Assessment and Evaluation of Mobilities for Diffusion in the bcc Cr-Mo-Fe System Greta Lindwall and Karin Frisk

JPEDAV (2012) 33:375 389 DOI: 10.1007/s11669-012-0107-z 1547-7037 ÓASM International Assessment and Evaluation of Mobilities for Diffusion in the bcc Cr-Mo-Fe System Greta Lindwall and Karin Frisk (Submitted May 15, 2012; in revised form July 23, 2012) An assessment of the diffusion mobility functions for diffusion in the bcc Cr-Fe-Mo system is presented. The optimization of the mobility parameters is performed by utilizing experimental diffusion data available in literature. New diffusion data for the Mo diffusion is produced by a diffusion couple experiment and is accounted for in the optimization. Agreement between calculated and measured diffusion coefficients is found. An M 6 C coarsening experiment is performed and the measured coarsening rate can be reproduced by diffusion calculation using the DICTRA software and the developed kinetic description. The mobility parameters are shown to have a strong influence on the calculated coarsening rate. Keywords 1. Introduction assessment, coarsening, DICTRA modeling, Mo carbides, multicomponent diffusion, tool steels During tool steel processing and for tool steels in service, multicomponent diffusion is often a decisive factor for the microstructure evolution. In order to understand and predict how the microstructure will develop; e.g. during austenitizing when primary precipitates dissolve or during tempering when secondary precipitates form and coarsen, it is important to have information about the diffusion characteristics of the steel of question. A way to predict the behaviour of the microstructure during processing is by kinetic calculations, and for these multicomponent kinetic descriptions of the participating elements are needed. The DICTRA software [1] is a software for multicomponent diffusional reactions. It accounts for the multicomponent effects in the system by coupling thermodynamic data with kinetic data stored in databases. The accuracy of the simulation outcome is strongly dependent on the quality of the databases. In the case of tool steels a lot of work has been put into the development of the thermodynamic databases, see e.g., [2] whereas development of kinetic databases for typical tool steel systems is limited. However, some reported assessments exist; Jönsson [3] performed assessments for the diffusion of several elements in both the fcc and the bcc phases with focus on iron based systems, Bratberg et al. [4] revised the kinetic description of Cr and V diffusion in the fcc phase and Lindwall and Frisk [5] assessed parameters for the diffusion in the bcc V-Cr-Fe system. The latter work focused on the V diffusion. Vanadium is the main precipitate Greta Lindwall and Karin Frisk, Swerea KIMAB, P.O. Box 7047 16407 Kista, Sweden. Contact e-mail: greta.lindwall@swerea.se. forming element in many tool steel grades, and the V diffusion has, therefore, an important influence on the precipitation kinetics. For the same reason is the diffusion of Mo of importance. For example, the M 6 C carbide is rich in Mo as well as the cubic MC carbide and the hexagonal M 2 C carbide. All these carbides are precipitate phases typically present in tool steels. Another alloying element common for tool steel is Cr which may take part in carbide formation but also dissolves to a large extent in the matrix. Consequently, a proper kinetic description of the bcc Mo-Cr-Fe system is expected to be valuable for accurate calculation of the precipitation kinetics. Some experimental diffusion data for the Mo diffusion is available in literature but, to the authors knowledge, there are no assessments published. In addition, the need of an assessed kinetic description in the case of Mo diffusion in the bcc phase has been pointed out in the case of creep resistant Cr steels. [6,7] The aim of the current work was to assess the diffusion parameters important for the bcc Mo-Cr-Fe system with the focus on the Mo diffusion. Experimental diffusion data found in literature was accounted for. New diffusion data was produced by a diffusion couple experiment and used for parameter optimization. The assessed parameters were validated by comparing calculated diffusion coefficients with measured coefficients. Further, a coarsening experiment was performed and the measured M 6 C coarsening rate was compared with a DICTRA simulation utilizing the new kinetic description. 2. Modelling 2.1 Diffusion Mobility Modelling A mathematical treatment of a diffusion phenomenon in a multicomponent alloy results in a system of coupled partial differential equations. To arrive there, the continuity equation is combined with the Onsager extension of Fick s second law of diffusion, [8-11] Journal of Phase Equilibria and Diffusion Vol. 33 No. 5 2012 375

Section I: Basic and Applied Research J k ¼ Xn 1 i¼1 D n @c j kj @z ; ðeq 1Þ where J k is the flux of component k through a unit area per unit time, c j is the concentration of component j, z is the space coordinate and D n kj is the diffusion coefficient matrix. In the volume fixed frame of reference the diffusion coefficient matrix, D n kj, is expressed as D n kj ¼ D kj D kn ðeq 2Þ when j is a substitutional component and as D n kj ¼ D kj ðeq 3Þ when j is an interstitial component. It is assumed that all the substiutionals have equal partial molar volumes and that only substitutional components have a volume contribution. [12] The diffusivities, D kj, in turn, are products of two factors, D kj ¼ Xn @l L i ki ; ðeq 4Þ @c i¼1 j one pure thermodynamic, @l i =@c j, and one kinetic, Lki. The kinetic term can be viewed as a proportionality factor dependent on the atomic mobility, M, of each element. [12] In a kinetic database, necessary for most diffusion simulations, expressions for the mobilities are stored instead of expressions for the diffusivities, as proposed by Andersson and Ågren. [12] The atomic mobility of a component, B, in a given phase can be separated into a frequency part, MB 0, and in an activation term, Q B. [12] M B ¼ MB 0 exp Q B 1 RT RT Xmag ; ðeq 5Þ where R is the molar gas constant and T is the temperature. The ferromagnetic transition is accounted for by the factor X mag which is a function of the alloy composition. [13,14] The frequency part and the activation energy part are generally dependent on temperature, pressure and composition where the composition dependency, when utilizing the CALPHAD method, can be expanded as a Redlich-Kister polynomial, [12] U B ¼ X x i U i B þ X " # X X m x i x r j U i;j B ðx i x j Þ r : ðeq 6Þ i i j>i r¼0 Here U B represents H B ¼ lnðmb 0Þ or Q B and x i and x j are the mole fraction of the component i and j, respectively. The parameters U i B and r U i;j B are the quantities stored in the kinetic database and are generally functions linearly dependent on temperature. Values for them are extracted from experimental information by optimization procedures. Depending on the level of ambition and the amount of experimental information available for the system of question, the functions may look quite different. It should be noted, that if all U i B parameters have the same value and all r U i;j B parameters are zero, U B and the corresponding mobility for component B, are not composition dependent. If the binary interaction parameters of first order are included, Eq 6 takes the following form in the case of the ternary system, Cr-Fe-Mo; U B ¼ x Cr U Cr B þ x FeU Fe B þ x MoU Mo B þ x Crx 0 Fe U Cr;Fe B þ x Cr x Mo 0 U Cr;Mo B þ x Fe x Mo 0 U Fe;Mo B ðeq 7Þ where B is either Cr, Fe or Mo. In the current work, values for all end-members of zeroth order U Cr B ; UFe B ; UMo B which had not already been evaluated by Jönsson, [3] were assessed. For the cases where sufficient experimental data in literature existed, some first order parameters ( 0 U Fe;Mo Mo and 0 U Fe;Mo Fe ) are also assessed. The first order parameter, 0 U Cr;Fe Mo, was assessed by accounting for experimental information obtained by a diffusion couple experiment. The remaining first order parameters ( 0 U Cr;Mo Mo ; 0 U Cr;Mo Cr ; 0 U Fe;Mo Cr and 0 U Cr;Mo Fe ) were not assessed and are, therefore, set to zero. The parameters r U Cr;Fe Fe and r U Cr;Fe Cr had been assessed earlier by Jönsson. [3] Most of the assessed parameters were obtained by using the optimization routine included in the DICTRA package [15] which makes it possible to account for experimental information in the form of diffusion coefficient data; i.e. tracer or self-diffusion coefficient data, intrinsic diffusion coefficient data and chemical diffusion coefficient data. An optimization starts with an initial calculation of the diffusion coefficients using estimates for the mobility parameters. The calculated coefficients are then compared to the experimental sets of data and based on this comparison the mobilities are optimized by a least squares error method to obtain best agreement possible. When there exists several sets of experimental data for the same diffusion coefficients, they may be given different weight depending on the uncertainty associated with the experiment and thus, scale their contribution during the optimization. In order to account for the diffusion data produced by the diffusion couple experiment an optimization script developed by Höglund [16] was used. It makes it possible to use Matlab in conjunction with DICTRA via the Thermo-Calc- Matlab Toolbox. [17] By comparing the calculated concentration profiles with measured profiles after heat treatment of a diffusion couple, mobility parameter values can be extracted. The same method has been utilized by Campbell [18] when developing a diffusion mobility database for Ni base superalloys. Instead of a Matlab code, they used Mathematica and Python scripts in conjunction with DICTRA. Assessed mobility parameters implemented in a diffusion mobility database need to be validated in order to determine their capability to describe various diffusion phenomena when utilized in simulation. Campbell et al., [19-21] while developing of a 10-component Ni base superalloy database, validated them by comparing activation energies with diffusion correlation published in literature, [22] by comparing calculated ternary and quaternary diffusion coefficients with measured values, [19] and by comparing calculations with single-phase [20] and multiphase [21] Ni base superalloy 376 Journal of Phase Equilibria and Diffusion Vol. 33 No. 5 2012

diffusion couple experiments. Comparisons of experimentally measured precipitate coarsening rates with simulated rates have also been used to validate mobility parameters. [4,5] In the current work the assessed parameters were validated by comparing calculated ternary diffusion coefficients with experimentally measured coefficients and by comparing a calculated coarsening rate of a Mo rich M 6 C carbide with its measured coarsening rate. For the latter comparison a coarsening experiment was performed. 2.2 DICTRA models A system of coupled differential equations describing a diffusional behaviour in a multicomponent alloy can in rare cases be solved analytically. For most cases, however, numerical methods are necessary and the DICTRA software [1] is aimed for such cases. The software includes different models of which the single-phase model and the coarsening model were applied in the current work. For simulation of the concentration profile evolution in the case of interdiffusion in a diffusion couple where both couple parts have the same phase structure, the singlephase model is appropriate. This is the most basic model in DICTRA and the continuity equation is solved numerically by a method based on the Galerkin method. [23,24] Oswald ripening of precipitates [25] is described by the coarsening model in DICTRA. [26] It is based on the Lifshitz, Slyozov and Wagner theory [27,28] and assumes that the maximum particle in precipitate distribution is approximately 1.5 times the average sized particle. The driving force for the coarsening is assumed to be a contribution, DG m, to the Gibbs free energy function. It is dependent on the interfacial energy, r, and is given by DG m ¼ 2rV m ; ðeq 8Þ r where V m is the molar volume and r is the radius of the precipitate phase. The model assumes that the calculations can be performed on one spherical precipitate of maximum size embedded in a spherical matrix cell and that local equilibrium holds at the phase boundary between the particle and the matrix taking the interfacial energy into account. At the outer cell boundary the composition is in equilibrium with an average sized particle see Fig. 1. Since the particle of maximal size will have a smaller energy contribution, see Eq 8, a shift in the equilibrium occurs and a concentration gradient in the matrix cell is generated in such a way that the maximum sized particle coarsens. Simultaneously, the matrix cell grows in order to maintain the total composition of the cell. To set up a coarsening simulation some initial conditions are needed; i.e. the chemical equilibrium compositions of the phases which can be obtained by thermodynamic calculations and a value for the initial size of the precipitate. The initial precipitate size determines the initial size of the matrix cell. In addition, a value for the interfacial energy is needed which should be between 0.1 and 1.0 J/m 2. Fig. 1 A schematic sketch of the DICTRA coarsening model. At the interface between the particle and the matrix the composition is obtained by assuming local equilibrium modified by the addition of DG m (r max ) to the Gibbs free energy function of the particle. At the outer cell boundary the composition is obtained by assuming local equilibrium modified by (DG m (Æræ)) 3. Assessments 3.1 The Fe-Mo System The self-diffusivity of Fe in bcc Fe has been studied extensively and an accurate assessment has been performed decades ago. [29] Therefore, in the case of the binary bcc Mo- Fe system, the parameters which remained to be assessed were the self-diffusivity of Mo in bcc Mo, the diffusivity of Mo in bcc Fe and the diffusivity of Fe in bcc Mo. For the self-diffusion coefficient of Mo in bcc Mo a number of measurements have been reported [30-35] of which some [30-34] has been reviewed by Maier et al. [35] Due to considerable discrepancies between the measurements and to the fairly limited temperature intervals studied, Maier et al. [35] performed additional investigations of the radioactive isotope 99 Mo in single crystal specimens over the temperature range 1360-2773 K. They concluded that their data was in agreement with the data reported by Borisov et al. [30] and Pavlinov and Bykov, [34] but found poor agreement with the data reported by Danneberg and Krautz [32] and by Askill and Tomlin. [33] In Fig. 2 the temperature dependence of the measured diffusion coefficients (symbols and dashed lines) is shown together with the calculated diffusion coefficients (solid line) after the current assessment. The experimental data reported by Maier et al. [35] were given the highest weight in the assessment. Nitta et al. [36] have reviewed published data concerning the tracer diffusion of Mo in bcc Fe, and found it necessary to re-determine the coefficient in the temperature interval 833-1163 K; i.e. the whole temperature interval of a-fe across the Curie temperature. By the use of the sputtermicrosectioning technique, they studied the 99 Mo isotope and concluded that its diffusion coefficient was smaller, in Journal of Phase Equilibria and Diffusion Vol. 33 No. 5 2012 377

Section I: Basic and Applied Research Fig. 2 The logarithm of the tracer coefficient of Mo in a bcc Mo matrix (solid line) as a function of temperature in comparison with measured coefficients (symbols and dashed lines) Fig. 3 The logarithm of the tracer coefficient of Mo in a bcc Fe matrix (solid line) as a function of temperature in comparison with measured coefficients (symbols and dashed lines) the temperature interval of question, than those obtained by Borisov et al., [37] Kucera et al. [38] and Alberry and Haworth [39] which had used conventional mechanical sectioning techniques to determine the penetration profile of the diffusing isotope. The data by Nitta et al. [36] was accounted for in the assessment of the Mo mobility in bcc Fe and the resulting calculated diffusion coefficient as a function of temperature is shown in Fig. 3 in comparison to the experimental measured coefficients. The higher temperature data points by Alberry and Haworth [39] are compatible with the data by Nitta et al. and were also included in the assessment. For the tracer diffusion coefficient of Fe diffusion in bcc Mo two sets of experimental data were considered, [40,41] which both concerned the 59 Fe isotope and covered the temperature interval 1173-1473 and 1273-1623 K. Due to the spread in the results reported by Lesage and Huntz [41] and the uncertainty whether diffusion along high diffusivity paths had contributed to the measured volume diffusion coefficient or not, the data by Nohara and Hirano [40] was given the decisive weight in the assessment. In Fig. 4 the calculated diffusion coefficient (solid line) is shown in comparison with the experimental measurements (symbols). All the assessed parameters are listed in Table 1. Two experimental studies on diffusion of Mo and Fe in bcc Mo-Fe alloys have also been reported [42,43] and hence, it was possible to assess values for Mo-Fe interactions parameters. In one of the works, by Nitta et al., [42] the tracer diffusion coefficients of the 59 Fe and the 99 Mo isotope in high-purity Fe-Mo alloys, containing 0.4 at.% Mo and 1.5 at.% Mo have been determined in the temperature interval 823-1173 K. There, they used the sputtermicrosectioning technique and found an increase of the Fe Fig. 4 The logarithm of the tracer coefficient of Fe in a bcc Mo matrix (solid line) as a function of temperature in comparison with measured coefficients (symbols and dashed lines) diffusion with increasing Mo content whereas the diffusion coefficient of Mo decreased with increasing solute content. The work by Nohara and Hirano [43] concerned the interdiffusion in the Fe-Mo system in the temperature range 1073-1673 K. They used diffusion couples and Matano analysis to determine the chemical interdiffusion coefficient and 378 Journal of Phase Equilibria and Diffusion Vol. 33 No. 5 2012

Table 1 Assessment results for the diffusion in the bcc Fe-Mo system Parameter Value References Mobility of Mo Q Mo H Mo Q Fe H Fe 0 Q Fe;Mo Mo +491311 This work Mo 58.707/R This work Mo +214657 This work Mo 83.279/R This work Mo 272835 This work 0 H Fe;Mo Mo 306.487/R This work Mobility of Fe Q Mo Fe +347797 This work H Mo Q Fe H Fe 0 Q Fe;Mo Fe 91.614/R This work Fe +218000 [3] Fe 83.036/R [3] Fe 112767 This work 0 H Fe;Mo Fe +37.942/R This work found it to decrease with increasing Mo content at each studied temperature. Assessment, where both data sets were given equal weight, resulted in the interaction parameters listed in Table 1. The temperature dependencies of the calculated tracer coefficients (solid lines) are compared to the measured coefficients (symbols) in Fig. 5 and 6; for the 0.4 at.% Mo-Fe alloy (Fig. 5a, 6a) and for the 1.5 at.% Mo-Fe alloy (Fig. 5b, 6b), respectively. The calculated chemical interdiffusion coefficients (solid lines), at different temperatures, as a function of Mo content are shown in Fig. 7 in comparison to measured coefficients (symbols). 3.2 The Cr-Mo System In the case of the binary Cr-Mo system the diffusion data reported in literature is sparse. The work by Gruzin and Zemskii, [44] who had studied the temperature dependence of the diffusion of the 99 Mo isotope in bcc Cr by the method of stripping layers and integral radioactivity measurements in the temperature interval 1373-1693 K, was used to assess values of the mobility parameters for Mo diffusion, see Table 2 and Fig. 8(a). For the mobility of Cr in bcc Mo two works [45,46] are considered. Mulyakaev et al. [46] studied the isotope 51 Cr diffusion in single crystal bcc Mo samples for the temperature interval from 1273 up to 1423 K. This data was given the dominant weight in the assessment. The data of Borisov et al. [45] were given a low weight due to the lack of experimental description. However, at higher temperatures, both data sets are compatible, see Fig. 8(b). The assessed parameters are listed in Table 2. The mobility parameters for the mobility of Cr in a bcc Cr matrix have been critically assessed by Jönsson [29] and are not reassessed here. 3.3 The Cr-Fe-Mo System Since accurate calculation of the coarsening rates for coarsening of Mo rich precipitates, such as the M 6 C carbide, was an aim of the current work, the Mo diffusion in the ternary bcc Cr-Fe system was also evaluated. The decision to focus on the Mo diffusion is based on the multicomponent coarsening approximation [47] which states that the elements that have the combination of the largest difference in composition between the precipitate and the matrix and lowest diffusion coefficient control the coarsening rate. Fig. 5 (a) The logarithm of the tracer coefficient of Mo in a bcc 0.4 at.%mo-fe matrix. (b) The logarithm of the tracer coefficient of Mo in a bcc 1.5 at.%mo-fe matrix Journal of Phase Equilibria and Diffusion Vol. 33 No. 5 2012 379

Section I: Basic and Applied Research Fig. 6 (a) The logarithm of the tracer coefficient of Fe in a bcc 0.4 at.% Mo-Fe matrix. (b) The logarithm of the tracer coefficient of Fe in a bcc 1.5 at.% Mo-Fe matrix Table 2 Assessment results for the diffusion in the bcc Cr-Fe-Mo system Parameter Value Reference Mobility of Mo Q Cr H Cr 0 Q Cr;Fe Mo +242625 This work Mo 125.8/R This work Mo 161371 This work 0 H Cr;Fe Mo 0 This work Mobility of Cr Q Cr Cr +407000 [3] H Cr Q Mo H Mo Cr 35.6/R [3] Cr +289338 This work Cr 107.8/R This work Fig. 7 The logarithm of the chemical diffusion coefficient of Mo in a bcc Mo-Fe alloy (solid lines) as a function of Mo concentration for seven different temperatures in comparison with experimental measurements (symbols) In the case of the bcc C-Cr-Fe-Mo system and the M 6 C carbide, this element is Mo. Due to the important influence of this parameter on coarsening calculations and due to limited amount of experimental diffusion coefficient data available in the literature, new diffusion data was produced by a diffusion couple experiment. For the diffusion couple, two model materials were fabricated. The diffusion of Mo in the bcc Fe matrix in the presence of Cr was in focus and hence, the model alloys were designed to have a ferritic matrix at 1423 K, alloyed with Cr and Mo. Two different Mo contents were selected for the couple parts in order to cause interdiffusion of Mo. In Table 3, the model alloy compositions are listed. The same strategy has been applied by Lindwall et al. [48] when investigating the diffusion of V in a bcc Fe matrix alloyed with Co. The model alloys were produced by casting; Fe (99.98% purity, metals basis), Cr (electrolytic metal) and Mo (99.8% purity) were weighed into a total weight of 50 g and placed in an Al oxide crucible. Induction heating of the elements until melting was performed in vacuum in an electric furnace followed by casting in cold Cu crucible into 10 mm diameter bars. First, the couple parts were homogenized approximately 4 h at 1423 K. The homogenized alloys were 380 Journal of Phase Equilibria and Diffusion Vol. 33 No. 5 2012

Fig. 8 (a) The logarithm of the tracer coefficient of Mo in a bcc Cr matrix as a function of temperature in comparison with measured coefficients (symbols). (b) The logarithm of the tracer coefficient of Cr in a bcc Mo matrix as a function of temperature in comparison with measured coefficients (symbols) Table 3 in wt.% Compositions of the diffusion couple parts Part 1 wt.% Part 2 wt.% Mo 0.26 1.55 Cr 14.1 14.0 Fe Balance Balance Fig. 10 The measured Mo concentration profiles in comparison to the calculated Mo profile applying the kinetic description including assessed binary mobility parameters for the bcc Mo-Fe system and the bcc Cr-Mo system Fig. 9 Micrograph of the cross-section of the interdiffusion region of the diffusion couple then cut into circular disks and polished. For the heat treatment of the diffusion couple a high speed quenching dilatometer was used [49] which allows for fast and controlled heating to the selected heat treatment temperature and fast and controlled cooling with nitrogen gas. The dilatometer setup made it possible to press the two couple parts together and the applied stress was sufficient to keep the parts close enough so that interdiffusion occurred. The compressive stress was, however, too low for any influence Journal of Phase Equilibria and Diffusion Vol. 33 No. 5 2012 381

Section I: Basic and Applied Research on the diffusivity to be expected. The holding time at the heat treatment temperature (1423 K) was 10 h. The heat treated diffusion couple was cut into halves for exposure of the interdiffusion region. The surface was prepared for analysis in a JEOL 7000F scanning electron microscope by grinding and polishing. Measurements of the elemental concentration profiles over the interface were performed by combined EDS (Fe and Cr) and WDS (Mo) analysis. Fig. 11 The measured Mo concentration profiles in comparison to the calculated Mo profile after optimization of a mobility parameter for Mo diffusion in the bcc Cr-Fe system A micrograph of a cross-section is shown in Fig. 9. Some precipitates have formed and can be seen in the microstructure. The fraction of precipitates is small and they are sparsely distributed and hence, the position for the line scans could be chosen so that the measurements were predominantly performed in the matrix. In Fig. 10 the measured Mo concentration profiles in wt.% is shown (symbols) as a function of distance. The measurements showed some scattering, especially on the Mo rich side, but was judged to be within acceptable limits (±0.10 wt.%) regarding WDS measurements. The measured Mo concentration profile is compared to the calculated profile (solid line) in Fig. 10. The kinetic description used for the calculations includes the assessments of the Fe-Mo system and the Cr-Mo system, see previous sections. From the comparison it was evident that the calculation did not reproduce the experimental results and indicated a need of a further assessment of the mobility function for the Mo diffusion in the bcc Fe matrix in the presence of Cr. In order to make use of the diffusion information obtained in the diffusion couple experiment the Thermo- Calc-Matlab Toolbox script by Höglund was used. [16] Since the diffusion couple was designed to evaluate the interdiffusion of Mo and since the diffusion of Fe, both in the bcc Cr-Fe system [3] and in the bcc Fe-Mo system (current work), had been accounted for, the critical parameter left to assess was the mobility of Mo in a bcc Cr-Fe matrix. The optimization procedure can be described as follows; an initial concentration profile is calculated with DICTRA for the given temperature and time conditions using the initial kinetic description and a guessed initial value for the Fig. 12 The calculated tracer diffusion coefficient of Mo in the bcc Cr-Fe system for different Cr contents as a function of temperature compared to the measured coefficients [50] (a) before the assessments and (b) after the assessments of the bcc Cr-Fe-Mo system. A K value is added to the coefficients in order to simplify comparisons between experiments and calculations 382 Journal of Phase Equilibria and Diffusion Vol. 33 No. 5 2012

mobility to be optimized. An initial position for the interface is also assumed. The calculated profile is transferred to Matlab where it is compared to the experimental concentration profile by calculation of the average error between them. This error is then minimized by a least square fit of a function depending on both the mobility and the position of the initial interface which both are varied until minimum of the error is reached. The mobility value is then transferred back to DICTRA and included in the kinetic description utilized for a new calculation of the concentration profile. The procedure is repeated until the calculated concentration profile is in acceptable agreement with the experimental measurement. For the current example, the best agreement was found for 0 Q Cr;Fe Mo ¼ 161371, see Table 2. The final calculated concentration profile is shown in Fig. 11 together with the measured profile. 4. Evaluation of the Assessments 4.1 Diffusion Coefficients As a first validation of the assessed kinetic description of the bcc Mo-Cr-Fe system, the Mo tracer diffusion coefficients for different Cr-Fe concentrations were calculated and compared with the measured coefficients reported on in literature. [38,50] Experimental data has been reported on by Čermák et al. [50] where the low-temperature tracer diffusion of the isotope 99 Mo in bcc Cr-Fe alloys had been measured. They used the serial sectioning method in the temperature interval 743-1123 K and for Cr concentrations from 8 up to 12.25 wt.%. At higher temperatures, from 931 to 1373 K, measurements have been reported by Kučera et al. [38] Their investigations also concerned the tracer diffusion of the 99 Mo isotope in bcc Cr-Fe alloys with Cr concentrations ranging from 0 up to 25 wt.%. The results of the calculated diffusion coefficients together with the measured coefficients are shown in Fig. 12(b) and 13(b). In comparison, the diffusion coefficients were calculated using the commercial diffusion mobility database MOB1, [51] see Fig. 12(a) and 13(a). The kinetic description therein includes no composition dependency of the Mo mobility function; i.e. all r U i;j Mo are zero and all the U i Mo parameters are given the same approximate values. It can be concluded that the assessment of the bcc Cr-Fe-Mo system improved the agreement between calculated and measured diffusion coefficient values. It should be Fig. 13 The calculated tracer diffusion coefficient of Mo in the bcc Cr-Fe system for different Cr contents as a function of temperature compared to the measured coefficient [38] (a) before the assessments and (b) after the assessments of the bcc Cr-Fe-Mo system. A value K is added to the coefficients in order to simplify comparisons between experiments and calculations Table 4 Chemical composition in wt.% for the coarsening model alloy C Cr Mo Si Mn P S Ni V Nb Co B 0.12 7.4 12.0 0.06 0.002 0.024 0.009 0.007 0.04 0.007 0.013 0.003 Journal of Phase Equilibria and Diffusion Vol. 33 No. 5 2012 383

Section I: Basic and Applied Research noted that the measured diffusion coefficients were not accounted for in the assessment of the parameters; i.e. the comparisons can be considered as an independent evaluation of the assessed parameters. 4.2 Coarsening of M 6 C Carbides As a second validation of the assessments, a coarsening experiment was performed and compared with a DICTRA calculation. For the experiment a model alloy was fabricated. The chemical composition was selected by thermodynamic calculations with the aim to have an equilibrium microstructure consisting of Mo rich M 6 C carbides embedded in a bcc Fe matrix alloyed with Cr and Mo at 1423 K. Fig. 14 Solidification structure of the model alloy after casting The model alloy was produced by casting in the same way as the diffusion couple alloys as described in the previous section. The obtained chemical composition of the alloy is listed in Table 4. A micrograph of the microstructure after casting, and before the coarsening heat treatment, is shown in Fig. 14. The grain boundaries are clearly visible and the accumulation of alloying elements, probably mainly Mo, to the grain boundaries is evident. To avoid the risk of oxidation and loss of elements due to evaporation during heat treatment, samples of alloy were vacuum sealed in quarts capsules. Two sets of samples were prepared; one to be heat treated 24 h at 1423 K to obtain an initial carbide size distribution, and one to be heat treated 500 h at 1423 K to obtain a coarsened carbide size distribution. In this way, conclusion about the carbide coarsening rate could be drawn. The heat treated samples were quenched in water, cut in halves and prepared for SEM analysis by grinding and polishing. The analyses consisted of EDS/WDS measurements of the phase compositions to compare with thermodynamic calculations and image analysis of micrographs of the microstructure to obtain information about the carbide sizes. Examples of the microstructure are shown in Fig. 15 for the sample heat treated 24 h and in Fig. 16 for the sample heat treated 500 h. For both heat treatment times there existed two classes of carbides; the ones located on the former grain boundaries (Fig. 15b, 16b) and the ones located insides the grains (Fig. 15a, 16a). For the imaging, a JEOL 7000F electron microscope was used. It is equipped with the software INCA Feature which provides a method for detection of the carbides in the imaging, and facilitates classification of the carbide sizes by morphology. Twenty images (magnification 91000) of each sample were analyzed. The locations for the images were Fig. 15 (a) Micrograph of the microstructure after heat treatment 24 h at 1423 K for the magnification 91000. (b) Micrograph of the microstructure after heat treatment 24 h at 1423 K for the magnification 9500 384 Journal of Phase Equilibria and Diffusion Vol. 33 No. 5 2012

Fig. 16 (a) Micrograph of the microstructure in grains after heat treatment 500 h at 1423 K for the magnification 91000. (b) Micrograph of the microstructure at grain boundaries after heat treatment 500 h at 1423 K for the magnification 9500 Fig. 17 The size distribution as a function of the average carbide equivalent radius after heat treatment 24 h at 1423 K Fig. 18 The size distribution as a function of the average carbide equivalent radius after heat treatment 500 h at 1423 K chosen randomly on the sample surfaces in order to obtain size distributions as representative as possible. The carbides were characterized by their area fraction, number and two-dimensional size. Figures 17 and 18 present the two-dimensional carbide size distributions in the form of histograms showing the frequency of carbides per count as a function of size class. The size class is represented by the circle area equivalent radius, R Eq. From the R Eq data the average carbide radii were determined and are listed in Table 5 along with the number of measured carbides and the total carbide area fraction. The measured phase compositions are listed in Tables 6 and 7 in comparison to the calculated equilibrium phase compositions at 1423 K. The phase compositions were measured for both heat treatment times. The compositions of the phases after 500 h of heat treatment had not changed considerably to the compositions measured after 24 h of heat treatment, see Tables 6 and 7. In addition, the measured area fractions, see Table 5, were approximately the same for both heat treatment times. Based on this information, it was decided that the carbides after 24 h at 1423 K had approximately reached their equilibrium composition and phase fraction and hence, justified the use of these carbide sizes as the start condition for the coarsening rate investigation. The measured average carbide radius after heat treatment 24 h determined the initial carbide radius and was used to calculate the initial size of the matrix cell size for the coarsening simulation. The input phase compositions were given by the thermodynamic calculations, see Tables 6 and 7. Journal of Phase Equilibria and Diffusion Vol. 33 No. 5 2012 385

Section I: Basic and Applied Research Table 5 The estimated average carbide diameter after 24 and 500 h of heat treatment at 1423 K 24 h at 1423 K 500 h at 1423 K Area frac., % N R Eq, lm Area frac., % N R Eq, lm M 6 C 4.0 ± 0.4 11289 0.57 ± 0.01 4.2 ± 1.6 740 2.21 ± 0.22 The given errors are estimates of 95% confidence interval of the expectation of the radius; R Eq ± dr, where r is the standard deviation, d=1.969n 1/2 and N is the number of analyzed carbides. The area fraction is the mean fraction for the 20 analyzed SEM images for each heat treatment time Table 6 The measured carbide compositions in wt.% in comparison with the calculated carbide composition at 1423 K Measured, wt.% 24 h at 1423 K 500 h at 1423 K Carbide composition Calculated Mean Std. dev. Min/max mean Std. dev. Min/max C 2.7 3.5 0.15 3.36/3.63 3.7 0.08 3.65/3.81 Cr 5.3 5.4 0.05 5.36/5.44 5.4 0.11 5.28/5.48 Mo 55.6 56.5 0.08 56.42/56.57 56.5 0.47 55.98/56.87 Fe 36.4 34.6 0.25 34.36/34.85 34.4 0.41 34.15/34.88 Table 7 1423 K The measured matrix compositions in wt.% in comparison with the calculated matrix composition at Matrix composition Calculated Measured, wt.% 24 h at 1423 K 500 h at 1423 K Mean Std. dev. Min/max Mean Std. dev. Min/max C 0.03 ÆÆÆ ÆÆÆ ÆÆÆ ÆÆÆ ÆÆÆ ÆÆÆ Cr 7.5 6.7 0.19 6.57/6.84 6.8 0.16 6.68/6.90 Mo 10.2 12.4 0.40 12.09/12.66 12.0 0.95 80.14/81.49 Fe 82.0 80.8 0.60 80.17/81.02 80.8 0.81 11.42/12.57 The initial condition left to be set was the value of the interfacial energy of the M 6 C carbide, a value which often is regarded as a fitting parameter when all other factors influencing the coarsening rate are determined; i.e. the thermodynamic and the kinetic descriptions for the system of question. In Fig. 19, the average carbide radius as a function of time is shown for simulations where the interfacial energy is set to 0.1, 0.2 and 1.0 J/m 2. The calculated results (solid lines) are shown in comparison to the measured average carbide radius (symbols). It could be concluded that an interfacial energy value between 0.1 and 0.2 J/m 2 reproduced the measured coarsening rate. This interfacial energy value is a bit lower than, but in the same range, as the value found appropriate by Fujita et al. [52,53] when modelling the precipitation sequence in ferritic stainless steels. They concluded that their experimental results for the M 6 C carbide could be reproduced by the use of an interfacial energy of 0.26 or 0.33 J/m 2. It should be mentioned that the study by Fujita et al. concerned a Nb and Fe rich M 6 C carbide (Nb 3 Fe 3 C). In Fig. 20, the simulation results where the kinetic description as it was before the Cr-Fe-Mo assessment was applied; i.e. with the MOB1 database, [51] are shown. The coarsening rate was overestimated by the calculations and for agreement with the measured coarsening rate an interfacial energy markedly lower than 0.1 J/m 2 was needed (0.03 J/m 2 ) which seems less realistic. 5. Concluding Remarks The current work concerned assessments of the mobility functions for diffusion in the bcc Cr-Fe-Mo system and 386 Journal of Phase Equilibria and Diffusion Vol. 33 No. 5 2012

Fig. 19 The simulated coarsening behaviour (solid lines) for the model alloy carbide for different interfacial energies in comparison to the measured average carbide size (symbols). The interfacial energy is 0.1, 0.2 and 1.0 J/m 2 points out the importance of an accurate kinetic description in order to obtain confident results when simulating diffusion phenomena in multicomponent alloy. The assessments were performed by accounting for experimental diffusion coefficients available in literature. In the case of an accurate assessment of the Mo diffusion in the bcc Cr-Fe phase, the published diffusion data was judged to be too sparse and, therefore, new diffusion data was produced by a diffusion couple experiment. The method provides a possibility to directly make use of the diffusion information meanwhile avoiding some of the possibly arising errors associated with traditional assessment using diffusion coefficient data; most experimentally determined coefficients have, to a certain extent, already been subjected to some assessment. The diffusion couple method can also easily be extended to systems containing higher number of components for which diffusion coefficient data seldom exists. Limited number of experimental measurements or large scattering between different published works influence the accuracy of the assessments, and determining the error associated with the Mo mobility assessment is difficult. Consequently, validation of the assessed kinetic database is important. In the current work, the database was validated by comparing calculated multicomponent diffusion coefficients to measured diffusion coefficients found in literature, and by comparing a calculated coarsening rate of Mo rich M 6 C carbides to the measured coarsening rate. An acceptable agreement between the measured and calculated diffusion coefficients could be concluded. It was also shown that the mobility function of Mo diffusion influenced the coarsening Fig. 20 The simulated coarsening behaviour (solid lines) for the model alloy carbide utilizing the kinetic description as it was before the assessment of the bcc Cr-Fe-Mo system; i.e. MOB1. [51] The interfacial energy is 0.03, 0.1 and 0.2 J/m 2 rate for the Mo rich precipitate. With the new kinetic description of the bcc Cr-Fe-Mo system, the measured coarsening rate could be reproduced by calculation using reasonable values for the interfacial energy. This result is of importance when, e.g., applying kinetic calculations for growth and coarsening of secondary precipitates during heat treatment of Mo and Cr alloyed tool steels. Acknowledgments This work has been financed by Böhler Edelstahl and Uddeholms AB. The authors dedicate thanks to the research committee; Ingo Siller and Odd Sandberg, for their interest and support. Dr. Lars Höglund is acknowledged for his help, the permission to use his Matlab scripts and for critical reading of the manuscript. Thanks are also dedicated to Profs. Malin Selleby and John Ågren for their comments. References 1. J.-O. Andersson, T. Helander, L. Höglund, P. Shi, and B. Sundman, Thermo-Calc and DICTRA, Computational Tools for Materials Science, Calphad, 2002, 26(2), p 273-312 2. J. Bratberg, Investigation and Modification of Carbides Subsystems in Multicomponent Fe-C-Co-Cr-Mo-Si-V-W Systems, Z. Metallkd., 2005, 96(4), p 335-344 3. B. Jönsson, Mobilities in Multicomponent Alloys and Simulation of Diffusional Phase Transformation, Computer Assisted Materials Design (COMMP-93), Tokyo, Sept 6-9, 1993, p 320-325 4. J. Bratberg, K. Frisk, and J. Ågren, Diffusion Simulations of MC and M7C3 Carbide Coarsening in bcc and fcc Matrix Journal of Phase Equilibria and Diffusion Vol. 33 No. 5 2012 387

Section I: Basic and Applied Research Utilizing New Thermodynamic and Kinetic Description, Mater. Sci. Technol., 2008, 24(6), p 695-704 5. G. Lindwall and K. Frisk, Assessment and Evaluation of the Mobilities for Diffusion in the bcc Cr-V-Fe System, J. Phase Equilib. Diffus., 2009, 30(4), p 323-333 6. Å. Gustafson and M. Hättestrand, Coarsening of Precipitates in an Advanced Creep Resistant 9% Cr steel Quantitative Microscopy and Simulations, Mater. Sci. Eng. A, 2002, 333, p 279-286 7. H. Magnusson and R. Sandström, The Role of Dislocation Climb Across Particles at Creep Conditions in 9 to 12 Pct Cr Steels, Metall. Mater. Trans. A, 2007, 38(10), p 2428-2434 8. A. Fick, Ueber diffusion, Ann. Phys., 1855, 170(1), p 59-86 9. A. Fick, On Liquid Diffusion, Philos. Mag. A, 1855, 10(63), p 30-39 10. L. Onsager, Reciprocal Relations in Irreversible Processes I, Phys. Rev., 1931, 37, p 405-426 11. L. Onsager, Reciprocal Relations in Irreversible Processes II, Phys. Rev., 1931, 38, p 2265-2279 12. J.-O. Andersson and J. Ågren, Models for Numerical Treatments of Multicomponent Diffusion in Simple Phases, J. Appl. Phys., 1992, 72(4), p 1350-1355 13. B. Jönsson, On Ferromagnetic Ordering and Lattice Diffusion: A Simple Model, Z. Metallkd., 1992, 83(5), p 349-355 14. B. Jönsson, Ferromagnetic Ordering and Diffusion of Carbon and Nitrogen in bcc Cr-Fe-Ni Alloys, Z. Metallkd., 1994, 85(7), p 498-501 15. B. Jansson, Evaluation of Parameters in Thermochemical Models Using Different Types of Experimental Data Simultaneously, Report TRITA-MAC-0234, Royal Institute of Technology, Stockholm, 1984 16. L. Höglund, Optimization of Mobility Parameters from Experimental Composition Profiles, Personal Communication, KTH, Stockholm, 2010 17. L. Höglund, DICTRA-ToolBox, Thermo-Calc AB, Stockholm, 2004 18. C.E. Campbell, A New Technique for Evaluating Diffusion Mobility Parameters, J. Phase Equilib. Diffus., 2005, 26(5), p 435-440 19. C.E. Campbell, W.J. Boettinger, and U.R. Kattner, Development of a Diffusion Mobility Database for Ni-Base Superalloys, Acta Mater., 2002, 50(4), p 775-792 20. C.E. Campbell, W.J. Boettinger, T. Hansen, P. Merewether, and B.A. Mueller, Complex Inorganic Solids, Structural, Stability and Magnetic Properties of Alloys, P.E.A. Urchin, A. Gonis, K. Rajan, and A. Meike, Eds., Springer, New York, 2005 21. C.E. Campbell, J.-C. Zhao, and M.F. Henry, Examination of Ni-Base Superalloy Diffusion Couples Containing Multiphase Regions, Mater. Sci. Eng. A, 2005, 407(1-2), p 135-146 22. A.M. Brown and M.F. Ashby, Correlations for Diffusion Constants, Acta Met., 1980, 28(8), p 1085-1101 23. J. Ågren, Numerical Treatment of Diffusional Reactions in Multicomponent Alloys, J. Phys. Chem. Solids, 1982, 43(4), p 385-391 24. O.Z. Zienkiewizc, The Finite Element Method, 3rd ed., McGraw-Hill, London, 1977 25. W. Oswald, Über die vermeintliche Isomerie des roten und gelben Quecksilberoxyds und die Oberflächenspannung fester Körper, Z. Phys. Chem., 1900, 34, p 495-503 26. Å. Gustafsson, L. Höglund, and J. Ågren, Advanced Heat Resistant Steels for Power Generation, R. Viswananthan and J. Nutting, Ed., IOM Communications Ltd., London, 1999, p 270-276 27. I.M. Lifshitz and V.V. Slyozov, The Kinetics of Precipitation from Supersatured Solid Solutions, J. Phys. Chem. Solids, 1961, 19(1-2), p 35-50 28. C. Wagner, Theorie der Alterung von Niederschlugen durch Umlosen, Z. Electrochem., 1956, 65, p 581-591 29. B. Jönsson, Assessment of the Mobilities of Cr, Fe and Ni in bcc Cr-Fe-Ni Alloys, ISIJ Int., 1995, 35(11), p 1415-1421 30. Y.V. Borisov, P.L. Gruzin, L.V. Pavliniv, and G.B. Fedorov, Self-Diffusion of Molybdenum, Metall. Pure Met. Sci., 1959, 1, p 213 31. M.B. Bronfin, S.Z. Bokshtein, and A.A. Zhukhovitsky, Self-Diffusion in Molybdenum, Zavod. Lab., 1960, 26(7), p 828-830 32. W. Danneberg and E. Krautz, Selbstdiffusionsuntersuchungen in Molybdän, Z. Naturforsch., 1961, 16(9), p 854-857 33. J. Askill and D.H. Tomlin, Self-Diffusion in Molybdenum, Philos. Mag., 1963, 8(90), p 997-1001 34. L.V. Pavlinov and V.N. Bykov, Self-Diffusion in Molybdenum, Fiz. Metall. Metalloved., 1964, 18, p 459-461, in Russian 35. K. Maier, H. Mehrer, and G. Rein, Self-Diffusion in Molybdenum, Z. Metallkd., 1979, 70(4), p 271-276 36. H. Nitta, T. Yamamoto, R. Kanno, K. Takasawa, T. Iida, Y. Yamazaki, S. Ogu, and Y. Iijima, Diffusion of Molybdenum in a-iron, Acta Mater., 2002, 50(16), p 4117-4125 37. V.T. Borisov, V.M. Golikov, and G.V. Sherbedinskiy, Diffusion of Molybdenum in Iron and in an Iron-Molybdenum Alloy, Phys. Met. Metallogr., 1966, 22, p 159-160 38. J. Kučera, B. Million, and K. Cíha, Diffusion of Molybdenum, Tungsten and Vanadium in Fe-Cr Alloys with a B.C.C. Lattice, Kov. Mater., 1969, 7(2), p 97-107 39. P.J. Alberry and C.W. Haworth, Interdiffusion of Cr, Mo and W in Iron, Met. Sci., 1974, 8(1), p 407-412 40. K. Nohara and K. Hirano, Impurity Diffusion of Iron in Molybdenum, J. Jpn. Inst. Met., 1973, 37, p 731-736 41. B. Lesage and A.M. Huntz, Diffusion of 59 Fe and 35 S in the Bulk and Grain Boundaries of Mo-Dependence on the Structure, J. Less-Common Met., 1974, 38(1-2), p 149-165 42. H. Nitta, K. Miura, and Y. Iijima, Self-Diffusion in Iron-Based Fe-Mo Alloys, Acta Mater., 2006, 54(10), p 2833-2847 43. K. Nohara and K. Hirano, Interdiffusion and Reaction Diffusion in the Fe-Mo System, J. Jpn. Inst. Met., 1977, 63(6), p 926-935, in Japanese 44. P.L. Gruzin and S.V. Zemskii, Study of Diffusion of Carbon and Molybdenum in Chromium, Metall. Metalloved. Chist. Metalloved., 1963, 4, p 243-250 45. Y.V. Borisov, P.L Gruzin, and S.V. Zemskii, Diffusion of Some Elements in Refractory Metals, Problems of Metal Science and Physics of Metals, Vol 8, Metallurgiya, Moscow, 1964, p 321-325, in Russian 46. L.M. Mulyakaev, G.V. Shcherbedinskii, and G.N. Dubinin, Bulk and Boundary Diffusion of Chromium in Molybdenum, Met. Sci. Heat. Treat., 1971, 13(8), p 665-668 47. J. Ågren, M.T. Clavaguera-Mora, J. Golcheski, G. Inden, H. Kumar, and C. Sigli, Application of Computational Thermodynamics to Phase Transformation Nucleation and Coarsening, Calphad, 2000, 24(1), p 41-54 48. G. Lindwall, K. Frisk, O. Sandberg, and I. Siller, Diffusion Calculations as a Tool for Tool Steel Design, Proceedings of the 8th International Tooling Conference, RWTH Aachen University, Aachen, 2009, p 1063-1070 49. Dilatometer DIL 805A/D, Bähr-Thermoanalyse GmbH, Hüllhort, 2005 50. J. Čermák, J. Růžičková, and A. Pokorná, Low-Temperature Tracer Diffusion of 99 Mo, 48 V and 181 W in Fe-Cr Ferritic Alloys, Acta Mater., 1996, 44(12), p 5007-5019 51. Diffusion Mobility Database MOB1, Thermo-Calc AB, Stockholm, 2008 388 Journal of Phase Equilibria and Diffusion Vol. 33 No. 5 2012

52. N. Fujita, M. Kikuchi, and H. Bhadeshia, Modeling M 6 C Precipitation in Nb-Alloyed Ferritic Stainless Steel, Metall. Mater. Trans. A, 2002, 33(11), p 3339-3347 53. N. Fujita, H. Bhadeshia, and M. Kikuchi, Precipitation Sequence in Nb-Alloyed Ferritic Stainless Steel, Model. Simul. Mater. Sci. Eng., 2004, 12, p 273-284 Journal of Phase Equilibria and Diffusion Vol. 33 No. 5 2012 389