A novel computational framework for establishment of atomic mobility database directly from composition profiles and its uncertainty quantification

doi:10.1016/j.jmst.2019.12.038

摘要/Abstract

Abstract:

In this work, a novel computational framework for establishment of atomic mobility database directly from the experimental composition profiles and its uncertainty quantification was developed by merging the Bayesian inference with the Markov chain Monte Carlo algorithm into the latest version of the HitDIC software. By treating the simulation of composition profiles with the composition-dependent coefficients as the forward problem, the inverse coefficient problem that provides the potential way to compute the atomic mobilities directly from composition profiles can be postulated. The values and uncertainties of the atomic mobility parameters of interest were assessed by means of Bayesian inference, where the composition profiles were consumed directly. Benchmark tests that consider the number of diffusion couples and the noise levels were conducted. Practical application of the current framework in determination of atomic mobility descriptions of fcc Ni-Ta and Ni-Al-Ta alloys was performed. Further discussion about the results of the benchmark tests and practical study case indicated that the present computational framework together with numbers of composition profiles from the multiple diffusion couples can help to establish the high-quality atomic mobility database of the target multicomponent alloys.

Key words: Atomic mobility, Uncertainty, HitDIC, Bayesian inference, Multicomponent alloys

. [J]. 材料科学与技术, 2020, 48(0): 163-174.

Jing Zhong, Lijun Zhang, Xiaoke Wu, Li Chen, Chunming Deng. A novel computational framework for establishment of atomic mobility database directly from composition profiles and its uncertainty quantification[J]. J. Mater. Sci. Technol., 2020, 48(0): 163-174.

图/表 18

Fig. 1. Common and newly direct strategies for the assessment of the atomic mobility descriptions and their uncertainties.

Fig. 2. Relations between the atomic mobility descriptions and various diffusion coefficients.

Fig. 3. Illustration of the strategy to evaluate the atomic mobility parameters directly from the composition profiles.

Table 1 Preset atomic mobility parameters in a hypothetical binary system.

Parameter	Value	Optimization notation
$Φ_{A}^{ A,0}$	-150000-50T	Fixed
$Φ_{A}^{ B,0}$	-150000-50T	Fixed
$Φ_{B}^{ A,0}$	-150000-50T	Fixed
$Φ_{B}^{ B,0}$	-150000-50T	Fixed
$Φ_{A}^{ A,B,0}$	-80000-50T	1000A₀+B₀T
$Φ_{B}^{ A,B,0}$	-80000-50T	1000A₁+B₁T

Table 2 Posterior estimation of atomic mobility parameters for the hypothetical binary system.

Noise Level	Couple Number	A₀	A₁	B₀	B₁
δ=0	1 (I)	-41.18	-110.33	-102.53	-134.66
	2 (II)	-122.07	-121.45	-97.69	-9.34
	3 (III)	-76.18	-96.59	-103.71	-33.29
δ=0.001	1 (I)	-107.95	-133.27	-15.60	17.33
	2 (II)	-174.47	-172.29	42.29	38.04
	3 (III)	-106.08	-93.40	-25.38	-37.45
δ=0.01	1 (I)	1.41	-179.40	-169.48	92.59
	2 (II)	-134.77	-199.63	0.34	70.64
	3 (III)	-153.12	-105.73	-125.77	-22.83
True value		-80	-80	-50	-50

Fig. 4. Histogram of the distribution of the concerned parameters, where 0, 1, 2 and 3 in the legends stand for A0, B0, A1 and B1, respectively. The mean values of the parameters are denoted with the solid lines and the confidence intervals with the quantile of [0.16, 0.84] are plotted with the dashed lines.

Fig. 5. Comparison between the simulated composition profiles and the experimental results for the diffusion couples of the hypothetical binary systems: (a)?(c) the optimized results with one diffusion couple, (d)?(f) for two diffusion couples and (g)?(i) for three diffusion couples.

Table 3 Preset atomic mobility parameters in a hypothetical ternary system.

Parameter	Value	Optimization notation
$Φ_{A}^{ A,0} $	-125000-88T	Fixed
$Φ_{A}^{ B,0} $	-125000-88T	Fixed
$Φ_{A}^{ C,0} $	-125000-88T	Fixed
$Φ_{B}^{ A,0} $	-125000-88T	Fixed
$Φ_{B}^{ B,0} $	-125000-88T	Fixed
$Φ_{B}^{ C,0} $	-125000-88T	Fixed
$Φ_{ C }^{ A,0} $	-125000-88T	Fixed
$Φ_{ C }^{ B,0} $	-125000-88T	Fixed
$Φ_{ C }^{ C,0} $	-125000-88T	Fixed
$Φ_{ A }^{ B,C,0} $	-50000	1000A₀
$Φ_{ B}^{ A,C,0} $	-50000	1000A₁
$Φ_{ C}^{ A,C,0} $	-50000	1000A₂

Table 4 Posterior estimation of atomic mobility parameters for the hypothetical ternary system.

Noise Level	Couples	A₀	A₁	A₂
δ=0	2 (I)	-50.93	-51.59	-56.21
	4 (II)	-50.46	-52.22	-51.95
	8 (III)	-50.18	-52.61	-52.34
δ=0.001	2 (I)	-49.44	-50.51	-105.82
	4 (II)	-49.95	-52.79	-52.34
	8 (III)	-49.99	-53.11	-54.26
δ=0.01	2 (I)	-53.71	-50.03	-67.04
	4 (II)	-50.46	-50.33	-51.25
	8 (III)	-48.09	-53.45	-62.13
True value		-50	-50	-50

Fig. 6. Histogram of the distribution of the concerned parameters, where 0, 1 and 2 in the legends stand for A0, A1 and A2, respectively. The mean values of the parameters are denoted with the solid lines and the confidence intervals with the quantile of [0.16, 0.84] are plotted with dashed lines.

Fig. 7. Model-predicted diffusion paths due to the optimization results in the hypothetical ternary system: (a) noise level δ=0; (b) noise level δ=0.001; (c) noise level δ=0.01.

Table 5 List of atomic mobility parameters of fcc phase in the Ni-Al-Ta system assessed in the present work and also taken from the literature.

Parameter	Value	Reference
$Φ_{Ni}^{Ni,0}$	-271378-81.79T	[34]
$Φ_{Ni}^{Al,0}$	-144600-64.85T	[34]
$Φ_{Ni}^{Ta,0}$	-265535-75.14T	[30]
$Φ_{Al}^{Ni,0}$	-268381-71.04T	[34]
$Φ_{Al}^{Al,0}$	-123111.56-97.34T	[34]
$Φ_{Al}^{Ta,0}$	-265535-75.14T	[30]
$Φ_{Ta}^{ Ni,0} $	-276000-72.8T	[30]
$Φ_{Ta}^{ Al,0}$	-123111.56-97.34T	[30]
$Φ_{Ta}^{ Ta,0}$	-265535-75.14T	[30]
$Φ_{Al}^{ Al,Ni,0}$	-27571	[34]
$Φ_{Al}^{ Ni,Ta,0}$	$-604243.97_{-15648.03}^{+15557.70}$	This work
$Φ_{Ta}^{ Al,Ni,0}$	$-346170.48_{-21561.40}^{+21316.50}$	This work
$Φ_{Ta}^{ Ni,Ta,0}$	$-458378.53_{-13157.47}^{+13074.30}$	This work

Fig. 8. Corner plot of the atomic mobility parameters of fcc Ni-Al-Ta alloy sampling with 10 Markov chains (1000 iterations for burn-in period).

Fig. 9. Simulated composition profiles of fcc Ni-Ta diffusion couples at different temperatures, compared with the experimental data [30] (denoted in points), and the model-predicted ones (denoted in dashed lines) due to the atomic mobilities assessed by Chen et al. [30] and DICTRA.

Fig. 10. Comparison between the calculated interdiffusion coefficients ($\tilde{D}_{\text{TaTa}}^{\text{Ni}}$) due to the presently obtained atomic mobilities and the experimental results from Ref. [37]. The presently evaluated $\tilde{D}_{\text{TaTa}}^{\text{Ni}}$ based on the experimental composition profiles [30] and Boltzmann-Matano method in combination with the distribution functions [35] are also superimposed for comparison.

Fig. 11. Comparison of the simulated composition profiles and the experimental data [40] for the diffusion couples of Ni-Al-Ta systems: (a-1)?(a-3) Composition profiles at 1473 K; (b-1)?(b-3) Composition profiles at 1523 K; (c-1)?(c-3) Composition profiles at 1573 K.

Fig. 12. Comparison of the interdiffusion coefficients calculated by Matano-Kirkaldy methods (denoted as M-K) and the numerical inverse method (denoted as HitDIC).

Fig. 13. Histogram of the distribution of the rescaled concerned parameter where 0 in the legend stands for $\Phi _{\text{Ta}}^{\text{Ni},\text{Ta},0}$/1000.

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