlations, such as that given byGerman [12]:

log10(V%) = −0.237 + 0.0336 ∗ TDP − 1.74e

− 4 ∗ TDP2 + 5.05e − 7 ∗ TDP3

where TDP is the dew point in ◦C, and V% is the volumefraction of water as a percentage (which is equal to its partialpressure). The set of lines plotted in Fig. 3(a) [13,14] showsthe value of the oxygen activity in several N2–H2 atmospheresas a function of the water content. Auxiliary axes for the dew


Fig. 3. Oxygen activity for different N2/H2 sintering atmospheres at 1100 ◦C, (a) without and (b) with the addition of 0.1% CH4, as a function of the dew point ofthe initial atmospheres (before equilibrium is established) [13].

point and the pH2O/pH2 ratio have also been plotted in the samediagram. The calculations have been done using the SGTE puresubstance database (SSUB) [15], at a temperature of 1100 ◦C,which is a typical value of sintering temperature.

The abscissa of Fig. 3(a) (inlet pH2O) represents thewater content of the atmosphere before entering the furnace,i.e. before it reaches equilibrium with the remaining species.However, for these N2–H2 atmospheres, this value is almostthe same as the actual water partial pressure in the atmosphereat the temperature of the calculation.

The calculation of these curves has been performed usingthe following procedure: it is convenient first to redefine thecomponents; instead of H, O, N, we use H2O, N2, H2. Theconditions for the calculation are a total pressure of 1 bar,temperature, a total amount of 1 mole, N2/H2 ratio (a differentconstant value for each curve: 99/1, 95/5, 90/10, 60/40, 20/80and 1/99), and a certain oxygen activity. The oxygen activityis varied and the calculation repeated several times, i.e. thevalue is used as the “stepping” condition in the calculation.The abscissa (inlet pH2O) is the mole fraction of the componentH2O which, as mentioned above, is almost the same value asthe actual mole fraction of the species H2O in the gas mixtureat equilibrium.

Additions of hydrocarbons such as CH4 to the sinteringatmospheres are intended to control the carbon content ofthe sintered parts. However, even small quantities of CH4can change the oxygen potential of the atmosphere. This wasillustrated by Ortiz et al. [13,16] and is shown in Fig. 3(b)for several N2–H2 sintering atmospheres with a small additionof CH4 (0.1%), compared to the same atmospheres withoutmethane (Fig. 3(a), and the dotted lines in Fig. 3(b)), at thesame temperature of 1100 ◦C. Several features can be stressedin this Fig. 3(b): first of all, the abscissa represents the watercontent of the gases entering the furnace, i.e. before reachingthe equilibrium; if one represents the equilibrium partial

pressure of water in the atmosphere, one could get just the sameset of lines as in Fig. 3(a), giving a false impression that CH4 isnot affecting the oxygen potential of the atmosphere.

Secondly, a breakdown point can be clearly seen in Fig. 3(b),at a water content that is the same as the CH4 content ofthe gas. For sintering atmospheres with water content higherthan that value, this hydrocarbon has almost no effect onthe oxygen potential; hence the reduction capability of theseatmospheres is controlled by the water content, as if no CH4were present. However, when the water content is lower thanthat of CH4, the oxygen potential is sharply reduced by severalorders of magnitude. For atmospheres with a low H2 content,saturation is produced due to the carbon activity reaching valueshigher than unity (labeled “sooting” in Fig. 3(b)); here, the gascomposition (the CO/CO2 ratio, and hence the oxygen activity)is controlled by the Boudouard reaction,

C + CO2 = 2CO.

The lines plotted in Fig. 3(b) have been calculated following thesame approach as with Fig. 3(a): the components defined areCH4, H2O, N2 and H2, instead of C, H, O and N. The conditionsfor the calculation are a total pressure of 1 bar, temperature, atotal amount of 1 mole, N2/H2 ratio (a different value for eachcurve: 99/1, 95/5, 90/10, and so on), the CH4 quantity (molefraction 0.001), and a certain oxygen activity, whose value isused as the “stepping” condition in the calculation.

In addition, several horizontal lines have been plotted inFig. 3(a) and (b). They represent the oxygen potential formetal/oxide equilibria, of several metals used in sintering (Fe,Cr, Mn and the powder Astalloy CrM), at the temperature ofthe calculation. They give an indication of the atmosphere’scapability for the reduction of the metal oxides.

The carbon activity of these atmospheres is represented inFig. 4, showing the conditions for methane decomposition withcarbon formation (sooting).


Fig. 4. Calculated carbon activity for N2–H2 atmospheres with 0.1% CH4, at1100 ◦C as a function of the water content, represented also as the dew point ofthe atmosphere [13].

Finally, this analysis can be done at different temperatures,as is represented in Fig. 5 for two values of the dew point(−45 and −15 ◦C), and for different N2/H2 ratios, expressedas N2 partial pressure. According to this, atmospheres that aresupposed to be extremely dry (i.e. with a very low dew point,−45 ◦C), are more likely to experience CH4 decomposition andhence sooting.

Calculations such as those shown here allow the controlof both the oxygen and the carbon potential in sinteringatmospheres, thanks to changes in the gas composition.

2.1.5. Interface control in PVD of cemented carbideToday, about 60% of cemented carbide cutting tools are

CVD coated [17] including an Al2O3 layer for excellent wear-resistance. There are typically two forms of Al2O3 phases,i.e. the stable α- and meta-stable κ-Al2O3 phases. Althoughthe α-Al2O3 phase shows excellent high-temperature stability,it is difficult to form and has relatively poor adhesion to theinner non-oxide layer compared with the κ-Al2O3 phase. Topromote the formation of α-Al2O3 and improve its adhesion tothe inner non-oxide layer, a bonding layer was deposited beforethe deposition of the Al2O3 [18,19]. In order to fully understandphase stability in this bonding layer in the CVD coatingof cemented carbide cutting tools, detailed thermodynamiccalculations were carried out to investigate the effects ofvarious processing parameters [20]. Feeding gases flowedupwards in the chamber at 60 torr pressure with a temperatureof 970 ◦C at the bottom and 1000 ◦C at the top of thechamber [20]. The input gas mixture consisted of H2 (86%volume), CH4 (4%), CO2 (1.7%), N2 (8%), and TiCl4 (0.3%).These experimental conditions were used as initial conditionsin the present thermodynamic calculations using the Thermo-Calc program [21] and the SGTE substance database [22]

combined with the FEDAT database of TCAB [23]. As a firstapproximation, the Gibbs energy, Gfcc

TiO, for the end-memberof TiO in the Ti–O fcc phase and the interaction parameterbetween oxygen and vacancy were estimated from those of thehcp phase in the Ti–O binary system [24].

The effects of individual processing parameters, i.e. gas flowratios, temperature, and pressure, and their combinations, onthe phase stability were investigated by systematically varyingtheir values. Equilibrium calculation resulted in three-phaseequilibrium of gas, fcc, and graphite, with the composition offcc being Ti(C0.016N0.984) with little oxygen, typically calledthe TiCNO phase in the literature, and the CH4 concentration inthe gas phase being 0.07%. It shows that over 98% of methanehas been consumed. If graphite is assumed to be difficult toform, as one does usually, the formation of the fcc-TiCNOphase would be observed, in agreement with the suggestedphase in the literature [19,25].

However, in the experimental investigations, it was foundthat, under typical CVD processing conditions for thebonding layer, four types of titanium oxides, i.e. TiO2,Ti4O7, Ti3O5, and Ti2O3, were observed from the X-raydiffraction (XRD) analysis by extending the deposition timefor the bonding layer instead of the fcc-TiCNO phase [20].To understand the difference between the calculations andexperimental observations, the thermodynamic driving forcefor the formation of graphite, defined as Dgra = −�G

RT with�G being the Gibbs free energy change for the formation ofgraphite, was evaluated. It was found that only the flow ratiosof CH4 and CO2 change the sign of Dgra, as shown in Fig. 6.If the flow ratio of CH4 is higher than 1.8%, the driving forcefor the formation of graphite will be positive, which means thatthe graphite will form from the gas phase. On the other hand,graphite will not form if the CH4 flow ratio is lower than 1.8%.For CO2, the zero driving force point is around 3.7% with thehigher CO2 flow ratio preventing the formation of graphite.

Consequently, the phase diagram of the CH4 flow ratio andtemperature was calculated and plotted in Fig. 7. It shouldbe noted that there is no stable phase region for the Ti2O3compound in Fig. 7 to explain the formation of the Ti2O3compound in one of the samples at the top of the CVD chamber.This suggests that the formation of the mixtures of titaniumoxides is not only influenced by the flow ratio of CH4 but alsoby other factors, which are discussed in detail in Ref. [26].

To investigate the decomposition of CH4, Fulcheri andSchwob [27] considered the following chemical reactionCH4 → C+2H2. This reaction is endothermic and the standarddecomposition enthalpy is: �H 0 = 74.6 kJ/mol. Fulcheri andSchwob plotted the energy supply needed for the above reactionas a function of temperature and concluded that the temperatureshould be between 1000 ◦C and 2000 ◦C for the reaction to takeplace. Moradov mentioned that the processing temperatures forthis decomposition reaction is 1400 ◦C or higher without ametal catalyst [28].

2.1.6. Sintering of Si3N4–SiC ceramicsCeramics based on non-oxide compounds such as Si3N4 and

SiC are suitable candidates for high-temperature engineering


Fig. 5. Calculated carbon activity (ref. state: graphite) for N2–H2 mixtures containing different quantities of CH4 ((a) 1%; (b) and (c) 0.1%) and H2O (dew pointsof (a) and (b) −45; (c) −15 ◦C), as a function of temperature [16].

applications. SiC-reinforced Si3N4 composites are fabricatedby addition of whiskers, fibers and platelets of SiC to Si3N4.Also, the use of precursor polymers has attracted wide attentionas an alternative approach to produce Si3N4–SiC ceramics andoffers a number of advantages compared to classic powdertechnology [29]. With the synthesis of Si–C–N ceramics fromprecursor polymers, one can control the materials composition,structures and properties on an atomic scale. To predict andcontrol the sintering conditions for all types of Si3N4–SiCceramics, thermochemical calculations can be used.

The stability and decomposition of silicon carbide–siliconnitride (Si3N4–SiC) ceramics during the sintering process wereinvestigated by thermodynamic calculations. Thermodynamicdescriptions for the Si–C and Si–N systems were acceptedfrom the literature [30,31]. The data for C–N gas species weretaken from the database of the Scientific Group ThermodataEurope (SGTE) [15]. Isothermal sections were calculated by

extrapolation from the binary subsystems [32]. Fig. 8 showsthe isothermal section in the Si–C–N system applicable fortemperatures 1757 K < T < 2114 K and a total pressureof 1 bar. The composition of a precursor-derived Si–C–Nceramic (PHMS) is indicated for discussion in the followingsections. The isothermal section shows a silicon nitride–siliconcarbide–nitrogen gas (Si3N4–SiC–N2) phase field indicatingthe stability of Si3N4–SiC ceramics. At temperatures lower than1757 K, SiC and nitrogen react to form Si3N4 and graphite. Attemperatures higher than 2114 K, silicon nitride decomposesand forms liquid silicon and nitrogen. A detailed analysis ofthe system thermodynamics shows that the nitrogen partialpressure has to be changed simultaneously with the heating orcooling of the furnace during the sintering process to stabilizethe microstructure of the ceramic materials. The calculated“pN2-temperature” potential diagram in Fig. 9 provides generalguidelines for the correct sintering conditions. The stability area


Fig. 6. Effect of gas flow ratios on the driving force of graphite.

Fig. 7. Calculated phase diagram using temperature and CH4 flow ratio.

of Si3N4–SiC ceramics is indicated in the diagram. It showsthat, during temperature decrease from a sintering temperatureof, e.g., 2173 K (1900 ◦C) to room temperature, the nitrogenpressure has to be decreased in a well-defined and continuousway to avoid any phase reactions. If the nitrogen pressureis kept constant during cooling, silicon carbide and nitrogengas react to form free carbon, deteriorating the good materialsperformance. The next step of such a thermochemical analysishas to consider the influence of the most common sintering aidssuch as Al2O3 and Y2O3 on the phase reactions. Generally, thesinter process for technical ceramics and composites requires avery careful analysis of the partial pressures of the gas speciesand their interaction with condensed phases. Thermochemicalcalculations efficiently support this approach.

Fig. 8. Isothermal, isobaric (1 bar) section in the Si–C–N system valid for1757 K < T < 2114 K. The composition of an amorphous PHMS precursor-derived ceramic (see Fig. 9) is indicated.

Fig. 9. Potential phase diagram for the Si–C–N system valid, for example, forPHMS-derived ceramics with C:Si < 1.

2.1.7. Prediction of the high-temperature stability and decom-position of Si–C–N ceramics derived from precursors

Precursor-derived Si–C–N ceramics are X-ray amorphousup to temperatures of about 1773 K (1500 ◦C). Athigher temperatures, the materials crystallize, decomposeand cannot be used as an engineering material. Precursor-derived amorphous Si–C–N ceramics of varying compositioncan be produced by polymer thermolysis. It is, however,too expensive and time consuming to investigate the high-temperature behavior of all possible ceramic compositions justby experimental methods. Thermodynamic simulations cansupport efficient materials characterization. It is, for example,possible to simulate the thermal gravimetrical analysis (TGA)of Si–C–N materials by means of calculated phase fractiondiagrams, as shown in Fig. 10 [32]. This diagram showsa Si–C–N precursor ceramic (PHMS) phase reaction at atemperature of 1757 K, where silicon nitride reacts withcarbon to form SiC accompanied by the formation of nitrogengas. A mass loss of 13.7 wt.% is expected. The simulationshows an expected second mass loss of 15.1 wt.% at 2114 Kdue to the decomposition of silicon nitride forming silicon


Fig. 10. Calculated phase fraction diagram in the Si–C–N system for PHMS-derived ceramics (C:Si < 1).

and nitrogen gas. These simulated weight losses are in goodagreement with the results of the thermal gravimetrical analysisFig. 11. However, the TGA shows that the first reactionoccurs at significantly higher temperatures than expectedfrom the simulation. The deviation can be explained byconsidering the amorphous microstructure of precursor-derivedceramics. However, no thermodynamic data is available for theamorphous Si–C–N materials. Therefore, all thermodynamiccalculations are performed for materials assuming crystallizedmicrostructures composed of silicon nitride, silicon carbideand graphite at temperatures below 1757 K. Nevertheless,important quantitative information on expected TGA resultscan be derived from the calculations. Similar calculations werealso made for Si–B–C–N precursor-derived ceramics to analyzetheir high-temperature stabilities [33].

Fig. 11. TGA/DTA measurement of PHMS precursor-derived ceramic (BN crucible, nitrogen atmosphere, heating rates: RT–1000 K, 10 K/min; 1000–2000 K,5 K/min).

2.1.8. Simulation of the thermolysis of metallorganic precur-sors for Si–C–N ceramics

The simulation of the thermolysis of metallorganicprecursors to produce amorphous Si–C–N ceramics requiresthe calculation of the reaction paths in the Si–C–N–H system.A database was developed for this quaternary system bythermodynamic optimization [33]. Gas species descriptionswere accepted from the database of the Scientific GroupThermodata Europe (SGTE) [15]. Various gas species formduring the thermolysis of Si–C–N–H precursor polymers, e.g.Si, Si2, Si3, N2, Si2C, SiC2, SiN, Si2N, CN, (CN)2, H2,CH4, and NH3. The thermolysis is usually carried out in anopen system reactor and flowing gas atmosphere, whereasthe calculations are performed for a closed system and staticatmosphere conditions. Nevertheless, the composition of thefinal amorphous ceramic Si–C–N reaction products could bederived in good accordance with the experimental results. Thedecomposition of metal-organic precursor polymers containing50 wt.% of hydrogen was simulated. The calculated isothermalsection for 573 K and 50 wt.% hydrogen is shown in Fig. 12.The polymer compositions are indicated. An extended gasphase–silicon nitride field is detected. At this temperature, thegas phase mainly consists of methane. The composition of thegas phase forming with progressing precursor decomposition(increasing temperature) was simulated. During thermolysis, acontinuous change in the condensed material composition, aswell as of the gas phase composition, occurs. The consequenceof these reactions are curved reaction paths due to thecontinuous change in the ratio of the gaseous species, startingin the space of the Si–C–N–H concentration tetrahedron andending on the tie-lines graphite–silicon nitride or the tie trianglecarbon–silicon nitride–silicon carbide. For the given example,the carbodiimide-based PHMS precursor, Fig. 13(a), showsthe fractions of the condensed phases. Fig. 13(b) presentsthe corresponding gas phase composition. At temperatureshigher than 773 K, methane and, additionally, hydrogen areformed. At a thermolysis temperature of 1323 K, hydrogenis the dominating gas species. The precursor materials withsilicon deficits (with respect to the Si:N ratio of 0.75 forSi3N4) additionally form small amounts of nitrogen and


Fig. 12. Isothermal section (573 K) at 50 at.% hydrogen in the Si–C–N–H system with compositions of different precursor materials indicated.

(a) Phase fractions of the solid products. (b) Gas phase composition.

Fig. 13. Results of thermodynamic calculations for the PHMS precursor (11.5 Si, 15.7 C, 10.5 N, 62.3 H; at.%): (a) phase fractions of the solid products; (b) gasphase composition.

Fce

ammonia. All precursors completely lose hydrogen during thethermolysis. The resulting amorphous materials compositionsare located in the Si–C–N subsystem, as indicated for thedecomposition of the PHMS precursor material Fig. 14. Thematerials composition is located on the carbon–Si3N4 tie line.The experimentally derived ceramic thermolysis product isX-ray amorphous, but the bulk composition is similar to thecalculated composition.

2.1.9. Oxidation of γ -TiAl materials

The oxidation of γ -TiAl based alloys in air and variousother atmospheres has been extensively studied experimentallyover the past few decades. The reaction kinetics andcharacter of the developing oxide layers and subscale zonesare influenced by many alloy properties such as grainsize and surface roughness. To clearly separate the effectsof the thermochemical conditions during oxidation from

ig. 14. Compositions of solid products of PHMS precursor thermolysisalculated for temperatures 298–1323 K, displayed together with the phasequilibria in the Si–C–N system for 1687 K < T < 1757 K.


such alloy characteristics, thermodynamic calculations werecarried out. These calculations serve as a reference for adeeper understanding of the reaction kinetics and phasedevelopment [34].

Using the thermodynamic database for the Ti–Al–O systemdeveloped by [35,36], it is possible to calculate potentialphase diagrams and compare them with results from TEMinvestigations of the surface of oxidized γ -TiAl materials.As known from experiments, oxygen and aluminum are themain diffusing components during the formation of a sub-scalezone below an oxide layer on the alloy surface. Therefore, thechemical potentials of the diffusing components Al and O wereused as axes for the calculated potential diagram at 1173 K.Phase equilibria are composed of γ -TiAl, α2-Ti3Al, Al2O3 andthe ternary X-phase. Based on this diagram, the qualitativeoxidation behavior of γ -TiAl in air can be derived. Theformation of the sub-scale zone can be described by assumingtime-dependent changing μAl/μO ratios. The nitrogen effectwas neglected as a first approximation (Fig. 15). This approachcan be used as a first step in the analysis of the phase evolutionof the diffusion-controlled process of γ -TiAl oxidation for tworeasons: (1) chemical potential gradients are the driving forcesfor diffusion, and (2) local equilibrium at the interfaces of thephases can be assumed. For the sake of simplicity, a linearvariation in the chemical potentials along the diffusion zone canbe assumed. The change in the chemical potentials of aluminumand oxygen through the diffusion zone can then be described bythe iterative formulae μi+1(Al)(t) = μi (Al) + �μ(Al)(t) andμi+1(O)(t) = μi (O) + �μ(O), respectively, with �μ(O) > 0and �μ(Al) < 0, where the index i is equivalent to a distancewithin the alloy. Oxygen is a fast-diffusing component, andthe time dependence of the step value �μ(O) was neglected.The step value for aluminum can then be related to �μ(O)

by a time-dependent model parameter q(t): �μ(O)/�μ(Al)(t)= q(t) with ∞ < q(t) < 0. The parameter q(t) defines thepaths indicated in the potential phase diagram in Fig. 8 at atime t . It defines the slope of a straight line which starts atrefμ, assuming very small oxygen impurities in γ -TiAl. Withthese assumptions, the oxidation process of γ -TiAl can bedescribed as a series of successive local equilibrium states t1to t4. Path t4 defines the start of the oxidation with γ -TiAl(O)and Al2O3 in equilibrium. Path t2 predicts the evolution of theinterfaces Al2O3/X and γ -TiAl(O)/X in good accordance withthe surface layers found after 100 h of oxidation. The oxidationexperiments show that, after 250 h, α2–Ti3Al has additionallyformed, which is simulated by path t3. Path t4 predicts the finalstate of oxidation at which the X-phase disappeared completelyafter extended time periods of oxidation.

2.1.10. Interface reaction of yttrium silicates with SiC-coatedC/C–SiC composites in heat shield applications

Yttrium silicate coatings are potential candidates forthe oxidation protection of SiC-pre-coated C/C–SiC heatshields for space shuttle systems. Plasma wind tunnel testsshowed that they can withstand temperatures of 1650 ◦C,but at temperatures higher than 1750 ◦C the coatings failbecause of blister formations and subsequent spallation of the

Fig. 15. Light optical images of the γ -TiAl oxidation zone and the Ti–Al–Opotential diagram (1173 K, p = 1 bar) with indicated diffusion paths.

Y-silicate coatings. To simulate the phase reactions, aquaternary Y–Si–C–O database was developed and usedfor application orientated calculations [37]. The isothermalpotential diagram in the Y–Si–C–O system was calculatedusing CO- and SiO-partial pressures as axes Fig. 16.Preliminary calculations had shown that CO and SiO arethe main gas species formed from interfacial reactions. Thethree-phase equilibrium field for gas, SiC and Y2SiO5 wascalculated (by fixing them as stable phases). This three-phasefield exists between lines of four-phase equilibria: (a) gas, SiC,Y2SiO5, Y2Si2O7; and (b) gas, SiC, Y2SiO5 and graphite. Atthe intersection point of these four phase lines, the five-phaseequilibrium of gas, SiC, Y2SiO5, Y2Si2O7 and graphite exists,representing the heat shield coating system under investigation.The gas phase consists mainly of carbon monoxide, and itspressure increases significantly with temperature, as shown inFig. 17. Blister formation and spallation of the coating can beexplained by the interface reactions and the accompanied gasphase formation and pressure increase.

2.2. Time-dependent problems

When the time available for the processes to occur is notlong enough for equilibrium to occur, either a time-dependentsolution is used or the equilibrium information has to beconsidered with additional care. This section discusses someexamples of the application of computational thermodynamicsto time-dependent problems.


Fig. 16. Potential diagrams in the Y–Si–C–O system for 1853 K, 1903 K, and2000 K.

Fig. 17. Temperature-dependent vapor pressure in equilibrium with SiC,Y2SiO5, Y2Si2O7 and carbon.

2.2.1. Segregation in steel

Solidification rarely occurs in equilibrium conditions inindustrial processes. The way that solute redistributes duringthis process is very important in defining the properties ofsteels. The first kind of information that can be obtained,using computational thermodynamics, is the equilibrium phaseand phase fraction. Actual industrial results frequently deviatefrom this, due to non-equilibrium solidification. Algorithmsfor the calculation of segregation, assuming that completehomogenization occurs in the liquid (Scheil model) and nodiffusion occurs in the solid, have been implemented incomputational thermodynamic software (e.g. [21]). Recently,

Fig. 18(a). Solidification modeling of M2 high-speed steel using differenttechniques. Experimental values of T solidus � are indicated [38].

the option of assuming that interstitial elements are completelyhomogenized in both liquid and solid while substitutionalelements follow the Scheil model (i.e. only completehomogenization in the liquid) has also been also implementedas the “partial equilibrium” model [38]. These three models canbe implemented in computational thermodynamics softwarewithout explicitly considering time as a variable. These modelsare useful in several cases, depending on what information isnecessary. Non-equilibrium phases and the effective solidustemperature in the industrial solidification of high-alloy steels,for instance, can be estimated with reasonable precision usingthese simplifications [38].

Fig. 18(a) presents the results of the calculations performedby Chen and Sundman [38] for M2 tool steel using thesethree models. It is evident that the partial equilibrium modelpredicts correctly the phase sequence as well as the solidustemperature. These models, however, are not always accurateenough when dealing with structural steels solidified undercontinuous casting conditions or in the predicition of the actualprofiles of chemical composition in the segregated solid. In thiscase, it is essential to consider mass transfer by diffusion, inboth the solid and the liquid phase during solidification. Theresults of this type of calculation using software that couplescomputational thermodynamics with the solution of diffusionequations (DICTRA) [21] are presented in Fig. 18(b). Theagreement with the experimental data of Ueshima et al. [39]is evident.

2.2.2. DICTRA simulation of carburizing of stainless steel(a) Definition of the problem. Carbon is usually considered asa harmful impurity in austenitic stainless steels because it mayreact with chromium to form Cr-rich carbides and eventuallycause a loss in corrosion resistance. It is thus necessary to keepthe carbon level very low by metallurgical processing in thesteel plant or by adding strong carbide formers like Ti and Nb


Fig. 18(b). Calculated Mn distribution in cast steel containing %C = 0.13,%Si = 0.35, %Mn = 1.52, considering diffusion in both liquid and solidphases (DICTRA) [6]. Experimental data from [39].

Table 3Composition of stainless steel (mass%)

Cr Ni Ti Fe

18 10 1 bal

to the steel. These will react with C to form very stable MCcarbides, and the residual carbon content in solution will be solow that no Cr-carbides form.

If stainless steel is welded to low-alloy steel, it may becarburized by the low-alloy steel and gain much increasedcarbon content and, as a result, there will be unwantedcarbide formation. That problem was considered by Helanderand Agren [40]. Exposure of stainless steel in a carbon-richatmosphere, e.g. a gas containing hydrocarbons, may alsolead to carburizing. This situation will be considered in thepresent example. We shall consider a stainless-steel plate, witha thickness of 6 mm, with the initial composition given inTable 3. The initial carbon content is assumed to be very low.

We consider an atmosphere having a carbon activity ofunity relative to graphite, i.e. the gas is on the soot limit. Thetemperature is 800 ◦C (1073 K) and we consider an exposuretime of 1000 h, i.e. 6 weeks. The question is: what will happento the steel?(b) Simplifications. We first rephrase the question into athermodynamic/kinetic question. How will the amount ofcarbon and carbides vary with time and distance from the steelsurface? We shall make three different simulations with theDICTRA code [41].(c) Thermodynamic formulation of the problem. In the firstsimulation, we consider carburizing of pure Fe in the austeniticstate. A fixed carbon activity of unity relative graphite is takenas the boundary condition, and the size of the system is takenas 3 mm. The result is shown in Fig. 19.

Fig. 19. Carbon content profiles at the times indicated during carburizing ofpure iron at 800 ◦C.

Fig. 20. Carbon content profiles at the times indicated during carburizing ofstainless steel at 800 ◦C. Single-phase austenite is assumed.

The assumed maximum distance (3 mm) is treated as theboundary condition defining a closed system. As can be seen,there is a massive carburization and the carbon content after1000 h has almost leveled out at the level corresponding toequilibrium with the atmosphere.

In the next simulation, we shall take into account thealloying, but still assume that we have a single-phase austeniticstructure. We then use mixed boundary conditions on the left-hand side with the carbon activity unity relative to graphite andzero flux for the other elements. The result is shown in Fig. 20.

The carbon content at the surface is now much higher dueto the presence of the alloy elements Cr and Ti, which lowerthe carbon activity for a given carbon content, and the profiles


Fig. 21. Carbon content profiles at the times indicated during carburizing ofstainless steel at 800 ◦C. Carbide formation is taken into account.

have a different shape due to the alloying effect on carbondiffusion. Finally, in the third simulation, we allow the MCcarbide, M23C6 and M7C3 to form. We have used the dispersemodel in DICTRA and enter these phases as spheroidal phases.We have used a so-called labyrinth factor that is f 2, where f isthe volume fraction of austenite. As can be seen in Fig. 21, thecarbon profile again looks quite different. The correspondingprofiles showing the fraction of phases are shown in Fig. 22.Close to the surface, the carbon content is highest, we haveM7C3 and then, at lower C content, we have M23C6. TheMC carbide is formed in addition to the other two carbides,but not in very high amounts. It may also be interesting toinvestigate the residual Cr content in austenite in order to seeif the corrosion resistance is sufficient. As a rule of thumb,one requires 11–12 mass% Cr dissolved in austenite to havesatisfactory corrosion resistance. These series of curves areshown in Fig. 23.

The disperse model in DICTRA is based on the assumptionthat the disperse phases, i.e. the carbides, occur with acomposition and amount that are given by equilibrium forthe local composition. The model does not take into accountany microstructural features, but only thermodynamics anddiffusional mobilities. In the present simulation, the Thermo-Calc databases TCFE3 and MOB2 were used [21]. It shouldbe emphasized that, in reality, it would take some timefor the carbides to form and, moreover, one often observesthat they appear preferentially at grain boundaries. A moredetailed simulation, e.g. by phase-field techniques, wouldrequire much additional input information and need muchlonger computation times.

2.2.3. Describing crystallization kinetics — the influence ofinterfacial energy and diffusion data

Very interesting properties can be obtained with thenanocrystallization of metallic glasses obtained by controlledheat treatment. Typically, in these nanostructures, precipitate

Fig. 22. Weight fraction of carbides at the same time as in Fig. 21 duringcarburizing of stainless steel at 800 ◦C.

Fig. 23. Cr content in austenite at the times indicated. Cr content lower thandashed line corresponds to insufficient corrosion resistant.

sizes range between 5 nm and 50 nm embedded in an amor-phous matrix with nanocrystal volume fractions of 30–80%(particle densities of 1022–1028 m−3). The precipitation of theprimary phase occurs in a supersaturated/undercooled liquidsolution, where growth is usually limited by the diffusion of so-lute atoms. The driving force is the determining factor neededto evaluate the rate of the transformation and can be calcu-lated by CALPHAD methods. To model the time evolution ofthe structure, other quantities such as diffusion coefficients andinterfacial energies are necessary. In the present example, theseare estimated indirectly from experimental kinetic data.


At the onset of primary crystal precipitation, the nucleationfrequency, I (T ) (assumed homogeneous), is given by [42]:

I (T ) = Io(T ) · exp

[− 16πσ 3

3RT (�GT )2

](1)

Io(T ) = 8Nv D(T )

(�a)2

( σ

RT

)1/2(2)

where D is the diffusion coefficient, σ is the molarcrystal–liquid interfacial energy, Nv is the number of nucleationsites per unit volume in the liquid, a is the mean atomic radius,� is the mean interfacial thickness in units of the mean atomicradius [43], and �GT , for primary precipitation, is calculatedby [44–46]:

�GT = Σ x xti (μxt

i − μliqi ) (3)

where xφi (i = 1, 2, . . . , c) denote the atomic fractions of each

component i in phase φ (φ = xt → crystal; liq → liquid)and μ

φi denote their chemical potential at temperature T . Nuclei

form with a critical radius r∗ given by

r∗ = 2σa

�GT. (4)

Interface controlled growth will lead to a growth rate u(T )

given by

u(T ) = 2λD(T )

�2a

[1 − exp

(−�GT

RT

)]. (5)

Here, λ is the product of the fraction of surface sites whereatoms are preferentially added and the length of the interfacein units of the mean atomic radius.

The kinetics of the process can be viewed in the followingmanner. At any time, t , the transformation proceeded upto a fraction, ξ · f1, of the total volume, V , where f1 isthe total volume fraction available for primary crystallisationand ξ is the actual degree of advancement of the process.The Kolmogorov–Johnson–Mehl–Avrami formalism [47–50]allows the evaluation of the rate of the transformation, sinceit has been shown [43] that the Avrami equation

ξ = 1 − exp(−ξex) (6)

where ξex is the extended degree of advancement of thetransformation, remains valid for a primary crystallisationprocess, provided that the temperature dependence of f1 may beneglected in the transformation temperature interval. The meanconcentration of the matrix is then given by

cti = cxt

i + coi − cxt

i

1 − ξ f1(7)

where coi is the initial composition of the parent phase.

That is, the nucleation frequency depends on the degreeof advancement of the process. In a first-order approach [43],the expected decrease in the actual nucleation frequencyis evaluated by multiplying the homogeneous nucleationfrequency I (T ) from Eq. (3) by the factor ϕ[x(t)] given by

ϕ(x) = 1 − x(t)

1 − ξx(t). (8)

Fig. 24. Establishment of local equilibrium at the interface of the primarygrains.

From the point of view of growth, a still supersaturated matrixsurrounds the nuclei just formed. That is, there is an initialgrowth transient, for each individual grain, in which the localequilibrium between the emerging crystal and the liquid aheadof the crystal–liquid interface is established (with compositionc∗), as shown schematically in Fig. 24. Afterwards, diffusionbecomes the controlling mechanism. During the growthtransient, the interplay between interface- and diffusion-limitedgrowth acts in a way to increase the importance of thelatter mechanism until it becomes preponderant. The simplestapproach is to assume that the transient is interface-controlled,whereas the second stage of individual grain growth, in theextended view, i.e. neglecting spatial impingement betweengrains, is diffusion-controlled [51].

Since both D and �GT in Eq. (5) change in the course of thetransformation, the interface-controlled growth rate is evaluatedby multiplying the growth rate obtained from Eq. (5) by thefactor ϕ[x(t)] from Eq. (8).

When solute fluxes through the interface are driven bysteady-state diffusion, the growth rate of an isolated growinggrain with radius R under spherical symmetry in this steady-state regime is given, for a binary system, by [43],

dR

dt= Ds f1

R{ϕ[x(t)]}2 (9)

where ϕ2 accounts for soft impingement (overlapping ofdiffusion fields).

In a multicomponent system, each of the elements hasits own diffusion coefficient, and consequently the differentelements will generate different concentration gradients. Thegeneral solution in this case has been reported [3]. In thefollowing, Ds will be considered as an effective diffusioncoefficient to deal with multicomponent systems.

As an example the primary nanocrystallization of aFINEMET alloy glass with composition Fe73.5Cu1Nb3Si17.5B5is discussed in terms of these considerations. The goal is toestablish the limits inherent to the approach to get estimatedvalues of both the interfacial energy and the apparent diffusioncoefficients, D and Ds , from experimental kinetic data.

The details of alloy preparation and characterization of thenucleation and growth processes are given elsewhere [43].


FgD

pDn

D

HdT(

ctσ

aahq

abitva

cffar

ig. 25. Selected values of the nucleation frequency (I ), interface controlledrowth rate (u), and effective diffusion coefficient (Ds) that fit experimentalSC data.

To describe the kinetics of the overall nanocrystallizationrocess, the temperature dependence of D(T ) or, equivalently,s(T ), is given by an Vogel–Fulcher relationship [52,53],

amely

(T ) = Do · T · exp{−B/(T − To)}. (10)

ere, Do, B and To are empirical fitting parameters used toescribe the sharp increase of diffusivity above glass transition.he ratio B/To is often used to differentiate between strongB/To ∼ 100) and fragile (B/To ∼ 5) glasses.

The adjustment of the general trend in the calorimetricurves depends mostly on the mechanisms involved in theransformation. The problem is that neither the values of

nor those of B and To (or, equivalently, the apparentctivation energy of D) are well known, and the temperaturest which transformation occurs, and their shift with changingeating rate, are very dependent on the precise values of theseuantities.

The modelling and calculation of the transformation rates a function of temperature at various heating rates haseen performed to determine the optimum range of values fornterfacial energy, D, and Ds , to get reasonable agreement withhe experimental continuous heating DSC curves. All otheralues used in the calculation have been maintained constantt the values already published [43].

Fig. 25 shows the calculated nucleation frequency, interfaceontrolled growth rate and effective diffusion coefficient as aunction of temperature for the two limiting values consideredor the interfacial energy. Agreement with experimental points,lso shown in this figure, may lead to a quite substantial widerange of values than those quoted, because of the uncertainty

Fig. 26. Comparison of the calculated and experimental temperaturedependence of the transformation rate, assuming that σs = 0.131 J m−2 andadjusting D and Ds values accordingly.

Fig. 27. Comparison of the calculated and experimental temperaturedependence of the transformation rate, assuming that σs = 0.136 J m−2 andadjusting D and Ds values accordingly.

inherent in the measurements of I and u at a given temperatureby transmission electron microscopy (TEM). In that respect,continuous heating DSC curves constitute a unique set ofkinetic data exploring a wide temperature range that, even withsome uncertainty, are accurate enough to fix the respectivevalues of σ , B and To required to get agreement of the modelledkinetic process.

Figs. 26 and 27 show the calculated transformation rate forthe two limiting values of σs = σ · [V 2

m · NA]−1/3, where Vm isthe molar volume and NA is Avogadro’s number. In each figure,the experimental transformation rate obtained by DSC is alsoincluded to allow easy comparison with calculated values forheating rates varying from 1.25 K min−1 to 80 K min−1.

As can be deduced from inspection of Fig. 26, when theinterfacial energy changes from 0.131 J m−2 to 0.136 J m−2,the homogeneous nucleation rate decreases substantially and,consequently, the values of u and Ds (or D) have to increase tocompensate such a decrease. At the same time, the apparent


activation energy of the overall primary nanocrystallizationhas to be maintained. This is the main reason for the narrowrange of deduced values for σ that are consistent with theexperimental transformation rates. Consequently, the indicatedmodelling procedure is quite suitable for obtaining indirectevaluation of the interfacial energy between the nanocrystals ofα-(Fe,Si) with DO3 structure embedded in a disordered matrixwith global Fe73.5Cu1Nb3Si17.5B5 composition.

It is worth mentioning that there is a spread in theexperimental values obtained, clearly seen in Figs. 26 and 27when several curves are shown for the kinetic transformationat a fixed heating rate. However, it is clear that the modellingis better at reproducing the earlier part of the transformationthan its end. To obtain better results in the later stages probablyrequires a more refined analysis of the impinging stage fordiffusion-controlled growth. To improve the analysis, reliablevalues of the supersaturation in the remaining matrix are alsoneeded. Furthermore, it is important to notice the limitationin the coupled derivation of parameters from the kinetic data;clearly, the ability to make predictions would greatly benefitfrom an independent determination of values for interfacialenergies as well as diffusion data.

3. Limitations to current applications — evaluatinginterfacial energies

The examples above show that progress in computationalthermodynamics is making it possible to solve materials-related problems with a complexity that could only be tackledconceptually a decade ago. However, as one proceeds toformulate and solve these problems, the limited extent of thedata that is available becomes evident. The last few decadeshave seen a dramatic evolution in the amount and quality ofassessed thermodynamic data that is included in the availabledatabases [21,54]. More recently, the same has happened withdiffusion data needed to address transformations in alloys [21,55]. Bulk thermodynamic and diffusion data are insufficient,however, to properly predict important microstructure features,as well as their evolution. In particular, interfacial andstrain energies are important in defining phase nucleationand morphology. The following section demonstrates howtools currently available in computational thermodynamics andrelated fields can contribute to the quantification of theseimportant variables.

3.1. Computation of interfacial energies using thermodynamic,first-principles and semi-empirical atomistic approaches

Nucleation kinetics is one of the key issues in the theory ofphase transformations where a proper quantitative approach islimited by the lack of data for interfacial energy between thematrix and precipitating phases. Recent progress in atomisticapproaches shows some feasibility of providing the necessaryinformation. In the present review, an attempt will be madeto compute the interfacial energy between γ (Ni-rich fcc) andγ ′ (L12Ni3Al) phases in the Ni–Al binary system using threedifferent approaches, and then to compare the results. Themethods used in the present work are:

(1) a classical thermodynamic approach;(2) a first-principles calculation; and(3) a semi-empirical atomistic approach based on the modified

embedded atom method (MEAM) interatomic potential.

3.1.1. Thermodynamic method for the calculation of interfacialenergies for γ /γ ′ phases in binary and multi-component nickel-base alloys

The overall value of any interfacial energy contains manycontributory factors, including:

(a) a chemical contribution;(b) a strain energy contribution;(c) the presence of dislocation arrays;(d) the presence of chemical segregation.

The γ /γ ′ phase transformation in nickel alloys exhibitsa small lattice misfit, which minimises any strain energycontribution. This, in turn, also minimises the presence ofinterface dislocations, which reduces the potential degree ofsegregation to the boundary. We can therefore expect that theinterfacial energy for this transformation in such systems willbe dominated by the chemical term.

The simplest thermodynamic approach for the calculation ofinterface energies follows the work of [42], who assumed that,in the case of liquid–solid interfaces, the value of σ for puremetals scales with the difference in enthalpy:

σ = α�Hm. (11)

This approach has also been used successfully in studiesrelated to the relative nucleation of competing phases from theliquid [56].

A theoretical justification for such an approach has beengiven by [57], which relates α to z∗ (the number of atomsper unit area of the interface), N∗ (the number of cross bondsper atom at the interface), and z (the co-ordination numberof nearest neighbours in the lattice, which equals 12 in FCCstructures). Eq. (11) then becomes:

σ = z∗N∗

z No�Hm (12)

where No is Avogadro’s number and �Hm is the molar enthalpyof solution of one mole of γ ′ in the γ matrix in equilibrium atthe coarsening temperature. A similar approach has been usedmore recently by [58].

Eq. (12) should be considered only a first approximation,with the following limitations. Firstly the derivation uses onlyfirst-nearest-neighbour energies, and secondly it considers thatthe particle is only bounded by {111} interfaces. The potentialimportance of second-nearest-neighbour interactions has beenstressed by [59], and related studies on anti-phase boundaryenergies have shown that third-neighbour energies may alsobe needed [60]. However, first-principles calculations indicatethat, for Ni3Al and Ni3Ti, the third-nearest-neighbour energy isabout 1/8th of the value of the magnitude of the first-nearest-neighbour energy. When combined with the relative numberof such neighbours, this much reduces the difference thatmight be expected from taking extra neighbours into account.


Table 4Comparison of calculated and experimental values for the {100} interfacialenergy of binary Al–Ni alloys

Temperature(K)

Orientation(hkl)

Surface energy(mJ m−2)

Method Reference

– 17–20 [70]– 42–80 [64]– 16 Nucleation [71]– 6–8 Coarsening [66]900/1000 {100} 14 Coarsening [72]1073 20–24 Calculated [62]

Table 5Coherent interface energy calculations for γ –gamma prime phases in nickel-base superalloys at 1073 K [62]

Alloy Temperature (K) Energy (mJ m−2)

Nimonic 105 1073 58Udimet 700 1073 71IN 738 1073 76Nimonic PK 33 1073 75Nimonic 80A 1073 61Nimonic 90 1073 72Nimonic PE 16 1073 91

Table 6Calculated effect of temperature on the interfacial energy of nickel basesuperalloys [62]

Alloy Temperature (K) Energy (mJ m−2)

Nimonic 115 1073 65Nimonic 115 1173 63Nimonic 115 1273 61Nimonic 115 1373 58Nimonic 115 1473 55

Objections can also be raised concerning the approximationsinvolved in calculating the relevant values of �H from theavailable thermodynamic data, especially in multi-componentsystems [61]. The method used to derive the calculated valuesquoted in this section [62] involves perturbing the temperatureby �T and calculating the driving force that now existsbetween the phases of the same composition that were inequilibrium at temperature T . The relevant �H is then obtainedfrom (dG/dT ) × T , which can be considered a reasonableapproximation if �T is small and should be at least as validas the methods used by [57] and [58].

Using Eq. (12) yields values for σ of 20–24 mJ m−2 forNi–Al binary alloys and 91 mJ m−2 for the multi-componentsuperalloy Nimonic PE16. These are consistent with the generalmagnitude of the values previous reported in the literature (seeTables 4 and 5). For further details regarding the superalloyslisted in Table 5 see [62], and for PE16, in particular, see [63].Note that it is one of the advantages of the the thermodynamicmethod that it can also calculate the expected magnitude of theeffect of temperature on the interfacial energy in such systems(Table 6).

It should be noted that the literature values for σ in binaryNi–Al alloys cover a surprising range (Tables 4–6). Most of

these have been obtained by back-calculation from coarseningrate experiments and it is therefore necessary to look in moredetail at the equations used to relate coarsening to the interfacialenergy. A general form of the coarsening equation is givenby [64]:

k =[

8DeffVmσ

9Gα′′m (Nβ − Nα)2

]1/3

(13)

where k is the rate constant, D is the effective diffusionconstant, Vm is the molar volume, Nα,β is the mole fractionsof the relevant phases, and Gα′′

m is the second differential ofthe free energy versus composition curve. For an ideal binarysolution, this can be simplified to:

k =[

8Deffσ Nα(1 − Nα)Vm

9(Nβ − Nα)2 RT

]1/3

. (14)

If it is assumed that the solubility limit is very low, the equationcan be simplified further:

k =[

8Deffσ NαVm

9RT

]1/3

. (15)

Many authors have used Eq. (15) despite the fact that a systemsuch as Ni–Al exhibits appreciable solid solubility and is farfrom ideal. The use of Eq. (15) has been justified by [64]on the grounds that initial calculations of Gα′′

m in the Ni–Alsystem (based on the thermodynamic data of [65]) appearedto indicate a low deviation from ideal behaviour in nickel-richNi–Al alloys. It was also considered that error bars in the valuesof the other input parameters did not justify being over-preciseas far as phase diagram data is concerned. This was possiblyjustified at that time, but more accurate free energies are nowavailable. It is therefore interesting to note that calculatedvalues of Gα

m using current characterizations of this systemindicate appreciable deviations from ideality, so it is preferableto use Eq. (13).

Values of Gαm for binary alloys of Ni–Al and Ni–Ti that have

been derived [62] using the Thermotech Nickel database areindicated in Fig. 28. It is obvious that values of σ calculatedby Eq. (14) or Eq. (15) will be reduced in proportion to thevalue of Gα

m , which could be one of the reasons for the widespread of values obtained in the literature (especially the verylow values quoted by [66]). For further self-consistency, thecalculated values of Nα and Nβ should come from the samedata set used for the determination of �Hm.

For multi-component alloys, it is not so easy to calculateGα

m , and so the compromise procedure adopted here is to usean average value of Gα

m from Fig. 28, assuming that the sum of(Al + Ti) in the alloy is the most important controlling factor.Clearly, the accuracy of any calculated value of σ will alsodepend on using a self-consistent set of diffusion coefficientsand molar volumes for the other parameters in Eqs. (13)–(15).As a further exercise in self-consistency, it is possible toplot the derived values of σ against the lattice mismatch atthe appropriate temperature, as programmes such as JMatProcan provide integrated access to all the required properties,


Fig. 28. Ratio of calculated non-ideal value of Gα′′m to ideal value of Gα′′

m , forNi–Al and Ni–Ti binary alloys.

including their temperature dependence. For a more detailedanalysis, see [62].

The above results should be set in the context ofother interfacial energies, especially for interfaces involvingcarbides in steels, which exhibit considerably higher values(250–550 mJ m−1) [67,68]. This is not surprising, as thesevalues also contain contributions from a higher misfit energyas well as grain-boundary dislocations. As a general approachto calculating the energy of complex interfaces is still in itsinfancy, such energies tend to have large error bars and are stillmostly obtained by fitting procedure [69].

3.1.2. First-principles calculations for the interfacial energiesbetween γ ′-Ni3Al and pure Ni

The interfacial energies between γ ′-Ni3Al and γ -Ni at zeroKelvin have been calculated recently [73,74]. These authorsused the Vienna ab initio simulation package (VASP) [75]with ultrasoft pseudopotentials and the generalized gradientapproximation (GGA) [76]. Energy cutoff was determinedby the choice of “high accuracy” in the VASP. The spin

Fig. 29. Supercell used to model the 100 interfacial boundary between γ ′-Ni3Al and γ -Ni.

Fig. 30. Supercell used to model the 110 interfacial boundary between γ ′-Ni3Al and γ -Ni.

polarization calculation adopted in the present work means thatthe magnetization is considered.

Two types of interfaces were considered: (001) with contains4 cubic L12 γ ′-Ni3Al unit cells and 4 cubic fcc γ -Ni cell(Fig. 29), and (011) with 4 L12 γ ′-Ni3Al unit cells and 4 fccγ -Ni cell cut in the [011] direction (Fig. 30). The 12 × 12 × 6and 8 × 12 × 8 Monkhost k points are used for (001) and(011) interfaces, respectively, to provide a higher accuracy inthe [001] direction.

There are two steps in calculating the interfacial energies asfollows:

(1) Calculate the total energy of the supercell with full atomicrelaxations, Etot(a, b, c), with a, b, and c representing therelaxed lattice parameters and the interface is coincidentwith b and c.

(2) Using a supercell with the same size as step 1, calculate thetotal energies for the single γ ′ or γ phase using the fixedlattice parameters b and c derived in step 1 but allowing ato be relaxed. Their total energies are denoted by Eγ ′(a)

and Eγ (a), respectively.

The interface energy then can be calculated as:

σ ={

Etot(a, b, c) − 1

2

[Eγ ′(a) + Eγ (a)

]}/2S (16)

where S represents the area of the interface.The calculated energies are summarized in Table 7. Our

calculated interfacial energies are 39.6 mJ m−2 for the (001)interface and 63.8 mJ m−2 for the (011) interface. Wolvertonand Zunger [77] calculated the formation energy of Ni7Al in theD7 structure that can be viewed as alternating planes of Ni3Aland Ni4 in the [111] direction. By this calculation, they gave anestimated energy of 33 mJ m−2 that is considerably lower thanthe value of 63 mJ m−2 for the (100) direction by Price andCooper [78] calculated using the LMTO method.


Table 7

The calculated interfacial energy between γ ′-Ni3Al and γ -Ni (mJ m−2)

Source {100} {110}This work 39.6 63.8Price and Cooper [78] 63

Table 8Calculated enthalpy of formation, �H f (J/mol), lattice parameter, a (A), bulkmodulus, B (GPa) and elastic constants, C11, C12, C44 etc. (GPa) of L12Ni3Aland B2 NiAl compounds using the present MEAM interatomic potential, incomparison with experimental data

Property Ni3Al NiAlMEAM Experimental MEAM Experimental

�H f 42 100 42 100a * 65 300 64 000–66 100a

*a 3.567 3.567b * 2.866 2.886b

B 181 177c 160 158c

C11 254 230c 200 199c *C12 145 150c 140 137c *C44 136 131c 117 116c

The property values that were used for parameter optimization are marked with“*”.

a [82].b [83].c [84].

3.1.3. Calculation of interfacial energies between γ ′-Ni3Al andγ -Ni solid solutions

An attempt to compute the interfacial energy betweenγ ′-Ni3Al and γ -Ni has now been made using an atomisticapproach (molecular statics) based on a semi-empiricalinteratomic potential. The interatomic potential used wasthe second-nearest-neighbor modified embedded atom method(2NN MEAM) potential [79,80]. The MEAM for an alloysystem is based on the MEAM potentials for composingelements. In the present work, the (2NN) MEAM parametersfor Al and Ni were taken from Lee et al. [81] without anymodification. In the MEAM [81], the values of nine modelparameters should be determined to describe a binary alloysystem. Out of the nine model parameters, the followingthree, Ec, re, B , are the most decisive. The Ec, re andB parameters give effect to enthalpies of formation, latticeparameters and bulk modulii of individual phases in the relevantalloy system, respectively. Ideally, it is expected that thismethod should reproduce all the known physical propertiesof individual phases using one set of model parameters. Inpractice, some phases with relatively simple atomic structuresare selected, and the parameter values are determined by fittingto the known physical properties of the selected phases. Inthe present study, for the Al–Ni alloys system, the B2-orderedNiAl compound and the L12-ordered Ni3Al compound wereselected, and experimental data on the enthalpies of formation,lattice parameters, and elastic constants for the two phases wereused to determine the model parameter values. In Table 8, theproperty values for NiAl and Ni3Al phases calculated using thepresent MEAM potential are compared with experimental data.The property values that were used for parameter optimizationare marked with a “*”.

Table 9Calculated (100) interfacial energy at 0 K between Ni3Al and γ -Ni with variousAl contents (mJ m−2)

at.% Al in γ -Ni (100) interfacial energy

0 −435 −2

10 3415 57

For the calculation of interfacial energy, a supercellcomposed of equal amounts of γ -Ni and Ni3Al and involvingan interface between the two phases at a given orientationis prepared, as has been shown in Figs. 29 and 30. Whenthe γ -Ni is not pure Ni, in order to minimize the statisticalerror, a large size of supercell (40 × 20 × 20 unit cells) isused. Then, the internal energy of the supercell is calculatedallowing atomic relaxation. This energy is compared with theaverage energies of γ -Ni and Ni3Al samples of the same size,calculated by maintaining the same lattice parameters withthe supercell in the two directions parallel to the interfacebut allowing relaxation into the direction perpendicular to theinterface. To remove any surface effects, a three-dimensionalperiodic boundary condition is applied in all calculations. Dueto the periodic boundary condition, it should be regarded thatthe supercell shown in Figs. 29 and 30 involves two interfaces.The interfacial energy can then be obtained by the followingequation:

σ ={

Esupercell − 1

2

[ENi + ENi3Al

]}/2A (17)

where A represents the area of the interface. It was foundthat the interfacial energy of the (100) interface calculated inthis way strongly depends on the composition of γ -Ni (seeTable 9). The individual calculated values in this Table areaverage values of ten calculations using differently generatedrandom solutions of the fcc γ -Ni phase. The negative valuecalculated for the interfacial energy between pure nickel andNi3Al is another way of expressing that, in the presence ofpure nickel, Ni3Al would dissolve to form a solid solution oflower energy than the two-phase mixture. The Al content ofγ -Ni in equilibrium with Ni3Al at 1073 K is about 15 at.%.Therefore, if the temperature dependence of the interfacialenthalpy can be assumed to be small, then the calculated valuefor the Ni–15%Al can be regarded to be close to the interfacialenthalpy between Ni3Al and γ -Ni at 1073 K. It should be notedhere that the quantity computed in this study is the interfacialenthalpy not the free energy. Because the interfacial entropywould have positive values, the calculated values in Table 9should be regarded as upper limits of interfacial energies atindividual temperatures where the equilibrium Al contentsin γ -Ni with Ni3Al correspond to the given Al percentagevalues.

3.1.4. Concluding remarks on calculation of interfacialenergies

All three approaches described here yield low interfacialenergy values (<100 mJ m−2) for the interfaces between


γ ′-Ni3Al and γ -Ni, in qualitative agreement with experimentalinformation (6–80 mJ m−2). In a situation where accurateexperimental data are not available and current first-principlescalculations can show a scatter of more than 50%, it doesnot seem reasonable to compare the accuracy of individualapproaches. However, each method has its own definiteadvantages. The thermodynamic approach can provide agood estimation for wide range of multi-component materialssystems. The semi-empirical approach can consider anisotropyof interfaces, misfit strain effects, segregation effects, as wellas the chemical effects, and can be used for simulations of theearly stages of nucleation. As the most fundamental approach,the first-principles calculation [85] should ultimately provide areference value for the other approaches.

4. Conclusions

The examples presented in this paper show that computa-tional thermodynamics has evolved to a point where complexmaterial thermodynamics problems can be solved with relativeease. Furthermore, the coupling of diffusion modeling with theconventional computational thermodynamics is making possi-ble the solution of many problems that are important to the un-derstanding of phase evolution in materials. One can foresee theavailability of tools that efficiently predict microstructure with aminimum of guesswork and fitting constants, but two importantquestions must be considered. Firstly, although computationalthermodynamics is now a mature discipline, its disseminationas an everyday tool to the materials engineer has been slow.One possible reason is the fact that a significant part of thesyllabusi for thermodynamics courses in engineering educationtoday still focus on problem-solving techniques that pre-datecomputational thermodynamics and are self-limiting in the ap-plications that can then be handled. One possible way of con-tributing to the reduction of this gap would be a discussion ofhow to change engineering curricula in order to include morefamiliarity with the modern calculation methods now available,without neglecting the essential axioms of thermodynamics.These clearly remain unchanged, regardless of the calculationtools available.

A second important observation relates to the need fordata. Thermodynamic methods rely on data. As computationalthermodynamics evolves from “bulk thermodynamics” to theprediction of microstructure and its evolution, more extensivedata on important properties such as interfacial and strainenergies will be necessary. While proven approaches forderiving these values from the application of kinetic theories toexperimental data might continue to be used, it will seriouslylimit the ability to make predictions and extrapolations, oneof the strengths of computational thermodynamics. Moreadvanced calculation techniques, as discussed in the last sectionof this paper, might be the only solution to derive the necessaryproperties and also avoid interference by other unforseenfactors.

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