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Atomic mobilities, diffusivities and their kinetic implications for U–X (X=Ti , Nb and Mo) bcc alloys
CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 37 (2012) 49–56
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CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry journal homepage: www.elsevier.com/locate/calphad
Atomic mobilities, diffusivities and their kinetic implications for U–X (X = Ti, Nb and Mo) bcc alloys Yajun Liu a,∗ , Di Yu b , Yong Du c , Guang Sheng d , Zhaohui Long e , Jiang Wang f , Lijun Zhang c a
Western Transportation Institute, Montana State University, Bozeman, MT, 59715, USA
b
Computer Science and Engineering, University of South Florida, Tampa, FL, 33620, USA
c
State Key Laboratory of Powder Metallurgy, Central South University, Changsha, Hunan, 410083, PR China
d
Scientific Forming Technologies Corporation, Columbus, OH, 43235, USA
e
School of Mechanical Engineering, Xiangtan University, Xiangtan, Hunan, 411105, PR China
f
School of Materials Science and Engineering, Guilin University of Electronic Technology, Guilin, Guangxi, 541004, PR China
article
info
Article history: Received 15 November 2011 Received in revised form 4 January 2012 Accepted 20 January 2012 Available online 15 February 2012 Keywords: Diffusion Mobility CALPHAD DICTRA bcc U–X (X = Ti, Nb and Mo) alloys
abstract Based on various kinds of diffusivities as well as the thermodynamic descriptions within the CALPHAD framework, the atomic mobilities of U, Ti, Nb and Mo are explored in this work with the DICTRA software. The mobility end-members are evaluated from the impurity diffusivities as well as the extrapolated interdiffusivities, while the interaction parameters for atomic mobilities are determined from the tracer diffusivities, the intrinsic diffusivities and the interdiffusivities. The reliability of such purely kinetic quantities is carefully verified by the comparison between the calculated and experimentally measured quantities, including the concentration profiles in Ti/U diffusion couples. This work is established to provide fundamental information for U-based alloy design when the kinetics of microstructure evolution is of prime concern. © 2012 Elsevier Ltd. All rights reserved.
1. Introduction In recent years, the development of next-generation novel alloys that feature superior properties is highly demanded due to the limited efficiency and reliability of existing materials. However, to design such materials used for advanced applications, basic information and subsequent profound analysis, such as qualities of different candidates, are necessary. Traditional materials design is solely based on the trial-and-error approach. To overcome this hurdle and supplement purely experimental approaches, CALPHAD-based thermodynamics and kinetics have become a valuable approach for materials scientists/engineers to tailor materials behavior, thereby playing a key role in the development of new materials and optimization of existing ones [1,2]. In order to address safety issues related to nuclear power, the US Department of Energy initiated the RERTR program in 1978 to utilize nuclear research reactors with low enriched uranium (LEU) [3,4]. U–Mo alloys have been identified as a high performance fuel based on their high density and the stability of
γ -U under irradiation [5,6]. However, the application of U–Mo alloys has been hampered by the interaction with its Al alloy cladding, leading to the formation of intermetallic phases that should be controlled in order to enhance the functionality and the service life of such fuels [7,8]. It was reported that the addition of Ti, Nb and Zr in U–Mo alloys can mitigate such detrimental diffusional interactions, which are chosen based on their potential ability to suppress the formation of detrimental phases [9]. In order to explore the microstructure evolution of such alloys, the kinetic quantities are thus of great concern. From a fundamental point of view, the atomic mobilities are necessary to simulate and understand how the microstructure will evolve upon exposure to extreme conditions. To this end, CALPHAD can provide necessary information to facilitate alloy design, when the atomic mobilities are concurrently combined with thermodynamic description based on the CALPHAD framework. For the establishment of a general mobility database for U-based alloys, this work is thus established to provide the atomic mobilities for U, Ti, Nb and Mo for U–X (X = Ti, Nb and Mo) bcc alloys with the CALPHAD technique. 2. Model description
∗
Corresponding author. Tel.: +1 404 513 1544. E-mail addresses: [email protected], [email protected] (Y. Liu).
0364-5916/$ – see front matter © 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.calphad.2012.01.007
The temporal and spatial evolution of element i in an ncomponent system can be generally described by the following
50
Y. Liu et al. / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 37 (2012) 49–56
mass-conservation law: ∂ Ci + ∇ · J˜iN = 0 (1) ∂t where Ci is the concentration of element i with the unit of moles per volume; t is time; J˜iN stands for the flux of element i defined with respect to the number-fixed frame of reference, which is a function of the n − 1 independent concentration gradients expressed by: n −1
J˜iN = −
Dnij ∇ Cj ,
(2)
j=1
where Dnij denotes the interdiffusion coefficient given by Andersson and Ågren [10]: Dnij
n ∂µk ∂µk − , = (δik − xi )xk Mk ∂ xj ∂ xn k=1
(3)
where δik is the Kronecker delta (δik = 1 if i = k; otherwise δik = 0); xi and xk denote the molar fractions for elements i and k, respectively; µk is the chemical potential of element k; Mk is the atomic mobility of element k, which is given in the CALPHAD framework as [11] Mk =
1
exp
RT
−Qk∗ + RT ln(Mk0 )
=
RT
1 RT
exp
Φk RT
,
(4)
where Mk0 stands for the frequency factor; Qk∗ is the activation energy; R is the gas constant; T is temperature; Based on the CALPHAD framework, Φk depends on the composition and temperature, expressed by the Redlich–Kister polynomials with [11]:
Φk =
xi Φki
+
i
i
Φki
r
xi xj
j >i
r
i ,j Φk
3.2. The U–Nb binary system
(xi − xj )
r
,
(5)
r
i ,j Φk
where and are the mobility parameters for the endmembers and the interactions, respectively. It is noted that the interdiffusion fluxes are thus defined in the number-fixed frame of reference and the interdiffusion coefficients depict how the atoms mix in this frame of reference. In order to gain an in-depth understanding of the movement of atoms for the system of interest, the intrinsic diffusion coefficient with respect to the lattice frame of reference is needed, which is given by Liu et al. [11]: I
Dnij
= xi Mi
∂µi ∂µi − ∂ xj ∂ xn
,
(6)
where I Dnij is the intrinsic diffusion coefficient of element i used to correlate the flux of element i and the concentration gradient of element j with element n being the dependent one. In order to respect the thermodynamic and kinetic quantities separately, the intrinsic diffusion coefficient of element i can be alternatively given by Liu et al. [11]: I
Dnij = D∗i Fij ,
(7)
∗
where Di is the tracer diffusion coefficient of element i which can be calculated by D∗i = RTMi ; Fij is the thermodynamic factor that is correlated with the gradients of chemical potentials given by Liu et al. [11]: Fij =
xi RT
∂µi ∂µi − ∂ xj ∂ xn
.
Bochvar et al. [12] studied the self-diffusion coefficients of bcc U from 1073 to 1323 K with the serial sectioning method. Using diffusion couples made of natural U and U enriched with 20% 235 U and the residual activity method, Adda and Kirianenko [13] investigated the self-diffusion coefficients of bcc U in the temperature range 1073–1313 K. The diffusion of 235 U in bcc polycrystalline U was also characterized by Rothman et al. [14] from 1077 to 1343 K, who utilized the lathe sectioning technique. Keroulas et al. [15] explored the impurity diffusion of 234 U in bcc Ti with an improved technique that was designed to examine U concentrations in metals 104–105 times smaller than can be detected by autoradiography of 234 U. Such a technique depends on recording fission fragments in dielectrics, and the temperatures selected were 1298, 1253, 1213 and 1188 K. Pavlinov and Nakaneshnikov [16] studied the impurity diffusion coefficients of 235 U in polycrystalline bcc Ti from 1188 to 1473 K, where 235 U was vapor deposited and the residual activity and absorption technique was utilized. Fedorov and Smirnov [17] measured the impurity diffusion of 235 U in bcc Ti between 1182 and 1775 K, but the techniques and details were not reported to show how to determine such quantities. The interdiffusion for bcc Ti–U alloys was studied by Adda and Philibert [18] with various techniques, including metallography, microhardness and the Castaing microanalyser. The interdiffusion coefficients were evaluated for the U/Ti diffusion couples annealed at 1348, 1323, 1273 and 1223 K, and the results were presented in the U concentration between 0.05 and 0.9. In addition, the Kirkendall effect was observed and the intrinsic diffusion coefficients of U and Ti were determined with the Darken relation.
(8)
3. Experimental information 3.1. The U–Ti binary system There have been several reports for the self-diffusion of bcc U. With vapor-deposited 233 U isotopes on U polycrystals,
There have been many measurements for the impurity diffusion of U in bcc Nb and Nb in bcc U. Using 235 U isotopes that were vapor deposited and the subsequent characterization with the residual activity technique, Pavlinov et al. [19] investigated the impurity diffusion of U in polycrystalline bcc Nb from 1773 to 2273 K. Through vapor deposition of 235 U and the serial sectioning technique, Fedorov et al. [20] investigated the impurity diffusion of U in polycrystalline bcc U from 1923 to 2273 K. The impurity diffusion coefficients of Nb in bcc U were studied by Peterson and Rothman [21] using 95 Nb and the lathe sectioning technique, where the temperature varied from 1064 to 1375 K and the polycrystalline bcc U was employed. The tracer diffusion coefficients of 235 U and 95 Nb in bcc U–Nb alloys were investigated by Fedorov et al. [22] with the method of stripping off layers and measuring the integrated radioactivity of the remainder of the species. The diffusion coefficients of U and Nb were evaluated across a wide temperature regime with the concentration of Nb varied from 0 to 100 at.%. Generally, if the tracer diffusion coefficients were plotted in the Arrhenius format, the linear relations for U and Nb were found. Interdiffusion studies in the U–Nb binary system were conducted by Peterson and Ogilvie [23,24], using U/Nb diffusion couples and an electron microbeam probe. The temperatures adopted were 1269, 1165 and 1073 K, respectively and the concentration curves in the U/Nb diffusion couples were recorded. However, the metallography analysis showed that an intermetallic phase that is not shown in the current U–Nb binary phase diagram was detected. The interdiffusion coefficients were calculated based on the Matano method, and the intrinsic diffusion coefficients of U and Nb were obtained from the velocity of the marker interface. Using diffusion couples and the electron microprobe analysis, the interdiffusion behavior in bcc U–Nb alloys was also explored by Fedorov and Smirnov [25] at 1353 K, where the Matano method was adopted to retrieve such values from concentration distribution within diffusion couples.
Y. Liu et al. / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 37 (2012) 49–56
51
Table 1 Mobility parameters for U, Ti, Nb and Mo in bcc U–X (X = Ti, Nb and Mo) alloys (all in SI units). Phase
Model
Mobility
Parameters
ΦUU:Va = −11 2127.11 − 132.24T ΦUTi:Va = −135 965.41 − 129.38T ΦUNb:Va = −387 210.79 − 127.09T U
ΦUMo:Va = −361 400.79 − 168.55T 0
ΦUMo,U:Va = 119 872.45
0
ΦUTi,U:Va = −69 983.21
1
ΦUTi,U:Va = −40 324.14
0
ΦUNb,U:Va = 60 032.48
ΦTiTi:Va = −151 989.95 − 127.37T [29] bcc
(U, Ti, Nb, Mo)1 (Va)3
Ti
ΦTiU:Va = −115 326.47 − 139.42T 0
ΦTiTi,U:Va = −60 025.36
1
ΦTiTi,U:Va = −30 131.24
Nb:Va ΦNb = −395 598.95 − 82.03T [30]
Nb
U:Va ΦNb = −107 072.73 − 181.63T 0
Nb,U:Va ΦNb = −30 576.42
1
Nb,U:Va ΦNb = −119 872.45
Mo:Va ΦMo = −479 740.87 − 63.98T [31]
Mo
U :Va ΦMo = −149 876.35 − 110.83T 0
3.3. The U–Mo binary system There have been limited reports for the impurity diffusion of U in bcc Mo. Fedorov et al. [20] reported the impurity diffusion coefficients of 235 U in bcc Mo from 1973 to 2473 K, and the reported values show the Arrhenius behavior. However, the technique and experimental details were not mentioned. Pavlinov and Nakonechnikov [19] studied the diffusion of 235 U in bcc Mo from 1773 to 2273 K, using the residual activity technique. The diffusion of U in bcc Mo is characterized by a high activation energy, which is attributable to the high melting point of Mo. The tracer diffusion of 235 U and 99 Mo was investigated by Fedorov and Smirnov [25] for U–Mo alloys with the Mo concentration up to 30 at.% in a wide temperature range from 1173 to 1373 K. However, the technique which was used in their experiments was not mentioned. Adda and Philibert [26] conducted a very complete study of the interdiffusion in U–Mo alloys, using pressure bonded diffusion couples made of pure U and an alloy of U-30 at.% Mo. The annealing temperature varied from 1323 to 1123 K, and the concentration gradients were determined with the electron microbeam probe. The intrinsic diffusion coefficients were also measured with the aid of markers and the interdiffusion coefficients for the marker interface. 4. Results and discussion DICTRA is a numeric software package that is designed to solve diffusion-controlled problems. This software is formulated within the CALPHAD framework, in which the numeric procedure is simultaneously interfaced with Thermo-calc to retrieve necessary thermodynamic information. In this work, the atomic mobilities for U, Ti, Nb and Mo are inversely explored in the Parrot module in DICTRA, based on the experimental data reported in the literature as well as the thermodynamic descriptions for U–Ti [27], U–Nb [28] and U–Mo [27]. During the assessment, the self-diffusion and impurity diffusion coefficients are evaluated first with the aim to establish mobility end-members that are necessary to span the mobility space. Second, the tracer diffusion
Mo,U:Va ΦMo = 79 325.84 + 32.45T
coefficients and intrinsic diffusion coefficients are taken into consideration to evaluate the interaction parameters for each element. The quality of such tracer diffusion coefficients and intrinsic diffusion coefficients used in the Parrot module is judged by their extrapolation ability to arrive at self-diffusion coefficients and impurity diffusion coefficients. Third, the interdiffusion coefficients are used to further evaluate the interaction parameters so as to reproduce the change of interdiffusivities with respect to composition. Finally, the parameters for the atomic mobilities are carefully adjusted so as to represent a great majority of the experimental data reported in the literature. The atomic mobilities obtained in this work are presented in Table 1, where the reported mobility end-members for Ti in bcc Ti [29], Nb in bcc Nb [30] and Mo in bcc Mo [31] are also given. 4.1. The U–Ti binary system The thermodynamic description of the U–Ti binary system is taken from the assessment of Berche et al. [27]. In the U–Ti binary phase diagram, U and Ti can form a continuous solid solution for the bcc phase over a wide temperature regime, allowing the diffusion characteristics for impurity diffusion, intrinsic diffusion and interdiffusion to be well explored. The mobility end-member for self-diffusion of bcc U is evaluated from the experimental reports from [12–14], where the experimental data are pretty self-consistent, and the results are given in Fig. 1. The mobility end-member for U to diffuse in bcc Ti is obtained with the impurity diffusion coefficients reported by Keroulas et al. [15], Pavlinov and Nakaneshnikov [16], and Fedorov and Smirnov [17] as well as the extrapolated interdiffusion coefficients given by Adda and Philibert [18] near the Ti edge, the results of which are given in Fig. 2 along with calculated values with the atomic mobilities presented in this work. On the other hand, there has been no experimental report for the impurity diffusion of Ti in bcc U. In order to establish the atomic mobilities for Ti in bcc U, the mobility end-member is obtained by evaluation of the extrapolated impurity diffusion coefficients from
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Y. Liu et al. / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 37 (2012) 49–56
Fig. 1. Calculated and experimentally measured self-diffusion coefficients of U in bcc U.
Fig. 3. Calculated and experimentally measured intrinsic diffusion coefficients of U in bcc U–Ti alloys.
Fig. 2. Calculated and experimentally measured impurity diffusion coefficients of U in bcc Ti.
Fig. 4. Calculated and experimentally measured intrinsic diffusion coefficients of Ti in bcc U–Ti alloys.
the interdiffusion coefficients reported by Adda and Philibert [18] around the U-rich edge. The calculated intrinsic diffusion coefficients for U and Ti are presented in Figs. 3 and 4, along with the experimental values from [18]. Evidently, the intrinsic diffusion coefficients for such two elements in the Ti–U bcc alloys generally form a valley against the composition around 40 at.% U. As is evident in Figs. 3 and 4, there is good agreement between the calculated and experimentally measured data. The change of interdiffusion coefficients against the concentration also features the same characteristics. As shown in Fig. 5, the calculated interdiffusion coefficients are characterized by a valley around the Ti-rich edge, which is consistent with the experimental tendency observed by Adda and Philibert [18]. The agreement is acceptable within experimental error. In order to check the quality of atomic mobilities as well as the thermodynamic description used in this work, it is also beneficial to simulate concentration profiles for diffusion couples, the results of which can thus be compared with experimental values. Two computational simulations are conducted in this work for
U/Ti diffusion couples with DICTRA. One is annealed at 1323 K for 86 400 s and the other annealed at 1223 K for 86 400 s. The comparisons between the experimental data from [18] and modelpredicted concentration curves are given in Figs. 6 and 7, where the agreement is excellent. It is noted that the concentration curves are generally asymmetric with respect to the Matano interface, with a slightly longer tail on the U-rich side, which indicates higher impurity diffusion coefficients around the U-rich edge. 4.2. The U–Nb binary system The thermodynamic description of the U–Nb binary system is taken from the optimization of Liu et al. [28]. In the U–Nb binary phase diagram, the bcc phase is characterized by spinodal decomposition and a continuous solid solution over a large temperature regime. The calculated impurity diffusion coefficients of U in bcc Nb are given in Fig. 8 with the experimental data from [19,20] and the extrapolated values from the interdiffusion coefficients reported
Y. Liu et al. / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 37 (2012) 49–56
53
Fig. 5. Calculated and experimentally measured interdiffusion coefficients in bcc U–Ti alloys at various temperatures.
Fig. 7. Calculated and experimentally measured U concentration in one U/Ti diffusion couple annealed at 1223 K for 86 400 s.
Fig. 6. Calculated and experimentally measured U concentration in one U/Ti diffusion couple annealed at 1323 K for 86 400 s.
Fig. 8. Calculated and experimentally measured impurity diffusion coefficients of U in bcc Nb.
by Fedorov and Smirnov [25]. It is noted that the data by Pavlinov et al. [19] in Fig. 8 are higher by two orders of magnitude, compared with those values given by Fedorov et al. [20] and Fedorov and Smirnov [25]. These two sets of experimental values from [20, 25] are generally consistent with each other, allowing the mobility end-member for U in bcc Nb to be valid over a large temperature regime. The calculated impurity diffusion coefficients of Nb in bcc U are given in Fig. 9, together with the experimental data from [21,22] and the extrapolated values from the interdiffusion coefficients given by Peterson and Ogilvie [23], and Fedorov and Smirnov [25]. The experimental results reported by Peterson and Rothman [21] differ significantly from those values reported by Fedorov and Smirnov [22], Peterson and Ogilvie [23] and Fedorov and Smirnov [25]. As is evident, experimental values from [22,23,25] are generally in good agreement, resulting in a well-established mobility end-member for Nb in bcc U. The tracer diffusion coefficients of U and Nb are explored in this work with the aid of experimental report from [13,22], as given in Figs. 10 and 11. It is evident that the atomic mobilities
provided in this work allow the U and Nb motilities to be well represented across a wide temperature and concentration regime. The U and Nb atomic mobilities generally decrease with the decrease of U concentration. The interdiffusion coefficients for U–Nb binary alloys are also explored in this work, and the calculated and experimentally measured values from [25] at 1273 K are given in Fig. 12. It is noted that the interdiffusion coefficients generally decrease with the concentration of Nb and the curve is characterized by a curvature change around 50 at.% Nb, which is resulted from the thermodynamic factor involved in the interdiffusion coefficients. When the temperature is further decreased from 1273 K and the alloys maintain a composition around 50 at.% Nb, the negative thermodynamic factor will also lead to negative interdiffusion coefficients. Unlike the activation enthalpies for tracer diffusion coefficients, the activation enthalpies for interdiffusion coefficients are dependent on thermodynamics as well as kinetics. The calculated activation enthalpies for the interdiffusion coefficients are presented in Fig. 13, where the reported values from [22,23,25] can be well
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Y. Liu et al. / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 37 (2012) 49–56
Fig. 9. Calculated and experimentally measured impurity diffusion coefficients of Nb in bcc U.
Fig. 11. Calculated and experimentally measured U tracer diffusion coefficients for the U–Nb bcc alloys.
Fig. 10. Calculated and experimentally measured Nb tracer diffusion coefficients for the U–Nb bcc alloys.
Fig. 12. Calculated and experimentally measured interdiffusion coefficients for the U–Nb bcc alloys.
represented. In general, the activation enthalpies increase with the addition of Nb over the whole concentration range, and the experimental values from different authors generally distribute around the calculated line.
A careful evaluation indicates that they almost share the same slope, i.e. the activation enthalpies for the impurity diffusion coefficients of U in bcc Nb, but the intersect is different, i.e. the frequency factor for the impurity diffusion coefficients. In this work, there is no way to judge which one is more reliable. Preference is given to the work of Fedorov et al. [20], as their data quality has been verified in Figs. 2, 8 and 9. The mobility endmember for Mo in bcc U is based on the extrapolated interdiffusion coefficients measured by Adda and Philibert [26]. Although there are tracer diffusion coefficients of U and Mo in U–Mo alloys from [25], the magnitude of such data does not allow the physical significance to be acknowledged. As such, they were not accepted in this work. However, the U tracer diffusion coefficients from [13] show reasonable order of magnitude, which can be seen for the comparison in Fig. 15. The calculated intrinsic diffusion coefficients of Mo in bcc Mo–U alloys are given in Fig. 16, together with the experimental values from [26] at 1273, 1223, and 1123 K. It is evident that the intrinsic diffusion coefficients of Mo generally increase with
4.3. The U–Mo binary system The thermodynamic description of the U–Mo binary system is taken from the work of Berche et al. [27]. Compared with the U–Nb phase diagram, the bcc phase in the U–Mo system shares different characteristics. The bcc phase is characterized by spinodal decomposition over the whole temperature regime within which the bcc phase is stable in the U–Mo binary system. Thus, the single phase region for bcc phase is narrow and relevant experimental work may experience great challenge. Fig. 14 is the model-predicted temperature dependence of the impurity diffusion coefficients of U in bcc Mo compared with the corresponding experimental data from [19,20]. It is noted that such two sets of experimental data differ greatly from the magnitude.
Y. Liu et al. / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 37 (2012) 49–56
Fig. 13. Calculated and experimentally measured activation enthalpies for interdiffusion coefficients for U–Nb bcc alloys.
Fig. 14. Calculated and experimentally measured impurity diffusion coefficients of U in bcc Mo.
the U content, which is consistent with the change of the solidus curve in the U–Mo binary phase diagram. As can be seen, the solidus curve generally decreases with the addition of U. The same tendency can be evident in Fig. 17, where the calculated and experimentally measured interdiffusion coefficients generally show a good agreement at four temperatures (1323, 1273, 1223 and 1123 K), considering the experimental errors and the difficulty to obtain suitable concentration gradients in evaluating the interdiffusion coefficients. The activation enthalpies for the interdiffusion coefficients are given in Fig. 18, where the calculated line shows a tendency to decrease with the addition of U. It is noted that the experimental data from [26] feature significant deviation from the calculated line which is believed to be more reliable. As mentioned above, the tracer diffusion coefficients from [25] are not taken into consideration in this work, as the magnitude of such diffusivities does not allow the self-diffusion coefficients of bcc U to be extrapolated. This fact can be further verified by the calculated interdiffusion coefficients with the use of thermodynamic description from [27], the tracer diffusion
55
Fig. 15. Calculated and experimentally measured tracer diffusion coefficients of U in bcc U-10 at.% Mo alloys.
Fig. 16. Calculated and experimentally measured intrinsic diffusion coefficients of Mo in bcc U–Mo alloys.
coefficients from [25] and the Darken formula. The calculated interdiffusion coefficients with such a combination in the region of high U content differ substantially from the experimental data reported by Adda and Philibert [26] by more than one order of magnitude. 5. Conclusions Within the CALPHAD framework, the atomic mobilities for U, Ti, Nb and Mo are explored in this work in order to aid the design for novel nuclear materials. The thermodynamic descriptions of the U–Ti, U–Nb and U–Mo binary systems published in the literature are utilized in this work to provide thermodynamic factors that are necessary to generate intrinsic diffusion coefficients and interdiffusion coefficients. In order to establish high-quality atomic mobilities for such elements, the self-diffusion coefficients, impurity diffusion coefficients, tracer diffusion coefficients, intrinsic diffusion coefficients, and interdiffusion coefficients are concurrently taken into consideration. The calculated diffusion coefficients are
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Y. Liu et al. / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 37 (2012) 49–56
the calculated concentration distribution of Ti is pretty close to the measured quantities. In general, the atomic mobilities provided in this work show a satisfactory behavior in studying diffusionrelated phenomena, when such fundamental quantities are combined with thermodynamic descriptions. The results of this work are beneficial to explore novel nuclear alloys when cost reduction is the first consideration. Appendix. Supplementary data Supplementary material related to this article can be found online at doi:10.1016/j.calphad.2012.01.007. References
Fig. 17. Calculated and experimentally measured interdiffusion coefficients for U–Mo bcc alloys.
Fig. 18. Calculated and experimentally measured activation enthalpies for interdiffusion in bcc U–Mo alloys.
then compared with the measured ones, where a satisfactory agreement is evident. Computational simulations are performed for the U/Ti diffusion couples annealed at two temperatures, where
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CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry journal homepage: www.elsevier.com/locate/calphad
Atomic mobilities, diffusivities and their kinetic implications for U–X (X = Ti, Nb and Mo) bcc alloys Yajun Liu a,∗ , Di Yu b , Yong Du c , Guang Sheng d , Zhaohui Long e , Jiang Wang f , Lijun Zhang c a
Western Transportation Institute, Montana State University, Bozeman, MT, 59715, USA
b
Computer Science and Engineering, University of South Florida, Tampa, FL, 33620, USA
c
State Key Laboratory of Powder Metallurgy, Central South University, Changsha, Hunan, 410083, PR China
d
Scientific Forming Technologies Corporation, Columbus, OH, 43235, USA
e
School of Mechanical Engineering, Xiangtan University, Xiangtan, Hunan, 411105, PR China
f
School of Materials Science and Engineering, Guilin University of Electronic Technology, Guilin, Guangxi, 541004, PR China
article
info
Article history: Received 15 November 2011 Received in revised form 4 January 2012 Accepted 20 January 2012 Available online 15 February 2012 Keywords: Diffusion Mobility CALPHAD DICTRA bcc U–X (X = Ti, Nb and Mo) alloys
abstract Based on various kinds of diffusivities as well as the thermodynamic descriptions within the CALPHAD framework, the atomic mobilities of U, Ti, Nb and Mo are explored in this work with the DICTRA software. The mobility end-members are evaluated from the impurity diffusivities as well as the extrapolated interdiffusivities, while the interaction parameters for atomic mobilities are determined from the tracer diffusivities, the intrinsic diffusivities and the interdiffusivities. The reliability of such purely kinetic quantities is carefully verified by the comparison between the calculated and experimentally measured quantities, including the concentration profiles in Ti/U diffusion couples. This work is established to provide fundamental information for U-based alloy design when the kinetics of microstructure evolution is of prime concern. © 2012 Elsevier Ltd. All rights reserved.
1. Introduction In recent years, the development of next-generation novel alloys that feature superior properties is highly demanded due to the limited efficiency and reliability of existing materials. However, to design such materials used for advanced applications, basic information and subsequent profound analysis, such as qualities of different candidates, are necessary. Traditional materials design is solely based on the trial-and-error approach. To overcome this hurdle and supplement purely experimental approaches, CALPHAD-based thermodynamics and kinetics have become a valuable approach for materials scientists/engineers to tailor materials behavior, thereby playing a key role in the development of new materials and optimization of existing ones [1,2]. In order to address safety issues related to nuclear power, the US Department of Energy initiated the RERTR program in 1978 to utilize nuclear research reactors with low enriched uranium (LEU) [3,4]. U–Mo alloys have been identified as a high performance fuel based on their high density and the stability of
γ -U under irradiation [5,6]. However, the application of U–Mo alloys has been hampered by the interaction with its Al alloy cladding, leading to the formation of intermetallic phases that should be controlled in order to enhance the functionality and the service life of such fuels [7,8]. It was reported that the addition of Ti, Nb and Zr in U–Mo alloys can mitigate such detrimental diffusional interactions, which are chosen based on their potential ability to suppress the formation of detrimental phases [9]. In order to explore the microstructure evolution of such alloys, the kinetic quantities are thus of great concern. From a fundamental point of view, the atomic mobilities are necessary to simulate and understand how the microstructure will evolve upon exposure to extreme conditions. To this end, CALPHAD can provide necessary information to facilitate alloy design, when the atomic mobilities are concurrently combined with thermodynamic description based on the CALPHAD framework. For the establishment of a general mobility database for U-based alloys, this work is thus established to provide the atomic mobilities for U, Ti, Nb and Mo for U–X (X = Ti, Nb and Mo) bcc alloys with the CALPHAD technique. 2. Model description
∗
Corresponding author. Tel.: +1 404 513 1544. E-mail addresses: [email protected], [email protected] (Y. Liu).
0364-5916/$ – see front matter © 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.calphad.2012.01.007
The temporal and spatial evolution of element i in an ncomponent system can be generally described by the following
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Y. Liu et al. / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 37 (2012) 49–56
mass-conservation law: ∂ Ci + ∇ · J˜iN = 0 (1) ∂t where Ci is the concentration of element i with the unit of moles per volume; t is time; J˜iN stands for the flux of element i defined with respect to the number-fixed frame of reference, which is a function of the n − 1 independent concentration gradients expressed by: n −1
J˜iN = −
Dnij ∇ Cj ,
(2)
j=1
where Dnij denotes the interdiffusion coefficient given by Andersson and Ågren [10]: Dnij
n ∂µk ∂µk − , = (δik − xi )xk Mk ∂ xj ∂ xn k=1
(3)
where δik is the Kronecker delta (δik = 1 if i = k; otherwise δik = 0); xi and xk denote the molar fractions for elements i and k, respectively; µk is the chemical potential of element k; Mk is the atomic mobility of element k, which is given in the CALPHAD framework as [11] Mk =
1
exp
RT
−Qk∗ + RT ln(Mk0 )
=
RT
1 RT
exp
Φk RT
,
(4)
where Mk0 stands for the frequency factor; Qk∗ is the activation energy; R is the gas constant; T is temperature; Based on the CALPHAD framework, Φk depends on the composition and temperature, expressed by the Redlich–Kister polynomials with [11]:
Φk =
xi Φki
+
i
i
Φki
r
xi xj
j >i
r
i ,j Φk
3.2. The U–Nb binary system
(xi − xj )
r
,
(5)
r
i ,j Φk
where and are the mobility parameters for the endmembers and the interactions, respectively. It is noted that the interdiffusion fluxes are thus defined in the number-fixed frame of reference and the interdiffusion coefficients depict how the atoms mix in this frame of reference. In order to gain an in-depth understanding of the movement of atoms for the system of interest, the intrinsic diffusion coefficient with respect to the lattice frame of reference is needed, which is given by Liu et al. [11]: I
Dnij
= xi Mi
∂µi ∂µi − ∂ xj ∂ xn
,
(6)
where I Dnij is the intrinsic diffusion coefficient of element i used to correlate the flux of element i and the concentration gradient of element j with element n being the dependent one. In order to respect the thermodynamic and kinetic quantities separately, the intrinsic diffusion coefficient of element i can be alternatively given by Liu et al. [11]: I
Dnij = D∗i Fij ,
(7)
∗
where Di is the tracer diffusion coefficient of element i which can be calculated by D∗i = RTMi ; Fij is the thermodynamic factor that is correlated with the gradients of chemical potentials given by Liu et al. [11]: Fij =
xi RT
∂µi ∂µi − ∂ xj ∂ xn
.
Bochvar et al. [12] studied the self-diffusion coefficients of bcc U from 1073 to 1323 K with the serial sectioning method. Using diffusion couples made of natural U and U enriched with 20% 235 U and the residual activity method, Adda and Kirianenko [13] investigated the self-diffusion coefficients of bcc U in the temperature range 1073–1313 K. The diffusion of 235 U in bcc polycrystalline U was also characterized by Rothman et al. [14] from 1077 to 1343 K, who utilized the lathe sectioning technique. Keroulas et al. [15] explored the impurity diffusion of 234 U in bcc Ti with an improved technique that was designed to examine U concentrations in metals 104–105 times smaller than can be detected by autoradiography of 234 U. Such a technique depends on recording fission fragments in dielectrics, and the temperatures selected were 1298, 1253, 1213 and 1188 K. Pavlinov and Nakaneshnikov [16] studied the impurity diffusion coefficients of 235 U in polycrystalline bcc Ti from 1188 to 1473 K, where 235 U was vapor deposited and the residual activity and absorption technique was utilized. Fedorov and Smirnov [17] measured the impurity diffusion of 235 U in bcc Ti between 1182 and 1775 K, but the techniques and details were not reported to show how to determine such quantities. The interdiffusion for bcc Ti–U alloys was studied by Adda and Philibert [18] with various techniques, including metallography, microhardness and the Castaing microanalyser. The interdiffusion coefficients were evaluated for the U/Ti diffusion couples annealed at 1348, 1323, 1273 and 1223 K, and the results were presented in the U concentration between 0.05 and 0.9. In addition, the Kirkendall effect was observed and the intrinsic diffusion coefficients of U and Ti were determined with the Darken relation.
(8)
3. Experimental information 3.1. The U–Ti binary system There have been several reports for the self-diffusion of bcc U. With vapor-deposited 233 U isotopes on U polycrystals,
There have been many measurements for the impurity diffusion of U in bcc Nb and Nb in bcc U. Using 235 U isotopes that were vapor deposited and the subsequent characterization with the residual activity technique, Pavlinov et al. [19] investigated the impurity diffusion of U in polycrystalline bcc Nb from 1773 to 2273 K. Through vapor deposition of 235 U and the serial sectioning technique, Fedorov et al. [20] investigated the impurity diffusion of U in polycrystalline bcc U from 1923 to 2273 K. The impurity diffusion coefficients of Nb in bcc U were studied by Peterson and Rothman [21] using 95 Nb and the lathe sectioning technique, where the temperature varied from 1064 to 1375 K and the polycrystalline bcc U was employed. The tracer diffusion coefficients of 235 U and 95 Nb in bcc U–Nb alloys were investigated by Fedorov et al. [22] with the method of stripping off layers and measuring the integrated radioactivity of the remainder of the species. The diffusion coefficients of U and Nb were evaluated across a wide temperature regime with the concentration of Nb varied from 0 to 100 at.%. Generally, if the tracer diffusion coefficients were plotted in the Arrhenius format, the linear relations for U and Nb were found. Interdiffusion studies in the U–Nb binary system were conducted by Peterson and Ogilvie [23,24], using U/Nb diffusion couples and an electron microbeam probe. The temperatures adopted were 1269, 1165 and 1073 K, respectively and the concentration curves in the U/Nb diffusion couples were recorded. However, the metallography analysis showed that an intermetallic phase that is not shown in the current U–Nb binary phase diagram was detected. The interdiffusion coefficients were calculated based on the Matano method, and the intrinsic diffusion coefficients of U and Nb were obtained from the velocity of the marker interface. Using diffusion couples and the electron microprobe analysis, the interdiffusion behavior in bcc U–Nb alloys was also explored by Fedorov and Smirnov [25] at 1353 K, where the Matano method was adopted to retrieve such values from concentration distribution within diffusion couples.
Y. Liu et al. / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 37 (2012) 49–56
51
Table 1 Mobility parameters for U, Ti, Nb and Mo in bcc U–X (X = Ti, Nb and Mo) alloys (all in SI units). Phase
Model
Mobility
Parameters
ΦUU:Va = −11 2127.11 − 132.24T ΦUTi:Va = −135 965.41 − 129.38T ΦUNb:Va = −387 210.79 − 127.09T U
ΦUMo:Va = −361 400.79 − 168.55T 0
ΦUMo,U:Va = 119 872.45
0
ΦUTi,U:Va = −69 983.21
1
ΦUTi,U:Va = −40 324.14
0
ΦUNb,U:Va = 60 032.48
ΦTiTi:Va = −151 989.95 − 127.37T [29] bcc
(U, Ti, Nb, Mo)1 (Va)3
Ti
ΦTiU:Va = −115 326.47 − 139.42T 0
ΦTiTi,U:Va = −60 025.36
1
ΦTiTi,U:Va = −30 131.24
Nb:Va ΦNb = −395 598.95 − 82.03T [30]
Nb
U:Va ΦNb = −107 072.73 − 181.63T 0
Nb,U:Va ΦNb = −30 576.42
1
Nb,U:Va ΦNb = −119 872.45
Mo:Va ΦMo = −479 740.87 − 63.98T [31]
Mo
U :Va ΦMo = −149 876.35 − 110.83T 0
3.3. The U–Mo binary system There have been limited reports for the impurity diffusion of U in bcc Mo. Fedorov et al. [20] reported the impurity diffusion coefficients of 235 U in bcc Mo from 1973 to 2473 K, and the reported values show the Arrhenius behavior. However, the technique and experimental details were not mentioned. Pavlinov and Nakonechnikov [19] studied the diffusion of 235 U in bcc Mo from 1773 to 2273 K, using the residual activity technique. The diffusion of U in bcc Mo is characterized by a high activation energy, which is attributable to the high melting point of Mo. The tracer diffusion of 235 U and 99 Mo was investigated by Fedorov and Smirnov [25] for U–Mo alloys with the Mo concentration up to 30 at.% in a wide temperature range from 1173 to 1373 K. However, the technique which was used in their experiments was not mentioned. Adda and Philibert [26] conducted a very complete study of the interdiffusion in U–Mo alloys, using pressure bonded diffusion couples made of pure U and an alloy of U-30 at.% Mo. The annealing temperature varied from 1323 to 1123 K, and the concentration gradients were determined with the electron microbeam probe. The intrinsic diffusion coefficients were also measured with the aid of markers and the interdiffusion coefficients for the marker interface. 4. Results and discussion DICTRA is a numeric software package that is designed to solve diffusion-controlled problems. This software is formulated within the CALPHAD framework, in which the numeric procedure is simultaneously interfaced with Thermo-calc to retrieve necessary thermodynamic information. In this work, the atomic mobilities for U, Ti, Nb and Mo are inversely explored in the Parrot module in DICTRA, based on the experimental data reported in the literature as well as the thermodynamic descriptions for U–Ti [27], U–Nb [28] and U–Mo [27]. During the assessment, the self-diffusion and impurity diffusion coefficients are evaluated first with the aim to establish mobility end-members that are necessary to span the mobility space. Second, the tracer diffusion
Mo,U:Va ΦMo = 79 325.84 + 32.45T
coefficients and intrinsic diffusion coefficients are taken into consideration to evaluate the interaction parameters for each element. The quality of such tracer diffusion coefficients and intrinsic diffusion coefficients used in the Parrot module is judged by their extrapolation ability to arrive at self-diffusion coefficients and impurity diffusion coefficients. Third, the interdiffusion coefficients are used to further evaluate the interaction parameters so as to reproduce the change of interdiffusivities with respect to composition. Finally, the parameters for the atomic mobilities are carefully adjusted so as to represent a great majority of the experimental data reported in the literature. The atomic mobilities obtained in this work are presented in Table 1, where the reported mobility end-members for Ti in bcc Ti [29], Nb in bcc Nb [30] and Mo in bcc Mo [31] are also given. 4.1. The U–Ti binary system The thermodynamic description of the U–Ti binary system is taken from the assessment of Berche et al. [27]. In the U–Ti binary phase diagram, U and Ti can form a continuous solid solution for the bcc phase over a wide temperature regime, allowing the diffusion characteristics for impurity diffusion, intrinsic diffusion and interdiffusion to be well explored. The mobility end-member for self-diffusion of bcc U is evaluated from the experimental reports from [12–14], where the experimental data are pretty self-consistent, and the results are given in Fig. 1. The mobility end-member for U to diffuse in bcc Ti is obtained with the impurity diffusion coefficients reported by Keroulas et al. [15], Pavlinov and Nakaneshnikov [16], and Fedorov and Smirnov [17] as well as the extrapolated interdiffusion coefficients given by Adda and Philibert [18] near the Ti edge, the results of which are given in Fig. 2 along with calculated values with the atomic mobilities presented in this work. On the other hand, there has been no experimental report for the impurity diffusion of Ti in bcc U. In order to establish the atomic mobilities for Ti in bcc U, the mobility end-member is obtained by evaluation of the extrapolated impurity diffusion coefficients from
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Y. Liu et al. / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 37 (2012) 49–56
Fig. 1. Calculated and experimentally measured self-diffusion coefficients of U in bcc U.
Fig. 3. Calculated and experimentally measured intrinsic diffusion coefficients of U in bcc U–Ti alloys.
Fig. 2. Calculated and experimentally measured impurity diffusion coefficients of U in bcc Ti.
Fig. 4. Calculated and experimentally measured intrinsic diffusion coefficients of Ti in bcc U–Ti alloys.
the interdiffusion coefficients reported by Adda and Philibert [18] around the U-rich edge. The calculated intrinsic diffusion coefficients for U and Ti are presented in Figs. 3 and 4, along with the experimental values from [18]. Evidently, the intrinsic diffusion coefficients for such two elements in the Ti–U bcc alloys generally form a valley against the composition around 40 at.% U. As is evident in Figs. 3 and 4, there is good agreement between the calculated and experimentally measured data. The change of interdiffusion coefficients against the concentration also features the same characteristics. As shown in Fig. 5, the calculated interdiffusion coefficients are characterized by a valley around the Ti-rich edge, which is consistent with the experimental tendency observed by Adda and Philibert [18]. The agreement is acceptable within experimental error. In order to check the quality of atomic mobilities as well as the thermodynamic description used in this work, it is also beneficial to simulate concentration profiles for diffusion couples, the results of which can thus be compared with experimental values. Two computational simulations are conducted in this work for
U/Ti diffusion couples with DICTRA. One is annealed at 1323 K for 86 400 s and the other annealed at 1223 K for 86 400 s. The comparisons between the experimental data from [18] and modelpredicted concentration curves are given in Figs. 6 and 7, where the agreement is excellent. It is noted that the concentration curves are generally asymmetric with respect to the Matano interface, with a slightly longer tail on the U-rich side, which indicates higher impurity diffusion coefficients around the U-rich edge. 4.2. The U–Nb binary system The thermodynamic description of the U–Nb binary system is taken from the optimization of Liu et al. [28]. In the U–Nb binary phase diagram, the bcc phase is characterized by spinodal decomposition and a continuous solid solution over a large temperature regime. The calculated impurity diffusion coefficients of U in bcc Nb are given in Fig. 8 with the experimental data from [19,20] and the extrapolated values from the interdiffusion coefficients reported
Y. Liu et al. / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 37 (2012) 49–56
53
Fig. 5. Calculated and experimentally measured interdiffusion coefficients in bcc U–Ti alloys at various temperatures.
Fig. 7. Calculated and experimentally measured U concentration in one U/Ti diffusion couple annealed at 1223 K for 86 400 s.
Fig. 6. Calculated and experimentally measured U concentration in one U/Ti diffusion couple annealed at 1323 K for 86 400 s.
Fig. 8. Calculated and experimentally measured impurity diffusion coefficients of U in bcc Nb.
by Fedorov and Smirnov [25]. It is noted that the data by Pavlinov et al. [19] in Fig. 8 are higher by two orders of magnitude, compared with those values given by Fedorov et al. [20] and Fedorov and Smirnov [25]. These two sets of experimental values from [20, 25] are generally consistent with each other, allowing the mobility end-member for U in bcc Nb to be valid over a large temperature regime. The calculated impurity diffusion coefficients of Nb in bcc U are given in Fig. 9, together with the experimental data from [21,22] and the extrapolated values from the interdiffusion coefficients given by Peterson and Ogilvie [23], and Fedorov and Smirnov [25]. The experimental results reported by Peterson and Rothman [21] differ significantly from those values reported by Fedorov and Smirnov [22], Peterson and Ogilvie [23] and Fedorov and Smirnov [25]. As is evident, experimental values from [22,23,25] are generally in good agreement, resulting in a well-established mobility end-member for Nb in bcc U. The tracer diffusion coefficients of U and Nb are explored in this work with the aid of experimental report from [13,22], as given in Figs. 10 and 11. It is evident that the atomic mobilities
provided in this work allow the U and Nb motilities to be well represented across a wide temperature and concentration regime. The U and Nb atomic mobilities generally decrease with the decrease of U concentration. The interdiffusion coefficients for U–Nb binary alloys are also explored in this work, and the calculated and experimentally measured values from [25] at 1273 K are given in Fig. 12. It is noted that the interdiffusion coefficients generally decrease with the concentration of Nb and the curve is characterized by a curvature change around 50 at.% Nb, which is resulted from the thermodynamic factor involved in the interdiffusion coefficients. When the temperature is further decreased from 1273 K and the alloys maintain a composition around 50 at.% Nb, the negative thermodynamic factor will also lead to negative interdiffusion coefficients. Unlike the activation enthalpies for tracer diffusion coefficients, the activation enthalpies for interdiffusion coefficients are dependent on thermodynamics as well as kinetics. The calculated activation enthalpies for the interdiffusion coefficients are presented in Fig. 13, where the reported values from [22,23,25] can be well
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Y. Liu et al. / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 37 (2012) 49–56
Fig. 9. Calculated and experimentally measured impurity diffusion coefficients of Nb in bcc U.
Fig. 11. Calculated and experimentally measured U tracer diffusion coefficients for the U–Nb bcc alloys.
Fig. 10. Calculated and experimentally measured Nb tracer diffusion coefficients for the U–Nb bcc alloys.
Fig. 12. Calculated and experimentally measured interdiffusion coefficients for the U–Nb bcc alloys.
represented. In general, the activation enthalpies increase with the addition of Nb over the whole concentration range, and the experimental values from different authors generally distribute around the calculated line.
A careful evaluation indicates that they almost share the same slope, i.e. the activation enthalpies for the impurity diffusion coefficients of U in bcc Nb, but the intersect is different, i.e. the frequency factor for the impurity diffusion coefficients. In this work, there is no way to judge which one is more reliable. Preference is given to the work of Fedorov et al. [20], as their data quality has been verified in Figs. 2, 8 and 9. The mobility endmember for Mo in bcc U is based on the extrapolated interdiffusion coefficients measured by Adda and Philibert [26]. Although there are tracer diffusion coefficients of U and Mo in U–Mo alloys from [25], the magnitude of such data does not allow the physical significance to be acknowledged. As such, they were not accepted in this work. However, the U tracer diffusion coefficients from [13] show reasonable order of magnitude, which can be seen for the comparison in Fig. 15. The calculated intrinsic diffusion coefficients of Mo in bcc Mo–U alloys are given in Fig. 16, together with the experimental values from [26] at 1273, 1223, and 1123 K. It is evident that the intrinsic diffusion coefficients of Mo generally increase with
4.3. The U–Mo binary system The thermodynamic description of the U–Mo binary system is taken from the work of Berche et al. [27]. Compared with the U–Nb phase diagram, the bcc phase in the U–Mo system shares different characteristics. The bcc phase is characterized by spinodal decomposition over the whole temperature regime within which the bcc phase is stable in the U–Mo binary system. Thus, the single phase region for bcc phase is narrow and relevant experimental work may experience great challenge. Fig. 14 is the model-predicted temperature dependence of the impurity diffusion coefficients of U in bcc Mo compared with the corresponding experimental data from [19,20]. It is noted that such two sets of experimental data differ greatly from the magnitude.
Y. Liu et al. / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 37 (2012) 49–56
Fig. 13. Calculated and experimentally measured activation enthalpies for interdiffusion coefficients for U–Nb bcc alloys.
Fig. 14. Calculated and experimentally measured impurity diffusion coefficients of U in bcc Mo.
the U content, which is consistent with the change of the solidus curve in the U–Mo binary phase diagram. As can be seen, the solidus curve generally decreases with the addition of U. The same tendency can be evident in Fig. 17, where the calculated and experimentally measured interdiffusion coefficients generally show a good agreement at four temperatures (1323, 1273, 1223 and 1123 K), considering the experimental errors and the difficulty to obtain suitable concentration gradients in evaluating the interdiffusion coefficients. The activation enthalpies for the interdiffusion coefficients are given in Fig. 18, where the calculated line shows a tendency to decrease with the addition of U. It is noted that the experimental data from [26] feature significant deviation from the calculated line which is believed to be more reliable. As mentioned above, the tracer diffusion coefficients from [25] are not taken into consideration in this work, as the magnitude of such diffusivities does not allow the self-diffusion coefficients of bcc U to be extrapolated. This fact can be further verified by the calculated interdiffusion coefficients with the use of thermodynamic description from [27], the tracer diffusion
55
Fig. 15. Calculated and experimentally measured tracer diffusion coefficients of U in bcc U-10 at.% Mo alloys.
Fig. 16. Calculated and experimentally measured intrinsic diffusion coefficients of Mo in bcc U–Mo alloys.
coefficients from [25] and the Darken formula. The calculated interdiffusion coefficients with such a combination in the region of high U content differ substantially from the experimental data reported by Adda and Philibert [26] by more than one order of magnitude. 5. Conclusions Within the CALPHAD framework, the atomic mobilities for U, Ti, Nb and Mo are explored in this work in order to aid the design for novel nuclear materials. The thermodynamic descriptions of the U–Ti, U–Nb and U–Mo binary systems published in the literature are utilized in this work to provide thermodynamic factors that are necessary to generate intrinsic diffusion coefficients and interdiffusion coefficients. In order to establish high-quality atomic mobilities for such elements, the self-diffusion coefficients, impurity diffusion coefficients, tracer diffusion coefficients, intrinsic diffusion coefficients, and interdiffusion coefficients are concurrently taken into consideration. The calculated diffusion coefficients are
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Y. Liu et al. / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 37 (2012) 49–56
the calculated concentration distribution of Ti is pretty close to the measured quantities. In general, the atomic mobilities provided in this work show a satisfactory behavior in studying diffusionrelated phenomena, when such fundamental quantities are combined with thermodynamic descriptions. The results of this work are beneficial to explore novel nuclear alloys when cost reduction is the first consideration. Appendix. Supplementary data Supplementary material related to this article can be found online at doi:10.1016/j.calphad.2012.01.007. References
Fig. 17. Calculated and experimentally measured interdiffusion coefficients for U–Mo bcc alloys.
Fig. 18. Calculated and experimentally measured activation enthalpies for interdiffusion in bcc U–Mo alloys.
then compared with the measured ones, where a satisfactory agreement is evident. Computational simulations are performed for the U/Ti diffusion couples annealed at two temperatures, where
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