Chin. Sci. Bull. (2014) 59(15):1672–1683 DOI 10.1007/s11434-014-0127-7

csb.scichina.com www.springer.com/scp

Review

Materials Science

Computational diffusion kinetics and its applications in study and design of rare metallic materials Yuwen Cui · Guanglong Xu · Yi Chen · Bin Tang · Jinshan Li · Lian Zhou

Received: 14 April 2013 / Accepted: 28 June 2013 / Published online: 12 February 2014 © Science China Press and Springer-Verlag Berlin Heidelberg 2014

Abstract Computational diffusion kinetics (CDK), with a spirit of and being coupled with the computational thermodynamics (CT, or called as the CALPHAD technique), plays increasingly important role in the alloy design/optimization and microstructure control during the processing of advanced metallic materials. This paper is to highlight recent progress of CDK in research with great focus on novel Ti and Zr alloys, which was largely performed in the authors’ group. It ends with one representative example of the applications of CDK, coupled with CT, quantitative phase field, and three-dimensional (3D) statistical calculation, in designing the heattreatment schedule for the dual phase (α β) Ti–6Al–4V alloys. Keywords Computational diffusion kinetics · Computational thermodynamics · Phase field · Microstructure · Rare metallic materials

1 Introduction Diffusion is the transport of matter from one point to another by thermal motion of atoms or molecules. Diffusion governs SPECIAL ISSUE: Materials Genome Y. Cui (&) · G. Xu · Y. Chen Computational Alloy Design Group, IMDEA Materials Institute, C/Eric Kandel 2, 28906 Getafe, Madrid, Spain e-mail: [email protected] G. Xu Facultad de Informa´tica, Universidad Polite´cnica de Madrid, 28040 Madrid, Spain Y. Chen · B. Tang · J. Li · L. Zhou State Key Laboratory of Solidification Processing, Northwestern Polytechnical University, Xi’an 710072, China

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most of microstructural changes like phase transformations, homogenization, solidification, sintering, recrystallisation, high-temperature creep, thermal oxidation, hydrogen embrittlement, etc. It also plays a key role in performance and stability during preparation, processing, and heat treatment of materials at elevated temperatures. It is also known that tactically suppressing or preventing diffusion results in stabilization of metastable phases and microstructures, which can provide the obtained materials with unique mechanical properties. As one of the most successful computational materials’ techniques in areas of the engineering materials, computational diffusion kinetics (CDK), has often been integrated with computational thermodynamics (CT, also called as the CALPHAD approach) to form a reliable thermo-kinetic base for the mesoscale microstructure modeling (in particular, phase-field and Landau models). Such an integrated computational materials approach has the ability to quantitatively capture complexity in homogeneous microstructural change and to act significantly more than what the individual techniques can do (i.e., phase diagram/ thermodynamics and diffusion phenomena, respectively), thereby playing increasingly important role in alloy design/ optimization and microstructural control during the processing of advanced metallic materials [1, 2]. In the past, most diffusion researches were accomplished by extensive laboratory testings that are often costly and time consuming, and diffusion data were documented and classified into tracer diffusion coefficient (which includes self-diffusion and impurity diffusion), intrinsic diffusion coefficient, and interdiffusion coefficient [3]. Implementation of all types of diffusion data becomes very tedious and hardly practicable particularly for the quaternary and multicomponent alloy systems, for which the average effective interdiffusion (AEI) coefficient [4], has been proposed as an approximation by averaging over

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individual components along the specific trajectories (path) of diffusion route. The AEI helps us to study the interdiffusion behaviors of the individual components and characterize the concentration profiles of multicomponent diffusion couple; however, it is not really helpful in a materials modeling. Based on the absolute reaction rate theory, CDK has emerged as an efficient approach to simulate the diffusion-controlled process, in which diffusional mobility (also called as atomic mobility), rather than diffusion coefficient, is used as a base, which can be readily assessed by fitting to experimental diffusion data in conjunction with CT. Similar to the thermodynamic parameters of the phenomenological CT technique, the mobility parameters in CDK (either activation energy or frequency factor) is assumed to be composition dependent and can be expressed by the Redlich– Kister polynomial in composition [5, 6]. As a result, CDK is particularly practical for creating kinetic databases and diffusion modeling of multicomponent systems because it simply defines one unique mobility for each component in multicomponent systems and can be extrapolated from binary to ternary and to higher-order alloy systems with a high degree of accuracy for many commercially important alloy systems that contain more than a dozen elements. It is also because CDK is in most cases performed in conjunction with the CTbased thermodynamic database to obtain sufficiently reliable thermodynamic quantities. So far, CDK has been successfully applied to many processes as diverse as nitridation and carbonization [7], diffusion couple and welding [8], coarsening and dissolution of precipitates [9], solidification and casting [10], etc. In recent years, it has been extended to investigate the microstability of Kirkendall plane and void formation having promising potential in prediction of impurity and defects segregation along the interfaces during coating and diffusion welding processing [11]. More importantly, its coupling with the mesoscale modeling of microstructure for providing realistic materials parameters has enabled phasefield and Landau Modeling to be developed toward a quantitative modeling tool [12, 13], which in turn matures CDK and stimulates the development of integrated thermo-kinetic database when it is being interfaced with CT. As the feasibility and success of CDK for multicomponent alloy systems, there have been many studies focusing on development of mobility database for application of materials engineering, particularly in Europe, USA and China. CDK was initialized at KTH (Stockholm, Sweden) via joint efforts with the Max-Planck Institute fu¨r Eisenforschung (Du¨sseldorf, Germany), which resulted in the commercial software DICTRA [14], now operated by Thermo-Calc Company. The company developed the MOB2, MOBNi, and MOBAl databases (and their upgraded versions) specialized in steels, nickel-based superalloys, and aluminum-based alloy systems, respectively. IMDEA materials institute (Madrid,

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Spain) is another institution in Europe that is active in the research of Co-based superalloys and rare metallic materials [15]. Thermodynamics and Kinetics Group at NIST is well known for its mobility database developed for the Ni-based super alloys where the description of Ni–Al–Cr is widely acting as a mobility template [16]. Central South University (Changsha, China) is leading the research of the mobility database for the Al-based alloy systems [17]. This review presents the research of the authors mainly at the IMDEA Materials Institute on the CDK topic with focus on novel Ti and Zr alloys covering all aspects of thermodynamics, diffusion kinetics, and microstructure development. It ends with one representative example that successfully integrates the three techniques, further in conjunction with three-dimensional (3D) statistical calculation, for designing heattreatment schedule for the dual phase (α β) Ti-6-4 alloys.

2 Diffusional mobility 2.1 Diffusivity and mobility As stated before, there need to be all three kinds of diffusion coefficients for describing a full diffusion picture, including tracer, intrinsic and interdiffusion coefficients, as well as their variations in composition and temperature. It becomes very tedious and impracticable without using the diffusional mobility, especially for multicomponent alloy systems. The incorporation of the mobility in CDK and its coupling with CT offers an efficient approach for diffusion research and modeling. Based on absolute-reaction rate theory that is accepted by CDK, the diffusional mobility MB of species B is expressed as a function of temperature T (K) [5, 18, 19],
 QB 1 mag 0 MB ¼ MB exp  C; ð1Þ RT RT where QB is the activation energy, MB0 is the frequency factor, R is the gas constant, and mag C is the magnetic ordering factor (in case when there is no magnetic effect on the diffusional mobility, mag C ¼ 1). Both the parameters, RT ln MB0 and QB, are composition- and temperaturedependent. As the spirit of the CT technique, the composition dependence of the mobility parameters Qi (RT ln MB0 and QB) can be expressed in terms of the Redlich–Kister polynomial [20]: " # m X X XX p r r p;q Qi ¼ xp Q i þ xp xq Qi ðxp  xq Þ p

þ

p

XXX p

q[p v[q

q[p

r¼0

h i xp xq xv vspqv s Qp;q;v þ    ; ðs ¼ p; q; vÞ; i ð2Þ

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Table 1 Diffusion coefficients and mobility Diffusion coefficients Tracer diffusion Intrinsic diffusion Interdiffusion

General diffusion theory   D ¼ D0hexp Q RT i log c Di ¼ Di 1 þ oolog C

CDK

~ ¼ xA DB þ xB DA D

Dkj ¼

D*k = RTMk i

k Dkj ¼ xk Mk ol oxk

n  P i¼1

i dik  xi Mi ol ol Vm



leads to shrinkage of the specimen, while a positive deviation will cause swelling of the specimen. When the total volume of diffusion couple changes with interdiffusion reaction, it will bring about systematic error during the study of interdiffusion in diffusion couple experiments; however, such an error for diffusion in substitutional alloys is hardly beyond 5 %, which therefore is generally ignored.

j

2.3 Assessment of mobility parameters where xp is the mole fraction of species p, Qpi is the endmember value Qi of species i in pure species p, rQp,q i , and sQp,q,v are the binary and ternary interaction paramei ters, and vspqv is given by vspqv = xs (1 − xp − xq − xv)/3. The mobility parameters, Qpi , rQp,q and sQp,q,v , can be numerii i cally assessed by fitting to the experimental diffusion coefficients. 2.2 Relation between diffusion coefficients and diffusional mobility All three kinds of diffusion coefficients can be reasonably yielded from mobility. The tracer diffusion coefficient Di is related to the diffusional mobility by the relation D*i = RTMi, the intrinsic diffusion coefficient with a constant molar volume is given by Eq. (3): D ¼ xi M i

oli ; oxi

ð3Þ

where μi is the chemical potential of species i, and the interdiffusion coefficient can be derived from
 n1 X oli oli n ~ Dkj ¼ ðdik  xk Þxi Mi  ; ð4Þ oxj oxn i¼1 where the Kronecker delta δik is 1 when i = k and is 0 otherwise. See Table 1 for the relations between the diffusion coefficients and mobility. Diffusion, in most of the real alloy systems, is affected by a change in total volume. The effect is often very minor but could be considerable as the molar volume deviates from the ideal case. For a binary system of species A and B, the molar volume Vm is expressed by the partial molar volumes VA and VB of the components A and B by [21] Vm ¼ xA VA þ xB VB : ð5Þ In case VA and VB are fixed, either VA = VB = Vm or VA ≠ VB but constant for all compositions, the total volume does not change, and there will be no change in the length of the diffusion couple sample after annealing. However, in almost all real systems, the molar volume deviates from ideality so that it will result into change in the dimension of the diffusion couple specimen. If there is a negative deviation from ideality, then the total volume will decrease that

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Figure 1 shows the frames of CT and CDK, where the blocks in gray stand for the tasks of CT and those in white are for CDK. As can been shown, CT and CDK, on the one hand, enable accurate prediction of thermodynamic, physical, and kinetic properties of materials, and as a whole, form a base of materials for quantitative modeling of microstructure on the other hand. As the thermodynamic parameters for CT, the diffusional mobility of multicomponent alloys can be appropriately extrapolated from the parameters of lower-order systems that have been obtained from the critical assessment. The assessment procedure of mobility parameters is schematically illustrated in Fig. 2. First, retrieve the thermodynamic database from the literature for the topic alloy systems, and collect and evaluate available diffusion data. For these alloy systems without experimental diffusion data or those that are hardly/ impossible to determine, the theoretical calculations (The first principle calculation being particularly useful.) can be used as alternative resource, whereas the semi-empirical relations help reduce the number of adjustable parameters. Second, assess the mobility parameters that are defined by the model (Eq. (2)) and ordering effect(s) by fitting to the evaluated and selected diffusion data, and then develop the corresponding mobility database. Third, validate the developed mobility database by comparing the calculated diffusion coefficients with experimental points and then for other in-depth diffusion behavior resulting from interdiffusion. Iterate the steps II and II until the best agreement is reached. Finally, apply the mobility database to mesoscale modeling of microstructure and engineering applications. 2.4 Simulation of diffusion experiment Diffusion profile is the composition of atoms/molecules of interest as function of position in diffusion zone (of diffusion couple) usually acquired from electron probe microanalyzer (EPMA) analysis. Simulating diffusion profile and other complex behaviors resulting from interdiffusion (i.e., diffusion path, Kirkendall velocity, and lattice plane displacement) and validating against the experimental EPMA data are the other key issues for validation of a mobility database in CDK. In the case of diffusion couple where the partial molar volume is

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Gibbs energy

Solve fluxEQ

Diffusion coefficients

Solving diffusion Eq

Moving

∂ 2G ∂ xi ∂ x j

Data Kinetics

interface

Thermodynamics

Single phase

Fig. 1 (Color online) Frame of CT and CDK

occurring in a diffusion experiment becomes a critical task of validation of the mobility database in CDK. If diffusion occurs by a vacancy mechanism, the nonuniform velocity of the inert markers v, with respect to the laboratory-fixed frame of reference, is dependent on the difference of the intrinsic diffusion fluxes of the species:

Assessment of mobility parameters

Evaluation of diffusion data

r  v ¼ Vm r  ðJA þ JB Þ:

Atomic mobility Design & Application

When expressing in the diffusivities and the pintrinsic ffi Boltzmann variable k ¼ z t , it yields v¼

Validation of diffusion experiment

Fig. 2 Development of diffusional mobility database

assumed to be constant (see Sect. 2.2), the variation of composition with diffusion time t in the diffusion zone can be described by the equation of continuity: 1 oxi þ r  J~i ¼ 0: Vm ot

ð7Þ

ð6Þ

Equation (6) can be solved numerically, corresponding to different initial conditions and boundary conditions. The solution expresses the form of the concentration profile. The Kirkendall effect was named after the famous experiment by Smigelkas and Kirkendall [22] revealing the inequality of diffusivities of the species in the copper/brass diffusion couple by placing molybdenum wire as an inert marker. Two conclusions were drawn which had enormous impact on the solid-state diffusion theory, i.e., zinc diffuses more rapidly than copper, and diffusion occurs by a vacancy mechanism. Representing the Kirkendall effect

Vm ðDB  DA Þ oxB pffi : ok t

ð8Þ

As the above equation shows, the motion of the inert markers depends on the differences of the intrinsic diffusivities of the species and composition gradient; more specifically, it will move toward one end of the diffusion couple. In a diffusion-controlled interaction, the Kirkendall plane is the only plane that stays at a constant composition during the whole diffusion annealing, and it moves parabolically in time with a velocity [23, 24]: vk ¼

dz zk  zk0 zk ¼ ¼ ; dt 2t 2t

ð9Þ

where zk and zk0 are the positions of the Kirkendall plane at times t = t and t = 0, respectively. As proposed [23, 24], the Kirkendall plane can be positioned by finding a crossover point between the velocity straight line and the Kirkendall curve in the Kirkendall velocity construction (KVC), i.e., a plot of the velocity of the inert markers as a function of the distance; see the details in Fig. 3. The microstability of the Kirkendall plane can be then measured by studying the slope of the intersection in a KVC, i.e., if it is an intersection with a negative gradient, then there is a stable Kirkendall plane (Fig. 3a); otherwise, it is unstable Kirkendall plane (Fig. 3b). In case when they intersect by more than one time, the

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Fig. 3 Schematic velocity diagram of a hypothetical diffusion couple. a The straight line 2tvk = Xk intersects the velocity curve 2tv-X at a point with a negative gradient and b intersects at a point with a positive gradient

multiple Kirkendall planes can thereby be developed. The lattice plane displacement can be computed from the velocity of the inert markers by integrating Eq. (9): oy y  2vt ¼ ; oz0 z0

ð10Þ

where z0 is the position of the original location of the markers, and y is the placement of the markers relative to z 0.

3 Diffusion kinetics modeling Most of the properties and performances of advanced rare metallic materials, represented by Ti- and Zr-based alloys, are governed by alloying elements and phase transformations occurring in these alloy systems. As Banerjee and Mukhopadhyay [25] reviewed, the phase transformations in Ti- and Zr-based alloys are so rich that each type of phase transformation can be found, ranging from diffusional (short and long range) controlled to displacive (strain and shuffle) dominated, and most of these

transformations are closely associated with (or implicitly via coupling) with the diffusion processes of alloying elements. This fact makes CDK particularly helpful in the research and development of rare metallic materials, which made us to write this paper. 3.1 Diffusion coefficients The diffusion coefficients are among the fundamental quantities of designing and scheduling the processing parameters like heat-treatment temperature and time. As stated above, all types of diffusion coefficients can readily and reliably be derived from the mobility database in CDK, which in turn can be used to validate the mobility database itself. Figure 4 presents the good agreement between the calculated and the experimental Ti tracer diffusion coefficients in the Ti–Zr bcc binary alloys (Fig. 4a) [26–28] and those of the interdiffusion coefficients (Fig. 4b) [29]. Representative example for ternary diffusion can be seen in Fig. 5 for the interdiffusion coefficients in the Ti–Al–V ternary bcc alloys [30, 31] (Fig. 5a) and Co–Fe–Ni [32] (Fig. 5b).

Fig. 4 The calculated a impurity diffusion coefficients of Ti in β-Zr and b interdiffusion coefficient of the bcc Ti–Zr alloys

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Fig. 5 The calculated ternary interdiffusion coefficients of a Ti–Al–V and b Co–Fe–Ni alloys compared with the experimental values

3.2 Anomalous diffusion It is well known that for some face-centered cubic (fcc) metals, e.g., silver, there is a slight curvature in the Arrhenius plots which has been attributed to a contribution to diffusion from divacancy and a minor from vacancy double jumps when it is very close to melting temperature [3]. The behavior of self-diffusion in bcc metals is quite varied, while there are some linear or slightly different curves, yet a large group of anomalous metals including Ti, Zr, Hf, γ-U, and ε-Pu shows a strong curvature and anomalously low values of D0 and Q. For Ti and Zr metals, enhancement of diffusivity by the dislocations induced from the α → β phase transformation and/or segregated impurities has been strongly supported [3], and the presence of the ω embryos as activated complexes of diffusion may also be a plus. Liu et al. [33–35] assessed the

mobilities of the Ti–V, Ti–Nb, Ti–Mo, and Ti–Zr binary bcc alloys, however, by describing the anomaly with a piecewise polynomial. It is more physical to use a nonlinear relation: D ¼ D1 expðQ1 =RT Þ þ D2 expðQ2 =RT Þ;

ð11Þ

from which to reasonably extrapolate to binary and then to higher alloy systems. Figure 6 illustrates the calculated Al impurity diffusion coefficient in bcc-Ti [31, 36–39] and the Ti tracer diffusion in the Ti–Al binary bcc alloys [40, 41]. For the intermetallics like TiAl and Ti3Al [42], distinct diffusion mechanism is dominant in different regimes of temperature and composition of the intermetallics or by a combination of two or more mechanisms, which have to be dealt with by the same way. Magnetic and chemical orderings both remarkably suppress diffusion leading to anomalous diffusion too. This

Fig. 6 Arrhenius plot of a Cr impurity diffusion in body centered cubic (bcc) Ti and b Ti tracer diffusion in bcc Ti–Al alloys compared with the experimental data

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can be understood from the fact that the nearest-neighbor jump in an ordered state would cause an atom to jump into the wrong lattice position decreasing the degree of order; therefore, diffusion can only occur via complex mechanism involving several consecutive jumps that can restore the ordering after a complete jump cycle. Jonsson [43] introduced the ferromagnetic diffusion model proposed by Braun and Feller-Kniepmeier [44] into CDK, and expressed the activation energy of diffusion in ferromagnetic state by


 6 1 P mag Qk ¼ Qk þ a2 DH  ; ð12Þ QPk RT where Qpk is the activation energy of diffusion in the paramagnetic state, mag DH is the magnetic enthalpy that can be readily retrieved from CT, and α2 is the materials parameter. The expression has been extended to describe the composition dependence of the ferromagnetic effect on diffusion; see Fig. 7 for the diffusion in the Co–Al binary fcc alloys [15, 45, 46]. The diffusion mechanisms in the chemically ordered state are generally complex, including [3] the complex jump cycle, triple defect, and antistructure bridge [47], etc. When comparing disordered and ordered alloys at the same temperature, the diffusion coefficient may be several orders of magnitude lower in the ordered alloy. Based on the work by Girifalco [48] on the use of a long-range order parameter to describe the effects of ordering on diffusion, ˚ gren [49] developed a generalized pheHelander and A nomenological model for diffusion in the ordered phases, which expresses the activation energy as a function of the degree of ordering and derives the thermodynamic factor and the degree of ordering from a CALPHAD-based thermodynamic description at a given temperature:   XX a b DQord DQord ð13Þ k ¼ kij yi yj  xi xj ; i

i6¼j

is an ordering contribution to the activation where DQord k energy, y is the site fraction of component, and DQord kij is the contribution to the activation energy for component k due to the chemical ordering of the i–j atoms. Cui et al. [50] applied this model and performed a thorough study of the diffusion in the B2 phase of the Fe–Al binary system [51, 52] (Fig. 7b). 3.3 Diffusion couple experiment Evaluation of the assessed diffusional mobility includes not only the comparison with the experimental diffusion coefficients, but also the comparison between the predicted and the observed in-depth diffusion-controlled behaviors. Such modeling offers practical significance to representing and understanding of interdiffusion processes occurring in alloy homogenization, surface modification, microsegregation, growth and coarsening, gradient sintering, welding and solidification, etc. Figure 8a shows the simulated concentration profile of the Ti–Al–V ternary diffusion couple [41]; Fig. 8b for the simulated diffusion paths of a number of the Ti–Al–Cr diffusion couples [31]; and Fig. 8c for the simulated solidification curves [53] of 5052 Alloy under different cooling conditions, including the Scheil model and others [54–56]. Nowadays, the thermodynamic and mobility databases have been often integrated and then used to predict and analyze quantitatively the Kirkendall effect, the microstability of the Kirkendall plane, and the lattice plane displacement resulting from interdiffusion. This predictive CDK skill has reached a stage of maturity for practical application in the prediction of microstructural development (particularly the occurrence of the Kirkendall voids, the microsegregation of impurities at the Kirkendall plane, and its microstability, etc.) and thus the lifetime of welding and coating during the treatment and service of the parts.

Fig. 7 (Color online) The calculated interdiffusion coefficients of a Co–Al fcc alloys and b Fe–Al B2 alloys

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Fig. 8 (Color online) Comparison between the simulated and experimental values (symbols) a diffusion profile of Ti–Al–V ternary diffusion couple; b diffusion paths for the Ti–Al–Cr ternary couples at the temperature 1,473 K; and c solidification curves of Alloy 5052

Fig. 9 a Intrinsic diffusion coefficients and interdiffusion coefficient versus xMo at 1,773 K in Ti–Mo alloys; b KVC for the Ti/Mo diffusion couple at 1,773 K; and c Kirkendall shift versus time. Symbols represent the experiment data and solid curves are the calculated results

Figure 9a illustrates the calculated variation of the intrinsic diffusion coefficients of Ti and Mo with alloy composition at 1,773 K, proving that Ti is an intrinsically faster diffuser over the entire composition range and in turn the Kirkendall (inert) markers, if they occur, move toward the Ti side [28, 57]. Figure 9b presents the KVC of the Ti/Mo diffusion couple after annealing at 1,773 K for 10,500 s. Note that the reduced velocity curve intersects the straight line v = z/2t, determined by Eq. (7), at a point (about 15 at.% Mo) with a negative gradient; as a consequence, there is a stable Kirkendall plane for the Ti/Mo couple at 1,773 K, namely, the Kirkendall marker remains sharply concentrated during the process which is prone to segregating the impurities and structural defects. Figure 9c is the simulated Kirkendall shift of the Ti/Ti15 wt%Mo/Ti couple as a function of square root of the diffusion time where a linear relation suggests that the markers move parabolically in time from t = 0 [28, 57]. The work of Campbell et al. [58] demonstrates the appropriate predictability of CDK in the occurrence of the Kirkendall void and its most likely position in the diffusion zone of Nibase superalloys.

4 Applications in microstructural modeling and alloy design 4.1 Quantitative phase-field modeling for Zr-based alloys Phenomenologically representing the lattice collapse mechanism, the phase-field model integrated with CT and CDK can be used to simulate and study the phase transformation kinetics, transformation sequence (i.e., from pretransition structure to a thermal ω to isothermal ω), and microstructural evolution during the β → ω phase transformation in the Ti- and Zr-based alloys [59– 61]. The simulations in Fig. 10a, b represent such a representative snapshot of transformation sequence and microstructure under different conditions in these Ti and Zr rare metallic alloys. In the phase-field simulations of the isothermal ω phase transformation, the chemical mobility Mkj l has to be employed in the Cahn–Hilliard kinetic equation, which can be expressed in the multiphase diffusional mobility Ml by

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Fig. 10 Phase transformation sequence and microstructure evolution of the Zr–Nb alloys simulated by coupling CT/diffusion kinetics with phase field model. a Pre-transition effect; b athermal ω phase; c isothermal ω phase; and d ω phase transformation kinetics for the Zr-8 at.% Nb alloy aging at different temperatures

Mlkj ¼

n   1 X djl  Xj ðdlk  Xk ÞXl Ml ; Vm l

ð14Þ

in where Ml itself is a function of the diffusional  1g  mobility g the concept of CDK: Ml ¼ Ml- þ Mlb  MlMlb [12]. Numerically solving the kinetic equations under different treatment conditions leads to a quantitative simulation of microstructural evolution and transformation kinetics; see Fig. 10c for those of the isothermal ω transformation in the Zr–Nb alloys. 4.2 Design of ultrafine grain Ti-64 Many of the important engineering materials sustain twophase microstructure in which both phases may evolve, i.e., through growth and/or coarsening, under processing or service conditions, and, which in turn, affects the properties of these materials. Yet, the nontrivial study of the grain growth and coarsening in two-phase materials largely relies on a large number of experimental works and semi-

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empirical calculations [62, 63]. The development of ultrafine-grained dual-phase Ti alloys serves as one of the hot research topics in light metals and alloys. The bulk multicomponent Ti alloys subjected to severe plastic deformation processing methods like multiaxial forging (MAF) were shown to exhibit the submicron grains and thus superior mechanical properties [64, 65]. However, the thermal stability and growth/coarsening mechanism of the sub-micron two-phase grains under the subsequent static and/or dynamic conditions remain unclear. This paper provided one example of recent modeling successes in quantitative understanding of the mechanism of the growth/coarsening of ultrafine-grained two-phase Ti–6Al– 4V alloys and designing of the subsequent heat-treatment schedules [13]. A quantitative phase-field model (QPF), as applied to αphase grain growth with pinning of β-phase for Ti–6Al–4V, is illustrated in Fig. 11. The model has a feature to directly use scanning electron microscopy (SEM) images as starting microstructure (Fig. 11a) and has an ability to perform

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microstructural evolution at experimentally relevant length and time scales, based on the facts that the Kim-Kim-Suzuki model [66] was employed to remove length scale limit, and that real-time thermo-kinetic databases of CT and CDK were implemented to instantly provide accurate thermodynamic and diffusional mobility properties. The QPF modeling (Fig. 11b) confirms that α-grain growth with β-phase particles is governed by a bulk-boundary mixed diffusion mechanism. By introducing the bulk–boundary mixed diffusion and allowing adjustable interfacial energies for αgrain boundary and α/β-phase boundary, the model allows prediction of effects of high diffusion path and interfacial energies on morphological change and grain growth kinetics. With quantitative parametric QPF virtual experiments, some important material parameters for Ti-64, such as boundary energies, interfacial mobility, kinetic data of α growth/dissolution and coarsening, were obtained. Due to the limiting computational resources, the large scale 3D phase-field modeling grain growth coupled with long-range diffusion process is yet to be applied to any engineering design and optimization of new materials, and this is particularly true for multiphase and/or multicomponent engineering materials. This challenge can be greatly met by further coupling the QPF with statistical fast acting (FA) model [13, 67]. Simply speaking, using the important

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real materials obtained from the QPF as inputs to FA via a generalized statistical growth kinetic equation of dual or triple phase materials, we have ( "P   #) 2 o/ai o j /j Rj 1 Rj  1=Ri  P=2 a P ¼ mca a /i ; 2 oRi ot j /j Rj ð15Þ where φi is a topological distribution function describing the number of grains of a grain size class i, Ri is the equivalent radius, m is the grain boundary mobility, γ is the grain boundary energy, and P is the total pinning force of all dispersoids. The coupling between QPF and FA, on the one hand, allows for precise 3D statistical prediction of the grain size and its distribution during the grain growth/ coarsening process of two-phase materials under the static and dynamic heat-treatment conditions at experimentally relevant time scale (Fig. 11c); it enables an integrated materials database including thermodynamics, kinetics, and properties to be built, on the other hand. Such an integrated database, along with the integration technique between different computational modeling techniques, is playing a critical role in Materials Genome Initiative, a multi-stakeholder effort newly proposed to develop an infrastructure to accelerate materials’ discovery and

Fig. 11 Integrated experimentation-CT-CDK-phase field-statistical calculation for dual-phase Ti-6Al-4V alloys. a SEM-based starting microstructure; b microstructure evolution by integrated CT-CDK-phase field model; and c 3D statistical calculation for dual-phase materials

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deployment (http://www.whitehouse.gov/blog/2011/06/24/ materials-genome-initiative-renaissance-american-manufa cturing).

5 Conclusion CDK, working with the diffusional mobility (rather than diffusion coefficients) and the same extrapolation method as CT does, has emerged as one of the most successful techniques of computational materials in the alloy design/ optimization and microstructural control during the processing of advanced metallic materials. This paper highlighted the recent progress of the CDK research with great focus on novel Ti and Zr alloys, which was largely performed by the authors’ group, and the main conclusions can be drawn as below: (1)

(2)

(3)

Fully coupled with CT, CDK allows precise prediction of all kinds of diffusion coefficients and interdiffusion phenomena including diffusion anomalies in ferromagnetic and chemical ordering structures, thereby becoming the most efficient approach for diffusion research; CDK has been widely impressed with a straightforward way for analyzing and understanding the experimental findings; yet, it plays an increasing role in designing and optimizing alloy compositions and new heattreatment and processing conditions, including predictions of solute (re)distribution, diffusion path, composition/shift/velocity/microstability of Kirkendall plane, occurrence and position of Kirkendall voids in diffusion zone, etc.; Recent successful example shows that integrating CT, CDK, and phase-field model enables the QPF, and further coupling with 3D statistical calculation not only allows for prediction of grain growth mechanism and kinetics, but also permits computer-aided design of heat-treatment schedule for the ultrafine-grained two-phase (α β) Ti–6Al–4V commercial alloys.

Acknowledgments This work was supported by the State Key Laboratory of Solidification and Casting, Northwestern Polytechnical University (SKLSP200906), and the Program of Introducing Talents of Discipline to Universities (B08040). Yuwen Cui would like to acknowledge the support under the AMAROUT-II Program.

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